J. Phys. Chem. 1993,97, 42564258
4256
Irreversible Adsorption on Nonuniform Surfaces: The Random Site Model Xuezhi Jin,+ N.-H.Linda Wang,? Gilles Tarjus,s and Julian Talbot'qt School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, and Laboratoire de Physique ThCorique des Liquides, Uniuersitd Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France Received: February 9, 1993; In Final Form: March 15, 1993
The random sequential adsorption (RSA) process provides a starting point for the development of quantitative descriptions of the irreversible adsorption of proteins and other large molecules. To date, none of the numerous theoretical and computational studies of RSA have considered the effect of nonuniform aperiodic substrates which are present in many physical systems. We study the irreversible adsorption of spherical particles adsorbing on randomly distributed p i n t sites. There exists an unexpected, yet simple, mapping between this process and the corresponding adsorption on a smooth, continuous surface.
Many proteins and other large molecules are known to adsorb irreversibly from solution on solid substrates. While the description of the adsorption process is relatively straightforward at short times, the many-body effects which become increasingly significant as additional molecules crowd together on the surface greatly complicate the development of theoretical approaches. One simple model which does incorporate the surface exclusion effect is the random sequential adsorption (or, more generally, addition) (RSA) process. At this level of description, the protein molecules are represented by impenetrable spheres which adsorb sequentiallyin randomly selected locations on a uniform surface. If a trial position results in an overlap with a preadsorbed particle, the trial is abandoned. Otherwise, the particle remains fixed in place. The surface fills rapidly at first, but very slowly in the latter stages. The "jamming limit" coverage, corresponding to the state in which it is impossible to add additional spheres in any location, is 54.7%, as has been found in a number of computer simulation studies.'-2 The belief that even simple RSA is a reasonable model for irreversible adsorption processes is supported by a number of experimentalstudies. For example, Onoda and Liniger3measured the saturation coverage resulting from the adsorption of latex spheres on a silica surface. The observed value, 55%, is in good agreement with the value quoted above. Various extensions and generalizations of the simple model are possible. For example, there have been a number of studies of the RSA of nonspherical particles, which may be more realistic for some proteins such as f i b r i n ~ g e n . ~Other ? ~ models incorporate more realistic transport mechanisms of the adsorbent from the bulk to the surface. In diffusion RSA (DRSA), for example, the adsorbing particles undergo a Brownian motion above the substrates6 Interestingly, the saturation coverage of the DRSA process is virtually indistinguishablefrom that of the simpler RSA model.6 To our knowledge, none of the numerous studies of RSA and related processes have considered the effect of a nonuniform aperiodic substrate. Yet in a number of adsorption processes, the substrate is manifestly heterogeneous. In particular, the motivation for the present work is provided by affinity chromatography,in which the adsorbent has immobilized ligands that bind selectively with a desired solute or a class of solute^.^ This technique has been widely used for separating proteins, enzymes, and other biological materials.8 We propose the following model to describe the adsorption process occurring in affinity chromatography. The ligands are represented by hard disks of diameter d. A total of Ns disks are placed on a planar surface of area L2 to generate a disordered Purdue University. Pierre et Marie Curie 4.
