the nonuniform wertheim ornstein-zernike equation: density profiles and pair correlation functions...

PHYSICA ELSEVIER Physica A 244 (1996) 147 163

The nonuniform Wertheim Ornstein-Zernike equation: Density profiles and pair correlation functions of a dimerizing hard sphere near a hard wall

D o u g l a s H e n d e r s o n a'*, O r e s t P iz io u, S tefan S o k o l o w s k i c, Andr i j T r o k h y m c h u k b" 1

Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA b lnstituto de Quimica de la UNAM, Cireuito Exterior, Coyoacgm 04510, MOxico D.F., Mexico

c Department.for the Modelling of Physico-Chemical Processes, FaculO' ~[" Chemisto', MCS Unie~ersity, 200-31 Lublin, Poland

Abstract

A dimerizing model of hard spheres near a hard wall is studied using the inhomogeneous Wertheim Ornstein-Zernike (WOZ2) equation and the inhomogeneous associative Percus-Yevick (APY2) closure. Dimerization occurs because of the directional associative interactions. A mutually consistent set of density profiles and inhomogeneous pair correlation functions are investigated. Also we obtain the dependence of the association constant on the distance of the fluid particles from the surface. This property is unavailable from singlet integral equations. The application of the WOZ2 equation is an important and promising step in the theory of inhomogeneous chemically associating fluids. This theory allows a lateral ordering of the fluid particles; further, it makes the study phase transformations at the fluid-solid interface, in the presence of chemical association, possible. The combined effects of bulk and surface chemical association can be studied by this method. We expect that a description of wetting phenomena of chemically associating fluids at solid surfaces can be achieved with the WOZ2 equation by using more sophisticated models and more adequate closures.

Dedicated to Ben Widom on the occasion of his 70th birthday

1. Introduction

Ben W i d o m has influenced Chemical Physics profoundly as well as being a wise and helpful friend. The authors are not alone in believing that his surname might be more

* Corresponding author. E-mail: [email protected]. I Permanent address: Institute for Condensed Matter Physics, National Academy of Sciences of the Ukraine, Lviv 11, Ukraine.

0378-4371/97/$17.00 Copyright c~5:) 1997 Elsevier Science B.V. All rights reserved PII S03 7 8 - 4 3 7 1 ( 9 7 ) 0 0 2 3 7 - 9

148 D, Henderson et al. / Physica A 244 (1997) 14~163

appropriately Wisdom. DH probably met Ben first at a Gordon Conference in 1963 but certainly at a conference in Pittsburg in 1964. Ben has always been physically and intellectually youthful. At the time of the Pittsburgh conference, DH thought of him as an exact contemporary. No doubt, this amused Ben. Now perhaps, an observer might think of Ben as the younger of the two. In any case, we can look forward to many more important contributions from him. It is a pleasure to salute Ben at the midpoint of his career.

The problem of describing chemical association in inhomogeneous fluids is one of the fundamental issues whose understanding is necessary for the development of a microscopic theory of adsorption and heterogeneous catalysis [1,2]. Also, surface phase transitions and other phenomena in these fluids are of interest. However, to include the effects of association between the fluid species as well as between them and the confining surfaces makes the already difficult description of inhomogeneous fluids even more difficult.

The statistical mechanical theory of homogeneous associating fluids has developed rapidly during the last decade and now has a strong fundamental basis. One of the most successful approaches for the investigation of chemical association phenomena is the theory of Wertheim which comprises the multidensity Ornstein-Zernike integral equation and thermodynamic perturbation theory [3 6]. This approach has been applied, starting from the simple dimerizing hard spheres through polymerizing fluids and to models with many site-site and site-center associations, as well as to mixtures (see, e.g., [7-15]). In addition, there has been significant progress in simulation studies of associating fluids [16-21].

