Temperature-Induced Growth of Wormlike Copolymer Micelles

Above about 41 °C, the micellar aggregates become denser as the cloud point is ... contribution from the stretched PPO blocks to the free energy per ...
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Langmuir 2005, 21, 4904-4911

Temperature-Induced Growth of Wormlike Copolymer Micelles M. Duval, G. Waton, and F. Schosseler* Institut Charles Sadron, UPR22 (CNRS-ULP), 6 rue Boussingault, 67083 Strasbourg Cedex, France Received January 21, 2005. In Final Form: March 16, 2005 We study the temperature-induced growth of polymer micelles based on Pluronic P84 in brine (2 M NaCl) using small-angle neutron scattering, static and dynamic light scattering, and viscometry as a function of temperature and polymer concentration. Spherical micelles below 30 °C are shown to grow between about 30 and 40 °C into wormlike micelles long enough to enter the semidilute regime for polymer volume fraction larger than 0.005. The entanglements in this regime are responsible for a huge increase in the viscosity. Above about 41 °C, the micellar aggregates become denser as the cloud point is approached and the viscosity drops.

Triblock copolymers PEO-PPO-PEO based on poly(ethylene oxide) (PEO) and poly(propylene oxide) (PPO) are widely used due to their amphiphilic behavior in aqueous solutions and to their inocuity. Water is a selective solvent for the PEO blocks in the temperature range 0-100 °C, and their hydrophilic/hydrophobic balance can be varied by changing the PEO/PPO molar ratio and the temperature.1-3 Above the critical micelle concentration (cmc) and the critical micellar temperature (cmt), the copolymer chains form micelles with a PPO core and a PEO corona.4 The cmc and the cmt are dependent on the total molecular weight1-3 and on the concentration of added components like, e.g., monovalent salts.5-7 The structure of the micelles has been studied in detail with scattering techniques.4 At the transition, the most common scenario for the unimers is to form spherical micelles with the aggregation number being an increasing function of the temperature.8,9 As the radius of the core increases, there is a growing entropic contribution from the stretched PPO blocks to the free energy per micelle. Hence the aggregation number cannot increase indefinitely and a spheroidal or cylindrical shape can be expected8,9 if the size of the PEO blocks is not too large compared to that of the PPO block.10 This is basically the same packing parameter criterion as the one used for conventional surfactants.11 Clear experimental evidence for the growth of elongated micelles has been provided by scattering techniques for the commercial Pluronic copolymers P85 (Mw ) 4600 g/mol, PEO weight fraction ≈

0.5)12-14 and P84 (Mw ) 4200 g/mol, PEO weight fraction ≈0.4).15-18 On the other hand, samples with about the same length of PPO block as P84 and P85 but higher or lower PEO weight fraction do not appear to form elongated micelles.12,17,19 This micellar growth followed by the progressive building of a transient network of entangled wormlike polymeric micelles was held responsible for the spectacular increase in viscosity recently reported for P84 solutions:16,18 upon heating a solution with about 0.04 w/w polymer concentration in 2 M NaCl/H2O solvent, an increase of the zero shear viscosity by a factor about 105 was observed between T ≈ 30 °C and T ≈ 40.5 °C, before a drop in viscosity upon further heating to the cloud point at T ≈ 43 °C.18 This behavior remained unnoticed in previous rheological studies on P84 in pure water.20 Instead the observed moderate increase in viscosity could be accounted for by a model of sticky spherical micelles20 with parameters measured independently in small-angle neutron scattering (SANS) experiments.21 Even in the case of P85 solutions undergoing a sphere-to-rod transition, only moderate viscosity increases by typically 1 order of magnitude have been reported at low concentrations.14 Motivated by this intriguing peculiarity of P84 in brine solutions, we have performed an extensive characterization of the micellar growth in these systems as a function of polymer concentration and temperature, by combining static and dynamic light scattering, SANS, and viscometric measurements. In this way, we aim to check the consistency of the picture of entangled wormlike polymeric micelles and to understand the origin of the maximum in

(1) Wanka, G.; Hoffmann, H.; Ulbricht, W. Colloid Polym. Sci. 1990, 268, 101 and references therein. (2) Alexandridis, P.; Holzwarth, J. F., Hatton, T. A. Macromolecules, 1994, 27, 2414 and references therein. (3) Kozlov, M. Y.; Melik-Nubarov, N. S.; Batrakova, E. V.; Kabanov, A. Macromolecules 2000, 33, 3305 and references therein. (4) for a recent review, see Mortensen, K. Colloids Surf., A 2001, 183-185, 277. (5) Almgren, M.; Alsins, J.; Bahadur, P. Langmuir 1991, 7, 446. (6) Bahadur, P.; Li, P.; Almgren, M.; Brown, W. Langmuir 1992, 8, 1903. (7) Alexandridis, P.; Holzwarth, J. F. Langmuir 1997, 13, 6074. (8) Brown, W.; Schille´n, K.; Almgren, M.; Hvidt, S.; Bahadur, P. J. Phys. Chem. 1991, 95, 1850. (9) Mortensen, K.; Pedersen, J. S. Macromolecules 1993, 26, 805. (10) Linse, P. J. Phys. Chem. 1993, 97, 13896. (11) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: San Diego, CA, 1991.

