Langmuir 1995,11,30-36
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Articles Dynamic Light Scattering of CTABlNaSal Threadlike Micelles in a Semidilute Regime. 3. Dynamical Coupling between Concentration Fluctuation and Stress Norio Nemoto,*tt Mitsue Kuwahara,s Ming-L. Yao,g and Kunihiro Osaki$ Department of Applied Physics, Kyushu University, Hakozaki, Fukuoka 812,Japan, Institute for Chemical Research, Kyoto University, Uji, Kyoto 611,Japan, and Rheometrics Fareast Co.,Shinagawa, Tokyo 141,Japan Received February 25, 1994. I n Final Form: September 21, 1994@ Dynamic light scattering measurements are made on viscoelastic networks formedby threadlike micelles of cetyltrimethylammonium bromide (CTAJ3) in aqueous sodium salicylate (Nasal) solutions at three temperatures T of 25, 33, and 40 "C. Dynamic shear moduli of the same samples are also measured in to 1 x lo2rad s-l at T = 25,33,40,50,and 60 "C. The surfactant the angular frequency range from 1x concentration CDof the samples is k e d at 0.1 M and a ratio of the salt concentration Cs to CD is varied from 0.8 to 6. The time correlation function A,(t) of light intensity scattered from the solutions with 0.8 5 CS/CD 5 4 exhibits the bimodal distribution of the decay rate r. The fast and the slowly decaying
components are assigned as the diffusivegel mode and the relaxation mode,respectively,from the scattering vector dependences of the first cumulant Ti (i = f, s) estimated from respective time profiles of the decay curves at short and long times. The dynamical correlation length (H calculated from D,= rf/q2is found to take a constant value of 10.5 & 1.5nm irrespective of C&'D and T for 1 5 C&'D 5 6. The characteristic for the slow mode agrees with the mechanical relaxation time ZM which is relaxation time zs (=rS-l) obtained by a fit of the dynamic shear modulus data of the same solutions to a Maxwell model with the single relaxation time ZM. The results demonstrate that concentration fluctuation in the micellar network decays by local cooperative diffusion of network strands at the short time as well as by the slow relaxation of elastic stress generated by the network deformation due to the mass flow.
Introduction A cationic surfactant of cetyltrimethylammonium bromide (CTAB) forms a threadlike micellar structure in aqueous sodium salicylate (Nasal) solutions and exhibits prominent viscoelastic behavior at low surfactant concentrations CD. During the last couple of years, we have performed systematic studies on dynamics of the CTABI NaSaVW micelle network (W stands for water) formed above CD= 0.006 M using dynamic light scattering (DLS), forced Rayleigh scattering(FRS),and dynamic viscoelastic (DVE) and clarified differences in dynamical behavior between the micellar network and the entanglement network of high molecular weight flexible polymer chains.'-1°
* To whom correspondence should be addressed. +
Kyushu University.
* Kyoto University.
Rheometrics Fareast Co. Abstract published in Advance A C S Abstracts, December 1, 1994. (1)Yamamura, T.;Kusaka, T.;Takatori, E.; Inoue, T.; Nemoto, N.; Osaki, K.; Shikata, T.;Kotaka, T. Nihon Reorogi Gakkaishi 1991,19, 45 (in Japanese). (2)Nemoto, N.;Yamamura,T.; Osaki, K; Shikata,T.Langmuir1991, 7,2607. (3)Nemoto, N.; Kuwahara, M. Langmuir 1993,9,419. Nemoto, N.; Hirayama, T.; Osaki, KNihon Reorogi (4)Kuwahara, M.; Gakkaishi 1994,22,57 (in Japanese). (5)Nemoto, N.; Kuwahara, M. Colloid Polym. Sci. 1994,272,846. (6)Koike, A.;Yamamura, T.; Nemoto, N. Colloid Polym. Sci. 1994, 272,955. (7)Doi, M.; Edwards, S.F. The Theory ofpolymer Dynamics: Oxford University Press: Oxford, 1986. (8)Lodge, T. P.; Rotstein, N. A.; Prager, S. Adu. Chem. Phys. 1990, 79, 1. (9)Nemoto, N. In Polymer Rheology and Processing; Collyer, A. A., Utracki, L.A.,Eds.: Elsevier: London, 1990;p 3. 8