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0022-3654/93/2091-4256%04.00/0
configuration of sites at a density ps. The configurations can be realized in a number of ways. For example, one may quench an equilibrium fluid of hard disks. The adsorption process is then modeled by attempting to add sequentially hard spheres of diameter IJ at the center of each site (correspondingto adsorption on a ligand). New particles cannot overlap with any preadsorbed spheres and remain fixed in place once accepted. The simplest case of this model occurs when o l d < 1 for which a particle cannot be excluded from a site by the presence of other particles on any other site, but only by a particle already on the given site. Obviously, all sites will be eventuallyoccupied and 8, = a,where a = m2p,/4 is a dimensionless site density. The adsorption kinetics in this case are described exactly by the Langmuir model. A more interesting and more realistic situation is for which u >> d. In this limit, the adsorption sites approach points and may be considered as completely randomly distributed (strictly, only in thelimit d-0); i.e., theconfigurationof sitescorresponds to that of an ideal two-dimensional gas. However, since u remains finite, nontrivial exclusion effects are present between the adsorbent species. In this Letter, we present an investigationof the random sequential adsorption of spherical particles (or disks) on point sites which have been randomly placed on a planar surface, or RSA-RS (RSA-random sites) in abbreviated form. In computer simulations of the process, the substrate is first generated by placing Ns p i n t sites randomly and uniformly on a square cell of side L. In the adsorption process, one of the NS sites is selected at random. If a sphere of diameter IJ can be centered on this site without overlapping with any previously adsorbed spheres (the usual periodic boundary conditions are imposed), the attempt is accepted;otherwise, a new site isselected. A number of properties may be computed; in particular, we are interested in the time-dependent surface coverage and its dependence on a. The elapsed time is clearly proportional to the number of attempts, NA,which have been made toadsorb spheres. Specifically, we define the dimensionless time as
To obtain reliable estimatesfor 8(t, a ) ,it is necessary to average over both a number of site configurations (at fixed a) and, for each site configuration, a number of adsorption processes. Some results are shown in Figure 1. For very small values of a,for which the site-site separation generally exceeds IJ, there are no exclusion effects, and e(-, a)= a + @a2). At the other extreme of a dense covering of sites, the surface appears continuous to the adsorbent, and one then recovers the usual RSA result, e(-, -) = 0.541. Further consideration of the RSA-RS model suggests an 0 1993 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 97, No. 17, 1993 4257
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Figure 1. Time-dependent coverage for an RSA process on the random site model for different values of the dimensionless site density, a. The dimensionlesstime, r, iscomputed from thecumulativesimulationattempts according to (1).
intimateconnection with the RSA process on a continuoussurface, RSA-C. More specifically, we argue that for any a there exists an exact mapping between configurations of the RSA-RS and RSA-C processes at the same density. Therefore, all properties of the RSA-RS process may be derived from a knowledge of only the behavior of the RSA-C process. To develop fully our argument, some additional concepts and definitions are required. Figure 2a shows a configuration of hard spheres of diameter u produced by a RSA-C process. If a total NA attempts have been made, the elapsed (dimensionless) time, 7 , is given by (1) (with t replaced by 7 ) . The shaded area represents the exclusion surface, i.e., area that is inaccessible to the center of an additional sphere. The kinetics of the adsorption processes are governed by d0cld.r = 4 C ( f l C ( 7 N (2) where &(ec) is the fraction of the surface which is available. Similarly, in the RSA-RS model we have
where &(& a) is the fraction of available sites. With these definitions in hand, we consider again the configuration shown in Figure 2a. The connection with the RSA-RS model is made by associating an adsorption "site" with each attempted placement of a new sphere. Clearly, there is a site at the center of each adsorbed sphere. In addition, there are a number NA - N sites lying in the exclusion area which correspond to failed attempts. The resultingconfiguration of adsorbed particles and sites, shown in Figure 2b, does not correspond directly to a RSA-RS configuration, since all sites are unavailable for the adsorption of a new sphere. We therefore add an additional Ns - NA sites randomly to the configuration, as shown in Figure 2c. Of these, a number (Ns- NA)&(7)are, on average,available,i.e., lieoutside the exclusion surface. The fraction of the total Ns sites which is available is then [(Ns - N A ) / & ] & ( ~ ) . We now claim that this configuration is characteristic of configurations generated by an RSA-RS process. Therefore
where we have used the definitions of a and 7 . From the kinetic equations (3) and (4) we have
Since, by construction, the coverage resulting from the RSA-RS
Figure 2. (a) A configuration of hard spheres generated from a RSA process on a continuous surface. Each sphere has a relative (projected) area of 0.006 14, and 200 attempted placements, corresponding to a reduced time of 7 = 1.2, have been made. Fifty-four attempts were successful, leading to a coverage of 0.332. (b) The same configuration as in (a) showing in addition the exclusion circles associated with each adsorbed sphere (shaded areas) and the location of the 200 attempts (*). Note that all the attempts are contained within the exclusion surface. (c) shows a configuration which has been generated from (b) by placing 800 additional sites ( 0 )randomly on the surface. These, together with the locations of the attempts shown in (b), form the RSA-RS configuration. The total of IO00 sites corresponds to a dimensionless site density of a = 6.14.
