J. Phys. Chem. 1983, 87, 347-351
not yet been tested experimentally.
NFE
Acknowledgment. Thanks are due to Neos Company for the samples of NF and NFE and to Nikko Chemicals Company for samples of DE9, DE25, and HE25.
SAS
Symbols DEm polyoxyethylene dodecyl ether, viz., CI2Hz50(CH&HzO),H anionic fluorocarbon surfactant, viz., C9Fl70CBNF H4S03Na (31) Wheeler, J. C.; Widom, B. J. Am. Chem. Soc. 1968, 90,3064.
347
nonionic fluorocarbon surfactant, viz., CgF,,O(CH2CH20)18.4CH3
sodium alkyl sulfate, viz., C,H2,+1S04Na mole fraction of surfactant i in unmicellized molxbi ecules of two surfactants critical mole fraction of hydrocarbon surfactant xcH Xi mole fraction of surfactant i in the system Xmi mole fraction of surfactant i in mixed micelles Tc upper critical solution temperature R&tW NO.SDS, 151-21-3;STrS,3026-63-9; STS, 1191-50-0; DE5,30&95-6; DE7,305597-8; DE9,305599-0; DE25,9002-92-0; HE25, 9004-95-9; NaC1, 7647-14-5; NF, 70829-87-7; NFE, 83731-88-8.
Contribution to the Theory of ReactionIDiffusion Kinetics in Spatially Heterogeneous Mediat Stephen E. Webber Department of Chemisby, UniverstYy of Texas, Austin, Texas 78712 (Recelved: July 8, 1982; In Final Form: September 28, 1982)
A matrix formulation of the linearized kinetic equations that govern simultaneous diffusion and reaction in a spatially inhomogeneousmedium (e.g., micelle or polymer solutions) is presented. For a special model the results are equivalent to earlier solutions (Infelta et al. J.Phys. Chem. 1974, 78, 190. Tachiya, Chem. Phys. Lett. 1975,33,289) but the matrix method may be extended to less restricted models. Several examples are discussed. In all cases numerical solution of the eigenvalues, eigenvectors for a band matrix is required.
I. Introduction One of the long-standing problems in chemical kinetics is to calculate the rate of a process in a system that is not spatially homogeneous. One of the most famous calculations in physical chemistry is the rate of coagulation of colloid particles by Smoluchowski in 1915,’ which has subsequently influenced the elementary application of the diffusion equations to chemically reacting systems. In a well-known review article Noyes2 discussed the early transient behavior of a system with an initially nonequilibrium distribution of reactants, such as might be the case for a system excited by a short burst of radiation. There have been quite a few treatments of this problem for excited-state processes, since the lifetime of the excited state is often short compared to the time required for diffusion to “homogenize” the system. Our interest in this problem arose from experimental studies of excited-state processes in polymer coils in solution. At sufficiently low concentration the polymer coils are independent of each other, but because of the inherent coil density, the concentration of polymer bound chromophore in the vicinity of the coil can be rather high (e.g., >lo-, M). Consequently, a polymer solution is a typical example of a spatially inhomogeneous system in which the region of inhomogeneity may be large compared to the diffusion length of a small molecule during the lifetime of an excited-state species (estimated by & = (SDT#/~ where D is t.he sum of the diffusion constants for the polymer bound species and the small molecule reactant and 73 is the lifetime of the excited-state species). The model to be presented in the next section grew out of the following ‘This work was reported at the 37th Southwest Regional American Chemical Society Meeting, Dec 9-11, 1981, San Antonio, T X . 0022-3654/83/2087-0347$01.50/0
consideration: under what circumstances might one observe complex kinetic behavior in spatially inhomogeneous systems, and under what circumstances can a complex system exhibit simple behavior? While the model to be presented is greatly simplified, some of the results are illustrative and could be extended to a variety of physical systems. In sections 111 and IV some of the ideas illustrated in the next section will be applied to more complex models. A great deal of effort has been devoted to the kinetics of processes involving micelles, which represent the premiere example of a spatially inhomogeneous system. The model discussed in section I1 was f i t presented by Infelta et alS3and Tachiya4 and has been used to analyze experimental quenching results for micelle-bound probes by many worker^."^ The main contribution of this model is that a complex set of diffusion-reaction equations is replaced by pseudo-first-order rate equations written in terms of the occupancy number of a spatially inhomogeneous region of space by a reactant. This linearization of the complete diffusion/reaction equations allows a much more transparent analysis of what is a very complex sys(1) Smoluchowski, M. V. Ann. Phys. 1915,48,1103. 2.Phys. Chem. 1917, 92, 129. (2) Noyes, R. M. Prog. React. Kinet. 1961, 1, 129. (3) Infelta, P. P.; Cratzel, M.; Thomas, J. K. J.Phys. Chem. 1974, 78, 1 on
(4) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (5) (a) Atik, S. S.; Singer, L. A. Chem. Phys. Lett. 1978,59, 519. (b) Ibid. 1979, 66, 234. (6) (a) Dederen, J. C.; Van der Auweraer, M.; DeSchryver, F. C. J . Phys. Chem. 1981,85,1198. (b) Dederen, J. C.; Van der Auweraer, M.; DeSchryver, F. C. Chem. Phys. Lett. 1979,68, 451. ( 7 ) (a) Infelta, P. P. Chem. Phys. Lett. 1979,61,88. (b)Maestri, M.; Infelta, P. P.; Gratzel, M. J. Chem. Phys. 1979, 69, 1522.
