J. Phys. Chem. 1983,87,2956-2959
2956
Adsorption of Gases and Vapors on a Solid Surface Douglas Henderson" IEM Research Laboratow, San Jose, Callfornia 95193
and Ian K. Snook Department of Applled Physics, Royal Melbourne Institute of Technology, Melbourne, Victoria, Australia (Received: December 16, 1982)
The adsorption of gases and vapors on a planar surface is examined by a simple van der Waals-like theory and by the mean spherical approximation (MSA). For simplicity, the intermolecular potential of the gas molecules is assumed to be a hard-sphere potential with a Yukawa tail and the interaction of the molecules with the surface is assumed to be a hard repulsion with an exponential tail. If the gas molecules are strongly adsorbed, sharp positive maxima similar to the experimental results of Specovius and Findenegg are seen in the critical region in both the van der Waals and MSA calculations. For weakly adsorbed gases, the MSA predicts negative values for the adsorption including negative extrema in the critical region. Such isotherms have not been observed experimentally but might be seen in carefully chosen systems.
Introduction Specovius and Findenegg' and Loring2have measured adsorption isotherms of ethylene and propane adsorbed on graphite and have found interesting features such as maxima and divergences in the critical region. Similar isotherms have been found in the computer simulation of van Megen and Snook.3 These divergences have also been predicted by the mean spherical approximation (MSA) for the adsorption isotherms of a Yukawa fluid adsorbed by a hard wall with an attractive exponential p ~ t e n t i a l(although ~,~ adsorption isotherms were not displayed in these papers). In this paper we will display some of these results and also develop a simple theory, based on the van der Waals theory of fluids, which qualitatively accounts for some of these results. We develop this van der Waals theory of adsorption first.
van der Waals Theory We regard the gas-surface system as a solution of a single giant molecule (the surface) and a solvent (the adsorbed gas). We use the index 1for the gas molecules and 2 for the giant molecule. Generally, we will not index any functions or properties which are solely those of the gas. We define gl,(x) = 1 + hlz(x)to be the radial distribution function for a giant molecule-gas molecule pair. The distance x is measured from the distance of closest approach of a gas molecule to the giant molecule so that the distance between the centers of a gas molecule and a giant molecule is R12 x , where Rlz = (a + u z ) / 2 and u and uz are the diameters of the gas and giant molecules, respectively. Thus, p ( x ) = pg&) is the density profile of the gas molecules near the giant molecule and the adsorption isotherm is obtained from
-
where p = N / V is the bulk density. Using the OrnsteinZernike equation and the fact that p z 0, we have ~
PShlZ d r =
1
- 1 - pap/apz
~ dr2 1 2
1 - p$c dr
paP/aP
(2)
where the c's are the direct correlation functions and p = l / k T . Each of the quantities in eq 2 is evaluated in the limit p z 0. Now h,, = -1 inside Rlz. Thus
-
where we have used the fact that Rlz is large. Hence
In the van der Waals theory, the pressure is given by
P = PO
-
(5)
CPiPjaij ij
where
aL, = -2aJmu,,(r)rZ dr I
and p o is the hard-sphere pressure. For the moment, use the original van der Waals expression n
+
(1) J. Specovius and G. H. Findenegg, Ber. Bunsenges. Phys. Chem., 84, 690 (1980). (2) R. Loring, Dissertation, Bochum, 1979. (3) W. van Megen and I. K. Snook, Mol. Phys., 45, 629 (1982).
E. Waisman, D. Henderson, and J. L. Lebowitz, Mol. Phys., 32, 1373 (1976); D. Henderson, J. L. Lebowitz, L. Blum, and E. Waisman, ibid., 39, 47 (1980). ( 5 ) N. E. Thompson, D. J. Isbister, R. J. Bearman, and B. C. Freasier,
lim 1 - pP2-Q
p2b2- 2pb12
ap =
~ P Z
(1- pb)'
+ 2 P ~ a ~ (10) ~
Now b12 = 2 ~ ~ and ~ ~ 1 3
(4)
Mol. Phys., 39, 27 (1980).
a12 = - 2 7 r R , 2 2 ~ ~ ~ & ) d=x -27rRlZ2a',,
(11)
Hence, subtracting the volume terms (of order RlZ3),taking
0022-3654/83/2087-2956$0 1.50/0 0 1983 American Chemical Society
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2957
~
t
1
insteady of the correct result
A method of overcoming this difficulty at low densities will be developed in the next section in connection with a more sophisticated but physically less transparent theory of multilayer adsorption. Mean Spherical Approximation The MSA has been applied to a fluid near a hard This approximation is based upon the Ornstein-Zernike (OZ) relation h12 = c12 + Phl2*C 0
0.2
0.4
0.6
PO3
where c and c12 are the direct correlation functions for bulk and wall-fluid systems. We couple eq 16 with the exact result
Figure 1. Adsorption isotherms calculated from eq 24 with clz = 5c and A,, = X = 1.8. The curves are labeled with the value of T / T , .
