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The structure of the liquid-vapor interface Rensink, Ronald Andy 1982

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THE STRUCTURE OF THE LIQUID-VAPOR INTERFACE by RONALD ANDY RENSINK B.Sc, University of Waterloo, 1979 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1982 © Ronald Andy Rensink, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date March 17, 1982 DE-6 (3/81) i i Abstract This thesis presents a review of the theories currently being used to describe the structure of the liquid-vapor interface. The f l u i d s considered are those consisting of "argon-like" mole-cules, which have r a d i a l l y symmetric potentials. "Wall e f f e c t s " upon the interface are assumed to be n e g l i g i b l e . The derivations of several theories have been recast into forms which depend upon a few common p r i n c i p l e s . The technique of functional d i f f e r e n t i a t i o n i s extensively used in this re-gard. This allows each theory to be i d e n t i f i e d with one of three d i f f e r e n t approaches: mean-field theory, integro-differen-t i a l equations, or fluctuation theory. Good agreement i s found between the results of theories within each c l a s s . The three approaches are shown to be d i f -ferent aspects of a single consistent model of the liquid-vapor interface. i i i Table of Contents Abstract i i L i s t of Tables v L i s t of Figures v i Acknowledgement v i i Chapter 1. Introduction 1 Chapter 2. General Formulation of Mean-Field Theory 6 2.1 Basic Thermodynamic Quantities 7 2.2 Mean-field Theory 9 2.3 Uniform F l u i d - Equation of State 12 2.4 D i f f e r e n t i a l Description of Non-Uniform Fluids 16 2.5 Thermodynamic Properties of Non-Uniform Systems 22 a) pressure tensor 22 b) surface tension 24 Chapter 3. Mean-field Theories 26 3.1 Separation of Potential 27 3.2 Mean-field Theories 31 a) Toxvaerd perturbation theory 31 b) generalized van der Waals theory 35 c) modified van der Waals theory 38 3.3 Simulation and Experiment 40 a) computer simulation 40 b) simulation vs experiment 43 Chapter 4. I n t e g r o - d i f f e r e n t i a l Equations 44 4.1 P r o f i l e Equations 45 a) general formulation 45 b) p r o f i l e "wings" 47 4.2 Kirkwood-Buff Theory 50 4.3 Triezenberg-Zwanzig Theory 52 4.4 Numerical Solutions 55 a) Toxvaerd 55 b) Co, Kozak, and Luks 58 c) Salter and Davis 60 4.5 Relation to Mean-field Theory 61 4.6 Long-ranged Correlations 64 a) existence of long-ranged correlations 64 b) Fourier transforms 67 c) simulation 68 Chapter 5. Fluctuation Theories 5.1 B u f f - L o v e t t - S t i l l i n g e r Theory a) general formulation b) interface width and surface tension c) long-range correlations 5.2 Column P a r t i t i o n Function a) column p a r t i t i o n function b) density p r o f i l e c) correction to surface tension 5.3 Sobrino-Peternelj Theory a) general formulation b) correction to surface tension c) correlations and interface width 5.4 Relation amongst the theories Bibliography Appendix A. Elementary S t a t i s t i c a l Mechanics of the Liquid State A.1 Density Functions a) canonical ensemble b) grand canonical ensemble A.2 D i s t r i b u t i o n Functions a) general formulation b) r e l a t i o n to thermodynamic properties c) Born-Green-Yvon hierarchy A.3 Direct Correlation Function a) general formulation b) r a d i a l direct c o r r e l a t i o n function A.4 Generalized Functions a) density functions b) U r s e l l functions c) d i r e c t c o r r e l a t i o n function Appendix B. On the Shape of the Equilibrium Density P r o f i l e a) constant density p r o f i l e b) s t r i c t l y decreasing p r o f i l e V L i s t of Tables Number T i t l e Page I . Properties of argon interface (perturbation theory) 34 II Comparison of GVDW and perturbation theories 37 III Properties of argon interface (MVDW theory) 39 IV Properties of argon interface (simulation) 42 V Properties of " i n t e g r o - d i f f e r e n t i a l " interface 58 v i L i s t of Figures Number T i t l e Page 1 equation of state for uniform f l u i d 15 2 Maxwell construction 21 3 interface density p r o f i l e 22 4 square-well potential 27 5 Lennard-Jones potential 28 6 argon interface p r o f i l e 37 7 "wings"of the p r o f i l e 49 8 geometry of c y l i n d r i c a l subsystem 50 9 low-k divergence of horizontal correlations 69 10 fluctuation of the i n t r i n s i c p r o f i l e 88 11 behaviour of surface tension corrections 89 12 t y p i c a l graph of g(r ; i) for uniform f l u i d 101 13 t y p i c a l graph of c(rlz) for uniform f l u i d 108 14 free energy inequality 116 15 p r o f i l e construction 117 Acknowledgement I would l i k e to acknowledge the kind assistance given by Drs. Luis de Sobrino and Joze Peternelj in the course of preparing th i s thesis. I also wish to thank Ms. Lore Hoffmann for help in guiding me through the bureaucratic maze. F i n a l l y , acknowledge-ment i s made of f i n a n c i a l assistance received from the Natural Sciences and Engineering Research Council (NSERC). 1 Chapter 1 Introduction One of the more challenging problems in equilibrium thermo-dynamics i s the determination of the thermodynamic properties of the liquid-vapor interface. In p r i n c i p l e , a l l information about an N-particle ( c l a s s i c a l ) system i s contained in the p a r t i t i o n function where h i s the Planck c o n s t a n t , ^ the Boltzmann factor (ksT)~J co the "phase space" volume, and H(p,r) the hamiltonian of the system. For l i q u i d s and dense gases, direct evaluation of Q i s seldom tractable - the molecules have neither weak interactions nor periodic positions, thereby rul i n g out s i m p l i f i c a t i o n s of the type available to "ideal-gas" and " s o l i d - s t a t e " systems. The purpose of t h i s thesis i s to present a review of the theories currently being used to treat t h i s problem. The d e r i -vations of several theories have been recast into d i f f e r e n t (and hopefully simpler) forms which depend upon one or two common p r i n c i p l e s . This enables each of the major theories to be iden-t i f i e d with one of three d i f f e r e n t approaches to the problem. The three classes of theories so formed are shown to be d i f f e r -ent aspects of a single, consistent model of the liquid-vapor i n t e r f a c e . H i s t o r i c a l l y , early work on the liquid-vapor interface was c a r r i e d out by investigators such as Laplace, Young, and Maxwell. Much of modern theory, however, has i t s o r i g i n in the work of van der Waals [1] in 1893. Later studies by workers 2 such as H i l l [2] and Plesner and Platz [3] developed the theory to the point where the descriptions of surface tension agreed well with experimental data on inert gases. The advent of high-speed computers in the late 1960's caused a renewal of interest in the subject; computer simulation of non-uniform systems pro-vided detailed information on interface structure and surface tension. In spite of their increased quantitative accuracy, however, many of the "modern" theories retain a s i m p l i c i t y of form comparable to that of the van der Waals model. The theories reviewed here assume an N-particle (canonical) system of fixed volume V and temperature T. The hamiltonian i s of the form where m i s the p a r t i c l e mass, and U(r, / W )=U(r , , r 2 , . .. , fN ) is the interaction p o t e n t i a l . Further, U(r,/Ai,) i s assumed to be the sum of pairwise additive, r a d i a l l y symmetric pair potentials, v i z . , U t ? „ N ) - 2: U ( ri}) , with rv =/r (-r/. Consideration i s r e s t r i c t e d to "argon-like" molecules, which have an intermolecular potential of the form U H being a short-ranged, repulsive "core" p o t e n t i a l , and U A being a long-ranged, weakly a t t r a c t i v e p o t e n t i a l . For s i m p l i c i t y , the system i s assumed to be confined to a rectangular parallelopipedal container of fixed dimensions L^xL^xLj. The edges of the container are aligned with the x, y, 3 and z s p a t i a l axes. Each dimension L ; i s assumed to be much larger than the range of the molecular p o t e n t i a l . The container walls are assumed to produce no "wall e f f e c t s " , i . e . , the pre-sence of a nearby wall w i l l not affect any property of the f l u i d . For the study of planar interfaces, these assumptions concerning the nature of the container c l e a r l y e n t a i l l i t t l e loss of generality. Three major approaches have been used to determine the struc-ture of the interface. The f i r s t i s "mean-field" theory, which replaces the at t r a c t i v e part of the potential by i t s average value <UA (? i / a /)>. This "mean-field approximation" leads to with Q„ being the p a r t i t i o n function of the "reference" system of repulsive p a r t i c l e s [4] , The free energy F of the system can then be written where General consequences of mean-field theory are surveyed in chapter 2. The existence of a stable two-phase f l u i d system is discussed, and some general thermodynamic properties of the interface are b r i e f l y described. Chapter 3 examines in d e t a i l the theories of Toxvaerd [5] , Abraham [6][7] , and Bongiorno and Davis [4][8] , which are representative of the mean-field approach. Their description of the interface p r o f i l e and sur-4 face tension are found to agree well with simulation. A second way to determine the properties of the interface is through the use of the "correlation functions" g(f" 1,r 2) and c ( r 1 , f 2 ) . These functions describe the response of the system to perturbations in external potential and density (appendix A), and form the basis of the theories reviewed in chapter 4. The theory of Kirkwood and Buff [9] develops a set of integro-d i f f e r e n t i a l equations based upon g ( ? i , r " 2 ) ; the theory of Triezenberg and Zwanzig [10] develops a " p a r a l l e l " set of equa-tions based upon c ( r , , r 2 ) . The "inverse" nature of g ( r 1 f r 2 ) and c ( r 1 f r 2 ) e n t a i l s that these descriptions are equivalent. These i n t e g r o - d i f f e r e n t i a l equations y i e l d interface p r o f i l e s and sur-face tensions similar to those of the mean-field theories. Indeed, the " i n t e g r o - d i f f e r e n t i a l " description i s shown to reduce to the mean-field description when the mean-field approx-imation i s made and fluctuations are kept small. The t h i r d approach i s "fluctuation theory", which determines the e f f e c t of imposing density fluctuations upon an " i n t r i n s i c " or "bare" liquid-vapor system. Since U(r ) / W) i s a function of the p a r t i c l e positions, the free energy F becomes a function of the p a r t i c l e d i s t r i b u t i o n {n}. The p a r t i t i o n function can therefore be written Q = jE 1 e x / 9 I ~/3F i where {n}0 i s the most probable d i s t r i b u t i o n , and {v}={n}-{n}0 5 is a density fluctuation about {n} 0. The primes indicate that summation i s r e s t r i c t e d to d i s t r i b u t i o n s which conserve mass. Chapter 5 reviews the theories of Buff, Lovett, and S t i l l i n g e r [11], Weeks [12], and Sobrino and Peternelj [13], which use fluctuation theory to evaluate corrections to the " i n t r i n s i c " interface p r o f i l e and surface tension. The ap-pearance of long-ranged horizontal correlations in the interface region i s explained in terms of c a p i l l a r y wave fluctuations of the interface. By i d e n t i f y i n g the most probable d i s t r i b u t i o n with that described by the mean-field and i n t e g r o - d i f f e r e n t i a l theories, the various approaches can be combined to form a con-sistent description of the properties of the liquid-vapor inter-face. 6 Chapter 2 General Formulation of Mean-Field Theory This chapter presents a discussion of some general aspects of mean-field theory. Section 2.1 provides a brief review of the thermodynamics of non-uniform systems, with emphasis being placed upon the role of the density functions. Section 2.2 uses thi s formalism to develop a "mean-field" theory of f l u i d s . The basic assumptions of the theory, together with their physical interpretation, are examined. The equation of state of a uniform f l u i d i s derived in sec-tion 2.3. The thermodynamic s t a b i l i t y of such systems i s d i s -cussed. Non-uniform f l u i d s with small variations in density are considered in section 2.4, where a " d i f f e r e n t i a l " expression for the free energy i s developed. The general c h a r a c t e r i s t i c s of a planar liquid-vapor interface are presented. Section 2.5 examines some of the thermodynamic properties a r i s i n g from the density gradient of the interface - in particu-l a r , the existence of an anisotropic pressure tensor and a sur-face tension. 7 2.1 Basic Thermodynamic Quantities A uniform system can be characterized by a set of independent extensive parameters, together with the system energy E [14]. For the f l u i d s considered here, the basic extensive parameters are taken to be the volume V, p a r t i c l e number N, and entropy S. The associated intensive parameters are the pressure p, chemical potential JJ., and temperature T. The t o t a l energy i s £ > -pV * JJLN + TS . (1) For systems considered in th i s work, V, N, and T are assumed to be held constant. Defining the (Helmholtz) free energy F by F • £ - TS , (2) the conditions governing the state of stable equilibrium become & F = o 6 3 F > o j i . e . , the free energy i s a minimum. Generalizing to non-uniform systems, i t i s assumed that for any extensive parameter X, there exists a corresponding density function x(r) such that i ) X =* Xj "X ( r ) dF j d? a d V for any s p a t i a l configuration that the system may take, and i i ) a l l basic thermodynamic relations remain v a l i d for the corresponding density functions. 8 Although the existence of a (par t i c l e ) density n(r) can be e a s i l y admitted, the postulate has been questioned in regards to i t s use for quantities such as F [15], which may have s i g n i f i -cant non-local contributions. No sat i s f a c t o r y alternatives have yet been found for general use in f l u i d systems, however. The results obtained from theories depending upon th i s postulate are in f a i r agreement with experiment (chapter 3); assuming the existence of a free energy density f ( r ) should be suitable for present purposes. The density functions are often functionals of the p a r t i c l e density of the entire system, and are denoted x(r;{n}) when thi s i s to be emphasized. If the p a r t i c l e density has a uniform value n throughout the system, the density function w i l l be denoted more simply by x(r;n). Generalizing from uniform systems, the stable equilibrium state of a non-uniform f l u i d i s taken to be that which minimizes the t o t a l free energy F. This p r i n c i p l e forms the basis of the mean-field approach. 9 2.2 Mean-field Theory The . f l u i d s considered in t h i s work are those consisting of "argon-like" molecules. These are defined to be molecules with a r a d i a l intermolecular potential U(/r,-r a| ) which i s strongly repulsive at small distances of separation, and weakly attrac-tive when r„ = | r , - r a | i s large. Calculation of the free energy of such f l u i d s i s done using perturbation theory, since U(r,a) can be considered the sum of a short-ranged repulsive "reference" potential U„(r„) and an a t t r a c t i v e "perturbation" U A ( r / a ) . Let FK denote the free energy of a "reference" system composed of p a r t i c l e s with potential U*(ryij<; with a p a r t i c l e d i s t r i b u t i o n and temperature equal to that of the "subject" system. The t o t a l free energy F can then be expressed by a series The fundamental assumption of mean-field theory i s that the f i r s t - o r d e r "perturbation" e f f e c t i s s u f f i c i e n t to determine F. Thus, F = F/t *• j t F3) (r, , ra) UA ( ria) d? l/3 , (2) where g R ( f , , r 2 ) i s the pair d i s t r i b u t i o n function of the re-ference f l u i d (appendix A), defined by In eq.