advances in chemical physics€¦ · advances in chemical physics edited by ... universitc libre de...

Advances in CHEMICAL PHYSICS

Edited by

I. PRIGOGINE Center for Studies in Statistical Mechanics and Complex Systems

The University of Texas Austin, Texas

and International Solvay Institutes UniversitC Libre de Bruxelles

Brussels, Belgium

and

STUART A. RICE Department of Chemistry

and The James Franck Institute The University of Chicago

Chicago, Illinois

VOLUME 102

AN INTERSCIENCE@ PUBLICATION JOHN WILEY & SONS

NEW YORK CHICHESTER WEINHEIM BRISBANE SINGAPORE TORONTO

ADVANCES IN CHEMICAL PHYSICS VOLUME 102

EDITORIAL BOARD

BRUCE J. BERNE, Department of Chemistry, Columbia University, New York, New York,

KURT BNDER, Institut f i r Physik, Johannes Gutenberg-Universitat Mainz, Mainz, Ger-

A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State Univer-

DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, Califor-

M.S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K. WILLIAM T. COFFEY, Department of Microelectronics and Electrical Engineering, Trinity

College, University of Dublin, Dublin, Ireland F. FLEMING CRIM, Department o f Chemistry, University of Wisconsin, Madison, Wiscon-

sin, U.S.A. ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indi-

ana, U.S.A. GRAHAM R. FLEMING, Department of Chemistry, The University of Chicago, Chicago, Illi-

nois, U.S.A. KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois,

U.S.A. PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Brussels, Bel-

gium ERIC J. HELLER, Institute for Theoretical Atomic and Molecular Physics, Harvard-Smith-

sonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A. ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania,

Philadelphia, Pennsylvania, U.S.A. R. KOSLOFF, The Fritz Haber Research Center for Molecular Dynamics and Department

of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology,

Pasadena, California, U.S.A. G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, Universite Libre de

Bruxelles, Brussels, Belgium THOMAS €? RUSSELL, Department of Polymer Science, University of Massachusetts,

Amherst, Massachusetts, U.S.A. DONALD G. TRUHLAR, Department of Chemistry, University of Minnesota, Minneapolis,

Minnesota, U.S.A. JOHN D. WEEKS, Institute for Physical Science and Technology and Department of Chem-

istry, University of Maryland, College Park, Maryland, U.S.A. PETER G. WOLYNES, Department of Chemistry, School of Chemical Sciences, University

of Illinois, Urbana, Illinois, U.S.A.

U.S.A.

many

sity, University Park, Pennsylvania, U.S.A.

nia, U.S.A.

Advances in CHEMICAL PHYSICS

Edited by

I. PRIGOGINE Center for Studies in Statistical Mechanics and Complex Systems

The University of Texas Austin, Texas

and International Solvay Institutes UniversitC Libre de Bruxelles

Brussels, Belgium

and

STUART A. RICE Department of Chemistry

and The James Franck Institute The University of Chicago

Chicago, Illinois

VOLUME 102

AN INTERSCIENCE@ PUBLICATION JOHN WILEY & SONS

NEW YORK CHICHESTER WEINHEIM BRISBANE SINGAPORE TORONTO

This text is printed on acid-free paper.

An Interscience" Publication

Copyright 0 1997 by John Wiley & Sons, Inc.

All rights reserved. Published simultaneously in Canada.

Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc.

