Langmuir 2009, 25, 2467-2472
2467
Nonlinear Rheology of Surfactant Wormlike Micelles Bridged by Telechelic Polymers Herve´ Tabuteau,† Laurence Ramos,*,† Kaori Nakaya-Yaegashi,‡ Masayuki Imai,‡ and Christian Ligoure*,† Laboratoire des Colloı¨des, Verres et Nanomate´riaux (UMR CNRS-UMII 5587), cc26, UniVersite´ Montpellier II, 34 095 Montpellier Cedex 5, France, and Department of Physics, Faculty of Sciences, Ochanomizu UniVersity, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-0012, Japan ReceiVed October 7, 2008. ReVised Manuscript ReceiVed December 12, 2008 We have investigated the nonlinear rheology of a soft composite transient network made of a solution of surfactant wormlike micelles (WM) in the semidilute regime that are reversibly bridged by telechelic polymers. The samples are well described, in the linear regime, as two Maxwell fluids components blends, characterized by two markedly different characteristic times. The slow mode is mainly related to the transient network of entangled WM, and the fast mode to the network of telechelic chains. In this paper we investigate the nonlinear viscoelasticity and show that the nonlinear behavior reflects as well the behavior of two coupled networks. On one hand, stress relaxation experiments and time-resolved stress response following the application of a constant shear rate show that, in the weakly nonlinear regime, these novel composite networks stiffen. A fourfold increase of the elastic modulus with respect to the linear value is reached for strain amplitude of about 200%. This strain hardening is due to the nonlinear stretching of the telechelic polymer chains. On the other hand, the samples exhibit shear banding in the highly nonlinear regime, similarly to pure semidilute solutions of WM.
* To whom correspondence should be addressed. E-mail: ligoure@ lcvn.univ-montp2.fr (C.L.),
[email protected] (L.R.). † Universite´ Montpellier II. ‡ Ochanomizu University.
anchor its two stickers into one given surfactant assembly and form a loop, or bridge two distinct surfactant assemblies. We have shown that this soft composite material behaves as a near perfect blend of two Maxwell fluids.8 The slower mode is associated with the transient network of entangled wormlike micelles, and the faster one to the network of telechelic chains. In this paper we report on the nonlinear rheology of solutions of entangled wormlike micelles bridged by telechelic polymers. Because of the presence of two networks in the system, we expect to probe the nonlinear properties of both telechelic chains and wormlike micelles. For networks of telechelic chains, most of the experimental findings have focused on steady shear experiments.11-13Shear thickening is observed at the onset of the nonlinear regime followed by shear thinning at larger shear rates. Shear thickening is mainly caused by the nonlinear elasticity of individual polymeric chains that become overstretched by the imposed shear stress.14-16 For the shear-thinning behavior, the more plausible mechanism is the decreasing dependence of the detachment time with the tension acting on the active chains.17 Nonlinear transient shear18-20 or extensional20 rheology of telechelic polymers in the network regime has also been reported and explained by the stress-induced hardening of the network. On the other hand,
(1) In, M. Linear rheology of aqueous solutions of wormlike micelles. In Giant micelles properties and applications; Zana, R., Kaler, E. W., Eds.; CRC Press: New York, 2007; Vol. 140, pp 249-288. (2) Pfeiffer, D. G. Polymer 1990, 31, 2353–2360. (3) Shashkina, J. A.; Philippova, O. E.; Zaroslov, Y. D.; Khokhlov, A. R.; Pryakhina, T. A.; Blagodatskikh, I. V. Langmuir 2005, 21(4), 1524–1530. (4) Couillet, I.; Hughes, T.; Maitland, G.; Candau, F. Macromolecules 2005, 38(12), 5271–5282. (5) Penott-Chang, E. K.; Gouveia, L.; Fernandez, I. J.; Muller, A. J.; DiazBarrios, A.; Saez, A. Colloids Surf., A 2007, 295(1-3), 99–106. (6) Lee, J. H.; Gustin, J. P.; Chen, T. H.; Payne, G. F.; Raghavan, S. R. Langmuir 2005, 21(1), 26–33. (7) Ramos, L.; Ligoure, C. Macromolecules 2007, 40(4), 1248–1251. (8) Nakaya-Yaegashi, K.; Ramos, L.; Tabuteau, H.; Ligoure, C. J. Rheol. 2008, 52(2), 359–377. (9) Yoshida, T.; Taribagil, R.; Hillmyer, M. A.; Lodge, T. P. Macromolecules 2007, 40(5), 1615–1623. (10) Lodge, T. P.; Taribagil, R.; Yoshida, T.; Hillmyer, M. A. Macromolecules 2007, 40(13), 4728–4731.
