2348
Langmuir 1997, 13, 2348-2353
Exchange Kinetics in Spherical Geometry Yongmei Wang Department of Chemistry, North Carolina A & T State University, Greensboro, North Carolina 27411
Randy G. Diermeier† and Raj Rajagopalan*,†,‡,§ Department of Chemical Engineering and Department of Physics, University of Houston, Houston, Texas 77204-4792 Received August 5, 1996. In Final Form: December 19, 1996X The decay of the surface concentration Γ(t) of polymers undergoing exchange kinetics between a spherical adsorbent and the bulk solution is studied. The influence of bulk diffusion and the radius R of the spherical adsorbent on the decay rate has been examined. It is shown that the influence of bulk diffusion is reduced when the radius of the adsorbent particle is small. However, in contrast to the case of planar adsorbents, for which a simple-exponential decay of Γ(t) corresponds to detachment-controlled desorption, a simpleexponential decay in the case of spherical adsorbents is not necessarily an indication of detachmentcontrolled decay. It is shown that when R is small (relative to the recapture length, Q, given by the ratio of the coefficients of reattachment and detachment), the decay of Γ(t) is close to a simple-exponential function with a decay lifetime τ ≈ τdet + QR/D, where τdet is the detachment lifetime of the polymer from the surface and D is the diffusion coefficient of the polymer in solution. Only when R is small enough so that QR/D < τdet, does the measured decay lifetime become the true detachment lifetime. The results may also be extended to understand the exchange rate of surfactant molecules among micelles or bilayers.
I. Introduction Experimental measurements of the rate of exchange such as the exchange of adsorbed polymers with polymers in the bulk solution are often made1-6 to obtain dynamic information on the systems. Related examples include the exchange of diblock copolymer chains between polymer micelles (measured, for example, using the fluorescencelabeling method7) and the exchange of small surfactant molecules among micelles or bilayers. In most of these studies, the measured decay is often assumed to be controlled solely by either the rate of detachment of polymers from the adsorbing surface in the case of exchange of adsorbed polymers or the rate of extraction of a single surfactant chain from the micelle in the case of exchange of surfactants between micelles. The possibility of bulk diffusion affecting the measured decay has been considered rarely. Granick and co-workers4,5 have made extensive experimental measurements of polymer exchange kinetics and have reported observing a stretchedexponential decay of the surface concentration of the polymers. This phenomenon has puzzled many investigators and has received several explanations.4,8 We have recently shown that the observed stretched-exponential †
Department of Chemical Engineering, University of Houston. Department of Physics, University of Houston. § Current address: Department of Chemical Engineering, University of Florida, Gainesville, FL 32611-6005. E-mail: Raj@ Eng.UFL.edu. X Abstract published in Advance ACS Abstracts, March 1, 1997. ‡
(1) Pefferkorn, E.; Carroy, A.; Varoqui, R. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 1997. (2) Pefferkorn, E.; Haouam, A.; Varoqui, R. Macromolecules 1989, 22, 2677. (3) Van der Beek, G. P.; Cohen Stuart, M. A.; Fleer, G. J. Macromolecules 1991, 24, 3553. (4) Douglas, J. F.; Johnson, H. E.; Granick, S. Science 1993, 262, 2010. (5) Johnson, H. E.; Douglas, J. F.; Granick, S. Phys. Rev. Lett. 1993, 70, 3267. (6) Dijt, J. C.; Cohen Stuart, A.; Fleer, G. J. Macromolecules 1994, 27, 3229. (7) Wang, Y.; Kausch, C. M.; Chun, M.; Quirk, R. P.; Mattice, W. L. Macromolecules 1995, 28, 904. (8) Chakraborty, A. K.; Adriani, P. M. Macromolecules 1992, 25, 2470.
