Relaxation in Micellar Solutions
the series is obvious from the fact that the ratio of the partition functions of the activated complex and the molecule Z+/Z is constant for an homologous series if the assumption of identical activated complex is maintained.
Acknowledgment. This research was supported by the U.S.-Israel Science Foundation. References and Notes (1) (a) S.Chervinsky and 1. Oref, J. Phys. Chem., 79, 1050 (1975); (b) ibid., 81, 1967 (1977). (2) A. N. KO and B. S. Rabinovitch, J . Chem. Phys., 66, 3174 (1977). (3) J. F. Meagher, K. J. Chao, J. R. Barker, and B. S.Rabinovitch, J. Phys. Chem., 78, 2535 (1974).
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978 2037
(4) E. A. Hardwidge, 8. S. Rabinovitch, and R. C. Ireton, J. Chem. Phys., 58, 340 (1973). (5) R. Kubo, "Statistical Mechanics", North-Holland Publishing Co., Amsterdam, 1965. (6) . . (a) M. R. Hoare and Th. W. Buiiarok, J. Chem. Phvs., 52, 113 (1970): . . (b) M. R. Hoare, ibid., 52, 5685 (1970). (7) N. B. Slater, "Theory of Unimolecular Reactions", Cornell University Press, Ithaca, N.Y., 1959, p 119. (8) W. Forst, "Theory of Unimolecular Reactions", Acedemlc Press, New York, N.Y., 1973, p 15. (9) F. H. Dorer and B. S.Rabinovitch, J. Phys. Chem., 69, 1973 (1965). (10) S. W. Benson and H. E. O'Neal, Natl. Stand. Ref. Data Ser., Natl. Bur. Stand., No. 21 (1970). (11) Some frequencies in the complex might be slightly lower due to the increase in the mass of the tail but the effect will be very small for large enough consecutive members.
Chemical Relaxation Studies in Micellar Solutions of Dodecylpyridinium Halides Tohru Inoue,* Ryolchl Tashiro, Yoko Shibuya, and Ryosuke Shimozawa Department of Chemistry, Faculty of Science, Fukuoka University, Fukuoka 8 14, Japan (Received January 16, 1978; Revlsed Manuscript Received June 23, 1978) Publication costs assisted by Fukuoka University
Chemical relaxation measurements in aqueous solutions of dodecylpyridinium halides (DPX: X = I, Br, C1) were carried out by temperature-jump and pressure-jump techniques. Under some fortunate conditions two relaxation processes could be observed by temperature-jump in DPI solutions. Among the three DPX, remarkable differences in the concentration dependence of the reciprocal of slow relaxation time, 72-1, and in the effect of added salt on 72-1 were observed. In DPI solutions T ~ increased - ~ almost linearly with the surfactant concentration, and also increased with the addition of KI. In contrast to DPI, in the case of DPBr and DPC1, 72-1 showed a concentration dependence having a maximum in the pressure-jump concentration range, and decreased with the amount of added salts at low concentrationsof added KBr or KC1, whereas by the addition of salts at high concentration 7c1showed the same behavior as that of DPI. These differences in the behavior of T ~ among - ~ the three DPX were well explained by assuming a single rate-determining step in the micellization-dissolution multiple step process and by introducing a parameter which represents the degree of counterion association on the micellar surface.