Library of Congress Catalog Number: 58-9935

ISBN 0-47 I - 19 144-2

Printed in the United States of America

I 0 9 8 7 6 5 4 3 2 1

CONTRIBUTORS TO VOLUME 102

JENS ALS-NIELSEN, H.C. 0rsted Laboratory, Niels Bohr Institute, Copenhagen,

MARC BAUS, Faculte des Sciences, Universite Libre de Bruxelles, Brussels, Bel-

JOHN S. DAHLER, Department of Chemical Engineering and Materials Science,

JACK F. DOUGLAS, Polymers Division, National Institute of Standards and Tech-

KRlSTlAN KJAER, Department of Solid State Physics, Riser National Laboratory,

S. HUTZLER, Physics Department, Trinity College, Dublin, Ireland MEIR LAHAV, Department of Materials and Interfaces, The Weizmann Institute of

Science, Rehovot, Israel LESLIE LEISEROWITZ, Department of Materials and Interfaces, The Weizmann In-

stitute of Science, Rehovot, Israel RONALD LOVETT, Department of Chemistry, Washington University, St. Louis,

Missouri E. PETERS, Shell Research and Technology Centre, Amsterdam, The Netherlands;

present address: Laboratory of Aero and Hydrodynamics, TU Delft, The Netherlands

RONIT POPOVITZ-B~Ro, Department of Materials and Interfaces, The Weizmann Institute of Science, Rehovot, Israel

G. VERBIST, Shell Research and Technology Centre, Amsterdam, The Nether- lands

ELIGIUSZ WAJNRYB, Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota; permanent address: In- stitute of Fundamental Technological Research, Polish Academy of Sci- ences, Warsaw, Poland

Denmark

gium

University of Minnesota, Minneapolis, Minnesota

nology, Gaithersburg, Maryland

Roskilde, Denmark

D. WEAIRE, Physics Department, Trinity College, Dublin, Ireland ISABELLE WEISSBUCH, Department of Materials and Interfaces, The Weizmann

Institute of Science, Rehovot, Israel

V

INTRODUCTION

Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.

I . PRICOCINE STUART A. RICE

vii

CONTENTS

THE THERMODYNAMIC FORCES IN AN INTERFACE

By Ronald Lovett and Marc Baus

MOLECULAR SELF-ASSEMBLY INTO CRYSTALS AT AIR-LIQUID INTERFACES 39

By Isabelle Weissbuch, Ronit Popovitz-Biro, Meir Lahav, Leslie Leiserowitz, Kristian Kjael; and Jens Als-Nielsen

SOME APPLICATIONS OF FRACTIONAL CALCULUS TO POLYMER SCIENCE 121

By Jack F Douglas

THE NEWTONIAN VISCOSITY OF A MODERATELY DENSE SUSPENSION 193

By Eligiusz Wajnryb and John S. Dahler

A REVIEW O F FOAM DRAINAGE 315

By D. Weaire, S. Hutzlel; G. Verbist, and E. Peters

AUTHOR INDEX

SUBJECT INDEX

375

383

ix

THE THERMODYNAMIC FORCES IN AN INTERFACE

RONALD LOVETT

Department of Chemistry, Washington University, St. Louis, M O 631 30

MARC BAUS

Facultk des Sciences, C.P. 231, UniversitC Libre de Bruxelles, B-1050 Brussels, Belgium

CONTENTS

I. Introduction 11. The Force Problem

111. The Solution IV. Density Functional Theory

VI. Molecular Realizations VII. The Cylindrical Interface

VIII. Young’s Model and the Surface Tension IX. The Free Energy Density X. The Stress Tensor Exposed

XI. Conclusions

V. Thermodynamics

Acknowledgments References

I. INTRODUCTION

The surface tension values tabulated in handbooks give a tension acting on a (geometric) surface, a macroscopic scale view of the force located in a phase boundary. On a molecular scale, properties change continuously

Advances in Chemical Physics, Volume 102, Edited by I. Prigogine and Stuart A. Rice. ISBN 0-471-19144-2 0 1997 John Wiley & Sons, Inc.

1

2 RONALD LOVETT AND MARC BAUS

through a transition region, and the answer to the question of where this tension is located must be some continuous force distribution. Since the characteristic length (thickness) of the transition region is the same order of magnitude as the correlation length t in the system, 8 gives the length scale on which the force distribution varies.