0 1983 American Chemical Society
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The Journal of Physical Chemistry, Vol. 87, No. 2, 1983
Webber
(d/dt)pn(t) = [P$n-l(t) - ( P a + nbd)pn(t) + ( n + 1)P@n+l(t)l - (ko + nk,)p,(t) ( 1 ) The first term is from diffusive behavior only. If we take ko = k, = 0 (i.e., no reactive site in the cell) the steady-state solution to (1)is easily obtained: Figure 1. Schematic representation of the model of section 11. A reactive site ( * ) resides within the cell, and the cell exchanges the reactant Q with the solution at rates &, p,,. The reaction rate of Q with is assumed to be directly proportional to the number of Q molecules within the cell (reaction rate = nkq). The reactive site decays unimolecularly with rate constant k o.
tem. As Tachiya4 has pointed out, a rigorous analytical solution exists for the simplest linearized diffusion/reaction model. What we would like to point out in this paper is that the same ideas can be applied to more complex diffusion/reaction models and, by numerically solving the resulting matrix equations, obtain the time dependence. Since the numerical solution of large matrix equations has become routine, one is not limited to oversimplified models simply because an analytical solution exists. Thus we will show how one could deal with the problem of diffusion of a reactant to an inhomogeneous region (e.g., a micelle or polymer coil), followed by diffusion of the reactant within this region to a reactive site. This added feature is quite obviously of importance in certain physical systems but is not easy to deal with m a t h e m a t i ~ a l l y . ~ ~ ~ 11. Simplified Diffusion/Reaction Model for Spatially Inhomogeneous System The physical idea behind this model is illustrated in Figure 1, in which the processes of diffusion into or out of a region of space (e.g., a polymer coil or micelle) of a small quencher molecule (Q) occurs continuously. In the following we will refer to the region of space in which reaction will occur as the “cell”. We are taking the Q occupation number of each region to change by f l for each elementary step (Le., we are ignoring simultaneous diffusion of two or more Q molecules into or out of the cell). We are ignoring any specific interaction of the Q molecules with each other or with the cell, such that the average Q occupancy of the cells will be given by a Poisson distribution (see below). Treating the kinetics of cells as illustrated in Figure 1 leads to a simple “ladder” type theory. In the equations to follow we use the following definitions: p , ( t ) is the probability at time t of a reactive cell containing n of the Q molecules. Pa is the rate at which a cell accepts a Q molecule from the bulk solution; assumed to be proportional to the concentration of Q in the bulk solution, Le., Pa 0: [Q]. This is an important assumption in the original m0del~3~ and neglects correlations of the local concentration in the solution phase with the occupation number of the cell. P d is the intrinsic rate of losing a Q molecule to the bulk solution; for a cell containing n of the Q molecules the rate of loss will be n&. k, is the unimolecular rate of deactivation of a reactive cell. k is the intrinsic rate of quenching (or other reaction) of i$with the reactive site within a cell; for a cell containing n of the Q molecules the quenching rate will be nk,. With these symbols and assumptions of the model the equation for the p , ( t ) quantities can be written: (8)See Van der Auweraer et al. for a rather detailed discussion of intramicellar diffusion on fluorescence quenching of solubilized molecules, as well as references to earlier work by this group. Van der Auweraer, M.; Dederen, J. C.; Gelade, E.; DeSchryver, F. C. J. Chem. Phys. 1981, 74,1140. (9) Gosele, U.; Klein, U. K. A.; Hauser, M. Chem. Phys. Lett. 1979,68, 291.