S0743-7463(96)00766-4 CCC: $14.00
behavior arises from slow bulk diffusion.9 In the case of strong adsorption, as is the case in most instances of polymer adsorption, the so-called recapture length, Q, is large, where Q is defined as Q ) Γeq/Ceq, Γeq and Ceq being the equilibrium surface and bulk concentrations. The recapture length in polymer adsorption is typically of the order of micrometers. Polymer chains that leave the surface have to diffuse over a distance that is greater than Q to avoid being recaptured by the adsorbing surface. Hence, when the diffusion in the bulk is slow, the desorbed chains have a higher probability of reattachment and, therefore, the influence of diffusion on the decay can be significant. More discussion on the observation of the stretched-exponential behavior will be presented in the next section. As one might expect, the extent of the influence of diffusion on the decay rates of exchange kinetics depends upon the geometry of the adsorbing surface (the substrate). The commonly used substrates are either planar surfaces or spherical colloidal beads. Experimental measurements of exchange kinetics of polymers adsorbed on colloidal beads were first made by Pefferkorn et al.,1,2 who have shown that there are exchanges between adsorbed polymers and polymers in the bulk solution. However, the decay rates measured in these studies were equated simply to the detachment rates of polymers from the surface. It is not clear if and how much the diffusional resistance in the bulk phase has influenced the measured decays in the above studies. Therefore, in this report, we examine the interplay between the diffusion and the detachment steps in exchange kinetics in spherical geometry. Our objective here is to develop the criteria necessary for the interpretation of experimental kinetic data on the basis of spherical adsorbents so that the rate parameters can be determined appropriately from experimental measurements. In particular, we find that as the radius R of the spherical adsorbent decreases, the decay of the surface concentration can change from diffusion-controlled to detachmentcontrolled. However, there is a range of R for which the (9) Wang, Y.; Rajagopalan, R.; Mattice, W. L. Macromolecules 1995, 28, 7058.
© 1997 American Chemical Society
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Langmuir, Vol. 13, No. 8, 1997 2349
decay is essentially simple-exponential but the decay lifetime can be much greater than the detachment lifetime. We will present criteria for determining when the curvature of the substrate is important and when the decay will approach the true detachment-controlled limit in the case of spherical substrates. Before we proceed to the theoretical models and results, a brief description of the types of experiments of interest in this context is in order. In this report, we shall be concerned with the types of exchange kinetic measurements similar to the ones reported in refs 1 and 2. In such experiments, the polymer chains initially adsorbed on a substrate are labeled appropriately to distinguish them from the chains in the bulk. The labels are chosen such that they do not influence the adsorption preference of the polymers. At time t ) 0 the adsorbed (i.e., labeled) polymers are in equilibrium with the (unlabeled) polymers in the bulk solution. For t > 0 the experiment monitors the amount of labeled polymers still adsorbed on the surface. We assume that the detachment and reattachment rate coefficients are given by kd and ka, respectively. The detachment and attachment rate coefficients satisfy the following equilibrium condition:
kaCeq ) kdΓeq
(1)
where Γeq and Ceq are the equilibrium surface concentration (which is also the magnitude of Γ of labeled polymers at time zero) and the equilibrium bulk concentration, respectively. A detached labeled polymer can readsorb on the surface if it does not diffuse away from the surface. The kinetic model presented in the next section accounts for such a possibility. This paper is organized as follows. In section II.A, we review the results for the planar geometry and discuss in detail the behavior of the decay of Γ(t) in the ‘tail’ region (i.e., for large times). Section II.B then presents the results for the spherical geometry. This is followed by a brief treatment of exchange kinetics for planar substrates in a finite domain (section II.C). We then conclude the paper with an example showing how one can extend the results obtained here to study the exchange of molecules between micelles. II. Models, Results, and Discussion II.A. Desorption from a Planar Surface in a SemiInfinite Domain. We first review the results for the desorption from a planar surface into a solution in a semiinfinite domain,9 as this will be useful in the subsequent discussion on spherical geometry. The basic equations governing the decay of the surface concentration of labeled polymers during the exchange with unlabeled polymers in a bulk solution are given by