The theory of inhomogeneous chemically associating fluids (ICAFs) is less developed than that of homogeneous CAFs. Several attempts has been undertaken to extend the so-called singlet level integral equation theory of Henderson, Abraham and Barker (HAB) [22,23] to associating fluids [24-29]. However, this simple method provides, by construction, only the density profiles of the particles while the inhomogeneous pair correlation functions are unavailable in this approach. It is possible that what appears to be a recent decrease in the number of applications that use this approach demonstrates an absence of progress in this line of study. An inability to investigate those phenomena that require the inhomogeneous pcfs is a drawback.

Kierlik and Rosinberg in [30-32] have overcome some of the problems that are encountered by integral equations by proposing a perturbational density functional (KRPDF) approach for ICAFs [30~33]. This approach is more sophisticated than the singlet level integral equations. It combines the ideas of the density functional method for inhomogeneous fluids [34] with Wertheim's theory for bulk association [3-6]. A comparison with computer simulation data indicates that this procedure is successful for nonoverlapping associating hard spheres near a hard wall and in a slit-like pore. As documented by Kierlik and Rosinberg in [32], it is difficult to apply this method for overlapping associating monomers. We are not aware of any studies using the KRPDF approach for models other than tangent hard spheres and systems with

D. Henderson et al. / Physica A 244 (1997) 147 163 149

hard core type forces between the fluid molecules and the surfaces. It is possible that the K R P D F approach will encounter additional difficulties in applications to systems with attractive interactions, as occurs with density functional approaches for nonassociating inhomogeneous fluids. More study is required.

In this communication, we make a qualitative step forward in the study of ICAFs by integral equations. We present the solution of the nonuniform Wertheim's Ornstein-Zernike equation. In the theory of nonassociating fluids, second-order (inhomogeneous) OZ equations have been used by many authors [35-41]. Important results have been obtained. In the study of a Lennard Jones fluid in contact with a hard wall, it was shown that the second-order OZ yields a qualitatively correct description of the wetting behavior of a fluid near a hard wall. Moreover, this theory has been applied to the primitive model of electrolyte solutions [37], in which the association of ions is an important phenomenon and to overlapping hard spheres [41], In spite of the success of Wertheim's theory for homogeneous systems, to our knowledge there have been no investigations involving the second-order WOZ equation (WOZ2) for inhomogeneous fluids.

We expect that the application of the WOZ2 equation would be important not just by making the pcfs available. The application of this method may have interesting consequences for the description of surface phenomena. In particular, we expect to be able to describe the influence of association on the vapour-liquid equilibrium near a surface by implementing the closures that contain associative interactions and more sophisticated bridge functions, if necessary. Also, investigations of association with the surface are a promising area for the procedure outlined here. However, in this exploratory communication, the solution of the nonuniform WOZ equation is given for the simplest hard-core model of a ICAF. With few exceptions, we restrict our attention to the simple associative nonuniform Percus Yevick approximation (APY2), which seems natural for hard sphere models. The implementation of other closures, such as the AMSA2 and AHNC2, that are the extensions of their counter- parts for homogeneous associating fluids [3-6, 9,12], will be considered elsewhere. It is to be expected that these alternative approximations will be suitable for models with longer range interactions. Also, because of the simplicity of the nonassociative interactions considered here, we restrict our attention to the dimerization equilibrium. The polymerization of the species is a technical issue which can be included later without qualitative complications.

2. Model and procedure

For simplicity, we restrict our attention to models of ICAFs with hard sphere interparticle interactions that also involve the possibility of the dimerization of the particles. The fluid is in contact with an idealized surface, in the form of a hard wall. To avoid unnecessary repetition in the description of the model and the details of our numerical procedure, we refer the reader to the original publications cited below.

150 D. Henderson et aL / Physica A 244 (1997) 147-163

Consider the one-component model for a dimerizing fluid A + A ~ A2 proposed by Wertheim [5] and generalized by Zhou and Stell for interpenetrating cores [42]. The interaction between the fluid particles is taken in the form

uo,2) = U.o.(rl2) + U°s(Xl2), (1)

where x12 = 1r12 + d(O~) - 11(02)1, d(f2) denotes the position and orientation of the attractive interaction site embedded into the core defined by the nonassociative term

and

l oo, r <L, U,o,(r)= D, L < r < 1,

0, r > l ,

(2)

= ~ - e " " x~<a, U.~(x) (3)

O, x > a .