(2), the r e s t r i c t i o n r,#r 2 i s replaced by requiring 10 U A(0)=0. The term F in eq.(3) represents the eff e c t of the short-ranged interactions; the second term represents the effect of the long-ranged interactions. Corrections due to three-body potentials have been neglected. S u f f i c i e n t and necessary conditions for the v a l i d i t y of eq.(2) are [8] i) the pair d i s t r i b u t i o n function of the reference f l u i d i s the same as that of the subject f l u i d , and i i ) the entropy S R of the reference f l u i d must be equal to the entropy S of the subject f l u i d . The mean-field hypothesis is therefore equivalent to the assump-tion that the short-ranged repulsive forces completely determine the structure of the f l u i d . This assumption has been found ac-curate for uniform f l u i d s of density greater than twice the c r i -t i c a l density; at lower densities, i t becomes increasingly inac-curate [16]. To proceed further, a second hypothesis i s made, v i z . , F* = J U W r ; ; J? , (3) where f ^ ( n ( f ) ) i s the free energy density of a uniform reference system of density n=n(r). Since the reference potential has an extremely short range (approximately a molecular diameter), con-sidering f A ( n ( f ) ) to be a "mass- add i t i v e " function ( i . e . , en-t i r e l y l o c a l in origin) should e n t a i l few inaccuracies [4]. The free energy density of a "mean-field" f l u i d can therefore be accurately represented by the int e g r a l expression 11 Due t o the l a c k of knowledge about the d i s t r i b u t i o n f u n c t i o n of a non-uniform f l u i d , a " l o c a l - c o r r e l a t i o n " assumption i s added t o m e a n - f i e l d t h e o r y . T h i s a s s u m p t i o n s e t s i . e . , t he d i s t r i b u t i o n f u n c t i o n i s t h a t of a u n i f o r m f l u i d a t d e n s i t y n ( r + ) , where r* i s some l i n e a r c o m b i n a t i o n of r , and r 2 . The argument " r ( J " i n d i c a t e s t h a t o n l y the d i s t a n c e of s e p a r a t i o n i s r e l e v a n t f o r such a u n i f o r m f l u i d . The a c c u r a c y of the " l o c a l - c o r r e l a t i o n " a s s s u m p t i o n i s not known i n g e n e r a l , but p r o v i d e d t h a t r + i s c a r e f u l l y chosen, eq.(5) s h o u l d be a good a p p r o x i m a t i o n when the f l u i d has o n l y s m a l l d e n s i t y g r a d i e n t s . The e q u i l i b r i u m p a r t i c l e d e n s i t y n 0 ( r ) i s d e t e r m i n e d by the re q u i r e m e n t t h a t i t m i n i m i z e F, s u b j e c t t o the c o n s t r a i n t of mass c o n s e r v a t i o n J r> ( ?) C(F = N . T h i s can be r e c a s t t o r e q u i r e n 0 ( r ) t o m i n i m i z e t h e f u n c t i o n a l F~ - cc J * (F) d? where « i s a L a g r a n g i a n m u l t i p l i e r . For a system i n c h e m i c a l e q u i l i b r i u m , JLE-. _ cc = ( Fj = o , (6) showing t h a t « i s j u s t the c h e m i c a l p o t e n t i a l , which i s c o n s t a n t t h r o u g h o u t the system. S u b s t i t u t i o n of eqs.(2) and (3) i n t o (6) 12 y i e l d s a>*(F.) " «/*. an integral equation for the equilibrium p a r t i c l e d e n s i t ym 0 ( P ) . 2.3 Uniform F l u i d - Equation of State One solution of eq. ( 2 . 2 . 7 ) i s the constant-density d i s t r i b u -tion n 0(r)=n. A s u f f i c i e n t condition for the existence of t h i s solution i s that WA In) = / UA ( ru ) fc(r,ai») <J?,Z be a smooth, bounded function of n. The function g^(r, a ;n) i s a bounded function of r y i, with asymptotic value g(r / - 2 = a>;n) = 1 . For a l l models studied, i t i s also a continuous function of n [ 1 6 ] [ 1 7 ] , The models of \JA (r ( i) commonly used (chapter 3) are bounded, and approach zero faster than ( l / r J a ) a in the long-dis-tance l i m i t . For such models, a constant-density solution can e x i s t . From e q . ( 2 . 2 . 4 ) , the free energy density of a uniform system is The equation of state i s determined by way of eq. ( 2 . 1 . 2 ) , which becomes 13 p (n) = in p. (in) ~ f (m) = + £ + ± (d**-)*3 , (2) where p^(n) i s the pressure due to the repulsive part of the po t e n t i a l . This pressure is calculated -via the v i r i a l theorem (appendix A). Eq.(A.2.lf) becomes S>h l»>- - - ^ J ^ ( ^ ^ ) ( - ^ J ^ } ) ^ A « . <3> Combining eqs.(2) and (3), the equation of state i s •y- /.ifa n. ) . (4) dm ^ a. / D i f f e r e n t i o n of eq.(2), together with the use of the identity Ju (n) = f ' (n) , gives / i U n ) = £ , (5) the prime denoting d i f f e r e n t i a t i o n with respect to the argument, in t h i s case n. Substituting eq.(4) into (5), Various analytic models have shown 2q/dn to be posit i v e in the 4 14 region of the repulsive forces, and n e g l i g i b l e elsewhere [16][17]. It i s assumed here that 2g/2>n i s such that the l a s t term of eq.(7) i s a monotonically increasing function of n, the second term becoming a s t r i c t l y increasing function of n. This gives r i s e to a p o s i t i v e function /3p^(n) with a unique point of zero slope (figure 1). The assumption thus ensures that the reference system w i l l be stable at any density, since the com-p r e s s i b i l i t y 1/p(n) remains p o s i t i v e . A very good approximation of g * ( r 1 , r 2 ) i s [17] where y(r,,r 2;{n}) i s the Percus-Yevick (PY) function (appendix A). The form of /3/V(n) thus depends upon temperature in the manner depicted in figure 1, the reference system remaining stable for a l l temperatures. Since the potential U^tr^) is bounded, and dq/dn i s non-zero only for small r,a, the terms/OW^4 and t^fnW^  w i l l be much smaller than the l a s t term of eq.(7). Thus, /?/*'( n) i s e s s e n t i a l l y the function /3//s(n) translated by an amount /3VlA(n), as shown in f igure 1. The c r i t i c a l temperature T c i s defined by the condition that the isotherm /?e/u.'(n) has a unique zero, the c r i t i c a l density n e . Above T e, ^ ( n ) i s s t r i c t l y increasing; any constant-density solution of eq. (2.2.7) i s unique. Since /i'(n) i s always posi-t i v e , such solutions are also thermodynamically stable. Indeed, for a fixed-volume system, these uniform densities y i e l d the f i g u r e 1 - e q u a t i o n of s t a t e f o r u n i f o r m f l u i d g l o b a l minimum of F ( s e c t i o n 2.4). 16 Below T c, the s t a b i l i t y of a uniform f l u i d depends upon i t s density. If n is such that yu'(n)>0, the f l u i d i s thermodynami-c a l l y stable, even though i t s free energy may only be at a l o c a l minimum. For a value of N such that jX\n)<>§, a uniform system is thermodynamically unstable; the equilibrium density must be non-uniform. A more detailed approach which can describe such non-uniform systems is developed in the next section. 2.4 D i f f e r e n t i a l Description of Non-uniform Fluids For non-uniform f l u i d s with small density variations, the integral equations (2.2.4) and (2.2.7) can be approximated by d i f f e r e n t i a l equations. From eqs.(2.2.2) and (2.2.3), £S»*(n)[ft(n*i»(?*))-Jjt(rtiih(r,))]UA(r,a)ctF,/j ( D where •?(»(?))= (n( F)) * f WA (*LF)) « a (F) , (2) Since n(r) and g(r / a;n(f)) are assumed to be continuous functions of their arguments, the differences in eq.(l) can be expanded in Taylor's s e r i e s . Due to the independence o f ' f ( r ) upon the d i r e -ction of the gradient, only scalar terms involving the operator V can appear. Since wall e f f e c t s are assumed to vanish, the density gradient i s p a r a l l e l to the walls; the divergence 17 theorem can be employed to eliminate the V^n terms [18]. The density i s assumed to vary s u f f i c i e n t l y slowly to cause V vn (and higher orders) to become vanishingly small. Eq.(l) then yields the " d i f f e r e n t i a l " free energy density {(F) = f (»(F)) * b (»(F))[\7* (F)J*, (3) where b(n) i s some p o s i t i v e , f i n i t e function of n. Via eq.(2.3.2), f(n) can be interpreted as the free energy density of an "adjunct" f l u i d , constrained to remain at a con-stant density n=n(r), and with temperature equal to that of the "subject" f l u i d . This term represents the contribution of the (l o c a l l y ) uniform f l u i d . The second term of eq.(3) represents a "gradient" energy, providing the correction required when lo c a l inhomogeneities are present. Eq.(3) was f i r s t derived by van der Waals [1], who considered b(n) to be a constant. A more general treatment, considering l-j(n) to be some positive function, was lat e r done by Cahn and H i l l i a r d [18]. From eqs.(2.2.6) and (3), the " d i f f e r e n t i a l " equation of the equilibrium p a r t i c l e density is l'( »0(?))[vr,JlFj]* + I t ( n j F ) ) v \ ( F ) = / * 11\{ ?)) - j * . (4) For s i m p l i c i t y , consideration w i l l be r e s t r i c t e d to planar den-s i t y p r o f i l e s along the z-direction, i . e . , n(r)=n(z). Eq.(4) then reduces to 18 ul* + all*) = fit ( » ) - ^ . (5) A p h y s i c a l i n t e r p r e t a t i o n c an be g i v e n t o t h i s e q u a t i o n by r e -g a r d i n g t h e e n e r g y d i f f e r e n c e ^ ( n ) - / / t o be t h e " c a u s e " o f t h e i n h o m o g e n e i t y , n j ( z ) as t h e " r e s p o n s e " , and b ( n ) as t h e " i n f l u e n c e f a c t o r " , w h i c h d e t e r m i n e s how much g r a d i e n t i s needed t o make up t h e e n e r g y d i f f e r e n c e [ 8 ] . M u l t i p l y i n g e q . ( 5 ) by n ^ ( z ) and i n t e g r a t i n g , • ».(*) t> (hllh^BlJ* = J[/t(nJZ)) - Mid* * c 0»s<. ; (6) o an a l t e r n a t e form of t h e p r o f i l e e q u a t i o n . To s t u d y t h e i n t e r f a c e between two f l u i d p h a s e s , r e g i o n s o f ( e s s e n t i a l l y ) u n i f o r m d e n s i t y a r e assumed t o e x i s t a t e i t h e r end of t h e c o n t a i n e r . E a c h r e g i o n i s r e f e r r e d t o a s a " b u l k p h a s e " o f t h e s y s t e m . The d e n s i t y of t h e upper phase ( g r e a t e r z) i s d e n o t e d n,-; t h e . d e n s i t y o f t h e l o w e r phase i s d e n o t e d n L . I f nc^NL' t h e n by c o n v e n t i o n nL>nc>, i . e . , t h e d e n s e r " l i q u i d " phase i s l o c a t e d below t h e " g a s e o u s " o r " v a p o r " p h a s e . S i n c e t h e p h a s e s a r e v i r t u a l l y u n i f o r m , t h e b o u n d a r y c o n d i -t i o n s f o r t h e s o l u t i o n s o f e q . ( 5 ) must be t h a t n ' ( z ) a p p r o a c h z e r o a t b o t h e x t r e m i t i e s . The s y s t e m t h e n has d e n s i t y g r a d i e n t s p a r a l l e l t o t h e w a l l s , a s was r e q u i r e d i n t h e d e r i v a t i o n o f e q . ( 3 ) . U s i n g e q s . ( 5 ) and ( 6 ) , t h e bou n d a r y c o n d i t i o n s c a n be r e s t a t e d fi (mL) = fi I "e, ) = (7) 19 and ° - S [ /te>) - /*] c/» . (8) Equation (8) i s the Maxwell construction, which determines the value of ju for the two-phase system (figure 2). Using eq.(2.3.2), th i s can be rewritten o - p (n,.) - pi . (9) The boundary conditions thus imply the existence of thermo-dynamic equilibrium between the two phases. If n L and are d i f f e r e n t , conditions (7) and (8) uniquely determine their values at any given temperature. Of the possible solutions of eq.(5), only monotonic p r o f i l e s are considered, since these y i e l d the minimum of the free ener-gy, when subject to the constraint n(r)=n(z) (appendix B). Using the boundary conditions at the lower end of the container, eq.(6) can be written [4] f < - f(*^ ) - /A (*J*)- r>L) t (10) which becomes cJ»0 = _ f -f(h) - f(hL) - yu * L ) -j ( 1 1 ) The sign convention of eq.(1l) i s chosen to lead to a (possibly) denser phase in the lower end of the container. If the regions at the extremities have the same density, the f l u i d must be uniform throughout the container. These are the 20 "single-phase" systems discussed in section 2.3. If T>TC , or n<-V>N>ntV, phases of d i f f e r e n t densities are not possible; the uniform d i s t r i b u t i o n alone can minimize the free energy (appen-dix B). Since t h i s i s a global minimum, these f l u i d s must be stable. For T<TC and nqV<N<ntV, a "two-phase" solution i s possible. In such a system, the phase of density n a corresponds to a "gaseous" or "vapor" phase; the phase of density n L corresponds to a denser " l i q u i d " phase. Equation (11) then describes the "interface" between the two phases. Since the width of the interface i s an i n t r i n s i c property of the f l u i d (see below), i t s free energy i s proportional to the area A of the container, whereas the t o t a l free energy i s proportional to the volume V. For s u f f i c i e n t l y large bulk phases, F approaches ^ = A J ^ ( " l*» etz , where the two-phase function £ (») = rLzJZJL f(*L) * - * f . (12) From figure 2, f a(n)£f(n) in the range n q5n<n L. Thus, i f the bulk phases are large, the two-phase solution y i e l d s the global minimum of F. Single-phase f l u i d s can exist in thi s range i f /u'(n)>0, but they are metastable. The p r o f i l e of the liquid-vapor interface described by eq.(5) i s sketched in figure 3. A d i s t i n c t interface region i s e v i -dent. H 1 1 1 1 > figure 2 - Maxwell construction It is of interest to note that eq.(5) i s invariant under ver-t i c a l t r a n s l a t i o n . Provided that the interface remains far 22 — > z figure 3 - interface density p r o f i l e enough away from the fixed walls to maintain the boundary condi-tions, i t s shape i s therefore independent of the amount of subs-tance in each phase [19]; i t i s an i n t r i n s i c property of the f l u i d . 2.5 Thermodynamic Properties of Non-uniform Systems Although temperature and chemical potential are constant throughout the system, the presence of density gradients leads to thermodynamic quantities not found in uniform systems. This section presents a discussion of the anisotropic pressure tensor and surface tension found in liquid-vapor systems. a) pressure tensor Addition of an a r b i t r a r y external potential 5B(r) to the system causes the p r o f i l e equation (2.4.4) to become fL -fi (njr)) + b(h)[x7»J* +• a £ ( * ) P a * . = T(F). ( D Multiplying by Vn, t h i s becomes 23 P . = - •nJF) V Se (F) , (2) the statement of m e c h a n i c a l e q u i l i b r i u m , w i t h p r e s s u r e t e n s o r [19] P = [/i(h0(?))~ -f(hjr))-nJr)¥(F)]lL * 2 £ ( n ) [ v * V h - £ ( * f ] L ] * B , (3) where JB i s a symmetric t e n s o r such t h a t V- B = o • IB (!£.' o) - o . (4) For p l a n a r p r o f i l e s , w i t h fP(r)-*0, eqs.(3) and (4) l e a d t o a p r e s s u r e p^Cz) normal t o the i n t e r f a c e From e q . ( 2 . 4 . 5 ) , p^(z)=0. Thus, p ^ z ) i s a c o n s t a n t , e q u a l t o the p r e s s u r e py i n the b u l k phases p„(z) = p± . (6) The c o n s t a n c y of P/v(z) i s j u s t the c o n d i t i o n • of h y d r o s t a t i c e q u i l i b r i u m of the f l u i d . S i n c e the system i s symmetric i n the (x,y) p l a n e , eqs.(3) and (4) a l s o l e a d t o a p r e s s u r e p T ( z ) tangent t o th e i n t e r f a c e . From the thermodynamics of the system undergoing a s m a l l d e f o r -mation of a v e r t i c a l w a l l , pT ( ? ) = yUnJi) - f ( * ) , (7) and t h e r e f o r e , 2B =0. 24 The pressure tensor for the liquid-vapor system i s therefore P - [ r »j e; - f (»J*»J 1L + 3. b(»0(z))[ Vnjz) t?n (?) - ± ( > ) * 1L ] , (8 ) which i s isotropic only in the bulk phases of the system. b) surface tension Consider a system located between the planes z=a and z=a+L. The surface tension y i s defined to be [20] C4.*L r = / t p„(D - p r U ) } otz . (9) « If the l i m i t s of integration extend to the homogeneous regions, y becomes an i n t r i n s i c property of the system. This i s s i g n i -f i e d formally by writing CO - CP the "mechanical" d e f i n i t i o n of surface tension. From eqs.