3470
Langmuir 1994,10, 3470-3476
Rheo-Optical Behavior of Wormlike Micelles Toshiyuki Shikata,**tSam J. Dahman,S and Dale S. Pearson$>§ Department of Macromolecular Science, Faculty of Science, Osaka University, Toyonaka Osaka 560, Japan, and Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, California 93106 Received February 25, 1994. I n Final Form: June 27, 1994@ A cationic surfactant, cetyltrimethylammoniumbromide, CTAB, forms long and stable threadlike or wormlike micelles in aqueous solution with sodium salicylate,Nasal. This system makes an entanglement network similarto concentratedpolymer systemsand showsprofound viscoelasticbehavior highly dependent on concentrationof CTAB, CD,and Nasal, CS,and the system also is strongly flow birefringent. The shear stress, a,,, and the normal stress difference,,a - ,a were determined as functions of shear rates, 7. Flow birefringence measurements were carried out over almost the same shear rate range with an apparatus capable of simultaneously measuring birefringence, An, and orientational angle, x. The data obtained indicate that the refractive index tensor and the stress tensor are linearly related, An sin(2~)/2 = Cu,, and An cos(2x) = C(u, - uyy),by a stress-optical coefficient C that is approximately -3.1 x cm2 dyn-l and is essentially independent of CD and CS.This linearity suggests that the origin of elasticity of the wormlike micellar system is the orientation of micellar portions between entanglement points as well as a Gaussian chain network. The sign and magnitude of C are consistent with the proposed structure in which most CTA+ ions are arranged in a radial pattern with their molecular axis mostly perpendicular to the axis of the wormlike micelle. We also estimated the persistence length of the wormlike micelle as q = 26 nm through flow birefringence data.
Introduction reason for the strong elastic properties like those of polymer systems, whereas the relaxation mechanism in Various kinds of surfactant or detergent molecules form the wormlike micellar system is not controlled by the very long and stable threadlike or wormlike micelles in diffusion process of micelles among themselves, which aqueous systems with or without some occurs in the polymeric systems,11J2 but should be Especially aqueous systems consisting of a cationic surfactant such as cetyltrimethylammonium b r ~ m i d e ~ - ~ controlled by a completely different mechanism which reflects unique characteristics of the dynamics and or cetylpyridinium bromide3and some salts such as sodium structure of the wormlike m i ~ e l l e . ~ > ~ J ~ J ~ salicylate show very unique viscoelastic behavior. The physical origin and essential features of elasticity In those systems, the wormlike micelles sometimes in the micellar system are not fully understood. The first behave like polymer systems; for example, they make very possibility is that portions of the wormlike micelle between condensed entanglement networks,2-8 and the motion of entanglement points have sufficient contour length that entangling micelles detected by dynamic light scattering they can bend or fold rather freely and behave as Gaussian showed gellike behavior just similar to that observed in chains without changing their contour length, as is entangling polymeric s y s t e m ~ . ~ Furthermore, J~ the observed in the polymer system;15J6excess orientation or f a ~ t ~ zthat ~ , the ~ - plateau ~ modulus detected by a dynamic loss of entropy of the micellar portion caused by strain visoelastic measurement of entangling micellar systems generates elastic response. In this case, the refractive is approximately proportional to the square of the index tensor must be proportional to the stress tensor, concentration of surfactant molecules is essentially idenand the so-called stress-optical law15J6must hold in a tical to that observed in polymeric systems.ll On the other rather wide strain range, depending on the length of the hand, the relaxation spectrum in the terminal zone of the entangling micellar portion. entangling micellar system could be well expressed with only one single spikelike s p e c t r ~ m , but ~ ~ that ~ ~ ~in- the ~ Another possibility is that the portion of wormlike entangling polymer system is expressed with a very broad micelle between entanglement points is rather stretched box type spectrum.ll even a t the equilibrium state, and it could be really These similarities and differences mean that the extended, elongating its contour length with a certain entanglement among the wormlike micelles is the essential elastic constant. If this were the main contribution to the stress of the wormlike micellar system, the stress-optical + Osaka University. law would not hold in a wide strain range, because the University of California. refractive index tensor would not be a linear function of * Deceased. the stress tensor in a wide range, and the wormlike micelle Abstract published in Advance A C S Abstracts, September 1, 1994. would be destroyed by a strain larger than a certain critical (1)Gravsholt, S. J. Colloid Interface Sei. 1976,57,575. magnitude. (2)Candau, S.J.;Hirsh, E.; Zana, R.; Delsani, M. Langmuir 1989, Rheo-optical measurements, especially the flow bire5, 1225. (3) Rehage, H.; Hoffhann, H. Mol. Phys. 1989,5,1225. fringence measurement, should be the most powerful and (4)Shikata, T.;Hirata, H.; Kotaka, T. Langmuir 1987,3, 1081. useful technique to examine the relationship between the @
(5)Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988,4 , 354. (6)Shikata, T.;Hirata, H.; Kotaka, T. Langmuir 1989,5,398. (7)Shikata, T.;Hirata, H.; Takatori, E.; Osaki, K. Non-Newtonian Fluid Mech. 1988,28,171. (8)Shikata, T.; Kotaka, T. J.Non-cryst. Solids 1991,131-133,831. (9)Ng, S. C.; Gan, L. M.; Chew, C. H. Colloid Polym. Sci. 1992,270, 64. (10)Nemoto, N.; Kuwahara, M. Langmuir 1993, 9, 419. (11)Ferry, J.D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980.
