摘要:

FARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 59 1975 Physical Adsorption in Condensed Phases THE FARADAY DIVISION CHEMICAL SQCIETY LONDONFARADAY DISCUSSIONS OF THE CHEMICAL SOCIETY NO. 59 1975 Physical Adsorption in Condensed Phases THE FARADAY DIVISION CHEMICAL SQCIETY LONDONA GENERAL DISCUSSION ON Physical Adsorption in Condensed Phases 2nd, 3rd and 4th April, 1975 A GENERAL DISCUSSION on Physical Adsorption in Condensed Phases was held in the Department of Chemistry at the University of Bristol on 2nd, 3rd and 4th April, 1974. The President of the Faraday Division, Professor T. M. Sugden, F.R.S., was in the chair : about 150 Fellows of the Faraday Division and visitors from overseas attended the meeting. Among the overseas visitors were : Miss D. Baumer, West Germany Dr.K. Brauer, West Germany Dr. J. A. De Feyter, The Netherlands Prof. G. H. Findenegg, West Germany Prof. K. E. Gubbins, U.S.A. Mr. J. M. Haile, U.S.A. Dr. F. Th. Hesselink, The Netherlands Dr. D. Horn, West Germany Dr. R. 0. James, Australia Mr. H. Kern, West Germany Mr. L. K. Koopal, The Netherlands Mr. J. Krueger, West Germany Dr. J. E. Lane, Australia Prof. J. Lyklema, The Netherlands Dr. M. Millot, France Dr. D. J. Mitchell, Australia Prof. B. A. Morrow, Canada Prof. B. W. Ninham, Australia Dr. J. W. Perram, Australia Mr. H. Petry, West Germany Dr. P. C. Scholten, The Netherlands Prof. A. Silberberg, Israel Prof. C. A. Smolders, The Netherlands Dr. H. G. Smolka, West Germany Mr. J. Specovius, West Germany Dr. L. Ter-Minassian-Saraga, France Prof. L.Viet, France Mrs. M. Vignes-Adler, France Dr. H. J. Vledder, The Netherlands Dr. S. G. Whittington, Canada Mr. A. M . J. Spruijt, The Netherlands Dr. R. R. Stromberg, U.S.A.ISBN: 0 85186 858 4 ISSN: 0 301-7249 0 The Chemical Society and Contributors 1976 Printed in Great Britain at the University Press, AberdeenCONTENTS 7 Introduction by D. H . Everett 13 22 29 38 46 55 63 76 88 97 109 117 127 Self-Energy in Adsorption by J. Mahanty and B. W. Ninham A Computer Simulation of the GaslLiquid Surface by G. A. Chapela, G. Saville and J. S. Rowlinson Structure of the Liquid/ Vapour and LiquidlSolid Interfaces by J. W. Perram and L. R. White Adsorption of Fluids: Simple Theories for the Density Profile in a Fluid near an Adsorbing Surface by G. H .Findenegg and J. Fischer GENERAL DIscussIsN-Dr. S. Levine, Dr. D. Clian, Dr. P. Richmond, Prof. B. W. Ninham, Prof. D. H. Everett, Mr. G. A. Chapela, Dr. G. Saville, Prof. J. S. Rowlinson, Prof. S . G. Whittington, Dr. M. Lal, Dr. J. E. Lane, Dr. N. G. Parsonage, Dr. S. Toxvaerd, Prof. G. H. Findenegg, Dr. C. A. Croxton, Dr. J. W. Perram, Dr. L. R. White. Experimental Test of a Mdtilayer Model of a Regular Solution/ Vapour Interface by J. E. Lane Ahorption of Surface Active Agents in a Non-Aqueous Solvent b y A. Couper, G. P. Gladden and B. T. Ingram Adsorption Kinetics in Micellar Systems by J. Lucassen GENERAL DIscussroN-Prof. A. Silberberg, Dr. R. Parsons, Dr. J. F. Padday, Dr. J. E. Lane, Prof. K. E. Gubbins, Dr. E. Dickinson, Dr. A. Couper, Prof. D. H. Everett, Mr.R. C . Watkins, Dr. R. Parsons, Dr. Th. F. Tadros, Dr. B. T. Ingram. Adsorption and Structuring Phenomena at the SolidlLiquid Interface by C. E. Brown, D. H. Everett, A. V. Powell and P. E. Thorne Adsorption of Polycyclic Aromatic Hydrocarbons onto Graphite b y A. J. Groszek Infrared Studies of Adsorption at the SolidlLiquid Interface by K. Marshall and C. H. Rochester. GENERAL DIscussIoN-Prof. J. Lyklema, Prof. D. H. Everett, Dr. B. Vincent, Prof. J. S . Rowlinson, Prof. G. H. Findenegg, Mr. H. Kern, Mr. R. C . Watkins, Dr. C . E. Brown, Prof. A. M. Posner, Dr. P. Richmond, Prof. E. A. Morrow, Dr. C. H. Rochester. 133 Dipolar Eflects in Adsorption from Mixtures by S. Levine, K. Robinson, A. L. Smith and A. C. Brett6 CONTENTS 142 Analysis of Models of Adsorption of Metal Ions at Oxidel Water Interfaces by R.0. James, P. J. Stiglich and T. W. Healy 157 The Eflect of pH on the Adsorption of Sodium Dodecyl Sulphonate at the Alumina/ Water Interface by D. W. Fuerstenau and T. Wakamatsu 169 GENERAL DIscussIoN-Dr. P. Richmond, Dr. S. Levine, Prof. J. Lyklema, Dr. R. Parsons, Prof. G. M. Bell, Dr. B. Vincent, Dr. J. F. Padday, Dr. R. 0. James, Dr. P. Stiglich, Dr. T. W. Healy, Prof. A. M. Posner, Dr. M. A. Malati, Dr. D. E. Yates, Dr. L. Ter-Minassian-Saraga, Dr. Th. F. Tadros. 181 Phase Transitions in Adsorbed Polymer Systems by D. Chan, D. J. Mitchell and L. White 189 Conigurational State of Adsorbed Chain Molecules: Behaviour of Terminally Anchored Chains by A. T. Clark, M. Lal, M. A. Turpin and K. A. Richardson 196 GENERAL DIscussIoN-Dr.S. Levine, Dr. D. Chan, Dr. D. J. Mitchell, Dr. L. White, Dr. F. Th. Hesselink, Mr. R. C. Watkins, Prof. G. M. Bell, Prof. A. Silberberg, Dr. M. Lal, Prof. S. G. Whittington, Dr. J. E. Lane, Mr. R. G. Linford. 203 Structure and Properties of Macromolecular Surface Phases by A. Silberberg 209 Adsorption and Desorption Rates of Polystyrene on Flat Surfaces by W. H. Grant, L. E. Smith and R. R. Stromberg 218 Dynamic and Static Properties of Proteins Adsorbed at the AirJWater Interface by J. Benjamins, J. A. de Feijter, M. T. A. Evans, D. E. Graham and M. C. Phillips 230 Characterization of Polymers in the Adsorbed State by Double Layer Measure- ments. The Silver Iodide+Poly(vinyl Alcohol) System by L. K. Koopal and J. Lyklema 242 Eflect of Polymer Adsorption on the Properties of the Electrical Double Layer by B. V. Kavanagh, A. M. Posner and J. P. Quirk 250 GENERAL DIscussIoN-Dr. M. Lal, Dr. F. Th. Hesselink, Prof. Viet, Dr. I. D. Robb, Dr. Th. F. Tadros, Prof. J. Lyklema, Dr. M. C. Phillips, Dr. J. E. Lane, Dr. A. Silberberg, Dr. J. A. de Feijter, Dr. D. E. Graham, Dr. R. Parsons, Dr. B. Vincent, Mr. L. K. Koopal, Dr. T. W. Healy, Prof. A. M. Posner, Dr. B. V. Kavanagh, Prof. J. P. Quirk, Dr. M. J. Garvey. 262 AUTHOR INDEX

ISSN:0301-7249    

DOI:10.1039/DC9755900001

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Physical Adsorption in Condensed Phases : Introductory Paper BY D. H. EVERETT Received 5th May, 1975 In selecting topics for Faraday General Discussions one objective has been to identify an area of physical science in which new developments appear to be imminent, and so to set the stage for, and give impetus to, fresh advances. Looking back over the past seventy years one can pick out many examples of Faraday Discussions which have indeed proved to have been milestones in the evolution of a subject. It is also true that since many areas of science seem to evolve in a cyclic fashion, one may find several Discussions ten or twenty years apart which have returned to, and given new life, to a particular field of study. One may cite, for example, the electrolyte discussions of 1927,1957 and that planned for 1977; and those on liquids of 1936,1953 and 1967.The present Discussion, however, does not form part of such a sequence. In part at least this is because adsorption in liquid systems is relevant to a wide range of phenomena so that various aspects of the subject have formed part of many earlier Discussions; in particular the problems of ionic adsorption have been dealt with under the umbrella of the Electrical Double Layer.* Developments over the past few years, and foreseen in the near future, led the Colloid and Surface Chemistry Group of the Faraday Division to the view that a broadly based Discussion on the fundamental aspects of adsorption would be timely. Because of the wide range of topics which could have been included, it was decided to exclude work specifically concerned with the physical adsorption of gases by solids and to limit discussion to problems relevant to gas/liquid, liquid/liquid and liquid/ solid interfaces.In the event, papers in the first and third areas only were submitted, although many of the basic principles to be discussed here will be equally applicable to liquid/liquid interfaces. The preambles to several of the papers in this Discussion emphasise that work on adsorption in condensed systems is not only of considerable fundamental importance and interest in its own right, but is also very relevant in a wide range of problems in colloid science. It is clear indeed that the motivation for much of the work to be reported at this meeting stems from this consideration.The particular areas of interest relate first to vapour/liquid interfaces, which play a fundamental role in the formation and stability of foams, in controlling the drainage of liquid from or through porous media, and an understanding of which will contribute to knowledge of the properties of liquid/liquid interfaces of biological importance. Secondly, we shall be considering adsorption at liquid/solid interfaces, which determines among other things the properties of thin films, and the phenomena of lubrication and colloidal stability. The subject matter of this Discussion can alternatively be grouped according to the nature of the fluid phase: pure components, solutions of organic substances, surfactant and ionic aqueous solutions and polymer solutions.All of these, in their different ways, have considerable relevance to practical problems of industrial and environmental importance. * Trans. Faraday SOC., 1951, 47, 409-414 (abstracts only). 78 PHYSICAL ADSORPTION I N CONDENSED PHASES In this introductory paper I should like to outline how, in my view, the themes to be discussed are inter-related, and how work on them may be stimulated by this meet- ing. Let us start at the beginning and reflect on the fact that our understanding of the basic concept of surface tension is still lamentably incomplete 170 years after its importance was first realised by Young and Laplace. That surface tension was a manifestation of the asymmetry of intermolecular forces acting upon surface molecules was appreciated by Laplace.But in the absence of satisfactory theories of these forces-especially in dense systems-detailed understanding of the relationship between molecular structure and surface tension, both in pure liquids and solutions has proceeded only slowly. Following the elucidation of the nature of the forces between individual molecules by Debye, Keesom and London (which confirmed the empirical formulation of Mie later exploited by Lennard-Jones) the intermolecular energy of condensed systems was evaluated by pair-wise summation, and this procedure has formed the basis of theories of the bulk liquid state, and of Hamaker’s treatment of interparticle forces. This emphasis on energies at the molecular scale has earned for this theoretical approach the appellation ‘‘ the microscopic or Hamaker theory ”.In contrast, the alternative approach via quantum electrodynamics, pioneered by Lifshitz, has enabled the energies of molecular assemblies (regarded as continuous media) to be calculated from the electromagnetic properties of the individual bulk phases. This theory again has earlier roots in Maxwell’s work, and indeed current applications are often expressed in semi-classical rather than fully quantum mechanical terms : this approach is designated the “ macroscopic, or Lifshitz, theory”. The idea that electromagnetic properties, intermolecular forces and surface tension were closely linked has also a long history-as one example of this we may quote the correlations sought for, and found, between the parachor and molar refractivities.In recent years a certain rivalry between these two aspects of the problem has developed and it has been questioned whether they are compatible and whether it is justifiable to apply continuum theory at very close separations where the detailed molecular structure of one phase is “ seen ” by molecules in the other. It turns out, as Ninham and his co-workers have shown in their prolific work of the last few years, that the term “ continuum theory ” is not really justified, for the underlying ideas can also be applied to molecules which can be represented by a region of space having an inhomogeneous polarisability conveniently described by a Gaussian distribution whose width is a measure of the size of the molecule : this leads to the concept of the self-energy of a molecule.Ninham’s paper in this Discussion reviews some of the progress which has been made in this area. Although the specific applications outlined in this paper are concerned mainly with gas/solid interactions, we may expect further developments dealing with liquid/solid systems in more detail, especially if, as is foreshadowed, the theory will be essentially parameter-free. However, the continuum theory pays minimum attention to the statistical mechanical aspects of the distribution of molecules in interfacial regions. Yet the properties of interfaces are largely dependent on their structure in molecular terms. Unfortunately, there is little prospect of experimental data on surface properties providing more than broad indications of this structure. Thus the measured quantities such as surface tension or adsorption correspond to integrals of excess quantities taken through the interface: surface tension results from the distribution of excess tangential stress through the surface while the adsorption is the integral of the excess amount of matter in the interface.A given distribution of stress or matter leads unambiguously to a definite value for the surface tension or adsorption; but a singleD. H. EVERETT 9 measurement of either can tell us little about the distribution: in principle a given surface property can arise from different detailed distributions in the surface. Indeed in non-equilibrium surface states such a situation may arise in practice. It is only in the equilibrium state-where nature decides the distribution-that a one-to-one relation exists.Much effort has been devoted to the theoretical study of the distribution of matter and stress in the gas/liquid interface, using thermodynamic and statistical mechanical arguments. The simplest theories consider an abrupt step-function in density at the surface; more detailed theories deduce that the density falls monotonically from that of the liquid to that of the vapour over a few molecular diameters. More recently, following their successful application to the elucidation of the structure of bulk liquids, Monte Carlo and molecular dynamic techniques have been used to check other approaches. A major problem which emerges is that of deciding whether or not molecular layering (leading to an oscillatory density distribution function through the surface) occurs at the vapour/liquid interface: the papers by Rowlinson and his co-workers, and by Perram and White, address themselves, inter alia, to this problem.Findenegg and Fischer consider the similar problem for a fluid near a solid wall. It is clear that, whatever their detailed structure, “ surface layers” even in a simple system must extend in a direction normal to the surface for distances corres- ponding to several molecular diameters. As indicated above, this is to be expected on theoretical grounds, but is also in many instances a necessary qualitative deduction from experiment : analysis of adsorption measurements, using realistic estimates of molecular size, shows that the data cannot be accounted for without assuming multi- layer adsorption.Although much experimental work on adsorption from solution has been analysed in terms of models which assume that all composition changes occur within one molecular layer (i.e., monolayer theories) it has been known for some time that, except possibly for ideal solutions, such models are unrealistic and in most cases are thermodynamically inconsistent. Nevertheless they provide a convenient basis for the correlation of data, and as such have proved useful. But it is important to assess the validity of attempts to establish more realistic multi-layer models : several theories are available but comparison with experiment is not always easy because of the difficulty of knowing with adequate certainty the appropriate values of the various parameters involved.Furthermore, the body of accurate experimental data with which to test the theories is woefully inadequate. It is important, therefore, to have presented here Lane’s careful measurements and analysis of the properties of the vapour/liquid interface for the cyclohexane + carbon tetrachloride system. That we are still unable satisfactorily to account in detail for the surface properties of this system represents a major challenge. The papers from Edmonds and McLure and from Couper, Gladden and Ingram deal with rather more complex systems. The former authors provide interesting thermodynamic data which in due course will provide tests of more elaborate theories : in particular they show that, as with bulk properties, fluorocarbon + hydrocarbon mixtures exhibit some unusual surface properties.The systems studied by Couper et al. form micelles in the bulk solution, and measurements of surface tension provide a convenient and precise method of studying such phenomena. Similar types of system have been studied by Lucassen: here the measurement of surface properties has been applied to a study of the kinetics of micellar aggregation. These two papers illustrate the way in which studies of adsorption, through surface tension measurements, can be applied to other problems. Evidence that under certain conditions quite major structural changes can occur10 PHYSICAL ADSORPTION IN CONDENSED PHASES at the solid/liquid interface has been accumulating for several years and the paper from Brown et al. collects together and analyses information on alkane/carbon, alkanoll carbon and various solution/carbon interfaces.For pure liquids the adsorption data of Findenegg and the calorimetric data of Thorne lead to clear and qualitatively similar conclusions about the thickness of the structured layers of n-alkanes and n-alkanols at the graphite surface: these layers exhibit properties closely similar to the solid material. Similar conclusions result from measurements of adsorption by graphite from n-alkane and n-alkanol mixtures by Bown and by Brown. Perhaps the most significant feature of this work, which is closely related to later papers on polymer adsorption, is the dependence of these effects on the length of the hydrocarbon carbon chain. It is also interesting and important to discover that chain-branching, in at least one instance, eliminates the structuring, as does the substitution of a less well ordered (" non-graphitised ") surface for one consisting essentially of basal planes.Groszek's paper on the adsorption of condensed ring hydrocarbons by carbon surfaces exhibits similar features which are still not well understood. One might hope to be able to learn more about the interface from spectroscopic and similar techniques. Some of these (e.g., ellipsometry) while giving general information about the thickness of surface layers are of limited use, since to interpret the data it is first necessary to introduce a simple model of the interfacial structure, and what is deduced is a parameter dependent upon some average properties.N.M.R. methods using isotopically labelled compounds may prove more valuable since it may be possible to identify those groups in the molecule which are perturbed in the adsorbed state. But this technique is still being developed and, because of the small fraction of the molecules in the system which are in the interface, makes considerable demands on high sensitivity. Nevertheless, one may look forward to progress in the measurement and in the interpretation of n.m.r. data relating to surface phases. The use of infra-red spectroscopy, already widely used for gas/solid adsorption has only recently been extended to the solid/liquid interface. Marshall and Rochester's paper illustrates some of the possible applications of this technique: it is, of course, particularly useful (as with gas adsorption) for those systems involving silica where strong hydrogen bond interactions between the adsorbate and the adsorbent lead to substantial modifications of the infra-red spectrum of both the solid and adsorbate.The second half of the Discussion moves on to generally more complex problems, in which there is a strong interplay between forces of different kinds-dipolar and electrostatic forces being added to dispersion forces. Three groups of systems are represented here; first, those in which two dipolar species compete for adsorption on an electrically charged interface, and in doing so modify the potential drop across the surface layer ; secondly, those in which the solid surface itself can take part in ionic exchange or dissociation equilibria ; and, thirdly, those in which the adsorption of an ionic surfactant is influenced by pW.These papers are clearly important and relevant to practical problems involving oxide/aqueous solution interfaces. The concept of the formation of" hemimicelles " at a surface may be relevant to the structural changes at the graphite/alkane interface mentioned above. These pagers lead on naturally to the major problem of polymer adsorption, to consideration of the conformation of adsorbed polymers, and of the phase changes which they can undergo. Silberberg's paper introduces the general problems of polymer adsorption and directs attention to the distorting effect of a surface upon the segment distribution function and as a consequence on the centre-of-gravity con- centration function: it would be interesting in connection with studies of alkane adsorption to know to what extent the results for polymers may be applied to shorter chains.For example, the thickness of adsorbed layers of alkanes seems to be aD. H . EVERETT 11 linear function of chain length for n > 6 , while for polymers the thickness is pro- portional to the radius of gyration which varies with n*. Chan, Mitchell and White make use of the methods which have been so successful in the hands of Edwards and de Gennes for polymer solutions, in conjunction with the " continuum '' approach to intermolecular forces to show how a " phase change " in the adsorbed polymer can occur depending upon temperature or solvent composi- tion; Clark, Lal, Turpin and Richardson apply the Monte Carlo technique to the particular problem of the configuration of a polymer molecule, one end of which is terminally anchored, to throw further light on the effect of solvent/polymer interaction on conformation.One of the major experimental problems in the study of polymer adsorption arises from the slow attainment of an equilibrium conformation; and the even longer time needed to reach equilibrium on desorption. That adsorption proceeds in two steps- initial adsorption of one or more segments by collision with the surface, followed by rearrangement of the chain conformation-is investigated by Grant, Smith and Stromberg and reported here. In the final three papers we enter, I think, even more difficult areas. Protein adsorption even at the waterlair interface is a complex phenomenon, depending not only on the nature of the protein but also on the electrical state of the molecule.Until the simpler systems are understood, we shall be a long way from a realistic theory of the more general case of proteins at liquid/liquid interfaces and in membranes. The paper from two Unilever groups on casein adsorption provides some interesting data, some of which are not at first sight easy to understand. In particular, I antici- pate that some will wish to enquire more deeply into the apparent failure of the data to conform to the Gibbs adsorption equation. Studies of electrical double layers have played a major role in the elucidation of the colloidal stability of hydrophobic dispersions : and even though the problem has not been solved fully in simple systems, bolder experimentalists, recognising many practically important systems, have begun to examine the interplay between the structure of double layers and the adsorption of non-ionic polymers.Koopal and Lyklema have worked with one classical dispersion-silver iodide-while Kavanagh, Posner and Quirk have studied another-gibbsite. Both conclude that the polymers form relatively thick layers and that these layers can account for the influence of such polymers on colloidal properties. We may hope that from this Discussion will emerge a clarification of a range of problems, and an appreciation of the way in which further progress in others can usefully be pursued. But one must stress that the papers in this Discussion only take us part-way to the elucidation of the nature of interparticle forces and their dependence on adsorption phenomena.For all the papers in this Discussion are concerned with adsorption on a single surface-what is important in determining colloidal stability is the adsorption between two neighbouring surfaces. The basic problems involved require answers to the following questions. (i) Does the adsorption depend upon the distance between the surfaces? (ii) If so, how rapidly is adsorption equilibrium achieved when the separation changes? (iii) If the adsorption does not change (either because the equilibrium adsorption is insensitive to separation, or because the time constant for equilibration is too long in relation to the time scale of the process concerned), how do the forces between the adsorbed layers themselves influence the overall interparticle energy ? The first question is important because of the relationship between the effect of separation on the adsorption isotherm and interparticle energy as shown in separate thermodynamic treatments of Hall, and of Ash, Radke and myself.The relationship12 PHYSICAL ADSORPTION I N CONDENSED PHASES between the interparticle energy (Vp(c)) at a given separation (h) in a c-component solution, and that at the same separation in pure solvent (1) is linked to the change in the relative adsorption of the various species with separation and with bulk chemical potentials pf through the equation i=c Pul The primary information needed is that on the relative adsorption isotherm--I'i,l as a function of pi-at infinite separation and at separation h for each of the components.We have shown incidentally that this equation applies both to non-ionic systems and electrolyte solutions provided proper attention is paid to the electrical state of the surfaces: in the latter case application of a simple point charge model of ionic adsorption leads to equations essentially identical to the DLVO results. However, apart from this case, little theoretical or experimental attention has been paid to the evaluation of adsorption isotherms as a function of the separation between adsorbing surfaces; this I believe to be one of the more important tasks in the future. The problem of the kinetics of adsorption, too, needs further study, since in many instances the duration of an encounter between two particles undergoing Brownian motion may be too short for adsorption equilibrium to be achieved. In this case the interparticle forces are those corresponding to particles coated with a layer of adsorbed material-and this is a problem which has been tackled both in terms of the Hamaker approach by Vold, and of the Lifshitz approach by Ninham. However, even though the magnitude of the adsorption may not change with separation, the conformation of the adsorbed layers (especially if they are polymeric) may: the problem here is that of the so-called steric stabilisation upon which much work has already been done- but where a much deeper analysis still has to be attempted. In conclusion, it is clear that our Discussion will cover much ground-but that there are links between the various topics which evolve from apparently simple problems to which we may soon expect satisfactory answers, to very much more complex problems which represent a substantial challenge. One may hope that the stimulus given by this Discussion will lead to substantial progress in the future. Note that the paper by Edmonds and McLure, referred to by Prof. Everett on p. 9 and by Dr. Dickensoii on p. 89, was read and discussed at the meeting but was:not submitted for printing. Ed.

