Langmuir 1994,10, 1663-1666
1663
New Equations for Multicomponent Adsorption Kinetics J. Talbot,’ X. Jin, and N.-H. L. Wang School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
Received October 21, 1993. In Final Form: April 25, 1994’ We present a molecular theory for multicomponentadsorption kineticswhich assumes that the transient configurations of adsorbed particles can be approximated by the corresponding equilibriumconfigurations at the same coverage. Rate equations are then derived from a scaled particle theory (SPT)equation of state for a two-dimensional fluid mixture of hard convex particles. The predictions of the theory are in excellent agreement with the results of a computer simulation of the competitive adsorption of two species of d i s h of different diameters.
I Introduction The quantitative description of the adsorption kinetics and isotherms of mixtures is asubject of immense practical importance, particularly in the area of separations. The most widely used models are undoubtedly those having a Langmuirian basis. For example, the rate of adsorption of species i is given by
where 8i is the partial coverage of component i , 8 is the total coverage, 8 = E&,and k , i and kd,i are the adsorption
and desorption rate constants of component i, respectively. This equation, or a modified form, has been used to model the competitive adsorption of different proteins,’I2as well as the adsorption of single proteins with changes of conformation and/or orientation on adsorption.'^^ While this equation is valid when each molecule is associated with one site, and moleculesoccupyingdifferent sites do not interact, it is seriously in error for large molecules adsorbing on a continuous ~ u r f a c e .An ~ obvious failing of eq 1is that it does not account for the size of the adsorbing molecule: for equal adsorption and desorption rate constants, large molecules adsorb a t the same rate as do smaller ones a t a given total coverage, 8. In this Letter, we present new equations for multicomponent adsorption kinetics, which are a considerable improvement over the Langmuir equations. Our approach is based on the followingassumptions. The solutes are assumed to interact via short range force, so that they may be regarded as hard impenetrable particles. We also assume that the adsorption is restricted to a t most one monolayer. In the absence of adsorbed particles, solutes adsorb at a rate ka,ici, and with increasing surface coverage the adsorption rate is reduced as a result of the geometrical blocking effects. This effect, for species i, is represented by the available surface function #i(ej), so that the generalized kinetic equations may be written as
To make these ideas more precise, we consider first a monocomponent system of spherical hard particles of Abstract published in Advance ACS Abstracts, June 1, 1994. (1) Beissinger, R. L.; Leonard, E. F. J. Colloid Interface Sci. 1982,85, 521. ( 2 ) Cuypers, P. A.; Willems, G . M.; Hemker, H. C.; Hermens, W. Th. Ann. N . Y. Acad. Sci. 1987,516,244. (3) Soderquist,M. E.; Walton, A. G . J. Colloid Interface Sci. 1980,75, 386. (4) Schaaf,P.;Talbot, J. Phys. Rev. Lett. 1989,62,175; J. Chem. Phys. 1989,91, 4401.
diameter Q adsorbing on a continuous planar surface of area L2. We use p to denote the density of adsorbed particles, and 8 = ms2p/4 as the corresponding coverage. The available surface function, 4(8), is the probability that a new particle introduced randomly in the surface configuration does not overlap with any preadsorbed particles. Equivalently, it is the fraction of the surface into which the center of a new sphere may be placed so that no overlap occurs. In general, 4(8) depends in a nontrivial way on the distribution of particles on the surface, which in turn depends on the nature of the adsorption process. We note that an equation related to 2 has been proposed by Ward et al. (for the adsorption of a single ~ o m p o n e n t )within ~ the framework of their “Statistical Rate Theory of Interfacial Transport”? However, this treatment leads to a definite relationship between the adsorption and the desorption rate constants, which is not required in our approach. When kd = 0 and in the absence of surface diffusion, the adsorption is completely irreversible, and @(e)may be determined within the framework of the randomsequential adsorption (RSA) model. In ita simplest form, RSA involves the sequential, random placement of particles on a surface without overlap and without relaxation. The process has an infinite memory, and the resulting configurations are distinct from those a t equilibrium. If desorption and/or surface diffusion is permitted, the surface configuration relaxes to equilibrium. To describe the latter, we exploit the potential distribution method of Widom7 to relate the available surface function to the residual chemical potential of the component p.ciR in the two-dimensional mixture In diq = -piR/kT
(3)
where Tis the absolute temperature and k is Boltzmann‘s constant. At a given coverage, 489 is greater than 4*A. At low coverages, the difference is very small: indeed if the available surface functions are expanded as power series in the coverage, the coefficients are identical up to the second order.8 At higher coverages, the differences are approaching zero at relatively more pronounced, with dRSA low coverages (0.547 for monocomponent hard spheres adsorbing on a plane), while there is a finite probability to insert additional particles into equilibrium configurations up to much higher densities. Consider next an adsorption-desorption process on an initially empty surface. The system approaches a steady state in which the surface configurations are a t equilibrium. (5) Ward, C. A.; Findlay, R. D. J. Chem. Phys. 1982, 76, 6615. (6) Ward, C. A.; Findaly,R. D.;Rizk, M. J. Chem.Phys. 1982,76,5599. (7) Widom, B. J. Chem. Phys. 1963,39, 2808. (8)Widom, B. J. Chem. Phys. 1966,44, 3888.
