SANS and Rheology Study of Aqueous Solutions and Gels Containing

Dynamic temperature sweep experiments were carried out using a TA Instruments AR500 controlled stress rheometer fitted with a Peltier temperature cont...
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Langmuir 2003, 19, 3229-3235

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SANS and Rheology Study of Aqueous Solutions and Gels Containing Highly Swollen Diblock Copolymer Micelles V. Castelletto* and I. W. Hamley Department of Chemistry, University of Leeds, Leeds LS2 9JT, U.K.

R. J. English Centre for Water Soluble Polymers, North East Wales Institute, Plas Coch Campus, Mold Road, Wrexham LL11 2AW, U.K.

W. Mingvanish Department of Chemistry, Faculty of Science, King Mongkut’s University of Technology Thonburi, 91 Prachautid Road, Bang Mod, Bangkok 10140, Thailand Received November 27, 2002. In Final Form: January 22, 2003 Aqueous micellar solutions and gels of the diblock copolymer E398B19 (E ) oxyethylene, B ) oxybutylene) have been investigated by rheology and small-angle neutron scattering (SANS) in the liquid, soft gel, and hard gel phases. Rheology experiments were used to confirm the microstructure and flow behavior of the system in the vicinity of the liquid-soft gel boundary previously located, as well as to identify a new liquid-hard gel transition at low temperatures. The dimensions of the micelles in the liquid phase were obtained from the micellar form factor described by a core/shell model. The parameters extracted from the modeling were in good agreement with previous results for the same system from light scattering. The micellar radius was found to decrease with temperature in the liquid phase, as a consequence of the contraction of the corona chains, caused by a decrease in solvent quality for poly(oxyethylene) at high temperature. The shape of the corona as a whole was preserved on increasing temperature, only undergoing a reduction in width. The equilibrium phase is fluid at volume fractions lower than the volume fraction for freezing of hard spheres, φf ) 0.494, in good agreement with a system of hard spheres. In the hard gel, the micelles are arranged in a bcc structure, as expected for micelles with lengthy E blocks.

1. Introduction Block copolymers self-assemble into micellar aggregates in a selective solvent due to their dual hydrophilic/ hydrophobic nature. If the volume fraction of micelles is sufficiently high, crystallization can occur when the micelles pack into a regular array.1-7 The crystal phase is sometimes termed “hard gel” (based on the existence of a finite yield stress and a dynamic elastic modulus G′ > 104 Pa).8 In solution, block copolymer micelles are usually spherical and they can be viewed as model colloidal particles. The effective interaction potential between the micelles varies as a function of the relative size of the micellar core and corona, which in turn depends on the composition and/or molecular weight of the copolymer. Differences in the effective intermicellar potential lead to the possibility of ordering in body-centered cubic (bcc) or * Author for correspondence. (1) Watanabe, H.; Kotaka, T.; Hashimoto, T.; Shibayama, M.; Kawai, H. J. Rheol. 1982, 26, 153. (2) Hashimoto, T.; Shibayama, M.; Kawai, H. Macromolecules 1983, 16, 1093. (3) Mortensen, K.; Brown, W.; Norde´n, B. Phys. Rev. Lett. 1992, 68, 2340. (4) Mortensen, K.; Pedersen, J. S. Macromolecules 1993, 26, 805. (5) McConnell, G. A.; Gast, A. P.; Huang, J. S.; Smith, S. D. Phys. Rev. Lett. 1993, 71, 2102. (6) McConnell, G. A.; Lin, M. Y.; Gast, A. P. Macromolecules 1995, 28, 6754. (7) Hamley, I. W.; Fairclough, J. P. A.; Ryan, A. J.; Ryu, C. Y.; Lodge, T. P.; Gleeson, A. J.; Pedersen, J. S. Macromolecules 1998, 31, 1188. (8) Hvidt, S.; Jørgensen, E. B.; Schille´n, K.; Brown, W. J. Phys. Chem. 1994, 98, 12320.