Here ea~ and a are the association energy and range of attraction, respectively, L is the bonding length, D is the height of the square mound. The particle diameter is taken as the length unit. The parameter D satisfies the condition: e x p ( - / 3 D ) _~ 0. The geometric parameters of the interaction are subject to the restriction L < 2d + a < L - (2 - x/~)d to ensure steric saturation at the dimer level [5]. We choose the following set of parameters in this work: L = 0.9, d = 0.45 and a = 0.1.

The most important ingredient that is necessary in the following derivation is the associative "Mayer function". This characterizes the bonding effects and is defined as

I-5,43]

F.Arl2)=fd~lfdO2exp[-!3U.o.(rl2)]{exp[-,6U.~(Xl2)]-l} = ~exp([3eas)(2a - 2d + r12)(a + 2d - rle)2/24dZr12, L < r12 < 2d + a

t0 , otherwise.

(4)

The model, defined by Eqs. (1)-(4), is considered near an impenetrable surface

or, z~<O, (5) Ul(Z) = O, z > O ,

where z is the distance between the fluid particle and the wall. The WOZ2 equation for the partial correlation functions reads [3,4]

h~p(1, 2) -- c~p(1, 2) = ~ fd3c=.(1, 3)~ruv(3)hvp(3 , 2 ) , / i ,V

(6)

D. Henderson et al. / Physica A 244 (1997) 147-163 151

where the Greek indices take the value 0 or 1 for the undimerized and dimerized

species, respectively. The partial pcfs and partial direct correlation functions (dcfs) are

denoted, as usual by h,,(l, 2) and c,~( 1,2). The elements of the matrix of densities atll.(i)

are [3,5]

(7)

where the density profile (DP) of the particles, p(i), consists of the partial DPs p,(i)

(p(i) = PO(i) + PI(i)) with x = 0 and 1 for the undimerized and dimerized states of

particles, respectively.

According to [3,4], the partial DPs are

h(i) = <exPC - B~l(i)lexp[co(i)l = yO(i)exp[ - /XI,(i)],

~~(4 = bdi)cl(i) = _vlG)expC - NJIG)I 3 (8)

where t’ denotes the activity; c,(i) and y,(i) are the one-particle partial dcf and cavity

distribution function, respectively. The two-body dcfs are related to the one-particle

dcf as follows [3,4]:

coo(i,j)=$$$; c,,(i.j)=$#. 1 1

These relations can be written as one equation

Vjc,tj) = s

di[Viol(i)lco,(i,j),

or via the cavity functions as follows

Vcdj) = V ln.h(j); h(j)

Vc,(j) = v - [ 1 h(j) ’

(9)

(10)

(11)

Eq. (10) is the analogue of the Lovett-Mou-Buff-Wertheim (LMBW) equation

[44,45] for the partial one-particle dcfs. However, it is convenient, for a hard wall

interaction, to use the equivalent equation of Triezenberg and Zwanzig (TZ) [46] that

contains the inhomogeneous pair correlation functions. As in the theory of

inhomogeneous nonassociating fluids, this equivalence is established through the OZ

equation. However, for associating fluids, we must use the WOZ2 equation (5) which

can be written as

1 jd3 [S,,S(l, 3) - C,,(l, 3)1[&,6(3,2) + H,,,(3,2)1 = 6,,6(1,2). ”

(12)

152 D. Henderson et al. /Physica A 244 (1997) 147 163

In Eq. (12) we have introduced the functions

q~(1, 2) = ~ (p~.(1, 2)cr.~(2), (13) ,u

where q~ -= (C, H) and ~0 = (c, h). As usual, 6,p and 6 (i , j) denote the Kronecker symbol and delta function, respectively. With this result, Eq. (l 1) reduces to

and

Vco(j) = Vlnyo( j )

= f d i { h o o ( i , j ) p ( i ) V [ - f lU~(i)] - h ~ o ( i , j ) p o ( i ) V l n y o ( i ) } , J