@
0743-746319512411-0030$09.00/0
A DLS study on the CTAEVNaSaVW system with CD 2 0.006 M and a constant ratio of the salt concentration CS to CD,C&D = 1,among those studies, provided us with a new information that the normalized time correlation functionA,(t) of light intensity scattered from the samples exhibited the bimodal distribution of the decay rate I'for CDL 0.1 M, i.e. the presence of a slowly decaying mode as well as the normal diffusive gel mode. The first cumulant Ts characteristic of the slow mode was found independent of the scattering vector q at low q(=lql) and the inverse of TS roughly agreed with the mechanical relaxation time ZM obtained by a fit of the dynamic shear modulus data of the same solutions to a Maxwell model with one relaxation time Z M . ~ Similar results have been reported on the same system by Brown et al.ll The slow relaxation mode was also detected for the semidilute and concentrated solutions of macromolecules in poor or theta sol~ent.~~-~~ There are two theoretical ideas for interpretation of the slow mode.18-22 Wang predicts that the difference in partial specific volumes of the polymer and the solvent (10)Nemoto, N.; Kishine, M.; Inoue, T.; Osaki, K. Macromolecules 1990,23,659;1991,24,1648. (11)Brown, W.; Johansson, K.; Almagren, M. J . Phys. Chem. 1989, 93,5888. (12)Amis, E. J.; Han, C. C. Polym. Commun. 1982,23,1043. (13)Mathies, P.;Mouttet, C.; Weisbuch, G . J. J . Phys. (Paris) 1980, 41,519. (14)Nemoto, N.; Makita, Y.; Tsunashima, Y.; Kurata, M. Mucromolecules 1984,17,2629. (15)Adam, M.; Delsanti, M. Macromolecules 1986,18,1760. (16)Brown, W.; Nicolai, T.; Hvidt, S.; Stepanek, P. Macromolecules 1990,23,357. (17)Nicolai,T.; Brown, W.; Hvidt, S.;Heller, KMucromolecules 1990, 23,5088. (18)Wang, C. H. J . Chem. Phys. 1991,95,3788. (19)Wang, C. H. Macromolecules 1992,25,1524.
0 1995 American Chemical Society
DLS of CTABlNaSal Micelles gives rise to dynamical coupling between concentration fluctuation and density fluctuation and then the viscoelastic mode appears in the DLS power spectrum.18J9 Another idea is that mass flow or concentrationfluctuation in the gel-like solutions induces deformation of the viscoelastic network and consequently concentration fluctuation may slowly decay out as the elastic stress generated by the network deformation relaxes.15v20-22 These two ideas are essentially incompatible with each other and the physics underlying the slow mode is under issue in the DLS study of viscoelastic polymeric liquids. In an earlier report,6we tentatively discussed the slow mode observed in the CTAEUNaSaVW system with CD2 0.1 M in terms of the latter dynamical coupling model. However, the rough agreement between rsand TM suggests that a more detailed study should be done in order to prove the presence of the dynamical coupling in this threadlike micelle system. On the other hand, one big advantage of the micellar network compared with polymer entanglement network is that the former shows the single relaxation behavior, whereas the latter shows the viscoelastic behavior with a very broad distribution of relaxation times. In the case of single relaxation behavior, the theory is considerably simplified and a more unambiguous conclusioncan be drawn from detailed comparison between theoretical predictions and experimental data. This paper reports results of DLS and DVE measurements on a series of the CTAB/NaSal micellar solutions with a fured value of CD= 0.1 M and varying C$CD from 0.8 t o 6 in the temperature range from 25 to 60 "C. The results prove the presence of the dynamical coupling in this micellar network.