In an 1875 review of Young and Laplace’s phenomenological description of the surface tension, Maxwell [l] lamented the absence of a molecular identification of the forces in an interface. Van der Waals’ [2] theory of the continuity of states provided the first such identification. Increasingly detailed searches for a mechanical definition of the “local force” in a non- uniform fluid, however, have only given rise to controversy. The book of Rowlinson and Widom [3], for example, provides an excellent review of work on fluid interfaces, but it bases its molecular theories for the forces in an interface on a (mechanical) stress tensor paradigm that it clearly states is flawed. We illustrate the ambiguity of this approach in Section I1 with a simple calculation.

The thermodynamic description of curved interfaces is particularly con- fused, for the basic dejinition of the surface tension of a curved interface presumes knowledge about a local force distribution. This problem is important since curved interfaces are present in liquid drops, in bubbles, in emulsions and colloidal suspensions, in liquid films adsorbed on rough solid surfaces or in porous materials, and almost all tabulated surface tension values are deduced from observations on curved interfaces. There are other surface properties, such as the spontaneous curvature of an interface with a complex molecular structure, whose understanding clearly requires the local force distribution.

The fundamental problems associated with defining the thermodynamic forces in interfaces on a spatial scale approximately equal to 8 is the theme of this chapter. We identify the crucial confusion in the definition of a local force in Section I11 and this leads directly to a thermodynamic definition of the force distribution with a clear, direct physical interpretation. The techni- cal prescription of the force distribution is elaborated in Section IV and a second physical interpretation of this force distribution is advanced in Section V.

In fact, Sections 11-V really address thermodynamic questions. Only in Section VI is a molecular realization for the results of these sections developed. In describing the cylindrical interface in Section VI, we encounter a system in which the two principal radii of curvature are different, in which the molecular structure in the interface is different in different directions. A thermodynamic argument is given in Section VII, however, that the force distribution is nevertheless isotropic.

From the perspective of our definition of the local force distribution, we

THE THERMODYNAMIC FORCES IN AN INTERFACE 3

ask in SectionS VIII and IX what phenomenological descriptions can be given for the surface tension and the local free energy density. Finally, since the stress tensor paradigm has dominated recent work in this area, we review in Section X the physical ambiguity of this mechanical picture and describe an error that is routinely made in constructing molecular expressions for the stress tensor. A summary of the present status of the local force problem is given in Section XI.

11. THE FORCE PROBLEM

Let us consider a system of N structureless particules in a volume V inter- acting through pairwise additive forces, that is, a system with Hamiltonian

N p? H = c + c u(rij)

i = l 2m i < j

The pair potential u(rij) is a two-particle field, since it assigns a potential energy to every pair of positions r i , r j . It is the finite range of u(rij) that makes e > 0.

We consider the temperature T and the average density p = N / V to be adjusted so that the system has coexisting liquid and vapor phases. The “interface” of interest is the transtion region between these two phases. Since the force distribution we seek describes how the force varies with position in the interface, it is a one-particle field. Theforce problem is, How can we construct a one-particle force when all the intermolecular forces arise from two-particle forces? How can the forces associated with u(rij) be localized ?

The central question is subtle because, for a system at equilibrium, the average force on matter at any r must be zero. We can create a nonzero force, however, by dividing a system (in our mind) into two parts and asking for the force that the particles on one side of a (imagined) dividing surface exert on the particles on the opposite side [4]. Imagine the forces across a plane at x = 0 (see Fig. 1). The total force of the matter in x < 0 on that in x > 0 is

with p2(r1, r2) the pair number density or two-particle distribution function. For a bulk phase, p2(r,, r2) -+ p2(r12) and the integral over r2 in Eq. (2.2)


Figure 1. The molecules in x < 0 do exert a force on those in x 2 0.