(2.4.3), (5), and (7), the " d i f f e r e n t i a l " form i s CO r = SL f £ (»Jt»L ml (*)J2 di . (11) - CD The Gibbs dividing surface z„ i s defined to be such that the excess mass vanishes, * « / C D O » J [ ^ ( i j - ^ j c / z + J [ r>{ i) - nq 3 di . (12) Taking zN to be the dividing surface, eqs.(6), (7), and (9) y i e l d 2 / V oo 7= f L fd)- fl*i)] + J[ f(i) - ?<r>i)]di (13) - c o t / v 25 i . e . , y becomes the s u p e r f i c i a l free energy density. For con-venience, 2^ i s usually set to zero. Another expression for y can be obtained by considering the reversible work required to produce a volume-conserving deforma-tion of the container. Denoting the change in cross-sectional area by A(z), the change in free energy is* «.* L A F = J C pN (*) - />r te)J d (g) . (14) a It i s evident from the d e f i n i t i o n of z„ that any deformation preserving both N and V must leave z^ unchanged. Assuming that A(z) i s constant in the v i c i n i t y of the interface, eqs.(9) and (14) lead to I . (15) ( &A / N,V.T J where A i s the area of the Gibbs equimolar dividing surface. Integrating eq.(6), and using eqs.(2.4.3) and (10), the free energy of the liquid-vapor system becomes F = JJLN - pk V * yA (16) the interface free energy contributing to F in a natural fashion [19]. 26 Chapter 3 Mean-field Theories In order to predict interface properties, the "general" mean-f i e l d theory requires f^(f,) and g ^ ( r 1 f r 2 ) to be s p e c i f i e d . This chapter reviews some of the approximations used for these quan-t i t i e s , and compares the results with computer simulations and experiment. Section 3.1 discusses the use of the "square-well" and Lennard-Jones "6-12" potentials for modelling uniform f l u i d s . The Barker-Henderson [21] and Weeks-Chandler-Anderson [16] schemes for separation of the potential are presented. Section 3.2 reviews the theories of Toxvaerd [5], Abraham [6], and Bongiorno and Davis [4], which generalize the separa-tion schemes to describe non-uniform f l u i d s . Their results for interface width and surface tension are compared to each other and to simulations. Good agreement i s found throughout the tem-perature range of liquid-vapor coexistence. A brief review of computer simulation i s given in section 3.3. The r e l a t i o n between the results of theory, simulation, and experiment on " r e a l " f l u i d s i s discussed. 27 3.1 Separation of Potential The s p e c i f i c a t i o n of the intermolecular potential and i t s subsequent separation into "reference" and "perturbation" poten-t i a l s are important concerns in mean-field theory, since these determine the form of the free energy density * I r,) = fj,(nlF,)) i- £ j UA ( K / A ) fcCF.,?,) etf=a . The form of f(r,) in turn determines the accuracy of the mean-f i e l d approximations. The simplest model of U(r,a) i s the "square-well" potential U s < v(r, a), shown in figure 4 u I I figure 4 - square-well potential The parameter cr, represents the (hard-core) diameter of the par-t i c l e , and (Tj_ the range of an a t t r a c t i v e potential of strength €,. The square-well potential is a crude approximation of actual intermolecular potentials, but i t can lead to good descriptions of the thermodynamic behaviour of inert gases [22]. The major advantage of eq.(l) is i t s considerable s i m p l i f i c a t i o n of nu-merical c a l c u l a t i o n s . 28 A more r e a l i s t i c model i s the Lennard-Jones " 6 - 1 2 " p o t e n t i a l , shown in figure 5. U"(r,z) = H ^ i f J ^ - l ^ j ' ] (2) figure. 5 - Lennard-Jones potential The parameter cr governs the range of the po t e n t i a l , and € i t s strength. Although use of the 6 -12 potential gives poor agree-ment with spectroscopic and scattering data, careful choice of o-and e leads to good approximations of the thermodynamic beha-viour of inert gases [ 2 3 ] . Further, the s i m p l i c i t y of the 6 -12 potential favors i t s use in computer simulations of argon-like f l u i d s . Uniform f l u i d s have been studied using two d i f f e r e n t separa-tion schemes. The oldest is"the Barker-Henderson (BH) scheme [ 2 1 ] , which separates U(r / a) into p o s i t i v e and negative parts (3) I tr,3 ) = O ; rlx < cr. In the BH theory, f*(n) and g^(r*,,r 2) are taken to be those of a reference system consisting of hard spheres (<r, = cr,) 29 (»> = fs <»> ( 4 ) where the e f f e c t i v e hard-sphere diameter 5 i s given by [21] 6 = J ( l - exp C -/3 U (r,t)3) dru . (6) Results obtained from theories using the BH separation scheme have a slow convergence in U 4 ( r ; i ) . The second-order term in \JA{r,z) (section 2.2) i s generally required for results to agree with experiment [7][24]. The BH expression for t i n ) becomes [21] where (8>n/9p)s i s the compressibilty of the hard-sphere f l u i d . Since (<?n/<9p) <x <Ana'>, the added term represents the mean-square contribution from energy fluctuations [8], Pressures and internal energies are calculated by numerical d i f f e r e n t i a t i o n of F, P - - 2^ : (8) Using the Percus-Yevick free energy density, compressibility, and d i s t r i b u t i o n function together with the Lennard-Jones poten-t i a l in eq.(7), calculations of p and AE agree well with experi-30 merit [21]. The free energy, however, i s not accurately given by BH theory. Equation (7) leads to values 10% below those ob-tained from simulations [16], The second separation scheme i s that of Weeks, Chandler, and Anderson (WCA) [16]. This theory emphasizes the role of inter-molecular forces, separating the potential into (positive) re-pulsive and (negative) a t t r a c t i v e parts " ( - U ( »-„) ; r,z < yv = o ; where U(r v)<U(r^) for a l l ri:t. (10) The WCA scheme sets f * ( * ; = (11) ?*(p'.p*> = ^ f F - ^ ( 1 2 ) The effective, hard-sphere diameter d i s given by [16] J Cr.fri t»V elFv = Jjtl(F..?iil»\)dF,t (13) The value of d i s v i r t u a l l y independent of n [25]. 31 Results obtained from WCA theory have a good convergence in u/q(r/*) ~ a second-order term i s not necessary [7], Pressures and internal energies are calculated from the v i r i a l theorem (appendix A) /v - " - i^hr^.^->»)77n(/*"<"«))*f>* ( 1 4 ) *JL = J2? J y^'i?,,^; U dFu . (15) The results of WCA theory are in excellent agreement with simu-lation s [16], esp e c i a l l y at high densities (above 2n c), where repulsive e f f e c t s dominate. At high densities, the free energy calculations also agree well with simulations. As density de-creases, so does th i s agreement, although i t always remains at least as good as that of the BH scheme [16]. 3.2 Mean-field Theories a) Toxvaerd perturbation theory The extension of BH theory to non-uniform f l u i d s was f i r s t done by Toxvaerd [5], who used perturbation theory to derive the expression - 4 »(FjJ»(rt)(i£^)fo(r,.?t)UA9llru)jFt (1) where c)n(r)/c>p i s the l o c a l compressibility of the f l u i d . The d i s t r i b u t i o n function i s approximated by 32 The form of g<y(r/;i) used i s e s s e n t i a l l y that of the PY d i s t r i b u -tion function. Although the function i s asymmetrical in i t s arguments, t h i s should not lead to serious error; eq.(2) is ac-curate for small rl3 , where the theory is most sensitive to the form of g*(r , ,r 2) [6] . The l o c a l compressibility i s set equal to that of a hard-sphere f l u i d of density n ( r ) . Using the PY compressibility equation [26], this quantity can be accurately determined, and is unlikely to be a source of error [5]. The Toxvaerd theory uses the PY free energy density = »kBr[ ]J2_J£L!l±£L -h I *KT)] ,(3) where ^ (n;<5) = ^ n , and I(T) i s a density-independent term. This choice of fj^n) i s not the most accurate one, but since the d i s t r i b u t i o n function and compressibility are calculated from the same theory, «use of eq.(3) should lead to self-consistent results [5]. The structure of the interface i s determined by minimizing the excess free energy. Taking the div i d i n g surface to be the Gibbs equimolar surface z^, t h i s becomes equivalent to minimi-zing the surface tension (section 2.5). From eq.(2.5.l3), the quantity to be minimized i s L X = Citr, f J f(z)dt - L( ?(»L) + ] L ~* a L - L J (4) 33 For computational purposes, L i s set to 15<5. The Toxvaerd theory assumes a density p r o f i l e of the general form (Fermi function) nc?; = lr>L exl° 1 a ( b ~ ^ J * (5) e*p [ a (b - ?)J * I b = ( - ai' ) L» a. cx = cc i ; Z < b = cc z • S 1 b , with a, and a 2 being parameters adjusted to minimize y. Eq.(5) is used for the p r o f i l e because i t i s similar to the form of the " d i f f e r e n t i a l " p r o f i l e (section 3.2c). Further, i t s exponential behaviour in the "wings" of the interface (where the gradient i s small) agrees well with the exact solution of the " i n t e g r a l " equation (section 4.1a). The width w of the interface can be defined in several ways [27]. Owing to the generality of i t s application, the measure used here w i l l usually be the "10-90" width wx. This i s defined to be the distance separating the points n(z)=n^+0.1An and n (z) =n,+0 . 9An, where An=nt-n<i. For the p r o f i l e of eq.(5), = cr U ? (±, + £J . (6) 34 Table I - Properties of argon interface (perturbation theory) T(K) a, (A) 84.0 1 .55 2.95 7. 35 _ 16.18 18.5 89.95 1 .50 2.50 7. 97 1 4.08 14.58 (16.6) 101 .83 1 .40 2.00 9. 07 11.03 1 1 .43 119.80 1.15 1 .35 12. 03 6.78 7.09 ( 7.8) 143.76 0.65 0.75 21 . 45 2.10 2.27 ( 3.4) •y in units of erg/cm* YTOX- Toxvaerd; «/k« = 119.8K, *=3.405A [5] •ytmp- Lee,Barker ,and Pound; «/k 8=1l9.4K, a = 3.40A [28] Tv*- MC simulation; <r/k* = 1 1 9.8K, cr = 3.405A [29] - brackets denote quantities estimated by interpolation Results for an argon-like f l u i d are presented in Table I. The bulk densities nL and n, are calculated from the Percus-Yevick equation of state; the parameters <A S=119.8K and o-=3.405A are chosen to give agreement between calculated and experimental pVT properties of argon. The equilibrium values of >S and Wo are determined for temperatures between the t r i p l e point of argon (83.86K) and i t s c r i t i c a l point (150.72K). The Toxvaerd theory produces results in good agreement with simulations. The predicted interface p r o f i l e s are similar to those of simulations [5], but surface tensions are somewhat lower. A l a t e r study by Lee, Barker, and Pound [28] replaced the PY equation of state with that of Carnahan and St a r l i n g [30]. The free energy density becomes f = kaT[ 3 In A + In » - i + V*(»J*)-3*'(»;*> ) (7) L C i - ?(us))3 J 35 where the thermal wavelength A = (27rh//mkBT)'/i. Using t h i s equa-tion of state - which is the best available for uniform f l u i d s -the predicted p r o f i l e s are the same as those of Toxvaerd, while the surface tensions are s l i g h t l y higher (Table I ) . Thus, the general formalism leads to results which apparently are not very sensitive to the form of f*(n). b) generalized van der Waals' theory The extension of WCA theory to non-uniform f l u i d s was f i r s t accomplished by Abraham [6][7], using an approach similar to that of Toxvaerd. The general form of the free energy i s the free energy density of eq.(2.4.1) becomes the (van der Waals) in t e g r a l form The theory i s referred to as the generalized van der Waals (GVDW) theory. The d i s t r i b u t i o n function used i s similar to that of eq. (3.1 . IX), + i Kir*)J^l Fa> U(rl7) . (2.2.2) Choosing an asymmetrical d i s t r i b u t i o n function % (7. ,7* ;M) - y?(r, ; M ( F, )) (9) 36 where d is given by eg.(3.1.13). As in the Toxvaerd theory, an asymmetry exists in the arguments of g> ?(r 1,r 2), but thi s is not f e l t to be a serious objection [6]. The GVDW theory uses the Carnahan-Star1ing form of the free energy density (eq.(7)). The interface p r o f i l e i s determined by minimizing the excess free energy in a fashion similar to that of Toxvaerd. Since variations in the parameter change the excess free energy by less than 1% [5], the GVDW theory uses a si m p l i f i e d p r o f i l e *(z) = >IL e x p ( - « . z ] * h q ^ ^ ex/s L - a.Bz] + l with a single adjustable parameter a 0 . Table II compares the results of Abraham and Toxvaerd for the case of an argon-like f l u i d . The parameters £/k B=1l9.8K and cr=3.405A for both studies. In general, the GVDW results agree better with simulation than do those of perturbation theory. This i s largely due to the use of the WCA d i s t r i b u t i o n function. At temperatures far below the c r i t i c a l temperature, the liguid-phase d i s t r i b u t i o n function dominates the structure of the interface [25]. Since the WCA approximation i s very accurate at these (liguid) densi-t i e s , eg.(9) i s a very good approximation of g w ( r 1 , r 2 ) near the t r i p l e point. 37 T a b l e I I - C o m p a r i s o n of GVDW and p e r t u r b a t i o n t h e o r i e s T ( K ) a a -KB "Yrax "^LtP 84.0 2.5 6.0 7.35 20.26 16.18 (18.5) 89.9 2.5 6.0 7.97 17.0 14.13 14.58 (16.6) y i n u n i t s o f erg/cm 3 -^ 8 ' w » 8 - Abraham [6] [7] » O J [ , W T O X ~ T o x v a e r d [5] •y^ar- L e e , Baker ,and Pound [28] >i/~ - MC s i m u l a t i o n [29] - b r a c k e t s i n d i c a t e q u a n t i t y e s t i m a t e d by i n t e r p o l a t i o n A c o m p a r i s o n of t h e GVDW and s i m u l a t e d p r o f i l e s i s shown i n f i g u r e 6. The o s c i l l a t i o n s of t h e s i m u l a t i o n p r o f i l e a r e an a r t i f a c t o f t h e s i m u l a t i o n p r o c e d u r e [ 3 1 ] . — 6 V D W theory — 8<5Y -theory - - A A C s C*»•> u . I a t t ' o h 7~ - &H K f i g u r e 6 - a r g o n i n t e r f a c e p r o f i l e [ 6 ] [ 3 8 ] 38 c) modified van der Waals theory Bongiorno and Davis [4] studied the liquid-vapor interface by way of the d i f f e r e n t i a l free energy density •fir) = fl*(r)) + & (h(r)) [xrnj*. (2.4.3) Using the WCA approximation scheme, a "modified van der Waals" (MVDW) theory i s developed by setting <»> = -p;s (*) . (12) Neglecting terms of order Vn, eqs.(2.4.2) and (2.4.1) y i e l d fit,) = * i »a(F.)jg~(ruiK{±^))UAlrl9) alrx (13) -st?^, ^A' J^hr^-^(^^ u^^)<^  (14) the divergence theorem being employed to evaluate eq.(14). Except for the s p e c i f i c a t i o n of a symmetric d i s t r i b u t i o n func-tio n , MVDW theory i s e s s e n t i a l l y the d i f f e r e n t i a l version of the GVDW theory; both should give similar results near the c r i t i c a l temperature. The interface p r o f i l e i s calculated numerically from sL»*. = - [ - {("LI ~ 1 (2.4.11) The values of ,nt and n,, are determined from the requirement of thermodynamic equilibrium between the two bulk phases 39 The surface tension is calculated from the general formula c o y ~ A / %> (»'*))[ »'(i)]3 . (2.5.11) - CO Using eq.(2.4.11), t h i s can be rewritten y = a / I %>(»> ( - J J L K - f (mL) + / " ^ ) J J» , (15) "« which i s independent of density p r o f i l e . The results of MVDW theory for an argon-like f l u i d are shown in Table I I I . The parameters d/ka=116.41K and er = 3.37A optimize agreement between experiment and ca l c u l a t i o n when g y ? ( r 1 , r 2 ) i s used [32]. Table III - Properties of argon interface (MVDW theory) T(K) wD~(A) y» 84.0 21.6 18.5 1 3.45 89.95 ( 7.5) 18.61 (16.6) 1 1 .88 101.83 ( 8.5) 14.61 9.0 119.78 (11.0) 9.36 ( 7.8) 4.97 130.0 (13.2) 6.72 143.76 (21.0) 3.62 ( 3.4) 0.75 - brackets indicate estimated quantities - y in units of erg/cm a •y* - MVDW theory; */ka = 1 1 6.41K, o-=3.37A [4] -ys,„- MC simulation; e/k B=1l9.8K, <r=3.405A [29] ye*p- experimental value [28] (2.4.9) (2.4.7) 40 The generated p r o f i l e s are very similar to those of the integral theories, the differences being almost negl i g i b l e [4][8]. The surface tensions are higher than those of the other theories, due in part to the d i f f e r e n t form used for iHr,^). The agreement between the integral and d i f f e r e n t i a l theories at a l l temperatures points out the dominance of the (Vnj 2 term in the Taylor expansion of eq.