J . Phys. Chem. 1988, 92, 4824-4825
4824
Solvent and ligation effects can change the ordering of these orbitals in metalloporphyrins and Davis et aL3have suggested that the two nearly degenerate ground states are mixed for Chl a*+. The results presented here suggest that there is a reordering of the energy levels for Chl a in the solvents used so that D, is the ground-state designation. The molecular orbital electron density map of DI4’ is consistent with the observed R R vibrational shifts, Le., downshift of v(C,C,) and possibly u(CbCb)vibrations, changes in the C,-N region, and conjugation of u(Cloa=O) into the T system. (47) Petke, J. D.; Maggiora, G. M.; Shipman, L.; Christoffersen, R. E. Photochem. Photobiol. 1979, 30, 203-223.
The most significant change that we have identified in the R R spectrum of the Chl a cation radical is centered at the isocyclic fifth ring and may result from enolization of the C9 ketone group. Work currently underway is aimed at providing further insight into the precise nature of the structural changes that occur with cation radical formation. RR studies of Chl a derivatives should aid in the determination of the changes that occur at the ring V ester and ketone groups. Acknowledgment. This work was supported by the Department of Energy (Grant DE-FG02-84ER13261) and the National Institutes of Health (Grant GM 35108). We thank Dr. Henry Crespi for the kind gift of d-Chl a and Dr. Laura A. Andersson for helpful discussions.
Random Sequential Adsorption of Hard Rods on a One-Dimensional Continuum Surface P. Schaaft and H. Reiss* Department of Chemistry and Biochemistry, University of California, Los Angeles. Los Angeles, California 90024 (Received: May 26, 1988)
The theory for the kinetics of the random sequential adsorption (RSA) of hard rods on a line is formulated in such a manner that the time rate of change of uncovered intervals, as species, is expressed in terms of only the consumption of such species. This formulation leads to equations that are somewhat simpler than those of previous formulations, and a solution that is identical with that obtained by previous authors. The new formulation appears to be more effective than previous ones in suggesting an approximation for dealing with RSA in multidimensional systems.
Introduction The problem of random sequential addition (RSA) has attracted considerable attention during the past few years.’“ When focused on the irreversible adsorption of immobile species on a surface, it is called random sequential adsorption. The rules governing such a process are the following: once a particle is adsorbed it can never desorb (irreversibility); the particles cannot diffuse on the surface and they are never allowed to overlap one with each other. The salient feature of the process is of course not so much described by the word “sequential” as by the terms “immobile” and ”irreversible”. These terms emphasize the fact that the process is entirely non-Markovian and possesses an infinite memory. The mathematics designed to treat it must therefore match this fact-a distinctly nontrivial requirement. Important applications of RSA are (1) the modeling of oxidation and other chemical reactions along a polymer chain,5 and (2) the adsorption of proteins2 and latexes6 on solid surfaces. Theoretical approaches to the problem have involved both lattice7,* and continuumg models, and in some cases continuum results have been derived from lattice treatments by allowing the lattice parameter to pass to zero.5 In spite of the number of papers devoted to this problem, exact analytical solutions describing the time evolution of the addition process have been obtained only for the one-dimensional although a number of author^'*^*^ have successfully derived the asymptotic form of this time evolution for two-dimensional cases, as well as the corresponding exact degree of coverage at the “jamming limit”, a necessary consequence of the immobility of the adsorbed entities. For the case of hard rods in the one-dimensional continuum, the analytical solution has been obtained by first solving the problem on a lattice and then passing to the c o n t i n u ~ m or , ~ by working in the continuum d i r e ~ t l y .In ~ both of these approaches, intervals of uncovered surface large enough to accept an additional hard rod are treated as different kinetic species identified by their +Permanent address: Institut Charles Sadron, (CRM-EAHP), CNRS, Strasbourg, France.
0022-3654/88/2092-4824$01.50/0
precise lengths. As such they can be both “produced” and “consumed” by the adsorption process, and the corresponding kinetic equations that describe the time evolution of their “concentrations” reflect these possibilities in the appearance of both “product” and “reactant” terms. A necessary consequence of this is the emergence of kinetic equations of considerable complexity, from which it is difficult to extract approximate methods for dealing with the continuum problem in two or more dimensions. To reduce some of this complexity we formulate and solve below kinetic equations involving defined species which are only consumed; Le., they are never produced by the adsorption process. Of course, we arrive at the same result as previous authors for the time evolution of the one-dimensional continuum process. Not only is the treatment simpler (to our taste), but there is good reason to assume that it will more easily point the way to useful approximations for the treatment of multidimensional systems (indeed it already has been useful, in this respect, for the case of lattice systemslO).
Formulation and Solution As indicated above, the random sequential adsorption of hard rods of length a on a continuous one-dimensional surface has been studied by Gonzales et aL5 and by W i d ~ mand , ~ exact analytical expressions of the time evolution have been obtained. We address the same problem here, using a somewhat different approach, originally developed by Cohen and Reiss” for the study of the (1) Hinrischsen, E. L.; Feder, J.; Jsssang, T. J . Stat. Phys. 1986,44, 793. (2) Feder, J.; Giaever, I. J . Colloid Interface Sci. 1980, 78, 144. (3) Pomeau, Y. J . Phys. A: Math. Gen. 1980, 13, L193. (4) Swendsen, R. H. Phys. Reu. 1981, A24, 504. (5) Gonziles, J. J.; Hemmer, P. C.; Hsye, J. S.Chem. Phys. 1974, 3, 228. (6) Onoda, G.Y.; Liniger, E. G. Phys. Reu. A . 1986, 33, 715. (7) Nord, R. S.;Evans, J. W. J . Chem. Phys. 1985, 82, 2795. (8) McKenzie, J. K. J. Chem. Phys. 1962, 37, 723. (9) Widom, B. J . Chem. Phys. 1966, 44, 3888. (10) Schaaf, P.; Talbot, J.; Rabeony, H. M.; Reiss, H. J . Phys. Chem., following paper in this issue. (11) Cohen, E. R.; Reiss, H. J . Chem. Phys. 1963, 38, 680.