(12) Mortensen, K.; Brown, W. Macromolecules 1993, 26, 4128. (13) Schille´n, K.; Brown, W.; Johnsen, R. M. Macromolecules 1994, 27, 4825. (14) Jørgensen, E. B.; Hvidt, S.; Brown, W.; Schilln, K. Macromolecules 1997, 30, 2355. (15) Liu, Y.; Chen, S.-H.; Huang, J. S. Macromolecules 1998, 31, 6226. (16) Michels, B.; Waton, G.; Zana, R. Colloids Surf., A 2001, 183185, 55. (17) Aswal, V. K.; Goyal, P. S.; Kohlbrecher, J.; Bahadur, P. Chem. Phys. Lett. 2001, 349, 458. (18) Waton, G.; Michels, B.; Steyer, A.; Schosseler, F. Macromolecules 2004, 37, 2313. (19) Jain, N. J.; Aswal, V. K.; Goyal, P. S.; Bahadur, P. J. Phys. Chem. B 1998, 102, 8452. (20) Liu, Y.; Chen, S.-H.; Huang, J. S. Phys. Rev. E 1996, 54, 1698. (21) Liu, Y.; Chen, S.-H.; Huang, J. S. Macromolecules 1998, 31, 2236.

I. Introduction

10.1021/la050177c CCC: $30.25 © 2005 American Chemical Society Published on Web 04/23/2005

Growth of Copolymer Micelles

the viscosity vs temperature curve. After the Experimental Section, the paper will present first the structure of the micellar solutions, then their dynamics, and finally their rheological properties. In the final section, we discuss the results and show that a consistent picture is indeed obtained if the temperature induces the growth of giant wormlike micelles that are more or less entangled depending on the concentration. In this picture, the viscosity drop above T ≈ 40 °C can be explained by a collapse of these wormlike micelles. II. Experimental Section Samples. We used without further purification the commercial Pluronic P84 supplied by BASF. The total molecular weight is 4200 g/mol, and the nominal stoechiometry is PEO19- PPO43 -PEO19. In the absence of added salt, the cmt and the cloud point temperatures are around 20 and 75 °C, respectively. With 2 M added NaCl, the microcalorimetric measurements16 show that, for a 0.04 w/w concentration, the cmt and the cloud point are at about 2 and 43 °C, respectively. The onset of micellar growth is shifted down from 55 to about 30 °C.16 In this temperature range the temperature regulation of the solution is easier and the evaporation is minor. Stock solutions were prepared at C ) 0.04 g/cm3 and then stored in the dark at 4 °C for further use. Solutions at lower concentration were prepared by dilution of the stock solution with the brine on a weight fraction basis. The average density of the copolymer (see below) is close to that of the brine and concentrations in weight fraction are very close to concentrations in g/cm3. They are also equivalent to the volume fraction φ of the polymer with a relative error less than 0.05. Except for the special case of the normalization of scattering intensities, we will not distinguish in the text the numerical values of φ and C. SANS. Experiments were performed on the spectrometer D11 at the Institut Laue Langevin, Grenoble. A unique wavelength of the incident neutrons, λ ) 13 Å, and three different distances D between sample and detector were used, D(m) ) 1.5, 5, and 21, thus covering a domain in transfer wavevector q between 1.42 × 10-3 and 0.115 Å-1. The samples were dissolved in 2 M NaCl/D2O to provide a good contrast. They were measured in rectangular Hellma cells with a 5 mm optical path for different temperatures. To provide enough equilibrium time, they were placed in advance in a thermostated bath at the temperature of study about 3 h before the measurement. Microcalorimetric experiments on 0.04 g/cm3 in 2 M NaCl/D2O ensured that no noticeable shift of the characteristic transition temperatures was induced by the deuteration of the solvent. The data were treated according to standard procedures and put on an absolute intensity scale by use of a H2O standard.22,23 The absolute scattering cross section per unit volume I(q) is related to the structure function 2 2 S(q), where KSANS is the contrast factor. S(q) by I(q) ) KSANS The contrast factor in the SANS experiments can be estimated with only a reasonable accuracy. The specific volumes of PEO and PPO depend on the temperature24,25 and should also depend on the salt concentration. Specific volume data in the presence of added NaCl have not been published as far as we know. Therefore we have adopted for all temperatures the unique values VPEO ≈ 0.84 cm3/g and VPPO ≈ 1.02 cm3/g, estimated from data measured in pure H2O.25 This yields the values for the scattering length densities FPEO ≈ 0.674 × 1010 cm-2 and FPPO ≈ 0.337 × 1010 cm-2. The scattering length density of pure D2O at 20 °C is FD2O ≈ 6.32 × 1010 cm-2, but we have to take into account the contribution of the salt. To do this appropriately, the electro-

(22) Lindner, P. J. Appl. Crystallogr. 2000, 33, 807. (23) For the differential scattering cross section of water, a calibration relation different from the one in ref 22 was used to account for the change of detector on the D11 instrument in May 1999: dΣ/dΩ (cm-1) ) 0.74158 + 0.022078λ + (4.7932 × 10-4)λ2 + (6.3763 × 10-5)λ3 (P. Lindner, private communication). (24) Armstrong, J. K.; Parsonage, J.; Chowdhry, B.; Leharne, S.; Mitchell, J.; Beezer, A.; Lo¨hner, K.; Laggner, P. J. Phys. Chem. 1993, 97, 3904. (25) Sommer, C.; Pedersen, J. S.; Stein, P. C. J. Phys. Chem. B 2004, 108, 6242.

