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J. Phys. Chem. 1980, 84, 1536-1540
Kinetics of Micellization of Triton X-I00 in Aqueous Solutions C . 4 . Herrmann and M. Kahlwelt” Max-Pianck-Institut fur biophysikalische Chemie, 0-3400 Gottingen, West Germany (Received October 10, 1979) Publication costs assisted by Max-Pianck-Geseilschaft
As a continuation of investigations on the influence of nonionic amphiphilic substances on the mutual solubility of p-xylene and water, we have measured the relative turbidity of the homogeneous solution as function of composition, using a polyglycol ether (P4,3)as the third component. The results confirm our earlier X-ray measurements qualitatively, showing that the microstructure of such solutions is manifold and changes gradually from one configuration to another. In order to study the kinetics of formation of different types of aggregates, we have started with the kinetics of micellization of Triton X-100 in diluted aqueous solutions. The experimenb were performed with T-jump and p-jump perturbations, using a fluorescence indicator for detection. It turned out that in this system the concentration dependence of the time constant of the fast relaxation process ( T J and the relative amplitude of the slow relaxation process (A2)are satisfactorily described by the theory, whereas the concentration dependence of the time constant of the slow relaxation process ( T J shows considerable discrepancy with theoretical prediction.
Introduction In a preceding paper1 we have studied the influence of some nonionic amphiphilic liquids on the mutual solubility of p-xylene and water. For this purpose we have chosen polyglycol ethers, which may be written as Pij E H[CHz]i[O(CH2)2],0H where the oleophilicity (i) as well as the hydrophilicity 0‘) can be increased stepwise and independently. The investigations were carried out with i ranging from 1to 5 and j ranging from 0 to 4, and with Triton X-100. As predicted by thermodynamic considerations,2 increasing amphiphilicity of the third component leads to an increasing mutual solubility of “oil” and water. The polyglycol ethers with low i and j do not seem to form micelles “by themselves” in the water region. The increased solubility of p-xylene in that region may then be explained by a solvation of the oil molecules by the oleophilic groups of the Pij molecules, the hydrophilic groups keeping the solvates in solution. In the oil-rich region, on the other hand, the increased solubility of the water molecules is caused by their solvation by the hydrophilic groups of the Pijmolecules, the oleophilic groups keeping the solvates in solution. X-ray small-angle scattering measurements1 indeed confirm the existence of spherical aggregates in both regions close to the miscibility gap. Close to the critical point, one finds critical fluctuations. In the C region, X-ray measurements indicate the existence of quasi-lamellar aggregates of Pij molecules, similar to the alcohol aggregates found in concentrated wateralcohol mixtures. Turbidity Measurements The microscopic structure of these ternary systems thus seems to be manifold and to change gradually from one configuration to another one. This is demonstrated in Figure 1, where in (a) we have plotted the miscibility gap as well as some lines of equal turbidity of the homogeneous solution at 25 “C, A denoting p-xylene, B, water, and C, P43.In parts b-f of Figure 1we have plotted the relative turbidity (Illo)along some lines in this phase diagram, Io being the turbidity for C, i.e., of pure P43. Figure l b shows the turbidity along the miscibility gap, with a pronounced maximum at the critical point. Figure ICshows the turbidity along the B-C side with a pro0022-3654/80/2084-1536$01 .OO/O
nounced maximum a t low mole fractions of P43,which is also found in the viscosity of the solution, similar to that in water-alcohol mixtures at low alcohol concentrations. Figure If shows the turbidity along the A-C side with a rather smooth monotonical decrease. Figure I d shows the turbidity along the line between the critical point and C, with a steep increase near the critical point. As one can see from Figure la, this line proceeds through a valley between the B-C side and a region of increased turbidity in the P43-ri~h region. This region is shown in Figure le, which shows the turbidity along the line between the point xA/xB = 3/2 and C. With increasing oil concentration, this lamellar structure seems to break down and yields spherical aggregates found in the oil-rich region. Kinetics of Formation of Triton X-100Micelles Figure 2 shows a schematic representation of the water