(2.4.l). Although the higher-order terms- are s i g n i f i c a n t in the'wings of the interface, their con-t r i b u t i o n to the t o t a l free energy i s small [8]. The d i f f e r -e n t i a l mean-field theory therefore remains accurate throughout the temperature range of liquid-vapor coexistence. 3.3 Simulation and Experiment a) computer simulation Two types of computer simulation are commonly used - Monte Carlo (MC) and molecular dynamics (MD) [23]. The f i r s t method is a stochastic process which estimates canonical averages. One p a r t i c l e (out of several thousand) i s moved at a time; each con-figuration i s weighted by the Boltzmann factor. For the system of p a r t i c l e s interacting via pair potentials, t h i s factor i s exp[-/?/ 2XU(r { /) ]. The MD approach simulates the motion of (several thousand) p a r t i c l e s by numerically solving the equation of motion, which for the systems considered i s 41 = S - V Ulr.-j) (1 ) To reduce computation time, the potential i s truncated at a radius r r =2 . 5 a [ 33 ] [ 34 ]. Using the "truncated" potential Ur ( ru) = U ( r,z) - U ( rT) • r„ < rT = O ; ru >, rT ( 2 ) in eq.(l) prevents p a r t i c l e s from receiving an impulse upon crossing the radius r r . For uniform f l u i d s , the loss of free energy due to use of U r(r / ; i) can be calculated; for non-uniform f l u i d s , the corrections required are not completely known [ 3 4 ] . Both methods use a small external potential to break the sym-metry of the system and produce a density p r o f i l e along the z-d i r e c t i o n . The effect of the external potential upon the in t e r -face i s regarded as ne g l i g i b l e [ 3 4 ] . The surface tension i s obtained by canonical averaging •X = ( i S - 3^<j* *JUlr*j) \ (4.2..4b) which i s an exact equation. Chapela et al.[29] simulated systems of various sizes at d i f -ferent temperatures. The results are shown in Table IV. The p r o f i l e s generated are similar to that shown in figure 6, and are e s s e n t i a l l y the same using either method of simulation.. The truncation of the potential apparently has l i t t l e e f fect upon the density p r o f i l e . 42 Table IV - Properties of argon interface (simulation) [29] Type (N) w* r * MC ( 255) 0 .701 1 .69 1 .30 MD (1020) 0 .699 2 .02 1.10 MD (4080) 0 .701 2 . 1 1 1.10 MD ( 255) 0 .708 1 .78 1 .07 MD ( 255) 0 .759 1 .91 0.90 MD (1020) 0 .785 2 .22 0.83 MD ( 255) 0 .823 2 .03 0.78 MD (1020) 0 .836 2 .66 0.74 MC ( 255) 0 .918 2 .44 0.67 MC ( 255) 1 . 127 3 .35 0.35 T* = k a T A w* =wyo-- dimensions of container N= 225 - 5<rx5o-x25o-N=1 020 " 1 O c r X l 0 * X 2 5 c r N=4080 - 20 £ r x20ax25c r The surface tensions determined by the two methods are some-what d i f f e r e n t . This i s to be expected, since y depends strong-ly upon the form of U(r ; J) (section 3.3b). The difference is not large, however; the results of the two methods are f a i r l y consi-stent with each other. An interesting observation i s that an increase in system size ( i . e . , cross-section) causes a corresponding increase in w~ and decrease in y. The asymptotic l i m i t s appear to be reached by the N=4080 system [29]. Thus, for s u f f i c i e n t l y small systems, these " i n t r i n s i c " quantities depend upon the area of the in t e r -face . 43 b) simulation vs experiment In t h i s chapter, the mean-field p r o f i l e s and surface tensions have been compared with those of simulations, rather than with experiment. One reason for thi s is the i n a b i l i t y of present-day experimental techniques to accurately measure the interface pro-perties [35], The p r o f i l e s generated by simulation do not have this disadvantage; they can be compared d i r e c t l y with theoreti-ca l p r o f i l e s . Another consideration i s the strong dependence of upon the form of the intermolecular p o t e n t i a l . For example, replacing the "6-12" potential with the more accurate Barker potential [36], Lee, Barker, and Pound [28] found that the calculated sur-face tension increased by 0.5-1.2 erg/cm2"; i f t r i p l e t interac-tions were included, y decreased by 1.4-4.0 erg/cm2-. Singh and Abraham [37] report the truncation of the Lennard-Jones poten-t i a l causes y to decrease by 7.5 erg/cm a at the t r i p l e point. This s e n s i t i v i t y to the form of the potential accounts for much of the discrepency between calculated and experimental values, esp e c i a l l y near Tc (Table I I I ) . The experimental results i n d i -cate that mean-field theory i s a good approximation. Unless the intermolecular forces are well known, however, agreement between experiment and theory can have no further s i g n i f i c a n c e . The mean-field formalism i s best tested against the results of simu-la t i o n s , which use "molecules" of known po t e n t i a l . 44 Chapter 4 I n t e g r o - d i f f e r e n t i a l Equations An alternative method of investigating non-uniform f l u i d s i s by way of i n t e g r o - d i f f e r e n t i a l equations, which are based upon the d e f i n i t i o n s of the c o r r e l a t i o n functions g ( r , , r a ) and c(r , , r , ). This chapter presents a discussion of these equa-tions, together with some approximation schemes employed to solve them. Section 4.1 develops three related i n t e g r o - d i f f e r e n t i a l equa-tions for the density p r o f i l e . The Kirkwood-Buff . [9] theory i s derived in section 4.2, and the Triezenberg-Zwanzig [10] theory in section 4.3. These two theories give r i s e to " p a r a l l e l " sets of equations, d i f f e r i n g only in the choice of c o r r e l a t i o n func-tion used. The theories are shown to be equivalent to each other in the mean-field approximation; the p r o f i l e equations are exactly those discussed in section 4.1. The numerical studies of Toxvaerd [38], Co, Kozak, and Luks [39], and Salter and Davis [32] are reviewed in section 4.4. Good agreement i s found between the results of the "integro-d i f f e r e n t i a l " approach, mean-field theory, and simulation. Section 4.5 shows that t h i s agreement is not fortuitous; the accuracy of the mean-field ' approximation necessarily e n t a i l s agreement between a l l three sets of r e s u l t s . Section 4.6 discusses the appearance of long-ranged c o r r e l a -tions in the interface as the external f i e l d goes to zero. The behaviour of these correlations i s found to agree with computer 45 simulation. 4.1 P r o f i l e Equations a) general formulation For a system with fixed N, V, and T, the p a r t i c l e density is a functional of the external potential SP(r) (cf. eq.(2.5.l)) (1 ) Conversely (appendix A), ¥(r) is determined uniquely by the equilibrium d i s t r i b u t i o n n 0 ( r ) , <£ C F, ) = <£ i F, • { » J ) (2) The system can therefore be described either in terms of n 0 ( f ) or in terms of 5?(r). The t r a n s l a t i o n a l invariance of t h i s r e l a -tion e n t a i l s that i f 5f(r) i s displaced an amount fi, then n 0 ( f ) w i l l be s i m i l a r l y displaced, i-.e., [40] • iSflF+fi))! - »*(r,->fi ; I 1£(F)} ) , (3) and conversely »J F+firf) = ¥ ( •> fi ; f . (4) For small f i , the density can be expanded »o(r;*2) » + f -fH f P '- [¥<r-2+fi)- Vl^)]elPz + 0-<&*e>).(5) ' bSP(Ft) 46 Using the d e f i n i t i o n of the U r s e l l function u ( r , , r 2 ) (appendix A) i-*Jpd = -/?/ u^,,?,; + S(r,-*t)] (A.4.8) = ~/3[ f(r,,-?z)-/J + mtr9)S(r,-rz)} , eq.(5) becomes, in the l i m i t /5-*0, A similar expansion of Sf(r) can be done using the d i r e c t cor-r e l a t i o n function c ( r , , r 2 ) (appendix A), defined by (A.4.16) where i s the fugacity of the system. Carrying out the expan-sion, which becomes, in the l i m i t /3-»0, V[l» (J*LZ>) + /3*(F,)] - J" c ( F , , ?, ) V fi0( r a; a^ pj , (8) the "generalized Euler" (GE) equation. Eqs.(6) and (8) are i n -verse relations connecting the p a r t i c l e density, external poten-t i a l , and c o r r e l a t i o n functions. They were derived independent-ly by Lovett, Mou, and Buff [40], and Wertheim [41]. From the d e f i n i t i o n of c(r,;{n}) in terms of c ( r , ,r a ;{n}) (appendix A), eq.(8) becomes 47 The terms on the l e f t side of eq.(9) represent the forces exerted upon a p a r t i c l e by the external f i e l d plus thermal (kinetic) contributions [20]. The function k aTc(r,) thus repre-sents the e f f e c t i v e s i n g l e - p a r t i c l e " i n t e r n a l " potential due to the interactions among the p a r t i c l e s in the system. Thus, for a system of p a r t i c l e s interacting via pair potentials, Vc(F,) = Jf ( Ft) V U C *-u) alFt . (10) Combining eqs.(9) and (10), X7m0<F,) = -/3*.(F,)V2 (F.) ~/3J >~>. ( F,) na < F,) y (?,,?, )KTU (r.Jel?, ,(11) the f i r s t member of the Born-Green-Yvon (BGY) hierarchy, which can be derived from the d e f i n i t i o n of the p a r t i t i o n function (appendix A). If the f l u i d i s composed of p a r t i c l e s with pair potentials only, eqs.(8) and (11) are equivalent. b) p r o f i l e "wings" For a system with a planar density p r o f i l e , and with f?(?)=0, eq.(11) reduces to ml (Z) = y3 j <**>y < 7>> F* > ^ i . LA'tr,;) elFti (12) An analytic solution can be obtained for t h i s equation in the "wings" of the p r o f i l e , where gradients are small. Since these regions are near the bulk phases of the system, g ( r , , r 2 ) can be 48 c l o s e l y approximated by g ( r ( J ; n b ) , where n t« {n^n,}. Expanding n 0 ( r ) in eq.(12) in a Taylor's series, and neglecting terms of order f v n , *>• cej = >•><,cz,)[ <*, a,) + (*,)J , (13) the " d i f f e r e n t i a l " version of eq.(12). The values of and are » CD From the boundary conditions £ { m * • ( ? ) = I k the solution i s of exponential character in the wings of the p r o f i l e , with a decay length [20] Since g(r,a;n) increases with n for small rlz, where U'(r,2) i s greatest (section 2.2), the p r o f i l e has a less rapid decrease in the wing near the bulk l i q u i d phase (figure 7). This asymmetry r e f l e c t s the larger range of correlations in the l i q u i d phase. A similar treatment can be done for e q . ( 8 ) , which depends upon the di r e c t c o r r e l a t i o n function. The resultant p r o f i l e i s also exponential in the wings of the interface; the decay length i s [ 2 0 ] J *3 1 ** 49 N figure 7 - "wings" of the p r o f i l e " where co do = Vzr / c C r n ; k ) rn drlx o so a Although the GE and BGY equations are equivalent for f l u i d s with pair potentials, their d i f f e r e n t i a l p r o f i l e s are not exactly the same - the bulk-phase approximations of the correlations do not have the same degree of accuracy [20]. 50 4.2 Kirkwood-Buff Theory A set of integral equations dependent upon the d i s t r i b u t i o n function g ( r , , r j ) was derived by Kirkwood and Buff [9], via a theory based upon the generalized v i r i a l theorem (appendix A). The f l u i d i s assumed to be composed of p a r t i c l e s interacting via pair potentials. Its p r o f i l e is assumed to be planar. Consider a c y l i n d r i c a l subsystem of the f l u i d , with cross-section A s, length Ls , and orientation p a r a l l e l to the x-axis (figure 8). The dimensions of As are taken to be much smaller than Ls. Since the molecules have v i r t u a l l y only one degree of figure 8 - geometry of c y l i n d r i c a l subsystem t r a n s l a t i o n a l freedom, the v i r i a l of the subsystem becomes ys = < S - -<»\kBT ( D The geometry of the subsystem e n t a i l s that the contribution to v due to pressure i s - pr (?) J F-oLA = -pTlz)AiLs J 51 while the intermolecular forces contribute Summing the contributions to the v i r i a l , the pressure tangent to the interface can be expressed as pr(z,)~ ri0(z,)kBT - ^ J m„ (z,) (o„ (Z2)J( ?,, ?*) U \r„) dr/3 . ( 2 ) Due to the hydrostatic equilibrium of the f l u i d , the pressure normal to the interface i s constant; a similar development then y i e l d s Eqs.(2) and (3) form the basis of the Kirkwood-Buff (KB) theory. The density p r o f i l e i s determined from eq.(3),.which i s an inte g r a l equation for n 0 ( z ) , since p*, i s a constant. D i f f e r e n t i a t i o n of eq.(3), using the symmetry of g ( f 1 f r 2 ) , y i e l d s [20] the BGY equation (4.1.12). The p r o f i l e s of KB theory and the BGY equation are therefore i d e n t i c a l . The surface tension of the interface i s calculated by substi-tution of eqs.(2) and (3) into eq.(2.5.l0), the mechanical de-f i n i t i o n of Thus, y = ''*f " . ( l . ) " o { * z ) j ( r ; . * * ) W ( r . . ) d K n d * l (4a) ^ J ~3*'* ^ ( ^ « , ( B , ) o ( ? „ F j W V r „ ) ^ ^ .(4b) 52 For systems of p a r t i c l e s with pair potentials eqs.(4a) and (4b) are exact. 4.3 Triezenberg-Zwanzig Theory By investigating the response of a system to density "perturbations", Triezenberg and Zwanzig [10] developed a paral-l e l set of integral equations, dependent upon the direc t c o r r e l -ation function c ( r r 1 , r 2 ) . The f l u i d under consideration has a planar density p r o f i l e , interface area A0=LxL, and Gibbs surface Z / V = 0. The free energy required to form a small density perturbation v(r) in the equilibrium d i s t r i b u t i o n i s , to order , From eq.(2.2.6) and the conservation of mass, the f i r s t term disappears; eq.(1) becomes (1 ) A F = --L f c " ( \r, ,r ) v ( ir,) v ( K , ) cLir, o/e-a 2/3 J (2) where (appendix A) = C ( F,,FJ - J-j6(r.-rt) . (A.4. 16) In the Triezenberg-Zwanzig (TZ) theory, each density pertur-bation i s associated with a d i s t o r t i o n s(x,y) of the dividing surface. The function g(x,y) i s defined by 53 O = J I *•>„ li) + X>(r)-K±]dz + J [ H0 ( H ; f v(r ) - tnq J di . If 2^=0, c(x,y) becomes the "generalized" d i v i d i n g surface of the system; symmetry in the (x,y) plane i s not required. For small perturbations, the surface deviates only s l i g h t l y from pl a n a r i t y ; v i r ) is associated with a v e r t i c a l s h i f t of the prof i l e It i s assumed that the d i s t o r t i o n s can be represented by de-coupled surface waves t (4) where s=xi+ y 5 , and the hacek denotes the two-dimensional Fourier transform. From the periodic boundary conditions g(0,y)=£(L,y) and £(x,0)=£(x,L) In terms of the Fourier transforms, eq.(2) becomes [ 1 0 ] AF = ~4^r S I irik)/*/** (z.) («r*) c - ( T c , d z , dz2 a./3 t J For small k, c~(Tt,z 1 f z 2 ) can be expanded (5) (6) where 54 and therefore <^.(z,,2 3) = c~(z,te2) • i . (7) In the l i m i t k - 0 , the free energy change i s zero. Eq.(6) then y i e l d s [ 1 0 ] the GE equation (4 . 1.8). The surface tension i s calculated by considering the work done to d i s t o r t the interface. For small d i s t o r t i o n s , AF - -y0(A-Ao) + '/a * v , £ a* j I V(sil7 dl Matching the second-order terms of eq.(6) and (8), the TZ expression for surface tension. Unlike the KB equation, eq.(9) does not depend upon the molecules interacting via pair p o t e n t i a l s . When the f l u i d i s composed of p a r t i c l e s with pairwise inte r a -ctions, the KB and TZ results agree c l o s e l y . From the equiva-lence of the BGY and GE p r o f i l e s (section 4.1), the KB and TZ theories must generate i d e n t i c a l p r o f i l e s . 55 Although the KB and TZ expressions for surface tension have not yet been shown to be equivalent [20], they must give similar results when the mean-field approximation i s used. Repeated p a r t i a l integration of the KB equation (4.2.4b) yiel d s [42] ^ = vv J s,, j ( P.^iUl",,)] al?, Jz, (10) From eq.(2.2.2), which establishes that the KB and TZ equations for surface ten-sion should give similar results provided that the f l u i d is com-posed,of p a r t i c l e s interacting only via pair p o t e n t i a l s . 