Langmuir, Vol. 21, No. 11, 2005 4905 striction exerted by the hydrated Na+ ion must be considered.26 The number of water molecules involved in the hydration shell is still under debate,27-29 and we used a typical hydration value of 5 for an estimated total volume of the hydrated ion of 68.7 cm3/mol.27 The Cl- ion can be assigned a molar volume about 22.8 cm3/mol.31,32 Using these values, we obtain Fbrine ≈ 6.48 × 1010 cm-2 for the scattering length density of the brine. In view of this large value compared to those of the polymer constituents and considering the scope of this paper, we used a mean value Fpol ≈ 0.456 × 1010 cm-2 calculated with an average specific volume of the chains of Vpol ≈ 0.95 cm3/g, estimated on the basis of the weight fraction of the constituents in the whole chain. A simple estimation for the error introduced by this approximation can be obtained by using a two-shell model for the micelles33 and assuming constant polymer densities in the core (PPO) and the corona (PEO). Then the scattering intensity at q ) 0 is simply proportional to [(FPEO - Fbrine) + φcore(FPPO - FPEO)]2, where φcore is the volume fraction occupied by the core, which should be close to the weight fraction of PPO in this model and thus about 0.6 for P84. The preceding expression can be evaluated with either the above values of FPEO and FPPO or FPEO ) FPPO ) Fpol, and it turns out that the relative difference is less than 0.006. Liu et al.21 used a much more elaborate model allowing for a polymer concentration profile in the corona and considered also that the possible intermixing of the PEO and the PPO in the core and the corona was not measurable, due to the too similar scattering length densities of both polymer species compared to that of the deuterated solvent. We will use in the following the value KSANS ) Fpol - Fsolv ≈ -6.03 × 1010 cm-2. Light Scattering. The static (SLS) and dynamic (DLS) measurements were performed in the angular range 22° < θ < 145° using the ALV/DLS/SLS-5020F experimental setup (ALVLaser Vertriebsgesellshaft mbH, Langen, Germany) consisting of an He-Ne laser (22 mW, λ0 ) 6328 Å), a compact ALV/CGS-8 Goniometer system, and an ALV-5000 autocorrelator. This angular range corresponds to scattering vectors q between 5 × 10-4 and 2.5 × 10-3 Å-1. The solutions used in this study were cleaned by filtration at room temperature through membrane filters with a pore size of 0.45 µm (Millex LCR hydrophilic; membrane, poly(tetrafluoroethylene); Millipore Co., Molsheim, France) into cylindrical glass cells with a diameter of 10 mm. The values of the absolute scattering cross section I(q) were obtained through the calibration with a toluene standard (Rayleigh ratio ≈ 1.368 × 10-5 cm-1). The structure function S(q) can again be calculated from I(q) by using the appropriate contrast factor K2LS where

dn (dC )/V

KLS ) (2π/λ0)ntol

pol

with ntol ≈ 1.5 being the refractive index of toluene and dn/dC an average refractive index increment of the copolymer relative to the solvent. This expression of KLS includes the required correction of scattering volume for cylindrical cuvettes and is valid for vertically polarized incident light. In defining KLS, we have neglected the small differences in the refractive index increment of each component of the copolymers. This approximation is consistent with that used to estimate KSANS. We will neglect as well in this paper the weak variation of the refractive index increment with temperature15 and adopt the mean value dn/dC ≈ 0.12 cm3/g measured for P84 in H2O solutions.15 The normalized autocorrelation functions g(2)(q,t) of the light scattering intensity were analyzed following34 (26) Heinrich, M. The`se Universite´ Louis Pasteur, Strasbourg, 1998. (27) Zana, R.; Yeager, E. In L’eau et les syste` mes biologiques, Colloques internationaux du CNRS, Editions du CNRS, Paris, 1976. (28) Ohtaki, H.; Radnai, T. Chem. Rev. 1993, 93, 1157. (29) Rempe, S. B.; Pratt, L. R. Fluid Phase Equilib. 2001, 183-184, 121. (30) Carrillo-Tripp, M.; Saint-Martin, H.; Ortega-Blake, I. J. Chem. Phys. 2004, 118, 7062. (31) Zana, R.; Yeager, E. J. Phys. Chem. 1967, 71, 521. (32) Millero, F. J. Chem. Rev. 1971, 71, 147. (33) Hartley, G. S. Q. Rev. Chem. Soc. 1948, 2, 152.

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∫ exp(-t/τ)P(τ) dt]

2

using the CONTIN routine35 built in the ALV software, where t is the time, P(τ) is the distribution function of the relaxation times τ, and β ≈ 1 is the coherence factor of the instrument. An apparent hydrodynamic length Rh,app in solution was calculated through the Stokes-Einstein relationship

Rh,app ) q2

kBT τ(q) 6πηs

where kB is the Boltzmann constant, T the absolute temperature, and ηs the viscosity of the brine. It is about 1.21 ( 0.02 times that of pure water at the same temperature between 18 and 40 °C.36 For each set of experimental conditions, several measurements were done and analyzed to test the robustness of the fitted distribution of relaxation times. Rheology. Measurements were performed using concentric cylinders geometry (Couette). A home-built thermostated cell was installed on a regular rheometer (Thermo Haake, RS1). A full description of the cell was given elsewhere.18 The inner cylinder is rotating, and the outer static cylinder is surrounded by a cylindric bath with circulating water. The temperature is measured by a Pt probe located inside the massive aluminum base of the bath. The rheometer is by construction a stresscontrolled device but can be operated by software in the shearcontrolled mode, allowing a great versatility in experiments. Inertia corrections37 included in the software operating the rheometer have been applied when necessary. The shear viscosity was measured in the stress-controlled mode (σ ) 0.2 Pa) for the highest concentrations. However for the more dilute solutions below 0.015 w/w, applying a constant stress resulted in a high initial shear rate that was found to decrease progressively in time. On the other hand, applying a small constant shear rate about 0.08 s-1 gave a steady behavior and more reproducible results. Thus it is likely that these samples exhibit some shear thickening phenomena that we leave for future work. Typically the temperature variation of the shear viscosity was measured with temperature increments of 5, 1, or 0.5 °C, depending on the steepness of the viscosity variation at the given T. The change in temperature was fairly quick (about 1 or 2 min), but the system was allowed to rest for 15 min before the measurement of the viscosity was performed. In these conditions, a steady behavior could generally be obtained within the 15 min imparted to the rheometric test itself. For a few experimental conditions, we checked that the rheometric tests were performed in the Newtonian regime.

