From Hard Spheres to Soft Spheres: The Effect of Copolymer

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From Hard Spheres to Soft Spheres: The Effect of Copolymer Composition on the Structure of Micellar Cubic Phases Formed by Diblock Copolymers in Aqueous Solution Ian W. Hamley* and Christophe Daniel School of Chemistry, University of Leeds, Leeds, LS2 9JT, U.K.

Withawat Mingvanish, Shao-Min Mai, and Colin Booth Department of Chemistry, University of Manchester, Manchester, M13 9PL, U.K.

Loic Messe and Anthony J. Ryan Department of Chemistry, University of Sheffield, Sheffield, S3 7HF, U.K. Received July 30, 1999. In Final Form: November 10, 1999 Small-angle X-ray scattering (SAXS) was used to determine the packing of micelles in cubic phases formed by poly(oxyethylene)-poly(oxybutylene) (EB) diblock copolymers in water. SAXS with large amplitude oscillatory shear was used to identify structures formed by the highly asymmetric molecules E96B18, E184B18, E315B17, and E398B19 (where the subscripts denote the number of repeats). At a constant temperature (20 °C) and concentration (10 wt %), we find that the two copolymers with shorter hydrophilic blocks form face-centered cubic (fcc) gels, whereas the two copolymers with longer corona-forming blocks form bodycentered cubic (bcc) gels. SAXS also confirmed that the sols formed at lower copolymer concentrations are micellar liquids. Our results for the gel structures are in accord with the observation that micelles with relatively short coronas behave as hard spheres, and pack in a fcc structure, whereas micelles with large coronas act like soft spheres and pack in a bcc array. This is confirmed by assembling a phase diagram as a function of copolymer asymmetry and concentration using results from the four copolymers discussed here and a series of gels of other EB diblocks studied previously by us.

1. Introduction Block copolymers behave as amphiphiles in a selective solvent, i.e., at a fixed temperature they form micelles above a critical micelle concentration.1 Considering XY diblocks in solvent, S, selective for X, their association behavior is controlled by three binary χ parameters, χXS, χYS, and χXY. Most homopolymers in solution exhibit UCST behavior, for which χ decreases with temperature. However, poly(oxyethylene) in water is unusual in showing LCST behavior. Consequently, the temperature dependence of micelle size and association number of block copolymers containing poly(oxyethylene) differs from that of most others. This leads to the interesting observation, at sufficiently high copolymer concentration, of thermally induced transitions from micellar solution to gel and from gel back to solution, as observed for example for poly(oxyethylene)-poly(oxybutylene) (EB) diblocks in water.2-4 Starting from a high-temperature sol and lowering the temperature, the E corona expands and the micellar radius increases leading to a larger excluded volume and gelation at a critical volume fraction (e.g., Φ ) 0.68 for a bodycentered cubic structure or Φ ) 0.74 for a face-centered cubic structure). However, as the temperature is lowered, micellar dissociation becomes increasingly important.2 (1) Hamley, I. W. The Physics of Block Copolymers; Oxford University Press: Oxford, U.K., 1998. (2) Bedells, A. D.; Arafeh, R. M.; Yang, Z.; Attwood, D.; Padget, J. C.; Price, C.; Booth, C. J. Chem. Soc., Faraday Trans. 1993, 89, 1243. (3) Deng, N.-J.; Luo, Y.-Z.; Tanodekaew, S.; Bingham, N.; Attwood, D.; Booth, C. J. Polym. Sci. B, Polym. Phys. 1995, 33, 1085. (4) Li, H.; Yu, G.-E.; Price, C.; Booth, C.; Hecht, E.; Hoffmann, H. Macromolecules 1997, 30, 1347.

Eventually this leads to the breakup of the gel, and the system becomes a mobile sol again. A similar effect has been noted for poly(oxyethylene)-poly(oxypropylene)poly(oxyethylene) triblocks,5,6 and the dissociation effect has been traced to the standard enthalpy of micellization tending to zero as concentration is increased.7 The term “gel” refers to the rheology of the solutions, the gel state being defined by a finite yield stress. It does not refer to bridging of copolymer chains in a network, since this is not possible for micelles formed by diblocks. So-called hard gels are empirically defined to have a dynamic shear modulus G′ > 103 Pa.6 These hard gels are now firmly established to be cubic arrangements of micelles, either face-centered cubic (fcc) or body-centered cubic (bcc).1 Whether a face-centered or body-centered arrangement is favored depends on the effective interaction potential between the micelles.8,9 For micelles with short corona blocks relative to the core, a face-centered cubic structure is favored. Such micelles behave as hard spheres, in analogy with sterically stabilized colloids.8,9 On the other hand, diblocks forming micelles with large corona blocks relative to the core behave as soft spheres, i.e., the intermolecular potential is more long-range.8,9 In (5) Wanka, G.; Hoffmann, H.; Ulbricht, W. Macromolecules 1994, 27, 4145. (6) Hvidt, S.; Jørgensen, E. B.; Brown, W.; Schille´n, K. J. Phys. Chem. 1994, 98, 12320. (7) Yu, G.-E.; Deng, Y.-L.; Dalton, S.; Wang, Q.-G.; Attwood, D.; Price, C.; Booth, C. J. Chem. Soc., Faraday Trans. 1992, 88, 2537. (8) McConnell, G. A.; Gast, A. P.; Huang, J. S.; Smith, S. D. Phys. Rev. Lett. 1993, 71, 2102. (9) Gast, A. P. Langmuir 1996, 12, 4060.