4.4 Numerical Solutions a) Toxvaerd One of the e a r l i e s t numerical solutions was obtained by Toxvaerd [38], for a f l u i d of argon-like p a r t i c l e s . The approach i s based upon the BGY equation, written in c y l i n d r i c a l coordinates [33], aa oo *l ( 2 J = a.Tr/3 m0( i,)J n I z, + c/zl;i J g (F,,P2) W( r,7) d r n ( 1 ) Upon integration, t h i s becomes 11/ CO CO too -co IZIII where the l i m i t s of integration of z, depend upon the choice of 56 "reference" bulk density n t. Once some suitable approximation for g ( r 1 f r 2 ) has been selected, an i t e r a t i v e procedure can be used to solve for n 0 ( z ) . A t r i a l density function with appro-priate boundary conditions i s placed into the right side of eq.(2), generating a new density function. Some li n e a r combina-tion of the old and new functions is placed into the right side of eq.(2), generating yet another density function; the proce-dure i s repeated u n t i l a consistent (stable) solution i s found. Unless the bulk densities are chosen to be consistent with n(r) and g ( f , , r 2 ) computations for the model p o t e n t i a l , i t e r a t i o n s w i l l not necessarily conserve p a r t i c l e number; solutions w i l l diverge towards the l i q u i d or vapor densities [39] [ 4 3 ] . Some of the early attempts at numerical solution did not take th i s last condition into account [44] [ 4 5 ] ; their solutions cannot be ac-curate . The Toxvaerd study modelled p a r t i c l e s by a Lennard-Jones "6-12" p o t e n t i a l , setting the parameters ^/k 8=ll9.8K and o-=3.405A to be consistent with those of the mean-field study (section 3.2a). Defining an " e f f e c t i v e density" n f ( z 1 : z 2 ) by hE ( E , •  ? , ) = 6 ) * ( I - B ) m ( ; o £ & ( 1 } the d i s t r i b u t i o n function i s chosen to have the general form = ~ q ( r l t ; nL) * *±zJ2±l±iil*L c, ( r,t • „<) (3) a l i n e a r combination of the "bulk" functions. The accuracy of th i s form of g ( f 1 f r 2 ) i s not known in general; however, g ( r 1 , r 2 ) does have the correct l i m i t i n g values as n(z)-+nfc. The values of 57 n L and n 4 are determined by the PY equation of state [38], Consistent with t h i s , the PY solution (section 3.2a) is used for g(r„;n u). The function g(r„;n<.) i s approximated by exp[-/3 U ( r w ) ] , which i s accurate only for low d e n s i t i e s . The "reference" density n^ i s chosen to be n t. Since i t e r a -tion does not conserve p a r t i c l e number, the p r o f i l e i s shifted after each i t e r a t i o n to compensate for t h i s . An a r t i f a c t of th i s technique is that stable solutions are generated only i f 0.7<e<0.8 [38]. Although t h i s approach thus requires an unphy-s i c a l ( i . e . , asymmetrical) d i s t r i b u t i o n function, t h i s i s not a serious objection provided that the density varies s u f f i c i e n t l y slowly (cf. section 3.2b). The stable solutions of the BGY equation are smooth monotonic p r o f i l e s , similar to those of mean-field theory (figure 6). Interface widths are given in Table V. The surface tensions of the p r o f i l e s , calculated via eq.(4.2.4a), are also given in Table V. A l a t e r study by Toxvaerd [43] eliminated the need to s h i f t the p r o f i l e s at each i t e r a t i o n , by choosing the bulk densities to be consistent with the Lennard-Jones potential used. A sym-metric d i s t r i b u t i o n function was employed. The resulting pro-f i l e s were v i r t u a l l y . i d e n t i c a l with those of the e a r l i e r study. Thus, although the technique for conserving p a r t i c l e number i s somewhat ad hoc, i t does lead to r e l i a b l e r e s u l t s . 58 Table V - Properties of the " i n t e g r o - d i f f e r e n t i a l " interface T(K) wTQ,(&) (A) >rox ~Xt*i "Kn 89.85 ( 9 . 0 ) - 8.7 - 11.6 101.83 (11.0) - 7.0 - 9.2 102.67 - (10.4) - 6.8 119.8 (14.0) - 4.4 - 5.5 125.48 - (16.6) 3.1 141.17 - (23.2) - 1.0 143.76 - - 1.6 - 1.1 - widths estimated from graphs - r in units of erg/cm 1 ?W '"TO* " Toxvaerd [38] T'CAL r w « u ~ Co,Kozak,and Luks [39] " V S D - Salter and Davis [32] b) Co, Kozak, and Luks A s l i g h t l y d i f f e r e n t method of solving the BGY equation was developed by Co, Kozak, and Luks [39], who obtained solutions for argon-like f l u i d s composed of p a r t i c l e s with square-well potential U S " ( r l 3 ) = CO ; r„ $ cr, = - 6 , ; tr, < r , 7 < c r , = o i rn >, a 3 . (4) The parameters | i = 1 . 8 5 , *./k8 =56.47K, and cr = 3 . 5 6 5 A were chosen to give good agreement with experimental data on inert gases [22]. The d i s t r i b u t i o n function has the general form of 59 equation (3), with 6=0.5. This produces a function symmetric in i t s arguments The values of n L and n,, are obtained from experimental data. The square-well "bulk" d i s t r i b u t i o n functions are calculated by an i t e r a t i o n procedure [22] involving the second member of the BGY hierarchy - kaT V, y(F,,ra) = V, U ( r,, rt) * Jv, U (F,t ?,) n (rjyl?,. r3 )j (r,. r,) dri ,(A.2.22) where the superposition approximation (appendix A) has been used. Once g ( r 1 , r 2 ) has been calculated for the system, the density p r o f i l e i s obtained by an i t e r a t i v e method similar to that of Toxvaerd (section 4.4a). To avoid divergences, both bulk densi-t i e s are used as "references". At each i t e r a t i o n , p r o f i l e s are generated for both values of n^ in eq.(2); the two new p r o f i l e s are subsequently averaged. This causes convergence to a smooth monotonic p r o f i l e similar to figure 8, but with a narrower interface width (Table V). The interface widths are in general smaller than those of Toxvaerd, but Toxvaerd [43] in a later study showed t h i s to be a result of the form of the potentials used. 60 c) Salter and Davis Salter and Davis [32] obtained the surface tension of several " i n e r t " f l u i d s by use of the KB equation (4.2.4a). The inte r -molecular potential i s modelled by the Lennard-Jones potential, with parameters e/kB=116.41K and <*=3.37A chosen to match those of the MVDW theory (section 3.2c). The d i s t r i b u t i o n function i s also taken to be that of MVDW theory, = yrc*«> *(Nr^)] > ( 3 . 2 . 1 0 ) where d is given by eq.(3.1.11). A v a r i a t i o n a l approach similar to that of Toxvaerd (section3.2a) was used to determine the equilibrium d i s t r i b u -t i o n , with a " t r i a l " p r o f i l e ex/o [ « ( b - B ) ] + I b = ( «;' - <za-') t r , ^  (3.2.5) cc = cc, j z < b = ; * I b a, and a 2 being adjustable parameters. The values of a, and a 2 are chosen to minimize the surface tension, which i s calculated via eq.(4.X.4a). The Salter-Davis approach y i e l d s surface tensions having the best agreement with simulation and experiment (Tables IV, V). This is largely due to the accuracy of the WCA approximation for g ( f , , r 2 ) , which i s good when n i s large. Near the c r i t i c a l tem-perature, a l l d i s t r i b u t i o n functions have inaccuracies, and re-61 suits cannot be expected to agree well with simulations or experiment. 4.5 Relation to Mean-field Theory The KB/TZ i n t e g r o - d i f f e r e n t i a l equat-ions can be related to mean-field theory v i a the generalized v i r i a l theorem (cf. section 4.2). For the system considered here, eq.(A.2.lO) becomes BT = £ kaTj F *£j*(?,)n(?t)j(F,lP-x)U(ir,7)elP„z. O) Divide the intermolecular potential into the sum of a strong repulsive potential U ) D(r / J) and a weak a t t r a c t i v e potential U„(r / i 2). The free energy of the "reference" system of repulsive p a r t i c l e s (section 2.2) i s F / ? = J kBT j n(F)dF + i / ^(F^tnlFJ^^iF.F^LA^r^ol^-T^^) Employing the mean-field assumptions (section 2.2) i) s = m 3 IF, ,Ft) = jK { r, , Ft) M i * J ?n ( »IF}) JF j (3) equations (1) and (2) y i e l d F= J f« (*>(FJ) dF + ~ fn (rJ»(F2)j^(r,,rx)UA(Kl1_')etF,/l the mean-field expression for the free energy of the system. Thus, when the conditions of eq.(3) hold, and fluctuations in the f l u i d are small, the mean-field and i n t e g r o - d i f f e r e n t i a l 62 results must coincide. A high degree of accuracy of the mean-field assumptions w i l l not only lead to good agreement between mean-field results and simulation, but also between these results and the integro-d i f f e r e n t i a l results as well. The good agreement of the various numerical results (section 4.4) thus provides further evidence of the high accuracy of the mean-field approximation for argon-l i k e f l u i d s . The close r e l a t i o n between the integral expressions leads to a d i f f e r e n t i a l form of the i n t e g r o - d i f f e r e n t i a l theory, v a l i d when variations in the density of the f l u i d are small. Expanding n(z) in the KB equation (4.2.4b) about z, neglecting terms of order C7Vn, and using the divergence theorem to e l i -minate the F an term (cf. section 3.3c), -y= a | £ x (^u)Ih;iEjJ ao(z . (4) If the " l o c a l c o r r e l a t i o n " assumption is made, then [8] - in &r[Trhll"*'>»r,; J r u (5> Replacing S(n) b y S M n ) in eq. (2.4.5) yields the d i f f e r e n t i a l KB p r o f i l e equation lx '(h) f ho + a.%z l«) *.n(e) = /* ( - yu . (6) 63 Bongiorno and D a v i s [8] s o l v e d eq.(6) i n a manner s i m i l a r t o t h a t used f o r the MVDW t h e o r y . The p r o f i l e s g e n e r a t e d from the d i f f e r e n t i a l KB t h e o r y were almost i d e n t i c a l t o t h o s e of the d i f f e r e n t i a l m e a n - f i e l d t h e o r y . A s i m i l a r development can be c a r r i e d out f o r TZ t h e o r y [ 8 ] . N e g l e c t i n g terms of o r d e r V vn, eq.(4.3.9) y i e l d s y = a / dz > (7) where - L -r r , Due t o the n a t u r e of the a p p r o x i m a t i o n s made, the " l o c a l c o r r e l a t i o n " assumption f o r c ( r 1 f r 2 ) i s not needed; the q u a n t i t y V'tn) can i n a l l g e n e r a l i t y be taken t o be the second moment of the m o l e c u l a r c o r r e l a t i o n s . A " d i f f e r e n t i a l " TZ p r o f i l e e q u a t i o n s i m i l a r t o eq.(6) can be d e v e l o p e d . S i n c e t h e r e i s no adequate t h e o r y of c ( r , a ; n ) a t h i g h d e n s i t i e s [ 8 ] , such a p r o f i l e has not been s t u d i e d . 64 4.6 Long-ranged Correlations Although compatible with the mean-field description, the i n t e g r o - d i f f e r e n t i a l equations lead to a phenomenon not des-cribed by the mean-field formalism: the existence of long-ranged correlations in the interface region. The presence of these cor r e l a t i o n s in liquid-vapor systems was f i r s t noticed by Lincoln, Co, and Luks [39] for systems of "square-well" par-t i c l e s . A la t e r investigation by Wertheim [41] showed that the long-ranged correlations occur in the interface region, and are common to a l l liquid-vapor systems. a) existence of long-ranged correlations For a planar system with graviational potential Sf(r)=mgz, the p r o f i l e equations (4.1.6) and (4.1.8) become [41] = -fn.,(ii)c;(z,tztl)alz-i (1) = -/3 h n j f u i ( z l j i i ) d z 2 . (2) In the l i m i t g-*0, the value of n 0(z) remains non-zero in the interface region (sections 4.2-4.3). The integral on the right side of eq.(2) must therefore diverge as g"', indicating the growth of long-ranged c o r r e l a t i o n s . Another approach i s via the v a r i a t i o n a l d e f i n i t i o n of the 65 cor r e l a t i o n functions (appendix A), which lead to f 6 (-0 *(?.)) d F = f c - f F ^ . j u ^ K i . r . J ^ , (3) &»{r3) &(-/3&(F3)) J = <5 ( r, - F3 ) The corresponding Fourier transforms also obey the "inverse" r e l a t i o n [20] (4) Expanding the transforms in powers of k, and equating the terms in k° and k* , (6) the. symmetrical relations of the c o r r e l a t i o n moments [20]. Using eqs.(4.3.9), (2) and (5), (7) Since v remains f i n i t e as g-»0, the integral on the right side of eq.(7) must diverge as g"a. Eqs.(2) and (7) indicate the exis-tence of long-ranged correlations of co r r e l a t i o n length a divergent .quantity. The behaviour of the correlations can be described in the l i m i t g-0; eqs.O) and (2) become the "eigenfunction" equations 66 The eigenf unctions nj (z)/[ n0( z) ] / < J and [ n ; ( z ) ] v * are assumed to have a d e f i n i t e l i m i t as g-*0. Further, both n„(z) and [n<;(z)]* are assumed to be integrable. The solution n 0(z) is assumed to be unique. Since the kernel K(z,,z 2) i s symmetric, H i l b e r t -Schmidt theory [46] yiel d s where fiz) i s the eigenfunction of the equation. Eqs.(8) and (9) then y i e l d C O ( E , , 2 J ) = * » ( ' * > (11) I3 ma(z.) »0 ( z aj u.a e ?j = (12) /3 my £ H where Since u ( r , , r 2 ) i s not pointwise divergent, the divergence of u 0 ( z , ,z 2) as g-*0 must result from long-ranged horizontal corre-l a t i o n s . Long-ranged v e r t i c a l correlations are ruled out by the similar divergence of both u 0 ( z 1 r z 2 ) and i t s integral over z, eq.(2). Therefore, eq.(12) describes horizontal correlations r e s t r i c t e d to the interface region, where np(z)=0. From the non-divergence of eq.(1l), these correlations must be non-local in o r i g i n . 67 b) Fourier transforms E q s . ( l l ) and (12) can be considered the k=0 l i m i t s of the two-dimensional Fourier transforms cCR,z 1 fz 2) and u ( E , z 1 f z 2 ) . The form of these transforms (for the case g-+0) can be calcu-lated by rewriting eqs.(l) and (2) as *0(z,) -/3*y,j f [ u (l.z.i, ) - u0^.jsi)~j dzf = ~/3>~>CJ { u. (t^l/^t )dzI .( 1 4) In a manner similar to that of part (a), these can be rewritten in the "eigenfunction" forms similar to eqs.(8) and (9); the Hilbert-Schmidt equation (10) leads to * /3 r>,j I an */3 yy,jf [ k* czle,tgt) + ...] eli,6lii (16) Using eqs.(4.3.9) and (7), these become, to order k a [41] c < * ' « " * « ) - h . C ^ . t w I l ' ^ J ( 1 7 ) Ho (i.) M „ ' ( 2Z) (18) The g-»0 l i m i t of eq.(l8) i s u(ki„*t)= (19) The amplitude of eq.(l9) goes as k^T/^ , suggesting that the 68 long-ranged correlations have their o r i g i n in c a p i l l a r y surface waves [35]. c) simulation The existence of long-ranged correlations in the interface was f i r s t shown in a direc t fashion by*Kalos, Percus, and Rao [35], who v e r i f i e d eq . ( l 9 ) by computer simulation. The quantity u(Tt rz,,z 2) i s determined via the rel a t i o n < W*,8„*,) = « ( R , H . . O + * ( i . ) S ( i , - i a ) . (20) To calculate horizontal c o r r e l a t i o n s , consideration i s given to a f l a t , horizontal subsystem of thickness Az. For Az small, eq .(20) y i e l d s * HU)*^[*lLfe±L + l i ] . (21) The quantity computed i s < Z L it-L?,-^)))/(N(az)) = TTuT^i ] " V * , * , , ? , ) ^ , ^ J (22) with i and j running over the N(Az) p a r t i c l e s of the subsystem, and brackets denoting the average over a l l configurations. Combining eqs . ( 2 l ) and (22), u ^ z , j ) = ^[((%**ptiK-lr.-'-Fjfi>/<Niaz})) - i j (23) The simulated system was composed of 1024 "Lennard-Jones" p a r t i c l e s , confined to a container of dimensions 6.78crx6.78o-x56.24<r. The reduced temperature T*=kBT/e =0.7, cor-69 r e s p o n d i n g t o t h a t of an a r g o n f l u i d n e a r i t s t r i p l e p o i n t . The m o l e c u l a r d y n a m i c s s i m u l a t i o n ( s e c t i o n 3.3) was u s e d . The r e s u l t s o f s i m u l a t i o n a g r e e w e l l w i t h e q . ( l 9 ) . W i t h i n t h e l i q u i d p h a s e , t h e low-k d i v e r g e n c e i s a b s e n t ; a t m i d - i n t e r -f a c e ( z = z w ) , t h e low-k d i v e r g e n c e does i n d e e d a p p e a r ( f i g u r e 9 ) . F o r t h e s y s t e m s t u d i e d , t h e r e d u c e d q u a n t i t i e s c a l c u l a t e d a r e >-*=0.7, and n* ( z m ) = 0 . 