(14)

Vcl (j) = V [ _ y ~ j

= f d i { h o x ( i , j ) p ( i ) V [ - f lUx(i)] - h a x ( i , j ) p o ( i ) V l n y o ( i ) } . (15)

The theory, consisting of Eqs. (6) and (10) or (6) and (14) and (15), must be supple- mented by a closure relation. We use the APY2 approximation, which reads

y~t~(1, 2) = C5,o6po + h,~(1,2) - c,~(1, 2), (16)

where the partial cavity pair distribution functions y~(1, 2) are related to the partial pcfs of the dimerizing model as follows [3-5]:

6~o6po + h ~ ( i , j ) = c-~V"""(r 'J)y~(i , j ) + F,~(rij)Yoo(i , j)(1 - 6~o)(1 - C~po) • (17)

In the following section, to give a more complete picture of the previous theoretical developments, we present some results from the associative analogue of the HAB equation proposed recently by Holovko and Vakarin [27]. This equation has been developed by inserting a nonassociative giant single particle (w) into an associating fluid (in the spirit of HAB theory) and using Wertheim's first-order OZ equation. In this approach, the density profile is considered as the limiting value of the fluid giant particle total pair correlation function. The degree of dimerization is determined by the solution for the homogeneous fluid. The associative PY closure has been applied for both the fluid fluid and fluid-giant particle correlations in the equation for the density profile [27]

t" h{W(r,2) - c~W(r,2) = ~ PYu~ j d r 3 c 2 ( r , 3 ) h { W ( r 3 2 ) , (18)

where hfW(r12) and cfW(r12) are the partial pair and direct correlation functions between a fluid particle in the bonded state ~ (~ = 0, 1) and the giant particle. The

D. Henderson et al. /Physica A 244 (1997) 147 163 153

elements of the matrix of densities, p~~, follow from the solution of the bulk problem [5]. The DP in this formulation are defined as follows

p(r) = polo(1 + MoW(r)) + p lo lh{ ' ( r ) . (19)

Another possible method for the development of a singlet level theory for inhomogeneous chemically associating fluids uses the associative LMBW equation, in the form of Eqs. (14) and (15). As does the HAB approach, this procedure employs the solution of the bulk associating fluid. For a nonassociating fluid, this procedure is exactly equivalent to the HAB approach with the HNC closure. For an associating fluid, the situation is more complex and the relation of this alternative procedure (if any) to the HAB procedure is not established. We shall refer to this approach as the ALMBW singlet theory. For comparison with the other approaches, some results obtained from this approach are presented.

We now describe briefly our numerical algorithm for the solution of the WOZ2 equation with the associative LMBW supplement. The presence of an associative interaction alters the details of the numerical algorithm somewhat. The resulting scheme is rather demanding computationally. The numerical algorithm for the solution of the WOZ2 equation together with the LMBW equation, transformed to the TZ form, and the APY2 closure uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations in the WOZ2 equation reduces to a sum of one-dimensional interations. The details of this procedure can be found in Refs. [35 39]. The grid size in the normal direction is Az = 0.05 and 100 terms in the Fourier-Bessel expansion are included. However, if the association energy is high, we must include more terms in this expansion to obtain smooth curves for the density profiles and the inhomogeneous pair correlation functions. We assume that the system reaches its bulk counterpart when z = 3. Our previous experience in the solution of the OZ2 equation [35 41] indicates that this is adequate. Obviously. there are situations (near critical points in fluids with attractive forces, for example) where this choice may be inadequate. Some of the calculations have been performed using the NSF funded BYU Computational Chemistry Silicon Graphics Power Challenge computer.

3. Results and discussion

We turn our attention to the results of the WOZ2/APY2 approximation. The associating fluid is considered at three bulk densities, namely, at p = 0.1, 0.4, 0.6, and for three different values of the association energy /~s,,., [~:,= = 2.0, [~:,,., = 10.0, and [~s,,,= = 12.0. However, we have observed that the results for [~;,,, = 2.0 are very close to the case of a hard sphere fluid in contact with a hard wall because of the negligible fluid dimerization. Therefore, these data are equivalent to a hard sphere fluid at a hard wall.