corner of the phase diagram of systems which form micelles “by themselves”. Curve a represents the cmc curve, starting at the cmc in pure water. As oil is added, the cmc decreases, until the curve intersects with the solubility curve of the oil (curve b). Curve b, on the other hand, starts a t the solubility of oil in pure water. As an amphiphilic substance is added, the solubility increases slightly, until at the intersection with the cmc curve it shows a break, the solubility now increasing much stronger with xc, being caused by the solubilization of the oil molecules by the micelles. However, if one chooses substances with weaker amphiphilicity, which do not form micelles by themselves, the solubility of the oil is also increased. This tendency increases with increasing amphiphilicity, i.e., increasing i and j (see Figure 2 in ref l),as does the tendency for the formation of micelles, until it becomes meaningful to speak of a cmc. The cmc will decrease with increasing oleophilicity. The better defined the cmc curve becomes, the stronger the curvature of the solubility curve a t the intersection of the two curves, until one finds a rather sharp break. Correspondingly, one has to expect a continuous transition of the shape of the solubility curve from alcohols 0’ = 0) to typical surfactants. In order to study the kinetics of formation of the different types of aggregates in the ternary systems we started with the investigation of the formation of Triton X-100 micelles in pure water, choosing this substance as a representative of a Pij with strong amphiphilicity. 0 1980 American Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980 1537
Micellization of Triton X-100
B
1000 -
I
L
A-B
-
C
A
B
'B
t
1-c
--
B
xc
C
0 A-C
" \
A
-
C
xc
Figure 1. Turbidity of the system p-xylene (A)-HzO (B)-P43 at 25 'C: (a) lines of equal turbidity; (b) relative turbidity I l l , along the miscibility ,, corner; (e) I I I , from (2) at the miscibility gap to the gap; (c) I I I , along the HzO-P,, side; (d) I l l , from (1) at the miscibility gap to the P ., P,, corner; (f) I I I , along the p-xylene-P,, side. I , is the turbidity of pure P
B
pendence of the amplitudes has been analyzed by Teubner, Diekmann, and Kahlweit.' The theory predicts two relaxation processes: the first arises from a rapid rearrangement of the size distribution of the proper micelles and the second arises from the slower formation of new micelles. For the dependence of the time constant r1 of the fast process, the theory predicts a linear increase of (l/rl) with the total surfactant concentration: 1/71 = (6,/U2)[1
Flgure 2. Water corner of the phase diagram of the ternary system oil-water-surfactant (schematically): (a) cmc curve; (b) miscibility gap
The results for other amphiphilic substances with weaker amphiphilicity will be published in a forthcoming paper, togethLer with results in a ternary system, i.e., in the region just below the cmc curve in Figure 2. The kinetics of micellization of ionic surfactants in diluted aqueous solutions has been studied extensively, both experimentally and the~retically,~ whereas for nonionic surfactants oinly very few data have been p ~ b l i s h e d . ~ The ,~ basis for the analysis of the time constants is the theory by Anianssoin and whereas the concentration de-
+ (U2/m)X]
(1)
where b, denotes the mean dissociation rate constant of the proper micelles, 0 the width of their size distribution, m their mean aggregation number, and X ( N - Nl)/N1 (2) the ratio between the number of monomers incorporated into micelles and that of free monomers, with N denoting the total number density of surfactant molecules and N1 that of free surfactant molecules. For the dependence of the time constant r2of the second slower process, the theory predicts3
1588
020,
-
We recall that during the fast relaxation process only the mean aggregation number m of the micelles changes, whereas during the slow process both 2 and m may change. Since experiments show that the fluorescence of the solution changes during both processes, we conclude that the equilibrium constants Kiare not only functions of the intensive variables but also of the mean aggregation number. Multiplica44on of eq 8 from j = 1 to j = i yields
0.15 21
1
Herrmann and Kahlweit
The Journal of Physical Chemistry, Vol. 84, No. 12, 1980
0.10
=-----,
0.05
0.74
1
0
3
2
I
4
Since for physical reasons Kimust decrease with increasing i, we set as approximation
5
- x Figure 3. Concentration dependence of tlme constant of slow relaxation processes (theoretically). y (1/7,)(dlMb,m2)vs. X with v as a
Ki= Kl/i which yields
parameter. (r*/rfi) = 14, rfi = 85.
