3465
J. Phys. Chem. 1988, 92, 3465-3470
Ultracentrifugal Sedimentation of Micelles of Dodecyldimethylammonium Chloride in Solution Subject to the Sphere-Rod Equilibrium Shoichi Ikeda,* Sumio Ozeki, Department of Chemistry, Faculty of Science, Nagoya University, Chikusa, Nagoya 464, Japan
and Kinji Kakiuchi Institute f o r Protein Research, Osaka University, Suita, Osaka 565, Japan (Received: February 25, 1987; In Final Form: July 22, 1987)
Sedimentation coefficients of micelles of dodecyldimethylammonium chloride in aqueous solutions have been measured at different NaCl concentrations at 25 OC. The sedimentation of micelles takes place as flotation of a schlieren trough from the bottom toward the meniscus. At low NaCl concentrations the trough rises up with constant velocity, independent of the micelle concentration, and can be assigned to the spherical micelles. At high NaCl concentrations the trough is sharp, and its sedimentation coefficient is generally large in magnitude and varies anomalously with the micelle concentration. The fast, sharp trough is caused by the sedimentation of rodlike micelles dissociating behind the leading edge of the boundary and the strong dependence of the sedimentation coefficient of a rodlike micelle on its concentration. The Gilbert-Fujita theory of sedimentation for a chemically reacting system has been developed in such a way as to express the sedimentation coefficient of the observed trough in terms of the sedimentation coefficients of spherical and rodlike micelles. Assuming values of the sedimentation coefficient of a spherical micelle, the theory has been applied to derive the sedimentation coefficient of a rodlike micelle, giving reasonable values.
Introduction The sphere-rod transition of surfactant micelles in aqueous solutions has been investigated in detail mainly by means of static light scattering.',* According to our earlier work,3 dodecyldimethylammonium chloride (DDAC) forms spherical micelles in aqueous NaCl solutions when the NaCl concentration is lower than 0.80 M, while it is in either spherical or rodlike micelles when the NaCl concentration exceeds 0.80 M. In water or in the presence of NaCl less than 0.80 M, the Debye plot of light scattering gives a straight line with a positive slope, indicating repulsive interaction between spherical micelles, and at the threshold NaCl concentration where the repulsion and the attraction between spherical micelles are balanced, the Debye plot is horizontal. The Debye plot above 0.80 M NaCl, however, decreases with increasing micelle concentration and reaches a constant value characteristic of the given NaCl concentration, showing a concentration-dependent sphere-rod equilibrium of micelles. Similar salt-induced sphere-rod transitions of ionic micelles have been observed for dodecyldimethylammonium bromide4 and for higher alkyltrimethylammonium halide^^-^ in aqueous salt solutions. Hydrodynamic properties of micellar solutions also reflect the micelle size and shape sensitively. The intrinsic viscosity of aqueous NaCl solutions of DDAC was measured at different NaCl concentrations, and its values at high NaCl concentrations were correlated with the molecular weight of rodlike micelle^.^ The translational diffusion coefficients of rodlike micelles in 4.00 M NaCl as well as in more dilute NaCl were deduced from dynamic light scattering.10 These properties clearly showed the formation of rodlike micelles and also their considerable flexibility.
In the present work we report our experimental work on ultracentrifugal sedimentation of DDAC micelles in aqueous NaCl solutions and discuss the results on the basis of a concentrationdependent sphere-rod equilibrium of micelles, in order to determine the sedimentation coefficients of micelles at different NaCl concentrations.
