- Email: [email protected]
The dissolution of micelles in relaxation kinetics
The Dissolution of Micelles in Relaxation Kinetics INTRODUCTION Muller (1) first called attention to the remarkable situation in which the same elementary reaction steps o f monomer addition to or extraction from micelles give rise to relaxation processes in two time domains differing by several orders of magnitude. He attributed the faster process to simple single monomer addition and extraction, and the slower process to a cascade o f such events in which a micelle would eventually dissolve completely into monomers. A number o f authors have developed quantitative variations on this theme (2-5). In many recent developments, it has been repeatedly asserted that during the fast process, no micelle dissolution at all can occur, i.e., the fast process proceeds at constant molar concentration of micelles, and considerable discussion over this point has appeared in the recent literature (6-8). A simple straightforward test o f this hypothesis is developed here directly from the differential equations of kinetics.
in the first parentheses of dA~+l/dt. From this simple result, it is clear that the total number of micelles will be constant only under the unusual accidental condition that the concentrations near the minimum in the initial equilibrium distribution correspond to the condition of final equilibrium in the p e r t u r b e d state, but otherwise generally only if the distribution has a zero minimum at A~ and Aj-1. If this zero concentration prevails initially, however, then there can be no direct flow o f material through the region of the minimum, which prohibits the type of sequential, stepwise transfer assumed by Aniansson et al. (2, 3, 5) in their steady-state treatment of the slower relaxation process. In their treatment o f the kinetics, these authors have also assumed that the rate constants for addition of monomer or removal of m o n o m e r are independent of the size o f the micelle. If the same assumption is made here, we may write these rate constants as k~ and k~ in the initial equilibrium state, and as ks and kb in the perturbed state. If primed concentration variables also refer specifically to the initial equilibrium state, then Eq. [1] becomes, at zero time, when the perturbation ends,
THEORY We define a general molar distribution under an initial condition of equilibrium, A0, A1, A2 . . . . An, in which the subscript by convention designates the number of monomer units added to one original monomer unit, e.g., A0 is the monomer itself. The rate of change of molar concentration of a general oligomer, A~, is given by
[3]
This can be written, by the conditions for initial equilibrium, (dAjdt)o = (ks/K' - kb)([A~] - [A~+I])
[4]
Here K = kf/kb and K ' = k'flk'b. When Eq. [4] is summed from i = j to i = n to get the initial rate o f micelle dissolution, we find that for this sum to equal zero (constant number o f micelles), it is necessary that we have the same succession o f negative first differences of concentration in the right-hand side of Eq. [4] below the micelle peak as we have positive first differences o f concentration above the micelle peak. This implies that complete symmetry in the molar distribution o f micelles is required to equate the sum over all micelles of Eq. [4] to zero. Conversely, once we have a completely symmetrical molar distribution o f micelles, this requires that the sum of Eq. [4] over the micellar region is zero. This in turn requires that Eq. [2] equals zero at zero time. We have already shown that this implies in general
[1]
Here the k-values all represent rate constants in the perturbed condition of the system. If the distribution has a minimum molar concentration at i = j , then we might define the micellar region as every species from j - m e r up to the largest micelle, n-mer. To find the rate of change of the total number of moles o f micelles we need only determine the sum of all terms of the type of Eq. [1] from i = j to i = n. Thus, ~ dAi/dt = kj_lz[Ao][Aj_l] - kj,j-l[A~].
+ (-ks[A~][A;] + kb[A~+,]).
= ( k b l K ' ) ( K - K')([A~] - [A'+,]).
d A J d t = (ki-l.~[Ao][Ai 1] - k~,~-l[Ai]) + (-ki,~+l[Ao][Ai] + ki+a,i[Ai+l]).
