Structure of hardparticle fluids near a hard wall. IV. y w (z,θ) for homonuclear harddiatomicsS. Labík, A. Malijevský, and W. R. Smith Citation: The Journal of Chemical Physics 94, 2232 (1991); doi: 10.1063/1.459893 View online: http://dx.doi.org/10.1063/1.459893 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/94/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Background and bridge functions for the homonuclear hard diatomic fluid J. Chem. Phys. 90, 7330 (1989); 10.1063/1.456212 Structure of hardparticle fluids near a hard wall. III. y w (z) for hardsphere mixtures J. Chem. Phys. 88, 3893 (1988); 10.1063/1.453838 Structure of hard particle fluids near a hard wall. II. y w (z) for hard spheres J. Chem. Phys. 88, 1944 (1988); 10.1063/1.454118 Structure of hard particle fluids near a hard wall. I. y w (z) for hard disks J. Chem. Phys. 86, 5783 (1987); 10.1063/1.452507 Structure of a simple fluid near a wall. I. Structure near a hard wall J. Chem. Phys. 69, 5450 (1978); 10.1063/1.436536

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Structure of hard-particle fluids near a hard wall. IV. Y w(z,e) for homonuclear hard diatomics

s. Labfk,a) A. Malijevsky,a) and W. R. Smith Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada NI G 2 WI

(Received 24 July 1990; accepted 23 October 1990)

Homonuclear hard diatomics in the vicinity of a hard wall have been studied using Monte Carlo simulations. During the simulations the cavity distribution functions for parallel and perpendicular orientations with respect to the walland their Legendre expansion coefficients were calculated. It is demonstrated that the Legendre expansion is rapidly convergent and only several coefficients are required to describe the wall-diatomic correlation function within the simulation error. In addition, two simple methods for theoretical prediction of the cavity distribution function at small distances are compared with the simulation data. These approaches require as input quantities only thermodynamic data for the bulk homogeneous fluid. Both methods yield good accuracy for the cavity correlation function at the specific orientations indicated and for the first two Legendre coefficients, but fail to accurately describe higher Legendre coefficients.

I. INTRODUCTION

The primary quantity describing the structure of a fluid in the vicinity ofa wall is the density profile,pw (z,D.), giving the dependence of the local density on the wall-particle dis­tance z, and orientation D.. A related quantity is the pair correlation function (PCF), gw (z,D.) = Pw (z,D.)lp, where p is the average bulk fluid density. At distances less than the contact distance of a hard particle with a hard wall, Pw (z,D.) and gw (z,D.) are identically equal to zero. At such distanoces, the primary structural quantity of interest is the cavity correlation function (CCF), Yw(z,D.) = gw (z,D.)exp(uwlkn. The term "cavity" refers to a par-

ticle which interacts with all the fluid particles but not with the hard wall. y w (z,D.) is, in general, a much smoother func­tion thangw (z,D.). In addition to being of interest in its own right, it can also be used as reference quantity in perturba­tion theories.

Unlike the case of spherically symmetric particles, which have been intensively studied by both computer simu­lations and theoretical approaches (see., e.g., I for a review), hard nonspherical molecules in the vicinity of a hard wall have received only scant attention. Sullivan et al. 2 reported the wall-center and wall-site correlation functions of hard diatomics using computer simulations and the Reference In­teraction Site Model (RISM); Dickman and Hall simulated hard chain-like molecules3 and Nezbeda et al.4 published a computer simulation study of the wall-site correlation func­tions of hard heteronuclear diatomics, linear triatomics, and their mixtures with hard spheres. Sokolowski and Patryki­jewS and Holyst6 studied wall-nonspherical-particle proper­ties on a theoretical basis. Thus, only limited information concerning wall-nonspherical-particle structure is currently available.

., Permanent address: Department of Physical Chemistry, Institute of Chemical Technology, Prague [6628, Czechoslovakia.

In previous papers of this series 7-9 (cited hereafter as papers I-III) we described Monte Carlo (MC) computer simulations and possible theoretical approaches to the calcu­lationof Yw (z) for hard disks, pure hard spheres and mix­tures of hard spheres, respectively. We have shown that us­ing only thermodynamic data, Yw (z) can be accurately calculated in the wall-cavity overlap region and reasonably extrapolated to small distances from the wall. The principal aim of this paper is to extend the previous studies to non­spherical molecules, and in particular to homonuclear hard diatomics (HD).