can be written as an integral over rlZ = r2 - rl,

with dx = Ly L, the area of the imagined plane at x = 0. Equation (2.3) gives a force that is (strictly speaking, asymptotically as dX + 00) pro- portional to dx . Adding the momentum/ideal gas contribution to the force per unit area gives

dr p2(r)ru‘(r) (2.4)

the virial expression for the pressure p .

and the analogue of Eq. (2.3) is In the presence of a planar interface at z FZ 0, p2(r1, r2) -, p2(z1, z 2 , rI2),


Similarly, the same assessment for the force acting across the z = z1 plane gives

and averaging this over all z1 gives

If yplanar is the surface tension of the interface at z % 0, we expect

(2.7)

from which we can deduce [ 5 ]

the so-called Kirkwood-Buff formula [6] for the surface tension. This approach to introducing a force has two problems. First, while the

total force of the matter on one side of an imagined plane on the matter on the other side is correct, there is no basis for saying where the force acts. Summing over nonlocal forces does not localize where the force acts. The basic two-particle + one-particle projection problem is not addressed. The success (validity) of Eqs. (2.4) and (2.8) reflects the fact that only the total force is required to determine p or yplanar. There is no explicit assertion in Eq. (2.7) about where p or yplanar act. In fact, effects associated with the walls of the system are removed from Eqs. (2 .4) and (2 .8) by taking the L,, L,, L, -+ co limit. Of course the integrand of dz, in Eq. (2.8) is obviously only nonzero at z1 where the local structure is anisotropic, a region of size t, but there is no localization of the force on a finer scale. Similarly, simple thermodynamic processes such as changing L, or L, can be described without localizing the forces. But for more local deformations of a planar interface or for any deformation of a curved interface it is necessary to know where the forces actually act.

6 RONALD LOWTT AND MARC BAUS

We have only given the “force across an imaginary surface” argument heuristically. In fact, this argument is the standard argument in the literature, but it is usually given an air of sophistication by formulating it in terms of a stress tensor [ 3 ] . The inability to localize forces is intrinsic to the stress tensor formulation, however, and this has kept stress tensor based arguments in a cloud of controversy. This classical formulation is reviewed in Section X. In Section 111, we show that a clearer formulation of the physical questions leads directly to an intuitively transparent identification of the forces in an interface.

111. THE SOLUTION

There is a second problem with identifying the forces in an interface by imagining the forces across a geometric surface. This division of the system into two parts does not correspond to any real experiment. In a real experiment there is a partitioning of the universe, but it is a partitioning into “the system” and “the environment.” The system exerts forces on the environment and it is these forces that we, the environment, observe.

A more realistic Hamiltonian is Eq. (2.1) with terms added to confine the particles to 0 I x I L. Suppose, as a simple example, that

with 41(x) shown in Fig. 2. The added one-particle terms confine the particles to 0 I x I L. Of course, 4 1 ( x ) is created by some external apparatus exerting a force

on the ith particle and it is this force that we would observe. While the precise value of Eq. (3.2) depends on the explicit molecular configuration, x l , . . . , x N , some configuration averaging is always done in an experiment

Figure 2. A single-particle potential that will confine molecules to x 2 0.


and we will invoke some equilibrium ensemble average ( ) here. In this case,

just depends on L and the (average) force on the wall at x = L, is

We can rephrase this argument in a thermodynamic style by evaluating the work associated with a "virtual" displacement. If we calculate the (Helmholtz, say) free energy A of our system, it depends on (among other things) L,

A = A(L) (3.3)

If we displace L + L + u as shown in Fig. 3(a),

A(L) + A(L + u) = A(L) - uF(L) + O(u2) (3.4)

with F(L) the thermodynamic force conjugate to u,

To localize this force, all that is necessary is to localize the test or virtual displacement u as shown in Fig. 3(b). Let u = u(z). Now A = A(L, [u]) , a functional of u(z).

