pubs.acs.org/Langmuir © 2009 American Chemical Society
Transient Network Theory of Wormlike Micelles: Topological Force Accelerates Relaxation Fumihiko Tanaka* Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Katsura, Kyoto 615-8510, Japan Received October 7, 2009. Revised Manuscript Received November 15, 2009 A new transient network theory is developed to study the rheological properties of entangled wormlike micelles. The dynamic mechanical moduli are calculated on the basis of the model network in which local structure of the wormlike micelles is represented by the entangled loops. These loops interact with each other by statistical force due to the topological constraints. They are assumed to pass through each other within a finite time to release the internal stress (phantom chain-crossing model). The high frequency plateau modulus G¥ and the relaxation time τ as functions of the micellar concentration are calculated and compared with the experimental data on the aqueous solutions of cetyltrimethylammonium bromide (CTAB) mixed with ionic aromatic compound sodium salicylate (NaSal). It turns out that creation and annihilation of the entanglements are strongly coupled to the topological force. The rheological relaxation time decreases with the concentration of the micelles because the average radius (correlation length ξ) of the entangled loops, and their contour length L decrease with the CTAB concentration. Hence, the topological force is amplified by the increase in the concentration and, as a result, accelerates the stress relaxation by chain-crossing.
1. Introduction It is well-known that certain cationic surfactants form long flexible wormlike micelles in aqueous solutions when salts are mixed. Typical example are cetylpyridinium bromide (CPyB) or cetylpyridinium cloride (CPyC)1-4 with added NaBr, NaCl, cetyltrimethylammonium bromide (CTAB),5-11 or cetyltrimethylammonium cloride (CTAC)12 mixed with ionic aromatic compounds such as sodium salicylate (NaSal) or phthalimide potassium salt (PhIK). In the latter systems, one-to-one complexes of CTAB (or CTAC) and NaSal form long cylindrical micelles of uniform diameter of about 5-10 nm.5,12 Solutions of wormlike micelles of cetyltrimetylammonium tosylate (CTAT) in water have also been studied.13,14 Even at surfactant concentrations c as low as 10-2 mol L-1, the threadlike micelles grow long and are mutually entangled, so that the solutions show characteristic viscoelastic properties. Although entangled threadlike micelles resemble concentrated polymer solutions with high viscoelasticity in which polymer chains are entangled, the relaxation mechanism of CTAB/NaSal/H2O system is quite different from that of polymer solutions.6-9 For instance, the micellar solutions behave like a Maxwell fluid with a single relaxation time in a wide concentration region. *Corresponding author. (1) Porte, G.; Appell, J.; Poggi, Y. J. Phys. Chem. 1980, 84, 3105. (2) Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511. (3) Appell, J.; Porte, G. J. Colloid Interface Sci. 1981, 81, 85. (4) Appell, J.; Porte, G.; Poggi, Y. J. Colloid Interface Sci. 1981, 87, 492. (5) Sakaguchi, Y.; Shikata, T.; Urakami, H.; Tamura, A.; Hirata, H. J. Electron Microsc. 1987, 36, 168. (6) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081. (7) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 354. (8) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1989, 5, 398. (9) Shikata, T.; Hirata, H.; Kotaka, T. J. Phys. Chem. 1990, 94, 3702. (10) Wheeler, E. K.; Izu, P.; Fuller, G. G. Rheol. Acta 1996, 35, 139. (11) Inoue, T.; Inoue, Y.; Watanabe, H. Langmuir 2005, 21, 1201. (12) Clausen, T. M.; Vinson, P. K.; Minter, J. R.; Davis, H. T.; Talmon, Y.; Miller, W. G. J. Phys. Chem. 1992, 96, 474. (13) Soltero, J. F. A.; Puig, J. E.; Manero, O. Langmuir 1996, 12, 2654. (14) Soltero, J. F. A.; Bautista, F.; Puig, J. E.; Manero, O. Langmuir 1999, 15, 1604.
5374 DOI: 10.1021/la903790g
The relaxation time of the stoichiometric mixture of CTAB and NaSal with the ratio 1:1 decreases with the concentration instead of increasing.6,7 It also depends sensitively on the concentration cs* of the free NaSal remaining in the solutions under addition of excessive NaSal.7 To explain these anomalous viscoelastic behavior, Shikata7 proposed a quasi-network theory in which the threadlike micelles may dissipate stress by annihilation/creation of the entanglements through mutual chain passing like phantom chains,and the free NaSal ions work as catalyst for the scission reaction for the chain passing. Other attempt to elucidate the static and linear viscoelastic properties has been done on the basis of the idea of reptation of living polymers.15 The threadlike micelles are assumed to reptate under reversible scission/recombination reactions.It was shown that if the reaction time τbreak for the scission of a chain (assumed to be proportional to the reciprocal mean length L of the micelles) and for the recombination (assumed to be the same order of τbreak) are both shorter than the reptation time τrep, the stress relaxation obeys a single exponential form with the decay time τ = (τbreakτrep)1/2. As far as the concentration dependence is concerned, the terminal relaxation time obeys a scaling law τ ∼φb (b = 3/2) if the semiempirical relation τrep ∼ L3φ2 is employed. The thermal and rhelogical properties are reviewed by Cates.15 The experiments on CTAB/NaSal/H2O with 1:1 molar ratio, however, showed that the relaxation time decreases with the concentration in the range c = 10-2-10-1 mol L-1.6,7 Also, DLS study of cetylpyridium chrorate (CPClO3) with added sodium chlorate (NaClO3) showed the anomalous decrease of the relaxation time in the semiconcentrated region under a fixed high salt concentration ([NaClO3]=1 M).16 In other systems such as CTAT in water,13 the relaxation time as a function of the concentration first shows a peak and then decreases. The curve shifts to lower value keeping the same shape. Similar (15) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (16) Buhler, E.; Munch, J. P.; Candau, S. J. Europhys. Lett. 1996, 34, 251.