III. Results 1. Static Scattering. Figure 1 shows the evolution with temperature of the SLS intensity I(q) for a solution with polymer concentration C ) 5 × 10-3 w/w. For all the investigated concentrations (C ) 10-3, 5 × 10-3, 4 × 10-2 w/w), the general trend is the same, i.e., an increase of the average intensity level with temperature whereas the q dependence of I(q) becomes stronger. This behavior is more pronounced as the concentration decreases. More quantitative information can be obtained through the Zimm analysis38 of the inverse scattering intensity (Figure 2). In the q window where the Zimm representation gives a straight line, I-1(q) can be fitted as I-1(q) ) I0-1(1 + q2R2app/3), qRapp < 1, yielding the intensity extrapolated to q ) 0, I0, and an apparent radius of gyration, Rapp. It can be noted that, for φ ) 5 × 10-3, the Zimm behavior (34) Berne, B. J.; Pecora R. Dynamic Light Scattering with Applications to Chemistry, Biology and Physics; Wiley-Interscience: New York, 1976. (35) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 229. (36) International Critical Tables of Numerical Data, Physics, Chemistry and Technology, 1st electronic ed.; Washburn, E. W., Ed.; Knovel: Norwich, NY, 2003; Vol. V, p 15. (37) Krieger, I. J. Rheol. 1990, 34, 471. (38) Zimm, B. H. J. Chem. Phys. 1948, 16, 1093.

Figure 1. Variation with temperature of the light intensity I(q) scattered from a solution with φ ) 5 × 10-3. The curves are labeled with the temperature in °C.

Figure 2. Zimm representation of a few intensity curves (φ ) 5 × 10-3). The straight lines correspond to fits in the Guinier regime. The label of each curve is the temperature in °C.

extends through the whole q range at T ) 33.9 °C and is restricted to a narrower window at T ) 39.8 °C, with a downward departure of the data points at large q values. At T ) 42.7 °C, the deviation from the Zimm behavior is an upward curvature. This change of curvature will be discussed later on. Figure 3 displays the temperature variation of I0 (Figure 3a) and Rapp (Figure 3b). Both parameters show first an initial moderate increase up to temperatures about 30 °C and then a much steeper increase. This is less visible in Figure 3b because the static light scattering (SLS) technique does not allow one to measure accurately Rapp values below 200 Å. For such small values, the angular variation of the inverse scattering intensity is lower than 8% in the q window explored by our setup. Therefore the corresponding data below 30 °C are dispersed. In Figure 3a, the limiting slopes describing the regimes of moderate and steep increases can be used to define the onset of steep micellar growth at T ) Tmg. This temperature is a decreasing function of the polymer concentration. The regime of steep intensity increase extends up to T ≈ 40 °C for the most dilute solution (10-3 w/w). This is true as well for the increase of Rapp (not shown in Figure 3b for the sake of clarity). On the other hand, as the polymer concentration increases the growth in I0 and Rapp tends to slow below 40 °C. Finally for temperatures above 40 °C, very different behaviors are observed according to the concentration. In the most concentrated solution (0.04 w/w), the turbidity increases strongly, hence probably the decrease of I0, but Rapp does not change. The present experimental setup does

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Figure 3. Variation with temperature of the parameters obtained from Zimm analysis: (a) intensity extrapolated to q ) 0; (b) apparent radius of gyration.

Figure 4. Plot of the light intensity extrapolated to q ) 0 vs the apparent radius of gyration. These quantities have been measured as a function of temperature in the interval 18 < T (°C) < 47.2 (data for φ ) 5 × 10-3 in Figure 3). The straight line is a power law fit with the indicated slope.

not allow one to measure the turbidity, but experiments on the setup described in ref 18 show that the transmission of a 2 mm thick sample decreases from about 1 at T ) 20 °C to 0.66 at T ) 44.5 °C. Thus multiple scattering effects are likely for the 10 mm cell used here. Due to this limitation, we did not pursue a detailed investigation in this temperature range for the higher concentration. For the intermediate concentration (5 × 10-3 w/w), the intensity displays an abrupt increase followed by a moderate one while the apparent radius of gyration first decreases and then becomes level. Finally, in the most dilute solution (10-3 w/w), both the intensity and the radius of gyration drop down at 41.8 °C. Inspection of the scattering cell showed the existence of sedimentation. A plot of I0 vs Rapp (Figure 4) reflects the three regimes of micellar growth discussed up to now for the solution with C ) 5 × 10-3. Below T ) 30 °C, the measured Rapp values are too small to be reliable (Rapp < 200 Å). Then up to T ≈ 40 °C, the data can be described by a power law, I0 ∼ R1.58(0.05 . Above T ≈ 40 °C, the opposite monotonic app variations of I0 and Rapp result in a strong departure from the power law observed between 30 and 40 °C. When C ) 10-3 w/w, the same power law behavior is observed in the second regime with an exponent 1.72 ( 0.07. Finally, Figure 5 completes the picture for the growing micellar aggregates by combining SLS and SANS data measured for three temperatures. Measurements were done on a solution of P84 in D2O/2 M NaCl at a volume fraction φ ) 5 × 10-3. At this concentration, the aggregates