= A(L, [O]) - dz F ( L , Z)U(Z) + @(u2) s with

(3.7)

a local force acting on the x = L boundry.


x = L + L + u

X

I (a) ( b )

Figure 3. (a) The displacement of a wall at x = L to L + u is associated with an exchange of energy between the system and the environment. (b) The response to a local displacement u(z) identifies a local force F(L, z) acting on the wall.

The forces associated with real walls are more complex, of course. A more realistic wall, for example, would be generated by

with Rk the location of a particle of the wall (still the environment) that interacts with the particles in the system through uw(Ir - R I). But the experiment of interest is a simple displacement, Rk + Rk + u, or a more complex deformation,

The one-particle field thermodynamically conjugate to the one-particle field u(R) gives

(3.10)

as a local force. There are many ways to project the two-particle field u ( r l Z ) into a one-

particle field f(r). It is impossible to say that one is more correct than another without specifying the use that is to be made of the one-particle field. We resolve this problem by specifying that the one-particle force is to be used to calculate the free energy change for a family of deformations


that can be described with a one-particle displacement field u(Y),

A[u] = 4 0 1 - drf(r) . u(r) + O(u2) s (3.1 1)

Thus Eq. (3.10) identifies the force thermodynamically conjugate to u(v). This definition forf(r) is just a local generalization of the experimental definition of the force. Of course, formal definitions need not be closely related to actual measurements. It is a long argument from the definition of the pressure as “a force conjugate to the thermodynamic coordinate V ” to the deduction of p from a measurement of the height of a fluid in a manometer. But Eq. (3.10) is, at least in principle, an operational definition and Eq. (3.11) states the question to which Eq. (3.10) is an answer. In Section IV, we extend this definition to more sophisticated walls. In Section V, we note how the interpretation given to this force varies with the direction of u(u).

IV. DENSITY FUNCTIONAL THEORY

We indicated in Section I11 how the local force distribution on a wall can be defined. For a wall, the partitioning of the universe into two parts is straightforward and the identification of the force on the wall as an observable is reasonable. But the force that the wall exerts on the system changes the local structure in a layer of thickness approximately equal to 8 at the wall [S]. The force distribution on a wall that intersects an interface depends explicitly on the details of the forces exerted by the wall.

Can the force distribution away from a wall be defined? Can we discuss the force distribution in a spherical drop where there is no intersection of the interface with a wall? Since this force distribution would be independent of the forces actually present at the walls, we have called this the intrinsic force distribution in an interface. We now show that density functional theory allows us to extend the ideas of the previous section to the intrinsic interface.

The tradition in mechanics is to “specify” a many-particle system by decomposing the interaction potential

into one-particle, two-particle, three-particle, . . . , terms. Thus the free energy is a functional of one-particle, two-particle, . . . fields. While this is


natural in a mechanical context, it is often convenient to consider the density pl(rl) as an independent one-particle field. That this could be done is an implicit assumption of all mean field or van der Waals theories of phase equilibrium.

Schwinger’s [7] formulation of quantum field theory showed the advantage of regarding many-particle Green functions as functionals of the fields in Eq. (4.1), particularly as a functional of (pl(r). The thermodynamic extension of this is the generation of the n-particle number density or distribution function pn(rl, . . . , v,) by functionally differentiating the free energy with respect to (successively) +l(rl), (pl(rz), ..., (pl(v,). The idea that one could replace 41(r) with pl(r) was implicit in early series expansions of Yvon [S], but the first explicit recognition that a transformation of the independent field was being made was given by Percus [9]. Percus [lo] subse- quently showed that the formal assumption that this could be done opened the door to very simple constructions of functional expressions for the free energy and approximate integral equations.

While variational formulations and shifts of independent fields were developed in this milieu (see [ll], e.g.), the first formal proof that one could take pl (r ) as an independent field was given by Hohenberg and Kohn [12] for the ground state of a quantum system. This proof was extended to the free energy of equilibrium states by Mermin [13]. Since the original “proof by contradiction” argument is not very suggestive physically and we need a generalization of the original argument, we will recast the argument in thermodynamic terms [ 141.