Published on Web 12/07/2009
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Figure 1. Model network of entangled loops as building blocks representing the solutions of wormlike micelles. The contour length L of each loop is related to the correlation length ξ of the solution through the relation ξ2 = L.
nonmonotonic behavior of the relaxation time as a function of the surfactant concentration was also observed in nonionic surfactants CnH2nþ1(OC2H4)mOH (referred to as CnEm) in wtaer.17 It was pointed out that these surfactants-salt mixtures can roughly be classified in three categories:18 highly hydrophobic surfactants with low counterion binding, less hydrophobic surfactants with enhanced counterion binding, and moderately hydrophobic surfactants with strong counterion binding. In the first two categories, reptation-reversible scission mechanism takes place for the stress relaxation. In the second category, reptation model for multiply connected networks proposed by Lequeux19 may be applicable. But in the third category, the networks are saturated, and hence reptation is impossible. To explain the anomalous decrease of the relaxation time, and the concentration dependence of the plateau modulus, we refine in this paper Shikata’s idea of phantom chain passing in terms of the transient network theory by which we can actually quantitatively evaluate the linear response, nonlinear viscosity, strain hardening, stress overshoot, etc. We specifically focus on the semiconcentrated region (c=10-2-10-1 mol L-1) of the aqueous solutions of CTAB/NaSal with the stoichiometric molar ratio, where counterions are strongly bound to the surfactant ions. The average length of the cylindrical wormlike micelles is practically infinite. Micelles are mutually entangled, but have no junctions. We introduce the creation and annihilation rates of the entanglements, which are coupled with the topological force caused by the entanglements. We show that the chain passing is enhanced by the topological force if the coupling is sufficiently strong, and hence the stress relaxation is accelerated.
2. Transient Network Theory of Wormlike Micelles The left figure in Figure 1 shows a TEM photograph5 of aqueous solutions of cetyltrimethylammonium bromide (CTAB) (concentration c = 0.02 mol L-1) mixed with sodium salicylate (NaSal) at the stoichiometric composition of 1:1. The micelles form long threadlike entangled strings with about 20 nm persistence length. The diameter a of the micellar cylinder is about 5-10 nm, while the mean mesh size ξ (correlation length) of the network is 100 nm at this concentration. Micelles are so long that the end part of the cylinders cannot be found. The micellar chains are mutually entangled but there are no direct cross-links. These fibrillar networks are subjected to random thermal motion in the solution, and flow under external force like other common viscoelastic liquids. To describe such a network with many localized entanglements without chain ends, we introduce a model network consisting of (17) Kato, T.; Nozu, D. J. Mol. Liq. 2001, 90, 167. (18) Oelschlaeger, Cl.; Waton, G.; Candau, S. J. Langmuir 2003, 19, 10495. (19) Lequeux, F. Europhys. Lett. 1992, 19, 675.
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Figure 2. Possible paths of chain reaction. In the chain passing, a pair of entangled loops is separated by chain passing. In the chain exchange, a pair of loops is merged to form a larger loop. In the chain merger, separated loops merge into a double loop. All reactions take place through a intermediate excited state where micellar threads merge in forming a cross-link.