0743-7463/94/2410-1663$04.50/00 1994 American Chemical Society
Letters
1664 Langmuir, Vol. 10, No. 6, 1994
\ 0.2
0.0 0.0
0.2
0.4
0.6
Coverage, 6
Figure 1. Available surface function for the adsorption of hard spheres (or disks) on a planar surface. Both the equilibrium (top) and irreversible (RSA) (bottom) lines are computed from equations proposed by Schaaf and Talbot.' The middle lines show the transient kineticsfor an adsorption-desorptionprocess with kdk.c = 5,10,20, and 30 (topto bottom) and were obtained from a computer simulation.
As shown in Figure 1, the transient 4 for the adsorptiondesorption process lies between the equilibrium and RSA values; the intermediate configurations are therefore not equilibrium ones. A truly reversible process (in the thermodynamic sense) could be realized by quasi-statically decreasing the desorption rate, k d , from an initially large value for a solution in contact with a solid surface, for example by varying the temperature, so that the intermediate configurations are always equilibrium ones, i.e. they are described by the upper curve of Figure 1. We now review the state of knowledge for 4 in the three cases: RSA, transient, and equilibrium (the latter is, of course, just a limiting case of the second). Exact descriptions of RSA and equilibrium configurations are available only in one dimension. However, very accurate approximations for &MA have been developed for a number of systems includingspheres4*9andnonspherical particles.l0 The transient available surface function has not been determined exactly, even in one dimension. However, Tarjus et al." have computed the coverage expansion for the adsorption-desorption of spheres on a plane up to the third order. The expansion has the correct behavior, i.e. it evolves from near the RSA value at short times to equilibrium at longer times. The equation is restricted to monocomponent systems and the coverage range over which it is valid is quite limited.11 We note that diffusional relaxation, i.e. RSA plus surface diffusion, has been considered in the one-dimensional case.12 As far as equilibrium systems are concerned, a number of accurate equations of state exist for the hard disk fluid. The curve shown in Figure 1was computed from that proposed by Hendersonle because of its simple form, although even more accurate equations have been devised.17J8 The extension of these concepts to multicomponent systems is straightforward: 4i((Oj])is the available surface for component i and the generalized multicomponent Langmuir equation takes the form of eq 2. For finite adsorption and desorption rates, the development of theories for the transient 4:s is a challenging problem: even the corresponding functions for the RSA case are ~~
~~
(9) Evans, J. W. Phys. Rev. Lett. 1989, 62, 2641. (10) Ricci, S. M.; Talbot, J.; Tarjus, G.; Viot, P. J. Chem. Phys. 1992, 97. - . , 5219. - -- -. (11) Tarjus, G.; Schaaf, P.; Talbot, J. J. Chem. Phys. 1990,93,8352.
(12) Privman, V.; Barma, M. J. Chem. Phys. 1992,97,6714.