face-centered cubic (fcc) structures.5,6,9 Micelles that act as hard spheres pack into fcc cubic arrays whereas softer interaction potentials favor a bcc structure.5,10 Highly swollen diblock copolymer micelles are expected to behave as soft spheres and pack in a bcc structure at sufficiently high polymer concentration. This has previously been confirmed for aqueous solutions of the highly asymmetric diblock copolymer E398B19 (E ) oxyethylene, B ) oxybutylene, the subscripts indicating the number of repeats).11 The dimensions of the micelles of E398B19, determined from static and dynamic light scattering, have already been reported in the literature.12 Rheology and mobility experiments have also been undertaken to determine the sol-gel phase diagram for this system.11 In particular, these studies led to an interesting feature, that is, the identification of a “soft gel” phase between micellar liquid and gel phases.11 Soft gels may be distinguished from sols because they possess a small but finite yield stress, and they can be distinguished from hard gels because the dynamic elastic modulus is smaller.11 The structure of this soft gel phase has been the subject of recent controversysit has been suggested that soft gels observed in block copolymer solutions result from ag(9) Hamley, I. W.; Daniel, C.; Mingvanish, W.; Mai, S.-M.; Booth, C.; Messe, L.; Ryan, A. J. Langmuir 2000, 16, 2508. (10) Gast, A. P. Langmuir 1996, 12, 4060. (11) Kelarakis, A.; Mingvanish, W.; Daniel, C.; Li, H.; Havredaki, V.; Booth, C.; Hamley, I. W.; Ryan, A. J. Phys. Chem., Chem. Phys. 2000, 2, 2755. (12) Mingvanish, W.; Mai, S.-M.; Heatley, F.; Booth, C.; Attwood, D. J. Phys. Chem. B 1999, 103, 11269.

10.1021/la026917o CCC: $25.00 © 2003 American Chemical Society Published on Web 03/13/2003

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gregation of spherical micelles via percolation into fractal structures13-15 or, alternatively, that they correspond to highly defective solids.11 In a complementary study to that here, we have performed small-angle neutron scattering (SANS) experiments on dilute solutions of E398B19 in deuterium oxide (D2O) to probe the structure of micelles as a function of temperature and concentration and to determine the density profile of the corona “brush” chains, which were found to be highly swollen.16 In this work we extend these studies of the E398B19/water system, investigating the phase diagram and flow behavior of this system in order to probe its microstructure as a function of temperature and concentration in the vicinity of the liquid-soft gel transition and the soft gel-hard gel transition. New rheological studies of the linear viscoelastic behavior of the diblock solutions confirm the liquid-soft gel boundary previously identified.11 In addition, the low-temperature boundary between liquid and hard gel phases is located for the first time. Modeling of SANS data provides information on micelle dimensions as a function of temperature and concentration. The effective volume fraction of micelles is determined from the model structure factor.

the sample was prevented by coating the exposed edge with a thin film of low-viscosity PDMS fluid (Dow Corning, 10 cS). During each temperature sweep, the stress amplitude was varied so as to maintain a constant strain amplitude, within the linear region of the sample. Linearity was verified independently by carrying out dynamic amplitude sweeps at several temperatures. Small-Angle Neutron Scattering. Experiments were conducted at the Laboratoire Le´on Brillouin (CEA-CNRS), using the SANS instrument PAXY with a two-dimensional detector, comprising 128 × 128 cells of 5 × 5 mm2 size. The wavelength λ ) 12 Å and the sample-detector distance 3.5 m were chosen to measure the full SANS curve. Measurements were carried out for temperatures ranging between 25 and 75 °C. Solutions were poured into sealed 1 mm thick standard quartz cuvettes. Gels were mounted in sealed 1 mm thick aluminum cells with diamond windows. Temperature control was achieved using a water bath. The data from the two-dimensional area detector were converted into one-dimensional intensity profiles by radial averaging. The SANS data were then corrected to allow for sample transmission and background scattering (using a D2O sample as reference). Finally, SANS curves were corrected for detector cell efficiency and calibrated to absolute units using a graphite secondary standard.17 The resulting solvent-corrected and calibrated intensity is denoted Ic(q).

2. Experimental Section

Small-Angle Neutron Scattering. The coherent part of the SANS intensity from an homogeneous solution of monodisperse spherical micelles, Ic(q), can be written in the decoupling approximation, as18