(11) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. J. Rheol. 1993, 37(4), 695–726. (12) Tam, K. C.; Jenkins, R. D.; Winnik, M. A.; Bassett, D. R. Macromolecules 1998, 31(13), 4149–4159. (13) Xu, B.; Yekta, A.; Li, L.; Masoumi, Z.; Winnik, M. A. Colloids Surf., A 1996, 112(2-3), 239–250. (14) Koga, T.; Tanaka, F. Eur. Phys. J. E 2005, 17(2), 115–118. (15) Indei, T.; Koga, T.; Tanaka, F. Macromol. Rapid Commun. 2005, 26(9), 701–706. (16) Marrucci, G.; Bhargava, S.; Cooper, S. L. Macromolecules 1993, 26(24), 6483–6488. (17) Tanaka, F.; Edwards, S. F. Macromolecules 1992, 25(5), 1516–1523. (18) Serero, Y.; Jacobsen, V.; Berret, J. F.; May, R. Macromolecules 2000, 33(5), 1841–1847. (19) Berret, J. F.; Sereo, Y.; Winkelman, B.; Calvet, D.; Collet, A.; Viguier, M. J. Rheol. 2001, 45(2), 477–492. (20) Tripathi, A.; Tam, K. C.; McKinley, G. H. Macromolecules 2006, 39(5), 1981–1999.
1. Introduction Aqueous solutions of surfactant wormlike micelles mixed with hydrophobically modified polymers constitute a class of soft composite materials with remarkable rheological properties.1 Because the polymers bridge the wormlike micelles through the few hydrophobic stickers they carry along their backbone, a strong synergetic enhancement of the viscosity with respect to the binary solutions of each component2-6 is measured. Very recently, solutions of wormlike micelles mixed with telechelic polymers, where a hydrophobic sticker is grafted to each extremity of a hydrophilic backbone, have been investigated.7-10 Due to the simpler polymer architecture, the structure of networks of wormlike micelles and telechelic polymers is in principle less complex than that of networks formed by hydrophobically modified copolymers and wormlike micelles. This renders the modeling of the structure formed by wormlike micelles and telechelic polymer easier. Very generally, when a telechelic polymer is added to a solution of wormlike micelles, it can either
10.1021/la803304z CCC: $40.75 2009 American Chemical Society Published on Web 01/23/2009
2468 Langmuir, Vol. 25, No. 4, 2009
shear-banding flow, as recently reviewed by Puig et al.,21 is the most important and intriguing manifestation of the nonlinear rheology of wormlike micelles. Shear banding appears as a discontinuity in the shear stress vs shear rate flow curve, in which a stress plateau develops between two critical shear rates. The most commonly invoked mechanism for shear banding is the shear-induced transition, usually attributed to a constitutive instability of purely mechanical origin along the negative slope of the flow curve. Recently Hu and Lips22 demonstrated conclusively that, for the same experimental system of wormlike micelles as the one we have investigated, shear banding results from a mechanical instability triggered by chain disentanglement. The paper is organized as follows. Section 2 is devoted to materials and methods. In section 3, we briefly summarize the main linear rheological properties of the system reported in ref 8. In section 4 we report on the stationary shear stress-shear rate flow curves and discuss the results. Section 5 presents the results of the transient tests we performed: stress relaxation experiments and time-resolved stress response following the application of a constant shear rate. We show that these experiments allow us to probe the nonlinear elasticity of the telechelic chains.