Experimental Section Materials. Twice-recrystallized cetyltrimethylammonium bromide (CTAB) (Nacalai Tesque) and special-grade sodium salicylate (Nasal) (Nacalai Tesque) were used in this study as cationic surfactant and salt samples, respectively. Dust-free purified water (resistance R > 16 MQ)was used as solvent. The samples were prepared by mixingprescribed amounts of aqueous solutions of CTAB and Nasal, and equilibriating at 60 "C for an hour and then at room temperature for at least 5 days. The surfactant concentration CDof eight solutions tested was fixed , at 0.1 Mand ratios ofthe salt concentration Cs to CD,C ~ C Dwere 0.8, 1 .O, 1.2,1.5,2,3,4,and 6. For dynamic light scattering, a part of the solutions was made optically clean by filtering with a Millipore filter (nominalpore size, 0.22pm) and equilibriated in dynamic light scattering cells for about 1 week. Methods. Dynamic light scattering (DLS) measurements were made with an instrument reported elsewhere.3 Avertically polarized single frequency 488-nm line of an argon ion laser (Spectra Physics, Beamlock 2060) was used as a light source with an output power of 600 mW. The normalized time correlation function A,(t) of the vertical component of the light intensity scattered from the solutions was measured using the digital correlators (Malvern and Otsuka Electronics) at 12 fixed scattering angles ranging from 10.4"to 150". The experiments were conducted at three temperatures of 25,33, and 40 "C. Dynamic viscoelastic (DVE)measurements were made with a Couette-type rheometer (Rheometrics, Fluid Spectrometer FRSII). The storage and the loss shear moduli, G(o)and G ( w ) , of the solutions were measured over the angular frequency w range from 1 x 10-2to 1 x 102 rad s-l at five temperatures of 25,33,40,50,and 60 "C. Reliable G* data were not obtained for some solutions at 60 "C. Results and Discussion 1. Dynamic Light Scattering (DLS). 1.1. Time Profiles of A,@). Preliminary DLS measurements (20)Brochard, F.; de Gennes, P.-G. Macromolecules 1977,10, 1157. (21)Brochard, F. J. Phys.(Paris)1983,44, 39. (22)Doi, M.;Onuki, A. J.Phys.11 1992,2,1631.
Langmuir, Vol. 11, No. 1, 1995 31
"d
,
Ill.d
102
71 0 6 1
1 I
,,,d
1
T/BS Figure 1. Time profiles of the normalized time correlation functionA,(t) for the CTAB/NaSal/W system with CD= 0.1 M and C ~ C = D1.2 at 25 "C. Scattering angles 8 are 15")40°, and 150"as indicated in the figure. The curves are composite ones obtained by superposition of three rawA,(t) data with different sampling times to each other.
revealed that time profiles of A,(t) of the CTABMaSal micelle solutions a t 25 "C were surely represented by the bimodal distribution of the decay rate r, being in agreement with the earlier finding reported in ref 6. Since our correlator is the one with simple linear channel spacing (the maximum channel number of 10241,a single runwith one sampling time A z could not provide A,(t) data sufficient for precise data analysis over a very broad r range. Therefore we measuredA,(t) repeatedly by varying A t by more than 2 orders of magnitude and obtained a composite curve from superposition of a couple of A,(t) data to each other. Figure 1shows, as typical examples, composite curves thus obtained for the sample with C ~ C D = 1.2 a t 25 "C a t three scattering angles 8 of 15,40, and 150". As is seen from the figure, time profiles ofA,(t) are represented by the bimodal distribution of the decay rate. Since the two modes are well separated in the time scale, we applied the cumulant analysis to the data a t the short and the long time domains, respectively, to obtain the first cumulant Tf and rs(rf>> r,)characteristic of the fast and the slow modes. A&) = p
+ Aijg,'"(t)12
(1)
Here Ai (i = f, s) is the amplitude of the ith mode and gJ1)(t) is the time correlation function of the scattered electric field a t the scattering vector q (q IqI = (4nIA) sin(8/2)). The baseline /? was put equal to unity for an estimate of rsand was taken as (1 A,) for an estimate of Tf. Eventually (1 A,) was found very close to the stationary value of&@)appearing during the intermediate time interval. It is to be noted that Af was always much larger than As and that Ap'(A, Af) increases with increasing q. The data were also analyzed with the histogram method. Integration of respective peaks gave values of Tf and rawhich were found to be in agreement with those from the cumulant analysis within experimental uncertainty. Exactly the same analysis was applicable to A,(t) data of other samples except the one with C ~ C = D 6 to which only the fast mode was observed. The bimodal distribution
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Nemoto et al.
