Statistical mechanics of adsorption: Summation of the virial expansion for the two-dimensional equation of state

Statistical Mechanics of Adsorption" Summation of the Viriai Expansion for the Two-Dimensional Equation of State S. SOKOLOWSKI Department o f Theoretical Chemistry, Institute of Chemistry, Maria Curie-Sk,,rodowska University, 20031 Lublin, Nowotki 12, Poland Received N o v e m b e r 23, 1976; accepted February i7, 1978 The Fulifiski procedure of summation of the virial expansion for the three-dimensional equation of state is used to obtain adsorption isotherm. INTRODUCTION

The theoretical description of a two-dimensional fluid interests many investigators because of its relation to the large class of interfacial phenomena, which are now extensively studied for their practical importance. One of the methods of treatment of interfacial phenomena is a mental division of the whole bulk system into three subsystems: two neighboring bulk phases and a two-dimensional phase between them. Attempts to describe the two-dimensional mobile phases have been made previously. Devonshire (1) was the first to modify the cell model to the two-dimensional case. Next, the two-dimensional analog of the van der Waals fluid was investigated by a number of authors (2). Van der Waals-type two-dimensional equations of state have recently been proposed by Bergman (3) and Ross and Morrison (4). More interesting and extensive investigations on this field have been reported by Andrews (5). In recent years, attention has been given to the computation of the thermodynamic functions of three-dimensional fluids (68). Surprisingly, it appears that the twodimensional counterparts of many theories developed for bulk phases involve a greater degree of complexity than that for ordinary bulk phases. Actually, there is no analytical solution for the radial distribution function even for a two-dimensional system of hard

disks. For this reason, the direct adaptation of such a convenient and rigorous theory, proposed by Andersen et al. (9) and by Verlet and Weis (10), to two dimensions is very tedious (11, 12). Looking for some other theoretical possibilities we applied the procedure of summation of the two-dimensional virial equation of state proposed by Fulifiski (13) to three-dimensional phases. The resulting first-order equations for the compressibility factor and the adsorption isotherm contain contributions to the virial coefficients from simple polygons (rings). It is evident that this simple theory is inferior to the Percus-Yevick or Hypernetted Chain theories (14, 15), which take into account a greater number of diagrams contributing to the virial coefficients. However, it is possible to generalize the proposed theory by: (i) summation up to infinity of other classes of graphs; and (ii) application of the van der Waals-type equation, p = pa + p ( p , T ) , where Pa is the pressure of hard-disk fluid of diameter d, p is the two-dimensional density, and ~ takes into account the attractive interactions in the system. THEORY

General

We shall consider a one-component twodimensional fluid with two-body central forces u(lr~ - rjl) = u~j. In this case the I

I

399

Journal of Colloid and Interface Science, Vol. 66, No. 3, October 1, 1978

0021-9797/78/0663-0399502.00/0 Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved.

400

S. SOKOLOWSKI

total potential energy of the system of N molecules is given by

A/3,~=lim m>l

1

s--,o~ 2S

N

u~ = ~ u,j.

XI

[1]

i
The thermodynamic limit (i.e., N---> ~, S ---> ~, N / S = finite, where S is the surface area) is assumed. Thus, the two-dimensional virial equation of state (which itself is related to the grand canonical partition function) has the form,

p

~,

~

m=lm+

flmp m+',

[23

I

2!Bz =

~

~

:

"f(m+l)l' [5]

S---~o~

Introduce the Fourier transform y(q) of the Mayer function f ( r ) ,

(')'f

f(r) =

-~

1

where the irreducible cluster integral, tim, and the virial coefficient, Bin, are expressed in terms of m + 1 point-biconnected graphs (15). For example, 1!/3

°"

[6]

so that

m -

"drta-Flfl2

Y(q) = f dre-iqrf(r),

P - p + ~ Bmp m+l kB T m=1 =

dr1""

"'"

dqe-iqry(q).

[7]

Substituting the last equation into Eq. [5], we obtain: m>~

87r2

,

dq[Y(q)]m+a'

[8]

1 /31

,

-..4

47rz y(0).

Thus, the sum of contributions from all simple polygons (up to infinity) is easily calculated: 11(p, T)

etc., In the above diagram each line joining two circles represents the Mayer function: f j = exp(-u;JkB T) - 1.