(12)Doi, M.; Edwards, S. F. The Theory ofPo1ymerDynamic.s;Oxford University Press: Oxford, 1986. (13)Cates, M.E. Macromolecules 1987,20,2289. (14)Cates, M.E.Macromolecules 1988,21,256. (15)Flory, P. J. Statistical Mechanics ofchain Molecules;Interscience Publishers: New York, 1969. (16)Treloar, L.R. G. The Physics of Rubber Elasticity; Clarendon Press: Oxford, 1975.
0743-7463/94/2410-3470~04.50/0 0 1994 American Chemical Society
Rheo-Optical Behavior of Wormlike Micelles
Langmuir, Vol. 10,No. 10, 1994 3471
Table 1. Several Rheological and Structural Parameters for Wormlike Micellar Solutions CDfM
cs*m
?IS
Ye1
Ym
Yc2
Me"
0.01
0.2 0.2 0.05 0.2 0.3
-2 -2 2-3 -2 -0.3
9 6 2 3 2.5
0.1 M decreased with applied strain larger than 4. However, such a remarkable difference was not identified in the linear viscoelastic behavior.' From these, it is likely that mechanical and/or structural features of the wormlike micelle at CS* > 0.1 M are slightly different from those a t CS* < 0.1 M, and the difference could not be detected by viscoelastic measurements and flow birefringence in the linear region, whereas it could be obvious in the nonlinear region.
Discussion The Stress Optical Coefficient. As we have pointed out previously, the stress-optical law holds well in the wormlike micellar system not only under the weak flow but also up to the (&-dependent maximum strain, ym, even under the strong flow, and the stress-optical coefficient,C , could be evaluated as -3.1 x 10-8cm2dyn-l. Rehage and H o h a n n also reported quite similar C values for several kinds of cationic surfactant system^.^ According to the theoretical interpretation of the stress optical-law in polymeric ~ y s t e m s , ~the ~ J validity ~ J ~ of the stress optical-law is based on the Gaussian behavior of component chains; therefore, if the origin of the elasticity of the wormlike micellar system is Gaussian characters of the wormlike micelle, deviation from the stress-optical law under the strong flow condition seen in Figure 6 would mean that statistical and dynamic features ofthe wormlike micelle in this system are no longer approximated by the Gaussian chain statistics. Practically, the relationship between stresses and time (or shear strain) in Figure 6 qualitatively looks like that of a simple free joined chain mode115J6for a polymeric system. The stress predicted with the chain model shows linear response against strain, and the stress-optical law holds in small strain or the Gaussian behavior range of the chain. However, in the large stain range the stress dramatically increases, according to the inverse Langevin function of strain, while the optical component n, = An sin(2~)/2 shows a maximum a t certain strain and decreases to zero a t full stretch state because x approaches to zero, and the stress-optical law no longer holds. These strongly suggest that the origin of the elasticity of the wormlike micellar systems might result from excessive orientation of micellar portions between the entanglement points due to the Gaussian dynamics as well as polymeric systems or rubber. Therefore, we here assume that the wormlike micelle has Gaussian statistics and dynamics in a not so elongated state or under the weak flow condition, and we consider the stress-optical coefficient of the wormlike micelle with a theoretical model based on the idea for polymeric rubber systems. The stress-optical coefficient, C , for a linear flexible free jointed (Gaussian) chain network system had been calculated theoretically as f 0 1 l o w s ~ ~ J ~ J ~
C=
+
2n(n2 212 A a 45nk,T
(3)
where n, Aa, and kg are the mean value of the refractive index, anisotropy of polarizability of an equivalent random walk (Kuhn) segment, and Boltzman's constant, respec-
Shikata et al.
