摘要:

345 Colloidal Interactions during the Precipitation of Uniform Submicrometer Particles J-L. Look, G. H. Bogush and C. F. Zukoski 359 General Discussion 385 List of Posters 389 Index of Names345 Colloidal Interactions during the Precipitation of Uniform Submicrometer Particles J-L. Look, G. H. Bogush and C. F. Zukoski 359 General Discussion 385 List of Posters 389 Index of Names

ISSN:0301-7249    

DOI:10.1039/DC99090FX001

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

GENERAL DISCUSSIONS OF THE FARADAY SOCIETY/FARADAY DISCUSSIONS O F THE CHEMICAL SOCIETY Date Subject 1907 Osmotic Pressure 1907 Hydrates in Solution 1910 The Constitution of Water 191 1 High Temperature Work 1912 Magnetic Properties of Alloys 1913 Colloids and their Viscosity 1913 1913 The Passivity of Metals I9 I4 Optical Rotatory Power 1914 The Hardening of Metals I9 I 5 The Transformation of Pure Iron 1916 Methods and Appliances for the Attainment of High Temperatures in a Laboratory 1916 Refractory Materials 1917 Training and Work of the Chemical Engineer 1917 Osmotic Pressure I9 I7 Pyrometers and Pyrometry 1918 The Setting of Cements and Plasters 191 8 Electrical Furnaces 191 8 Co-ordination of Scientific Publication 1918 The Occlusion of Gases by Metals 1919 The Present Position of the Theory of Ionization 1919 The Examination of Materials by X-Rays 1920 The Microscope: Its Design, Construction and Applications I920 Basic Slags: Their Production and Utilization in Agriculture 1920 Physics and Chemistry of Colloids 1920 Electrodeposition and Electroplating 1921 Capillarity 1921 The Failure of Metals under Internal and Prolonged Stress I92 I Physico-Chemical Problems Relating to the Soil 1921 Catalysis with special reference to Newer Theories of Chemical Action 1922 Some Properties of Powders with special reference to Grading by Elutriation 1922 The Generation and Utilization of Cold 1923 Alloys Resistant to Corrosion 1923 The Physical Chemistry of the Photographic Process 1923 The Electronic Theory of Valency 1923 Electrode Reactions and Equilibria I923 Atmospheric Corrosion.First Report 1924 Investigation on Oppau Ammonium Sulphate-Nitrate 1924 Fluxes and Slags in Metal Melting and Working 1924 Physical and Physico-Chemical Problems relating to Textile Fibres 1924 The Physical Chemistry of Igneous Rock Formation 1924 Base Exchange in Soils 1925 The Physical Chemistry of Steel-Making Processes 1925 Photochemical Reactions in Liquids and Gases 1926 Explosive Reactions in Gaseous Media 1926 Physical Phenomena at Interfaces, with special reference to Molecular 1927 Atmospheric Corrosion. Second Report 1927 The Theory of Strong Electrolytes 1927 Cohesion and Related Problems 1928 Homogeneous Catalysis 1929 Crystal Structure and Chemical Constitution 1929 Atmospheric Corrosion of Metals.Third Report 1929 Molecular Spectra and Molecular Structure I930 Colloid Science Applied to Biology The Corrosion of Iron and Steel Orientation Volume Trans. 3* 3* 6* 7* 8* 9* 9* 9* I o* 1 1 I 0; 12* 12* 13* 13* 13* 14* 14* 14* 14* 15* 15* 16* 16* 16* 16* 17* 17* 17* 17* 18* 18* 19* 19* 19* 19* 19* 20* 20* 20* 20* 21 21* 22* 20; 22* 23* 23* 24 * 24 * 25* 25* 26* 26Dare I93 I I932 1932 I933 1933 934 934 935 93 5 936 936 93 7 937 938 938 I939 I939 I940 1941 1941 I942 1943 1944 1945 I945 1946 I946 I947 1947 1947 I947 1948 1948 I949 1949 I949 1950 1950 1950 1950 1951 1951 I952 1952 1952 1953 I953 1954 I954 I955 1955 1956 1956 I957 1958 1957 1958 I959 1959 1960 I960 1961 1961 I962 1962 1963 Faraday Discussions of the Chemical Society Su hjecr Photochemical Processes The Adsorption of Gases by Solids The Colloid Aspect of Textile Materials 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 Structure and Molecular Forces in ( a ) Pure Liquids, and ( h ) Solutions The Properties and Functions of Membranes, Natural and Artificial Reaction Kinetics Chemical Reactions Involving Solids Luminescence Hydrocarbon Chemistry The Electrical Double Layer (owing to the outbreak of war the meeting was The Hydrogen Bond The Oil-Water Interface The Mechanism and Chemical Kinetics of Organic Reactions in Liquid The 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 Sociite de Chimie Physique at Bordeaux) Colloidal Electrolytes and Solutions The Interaction of Water and Porous Materials The Physical Chemistry of Process Metallurgy Crystal Growth Lipo-proteins Chromatographic Analysis Heterogeneous Catalysis Physico-chemical Properties and Behaviour of Nuclear Acids Spectroscopy and Molecular Structure and Optical Methods of Investigating Electrical Double Layer Hydrocarbons The Size and Shape Factor in Colloidal Systems Radiation Chemistry The Physical Chemistry of Proteins The Reactivity of Free Radicals The Equilibrium Properties of Solutions o n Non-electrolytes The Physical Chemistry of Dyeing and Tanning The Study of Fast Reactions Coagulation and Flocculation Microwave and Radio-frequency Spectroscopy Physical Chemistry of Enzymes Membrane Phenomena Physical Chemistry of Processes at High Pressures Molecular Mechanism of Rate Processes in Solids Interactions in Ionic Solutions Configurations and Interactions of Macromolecules and Liquid Crystals Ions of the Transition Elements Energy Transfer with special reference to Biological Systems Crystal Imperfections and the Chemical Reactivity of Solids Oxidation- Reduction Reactions in Ionizing Solvents The Physical Chemistry of Aerosols Radiation Effects in Inorganic Solids The Structure and Properties of Ionic Melts Inelastic Collisions of Atoms and Simple Molecules High Resolution Nulcear Magnetic Resonance The Structure of Electronically Excited Species in the Gas Phase abandoned, but the papers were printed in the Transactions) Systems Published by Butterworths Scientific Publications, Ltd Cell Structure Volume 27' 28* 29 29* 30* 30* 31* 31* 32* 32* 33* 33* 34* 33* 35* 35* 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* 1 I * 12* 13 14 15* 16* 17* 18* 19 20 2 I* 22 23 24 25 26 27 28 29 30 31 32* 33* 34 35Date 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 197 1 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 1980 1980 198 1 1981 1982 1982 1983 1983 1984 1984 1985 1985 1986 1986 1987 1987 1988 1988 1989 1989 1990 Faraday Discussions of the Chemical Society Subject Fundamental Processes in Radiation Chemistry Chemical Reactions in the Atmosphere Dislocations in Solids The Kinetics of Proton Transfer Processes Intermolecular Forces The Role of the Adsorbed 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 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 Precipitation Potential Energy Surfaces Radiation Effects in Liquids and Solids Ion- Ion and ,Ion-Solvent Interactions Colloid Stability Structure and Motion in Molecular Liquids Kinetics of State Selected Species Organization of Macromolecules in the Condensed Phase Phase Transitions in Molecular Solids Photoelectrochemistry High 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 Oxidation Volume 36 37 38 39 40 41 * 42* 43 44 45 46 47 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 Struzure of Surfaces a i d Interfices as studied using Synchrotron Radiation 89 * Not available; for current information on prices etc., of available volumes, please contact the Marketing Oficer, Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 4WF, stating whether or not you are a member of the Society.

ISSN:0301-7249    

DOI:10.1039/DC990900X003

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1990, 90, 17-29 Electric-field-induced Aggregation in Dilute Colloidal Suspensions Paul M. Adriani and Alice P. Gast Department of Chemical Engineering, Stanford University, Stanford, CA 94305-5025, USA Electric-field-induced chain formation in dilute, non-aqueous suspensions of sterically stabilized, 1 Fm, poly(methy1 methacrylate) (PMMA) latices are investigated. Optical microscopy and digital image analysis provide the chain-length distribution. We find that the particles carry a charge sufficient to inhibit field-induced aggregation. Equilibrium predictions of chain aggre- gation incorporating a screened Coulombic repulsion and field-induced dipole attraction agree well with experimental observations near the onset of aggregation; chain formation becomes diffusion limited above the thresh- old field strength.1. Introduction Colloidal particles subjected to an applied electric field interact via attractive forces due to particle polarization. Dilute suspensions in strong fields typically aggregate into chain-like structures, while concentrated suspensions form a dense complex network. The strength of the attraction is controlled by the magnitude of the applied field and aggregation is reversible upon its removal. Network formation in concentrated sus- pensions can produce large changes in rheological properties; such suspensions are called electrorheological fluids. There is a growing interest in applying electrorheological fluids to a number of devices including electromechanical transducers and vibration isolation elements.''2 Owing to their potential technological importance, electrorheological fluids have been the subject of a number of studies employing a variety of techniques including rheological optical experiment^,^,^ computer simulation^,^'^ and theo- retical models.8-10 Two recent reviews provide an overview of electrorheological fluids and their application.""2 Analogous field-induced flow transitions occurring in ferro- fluids, dispersions of magnetic particles, have also been the subject of much investi- gation.I3 Efforts to model electrorheological fluids are impeded by the concentrated nature of the suspension.The pair-interaction potential provides the basis for fundamental models of suspension behaviour. Strong attractive forces induce particle aggregation, but it is both the interaction forces and the resulting structure that are responsible for the field-induced rheological changes.I4 Determining the precise nature of the particle interactions is difficult in concentrated suspensions where the highly complex structure and the enormous number of particle interactions complicate experimental interpreta- tion.As a step towards understanding the mechanism of the electrorheological effect, we investigate field-induced forces between particles in dilute suspension where one can study isolated particles associating into chains. Electric-field-induced chain forma- tion has been studied in biological cell suspension^^^"^ and in aqueous suspensions of polymer l a t i c e ~ . ~ ~ , ~ ~ In this paper we present a microscopic study of electric-field-induced aggregation in a dilute non-aqueous suspension of polymer latices.We compare aggregate size and 1718 Field-induced Aggregation number distribution to theoretical predictions from a model particle interaction potential and an equilibrium statistical mechanical theory for chain formation in dilute sus- pensions. The interaction potential includes a field-induced dipole attraction; however, the high dipole interaction strength required for aggregation suggested the presence of a repulsive force between the particles. We find that despite the low dielectric environ- ment, the particles carry a significant charge, which we characterize via electrophoresis. The inclusion of an electrostatic repulsion with the dipole attraction yields reasonable agreement between theory and experiment near the onset of aggregation. A transition to transport-limited aggregation occurs at elevated field strengths, indicating the need for additional studies of non-equilibrium chain formation.2. Particle Interactions We first consider particle interactions in the absence of an electric field. A steric layer grafted to the particle surface provides a short-range repulsion sufficient to prevent irreversible particle aggregation due to Van der Waals forces. Since the steric layer repulsion is powerful over only a very short range, we can model it with an effective hard-sphere repulsion. Particle charge creates a Coulombic repulsion screened by ions in the solvent. This electrostatic repulsion can be represented in the linear superposition approximation as a Yukawa potential," where k,T is the thermal energy, a is the particle radius, and r is the centre-to-centre distance between particles scaled on the diameter, 2a.The Debye length, K - ' , reflects the effective range of the Coulombic repulsion screened by the ions in solution. The magnitude of the electrostatic repulsion, scaled by the parameter A, takes two forms, 1 a W: constant potential A =i (7) [ Q2/(l + Ka)' constant charge where the Bjerrum length I = e2/4mo&,kB T is the characteristic range of interaction between two charges e in a medium of relative permittivity E , ; E~ is the vacuum permittivity. The first expression applies to particles maintaining a constant surface potential, qo = +oe/ k , T, as they approach one another.We approximate the surface potential, +b0, with the zeta potential measured from the particle electrophoretic mobility. The second form of the electrostatic repulsion applies to particles with constant surface charge Q = ZZ/a, where 2 is the number of charges on the particle. When the surface potential is low, the charge is related to the potential as Q = (1 + Ka)Wo. Note that the Bjerrum length is greatly extended in a low relative permittivity solvent, thus the relative magnitudes of charge, zeta potential and interaction strength typical of aqueous sus- pensions are quite different in non-aqueous suspensions. The range of the particle electrostatic repulsion depends on the ion concentration, No = 4v2a2no, through the Debye screening length K - ~ , as14 2No 3Q4 (Kal2=-+- 1-4 1 - 4 (3) where 4 is the particle volume fraction.The second term is the contribution from the Q counter-ions associated with each particle to maintain electroneutrality in the sus- pension. In non-polar solvents the ionic species are unknown and their concentrations are extremely low. We approximate the ionic concentration from the suspension conduc- tivity, a, as a = 2e20no where we assume univalent ions with mobility, o, dominated by the fastest species and related to the ion diffusivity by D = wk,T."P. M. Adriani and A. P. Gast 19 The linear superposition approximation (LSA) in eqn ( 1 ) becomes a poor approxima- tion to the exact numerical solution of the non-linear Poisson-Boltzmann equation as the particles approach c ~ n t a c t .' ~ The magnitude of error depends on the degree of charge regulation on the particle surface; in the limit of constant surface charge the exact repulsion exceeds the LSA force at small separations by a factor of four for Ka = 2, while for constant surface potential the exact repulsion near contact is half that of the LSA force. An alternative to the LSA is the Derjaguin approximation, which agrees well with the exact solution for particles with constant potential but diverges for constant charge.lg Since the degree of charge regulation is unknown in our system and to retain a potential amenable to statistical mechanical analysis, we choose the LSA. The field-induced particle interactions are highly complex for d.c.or low-frequency a.c. owing to the mobility of the charged particles and counter-ions. Particle interactions simplify considerably for high-frequency a.c. fields. The particles and the counter-ion cloud can no longer follow an applied a.c. field at frequencies higher than 1- 10 ~ H z , ~ ~ ~ ~ ~ thus particle electrophoresis is suppressed and the counter-ion cloud retains its zero-field distribution. The field-induced interaction then simplifies to a dipole attraction arising from the difference in polarizability (or relative permittivity) between the particles and the solvent. The dipole moment of an isolated sphere induced by an electric field E ( t), po( t ) = 4fl~,&,pu~E( t), depends on the difference between the relative permittivities of the particle, E ~ , and the .solvent, E , , through /3 = ( E~ - E , ) / ( cp + 2 ~ , ) where a is the sphere radius.The upper limit, p = 1, represents conductive particles, whereas the lower limit, p = -1/2, applies to spheres much less polarizable than the solvent as in an aqueous suspension of polystyrene latices.17918 The dipole moment is modified by the interaction between a dipole and the field produced by a nearby second dipole.' Higher-order multipoles are induced as however, both of these effects contribute to a higher order in /3 and can be neglected for a pair of spheres if lpl<0.5 and for the extreme case of an infinite chain of spheres if < 0.2.22 We assume the relative permittivity of the PMMA latices in this study to be the same as bulk PMMA, 3.0 in the kHz range, while that of hexane is 1.9.These relative permittivities yield p =0.16, a value small enough to render higher-order interaction terms negligible. The dipole strength for two spheres at contact with dipoles oriented head to tail is p = po[l +p/4+ O(p')]; neglect- ing the higher-order terms constitutes a 4% error in p for this system. The time-averaged potential energy of interaction udip( r, 8) between two hard spheres with aligned dipoles of moment p = ( p ( t))r,,,s. is where 8 is the angle between the dipole moment vector and the centre-to-centre vector r, r is scaled on the particle diameter 2a, P2(x) = (3x2-1)/2 is the second Legendre polynomial, and A = p2/ T T T T E ~ E , ~ ~ ~ k , T is the dimensionless dipole interaction strength. The attraction between aligned dipole moments is maximized when the dipoles are positioned head to tail with the contact potential energy, 2A, scaling as the square of the applied electric field.A peak-to-peak applied potential of 800V yields E,,,,,= 0.50 V pm-' and A = 4.7, or a maximum energy at contact of 9.4kBT. Combining the dipole attraction with the electrostatic repulsion yields r < l k,T A/rexp[2~a(r-l)]-(2A/r~)P,(cos 8) r 2 1 a potential energy whose minimum is no longer at hard-sphere contact with a magnitude significantly lower than the dipole contact energy, -2A. Typically with A = 10, Ka = 3, and A ranging from 4 to 10, the potential-energy minimum varies from -2.4 to -10kB7', while the dipole contact energy is -8 to -20kBT. ( 5 ) -=[ ub-9 6 ) a20 Field-induced Aggregation 3.Equilibrium Model for Chain Formation Particles having aligned dipole moments will aggregate into chains. We are interested in chain formation by particles having induced dipole moments aligned in an applied electric field. A similar problem occurs in chain formation among particles with per- manent dipole moments fully aligned in the limit of strong applied fields. An equilibrium theory of chain formation for hard spheres with permanent dipoles has been developed previo~sly.~~-*~ In the limit of strong fields, this theory can be applied to the induced dipole problem. In this section we extend these results to include electrostatic repulsion and we derive modified expressions for the chain-length distribution. At very low concentrations, only the equilibrium between singlets and doublets needs be considered.The singlet and doublet number densities p1 and p2 are governed by the partition function of each state q1 and q2 via the relation and by particle conservation P1+2P2= P (7) where p is the total number density of particles. Solving eqn (6) and (7) for p1 and p2 to O(p2) yields The osmotic pressure of the particles, neglecting chain-chain interactions, is II = (2pi)kBT; therefore, to O ( p 2 ) we have n -- - P1 +P2 = P -B,P2 k B T (9) where we can now identify -B, as the second virial coefficient. The singlet partition function accounts for the translational degrees of freedom with q1 = V/A3, where A = (h2/2.rrrnkBT)”2 is the thermal wavelength, h is Planck’s constant, and rn is the particle mass.The doublet partition function is a sum over all bound states of a particle pair. In previous studies on strong dipolar interaction^,^'-^^ the relevant particle configurations are highly localized, and the pair interaction potential was expanded in a Taylor series about the potential-energy minimum. This expansion is not useful for the present interaction potential, since the energy minimum moves and becomes quite broad as A is varied. The potential-energy minimum is given implicitly by 6h - = d i m ( 1 + 2 ~ a d , ~ , ) exp [ - 2 ~ a ( dmin - I ) ] A if dmin > 1 otherwise The shape of the interaction potential near the energy minimum varies with the relative magnitude of the electrostatic repulsion to the dipole attraction rendering simple approxi- mations unreliable.We therefore evaluate the doublet partition function numerically employing the full potential. The doublet partition function is proportional to the configuration integral over all bonded states of a particle pair. The integral is dominated by two potential-energy minima, at r = dmin , 8 = 0 and T. By symmetry we need only integrate over a hemisphereP. M. Adriani and A. P. Gast 21 to give exp [ - u ( r ) / k , T ] d+ sin 8 d8r2 dr. (11) Substitution of the pair potential, eqn ( 5 ) , and recognizing that sin 8 = 8 and P,(cos 8) == 1 - 302/2, in the region near 8 = 0, permits the angular integrals to be evaluated analyti- cally. The choice of upper limit on 8 is irrelevant for large dipole strengths. We choose the upper limit of d 2 to preserve the correct form in the absence of a dipole. We thus obtain VB, 42 =-yT I 1 (12) r3 I:' 6A ~ , = 2 7 ~ ( 2 a ) ~ exp [-Urep(r)/kBT] exp ( 2 ~ / ~ ~ ) - [ i - e x p (-6A/r3)]r2dr where ureP(r) is the electrostatic repulsion energy, and r is dimensionless on 2a.The form of eqn (12) holds for any potential having an isotropic repulsion, and reduces to previous results for B, in the limit of large A and a hard-sphere The choice of integration limits, rl and r 2 , corresponds to the definition of a bonded state. The doublet partition coefficient is only sensitive to these limits when the attractions become very weak. A reasonable definition of a doublet includes those particles falling within the nearest-neighbour shell 1 < r < 2. This choice applies well to both weak and strong particle interactions and corresponds to our criterion for counting doublets in our video images as described below.This theory of singlet-doublet equilibrium can be extended to consider the equili- brium of chains of all length^.^^-^' In general, eqn (6) becomes and the conservation equation is Detailed calculations of the chain partition function, q,, have been carried out for hard spheres with dipole^.'^,^^ All relevant short-range effects are incorporated in the doublet equilibrium constant, B,, and in the equilibrium particle spacing, dmin . The screened Coulombic repulsion decays exponentially making the electrostatic interactions negli- gible beyond nearest neighbours. Since there are no new long-range interactions, the chain partition function remains the same as for hard spheres [eqn (3.7) of ref.(24)] with eqn (12) for B, and the substitution A + A' = A/dLin, where dmin is determined from eqn (10). The calculated partition functions yield the number densities of all chain lengths. From this chain-length distribution one can calculate associated quantities such as the number fraction of singlets or the average chain length. Equilibration produces an average chain length near unity for small dipole strengths increasing rapidly above a critical threshold. In fig. 1 we compare the predictions for singlet-doublet equilibrium with full equilibrium among chains of all lengths employing parameters appropriate to our experimental system. The average chain length increases more abruptly for the full equilibrium calculations, as the equilibrium shifts toward long chains.The onset of aggregation in the singlet-doublet equilibrium occurs at the same field strength as the full equilibrium, but the average chain length increases more gradually. The full equilibrium model for chain length distribution is applicable when the potential-energy minimum is on the order of a few kBT and there is sufficient time for equilibration. In dilute suspensions, when equilibration time is limited, the singlet- doublet model is more appropriate. When attractions due to large applied fields exceed 6 k,T, aggregation becomes irreversible on timescales comparable to our experiments Cnp, = p . (14)22 Field-induced Aggregation n 1.6 1.4 1.2 1 .o ._ 0 250 500 750 1000 1250 1500 V,-,P Fig.1. Number average chain length predicted from eqn (8) and (12) against the peak-to-peak applied voltage for particle interaction parameters A = 10, K a = 3, and volume fractions 4 = 0.0067 and 0.0041, respectively, for singlet-doublet equilibrium (- and - - -), and equilibrium of all chain lengths ( * + - and - - - - -). and a non-equilibrium description of chain formation such as the Smoluchowski equation is more appr~priate.~ 4. Experimental We study aggregation of 0.98 pm diameter poly(methy1 methacrylate) (PMMA) latices suspended in hexane at volume fractions in the range 0.001-0.01. The particle synthesis and characterization are described e l s e ~ h e r e . ~ ~ - * ~ A covalently bound poly( 12-hydroxy- stearic acid) steric layer, ca. 0.01 pm thick, provides a strong, short-range repulsion to ensure stabilization against aggregation due to van der Waals attractions.We observe aggregation between two thin electrodes through a 20x objective on a Zeiss Axioplan optical microscope in transmission. The objective working distance allows us to focus at any level within the cell, while the depth of field is comparable to one particle diameter. The electrode cell, comprising a machined acrylic substrate, flat copper electrodes, and a glass cover slip is sealed with silicone grease and compressed together by small screws. A 560 pm gap between the 420 pm thick electrodes provides a reasonably homogeneous electric field near the middle of the cell. Segments of 610 pm outer diameter polyethylene tubing squeezed into the gap permit injection of the suspension while minimizing solvent evaporation. We maintain a slight convection parallel to the electrodes to provide a continuous supply of particles to observe.Our observation time is limited by a slow particle drift toward the cell walls due to dielectrophoresis. The dielectrophoretic force, Fdiel = p, - V E == pa3V E 2, induces the motion of a dipole in a non-uniform electric field. The dielectrophoretic velocity increases with the square of the field gradient. At field strengths near the onset of aggregation, dielectrophoresis is negligibly slow compared to aggregation, convection, and observation times. Doubling the field strength increases the dielectrophoretic velocity by a factor of four, reducing the dielectrophoresis timescale to a few minutes.Since the dielectrophoretic force is independent of the sign of the applied field, it is not suppressed by increasing the frequency of the applied field. This force is proportionalP. M. Adriani and A. P. Gas? 23 to the number of dipoles in a chain, thus long chains drift more rapidly to the cell walls. This effect is minimized by making the applied field as homogeneous as possible. In our electrode geometry, the field is most homogeneous at depths near the middle of our cell; we therefore focus in this region for our microscopic observations. To prevent particle accumulation at the electrode surface due to electrophoresis, dielectrophoresis, or image forces, we wrap each electrode with a 100 pm layer of Teflon tape to act as a particle barrier. The dielectric constant of Teflon, 2.1, is similar to that of hexane, 1.9, and the tape is uniform in thickness, so the field between the electrodes should not be significantly perturbed by the Teflon layer.To induce aggregation, we subject the suspension to a sinusoidal ax. field at a frequency of 3 kHz and a peak-to-peak potential, Vp-p, in the range 0-1200 V from a Tektronix model FG504 40 MHz function generator with a Trek model 610B high-voltage amplifier. The potential is monitored at the generator, amplifier, and electrodes on a Hewlett-Packard model 54501A digitizing oscilloscope. We record suspension behaviour on a Sony model VO-5800H video cassette recorder digitizing individual frames on an Imaging Technology Series 100 real-time digital image processing board in an IBM PC AT.We analyse images with IMAGELAB and IMAGETOOL image processing software by Werner Frei Associates. We typically record aggregation experiments for one minute at a given applied field with approximately 100 particles in the field of view. Particles move across the field of view on a timescale of 5-10 s owing to slow convection parallel to the electrodes produced in order to observe independent sets of particles. From each minute of recorded observations, we digitize and average five frames spaced 10 s apart. On longer timescales particle sedimentation and dielec- trophoresis become significant, so the cell is occasionally flushed with fresh suspension. We measure suspension conductivity, a, in a stainless steel Couette cell of a 13 mm cylinder with a 12 mm radius and a gap of ca.0.5 mm. We need large electrodes and a small gap to measure the low suspension conductivity. The conductance, C, across the gap is measured with a Yellow Springs Instrument Co. Model 35 Conductance Meter. Conductivity is proportional to conductance, a = k , C, with the cell constant, k , = 0.046 cm-', determined by the cell geometry and calibrated with a salt solution in a YSI model 3403 conductivity cell of a known cell constant. We found that suspension conductivity depended on sample age owing to the accumulation of ionic impurities. In order to maintain reproducible solution ionic strengths, we saturated our hexane with sodium acetate. We then found the conductivity of the PMMA suspensions in saturated hexane, a = 5 x R-' m-', to reflect an ionic concentration of no ==: 5 x 1013 cm-3 (= 1 x mol dmP3) assuming univalent ions of diffusivity D = (0.5-1) x lop9 m2 s-'.This ionic strength corresponds to a screening length of 2.6 < Ka < 3.7; the screening due to counter-ions alone would contribute Ka = 0.2 for a particle volume fraction 4 = 0.01. We assume K a = 3 for our theoretical predictions of suspension aggregation, except for one suspension with conductivity a = 6.4 x We measure the particle electrophoretic mobility in the same electrode cell for the aggregation experiments. Applying a weak, low-frequency (f = 0.5 Hz) sinusoidal a.c. field R-' m-', and 3.1 < K a < 4.3, where we assume Ka = 3.7. E ( t ) = (Ep-p/2) cos (2Tft) where Ep-p is the peak-to-peak amplitude of the field, typically Ep-p = 20 Vp-,/560 pm = 0.036 V pm-'.We follow particle oscillations z ( t ) = zO+(zp-,/2) sin (2r-t) where z is in the direction of the applied field and zP-, is the amplitude of oscillation, to determine the particle velocity U ( t ) = z'(?) = (2Tf)(ZP-,/2) cos ( 2 T f f )24 Field- induced Aggregation and the electrophoretic mobility We then calculate the zeta potential from the particle m~bility,~' assuming monovalent ions and molar ionic conductances of 70 cm2 K' mol-', equivalent to an ion diffusivity of 1 x m2 s-l. Again, to prepare suspensions with a controlled zeta potential, we saturate the solution with sodium acetate. After such contact, the measured particle mobility is 1.6 x lop9 m2 V-' s-' and the zeta potential is 40 mV, corresponding to a coefficient for the electrostatic repulsion [eqn (2)] of A = 10.This zeta potential corre- sponds to 100 charges per particle and an extremely low surface charge density of 3 x lop5 nm-*. The suspension with cp = 0.0028 has a lower electrophoretic mobility, 0.9 x m2 V-' s-', reflecting a weaker electrostatic repulsion of A = 3. This is the suspension having a higher conductivity, owing to a larger concentration of ionic impurities in the solution. 5. Results We quantify suspension images by counting the number and length of chains to determine a chain-length distribution and average chain length. The image analysis procedure begins with a 464 x 512 pixel grey scale image; each pixel has an intensity value ranging from 0 for black to 255 for white.The pixels are wider than they are tall with an aspect ratio of 1.2. Calibration of the video image with a microscope grid yields a scale of 2-06 vertical pixels pm-'. Two sets of particles are visible in the original image. The particles within the focal plane appear bright while dark particles appear from other planes. Since the depth of field is comparable to the particle diameter we can calculate the apparent particle volume fraction from the number of bright particles. We employ this relation to cut the image intensity histogram at a value appropriate to include only the bright particles. Variations in illumination are corrected by subtracting a smoothed image from the original to equalize the intensity histogram. We count particle chains by searching for vertical strings of bright pixels.Requiring a continuous line of bright pixels is too strict; permitting breaks of one or two pixels captures the chains visible in the original image. A chain of 1-3 pixels corresponds to a singlet, 4-6 to a doublet, and so on. The physical particle size is 2.06 pixels, but since the image is somewhat larger, a length criterion of 3 pixels is best for distinguishing singlets and doublets; this may lead to a slight underestimate of the length of longer chains. We show the singlet and doublet number fractions for suspensions of volume fraction 0.0067 and 0.0041 in fig. 2 ( a ) and ( b ) . Below 750 V, the number fractions of singlets and doublets correspond well to the theoretical prediction for nearest neighbours in a Yukawa fluid.The expected number of doublets would be about eight times greater for hard spheres without electrostatic repulsion. Doublet formation begins near 750 VP+ for the 0.0067 volume fraction suspension and around 850 VP+ for cp = 0.0041. The singlet and doublet fractions change with applied field in reasonable agreement with the theoretical predictions of the singlet-doublet equilibrium model until the highest field strengths where the experimental points begin to saturate. The saturation occurs at a lower field strength for more dilute suspensions suggesting that the aggregation becomes diffusion limited. The theoretical predictions of singlet and doublet number fraction show the effect of 10% uncertainty in the inverse Debye screening length, K ; a 10% uncertainty in the electrostatic repulsion parameter, A, is of comparable con- sequence.The experimental number fractions of singlets and doublets sum to unity near the onset of aggregation, but triplets and higher n-mers account for 10% of the aggregates at the highest field strength. The presence of longer chains should be more pronouncedP. M. Adriani and A. P. Gast 25 250 500 750 1000 1250 1500 0 Fig. 2. Theoretical and experimental chain number fractions of (a) singlets and ( b ) doublets against the peak-to-peak applied potential for particle interaction parameters A = 10, KU = 3 * 0.3, volume fraction 4 = 0.0067 (- and 0), and volume fraction 4 = 0.0941 (- - - and 0). at higher particle volume fractions. The presence of triplets and higher n-mers does not explain the slight saturation in singlet number fraction for 0.0041 since the effect of longer chains is to reduce the number fraction of singlets not to increase it.The singlet and doublet number fractions for volume fractions of 0.0028 and 0.0014 are displayed in fig. 3(a) and (b). As before, the distributions agree well with those predicted theoretically up to a field strength where doublets comprise ca. 10% of the suspension. Again we observe a saturation reflecting a transition to diffusion-limited kinetics, a process naturally occurring at lower fleld strengths in more dilute suspensions. The profound influence of the electrostatic repulsion is evident in our results. While our addition of sodium acetate facilitated controi over the suspension ionic strengths, the suspension having a volume fraction, cp = 0.0028, had a slightly higher conductivity.This results in a lower particle surface charge, a smaller electrostatic repulsion, A = 3 , and enhanced screening, Ka = 3.7. The aggregation threshold predicted for these condi- tions is substantially reduced to about 800 Vp-p(A = 4.7) from the 1000 Vp-p ( A = 7.4) required for aggregation when A = 10 and KU = 3. If the electrostatic repulsion were negligible ( A = 0), the required potential would be only 700 Vp-p (A = 3.6). Thus the electrostatic repulsion has enormous influence on the field required to produce aggregates. The electrostatic repulsion is also evident in observations of aggregate disruption after removal of electric field. Studying particle chains we find that they dissociate immediately upon removal of the applied field.Particles move apart initially via ballistic motions and then continue on Brownian trajectories beyond one diameter.26 Field- induced Agg reg a t io n " 0 250 500 750 1000 1250 1500 V,-,/V Fig. 3. Theoretical and experimental chain number fractions of ( a ) singlets and ( b ) doublets against the peak-to-peak applied potential for particle interaction parameters A = 3, ~a = 3.7 * 0.3, and volume fraction qh = 0.0028 (- and 0); and A = 10, KU = 3 k0.3, and qh = 0.0014 (- - - and 0). The average chain lengths in fig. 4 ( a ) and (b) are near unity at low field strengths for all particle volume fractions. At the onset of aggregation the increasing number of doublets causes the average chain length to rise.The average chain length saturates at fi = 1.2 for the dilute volume fractions 0.0028 and 0.0014. Average chain-length predic- tions agree well at low fields and near the onset of aggregation but diverge as the experimental data saturate. We discuss this saturation in terms of equilibration and diffusion times below. 6. Discussion Our theoretical predictions of chain aggregation incorporate electrostatic repulsion and dipole attraction into the pair-interaction potential. The agreement with experimental measurements of chain formation near the onset of aggregation suggest that we have a reasonable representation of the particle interaction potential. We show in fig. 5 the particle interaction potentials at the theoretical onset of aggregation for our four experimental volume fractions.The magnitude of the energy minimum required for aggregation varies from 3 to 6k,T as volume fraction decreases from 0.0067 to 0.0014. The 0.0028 volume fraction suspension conforms to this trend despite its lower electro- static repulsion and higher ionic screening compared to the other suspensions. Chain formation is an equilibrium process provided the magnitude of the interaction potential remains moderate. At elevated field strengths, the magnitude of the potential-P. M. Adriani and A. P. Gust 27 0 250 500 750 1000 1250 1500 V,-*IV Fig. 4. Theoretical and experimental number average chain length against the peak-to-peak applied potential for the same particle interaction parameters as in fig. 2 and 3 and particle volume fractions ( a ) 4 = 0.0067 (- and 0) and 4 = 0.0041 (--- and 0 ) and ( b ) 4 = 0.0028 ( - * - and A) and 4 = 0.0014 (- - - - - and 0).energy minimum I urnin( exceeds 6- 10kB T rendering equilibration slow. The characteristic equilibration time, t,, = 67r77a3/ kB T exp ( - urnin/ kB T ) , where 77 is the solvent vis~osity,~' increases from ca. 5 s near the onset of doublet formation to 500 s where saturation occurs. Thus we find that doublet formation becomes a diffusion-limited, non-equili- brium process above ca. 1000 V for A = 10 and K a = 3. Diffusion-limited chain formation is a slow process in dilute suspensions. The characteristic time for doublet formation with no repulsive energy barrier and a short- range attraction is tdblt = 7r77a3/ 4kB T3' Under our most dilute conditions, 9 = 0.0014, and the characteristic time for initial doublet formation is 20s.This time increases as the aggregation proceeds and the population of single particles diminishes. Convection of chains to the cell walls due to electric-field gradients prevents studies of chain formation over long periods; this limitation motivates future improvements in our experimental design to minimize electric-field gradients. We note that the electrostatic repulsion evident in these suspensions is somewhat surprising since the relative permittivity of the solvent is so low. The possibility of significant electrostatic repulsions in otherwise nearly hard-sphere suspensions can have important implications for interpretation of order-disorder transition^^^ as well as rheological behaviour. In a previous study, Ottewill and co-workers28 measured the scattering structure factor of similar 30 nm diameter PMMA particles suspended in dodecane. By comparing the structure factor with that derived from the mean spherical approximation for particles interacting via a Yukawa potential, the authors determined28 Field-induced Aggregation 2l r Fig.5. Theoretical particle interaction potential for aligned dipoles (6=0) at the onset of aggregation for the same interaction parameters as in fig. 2 and 3 and particle volume fractions 4 = 0.0014 (-), 4 = 0.0028 (- - -), 4 = 0.0041 ( - - . ), and 4 = 0.0067 (- - - - -). that a repulsion acting over 10nm best described their particles. This repulsion was attributed to the PHS steric stabilization layer; however, the range of the repulsion varied substantially with particle volume fraction, and it was supposed that this reflected changes in the thickness of the steric layer.In light of the results of this investigation, we suggest that a moderate electrostatic repulsion may play a role in these suspensions. Further studies of PMMA latices coated with PHS should address this possibility. 7. Conclusions We have presented a study of electric-field-induced aggregation in a non-aqueous colloidal suspension. We observed 1 pm PMMA particles in dilute suspension in hexane subjected to fields on the order of 500 V mm-I. The extreme fields required to effect chain formation and the rapid disruption of chains upon removal of the field both reflect the electrostatic repulsion operative in these suspensions.This electrostatic repulsion in otherwise model non-aqueous suspensions may have profound implications for systematic studies. We focused our attention on dilute suspensions to provide a test of our proposed interaction potential energy. In very dilute suspensions, chain formation can be approxi- mated by singlet-doublet equilibrium. A model of equilibrium doublet formation accurately reflects the population distribution and average chain length around the aggregation threshold. This suggests that we may model the interparticle interactions in these suspensions via an electrostatic repulsion added to a field-induced dipole attraction. This provides the basis for further studies of more concentrated suspensions.In elevated fields the aggregation process is no longer represented by an equilibrium population distribution and, at low particle concentrations, chain formation becomes diffusion limited. These indications provide motivation for future investigation of non-equilibrium field-induced aggregation and structure formation at higher particle concentrations.P. M. Adriani and A. P. Cast 29 This work was supported in part by Lord Corp., Ford Motor Co. and by IBM. We thank R. Ottewill for his generous gift of the polymer latices used in this study, C. Steinmetz for his donation of a digitizing oscilloscope, G. Homsy for his loan of image analysis equipment, and K. Smith for her help in measuring suspension conductivity. References 1 W.M. Winslow, J. Appl. Phys., 1949, 20, 1137. 2 Z. P. Shulman, R. G. Gorodkin, E. V. Korobko and V. K. Gleb, J. Non-Newtonian Fluid Mech., 1981, 3 D. L. Klass and T. W. Martinek, J. Appl. Phys., 1967, 38, 67, 75. 4 G. G. Fuller, Annu. Rev. Fluid Mech., 1990, 22, 387. 5 G. G. Fuller, K. Smith and W. R. Burghardt, submitted. 6 P. Bailey, D. G. Gillies, D. M. Heyes and L. H. Sutcliffe, Mol. Sim., 1989, 4, 137. 7 S. Miyazima, P. Meakin and F. Family, Phys. Rev. A, 1987, 36, 1421. 8 P. M. Adriani and A. P. Gast, Phys. Fluids, 1988, 31, 2757. 9 D. J. Klingenberg and C. F. Zukoski IV, Langmuir, 1990, 6, 15. 8, 29. 10 R. Tao, J. T. Woestman and N. K. Jaggi, Appl. Phys. Lett., 1989, 55, 1844. 11 A. P. Gast and C. F. Zukoski, Adv. Colloid Interface Sci., 1989, 30, 153. 12 H. Block and J. P. Kelly, J. Phys. 0, 1988, 21, 1661. 13 For a bibliography on ferrofluids, see S. W. Charles and R. E. Rosensweig, J. Magn. Magn. Mater., 14 W. B. Russel, in Theory of Dispersed Multiphase Flow, ed. R. E. Meyer (Academic Press, New York, 15 H. P. Schwan and L. D. Sher, J. Electrochem., 1969, 116, 22C. 16 S. Takashima and H. P. Schwan, Biophys. J., 1985,47, 513. 17 F. Richetti, J. Prost and N. A. Clark, in Physics of Complex and Macromolecular Fluids, ed. S . A. Safran and N. A. Clark (John Wiley & Sons, New York, 1987), pp. 387-411. 18 S. Fraden, A. J. Hurd and R. B. Meyer, Phys. Rev. Lett., 1989, 63, 2373. 19 A. B. Glendinning and W. B. Russel, J. Colloid Interface Sci., 1983, 93, 95. 20 K. H. Lim and E. I. Franses, J. Colloid Interface Sci., 1986, 110, 201. 21 D. J. Jeffrey, Proc. R. SOC. London, Ser. A, 1973, 335, 355. 22 T. B. Jones and R. D. Miller, J. Elecrrostar., 1990, in press. 23 P. C. Jordan, Mol. Phys., 1979, 38, 769. 24 P. C. Jordan, Mol. Phys., 1973, 25, 961. 25 P. G. de Gennes and P. Pincus, Phys. Kondens. Mater., 1970, 11, 189. 26 R. H. Ottewill, Langmuir, 1989, 5, 4. 27 L. Antl, J. W. Goodwin, R. D. Hill, R. H. Ottewill, S. W. Owens and S. Papworth, Colloid SurJ, 1986, 28 I. Markovic, R. H. Ottewill and S. M. Underwood, Langmuir, 1986, 2, 625. 29 D. J. Cebula, J. W. Goodwin, R. H. Ottewill, G. Jenkin and J. Tabony, Colloid Polym. Sci., 1983, 261, 30 R. J. Hunter, The Zeta Potential in Colloid Science (Academic Press, New York, 1981), pp. 100-112. 31 W. B. Russel, D. A. Saville and W. R. Schowalter, Colloidal Dispersions, (Cambridge University Press, 32 W. van Megen and P. N. Pusey, Nature, 1986,320, 340. 1983, 39, 190. 1983), pp. 1-34. 17, 67. 555. New York, 1989), pp. 270, 329. Paper 01022666; Received 17th May, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000017

