Surface tension of a molecular fluid Many body potential t

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MOLECULAR PHYSICS, 1981, VOL. 43, No. 5, 1035-1041 Surface tension of a molecular fluid Many body potential t by MARTIN GRANT and RASHMI C.

1 MOLECULAR PHYSICS, 1981, VOL. 43, No. 5, Surface tension of a molecular fluid Many body potential t by MARTIN GRANT and RASHMI C. DESAI Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 (Received 15 January 1981 ; accepted 9 March 1981) We derive a general expression for surface tension in a dense molecular fluid with an arbitrary many body interaction. This formula reduces to that of Gray and Gubbins under the assumption of pairwise additivity. We then show how this general expression is transformed into a sum rule expression and also into an expression involving the direct correlation function. These last two expressions are analogous to the simple fluid surface tension formulas of Jhon, Desai and Dallier and Yvon-Triezenberg-Zwanzig respectively. We also discuss the case of a fluid in d dimensions. 1. INTRODUCTION In recent years, there has been a surge of renewed interest in the statistical mechanics of liquid-vapour interfaces [1, 2]. In this paper, we shall consider some formal aspects related to the preeminent physical quantity in this area, surface tension. In particular we shall concern ourselves with the roles of many body potentials and the external field in the theory of surface tension of a molecular fluid. In a recent paper [3], hereafter referred to as I, we derived a general statistical mechanical expression for the surface tension of a simple fluid with an arbitrary many body interaction potential. This expression reduces to the well known Kirkwood-Buff formula [4] if we restrict ourselves to pairwise additive potentials. In addition, through the use of dynamical sum rules, we showed that the general expression is exactly equivalent to the Yvon-Triezenberg-Zwanzig [5] fluctuation formula for surface tension. However, this work was restricted to simple fluids. In this paper, we extend our treatment to the case of molecular fluids. " The Kirkwood-Buff theory of surface tension for simple fluids has been extended to molecular fluids by Gray and Gubbins [6]. As with Kirkwood and Buff's work though, their work was restricted to pairwise additive potentials. In the following we do not make this assumption and thus.derive a general expression, (14), applicable to molecular fluids which reduces to the result of Gray and Gubbins for pairwise additive potentials. As in I we then introduce sum rules to show that this general expression is equivalent to a correlation function expression for surface tension. This correlation function expression, (20), is analogous to that of Jhon et al. [2], applicable t Supported by NSERC of Canada /81/ $ Taylor & Francis Ltd

2 -, 1036 M. Grant and R. Desai to simple fluids. Then by evaluating the sum rules of this expression in the manner of Schofield [7], we finally transform our general expression for molecular fluids into a form, (22), which is analogous to the Yvon-Triezenberg- Zwanzig formula involving the direct correlation function. We have also extended the treatment of I to liquid-vapour systems in d dimensions. The extension here is quite straightforward so we only state results, (24) and (25), noting however some formal considerations involving interface stability in d dimensions. 2. DERIVATION AND TRANSFORMATION OF THE GENERAL FORMULA FOR SURFACE TENSION Let us consider a liquid-vapour system contained in a cubical container of volume s = L 3 with the planar interface in the X1X 2 plane, and described by a hamihonian H= N Z Ti+ N Z 1 m! Z' Vm( ri,~,''" rq, wi, wj, Wq)" i=1 m=l i,j... q = T+ V. (1) Thus, we have a system of N rigid molecules where r I, T i and W i respectively denote the centre of mass position, kinetic energy and set of Euler angles for the hh molecule. The prime on the-summation over i, j,... q means that the i, j,... q are m distinct molecules, with 1/m! ensuring that each given group of m particles is counted only once. The term in V with m= 1 is the only external field contribution; in the infinite volume limit this stabilizes the Gibb's dividing surface in the X1X 2 plane [8], at X 3 = 0. We proceed to evaluate the surface tension through its well-known definition as a thermodynamic derivative [9] a = - (2) ~A p,n where F is the free energy, A is the surface area, and fl is the inverse temperature measured in energy units. We relate the free energy to the hamiltonian through the classical canonical partition function Z=exp (-flf)=c ~... ~ dar 1... a3rn a, Wl... a3 WNexp 03V), (3) where C is a constant coming from T in (1). Since it is separable from V in the hamihonian, we need not discuss its specific form in terms of translational and rotational contributions. To evaluate (2) from (3), physically we are looking at a change in the free energy during an isothermal process which dilates the system in the 7(2 direction while compressing the system in the X 3 direction so as to keep the volume constant. With this in mind we make the following scaling transformation [10] which puts the area dependence of the integrals into the hamihonian explicitly in terms of our ' boundary conditions ' described above. The particle