I. Introduction the linear dependence of appears when a small amount of long-chain alcohol is added to the original surfactant Since the temperature-jump technique had been applied ~ a m p l e . ~ Now, ~ , ~ Jit~ is known that the relaxation time to the surfactant solution by Kresheck et al.,l chemical associated with the micelle formation-dissolution process relaxation techniques have been extensively used to study is influenced drastically by the presence of surface active the kinetics of micelle formation, and a number of papers impurities. have been published in this field.l-12 Now it is well known Kinetics of micelle formation of dodecylpyridinium salts that in a surfactant solution there exist two relaxation (DPX) were already studied by Lang et ala8and by processes associated with micelle-monomer equilibrium. Hoffmann et aL9using various relaxation techniques. Due The faster one, designated by relaxation time T ~has , been to the facts described above it seems important to reinobserved by ultrasonic absorption or the shock-tube vestigate the relaxation in DPX systems with respect to method. On the other hand, the slower one, designated its concentration dependence. Therefore, we carried out by relaxation time r2, has been observed by temperarelaxation measurements with DPX (X = I, Br, C1) by ture-jump (T-jump) or pressure-jump (P-jump) methods. T-jump and P-jump methods using highly pure samples. Important information about the mechanism of micelle As a result, there were observed remarkable differences formation is obtainable from the concentration dependence in the effect of surfactant concentration and of added of the relaxation times. For the fast process, a linear inorganic salts on 72-1 among these DP salts. In this work dependence of T ~ on - ~the surfactant concentration has been obtained with various surfactant ~ y s t e m s , ~ ~ ~ * ' - lwe l will report the results and also propose a new model for the micelle formation mechanism which explains the whereas for the slow process, inconsistent concentration present experimental results. dependence has been reported by different workers even with the same s ~ r f a c t a n t . ~ ~ ~In* '1966, J ~ J ~Kresheck et al. 11. Experimental Section reported that T ~ obtained - ~ by the T-jump method in dodecylpyridinium iodide (DPI) solution increased linearly Materials. All the surfactants used in this study were with surfactant c0ncentration.l During the following synthesized and purified under carefully controlled conseveral years, linear dependence of 72-1 on concentration ditiohs. was reported for other surfactant system^.^,^^,^ Recently, Dodecylpyridinium Bromide (DPBr). Dodecyl alcohol however, it has become evident by P-jump measurement which was purified by successive distillation was converted with sodium alkyl sulfates that r2-l is almost independent to dodecyl bromide according to the procedure described of the concentration when a very pure sample is used, and in ref 13. After purification the product was identified as 0 1978 American Chemical Society
2038
The Journal of Physical Chemistry, Vol. 82, No. 78, 1978
Inoue et al.
TABLE I: Critical Micelle Concentration (cmc) of Dodecylpyridinium Halides (DPX) mater- cmc,a meth- teomp, ial 10-3 M odf cmc,b lO-'M C DPI
6.1 5.7 5.0 5.5 3.0' 1.7d
l.le DPBr DPCl
11.6 11.7 11.5 14.5 17.6 17.8 19.6
EC EC ST
uv
UV UV
uv ST EC
uv ST uv EC EC
40 25 20 20 20 20 20 24 25 19 23 23 25 5
5.60 (LS, 25 C) 5.70 (ST, 25 ''2) 2.94 (LS, 25 C)' 1.80 (ST, 25 C)d
12.O(ST, 2 5 ° C ) 11.6 (LS, 25 "C) 14.6 (ST,25 C) 3
4
5
14.7 (13, 2 5 ° C )
This work. Reference 14. ' In 0.01 M KI solution. In 0.02 M KI solution. e In 0.05 M KI solution. f EC = electroconductivity, ST = surface tension, UV = UV absorption, LS = light scattering. a
a single peak in a gas chromatogram. Dodecyl bromide was heated with 20 mol % excess pyridine at 120 "C for 6 h to give crude DPBr which was purified as follows. The crude DPBr was recrystallized four times from acetone to give a light brown crystal, which was extracted for 72 h with ether to remove any trace of dodecyl alcohol or dodecyl bromide. The crystal was dissolved in warm methanol, mixed with charcoal, and filtered. The colorless filtrate obtained gave a white crystal on evaporation. The crystal was recrystallized three times from acetone and dried in vacuo. Dodecylpyridinium Iodide (DPI). DPI was prepared from the DPBr obtained above by crystallization twice from saturated KI solution and twice from water. The product was dried in vacuo, and finally a pale yellow crystal was obtained. Dodecylpyridinium Chloride (DPC1).Dodecyl chloride, obtained by the reaction of purified dodecyl alcohol with thionyl chloride, was heated with 20 mol % excess pyridine for 12 h a t 120 "C to give crude DPCl. The product was purified by the same procedure used for DPBr. Inorganic salts used as additives were of reagent grade and used without further purification. Critical Micelle Concentration (cmc). The cmc's of DPX were determined by UV absorption, surface tension, and electroconductivity, and the values agreed well with those in the literature.14J5 They are summarized in Table
I. Temperature-Jump (T-Jump)Technique. A singlebeam spectrophotometric T-jump apparatus was used in this study. A jump in temperature of 4.0 "C could be obtained by discharging a 0.1-pF capacitor charged to 20 kV through the sample solution. An UV absorption characteristic of the DPX micelle was used to detect micellar relaxation. For recording the relaxation signal either an oscilloscope or a digital memory was used. In some cases the signal-to-noise ratio was so poor that reliable analysis was difficult. To avoid this difficulty, the temperature jump was repeated 4-9 times under the same condition and each signal was stored and averaged in an averaging apparatus, thus S / N ratio of a signal was improved. Pressure-Jump (P-Jump)Technique. A P-jump apparatus with conductivity detection was used in this study. Increasing hydraulic pressure was applied to the sample cell. When the pressure reached about 150 atm a brass diaphram burst and an abrupt decrease of the pressure
7
6
co
(
9 1 0 1 1 1 2
3
10-~ M
)
Flgure 1. Plot of T2-' observed by P-jump measurement vs. total DPI concentration, C,, at 25 'C: (0)no added KI; ( 0 )CKI = 1 X M; ( 0 )CKI = 5 X M; ( 0 )CKI = 1 X lo-' M; (e)CKI = 2 X
M.