dΓ(t) ) -kdΓ(t) + kaCs(0,t) dt
(2)
∂C(z,t) ∂2C(z,t) (0 < z < ∞) )D ∂t ∂z2
(3)
where Γ(t) is the concentration of the labeled chains on the surface per unit area, C(z,t) is the concentration of the labeled chains in the bulk solution, Cs(0,t) is the concentration C(z,t) at z ) 0, and kd and ka are the detachment and the attachment rate coefficients, respectively. During the exchange of chains between the substrate and the solution, the adsorbed layer (consisting of both labeled and unlabeled chains) is in equilibrium with the bulk
solution. Hence ka and kd can be considered to be independent of time. The initial conditions for the above set of equations are given by Γ(0) ) Γ0 and C(z,0) ) 0. One of the boundary conditions is dictated by conservation of mass at the surface and is given by
dΓ(t) ∂C |z)0 ) ∂z dt
D
(4)
while the second, for concentration at z f ∞, is simply
C(∞,t) ) 0
(5)
The solution for Γ(t) under the above conditions is given by
R2 R1 Γ(t) ) f(R1xt) + f(R xt) Γ0 (R2 - R1) (R1 - R2) 2
(6)
where f(w) ) exp(w2) erfc(w), in which erfc(w) is the complementary error function. The parameters -R1 and -R2 are the roots of the following equation
y2 +
ka
y + kd ) 0
(7)
- xk2a/4D - kd
(8)
xD
and may be written as
R1,R2 )
ka 2xD
The solution for Γ(t) differs from eq 6 when R1 ) R2 (see ref 9). The decay of Γ(t) according to the above solution can vary from a simple-exponential function to an essentially stretched-exponential function (exp(-tβ), β < 1). If one defines Q ) Γeq/Ceq ) ka/kd and δ ) Q2kd/4D, R1 and R2 can be rewritten as
R1,R2 ) xkd(xδ - ix1 - δ)
(9)
where δ is a dimensionless parameter whose magnitude determines the behavior of the decay. If δ , 1, Γ(t) is close to a simple-exponential decay, whereas Γ(t) exhibits stretched-exponential behavior for δ g 1. The physical significance of the above results can be explained as follows. The time associated with the detachment process is τdet ) 1/kd, and the time associated with diffusion over a distance Q is τdiff ) Q2/4D. The typical value of τdiff for polymers is of the order of seconds when Q is in the range of micrometers. The value of τdet varies with different polymers and different substrates. For example, for polystyrene physisorbed on a silicon surface, τdet is probably of the order of seconds to minutes.6 The relative magnitude (δ ) τdiff/τdet) of these two time scales determines the measured decay. When τdet . τdiff (approximately τdet g 100 τdiff), the decay is close to a simpleexponential function during the early part of the decay (that is, when Γ(t)/Γ0 decreases from 1 to 1/e), since detachment of the polymer from the substrate controls the rate of change of the surface concentration. However, when τdet is comparable to τdiff, the decay of Γ(t) is influenced by diffusion and exhibits a stretched-exponential behavior. Moreover, the time needed for Γ(t)/Γ0 to decay from 1 to 0.05 can extend over times of the order of several thousand τdiff. Usually experimental measurements of the normalized surface concentration can be made down to 0.01. Figure 1 presents several sample decays that cover the range where the normalized Γ decreases from
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of the sphere of radius R, Γ(t) is the amount of labeled chains adsorbed on the spherical surface, C(r,t) is the concentration of labeled chains in the bulk solution, and Cs(R,t) is the concentration of labeled chains at the adsorbing surface. The initial conditions are Γ(0) ) Γ0 and C(r,0) ) 0, and the boundary condition at R corresponding to mass conservation at the surface is
dΓ(t) ∂C ) | ∂r r)R dt
D
(12)
The second boundary condition, in the bulk, is
C(∞,t) ) 0
Figure 1. Surface concentration versus time with planar substrates for τdiff ) 10 (Q ) 2 and D ) 0.1). All variables should be in self-consistent units: (a) τdet ) 1000; (b) τdet ) 100; τdet ) 10.