3 coordinates are scaled as follows Sur[ace tension o~ a molecular [luid 1037 = ( Xl i*, Ax2 i*, --~ ~ x3 i*, (4) which implies that the integrals over x~ i* and x3 i* are dimensionless (without loss of generality x 1 * integrals are chosen dimensionless). It is then straightforward to show from (2), (3) and (4) that "= T~' (5) where the angular brackets denote a canonical ensemble average. In (5) the total potential V is to be a function of the starred coordinates for the differentiation with respect to area. In order to evaluate a for molecular fluids, we first note that and so that Then from (1) we have ~---~ x~'* = - ~ x~i* ~- 7' X i ~-~ (Axz ~* ) = x ~'* -- A (6 b) ~V 1 ~-~= ~ ~ (xs k V,k-xa k Vak)V. (6 c) N with 1(1 a= E am (7) m=l ) t, J,... q Now we consider the intrinsic surface tension (m~>2), [11], using the m body distribution function for a molecular fluid : n~(m,... Rm, ~1,... to") =<ij~,.' q 8(Rl-ri)...~(Rm-rq) (8) in which R i and to i are the field coordinates. So we can rewrite (7), using the fact that n m and V m are symmetric functions of their arguments, as 1 nm(~l... Rm,~l... ~m){(2~1_~71~1) c%-2a(m- 1)! + (x~ v~ ~- X~ ~)} v~(ri,... lira, ~;,...,~m), (9) where the integrations over all the arguments of nm and V m, now functions of field coordinates, are implicit ; we denote this by putting bars over them.

4 1038 M. Grant and R. Desai It is convenient to express V m as a functions of R ij- R i- RJ, the relative position coordinates of which 3m-3 are independent due to translational invariance. We also introduce the spherical coordinates RiJ=(R ij, 0 i3, ~ij) which are related to the cartesian basis by Xt ij = R ij cos r sin 0 ij, ] X2 ii -~ R ii sin r sin OiJ' I (10) Xa i~ = R is cos 0 ij. Then for a typical gradient term we have and V21 v~ VX212 ~ sin r COS 0 TM ~ COS r [.~-i~ ~R12 ~ R12 ~012 + R12 sin 012 ~r +,:3 \~-v ~Rlr ~ al, ~01---~ Rl~cos 01, ~1~ V,. (11) V'lV~=L~-~ ~R12 R TM ~012~-r=3A~ \-R~R1, Rlr ~01r v~. (12) /_J It is then straightforward to show that 1 F(x212)2- (x,12)'-] OVm am=2a(m_2)! nm L ~-~ _l br bv, n -t 4A(m-2)! nmsin cos 01~--~-~+C~, (13) where in the second term we have put sin2r (Since (X212) 2- (Xa12)2-> 89 because of the particular broken symmetry of the problem.) Also in (13), This term is an artifact of the derivation which does not result from the broken symmetry of the problem and so is exactly zero. It results from the particular choice of scaling the area with the x 2 coordinate in (4). Other choices of scaling leave this term exactly zero. We can physically see that it has no contribution by rotating the system in the X11~ X212 plane and noting that this term vanishes for some values of r so it must be zero for all values of r Such an argument is implicit in [13], but we exhibit explicitly the functional dependence here because we use this argument later in this section. So, for m I> 2 1 nm(,1,.m r ~,)m )[ ('~212)R I_~!~312)2] OVm (~m=2a(m_ 2 )!... - _-=-- 0~) A(m'2)! nm(rl' "'" ~m, ~1, ~) sin 0 TM eos 0 TM 0Vm... b~l~, (14)