35 . The g r a d i e n t i s n o t a c c u r a t e l y d e t e r -mined a t z^, but s e t t i n g n*'(z m) = 0.50 ( w i t h i n t h e m a r g i n of e r r o r ) c a u s e s t h e c u r v e of e q . ( 1 9 ) t o a g r e e w i t h t h e r e s u l t s of s i m u l a t i o n ( f i g u r e 9 ) . A ^ f r O I 2 k f i g u r e 9 - low-k d i v e r g e n c e o f h o r i z o n t a l c o r r e l a t i o n s The c o e f f i c i e n t o f e q . ( l 9 ) i s n o t p r e c i s e l y d e t e r m i n e d , but .the 1/k 1 d i v e r g e n c e o f t h e c o r r e l a t i o n s i s w e l l e s t a b l i s h e d by t h e s i m u l a t i o n . 70 Chapter 5 Fluctuation Theories This chapter reviews several " f l u c t u a t i o n " theories, which determine the effect of imposing density fluctuations upon an " i n t r i n s i c " liquid-vapor system. Of p a r t i c u l a r concern are the e f f e c t s of interface fluctuations upon the surface tension and interface width. The simplest fluctuation theory, that of Buff, Lovett, and S t i l l i n g e r [11], i s presented in section 5.1. The more refined approach of Weeks [12] i s presented in section 5.2. Both of these theories consider the fluctuations as composed of c a p i l -lary waves t r a v e l l i n g along the interface. In addition to a l -tering interface width'and surface tension, the c a p i l l a r y waves are shown to be the o r i g i n of the long-ranged correlations d i s -cussed in section 4 . 6 . Section 5.3 presents the theory of Sobrino and Peternelj [13], which analyzes density fluctuations via an eigenvalue problem. Corrections to the surface tension from fluctuations other than c a p i l l a r y waves are described. The i d e n t i f i c a t i o n of the " i n t r i n s i c " system with that described by the "mean-field" theories i s made. A brief discussion of the r e l a t i o n amongst the various f l u c -tuation theories i s given in section 5.4. The r e l a t i o n between the mean-field and fluctuation theories is also b r i e f l y exa-mined, showing how the two classes of theory combine in a consi-stent way to describe the various properties of the liquid-vapor 71 interface 5.1 B u f f - L o v e t t - S t i l l i n g e r Theory a) general formulation One of the e a r l i e s t fluctuation theories was that of Buff, Lovett, and S t i l l i n g e r [11], who studied the effect of c a p i l l a r y waves upon a bare interface. The theory assumes a system with an i n t r i n s i c surface tension %, and Gibbs surface zN set to zero. The area of the bare interface i s denoted by A 0=LxL. In the Buff-Lovett-Sti11inger (BLS) approach, each f l u c t u -ation i s assumed to be associated with a d i s t o r t i o n g(s) of the dividing surface. It i s further assumed that Q ( S ) can be repre-sented by decoupled harmonic waves (cf. section 4.3) SCS) = 2 « B [ <*-s] (4.3.4) where T - ^ ( ^ « T 4- y>,j) ; * ? X / ^ (r & . (4.3.5) The work required to create a small density fluctuation is thus The density fluctuations of the system lead to a " c a p i l l a r y wave" p a r t i t i o n function Q c = KJTT al?(k)ex?> [ f / Xttj/2 (74 k* + )] } (2) where K i s a normalizing factor. 72 The c a p i l l a r y waves imposed upon t h e d i v i d i n g s u r f a c e g i v e i t a w i d t h wc . Due t o t h e f l u c t u a t i n g n a t u r e of t h e s u r f a c e , a n a t u r a l measure of i t s w i d t h i s t h e rms v a l u e w"-"'. The "10-90" w i d t h w* = 2.56W""f. S i n c e t h e waves a r e d e c o u p l e d , [6] u/-t* = ( u / - c " " ' J 2 = < = r (l?(U)l*) = J z a S ( r^jar, + -y0 k*)~' . (3) U s i n g t h e r e l a t i o n 1 =An„ An y = ( L 4 f f ) J Ak x A k y , and l e t t i n g L become l a r g e , e q . ( 3 ) y i e l d s an i n t e g r a l , y i e l d i n g t h e upper and l o w e r c u t - o f f wave numbers d e n o t e d by k + and k. r e s p e c t i v e l y . The v a l u e s of t h e s e two p a r a m e t e r s a r e not f i x e d by t h e t h e o r y ; t h e y must be d e t e r m i n e d f r o m o t h e r c o n s i d e r a -t i o n s . The s u r f a c e t e n s i o n of t h e i n t e r f a c e i s m o d i f i e d by t h e c a -p i l l a r y waves. E v a l u a t i n g t h e i n t e g r a l s o f e q . ( 2 ) , Q c = /< 77" f -£l*^L ( ^ A H + ra k ')] (5) where j i s a m u l t i p l e of a h a l f - i n t e g e r , w i t h a v a l u e d e p e n d i n g upon t h e d e f i n i t i o n o f K ( i . e . , t h r o u g h d i m e n s i o n a l a n a l y s i s ) . The f r e e e n e r g y o f t h e s y s t e m i s F= ~ <?& = -LJZJI + -L- S L* [/3(AO)J (r*aA* + K.k>j\ (6) /3 /3 2/3 T< L ' J -C a l c u l a t i n g t h e s u r f a c e t e n s i o n v i a e q . ( 2 . 5 . 1 5 ) , and u s i n g e q . ( 4 . 3 . 5 ) , 73 Identifying the f i r s t term on the right side of eq . ( 7 ) with the i n t r i n s i c surface tension, the correction A'K to surface tension becomes The proper normalization of eq . ( 5 ) i s not determined by the theory. The o r i g i n a l study by BLS set j=-. Later, Davis [47 ] took j= T/z ,'and Evans [ 20 ] set j = - 1 . b) interface width and surface tension The e f f e c t of density fluctuations upon the bare interface cannot be precisely determined u n t i l k + and k_ are specified.. This s p e c i f i c a t i o n i s independent of BLS theory; several d i f f e r -ent choices of the cut-off parameter have been made. In the o r i g i n a l treatment by BLS, k + was chosen to be inver-sely proportional to the interface width w (k= w/w'"•") in order to "maintain the i n t e g r i t y of c a p i l l a r y waves as c o l l e c t i v e coordinates" [ 1 1 ] , The value of k- i s taken to be zero. Since BLS consider the bare interface to have no width ( i . e . , i t i s a sudden discontinuity in density), the width w is just that of the dividing surface (wc ). This leads to a transcendental equation for wc (and thus for k + ) . A l a t e r adaptation (to f i n i t e interface areas) set k_=2rr/L [ 4 7 ] , since t h i s i s the smallest non-zero wave vector which can be sensibly defined. 74 Davis [47] set k^=2w/w0, where w0 i s the i n t r i n s i c width of the interface, assumed to be that of mean-field theory. The lower cut-off wave vector k_ was set to 2n/L., Evans [20] set k+=2n/a-, where <r i s the "Lennard-Jones" mole-cular diameter (section 3.1). The value of k_ was 2rr/L. Such a set of cut-off parameters is independent of temperature. Due to the logarithmic character of eq.(4), a precise speci-f i c a t i o n of k + i s not required to determine wc ; a l l of the pre-ceding s p e c i f i c a t i o n s lead to approximately the same value. For example, suppose L=1cm, ?£ = 50 dyn/cm*, T=300K, g=0, and w0 = 5A\ The BLS sp e c i f i c a t i o n s y i e l d wc=4.8A; the sp e c i f i c a t i o n s of Davis lead to we=4.7A" [47]. Away from the c r i t i c a l point, tT(^?) = 1 (section 3.2); the l i m i t s chosen by Evans w i l l also lead to approximately the same result for wc. This i n s e n s i t i v i t y to k + i s largely due to the dominance of larger wavelengths; over 75% of the contributions to wt come from waves of wavelength larger than 100Wo [47], Equation (4) has the two l i m i t i n g forms: i) g=0, L*oo S = 4*V. ( f c ) ' (9) i i ) g*0, L= oo ^ - £ggr) • (10) The divergence of eq.(9) as L-*a> r e f l e c t s the dominance of the 75 l a r g e r wavelengths i n d e t e r m i n i n g the i n t e r f a c e w i d t h . T h i s d i v e r g e n c e i s removed i n e q . ( l O ) , s i n c e i t now r e q u i r e s work t o d i s p l a c e the Gibbs s u r f a c e a f i n i t e amount. I f T=300K, mAn=1 g/cm 3, w0=5A, and g=980 cm/sec*, the c u t - o f f parameters of D a v i s t o g e t h e r w i t h eq.(9) y i e l d wc=4.68&. I f both g-*0 and L-* oo, then wc b e g i n s t o d i v e r g e l o g a r i t h m i -c a l l y . T h i s i s not a r i g o r o u s r e s u l t , however, s i n c e e q . ( l ) i s v a l i d o n l y f o r s m a l l f l u c t u a t i o n s . A d a p t i n g f l u c t u a t i o n t h e o r y t o accomodate l a r g e , f i n i t e f l u c t u a t i o n s has not y e t been accom-p l i s h e d [ 4 7 ] . The c o r r e c t i o n t o the s u r f a c e t e n s i o n ( e q . ( 8 ) ) remains f i n i t e as g-*0 and L-*a>. In c o n t r a s t t o wc , the s u r f a c e t e n s i o n . c o r r e c -t i o n A-yc i s v e r y s e n s i t i v e t o the s p e c i f i c a t i o n of k +; f u r t h e r , the c o r r e c t value- of j cannot be de t e r m i n e d w i t h BLS t h e o r y . The v a l u e of LVC i s t h e r e f o r e not w e l l e s t a b l i s h e d by c a p i l l a r y wave t h e o r y . For argon near i t s t r i p l e p o i n t , the h i g h e s t v a l u e of Ayc i s t h a t c a l c u l a t e d by Evans: A>£=32 dyn/cm* [ 2 0 ] . C l e a r l y , the c o r r e c t i o n A>£ has a s i g n i f i c a n t e f f e c t upon the s u r f a c e t e n s i o n ; a p r e c i s e d e t e r m i n a t i o n of ky. i s ne c e s s a r y i f BLS t h e o r y i s t o make a c c u r a t e d e s c r i p t i o n s of the i n t e r f a c e . The g r a v i t a t i o n a l c o n t r i b u t i o n '/z mgAnw* i s of the or d e r of 10"'1 dyn/cm* f o r g=980 cm/sec* and the f l u i d away from the c r i t i -c a l p o i n t ; t h i s c o n t r i b u t i o n can u s u a l l y be n e g l e c t e d . 76 c) long-ranged correlations To determine the possible contribution of the c a p i l l a r y waves to the long-ranged correlations in the interface (section 4.6), consider a liquid-vapor system with i n t r i n s i c p r o f i l e n 0 ( z ) , in a g r a v i t a t i o n a l f i e l d g. Applying a perturbation A¥(r) to the external p o t e n t i a l , the change in energy i s 4 f = -y,(A-A,)* ^ ^ y 4 ^ J / ?(s)/2 ols + fhji-t(S))&V(?) dF = 2 . / 5 Y / U / * ( mj&n * -y0 k*) + J *Jl) A<*(F)ctF - <p n9'{t)A<e(F) expl-ik-j] dr t for small disturbances of the system. The p a r t i t i o n function of the system changes by a factor AQ AQ r K 'J TT df(H) z*P I - ££± ( lf(t<)lz ( *r,jA» + r„k * )) -X? Z: ? (K) f r,'B (2) a. <?(F) e x o £ -itr.-s J dr] t P L 3.A0(»»j&* + Kk*) •* -> ( 1 2) where K' and K" are normalization factors. The change in free energy is therefore /3aF = -l*AQ = v» + /*•*• (S»;(*iav<F>**pL£*-H)x ( 13 ) From the v a r i a t i o n a l d e f i n i t i o n of the U r s e l l function (ap-pendix A), ( ' z ) "/3» 6U(r.jSU(rJ ~ i/3** * Kk* ( 1 4 ) 77 and therefore [20] for small k, " ' ' " - J ^ ' ( ? ! ) (15) Matching terms in powers of k with eq.(4.6.l8), i t can be seen that the long-ranged horizontal correlations have their o r i g i n in c a p i l l a r y waves [20]. 5.2 Column P a r t i t i o n Function A more detailed model of interface fluctuations was given by Weeks [12]. It divides a container of dimensions LxLxL into M v e r t i c a l columns of height L and area AxAy=La'/Ma'= A x , where A is set to be approximately equal to the co r r e l a t i o n length of the p a r t i c l e s in the l i q u i d phase. When a l l columns have the same number of p a r t i c l e s , a " f l a t " surface i s formed; the p a r t i c l e d i s t r i b u t i o n i s that which mini-mizes F. The effect of p a r t i c l e fluctuations upon th i s surface is examined through the use of the column p a r t i t i o n function (CPF). a) column p a r t i t i o n function Each column i s assumed to have an i n t r i n s i c interface width wp and surface tension yp . The location of the center of column (ab) i s S a i = CLJ. t v bAJ ; a.,b £ ~Z The number of p a r t i c l e s in column ( i j ) i s denoted and the 78 corresponding Gibbs surface for the column i s = | j N;i/vP - _ j, 7 (1) 'J 1 ^ L - a. J where V P= / L . The shape of the dividing surface i s given by the set of occupation numbers {N,y}; when no fluctuations are pre-sent, the surface is " f l a t " , corresponding to the set {N}, where N=N/Ma. The "column p a r t i t i o n function" of such a system can be writ-ten where Q[{N,y}] is the product of the p a r t i t i o n functions of the single columns, and the prime indicates that the summation i s subject to the constraint £ / A / / , - / = A / . (3) •J Weeks assumes that fluctuations in the bulk phases do not aff e c t the surface tension; the only effect upon -y i s due to fluctuations of the interface. Thus Qcol - Q L (NiJ ^ , (4) where Q[{N}] i s the p a r t i t i o n function of the " f l a t " interface system, and AQX = X ' exp [-/? ( <y { (AT.-J)) - r(f*}))l-*J , (5) where >/({N,y})L* denotes the energy of the configuration corres-79 ponding to {N,-,.}. The work required to form a distorted (dividing) surface s(x,y) i s (cf. eq.(4.3.8)) for small d i s t o r t i o n s . Only the nearest-neighbour columns are considered; thus g.. i t =£,v<s y.„t i s subject to the r e s t r i c t i o n l6l + U/=], The last term on the right side of eq.(6) i s twice as large as that used by Weeks; however, i t seems plausible to use the average of the squares of the p a r t i a l d erivatives. Because the horizontal dimensions of the columns have appro-ximately the same size as the c o r r e l a t i o n length of the mole-cules in the l i q u i d phase, the surface of each column should not have any s i g n i f i c a n t d i s t o r t i o n s ; eq.(6) should be quite ac-curate for describing the contributions due to surface fluctua-tions. From e q s . ( l ) , (5), and (6), where H^ =N,y-N. To evaluate AQX , use i s made of the normal coor-dinates H = £ H.j exp [ - i ( s(k„ + S;k~)] Htj = 4" e*pL < ( * S ;*-)J (8) 80 where the wave vectors are of the form * i * ; - ^ « * < 4?- . (9) The constraint of mass conservation is represented by exclusion of the k=0 component. Since S /v * = E //v_/ a (10) and = . - ^  (/-ex / 3 [ - ^ ^ ^ ^ / < « ) J j(ex / > [ - , ( s , / c»,^ >^;jJ j(ll) equation (7) becomes where YCkP ) = 4-2 [ cos ( k^i')+cos ( k h ^  ) ]. From eqs.(4) and (12), the column p a r t i t i o n function can be written QcoL = Q l l X t n r [ ^ J f n * K Yi**))]"'* ( 1 3 ) 81 b) density p r o f i l e The effect of fluctuations upon the i n t r i n s i c p r o f i l e i s de-termined by examining the behaviour of a column far from a wall. The probability of column (ab) having =H i s [ 12 ] P ^ . t - < 6 ( Hub - H j ) . (14) Since £ ( H ) cc J &xp [ c H<*] du - (15) the pr o b a b i l i t y is «J<*« & (^S^ay^j)(-i^-)-^ L-'- H] (16) Via e q . ( 1 ) , the probability of the column (ab) having a Gibbs surface of height Z i s = /h^Z (17) where ^ * = -i - . 21 '— -y—- (18) For small k (which provides the major contribution to v^), Y{KJ?)*kzjez , and the CPF result ( e q . d 8 ) ) becomes i d e n t i c a l to that of BLS theory (eq.(5.1 . 3 ) ) . The CPF treatment i s e s s e n t i a l l y a refinement of the c a p i l -lary wave theory; the cut-off wave vectors ky. and k_ are again l e f t unspecified. Weeks takes k + = and k_=:?V/_ (eq.(9)), but these cut-offs are not implied by the theory. o 82 The density p r o f i l e produced by the fluctuations may be de-termined in the case of an " i n t r i n s i c " p r o f i l e of the form np (2) = hL j i - fy = ; 2 > *,>• (19) The "equilibrium" p r o f i l e n 0(z) i s therefore -co i = i ("L * »<) - ^ ^ - ^ ) e r f ( ? / i u , J . (20) c) correction to surface tension From eq..( 1 3 ) , A F m - M l h A Q x f ^ l T f i t t ^ J * * * ' * * * * * * * ) ! • ( 2 1 ) Thus, defining the surface tension in terms of the excess free energy, the small-k approximation y i e l d s - 4 f " £ „ '"/WfefV 1 ~3*nt'-Kk>J>2 (22) where k+=1I/j has been used in eq.(23). Neglecting the gravita-t i o n a l contribution (generally i n s i g n i f i c a n t away from the c r i -t i c a l point), the correction A~ye becomes - $*> I <•>(•&&) - • (24) 83 5.3 S o b r i n o - P e t e r n e l j T h e o r y a) g e n e r a l f o r m u l a t i o n A g e n e r a l t r e a t m e n t o f t h e r o l e o f d e n s i t y f l u c t u a t i o n s upon t h e " i n t r i n s i c " i n t e r f a c e was f i r s t g i v e n by S o b r i n o and P e t e r n e l j [ 1 3 ] . The s y s t e m under c o n s i d e r a t i o n i s c o n f i n e d t o a c o n t a i n e r of h o r i z o n t a l d i m e n s i o n s L x L , w i t h v e r t i c a l b o u n d a r i e s a t z=-B, z=A; t h e G i b b s s u r f a c e z ^ i s s e t t o z e r o . The f r e e e n e r g y o f t h e f l u i d i s assumed t o be of t h e " d i f f e r e n t i a l " form ( c f . e q . ( 2 . 