ISSN:0301-7249    

DOI:10.1039/DC9755900007

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Self-energy in Adsorption JAGADISHWAR MAHANTY Department of Theoretical Physics BARRY W. NINHAM* Department of Applied Mathematics, Research School of Physical Sciences, The Australian National University, Canberra, A.C.T. 2600, Australia AND Receiued 10th December, 1974 The concept of self-energy of a molecule, the dispersion analogue of the Born self-energy of an ion, is developed. The use of this concept in theories of adsorption and interfacial energies is discussed. When molecular size is taken into account, the theories of Lifshitz, Brunauer, Emmett and Teller, restricted adsorption and the Hill theory all emerge as special cases. In general a simple power law of the isotherm is not always appropriate for multilayer adsorption. The concept of self-energy takes on significance whenever an object has a finite extent or is delocalised.For then the abstraction that the object can be considered separately from its surroundings becomes philosophically tenuous, as one part of the object can consider its other parts to belong to the rest of the world: hence, perhaps, the uncertainty principle. No difficulty occurs if the environment or object are immutable. If the opposite holds, the reaction of the (changed) environment to the object will be different and the sev-energy due to this reaction field will be different. The shift in self-energy due to radiative corrections to energy levels is a central problem of quantum electrodynamics. The Born 9 (electrostatic) self- energy of an ion is important in electrolyte theory, physical adsorption and in the migration of ions through membranes.Debye-Huckel theory results from the change of Born energy due to other ions. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager’s theory of the dielectric constant for dipolar fluids. Equally, in any theory for the surface energy of water, or adsorption of dipolar molecules, the self-energy of a dipole as a function of its distance from the interface must be invoked. In adsorption proper the same self-energy for an ion appears in the partition function. Once this is determined, so is the adsorption isotherm, and the change in interfacial tension due to dissolved ions.5 The point of this preamble is to stress that new developments reviewed here, which may appear mathematically abstruse or obtuse, stem from one fundamental physical concept.We shall focus attention on consequences for dispersion forces and adsorption which come about provided we admit that molecules are not points. Semi-classical aspects of such radiative effects have received a fair amount of attention lately: and the desirability of further work is clear. To fix ideas, consider physisorption from a gas. The self-energy problem arises as follows. Recall that for many layer adsorption continuum theory gives an isotherm ’-lo P l (noti-retarded); In - cc - (retarded), P , l4 1314 SELF-E NER G Y IN ADS 0 R P TI ON where AH is interpreted as in Lifshitz t h e ~ r y . ~ This form had been in dispute, but recent comparison of experiment l1 and theory l2, l3 for liquid helium leave little doubt that Lifshitz theory is quantitatively correct here, in spite of continuing com- putational difficulties due to incomplete spectral data.Additional support comes from a priori calculations on spreading of hydrocarbons on water 14* l5 and on sapphire.16 However, for very thin layers continuum theory breaks down, and a microscopic theory is necessary. 7-1 We require the molecular partition function exp( - E,/kT) where EI is the energy required to bring an adsorbate molecule from the jth absorbed layer to the gas phase. Because of the divergence in dispersion energy ( c ~ l Z / ~ ) near an interface, the molecular partition function had to be treated as a phenomenological term, severely limiting predictive capabilities. B.E.T. theory gives ln(P/P,) GC 1 /Z, and although it has seen good service, appears impossible to reconcile with eqn (1). The divergence difficulty is not peculiar to adsorption.Theories of interfacial tension, interaction energies (via either macroscopic or micro- scopic approaches), transport in inhomogeneous media, polymer adsorption and configurational problems, all require resolution of this problem. The introduction of finite size into the description of a molecule leads to a finite value for dispersion energy and the difficulty disappears. A parameter free theory of adsorption which reconciles existing theories can then be constructed. MOLECULAR SIZE A N D SELF-ENERGY The dispersion self-energy of a molecule of finite size can be defined 20-22 as the change in its energy due to its coupling with the electromagnetic field, or equivalently, as the change in zero point energy of the field due to its coupling with the oscillating dipole moment it induces on the molecule.We need to characterise molecular size, a notion which is hardly new. London 2 3 extended his theory of dispersion forces between point molecules to include extended electronic oscillators. More recently dispersion forces between large molecules has received much attention.24 We consider a single atom centred at R interacting with the radiation field, and follow an approach developed in several earlier papers.2o* 21 In Lorentz gauge, after Fourier transformation with respect to time, Maxwell’s equations become 1 (3) To close the equations we need a relation between the dipole moment density p(r, w) and electric field E(r, co), which for point molecules within the framework of linear response theory is E(P, co) = a(o)E(R, o)6(r-R), (4) where a(o) is the polarisability tensor of the molecule.This relation cannot be strictly correct, because the dipole is spread out over a region of the order of the volume of the molecule and in a semi-classical formulation it can be shown 20* 2 1 that the actual relation is ( 5 ) where ar(r-R, w ) is peaked around R with a range of the order of the size of the molecules, and can be given an explicit expression in terms of a sum over matrix p(r, a) = a(r-R, w)E(R, a),J . MAHANTY A N D B . W. NINHAM 15 elements of atomic wavefunctions. That is the end of the matter, for the solution of (2) is now G(r-r’, w)a(r’-R; co)E(R, co) d3r‘ s 4nio A(r,co) = -- C with G(r-r’ ; co) the diadic Green function solution of i.e. Putting (6) into (3), and taking r = R, we have a secular equation for the perturbed frequencies of the electromagnetic field : II+4n6(R9R; o)l = 0; 6 ( R , R ; o) = S d 3 r ” ~ , + V r V r ) G ( r r ” ; co)a(r”-r’; co).(9) Immediately then, the dispersion self-energy of the atom is d odw - lnlI+4nQ(R, R; o)l N 2fi d( trace 6 ( R , R; it). (10) fi ”=Gf du;, The effect of the spread in polarisation density is to make B(R, R ; is) convergent, and it is this spread, analogous to the current distribution in a two-level atom 2 5 which gives convergent results for radiative corrections to atomic energy levels. Now assume, for computational convenience only, that the atom is isotropic and that a(r, co) = la(co)f(r) where the form factorf(r) is a peaked function and take (a is the size of the atomic system) f ( r ) = In the non-retarded limit, we have r m - 1 exp( - 5) +a a2 * L I C 4 d< a(i5) N - rydberg (H atom) J n E, E! ~ n3a3 J o so that the dispersion self-energy is of the same order of magnitude as the binding energy, but the opposite sign.The same formalism permits extension to two or more atoms. The interaction energy, the difference between the complete energy of the coupled system and sum of dispersion self-energies of two isolated atoms reduces to London-Casimir results for IR1-Rzl % a, but remains finite at zero separation. For like atoms this energy is of the order of the binding energy of the molecule formed by them.The same concepts can be used to develop a simple semi-classical estimate of the Lamb shift in hydrogen,26 and to explain the differences in binding energy (face centred cubic versus hexagonal close packed) of rate gas crystals. 9 * A slightly different approach can be based on charge rather than polarisation distribution, and the formalism16 SELF-ENERGY IN ADSORPTION extended to include quadrupole and octopole effects.” Interestingly, for a three- dimensional oscillator form factor, the net potential curve appears to show a deepening of the “ bowl ” part of the potential which is necessary to fit the thermodynamic pro- perties of argon calculated by Monte Carlo For a many body system the same formalism holds. The sole difference is that the complete expression for dispersion energy now becomes 21 a 3Nx 3N matrix, the 0th submatrix of which Gi(Ri, Rj; co).If the molecules are considered to constitute a material medium, the same kind of analysis can be carried through to work out the dispersion energy, and interaction energy between different parts of the system in terms of the macro- scopic fields in a dielectric medium. The entire N-particle dispersion energy can be shown to be 21 E,(N) = - 47ci d o In1 I - 4n6‘M)I, where WM) is a 3N x 3N matrix with 3 x 3 elements 6iM)(R,, Rj ; a) constructed from Maxwell’s equations. We can now construct a theory of surface 2o and interaction 21 energies which goes beyond and substantiates those of other authors. 30-33 Liftshitz theory emerges in the limit that separation of bodies is greater than the size of any molecules, and the rigorous result that the Lifshitz energy between two bodies is the difference between the surface energy of the whole system and the sum of surface energies of the bodies taken in isolation follows.SELF-ENERGY OF A MOLECULE NEAR AN INTERFACE In an inhomogeneous medium, self-energy depends on position, and this change with position provides a force field acting on a molecule. To lowest order in WM> the expansion of eqn (13) gives E,(N as a sum of self-energies of individual molecules, that of the Zth being (14 E,(E) 21 - - 4dco trace 6(M)(Rl, R,; 0). i J The general expression is much more complicated, and this potential is analogous to that used in the theory of the inhomogeneous electron 35 For illustration, we exhibit this potential for a molecule near an interface of two dielectrics (1,2), in the non-retarded limit.Take the z axis perpendicular to the interface and medium 1 to the left. Then if r’ denotes source and r field point, we have ( 1 9 WM)(r, r‘ ; w) = -V,VptJ G(M)(~, r”)cI(r”-r’ ; co) d3r” with V2GCM)(r, Y‘) = Id(r-r’). Since we are now concerned with macroscopic fields, the boundary conditions are continuity of G(M) and ~ 3 G ( ~ ) / d z , and explicitly with 6(z) and sgn(z) step and signum functions and AI2 = ( E ~ - E ~ ) / ( E ~ + E ~ ) . This result is familiar. Recall that - 4 ~ G ( ~ ) ( r , r‘) is the electrostatic potential at r due to a unit charge at r‘, which includes direct and image potential, so that -44nV,V,GM)(r, r’) * This extension has been carried out by D.D. Richardson, J. Phys. A, 1975, 8, 1828.J . MAHANTY AND B. W. NINHAM 17 is just the field at Y due to a dipole at P'. the interface we find For a gaussian molecule at distance z froin E,(z) = k3 P a so dr . ( i e ) ~ ( ~ + 6 ~ ) - A ~ 2 ( ~ - 6 ~ ) x { exp( - $) - '+ erfc (t) + G3 ~ ' " " (exp - ( t 2 ) - exp - la)^) dt}] (1 8) n3a3 .s2 0 2A * .(it). dt-, ( z + +a) 1 1 1 --s -+ n+a EJr d t u(i<){-(-+-)-G(L-')}; 2 E l E2 3 E2 E l (z = 0). (19) Further, for large z, izl & a, the force on the molecule is the well-known result, e.g. ref. (36) It turns out that this is a good asymptotic representation when z 2 2a, but for z -+ 0 force and energy tend to a finite value. Such expressions provide very easily the change in interfacial tension at a liquid interface due to dissolved (non-ionic) molecules.PHYSICAL ADSORPTION We return to physical adsorption and the reconciliation of Langmuir, B.E.T. l7 and Lifshitz isotherms, and follow the formulation of de Boer 38 illustrated in fig. 1-3. Following an earlier paper 39 for a molecule MI in the first layer, with the zero of energy the self-energy in the gas phase at z = co, we have from eqn (18) A1 3 Ei(a) = -- tic' Jm d< a3(i&--, 7ra3 E3 where c1 z 0.2 is a constant whose value depends weakly on the form factor. the alternative model of fig. 3 eqn (1 9) gives the corresponding result. For The difference FIG. 1.-Schematic representation of adsorbed layers. Molecules MI, M2, M3 . . . sit on filled layers with dielectric properties of the bulk adsorbate18 SELF-ENERGY I N ADSORPTION Qint = -(Ead-Egas) = E:l)(O)-E,'l)(a) gives the activation energy which must be taken up by an adsorbed molecule to allow it to move over the surface, and can be estimated using typical U.V.data. For molecule Mj in the jth layer, we assign the Mi E3 E3 U € 2 E l zi = z j = ( 2 j - l ) a v 1 f , = O M I € 1 FIG. 2.-Model for calculation of self-energies. FIG. 3.-An alternative model. Here the adsorbed molecule nestles closely into the bulk adsorbate or liquid layer. underlying adsorbate dielectric properties of the bulk liquid. The corresponding self-energy which includes multiple imaging is much more complicated, but to lowest order in cx consists of (1) the energy of adsorption from the gasfphase onto an infinite medium 2, and (2) the interaction energy of a molecule 3 with medium 1 across a medium 2 of thickness (2j- 1)a.The last term gives the l/Z3 isotherm for thick films and is absent from B.E.T. theory. The first term does not appear in the theories of Halsey and Tgnoring entropic effects, the isotherm can be written in de Boer's notation 38 as j The symbols have their usual meaning and Q j = - Eik) is the heat of adsorption in thejth layer. With as above, it can be shown 39 that for largej, Q j comprises 4 terms: (a) the contribution to the surface energy/molecule of medium 2 at the interface [12]; (b) at the interface [23]; (c) the energy of interaction/molecule of a molecule at the surface of medium 3 with 1 across 2 ; (d) the energy of condensation required to take thejth molecule from gas to bulk liquid phase.We now do the sums and find (I) the Lifshitz isotherm emerges automatically under the condition I % ln(p,/p) & 1 / &, where k = 119 depends on molecular size. The second inequality is automatically satisfied by a continuum theory, but if it is reversed, and the conditions under which this can happen are clear, then (11) the isotherm becomes identical with B.E.T. theory : J . MAHANTY AND B . W. NINHAM 0- K The difference is that present theory contains only one parameter, go, the number of adsorption sites, rather than three. Evidently, depending on molecular size, pressure and dielectric properties, a wide range of behaviour can be expected in principle even for the special case of dispersion forces, and attempts to fit isotherms to a power law may be inappropriate.In general the more complicated theoretical expressions should be used. (111) At lower pressures restricted adsorption can, but need not necessarily emerge automatically. A step-wise isotherm need not indicate composite surfaces and depends on polarisability, size and dielectric properties in a predictable manner. (IV) We have ignored lateral interactions in the first layer. If the entropy of the first layer is included, and we make two extreme assumptions: (a) neglect terms in Eil) which involve the influence of the substrate, and (b) assume that at coverage 8 = o/ao the adsorbed layer has dielectric properties 6; = (1-8)+& where g2 is the bulk adsorbate dielectric constant, then the Hill isotherm also emerges.In general the Hill isotherm 37 is too drastic an approximation, and the formalism above permits the inclusion of the nature of the substrate which strongly affects the strength of the lateral interaction and condensation. We remark further on this point in the concluding paragraph. ELECTROLYTES AND CONDUCTION PROCESSES For completeness we remark briefly on effects of mobile ions or charge carriers (metals) in adsorption on electrically neutral surfaces. (For charged surfaces, while some answers have been obtained,40 much work remains to be done.) At finite temperature eqn (1 3) becomes the free energy arising out of dispersion interaction : 2nnkT C,, = T , 00 F = kT c' In D(i5,); n = O where D(iC,) is the secular determinant.With free charges present, the Green function which occurs in D(i5,) must be modified; e.g. in electrolytes if the source oscillates with high frequency, the ions are too massive to follow the field and con- tribute to the dielectric constant, but at low frequencies the ions respond to the field and modify dispersion interactions. For electrolytes, only the n = 0 term is modified. Many such problems can be handled through the linearised Poisson-Boltzmann For the term in zero frequency rather than (16) we have to satisfy Poisson's equation where the sum is over all ionic species of charge e, and density n, = n,(O) exp( - e+(r)/kT.20 SELF-ENERGY I N ADSORPTION Linearisation yields (27) Repetition of the same analysis then gives the change in interaction free energy between two molecules due to electrolyte as 22 (V" - rc;)+(v, v') = - 4n6(r- r'), l/rcD = Debye length.{6[exp( - ~ D R ) - 11 f AF(R) = -- kTa2(0) 2&2(0)R6 and a temperature independent change in the self-free energy of a neutral molecule which is 22 Electrostatic contributions to the free energy of an ion can be discussed within the same framework. This self-energy is not due to dispersion interactions, but due to interactions of the ionic charge cloud with itself in the presence of other ions, and is where p v is a form factor for the ion. Together with (27) this expression gives immediately the Born energy and Debye-Hiickel theory. With the interface problem, the self-energy of an ion at distance z can easily be calculated, and leads to an extension of Onsager-Samaras theory for the change in surface tension of water due to dissolved electrolyte.CONCLUSION In general in adsorption problems, both electrostatic (in double layers) and dispersion self-energies occur in the molecular partition function and are often equally important. It is usual to ignore one, or the other. The dispersion self- energy concept provides a unification of existing theories of physical adsorption. It goes somewhat further, for the distinction between physisorption and chemisorption now becomes much less clear, and there is some possibility that semi-classical ap- proaches of the sort outlined, with decoration, may provide a useful quantitative picture complementary to present theories of chemical binding. Dispersion self- energy has been used with some considerable success to predict theta points in polymer solutions in terms of dielectric proper tie^,^^ and phase changes in polymer adsorp- t i ~ n .~ ~ Finally we remark that few layer, lateral interaction, 2-dimensional con- densation problems in adsorption are accessible by these techniques. It will be normally sufficient to recognise that the first few layers have anisotropic dielectric properties. Extension of the formalism using formulae which give the anisotropy as a function of coverage, derived el~ewhere,~~ will provide the appropriate isotherms- * M. Born, Z. Phys., 1920,1,45. R. A. Robinson and R. H. Stokes, Efectrolvte Solutions (Butterworth, London, 2nd edn., 1959). P. Debye and E. Huckel, Phys. Z., 1923, 24, 185.L. Onsager, J. Amer. Chern. SOC., 1936,58, 1486. L. Onsager and N. T. Samaras, J. Chem. Phys., 1934,3, 528. M. 0. Scully and M. Sargent. Phys. Today, 1972, 25, 38. G. Halsey, J. Chem. Phys.. 1948, 16, 25. M. A Cook, J. Amer. Chem. SOC., 1968,70,2925. ' I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, Ado. Phys.. 1961, 10, 165.J. MAHANTY AND B . W. NINHAM 21 lo M . Dole, J. Chem. Phys., 1948, 16, 25. l2 P. Richmond and B. W. Ninham, J. Low Temp. Phys., 1971,5, 177. l3 E. S. Sabisky and C. H. Anderson, Phys. Rev. A, 1973,7,790. l4 P. Richmond, B. W. Ninham and R. H. Ottewill, J. Colloid Interface Sci., 1973, 45, 69. l 5 P. M. Kruglyakov, KolloidZhur., 1974, 36, 160. C. H. Anderson and E. S. Sabisky, Phys. Rev. Letters, 1970,24, 1049. P.Richmond, personal communication, 1974; embodied in Research Report, Unilever Research, Port Sunlight Laboratory. l7 S . Brunauer, P. H. Emmett and E. Teller, J. Amer. Chem. Soc., 1938, 60, 309. D. M. Young and A. D. Crowell, Physical Adsorption of Gases (Butterworth, London, 1962), chapter 5. l 9 S. Brunauer, L. E. Copeland and D. L. Kantro, in The Solid-Gas Znterface, ed. E. A. Flood (Dekker, New York, 1969, p. 77. 'O J. Mahanty and B. W. Ninham, J. Chem. Phys., 1973,59,6157. J. Mahanty and B. W. Ninham, J.C.S. Faraday ZI, 1975,71, 119. 22 J. Mahanty and B. W. Ninham, Dispersion Forces (Academic Press, London, 1976), chap. 4. 23 F. London, J. Chem. Phys., 1942,46,305. 24 B. Linder and D. A. Rabenold, Adu. Quantum Chern., 1972, 6, 103. [These authors give a unified treatment of dispersion forces between two molecules of arbitrary sizes and electron delocalisations. Based on linear response theory their approach achieves a synthesis of earlier approaches based on (1) charge fluctuations on individual molecules (A. D.McLachlan, Proc. Roy. SOC. A, 1963,271,387; 274,80; B. Linder, J. Chern. Phys., 1964,40,2003; H. Jehle, Adu. Quantum Chem., 1965,2, 195), (2) collective properties of the interacting bodies (S. Lundqvist and A. Sjolander, Arkiu Fys., 1964,26,17; G. D. Mahan, J. Chem. Phys., 1965,43,1569; A. A. Lucas, Physica, 1967, 35, 353), (3) collective properties of different parts of the interacting bodies as implied by the theories of E. M. Lifshitz, Soviet Phys. JETP, 1956, 2, 73 or of D. Langbein, Theory of uan der Wmls Attraction (Springer, Berlin, 1974).We follow the field point of view developed in ref. (22).] 2 5 M. D. Crisp and E. T. Jaynes, Phys. Rev., 1969, 179, 1253. 26 J. Mahanty, Nuovo Cimento, 1974,22B, 110. 27 J. Mahanty and D. D. Richardson, J. Phys. C, 1975, 8, 1322. 28 K. F. Niebel and J. A. Venables, Proc. Roy. SOC. A, 1974,336, 365. 29 J. A. Barker, R. A. Fisher and R. 0. Watts, Mol. Phys., 1971,21,657. 30 F. M. Fowkes, J. Colloid Interface Sci., 1968, 28, 293. 31 R. A. Craig, Phys. Rev. B, 1972, 6, 1134; J. Chem. Phys., 1973,58, 2988. 32 D. J. Mitchell and P. Richmond, J. Colloid Interface Sci., 1974, 46, 118. 33 J. N. Israelachvili, J.C.S. Faraday 11, 1973, 69, 1729. 34 W. Kohn and L. J. Scham, Phys. Rev. A, 1965, 140, 113. 35 A. M. Beattie, J. C. Stoddart and N. H. March, Proc. Roy, SOC. A, 1971, 326, 97. 36 J. N. Israelachvili, Proc. Roy. SOC. A, 1972, 331, 39; B. V. Derjaguin, I. E. Dzyaloshinsky, 37 T. L. Hill, J. Chem. SOC., 1946,14,441; 1967, 15, 767. 38 J. H. de Boer, The Dynumical Chructer of Adsorption (Oxford Univ. Press, London, 1968). 39 J. Mahanty and B. W. Ninham, J.C.S. Faraduy 11, 1974,70, 637. 40 C. J. Barnes and B. Davies, J.C.S. Faraday 11, 1975, 71, 1667. 41 V. N. Gorelkin and V. P. Smilga, Kolloid Zhur., 1972, 34, No. 10; Doklady Akad. Nauk S.S.S.R., 1972,208,635; Soviet Phys. JETP, 1973,36,761. 42 V. P. Smilga and V. N. Gorelkin, Research in Surface Forces, ed. B. V. Deryaguin (Consultants Bureau, New York, 1971), vol. 3 (in Russian, Nauka Press, MOSCOW, 1967). 43 B. Davies and B. W. Ninham, J. Chem. Phys., 1972,56,5797. 44 D. Chan and B. W. Ninham, J.C.S. Faraday 11, 1974,70, 586. 45 D. Chan, D. J. Mitchell and L. R. White, Faraday Disc. Chem. SOC., 1975, 59, 181. 46 B. W. Ninham and R. A. Sammut, J. Theor. Biol., 1974, submitted. M. M. Koptelova and L. P. Pitayevsky, Disc. Faraday Soc., 1965,40,246.