flexible loops made up of the cylindrical micelles with dimeter a and contour length L (Figure 1 right). Micelles do not form loops in the actual solution, but loops are considered as the building blocks of the networks, like the concentration blobs in polymer solutions,20 which are under so strong topological constraints that cooperative movement is allowed only within the scale of the correlation length. Representation of the network in terms of such closed loops enables to classify the local topological relations of the micelles rigorously. The contour length L of a loop is adjusted to fit the correlation length ξ of the actual solution. To do this, the relation ξ = Lν between the mean radius of gyration and the total length of the loop is assumed where ν=3/5 is Flory’s exponent of the excluded volume effect. Practically, in the present primitive stage of modeling, we neglect the excluded volume effect, and use the index ν=1/2 for the Gaussian chains. Because it is known from many studies that the solutions behave like a Maxwell fluid with a single relaxation time τ, the loops are allowed to pass through each other within a finite lifetime through the process of merging and separating of the micellar segments. To see how the topological relations of the loops change under external force (or deformation) and/or thermal motion, we arbitrarily pick up a pair of loops in the system and see whether they are entangled or separated. The topological relation of the pair may be classified by the topological invariants such as Gauss linking number, Alexander polynomials, Jones polynomials etc.21 To simplify the problem, we consider Gauss linking number G only throughout this paper.22,23 The entangled states have nonzero linking number |G|=1, 2, ..., while disentangled states have G=0. There are three possible reaction paths caused in the chain reactions: chain passing, chain exchange, and chain merger (Figure 2). Chain passing is the process in which an entangled pair is separated, or vice versa, by mutual passing of the chain segments after contact. Chain exchange is the process in which an entangled pair merges at the contact point to form a new loop of double size and vice versa. Chain merger is the process in which a separated pair merges at the contact point to form a new loop (20) de Gennes, P. G. Scaling Concept in Polymer Physics; Cornell Univ. Press: Ithaca, NY, and London, 1979; Chapter 3. (21) Vologodskii, A. V.; Lukashin, A. V.; Frank-Kamenetskii, M. D. Sov. Phys. JETP 1975, 40, 932. (22) Tanaka, F. Prog. Theor. Phys. 1982, 68, 148; 164. (23) Tanaka, F. J. Chem. Phys. 1987, 87, 4201.
DOI: 10.1021/la903790g
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Taking the sum of the coupled eqs 2.1, we find the chain conservation law DFðr, tÞ þ r 3 ðvðtÞFðr, tÞÞ ¼ 0 Dt Figure 3. Transition between entangled and unentangled states. The vector connecting the centers of mass of the two loops is r. The reaction rates are designated as β(r) for separation, and R(r) for linking.
of double size. All these processes are accompanied by the intermediate excited state with a contact point during the transition. When a micellar thread collides with another one in the entangled network, it can form a transient cross-link, and two threads either separate on the other side or exchange segments. Such ghostlike crossing for stress relaxation was first proposed by Shikata7,8 for CTAB and then examined by Appell et al.24 for other micelles CPyClO3. In this paper, we focus on the chain passing only and neglect the formation of double loops. Let ψe (r, t) ΣG6¼0ψG (r, t) be the number of entangled loop pairs in the unit volume of the solution whose centers of masses are specified by the vector r, and let ψ0 (r, t) be the number of separated loops whose centers of masses are specified by the vector r (Figure 3). Our transient network model starts with the time-development equations Dψe ðr; tÞ þ r 3 ðve ðtÞψe ðr; tÞÞ ¼ -βðrÞψe ðr; tÞ þ RðrÞψ0 ðr; tÞ Dt ð2:1aÞ Dψ0 ðr; tÞ þ r 3 ðv0 ðtÞψ0 ðr; tÞÞ ¼ βðrÞψe ðr; tÞ -RðrÞψ0 ðr; tÞ Dt ð2:1bÞ where the right-hand sides of the equations describe the transitions between entangled and distangled states; β(r) is the probability per unit time for an entangled pair with the chain vector r to be separated into disentangled ones with the same r, and R(r) is the probability per unit time for a separated pair with the chain vector r to be entangled. The second term (convection term) on the left-hand side in each eq represents the pair under consideration is deformed by the external force. For simplicity, the chain vector r of an entangled pair is assumed to be deformed affinely to the time-dependent macroscopic deformation tensor λˆ (t), so that we have the relation26,27 ve(t)=v(t), where ! d ^λ ^-1 λ vðtÞ 3r dt 3
ð2:2Þ
The interaction with other loops in the vicinity is taken into account as the affine deformation assumption. As for the chain vector r of a separated pair, it is not clear whether it deforms affinely or not, but we follow the study on the elasticity of entangled rubber networks by Graessley and Pearson25 (referred to as GP), and assume that it also obeys v0 =v(t) due to the strong constraints by other surrounding loops which are entangled with each loop of the pair. (24) Appell, J.; Porte, G.; Khatory, A.; Kern, F.; Candau, S. J. J. Phys. II Fr. 1992, 2, 1045. (25) Graessley, W. W.; Pearson, D. S. J. Chem. Phys. 1977, 66, 3363. (26) Tanaka, F.; Edwards, S. F. Macromolecules 1992, 25, 1516. (27) Tanaka, F.; Edwards, S. F. J. Non-Newtonian Fluid Mech. 1992, 43, 247; 272; 289.