3474 Langmuir, Vol. 10, No. 10, 1994
tively: Aa = al - a2,ai is the polarizability in the direction along the backbone of the chain, and az is that in the direction perpendicular to the backbone. The above equation is calculated under the condition of index matching that the average refractive index of dissolved chains is equal to that of the surrounding medium. Supposing eq 3 is valid in the wormlike micellar system allows the optical anisotropy of the Kuhn segment of the wormlike micelle to be estimated as Aa = -8.5 x cm3from C = -3.1 x loT8dyr-l cm2and n = 1.3 (water). This value is exactly the same as that for the wormlike micelle of cetylpridinium salicylate reported by S ~ h o r r ~ with a technique of electric birefringence. In the Gaussian chain system, the optical anisotropy of Kuhn segments could be determined by both optical anisotropy of a monomer of the flexible chain, Aao, and the equivalent Kuhn length, 1~ = (r2)/L,of the chain, or persistence length, q = (r2)/(2L), as eq 4 shows.15 L is the contour length of the chain, and (r2)1/2 is the mean square end-to-end distance of the chain under the unperturbed state:
aMonomer of the Micelle
a;
PXi5"K:,t
OSnm
Cross Section
CI
x
Figure 7. Schematic representation of a structural model for a~ monomer of the wormlike micelle. ~ ~
L 0.05
where A is the length of the monomer of the chain. Thus, if one could estimate both values of hao and A, the persistance length, q , of the chain would be determined. In the entangling wormlike micellar system, the discussion above should be satisfied by replacing L and (rz)l'z with the contour length, Le,and the mean square distance, (rez)1/2, of the micellar portion between entanglement points along the same micelle, respectively. Monomers and Kuhn Segments of the Wormlike Micelle. Birefringence of the wormlike micellar system, An, could be expressed as a product of the order parameter, S, of micellar monomers and the characteristic birefringence, Ano, of the perfect aligned system. And Ano is proportional to the optical anisotropy of the micellar monomer,23 An = SAn' Ano =
27c(n2
(5)
+ 2)2~,Aa0 9n
where emis the number density of the micellar monomer. Here, we propose a simple structural model for the wormlike micelle. The wormlike micelle in our model is constructed by disk shaped monomers consisting of m CTA+Sal- complexes, as shown schematically in Figure 7. With this simple model we could obtain the relation between Ano and AaO as Ano =
2dn2
+ 2)2NAC,Aa0 9000nm
(7)
where N A is Avogadro's number. Our image of global shape ofthe wormlike micelle is just like the Porod-Kratky wormlike chain model,15which has continuous curvature and is described by only two parameters of contour length and persistence length. When the Porod-Kratky wormlike chain is sufficiently long, the statistical character of the chain could be completely described by the Gaussian statistics again with the equivalent Kuhn length, 1~ = 2q. Now, we are concerned with a critical conditionjust before (22) Schorr, W.Ph.D. Thesis, University of Bayreuth, 1982. (23) Stein, R. S.;Tobolsky, A. V. Textile Res. J . 1948,18, 201
0.1
0.15
OO
(4)
CD
Figure 8. Relationshipbetween the highest birefringence,Ano, at the critical shear strain (see text) and the concentration,CD, of CTAB.