2. Experimental Section 2.1. Experimental Systems. We use surfactant solutions composed of a mixture of cetylpyridinium chloride [H3C(CH2)15]C5H4N+Cl- (CpCl) and sodium salicylate (NaSal), with a constant molar ratio [NaSal]/[CpCl] ) 0.5, diluted in brine [NaCl] ) 0.5 M. This system is known to form wormlike micelles even at low surfactant concentration.23 NaSal and NaCl are used as received, and CpCl (from Fluka) is purified by successive recrystallizations in water and acetone. We add to the wormlike micelles solution triblock copolymers, which are synthesized in our laboratory. The polymer is a water-soluble poly(ethylene oxide) (PEO) block which has been hydrophobically modified and purified using the method described in refs 24 and 25. The molecular weight of the starting PEO is determined by size-exclusion chromatography. The hydrophobically modified PEO contains an isocyanate group between the alkyl chain and the ethylene oxide chain. Triblock “telechelic” polymers, C18-PEO10K-C18 have been prepared with a C18H37 aliphatic chain grafted to each extremity of the central PEO chain of molecular weight 10 000 g/mol. After modification, the degrees of substitution of the hydroxyl groups were determined by using NMR and were found to be larger than 98%. The radius of gyration of the POE block is 37 Å. The samples are prepared by weight. We first incorporate the surfactant CpCl and the hydrophobically modified PEO in brine until complete dissolution of the polymer (this requires 1 day or more depending on the polymer concentration). After addition of NaSal to the homogeneous mixture, the sample is stirred several times for homogenization and then left undisturbed at 30 °C for several days. The samples are characterized by the mass fraction of surfactant φ ) (mCPCl + mSal)/mtot, where mCpCl, mSal, and mtot are, respectively, the mass of CpCl, the mass of sodium salicylate, and the total mass of the sample, and by the sticker (C18H37) over surfactant molar ratio β. In our experiments, the mass fraction of surfactant is fixed at φ ) 9%, and the molar ratio of stickers over surfactant, β, varies between 0 and 2%. The structure of the samples has been probed by small-angle neutron-scattering experiments. For the range of experimental values for φ and β investigated here, we have shown that the samples consist of a solution of entangled wormlike micelles of radius 21 Å linked and decorated by the triblock copolymers.7 (21) Puig, J. E.; Bautista, F.; Soltero, J. F. A.; Manero, O. Nonlinear rheology of giant micelles. In GIANT MICELLES, Properties and Applications; Zana, R., Kaler, E. W., Eds.; CRC Press: New York, 2007; Vol. 140, pp 289-322. (22) Hu, Y. T.; Lips, A. J. Rheol. 2005, 49(5), 1001–1027. (23) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92(16), 4712–4719. (25) Vorobyova, O.; Yekta, A.; Winnik, M. A.; Lau, W. Macromolecules 1998, 31(25), 8998–9007.
Tabuteau et al. 2.2. Rheology Measurements. The flow curves, stress, σ, vs shear rate, γ˙ , obtained by imposing either σ or γ˙ , were determined with a stress-controlled rheometer (Paar Physica UDS 200) in Couette geometry at a temperature of T ) 23 °C. All other rheological measurements were performed with an Ares (RFS III) straincontrolled rheometer equipped with a cone and plate geometry at T ) 30 °C. Step-strain experiments were carried out for strain amplitude up to γ ) 100%. This ensures that the applied strain is obtained in a time period shorter than the relaxation time of the telechelic network chains, τfast (in the range of parameters, φ, β, investigated here, τfast varies between 0.03 and 0.19 s). We also performed start-up experiments, during which a constant rate is imposed and the stress is monitored as a function of time or strain. Finally, flow curves, σ vs γ˙ , corresponding to stationary states were determined. To this end, we imposed a step rate and we monitor the evolution of the corresponding stress. We took the stress value when the flow reaches a steady state, which is obtained in about 30 s in the linear regime and in about 1 h in the plateau regime.