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1990,90, 31-40 Phase-transition Phenomena in Colloidal Systems with Attractive and Repulsive Particle Interactions Agienus Vrij," Marcel H. G. M. Penders, Piet W. ROUW, Cornelis G. de Kruif, Jan K. G. Dhont, Carla Smits and Henk N. W. Lekkerkerker Van't H o f laboratory, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands We discuss certain aspects of phase transitions in colloidal systems with attractive or repulsive particle interactions. The colloidal systems studied are dispersions of spherical particles consisting of an amorphous silica core, coated with a variety of stabilizing layers, in organic solvents. The interaction may be varied from (steeply) repulsive to (deeply) attractive, by an appropri- ate choice of the stabilizing coating, the temperature and the solvent.In systems with an attractive interaction potential, a separation into two liquid- like phases which differ in concentration is observed. The location of the spinodal associated with this demining process is measured with pulse- induced critical light scattering. If the interaction potential is repulsive, crystallization is observed. The rate of formation of crystallites as a function of the concentration of the colloidal particles is studied by means of time- resolved light scattering. Colloidal systems exhibit phase transitions which are also known for molecular/ atomic systems. In systems consisting of spherical Brownian particles, liquid-liquid phase separation and crystallization may occur. Also gel and glass transitions are found.Moreover, in systems containing rod-like Brownian particles, nematic, smectic and crystalline phases are observed. A major advantage for the experimental study of phase equilibria and phase-separation kinetics in colloidal systems over molecular systems is the length- and time-scales that are involved. In colloidal systems length-scales are such that light scattering is an appropriate technique to probe microstructures and timescales are so large that it is fairly easy to perform meaningful time-resolved experiments during a phase transition. It is thus feasible to gain an understanding on the microscopic level of phase equilibria and phase-separation kinetics using light scattering on colloidal systems. For colloidal systems it is possible to vary the pair-interaction potential more or less systematically.This can be done in several ways. As is well known, in the case of charged colloidal particles in water, the interaction potential can be varied by the addition of salt. For colloidal particles in apolar organic solvents on the surface of which polymer chains are grafted, the pair-interaction potential can be varied by either choosing a different polymer chain, changing the solvent or varying the temperature. In mixtures of apolar and polar solvents particles may be slightly charged, giving rise to a screened Coulomb type of interaction potential. There is one complication in the comparison of results for phase-transition kinetics in colloidal systems and in molecular systems. In colloidal systems hydrodynamic interactions will influence the phase-separation kinetics.Since these hydrodynamic interactions are not pair-wise additive, a theoretical treatment of phase-separation kinetics in colloidal systems at higher volume fractions will be extremely difficult. At larger volume fractions it is therefore probably more sensible to predict general trends rather than to try to calculate experimentally measurable kinetic parameters in detail. 3132 Phase-transition Phenomena In this paper we describe some of our experimental observations concerning the location of the spinodal and crystallization of colloidal systems. The location of the spinodal is measured using pulse-induced critical scattering (PICS). The kinetics of crystallization are studied using a specially designed light-scattering apparatus which enables us to perform time-resolved experiments. Attractive and Repulsive Interactions in Colloidal Systems In colloidal systems consisting of particles stabilized by a protective layer of (long-) chain molecules and having a particle core composed of material with a refractive index comparable to that of the solvent, the van der Waals-London attraction forces between the cores are small.’ The interaction forces are therefore dominated by the chain-chain and chain-solvent interactions of the opposing protective layers of the touching particle surfaces, which may range from repulsive to attractive depending on the ‘quality’ of the solvent.Interactions due to protective layers of long-chain molecules were already investi- gated a long time ago.’ Studies on layers of short linear C,,-alkane chains were reported for repulsive3 and more recently also for attractive forces and the transition between them.3-5 In view of the close packing of the stabilizing layers grafted onto the surface, two particles can hardly interpenetrate without imposing appreciable steric hindrance on the terminal chain segments.As a consequence, the interaction potential must become repulsive even at small overlap. Also the solvent molecules play a role here through ‘solvation’ interactions. If there is a preferential solvation of chain (ends) by solvent molecules, repulsive forces will already be felt before the bare chain segments are actually in contact, because the removal of solvent molecules requires work.The interparticle distance where the repulsive force becomes large is much larger than the thickness of the overlap region. Therefore these forces can be described as a hard-sphere repulsion, characterized by a hard-sphere diameter, whenever the chain molecules are small in comparison with the particle core diameter. This has turned out to be a very adequate model potential for dispersions of our silica particles in cy~lohexane.~.’ For longer polymer chains the repulsion is softer. This feature plays an important role in the kinetics of crystallization in these systems.x If, however, there is a preferential solvation of chain segments by other chain segments instead of solvent molecules, effective attraction forces result when two opposing particle surface layers approach each other.The details of these contributions must come from proper averaging of solvation forces as calculated’ and with smooth macroscopic surfaces. These attractive interactions may give rise to spinodal decompo- sition. Attractive Interactions and Demixing To describe attractions due to a preferential solvation of chain segments by other chain segments, an entropic and an energetic component to the local mixing process of chain segments and solvent molecules, by analogy with the Flory and Krigbaum modelI2 for polymer segments, are taken into account. For the interaction potential V( r ) between two particles the following form is used, L ( r ) -- (: Here kB is Boltzmann’s constant, T is the temperature, L( r ) is proportional to the overlap volume of the polymer-coating layer at interparticle separation r and 8 is the thetaA.Vrij et al. 33 A Fig. 1. The square-well interaction potential. The width of the well is A, its depth is E and the hard-core diameter is u. temperature of the chain/solvent combination. The function L( r ) is non-zero in a short r range only. For a good solvent 0 << T, and V ( r) is strongly positive. This implies that the bare hard-sphere diameter is just slightly increased, as mentioned before. In a poor solvent 0 >> T. For 8 > T, V ( r ) is an attractive well in a small range where L( r) is non-zero. The attractive well disappears for T 2 0. The question now arises as to which model potential may be used adequately to describe such a supramolecular fluid of adhesive hard spheres.Since it may be expected that, because of the narrowness of the attractive well, the actual shape of the well will not be very important, one may choose the potential that renders the simplest statistical thermodynamic connection between microscopic and macroscopic properties. Several model potentials consisting of a steep repulsion followed by a short-range attraction are possible. For a physically realistic potential, at least three model parameters are needed: the diameter of the hard core and the depth and range of the attractive well. A commonly used model fulfilling these requirements is the square-well potential, defined as, V( r ) = 00; r < u -&; a < r < u + A (2) - = 0; a + A s r This potential is sketched in fig.1. Here u is the hard-core diameter, E a 0 is the depth of the well and A is the range or width of the well. A relatively straightforward characterization of the interaction forces can be obtained from the osmotic pressure derivative alT/ap, where, B2 = 2 ~ \ ~ ' d r r2[ 1 - exp ( -=)I. V( r ) (4) Here Il is the osmotic pressure, p is the particle number density and B2 is the second34 Phase-transition Phenomena virial coefficient. In the case of the square-well model [eqn (2)] it is easily seen that, where a = 2 4 - exp - CT "[ (k;T)-']' It follows that where 4 = ( .rr/6)po3 is the volume fraction of hard-core material. The second virial coefficient is a difficult object to measure experimentally because one must perform experiments on dilute solutions with a volume fraction of say <0.05.Static light- scattering measurements, for example, are just not accurate enough for this goal. Fortunately, however, LY can also be deduced from measurements of the particle diffusion coefficient. Diffusion coefficients from dynamic light-scattering measurements do not require inaccurate absolute intensity measurements. For the diffusion coefficient it is found that5713714 with k,, = 1.45 -0.56a = 1.45 - 13.44 (9) Here Do is the diffusion coefficient at infinite dilution. It should be mentioned that this simple relationship for a. applies only for a 'sticky' hard sphere, i.e. a very short-range attractive well preceding the hard-core repulsion. As was mentioned before, this is the case in the systems used here. At each volume fraction the proportionality constant k,, in eqn (8) contains a constant term and an additive temperature-dependent term of the exponential type.A plot of the function a( T ) , defined as a( T) = kBT/67n7( T)D( T) (10) versus the temperature T is plotted in fig. 2 for a stearyl silica coded SP23, in benzene. These particles have a hydrodynamic radius of 32 nm. The three curves correspond to three volume fractions 0.0128, 0.024 and 0.0355. In eqn (10) q( T ) is the temperature- dependent viscosity of the solvent. The displayed curves show a common point of intersection and a steep increase in a small temperature range of a few degrees around T = 3 10 K. The point of intersection is located at a temperature where no net interactions are felt. According to eqn (9) this happens for (Y = 1.465/0.56.The increase of a ( T) is due to the rising attraction forces. The steepness of the curve is in fact consistent with the exponential dependence given in eqn (9). The drawn lines were calculated4 assuming A=O.3 nm, and using L and 8 as fitting parameters; L=94.7 and 8=322.4 K. Other somewhat arbitrary choices of A lead to similar values of L and 8. If A is changed from 0.1 to 0.9 nm, L changes from 96 to 90 and 8 changes from 326 to 319 K. Below a certain temperature threshold the value of a ( T ) suddenly drops because of actual phase separation of the silica dispersion into a concentrated and a dilute phase. JansenI5 found in his phase-separation studies that often the concentration of silica particles in the concentrated phase decreased with decreasing temperature in contradic- tion with theoretical predictions based on a simple gas-liquid type of phase transition.A.Vrzj et al. 35 43, 1 28 1 I I 1 300 310 320 T / K Fig. 2. The quantity a( T ) , defined in eqn (lo), as a function of the temperature for three volume fractions: +, 0.0128; 0, 0.0240; A, 0.0355. He found the trend that the lower T is, the lower the particle concentration becomes. The phases often had a gel-like appearance. It is for this reason that we started a new investigation of instability against phase separation by looking at states where the system is not yet phase-separated, but still homogeneous, i e . by observing the spinodal curve with the technique of pulse-induced critical scattering. The technique is well known in the polymer field and is essentially a measurement of the critical light scattering upon fast cooling of the sample before any phase separation has occurred. In fig.3 the spinodal temperature vs. the volume fraction is plotted for the silica coded SJ4 in benzene (these particles have a hydrodynamic radius of 25nm). An important conclusion that can be drawn from this plot is that the measured spinodal curve has a 'parabolic' shape, in agreement with theoretical predictions of spinodal curves of the liquid-gas type. Thus no anomalous shape is found here. The drawn line is calculated using the so-called adhesive hard-sphere (AHS) or sticky hard-sphere model. This liquid-state model has been investigated with MC ~imulations~~"' and seems especially suited for particles with a very narrow attractive well as in our silica dispersions.The AHS model is a limiting case of a narrow attractive well at a hard-sphere surface, and was introduced by BaxterI8 because it allows an exact solution of the approximate integral equation of Percus and Yevick for the liquid state. At the spinodal temperature the isothermal compressibility dp/aII goes to infinity. For the AHS model this means that, with36 Phase- tra nsition Phenomena 20 18 u < 16 h 14 12 1 1 I I I I 0.10 0.20 0.30 4 Fig. 3. The experimental and calculated AHS spinodal curve. Here T~ is a stickiness parameter which is equal to 2 / a and is related to the temperature according to eqn (6). The line drawn in fig. 3 is calculated using eqn (l), (2), (6), (1 l ) , (12), with L = 23.1 and 8 = 355 K, used as fitting parameters.The agreement between the experimental and theoretical spinodal in fig. 3 is quite reasonable. Repulsive Interactions and Crystallization In recent years crystallization has been observed in colloidal systems of sterically stabilized particles in organic solvents. It appears that the rate of crystallization is strongly dependent on the concentration” and the rate of Also the nature of the stabilizing coating plays an important role.* By coating the silica cores with different polymer chains a more or less soft repulsive potential can be realized in addition to a hard-core repulsion. We found that the rate of crystallization is extremely sensitive to the steepness of the repulsive potential.* The observations described in ref.(8) are qualitative. In this section we describe the first experiments in a quantitative light-scattering study of the kinetics of crystallization. The system on which these experiments are performed is a silica coated with a thin layer of y-methacryloxypropyltrimethoxysilane- silicaz3 (TPM-silica). The solvent is a mixture of ethanol and toluene with a composition such that its refractive index matches that of the particles. The particles are slightly charged, giving rise to a screened Coulomb type of interaction potential. The Debye length is estimated to be ca. 100 nm.24 The hard-core radius is 160 nm. After the chemical synthesis and purification of the particles, a dilute dispersion in ethanol-toluene is prepared, which is then centrifuged.The glassy sediment is redis- persed, using a vortex mixer. Immediately after the redispersion, light-scattering curves are recorded. After completion of the crystallization process we found that shaking the sample with a vortex mixer, to within experimental error, the same nucleation/crystalliz- ation growth curve was reproduced as that recorded right after redispersion of the centrifuged sample. Contrary to vortex mixing, sonification of samples was found to lead to immediate crystallization. Apparently, sonification does not destroy crystallites,A. Vrij et al. 37 lnterfoce m LASER P 2 Fig. 4. A sketch of the static light-scattering apparatus. M1 and M2 are two mirrors used to align the laser beam. L is a lens to make the beam parallel.PH1 and PH2 are two pinholes with a diameter of 3 mm. BE is a beam expander and DIA a diaphragm which can be used to adjust the scattering volume. TU1-4 are beamguides to avoid any light other than the scattered light from the sample being detected. BS is a beamstop and PF1 is a polarization filter. PF2 is a circular sheet of polarization filter which is positioned in front of the optical-fibre channel inlets FI. RT and RF are the radii of the thermostatting vat and the circular array of the fibre channel inlets, respectively. The thermostatting vat serves also as a lens to focus the scattered radiation onto the fibre-channel inlets. The fibre-channel outlets FO are imaged onto a diode camera CA by means of a Nikon objective, LN. The camera operates at 5 "C and must be flushed with dry nitrogen to prevent condensation on the diode array chip.but even seems to enhance their rate of formation. Vortex mixing was found to destroy the nuclei. Light-scattering curves were recorded with an apparatus which is specially designed for time-resolved measurements. For the experimental study of the kinetics of phase transitions on a timescale of say 10s or less, scattered intensities at all angles simul- taneously are needed. There is just no time to scan the relevant scattering angles with a single detector. The principle of the light-scattering apparatus is sketched in fig. 4.25726 The thermostatted toluene bath acts also as a lens as to focus the scattered light onto the inlets of 140 optical fibre channels. The fibre channel outlets are arranged in a rectangular array, which is imaged onto an array of 512 diodes.The time resolution of the diode array camera (model M1452A from EGG) is 10 ms and its dynamic range is 214. The intensity of each fibre channel outlet spreads over two diodes. What is observed then is a series of 140 peaks on the diode array, where the area of each peak corresponds to the intensity which is scattered at the corresponding scattering angle. The time evolution of the scattering curves of TPM-silica dispersions was recorded in this way at several overall concentrations. Three such experiments are shown in fig. 5, for overall concentrations of 0.280 ( a ) , 0.294 ( b ) and 0.329 g cm-3 ( c ) . In fig. 5(a j the experimentally measured points are plotted in the first and the last scattering curve only.For all other scattering curves the actually measured points are not indicated for clarity, but just connected by straight lines. As can be seen from this figure, the relative scatter in the Bragg-peak height and the growth rate of the Bragg-peak intensity are both functions of the overall concentration. Apart from experimental errors, the scatter in the Bragg-peak height is due to the finite number of crystallites in the scattering volume (which is 0.12 cm'). The intensity over the Debye-Scherrer ring is not a constant, but varies depending on the number of crystallites in the scattering volume and their size. The relative standard deviation of the Bragg-peak height varied enormously with the overall concentration of colloidal particles.The scattering curves shown here are0.60 2.00 4.00 6.00 80.00 h Y v) .- C @ 2 40.00 .r 0.00- 0 .00 2.00 0.00 2.00 4.00 6.00 k 2 / 1 0 - l ~ m-’ 4.00 6.00 k ’ l m-? Fig. 5. Light-scattering curves for TPM silica at three overall concentrations: ( a ) 0.280, ( b ) 0.294 and ( c ) 0.329 g ~ r n - ~ . I is the intensity in (for all figures the same) arbitrary units, k is the scattering vector and t is the time. The drawn line is the Bragg-peak height as a function of time.A. Vrij et al. 20 c) E c 1- U 15 .- - d cd x U 10 5 39 ' T 25 / h I 4 I I I l a' I 1 I I 8 , I , * ; * o I , +$ * , .g-- + * , '---d."--- I I I I 0.26 0.28 0.30 0.32 0.34 concentration/g cmP3 Fig. 6. The crystallization rate as a function of the overall concentration.averages obtained by carefully rotating the cuvette during 12s, the time over which scattered intensities are collected. The actual variation of the intensity over the Debye- Scherrer ring is therefore much larger than the scatter in the intensities in fig. 5 would suggest. The curves showing the Bragg-peak height us. time are also given in fig. 5 as projections on the intensity us. time plane. We have used the rate of increase of the Bragg-peak height as calculated from these curves as a measure for the crystallization rate. The crystallization rates thus determined, for several overall concentrations, are collected in fig. 6. It is clear that the crystallization rate is a sensitive function of the overall concentration and displays a distinct maximum. Such a dependence was already observed qualitatively by Pusey and van Megen.19 The standard deviation in the Bragg-peak height for a given scattering volume contains both information on the size and the number concentration of crystallites in the sample.The width of the Bragg-peak is a measure for the size of the crystallites. We are currently measuring these quantities again as a function of the overall concentra- tion. The scattering-angle resolution in these experiments is much better than in the experiments shown here (which was lo). Summary In the colloidal systems consisting of spherical Brownian particles with an attractive pair-interaction potential, spinodal instabilities are observed. The location of the spinodal is measured with pulse-induced critical scattering.It is found that for the system of silica spheres with a short-range attractive potential that was used in our experiments, the location of the spinodal is well described by Baxter's adhesive hard- sphere model. Colloidal systems consisting of spherical particles with a soft repulsive pair-interac- tion potential crystallize at certain concentrations. The rate of crystallization as a function of the concentration shows, at least for the system under investigation in the40 Phase-transition Phenomena present paper, one optimum within a narrow concentration range. We are currently investigating the number concentration and the size of the crystallites as a function of the concentration. References 1 A. A. CaljC, W. G. M. Agterof and A. Vrii, in Micellization, Solubilization and Microemulsions, ed.K. L. Mittal (Plenum Press, New York, 1977), vol. 2. 2 D. H. Napper, in Polymer Stabilization of Colloidal Dispersions (Academic Press, London, 1983). 3 J. W. Jansen, C. G. de Kruif and A. Vrij, J. Colloid Interface Sci., 1986, 114, 471. 4 P. W. Rouw, C. G. de Kruif, J. Chem. Phys., 1988, 88, 7711. 5 R. Finsy, A. Devriese and H. N. W. Lekkerkerker, .I. Chem. Soc., Faraday Trans. 2, 1980, 76, 767. 6 A. K. van Helden and A. Vrij, J. Colloid Interface Sci., 1980, 78, 312. 7 C. G. de Kruif, W. J. Briels, R. P. May and A. Vrij, Langmuir, 1988, 4, 668. 8 C. Smits, W. J. Briels, J. K. G . Dhont and H. N. W. Lekkerkerker, Prog. Colloid Pol.ym. Sci., 1989, 79, 9 D. Henderson, J. Colloid Interface Sci., 1988, 121, 486. 287. 10 C. E. Herder, B. W. Ninham and H. K. Christenson, J. Chem. Phys., 1989, 90, 5801. 11 J. N. Israelachvili, S. J. Kott, M. K. Gee and T. A. Witten, Langmuir, 1989, 5, 1111. 12 P. J. Flory and W. R. Krigbaum, J. Chem. Phys., 1950, 18, 1086. 13 G. K. Batchelor, J. Fluid. Mech., 1972, 52, 245. 14 B. Cichocki and B. U. Felderhof, .I. Chern. Ph-vs., 1988, 89, 1049. 15 J. W. Jansen, C . G. de Kruif and A. Vrij, J. Colloid Interface Sci., 1986, 114, 481. 16 W. G . T. Kranendonk and D. Frenkel, Mol. Pbys., 1988, 64, 403. 17 N. A. Seaton and E. D. Glandt, J. Chem. Phys., 1987, 87, 1785. 18 R. J. Baxter, J. Chem. Phps., 1968, 49, 2770. 19 P. N. Pusey and W. van Megen, Nature (London), 1986, 320, 340. 20 C. G. de Kruif, P. W. Rouw, J. W. Jansen and A. Vrij, J. Phys. C3., 1985, 46, 295. 21 S. Emmett, S. D. Lubetkin and B. Vincent, Colloid Surf.', 1989, 42, 139. 22 K. E. Davis, W. B. Russel and W. J. Glantschnig, Science, 1989, 245, 507. 23 A. P. Philipse and A. Vrij, J. Colloid Interface Sci., 1989, 128, 121. 24 A. P. Philipse and A. Vrij, J. Chem. Phys., 1988, 88, 6459. 25 H. 0. Moser, 0. Fromhein, F. Herrmann and H. Versmold, J. Phps. Chem., 1988, 92, 6723. 26 J. K. G. Dhont, G. Harder, C. G. van der Werf, H. N. W. Lekkerkerker, H. Breiner and H. 0. Moser, submitted. Paper 0/02321C; Received 21st Maji, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000031