I - " I I ! /P
U
3 0
0
0
2
4
O /
I
I
I
I
l
l
I
I
6
8
10
12
14
16
18
20
co
(
M
)
Figure 2. Plot of T2-' observed by T-jump measurement vs. total DPI concentration, Co,at final temperature of 29 'C: (0)CKI= 1 X IO-* M; (0)CKI = 2 x lo-* M; (0)CKI = 3 x lo-' M; (0)CKI 5 x lo-' M.
from 150 to 1atm was obtained within a period of 100 ps. The relaxation curve was obtained by recording the output signal using either an oscilloscope or a digital memory equipped with a rectifier.
111. Experimental Results In the present experiments only the slower of the two relaxation processes could be observed except in the case of DPI in which two processes were observed by the Tjump technique at high added KI concentrations. Evidence for the existence of a fast process has been obtained from ultrasonic absorption5cor the shock-tube methodag Reciprocals of the slow relaxation time, r f l , in DPI solution obtained by P-jump and T-jump measurements are given in Figures 1 and 2, respectively, as a function of the total DPI concentration, Co. As seen in Figure 1,T ~ increases linearly with Co although deviation from linearity becomes appreciable a t concentrations close to the cmc. At high DPI concentrations, the values of rL1 flatten out as seen in Figure 2. Furthermore, the addition of KI to the surfactant solution increases the slope of the r2-l VS. Coplot. Here, it is worth noting that, when increasing the DPI concentration, the amplitude of the relaxation curve diminished, disappeared, and then appeared. The concentration range of the disappearance of the relaxation amplitude is shown by dashed line in Figure 2. Taking account of the well-controlled purity of the sample used in this study, it seems that a linear concentration de-
~
The Journal of Physlcal Chemistry, Vol. 82,No. 18, 1978 2039
Relaxation in Micellar Solutions
5 L
'
1,o
I
1
I
1,5
2,o
2,5
co
(
M
]
)
Flgure 3. Plot of Ti-' observed by T-jump measurement vs. total DPI concentratlon, C ,,, at a final temperature of 29 OC: (0)CKI = 3 X M; (e)CK, = 5 X IO-' M.
e--
1,5
&/
2,O
2.5
CO
3,O
lo-* M
(
3.5
)
Flgure 5. Plot of T2-' observed by P-jump measurement vs. total DPCl concentration, Co,at 5 OC: (0)no added KCI; ( 0 )CKcl = 5 X M. M; (e)CKCl = 1 X
co
(
M
)
Flgure 4. Plot of 7 ~ vs. ' total DPBr concentration, Co, at 21'C: circles, T-jump; squares, P-jump; (0, 0)no added KBr; (El) CK&.= 2 X M; (61) CKp = 5 X M; (0, m) C K B=~ 1 X IO-' M; (El) C K B=~ M; (a)CKB~= 5 X 1.5 X 10- M; ( 0 )CKB, = 2 X M; (e) CKBr = 1 X lo-' M.
pendence of T ~ cannot - ~ necessarily be attributed to the presence of an impurity as has been claimed for alkyl sulfate^.^^*^ The variation of T ~ for - ~DPI in 0.03 and 0.05 M KI solutions obtained from T-jump measurements is shown in Figure 3. Comparing the present results with those by Lang et a1.,8 T ~ is- in ~ rather good agreement, whereas the concentration dependence of 72-l is similar in general features but the magnitudes of rfl are about three to four times larger in the present study than in those by Lang et al. A plot of r2-l vs. Co for DPBr obtained by P-jump and T-jump measurements is shown in Figure 4. When the - ~a maximum in the amount of added KBr is small, T ~ has concentration range effective for the P-jump technique, and with an increasing amount of added KBr, 72-1 tends to decrease in magnitude. On the other hand, when KBr is added at high concentration, the general behavior of r2-l becomes similar to that of DPI, i.e., 72-l shows a linear dependence on Co and increases with the concentration of added salt. These results for DPBr are in agreement with some results already reported by Lang et a1.8 and Hoffmann et al.9 with respect to both the concentration dependence and the magnitude of r2-I. According to Mukerjee and Ray,16 the degree of counterion association on the micellar surface is very large for DPI micelle. For DPBr it is thought that the degree of counterion association on the micellar surface may increase with added KBr. Therefore, the results in Figure 4 suggest that a linear concentration dependence of r2-l appears when the degree of counterion association becomes
0' 0
I
I
I
5
10
CO
(
15 M
I
20
)
Flgure 6. Plot of 7 2 - I observed by T-jump measurement vs. total DPCl concentration, Co, at a flnalJemperature of 16 'C: (0)CKc,= 2 X lo-' M; (0) CKcl = 5 X M; (e)CKcl = 1 X lo-' M; ( 0 )CKCl = 2 X 10-i M.