3. Linear Rheology The linear rheology of this composite material has been studied previously by us8 and is summarized below. Solutions of bridged wormlike micelles can be described as two Maxwell fluids components blends, characterized by two markedly different characteristic times, τfast and τslow, and elastic moduli, Gfast and Gslow, with Gfast . Gslow. The two modes are found to be strongly coupled. The slow mode is associated with the transient network of entangled wormlike micelles, whose relaxation can be well understood in the framework of the breakable reptation theory.26 However, the presence of transient junctions, i.e., the telechelic bridges, between the micelles slows down the relaxation process in qualitative agreement with the sticky reptation model.27 The fast mode is associated with the transient network formed by the telechelic chains. The features of this network are qualitatively well described by the transient network theory:17 the temperature dependence of the relaxation time follows an Arrhenius law, and the elastic modulus is proportional to the density of telechelic chains. From the proportionality factor, the ratio between the elastically active chains (the ones that bridge two micelles) and inactive chains (the ones that form a loop) has been evaluated. Interestingly, we have also shown that this transient network exists at arbitrary low polymer concentration without any percolation transition, contrary to all other experimental realizations of transient networks made of telechelic polymers. The reason is that the network of telechelic polymers is supported by the substrate network of entangled micelles, which has a much longer relaxation time. Linear rheological data (relaxation times and plateau moduli) of all samples investigated here are reported in Table 1.
4. Stationary Shear Stress-Shear Rate Flow Curves We show in Figure 1a the flow curves measured in the stationary states for samples with a fixed surfactant concentration φ ) 9%, and various amount of copolymer, β. The features of the flow curves are maintained whatever the polymer concentrations and are the same as those of a solution of entangled wormlike micelles. In the low shear rate regime, the stress is proportional to the shear rate as expected for a Maxwell fluid in a stationary state. This regime is followed, for a shear rate larger than γ˙ p, by a quasi-plateau of the stress. Very generally, for semidilute solutions (26) Cates, M. E. Macromolecules 1987, 20(9), 2289–2296. (27) Leibler, L.; Rubinstein, M.; Colby, R. H. Macromolecules 1991, 24(16), 4701–4707.
Nonlinear Rheology of Micelles Bridged by Polymers
Langmuir, Vol. 25, No. 4, 2009 2469
Table 1. Linear Viscoelastic Properties (Plateau Shear Moduli and Relaxation Times) of the Samplesa temp (°C) 23
30
β (%)
Gslow (Pa)
τslow (s)
Gfast (Pa)
τfast (s)
0 (naked micelles) 0.1 0.25 0.5 0.75 1 2 0 (naked micelles) 0.25 0.75 2
130 152 159 191 211 252 344 145 168 199 242
0.58 0.83 0.95 1.6 2.09 2.02 4.39 0.09 0.17 0.35 1.29
89 163 286 406 475 1089 58 200 692
0.04 0.05 0.07 0.09 0.1 0.19 0.03 0.04 0.1
a
Data were obtained by fitting the frequency dependence of the complex moduli with a two Maxwell fluids blend model (β * 0, bridged micelles) or with a Maxwell model (β ) 0, naked micelles).
Figure 1. (a) Shear stress vs shear rate in steady state for samples with different amounts of polymer (0 < β < 2%). (b) Same data as in Figure 1a with the rate γ˙ scaled by the relaxation time of the wormlike micelles, τslow, and the stress σ scaled by the shear modulus of the wormlike micelles, Gslow.
of wormlike micelles, the onset of the plateau marks the onset of shear banding.28 Data obtained at high shear rates (i.e., after escape of the plateau) are not very reliable and are not presented here. Noticeably, the value of the stress plateau, σp, does not vary with β and is equal to the value obtained for the solution of pure wormlike micelles (in the absence of telechelic chains) within 5%. By contrast, the critical shear rate, γ˙ p, is reported to lower values as the quantity of copolymer increases. The flow curves are represented using normalized units28 in Figure 1b, where the steady shear stress normalized by the elastic modulus of the network of wormlike micelles, Gslow, is plotted as function of the normalized shear rate γ˙ pτslow. The β dependence of the normalized values, σp/Gslow and γ˙ pτslow, are plotted in Figure 2. Both σp/Gslow and γ˙ pτslow decrease steadily but weakly (by less (28) Berret, J. F. Rheology of wormlike micelles: equilibrium properties and shear banding transition. In Molecular Gels; Weiss, R., Terech, P., Eds.; Elsevier: Dordrecht, 2005; pp 235-275.
