Molecular-Based Study of Fluids - ACS Publications - American

6 Fluids at Interfaces JOHANN FISCHER

Downloaded by MONASH UNIV on September 15, 2015 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch006

Ruhr-Universität, Institut für Thermo- und Fluiddynamik, D-4630 Bochum, Federal Republic of Germany Three topics are considered in this chapter: gas adsorption on solid surfaces, the free liquid surface, and a liquid in contact with a wall. They are treated theoretically, and results are compared with experiments and simulations. A review of virial expansions and of the application of the first equation of the Born-Green-Yvon hierarchy for inhomogeneous fluids is given. For the case of gas adsorption, the structure of a fluid adsorbed on a plane surface at supercritical and subcritical temperatures is shown, together with adsorption isotherms. Adsorption in pores is dealt with in a simple model. From the study of the liquid-gas interface the coexisting densities, the surface tension, and the surface thickness are obtained. Finally, the structure of a liquid close to a wall is discussed.

T

H E EQUILIBRIUM PROPERTIES O F FLUIDS AT INTERFACES are of practical

importance in many engineering processes. While molecular theory will hardly be able to make quantitative predictions for all real situations, it can help us at least in understanding many of the interface phenomena. In order to achieve the latter goal we will keep the model of the fluid molecules and the solid surfaces as simple as possible—we consider only Lennard-Jones or hard-sphere molecules and unstructured walls with forces perpendicular to the surface—and concentrate on some physically interesting situations. Statistical mechanics of inhomogeneous fluids is now in development and different theoretical methods are in competition. Here, we will briefly describe the virial expansion and the use of the first equation of the Born-Green-Yvon hierarchy, as these will subsequently be used in treating the model systems. A comparison with the density functional method closes the first section. Theory Consider a fluid of N identical spherical particles which neither form a free boundary nor are in contact with a smooth solid surface at tem0065-2393/83/0204-0139$06.00/0 © 1983 American Chemical Society In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

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MOLECULAR-BASED STUDY O F FLUIDS

perature T. The potential between two fluid atoms is u(r ) = u and the potential that a fluid atom experiences from the solid surface is u (r ) = MJ. We are mainly interested in the local density n(r), which is the same as the one-particle distribution function and is given by ik

ih

s

n(r ) = (N/Z ) j exp |-p[S x

N

+ 2

j dr . . . d r 2

{

(1)

N

where Z denotes the configurational partition function and (3 = 1/kT. For evaluating the local density in the case of a gas in contact with a wall we can think of a virial expansion. Technically we start from Equation 1 and use van Kampen's method (I), which results in (2)


N

n(r ) = n exp {-(3ul}[l + v^rjnj, + v (r )n| + . . .] x

2

b

(2)

1

where n is the gas density far away from the solid surface. In that expansion the coefficient v describes the interaction of (i +1) particles among themselves and with the wall. Expressions for v and v are given elsewhere (2). If the kinetic energy of the gas molecules is small compared with the adsorptive potential of the wall, then the molecules tend to sit in-a layer close to the wall. Thus, even for low bulk gas densities, the local density in the adsorbed layer may become so high that one has to consider the simultaneous interaction of many particles. In that case, the expansion shown in Equation 2 can no longer be used. Concluding, we can say that a virial expansion is expected to be useful at high temperatures and low densities. Another route for evaluating the local density n(r) is the first equation of the Born-Green-Yvon hierarchy. In that connection we consider the probability of finding simultaneously two particles in the volume elements dr and dr . We denote that probability as n(r^) n(r ) g(r r ) dr dr . The function g(r ,r ) is called the pair correlation function. Now, by differentiating Equation 1 with respect to the local coordinate r and denoting that differentiation by V we obtain, after rearranging b

{

2

x

Y

2

2

2

l

1?

2

x

2

:

1 ?

V ln n W = - V x

x

$u\ + j (- V _ ! pu ) g(r r ) n(r ) dr 12

1?

2

2

2

(3)

This is the first equation of the Born-Green-Yvon hierarchy, (the BGY equation). The first term on its right-hand side is the force exerted by the wall on a particle at r The integral is the mean force that a particle feels from the other fluid particles. These forces are balanced by the density gradient on the left-hand side. In order to calculate the local density from that equation one has to use an approximation for the pair correlation function. v

In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

6.

FISCHER

141

Fluids at Interfaces

Different approximations for the pair correlation function in connection with the BGY equation are possible. In a previous article (3) the author and a co-worker had the goal of making a physically reasonable approximation which, on the other hand, should be simple enough that numerical solutions were readily attainable. We split the potential u between two fluid particles into a short-range repulsive part and an attractive part. Hence the mean force in Equation 3 splits into a mean repulsive and a mean attractive force. In the mean attractive force we neglect any correlations. The mean repulsive force is treated in the hardsphere approximation, the pair correlation function being taken as that of a homogeneous hard-sphere system at a mean density, which is obtained by averaging the local density over the volume of a molecule. Contrary to the virial expansion, this is not a systematic but an ad hoc approximation scheme. The BGY method, however, has a much larger range of applications. Moreover, a recent investigation (4) using a more sophisticated approximation for the pair correlation function has confirmed that the above described method yields at least qualitatively correct results. The first approach to fluids at interfaces was originated by van der Waals and is called, in its modern version, density functional theory. The basic idea is to write the Helmholtz energy A of the system as a function of the local density n(r) and the direct correlation function c(r r )


ik

ly

A = 9[n(r), c(r r )] 1;

2

2

(4)

Instead of the direct correlation function the pair correlation function may also be used. It must be stated that all the functions used are only approximate expressions. After making suitable approximations for the correlation function, one gets an equation for the local density by minimizing the Helmholtz energy. A review of such approaches can be found elsewhere (5). In comparing the BGY with the density functional approach we learn that in both methods an approximation for the correlation function has to be made. The starting equation in the BGY approach, however, is exact, while the expression for the density functional is always an approximation. On the other hand, our BGY method requires some numerical calculations. This is not necessarily the case in the density functional theory. For slowly varying density profiles, for example, gradient expansions can be made, which greatly facilitates the evaluation (5). Gas Adsorption

on Plane Solid Surfaces

A simple model for physical adsorption is that of spherical fluid particles in contact with a plane, structureless wall. In nature corre-

In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

142

MOLECULAR-BASED STUDY O F FLUIDS

sponding systems are that of argon or krypton adsorbed on the basal plane of graphite, for which accurate measurements have been made (6, 8). We assume the fluid particles to interact through a 6-12 LennardJones potential and to be in contact with a plane 3-9 wall


u'(z) = \W*

Egs

[(cr /z) gs

9

- (

Molecular-Based Study of Fluids - ACS Publications - American

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