( d A j d t ) o = (ks[A'o][A'_l] - kb[A[])
[2]
i=J
Equation [2] follows directly from Eq. [1] as a consequence of the fact that the term in the second parentheses o f d A J d t always adds out with the term 274
0021-9797/80/010274-02502.00/0 Copyright © 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal o f Colloid and Interface Science, Vol. 73, No. 1, January •980
NOTES the existence of a zero minimum of concentration in the overall distribution, unless the system already is at equilibrium in the perturbed state (in which case we should not be discussing kinetics). Having pointed out these generalities, we find it of special interest to draw attention to the fact that in the original Aniansson theory (2, 3, 5), the molar distribution of micellar species was assumed a priori to be symmetrical, e.g., a Gaussian error function. The direct consequence of this assumption is that there is indeed no dissolution of micelles during the rapid relaxation process. Subsequently, however, there will be no flow of material through the region of the consequential zero minimum in the overall distribution, a direct contradiction of the assumed steadystate condition for the slow process. Thus, some of the initial assumptions of this theory are internally inconsistent with each other. SUMMARY For a slow steady-state process involving sequential stepwise micelle dissolution to exist at all, the distribution of micelles must be somewhat skew, and the dissolution process, although very slow, must actually begin at the time of perturbation, together with the fast relaxation process. Thus, during the fast process, the total number of micelles is changing with time, and, in all likelihood, the rate of this dissolution process itself, although always small, is a maximum at zero time. This brings the picture into better accord with the general thesis of relaxation kinetics, that all coupled processes start together at zero time and occur in parallel. A recent extension of theory (9) has considered more general micellar distributions. The present type of symmetry
275
argument is not at all limited to the case where each forward rate constant and reverse rate constant is independent of micelle size; it has also been developed for a more general micelle distribution (10, 11). REFERENCES 1. Muller, N., J. Phys. Chem., 76, 3017 (1972). 2. Aniansson, E. A. G., and Wall, S. N., J. Phys. Chem. 78, 1024 (1974). 3. Aniansson, E. A. G., and Wall, S. N., J. Phys. Chem. 79, 857 (1975). 4. Lang, J., Tondre, C., Zana, R., Bauer, R., Hoffmann, H., and Ulbricht, W., J. Phys. Chem. 79, 276 (1975). 5. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 !1976). 6. Aniansson, E. A. G., and Wall, S. N., Ber. Bunsenges, Phys. Chem. 81, 1293 (1977). 7. Chan, S. K., and Kahlweit, M., Ber. Bunsenges. Phys. Chem. 81, 1294 (1977). 8. Hoffmann, H., Ber. Bunsenges. Phys. Chem. 82, 988 (1978). 9. Almgren, M., Aniansson, E. A. G., and HolmLker, K., Chem. Phys. 19, 1 (1977). 10. Kegeles, G., J. Phys. Chem. 83, 1728 (1979). 11. Kegeles, G., Arch. Biochem. Biophys., in press. GERSON KEGELES
Section of Biochemistry and Biophysics Biological Sciences Group The University of Connecticut Storrs, Connecticut 06268 Received June 4, 1979; accepted July 12, 1979
Journal of Colloid and Interface Science, Vo|. 73, No. 1, January 1980
in the first parentheses of dA~+l/dt. From this simple result, it is clear that the total number of micelles will be constant only under the unusual accidental condition that the concentrations near the minimum in the initial equilibrium distribution correspond to the condition of final equilibrium in the p e r t u r b e d state, but otherwise generally only if the distribution has a zero minimum at A~ and Aj-1. If this zero concentration prevails initially, however, then there can be no direct flow o f material through the region of the minimum, which prohibits the type of sequential, stepwise transfer assumed by Aniansson et al. (2, 3, 5) in their steady-state treatment of the slower relaxation process. In their treatment o f the kinetics, these authors have also assumed that the rate constants for addition of monomer or removal of m o n o m e r are independent of the size o f the micelle. If the same assumption is made here, we may write these rate constants as k~ and k~ in the initial equilibrium state, and as ks and kb in the perturbed state. If primed concentration variables also refer specifically to the initial equilibrium state, then Eq. [1] becomes, at zero time, when the perturbation ends,
THEORY We define a general molar distribution under an initial condition of equilibrium, A0, A1, A2 . . . . An, in which the subscript by convention designates the number of monomer units added to one original monomer unit, e.g., A0 is the monomer itself. The rate of change of molar concentration of a general oligomer, A~, is given by
[3]
This can be written, by the conditions for initial equilibrium, (dAjdt)o = (ks/K' - kb)([A~] - [A~+I])
[4]
Here K = kf/kb and K ' = k'flk'b. When Eq. [4] is summed from i = j to i = n to get the initial rate o f micelle dissolution, we find that for this sum to equal zero (constant number o f micelles), it is necessary that we have the same succession o f negative first differences of concentration in the right-hand side of Eq. [4] below the micelle peak as we have positive first differences o f concentration above the micelle peak. This implies that complete symmetry in the molar distribution o f micelles is required to equate the sum over all micelles of Eq. [4] to zero. Conversely, once we have a completely symmetrical molar distribution o f micelles, this requires that the sum of Eq. [4] over the micellar region is zero. This in turn requires that Eq. [2] equals zero at zero time. We have already shown that this implies in general
[1]
Here the k-values all represent rate constants in the perturbed condition of the system. If the distribution has a minimum molar concentration at i = j , then we might define the micellar region as every species from j - m e r up to the largest micelle, n-mer. To find the rate of change of the total number of moles o f micelles we need only determine the sum of all terms of the type of Eq. [1] from i = j to i = n. Thus, ~ dAi/dt = kj_lz[Ao][Aj_l] - kj,j-l[A~].