1-5, = (pa/pd)”(l/n!)e-8./8d = [ ( i i ) n / n ! ] e - n (2) where m
fi
=
npn =
(Pa/Pd)
n=O
= PQJ’c
(3)
Equation 2 is the familiar Poisson distribution, valid for a random occupation of a given volume in a macroscopically homogeneous solution, and fi is the average occupancy number of a cell. This latter must also be equal to the product of the Q number density (pQ) and the cell volume (VC).
The above equations and those to follow depend on the ratio of the rates of Pa, P d , and k,. It is interesting to consider the rates Pa, P d , which must be related to uc, DQ (the diffusion constant of Q) and the number density of Q molecules. If we use the elementary theory of liquidstate collisions1° to estimate the rate of Q molecule encounters with the cell we obtain
Pa = ~ T R ~ S-lQ ~ Q
(4)
where R, is the radius of the (assumed) spherical cell. From eq 3 @d
=
= (3D,/RC2)S-l ( 5 )
(Pa/PQUc) = ( 4 T R 9 Q P Q )/ ( p Q ( 4 a / 3 ) R , 3 )
where u, = ( 4 ~ / 3 ) R , 3was assumed. Thus P d may be taken as the inverse of the average time for a Q inside the cell to diffuse into the bulk solvent. If there exists a specific solvation of Q inside the cell, then the number density inside the cell (= pQceu) is not equal to the number density in the bulk solution (= pQBoh). Thus in eq 3 and 4 one must substitute pOwu and pQWh for pQ, respectively. This requires the multiplication of the right-hand side of eq 5 by the factor pQsoln/pQcel1. The form of the decay terms in eq 1 requires some comment. It is assumed that the reactive site in all cells has an intrinsic unimolecular decay rate constant, k,,. Furthermore, it is assumed that the rate of quenching by Q scales simply as the number of quencher molecules inside the cell. The former assumption is reasonable in the context of the model. The latter is more subtle but would not be expected to hold exactly for most physical cases. Since, in general, the reactive site can be located anywhere in the cell volume, there are obviously situations in which one Q species in a cell would be more efficient than two (or more) located initially more distant from the reactive site. Averaging over these different possible configurations greatly complicates the equations and the intuitive simplicity of eq 1 is lost. We will deal with this situation in a later section, but for now we will deal with eq 1 as it stands, keeping in mind the above caveats. The set of equations like eq 1can be written as a matrix equation (d/dt)b(t) = [-lko
+ R]B(t)
(6)
in which B ( t ) is a vector containing the quantities p , ( t ) , (10) See, for example, Berry, R. S.; Rice, S. A.; Ross, J. “Physical Chemistry”; Wiley: New York, 1980; Section 30.8, eq 30.108.
The Journal of Physical Chemistry, Vol. 87, No. 2, 1983
Reaction/Diffusion Kinetics in Heterogeneous Media
349
I
"1 2
.5
1
2
5
10 20
50
,
,
,
,
.2
.5
1
2
B,
~
5
,
,
10 20
Ko/Kq
= .25
B,/Bd
= 2
,I 50
Ba
Flgure 2. (a) Plot of four lowest eigenvalues of R matrix (seetext) for the case fla/&, = 2 and k , = k, = 1 as a function of Pa. The limiting values given by eq 11 and 13 are indicated by the dashed lines. (b) Plot of the magnitude of c, for i = 1-4 as a function of 8, (same parameters as (a)).
1 is the unity matrix, and R is a relaxation matrix of the form I
.
Kqt
Flgure 3. Plot of ( P ( t ) )as function of k,t for different values of &, Pd,k,, and k,. Strongly nonexponentil decay occurs when &,/Pa is much larger than unity but essentially exponential decay occurs if this ratio is less than unity.
.
