Temperature Dependence of the Thermodynamics and Kinetics of

A. Zilman,† S. A. Safran,*,† T. Sottmann,‡ and R. Strey‡. Department of Materials and Interfaces, Weizmann Institute, Rehovot, Israel 76100, a...
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Langmuir 2004, 20, 2199-2207

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Temperature Dependence of the Thermodynamics and Kinetics of Micellar Solutions A. Zilman,† S. A. Safran,*,† T. Sottmann,‡ and R. Strey‡ Department of Materials and Interfaces, Weizmann Institute, Rehovot, Israel 76100, and Institute of Physical Chemistry, University Cologne, D-50939 Cologne, Germany Received October 22, 2003. In Final Form: December 30, 2003 We predict theoretically the thermodynamics and relaxation kinetics of solutions of cylindrical branched micelles. Using a recently developed theory in combination with the experimental data, we explain the unusual, inverted temperature dependence of the phase separation observed in wormlike micelles and dilute microemulsions. We extend the model to treat the temperature dependence of the relaxation kinetics and explain the observations.

1. Introduction Surface active materials are major building blocks of a great many physical, chemical, and biological systems. They are important for the stabilization of water and oil solutions, forming microemulsions, that are a staple of many industrial processes.1 Control of the dynamic and static properties of microemulsions is a key goal in food processing, lubricant production, and the petrochemical industries. On the biological side, phospholipid surfactants comprise up to 80% of the dry mass of living organisms, because they form the membranes of biological cells.2 Despite their apparent simplicity, surfactant molecules are known to form a rich variety of structures in solutions.1 The main driving force for structure formation in surfactant (or surfactant and oil) solutions in water is the hydrophobic interaction, which acts to sequester the hydrophobic chain ends from water via formation of selfassembled structures, such as micelles or membranes.3 Despite the intensive theoretical and experimental research into the nature of the processes responsible for the formation and dynamics of structures formed in surfactant solutions, some questions still remain elusive. A typical phase diagram of a binary surfactant in water system can be found in ref 4. Similar phase diagrams have been measured in a wide variety of surfactant solutions and microemulsions.5,6 At low surfactant concentrations, the surfactant molecules are dispersed in solution. As the surfactant fraction is increased and exceeds the critical micelle concentration (cmc), the surfactant molecules begin to aggregate into spherical micelles. The micellization transition has been extensively studied, and we shall not be concerned with it here. As the fraction (concentration) of the surfactant or the † ‡

Weizmann Institute. University Cologne.

(1) Statistical Thermodynamics of Surfaces and Interfaces; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Addison-Wesley: Reading, MA, 1994. (2) Structure and Dynamics of Membranes; Lipowsky, R., Sackmann, E., Eds.; Elsevier: New York, 1995. (3) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1991. (4) Strey, R.; Pakusch, A. In Proceedings of the 5th International Symposium on Surfactants in Solutions; Mittal, K. L., Bothorel, P., Eds.; Plenum: New York, 1986; p 465. (5) Chen, S. H.; Rouch, J.; Sciortino, F.; Tartaglia, P. J. Phys.: Condens. Matter 1994, 6, 10855. (6) Khatory, A.; Lequeux, F.; Kern, F.; Candau, S. J. Langmuir 1993, 9, 1456.

temperature is increased still further, the spherical micelles begin to coalesce and form long cylindrical micelles. The radius of the spherical and cylindrical aggregates was found to remain constant throughout the phase diagram.4,6,7 As the temperature is increased, the cylindrical micelles begin to branch and eventually form a connected network. Originally motivated by rheological studies of solutions of wormlike micelles6 and conductivity5 measurements in dilute oil-in-water microemulsions, network formation has been also predicted theoretically.7-10 Networks have also been directly observed by cryogenic electron microscopy11,12 in both microemulsion and micellar systems. At still higher temperatures, the system becomes unstable and phase separates into a dense connected network in equilibrium with a dilute phase.4,5,7,10-13 At very high concentrations, liquid-crystalline, lamellar phases and other ordered phases appear. Phase separation in self-assembled networks and related systems has been extensively studied recently.7-10,14,15 An unusual feature of phase separation observed in nonionic micellar and microemulsion systems, which has still not received a quantitative explanation, is that the transition occurs as the temperature is increased,4-6 in contrast to the conventional phase separation paradigm, where the phase separation occurs as the temperature is decreased. In the usual case, the energy of interaction between the constituents dominates the entropy, and a condensed phase is stabilized as the temperature is lowered. This is not the case in the microemulsion and micellar systems, and we show here that the temperature dependence of the phase separation arises from the temperature dependence of the configurational entropy of the network. In this paper, we focus on the micellar system hexaethyleneglycol monododecyl ether (C12E6) surfactant in water, which has been well characterized experimentally.4 (7) Tlusty, T.; Safran, S. A.; Strey, R. Phys. Rev. Lett. 2000, 84, 1244. (8) Drye, T.; Cates, M. E. J. Chem. Phys. 1992, 96, 1367. (9) Lequeux, F.; Elleuch, K.; Pfeuty, P. J. Phys. II 1995, 5, 465. (10) Zilman, A. G.; Safran, S. A. Phys. Rev. E 2002, 66, 051107. (11) Berheim-Groswasser, A.; Tlusty, T.; Safran, S. A.; Talmon, Y. Langmuir 1999, 15, 5448. (12) Berheim-Groswasser, A.; Wachtel, E.; Talmon, Y. Langmuir 2000, 16, 4131. (13) Strey, R.; Glatter, O.; Schubert, K. V.; Kaler, E. W. J. Chem. Phys. 1996, 105, 1175. (14) Cristobal, G.; et al. Physica A 1999, 268, 50. (15) Zilman, A.; Safran, S. A. Europhys. Lett. 2003, 63, 139.