known approximately only in certain special cases, for example a binary mixture in which the particles have greatly differing diameters.13 The adsorption of polydisperse mixtures has also been studied by ~ i m u l a t i o n ~ ~ J ~ and the~retically.'~In the latter approach the exclusion effects were accounted for only to the first order in density. The idea of the present approach is embodied in Figure 1. As long as the coverage is not too high, the transient curve is well-approximated by that corresponding to equilibrium. Indeed, the two are equal to second order in the coverage. Therefore, we can develop approximate adsorption kinetic equations by replacing the s : 4 in eq 2 with the equilibrium values. We stress that this replacement is not in general valid and it is justified only because the RSA and equilibrium curves are quite similar. Boubllklg has derived an equation of state for a twodimensional fluid of hard convex particles using scaled particle theory (SPT).20Ageneralization of this treatment to an arbitrary mixture of such particles gives, for the chemical potential of component iZ1
where Si is the circumference and Ai is the (projected) area of component i, and the summations extend over the N components of the mixture. For the special case of a two-component mixture of hard disks of diameters u1 and u2 (Si = mi and Ai = 7ru:/4) the equations reduce to the result previously obtained by Lebowitz et al.,, With the help of eq 3, we present this case in available surfaceexplicit form:
ex,[
-
34
+ y-l(y-l + 218, - (e, 1 - e, - e,
4 , =~ (1- e, - e,) x 38,
exP[ -
+ Y(T + 210, -
1 - 4 - 82
3
(5)
]
(6)
+ ~-9,)~
(1 - e, - e,),
+ re,)2 (1 -e1 - e,l2 (0,
where y = uz/u1 is the size ratio. Note that these equations resemble the Langmuir equations but include in addition the nontrivial exponential factor. The approximate kinetic equations for the binary mixture in terms of the available surface functions are
(13) Talbot, J.; Schaaf, P. Phys. Rev. A 1989,40,422. (14) Meakin, P.; Jullien, R. Phys. Rev. A 1992,46, 2029. (15) Muralidhar, R.; Talbot, J. AZChE J. 1993,39, 1322. (16) Henderson, D. Mol. Phys. 1975,30,971. (17) Erpenbeck, J. J.; Luban, M. Phys. Rev. A. 1985,32, 2920. (18) Tobochnik, J.; Chapin, P. M. J. Chem. Phys. 1988,88, 5824. (19) Boublfk, T. Mol. Phys. 1975,29,429. (20) hiss, H.; Frisch, H. L.; Lebowitz, J. L. J. Chem. Phys. 1959,31, 369. (21) Talbot, J. Ph.D. Thesis, Southampton University, U.K., 1985. (22) Lebowitz, J. L.;Helfand,E.;Praestgaard,E. J. Chem.Phys. 1965, 34, 1037.
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Langmuir, Vol. 10, No. 6,1994 1665
The adsorption isotherms of various gases on solid surfaces (obtained by setting the rates in the above equations equal to zero) has been modeled using SPT equations of state by FindeneggeZ3In the following, we test the validity of our approach for the kinetics by comparing the theoretical equations for the adsorption-desorption process of a binary mixture of disks with computer simulations.
I1 Simulation A number of simulation studies of the RSA of mixtures have been rep0rted.~~J5 The novelty of the present study is that it allows for the possibility of desorption. Trial positions for new solutes are selected randomly and uniformly in a cell of side L with periodic boundary conditions to eliminate edge effects. The trial particle is accepted only if it does not overlap with any of the adsorbed particles. Surface diffusion is not permitted an adsorbed particle is fixed in place unless it desorbs. A species is selected as the reference component (say 1). The size of the system is specified by the relative area of a particle of We this species to the simulation cell area, r a = ira1~/4L~. used a value of 4.37 X 10" for this parameter in the simulations reported here. This corresponds to systems that are large enough so that finite size effects are negligible. At each simulation step, an attempt is made to add one particle of type 1,and ka,iCi/ka,lClattempts are made to add species i, since the numerator represents the effective collision rate of species i with the surface. It is easy to see from eq 2 by considering the initial situation of zero coverage, that one simulation step corresponds to a real time interval of
so that t = NSbpAt,where NskPis the number of simulation steps (or the number of attempts that have been made to add species 1). We next define the reduced simulation time as 2 TU1
t* = -N,,
4L2
so that t* = tk,,c, Desorption is simulated as follows. At the end of each step, each solute is removed from the surface with probability
where k*d,i = kd,i/(k,lCl). This probability is consistent with the selected time units and eq 2. Each adsorbed solute desorbs independently and with constant probability regardless of how long it has been adsorbed. For each adsorbed solute, a uniform random number E (0 < 5 < 1)is generated. If ( < Pdw, then the solute is removed from the surface. Otherwise it stays in place. The adsorption-desorption process is ongoing until equilibrium is reached. We have applied this algorithm to study the adsorption of a two-component mixture of hard disks. The surface (23)Findenegg, G. H. In Fundamentals of Adsorption; Myera, A. L., Belfort, G., Eds.; Engineering Foundation: New York, 1983; p 207.