5376 DOI: 10.1021/la903790g
ð2:3Þ
if ve =v0 v, where Fðr, tÞ ψe ðr, tÞ þ ψ0 ðr, tÞ
ð2:4Þ
is the total number of loop pairs whose separation vector is specified by r. For a uniform distribution of loops, F(r, t) is a constant F. Since F is the number density of loop pairs of the length L, it is proportional to F ∼ ðc=LÞ2 ∼ c2 ξ -2=ν
ð2:5Þ
where c is the micellar concentration. In the extreme case where there is no transition between the two state, and hence R(r) = β(r) = 0, the entire problem reduces to GP.25 In GP, the affine deformation of the topologically classified loops was assumed to calculate the chain entropy due to the entanglement constraints. The stress-strain curve was calculated and the initial Young modulus was estimated. In the present study, we introduce finite transition probability between the loops of different topology to allow the network to relax to the equilibrium state, and also to allow it flow under external forces. Therefore, our model can be regarded as the transient network counterpart of GP. Let us consider next the case opposite to GP where there is reaction, but no convection. Then, we have a chemical reaction kinetics Dψe ðr; tÞ ¼ -βðrÞψe ðr; tÞ þ RðrÞψ0 ðr; tÞ Dt
ð2:6aÞ
Dψ0 ðr; tÞ ¼ βðrÞψe ðr; tÞ -RðrÞψ0 ðr; tÞ Dt
ð2:6bÞ
The relaxation time τ by this chemical reaction is given by τðrÞ ¼ 1=ðRðrÞ þ βðrÞÞ ¼ 1=βðrÞð1 þ KðrÞÞ
ð2:7Þ
for a loop pair separated by the distance r, where KðrÞ RðrÞ=βðrÞ
ð2:8Þ
In more complete analysis given below, we show that this form is combined with the topological force and averaged over the probability of entanglements to find the rheological relaxation time. At this stage, we can see easily that our present model is a special case of the two-state transient network model in which chains take either A-state or B-state. The reaction between them AðrÞ h BðrÞ
ð2:9Þ
is allowed. The time development of the number of chains obeys DψA ðr; tÞ þ r 3 ðvA ðtÞψA ðr; tÞÞ ¼ -βðrÞψA ðr; tÞ þ RðrÞψB ðr; tÞ Dt ð2:10aÞ DψB ðr; tÞ þ r 3 ðvB ðtÞψB ðr; tÞÞ Dt ¼ βðrÞψA ðr; tÞ -RðrÞψB ðr; tÞ
ð2:10bÞ
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The A-state in the present loop problem is the entangled state, while the B-state is the separated one. In our previous transient network theory26-29 developed for the water-soluble telechelic polymers carrying hydrophobic short chains at the chain ends, the A-state corresponds to the bridge chain connecting the micellar junctions, while B-state is the dangling chain whose one end is separated from the junction and free to move. In the affine network theory of such telechelic polymers ! d ^λ ^-1 λ vA ¼ 3r dt 3
KðrÞ ¼ Φe ðrÞ=Φ0 ðrÞ
3. Equilibrium Distribution Function and the Detailed Balance Condition Let Φe(r) be the fraction of entangled pairs (the probability for a pair of loops placed at the separation vector r to be entangled), and let Φ0(r) = 1-Φe(r) be that of the separated pairs. They depend on the distance r between the mass centers of the pairs. If we assume the uniform spatial distribution of the loops, the equilibrium distribution functions are given by ð3:1Þ
for i=e, 0, where F is the total number of loop pairs, and Z
r
Φi ðrÞ ¼ Ci exp½ -
ðf i =kB TÞ dr
RðrÞ ¼ KðrÞβðrÞ
ð3:7Þ
once the dissociation rate β(r) is found, where
as in 2.2, and vB=0. The special case of the constant β was studied by Green and Tobolsky31 and Lodge32 and is referred to as the GT limit. The effect of the chain stretching was studied by Yamamoto33 by introducing the r-dependence of the decay rate.
ψoi ðrÞ ¼ FΦi ðrÞ
must be fulfilled. Because we can find the equilibrium distribution functions Φi(r) on the basis of the equilibrium statistical mechanics, we regard this condition to be given to the kinetic coefficients R and β. The recombination rate should then be given by 2.8, or
ð3:2Þ
ð3:8Þ
is the ratio of the equilibrium distribution functions. This ratio gives the equilibrium constant of the reaction 2.9.