ISSN:0301-7249    

DOI:10.1039/DC9755900013

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Computer Simulation of the Gas/Liquid Surface BY GUSTAVO A. CHAPELA AND GRAHAM SAVILLE Department of Chemical Engineering and Chemical Technology, Imperial College, London, S.W.7 AND JOHN S. ROWLINSON Physical Chemistry Laboratory, South Parks Road, Oxford Received 13th January, 1975 The gas/liquid surface of a system of 255 Lennard-Jones (12,6) molecules has been simulated by Monte Carlo sequences at three reduced temperatures which span most of the liquid range. The results at the two higher temperatures show a monotonic profile of density as a function of height, but those at a temperature close to the triple point suggest that the liquid just below the surface can form layers; that is, that local density is an oscillatory function of height. This phenomenon is compared with similar previous results and it is concluded that it may well be the consequence of the constraints applied to the simulated system rather than an inherent property of a free liquid surface.Preliminary densities for a mixture have also been calculated. Molecular theories of surface tension and adsorption relate these macroscopic properties rigorously to the singlet and pair distribution functions of molecules and the variation of these functions across the gas/liquid interface. They provide, however, only approximate methods for the a priori calculation of the distribution functions, and one way of checking these approximations is to simulate the behaviour of a system by molecular dynamic or Monte Carlo techniques. Such methods have been used successfully to study the molecular distribution functions in bulk liquids for nearly twenty years but the study of surfaces is more recent and more difficult.The most obvious difficulty is the choice of a realistic constraint which fixes a plane surface in space without disturbing its shape or density profile. In the real world this constraint is provided by the gravitational potential, but this is ineffectual in a system containing 100-1000 molecules, which is as large as can be handled on a computer. Over a distance of ten molecular diameters the change in the earth’s gravitational potential is negligible compared with the intermolecular potential energy. The constraints that have been chosen for computer simulation have been either an artificially strong gravitational potential confined, perhaps, to one part of the system, or the application to the “ bottom ” of the cell of an external intermolecular potential which is chosen to represent, as realistically as is convenient, the field provided by the bulk liquid. An additional constraint, whose distorting effect is hard to estimate, is that imposed by the use of cyclic boundary conditions in the two directions, x and y , parallel to the surface.If the simulated cell has a width L in the x and y directions then this constraint requires that the surface be horizontal on a square grid of spacing L. It is known that a free liquid surface oscillates vertically to give surface waves of a length comparable with the wavelength of visible light (“ ripplons ”), and there is no reason to suppose that there are not shorter waves of 22G.A . CHAPELA, G. SAVILLE A N D J. S. ROWLENSON 23 all lengths down to that of the molecular spacing. The systems simulated on the computer suppress all oscillations whose wavelength is longer than L. In view of these difficulties it is, perhaps, not surprising that the simulations of similar systems, but with different constraints, have not always led to the same conclusions. In particular it is uncertain whether, or to what degree, there is a " layering " in the liquid just below the surface. This term is used to describe an oscillatory variation of density in the z, or vertical, direction with a period of about one molecular diameter. The results in this paper throw some light on this point, without settling it conclusively, and will be supplemented by further results, given at the meeting.MONTE CARL0 SIMULATION The present study is a Monte Carlo simulation of a system of 255 molecules between each pair of which there is a Lennard-Jones (1 2,6) potential u(r), characterised by the usual parameters of a collision diameter o [that is, u(a) = 01 and a depth E (that is, umi, = -8). The system is confined to a rectangular prism 50 square on its base and 250 in the vertical, or z, direction. Cyclic coordinates are used in the x and y directions and all potentials terminated at r = 2.50 [where u(r) = -0.016~1 so that a molecule cannot interact directly with its image. Potentials are set positive infinite for r < 0.80 to avoid computational difficulties. Below the base of the cell, z = 0, there is a simple representation of the bulk liquid obtained by assuming that each molecule occupies all positions at random.From this continuum representation there is emitted into the cell (z > 0) a Lennard-Jones (9,3) potential given by,l u(z) = p ~ ~ [ ( ~ ~ - ~ ~ ] (0.80 < z < 2.50) where the subscript w denotes " wall " and where ew/8 = (10/3)* O ~ / O = (2/15)*. The reduced density of the fluid, p, is (Na3/V). The top wall of the cell (z = 250) is similarly represented but, because of the lower density of the gas, this representation differs little from a plane infinitely repulsive wall at z = (25.0-0.8)~. Three Monte Carlo sequences have been generated, at reduced temperatures of z = kT/e of 0.701, 0.918 and 1.127. These temperatures are, respectively, close to that of the triple point, at a moderately high vapour pressure, and fairly close to the critical point.If argon is conventionally represented by a (12,6) potential with Elk = 119.8 K, then these temperatures correspond to 84, 110 and 135 K. The sequence at z = 0.701 was started at a randomly chosen configuration whose overall volume was chosen to match the expected liquid density. The first 1.5 x lo6 con- figurations were used to establish equilibrium, the approach to which was monitored by recording the total configuration energy of the fluid, the liquid and gas wall energies, and the density profile, p(z). The same properties and the surface tension were then calculated at each temperature during sequences of equilibrium states of from 3 to 6 x lo6 steps.The statistical average which gives the surface tension is where A is the area of the cell (250~) and the prime denotes differentiation. By symmetry, this sum vanishes in the isotropic fluids and so the whole contribution to y comes from the surface layer. To avoid the effects of the wall the sum was taken only over those molecules for whose centres z 2 60.24 COMPUTER SIMULATION OF THE GAS/LIQUID SURFACE A sequence for an equimolar mixture was generated at a temperature equivalent to 2, = 0.918 and Zb = 0.701 ; that is, for two molecules for which &,,/ebb = 0.763. The diameters were assumed to be equal, cr, = bab = bbb, and the cross-energy to be given by Berthelot’s rule &,b = (&,,Ebb)*. The walls at top and bottom were taken to be composed of a single substance whose properties were those calculated from the van der Waals 1-fluid appr~ximation.~ RESULTS The primary results are the density profiles, p(z), for each sequence. Those for the single component system are shown in fig.1-3. Near the bottom of the cell, z = 0, the density has a strong maximum caused by the packing of the first layer of 210 FIG. 1.-The density profile, pas a function of the height, z at a temperature of T = 0.701 for a r m of 5.6 x 106 configurations. z i o FIG. 2.-The density profile, p as a function of the height, z at a temperature of 7 = 0.918 for a run of 5.6 x 106 configurations.G . A. CHAPELA, G . SAVILLE AND J . S . ROWLINSON 25 molecules in the liquid against the flat repulsive wall potential of eqn (1).This peak is followed by one or two others which resemble the usual oscillations of g(r), the pair distribution functions in bulk liquid. At the two higher temperatures these oscillations f ~ ~ " " " ~ 1 1 ' 1 ' ' ' 5 10 15 4 0 FIG. 3.-The density profile, p as a function of the height, z at a temperature of T = 1.127 for a run of 3.5 x lo6 configurations. disappear beyond about z = 30 and the bulk liquid is then of constant, if slightly "noisy", density until the surface is reached. This is marked by a smooth and apparently symmetrical fall of density from that of the liquid to that of the gas. At z = 1.127 this fall is spread over about 5a, at z = 0.918 over about 40, and at z = 0.701 over about 3 0 . The density profile at the lowest temperature differs qualitatively from the other two in that it shows pronounced layers out to a distance of 100, a distance which would be presumed to be beyond the range of influence of the wall.The density of the bulk phases is readily found by averaging over the horizontal The discussion of these is deferred to the next section. 1 I I I 1 I I t 0.3 0.5 0.7 0.9 P FIG. 4.-The density of the bulk phases as found by simulation of systems of one&hase by Hansen and Verlet l 1 (triangles), and by simulation of systems with an interface by Liu lo (O), by Lee et aL9 (O), and in this work (0).26 COMPUTER SIMULATION OF THE GAS/LIQUID SURFACE (or oscillatory) portions of p(z). These densities are compared in fig. 4 with those obtained by others for a (12,6) fluid from simulations on systems both with and without a gas/liquid surface. The agreement is good.The surface tension is subject to greater statistical error and, as is shown in fig. 5, the agreement between different sets of computer results is satisfactory but not as good as for density. The first results for the mixture (1.2 x lo6 configurations) show that the molecule of lower E moves preferentially to the surface. The value of z at which p has fallen half-way from its value in the liquid to that in the gas is greater for the component of lower energy by about 0.50. These first results show no layering of the sum of the two densities, [p,(z) + pb(z)], but a degree of out-of-phase oscillation between them. Since, however, the run started from a randomly uniform local composition this partial layering may be a consequence of the initial movement of one component to the surface.Lattice models do not show this partial la~ering.~. A longer run is needed to resolve this point and it is hoped to present further evidence at the meeting. 2;o- 1:5- b4 * x $ 1.0- 0s - \ ”\ 0 \ 0 I I I I I I I I 0.6 0.8 I .o 1.2 1.4 7 FIG. 5.-The surface tension y as a function of temperature from the simulations of Liu lo (O), Lee et aL9 (0) and in this work (0). The line is the result of the perturbation calculations of Lee et al. THE EVIDENCE FOR LAYERS The first simulation to show strong layering was a molecular dynamic sequence with 200 molecules, made by Croxton and Ferrier.6 Since this was for a two- dimensional array it is uncertain evidence of the behaviour of real liquids, but Croxton ’ has since argued that liquid metals, in particular, might be expected to have a layered structure.A molecular dynamic run by Opitz * on 300 Lennard-Jones molecules in three dimensions at (apparently) z = 1.03 shows a bulk liquid which might have feeble layers of a spacing of about 2.50. However, the volume of the bulk liquid is not sufficient to show conclusively that the weak oscillation is not random. The evidence in fig. 3 of this work (T = 1.127) is that there are no oscillations of such long wave- length at these high temperatures.G, A. CHAPELA, G. SAVILLE AND J . S. ROWLINSON 27 There have been two recent Monte Carlo simulations of Lennard-Jones systems which should be closely comparable with the work reported here. Lee, Barker and Pound placed 1000 molecules in a rectangular prism whose base was 6.540 square and whose height was 400.The liquid was confined to the centre of the prism by a strong " gravitational '' potential which dissuaded the molecules from entering the parts of the prism near the ends, that is z < 20 and z > 380. At a temperature of z = 0.7035 they obtained two surfaces whose density profile at the surface resembled that of fig. 1. However below both surfaces the liquid was strongly layered; in each case there were about 8 maxima and minima in p(z), the separation of the maxima was close to CT and the peak-to-trough height was 0.2~-0.3~. Liu lo chose a system of 129 molecules without a gravitational field, and, as in this work, confined the liquid to one end of the prism by an intermolecular potential.His was less realistic, however, since it consisted of a square-well of arbitrary depth and width. He made runs at z = 0.75,0.90, 1.10 and 1.30, and claimed to find a " striking layered structure " at the two lower temperatures. It is, however, hard to see anything but random noise in his broad histograms, and the peak-to-trough heights of these are no more than 0.05~. The results of Optiz, of Liu and those shown in fig. 2 and 3 suggest that layering is not present at high reduced temperatures. If it is a true property of a gaslliquid surface, that is, one not induced by the constraints necessary for computer simulation, then it occurs only at low reduced temperatures. The three results at the lowest temperatures (Lee et al.z = 0.7035 ; Liu, z = 0.750 ; and fig. 1, z = 0.701) are all close to the triple point, which has been put at z = 0.68 i- 0.02 by Hansen and Verlet l 1 after extensive simulation of the bulk liquid and solid. In no case however were the x and y dimensions of the cell an integral multiple of the crystal lattice spacing, and, at least in our case, even the lowest layer of molecules appeared to be arranged without long-range order in the x-y plane. Moreover our liquid density, p = 0.81, is appreciably below Hansen and Verlet's estimate of 0.85+_0.01 for the freezing liquid. We conclude that the layering is not likely to be the onset of crystallization, but is a property of the constrained fluid systems under study. It is hard on this evidence to settle the question of whether or not the layers would persist in the absence of the constraints, but there are some indications that they would not.Thus our results and those of Lee et al. show oscillations which differ in amplitude by a factor of 2. We apply the constraint to the bulk liquid and the amplitudes decreases slowly as z increases. The last recognizable peak in fig. 1 is at z = 9.80 whilst the centre of the interface is at z = 12.80. Lee et ul. apply the constraint to the gas side of the interface and so ensure that their surface is even more closely confined to a plane than is imposed by the cyclic coordinates in the x and y directions. Their oscillations appear to start in the interface, to reach a maximum at a depth of about 40 below it and then to die away in the bulk liquid. The position and size of the oscillations appear therefore to depend on how the constraints are applied and, in particular, to depend on how strongly the surface is constrained to local planarity.It is possible that if all restriction on planarity could be lifted then the layers would disappear. This hypothesis could be tested by re- peating the runs in cells with substantially larger x and y dimensions. We are indebted to Dr. N. G. Parsonage for advice on the Monte Carlo simulation of surfaces, to Dr. J. A. Barker for an early copy of his paper, and G. A. C . thanks CONACYT (Mexico) for their support.28 COMPUTER SIMULATION OF THE CAS/LIQUID SURFACE ' D. M. Young and A. D. Crowell, Physical Adsorption of Gases (Butterworth, London, 1962), * F. P. Buff, 2. Elektruchem., 1952, 56, 3 1 1 . chap. 1. T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans Faruduy Suc., 1968,64, 1447. S . Ono and S. Kondo, Handbuch der Physik (Springer, Berlin, 1960), vol. 10, p. 262 et seq. J. E. Lane and C. H. T. Johnson, Austral. J. Chem., 1967,20, 61 1. C. A. Croxton and R. P. Ferrier, J. Php. C, 1971, 4, 2447. ' C. A. Croxton, Liquid State Physics (Cambridge U.P., 1974), chap. 4. * A. C. L. Opitz, Plzys. Letters, 1974, 474 439. J. K. Lee, J. A. Barker and G. M. Pound, J. Chem. Phys., 1974, 60, 1976. J. P. Hansen and L. Verlet, Phys. Rev., 1469, 684, 151. lo K. S. Liu, J. Chem. Phys., 1974, 60,4226.

ISSN:0301-7249    

DOI:10.1039/DC9755900022

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Structure of the LiquidiVapour and Liquid/Solid Interfaces BY JOHN W. PERRAM AND LEE R. WHITE* Department of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT. 2600, Australia Received 8th January, 1975 A theoretical method of determining the density profile p(')(z) at liquid/vapour and liquid/solid interfaces is developed, based on a procedure of Helfand, Frisch and Lebowitz. The density profile for a given fluid is shown to be related to the value of the pair distribution function &)(r) for a two component fluid in the limit that the molecular radius of the second component becomes infinitely large (i.e. macroscopic) and its density vanishes. The density profile for hard spheres against a hard interface is obtained in the P.Y.approximation. The extension to the liquid/vapour interface is discussed and the free-interfacial density profile for a fluid with an attractive inverse sixth power pair potential is displayed. A comparison is made with other modern theories. 1. INTRODUCTION The structure of fluids in inhomogeneous regions has been intensely studied. lml The principal methods of investigation being quasi-thermodynamic arguments, 1-6 integral equations 7-1 and computer simulations. l2-I7 Fundamental to all three approaches is the singlet density p ( l ) (z) which varies with some co-ordinate z in the interfacial region. All three approaches have been described by Croxton,18 but a brief description of each is given here. 1.01 :8 -6 -4 -2 0 2 4 6 8 Z FIG. 1 .-Typical monotonic density profile obtained by the constant chemical potential approach.T/Tc = 0.50 (A), 0.80 (B), 0.98 (C). The quasi-thermodynamic theories are familiar to workers in colloid science and electrochemistry, although the crude formulae used there have been considerably refined. Fundamentally, a local chemical potential p(z) = p[p(z), z] is defined. It is a function of the local density p(z) and is usually written p = p' + kTln a[p(z)] + u(z) 2930 STRUCTURES OF INTERFACES where po is at some reference state, a[p(z)] is the local activity, usually calculated from one of the hard sphere equations of state,lg* 2o and u(z) is the interaction of an atom at z with the inhomogeneity (and hence is a functional of ~ ( 2 ) ' ) . By requiring p(z) to be constant, a non-linear integral equation for p(z) has been ~ b t a i n e d , ~ ' ~ which has been solved to yield p(z).For all theories of this type, p(z) is a monotonic function of z,18 as shown in fig. 1. The starting point for most of the integral equation theories is the first order Born-Green equation, relating p'l' ( r ) to the (inhomogeneous) pair correlation function pC2) (rl, r,), viz. kTVlp'l' (u)+ JVl@(lrl - - U ' ~ I ) ~ ' ~ ' (rl, r2)drz = 0. (1 -2) Various closures 3* * * l1 permitting solution of this equation have been suggested. Toxvaerd writes Pt2) (r1, r2) = aP(2) 0.12lp(l) (0 + (1 - 4P'2' (r12lp(l) @,I) (1 -3) where p ( 2 ) (r121p) is the radial distribution function for the unstable liquid at uniform density p . With a further approximation on pc2) (rlp), eqn (1.2) is solved to yield a monotonic profile for p'l) (r) (see fig.1). The approximations used are difficult to assess. Nazarian has used two alternative closure approximations (1.4) and p'2' (rl, r2) = p&m (rl2)e(z1 +z2)+p$%oUR (r12)@-(zl +z2)] I 2 FIG. 2.-Oscillatory density profiles obtained by solving an approximate BBGY equation for p(') (2). I, Nazarian approximation (1.4) ; 11, Nazarian approximation (1.5) ; 111, Croxton and Ferrier " bootstrap " function solution.J . W. PERRAM AND L. R . WHITE 31 and solved (1.2) for a liquid argon potential (90 K) to obtain strong density oscilla- tions as shown in fig. 2 for both closures. Croxton and Ferrier * have used an unusual closure relation where gCz) (rl, v,) is replaced by its bulk liquid value and a function @(z2) is introduced multiplying the potential <D(r12) in (1.2).The function is chosen so that it effectively creates a free surface by cutting off the interaction between par- ticles as a function of the z component of the second particle. Croxton and Ferrier apply this procedure to a liquid argon (at its triple point) potential to obtain the mild density oscillation as shown in fig. 2 which should disappear as the critical point is approached. Although both methods give density oscillations, the method of Nazarian shows much larger effects. A controversy arises. Does p{l) (z) exhibit an oscillatory nature near the interface and if so how is such behaviour modified by the approach to the critical temperature ? As has frequently been the case,21* 22 when approximate statistical theories conflict, the best way of testing them is by some type of computer simulation.This has proved invaluable in the study of homogeneous fluids,18 but removal of one of the periodic boundary conditions means that this is not as straightforward as it might seem. The first study l3 was a molecular dynamics simulation of a two dimensional interface of particles confined to the surface of a cylinder. A random field was applied to particles at the base and the transition profile generated. Again, density oscillations were reported. Another method is to simulate a thin film,15 so that two interfaces are generated. Liu l7 has recently “ created ” a single interface by placing an attractive potential at one end of the computer “ box ”.In this way, condensation is forced on the system and again an oscillatory profile is obtained. However, Rowlinson l 6 reports much less pronounced oscillations. We present here a theory in which we aim to unify the various approaches to the liquid surface. The method is based on an idea of Helfand, Frisch and Lebowit~.,~ Consider a two component system where the interaction between type 1 particles is via a pair potential u(r12) and between type 1 and 2 particles by a potential @(z) where z = Ir, -r2( - B’ where 0’ is the hard sphere radius of type 2 particles. In such a system the thermodynamics is completely specified by a knowledge of the distribution functions g$$), &), g\$) (z) all three of which are functions of the number densities p and p’ of the two particle types.If we now take the limit u’--+co while taking p‘--+0 then we have effectively created an interface at z = 0 of particles of type 1 with an interaction potential @(z) with the interface. One has then for the density dis- tribution of type 1 particles away from the interface This method produces density oscillations as well. where gC2) (rlJ is the bulk distribution function at density p of particles of type 1. If we took as the interfacial potential @(z) = co if z < 0 then the limiting procedure of eqn (1.6) would yield the density profile for the liquid against a hard impenetrable interface with which it interacts via the potential $(z). Provided 4(z) has a short range repulsive component (i.e. the liquid particles have = 4(z) if z > 0 (1.8)32 STRUCTURE OF INTERFACES some sort of repulsive core, it is obvious that such an interface, with p"' (2) given by (1.6), will exhibit density oscillations.These oscillations will be smoothed by softening the repulsive core but will be little affected 2 5 by changes in the attractive part of 4(z)* The preceding argument does not enable us to say much about the liquid/vapour interface, however, and it is at this interface where the controversy exists. Suppose that type 2 particles are vacuum bubbles of diameter o', assumed large. Let us consider a liquid particle near the bubble. The force of interaction of the particle with the interfaces is always, for real fluids, in the direction of the bulk liquid. This may be interpreted as a repulsive force between the bubble and the particle.The potential energy of interaction @(z) is plotted schematically in fig. 3, together with its Boltzmann factor. @(z) must be constant as Izl-+ co the difference @( - co) - @(a) is the molecular heat of vaporization and is a function of density and temperature. 0 2 3 FIG. 3.-A schematic representation of the potential of interaction @(z) of an atom in a liquid with the rest of the liquid as it moves through the liquid/vapour interface (A). The associated Boltzmann factor exp[- p<D(z)] is also displayed (B). C represents the heat of vaporization. A soft potential @(z) of the form shown in fig. 3 would lead to a smoothing of the density profile. As the critical temperature is approached from below the heat of vaporization will tend to zero and the will be a temperature (5 T,) at which the density profile becomes monotonic.With the above approach, we can see how the density profile is a function of the softness of the interfacial interaction potential. The choice of a hard potential @(z) or a closure approximation which amounts to such a choice will give rise to strong density oscillations. Correspondingly, a closure which amounts to the choice of a soft potential @(z) will give rise to monotonic or, at least, weakly oscillatory density profiles. The approach outlined above can be applied to the pair distribution function pC2) (zl, z2, r12). If &)I (zi, z2, r12) is the triplet correlation function in the mixture between a type 2 particle and two type 1 particles, thenJ . W.PERRAM AND L . R . WHITE 33 In the mixture, the triplet correlation function g$$)l can be replaced by the super- position approximation 2 5 Therefore, in the limit, where we have invoked the results (1.6) and (1.7). Thus the result (1.11) for the pair correlation function at an interface which was suggested by Green 26 and used by Berry and Reznek 27 can be seen as an approximation of the same order of mag- nitude as the superposition approximation in two component mixtures of very unequal size. Further since the approximation (1.1 1) is now couched in the familiar terms of a superposition approximation, one can systematically correct the approxi- mation in a statistically rigorous way.28* 29 The degree of approximation can also be tested in an average way by invoking the relation between neighbouring orders of the hierarchy of distribution functions. Here the appropriate relation is 9131’1 (21, 22, Y l 2 ) = 9122) (2119122) (22)gW (r12).PC2’ (21, 22, r12) = P‘l’ (Zl>P“’ ( z M 2 ’ ($12) (1.10) (1.11) - 1 = 1 P‘l’ (22) W2’ (21, 22, G2)- w 2 (1.12) which should be obeyed for all zl. Thus f ( z J = -J P(1)(z2)c9(2)(% 2 2 , r 1 2 ) - 11 dr2 V may be plotted and its deviation from unity used as a measure of the degree of approximation involved in g(2) (zl, z2, r12). In the next section, we exhibit density profiles for a number of interfaces in the Percus-Yevick approximation. 2. PERCUS-YEVICK EQUATION FOR THE DENSITY PROFILE We consider an M+ 1 component system of particles, with numbers N j of type j , j = 1, . . .M+ 1, enclosed in a volume V.Each particle has a core of diameter X,, so that the distances of closest approach Rl, are assumed to be R,, = +(& + Rj). It is our intention to allow &+I to become much larger than other R,, at the same time allowing pM+1 = N M + l/Vto tend to zero. This creates an interface in the form of a macroscopic spherical cavity. The particles interact through potential functions Uafi (r) of the form Uafi(r) = +a r < Ra, = u$’(r) r > Rap We investigate various forms for the Uafi (r). The total correlation functions h.8 ( r ) and direct correlation functions cap ( r ) are connected by the Ornstein-Zernike equations 2o M+1 y= 1 ha,fi(r) = ca,p(r)+ c P y J ca,y(~s~)hy,fi(~r-s~> ds (2.1) which, when supplemented by the Percus-Yevick approximation ha,fi(r) = - 1 +~2fi(r)/ll -exp(uap/WI (2.2) allow (in principle) the calculation of all the correlation functions.For the case when u$) ( r ) = 0, these equations have the happy feature of an analytic solution.30 Let us focus attention on h ~ + l , j . If R M + ~ S R,, then the distribution function 59-B34 STRUCTURE OF INTERFACES g M + 1 , j = 1 + hM+ l , j , when multiplied by p j = N,/Y is a measure of the variation in density as we move away from our central (M+ 1)-type particle. The question as to when our ( M f 1) particle becomes macroscopic is a matter of semantics, but we shall study this by considering a range of values of up to a factor of 40 times other particle diameters. A very convenient numerical method has been given 24 for the numerical evaluation of the functions hij for hard sphere mixtures, and there is no doubt that h M + l , j ( r ) will be oscillatory.It is however, reasonable to ask whether these oscillations are a result of the hardness of the cores. Fortunately, methods have been given 31 for the easy modification of the h a p ( r ) to account for the softness of the core. An effective hard sphere diameter RZj is defined by R,*B = (1 - exp[ - uaB/kT]) dr. (2.3) So" Now for all potentials, although the functions g a b (Y) may be discontinuous, the function Ydls(d, defined by Y a p ( r ) = gus (Y) ex~[ua~lkTl Y,*B(d = -c,*,<d r < K s (2.4) is continuous. For effective hard spheres = s,*a(r> r > R$. Then, gives good estimates of the correlation functions as a function of distance.Fra. 4.-The mixed radial distribution function glz(r) (i.e., density profile) for a system of hard spheres (po3 = 0.85) against a hard sphere (p' = 0) with u' = Ma.J . W. PERRAM AND L. R. WHITE 35 From the above, we can see how surface density oscillations can arise, even when the j, N+ 1 exclusion is quite soft (of the order of 1-2 f i j values). If Rj < RM+l then RT, M+l % Aj, M+f very closely, and so the functions y , y* are essentially the same. Thus outside the range of the soft potential the form of the functions has (r), hzp (r) are the same. We first examine the limiting process of eqn (1.6) for a system of hard spheres, bulk density po3 = 0.85, against an impenetrable interface. We obtain, numerically, the solution of eqn (2.1) in the Percus-Yevick approximation for a mixture of hard spheres at the above density and hard spheres of radius o’(o/o’ 4 1) at zero number density.In fig. 4, the value of hI2 (2) is plotted for ~ / o ’ = 1/40. The results for a/o’ = 1/25, 1/20, 1/15 are scarcely distinguishable from the exhibited curve. We can examine the limiting process in more detail however. 4.0 I 1 I L 1 0 0. I 0 . 2 (ula’) FIG. 5.-The peak heights of the first three maxima in the density profile for a hard sphere system as a function of u/u’. A, first maximum ; B, second maximum ; C, third maximum. In fig. 5, the height of the first, second and third peaks are plotted as a function of (ole'). For the mixed hard sphere system it is easily shown that where n 3 q = -pa . 6 With pa3 = 0.85 we see that lim h1,(a/2) = 5.137.p’+O U’-+ Q) A linear extrapolation of the first peak height for the points corresponding to c/o’ = 1/40, 1/25, 1/20, 1/15 passes through c/o’ = 0 at h12 (1) = 5.14. Similar extra- polations for the second and third peak heights also seem indicated due to the linearity in the region o/o’ < 0.1. Thus very accurate numerical estimates of the limit (1.6) may be obtained by linear extrapolation from two or three h12 (2) curves calculated with (c/o’) < 0.1.36 STRUCTURE OF INTERFACES 3. THE LIQUID/VAPOUR INTERFACE We combine here the idea of the self consistent energy profile 4(z) mentioned in the Introduction with the equations (2.3)-(2.5) which give a prescription for the modi- fication of interfacial density. Suppose the small particles interact via the potential ull(r) = +a r -= Rll = -&(R11/r)6 r > R11 and suppose that the density profile pl(r) is given by Pl@) = Pls221’(r) then the energy of interaction of a particle 1 at r with every other particle is n Then +(P) may be defined as 4(r) = V(r)- V(m).(3.2) (3.4) If we replace u; by 4 in eqn (2.3) to calculate Rzl and use (2.5) to determine g$i), we obtain vW) + +(r) = p 1 J y~(rz>exp[ - BM,)I Id 1 1(1r - r2 ~)g\?(ir - r2 I) (3.5) vz which is the self-consistent equation for the energy profile +(T). The key question is however whether or not iteration of the equations produces a convergent answer, and what is the nature of p l ( r ) derived from it. ( z l 4 FIG. 6.-The density profile p(l)(z)/p~ at the liquid/vapour interfacelfor a system of particles (po3 = 0.85) with a hard sphere core plus an attractive interaction -E(o/r)6, for ElkT = 1.5.J . W.PERRAM AND L. R. WHITE 37 As a rough guess, we would suspect that at low temperatures, as measured by e/kT, stable solutions could be obtained, because the self-consistent potential will be quite hard. Such is indeed the case, for fig. 6 shows p l ( r ) in the vicinity of the interface for EIkT = 1.5 and density plR:l = 0.8. (The results of Lee, Barker and Pound, although for a different potential, correspond roughly to e/kT = 1.4 and pRfl = 0.81). We note that the density exhibits pronounced oscillations. We have also studied i$kT = 1.25, at the same density, and find similar, but less pronounced oscillations. T. L. Hill, J.Chem. Phys., 1952, 20, 141. I. W. Planer and 0. Platz, J. Chem. Phys., 1968,48, 5361. S . Toxvaerd, J. Chern. Phys., 1972,57,4092. S. Toxvaerd, J. Chem. Phys., 1971, 55, 3116. €3. W. Ninham and J. Mahanty, J. C. S. Faraday I& 1975,71. G. H. Findenegg, personal communication. J. G. Kirkwood and F. P. Buff, J. Chem. Phys., 1949,17, 338. * C. A. Croxton and R. P. Ferrier, J. Phys. C., 1971,4,1909; Phil. Mag., 1971,24,489. !a C. A. Croxton and R. P. Ferrier, J. Phys. C., 1971,4, 1921 ; Phil. Mag., 1971, 24,493. lo J. W. Perram and L. R. White, Nature, to be published, l1 G. M. Nazarian, J. Chem. Phys., 1972,56, 1409. l2 S. Toxvaerd, Mol. Phys., 1973, 26, 91. l3 C. A. Croxton and R. P. Ferrier, J. Phys. C., 1971, 4, 2447; Phys. Letters, 1971, 35A, 330. l2 S. Toxvaerd, Mol. Phys., 1973, 26, 91, l4 J. D. Bernal, Proc. Roy. SOC. A, 1964, 280,299. l5 J. K. Lee, J. A. Barker and G. M. Pound, J. Chem. Phys., 1974,60, 1976. l6 J. S. Rowlinson, personal communication. l7 K. S . Liu, J. Chem. Phys., 1974,60,4226. l9 L. Tonks, Phys. Rev., 1936,50,955. 2o H. Reiss, H. L. Frisch and J. L. Lebowitz, J. Chem. Php., 1959, 31, 369. 21 R. 0. Watts, Mol. Phys., 1974, 28, 1069. 22 A. Ralman and F. H. Stilfinger, J. Chem. Phys., 1971, 55, 3336 ; 1972,57, 1281. 23 E. Helfand, H. L. Frisch and J. L. Lebowitz, J. Chem. Phys., 1961,34, 1037. 24 F. Kohler, J. W. Perram and L. R. White, Chem. Phys. Letters, 1974, in press. 25 S. A. Rice and P. Gray, The Statistical Mechanics of Simple Liquids (Interscience, New York, 26 H. S . Green, Handbook of Physics, 1960,10,79. 27 M . V. Berry and S. R. Reznek, J. Phys. A, 1971,4, 77. 28 J. S. Rowlinson, Mol. Phys., 1964, 6, 591. 29 M. J. D. Powell, Mol. Phys., 1964, 7 , 591. 30 R. J. Baxter, J. Chem. Phys., 1970,52, 4559. 31 H. C. Anderson, D. Chandler and J. D. Weeks, J. Chem. Phys., 1972, 56, 3812. C. A. Croxton, Liquid State Physics-A Statistical Mechanical Introduction (Cambridge U.P., 1974). 1965).