Figure 5. Temperature variation of the structure function S(q) for φ ) 5 × 10-3. Each curve combines SANS and light scattering data. See text for details.

are always in the dilute regime whatever the temperature. In particular, the maximum zero-shear viscosity is 1 order of magnitude lower than the value extrapolated from the values measured at higher concentrations where the elongated micelles become entangled (see below). The structure functions S(q) were calculated from the absolute scattering cross section per unit volume as explained above. The rather good superposition of the SLS and SANS data in their overlapping q windows shows that our estimation of the contrast factors is not too bad. It confirms also that it is a reasonable approximation to consider the polymer micelles as one-component particles in the contrast conditions used here. It would not be the case in small-angle X-ray scattering experiments.25 It can be noticed that the worse superposition is obtained for T ≈ 34 °C, in the regime where the growth of I0 and Rapp is the steepest. Thus a small difference in the actual temperature during the SLS and SANS experiments could well explain this mismatch. We tried to match in the SLS experiments the thermal protocol used in SANS experiments and found that the scattering intensity reached an equilibrium value about 15 min after the temperature at this concentration. However, repeating the temperature from different initial temperatures or after different cooling histories revealed some sensitivity of the scattering intensity to the whole thermal history. The error bars in Figure 5 for the SLS results at T ) 34.1 °C are reflecting the variations recorded upon varying the thermal history. In the dilute solution investigated here, we can neglect the interactions between aggregates. If homogeneous

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aggregates are assumed, then the structure function is simply given by S(q) ) AφVP(q), where φ is the volume fraction occupied by the aggregates, V their volume, P(q) their form factor (P(0) ) 1), and A a dimensionless constant that should be close to 1 if the contrast has been correctly estimated and the model used for P(q) is fully consistent with the data. At T ) 25.0 °C, a value Rapp ≈ 41 Å is obtained from a Zimm analysis. Using for P(q) the form factor of homogeneous spheres with radius R, a reasonable fit of the data can be obtained on the whole q range with R ) x5/3 Rapp ≈ 53 Å and A ≈ 0.77.39 The latter value is not surprising since the spherical micelles are not homogeneous and there is a radial polymer concentration profile.21 This means that our expression for the structure function is only an approximation that provides order of magnitude for the dimensions of the polymer micelles. More elaborate models have been used with consistent results20,21,40 but are out of the scope of the present paper. Our values are nevertheless consistent with a core radius about 38-44 Å, as reported by Liu et al.21 Finally the aggregation number N ≈ 77 can also be estimated from the volume V of the spheres and the average specific volume Vpol. As the temperature increases, the shape of S(q) is evolving and, at 40.0 °C, the characteristic form factor of wormlike chains is obtained.41 The vertical arrow in Figure 5 indicates the cross over at q ≈ 5.5 × 10-3 Å-1 between the q-1 behavior at higher q and a larger slope at smaller q values. This cross over is a measure of the persistence length l ≈ 640 Å since it occurs for ql ) 3.5.42 Then the classical Benoit-Doty formula43

R02/l2 )

[

x 3 6 6 1 - + 2 - 3(1 - e-x) 3 x x x

Figure 6. Typical distributions of relaxation times measured in dynamic light scattering experiments.

]

x ) L/l allows one to estimate the contour length of the micelles from the values of l and of the radius of gyration Rapp ≈ 1610 Å for these data. In the above formula, R02 is the squared radius of gyration for Gaussian statistics. We get LG ≈ 14200 Å for Gaussian chains and a smaller value Lexcl ≈ 12300 Å if their swelling by excluded volume effects is allowed.41 The decay of S(q) at large q values is governed by the finite cross section of the micelles. We can get an estimate for the transverse apparent radius of gyration Rt,app by fitting the quantity qS(q) as qS(q) ) 2 /2)39,42 for q values above 5.5 × 10-3 Å. B exp(-q2Rt,app This yields Rt,app ≈ 35 Å, which would correspond to a radius Rt ) 21/2Rt,app ≈ 50 Å for an homogeneous cylinder. Finally we can also try to adjust the data for q > 5.5 × 10-3 Å with the regular form factor of an homogeneous straight cylinder with radius Rcyl and length Lcyl. The values Rcyl ≈ 45 Å, Lcyl ≈ 12000 Å, and A ≈ 0.77 provide both a fair description of the data at large q and an intercept at q ) 0 consistent with the experimental value,39 from which we can estimate N ≈ 6350 at T ) 40.0 °C. With these estimations, the ratio Rcyl/l is about 0.07, rather different from the value 0.2 used by Pedersen and (39) The fits to the experimental data are available as Supporting Information via the Internet at http://pubs.acs.org. (40) Pedersen, J. S.; Svaneborg, C. Curr. Opin. Colloid Interface Sci. 2002, 7, 158 and references therein. (41) Pedersen, J. S.; Schurtenberger, P. Macromolecules 1996, 29, 7602. (42) Rawiso, M.; Duplessix, R.; Picot, C. Macromolecules 1987, 20, 630. (43) Benoit, H.; Doty, J. Phys. Chem. 1953, 57, 958.

Figure 7. Variation with temperature of the exponent R characterizing the scaling of the main relaxation time with scattering wavevector.