154 D. Henderson et al. / Physica A 244 (1997) 147 163

0.14

0.12

0.10

~- 0.08

0.06

0.04

0.02

0.00 i

0 0.5 i i i i i i

1 1.5 2 2.5 3 3.5 z

a ] 0.14

0.12

0.10

~- 0.08 "E

0.06

0.04

0.02

0 4

,., ,_ ,-- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i i i i i | i

0.5 1 1.5 2 2.5 3 3.5 4 z

0.14

0.12

0.10

~- 0.08

0,06

0.04

0.02

0.00

i , ................................................ ,..,-,

.•,-,

i i i i i i

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4

z

Fig. 1. The density profiles, p(z), of dimerizing hard spheres at the density p = 0.1 near a hard wall. The association energy fle,s is 2.0 (a) 10.0 (b) and 12.0 (c). The results follow from the APY2 closure (solid lines) and associative analogue of the HAB equation in the APY-APY1 approximation (dashed line). In (b) and (c), the lower solid curves are for the density of particles participating in bonds pl(z).

Consider first the results for the DPs from the second order integral equations, i.e. from WOZ2/APY2 closure, and from the associative analogue of the HAB approach,

i.e. the 'singlet' theory for Wertheim's model in the APY/APY1 approximation [27]. A comparison of the DPs from both theories is given in Figs. 1-3.

The curves presented in Fig. 1 are for p = 0.1. The singlet and pair theory yield coincident DPs at low density and at negligible dimerization, i.e. for fle,s = 2.0. However, at this low density, the difference between the two theoretical approximations increases as the degree of dimerization increases. Both theories exhibit the effect of the expulsion of the dimers from the vicinity of the wall (depletion of the profiles). However, in the pair theory these trends are stronger. The cusp on the density profiles, observed at z = L, is less pronounced in the pair theory than in the singlet level theory. In contrast to the singlet approach, the pair level theory also gives the density profiles of the particles participating in bonds, p~ (z). Moreover, the pair theory does not require the degree of dimerization as an input. It is clear from a comparison of Figs. 1 (b) and (c) that increasing association leads to a stronger depletion of pl (z) near


1.0

0.8

0.6

0.4

0.2

0.0 i i i i i i N

0 0.5 1 1.5 2 2.5 3 3.5 Z

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

'\\ • \

\

i i i i i i i

0.5 1 1.5 2 2.5 3 3.5 Z

0.7

0.6

0.5

-~- 0.4

0.3

0.2

0.1

0 0 0.5

\ \

\

, , "

. , , , -

i i i i i i

1 1.5 2 2.5 3 3.5 4 Z

Fig. 2. The same as in Fig. 1, but at bu lk fluid densi ty p = 0.4.

a wall. The asymptotic value of p~(z) gives the bulk number density of the bonded particles. At low bulk density, the influence of the wall on p~(z), as well as on p(z), does not extend farther than z = 1. All the important events occur close to the surface.

It is well-known that for hard spheres, the DPs from the singlet PY approximation are determined by the bulk pair correlations. The PY2 approximation is superior to the singlet method with simple closures, especially in the vicinity of a wall [35 37]. We expect similar trends for associating fluids.

In Fig. 2 we present the results for the DPs for an associating fluid at the density p = 0.4 at different degrees of dimerization. For/~E,s = 2.0, we observe behavior that is very similar to a hard sphere fluid in contact with a hard wall. The contact value of the DP from the pair theory is higher than from the singlet theory. It is known that, with the PY closure, the latter approach underestimates the contact value, especially at intermediate and high densities. Thus, we are confident of the higher accuracy of the pair theory results. The DP at this density indicates the formation of a second layer of particles. For higher dimerization, ~,~ = 10.0 (cf. Fig. 2b), the contact values from both approaches become lower. However, the trend for depletion is stronger for the pair theory. At still higher dimerization,/?~,s = 12.0, the contact value of the DP from

