摘要:

OFFICERS AND COUNCIL OF THE FARADAY DIVISION 1995-96 President Prof. H. M. Frey (Reading) Vice-Presidents who have held office as President Prof. A. D. Buckingham (Cambridge) Prof. N. Sheppard (Norwich) Prof. R. H. Ottewill (Bristo1)Gray (Cambridge) Prof. J. P. Simons (Oxford) Prof. R. Parsons (Southampton) Vice-Presidents Prof. A. Carrington (Southampton) Prof. M. W. Roberts (Cardiff) Prof. M. A. Chesters (Nottingham) Prof. I. W. M. Smith (Birmingham) Prof. R. N. Dixon (Bristol) Prof. F. S. Stone (Bath) Prof. M. J. Pilling (Leeds) Prof. M. N. R. Ashfold ’ Ordinary Members Or. D. C. Clary (Cambridge) Prof. A. J. Stace (Sussex) Prof. P. W. Fowler (Exeter) Prof. Sir John Meurig Thomas (London) Prof. R. K. Harris (Durham) Prof. R. P. Townsend (Port Sunlight) Dr.S. L. Price (London) Prof. A. Zecchina (Turin) Prof. S. K. Scott (Leeds) Honorary Secretary Prof M. J. Pilling (Leeds) Honorary Treasurer Prof. F.S. Stone (Bath) Secretary Mrs. A. C. Bennett Faraday Editorial Board Prof. M. N. R. Ashfold (Bristol) (Chairman) Prof. A. R. Hillman (Leicester) Prof. J. A. Beswick (Paris) Dr. J. Holzwarth (Berlin) Dr. D. C. Clary (Cambridge) Prof. D. Langevin (Bordeaux) Dr. L. R. Fisher (Bristol) Prof. S. K. Scott (Leeds) Prof. B. E. Hayden (Southampton) Dr. R. K. Thomas (Oxford) Prof. J. S. Higgins (London) Scientific Editor: Prof. A. R. Hillman Managing Editor: Dr. R. A. Whitelock Production Editor: Mrs. S. Shah Assistant Production Editors: Dr. J. C. Thorn, Dr. J. S. Humphrey The Faraday Division of the Royal Society of Chemistry, previously The Faraday Society, founded in 1903 to promote the study of Sciences lying between Chemistry, Physics and Biology Faraday Discussions (ISSN 0301 -7249) is published triannually by the Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 4WF, England.1995 Annual subscription rate EC f 207.00, Rest of World E2 17.00, USA $380.00, including air-speeded delivery, Canada f217 + GST. Change of address and orders with payment in advance to: The Royal Society of Chemistry, Turpin Distribution Services Ltd., Blackhorse Road, Letchworth, Herts SG6 lHN, UK. NB Turpin Distribution Services Ltd., is wholly owned by the Royal Society of Chemistry. Customers should make payments by cheque in sterling payable on a UK clearing bank or in US dollars payable on a US clearing bank. Air freight and mailing in the USA by Publications Expediting Inc., 200 Meacham Avenue, Elmont, NY 1103. Periodicals postage paid at Jamaica, NY 11431. USA Postmaster: send address changes to Faraday Discussions, Publications Expediting Inc., 200 Meacham Avenue, Elmont, NY 11003. All other despatches outside the UK by Bulk Airmail within Europe, Accelerated Surface Post outside Europe. PRINTED IN THE UK. 0The Royal Society of Chemistry 1995. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form, or by any means, electronic, mechanical, photographic, recording, or otherwise, without prior permission of the publishers.

ISSN:1359-6640    

DOI:10.1039/FD99501FX001

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

General Discussions of the Farada y Society/ Fa raday D isc ussions of the Chemical Society Date Subject Volume 1907 Osmotic Pressure Trans. 3* 1907 Hydrates in Solution 3* 1910 The Constitution of Water 6* 191 1 High Temperature Work 7* 1912 Magnetic Properties of Alloys 8* 1913 Colloids and their Viscosity 9* 1913 The Corrosion of Iron and Steel 9* 1913 The Passivity of Metals 9* 1914 Optical Rotary Power 10* 1914 The Hardening of Metals 10* 1915 The Transformation of Pure Iron 11" 1916 Methods and Appliances for the Attainment of High Temperatures in a Laboratory 12* 1916 Refractory Materials 12* 1917 Training and Work of the Chemical Engineer 13* 1917 Osmotic Pressure 13* 1917 Pyrometers and Pyrometry 13* 1918 The Setting of Cements and Plasters 14* 1918 Electric Furnaces 14* 1918 Co-ordination of Scientific Publication 14* 1918 The Occlusion of Gases by Metals 14* 1919 The Present Position of the Theory of Ionization 15* 1919 The Examination of Materials by X-Rays 15* 1920 The Microscope : Its Design, Construction and Applications 16* 1920 Basic Slags : Their Production and Utilization in Agriculture 16* 1920 Physics and Chemistry of Colloids 16* 1920 Electrodeposition and Electroplating 16* 1921 Capillarity 17* 1921 The Failure of Metals under Internal and Prolonged Stress 17* 1921 Physico-Chemical Problems Relating to the Soil 17* 1921 Catalysis with special reference to Newer Theories of Chemical Action 17* 1922 Some Properties of Powders with special reference to Grading by Elutriation 18* 1922 The Generation and Utilization of Cold 18* 1923 Alloys Resistant to Corrosion 19* 1923 The Physical Chemistry of the Photographic Process 19* 1923 The Electronic Theory of Valency 19* 1923 Electrode Reactions and Equilibria 19* 1923 Atmospheric Corrosion.First Report 19* 1924 Investigation on Oppau Ammonium Sulphate-Nitrate 20* 1924 Fluxes and Slags in Metal Melting and Working 20* 1924 Physical and Physico-Chemical Problems relating to Textile Fibres 20* 1924 The Physical Chemistry of Igneous Rock Formation 20* 1924 Base Exchange in Soils 20* 1925 The Physical Chemistry of Steel-Making Processes 21* 1925 Photochemical Reactions of Liquids and Gases 21* 1926 Explosive Reactions in Gaseous Media 22* 1926 Physical Phenomena at Interfaces, with special reference to Molecular Orientation 22* 1927 Atmospheric Corrosion, Second Report 23* 1927 The Theory of Strong Electrolytes 23* 1927 Cohesion and Related Problems 24* 1928 Homogeneous Catalysis 24* 1929 Crystal Structure and Chemical Constitution 25* 1929 Atmospheric Corrosion of Metals, Third Report 25* 1929 Molecular Spectra and Molecular Structure 26* 1930 Colloid Science Applied to Biology 26* 1931 Photochemical Processes 27* 1932 The Adsorption of Gases by Solids 28* 1932 The Colloid Aspect of Textile Materials 29 Faraday Discussions Date 1933 1933 1934 1934 1935 1935 1936 1936 1937 1937 1938 1938 1939 1939 1940 1941 1941 1942 1943 1944 1945 1945 1946 1946 1947 1947 1947 1947 1948 1948 1949 1949 1949 1950 1950 1950 1950 Electrical Double Layer 1951 Hydrocarbons195 1 The Size and Shape Factor in Colloidal Systems 1952 Radiation Chemistry 1952 The Physical Chemistry of Proteins 1952 The Reactivity of Free Radicals 1953 The Equilibrium Properties of Solutions on Non-electrolytes1953 The Physical Chemistry of Dyeing and Tanning 1954 The Study of Fast Reactions 1954 Coagulation and Flocculation 1955 Microwave and Radio-frequency Spectroscopy 1955 Physical Chemistry of Enzymes 1956 Membrane Phenomena 1956 Physical Chemistry of Processes at High Pressures 1957 Molecular Mechanism of Rate Processes in Solids 1957 Interactions in Ionic Solutions 1958 Configurations and Interactions of Macromolecules and Liquid Crystals 1958 Ions of the Transition Elements 1959 Energy Transfer with special reference to Biological Systems 1959 Crystal Imperfections and the Chemical Reactivity of Solids 1960 Oxidation-Reduction Reactions in Ionizing Solvents 1960 The Physical Chemistry of Aerosols 1961 Radiation Effects in Inorganic Solids 1961 The Structure and Properties of Ionic Melts 1962 Inelastic Collisions of Atoms and Simple Molecules 1962 High Resolution Nuclear Magnetic Resonance 1963 The Structure of Electronically Excited Species in the Gas Phase 1963 Fundamental Processes in Radiation Chemistry 1964 Chemical Reactions in the Atmosphere Subject Liquid Crystals and Anisotropic Melts Free Radicals Dipole Moments Colloidal Electrolytes The Structure of Metallic Coatings, Films and Surfaces The Phenomena of Polymerization and Condensation Disperse Systems in Gases: Dust, Smoke and Fog Volume 29* 30* 30* 31* 31* 32* 32* Structure and Molecular Forces in (a)Pure Liquids, and (b) Solutions The Properties and Function of Membranes, Natural and Artificial Reaction Kinetics Chemical Reactions Involving Solids Luminescence Hydrocarbon Chemistry The Electrical Double Layer (owing to the outbreak of the war the meeting w abandoned, but the papers were printed in the Transactions)The Hydrogen Bond The Oil-Water Interface The Mechanism and Chemical Kinetics of Organic Reactions in Liquid SystemsThe Structure and Reactions of Rubber Modes of Drug Action Molecular Weight and Molecular Weight Distribution in High Polymers (Joint Meeting with the Plastics Group, Society of Chemical Industry) The Application of Infra-red Spectra to Chemical Problems Oxidation Dielectrics Swelling and Shrinking Electrode Processes The Labile Molecule Surface Chemistry (Jointly with the Societe de Chimie Physique at Bordeaux Published by Butterworths Scientific Publications Ltd Colloidal Electrolytes and Solutions The Interaction of Water and Porous Materials The Physical Chemistry of Process Metallurgy Crystal Growth Lipo-proteinsChromatographic Analysis Heterogeneous Catalysis Physico-chemical Properties and Behaviour of Nuclear Acids Spectroscopy and Molecular Structure and Optical Methods of Investigating Cell Structure 33* 33* 34* 34* 35* 35* !as 35* 36* 37* 37* 38 39* 40* 41* 42* 42 A* 42 B* Disc.1* 2 Trans. 43* Disc. 3 4* 5* 6 7* 8* Trans. 46* Disc. 9* Trans. 47* Disc. 10* 11* 12* 13 14 15* 16* 17* 18* 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33* 34 35 36 37 Faraday Discussions Date 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 1981 198 1 1982 1982 1983 1983 1984 1984 1985 1985 1986 1986 1987 1987 1988 1988 1989 1989 1990 1990 1991 1991 1992 1992 1993 1993 1994 1994 1994 1995 Subject Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Absorbed State in Heterogeneous Catalysis Colloid Stability in Aqueous and Non-aqueous Media The Structure and Properties of Liquids Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Oxidation Bonding in Metallo-organic Compounds Motions in Molecular Crystals Polymer Solutions The Vitreous State Electrical Conduction in Organic Solids Surface Chemistry of Oxides Reactions of Small Molecules in Excited States The Photoelectron Spectroscopy of Molecules Molecular Beam Scattering Intermediates in Electrochemical Reactions Gels and Gelling Processes Photo-effects in Adsorbed Species Physical Adsorption in Condensed Phases Electron Spectroscopy of Solids and Surfaces PrecipitationPotential Energy Surfaces Radiation Effects in Liquids and Solids Ion-Ion and Ion-Solvent Interactions Colloid Stability Structures and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids PhotoelectrochemistryHigh Resolution Spectroscopy Selectivity in Heterogeneous Catalysis Van der Waals Molecules Electron and Proton Transfer Intramolecular Kinetics Concentrated Colloidal Dispersions Interfacial Kinetics in Solution Radicals in Condensed Phases Polymer Liquid Crystals Physical Interactions and Energy Exchange at the Gas-Solid Interface Lipid Vesicles and Membranes Dynamics of Molecular Photofragmentation Brownian Motion Dynamics of Elementary Gas-phase Reactions Solvation Spectroscopy at Low Temperatures Catalysis by Well Characterised Materials Charge Transfer in Polymeric Systems Structure of Surfaces and Interfaces as studied using Synchrotron Radiation Colloidal Dispersions Structure and Dynamics of Reactive Transition States The Chemistry and Physics of Small Metallic Particles Structure and Activity of Enzymes The Liquid/Solid Interface at High Resolution Crystal Growth Dynamics at the Gas/Solid Interface Structure and Dynamics of Van der Waals Complexes Polymers at Surfaces and Interfaces Vibrational Optical Activity: From Fundamentals to Biological Applications Atmospheric Chemistry : Measurements, Mechanisms and Models Volume 38 39 40 41* 42* 43 44 45 46 41 48 49* 50* 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65* 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88* 89 90 91 92 93 94 95 96 97 98 99 100 * Not available; for current information on prices etc., of available volumes.please contact the Marketing Oflcer, Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB 4WF, stating whether or not you are a member of the Society.

ISSN:1359-6640    

DOI:10.1039/FD995010X003

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,101,51-64 Brownian Dynamics Simulation of Particle Gel Formation: From Argon to Yoghurt Bert H. Bijsterbosch and Martin T. A. Bos Department of Physical and Colloid Chemistry, Wageningen Agricultural University, Dreijenplein 6,6703 HB Wageningen, The Netherlands Eric Dickinson* Procter Department of Food Science, University of Leeds, Leeds, UK LS2 9JT Joost H. J. van Opheusden Department of Agricultural Engineering and Physics, Wageningen Agricultural University, Bomenweg 4,6703 HD Wageningen, The Netherlands Pieter Walstra Department of Food Science, Wageningen Agricultural University, P.O. Box 8129, 6700 E V Wageningen, The Netherlands The influence of interparticle interactions on the fractal structure of gels formed from networks of aggregated spherical particles is investigated by Brownian dynamics computer simulation.In moderately concentrated systems of particles interacting with non-bonded Lennard-Jones inter-actions, restructuring of the network towards a phase-separated state leads to time-dependent changes in the primary cluster mass and in the intermediate-range fractal dimensionality. Using a colloid-type interaction potential of shorter attractive range leads to the same coarse network struc- ture but a slower rate of restructuring. Networks derived from simulations incorporating flexible irreversible bond formation and repulsive interparticle interactions have a polymer gel character with regular spacings between chains and a small average pore size. Systems exhibiting both bonding and attractive non-bonding interparticle forces can produce permanent fine or coarse microstructures depending on the relative rates of cross-linking and phase separation.Structural features of the simulated gels have much in common with model food particle gels formed from aggregated protein par- ticles. A particle gel is a soft, elastic, solvent-rich material made from a network of aggregated colloidal particles. Food colloids such as yoghurt, curd and margarine are examples of particle gels; the ‘particles’ may consist of protein molecules (e.g.whey proteins), protein aggregates (e.g. casein micelles), protein-coated emulsion droplets, or fat crystals.’ The formation of a dairy-type particle gel from a dispersion of milk protein ‘particles’ may be induced in a number of different ways (e.g.addition of acid, enzyme action, heating) depending on the conditions and the type(s) of protein involved.It is well known experi- mentally that a small change in processing conditions (like pH or temperature) can 51 Brownian Dynamics Simulation produce a large change in the kinetics and mechanism of aggregation, and also in the microstructure and texture of the resulting gel. The structure of a particle gel is typically rather coarse compared with the typically fine network structure of a macromolecular gel such as gelatin. In mechanical terms, the particle gel is much more brittle, and it tends to have a shorter linear elastic region and a smaller fracture train.^'^ While the material is necessarily uniform on the molecular scale (i.e.at distances smaller than the particle size) and also on the macroscopic scale (i.e.at the scale of specimens used in laboratory testing), the gel structure over some intermediate length scale has been recognized as being characterized by a fractal-type scaling behavio~r.~ Intuitive arguments suggest395 that this fractal scaling has its origins in the fractal-type structure of precursor aggregates which join together to make up the percolating particle network. Additionally, there is the possibility that processes of gel restructuring may generate (different) fractal scaling behaviour. At its lower limit, the fractal scaling regime is expected to merge into a region of short-range liquid-like oscil- latory structure like that found in simple liquids or stable concentrated dispersions.6 The overall microstructure is likely to be ultimately dependent on the overall particle volume fraction and on the detailed nature of the particle-particle interactions during and after gelati~n.~ Computer simulation studies have shown8-'' that the large stringy aggregates pro- duced by irreversible cluster-cluster coagulation of colloidal particles at very low volume fraction are characterized by a single fractal dimensionality, d, , when examined on length scales much larger than the individual particle size.In particular, it has been demonstrated that in three-dimensions there are two limiting fractal dimensionalities corresponding to diffusion-limited aggregation (d, = 1.Q where clusters stick imme- diately on contact, and reaction-limited aggregation (d, z2.1), where the sticking prob- ability is vanishingly small.The switch from diffusion-limited behaviour to reaction-limited behaviour on changing the experimental conditions (and hence the interparticle interactions) has been confirmed experimentally on model systems.' '-I7 (In practice, there is usually a transition from reaction-limited to diffusion-limited fractal structure with increasing aggregate size.) Simulated (non-fractal) structure on short length scales (of the order of the individual particle diameter) is necessarily determined by excluded volume effects and short-range repulsive interactions.The presence of weak or moderate attractive interparticle forces between colliding aggregates may allow any associating clusters to 'anneal ' somewhat prior to irreversible 'freezing in' of the per- manent network str~cture.'~-~~-~' Values of d, inferred from experiment^^-^ on casein particle gels are substantially higher than the idealized values quoted above for diffusion-limited or reaction-limited aggregation, possibly due to effects of aggregate restructuring and/or aggregate interpenetration at the finite particle concentration. Aggregate restructuring has been demonstrated experimentally.2 The simulation studies cited so far mainly refer to very low particle concentrations. At moderate volume fractions, 4, more appropriate to experimental gelation, it has already been noted63l9 that the simulated d, (inferred from the scaling of the aggregate radius of gyration) deviates significantly from the infinite dilution behaviour.While experimental studies on casein particle gels have indicated2 consistent fractal scaling behaviour up to volume fractions as high as 0.3-0.35, it is clear that eventually, at high particle concentrations, the fractal scaling region must disappear altogether. The ques- tion arises as to what are the conditions under which particle gels can usefully by described as 'fractal'. Another related question is: how does the structure (fractal or otherwise) depend on the particle concentration and the interparticle interactions? There have been few large-scale systematic simulation studies of the effect of # on the structure of colloidal aggregates and particle gels.One recent exception is an off-lattice study of diffusion-limited cluster aggregation in two dimensions in which it is suggested2' that there exist two distinct fractal scaling regimes. On a relatively short B. H. Bgsterbosch et al. length scale (from ca. four particle diameters up to some characteristic structural corre- lation length, t),the authors propose22 a fractal dimensionality which is the same as the high dilution value (d, = 1.45); on longer length scales there is another fractal dimen- sionality which gradually increases with # from 1.45 up to 2.0 (the Euclidean dimension- ality of the simulation space).The reason why d, does not immediately increase to 2.0 at length scales larger than 5 is attributed22 to interpenetration and/or packing of the ‘fractal blobs’. For 4 > ca. 0.25, 5 becomes less than four particle diameters, and so the short-range scaling region disappears altogether. The aggregation process is then said to be ‘space-limited’; that is, there is simply no space available in the system to build up proper fractal structures. In systems with attractive interactions between the particles, another issue to be considered is (reversible) flocculation or phase ~eparation.~~ At finite #, when the inter- action strength is weak (E % kT),the simulated system consists of reversibly formed flocs with an equilibrium fractal-type structure over medium length scales.24 Beyond a certain percolation threshold # > #*, such a system can be regarded as a very weak particle gel,25 though the significance of the network is now really more geometrical than struc- tural or mechanical.By analogy with a molecular fluid, as the interparticle attraction gets a little stronger (E = few kT),the flocculated dispersion becomes thermodynamically unstable with respect to phase separation into coexisting gas-like and liquid-like colloi- dal phases.26 This leads to growth of dense non-fractal regions of liquid-like disorder by a process of spinodal decomp~sition~~ and ultimately in real systems (under gravity) to macroscopic colloid-type phase separation. When the interparticle attraction is much stronger (E %-kT), the system is still thermodynamically unstable with respect to phase separation (though the favoured condensed phase may now be a colloidal crystal2* rather than a colloidal liquid).The rate of coarsening of the initially strongly flocculated percolating network structure may now be rather small because of the slowing down of local structural rearrangements due to the combination of strong short-range inter- actions and a relatively high local density. In a situation where genuinely irreversible aggregation (ie.permanent particle-particle bonding) takes place between flocculating particles, the fully phase-separated state is never reached because the growing domains become geometrically ‘pinned’ by the cro~s-linking.~~~~ In practice, essentially the same situation can be envisaged in a non-bonding system for which the particle-particle attractive energy is very strong (say > 10 kT).That is, owing to a geometrical pinning mechanism, a metastable glassy structure becomes ‘frozen in’ over timescales long com- pared with the simulation (or experimental) timescale. In this paper we investigate the effect of the strength of interparticle interactions on the microstructure of particle gels generated by Brownian dynamics simulation. Empha- sis is placed on the structure of percolating systems exhibiting both fractal growth and phase separation. Three different classes of model are considered. Model A (the ‘argon gel ’ model) consists of spherical particles which interact with pairwise additive Lennard- Jones potentials of variable attractive well depth.30 Model B (the ‘bonding’ model) con- sists of particles whose pairs can interact with weak attractive or repulsive forces, but which can also become linked together via flexible irreversible bond^.^^^^ Model C (the ‘colloid’ model) is similar to model A except that the Lennard-Jones interaction is replaced by a shorter-range potential more appropriate to colloidal particles.In each case, after starting off from a pseudo-random distribution, the simulated system of par- ticles becomes aggregated into a dense gel-like network. Simulation Methodology Consider a system of N spherical particles each of mass rn interacting with a pairwise potential u(r) where r is the centre-to-centre separation.The translational Brownian Brownian Dynamics Simulation motion is described by a general Langevin-type equation of the form32 where t is time, vi is the velocity component of one particle in one direction i, and the sum is over all 3N translational degrees of freedom. The right-hand side of eqn. (1) is a sum of three forces: a hydrodynamic force which depends on a set of friction coefficients {cij>, a systematic force Fi which is the sum over all the interparticle forces f(r)= -du(r)/dr acting in direction i, and a stochastic force which depends on a set of Gaussian random numbers (xi>and another set of coefficients {aij) related to {Cij> by When interparticle hydrodynamic interactions are neglected (as here), the configuration- dependent friction tensor ciiis simply replaced by a constant friction coefficient r = 6nqa (3) where q is the medium viscosity and a is the particle radius.The Langevin equation then reduces to m(dv,/dt) = -cvi + Fi+ Si; i = 1.. .3N (4) where Siis the fluctuating force describing random-walk motion in direction i for a particle with an Einstein-type diffusion coefficient D = kT/[ (5) A Brownian dynamics 'moving-on' algorithm based on eq n. (4) has the form &(At) = [-'FiAt + R,(D,At); i = 1...3N (6) where Ar,(At) is the overall change in particle position in direction i during the timestep At, and Rj(D, At) is the random translational displacement of a particle with diffusion coefficient D during the time interval At.An important requirement for eqn. (6)to be valid is that At should be large compared with the characteristic timescale of solvent molecular motion, but also small enough to ensure that the force Firemains effectively constant when the particle position changes from ri(t) to ri(t + At). In simulations of type B where some individual particles become bonded (or other- wise irreversibly aggregated together), it is also necessary to monitor their rotational motion in order to describe the multi-particle dynamical behaviour properly. A Langevin-type equation analogous to eqn. (1) can be written down33 for the rotational Brownian motion. In the absence of interparticle hydrodynamic interactions, the rota- tional and translational motions are de~oupled.~~So the 'moving-on ' routine for torque-free independent particle rotation has the simple form AOi(At) = RS(DR,At); i = 1.. .3N (7) where Aei = Ry is the random rotational displacement in direction i for a particle with rotational diffusion Coefficient DR= 3D/4a2,and the sum now is over all 3N rotational degrees of freedom. In model A we assume3* that particle pairs interact with a Lennard-Jones potential of the form -u(r)= 4~[(cr/r)'~(~/r)~] (8) where cr is the collision diameter [i.e. the value of r where u(r)= 01 and E is the maximum attractive energy [ix.u(r)= --E andf(r) = 0 at rmin= 2'/6a]. The potential is B. H. Bijsterbosch et al. set to zero for r > 2.5 c to improve computational efficiency.The hydrodynamic radius of the particle is arbitrarily set at a = f G. In model B we assume7 that particles can form irreversible flexible linkages with other particles that approach to within a certain bonding distance (i.e. r < dbond). The rate of reaction is expressed in terms of a probability, Pbond, that a bond will form during time interval At, Once a bond is formed, it cannot be broken and the angle(s) between two (or more) bonds on the same particle cannot change. The subsequent rela- tive particle motion is restricted such that the surface-to-surface bond length does not exceed the maximum specified bond length hbond = dbnd -2a. The generation of new configurations is subject to both the excluded volume and the maximum bond length constraints.In addition to being able to form permanent bonds, the model B particle pairs interact with a distance-dependent force of the form where w is an interaction strength parameter, which can be attractive (w > 0) or repul- sive (w < 0), and d,,, is the maximum range of the particle-particle interaction. In model C we assume that, when the distance-dependent force is attractive [i.e. f(r) < 01, the particle pairs interact with a shifted version of the potential form u(r)= (c1r)-I2 -c2rV2+ cg exp(-c,r); r > rmin (10) with coefficients cl, c2 and c3 set to give u(r)= --E at r = rmin.When the force is repul- sive [f(r)> 01, the model C particles are assumed to interact with the repulsive part of the Lennard-Jones potential [eqn.(S)]. The energy and force are both continuous func- tions of r, but, unlike the full Lennard-Jones potential, the model C interaction has a shorter finite attractive range [ie. u(r)= 0 andf(r) = 0 for r/a > 1.22051. (In practice, the attractive range may be as short as r/a = 1.01 for pm-sized particles, and so the range of the assumed model C potential may actually be realistic only for rather small colloidal particles.) Fractal Analysis In a particle gel formed by irreversible aggregation of spherical particles, three spatial scales of structure may be expected? (i) short-range order from packing and excluded volume effects, (ii) medium-range disorder associated with the ramified structure of the aggregating clusters, and (iii) long-range uniformity for a homogeneous material.In terms of the particle-particle distribution function g(r), the three regions are associated with (i) strong damped liquid-like oscillations out to a few particle diameters (r < ro),(ii) a fractal scaling regime with g(r) -(r/5)df-3; ro d r < 5 (11) and (iii) a non-fractal uniform structure with g(r) = 1 beyond some characteristic corre- lation length <. In estimating an effective d,from the simulations, it is convenient to smooth out the short-range oscillations in g(r)by calculating the integrated pair distribution function n(r) = 4~p, s2g(s)ds (12)sb where po is the average particle number density and n(r) is the average number of par- ticles within range r of another particle.d, can be determined from the slope of the linear region of a plot of log n(r)us. log r. For large length scales (r > 5) we have n(r)-r3 and Brownian Dynamics Simulation df = 3. The d, of the intermediate regime is only one aspect of the gel structure. Also important is the scaling prefactor no in n(r) = no(r/ro)df (13) where ro is some arbitrary length scale. If we take ro as the primary particle radius (i.e. 0/2 in models A and C, or a in model B), no becomes the rescaled mass of the primary particle. The value of no is a measure of the average number of particles in the primary clusters from which the fractal scaling regime is built, and a large value of no indicates a coarse microstructure. On the other hand, for large values of r, one must have n(r) = dWo)3 (14) We find no = + if we artificially extend the homogeneous scaling regime to the primary particle scale.Low values of no (possibly even less than unity) imply open structures. The crossover between the fractal scaling regime and the homogeneous regime takes place at the correlation length [: For ‘classical ’ cluster-cluster aggregation (no local reorganization or phase separation) with ro = a = a/2 and no = 1, eqn. (16) reduces to3 4(df-3)-l (17) Simulation Results and Discussion Brownian dynamics simulations have been carried out3’ in three dimensions with periodic boundary conditions on systems of model A particles with parameter values corresponding to N = 10o0, a = 1 pm, = 1 mPa s, T = 298 K and At = 3.44 ms.Per- colating networks were generated at volume fractions above + = 0.07. The aggregated structures were analysed after various times up to t = 1.1 x lo3 s (i.e. 3.2 x lo5 timesteps) in simulation runs on systems with well depths of E = 2 kT or 4 kT and volume fractions of + = 0.09, 0.1 1 or 0.13. The following quantities were calculated: g(r), n(r),d, and no. Fig. 1 shows derived values of no and d, as a function of 4 and the number of timesteps, nstep.Note that there is no universal fractal scaling behaviour in these systems. The systems are thermodynamically unstable with respect to gas-liquid phase separation. With increasing simulation time, there is a tendency for no to increase, indi- cating a process of short-range densification; there is also a tendency for df to decrease, indicating a process of medium-range stringly cluster formation.A qualitatively similar effect of aggregate ageing on gel structure has recently been reported by van Garderen et a2.22An increase in the maximum attractive energy of the Lennard-Jones potential from E = 2 kT to E = 4 kT leads to a substantial increase in no and a substantial reduction in df . The time evolution of the primary close-packed cluster radius seems to be relatively insensitive to #. The same is certainly not true, however, for the time evolution of d,, especially for the system with E = 4 kT. Under conditions of thermodynamic phase separation, the percolating clusters have a transient character. The process of short-range densification induces the formation of a coarser blob-like structure with larger voids and the elongation and/or breaking of some existing aggregate branches.This is in accordance with a simultaneous increase in no and decrease in df . We explain the decrease in d, by noting that it is more likely that particles necessary for the increase in no are moved from intermediate length scales (the fractal scaling regime) than from long length scales (the homogeneous region). Therefore B. H. Bijsterbosch et al. 'I 6-5-4-3-2-0 1 1.5 2 2.5 3 3.5 1~-~nstep (b) 2.25 2 df 1.75 1.5 1.25 1 1.5 2 2.5 3 3.5 1~-~nstep Fig. 1 Time-dependent structure of simulated model A particle gels. The plots show fitted values of (a) no and (b) d, as a function of E and 4.The quantities no and d, are plotted against nStep;0, 4 = 0.09; 0,4 = 0.11;0,4= 0.13. Open symbols: E = 2 kT; filled symbols, E = 4 kT. the structure on intermediate length scales must become more stringy (i.e.a lower value of df). Increasing the well depth increases the rate of formation of the initial percolated network and also the rate of reorganization (though not necessarily to the same extent). As expected, for larger values of E, the reorganized clusters become more closely packed on short length scales. It is likely that the time-dependent evolving structure will depend on the shape of the interparticle potential as well as on the well depth. The (molecular) Lennard-Jones potential strongly favours local structural reorganization because it has a rather broad interaction range.A more realistic colloidal potential, for instance one based on the well known DLVO potential form,' has a steeper repulsion and a much shorter attractive range (relative to the particle size), i.e. more like model C [eqn. (lo)]. We turn next, however, to the situation in which the extent of thermodynamically driven structural reorganization is permanently restricted by irreversible bond forma- tion. Brownian dynamics simulations have been carried out7 in two dimensions on periodic systems of model B particles with N = 400, Cp = 0.35,a = 1,d,,, = 4, hbond= 0.3, D = 1.0 and At = 0.005. Structures were typically analysed at t = 150 (i.e. after Brownian Dynamics Simulation (a Fig.2 Characteristics of two-dimensional model B particle gels generated as a function of w and Pbond: (a) N,,, (b) nbond (c) S, (arbitrary units) B. H. Bvsterbosch et al. Fig. 3 Snaphot picture of three-dimensional network structure produced from simulated aggre-gation of model B particles (after 2.1 x lo5 timesteps) with t$ = 0.05, w = 0 and Pbond= 1.0. The darkest particles lie furthest away from the observer. 3 x lo4 timesteps) for various simulations runs with values of the interaction strength parameter set in the range -5 < w < 2 and values of the bonding probability set in the range < Pbond < 1.0.The following quantities were calculated: N,,,, the number of particles in the largest aggregate; nbond, the average number of bonds per particle and Sp,the average pore size.To simulate a permanent cross-linked gel, it is necessary for all or most of the par- ticles to become part of a single large aggregate that spans the basic simulation cell. We see in Fig. 2(a) that this condition is satisfied (i.e.N,,, -+ N) for most the values of w and l’bond investigated. (Simulation times longer than t = 150 are required to generate a percolated network if the bonding probability is very low or if the interparticle inter- action is very strongly repulsive.) The quantity nbond is a measure of the mean local coordination number of the particles in the network. The data in Fig. 2(b) show that there is a systematic increase in nbond with increasing w.This is because gel systems with substantial repulsive interactions (w < -3) contain wiggly linear chains of low average particle coordination number (nbond + 2), whereas gel systems with attractive inter- actions (w 3 1) contain close-packed regions of relatively high average coordination number (nbond > 3). Fig. 2(c) shows that there is a large increase in s, as the interaction parameter changes from repulsive to attractive. Reduction in Pbond favours a fine-pore structure when the particle-particle interaction force is repulsive, but the opposite is the case when the force is attractive. Substantial net repulsive particle interactions lead to strong chain-chain repulsion during cross-linking. This inhibits multi-particle bonding, maximizes particle-solvent interactions, and generates a maze-like pore structure with average spacing of the order of the range of the interparticle repulsion.Because single bonds are quite flexible in this model, the gel structure generated with w 40 has some of the mechanical and swelling characteristics of a typical polymer gel. That is, owing to the flexibility of the highly solvated chains, so long as the individual particle segments on the chains are reasonably small (say 10 nm), one expects a substantial entropic contribution to the macroscopic elastic properties of the gel. In contrast, attractive particle interactions cause flocculation into close-packed clusters prior to irreversible cross-linking. This encourages multi- particle bonding (nbond increases), maximizes particle-particle interactions at the expense Brownian Dynamics Simulation of particle-solvent interactions, and generates a pore structure with average spacing very much greater than the monomer particle size.Such a coarse rigid gel would be expected to have macroscopic elastic behaviour dominated by energetic factor^.^ On the other hand, if the particles are very reactive (Pbond -I), they do not have suficient time to rearrange into close-packed clusters prior to bonding; the structure of the gel is now entirely kinetically controlled. It is interesting to note that, in this case, even though the rate of bonding (ccPbond) is higher, there is a lower overall density of cross-links (ccnbond) in the final network.This also has implications for gel elasticity. Computations have also been carried out in three dimensions on systems of model B particles with N = 1000, d,,, = 3a, hbond = O.lu, D = 1.0, At = 0.001 and 6 = 0.05, 0.07 and 0.1. Structures were typically analysed at various times up to t = 50 (ie. 5 x lo4 timesteps) for simulation runs with Pbond = 1.0 and w = -1, 0 or + 1. Based on visual inspection of the pictures of the gel structures, the trends of behaviour seem to be in good qualitative agreement with the previous two-dimensional results. Fig. 3 shows a pictorial representation of the gel structure obtained with w = 0 and 6 = 0.05 after t = 10. The plot of log n(r) us. log r in Fig. 4 for the same system shows a rather limited fractal scaling regime (d, z 1.9 up to z 7a).In contrast to model A, we see that, even at low volume fraction, model B with rapid cross-linking (Pbond = 1.0) and no inter- actions (w = 0) produces a thin-stranded network structure (no = 0.5) having a fine-pore distribution but no dominant fractal character in the scaling of n(r). The kinetic balance between phase separation and cross-linking may also influence the microstructure of mixed particle ge1s.j' To investigate such behaviour, we have extended the model B simulations to a binary two-dimensional mixture of equal sized particles (1 and 2). The system is characterized by three separate pair parameters wll, w22 and w12 corresponding to the 1-1, 2-2 and 1-2 interactions and a contant Pbond for all the reacting pairs.Fig. 5 shows some results for an equimolar mixture with N = 720,6 = 0.35, d,,, = 4.2, hbond= 0.30, D = 1.0, At = 0.005, t = 250, WI1 = 0, W22 = 0 and w12 = -4. The relative density of 1-2 particle pairs, n12(r),is plotted against Y for three different values of Pbond. When the rate of cross-linking is high (Pbond = lo-'), the func- tion nI2(r) has sharp peaks at r z2a, 4a and 6a, corresponding to a mixed gel network with little compositional heterogeneity on length scales larger than the particle diameter. When the cross-linking rate is an order of magnitude lower (Pbond = the begin- nings of local phase separation start to become evident in the n12(r) data: the height of the peak at r z 2a is greatly reduced, indicating a considerable loss of 1-2 nearest 100 s10 1 1 10 rla Fig.4 Scaling analysis of n(r) for system illustrated in Fig. 3. The points are the simulation data. The slope of the fitted straight line on the log-log plot gives the intermediate range d, = 1.9. B. H. Bijsterbosch et al. 300 200 h v2J 100 0 2 4 6 8 10 ria Fig. 5 Influence of rate of cross-linking on the two-dimensional structure of mixed model B par-ticle gels with wll =w22=0 and wI2 = -4. The number of unlike particle pairs (in interval 0.2a), n12(r), is plotted against r: a,Pbond = 10-I; 0,Pbnd = lo-’; A,Phnd= lo-’. neighbour pairs at the expense of 1-1 or 2-2 pairs. When the cross-linking rate is very low (Pbond= low5),the function n,,(r) is effectively zero for r <44 and it increases monotonically with r thereafter (within the statistical error). This plot is consistent with rather extensive phase separation in an aggregation system exhibiting liquid-liquid incompatibility (segregation) on a timescale short compared with the timescale for par- ticle cross-linking.Taken together with detailed simulation snapshot^,^' the data in Fig. 5 are a reflection of the fact that, once the size of bonded clusters becomes similar to that of micro-phase-separating domains, the process of compositional reorganization is halted. A similar type of behaviour is expected in three dimensions, albeit with a much greater degree of self-connectivity (bicontinuous structure) amongst the ‘frozen-in’ domains of similar composition.Finally, we report preliminary results in three dimensions for systems of model C particles with values of well depth E in the range 2-10 kT and volume fractions in the range 0.07 d 4 d 0.15. The model C interaction potential is of much shorter attractive Fig. 6 Snapshot picture of three-dimensional network structure produced from simulated aggre- gation of model C particles (after 1.9 x lo6 timesteps) with 4 =0.11 and E =4 kT Brownian Dynamics Simulation 1 I IIIII1000 1 10 r/aFig. 7 Scaling analysis of n(r) for system illustrated in Fig. 6. The points are the simulation data. The slope of the fitted straight line on the log-log plot gives the intermediate range d, = 1.9. range (with respect to the particle size) than the model A Lennard-Jones potential.Fig. 6 is a pictorial representation of the gel structure obtained after a simulation time of nstep= lo5 in a system with 6 = 0.112 and E = 4 kT. Fig. 7 is a plot of log n(r) us. log r for the same system; the straight line fit gives an effective d, = 1.9 and no = 2.0. These latter values are in fact rather close to those obtained with model A under similar condi- tions (see Fig. 1). Note that the structures illustrated in Fig. 4 and 6, while looking quite different by eye, both have the same intermediate range df. In contrast to the fine particle-bonded structure (no= 0.5) illustrated in Fig. 4, the picture in Fig. 6 is typical of a coarse phase- separated gel (no = 2) in the early stages of cluster densification.The very different appearance of the two network structures in terms of porosity and strand thickness indicates the danger of blindly adopting the fractal dimensionality as a single parameter to describe the structure of simulated particle gels. (Presumably this comment is equally applicable to some real particle gels.) As we are in the two-phase region, and there are no bonds forming to ‘cement’ the network permanently, the model C gel structure is undoubtedly time-dependent but, unlike the Lennard-Jones ‘argon gel’, the rate of restructuring of this ‘colloid gel’ is quite slow compared with the total simulation time. The structure also appears to be more senstive to 6 than to the well depth. Increasing 4 to 0.132 leads to an increase in the fitted value of d, to ca.2.1 (for either E = 2 kT or E = 10 kT). Conclusions The particle gel simulations reported in this paper have explored the effects of particle interactions and volume fraction on network structure. Whereas structures formed by ‘classical’ irreversible cluster4uster aggregation at low particle concentrations are characterized by a constant time-invariant fractal dimensionality on intermediate length scales (d, in the range 1.8-2.1 depending on the sticking probability), this in not the case with more complicated forms of interactions at higher concentrations. For such systems the value of d,, obtained for instance by fitting the integrated pair distribution function n(r) over an intermediate range of r values, does not give by itself a full description of the microstructure of the particle gel.It is clear that other structural parameters must also be considered, such as the short-scale primary cluster mass or the average pore size. B. H. Bijsterbosch et al. When the volume fraction in the simulated system reaches 4 x 0.3 or above, we cease to find a convincing fractal scaling regime at all. With moderate or strong (non-bonding) attractive interactions between the particles, any inferred fractal character is necessarily time-dependent because the system is ultima- tely thermodynamically unstable with respect to gas-liquid phase separation (or liquid- liquid phase separation in a segregating binary system). The rate of restructuring, as measured by the rate of increase of no, depends on the range of the attractive inter- actions.It is faster for a hypothetical argon gel composed of ‘slippy’ Lennard-Jones molecules than for a model yoghurt-like colloid gel composed of ‘sticky’ protein par- ticles (casein micelles). Real systems like those containing milk protein particles often have strong molecular interactions generated during restructuring. Nevertheless, despite this complexity, for reasons that are not fully clear, the macroscopic properties of such systems (permeability, elastic modulus, etc.) have been ~hown~-~ to be well described by single parameter fractal scaling over a wide range of volume fractions. Unfortunately, we have to conclude that, so far, we have not been able to establish a convincing quantitat- ive link between the ‘experimental ’ fractal dimensionalities determined previously for casein particle gel^^-^ and the restructuring processes generated in our simple particle gel simulations.The bonding model represents the extreme case in which permanent cross-links are formed leading eventually to a strong covalently bonded network. A fine-pore cross- linked structure is produced when the interparticle pair interaction is neutral or (especially) net repulsive. The resulting microstructure may actually possess little or no fractal character, particularly when 4 is high and the spacing between particle chains is rather narrow and regular. In such a system, once considerable cross-linking has occurred between repulsively interacting particles, there is little aggregate diffusion, and restructuring becomes limited to slow permanent ‘handshakes’ between flexible extend- ing arms.Such cross-links may not have much effect on the structural scaling behaviour, but they could be extremely important in relation to the rheology of the ageing gel. This contrasts with the possibility of extensive time-dependent structural changes in the non- bonding situation, where some initially formed aggregates may thin out or break up, whilst others coarsen under the persistent drive towards phase separation. On the other hand, the combination of bond formation and attractive (non-bonded) interactions ulti- mately produces a constant microstructure of variable coarseness (depending on the relative rates of particle bonding and cluster compactification). This is due to the growth of coarsening domains eventually being halted in its tracks by the increasing accumula- tion of structure-reinforcing cross-links.We acknowledge valuable discussions with T. van Vliet during the course of this project. ED. is grateful to Wageningen Agricultural University for the award of a Visiting Senior Fellowship during the period in which some of this research was carried out. References 1 E. Dickinson, An Introduction to Food Colloids, Oxford Univerity Press, Oxford, 1992. 2 L. G. B. Bremer, Ph.D. Thesis, Wageningen Agricultural University, 1992. 3 P. Walstra, T. van Vliet and L. G. B. Bremer, in Food Polymers, Gels and Colloids, ed. E.Dickinson, Royal Society of Chemistry, Cambridge, 1991, p. 369. 4 L. G. B. Bremer, B. H. Bijsterbosch, R. Schrijvers, T. van Vliet and P. Walstra, Colloids Surf., 1990, 51, 159. 5 L. G. Bremer, B. H. Bijsterbosch, P. Walstra and T. van Vliet, Adv. Colloid Interjiace Sci., 1993,46, 117. 6 E. Dickinson, J. Colloid Interface Sci., 1987, 118,286. 7 E. Dickinson, J. Chem.SOC.,Faraday Trans., 1994,90, 173. 8 H. J. Herrmann, Phys. Rep., 1986,136,153. 9 R. Jullien, Contemp. Phys., 1987,243,477. Brownian Dynamics Simulation 10 P. Meakin, in Phase Transitions and Critical Phenomena, ed. C. Domb and J. L. Lebowitz, Academic Press, London, 1988, vol. 12, p. 335. 11 D. A. Weitz, J. S. Huang, M. Y.Lin and J. Sung, Phys. Rev. Lett., 1984,53, 1657; 1985,54, 1416.12 C. Aubert and D. S. Cannell, Phys. Rev. Lett., 1986,56, 738. 13 G. Bolle, C. Cametti, P. Codasterfano and P. Tartaglia, Phys. Rev. A, 1987,35, 837. 14 M. Y. Lin, R. Klein, H. M. Lindsay, D. A. Weitz, R. C. Ball and P. Meakin, J. Colloid Interface Sci., 1990,137,263. 15 P. W. J. G. Wijnen, T. P. M. Beelen, C. P. J. Rummens and R. A. van Santen, J. Non-Cryst. Solids, 1991,136,119. 16 M. Carpinetti and M. Giglio, Adv. Colloid Interface Sci., 1993,46, 73. 17 P. W. Zhu and D. H. Napper, Phys. Rev. E, 1994,50,1360. 18 G. C. Ansell and E. Dickinson, Phys. Rev. A, 1987,35,2349. 19 G. C. Ansell and E. Dickinson, Faraday Discuss. Chem. SOC., 1987,83, 167. 20 E. Dickinson and C. Elvingson, J. Chem. SOC., Faraday Trans.2, 1988,84, 775. 21 P. Meakin, J. Colloid Interface Sci., 1990, 134, 235. 22 H. F. van Garderen, W. H. Dokter, T. P. M. Beelen, R. A. van Santen, E. Pantos, M. A. J. Michels and P. A. J. Hilbers, J. Chem. Phys., 1995, 102,480. 23 E. Dickinson, Annu. Rep. C, Royal Society of Chemistry, London, 1983, p. 3. 24 E. Dickinson, C. Elvingson and S. R. Euston, J. Chem. SOC., Faraday Trans. 2, 1989,85,891. 25 E. Dickinson, Chem. Ind., 1990, 595. 26 J. A. Long, D. W. Osmond and B. Vincent, J. Colloid Interface Sci., 1973,42, 545. 27 J. W. Cahn, J. Chem. Phys., 1965,42,93. 28 P. Pieranski, Contemp. Phys., 1983,24,25. 29 F. Sciortino, R. Bansil, H. E. Stanley and P. Alstrram, Phys. Rev. E, 1993,47,4615. 30 M. T. A. Bos and J. H. J. van Opheusden, in preparation. 31 E. Dickinson, J. Chem. SOC., Faraday Trans., 1995,91,51. 32 E. Dickinson, Chem. SOC. Rev., 1985,14,421. 33 P. G. Wolynes and J. M. Deutch, J. Chem. Phys., 1977,67, 733. 34 E. Dickinson, S. A. Allison and J. A. McCammon, J. Chem. SOC., Faraday Trans. 2, 1985,81,591. Paper 5J03291A; Received 22nd May, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100051