coverages, 8,81, 82 as a function of time, and the available surface functions as a function of coverage were computed. Examples of the simulation data are shown in Figure 2. We first comment on the qualitative behavior. Most striking is the fact that in all four cases, the coverage of one of the components (component 1) has a local maximium. The explanation of this behavior follows from the kinetic equations. As the adsorption proceeds, 81increases while 41 decreases, so that d&/dt decreases. After some time, the net adsorption rate of component 1 is zero. However, the second component 2 continues to adsorb and dl continues to decrease. The solute 1then displays a net desorption (d8l/dt < 01, and the system eventually approaches an equilibrium in which the adsorption rates for both species are zero. Thus, this simple model provides a possible explanation for the Vroman e f f e ~ t ~in ~ twhich 2 one initially adsorbed protein is displaced by another. For the systems shown in parta a and b of Figure 2, the rate constants are the same for both components. The displacement is thus totally attributed to the size difference, with the large component displaying a local maximum. It is worth emphasizing that the behavior displayed by systems a and b is totally different from that predicted by the Langmuir equation, both quantitatively and qualitatively. Indeed, the multicomponent Langmuir equations are expected to be a particularly poor approximation in this situation since they do not exhibit a size-dependent adsorption rate, i.e., for equal concentrations and adsorption and desorption rates, large particles adsorb a t the same rate as smaller ones! Figure 2c, in which the two components have the same size,also displays a local maximum for the coverage of component 1,but in this case it is the result of the difference in the adsorption rate constants. A second issue is how well the approximate kinetic equations describe the simulation data. We have integrated eqs 7 and 8 with the appropriate values of the adsorption and desorption parameters, k , i and kd,i, and size ratio, y, numerically to obtain the time-dependent coverages. As can be seen from the figure, the agreement between the theory and simulation is excellent, particularly in parts a and b of Figure 2. This agreement is all the more remarkable considering that two different approximations are inherent in the equations. The SPT equations of state on which the theory is based are themselves approximate, and are expected to deteriorate in quality with increasing surface coverage. Secondly, as already discussed, the surface configurations are not equilibrium configurations except a t the final steady state when the net adsorption of both components is zero. As might be expected, the equilibrium coverages are described more accurately than the transient values. The largest differences between the theory and simulation occur around the local maxima. Slightly larger discrepancies are observed in Figure 212, which has larger values of the equilibrium coverages. In summary, according to the new kinetic equations the adsorption rate of a component in a mixture depends not only on the adsorption rate constants and the total surface coverage, as in the Langmuir equation, but also on the size of the competing solutes and the composition of the adsorbed layer. For values of kd,i that are not too small, the proposed equations describe computer simulations of a reversible adsorption process with high accuracy. An improved description would be desirable if the desorption (24) Vroman, L.; Adams, A. L.; Fiacher, G. C.; Munoz, P. C. Blood 1980, 55, 156.
1666 Langmuir, Vol. 10, No. 6,1994
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Figure 2. Partial coverages of components 1 (6'1) and 2 (6'2) as a function of the reduced simulation time, t* (eq lo), for the adsorptiondesorption of disks of diameters 6 1 and 62. The points show simulation data and the solid lines were obtained by integratingthe rate equations. (a, top left) 62/61 = 0.5, k&2/k41c1 = 1 and k*a1= k*d,z = 0.25. 01q = 0.097,6'p= 0.208. (b, top right) udal = 0.25, k&dk,lcl = 1 and k*d,i = k*da = 0.25. 01m = 0.049,6'P = 0.124. (C, bottom left) 6 2 / 6 1 = 1.0, k&2/k,ici 0.25 and k*d,i 0.2, k*d,2 = 0.05. 81q = 0.208, 6 ' 2 9 = 0.208. (d, bottom right) 6 2 / 6 1 = 0.25, k,2~2/k,lci = 1 and k*d,l = 0.01, k*dg = 0.1. 6'1W = 0.040,6'zq = 0.198. The equilibrium coverages were found from the stationary solutions of eqs 7 and 8.
rates are small. In such cases, the adsorption process may appear irreversible on the timescale of observation and equations developed for the RSA process could be used. We believe that a large amount of experimental data could be profitably reanalyzed using the new equations. Although we have applied them only in the case of a twocomponent mixture of hard disks, they are considerably more general. They are readily extended to any number of components and describe particles of arbitrary convex shape. The equations may also find application in the
description of molecules that adsorb in two or more orientations, each with varying projected areas. Our treatment does, of course, assume an energetically homogeneous surface. Future refinements and extensions of the theory should allow for the possibility of surface heterogeneities, which are of common occurrence.
Acknowledgment. The support from NSF GER9024174 and Rohm and Haas Company is gratefully
acknowledged.
