4006
J . Phys. Chem. 1988, 92, 4006-4007 We have fitted aii,b,, and cii to the p-V-T data of CC14 and CS2 reported in Table IV. The parameters are given in Table VII, and the mean standard deviations in pressure were f0.5 bar for CC1, and f 0 . 3 bar for CS2. The parameters 0 and {, also given in Table VII, were fitted to the p-V-T data of the mixture of composition x = 0.4972, with a standard deviation in pressure of f0.2 bar. Figure 9 shows the experimental and calculated densities of the mixture using the parameters of Table VII. It can be observed that the agreement is very satisfactory, confirming the flexibility of Deiters' EoS in representing p-V-T and phase equilibria of fluid mixtures. Acknowledgment. This work was supported in part by the USA-Spain Committee for Scientific Cooperation under grant CCB84-003. Registry No. CCl,, 56-23-5; CS2, 75-15-0
~~~
(49) Deiters, U.; Swaid, I. Ber. Bunsen-Ges. Phys. &em. 1984, 88, 791, (50) Deiters, U. Fluid Phase Eguilib. 1983, 12, 193.
Supplementary Material Available: Table of raw data (pressure p and period of vibration T ) (6 pages). Ordering information is given on any current masthead page.
COMMENTS Molecule-Mlcropore Interaction Potentials Sir: Equilibrium and transport properties of fluids confined inside micropores (whose characteristic dimension is of the order of a few collision diameters) is important in a variety of systems, including biological and synthetic membranes, selective catalysts, and various porous media. Several theoretical' and computer simulation2 studies of microporous systems have been reported in recent years. In the former, the interaction between the fluid molecules and the micropore walls have been taken into account by introducing idealized potentials (e.g., the hard-sphere model), which greatly simplify the approximate theoretical treatment. More realistic potentials were used in some of the computer simulation studies of simple fluids confined inside slit micropores (two parallel semiinfinite flat solids), which include the summed 10,4 interaction potentials of Crowel13 and Steele.4 Anticipating forthcoming computer simulations of fluids in micropores of polygonal and circular cross sections, we have derived interaction potentials for these geometries. Our results may be regarded as extensions of the Crowell summed 10,4 potential in which the lattice periodicity is neglected, and the solid atoms are distributed uniformly on parallel mathematical planes. See Feke et al.5 for formulas for the interaction of spherical particles using Lennard-Jones m,n potentials in general; we use herein only the 12,6 potential.
(1) Post, A. J.; Glandt, E. D. J . Colloid Interface Sci. 1985, 108, 31 and references cited therein. (2) Subramanian, G.; Davis, H. T. Mol. Phys. 1979, 38, 1061. Snook, I . K.; van Megen, W. J . Chem. Phys. 1979, 70, 3099. Snook, I. K.; van Megen, W. J . Chem. Phys. 1980, 72, 2907. Magda, J. J.; Tirrell, M.; Davis, H. T. J . Chem. Phys. 1985, 8 3 , 1888. (3) Crowell, A. D. J. Chem. Phys. 1954, 22, 1397. (4) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: New York, 1974; Chapter 2. (5) Feke, D. L.; Prabhu, N. D.; Mann, J. A,, Jr.; Mann, J. A. I11 J . Phys. Chem. 1984, 88, 5735.
Figure
Definition of the coordinate system and notation use in eq 3 .
The potential energy of interaction between a test fluid molecule and a single atom within the solid is assumed to be given by the Lennard-Jones expression
where r is the interatomic separation, u the separation at which the potential becomes zero (collision diameter), and t the depth of the potential well. Assuming that the atoms of the solid are distributed continuously up to and on a sequence of parallel surfaces that form the pore wall, then the interaction potential of the test fluid molecule with one of these surfaces of area A and number density n is given by U = I d a nu(r) = 4 e n L d a
or, in dimensionless form
0022-3654/88/2092-4006$01 .SO10 0 1988 American Chemical Society
[(
:)I2
-
(
:)6]
The Journal of Physical Chemistry, Vol. 92, No. 13, 1988 4007
Comments
X *-
= t
\ I O
L
15
20
25
30
I 01
Figure 2. Plot of the potentials (4) and ( 5 ) scaled with n*. The exact formula (4) depends explicitly on the cylinder radius R*. T o the resolution of the plot, the asymptotic formula (5) follows the exact formula (4) for R* = 20. However, the plot of the exact formula shows a significantly lower minimum for R* = 3. The location of the minimum was the same for both cases, R* = 1.008.