large. More detailed discussion and analysis will be given in section IV. The variation of ~ 2 -for l DPCl with C, is given in Figures 5 and 6 measured by P-jump and T-jump techniques, respectively. It can be seen from Figure 5 that T ~ de- ~ creases monotonously with Co in the absence of KC1, and with addition KC1 a maximum appears. The behavior of r2-l for DPCl with KC1 at higher concentrations as shown in Figure 6 is similar to that for DPBr with KBr, however the magnitude of the effect of added salt on r2-I is small compared with that for DPBr.
IV. Discussion So far several models for the mechanism of micelle formation have been proposed to explain the relaxation data obtained in surfactant s o l ~ t i o n . ~ ~ For ~ ~ the ~~~~JJ~assignment of the two relaxation processes the following concept is now generally a c ~ e p t e d : ~ t * J ~the J ~fast J ~ process is attributed to the perturbation of the exchange equilibrium of monomer between micellar and bulk phases, and the slow one to that of the micellization-dissolution equilibrium. The expressions for the relaxation time r1 for the exchange process have been derived by several authors under various assumptions, and equations of the type
2040
Inoue et at.
The Journal of Physical Chemistry, Vol. 82,No. 18, 1978
rl-l
= k-
(
+ /3
N
K
Co - cmc cmc
( n - l)Al + pX, 6A,-,Xp kn+
were ~ b t a i n e d , ~where J ~ J ~k~ is the rate constant for the dissociation of one monomer from the micelle, Co is the total concentration of the surfactant, and N and /3 are constants having different values according to the kinetic models. This equation predicts a linear dependence of r1-l on Cot and this prediction is realized by experimental results. On the other hand, for the slow process any model proposed should reasonably explain the concentration dependence of showing the fairly complicated behavior for the dodecylpyridinium salts obtained in this study. In considering the mechanism for micellization it may be useful to note the following experimental findings in the present study: (i) Different counterions give a different concentration dependence of r2-l for dodecylpyridinium salts. (ii) By adding KBr at high concentration, DPBr shows the same behavior of r2-l as DPI. (iii) The effects of added salts on r2-l are very large, and they are positive in sign for DPI and negative for DPBr and DPC1. Aniansson et al.1°J9proposed a diffusion model for the slow process in which micelle formation and dissolution is treated as a steady flow of aggregates through a bottle neck of intermediate nuclei. They obtained the expression of r2-l as follows: r2-1
N
m2 R([AJ + o2cJ
(1)
where [A,] is the equilibrium concentration of the surfactant monomer, m the average aggregation number, c3 the concentration of micellar particle, (T the variance of the distribution curve of the micellar aggregation number, and R a composite term expressed by
R = Cl/kl[A,I
(2)
1
The parameter R cannot be explicitly given without further consideration and assumptions than those implied in deriving eq 1. By assuming only one type of nucleus, eq 1 can be transformed into r2-l
1
E
k-cim2 [A,] +
An-,Xp + A1 eAnXp kn-
A,Xp
+ (m
-
n)A,
fast slow
KZ
+ (q - p)X, * A,X,
(4) (5) fast (6)
where A, represents a single surfactant ion, X, a counterion in solution, A,X, a stable micelle consisting of m surfactant ions and q counterions, and AnXpan intermediate aggregate consisting of n surfactant ions and p counterions. It is assumed here that step 5, which corresponds to the formation of A,X , is the slowest rate-determining step, and that all the otier steps in the process equilibrate much more rapidly, and also that the exchange equilibrium of the counterion between aggregates and the solution phase is established very rapidly. Then, the concentrations of the intermediate species appearing in step 5 are expressed in terms of equilibrium constants and the concentrations of stable species in steps 4 and 6. The rate of micelle formation which is equal to that of the formation of AnXpleads to d[A,X,]/dt = d[AnXp]/dt = kn+[AlI[An-,XpI - k;[AnXpI = kn+K1[A13"[x,lp- kn-K2[AmXql [A1ln-"[XJp-q (7) where k,+ and k; are the forward and backward rate constants for the slow step, and K1 and K2 are the equilibrium constants expressed by