Figure 2. Stress plateau, σp, scaled by the shear modulus of the wormlike micelles, Gslow (circles) and the reduced plateau rate γ˙ pτslow (squares) as a function of the amount of polymer, β. The data were determined from the flow curves shown in Figure 1.
than a factor of 3) with β. Interestingly, Berret et al.29 have shown by varying the surfactant concentration for the same system of semidilute pure surfactant micelles as the one investigated here that stress plateaus are found in the flow curve below the critical conditions σp/G0 ) 0.90 ( 0.05 and γ˙ τR ) 3.0 ( 0.5, where G0 and τR are, respectively, the elastic plateau and the relaxation time for a semidilute solution of wormlike micelles. Our samples fulfill these conditions as shown in Figures 1b and 2, provided that we use Gslow and τslow, which increase with β,8 as normalized units. Our measurements suggest that flow curves for bridged wormlike micelles are essentially controlled by the behavior of the network of wormlike micelles, whose elastic modulus is increased and whose relaxation is slowed down due to the coupling with the telechelic polymers. In agreement with this slowing down, the onset of the plateau associated with the shear banding occurs at smaller shear rates. Hence, the flow curves essentially reflect the “standard behavior” of semidilute wormlike micelles solutions: Maxwellian linear behavior and shear-banding transition associated with nonlinear stress plateau. This is expected considering the time scales involved. In fact, the relaxation time of the network of telechelic chains, τfast, varies between 0.045 and 0.19 s when β varies between 0.1 and 2% and is more than 1 order of magnitude smaller than the one related to the micelles, τslow (τslow lies in the range 0.83- 4.39 s). As a consequence, in order to probe a direct effect of the telechelic chains, one should impose a shear rate of the order of 1/τfast, which ranges from 5 s-1 for β ) 2% to 20 s-1 for β ) 0.25%, hence much larger than the shear rate at which shear banding occurs.
5. Probing the Nonlinear Elasticity of the Telechelic Chains In order to probe the nonlinear properties of the composite materials associated with the telechelic chains, rheological quantities must be measured on a sufficiently short time scale, i.e., on a time scale smaller than τfast, such that the network of telechelic chains has not relaxed, and/or for shear rate larger than 1/τfast. In the following we describe two different rheological tests that allow one to probe the onset of nonlinearity for the telechelic chains. 5.1. Stress Relaxation Experiments. The stress relaxations have been measured for strain amplitudes, γ, ranging from 10 to 200% for three different samples with copolymer contents β ) 0.25%, 0.75%, and 2%. As detailed in the Experimental Section, (29) Berret, J. F.; Porte, G.; Decruppe, J. P. Phys. ReV. E 1997, 55(2), 1668– 1676.
2470 Langmuir, Vol. 25, No. 4, 2009
Tabuteau et al.
Figure 3. Stress relaxation curves after different steps strain (30-90%) for a sample with β ) 2%. The solid lines correspond to fits of the data by two Maxwell components in parallel, and the symbols are data points.