Experimental Section The sample of DDAC was the same as previously prepared and u~ed.~,~ Ultracentrifugal sedimentation of surfactant solutions was performed on a Spinco E ultracentrifuge, using a standard aluminum cell having a centerpiece of 12-mm thickness or, together with it, putting a cell of the same type but having a wedge-shaped window at the position of counterbalance. The cell was loaded on an An-D rotor, and the temperature was kept at 25 "C. The rotor speed was chosen as 52600 rpm for 0.20-3.00 M NaCl solutions and as 31 410 rpm for 4.00 M NaCl solutions. The schlieren optical system with a phase-cut plate was inserted in the light path. Photographs of the schlieren pattern were taken a t appropriate time intervals. The sedimentation boundary appeared as a trough, instead of a peak, and it rose up with time, instead of sedimenting down. Its radial position on the photographs with a magnification factor of 20.8 was read on a Fuji Minicopy Reader Q4A. The sedimentation coefficient, s (S), of the trough was determined from its radial distance, r1 (cm), from the center of rotation at a time, t (s), after the rotor attained a constant speed. If the angular velocity is w (s-l), equal to (2a/60)(rpm), then the sedimentation proceeds according to In rl = In rb
(1) Ikeda, S.In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum: New York, 1984; Vol. 11, p 825. (2) Imae, T.; Ikeda, S. In Surfactants in Solution; Mittal, K. L., Chattoraj, D. K., Eds.; Plenum: New York, in press. (3) Ikeda, S.; Ozeki, S.;Tsunoda, M. J. Colloid Interface Sci. 1980, 73, 27. (4) Ozeki, S.;Ikeda, S. Colloid Polym. Sci. 1984, 262, 409. ( 5 ) Ozeki, S.; Ikeda, S. J . Colloid Interface Sci. 1982, 87, 424. (6) Imae, T.; Kamiya, R.; Ikeda, S. J. Colloid Interface Sci. 1985, 108, 215. (7) Imae, T.; Ikeda, S. J. Phys. Chem. 1986, 90, 5216. (8) Imae, T.; Ikeda, S.Colloid Polym. Sci. 1987, 265, 1090. (9) Ozeki, S.; Ikeda, S. J. Colloid Interface Sci. 1980, 77, 219. (10) Flamberg, A.; Pecora, R. J. Phys. Chem. 1984, 88, 3026.
0022-3654/88/2092-3465$01,50/0
+ su2t
(1)
where rb is the distance of the bottom of the solution from the center of rotation. The sedimentation boundary rising upward as a trough means flotation of micelles under the ultracentrifugal field, thus giving negative values to the sedimentation coefficient. The sedimentation boundary had either of two patterns, depending on the NaCl concentration. They are illustrated in Figure 1. The sedimentation velocity of solute particles, or of DDAC micelles here, depends on the viscosity coefficient and the density of solvent, or of aqueous NaCl solutions here, through the friction and buoyancy of the particle. Thus, it is usual to reduce the value 0 1988 American Chemical Society
,-lo
3466 The Journal of Physical Chemistry. Vol. 92. No. 12, 1988
Ikeda et al.
0
-n- 2 0
-3 0 a
Figure 1. Sedimentation patterns of micellar solutions of DDAC at 25 'C. (a) 0.20 M NaCI: top. 0.954 g dL-'; bottom. 1.470 g dL-'; time. 87 min a1 52640 rpm: diagonal slit. 60°. (b) 2.00 M NaCI: 0.375 g dL-': time, 39 min at 52640 rpm: diagonal slit. 70'.
1
0
b
3
2 C(lO-~~Cln~~)
F i m2. Sedimentation avlfieient of the trounh as a luna m of DDAC Eokenlralion at different NaCl cnncentrations-C, (M):0.0.20 I0.50 A. 1.00 0. 2.00. 0. 3.00 A. 4.00.
of sedimentation coefficient in a given solvent to that in water, both at 25 OC here. If the solvent has a viscosity coefficient. no. and a density, pm without any suffix for the given solvent and with a suffix. w. for water, then the sedimentation coefficient in water, s. can be obtained from that in the given solvent, s by s,=s--
no 1 - ~ P o , w n0.w
I
- DPO
where V is the partial specific volume of the solute. The partial specific volume of DDAC micelles was found to be 1.092 cm' g-'? and it is assumed to have a value independent of the NaCl concentration or of whether the micelle is spherical or rodlike.