[AnXpl [AII"-"[XJ~-~ (8) [All n-l [x,Ip [A,XqI The brackets in these equations represent the concentration of the enclosed species. By substituting the relations & = & + A& in eq 7, where represents the concentration of the ith species and A& a small deviation of ti from its equilibrium value ti,and by linearizing according to the condition of a small perturbation, the rate equation leads to K1 =
[An-lXpI
K2 =
(3) (T2c3
where ci is the concentration of the micellar nucleus, and k- the dissociation rate constant of a surfactant monomer from the nucleus. Based on eq 3, the effect of counterion on rcl was interpreted in some detail by Hoffmann et aL93' Thus far models for the micelle formation mechanism have been proposed neglecting the participation of counterion, except that proposed by Kahlweit and coworkers.ll For ionic surfactants it may be indispensable to take into account the participation of counterion, since the static and kinetic nature of the micelle may be strongly affected by counterion. It may be reasonably assumed that micelles are formed from (or broken up into) monomers by stepwise bimolecular association (or dissociation) reactions. On the other hand, it is well established experimentally that the slow relaxation process is characterized by one distinct relaxation time which is well separated from the other. It may be allowed from this fact to assume that there exists a single rate-determining step in the sequence of stepwise bimolecular reactions. Hence, the following mechanism was assumed for the micellization-dissolution process taking into account the counterion:
Mass balance requirements for the surfactant ion and counterion are A[A,] mA[A,X,] N 0 (loa)
+
Nx,l + qA[A,X,I
N
0
(lob)
where the contribution from the intermediate aggregates is assumed to be neglected. Combination of eq 10a and 10b with eq 9 leads to
Thus, the reciprocal of the relaxation time T~ is given by the coefficient of A[A,X,] on the right-hand side of eq 11. Using the relations
The Journal of Physical Chemistty, Vol. 82, No. 18, 1978 2041
Relaxation in Micellar Solutions
10
in order to express r2-l in terms of the total concentration of the surfactact, Co and the equilibrium concentration of a single ion, [A,], one finally obtains for r2-l
mr2(x - 1) ((1- r)x + ry+l
+ ((1- r)x1 + r)rs
where following replacements were made (0 5 r 5 1) r = q/m = p / n
(14b) (14c)
r represents the ratio of the number of counterions bound on the micellar surface to that of surfactant ions in the micelle or, in other words, 1- r corresponds to the degree of ionization of the micelle. Furthermore, it was assumed here that the degree of ionization of A,X was approximately equal to that of the stable micefle. Since the aggregation number m is generally of the order of lo2,the last term on the right-hand side of eq 13 can be neglected except when x is close to 1, Le., Co is close to the cmc. Therefore =
c
[
+ r2((1- r)x + rJ"+l ((1- r)x + rJ's -
x-l
2.0
2.5
X
s=m-n x = Co/[A,] N _ Co/cmc
72-1
1,5
1,o
(144
]
Figure 7. Plot of 7;' vs. xfor DPBr. The data points are those obtained by P-jump experiments. The solid lines are theoretical curves plotted according to eq 15: (0, 1) 15 O C ; (0, 2) 10 O C , (0,3) 5 OC. The values of parameters used for the theoretical curves are r = 0.63,s = 10.2 (15"C),13.6 (10"C),13.1 (5 "C),and C = 4.24 X lo3 (15 oc), 2.30 x 103 ( i o o c ) , 5.47 x 102 (5 o c ) .