For linear molecules, the wall-particle correlation func­tion depends on the distance from the wall and on a single orientational angle. For symmetric molecules the wall-cen­ter coordinate frame is probably the simplest choice, and gw (z,D.) =gw (z,e) can be expanded in the Legendre series

gw (z,e) = I gwk (z).J2k + IPk (cos e), '-1) k=O

where Z is the distance of the center of the HD from the wall and e is the angle between the vector connecting the sites of molecule and the vector normal to the wall (see Fig. 1), and where Pk (cos e) is the Legendre polynomial of degree k. We remark that the origin of the z coordinate used in this paper differs from that used in papers I-III. Here, the position of the wall is at z = 0 and the closest approach of the HD corre­sponds to z = 0.5u, where uis the diameter of a sphere form­ingthe HD.

The expansion coefficients, gwk (z), are related to gw(z,e) by

.J2k + 1 II gwk(Z) = gw (z,e)Pk (cos e)d(cos e). 2 _I

(2)

We remark that using definition (2), guD (z) is identical to the angular average of the wall-center correlation function, gw (z), and due to the symmetry of the molecules all coeffi-

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Labik, Malijevsky, and Smith: Fluids near a hard wall. IV 2233

cients with odd indices are identically zero. The higher coef­ficients are conveniently expressed in the reduced form

g~k (z) = gwk (z)/g lLO (z) =g';'k (z)/gw (z), (3)

since the reduced coefficients generally show a much simpler dependence on distance thangwk (z) themselves.

An alternative description of the wall-particle correla­tion function is obtained using the wall-site coordinate frame and the corresponding function is denoted by Gw (Z,e). Z is the distance of a sphere from the wall and e is defined as in the wall-center coordinate frame. The Legendre expansion of Gw (Z,e) has the same form as (I) and the wall-site ex­pansion coefficients, GWk (Z), are defined by the analogue of Eq. (2). The leading term in the expansion, GILO (Z) =Gw (Z), is the average wall-site correlation func­tion, and all Legendre coefficients are nonvanishing in gen­eral.

It is well known that the expansion (I) converges rea­sonably rapidly only for smooth functions. The correlation function of the hard-wall-hard-particle pair has ajump dis­continuity at distances and angles corresponding to wall­particle contact, and its expansion (I) is thus extremely slowly convergent in both the wall-center and the wall-site frames. The cavity correlation function,yw (z,e), on the oth­er hand, is a smooth function, and its Legendre expansion is thus much more rapidly convergent

Yw(z,e) = f YWk(z)~2k + IP~cos 0). (4) k=O

It may be expected that only several terms in this expansion are required to achieve a precise description of the entire angular dependence.

In Sec. II of this paper, we describe our computer simu­lation methodology and we test the consistency of our calcu­lations via the comparison of thermodynamic data calculat­ed from different routes. Section III is devoted to the extension of the theoretical predictions of y w (z,e) proposed and tested in previous papers of this series. In Sec. IV we present MC results for the structure and we compare these with the predictions obtained from the theoretical ap­proaches described in Sec. III. We also examine the conver­gence of the expansion of y w (z,e), and the possibility of the calculation of the complete wall-center or the wall-site corre­lation functions from their Legendre series expansions.

II. MONTE CARLO SIMULATIONS

All simulations presented in this paper were performed using the MC method. The algorithm used is in principle a combination of previous treatments used for the hard-wall­hard-sphere calculation9 and for the bulk fluid simulations ofHD.IO,11 The NVTensemble was used with a rectangular box (prism) of size I X 1 X 3 units. The ends of the prism were hard walls and periodic boundary conditions were used in directions parallel to the other faces. The'number of hard diatomics, N, was fixed at 768 and the system density was adjusted by changing the size of the RD. We performed sim­ulations for three bulk densities near the overall bulk densi­ties Nfi3 IV = 0.2, 0.3, and 0.4 at the fixed RD elongation L * = L ((7 = 0.6. The initial configuration for the highest

density was formed as a copy of three identical cells of unit size obtained from our previous calculations for the homoge­neous fluid,11 and about 3 million configurations ,were ini­tially generated to reach equilibrium. For lower densities the final configuration of the next higher density was used as a starting point and the same number of initial configurations were generated to equilibrate the system. Our final results were obtained by averaging over an additional 15 to 29 mil­lion configurations. We remark that we have used longer simulation runs in comparison with bulk fluid simulations, and a greater sampling at higher densities in order to elimi­nate correlations between subsequent analyses, which occur as a result of the tendency of the simulations to preserve the structure in the vicinity of the wall. The relatively long runs used resulted in a reasonable precision of the calculated har­monic coefficients. For a given simulation length, the wall­particle correlation functions are generally less precise than the bulk fluid correlation functions. Contrary to the bulK fluid case where N(N - 1)/2 pairs contribute to the averag­ing for each analyzed configuration, in the wall-particle cal­culation only N measurements contribute.