+ (-ks[A~][A;] + kb[A~+,]).
= ( k b l K ' ) ( K - K')([A~] - [A'+,]).
d A J d t = (ki-l.~[Ao][Ai 1] - k~,~-l[Ai]) + (-ki,~+l[Ao][Ai] + ki+a,i[Ai+l]).
( d A j d t ) o = (ks[A'o][A'_l] - kb[A[])
[2]
i=J
Equation [2] follows directly from Eq. [1] as a consequence of the fact that the term in the second parentheses o f d A J d t always adds out with the term 274
0021-9797/80/010274-02502.00/0 Copyright © 1980 by Academic Press, Inc. All rights of reproduction in any form reserved.
Journal o f Colloid and Interface Science, Vol. 73, No. 1, January •980
NOTES the existence of a zero minimum of concentration in the overall distribution, unless the system already is at equilibrium in the perturbed state (in which case we should not be discussing kinetics). Having pointed out these generalities, we find it of special interest to draw attention to the fact that in the original Aniansson theory (2, 3, 5), the molar distribution of micellar species was assumed a priori to be symmetrical, e.g., a Gaussian error function. The direct consequence of this assumption is that there is indeed no dissolution of micelles during the rapid relaxation process. Subsequently, however, there will be no flow of material through the region of the consequential zero minimum in the overall distribution, a direct contradiction of the assumed steadystate condition for the slow process. Thus, some of the initial assumptions of this theory are internally inconsistent with each other. SUMMARY For a slow steady-state process involving sequential stepwise micelle dissolution to exist at all, the distribution of micelles must be somewhat skew, and the dissolution process, although very slow, must actually begin at the time of perturbation, together with the fast relaxation process. Thus, during the fast process, the total number of micelles is changing with time, and, in all likelihood, the rate of this dissolution process itself, although always small, is a maximum at zero time. This brings the picture into better accord with the general thesis of relaxation kinetics, that all coupled processes start together at zero time and occur in parallel. A recent extension of theory (9) has considered more general micellar distributions. The present type of symmetry
275
argument is not at all limited to the case where each forward rate constant and reverse rate constant is independent of micelle size; it has also been developed for a more general micelle distribution (10, 11). REFERENCES 1. Muller, N., J. Phys. Chem., 76, 3017 (1972). 2. Aniansson, E. A. G., and Wall, S. N., J. Phys. Chem. 78, 1024 (1974). 3. Aniansson, E. A. G., and Wall, S. N., J. Phys. Chem. 79, 857 (1975). 4. Lang, J., Tondre, C., Zana, R., Bauer, R., Hoffmann, H., and Ulbricht, W., J. Phys. Chem. 79, 276 (1975). 5. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, I., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 !1976). 6. Aniansson, E. A. G., and Wall, S. N., Ber. Bunsenges, Phys. Chem. 81, 1293 (1977). 7. Chan, S. K., and Kahlweit, M., Ber. Bunsenges. Phys. Chem. 81, 1294 (1977). 8. Hoffmann, H., Ber. Bunsenges. Phys. Chem. 82, 988 (1978). 9. Almgren, M., Aniansson, E. A. G., and HolmLker, K., Chem. Phys. 19, 1 (1977). 10. Kegeles, G., J. Phys. Chem. 83, 1728 (1979). 11. Kegeles, G., Arch. Biochem. Biophys., in press. GERSON KEGELES
Section of Biochemistry and Biophysics Biological Sciences Group The University of Connecticut Storrs, Connecticut 06268 Received June 4, 1979; accepted July 12, 1979
Journal of Colloid and Interface Science, Vo|. 73, No. 1, January 1980