Samples and Characterization. The copolymer E398B19 is identical to that used earlier, the details of the synthesis of which are provided elsewhere.12 Diblock E398B19 has a narrow molecular weight distribution, the ratio of mass average to number average molar mass being Mw/Mn ) 1.05, determined by GPC on the basis of poly(oxyethylene) calibrants, where Mn ) 18 900 g mol-1 was determined by analyzing 13C NMR spectra. Solutions were prepared by mixing the appropriate amount of polymer with D2O or H2O for SANS or rheology experiments, respectively (no differences in phase behavior were reported when using the two solvents), and storing for a period of days at low temperature (4 °C). More concentrated samples were mixed by heating the sample to 60 °C, to reduce the viscosity and improve homogenization. Rheology. Rheological characterization was performed for samples with concentrations between 3.25 and 5.0 wt % E398B19. Solutions were stored at 4 °C prior to characterization. Isothermal experiments were carried out on a Rheometric Scientific ARES controlled deformation rheometer fitted with a dual range (10/ 100 g cm) force rebalanced transducer. The temperature of the sample was controlled via an external circulator bath. Drying at the sample edge was prevented by the use of a “solvent trap” device that maintained an atmosphere saturated with water vapor above the sample. A concentric cylinder geometry (outer radius 34 mm, inner radius 32 mm) was used for samples containing lower concentrations of polymer. Samples containing higher polymer concentrations were measured using a cone and plate (diameter 50 mm, cone angle 0.04 rad). When using the cone and plate, residual stresses arising from compression of the sample were allowed to relax completely prior to commencing the experiment. This was ensured by monitoring the decay of the normal force exerted by the sample. All experiments were conducted at a strain amplitude corresponding to the linear viscoelastic region of the samples. Dynamic temperature sweep experiments were carried out using a TA Instruments AR500 controlled stress rheometer fitted with a Peltier temperature control system. A parallel plate geometry was used (40 mm diameter, 1 mm gap), the rate of heating was 0.5 °C min-1, and the frequency of oscillation was ω ) 1.0 rad s-1. Drying out of (13) Li, H.; Yu, G.-E.; Price, C.; Booth, C.; Hecht, E.; Hoffmann, H. Macromolecules 1997, 30, 1347. (14) Lobry, L.; Micali, N.; Mallamace, F.; Liao, C.; Chen, S.-H. Phys. Rev. E 1999, 60, 7076. (15) Chen, S.-H.; Liao, C.; Fratini, E.; Baglioni, P.; Mallamace, F. Colloids Surf., A 2001, 183-185, 95. (16) Castelletto, V.; Hamley, I. W.; Pedersen, J. S. J. Chem. Phys. 2002, 117, 8124.

3. Theory

Ic(q) ) np〈|A(q)|2〉S(q)

(1)

where np ) NA[φ]n j is the number density of scatterers (here NA denotes the Avogadro number, [φ] is the molar concentration, and n j is the micellar aggregation number), A(q) is the monodisperse intraparticle scattering amplitude, S(q) is the monodisperse intermicellar structure factor, and q is the scattering vector magnitude given by q ) 4π sin θ/λ, where 2θ is the scattering angle. In our work we considered a spherical core/shell micellar model. According to this model, A(q) assumes the form19

A(q) ) Vcore(Fcore - Fshell)Acore(q) + Vmic(Fshell - FS)Ashell(q) (2) Here Acore(q) and Ashell(q) are the scattering amplitudes of core and shell, respectively, and Vcore ) (4πRcore3)/3 (Rcore denotes the radius of the core) and Vmic ) (4πRmic3)/3 (Rmic ) Rcore + D, where D is the corona width) are the overall volumes of the core and micelle, respectively. The terms Fcore, Fshell, and FS in eq 2 are the scattering length densities of the core, shell, and solvent, respectively. Allowing for a degree of solvent penetration into the micellar core, the micellar aggregation number, n j , can be written as a function of the volume fraction of solvent contained inside the core, fcore, and the micellar core radius, Rcore: 3

n j)

4π (1 - fcore)Rcore 3 VB

(3)

where VB ) 2346.5 Å3 is the volume of the B19 block. (17) Cotton, J. P. Initial Data Treatment. In Neutron, X-ray and Light Scattering: Introduction to an Investigative Tool for Colloidal and Polymeric Systems; Lindner, P., Zemb, T., Eds.; North-Holland: Amsterdam, 1991. (18) Kotlarchyk, M.; Chen, S.-H. J. Chem. Phys. 1983, 79, 2461. (19) Willner, L.; Poppe, A.; Allgaier, J.; Monkenbusch, M.; Lindner, P.; Richter, D. Europhys. Lett. 2000, 51, 628.

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Our model assumes that the density profile is homogeneous in the core and inhomogeneous in the corona. Therefore, the core scattering amplitude in eq 2 is that of a homogeneous sphere with radius Rcore, given by20

Acore(q) )

3[sin(qRcore) - qRcore cos(qRcore)] (qRcore)3

(4)

while the shell scattering amplitude is written as a function of the density profile in the corona n(r):

Ashell(q) )

∫RR

n(r) sin(qr)r2 dr/qr

mic

core

∫R

Rmic core

2

(5)

n(r)r dr

Figure 1. Phase boundaries for aqueous solutions of block copolymer E398B19 obtained from rheological experiments: (b) data taken from ref 11; (0) data presented in this work. The broken lines are guides to the eye.

Here r is the radial coordinate measured from the center of the micelle. In this work we considered a general power law for n(r)19