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1990, 90, 41-55 Use of Viscoelastic Measurements for Investigating the Stability/ Flocculation of Concentrated Dispersions Tharwat F. Tadros” and Andrew Hopkinson? ICI Agrochemicals, Jealotts Hill Research Station, Bracknell, Berkshire RG12 6E Y, UK The use of viscoelastic measurements for studying stability/ flocculation of concentrated dispersions is discussed. With electrostatically stabilised dis- persions, the system becomes predominantly elastic when significant double- layer overlap occurs. This was demonstrated using polystyrene latex disper- sions in and mol dm-3 NaC1. Plots of the elastic modulus G’ versus surface-to-surface separation distance h showed a rapid increase when h was less than twice the double-layer thickness. The experimental G’ values were compared with theoretical values calculated from the second differential of the interaction energy us.distance relationship. With sterically stabilised latex dispersion [containing grafted poly(ethy1ene oxide) chains], predominantly elastic response was also obtained when h became less than twice the adsorbed layer thickness (26). The G’ vs. volume fraction curves were converted to G’ vs. h and this was compared with the values of G’ calculated from direct force, F, vs. distance curves. In both cases there was a rapid increase in G’ with decrease in h when h < 26. Viscoelastic measurements could also be applied for flocculated disper- sions. With weakly flocculated systems such as those obtained by addition of a free non-adsorbing polymer (depletion flocculation), the systems showed pronounced non-Newtonian behaviour above a critical volume fraction of free polymer (4;).The occurrence of this behaviour decreased with increas- ing molecular weight of the free polymer. The extrapolated yield stress was used to calculate the energy of separation between particles, Esep, in the flocculated dispersion. Esep was compared with the theoretical value of the free energy of depletion, Gdep, that was calculated using available theories. With strongly flocculated dispersions, scaling concepts could be applied and the power exponent in # (G’- # n ) could be used as a measure of the strength of flocculation. From G’ and the critical strain y,, above which non-linear response is obtained, the cohesive energy of the flocculated structure, E,, was calculated.log-log plots of E, versus # were used to cbtain the power exponent in # and this should give a measure of the strength of flocculation. Concentrated dispersions, both of the aqueous and non-aqueous types, find applications in many industrial systems of which the following are worth mentioning: paints, dyestuffs, pigments, paper coatings, printing inks, detergents, ceramics, cosmetics, pharmaceutical and agrochemical formulations. In all these systems, it is essential to control the stability/flocculation of the dispersion during its preparation, subsequent storage and on application. This requires control of the interparticle interaction forces between the particles. Four different types of such interactions may be distinguished: ’ hard-sphere, electrostatic (soft), steric and van der Waals.In practice it is possible to obtain dispersions with various combinations of these interactions. By controlling the nature t Present address: Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW. 4142 Stability/ Flocculation of Concentrated Dispersions and magnitude of these interaction forces one can control the bulk properties of these dispersions. Investigations of the stability/ flocculation of concentrated dispersions are by no means simple, since measurements need to be carried out without diluting the system. With model systems it is possible to apply scattering methods for such investigations, of which small-angle neutron scattering SANS is by far the most appropriate.2 Informa- tion on the microstructure of the particles and their arrangement in space can be obtained.Clearly these studies form the basis of any quantitative relation between the particle interaction potential and the bulk properties of the system. However, such measurements of SANS are accessible in a small number of laboratories and they can be applied only to model systems. Viscoelastic measurements offer an alternative procedure for investigation of interpar- ticle interactions3 and hence they give direct information on the stability/flocculation of the concentrated dispersion. Although less rigorous and less quantitative than SANS, such measurements are available in many laboratories and they do not require any special preparation of the dispersion.It is the objective of this paper to illustrate the usefulness of these measurements in studying the properties of concentrated dispersions. Various systems will be discussed to illustrate the general applicability of these measure- ments. These systems will range from electrostatically and sterically stabilised disper- sions to flocculated systems. The latter will include both weakly and strongly flocculated (coagulated) dispersions. To simplify the analysis, model latex dispersions that can be prepared fairly monodisperse were used. Two different types of rheological measure- ments were carried out, mainly dynamic (oscillatory) measurements, and steady-state (shear stress-shear rate) measurements. Experimental Latex Preparation Two main polystyrene latex dispersions were prepared. The first was an electrostatically stabilised latex that was prepared using the surfactant-free emulsion polymerisation technique described by Goodwin et aZ.4 The latex was extensively dialysed and then concentrated.The particles were fairly monodisperse, with an average radius of 700 nm as determined by photon correlation spectroscopy (Malvern PCS 47000) and the Coulter Counter (standard deviation <5%). This latex was also used for studying steric interac- tion by coating the particles with a physically adsorbed poly(viny1 alcohol) layer (PVA). The weight-average molecular weight of the polymer, M,, was 45 000 with 12% acetate groups. The adsorption isotherm of the polymer on the latex was determined at 21-23 "C and this gave a plateau value of 2.8 * 0.4 mg m-2, in good agreement with the results obtained b e f ~ r e .~ The adsorbed layer thickness of the PVA layer was determined using PCS to be 46 nm.6 The concentrated sterically stabilised latex was prepared by adding PVA solution to the dilute latex which was stirred overnight to ensure complete adsorption. NaCl solution was then added to the latex to give lo-* mol dmP3 electrolyte in order to suppress any contribution from double-layer effects. The resulting latex was then concentrated by centrifugation. The supernatant solution was used to dilute the concentrated sediment to give suspensions of various volume fractions. The second polystyrene latex dispersion was a sterically stabilised system containing grafted poly( ethylene oxide) (PEO).It was prepared using a dispersion polymerisation technique in water-alcohol mixtures.' The basic recipe consisted of a styrene monomer, a macromonomer of PEO ( Mw = 200) methacrylate, 2,2'-azo-bis( 2-methylpropion- onitrile) and peroxide inhibitors and a water-alcohol mixture (2: 3 ratio). Details of the preparation of this latex were given elsewhere.' The z-average particle size of theTh. F. Tadros and A. Hopkinson 43 latex was determined by PCS. This showed that the latex was fairly monodisperse with an average radius of 175 nm. The adsorbed layer thickness 8 h was obtained from the relative viscosity, as described in detail before.8 8 h was found to decrease with increasing volume fraction, 4, of the suspension, owing to the possible interpenetration and compressure of the layers.A summary of the 8 h values as a function of 4 is given below. 4 0.33 0.38 0.43 0.46 0.50 0.52 0.54 0.57 S h 20.5 19.3 17.0 15.1 15.0 13.3 11.1 9.2 Free Polymer for Depletion Flocculation Two types of polymer were used for investigation of the effect of addition of dissolved polymer on the rheology of the dispersions. These were poly(ethy1ene oxide) (PEO) with molecular weights of 20 000, 35 000, 90 000 and hydroxylethyl cellulose (HEC) with molecular weight of 70 000, 124 000, 223 000. Both polymers were commercial materials that were used as received. Rheological Measurements Dynamic ( Oscillatory) Measurements A Bohlin VOR rheometer (Bohlin Reologie, Lund, Sweden) interfaced with facit DTC2 or IBM microcomputer was used for such measurements.The instrument can operate in the frequency range 10-3-20 Hz and has interchangeable torsion bars covering a wide range of sensitivities. A coaxial cylinder (C25) with a moving cup of radius 27.5 nm and a fixed bob of radius 25.0 nm was used. The Bohlin VOR rheometer is a constant- strain instrument which performs oscillatory tests by turning the cup back and forth in a sinusoidal manner. The shear stress in the sample is measured by measuring the deflection in the bob which is connected to the interchangeable torsion bars. The phase-angle shift is computed from the time displacent between the sine waves of stress and strain ( A t ) , i.e. S = wAt, where w is the frequency in rad s-’ ( w = 27~v where v is the frequency in Hz).The complex modulus G*, storage modulus G’ and a loss modulus G” are calculated from the stress and strain amplitudes ( T ~ and yo, respectively) and phase-angle shift S : G* = To/ Yo GI= G” cos S In oscillatory measurements, one initially fixes the frequency and measures the rheological parameters as a function of strain amplitude. This enables one to obtain the linear viscoelastic region, where G*, G’ and G” are independent of strain, at any given frequency. Above a critical strain y c , G* and G’ start to decrease with increasing applied strain (non-linear response) whilst G” begins to increase. Once the linear region is indicated, then measurements are made as a function of frequency at a fixed amplitude. As we will see later, many systems show only a linear region at very low amplitudes, and therefore measurements need to be made at low deformations.Shear Modulus Measurements These were performed by using the Rank pulse shearometer (Rank Brothers, Bottisham, Cambridge). This is based on the model originally described by van Olphen’ and later44 Stability/ Flocculation of Concentrated Dispersions developed by Goodwin and Smith.” The suspension is placed in a cell that is fitted with two parallel plates or Perspex discs, whose distance of separation, d, can be changed by using a micrometer attached to one of the plates. Each plate is connected to a piezo-electric crystal (LiCl, transducer). An electric generator is connected to the bottom crystal and is used to initiate a small amplitude (ca. rad) strair, with high frequency (ca.200 Hz). This produces a shear wave that propagates through the dispersion and is detected at the upper crystal. The latter is connected to a microcomputer, enabling one to display the damping of the shear wave. By measuring the time t , for shear wave maximum to go to a minimum, at a given distance d between the plates, one can calculate the shear wave the shear wave velocity u. t, is usually measured as a function of d, and u is obtained from the slope of the linear plot of d us. t,. The shear modulus G, (which is the elastic modulus at a frequency of ca. 200 Hz) is given by where p is the density of the suspension. G,= u2p ( 6 ) Steady -s ta te Measu rem en ts Shear stress ( r ) vs. shear rate curves (Y) curves were obtained by using a Haake- Rotovisco rotational viscometer fitted with concentric cyclinder platens.For Newtonian systems, the viscosity 37 was obtained from the slope of the linear curve between r and 7. For non-Newtonian systems, which showed pseudoplastic flow, the data were ana- lysed by using the Bingham,” or Casson’s,12 equation: where rp is the extrapolated (to j = O ) yield stress and qPl the apparent or plastic viscosity, that is the slope of the linear portion of the r vs. y curve, r, is the Casson yield value, and 37, is the Casson viscosity. Results and Discussion Electrostatically Stabilised Polystyrene Latex Dispersions Strain-sweep measurements at a frequency of 1 Hz showed a continuous reduction in the values of the moduli with the applied strain and it was not possible to identify the linear region, as this was at strains lower than 0.004 for lo-’ mol dmP3 NaCl and 0.01 for lov3 mol dm-3 NaCl.Measurements were made at fixed low strains that are as near as possible to the linear region. Within this strain range plots of G”, G’ and G” us. v showed some dependence on the applied frequency, particularly at low # values. The results will be published in detail e1~ewhere.l~ The variation of G”, G’ and G” with # (at v = 1 Hz and yo 0.004 or 0.01) at the two NaCl concentrations (loP5 and mol dm-3) showed the expected behaviour of increase of modulus with increasing #, and there were significant differences between the values obtained in and mol dmP3 NaC1. These plots have been published before14 and the results showed the following trends.At any given # value, the values of the moduli are orders of magnitude lower in 10P3mol dm-3 NaCl when compared with the results in lop5 mol dmP3 NaCl. Within the volume fraction range studied in lop5 mol dm-3 (0.465- 0.525), G’> GI’ over the whole frequency range studied) (1OP5-5 Hz). In contrast, in mol dm-3, G’ were either close to or lower than G”, within the 4 range 0.253-0.566. The above results can be qualitatively explained in terms of the double-layer interac- tion between the particles. This is best illustrated from plots of G’ us. h, the surface to surface distance between the particles in the dispersion, i.e. h = 2a[(4,/+)i/3 - 11 (9)Th. F. Tadros and A. Hopkinson 0 0 10 45 50 100 150 200 h/nm Fig. 1. Variation of G' with h at two NaCl concentrations ( triangles are calculated values in and mol dm-3 NaCl).rnol dm-3 NaC1) closed mol dm-3 NaCl using data from ref. (15) acd (16). 0, Gk,, (lop5 mol dm-3 NaC1); A, G:heor; 0, GLxp where 4m is a constant that is charactristic of the type of array, e.g. 0.74 for hexagonal or face-centred cubic array, 0.68 for body-centred cubic, 0.64 for random arrangements of particles. For the present calculation of h a value of &, = 0.68 was used. The plots of G' vs. h are shown in fig. 1. G' increases rapidly with decreasing h, as expected from double-layer interaction. With the dispersion in mol dm-3, the double-layer extension ( 1 / ~ ) is 100 nm and hence one would expect a rapid increase in the modulus at h < 200, as found experimentally. Under these conditions, the dispersions are highly elastic as a result of the strong repulsive force between the particles.With the dispersions in lo-* mol dmP3, ( 1 / ~ ) is only 10 nm and hence a high modulus will be reached only at h separation distances comparable to 20 nm. This separation distance was not reached at the highest volume fraction studied, namely 0.566. Indeed at this 4 value the dispersion is less elastic than viscous and the G' value is still low (176 Pa). To obtain highly elastic dispersions in mol dm-3, one needs to go to much higher volume fractions (>0.6). It is perhaps useful to compare the experimental G' values with theoretical values that may be obtained using the analysis of Goodwin and co-worker~.'~*'~ These authors have theoretically related the high-frequency limit of the storage modulus (the shear46 Stability1 Flocculation of Concentrated Dispersions modulus) to the interparticle force as a function of volume fraction, and thereby of interparticle distance.The following expression was derived for the theoretical shear modulus: where cy = (3/32)4,n (n being the coordination number), R is the centre-to-centre separation for the particles (R = 2a + h ) and VT is the total energy of interaction between the particle. V - is given by the expression 4.rr~~,a’rl/; exp[-K(R-2a)] R VT = where E is the permittivity of the medium, E, that of the free space and K is the Debye-Huckel parameter. By differentiating eqn (1 1 ) twice, Goodwin and co- w o r k e r ~ ~ ~ , ~ ~ obtained the following expression for Gth (where KU < 10): G:,, = 4 m a ~ s , a ~ r l / ~ exp-[K(R-2a)] G:, were calculated for the dispersions in lo-’ mol dmP3 ( K a < 10) since these were highly elastic and the modulus showed little dependence on frequency.In these calcula- tions a was taken to be 0.833. The results of calculation are shown in fig. 1 (closed triangles). It can be seen that the theoretical values increases less rapidly with decrease of h than the experimental values. This implies that the simple model suggested by Goodwin and c o - w o r k e r ~ ’ ~ ~ ~ ~ is probably only applicable within a limited range of separation distances. There is, as yet, no theory that can accurately predict the elastic modulus under these conditions of significant overlap between the double layers.Another useful way of describing the interactions in concentrated dispersions is to apply scaling concepts. These have been successfully applied for the aggregation behaviour of concentrated dispersions. *’”’ Generally speaking, the modulus of a disper- sion scales with the volume fraction 4 with an exponent n, G‘= k4“. (13) The power n can be obtained from a log-log plot of G’ vs. 4. These plots are usually linear above a critical 4 value that denotes the onset of gel formation. For the above latex dispersions in mol dmP3 NaCl, the value of n, obtained from these linear plots, was in the region 20-30. These values are significantly higher than those obtained with flocculated suspension, as will be discussed later. Again, no theories are available for predicting the value of the exponent and as such it can be used only in a qualitative manner to describe the strong repulsive nature of the latex dispersions. and Sterically Stabilised Dispersions With polystyrene latex dispersions containing grafted PEO chains, the relative viscosity vs.volume fraction relationship was used to obtain the adsorbed layer thickness (6) as a function of 4. Details of the results obtained were published elsewhere’ and these showed a reduction of 6 from a value of 20.5 nm at 4 = 0.33 ( +eR = 0.460) to a value of 9.2 nm at 4 = 0.57 ( +efi = 0.665) (see Experimental). These results indicate that by increasing 4, compression of the PEO chains occurs as the particles approach each other very closely in a concentrated dispersion. This compression is reflected in the elastic properties of the dispersions as discussed below.Th. F.Tadros and A. Hopkinson 47 Plots of G”, G‘ and G” vs. 4 (published elsewhere*) showed an interesting transition as the volume fraction of the dispersion was increased. At 4 < 0.5, G”> G’, i.e. the dispersion showed a predominantly viscous response within the frequency range 1 O-*- 1 Hz. This is not surprising since at 4 < 0.5 the surface-to-surface distance of separation between the particles, h, is less than 28 and hence the interactions between these terminally anchored PEO chains are weak. These chains are only slightly compressed (8 decreases from 20.5 to 15.0 nm as 4 increases from 0.33 to 0.48). Such compression probably occurs with the larger PEO chains (note that grafted PEO chains are polydis- perse) and such a small effect does not lead to a predominantly elastic response within the frequency range studied.However, when 4 > 0.5, G’> G” and the magnitude of the moduli starts to rise strongly with further increase in 4. For example, when 4 is increased from 0.5 to 0.575, G’ at v = 1 Hz increases by an order of magnitude, and on further increase of 4 to 0.585, G’ increases by another two orders of magnitude. Moreover, at such high 4 values, the moduli tend to show little dependence on frequency within the range studied. In other words, the dispersion behaves as a near-elastic gel, as a result of the strong steric interaction between the PEO chains. The latter may undergo interpenetration with the peripheries of the larger PEO chain and further compression of the whole grafted polymer layers may occur. Indeed at 4 = 0.585, G” == GI= 4.8 x lo3 Pa and q’= 8.82 x lo3 Pa s, whereas at 4 = 0.65, G” = G’ = 1.12 x lo5 Pa and q‘ = 1.6 x lo5 Pa s, i.e.the latex behaves as a near-elastic solid. The above viscoelastic results clearly show their value in studying steric interaction in concentrated dispersions. It is possible to convert the modulus vs. 4 curves into G’ vs. h curves using eqn (9) and compare these with the F vs. h curves obtained from direct force meas~rements.~~ In these calculations a value of 4m = 0.68 was used. These F vs. h results can be converted to GYh results using a similar analysis to that of Goodwin and c o - ~ o r k e r s . ~ ~ ~ ’ ~ These calculations were recently obtained by Costello2’ and will be published in detail in the near future.21 A summary of the results obtained is shown in fig.2 which gives the measured G’ value and the theoretical values as a function of h. Although the values of G’ and Gbh vary considerably, the general trend is the same, namely a rapid increase in the elastic modulus with decrease in h, when the latter is less than 30 nm. As discussed above, when h < 30 nm, interpenetration and/or compress- ion of the PEO chains occurs. Thus, viscoelastic measurements can give a quantitative measure of the steric interaction between particles in a concentrated dispersion. It is also possible to apply scaling concepts to concentrated sterically stabilised dispersions. Recent results obtained in our laboratory” using similar latex with grafted PEO showed that GI- 430.The exponent in 4 is high and of the same order as that obtained for electrostatically stabilised dispersions. This high value of n is common in dispersions, where the net force is repulsive, but as mentioned before there are no theories that could predict such scaling factors. Similar results were obtained for physically adsorbed polymers on polystyrene latex.23 For example, results using poly(viny1 alcohol) (with M , = 45 000, S = 46 nm) on 700 nm polystyrene latex showed a rapid increase in G* and G’ at 4 > 0.53, whereas GI’ remained fairly At 4 > 0.53, h < 28, when one assumes a value of 4m = 0.64, i.e. random packing. The latter is more probable with dispersions containing long dangling tails.Under these conditions, elastic interaction between the long, dangling PVA chains occurs resulting in a predominantly elastic response for the dispersions. Weakly Flocculated Dispersions These are best exemplified by sterically stabilised dispersions to which a free (non- adsorbing) polymer is added in the continuous phase. Two such systems were investi- gated, namely, addition of poly( ethylene oxide) ( PEO)24 and hydroxyethyl cellulose (HEC)25 to a polystyrene latex dispersion with grafted PEO (the latex volume fraction48 Stability1 Flocculation of Concentrated Dispersions I I I 10 20 30 h/nm Fig. 2. Variation of G' with h for a sterically stabilised latex dispersion: e, calculated from F / h relationship; 0, experimental data for G', M , (PEO) = 2000.was kept constant at 0.3). Fig. 3 shows the variation of T~ and G, with free-polymer volume fraction &, for the three PEO polymer studies, whereas fig. 4 shows the variation of T~ with & for the three HEC sample studies. Above a critical &, value the rheological parameters increase with increase of &,. This critical value is denoted by @;, i.e. the free-polymer volume fraction above which depletion flocculation results in a structured system with measurable yield value and modulus. It is useful to compare ++ with the semidilute polymer volume fraction, which can be calculated for the equation where R, is the radius of gyration of the polymer that could be calculated from the intrinsic viscosity, using the Stockmayer-Fixman relationship,'6 6 is a constant that is equal to 5.63 for hexagonal close packing of the polymer coil, N,, is the Avogadro constant and p is the density of the polymer.A summary of the values of 4; and 4; and R, is given in table 1. 4; values are significantly higher than q5;, particularly with HEC. However, calculation of 4; using eqn (14) is approximate and based on a crude model of hexagonally close-packed polymer coils behaving as hard spheres. As discussed before27728 it is possible to relate the Bingham yield stress rP to the interparticle interaction. The latter may be equated to the amount of energy required to separate the flocs into single i.e. Tp = NEsep (15) where N is the total number of contacts between particles in flocs and Esep is the energy required to break each contact.The total number of contacts, N, may be related to theTh. F. Tadros and A. Hopkinson 49 2000 - ( a ) 1600 - a“ 1200 1 0” 1 800 - 400 - I . . > I . . . I . . . 0 0.02 0.04 0.06 0 0 0 0.02 0.04 0.06 0.08 4 P Fig. 3. Variation of G, and rp with volume fraction of free polymer (PEO) for a latex dispersion (4 = 0.3). M,(PEO): 0, 20 000; A, 35 000; 0, 90 000. volume fraction, 4, and the average number of contacts per particle (the coordination number), n, by Combining eqn (15) and (16) one obtains 3 4nEsep rp =-. 8 r a 3 Thus, Esep can be calculated from 4, provided a value can be assigned for n. The maximum value of n is probably 8, which is the average number of contacts in a floc for random close packing. However, recent work on the structure of aggregates indicates that quite open structures often arise in which n would be lower than 8 .It is also likely that n may decrease with increase in free-polymer concentration as more open flocs are produced at higher values of &. However, for the sake of comparison, two values of n were assumed, namely 8 and 4, and these were kept constant at all 4,, values. Another assumption that has to be made for calculating Esep from T~ is that above the yield point all contacts are broken. This assumption is probably justified with weakly flocculated systems, whereby the floc can be reversibly broken under high shear, resulting50 Stability/ Flocculation of Concentrated Dispersions l o t P P U 0 0.002 0.004 0.006 0.008 0.01 0.012 &J Fig. 4.ra us. #, for hydroxy ethyl cellulose with various molecular weights for a polystyrene latex dispersion (# = 0.3). Mw: 0, 70 000; A, 124 000; 0, 223 000. Table 1. #i7 #: and R, for the polymers used in depletion flocculation ( a ) poly( ethylene oxide) (PEO) 20 000 0.02 0.03 5 5.52 35 000 0.01 0.024 7.59 90 000 0.005 0.0 12 12.9 ( b ) hydroxy(ethy1 cellulose) (HEC) 70 000 0.0035 0.012 11.9 124 000 0.002 0.007 17.1 223 000 0.001 0.004 24.8 in the formation of primary units. Evidence that this is the case is obtained from the relatively small dependence of plastic viscosity on the free polymer concentration. The results of the calculation of Esep from T~ on the basis of the above assumptions are given in table 2 for the PEO system and in table 3 for the HEC system. These values of Esep may be equated to the free energy of flocculation due to depletion Gdep.The latter can be calculated using Asakura’s and 00sawa’s~~ or Fleer et aZ.’s3* theory. In the first case, Gdep/ kB T = - ( 3 / 2 ) 42pX2; 0 < x < 1 (14) where #2 is the volume concentration of the polymer, p = a / A , where A is the depletion thickness (equal to R,) and x = [A - (h/2)/A].Th. F. Tadros and A. Hopkinson 51 Table 2. Summary of the results for PEO of Esep calculated from the experimental T~ values and Gdep calculated using Asakura and 00sawa’s*~ (AO) and Fleer et aL’s ( FSV)30 models Esepl k0 T Gdep/ k0 c,bp T ~ / N ~ - ~ n = 8 n = 4 A 0 FSV 0.025 0.03 0.04 0.06 0.08 0.015 0.02 0.03 0.04 0.01 0.01 5 0.02 0.025 2.0 2.8 3.8 5.8 13.1 2.3 4.4 7.0 11.7 1.2 2.8 4.4 5.9 ( a ) PEO, M, = 20 000 9.1 18.2 12.7 25.4 17.3 34.6 26.4 52.8 59.6 111.2 ( b ) PEO, M, = 35 000 10.5 21 .o 20.0 40.0 31.9 63.8 53.2 106.0 (c) PEO, M , = 90 000 5.5 11.0 12.7 24.5 20.0 40.0 25.9 53.8 25.3 30.3 40.5 60.7 80.9 16.4 21.8 32.7 43.6 12.3 18.4 24.5 30.6 81.9 102.9 149.5 261.1 397.4 54.1 78.0 135.0 203.0 48.4 85.6 131.3 185.7 Fleer et aZ.-’” gave the following expression for Gdep: where v, is the molecular volume of the solvent, pl is the chemical potential of the solvent at a volume fraction c$p of the free polymer and py the corresponding value at &, = 0.(pl - /LO) can be calculated from c$p using the expression31 where n2 is the number of polymer segments and x is the Flory-Higgins interaction parameter which for PEO is 0.473 and for HEC is 0.47.Results of calculation of Gdep based on the above two theories are given in tables 2 and 3. For the PEO system the calculated values based on Fleer et aZ.’s3’ model deviate from the experimental Esep values, wherever reasonable agreement is obtained using Asakura and Oosawa’s In contrast with HEC, better agreement is obtained using Fleer et d ’ s model,-’” particularly at high M , values. Strongly Flocculated Systems Two main systems were investigated. In the first case, a sterically stabilised dispersion (polystyrene latex with physically adsorbed PVA) was flocculated by making the disper- sion medium a poor solvent for the chains. This was obtained by addition of electrolyte (KC1 or Na2S0,) or heating the dispersion at constant electrolyte concentration.In the second case, an electrostatically stabilised dispersion was flocculated by addition of 0.2 mol drn--’ NaCl. Details of the results obtained will be published elsewhere.32 As an illustration, fig. 5 shows the variation of G*, G’ and G” for a PVA-coated latex52 Stability1 Flocculation of Concentrated Dispersions Table 3. Summary of the results for HEC of Esep calculated from the experimental rs values and Gdep calculated from theory ~~ ~~ Esep/ kB Gdep/ kB c#+, rs/Nm-’ n = 8 n = 4 A 0 FSV 0.002 0.003 0.004 0.006 0.008 0.010 0.0 12 0.001 0.002 0.003 0.004 0.005 0.007 0.010 0.001 0.002 0.003 0.004 0.005 0.007 0.010 0.05 0.6 0.7 2.2 3.4 5.1 8.3 0.5 0.9 1.6 2.7 4.0 5.7 10.0 0.4 1.9 4.9 7.9 10.7 20.0 25.4 (a) M,., = 70 000 2.3 4.6 2.8 5.6 3.2 6.4 10.0 20.0 15.5 30.9 23.2 46.4 37.7 75.4 (b) Mw= 124000 2.3 4.6 4.1 8.2 7.3 14.5 12.3 24.5 18.2 36.4 25.9 51.8 45.5 90.9 ( c ) Mw = 223 000 1.8 3.6 8.7 17.3 22.3 44.5 35.9 71.9 48.7 97.3 90.9 181.8 115.5 23 1 .O 2.7 4.0 5.4 8.0 10.7 13.4 16.0 1.6 3.1 4.7 6.2 7.8 10.9 15.6 1.8 3.7 5.5 7.3 9.2 12.8 18.3 28.6 43.4 58.5 89.7 122.2 156.0 191.0 17.0 34.6 53.0 72.0 91.8 133.3 200.7 20.7 42.9 66.7 91.8 118.5 176.3 274.2 Fig.5. Variation of G* ( x), G’ (0) and G” (A) with CNu2S04 for a PVA-coated latex dispersion. Cp = 0.5, o = 1 Hz, y = 0.01.771. F. Tadros and A. Hopkinson 53 G " 1 10 100 2' 50 0 a" I- \ 1 1 1 G & * " J 0.1 0.2 0.5 1 2 5 10 20 50 a" ... 0 0.5 1 2 5 10 20 50 Yo/ 1 Fig. 6. Strain-sweep results for a flocculated latex ( CNaCl = 0.2 mol dm-3) at various volume fractions.( a ) 4 = 0.346, (b) 4 = 0.205, ( c ) 4 = 0.121, jd) 4 = 0.065. dispersion with 4 = 0.5, as a function of Na,SO, Concentration. The moduli (particularly G* and G') initially decrease with increasing CNaZSo4, reach a minimum and then suddenly increase above a critical electrolyte concentration. The initial reduction is due to the reduction in the adsorbed layer thickness, as a result of a reduction of solvency with an increase in the electrolyte concentration. However, above the critical flocculation concentration of Na2S04 (0.15 mol dm-3) all moduli show a rapid increase. This critical flocculation concentration marks the onset of incipient flocculation, i. e. the 8-point for the chain. Incipient flocculation of sterically stabilised dispersions can also be obtained by increasing the temperature at constant electrolyte concentration.This was demon- strated by using a dispersion in 0.15 mol dm-3 Na2S04 and changing the temperature from 15 to 30 "C. At 15 "C, the dispersion was stable and gave low modulus values, which decreased slightly with increase in temperature (due to reduction of solvency and collapse of the PVA chain), but above 25 "C there was a rapid increase in the modulus with increase in temperature. This temperature denotes the critical flocculation tem- perature (CFT) of the dispersion. One important finding from this work was that on cooling the flocculated dispersion, i.e. from 30 to 15 "C, the modulus values did not reach their initial low values, i.e. restabilisation did not occur in full.In other words, flocculation of such sterically stabilised dispersion is not completely reversible. The second flocculated system studied was that based on an electrostatically stabilised latex to which 0.2 mol dm-3 NaCl was added. Measurements were carried out as a function of strain amplitude at various 4 values, as illustrated in fig. 6. The structure of such coagulated suspensions becomes partially broken down above a critical strain (deformation) value that depends on 4. It can be seen that above a critical strain, ycr, the response is non-linear. A frequency sweep in the linear region, shows that G* = G', and G"= 0, and there was hardly any dependence on frequency in the range 1OW2-1 Hz.54 A log-log plot of G’ vs. 4 gave the following scaling equation: Stability/ Flocculation of Concentrated Dispersions G’ = 1.98 x 10746.0.(17) The above exponent is higher than most of the reported values in the literature. Many author^^^-^^ have reported that the exponent for flocculated suspension is in the range 2.0-3.5. However, the value of the exponent depends to some extent on the treatment to which a coagulated suspension has been subjected before the measurements were made. Recently, Ball36 used a novel method for calculating the elasticity of individual fractal clusters, and an exponent of 4.50*0.2 can be predicted for chemically limited aggregation in three dimensions and of 3.5 f 0.2 for diffusion-limited aggregation. The power produced by Ball’s theory is lower than that found experimentally in our investiga- tion.The higher experimental power in n is indicative of a strong flocculated structure as recently found by Sonntag and Several other flocculated systems were studied in our laboratory to obtain the power exponents. This first was a PVA-coated latex that was flocculated using KCl(2 mol dm-3) at 25 “C. This system gave the following power-law relation: G’= 1.2 x 10645.5. (18) The exponent in 4 is lower than that for the coagulated latex suspension using NaCl. The second system was a PVA-coated latex which was flocculated by heating to 50 “C at constant KCl concentration (0.2 mol dmP3). The power-law relationship was: G’= 5.95 x 10’4~.0. (19) The exponent is significantly less than that obtained for NaCl-flocculated bare latex or KC1-flocculated PVA-coated latex.This indicates that flocculation by heating a sterically stabilised latex produces structures which may be different from those obtained using high concentration of electrolyte. These structures are also different from those obtained by coagulation of an electrostatically stabilised latex dispersion. From a knowledge of y, and G’, one can calculate the cohesive energy, E,, of the flocculated structure. E, is related to the stress in the flocculated structure a, by the following equation:38 Since (T,= ycG’, then E, = joy‘ ycG’ d y = (1/2)yfG’. log-log plots of E, vs. 4 are linear, and the exponent n may be used as an index for strength of flocculation. For example, for the coagulated base latex suspension, n was found to be 9.1, which is indicative of a deep primary minimum coagulation process that involves hundreds of k , T units.Conclusions Viscoelastic measurements offer a powerful tool for investigating the stability/floccula- tion of concentrated dispersions. With stable systems, repulsion can be studied by measurements of the elastic modulus G’ as a function of volume fraction 4 while fixing other parameters of the system, e.g. particle size, electrolyte concentration and adsorbed layer thickness. The G’ vs. 4 curves can be compared with G’ vs. 6 results obtained using direct force measurements. These rheological measurements can also be applied to study interactions in flocculated system. With a weakly flocculated dispersion it isTh. F. Tadros and A. Hopkinson 5 5 possible to obtain the energy of interaction between the particles and compare this with the free energy of attraction, e.g.with depletion flocculation. With strongly flocculated systems, scaling concepts may be applied to obtain the power exponent for the depen- dence of G' on #. This exponent may be used as a qualitative index for the strength of flocculation. References 1 R. H. Ottewill, in Concentrated Dispersions, ed. J. W. Goodwin, Royal Society of Chemistry Publication, 2 R. H. Ottewill, in Future Directions in Polymer Colloids. ed. M. S . El-Aasser and R. M. Fitch, Nato 3 Th. F. Tadros, in Flocculation and Dewatering, ed. B. M. Mougdil and B. J. Scheiner (United Engineering 4 J. W. Goodwin, J. Hearn, C. C. Ho and R. H. Ottewill, Colloid Polym. Sci., 1974, 252, 464. 5 M. J. Gamey, Th. F.Tadros and B. Vincent, J. Colloid Interface Sci., 1974, 49, 57. 6 Th. van den Boomgaard, T. A. King, Th. F. Tadros, H. Tang and B. Vincent, J. Colloid Interface Sci., 7 C. Bromley, Colloids Surf, 1985, 17, 1. 8 C. Prestidge and Th. F. Tadros, J. Colloid Interface Sci., 1988, 124, 660. 9 H. Van Olphen, Clay Clay Miner., 1956, 4, 68; 1958, 6, 106. no. 43 (RSC, London, 1982), chap. 9, pp. 197-217. AS1 Ser., Ser. E Applied Sciences no. 138 (Martinus Nijhoff, Dordrecht, 1987), pp. 253-275. Trustees Inc., Place, 1989), pp. 43-87. 1978, 66, 68. 10 J. W. Goodwin and R. W. Smith, Faraday Discuss. Chem. SOC., 1974, 57, 126. 11 R. W. Whorlow, Rheological Techniques (Ellis Horwood, Chichester, 1980), p. 30. 12 M. Casson, in Rheology of Disperse Systems, ed. C. C. Hill (Pergammon Press, Oxford, 1959), p. 84. 13 A. Hopkinson and Th. F. Tadros, to be published. 14 Th. F. Tadros, Langrnuir, 1990, 6, 28. 15 J. W. Goodwin and A, M. Khider, Colloid and Interface Science, ed. M. Kerker (Academic Press, New 16 R. Buscall, J. W. Goodwin, M. W. Hawkins and R. H. Ottewill, J. Chem. SOC., Faraday Trans. 1, 1982, 17 W. D. Brown, PhD Thesis (University of Cambridge, 1987). 18 R. Buscall, P. D. Mills, J. W. Goodwin and D. W. Lawson, J. Chern. Soc,, Faraday Trans. 1, 1988,84, 19 B. A. De L. Costello, P. F. Luckham and Th. F. Tadros, Colloids Surf, 1988/1989, 34, 301. 20 B. A. De L. Costello, PhD Thesis (Imperial College, in preparation). 21 B. A. De L. Costello, P. F. Luckham and Th. F. Tadros, to be published. 22 R. C. Navarette and Th. F. Tadros, to be published. 23 A. Hopkinson and Th. F. Tadros, to be published. 24 C. Prestidge and Th. F. Tadros, Colloids Surf, 1988, 31, 325. 25 M. Gover and Th. F. Tadros, to be published. 26 W. H. Stockmayer and M. Fixman, J. Polym. Sci. Part C, 1963, 1, 137. 27 P. F. Luckham, B. Vincent and Th. F. Tadros, Colloids Surf, 1983, 6, 101. 28 D. Heath and Th. F. Tadros, Faraday Discuss. Chem. SOC., 1983, 76, 203. 29 S. Asakura and F. Oosawa, J. Chem. Phys., 1954, 22, 1255; J. Polym. Sci., 1958, 33, 245. 30 G. J . Fleer, J. H. M. H. Scheutjens and B. Vincent, ACS Symp. Ser., 1984, 240, 245. 31 P. J. Flory, Principles ofPolymer Chemistry (Cornell University Press, Ithaca, 1953), p. 511. 32 A. Hopkinson and Th. F. Tadros, to be published. 33 A. Zosel, Rheol. Acta, 1982, 21, 72. 34 R. Buscall, I. J. M. McGowan, P. D. Mills, R. F. Sutton, L. F. White and L. F. Yates, J. Non-Newtonian 35 W. B. Russel, Powder Technol., 1987, 51, 15. 36 R. Ball, personal communication. 37 R. C. Sonntag and W. B. Russel, J. Colloid Interface Sci., 1958, 116, 414. 38 J. D. F. Ramsay, J. Colloid Inreflace Sci., 1986, 109, 449. York, 1976), vol. IV, p. 529. 78, 2889. 4249. Fluid Mech., 1987, 24, 183. Paper 0/02267E; Received 18th May, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000041