where M denotes a proportionality constant, 6, the mean dissociation rate constant in the minimum of the size distribution, d the effective width of this rate determining region, and
-
m2 = f i 2
+ a2
(4) The exponent u is defined as ratio between the aggregation number of the nucleus n and the mean aggregation number of the micelles: v
= n/fi
2 / m Iv I1
(5)
Assuming the first factors to depend only weakly on X (if at all, d should decrease with increasing X ) one may then plot
+ (m/mz)X-l 1 d - -= X U M6,Z 1 + (aZ/rn)X 1
Y
(6)
72
ni Ki = K1i/i!
j=1
With this assumption one obtains for the ratio of incorporated indicator molecules (C - lo) to free indicator molecules3
(r-
r
r01/ro
=
=I
F = c(r- lo)
where N7 denotes the number density of monomers at the cmc. The shape of these curves depends mainly on the value of u. In Figure 3 we have plotted y (see eq 6) vs. X with rfi = 85, a = 35, Le., (film2)= and ( a 2 / f i )= 14, with u as a parameter. For low values of u, the curve decreases monotonically with increasing X , while for higher values of v it first passes a minimum, which is followed by a maximum and a monotonical decrease. Both results are strictly valid only for nonionic surfactants. The dependence of the amplitudes of the relaxation processes depends on the method of perturbation as well 88 on that of detection. In the case of nonionic surfactaqts, the detection by fluorescence indicators seems to be the most appropriate one, whereas the perturbation may be carried out either by T-jump or p-jump or in a stoppedflow apparatus. Assuming that each micelle may incorporate one or more indicator molecules, we write the law of mass action as
where Zi denotes the number density of micelles with i indicator molecules per micelle and C0 that of the free indicator molecules in solution.
(13)
where C is a proportionality constant, one finds by inserting eq 12
F = C{2Kl(1 f 2KJ-l (7)
(12)
where denotes the total number of indicator molecules and 2 the total number of micelles per unit volume of the solution. We recall that K1 is independent of { but apparently a function of m. Setting for the excess fluorescence per unit volume of solution (in arbitrary units)
vs. X , where X may be evaluated according to
x = (N - N,)/N,
(10)
(14)
Equation 14 predicts that with constant surfactant concentration, Le., constant 2, F should increase linearly with l, By measuring F vs. {with different 2 one may then evaluate the two parameters C and K1. The relative amplitude of the slow relaxation process A2 is then given by3
where AZ/Zis determined by the method of perturbation. A derivation of the corresponding expression for Al is at present not possible, since an independent petermination of the dependence of K1 on m at constant 2 and constant intensive variables is hardly feasible. The first factor in eq 15 is positive and approaches zero for X and constant {. If the perturbation is carried out by a T-jump, one finds for the second factors
-
QJ
A2
where AHmdenotes the enthalpy of reaction at i = m,AR the mean reaction enthalpy of the reactions 2 Ii Ia, and l6Tl the T-jump.
Mlcellizatlon of Trlton X-100
The Journal of Physlcal Chemistty, Vol. 84, No. 12, 1980 1539
I
I
I
30
40
'0 5
0
0.2
-
04
0.6
08
c [10-6mol cm-3~
Flgure 4. Fluoirescence Fd in arbitrary units vs. total concentration c of surfactant in the presence of 4 X l o 8 mol cmm3MgANS at 10 OC:
(0) without NaCI; (0) 1X
mol ~ r n NaCI. -~
Equation 16 predicts that (at low X ) A2 has the same sign as (a ln cmc/d'T),. It further predicts that A2 changes sign, when the first bracket vanishes, Le., A2 = A2 = 0 for
0 0
10
20
9
["I
Flgure 5. cmc cy vs. temperature 0 In the presence of 4 X lov8mol mol ~ m NaCi. - ~ cm-3 MgANS: ( 0 )wlthout NaCI; (0) 1X
Equation 17, however, is meaningful only if AHm_andAI? are of the same sign, and, further if IAHJ 2 ImI. It can be shown3that, at this particular concentration, the slow rehration process vanishes altogether, i.e., the fast relaxation p,rocess leads to the final equilibrium.