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Paraday Discuss., 1995,101,65-76 Gelation in Colloid-Polymer Mixtures Wilson C. K. Poon, Angus D. Pirie and Peter N. Pusey Department of Physics and Astronomy, The University of Edinburgh, MayJeld Road, Edinburgh, UK EH9 352 Moderate concentrations of a small non-adsorbing polymer cause a suspen- sion of colloidal particles to phase separate into coexisting colloidal fluid and crystal via the ‘depletion’ mechanism. At higher polymer concentra- tions, crystallization is suppressed, and a variety of non-equilibrium aggre- gation behaviour is observed. We report the results of small-angle laser light scattering studies of aggregation in a model system: colloidal PMMA- polystyrene. In all cases, ‘rings’ in the small-angle scattering are observed. At intermediate times, the scattering S(Q, t)from any one sample collapses onto a master curve under the scaling ansatz S(Q)-+[Q,(t)]’S[Q/Q,(t)], where Q,(t) is the (time-dependent) position of the small-angle scattering peak.Just across a ‘non-equilibrium boundary’ and at moderate colloid volume fractions ($ 2 O.l),the peak collapses continuously and completely. The exponent d was found to be 3, and the master curve takes the Fur- ukawa form; the peak position Q,(t) scales approximately as t-‘I4and t-l at short and long times, respectively, behaviour reminiscent of classic spino- dal decomposition in fluids. At higher polymer concentrations, d decreases from 3, saturating at d z 1.7, the fractal dimension of aggregates formed from diffusion-limited cluster aggregation (DLCA).In the latter case, where aggregation has led to a system-spanning ‘gel’, the small-angle ring appears more or less ‘frozen’ for a finite period of time (minutes to hours). Rapid collapse of the gel structure follows and the small-angle ring disappears in a matter of seconds. At lower colloid volume fractions, $ x 0.02, and just across the non-equilibrium boundary, a latency period elapses before a small angle ring becomes visible, whose position remains roughly constant while it brightens in time, behaviour consistent with classic nucleation. We suggest that non-equilibrium behaviour is ‘switched on’ by a hidden, meta- stable gas-liquid binodal. Different regimes of aggregation behaviour are controlled by the nucleation-spinodal cross-over and the transient perco- lation line within this binodal. 1 Introduction It is well known’ that the addition of enough non-adsorbing polymer to an otherwise stable colloidal suspension can cause phase separation via the depletion mechanism: exclusion of polymer from the region between two nearby particles gives rise to an unbalanced osmotic pressure pushing the particles together, resulting in an attractive depletion potential, Udep.For the simple case of hard spheres with added polymer the total potential between two particles is given by: ; for r < Q U(r)= -IIp I/overlap = Udep; for Q < r < CT + 2r, (1)[o+m ; for r > Q + 2r, where B = 2a is the particle diameter and IIpis the osmotic pressure of the polymer.Kverlapis the volume of the overlapping depletion zones between two particles at an 65 Gelation in Colloid-Polymer Mixtures intercentre separation of r. Explicitly where denotes the ratio of the radius of gyration of the polymer, r8, to the radius of the colloidal particle, a. Simple predicts that the topology of the phase diagram depends sensitively on <. For small polymers added to hard-sphere colloids, < 5 0.3, the only effect of the polymer is to expand the region of colloidal fluid-crystal coexistence, which occurs in the range of colloid volume fractions & = 0.494 c $ < 0.545 = & in a pure hard- sphere suspension. For larger polymers, colloidal liquid-gas coexistence becomes pos- sible and a critical point appears on the phase diagram.These theoretical predictions have received significant confirmation in one particular model system : colloidal poly(methy1 methacrylate) (PMMA) sterically stabilised by chemically grafted poly( 12-hydroxystearic acid) with added random-coil polystyrene dispersed in hydrocarbon sol- vent~.~~~Computer simulations also support this picture.6 Here, we concentrate on the behaviour of a colloid-polymer mixture with small size ratio 5 0.1. The sterically stabilised PMMA particles have radius R = 238 f5 nm, with a polydispersity of ca. 5%. Polystyrene of molecular weight M, = 370000 with MJM, < 1.1 was used. In cis-decalin at room temperature, dynamic light scattering measurements give rgz 19 nm, so that 5 x 0.08.The equilibrium phase behaviour of this system has been given bef~re,~ and is repro- duced here for reference, Fig. 1, where the axes are the colloid volume fraction 4 and the (weight) concentration of polymer, c, . At c, = 0, the usual hard-sphere disorder-order, or freezing, transition is observed: the states of the pure colloidal suspension are colloi- dal fluid for 4 < & = 0.494, coexisting fluid and polycrystal for c#+ < $ < 0.545 = &, A A A 0.004 0.002 4 -:xu\ 0.000 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 @ Fig. 1 Phase diagram of PMMA particles (volume fraction (6) and polystyrene (weight concentra- tion cp) in cis-decalin. Circles and ‘F’ represent one-phase colloidal fluid; squares and ‘F + C’ represent coexistence of fluid and crystal; triangles and ‘Neq’ represent samples in which homoge- neous nucleation is suppressed (non-equilibrium states).(Adapted from ref. 4, 4.u. for further details concerning the high-+ region not discussed here.) W.C.K. Poon, A. D. Pirie and P. N. Pusey colloidal polycrystal for #M -= # ,< 0.58 and colloidal glass for # 20.58. As predicted by theory, the addition of a polymer leads to a strong broadening of the fluid-crystal coexistence region : at any colloid concentration 0 < # < &, enough polymer causes phase separation into coexisting colloidal fluid and crystal states. At even higher polymer concentrations, however, crystallization is apparently suppressed. Instead, a variety of non-equilibrium behaviour and states are observed, which form the subject of the present paper.A preliminary report of these non-equilibrium states has been published earlier.' We have investigated extensively a number of samples covering a range of colloid volume fractions and polymer concentrations using small-angle light scattering. For the purpose of light scattering experiments, a mixture of roughly 1 : 3 (by volume) tetralin and cis-decalin was used as the dispersion medium, matching the refractive index of the PMMA particles. This results in a slight (downward) movement of the various bound- aries relative to the corresponding system in cis-decalin shown in Fig. 1: presumably the polymer coils are slightly expanded by tetralin, which is a better solvent.2 Experiments 2.1 Direct Observations First, we describe direct observation of the samples, left undisturbed after extensive tum- bling to randomise the particle positions. Samples in the fluid region of the phase diagram showed no change over a day or two, followed by slow gravitational settling of the particles over a period of several days. In samples with compositions just into the fluid-crystal coexistence region of the phase diagram, cp > cyex(#), small iridescent crys- tallites were observed to nucleate homogeneously throughout the sample volume after a few hours. These crystallites then grew and, over a day or two, settled under gravity, leaving a well defined boundary separating a colloidal fluid upper phase from a poly-crystalline lower phase.Again, particles in the colloidal fluid upper phase then settled under gravity over a period of several days. As mentioned above, homogeneous nucleation is suppressed in samples with a polymer concentration above a certain volume-fraction dependent threshold (Fig. 2) the 'non-equilibrium boundary', cEeq(#). In all samples with c, 2c:"~ and 4 50.3, particles settled under gravity, separated by sharp boundaries from clear supernatant, at rates much greater than those of the colloidal fluids described above. However, with increas- ing polymer concentration, these samples exhibited periods of 'latency' before settling was noticeable. At the cessation of this rapid settling, all the particles appeared to be in the sediment, whose height suggested that it had a ramified structure.Over the course of weeks, this ramified sediment compacted to a final height consistent with volume frac- tion 20.6, i.e. approaching random close packing. This compact sediment ultimately crystallised after a few months. The latency period increases sharply with cp and 4, so that samples with high polymer concentration or colloid volume fraction may never sediment over the duration of our observations. We mention that the sedimentation behaviour described here is similar to that observed by Grant and Russel8 in colloidal silica dispersed in hexadecane at low tem- peratures, by Lekkerkerker and co-workersg in colloidal silica with added polystyrene or poly(dimethy1 siloxane), and by Parker et aZ.1° in oil-in-water emulsions with added hydroxy-ethyl cellulose. 2.2 Small-angle Light Scattering Scattering of laser light by the samples, at angles up to ca.10 degrees, was observed on a screen by a video camera. All samples above the non-equilibrium boundary ci"q(4) exhibited small-angle scattering 'rings ', which brightened and collapsed as a function of time after mixing the samples. Integration of the scattering at fixed scattering angles (ie. Gelation in Colloid-Polymer Mixtures "T T-@ Fig. 2 Non-equilibrium 'phase diagram', showing S, N and T regions. Symbols have the same meaning as in Fig. 1, except that circled triangles represent non-equilibrium samples studied by small-angle light scattering. The subscript to 'S' is the dynamic scaling exponent, d, eqn.(4), found for that sample. The various boundaries, defined in the text, are cr'"(4) (bold line), cFq(4)(thin line), ci(#) (long dashes) and ci(4) (short dashes). around annuli centred on the straight-through beam) then gave the average scattered intensity as a function of the scattering vector Q, I(Q). One set of time-dependent inten- sity curves, I(Q, t),is shown in Fig. 3(a). These observations are similar to those made by Rouw et al. in model 'sticky hard sphere' colloidal suspensions," by Bibette et al. in gelling emulsion droplets" and by Giglio and co-workers in the salt-induced floccu- lation of charged colloid^.'^ We found that for the index-matching conditions used in this work, the single-particle form factors at low Q were practically constant over the relative range of Q.The scattered intensity Z(Q) can therefore be taken to represent the structure factor, S(Q).Consider, first, a sequence of samples with 4 x0.1 and increasing polymer concen- tration (circled triangles in Fig. 2). The non-equilibrium boundary for 4 = 0.1 occurred at c? w 3.7 x g ~m-~. A sample just across the non-equilibrium boundary, cp xcieq,gave small-angle scat- tering rings that brightened and collapsed continuously to leave peaked forward scat- tering over a matter of minutes, Fig. 3(a).Rapidly fluctuating speckles were observed in the scattering pattern during the whole period of the ring collapse. The time-dependent Fig. 3 Small-angle light scattering.(a)Selection of time-dependent scattered intensity, I(Q, t), for a sample at 4 w 0.1, with cp just across the non-equilibrium boundary (the circled data point at 4 x 0.1 labelled S(3jin Fig. 2). Time (in s) after cessation of tumbling is indicated alongside each curve. (b) The time-dependent peak position, Q,(t), for a sequence of samples with increasing polymer concentrations (the circled set of data points with 4 x 0.1 in Fig. 2). Polymer concentra- tion increases in the order 0,0,A, 0,*. (c) Dynamic scaling of the intermediate-time (27-63 s) scattering patterns for the sample in (a),with scaling exponent d = 3. The dotted line is a fit to the Furukawa form. (d)The scaling exponent d as a function of polymer concentration, cp,for the set of samples described in (b).The polymer concentration is given in terms of the depletion potential at contact calculated using eqn.(1) with 11, approximated by Van't Hoffs Law. (e)I(Q, t) for a sample at 4 w 0.02 and cp z 5.5 x lop3g ~rn-~ (the circled data point labelled N at 4 w 0.02 in Fig. 2). The inset shows a log-linear plot of the intensity of the peak as a function of time. W.C. K. Poon, A. D. Pirie and P. N. Pusey 69 50.00 164 as 0.000.00L 10 -'I 10 100 1000 Qa time/s 3.00- b2.00 - 1.00 - / %', Furukawa ---__.-_0.00 1, ,I, 1,,,,,,1,,,,,,,,,,,,,,,,,(,,,,,,,,,1,,,,,,,,, 0.00 1.00 2.00 3.00 4.00 5. 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.io Q10, UdedkT i50.00 200 9 1 I 30.00 a Y 6 G$20.00 I 10.00 : 0.00 & 0.00 0 Qa Gelation in Colloid-Polymer Mixtures peak position, Q,(t), scaled approximately as t-0-25 at early times, but crossed over to an approximate t-’ scaling at late times, Fig.3(b).At intermediate times, the low-angle scattering was found to exhibit dynamic scaling, i.e. the function had a time-invariant form, Fig. 3(c),with the fitted value of d = 3. This pattern of behav- iour is reminiscent of spinodal decomposition in simple fluids and polymers. 14*’ Indeed, the scaling function F(Q/Q,) was found to fit the ‘universal’ form proposed for Fur- ukawa,’ Fig. 3(c).We will call this ‘spinodal-like’ (S)behaviour. Samples with higher polymer concentrations continue to exhibit continuously col- lapsing and brightening rings, with scaling behaviour, eqn.(4), at intermediate times. Again, rapidly fluctuating speckles were observed throughout. However, the exponent d was found to decrease with increasing c,. The dependence of d on c, is shown in Fig. 3(4.Consistent with this finding, the shape of the scaling function P(Q/QA could no longer be fitted to the Furukawa form. Nevertheless, we will also label this type of behaviour as ‘spinodal-like’ (S), nomenclature used by Giglio and co-workers to describe similar light scattering findings in salt-induced aggregation of colloids.’ Observed under low magnification in a phase-contrast microscope, samples showing S behaviour displayed interconnected domain patterns strikingly similar to the visual appearance of systems undergoing classic spinodal decomposition (real materials or computer simulations) (see, e.g.Fig. 1.5 in ref. 16). The collapse of the small-angle scattering peak in samples with polymer concentra- tions above an upper threshold, ci(4) (w5x g cmW3 at (b = 0.1) is incomplete. In these samples, after initial collapse, the ‘ring’ appeared ‘frozen’ to the naked eye; a plot of the peak position as a function of time [Fig. 3(b)] showed that there was a significant slowing down of the rate of collapse after the first few minutes. Dynamic scaling, eqn. (4), was found to hold with d = 1.7 _+ 0.2, the exponent for diffusion-limited cluster aggre- gation (DLCA),17 in all samples with cp 2cf. Temporal fluctuations in the speckles on this ‘frozen ring’ were very slow.Preliminary rheological measurements of samples at this stage of their evolution gave finite yield stresses. After a fixed period of time, roughly coincident with the onset of gravitational settling of the particles described in the pre- vious section, the fluctuation of the speckles suddenly speeded up and the rings disap- peared within seconds, leaving peaked forward scattering. We shall label this type of behaviour as ‘transient gelation’ (T). Sand T behaviour were also observed in samples with increasing c, at (b w 0.2. No S behaviour was observed at (b z0.4; instead, T behaviour was found immediately across the non-equilibrium boundary. The small-angle scattering behaviour of a sample just across the equilibrium boundary at q5 = 0.02 was found to belong to a third category, Fig.3(e).A small-angle scattering ring only appeared after an initial latency period. The position of this ring stayed approximately constant initially, while the intensity increased exponentially [inset, Fig. 3(e)].This behaviour is somewhat reminiscent of the predic- tions of the linear Cahn theory for spinodal decomposition.16 However, we do not expect a latency period in any spinodal scenario. Moreover, direct observation under low magnification in a phase-contrast microscope showed disconnected droplets, remi- niscent of samples undergoing phase separation by nucleation and growth (see, e.g. Fig. 1.5 of ref. 16). We therefore propose to label this behaviour ‘nucleation-like’ (N).Above a threshold polymer concentration, cf ,however, Sbehaviour was again observed.The different kinds of non-equilibrium behaviour classified according tosmall angle light scattering are summarised in Fig. 2 in the (b-c, plane. None of the samples with composition lying above the non-equilibrium boundary, c”pq((b), show crystal-fluid phase separation via homogeneous nucleation. Samples above the uppermost boundary, ci(4),show transient gelation. Between ci(4) and ~;~q((b)there is apparently another boundary, ci((b)separating regions of Sand N behaviour. W. C.K. Poon, A. D. Pirie and P. N. Pusey 3 Discussion 3.1 Diffusion-limited Cluster Aggregation The most natural starting point for the interpretation of the results presented above is the idea of DLCA." Under the influence of a strong interparticle attraction induced by the depletion mechanism, diffusing particles and clusters bond when they touch.Indeed, a simple computer simulation of this process (off-lattice18 or on-latti~e'~), in which the diffusion coefficient of a cluster was scaled realistically as the inverse of its radius of gyration, gave a brightening and collapsing ring in the structure factor at small angles in both two and three dimensions. The decreasing peak position, Qm(t), corresponds to growing characteristic cluster size in the aggregating system, R(t)-Q, '(t).If in a scaling regime R(t) is also the typical distance over which a cluster has to diffuse in order to touch (and therefore bond with) another cluster, then the diffusive dynamics should evolve according to t--R2(t)-R3(t)D where, in the last relation, we have assumed that the diffusion coefficient D for a cluster of radius R scales as R-l.According to this argument, given e.g. by Binder and Stauf- fer,20 we expect the peak position to evolve according to Q,(t) -R-'(t) -t-'/3 Our simulations provide significant confirmation of this prediction. However, hydrodynamic interactions are neglected in the argument above, as well as in our simu- lations. Hydrodynamic interactions are important on two accounts. As the effective volume fraction of clusters increases, the diffusion rate of each cluster is reduced. Lubri- cation forces between two approaching clusters also reduce cluster mobility.Both of these effects are expected to give slower aggregation dynamics. Our data in Fig. 3(b) show that, at early times, Qm(t) -t-ll4 for all samples, i.e. the scaling is indeed slower than diffusion without hydrodynamic interactions. In fact, Siggia2' has shown that, in simple fluids, lubrication forces lead to a scaling close to R(t)-t1I4. It is not clear, however, whether his arguments could be applied without modification to clustering colloids. In any .case, Siggia's treatment neglects the concentration dependence of the cluster diffusion coeficien t. This growth of clusters cannot continue for ever. At some point they touch and span space, gelation occurs. At this point, further collapse of the small-angle peak in the structure factor is arrested, Q,(t) -to, a frozen ring.This describes closely what is observed for all samples with the highest polymer concentrations, cp > ci($). The scat- tering from samples with cp2ci($) obeys the dynamic scaling law, eqn. (4), with the exponent d = 1.7. This is consistent with the presence of clusters with a fractal internal structure characterised by a fractal exponent of 1.7, the exponent expected for DLCA in three dimensions." However, a frozen ring at Qa -O(10-') implies that the fractal structure only applies over a length range of, at most, a single decade, so that the use of a single fractal exponent to characterise the gel structure is clearly inappropriate.18 At low concentrations, a simple argument can be used to obtain a scaling relation- ship between cluster size immediately prior to gelation and the initial particle concentra- tion, 4.Assuming a fractal dimension, df, for the individual clusters, and a compact packing of these clusters to fill space, the typical radius of clusters at gelation Rgel [or the inverse of the position of the frozen small-angle ring, 2n(QE')-'] is given by the relation (see, e.g. ref. 12). Gelation in Colloid-Polymer Mixtures This scaling relation has been tested by Bibette et a1.l' for particle gels formed from purely liquid emulsion droplets. These authors found that eqn. (7) holds at 4 < 0.01. Systematic deviations were, however, observed at higher volume fractions; the frozen ring positions tended to a constant.We observed frozen rings at d = 0.1. 0.2. 0.4. Just across the transient gelation boundary, c, 2 ci(+), the frozen ring positions were Qg'ax 0.2 in all three cases. This compares well with the data of Bibette et a1.,12 who found that QK'a M 0.2 at # 5 0.1. Presumably the breakdown of eqn. (7) is due to the interpenetration of clusters at high densities (as observed, e.g. in the real-space pictures in ref. 18 and 19),which is not taken into account by the simple argument leading to the scaling relation, eqn. (7), between $ and df . The bonds formed between particles owing to the depletion mechanism are not per- manent. The depth of the depletion potential is of the order of a few k,T in the region of the phase diagram under consideration [cJ: horizontal axis in Fig.3(d)]. Diffusion-limited aggregation, which leads to ramified clusters and ultimately (in a large enough system) gelation, therefore competes with thermal rearrangement. This competition is clearly seen in Fig. 3(b). Initially, only diffusive aggregation is important ;approximate t-1/4 scaling is found at early times in all cases. In samples with the highest polymer concentrations, c, > ci , diffusion-limited aggregation 'wins'; Q,(t) crosses over to an approximate to dependence (a frozen ring), indicating gelation. At lower polymer con- centrations, thermal rearrangement ultimately dominates, and frozen rings are not observed. Presumably clusters compact while forming and fail to span space.22 The fractal dimension of these clusters depend on cp, and therefore on the depletion poten- tial.The observed dependence, Fig. 3(4, compares well with the experiments and simula- tions of Shih et al.,23who investigated in detail the effect of finite bond strength on DLCA. The growing and compacting clusters undergo fast gravitational sedimentation, as observed. Even in samples which show frozen small-angle scattering rings, in which thermal rearrangement is not fast enough to prevent gelation, thermally driven structural changes must still occur, albeit on a very slow timescale. Preliminary dynamic light scattering data' support this conjecture. Thermal rearrangement ultimately leads to the disintegration of the gel structure, which results, presumably, in the beginning of the fast gravitational settling behaviour already described.3.2 Onset of Aggregation The discussion, up to this point, has been based on the idea of the competition between diffusion-limited aggregation due to the depletion attraction, and thermally driven rearrangements. What this scheme has not been able to explain is the sharp onset of non-equilibrium behaviour : the boundary between homogeneous nucleation to give coexisting colloidal fluid and crystal phases, c$"'(4) < c, < c:"~(+),and non-equilibrium aggregation, c, > c:"~(+), is sharply defined and experimentally highly reproducible. We therefore need to explain the position of the non-equilibrium line, cteq(#). A brightening and collapsing small-angle scattering ring is reminiscent of the behav- iour of simple mixtures (binary liquids, alloys) undergoing spinodal decomposition.l6 Indeed, we want to suggest that there is an underlying connection. Following our earlier theoretical work3 we write the Helmholtz energy, F, of a colloid-polymer mixture as a function of the colloid volume fraction, 4, and the polymer chemical potential, p,,? F = F(#, p,). F can be calculated within a mean-field framework for a disordered t The chemical potential, pp, which determines the polymer fugacity, must be equal in any coexisting phases. Using this variable reduces the number of equations to be solved in calculating the phase diagram. See ref. 3 for the relation between ppand the experimental variable cp. W. C.K. Poon, A. D. Pirie and P. N. Pusey # Fig. 4 Schematic Helmholtz energy diagrams, F(4, pp).(a)Low pp. Any homogeneous fluid with composition between 4f and $c will phase separate into coexisting fluid and crystal phases with those volume fractions (the common tangent construction). (b)At high pp, the fluid branch has a double-minimum structure. The common tangent construction can be used to trace out both the equilibrium phase boundary, 4f and 4c,and the metastable gas-liquid binodal, 4gand (c) The metastable gas-liquid binodal c:q(#) and the equilibrium fluid-crystal boundary cr‘“(4) as calcu- lated using the theory of Lekkerkerker et aL3 The hatched area, c:, is approximately halfway between the (metastable) gas-liquid binodal and spinodal calculated by the mean-field theory of Lekkerkerker et aL3 The position of the transient gelation line, cf ,has been estimated from ref.25. arrangement of colloids and polymers, the ‘fluid branch’, and an ordered arrangement of colloids with polymers randomly dispersed, the ‘crystal branch’. At low polymer chemical potentials, the fluid and crystal branches each show a single minimum, Fig. 4(a). This gives rise to single-phase fluid, fluid-crystal coexistence, or single-phase crystal. The colloid concentrations in coexisting fluid and crystal phases are obtained by the ‘common tangent construction’ (see, e.g. ref. 24). At higher polymer concentrations, however, the fluid branch 01 the Helmholtz energy shows a ‘double-minimum’ structure, Fig. 4(b). At larger polymer to colloid size ratios, this double minimum can give rise to a region of colloid gas-liquid coexistence in the phase diagram.’v5 For small polymers, however, the theory predicts only separation into fluid and crystal phases. Nevertheless, the ‘metastable gas-liquid binodal’, which is ‘buried’ within the fluid-crystal coexistence region predicted by equilibrium thermody- namics, can still be traced out. The result, from the theory we developed earlier,3 is shown in Fig. 4(c). We suggest that the metastable gas-liquid binodal should be identi- fied with the experimental non-equilibrium boundary, ~:‘~(4).For polymer concentrations in the region cT“”(4)< c, < cieq(4),the homogeneous fluid state is stable against small, local density fluctuations.Fluctuations into the Gelation in Colloid-Polymer Mixtures ordered structure larger than a critical size will grow. Fluctuations into denser but still amorphous arrangements will decay. Thus we observe homogeneous nucleation of col- however, fluctuation into the loidal crystals. At higher polymer concentrations, c, > c;~~, nearby local (liquid) minimum with a disordered structure becomes possible. In other words, at c, > cYq, local exploration of phase space is controlled by the double- minimum structure of the fluid branch of the Helmholtz energy. We therefore suggest that it is the crossing of the metastable gas-liquid binodal which switches on non- equilibrium aggregation behaviour. If this is the case, we may expect to find that the initial behaviour of homogeneous fluids above cFq is similar to simple fluid systems undergoing phase separation.In par- ticular, we expect to see ‘nucleation’ close to the binodal and ‘spinodal decomposition’ further in. Moreover, the lattice-gas simulations of Binder and co-worker~~~ suggest that deep inside a region of two-phase coexistence in a simple fluid there should be a cross over from spinodal decomposition to ‘transient (or dynamic) percolation’. Some time after quenching, the ‘second phase’ spans space. The space-spanning structure, a ‘gel’, lasts for a finite time, z, and then collapses. The gel life time, z, increases dramatically with the depth of quench. All three types of kinetic behaviour, nucleation, spinodal decomposition, and transient percolation (or, more appropriately for colloidal systems, transient gelation), are observed in our system according to the small-angle light scat- tering evidence presented in Section 2.2.Immediately above cieq(4),which in our scheme is traced out by the metastable fluid-fluid binodal, we find N behaviour, a delay before a small-angle scattering ring, and micrographs showing disconnected droplets. The cross-over from nucleation to spinodal behaviour is expected to occur, for systems with short-range forces, in a narrow region roughly halfway between the mean-field spinodal and binodal,26 giving the hatched area in Fig. qc), which we suggest should be identified with the experimental boundary c:($).Any homogeneous fluid with composition above c:(4) is unstable towards local fluctuations, thus initiating rapid (diffusion-limited) aggregation under the depletion attraction. Just across ci(+), thermal rearrangements are fast enough to give compact aggregates, so that the dynamic scaling ansatz, eqn. (4),applies at intermediate times with exponent d = 3. At higher polymer concentrations, aggregates become pro- gressively more ramified, so that d decreases from 3. Throughout the tegion ~$4)< c, < ci(4), however, thermal rearrangements are still fast enough to ‘win’ over aggre- gation, giving rise to S behaviour. Furthermore, in many fluid systems, a speeding up of the dynamics at late times is observed, often reaching ca. t-’ asymtotically.This is indeed is observed, Fig. 3(b), for our S samples, although it is not clear whether the surface tension type arguments given by Siggia2’ apply in our system. Finally, at the highest polymer concentrations, above ci(d),aggregation wins over thermal rearrange- ments, and we expect transient gelation. The transient percolation line given in ref. 25 can be mapped onto our phase diagram. Fig. 4(c),which we suggest should give ci(4). When the transient percolation regime is approached we expect the fractal structure of the ramified aggregates to approach that given by classic DLCA, with fractal exponent df = 1.7.’’ Intermediate-time dynamic scaling, eqn. (4),with d = 1.7, is consistent with this expectation. The predicted ‘kinetic map’ in Fig.4(c)is at least topologically consistent with the experimentally determined regions of small-angle scattering behaviour shown in Fig. 2. There remains, however, significant disagreement between prediction and experiment as to the absolute positions of the various boundaries. Possible reasons for this discrepancy in the case of the equilibrium boundary, cFex(4),have been discussed previously>35 the mean-field nature of the theory used and the approximate treatment of the polymer (for a theory which includes the effect of polymer non-ideality up to second virial level see ref. 27) being the main factors. These factors will presumably also affect the calculation of the metastable fluid-fluid binodal. There is apparently no theory for predicting accu- W.C. K. Poon, A. D. Pirie and P. N. Pusey rately either the cross-over from ‘nucleation’ to ‘spinodal’ or the ‘dynamic percolation’ (or transient gelation) line. (It is possible2* that, in our case, the latter could be mapped on to the ‘percolation line’ cal~ulated~~ for the Baxter sticky hard-sphere system by matching the second virial coefficients of the sticky hard-sphere potential and the deple- tion potential.) Of course, the correct description of metastable states in general is a lively subject of debate. Our findings here are a contribution to this discussion. 4 Conclusion We have presented a plausible and self-consistent scheme to explain the onset of non- equilibrium aggregation of colloid-polymer mixtures which, according to equilibrium thermodynamics, should phase separate into coexisting colloidal fluid and crystal.We suggest that the non-equilibrium behaviour is ‘switched on’ by the presence of a hidden metastable fluid-fluid binodal. Different regimes of non-equilibrium aggregation, observed by small-angle light scattering, are then controlled by positions of the nucleation-spinodal cross-over and the transient percolation line inside this binodal. The practice of relating non-equilibrium behaviour to metastable phase boundaries is a common one in metallurgy (see, e.g. Section 10.1.6in ref. 24). The scheme we have suggested is chiefly successful in giving a semiquantitative pre- diction of the positions of the various boundaries in the non-equilibrium map [Fig.qc), compare Fig. 21 as well as suggesting a plausible explanation for the small-angle light scattering observations. The ‘S’ regime in which the small-angle peak scales in a spinodal-like manner but with fractal exponents is still to be studied in detail, especially in relation to sporadic claims of such ‘fractal’ behaviour in early-stage spinodal decom- position in other systems.30 The role of gravity in the formation of aggregates in the N and S region, in fixing the position of the transient gelation line ci(#), and in determin- ing the final structure of the sediment for all sample with composition above c:“, has not yet been investigated. The precise mechanism by which the transient gel breaks up is still unclear. In connection with the role of gravity and the break-up of the gel structure, we mention the very recent work in those directions of Allain et aL31 on aggregating calcium carbonate colloids.A.D.P. thanks the UK EPSRC and Unilever Research, Port Sunlight Laboratory for financial support. W.C.K.P. thanks the Rideal Trust and the University of Edinburgh for travel grants to attend the Faraday Discussion. Part of the equipment used has been funded by the Royal Society, London. We thank Mike Cates, Mark Haw, Henk Lekker- kerker and Patrick Warren for extensive and helpful discussions on various aspects of this work. References 1 W. C. K. Poon and P. N. Pusey, in Observation, Prediction and Simulation of Phase Transitions in Complex Fluids, ed. M. Baus, L.F. Rull and J-P. Ryckaert, Kluwer, Dordrecht, 1995, p. 3. 2 A. P. Gast, C. K. Hall and W. B. Russel, J. Colloid Interface Sci., 1983,96,251. 3 H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants and P. B. Warren, Europhys. Lett., 1992,20, 559. 4 W.C. K. Poon, J. S. Selfe, M. B. Robertson, S. M. Ilett, A. D. Pirie and P. N. Pusey, J. Phys. II, 1993,3, 1075. 5 S. M. Ilett, A. Orrock, W. C. K. Poon and P. N. Pusey, Phys. Rev. E, 199551, 1344. 6 E. J. Meijer and D. Frenkel, J. Chem. Phys., 1994,100,6873. 7 P. N. Pusey, A. D. Pirie and W. C. K. Poon, Physica A, 1993,201,322. 8 M. C. Grant and W. B. Russel, Phys. Rev. E, 1993,47,2606. 9 C. Smits, B. van der Most, J. K. G. Dhont and H. N. W. Lekkerkerker, Adv. Colloid Interface Sci., 1992, 42, 33.10 A. Parker, P. A. Bunning, K. Ng and M. M. Robins, Food Hydrocolloids, 1995, in the press. Gelation in Colloid-Polymer Mixtures 11 P. W. ROUW, A. T. J. M. Woutersen, B. J. Ackerson and C. G. de Kruif, Physica A, 1989,156,876. 12 J. Bibette, T. G. Mason, Hu Gang and D. A. Weitz, Phys. Rev. Lett., 1992,69,981. 13 M. Carpineti and M. Giglio, Phys. Rev. Lett., 1992,68, 3327; Phys. Rev. Lett., 1993, 70, 3828. 14 H. L. Snyder and P. Meakin, J. Chem. Phys., 1983,79,5588. 15 H. Furukawa, Adv. Phys., 1985,35,703. 16 J. M. Gunton, M. San Miguel and P. S. Sahni, in Phase Transitions and Critical Phenomena, ed. C. Domb and J. L. Lebowitz. Academic Press, London, 1983, vol. 8, ch. 3. 17 T. Vicsek, Fractal Growth Phenomena, World Scientific, 2nd edition, 1994.18 M. Haw, W. C. K. Poon and P. N. Pusey, Physica A, 1994,208,8. 19 M. Haw, M. Sievwright, W. C. K. Poon and P. N. Pusey, Physica A, 1995,217,231. 20 K. Binder and D. Stauffer, Phys. Rev. Lett., 1974,33, 1006. 21 E. R. Siggia, Phys. Rev. A, 1979,20, 595. 22 M. D. Haw, M. Sievwright, W. C. K. Poon and P. N. Pusey, Adv. Colloid Interface Sci., 1995,62, 1. 23 W. Y. Shih, J. Liu, W. H. Shih and I. Aksay, J. Stat. Phys., 1991,62,961. 24 R. T. DeHoff, Thermodynamics in Material Science, McGraw Hill, New York, 1992. 25 S. Hayward, D. W. Heermann and K. Binder, J. Stat. Phys., 1987, 49, 1053; G. Lironis, D. W. Heer- mann and K. Binder, J. Phys. A: Math. Gen., 1990,23, L329. 26 K. Binder, Rep. Prog. Phys., 1987,50,783. 27 P. B. Warren, S. M. Ilett and W. C. K. Poon, Phys. Rev. E, 1995,52, 5205. 28 H. N. W. Lekkerkerker, J. K. G. Dhont, H. Verduin, C. Smits and J. S. van Duijneveldt, Physica A, 1995, 213, 18. 29 Y. C. Chiew and E. D. Glandt, J. Phys. A: Math. Gen., 1983,16,2599. 30 D. W. Schaefer, B. C. Bunker and J. P. Wilcoxon, Proc. R. SOC.London A, 1989,423, 35. 31 C. Allain, M. Cloitre and M. Wafra, Phys. Rev. Lett., 1995, 74, 1478. Paper 5/03286E; Received 22nd May, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100065