Schurtenberger in their Monte Carlo simulations.41 Therefore we cannot use their interpolation functions to analyze more quantitatively our data. 2. Dynamic Light Scattering. The CONTIN analysis of the intensity correlation functions results most of the time in one well-defined relaxation mode. Although this mode represents usually more than 90% of the decay, its shape can vary significantly according to the experimental conditions (Figure 6a,b,e). In some cases (Figure 6c,d), small amplitude peaks corresponding to relaxations faster or slower than the main relaxation can be present. However, although these modes appear rather reproducible for one set of experimental parameters {C, T, q}, it was not possible to obtain a significant characterization of their variation with one of these parameters. As an example, Figure 6b,c shows that the two additional relaxations that are evident for the scattering angle θ ) 50° vanished for θ ) 70°, probably resulting in the very broad single relaxation mode. We have therefore restricted our analysis to the main decay time that scales as τ ∼ q-R, where the exponent R is about 2 for C ) 0.04 w/w. The most significant result of the DLS experiments is the variation with temperature of the exponent R for C ) 10-3 and for C ) 5 × 10-3 w/w (Figure 7). The dispersion of the data reflects more the fluctuations linked to the thermal history before each measurement than the error on the values of R that are typically less than 0.05. In particular, at T ≈ 34 °C, the fluctuations in R reflect the fluctuations in the SLS intensity depicted in Figure 5.

Growth of Copolymer Micelles

Figure 8. Variation with temperature of the apparent hydrodynamic length deduced from DLS decay time at a given scattering angle by using the Stokes-Einstein relationship. See text for details.

Figure 9. Variation of the zero-shear viscosity measured as a function of temperature for different polymer concentrations.

The continuous line in Figure 7 is a curve fit that attempts to smooth these fluctuations. Starting from R ≈ 2 below T ≈ 34 °C, the exponent increases up to a maximum about 3 in the interval 34 < T < 40, and then decreases back to about 2.2 (40 < T < 47). When R * 2, it is only possible to define an apparent diffusion coefficient Dapp(q) ) [q2τ(q)]-1 and, through the Stokes-Einstein relation (see Experimental Section), an apparent hydrodynamic length Rh,app(q). This quantity is plotted in Figure 8 as a function of the temperature for two scattering angles, θ ) 40° and θ ) 130°. For the most concentrated solution (C ) 0.04 w/w, closed symbols), since R ≈ 2 at all temperatures, the values of Rh,app coincide and define a true hydrodynamic length, which is about 65 Å below about 27 °C and increases up to about 220 Å at T ≈ 32 °C where it stays at this level up to T ≈ 40 °C. For C ) 5 × 10-3 w/w, R ≈ 2 up to T ≈ 34 °C (Figure 7) and Rh,app does not depend on the scattering angle in this range of temperatures. It is about 71 Å below 30 °C. At T ≈ 34 °C, Rh,app has increased to 300 Å and, upon further increase of the temperature, Rh,app(θ ) 40°) and Rh,app(θ ) 130°) split more and more as the value T ≈ 40 °C is approached. Above this temperature, the decrease in R reduces the difference between the two Rh,app values. For the purpose of the discussion later on, we note that, at T ≈ 34 °C, Rapp ≈ 500 Å, which would correspond to Lcyl ≈ 1750 Å and Lexcl ≈ LG ≈ 2300 Å, if l ≈ 640 Å is assumed to hold at this temperature. Finally, at C ) 10-3 w/w, the general behavior is the same as at C ) 5 × 10-3 w/w, with Rh,app ≈ 78 Å below 30 °C.

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Figure 10. Variation as a function of polymer concentration of the zero-shear viscosity interpolated at T ) 40.5 °C from the data in Figure 9.

3. Rheometry. Figure 9 shows the zero shear viscosity of the P84 solutions as a function of temperature. The behavior that was previously reported for C ) 0.04 w/w18 is seen to be reproduced qualitatively at all concentrations down to C ) 5 × 10-3 w/w. The viscosity curves start to take off at T ≈ 30 °C and increase to a maximum for T ≈ 40 °C before decreasing by a large factor for a further increase of 3 °C in temperature. The amplitude of the viscosity changes increases considerably with the polymer concentration: for C ) 0.04, the increase factor is nearly 105 and the drop factor is larger than 20. In Figure 10, we have plotted as a function of concentration the viscosity values interpolated at T ) 40.5 °C, close to the maximum. Except for the lower concentration (C ) 5 × 10-3 w/w), the data are well described by a power law η0 ∼ C3.25(0.05. IV. Discussion and Conclusions We discuss first the behavior for temperatures below about 40 °C. In this regime, there is convincing evidence for the growth of large aggregates above T ≈ 30 °C, provided by the increase of I0 and Rapp (Figures 1 and 3). As already noted by Michels et al.,16 the temperature of the onset of micellar growth decreases weakly as the polymer concentration increases (Figure 3a) and this corresponds to a weaker entropic penalty associated with aggregation at higher concentration. A first hint to the shape of these aggregates is given by the relationship between I0 and Rapp (Figure 4). Assuming no dramatic variation of the refractive index increment in the interval 30 < T (°C) < 40, for the most dilute solutions, I0 is proportional to the weight-average molecular weight of the aggregates and R2app is close to the z-average of their squared radius of gyration.44 For monodisperse polymer chains in good solvent conditions, the slope in Figure 4 should be 1/0.588 ≈ 1.70,45 compared with 1.58 ( 0.05 (φ ) 5 × 10-3) and 1.72 ( 0.07 (φ ) 10-3) for the present micellar aggregates. Although polydispersity effects as well as small shifts of the refractive index increment with temperature can affect the experimental value of the exponent, the similarity of the values bears enough significance to exclude rigid rod behavior. It is easily checked that even an exponential distribution of lengths n(L) ) exp(-L/L h )/L h does not change this conclusion. Since the moments of this distribution are simply p 〈Lp〉 ) ∫+∞ h p, it follows that the scaling I0 0 n(L)L dL ) p!L ∼ Rapp should be still obeyed for rigid rods. (44) Zimm, B. H. J. Chem. Phys. 1948, 16, 1099.