2o
N
r0
(15)
10
t
c/
where
C = mk,-K2/[Al]s(r+1) Equation 15 predicts that a plot of r2-I vs. x has a maximum in general, and in the limiting case in which r is equal to 0 or 1 r2-,shows a linear dependence on x . As mentioned in section 111, it was reported that the degree of counterion association for the DPI micelle is very large. Therefore, the r value for DPI may be close to 1and hence r2-l will increase linearly with C,. For DPBr, it is thought that r approaches 1only when a large concentration of KBr is added. In fact, Anacker and GhoseZ1reported that the value of r for cetylpyridinium bromide in 0.2 M KBr is 0.95. Therefore, the change in the behavior of rZ-, for DPBr induced by KBr addition may be attributed to the variation of r. Equation 15 also predicts a linear increase of r2-I with Co for nonionic surfactant providing r is equal to 1. This has been found experimentally by some workers.2cJ1 Equation 15 can be rewritten in the form
-
Comparing eq 16 with eq 3, it may be thought that [A,X,] plays a role similar to ci, and [A,X,] and ci are important factors which control the slow relaxation time. The three parameters r, s, and C in eq 15 are thought to be functions of x. However, they may be assumed constant in the low concentration range except near the cmc. The value of r can be estimated as the slope of log cmc vs. log C, plot, where C, is total counterion concentrationaZ2Using the r value obtained from this relation eq 15 was fitted to the present experimental data. It is well
1.5
1.0 X
Flgure 8. Plot of T*-' vs. xfor SDS. The data points are taken from ref 12. The solid lines are theoretical curves plotted according to eq 15: (0,1)25 O C ; (0, 2)20 OC; (0,3) 15 OC. The values of parameters used for the theoretical curves are r = 0.46,s = 13.3 (25 "C),15.2 (20 OC), 17.5 (15 "C),and C = 1.22 X lo4 (25 "C),6.29 X lo3 (20 OC), 3.15 X (15 "C).
lo3
known that in the conventional treatment, [A,] can be assumed conjtant and equal to the cmc above the c ~ c . ~ ~ Hence, for [A,] the cmc values were used and the values of s and C were computed to give a best fit to the experimental data by the least-squares method. In this computation, a few experimental points which appeared scattered were excluded. The results for DPBr at various temperatures are shown in Figure 7. Here, r was estimated from the cmc data by Ford et al.14 The theoretical curve shows a good agreement with experimental points except near the cmc. The appreciable deviation at concentrations close to the cmc may be ascribed to the following: (i) In deriving eq 15, an approximation for mass balance (eq 10 and 12) was used. This approximation holds at concentrations well above the cmc. (ii) The theoretical curves were calculated under the assumption of constant C, i.e., constant [A,]. This assumption may not be the case-near the cmc. According to Kahlweit and co-workers,ll [A,] may be regarded as constant and equal to the cmc only when Co is higher than a definite concentration somewhat above the cmc. It may be interesting to apply eq 15 to another surfactant, sodium dodecyl sulfate (SDS),which shows a quite
2042
The Journal of Physical Chemistry, Vol. 82, No. 18, 1978
Inoue et al.
TABLE 11: Values of Parameters Used for the Computation of the Theoretical Curves ~
r 0.95 0.95 0.95 0.95 0.95 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.63 0.49 0.49 0.46c 0.46 0.46
material DPI
DPBr
DPCI SDS
a
From ref 14.
c, s-‘ 2.37 X 10’ 2.37 X IO’ 2.37 X 10’ 2.37 x 10’ 2.37 X 10’ 5.47 x 10’ 2.30 x 103 4.24 x 103 1.53 x 104 1.53 x 104 1.53 x 104 1.53 x 104 1-53 x 104 6.33 x 104 6.33 x 104 3.15 x 103 6.29 x 103 1.22 x 104
S
5 5 5 5 5 13.1 13.6 10.2 12.5 12.5 12.5 12.5 12.5 14.1 14.1 17.5 15.2 13.3
From ref 15.
1x81 -
[xadl
~~
lom3M
10- M
6.7 6.0 5.4 4.0 2.7 11.5 11.2 11.0 11.2 10.75 10.15 9.3 8.6 19.6 17.6 8.4 8.2 8.1
0 1 2 5 10 0 0 0 0 2 5 10 15 0. 10 0 0 0
M
cmc,
temp, “ C
5.7
25 25 25 25 25 5 10 15 21 21 21 21 21 5 5 15 20 25
3.ga 3.0 ll.Sb 11.2b 11.0b 11.2b
19.6 8.4 8.2
8.1
From ref 22.