instability occurs under the strong flow condition. Because the system shows x very close to zero a t the condition, the order parameter, S,ofthe micelle should be approximately equal to unity. Thus, the highest value of An a t the condition could be considered as Ano. Figure 5 is one typical example, and we could evaluate Ano around t = 4 s for a solution with CD= 0.03 M. According to these considerations, we could determine AaOlm x -8.0 x cm3 from a slope of the plot between Ano and CD at the critical condition, as shown in Figure 8. The number of the CTA+Sal- complex, m, could be roughly estimated as follows. A drumstick shape object in the right side of Figure 7 represents the CTA+Salcomplex including repulsive interaction radii of each atoms. In this model we assume that CTA+has a n almost fully extended conformation, and this assumption is not contradictory to results of electron microscopic observation,24,25light ~ c a t t e r i n g and , ~ ~ small ~ ~ ~ angle neutron scattering experimentsz8on the wormlike micellar system. Although Sal- anions are located around the place close to ammonium groups of CTA+, keeping the direction of OH and COO- groups toward the water phase, motion of Sal- is rather fast in the m i ~ e l l etherefore, ;~ it is rather hard to estimate the shape of the CTA+Sal- complex correctly. To estimate the height, h , and head size, w (= A), of the CAT'Sal- complex, we referred to Imae's data for spherical micelles of a CTASal:NaSal/W s y ~ t e m , ~ ~ ~ ~ ~ and h * 2.4 nm and w x 0.85 nm were obtained. The circle in Figure 7 represents the cross section of the wormlike micelle, and m could be evaluated as m = 2nhIw x 18. Then, AaO = - 1.4 x cm3was obtained. Finally, we could determine the persistence length as q x 26 nm by means of eq 4 with values of m 18 and ilx 0.85 nm; therefore, the Kuhn length of the micelle could be also (24)Vinson, P. K.;Talmon, Y. J. Colloid Interface Sci. 1989,133, 288.
(25) Clausen,T.M.;Vinson, P.K.;Minter, J. R.; Davis, H.T.;Talman, Y.; Miller, W. G. J . Phys. Chem. 1992,96,474. (26) Imae, T.;Ikeda, S.J . Phys. Chem. 1986,90, 5216. (27) Imae, T. J . Phys. Chem. 1990,94,5953. (28) Herbst, L.;Kalus, J.; Schmelzer, U. J . Phys. Chem. 1993,97, 7774.
Rheo-Optical Behavior of Wormlike Micelles
Langmuir, Vol. 10, No. 10, 1994 3475
Table 2. Summary of the Persistence Length,q, and the Molecular Weight per Unit Length,ML,of Various Wormlike Micelles CD
cs 4
ML
CTAB:NaSal/W
CTASal:NaSal/W
CTAB:NaBr/W
CPB:NaBr/W
0.01-0.1 M 0.03-0.4 M (this study) 26 nm 8900 nm-'
2.24 1 0 - 3 ~ 0.1 M (Imae2') 110-150 nm 11000-12000 nm-'
4.7 x 10-9-8.0 x 10°3M 0.3-0.5 M (Imae et a1.2'3 42-53 nm
6.0 x 1 0 - 3 ~ 0.2-0.8 M (Appe1lZ9) 20f5nm
estimated as about 52 nm. Furthermore, we could estimate molecular weight per unit length of the micelle as M L sz 8900 nm-l with the same m and A. For comparison, we summarized q and M Lfor various kinds of wormlike micelles obtained by different groups a t similar conditions in Table 2. Imae evaluated q and M L from a dilute aqueous solution of CTASal with 0.1 M NaSal with the light scattering t e c h n i q ~ e . ~Agreement ' between Imae's M Land ours is not bad, but Imae's q is considerably larger than ours. There are two ways to explain this difference in q. The first one is that q is strongly dependent on CDand the micelle in our systems is much more flexible than her condition in which the micelle does not form the entanglement network. The other is that the method we used or she used or both of them included serious error in the determining procedure. Imae et a1.26also estimated q for the CTAB:NaBr/W system, and they found 42-53 nm, dependent on the concentrations of both CTAB and NaBr. Appell et al.29 estimated q of a wormlike micelle of cetylpyridinium bromide, CPB, with NaBr by means of light scattering and magnetic birefringence, and they obtained q = 20 f 5 nm. Interestingly, their q is very close to ours. It is well known that the slowest relaxation mechanism of the CTAB:NaSaVW system and systems with a simple salt such as CTAB:NaBr/W or CPB:NaBr/W is somehow different, and Cates' mode113J4for the slowest relaxation mode was attempted to be applied to systems with simple salt^.^,^,^ However, it is not clear whether there is a difference in short time character such as persistence length of the CTAB:NaSaVW and systems with simple salts. It is interesting to note that Appell et al.29 employed a relationship between the mean radius of gyration and the mean hydrodynamic radius of the wormlike micelle to estimate q, and if one applied Imae's data for the CTAB:NaSal systemz7to the relationship, q = 20-30 nm could be obtained. Olsson and his co-workers extensively investigated the nuclear magnetic resonance, NMR, response of aqueous CTASal solutions, and they reported that micelles formed in aqueous solution behaved as rather rigid rods up to a concentration of 1 mM, a t which the long micelles started entanglement; however, the persistence length of the micelles was dramatically reduced by adding NaSaL30 Since our systems contain a lot of NaSal, the fact that the persistence length of our wormlike micelles is rather short does not contradict their conclusion. Some electron micrograph^^^^^^^^ of the wormlike micelles of the CTAB: NaSaYW system present rather extended images of the micelles, and they make one imagine a very long persistence length for the wormlike micelles. The image must show the real cylindrical structure of the wormlike micelles; however, it would not describe real conformation of the wormlike micelles in solution even a t a very low (29)Appell, J.; Porte, G. J. Colloid Interface Sci. 1982, 87,492. (30) Olsson, U.; Soderman, 0.;Guering, P. J.Phys. Chem. 1986,90, 2523. (31) Shikata, T.; Sakaiguchi, Y.; Urakami, H.; Tamura, A.; Hirata, H. J . Colloid Interface Sci. 1987, 119, 291. (32) Vinson, P. K.; Talmon, Y. J. Colloid Interface Sci. 1989, 133, 288.