only the data for γ smaller than 100% are relevant, and thus have been analyzed. Data are shown for different strain amplitudes in Figure 3, ranging from 30 to 90%, for a sample with β ) 2%. The chosen semilogarithmic representation allows one to visualize the two distinct relaxation processes. Figure 3 shows that, for the range of strain amplitude γ investigated, the short-time behavior (associated with the transient network of telechelic chains) varies with γ, while the long-time behavior, associated with the wormlike micelles, appears independent of the strain amplitude, since we obtained an almost perfect collapse of the different curves for times larger than 0.1 s. In order to quantify the strain dependence of the stress relaxation, we fitted the stress relaxation with a double exponential. Given the collapse of the data at long time, we have chosen to fix the two parameters related to the second relaxation process, Gslow and τslow, to their values measured in the linear regime (see Table 1) and keep only the characteristics of the first relaxation process (Gfast and τfast) as fitting parameters. In all cases, good fits are obtained, as shown in Figure 3, where the lines are the best fits and the symbols the experimental data points. The strain dependences of, respectively, Gfast and τfast are shown for the three samples in Figure 4a and 4b, respectively, where the two parameters have been scaled by their numerical values in the linear regime. Noticeably, we find that all the elastic modulus data for various amounts of copolymer collapse onto a single master curve. Moreover, above a critical strain γc of the order of 70%, a pronounced strain hardening is measured, as revealed by a continuous growth of the elastic modulus with γ. We measure for γ ) 100% a value about 1.8 times bigger than the value in the linear regime. The experimental results for the characteristic times are not so clear. This presumably reflects the fact that the characteristic times are very small (of the order of 0.1 s) and their determination is not accurate enough. A collapse between the data obtained for different copolymer concentration is not observed. Nevertheless, two features are shared by all the samples: below a critical strain of the order of 30%, a clear linear regime is observed in which the data are independent of the applied strain. Above a critical strain, τfast is measured to decrease. Data for β ) 0.75 and 2% follow a similar trend and decrease by about 25% for a strain of 100%. The data are noisier for β ) 0.25%, and the decrease is stronger. This can be attributed to more uncertainty in the experimental determination of τfast because of the lower absolute value of τfast, implying that the number of data points used to fit the data is sensitively reduced. 5.2. Transient Behavior upon Application of a High Shear Rate. Additional information on the strain hardening can be obtained from the time-resolved stress response following the application of a constant shear rate. Shear rates higher than 1/τfast are considered here. We show in Figure 5a the time evolution
Figure 4. (a) Variation of the elastic modulus of the fast mode, Gfast, linear scaled by its value in the linear regime,Gfast , as a function of the applied strain γ for different polymer contents β ) 0.25% (squares, linear linear Gfast ) 58 Pa), 0.75% (triangles, Gfast ) 200 Pa), and 2% (circles, linear Gfast ) 692 Pa). (b) Variation of the associated relaxation time τfast linear scaled by its value in the linear regime τfast , β ) 0.25% (squares, linear linear τfast ) 0.029s), 0.75% (triangles, τfast ) 0.035s), and 2% (circles, linear τfast ) 0.1s), as a function of the applied strain γ. The symbols are the same as in Figure 4a.
Figure 5. (a) Transient stress measurements obtained from start-up experiments for various applied shear rates (10-40 s-1) for a sample with β ) 2%. (b) Stress plotted against strain γ for the same data as in (a), with γ ) γ˙ t.
of the stress when a constant shear rate γ˙ , ranging from 10 to 40 s-1, is imposed. The data obtained for the various shear rates are qualitatively similar. They display a pronounced stress
Nonlinear Rheology of Micelles Bridged by Polymers
Langmuir, Vol. 25, No. 4, 2009 2471
collapse between the data points measured using the stress relaxation tests and the results derived here from the differential modulus. Hence, thanks to the differential modulus analysis, we have been able to extend the range of strain amplitude to ∼200%, while it was restricted to ∼100% in the stress relaxation linear for experiments. In the same figure, we have plotted Gslow/Gslow naked micelles, obtained from stress relaxation experiments, which does not show any variation up to a strain amplitude of ∼200%.