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1990, 90, 57-75 GENERAL DISCUSSION Prof. C. F. Zukoski (University of Illinois) addressed Prof. Gast: Is it necessary to consider how far ions move in half a cycle? The average drift velocity (U) of an ion can be calculated from (U)=oF, where F is the electrical force, eE, and o can be estimated from o = D / k g T . Then at lo6 V m-', (U) =r 3.8 cm s-l. For coated electrodes, no loss current can pass between the electrodes. As the relaxation frequency of the ions can be estimated from a/ toEo =: 2 x lo4 s-', ions can respond to a frequency of 3 x lo3 s-l. The question then is how far ions can move at a frequency of 3 x lo3 s-'. This can be estimated from the time for half a cycle (1.7 x s) and the average drift velocity of the ions. From these calculations one finds that the distance travelled at 500 V across 560 pm is ca.5.7 pm. Thus near the fields where the doublet concentration saturates, ions can move several particle diameters. As it is improbable that there will be any electrochemistry occurring at the electrodes, the field is unlikely to be uniform across the gap. Is it possible that the saturation in doublet concentration is due to relaxation of the field due to ions migrating to the electrode? Prof. A. P. Gast (Stanford University) replied: This is an interesting possibility that we had not considered. The characteristic length for ion diffusion within one cycle we calculate from l2 = D / o with an ion diffusivity of D = 1 x m2 s-' and a frequency of 3000 s-', is 0.6 pm, essentially one particle radius.From your equation for the drift velocity with ( U) = DeE/k,T we calculate a velocity of 0.038 m s-' for 1000 V mm-'. This provides a distance travelled in one half cycle, 2 / w = 1.7 x of 6.5 pm, a distance large compared to a particle but small compared to the whole gap. The saturation in electric field could have electrostatic origins such as the one you suggest, however, we have no evidence for large field gradients in our cell. We could only unambiguously test this hypothesis with a better cell design and further studies at higher frequencies. Prof. D. A. Saville (Princeton Uniuersity) said: We have studied the motion of small fluid globules in apolar liquids ( E d 10) at high field strengths (> 1000 V cm-') and find some evidence of field-induced charging. In such cases, the particle charge inferred from low-field-strength mobility measurements may not appreciably change at higher fields.Did you find any evidence of field-induced charging? Could such an effect be responsible for the 'saturation' shown in, e.g., fig. 3 and 4? Prof. Gast replied: Electric field-induced charging could provide an explanation for the saturation behaviour observed in our experiments. Note that the saturation did occur at similar field strengths for different volume fractions. Surely more experiments are in order to verify the mechanisms. It would be quite interesting to measure the electrophoretic mobility of our particles in strong fields to look for field-induced charging. One could perhaps make such a measurement by applying a high-voltage AC field with a DC bias.A better cell design would probably be required to avoid contributions from dielectrophoresis occurring under high field strengths. We have seen evidence for saturation effects in dielectrophoretic mobility measurements of glass beads in oils. Prof. S. P. Stoylov (Bulgarian Academy of Science, Sojia) asked Prof. Gast: First, in view of the numerous results which indicate a large induced-dipole moment of a non-dielectric (interfacial) nature for latex dispersions in aqueous media, is it not possible that the induced-dipole moment measured by you is of a similar nature? A way to 5758 General Discussion investigate this would be to make measurements at frequencies much lower or much higher than 3 kHz. Have you made such measurements? Secondly, have you measured the distribution of the electric field strength in your cell? You state that the Teflon tape covering your electrodes has the same relative permittivity as your solvent and that therefore the tape does not perturb your electric field.This will be correct only when the electric conductivities are equal. Electro-induced lattice interactions have been described by Vorobeva' and by Sokerov, Vorobeva and Stoylov' followed in the first case microscopically and in the second case by light scattering. 1 T. Vorobeva, Thesis, Moscow, 1967. 2 S. Sokerov, T. Vorobeva and S. P. Stoylov, Polym. Sci. Symp., 1974, 44, 147. Prof. Gast responded: We have no evidence for large contributions to our dipole moment due to interfacial effects.Owing to the good agreement with a dielectric dipole model we were not compelled to consider the interfacial dipole. It would be fruitful in the future to make measurements such as ours at a wide range of frequencies to determine whether this mechanism is important in low-relative-permittivity fluids. We found no difference between Teflon-coated and uncovered copper electrodes. We preferred the former simply because they provided a better seal with the cell walls and they reduced particle deposition on the electrodes. A simple calculation indicates only a minor perturbation to the electric field for a layer with relative permittivity so close to that of the solvent. We did not see any major distortions of the electric field across our cell. Dr B. Vincent (University of Bristol) said: As shown in fig.5 of your paper, with increasing particle volume fraction of the dispersion, the critical depth of the energy minimum (and hence the applied electric field) for the onset of dimer formation decreases. This is to be expected on thermodynamic grounds, in terms of the net energy/entropy balance in the system.' However, the experimental observation times were rather short (ca. 1 min). Are you sure that equilibrium has indeed been established with this timescale, because our experience of weak, reversible flocculation is that quite long timescales may be necessary to establish equilibrium? 1 J. A. Long, D. W. J. Osmond and B. Vincent, J. Colloid Interface Sci., 1973, 42, 545. Prof. Gast replied: Yes, the critical interaction energy to produce doublets increases with volume fraction as anticipated for a phase transition.Near this threshold we are observing equilibrium doublet formation; the formation of larger phases, such as those you have observed, could be much slower. As mentioned in the text, the time characteris- tic of this equilibrium, t,, = 6.rrq-a3/kgT exp (-umi,/kBT) is ca. 5 s near the onset of doublet formation. It is interesting to note, however, that the failure of the equilibrium model occurs at the same potential-energy minimum for each sample corresponding to an equilibrium time of ca. 500s. Prof. J. Lyklema and Dr P. C. van der Hoeven ( Wageningen University) commented: In the paper by Adriani and Gast the authors conclude, somewhat to their surprise, that in media of low relative permittivity, electrostatic repulsions may act beyond the range of stabilising layers.We have found similar phenomena. The stability of concen- trated dispersions of a number of inorganic solids and zeolites in pure non-ionic surfactants, in the presence of dodecylbenzenesulphonic acid, appeared to be entirely of electrostatic origin. Measurements to prove this included rheology, analyses of sediment structures and electrokinetics. Admittedly, in our systems the likelihood of electrostatic stabilisation is somewhat greater, because our relative permittivity wasGeneral Discussion 59 140 120 - > 100 - 'm -E 80 1 2 60 U z 40 20 -6 -5 -4 -3 -2 -1 0 log (% Ca octanoate) Fig. 1. Electrophoretic mobility against log (% calcium octanoate) for PMMA- PHS particles, diameter 614 nm, in dodecane.higher (ca. 6) and because of the presence of a 'candidate' charge-determining species. However, the repulsive forces must be substantial since systems containing over 20% of the solid phase can be stabilised. It may perhaps be concluded that electrostatic stabilisation in systems of low polarity is more common than is usually thought. Dr P. Bartlett (University of Bristol) said: I am surprised by the high electrophoretic mobility of 1.6 x m2 V-' s-I quoted by Prof. Gast for her PMMA spheres. I have made measurements on similar PMMA spheres of 670 nm in diameter in pure cis-decalin and find mobilities typically two orders of magnitude lower ( ~ 2 x lo-" m2 V-' s-I). I wonder if Prof. Gast could comment on the mobilities of her particles in pure hexane prior to the addition of sodium acetate? Prof.Gast responded: The particles had an electrophoretic mobility of the same order of magnitude before the addition of sodium acetate but it was difficult to reproduce. Freshly made samples had quite variable mobilities and conductivities presumably due to variable concentrations of ionic impurities. You will note that while the addition of sodium acetate helped to control the mobility, one sample had a higher ionic content and a lower mobility. Prof. R. H. Ottewill and Mr A. Schofield (University of Bristol) commented: This paper provides some nice results in the difficult field of electrical effects on colloidal particles in non-aqueous media. In laser electrophoresis studies on poly( methyl methacrylate) (PMMA) particles stabilised by poly( 12-hydroxystearic acid) (PHS) we have found the basic particles to have a weak positive charge.There has been some variation between samples and most examinations have been carried out in dodecane. The weak positive charge, however, can be considerably enhanced by the addition of small amounts of calcium octanoate. Fig. 1 illustrates some electrophoretic mobility results obtained as a function of calcium octanoate concentration. From these it can60 Genera 1 Discussion be seen that initially the electrophoretic mobility becomes more positive with increase in calcium octanoate concentration and then decreases. The positive charge is probably a consequence of adsorption of [Ca octanoateJ+ or Ca2+ by attraction to the ester linkages on the poly(l2-hydroxystearic acid) chains. The reason for the maximum is not completely clear but preliminary conductance measurements have provided some evidence for the self-association of calcium octanoate in this region.Small-angle neutron scattering measurements have also provided evidence for charg- ing of the latex particles on the addition of calcium octanoate.' 1 R. H. Ottewill, A. R. Rennie and A. Schofield, Prog. Colloid Polyrn. Sci., 1990, 81, 1. Prof. G. Frens (University of Delft) said: Could the effects which Prof. Gast has described be useful in the technology of electrophotography, where particles are attracted by, and deposited on, a charge pattern on a surface? The problem is to obtain deposits of more than one particle in thickness.For electrophotography one uses suspensions of electrically stabilised particles in dielectric liquids of low E.' The problem, as in electrophoretic deposition, is that particles in the second and further layers of the deposit must be fixed in their positions by some form of coagulation. Could the energy minima for two particles in an electric field be used to overcome their repulsion and give irreversible coagulation in the deposit? What field strength must be reached in the liquid to obtain such a result? 1 S. Stotz, J. Colloid Interface Sci., 1978, 65, 118. Prof. Gast replied: The interactions producing our particle chains are purely field- induced and thus vanish upon removal of the field. Such a field-induced dipolar interaction may, however, be sufficient to overcome a repulsive barrier and leave particles bound in a primary minimum after the field is removed.We have seen such effects with magnetically polarisable particles which can be made to remain in chains in the absence of the external field. Of course, the field strength required to produce such an effect will depend on the size of the repulsive barrier to be surmounted. The purely field-induced dipolar interaction energy scales with the square of the dipole moment as P2 U - 4.rr~~~,8a'li,T and to a first approximation the dipole moment scales with the electric field strength as while with multibody interactions an enhancement of the dipole strength over the isolated particle value is seen.' Thus, with knowledge of the stabilising forces in the suspension or with careful deposition experiments such a field-induced irreversible aggregation could be achieved.1 P. Adriani and A. P. Gast, Phys. Fluids, 1988, 31, 2757. Prof. G. Cevc ( University of Munich) (communicated): Looking at the experimental values for the chain number fractions as a function of volume fraction in the range of high peak-to-peak applied potentials (near the saturation region) I notice that the absolute numbers vary non-monotonically: i.e. for small q5 they decrease and for higher 4 they increase with the value of 4. Can you explain this phenomenon? Prof. Gast replied: We have looked at the number fractions of single particles where we believe the saturation phenomena to begin (by interpolating between available data points, not by extrapolation to higher fields).We find that the singlet volume fraction at saturation varies monotonically but not in the linear fashion that one would expectGeneral Discussion 61 if the experimental time was constant and the doublet formation time was tdb= 7rr]a3/4kgT. This deviation may be due to the fact that the limiting transport process may not be random diffusion but rather diffusion in a restricted space essentially transverse to the field. It is also useful to reiterate that the volume fraction 4 = 0.0028 sample did have a different ionic strength. Prof. Saville (communicated): The authors carry out experiments at a frequency of 3 kHz and argue that at this frequency the particles and their counter-ion cloud can no longer follow the imposed field.Accordingly, they ignore the dipole due to the particle charge and counter-ions. I believe that this dipole may be significant for the following reasons. One way of estimating a characteristic relaxation frequency is to use a timescale based on the distance ions must move to follow the imposed field, the particle radius plus Debye thickness, a + 1 / ~ , and the ion diffusivity, D. Then, as argued by DeLacey and White,' the characteristic frequency is approximately D ( u K ) ~ / u * ( 1 + U K ) ~ . For a 1 pm particle with U K =J 3 and D = m2 s-I, the characteristic frequency is slightly more than 2 kHz. Numerical calculations by DeLacey and White show that at such frequencies relaxation is not complete and a significant dipole still exists.Although estimates of this sort are imprecise, the operating frequency in these experiments is of the same order of magnitude as the relaxation frequency, so it appears that the induced dipole due to particle charge ought to be included in the analysis. For the problem under consideration the dipole due to particle charge, estimated as Q(a + K - ~ ) = C m, is larger than the polarization dipole due to differences in relative permittivity. If, as these estimates indicate, the particle charge-counter-ion cloud dipole is significant, what effect will this have on the theory of the equilibrium chain length? Will the agreement between theory and experiment be altered substantially? 1 E. H. B. DeLacey and L. R. White, J. Chem. SOC., Furuduy Trans. 2, 1981, 77, 2007. Prof.Gast replied: Your are correct in pointing out that the relaxation time for our ionic double layer (based on the estimates of ionic strength from conductivity) is comparable to the frequency of our alternating current field within numerical factors. Certainly it would be interesting to do such an experiment at much higher frequencies; however, our experimental arrangement precluded this. It is not clear how the dipole due to the distortion of the double layer contributes to our calculation of chain length and doublet equilibrium. As you point out, an upper bound on the magnitude of this dipole, Q(a + K - ' ) , is in the region of C m based on our estimate of the zeta potential and ionic strength. The dipole due to dielectric differences, p = 4 7 7 ~ ~ s , a ~ P E , is of the same order of magnitude, 7 .6 ~ for E = 1000V/560pm7 c0= 8.85 x C V-' m-', E, = 1.9, a = 0.50 pm and p = 0.162 for this system. If these estimates are correct, this dipole will indeed affect our predictions by adding an additional attraction between our particles. The effect of a double-layer dipole on our results depends on the field dependence of the dipole; this may be non-linear and even saturating at elevated field strengths. It would be interesting to learn, however, how two such distorted double layers interact upon close approach of the particles and how this dipole persists around a chain of particles. Surely the point-dipole approximations are no longer valid. It could be that the short-range attraction is not so severely altered thus providing the agreement between theory and experiment.Dr R. Buscall (ICI, Runcorn) (communicated): A number of years ago ICI provided Dr Block of Liverpool University with a sample of a similar latex in dodecane. Dr Block was performing dielectric spectroscopy on flowing polymer solutions and he wanted to compare the behaviour of a dispersion. The sample was found to show a very strong electrorheological response. This was unexpected and so several, larger62 General Discussion samples were synthesized for further study. These, however, failed to show any response whatsoever, even at very high field strengths. It was suspected that the response shown by the first sample may have been caused by some polar impurity, e.g. water. However, the addition of water and other polar liquids in trace amounts failed to produce an effect, as did copolymerisation of polar monomers with the MMA.Does Prof. Gast have any comments regarding what type of contaminant might produce such a response; is it correct to say that it would have to be surface-active and highly polarisable? Prof. Gast responded: The electrorheological response we find with the 1.0 pm poly(methy1 methacrylate) spheres is quite weak. As we show in the paper, the electro- rheological tendency is well described by material parameters expected for PMMA in hexane and thus requires no impurity to account for the effect. The fact that we do see a weak positive charge on the particles does, however, indicate the presence of an unknown impurity. The effect of charged impurities on electrorheology is not obvious, and, judging from the nature of this discussion, not fully understood or agreed upon.It is quite likely that residual surface-active impurities may influence such systems. The observation of a very strong electrorheological effect in such systems would, as you point out, imply the presence of a material having a very high relative permittivity. The addition of water, in general, may degrade the electrorheological effect by increasing the conductivity in the suspension leading to power consumption and heating. Prof. N. Ise (Kyoto Uniuersity) turned to Dr Dhont: Spinodal decomposition takes place when the second derivative of the free energy of the system with respect to concentration is negative. This implies that the concentration dependence of activity must be negative.Our previous measurements of various linear polyelectrolytes showed that the solute activity increased with increasing concentration. (For a convenient review, see ref. 1.) Furthermore, our measurements of ionic latex suspensions showed that the activity of the latex was practically independent of its own concentration.* We have never measured the activity of the silica particle suspension. Do you have experimental activity data for silica suspensions? 1 N. Ise, Ado. Polym. Sci, 1971, 7, 536. 2 M. Sugimura, T. Okubo, N. Ise and S. Yokogama, J. Am. Chem. Soc., 1984, 106, 5069. Mr M. H. G. M. Penders, Dr J. K. G. Dhont and Prof. A. Vrij (University of Utrecht) replied: The divergence of the measured structure factor at small wavevectors with decreasing temperature implies that the osmotic compressibility of the system of Brownian particles which scatter light, goes to infinity.This system of Brownian particles becomes unstable. A vanishing and, in the unstable region, negative reciprocal osmotic compressibility is equivalent to what you call ‘a negative concentration dependence of the activity’. Structure-factor measurements at small wavevectors thus probe the activity. We did not measure the osmotic compressibility using different experimental techniques besides light scattering. It is not possible to use electrochemical or isopiestic methods in our systems. Our systems are not of an ionic nature and the molecular weight of the particles is far too high to measure any change in the solvent activity in the concentration range we measured here.Prof. Ise (communicated ): You obtained the osmotic compressibility by extrapolating the structure factor S ( (I) to Q = 0, where Q is the scattering vector. This extrapolation is valid if the solution is homogeneous. There is direct experimental evidence available showing that solutions of ionic polymers and suspensions of ionic latex particles are not homogeneous. (See for example, ref. 1.) I presume that this is also the case with ionic silica particle suspensions. If so, there should be a sharp upturn of the structureGeneral Discussion 63 factor at very low Q regions, which would invalidate the extrapolation to Q = O . Therefore, I strongly suspect that the authors have been observing the spinodal decompo- sition. In this respect, I should draw the authors’ attention to similar phase-separation phenomena reported for ionic latex suspensions by Arora et aZ.* The height and width of the Bragg peak depend on three factors, namely the Debye- Waller effect, the paracrystalline distortion and the size of the ordered structures. Therefore, it is difficult to accept the authors’ assumption that rate of increase of the Bragg peak height is a measure of the crystallisation rate. According to our microscopy study on latex suspensions,’ the oscillatory motion of latex particles at lattice points (which corresponds to the Debye- Waller effect) is substantial at low latex concentrations and becomes less and less so as the concentration increases.It is also highly likely that the paracrystalline distortion becomes less and less pronounced with increasing con- centration. Thus it is not clear to me why the authors observed practically the same scattering intensity at t = 0 for three different concentrations (fig.5 , their paper) with the intensity of the 0.329gcm-’ sample reaching the largest value among the three concentrations afterwards. Could the sedimentation effect of the silica particles (radius, 1600 A) be ignored in these experiments? The authors observed crystallites with well defined sizes in solution. Similar structures have been reported for ionic latex particle suspensions by several author^.^ We inferred the existence of such localised structures for ionic polymer solutions on the basis of scattering experiments.’ The authors’ explanation of the existence of the crystallites in terms of repulsive interactions is not acceptable.How can the authors explain the formation of crystallites of definite sizes in solution, if the particles interact only via repulsion? How can the particles at the boundary stay there if they feel purely repulsive interactions from the particles inside? 1 N. Ise, H. Matsuoka, K. Ito and H. Yoshida, Furuday Discuss. Chem. Soc., this Discussion. 2 A. K. Arora, B. V. R. Tata, A. K. Sood and R. Kesavamoorthy, Phys. Rev. Lett., 1988, 60, 2438. 3 K. Ito, H. Nakamura, H. Yoshida and N. Ise, J. Am. Chem. Soc., 1988, 110, 6955. 4 A. Kose, M. Ozaki, K. Takano, Y. Kobayashi and S. Hachisu, J. Colloid Interface Sci., 1973, 44, 330; N. A. Clark, B.J. Ackerson and A. J. Hurd, Phys. Rev. Lett., 1983, 50, 1459; T. Yoshiyama and I. Sogami, Lungrnuir, 1987,3, 851; K. Ito, H. Nakamura, H. Yoshida and N. Ise, J. Am. Chem. Soc., 1988, 110, 6955. 5 N. Ise and T. Okubo, Acc. Chem. Res., 1980, 13, 303; N. Ise, Angew. Chem., Int. Ed. Engl., 1986, 25, 323. Mr Penders, Dr Dhont and Prof. Vrij responded: The silica systems used for the PICS measurements, presented in our paper, are not of an ionic nature and therefore the behaviour of these systems cannot be compared to the behaviour of the suspensions of ionic latex particles and solutions of ionic polymers you mention in your comment. The upswing of the structure factor at small wavevectors is a result of an increasing correlation length, or equivalently, due to an increase of the osmotic compressibility (just as in molecular or atomic systems).This upswing is monotonic as a function of the wavevector. With decreasing wavevector, it follows from SANS experiments’ that there is a monotonic increase of S(Q) for silica systems dispersed in benzene. As the temperature is lowered this increase is more steep, as it should be at the approach of the spinodal curve from the stable region in the phase diagram. In a decomposing system on the other hand, the intensity develops a maximum at very small wavevectors which increases with time, as described by the Cahn- Hilliard theory. Such a time dependence of the intensity was not observed in the experiments described in our present paper. 1 C. G. de Kruif, P. W. Rouw, W. J. Briels, M.H. G. Duits, A. Vrij and R. P. May, Lungmuir, 1989, 5,422. Dr Dhont continued: What is measured at t=O is the structure factor of the (metastable) fluid. From its primary maximum Bragg peaks start to grow. In your64 General Discussion remarks you seem to imply that there is a relation between the height of the primary maximum of the structure factor of the metastable fluid and the height of the measured Bragg peak after completion of crystal growth. This is certainly a complicated relation which is not yet fully understood: a statistical-mechanical theory for the kinetics of crystallisation is not available. Sedimentation of Brownian particles in the fluid phase is only noticeable after ca. 2 weeks (in samples which are dilute enough or so concentrated that no crystallisation occurs), and therefore plays no role during the crystallization process which takes at most ca.1.5 h. However, crystallites, once formed, do sediment during their growth up to a level that remains unchanged over more than a day or two, after which separated crystals start to coalesce without further noticeable change of the volume occupied by the crystalline phase. Thus, the nucleation kinetics are not affected by sedimentation processes, since sedimentation simply does not occur, but the details of crystal-growth kinetics might be influenced a little owing to sedimentation of crystals. As was mentioned above, the separated crystals which are grown out of the indepen- dently formed nuclei, start to coalesce after a day or two. The final thermodynamically stable state is therefore not the long-lived state occurring immediately after crystal growth stops, but is a state where a single large crystal is in coexistence with a fluid phase.This coalescence process is probably driven, in part, by gravity forces. The experiments described in the paper are all restricted to times where nucleation and subsequent crystal growth appear, before coalescence occurs. We disagree, totally, with Prof. Ise’s view that crystallisation can occur only in coexistence with a fluid phase for systems with an attractive component in the pair- interaction potential. The total force on a particle is not the sum of all the pair-potential interaction forces from the other particles in the system, as is incorrectly assumed in Prof.Ise’s question. There is an additional force on a particle j , which is called the Brownian force FLr, and is given by where PN is the probability density function of the position coordinates r l , . . . , r N of the N particles in the system, and V j is the gradient operator with respect to 5, the position coordinate of the j t h particle. The total force experienced by this particle is therefore where V is usually taken to be equal to a sum of pair-interaction potentials. The equilibrium form of PN, for a canonical ensemble, is obtained by minimising the free energy (with constraints pertaining to the canonical ensemble), and is, of course, the Boltzmann factor, Substituting this in eqn (2) gives, Fljj,, = 0; in equilibrium (4) as it should be. In his argument, Prof.Ise overlooks the Brownian force in eqn. (2), which amounts, in thermodynamic terms, to saying that the system will minimise its total potential (plus kinetic) energy, instead of its free energy. A particle on the boundary of a crystallite will not be pushed into the fluid phase, rendering an unstable crystallite, just because the potential energy of the system then becomes smaller. The potential energy is decreased in this event but the free energy is increased. In terms of forces, the total potential interaction force will tend to push a particle into the fluid phase, but the Brownian force will do the opposite; in equilibrium these two forces cancel, accordingGeneral Discussion 65 to eqn (4). We note here that computer simulations and density functional theory calculations do predict crystal-fluid coexistence for systems with a purely repulsive pair-interaction potential.From the thermodynamic point of view, crystallisation in these systems can be understood as follows. The total potential energy will probably increase on formation of a crystallite (with a higher overall density than the fluid phase) and therefore counteracts crystallisation. The entropy can be thought of as having two distinct contributions, a ‘configurational’ entropy, which decreases as a crystal is formed and which therefore also counteracts crystallisation, and a ‘translational’ entropy, related to the free volume which is available for each particle. In an ordered structure, the space available for each particle is larger than in a disordered structure. Therefore, this contribution to the entropy will favour crystallisation.At a high enough volume fraction in the ‘disordered’ fluid phase, the translational-entropy contribution to the free energy will be larger than the potential-energy and configurational-entropy contributions, so that crystals will be formed. The formation of crystals continues up to a point where the volume fraction of the remaining fluid is lowered such that the free energy per particle in the fluid phase equals that of the crystal phase. Prof. Ise (communicated): I was very much puzzled by the responses of Dr Dhont and co-workers for various reasons. While we are fully aware of the uncertainty about the ionic state of silica particles, the authors clearly reported in their paper, ‘the sample was slightly charged, giving rise to a screened Coulomb type of interaction potential. The Debye length is estimated to be eu.100 nm.’ Now, in the response to my comment, they have said that the samples are not of an ionic nature. These descriptions are contradictory. I would like to ask the authors how they came to these diametrically opposed conclusions. I know from our discussion in the meeting how the osmotic compressibility was determined. The reference to the SANS experiment is not relevant to the present argument. The point in question is how reliable the S ( Q ) value determined by the authors’ extrapolation can be. My question arises not only from the observation of inhomogeneity in highly charged polymeric systems mentioned in my previous comment but also from the observed sharp upturn of SAXS curves for low-charge-density ionomer films.’ Even though one takes into account the fact that the upturn was obtained in solid without solvent, the presence of a very small number of ionic groups might render inaccurate the extrapolation of S ( Q ) to Q = 0.Therefore it is necessary to estimate the osmotic compressibility by an independent method, if the authors are to claim that they are observing spinodal decomposition. (Note that the isopiestic method or solvent vapour pressure measurement can be used, in principle, whether the systems are ionic or non-ionic, while the electrochemical methods are of course not feasible in non-ionic cases.) It is not clear to me what the authors want to imply by ‘a relation between the height of the primary maximum of the structure factor of the metastable fluid and the height of the measured Bragg peak after completion of crystal growth’.All that I wished to emphasise was that the peak height is determined by the three factors according to the principles of scattering and cannot generally be a measure of the size of the crystallites only. Therefore the authors may be correct when the other two factors, the Debye-Waller effect and the paracrystalline distortion, are practically constant or do not become less enhanced during the crystallisation process. However can the authors be sure about this? The final point, on which the authors claimed to disagree totally with our view, was mentioned in my lecture. As far as I know, most of the computer simulation, for example, was carried out with a purely repulsive potential.The same calculation has not been carried out with a more realistic potential containing repulsive plus attractive com- ponents, except by Tata et uZ.* No one has yet demonstrated how satisfactory it is to take into consideration the attractive components. Certainly, the explanation advanced66 General Discussion by these authors and many others in terms of the repulsive potential and positive translational entropy increase is not the only possibility. 1 R. A. Register, A. Sen, R. A. Weiss and S. A. Cooper, Macromolecules, 1989, 22, 2224. 2 B. V. R. Tata, A. K. Sood and R. Kesavamoorthy, Prarnana-J. Phys., 1990, 34, 23; B. V. R. Tata, A. K. Arora and M. C.Valsakumar, submitted for publication; A. K. Sood, Solid State Physics, ed. H. Ehrenreich and D. Turnbull (Academic Press, New York, 1990), in press. Dr Dhont responded: It is clearly stated in our paper that the PICS measurements were performed on stearylsilica, which is non-ionic. The quoted lines, ‘the sample was slightly charged, giving rise to a screened Coulomb type of interaction potential. The Debye length is estimated to be ca. 100nm’, refers to TPM-silica, and this is certainly ionic. Certainly, besides the size of the crystals, the measured Bragg peak heights are functions of the mean-square displacements of particles in the crystal and possible defaults in the crystal structure. In a (semi-)quantitative interpretation of the data in terms of crystal growth rates one should try to take these things into account.Probably a cleaner quantity from which crystal growth rates can be determined is the integrated intensity of a Bragg peak. Prof. Ise’s point was that with only repulsive pair interactions no crystallisation can occur. The mentioned computer simulations do include only repulsive interactions and do show crystallisation. Prof. Ise (communicated): Your answer reminds me of an anecdote discussed by M. Eigen.’ The experiment (simulation) must be done without cutting the legs of cockroaches (in other words, without ignoring the attractive component), if the correct answer is to be sought. Thus I have to disagree with the authors. 1 M. Eigen and R. Winkler, Das Spiel, Naturgesetze sfeuern den Zufall (Piper, Munich, 1976), chap.17; R. Kimber and R. Kimber, Law of the Games (Penguin, New York, 1983). Dr Dhont replied: The anecdote of M. Eigen does not apply to this discussion: Prof. Ise’s view is that crystallisation cannot occur in systems with a purely repulsive pair- interaction potential, in contradiction with the experiment (simulation). Prof. W. B. Russel (Princeton University) addressed Dr Dhont: Your presentation introduced data on the number of crystallites, showing a maximum as a function of concentration. If your data on crystallisation rate are normalised, i.e. reduced to a rate per crystallite, does the maximum in fig. 6 of your paper persist? Dr Dhont replied: In order to obtain the growth rate of a single crystal, the crystallisation growth rate should be divided by the intensity of light scattered from a single crystallite of size ca.half that of the crystallites after completion of the crystallisa- tion process, and the number density of the crystals. The results shown in the presentation were obtained two months ago. These kinds of calculation are therefore yet to be performed. Prof. Russel continued: Fig. 2 shows the spinodal from Baxter’s model plus the limited calculations of Haymet for the fluid-solid transition.’ Continued to lower T this transition might transcend the fluid-fluid as the equilibrium process. The question is whether the decomposition observed leads to a fluid-like dense phase or a solid as established for other systems with short-range attractions. 1 N. A. Seaton and E. D. Glandt, J. Chem.Phys., 1987, 86, 468; Y. C. Chiew and E. D. Glandt, J. Phys. A, 1983, 16, 2599; S. A. Safran, 1. Webman and G. S. Crest, Phys. Reti. A, 1985, 32, 506; S. I. Smithline and A. T. S. Haymet, J. Chern. Phys., 1985,83, 4103.General Discussion 67 2.0 1.0 0.5 7 0.2 0.1 \ \ I / 1 I I I I I I 0.00 0.20 0.40 0.60 77 Fig. 2. Phase diagram for spheres with adhesion, T - ~ , and volume fraction, 7: spinodal and fluid-solid transition (-- - - ), percolation transition (-, * - ) . I Mr Penders replied: Our measurements have been performed in the metastable region of the phase diagram. It was ensured that all the samples rested in the metastable region only for a short time ( t = 2 0 s ) by moving the samples quickly from a ‘high’- temperature air bath to a ‘low’-temperature silicone oil bath in the neighbourhood of the spinodal curve (the duration of the ‘thermal pulse’ was ca.0.2 s). In this way it is possible to approach the spinodal curve very closely before phase separation takes place. During the measurements in the metastable region our samples stayed homogeneous and we observed no phase transitions like the percolation line described by Seaton and Glandt, Chiew and Glandt, Safran, Webman and Grest, and the fluid-solid transition described by Smithline and Haymet. The results shown in fig. 3 of our paper support support the fact that the experimentally determined spinodal is in reasonable agreement with the spinodal curve following from the adhesive hard-sphere model of Baxter. From these experiments performed in the metastable region there are no indications that the fluid-solid transition might transcend the transition determined by the spinodal line according to Baxter.In the unstable region the sample separates into two phases spontaneously. The question may then arise whether the dense phase is fluid-like or more glass-like. This may be temperature- and time-dependent. Further research is necessary to elucidate the behaviour of the separated phases in our silica dispersions. Mr Penders continued: In relation to the paper by Vrij and co-workers, in which the importance of solvation interactions is discussed, I wish to present some recent results of our work regarding solvation forces (M. H. G. M. Penders, A. Imhof and A. Vrij). In many theories of colloidal systems the solvent is often regarded as a continuous background.More recently, the finite size of the solvent molecules was taken into account by Hansen et al.’ and by our group.* By considering a binary mixture of large hard spheres (2) in a ‘solvent’ of small hard spheres ( 1 ) a system of colloidal particles in a solvent was simulated. Based on calculations of the osmotic compressibility of the large particles versus the volume fraction of the large particles it can be concluded that the effective repulsion between the large particles decreases owing to the presence of68 General Discussion -2.0 -‘*O- 0.0 0.2 0.4 0.6 4 Fig. 3. In 1/ S( Q = 0 ) versus c,b as a function of T~ for a binary mixture of large and small spheres (diameter ratio 160/1). The PY curve ( a ) represents the In l/S(Q = 0) versus 4 plot for monodis- perse hard spheres dispersed in a solvent which is regarded as a continuous background.T], = 00 ( b ) , 2.0 (c), 1.0 ( d ) , 0.6 ( e ) and 0.5 (f). small solvent particles. The use of such a hard-sphere solvent is not so realistic because attractive forces are totally neglected. The next step is to introduce one adhesive parameter to mimic, in an approximate way, the influence of attractive forces on the effective interactions between the colloidal particles. This will be accomplished by using the Baxter theory of adhesive hard spheres3 as shown by B a r b ~ y . ~ By introducing an attraction between two solvent particles, a ‘poor’ solvent can be modelled ( T , ~ < 00; T~~ = T~~ = 00; the reciprocal value of T , , ~ measures the adhesive strength between Q and p ) .The influence of T~~ on the compressibility [or S( Q = 0), the structure factor at zero wavevector, which is proportional to the compressibility] as a function of the volume fraction, 4, of the colloid particles (2) is presented in fig. 3 and 4. The volume fraction of the solvent particles ( 1 ) is 41 = @;( 1 - 4 ) with 4: = 0.4. From these figures it can be seen that the initial slope of the In [ 1/S( Q = O)] versus 4 plot decreases as TI1 decreases. This slope is proportional to the second virial coefficient, BZ, and is negative when T~~ < 0.5. At lower T~~ values ( T l l S 0.3) there exists a volume fraction, 4 (see fig. 4), at which the compressibility goes to infinity and phase separation may occur.A ‘good’ solvent can be modelled by increasing the attraction between small (solvent) and large (colloidal) particles ( T~~ < a; T~~ = T~~ = 00). The influence of T~~ on S ( = 0 ) as a function of a 4 is shown in fig. 5 . The initial slope of the In [ 1/S( Q = O)] versus 4 plot increases as T~~ decreases indicating that the effective repulsion between two large particles increases. At T~~ = 150 the plot matches the Percus-Yevick curve of hard spheres, where the solvent was treated as a continuous background. At lower T~~ values there is a larger effective repulsion compared to the PY curve. It can be concluded that the solvent quality can be altered by varying the interactions between small and large particles (simulating a good solvent) or varying the interaction between two solvent particles (simulating a poor solvent).General Discussion 69 0.0 n 0 -5.0 II 0 w v \ CI - c -10.0 -15.0 -20.0 ' I I 1 I I I 0.0 0.2 0.4 0.6 4 Fig.4. In l / S ( Q = 0) versus 4 at low T~~ values for a binary mixture of large and small spheres (diameter ratio 160/1). ( a ) PY, T~~ = 0.4 ( b ) , 0.3 ( c ) , 0.25 (d) and 0.20 (e). 4 Fig. 5. In 1/S( Q = 0) versus 4 as a function of T~~ for a binary mixture of large and small spheres (diameter ratio 160/1). (c) PY, 712 =20 ( a ) , 50 ( b ) , 200 (d), 500 (e) and co (f). Recently some structure-factor calculations have been performed. The results of these will be given in a following publication. 1 T. Biben and J. P. Hansen, Europhys. Lett., 1990, 12, 347. 2 A. Vrij, J. W. Jansen, J.K. G. Dhont, C. Pathmamanoharan, M. M. Kops-Werkhoven and H. M. Fijnautj, Faraday Discuss. Chem. Soc., 1983, 76, 19. 3 R. J. Baxter, J. Chem. Phys., 1968, 49, 2770. 4 B. Barboy, Chem. Phys., 1975, 11, 357.70 General Discussion Fig. 6. Model for two interacting core-5hell particles. A, and A, are the Hamaker constants for the core and the medium, respectively; A, is the mean Hamaker constant for the composite sheath. Dr Vincent said: The authors have chosen the Flory-Krigbaum segment-mixing model to account for the attractive interactions in poor solvency conditions between C1,-coated silica particles. I think this is unrealistic because the packing density of the C18 chains is too high to permit significant overlap. Moreover, I agree with Prof. Russel that an interaction requiring sheath overlap is unlikely to lead to vapour-liquid-type colloidal phase transitions, as reported.In an earlier paper’ we demonstrated that such phase transitions in these types of system could be interpreted in terms of longer-range van der Waals interactions between (non-contacting) particles. The segment-solvent interactions still play a significant role (in terms of the Flory x parameter) as shown in the following derivation* for the long-range van der Waals interaction ( VA) between two core-shell particles, as illustrated in fig. 6. As b --+ 0, the first term in the square brackets dominates. Also, designating A, and A, a? the Hamaker constants for the ‘polymer’ and solvent, respectively, A2 = A,,’ and A3 = [ + ( 1 - #,)A, 1: where 4, is the average segment volume fraction in the sheath.Hence, 1/2 2 Using a lattice model for the shell, and assuming only van der Waals forces between segments and solvent molecules (ie. no ‘associative’ bonds such as H bonds), then it may readily be shown2 that where z is the lattice coordination number ( z = 6 for a cubic lattice), and xu is the enthalpic part of x. Thus, For athermal mixtures (xu = 0) and/or &-+ 0, V, -+ 0, as expected. However, for x -xu = 0.5 and & --+ 1, as in the present case, VA( b) is significant. For example, for a = 3 0 n m a n d z = 6 , VA-13k,Tat b=0.3nmand 1.3kBTat b=3nm. 1 J. Edwards, D. H. Everett, T. O’Sullivan, I . Pangalon and B. Vincent, J. Chem. Soc., Faraday Trans. I , 2 B. Vincent, J. Colloid Interface Sci., 1973, 42, 262.1984, 80, 2599. Mr Penders replied: What we think in physical terms about the interactions between our silica particles in a solvent is carefully described in our paper.General Discussion 71 As a detailed understanding of the local liquid structure between particle surfaces is lacking, it is not possible at the moment to write down an expression for the effective pair potential between the particles from first principles. (Very recently, however, we made an attempt to take a first step in this direction: see the discussion contribution of Penders and Vrij. There it is shown, from first principles but with an approximate model, that colloidal particles that do not show direct interactions can, nevertheless, show indirect attractions, by the presence of a solvent consisting of attractive molecules.) As a compromise we have chosen a square well with a well depth that contains elements of an entropic and of an energetic nature, which balance each other at a temperature T = 8, i.e.V(r, T ) / k , T = - L ( r ) = 0; T > 8 We think that it will be possible to express the results of any detailed model in this form, when it is used near the temperature where the contribution of the surface layers to the second virial coefficient is zero, or 1; r2{ 1 - exp [ - V ( r, T ) / k , TI} dr = 0. For T = 8, we expect that the exponent in the integral will be proportional to [( 8/ T) - 13. This is corroborated by the experimental results shown in fig. 2 of our paper, which show an ‘exponential upswing’. The use of the expression in our paper for V ( r ) does not imply the use of the lattice model for polymer segments in solution.Also Flory and Krigbaum discuss that such a direct connection is not necessary, and they introduce instead of the lattice parameters jy and 1/2, the auxiliary parameters K and +, with 8 = TK/+. We prefer not to invoke macroscopic van der Waals-London forces at this level. It is not permissible, we believe, to apply these equations at microscopic distances of the order of 1 A. Further, contributions of solvation forces are neglected, as these are unrealistic for such small distances between the particle surfaces. Of course, for distances larger than the range of local solvation forces, one may use macroscopic van der Waals-London forces but the effects of these will be negligible for the small particles used here, which have only small differences in refractive index between the particle core and solvent.Dr Bartlett said: Could Dr Dhont comment on the structure of the crystalline phase formed by TPM-stabilised silica? His system might be expected to show a rather interesting crystal structure intermediate between f.c.c., the stable high-density solid phase for long-range (screened Coulombic) potentials,’ and the random stacked structure observed2 in sterically stabilised colloids where the potential is very short-ranged. 1 E. B. Sirota, H. D. Ou-Yang, S. K. Sinha, P. M. Chaikin, J. D. Axe and Y. Fujii, Phys. Rev. Left., 1989, 2 P. N. Pusey, W. van Megen, P. Bartlett, B. J. Ackerson, J. G. Rarity and S . M. Underwood, Phys.Rev. 62, 1524. Lett., 1989, 63, 2153. Dr Dhont replied: The scattering curves in the paper exhibit only the (111) Bragg peak of an f.c.c. and/or h.c.p. structure, owing to the fact that the form factor of the particles is extremely small beyond that peak. The particles are refractive index matched up to the third decimal point. Under slightly different conditions, two additional Bragg peaks are found, which are located at the (200) and (220) positions of an f.c.c. structure. The (200) peak is much less intense than the (220) peak, indicating that the structure72 General Discussion is some random mixture of f.c.c./h.c.p.-wise stacked hexagonally packed planes. The volume fraction at which the system crystallises (ca. 0.18) is still too large for a b.c.c.structure to be more stable than f.c.c./ h.c.p. structures. Prof. Cevc asked: In your text you mention that ultrasonication does not destroy the nuclei for your phase transitions whereas vortex mixing is devastating in this respect. Can you explain this, at first glance, counter-intuitive observation? Dr Dhont replied: An explanation would be that the disturbances created by the ultrasonic sound waves are of an amplitude and wavelength that matches with number- density fluctuations leading to the formation of crystallites. The sound waves could then be considered as an external force field that helps the system to find its way to the crystalline state. Dr Vincent turned to Dr Tadros: With regard to the depletion experiments, you indicate much better agreement between the values of Esep (calculated from T~ values) and values estimated from FSV theory, in the cases where HEC is the added polymer compared to when PEO is added.Since the polystyrene particles you used had grafted PEO chains on them, it would be more appropriate to use the soft-sphere model developed by ourselves,’ rather than the hard-sphere model implicit in the FSV theory. The soft-sphere model predicts weaker depletion interactions, especially in the case where the grafted and free polymers are of the same type, owing to some interpenetration of the two. However, when the two polymers are incompatible (e.g. PEO and HEC), the depletion interaction is stronger and closer to the hard-sphere case. This may account for the discrepancies you observe.1 A. Jones and B. Vincent, Colloids Surf.‘, 1989, 42, 113. Dr Th. F. Tadros ( I C I Jealotts Hill, Bracknell) replied: I agree with you that for our system with grafted PEO, we should take into consideration the interaction between free PEO and the grafted PEO chains. This, as I mentioned, explains the discrepancy between Esep calculated from T~ and Gdep calculated using FSV theory. As you men- tioned, HEC is probably incompatible with PEO and hence better agreement is obtained in this case using the hard-sphere theory. I am still not certain, however, that introducing the ‘softness’ of interaction suggested by Jones and Vincent will be sufficient to bring the experimental Esep value into line with theory. As you observe in table 2 of our paper, the discrepancy is very large and hence agreement with a modified theory is unlikely. Prof.Zukoski remarked: The G, data for charge-stabilised spheres are not in keeping with a lot of data published on charge-stabilised suspensions where the scaling theories of Goodwin et al. have been shown to be very accurate. I am concerned about the authors’ statement that there are no models accurate for G,. Dr Tadros responded: The agreement between experimental G, values and those calculated using the scaling theories of Goodwin et al., was obtained for relatively small latices. In this case, regular packing occurs as indicated by iridescence. In our system, the particles were larger (700 nm radius) and regular packing may not occur. In addition, with large particles contributions from van der Waals attractions at high 4 values may be considered.It is not, therefore, surprising that disagreement between GLxp and G&,eor is obtained. We are currently studying a latex system consisting of smaller particles to see whether the agreement can be restored.General Discussion 7 3 Prof. Zukoski then said: This is a comment referring to the use of strain sweeps to determine characteristic energies of strongly interacting systems. We have studied ordered charge-stabilised latex suspensions. Strain sweeps show plateau storage moduli up to strains of ca.0.04. For larger strains, the storage moduli decay. Defining a characteristic strain, yy, as the start of the decay we find that the product of Go and yy gives a measure of the static yield stress, as characterised by a plateau stress seen at low shear rates, and the stress at which the viscosity diverges, as determined from constant stress measurements. We find that these three measurements give the same yield stress over a wide volume-fraction range.Dr Tadros replied: Your analysis is quite interesting and would suggest a possibility of calculating the yield stress from dynamic measurements. We have many systems in our laboratories on which both dynamic and constant stress measurements can be made. We will certainly check whether such agreement is observed. Prof. Russel addressed Dr Tadros: Goodwin and Russel et al. have successfully correlated a rather substantial set a data for G’ with charged latices over a broad range of ionic strengths and volume fractions using the model represented by eqn (12) of your paper.Your data fall in a similar range of dimensionless separation (see fig. 7) but do not compare well with the rest. The reason might lie as easily with the strain amplitude and frequency dependence of your data as with the correlation. The data in fig. 2 of your paper are limited and perplexingly scattered. The correlation based on eqn (12) is demonstrated to be valid, with the maximum packing fraction as an adjustable parameter, by data from Mewis and co-workers yet to be published, but not by yours. With the weakly flocculated dispersions you do not demonstrate that: ( a ) the data is reproducible and free of history dependence; ( b ) the extrapolation to T~ is accurate; (c) T~ is linear in 4 as required by eqn (16) of your paper; or ( d ) the flocs break down to individual particles.Hence the calculations of Esep/ kB T have little credibility. For strongly flocculated dispersions the power-law dependence of G’ on 4 is commonly observed and the exponent is recognised to depend on the structure. However, I know of no evidence that n increases with the strength of attraction as asserted. So with the current state of theory, I suggest that rheological measurements can be reliably inverted to yield the interparticle potential only for closely packed particles. Dr Tadros replied: As I mentioned in the paper, we obtained only a small linear region at very low strains. This is due to the large particle size of our latex system. This, as you suggested, may explain the discrepancy between Gd,, and G:heor.We are currently carrying out experiments using latices of various sizes in order to establish the reason behind this discrepancy. I agree with you that the data shown in fig. 2 of our paper are limited. However, we have recently obtained more results using latices of various sizes. The general trend is still the same. Clearly if one uses the maximum packing fraction as an adjustable parameter, better agreement may be obtained. In response to your third point, ( a ) the data are certainly reproducible and free of history dependence; ( b ) the extrapolation to T~ is accurate since we obtained straight lines above jcrit; ( c ) and ( d ) the weakly flocculated structure is broken to individual particles as evidenced by the low qpl obtained which is very close to that in the absence of added free polymer.I agree with you that the power dependence of G’ on 4’ is related to the structure of the flocculated system. I have perhaps overstated in our paper that n depends on floc strength. This may not be the case and we are currently checking more systems.74 General Discussion 10' h E -8 lo-' * o o o o 0 * * * * * A I A A I 0 I 8' +. - lo-* * * ! lo-' loo 10' l o 2 4red Fig. 7. Correlation of static shear modulus for polystyrene latices in electrolyte solutions (Russel, Saville and Schowalter, Colloidal Dispersions (Cambridge University Press, Cambridge, 1989), p. 475. Finally, I agree with you that concordance between theory and experiment is obtained only for closely packed particles, which unfortunately is not the case with many practical systems.Mr A. E. Duisterwinkel (Derft University) commented: When you equate the Bingham yield stress, T ~ , to the interparticle interaction, you assume that all bonds are separated, i.e. T~ = NEsep. However, as shown by Prof. Ottewill for non-interacting particles and as assumed by Stein et al.' (for interacting particles), shear planes are formed at high shear. Thus, only 1/3 of the bonds must be separated, i.e. T~ = fNEsep. With this equation, a much better agreement between experiment and theory is found. 1 F. W. A. M. Schreuder and H. N. Stein, Rheol. Ada, 1987, 26, 45. Dr Tadros replied: I agree that shear planes can be produced at high shear rates, however, whether this occurs with weakly flocculated systems is difficult to ascertain. If one agrees with your picture then a factor of 1/3 for N would produce better agreementGeneral Discussion 75 with FSV theory for the PEO system. I will certainly consider your suggestion in future calculations. In addition, we will carry out experiments to ensure that layering occurs at high shear rates. Dr H. J. Ploehn (Texas A&M University) said: My question is primarily concerned with your interpretation of the linear viscoelastic behaviour of sterically stabilised suspensions. The dominance of the elastic modulus over the loss modulus for volume fractions greater than 0.50 is attributed to compression of the stabilising PEO chains. Is this an oversimplification of the physical situation? There is also an indirect effect: interactions of the stabilising layers hinder particle self-diffusion and increase the characteristic time for microstructural rearrangement in the suspension. When this time is shorter than the experimental time (the reciprocal of the frequency of the applied deformation), then the elastic modulus varies with frequency; when the suspension’s characteristic time is very long, then the elastic modulus becomes independent of frequency, at least for the ‘frequency window’ provided by the Bohlin VOR. Linear viscoelastic measurements of the modulus of hard-sphere silica suspensions’ and effective hard-sphere PMMA suspensions* show that the elastic modulus becomes greater than the loss modulus at particle volume fractions of ca. 0.50. This is similar to what you observe. Do your measurements indicate ‘softness’ in the interactions of the PEO layers, or can the increase in elastic modulus be ascribed simply to the increase in volume fraction beyond 0.50, with the PEO layer providing an effective hard-sphere interaction ? 1 J. C. van der Werff, C. G. de Kruif, C. Blom and J. Mellema, Phys. Rev. A, 1989, 39, 795. 2 W. J. Frith, Ph.D. Thesis (Catholic University, Leuven, 1986). Dr Tadros responded: I agree with you that the interpretation based solely on interpenetration and/ or compression of the chains is an oversimplification. As you stated, interactions of the stabilising layers hinder self-diffusion and that probably explains the lack of dependence of G‘ on frequency. Our results indicate ‘softness’ in the interactions of PEO layers, since they give large thicknesses (hydrodynamic) of the order of 10-20 nm. I would not, therefore, treat our particles as hard spheres.