10.1021/la0359732 CCC: $27.50 © 2004 American Chemical Society Published on Web 02/07/2004

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We use a recently developed theory of phase separation in networks10,15 in combination with the available experimental data4,16,19 to show that the unusual, inverted temperature dependence of the phase separation region in micellar surfactant solutions can be explained by the temperature dependence of the spontaneous curvature of the surfactant. We then extend the theory to treat the nonequilibrium properties and the temperature dependence of the relaxation times of solutions of equilibrium micelles. The relaxation kinetics of linear cylindrical micelles has been treated in refs 20 and 21 and applied to ionic surfactant solutions; see ref 22 for a review. We generalize the theory of refs 20 and 21 to the case of branched chains. We show that just as is the case for the phase diagram, the relaxation times in the system are sensitive to the temperature dependence of the surfactant spontaneous curvature. Both the thermodynamic and structural predictions, along with the relaxation phenomena, are in quantitative agreement with experimental data of ref 4. Most of the parameters used in our fits of the data on micelles are obtained from experiments on microemulsion systems, and the good fits obtained here indicate that the curvature energy concept can be extended to treat micellar structures. 2. Thermodynamics As described in the previous section, for certain values of parameters such as the concentration and the temperature, binary solutions of surfactants in water selfassemble into cylindrical aggregates that either can be finite with end-caps or can branch to form an extended network structure. The system of cylindrical micellar aggregates can be viewed as a solution of self-assembled chains. The aggregate radius R plays the role of the “monomer size”, and spherical micelles can be thought of as chains of unit length. The radius of the tubes is determined by optimization of the total bending and compression energy of the surfactant layer comprising a micelle. This fixes the radius as can be seen in a simple physical model that considers total energy, per unit length, of a cylindrical micelle. Denoting the surfactant head size as a and the size of the tail as d, the energy of a micelle of radius R is 4πR2[κ(1/R - c0)2 + κ1/a2(R/d - 1)2] where κ is the bending rigidity of the surfactant monolayer, c0 is the spontaneous curvature, and κ1 is the cost of stretching the surfactant chains from their equilibrium length d.23,24 We will be concerned with the case when the spontaneous curvature is positive, that is, the surfactant layer tends to curve toward the oil side. The molecular stretching constant, κ1, is typically much larger than the bending rigidity κ.23 Therefore, the deviation of the micellar radius R from the equilibrium chain length d is negligible. Similarly, in the case of dilute microemulsions, the micellar radius is determined by the ratio of the oil and surfactant volume fractions: R = dφoil/φs. (16) Strey, R. Colloid Polym. Sci. 1994, 272, 1005. (17) Sottman, T.; Strey, R. J. Chem. Phys. 1997, 106, 8606. (18) Strey, R. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 182. (19) Hellweg, T.; Langevin, D. Phys. Rev. E 1998, 57, 6825. (20) Turner, M. S.; Cates, M. E. J. Phys. II (France) 1992, 2, 503. (21) Marques, C. M.; Turner, M. S.; Cates, M. E. J. Non-Cryst. Solids 1994, 172, 1168. (22) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (23) Safran, S. A. Statistical Thermodynamics of Surfaces and Interfaces; Addison-Wesley: Reading, MA, 1994 and Westview Press: Boulder, CO, 2003. (24) Bossev, D. P.; Kline, S. R.; Israelachvili, J. N.; Paulaitis, M. E. Langumir 2001, 17, 7728.

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Phenomenological, mean field considerations and more rigorous theories7,8,10,15,25 predict that the free energy of a solution of self-assembled chains far above the sphereto-cylinder transition, with a concentration φe of ends and a concentration φj of 3-fold junctions, is given by

f(φ, φe)/T ) (1 - φ) ln(1 - φ) + φe(ln φe - 1) + 3 1 φee(T) + φj(ln φj - 1) + φjj(T) - φj ln φ - φe ln φ 2 2 (1) where e(T) and j(T) are the end and junction energies, in units of kBT. For micelles, the volume fraction of tubes, φ, is the volume fraction of surfactant that is contained in the tubes and excludes surfactant molecules in spherical micelles. The first term in eq 1 is the excluded volume of the chains. The next four terms are the free energies of the “gas” of the ends and junctions, respectively, reflecting the translational entropy of the junctions and the ends. The next to last term is the reduction in the configurational entropy of the chains and the gas of junctions, due to the fact that each junction confines an end and an internal monomer of a chain to the same point in space. Analogously, the last term reflects the fact that two ends are constrained to a single chain. Minimizing this expression over φe and φj, one finds the equilibrium densities of the ends and junctions to be given by

φe ) φ1/2e-e(T) and φj ) φ3/2e-j(T)

(2)

Thus, the equilibrium free energy of the binary system far above the cmc is

F(φ, e, j) ) (1 - φ) ln(1 - φ) - φ1/2e-e(T) - φ3/2e-j(T) (3) Equation 2 shows that the numbers of ends and junctions strongly depend on their respective energies, e(T) and j(T), and therefore on temperature via the temperature dependence of the energies of the ends and the junctions, relative to the cylinder energy. The parameters e(T) and j(T) are determined by the bending energy of the surfactant layer and differ for the semispherical end-cap and the saddlelike junction geometries. As is well-known,23 the bending energy of a surfactant layer of bending rigidity κ, spontaneous curvature c0, and saddle-splay modulus κj is given by the Helfrich expression

[(

)

1 1 1 + - c0 E) I κ 2 R1 R2

2

]

1 1 + κj R1 R2

where the integration is over the surface of the layer, and R1 and R2 are the local radii of curvature. These concepts can be extended for a semiquantitative treatment of micelles as well.1 To calculate the end-cap energy, one must calculate the free energy difference between a surfactant molecule in the end-cap region and in the cylindrical part of a micelle. Thus,

) ] + 2πTκj

2 12 κ 11 - c0 - c0 e(T) ) 2π R2 T 2R 2R

[(

)

(

2

κj κ ) 4π (βe + Rec0R) + 2π T T

(4)

where Re ) -0.5, βe ) 3/4, and κ is the bending rigidity (25) Kindt, J. T. J. Phys. Chem. B 2002, 106, 8223.