Rrsultr At low NaCl concentrations where only spherical micelles are present besides monomers. the sedimentation velocity is slow so that it takes about 30 min to have a trough split from the bottom at 52600 rpm. Thiscan beseen in Figure la. The dimentation boundary lwks normal. A micellar solution usually accompanies a small hollow (hump) behind the main sedimentation trough (peak) owing to the monomer-micelle equilibrium
mD 2 D, (3) where D is the monomer and D, is the spherical micelle having aggregation number m. The micelle dissociates into monomers behind the trough (peak). where the surfactant concentration is lowered." On the other hand, at high NaCl concentrations where rodlike micelles are formed h i d e s spherical micella. the sedimentation boundary is very sharp, and it trails a tail toward the bottom. as Seen in Figure Ib. The sharp boundary must be caused by a strong dependence of the sedimentation mfficient of the rodlike micelle on the concentration. because the sedimentation velocity is higher behind the rising boundary. but it is lower ahead of the boundary.'z The trailing of the boundary behind the trough (peak) can be ascribed to the occurrence of the sphererod equilibrium of micelles nD, 0,. (4) where , D is the rodlike micelle having aggregation number mn. The rodlike micelle dissociates into spherical micella behind the trough (peak). where the micelle concentration is lowered." At high NaCl concentrations the base line deviated from the horizontal line and was curved upward at both meniscus and bottom. because of the accumulation of micelle and NaCI. respectively. there. In 4.00 M NaCl solutions turbidity was sometimes. but not always, developed as swn as the solution was subject to ultracentrifugation. when the DDAC concentration was higher than 0.30 g dL-'. ( I I ) Ikda. S.;hkiuehi. K.J. Collotd hrrrfme Sri. 1967. 23. 134. (12) Fujita. H. J. Chrm. Phys. 1956. 24. 1084. (13) Gilbert. G . A. Dirrurr. Forodoy Sm. t9S3. I3. 159.
0.50
25200
-1.74
-I 42
IS ...7
1... 5.6
1.00 2.00 3.00 4.00
27500 30800 32900 34500
-2.10 (-2.53) (-3.05) (-3.20)
-1.52 (-1.64) (-1.73) (-1.81)
15.4 14.9 14.7 14.6
15.9 16.5 16.7 16.7
Figure 2 shows the sedimentation coefficient of the trough of the boundary as a function of DDAC concentration. The sedimentation coefficients in 0.20 and 0.50 M NaCl are scarecly dependent on the DDAC concentration, and the sedimentation behavior in 0.20 and 0.50 M NaCl can be attributed to the spherical micelles. Values of the sedimentation coefficient, sm0, are given in Table 1. In 1.00 M NaCl the sedimentation coefficient is slightly but anomalously dependent on the DDAC concentration. The limiting values of sedimentation coefficient can be obtained by its extrapolation to the critical micelle concentration, as given in Table 1. The sedimentation coefficient has a minimum value, -2.80 S, a t 1.20 g dL-I. At higher NaCl concentrations the magnitude of the sedimentation coefficient is higher. and its concentration dependence is stronger and more anomalous. Its limiting value cannot be determined by the extrapolation to the critical micelle concentration. It has a clear minimum at an intermediate concentration: -7.0 S at 0.47 g dL-' in 2.00 M NaCI, -16.2 S a t 0.35 g dL-' in 3.00 M NaCI. and -39.3 S at 0.34 g dL-' in 4.00 M NaCI. At NaCl concentrations higher than 0.80 M,rodlike micella are formed in equilibrium with spherical micelles, and the equilibrium shifts toward either way, depending on the micelle concentration.) Thus, the anomalous dependence of the sedimentation coefficient on the concentration must be related to the concentration-dependent equilibrium between these two kinds of micelles and the specific properties or the concentration dependence of sedimentation of rodlike micelles. Discussion ( A ) Sedimenrarion Coefficients of Spherical Micelles. Our
light scattering measurements on aqueous solutions of spherical micelles of DDAC a t low NaCl concentrations3 have shown that a linear double logarithmic relation log M, = 0.163 log (C, C,) 4.44 (5)
+
+
holds between the micelle molecular weight, M,. and the ionic strength or the CI- concentration Co + C,, where Cois the critical micelle concentration and C, is the NaCl concentration. both in molar units. It is well-known that for a homologous series of polymers the sedimentation coefficient and the molecular weight are linearly
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3461
Sedimentation of DDAC Micelles in Solution 0 6
7
- o l
g0
I
4 2
I
4 6
4 4
Figure 3. Double logarithmic plots of -sm0 and -s,O against M , for DDAC micelles: 0, -sm0; 0, -s,O. 0 , the point plotted by extrapolation of the straight line to the molecular weight of spherical micelle.