4. Dynamic Mechanical Moduli To study linear response of the network under small oscillatory shear flow along x axis, we consider the deformation tensor 2
3 1 εeiωt 0 ^λðtÞ ¼ 4 0 1 05 0 0 1
ð4:1Þ
where ω is the frequency of oscillation, and the amplitude ε of the oscillation is assumed to be small. The affine velocity vector 2.2 is then given by 2
3 iεωyeiωt 5 vðtÞ ¼ 4 0 0
ð4:2Þ
Substituting this form into 2.1, we find
0
are their equilibrium profiles. The front factor Ci are the normalization constants, and f i =kB T -r ln Φi ðrÞ
ð3:3Þ
are the tensions due to the change in the distance of the pair in the state i. At a given position vector r, the chain takes either entangled state or disentangled state. Therefore, we have the identity Φe ðrÞ þ Φ0 ðrÞ ¼ 1
ð3:4Þ
as in GP, and the relation f e ðrÞΦe ðrÞ þ f 0 ðrÞΦ0 ðrÞ ¼ 0
ð3:5Þ
holds for the forces. The normalization constants Ci are decided to fulfill the relation 3.4. In order to ensure the existence of the equilibrium state, the detailed balance condition βðrÞΦe ðrÞ ¼ RðrÞΦ0 ðrÞ
and similar equation for ψ0 (r, t). To linearize these equations, we introduce the deviations ηi (r,t) from the equilibrium state by ψi ðr, tÞ ψoi ðrÞ½1 þ εηi ðr, tÞ
ð4:4Þ
for i = e,0, where ψi(r) is the equilibrium distribution function. On substitution, we confirm 3.1 from the 0-th order equation, and obtain Dηi ðr, tÞ - iεωyeiωt fi, x =kB T ¼ -½RðrÞ þ βðrÞηi ðr, tÞ Dt
ð4:5Þ
from the first order equation, where fi, x ðrÞ=kB T -
ð3:6Þ
(28) Tanaka, F.; Koga, T. Macromolecules 2006, 39, 5913. (29) Koga, T.; Tanaka, F.; Kaneda, I.; Winnik, F. M. Langmuir 2009, 25, 8626. (30) Candau, S. J.; Hirsch, E.; Zana, R.; Delsanti, M. Langmuir 1989, 2, 1225. (31) Green, M. S.; Tobolsky, A. V. J. Chem. Phys. 1946, 14, 80. (32) Lodge, A. S. Trans. Faraday Soc 1956, 52, 120. (33) Yamamoto, M. J. Phys. Soc. Jpn. 1956, 11, 413. Yamamoto, M. J. Phys. Soc. Jpn. 1957, 12, 1148. Yamamoto, M. J. Phys. Soc. Jpn. 1957, 13, 1200.
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Dψe ðr, tÞ Dψ ðr, tÞ þ iεωyeiωt e ¼ -½RðrÞ þ βðrÞψe ðr, tÞ þ RðrÞF Dt Dx ð4:3Þ
D½ln Φi ðrÞ Dx
ð4:6Þ
are the x component of the forces 3.3. To find the stationary oscillatory solution, we assume the form ηi (r, t) = Qi(r)eiωt. Substituting into the above equations and solving for the amplitudes Qi, we find ηi ðr, tÞ ¼
iωyfi, x eiωt kB T½RðrÞ þ βðrÞ þ iω DOI: 10.1021/la903790g
ð4:7Þ 5377
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The (i,j) component of the stress can be calculated by the relation26 Z σ i, j ðtÞa3 ¼
dr ½ðxi fe, j Þψe ðr, tÞ þ ðxi f0, j Þψ0 ðr, tÞ
ð4:8Þ
invariant, and discard all other invariants in hoping that it effectively capture the essential feature of the topological entanglements. The equilibrium distribution for the entangled loops with G1,2= m can then be found by Φm ðrÞ ÆδðG1, 2 -mÞæ
For the linear response, we have Z σ x, y ðtÞa3 ¼ εF
dr ½ðxfe, y ÞΦe ðrÞηe ðr, tÞ þ ðxf0, y ÞΦ0 ðrÞη0 ðr, tÞ ð4:9Þ
for m = 0, (1, (2, ..., where Æ...æ indicates the average over all conformations of the loops separated by the vector r.22 For the entangled pair, we have Φe ðrÞ ¼
The complex modulus is defined by the response to the deformation λx,y =εeiωt in the form σ x, y ðtÞa3 ¼ εGðωÞeiωt
ð4:10Þ
Substituting 4.8 into the stress 4.9, and comparing with the definition of modulus, we find
Z
dr Φe ðrÞðxfe, y Þ
GðωÞ ¼ F
½1 þ KðrÞiω ðyfe, x =kB TÞ RðrÞ þ βðrÞ þ iω
G0 ðωÞ ¼ F
"
Z
½1 þ KðrÞω
dr Φe ðrÞðxfe, y Þ ðyfe, x =kB TÞ ω2 þ ½RðrÞþβðrÞ2
Φ0 ðrÞ ¼ 1 -Φe ðrÞ
Z
#
½1þKðrÞ2 βω dr Φe ðrÞðxfe, y Þ ðyfe, x =kB TÞ ω2 þ ½RðrÞþβðrÞ2
ð4:13Þ for the loss modulus. The relation 3.5 has been used to eliminate Φ0(r) and f0 in favor of Φe(r) and fe.