ISSN:0301-7249    

DOI:10.1039/DC9755900029

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Adsorption of Fluids: Simple Theories for the Density Profile in a Fluid near an Adsorbing Surface BY GERHARD H. FINDENEGG* Lehrstuhl f. Physikalische Chemie 11, Ruhr-Universitat Bochum, D-463 Bochum, BRD, Postfach 2148 JOHANN FISCHER AND Dept. of Theoretical Physics, Ruhr-University, 463 Bochum, BRD Received 23rd December, 1974 Two simple theories of fluid adsorption are discussed. One starts from the first equation of the BBGKY-hierarchy whereas the other considers a reference system in which the fluid is bounded by a hard wall, and the attractive potential is added with a coupling parameter. For the latter theory numerical results are presented for the density profile and the integrated surface excess over a wide range of fluid densities. These results are compared with those of asymptotic theories.The problem of the density profile of the reference system is considered. Gas adsorption at or below the normal boiling point of an adsorptive can usually be considered as condensation of the vapour under the influence of the attractive potential of the surface. The density of such an adsorbed layer will be more or less uniform up to its outer surface where it falls off sharply to the density of the vapour phase. Such a model is less realistic at higher temperatures, and above the critical temperature there can be no discontinuity in the density of the fluid. Rather, the density will increase smoothly towards the surface. At low densities, when the fluid behaves as a perfect gas, the local density at a point Y near the surface, pcl)(r), is simply the bulk density po times a Boltzrnann-factor,l where u,(Y) is the interaction potential between an isolated molecule at Y and the solid.If u,(r) is only a function of the distance z from the surface, pcl'(r) = pcll(z), the amount of adsorption per unit area is given by Inserting eqn (1) into (2) leads to Henry's law, i.e., adsorption along an isotherm increases linearly with the pressure of the gas. At higher densities deviations from this law occur. It has been established by measurements at higher pressures that the adsorption isotherms pass through a maximum at pressures at which the fluid has about the critical (or a somewhat lower) density. Qualitatively, this behaviour is readily explained by the fact that the density near the surface cannot increase indefinitely ; hence, the surface excess decreases as the bulk density gradually approaches that near the surface.To obtain a quantitative isotherm equation it is necessary to have a theory for p'l'(z) which is applicable over an extended range of densities of the fluid. 38G . H. FINDENEGG AND J . FISCHER 39 Analytical expressions for the density profile in a liquid at relatively large distances from a wall have been derived by Steele and by Kuni and Ru~anov.~ If these are inserted into eqn (2) an isotherm equation r ( p ) is obtained which reproduces qualita- tively the main features of high pressure adsorption isotherms over a range of tem- perature~,~ even though it is based on an extrapolation of the asymptotic expression for pcll(z). It is of interest, therefore, to investigate the density profile close to the wall as a function of the bulk density and temperature.Below we discuss two simple theories for pcl’(z) which are derived from the same starting point along different ways. Numerical solutions are presented for one of them and the results are compared with the asymptotic expressions. THEORY We consider the following system: a fluid consisting of N spherical molecules is bounded by a structureless wall in the plane z = 0 that exerts attractive forces on the individual molecules. The potential energy of a molecule at a distance z from the wall is (cf. fig. 1) u,(z) = 00 for z < zo (34 u,(z) = u,(zo)(zo/z)3 for z 2 zo (3b) where zo denotes the collision diameter between the molecule and the wall, and the z - ~ dependence arises from the integration over the dispersion potential of the mole- cule with volume elements in the half-space of the wall.FIG. 1.-Two particles at different distances z from the wall. The spheres with radius r, indicate the range of the total correlation function ho. The potential of the wall us(z) and the collision diameter zo are also indicated. The first way to obtain the density profile in the fluid is by considering the first equation of the BBGKY-hierarchy for the above system. Let u(r,J be the pair potential between two fiuid molecules, then the total potential energy of the system is Hence the configurational partition function is ZN = I exp[ - E/kT] drl . . . dr,,40 ADSORPTION OF FLUIDS and the density profile (All configurational integrations are to be taken over the half-space z 3 zo.) Differ- entiation of this expression leads to where the usual definition of the pair correlation function g(rl, r,) has been used.To close the BBGKY-hierarchy one may assume that g(rl, r2) can be replaced everywhere by the pair correlation function go(rl, r,) in the bulk fluid. To proceed it seems indispensable for higher densities to take go(rl, r2) from some theoretical work and to evaluate the resulting integrodifferential equation numerically. For low densities we may use the approximation which gives % h 3 y2) = exPC - u(r1 ,)/kT3, Q O h , r,) -v1- = hgo(r19 r2) = VlhO(b r2), Nr1, r2> = dr1, r2) - 1 (5) ( u:3 where the total correlation function is used. We then obtain or h~[p(~)(r,)/p~] = -'%I+ p(')(r2)hO(rl, r2) dr,-p, h,(oo, r,) dr,.s s (6) In the second integral (rl = co) the volume of integration is effectively determined by the range of ho (rm in fig. 1) whereas in the first integral the wall causes a cut-off. A second method is to construct a reference system where the wall exerts no attractive forces on the molecules. The attractive potential is then added with a coupling parameter 5. The density profile can then be written as and again differentiation gives Formal integration then yields Here, pcl)(r, = 1) is the density at r when the attractive potential u,(r) is acliug. When the wall exerts no attractive forces on the particles of the fluid the densityG. H. FINDENEGG AND J . FISCHER 41 p"'(r, 5 = 0) will still deviate from po, as a consequence of packing effects near the wall, and also because of the unbalanced attractive potential of the fluid (as in the surface region of a liquid).who circumvented the last-mentioned difficulties by choosing a reference system (5 = 0) such that an imaginary boundary plane is drawn through a large sample of the fluid and the coupling parameter transforms the fluid in one half-space into the wall. With this choice of reference state us@) in eqn (7) is to be replaced by a perturbation potential where ufl(r) is the potential for a molecule at r (z>zo) with the reference fluid at z < 0. To proceed from eqn (7) one assumes that h(rl, r2, <) can be replaced by the total correlation function of the bulk fluid ho(rl, r2). In order to obtain a simple approximate equation for p<l'(r) Steele also replaces p"'(rY <) by p o and up(r2) by up(rl) and takes the integral over the fluid volume (z > z,) as an integral over an infinite volume.Thus Nearly the same method as outlined above has been proposed by Steele ulsr) = USW - ufl(r)Y = -'&I[ kT 1 + p o Jrn dr, h0(r1, r2)] where the well-known compressibility equation Po j h(l.12) dh2 = PokT- 1 (9) has been used (K is the isothermal compressibility of the bulk fluid). The resulting density profile differs from eqn (1) only by the thermodynamic factor pokTI. From the assumptions that have been made it seems clear that eqn (8) should hold far away from the wall. There up(z) will be small compared to ( p O ~ ) - l and eqn (8) can be approximated by This represents the leading term of another asymptotic theory which was derived in a different way by Kuni and Ru~anov.~ If this expression is used (in spite of its asymptotic character) to calculate the total surface excess [eqn (2)] the adsorption isotherms are in qualitative agreement with experimental res~lts.~ The asymptotic density profile can be checked by calculating p("(z) by less restrictive assumptions.In principle, eqn (4) would be preferred for this purpose and some calculations for low densities were made with eqn (6). The starting point for most of our calculations is, however, eqn (7) with the following assumptions: (i) h(rl, p i , c) is replaced by ho(rl 2) ; (ii) p(')(r, 5) depends linearly on <, i.e. (10) The density profile ~ ' ~ ' ( r , < = 0) could be obtained from eqn (4) with us@) = 0 but this involves solving an integro-differential equation.For the present work we assume (iii) that p"'(z) - p* = - p$KUp(Z). pc1+, <) = (1 - &P(v, = 0) + {p{l)(r, 5 = 1). pc1)(r, = 0) = po. (1 1)42 ADSORPTION OF FLUIDS With these approximations we obtain Eqn (12) refers to a reference state in which the fluid is bounded by a hard wall and the attractive potential of eqn (3b) is added by the coupling parameter c. Eqn (8) is based on a continuous fluid as a reference state and the coupling parameter trans- forms one half space of the fluid into the solid, thereby adding the potential up. However, the assumptions in the derivation of these two equations can be made with either of these models and yield formally the same relations, the only difference being that up is replaced by us and vice versa.We prefer the former reference state since this coupling procedure seems to be better defined. Thus we compare eqn (12) with the asymptotic equation ln[p(l)(zl)/po] = - %(p,kTK). RESULTS A N D DISCUSSION Eqn (12) and (13) were evaluated numerically for a fluid whose particles interact by a Lennard-Jones potential u(r) = 4E[(;)12-(q)6]. 0.2. I , I.- I _ L IIL-.*___- 10 1.5 21) 2.5 11) 1.5 2.0 2.5 3.0 1.0 1.5 2.0 25 30 FIG. 2.-Density profiles p(')(z) in reduced units plotted against z/zo for different bulk densities p and temperatures T. The solid curves show the results from the numerical evaluation of eqn (12), the dashed curves correspond to the asymptotic theory, eqn (13).Tabulated go(r) data have been used at pa3 = 0.45 and 0.65; at pa3 = 0.1, go@) is obtained from the low density approximation, 2120 eqn (5).G . H. FINDENEGG AND J . FISCHER 43 For the solid-gas potential, eqn (3), zo = cr and u,(zo) = - 2 ~ were taken. These values correspond roughly to the interaction of argon with a graphite surface. The pair correlation function for low densities is given by eqn (5). For high densities tabulated values of go@) for a Lennard-Jones fluid obtained by molecular dynamics calculations were used. The thermodynamic quantity pokTic that appears in the asymptotic expression, eqn (13), is obtained from the total correlation function ho through eqn (9). The integrals in eqn (9) and (12) were computed for r < r,, where r, corresponds to the zero of ho(r) next to 5.00.0 0.2 0.4 0.6 0.8 PO3 FIG. 3.-(a) Adsorption l' in reduced units plotted against bulk density p at T = 2.84~lk. The solid line and the filled circles are obtained from eqn (12). The dashed line and the open circles show the results of the asymptotic theory, eqn (13). The adsorption isotherm for argon according to eqn (13) is also indicated (dotted curve). (b) The thermodynamic quantity pok?"x as obtained from the compressibility eqn (9) using tabulated go(r) data6 (full circles) and using the low density go(r) according to eqn (5) (full line). The dotted curve is calculated from tabulated PVTdata of argon.' In principle, a comparison of the density profiles following from the asymptotic expression (13) in combination with eqn (9), and from eqn (12), respectively, should be consistent even if the go(r) data employed are not entirely self-consistent.How- ever, eqn (9) is sensitive to minor variations in go(r) at large r and significant errors in pokTic (and, therefore, also in the asymptotic density profile) arise from an in- correct interpolation of the tabulated gO(r) data. Nevertheless, our procedure seems to be sufficiently accurate to be used to discover systematic deviations of the asymptotic expression (13) from eqn (12). This is also indicated by the fact, that the values of pokTK which are obtained from the tabulated go@) data with the help of eqn (9) are in reasonable agreement with the experimental values for argon (as one should indeed expect 6, if the parameters cr = 0.3405 nm, Elk = 119.8 K are used for the Lennard- Jones potential of argon.This is shown in the lower part of fig. 3 and 4. The curves44 ADSORPTION OF FLUIDS for argon were obtained from tabulated PYT data over an extended range of pres- sures.' These graphs also indicate the range of densities in which the low-density- approximation for go(r) [eqn (93 may be employed. \ \ 0 1 \ \ \ \ . \ \ \ \ \ \ \ \ '\. \ \ '. i .'\ 0.5 1 ;. . 0.2 0.4 0.6 0.8 i"' " 1 ' 0 PO3 FIG. 4.-A similar plot to that in fig. 3 for the isotherm at T = 1.584k. A comparison of density profiles according to eqn (12) and (13), respectively, is shown in fig. 2. The critical constants for a Lennard-Jones fluid are : Tc = 1.36~/k, pc = 0 . 3 6 r 3 (the experimental values for argon are somewhat lower: T .= 1.26 E/k; pc = 0.32 r3). The profiles in the upper row of fig. 2 therefore correspond to temperatures above 2Tc, those in the lower row to an isotherm relatively close to the critical temperature. At low temperatures and high densities the density profile of eqn (12) exhibits distinct maxima near z/z, = 2 and 3, indicating a tendency towards a layer-like structure near the wall. This mainly reflects the damped oscillations in the pair function g,(r) at higher densities, but a flat maximum in p{')(z) near z/z, = 2 is also obtained at low densities, where go(r) is approximated by the single-peaked function exp[ - u(r)/kT] of eqn (5). This is indicated in fig. 2 in the profile at pOo3 = 0.1 and The adsorption f was obtained from the density profiles by numerical integration between 1 < (z/z,) and (zlz,) < 5 plus a correction for the excess density at (z/z,) > 5.Some results along isotherms at 2.84 Elk and 1.58 E/k, covering a wide range of densities up to the triple point density of the liquid (0.84 r3 for argon), are plotted in fig. 3 and 4. The following conclusions emerge. (a) In comparison with the numerical computation of eqn (12) the asymptotic theory predicts a steeper increase of r at low densities and a steeper decrease of r at high densities. kT/& = 1.58.G . H. FINDENEGG AND J . FISCHER 45 (6) At high temperatures (T 3 2T, ; see fig. 3) the asymptotic theory represents a reasonable approximation to eqn (12) at low and moderate densities but seems to give too low a surface excess at high densities.The large scatter in the values of rasympl reflects the sensitivity of the coefficient pokTrc to errors in the pair function (c) The isotherm in fig. 4, corresponding to a temperature not far above T’, shows strong adsorption at around the critical density. However, the asymptotic expression (13) is not applicable in the critical region.8 Sufficiently far away from the critical density the results are qualitatively in agreement with those at higher temperatures. It remains to check to what extent these results are influenced by our assumptions. As an alternative to assuming eqn (10) we replaced pcl’(r, c) in the integral by p{’j(r, = 1), but this had no significant influence on the numerical results. A more serious error is introduced by replacing pcl’(r, ( = 0) on the left-hand side of eqn (7) by po [eqn (ll)] : for low densities pc”(r, < = 0) can be calculated by use of eqn (6) taking u,(z) = 0 (z 3 zo).It was found that close to the wall the correct reference density falls below po by up to 20 % for po = 0.1 r3 (at T = 1.58 e/k). Thus our results are probably based on too high values of the reference profile and should be too high; this should not affect the comparison of eqn (12) and (13) since both are based on the same reference profile. As was already mentioned some preliminary computations at low densities were also made for eqn (6). The resulting values of I‘ fall below those of eqn (12), probably as a consequence of the incorrect reference density profile used in eqn (12).go(r). CONCLUSION In this paper we have compared the density profile and adsorption r of the asymptotic theory [eqn (13)] with results obtained from eqn (12) which was derived using less restrictive assumptions. The main results are : (i) along an isotherm the asymptotic theory gives too great an increase in r at subcritical densities and too steep a decrease of r at higher densities ; (ii) even at low densities eqn (12) exhibits a tendency towards a layer-wise structured density profile, whereas the asymptotic theory yields monotonic behaviour if the potential u,(z) is monotonic. The most serious assumption in the derivation of eqn (1 2) seems to be introduced by replacing the density profile of the reference state (fluid bounded by hard wall without attractive potential) by the bulk density po. To clarify this point evaluation of eqn (4) is desirable. W. A. Steele, The Solid-Gas Interface, ed. E. A. Flood (Dekker, New York, 1966), vol. 1, chap. 10; see also W. A. Steele, The Interaction of Gases with Solid Surfaces (Pergamon, Oxford, 1 974). P. G. Menon, Chem. Rev., 1968, 68, 277; see also P. G. Menon, Aduunces in High Pressure Research, ed. R. S . Bradley (Academic Press, New York, 1969), vol. 3, p. 313. F. M. Kuni and A. I. Rusanov, Russ. J. Phys. Chem., 1968,42,443, 621. G . H. Findenegg, Ber. Bunsenges. phys. Chem., 1974, 78, 1281. F. Kohler, The Liquid Stare (Verlag Chemie, Weinheim, 1972), chap. 4 and 7. L. Verlet, Phys. Rev., 1968, 165, 201. F. Din, Thermodynamic Functions of Gases, ed. F. Din (Butterworth, London, 1962), voi. 2. V. L. Kuz’min, F. M. Kuni and A. I. Rusanov, Rum. J. Phys. Chem., 1972, 46, 1032. Note added in proof: In fig. 3 and 4 the scale of ra2 is to be reduced by dividing the displayed values by 20.