In thermodynamics, the Helmholtz free energy A(T, V , N ) of a uniform system fixes the pressure through

Can the derived quantity p be taken as an independent variable? The answere is yes. Further, if we trade A(T, V , N ) + G(T, p , N ) = A(T, V ( T , p , N) , N ) + pV( T , p , N) , Eq. (4.2) is replaced by

The key step in justifying this transformation is the proof [ lS, 161 that the relation Eq. (4.2) can be inverted; that we can solve p = p(T, V , N ) for V = V ( T , p , N ) . Invertibility is guaranteed if the curvature of A(T, V , N ) as a function of V is sign definite, for this means that a tangent of slope - p can be constructed at only one value of V , as shown in Fig. 4. In fact, the


Figure 4. Because the curvature of A(T, V, only one tangent to the curve of this function with any prescribed slope -p.

as a function of V is sign definite, there can be

second law of thermodynamics guarantees that A(T, V , N ) must have a sign definite curvature in all its arguments.

The extension of this kind of argument to the molecular scale structure of an equilibrium system is immediate. In addition to being a function of the usual thermodynamic variables, the free energy A = A(T, N , [ 4 J , . . .) is a functional of the one-particle, two-particle, . . . fields 41(r), 42k1, r21, . . ., and

identifies the n-particle distribution function as the function that is thermodynamically conjugate to the potential 4,(r1, . . . , rn). Just as the curvature of A with respect to T , V , or N is sign definite, the curvature of A with respect to the functional arguments 41(r1), 42(r1, rz), . . . is also sign definite. Again, this is required by the second law although, for the analytically insis- tent, one can give a “proof” in terms of (Gibbsian) equilibrium ensemble averages using the Gibbs inequality [13, 141. The consequence of this sign definiteness is that the relation (4.4) can be inverted. Only one 4,(r1, . . . , r,) can lead to any pn(rl, ..., r,). The function pn(rl, ..., r,) can replace 4,(rl, . . . , r,) as an independent field. Specifying a thermodynamic state by giving T and the fields pl(rl), p2(r1, rz), ..., implicitly fixes all the fields in the Hamiltonian.

In macroscopic thermodynamics, we often parametrize a family of states for which we can calculate a free energy and then identify the “correct” value for the parameter as that associated with the lowest free energy. There is an important example of this parametrization at the molecular level. Associated with a change 41(r) -+ pl(r) of the independent field is a change


A[41] + V[pl] in the free energy. The parameter V[pl] is generated by the functional Legendre tranqorm

with 41(r, [ p J ) determined by inverting the n = 1 instance implicit q51(r) fixed by

(4.5)

of Eq. (4.4). An

is associated with each pl (r ) .

the functional If we know V[pl] and prescribe an external field 41(r), we can construct

[something like A[4J see Eq. (4.5)] with a parametric field pl(r). The field p l ( r ) , which minimizes this “free energy,” is the p l ( r ) field actually associated with the prescribed 41(r). The typical mean field or van der Waals theory for pl(r) may be thought of as realizing this program with an approximate functional for V[pl]. Admitting more molecular structure into V[pl] provides a more molecular theory for pl(r). An extensive industry has developed around this methodology. Although the term “density functional theory” is often used loosely to describe this approximation methodology in quanta1 fermion systems or in classical fluids, it is only the formal result, that the pn(rl, . . . , r”), n 2 1, fix the thermodynamic state, which is important here.

Now consider the force distribution in the spherical interface associated with a liquid drop. As the Hamiltonian H for this system contains no coupling between the interface and the environment, this force distribution is not observable. But suppose we center the drop at the origin and “derive” a new system by the rule

,..., rn) ifz,>O k = l , ..., n Pn(rl, . . . , r,) = { ~ ( ~ l if any zk < 0 k = 1, ..., n (4.6)

Even if H [as in Eq. (2.1)] only contains pair forces, R will have n-particle forces for all n 2 1. These complex forces are equivalent to the forces that the matter of the original system in z < 0 exerts on the matter in z 2 0. But


the structure induced by R is simple and the forces acting in the interface have been converted into external, observable forces at the z = 0 plane.