4 . 3 . ) ) •F(F) = f (h(r)) + *C (V">* , ( 1 ) where <* i s a c o n s t a n t . I f t h e v a l u e o f oi. i s c h o s e n c a r e f u l l y , t h i s a p p r o x i m a t i o n i s q u i t e a c c u r a t e f o r d e t e r m i n i n g t h e i n t e r -f a c e s t r u c t u r e [ 4 8 ] , and has t h e a d v a n t a g e o f b e i n g a n a l y t i c a l l y s i m p l e . The f r e e e n e r g y of t h e s y s t e m w h i c h has a d e n s i t y n ( r ) c an be w r i t t e n F [ h ( r ; ) = F[n0(r)] ' j f , ,., »(r.) dr,/9 (2) where n 0 ( f ) i s t h e d e n s i t y d i s t r i b u t i o n w h i c h m i n i m i z e s F, and v(f)=n(f)-n0(r) i s a d e n s i t y " f l u c t u a t i o n " a b o u t n 0 ( f ) . The d e n s i t y n 0 ( ? ) i s i d e n t i f i e d w i t h t h e m e a n - f i e l d p r o f i l e of c h a p -t e r s 2-3. The f i r s t - o r d e r t erm o f e q . ( 2 ) v a n i s h e s b e c a u s e o f t h e c o n s e r v a t i o n of mass j V(F) olF = o . (3) 84 together with the constancy of the chemical p o t e n t i a l . Assuming that the fluctuations about n 0 ( r ) are small, the second term on the right of eq.(2) thus represents the correction AF[v] due to the fluctuation v(f). For the system under consideration, eq.(l) implies To determine the interface free energy FX , i t i s necessary to consider a system similar to that described above, but con-taining two uniform phases of l i q u i d and vapor, separated by a wall at z = 0 . Denoting the free energies of the l i q u i d and vapor phases by F t [ n L ] and F^fn,] respectively, the interface p a r t i t i o n function Q X can be written [ 1 3 ] where the fluctuations and free energy corrections are defined analogously to those of the entire system. In a manner similar to CPF theory, the fluctuations considered in eq.(5) are not continuous; rather, they are fluctuations in the occupation num-bers of "van Kampen" c e l l s of dimension JPXJ?X.J2, where Jt. i s of the order of a bulk co r r e l a t i o n length. The treatment i s e s s e n t i a l -ly a three-dimensional generalization of the CPF approach. Since the (interface) excess free energy density i s equal to the surface tension when z/V = 0 (section 2.5) AFivJ = £J [ fg}) V*(P) * 5 OL [ X7v]* J dp (4) e x 85 r[»0J - FL ZV,4} - F« = -y0 L* , (6) where % i s the mean-field surface tension. From eqs.(5) and (6), therefore, - L«l £ exp{-/3&Fi L^J]]- exp [-/SaF^t^]]]] m (7) The Sobrino-Peternelj (SP) theory evaluates eq.(7) by expres-sing the v(r) as a series V(FJ = (^ ) ; (8) the ^ ( r ) being orthogonal eigenfunctions of the eigenvalue problem f " U ? ) ) ^ - 2 oc <Pp - Xp <=Pp ; Z-Vf,, =0 , (9) where n i s the normal of the container wall (cf. section 2.0). The eigenvalues and eigenfunctions are of the form [13] A,,* = • M i J - 2 ^ i l i x l j * *r + , (10) <fiJk = y A ( i ) co,(±L2L)c.oS ( J Z L ) . (11) where vk and yk result from the eigenvalue problem ?"(»UJ) rk (a) - X « V A n ( t ) - VA-ft/i) . (12) / The corresponding eigenvalues and eigenfunctions of the l i q u i d and vapor phases are obtained in a similar fashion; the v£ and y£ result from [13] 86 V*h(*) - - v£ aj 0 3 ) = f " ^ , ; + ; ^ = f"fn,)+ . (14) Sobrino and Peternelj set i A 1^=L/i ;from eqs.(lO) and (11), i t is seen that this i s equivalent to setting ki. = ^/j ,k. = rr/L. Like the other fluctuation theories, these cut-off parameters are somewhat a r b i t r a r y . The density fluctuations v ( f ) , vu(f) and v ^ i r ) are expanded in terms of the appropriate eigenfunctions. The term which cor-rects the vk to account for eq.(3) does not depend on L [13], and in the l i m i t of large L, i t can be neglected. Thus, F r can be written ^ - ^ L Z + # f l E L"l& t"* + S<J} ( 1 5 ) where S.y = ^  i^^.y)] the prime indicating the exclusion of the i=j=0 term. The value of m i s the number of discrete points in the spectrum of the operator of eq.(9). The other values of k form a continuum, whose contribution to the free energy is given by Su [13]. 87 b) corrections to the surface tension To ascertain the eff e c t of fluctuations upon the surface ten-sion, i t i s necessary to specify the form of f' ( n ) . One choice i s the "Morse p o t e n t i a l " where .us* = ^  J" a?* /»7?;/de . (17) Setting the parameter f=1.3 yi e l d s a free energy density which produces a mean-field p r o f i l e of correct interface width and surface tension [13], The correction to the surface tension, A % , i s defined to be F r/L*-7i. The sums of eq.(l5) are evaluated f i r s t for k=0. The eigenfunction v0 i s v„ " JLZLSL. Am , which i s associated with the (normalized) eigenfunction The functions f./o thus correspond to c a p i l l a r y wave fluctuations (cf. sections 5.1, 5.2), which leave the i n t r i n s i c p r o f i l e un-changed. The correction A % due to these c a p i l l a r y waves i s [13] *V* - ^ [<•- * -+ J^%-[Lirx j>u + 6l<kL) + > (18) 88 where x=Vwa, G a = , e, =0.090, and e i = 1.22. The la s t term of eq.(18) represents the contribution A due to gravity, and i s generally n e g l i g i b l e away from the c r i t i c a l point" (section 5.1). For the "Morse p o t e n t i a l " model, there are two discrete e i -genvalues. As discussed above, the f i r s t (k=0) eigenvalue cor-responds to c a p i l l a r y wave fluctuations. The second eigenvalue corresponds to (localized) fluctuations of the p r o f i l e i t s e l f (figure 10). Together with the corrections stemming from the d i s t o r t i o n of the fluctuations due to the presence of the i n t e r -face (corresponding to the continuous range of the eigenvalues), these fluctuations y i e l d a "non-capillary" correction Ayn [13] A h, O figure 10 - fluctuation of the i n t r i n s i c p r o f i l e ( 1 9 ) 89 For an argon-like mean-field f l u i d near the t r i p l e point (T= 8 4 K ) , An = 0 .85o - " 5 , <r=3.4A, % = 18 dyn/cm* , and wa. = 0 .57cr [13]. Estimating the co r r e l a t i o n length Jl =3<r, the surface tension cor-rections are found to be Ayc*-2.3 dyn/cm1, A?£*-2.4 dyn/cm*-[13]. Non-capillary contributions are therefore of importance when correcting for fluctuations. The behavior of &~rc is quite similar to that of the CPF re-sults (section 5.2) in which only the c a p i l l a r y fluctuations are considered. This i s to be expected, since both are due to the " c a p i l l a r y wave" fluctuations of the i n t r i n s i c interface. Indeed, the q u a l i t a t i v e behavior of Ay i s similar for both the CPF and SP theories (figure 11). figure 11 - behaviour of surface tension corrections 90 c) correlations and interface width Via eq.(8), the p a r t i t i o n function of the liquid-vapor system can be written Q = K exp [ -/3F Cr>0l] JJT dap exF [ -/*/z a* Ap ] } (20) where K i s a normalization factor. Since the eigenfunctions are orthogonal, = " J " Srr' > ( 2 1 ) and therefore = Z: ( (22) k x ' k Because the corrections to the \ p due to eq.(3) can be neglected for large L, <v(?.)»(r,)\ - ^ TAI*.)-**^) ^ T (*£*>™W . (23) Evaluation of eq.(23) for k>1 y i e l d s [13] ( W ' ' ( l , ; * ^ T ^ l WT^JU^l) > (24) showing that the horizontal c o r r e l a t i o n length of these terms is of the order of £ , the bulk c o r r e l a t i o n length. The case k=0 requires separate consideration, since vQ vanishes as g-»0. Careful analysis shows [13] (\>(r,)^(r,)> = ^ [ S ( l ? . 4 ! , i ) + Sds,-?,,)) , (25) 91 where S(.-x) = S l c o s 1 i 7 r n C^ L) . c = _k r^£ \ V C L - > V i ' ' J a« The " v e r t i c a l " correlations of the k£1 terms are very small [13]; thus, for distances far beyond the bulk correlation length, <v(f)v(r)>~<v(f)v(f)>0 . The correlations can be calculated for certain l i m i t i n g cases. For the case g->0 (L a«G~*), Sfrx) - -f-* ( 1 - J2L + *± - ^ L. L» ( I - exp [-air-x/Ll) , (26) and thus eq.(25) leads to long-ranged correlations limited only by the size of the walls. In the l i m i t L-» co, the SP theory y i e l d s S(^) * I^==. ^ p [-re,*] (27) which has a cor r e l a t i o n length of (>£/mgAn) a, i d e n t i c a l to that of the " c a p i l l a r y wave" theories (cf. sections 4.6, 5.1). The ef f e c t of fluctuations upon the interface width is deter-mined by examining the displacement Q(S) of the dividing sur-face , = L*'*»* J" d P • ( 2 8 ) Since only the k=0 eigenfunctions have long-ranged correlations, * i h ^ J < W F J wp a ;> a , (29) i . e . , fluctuations other than those of the c a p i l l a r y waves have l i t t l e e f f e c t upon the - interface width. 92 For the case g-»0 (L 3«G~ a), eqs.(25), (26), and (29) y i e l d [13] '<**> - S ^ r . (30) which for large L i s v i r t u a l l y i d e n t i c a l to that of the BLS and CPF theories (eq. ( 5 . 1 . 9) ) . For the l i m i t L-»OD, eqs.(25), (27), and (29) y i e l d [13] which agrees well with the BLS and CPF results (eq.(5.1.10)) for small since SP theory sets k + = rr/^e . 5.4 Relation amongst the theories The "fluctuation theories" described in t h i s chapter form a class of theories which are e n t i r e l y compatible amongst them-selves; the differences which aris e stem largely from d i f -ferences in complexity of treatment. The BLS theory i s perhaps the simplest possible description of the 'effect of fluctuations upon an i n t r i n s i c p r o f i l e , treating them only as decoupled ca-p i l l a r y waves. The CPF theory follows the same general ideas, but i t i s more refined in i t s treatment, separating the system into an array of v e r t i c a l columns. This refinement leads to several added insights into the effect of c a p i l l a r y fluctua-tions. The SP theory c a r r i e s t h i s subdivision further, dividing the system into c e l l s . In addition to describing the effects of c a p i l l a r y waves, i t uncovers additional e f f e c t s due to p r o f i l e 93 fluctuations, and due to d i s t o r t i o n s of the bulk density f l u c -tuations, caused by the presence of the interface. The major source of disagreement among the results of the theories is the choice of cut-off parameters, which are independent of the theories themselves. The SP theory, although the most comprehensive of the f l u c t u -ation theories, does lack an element of generality, since the c o e f f i c i e n t of eq.(5.3.1) i s a constant. However, eq.(5.3.l) is a f a i r l y accurate approximation of f ( r ) ; a generalization of SP theory to non-constant c o e f f i c i e n t s would probably y i e l d only s l i g h t corrections to the main r e s u l t . The fluctuation theories describe corrections to properties of an " i n t r i n s i c " interface p r o f i l e . From section 5.3, i t i s seen that t h i s p r o f i l e i s exactly that described by the mean-f i e l d (or i n t e g r o - d i f f e r e n t i a l ) theories. The fluctuation theories are thus the second-order corrections of the mean-field r e s u l t s . This should be s u f f i c i e n t for the description of a l l liquid-vapor systems, except those near the c r i t i c a l point, where fluctuations become large and corrections beyond second-order terms are required. A t h e o r e t i c a l treatment which combines the two classes of theories in the manner described above yie l d s results in good agreement with simulation. For example, the MVDW value of the ( i n t r i n s i c ) surface tension of an "argon-like" f l u i d at T=84K is 21.6 dyn/cm3- (Table I I I ) . Adding the corrections of section 5.3 to t h i s figure y i e l d s an "equilibrium" surface tension of 94 16.9 dyn/cma, which i s in good agreement with the simulation value of 18.5 dyn/cm3. Indeed, because of the small interface area of the simulated system, t h i s simulation value i s too high; l e t t i n g L-?o°, i t should decrease somewhat, an eff e c t which has been observed (section 3.3a). The behaviour of interface width and surface tension as L-^oo in simulated systems i s in agreement with the results of the fluctuation theories [12][29]. 95 Bibliography 1) J.D. van der Waals; translation by J.S. Rowlinson, J. Stat. Phys., .20, 1 97 (1978). 2) T.L. H i l l , J . Phys. Chem., j56_, 526 (1952); J. Chem. Phys., 20, 141 (1952); J. Chem. Phys., 30_, 1521 (1959). 3) I.W. Plesner, J . Chem. Phys., 4_0, 1510 (1964); I.W. Plesner and 0. Platz, J. Chem. Phys., 48, 5361 (1968). 4) V. Bongiorno and H.T. Davis, Phys. Rev. A, J_2_, 2213 (1975). 5) S. Toxvaerd, J. Chem. Phys., _55_, 3116 (1971). 6) F.F. Abraham, Physics Reports, .53, 94 (1979). 7) F.F. Abraham, J. Chem. Phys., 63., 157 (1975). 8) V. Bongiorno, L.E. Scriven, and H.T. Davis, J. C o l l . and Interf. 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Pathria, S t a t i s t i c a l Mechanics , Pergamon (Oxford, 1972T; 50) J.K. Percus, in The Equilibrium Theory of C l a s s i c a l Fluids , H. Fri s c h and J. Lebowitz, eds., Benjamin (Reading, Mass., 1964). 51) C.A. Croxton, Liquid State Physics , Cambridge University Press (Cambridge, 1974). 52) L. Rosenfeld, Theory of Electrons , Dover (New York, 1965). ~~ 53) J.G. Kirkwood, J. Chem. Phys., _3, 300 (1935). 54) I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall (Englewood C l i f f s , N.J., 1968). ~~ 98 Appendix A Elementary S t a t i s t i c a l Mechanics  of the Liquid State The study of the thermodynamics of non-uniform f l u i d s re-quires a formalism which i s not widely known. This appendix reviews some of the basic concepts of s t a t i s t i c a l mechanics, developing from them the essential parts of th i s formalism. A .1 Density Functions a) canonical ensemble A system consisting of a large number N of p a r t i c l e s of mass m, interacting via a potential U(r , ,? 2, . . . , r N )=U(r,/A/) , has a p a r t i t i o n function of the form [49] QN = •fijj^z j <s.xp [-/3 ( p«-Va.*n •+ U ( dF,/N alpl/JV where Z „ i s the N-particle configuration i n t e g r a l , and A =h/(2ffmkQT)'/a- is the thermal wavelength. Since the pro b a b i l i t y of a system to have a given s p a t i a l configuration i s proportional to exp[-/JU( r, / v) ] , the (canonical ensemble) s - p a r t i c l e density function can be defined »"y">> - TN^TJT —-This function gives the p r o b a b i l i t y of simultaneously finding ( i d e n t i c a l ) p a r t i c l e s at each of the positions F, , r 2 , . . . , r s . 99 b) grand canonical ensemble A system of fixed volume V and temperature T, open to par' t i c l e exchange, has a p a r t i t i o n function of the form [49] Q = _ _ ± ^ — exp [ -/3 JUL N J A/ A / / _A 1 A / A y /v I <T where / * i s the chemical p o t e n t i a l , and ^ =exp[/3/-<-31nA] i s the fugacity of the system. The (grand ensemble) s - p a r t i c l e density function is defined by h ( r , A j = 2: \n„ ( F,/s ) P(A/) . U) A / ^ S The p r o b a b i l i t y P(N) = Z*> ; thus eq.(4) can be rewritten A/7 3 m(^>/t) = "S. > J V m^ir./s) Z „ . (5) The s i n g l e - p a r t i c l e density i s normalized to J *(F) - < A / > , (6) the brackets denoting the ensemble average. 100 A.2 D i s t r i b u t i o n Functions  a) general formulation The s t a t i s t i c a l description of the l i q u i d state i s accomp-lished via the concept of c o r r e l a t i o n , which is given mathemati-c a l l y by the U r s e l l c o rrelation function [50] u (7<, Fi) - * ( p< - P2) - * (f •) * ( r 2 ) O) or equivalently, by the two-particle density function n(r, ,r 2) i t s e l f . Far beyond the range of the molecular potential, the s p a t i a l ordering deteriorates, i . e . , Linn r, , rz ) = tolr,)^(rz) } (2) and the correlations tend to zero accordingly. The functions n ( r , , r 2 ) and u ( r , , r 2 ) have important roles in the description of the thermodynamic properties of the l i q u i d state (see below). However, i t i s often more convenient to work with their dimensionless counterparts instead. The f i r s t of these i s the s - p a r t i c l e d i s t r i b u t i o n function [51] j(Fl/s) = . (3) J 7T to(F;) i When g ( r , , r 2 ) depends only upon the distance r / z=|r,-r 2|, i t i s known as the r a d i a l d i s t r i b u t i o n function g ( r y i ) . A represen-tative graph of g(r 7 i) for a uniform f l u i d i s shown in figure 12. 