related in a double logarithmic scale. In accordance with this, the logarithm of the negative sedimentation coefficient is linearly dependent on the logarithm of the molecular weight for the spherical micelles at low NaCl concentrations, as shown in Figure 3. The relationship is expressed by
If we assume that both eq 5 and 6 are valid for higher molecular weights, where spherical micelles are in equilibrium with rodlike micelles, then the sedimentation coefficient of spherical micelles can be evaluated at higher NaCl concentrations from their extrapolation. These results are given in Table I. By means of eq 2, we can convert the values of sedimentation coefficients in aqueous NaCl solutions into those in water, as given in Table I. These values are also plotted in Figure 3, which again give a linear double logarithmic relation with the molecular weight. This relationship is expressed by
I-S,,~~~= 0.78 log M , - 3.27
(7)
If the spherical micelle were a rigid sphere, the coefficient of eq 7 should be 0.67, but the present result would suggest deviation of micelle shape from it. ( B ) Weight Fractions of Spherical and Rodlike Micelles. At NaCl concentrations higher than 0.80 M, both spherical and rodlike micelles are formed above the critical micelle concentration. If the molecular weights of monomer, spherical micelle, and rodlike micelle are M I , M,, and M,,, respectively, and their weight fractions are w,,w,, and w,, respectively, then the weight-average is equal to molecular weight of total surfactant, Mw’, M,’ = Mlwl
+ Mmwm+ M,,w,,
(8)
From the excess of the reduced intensity of light scattered at zero angle, ARo, a t a given surfactant concentration, c (g ~ m - ~we ), can estimate the weight-average molecular weight of surfactant as M,‘ = ARo/Kc
(9)
where K is the optical constant of light ~ c a t t e r i n g . ~ Having the weight-average molecular weight, Mw’,as a function of concentration, c, we can now calculate the weight fraction of monomer, wl, by means of Adams’ equation:I4
In Figure 4 the values of w1are plotted against concentration for 1 .OO and 2.00 M NaCl solutions. It is found that the monomer concentration CI
=
CW]
(14) Adams, E. T., Jr. Biochemistry 1965, 4, 1646.
1
o4HK
log l l m
log
op-
(11)
1 0
20
10
c(I~I-~scm-~)
.,
Figure 4. Plots of weight fractions of monomer, spherical, and rodlike micelles against total concentration: (A) 1.00 M NaCI, (B) 2.00 M NaCI. w , ; 0, w,; 0 , wmn.
is equal to the critical micelle concentration, co (g ~ m - ~when ), the concentration is higher than co. On the other hand, the Debye plot of light scattering gives the weight-average molecular weight of micelles, M , by means of
provided that the second virial coefficient is negligibly small. Here R , and Rooare the reduced intensities of light scattered at zero angle from the surfactant solutions of concentrations c and co, respectively. We can calculate the molecular weight of spherical micelles by means of eq 5, and also we know the molecular weight of rcdlike micelles, which follows the relationship3 log M,, = 2.64 log (Co
+ C,) + 4.70
(13)
Then we can obtain the weight concentrations of these two kinds of micelles by
The weight fractions of spherical and rodlike micelles are given by w, = c,/c (16) wmn
=
Cmn/C
(17)
In Figure 4 the values of w, and w,, are also plotted as functions of concentration for 1.00 and 2.00 M NaCl solutions. ( C ) Sedimentation Coefficients of Rodlike Micelles. It has been established that the sedimentation boundary of a chemically interacting “monomer”-”polymer” system generally consists of two components (peaks or troughs), one being slower and sharp and the other being faster and broad, if the “polymer” is larger than a “dimer”.15-1s Here the ”monomer” is not the surfactant monomer, but it is a macromolecule subject to observable sedimentation under an ultracentrifugal field. The slower component sediments with a sedimentation coefficient of the “monomer”, and (15) Gilbert, G. A. Discuss. Faraday SOC.1955, 20, 68. (16) Gilbert, G. A. Proc. R . SOC.London, A 1959, A250, 377. (1 7) Fujita, H . Mathematical Theory of Sedimentation Analysis; Academic: New York, 1962; pp 204-215. (18) Cam, J. R. Interacting Macromolecules; Academic: New York, 1970; pp 93-117.