1 to ∼0.04. In these sample calculations, we have fixed τdiff at 10 and have varied τdet over the set 100τdiff, 10τdiff, and τdiff. The curves for τdet < τdiff will be indistinguishable from the one for τdet ) τdiff. If we assume that τdiff ) 10 corresponds to 10 s in real time, then the time range in Figure 1 will correspond to the usual time window accessible in experimental measurements such as those in refs 4 and 5. One thus sees that curves b and c in Figure 1 exhibit a stretched-exponential behavior in the time window corresponding to the ones for the data reported in refs 4 and 5. Usually τdiff can be of the order of seconds to minutes. Hence the stretched-exponential behavior for Γ(t) reported in refs 4 and 5 can easily be caused by the diffusional resistance of the polymer chains in the bulk. Even the decay under detachment-controlled conditions has a long nonexponential tail, as can be seen in curve a. The physical reason for the presence of a slow and nonexponential tail in the decay of Γ(t) can be explained as follows. The rate of transport through diffusion is proportional to the concentration gradient. Initially the bulk solution has no labeled chains and therefore the transport of labeled chains away from the surface region is relatively fast. At later times, as more and more labeled chains desorb from the surface, the concentration of the labeled chains in the bulk builds up and the concentration gradient decreases. Therefore, the transport becomes slower, which leads to a slow nonexponential decay of Γ(t). This also underscores the point that if other transport mechanisms are present (e.g., continual removal of the solution from the cell), the decay will not extend over such a long period of time. II.B. Desorption from a Spherical Surface in an Infinite Volume. Now we turn to study the central problem in this report, exchange kinetics in spherical geometry. The physical situation under consideration is similar to the previous case, except that polymers are now adsorbed on a spherical surface. We again label the originally adsorbed polymer chains in order to differentiate them from those in the bulk at t ) 0. The equations that govern the decay of the labeled chains on the surface are
dΓ(t) ) -kdΓ(t) + kaCs(R,t) dt
[
]
∂C(r,t) 1 ∂ 2∂C(r,t) r )D 2 ∂t ∂r r ∂r
(R < r < ∞)
(10) (11)
where the origin of the coordinate is fixed at the center
(13)
The above equations can be solved using the Laplace transform:
R1(R1 - A2) Γ(t) ) f(R xt) + Γ0 (R3 - R1)(R2 - R1) 1 R2(R2 - A2) R3(R3 - A2) f(R2xt) + f(R xt) (R1 - R2)(R3 - R2) (R1 - R3)(R2 - R3) 3 (14) in which the function f(Rxt) is of the same form as in eq 6. The parameter A2 is defined below, and the three parameters -R1, -R2, and -R3 are the roots of the following cubic equation:
y3 + A2y2 + A1y + A0 ) 0
(15)
with
A0 ) kdxD/R; A1 ) kd; A2 ) ka/xD + xD/R (16) The following results for the R’s can be found in standard mathematical handbooks:
A2 - (s1 + s2) 3
(17)
R2 )
A2 1 ix3 + (s1 + s2) (s - s2) 3 2 2 1
(18)
R3 )
A2 1 ix3 + (s1 + s2) + (s - s2) 3 2 2 1
(19)
R1 )
with
s1 ) [p + (q3 + p2)1/2]1/3 and s2 ) [p - (q3 + p2)1/2]1/3 (20) where
1 1 p ) (A1A2 - 3A0) - A32 6 27
(21)
1 1 q ) A1 - A22 3 9
(22)
The sign of ∆ ) q3 + p2 determines the nature of the three roots. When ∆ < 0, R1, R2, and R3 are all real. When ∆ > 0, R1 is real and R2 and R3 are complex conjugates. Figure 2 presents several sample calculations of Γ(t) for different values of R with ka, kd, and D constant. The calculation shows that the decay from a spherical surface (denoted, for convenience, by ΓSp(t)) is always faster than the decay from a planar surface (denoted by ΓPl(t)). Figure
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Langmuir, Vol. 13, No. 8, 1997 2351
Figure 2. Surface concentration versus time for a spherical bead of radius R for kd ) 0.01, ka ) 0.02, and D ) 0.001 (δ ) 10): (a) planar surface (R ) ∞); (b) R ) 10; (c) R ) 1; (d) R ) 0.1; (e) simple-exponential function exp(-kdt).