n(r) ) r-x/(1 + exp[(r - Rmic)/σF]), Rcore e r e Rmic (6) where σF is a profile width parameter and x is constant. In a dilute solution the interactions among micelles are negligible, and the scattering is solely due to the particle scattering (form factor). We found that the interactions between the micelles could be modeled using a structure factor corresponding either to an interaction potential for soft spheres, or to the hard sphere potential. A long-range tail in the potential is required for soft spheres to pack in a bcc structure,5,10 and therefore it appears inconsistent to use a hard sphere structure factor for E398B19 micelles. However, we found that the data are not sensitive to the details of the structure factor (due to the limited q-range accessed), and therefore, we used a model with the minimum number of parameters based on interacting hard spheres. The analytical expression of the hard spheres structure factor has already been given elsewhere21 and is omitted here. The model used to fit the SANS data thus includes in total eight fitting parameters: six describing P(q) and two describing S(q). The six parameters related to P(q) are the volume fraction of solvent contained inside the core, fcore, the core radius, Rcore, the micellar radius, Rmic, the fraction (Fcore - Fshell)/(Fshell - FS), the exponent x, and the width, σF. The aggregation number is calculated from eq 3. The parameters which describe S(q) are the volume fraction of equivalent hard spheres, φ, and the effective hard sphere radius, Reff. To fit the SANS curves, the modeled intensity was also convoluted with the resolution function for wavelength spread (∆λ/λ ) 0.1) and collimation effects. 4. Results and Discussion Rheology. The results of rheological experiments are summarized in Figure 1 in the form of a “phase diagram”, with each point representing the position of a phase boundary at a given polymer concentration and temperature. Results from the present study are plotted along with data from the previous study of Kelarakis et al.11 Good agreement is obtained for both the liquid-soft gel and soft gel-hard gel phase boundaries. At temperatures above 30 °C the system undergoes a transition from a liquid micellar phase (micelles with no translational (20) Rayleigh, Lord. Proc. R. Soc. (London) 1911, A-84, 25. (21) Ashcroft, N. W.; Lekner, J. Phys. Rev. 1966, 145, 83.

Figure 2. Dynamic temperature sweep data for (4) 3.75, (9) 4, (O) 4.5, and (1) 5 wt % E398B19. Experiments were carried out at the heating rate 0.5 °C min-1. Dotted lines mark the positions of the microstructural transitions between liquid-soft gel and hard gel-soft gel.

order), through a soft gel, to a hard gel (bcc), as the concentration of E398B19 is increased. We also detected a distinct liquid-hard gel transition at lower temperatures, which has not previously been reported for E398B19 solutions. An intermediate soft gel phase was not found, although this may simply be due to the sensitivity of the rheometer. A more detailed discussion of the rheological criteria used to pinpoint the positions of the various phase transitions will be presented below, considering particularly the microstructural changes accompanying the liquid-soft gel and liquid-hard gel phase transitions. Dynamic temperature sweep experiments were initially conducted to locate phase boundaries for various concentrations of E398B19. These data are plotted in Figure 2, which shows G′ versus temperature. At low temperatures each of the solutions behaves as a Newtonian fluid of low viscosity. In this case, the phase angle is ∼90° and correspondingly G′ ) 0. On heating, the transition to the hard gel phase is marked by an abrupt increase in the storage modulus over the temperature range 2-3 °C. For E398B19 solutions between 4 and 5 wt %, G′ increases from zero to around 500 Pa, the onset of the liquid-hard gel transition shifting to higher temperatures as the polymer concentration is reducedsconsistent with the negative slope of the liquid-hard gel boundary plotted in Figure 1. On further heating and at concentrations above 4% the storage modulus is seen to increase and pass through a maximum. Further heating causes a steady decrease in the modulus. The maximum in G′ is seen to occur at ∼20 °C for a 5 wt % solution of the polymer and shifts to around ∼16 °C for a 4 wt % solution. Note that the maximum in G′ attained varies from ∼900 Pa at 4 wt % E398B19 to ∼1400 Pa for a 5 wt % solution, which is somewhat lower than