32 Langmuir, Vol. 11,No. 1, 1995
Figure 2. First cumulant rffrom the fast mode divided by the square of the scattering vector q and plotted against q2 for the six samples with various salt concentrations. The plot shows that rdq2is independent of q2. Vdues of C&'D are ( 0 )0.8, (0) 1.0, ( A ) 1.2, (+) 1.5, (V)2.0, and ( 0 )6.0. Table 1. Results of DLS Measurements on the CTAB/ NaSaYW System
CdCD 0.8 1.0 1.2 1.5 2.0 3.0 4.0 6.0
DdiO-7 cm2 5-1 T=25 T=33 T=40 "c "C "C 3.1 2.1 2.0 2.1 2.0 2.5 2.5 2.2
3.8 2.6 3.0 2.9 2.8 3.0 3.0 2.7
5.7 3.5 3.6 3.1 3.5 3.3 3.6 2.8
T=25 "C
rJs-1 T=33 "C
T=40 "C
0.33 1.1 1.4 0.60 0.26 0.95 5.0
1.2 5.9 14 5.0 2.2 4.2 27
3.2 15 36 15 12 11 56
ofthe decay rate was found to be the better representation for describing time profiles of A&) of the seven samples at elevated temperatures of 33 and 40 "C, but lower amplitude for the slow mode made determination of T, and A, a bit more difficult. 1.2. The Fast Mode. Figure 2 shows a plot of T f / q 2 of six samples at T = 25 "C against the square of the scatteringvector. The data Tf/q2 of the two samples with CS/CD= 3 and 4 are not shown for clarity of the figure. The Tflq2is independent ofq2to an experimental accuracy of 10% over the whole range of 8 from 10.4" to 150" measured, indicating that the fast mode is the diffusion mode with the cooperative diffusion coefficient D,as a n average of rf/q2.The values are listed in Table 1. From the table, we see that D, takes a constant value for CS/CD z 1. The D,value a t CS/CD = 0.8 is slightly larger than the constant value at higher CS/CD. This may be related to the fact that the threadlike structure of the CTABI Nasal micelle is stabilized by formation of a neutral 1:1 complex of CTA+and Sal- ions. The bulk elastic modulus of the network formed under a n insufficient amount of Sal- ions, then, may slightly increase due to electrostatic repulsive interaction between micellar network strands, resulting in a n increase of D,.6 Since an increase in D, with increasing T may be attributed to temperature dependence of local friction in the solution, we calculated the dynamical correlation length & from D,using eq 4. 6H
= kBT16Z
rP,
(4)
Here, a s the solvent viscosity 7,) we used viscosity values of Nasal-water mixtures accounting for a decrease in C ,
Figure 3. Plot of the dynamical correlationlength [H against C&D. The l j takes ~ a constantvalue of 10.5 f 1.5 nm for I 1 in the range of T from 25 to 40 "C. Symbols for Tare (0) 25, (0) 33, and (A) 40 "C. of the solution due to incorporation of salicylate ions into the m i ~ e l l e s .A~plot of & vs C ~ C in D Figure 3 shows that EH is independent of T and takes a constant value of 10.5 f 1.5 nm for CS/CD2 1. Values ofq covered with the DLS experiment through 8 = 10.4-150" ranges from 3.17 x nm-l. Thus q 5 H < 1 is satisfied for all to 3.31 x 8, which is consistent with constancyofTdq2over the whole range of q as exemplified in Figure 2. The above results are quite in contrast to those observed for the CTAB/NaSal micelles with lower surfactant concentration of CD = 0.01 M and varying CS/CDfrom 1 to 41.3 For the latter samples, Tdq2 was strongly dependent on q due to a n additional contribution of intramolecular motions of the network strands, and EH was dependent on CS/CDin a complicated manner and appreciably decreased with increasing T . At CD= 0.1 M, the long threadlike micelles well overlap one another and form a stable gel-like network whose shorttime dynamics is hardly affected by either addition of the salt or thermal agitation. It will be shown below, however, that both CS and T severely affect the long time-scale dynamics of this micelle system in a complicated manner. 