5. Entanglement Probability and the Topological Force
σðrÞ Æm2 æ1=2
Z Z
r10 ðs1 Þ r20 ðs2 Þ 3 ðr1 ðs1 Þ -r2 ðs2 ÞÞ jr1 ðs1 Þ -r2 ðs2 Þj3
2
σðrÞ ¼ σ 0 e -r =ξ 2
ð5:6Þ
σ0 ¼ C1 ðL=aÞ1=2 = C1 ðξ=aÞ
ð5:7Þ
The microscopic length scale a is the size of the repeat unit of a chain, and it corresponds to the diameter of the micellar cylinder in the present study. The numerical factor C1 in the amplitude σ0 is of order unity, and depends on the lower cutoff of the length scale which is introduced to avoid unphysical divergence of the integral.22 We found that C1 =0.75-0.80 gave a good fit of the experimental second virial coefficient of the solutions of cyclic polystyrene.23 The entanglement probability distribution can be described by 2 1 2 Φm ðrÞ ¼ pffiffiffiffiffiffi e -m =2σðrÞ 2πσðrÞ
ð5:8Þ
for m 6¼ 0, and hence ds1 ds2 ð5:1Þ
where ri (si) (0 e si e L) is the position vector of the loop i at the contour length si, ri0 (si) is its derivative. Since the Gauss linking number takes integer value 0, ( 1, ( 2, ... and is a topological invariant, we can classify the topological entanglements of a pair of loops. However, it is not sufficient by itself to perfectly specify the topological classes. For instance, the class with G=0 is supposed to give disentangled loops, but is known to include not only separated loops but also some entangled loops. The class with finite G1,2 = m also includes several different types of entanglements. We take, however, G as the major 5378 DOI: 10.1021/la903790g
ð5:5Þ
This is the mean square linking number for a pair of loops with the contour length L whose centers of mass are placed at the fixed separation vector r. If we assume the chains to be Gaussian, it is explicitly given by
To find the equilibrium distribution for the specified topology of the loops, let us consider Gauss linking number 1 G1, 2 4π
ð5:4Þ
where ξ2 =4Æs2æ/3 (Æs2æ is the radius of gyration of a loop), and
for the storage modulus, and taking the imaginary part, G00 ðωÞ ¼ F
ð5:3Þ
In our previous paper22,23 for the study of entanglement probability, we introduced the topological moment
ð4:12Þ "
Φm ðrÞ
and for the disentangled pair, we have
#
2
X m6¼0
ð4:11Þ We can now readily see that the chemical relaxation time 2.7 is averaged by the probability for the encounter of the micellar segments, which depends on the entanglement probability and the topological force. Taking the real part, we find
ð5:2Þ
Φe ðrÞ ¼
X
Φm ðrÞ = 1 -erfð1=σðrÞÞ
ð5:9Þ
m6¼0
where
rffiffiffi Z x 2 2 e -t =2 dt erfðxÞ π 0
ð5:10Þ
is the Gauss error function. The probability for the pair to be disentangled is then given by Φ0 ðrÞ ¼ 1 -½1 -erfð1=σðrÞ ¼ erfð1=σðrÞÞ
ð5:11Þ
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Figure 4. (a) Probability Φe(r) for a pair of Gaussian loops with contour length L ∼ ξ2 to be entangled when their centers of mass are placed at a distance r. The correlation length ξ is changed from curve to curve, while the microscopic length a is kept constant. (b) Topological force working on the entangled loop pair plotted against the scaled distance between the mass centers. Loops with smaller ξ have stronger forces when compared at the same scaled distance because the loops are more easily brought to contact.
Hence, the ratio (equilibrium constant) is 1 -erfð1=σðrÞÞ KðrÞ ¼ erfð1=σðrÞÞ
Z
¥
r4 dr Φe ðrÞf~e ðrÞ2
0
ð5:12Þ
ð5:13Þ
for the dimensionless topological force working between entangled loops whose centers of mass are separated by the distance r. Because σ0 is proportional to (L/a)1/2, and hence to ξ/a, the force, when compared at the same scaled distance r/ξ, becomes stronger in proportion to ξ-1 ∼ c with the concentration. Figure 4a shows some typical profiles of the entanglement probability as functions of the distance between the loops. The distance r is measured by the correlation length (mean radius of gyration). The ratio of ξ relative to the microscopic length a is varied from curve to curve. The contour length of the loop is given by L=ξ2/a. As the chain becomes longer, the probability for entanglement increases when it is compared at a given distance. Figure 4b shows the topological force derived from the entanglement probability. Since Φe(r) is a function of the distance r only, the force fe(r) is isotropic. The x component of the force which is necessary to calculate the stress is x fe ðrÞ ð5:14Þ fe, x ¼ r
6. Comparison with the Experiments Substituting the equilibrium probability for entanglements into the complex modulus 4.12 and 4.13, and measure all lengths in the unit of the correlation length ξ, we find Z ¥ ½1 þ KðrÞω2 3 0 r4 dr Φe ðrÞf~e ðrÞ2 G ðωÞ=Fξ kB T ¼ C2 ω2 þ βðrÞ2 ½1þKðrÞ2 0 ð6:1Þ Langmuir 2010, 26(8), 5374–5381
for the storage modulus, and G00 ðωÞ=Fξ3 kB T ¼ C2
Taking the derivative of the entanglement probability, we find explicitly 2 fe ðrÞξ 2r e -1=2σðrÞ pffiffiffiffiffiffi f~e ðrÞ ¼ kB T ξ 2πσðrÞΦe ðrÞ
Figure 5. Effect of the micellar concentration. The diameter of a micellar cylinder remains the same, while the correlation length decreases with the concentration. Hence the loops are more easily brought to contact at higher concentration. The topological force, when compared at the same scaled distance r/ξ, becomes stronger with the concentration.