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of the surfactant layer. The saddle-splay modulus κj contributes the constant term, by the Gauss-Bonnet theorem.23 More refined numerical calculations26 verify this formula and can be used to compute the junction free energy, j:

κ κj j(T) = 4π (Rjc0R + βj) - 2π T T

(5)

where, depending on the approximation, Rj = 0.8-1.3 and -βj = 0.13-0.6.26 It is interesting to observe that both the junction and end-cap energies depend linearly on the spontaneous curvature through the dimensionless term c0R. We also note that the saddle-splay term gives a negative contribution to the junction energy, due to the saddlelike shape of a junction. One can assume that the temperature dependence of κ and κj can be neglected in the temperature range of interest, since many other properties of microemulsions can be accounted for by considering only the variation of the spontaneous curvature with temperature.16,19 Consistent with those findings, we assume that the primary temperature dependence comes from the variation of the spontaneous curvature, c0, with temperature that has been determined experimentally16 to vary as c0 ) c*(T h - T). The microscopic reason for this dependence of the spontaneous curvature on the temperature is not understood in a detailed manner but has been conjectured to arise from changes in the headgroup hydration and of the packing volume of the surfactant chains with temperature. For the ternary system of water-n-alkane-C12E5, the coefficient c* in the expression for the spontaneous curvature was found to be almost constant, independent of the size of the headgroup, j: c* = 1.2 × 10-3 Å-1 K-1.17 As a first approximation, we use this value for c* for the binary system water-C12E6. The temperature, T h , in the binary system can be identified with the temperature where the dilute lamellar phase (LR) extends to the lowest surfactant concentration. Extrapolating from T h ) 48 °C for water-C12E4 and T h ) 58 °C for water-C12E5,18 we estimate T h ) 68 °C for the water-C12E6 system treated in the present paper. In accord with the available experimental values for the systems C12Ej (j ) 5, 6),17,19 we take κ ) 1.00T ) 0.87T h and κj ) -0.46κ for the system water-C12E6. The negative value of κj reflects the tendency of the monolayer to prefer spherical geometry over the saddles. At temperatures close to T h , the preferred spontaneous curvature is near zero, and the system prefers to form locally flat structures such as bilayers and vesicles. Because the spontaneous curvature, c0, increases as the temperature is decreased, we find that for temperatures lower than T h , the spontaneous curvature prefers water-in-oil interfaces. This will cause the spherical end-caps of cylindrical micelles to have a higher energy cost as T f T h , as we discuss below. As seen from eqs 4 and 5, the end energy increases with the temperature, in contrast to the energy of a junction that decreases as the temperature is increased. The temperature dependence of the energies of junctions and ends is summarized in Figure 1. At low temperatures, the energetic cost of junctions is high and the number of junctions in equilibrium is low. This is because the spontaneous curvature, c0, decreases with increasing temperature, T, and high spontaneous curvature stabilizes the semispherical end-caps. In contrast, junctions, that have a relatively flat region in their middle, are favored (26) Tlusty, T.; Safran, S. A. J. Phys.: Condens. Matter 2000, 12, A253.

Figure 1. Variation of the reduced end energy e/T h and the reduced junction energy j/T h with temperature. The end energy increases with temperature, while the junction energy decreases.

by small values of c0. Therefore, as the temperature is increased, junctions and their associated branching points become more abundant leading to the formation of a connected network. 2.1. Sphere-to-Cylinder Transition. The sphere-to cylinder transition is, in many respects, akin to the polymerization transition which has been discussed in the context of “living polymers”.27,28 In general, there is a balance between the translational entropy of the aggregates, which favors small aggregates such as spheres, and the increase of the end energy with temperature, which favors long tubes (which, by analogy with the polymerization transition, we will also denote as chains). Above a certain critical temperature (that varies with concentration), the system begins to form long cylinders; longer cylinders have fewer end-caps whose energy cost becomes large as the temperature is increased. The polymerization transition has been extensively studied, and we will not be concerned with its detailed characteristics here (cf., for example, ref 28); instead, we shall use a rough estimate as described below (see ref 23). At low temperatures, the spontaneous curvature is relatively large, and the semispherical end-caps are energetically preferred. Thus, at low temperatures the system consists of spherical micelles of radius R, determined by the size of the surfactant chains. As the temperature is increased, the spontaneous curvature decreases and the semispherical end-cap energy (related to the cost of bending on the length scale R that is fixed at the surfactant size in a micelle) increases, as in eq 4. It therefore becomes energetically favorable for the system to make fewer end-caps, and the self-assembly becomes dominated by long cylinders. This structure reduces the number of energetically costly end-caps, at the cost of the reduction of the translational entropy of the micelles. Neglecting the junctions, the mean length of cylindrical micelles, L h , can be expressed in units of the cylinder radius, R, that defines the basic size of the “monomers” that comprise the self-assembling chains, as L h /R ) 2φ/φe = 2φ1/2ee(T). Therefore, the mean length decreases as the (27) We emphasize here that this sphere-to-cylinder transition is not the cmc transition that occurs at much lower surfactant concentrations. (28) Wheeler, J. C.; Pfeuty, P. Phys. Rev. A 1981, 24, 1050.