1.3. The Slow Mode. The characteristic decay rate T, of the slow mode is plotted against q for four samples with CS/CD= 1, 1.2, 1.5, and 3 a t T = 25 "C in Figure 4. The data confirm that Tsis independent of q and indicate that the slow mode is the relaxation mode. The T, of the other three samples with CS/CD= 0.8,2, and 4 was also found independent ofq. A plot of T,against C ~ C at D three temperatures given in Figure 5 shows that T,is dependent on CS/CDin a very complicated manner. Here solid curves are empirically drawn a s a guide for the eye. At 25 "C, T,first increases with increasing salt concentration, takes a maximum a t CS/CD= 1.2, decreases with CJCD until it takes a minimum at CS/CD = 2, and then increases once again. The unique CS/CD dependence of T, appears to persist a t elevated temperatures of 33 and 40 "C, while the height and the depth at the extremes gradually become smaller. The T, looks to increase monotonically with increasing T a t the same salt concentration. The inverse of T, may be taken as the relaxation time t, which characterizes the slow decay of concentration fluctuation and should be compared with the mechanical relaxation time t~ obtained from the dynamic viscoelastic measurements on the same CTAEVNaSaW system. Similar complicated variation in T, with CD was found for the same CTAEXNaSal system with an equimolar ratio of CTAB and Nasal by Brown et al.ll and our group6 and
DLS of CTABINaSal Micelles '
1
Langmuir, Vol. 11, No. 1, 1995 33 I
I
00
i
0.2-
'
I
1
I
I
I
T
i
A A
T
T2 P
:\
0.1 -
0O0 2 1
00000
0 1
0
0
0 1
1
,
1
,
1
4
2
0
q I I 05cm-1 Figure 4. Plot of the first cumulant rsfrom the slow mode against q for the four sampleswith various salt concentrations. The plot showsthat the slow mode is the relaxation mode. Values 3. for CS/CDare (0)1, ( 0 )1.2,(A) 1.5, and (0)
0
I
I
I
4
2
g i 1Oscm-l Figure 6. Dependence of the relative amplitude of the slow mode A$@, + Af) on the scattering vector for three samples. Solid curves are empiricallydrawn as a guide for the eye. Values 4. for CS/CDare (0)1,(A) 2, and (0) 102
'
' """1
L
1
16'
2
1
cS/c
L
6
0
Figure 6. Dependence of rson CS/CDat three temperatures. Solid curves are empirically drawn as a guide for the eye to show that r. is dependent on CS/CDin a complicted manner. Symbols for T are the same as in Figure 3. lo-'
was discussed in relation to their dynamic viscoelastic behaviors in a previous report.6 Figure 6 shows the scattering vector dependence of the relative amplitude of the slow mode, A$@, Af), for the three samples with C ~ C = D 1, 2, and 4 at 25 "C. An experimental uncertainty for A, was about 0.02. The relative amplitude looks to decrease linearly with a n increase in q for the two samples with C ~ C = D 1and 2. For the sample with C ~ C = D4, it tends to decrease with q at low q but appears to level off a t higher q. It may be also noticed that A$(A, Af) is a decreasing function of the salt concentration. 2. Dynamic Viscoelasticity (DVE). 2.1. Analysis of Dynamic Modulus Data. In Figure 7, we show the angular frequency w dependence of the storage and the loss shear moduli, G ( w ) and G ( w ) , of the eight samples with C&D from 0.8 to 6 a t 25 "C. The G' and G" are proportional to o2and o in the low o region, respectively, which are characteristic of the steady flow behavior.23With a n increase in o,frequency dependences of G and G" of the seven samples with C ~ C from D 0.8 to 4 are characterized by leveling-off of G and a maximum of G . The G appears to increase slightly a t the high w end. Such
+
+
(23)Ferry, J. D.Viscoelastic Properties of Polymers; John Wiley & Sons, Inc.: New York, 1980.
u 100
102
a/rad s-' Figure 7. Angular frequencyo dependenceof the storage and the loss shear moduli, G'(o)and G ( o ) ,of the eight samples at 33 "c. Values for C&D are ( 0 )0.8, (A) 1, (+) 1.2, (0) 1.5, (0) 2, (v)3, (0)4, and ( x ) 6. The plateau modulus GN and the maximum relaxation time ZM are obtained by fitting eq 5 of the Maxwell model to respective G and G data. frequency dependences suggest that the relaxation behavior of the CTABNaSal micelles on the low o side can be described with a Maxwell model with one relaxation time ZM and the plateau modulus G'N
We evaluated 6and t~ by fitting eq 5 to the respective G and G data. We also estimated 6and ZM ofthe sample with the highest salt concentration of C ~ C=D6 from slopes of G and G at the low w end assuming that G and G may be represented as G' = & W ~ Z M ~and G = G$,WNM in
34 Langmuir, Vol. 11, No. 1, 1995
Nemoto et al.