In order to test the consistency of our. results we calcu­lated the bulk density using two independent methods. First, the bulk density was calculated as an average density in the cube in the middle third of the prism. Second, the bulk den­sity was calculated using the exact relation for the pressure at the wall

PV I(NkT) = 2Gw (0/2) (5)

obtained using recently derived relation for general fused­hard-sphere fluids at a hard walV and the equation of state 12

PV I(NkT) = I + 1.45999371 + 2.171 07712

- 2.515 93713

,

- (1 - 'r7) 3

(6)

where the packing fraction 71, is given by 71 = Nfi3( 1 + 3L *12 - L *3/2)IV. The bulk density was ob­

tained by substituting Eq. (5) in (6) using values of Gw (0'12) extrapolated from the MC data, and solving the resulting cubic equation. The resulting values of density, to­gether with the other numerical details of our MC simula­tions, are given in Table I. The mutual agreement of both methods confirms the consistency of our results.

The pair correlation function was calculated in both the wall-center and wall-site coordinate frames and the coeffi-

TABLE I. Parameters of the simulations and the computed bulk fluid den­sities. a is the diameter of the spheres forming the DB, Nc is the number of configurations generated, N, is the sampling interval between subsequent analyses of the system, and P a and P b are the bulk densities obtained (rom the middle third of the simulation box and from solving Eqs. (5) and (6), respectively.

Run a nc N, Po Ph

A 0.0927 15 X 10" 3072 0.198 ± 0.02 0.194 ± 0.04 B 0.1067 23Xl0" 4608 0,299 ± 0.01 0.299 ± 0.03 C 0.1176 27X 10" 6144 0.398 ± 0.01 0.393 ± 0.05 D 0.1176 29XlO" 6144 0.399 ± 0.01 0.394 ± 0.05

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2234 Labik, Malijevsky, and Smith: Fluids near a hard wall. IV

cients gwO Cz), g:2 (z), g:!A (z), g:!A (z), GwO (Z), G:1 (Z), and G ~2 (Z) were calculated. In order to further test the reliability of the data we performed two independent simula­tion runs at the same overall box density near Ncr IV = 0.4. The excellent mutual agreement of the results of both runs can be seen from Table I (simulations C and D) and the figures shown later in Sec. IV.

We obtained the cavity correlation function using a method similar to that used in paper II. The method is based on the fact thatyw (z,O) is proportional to the ensemble aver­age of the probability of the successful insertion of a cavity at the given distance and orientation. The proportionality con­stant is~, the CCF of a HD completely inside the wall, given by

y~ = exp(llfl../kT) , (7)

where Ilfl../kT is the reduced excess chemical potential of hard diatomics at the bulk density. The insertion probability was obtained as follows. For each analyzed configuration we randomly chose one hundred points which define one hundred lines perpendicular to the walls. On these lines we attempted to insert a cavity at the paralIel, perpendicular and 10 randomly selected orientations at 192 distances rang­ing from - 0.8u (corresponding to the cavity being fully immersed in the wall at the perpendicular orientation) to 4u. For each distance, the average probabilities of successful cavity insertion at the parallel and perpendicular orienta~ tions yield "slices" of Yw (z,O) at these orientations. The average insertion probability over the 10 random orienta­tions is proportional to the expansion coefficientywO (z) and the higher reduced Legendre coefficients are l3

Y:k(Z) =~2k+ 1 (Pk(cos e», where ( ... ) denotes the average value over all successful insertions at the given dis­tance. In this manner, we calculated Y~2 (z), Y~>4 (z), andY:6 (z).