5 Sur[ace tension of a molecular [luid 1039 where integrations over all the barred variables, including the Euler angles, are implicit. For potentials which are pairwise additive this reduces to the expression due to Gray and Gubbins [6]. Now we show how this general expression can be put into a correlation function form through the use of sum rules. In the same way as in our previous calculations in I we have derived the force-force sum rule, which has as its non-local part (i.e. that part which does not vary as ~(R- R') or VS(R- R')).... ~--1 t - (~m--2) (gi( R, to)gi( R, to ))non loc~1- (m_2)! nm( R, R', ~1,... Rm-2, to, to, to1,... X ViV' i Vm(R, R', ~1,... ~ra-2, to, to', ~)1... (~ra-2) (15) where gi(r, to)= ~.pi ~ 8(R-r~)8(to-WJ) is the ith component of the local J momentum density, and integrations over the barred variables, as well as an implicit summation over m, are implied. We expect a straightforward generalization to molecular fluids of Jhon et al.'s correlation function form of surface tension [2]. Therefore we suggest the sum rule form of surface tension for a molecular fluid is 2~ [(~212)2(g3(~1, t~!)g3(~2, ~2))- (~312)2(g2(~1, ~,}1)~2(~2 ' ~.~2))] "--nm [(.,~212) 2 V31 V32- (X812) 2 V21 V22]Vm, (16) 2A(m-2)! wherein integrations are implicit over the barred variables including the Euler angles and a summation is implicit over m. So we now take two derivatives of V m, transforming (16) from a cartesian to a spherical coordinate system. For simplicity, we do not display the terms involving derivatives with respect to ~, since the algebra become tedious ; but they vanish for the same reason as in the above derivation of am" Also as in I we make use of the constructed symmetry of the operator on the right-hand side of (16) with respectto the different field points, i.e. terms involving sin 0 lr and cos 01~ with r # 2 vanish. Keeping this in mind we expand the expression above in spherical coordinates to get the following candidates for non-zero contribution : nm [(~ 12)2 V31 V32 -- (.,Y312)2 V21 V22] Vr a 2A(m-2)! n m f(.x212')2 - (ff~,:112) 2) 3Vm 3nm 2A(m-2)!<- - > - ~i~ [--- J~q4A(m-2)! sin 012 cos 012 with cot 3Vm Q1 = and 38 Vm ~2 Vm 92 = --2 sin s ~12/~12 sin 012 cos ~12 sin 2 ~12 COS 2~ TM t~ TM 30 TM (~012)2" 3v, (17) (18) (19)

1040 M. Grant and R. Desai The term Q1 is zero for the following reason. The form of the operator of (16) is such that if X2~--~X3, the operator is reflected through the origin.

6 1040 M. Grant and R. Desai The term Q1 is zero for the following reason. The form of the operator of (16) is such that if X2~--~X3, the operator is reflected through the origin. For the system we are considering here with planar symmetry in the X1X 2 plane, this is physically equivalent to rotating 0 by rr/2. All the terms in (16) do exhibit this symmetry except Q1, so Qt = 0. Q2 is also zero by virtue of the argument following (13). Physically this occurs because this term does not result from symmetry breaking. The nonvanishing contributions to the correlation function expression in (16) arise from symmetry breaking. So we have from (i6) and (17) O" = ~A-A ~ [("~212)$(g3( ~1, ~'~1)g3(~2, ~2))- (X812)2(g2( ~1, 0.~1)g2(~2, ~$))]. (20) Now we can proceed as we did in I, following the approach of Schofield [7]. First we obtain, analogously to the manipulations in I ~(R, co)=v 9 8S(R, to)-$nl(r, CO)VVI(R, co), (21) where 8B-B-<B>, and S is the microscopic stress tensor with the explicit external field contribution removed. We insert this into the correlation function expression, noting that terms arising from the first piece ' squared ' are not symmetry breaking and so have no contribution.. Terms arising from the second piece ' squared ', and ' cross terms ' do arise from symmetry breaking however. After this is done we relate Vt/1 and V V 1 through the use of functional derivatives, (see, for example, the recent paper of Gubbins [12]). Then we make use of a generalization of the important identity, derived by Schofield ((22), [7]), for molecular fluids to obtain : 3-1 c~ =--4- I S ax31 d~,.~ I I dx3 s d3 '~ V81 nl( X31, col)v32 nl( X82, cos) x ~ d s pps C(p ; X31, X3 s, col, cos). (22) This is the analogue of the Yvon-Triezenberg-Zwanzig formula to molecular fluids [13], where p is the two-dimensional vector along the interface and C( v ; X3X, X3S, to1, cos) is the direct correlation function for a molecular fluid given by C(R1, RS col, cos) = n1-1(r1, col)~(r1 _ RS)~(col _ cos) _ (Sn(R1, col)c~n(rs ' cos))-1. (23) 3. Co.NCLUSIONS To summarize, we have introduced a generalized formula for surface tension in a dense molecular fluid. We then showed that this expression, which was explicitly dependent upon all the many body interactions, was equivalent to a correlation function expression involving sum rules, and then equivalent to a formula involving the direct correlation function. The physical reason behind this result is that the translational symmetry breaking for the liquid-vapour interface is assumed to be due to a single particle external field.

