3000
J. Phys. Chem. B 2009, 113, 3000–3007
Transient Gelation and Glass Formation of Reversibly Cross-linked Polymeric Micelles Fre´de´ric Renou, Taco Nicolai,* Lazhar Benyahia, and Erwan Nicol Polyme`res, Colloı¨des, Interfaces, UMR CNRS 6120, UniVersite´ du Maine, 72085 Le Mans Cedex 9, France ReceiVed: NoVember 14, 2008; ReVised Manuscript ReceiVed: December 19, 2008
Poly(ethylene oxide) (PEO) end-capped with a hexadecyl group at one (R-PEO) or both ends (R,ω-PEO) are highly asymmetric diblock or triblock copolymers that form spherical micelles in aqueous solution. R,ωPEO can bridge between two micelles leading to reversible association of the micelles. At a first critical concentration (Cp), the micelles percolate and a transient network is formed with an elastic modulus determined by the concentration of R,ω-PEO. Cp increases with increasing fraction of R-PEO and is insensitive to the temperature. At a second critical concentration (Cc), a liquid-solid transition occurs. Cc is independent of the fraction of R-PEO and increases with increasing temperature. There are indications that the solid state is formed by nucleation and growth of domains of dynamically arrested micelles. The properties of the transient network are almost the same in the liquid and in the solid state. Introduction Poly(ethylene oxide) (PEO) chains functionalized at one (RPEO) or both ends (R,ω-PEO) with an aliphatic group can be considered as highly asymmetric block copolymers of which the association behavior has been studied extensively.1-3 The properties of PEO end-functionalized with alkyl groups in aqueous solution have been reviewed comprehensively by Winnik and Yekta.4 For a more recent review see also ref 5. Similar behavior was found for PEO end-capped with fluorinated alkyl groups.6 In aqueous solution the hydrophobic groups self-assemble above a critical concentration to form the cores of spherical micelles. The number of hydrophobic groups in the core (p) increases with increasing hydrophobicity but is insensitive to the concentration7 except at very high concentrations where p increases with increasing concentration.8,9 Close to the overlap concentration the viscosity of R-PEO micelles increases steeply with increasing concentration due to crowding. But whereas for star polymers10 and hard spheres11 the increase of the viscosity is continuous, for polymeric micelles it is interrupted by a discontinuous jump to solidlike behavior if the aggregation number is sufficiently high.8 The solid is formed because close packed micelles jam and can no longer diffuse. For PEO micelles, the liquid-solid transition can be induced either by increasing the concentration at a critical value Cc or by decreasing the temperature at a critical temperature Tc.8,12,13 In both cases, the effective volume fraction (φe) of the micelles increases. A crystalline order of the micelles is observed in the solid state.8,9,12,14 The fraction of crystalline order increases with increasing φe and is very small at the critical value where the transition occurs. By comparing rheology and small-angle X-ray scattering during the liquid-solid transition, it was shown that initially a disordered solid is formed, and that the isotropic liquid crystalline phase of micelles is formed more slowly in the solid state.8 This means that the solidlike behavior correlates with, but is not caused by, the formation of crystalline order. For R,ω-PEO, both end-groups are situated in a micellar core. The number of hydrophobic groups in the core (p) is the same * To whom correspondence should be addressed.
for R-PEO and R,ω-PEO with the same end-groups if the length of the R,ω-PEO chain is twice that of the R-PEO chain.7,15 In mixtures of R-PEO and R,ω-PEO, mixed micelles are formed with the same aggregation number at all ratios.15 In dilute solution, the difunctionalized chains loop back to form the petals of flowerlike micelles, but when different micelles encounter each other they can form bridges between different micelles. The additional degrees of freedom of end-groups that can enter other micellar cores increase the entropy of the system.16 Since the enthalpy doesn’t change between the loop and bridge configuration, micelles containing difunctionalized chains feel an attractive interaction that is of purely entropic origin. Micelles containing difunctionalized chains can be viewed as adhesive spheres with a short-range attractive potential.15-17 The strength of the attraction is proportional to the number of difunctionalized chains per micelle that can be controlled either by varying the length of the hydrophobic block or by varying the ratio of di- to monofunctionalized chains. If attraction is strong, it leads to phase separation into a dense phase of highly close-packed micelles cross-linked by bridging chains and a dilute phase.15,17,18 At high densities, a liquid-solid transition of the densely cross-linked micelles occurs15 similarly to that observed for R-PEO micelles. Cross-linked micelles show crystalline order in the solid state similar to that of unconnected micelles.9 If the attraction