ISSN:0301-7249    

DOI:10.1039/DC9755900038

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

GENERAL DISCUSSION Dr. S . Levine (Manchester) (part11 communicated): Mahanty and Ninham mention the Born theory of the self-energy of an ion and the Onsager theory of the dielectric constant of a polar liquid, both of which have the following features in common. A finite size, which must be regarded as empirical, is assigned to the ion or dipole and the immediate surroundings are treated as a continuous dielectric medium. This medium is not necessarily homogeneous since such effects as electrostriction and dielectric saturation have been considered by various authors. All these refinements draw upon macroscopic theories and involve adjustable parameters. Such theories have been very useful in the past but sooner or later they need to be superseded by theories in which the properties of the molecules in the immediate neighbourhood of the ion or dipole are explicitly taken into account.This would involve both quantum mechanics and statistical mechanics. Mahanty and Ninham have formulated a theory of the self-energy of a molecule associated with the London dispersion forces. It seems to me that a pertinent question is whether limitations similar to those mentioned above exist in their theory for the condensed state of matter. The Lifshitz theory and its more recent develop- ments by the Canberra school and others undoubtedly constitute a major advance in the theory of the van der Waals interactions in condensed media. But it is basically a macroscopic theory, which seeks to determine the normal (collective) modes of electromagnetic fluctuations and such modes are dependent on the geometry of the system under consideration. If, for example, one were to use the model of a ffuctuat- ing dipole in a cavity, then the fluctuation modes in the surrounding medium would depend on the size and shape of the cavity and also on the position of the dipole within the cavity.A similar situation exists with the electrostatic theory of the self- energy of an ion or dipole in a cavity. Mahanty and Ninham consider in an ingenious manner the coupled electromagnetic fluctuations of a system of molecules, each of which is characterized by a polarisability tensor. Size and shape factors (radius a and spherical symmetry respectively) and also intensity factor (a(o)) for a single molecule are built into this tensor.In applying their method to a condensed medium, the polarisability a(o) has to be related to the macroscopic dielectric susceptibility through the density. This is basically a problem in statistical mechanics, an approximate solution for which is obtained precisely by imagining a typical molecule to be located inside a cavity. In the case of adsorption a similar problem will arise at high surface coverage, suck as in the case of multilayer adsorption. It seems to me that there will always be a certain stage where quantum mechanics and statistical mechanics must take over from fluctuation theory of dispersion forces. Drs. D. Chan and P. Richmond (Unilever, Port Sunlight) (communicated): Application of the self energy concept is well illustrated by the following calculation of isosteric heats of adsorption for n-alkanes and n-1-alcohols on graphite.Using elementary statistical mechanics it can be shown that the isosteric heat of adsorption is dz exp( - AF(z)/kT)AF(z) fo dz(exp( -AF(z)/kT) - 1) q = kT+ (1) 46GENERAL DISCUSSION 47 where AF(z) is the change in self energy of adsorbed molecules when they are brought from bulk solution up to a distance z from the (planar) interface. To calculate AF(z) using a microscopic model is a formidable task and we circumvent this many body problem by an alternative approach based on the Lifshitz continuum theory. Now for relatively small molecules, the dominant contribution to the Lifshitz interaction free energy with an interface arises from dipole fluctuations. It may then be shown, if we neglect retardation effects, that - kT = rCn)A(iCn) 3 A2(iCn)a2(iT,) (2) ~ ~ , ( i ~ , , ) ~ 3 + G c:(icn)z6 FI(4 = where &*(iClJ - ES(i5ll) &w(iCn) + &s(iCn)’ A(i5ll) = E, and E, are dielectric permeabilities for the wall and solvent respectively evaluated at imaginary frequencies ic, = 2nnkT/k.a(i5,) is the polarizability of the solvent molecule. The sum extends over zero and positive integers and the term for n = 0 is weighted with factor 3. For values of z greater than a, where a is the molecular diameter, the second term is seen to be small when we note that A 5 1 and o! N a. I I I I 1 I I I 2 4 6 a NC represent the spread of experimental data in the literature. FIG. 1.-The isosteric heat for n-alkanes as a function of number of carbon atoms.The bars In principle we should include quadrupole terms as well as Born repulsion effects in the interaction energy function, however, we do not as yet have a satisfactory theory of these terms and assume that the function FI(z) holds down to a cut off distance. Below this distance we suppose the adsorbed molecules “ feel ” an effective hard core of size a. We then have from (1) and (2)48 GENERAL DISCUSSION q = kT We have computed this expression for n-alkanes and n-alkan-1-01s adsorbed on graphite. The dielectric data for graphite were constructed from measurements made by Taft and Philipp.2 The adsorbed molecules were treated as simple points, no structure was included except that the cut off parameter a was taken to be 2.1 A in each case.The polarizabilities for the alkanes and the high frequency component for the alcohols were obtained from bulk dielectric data (refractive indices and 2 4 6 NC FIG. 2.-The isosteric heat for n-alkan-1-01s. The dotted line represents the contribution from opticallultra-violet frequencies. The solid line includes the zero frequency contribution from the polar -OH group. The dots denote experimental data. ionization potentials) in conjunction with the Clausius-Mossotti relation. Static dielectric constants for the alcohols were also obtained using relations for OH groups given in ref. (2). The results are shown in the figures. In fig. 1 the bars denote the spread in experimental data quoted in the literature. In fig. 2, the dotted line denotes the high frequency contribution only; the solid line includes in addition the zero frequency term.The dots denote experimental points." Quite good agreement is obtained, although we remark that the results are fairly sensitive to variations in a. None the less it is clear that Q may be fixed by matching at one point only. Prof. B. W. Ninham (Canberra) said : In informal discussions, Levine has asked whether our methods are claimed to supplant statistical mechanics. Plainly the answer is No. He also asked how we can use Maxwell's equations, which are macroscopic and do not hold on a microscopic level, to give a description of microscopic phenomena involving particles. V. A. Parsegian and B. W. Ninham, J. Colloid Interface Sci., 1972, 37, 332. E. A. Taft and H. R. Philipp, Phys. Reu., 1965,138, 197.Handbook of Chemistry and Physics (The Chemical Rubber Co., Ohio, 50th edn., 1972). A. V. Kiselcv, Disc. Faraday Soc., 1965, 40, 205.GENERAL DISCUSSION 49 Here I would simply say that Maxwell’s equations do hold at a microscopic level. Essentially we have to justify our identification of the Green’s function solutions of the macroscopic Maxwell equations with the screened Green’s function solutions of the microscopic equations. There is, unfortunately, no rigorous justification : the identification claims strong plausibility only when dispersion forces are involved. That is where the statistical mechanics has gone : indeed that seems to me the central problem of statistical mechanics. It might be useful to refer to section 5 of ref. (21), Langbein’s book, ref.(24) and the paper by M. J. Renne (Physica, 1971, 56, 125). Ref. (44) and surface energy calculations appear to provide support. Levine also asked how a dielectric constant could be assigned to a single layer. In this connection, one can remark that formulae for the eflectiue dielectric constants of a layer (which are anisotropic) have been developed as a function of the spacing of the molecules-see ref. (46). It is basically a convenient trick which one comes to after extending Rayleigh’s original substantiation of the Lorentz-Lorenz formula (Phil. Mag., 1892, 34, 481). Prof. D. H. Everett (Bristol) said: Mahanty and Ninham say that their eqn (22) is derived by ignoring entropy effects. However, this equation, essentially similar to that derived by de Boer but allowing the properties of successive adsorbed layers to differ, is based on a kinetic argument which implies the introduction of configurational entropy terms.This equation reduces, in special cases, to the B.E.T. equation and to the Langmuir equation, both of which, as their derivation from statistical mechanics shows, involve consideration of configurational entropies. One may also question the validity of the derivation of the so-called Hill isotherm [eqn (24)] given in more detail in a previous paper : this is said to follow from (22) when the entropy of the first layer is included. However, if as suggested above this entropy is already implicit in (22), then its relationship to (23) is not as simple as indicated by the authors. Prof. B. W.Ninham (Canberra) said: I agree with Everett. Also I confess that the derivation of the Hill isotherm as given here is a bit of a fiddle. The important point, however, is that when the effective dielectric constants of the adsorbed layer are assigned, as in ref. (46), one builds in the lateral interactions easily and correctly by our techniques. Mr. G. A. Chapela and Dr. G. Saville (Imperial College, London) and Prof. J. S . Rowlinson (Oxford University) presented computer results which supplemented those in their printed paper: The Monte Carlo run at z = 0.701 has been complemented by a molecular dynamic run of 38 900 moves at z = 0.71. This has an interfacial profile which is indistinguishable from that of the Monte Carlo run and a surface tension of (ycr2/kT) = 1.55, which is somewhat lower than the figure of 1.80 obtained from 9.8 x lo6 Monte Carlo configurations, but within the combined statistical error.Both results confirm the perturbation calculation shown in fig. 5. The most important difference between the two methods of simulation is that the molecular dynamic run shows no oscillations of density below the interface. This absence confirms the tentative view proposed in our paper that the oscillations are artefacts of the Monte Carlo simulation. This view is confirmed further by the fact that Barker and colleagues have now withdrawn their claim that these oscillations are a true feature of an equilibrium surface. They repeated their earlier work on a double-sided film of liquid but without the artificial gravitational field, with more J.Mahanty and B. Ninham, J.C.S. Firidiy 11, 1974, 70, 637. F. F. Abraham, D. E. Schreiber and J. A. Barker, J. Chern. Phy.~., 1975, 62, 1958.50 GENERAL DISCUSSION preliminary moves to establish equilibrium and with only 256 molecules. A run of 6.2 x lo6 configurations showed smaller oscillations than before, which were decaying steadily as the run proceeded. They assume, therefore, that these would vanish when true equilibrium was attained. What is still obscure is why systems with an interface take so much longer than bulk fluids to reach equi- librium, why Monte Carlo simulation seems more prone to produce oscillations than molecular dynamics, and which of the constraints imposed in both kinds of simulation is responsible for their appearance. Preliminary results have also been obtained for the mixture described in the paper. These were obtained by Monte Carlo simulation (7.7 x lo6 configurations) but show no oscillations of density.The profile of component a extends further into the vapour than that for component b by about one diameter. The Gibbs absorption agrees, within statistical error, with that calculated from the change of surface tension with composition. We believe, with them, that the oscillations are artefacts. Prof. S. 6. Whittington (Ontario) said : Rowlinson suggests that the oscillations in the density may be due to the periodic boundary conditions in the xy plane, used in his Metropolis experiments. If this is the cause of the oscillations they should depend on the size of the periodic rectangle and I would like to ask if Rowlinson has varied these dimensions in his calculations ? Prof.J. S. Rowlinson (Oxford) said: I agree with Whittington that it should be possible to see if it is the repetition of molecular coordinates in the x and y directions which leads to the oscillations of the density p(z) by changing the x and y dimensions of the prism. We tried this by increasing these dimensions from 5 to 7 or 10 diameters, thus more than doubling the number of molecules, but unfortunately the Monte Carlo programme then ran too slowly for it to be practicable. Dr. M. La1 (Unilever, Port Sunlight) said: In the paper by Chapela, Saville and Rowlinson the gas/liquid surface has been idealized as an assembly of a given number of atoms interacting with a plane surface. In real systems, however, there would occur an exchange of atoms between the surface and the bulk.Should such an exchange have any effect on the density profile of the interface? Have the authors considered simulating the interface assuming the grand canonical ensemble approach ? Prof. J. S. Rowlinson (Oxford) said : There was no interchange of molecules across the basal plane (z = 0) and this accounted for the strong oscillations at low values of (z/c). Only at the lowest temperature, and for the Monte Carlo run, did these propagate far into the bulk liquid, and this is the effect which we believe to be spurious. We did try to use a grand-canonical ensemble, in which molecules can be created and annihilated, as well as moved, but it was not possible to choose the chemical potential so that the cell did not fill completely or empty completely nor, if it had been possible, is the chemical potential a parameter which would have fixed the relative amounts of liquid and vapour.The same would, no doubt, have happened with a constant pressure ensemble; both were suitable for studying the adsorption of gases on solids but not for studying the surface of a pure liquid. Dr. J. E. Lane (CSIRO, Australia) said: Have Rowlinson and his colleagues determined the distribution of stress within and parallel to the surface region? ThisGENERAL DISCUSSION 51 information would fix the position of the surface of tension, and the surface excess(es), with respect to this particular dividing surface can then be determined from the density profile(s). Knowledge of these excesses, together with the partial molal volumes of the components in the bulk phases, enables the variation of the surface tension with curvature, and the variation with position in a gravitational or centrifugal field to be obtained through thermodynamic relationships.The stress distribution would also be invaluable in formulating new models of a liquid/vapour interface. Prof. J. S. Rowlinson (Oxford) said: Several suggestions for further work might be practicable, but only at the cost of much greater computing time or of large statistical errors. I agree with Lane that it should be possible to locate the surface of tension by computing the contribution to y from each layer of liquid, interface and gas.This information was in principle already available for each zone of thickness 0 from (z/a) = 6 outwards, but longer runs would be needed if the averages were to be steady enough to be useful. Dr. Levine’s point about the constancy of the chemical potential throughout the interface could be tested in the molecular dynamics run (which is a method of simula- tion which does not pre-suppose the truth of thermodynamics) by taking the ensemble average WP[ - c uio(z)lW> i where u&) is the energy of interaction of a test molecule, 0, at height z, with molecule i which may be at any height. In taking this average the test molecule is supposed to occupy all positions at random in the plane z, but not to cause the other molecules, i, to relax from their previously chosen equilibrium configuration.Widom has shown that such an average is proportional to exp[-p(z)/kT]. The method has previously been used in bulk fluids at densities up to about one-third of that of the liquid at the triple point, but here some preliminary trials on the Monte Carlo runs showed that it led to bad statistics even in the outer part of the interface. We hope to take this up again if we can get sufficient computing time. McLure and Everett ask if it would be possible to simulate the behaviour of a small spherical drop and so test the supposed dependence of y on curvature. This should in principle be possible since others have studied nucleation by simulation, but again very long runs would be needed to get y to the required accuracy. Machine methods are not suitable for the critical region since the correlation length is inevitably larger than the cell size.It is therefore impossible to answer McLure’s question about the critical index which governs the vanishing of y at TC. The experimental value is about 1.27 and the classical, or van der Waals value, is 3. The results in fig. 5 of our paper couId not be used to discriminate between these figures. Dr. N. G. Parsonage (Imperial College, London) said: Would the oscillations in the density still disappear if they were referred to the distance from the liquid/vapour interface? (It would, of course, be necessary to define the position of that interface by a Gibbs-type construction). The position of the interface would fluctuate and this would tend to even out the oscillations.Fluctuations in position of about a/2 would be required. The fluctuation in position of the interface would probably be slow because it involves the concerted action of a large number of molecules. This could explain why in this work and that of Barker and co-workers the oscillations tended to disappear in very long runs. B. Widom, J. Chem. Phys., 1963,39,2808.52 GENERAL DISCUSSION Prof. J. S. Rowlinson (Oxford) said : Parsonage’s suggestion that the positions of the oscillations should be measured from the Gibbs surface rather than in a ‘‘ labora- tory ” frame of reference is a good one and may account for the slow disappearance of the oscillations found in the very recent paper of Abraham, Schreiber and Barker. Here the interfaces were free to move slowly in laboratory coordinates, and seem to have moved by 1.3 Q in 6 x lo6 configurations.Our Monte Carlo oscillations at z = 0.701 are more stable and it would make no difference here if their positions were measured from the Gibbs surface since this moved by less than 0.1 Q in 8 x 106 configurations. Dr. S. Toxvaerd (Copenhagen) said : Recently some statistical mechanical calcula- tions have appeared which show a solid-like ordering of the interface between a simple liquid and its corresponding gas.l- The calculations are based on the Born-Green (BG) equation in the density p(r) and are mentioned in the paper by Perram and White. However, these profiles (Perram and White, fig. 2) cannot be interpreted as the density in the fluid interface. In the work by Croxton et aL1 the energy operator @(z) is density dependent.The first iteration cycle has changed the position of the Gibbs (equimolecular) dividing surface towards the liquid. Thus, <D[p(z)] has to be corrected before the next iteration cycle. In ref. (2) the iteration technique used for numerical calculation of p(z) was not given but, using the technique described el~ewhere,~ we find that the approximation of g(rl v,) used by Nazarian does not give an oscillating interfacial profile. The iterations diverge. However, the BG equation has a monotonic solution of p(z) for a particular choice of g(rl, r2).3 With regard to the calculations of p(z) performed by Perram and White, the density profile obtained is the mean density of the solvent (1) up to a (big) particle (2) with variable radius given by formulae (2.3) [and (3.5)] and an attractive, soft and static potential $(z).This density profile differs from the monotonic density profile in a fluid interface (Rowlinson, fig. 1) by having an oscillating density variation up to the soft wall (particle 2). That a static (attractive) potential will explain the layered structure is also seen from computer experiment^.^^' However, the computer experiments all give a monotonic density variation in the liquid/vapour interface. Prof. J. S. Rowlinson (Oxford) said: The apparently spurious oscillations of density shown by some Monte Carlo calculations show the danger of imposing too great a degree of smoothness on the gas/liquid interface. If this is done by packing against a hard surface then strong oscillations are, of course, inevitable.I think, therefore, that it is dangerous to interpret the situation at the surface of a bubble as “ repulsive force between the bubble and the particle ”. The translation of this analogy into statistical mechanics can lead to equations for the distribution functions at an arti- ficially smooth surface. It is probably unimportant whether the sufiace is plane as in earlier work, or curved as in the paper of Perram and White, what is important is its smoothness and I think that it is this feature which has led to the almost certainly false oscillations found in many theoretical calculations. Prof. G. H. Findenegg (Ruhr- University, Bochurn) said : In perturbation theories C. A. Croxton and R.P. Ferrier, J. Phys. C, 1971, 4, 1909. G. M. Nazarian, J. Chem. Phys., 1972,56, 1408. S. Toxvaerd, MoZ. Phys., 1973, 26, 91. K. S. Liu, J. Clzem. Phys., 1974, 60, 4226. S. Toxvaerd, J. Chem. Phys., 1975,62, 1589. F. F. Abraham, D. E. Schreiber and J. A. Barker, J. Chem. Phys., 1975, 62,1958. G . A. Chapela, G. Saville and J. S . Rowlinson, this Discussion.GENERAL DISCUSSION 53 the density profile in a fluid near an adsorbing wall is expressed relative to that of a reference system. The reference system used by Steele and by Kuni and Rusanov corresponds to the situation at the lower boundary (z = 0) of the system considered by Chapela et al. If their Monte Carlo results are sufficiently accurate it would be useful to compute the (small) surface excess for this reference system.Mr. G. A. Chapela (Imperial College, London) and Prof. J. S . Rowlinson (Oxford University) said: Our results at low values of (z/a) could be analysed to study the adsorption of a liquid against a plane surface, as Findenegg proposes, although the runs were not made with this purpose in mind. They would, however, give him information only about a liquid adsorbed on a homogeneous infinite slab of the same density and intermolecular potential as the bulk liquid, but if this was of use as a reference system then it should be possible to extract it. Dr. C. A. Croxton (Newcastle, N.S. W., Australia) said: In the discussion of stable density oscillations in the single particle distribution, the observation should be made that such a structured transition zone bears important implications for the surface thermodynamics parameters.In particular, it bears the implication of an inversion in the surface tension-temperature curve-y(T) actually showing a positive slope at temperatures just above the.triple point. Certainly this is at variance with probably all experimental evidence in the case of the Lennard-Jones systems, and it would appear that the single particle distribution should be monotonic, in agreement with Rowlinson’s extended simulations. What is disturbing, however, is the ease with which spurious oscillations develop in the single particle distribution on the basis of both machine simulation and BGY-PY calculations. Such spurious oscillations undermine confidence in the application of these techniques and attention should be focused on this disconcerting feature of the determinations.A number of machine simulations in 2 and 3 dimensions have now been performed and in virtually every case oscillations have developed. The inhibition of major structural re-organisation is a notorious weak point in the simulation schemes, and one wonders to what extent the suppression of long wavelength density fluctuations in these microscopic assemblies is responsible for the important qualitative discrepancy in the results. In the case of the statistical mechanical analyses the specification of ~ ( ~ ~ ( z ) inevitably involves the anisoptropic pair distribution P(~)(Z, v ) for which some form of closure has to be adopted either explicitly in the case of the BGY analyses, or implicitly in the case of PY.Specification of P ( ~ > ( Z , r) implies specification of the surface constraining field, and, if this is too “ hard ”, oscillations are induced into the single particle distribution. Conversely, a soft constraining field results in a monotonic profile. The role of the closure device urgently needs assessment: closures at present are adopted on a largely ad koc basis. The diameter of the vacuum bubble in the paper by Perram and White is -25 atomic diameters, and I am concerned that the pronounced oscillations arise primarily from the very strong surface field which develops at the surface of such a small bubble. The extent to which such oscillations can be taken to be representative of a planar interface, that is a bubble of infinite diameter, is not clear. The results would, however, be of great interest in cavitation studies. The development of pronounced oscillations in the case of certain liquid metal systems appears to be a real possibility. The very strong, short-range, attractive interaction between ions, the long range Friedel oscillations, if they persist at the liquid surface, and Lang-Kohn oscillations in the electronic distribution arising54 GENERAL DISCUSSION from the quantum interference effect may well conspire to induce stable density oscillations in the single particle distribution. In the case of liquid metals there is evidence of inversion in the y(T) characteristic for certain systems. Again, the resolution of the closure difficulty becomes of central importance if a theoretical understanding of oscillatory profiles is to be achieved. Dr. J. W. Perram and Dr. L. R. White (Australian National University) said: Although the assumption of a smooth interface could cause oscillations in the density, it need not do so. According to the prescription given by us, if the self-consistent repulsion between the atom and the bubble were " soft " enough, then the oscilla- tions would be damped out. The later simulations by Abraham et d.,' where many more configurations were taken, produced monotonic density profiles, although the interface was severely constrained. F. F. Abraham, D. E. Schreiber and J. A. Barker, J. Chern. Phys. 1975, 62, 1958.