156 D. Henderson et al. / Physica A 244 (1997) 147-163

2.5

2.0

1.5

1.0

0.5

0.0 i i , i J

0.5 1 1.5 2 2.5 z

i i

3 3.5

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0

L

\ \ \

i i i i i i i

0.5 1 1.5 2 2.5 3 3.5 z

1.6

1 . 4

1.2

1.0

0.8

0.6

0.4

\ \ \ \ \

- _ : _ - - _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.2 . . . . . . . 0 0.5 1 1.5 2 2.5 3 3.5 4

Z

Fig. 3. T h e s a m e as in Fig. 1, bu t a t b u l k f luid d e n s i t y p = 0.6.

the APY2 is even lower than the bulk density. The singlet approach does not shows this physically reasonable behavior. The depletion of the profile leading to a lower contact value of the DP in comparison with the bulk density has been observed by Kierlik et al. in [32,33]. Also, we observe a smoothing of the layering behavior of the D P with increasing dimerization. These trends follow straightforwardly from entropic arguments.

In Fig. 3 we show the results for the DPs at the density p = 0.6. As expected, the difference between the DPs from the APY2 and singlet approach increases with increasing density. The largest difference is observed at contact (yielding different predictions for the pressure) and within the region of the first minimum. A weak layering of the fluid is seen in Fig. 3a. Switching on the association (/~as = 12.0) leads to a smoothing of the layered structure in the interfacial region and to a significantly lower contact value for the DP. A weak layering in Pl (z) is observed but is expected to increase at higher fluid densities. This effect may influence wetting and layering transitions for the models with chemical association and attractive particle-particle and particle-wall interactions.


1.8

1.6

1.4

1.2

1.0 "E

0,8

0.6

0,4

0.2

\

\ \

i i i i

0 0.5 1 1.5 2 Z

i i i

2.5 3 3.5 4

Fig. 4. A comparison of the density profiles from the APY2 approximation (solid line), from the associative analogue of the HAB equation in the APY/APY approximation (dashed line) and the singlet ALMBW/APY equation (dotted line) for p = 0.6 and [:~e.~ = 10. The upper curves for p(z) whereas lower curves are for pl(z).

The 'singlet' level results above follow from the associative analogue of the HAB integral equation. As we mentioned in the theoretical section, another approach is the use of the ALMBW equation with the bulk APY (or from any other approximation) pair correlation functions taken as an input. The results for the DPs that follow from the ALMBW/APY approximation are compared with the APY2 and the HAB APY/APY results in Fig. 4. We have chosen the 'difficult' state of a hard sphere fluid characterized both by both high dimerization and high density, /h:,.~ = 10.0 and p = 0.6. The ALMBW/APY profiles are much closer to those of the more sophisticated APY2 theory than are HAB APY/APY results. Moreover, the component of the profile that describes the behavior of the bonded particles with respect to the wall, pl(z), from the ALMBW/APY theory has similar shape to the pl(z) from the APY2 approach. It would be interesting to compare the LMBW/APY results with those of the HAB AHNC/APY approach. Verifying the accuracy of these approaches at a quantitative level requires simulational studies. These are now in progress in our laboratory. However, there is little doubt that the pair level theory is successful for the DPs of chemically associating fluids. However, this is obtained at the cost of more massive numerical work.

In Fig. 5 we show the degree of dimerization of the fluid, Z(z) -- p~(z)/p(z), versus the distance from the wall for intermediate (p = 0.4) and high densities (p = 0.6) and for two degrees of dimerization. The asymptotic value of the degree of dimerization determines the bulk dimerization. Near the wall, for both densities at the chosen value of the association energy, X(z) is lower near the wall than in the bulk. It is lower for the lower density, as expected. A cusp in Z(z) is observed at the distance corresponding to the bonding length. At larger distances, Z(z) is smooth and tends to the bulk value. For the higher density, p = 0.6, there is a slight evidence of a maximum corresponding to a second layer for [3S,,s = 10.0; however, at /h:,~ = 12.0 the Z(z) values are almost