½1þKðrÞ2 βω ω2 þ βðrÞ2 ½1þKðrÞ2 ð6:2Þ
for the loss modulus after integration over the angle variables. The numerical factor C2 appears from the angular integral28 Z C2 ¼
ðsin2 θsin φ cos φÞ2 sin θ dθ dφ ¼
4π 15
ð6:3Þ
The factor Fξ3 is the number of loop pairs in the spatial volume ξ3, and is scaled as Fξ3 = ðc=LÞ2 ξ3 = c2 ξ3 -2=ν = c2 -ð3ν -2Þ=ð3ν -1Þ
ð6:4Þ
-ν/(3ν-1)
for the correlation length if we apply the scaling law ξ = c of semidilute polymer solutions.20 For the excluded volume chains, we have ν=3/5, so that Fξ3 = c2.25. For Gaussian chains, we have ν=1/2, so that Fξ3 = c3. To compare with the experimental data, let us consider the effect of concentration. When the concentrations of CTAB and NaSal are increased with their molar ratio kept at 1:1, the correlation length decreases. If we apply the scaling law of the semidilute polymer solutions ξ/a = c-3/4 (=c-1 for a Gaussian chain), the ratio ξ/a decreases with the concentration, so that the topological force start to work at shorter distance as is seen from Figure 4b. The situation is schematically illustrated in Figure 5. In the concentrated solutions, the topological force produces strong force between the micellar segments in contact, and hence it accelerates the chain passing. This is the main reason for the experimental observation that the rheological relaxation time decreases with the micellar concentration. In order to show the acceleration mechanism in more detail with regard to the dynamic mechanical modulus, let us assume that the decay rate β(r) of the entangled state is coupled to the force in the form βðrÞ ¼ β0 ½1 þ g1 fe ðrÞ2
ð6:5Þ
as in our previous study of transient networks,28 where β0 is a certain microscopic time scale, g1 = g1(T) is the temperature DOI: 10.1021/la903790g
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Figure 7. Cole-Cole plot of the complex modulus for (a) g2 = 0 and (b) g2 = 100. Figure 6. Dynamic mechanical modulus with varied concentration in the range ξ/a = 10-80. (a) (g2 = 0) There is no coupling between chain crossing rate and the topological force. (b) (g2 = 100) Loops are entangled and disentangled by the topological force.
dependent coupling constant. The time scale β0-1 is governed by the activation energy ΔE for the chain passing. It was experimentally estimated to be 25.4 kcal mol-1 from the Arrhenius plot for CTAB/KBr brine,30 21.8 kcal mol-1 for CTAT.13 By scaling the topological force, 6.5 can be written as βðrÞ ¼ β0 ½1 þ gf~e ðrÞ2
ð6:6Þ
where g g2 ðTÞ=ξ2
ð6:7Þ
is the dimensionless coupling constant. The temperature factor g2(T) g1(T)(kBT)2 is separated. Because g1(T) is expected to increase with the temperature, g2 is also expected to increase with the temperature.Thus, we can see that micelles pass through each other more easily at higher temperatures. If there is no coupling g=0 as GT limit in the literature,31,32 then there is no mechanism of acceleration. The relaxation time stays constant with the concentration. If g is sufficiently large, however, oscillatory deformation brings the smaller loops into contact with each other more easily, so that the topological force promotes the micellar segment to pass through to relax the stresses. Figure 6 shows the storage and loss moduli for the case of (a) no coupling g2 = 0 and (b) g2 = 100. The concentration is varied from curve to curve from ξ/a = 80 (low concentration) to 10 (high concentration). It is clearly seen that the relaxation time, as estimated from the peak frequency of the loss moduli, stays constant with concentration in part a, while it decreases in part b. Figure 7 draws the corresponding Cole-Cole plot of the complex modulus in Figure 6. Part a (g2 = 0) obeys the semicircular law with a single relaxation time β0-1, while part b deviates from it because the chain passing rate depends on the distance between the micellar loops due to the chain-force coupling. The semicircles are flattened as a manifestation of the nonlinear chain stretching. Shear thickening is possibly observed under such condition.29 Figure 8 shows the high frequency plateau value G¥ of the storage modulus Z ¥ r4 dr Φe ðrÞf~e ðrÞ2 ½1 þ KðrÞ ð6:8Þ G¥ =C2 kB T ¼ Fξ3 0
as a function of the concentration. It is independent of the coupling constant g as is evident from this equation. The 5380 DOI: 10.1021/la903790g
Figure 8. High frequency plateau modulus log(G¥/kBT), the zerofrequency viscosity ln η0 and the relaxation time ln τm (τm η0/G¥) plotted against the logarithmic concentration log c of CTAB: (a) no force coupling g2 = 0; (b) strong coupling g2 = 100. Vertical axis is logarithmic scale with arbitrary shift constant.