different concentration dependence of r2-lcompared with DPX. The general feature is well simulated by eq 15 as seen in Figure 8. Here also an appreciable deviation is seen at concentrations near the cmc. In systems with an added salt which has common ion with the counterion of the surfactant, the mass balance becomes, instead of eq 12b
co
~~
[Ail or [A,’],
Cqx,
+q [ G ]
(17)
-
,
i v) w
“7
3 v
where [&dl represents the concentration of added salt. Equations 11, 12a, and 17 lead to the following expression for r2-l: r2-1
[
= fs(r+l)C
x-1
((1- r)x
+ r + xS)rs + r2 ((1- r)x
-
+r+
]
(18) x,]~~+~
where x =
ri I
cO/[A,’I
f = [Al]/[Al’]
xs = [xaJ/[Al’I N cmc/cmc’
and the prime designates the quantity in the presence of added salt. In deriving eq 18 it was assumed that rn, k;, Kz,r, and s are invariant with [&]. In general, they may be functions of [Xad],however, they can be considered constant at such low added salt concentrations as the conductivity P-jump technique is effective. Since f is larger than 1,this factor increases rZ-l, On the other hand, the factor x, in eq 18 decreases r2-l. Hence, from eq 18 it is expected that r2-l is increased or decreased by the addition of salt as a result of the competition of these two factors. When the extent of depression of the cmc caused by added salt is large, i.e., the slope of log cmc vs. log Co plot is large, or in other words the r value is large, rrl may increase with the addition of salt. (This is the situation for DPI.) On the contrary, when the r value is small, r2-l may be decreased by the presence of added salt. (This is the situation for DPBr and DPC1.) Theoretical curve calculated from eq 18 was compared with the present experimental data for dodecylpyridinium halides with added salts. Figure 9 shows the results for DPBr at 21 “C. In this figure, solid curves were theoretical ones from eq 18. The values of r, s, and C in eq 18 were assumed to be the same as those obtained by the best fit of eq 15 without added
1,o
la5
2,o X
Figure 9, Plot of ’ ;7 vs. x for DPBr at 21 O C . The data points are those obtained by P-jump experiments. The solid lines are theoretical curves plotted according to eq 15 (1) and eq 18 (2-5): (0, 1)no added KBr; (8,2)CKm 2 X lob3M; (e,3 ) CK3 = 5 X M; ( 0 , 4 ) CKk = 1 X lo-* M; (0, 5) CKBr= 1.6 X 10- M.
salt. Although the deviations of the experimental points from the curves are appreciable in the low concentration range, the variations of r;l with added salt concentrations are in good agreement with those obtained by eq 18. Similar comparisons of the theoretical curves with experimental data are given in Figures 10 and 11 for DPCl and DPI, respectively. For DPC1, r was estimated from the cmc data by Ford et al.14 For DPI, eq 18 could not reproduce the experimental data when [A,] was replaced by the cmc and r = 0.85, which was obtained from the log cmc vs. log C, plot. Instead of that, the best fit calculation was carried out using the [Al] value taken as the concentration at which extrapolated straight lines in Figure 1 intersect the abscissa. The parameters were determined as r = 0.95 and s = 5. From Figures 10 and 11 it is seen that the variations of r2-l with added salt concentration are successfully explained by eq 18. The values of r, s, [Al], and [Ai] used for the computation of the theoretical curves are summarized in Table I1 together with the experimental values of the cmc. As described above, assuming a single rate-determining step in a sequence of association-dissociation reactions for the micellization-dissolution process, and introducing a parameter r for the degree of counterion association on the
The Journal of Pbysical Chemistry, Vol. 82, No. 18, 1978 2043
Relaxation in Micellar Solutions
and the rate-determining step, were about 13 for DPBr and DPC1, and 5 for DPI, respectively. These values lead to aggregation numbers for the micellar nucleus which are considerably different from the values near 10 reported by Hoffmann et al. based on the Aniansson-Wall model.9J0320These differences may be attributed to the difference in approaches, and a more detailed discussion needs much more information on the size distribution of oligomers than available at the present time. We estimaled the s values under the assumption of constant [A,]. According to the mass action model it was shown that [A,] changes slightly with total surfactant concentration, especially for the partially associated micelle.24 If this variation of [A,] is taken into the computation, somewhat modified s values may be obtained.
r = 0,119 s = 14.1
e,
C = 6,33x104
1,5
1,o
References and Notes
X
Figure 10. Plot of T2-' vs. x f o r DPCl at 5 OC. The data points are those obtained by P-jump experiments. The solid lines are theoletical curves plotted according to eq 15 (1) and eq 18 (2): (0, 1) no added KCI; (0,2) CKCI= 1 x lo-* M.