is weaker, transient clusters are formed that increase in size with increasing concentration, and a transient network is formed at the percolation concentration (Cp). Cp decreases with increasing attraction until it crosses the binodal and phase separation occurs. The elastic modulus of the network increases steeply with increasing concentration as more bridging chains become elastically active until all R,ω-PEO chains participate. The high frequency shear modulus of the fully crosslinked network is determined by the number of the bridging chains and increases linearly with the polymer concentration.15 The terminal relaxation of the shear modulus is characterized by a single relaxation time (τ) that is determined by the lifetime of the bridges. τ increases with decreasing temperature and increasing length of the hydrophobic block but is independent of the concentration.19,20 At higher frequencies, the relaxation modes of the transient network are probed.21,22 A recent
10.1021/jp8100442 CCC: $40.75 2009 American Chemical Society Published on Web 02/19/2009
Reversibly Cross-linked Polymeric Micelles theoretical description of transient gels formed by difunctionalized polymers was given by Indei,23 who also discussed earlier theories. Transient networks and solids (often called hard gels) have also been observed for PEO end-capped with oxybutylene (B-E-B),24,25 oxypropylene,26 or styrene oxide (S-E-S).27 Interestingly, the relaxation of the transient network at high frequencies could be observed even in the solid state. Ricardo et al. compared the state diagram of S-E-S with that of B-E-B and PEO end-capped with alkyl groups (C-E-C) and found similar behavior for B-E-B but different behavior for the latter. Notably C-E-C phase separated at all temperatures at lower volume fractions, while S-E-S and B-E-B only phase separated at higher temperatures. These authors also investigated the effect of mixing mono- and difunctionalized chains.28,29 The objective of the work reported here was to investigate in detail the effect of cross-linking on the liquid-solid transition for PEO end-capped with hexadecyl at one or both ends. The molar mass of the PEO chain was 4.5 kg/mol for R-PEO and 9.0 kg/mol for R,ω-PEO, that is, the hydrophobic-hydrophilic balance was equal. The properties of almost the same system at lower concentrations have already been studied in detail.15 In dilute solutions the aggregation number of the pure or mixed micelles was the same and independent of the temperature. The effect of attraction can thus be studied at constant p by varying the ratio of R-PEO and R,ω-PEO. By adding R-PEO, the attraction was reduced and phase separation could be avoided. Experimental Methods Materials. Commercial PEO samples with weight-average molar mass (Mw) of 4.5 kg/mol containing a hydroxyl group at one end and with Mw ) 9.0 kg/mol containing hydroxyl group at both ends were purchased from Aldrich. Mw and the polydispersity index (Mw/Mn ) 1.05) were determined by size exclusion chromatography (SEC). The SEC results showed that a small fraction of monofunctional PEO contained about 8% w/w of difunctionalized chains with twice the molar mass. Hexadecylbromide was purchased from Aldrich and had a purity grade of 97%. The synthesis of the copolymers for the present investigation was described in ref 30. We note that for the sample studied here the hexadecyl group was connected to PEO as an ether, while in ref15 it was connected as an ester. Size exclusion chromatography in tetrahydrofuran (THF) showed that the functionalization had no influence on the molar mass distribution. 1H NMR showed that quantitative functionalization was attained within the experimental error of 5%. Clear solutions of functionalized PEO were obtained in demineralized water (Millipore) at pH 7 after stirring for several hours at 90 °C. The total PEO concentration (C) was calculated using a density of 1.15 kg/L.31 The small temperature dependence of the density was neglected. Functionalized PEO was soluble up to C ) 750 g/L. Methods. Rheology measurements were done using a stresscontrolled rheometer (AR2000, TA Instruments) with a cone and plate geometry (diameter 2 cm and angle 4° or 4 cm and 2°). The temperature was controlled by a Peltier system. Solvent evaporation was avoided by covering the geometry with mineral oil. Oscillatory shear measurements were done with an imposed stress of 1 Pa to determine the storage (G′) and loss moduli (G′′). Measurements were in the linear response regime for the solids and the liquids. For the liquids the dynamic viscosity obtained was the same as the viscosity obtained from flow measurements with an imposed stress of 1 Pa.
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3001
Figure 1. Concentration dependence of Tc for solutions of R-PEO and R,ω-PEO with different fractions of R,ω-PEO. Error bars indicate the difference between the temperature where the storage modulus of the solid starts to decrease and where the viscosity becomes equal to that of the liquid during heating at a rate of 5 °C/min. The solid line is a guide to the eyes.