3468 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
the faster component migrates with a sedimentation coefficient approximately equal to the weight-average sedimentation coefficient of “monomer” and “polymer”, based on their initial weight fractions. This result will be demonstrated in the Appendix, on the basis of Fujita’s the0ry.I’ It has been accepted that a spherical micelle has the structure with the hydrocarbon moiety inside and the polar or ionic head groups outside, so that its size may be primarily determined by the geometrical r e q u ~ r e m e n t . l ~The . ~ ~size distribution of spherical micelles is so narrow that the spherical micelle may be regarded as monodisperse, and the micelle formation can be expressed by eq 3. On the other hand, there has been some controversy about the size distribution of rodlike micelles. We have shown by two ways of analysis of light scattering2’B22that the ratio of weight-average molecular weight to number-average molecular weight for micelles approaches unity as the micelle concentration exceeds far beyond the critical micelle concentration, thus indicating a narrow distribution of size of rodlike micelles. From a theoretical consideration, we have also reached the same conclusion that the size or length of rodlike micelles follows a Poisson distribution, which is narrow enough, because a rodlike micelle consists of rapidly exchangeable or nonlocalized constituent molecules or ions.23 In this approximation, the sphere-rod equilibrium of micelles can be expressed by eq 4. We may then apply the Gilbert-Fujita theory for the micellar solutions of DDAC containing spherical and rodlike micelles, by identifying them with the “monomer” and the “polymer”, respectively. The total micelle concentration is given by c - cl, where the value of the monomer concentration, cl, is equal to the critical micelle concentration, co. Thus, the weight concentrations of spherical and rodlike micelles are represented by c, and c,, and the total micelle concentration is c - co = c , -+ c,,
(18)
Since values of the sedimentation coefficients of spherical micelles observed at low NaCl concentrations are low, it seems unlikely that the slower, sharp trough assignable to the spherical micelle would appear and split from the bottom within the time of ultracentrifugation examined at high NaCl concentrations. Thus, the observed sharp trough of the sedimentation boundary must be assigned to the faster, broad trough, and its observed sharpness must be a result of strong concentration dependence of the sedimentation coefficient of rodlike micelles, as was referred to previous1y.l2 If the initial concentrations of spherical and rodlike micelles are c, and c,, and their sedimentation coefficients are represented by s, and s, the faster trough of the sedimentation boundary should have a sedimentation coefficient
Ikeda et al.
2
1
cmn
Figure 5. Plot of c,,/[s(c
- c), - smoc,]
3
(i0-2grm-3)
C, (M): B, 1.00; 0, 2.00;
vs .,c,
0 , 3.00; 0,4.00.