[4]

The usual procedure of calculating the virial coefficients corresponds to the horizontal summation of the rows in Eq. [3]. In contrast, the procedure used here corresponds to the vertical summation of the columns (13). The simplest approximation consists of neglecting all graphs of the expansion [2] except the simple polygons, i.e., taking into account the first column from Eq. [3]. Now, let us consider a particular simple polygon built of m black circles. The contribution to /3~ from this graph is given by: Journal of Colloid and Interface Science, Vol. 66, No. 3, October 1, 1978

8~ ~

dq

l - ~(q)P

+ In ([1 - PT(q) l)} •

[9]

The two-dimensional equation of state in this approximation reads P

k~T

-

p + I~(p,T).

[10]

In order to derive the adsorption isotherm, we assume that the adsorption can be explained as a chemical equilibrium between a three-dimensional gas and a system

STATISTICAL MECHANICS OF ADSORPTION of particles which vibrate independent of one another in a direction perpendicular to the surface. We can treat the vibrations in " E i n s t e i n approximation," so they contribute to the chemical potential of the adsorbed phase with a temperature-dependent but density-independent term. Making use of the Gibbs relation,

P _ P a + I I ( p , T ) _ ild(p)" lq3T PmT

1 + ~/8

Z where fg denotes the fugacity of the threedimensional gas, from Eqs. [9] and [10] we obtain [12]

where H(p,T) -

1

87r2

1

PY(q)

-

and KH is H e n r y ' s constant.

- [H(p,T)

Pa

3

kBT

-

"q

In

1

1

8a

4a(1 - ~)

a(1 - ~7)

+

1

8a

In (1 - ~),

a = 7r/4.

[19]

RESULTS AND DISCUSSION We consider now the simple case when f ( r ) = II(r/~),

P + 02 -

[15]

1 -p

The last two terms in Eq. [14] accounted for the attractive interactions in the system. We shall propose some generalization of Eq. [14], namely,

[18]

In Oa/Ku

1 -

where Pa is the two-dimensional pressure of a hard-disk system, the quantities of which will be further denoted by the subscript " d . " In the approximation proposed by Ross and Morrison, the hard-disk virial coefficients were assumed to be equal to 2.0. Thus, the hard-disk virial equation of state summed to infinity reads

- Ha(p)]},

where t9a is the adsorption isotherm for a hard-disk system:

+ (B1 - 2)p 2 + ( B 2 - 2)p 3, [14]

- -

[17]

fo = Oa exp{-[y(0) - Ya(o)]P

The two-dimensional equation of state recently proposed by Ross and Morrison (4) h a s the following form: Pa

(1 - ~/)2 '

where V = rcpd2/4, and Z is the two-dimensional compressibility factor Z = p/kB TO. Expansion of the above equation in powers of V gives coefficients which are, through ~fl, near approximation to the exact virial coefficients (16, 18). In addition, this equation of state agrees well with computer experiments and its accuracy is as good as Pad6's approximation of Ree and H o o v e r (18). The adsorption isotherm corresponding to Eqs. [16] and [17] is

The van der Waals E q u a t i o n o f S t a t e

_

[16]

One key ingredient in such a treatment of two-dimensional fluids is an accurate equation of state for hard disks. T w o years ago H e n d e r s o n (16) obtained a useful and accurate equation of state by starting with the scaled particle theory (17):

dlnf~- ~T

fg = KHR exp[--y(0)p -- H ( p , T ) ] ,

401

[20]

where H(x) is the step function: H(x)=

~1

for

10 for

[x[ < 1

Ixl > 1.

[21]

This corresponds to the following funcJournal of Colloid and Interface Science, Vol. 66, No. 3, October 1, 1978

402

S. SOKOLOWSKI F o r p > Pmax both integrals can be treated as an analytical continuation of considered approximation. H o w e v e r , in this region o f p the integrands are singular at some values of q. According to Fulifiski (13), we interpret these singular integrals as their Cauchy principal values. The numerical calculations o f the twodimensional equation of state were perf o r m e d f o r the L e n n a r d - J o n e s 1 2 - 6 potential:

t.0 P

!/

/

06

02

°

0.1

~

u(r)= 4I(!)6-

°

o.~

o.'~

~

FIG. 1. T h e two-dimensional equation o f state comp u t e d from the data of adsorption of k r y p t o n on graphite at T = 0.651. The solid line d e n o t e s the theoretical c u r v e obtained from Eq. [10], the d a s h e d line d e n o t e s the results o f molecular d y n a m i c s by T o x v a e r d (21), and the filled circles are the experimental points.