,-. i I
V
N d 0 v
3
N
1,o
1,5
2,o
X
Flgure 11. Plot of T*-' vs. x for DPI at 25 OC. The data points are those obtained by P-jump experiments. The solid lines are theoretical curves plotted according to eq 15 (1) and eq 18 (2-5): (0, 1) no added KI; (a,2) cKI= 1 x 10-3 M; (0,3) cKI= 2 x 10-3 M; (a,4) cKI = 5 x 10-3 M; ( 0 ,5) cKI = I x IO-* M.
micellar surface, the concentration dependence of T ~ and - ~ the added salt effect on ~ 2 -can l be explained consistently for dodecylpyridinium halides. Furthermore, using the r values which seems to be reasonable for dodecylpyridinium &Its an agreement between theory and experiment was obtained. On the other hand, values for the parameter s, which represent the distance between the stable micelle
(1) G. C. Kresheck, E. Hamori, G. Davenport, and H. A. Scheraga, J . Am. Cbem. Soc., 88, 246 (1966). (2) (a) B. C. Bennion, L. K. J. Tong, L. P. Holmes, and E. M. Eyring, J . Pbvs. Cbem.. 73. 3288 (1969): (b) B. C. Bennion. and E. M. Evrina. J. kolbklInterface Sci.,32, 286 (1970); (c) J. Lang and E. M. Eyring, J. Polym. Sci., Part A-2, 10, 89 (1972). (3) (a) E. Graber, J. Lang, and R. Zana, Kolloid-Z. Z. Polym., 238, 470 (1970): (b) E. Graber and R. Zana. ibid., 238, 479 (1970). (4) (a) K. Takeda and T. Yasunaga, J . Colloid Interface Sei., 40, 127 (1972); (b) K. Takeda and T. Yasunaga, ibid., 45, 406 (1973); (c) T. Yasunaga, K. Takeda, N. Tatsumoto, and H. Uehara, "Chemical and Biological Applications of Relaxation Spectrometry",E. WynJones, Ed., D. Reidel, Boston, 1975, p 143. (5) (a) P. J. Sams, E. Wyn-Jones, and J. Rassing, Chem. Phys, Lett., 13, 233 (1972); (b) J. Rassing and E. WynJones, [bid, 21, 93 (1973); (c) J. Rassing, P. J. Sams, and E. WynJones, J. Cbem. Soc., Fara&y Trans. 2, 70, 1247 (1974). (6) T. Janjic and H. Hoffmann, Z. Pbys. Chem., 86, 322 (1973). (7) R. Folger, H. Hoffmann, and W. Ulbricht, Ber. Bunsenges. Phys. Cham.. 78. 986 (1974). (8) J. Lang] C. Tondre, R. Zana, R. Bauer, H. Hoffmann, and W. Ulbricht, J. Pbys. Cbem., 79 276 (1975). (9) H. Hoffmann, R. Nagel, G. Platz, and W. Uibricht, Co/ioidPolym.Sci., 254, 812 (1976). (10) E. A. G. Aniansson, S.N. Wall, M. Almgren, H. Hoffmann, I.Kielmann, W. Ulbricht, R. Zana, J. Lang, and C. Tondre, J . Phys. Chem., BO, 905 (1976). (11) S.-K. Chan, U. Herrmann, W. Ostner, and M. Kahlweii, Ber. Bunsenges. Pbys. Cbem., 81, 60, 396 (1977). (12) T. Inoue, Y. Shibuya, and R. Shimozawa, J. ColloidInterface Sci., in press. (13) "Organic Synthesis", Collect. Vol. I, 1956, p 29. (14) W. P. J. Ford, R. H. Ottewill, and H. C. Parreira, J . ColloidInferface Sci., 21, 522 (1966). (15) J. E. Adderson and H. Tavlor. J. Collokd Interface Sci.. 19. 495 (1964). (16) P. Mukerjee and A. Ray, J . Phys. Cbem., 70, 2150 (1966). (17) T. Nakagawa, Colloid Polym. Sci., 252, 56 (1974). (18) A. H. Colen, J. Pbys. Chem., 78, 1676 (1974). (19) (a) E. A. G. Aniansson and S. N. Wall, J . Pbys. Cbem., 78, 1024 (1974); (b) ibid., 79, 857 (1975). (20) H. Hoffrlann, H. Nusslein, and W. Ulbricht, "Micellization, Solubilization and Microemulsions", K. L. Mittal, Ed., Vol. 1, Plenum Press, New York, N.Y., 1977, p 263. (21) E. W. Anacker and H. M. Ghose, J. Am. Chem. Soc., 90, 3161 (1968). (22) M. L. Corrin, J. Colloid Sci., 3, 333 (1948). (23) K. Shinoda, T. Nakagawa, B. Tamamushi, and T. Isemura, "Colloidal Surfactants", Academic Press, New York, N.Y., 1963. (24) K. J. Mysels, J . Colloid Sei., 10, 507 (1955).