6. Discussion
Figure 6. Variation of the elastic modulus of the fast mode, Gfast, scaled linear by its value in the linear regime Gfast , as a function of the applied strain γ. The hollow symbols correspond to the data derived from the differential modulus for a sample with β ) 2% for various applied shear rates (10, 12, 14, 20, 30, and 40 s-1), and the filled diamond-shaped symbols are data determined through stress relaxation experiments G(t) and are identical to those plotted in Figure 4a. The black line corresponds to the best fit of G(t) (see main text). For comparison, the variation of the elastic modulus of naked micelles obtained from stress relaxation experiments is also shown (crosses).
overshoot before reaching a steady-state value. The time at the maximum stress overshoot is shifted toward smaller value as γ˙ increases, while the stress in steady-state flow weakly increases with the imposed shear rate. Interestingly, we found that the different curves up to the stress overshoot approximately collapse on a single master curve when the stress is plotted as a function of the applied strain, γ ) γ˙ t, where t is the time elapsed since the beginning of the test (Figure 5b). In all cases, we measured a time at the maximum stress overshoot smaller than the fast relaxation time τfast (for β ) 2%, T ) 23 °C, τfast ) 0.19 s). The maximum of the stress overshoot is found for a strain equal to (220 ( 10)%. Experiments with pure wormlike micelle solutions show a linear behavior up to strain larger than 250%.30 Hence we expect that the nonlinear elasticity observed here is only due to the telechelic chains. The differential modulus, K ) (∂σ)/(∂γ), can be computed as a function of the strain for the different applied shear rates. K provides quantitative information on the nonlinear elasticity of the material. In order to compare the nonlinear behavior of the telechelic chains as obtained by stress relaxation experiments (Figure 4b) and by the present start-up experiments, one needs first to subtract from the numerical values of K the values of the linear elastic modulus for the network of linear linear ) 344 Pa). So we identify K - Gslow wormlike micelles (Gslow as the elastic modulus of the network formed by the telechelic chains. To directly compare with the data measured thanks to linear the stress relaxation experiments, we have normalized K - Gslow linear by the linear elastic modulus of the telechelic chains, Glinear (G fast fast linear ) 1089 Pa) and plotted the strain dependence of (K - Gslow )/ linear for the various imposed shear rates in Figure 6. We note Gfast that the differential modulus is ill-defined at small strains because of the lack of data points. We have therefore only represented linear linear )/Gfast is significantly >0. data points for which (K - Gslow With this method, we have been able to investigate the nonlinear elastic properties for strain amplitudes ranging from 80 to 200%. We found a continuous growth of the elastic modulus which reaches values about 4 times larger than that in the linear regime. To allow a direct comparison, we have also plotted in Figure 6 the results derived from the stress relaxation experiments for the same sample. Remarkably, we found that the two experimental configurations give consistent results, as shown by a reasonable (30) Berret, J. F. Langmuir 1997, 13(8), 2227–2234.
The transient results clearly show that we probed strain hardening of the composite system, for time shorter than τfast, and that this phenomenon is only related to the network of telechelic polymers: indeed, in the range of strain amplitude or strain rate investigated, the network formed by the entangled micelles remains in the linear regime. Moreover, whereas the linear , strongly varies with elastic modulus in the linear regime, Gfast linear varies between 58 the quantity of telechelic polymer, β (Gfast and 692 Pa at T ) 30 °C when β varies between 0.25 and 2%), the γ-dependence of the elastic modulus, once rescaled by its value in the linear regime, is independent of β (Figure 4a). This strongly suggests that the measured strain hardening probes a property of individual telechelic chains and results from the nonlinear elastic response of the active telechelic chains to elongational deformation. Similar results have been previously reported for elastically active chains in transient networks of telechelic polymers in water.12,18,19,31 In these references, the critical strain and the increase of the elastic modulus measured were of the same order of magnitude as those measured by us in our composite networks. It should be noticed that networks of usual polymer chains exhibit