ISSN:0301-7249    

DOI:10.1039/DC9755900046

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Experimental Test of a Multilayer Model of a Regular Solution Napour Interface BY J. E. LANE CSIRO Division of Applied Organic Chemistry, G.P.O. Box 433 1 , Melbourne 3001, Australia Received 10th December, 1974 Surface tensions and adsorptions at the surface are reported for the system cyclohexane+carbon tetrachloride over a range of compositions and temperatures. The experimental data are compared with the values of these properties obtained with a multilayer model of the regular solution/vapour interface. Model and experiment are in good agreement for the adsorption, but not for the surface tension. The quasi-crystalline models of a surface, usually based on some form of the strictly regular solution approach,l have been popular in both the monolayer 2-4 and the multilayer 5-7 forms, and have been adapted to describe the solid/solution interface,'-l l the solid/vapour interface, polymer adsorption ** 9* and the stability of colloidal dispersions.The predictions of these models are generally in qualitative agreement with experimental data and indicate a wide variety of observed phenomena such as the formation of surface azeotropes 7* l1 and the multiple surface transitions often observed at the solid/vapour interface. The magnitude of the quantitative difference between model and experiment is less well established. This lack of information is a hindrance to any critical examination of the assumptions used in the development of the models. The particular multilayer model examined in this paper has been described in detail.' It assumes that the molecules of the solution components are approximately spherical, of similar molar volumes, and non-polar and symmetrical with respect to interactions with the nearest neighbour molecules.It is necessary to know the excess Gibbs function G: for mixing of the solution components, which should be an approximately symmetrical function of the mole fraction of one component.l' The system cyclohexane (component A) + carbon tetrachloride (component B) satisfies these requirements reasonably well. The molar volumes differ by approxi- mately 12 %,16 the dipole moment of cyclohexane in carbon tetrachloride is zero,17 and carbon tetrachloride has only an octopole and higher moments. The G: data are available l8 for the temperature range 40-70°C, and this can be extended to lower temperatures using experimental heats of mixing.EXPERIMENTAL AnaIaR carbon tetrachloride was further purified by distillation using a reflux column having the equivalent of 100 theoretical plates at total reflux.20 The distilled material showed a single peak when checked with a gas chromatograph having a column 45 ft long packed with 20-28 % of normal hexadecane on a 60/70 acid washed silanised Chromosil G. The densities at 20,25 and 30°C are 1594.1 1,1584.30 and 1574.59 kg M--~ in agreement with a number of values tabulated by Timmermans.21 The density at 30°C is in excellent agreement with the value given by Wood and Gray l6 for air-saturated carbon tetrachloride. The 5556 MULTILAYER MODEL surface tension at 25°C was found to be 26.20 mN m-l which is a little higher than the value 26.14 mN m-l reported by Lam and Benson.22 Merck G.R.grade cyclohexane, dried with molecular sieves type 5A, gave a single peak using the gas chromatographic column described above. The densities at 20, 25 and 30°C were respectively 778.67,773.81 and 769.12 kg m-3, again in agreement with values tabulated by Timmermans.21 The surface tension at 25°C was 24.38 mN m-l, in excellent agreement with the values reported by Lam and Benson 22 and Chaudri et aZ.23 Densities were measured using single stem pycnometers, calibrated with de-ionized water further purified by distillation from alkaline potassium permanganate solution, then given a further straight distillation. The density of water was obtained from the table of Bigg.24 Surface tensions were measured with a twin capillary rise apparatus in which the capillary was connected at the top and bottom to the wider tube.The top of the wide tube was fitted with a screw cap having a Teflon seal which allowed the whole apparatus to be sub- merged in a thermostatted water bath. The radius of the wide tube (precision bore) was measured directly at 25°C as 0.790 6 cm. The radius of the capillary, 0.019 430 cm at 25°C was obtained by calibration with water, purified as described above, taking the surface tension to be 72.01 mNm-l at 25°C.25 The radius was checked by measuring the surface tension of benzene purified by the method of Harkins and Brown.26 This was 28.21 mN m--' at 25"C, in agreement with the value given by Young and hark in^.^' The radii of the tubes were corrected for temperature changes using the linear coefficient of thermal expansion of glass 3 .2 ~ K-1.28 All calculations of surface tension were made by a method described earlier.29 RESULTS The surface tension and density of the two pure components and 9 intermediate compositions were determined at 19 temperatures between 7.58 and 40.00"C. The error in the surface tension values was estimated to be f0.03 mN m-l for tempera- tures between 16.00 and 40.0O0C, the temperatures being maintained to $.O.O5"C. Below 16°C the error increased to k0.09 mN m-l at 7.58"C. The increased error was due to a number of factors, including poorer control of the thermostat and a fault subsequently found in the thermometer. Representative surface tension data, as a function of the mole fraction x of cyclohexane, at 7.58, 16.04, 30.00 and 40.00"C are shown in fig.1.' The error in 27 4 E 23 I I I I I I I c 0.0 0.2 0.4 0.6 0.8 1.0 Xcycio hexane FIG. 1.-Surface tension o/mN m-I, for the system cyclohexane+carbon tetrachloride, as a function of mole fraction x of cyclohexane at 7.58, 16.04,30.00 and 40.00°C. Continuous curves show values for multilayer model, experimental points represented by and + . * A table of the complete surface tension data is available on request from the author.J. E. LANE 57 composition is too small to indicate in the figure but the error in surface tension at the three higher temperatures is equivalent to the radius of the circular symbols. The larger error in the data at 7.58"C is indicated by the vertical section of the crosses.The surface tension results at 25°C are shown in greater detail in fig. 2, and again the I I I I I I 1 I 1 I J 0.0 0.2 0.4 0.6 0.8 I .o xc yclohexane FIG. 2.-Surface tension o/mN m-l, for the system cyclohexane+ carbon tetrachloride; as a function of mole fraction x of cyclohexane at 25°C. Continuous curve represents multilayer model with fixed lattice size, broken curve represents multilayer model with variable lattice size, experimental points shown by +. vertical section of the crosses indicates the error in the surface tension measurement. It was found that at all temperatures the surface tension Q, as a function of the mole fraction x of cyclohexane, could be represented to the precision of the measurements by the parabolic relation, where the coefficients aT, PT and yT are temperature dependent.The surface excesses per unit area, ri and rb of cyclohexane and carbon tetra- chloride, with respect to an arbitrary Gibbs dividing surface, are related to surface tension by the adsorption isotherm 0 = C I T + P T X + Y T X ~ (1) where pi is the chemical potential of component i, s ' ~ is the surface excess entropy per unit area, and T is the thermodynamic temperature. The position of the dividing surface can be fixed by a number of different conventions, but a convenient one for comparison with the multilayer model is defined by T A + T B = 0 (3) the omission of the primes indicating the use of this particular dividing surface. Combining (2), (3) and the Gibbs-Duhem relation for bulk phases gives, on re- arrangement, r A = - (l - x > ( a a / a x > T / ( a ~ A / a x ) , .(4) The experimentally derived excess Gibbs function G," for the mixing of cyclohexane58 MULTILAYER MODEL with carbon tetrachloride can be fitted as a function of x by the expression G: = X ( 1 - X ) ( a , 4- bTX 4- CTX) where aT, bT and cT are temperature dependent coefficients. Differentiating G: with respect to x gives RT + X(1- X)(2(b~ - a,) + 6X(+ - b T ) - 1 2 c T x 2 ) ( 6 ) - (%jT = X Combination of eqn (l), (5) and (6) then gives the adsorption as (7) rA = - ~ ( ~ - ~ ) ( P T + ~ Y T x ) RT+x(l-~)(2(b,-a,)+6X(C~- b T ) - 12cTx2]>' The values of GE as a function of composition were measured for this system between the temperatures of 40 and 70°C by Scatchard et a2.I8 Values of Gz at lower temperatures can be obtained by using the heat of mixing as a function of composition as measured by Ewing and Marsh I9 at temperatures of 15, 25 and 45°C.At any particular concentration the heat of mixing H," can be expressed as a linear function of temper- ature by H," = A,+B,T, the coefficients A , and B, being concentration dependent. Straightforward thermodynamic arguments show that G: as a function of temperature must then satisfy G: = A, -B,T In T f C,T. The values of A , and B, are obtained from the heats of mixing, and C, from the experimental values of Gt at 40°C. Using this procedure, the coefficients aT, bT and cT of eqn (5) were found to be 295.88, -31.01 and 8.292 J rnol-' at 25.00"C. The coefficients PT and yT at 25.00"C were obtained from the surface tension data as -2.707 and 0.8806 mN m-l respectively.Substitution of these coefficients into eqn (7) gives rA as a function of solution comp- osition at 25.OO0C, and this is shown by curve A in fig. 3. 2 .o c1 E 0.0 0.0 0.2 0.4 0.6 0.8 I .O xcyciohexane Fm. 3.-Adsorption lo7 x rA/mol nr2, for the system cyclohexane+ carbon tetrachloride, as a function of mole fraction x of cyclohexane at 25°C. Curve A represents experimental values, curve B the multilayer model with fixed lattice size, curve C the multilayer model with variable lattice size. The surface excess entropy s ' ~ can be obtained from (2) and (3) if the partial derivatives (apJaT), are known. It is more convenient for the present purpose to use an alternative entropy function, For a system with i phases, j types of interface and k components, changes in the Helmholtz free energy F of the system are given by 30 dF = -S dT-C Pi dV'+C C T ~ dAj+C pk dnk.(8) i j kJ. E. LANE 59 The second differential coefficient of F with respect to T and AJ gives the Maxwell type relation, The term on the left-hand side of (9) represents the increase in entropy of the whole system on increasing the area of the interface J without change of temperature, volume of the phases, areas of the other interfaces and numbers of moles of the various components. The temperature coefficient of surface tension at constant composition was obtained by linear regression from the experimental data over the temperature range :2.55-48.OO0C- (the data--& lwwer- ~ ~ p c - r a t u ~ e s w ~ e - m ~ as&- be~ami--of their iwer precision).The standard deviation was less than 0.04 mN m-l K-l at all composi- tions. The results are shown in fig. 4, the experimental error being indicated by the lengths of the vertical section of the crosses. The error may appear in fig. 4 to be large but represents a relative error of only 3 %. -0.120 -0.130 0.0 02 0.4 0.6 0.8 1.0 Xc yclohexana FIG. 4 . T h e temperature dependence of surface tension (aa/aT)x/mN m-l K-I, for the system cyclohexane+carbon tetrachloride as a function of the mole fraction x of cyclohexane. The con- tinuous curve represents the multilayer model with fixed lattice size and the experimental points are shown as +. DISCUSSION The multilayer model of a surface using the quasi-chemical (QC) approximation has been described in detail.' The important relationships are repeated here because some can be put in a simpler form than given before, and there are several typograph- ical errors in the equations of the original paper.The model is quasi-crystalline with layers of lattice sites numbered 1,2, 3, . . ., r, the first layer is adjacent to the vapour phase and the rth layer is in the homogeneous part of the solution phase. Three parameters are used, all fixed by reference to the physical properties of the solution. The interchange free energy w of the strictly regular solution model of a solution 1* l5 is determined by the excess Gibbs function G," of mixing for the solution phase. For the hexagonal close-packed lattice, the number of moles of lattice sites nu in unit area of the first layer is obtained from the molar volume V, of the solution as where L is Avogadro's number.There is usually a variation of molar volume with concentration, so an average of the molar volumes of the two pure components is used to determine nu. nu = L - + v , ~60 MULTILAYER MODEL The surface layer parameter u, although rigorously defined in terms of inter- molecular interactions, is obtained from the surface tensions QX and Q; of the two pure components A and B respectively as u = (a~-o~)l(n*L). (11) The mole fraction xi of component A in layer i is obtained as the simultaneous solu- tion of the set of non-linear equations for i = 1,2, 3, . . ., r where a, = I forj = i, ai = m forj # i, b = 1 for i = 1 and b = 0 for i # 1, z is the coordination number of the lattice, lz is the number of nearest neighbour molecules that each molecule has in the same layer as itself, and mz is the number of nearest neighbour molecules in the adjacent layer.The term Zfj is the most probable pro- portion of nearest neighbour pairs having an A molecule in layer i and a B molecule in layer j . The term x' is the usual term for unlike nearest neighbour pairs in the QC regular solution mode1.l' l5 The terms El, and ff are found as solutions of the equa- tions (x - Z)(l- x - Z))ln2 = exp(2w/kT). (14) The adsorption is given as r rA = nu C (xi--) i= 1 and the surface tension Q of the solution as The surface tensions calculated from the multilayer model at 7.58, 16.04,30.00 and 40.00°C are shown as the continuous curves in fig.1. The difference between model and experiment is more readily seen in fig. 2 where the continuous curve shows the surface tension of the model at 25.0O0C, using w/kT = 0.1 14, vlkT = -0.136 and n" = 5.40 x mmol m-2. The definition of u in eqn (11) ensures that model and experiment are in agreement for the two pure liquids. At the intermediate composi- tions the surface tension for the model tends to be higher than the experimental value, and for most mixture compositions the disagreement is greater than the experimental error. The exception to this behaviour is the measurement at 7.58"C but this is possibly due to the greater experimental error at this temperature. The adsorption, calculated through eqn (15) at 25°C is shown as a function of composition by the curve B in fig.3. The agreement with the experimental curve A is very good. This result is not surprising and should apply to most systems where GE is an approximately symmetrical function of x and thus the QC approximation forJ. E. LANE 61 (apA/ax), will be close to the experimental value. The reason for the agreement is that the multilayer model is completely consistent with the adsorption isotherm.' If eqn (4) is rearranged and integrated over the whole concentration range, it gives Thus the definition of v by eqn (1 1) automatically ensures that the integral on the right hand side of eqn (17) is identical for both the model and the experimental adsorptions, Because (apA/a& must always be positive for thermodynamic stability of the ~ y t e r n , ~ ~ then if the model has a lower value of r A than the experimental value over one section of the concentration range, it must be larger over the remaining concentration range in order to satisfy eqn (17).The derivative (do/8T), for the model is shown as a function of concentration by the continuous curve in fig. 4. The agreement between model and experiment for this property indicates that if the solution surface tension for the model is higher than the experimental value for one temperature, it will also be higher at other tempera- tures. It reinforces the argument that the apparent agreement for surface tension between model and experiment at 7.58"C is a result of the greater experimental error at this temperature.In making this comparison between model and experiment, the formal parameters have not been treated as adjustable. It is possible to force reasonable agreement for both surface tension and adsorption by arbitrarily increasing the value of nu but the model then loses its physical significance. Hoar and Melf~rd,~ in discussing a monolayer model, suggested that nu should be treated as a function of x, thus varying with changes in the molar volume of the solution. This suggestion was tried using a parabolic fit of density against composi- tion to determine the value of nu at a particular concentration. The surface tension at 25°C is shown by the broken curve in fig. 2 and agreement between model and experiment is improved over the model with fixed nu.The corresponding adsorption calculation is shown as curve C in fig. 3, and the result is disastrous, showing aspurious surface azeotrope at x = 0.38. The reason for this strange behaviour is the result of coincidence. Although the surface tension per unit area of cyclohexane is lower than that of carbon tetrachloride, the greater molecular density of carbon tetrachloride makes the surface tension on a molecular surface area basis marginally lower for carbon tetrachloride than for cyclohexane, whereas v/kT = -0.136 for the fixed lattice at 25"C, v/kT = 0.0068 for the variable lattice (in both cases treating cyclo- hexane as component A). As w/kT = 0.114, the absolute value of v/kT for the variable lattice satisfies the condition for azeotrope formation.' This example shows the necessity of using a model only under the conditions on which its development is based.The adsorption isotherm requires that changes in surface tension with bulk composition be a function of the surface excess of a component, and not the actual numbers of molecules in unit area of a surface layer. The higher values of surface tension given by the model suggest that some relaxa- tion mechanism has been neglected. This could be the assumptions that the lattice is always fully occupied and that changes in the vapour composition with changes in solution composition do not influence the surface properties of the solution. Some preliminary calculations using a hole model of a quasi-lattice liquid/vapour interface suggest that both of these assumptions are incorrect.A further criticism of the model is the use of nearest neighbour energies of inter- action rather than nearest neighbour force of interaction to calculate surface tension62 MULTILAYER MODEL (as is used for Monte Carlo type calculations 3 2 9 33). Molecules with equal nearest neighbour energies of interaction may not interact with equal intermolecular force as the shape of the energy/distance of separation curve may differ. I acknowledge the experimental work carried out by P. Chua and M. Weeks and the advice and assistance with the purification of the solvents given by Dr. I. Brown. E. A. Guggenheim, Mixtures (Oxford University Press, Oxford, 1952). E. A. Guggenheim, Trans. Faraday SOC., 1945, 41, 150. T. P. Hoar and D.A. Melford, Trans. Faraday SOC., 1957,53, 315. G. Delmas and D. Patterson, J. Phys. Chem., 1960, 64,1827. R. Defay and I. Prigogine, Trans. Faraday SOC., 1950,46, 199. S. Ono and S . Kondo, Handbuch der Physik, ed. S . Flugge (Springer-Verlag, Berlin, 1960), vol. X, p. 240. S. G. Ash, D. H. Everett and G. H. Findenegg, Trans. Furday SOC., 1968, 64,2639. S. G. Ash, D. H. Everett and G. H. Findenegg, Trans. Furuduy Soc., 1968,64,2645. D. H. Everett, Trans. Faraday SOC., 1964, 60, 1803. D. H. Everett, Trans. Faraday SOC., 1965,61,2478. l2 J. E. Lane, Preprinfs 48th National Colloid Symposium, The University of Texas at Austin, Austin Texas, June 24-26, 1974 (Division of Colloid and Surface Chemistry, American Chem- ical Society, Austin, 1974), p. 246. ' J. E. Lane, Austral.J. Chem., 1968, 21, 827. l3 R. J. Roe, J. Chem. Phys., 1974,60,4192. I4 S . G. Ash, J.C.S. Faraday 11, 1974,70,895. E. A. Guggenheim, Applications of' Statistical Mechanics (Oxford University Press, Oxford, 1966), chap. 6, p. 80. l6 S. E. Wood and J. A. Gray, J. Amer. Chem. SOC., 1952, 74, 3729. C. G. Le Fhvre and R. J. W. Le Fkvre, J. Chem. SOC., 1956,3549. G . Scatchard, S. E. Wood and J. M. Mochel, J. Phys. Chem., 1939,61,3206. M. B. Ewing and K. N. Marsh, J. Chem. Thermodynamics, 1970,2, 351. 2o I. Brown and A. H. Ewald, Austral. J. Sci. Res. A, 1951, 4, 198. 21 J. Timmermans, Physico-chemical Constants ofpure Organic Compounds (Elsevier, Amsterdam 22 V. T. Lam and G. C. Benson, Canad. J. Chem., 1970,24,3773. 23 M. M. Chaudri, P. K. Katli and M. N. Baliga, Trans. Faraday SOC., 1959,55, 2013. 24P H. Bigg, Brit. J. Appl. Phys., 1967, 18, 521. 25 W. D. Harkins and A. E. Alexander, Technique of Organic Chemistry, ed. A. Weissberger 26 W. D. Harkins and F. E. Brown, J. Amer. Chem. SOC., 1919, 41,499. dam, 1965), vol. 2, p. 157 and 184. (Interscience, New York, 3rd edn., 1959), vol. 1, part 1, chap. 14, p. 772. T. F. Young and W. D. Harkins, International Critical Tables, ed. E. W. Washburn (McGraw Hill, New York, 1928), vol. 4, p. 454. 28 N. A. Lange and G. M. Forker, editors, Handbook of Chemistry (McGraw Hill, New York, 10th edn. rev., 1967), p. 856. 29 J. E. Lane, J. ColIoid Interface Sci., 1973, 42, 145. 30 J. W. Gibbs, Collected Works (Dover, New York 1961), vol. 1, p. 231. 31 I. Prigogine and R. Defay, Chemical Thermodynamics, trans. D. H. Everett (Longmans Green, 32 0. K. Rice, Statistical Mechanics, Thermodynamics and Kinetics, (Freeman, San Francisco, 33 J. K. Lee, J. A. Barker and G. M. Pound, J. Chent. Phys., 1974,60, 1976. London, 1954), chap. 15, p. 220. 1967), p. 345.