If we now introduce a displacement field u(r) = u(x, y)i, the conjugate thermodynamic force will be the local force distribution in the z = 0 plane at x, y. Under such a deformation,

The functional chain rule

and Eq. (4.4) may be applied to Eq. (4.7) to get

The potential &k(rl, . . . , rk) has two physically distinct sources, the intermolecular interaction of k particles and the coupling between k particles and the environment. These are naturally labeled the internal interactions and the external interactions and

The deformation u(r) only changes bFt(rl , . . . , rk),


so the linear response to u(r) is

I- -

(4.12)

with

By symmetry,f(x,, yl) =f(,/s:); this is just the radial distribution of the force in the spherical interface. The nonzero values for pk(rl, . . . , rk) are just the values of pk(rl, . . . , yk) of the original system, so the only significance of the bar is the restriction of the integration domains to zi 2 0.

Equation (4.13) gives a formal definition of the force distribution in the spherical interface in terms of the #y t ( r l , . . . , r k ) , but these fields are only known implicitly. Fortunately, there is a general relation in equilibrium statistical mechanics [17,18],

(4.14)


For the pair additive force model Eq. (2.1), the only internal potential is 4p(rl, rz ) = u(r12) and Eq. (4.14) is equivalent to

which can be used to rewrite Eq. (4.13) as

Equation (4.14), the first equation in the Yvon-Born-Green hierarchy, asserts physically that the local force on matter at r1 must be zero at equilibrium. If this force is a sum of external and internal forces, the sum of the external forces-needed in Eq. (4.13Fmust be the negative of the sum of the internal forces as given in Eq. (4.16). All quantities in Eq. (4.16) are known and the only reason a nonzero result is found is that the integration domain is restricted. The force calculated is that due to the particles in z < 0. The projection into a one-particle field is again achieved by introducing u(x, y) and Eq. (4.12) identifies the context in which this force distribution is useful. That is, the result of this “density functional” argument is that the force distribution in an intrinsic interface-an interface unper- turbed by other forces-an be deduced from an expression that is a partic- ular realization of looking at the forces on both sides of an imaginary dividing surface. We have selected the realization of this, however, which can be used to determine the free energy change associated with a particu- lar class of deformations.

When we observe that p = p(T, V , N ) can be inverted to obtain V = V(T , p , N) , there is one reservation: The proposed V must be realiz- able. If we ask, “What p will be associated with a I/ < O?” the answer is, “None.” It is possible that the dramatic conversion of intermolecular forces into external (and hence observable) forces might not actually be possible. There might not be c # ~ ~ ( r ~ ) , 42(r1, r2), . . . fields, which produce the structure Eq. (4.6). The advantage of the 4,(r1, . . . , rk) fields introduced in this section is that they lead to a simple picture. The “derived structure is defined in Eq. (4.6), the deformation enters Eq. (4.9) and the linear response calculation leading to Eq. (4.13) is straightforward/mechanical. All that remains


is to use Eq. (4.14) to reexpress Eq. (4.13) in terms of known intrinsic structure factors. But the structures and deformations that we introduced are dramatically different from realistic laboratory structures and displacements.