concentration dependence of the integral appears from the ratio ξ/a through the topological moment 5.6. Although f~e in the integrand of the storage modulus is proportional to (a/ξ) = c for a Gaussian chain due to the binary collision, the overlap of the entanglement probability Φe(r) and the force f~e (r)2 decreases with the concentration, and hence the integral is a decreasing function of c. Taking the fact that the loop density Fξ3 is proportional to c3, we find the slope of G¥/kBT is approximately equal to 2 in agreement with the experiment. Similarly, the zero-frequency viscosity η0 limωf0 G00 (ω)/ω can be calculated by Z
¥
η0 =C2 kB T ¼ Fξ3
r4 dr Φe ðrÞf~e ðrÞ2 =βðrÞ
ð6:9Þ
0
The mean relaxation time defined by the ratio τm η0/G¥ is also plotted in the figures. For g2=0, it is a constant, while for g2= 100, it decreases with the concentration. Figure 9 shows the theoretical relaxation time found from the peak frequency of the loss modulus (solid lines) plotted against the concentration of CTAB, and compared with the experimental data by Shikata6,7 (circles). The coupling constant g2 between the chain crossing rate β(r) and the topological force fe(r) due to entanglements is changed from curve to curve. The experimental data are vertically shifted because β0 is unknown. If there is no coupling (g2 = 0), the relaxation time stays constant with the concentration. If g2 is finite, the topological force enhances the chain crossing rate, and as a result the relaxation time decreases with the concentration because the micelles are driven by the force and have more chance to pass through each other. In other words, topological force accelerates the relaxation of the stress stored in the network. Langmuir 2010, 26(8), 5374–5381
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Figure 9. The rheological relaxation time log(τβ0) plotted against the logarithmic concentration log c of CTAB. The force-disentaglement coupling constant g2 is varied from curve to curve. The relaxation time τ is estimated from the peak frequency of the loss modulus, and scaled by the reciprocal rate β0-1.
In the experiment of CTAB/NaSal, the relaxation time is observed to increase in the higher concentration region above c=10-1 mol L-1 as indicated by the black circles in Figure 9. Such increase of τ in the higher concentration region is however not necessarily the case for other salts.
7. Conclusions and Discussion We have developed a new transient network model for the study of rheological properties of wormlike micelles. Because the end parts of the micellar cylinders are not found in the experiments, the wormlike micelles are practically infinitely long. They form entangled networks with no free ends. The structural unit of motion should therefore be the part of a threadlike cylinder of the contour length L contained in the three-dimensional space of the correlation length ξ. Like the concentration blobs in polymer solutions, such segmental parts make correlated motion by thermal agitation and under external force. The observed increase in the diffusion constant8,17 with CTAB concentration is the manifestation of the cooperative motion in the unit of blobs. To describe the constraints by the entanglements between the blobs, we replaced them by the loops of worms with the average
Langmuir 2010, 26(8), 5374–5381
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dimeter of ξ. The topological invariants for the loops can then be applied to find the stress originating in stretching of the entangled loops, and hence enables to estimate viscoelastic properties of the solutions. When the force exceeds the threshold, the micellar loops are locally merged and separated, hence resulting in chain passing. In the extreme limit where there is no chain passing, the topological relation is strictly conserved. Our model reduces to the theoretical model of the entangled rubbers developed by Graessley and Pearson.25 Studies of the elongational flows within the present theoretical framework will lead to more direct connections to their work. We focused on the linear viscoelasticity in this paper. The complex modulus was calculated on the basis of the entangled but mutually passable loops. The high frequency storage modulus and the rheological relaxation time are calculated as functions of the concentration of the micelles, and compared with the experimental data, thereby we focused on the anomalous decrease of the relaxation time with the micellar concentration in the region c= 10-2-10-1 mol L-1. In the experiments of the stoichiometric mixtures of CTAB and NaSal, the relaxation time changes nonmonotonically; it increases in the higher concentration region c=10-1-1 mol L-1. However, such increase was not observed in other salt such as sodium paratoluene sulfonate. We confined our study to the stoichiometric mixture of CTAB/ NaSal in this paper. The effect of excess NaSal has been extensively studied in the experiments.6,7 The electrostatic interaction affects the binding of counterions, and hence the relaxation time changes in more complex way with the salt concentration. Further refinement of the present model would involve the modification of the microscopic time scale β0-1 by the excess NaSal, as well as the effect of temperature. Acknowledgment. The author would like to thank Professor T. Shikata for his many valuable comments on the rheology experiments of wormlike micelles, and for providing the author the numerical data of the relaxation time in Figure 9. This work is partly supported by a Grant-in-Aid for Scientific Research on Priority Areas “Soft Matter Physics” from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and partly supported by a Grant-in-Aid for Scientific Research (B) from the Japan Society for the Promotion of Science under Grant No. 19350057. I wish to acknowledge their support.
DOI: 10.1021/la903790g
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