Surface tension of a molecular fluid 1041 As to possible further work, an explicit calculation of our general expression, including a three body potential, may be possible in the manner of Halle et

7 Surface tension of a molecular fluid 1041 As to possible further work, an explicit calculation of our general expression, including a three body potential, may be possible in the manner of Halle et al. [14]. One extension which we have carried out, and will just mention in passing, is the case of a simple fluid in d dimensions. This calculation is very similar to that presented in I. For d> 1, the generalized surface tension is 1 nrn(r1,... ~m) ~(R12)2_(2d12)2d~ OVra(R1,... ~m) (24) a(a) = 2(m - 2)~ L a-1 ( (--d~- l--~ J -~R i~ ' where a summation over m>~2, and integration over the barred variables is implied, n m is the m body distribution function, V m is the m body interaction in a d-dimensional simple fluid hamiltonian analogous to (1), and L is the ' edge length ' of the d-dimensional system. This can be transformed as in I to with /3-1 a(a)= $ dxa I dx'a Van(Xa)V'a n(x'a) I da-1 pp2 4 ; (25) c(a)(r, R')= 3(R- R') (3n(R)3n(R,))_l, (26) n(x ) the direct correlation function for this system. Arguments about the stability of such d-dimension interfaces, in the manner of Weeks [8], lead to the physically obvious conclusion that interfaces with d> 1 are stable, and the greater the dimensionality, the greater the stability. REFERENCES [1] EVANS, R., 1979, Adv. Phys., 28, 143. [2] (a) JHON, M. S., DAHLER, J. S., and DESAI, R. C., 1981, Adv. Chem. Phys., 46, 279. (b) JHON, M. S., DESAI, R. C., and DAHLER, J. S., 1979, J. chem. Phys., 70, (c) JHON, M. S., DESAI, R. C., and DAHLER, J. S., 1978, Chem. Phys. Lett., 56, 151. [3] GRANT, M., and DESAI, R. C., 1980, J. chem. Phys., 72, This paper is referred to as I in the text. [4] KIRKWOOD, J. C=, and BUFF, F. P., 1949, J. chem. Phys., 17, 338. [5] (a) YVON, J., 1948, Proceedings of the IUPAP Symposium on Thermodynamics, Brussels, p. 9. (b) TRIEZENBERC, D. G., and ZWANZI~, R., 1972, Phys. Rev. Lett., 28, [6] GRAY, C. G., and GOBmNS, K. E., 1975, Molec. Phys., 30, 179. [7] SCHOFIELD, P., 1979, Chem. Phys. Lett., 62, 413. [8] WEEKS, J. D., 1977, J. chem. Phys., 67, [9] We note for completeness that we could just as easily evaluate a in a similar manner through its 'mechanical definition': a= J dx3[pi~--pt(x3)], where PN and PT(x3) are the pressures normal and tangential to the surface. [10] BUFF, F. P., 1952, Z. Electrochem., 56, 311. [11] For brevity we do not discuss the case m=l. The discussion is analogous to that for a simple fluid in I. [12] GOBmNS, K., 1980, Chem. Phys. Lett., 76, 329. [13] Equation (18), [3] should be multiplied by a factor of 1/4 on its right-hand side. [14] HALLE, J. M., GRAY, C. G., and GUBBINS, K. E., 1976, J. chem. Phys., 64, 2569.

7 To solve numerically the equation of motion, we use the velocity Verlet or leap frog algorithm. _ V i n = F i n m i (F.5) For time step, we approxim

7 To solve numerically the equation of motion, we use the velocity Verlet or leap frog algorithm. _ V i n = F i n m i (F.5) For time step, we approxim 69 Appendix F Molecular Dynamics F. Introduction In this chapter, we deal with the theories and techniques used in molecular dynamics simulation. The fundamental dynamics equations of any system is the