0.9

0.8

0.7

~" 0.6

0.5

0.4

0.3 0 0.5

Y / ,"

/," / ~ . . . . . . ,," I

,' i ,," /

I ," / . I

i i i i i i

1 1.5 2 2.5 3 3.5 4 Z

Fig. 5. The dependence of the fraction of dimerized species X(z) on the distance from the wall for fl~a, = 12 for p = 0.6 (solid line); [3ea~ = 10 for p = 0.6 (dotted line) and fle,s = 10 for p = 0.4 (dashed line).

constant for z > 1, due to a very high dimerization. The wall is chemically inert and does not influence the processes of association in any way except geometric

confinement. We now discuss the inhomogeneous pair distribution functions (IPDFs) from the

APY2 approximation. The lower level equations do not distinguish between the

homogeneous PDFs and the IPDFs. Below, we compare three functions, namely

9bulk(t'lZ) which follows from I P D F g ( Z l , Z 2 , R 1 2 ) for Z1,Z2"---~ (30; 9 ( Z l = 0 , Z 2 = 0, R 1 2 ) ~- gll and 9(z l = O, z2, R l~ = O) =-- 9 . . The intermolecular parts of these functions are shown in Figs. 6~8. Fig. 6 contains the results for p = 0.1 at/3e~s = 2.0

and /~as = 12.0. For /3e,s = 2.0, we observe behavior similar to that of a hard

sphere fluid [-35-37]. The function 9b,~k(r12) is almost the same as 9± whereas 911 is lower at the contact and for R12 < 2. At low density, these functions differ from unity at small interparticle separations. For high dimerization,/~e,~ = 10.0, we observe a different sequences of these curves. The lowest function is 9±, and the upper

curve is gti- This 9it curve for both association energies is closer to the ideal gas behavior, i.e. 9 - 1, so the in-plane correlations are weaker for a hard sphere fluid as well as for a chemically associating fluid near a hard wall. In the case of a chemically

associating fluid this behavior is due to the depletion of dimerized species from the contact layer.

In Fig. 7 we presented a set of IPDFs for an intermediate density fluid, p = 0.4 for /3eos = 2.0 and/3~,~ = 12.0, in parts a, and b, respectively. F o r / ~ = 2.0 at this density (Fig. 7(a)), all the curves are close to each other. The sequence of curves at contact is similar to the case of low density, p = 0.1, cf. Fig. 6(a). Contacts between particles in a parallel orientation are slightly less probable than in a perpendicular orientation. For high association energy,//~,~ = 12.0 (Fig. 7(b)), the correlations in the bulk and for different orientations of the particles close to the surface are different. The bulk function, gbutk(r12), has a maximum in addition to the maximum corresponding to contact. Only for the function 9± is the secondary peak weakly pronounced. The

D. Henderson et al. / Physiea A 244 (1997) 147-163 159

cq

1.1 5 , - -

1 . 1 0

I , 0 5

1.00

0 . 9 5

i

\ \

i I L I i

2 3

r l 2

1,1 , i i

b

1.0

0.9

CY~

0.8

0 . 7 1 2 3 4

r-1 2

Fig. 6. The i n h o m o g e n e o u s pair d is t r ibut ion funct ions for the bulk fluid gb~lk(r12) (solid line), 9(zl = 0, 2" 2 = 0, R12) ~ gkl (dashed line) and 9(zl = O, z2, R 1 2 = 0) ~ 91 (dot ted line). The value of the associa t ion energy is fle,~ = 2.0 (a) and flea~ = 12 (b). T h e bulk fluid densi ty is p = 0.1.

minimum of g± is shifted towards larger interparticle separations than in gbulk(rl2),

because of the difference in the pair correlations in the bulk fluid and the pair correlations between particles belonging to different layers from the wall. The parallel pair correlation function decays smoothly and does not extend to large separations. If the wall would be chemically active then one should expect manifestations of the Langmuir Hinshelwood mechanism for a chemical reaction between the adsorbed particles [1,2]. For this association energy, we still observe an enhanced parallel pair correlation function between adsorbed particles in comparison to the bulk homogeneous fluid.