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This is confirmed by the shape of the form factor measured at T ) 40.0 °C (Figure 5). This shape is typical for wormlike chains41 and is consistent with l ≈ 640 Å and Lexcl ≈ 12300 Å. The aggregates are large enough at this temperature (Lexcl/l ≈ 20) to exhibit a coillike conformation rather than a rodlike shape. One could therefore expect a rodlike behavior at lower temperatures, where the aggregates are shorter. Indeed the S(q) measured at T ) 34.1 °C can be reasonably described by the form factor of a straight homogeneous cylinder with Lcyl ≈ 1300 Å, Rcyl ≈ 42 Å and A ) 0.77. This would yield a radius of gyration about 380 Å not too far from the experimental value Rapp ≈ 530 Å if polydispersity effects are kept in mind: the latter value is an experimental z-average while Lcyl is roughly estimated without any polydispersity effects included. However, the power law in Figure 4 describes the data in the whole range of Rapp values as soon as micellar growth has started and this feature seems to exclude a rodlike behavior at T ) 34.1 °C, right in the middle of the growth regime. Dynamic light scattering results appear to support as well a wormlike chain behavior. The dynamics of rigid rods in dilute solutions are linked to rotational as well as translational diffusion.46 In the limit qLcyl , 1, the DLS experiments should show two decay rates,47 the slower one being proportional to q2, with a diffusion constant Γt given by a weighted sum of the two diffusion constants associated with displacements along (Γpar) and across (Γperp) the rod axis, Γpar ∼ Γperp ∼L-1 cyl . The faster decay rate includes the rotational motion of the rods and is given by Γrot + q2Γt, with Γrot ∼ L-3 cyl . A nonzero depolarized scattering intensity should be measurable, and its autocorrelation function should exhibit a single q independent decay rate related to Γrot. This behavior was checked experimentally for P85 aqueous solutions.13 In the P84 solutions, we were unable to measure any depolarized light scattering intensity but this could simply be due to the weak incident intensity. More significantly, the CONTIN analysis of the intensity correlation functions did not reveal two distinct decay rates with the behavior expected for rodlike aggregates, even in conditions very favorable for their observation, e.g., φ ) 10-3 and T ) 34.7 °C. Instead, a single mode decay was observed most of the time, with spurious, nonsystematic although reproducible, small additional contributions (Figure 6). As the aggregates grow, the value of the exponent R measuring the q dependence of τ increases progressively from 2 to about 3, which is the value expected when the internal relaxation modes of flexible chains are probed by DLS experiments in the regime qRapp > 1.46 Thus it is likely that intermediate R values between 2 and 3 reflect simply the polydispersity in the samples, until at T ≈ 40 °C, most of the wormlike chains are large enough for their internal dynamics to be probed in the DLS experiments and R ≈ 3. At the same temperature, the relaxation modulus measured in step strain experiments becomes a single exponential as shown by previous work performed at φ ) 0.04.18 Cates has modeled the viscoelastic behavior of entangled living polymer solutions48,49 and has shown that this feature corresponds to the limiting case where the breaking time τb ∼ L h -1 of chains with average length L h is much shorter than their disengagement time (45) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (46) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: Oxford, 1986. (47) Pecora, R. J. J. Chem. Phys. 1968, 48, 4126. (48) Cates, M. Macromolecules 1987, 20, 2289. (49) Cates, M. J. Phys. (Paris) 1988, 49, 1593.

Duval et al.

τd ∼ L h 3φ3/2.46 As τb/τd ∼ L h -4, this limit can be reached as the micelles grow and η0 should then scale as η0 ∼ G0(τbτd)1/2,48,49 where G0 ∼ φ9/4 is the elastic modulus.45 The average length L h of the living polymers can be obtained from a simple model for the breaking and the recombination kinetics, where the breaking rate is proportional to the length of the chains and the recombination rate of two chains is proportional to the product of their concentrations. It follows48,49 that the size distribution n(L) of the chains decreases exponentially n(L) ∼ exp(-L/L h )/L h with L h ∼ φ1/2. Inserting this result into the expression for η0, we get the now classical result η0 ∼ φ7/2 in rather good agreement with our experimental value. This picture is however oversimplified as far as the distribution of lengths is concerned. The fact that the copolymer chains aggregate first in spherical micelles that grow subsequently under suitable conditions has a consequence for the size distribution of the aggregates. This has been worked extensively in the case of short surfactant molecules. The simpler phenomenological models50 assume a cylindrical body and hemispherical end caps with the same radius and a lower standard chemical potential for the surfactant molecules in the cylindrical body than in the hemispherical caps. They predict a monotonic growth of the micelles above the cmc and a smooth distribution of sizes with a single maximum shifting to larger sizes with increasing concentration. Porte et al.51 showed that the very different local structure of the molecules according to their local environment can result in an energy barrier, associated with the creation of unfavorable end caps and the subsequent micellar growth. They proposed that the growing micelles, although retaining a cylindrical symmetry, should have spherical end caps with a larger radius than the cylindrical body. May and Ben-Shaul gave recently52 a molecular theory of the sphere to rod transition in dilute micellar solutions taking into account the molecular arrangement as a function of local curvature of the micelles. They calculated the size distribution function as a function of concentration and showed distinctly that the onset of micellar growth takes place at a critical concentration and that there is a gap in the size distribution above this concentration, i.e., small aggregates coexist with much larger micelles. Such features might be conserved in the case of polymer micelles where the growth is induced by the temperature. Finally it has also been proposed53 that the initial growth occurs by a fusion of spherical micelles with other spherical or short spherocylindrical micelles. In this case, an exponential distribution of lengths is expected from the very beginning of the growth process up to the onset of entanglements. The influence of these various distributions of lengths on the scaling behavior of η0 with the concentration has still to be established theoretically. Further work is clearly needed on this point. Finally we note that the zero-shear viscosity at φ ) 5 × 10-3 stays out of the power law behavior valid for higher concentrations. This suggests that the wormlike chains are not long enough to be entangled at this concentration. The existence of a cross over in this range of concentration was inferred from previous light scattering experiments on the same system.16 It is confirmed here by the behavior of I0, Rapp, and Rh,app (Figures 3 and 8). The saturation (50) Ben-Shaul, A. In Micelles, Membranes, Microemulsions, and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer: Berlin, 1994. (51) Porte, G.; Poggi, Y.; Appel, J.; Maret, G. J. Phys. Chem. 1984, 88, 5713. (52) May, S.; Ben-Shaul, A. J. Phys. Chem. B 2001, 105, 630. (53) Ilgenfritz, G.; Schneider, R.; Grell, E.; Lewitzki, E.; Ruf, H. Langmuir 2004, 20, 1620.