-
the limit R12 we obtain
m,
and dividing by the surface area 4rR1?,
where the prime has been dropped from a'l2 for a simpler notation. We originally obtained eq 12 in 1976.6 It has been obtained earlier by Steele' and by Kuni and Rusano@ from an examination of the asymptotic form of the density profile at large distances from the surface. Equation 1 2 results if this asymptotic form makes the dominant contribution to r (i.e., the van der Waal's approximation). Very recently, eq 12 has been obtained by Stott et aL9 If the Percus-Yevick hard-sphere mixture equation of state is used in place of eq 7, we obtain6 3dl
+ 2n)
3n(l + 2n)
hlz(x) = -1
x
0
c12 = Klz exp(-z12x)
x
>0
(18)
We obtain c ( r ) from the MSA or generalized MSA (GMSA) for the bulk fluid, where the OZ relation h = c ph*c (19)
+
is coupled with h(r) = -1
ru
r
(21)
Equations 16-21 can be solved analytically. We consider a system of hard-sphere molecules with attractive Yukawa tails interacting with a hard wall with an attractive exponential tail. Thus, the pair potential is u(r) = r u
(22)
0 x
>0
(23)
With X = 1.8, the Yukawa fluid is qualitatively similar to a Lennard-Jones fluid with the same values of t and u. For example, with this value of A, k T c / t = 1.26, pcu3 = 0.31, and pcu3/e = 0.15. For simplicity, we take X12 = 1.8 also. If we choose K = P t and z = X, the MSA is being used to describe the bulk fluid. If K12and z12 are chosen by some theoretical criteria, then we may refer to our procedure as a GMSA. The theoretical criterion'O we use is to adjust z and K so that the compressibility and pressure equations (as well as the energy equation) all yield consistent and reliable results for the thermodynamics of the bulk system. If the GMSA is used, the bulk fluid need not be Yukawa fluid. The parameters K and z can be adjusted to parametize the thermodynamics of the bulk fluid. Equation 21 can be replaced by a sum of Yukawas, permitting a great deal of flexibility. For simplicity, there we restrict ourselves to a Yukawa fluid, eq 23, and one term in eq 21. If we choose K12= @elz = z12= X12 the MSA is being used to describe the interfacial region. This is the procedure (10) D. Henderson, E. Waisman, J. L. Lebowitz, and L. Blum, Mol. Phys., 35, 241 (1978).
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Henderson and Snook
The Journal of Physical Chemistry, Vol. 87,No. 15, 1983
we follow here. We could refer to this hybrid procedure as a MSA/GMSA theory. The use of a GMSA for the interfacial region to yield a GMSA/GMSA theory would be attractive if there were theoretical criteria to choose KI2 and z12. One criterion might be the contact value theorem1' but, in the absence of a useful second criterion, a GMSA does not seem feasible a t the moment. The adsorption isotherm is
P 0 97'
!
' 1 01
I
where a = (pdp/dp)'I2,C is given by eq 8-11 of the 1980 paper of Henderson et al.; and b is a bulk fluid parameter obtained from the GMSA. In the van der Waals limit, z12 -0 (25) C = K21/[a(l - r1PI and a result similar to eq 13 is obtained. Adsorption isotherms, obtained from eq 24 with t12 = 5t, are shown in Figure 1. They are qualitatively similar to those of Specovius and Findenegg and to those obtained from eq 13. One deficiency of the GMSA/MSA isotherms is they give eq 14 instead of eq 15 a t low densities. This is because of the linearization inherent in the MSA. This replacement of exp(-/3ulz(x)j by a truncated expansion is similar to the situation in the perturbation theory of bulk fluids. Barker and Henderson12have argued that, in perturbation theory, these higher-order terms in p are fluctuation terms, related to the compressibility of a hard-sphere fluid. This is qualitatively correct since these high-order terms are small at high densities, where the hard-sphere fluid is nearly incompressible. In their early papers, Barker and Henderson assumed that the p2 term in the free energy was proportional to kTdp/dpo, where p o is the hard-sphere pressure. Interestingly, one system for which the second-order free energy term can be calculated exactly is a lattice gas. For this system, the second-order free energy term is proportional to (kTdp/ dpo)2,where p o is the analogue of the hard-sphere system for this system, namely, a lattice gas with site exclusion. Presuming that
6
Figure 2. Adsorption isotherms calculated from eq 27 with c12 = 56 and A,, = X = 1.8. The curves are labeled with the value of T I T , .