a significant non-Gaussian hardening at strain well above 200%. In contrast, the hardening for our system is seen at strains ∼70% as for binary solutions of telechelic chains. A simple explanation can be proposed, which is based on the fact that the nodes of the telechelic network, i.e., the micelles, have a finite size and do not deform under stress. Hence, the strain experienced by the telechelic chains is larger than the global strain applied on the sample. The strain experienced by the telechelic chains can be evaluated simply. For a volume fraction of micelles φ ) 9%, one can estimate the mean distance between the nodes by assuming a local isotropic cubic order: φ ) (3π/4)(r/d)2, where d is the lattice parameter of the cubic lattice and r ) 21 Å is the radius of the micelles. One obtains d ≈ 107 Å, and hence an average distance between the surface of two adjacent nodes d - 2r ) 65 Å. Let us suppose now for simplicity that we apply a relative elongational deformation λ to the sample, which is supposed to deform in an affine way. Then a telechelic active chain will feel a relative deformation λchain ) λ(d/(d - 2r)) - (2r)/(d - 2r) ≈ λ/0.6 - 0.65 for φ ) 9%. So for λ ) 1.7 one gets λchain ) 2.2, a value close to the value for which network of usual chains exhibit hardening. A quantitative analysis of the strain hardening is a very difficult task. We used the simplest and more tractable model, i.e., the three-chains model32 of the non-Gaussian network, to compute Gfast, as done recently by Inoue et al.,33 who analyzed successfully the finite extensibility of a network of wormlike micelles in this framework. Unfortunately the three-chains model fails completely to reproduce the experimental strain hardening and predicts a much weaker increase of Gfast. This discrepancy is not surprising, (31) Pellens, L.; Corrales, R. G.; Mewis, J. J. Rheol. 2004, 48(2), 379–393. (32) Treloar, L. R. G. The Physics of Rubber Elasticity; Clarendon: Oxford, UK, 1958. (33) Inoue, T.; Inoue, Y.; Watanabe, H. Langmuir 2005, 21(4), 1201–1208.
2472 Langmuir, Vol. 25, No. 4, 2009
since the strain hardening coming from the telechelic chains is observed at strains well below 200%; it presumably originates, as argued above, from the finite size effect of the nodes. With simple symmetry arguments, one expects the modulus to scale linear ∝ (1 + aγ2). Such phenomenological functional as Gfast/Gfast form works reasonably well with our experimental data. The best fit shown in Figure 6 gives a value of 0.7 for a.
7. Conclusion We have described the nonlinear rheology exhibited by a double transient network formed by mixing wormlike micelles and telechelic polymers. This composite material exhibits two distinct nonlinear behaviors. The samples exhibit strain hardening (increase of the elastic modulus with the strain) on time scales shorter than the relaxation time of the network of telechelic chains, τfast. The strain hardening has been interpreted in terms of the non-Gaussian elastic response of telechelic chains to an elongational deformation. We mention that the structure of our experimental systems might present some interesting analogies with that of a network of biological rigid polymers, like actin, cross-linked with proteins. For this class of material, strain hardening is measured as well, but its amplitude is much more pronounced than for wormlike micelles bridged by telechelic
Tabuteau et al.
polymers.34,35 This might be related to the markedly different rigidities of the components. The features of the flow curves measured in the steady state present strong similarities with those observed in a solution of wormlike micelles, in the absence of telechelic polymers. In particular, a quasi-plateau of stress is observed above a critical shear rate, γ˙ p. We measure a continuous decrease of γ˙ p with the quantity of telechelic polymer, β. Consequently, adding telechelic polymer to a solution of entangled wormlike micelles allows a fine tuning not only on the linear viscoelasticity but also on the flow properties of surfactant wormlike micelles. Acknowledgment. We thank T. Phou and R. Aznard for polymer synthesis. This work was conducted in the framework of the Network of Excellence “Soft Matter Composites: An Approach to Nanoscale Functional Materials”, which is supported by the European Commission. This project has also been supported by ANR under contract ANR-06-BLAN-0097 “Tailored Transient Self-Assembled Networks (TSANET)”. LA803304Z (34) Gardel, M. L.; Shin, J. H.; MacKintosh, F. C.; Mahadevan, L.; Matsudaira, P.; Weitz, D. A. Science 2004, 304(5675), 1301–1305. (35) Lieleg, O.; Bausch, A. R. Phys. ReV. Lett. 2007, 99 (15).