concentration, and there must be many artificial factors in the procedure of making samples, drying them, and staining them. Thus, one must be very careful when any conformational information of the wormlike micelles is taken from the electron micrograph^.^^^^^ Estimation of Anisotropy of the Monomer. One can estimate the monomeric optical anisotropy, AaO, through the structural model we proposed in Figure 7. In that procedure, one needs information about polarizabilities, bl and bz, which are respectively a parallel component and a perpendicular component to the molecular axis of CAT+. As we mentioned previously, motion of Sal- in the micelle is so quick that the contribution of Sal- to optical anisotropy could be ignored. According to NMR experimental results by Soderman et al.,34the motion of three to four carbon atoms in the terminal portion of a methylene chain of CTA+ is rather quick, and rotation of methyl groups connected to a nitrogen atom is also fast. Moreover, anisotropy of polarizability of N-C and C-H bonds is very little,34so that we could ignore the entire contribution of the trimethylammonium group to the anisotropy. Thus, we might approximately regard CTA+ as a paraffin chain consisting of 12 methylenes from the viewpoint of anisotropy of polarizability. Since polarizabilities of a methylene unit could be quoted from the l i t e r a t ~ r ewe ,~~ could approximately estimate bl and bz of CTA+ in the cm3 and bz = 21.6 wormlike micelle as bl = 23.4 x x loTz4cm3. Furthermore, we could calculate aIoand azoof the monomer of the micelle in this manner:
alo= S,mb,
(8)
m
a: = S,{b,C cos[2@jlmI + j=l m
bzCC O S [ ( ~ C-/ ~( 2) 7 ~ j h ) I )(9) j=l
where S, is the order parameter of the backbone of methylene units and m is the number of CTA+Salcomplexes in the monomer disk. NMR relaxation experiment^^^ give direct information about the order parameter, S,, of methylenes in the micellar state and also in the liquid crystalline state. The magnitude ofthe order parameter of C-H (or C-D) bonds of methylene groups from a or the first position connected to the ammonium group up to the last third or fourth RZ -0.2, whereas those of position was reported a s SC-H last four to the end methyl decrease very rapidly, and (33) Clausen,T. M.;Vinson,P. K.; Minter, J. R.; Davis, H. T.; Talmon, Y.; Miller, W. G. J. Phys. Chem. 1992, 96,474. (34) Soderman,0.;Walderhaun, H.;Henriksson,U.;Stilb,P. J.Phys. Chem. 1986, 89,3693. In their original paper, the sign of the order parameter is positive; however, the average orientation of the C-D or C-H bond should be essentially perpendicular to the local director taken to be normal to the surface of the micelle, and the sign should the minus sign. This is discussed in be negative. Thus, we accepted detairin refs 37 and 38. (35) LeFevre, R. J. W. Adu. Phys. Org. Chem. 1965, 3 , 1. (36) Saundens, D. W. Trans. Faraday SOC.1957, 53, 860.
Shikata et al.