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

FU~U~UY 1995,101,77-91D~SCUSS., Application of Cascade Theory to the Description of Microphase-separated Biopol ymer Gels Allan H. Clark Unilever Research Laboratory, Colworth House, Sharnbrook, Bedford, UK MK44 1LQ Studies of the kinetics of gelation of an acid-modified waxy maize amylopec- tin have been performed using gel-time measurements and small- and wide- angle X-ray diffraction. The gel time-concentration relationship for gels set at 5°C showed a high negative C-s.2 power law dependence over most of the concentration range considered (ca. 14-55% w/w), with a decrease in this exponent to -7.4 as the gelation temperature increased to 20°C. The effective critical concentrations, at which the gel time-concentration relationships diverged, shifted from ca.14 to 25% w/w for this temperature increase. For the 5°C gels, changes in X-ray scattering continued over periods in excess of one month, with the kinetics of this process showing a c-4.1 concentration dependence. Comparison of the X-ray results at low and wide scattering angles showed a clear correlation between the growth of B-form starch crystallinity in the gels, and the aggregation of individual amylopectin molecules. From these, and some microscopical observations, a picture emerges of the gelation in such systems, involving the rate-determining formation of large semi-crystalline aggregates (of diameter 0.1-1.0 pm) which subsequently percolate. The simple cascade theory approach to biopolymer gelation will require substantial modification if it is to provide a realistic description of this situation. In previous articles,'-6 a cascade theory approach has been used to describe the basic features of biopolymer gelation.This theory was originally developed7 to treat the poly- functional condensation and gelation of synthetic polymer monomers in the absence of solvent. Its extension to highly solvated biopolymer gels has required the formal inclu- sion of the solute concentration in the analysis. In addition, the assumption that an elastically active network chain (EANC) contributes RT per mole of chains to the shear modulus of the gel has been generalised to aRT per mole, where a is a non-ideality parameter, itself likely to be temperature dependent, and usually significantly greater than one.In essence, the cascade approach to gelation corresponds closely to the earlier Flory-Stockmayer with gelation arising when a critical degree of cross-linking is reached, but it offers the advantages of a greater generality in the treatment of aggregating objects, and an elegant and general mathematical approach to the counting of elastically active chains. In the treatment of highly solvated gels, extension of the Flory-Stockmayer and cascade approaches to include solvent is crucial. For reversible biopolymer gelation in a solvent, Hermans" was probably the first to address this issue. He retained the Flory- Stockmayer approach to counting active chains, but used mass action to relate the degree of cross-linking of the polymer chains to the solute concentration.As usual, sites for cross-linking (or functionalities) were treated as independently, and randomly, react- ing species, and these were assumed to enter into an association equilibrium, based on a balance between a second-order cross-linking forward reaction and a first-order disso- ciation process. This assumption led to the concept of a critical gel concentration C,, 77 Microphase-separated Biopolymer Gels below which critical branching could not be achieved, and hence gel formation was impossible. Clearly, the Hermans approach was intended to treat networks based on reversible associations between flexible chain polymers carrying a large number of potential binding sites, but quite early on it was applied to describe the modulus-concentration behaviour of heat-set globular protein gels, by Bikbov and co-workers." The descrip- tion of data achieved was good, despite features such as (a) that the underlying protein molecules were unlikely to be even approximately random coils; (b) that they almost certainly had only a small number of binding sites for association; and (c) that the links formed between molecules were probably long-lived.This early success, and apparent flexibility of the approach, prompted Clark, Ross-Murphy and co-workers to explore the model further, and to apply it in a modified cascade theory form to other heat-set protein gels,' 92i4 polysaccharide gels1-3-5,6 and gelatin.' y2y5 Initially, the reversible char- acter of the model was retained, other limitations of the Hermans treatment, such as the restriction to flexible coils and high functionalities, being removed through the cascade theory adaptation.Subsequently, the restriction to an equilibrium cross-link association has been lifted4 by showing that essentially the same equations for the modulus- concentration relationship are obtained if gels are assumed to arise uia a competition between an irreversible cross-link reaction, and some first-order 'wastage' process destroying the availability of branching sites (such as cyclisation). The cascade approach has limitations, however, including a failure to account for certain types of experimental observations. It is a mean-field theory and, as such, is expected to fail to predict critical exponents associated with the sol-gel transition.Also, as a theory, it is constructed in terms of a number of parameters which, though poten- tially related to molecular properties, are generally treated empirically, and give the theory a phenomenological, rather than a statistical mechanical, character. Thirdly, and perhaps most seriously, from a practical point of view, is the fact that when it is present- ed in a kinetic form, the cascade approach fails significantly to describe the relationship between gel time and concentration, particularly for polysaccharide '2-14 In these polysaccharide systems, cross-linking between chains generally takes the form of a disorder-to-order conformational transition, followed by the lengthwise association of the ordered chains into so-called junction zones.The result is a fibrous network whose average strand thickness depends on such factors as solvent character and the degree of polysaccharide charge, and this situation seems somewhat removed from the network model envisaged by the original Hermans theory, or the newer cascade theory, approach. The failure of these approaches to describe the highly negative gel time- concentration power law relationships, found in practice for such gels, suggests an inadequacy to treat such systems (despite apparent successes in handling corresponding modulus-concen tra tion relations hips, ' me1 ting behaviours' and molecular weight effects6). This problematic situation regarding gel times, and gel networks involving chain association, can formally be remedied by generalising the mean-field theories to treat cross-linking in such cases as a molecular process involving the simultaneous association of a specific number of polymer chain segment^.^>'^-'^ However, the high order required in the kinetic equation for the cross-linking reaction, and its sometimes literal interpreta- tion in terms of junction zone chain composition (number of component chains), seem highly suspect.An alternative view is that the pairwise association process can be retained in such cases, provided that the rate constant for this event is allowed to be highly concentration de~endent.~ There could be several justifications for this.Solution non-ideality is one argument, but perhaps more importantly, since junction zone forma- tion is an ordering process closely related to crystallisation, it should be treated kinetically as such, and a nucleation-and-growth model adopted. Another factor is network inhomogeneity, i.e. microphase separation. The original cascade theory was A. H. Clark expected to apply to uniform (molecular) gel networks only. Its blind application to phase-separated gels may well be inappropriate. From the above it is clear that microcrystallisation and phase separation in gels (which often occur together) are likely to invalidate the simple cascade approach to gelation, or at the very least pose a considerable challenge to it. It is the purpose of this paper to examine this situation, and assess the problems involved, by selecting a thermo- reversible polysaccharide gel system which displays both polymer crystallisation and phase separation as intrinsic parts of its gelling mechanism.The kinetics of gelation of this system will be investigated both at a mechanical level, through the measurement of gel times, and at a structural level, through studies of molecular aggregation and crys- tallisation using X-ray diffraction. Deviations in behaviour in relation to the predictions of the simple cascade theory will be established, and the origins of these deviations considered. It will then be discussed whether the cascade theory can be used in a modi- fied form to treat such systems, or whether an entirely new approach is necessary.The biopolymer system used to test these ideas will be a degraded amylopectin made by acid hydrolysis of waxy maize starch at low moisture content. This material, being derived from waxy maize starch, should contain very little linear amylose, and its average molecular weight, though high (see below), is comparable to that of other poly- saccharides, and much less than that of conventional amylopectins. This form of amylo- pectin gels extremely slowly (taking from hours to days) on cooling solutions below ambient temperature, though the timescale depends very much on polymer concentra- tion and temperature. Highly opaque, yielding, thermoreversible gels are obtained, similar to those generated by so-called maltodextrins.These maltodextrins are currently of great importance to the food industry as fat-replacement materials and have been much studied.l5-I7 In a real sense, the material examined here can be thought of as a simple prototype form of these maltodextrins, and so the present work should provide not only a critical examination of the scope of the cascade approach to biopolymer gelation, but also a valuable insight into the mechanisms involved in the gelling of a particularly important class of polysaccharide food ingredient. Experimental Amylopectin Samples The amylopectins (samples 1 and 2) used in this work were members of a series of hydrolysed amylopectins referred to in a previous paper18 by Durrani and Donald. They were prepared by heating acidified waxy maize starch at very low moisture content for varying periods of time.Samples 1 and 2 referred to here correspond to samples B and E of ref. 18. In that article, Durrani and Donald report number-average and weight- average molecular weights from high-pressure liquid chromatography and multiple- angle laser-light scattering. For sample 1, these results were (2.0 f0.3) x lo5 and (1.20 & 0.04) x lo6, respectively, and for sample 2, (4.0 f0.1) x lo4 and (2.0 0.2) x 10'. In the experiments reported below, these materials were used without further purification. Gel-time Measurements Amylopectin solutions were made up in distilled water by dissolution at 90 "Cin a water bath (1 h). Solutions were made over the composition range 10.0-55.0% w/w at roughly 2% intervals.Solution pH ranged from 4.0 to 5.0 and was not further adjusted. Small amounts of sodium azide were added to the solutions to prevent bacterial attack. Ali- quots of the hot solutions (1 ml) were transferred into small glass phials, the phials further heated to 90 "C,then immersed in water at 5 "C within a constant-temperature Microphase-separated Biopolymer Gels fridge. Samples to be studied at ambient temperature (20°C) were left in contact with water in a temperature-controlled room. Solutions were observed periodically by inver- ting the tubes. The gel point was deemed to have been reached when solutions failed to flow with a level meniscus, and the corresponding gel time recorded.In a preliminary run, approximate values for gel times were established, then in subsequent experiments (three for sample 1 at 5°C and 20°C) these values were refined, and some general esti- mates of the uncertainties in the values made. Despite reheating samples in acid condi- tions, there was no evidence that gelling was significantly altered in these repeated gel-time studies, or in subsequent X-ray diffraction work, i.e. there was no reason to believe that significant further hydrolysis was occurring. In this work, samples were studied from the point of initial quenching to times of up to eight weeks, intervals of time ranging from minutes for the fastest gelling systems to days, and weeks, for the most protracted (or non-gelling) solutions.Small-angle X-Ray Scattering Measurements A selection of the samples prepared initially for gel-time measurements, and which were found to gel convincingly, were reheated and loaded into glass X-ray capillaries (1 mm diameter). The concentrations of interest to the study were C = 19.9, 25.6, 30.2, 34.3,40.6 and 44.2% w/w. The capillaries were sealed, reheated to 90°C and then loaded as quickly as possible into the sample holder of an Anton Parr Kratky compact small- angle X-ray scattering camera. The holder temperature was held at 5°C by means of a Peltier temperature-control unit. Diffraction experiments were performed using Cu-Ka X-rays (wavelength ca. 0.1543 nm) supplied by a Philips PW 1730 constant potential generator. The scattering experiment was performed under vacuum, and scattered inten- sity data measured using an INEL LPS 50 position-sensitive detector, associated elec- tronics, Varro Silena multichannel analyser and a standard personal computer.Data were collected over some 750 channels, corresponding to a q range of 0.91 x lo-' to 6.2 nm-', i.e. equivalent to a Bragg spacing range of 69-1 nm. Here q is the scattering vector defined as q = 4.n sin $/A, with 28 the scattering angle, and A the X-ray wave- length. In the experiments performed for a given sample, a first scatter curve was record- ed as soon as possible after introduction of the sample to the camera. Scattering was then recorded at regular intervals (e.g 10 min) for a period of up to 8 h, then thereafter, as much as possible, once a day for a period of up to 4 weeks.Each sample remained within the camera for the first 8 h, but was subsequently stored at 5°C in a fridge, and withdrawn for measurement when necessary. Individual scatter curves (i.e. at specific measurement times) were background-subtracted, scaled for concentration and exposure time, and printed out at ten channel intervals. Exposure times varied with concentration, and ranged from less than 300 s (for the most concentrated sample) to 800 s (for the least). Scatter curves were not corrected for the smearing influence of the slit collimation system. Wide-angle X-Ray Difiaction Samples covering the concentration range 14-55% w/w in 5% intervals were made up hot and then stored in a fridge at 5 "C for a period of 1 month. Small aliquots of these samples were subsequently transferred to the sample holder of a Philips PW 1050powder diffractometer, also held at 5 "C.The X-ray source was as described above, and intensity data collected over the angular range 20 = 5-35", at a scan rate of 1 degree min- '.Data were recorded digitally, data collection being under the control of a personal computer. Powder patterns for all the gels were plotted and compared, and traces for the empty sample holder also obtained. In further experiments, the hydrated-gel aliquots were left A. H. Clark to dry out for one week at ambient temperature in uncovered watch glasses. The dried materials were ground to fine powders, and their powder patterns recorded.Results and Discussion Gel-time Measurements Gel-time estimates are plotted in Fig. 1, for sample 1 at 5 and 20"C, and for sample 2 at 5°C. In all three cases, the data extend only over a limited concentration range. For sample 1 at 5 "C, for example, this extends from 16 to 46% w/w. A tentative estimate for the 14% sample also appears, but this is subject to a large uncertainty (ca. 50%),caused by its extreme sensitivity to the applied stress, a factor making it dificult to decide when gelation had occurred. Below 14%, no sample gelled within a timescale of over two months; only turbid suspensions were obtained. Solutions having concentrations >45% were extremely viscous, again making it difficult to judge when they had gelled on account of their slow flow, even in the hot state.For those gel times at 5"C, for which estimates have been presented in Fig. 1, the uncertainties are also considerable. For sample 1, the relative uncertainty of the results ranged from around 25%, at the extreme ends of the data set, to about 15% at the centre, these uncertainties being arrived at on the basis of repeats of the experiments themselves, and from how difficult it was to decide, in an individual case, when the gel point had been reached. Similar estimates apply to the other two data sets presented. Despite uncertainties, it is possible to arrive at some interesting conclusions from the data in Fig. 1. For sample 1, between concentrations of roughly 20 and 40% w/w, the log-log plot of gel time vs.concentration is close to linear, indicating a simple power law relationship. Linear regression confirms this, giving a slope of -5.2 0.2, and implying a roughly C-' dependence of the gel time for this system, Points outside this concentra- tion range are less certain, but if accepted, these would imply a steepening of the slope from the power law relationship at both extremes. At low concentrations, this would be expected, as a logarithmic divergence of the gel time is anticipated at the critical gel concentration (and indeed is predicted by the cascade approach4). A similar logarithmic divergence to shorter values (and ultimately to zero?) at high concentrations is not anticipated, however, and if real, will require explanation.It must be added that the cascade theory predicts4 a power-law exponent in the intermediate concentration region of -1.0, which is very different from the result of -5 found here. This also requires explanation. 110 c (Yo w/w) Fig. 1 Gel time us. concentration data for sample 1 (m, 5 "C;0, 5 "C)20 "C),and sample 2 (0, Microphase-separated Biopolymer Gels Similar power-law analyses can be made for the other two data sets in Fig. 1. For sample 1, studied at 20°C, all gel times become greatly increased on raising the tem- perature, values at the lower end of the concentration scale being most affected. A linear regression to all the data gives a good fit and a slope equal to -7.4 & 0.6, i.e. roughly a C-dependence. In addition, the critical concentration for gelation appears to rise from around 14% w/w at 5°C to roughly 25%, though again, good characterisation of this critical region of the gel time-concentration function was difficult, because of the extreme sensitivity of the gels to mechanical disturbance, and the fact that at this higher temperature, the gel time is rising very rapidly as the concentration falls.The effect of changing the average molecular weight, as demonstrated by the data for sample 2, also presented in Fig. 1, shows that a fall in molecular weight produces an overall lengthening of gel times, and possibly a fall in the power-law exponent which, in this case, was found to be -4.4 f0.2. In summary, it appears that for the modified amylopectins studied here, as for other polysaccharides, the cascade approach seriously underestimates the sensitivity of gel times to change in concentration, in the region of concentration where a power law applies.Secondly, this power law is sensitive to temperature changes, rising rapidly as temperature increases, but is less sensitive to molecular weight, which nonetheless has a significant overall effect on the absolute values of the gel times. Small-angle X-Ray Scattering Measurements Sol and Gel Scatter Curves Fig. 2 and 3 provide log-log displays of scatter curves measured for sample 1 solutions immediately after quenching, and later at the longest times considered (ca.4 weeks). The concentration range extends from 19.9 to 44.6% at intervals of ca. 5% (see Experimental).Sol scattering for a 1.0% w/w solution has been included in Fig. 2 to indicate the extent that the increased sample concentration depresses scattering at the smallest angles. Examination of the scattering at the lowest channels, for the 1.0% sol sample, using a Guinier plot [ln I(4) us. q2] reveals a curvature which is no doubt a consequence of the sample polydispersity. Taking the maximum slope of this plot (i.e. the slope at the smallest measured angle) and extrapolating this to infinite dilution (by including results at 2.0 and 5.0% w/w, not shown in Fig. 2) a maximum radius of gyra- m m 0.01 I I f 1 . ' 10 1160 ~1000 9(channel no.) Fig. 2 Scattered X-ray intensity us. q vector for amylopectin solutions immediately after quen- ching to 5°C (H, 1% w/w; 0, 19.9% w/w; A, 25.6% w/w; V,30.2% wfw +, 34.3% w/w; +,40.6%w/w; and x ,44.2%w/w) A.H.Clark t 9? fx lo\1 J h P- v 0.1 0.01 I I I I 1 10 100 1000 q (channel no.) Fig. 3 Gel scatteringcurves after 1 month at 5 "C.Symbols are as in Fig. 2. tion of ca. 8 nm was obtained. This is somewhat smaller than estimates of over 10 nm obtained by Durrani' for similar materials by synchrotron X-ray scattering. This differ-ence probably reflects access to a lower minimum q value in the synchrotron work, but may also be a consequence of neglect of slit-smearing effects in the present Kratky study. In any case, the sample polydispersity, coupled to the limited extent of the X-ray data, make any X-ray value for the radius an unknown type of average, and hence give only a rough estimate of the size of the amylopectin molecules present.From the shape of the sol scattering curves at low concentration, Durrani concluded' that the amylopectin molecules were non-spherical, particularly since attempts to fit this data to models involving distributions of spherical objects failed. From the measured slopes of the In I(q) us. In q plots (similar to those in Fig. 2) it was concluded that the amylopectin molecules were probably disc-like. The curves in Fig. 2 show a lower power law than described by Durrani," but this is probably because they have not been corrected for slit-collimation error. Since the present work is primarily concerned with the aggregation behaviour of the amylopectin, the issue of the structure of the primary particles will not be pursued further.Description of the particles as non-spherical, with diameters distributed about a value of 20 nm, will be accepted as ade-quate. The gel scatter curves after about 1 month at 5°C appear in Fig. 3. Not unex-pectedly, in view of the amylopectin aggregation which evidently takes place, these curves show a higher intensity at the lowest angles than the corresponding sol data, and fall slightly in intensity at larger angles. A significant feature also is the indication of a Bragg maximum at ca. 390 channels, which corresponds to a diffraction angle 28 of ca. 5.5" and, as is well known, is diagnostic of the presence of the B form of starch. This result is not surprising since such crystallinity has been demonstrated by others to be present in gels derived from unmodified amylopectin20*21and rnaltode~trins,'~-'~but what will be of interest in the present work is to use this Bragg peak as a means of monitoring the growth of crystallinity with time for each of the gelling systems, and to compare this behaviour across the range of concentrations studied.The gel scattering data at low angles are also of interest. Over the concentration range 19.9-34.3%, only small differences are observed in this angular region, scattering decreasing slightly as the concentration increases, but this effect becomes much more pronounced for the 40.6 and 44.2%systems.The lack of a strong concentration depen-dence of the scattering for samples up to at least 34.0%,coupled to the almost perfect superpossibility of all the data sets at higher angles, suggests a near constant short-range Microphase-separated Biopolymer Gels 7-6-h-5-UJ-g 4-a-II: F 0 2 4 6 8 10 12 14 16 In (tlmin) Fig. 4 Scattered intensity at 10 channels plotted against In t for the 25.6% (m), 34.3% (a)and 44.2% (A)w/w samples quenched to 5 "C. Continuous curves were constructed by smoothing and extrapolation. spatial environment for individual amylopectin molecules in these gels, and this could be consistent with a phase separation process which generates an almost constant local concentration of polymer in phase-separated regions.Homogeneous network formation, such as has been studied" by small-angle X-ray scattering for charged protein mol- ecules, usually generates larger differences in the scatter curves at the smallest angles than are observed here. This conclusion for the present gels is both consistent with a previous interpretation of small-angle X-ray scattering data for closely similar maltodex- trin gel^,'^-'^ and is, of course, also consistent with the highly opaque character of these materials. Time Dependence of Low-angle Scattering (Channels 10-40) Fig. 4 shows plots of scattered intensity at 10 channels against In t, for three of the systems studied (C = 25.6, 34.3 and 44.2% w/w). These three concentrations span the concentration range over which it was possible to make a convincing interpolation and extrapolation of the data (see continuous curves, also shown in Fig. 4).A tentative extrapolation of the corresponding 19.9% intensity-time curve was possible, but this was much less accurate, as a long time plateau value was less obviously achieved. Results similar to those of Fig. 4 were obtained using data at other channels (20, 30 and 40) and were treated in the same way. Initial analysis of the extrapolated intensity-time data involved plotting the curves in a log-linear form. This suggested that during the earliest stages of solution ageing, an exponential growth of intensity occurred, this growth later decreasing, and changing in form at longer times. For a given sample, results at different channel numbers all showed this behaviour, but had different rate constants in the exponential region.By analogy with studies of the spinodal decomposition in synthetic polymer solutions by light ~cattering,~~ one might feel tempted to invoke liquid-liquid phase separation as an initial step in the gelling of the present amylopectin solutions, but this seems unlikely as the processes involved are clearly very slow, and become even slower as the amylopectin concentration falls. An alternative approach to the data handling which was also undertaken extended the analysis to be performed for the wider angle Bragg diffraction peak, and treated the aggregation phenomenon as linked closely to polymer crystallisation from solution.This A. H. Clark required the application of the well known Avrami the~ry,~~.~’ and to achieve this, a kinetic progress variable F was defined by the relationship: 1 -F = [I(q, t = co)-I(q, t)]/[I(q,t = Go) -I(q, t = O)] where the intensity values are taken for a given sample (and for a specific choice of q, i.e. channel number) from extrapolated curves similar to those in Fig. 4. The quantity (1 -F) was then plotted us. In t, a typical result shown for the 30.2% system in Fig. 5, which includes data corresponding to four channel values (10, 20, 30 and 40). Inter-estingly, all four of these data sets fall closely on a single master curve, a result which was found also for the other five system concentrations considered (including the less certain 19.9% data).All six master plot superpositions were then individually averaged to produce the final six curves presented in Fig. 6. As can readily be demonstrated by the simple translation of these functions along the In t axis, they are similar (though not 1.0-0 0.8 -0.6 -% 8 F 0.0-1 -2 0 2 4 6 8 10 12 14 In (t/min) Fig. 5 (1 -F) function (for definition, see text) us. In t for the 30.2% gelling system. A super-20 (O),30 (A)and 40 (V).position of results is shown based on intensity data at channels 10(.), 1.0 -0.8-0.6: v 0.4 - 0.2 - 0.0 - 80.0.. + I I I 1 0 5 10 15 In (tlmin) Fig. 6 Average (1 -F) functions for all six gel concentrations (44.2-19.6% w/w, from left to right) plotted against In t.The open squares indicate the estimated gel times (and hence gel points) for each system concentration. 86 Microphase-separated Biopolymer Gels identical). The differences which occur may not be highly significant, however, when errors in the original extrapolations of the intensity-time data are realistically assessed, Quantitative examination of the master plots of Fig. 6 provided further information. For example, by plotting the times required for (1 -F) to reach 0.5 us. the correspond- ing concentrations, in log-log form, a power-law dependence on the concentration is established with an exponent equal to -4.1 0.2. This is not as negative a power law as was found earlier for the gel times, but, as can be seen from Fig.6 which also records the locations of gel times on each transition curve, this difference is to be expected, as the gel points do not (and are not expected to) occur at a constant value for (1 -F). The F value at which gelation occurs clearly moves to higher values of In t as the concentra- tion falls, and will eventually reach unity at the critical gel concentration. This observ- ation, which is consistent with cascade theory,2 emphasises an important difference between F [which measures the progress of the system from zero initial cross-linking (F = 0) to some concentration-dependent upper cross-linking limit (when F = l)] and the quantity a of the cascade theory. The latter is the fraction of potential cross-linking sites in the system which have reacted at any given time.Thus, at any concentration, a is expected to pass from zero to some upper limit (<1) dependent on the concentration and temperature. For simple homogeneous gels, involving random cross-linking, the gel point is expected to be reached when (and indeed if) a attains the value l/cf-l), where f,the so-called functionality, is the number of potential cross-linking sites per associating polymer molecule. Since F = 1.0 when the concentration-dependent, and therefore vari- able, upper limit of a is achieved, the gel threshold [a = l/(f-l)] is arrived at for different values of F as the concentration varies. A second quantitative analysis of the data in Fig. 6 involved the application of polymer crystallisation theory in the form of the Avrami model.To do this, the basic Avrami equation (1 -F) = exp( -kt") was least-squares fitted to the data of Fig. 6. Here k is a rate constant taking into account processes of both nucleation and multidimen- sional growth, and n is the Avrami exponent often interpreted in terms of this dimen- sionality. Such fitting showed that, whilst the initial portions of the master curves [i.e. (1 -F) in the range 1.0-0.51 could be satisfactorily described using the simple exponen- tial form of the Avrami equation (n = l), with rate constants k showing the expected C-4.1 concentration dependence, much poorer fits were achieved in all cases where the entire time dependence was included. In this case, the best fits achieved were for values of n < 1, and scattered about an average of ca.0.9 (minimum result, n = 0.74: maximum, n = 0.96). This failure of the Avrami model to describe the master plot data adequately was particularly evident at the longest times for each master plot, where the experimen- tal (1 -F) result fell off more slowly than could be accommodated by the theory. As Mandelkern26 has pointed out, however, such a situation is to be expected for polymer crystallisation from solution, as this is a much more complex process than crystallisation from the melt, i.e. the event originally addressed by the Avrami theory. In consequence, in the present case, no particular physical significance will be attached to the low n values which emerge, but it is worth noting that similarly low Avrami n values have been obtained in the past for starch systems, by authors studying processes of starch retr~gradation.~~-~' Time Dependence of the Bragg Difiraction Peak (Channels 300-500) The analysis of the last section focussed on small-angle scattering only which, consider- ing the 4 range involved, and the corresponding distance scale probed of ca.1-3 amylo-pectin diameters, can be assumed to be a local monitor of the progress of molecular aggregation. In applying the Avrami analysis to this data, the assumption was made implicitly that the growth of crystallinity correlated closely with this cross-linking event. Though this assumption seems to be supported by previous X-ray investigations2'V2' of A.H. Clark more conventional amylopectins, it may not be true in the present case, and must be examined further. The extent to which a correlation of this kind exists is now investi- gated by direct reference to the behaviour of the B-form Bragg peak at ca. 390 channels in the wider angle part of the small-angle X-ray scattering data. For each sample concentration studied, scatter curves at progressively increasing times were plotted over the angular range 300-500 channels, within which the Bragg peak occurs, and empirical monotonic backgrounds subtracted out. The areas of the residual diffraction peaks were then estimated, and results plotted against In t, as was done for the small-angle data in Fig. 4. The poorer quality of the Bragg peak data, however, and its tendency to level out less convincingly at long times, even in the case of the highest sample concentrations considered, meant that data extrapolation was less certain in this wide-angle situation. In practice, such extrapolation was attempted for the three highest concentration cases only, this data being then converted to a (1 -F) us.In t form, for the purpose of comparison with the corresponding master plots of Fig. 6. A typical result, in this case for the 40.6% system, appears in Fig. 7, which shows not only the original 40.6%master plot based on the small-angle data, but three estimates of the (1 -F') function for the Bragg peak area, based on slightly different extrapolation procedures. It is concluded that the growth of crystallisation within the gels does indeed correlate well with changes in small-angle scattering, particularly during the early stages of the gelling process, but this correlation may break down at the longest times, crys- tallisation seeming to continue when changes in the small angle X-ray scattering has ceased.This last conclusion is open to some doubt, however, not only because all curves in Fig. 7 are subject to an uncertainty relating to the extrapolation errors, but because the small-angle limit of the X-ray data represents a rather arbitrary equipment- determined cut-off. Not surprisingly, Avrami analyses of the Bragg peak (1 -F)us. In t functions, which were also attempted, led to much the same conclusions as were reached previously for the data of Fig.6. Again, n values were lower than unity, and fits to the data are not very good at long times. Wide-angle X-RayDfiaction Measurements A diffraction pattern for the dry modified waxy maize starch (prior to dissolution) is shown in Fig. 8, together with corresponding results for a 30.2% fully hydrated gel 1 1.o OOOOOOOQrl 0 0 0.8 8=I2 00.6 -=05 8 F 0 0.4 -=*0 -2 0 2 4 6 8 10 12 14 16 In (tlmin) Fig. 7 Estimates (0)of (1 -F) vs. In t for the 40.6% w/w system, based on three different extrapolations of Bragg peak (20 x 5.5") area vs. In t, are compared with the corresponding average (m) obtained from X-ray scattering at small angles Microphase-separated Biopolymer Gels 5 10 15 20 25 30 35 28I degrees Fig.8 28 wide-angle X-ray diffraction patterns for A, the 30.2% gel cured at 5 "C for 1 month; B, the same gel subsequently dried at ambient temperature for 1 week; and C, the powder pattern of the original undissolved modified waxy maize starch stored for one month (In t FZ 10.8),and for the same gel dried for a further week. Because modification of the waxy maize starch was carried out under dry conditions, this start- ing material retains the A-form crystallinity of the parent granular starch. The gel samples, however, show clear indications of the presence of the alternative (and anticipated) B-form, though, naturally, the signal to background ratio is low for the intact undried gel. Similar results were obtained for the whole range of gels studied (C = 14-55% w/w) with the gel crystallinity becoming more evident on increasing the amylopectin concentration.Some semi-quantitative analysis of the data was attempted. For example, the pat- terns from the dried gels, and the original amylopectin raw material, were divided into contributions from crystalline and amorphous diffraction, by hand-drawing empirical, but reasonable 'amorphous scattering' envelopes, and the proportion of crystallinity in these samples roughly assessed. The result of ca. 30.0% for the starting material was not particularly surprising for a native granular starch, but the fact that a similar estimate was found for the dried gels (29-35.0%, depending on the original amylopectin concentration) was less expected. If it is assumed that little additional crystallisation takes place on drying the gels at ambient temperature, this result suggests that, on gelation over a long period at 5 "C,recrystallisation of the modified amylopectin occurs to virtually the same level of the original granular starch (albeit in a different crystalline form).Although, in keeping with the Bragg peak results, some increase was found for the crystal content of the dried gels as a function of the gelling starch concentration, this was not large, a point which is perhaps surprising in the light of the very different starch concentrations giving rise to the hydrated gels, and the well established idea from cascade the~ry~*~ that the limiting amount of cross-linking (and hence crystal content) should vary considerably with the concentration of the network building species (i.e.limiting a depends strongly on C).That the starch content of the dried gels has probably A. H.Clark not been much altered by the drying seems confirmed by comparing measured areas for the prominent diffraction peak at 28 z 17", for the various hydrated gels, and for the corresponding powdered gels, after they have been normalised to account for the differ- ent concentrations. These normalised amplitudes varied insignificantly from one sample to another, showing that, to a rough approximation, the crystal content per unit starch mass was similar in all cases. Conclusions The present gel time-concentration results show that, in its simplest form, the cascade theory approach is unable to account for the behaviour of the current modified amylo- pectin system.The high negative power laws found in practice are much stronger func- tions of concentration than the C-I result predicted by the elementary the~ry.~ Some of this discrepancy might be accounted for by removing the assumption of solution ideal- ity, implicit in the simple cascade approach, and by a more careful choice of solute concentration units, The X-ray studies suggest that the problem is more fundamental, however, and relates to the occurrence of crystallisation as the source of molecular cross-linking. The course of such crystallisation is shown to follow a power law depen- dence on concentration, which is comparable to that found for the gel times, and the sensitivity of gel-time results to temperature would also be consistent with a rate-determining crystallisation event.As Mandelkern has pointed out,26 crystallisation of polymers from solution is a complex event based on nucleation and growth, which is likely to show a strong sensitivity to concentration and temperature variables. Use of the cascade description in this situation would seem only feasible if the rate constants intrinsic to this theory can be realistically reformulated to include such concentration and temperature effects. While such a reformulated cascade approach might apply to conventional poly- saccharide gels showing no real phase separation, i.e. gels based on a fibrous, but still molecular network, it is unlikely to be appropriate to the prssent amylopectin systems, which undoubtedly show phase separation in addition to polymer crystallisation.Simple observation shows that the present gels are highly opaque and, more crucially, exami- nation of these gels by both light and electron microscopy (results to be published separately) provides details of their phase-separated microstructures. Light microscopy, for example, carried out during gelation, indicates slow initial formation of sub-micrometre sized particles (0.1-1.0 pm) during the lag periods of the master curves of Fig. 5-7, which eventually become more concentrated, and associate, as the gel points are approached. Transmission electron microscopy confirms the presence of these par- ticles, but suggests that they are themselves porous, with a fine network structure arising from the association of the individual amylopectin monomers.The larger submicrometre particles are present in subgelling systems too, but here they fail to percolate through space, congregating instead into macroscopically phase-separated regions and, presum- ably, eventually precipitating from solution. Thus, it seems that the submicrometre par- ticles have only limited solubility, and are capable of both aggregation phenomena and ,solution demixing. Here again, a departure from the situation anticipated by the cascade theory description is indicated. For the present gels, it appears that the existence of a critical concentration has more to do with the percolation of a macroscopic phase, than .with the approach of an aggregation phenomenon to a critical threshold.The above description from microscopy is, of course, consistent with the present X-ray scattering curves from the gels, which suggest a rather constant local environment for individual amylopectin molecules, located as most of them appear to be within fairly similar large aggregates (irrespective of the initial solution composition). This situation is also consistent with the observation from wider angle diffraction that total crystallinity is not a strong function of the initial amylopectin concentration, and that it is quite 90 Microphase-separated Biopolymer Gels extensive. If most crystallinity develops during the formation of the primary particles, and much less is involved in the association process leading to percolation, this is exactly what is expected to be found.As is clear from Fig. 3, however, the increased particle density at higher concentrations, and the increased interaction between particles which accompanies this, eventually have an influence on the gel small-angle scattering, and probably also lead to somewhat higher levels of crystallinity. In keeping with this picture, the kinetics of gelation of the modified amylopectins are likely to be dominated by the kinetics of the development of the submicrometre subunits, and questions natu- rally arise concerning the nucleation and growth processes likely to underlie this, and whether, for example, heterogeneous nucleation plays a part.In this last context, it is interesting to note that the related maltodextrins15-’7 gel relatively much faster than the present materials. These substances are similar with respect to the amylopectin mol- ecules present but, in addition, they contain a low-molecular-weight amylose fraction. In consequence, it is tempting to assume that the linear material acts to nucleate the crys- tallisation processes associated with particle formation. In this respect, it is significant that the addition of low molecular weight amylose to the present amylopectins (which are relatively free of unbranched glucose polymers), has an accelerating influence on their gela tion. Returning finally to the original question posed by this work (i.e.whether the cascade approach can be modified to treat systems gelling by phase separation and crystallisation) the answer seems to be ‘possibly’ for the system examined above, but not necessarily in general, for all situations. Whilst the original cascade theory sought to be very general, and widely applicable, it is clear that its modification to treat the amylo- pectin gel systems studied here would have to be rather specific. In this case it would be necessary to formulate the kinetics of primary particle formation, and then superimpose on this the more conventional cascade ideas of random cross-linking of these subunits. Such a description would be in a sense ‘tailor-made’, and would probably not be imme- diately transferable to the formation of a microphase-separated heat-set globular protein gel (e.g.an ovalbumin gel). At this stage, at least three future actions seem possible regarding the development of the cascade description of biopolymer gelation. One is to pursue the extension of the original approach to treat molecular networks based on microcrystalline junction zones. This will require the introduction of some form of crys- tallisation kinetics. A second action is to develop a specific approach for the current modified amylopectin and its close relatives, and a third, and very broad option, is to pursue further the experimental studies of other types of phase-separated systems, to learn more of the diversity of possible gelation mechanisms. These are all worthwhile objectives, and future work will develop in these directions.I thank Mr. S. Sahota and Mr. M. Kirkland of the Colworth Laboratory, for communi- cating preliminary light and electron microscopy data for the gelling amylopectin systems. I am also grateful to the National Starch and Chemical Corporation, Bridge- water, New Jersey, for the gift of the amylopectin samples. References 1 A. H. Clark and S. B. Ross-Murphy, Br. Polym. J., 1985,17, 164. 2 A. H. Clark, in Food Structure and Behaviour, ed. J. M. V. Blanshard and P. J. Lillford, Academic Press, London, 1987, p. 13. 3 A. H. Clark, S. B. Ross-Murphy, K. Nishinari and M. Watase, in Physical Networks, Polymers and Gels, ed. W. Burchard and S.B. Ross-Murphy, Elsevier Applied Science, London, 1990, p. 209. 4 A. H. Clark, Polym. Gels Networks, 1993, 1, 139. 5 A. H. Clark, K. T. Evans and D. B. Farrer, Int. J. Biol. Macromol., 1994, 16, 125. 6 A. H. Clark, Carbohydr. Polym., 1994,23,247. 7 M. Gordon, Proc. R. SOC.London A, 1962,268,240. 8 P. J. Flory, J. Am. Chem. SOC., 1941,63,3083. A. H. Clark 9 W. H. Stockmayer, J. Chem. Phys., 1943,11,45. 10 J. R. Hermans, J. Polym. Sci., Part A: Polym. Chem., 1965,3, 1859. 11 T. M. Bikbov, V. Ya. Grinberg, Yu. A. Antonov, V. B. Tolstoguzov and H. Schmandke, Polym. Bull., 1979, 1, 865. 12 D. G. Oakenfull and A. Scott, in Gums and Stabilisers for the Food Industry-3, ed. G. 0.Phillips, D. J. Wedlock and P. A. Williams, Elsevier Applied Science, London, 1986, p. 465.13 D. G. Oakenfull and V. J. Morris, Chem. Ind., 1987, 16, 201. 14 D. G. Oakenfull and A. Scott, in Gums and Stabilisers for the Food Industry4, ed. G. 0.Phillips, D. J. Wedlock and P. A. Williams, IRL Press, Oxford, 1988, p. 127. 15 F. Reuther, P. Plietz, G. Damaschun, H-V. Purschel, R. Krober and F. Schierbaum, Colloid Polym. Sci., 1983,261,271. 16 F. Reuther, G. Damaschun, Ch. Gernat, F. Schierbaum, B. Kettlitz, S. Radosta and A. Nothnagel, Colloid Polym. Sci., 1984, 262, 643. 17 Ch. Gernat, F. Reuther, G. Damaschun and F. Schierbaum, Acta Polym., 1987,38,603. 18 C. M. Durrani and A. M. Donald, Polym. Gels and Networks, 1995,3, 1. 19 C. M. P. Durrani, Ph.D. Thesis, University of Cambridge, 1992. 20 S. G. Ring, Int. J. Biol. Macromol., 1985, 7, 253. 21 S. G. Ring, P. C. Colonna, K. J. I’Anson, M. J. Kalichevsky, M. J. Miles, V. J. Morris and P. D. Orford, Carbohydr. Res., 1987,162,277. 22 A. H. Clark and C. D. Tuffnell, Int. J.Peptide Protein Res., 1980,16, 339. 23 J. La1 and R. Bansil, Macromolecules, 1991, 24, 290. 24 M. Avrami, J. Chem. Phys., 1939,7, 1103. 25 M. Avrami, J. Chem. Phys., 1940,8,212. 26 L. Mandelkern, Crystallisation of Polymers, McGraw-Hill, New York, 1964, ch. 8. 27 R. G. McIver, D. W. E. Axford, K. K. Colwell and G. A. H. Elton, J. Sci. Food Agric., 1968,19,560. 28 T. Fearn and P. L. Russel, J.Sci.Food Agric., 1982,33, 537. 29 R. D. L. Marsh and J. M. V. Blanshard, Carbohydr. Polym., 1988,9, 301. Paper 51040545;Received 22nd June, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100077