4258 The Journal of Physical Chemistry, Vol. 97, No. 17, 1993 G5OI
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Figure 3. Result of applying the mapping, (6,8),to the curves in Figure 1. The coincidence of all curves suggests that the kinetics of a RSA process on the random site model can be deduced from a knowledge of the kinetics of an RSA process on a continuous surface.
process is equal to that of the RSA-C process,
e(t; C Y ) = Substituting (de/ar), with (dO/dr)(d~/dt), yields
(6)
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or 7 = a ( 1 - e-'/") (8) Equations 6 and 8 express the mapping between the RSA-RS and RSA-C processes. In particular, the saturation coverage of the RSA-RS model with a site density CY is equal to the coverage of an RSA process on a continuous surface at a dimensionless In Figure 3, we have shown the simulation results time 7 = CY. as a function of 7 = a( 1 - e-'/,). The coincidence of all curves supports the mapping between the two processes. For convenience, we present a semiempirical equation for this curveor, equivalently, the time-dependent coverage of the RSA-C process:
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1 0.3 136r2 0 . 4 5 ~ ~ (9) 1 1.82857 0 . 6 5 5 3 ~ ~7'/* The coefficients and form of the equation have been chosen to ec(r)= 0.547 (1
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Letters reproduce both the asymptotic and short-time behavior (to order ~3). Equation 9 describes the simulation results more accurately than any of the previously proposed fitting f ~ n c t i o n s .Equations ~ 6,8, and 9 provide a complete description of the adsorption kinetics for any value of a. A rigorous proof of the correspondence, as well as details of the structure of the adsorbed monolayer, may be obtained from a distribution function approach. The development uses concepts parallel to those applied to the RSA-C process.I0 We note that the simple mapping exists only for the case of randomly distributed sites. Nevertheless, we believe that the work presented here adds significance to previous studies of the RSA and related processes. In particular, the mapping applies in any dimension (with appropriate definitions of the densities). Therefore, we may immediately deduce the exact solution to the one-dimensional problem, Le., adsorption on randomly distributed point sites on a line, from the known analytic solution for the time-dependent density in the "parking problem".'I The three-dimensional process may be useful for generating models of random media.12 We believe that the model presented here can provide a reasonable description of the adsorption kinetics in affinity chromatography. It is certainly more accurate than those with a Langmuirian basis, except for systems with a very small CY value. The finite size of ligands in a real adsorbent will result in a distribution which is not strictly random. However, if the adsorbing species is large compared with the ligand, the random site model will be a very good approximation. Acknowledgment. The financial support of Rohm and Haas and NATO (Grant 890872) is gratefully acknowledged. References and Notes ( 1 ) Feder, J. J. Theor. Biol. 1980, 87, 237. (2) Meakin, P.; Jullien, R.Phys. Rev. A, in press. (3) Oncda, G. Y.;Liniger, E. G. Phys. Reu. A 1986, 33, 715. (4) Talbot, J.; Tarjus, G.; Schaaf, P. Phys. Rev. A 1989, 40, 4808. (5) Vigil, R. D.; Ziff, R. M. J . Chem. Phys. 1990, 93, 8270. (6) Senger, B.; Schaaf, P.; Johner, A.; Voegel,J.-C.; Schmitt, A.; Talbot, J. Phys. Rev. A 1991, 44, 6926. (7) Cuatrecasas, P.; Wilcheck, M.; Anfinsen, C. B. Proc. Natl. Acad. Sci. U.S.A. 1968, 61, 636. (8) Chase, H. A. Chem. Eng. Sci. 1984, 39, 1099. (9) Schaaf, P.; Talbot, J. J . Chem. Phys. 1989, 91, 4401. (10) Tarjus, G.;Schaaf, P.; Talbot, J. J . Stat. Phys. 1991, 63, 167. (1 1) Gonzblez, J. J.; Hemmer, P. C.; Hoye, J. S. Chem. Phys. 1974, 3, 288. (12) Given, J. A. Phys. Rev. A 1992, 45, 816.