III. APPROXIMATION FOR Yw(z,6)

In papers I and II we proved that the limiting slope of the CCF for cavities in the limit of a cavity fully immersed in the wall is given exactly by an expression involving the bulk pressure and a volume excluded by the cavity. Assuming that this relationship is also valid for larger distances we derived the zero-order approximation

In[Yw(z,e)ly~] = -PIlV(z,e)lkT, z>zo(e), (8)

where 11 V(z,e) is the excluded volume of a cavity located at the distance z and the orientation e (see Fig. 1) (i.e., the volume excluded to the centers of spheres on a DB by the cavity), and zoe e) is the greatest distance for which the en­tire cavity is fully immersed in the wall. Equation (8) gives satisfactory results only at z->zo(e) or in the low density limit. For higher distances and densities, we proposed two empirical corrections

In[yw (z,e)/~]

= -PIlV(z,e)lkT[l +a(z-zoCe»]; z>zo(e)

(9) and alternatively

wall .,,, ,

FIG 1. Definitions of the distance, orientation, and excluded volume of a HD cavity. z and Z are the respective distances from the wall of the cavity in the wall-center and in the wall-site coordinate frame and eis its orientation. The dashed line demarcates the volume excluded by the cavity used in Eqs. (8), (9), and (10).

In [Yw (z,e)ly~]

1 = - PIlV(z,O)lkT , z>zo(O), I-b [z-zoCe)]

CIa) whet:e a and b are density-dependent constants, whose val­ues may be determined from the pressure. Equation (5) gives the relation between the pressure and the average PCF at contact with the wall, Gw (uI2), which can be calculated fromyw (z,e), using

1 JI Gw (a/2) = - G",(uI2,e)d(cos e) 2 -I

= f Yw(uI2,e)d(cos e)

= f Yw (u/2 + L * cos 0 12,O)d(cos e). (11 )

Combination of Eqs. (5) and C 11) with (9) or (10) gives nonlinear equations for a or b which may be solved numeri­cally. Since both the pressure and y~ can be obtained from the equation of state ofthe HD fluid [Eq. (6)], the CCF can thus be predicted using HD thermodynamic data only.

We note in passing that another method of calculating Yw (z,e) exists in principle. 14

•15 This approach is similarly

based only on a knowledge of the equation of state. It uses the fact that the wall-cavity pair can be treated as a single super­posed particle and the problem of the CCF calculation is transferred to that of calculating the thermodynamic prop­erties of the superposed particle. The CCF in this case is given by9

In[yw (z,e)/~] = /It.tw -IlIlWDB (z,O), (12)

where Illl w is the excess chemical potential of the wall and Oil WDB (z,e) that of a particle formed by a wall-cavity pair. The difference in chemical potentials appearing in Eq. (12) can be obtained from a model in which the wall is taken as the limit of an infinitely diluted sphere of infinite size in a

J. Chem. Phys., Vol. 94, No.3, 1 February 1991

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Lablk, Malijevsky, and Smith: Fluids near a hard wall. IV 2235

mixture with hard diatomics. Unfortunately, such a calcula­tion requires a very precise_equation of state for highly non­convex hard bodies, which is not available at present, and we do not report any results based on this treatment.

IV. RESULTS AND DISCUSSION

In Fig. 2 we show results for In [Yw (z,O) Iy~] for the perpendicular and parallel orientations and for In [yu() (z)/y~1. It is seen that both profiles at fixed orienta­tions are monotonically decreasing functions which are al­most linear at distances near z = O. The nonlinearities at small distances result from the nonlinearity of /1 V(z,O) at these distances. The convex upwards shape at the largest distances corresponds to the onset of the first minimum of gw (z,O). At negative distances, the values for the perpendic­ular orientation are smaller than those for the parallel orien­tation, resulting from the fact that a cavity perpendicular to the wall has a greater excluded volume than a cavity parallel to the wall; at higher distances the opposite is the case. As expected, the angular average, YwO (z), displays values inter­mediate between both extreme orientations. Comparison of corresponding values at which the curves intersect the verti­cal dashed lines shows that the parallel orientation is pre­ferred to the perpendicular orientation at contact of the DB with the wall. The results of the approximate approaches are quite satisfactory. In the overlap regions (z<0.5a), the re­sults for the parallel orientation and the angular average are in agreement with the experimental data within the statisti­cal noise of the latter. In the perpendicular orientation the theoretical approaches underestimate the probability ofcav­ity insertion.

In Fig. 3, we compare the MC data with the results ob­tained from Eqs. (9) and (10) for the reduced coefficients Y!2 (z) andY~>4 (z). The Legendre coefficients, especially y~>4 (z), exhibit a rather complicated distance dependence, with several minima and maxima which arise from the dif­ferences in the orientational preferences discussed in the pre­vious paragraph. The values y~ (z) are generally about 5

-5

a.o ,

.. _-1--_.L~

0.0 0.5 1.0 -0.5 0.0 -10

-0.5 ·-0.5 0.5 1.0

zia zia Va

FIG. 2. Logarithm of the cavity-insertion probability at the perpendicular and the parallel orientations and the angular average In [YuO (z)/~,]. The points are the MC results, and the dashed and f~llines are the res~l~s from Eqs. (9) and (10), respectively. The numbers give the system denSIties and the dashed vertical lines denote the distances of closest approach of a HD to the wall.