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,101,93-104 Rheological Expression of Physical Gelation in Polymers Claudius Schwittay, Marian Mours and H. Henning Winter Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA Polymeric materials at the liquid-solid transition exhibit unusual simplicity and regularity in their relaxation pattern. This expresses itself in a self-similar relaxation modulus G(t)= St-” at long times A, -= t < co,where & is the characteristic time for the crossover to a different relaxation regime (e.g. crossover to glass transition or entanglement region). Rheological fea- tures of liquid-solid transitions are very similar for chemical and physical gelation: (1) broadening of the relaxation time spectrum, (2) divergence of the longest relaxation time (with an upper cut-off for physical gels) and (3) self-similar relaxation patterns, We have borrowed terminology from chemi- cal gelation and applied it to an example of physical gelation: the isothermal crystallization of isotactic polypropylene.The transition through the gel point has been investigated by dynamic mechanical experiments. The influ- ence of temperature and crystallization rate have been studied. The degree of crystallinity (estimated by the Avrami equation) at the gel point was very low (6-15% depending on the crystallization temperature). Introduction A cross-linking material is able to form molecular clusters which grow in size. A tran-sition from a liquid to a solid state occurs when the largest cluster diverges in size at a sufficiently high degree of cross-linking, pc.The material at this transition [gel point (GP)] is known as a critical gel. The long-range connectivity in the material can be arrived at by different mechanisms : (1) during chemical gelation permanent covalent bonds connect molecular strands into a three dimensional network while (2) in physical gelation the ‘bonds’ are of a temporary reversible nature (e.g. crystalline regions, hydro- gen bonding, ionic clusters, phase separated microdomains) and the average lifetime of such a bond is long compared with the observation time. The molecular weight distribu- tion diverges at the GP in chemically cross-linking systems but remains finite in physi- cally cross-linking materials.The difference between chemical and physical bonds can be clearly seen in a solvent extraction experimznt : it works well for identifying the GP of a chemically cross-linking system which before the GP (called ‘sol’) can be completely dissolved in suitable solvents and after the GP (‘gel’) cannot be completely dissolved because of the permanent network structure. This is different for a physical gel which can still be dissolved in suitable solvents, even beyond the physical GP. This makes it difficult to define the physical gel point. Rheologically, the GP (in chemically cross-linking systems) is defined as the instance during cross-linking at which the zero shear viscosity, uo, diverges and the equilibrium modulus, G,, is still zero.Before the GP, the zero shear viscosity may be measured in a constant rate experiment at low shear rates, while the equilibrium modulus after the GP is usually extrapolated from the long time response of a step strain experiment. The closer the material is to the GP the longer the stress needs to reach its steady state. Constant-rate experiments for measuring qo necessarily impose large strain, even at low 93 Rheological Expression of Physical Gelation in Polymers shear rates, which tends to interfere with the growth of the three dimensional structure. This is especially true for physical gelation since the bonds are reversible. Also, very near the GP the time to reach steady shear flow might be similar or longer than the lifetime of the physical bond, therefore the zero shear viscosity might remain finite at the GP because the material can still completely relax.Furthermore, transient experiments by which the zero shear viscosity before, and the equilibrium modulus after, the GP can be measured are not possible very near to the GP due to instrument limitations and can therefore only yield an approximate gel time. The viscosity measurement is therefore quite inconclusive. Critical gels exhibit an unusually simple and regular relaxation behaviour which expresses itself in a self-similar relaxation modulus which was experimentally discovered by Chambon and Winter :’ G(t)= St-” (1) where S is the gel stiffness and n the critical relaxation exponent. The self-similar relax- ation behaviour is valid in the terminal zone at long times, Lo < t < 00.The lower limit corresponds to some relaxation time that characterizes the crossover to a different relax- ation mechanism. This can be the glass transition or entanglement behaviour depending on the molecular weight of the precursor.2 Since the longest relaxation time diverges at the chemical gel point, the self-similar relaxation is valid to infinite time. The self-similar relaxation behaviour can also be expressed by a power-law relaxation time spectrum, known as a Chambon-Winter spectrum : The relaxation exponent may have values 0 < n c 1. Relaxation exponents <0 would lead to an unphysical increase in the relaxation modulus with time and the upper limit of n < 1 guarantees the divergence of the viscosity at the GP.Consequently, the dynamic storage and loss moduli, G’ and G”,scale with frequency, CI), at the GP: nnG’=--GI‘ -ST(1 -n)cos -d; l/& > w > 0 (3)tan 6 2 This leads to a loss tangent which is independent of frequency at the GP and closely related to the relaxation exponent n: nn tan 6 = tan -(4)2 The gel stiffness S was found to follow a simple power law equation3v4 S = GoA: (5) where Go is close to the equilibrium modulus of the fully cured material and Go Lo takes values in the order of the zero shear viscosity of the precursor. The frequency independence of the loss tangent is most easily and reliably used to determine the gel point.5 Dynamic mechanical experiments can be performed during the transition from liquid to solid.These experiments do not disturb the ongoing cross- linking process since usually only small strains are imposed on the materials. Many researchers use the crossover of G’ and G” as an indicator for the GP6which is wrong in general. The only exception in which the true GP and this crossover coincide is the critical gel for which the relaxation exponent n = 0.5.7 The independence of the loss tangent of frequency therefore provides the most reliable and generally valid method to determine the gel time. The method has been widely tested on chemically (e.g. PDMS,1,8 C. Schwittay et al. PU,' epoxies") and physically (e.g. PVC,ll thermoplastic elastomeric PPI2) cross-linking systems.Depending on the nature of these systems relaxation exponents span- ning almost the entire possible interval of 0 < n < 1were found. The importance of physical gelation (particularly crystallization) is, for instance, evident when considering the fact that many polymers solidify into a semicrystalline morphology. Their crystallization process, driven by thermodynamic forces, is hindered due to entanglements of the macromolecules and the crystallization kinetics is restricted by the molecular diffusion of the polymers. Therefore, crystalline lamellae and amorp- hous regions coexist in semicrystalline polymers. The formation of crystals during the crystallization process results in a decrease of molecular mobility since the crystalline regions act as cross-links which connect the molecules into a sample-spanning network.Dynamic mechanical experiments are known to be a sensitive measure of these pro- cesses. Injection moulding of semi-crystalline polymers is an example of a process which is dominated by crystallization. Particularly, the rate of crystallization is important. During injection of the molten polymer into the mould, a layer of polymer gets depos- ited at the cold walls of the mould where it starts solidifying while exposed to high shear stress due to ongoing injection. This is an important part of the process since the wall region later forms the surface of the manufactured product. A high shear stress during this crystallization might rupture the already solidified surface layers resulting in surface defects. This could be avoided if the crystallization behaviour were known together with its effect on the developing strength of the material.The polymer could be modified to adjust its crystallization behaviour or a different polymer could be used or developed. A suitable tool to monitor the time dependence of crystallization processes and thus to select the best material and processing conditions is rheology. The characteristic crys- tallization time for each material can be tailored to the given moulding circumstances. The stretching and shearing during the injection moulding have a large effect on crys- tallization.22 This will not be addressed in the following which is solely concerned with crystallization under static conditions.Objective The rheological analogy between physical and chemical gelation has been shown pre- viously for certain crystallizing polymers.' It is interesting to study the crystallization behaviour of a commercially important isotactic polypropylene to confirm this analogy. If the GP can be clearly defined and measured in rheological experiments it can be taken as a reference state for the development of the properties of the material. The gel time has to be measured and related to important crystallization conditions, e.g. temperature. Furthermore, the material stiffness at and after the gel point has to be studied and compared with structural parameters (crystal size, degree of crystallinity). It is important to show how rheology can be a powerful tool to study and monitor the crystallization behaviour of semi-crystalline polymers.Experimental The material used in this study was a commercial capacitor-grade isotactic poly- propylene (iPP). Polypropylene was selected not only because of its practical importance but also because it is known to exhibit slow crystallization kinetics (which makes it accessible to observation). iPP may achieve a high degree of crystallinity reported degrees of crystallinity up to 65-75%). The iPP used had the following charac- Rheological Expression of Physical Gelation in Polymers teri~tics:'~melt flow index (MFI) 4.3 g (10 min)-' at 230"C, M, = 79300, M, = 476 000. The glass transition temperature and the melting peak temperature were mea- sured using a Perkin Elmer differential scanning calorimeter (DSC-7).For this, the DSC sample was melted for 20 rnin at 200°C and then quenched in liquid nitrogen. The calorimetric scan was conducted at a heating rate of 15 K min-'. Experiments under the rapid crystallization conditions applied in industrial pro- cesses are extremely difficult to We therefore slowed down the crystallization process by setting the supercooling conditions accordingly. The rheological experiments were performed in a Rheometrics mechanical spectrometer RMS800 using parallel discs with a diameter of 25 mm. Samples were prepared by moulding the iPP pellets for 10 rnin at 200°C under vacuum in a Carver laboratory press.The samples were then allowed to cool slowly, still under vacuum. Moulded samples were stored in a vacuum oven at about 120 "C to prevent moisture absorption. Utilizing disposable fixtures on the RMS800, the sample was heated to 180°C. At this temperature the iPP is already molten. The sample disc was squeezed by ca. 30 pm in order to smooth the surface of the sample and the edge was carefully cut off. Then the samples were melted in the rheometer at 210°C for 20 min under nitrogen. iPP is thermally stable, with respect to molecular weight, up to 230°C and the breakdown to volatile products is insignificant below 300 "C in the absence of oxygen.15 The melting temperature in this study was well below these temperatures. After melting, the samples were cooled to the experimental temperature, still in a nitrogen atmosphere.The cooling took ca. 10-15 min since a temperature undershoot below the experimental temperature had to be avoided. Other- wise crystals would have formed faster than at the intended temperature,I6 thus short- ening the crystallization time considerably. The actual rheological experiment was started as soon as the experimental temperature was reached. This instance was taken to be the beginning of the crystallization (t = 0). The cooling time was not taken into account since a few min at a temperature some degrees higher than the actual testing temperature caused only negligible crystallization as will be shown below. For the same reason the experiment had to be started at the testing temperature and not several degrees higher.To avoid the slow and inaccurate response to programmed temperatures and temperature changes of the instrument's heating system, the temperature was con- trolled manually during the melting and cooling process. The gap setting was adjusted to accommodate the density changes during crystallization. Cyclic frequency sweeps were performed once the experimental temperature was reached. The frequency window between 0.01 rad s-l (at T = 146°C) or 0.1 rad s-' (at all other temperatures) and 200 rad s-was repeatedly scanned taking 3-6 data points per decade depending on the temperature. The experimental temperatures were T = 130, 135, 139, 146 and 151 "C. The experiment was stopped by the instrument as soon as the torque imposed on the transducer became too large due to the ongoing crystallization.The initial strain of ca. 10% (ensuring sufficiently high torque values) was manually decreased during the experiment to stay within the linear viscoelastic region and below the maximum torque of the instrument. The experimental data was analysed using IRIS17 and GELPRO" software. Experimental Results The DSC results are shown in Fig. 1. The peak in the thermogram was used to estimate the melting temperature, T, = 163 "C. A closer look at the temperature region between -50 and +50 "C revealed a glass transition temperature of < = -13"C. The master curves of the storage and loss moduli for the investigated polypropylene were determined at a reference temperature of ref= 140"C using time-temperature superposition1' as shown in Fig.2. Two frequency sweeps were performed above the C.Schwittay et al. -50 0 50 100 150 200 TI "C Fig. 1 DSC thermogram of a 9.5 mg iPP sample, conducted at a heating rate of 15 K min-' after quenching in liquid nitrogen. The melting peak can be clearly seen around T, = 163°C. x1 = 124.7 "C,x2 =173.7"C,peak =162.7"C,area =703.4 m. melting temperature at T = 171 and 181 "C, all others were taken below T,. The first frequency sweep of the CFS experiments after quenching to a temperature Kxp<T, approximately characterizes the melt behaviour since the crystallization process has not started to a substantial extent. o 130 A 135 .* Q 139 CII 140 1.146 x i59 0 171 u 181 102;-- 1 -2 I I Ill1 II I -1 I I I ill I/' 0 I I I IIIIII ' 1 r m I m 2 m r 3 10 10 10 10 10 10 a T olrad s-' Fig. 2 Master curve of isotactic polypropylene at Kef = 140"C. G' is represented by open symbols and G" by filled symbols. The frequency sweep data from different temperatures, given in the figure, are depicted by different symbols. Rheological Expression of Physical Gelation in Polymers 6 10 5x 10 3 10 ° ~1 2 1 10 I 1 0 5000 10000 15000 20000 25000 30000 35000 t /s Fig. 3 Cyclic frequency sweep data at a crystallization temperature of 146°C. Different fre- quencies are depicted by different symbols. Open symbols represented G, filled symbols represent G".1 (.ocl a 1 0 6000 12000 18000 24000 30000 tls Fig. 4 Loss tangent, tan 6, us. time during crystallization at KXp= 146°C. Different symbols depict different frequencies. The gel time can be estimated from the crossover of the lowest fre- quency data. C. Schwittay et al. 1 1~1 34 min o 6h i’min A 6h 20miri 0 6h 40min 9 6h 57min x 7h 25min 04 i Ii -1 10 1 I I I1(1(I( I 1 I1/11l( I I I11llll 1 I I IIllll 1 1 1 lllll~ -2 -1 0 1 2 3 10 10 10 10 10 10 o/rads-’ Fig. 5 Interpolated tan 6 us. frequency at different instances during the crystallization process (Kxp=146 “C).The estimated gel time is marked by the horizontal line. The relaxation exponent n is found to be equal to 0.78.A typical data set from the dynamic mechanical experiments is shown in Fig. 3 (Kxp=146“C).The plot shows the evolution of storage and loss modulus data at differ- ent frequencies with crystallization time. A long induction period is evident before the moduli start to rise sharply. At the beginning of the experiment, the low frequency storage modulus, G’, is smaller than the corresponding loss modulus, G”. As the crys- tallization proceeds, both moduli grow. Initially, G’ and G” increase in a nearly parallel manner but after some time the storage modulus starts to grow faster than the loss modulus. Eventually, G’ exceeds G” and, at the end, G’ dominates the experiment. The GP can be extracted by replotting the data in terms of the loss tangent tan S, Fig.4, to look for frequency independence. As the master curve reveals, the entanglement zone of this broadly distributed iPP extends to frequencies of about CL) =1 rad s-l. This is nearly true throughout the entire investigated temperature range due to values of aT near unity. Since the loss angle at GP is independent of the frequency only in the ter- minal zone, the frequency independence will show only in the range of co <1 rad s-l. The experimental frequency range is restricted by the equipment available and the experimental time in relation to the rate of change. Therefore the terminal frequency is about 0.1 <co <1 rad s-l for all temperatures except T =146°C. In Fig. 4, the GP is taken as the instance at which the tan 6 curves of the three lowest frequencies cross.The loss tangent could also be plotted us. frequency at interpolated times, Fig. 5. The time at which tan 6 is independent of the frequency at the low end of the frequency window is marked by a horizontal line. The gel times at the other crystallization temperatures were estimated in a similar way. Rheological Expression of Physical Gelation in Polymers 1o2 10' 5 c loo lo-' 10 15 20 25 30 35 degree of supercooling, ATIK Fig. 6 Semi-logarithmic plot of critical gel time us. degree of supercooling. The line represents an apparent linear fit of the data points. Data at 130°C (AT = 33 K) are estimated. Analysis and Discussion The logarithm of the critical gel time (estimated from the conditions tan S = constant) us.the degree of supercooling, AT = T, -Kxp,is depicted in Fig. 6. For Kxp= 130"C, the possible range of the critical gel time is shown since an exact value could not be obtained due to the fast crystallization rate. The data points can be fitted by an appar- ent linear function resulting in t, = A exp( -$) The fit results in parameter values of A = 215.6 h and B = 4.71 K. Thermodynamically, AT may be interpreted as the driving force of crystallization, large values resulting in a short crystallization time. B is an energy, normalized by the universal gas constant. The temperature range for which this equation is valid could not be manifested. It is evidently valid in the investigated temperature range of 130 < (T,,,/"C) < 151.Obvi-ously it does not apply to temperatures above T, where no gel can be formed. At temperatures significantly lower than 100"C the influence of the glass transition will increase. The diffusion of the macromolecules is hindered and spherulites are not able to form. Another uncertainty is the degree of supercooling itself. The thermodynamic equi- librium melting temperature, IT:, is not the same as the measured melting temperature, T,. Tz can be extrapolated using DSC data of melting peaks from material that crys- tallized at different temperatures.20921 Practically, the difference between T: and T, is of little importance. The thermodynamic melting point describes the melting temperature of an ideal infinite crystal without any defects.Polymers however are never able to form C. Schwittay et al. an ideal crystal due to entanglements, chain defects, etc. Furthermore, a different melting temperature would result in a purely horizontal shift in Fig. 6 not affecting the impor- tant parameter B. The degree of crystallinity at the GP can be estimated using the Avrami equation 1 -= exp(-KPA) (7) It combines the relative crystallinity by volume with a rate constant K and the crys- tallization time. The rate constant depends on crystallization temperature and the Avrami exponent is related to the crystal growth dimension.' The temperature 5922 dependence of polypropylene crystallization was studied by Kim et ~1.l~They measured of neat iPP at distinct temperatures as a function of the crystallization time.At a temperature of 129.7 "C they found the relative crystallinity to be zero after 1.5 min and Q.02after 5 min. At a temperature of 136.1"C, Vc remained nearly zero for more than 10 min. They evaluated an isothermal rate constant K for their calculated values of the Avrami exponent, nA, for nA between 2.45 and 2.74. A parameter set for K was extrapo- lated in this study using linear extrapolation fits in a log K 0s. T plot which lead to a simple equation for the temperature dependence of the rate constant K log K = A + BT,,, (8) with A = 96.99 and B = 0.25 K-'. The Avrami exponent nA was set to be nA = 2.5 = const. The gel time was taken from eqn. (6). The result is a nearly linear decreasing degree of crystallinity with increasing crystallization temperature as shown in Fig. 7.The error bars correspond to an error in the gel time of 1 min. The degree of crystallinity at the gel point is remarkably low and lies between 6 and 15%. The critical gel time was determined by plotting the loss angle 0s. frequency at inter- polated times. The time at which tan 6 is frequency independent in the terminal zone 0.20 .c1 .-!= 0a -a,cn 0.15 c (I! (?b s c.-t-.--a c u)e 0.10 a, .-c -a 2! 0.05 degree of supercooling,ATIK Fig. 7 Degree of crystallinity at the gel point us. degree of supercooling. Data calculated from eqn. (8). 102 Rheological Expression of Physical Gelation in Polymers was taken as the critical gel time.The frequencies were varied between 0.01 rad s-' (forKXp=146°C) or 0.1 rad s-' (for all other temperatures) and 200 rad s-l. Only a small part of the terminal zone is covered by this frequency range. Therefore the determi- nation of t, might lead to a systematic error since the real critical loss tangent might actually be found at shorter times and higher values of It. The effect of such an error is shown in Fig. 8. For T =146 "C the critical gel time when using a frequency range from co =0.1 rad s-' upwards is 417 min, when using the extended frequency range from co =0.01 rad s-', t, =380 min. Consequently the 'real' value of n is 0.78 (u>0.01 rad s-') instead of 0.65 (u>0.1 rad s-I).Measuring at lower frequencies, however, increases the experimental time further since At is approximately inversely proportional to o.This also increases the mutation numberg which is a measure for the total change of the measured property g during a single experiment of duration At. Long experimental times and high rates of change (here rates of crystallization) result in high mutation numbers. Rheological experiments usually require stable samples since the determination of dynamic properties are not instantane- ous. Recently it has become more and more common to examine materials with a changing structure such as polymers during gelation, phase transition, decomposition, polymerization, etc. since these materials have very distinct properties.A condition was that allowed measurements on samples with sufficiently slow mutation. (11:lo]c 6h 40min - + 6h 57min - X 7h 25min Oh 25min -1 10 I I I I IIIII I I 1 1 - -2 10 -1 10 10 wlrad s-' Fig. 8 Loss tangent, tan 6, us. frequency during crystallization at KXp=146°C. Different symbols depict the interpolation times given in the figure. The effect of a truncated frequency window on the estimation of t, is demonstrated with the two different lines (using urnin0.01 rad s-' and> urnin0.1 rad s-l, respectively, to determine t,).> C. Schwittay et al. Such experiments can be considered to be quasi-stable and therefore the mutation can be neglected. If this condition (small Nmu)is not satisfied, the rheometer may deliver false data or no data at all.The condition was checked for the dynamic experiments in this study. At the gel point, values for N,, were calculated (for G at w = 0.1 rad s-') to lie between 0.0045 =for qxp= 151 "C and 0.6565 for qXp 135"C.The mutation numbers for the loss modulus were less than half of the values determined for the corresponding storage modulus. Since the computed mutation numbers are smaller than one, it is safe to assume that the samples were quasi-stable. Dynamics data at crystallization temperatures below 135"C are not reliable, especially at low frequencies, since the crystallization rate is too high. Unfortunately, it is also impossible to measure at frequencies below 0.1 rad s-' for temperature below 140°C (this would result in N,, > 2).Therefore, the critical gel times determined from the tan S data at o > 0.1 rad s-' cannot be proven or possibly improved by measuring at fre- quencies further into the terminal region. It has to be assumed that the estimated gel times are too long (see Fig. 8). However since this is supposedly a systematic error it has no direct effect on the trends found in this study. Conclusions Cross-linking materials exhibit a self-similar relaxation pattern at the liquid-solid tran-sition which expresses itself in a power-law relaxation spectrum and modulus. The iso- thermal physical gelation (crystallization) of a semicrystalline polypropylene was found to behave similarly to chemical gelation.The dynamic moduli exhibited the same growth characteristics while changing from a viscous (G < G") to an elastic (G' > G") dominated phase. The dynamic moduli were followed at different crystallization tem- peratures over a wide range of frequencies. The time at which the material passes the gel point was found to decrease exponentially with the degree of supercooling below the equilibrium me1 t ing temperature. The degree of crystallinity at the gel point was estimated using the Avrami equation. It increases exponentially with the degree of supercooling. Very low relative crys- tallinities between 6 and 15% were found at the liquid-solid transition. This means that only a few physical cross-links are necessary to form a network that spans ('percolates') the entire sample.The crystallization rate greatly influences the rheological experiment. Low crys- tallization temperatures (below AT = 30 K) could not be investigated since the crys- tallization rate is too fast and the total change of investigated property is too large for reliable data to be obtained. However, rheology is shown to be a powerful tool to inspect and monitor crystallization processes. The project was supported by a grant from Raychem. M.M. gratefully acknowledges financial support of the German Academic Exchange Service (DAAD-Doktorandensti- pendium aus Mitteln des zweiten Hochschulsonderprogramms). The phenomenon of gel fracture as a reason for the occurrence of surface defects in injection moulding was introduced to us by Dr.Ye-Gang Lin. References 1 F. Chambon and H. H. Winter, Polym. Bull., 1985,13,499. 2 M. E. DeRosa and H. H. Winter, Rheol. Acta, 1994,33,220. 3 J. C. Scanlan and H. H. Winter, Macromolecules, 1991,24,47. 4 A. Izuka, H. H. Winter and T. Hashimoto, Macromolecules, 1992,25,2422. 5 E. E. Holly, S. Venkataraman, F. Chambon and H. H. Winter, J. Non-Newt. Fluid Mech., 1988,27, 17. 6 C. Y.M. Tung and P. J. Dynes, J. Appl. Polym. Sci., 1982,27,569. 7 H. H. Winter, Polym. Eng. Sci., 1987,27,22. Rheological Expression of Physical Gelation in Polymers 8 F. Chambon and H. H. Winter, J. Rheol., 1987,31,683. 9 H. H. Winter, P. Morganelli and F. Chambon, Macromolecules, 1988, 21, 532. 10 A. Apicella, P.Masi and L. Nicolais, Rheol. Acta, 1984,23,291. 11 K. te Nijenhuis and H. H. Winter, Macromolecules, 1989,22, 411. 12 Y. G. Lin, D. T. Mallin, J. C. W. Chien and H. H. Winter, Macromolecules, 1991,24, 850. 13 J. G. Cook, Handbook of Polyolefn Fibres, Merrow Publishing, England, 1967. 14 S. Piccarolo, J. Macromol. Sci.,Phys., 1992,31,501. 15 H. P. Frank, Polypropylene, Gordon and Breach Science, New York, 1968. 16 C. Y. Kim, Y. C. Kim and S. C. Kim, Polym. Eng. Sci., 1993,33, 1445. 17 H. H. Winter, M. Baumgartel and P. R. Soskey, in Techniques in Rheological Measurement, Chapman and Hall, London, 1993, ch. 5. 18 M. Mours, H. H. Winter, Rheol. Acta, 1994,33, 385. 19 J. D. Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1980. 20 J. I. Lauritzen and J. D. Hoffman, J. Res. Nut. Bur. Std, Sect. A, 1960,64, 73. 21 J. D. Hoffman and J. J. Weeks, J. Res. Nut. Bur. Std. Sect. A, 1962,66, 13. 22 G. Eder, H. Janeschitz-Kriegl and S. Liedauer, Prog. Polym. Sci., 1990, 15,629. 23 R. Hingmann, J. Rieger and M. Kersting, Macromolecules, 1995,28,3801. Paper 51033186; Received 23rd May, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100093