po3:c
is an apprpriate factor, we write as an approximate generalization of eq 24
-
- -b+ - -C (1 a
212
a
-
J+
Equation 27 is admittedly approximate. However, it does have the correct low density behavior but does not destroy the attractive features of eq 24 a t higher densities. In their later papers, Barker and Henderson have used more sophisticated approximations for the higher-order free energy terms. Similar refinements can be expected here also. However, eq 27 is good enough for our present purposes and will illustrate the possibilities of eq 24 and 27. Adsorption isotherms obtained from eq 27 with t12 = 5t are shown in Figures 2 and 3. The results are very similar (11) D. Henderson, L. Blum, and J. L. Lebowitz, J . Electroanal. Chem.. 102.. 315 (1979). . ~ . (12) J. A.Barker and D. Henderson, J. Chem. Phys., 47, 2856 (1967). ~~
Figure 3. Adsorption isotherms calculated from eq 27 with t 1 2= 5c and A,, = X = 1.8. The curves are labeled with the value of T I T , .
to Specovius and Findenegg. Note in particular that the density for which r is a maximum decreases as the temperature increases whereas the pressure for which r is a maximum has the opposite temperature dependence. Note also the crossings at high pressure in Figure 3. All of these features are in good agreement with the experimental results of Specovius and Findenegg. It should be emphasized that curves such as these are beyond the scope of such commonly used theories as the BET theory which assume the vapor or gas pressure to be small. Furthermore, the theory outlined here provides expressions which are not more difficult to use to analyze experimental data than the BET expressions. Although eq 13 and 24 give similar results for eI2 large compared to t, eq 24 contains a rich range of phenomena not seen in eq 13. Adsorption isotherms for e12 = 2t and t12 = t , respectively, are plotted in Figures 4 and 5. The adsorption can be negative as well as positive. For t12 = 2t, the adsorption changes sign a t very nearly the critical pressure so that there are rounded positive and negative extrema. A t exactly the critical temperature, there will be a sharp negative singularity. Presumably, one could
Adsorption of Gases and Vapors on a Solid Surface
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983 2959
0.2
t
0.1
r2
0.0 -1
U -0.1
I
I
1
I
0.2
0.1
I
I
0.3
pa3/€
Flgure 4. Adsorption isotherms calculated from eq 27 with ti, = 2e and A,, = A = 1.8. The curves are labeled wRh the value of T / T c .
find a set of parameters so that the adsorption changes sign at exactly the critical point so that there would be no singularities for T L T,. For t12 = E , the adsorption is clearly negative near the critical pressure so that the adsorption is large and negative (with sharp peaks) in the neighborhood of the critical point. For both t12 = 2t and t12 = e the adsorption again becomes positive at high densities and pressures. Perram and Smith and also Richmond have obtained13 isotherms similar to those displayed here using a sticky hard-sphere potential and the Percus-Yevick approximation. The present calculation has the advantage that the Yukawa and exponential potentials are more realistic than the sticky hard-sphere potential. Indeed by superposing several Yukawas or exponential potentials a given potential could be fit to any degree of accuracy.
Conclusions The MSA contains a rich range of adsorption phenomena not seen in simpler theories of adsorption. For strongly adsorbed gases, isotherms similar to those of Specovius and Findenegg are obtained. For weakly adsorbed gases, isotherms not yet seen experimentally are predicted. Whether such adsorption, with regions of negative adsorption and even negative singularities, are of physical interest is problematic. However, it might be of interest to measure the adsorption of some vapor such as mercury (13)J. W.Perram and E. R. Smith, Chem. Phys. Lett., 35,138(1975); 39,328(1976);h o c . R.SOC.London, Ser. A , 353,193(1977);E.R. Smith and J. W. Perram, J. Stat. Phys., 17,47(1977);P. Pichmond, J. Chem. SOC.,Faraday Trans. 2, 73,318 (1977).
Figure 5. Adsorption isotherms calculated from eq 27 with t I 2 = t and A,, = A = 1.8. The curves are labeled with the values of TIT,.
or gallium on a weakly adsorbing surface, such as glass or teflon, to see if isotherms such as those displayed in Figures 4 and 5 are seen. Adsorption of one species from a solution, especially near a critical point, may also yield interesting isotherms. The use of the MSA to describe the interfacial part of the problem means that the density profiles fail to satisfy the contact value theorem1' at high densities or pressures. We have argued4 that, as a result, at high densities, the MSA density profiles are at best qualitative. To some extent, these comments apply here. However, the situation is probably less serious here since our attention is directed mainly to smaller values of p and p and because the adsorption isotherms are integrals of the density profiles. One might expect the integration to smooth out the errors in the density profile. For strongly adsorbed gases, the van der Waals theory can be used to obtain a simple theory of adsorption which is qualitatively reasonable. However, the simple theory does not predict the negative adsorption of weakly adsorbed gases seen in the MSA calculations. Following Steeie,' eq 12 can be generalized to include negative adsorption by regarding a12as an integral of a perturbation potential which can be positive or negative depending on the relative strength of the particle-particle and particle-surface interactions. Finally, we note that the approach outlined here will not give the precise nature of the critical singularities for much the same reasons that approximate theories fail to give the precise nature of the bulk critical singularities. Acknowledgment. The authors are grateful for the stimulating comments of Dr. J. A. Barker. This work was supported in part by NSF Grant CHE80-01969 to D.H.