Prior to measuring the viscosity, the solutions were presheared at 90 °C (5 min at 300 s-1) in order to fully homogenize the systems. Subsequently, the temperature was reduced at a rate of 5 °C/min down to 5 °C during which the viscosity was measured using continuous or oscillatory shear with an imposed stress of 1 Pa. Both methods gave the same viscosity. The solid-liquid transition was determined by measuring G′ and G′′ during heating at a rate of 5 °C/min from 5 to 90 °C at 0.1 Hz with an imposed stress of 1 Pa. Results Liquid-Solid State Diagram. We have measured the storage (G′) and loss (G′′) shear moduli of aqueous solutions of R,ωPEO over a range of polymer concentrations (C) and temperatures (T). A liquid-solid transition was observed when decreasing the temperature or when increasing the concentration. It was characterized by a discontinuous change from a system that flowed when tilted to a system that was completely arrested. The kinetics of solid formation was very slow, see below, but melting of the solid was rapid. Measurement of G′ and G′′ during heating ramps of 5 °C/min yielded accurate estimates of Tc. Figure 1 shows the dependence of Tc on the concentration. At C ) 140 g/L, the system remained liquid even at 2 °C. At higher concentrations, Tc increased until it reached a maximum of about 55 °C at 250 g/L and remained stable up to at least 360 g/L. For comparison, we also show in Figure 1 the results for almost pure R-PEO and a mixture with a weight fraction F ) 0.54 of R,ω-PEO. The liquid-solid transition occurred at the same temperature for the same PEO micelle concentration independent of the fraction of bridging chains except at the lowest concentrations. We found that almost pure R-PEO formed a solid at 130 g/L, while pure R,ω-PEO did not down to 2 °C. The plateau of Tc for C > 250 g/L and the decrease observed at even higher concentrations for pure R-PEO32 were probably caused by a decreased hydration of PEO. Kelarakis et al.29 determined the transition for mixtures of B-E-B and B-E. These system are liquid at both low and high temperatures, because the hydrophobicity of the oxybutylene decreases with decreasing temperature. They found that the upper melting temperature increased weakly with increasing
3002 J. Phys. Chem. B, Vol. 113, No. 10, 2009
Renou et al. below the critical value (Cc ≈ 150 g/L at 20 °C), the system behaved as a viscous liquid and at higher concentrations as a solid. A relaxation process at high frequencies is clearly visible both in the liquid and the solid and does not depend strongly on the concentration. The concentration dependence of pure R,ω-PEO could not be explored over a broad concentration range because the system phase separated for C < 100 g/L. As mentioned above, the frequency dependence of G′ and G′′ of transient networks formed by hydrophobically end-capped PEO can be described by the so-called Maxwell model characterized by a single relaxation time (τ). The angular frequency (ω ) 2πf) dependence of the solid can be described by adding a frequency independent storage modulus (G′0)
Figure 2. Frequency dependence of the loss (open symbols) and storage (close symbols) shear modulus at different temperatures for R,ω-PEO at C ) 200 g/L. The solid lines represent fits to eqs 1 and 2.
G′ ) G∞
ω2τ2 + G′0 1 + ω2τ2
(1)
ωτ 1 + ω2τ2
(2)
G′′ ) G∞
where G∞ is the high frequency shear modulus of the transient network. If the elastic modulus of the transient network is caused by rubber elasticity of chains between cross-links, then33
G∞ ) νRT
Figure 3. Master curves of the frequency dependence of the loss (open symbols) and storage (filled symbols) shear modulus at Tref ) 20 °C for different concentrations of R,ω-PEO. The solid lines represent fits to eqs 1 and 2.
fraction of bridging chains, possibly because the length of the oxyethylene chain in B-E-B was somewhat larger than twice that in B-E. Visco-Elastic Relaxation. Figure 2 shows the frequency (f) dependence of G′ and G′′ for R,ω-PEO at C ) 200 g/L at different temperatures. At temperatures above the critical value (Tc ≈ 40 °C), the system behaved as a viscous liquid with G′ ∝ ω2 and G′′ ∝ ω at low frequencies. For T < Tc, the system behaved as a solid with an almost frequency independent G′ and a much smaller G′′. A relaxation process is clearly visible at high frequencies characterized by a relaxation time that increased with decreasing temperature. The data of the solids or of the liquids at different temperatures could be superimposed using horizontal shift factors. In this way master curves were obtained at different PEO concentrations with Tref ) 20 °C; see Figure 3. At concentrations
(3)
where ν is the molar concentration of elasticity effective chains, R the gas constant, and T the absolute temperature. The solid lines in Figures 2 and 3 represent fits to eqs 1 and 2. G′ is well described over the whole frequency range covered in the experiment, while G′′ is only well described at high frequencies for the solids. The weak increase of the loss modulus of the solid at low frequencies was already observed for R-PEO34 and indicates perhaps the presence of extremely slow relaxation processes in jammed micelles. G∞ depended very weakly on the temperature and increased approximately linearly with the polymer concentration in agreement with an earlier study on the same system.15 In the high concentration range covered in the experiment, we find that G∞)νRT if we take for ν the molar concentration of R,ωPEO. This was already shown in ref 15 for the liquid state and here we find that it is also true for the solid state. This means that the rubber elasticity of the transient network is the same in the solid and that jamming of the micelles simply leads to an additional frequency independent storage modulus G0. The origin of the elastic modulus of the solid is probably entropic due to constraints on the conformation of the PEO chains forming the corona.35 G0 is approximately proportional to the concentration, but is smaller than νRT. On the other hand, it is larger than RT times the number of micelles, which may be understood if the conformation of several, but not all, chains per micelle is correlated as was suggested by Watanabe on the basis of results on different polymeric micelles.35 We will show below that G0 is independent of the fraction R,ω-PEO implying that the constraints on the micelles are the same whether they are cross-linked or not. For small fractions of R,ω-PEO, G0 is larger than G∞. We have measured the relaxation time of the transient network for fixed concentrations as a function of the temperature both in the liquid and in the solid state. Figure 4 shows the results for 200 g/l and 300 g/l using an Arrhenius representation. The relaxation times were almost independent of the concentra-
Reversibly Cross-linked Polymeric Micelles
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3003
Figure 4. Relaxation time of R,ω-PEO solutions as a function of the inverse absolute temperature for different concentrations. The filled points indicate systems in the solid state.