ISSN:0301-7249    

DOI:10.1039/DC9909000057

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1990, 90, 77-90 Rheology of Aqueous Suspensions of Polystyrene Latex stabilized by Grafted Poly( Ethylene Oxide) Harry J. Ploehn Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, USA James W. Goodwin Department of Physical Chemistry, University of Bristol, Cantock's Close, Bristol BS8 1 TS A water-soluble carbodiimide has been used to end-graft aminated poly( ethy- lene oxide) (PEO) chemically onto colloidal polystyrene particles. Two particle sizes (115 and 347 nm diameter) and two PEO molecular weights (1 12 000 and 615 000 g mol-') were combined to give suspensions with four different ratios of polymer layer thickness to particle radius. Electrophoresis demonstrated that the PEO was grafted, not just adsorbed.Dynamic light scattering showed that the adsorbed and grafted layers had similar structures and that non-ionic surfactant perturbed the PEO configurations. Steady shear and oscillatory rheometry indicated that long-ranged polymeric forces between particles governed the variation of viscosity and storage modulus with applied stress and PS volume fraction. Hard-sphere and effective hard-sphere scaling helps rationalize the rheological behaviour in terms of the variation of the polymeric force among the different suspensions and hydrodynamic deformation of the polymer layers. Polymers are widely utilized to modify the properties of colloidal suspensions. The use of polymers to stabilize suspensions has been well documented,'32 and polymeric floccula- tion has been employed in several ii~dustries.~ Invariably, the polymeric additive also alters the bulk rheology of the colloidal suspension through at least one of three mechanisms: the polymer's influence on the state of aggregation of the suspension, more subtle alteration of the suspension microstructure via the polymer-mediated interparticle force, direct modification of the rheological properties of the suspending fluid.In some cases, the polymeric modifier is added primarily to control the bulk rheology. When the polymer is soluble in the suspending fluid, its effect on the system can be classified according to the polymer's interaction with the suspended particles: depletion, adsorp- tion and grafting modes have been studied e~tensively.~ Physical circumstances often dictate the role of the polymer and its mode of use in a given situation. In aqueous suspensions, electrostatic forces confer stability, so the polymer's role has most often been that of flocculant; through the adsorption mode, the polymer can promote aggregation by partially neutralizing the stabilizing charges and by forming bridge configurations between particles.In organic media, the dominance of the van der Waals attraction over electrostatic repulsion encourages the use of polymers as stabilizers; both adsorption and grafting modes have been employed. In some cases grafted stabilizers have an advantage: if the polymer has no affinity for the surface (relative to that of the solvent), then the possibility of bridging can be avoided. These generalizations, though, are becoming less valid as industrial needs and academic interests change. In particular, economic and environmental considerations favour less reliance on organic solvents and greater use of aqueous media.It would be 7778 Rheology of PEO-Stabilized PS Suspensions helpful to be able to use established techniques for polymeric stabilization in aqueous systems. Progress in this area requires further development of polymer stabilizer tech- nology, especially with regard to: (i) understanding the structure and behaviour of polymer layers, both in organic and aqueous colloidal suspensions; (ii) characterizing the effect of the polymer on the suspension microstructure and bulk properties; and (iii) identifying and understanding effects peculiar to aqueous systems, such as polymer- surfactant interactions, polymer electrostatic coupling, and so on.The rheology of polymer-stabilized suspensions depends on the Brownian motion of the particles, hydrodynamic interactions of the particles with the fluid and each other, and the total interparticle force. Early efforts focused on low molecular weight stabilizers of two types: block copolymers with an insoluble block, which absorbed on the growing particles and stabilized them during p~lymerization;~.~ and functionalized homopolymer, which was chemically grafted onto the particles in a post-polymerization The resultant polymer layers had high graft densities and low thicknesses, which produced a steep interparticle force that could be approximated by an effective hard-sphere interaction. Dynamic light scattering (DLS) meas~rements~ indicated that silica particles stabilized by octadecyl alcohol7 in cyclohexane interacted as hard spheres.Rheological measurements'o agreed with theoretical predictions for dilute hard spheres' and, at higher volume fractions, could be rationalized through 'Brownian hard-sphere scaling.I2 For poly( methylmethacrylate) (PMMA) particles stabilized by poly( 12-hydroxy- stearic acid), the stabilizer layer thickness could be distinguished from the particle radius.13 When the ratio of the thickness to the radius was small, the rheological displayed Brownian hard-sphere behaviour, provided the effective particle volume fraction was calculated from the particle radius plus the stabilizer thickness.However, the high- and low-shear limiting viscosities differed somewhat from those found in the silica suspensions. Hard-sphere behaviour was also observed for poly( vinyl chloride) plastisols in alkyl phthalate plasticizer^'^ and PMMA suspensions stabilized by poly( dimethylsiloxane) in low molecular weight silicone oils. l6 Deviations from hard-sphere scaling appeared" when the thickness : radius ratio became large, implying that deformation of the polymeric stabilizer was responsible. A number of s t u d i e ~ ' ~ * ' ~ - ~ ~ have characterized 'soft' long-range polymeric interactions in suspensions through rheological measurements. The details of the interaction, and hence the rheological behaviour, depended not only on the chemistry and molecular weight of the stabilizer and the thickness:radius ratio, but also on the composition of the medium its structure,16 and the temperat~re.'~ Few studies have examined the rheology of suspensions stabilized by high molecular weight polymer 19320*22 or stabilized suspensions in aqueous The objective of this work is to characterize polymer-mediated interparticle interac- tions in aqueous suspensions of polystyrene (PS) stabilized by high molecular weight poly(ethy1ene oxide) (PEO) through electrophoresis, dynamic light scattering, and rheological measurements.Utilizing a technique originally developed for grafting bio- chemical ligands onto colloidal substrates for agglutination tests,24 the PEO was chemi- cally end-grafted onto active sites on the surfaces of the PS particles in water.Elec- trophoresis measurements demonstrated that the PEO was anchored to the PS, not just adsorbed. Four samples, each having a different thickness : radius ratio, were prepared from two PEO molecular weights and two PS particle sizes. Dynamic light scattering and capillary viscometry measurements gave the hydrodynamic radius of the coated particles and confirmed that the PEO layer structure was typical of that for adsorbed polymers. 25,26 Steady shear and oscillatory rheometry provided data that reflected the influence of the polymer-mediated interparticle force on the suspension microstructure and bulk rheology ; differences between suspensions with varying degrees of particle softness were apparent.H. J. Ploehn and J. W. Goodwin 79 Experimental Materials Several suspensions of PS particles were prepared using standard emulsion polymeriza- tion techniques2’ and published recipes2* The polymerizations were initiated with 4,4’-azobis-4-cyanopentanoic acid ( Wako Chemicals GmbH) and produced particles stabilized by surface carboxyl groups.Suspension PSO was prepared without added surfactant; suspension PS1 resulted from the seeded growth of 31 nm PS particles, also without added surfactant; and suspension PS2 was prepared with surfactant, 12- dodecanoic acid. The suspensions were dialysed for more than a month against frequent changes of twice-distilled water. TEM gave particle diameters of 375, 115, and 347 nm for PSO, PS1, and PS2, respectively; a diffraction grating calibrated the measurements. Optical sizing of photographic images (> 500 per sample) yielded standard deviations in diameter of 4, 5 and 2% for PSO, PS1, and PS2, respectively. Conductometric titration of PSO resulted in a surface charge density of ca.0.5 pC crnp2. Three PEO samples were used as received: PEOO, a commercial grade sample with M , == 1.0 x lo5 g mol-’ and M,/Mn = 6.6; PEO1, a Polymer Labs standard with M , = 1.12 x lo5 g mol-’ and M,/M, = 1.03; PE02, another Polymer Labs standard with M , = 6.15 x lo5 g mol-’ and M,/M, = 1-10. All other reagents were BDH AnalaR grade. A water-soluble carbodiimide (WSC), 1-ethyl-3-( 3-dimethylaminopropyl) carbodiimide (Fluka), served as the linking agent in the grafting reaction. Monodisperse non-ionic surfactant hexaethylene glycol mono-n- dodecyl ether (Nikko Chemicals), henceforth denoted as C12E6, was used as received.Grafting Reactions To prepare the PEO for grafting, its hydroxyl end group was converted to an amine group. PEO was dissolved in warm toluene at concentrations of 0.5-5.0 x g cmP3 (depending on the molecular weight). A ten-fold molar excess of toluene-4-sulphonyl chloride and 0.5 cm3 pyridine were added, thereby replacing the PEO hydroxyl end groups with tosylate leaving groups and precipitating insoluble pyridinium hydro- chloride. Bubbling ammonia gas through the warm (303 K) stirred solution for 3-5 h gave end-aminated PEO (APEO). Most of the toluene was then evaporated off under vacuum. The APEO was dissolved in dioxane, freeze-dried to remove organic solvents, and redissolved in water at 0.5-5% by weight.Although the mechanism is rather c ~ m p l i c a t e d , ~ ~ the overall WSC grafting reaction is straightforward: WSC and APEO were added to the suspensions of carboxylated .PS particles, giving PS particles with APEO grafted through amide linkages plus a diurea byproduct. The APEO dosage, estimated from equilibrium adsorption data,26 was gauged to provide an adsorbed amount on the plateau of the adsorption isotherm; for 6.6 x lo5 g mol-’ PEO, a solution concentration of 2.0 x g cm-3 gave26 an adsorbed amount of 1.42 x lo-’ g cm-2. These values were used in conjunction with the particle diameter and volume fraction to compute the amount of APEO to be added to each suspension. To minimize bridging flocculation, the particles were added to stirred, diluted APEO solution so that the final particle volume fraction was 0.015.The sus- pensions were polymerically stabilized at this point. WSC was then added to give one molecule per 10 nm2 of particle surface, and the mixture was stirred overnight at 303 K. This procedure was followed for the following particle-polymer pairs: PS1-APEO1, PS1-APE02, PS2-APE01 and PS2-APE02. PS2-APE02 was grafted at 273 K. A slightly different procedure was followed for PSO-APEOO. N-hydroxybenzo- triazole was grafted onto PSO to form an intermediate active ester. In principle, the stability of this ester allows cleaning of the suspension, removing excess WSC which might otherwise damage the molecule to be grafted.. The benzotriazole ester is a good80 Rheology of PEO-Stabilized PS Suspensions leaving group, so the amide linkage is easily formed when APEO is added.Here, APE00 was mixed with uncleaned intermediate and stirred at 273 K. Sample Preparation Samples for electrophoretic mobility and DLS measurements were diluted by a factor of 5000 with filtered mol dm-3 sodium acetate buffer solution at pH 6.3; the pH was adjusted to other values through addition of sulphamic acid or sodium hydroxide. Other samples were diluted into buffer also containing 4.0 x Suspensions stabilized by the high molecular weight APEO, although not flocculated after centrifugation, could not easily be redispersed; sonication was not used for fear of degrading the APEO. Consequently, all samples for rheological measurements were treated with sufficient C12E6 surfactant to give monolayer coverage of the particles in equilibrium with 4 x lop4 mol dmp3 C12E6 in ~olution.~' Subsequent low-speed centrifu- gation removed any flocculated material.The samples were cleaned and concentrated through multiple cycles of centrifugation and redispersion into lop3 mol dmP3 sodium acetate and 4 x mol dmP3 C12E6. The buffer ionic strength was high enough to make the electrostatic double-layer thickness less than the APEO layer thickness, yet low enough to minimize perturbations of the APEO configurations. After centrifugation, the samples redispersed very easily. Particle volume fractions were determined by dry weight analysis, neglecting the weight of the APEO and taking 1.05 g cm-3 as the density of PS. mol dm-3 C12E6.Characterization Methods Particle electrophoretic mobility was measured with a Pen Kem System 3000 Automated Electrophoresis Apparatus. Mobility measurements were performed many times and averaged; measurements usually deviated less than 2.5% from the average. Dynamic light scattering measurements employed a Coherent mode-stabilized kryp- ton laser operating at a wavelength of 530.9 nm and a Malvern Instruments spectrometer with multibit correlator. Measured correlation functions were accurately fit by single exponential functions. The reciprocal of the decay time was plotted against the square of the scattering vector; the diffusion coefficient was extracted from the slope of the resulting line. The Stokes-Einstein equation then gave the particles' hydrodynamic radius.Capillary viscometry measurements were performed with a Ubbelohde capillary viscomerer thermostatted at 298.2 K. Typical flow times for the buffer-C12E6 solution were ca. 85 s. Measurements were repeated until the flow times agreed to within Q.1 s. All other rheological measurements utilized a Bohlin VOR rheometer fitted with one of three geometries: cone-and-plate, single-gap concentric cylinders, and double-gap concentric cylinders for high, medium, and low concentrations. Each sample was loaded into the rheometer and homogenized with shearing at 10 s-l for 2 min. The first sample, PS2-APE02, displayed drying problems at high particle concentrations. Drying was hindered by floating a small amount of silicone oil on top of subsequent samples. Excessive shear rates were avoided to prevent any mixing of oil with sample.Samples were recovered for reuse by washing with buffer-C12E6 solution. Low speed centrifuga- tion allowed removal of dried material and silicone oil. Electrophoresis measurements indicated that the particle mobility was not affected by exposure to the silicone oil. All measurements were completed within a week of the initial grafting reaction. At constant shear rate, the shear stress and viscosity reached steady-state values within 60 s of the start of the shear, and reported values reflect averages over 20-60 s. Shear stress and viscosity were reproducible for both increasing and decreasing shear-rate sweeps. Strain sweeps in oscillation (usually at 1 Hz) indicated that the storage modulusH.J. Ploehn and J. W. Goodwin 81 PSO-a-APE0 , PSO 0 1 2 3 4 5 mobility/ lo-' m2 V-' s- ' Fig. 1. Electrophoretic mobilities of PSO and PS2 particles without APEO layers (PSO, PS2), with adsorbed APEO layers (PSO-a-APEOO, PS2-a-APE02), and with grafted APEO layers (PSO-g- APEOO, PS2-g-APE02), both before (M) and after (UUII) treatment with C12E6 surfactant. was linear below 1% strain but strain-softened at higher amplitudes. The loss modulus was linear for strains up to 0.1%; at larger strains, the loss modulus first decreased, then exhibited a broad peak at ca. 5% strain. Results and Discussion Electrophoresis The steady-state velocity of a charged particle in an electric field depends on the balance between electric and viscous forces; the velocity divided by the field strength gives the electrophoretic mobility. Here, mobilities are used for comparison only; interpretation in terms of the electrostatic/polymeric environment around the particles3* is not the present concern.The mobility of PSO particles in mol dmW3 buffer varied from ca. - 4 ~ lo-* m' V-' s-l at pH 5.0 to -6 x lo-* m2 V-' s-' at pH 8.0. In the same buffer, PSO particles covered with adsorbed APEOO or grafted APEOO had nearly equal mobilities of ca. - 1 x lo--* m2 V-' s-', nearly independent of pH for values between 5 and 8. Since the mobility is inversely proportional to the effective particle radius, the hydrodynamic thickness of the APEO layers cannot account for the decrease in the magnitude of the mobility; disturbance of the electrostatic double layer by the APEO must be responsible.This phenomenon can be used to ascertain whether the APEO is grafted or merely adsorbed. In a displacement experiment, a displacer supplants adsorbed APEO, but grafted APEO remains near the surface. The particle mobilities should reflect this difference. The results of this test are shown in fig. 1; all mobilities were measured in mol dm-3 buffer. Samples of bare particles (PSO, PS2), particles with adsorbed APEO ( PSO-a-APEOO, PS2-a-APE02), and particles with grafted APEO ( PSO-g-APEOO, PS2-g-APE02) were treated with excess of C 12E6 surfactant. Before treatment, the mobilities of the adsorbed and grafted samples were comparable and relatively low. After treatment, the mobilities of the adsorbed samples rose to roughly that of treated bare particles, while the mobilities of the grafted samples rose slightly but remained low.The former result implies that the adsorbed APEO had been displaced, while the82 Rheology of PEO-Stabilized PS Suspensions Table 1. Dynamic light scattering results hydrodynamic layer thickness/nm PS- APEO preparation untreated added C12E6 change PS 1 -APE02 adsorbed 42 35 -7 PS 1 -APE02 grafted 54 25 - 29 PS2-APE02 grafted (273 K) 86 58 -28 PS2-APE01 adsorbed 9 12 +3 PS2-APE01 grafted 24 44 +20 latter suggests that grafted APEO was indeed grafted. The other possible factor, exposure to the WSC, should have been apparent in the mobilities before surfactant treatment. In addition, attempts were made to adsorb and to graft APEOO onto PSO which had been pretreated with C12E6.In each case, the mobility after exposure to APEOO equalled the initial mobility of bare PSO. Thus C12E6 prevented both adsorption and grafting of APEOO. Dynamic Light Scattering Measurements of the diffusivities of bare PS1 and PS2 led to hydrodynamic diameters of 119 and 362 nm, respectively; both values are within 5% of those determined through TEM. The difference between the hydrodynamic radii of APEO-coated and bare particles gave the hydrodynamic thickness of the APEO layer. Table 1 summarizes the results. For the smaller PS1 particles, grafting of APE02 produced somewhat thicker layers than adsorption alone; both values are lower than the 95-100 nm thicknesses for 6.6 x lo5 g mol-’ PEO adsorption onto PS reported previously,25926 Grafted APE02 on the PS2 was thicker than on PS1; perhaps the larger particle size or the different grafting temperature (used for this experiment only) were responsible. Adsorbed APEOl layers on PS2 were about 50% thinner than reported values,25326 but the grafted APEOl layers on PS2 were significantly thicker than the adsorbed layers.The effect of C12E6 surfactant on the hydrodynamic thickness of the APEO layers is not clear. C12E6 decreased the apparent thickness of APE02 layers, but the reduction was greater for grafted layers. Layers of APEOl appeared to increase in thickness after exposure to surfactant. While rationalization of these results is difficult, it is probably safe to say that the C12E6 did not completely displace adsorbed APEO, and that the C 12E6 significantly perturbs the configurations of APEO molecules remaining near the PS surface. Capillary Viscometry Saunders’ modification of Mooney’s equation32 relates the relative viscosity vr to the particle volume fraction q with the Einstein coefficient k, = 2.5 and cp, as the maximum packing fraction.For each sample, plots of the capillary viscometry data according to eqn (1) were linear. The intercept provided the effective hard-sphere scaling factor, f, defined throughH. J. Ploehn and J. W. Goodwin 83 Table 2. Capillary viscometry results PS-APE0 f a / S S/nm PS 1 -APE0 1 1.206 15.5 3.7 PS 1 -APE02 1.805 4.6 12.5 PS2-APE01 1.118 26.3 6.6 PS2-APE02 1.397 8.5 20.5 0.01 0.1 1 10 100 shear stress/Pa Fig. 2. Viscosity of the PS1-APE02 suspension as a function of shear stress and particle volume fraction.Particle volume fraction: 0, 0.254; 14, 0.269; A, 0.283; +, 0.295; 0, 0.348; 0, 0.369; x,0.400; 0, 0.433. where cpe is the effective hard-sphere volume fraction of the particle plus polymer layer, 6 is the layer thickness, and a is the particle radius. These results are presented in table 2. Large values off or small values of a / S indicate ‘softness’ in the interparticle force, that is less steepness in the force as a function of separation. For a particular particle size, softness increased with the molecular weight of the grafted polymer; for the same grafted polymer, softness decreased with particle size. The capillary viscometry data represent high shear, low cp limiting values of the hydrodynamic layer thickness, in contrast to the low-shear, low cp values provided by DLS.Steady-shear Viscometry For the four systems listed in table 2, the steady-shear viscosity was found to be a power-law function of shear rate at each particle volume fraction. Fig. 2 shows the viscosity of PS1-APE02 as a function of shear stress and volume fraction. The general features of the data are typical for all of the systems studied. At low 4, the viscosity reached limiting values at low stress but displayed shear thinning at higher stresses. Viscosities of suspensions with low volume fractions of larger PS2 particles approached limiting values at the highest measured stresses. At all stresses, viscosity increased with 4. Above a critical volume fraction ( 4 = 0.28 for PS1 suspensions, 0 = 0.37 for PS2 suspensions), the viscosity curves exhibited apparent yield stresses.PS2 suspension were iridescent (at rest) for values of 4 above the transition value.84 Rheology of PEO-Stabilized PS Suspensions 0.25 0.3 0.35 0.4 0.45 volume fraction Fig. 3. Relative viscosities of PS-APE0 suspensions as functions of volume fraction at two relative stresses: solid curves, ur = 0.0127; dashed curves, ur = 0.127. 0, PSI-APEO1; 0, PS1-APE02; E, PS2-APEOl; e, PS2-APE02. The changes in the steady shear rheology with increasing 4 characterize a transition from ‘liquid-like’ to ‘solid-like’ behaviour as the mean particle separation becomes comparable to the characteristic length of the interparticle force. The mean particle separation, h, scales as Thus h =: 35-40 nm and h =: 65-70 nm for the PS1 and PS2 suspensions, respectively, near the transition values of 4.At low shear rates, the microstructure depends on the balance between Brownian and interparticle forces: Brownian motion randomizes the microstructure of weakly interacting particles, giving liquid-like behaviour, but strong interactions promote short-range order and hinder particle self-diffusion, producing solid-like behaviour. Since the PS1 particles have a much higher diffusivity than the PS2 particles, the interparticle force must be greater in PS1 suspensions than in PS2 suspensions at the transition value of 4. Thus the mean separation at the transition should be smaller in the former. However, the liquid-solid transition is not sharp, and characterization of any true yield stress is problematic, so it is difficult to distinguish the difference between APE01 and APE02 layer interactions in this way.Data at larger stresses provide a better comparison of polymeric interparticle forces. Fig. 3 shows the relative viscosity of each system as a function of volume fraction for two values of the relative stress defined by ur = vu3/ kBT ( a is the dimensional shear stress). I f the particles were Brownian hard spheres, dimensional analysis” indicates that the four curves for each value of the relative stress (solid curves for a,=0.0127, dashed curves for (T, = O.127) should superimpose. Deviations from this limit reflect details of the interparticle force. At each relative stress, the viscosities of suspensions of smaller PS1 particles (open symbols) were greater than those of suspensions of larger PS2 particles at the sameH.J. Ploehn and J. W. Goodwin 85 volume fraction. Eqn (3) shows that the mean separation of PS1 particles was about one-third of that of PS2 particles at the same #. If PS1 and PS2 particles had force- separation curves with similar magnitudes, then the smaller mean separation of PS1 particles implies stronger interactions, thereby yielding the greater viscosities. Focusing on the solid curves for (T, = 0.0127, PS1-APE01 suspensions (open squares) had higher viscosities than PS1-APE02 suspensions (open circles), suggesting that the force between APEOl layers was greater than that between APE02 layers at the same separation. For PS2 particles, the viscosities of APEOl stabilized suspensions (solid squares) became greater than that of APE02 stabilized suspensions (solid circles) only for # > 0.34.Comparison of the pairs of curves for PS1 and PS2 imply that the interparticle force differs between small and large particles for at least one of the APEO coatings. Consider the dashed curves in fig. 3 for the higher relative stress, a,=O.127. Sus- pensions of larger PS2 particles (solid symbols) stabilized by APEOl and APE02 had comparable viscosities at all volume fractions. In this case, domination of the hydrody- namic force over the interparticle force may have rendered the viscosity less sensitive to the latter. If the polymeric forces were greater between the smaller PS1 particles for a given #, then at the same relative stress, the hydrodynamic force would be less important, and so the viscosity curves might vary with the polymeric force.This appears to be the case, because the viscosities of PS1-APE01 suspensions became greater than those of the PS1-APE02 suspensions as # increased. Notice the qualitative similarity between the shapes of the PS1 viscosity curves at a,=0.127 and those for PS2 at (T, = 0.0127 (solid symbols, solid curves). The 'crossover' occurred at a low stress for PS2 suspensions, but at higher stress for PS1 suspensions. In the limit of high shear stress, the hydrodynamic force dominates the polymeric force, and the polymer layers are significantly distorted from their equilibrium configura- tions. Capillary viscometry provides data in this limit (table 2).Plotting the data according to eqn (1) identifies the effective hard-sphere radius a + S and the scaling factor f [eqn (2)J In principle, the relative viscosity of an effective hard-sphere suspension is only a function of the effective volume fraction 4Se=f# and the effective relative stress =fa,. The relative viscosities of the four suspensions considered here should be identical functions of 4, at the same or,,, provided we neglect the variation of the polymeric force with a,,e. Deviations from effective hard-sphere behaviour then indicate deformation of the APEO layers by hydrodynamic forces. Assuming that the APEO layers suffered the maximum deformation in the capillary viscometer flow, the steady-shear viscosity curves would superimpose only at high values of In earlier ~ o r k , ' ~ ~ ' " ~ ' ~ the Krieger-Dougherty equation12 has successfully fit relative viscosity data for 4,<0.55.Using the value 4,=0.71 as indicated for hard spheres in the high-shear limit,'" this function is plotted in fig. 4. The experimental data, at constant but much lower a,,, had higher relative viscosities as a consequence of shear thinning. The solid curves in fig. 4 are for 0,,,=0.0142. Unlike the scaling in fig. 3, which brought curves for the same particle size into proximity, the effective hard-sphere scaling in fig. 4 brings curves for the same APEO molecular weight into proximity. The viscosity curves for APEOl-stabilized suspensions (squares) have the same general shape but do not superimpose. However, curves for APE02-stabilized suspensions nearly coincide.The better agreement of the viscosities of PS1-APE02 and PS2-APE02 suspensions suggests that these were closer to effective hard-sphere suspensions than those stabilized by APEO1. Two explanations are possible. First, the hydrodynamic force could have been stronger than the polymeric force between APE02 layers, thus making the particle more like effective hard spheres. However, the lack of superposition of the curves in86 Rheology of PEO-Stabilized PS Suspensions lo4 lo2 T T P 0 / 1 / / / d ./ 0 ./ 0 1 1 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 effective volume fraction Fig. 4. Relative viscosities of PS-APE0 suspensions as functions of effective volume fraction at two effective relative stresses: solid curves, u, = 0.0142; dashed curves, ur = 0.142.0, PS1-APEO1; 0, PS1-APE02; W, PS2-APEO1; a, PS2-APE02. The dotted curve is Krieger’s equation [eqn (4)] with 4m = 0.71. fig. 3 for or = 0.0127 (differing from only by the factorf) implies that hydrodynamic forces were not dominant. The second possible explanation is that APEOl and APE02 layers exhibited different degrees of compression at the same effective stress; the reasoning here is based on the presumption that the effective hard-sphere scaling used in fig. 4 is associated with maximum layer compression at high stresses. Relative to this limit, the better superposi- tion of the viscosity curves of APE02 suspensions suggests that APE02 layers were more deformed (compressed) than APEOl layers at the same CT,,,. Effective hard-sphere scaling seems to improve at a higher stress, ~,,,=0.142 (dashed curves in fig.4): the viscosities of APE02-stabilized suspensions (circles) agreed over a larger range of c $ ~ , and the viscosity curves for APEOl -stabilized suspensions (squares) moved closer toget her. Oscillatory Measurements Fig. 5 depicts the storage modulus of the PS1-APE02 suspension as a function of frequency ( w ) and volume fraction. The corresponding plots for the other suspensions had the same qualitative features. At low 4, the particles were well separated and the polymeric interactions were weak; thus the storage modulus was relatively small. At low frequency, the characteristic time for rearrangement of the suspension microstructure was much less than that of the oscillatory deformation, so the suspensions dissipated energy through particle diffusion rather than storing energy in polymeric interactions.At high frequency, though, the oscillation was faster than some of the diffusive modes of dissipation, so the storage modulus increased with frequency. Such behaviour characterizes liquid-like particle microstructure; however, the slope of the log ( G’)- log ( w ) plot was not 2 as expected for a true viscoelastic liquid. Even at volume fractions as low as 0.25, there must have been significant polymeric interactions between particles.H. J. Ploehn and J. W. Goodwin 87 0 M 2 Y v1 0.01 I I I , 1 1 1 1 1 I I , I , , , , I , I , , , 1 1 1 I 0.01 0.1 1 10 frequency/ Hz Fig. 5. Storage modulus of the PS1-APE02 suspension as a function of oscillation frequency and particle volume fraction.Particle volume fraction: *, 0.254; ., 0.269; 0, 0.283; A, 0.295; +, 0.318; A, 0.348; X, 0.369; 0, 0.400; 0, 0.433; 0, 0.496. The storage modulus increased with volume fraction as interactions became stronger and more numerous. The increased interactions hindered particle self-diffusion so that the spectrum of timescales for microstructural rearrangement shifted to longer times. The behaviour at high 4 was typical of a viscoelastic solid: the oscillation time was much faster than all diffusive relaxation processes, leaving the storage modulus indepen- dent of frequency. The transition from liquid-like to solid-like viscoelastic behaviour occurred at about the same critical volume fraction as was observed in the viscosity-shear stress results in fig.2. This agreement was found for each suspension. Dimensionless scaling of oscillatory data13 defines the relative storage modulus G’a3/ kB T and the relative frequency oa377,,,/ kB T with qm as the viscosity of the medium. The relative storage modulus of Brownian hard spheres thus depends only on the relative frequency and the volume fraction. Fig. 6 shows the relative storage modulus as a function of volume fraction for all of the suspensions at a constant relative frequency of 4.15 x The relative storage modulus of each suspension rose steeply at about the same volume fraction found for the viscoelastic liquid-solid transition in the steady shear data. For 4 < 0.40, the curves clearly do not superimpose, presumably due to differing polymeric interactions between particles.The relative moduli of suspensions of smaller PS1 particles were greater than those of larger PS2 particles. Again, this is a consequence of the smaller mean separations and stronger interactions of PS1 particles compared to PS2 particles at the same 4. For each particle size, the relative modulus of APEOl-stabilized suspensions (squares) was greater than that for those stabilized by APE02 (circles). This observation implies that the force between APE01 layers was greater than that between APE02 layers at the same separation. The viscosity data in fig. 3 (solid curves) and the modulus data in fig. 6 are consistent in support of this hypothesis. For 4 > 0.40, the relative storage modulus increased roughly exponentially with volume fraction.The moduli of PS1-APE02 and PS2-APE02 suspensions were par- ticularly close. As the mean separation became smaller, the particles interacted more strongly, increasing the moduli. If the polymeric force-separation functions in the two suspensions varied in a similar way over that range of separations, then the storage moduli would increase at the same rate. Alternatively, if the polymeric forces were steep enough, then the particles might have been acting as effective hard spheres; in88 Rheology of PEO-Stabilized 3 Q ; 0) Do 2 Y m 6, PS Suspensions -a 100 .- U - 0.01 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 m ~ 1 ’ 1 1 ~ 1 1 1 1 ~ 1 1 1 1 ~ 1 1 1 1 ~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ volume fraction Fig. 6. Relative storage moduli of PS-APE0 suspensions as functions of volume fraction at a relative frequency of 4.15 x lop5.0, PS 1 -APE01 ; 0, PS 1 -APE02; a, PS2-APE0 1 ; a, PS2-APE02. this case the moduli would depend only weakly on the details of the force-separation functions. The relative storage modulus of the PS2-APE0 1 suspension increased more rapidly than the other two suspensions in this volume fraction range. PS2-APE01 particles were arguably the ‘hardest’ according to the capillary viscosity data in table 2. These observations can be rationalized if the force between APE01 layers was a steeper function of separation than that between APE02 layers. Conclusions The rheological behaviour of PS suspensions stabilized by high molecular weight grafted APEO depends intimately upon the details of the grafting process, the resultant polymer layer structure, and the variation of the polymeric interparticle force with particle separation.The same can be said for suspensions stabilized by low molecular weight species, such as silica-octadecyl alcoh01~-~~ or PMMA-poly(hydroxystearic acid),I3 but the contrast between these ‘effective hard sphere’ suspensions and those studied here is noteworthy. The differences arise ultimately from the nature of the grafting process. Low molecular weight stabilizers usually are not adsorbed onto the particle substrate; instead, grafting is accomplished through extreme reaction conditions,” incorporation of the stabilizer during particle polymerization,l” or by strong adsorption of an insoluble block of a block Effective stabilization is achieved by maximizing the graft density, thereby forming a thin but dense stabilizer layer with minimum compressibility.These characteristics contribute to the success of effective hard-sphere scaling in ration- alizing the rheological behaviour. High molecular weight APEO, on the other hand, is readily adsorbed onto PS particles. The graft density is probably controlled by the adsorption behaviour; although reasonable, this conjecture has not been tested experimentally. The electrophoresis and DLS data given above imply that grafted and adsorbed APEO layers have similar structure, at least before treatment with C12E6 surfactant. Many studies 172y4 have shown that such polymer layers are diffuse and have density profiles that gradually decay withH.J. Ploehn and J. W. Goodwin 89 increasing distance from the particle surface. Two distinct implications for suspension rheology should be considered: the polymeric force gradually increases in magnitude over a long range as particle separation decreases; and the extended layers may be more susceptible to deformation by hydrodynamic forces. Certainly, at least the first effect is manifested in the rheological data presented in fig. 2-6. Relating the rheological behaviour to the details of the APEO layer structures is the prime objective, but the task is complicated by several factors. The main deficiency in the present study is the lack of data on grafted or adsorbed amounts of APEO per unit area of PS particle surface. Modelling studies4 have shown that adsorbed amount is the key factor linking polymer layer structure to polymeric forces.Independent measure- ment of adsorbed/grafted amounts would help clarify the details of the grafting mechan- ism (for example, the dependence of grafting efficiency on adsorbed amount) as well as assist in rationalizing the rheology. In particular, some of the differences among the viscosity curves in fig. 3 for a given APEO could be attributed to different graft densities. PS 1 was prepared without surfactant, while the PS2 polymerization included dodecanoic acid. Dialysis probably did not completely remove the surfactant from the PS2. If APEO adsorption and grafting were hindered by the surfactant, then the polymeric force might be less between large particles than between small particles for the same APEO stabilizer.Two other factors complicate the interpretation of the data presented here and merit further study. First, the behaviour of the suspensions during centrifugation and redisper- sion, and the electrophoresis and DLS measurements, indicated that the C 12E6 surfactant had a profound effect on the properties of the APEO-stabilized suspensions. The C12E6 could have associated with the APEO through a group-specific chemical interaction such as hydrogen bonding, or a less-specific physical interaction may have been respon- sible. Any further analysis of the macroscopic properties of the suspension that relies upon models of polymer layer structure must take the surfactant into account. A better understanding of such polymer-surfactant interactions would have an impact upon the performance of many materials, including paints, inks, oil recovery fluids, and consumer products.Secondly, electrostatic effects were generally ignored. This was probably a good approximation considering the low mobility of the coated particles and the ionic strength of the buffer. Even so, there may have been some dependence of APEO adsorption, grafting efficiency, or layer structure on the electrostatic environment around the PS particles. Coupling between the polymer layer and electrostatic double-layer structures becomes more important at very high and very low ionic strengths: in the former case, the finite volume occupied by the ions perturbs the polymer configurations, while in the latter, the electrostatic force has a range comparable to or greater than the polymeric force.Indeed, the polymeric and electrostatic components of the total interparticle force are more likely to be coupled than separable and additive. These considerations are vital for technologies involving interactions mediated by polyelectrolytes, enzymes, proteins, or other biological materials. H.J.P. gratefully acknowledges the financial support of the U.S. National Science Foundation under grant INT-8806327, and thanks all at the University of Bristol for their hospitality during his stay there. H.J.P. and J.W.G. thank Mr John Dimery and Dr Keith Ryan for their technical support and advice. References 1 D. H. Napper, Polymeric Stabilization of Colloidal Dispersions (Academic, New York, 1983).2 T. Sato and R. Ruch, Stabilization of Colloidal Dispersions by Polymer Adsorption (Marcel Dekker, New York, 1980). 3 Flocculation and Dewatering, ed. B. M. Moudgil and B. J. Scheiner (AIChE, New York, 1988).90 Rheology of PEO-Stabilized PS Suspensions 4 H. J. Ploehn and W. B. Russel, Adu. Chem. Eng., 1990, 15, 137. 5 Dispersion Polymerization in Organic Media, ed. K. E. J. Barnett (Wiley, New York, 1975). 6 R. J. R. Cairns, R. H. Ottewill, D. W. J. Osmond and I. Wagstaff, J. Colloid Interface Sci., 1976, 54, 45. 7 A. K. van Helden, J. W. Jansen and A. Vrij, J. Colloid Interface Sci., 1981, 81, 354. 8 K. Ryan, Chem. Ind., 6 June 1988. 9 A. Vrij, J. W. Jansen, J. K. G. Dhont, C. Pathmamanoharan, M. M. KopsWerkhoven and H. M. Fijnaut, Faraday Discuss.Chem. Soc., 1983, 76, 19. 10 C. G. de Kruif, E. M. F. van Iersel, A. Vrij and W. B. Russell, J. Chem. Phys., 1985, 83, 4717; J. Mellema, C. G. de Kruif, C. Blom and A. Vrij, Rheol. Acta, 1987, 26, 40; J. C. van der Werff and C. G. de Kruif, J. Rheol., 1989, 33, 421; J. C. van der Werff, C. G. de Kruif and J. K. G. Dhont, Physica A, 1989, 160, 205; J. C. van der Werff, C. G. de Kruif, C. Blom and J. Mellema, Phys. Rev. A, 1989 39, 795. 1 1 G. K. Batchelor, J. Fluid Mech., 1970, 41, 545; G. K. Batchelor and J. T. Green, J. Fluid Mech., 1972, 56, 401; G. K. Batchelor, J. Fluid Mech., 1977, 83, 97. 12. I. M. Krieger, Trans. SOC. Rheol., 1963, 7, 101; I. M. Krieger, Adu. Colloid Interface Sci., 1972, 3, 1 1 1 . 13 W. J. Frith, PhD Dissertation (Catholic University, Leuven, Belgium, 1986); W. J. Frith, J. Mewis and T. A. Strivens, Powder Technol., 1987, 51, 27; J. Mewis, W. J. Frith, T. A. Strivens and W. B. Russel, AIChE J., 1989, 35, 415. 14 T. A. Strivens, Colloid Polym. Sci., 1983, 261, 74; T. A. Strivens, Colloid Polym. Sci., 1987, 265, 553. 15 S. J. Willey and C. W. Macosko, J. Rheol., 1978, 22, 525. 16 G. N. Choi and I. M. Krieger, J. Colloid Interface Sci., 1986, 113, 94; 113, 101. 17 M. D. Croucher and T. H. Milkie, in The Efect of Polymers on Dispersion Properties, ed. Th. F. Tadros 18 M. C. A. Griffin, J. C. Price and W. C. Griffin, J. Colloid Interface Sci., 1989, 128, 223. 19 M. D. Croucher and T. H. Milkie, Faraday Discuss. Chem. SOC., 1983, 76, 261. 20 T. Milkie, K. Lok and M. D. Croucher, Colloid Polym. Sci., 1982, 260, 531. 21 C. Prestidge and Th. F. Tadros, J. Colloid Interface Sci., 1988, 124, 660. 22 M. A. Ansarifar and P. F. Luckham, Colloid Polym. Sci., 1982, 267, 736. 23 Th. F. Tadros, ACS Symp. Ser., 1984, 240, 411. 24 U.S. patent, Seragen, Inc. 25 T. Kato, K. Nakamura, M. Kawaguchi and A. Takahashi, Polym. J., 1981, 13, 1037; C. Cowell and B. Vincent, in The Efect of Polymers on Dispersion Properties, ed. Th. F. Tadros (Academic Press, New York, 1982), p. 263; J. A. Baker and J. C. Berg, Langmuir, 1988, 4, 1055. 26 M. A. Cohen Stuart, F. H. W. H. Waajen, T. Cosgrove, B. Vincent and T. L. Crowley, Macromolecules, 1984, 17, 1825. 27 J. W. Vanderhoff, H. J. van den Hul, R. J. M. Tausk and J. Th. G. Overbeek, in Clean Surfaces: Their Preparation and Characterization, ed. G. Goldfinger (Dekker, New York, 1970), p. 15. 28 J. W. Goodwin, R. H. Ottewill, R. Pelton, G. Vianello and D. E. Yates, Br. Polym. J., 1978, 10, 173. 29 J. March, Advanced Organic Chemistry (Wiley-Interscience, New York, 1985), p. 350, 372. 30 S. J. Partridge, Ph.D Dissertation (University of Bristol, 1987). 31 K. L. Koopal, V. Hlady and J. Lyklema, J. Colloid Inrerface Sci., 1988, 121, 49. 32 F. L. Saunders, J. Colloid Sci., 1961, 16, 13. (Academic Press, New York, 1982), p. 101. Paper 0/02262D; Received 17th May, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000077