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

FU~U~UY 1996,101, 105-123D~SCUSS., GENERAL DISCUSSION Prof. Hoffmann communicated to Prof, Keller: You defined a gel as a chemically or physically cross-linked swollen network. While this definition seems to be sufficient for many systems, it does not correctly describe some other systems. Some peculiar systems certainly have a swollen network character, but nobody would call them a gel, because the systems are very fluid. A typical example of such a system is an L,-phase of a surfactant in water. It has a sponge-like structure and can be swollen. It has, however, a zero-shear viscosity that is only a few times higher than the water viscosity. For the definition of a gel we should therefore always emphasize besides the network character the long structural relaxation or a yield stress value.Prof. Keller communicated in response: I do not think there is any contradiction between our respective views. The principal theme is ‘connectivity’ and not the methods by which this is diagnosed. It happens that in most cases the diagnosis is by a mechani- cal property, so the establishment of connectedness is signalled by a pronounced increase of the particular parameter chosen (e.g.modulus). Clearly, in the intriguing L, surfactant phases such mechanical criteria do not apply. However, a more important distinction is that the physical gels I quoted are in a non-equilibrium state, connected- ness being a geometric consequence of incomplete (and arrested) phase transformation. In contrast, the L, phase surfactants, as far as I know, are in an equilibrium state, where the connectedness (here ‘phase’ connectedness in my terminology) is an attribute of surface Gibbs free energy minimization.In view of the pertinence of the issue raised I have made appropriate reference to the subject matter in the full version of my Introductory Lecture, although it was not fea- tured in the oral presentation. Dr. Dumaine communicated : I handle a lot of polymeric solutions made by dilution of polymer blends, photoactive monomeric compounds and surfactants in organic sol- vents. These solutions may be regarded as either a solution or a microemulsion, and are used for providing thin photosensitive coats by spin coating. Can these polymeric solu- tions or microemulsions turn into gels through spinning, which rapidly induces concen- tration of solids, and cooling? Polymeric -----+ Polymeric solution \, ,,/ microemulsion In this context, what will be the best way to know if we get a coating of gel? Is it possible to get an intermediate compound between a gel and a coating? Usually, the coatings have several wt.% of organic solvent which stays in the polymer matrix up to TG;this behaviour is related to their sensitivity to absorption of chemical vapour.Is this absorptivity significant for the definition of a gel? 105 General Discussion Prof. Keller communicated in reply: The answer is ‘yes’, at least for solutions. In my experience complex gelation effects can appear under flow, particularly in the case of very high molecular weight materials.The effects are transient but may persist for a considerable time (minutes) after cessation of flow (particularly in the crystallizable poly- ethylene, where it can be hours). Such ‘gelation’ effects are manifested through a viscous gel-like layer forming along the walls of flow channels, which has a striking influence on the flow that brought it into being in the first place.’ In other instances it can affect solution behaviour long after flow has stopped.’ We attributed these layers to flow- induced liquid-liquid phase transformation initiated by an adsorption layer which can itself thicken as a result of flow (‘entanglement adsorption layers’ using our terminology). We have studied such effects extensively on PMMA and PS, and also on polyethyl- ene (PE).In PE the effects are also compounded with crystallization; some of the effects are well understood (shish-kebab crystal formation), while others (we assume associated with crystallization) are not, and merit exploration. In summary, this is a very rich, largely unexplained and, I believe, important field. The subject itself did not start with ourselves: its roots go far back, all referred to in ref. 1. Specifically, ref. 1-5 and 10, 11 of ref. 1 deal with observations on flow-induced gel formation (accompanying certain viscosity anomalies). Much later, the issue of flow- induced liquid-liquid phase separation was taken up again, but without explicit refer- ence to gelation and surfaces (ref.33, 34 in ref. 1). One particular consequence of thick, flow-induced layer formation along the walls of the flow channel is its effect on viscosity measurements of very high molecular weight polymers, which should impact on basic polymer science in a wider generality’ (note the unfortunate misprint in the Abstract of ref. 2: the word ‘pressure’ should read ‘presence’). The emphasis has been on events along surfaces. These surface effects, on continuing flow, can extend gradually inwards and may even pervade the full system with some striking consequences (e.g. top curve in Fig. 2 of ref. 1). It is also possible that at high strain rates they may occur even within the solution interior, irrespective of the surface. Where the polymer is crystallizable there is a vast range of further effects as already referred to above.1 P. J. Barham and A. Keller, Macromolecules, 1990,23,303. 2 P. J. Barham and A. Keller, Colloid Polym. Sci., 1989,267,494. Prof. Goodwin opened the discussion of Prof. Dickinson’s paper: What is the effect of the wall depth on the time evolution of the fractal structure? If the wall depth is shallow, will this result in a large amount of rearrangement so that the number of time steps in your submission may become unmanageably large? Prof. Dickinson replied: The first point to note, of course, is that true equilibrium is not reached for many of the gel structures generated by our computer simulations. What we produce are simulated assemblies of aggregated particles existing as quasi-steady- state networks or slowly evolving connected structures, To discuss quantitatively the effect of interparticle interactions on the kinetics of evolution of the gel structure, it seems appropriate to separate the case of the bonding model (B) from that of the two non-bonding models (A and C).In the former case, a permanent connected network is reached when most of the particles in the simulation have become linked together into a single large percolating aggregate which spans the basic simulation cell in each spatial direction. This typically takes tens of thousands of timesteps. Beyond this stage, further limited rearrangements do take place, but they are subject to severe restrictions on parti- cle movement imposed by the network of flexible but unbreakable cross-links.The rate of further network rearrangement depends on the bonding probability, Pbond,and the General Discussion interaction strength parameter, w. For a constant value of Pbond,the kinetics of approach to the final steady state (t -+ co) is fast for attractive non-bonded particle- particle interactions (w >> 0) and slow for repulsive interactions (w 40). For the non-bonded models, a gel-like aggregated structure is typically reached after just a few hundred timesteps, although we usually follow it for many tens of thousands of timesteps. The initial network is very much a non-equilibrium metastable glass-like structure analogous to a thermodynamically unstable fluid-like system that has been rapidly quenched to a temperature well below the coexistence line.Increasing the poten- tial well depth increases the rate at which the initial percolated network is formed. The subsequent rate of network restructuring is also dependent in the well depth (e.g. see Fig. 1 of our paper) and on the range of the interparticle attraction; it is slower for the ‘colloid’ model than for the ‘argon’ model, The equilibrium structure is typically a two-phase state, i.e. gas/liquid or gas/crystal depending on the strength of the inter- particle interaction). The true equilibrium state is not of interest here; in any case it is generally not reached over the timescale of our simulations. Dr. Clark asked: From the content of your paper, and from other recent pub- lications on the simulation of particle gelation, I have gained the impression that the concept of a fractal dimension, in the gel context at least, is no longer as clear cut and focused as it was once assumed to be.It appears that this quantity depends for its exact value on factors such as the distance scale of measurements, the concentration of par-ticles, the extent to which aggregation has progressed, etc. Is this a fair assessment of the situation and, if so, is it not time now to place much less emphasis on fractals where gels are concerned, and rather to focus attention on outputs of the simulations which are more directly relevant and important, such as the evolution of solid-like mechanical properties? Prof.Dickinson responded: According to our criterion, a particle gel has a fractal structure if there exists a distinct linear region in the log-log plot of the integrated pair distribution function, n(r), as a function of the particle pair separation, r. The slope of this linear region gives the effective fractal dimensionality df . Not all aggregated particle networks are fractal; whether any particular gel is fractal, or is not fractal, depends on the overall particle volume fraction, 4, and the detailed nature of the particle-particle interactions. Gels made at low 4 at the end of a classical cluster4uster aggregation process are certainly fractal in character, as are many gels which form at moderate 4 with relatively strong attractive non-bonded interactions.Generally speaking, the length of the fractal scaling regime is reduced with increasing 4, and a convincing fractal regime cannot usually be detected beyond 4 = 0.3. In addition, homogeneous networks simulated at relatively low 4 from bonding particles with repulsive non-bonded inter- particle forces do not have fractal structure. For those particle gels which do possess some fractal character, the value of df is certainly a useful parameter defining the nature of the disordered structure over a partic- ular length scale within the material, and indeed we have no better way of expressing such disordered scaling behaviour than by means of the concept of the fractal dimension(a1ity). However, what our simulations also show is that the single fractal parameter value is by no means sufficient to characterize the structure properly.This is graphically illustrated by the pair of three-dimensional pictures in Fig. 3 and 6 of our paper which have the same notional value of df but quite clearly very different micro- structures. Two other additional parameters which can usefully be specified are (i) the primary cluster mass, no, which defines the average size of the primary building blocks from which the fractal structure is built, and (ii) the correlation length 5 which defines the length scale beyond which the structure starts to pass over gradually into the homo- geneous regime (d, -+ 3). In addition to the set of parameters d,, no and 5, it may be General Discussion necessary also to define some other structural parameter(s) related to the spatial dis- tribution of voids (pores) within the network.Non-fractal homogeneous gels tend to be characterized by a rather uniform pore-size distribution, whereas phase-separated heter- ogeneous gels have a wide distribution of pore sizes. The most appropriate way of describing these important porosity characteristics has yet to be properly established. We are still working on this problem. Dr. Beelen said: In your paper it was stressed that aggregates have a fractal and a homogeneous part. During gelation the interconnected network of aggregates is formed by connections (inter-aggregate) of the homogeneous parts. For silica gels we observed, both by experiments (SAXS) and computer simulations, that the fractal parts keep growing, also after gelation.Did you observe this phenomenon also with your simula- tions? (Our simulations did not use a potential, but were based on cluster-cluster aggre- gation.) Did you observe shrinking effects with your simulation and was this shrinking based primarily upon this ‘collapse’ (or ‘dissolution’) of the homogeneous non-fractal part connecting the aggregates? Prof. Dickinson replied: It may be appropriate here to recall that the fractal scaling observed in our simulations carried out at moderate particle volume fractions (+ z 0.1) is a property of the whole interconnected particle network structure. This is very differ- ent from particle gels formed at very low volume fraction (say + + 0.01) during the late stages of classical cluster-cluster aggregation (CCA) where any fractal character can be regarded as being primarily due to the structure of the growing isolated aggregates.Another difference from the classical CCA case is that the growing aggregates in our simulations are subject to continuous rearrangements. Even at the very initial stages of our simulations, the local aggregate growth in the concentrated particle system is deter- mined by the local particle concentration gradient which in turn is determined by the growth of neighbouring aggregates. The simulated aggregate fractal structure is there- fore never at any stage the same as for CCA aggregates formed at high dilution. Because the particle density is relatively high, small aggregates start to join together at a very early stage in our simulations, and so, when the time is reached when a convincing fractal scaling regime first becomes evident, the system has already become connected into a percolating gel-like network.Hence, in contrast to the CCA situation, the fractal scaling which we observe should not be interpreted as being due to the fractal character of individual separate aggregates, but rather to a collective property of the whole evolv- ing interconnected assembly. A characteristic feature of the non-bonding simulations carried out with models A and C is a fractal structure which evolves with time due to local restructuring and short-range densification. Reorganization takes place because the system of strongly attractive colloidal spheres is mechanically unstable with respect to thermodynamic phase separation.Movement of material from the intermediate length scales to the short length scales under the influence of the strong attractive interparticle interactions leads to a reduction in d, and an increase in no. For a very large system in the absence of bonding, this restructuring would tend to continue indefinitely until the globally uniform gel were replaced by a macroscopic phase-separated state (no + a).In the presence of bonding (model B with w 0), the local restructuring and densification becomes frus- trated due to the topological constraints imposed by the network of permanent cross- links, and eventually restructuring is halted altogether.The inference from our simulations is that a real system which exhibits an evolving fractal structure with time is probably also exhibiting local restructuring under the influence of attractive non-bonded interactions. If this is what you mean by ‘shrinking’, then we certainly do observe the effect. When the structural evolution of the aggregated fractal structure eventually stops, General Discussion the behaviour can be explained in terms of the formation of a permanent network of strong interparticle bonds which resists any further restructuring. Mr. Bos added: If we define gel point as the time when one percolated cluster is formed, then yes, the scaling region expands after the gel point. This is because of reor- ganization: at small length scales the clusters are compacted, leading to longer scale inhomogeneities and a growth in the correlation length, and hence a growth of the scaling region.Prof. Khokhlov said: I also agree with your opinion and the opinion of Professor Clark that the attention to fractal dimensionality in these systems is probably over-emphasized. There are many other interesting structural motifs to be studied. From your pictures it is seen clearly that you have something like characteristic pore size and char- acteristic distance between the aggregates. In this connection, have you ever observed a maximum of the static structure factors in your system? In model B, especially where you have attraction plus irreversible bonding, I would expect that this might be the case in some situations.Prof. Dickinson replied: Maybe your assertion about our view of the importance of the fractal concept needs a little moderation. Any apparent differences in the degree of emphasis given to fractal dimensionality by the workers in the field of particle gels is probably largely explicable in terms of differences in the extent to which ideas of fractal scaling are useful in fitting data from the real experiments. Data obtained by a variety of experimental methods for casein gels, for instance, can be interpreted unambiguously in terms of a single fractal dimensionality over a wide range of particle concentrations, which is itself proper justification, I believe, for emphasizing the fractal character of the gel structure.That having been said, what our new simulations show is that other struc- tural parameters apart from d, are also required for fully characterizing the evolving gel structure. Depending on the particle volume fraction and interparticle interactions, the range of the fractal scaling regime may be short or long. In the former case it would be a clear mistake to give strong emphasis to the fractal dimensionality, but not in the latter case. Our work should be seen more as a refinement of the application of the fractal concept, rather than any substantial downgrading of our assessment of its importance in describing particle gel structure. While several of our sets of simulation conditions are certainly consistent with a maximum in the evolving static structure factor S(Q), the simulations also suggest that such a feature does not always have the same mechanistic origin.In general, a distinct maximum in the static structure factor of a gelling or gelled system at some value Q, would mean that the microstructure was made up of spatial entities with a characteristic size or regular spacing of the order of Q,'. The entities making up the partially ordered structure could consist of (i) regdarly spaced discrete fractal aggregates, (ii) dense phase- separating regions of characteristic average size, or (iii) uniform regularly spaced pores. What our simulations indicate is that the significance of such a maximum in the struc- ture factor is dependent on the particle volume fraction and the detailed nature of the interparticle interactions.Dr. Poon said : We have performed similar cluster aggregate simulations, in which we studied the time evolution of the structure factor 1 N N *S(Q)= 11ex~[IiQ (rj -rk)l jk In our first paper' we simulated irreversible DLCA off-lattice in 2D, and came to a conclusion similar to that in the present work, namely, that at high concentrations a General Discussion single fractal exponent did not give by itself a full description of the microstructure of the aggregating system. We showed that, in common with a number of recent experi- ments, a peak developed in the structure factor at low angles as aggregation proceeded, shifted to lower angles and became 'frozen' at a finite angle (or Q value) at the gel point.In a second paper we showed that the structure factor obeyed 'dynamic scaling' at intermediate times2 That is to say the function is time-invariant with d = 1.4 (in 2D) and d = 1.7 (in 3D). (Qm is the small-angle peak position.) We also calculated the real space pair distribution function, g(r), which showed a 'dip' below unity before returning to long-range homogeneity at R x 27r/Qm (see Fig. 1). The effects of finite bond energies (i.e. reversibility) are discussed in a third publication3 in which we study dynamic scaling and the local structure of aggregates. Has the time evolution of the structure factor, S(Q), been studied in your simula- tions? If so, was a small-angle peak discovered and did dynamic scaling hold? It would also be interesting to know if the pair distribution function [integrated to give n(r)in the a I I I I 6 h 0z4 2 0.0 0.5 1 .0 1.5 2.0 Qd3 r.I 2D: P = 0.1 L = 500 v t =so V h c V 2dQmv 0, t? 1 07 I 1 3 5 10 30 f Fig. 1 The small-angle peak in the structure factor, S(Q), from the early stage of a 2D lattice simulation with initial density, p = 0.1 (upper diagram), and the pair correlation function, g(r), at the same time. The position of the peak, Q,, and the length scale to which it corresponds, 2n/Q,, are indicated. General Discussion paper, eqn. (12)] showed a similar 'dip' indicating a depletion zone' surrounding growing clusters.1 M. D. Haw, W. C. K. Poon and P. N. Pusey, Physicu A, 1994,208,8. 2 M. D. Haw, M. Sievwright, W. C. K. Poon and P. N. Pusey, Physicu A, 1995,217,231. 3 M. D. Haw, M. Sievwright, W. C. K. Poon and P. N. Pusey, Ado. Colloid Interjiuce Sci., 1995,62, 1. Dr. Van Opheusden replied: We have studied the structure factor S(Q).The purpose of our investigation was to determine the nature of the early stage aggregation dynamics of the system, in order to test whether a spinodal decomposition model would be appro- priate. We have calculated the isotropic structure factor S(k) from the pair correlation function g(r) through S(k) = 1 + 4np r2[g(r)-13 sin(kr) drI" The time development of the structure factor for a 3D system of 1000 Lennard-Jones particles at volume fraction 3.75% is given in Fig.2. We observe a single peak developing at k/2n x 0.70-', where CT is the Lennard-Jones length scale parameter. The minimum of the pair potential is at r = 1.120, and the peak corresponds to pairs of particles aggregating into doublets at approximately that distance. This conclusion is corroborated by the results for different volume fractions, which show a peak developing at the same position, but with a different rate. We find that the maximal growth rate increases with volume fraction roughly linearly, the rate at 10% being a factor of 10 higher than that at 1%. This can be attributed to the higher number of pairs within cut-off range of the potential Essentially, we only see the development of the fluid peak, although instead of shift-ing to higher k values owing to cageing it shifts to lower values, indicating the develop- ment of larger structures.This could be due to the strong attraction we have used (E = 10k, T),which makes entropy effects in pair bonding rather unimportant. It would be interesting to test whether our various systems also exhibit the type of Furukawa scaling for the structure factor at later stages of the aggregation process as you have done, especially because in our models we include the possibility of local restructuring of the aggregates formed. Maybe it is appropriate to stress again that most of our results are obtained for structures that have already formed a gel, and that the development of fractal scaling regimes is probably very much due to this reorganization t= 0 l/ 43i -710 '/.,,,.I .1 , I 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 W2K Fig. 2 Early time development of the structure factor S(k) for an aggregating system of Lennard- Jones particles General Discussion process. We also include rotational diffusion, which does, in the free draining limit of our model at least, give more extended structures for the individual aggregates, which hence will more easily form a macroscopic cluster. I would expect to see the same ring phenomenon in our systems A and C, but with different type of scaling, due to rotation and reorganization, and no freezing-in at gelation. In our model B, with only mild reor- ganization, the picture will be similar, and only the rotational diffusion will give quanti- tatively different effects.The next part of your question refers to the dip in the pair correlation function, which you attribute to a depletion zone around growing clusters. In the context of growing clusters that could be a correct interpretation, but the problem is that we find a similar dip after gelation. Moreover, when, owing to reorganization, the dimensionality in the scaling regime diminishes, we find a deepening of the dip. I think the concept of a depletion zone is not correct, but a dip in g(r) is simply connected to the fractal scaling regime. The average number of particles n(r) around a given test particle is calculated from g(r)through n(r) =471p s2g(s)dsl The fractal scaling regime is defined as that range in which n(r)scales as n(r) -rdf (3) where d, is the fractal dimensionality.This implies that in the fractal scaling regime g(r) must scale as For fractal scaling (at d, < 3) that makes g(r)a decreasing function of r, which because of the normalization condition on g(r) l%[g(r)-l]r2 dr = 0 implies both a concentrated and a diluted regime. For the type of fractal scaling we find in our log(n) us. log(r) plots (Fig. 3), there is a kink discontinuity in n(r) at the correlation length <,where we cross over from the 1000 5001 200-100: 501 20-10: 5: 1 2-C------------------r-----7 1 23 5 10 20 r Fig. 3 Sample plot of the average number of particles in a sphere of radius r around a test particle in a fractal gel General Discussion fractal to the homogeneous scaling regime.Note that this is a sample plot only; the data in this graph are imaginary, and are not the results of a simulation. A kink in n(r) corresponds to a step in g(r),and because the derivative dn/dr increases at 5, so does g(r) [cc eqn. (4)]. In the homogeneous scaling regime g(r) = 1, which necessarily gives a dip below the cross-over point (Fig. 4). In real systems of course both kink and step discon- tinuity will be smoothed. It is probably appropriate to note that g(r) is a correlation function only, so the notion of a depletion zone is not quite correct. Because a pair correlation function has a particle at the origin each individual particle would be surrounded by such a depletion zone, and space would be filled with depleted region.A dip in g(r) corresponds to a decrease in the probability of finding particles at that particular distance, or rather the increase in probability of finding a void. The dip in the gelled system is associated with the average size of the voids in between the regions of concentrated particles, rather than with the size of the primary fractal clusters. In the late stages of gelation these will probably have disappeared altogether, and we are dealing with a space filling fractal network similar to the so-called Sierpinski sponges, rather than randomly packed DLCA-clusters. Prof. Djabourov opened the discussion of Dr. Poon’s paper: Do mechanical actions, such as vibrations or shear, have an influence on the non-equilibrium state? Dr.Poon replied: We have not so far studied explicitly the effects of vibrations or shear. However, it is possible that the shearing of growing clusters as they sediment under gravity may play a role. It appears that in any particular colloid-polymer mixture, transient gelation is not observed below a certain critical colloid volume frac- tion, +*. The value of +* increases if we keep the same polymer to colloid size ratio (thus roughly maintaining the same depletion potential range) but increase the particle size (thus increasing the gravitational Peclet number). A possible explanation for this observation has recently been suggested by Allain et al. (see ref.31 of our paper), which indeed involves gravity. Mr. Bos asked: Does the Furukawa scaling exponent change with time? I ask this because in our simulations we find decreasing scaling exponents for transient gels, and your interactions between particles suggest transient bonds. In the Euclidean space, as opposed to the Fourier space, we see a gradual change in scaling, as opposed to a collapse in a frozen ring. r Fig. 4 Pair correlation function for the fractal gel configuration from Fig. 3. Note that the data are only qualitatively correct. General Discussion Dr. Poon responded: Dynamic scaling holds in our system at intermediate times. By definition, df is a constant in this scaling ansatz. It is worth emphasizing, however, that we do observe d, to decrease as we increase the polymer concentration (and hence the depth of the depletion attraction between particles); see Fig.3(d) of our paper. Prof. Goodwin asked: Have you considered the effect of increasing a particle’s isome- try on the time to achieve equilibrium? This should introduce shear sensitivity to the process. Dr. Poon replied :We have not studied non-spherical particles. However, Lekkerker- ker and co-workers have studied the effect of non-adsorbing polymer on the phase behaviour of rod-like colloids. At high polymer concentrations they found non-equilibrium gel or glass states (some of which may be birefringent). 1 H. N. W. Lekkerkerker, P. Buining, J. Buitentius, G. J. Groege and A. Sroebants, in Observation, Predic- tion and Simulation of Phase Transitions in Complex Fluids,ed.M. Baus, L. F. Bull and J-P. Ryckaert, Kluwer, Dordrecht, 1995, p. 53. Prof. Goodwin communicated: I would like to draw the authors’ attention to, and seek their opinion of, the view expressed to me by Lekkerkerker as follows: In their very interesting work on the phase separation kinetics of colloidal mixtures Poon et al. suggest that ‘non-equilibrium behaviour is switched on by a metastable gas-liquid binodal’. I would like to suggest what is observed by the authors is in fact an example of the famous rule of stages formulated by Oswald almost a century ago.’ This rule asserts that the phase appearing first is not the thermodynamically stable phase but rather a metastable phase.In the case studied by Poon et al. this implies that the metastable gas-liquid tran-sition occurs before the fluid-crystal transition. Apparently, the colloidal liquid that is produced by the metastable gas-liquid transition crystallizes very slowly (if at all) giving rise to apparent non-equilibrium behaviour. In fact colloidal systems offer a fascinating possibility of making the metastable gas-liquid transition the stable one. As is well known already, a moderate degree of polydispersity (standard deviation diameter 8-10%) suppresses the fluid-crystal transition. So working with sufficient polydisperse samples with short-range deep attraction, the first transition encountered is the gas- liquid transition. Such a system was studied in detail by Verduin and Dhont,2 who found a phase transition scenario that is strikingly similar to that reported by Poon et al.