tion at high concentrations both in the solid and the liquid state, but decreased weakly at lower concentrations where only the liquid state was found. This was shown in more detail in ref 15. The relaxation time of the transient network increased with decreasing temperature both in the liquid and the solid state. In general, the decay of the transient network was faster in the liquid state than in the solid state. But just above Tc the decay was slightly slower than in the solid state just below Tc. There appears to be a jump to shorter relaxation times when the solid is formed. The system at 140 g/L is in the liquid state at all temperatures. Its relaxation times are close to that at 200 and 300 g/L in the solid state, but significantly smaller if the latter are in the liquid state. It was reported that in the liquid state the temperature dependence of τ could be described by the so-called Arrhenius equation: τ ∝ exp(-Ea/RT), with Ea the activation energy that increased with increasing length of the hydrophobic group.19,20 Here we find that Ea is larger at high temperatures than at low temperatures. The gradual change of the activation energy at low and high temperatures can also be clearly seen in the temperature dependence of the viscosity, see below. Viscoelasticity during the Liquid-Solid Transition. The liquid-solid transition is very fast for pure R-PEO except very close to Tc.32 However, for pure R,ω-PEO the transition is slow and takes more time when the temperature is decreased. The larger is the fraction of R,ω-PEO, the slower is the transition. A slow down of the transition was also found when nonfunctionalized PEO was added to R-PEO.32 Figure 5 shows G′ and G′′ as a function of time at different frequencies during the transition. After a lag period during which the moduli increased only very weakly, G′ increased steeply to reach a value much larger than G′′ and remained constant at later times. At lower frequencies, G′′ also increased, though more weakly, and reached a maximum when it crossed G′. At longer times, G′′ was very noisy. At higher frequencies, G′′ remained constant until it crossed G′ after which it decreased. Figure 6 shows the isochronal frequency dependence of G′ and G′′ at a few chosen times that were obtained by interpolation of the time dependence at different frequencies. In the steady state, the data could be fitted to eqs 1 and 2 and for comparison we have shown the fit to the steady state also in the plots at earlier times. It is clear that even after a short time, the relaxation could not be described by a single exponential decay. In addition, the terminal relaxation time was much slower than the relaxation of the transient network in the solid state. The
Figure 5. Evolution of the G′ (filled symbols) and G′′ (open symbols) during the liquid-solid transition of R,ω-PEO at 200 g/L and 5 °C.
Figure 6. Frequency dependence of the loss (open symbols) and storage (filled symbols) shear modulus at different times during the liquid-solid transition of R,ω-PEO at 200 g/L and 5 °C. Solid lines represent fits to eqs 1 and 2 of the steady state.
terminal relaxation time increased weakly during the lag time, followed by a rapid increase until it could no longer be detected in the frequency range covered in the experiment. During this fast change at low frequencies, the single relaxation mode at high frequencies became distinctly visible. After about 500 min, G′ and G′′ showed approximately the same power law dependence on the frequency, which is characteristic for a sol-gel transition.36 At later times, the values of G′ at low frequencies increased and those of G′′ decreased, until the characteristic behavior of the solid was obtained. Viscosity of the Liquid State. Because the solid formation was slow, the viscosity in the liquid state could be measured down to 2 °C during cooling ramps, even if the steady state
3004 J. Phys. Chem. B, Vol. 113, No. 10, 2009
Renou et al.
Figure 7. Temperature dependence of the viscosity of R,ω-PEO solutions at different concentrations. The solid lines represent fits to the Arrhenius equation at high and low temperatures. The filled symbols are values obtained from measurements in the steady state using η ) G∞τ.