The general solution to eq 6 is well-known
c(t) = e-kd&i%ie4t
(8)
i=l
where %i is the ith right eigenvector of R,Xi is the ith eigenvalue, and ci is a coefficient that properly accounts for the boundary conditions, Le. 6(0) = cciiii
(9)
i
The eigenvalues, eigenvectors of a tridiagonal matrix like eq 7 are fairly easy to obtain numerically, although we have not found a useful algebraic solution for general values of k,. However, there are some obvious limits for the solution to eq 6 that are confirmed by numerical solution. If the diffusion terms are small compared to k,, then the eigenvalues of R are approximately the diagonal terms nk, and the eigenvectors have the form (%Jj = 6ij (10) such that
p , ( t ) = p,(0)e-(kO+nkq)t
(11)
If p,(O) is given by a Poisson distribution (which implies no correlation between excitation and n) then a multiexponential decay of the following form is obtained:
p,(t) = [(ii)n/n!]e-Aexp(-(ko
+ nk,)t)
(12)
In the opposite extreme the diffusion terms are very large compared to k,. While awkward to demonstrate algebraically, the general solution (eq 8) is dominated by one of the eigenvalues, X1 = k,n, and the corresponding eigenvector. Thus in this case, after a short transient decay, we observe Pn(t) = Cl(%l)ne-(ko+fiW -a
(13)
where the constant a in eq 13 expresses the fact that at early times the decay of p,(t) may not be described by a single exponential. In eq 13 we have denoted the "dominant eigenvalue" of R as XI since the long time behavior of p , ( t ) is determined by the smallest eigenvalue of R.
The properties of the solutions discussed in eq 10-13 are illustrated in Figure 2. In Figure 2a we have plotted the four lowest eigenvalues of R (plus ko) as a function of &, for ba/@d = 2 (taking ko = k, = 1 in scaled units). As can be seen from the figure, for small 0,the eigenvalues are close to those given by eq 11 (i.e., ko + nk, = 1 + n). As 0,increases the lowest rate constant asymptotically approaches the value in eq 13, KO + iik, (= 3 for the assumed ko,k values). The other rate constants increase in magn i t d e very rapidly. The coefficients ci in eq 8 also depend strongly on pa. The magnitude lcil for i = 1-4 is plotted in Figure 2b (the sign of ci may be positive or negative, depending on ki). These lcil have been derived by assuming that p(0) obeys a Poisson distribution. For 0,Ik, all the lcil are significant (likewise lcil for i > 4 are significant, although tending to be smaller for higher i), which implies multiexponential decay in eq 8. For 0, > k, c1 dominates. For 0,> 5 the decay function, eq 8, becomes essentially a single exponential with rate ko + iik,. Obviously this behavior will depend on kq/Pa, but we have not explored this relationship numerically. In a typical experimental situation one observes the rate of decay of all excited or reactive cells, given by
( p ( t ) )= Cpn(t) i=O
(14)
which from eq 11 and 13 we expect to be multiexponential or singly exponential in the two limits described. Some calculations of the decay of ( P ( t ) )for several representative cases are presented in Figure 3. As discussed in the Introduction this set of equations also admits a closed analytical solution, first provided by Infelta et al.3 and derived by Tachiya4 using a generating function technique. The form of the solution is ( P ( t ) ) / ( P ( O )=) exp(-Alt - Az(l - e-A3t)]
(15)
where A1 = k0 + A2
+ kq)l
[Pakq/(Pd
= Pak:/[@d(Pd
+ kql21
A3 = P d + kq
(16)
(using the notation of the present paper). The agreement
350
Webber
The Journal of Physical Chemistty, Vol. 87, No. 2, 1983
between the formula given in eq 15 and the matrix solution is essentially exact so long as the R matrix is sufficiently where n’corresponds to the dilarge that p,,(O) is mension of the R matrix (stated differently, n‘is the largest occupation number considered in the matrix model). However, it is obviously quite convenient to fit eq 15 to experimental data and there would be no advantage at all to the matrix formulation if only this simplified model were considered. However, since the matrix formulation is capable of generalization to a variety of linearized models, it represents a useful contribution to the treatment of spatially inhomogeneous systems. We will consider two examples in the sections that follow. 111. Relation of t h e Model to Standard Collisional Rate Theory: T h e Two-State Model In this version of the previous model we take our cell to be very small (of molecular dimensions) with volume u,*. Because of the small volume we may safely assume that rt lolo s-l for an “allowed” reaction?. The general eigenvalues of the rate matrix of eq 17 are A1
A,
+ K1) + (1/2)[(KO - K1)2 + 4pa*~d*]”2 = -(1/2)(KO + K,) - (1/2)[(& - K1)’ + 4@a*pd*]1/2
= -(1/2)(Ko
(19) In the limit that kq* >> &* the following approximate forms are useful A, = -(ko + Pa*/2) + ... A2
= -(ko
+ kq* + pd* + Pa*/2) + ...