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the value of 104 Pa usually taken as characteristic of ordered crystalline phases formed by diblocks of this type.22 A further phase change is apparent at higher temperatures, which may be attributed to the hard gel-soft gel transition. In this case, there is an abrupt decrease in the storage modulus, again occurring over the temperature range 2-3 °C. In the soft gel phase present at elevated temperatures, the modulus is around 100 Pastypical for the soft gel phases of EmBn diblocks.23 At higher temperatures (>50 °C) it was impossible to maintain a linear response; hence, the data became unreliable. The abrupt reduction in the storage modulus at the hard gel-soft gel boundary has been attributed to the formation of a defective bcc phase. The microstructural changes underlying the apparent liquid-soft gel transition, occurring on heating solutions of the polymer prepared at concentrations less than 3.5%, are still the subject of some debate.11 In the present study, we observed a structural transition around 50 °C for a 3.25 wt % solution of E398B19 (Figure 1), with the system changing from a Newtonian liquid to a viscoelastic fluid with a G′ of ∼100 Pa at ω ) 1.0 rad s-1 (data not shown). It has been suggested that diblocks of this type may undergo a percolation transition on heating. In this respect, the collapse of the corona as the E-blocks are desolvated leads to an attractive intermicellar potential and the micelles assemble into a fractal network which spans the sample. At the gel point, both G′ and G′′ should exhibit the same power law frequency dependence G′ ∼ G′′ ∼ ω∆ (tan δ independent of frequency),24 where ∆ ) 0.72 is expected for a percolating system.25-27 If this criterion is not met, the microstructure may not be considered self-similar over a range of length scales and the accompanying structural transition may not be attributed to percolation. Unfortunately, the extreme strain sensitivity of microstructure precluded our obtaining reliable data showing the frequency dependence of the small deformation moduli at this temperature, and the existence of a percolation transition could not be examined. The negative slope of the boundary between liquid and soft gel phases in Figure 1 is consistent with that due to attractive interactions between micelles, as predicted by percolation theory.14 An alternative explanation is that the micelles form locally ordered crystalline grains rather than a percolation network.28 Changes in the dynamic shear modulus accompanying the liquid-hard gel transition for a 3.5 wt % solution of E398B19 are illustrated in Figure 3. At temperatures below 10 °C, the solution is a Newtonian fluid (G′ ) 0, G′′ ∼ ω), consistent with a low micellar volume fraction. The development of long-range order, as φ increases with increasing temperature, is apparent because the system undergoes a transition to a viscoelastic fluid. At T ) 10 °C, stresses in the system are still able to relax completely over relatively short time scales and a terminal response is seen with G′ ∼ ω and G′′ ∼ ω2. At T ) 12 °C, the micellar volume fraction is increased further and an elastic plateau (G′ > G′′) is apparent. We do not currently have SANS data covering these concentration and temperature ranges, (22) Hamley, I. W. Philos. Trans. R. Soc. London 2001, 359, 1017. (23) Kelarakis, A.; Havredaki, V.; Rekatas, C.; Mai, S.-M.; Attwood, D.; Booth, C.; Ryan, A. J.; Hamley, I. W.; Martini, L. G. A. Macromol. Chem. Phys. 2001, 202, 1345. (24) Winter, H. H.; Chambon, F. J. Rheol. 1986, 30, 367. (25) Derrida, B.; Stauffer, D.; Herrmann, H. J.; Vannimenus, J. J. Phys. Lett. 1983, 44, L701. (26) Herrmann, H. J.; Derrida, B.; Vannimenus, J. Phys. Rev. B 1984, 30, 4080. (27) de Gennes, P. G. C. R. Acad. Sci. Paris II 1978, 286, 131. (28) Castelletto, V.; Caillet, C.; Fundin, J.; Hamley, I. W.; Yang, Z.; Kelarakis, A. J. Chem. Phys. 2002, 116, 10947.

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Figure 3. Angular frequency dependence of G′ at (b) 10 and (2) 12 °C and G′′ at (]) 5, (0) 8, (4) 9, (O) 10, and (4) 12 °C for a 3.5% solution of E398B19. G′ is not shown for 5, 8, and 9 °C because it was zero.

Figure 4. Frequency sweep data for a 4% solution of E398B19 at 12 °C illustrating the linear viscoelastic behavior of the hard gel phase.

but it appears that long-range order is well developed at this temperature and the predominance of elastic modes of deformation indicates incipient freezing of the microstructure. Here the volume fraction of the ordered phase would be sufficient to create a sample-spanning structure capable of supporting stress over these experimental time scales. Increasing the polymer concentration further extends the plateau region to much longer time scales, as shown in Figure 4. Here the frequency dependence of the dynamic shear moduli for a 4 wt % solution of E398B19 is shown. In this case elastic modes of deformation persist, even at the longest experimentally accessible time scales. Such a response suggests that long-range order is well developed in the system, consistent with the crystallization of the micelles into an ordered bcc lattice. Thus, it appears that the dynamics of the microstructure are hindered progressively as the polymer concentration is increased at a constant temperature. Once the system freezes into a regular bcc array, stress relaxation occurs over extremely long time scales (∼104 s). In previous studies, this type of microstructure has been identified by the existence of a yield stress.11 We have performed stress relaxation experiments to probe the linear viscoelastic response of solutions of E398B19 in the hard gel region of the phase diagram, as this allows the range of time scales accessible experimentally to be extended compared to that for a dynamic frequency sweep. As can be seen in Figure 5, the magnitude of the shear modulus in the plateau zone exhibits very little increase over the range of polymer concentrations examined (3.75-5 wt %). However, the time scale required for relaxation of the stress increases considerably. It was possible to estimate the zero shear viscosity of each of the polymer solutions via integration of G(t) (the data cannot be described by a simple Maxwell model). The zero shear viscosity, η0, exhibited a very steep dependence on concentration in the hard gel region of the

Highly Swollen Diblock Copolymer Micelles

Figure 5. Stress relaxation of solutions of E398B19 at 12 °C, following the imposition of a step strain of 0.2%: (0) 3.75, (O) 4.0, (4) 4.5, and (3) 5.0% polymer. All data correspond to the hard gel region of the polymer solution phase diagram (Figure 1). The continuous lines are guides to the eye.