(Compare Fig. 2 of Verduin and Dhont with Fig. 2 of this paper.) 1 W. Ostwald, 2.Phys. Chem., 1897,22,289. 2 H. Verduin and J. K. G. Dhont, J. Colloid Interface Sci., 1995, 172,425. Dr. Poon communicated in reply: We thank Prof. Lekkerkerker for pointing out the work of Ostwald, and agree that our observations do appear to be an instance of the Ostwald rule of stages. Moreover, it seems reasonable to expect that if crystallization is indeed suppressed by polydispersity in a colloid-polymer mixture, our kinetic map in Fig. 2 would become very similar to that obtained by Verduin and Dhont. However, it is also possible that the addition of polymer to a polydisperse colloid may lead to crystallization of the larger particles (for which the depletion potential is deeper), i.e.fractionation may result (cf ref. 1).Investigation of polydispersity is underway in our laboratory. 1 J. Bibette, J. Colloid Interface Sci., 1991, 147,474. Prof. DuSek opened the discussion of Dr. Clark’s paper: You have raised an inter- esting question of whether the cascade theory can be modified to describe structure General Discussion 115 evolution during the microphase separation of gels. The cascade theory can be modified only at the level of the first shell substitution effect (possibly higher shell s.e., but then it becomes too complicated). Then only the distribution of the building fragments with respect to the number and type of bond they are engaged in develops with time, but otherwise the fragment combination remains random and the effect of long-range time and space correlation is lost.Perhaps some properties can be approximated satisfacto- rily but generally I do not consider the cascade theory appropriate for this purpose. Dr. Clark responded: Thank you for your comments. My present contribution on amylopectin gelation certainly points to the correctness of your conclusions. It would be very dificult to adapt the cascade approach to account fully for the experimental behav- iour of this system. At present I am working on a compromise solution which would involve combining kinetic development of primary crystalline particles through nucle- ation and growth, with random aggregation and gelation described by the simplest form of the cascade theory.Prof. Ross-Murphy said (Dr. Tobitani in part communicated): Your paper has very rightly emphasized the significance of a kinetic model connecting the gelation time to initial polymer concentration. Have you considered the effect of temperature on gelation time? Our unpublished measurements on heat set gels of bovine serum albumin (BSA) in Fig. 5 show that describing the concentration dependence alone is not sufficient, since both temperature and concentration can affect the gelation time. There appears to be a cross-term interaction between the two factors, since best fit lines through the data do not appear to be parallel.Fig. 5 illustrates this, and shows a final fit, in which c (concentration) follows a conventional kinetic scheme. In this model,' T (temperature) is assumed to follow a simple Arrhenius form. While both assumptions are undoubtedly too simplistic, we would also suggest that a more 'realistic' version would have far too many parameters to be helpful. 1 A. Tobitani, PhD Thesis, University of London, 1995. Dr. Clark responded: I have not studied the gel time-temperature relationship pre- dicted by cascade theory as closely as the corresponding gel time-concentration func-10' i 5 10 30 c (W/W"/*) Fig. 5 Gelation time plotted against concentration of BSA, for measurements at: 0,66; 0,61; and A. 56 "C General Discussion tion.It is implicit, however, in the latter' through the temperature dependences of appropriate rate constants, that log-log displays of gel time us. concentration data will show very much the behaviour you demonstrate in your figure. This is because of the universal form this function takes, i.e. t,k, = F(c/co), where k, is a r&e constant for bond dissociation, and co is the critical concentration. As temperature hicreases kb will always increase, and co will rise for a gel which melts at higher temperatures, and fall for the reverse situation. Your data certainly show the 56°C data set moving down as temperature increases, and probably to the right (ie. exothermic cross-link situation). This allows the apparent power law to increase, since the theoretical gel time curve diverges at co, and only shows a constant, and much lower, power law (-1) at c/co > 3.The cascade approach also anticipates the Arrhenius relationship which you say is implicit in your data for gel time us. temperature at constant concentration. Numerical calculations of In t, us. 1/T using the theory give straight lines over limited temperature ranges but, interestingly, predict that for a given temperature range, the apparent activa- tion energy will be a function of concentration. Finally, one should add the caution that where globular proteins are studied close to their unfolding temperatures, the rate of unfolding must be added to the kinetic description. This is not something I have included in the cascade model described here.1 A. H. Clark, Polymer Gels and Networks, 1993, 1, 139. Prof. Winter said :The temperature dependence seems to follow an exponential func- tion, very similar to what we saw with the crystallizing polypropylene (see our paper), while the concentration dependence seems to follow a power law. This is indicated from your plots, having a straight line in the semilog graph for the temperature dependence and straight lines in the log-log graph for the concentration dependence. What does this tell us about the underlying physics of these phenomena? Dr. Clark replied: The fact that these two dependences are explained by the cascade approach (see previous answer), suggests that this mean field description is capable of explaining (at least semi-quantitatively) much of the basic phenomenology of physical gelation.Since it is an adaptation of the original classical Flory-Stockmayer theory, modified to include the presence of solvent and reversible cross-linking, this success is probably to be expected. It should be admitted, however, that the model has limitations. It quite often fails to get quantitative aspects of the gelling phenomenon correct, such as the experimental power laws actually found for gel times in relation to concentration, and of course it is not expected to predict critical exponents correctly. The mean field character of the approach, and the extreme complexity of many real physical gelling systems, are no doubt the origins of this.Miss Kavanagh, Dr. Tobitani and Prof. Ross-Murphy communicated : One aspect which was not discussed in detail at the meeting, despite its importance, is the behaviour of gels around the critical gel concentration co.Despite the implication of certain fractal models, there is now no doubt that a co does exist and that this parameter is related only indirectly to the chain overlap concentration c*. Fig. 6 illustrates very careful mea- surements by Ikeda et d.' on gelatin gels close to, and below co. The solid line is a theoretical fit using the cascade model discussed by Clark in this paper, the dotted line is an empirical (polynomial) fit, and the critical concentration shown is that determined by Ikeda et al. The general question we would like to pose is, 'what actually happens below c,, ?' The data illustrated show that a region is reached where the modulus is apparently independent of concentration.Does this merely reflect the sensitivity of the instrument, or does it illustrate one expectation that the sol/gel boundary is intersected by the binodal? Is it also a general effect, or is it specific to gelatin gelation? In either case the General Discussion 117 observation may go someway towards explaining the low concentration ‘gel’ anomalies seen by ourselves and others for gelatin systems an order of magnitude and lower below 2CO . 1 T. Ikeda, M. Tokita, A. Tsutsumi and K. Hikichi, Jpn. J. Appl. Phys., 1989,28, 1639. 2 J-P. Busnel and S. B. Ross-Murphy, Znt. J. Bid. Mucromol., 1988,10, 121.Dr. Clark responded: My first reaction to Fig. 6 is that I am disappointed by the poor fit achieved by the cascade theory to the higher concentration data. It usually does much better than this. I am much less surprised, however, that deviations occur as one probes closer and closer to the critical concentration predicted by such higher concen- tration fits. Apart from the increased likelihood of experimental uncertainty playing a role in this situation, there are good theoretical reasons why gelling biopolymer systems could behave anomalously as concentration falls. The simple cascade theory approach assumes that the mechanism of cross-linking, and indeed the cross-linking entities them- selves, remain essentially the same as concentration falls, and neither of these assump- tions is certain to be valid.Secondly, the cascade model ignores the realistic possibility of solution demixing occurring together with gelation. This can lead to microphase- separated gels at higher concentrations and microgel suspensions/networks as concen- tration falls. I think that what you have found for gelatin gelation is likely to be a quite common feature of physical gelation, though the unequivocal demonstration of this will be quite taxing experimentally. Prof. Nishinari communicated : The kinetics of gelation of amylopectin should depend not only on the concentration and temperature but also on the molecular weight and degree of branching. Although you did not mention these factors in the concluding remarks on future research, I hope that you will take into account these points.You mentioned that the maltodextrins containing a low-molecular-weight amylose fraction gel faster than the amylopectin that you used. Leloup et a1.l studied the influence of the amylose/amylopectin ratio on gel properties, and concluded that amylose plays a domi- nant role in the gelation of the mixture of amylose-amylopectin, which is consistent with results of your ref. 20 and 21. Do you think that the addition of any amylose fraction promotes the gelation of amylopectin you used? It is interesting that you obtained the Avrami exponent smaller than unity. The experimental finding that the failure of the Avrami model became evident at the longest times seems to suggest that the fractions which form gels slowly deviate from the model.0.1 1 10 concentration (w/w%) Fig. 6 Shear modulus, G, plotted against gelatin concentration General Discussion At the constant concentration and temperature, how do the molecular weight and the degree of branching affect the kinetics of gelation? 1 V. M. Leloup, P.Colonna and A. Buleon, J. Cereal Sci., 1991, 13, 1. Dr. Clark communicated in reply : Variations in amylopectin branching character- istics were not studied in the present work, but this aspect of amylopectin is important and will, if possible, be included in future studies. Molecular weight is also important, as I demonstrate in Fig. 1 of my paper, which presents gel time concentration data for two molecular weight fractions of waxy maize amylopectin.As the molecular weight falls, gel times significantly lengthen under the same temperature and concentration conditions. In answer to your query about the influence of amylose, I can co9firm that addition of low molecular weight amylose segments to the waxy maize system dramatically shortens gel times, presumably through the amylose acting as a nucleating agent for the rate- determining crystallization process. The failure of the Avrami model probably relates to the fact that, in the present work, crystallization takes place from solution, not from the melt. At longer times depletion of solute from the solution surrounding the crystalline particles will have an influence. The low Avrami exponent I obtained seems to be typical of results found for starch retrogradation, in general, but I hesitate to provide an inter- pretation at this stage.Prof. Dugek opened the discussion of Prof. Winter’s paper : Structureless polymer solutions (like atactic polystyrene of high molecular weight in a good solvent) undergo a transition from a state of viscous liquid to an elastic solid (gel) when the concentration increases. Can a gel point be defined and determined or is the transition continuous? Prof. Winter said in reply: The liquid-solid transitions which we have studied so far, and there are quite a variety, have always exhibited a gradual increase of the longest relaxation time, divergence at the gel point, and a reduction again beyond the gel point.Samples at the gel point (critical gels) always relaxed with a power law spectrum at long times. I expect that this will also happen with the atactic polystyrene solution as the concentration, c, is increased. This argument is entirely based on the analogy between physical and chemical gelation. Physical gels developed the same relaxation patterns as were observed universally for chemical gels. If the analogy between physical and chemi- cal gels holds, then the atactic polystyrene sample with power law relaxation denotes the gel point concentration. To explore this in detail, it would be good to perform dynamic mechanical experi- ments on atactic polystyrene solutions if this has not already been done. The transition through the gel point is continuous as far as the dynamic mechanical data are con- cerned.At the gel point one can expect a power law relaxation time spectrum and use this for defining the gel point. An easy way of detecting the critical concentration would be repeated frequency sweeps on samples with increasing concentrations. The critical concentration c, for the gel point would be found (by interpolation) as the intersect of the tan 6(q, c) us. c curves when plotted for discrete frequencies wl, w2,w3,w4 etc. in the terminal frequency range. The resulting figure should look similar to Fig. 8 (later); however, the independent variable would be the concentration c (instead of the reaction time, t). Dr. In said: In the introduction of your paper you mention that the mass distribu- tion in physically gelling systems does not diverge at the critical point.Why do you expect a power law for the relaxation? In other words, are you sure that the constancy of tan 6 that you observe on a very narrow range of frequency, holds at low frequency? General Discussion This could be proved if you showed that tan 6 = tan(nz/2), but it is not clear whether you could determine the gel exponent n. Prof. Winter replied: The molecules do not change during physical gelation and the divergence of the mass distribution (which one is used to seeing in chemical gelation) does not occur. This takes away the classical definition of the gel point as the instant at which the largest molecular cluster diverges.However, if one neglects the type of connec- tion mechanism and focuses on the developing network structure, then physical gels look very similar to chemical gels. Rheology is probing this developing network connec- tivity and the liquid-solid transition in physically gelling systems is marked by the same power law relaxation which we are used to seeing in chemical gelation. Concerning your second question/comment, it is important to note that the power law relaxation occurs at low frequencies only. At higher frequencies, additional relax- ation mechanisms enter and the spectrum might adopt appropriate forms which are unaffected by the liquid-solid transition. The question remains of how low in frequency we have to go to observe the liquid-solid transition behaviour. In our study on PP, we barely make it into the terminal region, and it would be more convincing if the data would extend by one or two decades lower in frequency.Our result gives an upper possible value for the gel time. The solidification could have occurred earlier, but I do not think that this is possible since the degree of crystallinity is already extremely low at our proposed gel point. The relation which you would like to have tested, tan 6 = tan(nz/2), is satisfied by our data. One can see this when comparing the spacing of G‘ and G with the slope in the terminal region. If we were to do this study again, I would prefer a lower molecular weight PP since its low frequency behaviour would be easier to observe with our instruments.Prof. Ross-Murphy said: Your paper and other contributions at this meeting have demonstrated how improvements in instrumentation over the last 15 years have enabled small deformation rheological measurements to be made through the gel point. However, one limitation of such measurements is that measurements can only be made within the linear viscoelastic region, i.e. where G*, the complex storage modulus, and/or (3’ and G” respectively, its in-phase and out-of-phase components, are independent of the applied strain. Have you considered the effect of the strain dependence for physical gels close to the gel point? It can be argued that this linear viscoelastic strain ylin must be dependent on the proximity to the gel point, making the validity of such measurements very difficult to assess.Fig. 7 illustrates this tendency. The question that really needs to before gel point PiPC = 1 after gel point Fig. 7 Linear viscoelastic strain (in arbitrary units) plotted against p/p, (arbitrary scale), where p, is the gel point conversion (percolation threshold) General Discussion be answered is ‘what is the linear viscoelastic strain at the gel point’? As Fig. 7 devel-oped on the very reasonable hypothesis that ylin depends upon the reciprocal molecular weight of the sol fraction before gelation, and on the (amount of) gel fraction afterwards shows, this may be vanishingly small. It may be that the region illustrated is so minute that it is of no practical significance.However, we believe this is not so for physical gels, since the structure of these is necessarily more tenuous than that for covalently cross- linked systems. Such materials close to gelation are often closer to colloidal and particu- late systems. In these, the linear viscoelastic strain can be <0.001, rather than the values >ca. 0.1 found for covalent polymer networks. Prof. Winter said in reply: The question is difficult to investigate experimentally since the sample undergoes changes in time due to the physical gelation process. Chemi- cal gelation is more informative here, at least for systems in which the cross-linking reaction can be arrested near the gel point and strain effects can be tested. We produced stable polydimethylsiloxane critical gels in this fashion and studied the onset of non- linearity: strain levels up to two (y e 2) gave a linear viscoelastic response.’ In stable samples, the range of linear viscoelasticity is easy to measure even if it is very difficult to predict.One simply repeats shear (or extension) experiments at increasing strain levels and reduces the data according to the linear viscoelastic framework as described in the textbook by Ferry2 (for instance). Deviations from linearity show up at a characteristic strain or stress level at which the data cannot be collapsed any more. Physical gels, as you point out, are often less flexible and have a smaller range of linear viscoelasticity. To answer your question, one would have to repeat the gelation experiments at a set of discrete strain levels and then normalize with the strain.What we do instead is to run the gelation experiment under oscillatory shear once and then repeat it without flow until roughly the gel point so that gelation cannot be affected. Then we start the oscil- latory shear and compare the data from the two types of probing. This needs to be done for each individual system since strain effects are hard to predict. 1 S. K. Venkataramen and H. H. Winter, Rheol. Acta, 1990,29,423. 2 J. D. Ferry, Viscoelastic Properties ofPolymers, Wiley, New York,1980. Prof. Keller said: You determine the gel point, or in general perform rheology during the initial stages of crystallization while crystallization is still in progress.Are you not concerned that strain-induced orientation is affecting the crystallization itself? Prof. Winter replied: We keep the strain levels very small for our experiments. This is possible since PP is fairly viscous and the required torque level for our transducer is small. We did not expect an effect of small amplitude shear on the crystallization process but did not study this phenomenon. Our method would be ideal for exploring this effect of shear. When subjecting the sampie to large strain, we would expect drastic increase in crystallization rate as has been reported in the literature by Janeschitz-Kriegl,’ for instance. This phenomenon is especially important in our injection moulding example. 1 G. Eder, H. Janeschitz-Kriegel and S. Liedauer, Prog.Polym. Sci., 1990, 15, 629. Prof. Keller said: It does not follow that larger crystals formed at lower supercooling have larger ‘functionalities’. Normally, under those circumstances the chains will be crystalline to a larger extent by chain folding which being ‘intra’ molecular will reduce the functionality hence the gelation tendency. Prof. Winter replied: Let us look at the crystallization process at a low degree of supercooling where the crystal sizes are known to be largest. Here, we observe a long gel time t,, but the gel point is reached at a low degree of crystallinity (from applying Kim’s General Discussion 121 crystallization data’). The low degree of crystallinity at the gel point suggests that the few available crystals (per volume) are very effective in forming a network.By analogy to chemical gelation, we know that the gel point is reached at a lower extent of cross- linking when the functionality of the cross-links is increased. The PP crystals seem to be able to attach many tie chains and are, because of this, considered as cross-links of high functionality. A direct proof of this speculative argument is still needed. The proposed formation of large crystals of low functionality does not seem to occur here, but it certainly is an interesting phenomenon which should affect the rheology very strongly. 1 C. Y. Kim, Y. C. Kim and S. C. Kim, Polym. Eng. Sci., 1993,33,1445. Prof. Keller asked: At what stage did you measure the degree of crystallinity? The final degree of crystallinity (at which normally such measurements are-or can be- made) may be quite different from that which pertains to the time when the rheological measurements were made.Prof. Winter responded: The degree of crystallinity is so low at the gel point that it was not possible for us to measure it. We simply took crystallization data from the literature (as reported in the paper) and extrapolated to the early times at which the gelation occurred. It would be most interesting to learn more about these early stages of crystallization. Rheology is very powerful in showing the existence of a sample spanning cluster very early in the crystallization process but the clusters’ structure is still unknown. Prof. Djabourov said : You have investigated the crystallization of polypropylene in a supercooled state, using a commercial sample.In the supercooled state the nucleation is governed by the presence of impurities (even the walls of the vessels). You compared your data to the measurements of Kim et a1.l to derive the degree of crystallinity. How can you justify the comparison with the data obtained on a different sample? 1 C. Y. Kim, Y. C. Kim and S. C. Kim, Polym. Eng. Sci., 1993,33,1445. Prof. Winter replied: We cannot exclude the effects from the walls of the sample holders, but the samples had been cleaned from nucleating additives. This was also true for Kim et al. so that we expect similar crystallization behaviour. We tried to exclude effects of prehistory as much as possible by preheating the sample to a relatively high temperature before supercooling it.Prof. Djabourov said: In Fig. 7 of your paper you show that the degree of crys- tallinity at the gel point increases with the degree of supercooling. What is the physical meaning of this behaviour Prof. Winter responded: Our study’s focus was really on the rheological patterns of the solidification process. This gave us timescales and rates of change without revealing the underlying crystalline structure. A high degree of crystallinity at the gel point (see Fig. 7) tells us that the crystals are not very effective in connecting the flexible molecular strands into a network. This can be attributed to a relatively low functionality of the crystals, i.e.a small average number of tie chains per crystal. Dr. Howe communicated: What is the best method of measuring the gel point? Are there significant systematic differences between apparent gel times determined from (a) constant tan 8,(b) the G’ = G” crossover and (c)the onset of growth in G’? General Discussion 1lo-’ r-1000 2000 3000 4000 US Fig. 8 Evolution of loss tangent at several frequencies during cross-linking of PBD38 (T= 28 “C). The tan 6(0) intersect is the gel point. PBD38 is a nearly monodisperse polybutadiene with M, = 38000 and 7.7 wt.% vinyl content. The polymer is cross-linked at these vinyl groups with pbis(dimethylsily1)benzene as cross-linking agent. The reaction is catalysed by cis-dichlorobis(diethylsulfide)platinum(II).The experimental frequencies range from 1 s-’ to 100 s -’,with six frequencies per decade.Prof. Winter communicated in reply: There are many good methods for determining the gel point. They can be divided into two groups: (1)very well established are extrapo- lation methods in which the measured quantity diverges at the gel point. They allow us to narrow down the region of gelation from both sides of the gel point, but they fail in the immediate vicinity of the gel point. The most widely used extrapolation methods are the following : (i) solvent exposure of chemically cross-linking samples which dissolve before the gel point but merely swell beyond the gel point; very near to the gel point measurements fail since the swelling is strong enough to fracture the very fragile gel structure.(ii) Detection of the largest cluster with light scattering; the divergence of the largest cluster cannot be detected any more when its size becomes too large. Beyond the gel point, light scattering could be used to monitor the holes in the network. The deter- mination of the gel point through steady-state rheological properties, namely the steady shear viscosity and the equilibrium modulus, has been used to home in on the region of gelation: the main problem is the breakage of the molecular structure in these experi- ments which only allows detection of a liquid-solid transition on broken gels. (2) Direct measurement of the gel point: up to now we know only of one such method. It is based on the observation that the long time tail of the relaxation time spectrum adopts power law format at the gel point.’ This results in plots in which log G’(o)and log G”(o)us.log o are straight parallel lines at low frequencies. The gel point can be detected by a flat tan &m) or a tan S(m)intersect when plotted against conversion or reaction time.2 A newer example for such tan &intersect is shown in Fig. 8. With apparent gel times you probably mean some approximations to the gel point but not the real gel point. There have been many proposals and you mentioned some of them. (a) The constant tan d(o) should not be called apparent since it gives the real gel point (see above) when measured appropriately. (b) The G’ = G” crossover is far off in many gelation processes.Deviations are especially large for cross-linking of small mol-ecules and for most physical gels. (c) The growth of G’ cannot be used to determine the gel point since it starts much before the gel point. It also depends on the sensitivity of General Discussion 123 the instrument. This is evident from the many experimental data on chemical and physi- cal gels. 1 F. Chambon and H. H. Winter, Polym. Bull., 1985,13,499. 2 E. E. Holly, S. K. Venkataraman, F. Chambon and H. H. Winter, J. Non-Newtonian Fluid Mech., 1988, 27, 17. 3 H. H. Winter, Polym. Eng. Sci., 1987, 27, 1698. Dr. Howe communicated: What do you regard as the minimum practical frequency to be used when using the condition of constant tan 6 to determine the gel point? Prof.Winter communicated in reply: The choice of frequency is essential for the success of the experiment. Both, the lower and upper limit are important. You raised the question of the minimum frequency which is dictated by the reaction rate of the sample since the experimental time increases with l/w. During this experimental time, the sample keeps changing and the overall change during an experiment should not exceed a tolerance level. One might set this tolerance level at 10% change of the measured quantity during the taking of a single data point. This is manifested in the mutation number. On the other hand, the experiments have to be performed at sufficiently low fre- quency. As an example, cross-linking polybutadienes of high molecular weight respond strongly when probed at frequencies below w = l/A,,, where A,, is the longest relaxation time of the polybutadiene precursor.Little response to the cross-linking is seen at higher frequencies. This has direct implication for the tan 6 method which should only include the low frequency data. 1 H. H. Winter, P. Morganelli and F. Chambon, Macromolecules, 1988,21, 532. Dr. Howe communicated: Do you think that a material should have an equilibrium modulus to be correctly classified as a ‘gel’? If not, what is the minimum value of the longest relaxation time that would be sufficient for this classification? Prof. Winter communicated in response: A soft material with an equilibrium modulus satisfies the mechanical criteria for being a ‘gel’ (sometimes, other non-mechanical criteria are added). The existence of the equilibrium modulus implies that the connectivity of the gel is permanent. The gel might be sensitive to strain so that it is very difficult to determine its equilibrium modulus without breaking it. However, this does not interfere with its classification as a gel. More disputable is the definition of a soft material in which the network connectivity relaxes with a finite time. Many physical gels belong to this category. I suggest identify- ing physical gels by the gelation process by which they assemble. Physical gelation can be recognized by the increased connectivity and the long timescales which start to domi- nate the relaxation processes. The characteristic of (physical) gelation is this inherent increase of the longest relaxation time.