We may reformulate our thermodynamic definition in terms of a more realistic experiment, however. Rather than deriving a new system by remov- ing matter in z < 0, we can simply subject the system as it occurs in nature to a (virtual) displacement u(r) = u(x, y , z ) i with u(x, y , z ) only nonzero around z w 0. Phenomenologically, this displaces the system at x, y on the z = 0 plane by a distance approximately equal to J dz u(x, y , z). Therefore

4 u l - 4 0 1 = - d x d y dz u(x, y , z)f ,(x, Y , z )

with f z (x , y , 0) having the same interpretation as the f ( x , y ) in Eq. (4.13). The key point is that the linear response we seek is quite generally characterized by a Green function. Although the defining equation for the Green function is singular, the solution enables us to write the response to any deformation as an integral. The density functional argument using Eq. (4.6) is actually a construction of the Green function. If the resulting f (x , y ) is used to calculate the thermodynamic response to a realistic experiment it will give the same result that a direct imitation of the experiment would give, for all the formal steps in the calculation are just the same. We have simply inverted the order of “defining the experiment” and “calculating the result.”

V. THERMODYNAMICS

As a phase integral in the canonical ensemble partition function gives our only expZicit representation for A(T, N , [4J, [&I, . . .), we usually picture A(T, N , [41], [ 4 J , . . .) as a molecular quantity. The molecular picture is comforting because we lack thermodynamic experience with processes in which 41(r), rz), . . . are changed. We have only begun to realize such experiments in computer simulations. But all that is required from the molecular theory is the basic mechanical fact that the fields #1(~1), #z (~ l , rz), ... fix the thermodynamic state of a system, that these fields are independent thermodynamic variables. The “functional curvature” invoked is then a thermodynamic requirement. Whether a thermodynamic or molecular picture is being used is actually difficult to discern because the basic structures of thermodynamics and statistical mechanics are so


similar. The distinction in practice is simply one of interpretation [19]. The identification of the one-particle force distribution in the proceeding sections may be interpreted as a thermodynamic identification.

The picture developed in Section I11 can be summarized by saying that there is a thermodynamic response of the form Eq. (3.11) to any deformation field u(r),

ACul = ACO] - d r f ( r ) u(r) + s(u,) (5.1) f with

and that, for the pair additive force model Eq. (2.1), this can be evaluated as

In thermodynamics, however, we usually distinguish processes that alter V from those that alter N . We now extend this to a local level for interfaces in which the principal radii of curvature of the surfaces of constant density are the same at all points on the surfaces, that is, for interfaces with simple symmetries such as planar, cylindrical, and spherical interfaces.

We will use the thermodynamic notation in which thermodynamic deriv- atives of a function are presented as a linear response relation. Thus we write

in lieu of i3A/dT = -S, ... and describe 6T only vaguely as some (small) change in T. If we interpret 6u(r) as a physical displacement of matter, the corresponding change in


will be

6N = b dr 6p(r) = b dr[p(r - 6u(r)) - p(r)]

= - dr 6u(r) . Vp(r)

For a deformation to be associated with “ p 6V” type work only, it is necessary that 6 N in Eq. (5.6) be zero for any subvolume of the system. That is,

6u(r) * Vp(r) = 0 (5.7)

at all r. The deformations must be restricted to those that are tangential to the constant density surfaces in the system. Obviously, displacements paral- lel to constant density surfaces leave the local density invariant. But the direction of Vp(r) is also the natural choice for defining the normal direction in an interface. Thus, the tangential deformations satisfying Eq. (5.7) are just the deformations that we would use to define the tangential forces in the interface.

If 6u(v) is pictured as the displacement of the boundary au of some subvolume u, the deformation changes this subvolume by

6u = dS * 6u(r) = (5.8)

Equation (5.4) suggests that it is natural to define the density of volume change 6u(r) by

6u(r) = V 6u(r) (5.9)

This volume change per unit volume is dimensionless. The phenomenological definition of a local pressure would be in terms of

a force per unit area acting on the boundary aV of a system. For p(r) preserving deformations,

d S p(r) Su(r) = - dr V . [p(r) 6u(r)] (5.10) Jv 6 A = -

As stressed in Section IV, a force is only observable at the boundary. But, because we can convert any r to a boundary point by making a pn -+ pn change such as that shown in Eq. (4.6), we can use Eq. (5.10) to define p(r) at

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