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Figure 2. The predicted phase diagram of the binary surfactant (C12E6)/water mixture. The lower line is the predicted sphereto-cylinder transition; the corresponding experimental points from ref 4 are shown by triangles. The upper line is the predicted spinodal line of the phase separation between a dense network and excess dilute phase, along with the corresponding experimental points, shown as closed circles. The dashed line is the predicted boundary of the network region. Above this line, the system contains a network of branched cylindrical micelles, spanning the whole volume.

end-cap energy e(T) is reduced (i.e., as the temperature is lowered). We estimate that the sphere-to-cylinder transition occurs when the cylinder portion is reduced to zero and the structure contains only the two end-caps, that is, L h = 1, which occurs at the volume fraction equal to φ* that is given by

1 φ*1/2 ) e-e(T) 2

(6)

which defines a line in a (T, ln φ) plane:

-2e(T) - 2 ln 2 ) ln φ

(7)

We note that this transition is not a thermodynamic phase transition and is not accompanied by any singularities of the thermodynamic functions. Rather, it reflects the change in the structure of the micellar aggregates. Although this is not necessarily the case for the microemulsions where the sphere-to-cylinder transition can proceed through a coexistence region,29,30 the qualitative picture remains valid. The fit to the experimental data on the sphere-to-cylinder transition in micelles,4 using the available values of κ ) 1.00T ) 0.87T h and κj ) -0.46κ17,19 measured in independent experiments on microemulsions, determines the constants in the expression for the endcap energy, e, of eq 4 as Re ) -0.5 and βe ) 0.9. This fit uses a value Re ) -0.5 that is identical with the theoretical value, while the theoretical value of βe is 0.75 (see the discussion and eq 4). Using the theoretical value of βe ) 0.75 predicts a line for the sphere-to-cylinder transition that is parallel to the one shown in Figure 2 and about 15 °C higher. Similarly, measurements of the length scales in microemulsions16 determine that c0 ) c*(T h - T) where T h ) 68 °C and c* = 1.2 × 10-3 Å-1 K-1.16 The predicted (29) Safran, S. A.; Turkevich, L.; Pincus, P. J. Phys. (France) Lett. 1984, 45, L-69. (30) Blockhuis, E. M.; Sager, W. F. C. J. Chem. Phys. 2001, 115, 1073.

sphere-to-cylinder transition line is shown in Figure 2 along with the experimental points (triangular symbols). Note that as the temperature increases, the sphere-tocylinder transition is shifted to lower values of the surfactant volume fraction φ. Good agreement between theory and experiment is found for the micellar systems; all the parameters are fixed by other, independent measurements on microemulsion systems. The only fit parameters are small adjustments of the numerical constants R and β very close to their theoretically predicted values. It is not surprising that some adjustment is necessary in extrapolating the bending energy model to micellar structures, where at least one dimension of the system is not large compared to the surfactant size. Interestingly, we note that the predicted transition can also be used to understand the temperature and concentration dependence of the viscosity of these solutions as shown in Figure 3 of ref 18. As the concentration is increased, the sphere-to-cylinder transition occurs with decreasing temperature, consistent with the onset of the increase of the viscosity of the solution which is expected to be significantly larger for extended structures. 2.2. Network Formation and Phase Separation. At higher temperatures, the junction free energy decreases compared with the end-cap free energy, as seen from eqs 4 and 5 and Figure 2. The number of thermally generated junctions therefore increases, leading to the formation of larger and larger branched aggregates of tubes.24 When the number of junctions exceeds some critical value, a connected network spanning the entire volume of the system is formed. This process is purely geometrical and reflects a change in the connectivity of the system. It was first observed in water-in-oil microemulsions, that show high ionic conductivity when the connected network of water tubes is formed. As discussed below, in certain cases, network formation can be accompanied by a first-order thermodynamic transition. From very general geometric considerations,10,31,32 a connected network is formed when the number of junctions exceeds the critical value φj(φ, T) > (1/3)φe(φ, T), which determines a line in the (T, φ) plane, as shown by the dashed line in Figure 2. That is, for φ larger than φp(T) ) (1/3)ej(T)-e(T), a connected network exists in the system. As the number of junctions increases, they begin to play a role in the thermodynamics of the system, as follows from eqs 1 and 5. It has been shown that the increase in the number of junctions leads to a first-order phase separation between a dense connected network and a dilute phase of almost unbranched micelles.10,15 This transition is triggered by the increase in the configurational entropy of the large number of junctions, that overcompensates for the loss of the translational entropy of the connected chains in the dense phase. The spinodal line for this transition is determined by the condition ∂2f(φ)/ ∂φ2 ) 0 and can be calculated from eq 3. The critical temperature Tc is determined from the additional condition ∂3f(φ)/∂φ3 ) 0. In the low-density approximation (φ , 1), this can be solved analytically, and the critical temperature Tc is determined from

e-(1/2)(3j(Tc)-e(Tc)) ) 2

(8)

We have used eqs 4 and 5 for the end-cap and junction energies, respectively, and the experimental value for the critical temperature Tc ) 50 °C4 to estimate the numerical (31) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1981. (32) Lubensky, T. C.; Isaacson, J. Phys. Rev. Lett. 1978, 41, 829.