was solid. The results are shown for different R,ω-PEO concentrations in Figure 7 using an Arrhenius representation. At all concentrations, the temperature dependence could not be described using the same activation energy over the whole range. At high temperatures, we found Ea ≈ 80 kJ/mol and at low temperatures Ea ≈ 50 KJ/mol. The change of the activation energy is thus not related to the liquid-solid transition. For the similar system studied in ref 15. Ea was found to be 73 kJ/mol. The viscosity of liquids in the steady state can be calculated as η ) G∞τ and was equal to that found during the cooling ramp. However, when we use the same calculation for the solids in the steady state, we find much smaller values than observed during the cooling ramp. This is caused by the jump to smaller values of τ during the liquid-solid transition. Mixtures of Mono- and Difunctionalized PEO. A detailed study of mixtures of R-PEO and R,ω-PEO in the liquid state has been reported in ref 15. Here we extend this investigation to cover also the solid state. As was shown in Figure 1, the critical temperature of the liquid-solid transition was independent of F. Figure 8 shows the frequency dependence of the loss and storage shear modulus at 5 °C for different F at C ) 300 g/L. At all values of F, the samples behaved as solids with a constant low frequency shear modulus G0 that varied weakly, but not systematically, with F; see Figure 9. For systems endcapped with butylene oxide, a minimum was found for G′ as a function of the fraction of bridging chains.29 For all values of F, we observed the relaxation process of the transient network at high frequencies. G∞ decreased linearly with decreasing F and became small compared to G0 for small F where it could only be determined accurately by fitting to G′′. Figure 9 shows that G∞ ≈ νRT over the whole range of F if we take for ν the molar concentration of R,ω-PEO and consider that 8% of the PEO precursor chains used to synthesize R-PEO were difunctional. The relaxation time remained also almost the same. This means that at C ) 300 g/L a fully crosslinked transient network was formed even if only 8% of the chains could form bridges. The temperature dependence of the viscosity could be measured during cooling down to 2 °C also for F ) 0.54, but not for smaller F, because the liquid-solid transition occurred during the cooling ramp. For F ) 0.54, the temperature dependence was the same as for pure R,ω-PEO, but the absolute
Figure 8. Frequency dependence of the storage (a) and loss (b) shear modulus for different mixtures of R-PEO and R,ω-PEO at 300 g/L and 5 °C. The solid lines represent fits to eqs 1 and 2.
Figure 9. Dependence of G0 (squares) and G∞ (circles) on the fraction of R,ω-PEO in mixtures of R-PEO and R,ω-PEO at 300 g/L and 5 °C. The straight lines represents G∞ ) νRT and G0 ) 4 × 104 Pa.
values were smaller. For all samples the viscosity of the liquid state was equal to G∞τ. Figure 10 shows the viscosity as a function of the concentration for different values of F at 20 °C. As mentioned above, phase separation limits the investigation for pure R,ω-PEO to C > 100 g/L. For F ) 0.54 and F ) 0.17, η started to increase steeply at the percolation threshold (Cp). In ref 15, it was shown that the percolation threshold increased with decreasing F up to about 45 g/L at F ) 0.2, which was the lowest value used in that study. For the similar system studied here we find Cp ≈ 20
Reversibly Cross-linked Polymeric Micelles
Figure 10. Concentration dependence of the viscosity for mixtures of R-PEO and R,ω-PEO PEO micelles for different F at T ) 20 °C.
Figure 11. Diagram indicating the four different states of aqueous suspensions of PEO micelles at 20 °C: (I) transient clusters; (II) transient gel; (III) phase separation; (IV) solid state.
g/L at F ) 0.54 and Cp ≈ 50 g/L at F ) 0.17, consistent with the earlier results. The percolation threshold does not depend on the temperature since it is determined by the number of micelles and the number of bridging chains, which are independent of the temperature. At F ) 0.08, percolation was not observed in the liquid state, because Cp > Cc. The increase of the viscosity at F ) 0.08 is thus entirely due to crowding of the micelles and is comparable to that of star polymers. Discussion On the basis of the results reported here and in ref 15, we can draw a schematic state diagram of polymeric micelles for PEO end-capped with hexadecyl groups as a function of the interaction strength; see Figure 11. We have expressed the interaction in terms of the second virial coefficient in units of the particle volume (B2). B2 represents the effective excluded volume of the particles and B2 ) 4 for noninteracting hard spheres. The values of the second virial coefficient can be obtained from light scattering experiments usually in units of g2 · mol-1 · m3, as explained in ref 8. The corresponding values of B2 can be calculated by dividing the measured values at different F with the one obtained in the absence of attraction (F ) 0) and multiplying with 4. B2 is a useful parameter, because the association of particles with short-range attraction depends only very weakly on the interaction range when the interaction strength is expressed in terms of the second virial coefficient.37-39
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3005 At a given temperature, B2 could be varied without modifying the effective volume fraction by varying the fraction of difunctionalized chains which did not influence the aggregation number of the micelles. The effective volume fraction was also calculated from light scattering measurements at lower concentrations and was assumed to be proportional to the concentration. This assumption is not valid for φe > 1 however as SAXS showed that the aggregation number increased with increasing concentration in the solid state.8 When expressed in terms of B2 and φe, the state diagram depends little on the temperature except for T > 60 °C where the solid state is no longer formed. Of course, the relationship between F and B2 on one hand and C and φe on the other varies with the temperature. Four different domains can be distinguished: (I) transient clusters of micelles at low concentrations and weak attraction; (II) a transient network at higher concentrations and stronger attraction; (III) a binary state of a low density and a high density liquid; and (IV) a solid of jammed particles (glass, crystal) at high concentrations. The features of the state diagram are general for polymeric micelles with a sufficiently large aggregation number. The transition between a solution of transient aggregates (I) and a transient gel (II) can be determined without ambiguity by the appearance of a relaxation process at high frequencies. Alternatively, it can be determined by the sharp upward deviation of the viscosity from systems in the absence of bridging chains. The liquid-solid transition occurred at almost the same effective volume fraction even if the micelles formed a fully cross-linked transient network. A liquid-solid transition due to crowding was also observed for colloidal suspensions40 and multiarm star polymers.41 For hard spheres it was found that short-range attraction shifted the glass transition to slightly higher volume fractions.40 This is considered to be caused by transient clustering of the spheres that opens up more free space and thus reduces the effect of crowding. In experiments,42,43 the short-range attraction between the colloids was induced by a depletion interaction with added linear chains. The same effect was invoked to explain melting of the solid state of dense star polymer solutions after addition of nonfunctionalized chains.44 For PEO micelles, melting of the solid state could also be induced by adding linear chains.32,45 However, in the present work we found no significant effect of attraction induced by bridging on the critical temperature or concentration of the liquid-solid transition even though it was strong enough to cause phase separation at lower concentrations. The liquid-solid transition of polymeric micelles is well documented, but the mechanism that causes it is still not known. The observation that crystalline order is found in the solid state and not in the liquid state has sometimes been used to argue that crystallization causes the solidlike behavior.2 However, as mentioned in the introduction the fraction of ordered micelles increased with decreasing temperature well into the solid state without a significant change of the low frequency modulus. The definitive proof that the solid is not actually caused by crystallization was the finding that the solid was formed before the crystalline order, at least when the transition was fast.8 The structure factor of the system measured by small-angle X-ray scattering did not change significantly during the liquid-solid transition. The liquid-solid transition may thus be considered as a glass transition due to crowding of the micelles similarly to that observed for hard spheres and star polymers. However, contrary to the latter systems, for PEO micelles the transition is
3006 J. Phys. Chem. B, Vol. 113, No. 10, 2009 discontinuous in the sense that at a critical concentration or temperature of viscosity the system jumps to an immeasurably high value and stabile systems with intermediate viscosities can not be formed. For pure R-PEO, the transition is fast except close to the critical point,8 but it is slowed down when nonfunctionalized PEO is added.32 We found in this study that transient cross-linking of the micelles also led to strong slowing down of the liquid-solid transition. Apparently, the reorganization of the micelles involves the breaking of many cross-links, which is slower at lower temperatures. On the other hand, melting of the solid when heating above Tc was rapid in all systems. During the transition, the terminal relaxation process broadened until at a particular time the same power law frequency dependence of G′ and G′′ was observed at low frequencies. At later times, G′ increased and its frequency dependence weakened, while G′′ decreased. This behavior is similar to that observed during gelation of covalently cross-linking polymers.46,47 These systems show power law frequency dependence at the percolation threshold where a system spanning network is formed, which can be explained in terms of the percolation model.36 The origin of the behavior during the liquid-solid transition is different for PEO micelles because it was not caused by crosslinking. For R,ω-PEO the transient network is fully cross-linked both in the liquid and the solid state, and for R-PEO there are no cross-links, that is, bridging chains in either state. We speculate that domains of dynamically arrested micelles nucleate in the liquid. With time they grow in size and number and at some point percolate the whole space. Since the structure of the dynamically arrested domains is the same as that of the liquid domains, no signature of the transition is seen in scattering experiments. The properties of the transient network were almost the same in the liquid and the solid state. Its elastic modulus was in both states controlled by the number of bridging chains for concentrations much larger than the percolation threshold and was almost independent of the temperature. Cp increased with decreasing F, but if the concentration was high enough a fully cross-linked network could be formed even if the fraction of bridging chains was only about 8%. In most cases, the network was formed at lower concentrations than where the liquid-solid transition occurred at 20 °C. Only for the mixture with the smallest amount of R,ω-PEO was Cp > Cc. In order to relax the stress on a bridging chain, a chain-end needs to escape from the core of a micelle and subsequently relax its stretched configuration. The latter is much faster for the short chains used here so that the escape is the rate limiting factor. The relaxation of the transient network was well described by a single relaxation time in both the liquid and the solid states, but at a given temperature it was significantly shorter in the solid state. We have no explanation for the decrease of τ when the solid is formed. The temperature dependence of τ and thus the viscosity could not be described by a single activation energy over the broad range covered here. Ea was smaller for temperatures below about 40 °C than for higher temperatures. In earlier work, the temperature dependence was described by a single activation energy, but generally a smaller temperature range was covered.19,20 The strong increase of the viscosity with increasing concentration in the liquid state has two origins. The first one is the formation of a transient network leading to η ) τG∞ above the percolation concentration. The second one is crowding of the micelles above the overlap concentration. The first one domi-