Comparison with Experiments Material. As surfactant we have chosen Triton X-100. The commercial product (Rohm and Haas) was dissolved in ether, dried over Na2S04,and filtered through a membrane filter to remove ionic impurities. The solution was kept at -20 OC, the precipitating Triton separated in a thermostatecl centrifuge, and the remaining ether removed under vacuum at room temperature. Since the commercial product contains a certain distribution of components with different chain length, this procedure will in particular reduce the fraction of long-chain components. The relaxation experiments were carried out with a T-jump and a p-jump as perturbation, whereas for detection we have chosen the change of the quantum yield of the magnesium sialt of 8-amino-l-naphthalenesulfonic acid (MgANS) as indicator. The concentration of the indicator was 4 X mol ~ m - ~For . MgANS up to 1 X mol cm-?Ithe resulta were independent of the indicator concentratioii. The T-jump experiments were carried out in the presence of 1 X mol cm-4 NaC1. The cmc was determined by measuring the fluorescence of the solution vs. the surfactant concentration in the presence of MgANS, both with and without NaC1. These curves show il distinct change of the slope within a narrow Concentration range (Figure 4). The cmc was defined as the intersection of the two straight lines, which could be fitted to the experimental points below and above this transition range. We note that the slopes of these straight lines decrease with increasing temperature. Figure 5 shows the cmc as function of temperature. As one can see, the presence of NaCl increases its temperature dependence. The following experiments were carried out at 10 OC. ~, The cmc wasi taken to be c, = 3 X lo-' mol ~ m - which corresponds to a mole fraction x = 5.4 X The mole fraction, at which the system d20--Triton X-100 forms liquid crystals at this temperature, was determined to be x = 1.7 X (37 w t %) and is thus about three orders of magnitude higher than x , . ~
-
0
100
50
I
150
cm-31
Figure 0. Fluorescence F in arbitrary units vs. total number denslty lof MgANS at 10 O C . (1) 0.66 X loi8 cmS; (I) 1.48 X lo1' ~ m - ~([); 2 = 8.6 X 10l8 ~ r n - ~Fit . of eq 14 with C = 1.4 X and K , = 3.3 X ~rn-~.
=!!.
r=
The mean aggregation number A was determined by light scattering, using a KMX-6 low-angle photometer (Chromatix)? Triton X-100 turned out to be a nearly ideal scatterer up to X = 30, which manifests itself in a very low value for the second virial coefficient. At 10 "C the aggregation number of Triton X-100 in pure water was found to be rfi = 85, whereas in the presence of MgANS and NaCl it was found to be m = 110. These values are considerably lower than those published by Corti and Degiorgio.lo We presume this discrepancy to arise from the fact that the molecular weight of our surfactant was reduced by the method of purification, whereas the other authors apparently used a commercial product. To determine the equilibrium constant K1 in eq 15, we then measured the fluorescence of the solution vs. the indicator concentration, with 2, i.e., the surfactant concentration, as the parameter, again in the presence of NaC1. The results are shown in Figure 6. As one can see, the curves start linearly, but flatten out at higher MgANS concentrations. Since the latter may be caused by different reasons, e.g., that eq 10 is too poor an approximation, or that m decreases with increasing indicator concentration, or simply by quenching, we have restricted ourselves to the evaluation of the two parameters C and K1from the initial slopes. The full lines in Figure 6 represent the fit of eq 14 with C = 1.4 X cm3. cm3 and K1= 3.3 X Time Constant for the Fast Relaxation Process. The linear increase of l / r l with X as predicted by eq 1has been confirmed for quite a number of ionic systems. The only nonionic system hitherto investigated was H,O-Triton
The Journal of Physical Chemistty, Vol. 84, No. 12, 1980
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40
I
0 -I
I
0
I
- x
I
I
I
I
2
3
I
Herrmann and Kahlweit
/
1 4
Flgure 7. Concentrationdependence of the time constant for the fast relaxation process at 10 OC: ( 1 / ~ , )vs. X. The solid line is a fit of eq 1.
I
-6
-8' 0
'
,
I
1
I
2
3
1 4
1
5
- x Flgure 9. Concentratlon dependence of the relatlve amplitude of the slow relaxatlon process at 10 OC: A 2 vs. X. T-jump in the presence of 1 X mol om3 NaCI; (full line) fit of eq 15 with eq 16.
-1
0
I
I
I
I
I
I
2
3
4
5
6
- x
Figure 8. Concentration dependence of the time constant of slow relaxation process at 10 OC: ( 1 / ~ *vs. ) X. T-jump ([) and p-jump (I), both in the presence of 1 X mol ~ m NaCI: - ~ (solid line) p-jump without NaCI; (dotted line) ref 5.