Thermodynamics and Kinetics of Micellar Solutions

parameters Rj and βj in eq 5; this turns out to be very close to theoretical calculations of ref 26 (see below). The predicted phase stability boundary is shown in Figure 2 along with the experimental data (denoted as closed circles). Note that the theoretical line is for the spinodal (phase stability boundary), while the experimental points show the binodal (true phase separation region). However, in the region of the critical point, as shown in the figure, these two lines lie fairly close to each other. It is important to realize that in our model, there are no direct attractive interactions between the chains nor between the junction points. Thus, the phase separation is mainly entropic in nature.10,15 It occurs because the entropy of the junctions that are abundant in the dense phase overcompensates for the loss of the translational entropy of the chains. As discussed above, the energetic cost for junction formation decreases with temperature, so that the probability of junction formation is higher at higher temperatures. This explains why the phase separation occurs when the temperature is increased, in contrast to conventional, attraction-driven, first-order phase transitions, where the condensed phase is formed at low temperatures. As seen in Figure 2, our model accounts for both the cylinder-sphere transition and for the inverted temperature phase diagram as observed in the experiments. The only parameters in this fit are Rj and βj, whose fitted values (Rj ) 0.97, βj ) -0.5) are very close to those predicted from structural models described above, and the measured critical temperature. The fits can be improved by correcting for the difference in weight fraction versus volume fraction, but these fine points, as well as the precise values of the elastic constants and other parameters that should be used for micelles, should not be overemphasized since we are using an extrapolation of the bending energy for thin films (appropriate for microemulsions) to the case of micelles. It is still gratifying that all the qualitative trends and semiquantitative agreement are obtained, and we stress not so much the goodness of the fit as its consistency. The consistency of the fits to both the structural and thermodynamic data indicates that the phase separation is indeed associated with network formation, which was also directly observed by cryogenic transmission electron microscopy imaging.11,12 Further experiments with other surfactants that have different critical and sphere-tocylinder transition temperatures are needed, to determine fully the model parameters Rj and βj and to study the robustness of the theoretical predictions. 3. Kinetics 3.1. Cylindrical Micelles. Information about the characteristic relaxation times of a system undergoing a structural or phase change can be obtained from the approach to equilibrium of the neutron scattering intensity after the system is quenched from one temperature to another; this is a common technique for assessing the relaxation properties of soft matter systems.4,16 We focus here on experiments that measured the relaxation times4 after a temperature up-quench in a micellar solution of C12E6 and water. The subsequent increase of the light scattering intensity was then measured. Three distinct relaxation regimes were observed. First, very close to the critical temperature and concentration (the minimum of the upper curve of Figure 2), a wavelength-dependent mode is observed, whose relaxation rate scales like q3, where q is the scattering wavevector. This relaxation mode

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and its q-dependence are well described by well-known models of critical fluctuation dynamics.4,33 At lower temperatures, farther away from the critical region, a q-independent mode is observed. The relaxation rate of this mode is sensitive to both temperature and the surfactant volume fraction. For linear micelles, it has already been conjectured that this mode reflects structural rearrangements of the micelles, related to the relaxation of the numbers of ends and junctions.20 Over an extended temperature range, between the critical region near Tc and the wavevector-independent regime, the fluctuation relaxation is dominated by hydrodynamic effects (collective diffusion) and a q2 dependence is observed for the kinetics in this region. In this section, we predict theoretically the kinetics of approach to equilibrium in the region of the phase diagram where the system consists of linear, unbranched chains. We later use these theoretical predictions for the explanation of the relaxation T-jump experiments of ref 4, where the relaxation times to a new equilibrium state, following a sudden up-quench of the temperature, were studied. In the next section, we consider the effects of network formation and chain branching on the relaxation times. A similar model of the kinetics of linear living polymers was treated in refs 20-22 and later successfully applied to linear micelles of ionic surfactants.22 However, these works did not focus on the effects of network formation (branching) and the temperature dependence of the endcap and junction energies; this is a crucial effect for nonionic surfactants as we now show. As a first step, we neglect the junctions in the free energy eq 1; this is because the energy of the junctions in the region of interest is much higher than that of the ends. The free energy (per unit volume) of a solution of selfassembled chains with a fixed number density of chain ends, φe, is (see eq 1)

f(φ, φe)/T ) (1 - φ)(ln(1 - φ) - 1) + φe(ln φe - 1) 1 φ ln φ + φee(T) (9) 2 e where φ is the volume fraction of the monomers that make up the chains (in the case of wormlike micelles, this is the volume fraction of a surfactant in tubes, excluding that in the spherical micelles). The first two terms are the excluded volume of the chains and the entropy of an “ideal gas” of ends, the third term is the reduction of the entropy of the gas of ends due to the fact that two ends are constrained to belong to a single chain, and the last term is the energy of the ends. We note that if φe is expressed via the average polymer length N, φe ) 2φ/N, eq 9 reduces with an appropriate Legendre transform to the familiar Flory-Huggins theory for solutions of long chains.34 As before, the equilibrium number of end-caps is obtained by 1/2 -e(T) minimizing f(φ, φe) over φe to yield φeq . e ) φ e We now discuss the dynamics of the relaxation of the end-cap distribution in T-jump experiments. It is important to realize that the main effect of the temperature quench (or up-quench) is in the temperature dependence of the end energy, e(T). As was shown above, the temperature dependence of the spontaneous curvature, c0, implies that the end energy e(T) increases with increasing temperature (see eq 4). Therefore, the number of ends decreases, and the average chain length in units (33) Onuki, A. Phase Transition Dynamics; Cambridge University Press: New York, 2002. (34) Doi, M. Introduction to Polymer Physics; Oxford University Press: New York, 1996.