Table 2. Results of DVE Measurements on the CTAB/NaSavW System Cs/Cn 0.8
T=25"C
33°C
43 50 56 54
44 52 59 55 56 58 55 62
1.o 1.2 1.5 2.0 3.0 4.0 6.0
55 57
55 59
GNPa 40°C
ZM/s
50°C
60°C
T=25"C
33°C
40 "C
50 "C
60 "C
48 64 68 68 70 67 72
50 60
4 0.79 0.62 1.02 3.4 1.55 0.32 0.01
1.0 0.185 0.12 1.1 0.47 0.38 0.10 0.005
0.40 0.063 0.035 0.033 0.088 0.10 0.032 0.0027
0.11 0.016 0.008 0.007 0.011 0.015 O.Od7
0.025 0.006
44 56 66 66 59 60 59 65
69
0.0033
1
80
-
I
1
I
2
I
L
I
I
I
I
I
6
CJCD Figure 8. Plot of the plateau modulus (T~T)GN against G ~ C DThe . (TdT)GN takes a constant value of 60 =!c 8 Pa for c s / C ~t 1. Values for Tare ( 0 )25, ( 0 )33, (A) 40, (0) 50, and
V
10-31
(v)60 "C.
the flow region from eq 5. It is to be noted that the 6 value obtained by this procedure may not be equal to the plateau value even if the plateau were observed. The G and G curves a t higher temperatures were also fitted with eq 5 of the Maxwell model. Table 2 gives values of two characteristic quantities, GN and ZM. 2.2. Dependences of GN and CM on C ~ C D and T. Figure 8 shows salt concentration dependence of the plateau modulus 6 a t five temperatures from 25 to 60 "C. Here, small temperature correction of TdT as the reference temperature TO= 25 "C has been applied to & based on the assumption that the plateau of G originates from the nature of rubberlike elasticity of the CTABI NaSaVW network like chemically cross-linked flexible polymers.23 As is clear from the figure, (TdT)& looks to take a constant value of 60 f 8 Pa irrespective of C&D and T in the C&'D range from 1 to 6. The smaller (Td T)& a t C&D = 0.8 may be related to imperfectness of the network as is anticipated from the C&D value less than unity. The molecular weight Me between entanglement points was estimated as 1.95 x lo6from 6using eq 6 where the front factor was put equal to unity.
Me = CmRTIMN Here C, is the mass concentration of the CTAB/NaSal micelle. Using 1.17 x lo4 nm-l as the molecular weight per unit length of the micelle,24we obtain the contour length of the network strand L = 170 nm. A static light scattering study on dilute solutions of the CTABNaSal micelles reports that the persistence length Q ranges from 30 to 50 nm.25 Ifwe would assume that the micellar chains in the semidilute region were as stiff as in the dilute region, the number of the statistical segments between entanglement points, Ll2Q would be 1.7-2.8. This suggests that the network strands possibly take a pretty extended (24) Imae, T. J. Phys. Chem. 1990,94,5953. (25)Imae, T.; Abe, A,; Ikeda, S.J.Phys. Chem. 1988,92,1548
I
I
'
1
I
2
I
i
1
4
1
1
1
6
1
1
810
CdCD Figure 9. Dependence of the mechanical relaxation time TM on C&D. Solid curves are empirically drawn as a guide for the eye. Symbols for T are the same as in Figure 8.
structure and the theory of rubberlike elasticity used above might be inapplicable in a strict sense. The radius of gyration RGof the network strand is also calculable by making use of the well-known formula for the stiff chain26
R',
= (2Q)' (L'l6 - 114
+ 114L' - (11%'
') x
[1- exp(-2L')Il (7) where L' = Ll2Q. We obtained RG= 32-37 nm for Q = 30-50 nm. The ratio of RG to & from the fast mode discussed in the section 1.2 is about 3. This value may be compared with the corresponding digit of 5 obtained for semidilute solutions of polystyrene in benzene.14 The plateau in G is observed in the w range of 10-lo2 rad s-l. On the other hand, even the smallest decay rate Tfto be observed at the lowest scattering angle of 6 = 10.4"is about 3 x lo2s-l, which is larger than the largest w = lo2. Thus, concentration fluctuation in the fast mode decays out, due to the diffusive motion, in a faster time domain in which all network junction points only fluctuate around their equilibrium positions. The results of dynamic viscoelastic measurements may be regarded as an additional evidence for that the fast mode is the gel mode. The salt concentration dependence oft^ is shown in Figure 9. Solid curves in the figure are empirically drawn as a guide for the eye. Since only three reliable data were obtained a t 60 "C, the points are not connected. The data a t four temperatures from 25 to 50 "C show the unique CJCD dependence such as a minimum followed by a maximum, which was extensively discussed as a characteristic property ofthe CTAB/NaSal/W s y ~ t e m .Both ~,~~ (26) Fujita,H.PoZymerSolutions;Elsevier: Amsterdam, 1990;p 140. (27) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987,3, 1081; 1988,4 , 354.