Renou et al. nates for micelles with many difunctionalized chains that have a relatively low percolation concentration. The second one dominates when F is very small and Cp is larger than the overlap concentration. In the liquid state, the dynamics of the system slow down either because the effective volume fraction increases, or because the attraction strength increases. This situation is similar to that observed in experiments and computer simulations of colloidal suspensions.40 φe increases with increasing PEO concentration or decreasing temperature, while the attraction increases with increasing fraction of R,ω-PEO. Therefore for the polymeric micelles one can increase the lifetime of the transient bonds (τ) by increasing the alkyl length or decreasing temperature without changing the number of bridging chains and thus the interaction strength. For hard sphere systems, it has been suggested that strong attraction leads to a so-called attractive glass transition even at volume fractions much lower than in the absence of attraction at a critical value of the attraction strength.40 However, recent computer simulations have shown that the dynamics in systems with strong attraction depend on the lifetime of the transient bonds and that the system only becomes fully arrested when the bonds are irreversible.48,49 The effect of transient bond formation on the viscosity becomes important only close to and above Cp when the transient network is formed. In most experiments and simulations the lifetime of the bonds was increased by increasing the attraction strength, but when the attraction is strong Cp is situated below the binodal and the system phase separates before it percolates. In the one phase regime, that is, at lower attraction, the bond lifetime was most often small and therefore the effect of percolation on the dynamics of colloidal suspensions was small. In numerical simulations, the bond lifetime can be easily varied at constant average number of bonds and thus without changing the interaction energy. Such simulations show that the dynamics strongly depend on the bond lifetime if it is long.50,51 Experimentally, it is not so easy to control the bond lifetime independently of the number of bonds. In this sense, solutions of bridging micelles are good model systems to demonstrate that the dynamics are indeed controlled by the lifetime of the bonds. For the system studied here, the effect of percolation on the viscosity was already very strong, but the bond lifetime was still relatively short so that in all cases the liquid state was easily recognized by tilting the sample. However, the bond lifetime can be changed by modifying the hydrophobic end-group6 and dynamical arrest occurs at the percolation threshold if it becomes very long,. For such systems it will not be possible to detect the liquid-solid transition by a change of the dynamics. Conclusions Hydrophobically end-capped PEO chains form spherical micelles in aqueous solution. Mixtures of R-PEO and R,ω-PEO with the same hydrophobic group and the same chain length per group form mixed micelles with the same aggregation number independent of the ratio. R,ω-PEO chains can connect micelles by bridging, which leads to the formation of a crosslinked transient network above the percolation threshold. The elastic modulus of the fully cross-linked network is determined by the number of R,ω-PEO chains. The terminal relaxation time is determined by the lifetime of a bridge and varies little with the concentration or the fraction of R,ω-PEO. Closer to the percolation threshold defects of the network reduce both the elastic modulus and the relaxation time.
Reversibly Cross-linked Polymeric Micelles A liquid-solid transition occurred at a critical concentration that increased with increasing temperature. Transient crosslinking slowed the transition down but had no effect on the critical concentration or temperature. The evolution of the frequency dependence of the shear moduli during the transition was similar to that observed during the sol-gel transition of cross-linking polymers. The visco-elastic properties of the transient network were the same in the liquid and the solid state except that at a given temperature the relaxation time in the solid state was somewhat smaller. References and Notes (1) Alexandridis, P.; Lindman, B. Amphiphilic block copolymers; Self assembly and applications. Elsevier Science: Amsterdam, 2000. (2) Hamley, I. W. Block copolymers in dilute solution; Oxford University Press: Oxford, 1998. (3) Riess, G. Prog. Polym. Sci. 2003, 28, 1107–1170. (4) Winnik, M. A.; Yekta, A. Curr. Opin. Colloid Interface Sci. 1997, 2, 424–436. (5) Grassl, B.; Billon, L.; Borisov, O.; Francois, J. Polym. Int. 2006, 55, 1169–1176. (6) Berret, J. F.; Calvet, D.; Collett, A.; Viguier, M. Curr. Opin. Colloid Interface Sci. 2003, 8, 296–306. (7) Serero, Y.; Aznar, R.; Berret, J.-F.; Porte, G.; Calvet, D.; Collett, A.; Viguier, M. Phys. ReV. Lett. 1998, 81, 5584. (8) Nicolai, T.; Lafle`che, F.; Gibaud, A. Macromolecules 2004, 37, 8066–8071. (9) Francois, J.; Maitre, S.; Rawiso, M.; Sarazin, D.; Beinert, G.; Isel, F. Colloids Surf., A 1996, 112, 251. (10) Roovers, J. Macromolecules 1994, 27, 5359–5364. (11) Cheng, Z.; Zhu, J.; Chaikin, P. M.; Phan, S. E.; Russel, W. B. Phys. ReV. E. 2002, 65, 1–8. (12) Yamazaki, R.; Inomata, K.; Nose, T. Macromol. Chem. Phys. 2002, 203, 2322–2328. (13) Ameri, M.; Attwood, D.; Collett, J. H.; Booth, C. J. Chem. Soc., Faraday Trans. 1997, 93, 2545–2551. (14) Hamley, I. W.; Pople, J. A.; Ameri, M.; Attwood, D.; Booth, C. Colloids Surf., A 1998, 145, 185–190. (15) Lafle`che, F.; Durand, D.; Nicolai, T. Macromolecules 2003, 36, 1331–1340. (16) Semenov, A. N.; Joanny, J. F.; Khokhlov, A. R. Macromolecules 1995, 28, 1066–1075. (17) Pham, Q. T.; Russel, W. B.; Thibeault, J. C.; Lau, W. Macromolecules 1999, 32, 2996–3005. (18) Francois, J.; Beaudoin, E.; Borisov, O. Langmuir 2003, 19, 10011– 10018. (19) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. J. Rheol. 1993, 37, 695–726. (20) Lafle`che, F.; Durand, D.; Nicolai, T.; Gnanou, Y.; Taton, D. Macromolecules 2003, 36, 1341–1348. (21) Hough, L. A.; Ou-Yang, H. D. Phys. ReV. E. 2006, 73, 031802. (22) Ng, W. K.; Tam, K. C.; Jenkins, R. D. J. Rheol. 2000, 44, 137.