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Furuday Discuss. Chem. SOC., 1990,90, 91-106 Structure of a Binary Colloidal Suspension under Sheart Howard J. M. Hanley, John Pieper$ and Gerald C. Straty Thermophysics Division, National Institute of Standards and Technology, Boulder, CO 80303, USA Rex P. Hjelm Jr and Philip A. Seeger LANSCE, Physics Division Los Alamos National Laboratory, Los Alamos, NM 87545, USA Neutron scattering intensities from an aqueous mixture suspension of 91 nm polystyrene latex particles and 54nm silica particles are reported in the range 0.02 < Q/nm-' < 0.2, where Q is the momentum transfer. The sus- pension was dense at a mixture volume fraction of 0.15, and the poly- styrene/silica particle ratio was ca. 1.7 : 1. Results are given for the suspension at rest and under shear. The sheared data were obtained with a concentric cylinder shearing apparatus constructed and tested at the SANS facility of the National Institute of Standards and Technology and the pulsed neutron facility, LANSCE, of the Los Alamos National Laboratory.The design and operation of the cell is described. The shear-influenced behaviour of the mixture is compared with and contrasted to that of a pure polystyrene suspension that can form a crystal lattice in equilibrium, but which melts to a liquid-like structure under shear. A method is proposed to measure, by contrast matching or variation, the polystyrene and silica partial scattered intensities from the mixture suspension in H20-D20 solvents of different scattering-length densities. Estimates of the partial structure factors are given.Straty and co-workers',' have built an apparatus to investigate, by neutron scattering, a liquid under shear. The apparatus is versatile, rugged, designed to cover a wide range of shear rates and operating temperatures, and simple; the liquid is sheared between a stator and the rotating outer cylinder. We recently presented results on the microstructure (local particle order) of colloidal suspensions using the apparatus on the SANS (small-angle neutron scattering) facility at the National Institute of Standards and Technology (NIST)' and the pulsed neutron source of the Los Alamos Neutron Scattering Center (LANSCE).3 In this paper, data from a dense aqueous mixture of polystyrene latex spheres with approximate diameter (cps) 91 nm, and silica spheres with approximate diameter (csi) 54 nm are presented and compared and contrasted with the previous measurements.The species were selected deliberately to differ significantly in size. The mixture volume fraction was = 0.15 and the polystyrene-to-silica number density ratio was approximately 1.7 : 1. Several reasons make a mixture suspension an attractive candidate. First and foremost, pure colloidal suspensions have been studied extensively by light: X-rays,' and neutrons,' but not many authors have published on mixtures.6-'o The objective here was to investigate the microstructure of the mixture through the structure factor or the t Publication in part of NIST (formerly The National Bureau of Standards), not subject to copyright in $ Permanent address: Department of Physics, University of Colorado, Boulder, CO 80309, USA.the U.S.A. 9192 Binary Colloidal Suspension under Shear pair correlation function in the usual way. We wished, however, to distinguish the contribution of the polystyrene from that of the silica. To do this, the polystyrene or the silica are contrast matched3 from the mixture suspension in H20-D20 solvents of different scattering-length densities. The behaviour of a pure suspension or simple liquid under shear is very interesting. The phenomenon of shear-induced melting of a pure suspension is well documented by light4.” and neutron2312 scattering data, and the experiments are reinforced by computer-simulation results from model liquids.13-15 There has also been recent work on shear-induced ~rdering.’~~’’ By contrast, the behaviour of mixtures under shear is not known, and one of our objectives was to carry out some preliminary experiments.The paper is organized as follows. The basic equations for scattering from a colloidal suspension, and for the contrast matching from a mixture, are reviewed. The shearing apparatus is described and the sample preparation discussed. To set the stage for the mixture results, neutron scattered intensities from a pure suspension of the 91 nm polystyrene from the same batch sample used to make up the mixtures are presented first. We comment on the results in the context of the shear-induced melting phenomenon. The scattered intensities are then presented for the mixture at rest and under shear. Comparisons with the data from the pure sample are given. Estimates of the partial structure factor of the polystyrene and silica in the mixture are reported, and discussion remarks, conclusions and suggestions for further work complete the paper.Scattering Equations The basic expressions for the scattering from a suspension are outlined here for refer- ence.l8*l9 The scattered intensity ( I ) is proportional to the differential cross-section per unit volume of the sample, dC(Q)/da. Only the coherent contribution is of interest here. Q is the scattering wavevector for neutrons of wavelength A given by Q = ( 4 ~ / h ) sin e/2. ( 1 ) Here, 8 is the scattering angle, and Q = ko - k , ; ko and k , are the incident and scattered wavevectors, respectively, of the radiation; ko = 27r/h.If each colloidal particle is considered as a scattering centre, the expression for the cross-section is separated into terms for single particle and for interparticle particle scattering. For spherical particles where F ( Q ) is the form factor d w d f l = PIF(Q)I~S(Q) (2) F ( Q ) = C bjexp(iQ.Xj) and S ( Q ) is the structure factor, S(Q) = ( 1 ~ ~ ) (?? exp L ~ Q - ( ~ i - ~ i t ) ~ i i’ (3) (4) In eqn (2)-(4) Ri is the vector for the centre of mass of colloidal particle i, xj is the position of nucleus j relative to the centre of mass of particle i, p is the particle number density in volume V ( p = Np/ V ) , and bj is the scattering length of nucleus j . The measured intensity of interest is the intensity with respect to that of the medium of scattering length density, pm.In terms of pm, the form factor for a monodisperse system of spherical particles is where ps is the scattering-length density of the particle of volume radius R and volume, Vp = (4/3)nR3, and the function j ( Q R ) / ( Q R ) is F ( Q ) = V,(p,-p,)[3j(QR)/QR)I ( 5 ) j ( QR)/ QR) = (sin QR - QR cos Q R ) / ( Q R ) ~ . ( 6 )H. J. M. Hanrey et al. 93 The scattering-length densities are calculated from the operational definition where pmo' is the molecular density, and N ( j ) is the number of nuclei of type j per molecule of type k Thus, in terms of the particle volume fraction + = pV, we have Note that we can emphasize the role of the scattering-length density by writing eqn (8) as dC/da z Np(ps - pm)*I' (9) where I' contains all elements, other than those that depend directly on the scattering- length density, of the form factor and structure factor.In a typical experiment,20 the scattered intensity from the sample, I(sample), is measured at the volume fraction, +(sample), of interest. Then I(di1ute) is measured for a dilute sample at +(dilute) in the same apparatus under the same experimental conditions. The standard working expression to estimate the structure factor of a monodisperse single-component suspension follows S( Q ) = I(sample)c$(dilute)/[ I(dilute)+(sample)]. (10) Contrast Matching from Mixtures In the notation of eqn (9), the equation for a binary mixture can be written dx(Q)/damix z L C JNpJNqpp(mix)pq(mix)I~q; p7 = 132. (11) Pk(mix) = (pk - Pm); k = p7 4- (12) (13) dC(Q)ldap Np(pp - p m q ) I p p - (14) P 4 Here Suppose, therefore, pm is set equal to pq (designated pmq), eqn (11) becomes 2 1 Similarly, 2 1 dC( Q)/daq Nq(pq - Pmp) I q q * We have, therefore, eqn ( l l ) , (13) and (14) from which the total and the partial cross-sections can be extracted by appropriate manipulation of pm .Experimental Shearing Cell The apparatus is a Couette-type concentric cylinder fused quartz sample cell, coupled to a computer-controlled drive mechanism and a computer-controlled thermostat sys- Neutrons are scattered from the fluid in the annular gap between the cylinders (see fig. 1). A thermostatting fluid can be circulated within the inner stator, but is excluded from the neutron path by the sealed cross tube, see fig. 2. The outer cylinder is driven by a microprocessor-controlled d.c.servo motor, which is programmed for any desired combination of velocity, acceleration, delay time, direction of motion or rotation angle. We have verified that these variables are precise to within 0.1% for at least 24 h of continuous operation. A personal computer interfaces the motor and thermostatting controls with the operating and data acquisition system of the neutron facility, so that full automation of the experiment can be achieved if necessary.94 Binary Colloidal Suspension under Shear IB X Y Fig. 1. Schematic drawing of the shearing apparatus. M, drive motor; CR, cell rotor; CS, cell stator; TT, torque transducer; A, alignment coupling; X and Y, micrometer adjustment screws; Z, vertical adjustment. The neutron beam is directed to point B.The drive motor has an operating range from less than 0.006 r.p.m. to over 5400 r.p.m. delivering a maximum torque of ca. 5 N m. Drive system characteristics are shown in fig. 3. Shear rates of over 3.0 x lo4 Hz are possible with the present cell configuration; namely, the cell has a nominal mean diameter of 57 mm and an annular gap of 0.6 mm (1 r.p.m. is equivalent to a shear rate, G, of 4.89 Hz under these conditions). At present, the neutron beam is directed along a cell diameter if fluid is used as the thermostatting medium, but new cells are under construction that will allow the beam to be directed off centre. The gap width, and the range of attainable shear rates, will be adjustable by appropriate substitution of the stator and outer cell with others of different diameters.Sample Preparation Stock suspensions of a 30% mass fraction suspension of polystyrene in H20 and a 30% mass fraction suspension of silica (‘Ludox’?) in H 2 0 were available. According to the manufacturer’s specifications, ups = 91 nm and is ca. 7% polydisperse, and uSi = 50 nm and is essentially monodisperse. The estimates for polystyrene were confirmed by electron microscope imagery. Having scattering data, we now estimate the silica to be closer to 54nm in diameter. t The trade name is used to identify the product and does not imply any endorsement by NIST.H. J. M. Hanley et al. - sample region thermostatting fluid 5 E 4 z - 2 3 2- 2 2 1 Fig. 2. Details of the shearing cell. shear rate/ lo4 Hz 0.4 1.1 1.8 2.5 3.2 0 0 10 20 30 40 50 60 70 80 90 95 (1 200) (2400) (3600) (4800) rev.s-' (rev. min-') Fig. 3. Viscosity-torque relationship for the cell with its present configuration, see text. Two sets of sample solutions for the scattering experiments were prepared from the stocks; a polystyrene suspension, and mixtures of the polystyrene from the same polystyrene batch sample with silica in D20 and in D20-H20. Initial experiments with pure polystyrene were carried out in suspensions of volume fraction (4) 0.3, 0.085 and 0.06 in deionized water, the latter in a H20-D20 mixture. The suspensions were expected to be crystalline at these initial conditions based on the 4-ion concentration phase diagram determined from an X-ray study of the same batch sample.' To make these suspensions, the 30% polystyrene stock was deionized by tumbling with resin and diluted when necessary. The tumbling process was unsatisfactory in that the deionized 30% samples tended to aggregate over a period of days and had to be centrifuged or filtered before use.For later experiments, the polystyrene was96 Binary Colloidal Suspension under Shear outer cylinder 1 detector Y ,1z X Z Fig. 4. Neutron scattering geometry for the NIST SANS and LANSCE runs. Shear = du,/dy. purified by dialysis with an ion-exchange resin without agitation; which proved a more satisfactory procedure. For the mixture, the polystyrene solutions were subsequently dialysed with NaOH to replace the hydrogen counter-ions by Na+ to give a neutral pH in the polystyrene-silica suspensions.(The polystyrene-silica suspensions had to be pH neutral to ensure the silica component remained stable in suspension.) The silica was purified by filtration through a mixed-bed exchange resin and stabilized22 by adding NaOH solution until the pH reached ca. 8. Alternate solutions for each sample in D20 were prepared by dialysis against D20 in a closed container. Using measured values of the volume fractions of these treated samples, mixtures were made up for a mixture volume fraction +mix = 0.15 with ca. 64/36 polystyrene to silica number density ratio. The mixtures were pH 7-8 with an ion concentration of ca. 40 pmol ~ m - ~ . The mixture solvent was pure D20, or the appropriate D20-H20 proportion for contrast matching. A mixture with 28% D20 was prepared to match out the polystyrene, and a mixture with 60% D20 to match out the silica.The per cent D20 chosen corresponds to effective scattering-length densities (10'' cm-2) of 1.40 and 3.6 for the polystyrene and silica, given values of -0.56 and 6.4 for H20 and D20, respectively. Dilute solutions, 4 = 0.01, of pure polystyrene, pure silica and the mixture were prepared in D20. Scattering Measurements Scattering data were measured at the NIST and Los Alamos neutron facilities. Data were taken for the samples at rest and under shear with the scattering geometry of fig. 4 for all the experiments. In Cartesian coordinates the gradient, V, is in the y coordinate and the velocity, u, is along x. The shear-induced melting of the polystyrene suspensions was investigated at the NIST SANS facility.2 For these SANS measurements, a seven-beam converging pinhole collimation system23 illuminated a circle of ca.20mm diameter at the sample. The Q-range covered by the instrument's two-dimensional position-sensitive detector was 0.03-0.6 nm-I, given an incident wavelength of 1.2 nm, and a sample-to-detector distance of 3.6 m. Scattering intensities were recorded for the samples at rest with the cell stationary, and under shear with the outer cylinder rotating at speeds from 0.6 to 1500 r.p.m. The intensity data were corrected for the empty cell contribution, background scattering and sample transmission. We verified from previous that multiple scattering wouldH. J. M. Hanley et al. 97 ( C ) ( d ) Fig. 5. Scattered isointensity contours in the x-z plane from the 4 = 0.06 sample.( a ) from the system at rest; ( b ) from the system subjected to a shear rate G = 2.93 Hz; ( c ) G = 58.7 Hz; and ( d ) G = 176.0 Hz (see text). In the isointensity plots, the magnitude of Q ranges from Q = 0 at the centre of the plot to its maximum at the corner. be insignificant given the sample thickness here. The samples are strong scatterers, so the incoherent contribution to the scattering was taken to be negligible at the Q values of interest. The mixture was investigated initially3 on the time-of-flight (TOF) LQD instrument at LANSCE.25 The instrument was configured with a magnesium oxide filter to remove neutrons of wavelength shorter than ca. 0.15 nm from the beam.26 The moderator to sample distance and the sample to detector distance were 8.945 and 3.76 m, respectively. Data were collected into 147 TOF bins for wavelength determination. The width of each bin corresponded to A A / A = 1.6%.The raw data were corrected for the background and incoherent scattering of the medium and reduced to absolute units of differential scattering probability per unit solid angle as functions of Q by the methods described elsewhere.26327 The results are presented as differential cross-sections dC( Q ) / d n per unit volume in units of cm-'. Results Polystyrene We first display some of the results obtained for the pure polystyrene suspension from the NIST SANS experiments. Fig. 5 and 6, extracted from ref. (2) are the isointensity plots of the experimental scattering intensity in the x-z plane (fig.4) for the + = 0.06 and 0.30 samples, respectively. The sample was at rest in fig. 5 ( a ) . Six high-intensity spots are observed at Q == 0.05 nm-', with evidence of six less intense spots at Q == O.lOnm-'. As noted in ref. ( 2 ) , we interpret the inner grouping to arise from a two- dimensional hexagonally closed-packed layer with the close-packed direction along u. At this point we should state that we have no clear-cut way of knowing if the system at rest implies that the system is in equilibrium. Hess has observed28 that whether or not spots can be seen in the intensity pattern for an unsheared very dense suspension98 Binary Colloidal Suspension under Shear Fig. 6. Scattered isointensity contours in the x-z plane from the 4 = 0.30 sample; (a) from the system at rest; (b) from the system subjected to a shear rate of G = 58.7 Hz; ( c ) G = 293.4 Hz and ( d ) G = 1467.0 Hz (see text). depends on the time between the loading of the sample in the cell and the measurements.It is likely that a very dense unsheared colloidal suspension is polycrystalline. The mere act of introducing the suspension into a cell, or subjecting the system to a very low shear rate, defines a structure that may actually be that of the true equilibrium state. Fig. 5( 6) and (c) are typical of the intensity pattern from the sheared sample as the shear rate was increased. The suspension was subjected to shears of 2.93 Hz [fig. 5(b)], and 59 Hz [fig. 5(c)]. The inner six-fold symmetry is weakened with increasing shear and only four intensity maxima are seen, most probably because of a loss of correlation in the x-direction when the shearing motion forces one two-dimensional layer past another in the o direction.The intensity was liquid like at higher values of the shear, e.g. fig. 5 ( d ) for 176.0 Hz. The sequence was reversible and repeatable to within the resolution of the instrument. Isointensity contours for the 4 = 0.30 suspension are given in fig. 6. For the sus- pension at rest we surmise that the scattering pattern arises from the two-dimensional hexagonally close-packed layer along the o direction with little indication of structure in the gradient direction [fig. 6(a)]. Six-fold symmetry existed when the system was sheared, but the upper and lower spots became less intense with increasing shear.The crystal-like structure vanished, and the intensity pattern is that of a liquid [fig. 6(d)]. Polystyrene/Sitica Mixture Fig. 7(a) is the LANSCE LQD scattered intensity from pure polystyrene in D20 at rest, volume fraction # = 0.17. The pattern is consistent with fig. 5 and 6. Compare, however, with fig. 7(6), which is a typical pattern from the polystyrene-silica mixture in D20. The lattice-like order is not present. Possible structures in a mixture suspension are discussed by Lindsay and Chaikin6 who point out that a mixture structure may take considerable time to develop afterH. J. M. Hanley et al. 99 Fig, 7. (a) Typical scattered intensity from a dense pure polystyrene suspension, here at 4 = 0.17. The six intensity maxima are clear.The pattern is consistent with that from two-dimensional hexagonally closed-packed layers along the v direction [see fig. 5 and 6, and ref. (2)]. ( b ) Typical pattern from a mixture of polystyrene with silica. The intensity is liquid-like with no evidence of a lattice structure or anisotropy. All the mixture intensity patterns, for the mixture at rest and under shear, were of this general appearance. initial mixing of the components. Here, the time between sample preparation and the measurements was of the order of an hour. No special effort was made to agitate the mixture, or to allow the mixture to settle in the cell. The measurements for the sample at rest were made on the timescale of minutes. However, no observable lattice structure was induced when the mixture was subjected to a very low shear (much lower than would be expected to cause shear melting).Since we have argued that a very low shear may define any structure that is present for a pure suspension, it is very probable that the mixture does not have a definite lattice. Contrast Matching Fig. 8 gives the differential cross-sections versus Qa for pure polystyrene in D20 at a volume fraction 4 = 0.015, and for pure silica in D20 at # = 0.01 1. The heights of the possible maxima at Qa =r 0.04 are a function of the scattering-length densities and volume fraction difference and have been adjusted to be about equal. The diameter ups has been set at one for convenience. The dilute intensities were expected to scale with Qa.100 2500 2000 0" 1500 % .C Y m m 1000 500 Binary Colloidal Suspension under Shear 8 0 0 ra 8 9 0 0 0.05 0.1 0 0.1 5 0.20 Fig.8. Scattered intensities, expressed as the differential cross-section (,/cm-'), of dilute pure polystyrene and of dilute pure silica as a function of Qu with ups set to one. The diameter, asi, has been scaled to 0 . 6 5 ~ ~ ~ . 0, Polystyrene; 0, silica. 500 400 g 300 .- c) u 7 s b 200 100 0 0 0.05 0.10 0.1 5 0.20 Fig. 9. The differential cross-section from the 4 = 0.15 mixture (curve). U, The partial differential cross-sections of the polystyrene in the contrast-matched mixture. 0, The partial differential cross-sections of the silica in the contrast-matched mixture. Qu has been scaled as for the dilute pure samples. The positions of the peaks do not superimpose.H.J. M. Hanley et al. 3000 I 2000 h Y v) .- c i v 1000 x E E CI .I v) Y .- 0 -1 000 101 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17 0.19 Qu Fig. 10. Estimate of the polystyrene-silica intensity evaluated from eqn (1 l), using eqn (12) and (13) for the mixture at equilibrium (O), and under shear (0) at G = 147 Hz. We have assumed u12 = ( U ~ ~ U , ~ ) ~ ' ~ . The curves for the mixture and for the sheared mixture are given by the and 0, respectively. 2500 2000 .- 2 1500 % Y u v1 8 1000 0 0 0.1 0.2 Qp Fig. 11. Effect of shear on the cross-section of the mixture. The intensity falls at the Qa corresponding to the peak, and the drop is essentially independent of the applied shear rate. The curve is the run at equilibrium; +, sheared mixture at G = 147 Hz; 0, the mixture at G = 5.870 Hz.102 Binary Colloidal Suspension under Shear 0 500 400 .- 300 Y u aJ 0 0.1 Qu 0.2 100 0 0 0.1 Qff 0.2 Fig.12. ( a ) The cross-section of silica in the mixture at equilibrium (+) and under shear (0) at G = 146 Hz. No difference is observed. ( b ) The cross-section of polystyrene in the mixture at equilibrium (+) and under shear (0) at 146 Hz. Compare with fig. 11. Superposition of the curves with the ratio of aP,/asi=91/54 was poor, but better if aps/asi = 1.54, a ratio that is not inconsistent with the estimated uncertainties of the polystyrene and silica diameters. Fig. 9 represents the contrast variation measurements. Three curves are given: the cross-section of the mixture in D20 and the partial cross-sections of the polystyrene in the 60% D20 solvent and the silica in the 28% D20 solvent.The heights of the peaks have been adjusted to be approximately equal and Qa has been scaled as for the dilute pure samples, with aiix = (1.7a$ + azi)/2.7 and ups set at one. The polystyrene-poly- styrene and the silica-silica contributions to eqn (1 1) were calculated from the appropri- ate contrast-matched data, and the polystyrene-silica cross intensity estimated by sub-H. J, M. Hanley et al. 103 traction from the total mixture intensity (see fig. 10). Comments on these curves are deferred until the discussion. Scattering from the Sheared Mixture Although the definite shear-induced structures observed in sheared pure polystyrene suspensions were not seen in the polystyrene-silica mixtures, the maximum intensity was found to fall significantly when a shear was applied (fig.11). Furthermore, the drop was essentially independent of the applied shear rate to within the resolution of our instruments. Measuring the intensity from the contrast-matched sheared mixtures indicates that the drop comes exclusively from the polystyrene contribution (fig. 12). The cross-term for the sheared mixture was evaluated from eqn (11) and plotted in fig. 10. Remarks and Conclusion Neutron-scattered intensities were measured from a dense polystyrene-silica aqueous suspension at rest and under shear. Much of the work is preliminary but we have two definite, if at this stage qualitative, results. First, we were able to distinguish the scattered intensity from each of the two components by contrast matching selectively the com- ponents with an appropriately mixed H20-D20 solvent.Contrast matching or variation has been used to separate partial structure factors in biological and, using isotope substitution, is a well known technique to study the structure of alloy^.^' However, we have applied it to a liquid mixture. Secondly, the behaviour of the mixture under shear is definitely different from that of a pure suspension under shear. The scattered intensities from the mixture were compared with those from a pure polystyrene suspension that had a lattice structure at rest. The pure suspension lattice melted to a liquid structure under shear. We observed, however, that the polystyrene-silica mixture did not have a defined lattice at rest (see fig.7). Consequently, the mixture obviously did not display the phenomenon of shear-induced melting, but the scattered intensity fell (in the Q range investigated) when the mixture was sheared. Clearly, the shear- induced behaviour of mixtures requires further experimental work. A satisfactory conclusion to this study would be to estimate the structure of the mixture from the scattered intensities. We can write eqn (1 1) in terms of the Ashcroft- Langreth partial structure factors32 and assume that an eqn (10) is valid for each term. Thus, in principle the partial structure factor of species k is evaluated from eqn (10) where I(samp1e) is the contrast-matched intensity for k and I(di1ute) is the intensity of the form factor of pure k. For technical reasons, we were not satisfied that the dilute runs of fig.8 were of sufficient precision to be regarded as legitimate form-factor data. Very recently,33 however, some of the equilibrium LANSCE measurements were repeated on the same samples using the D11 spectrometer of the ILL, Grenoble. For these experiments, the mixtures were loaded into a standard quartz cell of pathlength 1 mm. The sample-detector distance was 20 m and the neutron wavelength was 1.2 nm giving an experimental Q range of 0.01 < Q/nm-' < 0.12. were placed on an absolute scale with respect to the water standard measured at 5 m. Corrections for the empty cell contribution and background scattering were made, but not for the contribution of the water solvent or incoherent scattering. The D11 data and the Los Alamos data were consistent to within the resolution of the instruments for the Q range of overlap.There are still some doubts that the dilute runs represented the form factors correctly; nevertheless, fig. 13 is an estimate of the polystyrene-polystyrene and silica-silica partial structure factors in the mixture. The apparent structure factor for polystyrene is con- sistent with other estimates from X-ray data' and our independent SANS from The ILL104 o o o o a " ~ Binary Colloidal Suspension under Shear 1 I@ l3a a a a ifl, Y , , , , , , I , ; , 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 r / u Fig. 14. The Fourier transforms of the estimated partial structure factors (polystyrene, 0; silica, 0) shown in fig. 13, plotted versus r / a with ap,=91 nm and asi=61 nm.The position of the silica peaks depends on the value selected for asi, which is uncertain to 10%. the same batch sample of the pure polystyrene. Fig. 14 shows the Fourier transforms of the polystyrene and silica curves. Since g ( r * ) = 1 +- I [ S ( W - 1 I ( W sin (Qar*) d ( Q 4 (15) 127r+r* where r* = r / a , these transforms are presented as approximations to the radial distribu- tion functions for polystyrene and silica in the mixture at rest. They suggest that theH. J. M. Hanley et al. 105 polystyrene structure at = 0.15 is very close to the structure of pure polystyrene at an equivalent volume fraction with a main peak in g ( r ) at ca. 137 nm, but that the silica has a weaker lattice with a peak at ca. 300nm. The numbers make sense but have to be regarded with caution.Furthermore, only the results from a system with one particular mixture volume fraction and one particular polystyrene-to-silica particle ratio are avail- able, so any general conclusions on the structure of a mixture are not possible at this time. Clearly the structure of the mixture will depend on factors such as the volume fraction, the particle size ratio, the number density ratio and the surface-charge densities. Overall, the contrast variation technique applied to suspension mixtures is satisfactory and has led to results that encourage further work. The structure data of the equilibrium suspension were not unexpected and were anti~ipated~~ by simulation of model l i q ~ i d s ~ ~ , ~ ~ and by simple theories of although the apparent long-range structure of the silica in the mixture was surprisingly evident.However, the way in which the mixture behaves under shear is new and will be investigated in more detail. The work of H. Hanley and G. C. Straty was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences. J. Pieper acknowledges a National Science Foundation Graduate Fellowship. The LQD of LANSCE is supported by the US Department of Energy and under contract W-7005-ENG-32 to the University of Califor- nia. We are very grateful to C. G. Glinka of NIST and P. Lindner of the ILL for their contribution to this work. References 1 G. C. Straty, NISTJ. Res., 1989, 94, 259. 2 G. C. Straty, H. J. M. Hanley and C. J. Glinka, J. Stat. Phys., 1990, in press.3 G. C. Straty, J. Pieper and H. J. M. Hanley, Mol. Phys., 1990, in press. 4 For example, see B. J. Ackerson and N. A. Clark, Physica, 1983, 118a, 221, and references therein. 5 E. B. Sirota, H. D. Ou-Yang, S. K. Sinha, P. M. Chaikin, J. D. Axe and Y. Fujii, Phys. Rev. Lett., 1989, 6 H. M. Lindsay and P. M. Chaikin, J. Chem. Phys., 1982, 76, 3774. 7 S. Yoshimura and S. Hachisu, Prog. Colloid Polym. Sci., 1983, 68, 59. 8 W-H. Shih and D. Stroud, J. Chem. Phys., 1984, 80, 4429. 9 W. Hart1 and H. Versmold, J. Chem. Phys., 80, 1387; B. V. R. Tata, Y. R. Kesavamoorthy and A. K. 62, 1524, and references therein. Sood, Mol. Phys., 1987, 61, 943. 10 T. Okubo, J. Chem. Phys., 1987, 87, 5528. 11 R. L. Hoffman, Trans. SOC. Rheol., 1972, 16, 155. 12 B. J. Ackerson, J. B. Hayter, N.A. Clark and L. Cotter, J. Chem. Phys., 1986, 84, 2344; S. Ashdown, I. Markovic, R. H. Ottewill, P. Lindner, R. C. Oberthur and A. R. Rennie, Langmuir, 1990, 6, 303. 13 H. J. M. Hanley, Lectures on Thermodynamics and Statistical Mechanics, ed. A. E. Gonzales and C. Varea (World Scientific Press, 1988), p. 109. 14 H. J. M. Hanley, J. C. Rainwater, N. A. Clark and B. J. Ackerson, J. Chem. Phys., 1983, 79, 4448. 15 T. Weider, M. L. Glasser and H. J. M. Hanley, 1990, in press. 16 B. J. Ackerson and P. N. Pusey, Phys. Rev. Lett., 1989, 61, 1033; B. J. Ackerson, C. G. De Kruif, 17 S. J. Johnson, C. G. De Kruiff and R. P. May, J. Chem. Phys., 1988, 89, 5909. 18 S-H. Chen and T-S. Lin, in Methods in Experimental Physics, Vol. 23, Part B, Neutron Scattering 19 R.H. Ottewill, in Colloidal Dispersions (The Royal SOC. Chem. Spec. Pub. no. 43, London, 1982). 20 For example D. J. Cebula, J. W. Goodwin, G. C. Jeffrey, R. H. Ottewill, A. Parentich and R. A. Richardson, Faraday Discuss. Chem. SOC., 1983, 76, 37. 21 P. Lindner and R. C. Oberthur, Reu. Phys. Appl., 1984, 19, 759. 22 J. D. F. Ramsay and B. 0. Booth, J. Chem. SOC., Faraday Trans. 1, 1983, 79, 173. 23 C. J. Glinka, J. M. Rowe and J. G. LaRock, J. Appl. Crystallogr., 1986, 19, 424. 24 C. J. Glinka, D. A. Aastuen, H. J. M. Hanley and G. C. Straty, unpublished data available from the 25 P. A. Seeger, R. P. Hjelm and M. Nutter, Mol. Cryst. Liq. Cryst., 1990, 180A, 101. 26 R. P. Hjelm, J. Appl. Crystallogr., 1988, 21, 618. 27 R. P. Hjelm and P. A. Seeger, Physics Conference Series, 97, 367 (IOP Publications, Bristol, UK, 1990). N. J. Wagner and W. B. Russel, J. Chem. Phys., 1989, 90, 3250. (Academic Press, London, 1987). Thermophysics Division, NIST, Boulder, CO.106 Binary Colloidal Suspension under Shear 28 S. Hess, personal communication, Tech. Univer., Berlin, 1990. 29 P-J. Derian, L. Belloni and M. Drifford, Europhys. Lett., 1988, 7 , 243. 30 H. B. Stuhrmann, Neutron Scattering, ed. W. Glasser (North-Holland, New York, 1988), p. 444. 31 For example, J. E. Enderby, D. M. North and P. A. Egelstaff, Philos. Mag., 1966, 14,961; S . Eisenberg, J-F. Jal, J. Dupuy, P. Chieux and W. Knoll, Philos. Mag. A , 1982, 46, 195; H. Ruppersberg, Phys. Chem. Liq., 1987, 17, 73. 32 N. W. Ashcroft and D. C. Langreth, Phys. Rev., 1967, 155, 682; 1967, 159, 500. 33 H. J. M. Hanley, G. C. Straty and P. Lindner, Physica, 1990, in press. 34 W. Schmaltz, T. Springer, J. Schelten and K. Ibel, J. Appl. Crystallogr., 1974, 7 , 96; K. Ibel, J. Appl. 35 I. Markovic and R. H. Ottewill, Colloid Polym. Sci., 1986, 264, 65; 454. 36 D. J. Evans and H. J. M. Hanley, Phys. Rev. A , 1979, 20, 1648. 37 M. Huber and J. F. Ely, NIST Tech. Note no. 1331, 1989. 38 D. Henderson and P. J. Leonard, in Physical Chemistry, An Advanced Treatise, Vol. viiiB/ Liquid State Crystallogr., 1976, 9, 296. (Academic Press, New York, 1971), chap. 7, p. 413. Paper 0/02323J; Received 17th May, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000091

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Faraday Discuss. Chem. Soc., 1990,90, 107-114 Hydration Force, Steric Force or Double-layer Force between Zwitterionic Surfaces? Ulrik Nilsson, Bengt Jonsson and Hhkan Wennerstrom Division of Physical Chemistry 1, Chemical Center, P.O. Box 124, S-22100 Lund, Sweden The force between two parallel planar zwitterionic surfaces has been calcu- lated using Monte Carlo computer simulations. The zwitterions are modelled as two oppositely charged hard spheres joined by a string of length L with parameters chosen to mimic a phospholipid system. All centres interact by a homogeneous Coulomb interaction and by a hard-sphere exclusion. The negative centres are anchored to the surface by a parabolic potential. For distances D between the surfaces, as defined by the location of the negative centres, that are larger than 2L there is an attractive force of the classical van der Waals type.When, on the other hand, D < 2L a strong repulsive force appears, which in the limit D << 2L is analogous to a double- layer force. Recently it was suggested (Israelachvili and Wennerstrom, Langmuir, 1990, 6, 873) that the repulsive so-called hydration force observed for biological lipid systems had its origin in confinements on surface excitations induced by a second surface. Here we demonstrate how this mechanism works in a particular microscopic model of the surface. Although still simplistic, several qualitative features of the force in the phospholipid systems are reproduced in the calculations. For example, a reduction of the size of the charged centres leads to a decrease in both the attractive and more significantly the repulsive forces.This mimics the observed difference between phosphatidyl choline and phosphatidyl ethanolamine. In the discussion of the net interaction between surfaces or colloidal particles a number of different mechanistic contributions have been introduced. These include the disper- sion force, the electrical double-layer force, steric forces, oscillatory packing forces, undulation forces, hydration (solvation) forces and hydrophobic forces.' For some of these forces the underlying molecular mechanism is well understood, while particularly for the hydration force and the hydrophobic force no consensus has been established regarding the origin of the effect. One way to systematize the description of the various force contributions is to use the Lifshitz theory as the conceptual basis.In this theory one calculates the interaction between two half-planes separated by a solvent using continuum electrodynamics. As a main result of the Lifshitz theory the asymptotic interaction emerges as determined solely by the frequency-dependent permittivities E ( o) of the respective media. Addi- tional molecular features are irrelevant. The Lifshitz theory is asymptotically exact and it can be said to describe the van der Waals interaction between (semi-) macroscopic bodies. Other contributions to the net force can thus be seen as deviations from the asymptotic behaviour. A large fraction of these deviations arises from the existence of a surface layer with chemical properties distinctly different from either of the bulk media.In colloidal systems it is the rule that such surface regions exist. In the spirit of the Lifshitz theory one could model the surface layer through a locally homogenous relative permittivity, E ~ ( w ) . This would affect the magnitude of the van 107108 Forces between Zwitterionic Surfaces der Waals force, but for a symmetrical situation with two identical surfaces a monotonic attractive interaction will always appear. What is then the origin of the repulsive forces? Different as they may seem, the electrical double-layer force, the undulation force and the steric force have some basic similarities. In all three cases the surface layer is seen as inhomogeneous in the normal direction, be it due to the counter-ion distribution, to the variation in the average density or due to the decay of the polymer segment density.Furthermore, the inhomogeneous distributions result from a competition between energetic contributions that tend to compress the surface layer and entropic effects which try to expand it. As a second surface approaches there is a confinement of configurational freedom lowering the entropy, which results in a net repulsive interaction. Recently Israelachvili and Wennerstrom2 argued that the hydration force, extensively studied for phospholipid and other amphiphilic systems, has an origin similar to the undulation and steric forces. The basic idea being that there are local excitations, ‘protrutions’, on the lipid bilayer surfaces and that these excitations can extend far enough to account for a force acting in the range 1-2 nm.An explicit expression for the force, F, per unit area, A, was derived F/A=na(D/A) exp(-D/A)[l-(l+D/A)exp(-D/h)]-’ (1) where n is the density of protruding groups, a the molecular restoring force for a protrusion, D the distance between the surfaces and A = k g T / a . The model leading to eqn (1) is a simplistic one and there is ample scope for improving the description of the molecular interactions while keeping the same concep- tual idea. In this paper we present one such attempted improvement, which brings out the analogy between the hydration force and not only the undulation force and the steric force but also the double-layer force. Model and Computational Procedure We consider a system of two parallel planar surfaces separated by a distance D.Confined to these surfaces are zwitterionic groups consisting of two spheres of radius r and with charges e and -e. The two spheres are joined with a centre-to-centre string of length, L, chosen as 0.5 nrn in the calculations (see fig. 1). All centres interact electrostatically with a Coulomb potential and the relative permittivity is chosen as 80. The centres also have a hard-sphere interaction so the distance of closest approach is 2r. Owing to the attached string distances larger than L are not allowed between centres of the same zwitterionic pairs. To fix the zwitterions to the surfaces the negative centres are confined with a potential V l ( z ) = k [ z - zo( i )] * + k’[ z - zo( i )] ; i = 1 , 2 where k and k‘ are suitably adjusted parameters.The linear term is introduced to give a symmetric distribution of the negative centres. The zwitterions are oriented at the surface by confining the positive charges through a potential This potential prevents the positive charges from penetrating behind the negative ones in any substantial way. For a particular choice of the parameters defined above and for the density of surface groups the model system is specified. The force between the two surfaces can be evaluated using v + ( z ) = k ( z - z0)’ for z < zo(l) or z > ~ ~ ( 2 ) . (3) where Az, = z,(2) - zo( 1).U. Nilsson, B Jonsson and H. Wennerstrom 109 D Fig. 1. The model system is composed of two parallel planar surfaces with confined zwitterionic groups.A zwitterion consists of two hard spheres of radius r and with charges e and -e. The maximum distance of separation between the spheres is L and the surfaces are separated a distance D. To evaluate the ensemble average we have performed Monte Carlo (MC) simulations using the Metropolis algorithm and the minimum-image convention in a manner described previ~usly.~ The particles outside the MC box were accounted for using a novel procedure. For each centre inside the MC box a corresponding mean distribLition is assumed to exist outside the box. On averaging this implies that we assume the same distribution outside the box as is obtained in the box during the simulation. The force between the two surfaces was calculated using two different methods depending on the separation distance D. For D > 2( L + r ) the two halves interact only through the electrostatic interaction. The force is then obtained by sampling the electro- static part of forces between centres on either side, i.e.(2 (electrostatic) . ) At shorter separations D s 2(L+ r ) there is also a hard-sphere contact between centres. It is difficult to obtain sufficient numerical accuracy to evaluate the anisotropic contact density. Instead the force was determined by evaluating which is directly obtained from the average densities p f ( 2 ) . same results within the statistical uncertainty. At intermediate separations the two procedures for calculating the force gave the110 0.02 0.015 0.01 0.005 0 - Forces between Zwitterionic Surfaces I I I I I ( b ) - - - 1 I I I I L x c 0 -0 Q) Y .- wl L 5 x c -a Y .- v1 L Q) D s 0.02 0.015 0.01 0.005 0 -14 -12 -10 - 8 - 6 - 4 - 2 Z l A z/ 8, Fig.2. (a) The number densities p- (-O-) and p+ (-) as a function of the normal distance from the surface. The surfaces are separated by 2.0 nm and the radius r = 0.20 nm. ( b ) The same as ( a ) but with r = 0.15 rim. For each separation the coefficient k’ in eqn (2) was adjusted to provide a closely symmetric distribution of the negative centres. The position of the surfaces was then determined by the location of the peak in the distribution function p - ( z ) for the negative centres. This is the procedure used to define the distance, 0, between surfaces. Results Simulations were performed choosing the length L of the string to be 0.5 nm.The density n of zwitterions was chosen as one per 0.65 nm2. Two different values of the radii of the charged centres r = 0.20 and 0.15 nm were used. In fig. 2 we show the density profiles for the negative and positive charges of a single surface ( D large). As seen, the negative charges have a narrow distribution p-( z ) around z = zo nm owing to the effect of the quadratic term in V-. Comparing fig. 2 ( a ) and ( b ) we see that the radii r have only a small effect on the distribution p - ( z ) .U. Nilsson, B Jonsson and H. Wennerstrom I 0 5: 0 * 9 . 0 0 -5- 0 111 J’ 1000 800 600 L. cd I, $ 400 4 --. v 0 - 0 0 0 0 0 0.00 0 - 200 I I I I L 0 5 10 15 20 25 D / A Fig. 3. The calculated pressure as a function of the distance D between the surfaces.0 r = 0.20 nm; 0, r=0.15 nm. The distribution function p+(z) for the positive centres shows a more interesting behaviour. For r = 0.20 nm there are two distinct peaks in p+(z). The one at the surface where p - ( z ) peaks is generated by the electrostatic interaction which works to keep the positive and negative centres close together. The peak at ca. 0.4 nm out from the surface is generated by the hard-sphere term. The larger the z value the more room is there for the positive centres to move. For r = 0.15 nm the excluded volume effect is less important and p+(z) contracts more towards the surface. The calculated distance dependence of the force per unit area is shown in fig. 3. For both values of r the model gives a long-range attraction and a short-range repulsion.With r=0.20 nm the forces are consistently larger in magnitude both in the attractive and the repulsive regimes. Discussion In addition to the reasons given in the Introduction there was a two-fold rationale for choosing the particular model system. Experimentally4” the most well characterized systems showing hydration forces are the phospholipids, particularly the phosphatidyl cholines, PC, and the phosphatidyl ethanolamines, PE. There are zwitterionic substances where the positive nitrogen is joined to the negative phosphate centre by three single bonds, which are free to rotate. The PCs have a large positive centre, where the nitrogen is covered by CH, groups, while for PE the nitrogen is a primary amine with a smaller effective radius.Modelling these systems by charged spheres attached by a string gives a somewhat too large configurational freedom to the headgroup, but to connect the centres by a stiff rod, as has previously been done,3 underestimates the flexibility. Secondly the use of a string constraint has a theoretically appealing feature. If the length of the string L exceeds the distance D between the surfaces, i.e. if D, the constraint on the distance between the negative and positive centres should be of no consequence. The system behaves as if the positive ions were free counter-ions and one has a situation with double-layer forces with counter-ions only.677 If on the other hand, if 2(L+ r ) < D there is no direct contact between the two surfaces and there is only a long-range electrostatic interaction.In such a situation there should be an attractive correlation force of the van der Waals type.*-”112 Forces between Zwitterionic Surfaces From this general argument that the force between the surfaces should go from an attractive van der Waals type to a repulsive double-layer-like one at short separations, we have immediately rationalized the qualitative features of fig. 3. As expected, the crossover from attraction to repulsion is found for separations slightly less than 2( L+ r ) where the two halves can first make hard-sphere contact. For the larger centres, r = 0.20 nm, the surface groups extend further out as seen in fig. 2. This has two consequences. On close approach the surfaces will affect one another more strongly than for r = 0.15 nm, resulting in a stronger repulsive force.Secondly, the isolated surface is more diffuse, perceivably resulting in a higher polarizability, which results in a strong correlation force at larger separations.' These results concerning the r dependence are in qualitative agreement with the experimental observation that the hydration force is considerably stronger for the PCs with their larger head groups than for PE. The model is clearly too crude to be expected to give a quantitative agreement with measured forces, but it is important to look at the general magnitudes. The strongly repulsive force obtained at shorter separations can be analysed with reference to two models that are opposite limiting cases. The double layer with counter- ions has been extensively studied only for a situation with negligible radius of the counter-ions.The force is in the Poisson- Boltzmann approximation where the dimensionless parameter s is obtained from the equation e2( n / A) D ~ ~ , T E ~ E , ' s tan ( s ) = Here E ~ E , is the relative permittivity and (n/A) the density of the charged groups on the surface. Another view of the model system is obtained by saying that charges are irrelevant and that the pressure is generated by the hard-sphere contacts. In such a case we can estimate the pressure using the Carnahan-Starling approximation. For hard spheres where c is the concentration of particles and 4 is the volume fraction occupied. In fig. 4 we have plotted the forces of eqn ( 5 ) and (7) compared to the simulated force.At shorter distances we can interpret the force in the more complex model system as partly a double-layer force and partly a hard-sphere steric force. Note also that the experimentally measured forces are of the same magnitude. However, there is nothing in eqn (5) and (7) that indicates an exponentially decaying force. In the simulation it is only the surface-surface interaction that is explicitly calculated. In the real system there is also a van der Waals component from the media, which we can write as F H - (bulk-bulk) = A 67r( D+ 6)' where H is the Hamaker constant, which for hydrocarbon-water-hydrocarbon is ca. 5 x lop2' J. Furthermore, the zero position for this van der Waals contribution is not quite specified and we use S = 2 r as a reasonable guess.The choice of 6 is important for the force at short separations. Using these numbers the F/A(bulk-bulk) term is small relative to the directly calculated forces at the shortest separations considered (for D = 0.4 nm; r = 0.20 nm;U. Nilsson, B Jonsson and H. Wennerstrom 113 D l A Fig. 4. The MC values of the pressure compared with the results from the Poisson-Boltzmann (PB) and the Carnahan-Starling (CS) equations [eqn ( 5 ) and (7) , respectively]. 0, r=0.20 nm; 0, r=0.15 nm; (-) PB; (. * - ) CS, r=0.20 nm; (- - -) CS, t = 0.15 nm. F/A(bulk-bulk) = 5 x lo5 N mP2) while it dominates at the largest separations. The explicitly calculated correlation contribution to the force is essentially of the van der Waals type (without dispersion) and the force component is F A - (surface-surface) = c1 D - ~ .(9) By fitting the simulation data at large distances the coefficient c1 can be estimated to c1 = -4 x J m2 for r = 0.15 nm. These values are of the same magnitude as found for a similar system in ref. (10) ( c , = - 1 x lop4' J m2). J m2 for r = 0.20 nm and c1 = -3 x Conclusions The computer simulations show that for a zwitterionic system with charged spheres connected by a string a strong repulsive force appears at short separations between surfaces. The repulsion is caused by a decrease in the configurational freedom of the surface groups induced by the presence of a second surface. The quantitative calculations show that for surfaces with a high degree of local molecular freedom, this effect can give a large contribution to the force.In the simulations we have concentrated on the configurations of the zwitterionic groups, while in a real system there are additional degrees of freedom that could contribute to the force. In addition the modelling of the electrostatic effects is simplistic since the surface region is really strongly dielectrically inhomogeneous. The repulsive force has features similar to both the classical electrical double-layer force and the steric repulsive force often observed for polymer-covered surfaces. Both these forces belong to a general group of repulsive forces of entropic origin and, as the present example demonstrates, they are more related than commonly recognized. References 1 J . N. Israelachvili, Intermolecular and Surface Forces (Academic Press, New York, 1985). 2 J. N. Israelachvili and H. Wennerstrom, Langrnuir, 1990, 6, 873.114 Forces between Zwitterionic Surfaces 3 M. Granfeldt, B. Jonsson and H. Wennerstrom, MoZ. Phys., 1988, 64, 129. 4 R. P. Rand and V. A. Parsegian, Biochim. Biophys. Acta, 1989, 988, 351. 5 J. Marra and J. N. Israelachvili, Biochemistry, 1985, 24, 4608. 6 S. Engstrom and H. Wennerstrom, J. Phys. Chem., 1978, 82, 2711. 7 L. Guldbrand, B. Jonsson, H. Wennerstrom and P. Linse, J. Chem. Phys., 1984, 80, 2221 8 B. Jonsson and H. Wennerstrom, Chem. Scr., 1985, 25, 117. 9 P. Attard, R. Kjellander and D. J. Mitchell, Chem. Phys. Lett., 1987, 139, 219. 10 P. Attard, R. Kjellander, D. J. Mitchell and B. Jonsson, J. Chem. Phys., 1988, 89, 1663. Paper 0/02319A; Received 24th May, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000107