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of the cylinder radius, L h /R ) 2φ/φe = 2φ1/2ee(T), increases with temperature for a fixed volume fraction φ. Assume that the system, initially at equilibrium at temperature T0, is suddenly quenched to a temperature T. The length distribution of the chains, along with the number of the end-caps in the system, will gradually change to a new equilibrium value. This equilibration process occurs via the spontaneous breaking and coalescence of the chains. A chain can break at any place along its length to form two additional end-caps, with an energy cost of 2e(T). The probability of such an event is therefore φe-Ee where Ee is the activation energy (in units of kBT) required for the formation of the intermediate “stalk” configuration required as part of the process of formation of two ends.35 On the other hand, two ends can coalesce in a collision, the probability of which is proportional to φe2e-Ee + 2e(T). Therefore, the number of ends evolves in time according to

dφe(t) 1 ) (φe-Ee - φe2(t)e-Ee + 2e(T)) dt τ0

(10)

where τ0 is the time required for end-cap formation and is determined by the microscopic properties of the surfactant end-caps and the molecular collision time. In the following, we consider Ee to be a constant independent of temperature since we assume that the primary temperature dependence is due to the temperature dependence of the end-cap energy. This assumption is a posteriori justified by our fits to the experimental data as shown below. The value of the activation energy, Ee, is complex, and its evaluation is beyond the scope of the present work. The dynamical equation for the end-cap density can also be derived from the dynamic equations describing the time evolution of the full length distribution of the cylinders, employed in refs 20 and 21. Starting from the initial condition, φ0e ) φ1/2e-e(T0), the system evolves toward the equilibrium state at the quenched temperature T, and at long times, the equilibrium density of end-caps is pro1/2 -e(T) portional to φeq . e ) φ e Equation 10 can be rewritten in terms of the dimensionless ratio F(t) ) φe(t)/φeq e :

dF(t) 1 ) (1 - F2(t)) dt τe

(11)

where τe ) τ0eEe/Leq ) τ0φ-1/2eEe-e(T). It follows that the characteristic relaxation time for the rearrangement of the end-caps is

τe ) τbφ-1/2e-e(T) where τb ) τ0eEe. This defines a new microscopic relaxation time that incorporates the effect of the activation energy of the intermediate state. Equation 11 can be solved analytically with the initial condition F ) F0 ) φ0e /φeq e ) ee(T)-e(T0) to predict the relaxation in time of the density of end-caps as

φe(t) ) φeq e F(t) F(t) ) 1 -

(12)

2e-2t/τe(1 - F0) e-t/τe(1 - F0) + (1 + F0)

which at long times, t . τe, becomes a pure exponential ∼ e-2t/τe. (35) Kozlovsky, Y.; Kozlov, M. M. Biophys. J. 2002, 82, 882.

We now relate the relaxation of the density of end-caps, φe(t), to the light scattering intensity, I(t), in the nonequilibrium system, where the number of ends has not yet relaxed to its equilibrium value of φeq e ; the instantaneous end-cap density is given by eq 12. The time scale of the relaxation of the end density is much larger than the time scale of the local density fluctuations which are measured in the static light scattering experiments. Therefore, at each time t, the system can be considered to be in a constrained equilibrium state, whose free energy is given by eq 9 with the end density given by eq 12, φe ) φ1/2e-e(T)F(t):

f(φ, t)/T ) (1 - φ)(ln(1 - φ) - 1) + φ1/2e-e(T)F(t)(ln F(t) - 1) (13) In the quasi-static approximation mentioned above, it then follows from the fluctuation-dissipation theorem34,36 that the static scattering intensity at time t is I(t) ) (∂2f(φ, t)/∂φ2)-1. Using eq 13, one finds

1 I(t) = 1 - φ-3/2e-e(T)F(t)(ln F(t) - 1) 4

(

-1

)

Expanding for long times, e-2t/τe , 1, yields

I(t) )

1 + 1 -3/2 -(T) 1+ φ e 4 φ-3/2e-(T) 1 e-2t/τe (14) 2(1 + φ-3/2e-(T))2(F - 1)2 0

Thus, the scattering intensity I(t) decays exponentially to its equilibrium value (1 + (1/4)φ-3/2e- e(T))-1 within the characteristic relaxation time

( )

1 1 φ*(T) 1 τe ) τbe-e(T)φ-1/2 ) τb 2 2 2 φ

1/2

(15)

where φ*(T) is the density at which the sphere-to-cylinder transition occurs at temperature T (see eq 6). The main temperature dependence of the relaxation time τe comes from the variation of the end energy, e(T), of eq 4 with temperature. As follows from eq 4, the end energy, e(T), increases with temperature. The relaxation time, τe ∼ e-e(T), therefore decreases with temperature, which agrees with the experimental data (see Figure 4). One consequence of eq 15 is that the relaxation time at the surfactant volume fraction φ and temperature T depends only on the ratio of φ to the sphere-to-cylinder transition concentration φ* at this temperature. We therefore predict that the measured relaxation times in different systems, with different material parameters, should collapse to a master curve, after appropriate scaling of the surfactant volume fraction φ by φ*. 3.2. Effect of Branching. As discussed above, the end energy, e(T), increases with temperature, in contrast to the junction energy, j(T), that decreases as the temperature is increased. Therefore, the density of junctions increases with temperature, compared to the end-cap density. At high temperatures, where there are many junctions in equilibrium, one must augment the theory of the previous section to take into account kinetics of junction formation and breaking. A junction can be created from the collision of an end-cap with an interior monomer (36) Landau, L.; Lifshitz, E. Statistical Mechanics, Vol. I; Pergamon Press: New York, 1980.