tion, y(q):

"y(q) = 2TrO~Ja(qa)/q,

[22]

where Ja is the Bessel function o f the first kind and zero order. Thus for a squarewell potential,

u(r) =

0

1 k

k,

[23]

f(r) = In(r/X) - n(r)]f0 - n ( r ) ,

[24]

where f0 = exp(--E0/kB T) - 1, we have Y(q) =

27rf0 [hJl(hq) - Jl(q)] q 277" -

- -

Jl(q).

q

[25]

The computation of integrals Ia and H calls for some additional remarks. These integrals represent the sums of some parts of the virial series. The radius of convergence is determined by the condition /~ p < min ]l/y(q) I = Pmax.

[26]

Journal of Colloid and Interface Science, Vol. 66, No. 3, October 1, 1978

(!)121

[27]

(reduced units e -- o- = 1 are used throughout). The method for computing Fourier transforms was that proposed by Linz (i9). Figure 1 shows the two-dimensional equation of state computed from the data on adsorption of krypton on graphite, reported by Thorny and Duval (20), in comparison with the theoretical curve calculated according to Eq. [10] and with the results of T o x v a e r d (21) obtained by using a molecular dynamics method. The values of parameters o- and e were taken from a paper by Toxvaerd: o- = 3.827 A, and E/kB = 164°K. The second series o f our numerical calculations concerned the application of Eq. [16]. The problem of determining the hard-disk diameter can be solved by using different criteria (22). Because there exists no analytical solution for the radial distribution function of hard disks, we shall discuss two simpler criteria: the B a r k e r H e n d e r s o n one (23) and the criterion proposed by Andrews (5). Both criteria lead to the temperature-dependent but density-independent hard-disk diameter. According to Barker and H e n d e r s o n (23), the hard-disk diameter is determined from the equation, dBn = I [1 - exp{-ul(r)/kB T}]dr,

[28]

where ul is the attractive part of the Lennard-Jones potential, whereas the criterion proposed by Andrews leads to the follow-

STATISTICAL MECHANICS OF ADSORPTION

403

Z L5

ing equation for d:

u(dA)= kBT.

[29]

Figure 2 shows the temperature dependence of d. It should be stressed that the criterion proposed by Andrews gives values too small in comparison with those for dBH. However, the two-dimensional compressibility factor calculated using dA gives values which are the nearest approximations to those of the computer experiments. Compare Figs. 3 and 4, which present the twodimensional equation of state at T = 1 and T = 0.95, respectively, to the computer experiments (24, 25). Additionally, in Figs. 3 and 4, the results obtained from the equations proposed by Ross and Morrison (4) and Andrews (5) are presented. Agreement of our equation with computer experiments at higher temperatures is good. Agreement with the data at low temperature is less satisfactory. The critical temperature lies between Tc C [0.64,0.66], whereas, on the basis of the Monte Carlo study (25), Tsien and Valleau estimated that Tc ~ [0.625,0.7] and, using the per-

,ili/ / ~

i i II/~/ i.o

~

/

FIG. 3. The two-dimensional compressibility factor at T = 0.95. T h e solid and long-dashed lines were calculated according to Eq. [16] for dA and dBH, respectively. T h e s h o r t - d a s h e d line was obtained by m e a n s of the Ross and Morrison (4) equation, w h e r e a s the d a s h - d o t line denotes the results of the M o n t e Carlo s t u d y (25).

turbational method, Steele (11) found that T~ = 0.60. CONCLUSIONS

The next terms in Eq. [2] are given by graphs containing more lines than points. The general procedure proposed by Fulifiski (13) makes possible summation of the more complicated subsets of graphs contributing to the virial coefficients, the simplest being the graphs composed of multiple points connected with each other through any number of chains, for example,

a 1.03

O.S7

;

iv

FXG. 2. T h e hard-disk diameter as a function of temperature. T h e solid line r e p r e s e n t s results calculated according to B a r k e r - H e n d e r s o n (23) criteria, w h e r e a s the d a s h e d line r e p r e s e n t s t h o s e results calculated according to A n d r e w s (5).

etc. (The multiple point is connected with more than two other points of the graph, and the chain is the sequence of lines joining two multiple points, without multiple points.) The summation of these terms proceeds in a fashion similar to that used previously. We do not present the details Journal of CoUoid and Interface Science, Vol. 66, No. 3, October 1, 1978

404

S. SOKOLOWSKI the equations proposed in this paper are easy to evaluate and have been presented because of their simplicity.