The results for the highest density studied, p = 0.6, are given in Fig. 8. For the case of negligible dimerization/3e,~ = 2.0, the model is similar to a hard sphere fluid at this density and the trends in the behavior of the DPs are as in the previous case, p = 0.4,


1.8

1.6

1.4

1.2

1.0

0.8

C]

i I , I i

2 3

rl 2

0,4

Cr~

1.5

1.4

1.,3

1.2

1.1

1.0

0.9

i

i,

i I i [

2 3

rl 2

b

Fig. 7. The same as in Fig. 6, but for p = 0.4.

cf. Fig. 7(a). Some changes in comparison to a hard sphere fluid occur for high

dimerization, i.e. for fiea~ = 12.0 )(Fig. 8(b)). This is the largest energy for which we have performed computations.

For high density, the correlations in the homogeneous fluid are strong. However, due to the high dimerization, the second maximum is quite low. The strongest correlations are observed at small interparticle separations. The correlations between

the particles located close to the confining surface are weaker. The weakest are the in-plane correlations; the normal correlations are stronger but are still smaller than in the bulk. However, all the distribution functions have cusps at z ~ 1.9, as a result of the strong dimerization. The peculiar behavior of the function, g~, at z ~ 1.4 can be explained as resulting from the correlations between the remaining monomers that are adsorbed at the wall with perpendicularly oriented dimers. As discussed above, a hard wall just limits the geometry. Dimerization in the bulk is more probable. Close to the

D. Henderson et al. /Physica A 244 (1997) 147 163 161

c'4

2.5

2.0

1.5

1.0

0.5

I

i i l

2 3

-q o

rl 2

O4

C

1.8

1.6

1.4

1.2

1.0

0.8

, . ' -

I

2

rl 2

i

b

3 4

Fig. 8. The same as in Fig. 6, but for p = 0.6.

surface, one can expect the formation of dimers between particles that are in contact with the wall those in layers more distant from the wall, i.e. the Eley-Rideal mecha-

nism [1,2]. In the case of a chemically active surface, the results can be expected to

change.

4. Conclusions

In summary, we have presented, for what we believe is the first time, the

solution of the second order Wertheim OZ equation for inhomogeneous fluids. Our computations have been performed for low, intermediate, and high bulk fluid

densities and for weakly and highly dimerized fluids. A comparison with associative analogue of HAB equations has been given. Moreover, in addition we have


calculated the density profiles using the associative LMBW/APY equations. The pair level theory is much greater utility because it provides mutually consistent density profiles and inhomogeneous pair correlation functions. It also permits an investigation of the dimerization probability as a function of the distance from a wall. It is known that, for nonassociating fluids, the pair level theory yields a much better description of the fluid density and correlation functions in the vicinity of the surface than do the singlet level approaches. We expect similar behavior for associating fluids. However, a quantitative verification of the accuracy of the proposed approach must await through computer simulation studies. This work is not in progress.

We would like to emphasize that the pair level theory for associating fluids should be applicable to the description of wetting and drying transitions near absorbing walls. An extension of the proposed algorithm is rather a technical issue. It would be possible and of interest to study the interplay of fluid-surface and fluid fluid attractive interactions in conjunction with association on these phenomena. An interesting extension may also include the study of inhomogeneous multisite models for polymerization as well as associating fluids in disordered porous media. The application of Wertheim's theory is not restricted to directional bonding. The second level WOZ2 equation can be applied to the models with spherically symmetric associative interactions [47].

Acknowledgements

This work was supported in parts by NSF grants CTS94-023584 and CHE96- 01971, by Cray Research. Inc. of Mexico under its University Research and Develop- ment Grant Program. O.P. acknowledges his fruitful discussions with M.C. Lozada Garcia. S.S. is grateful to the Instituito de Quimica de la UNAM, to the Departmento de Intercambio Academico and DGAPA de la UNAM for their financial support during his stay in Mexico.

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the nonuniform wertheim ornstein-zernike equation: density profiles and pair correlation functions...

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