0 1988 American Chemical Society
The Journal of Physical Chemistry, Vol. 92, No. 17, 1988 4825
Letters adsorption of “dimers” on a one-dimensional lattice. This approach has recently been extended by Schaaf and Reiss’O to the case of squares (each occupying four sites) on a simple square lattice, where a remarkably accurate, but still approximate, analytical result for the time evolution was obtained. In this latter approaches the main difference from previous methods lay in the definition of “species”. For example, in the case of the one-dimensional lattice, an uncovered interval, Le., a run of n ”empty” sites, was defined as a run of a t least n empty sites, it being possible that the run was part of a run of m empty sites with m > n. Similarly in the hard-square case, any twodimensional, connected group of designated empty sites (a possible species) could be part of a larger group of sites. We continue this definition of species in the present development. Thus we define by P(x,z) the probability that, at the reduced time z, one can find an interval of at least length x, within which there is no part of a hard rod of length a. The reduced time is given in terms of the actual time t by
where k ( t ) is the chance per unit time, at time t , that the center of a hard rod is adsorbed per unit length of uncovered surface. It is assumed that k(t) is independent of position on the surface. Suppose an interval of at least length x is known to be uncovered. The conditional probability qCy,z) that an adjacent interval of lengthy is also uncovered will be independent of x for x > a, since, if at time z the length x is uncovered, there could never have been any communication between the length y and the system on the other side of the length x because adsorption is both irreversible and immobile. In other words if length x is uncovered at time z it must always have been uncovered, and communication across the length could only have occurred by means of covering. We may therefore write
where, in the last step, we have used eq 3. Substituting eq 5 into eq 4 gives -dqCv)/dz = Y4b)
(6)
which can be integrated subject to q(y,O) = 1 to yield q ( j ) = e-yr
(7)
For x C a, eq 3 must be replaced by -dP(x)/dz = ( a - x ) P ( a )
+2
a+x
PCy) dy
1
(8)
(I
This equation is also easily explained. The first term on the right refers to the destruction of the interval when it is part of an uncovered interval of at least length a when the center of the rod being adsorbed lies within the range a - x. Correspondingly the second term on the right refers to the two ranges (on the two sides) of the interval a where an adsorbed particle can still overlap the interval of length x and destroy it. These ranges extend from a to a x on either side and involve uncovered intervals as large as a x, thus accounting for the limits on the integral in eq 8. Again, there are no “production“ terms in the equation. Using eq 2 and 7, the integrand in eq 8 may be written as
+ +
P(y) = P ( a )
(9)
so that the equation itself becomes -dP(x)/dz = ( a - x)P(a)
+ 2P(a)I
o+x ,-I*’)
a
dy (IO)
For x = a, eq 10 becomes
where, for simplicity, we omit the argument z. For x > a we can write -dP(x)/dz = (x
- a)P(x) + 2 1 * + ’ P ( y 3 dy’
(3)
This equation is easily explained. The first term on the right accounts for “destruction” of the interval of length x by the adsorption of a hard rod such that the entire rod lies within x ; i.e., the center of the rod can range through an interval of length (x - a). The second term on the right accounts for the destruction of the interval by processes in which the adsorbed rod lies partially outside of the interval. Clearly, the center of the rod can lie within an interval of length a/2, on either side of the length x, and still destroy the original interval. However, for the formation of this configuration it is necessary to have an original uncovered interval of at least length x a. This accounts for the limits in the integral on the right of eq 3. The factor of 2 preceding the integral accounts for the fact that this process of destruction can occur on both sides on the interval of length x. Notice that there are no ”production” terms on the right side of eq 3. For an interval of length x + y, eq 3 becomes (writing y ” = x y throughout) y’ - y under the integral and x
+
-
+
-d[P(x) qCv)l/dz = (x
+ Y - a ) P ( x ) 40.’) + 24Cv)Jx+s PCv’? dY” (4)
where we have used eq 2 both directly and in the form PO,’’+y) = PO,”) 40.’). The left side of eq 4 may be expressed as
which may be integrated, subject to the initial condition P(a,O) = 1, to yield
Finally, substitution of eq 12 into eq 8 and specialization of the result to x = 0, followed by integration, subject to the initial condition P(0,O) = 1, yields P(0,z) = 1 - J‘a exp[ 2J2\-}
dz”] dz’
(13)
Since P(0,z) represents the probability that a point, randomly chosen, is empty, it represents 1 - 8(z), where 8 is the fraction of surface covered at time z. This result for the evolution of the coverage with time was first obtained by Gondles et aL5 by passing from the discrete lattice to the continuum, and by W i d ~ mby ,~ working only in the continuum. However, as explained above, the new derivation involves no “production” of species and matches the underlying irreversible process more closely. As such it appears to be a better starting point for the design of approximate methods for dealing with multidimensional problems.’0 Acknowledgment. This work was supported by the National Science Foundation under NSF Grant DMR 8421383 and by the Centre National de la Recherche Scientifique.