101 z 9 ("•.«> f i g u r e 12 - t y p i c a l g r a p h of g(r,^) f o r u n i f o r m f l u i d A few g e n e r a l r e s t r i c t i o n s on g ( r ( i ) a r e [ 5 0 ] i ) g ( r , J > 0 i i ) l i m g(r,, ) = 1 i i i ) g ( r , J = 0 ;r,z< 5 i v ) g ( r ; i ) > 0 ; r / 4 > r ^ where S i s t h e m o l e c u l a r d i a m e t e r , and r w i s t h e mean i n t e r - , m o l e c u l a r s p a c i n g . The d e t e r m i n a t i o n o f t h e a n a l y t i c f o r m of g ( r / z ) has n o t been y e t a c c o m p l i s h e d . One common a p p r o x i m a t i o n scheme [17] i s t o s e p a r a t e g ( r ^ ) i n t o j(rn;r>) = y ( ru ; ») exp C-/3 U( r,,)) (4) where y ( r y i ; n ) i s t h e P e r c u s - Y e v i c k (PY) f u n c t i o n f o r a u n i f o r m f l u i d o f d e n s i t y n. An a n a l y t i c e x p r e s s i o n f o r y ( r , a ) c an be o b t a i n e d f o r c e r t a i n c l a s s e s of i n t e r m o l e c u l a r p o t e n t i a l s , i n c -l u d i n g t h e " h a r d - s p h e r e " p o t e n t i a l [ 2 6 ] . The PY a p p r o x i m a t i o n ( e q . ( 4 ) ) i s i n good agreement w i t h m o l e c u l a r d y n a m i c s s i m u l a -t i o n s on " L e n n a r d - J o n e s " f l u i d s ( s e c t i o n 3.3) [ 3 3 ] . 102 The second dimensionless function i s the pair correlation funct ion = j(7,,?t) - i . (6) When h ( r , , r 2) =h ( r ; i) , i t is known as the r a d i a l c o r r e l a t i o n func-t i o n ; from eq.(6), i t s properties are seen to be e s s e n t i a l l y the same as those of the r a d i a l d i s t r i b u t i o n function. b) r e l a t i o n to thermodynamic properties To establish the manner in which the thermodynamic properties of a system depend upon the d i s t r i b u t i o n function, consideration is given to the general theorem [49] ^ > - S^k.T , (7) where H is the hamiltonian of the system, and i s one of the 6N components of phase space. For a system of non-interacting p a r t i c l e s in zero external f i e l d , H0 = £ P i ' / a * . (8) Setting x,=p-, and summing eq.(7) over i , a <H0) = 3 < A/> kBT . (9) If the potential U(r,z) can be expressed as a sum of pair poten-1 03 t i a l s , the t o t a l energy can therefore be written E = l<N>kBT * £ f»(Fj*(?z) j l P t f J U(r,t) et?.,, . (10) Setting x, =r,; leads to the v i r i a l theorem of Clausius, which expresses the v i r i a l v of the system by [49] cr * < -Z. F„ - ^  ) = kBT . (11) The walls of the container contribute F . - O - - p f ^ ' d A - -3pV , (12) while the internal forces TU(r l A v) contribute =f~ j exp [ /3 U( rl/AJ)] Plz • V e x fo [ -/3 U ( P >//v)J » (P,, Pt ; o l r , / z . (13) Equations (11), (12), and (13) combine to y i e l d the equation of state p = <M)kaT + fexpl/SUiP.yJ] ntr,)r,(?j jfF^P^-Vexp ( 14 ) For systems in which p a r t i c l e s interact v i a pair potentials, eqs.dO) and (14) show that knowledge of g(f V i) i s s u f f i c i e n t to determine a l l thermodynamic properties of the system. 1 04 c) Born-Green-Yvon hierarchy For a system composed of p a r t i c l e s interacting via pair po-t e n t i a l s , v i z . , a d e f i n i t e hierarchy can be established among the d i s t r i b u t i o n functions [51]. Via eqs.(A.1.2) and (A.1.5), = -/3 * I ?,,s) t?( U ( rl/s) -/3/N °Zs f~ lV-s)[vU\r,,?'st,)eKp[-/3U(?,,At)]Jr„L/(]5) ' W N>, 5 ( A / - 5 / / J Substitution of the expression for n(f, / s < /) into eq.(l5) yields the Born-Green-Yvon (BGY) hierarchy V,»(rt/S)= -/3*(r„s) v, U(FI/S) - / 3 J v , U( » ( r , / s . . ) drs+l (16) which describes the relat i o n s h i p of the density functions with each other and with the intermolecular p o t e n t i a l . Setting s=1 yiel d s the f i r s t member of the BGY hierarchy V.mtr.) => -/31 Vti 7Z) V, U(F,tr^) dF2 , ( 1 7 ) while setting s=2 yi e l d s y(F-,,rt) = -/3»(F., rT)X7,U (r,tFJ - /3 fr, U I r> { r l / 3 ) d?j ( 1 8 ) 105 For a complete theory of the l i q u i d structure, a closure re-lati o n on the n(r, / s) i s needed. One of the most common approxi-mations is the Kirkwood closure r e l a t i o n [53]. Defining a quan-t i t y w K ( r , / s ) = - kBr LH JIF,/S) , 0 9) yields d C F > / s ) ' Z^Je.ri^ui^u?^ • (20) The quantity -VW,< (?./.£• ) can be interpreted as the average force on a molecule at point r,, with the s p a r t i c l e s assumed to be held at fixed positions. Thus, W^(r,/f) can be interpreted as the mean-force potential among the s p a r t i c l e s [51]. The closure r e l a t i o n on n(r,/ 3) i s obtained by assuming i. e . , that the superposition p r i n c i p l e i s v a l i d when applied to the mean-force potentials. Therefore, via eq.(!9), (21 ) Substitution of this expression into eq.(l8) yields -kBT V, Itn j ( F,t »% ) = jllr,, Ft ) V, U(r,lFt) + f\rU(F,,Ft)g(?„Fi)*(rt)_t)(Fi.Fa)etFi (22) for uniform f l u i d s . In the region of low-to-medium densities, the Kirkwood 1 06 approximation i s quite accurate [51]? eq.(22) provides a de-f i n i t e r e l a t i o n between g ( r l f r 2 ) and U ( r 1 f r 2 ) . Therefore, at these densities, knowledge of either g ( f , , r 2 ) of U ( f 1 , r 2 ) i s s u f f i c i e n t to determine the thermodynamic properties of the sys-tem. A.3 Direct Correlation Function a) general formulation The d i r e c t correlation function c ( r 1 , r 2 ) i s defined by the Ornstein-Zernike (OZ) equation [51] h l r , , r j = c ( r , , r j + J C(r,,?t)y,(P3)ln(F3tFi) d F3 . (1) Recursive substitution of h ( r , , r 2 ) into the integral on the right of eq.(1) leads to hir,^) = c(FttFt) + f c(F,.?3) *(r-j) c( ?3,rt) dF3 + J C » (rj) c( ? , / v J n (Fv) c ( F V j rz) dF3/H (2) One possible interpretation of c ( r , , f 2 ) i s to consider the t o t a l c o r r e l a t i o n h ( f , , r 2 ) as the sum of two e f f e c t s . The f i r s t is a " d i r e c t " c o r r e l a t i o n , resulting from the " d i r e c t " interaction of p a r t i c l e s 1 and 2.-'*The second contribution i s " i n d i r e c t " , the correlations brought about by interactions with the other par-t i c l e s , averaging over a l l their possible possible positions [51]. The effects of the 3^body, 4-body,...,N-body interactions are summed up to produce the t o t a l " i n d i r e c t " e f f e c t , repre-sented by the integral in e q . ( l ) . 107 The OZ equation determines c ( r , , r 2 ) whenever h ( r 1 , r 2 ) i s given. Conversely, since the two functions have formally simi-lar roles, knowledge of c ( r , , f 2 ) i s s u f f i c i e n t to determine h ( r , , r 2 ) . Thus, the d i r e c t c o r r e l a t i o n function i s merely an alternate way to describe the structure of a f l u i d , and i s en-t i r e l y equivalent to the approach involving h ( f 1 f r 2 ) . b) r a d i a l d i r e c t correlation function When c(r,,r 2)=c(r,J, i t is known as the r a d i a l direct c o r r e l -ation function. Via the OZ equation,, the form of cCr^ ) can be determined for distances greated than the range of h(r ; i) [52]. For a uniform f l u i d of density n, the " r a d i a l " version of eq.(1) becomes c ( rn) = h(trlt) - »f h ( t FI3I) c ( v r n ) d F 3 . (3) Expansion of c ( r a ) in a Taylor's series y i e l d s Beyond the range of h ( r / i ) , eqs.(3) and (4) y i e l d P J e ( " « ) = V ' ( ^ « ) , ' (5) where it3- = * Hz / b (* HB + i) (6) and (7) 108 The normalized solution to eq.(5) i s 7<* 1rr r„ where CB = j~ c ( r,4 ) at\ (8) (9) Eq.(8) shows that x i s a measure of the e f f e c t i v e range of the direct c o r r e l a t i o n function. From eq.(6), t h i s range i s seen to be of the order of a few molecular diameters. The a n a l y t i c a l form of c ( r 1 r r 2 ) can be approximated by the PY equation [33] = J ( i »)(<**/>[ /3U(rlt)J - 1 ) ) (10) which i s accurate for densities below the c r i t i c a l density n c. A representative graph of c ( r , , r 2 ) for a uniform f l u i d i s shown in figure 13. figure 13 - t y p i c a l graph of c ( r ) for uniform f l u i d 109 A.4 Generalized Functions Just as the pair density n ( r 1 r r 2 ) i s regarded as a special case of n(r,, y), so can the correlation functions u (r r / i) and c ( r / / 2 ) be regarded as special cases of generalized correlation functions u(f, / / v) and c ( r , / w ) . These generalized functions, a l -though having a p r o b a b i l i s t i c interpretation, can be more clear-ly understood as functional derivatives of the free energy of the system. The functional derivative i t s e l f i s a generalization of the p a r t i a l d erivative. For any functional J f y J = / F (-K , y (-K), y V x / j ct* the functional derivative i s defined to be [54] SJ 1^1 = 2F _ _J_ i c)F \ (1 ) The rules governing the use of functional derivatives are analo-gous to those governing the use of p a r t i a l d e r i v a t i v e s . a) density functions Addition of an m-body potential to the hamiltonian of the system produces a p a r t i t i o n function [50] Q [ c p ^ ] = z: fexp £-/3U(F„„) -/3?*,(r„„)]cfr,,„ '(2) and s - p a r t i c l e density function 110 Setting m=s, and applying functional d i f f e r e n t i a t i o n to eq.(2), Taking the l i m i t of small perturbations For the canonical ensemble, InQ =-/3F; thus, the s-particle density functions can be written *"•«> • -sifKj • ( 6 ) The function n{r,/s) can therefore be regarded as a measure of the dependence of the free energy upon the s - p a r t i c l e potential ^(f./j) [51-3. Via eq.(A.2.3), the s - p a r t i c l e d i s t r i b u t i o n function can be s i m i l a r l y expressed as 111 b) U r s e l l functions The v a r i a t i o n in density induced by a small perturbation can be expressed by S*(F, 19.) _ _/ £ (to ( F, i y,; Q f yj) _ j. C HIK.)Q) S Q C y, J For a canonical ensemble, th i s can be put into the form = U + (F, ,FT) showing that the variation of F i s divided into a singular l o c a l component plus a non-singular non-local component. The U r s e l l function u ( r , , r 2 ) can therefore be interpreted as the non-local c o r r e l a t i o n contribution to the change in F due to simultaneous changes in potential at points r, and r 3 [ 5 0 ] ; i t is generalized in a natural way by defining u(f,/ s) as the non-singular part of ^(r,/s) =" f j f i — j [ygjr] . (10) The pair c o r r e l a t i o n function i s s i m i l a r l y generalized by defining = (jTTTF^)) "(?,,sJ . ( n ) 1 1 2 c) d i r e c t correlation function To provide a v a r i a t i o n a l d e f i n i t i o n of the dire c t correlation function, i t i s necessary to establish that the s i n g l e - p a r t i c l e potential U(r) i s uniquely determined by the equilibrium density n 0 ( r ) . For the canonical ensemble, th i s can be proved by con-t r a d i c t i o n [20]. Assume that there exist two potentials U(r) and U(r)+AU(r), which give r i s e to the same density n 0 ( r ) . Denote the respective free energies by F, and F a . From the Schwarz inequality i t follows that From t h e . d e f i n i t i o n of the p a r t i t i o n function, eq.(12) yields Addition of eqs.(l3) and (14) yiel d s a contradiction; U(r) must be uniquely determined by n 0 ( r ) . Conversely, n 0 ( r ) i s also uniquely determined by U(r). The two functions thus provide a l t e r n a t i v e descriptions of the system. Using eqs.(A.3.l) and (8), the identity F, < Fz - J *(FJ A U (?) olf? (13) (14) f 6(~/3ce(?1)) y i e l d s the r e l a t i o n 1 1 3 The "equivalence" of u ( r 1 r r 2 ) and c ( f , , r 2 ) for use in describing correlations (cf. section A.3) becomes clearer - the two func-tions (or more precisely, u*(r 1,f 2) and c"(f,,r 2)) are "inverses" of each other. Via d i f f e r e n t i a t i o n of eq.(6), eq.(15) becomes 6*(-/3F) = - Z7T7 , (16) showing that c ( r , , r 2 ) i s the non-local c o r r e l a t i o n contribution to the work of creating simultaneous density variations at points r, and ^ [ 5 0 ] . The dire c t c o r r e l a t i o n function can there-fore be generalized by defining i t as the non-singular part of I-/3F 1 . (17) Al t e r n a t i v e l y , the generalized d i r e c t c o r r e l a t i o n function can be defined by where c ( F, J = / Q f l r , ) * in [ J24£fjL] • ( 1 9 ) Eq.(!9) can be rewritten in the more transparent form h(FJ = -3. zxp I -/3 *P(F.) * c ( r , ) J , . ( 2 0 ) showing that -ksTctr,) i s the e f f e c t i v e s i n g l e - p a r t i c l e poten-t i a l due to the interactions within the system [ 2 0 ] ; i t i s t h i s 1 1 4 "additional" potential which determines the the equilibrium den-s i t y . 1 1 5 Appendix B On the Shape of the Equilibrium Density P r o f i l e The purpose of thi s appendix i s to show that planar density p r o f i l e s which minimize F - subject to the constraint of mass conservation - are either constant or else s t r i c t l y decreasing (as z increases) . Generalizing from mean-field theory, the assumptions made are i) the p r o f i l e n(z) is piecewise smooth i i ) f ( z ) - t ( n ( z ) ) > 0 , i . e . , gradient contributions are non-negative (cf. eq.(2.4.3)) i i i ) JX' (n) i s either positive for a l l n (T>TC ) or else i s negative i f and only i f n i s within a closed i n t e r v a l [n,,n,] for T<TC. The behaviour of Jj(n) is similar to that shown in figures 1 and 2 of chapter 2. If T<TC, then the Maxwell construction (eq.(2.4.8)) uniquely defines the values n<-(<n,) and n L(>n 2). The behaviour of the mean-field f l u i d i s thus encompassed by the /<(n) so defined. 1 16 a) constant density p r o f i l e Consider a system for which T>Tt , or n^V>N>ntV. Using assumption ( i i ) , the free energy F obeys the inequality F > Af . Define the average density n A by O = / [ri(z) - nA Join . Since the value of/5'(n) is always p o s i t i v e , (1) (2) (3) the equality holding only for n(z)=n A (figure 14). From f W F(") figure 14 - free energy inequality 1 17 eqs.(1), (2), and (3) F > A f f(nA) c/e , ( 4 ) the minimum value of F being attained only with the constant-density d i s t r i b u t i o n n(z)=nA. b) s t r i c t l y decreasing p r o f i l e Consider a system with T<TC , and n<;V<N<n(.V. Taking the den-s i t y of the lower phase to be nt>n,,, the p r o f i l e can have a form l i m i t e d only by assumption (i) and the boundary conditions. A representative p r o f i l e i s shown in figure 15 + 1 — > figure 15 - p r o f i l e construction From figure 15, i t is evident that i f the p r o f i l e i s not s t r i c t l y decreasing, i t i s possible to define a density n x, and points z, and z 2 such that 2, h(i,) = m(l3) = mx (5) 1 18 The points z 1 and z 2 border a region containing a non-negative density gradient. The value of n r i s the average density of t h i s region. Using the two-phase function defined by eq.(2.4.12), F > A ] { (ri(z)) C/H > A ] dz . (6) From the d e f i n i t i o n of f 2(n), and eq.(5), 2 1 which is the free energy of the d i s t r i b u t i o n \ *~> L. - n c, I • " « « « « - ( ^ y * < - > ^ 1 5 - (8) Since t h i s d i s t r i b u t i o n preserves the volume and mass of the [z,,z 2] region, the free energy given by eq.(7) i s that of a p r o f i l e without this region, the mass having been moved into the bulk phases. Because of eq.(6), such a p r o f i l e has a lower free energy than the o r i g i n a l one. Repeating t h i s argument for a l l regions containing non-nega-ti v e gradients, the s t r i c t l y decreasing density p r o f i l e i s seen to minimize the free energy of the system. If the two bulk phases have the same density, then no state-ment can be made about the p r o f i l e with this method. However, i f the bulk phases are s u f f i c i e n t l y large, the "two-phase" 1 19 system (n^n,) has a lower free energy (section 2.4). Thus, the s t r i c t l y decreasing p r o f i l e provides an absolute minimum of F. 

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