ISSN:1359-6640    

DOI:10.1039/FD9950100105

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,101, 125-131 Supramolecular Structures in Polyelectrolyte Gels Alexei R. Khokhlov, Olga E. Philippova, Natalia L. Sitnikova and Sergey G. Starodubtsev Physics Department, Moscow State University, Moscow 117234,Russia Three physical situations involving the formation of microstructures in poly- electrolyte gels have been considered. (i) The microstructure emerging in weakly charged polyelectrolyte gels upon the addition of a poor solvent. The following possible reasons for the microstructure formation have been analysed : the polyelectrolyte/hydrophobic competition, the formation of ion pairs and their clustering into multiplets, and the partial vitrification. (ii) The ionomcr multiplet-type microstructures which appear in polyelectrolyte gels immersed in a solvent of medium polarity upon the increase of the degree of ionization of the gel.(iii) The microstructures which are formed in poly- electrolyte gels interacting with oppositely charged surfactants due to the aggregation of surfactants inside the network and quasicrystalline organiz- ation of the aggregates. Polyelectrolyte gels swollen in water are known to undergo an abrupt collapse tran- sition under small changes of external parameters. The collapse can be induced by a variation in temperature or by the addition of a poor solvent, low molecular weight salt, oppositely charged surfactants, linear polymers, multivalent ions etc.'-'The collapsed gels, as well as the swollen gels in the pretransitional region, may exhibit interesting supramolecular order.In this paper we consider three effects of this type. (i) Microstructures in Weakly Charged Polyelectrolyte Gels in a Poor Solvent When the gel collapse is induced by the addition of a poor organic solvent, three reasons for the microsegregation should be discussed. The first is connected with the so-called polyelectrolyte/hydrophobic competition, the effect which was first described theoretically by Borue and Erukhimovich" and further studied in some subsequent publications.' Suppose that the gel swelling in water is only weakly charged and that the solvent is poor for the uncharged monomer units, so that the polyelectrolyte gel swells only due to the presence of charges. Then, the uncharged parts of the chains tend to segregate from water.In principle, this can be done in two ways (Fig. 1). One possibility is just to adopt the conformation of a col- lapsed gel. But the obtained gain in the energy due to interaction of uncharged monomer units with each other may be smaller than the emerging loss of translational entropy of the counter-ions which are confined to a smaller volume of the collapsed gel. Thus, a compromise leading to the minimum in the Gibbs energy in some cases corre- sponds to a microsegregated structure with hydrophobic aggregates surrounded by highly water-swollen regions where most of the charged monomer units and counter- ions are accumulated (Fig. 1).The polymer-rich and water-swollen regions thus formed can be arranged in a periodic array. The second reason for the microsegregation is connected with the formation of ion pairs in the collapsed gels.This effect can be observed if the medium inside the collapsed 125 Supramolecular Structures in Polyelectrolyte Gels Fig. 1 Schematic representation of possible conformational changes in a polyelectrolyte gel arising from polyelectrolyte/hydrophobiccompetition polyelectrolyte network has a relatively low relative permittivity (due to the high con- centration of non-polar polymer chains). In this case some of the counter-ions lose their mobility and condense on the corresponding co-ions forming ion pairs. The ion pairs attract each other strongly due to the dipole-dipole interactions, resulting in the clus- tering of the ion pairs in the so-called ionomer m~ltiplets.'~~'~ This effect was con- sidered in a recent theoret.ica1 paper." It was shown that accounting for the possible formation of ion pairs for the collapsed gels in many cases leads to very significant changes in the properties of the collapse transition.The transition generally becomes sharper, and moreover two different states of collapsed gels (with and without ionomer multiplets) can be formed with the possibility of abrupt transitions between all the three states (swollen, collapsed without multiplets, collapsed with multiplets; we called this last statz the supercollapsed gel). Finally, another reason for microstructuring in the collapsed gels is the formation of glassy, kinetically frozen polymer-rich regions. Indeed, suppose that the pure polymer at a given temperature is in the glassy form (as it is at room temperature for most common gels with high swelling capacity : polyacrylic and polymethacrylic acids, polyacrylamide).In the course of the gel collapse, dense polymer nuclei can be formed which undergo the transition to the glassy state. After which the structure becomes kinetically frozen, and no further collapse is possible. For the case when the gel collapse is induced by the temperature change in a polar solvent, the polyelectrolyte/hydrophobiccompetition is known to be the main reason for the emerging microstructures. This was observed recently for weakly charged poly(acry1ic acid) gels'** ' and poly(N-isopropylacrylamide-co-acrylicacid) gels.20 By A.R. Khokhlov et al. means of small-angle neutron scattering (SANS) it was shown that a pronounced struc- tural inhomogeneity (reflected by the SANS peak as well as by the high value of the scattering exponent) appears already in the swollen gel near the point of the collapse transition. If the collapse is induced by the addition of a poor solvent all three reasons men- tioned above may be important. To show this the microstructure of polyacrylamide gels, containing 2, 5 and 10 mol% of cationic (diallyldimethylammonium bromide) or anionic (sodium or caesium methacrylate) units in water-ethanol mixtures was studied.2' The small-angle X-ray scattering (SAXS) intensity curve I(q) as a function of the scattering vector q for the gels did not show any scattering maxima.However, for highly charged poly(acry1amide-co-diallyldimethylammonium bromide) gels containing 10 mol% of positively charged units (PADADMAB-lo), the scattering exponents, p, evaluated according to the fitting I(q) zqPpin the range of 0.1 < (q/nm-') <0.47, were shown to increase abruptly from 1.7 to 3.3, together with the gel collapse (Fig. 2). The relatively high values of scattering exponents may indicate the presence of sharp inter- phase boundaries in the system. Since the value of p increases simultaneously with the gel collapse (when the volume fraction of non-polar polymer inside the gel becomes much higher leading to the enhanced probability of the formation of ion pairs), we conclude that the formation of ionomer multiplets is the main reason for the observed microheterogeneities for this case.It was shown that such microstructure can be formed only at sufficiently high con- centration of charged groups when the ion pairs easily aggregate in multiplets. Indeed, in contrast with PADADMAB-10 gel, the collapse of the PADADMAB-5 gel containing 5 mol% of charged units (at 52 vol.% of ethanol) has no effect on p. For this weakly charged gel the appearance of sharp interphase boundaries was found only at much higher ethanol content (Fig. 2). The water-swollen gels immersed in the solvent mixtures with high ethanol content do not reach the equilibrium collapsed state because of the vitrification of polymer-rich regions in the gel.This can be seen from Fig. 3 where the swelling curves for the 0 20 40 60 80 100 w(vol."h) Fig. 2 Scattering exponent p as a function of ethanol concentration (w) in water-ethanol mixtures for poly(acry1amide-co-diallyldimethylammonium bromide) gel, containing 10 mol% (1) and 5 mol% (2) charged units Supramolecular Structures in Polyelectrolyte Gels 1.o 0.5 -1 .o Fig. 3 Swelling ratio as a function of ethanol concentration in water-ethanol mixtures for poly(acry1amide-co-diallyldimethylammoniumbromide) gel, containing 5 mol% charged units : (1) swelling of dry gels, (2) collapse of swollen gels PADADMAB-5 gel are compared for the initially water-swollen and initially dry gel. It is clear that for initially water-swollen gels immersed in the solvent mixtures with ethanol content larger than 70 vol.% the non-equilibrium kinetically frozen structures emerge simultaneously with the rise of the scattering exponent from 1.7 to 3.5 in Fig.2. Thus, it is natural to associate the emerging supramolecular structure with the appear- ance of the glassy regions in the collapsed gel. The non-equilibrium gels were found to have the modulus of elasticity by two orders of magnitude higher than that of the equi- librium collapsed gels. Such high values of the elasticity modulus are reasonable if one assumes the existence of the glassy microaggregates. Hence in the collapsed polyelectrolyte gels in a poor solvent the interphase bound- aries (and thus some kind of microsegregation) are always present in the system.For the gels with high enough ionic content the sharp boundaries appear simultaneously with the gel collapse, and can be identified as connected with ionomer-type multiplet struc- ture, On the other hand, for the gels with low ionic content the sharp interphase bound- aries can be observed only at higher poor solvent concentration when kinetically frozen regions start to be formed. (ii) Microstructures in Polyelectrolyte Gels in Low Polar Solvents The existence of the supramolecular order in the collapsed polyelectrolyte gels caused by the ionomer multiplets can be further illustrated by the striking effect of the collapse of polyelectrolyte gel induced by additional ionization. This effect was first considered theoretically in ref.22. The main idea is the following. Suppose that we have a weak polyacid gel (e.g.polyacrylic or polymethacrylic acid) in a low polar solvent and we add some strong base to achieve a partial neutralization. A. R. Khokhlov et al. This leads to an increase in the amount of charged network units. As a result the gel should swell more due to the osmotic pressure of the counter-ions neutralizing the network charges. But simultaneously one must take into consideration the Gibbs energy of the potentially possible state of the collapsed gel with ionomer multiplet structure. The greater the number of charged monomer units, the more thermodynamically advan- tageous is this state because of the large energy gain due to the formation of ion pairs and their subsequent aggregation into multiplets.Finally, this leads to the sudden col- lapse of the gel at some definite degree of ionization. At higher ionization degrees the gel is always in the supercollapsed state with developed ionomer multiplet microstructure. This theoretical prediction was checked by studying the swelling behaviour of the poly(methacry1ic acid) gels with different degrees of ionization, a, in methanol. It was found that the gel swells at very low a and then collapses abruptly. At a further increase of o! the swelling ratio of the collapsed gel remains constant (Fig. 4). The gel collapse was assigned to the formation of ion pairs. This was supported by the results of conductivity measurements. It was shown that the gel swelling at low cc correlates well with the increase of the reduced conductivity, while the gel collapse is accompanied by a signifi- cant drop of the reduced conductivity.The last fact indicates that the ion pairs are formed. To study the aggregation of ion pairs in the multiplets the relative permittivity method was used. It was found that the relative permittivity increases with the decrease of the frequency of the applied electric field. The very high values of the relative permit- tivity at low frequencies evidence the formation of large aggregates with high polariza- bility, which can be regarded as ionomer multiplets. (iii) Supramolecular Structure of the Complexes of Polyelectrolyte Gel with Oppositely Charged Surfactants When the polyelectrolyte gel interacts with a sufficiently dilute solution of oppositely charged surfactants [surfactant concentration in this solution being much lower than the 0.0 0.2 0.4 0.6 0.8 10 a Fig. 4 Dependence of the relative mass of poly(methacry1ic acid) gel in methanol on the degree of ionization a Supramolecular Structures in Polyelectrolyte Gels critical micelle concentration (c.m.c.)], the surfactants are absorbed effectively by the gel with the formation of micelles inside the gels.23 We have explained this process theoreti- cally by taking into account the crucial role of the translational entropy of the gel co~nter-ions.~~The c.m.c.inside the gel turns out to be much lower than in the outside solution, since in the gel the charge of the surfactant micelles is compensated by the immobilized charges of the gel chains (contrary to the case of the solution where the compensation of the micellar charge by the counter-ions leads to a significant loss of their translational entropy).Simultaneously with the formation of micelles inside the gel, the gel collapses. The reason for this is the significant decrease of the exerting osmotic pressure of counter-ions inside the gels (counter-ions are now organized in micelles and so are no longer mobile). The SAXS investigation of the emerging supramolecular order in the collapsed cationic poly(diallyldimethylammonium chloride) gels interacting with oppositely charged sodium alkylsulfate surfactants shows a surprisingly perfect quasicrystalline ordering of the micelles inside the gels.25 The SAXS peaks observed are very narrow, corresponding to regions of practically perfect spatial organization of micelles of the size of at least 10 intermicellar distances (Fig.5). The position of the peaks corresponds to a body-centred cubic type of spatial arrangement of the micelles. The surprising feature is that the mesh size of the gel is much smaller than the distance over which the micelles are practically perfectly ordered, i.e. the gel chains do not disturb the order. The analogous results (narrow SAXS peaks corresponding to a high level of structur- al organization of surfactant aggregates in the gels) were also obtained for anionic poly(sodium acrylate) gels interacting with various cationic surfactants (alkyltrimethylammonium or alkylpyridinium halides).26 In this case the emerging supramolecular order was associated with the existence of lamellar surfactant phase.For both cationic gel-anionic surfactant and anionic gel-cationic surfactant com- plexes the values of the characteristic length correlated with the length of the surfactant molecules (and were somewhat smaller than the double length of the surfactant 1 0 1 2 3 4 q /nrn-’ Fig. 5 Intergrated SAXS profile of poly(dially1dimethylammonium chloride) gel-sodium tetra-decyl sulfate complex A. R. Khokhlov et al. molecule). The ratio between the amount of the charged groups of the network and the number of the surfactant ions in the gel phase has a little effect on the position of the SAXS peaks, but their sharpness changed considerably.This result demonstrates again that the structure is controlled by the optimum gel-surfactant complexes. Thus, polyelectrolyte gels can form the appropriate medium for the self-assembly of the surfactant systems. This fact could have interesting consequences for biological systems. The authors thank the Russian Foundation for Fundamental Research for financial support under Grant No 93-03-4187. O.P. and N.S. thank the International Science Foundation for financial support under Grant No M6T000. References 1 K. Dusek and W. Prins, Adv. Polym. Sci., 1969,6, 1. 2 T. Tanaka, Phys. Rev. Lett., 1978,40, 820.3 A. R. Khokhlov, Polymer, 1980,21, 376. 4 T. Tanaka, D. J. Fillmore, S. T. San, I. Nishio, G. Swislow and A. Shah, Phys. Rev. Lett., 1980,45, 1636. 5 M. Ilavsky, Macromolecules, 1982, 15, 782. 6 T. Hirokawa, T. Tanaka and E. Sato, Macromolecules, 1985,18,2782. 7 J. Ricka and T. Tanaka, Macromolecules, 1985,18, 83. 8 J. Ricka and T. Tanaka, Macromolecules, 184, 17, 2916. 9 A. R. Khokhlov, S. G. Starodubtzev and V. V. Vasilevskaya,Adv. Polym. Sci., 1993, 109, 123. 10 V. Borue and I. Erukhimovich, Macromolecules, 1988,21,3240. 11 J. F. Joanny and L. Leibler, J. Phys. (Paris), 1990,51, 545. 12 A. R. Khokhlov and I. A. Nyrkova, Macromolecules, 1992,25, 1493. 13 E. E. Dormidontova, I. Ya. Erukhimovich and A. R. Khokhlov, Makromol. Chem., Theory Simul., 1994, 3, 661.14 I. A. Nyrkova, A. R. Khokhlov and M. Doi, Macromolecules, 1994,27,4220. 15 A. Eisenberg, B. Hird and M. Moore, Macromolecules, 1990,23,4098. 16 I. A. Nyrkova, A. R. Khokhlov and M. Doi, Macromolecules, 1993,26,3601. 17 A. R. Khokhlov and E. Yu. Kramarenko, Makromol. Chem., Theory Simul., 1994,3,45. 18 F. Schosseler, F. Ilmain and S. J. Candau, Macromolecules, 1991, 24, 225. 19 F. Shosseler, A. Moussaid, J. P. Munch and S. J. Candau, J. Phys. ZZ, 1993,3, 573. 20 M. Shibayama, T. Tanaka and C. C. Han, J. Chem. Phys., 1992,97,6842. 21 0.E. Philippova, T. G. Pieper, N. S. Sitnikova, S. G. Starodoubtsev, A. R. Khokhlov and H. G. Kilian, Macromolecules, 1995, 28, 3925. 22 A. R. Khokhlov and E. Yu. Kramarenko, Macromolecules, in the press. 23 A. R. Khokhlov, E. Yu. Kramarenko, E. E. Makhaeva and S. G. Starodubtzev, Macromolecules, 1992, 25,4779. 24 A. R. Khokhlov, E. Yu. Kramarenko, E. E. Makhaeva and S. G. Starodubtsev, Makromol. Chem., Theory Simul., 1992,1, 105. 25 B. Chu, F. Yeh, E. L. Sokolov, S. G. Starodoubtsev and A. R. Khokhlov, Macromolecules, in the press. 26 Yu. V. Khandurina, A. T. Dembo, V. B. Rogacheva, A. B. Zezin and V. A. Kabanov, Polymer Sci. (USSR), 1994,36,189. Paper 5102683K; Received 27th April, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100125

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Faraday Discuss., 1995,101,133-146 Elastic Behaviour of Salt-free Polyelectrolyte Giovanni Nisato, Rachid Skouri, Frangois Schosseler, Jean-Pierre Munch and Sauveur Jean Candau Laboratoire d 'Ultrasons et de Dynamique des Fluides Complexes, URA 8.51, Universitk Louis Pasteur, 4 rue Blaise Pascal 67070 Strasbourg, France Poly(acry1ic acid) gels ionized after preparation for different preparation concentrations and degrees of cross-linking and ionization, have been studied as a function of the swelling ratio and at swelling equilibrium. Mea- surements performed as a function of the polymer concentration indicated that for low swelling ratios (<100) the shear modulus follows a scaling law with an exponent in agreement with classical theories. For a given concen- tration the shear modulus has been found to decrease with the degree of ionization as qualitatively predicted by a recent model.At swelling equi- librium the shear modulus varies linearly with the osmotic pressure of the counter-ions in a large concentration domain and for various cross-linking densities. The osmotic prefactor values, found both by mechanical and by light-scattering experiments, were in agreement with various independent measurements performed on poly(acry1ic acid) solutions. In this paper, we address an issue related to an intriguing result that we have recently reported, concerning the elasticity of partially ionized gels.'v2 More specifically, it was found that for an ionized poly(acry1ic acid) (PAA) gel at concentrations close to the concentrations of preparation, the introduction of electrostatic interactions lowers the shear modulus whereas it increases the osmotic compressional modulus.' The addition of salt screens these interactions and allows one to recover the moduli of unneutralized gek2A possible explanation for this effect is provided by the recent theory of Rubinstein et af.,3 based on a scaling theory for the configuration of polyelectrolyte chains in semi- dilute solutions4 and on the ideas proposed by Panyukov and Obukov et al.to relate network strand configuration and shear rnod~lus.~~~ In the limit of weak stretching of the polyelectrolyte networks, the shear modulus G is given by: Ck T R'2GzA-N R2 where C is the monomer concentration, N the average number of monomers per strand, kB the Boltzmann constant and 7'the temperature.R'2 and R2 are the mean-square end-to-end distances of, respectively, a strand in the state corresponding to the experi- mental conditions and a free chain of N monomers in the same state. For a given gel at fixed concentration C and in the absence of salt, R2 should be an increasing function of the ionization degree M: because of the swelling of the chains due to electrostatic interactions. According to eqn. (l),this leads to a decrease in the shear modulus and thus explains, at least qualitatively, the experimental results. As for the osmotic compressional modulus, it is simply related to the counter-ion entropy and therefore increases with a for a fixed polymer concentration.' The effect of G on a cannot be explained by classical approaches that assume that R2 corresponds to the 133 Elastic Behaviour of Salt-free Polyelectrolyte Gels mean-square end-to-end distance of a chain in the Gaussian state or in the preparation state, which implies that R2 is independent of However, the model of Rubinstein et al.does not account for all the experimental observations. In particular, it was found that both gels which are prepared from ionized monomers and neutral gels ionized after preparation exhibit a similar dependence of the shear modulus with According to eqn. (1) the shear modulus of a pre-ionized network in its preparation state should be identical to the modulus of an uncharged polymer which is equal to (C/N)k, T.However, it could be argued that the decrease of G observed in gels prepared from pre-ionized monomers is a consequence of a change of network topology varying with the degree of ionization of the monomers. The fact that the shear modulus is the same, within the experimental accuracy, for gels prepared from ionized monomers and gels subsequently ionized could then be considered as coin- cidental. Another experimental fact which is not taken into account by the model of Rubin- stein et al. is the lesser effect of the degree of ionization on the shear modulus as the degree of cross-linking is increased.2 For these reasons, it was suggested that the behaviour of the shear modulus was related to the change of the gel microstructure with electrostatic interactions.This hypothesis was supported by the light-scattering results which show that the frozen-in polymer fluctuations are strongly reduced by the presence of electrical Fur-thermore, neutron scattering experiments performed on gels revealed the presence of dense inhomogeneities of small size (<300 A) that can be associated with the frozen-in polymer concentration fl~ctuations.~~~~ The scattering data suggest that the polymer volume fraction associated with these inhomogeneities is a decreasing function of 01 and increases with the cross-linking density. Another prediction of the Rubinstein et al. model concerns the variation of the shear modulus upon swelling of a gel with a fixed ionization degree in the absence of salt.In the weak stretching limit this variation, obtained by using for R2 the result of the scaling theory of semi-dilute solutions of polyele~trolytes,~~~ is found to be: G M (M,/~)~/~C~/~ where a, refers to the degree of ionization in the preparation state, and is defined in the experimental section. This prediction is notably different from the result G M C1I3from the classical theory of Flory-Rehner.' ' Such a prediction, not yet tested experimentally to our knowledge, has its origin on a modification of the reference state of the gel, characterized by R2 and taken to be a semi-dilute solution of free chains with N mono-mers at the same concentration. This assumption leads to a C7/12dependence of G in the case of neutral gels swollen in good sol~ent.~'~ Osmotic deswelling experiments yielded values of the exponent of the power law G(C) between 0.33 and 0.6, which did not provide an unambiguous test of the basic assumption, Furthermore, another mecha- nism implying de-interpenetration of the network strands was invoked12 to explain exponents larger than 3, On the contrary, the theoretical predictions for polyelectrolyte gels differ consider- ably from the classical Flory's model as shown in eqn.(2), which is valid for both theta and good solvents. Thus, we can double-check the fundamental assumption by varying both a and C. We have performed measurements of G as a function of swelling degree for PAA gels of given preparation concentration C: and cross-link density rc with different ionization degrees (0.05 < a < 0.35).The cross-link density is defined as the molar concentrations ratio of bisacrylamide to acrylic acid units. The results show a behaviour of the gel modulus rather close to what was predicted by the classical theory. In order to get a more complete description of the swelling process, we have also performed light- scattering experiments. The device and the methods previously described' allow the G. Nisato et al. 135 determination of the distribution of scattered intensities on scanning through various positions in the gel. This provides a way of measuring the fluctuating component IFthat represents the dynamic fluctuations of concentration in the gel and the frozen-in com- ponent I, associated with the quasi-static spatial concentration fluctuations.14*' In this paper, we also consider the properties of polyelectrolyte gels in equilibrium with the solvent. In the absence of salt, the strong osmotic pressure due to the counter- ion entropy produces a strong stretching of the strands of the network. However, the model gives the very simple prediction that the swelling equilibrium is reached when the osmotic pressure is equal to the shear modulu~.~ We have measured the shear modulus of gels with different cross-link densities (5 x 3 r, < 4 x and different degrees of ionization (4.5 x < a < 0.25) at their maximum swelling. The obtained results have been compared with independent measurements of the osmotic pressure performed on PAA semi-dilute solutions' and light-scattering experi- ments.Experimental Section Sample Preparation The samples were obtained following the procedure described in previous papers.'.' 7~' Gels were prepared by radical copolymerization in an aqueous solution of acrylic acid and N-N'-methylenebis(acry1amide). The gelation reaction was initiated by ammonium peroxydisulfate. Gelifications were carried out in an oven at 70°C for at least 12 h, immediately after nitrogen was bubbled through the solution to remove the dissolved oxygen that strongly inhibits the radical reaction. Cylindrical glass tubes were immersed in the initial solution of monomers. Once the gelation was carried out, these tubes were taken out, the gels were carefully pushed out of the tubes with a piston and then cut.The transparent cylindrical samples were obtained without any visible defect. Their diameter was typically 7 mm, the aspect ratio (height/diameter) was 1 0.1. The degree of ionization a is defined as the ratio of the number of carboxylate groups to the total number of monomers. Since poly(acry1ic acid) is a weak acid, a can be varied over a wide range by changing the pH of the medium. The degree of ionization is a decreasing function of the polymer concentration, which has a finite value in aqueous solution due to the acid-basic equilibrium. For the concentrations of prep-aration used in this study (C,"ranging from 0.71 to 2.9 mol l-'), the dissociation of the polyacid is very low.Thus, the dissociation constant was approximated to: K, = 5.6 x the value corresponding to monomeric acrylic acid. The samples were prepared in pure aqueous solutions. This leads to a ranging from 4.5 x to 8.9 x in the preparation state. Higher ionization degrees (a 2 1 x were achieved by partial neutralization of the polyacid. The samples were swollen by a small amount of concen- trated NaOH solution in order to obtain the desired stoichiometric degree of neutral-ization f (post-ionization). We limited ourselves to neutralization degrees smaller than the Manning condensation threshold, so that we could assume f= a for the polymer concentration range studied. The gels were then either swollen to equilibrium by a large excess of pure water, or swollen by gradually adding small amounts of solvent so that the corresponding shear modulus measurements could be performed.Str&train Measurements Uniaxial compression measurements allow the determination of the shear modulus of a gel, provided that the volume of the sample is constant. This condition was fulfilled at the timescale of the experiments. Elastic Behaviour of Saltgree Polyelectrolyte Gels For relatively low swelling ratios (< 100 in our case), the simplified Mooney-Rivlin relation:" 0 = G(A -l/A2) (3) generally describes quite well the elastic behaviour at small deformation. Here A is the deformation ratio, cr is the stress per unit undeformed area and G is the shear modulus. Traditionally the ratio Q/(A -1/A2) is plotted us.and usually fits horizontal straight lines. For higher swelling ratios, Q is no longer a linear function of (A -l/A2). This behaviour has been reported before2.20v21 and can be interpreted in terms of non-Gaussian elasticity of polymer strands. An approach based on inverse Langevin functions20*21leads to the following type of expression: where A, and A, are either constants21 or functions of the state of the gel.,' The accu- rate study of the non-Gaussian elastic regime is beyond the scope of this paper, but this provides a basis for evaluating the elastic modulus. The samples were cylindrical gels, whose size varied following the swelling ratio (typical dimensions: from 8 mm to 4 cm, aspect ratio: 1 f.0.1).All the measurements were carried out at deformation ratio 0.8 < A < 1 with the apparatus described in a previous paper.' The swelling ratio Q, refers to the dry state. It was determined both gravimetrically and by measuring the dimensions of the samples. Fig. 1 shows plots of a/(A -1/A2) us. (A2 + l/A) for two gels at the swelling equi- librium. For the gel with the higher swelling degree one observes a deviation from the classical regime. Perturbations due to surface effects appear for small deformations Lie. a 0.85 0.90 0.93 0.95 0.97 l8O0O i l2O0O I 10000 8000 6000 1.88 1.9 1.92 1.94 1.96 1-98 Fig. 1 Representation of stress-strain data for elastic modulus measurements. The gel sample was prepared at C: = 1.42 mol I-' and r, = 0.02.The degree of ionization was 0.25. Degree of swell-ing: (0)Q =(.) 240 and Q = 27. Two series of measurements were performed on the same sample. Non-Gaussian elastic behaviour was observed at the higher swelling ratio [cf:eqn. (411. G. Nisato et al. 137 (A' + 1/A) w 21. In the strong swelling regime, G was defined as G' in eqn. (4); it was determined using the plot previously mentioned and also by evaluating the slope of 0 us. (A -1/A2) for small 1values.20 The two methods gave comparable results. The accuracy of the determination of G is ca. 10% in the low swelling regime and ca. 20% in the high stretching regime. Light-scattering Experiments The laser light-scattering device which was used has been described previously.13 The sample cell could be translated step by step (varying from 3.3 to 25 pm) by a motor so that the light intensity scattered from different volumes could be measured; the total displacement was 5 mm.The count rate for every probed volume was automatically acquired during an integration time of 5 s. Light intensities were normalized by the scattering intensity in toluene. The number of coherence areas n, over which the scattered light was gathered, is related to the coher- ence factor /3, which varies as n -Dynamical light-scattering experiments performed on a well known medium (latex) allowed an estimation of /3. In practice, /3 was varied by adjusting the size of two pinholes in front of the photodetector. Under our experimental conditions, the scattered light was collected over a single coherence area and /3 = 0.96.The variations of light intensity scattered by different volumes of a gel Igel,are shown in Fig. 2. The gel sample had the following characteristics: Y, = 0.02, CL = 7 x and C: = 1.11 mol I-'. As previously re~orted,'~.'~the ensemble-averaged intensity (I,,,), relative to the gel is significantly larger than Isol,the intensity scattered from a semi-dilute solution at the same concentration and degree of ionization. For a given scattering volume, Igelwas shown to be the sum of two terms, I, and I,. The first contribution to Igelarises from frozen fluctuations in polymer concentration; the second represents the dynamic fluctuations of concentration in the gel.On the time- scale investigated in a dynamic light-scattering experiment, I, can be considered as con- stant. On the other hand IF is independent of the scattering volume and is equal, within experimental accuracy, to the intensity scattered from the corresponding solution, Isole 60 1 I 50 i d ' 30 2z 20 10 <Igel'E 0 IF 0 200 400 600 800 1000 position (arb. units) Fig. 2 Variation of the light intensity scattered from different positions in a gel (C,"= 1.11 mol l-', o! = 7 x rc = 0.02). The solid line represents the ensemble-averaged intensity (Ige,)E. The dashed line corresponds to the fluctuating component I,. The intensity scattered from a semi-dilute solution at the same concentration and degree of ionization, Isol(not plotted), is inde- pendent of the scattering volume' 3,1 and is equal to I,.Elastic Behaviour of Salt-free Polyelectrolyte Gels On the other hand, I, varies considerably from one location in the gel to another as shown in Fig. 2. Theoretical Background We recall here the theoretical results of Rubinstein et aL3 that are needed to discuss our experiments. We consider the effect of swelling by pure water on the shear modulus of a given gel and the behaviour of the shear modulus of gels swollen at equilibrium in pure water. Polymer Concentration Dependence of the Shear Modulus of a Salt-free Gel The basic assumption is that the tension created by the osmotic pressure of the counter- ions is transmitted through the cross-links to the elastic chains, which behave as isolated chains with an applied force at their end-points.The resulting stretching occurs only at scale lengths larger than a stretching length 4, ,introduced by Pincus,22 that depends on the applied force F through 4, M k, TIF (5) Thz behaviour of the shear modulus as a function of the swelling degree depends on how 5, compares with the three following characteristic lengths of the system. (i) The correlation length (, beyond which all correlations are screened and the polyelectrolyte chain adopts a random walk conf~rmation.~~ (ii) The electrostatic screening length rB proportional to the classical Debye-Huckel length K-'. (iii) The size of the electrostatic blob D,below which the conformation is almost unperturbed by the electrostatic inter- actions.* In the absence of salt, both 4: and IC-'decrease with increasing polymer concentra- tion as C-'I2, whereas D is independent of the concentration and is fixed by the degree of ionization.* Rubinstein et aL3 consider three regimes for the polymer concentration dependence of G. At higher polymer concentration where cp> 5, the stretching is Gaussian and the model predicts: G x a-217~516 (6) The classical Flory-Rehner theory predicts a C1/3 dependence of G in the same regime. Upon swelling, the stretching remains Gaussian until 4, becomes equal to 4. Then 4, jumps discontinuously from 4 to D. This cross-over can be described by non-Gaussian stretching via the classical treatment involving the inverse Langevin function.' The modulus increases abruptly upon swelling.This occurs at a polymer concentration Ct given by: Below this concentration the gel experiences a stretching of linear assemblies of elec- trostatic blobs in which some remaining elasticity is stored. The modulus decreases with C with a dependence of C-'I6 in good solvent and C-'l3 in a theta solvent. Let us remark that while the two first regimes (ie.the decrease followed by a sharp increase of the modulus upon swelling) have been observed experimentally in several systems,20924*25the observation of the third regime has never been reported. This might be linked to the non-homogeneity and strand polydispersity of the gels that prevent the reaching of the stage where all strands are made of aligned electrostatic blobs.G. IVisato et al. Shear Modulus of Gels at Swelling Equilibrium Here we consider gels that have been allowed to swell in an excess of pure water. Following the well accepted assumption, the swelling equilibrium is reached when the osmotic pressure TI,,, is equal to the elastic pressure' 'n,,. In the absence of salt the osmotic pressure TIosm results essentially from the entropy of counter-ions and is given by26 Here 0,is the so-called 'osmotic factor' which has been written as 0,= 1 -Z&A, where A is the contour length of the chain between two consecutive charges and lB the Bjerrum length.26 In practice, one generally considers 0,as a prefactor which is not simply related to a.Under both extreme conditions of swelling, i.e. in the absence of salt or at high salt content, the shear modulus G follows a scaling law of the concentration: G = C' (9) Under these conditions, the elastic pressure is given by:12 with g = (+)/(1-I>. In the particular case where 1 = 5 one finds Il,, = G which is the classical result.' ' Therefore the behaviour of the shear modulus of gels at swelling equilibrium should parallel that of the osmotic pressure under the conditions where eqn. (9) applies. Experimental Results and Discussion Effect of Swelling on the Shear Modulus Fig. 3 shows, in log-log coordinates, the variations of the shear modulus with the degree of swelling Q for two series of gels with cross-linking densities of rc = 0.04 and 0.02, and various ionization degrees.One observes the behaviour already reported for poly- electrolyte gels,20 i.e. first a Gaussian stretching regime characterized by a decrease of G upon increasing Q, followed by an abrupt upturn associated with non-Gaussian elas- ticity. As usual we do not observe the very low concentration regime where G should be a decreasing function of Q. The best linear fits to the data in the weak stretching regime lead to values of the exponent of the power law ranging from -0.22 to -0.39. Looking now at the effect of the degree of ionization, it can be seen in Fig. 3 that G is a decreasing function of a for a given Q. In Fig. 4 we have reported the interpolated values of G as a function of a along with values found in previous experiments.2 The data show that G varies as a-* with x ranging from 0.13 to 0.21, which is not too far from the theoretical prediction (xth= $), especially considering that we have interpolated data for an arbitrary value of the swelling ratio.It is clear that the experimental dependence of G with C, differs significantly from the theoretical prediction [cf: eqn. (2)]. Also, Q corresponding to the minimum of G exhibits a much smaller variation with a than expected from eqn. (7). The experimental data do not show a strong variation of Ct in function of a. It must be remarked that the theory assumes the homogeneity of gels at scales larger than r. It is known that this is not generally the case and that the gels exhibit static long-range concentration fluctuations.Previous studies have shown that the electrical interactions tend to reduce these inhomogeneities. This is shown in Fig. 5, showing the Elastic Behaviour of Saltgree Polyelectrolyte Gels 0 3 0 E -j2 \ t32 0 0.25 10 100Q m U o! 0 0.05 0 0 0.15 0 0 0.25 0.35 10 Q 100 Fig. 3 (a) Shear modulus as a function of the swelling ratio. The samples were prepared at C: = 1.42 mol I-’, rc = 0.04. The power-law fits in the low swelling region give exponents between -0.22 and -0.31. The degrees of ionization are indicated in the figure. (b)Shear modulus for gels at Ci = 1.42 mol I-’ and rc = 0.02. The power-law fits in the low swelling region give exponents between -0.23 and -0.39.The degrees of ionization are indicated in the figure. effect of a on the fluctuating part I, and on the ensemble-averaged frozen component (Ic)E of the scattering intensity. For a gel with rc = 0.02 and Ci = 1.11 mol 1-l the observed decrease of I, upon increasing a is the same as that of a semi-dilute solution and corresponds to the decreasing osmotic compressibility. As for (I,)E, its behaviour clearly shows that the gels become much more homogeneous at the length-scales investi- gated by light scattering as the electrostatic interactions are enhanced. The effect of the swelling on I, and (Ic)E are shown in Fig. 6. The variation of I, with polymer concen- tration is linear and coincides within the experimental accuracy with that of semi-dilute polymer solutions. This linear variation of I, is the signature of the specific behaviour of G.Nisato et al.0 0.02 0 0.04 0 0.01 I I 0.1 1 a Fig. 4 Variation in log-log coordinates of the shear modulus as a function of the ionization ratio. 0)Moduli values extrapolated from the data in Fig. 3 (Q = 25, C, = 0.56 mol 1-l). (0,(0, m)data reported in ref. 2 (C, = 1 mol 1-'). Cross-linking ratios are indicated in the figure. The lines represent the best least-squares fit to the data. the osmotic pressure of polyelectrolyte systems that results predominantly from the entropy of the counter-ions. As for (IJE, it is found to be almost constant upon swell- ing of the gels. This behaviour is very different from that of the neutral gels for which (IJE is an increasing function of the degree of swelling.This has been attributed to an inhomogeneous swelling of the gel resulting from an expulsion of small clusters from the inside of larger ones.27 In polyelectrolyte gels the clusters seem to be smaller and denser (<300 A) than in neutral gels as revealed by SANS experiments." The fact that (Ic)E does not depend significantly on Q indicates another mechanism for the separation of 0 0 0 c 10-2 10-1 100 a Fig. 5 Effect of the degree of ionization on the frozen ((1JE)and fluctuating (I,) contributions to the scattered intensity. The gels were prepared at C: = 1.1 mol 1-', rc = 0.02. After ionization, C, = 1 mol I-'. 2.0.I I I Ill 1 1 1 l l l ~ l l l ~ l l l (a) - m - 1.5 - - H as 2Y't1.0 8 00.5 0 0.2 0.4 0.6 0.8 1.o 1.2 wmol I-' 1.2 (b1 01.0 -On 0 17 -0.8-Inn .-c c-2 0.6-Y 2-5" 0.4 0:ooo 0 O0 3 0.2-0 ll'llll'll'llli'll,II,, 0 0.2 0.4 0.6 0.8 1.o 1.2 clusters that would be non-trivially correlated, or a change of the gel structure in the course of swelling. One may wonder whether this micro-structure of the gel plays an important role in the thermodynamic properties? As far as the osmotic compressibility is concerned, the results of Fig. 6 show that to the first order it is not affected by the microstructure and arises only from the ionic pressure of counter-ions. As for the elasticity, one can argue that the shear modulus data reflect complicated averages between the elastic contribu- tions from the dense and more dilute region in the gel and that this would affect the a a 0 0 a a 1043 a 0 +O a x 0.05 0 + 0.15 o 0.25 lo3 -, ,OBI Q 1 I 1 ,,I 1 I I I I <0.01 L I I I 1 Fig.7 Shear modulus of salt-free polyelectrolyte gels of various initial polymer concentrations and cross-linking densities (cJ:Table 1) at swelling equilibrium as a function of the osmotic pres- sure of the counter-ions (k, TaC).The degrees of ionization are as indicated on the figure. scheme proposed by Rubinstein et aL3 In this respect, this model gives qualitatively a correct description of the data, more specifically concerning the effect of a on G and on Ct. However, if this were an effect of the microstructure, one would expect that upon increasing a the G(C)variation should approach the theoretical C5I6behaviour since the gel becomes more homogeneous.This is not observed in Fig. 3. Furthermore, it is quite striking that the exponent of G(C)is repeatedly found to be close to $, that is to say the value predicted by the classical theory based on the assumption that the reference state of the chain is Gaussian.’~~ An alternative explanation would be that the sensitivity of G on a derives partially or totally from a more fundamental effect associated with the inter-chain correlations. The positional correlations in polyelectrolyte systems are revealed by the presence of a peak in the SANS spectra.It has been suggested28 that there should be a strong coupling between the orientation and the separation of neighbouring chains, resulting in a local ‘pre-nematic’ clustering. Such an effect has been recently invoked to explain the lower- ing of the plateau modulus of semi-dilute worm-like micelles by electrostatic inter- action.29 When these interactions are screened out by addition of salt, the plateau modulus increases strongly. The similarity of behaviour between these systems and the polyelectrolyte gels suggest that the origin of the lowering of G by the electrostatic interactions is the same. A simple model derived by Elleuch and Lequeux3’ leads to the conclusion that an increase of the inter-chain correlations lowers the shear modulus.Moreover, the fact that the dependence of G on a is much less pronounced at high cross-link density is consistent with the model. A clear answer to these questions would be given by measure- ments of the plateau modulus of semi-dilute entangled solutions of flexible poly- electrolytes as a function of a. Gels at Swelling Equilibrium The swelling equilibrium condition is obtained by combining eqn. (8)-( 10): =nos,,, <D, k, TCa = gG Elastic Behaviour of Salt-free Polyelectrolyte Gels Table 1 Characteristic parameters of PAA gels investigated at swell- ing equilibrium a (%) Ci/mol 1-' rc (%) Q," G/Pa kBTaC/Pab 25 0.71 2 2050' 1000 4100 25 1.42 2 220 13100 38400 25 2.15 2 74 39200 113700 25 2.90 2 38 82000 264100 0.89 0.71 2 367 1100 820 0.63 1.42 2 22 14100 9500 0.51 2.15 2 9 38500 20000 0.45 2.90 2 6 72000 24100 0.63 1.42 0.5 123 2600 1700 0.63 1.42 1 44 6200 4900 0.63 1.42 2 22 14100 9500 0.63 1.42 4 12 28400 18400 25 1.42 0.5 1470 2500 5700 25 1.42 1 503 5800 16800 25 1.42 2 220 13100 3 8400 25 1.42 4 88 32800 96200 5 1.42 2 96 10300 17500 15 1.42 2 189 11800 26800 " Equilibrium degree of swelling.Osmotic pressure of the counter- ions. The bo re relation is universal and should apply independently of the c nditions of preparation and the cross-link density. Experiments performed on charged copolymers at small degrees of ionization (a < 0.025) led to results in agreement with the above predi~tion.~*~l-~~ 1.5 0 1 % X 0.5 Q0 x +o X 0.X0 + + 0 0 0.2 0.4 0.6 a Fig. 8 Variation of the osmotic prefactor QP with a. Our results, obtained from shear modulus (0) experiments in solutions, are compared with other data taken in gels and light-scattering (0)from the literature relative to PAA solutions : osmotic deswelling measurements'6 (e),osmotic pressure ( x ) and light-scattering (+) experiment^.^^ G. Nisato et al. Fig. 7 shows the variation of G with the product k,TCa for two series of gels with different cross-link densities, initial polymer concentration, and with a = 0.25 and u < (cf.Table 1). Also are plotted the results relative to gels with same values of Ci and rc but with a = 0.05 and 0.15.Except for one point the data fit reasonably onto two straight lines with slopes close to 1. The prefactor mP/g depends on a. If one assumes that g = 1, which corresponds to the value obtained for the classical affine deformation process, then one finds the values of (Dp reported on Fig. 8 for degrees of ionization greater than or equal to 0.05. For the lowest values of u, one finds CD, = 2.18, which seems to indicate that the osmotic pressure due to polymer interactions is no longer negligible. In Fig. 8 we also plotted data relative to osmotic pressure measurements and light scattering experiments in PAA solutions taken from ref. 34. On the same figure we show the value deduced from the light scattering results reported in Fig.6(a) and the results of direct measurements of the osmotic pressure through deswelling experiments independently carried out by Silberberg-Bouhnik et a1.I6 Conclusions The theoretical model of Rubinstein et al. describes qualitatively the experimental results. However, we found quantitative discrepancies concerning the dependences of G on a and C, and the effect of a on the cross-over concentration Ct between Gaussian and non-Gaussian stretching regimes. In the weak stretching regime, the exponents found for the power law dependences of G as a function of C are in better agreement with the classical approach than with this new model. The swelling equilibrium of neutralized gels swollen in pure water is correctly described by a linear dependence of G on the polymer concentration at the swelling equilibrium.This provides an experimental verification of the relation G = (Dpl-Iosm and an estimation of the osmotic prefactor QP. The latter was found to agree with experi- mental results previously reported in the literature for polyelectrolyte solutions. In order to clarify the physical phenomenon responsible for the dependence of G on a, a further investigation of the viscoelastic properties of solutions of entangled high molecular weight polyelectrolytes should be performed. We are grateful to M. Rubinstein and Y. Cohen for sending us copies of their papers prior to publication. We would also like to thank J. F. Joanny, F. Lequeux and J. Bastide for very useful discussions and A.Knaebel for her efficient help in the experi- mental work. References 1 F. Schosseler, F. Ilmain and S. J. Candau, Macromolecules, 1991,24,225. 2 R. Skouri, F. Schosseler,J. P. Munch and S. J. Candau, Macromolecules, 1995, 28, 197. 3 M. Rubinstein, R. Colby, A. V. Dobrynin and J. F. Joanny, in the press. 4 A. V. Dobrynin, R. H. Colby and M. Rubinstein, Macromolecules, 1995,28, 1851. 5 S. V. Panyukov, Sou. Phys. J. (Engl. Transl.), 1990,71, 372. 6 S. P. Obukhov, M. Rubinstein and R. H. Colby, Macromolecules, 1994,27, 3191. 7 K.Dusek and W. Prins, Adv. Polym. Sci., 1969,6, 1. 8 S. J. Candau, J. Bastide and M. Delsanti, Adv. Polym. Sci., 1982,44, 27. 9 A. Moussaid, S. J. Candau and J. G. H. Joosten, Macromolecules, 1994,27, 2102.10 F. Schosseler, R. Skouri, J. P. Munch and S. J. Candau, J. Phys. 11,1994,4, 1221. 11 P. J. Flory, The Principles of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953. 12 J. Bastide, S. J. Candau and L. Leibler, Macromolecules, 1980, 14, 719. 13 R. Skouri, J. P. Munch, F. Schosseler and S. J. Candau, Europhys. Lett., 1993, 23, 635. 14 P. N. Pusey and W. Van Megen, Physica A, 1989,157,705. 15 J. G. H. Joosten, J. L. McCarthy and P. N. Pusey, Macromolecules, 1991,24,6690. Elastic Behaviour of Salt-free Polyelectrolyte Gels 16 M. Silberberg-Bouhnik, R. Ory, I. Ladyzhinsky, S. Mizrahi and Y.Cohen, J. Polym. Sci. B, 1995, 30, 2269. 17 F. Schosseler, A. Moussaid, J. P. Munch and S. J. Candau, J. Phys. IZ, 1991, 1, 1197.18 A. Moussaid, J. P. Munch, F. Schosseler and S. J. Candau, J. Phys. ZZ, 1991, 1, 637. 19 L. R.G. Treloar, The Physics of Rubber Elasticity, Clarendon Press, Oxford, 1975. 20 U. P. Schroder and W. Oppermann, Makromol. Chem., Macromol. Symp., 1993,73,63. 21 J. Hasa, M. Ilavsky and K. Dusek, J. Polym. Sci.,Polym., Phys. Ed., 1975, 13,253. 22 P. Pincus, Macromolecules, 1976,9, 386. 23 P. G. de Gennes, P. Pincus, R. M. Velasco, F. Brochard, J. Phys. (Paris), 1976,37, 1461. 24 W. Oppermann, Die Angew. Makromol. Chem., 1984,1231124,229. 25 U. Anbergen and W. Oppermann, Polymer, 1990,31, 1854. 26 J. F. Joanny and P. Pincus, Polymer, 1980,21,274. 27 J. Bastide, L. Leibler and J. Prost, Macromolecules, 1990,23, 1821. 28 M. E. Cates, J. Phys. ZZ, 1992,2, 1109. 29 V. Schmitt and F. Lequeux, J. Phys. ZZ, 1995,5, 193. 30 K. Elleuch and F. Lequeux, submitted. 31 M. Ilavsky, Macromolecules, 1982,15,782. 32 M. Ilavsky, H. Hrouz, J. Polym. Bull., 1982, 8, 387. 33 M. Ilavsky, H. Hrouz, J. Polym. Bull., 1983,9, 159. 34 M. Nagasawa and A. Takahashi, in Light Scattering from Polymer Solutions, ed. M. B. Huglin, Aca- demic Press, London, 1972, p. 720. Paper 5/03288A; Received 22nd May, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100133