X-100.6 We have repeated these measurements, using a T-jump apparatus with optical detection. Figure 7 shows the results, which agree well with the earlier measurements. At low X , the results can be represented by a straight line, whereas at higher X they show an increasing positive deviation, which is also found in many ionic systems. Restricting the evaluation to the straight line, one obtains (6,/cr2) = 0.54 X lo3 s-l, (b,/m) = 7.66 X lo3 s-l, and thus (cr2/ m) = 14.2. With m = 110, this yields cr = 40 and b, = 8.4 X lo5 s-l. Time Constant for the Slow Relaxation Process. In previous measurements4i5of the concentration dependence of T, it was found that 1 / increases ~ ~ linearly from zero at X = 0. In our experiments we have extended the concentration range to higher X . Figure 8 shows the results. The experiments were carried out with a T-jump as well as with a p-jump, both in the presence of 1 X lo4 mol cm-3 NaC1, and with optical detection. As one can see, the two perturbation methods yield identical results. The linear increase of 1 / at~low~X was confirmed. At about X = 2, however, the linear increase is followed by an almost horizontal plateau. A comparison with the theoretical curves (Figure 3) shows that a fit is hardly possible. To obtain a plateau at high X, one would have to choose u = 1. This, however, leads to considerable discrepancies at low X . One could argue that the presence of NaCl may change the concentration dependence of 1 / ~as~is ,known from ionic systems. We have, therefore, repeated the measurements in a p-jump apparatus with optical detection but without NaC1. The result is shown in Figure 8 as a solid line. As one can see, the characteristic shape of the
curve does not change, although the level of the plateau decreases. Relative Amplitude of the Slow Relaxation Process. Figure 9 shows the concentration dependence of the relative amplitude A, of the slow relaxation process, measured with a T-jump apparatus with optical detection in the mol NaC1. presence of 1 X The curve shows the predicted shape: A, increases from negative values at X = 0, changes sign, and approaches m. zero again for X From zero at X , = 1.4 and (u2/m) = 14 one obtains from eq 17 AH,/m = 1.05, which is of the expected magnit~de.~ Using this number and ldTl = 2 K, we then fitted eq 15 with eq 16 to the experimental points, with (d In cmc/dT), as the parameter. The fit is shown in Figure 8 as a solid line and yielded (d In cmc/dT), = -13 X K-l. From Figure 5 , on the other hand, one finds (d In cmc/dT)p,T=m = -20 X K-l, which is in satisfactory agreement with the above value. We thus found that the concentration dependence of the time constant of the slow relaxation process disagrees considerably with the predicted shape, whereas the concentration dependence of the relative amplitude seems to be satisfactorily described by the theory.
-
References and Notes (1) C.4. Herrmann, U. Wurz, and M. Kahlweit in "Solution Chemistry of Surfactants", K. L. Mittal, Ed., Plenum Press, New York, 1979, Vol. 11, p 879. (2) C. Wagner, 2. Phys. Chem. (Leiprig), 132, 273 (1928). (3) M. Kahlweit and M. Teubner, Adv. Collold Interface Sci., In press. (4) J. Lang and E. M. Eyrlng, J . folym. Sci. A 2 , I O , 89 (1972). (5) J. Martell, Thesis, Gattingen, 1976. See S.-K. Chan, C.-U. Herrmann, W. Ostner, and M. KahlweB, Ber. Bunsenges. Phys. Chem., 81,396 (1977). (6) E. A. G. Aniansson and S. N. Wall, J . Phys. Chem., 78, 1024 (1974); 79, 857 (1975). (7) . . M. Teubner, S. Dlekmann, and M. Kahlweit, Ber. Bunsenges. fhys. Chem., 82, 1278 (1978). (8) See also P. Ekwall, Adv. Liquid Cryst., 1, 1 (197% Figure 138, p 137. The surfactant differs from Triton X-100 by an additional CH2 group in the oleophilic group. (9) C.-U. Herrmann and G. Klar, experimental details and further resuRs will be publlshed elsewhere. (10) M. Cortl and V. Deglorgio, Opt. Commun., 14, 358 (1975).