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Figure 8. (O) SANS data and fits using the model described in the text (solid line) for 3 wt % E398B19 solutions at (a) 25, (b) 45, (c) 55, and (d) 65 °C.

Figure 6. (O) SANS data and fits using the spherical core/ shell geometry for a 0.5 wt % solution at (a) 25 and (b) 75 °C.

Figure 9. Temperature dependence of (a) the radius of the hydrophobic core, Rcore, (b) the width of the hydrophilic corona, D, (c) the total radius, Rmic from the form factor, and (d) the effective total micellar radius, Reff, from the structure factor, extracted from SANS, using the hard sphere interaction for S(q), for (9) 0.5, (0) 2.5, (b) 3, and (3) 3.6 wt % E398B19.

Figure 7. (O) SANS data and fits using the model described in the text (solid line) for solutions at 35° C: (a) 2.5, (b) 3, (c) 3.6, and (d) 6 wt % E398B19 in D2O. The arrows indicate the positions of the measured reflections.

E398B19 phase diagram, being described approximately by a power law with an exponent of ∼15. Small-Angle Neutron Scattering. To study the structure of micelles in more detail, SANS experiments were performed and the data modeled as described in section 3. Figure 6 shows SANS intensity profiles for samples containing 0.5 wt % E398B19 at 25 and 75 °C together with model form factor fits. Figure 7 shows SANS intensity profiles for samples with concentrations between 2.5 and 6 wt % E398B19 at 35 °C, and Figure 8 shows data for samples containing 3 wt % E398B19 at temperatures between 25 and 65 °C. It was found that a satisfactory fit to the SANS data could be obtained by fixing x ) 1.2 and σF ) 10 Å in eq 6. Interactions between micelles become important for concentrations higher than 0.5 wt % (Figures 6-8), and the contribution of the structure factor maximum near q ) 0.0133 Å-1 increases as concentration increases. Figure 7d shows that at 35 °C and 6 wt % E398B19, which is in the

hard gel phase, the micelles are arranged in a long-range ordered structure (higher order reflections are present in the SANS profile). Here, eqs 1-6 are no longer suitable to model the SANS profile, which is not liquidlike. A representative example of the temperature dependence of the SANS profile in the liquid phase is shown in Figure 8 for 3 wt % E398B19. The effect of increasing temperature is mainly to increase the intensity at low q. There is a transition to soft gel at 60 °C for this samples the data for 65 °C therefore do not correspond to the liquid phase. Figure 9 shows the temperature dependence of Rcore, D, Rmic, and Reff obtained from the fits to the SANS data for solutions with four different concentrations. The volume fractions obtained (results not shown) were φ ) (0.36-0.21) with T ) (35-75) °C for 2.5 wt % E398B19, φ ) (0.46-0.32) with T ) (25-65) °C for 3 wt % E398B19, and φ ) (0.45-0.42) with T ) (35-45) °C for 3.6 wt % E398B19. The reduction of φ is driven by the decrease of Reff with temperature (Figure 9d). The total micellar radius, Rmic (Figure 9c), differs from the effective interaction radius, Reff (Figure 9d), by ∼(06.5)% for solutions with concentrations in the range (2.53.6) wt % E398B19. Differences of a similar magnitude are obtained when using a model with monodisperse form and structure factors, Rmic always being smaller than Reff.28-30 The range of the interaction potential is expected to be related to the width of the corona through X ) Reff/ Rmic ∼ 1.31 Using the results plotted in Figure 9c and d,

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Figure 10. Density profiles of the micellar corona, n(r)/n(Rcore), corresponding to the fits plotted (full line) in Figure 7b (Rcore ) 49 Å, Rmic ) 248 Å, and x ) 1.2) and (dotted line) in Figure 8c (Rcore ) 51 Å, Rmic ) 220 Å, and x ) 1.2).

