J . Phys. Chem. 1989, 93, 2192-2193
2192
TABLE 11: Dependence of p and K, on 1-Pentanol Concentration in DTAB (50 m d g of D,O) Micellar Solution
[pentanol], m 0.037 0.074 0.1 1 1 0.148
P 0.67 0.60 0.66 0.64
f 0.05 f 0.05 f 0.05
f 0.05
Kc 36 21 35 32
f8 f6 f8
fI
while the FT-PGSE self-diffusion method"I3 monitors the time-average mole fraction of solute diffusing with the micelles. The degrees of solubilization obtained with these two methods are in very good agreement, demonstrating that the two techniques measure the same quantity. Goto et aL30 found in gel filtration experiments that the apparent distribution coefficients of alkylparabens in sodium dodecyl sulfate micellar solutions depend on the concentration of the alkylparabens. Zana et aL3' in their study of alcohol solubilization in micellar solution found that the apparent distribution coefficient obtained with the solubility method is about one-quarter of the value obtained in the vapor pressure experiments that were performed at 100- to 1000-fold lower alcohol concentrations. These authors suggest that this discrepancy is due to the dependence of the apparent distribution coefficient on the alcohol concentration. We have measured the degree of solubilization, p , and the apparent distribution coefficient, Kc, of 1-pentanol in DTAB (50 mg/g of D20) micellar solution at four different 1-pentanol concentrations (Table 11). The K, values are calculated from the following equation:]' (30) Goto, A.; Endo, F. J . Colloid Interface Sei.1978, 66,26. (31) Zana, R.; Yiv, S.; Strazielle, C.; Lianos, P. J . Colloid Interface Sei. 1981, 80,208.
Kc
=
(%cellar
phase/ @aqueous phase
- PVaqueous phase/(l
- p)Vmicdlar phase
(5)
In the calculation of Kc we use 0.295 dm3/mol for the partial molar volume of the micellar DTAB.26 Our results show that in the concentration range studied p and Kc are constant within experimental error. The average value of the apparent distribution coefficient, Kc, is found to be 33 f 5, in good agreement with the value of 33 obtained from the application of a mass action model to measurements of heat of mixing at 25 "C by De Lisi et a1.26 In conclusion, the N M R paramagnetic relaxation technique provides a convenient, reliable way to study solubilization equilibria in micellar systems. This technique has considerable advantages over many other methods used to measure distribution coefficients. The values for the distribution coefficient obtained with this method agree with the values resulting from the FT-PGSE self-diffusion method. From the study of the dependence of the measured apparent distribution coefficient on solubilizate concentration we found that,for 1-pentanol in the concentration range of 0.037-0.148 m (mol/kg of D 2 0 ) the apparent distribution coefficients in DTAB solutions are constant within experimental error. In principle, this method can also be used in other microheterogeneous systems, including microemulsions, reverse micelles, liposomes, and polymer-surfactant solutions. The relaxation of other NMR-active nuclei can also be used to measure distribution coefficients of a number of different solubilizates.
Acknowledgment. This research was supported by the Natural Sciences and Engineering Research Council of Canada. N M R measurements were carried out in the Atlantic Region Magnetic Resonance Centre (ARMRC) at Dalhousie University.
Dlffusion and Nonlinear Reversible Trapping in a One-Dimensional Semiinfinite Model Membrane A. Prock* and W. P. Giering Chemistry Department, Metcalf Cenf e r For Science and Engineering, Boston University, Boston, Massachusetts 02215 (Received: December 13, 1988)
One-dimensional diffusion in the presence of nonlinear reversible trapping is described for the semiinfinite case-a special but experimentally important case. Although the results differ in quantitative aspects, they corroborate qualitatively the existence of sharp, slow-moving steps and pronounced inflections in the concentration profiles as recently reported.
Introduction In recent years a controversy has grown up about whether an electroactive dopant within a conducting polymer can be in thermodynamic equilibrium with dopant in an external phase. Points of contention are the observed disparity in desorption time vs absorption time for a given amount of dopant, and the existence of sharp steps and points of inflection in the concentration profiles of dopant within the conducting polymer. These observations have been interpreted by some researchers as casting doubt on the existence of a state of equilibrium. Reiss and co-workers1s2in addressing this problem have devised an equilibrium model where the interaction of the trap, T, with the dopant, A, is described chemically in terms of an equilibrium constant according to ( I ) Murphy, W. D.; Rabeony, H. M.; Reiss, H. J . Phys. Chem.1988, 92, 7007. (2) Kim, D.-u.; Reiss, H.; Rabeony, H. M. J . Phys. Chem.1988, 92, 2673.
0022-3654/89/2093-2192$01.50/0
A + T = AT, a = [A'][T']/[AT'] (1) where the prime refers to concentration normalized to total trap [AT]). A polymer sheet containing a concentration ([TI uniform concentration of homogeneous traps, immersed in the external phase, is treated as a one-dimensional diffusion problem where D turns out to be concentration dependent, vide infra. Numerical solution of the resulting partial differential equation' shows indeed that sharp steps and inflection points in the concentration profile result along with a disparity in absorption and desorption times and that the smaller a, the greater the disparity. This model requires one to solve a "stiff" differential equation,' i.e., where the diffusion coefficient is a sensitive function of material concentration. As the problem is posed, the equation, written in the usual distance and time variables, possesses singular points at the membrane boundaries, and a highly sophisticated method of integration must be employed. However, the concentration profiles presented' are virtually those for the semiinfinite problem. With this special but important case in mind we offer
+
0 1989 American Chemical Society
The Journal of Physical Chemistry, Vol. 93, No. 6, 1989 2193
Letters
' I1
l
i
0
i
i
I
0
0.1
0
0
0.2
2
2
Figure 1. Plot of normalized dopant concentration, c, vs z for the absorption case, with external uniform dopant concentration equal to trap concentration (c,, = l), and three different equilibrium constants. (a) a = (b) a = lo4; (c) a =
another, more tractable approach.