Growth of Copolymer Micelles

observed at φ ) 0.04 is typical of the behavior expected for semidilute solutions of long chains where these parameters should no longer depend on the chain length.45 At the two lowest concentrations, the growth of the chains can still be measured on the whole temperature range of the growth regime. We turn now to the discussion of the results at higher temperatures (T > 40.5 °C) where all the parameters discussed up to now show a marked change in behavior. A first drastic change is the curvature in the Zimm representation (Figure 2). The upward curvature observed at T ) 42.7 °C (φ ) 5 × 10-3) indicates changes in the conformation of the aggregates and/or their polydispersity.54,55 At least their conformation is different since I0 is still increasing while Rapp decreases (Figure 3), which results in a strong deviation from the power law behavior in the I0 vs Rapp plot (Figure 4). Strikingly the evolution of these parameters depends on the concentration, and kinetic effects are at work. At φ ) 10-3 the viscosity of the solution is very low and macroscopic phase separation occurs. At φ ) 5 × 10-3 a macroscopic phase separation is visible after a few days in H2O, with the polymer-rich phase at the bottom of the cell, while in D2O the polymer-rich phase accumulates at the top of the cell within hours. For the highest concentration, the viscosity is still very high and no macroscopic phase separation is visible within a few days. However the solution becomes more and more turbid with time. On the basis of these results it is clear that more compact and therefore less entangled objects are forming in solution above T ≈ 40.5 °C. This explains the viscosity drop. It is not yet possible to conclude whether the wormlike chains collapse and coalesce while keeping their cylindric shape or whether they form new structures like, e.g., microphase separated droplets reminiscent of the mesophases obtained in the concentrated regime. Future work in densitymatched solvent is planned to elucidate this aspect. Finally we want to compare our results with previous reports in the literature. It is somewhat surprising that the spectacular micellar growth found in the present study was not detected in earlier studies on P84. This is partly due to the scarcity of studies performed at small polymer volume fraction. In fact Liu et al. studied P84 in pure water,15,20,21 and the onset of micellar growth is then located at T ≈ 55 °C.16 Liu et al. measured at about this temperature a strong increase in the light intensity scattered from a solution with φ ≈ 0.03, associated with (54) Witten, T. A.; Scha¨fer, L. J. Chem. Phys. 1981, 74, 2582. (55) Schosseler, F.; Leibler, L. Macromolecules 1985, 18, 398.

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the onset of angular dependency without elaborating further.15 Most likely this is the same phenomenon as reported here. It is also striking that P85 solutions do not appear to form wormlike aggregates but rigid rods.13 Thus only a slightly longer PEO block seems to limit the size of the cylindrical aggregates and to prevent the onset of strong viscosifying effects. It might be so because the difference in the standard chemical potential for the PEO chains constrained in the cylindrical part with respect to the PEO chains constrained in the hemispherical caps becomes smaller for longer chains. Therefore less energy is gained by decreasing the number of end caps and the existence of a larger number of smaller aggregates is favored for entropic reasons. If only a difference in the end cap energy between P84 and P85 elongated micelles is involved, it should simply result in a different energetic prefactor in the scaling law relating L h and φ. A detailed comparison of P84 and P85 solutions would be worthwhile. To conclude, we have shown that the temperatureinduced growth of P84 spherical micelles yields wormlike polymer micelles that are long enough to behave like semiflexible polymers in solution. In particular, they show a transition, depending on temperature and concentration, from a dilute to a semidilute regime, where the wormlike micelles become entangled. The entanglements are responsible for a huge increase in the solution viscosity. Their effect is however controlled by the finite breaking time of the aggregates. Measurements of the size distribution of the aggregates would be very valuable. Above a critical temperature, the viscosity drop is associated with a densification of the aggregates in the solution and a decrease of the entanglements. The exact nature and morphology of these denser aggregates are still under investigation. Acknowledgment. It is a pleasure to thank B. Michels for interesting discussions and for his kind help at many steps of this study. We are grateful to our local contact at ILL, P. Lindner, for his kind assistance during the SANS experiments. We thank S. Soulimane for his experimental help during part of this study. We thank BASF company for kindly supplying the samples. Supporting Information Available: Figures showing the fits of the experimental data in Figure 5. This material is available free of charge via the Internet at http://pubs.acs.org. LA050177C