X was calculated to lie in the range 0.97-1.07. Usually, X g 1 is obtained when using a model with monodisperse form and structure factors.28-30 On the other hand, X < 1 might result from the particular characteristics of the E398B19 corona. This difference could be due to the “fuzziness” of the corona scattering profile, which could act as a polydispersity effect, eventually leading to an overestimated value of Rmic. Previous results from dynamic light scattering12 showed that the hydrodynamic radius Rh is 260, 265, and 245 Å for 25, 40, and 50 °C, respectively. As evident from Figure 9c and d, values of Reff and Rmic are in good agreement with these values of Rh. It is apparent that both Reff and Rmic decrease with temperature for 2.5-3.6 wt % solutions. A similar observation has been made for micelles formed by E86B10 in aqueous solution29 and results from a decrease in the size of the hydrophilic corona, D, that is larger than the increase in the core radius with temperature. The contraction of the corona chains with temperature is caused by a decrease in solvent quality for poly(oxyethylene) at high temperature. The dimensions of the B chains in the corona can be compared to previous results. Assuming a trans-planar conformation, the fully extended length of a B19 block is estimated to be 68.9 Å.32 Comparing Rcore in Figure 9a with this quantity shows that the B block is not fully extended in the micelle core. As already mentioned, the full set of fitted SANS curves could be modeled by fixing x ) 1.2 in eq 6, which is similar to the starlike density profile.19 Figure 10 displays the density profile of the corona for the 3 wt % E398B19 solution at 35 °C (Figure 7b) and 55 °C (Figure 8c). The shape of the corona as a whole remains constant with increasing temperature, only undergoing a reduction in its width. We have previously obtained fcore for E398B19 solutions under identical conditions to those studied here, using a form factor of a micelle with a spherical core and selfavoiding chains.16 The same values were used in this work, to reduce the number of fitting parameters. They have already been shown in Figure 4c of ref 16 and therefore will be omitted here. The micellar aggregation numbers (eq 3) obtained from the fixed fcore, together with the Rcore values shown in Figure 9a, were as follows: n j ) (116-182) with T ) (25-75) °C (29) Derici, L.; Ledger, S.; Mai, S.-M.; Booth, C.; Hamley, I. W.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 2773. (30) Castelletto, V.; Caillet, C.; Hamley, I. W.; Yang, Z. Phys. Rev. E 2002, 65, 050601(R). (31) Pedersen, J. S.; Svaneborg, C.; Almdal, K.; Hamley, I. W.; Young, R. N. Macromolecules, submitted. (32) Yang, Y.-W.; Tanodekaew, S.; Mai, S.-M.; Booth, C.; Ryan, A. J.; Bras, W.; Viras, K. Macromolecules 1995, 28, 6029.

Castelletto et al.

and 0.5 wt % E398B19, n j ) (91-182) with T ) (35-75) °C and 2.5 wt % E398B19, n j ) (103-153) with T ) (25-65) °C j ) (114-127) with T ) (35-45) and 3 wt % E398B19, and n °C and 3.6 wt % E398B19. At this stage, it should be mentioned that the expression in eq 1 calculated using the parameters mentioned in the text was found to be systematically below the experimentally observed Ic(q). This can be ascribed in part to the high aggregation numbers obtained from the model, which could lead to an underestimation of np. Aggregation numbers obtained from previous dynamic light scattering j ) 68, 83, and 83 Å for 25, 40, and 50 experiments are12 n °C, respectively. Using these values in eq 3 together with Rcore (Figure 9a), we obtain an average over all the concentrations fcore ) (0.67-0.65) for T ) (25-55) °C, which is higher than the average value fcore ) (0.55-0.48) for T ) (25-55) °C extracted from Figure 4b of ref 16. However, even using the aggregation number from light scattering, the normalization condition in eq 1 was not fulfilled. Therefore, we decided to relax the condition imposed by eq 1 and included an additional scaling factor, incorporated into np. Since np was a fitting parameter, it follows from eq 2 that the relative contrast ϑ ) (Fcore - Fshell)/(Fshell - FS) (instead of the absolute values (Fcore - Fshell) and (Fshell FS) themselves) was relevant to our modeling. This parameter can be used to determine the volume fraction of solvent in the corona. Values of ϑ obtained from the fits to the data could not be explained using the simplest assumption Fcore ) FB, Fshell ) FE, and FS ) FD (FB ) 0.199 × 1010 cm-2, FE ) 0.634 × 1010 cm-2, and FD ) 6.44 × 1010 cm-2 are the scattering length densities of the B-block, E-block, and deuterium oxide, respectively, calculated using the densities d ) 1.12 and 0.96 g/cm3 for the E-block and the B-block). Instead, an average scattering length density of the core and the shell was defined according to

Fcore ) (1 - fcore)FB + fcoreFD

(7)

Fshell ) (1 - fshell)FE + fshellFD

(8)

where fshell is the volume fraction of solvent contained inside the corona. The value of fshell calculated from eq 8 should be comparable to

fshell ) 1 -

n j VE 4 3 πD 3

(9)

calculated from the volume of the corona (VE ) 25 949.6 Å3 is the volume of a E398 block). Using ϑ obtained from our model according to eqs 7 and 8, we found that on average for all polymer concentrations fshell was in the range (0.98-0.95) for T ) (25-75) °C. On the other hand, eq 9 gave fshell ) (0.89-0.75) for T ) (2575) °C, using D and n j extracted from our modeling. Both results compare qualitatively well, regarding the expected decrease of fshell with temperature due to the reduced E-block solubility at high temperatures already discussed in relation to Figure 9c and d. The comparison of the parameters obtained using the form factor presented in this work with those previously obtained using a form factor of a micelle with a spherical core and self-avoiding chains has been undertaken in a previous work.16 Briefly, both models provide similar Reff and Rcore values, although it was found that φ was higher for the model with self-avoiding chains.