(20)
Inspection of eq 20 shows that A, is the slow rate, and solution of the eigenvector and boundary conditions reveals that cl(fl), dominates the relaxation of @(t), and consequently (P(t)). The other root corresponds to what is usually referred to as “static quenching”, i.e., those molecules at t = 0 that happen to be close together, described in this model by pl(0). Thus the bimolecular quenching rate constant from eq 20 (A,) is given by (see eq 4)
Kp = ( P a * / 2 p Q ) = 4a(RC*D/2) s-l cm3
(21)
where R,* is the radius of uc*. If Kq is to agree with standard rate theory D is the sum of the diffusion constant of Q and uc*, and R,* can be interpreted as twice the sum of the collision radii of uc* and Q. Thus this two state model easily accommodates “static” and normal diffusional quenching. There is a special case of the two state model we would like to discuss since it is germane to our previous work on polymer photophysics.” Suppose the reactive volume u,* is part of a larger cell that may contain variable numbers of Q molecules. The volume u,* is not isolated from the bulk solution by the larger cell, although a hindered diffusion to uc* may exist. We have proposed a model like
this for the quenching of a reactive species bound to a polymer chain, in which the polymer coil would play the role of the larger cell.lla We will make the following assumptions for this case: = ~TR$QPQ= rate of Q collisions with the cell of
volume u, (= (4a/3)RC3) (22a)
pa* = 13(uc*/uc) = ( C / P )= rate of collision with
u,*
(22b) where u,* = v,/P is assumed and P is the number of groups bound to the polymer coil (one of which is excited). Thus the sense of eq 22b is that the reactive volume uc* is as accessible to the solvent as any other part of the coil and hence the rate of Q collisions with u,* is just P’of the total rate of coi1-Q collisions. In our previous work it was pointed out that from standard theory the radius of gyration of a polymer coil is proportional to [7](a+1)/a,where a is the parameter in the Mark-Houwink equation (i.e., Ma a [q] and M and [7] are the molecular weight and intrinsic viscosity of the polymer, respectively). Using eq 22b we obtain the proportionality
pa* a
[7](a-2)/3a
(23)
which was found to agree reasonably well with the quenching rate of the triplet state of poly(2-vinylnaphthalene) samples of different molecular weights. Since the expression is a modified version of eq 4 we may refer to this model as a “free draining volume” model in the sense that the interior of u, has access to the bulk solvent.
IV. Generalized Diffusion/Reaction Model In the previous two sections we have considered models in which a reaction occurred upon entrance of molecule Q into either a cell volume (u,) or a smaller cell volume containing a reactive species (u,*). In this section we wish to consider the equations that govern a spatially inhomogeneous system in which each cell (volume u,) can contain a small cell (volume u,*) containing a reactive molecule. As in section I11 we assume that u,* is so small that there is negligible probability of occupancy by more than one quencher. In the present extension we will consider diffusion of Q molecules into uc* from the bulk solution and from u,. Thus this model is more appropriate for micelles and related phases where a reactive molecule could be located near the bulk solution, and hence be subject to direct collision from the solvent species, or diffusion within the cell can effect collisions with u,*. We use the following definitions: p(i,n;t)is the probability that volume u, will contain n Q molecules and that u,* will contain i Q molecules; the total number of Q molecules within the total volume u, + u,* is i + n, where i = 0 or 1. pa, Pa* are the rates at which u, and uc*, respectively, accept Q molecules from the bulk solution. &, &* are the rates at which u, and uc*, respectively, lose Q molecules into the bulk solution. Td, ya are the rates at which a Q molecule is transferred into u,* from u, or out of u,* into u,, respectively. As stated above we will take i to have only the values 0 and 1. Therefore we will be concerned only with states p(0,n;t) and p(1,n;t). A representation of the kinetic scheme embodied in this model is given in Figure 4. The set of equations governing the p(i,n;t)can be written in matrix form (like eq 6) except that the R matrix is not (11) (a) Pratte, J. F.; Noyes, W. A., Jr.; Webber, S.E. Polym. Photochem. 1980, I , 3. (b) Pratte, J. F.; Webber, s. E. Macromolecules 1982, 15, 417.
The Journal of Physical Chemistry, Vol. 87, No. 2, 1983 351
Reaction/Diffusion Kinetics in Heterogeneous Media
I
1
I
L.