Thermodynamics and Kinetics of Micellar Solutions

Langmuir, Vol. 20, No. 6, 2004 2205

of the chain. The probability of such a collision is ∼φeφ. Taking into account that breakup of a junction (into one end-cap and one chain) costs an energy e - j, we find that the density of junctions, φj(t), evolves in time as

dφj 1 ) (φφe(t)e-(Ee-e)(T) - φj(t)e-(Ej-j(T))) dt τ0

(16)

where τ0 is the microscopic time determined by the properties of the surfactant film. The energy Ej is the activation energy required for the fusion of an end with the bulk of a chain. We must therefore modify eq 11 to take into account that an end-cap can be annihilated by colliding with a chain, thereby forming a 3-fold junction. The junction formation rate directly reduces the end-cap formation rate, and we have

dφe(t) 1 -Ee 1 ) e (φ - φe2(t)) - e-Ej(φφe(t)ee(T) dt τ0 τ0

Changing again to the dimensionless variables Fe(t) ) φe(t)/ (φ1/2e-e(T)) and Fj(t) ) φj(t)/(φ3/2e-j(T)), we write

(18)

dFj(t) 1 ) (Fe(t) - Fj(t)) dt τj

δF(φ(r, t), φe(r, t), φj(r, t)) ∂φ(r, t) ) Γ∇2 ∂t δφ(r, t) where F(φ(r, t), φe(r, t), φj(r, t)) is the quasi-stationary free energy of eq 1.33 This analysis is outside the scope of the present work. In response to a temperature up-quench (as in ref 4), the number of ends eventually decreases, and the number of junctions increases, toward their respective equilibrium values. To study the approach to equilibrium, we write Fe ) 1 + ∆e and Fj ) 1 - ∆j and linearize eqs 18 near the fixed point to find

(

)

x(

2+

)

τ e τe + τj τej

2

-8

]

τe (2τe)-1 τj (20)

The eigenfrequencies τ(-1 determine the relaxation rates of the system. Using the expressions for τe, τj, and τej, following eq 18, one obtains that τe/τj ) φ-1/2ej(T)-e(T)eEe-Ej and τe/τej ) φ1/2eEe-Ej. Assuming that Ee - Ej is a constant independent of temperature (to be verified a posteriori by comparison to experimental data), the temperature dependence enters τ( only via τe/τj. Note that τe/τj decreases with temperature because the junction energy decreases compared to the end-cap energy. For all the temperatures of interest, the ratio τe/τj is larger than τe/τej. One can then expand eq 20 in the limit of small densities φ:

τ(-1 )

1 [2 + τe/τej + τe/τj ( 2τe

Thus, there is a qualitative change in the behavior for τ( at a temperature Tτ where 2 + τe/τej - τe/τj ) 0, that is, when the junction energy j becomes significantly lower than the e. For the particular system studied here, eqs 4 and 5 give Tτ = 40 °C. Neglecting the O(φ1/2) term, the expressions for τ( of eq 21 assume a particularly simple form:

τ- =

where τe-1 ) τ0-1φ1/2ee(T)-Ee, as in the preceding section; τej-1 ) τ0-1φee(T)-Ej, and τj-1 ) τ0-1ej(T)-Ej. This is a nonlinear system of first-order differential equations with a unique fixed point, Fe ) Fj ) 1. This fixed point is stable and corresponds to the equilibrium state, where both the end-cap and junction densities have relaxed to their equilibrium values φe ) φ1/2e-e(T) and φj ) φ3/2e-j(T). In the region of the phase diagram near the critical point, the spatial fluctuations of the monomer density, φ, become important. To study the fluctuation-related dynamics near the phase separation region, eqs 18 must be augmented by the mass diffusion equation for the monomer density

d∆e 1 1 2 ) -∆e + - ∆j dt τe τej τej

τe τe + ( τj τej

|2 + τe/τej - τe/τj|(1 + O(φ1/2))] (21)

φj(t)ej(T)) (17)

dFe(t) 1 1 ) (1 - Fe2(t)) - (Fe(t) - Fj(t)) dt τe τej

[

τ(-1 ) 2 +

(19)

d∆j 1 ) - (∆j + ∆e) dt τj The eigenfrequencies of this system that determine the characteristic relaxation rates are

{

τe/2 = τ0φ-1/2e-e(T)

for T < Tτ

-j(T)

for T > Tτ

τj = τ0e

{

τ τ+ = τj /2 e

(22)

for T < Tτ for T > Tτ

Remarkably, this means that below Tτ, the relaxation time of the slow mode, τ-, is practically unaffected by the junction kinetics. On the contrary, above Tτ, the relaxation time mainly reflects the rearrangement kinetics of the junctions and not of the ends. The plot of the ratios τe/τand τe/τ+, including the φ-dependence, is shown in Figure 3 for Ej ) Ee and φ ) 0.01. It shows that τ- increases exponentially (as τj = τ0e-j(T)) from τe/2 to very large values when the temperature rises above Tτ = 40 °C, in accord with eqs 22 (upper panel in Figure 3). On the other hand, the ratio τ-/τ+ is small at all temperatures, except in a narrow region around Tτ (lower panel in Figure 3)). Thus, due to the presence of the junctions, there are two relaxation modes in the system. One of them is “fast”, and its relaxation time is given by τ+; the other one is a “slow” mode, with the relaxation time τ-. As mentioned above, the slow mode time scales are determined mainly by the kinetics of the end-caps below Tτ and by the kinetics of junction formation above Tτ. In general, the relaxation of the end and junction densities does not follow a single-exponential decay law but is given by the sum of two exponentials: Fe,j(t) ) -t/τ+ -t/τAe,j + Ae,j . The constants Ae,j +e -e ( can be found from the initial conditions φ0e ) φ1/2e-e(T0) and φ0j ) φ3/2e-j(T0) which in the linearized variables become ∆0e ) ee(T)-e(T0) - 1 and ∆0j ) 1 - ej(T)-j(T0) ) -∆0e , respectively. The calculation gives

Ae( )

τ( (1 - τj/τ() τ( - τ -

Aj( ) Ae(/(1 - τj/τ()

2206

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Zilman et al.