ISSN:0301-7249    

DOI:10.1039/DC9755900055

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Adsorption of Surface Active Agents in a Non-aqueous Solvent BY A. COUPER,* G. P. GLADDENT AND B. T. INGRAM: Department of Physical Chemistry, University of Bristol, Bristol BS8 ITS Received 6th February, 1975 Measurements have been made of the surface tension of solutions of sodium dodecyl sulphate, dodecyltrimethylammonium bromide, decanol, and of n-dodecyl and n-tetradecyl hexaoxyethylene monoethers in pure formamide and in salt solutions. Adsorbed films of the ionic surfactants exert low surface pressures (about 15 mN m-1 near the solubility limit) and exhibit no discontinuity characteristic of micellar aggregation. The monoethers exert surface pressures up to 25 mN m-I and show a critical micelle concentration of 0.039 mol dm-3 (CI2E6) and 0.013 mol dm-3 (ClsEs) with aggregation numbers of about 40 and 64 respectively at 25°C.The lower surface activity and higher solubility of the surfactants in formamide compared with water indicate the lower stability of adsorbed and micellar states in the former solvent. Similarly the aggregation numbers of the micelles of monoethers in formamide are much lower than those in water. Extensive studies 1-3 of the adsorption and micellar aggregation of both ionic and non-ionic surfactants in water have led to the evaluation of the standard enthalpy and entropy of these processes. The principal factor determining the tendency to adsorption appears to be the decrease in free energy resulting from the association of the hydro- phobic parts of the surfactant molecules at the surface, and limited extensions have been made to solvents other than water 4-6 which are consistent with the theory that adsorption depends upon the difference in surface free energy between solvent and solute.Micellar aggregation in water of molecules, having an alkyl chain of n C atoms and a non-ionic hydrophilic group, involves a standard free energy change, AG:, which becomes more negative by -2.9 kJ mol-1 with each additional -CH2- group, and also becomes more negative as the polarity of the head-group diminishes. The standard entropy of micellization, AS:, is small and positive, increasing very slightly with increasing chain length. There is much evidence that 3 or 4 --CH2-- groups adjacent to the head group are not completely separated from the solvent when micelles are formed, and that they should be excluded from the incremental properties which are linear with hydrocarbon chain length.The increment of AGL per C atom seems to arise mainly from the greater cohesive forces between water molecules than those between water molecules and the hydrocarbon chain. Formamide, having a relative permittivity of 109.5 and surface tension of 58.5 mN m-l at 25"C, has sufficient cohesive force to favour adsorption and micellar aggregation, whilst its dielectric properties ensure that solutions of ionic surfactants and other salts have electrolyte properties closely resembling those of aqueous solutions.* t present address : Unilever Research Laboratory, 455 London Road, Isleworth, Middlesex. $ present address : Procter and Gamble Limited, Newcastle Technical Centre, Newcastle upon Tyne NE12 9TS.6364 SURFACTANTS I N NON-AQUEOUS SOLVENTS EXPERIMENTAL MATERIALS B.D.H. laboratory reagent formamide was dried, distilled twice at a low pressure of dry nitrogen, and finally purified by fractional freezing under dry nitrogen. About 10 % of the formamide was discarded in each of 5 cycles, and instead of an opaque mass of small crystals it finally yielded large transparent crystals melting at 2.55+0.Ol0C, in good agreement with the highest recorded value.9 The surface tension of this formamide was 58.5 mN m-l, whereas that of a sample freed from water and ionic material by passage through a molecular sieve and a mixed-bed ion exchange resin was 58.0+0.1 mN m-' and its melting point was in the range 2.3 to 2.4"C.The formamide was stored under dry Nz, and solutions were prepared and introduced into closed cells for surface tension measurement, ultracentrifugation, etc., in a dry box. n-Decanol was used as supplied by Eastman Kodak Limited. It was shown to be 98 % pure, the remainder probably being an isomer. n-Dodecyltrimethylammonium bromide, DTAB, was a sample prepared by Dr. J. W. Goodwin of this Department. It was purified by recrystallisation from moist acetone, and surface tension measurement revealed no minimum at the c.m.c. in aqueous solution. Sodium dodecyl sulphate, SDS, was prepared from Eastman Kodak n-dodecanol, shown by g.1.c. to be at least 99.9 % pure, by sulphonation, neutralization and recrystallisation. The product was purified by continuous extraction lo to remove dodecanol, until no minimum in surface tension at the c.m.c.could be detected. n-Dodecyl hexaoxyethylene glycol monoether, C1 2E6, was initially supplied as a 99 % pure sample by Procter and Gamble Limited. A purer sample was prepared by reaction of dodecyl bromide with the sodium derivative of triethylene glycol l1 to give C12E3, which, after purification, was chlorinated and converted to C12E6 by a further reaction with the sodium derivative of triethylene glycol. The product distilled off between 215 and 219°C at 5.3 N me2. An alternative method l2 was found to be more successful for C14E6. PP'-dichloro- diethyl ether was added to the sodium derivative of diethylene glycol at 110°C and the result- ing hexaoxyethylene glycol, E6, was distilled off.Tetradecyl bromide was then added to the sodium derivative of E6 at 120°C, and the C14E6 was extracted in xylene and purified by distillation. Both C12ES and C14E6 were further purified by chromatography over M 61 activated silica gel.13 MEASUREMENTS Solubilities were determined by measuring the temperature at which the cloudy suspension formed by rapid cooling of a solution in formamide, became clear on slow warming. Such values agreed with observations of the separation of solid on cooling to within 0.5"C. Osmotic coefficients of the solvent were determined cryoscopically using a Stantel F 23 thermistor sensitive to K. A seeding technique, which eliminated supercooling, allowed the measurement of the osmotic coefficient, gm, within 4 2 % at 0.15 mol kg-l and 4 10 % at 0.03 mol kg-l when its value approached unity.The extent of adsorption was obtained from measurements of the surface tensions of solutions, y, using photographs to determine the size and shape of hanging drops at equili- brium with their own saturated vapour. Fordham's tables l4 were used to derive y from selected diameters, and a detailed examination of both systematic and random sources of error showed that the total error was less than k0.24 %, i.e., 0.14 mN m-l for formamide. The temperature was controlled within kO.05 K. Pure liquids showed no variation of yo with the age of the drop up to 1 h, but solutions showed a small decrease, consistent with the diffusion of trace impurities to the interface. To allow for this a linear extrapolation to zero drop age was made, and this arbitrary procedure introduced some additional error, the total being estimated at less than 0.3 mN m-l.Molecular weights of micelles of C12E6 and C14E6 were obtained from sedimentation studies in a Beckman model E ultracentrifuge ; " sedimentation " was negative, and shouldA . COUPER, G. P . GLADDEN AND B . T. INGRAM 65 perhaps be termed " creaming " because the micelles were less dense than the solvent. The Archibald " approach to equilibrium '' technique l5 was used at 31 410 and 24 630 r.p.m. for CI2E6 and CI4E6 respectively, and sedimentation coefficients, s, were determined at 59 780 r.p.m. by layering surfactant solution containing micelles onto a solution just below the critical micelle concentration in a synthetic boundary cell.These measurements were combined with intrinsic viscosities measured in an Ostwald-type dilution viscometer modified to exclude atmospheric contamination, and densities from measurements in a small pyknometer (5 cm3) to derive micellar radii. The principal source of error in the molecular weight values in table 5, and micellar radii in table 6, was the difficulty of density measurement in the small volumes of solution available ; 0.005 % error in density leading to about 8 % in the expression 1 - ijp, where i j is the specific volume of solute, and p the solvent density. RESULTS AND DISCUSSION SURFACE PROPERTIES OF FORMAMIDE The temperature coefficient of y for formamide was -0.097t0.005 mN m-1 K-l, so that if the area per molecule in the surface was about 0.1 nm2, the excess surface entropy would be 5.9 J mol-1 K-I, and the total surface energy 87.4 mJ m-2.Thus the ratio of the surface energy to the energy of evaporation per molecule for formamide was about 0.08 compared with 0.19 and 0.37 for the unsymmetrical molecules CH30H and H,O and 0.74 for the symmetrical CCI4, giving an indication of the extent of specific orientation at the HCONH2 surface. The interfacial tension between formamide and n-decane at 25°C was 26.87f 0.04 inN m-I, and the mutually saturated surface tensions of formamide and decane were 58.5 mN m-I and 23.55 mN m-I respectively. Application of the relation- ship y12 = y1 +yz -24(y,y,)* gives a value of 0.754 for the parameter $I which is FIG. 1 .-Surface tension of solutions of surface active agents in formamide : 1, SDS in formamide ; 2, SDS in 0.1 mol dm-3 NaCl in formamide ; 3, SDS in 0.98 mol dm-3 NaCl in formamide ; 4, DTAB in formamide ; 5, DTAB in 5.8 mol dm-3 LiBr in formamide ; 6, decanol in formamide. 59-c66 SURFACTANTS I N NON-AQUEOUS SOLVENTS intermediate between the values for n-decane/water and n-decane and less polar solvents.Similarly, if the alternative form l7 y12 = y1+y2-2(ylyd,)* is used, where y: is the dispersion component of the surface tension of the polar liquid, we find yd = 32 mN m-l for formamide, compared with the value 19.8 mN m-l for water, whereas one might have expected from the physical properties of formamide a value nearer 28 mN m-l. The difference largely reflects the relative sizes and densities of the formamide and water molecules.The work of adhesion between formamide and decane, 55.2 mJ m-2, obtained from the surface and interfacial tensions, substantially exceeds the value of 48 mJ m-2 between water and decane, in agreement with the values obtained from contact angle measurements at paraffin wax surfaces of 57 mJ m-2 for formamide and 50 mJ m-2 for water at 25"C, confirming the greater affinity of hydrocarbon for formamide than for water. TABLE ME MEAN ACTIVITY COEFFICIENTS OF SDS IN FOFWAMIDE (25°C) concentrat ion osmotic coefficient mean activity mean activity mol kg- 1 of solvent coefficient (expt.) coefficient (calc.) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.9720 0.9675 0.9596 0.9612 0.9537 0.9548 0.9579 0.9583 0.9550 0.9516 0.9459 0.9412 0.9330 0.9242 0.9161 0.937 0.912 0.896 0.885 0.875 0.866 0.860 0.854 0.848 0.842 0.834 0.826 0.815 0.805 0.790 0.937 0.914 0.899 0.888 0.879 0.869 0.862 0.855 0.849 0.844 0.838 0.833 0.827 0.822 0.818 ADSORPTION OF SURFACE ACTIVE AGENTS AT THE SURFACE OF FORMAMIDE The variation of the surface tension of solutions of SDS, DTAB and decanol in formamide, shown in fig. 1 ? reveals no discontinuity characteristic of micellization up to the limit of solubility.The relative values of the initial gradient, g = (dn/dc),+o, where 7c is the surface pressure (=yo -y), are similar to those in water, but are much smaller, suggesting weaker adsorption in formamide. Cryoscopic determinations of the activity coefficients of SDS in formamide, listed in table 1, agreed almost within experimental error with values of the mean activity coefficient calculated from the Debye-Huckel equations for a 1 : 1 electrolyte up to 0.1 1 mol d ~ l l - ~ , but diverged slightly between 0.1 1 and 0.1 5 mol dm-3. In the absence of inorganic salt, the appropriate form of the Gibbs adsorption equation for a surface-active electrolyte R+X- is (1) in which ri!? is the adsorption of R+ relative to solvent (l), cRX is the concentration and y k the mean activity coefficient of solute.Since the Debye-Huckel equation fitted the experimental activity coefficients, it was convenient to use the former by substituting it in eqn (1) to give - dy = 2RT I'k\) d In cRxy+- , - dy = 2RT I'$[l- ac"( I + ~ * ) ~ ] d In c,, (2)A . COUPER, G.P. GLADDEN AND B . T. INGRAM 67 in which a, the constant in the Debye-Huckel equation, log y = -am*/( 1 + pm*), had the value 0.309 for formamide at its freezing point, and 0.356 and 0.360 at 25 and 40°C, and fl = 1.0. The area per molecule A' = 1 ITc1) differs significantly from the true area per mole- cule, A , in formamide, unlike the situation in water where A1 and A are virtually identical, because of the higher concentrations of solutions involved. They are related by the equation in which Al and A , are the areas occupied by solvent and solute molecules in the surface, and x1 and x2 their mole fractions. By assuming that the adsorbed solute was buried in solvent to a depth of about 0.6 nm, A, was estimated at 0.03 nm2, and A , was taken to be about 0.30 nm2.The true areas, A, were lower than the apparent areas Al, but did not differ by more than 10 %. The relative adsorption of SDS in 0.1 mol dm-3 NaCl was calculated without the activity coefficient term, because 1.0 1.5 2.0 2.5 areajnm2 molecule-l FIG. 2.-Surface pressure against area (T, A ) curves for SDS at the formamide/air interface (25°C) : 1, 0 formamide solution ; 2, 0.1 mol dm-3 NaCl in formamide ; 3, A 0.98 mol dm-3 NaCl in formamide. Harned's measurements on mixed electrolytes l9 suggested that its effect might be about equal and opposite to that of surface potential. The effect of ignoring these terms in 0.98 mol dm-3 NaCl was certainly negligible. Increasing the ionic strength of the formamide clearly decreased the electrostatic expansion of the film of SDS shown in the n against A curves of fig.2. However, the lowest area reached was about 0.80 nm2 at a surface pressure above 10 mN m-l, SO that SDS crystallised out before its chemical potential in formamide could generate a surface pressure sufficient for a close-packed monolayer. The experimental TC against68 SURFACTANTS IN NON-AQUEOUS SOLVENTS A curves were compared with the general equation of state for an ionised mono- layer 2 o in which the electrical pressure, ?re was calculated from the Gouy-Chapman theory, and the cohesive pressure, 7c, = 0.4 rn/As, was evaluated for m = 10, which gave the best account of this term as measured by the difference in surface pressure between formamide/air and formamide/decane interfaces. A , was estimated at 0.4 nm2.Agreement between calculated and observed surface pressures was better for the decane interface, where n, = 0 than for the air interface, where n:, (a negative quantity) amounted to almost half the resultant pressure at high TC values. DTAB was much more soluble in formamide than was SDS, although the differential heat of solution, determined from solubility measurements was 67 rfi: 8 kJ mol-l, only slightly greater than the value for SDS, 63 -f-4 kJ mol-1 compared with the much lower value of 29 kJ mol-' for SDS in water. The difference arose mainly therefore from the lower melting-point of DTAB. The same values for A , and A , in eqn (3) as for SDS were used to obtain true surface areas, A , which were derived without the activity coefficient term. The results in fig.3 indicate that the mono- layer u-as even more expanded in formamide than that of SDS, although n attained a (n: - n:, + 0.4 rn/As)(A - A,) = kT (4) I I 2 0.5 1.0 1.5 2.0 area/nm2 niolecule-l FIG. 3.-Surface pressure against area (n, A) curves for DTAB at the formamide/air interface (25°C) : 1, 0 formamide solution ; 2, 0 5.8 mol LiBr in formamide. higher value. To decrease the solubility of DTAB, 5.8 mol d111-~ LiBr was used, this being the only bromide sufficiently soluble for the purpose. The surface tension of this solution, without surfactant, was almost equal to that of water at the same temperature, and fig. 3 shows that A decreased to about 0.40 nm2 for 71: = 25 mN m-l, only slightly larger than the cross-sectional area of the head group.The high surface tension of the solvent and the extinction of n, at its high ionic strength combined to allow the close packing of solute. The adsorption of n-decanol at 25°C and 40°CA . COUPER, G . P. GLADDEN AND B . T. INGRAM 69 shown in fig. 4 indicates that a non-ionic surfactant in formamide without salt can attain an area of about 0.34 nm2 at about the same pressure. r b 0 . 5 1.0 1.5 2.0 2-5 area/nm2 molecule-' FIG. 4.-Surface pressure against area (n, A) curves for decanol at the formamide/air interface : 1, 0 25°C; 2, 0 40°C. The standard free energy of adsorption AGO, using standard states of 1 molecule ~ m - ~ in both surface and bulk phases, was calculated from g = (dy/dc),,o, using the equation in which the thickness of the surface layer, 6, was given the value 0.6 nm at the solution/air interface and 1 .O nm at solution/decane.The corresponding standard enthalpies, AH", and entropies, AS", were determined from the variation of adsorption over a small temperature range, and the probable error in their values is large. The effect of decreasing rc, by increasing the ionic strength of the formamide is evident from the increasing values of AGO in table 2 for both SDS and DTAB solutions. Using the incremental value of 1.45 kJ mol-l in AGO per CH2 group at the formamide/air interface,,l and the AGO value for decanol of - 15.6 kJ mol-l, it was estimated that the corresponding value for dodecanol would be - 18.5 kJ mol-l. The values for SDS in 0.98 mol dm-3 NaCl, - 17.0 kJ mol-l, and for DTAB in 5.8 mol dm-3 LiBr, - 17.7 kJ mol-l, show that at these ionic strengths the ionized surfactants have similar properties to those of the alcohol.The measured -AGO for adsorption of SDS at the solution/decane interface was larger than at the air interface, and if the difference is ascribed solely to the hydro- carbon chain it amounts to 0.27 kJ mol-1 per CH, group, and hence an incremental free energy of adsorption of 1.72 kJ mol-1 at the formamide/decane surface if any difference of the energy of adsorption of the polar head, at the two surfaces, is ignored. AHo for adsorption of SDS in 0.1 mol dm-3 NaCl in water was shown to be about - 10.5 kJ mol-1 between 20 and 40°C,22 very similar to the value, -8.7 kJ mol-1 in table 2 for formamide solution. Similarly ASo had a value in 0.1 mol dm-3 NaCl AGO = -RTln(g/kTd) (5)70 SURFACTANTS I N NON-AQUEOUS SOLVENTS in water of 67 J mol-I K-l which in terms of the standard states used here would have been about 34 J mol-I K-l, which is quite close to the value of 17 J mol-I K-l in table 2 for the formamide solution.The entropy change calculated for the loss of one degree of translational freedom 2 2 was -31 J mol-1 K-l, equivalent to about -63 J mol-1 K-l using our standard states, so there was an additional entropy gain of about 80 J mol-1 K-I, almost the same as that found in aqueous solution, which may result from desolvation of the ion upon adsorption. TABLE 2.-THERMODYNAMIC PARAMETERS FOR ADSORPTION AT FORMAMIDE SURFACES ads orbate interface SDS soln./air SDS 0.1 rnol dm-3 NaCl soln./air SDS 0.98 rnol dme3 NaCl soln./air SDS sohldecane SDS 0.1 mol dm-3 NaCl DTAB soln./air soln. /decane DTAB 5.8 rnol dm-3 LiBr soln. {air decanol soln./air ternp./OC 25 40 25 40 25 40 25 40 25 40 25 40 25 40 25 40 g/mN m-1 mol-1 dm3 203 101 329 277 1358 1107 932 410 2800 1400 59.2 52.0 1804 171 1 785 608 -AGO/ kJ mol-1 12.2k0.25 10.9+ 0.25 13.4f 0.25 13.51 0.25 17.0+ 0.25 17.2 0.25 14.7_+ 1.25 13.4+ 1.25 17.6_+ 1.25 16.4+ 1.25 9.2rfi: 0.25 9.2+ 0.25 17.710.3 18.2+ 0.3 15.610.25 15.6+ 0.25 -AHo/ ASa/ kJ mol- 1 J mol- 1 K- 1 36.116.6 -8Ok21 8.71 6.6 17+ 21 10.5+ 6.6 21 1 21 42f25 -92f92 38+25 -63192 7+6 8.5f20 3k10 50_+30 13&6 8 1 20 ADSORPTION A N D MICELLAR AGGREGATION OF C12E6 A N D C14E6 IN FORMAMIDE The surface tensions of solutions of C12E6 in formamide are shown in fig.5, in which curve 1 was obtained using the " 99 % pure " sample, with a minimum typical of a strongly adsorbed impurity. Curve 2, with a sharp discontinuity at the critical micelle concentration, was obtained with the later preparation, and similar curves were obtained with C14E6, and at 25, 30 and 40°C. mol dm-3 and the c.m.c. the linear surface tension graph of fig. 5 indicates a condensed adsorbed film, and the areas per molecule for the two com- pounds are listed in table 3. The size of the E6 terminal group may be estimated from Between TABLE 3.-AREA PER MOLECULE OF CONDENSED FILMS OF C ~ & B AND C ~ ~ E S arealnmz molecuIe-1 temp.l°C C12E6 C14ES 25 0.63 0.72 30 0.67 0.75 40 0.76 0.78 the experimentally determined diffusion radius, R,,, in water 23 of Carbowax 200, which was found to be 0.406 nm at 25"C, giving an estimated value of 0.482 nm for E6.This value corresponds to a sphere of volume 0.47 nm3, and the molecular volume of E6 is 0.416 nm3, so that not more than two solvent molecules could be involved. It is therefore reasonable to use the same value in formamide, correspond- ing to a cross-sectional area of 0.73 nm2. Comparison of this value with the experi-A. COUPER, G. P. GLADDEN AND B. T. INGRAM 71 mental values in table 3 suggests that the surfactant molecules were adsorbed in the: range 71 = 15 mNm-l to 24mN m-l, at the formamide surface as close-packed statistical coils of the soluble moiety as observed in bulk solvent. I,--..---- I -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 - 0 . 8 -0.6 -0.4 log, o(c/mol dm-3) Fro. 5.-Surface tension of Cl2E6 solutions in formamide (25°C) : 1,99 % pure CI2E6 ; 2, pure Cl2E6.- 6 . 0 - 5 . 0 &4 +d -4.0 E .- % -3.0 CJ c c! s 8 -2.0 0 - - 1.0 length of alkyl chain, n FIG. 6.-Variation of c.m.c. with chain length for CnE6 : 1, 0 CnE6 in water at 25"C3 ; 2, OCnE6 in formamide at 25°C; 3, A, OCnE6 in formamide at 30 and 4OoC,72 SURFACTANTS I N NON-AQUEOUS SOLVENTS The variation of c.m.c. in formamide, at three temperatures, with hydrocarbon at 25°C in fig. 6 in which different It is evident that the standard free energy chain length, n, is compared with that in water scales have been used for the two solvents. of micellization, AG;, defined by equation in which x& is the mole fraction at the c.m.c., can be extrapolated to AG; = 0 at a value of n between 6 and 7, compared with 4 for the micelles in water.This result suggests that when molecules of the type C,Es form micelles in formamide, two more CM2 groups retain their association with solvent, than in water. This is consistent with the equation of state for SDS adsorbed at the formamide surface, in which the cohesive pressure term involved two fewer CH2 groups than at the water surface, and with the higher solubility of hydrocarbons in formamide. The standard free energies of micellization of C12E6 and C14E6 are listed in table 4 together with the standard enthalpies and entropies determined from the variation of AG; with T. Values calculated for the phase separation model (a) were compared AG; = RTlnxS,, (6) TABLE 4.-THERMODYNAMICS OF MICELLTZATION IN FORMAMIDE substance temp./“C C12E6 25 C12E6 30 c.m.c./mol dm-3 0.0391 0.0444 0.0455 0.01 3 1 0.0140 0.0144 AG&/kJ mol-1 (a) -14.0 (6) -15.2 (c) -14.9 (a) - 15.9 (B) -15.2 (c) -15.1 (a) -16.4 (a) -18.7 (b) -18.3 (c) - 17.9 (a) - 18.9 (b) -18.5 (a) - 19.4 AH:/I<J mol-1 - 29.0 - 30.3 - 6.3 - 7.6 - 0.7 - 15.4 - 15.7 - 6.4 - 6.7 - 1.2 ASA/J mol-1 K-1 - 43 - 50 32 25 50 12 8.7 41 39 58 (a) calculated using phase separation model ; (b) calculated using the small systems ; (c) calculated using the mass action equilibrium m ~ d e l . ” ~ with those for the small systems using arbitrarily chosen values for the concentration of micelles at the c.m.c. (b) except at 40°C for which no micellar molecular weights could be obtained, and for the mass action equilibrium model ( c ) .~ ~ At 25”C, AG; for C12E6 in formamide was -16.0 kJ mol-1 (c.m.c. = 3.9 x mol dm-3) compared with -33 kJ mol-I in water (c.m.c. = 9 x mol dl~l-~), showing that the greater affinity of formamide for the hydrocarbon chain had diminished the tendency to form micelles. The incremental free energy per CH2 group was - 1.5 kJ mol-1 in formamide compared with -2.9 kJ mol-1 in water, closely parallel to the incremental adsorption free energies of - 1.45 kJ mol-I and -2.53 kJ mol-l respectively. However, the solubility of the surfactant was still largely dependent upon the head-group; a sample of CI2E3, prepared in the hope that it might show a conveniently lower c.m.c., proved to be virtually insoluble in formamide.In formamide AH: was negative, unlike the positive (endothermic) values for C12E6 and CI4E6 in water, and diminished with increasing chain length, and as the temperature rose. Consequently AS: was negative for C12E6 in formamide atA . COUPER, G . P . GLADDEN A N D B . T. INGRAM 73 25°C but became positive at higher temperatures, whilst for C14E6, AS: had a small positive value increasing with temperature. The entropy change in forming a micelle in formamide was much smaller than in water, 130 J mol-l K-l for C12E6 at 25°C and it is concluded that the " structuring " of formamide around the hydrocarbon of the monomeric surfactant was less than in water. This result is similar to the reduction in AS; observed when the glycol content in ethylene glycol/water solvent was increased.24 MICELLE MOLECULAR WEIGHTS A N D RADII Micelle molecular weights, determined using the Archibald approach to equili- brium technique, for C14E6 in formamide at three temperatures are shown in fig.7. The apparent variation of molecular weight with concentration resulted from the dependence of the sedimentation coefficient upon concentration, and so the molecular weights and aggregation numbers listed in table 5 were extrapolated to zero micelle concentration, i.e., to the c.m.c. 0.07 0.08 0.09 0.10 0.11 concentration/mol dm-3 FIG. 7.-Variation of apparent micelle molecular weight with concentration for CI4E6 in formamide : 1, 0 20°C; 2, A 25°C; 3, 0 30°C. The micelles were much smaller than those formed in water; e.g., at 25°C for C12E6 in formamide the molecular weight was about 1.37 x lo4, compared with about lo5 in If a spherical model of the micelle is constructed, having no void, but TABLE 5 .-MICELLE MOLECULAR WEIGHTS IN FORMAMIDE micelle compound temp.!'C mclecular weight aggregation number C12E6 20 18 300 41f3 25 13 680 30$.3 30 18 150 40-t 3 @14E6 20 27 150 5 7 1 5 25 30 360 64+ 5 30 31 440 66+ 574 SURFACTANTS I N NON-AQUEOUS SOLVENTS a radius equal to the length of the extended hydrocarbon chain, 1.60 nm for C12E6 and 1.85 nm for C14E6, the upper limit of the aggregation numbers would be 49 and 75 respectively. The increasing aggregation number of C14E6 with rising temperature is similar to the behaviour in water, but the low value for C12Es at 25°C obscured what is probably a similar trend for this compound.An anomalous concentration dependence was the source of this low value, and a similar anomaly was observed in the concentration dependence of the sedimentation coefficient for Cl2E6 at 25°C. Micellar radii calculated from the molecular weights and sedimentation co- efficients in three different ways, as shown in table 6, were based upon a spherical model and agree reasonably well, except for C12E6 at 25°C. A simple model for TABLE 6.-MICELLE RADII IN FORMAMIDE compound temp. 1°C Rslnm RAlnm RsAlnm Cl2E6 20 2.14+ 0.2 1.94+_ 0.2 lS9& 0.2 25 2.04f 0.2 1.74+ 0.2 1.31 k0.2 30 2.051 0.2 1.93 & 0.2 1.74k 0.2 G4E6 20 2.09+ 0.2 2.21 f0.2 1.89k0.2 25 2.38 & 0.2 2.29+ 0.2 2.14k0.3 30 2.3510.2 2.32+ 0.2 2.29k0.3 Rs, calculated from sedimentation coefficient and density ; RA, calculated from Archibald molecular weight and density ; RSA, calculated from Archibald molecular weight and sedimentation. the radius would be the sum of the extended hydrocarbon chain length and the diffusion radius, RD, of the head-group giving 2.08 and 2.33 nm for C12E6 and C14E6 respectively, in good agreement with experiment.Comparison between the maximum aggregation numbers, 49 and 75, and the numbers in table 5 suggest that the hydro- carbon chains may be a little less than fully extended, so that the radii Rs and RA in table 6 may represent the mean radii of the hydrodynamic particles. Using the values 2.08 nm for C12E6 and 2.33 nm for C14E6, the surface areas are 54.3 nm2 and 68 nm2 respectively, so that if the E6 head-group had the cross-sectional area, 0.73 nm2, the surface could accommodate 74 and 93 such groups.Viscosity measurements for C14E6 in formamide above the c.m.c. were used in the form (q/qc.m.c. - 1) for the specific viscosity, where q was the measured viscosity, and qc.m,c. that for the solution at the critical micelle concentration, to obtain the solvated volume of the micelle. This gave 88 nm3 for the solvated micelle compared with 51 nm3 for the unsolvated particle, corresponding to a radius of 2.30 nm, so that 41 % by volume of the micelle was represented by solvent. This is equivalent to 8 formamide molecules per ClQE6 molecule compared with about 25 water molecules in aqueous solvent.27* 28 Using the model proposed, of a voidless hydrocarbon sphere, of volume 17.1 nm3 for C1&6 and 26.5 nm3 for C1&, closely surrounded by 49 and 75 statistically coiled E6 residues respectively, in a spherical shell, there remains room to accommodate 9 HCONH2 molecules per C12E6 molecule and 6 HCONH2 molecules per CI4E6 in the solvated micelle.We thank Procter and Gamble Limited for providing studentships for G. P. G. and B. T. I. and for providing a sample of C12E6 ; we also thank B,P. Limited for a bursary to G. P. G. J. M. Corkill and J. F. Goodman, Adu. Colloid Interface Sci., 1969, 2, 297. T. Walker, J. Colloid Interface Sci., 1973, 45, 372. J. M. Corkill, J. F. Goodman and R. H. Ottewill, Truns. Furaday Soc.. 1961,57, 1627.A. COUPER, G. P. GLADDEN AND B. T. INGRAM 75 A. H. Ellison and W. A. Zisman, J. Phys. Chem., 1959, 63, 1121. N. L. Jarvis and W. A. Zisman, J. Phys. Chem., 1960, 64, 150, 157. D. C. Jones and L. Saunders, J. Chem. Soc., 1951, 2944. ’ J. W. Belton and M. G. Evans, Truns. Furuday SOC., 1945, 41,l. * J. M. Notley and M. Spiro, J. Phys. Chem., 1966, 70, 1502. G. F. Smith, J. Chem. Soc., 1931, 3257, l o S. P. Harrold, J. Colloid Sci., 1960, 15, 280. B. A. Mulley, Proc. 3rd Int. Congr. Surface Actiuity, 1960, 1, 31. B. A. Gringas and C. H. Bailey, Canad. J. Chem., 1958,36, 1302. l 3 J. M. Corkill, J. F. Goodman and R. H. Ottewill, Trans. Faraday SOC., 1967,57,1627. l4 S. Fordham, Proc. Roy. SOC. A, 1948, 194,l. l5 W. J. Archibald, J. Phys. Chem., 1947, 51, 1204. R. J. Good, Znd. Eng. Chem., 1970, 62, 54. l7 F. M. Fowkes, J. Phys. Chem., 1963, 67,2538. H. W. Fox and W. A. Zisman, J. Colloid Sci., 1952, 7,428. l9 H. S. Harned, J. Amer. Chem. SOC., 1926,48, 326. 2o J. T. Davies, Proc. Roy. SOC. A, 1951, 208, 224. 21 A. B. Taubman and S. I. Burshtein, KolfoidZhur., 1959, 20, 539. 22 J. J. Betts and B. A. Pethica, Tram. Faruday Soc., 1960,56, 1515. 23 A. Couper and R. F. T. Stepto, Trans. Faraday SOC., 1969, 65,2486. 24 D. G. Hall and B. A. Pethica, Non-Zonic Surfactants, ed. M. J. Schick (Academic Press, New 2 5 3. M. Corkill, J. F. Goodman and S. P. Harrold, Trans. Faraday Soc., 1964, 60, 202. 26 A. Ray and G. Nemethy, J. Phys. Chem., 1971,75809. 27 R. H. OttewilI, C. C. Storer and T. Walker, Trans. Faraday Soc., 1967, 63, 2796. 2B H. Schott, J. Colloid Interface Sci., 1967, 24, 193. York, 1967), vol. I, p. 516.