0.4 ..

L--;t-.--'-o';;--L...-'----l -0.5 0.0 0.5 1.0 -0.05L.._-+0.~5 --'-.,,10.""0 -'----"O.s\;-'--.l>;;J.O

Z/" zia

FIG. 3. The reduced Legendre coefficients Y~2 (z) and y~ (z). The notation is as for Fig. 2.

times smaller in magnitude than those ofy~ (z) . We remark that, although we have also calculated y~6 (z), its absolute magnitudes were less than 0.02 at all densities, comparable in magnitude to the scatter of the raw MC data. All these observations confirm the expected rapid convergence of the Legendre series of Yw (z,O). The theoretical approaches are only partially successful. y~2 (z) is described precisely with only minor discrepancies at the highest density. y~ (z) is predicted reasonably accurately everywhere at low density and at very low distances at all densities shown. The accura­cy of both theoretical methods deteriorates with increasing distance and density.

Figures 4 and 5 show the pair correlation function and the first two nonvanishing reduced expansion coefficients in the wall-center coordinate frame. The points are the original Me data (notice the two separate runs at density 0.4), and the full lines were obtained from Eq. (2), where gw (z,O) =Yw(z,O)expC - uwlkT) andyw(z,O) was calculated from

the summation of Eq. (4). The dashed curves correspond to the low density limit, i.e., to the expansion coefficients of

FIG. 4. The wall-center pair correlation function. The points are the MC simulation results, the full lines are results obtained from resummation of y", (z,e), and the dashed lines are the low-density limit.

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2236 Lablk, Malijevsky, and Smith: Fluids near a hard wall. IV

- .. ----,------- r-05

~.,

FIG. 5. The reduced Legendre coefficients It.:.2 (z) and gt,., (z) of the waIl­center pair correlation function. The notation is as for Fig. 4.

1.5 pa3 .. 0,3 pa3 _ 0.4

-N 1.0

g d J' 1

0.0

ZIa ZIa Va

FIG. 6. The wall-site pair correlation function. The points are the MC simu­lation results, the full lines are results obtained from the summation of the first four terms of the Legendre series of y w (z,8), the dotted lines are results obtained using the leading termy"" (z) only, and the dashed lines are the low density limit.

0.5

&: .J 0.0

-0.5

&: • ~ 0.0 ...

(!) '.

-0.5

Zla

~~.0.4

.... ' 2 Zla 2

Zla 2

FIG. 7. The reduced Legendre coefficients G~'1 (Z) and G~~ (Z) of the wall-site pair correlation function. The notation is as for Fig. 6.

15 •

10

5

0.0 0.5 1.0

cos(El)

FIG. 8. The orientation profile of Gu• (0/2,8) (at contact with the wall). The notation is as for Fig. 2.

exp( - uw/kT). It is thus seen that the y expansion yields very precise results for aIlgwl; (z) and that the higher order reduced coefficients tend to their correct low density limit, which again implies rapid convergence of the y expansion.

Similar results are obtained for the wall-site coordinate frame, shown in Figs. 6 and 7. The resummation of the y expansion is in perfect agreement with the Me data. Even the simplest possible approximation, truncating the yexpan­sion after the first spherically symmetric term, YuO' gives results which are in qualitative agreement with the simula­tion data. The Legendre series with two coefficients gives results indistinguishable from those obtained using the en­tire series.

In Fig. 8, the orientational profiles of Gw (Z,O) at con­tact with the wall (Z = 0'12) are shown. The points were obtained from resummation of the Legendre expansion of Yw (z,O) and the curves were obtained from the approxima­tions defined in Sec. III. The preferred molecular orientation at contact is parallel to the wall, which corresponds to the maxima at cos 0 = O. (See also the previous discussion con­cerning Fig. 2. ) The shallow minima observed near cos 0 = 0.75 correspond to the least favored orientation. The accuracy of the theoretical results seems to be quite sat­isfactory but they tend to underestimate the simulation re­sults for the parallel configuration, a tendency which in­creases with increasing density.

IA. Malijevsky, R. PospIsil, S. Labik, and W. R. Smith, Mol. Phys. (sub­mitted).

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Labik, Malijevsky, and Smith: Fluids near a hard wall. IV 2237

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