J. Phys. Chem. B, Vol. 113, No. 10, 2009 3007 (23) Indei, T. J. Chem. Phys. 2007, 127, 144904. (24) Kelarakis, A.; Yuan, X.; Mai, S. M.; Yang, Y.; Booth, C. Phys. Chem. Chem. Phys. 2003, 5, 2628. (25) Mistry, D.; Annable, T.; Yuan, X. F.; Booth, C. Langmuir 2006, 22, 2986–2992. (26) Mortensen, K.; Brown, W.; Jorgensen, E. Macromolecules 1994, 27, 5654–5666. (27) Ricardo, N. M. P. S.; Honorato, S. B.; Yang, Z.; Castelletto, V.; Hamley, I. W.; Yuan, X.; Attwood, D.; Booth, C. Langmuir 2004, 20, 4272– 4278. (28) Kelarakis, A.; Ming, X.; Yuan, X.; Booth, C. Langmuir 2004, 20, 2036–2038. (29) Kelarakis, A.; Havredaki, V.; Yuan, X.; Yang, Y.; Booth, C. J. Mater. Chem. 2003, 13, 2779–2784. (30) Renou, F.; Nicolai, T.; Nicol, E.; Benyahia, L. Langmuir 2009, 25 (1), 515-521. (31) Sommer, C.; Pedersen, J. S.; Stein, P. C. J. Phys. Chem. B 2004, 108, 6242–6249. (32) Renou, F.; Nicolai, T.; Benyahia, L. Macromolecules 2007, 40, 4626–4634. (33) Mark, J. E.; Erman, B. Rubberlike Elasticity: A Molecular Primer; Cambridge University Press: Cambridge, 2007. (34) Nicolai, T.; Benyahia, L. Macromolecules 2005, 38, 9794–9802. (35) Watanabe, H.; Kanaya, T.; Takahashi, Y. Macromolecules 2001, 34, 662–665. (36) Rubinstein, M.; Colby, R. H.; Gillmor, J. R. Dynamic scaling of polymer gelation. In Space time Organization in Macromolecular Fluids Springer Series in Chemical Physics; Tanaka, F., M.D.; Ohta, T., Eds.; Springer-Verlag: Berlin and Heidelberg, 1989; Vol. 51, pp 66-74. (37) Noro, M. G.; Frenkel, D. J. Chem. Phys. 2000, 113, 2941–2944. (38) Babu, S.; Gimel, J. C.; Nicolai, T. J. Chem. Phys. 2006, 125, 184512. (39) Foffi, G.; De Michele, C.; Sciortino, F.; Tartaglia, P. Phys. ReV. Lett. 2005, 94, 078301. (40) Sciortino, F.; Tartaglia, P. AdV. Phys. 2005, 54, 471–524. (41) Likos, C. N. Soft Matter 2006, 2, 478–498. (42) Eckert, T.; Bartsch, E. Phys. ReV. Lett. 2002, 89, 125701. (43) Pham, K. N.; Egelhaaf, S. U.; Pusey, P. N.; Poon, W. C. K. Phys. ReV. E 2004, 69, 054501–054507. (44) Stiakakis, E.; Vlassopoulos, D.; Likos, C. N.; Roovers, J.; Meier, G. Phys. ReV. Lett. 2002, 89, 208302–208301. (45) Yamazaki, R.; Numasawa, N.; Nose, T. Polymer 2004, 45, 6227– 6237. (46) Winter, H. H.; Chambon, F. J. Rheol. 1986, 30, 367–382. (47) Nicolai, T.; Randrianantoandro, H.; Prochazka, F.; Durand, D. Macromolecules 1997, 30, 5897–5904. (48) Babu, S.; Gimel, J.-C.; Nicolai, T. J. Chem. Phys. 2007, 127, 054501–054507. (49) Zaccarelli, E.; Saika-Voivod, I.; Buldyrev, S. V.; Moreno, A. J.; Tartaglia, P.; Sciortino, F. J. Chem. Phys. 2006, 124, 124901–124914. (50) Coniglio, A.; de Arcangelis, L.; del Gado, E.; Fierro, A.; Sator, N. J. Phys.: Condens. Matter 2004, 16, S4831–S4839. (51) Saika-Voivod, I.; Zaccarelli, E.; Sciortino, F.; Buldyrev, S. V.; Tartaglia, P. Phys. ReV. E 2004, 70, 141401–141408.
JP8100442