3476 Langmuir, Vol. 10, No. 10, 1994 these features are not so d e c t e d by size of micelles and the state of the phase.34 One can convert SC--Hto S, of the backbone of the methylene in this way:37,38S, = -2Sc-~. Thus, proposing m = 18 and S , = 0.4, we could cm3. This value estimate A a O= a10- azo= -5.0 x is negative, and its magnitude is rather close to our cm3). The minus experimental value ( A a o = -1.4 x sign in the calculated haoresults from a structural feature in the model in which CTA+ions are arranged in a radial pattern which their molecular axis perpendicular to the axis of the wormlike micelle. The estimation of S, above is based on the time scale ~ ) , and 3~ ofNMR experiments (typically shorter than there is still a problem whether one can accept the S, obtained by NMR relaxation as a n adequate one for optical anisotropy in our experimental time range. Rehage and Hoffmann also discussed the stress-optical coefficient, and they assumed a much smaller S, = 0.OEL3 One would need more precise information about the motion of all components in the wormlike micelle to improve the procedure for theoretical estimation of AaO. Although the above estimating procedure for AaO is quite rough and simple, the difference between calculated and experimental A a o is not bad; therefore, we might conclude that our experimental AaO value is acceptable for the optical anisotropy of the micellar monomer, and the evaluated persistence length, q , is also valid for the wormlike micellar system. The Full Stretch Condition of the Wormlike Micelle. We could also apply the idea of rubber elasticity12J5J6to the wormlike micellar system if the assumption of the Gaussian chain dynamics for the wormlike micelle were valid. Thus, we could estimate the molecular weight of the micellar portion between entanglement points, Me, and the average spacing, Ee, between entanglement points by use of well-known equations (eqs 10 and 11)in polymer rheo1ogy:l'
M e = -cRT
(10)
GNO
kBT 113
=
(2)
where c , R , and &O are the weight concentration of the micelle in g ~ m -the ~ ,gas constant (RT = NAkBT), and the plateau modulus in the high-frequency region, respectively. We estimated Me,Le = MJML,,and Ee a s a function of CDwith &O data in our previous papel.4 (Table 1). (37)Seeling, A.;Seeling, J. Biochemistry 1974,13, 4839. (38)Seeling,J.;Nieberberger,W. J.Am. Chem. SOC.1974,96,2069.
Characteristic behavior of the wormlike micelle a t the full stretch condition could be considered as follows. At the equilibrium condition, the average spacing between entanglements is te,while a t the full stretch condition the distance between the entanglement points should be Le; therefore, the ratio LJE, should mean the maximum elongation ratio of the wormlike micelle without any change in micellar structure except for conformational change. On the other hand, the full stretch condition of the micelle could be realized a t the second strain, yc2,just before occurrence of instability under the strong flow condition, and the maximum elongation ratio might be roughly estimated a s yc2, if slippage between micelles a t entanglement points never happened. Comparison between the ratio LJteand yc2is also shown in Table 1. The solution with CD= 0.1 and CS*= 0.05 M shows fair agreement between the ratio LdEe and yc2, whereas in other solutions yc2 is always larger than ther ratio LJEeby factor of 2-3. As we pointed out previously, there is some specific difference in micelles of solutions with CS* < 0.1 and CS* > 0.1 M. The difference would affect the strength of the micellar structure and also occurrence of the slippage a t the entanglement points. The wormlike micelles a t CS* < 0.1 M are not likely to experience slippage, but those a t CS* > 0.1 M are. At last, we must emphasize again that essential dynamic features of the wormlike micelle are well described by Gaussian chain dynamics in small strain and short-time range, and the wormlike micelle is rather flexible because the ratio ofpersistance length to the diameter of the micelle is only 5.
Conclusions We clarified that in an entangling wormlike micellar system the stress-optical law held well under the weak flow condition, and the stress-optical coeficient was C = -3.1 x cm2 dyn-'. However, the stress-optical law no longer held under the strong flow condition. These suggest that the origin of the elasticity of the wormlike micellar system under the weak flow condition is the same as for the polymer liquid; entropic elasticity resulted from Gaussian behavior ofthe wormlike micelle in a small strain range. We could estimate the optical anisotropy ofthe monomer cm3from birefringence of the micelle as A a o = -1.4 x data, and we also could evaluate the persistence length of the wormlike micelle as q = 26 nm through flow birefringence data. We proposed a n adequate structural model of a micellar monomer consisting of 18 CTA+Sal- complexes. The optical anisotropy of the monomer could be calculated with the proposed structuralmodel, and the sign and magnitude were very consistent with the experimentally obtained ones.