TABLE 11: Sedimentation Coefficient of Rodlike Micelles and Their Diffusion Coefficient Derived from It Dmn,wo,
C,, M 1.00 2.00 3.00 4.00
s,,,~~, S lo-’ cm2 s-I S k M,, ,:,s 56000 -3.94 0.37 -2.85 14.2 333000 -11.9 1.31 -7.74 6.49 1200000 -32.3 4.07 -18.8 4.36 2940000 -76.9 12.8 -43.4 4.12
R,,A 17.2 31.7 56.2 59.4
2, and also because the trough undergoes a strong sharpening effect, as seen in Figure lb. These two phenomena were also observed for aqueous (DzO) solutions of hexaoxyethylene dodecyl ether at different temperatures, which forms rodlike micelles in water at higher temperature^.^^,^^ Since the rodlike micelle sediments faster than the spherical micelle, the sedimentation front of the boundary should always accompany spherical micelles, besides the rodlike micelles. Then the sedimentation coefficient of the main trough might be subject to the Johnston-Ogston effe~t:~,~’if the sedimentation coefficient of the spherical micelle were concentration-dependent. However, our observation on spherical micelles indicates no concentration dependence of their sedimentation coefficient, i.e., eq 20. Then we may approximate the concentration dependence of the sedimentation coefficient of a rodlike micelle by Smn
0
=1 + kc,,
,,s
following the general f ~ r m u l a t i o n , ” ~where ~ ~ *:~,s* is the limiting sedimentation coefficient of a rodlike micelle and k is a constant. In this way we may write eq 19 as as shown in the Appendix. We assume that the sedimentation coefficient of spherical micelles does not depend on the concentration, as was observed at low NaCl concentrations
where sm0 is the limiting sedimentation coefficient of spherical micelle. On the other hand, the sedimentation coefficients of rodlike micelles should depend on the concentration, because the sedimentation coefficient of the observed trough changes with the micelle concentration in an anomalous way, as shown in Figure (19) (20) (21) (22) (23)
Tartar, H. V. J . Phys. Chem. 1955, 59, 1195. Tanford, C. J . Phys. Chem. 1972, 76, 3020. Ikeda, S . ; Hayashi, S.; Imae, T. J . Phys. Chem. 1981, 85, 106. Imae, T.; Ikeda, S . Colloid Polym. Sci. 1984, 262, 497. Ikeda, S . J . Phys. Chem. 1984, 88, 2144.
s(c - co)
- so,c,
Cmn
=--- ,s 0 1 kc,,
+
Since we know the weight fractions or concentrations of spherical and rodlike micelles as functions of total concentration, as given in Figure 4 or by means of eq 14 and 15, we can plot eq 22 in the form of a linear relation with c,:
(24) Ottewill, R. H.; Storer, C.; Walker, T. Tram. Faraday S o t . 1962,63, 2796. (25) Brown, W.; Johnson, R.; Stilbs, P.; Lindman, B. J . Phys. Chem. 1983, 87, 4548. (26) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961; pp 364-390. (27) Johnston, J. P.; Ogston, A. G.Trans. Faraday SOC.1946, 4 2 , 789. (28) Schachman, H. K. Ultracentrifugation in Biochemistry; Academic: New York, 1959; pp 90-103.
The Journal of Physical Chemistry, Vol. 92, No. 12, 1988 3469
Sedimentation of DDAC Micelles in Solution
I 6
5
4
log ”mn Figure 6. Double logarithmic plots of:,sand -S,,,”,~O DDAC micelles: 0,;:s-, 0,
7
against M,, for
Figure 5 shows the plots of the left-hand side of eq 23 against c,, at different NaCl concentrations. Straight lines were drawn, and from their intercepts and slopes the values of sm: and k were derived, as summarized in Table 11, together with the molecular weight of the rodlike micelle. Figure 6 shows the double logarithmic plots of the limiting sedimentation coefficient of rodlike micelles against the micelle molecular weight. They follow a linear relation:
The sedimentation coefficient of rodlike micelles in aqueous NaCl , : ,s can be converted into that in water, s , , , ~ ~ ,by eq solutions, 2. The double logarithmic plots of the limiting sedimentation coefficient of rodlike micelles in water against the micelle molecular weight are also given in Figure 6. The linear relation is expressed by