DLS of C T M I Nasal Micelles 10'1
1
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Langmuir, Vol. 11, No. 1, 1995 35 I
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It ? 1001
\\4
4
li \
I
10-31
I
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1I
2
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l
6
l
l
l
810
CdCD Figure 10. Comparison of the characteristic relaxation time t. of the slow mode with t~ at three temperatures of (0) 25, (0) 33, and (A) 40 "C. Filled symbols are for tsand open . curves are the same as empirically drawn ones for t ~ Solid for t~ in Figure 8. the minimum and the maximum appear to shift slightly to the right with a n increase in T , and the difference in ZM between two extremes diminishes with increasing T. Comparison of Ts in Figure 5 with ZM in Figure 9 strongly suggests that the slow mode observed in DLS is closely related to the global mechanical relaxation of the micellar network. The Nasal salt is considered to play two important roles in dynamics of the CTAB/NaSal/W system.6 First, a Salanion and a CTA+cation form a neutral 1:lcomplex, which is utilized as a basic material for formation of a stable threadlike micelle in addition to the intermolecular hydrophobic and electrostatic forces between surfactants. A small portion of Sal- ions, nevertheless, must be present in water under thermodynamical equilibrium and is supposed to make a n exchanging reaction with Sal residues inside the micelles. At low salt concentration close to C s / c , = 1, excess Sal- ions incorporated locally in the micelle deteriorate the micelle structure. With increasing C S ,the Nasal salt in water can also act as a conventional salt like NaBr, which is the second role. The former is very effective a t low CSand may give rise to a n increase in flexibility, whereas the latter as the salt-out effect may stiffen the micelle a t higher CS. The minimum of t~ may be brought by competition of the two opposing effects of the Nasal salt. With further increase in CS,the micelle becomes negatively charged not only by adsorption of Sal- ions on the micelle surface but also by their penetration into the interior of the micelle. Such disturbances by large amounts of excess Sal- ions on the micellar structure finally overcome the salt-out effect and may lead to a maximum of ZM. Then ZM may decrease once again, possibly due to breaking into shorter micelles. The above-mentioned qualitative picture seems applicable for the ZM behavior at elevated temperatures up to 50 "C. A rise of temperature simply loosens the micelle network structure itself. At 60 "C,however, breaking up of long micelles into short ones might induce destruction of the network at higher CS. 3. Comparison of z. and ZM. In a previous section, we notified that the slow decay ofconcentration fluctuation observed in DLS was the relaxation mode and could be characterized by the relaxation time ta(=rS-I).Figure 10 compares tswith the mechanical relaxation time ZM from DVE measurements. As is clear from the figure, good agreement between tsand ZM is obtained in spite of their
very complicated C ~ C dependences. D Therefore we may conclude that the slow mode found in DLS of the CTABI NaSal/W system is not a local but a global one which relaxes accompanied by the slow relaxation process of the viscoelastic network. The phenomenological theory developed by Doi and OnukiZ2explains the physics of the dynamical coupling between concentration fluctuation and elastic stress for the entangled polymer solutions as well as polymer blends. They argue that mass flow or concentration fluctuation in the system creates deformation of the viscoelastic network, which produces the gradient of the elastic stress. Then this gradient affects diffusive motion of polymer chains, resulting in slow decay of concentration fluctuation as the network stress relaxes. They predict that the normalized time correlation function A&) satisfies the following non-Markovian equation
Here, A&) = aA,(t)/at, rsis the decay rate for cooperative diffision without the viscoelastic effect, 7 the solution viscosity, G(t) the shear modulus a t time t, and Cve a characteristic length which determines the strength of dynamical coupling. Equation 8 is essentially a phenomenological equation obtained under two important assumptions that (1)the isotropic part ofthe network stress is zero and (2) the entanglement network is common to all polymers participating in the entanglement and moves with the tube velocity first proposed by Brochard.lg Equation 8 is considerably simplified in a case that the stress relaxation behavior is described by the Maxwell model with the single relaxation time t~ as G(t) = & exp(-tltM) in correspondence to the case of our CTABI Nasal system. Especially, under two conditions of r q t M * 1and r q t M >> q2Eve2,A&) is calculated to be a sum of two exponential functions