出版商:RSC    

年代:1995

数据来源: RSC

摘要:

Farday D~SCUSS.,1995,101, 147-158 Chemical Clusters in Polymer Networks Karel Dukk and Jan Somvarsky Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, 162 06 Prague 6, Czech Republic Chemical (topological) clusters in polymer networks have been defined as covalently bonded assemblies of units of a certain kind, for instance 'hard' units among the soft ones. For theoretical characterization, two methods have been used: (a) the theory of branching processes (TBP) by which the structures are generated from monomer units and (b) the generalized kinetic theory with a Monte-Carlo simulation of the process. The clusters are char- acterized by their average sizes and average number of issuing soft bonds (cluster functionality) and conditions where the clusters grow to infinity.Using the TBP, the characterization is based on classification of the bonds connecting units as hard +hard, hard +soft, soft +hard and soft +soft. The cluster size and functionality as a function of conversion are determined by the initial composition of the system, the rules of bond formation and the relative reactivities of functional groups. Under certain conditions, the hard clusters can grow to infinity. Method (b) allows us additionally to treat the long-range connectivity effects as well as to simulate the steric excluded volume effect by making the effective reactivity of a group dependent on the cluster size. The exclusion from the reaction of a larger fraction of groups in larger clusters makes the distribution narrower, shifts the gel point to higher conversions and causes the critical exponents to deviate from their classical values. The implications of the existence of hard clusters on equilibrium elasticity are discussed.In many crosslinking systems, the network is formed from units of different kinds (short and long, soft and hard etc.). By chemical reactions of functional groups, the units of one or the other kind can be combined into branched or crosslinked structures composed exclusively of hard or soft units, hard or soft chemical clusters. These clusters are called topological clusters because their size and shape are determined by the connectivity pattern of units of which the clusters are composed and not by their spatial arrange- ment.The spatial arrangement of units is also determined by non-bonding interactions between units and, therefore, the physical interactions effective during network forma- tion can influence the connectivity pattern and thus the size and shape of clusters. In the absence of spatial interactions discussed above, the structure of topological clusters depends on the initial molar ratios of the components, the functionality of the com- ponents, the reactivities of the functional groups and the history of network formation (staging). The existence of clusters is expected to affect the physical properties of the networks, such as their mechanical or scattering behaviour. The segmented polyurethanes (or other multiblock copolymers)' are linear ana-logues of three-dimensional chemical clusters.The arrangements of blocks in the copoly- mers (degree of polymerization or molecular weight of the blocks) as a function of composition, relative reactivities of functional groups and method of preparation can be characterized in a simple way by using first-order Markovian statistics.2 This way is, 147 Chemical Clusters in Polymer Networks however, rigorous only for isolated substitution effects. If long-range time correlations exist, such as those arising from repeating substitution effects or initiati~n,~.~this approach is no longer rigorous and a kinetic description of cluster formation has to be used. The existence of topological clusters in polyfunctional systems determined only by short-range range was proposed some time but theoretical descriptions were not offered.In this paper, methods of characterization of cluster sizes, functionality and cycle rank will be derived for cluster generation determined by (a) purely reactivity and history effects and (b) by combination of these effects with spatial correlations like diffu- sion or excluded volume. For case (a) the statistical theory of branching processes (TBP) will be adapted, whereas the more general cases of (a) and case (b) will be treated by a Monte-Carlo simulation of a generalized Smoluchowski coagulation process. The recently described basic algorithms of the Monte-Carlo simulation394 of network forma- tion will be used. Theoretical Theory of Branching Processes:Cluster Generation from Units Within the statistical theory of branching processes, the topological structures of units connected by covalent bonds are generated by combination of these units in different reaction states.The states of a unit differ in the type and number of bonds by which the unit is connected with other units (see, e.g. ref. 5).The basic information on this distribu- tion is conveniently described by a probability generating function (pgf) which describes the number fractions of units in different reaction states: Fon(Z, 2) = 1ni zi Fan, i(Zi) (1) I Zi = (Zil, zi2, . * .) (2) where ni are molar fractions of units i. Any such unit is characterized by the dummy variable Zi(each unit contributes to their number by 1, i.e.the exponent of 2 is 1); the variables zij in the pgfs of the component i (F0Ji(zi)characterize the number of bonds extending from the unit i to the unitsj, i +j. Thus, the basic pgf bears not only information on the types of units and bonds but, through the variables z, it is also ready for generation of multiunit structures. For the purpose of cluster generation, the units are distinguished according to whether they belong to the class of soft or hard units, and eqn. (1)is transformed into FOn(Z, Z) = C 'hi 'hi 'On, hi(zhi) + 1 n~jZ~j Sj(ZSi) (3)'On, I i where the vectors have the following meaning Zsj (Zsjh 9 zsjs) (4) Zsjh (z~jhl,Zsjhs 3 * * .) (5) The pgfs for the number of additional bonds issuing from units already bonded by one bond to another unit are given by the vector F(2,z) F(z, 2) [Fhh(z, 21, Fhs(z, Z), Fsh(Z, Z), F~S(z, Z)l (6) K.Dus'ek and J. Somvarsky The first subscript, e.g. si of Fsih.(q,j), identifies the unit si on which the unit hi is rooted. E.g. pgf FSihj(thj) is obtained by differentiation The generating function for the 'weight' fraction of molecules disregarding the difference in the type of units can be defined as m al in our case, w(z)= 1mhi zhi FOn, hi(uhi) + 1%j zsj FOn, sJusj) (11)i i where m are mass fractions of the corresponding units (in the present case of the degree of polymerization mi= nJ, and functions uhi u,,(z) usj usj(z) (12) bear the same subscripts as the variables z: Each f&hj is given by the following recursive equation: %ihj = zhj Fsihj(uhj) (15) Beyond the gel point, the extinction probability is the quantity by which the sol and gel fractions are characterized.The extinction probability is a vector (uhh? uhs, ush, @ss) (16) where, e.g. The values of the components of u are obtained by solution of the set of recursive equations of the type 'Sihj = Fsihj(Uhi) (19) In the approach outlined above all bonds are considered. The sol fraction, the number of elastically active network chains and other parameters characterizing the sol and gel are obtained in a standard way. For characterization of the size of chemical clusters or their functionality, only some types of bonds are relevant. Thus, for the degree polymerization or molecular weight of the hard clusters, only the h-h bonds are relevant, whereas for characterization of the functionality of clusters one counts the h-s bonds.This manipulation is demonstrated by the following scheme For cluster size distribution (averages) hard -+ hard Zhh = Zhh hard -+ Soft zhs = 1 Soft +hard Zsh = 1 soft -+ soft zss= 1 Chemical Clusters in Polymer Networks For functionality distribution (averages) hard -+ hard Zhh = Zhh hard + Soft zhs = zhs Soft + hard zsh = 1 soft + soft z,, = 1 For the elastic (soft) part hard -+ hard zhh = 1 hard + Soft zhs = zhs Soft + hard dsh = zsh soft + soft z,, = z,, The basic characteristics of the system, i.e. the number- and weight-average degrees of polymerization, the number- and weight-average functionalities (number of issuing h-s bonds) of hard clusters, depend on what structural unit is considered to be a cluster and what is not.Here, we will adopt a simple definition saying that any 'hard' unit, includ- ing the unreacted monomer, can be considered a hard cluster. This may be physically unreasonable and we may wish to consider as clusters only one or more hard units that have reacted at least with one functional group. Alternatively, one can define a hard cluster as containing at least one hard unit with functionality f> 2. For the general concept of derivation of the averages, these differences are irrelevant. Therefore, for calculating the degree-of-polymerization and functionality averages of hard clusters the pgf Fo, can be simplified as shown above (zsh = z,, = 1).The number of half-bonds between hard units is given by the value of the derivative of the F, with respect to the variables Zhihj and $Fg(l)is equal to the number of bonds between them. Therefore, The weight-average degree of polymerization, (P,), , can be calculated from the pgf wc(zh), which is given by the following equation: where The components of uiih are given by uiihj = zhj Fhih,(UEjh 9 Zh,s) (24) The weight-average degree of polymerization of hard clusters (P,), is obtained by dif- ferentiation where the types of hard and soft units are no longer distinguished: Zh, = zh2 = ' --Zh -... -= zhs.and Zhlsl = ' * = Zhzsl - K. Dus’ek and J.Somvarsky The number- and weight-average functionalities of clusters, given by the number of issuing soft chains from a cluster, depend on the definition of the hard cluster. If we consider all hard units, evidently the functionality of a cluster will increase starting from the value zero. The number-average functionality is given by the number of h-s bonds per number- average degree of polymerization of the cluster where (yc)wis defined as the number of issuing soft chains per weight-average degree of poly- merization of hard clusters: f*x and can be obtained by differentiation of pgf wc(zh, zhs): One can also derive the second-moment-average functionality defined by f.x This average can be derived from the number distribution given by the pgf Nc(Zh, zhs) Nc(Zh 9 zhs) = ngf zt ZLs x-9 f The pgf Nc(Zh, zhs) can be derived from the pgf w(zh, zhs) since &=o b It has been shown7 that for a single-parameter pgf, Fo(Z), the integral where can be solved by integration by parts, yielding for the number-fraction generating func- tion N(Z) the following expression: -N(Z) = <~~fl~FO“Z)I+~Xl)L-U(~)l21 (34) where Fo(l)is the derivative of F,(Z) with respect to Z for 2 = 1.ChemicaE Clusters in Polymer Networks Applying this procedure to number-fraction gf (31) one obtains 1 Nc(Zh 3 ‘hs) = (‘c>n 1nhi ‘h( ‘On, hi(6ih ‘hs) --2,c Fhih’On,hi(lbhihj uhjhi) (35) where Then, the second-moment functionality average, {fc), is given by The derivation of expression for the post-gel parameters, like the sol fraction, the number of elastically active network chains, or the number-average functionality of elas- tically active clusters, is analogous to the case of build-up of the whole network with the modifications already explained above.Remember that (1) for the sol fraction one should use the real mass fractions of units, mi = niMi/zniMi, (2) an elastically active network chain (EANC) is definition dependent: it may or may not include branch points within monomer units, or discard some short EANCs, (3) an elastically active cluster has an elastically effective functionality of 23 or higher, i.e. three or more (to infinity) soft chains issue from it, (4) the elastically effective functionality,f, , is the number-average of the number of issuing soft chains with infinite continuation per elastically active cluster.Crosslinking of bifunctional short (hard, component h,) and long (soft, component s) chains with a trifunctional crosslinker (hard, component h,) will serve as an example of a concrete system. Each unit bears functional groups of the same kind as are denoted by the same subscript as the units, The chains have functional groups of one kind and the crosslinker has groups of another kind and the following reactions are possible : s + h, -,sh, or h,s (37) h, + h2 -+h,h2 or h, h, (38) The pgf characterizing the composition of the system in terms of units in different reac- tion states reads In this pgf, ah,,ah2,a, are conversions of the respective groups and $s = -$h and 1 -a, = (1 -ahl)ks/khl,where k,/khl is the ratio of rate constant of like groups of soft (long) and hard (short) chains with the crosslinker groups.Relationships for parameters characterizing the hard clusters can be derived from this basic equation according to the recipe given above. Kinetic Generation of Clusters The kinetic simulation is the second approach to cluster growth simulation developed here. The cluster formation process is described by the generalized coagulation equa- tions of the Smoluchowski type. In this method of generation, the long-range connec- K. Dus’ek and J. gomvarsky tivity correlations determined by the chemical mechanism and kinetics are treated rigor- ously.Moreover, one can introduce the concept of the dependence of reactivity of a functional group on the structure (size, symmetry, cycle rank) of the cluster of which the group is a part.3*4*8Also translational and segmental diffusion control can be con- sidered. The usual method of solution of an infinite set of differential equations of chemical kinetics cannot be used for a generalized kernel of the Smoluchowski equations. The structure evolution is still described by temporal changes of concentrations of each dis- tinguishable species but a Monte-Carlo method has to be used. The criterion of dis- tinctness depends on our choice and on the capability of the computer and may go down to full topological information on the species.At present, we have restricted our- selves to the number of functional groups (I) and building units (x).~ The following reactions can be considered :intermolecular combination, cyclization and degradation. We will here consider only intermolecular reaction which, however, beyond the gel point and within the gel (largest molecule) gives rise to uncorrelated circuit closing and build-up of cycle rank (cf:ref. 4). Formation of a product can be described as a consecutive reaction involving (diffusion-controlled) transition into an activated complex and followed by a transition into a stable product. Generally, the rate constant for an intermolecular reaction of group A of molecule 1, characterized by composition [x,, Z,], with group B of molecule 2, characterized by [x,, Z2], depends on cornposition. The vectors x, Z describes the composition of the molecules in terms of units x and reactive groups 1.kAB. D kAB.C 7“1, Zll + “2 121 =(Cx1, I, ]A“, ,M)* ____+ kAB, -D where the rate constants depend on composition of molecules 1 and 2. kAB, d~i,kAB, D zip x2 9 12 9 -..) -kAB, -D = kAB, -D(xi, zi, ~2,I29 ... kAB, c kAB, c(xi, zi, x2 9 I2 9 - - .) Here we do not consider explicitly the control of the structural evolution by trans- lational or segmental diffusion, but we do consider a limited access to functional groups in the interior of larger clusters due to steric hindrance. Such an effect is assumed to control the network formation by free-radical copolymerization of polyvinyl mono- mer~.~ Ideally, when chemical kinetics based on the mass-action law are valid (no spatial long-range correlations), the rate constant for the contribution to the total reaction rate of molecules 1 and 2 by the reaction of their A and B groups reads kAB,C(xl, ‘1, x29 ‘2 9 .--)= kAB(zAlzB2 + zB1zA2) (41) The steric hindrance effect is introduced by exponents vl, v2 that are lower than unity The steric hindrance effect is here considered as a static effect, although it can also model the diffusion-controlled interpenetration of two reacting clusters.It is clear that for v < 1 the fraction of obstructed functional groups is larger for larger molecules (clusters) which corresponds to experimental results (c$, e.g., ref. 9). Disregarding details of the internal structure of clusters, one can put vk = v.The exponent v can be considered as characterizing the reactive surface of the mol- ecule: for v + l, all groups are accessible and the reactive surface is proportional to Chemical Clusters in Polymer Networks volume; for v -+ 2/3 the reactive surface is proportional to the (nearly spherical) molecule surface and only groups on the surface are accessible; for 2/3 < v < 1 the reactive sur- faces are corrugated with self-similar shapes. The value of v may change (decrease) with increasing compactness of the molecule, resulting, for instance, from intramolecular reactions. The Monte-Carlo procedure simulating network formation is explained in detail in ref. 3 and 4.Some changes have been made. To simulate evolution of hard clusters in evolving molecules, one would have to keep track of the internal structure of all clusters. This is possible but imposes more requirement on computer speed and memory. Instead of that, for the purpose of generation of hard clusters, the soft component is split into monofunctional fragments and the hard clusters are generated from these new initial components. Therefore, hard-cluster evolution is dependent only on the structure (size) of hard clusters and not on the size and composition of all the molecules in which the clusters are embedded. Discussion Evolution of clusters can be illustrated by crosslinking of mixtures of short and long chains. For a complete conversion of functional groups, the cluster size is fully charac- terized by the initial composition.For the general case of an $functional crosslinker and equal reactivities of the respective groups, the critical conversions for percolation through short or long chains, (ah)critand (aJcrit,respectively, are given by or where & and & are, respectively, the molar fractions of short and long chains. For f= 4, for example, the whole system gels at conversion a = ,/(1/3). For the fraction of short chains equal to 1/3, the infinite hard cluster is formed at conversion a = 1. For & = $s = 1/2, in the conversion range ,/(1/3) < a 6 J(2/3) finite hard clus- ters are interconnected to give an infinite network; for a 2 ,/(2/3) hard clusters are infinite and interpenetrated by soft chains (Fig.1). The effect of differences in reactivities of the like functional groups of short and long chains influences the cluster size evolution considerably (Fig. 2). If the reactivity of groups of short chains is higher by a factor of 10 (a difference in reactivity correspond- ing, for example, to primary and secondary OH groups in a reaction with isocyanate groups under catalysis by organotin compounds), the hara clusters grow bigger in the beginning compared with groups of equal reactivity for which the clusters are initially small and, for = 0.5 the value of (P,), diverges at a -+ 1. The effect of the size-dependent steric excluded volume effect modelled by decreasing exponent v in the kernel of the form (Z1Z2)" (see also ref.10) was examined using the Monte-Carlo simulation technique (Fig. 3). The decreasing exponent v is manifested by increasing critical conversion (percolation threshold) of hard clusters and a change in the slope of the log-log plot of the weight-average degree of polymerization, (P,),, or functionality, (fc),, of hard clusters against the distance from the critical point. By decreasing v, the population of the clusters becomes less heterogeneous, which causes delayed gelation. The critical exponent y of the proportionality (P,), =(ag- K. Dus'ek and J. gomvarsky I I finite infinite I hard hard I clusters clusters 0 0.2 0.4 0.6 0.8 1 (Y Fig. 1 Schematic representation of the growth of hard clusters in ideal crosslinking of a mixture of bifunctional long (soft) and short (hard) chains (fractions &, and ips, respectively) with a tetra- functional crosslinker (hard) as a function of conversion a.Stoichiometric system: $s = 1/2. increases with decreasing v from 1.0, to 1.55 and 1.85 f0.05 for v 1.0, 0.8, and 0.67, respectively. Its value becomes close to the percolation value (yperc= 1.795) for v = 0.7. It is important to note that the critical exponents were within the experimental error independent of the initial composition and differences in reactivities of the functional groups of the long and short chains. These two variables represent short-range inter- actions, whereas the variations in the exponent v affect the relative development of largest clusters. Of interest are the implications of hard cluster formation and existence on various physical properties.For instance, the equilibrium elastic modulus is proportional to the concentration of elastically active network chains. The number of elastically active chains decreases with increasing size of hard clusters since the short chains between crosslinks within the hard clusters are not deformed as a result of macroscopic strain. On the other hand, the number of chains issuing from an elastically active cluster cross- link increases and weakens the cluster fluctuation. According to the Flory-Ermanl ' junction-fluctuation theory, the equilibrium shear modulus of a network in the reference state, G,, is given by fe -2G, = RTC, = RT -Ve (45) fe where re is the effective cycle rank, R is the gas constant, T is the temperature in K,f, is the number-average functionality of the elastically active crosslink and v, is the concen- tration of elastically active network chains (EANC).Examining this equation, one can consider all active branch points (pointlike crosslinks) in the network as contributing to the concentration of EANCs (Ve)pintlike with branch points having effective functionality (fe)pointlike varying from 3 up to the chemical functionality J: Alternatively, one can consider the existence of hard clusters in which some branch points and the corresponding EANCs do not contribute to the rubber elasticity of the network [(VJcluster 3 (fe)clusterl.Generally, it holds that (felpointlike (felcluster Chemical Clusters in Polymer Networks 50 40 :3 0 P h!s 20 0 0.2 0.4 0.G 0.8 1 CL-Fig. 2 Effect of the composition and difference in relative reactivities of functional groups of long and short chain groups crosslinked with a trifunctional crosslinker on the weight-average degree of polymerization of hard clusters, (P,), and their second-moment-average functionality, (f,), . Stoichiometric system, no excluded volume effect. Coding of curves: xx-yy; xx fraction of short chains &, yy reactivity ratio of groups of short (k,) and long (kL) chains with the groups of the crosslinker -k&, . (a)(P,), ;(b)(f,),. and (Ve)pointlike 2 (Ve)cluster Applying the TBP to several concrete system, it was found that the following relation holds up to the percolation threshold of the hard clusters [Jelpointlike \Jelcluster which means that the decrease of the concentration of EANCs and increase of the effec- tive crosslink functionality are fully compensated.This also means that the cycle ranks should be the same. (ie)pointlike = (Ce)cluster (47) K. Dus'ek and J. Somvarsky 104 1o3 f h 0 102Qv 10 1 0.1 0.01 0.001 (kg -(1 50 40 30 (uh 0 % v 20 10 n 0 0.2 0.4 0.Ci 0.8 1 (Y Fig. 3 Effect of the excluded volume parameter (exponent v of the kernel) on the evolution of (P,), and (fc)2. For all curves, & = 1/2 and kdk, = 1. The coding of curves gives the value of the exponent v = 1.0,O.g and 0.67, respectively. (a)log-log plot (P,), us.the distance from the gel point a, -a; (b)(fc)2 us. a. This is indeed so because below the percolation threshold the cycle rank of the cluster itself is zero and it develops only after the threshold is surpassed. Only then > (le)cluster * The length of this paper does not allow us to make any detailed comparison of the predictions with experimental data. Only some summarizing remarks will be made : 1. The equilibrium moduli of polyurethane networks based on poly(tetramethy1ene oxide)12 are close to the prediction of the Flory-Erman junction-fluctuation theory for both the cluster and pointlike crosslinks approach below the calculated percolation threshold of the hard cluster.Above this threshold, no dramatic change has been Chemical Clusters in Polymer Networks observed, and only at higher content of the hard structure does the mechanical behav- iour deviate increasingly from the assumption that the equilibrium elasticity is con- trolled by the soft phase. 2. The clusters generated by the theory of branching processes by combination of units yield more or less randomly branched structures with fractal dimensions close to 2. Silica clusters developing during hydrolysis and condensation of tetraethoxysilane are easily observed by small-angle X-ray scattering (SAXS).' Clusters developing under acid conditions have a fractal dimension close to 2, which corresponds to randomly branched structures.Under alkaline conditions the growing clusters are more compact with fractal dimensions of ca. 2.5,so that kinetic modelling of excluded volume obstruct- ed structure growth is more appropriate. However, one has to include cyclization in the kinetic treatment. This is possible and relatively easy. 3. A SAXS study of polyurethane networks based on poly(oxypropylene)diol, tri- methylolpropane and 4,4'-diisocyanatodiphenylmethane (unpublished results) have revealed that there is a large difference between the scattering of samples prepared by one-stage and two-stage processes. In the one-stage process, the cluster size is controlled by the difference in the reactivities of the OH groups of TMP (primary) and PPD (secondary) which is not the case in the two-stage process.The networks prepared by the one-stage process exhibit much more scattering at smaller angles, indicating the presence of relatively much larger inhomogeneities. However, the size of the inhomogeneities increases with increasing off-stoichiometry (the excess of OH groups) which can be explained by the assistance of physical association to cluster formation. In conclusion, the simulation of topological inhomogeneities offers a reference base for physical investigation of inhomogeneities in polymer networks. Combination of simulation and physical investigation under different conditions can give an answer as to whether (a) spatial interactions affect the topological structure of the clusters, and (b) to what extent the transient interactions between topological structures affect the physi- cal response of the network.While the formalism of the theory of branching processes is sufficiently elaborated to deal with systems of any chemical complexity, kinetic simula- tion methods will be further elaborated to model the effects of diffusivities and obstruc- tion in various structural parts of the network on structure growth. The support of this work by the Grant Agency of the Czech Academy of Sciences (grant no. A4050508) and US-Czech Science and Technology Program (grant no. 920-34)is gratefully acknowledged. References 1 S. L. Cooper and R. W. Seymour, Macromolecules, 1973,6,49. 2 L. H. Peebles, Macromolecules, 1974,7, 873. 3 J. Somvarsky and K. DuSek, Polym.Bull., 1994,33, 369. 4 J. Somvarsky and K. DuSek, Polym. Bull., 1994,33,377. 5 K. DuSek and M. Ilavsky, Polym. Eng. Sci., 1979, 19, 246. 6 K. DuSek, in Telechelic polymers: Synthesis and Applications, ed. E. J. Goethals, CRC Press, Boca Raton, 1989, p. 289. 7 G. P. J. M. Tiemersma-Thoone, B. J. R. Scholtens, K. DuSek and M. Gordon, J. Polym. Sci.: Part B: Polym. Phys., 1991, 29, 463. 8 K. DuSek and J. Somvarsky, in Synthesis, Characterization and Theory of Polymeric Networks and Gels, ed. S. Aharoni, Plenum Press, New York, 1992, p. 283. 9 K. DuSek, in Development in Polymerization 3, ed. R. N. Haward, Applied Science, Barking, 1982, p. 143. 10 F. Leyrvaz and H. R. Tschudi, J. Phys. A, 1981,14,3389. 11 P. J. Flory and B. Erman, Macromolecules, 1982,15,800. 12 B. Nabeth, Ph.D. Thesis, INSA, Lyon, 1994; B. Nabeth, J-P. Pascault and K. DuSek, J. Polym. Sci., Part B: Polym. Phys., 1996, in the press. 13 D. W. Schaefer and K. D. Keefer, Muter. Res. Symp. Proc., 1986,73,277. Paper 5/03913D; Received 16th June, 1995

ISSN:1359-6640    

DOI:10.1039/FD9950100147

出版商:RSC    

年代:1995

数据来源: RSC