log ~ - s , , , ~=~0.68 ~ log M,, - 2.82 The coefficient, 0.68, of eq 25 for rodlike micelles has a value close to that, 0.67, for spherical micelles. If the rodlike micelle were an unperturbed random coil, the coefficient of eq 25 should be equal to O S , and if it were a perturbed random coil, the coefficient should be lowered further. We have to anticipate that our values derived for the limiting sedimentation coefficient must suffer from some errors, because of the assumptions and the theory applied. Among them, the complete monodispersity of both micelles and the neglect of the micellar equilibrium with monomer would have some effects on the final values. Furthermore, the theory and its result, eq 19, are based on the solution to the generalized Lamm equation with constant values of both sedimentation coefficients, but the concentration dependence of the sedimentation coefficient of the rodlike micelle is quite manifest in the observed sharpening effect. It should be noted that the overlap concentrations of rodlike micelles of DDAC are about 2.51, 0.35, and 0.17 g dL-’ in 2.00, 3.00, and 4.00 M NaC1, respectively, and they indicate that the rodlike micelles overlap together already at concentrations lower than those of the minimum sedimentation coefficient. (D) Application of the Svedberg Equation. Having found values of the sedimentation coefficient of micelles, we can now estimate the diffusion coefficient for both micelles by means of the Svedberg equation
M i=
RTS/,,O Q,W0(1 - UP0,w)
where the subscript i refers to m or mn and the superscript 0 represents the limiting quantity, and R is the gas constant. In Tables I and I1 values of the limiting diffusion coefficient of spherical and rodlike micelles are given. It is noted that the Svedberg equation postulates the monodispersity of solute particles. The hydrodynamically equivalent radius, R h , of the micelles can be derived from the Stokes-Einstein equation
where k is the Boltzmann constant. Values of Rh are also given in Tables I and 11. It seems likely that values of Rh for spherical micelles, less than 20 A, are too small. This means that the diffusion coefficient obtained has a too large value, probably because even the spherical micelle is not strictly monodisperse. Furthermore, owing to possible errors arising from various assumptions and treatments in deriving the sedimentation coefficient of rodlike micelles, together with the deviation from monodispersity, we would also have some question in applying the Svedberg equation.
Appendix Sedimentation Coefficient of the Sedimentation Boundary of Micellar Solutions in the Sphere-Rod Equilibrium. In treating this problem, we will apply the results of the solution of the continuity equation for sedimentation in chemically reacting systems consisting of “monomer” and “polymer”.I7 For this purpose, we modify our notations for concentrations here from those in the text as follows. We attach the superscripts 0 and p in order to represent the initial and the plateau concentrations, respectively, and express the concentrations at a given time, t , and at a given radial distance, r, by the symbols without the subscripts. For a micellar solution having both spherical and rodlike micelles, the total concentration of surfactant is given by c = co
+ c, +, ,c
(A- 1)
where co is the critical micelle concentration and is not influenced by the sedimentation. The monomeric surfactant does not sediment by the ultracentrifugal field, and the concentration-dependent sphere-rod equilibrium of micelle, eq 4, is assumed to be attained rapidly. It has been generally ~hown’~.’’ that, except for dimerization of the spherical micelles to form the rodlike micelle, the sedimentation boundary consists of two peaks (troughs) and that a slower peak (trough) is sharp and a faster peak (trough) is broad. The sedimentation coefficients of these two peaks (troughs), ss and sf, can be calculated from the solution to the continuity equation of the type of generalized Lamm equation, in which the sedimentation coefficient is given by
where s, and,,s are the sedimentation coefficients of spherical and rodlike micelles, respectively, and are assumed to be constant here. In order to obtain the sedimentation coefficients of the boundary, the continuity equation must be solved to derive the concentrations at a given time and a given distance. Under the assumption of negligible diffusion, it was solved analytically for a solution column of radial distances between r, and r,. The results give three regions of the solution column: (I) the solvent region, (11) the boundary between solvent and solution, and (111) the plateau region.” The sedimentation boundary is bounded by a slow, sharp edge and a fast, broad edge, as long as the total concentration exceeds a definite value given below. If the reduced concentration of spherical micelles, the reduced time of ultracentrifugation, and the reduced radial distance are introduced by
e,
=
=
64-31
2S,WZt
(‘4-4)
= 2 In ( r / r a )
(-4-5)
7
E
(C,/C,o)n-l
respectively, then the reduced concentrations at the three regions
3470 The Journal of Physical Chemistry, Vol. 92, No. 12, 1988
Ikeda et al.
can be derived. At an early stage of ultracentrifugation, Le., at T