/

it /

ACKNOWLEDGMENT The author wishes to express his gratitude to Professor J. Stecki for a helpful discussion.

LO

// ///,Y

t.g

REFERENCES

t.0

0'.~

d6

FIG. 4. The two-dimensional compressibility factor at T = 1.441. All lines, except the dash-dot one which was calculated from an equation developed by Andrews (5), have the same meaning as in Fig. 3. The filled circle denotes the result of a computer experiment (24).

of these calculations and write the final result only:

I2(p,T) = _½p2f dr {exp[qJ(r) - u(r)/k~T] × [1 + x(r)] - e x p [ - u ( r ) / k B T ] × [1 +

x(r) + qJ(r)] - tp(r)[x(r) + ½~b(r)]}, [30]

where x(r) -

1 fdqe-iqr I 1 --p-~q)J Y(q)P ]~ ' [31] (27"r)20

and

1 f dq e-iqr

0(r) - (27r)2p

y2(q)p2 PY(q) "

1-

[32]

It should be stressed that the threedimensional counterpart of the proposed theory gives, of course, worse results, compared to the Optimized Cluster Theory developed by Andersen et al. (7, 9), for example. The application of the latter to two-dimensional phases is, however, very tedious numerically (11, 12). In contrast, Journalof Colloidand InterfaceScience. Vol.66, No. 3, October 1. 1978

1. Devonshire, A. F., Proc. Roy. Soc. London A 163, 132 (1937). 2. Hill, T. L., "An Introduction to Statistical Mechanics." Addison-Wesley, Reading, Mass., 1960. 3. Bergman, E., J. Phys. Chem. 78, 405 (1974). 4. Ross, S., and Morrison, I. D., Surface Sci. 52, 103 (1975). 5. Andrews, F. C., J. Chem. Phys. 64, 1941 (1976). 6. Croxton, C. A., "Liquid State Physics--A Statistical Introduction." Cambridge University Press, London/New York, 1974. 7. Andersen, H. C., Chandler, D., and Weeks, J. D., Advan. Chem. Phys. 36, 105 (1976). 8. Mansori, G. A., and Canfield, F. B., Ind. Eng. Chem. 62, 12 (1970). 9. Andersen, H. C., and Chandler, D., J. Chem. Phys. 57, 1918 (1972). 10. Verlet, L., and Weis, J.-J., Phys. Rev. A5, 9393 (1972). 11. Steele, W. A., J. Chem. Phys. 65, 5256 (1976). 12. Rudzifiski, W., and Soko/owski, S., Acta Phys. Pol., in press, 1978. 13. Fulifiski, A., Acta Phys. Pol. A37, 177, 185 (1970); A40, 315 (1971). 14. Percus, J. K., and Yevick, G. J., Phys. Rev. 110, 1 (1958). 15. Stell, G., in "The Equilibrium Theory of Classical Fluids" (H. L. Frisch and J. Lebowitz, Eds.). Benjamin, New York, 1964. 16. Henderson, D., Mol. Phys. 30, 971 (1975). 17. Helfand, E., Frisch, H. L., and Lebowitz, J., J. Chem. Phys. 34, 1037 (1961). 18. Ree, F. H., and Hoover, H. G., J. Chem. Phys. 46, 418 (1967). 19. Linz, P., Math, Comp. 26, 509 (1972). 20. Thomy, A., and Duval, X., J. Chim. Phys. 67, 1101 (1970). 21. Toxvaerd, S., Mol. Phys. 30, 373 (1975). 22. Madden, W. G., and Fitts, D. D., Mol. Phys, 30, 809 (1975). 23. Barker, J. A., and Henderson, D., J. Chem, Phys. 47, 4714 (1967). 24. Fehder, P. L., J. Chem. Phys. 50, 2617 (1969). 25. Tsien, F., and Valleau, J. P., Mol. Phys. 27, 177 (1974).