ISSN:0301-7249    

DOI:10.1039/DC9755900063

出版商:RSC    

年代:1975

数据来源: RSC

摘要:

Adsorption Kinetics in Micellar Systems BY JACOB LUCASSEN Unilever Research, Port Sunlight Laboratory, Port Sunlight, Wirral, Merseyside L62 4XN Received 13th December, I974 The kinetics of adsorption and desorption to the air/water surface in submicellar surfactant solutions has been studied by means of a dynamic technique in which the.surface is subjected to small amplitude sinusoidal compression and expansion. In all cases studied so far a diffusion controlled mechanism was found, characterised by a frequency dependent surface tension variation and a phase difference between applied area change and resulting surface tension change of at most 45". When such experiments are carried out above the critical micellar concentration a characteristically different behaviour is observed with a much steeper frequency dependence of the surface tension change and with phase differences between 45 and 90".Experiments of this type, carried out for solutions of some nonionic surfactants can be quanti- tatively interpreted on the basis of a model in which the diffusional exchange of monomers between surface and bulk and the exchange between monomers in the intermicellar solution and micelles are consecutive processes. It is shown that establishment of micellar equilibrium under certain conditions can be a fairly slow process with relaxation times of the order of seconds, although the actual exchange of monomers with micelles could well be several orders of magnitude faster. Some implications of the presence of micelles on the general nature of dissipative processes near a surface are discussed.The resistance of fluid surfaces against deformation is an important factor in processes such as foaming, emulsification, liquid drainage from solids, and extraction. A measure for the resistance against compressional or dilational deformation is given by the surface dilational modulus This parameter, when measured in a small amplitude surface deformation experiment, should be expressed as a complex number as relaxation processes, such as diffusion of surfactant between bulk and surface, cause phase shifts between surface tension change and imposed area change. have shown that for all these systems a complete description of the dilational surface behaviour could be based upon a knowledge of an equilibrium surface equation of state com- bined with a diffusion exchange mechanism of surfactant monomers between the surface under deformation and the bulk solution.All these systems showed similar frequency dependences for 161 as well as for t,b. At high frequency, 1 ~ 1 approaches an upper limiting value and $ becomes zero. At low frequency $ approaches a limiting value of 45" and the slope in a log IEI against log frequency plot tends to 0.5. Above the critical micellar concentration of surfactant solutions, apart from the diffusional relaxation process between surface and bulk, micellar relaxation below the surface also has to be taken into account. Any change in subsurface monomer concentration resulting from surface area changes is bound to disturb the monomer- E = dy/d In A .(1) E E lel exp(it,b) (2) ,Measurements on a number of submicellar surfactant solutions 76J. LUCASSEN 77 micelle equilibrium and hence will set in motion a process of micellisation or de- micellisation. Evidence for the qualitative effect of the presence of micelles on surface dilational properties has been given previously.2 The phase angle I) reaches values between 45" and go", and the slope in log lei against log co plots exceeds the value of 0.5. In this paper is presented a first attempt at reaching a quantitative theory describing surface dilational properties in micellar systems. One problem is that, although much work has been carried out on micellisation kinetics in bulk s o l ~ t i o n s , ~ - ~ ~ no clear picture has emerged on its most likely mechanism.However, the different models used all lead to roughly similar relaxation time against concentration relation- ships and this warrants the adoption of a rather arbitrary model for the purpose of the present paper. Finally, the relation between increased phase angles in micellar systems on the one hand and dissipative processes leading to shock absorption and protection of thin liquid films in foams on the other are discussed. EXPERIMENTAL METHOD The method used for the measurement of surface dilational properties has been described previously.2 Briefly, an enclosed part of a surfactant solution's surface area is subjected to periodic compression and expansion of small amplitude and the ensuing variation of surface tension is measured. Conditions are chosen such that the surface deformation can be con- sidered uniform l 2 over the enclosed area.The surface dilational modulus is then character- ised by an absolute value, that is the ratio between amplitudes of surface tension change Ay and fractional area change AA/A, and by an angle @ reflecting the phase difference between surface tension change and imposed area change. Experiments were carried out at different frequencies of surface deformation, ranging between 10 and 0.05 cycles per minute. However, for the micellar systems in general only results at frequencies above 0.2 cycles per minute were sufficiently accurate for further analysis. MATERIALS The preparation, purity and surface chemical properties of the nonionic compounds CI2E6 and C14E6 have been described previously.2 The sulphobetaine, hexadecyldimethylammoniopropanesulphonate (HDPS) was pre- pared by reacting in ethyl acetate y-propane sultone with 99.5 % pure hexadecyldimethyl- amine. The compound was recrystallised once from isopropyl alcohol and twice from a 2 : 1 (v/v) acetone+isopropanol mixture.Purity was 100 %, according to mass spectro- metry. The compound had a Krafft point at 29°C. After heating concentrated solutions to well above this temperature, metastable solutions could be maintained at lower temperatures for at least a day without any sign of precipitate formation. Experiments on HDPS solut tions were carried out at 22, 28 and 35°C. At the two temperatures below the Krafft poin- the surface dilational behaviour of the metastable solutions was not found to be different from that at 35°C.None of the compounds displayed a minimum in their surface tension at the c.m.c., indicating a high degree of surface chemical purity. Water distilled from alkaline permanganate was used to prepare all solutions. Its surface tension at 25°C was 72.25 mN m-l. RESULTS Detailed data on equilibrium and dynamic surface properties of the nonionic com- pounds are given in ref. (2).78 KINETICS IN MICELLAR SYSTEMS 41 40- 3 9 - r( 38- x 37- - Fig. 1 shows the surface tension behaviour around the c.m.c. for the HDPS at three different temperatures. The c.m.c. appears to be virtually temperature- independent. - 4 2 t I 35i \ I I -8.5 - 8.0 -7.5 -7.0 - 6.5 log(concentrationlmo1 FIG. 1.-Surface tension of HDPS solutions in the neighbourhood of the c.m.c.0,22"C ; x ,28"C ; A, 35°C. The dynamic dilational behaviour for HDPS solutions below the c.m.c. did obey the simple diffusion exchange mechanism just as did the nonionic compounds des- cribed in ref. (2). No details of the analysis will be given here. The resulting para- meters are summarised in table 1. -8.5 -0.0 -7.5 -7.0 - 6.5 log(concentration/moI CM-~) FIG. Z.-Surface dilational modulus for HDPS solutions at different frequencies, as a function of concentration: x,lOc.p.m.; A,5c.p.m.; 0,Zc.p.m.; 0,lc.p.m.; \J,0.5c.p.m. Temperature 28°C. Drawn lines, theory.J. LUCASSEN 79 TABLE 1 108 x monomer concentration / compound temp./"C mol cm-3 ..e/s-' Do/cmZ 5-1 ~o/niN m-1 Q/S Cl&6 25 6.6 1.35 4 x 50 0.0046 C14E6 25 0.525 0.145 4x 60 0.025 HDPS 22 2.55 0.615 3.5x 65 0.18 HDPS 28 2.55 0.725 4 .0 ~ 65 0.13 HDPS 35 2.55 0.813 4.5x 65 0.037 Fig. 2 and 3 show the behaviour of the absolute value of the modulus IEI for HDPS solutions in the c.m.c. region at 28 and 35°C respectively, while fig. 4 and 5 reproduce similar data for the nonionic compounds CI2E6 and CI4E6. Finally fig. 6 illustrates the behaviour of the phase shift If/ for HDPS at 28°C. 1.5 h 7 1.0 E z E \ - w - v 8 0 . 5 c..l log(concentr ationlmol ~ r n - ~ ) FIG. 3.-As fig. 2. Temperature 35°C. C.M.C. \ ~\ 1 I I 3 I I I I -7.7 -7.6 -7.5 -7.4 -7.3 -7.2 -7.1 - 7 . 0 Iog(concentration/mol ~ r n - ~ ) FIG. 4.-As fig. 2, data for dodecyl hexaethylene glycol (G2Ed. Temperature 25°C.80 n I 4 E KINETICS I N MICELLAR SYSTEMS C.hI c.OB5 t \+ -0.7 -8.6 -8.5 -8.4 -8.3 - @ . 2 -8.1 -8.0 log(concentration/mol ~ r n - ~ ) FIG. 5.-As fig. 2, data for tetradecyl hexaethylene glycol (CI4E6). Temperature 25°C. 8 0 - 70 - 6 0 * 5 0 - 3 4 0 - 5% 3 0 - 20 - 10 - - 8.5 - 8 . 0 -7.5 -7.0 log(concentra tion/mol cm- 3, 10 c.p.rn. ; A, 5 c.p.m. ; 0, 2 c.p.m. Temperature 28°C. FIG. 6.-Phase angle, t j , for HDPS at different frequencies, as a function of concentration : x , DISCUSSION (a) THE EFFECT OF MICELLAR RELAXATION ON SURFACE DILATIONAL Measurements at concentrations above the c.m.c. of the surfactants under con- sideration gave phase angles above 45" and slopes in log IEI against log cr) graphs exceeding 0.5. This behaviour has been explained by a qualitative argument pre- viousIy.2 The model which we will adopt here to obtain a quantitative expression for the dilational modulus above the c.m.c.contains the following assumptions : (i) there is no surface excess of micelles and their concentration at equilibrium is uniform right up to the air/water surface; (ii) the intrinsic surface properties that is the surface tension, y , the surface P R 0 P E R T I E SJ. LUCASSEN 81 excess, F , and derived quantities such as the high frequency limiting value of the modulus -dy/d In r l4 remain constant above the c.m.c. ; (iii) the amplitude of the applied sinusoidal surface deformation is sufficiently small to permit linearisation of the problem ; (iv) micellar relaxation takes place according to the mechanism proposed by Kreschek et aZ.,4 in which the rate is limited by one slow decomposition step with rate constant k2.Micelles, by nature can be supposed to be non surface active. Although a certain measure of desorption due to electrostatic repulsion could be expected for micelles of anionic or cationic sur- factants, this does not apply to the nonionic surfactants discussed in this paper. When the surface tension is practically concentration-independent above the c.m.c., as found for the present systems, it seems obvious that all other intrinsic surface properties, prope1 ties not affected by diffusion exchange with the bulk liquid, will be concentration-independent as well. It is difficult to make predictions about the amplitude of the experiments being small enough to permit linearisation. For the systems discussed in the present paper most experiments were carried out at a relative area variation of about 5 %.Under these conditions the surface tension change resulting from a sinusoidal area change was sinusoidal as well and the dilational modulus proved to be independent of the amplitude. Both findings indicate that a linearised theory should be applicable. However, as pointed out before, other mechanisms, such as the highly unlikely one of an a-th order reaction appear to lead to roughly the same global relaxation behaviour and it seems warranted to choose a rather arbitrary mechanism for a first exploration of the area. By a method, similar to that used for submicellar systems, we then obtain the following expression for the surface dilational modulus (see Appendix I) : The first two of these assumptions seem to be plausible.The fourth assumption is the least likely to be a realistic one. (3) 1 {(I -ix)[(l +6(1 -ix)fl2 +ik(S2 - I)])* E = E~ l+(l-i)< [ 1 -ik+6(1 -ix)* where k k z / o , x = k(1 +p), 6 = [DM,/Do]+? ( f (dco/dT),/Do/2w, p = a2[cM/ c0],,= - c0 = - dy/d In r, i = \/ - 1 , with k2 the rate constant for the slow decompo- sition step,4 CI the number of monomers per micelle, C, the molar concentration of inicelles, DM the micellar diffusion coefficient, co the monomer concentration at the c.m.c., Do the monomer diffusion coefficient, ci, the angular frequency of the experi- ment. Eqn (3) covers the general case, taking into account monomer diffusion below the the surface, micellar diffusion and re-establishment of the monomer-micelle equi- librium.The parameter x can be related to the micellar relaxation time as it would be found from bulk relaxation experiment~.~-~. l4 This time, defined by where AcM and Ac, are deviations of micellar and monomer concentrations respect- ively from their equilibrium values, is found for the mechanism proposed by Kreschek et aL4 to obey82 KINETICS IN MICELLAR SYSTEMS In other words, values for x, to be obtained from a computer fit of experimental data to eqn (3) can be directly compared to values for (CUT)-' such as would emerge from experiments in which the deviation from equilibrium is uniform throughout the system and in which diffusion can be ignored. The second term inside the outer square brackets of eqn (3) can be considered as a " supply function ',, Fs, which reflects the ability of surface active material to exchange between surface and bulk during an experiment with a characteristic rate of the order 0: ((I -ix)[(I +S(l-ix)+)' +ik(J2 - I)])+ I-i/c+S(I-ix)+ F, = (1-i)C The higher the value of F,, the smaller lei will be compared to its high frequency value c0.It is of interest to consider the limiting behaviour of this " supply function ", F,, for very slow and for very fast micellisation. as to be expected for a submicellar systern.l the diffusion exchange process. For the latter case, where (8) transport plays a much more important role, as the monomers in the micelles are also able to exchange with the surface. The reduction of lei from its high fre- quency value can be expected to be greater, the larger the number of monomers per micelle and the larger the micellar concentration and diffusion coefficient.It should be stressed here, that the dilational modulus above the c.m.c. only vanishes completely for an experiment at infinitely low frequency. The supply of surfactant molecules which '' short-circuits " any surface tension change is always emanating from a layer of finite thickness, of the order ,/D/20.' Fig. 7 shows a typical example of the behaviour of [&I which could be expected in For the former case we have Fs = (1-i)C (7) Only the free monomers participate in Fs = (1 - 95 J(l+ P)(l +PS2), I J , - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 log(frequency /rad) FIG. 7.-Effect of micellisation on surface dilational modulus, as predicted by eqn (3).---. k, = 00; , . . . . ., kz = 0 ; -, k2 = 0.01. Example for wo = 1 ; DM/DO = 0.25; a = 50: C M = 0.02 co.J. LUCASSEN 83 the above-mentioned two extreme cases. The example refers to a concentration twice the c.m.c. for a hypothetical system with 50 monomers per micelle. The upper (dotted) line gives the behaviour of IEI for a very slow micellisation reaction and the lower (dashed) one represents fast micellisation. The ratio D,/Do is taken as 0.25.* It is obvious that in the frequency region where diffusion is important, the reduction of IEI is much larger in the latter than it is in the former case. In this case there is a transition from the dotted to the dashed curve in the frequency region where x m 1. In this region the slope in the log 161 against log o curve, as predicted beforq2 exceeds unity.Similar evidence, shown in fig. 8 shows that in the same frequency region the phase angle t,b exceeds the value of 45”, which was shown to be the maximum value for monomer diffusion exchange. The drawn line in fig. 7 represents the intermediate case, for k = lo-’(’. log(frequency /rad) data as for fig. 7. FIG. 8.-Effect of micellisation on the phase angle 4. --- , k2 = 0 ; -, k, = 0.01. Further (b) COMPARISON WITH EXPERIMENTAL EVIDENCE The agreement between theory, as expressed by eqn (3), and experimental evidence for the three surfactants discussed, was tested by means of a least square curve fitting programme. Apart from the values for 161 at various concentrations and frequencies, the following parameters had to be fed into this programme.(1) The equilibrium surfactant monomer concentration at the c.m.c., co. (2) The high frequency dilational modulus (3) The diffusion exchange parameter coo = orz = D/2(dc0/dr)’, both (2) and (3) at the c.m.c. The method used for obtaining the latter two parameters and for extrapolating them to the c.m.c. has been described previously.2* l4 (4) For all three compounds the number of monomers per micelle, a, was assumed to be 100.15 Actually the quality of the curve fit appeared to be only very weakly * It has been pointed out by McQueen and Hermans l3 that the ratio DM/& should be of the order a-*. The Stokes-Einstein relation predicts an inverse proportionality between particle radius and diffusion coefficient.= -(dy/d In r).84 KINETICS IN MICELLAR SYSTEMS dependent on a, so that it did not seem necessary to use a very accurate value for this parameter. Thus, for the parameter 6 , a value of 0.465 was adopted.13 The concentration dependence was introduced through eqn (3, which gives, with Ctot = CO + a c M , (9) (10) 7-l = kz(l -a+ac,,,/c,) z -k201+k2CIC/,,,CO = --7;1(1 fCtot/Cg). Examples of the curve fit for IEI are given by the drawn lines in fig. 2-5. Taking into consideration the large number of assumptions made and the wide range of frequencies and concentrations covered, the agreement between theory and experi- ment can be considered quite good. In principle a similar curve fit could be applied to the experimental values of the phase angle $. values was rather low and was therefore not used for further evaluation.In general, however, the behaviour of both Is1 and II/ above the c.m.c. was as to be expected from the theory. The model used here, therefore can certainly serve as a basis for a further refinement of the picture of coupled diffusion-niicellar relaxation. In the next paragraph we will discuss, on a preliminary basis, the significance of the results obtained from the least square fitting procedure. However, the experimental accuracy of this parameter at small (C) THE MECHANISM OF MICELLAR RELAXATION The curve fitting programme supplies values for the characteristic relaxation times T~ [see eqn (lo)]. They are presented in table 1, together with starting values, used for c0, co, wo and the diffusion coefficient Do.It appears that these relaxation times are of the order of a hundredth to a tenth of a second, certainly several orders of magnitude higher than would be expected for the exchange mechanism proper between monomers and micelles. lo Such a discrepancy has been apparent from all experiments on micellisation kinetics by various methods, except from the evidence obtained by n.m.r.169 l 7 It is of interest to note, that ro- values, as obtained by different techniques, including the present one, appear to increase with decreasing value of the c.m.c. The present systems have c.m.c. values (or monomer concentrations above the c.1n.c.) much lower than any other system studied before in this context and they also show the highest relaxation times found so far.This general trend suggests that the rate determining step in the process whereby micelles generate or absorb free niononiers is not the micellar decomposition, but diffusional transport from or towards micelles. Such a process should proceed at a higher rate, the larger is the monomer concentration. According to a very crude model, in which all micelles are considered stationary sources or sinks which are able to absorb or replenish all deviations from the monomer equilibrium concentration, To-values could be predicted which were about 2 orders of magnitude smaller than the experimental ones. Such a model, however, can be expected to give low estimates of relaxation times for two reasons. In the first place, micelles cannot be expected to act as infinitely “ deep ’’ sinks or infinitely “ high ” sources.Secondly, if they were to act as such in the first instance, it would be neces- sary that subsequently the micellar size distribution and the total number of micelles adapt themselves such as to obey the requirements of the various monomer-micelle equilibria. Such a rearrangement of micellar size and numbers would involve a “flow” throughout the whole of the micellar size distribution, as discussed by Aniansson et al.l1 As this will invoke monomer diffusion over much larger char- acteristic distances than the average separation between two micelles, the relaxationJ. LUCASSEN 85 times involved will be longer than would be expected from a simple sourcelsink model. The present scant knowledge of micellar size distributions does not seem to warrant a further evaluation of the model in this stage.( d ) THE EFFECT OF MICELLES O N DISSIPATIVE PROCESSES NEAR A SURFACE It has been shown previously ' 9 1 8 . l 9 that the lateral propagation over a liquid surface of locally applied surface tension changes strongly depends on the phase angle $. Such propagation is determined by the properties of longitudinal surface waves, for which the ratio between damping coefficient pL and wavenumber I C ~ is given by (1 1) PL/rcL = tan(nl8 + 1,b/2). This ratio determines the fraction of an initial surface tension variation which is still left after it has propagated over a distance d, equal to one wavelength : For an insoluble monolayer with $ = 0, this fraction is 0.075. For a micellar solution with $ = 80" it has decreased to as low a value as 3.5 x In other words, the presence of micelles can cause surface tension variations to propagate over much shorter distances than would be the case below the c.m.c.This effect of two-dimensional shock-absorption may well lead to a greater stability of free thin films under dynamic conditions. The effect of micelles on surface shock absorption is completely analogous to their effect on sound absorption. 5* In both cases micellisation-demicellisation will lead to an enhanced rate of energy dissipation. IArld=dlAr Id= 0 = exP( - 2nPL/KL). (12) (e) SUMMARISING REMARKS It has been shown in this paper how measurements of surface dilational properties carried out at various frequencies and concentrations can help to elucidate the kinetics of surfactant adsorption and desorption.When applied to micellar systems additional information can be obtained about micellar relaxation processes. Only compounds with a fairly low c.m.c. value can be studied in this way. For surfactants with a c.m.c. of the order of 10 mmol dm-3 or higher, the surface tension variations would be too low to allow accurate measurements. For such systems, with small diffusion relaxation times, a high frequency surface deformation such as could be obtained by capillary waves may well be used to study adsorption and micellisation kinetics. The technique used is of limited applicability. APPENDIX DERIVATION OF DILATIONAL MODULUS FOR MICELLAR SYSTEMS Variations in monomer and micelle concentration below the surface should obey the following relations : dco/8t = - ctk,C," -I- ak2cM + &a2co/dy2 dcM/at = klc; - k2cM+ &a2CM/dy2, (Al) (A2) in which kz can be identified with the slow step of the mechanism according to Kreschek et aI.4* l 3 We search for solutions of co and CM of the form86 KINETICS I N MICELLAR SYSTEMS For low amplitude disturbances, substitution of (A3) and (A4) into (Al) and (A2), and taking the equilibrium condition (cO);= - 03 = k 2 / k l ( c M ) y = - cn f d Y ) + P f d Y ) + Qf XY) = 0 f d Y ) + W d Y ) + Sf XY) = 0 (A51 (A61 (A71 into account leads to the following differential equations : which have a negative real part.The other two roots will not give vanishing values for fo andfM a t y = -a. The constants E, F, M and N now have to be eliminated by the boundary conditions for conservation of matter at the surface.For micelles, the assumption that they are not surface active, combined with their boundary condition : leads to Substituting this into eqn (A4) and combination with eqn (AS) gives fM(Y) = M[exp(nAy) -nA/n13 exp(nBY)l- (A331 An expression for the dilational modulus e is now obtained in a way analogous to the one used in ref. (1) : The ratio FIE is found by substitution of eqn (A8) and (A1 3) into eqn (A6) or (A7). The resulting expression should be valid at any value of y and therefore the coefficients of the exponential terms should each equal zero. This results in ( E + PM + QEn:) exp(nAy) +(F - PM nA/nB + QFni) exp(nBy) = 0 (A19J. LUCASSEN 87 Finally, from eqn (A10) we have R - P nini = - QS and Substitution of eqn (A16), (A17) and (A18) into eqn (A14), finally gives eqn (3) for the surface dilational modulus. Thanks are due to Mr. D. Giles for accurately performing the experiments and to Mrs. J. C. Savage for writing the least square curve fitting programme. J. Lucassen and M. van den Tempel, Chem. Eng. Sci., 1972,27, 1283. J. Lucassen and D. Giles, J.C.S. Furuday I, 1975,71,217. P. F. Mijnlieff and R. Ditmarsch, Nature, 1965, 208, 889. G. C. Kresheck, E. Hamori, G. Davenport and H. A. Scheraga, J. Amer. Chern. Soc., 1966,88, 246. E. Graber, J. Lang and R. Zana, Kolloid 2.2. Polynzere, 1970, 238,470. E. Graber and R. Zana, KoZloid 2.2. Polymere, 1970, 238,479. J. Rassing, P. J. Sams, E. Wyn-Jones, J.C.S. Faruday ZZ, 1973, 69, 180. ’ P. J. Sams, E. Wyn-Jones and J. Rassing, Chem. Phys. Letters, 1972, 13, 233. * J. Lang and E. M. Eyring, J. Polymer Sci. A-2, 1972, 10, 89. lo N. Muller, J. Phys. Chem., 1972, 76, 3017. l 1 E. A. G. Aniansson and S. N. Wall, J. Phys. Chem., 1974,78, 1024. l 2 J. Lucassen and G. T. Barnes, J.C.S. Faruday I, 1972, 68, 2129. l3 D. H. McQueen and J. J. Hermans, J. Colloid Interface Sci., 1972, 39, 389. l4 E. H. Lucassen-Reynders, J. Lucassen, P. R. Garrett, D. Gila and F. Hollway, Adv. Chem. Series, N.K. Adam Memorial Issue, 1975, to be published. l5 J. M. Corkill and T. Walker, J. Colloid Interface Sci., 1972, 39, 621. l6 H. Inoue and T. Nakagawa, J. Phys. Chem., 1966, 70, 108. l7 N. Muller and F. E. Platko, J. Phys. Chem., 1971, 75, 547. lS J. Lucassen, Truns. Faraduy Suc., 1968, 64, 2221, 2230. i9 J. Lucassen and M. van den Tempel, J. ColZoid Interface Sci., 1972, 41,491.

ISSN:0301-7249    

DOI:10.1039/DC9755900076

出版商:RSC    

年代:1975

数据来源: RSC