Highly Swollen Diblock Copolymer Micelles

Langmuir, Vol. 19, No. 8, 2003 3235

Considering the structure of the hard gel, Figure 7d shows the SANS intensity profile for the 6 wt % E398B19 gel typical of that obtained for temperatures and concentrations falling in the hard gel phase region in Figure 1. A spherically averaged powder diffraction pattern was observed, consisting of three diffraction rings. The position of these rings is in the ratio 1:(2)1/2:(3)1/2, in good agreement with previous small-angle scattering (SAXS) studies of the E398B19/water hard gel phase,11 and can be indexed as 110, 200, and 211 reflections, respectively, consistent with a bcc structure (Im3 h m symmetry). Assuming a cubic structure, the interplanar spacing, dhkl ) 2π/qhkl (qhkl ) position of the scattering peak with indices h, k, l), can be used to calculate the cell parameter, a, via

dhkl ) a/xh2 + k2 + l2

(10)

From the positions of the reflections observed in Figure 7d, a cell parameter a ) (492.2 ( 4.1) Å was calculated, leading to a distance between nearest neighbors d ) (426 ( 3.3) Å, from which it is possible to calculate a micellar radius, rm ) (213 ( 1.5) Å, which is similar to Reff and Rmic at 35 °C (Figure 9c and d). As already discussed, the hard sphere potential of interaction can be used to describe the structure factor in the limited q-range accessed. Attractive interactions are likely to be important, especially in the soft gel region, and could be calculated using Baxter’s sticky sphere model.33 This would introduce additional parameters into the description of the structure factor, which is not justified by the present data (further SANS experiments are planned to investigate this by extending the measurements to lower q). Turning now to the mechanism of the liquid-soft gel phase transition, it has previously been reported that gelation occurs via a percolation transition due to attractive interactions between micelles in aqueous solutions of triblock E13P30E13 (P ) oxypropylene), on the basis of the scaling of the dynamic shear moduli at the gel point and from modeling of SANS data using the sticky hard sphere model.14,15,34 As already mentioned, the decrease in the liquid-soft gel transition temperature with increasing concentration can be described by the percolation model14 as the effective attractive interactions between micelles increase with polymer concentration. The structure and rheology of solutions of E398B19 in water have previously been investigated by SAXS and rheometry.11 SAXS was used to explore the structure of the gel phase and to locate the soft gel-hard gel phase (33) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770. (34) Mallamace, F.; Chen, S.-H.; Liu, Y.; Lobry, L.; Micali, N. Physica A 1999, 266, 123.

transition boundaries, although no modeling of the SAXS curves was undertaken. The liquid-soft gel boundary was studied by rheometry, and it was tentatively identified as a percolation transition, in analogy to the E13P30E13 solutions,14 due to the similarity of the shape of the phase transition boundary in the two systems. An alternative interpretation is that the soft gel region corresponds to coexisting liquid and hard gel phases, as previously proposed for E87B18 solutions.28,30 Distinguishing between these possibilities requires measurements of the scaling of dynamic moduli with frequency in the vicinity of the transition and/or analysis of the structure factor using the sticky hard sphere model. 5. Conclusions This work represents an extension of prior studies of the microstructure and flow behavior of the E398B19/water system in the vicinity of the liquid-soft gel transition and the soft gel-hard gel transition. New rheological studies of the linear viscoelastic behavior of the diblock solutions identified a liquid-hard gel transition at low temperatures and confirmed the liquid-soft gel boundary previously located.11 Aqueous micellar solutions and gels of E398B19 diblock copolymer were also investigated by SANS. A model for a core/shell form factor and a hard sphere structure factor was fitted to the SANS curves in the liquid phase. It was found that the shape of the corona as a whole remains constant with increasing temperature, only undergoing a reduction in its width as a consequence of the contraction of the corona chains with temperature, caused by a decrease in solvent quality for poly(oxyethylene) at high temperature. Consequently, the micellar radius decreases with temperature. The dependence of the micellar radius with temperature was in good agreement with previous results for the same system from light scattering.12 We also compared our results with those obtained from modeling the same data with a spherical core and selfavoiding chains,16 and we found a reasonable agreement between the parameters extracted from both models. Detailed modeling of attractive contributions to the SANS curves was not possible, since the structure factor was not sensitive to these, in the q-range investigated. Further SANS experiments are planned for similar highly swollen B20E610 micelles in order to detect the influence of attractive interactions at very low scattering angles. Acknowledgment. This work was supported by EPSRC, U.K. (Grants GR/M51994 and GR/N22052), and LLB, France (Project No 6055). The authors thank Laurence Noirez for her assistance during SANS experiments at the LLB. LA026917O