where u* = U/c, r* = r / a , n* = nu2, and da* = da/a2. The total potential energy of the molecule is obtained by summing the last equation over the sequence of the parallel surfaces, with the appropriate values for A* and n*. 1 . Pores of Circular Cross Section. Using the notation of Figure 1, eq 2 gives V ( p * , R * ) = 16n*R*L+-dz* S r0d c p
I 51 + ! - -
(3)
where R* is the radius of the cylinder scaled with a and r*, = z * ~ p * 2 R*2 - 2p*R* cos 9. Integrating first over z*, next over cp, and using a transformation property of the hypergeometric function6 we obtain the expression
+
+
Case b
Case a
Figure 3. The coordinate systems and notation for pores of polygonal cross sections.
between the molecule and the infinite flat strips that define the bounding walls. Therefore the problem first relies on the computation of the interaction energy between a molecule and an infinite flat strip. If we use the notation of Figure 3, eq 2 becomes
where L* is the dimensionless width of the strip and r*, = X*, + P2+ Z*,. The limits xl*, x2* take on the values x l * = L* - x*, x2* = x* and x l * = x * , x2* = L* + x* for the cases a, b respectively. Performing the integrations results in the formula u*(z*,x*,L*) =
r 6
- (cos5 e, 5
+ coss e,)
4
- - (cos7 el 7
+ cos7 e,) + -91 (cos9 e, +
1
where the trigonometric functions are given by Here x* = R* - p* is the dimensionless distance of closest approach between the fluid molecule and the cylindrical surface, and F[a,P;y;z] denotes the hypergeometric series with parameters a 7
cos
e, = (L* - x * ) / ( z * 2 + (L* - x * ) 2 ) ” 2 = X*/(Z*,
+ x*2)1/2
cos el = -X*/(z*Z
+ x*2)l/2
cos
P, Y.6
Equation 4 becomes identical with the 10,4 potential of interaction between a Lennard-Jones molecule and an infinite flat surface
-
in the limit R* m. However, when R* is small, the depth of the potential well increases as a consequence of the pore curvature. A comparison of eq 4 and 5 is shown in Figure 2 for two values of R* and the same surface number density n*. Summing the interaction potential (4) over 10 cylindrical surfaces with the same spacing, surface number density and Lennard-Jones parameters as those used by Fischer et al.’-* reproduces their potential energy graphs obtained by numerical integration. (Note that the summation over 10 cylindrical surfaces is implied in ref 8.) 2. Pores of Polygonal Cross Sections. In this geometry the potential energy may be expressed as a sum over the interactions (6) Ryshik, I. M.; Gradstein, I. S. Tables of Series, Products, and Integrals; Veb Deutscher Verlog der Wissenschaften: Berlin, 1962. Whittaker, E. T.; Watson, G . N. A Course of Modern Analysis; Cambridge University Press: London, 1935. (7) Fischer, J.; Bohn, M.; Korner, B.; Findenegg, G. H. German Chem. Eng. 1983, 6, 84. (8) Fischer, J. In Molecular Bused Study of Fluids; Haile, J. M., Mansoori, G. A., Eds.; Advances in Chemistry Series 204; American Chemical Society: Washington, DC, 1983; Chapter 6.
e,
for case a, and
cos 6, = (L*
+ x*)/(z*, + (L* + x*)*)I/,
-
-
for case b. It may be verified that eq 7 reduces to the 10,4 interaction potential in the limit 8 , 0 and 0, 0. In summary, we have provided formulas for the computation of the interaction energy between molecules and pores of circular and polygonal cross section. These results may be useful in computer simulation studies of equilibrium and transport properties of fluids confined inside micropores. Computation of the potential through eq 4 is more effective than the numerical integration schemes used by the authors quoted herein. Acknowledgment. We acknowledge with gratitude support by the Office of Naval Research through an SRO contract with Case Western Reserve University. Department of Chemical Engineering Case Western Reserve University Cleveland. Ohio 44106
George J. Tjatjopoulos Donald L. Feke J. Adin Mann, Jr.*
Received: August 10, 1986; In Final Form: December 1, 1986