A,(t) = A, exp(-r,t)
+ A, exp(-t/tM)
(9)
The second term of the right-hand side of eq 9 accounts for the viscoelastic effect and is of the order of the shear modulus divided by the bulk modulus, thus being very small in a good solvent. Equation 9 correctly explains not only time profiles ofA,(t) shown in Figure 1but also that the fast and the slow modes are the diffusion one (rf= Ts =D& and the relaxation one = ta= t ~ )respectively. , For a more quantitativetest ofthe theory, we calculated A, using = 0.65 ~ ~ z t $for ~ aentangled , homopolymer solutions. The &, is the radius of the so-called blob and may be expressed as C b = 1.36~ in the semidilute regime. Since Ts (=rJhas been shown to be accurately proportional to q2, q2&e2/rqtMbecomes independent of q and is equal to 3.9n CH~&/~BT = 0.2 when (H = 10.5 nm and & = 60 Pa, being independent of CS/CD and T,are used. The theory, thus, gives a constant value of 0.17 as the relative amplitude of the slow mode for samples with any combination of q , C&D, and T. In comparing this prediction with the data shown in Figure 6 , we see that agreement between the theory and the experiment is satisfied only for the sample with C&'D= 1a t low q and cannot explain the q and T dependences of the relative amplitude. On the other hand, the theory gives a reasonable reason why we could not detect the slow mode
eve2
Nemoto et al.
36 Langmuir,Vol. 1 1 , No. 1 , 1995 to the CTABMaSal micellar network with CD= 0.01 M. At C&'D = 1, (H and 6are proportional to cD-0'45 and CD2,0-2,2, respectively.lP6 A decrease in CDfrom 0.1 to 0.01 M lowers q2&e2/r,rM by a factor of 5-7. Such a n extremely small amplitude makes detection of the slow mode very difficult in DLS of the low CD system. From a n experimental point of view, we used the linear correlator for the present DLS measurements and superposed two or three raw data with different sampling times to obtain A&) in Figure 1, which made determination of the relative amplitude ambiguous to some extent. Therefore it seems too early to discuss more about the discrepancy between the theory and the experiment. Fortunately we have recently found the slow mode for semidilute solutions of poly(viny1 a1cohol)horax and are making DLS measurements with a multiple t correlator which covers a broad dynamic range. We will report on results of numerical analysis of DLS data of this system using a general expression of eq 8 quite soon. As described in the Introduction, there is another theoretical idea which may explain the appearance of the viscoelastic mode in the DLS s p e c t r ~ m . ' ~At J ~ present we have no specific volume data of the CTAB/NaSal threadlike micelle, but a rough calculation indicates that there is a small difference in specific volume between the micelle and the solvent. Therefore the Wang theory seems also applicable for our data. As a concluding remark, we would like to touch briefly on molecular models which are proposed for interpretation of the unique dynamical behavior of the CTABMaSaVW
system. One model is a diffusion model of a rigid rodlike micelle in the semidilute solution in which ZM corresponds to the rotational relaxation time of the rod.2 This rod model is semiquantitatively successful in describing translational diffusion and mechanical relaxation behaviors of the network formed at low CDof 0.01 M.5 The other is the nontopological network (NTN) model proposed by Shikata et al.27 In the NTN model, threadlike micelles in the network are so long that diffusion rate of the micelle as a whole is negligiblysmall and displacements of network strands occur by the melting and recombination process of network strands a t junction points. The model takes into account the property, intrinsic of the treadlike micelle, that it is formed by intermolecular forces. Water is regarded as good solvent to the CTABMaSal micelle as implicitly assumed in the rod model. If this conjecture were true, the rod model predicts that the slow mode would be forced unobservable owing to the large bulk modulus. With the melting and recombination process, mechanical relaxation of the network occurs and a t the same time surfactant molecules lose their memories about what chains they had originally belonged to, resulting in decay of concentration fluctuation as the stress relaxes. The dynamical coupling found in the present work seems to favor the NTN model.
Acknowledgment. We are indebted to Professors M. Doi and A. Onuki for helpful discussions. LA940181G