2 also includes the decay of ΓPl(t) and a simple-exponential decay function given by exp(-kdt). The decays of ΓSp(t) for different values of R lie in between these two limiting functions. These numerical results demonstrate that the decay of ΓSp(t) can change from diffuion-controlled to detachment-controlled as the radius R decreases for otherwise identical conditions. It is of interest to determine the criteria for the above two limiting situations, namely, (i) when ΓSp(t) approaches ΓPl(t) (i.e., the surface of spherical adsorbents can be considered as ‘flat’) and (ii) when ΓSp(t) approaches the simple-exponential decay function exp(-kdt). To make the discussion simpler, we define a dimensionless time t′ ) kdt so that the three parameters R′1, R′2, and R′3 (R′ ) R/xkd) can be expressed in terms of two dimensionless parameters, δ ) Q2kd/4D and x ) Q/R. The cubic equation which determines the three roots - R′1, - R′2, and - R′3 then becomes
(
y3 + 2xδ +
)
x x )0 y2 + y + 2xδ 2xδ
(23)
[
1 1 2 3 (1 - δ) + (4δ - 5)x + 3 + x + x3 + 27 2δ 4δ 1 4 x (24) 16δ2
(
)
]
The sign of ∆ determines the nature of the three roots in spherical geometry, which in turn determines the characteristic behavior of ΓSp(t). Using the above expression, one can impose a further restriction on the value of R for the first limiting situation by requiring that the absolute magnitude of the first order term in x in eq 24 be small enough such that ∆ is essentially given by its value at x ) 0:
x
4Q. On the other hand, if δ is much less than 1, we have R > 5Q. These specifications provide quantitative measures for the following physically intuitive situation: i.e., the surface can be considered ‘flat’ when the surface curvature (represented by the magnitude of R) is much larger than the recapture length, Q. The surface curvature can also influence the adsorption properties such as the adsorbed amount per unit area or the thickness of the adsorbed layer at equilibrium. Such an influence arises from the modifications of the conformation of adsorbed polymers caused by the curvature of the adsorbing surface. However, this influence of surface curvature on the equilibrium adsorbed properties will be important only when the curvature is significant over the length scale of the size of a polymer chain, namely, its radius of gyration (typically around nanometers). In contrast, the influence of surface curvature on the decay of ΓSp(t) in exchange kinetics becomes important only when the curvature is significant over the length scale of the recapture length, Q. In most cases of polymer adsorption, the recapture length, Q (which is around micrometers), is much larger than the size of the polymer chains (which is around nanometers). Therefore a spherical adsorbing surface may be effectively ‘flat’ as far as the equilibrium adsorption properties are concerned but may not be ‘flat’ for the decay of exchange kinetics. II.B.2. Limiting Condition for ΓSp(t) f exp(-kdt). To find the condition for this limiting situation, we first note that if 2xδ , x/2xδ, the coefficient 2xδ in the y2-term may be neglected. In this case the three roots simplify to
(29)
which can also be rewritten as
QRkd ,1 D
(30)
This criterion specifies the condition corresponding to not only when ΓSp(t) is a simple-exponential decay function but also when the rate coefficient for the decay is kd. As discussed in section II.A (following eq 9), δ < 1 specifies the condition for ΓPl(t) to become simpleexponential and equivalently (in the case of planar substrates) detachment-controlled. As we will see shortly, the above holds good for spherical substrates for δ < 1. However, for δ > 1, ΓSp(t) can be essentially simpleexponential as R decreases but before the condition in eq 30 is satisfied (i.e., the decay of ΓSp(t) is not detachmentcontrolled). This is a consequence of the fact that as x increases, ∆ in eq 24 changes from negative (since δ > 1) to positive. When ∆ > 0, R′2 and R′3 are complex conjugates, and R′1 is real, positive, and large in magnitude. Hence the decay of ΓSp(t) is primarily controlled by the f(R′2xt′) and f(R′3xt′) terms. Since R′2 and R′3 are complex conjugates, the decay of ΓSp(t′) is close to a simpleexponential function with a decay rate, k′, approximately given by
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Figure 4. Curves a-d in Figure 2 over a wider range of time, t. Figure 3. Rate k′ versus x for δ ) 10: (a) numerical values based on eqs 17-22; (b) plot according to eq 32 (k′ ) x/(x + 4δ)). 2
k′ ) (Im(R′2))
(31)
in which Im(R′2) is the imaginary part of R′2 (this can be shown mathematically and has also been used in developing the criteria for the planar substrates). One sees from eq 24 that ∆ becomes positive approximately when x > (δ - 1)/(4δ - 5). If δ is much greater than 1, this transition occurs at x ∼ 1/4, almost independent of δ. Further, for x > 1, the rate coefficient, k′, is described well by the following relationship:
k′ ≈
x x + 4δ
(32)
Figure 3 presents the numerical values of k′ based on eqs 17-22 and eq 32. The limit of k′ when x f ∞ is 1, which corresponds to the case when ΓSp(t) approaches the limiting form exp(-kdt). Otherwise, the decay lifetime, τ, of ΓSp(t) is given by (according to eq 32)
τ ≈ τdet +
QR for x > 1 D
(33)
in which τdet ) 1/kd. The quantity QR/D usually cannot be greater than τdiff ) Q2/4D (defined earlier in section II.A) since the above equation is valid when R < Q (x > 1). One sees that the criterion specified in eq 30 for ΓSp(t) to approach the limiting form exp(-kdt) arises directly from eq 33. Only when QR/D , τdet (which is simply equivalent to eq 30) is the decay detachment-controlled and the measured τ ≈ τdet. The above criterion for the decay being detachmentcontrolled in the case of spherical geometry is consistent with the criterion developed in the case of planar geometry. For the latter case, the criterion for the decay being detachment-controlled is τdiff , τdet. Since QR/D is less than τdiff, QR/D , τdet. On the other hand, if τdiff > τdet (the ΓPl(t) is diffusion-controlled), then there is a range of R for which QR/D . τdet (QR/D < τdiff), and the decay is simpleexponential. The measured decay lifetime under such circumstances is then dominated by QR/D and is independent of τdet. In section III, we will show that this may be the case for the measured duration time of surfactant molecules in micelles. Moreover, in the previous section, we noted the presence of a long nonexponential tail in the decay of ΓPl(t) even when δ < 1 (i.e., detachment-controlled decay) with a normalized Γ/Γ0 greater than 0.01, thus measurable in experiments. In spherical geometry, this is no longer the
case. Figure 4 replots the results in Figure 2 over an extended range of time. One can see that for curves c and d the Γ/Γ0 value at large t is much less than 0.01. Therefore, the nonexponential tail will not be detected in experiments and curves c and d will appear to be essentially simple-exponential decays in experimental measurements. The important physical picture that emerges from the above discussion is the following: The physical situation leads to two time scales, namely, τdet (the detachment lifetime) and τdiff ) Q2/4D (the time for diffusive transport of molecules over a distance of the order of the recapture length, Q), and it is the relative magnitude of the two that determines the characteristic behavior of Γ(t). For τdet . τdiff, ΓPl(t) is detachment-controlled. For these conditions, ΓSp(t) is also detachment-controlled. For τdet e τdiff, ΓPl(t) becomes diffusion-controlled. For ΓSp(t), however, two different situations present themselves in this case. If the radius R of the spherical substrate is such that R > 4Q, then the substrate may be considered to be effectively planar. In such a case, ΓSp(t) is also diffusion-controlled. On the other hand, if R < Q, the decay will be simpleexponential with a decay lifetime given by τ ≈ τdet + QR/ D. The maximum value of QR/D is τdiff. Only when R is small enough so that QR/D , τdet will the measured lifetime be the true detachment-controlled decay time. II.C. Desorption from a Planar Surface in a Finite Domain. Now we briefly examine desorption from a planar surface in a finite domain. The physical situation under consideration here is similar to that in section II.A, but the volume of the bulk phase extends over only a finite length L. We would like to determine if the finite extension in length L would affect the measured decay. The equations that govern the decay of Γ(t) are the same as those in section II.A, with the exception of the second boundary condition specified in eq 4, which is now replaced by the zero-flux condition:
∂C )0 | ∂t z)L
D
(34)
The equations can again be solved using the Laplace transform, but the Laplace inversion can be done only numerically. Figure 5 presents numerically calculated decays of Γ(t) for several values of L. The early parts of the decays for the different L’s are the same, but the Γ(t)’s approach different plateau values for larger t’s. The plateau values are determined by the conservation of mass of the initially adsorbed chains and can be written as
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Langmuir, Vol. 13, No. 8, 1997 2353
N0 ) kaC0 k1 4πR20
(36)
where k1 is the rate of extracting one surfactant molecule from the micelle, ka is the rate of addition of one monomer to a micelle, and C0 is the free monomer concentration. The coefficient ka has the dimension of length/time, as in the cases of adsorption. We can define a recapture length, Q, as in the previous sections as follows:
Q)
Figure 5. Surface concentration versus time for a planar substrate in a finite domain in the z-direction for ka ) 0.0082, kd ) 0.00041, and D ) 0.013: (a) L ) 25; (b) L ) 50; (c) L ) 100.