Figure 4. The predicted relaxation times τ as a function of the volume fraction φ for different temperatures (solid lines) along with the experimental data (discrete symbols). All the fits are performed with only one parameter, τb ) τ0eEe, that determines the common microscopic time scale, which is taken to be τb ) 0.4 ms.

Figure 3. Scaled relaxation rates τe/(2τ+) and τe/(2τ-) as a function of the reduced temperature T/T h . Note the abrupt decrease in the relaxation rate around Tτ.

Thus, in addition to the fact that the fast mode term decays to equilibrium much faster than the slow mode, the amplitude of the slow mode, A- . A+, is always much higher than that of the fast mode, A+, because τ+ , τ-. To a good approximation, at experimentally relevant time scales, the relaxation of the ends and junctions should follow a single-exponential decay with the relaxation time τ-. As mentioned above (cf. eq 22) and shown in Figure 3, the slow relaxation time, τ-, increases exponentially as the temperature is increased above Tτ. Thus, our theory predicts a dramatic slowing down of the relaxation processes beyond Tτ, due to the presence of the junctions. The scattering intensity decay in time can be related to the relaxation of the end-caps and of the junctions via eq 1. Namely, as explained in the preceding section, according to the fluctuation-dissipation theorem, the scattering intensity I(t) is proportional to I(t) = (∂2F(φ, Fe(t), Fj(t))/∂φ2). Thus, at long times it follows a singleexponential decay with the relaxation time τ-. Therefore, as mentioned above, for temperatures below Tτ, the relaxation of the scattering intensity is practically not influenced by the junction kinetics and reflects only the rearrangement of the ends. In Figure 4, the predicted relaxation rate, τ--1, is plotted as a function of the density, φ, for different temperatures, along with the experimental data from ref 4. We assumed Ee ) Ej, and the microscopic time τb ) τ0eEe was taken to

be τb ) 0.4 ms to fit the experimental data. With this one fitting parameter, we can then predict the behavior of the relaxation time as a function of concentration for a wide range of temperatures. The agreement between the predicted and measured relaxation rates is very good up to the temperature T ∼ 35 °C. At high temperatures (T > 35 °C), the wavevector-independent mode which shifts to shorter time scales could not be detected experimentally (the theoretically predicted slow, wavevector-independent mode is estimated to shift to unobservably long time scales) and the collective diffusion mode that scales quadratically with the wavevector was the only mode that could be measured; this can be attributed to the slowing down of the relaxation times due to the critical fluctuations which begin to play a role as the temperature increases and the critical point is approached.4,33 4. Summary and Discussion One of the common structures formed by binary water/ surfactant mixtures is cylindrical micelles.6,37,38 These long cylindrical micelles can in turn branch, and in the predicted temperature and concentration range, they form a connected network. One of the peculiar features of water/ surfactant solutions is the “inverted” phase diagram, that shows a first-order phase separation as the temperature is increased, in contrast to conventional phase separations that are usually observed as the temperature is decreased. Using a recently developed general theory of phase separation in equilibrium networks,7,10,15 we have shown here that the unusual, inverted temperature dependence of the phase behavior of the wormlike micelles has its origin in the temperature dependence of the spontaneous curvature of the surfactant layers comprising the micelles. The energetic properties of the end-caps and of junctions can be treated within a bending theory of surfactant monolayers.23 At low temperatures, the preferred spontaneous curvature is high, favoring the semispherical endcaps and long, unbranched cylindrical micelles. As the (37) Schick, M.; Gompper, G. Phase Transitions and Critical Phenomena; Self-Assembling Amphiphilic Systems, Vol. 16; Academic Press: London, 1994. (38) Buhler, E.; Munch, J. P.; Candau, J. S. J. Phys. II 1995, 5, 765.

Thermodynamics and Kinetics of Micellar Solutions

temperature is increased, the spontaneous curvature decreases, which favors branching and, eventually, phase separation. Extending the treatment to take into account the kinetics of chain breaking and coalescence, along with the junction formation kinetics, enables us to predict the relaxation times related to the approach to equilibrium, as observed in T-jump experiments. Remarkably, we have shown that below a certain temperature Tτ, where the junctions become sufficiently numerous compared with the end-caps, the relaxation time is determined mainly by chain breaking and reassembly via end-cap fusion and is almost independent of the junction kinetics. However, for temperatures greater than Tτ, the principle relaxation mechanism arises from the kinetics of junction formation, and the end-caps are negligible. The theoretical predictions are in good agreement with the experimental findings of

Langmuir, Vol. 20, No. 6, 2004 2207

ref 4 where thermodynamics and kinetics of the binary system C12E6/water were studied. Although the theory was formally developed only for wormlike micelles, similar behavior should be expected in dilute tubular microemulsions.7,11,12 Additional experiments at higher temperatures and close to the region of the phase separation will allow one to identify additional mechanisms involved in the relaxation behavior of solutions of branched wormlike micelles. Acknowledgment. The authors are grateful to R. Granek for helpful discussions. This work was supported by the German-Israel Foundation, the Schmidt Minerva Center, and the PRF administered by the American Chemical Society. LA0359732