Figure 4. Schematic representation of the diffusion/reactlon system of section IV. Transition rates between different p(i,n)states indicated for first few members.
tridiagonal but is a band matrix as given by eq 24. In eq 24 K(O,n) = P a + P a * + P a * + nPd + n?’d K(l,n) = P a
+ Pd* + nod + kq* + ?’a
(25)
The general solution for this system of equations is exactly like eq 8 and 9 except that the vector b(t)has doubled in dimension because of the double indices n and i, and likewise R has twice the dimensionality. In addition, the R matrix in (24) is more difficult to solve than that of eq 7 because of the larger number of non-zero matrix elements parallel to the diagonal. The relationships between the various rate constants can be deduced in a fashion like eq 4 and 5. For example, if u,* could be populated only by diffusion from the bulk then (Pa*/Pd*)
=
uc*PQ
(26)
but since u,* may not necessarily be treated like a spherical volume in bulk solution, the form for Pa* may not be treated in general like eq 4. Likewise if the equilibrium were established between u, and u,* only (no diffusion to and from u,* from the bulk solution such that all P* quantities were zero) then the steady-state equations for occupancy would be -n~&(O,n) ~$(l,n-1) = dp(O,n)/dt = 0 (27)
+
with the solution o(1,n-l)
n(?’d/?’a)p(otn)
(28)
We may assume that p(1,n-1) is much smaller than p(0,n) because of the small volume of u,* (hence p(1,n-1) has no effect on the normalization of p(0,n)).Then the average occupancy of u,* for all cells (taking u, and u,* together) containing a total of n Q molecules is given by fi* = fl(1,n-l) = n(yd/?’a)P(O,n) = n(Uc*/uc)p(o,n) (29) where the last form in eq 29 is based on the volume ratio of u,* and u, (i.e., no specific binding of the Q molecule occurs in any u,*). Thus the relationship (?’d/?’a) = (vc*/uc) (30) is predicted. This is analogous to eq 26, and, as is the case
there, no further relationship can be drawn without specifying the assumed geometric properties of u,* and the nature of diffusion within u, (i.e., isotropic or anisotropic, etc.). Because of the greater complexity of R in eq 24 relative to eq 7 the general relation between the various rate constants is less obvious. Certain limiting behaviors can be deduced by inspection: (1)If Pa*, Pd* are very small and Y ~Yd , are very large the solution to the general equation should be like the simplified model of section 11. This would be appropriate to a reactive site at the interior of a cell. More realistically if Pa*, &* can be neglected relative to Y ~Yd , but ya, Yd are much smaller than Pa, Pd (i.e., exchange of quenchers with solvent faster than diffusion within the cell) then we would expect single exponential decay in which the average occupancy of the cell can be used, Le., like the two-state theory except that the number density of quenchers within u, may be taken to be fi/u,. (2) If Pa*, pd* are much larger than Td then the rate of decay due to quenching should be equivalent to the two-state theory of section 111. The behavior of ( P ( t ) )(eq 14) should be independent of Pa, Pd since the number of quenchers inside u, is irrelevant to the quenching process Yd. for small The main advantage to the present set of equations is that it provides a systematic way to approach bimolecular reaction systems in spatially inhomogeneous systems, in which diffusion is competitive with a reaction. The number of independent parameters are three (e.g., pa,Pa*, and Itq*) since the values of the inverse diffusive rates can be fixed by equilibrium considerations (Le., see eq 3, 26, and 30). (It is assumed that the equilibrium distribution of the Q molecules is either random or is known). It will often be the case that Itq* is known from small molecule analogues, or that the calculated rate constants are not sensitive to Itq* because it is so large relative to the diffusive rates.
V. Summary The linearized diffusion/reaction rate equation for a spatially inhomogeneous system has been written in the form of a matrix equation. For a simplified case the numerical solutions are equivalent to an analytical solution first presented by Infelta et ala3and T a ~ h i y a .It~ is demonstrated that the matrix formulation can be extended to other models yielding a straightforward numerical solution for cases that do not admit to a closed analytical solution. We believe this is a useful contribution to the theory of reaction kinetics in inhomogeneous media since a variety of fairly complex models for micelles has been proposed recently.8 Acknowledgment. We acknowledge useful discussions with Professor W. C. Gardiner of this Department. We have greatly benefitted from financial support by the Robert A. Welch Foundation (F-356) and the National Science Foundation (DMR8013709). We thank one of the referees for a number of helpful suggestions.