Results and Discussion The diffusion equation] is transformed with new variables z = 0 . 5 ~ / ( D ~ t )and I / ~ 0 = t1l2. Since the diffusion coefficient depends only on total dopant concentration, c, we seek a solution
that depends only on z and arrive at a total differential equation d/dz(D1 dc/dz) = -22 dc/dz where c = [A'] and a as1
+ [AT']
(2)
and D, (dimensionless) is related to c
Dl = 0.5[((~+ c - l ) / ( ( a
- c + I)'
+ 4 a ~ ) '+/ ~11
(3)
and z is related to the variables used in ref 1 ( E and T ) by z = f/2T'/2. The singular point is now at infinity. Letting y1 = c and y2 = DI dcldz, we obtain dYl/dZ = Y2/DI
(4)
dy,/dz = -2zy,/Di
(5)
with y,(O) equal to the normalized solute concentration, co, just inside the membrane surface and determined by the reversible equilibrium between the membrane and the external phase, and y2(0) is chosen to make c vanish at large z. The Adams-Moulton method of integration3 gives well-behaved solutions which converge ~ m o o t h l yeven , ~ for a = 1 X Owing to the scaling of this problem, for a given value of a , the resulting graph of concentration vs z suffices for any t , with the x coordinate scaled accordingly. Results are shown in Figure 1 for co equal to unity and for three values of the equilibrium constant a: 1 X 1X and 1 X These results are qualitatively similar to those in ref 1; for a = 0.1 (not shown), there is full agreement. For the common value of a = 1 X lo4 we find more rounding of the concentration profile and more extension into the membrane. The desorption problem can be treated in a similar manner. (Graphs are not shown here but have been computed.) We begin with a selected uniform dopant concentration, co, within the membrane and zero dopant concentration outside; y2(0) is chosen to make c approach co for large z, Le., as we go deeper into the membrane. Again, for a given value of a, a single graph of concentration vs z suffices for any selected time. In the case where c is greater than 1, the concentration profiles are most effectively studied by considering the extreme case of a = 0. In somewhat different form, this problem was discussed by her man^.',^ For clarity we discuss it here in t e r m s of t h e ( 3 ) Shampine, L. F.; Gordon, M. K. Computer Solutions of Ordinary Duferential Equations; W. H. Freeman: San Francisco, 1975. (4) Computations were performed in both single and double precision with no significant differences. In double precision the half-height position of the concentration profile (co = 1.0, a = lo4) already reaches fourth significant figure stability for a step length of lo4 (800 steps).
Figure 2. Plot of c vs z for absorption with co = 1.2. Solid curve: computation for a = IO-). Dotted curve: calculation for a = 0.
present formulation. For c > 1 then D1 = 1 (no trapping), and for c < 1 then Dl = 0 (complete trapping). Thus, the problem splits into two parts. For all z where c > 1, the solution is related to erf (z). For all other z, c = 0. The solutions join at point z, where c = 1. To find z, we set the particle flux, -D &/ax (c = 1, x = x,) equal to dx,/dt, reflecting total filling of the traps, and further require that when scaled to z the picture remain stationary, or z, stays constant. Write dc/dx = (dc/dz)(&/ex) = 1/(2(Dot)1/2(dc/dz). Then use dz, = 0. The flux condition at z, leads to -dc/dz = 22,. The A resulting equations for the point z, are (where c = O . ~ ( T ) I / ~erf ( 4 + co) 1 = 0 . 5 ( ~ ) ] / ' erf A (z,) + co (6) -dc/dz = -A exp(-z:)
= 22,
(7)
On solving for A and z, we arrive at A = 22, exp(zC2)
(8)
where z, satisfies the equation
(9) As a check of the computations, these analytical results were compared with those of direct computation for co = 1.2 with a = and cy = For a = the concentration profile is almost indistinguishable graphically from that for a = 0 (dotted graph in Figure 2). Even for a larger value of a, namely, as shown in the solid graph of Figure 2, the computed absorption graph still resembles the derived case for a = 0. We may compare the computed amount of solute absorbed with that calculated for a = 0 (the integral6 of c(z) from 0 to z,, equal to 22, exp(zC2)). a= and a = 0, the amount For the three values a = absorbed is a little larger for the first cy as expected, reaching a limit for a = 0, Le., 0.3403, 0.3382, and 0.3366, respectively. Again, note that owing to the scaling these numbers suffice for any given time interval.
Summary The work reported in this paper gives qualitative support to the results of Reiss et al.,] with some quantitative differences. Importantly, both sets of results show that a model of equilibrium trapping indeed leads to slow-moving dopant concentration profiles showing sharp inflections. The semiinfinite case leads to a total differential equation which is not only easy to integrate numerically but also provides a very convenient scaling. Acknowledgment. We gratefully acknowledge the Graduate School of Boston University for support of this work. We thank Professor Howard Reiss for his c o m m e n t s a n d criticism of t h i s paper. ( 5 ) Hermans, J. J. J . Colloid Sci. 1947, 2, 387. H. Reiss comments that indeed the step in concentration cannot be sharp unless CY equals zero. (Private communication.) ( 6 ) Abramowitz, M.; Stegun, 1. A. Handbook of Mathematical Functions; Dover: New York, 1965.