Γ(∞) Q ) Γ0 Q+L
(35)
In the case of most experimental studies, this value is negligible. In Monte Carlo simulations, this value may not be negligible9 if the length L is not sufficiently large. However, the influence of the finite dimension in the direction normal to the substrate is felt only at large times, and the rate of decay of Γ(t) before Γ(t) reaches the plateau value is the same as the one in the case of semi-infinite domain. Therefore, the results presented earlier in ref 9 for comparisons between Monte Carlo simulations and the solution for the semi-infinite-domain case are valid. III. Concluding Remarks We will now comment on the implication of the above results to the exchange of surfactant molecules (or copolymer chains) between two micelles (or polymer micelles). The decay measured in this type of exchange kinetics (i.e., the decay of the number of labeled chains in a micelle) is more complicated, since one has to consider the distance between the two micelles. In addition, the micelles may collide with each other and exchange surfactant molecules during such collisions. Therefore, we will not be able to calculate the exact decay in this type of exchange kinetics. However, the exchange of molecules in a micelle with free monomers is similar to what we have considered in section II.B. There is a similar recapture length due to the equilibration between molecules in a micelle with free monomers in the bulk solution. The flux of molecules leaving a micelle must balance the flux of molecules added to the micelle, if a micelle does not break apart over time. Assuming that N0 surfactant molecules form a spherical micelle with radius R0, we have the following relationship between k1 and ka:
N0 4πR20C0
(37)
where N0/4πR20 is the head-group density at the interface. As an example, consider micelles formed by sodium dodecyl sulfate in water, with a typical aggregation number around 70, a micellar radius of (assuming spherical shape) 18 Å, and a critical micelle concentration of 8 mM. The estimated value of Q for this case is 3.6 × 103 Å, two orders of magnitude larger than the radius of the micelle. Hence the decay of exchange of surfactants in a micelle with free monomers is likely to be a simple-exponential function. Using a typical value of diffusion coefficient for a surfactant, D ) 1 × 10-6 cm2/s, we estimate that τdiff ) 3.2 × 10-4 s and QR/D ) 6.0 × 10-6 s. The experimentally determined average lifetime of a surfactant molecule in a micelle is known10 to be around 10-5 s. This lifetime is very close to the value of QR/D estimated above. Hence we suspect that this average lifetime measured is really QR/D, not necessarily 1/k1. However, resolving the difference between the decay rate QR/D and the intrinsic lifetime τ ) 1/k1 in experiments is difficult for these system. In closing, we have examined the decay of surface concentration in exchange-kinetic experiments based on spherical adsorbing surfaces relative to the ones with planar surfaces. The influence of bulk diffusion is shown to be more pronounced in planar geometry than in spherical geometry. In the case of planar geometry, a simple-exponential decay of surface concentration indicates the dominance of the detachment step of the polymers from the surface. However, in the case of spherical adsorbents, this is not necessarily the case. It is shown that when R < Q, the decay of ΓSp(t) will be simple exponential, with a decay lifetime given by τ ) τdet + QR/D. If QR/D . τdet, then the measured lifetime is independent of τdet. Only when QR/D < τdet, will the measured lifetime be the true detachment-controlled decay lifetime. Thus by decreasing the radius of the spherical beads, one can reduce the influence of the transport step on the decay rate and can measure the true detachment rates. This is one of the advantages of experimental studies based on colloidal beads as substrates. Acknowledgment. We would like to thank the National Science Foundation and the Texas Advanced Technology and Research Program for partial support of the research reported here. LA9607666 (10) Israelachvili, J. N. Intermolecular & Surface Forces, 2nd ed.; Academic Press: New York, 1991.