出版商:RSC    

年代:1990

数据来源: RSC

摘要:

Furaduy Discuss. Chern. SOC., 1990, 90, 115-127 Rheology of Weakly Interacting Colloidal Particles at High Concentration Richard Buscall," Ian J. McGowan and Colin A. Mumme-Young ICI plc, Corporate Colloid Science Group, PO Box 11, The Heath, Runcorn, Cheshire WA7 4QE The paper describes an experimental study of the rheology of concentrated latex under conditions where a secondary miminum in the interparticle pair potential is expected. Particular attention is given to the Bingham stress uB derived from steady-shear-flow data, which is taken as a measure of the additional or extra stress arising from the interparticle attraction. The data are used to deduce a correlation describing the dependence of the Bingham stress on the strength of attraction, S, the latter being taken as proportional to the calculated depth of the secondary minimum, with the result uB= l ~ f ( 4 ) S ' .~ d - ~ , where k is a constant and d is the particle diameter. A weak attraction between colloidal particles can also be induced by the addition of non-adsorbing polymer. Data for systems of this type are analysed similarly and shown to support this dependence. Earlier data illustrating the effect of particle size and concentration on the storage modulus are also discussed. The dependence upon volume fraction is shown to be very similar to that of the Bingham stress and the variation of both properties is described well by an equation deduced from liquid-state theory on the assumption that the attractive forces do not greatly perturb the structure, this reads G,, uBK (442+ 2+3 - 44)/( 1 - 4)3.It is also shown, however, that for volume fractions 4>0.2 or so this is virtually indistinguishable from the concentration dependence of the modulus observed for particulate gels formed by coagulation in the primary minimum. It would thus appear that the concentration dependence of the modulus is very insensitive to local structure and the strength of the attraction. In contrast, this is not the case for the particle-size dependence. The effect of attractive forces on the low-shear and high-shear viscosity coefficients is also described. A central aim in dispersion rheology is to understand and quantify the effects of particle interactions on the bulk properties of concentrated colloidal dispersions. Progress towards this aim depends, in part, upon obtaining rheological data for model dispersions that are sufficiently well characterised for the particle interaction potentials to be estimated theoretically.Much of the recent work in this vein has been concerned with systems with purely repulsive interactions. Thus a fairly substantial body of data now exists concerning the shear viscosity and linear viscoelasticity of model systems for which the interactions can be regarded as hard sphere like.'-6 There are also data available illustrating the effect of soft electrostatic repulsion on the elasticity and flow of ordered latex.'-I3 In the case of attractive forces it is helpful at the outset to distinguish between weak attraction of the type associated with a secondary minimum in the DLVO potential, or with depletion flocculation, and the strong van der Waals attraction experienced by fully destabilised particles in the primary minimum.This distinction is helpful, if not essential, in part because there are problems with arriving at even an order-of-magnitude estimate of the van der Waals attraction between particles in 'contact', and also because there are expected to be qualitative as well as quantitative differences in rheology between 115116 Dispersion Rheology systems where the attraction is weak enough to cause reversible aggregation, and those where it is strong enough for diffusion to be eliminated entirely. Rheological data for irreversibly aggregated systems tend to be interpreted on a purely structural/geometric basis14-16 although an approach based on forces is favoured by The present paper is concerned with the effect of a weak secondary minimum on the rheology of concentrated latex and with a comparison of data for systems of this type with those obtained when the attraction derives from the presence of added, non-adsorbing polymer (depletion flocculation). It is well known that a secondary minimum in the potential-energy curve for aqueous electrocratic dispersions can be arranged by reducing the Debye length ( I D = K - ' ) by the addition of electrolyte to the point where the tail in the van der Waals attraction is no longer swamped by the electrostatic repulsion.The addition of judicious amounts of electrolyte alone does not, however, result in systems amenable to rheological investigation.This is because the primary maximum is also lowered to the point where rapid orthokinetic coagulation occurs when the dispersions are subjected to shear. However, this problem can be avoided following Qttewill and Walker" by means of the adsorption of monodisperse non-ionic surfactants having short ethoxylate chains, the idea being to create a steric barrier just thick enough to prevent access to the primary minimum. The thickness of the layer 6 may or may not itself determine the position and depth of the secondary minimum, depending upon the relative values of S and I D . This idea was exploited in an earlier carried out in collaboration with this laboratory. In this earlier work by Partridge et al. it was shown that the elastic moduli of the dispersions could be predicted quite accurately from an equation of Zwanzig and Mountain'l together with theoretical estimates of the interaction potential.This result might reasonably be taken to imply that theoretical DLVO potentials are good enough to provide a meaningful basis for the analysis of other rheological data obtained for weakly attracting systems. In their study Partridge et al.19*20 examined the effect of solid-phase volume fraction on the limiting shear modulus and sedimentation rate of three latices with mean particle diameters, d, between 0.97 and 1.9 pm. In addition, the steady-shear-flow behaviour of one of the latices was studied in detail. This paper describes a complementary study of the effect of particle size on shear flow in which the particle size is varied between 0.4 and 3.5 pm. The quantity of particular interest here is the Bingham stress; this is taken as a measure of the additional shearing stress arising from the interparticle attraction.In theory, varying the particle size causes the attraction to vary in near proportion, everything else being equal (fig. 1, below). The data, in conjunction with a simple dimensional argument, will be used to infer the dependence of the Bingham stress on the strength of attraction. The resulting correlation will then be compared to that derived from similar data obtained earlier for latex subjected to depletion flocculation. Experimental Surfactant-free polystyrene latices were prepared and purified by the usual method^.^^-'^ The pure monodisperse non-ionic surfactants used were hexaethyleneglycol mono-n- dodecyl ether (C,,E,) and octaethyleneglycol mono-n-dodecyl ether ( C,2E8) obtained from the Nikko Chemical Co.Ltd. (Tokyo). Monolayer coverage of the latex was achieved by means of titration using surface-tension measurements to detect the end point, and an adsorption isotherm given by Partridge" for a latex not too dissimilar in size as a guide. The flow curves were determined at a temperature of 298 K using a Bohlin CS controlled stress rheometer. Steady-state curves were obtained, starting at the lowest stress, by shearing at each stress until a constant shear rate was obtained.R. Buscall, I. J. McGowan and C. A. Mumme-Young 117 Interaction Potentials The theoretical interaction potentials employed in the discussion to follow were calcu- lated according to, U ( r ) = u A ( r , + UR( r , + US( r , (1) with the van der Waals attraction given by,25 where A(x) = x2+2x and x = ( r / d - 1).The Hamaker coefficient Aeff(r) is expected to depend upon distance by virtue of screening by electrolyte of the zero-frequency term and of retardation of the higher-frequency terms.26y27 Retardation was dealt with by means of the approximate method suggested by Mahanty and N i ~ ~ h a m , ~ ’ the effect of retardation was, however, rather insignificant around the secondary minimum. Aeff( r ) could almost equally as well have been taken to be constant with a value of 0.0064aJ (1.56 kBT at 298 K). This value agrees with that quoted by Hough and White26 except that it has been assumed that the zero-frequency term is fully screened out by the relatively high electrolyte concentrations used in this work.In contrast, Partridge et al. ignored screening and used a value of 2.31 k,T. The electrostatic potential was taken as, UR( r ) = .rrd&,tzo& In { 1 -t exp ( -KT) exp [ K ( d + 26)]} (3) which is a good approximation for large K d . (c16 here is the electric potential at the edge of the adsorbed layer and, following Partridge et al., this was taken as equal to the 6 potential. The steric potential was approximated by,28 rrdkBT - U,( r ) = - & ( 4 - ~ ) ( 2 6 + d - r)’; 6 < ( r - d) < 26. Vl (4) This expression assumes constant density radially in the adsorbed layer, and may be considered a reasonable approximation for thin layers.The precise choice of values used for the constants in eqn (4), that is, for the volume fraction of chains in the adsorbed layer C$27 the partial molecular volume of water V, and for the interaction strength x, is unimportant since the steric term rises very rapidly indeed at distances of closest approach less than 26. Nevertheless the following plausible values were used: 0.1, 3 x m3, 0.35 for C$2, V, and x, respectively. The values of 6, the adsorbed-layer thickness, were derived by equating 6 with the fully extended chain length of the surfactant molecules as obtained from Catalin models. For example, the value used for C,*E6 was 3.95 nm. This might be compared to the experimental value obtained by Ottewill and Walker” of 5-tO.5 nm and that obtained at the air/water interface by Corkill et al.29 of 4+0.5 nm.In order to facilitate comparison of the current results with earlier results for depletion-flocculated latex, an estimate of the depletion potential was required. From Asakura and Oosawa3’ this is given by 3 1 where d23 = (d +2Rg)/2; R, is the radius of gyration of the solution polymer and no, its osmotic pressure. The latter was calculated from118 Dispersion R h eology D? 8 9 10 1 1 12 13 I4 15 16 I7 18 (r-d)/nm Fig. 1. Reduced pair potentials at the secondary minimum. The left-hand curves illustrate the effect of particle diameter (0.6, 1, 3.5 pm) and layer thickness (E6, Es). The right-hand curves compares a depletion potential of similar minimum depth. In each case dref= 1 pm. where p is the polymer number density, M, its number-average molecular weight and A2 its second virial coefficient.For POE the latter is rather and the correction to ideality is very small over the concentration range of interest here. The calculated potential curves are shown in fig. 1. The interaction energy is plotted in the reduced form U(r)dref/dkBT with dref= 1 pm which largely scales out the effect of particle size. It can be seen that the position and depth of the secondary minimum does not depend upon the layer thickness, although the implication is that it would for slightly longer ethoxylate chains. Results and Discussion Secondary-minimum flocculation was found to produce flow curves showing significant shear thinning at volume fractions equal or greater than ca.0.3. The curves were very similar in form to those observed previously for weakly flocculated latex.19921 The curves were analysed in terms of the Bingham equation which has for higher stresses and shear rates, where uB is the Bingham stress, 9 is the shear rate and 77, is the so called plastic viscosity. The Bingham stress is shown as a function of volume fraction and particle diameter in fig. 2. It can be seen that uB increases exponentially with concentration, and shows no discernible dependence upon particle size. The results obtained for the two surfactants C12E6 and C12E8 agree within the general scatter as expected. At higher concentrations the data for 1 pm particles agree reasonably well with those for very similar sized particles generated by Partridge,” there is, however, some disagreement at lower concentrations.It is suspected that this may be due to differences in the rheometers used, the older Deer PDR81 rheometer used by Partridge being somewhat sensitive to small errors in the air-bearing bias when used to measure thin fluids at lower stresses, the newer rheometer used here is much better in this respect. The correspondingR. Buscall, I. J. McGowan and C. A. Mumme-Young 2.00 \ Y t 119 0.02 1 1 I I I I I I 0.20 0.30 0.40 0.50 0.60 0.70 0.80 4 Fig. 2. Semi-log plot of Bingham stress plotted against volume fraction for different layer thick- nesses (E8). Open symbols: C&6, (0) d = 3.5 pm; (0) d = 1.0 pm; (V) d = 0.6 pm; (0) 0.4 pm. (@) d = 1.0 pm and ClZE8. (. - .) Fit of the data of Partridge" for C1&6 and d = 0.97 pm.(-1 Eqn (12). Inset: log-log plot of the same data. 200 150 17r l o o 50 0 0.2 0.3 0.4 0.5 0.6 4 Fig. 3. Relative plastic viscosity plotted against volume fraction. (m) d = 3.5 pm; (0) d = 1.0 pm; (A) d = 0.6 pm; (0) d = 0.97 pm [from ref. (19)]. (-) Eqn (8) with 4max = 0.6 and 0.65. plastic viscosity data are plotted in fig. 3. The data are compared with curves calculated from the Dougherty-Krieger equation,' which, with +max==0.6, represents well the data for neutrally stable or hard sphere-like latice~.~ Note that for the smaller latices qm= qHs and so in these cases cB can be120 Dispersion Rheology regarded as a measure of the extra shear stress arising from the weak attraction. In the case of the 3.5 pm diameter particles the limiting viscosity is enhanced substantially, as is found in fully irreversibly aggregated systems.14 Given that the shearing forces on an N-particle cluster increase as the square of the particle size, whereas the attraction is only proportional to the first power (roughly), this result might be considered surpris- ing.It implies that the effect of increasing the attractive force is stronger than might have been expected. However, the secondary minimum is estimated to be ca. 25 kgT deep in this case (fig. I ) , as compared to ca. 6 and 3.5 kBT for the other two latices, it is thus suspected that the difference is indeed something to do with a transition from reversible to irreversible aggregation. Clearly in this case uB underestimates the ‘extra’ stress due to the attractive forces.Before the steady-flow data are discussed further it will be useful to re-examine the scaling of the storage-modulus data of Partridge et aL19,20 Given the reasonable assump- tion that the modulus is proportional to the second derivative of the pair potential, everything else being equal, dimensional considerations suggest that it should then scale with particle size and concentration as, given that the particle diameter, d, is the only relevant length scale. It should be emphasised that this result assumes implicitly no variation of microstructure with U ( Y). Partridge et al. measured the limiting storage modulus, G, for three latices having particle diameters of 0.97, 1.4 and 1.92 pm and for volume fractions between 0.3 and 0.6. The experimental data for each latex were compared with predictions based on an equation of the type derived by Zwanzig and Mountain21 for simple liquids, =-[oag(~)~ 3 4 2 T d z ; a2U z = - 2 r 57rd3 az d ’ In order to perform the comparison, Partridge et al.used qd as an adjustable parameter in the pair potential and assumed that the pair distribution function, g(z), could be approximated by that for the hard-sphere fluid. They found that near-perfect agreement between the observed and predicted moduli could be obtained by using plausible values for G6, i.e. values similar in magnitude to y potentials measured on diluted samples by means of electrophoresis. The quantitative comparison made by Partridge et al. is clearly rather a composite test of the theory since it requires not only that the form of the theoretical pair potential is essentially correct but also that the hard-sphere pair distribution function is a good approximation.Because of this it is useful to re-examine the results in the light of eqn (lo), but in a way which does not require a detailed pair potential to be constructed. Several points pertinent to the ensuing discussion of the steady-flow results will be made along the way. For a constant and/or small steric barrier thickness, 8, the pair potential, U ( r ) , for r = ( d +2S) is expected to be approximately proportional to the particle diameter, d, everything else being equal (see fig. l ) , as is its second derivative in respect of r. Given this, eqn (10) predicts that the modulus should be approximately independent of particle size provided that the scaled pair-distribution function g ( z ) is not sensitive to U and thus d.Fig. 4 shows that the data obtained by Partridge et al. are consistent with this idea. The lack of dependence of G, upon particle size might be contrasted with the scaling expected for hard spheres of G, oc d - 3 , that observed experimentally for non-equilibriumR. Buscall, I. J. McGowan and C. A. Mumme-Young 2000 I-- /@ / 1000 + 500 - a" \ Q 121 50 I 0.2 0.3 0.4 0.5 0.6 0.7 4J Fig. 4. Semi-log plot of wave-rigidity modulus against volume fraction. (0) d = 0.97 pm; (0) d = 1.4 pm; (A) d = 1.9 pm. (-) Eqn (12) [data taken from ref. (19)]. Inset: log-log plot of the same data. particulate gels14-16 formed from very strongly attracting ( i.e.coagulated) particles of something like G,a d-2.5, and the even stronger dependence observed for particulate gels in which the interactions are long-range From these comparisons it might be said that the lack of dependence on particle size is a curiosity associated with weak attractions. The integrand in eqn (10) is expected to peak sharply by virtue of the rapid fall-off of d2 U ( z ) / d z 2 together with the decay of g ( z ) away from the first peak. Consequently, the concentration dependence dictated by eqn (10) may be determined by that of the height of the first peak in g ( z ) , that is, in obvious notation, ~rnx +'gm(+). ( 1 1 ) Following Partridge et al. the hard-sphere form is adopted as a first approximation and, on using the Carnahan-Starling equation of and the virial theorem to obtain the value of gm(+), eqn ( 1 1 ) becomes Fig.4 shows that this agrees well with the experimental data implying that g , is approximately proportional (but not necessarily equal) to g:'. This might be thought surprising given, first, that potentials of the order of 10 k,T might be expected to perturb g ( z ) significantly at the lower concentrations and secondly, that any perturbation in g ( z ) should logically decrease with increasing concentration. The apparent agreement should not, however, be taken to mean that the structure is not significantly perturbed. Over the range of volume fraction for which G is available, the data can be fitted almost equally as well by a power law using an exponent of ca.3.7. The significance of this type of dependence is that it is shown by particulate gels formed by the coagulation of fully destabilised particles in the primary m i n i m ~ m . ' ~ - ' ~ , ~ ~ In systems of this latter type the attractive forces are strong enough to cause the aggregation to be fully irreversible122 Dispersion Rheology and there is no doubt that the short-range structure is significantly perturbed in the sense that the particles are in strict contact with their immediate neighbours at all volume fractions.Power-law behaviour is predicted at low-to-moderate volume fractions (of order 0.1) as a consequence of the fractal microstructure, but its apparent persistence right up to close packing has been a p ~ z z l e . ' ~ - ' ~ It is possible that a cross-over from a fractal microstructure to a compact microstructure would never be noticed, given less than precise data.This point will be discussed in more detail elsewhere in the context of a comparison of various different limiting models for the modulus of a particulate gel.35 However, models apart, the experiments would suggest that the concentration dependence of the modulus is not sensitive to short-range structure, nor to a notational cross-over from reversible to irreversible aggregation. This may well be true of other average properties, although dynamic properties like the low-shear viscosity might be expected to be more indicative. Note that the modulus appears to obey the simplest possible scaling consistent with the equation of Zwanzig and Mountain and dimensional analysis, depending ultimately upon a' U/ar2, the force constant for a single interparticle bond.Given this clear-cut empirical result, it is appropriate, in the absence of any theory, to compare the steady-flow results with the corresponding naive dimensional estimates. Dimensional considerations would suggest that the non-Newtonian part of the shear stress in steady shear flow should scale as a B ~ C d - ' , supposing it to be proportional to the (maximum) interparticle force. The scatter, notwithstanding fig. 2, shows clearly that this scaling does not apply. Fig. 2 also shows that aB exhibits a rather similar dependence on volume fraction to the modulus and thus eqn (12). The data are compared with the best-fit power-law curve in the inset, the fitted curve having a slope of 3.9.The data, however, unlike those for the modulus and in spite of the scatter, show what appears to be systematic curvature when plotted on a log-log basis. Eqn (12) provides the better fit. For the purposes of comparison, note that the dependence on concentration is somewhat stronger than the cubic dependence shown by coagulated systems.14-16 The observed lack of dependence of the Bingham stress on particle size is superficially consistent with a scaling form aBK L 2 ( d U / a r ) U", with E close to unity. This is not, however, sufficient to establish that the extra stress actually has this implied dependence. In order to confirm this, it would be necessary to examine also the effect of varying the secondary minimum depth at constant particle size.The ideal way to do this would be to vary the layer thickness in stages using the largest (3.5 pm) latex. Unfortunately, this is not an option in practice since the useful range of monodisperse alkanol-ethoxylates available is very limited, chain lengths greater than E8 being very difficult to synthesise and the homologues below E5 being insoluble as a result of a merging of the Kraf€t and cloud points. An alternative way of arranging weak attractions between colloidal particles is by means of depletion flocculation. That is, by means of the addition of non-adsorbing, soluble polymers. The depletion potential [eqn (5)] depends directly on the concentra- tion of polymer in solution [eqn (6)] and so the strength of attraction can be varied in this way.Experiments of this type had been performed earlier as part of a broad preliminary study of the rheology of depletion-flocculated latex.36 The results had not been analysed in any depth, however, because of a lack of any basis for scaling the data. The present study prompted a careful re-examination of these results, as did some relevant Brownian-dynamics simulations carried out recently by Heyes and M ~ K e n z i e ~ ~ who have demonstrated excellent agreement between a flow curve obtained from an explicit simulation of a particle-polymer mixture and one obtained by considering the particles alone and allowing them to interact via the equivalent Asakura-Oosawa potential. This result suggests that the depletion interaction can be represented adequately by an effective potential for both dynamic and static3' purposes.Comparison of rheological properties arising from depletion flocculation with those arising from other forms of weak flocculation would be instructive. The water-soluble polymer used0.5 ~ 0.4 -- 0.3 X a c 0.2 0.1 -- -- -- . 0.0 0.2 0.4 0.6 0.8 1 .o PEO concentration (% w/v) I t m-t I 123 Fig. 5. Partial phase diagram for latex/ POE mixtures comparing theoretical and observed floccula- tion concentrations; d = 0.3 pm; latex stabilised by CI4Ell (nominal). (0) Observed, (-) predicted. 1 0 6 6 I ! I I I I I lo-* lo-' 1 10 c/ Pa Fig. 6. log-log plot of relative viscosity against shear stress showing the effect of added poly(oxy- ethylene); d = 0.3 pm; 4 = 0.3; latex stabilised by C14Ell (nominal), polymer concentrations (%w/v): (e) 2.8, (0) 1.4, (m) 0.8, (A) 0.7, (0) 0.6, (A) 0.4, (0) 0.3, (0) 0.2, (7) 1.0.was poly(oxyethylene), having a viscosity-average molecular weight of ca. 100 000 and a number-average molecular weight of cu. 30 000. Fig. 5 compares the observed critical flocculation concentrations of POE with predictions made using second-order perturba- tion theory in the manner described by Gast et aL3' The good agreement gives confidence in the calculated potentials. Fig. 6 shows flow curves obtained for polystyrene latex containing various concentra- tions of poly(oxyethy1ene). The flow curves are unremarkable in shape except that the curves for the two highest concentrations show evidence of another shear-thinning124 Dispersion Rheology lo I b 1 -+-.-,-+- 16' 1 10 lo* const.x u,, Fig. 7. As for fig. 6 except against reduced stress (see text). c,(% w/v) 0.1 0.2 0.5 1 2 5 10 1 .. 10 -2a 1 2 5 10 20 50 Urninlk,' Fig. 8. log-log plot of Bingham stress against - Umi,/kBT. (0) Depletion flocculation. The line is derived from eqn (13), eqn (1)-(6) and a fit to the data in fig. 3 (see text). transition at lower stress. This latter feature is and it seems to be characteristic of weak flocculation at high particle and polymer concentrations, it will not, however, be discussed further here. Fig. 7 shows that the common shear-thinning transition can be scaled by shifting the curves along the stress axis. The reciprocal of the shift factor used to obtain the best superposition, oC, which because of the lack of dependence of the low-stress viscosity on polymer concentration is proportional and very nearly equal to the Bingham stress (this is not necessarily true, in general), is plotted against polymer concentration (upper scale) in fig.8. The fitted curve, shown by the broken line, conformsR. Buscall, I. J. McGowan and C. A. Mumme-Young 125 Fig. 9. Semi-log plot of low-shear relative viscosity against volume fraction. (0) Secondary- minimum flocculation [data from ref. (19)]; (0) depletion flocculation (note that the spread shown is non-systematic, i.e. it does not correlate with the strength of flocculation); (A) 1 pm PSL flocculated by means of the addition of sodium carboxymethyl cellulose [data taken from ref. (22)]. (-) Doolittle equation: qr = K exp [#/(Oh4 - 4 ) ] .to ocN ck; which, given that the second virial coefficient of POE in aqueous electrolyte solutions is small, in turn implies that the extra stress depends upon the strength of the attraction in an essentially identical fashion (lower scale). This is, of course, very similar to the dependence implied by the secondary-minimum experiments. The range of strengths of attraction encountered in the two sets of experiments are comparable on the basis of estimates made using eqn (1)-(6), the depth of the minima in the theoretical potential calculated for the secondary-minimum flocculation (SM) experiments varying between an estimated 2.5 and 23 kBT, and that for the depletion- flocculation (DF) experiments between ca. 2 and 30 kBT. In each case the extra stress is found to be at least consistent with a dependence upon particle size and strength of attraction (SK Urnin) of the form Given this apparently consistent scaling, it is tempting to consider whether there is any correspondence between the absolute magnitudes of the non-Newtonian stresses found in the two experiments. For example, can one set of data be predicted from eqn (13) with the constant of proportionality determined from the other via estimates of Urnin.The result of such an attempt is shown by the solid line in fig. 8. The agreement is good (although it is much better than expected, given the uncertainties involved in calculating the interaction potentials). At first sight, it appears that the extra stress depends only upon Urnin and not on the detailed form of the potential.This is not, however, a sound conclusion to draw since, for equal Urnin, the estimated SMF and DF potentials are very similar in other respects (fig. 7). This is a happy, but fortuitous, consequence of the choice of molecular weight of the polymer used in the DF experiments. Other comparisons can be made. For example, Partridge determined the 'zero-shear' relative viscosity for one particle size ( d = 0.97 pm) and a series of volume fractions. Data are also available for a similar sized latex flocculated by the addition of sodium carboxymethyl celluloseYa although in this case little is known about the particle attraction except that it is very weak. These results are replotted in fig. 9. Also shown 0°K S'.9d-2. (13)126 Dispersion Rheology are data taken from fig.7 for scaled stresses of the order of 0.1. These show the effect of varying S at constant volume fraction, whereas the other two sets illustrate the effect of varying the latter at constant S. The data suggest a single curve, implying that the low-shear viscosity depends upon volume fraction, but not upon particle size or the strength of the attraction (within limits). This is a rather remarkable result, further work is clearly required in order to confirm it and to establish its generality. The data in fig. 7 are compared to the Doolittle equation which appears to account well for the divergence of the viscosity of dispersions at the glass transition,6 this equation reads with &, = 0.64 and A = 1. The comparison suggests that the viscosity should diverge at a volume fraction of 0.64 which corresponds to random close packing.Conclusions The lack of dependence of modulus upon particle size seen by Partridge et al. implies that the static structure in concentrated, weakly interacting dispersions is not perturbed significantly by the interparticle attraction. The range of particle size examined by Partridge was admittedly rather narrow. It does, however, correspond to a doubling in the strength of attraction. The concentration dependence of the modulus, which is better defined that its size dependence, can be interpreted as supporting this conclusion. However, comparison with the behaviour of particulate gels formed by coagulation shows that the concentration dependence is very insensitive to short-range structure and the strength of attraction for volume fractions of 0.2 or more.It would appear that it is only the absolute value of the modulus which is sensitive. The modulus data, whilst showing very simple and predictable dependences upon particle size and concentration, are thus somewhat ambiguous. The position with the Bingham stress is no less compli- cated. It appears to show a dependence upon concentration very similar to that shown by the modulus and this alone might be taken as an indication of ‘simple’ behaviour. However, the dependence upon particle size observed does not accord with the corre- sponding idea that the extra stress should increase in direct proportion to the maximum force of attraction. It is difficult to comment on the reasons for this in the absence of a real idea of the mechanism of shear-thinning.The secondary-minimum experiments and the depletion-flocculation experiments appear to support a scaling with particle size and strength of attraction of the form, c ~ ~ = f ( # ) d - ~ S & , with E slightly less than two. It is immaterial whether the strength, S, is defined in terms of (minimum) energy or (maximum) force, given the similarity of the calculated potentials. It would be interesting to establish whether this correlation holds more generally with S defined in one way or the other. Obvious depletion- flocculation experiments come to mind where both the concentration and the molecular weight of the polymer are varied. The limited set of data for the low-shear viscosity suggest a unique curve independent of the mechanism of attraction and its precise strength (within limits fully irreversibly aggregated systems do not behave like this).The data presented and discussed are a rather sparse sample of the experimental space [+, d, Urnin, aU/dr, q( y), G, etc.] but they show perhaps enough interesting features to render further study of model dispersions of this type worthwhile. It may be possible to build up a picture based on experimental correlations that is not much more complicated than that found for hard-sphere-like dispersions. It is hoped that this suggestion is taken up.R. Buscall, I. J. McGowan and C. A. Mumme-Young 127 J. M. Shankey is thanked for performing some of the experiments on depletion- flocculated latex.References 1 I. M. Kreiger, Adv. Colloid Interface Sci., 1972, 3, 111. 2 J. Mewis, W. J. Frith, T. A. Strivens and W. B. Russel, AIChE J., 1989, 35, 415. 3 C. G. de Kruif, E. M. F. van Iersel, A. Vrij and W. B. Russel, Chem. Phys., 1985,83, 4717. 4 J. C. van der Werff and C. G. de Kruif, J. Rheol., 1989, 33, 421. 5 R. Buscall, Faraday Discuss. Chem. SOC., 1983, 76, 338. 6 L. Marshall and C. F. Zukoski, J. Phys. Chem., 1990, 94, 1164. 7 I. M. Kreiger and M. Eguiluz, Trans. SOC. Rheol., 1976, 20, 29. 8 R. Buscall, J. W. Goodwin, M. W. Hawkins and R. H. Ottewill, J. Chem. SOC., Faraday Trans. 1, 1982, 9 J. W. Goodwin, T. Gregory and J. A. Stile, Adv. Colloid Interface Sci., 1982, 17, 185. 78, 2873; 2889. 10 D. W. Benzing and W. B. Russel, J. Colloid Interface Sci., 1981, 83, 167; 178.1 1 R. Buscall, Colloids Sur-, 1985, 15, 1. 12 L-B. Chen and C. F. Zukoski, J. Chem. SOC., Faraday Trans., 1990, 86, 2629. 13 B. J. Ackerson, J. B. Hayter, N. A. Clark and L. Cotter, J. Chem. Phys., 1986,84, 2344. 14 R. Buscall, I. J. McGowan, P. D. A. Mills, R. F. Stewart, D. Sutton, L. R. White and G. E. Yates, 15 R. Buscall, P. D. A. Mills, J. W. Goodwin and D. W. Lawson, J. Chem. SOC. Faraday Trans. 1, 1988, 16 R. Buscall, in Structure, Dynamics and Equilibrium Properties of Colloidal Systems, ed. E. Wyn-Jones 17 R. J. Hunter, Adu. Colloid Interface Sci., 1982, 17, 197. 18 R. H. Ottewill and T. Walker, Kolloid 2. 2. Polym., 1968, 227, 108. 19 S. J. Partridge, Ph.D. Thesis (University of Bristol, 1985). 20 J. W. Goodwin, R. W. Hughes, S. J. Partridge and C. F. Zukoski, J. Chem. Phys., 1986, 85, 559. 21 R. W. Zwanig and R. D. Mountain, J. Chem. Phys., 1965, 43, 4464. 22 J. W. Goodwin, J. Hearn, C. C. Ho and R. H. Ottewill, Colloid Polym. Sci., 1974, 252, 464. 23 Y. Chung-li, J. W. Goodwin and R. H. Ottewill, Colloid Polym. Sci., 1976, 60, 163. 24 A. M. Homola, M. Inoue and A. A. Robertson, J. Appl. Polym. Sci., 1975, 19, 2077. 25 H. C. Hamaker, Physics, 1937, 4, 1058. 26 D. B. Hough and L. R. White, Adu. Colloid Interface Sci., 1980, 14, 3. 27 J. Mahanty and B. W. Ninham, Dispersion Forces (Academic Press, London, 1976), p. 149. 28 D. H. Napper, Polymeric Stabilisation of Colloidal Dispersions (Academic Press, London, 1983). 29 J. M. Corkill, J. F. Goodman and S. P. Harold, Trans. Faraday SOC., 1964, 60, 202. 30 S. Asakura and F. Oosawa, J. Chem. Phys., 1954, 22, 1255; J. Polym. Sci., 1958, 33, 183. 31 H. de Hek and A. Vrij, J. Colloid Interface Sci., 1981, 84, 409. 32 W. Polck and W. Burchard, Macromolecules, 1983, 16, 978. 33 N. Carnahan and K. Starling, J. Chem. Phys., 1970, 53, 600. 34 J. D. F. Ramsay, in Structure, Dynamics and Equilibrium Properties of Colloidal Systems, ed. E. Wyn-Jones 35 R. Buscall, J. Rheol., submitted. 36 D. Belbin, R. Buscall, C. A. Mumme-Young and J. M. Shankey, P.R.I. Reprint, 1985, 7, 1. 37 D. M. Heyes, private communication; D. M. Heyes, D. W. McKenzie and R. Buscall, J. Colloid Interface 38 A. P. Gast, C. K. Hall and W. B. Russel, Faraday Discuss. Chem. SOC., 1983, 76, 338. 39 A. Morton-Jones and R. Buscall, paper in preparation. 40 R. Buscall and I. J. McGowan, Faraday Discuss. Chem. SOC., 1983, 76, 277. J. Non-Newtonian Fluid Mech., 1987, 24, 183. 84, 4249. (Kluwer Academic Press, in press). (Kluwer Academic Press, in press). Sci., submitted. Paper 0/01391I; Received 29th March, 1990

ISSN:0301-7249    

DOI:10.1039/DC9909000115

出版商:RSC    

年代:1990

数据来源: RSC