10.1021/la991035j CCC: $19.00 © 2000 American Chemical Society Published on Web 01/20/2000

Copolymer Effect on Micellar Cubic Phases

fact, the observed face-centered cubic structure usually consists of sequences of ABAB... stacking of layers of hexagonally packed micelles (i.e., hexagonal close-packed (hcp) structure), in addition to the ABCABC... sequence of a perfect fcc structure.10-16 Similar observations have been made for sterically stabilized17,18 or charge stabilized19-21 colloidal particles. A detailed model for the structure factor from different stacking sequences of hexagonal-packed layers has also been described.22 Both fcc and hcp arrangements correspond to the same volume fraction occupied by spheres, Φ ) 0.74, and in general it will be difficult to distinguish which is the equilibrium state, especially when the system is subjected to external fields, such as shear. Here, we use the term fcc, since the observed diffraction patterns for the initial, unsheared, gels are consistent with this structure. However, it should be kept in mind that under shear, long-range fcc order is not observed, and the structure is better described by sequences of hexagonal close-packed layers. Large amplitude oscillatory shear (LAOS) has been extensively used to prepare aligned samples of block copolymer melts.1 For solutions of block copolymers, steady shear is the most widely used technique10-12,23-26 although previous work12,13,15,27-31 has exploited LAOS. Mortensen and co-workers23-25 have used steady shear to prepare aligned domains of gels of poly(oxyethylene)-poly(oxypropylene)-poly(oxyethylene) (EPE) triblock copolymers, the structure of which was elucidated using small-angle neutron scattering (SANS). In both E25P40E25 (Pluronic P85)23 and E97P39E97 (Pluronic F88),24,25 a bcc cubic phase was identified. SAXS has also been used to elucidate the complex rheology shown by solutions of F68 (E76P29E76) forming a bcc gel.26 The effect of shear on face-centered cubic structures has also been examined for Pluronic copolymers.11,12 The effect of steady shear, applied using a Couette cell, on the orientation of a face-centered cubic structure in E127P48E127 (Pluronic F108) was investigated (10) McConnell, G. A.; Lin, M. Y.; Gast, A. P. Macromolecules 1995, 28, 6754. (11) Berret, J.-F.; Molino, F.; Porte, G.; Diat, O.; Lindner, P. J. Phys., Condens. Matt. 1996, 8, 9513. (12) Diat, O.; Porte, G.; Berret, J.-F. Phys. Rev. B 1996, 54, 14869. (13) Pople, J. A.; Hamley, I. W.; Fairclough, J. P. A.; Ryan, A. J.; Komanschek, B. U.; Gleeson, A. J.; Yu, G.-E.; Booth, C. Macromolecules 1997, 30, 5721. (14) Molino, F. R.; Berret, J.-F.; Porte, G.; Diat, O.; Lindner, P. Eur. Phys. J. B 1998, 3, 59. (15) Hamley, I. W.; Pople, J. A.; Fairclough, J. P. A.; Terrill, N. J.; Ryan, A. J.; Booth, C.; Yu, G.-E.; Diat, O.; Almdal, K.; Mortensen, K.; Vigild, M. J. Chem. Phys. 1998, 108, 6929. (16) Hamley, I. W.; Pople, J. A.; Diat, O. Colloid Polym. Sci. 1998, 276, 446. (17) Pusey, P. N.; van Megen, W.; Bartlett, P.; Ackerson, B. J.; Rarity, J. G.; Underwood, S. M. Phys. Rev. Lett. 1989, 63, 2753. (18) Ackerson, B. J. J. Rheol. 1990, 34, 553. (19) Dux, C.; Versmold, H.; Reus, V.; Zemb, T.; Lindner, P. J. Chem. Phys. 1996, 104, 6369. (20) Dux, C.; Musa, S.; Reus, V.; Versmold, H.; Schwahn, D.; Lindner, P. J. Chem. Phys. 1998, 109, 2556. (21) Clarke, S. M.; Rennie, A. R.; Ottewill, R. H. Langmuir 1997, 13, 1964. (22) Loose, W.; Ackerson, B. J. J. Chem. Phys. 1994, 101, 7211. (23) Mortensen, K. Europhys. Lett. 1992, 19, 599. (24) Mortensen, K.; Brown, W.; Norde´n, B. Phys. Rev. Lett. 1992, 68, 2340. (25) Mortensen, K. Prog. Colloid Polym. Sci. 1993, 91, 69. (26) Eiser, E.; Molino, F.; Porte, G.; Diat, O. Submitted. (27) Hamley, I. W.; Pople, J. A.; Gleeson, A. J.; Komanschek, B. U.; Towns-Andrews, E. J. Appl. Cryst. 1998, 31, 881. (28) Hamley, I. W.; Pople, J. A.; Fairclough, J. P. A.; Ryan, A. J.; Booth, C.; Yang, Y.-W. Macromolecules 1998, 31, 3906. (29) Hamley, I. W.; Mortensen, K.; Yu, G.-E.; Booth, C. Macromolecules 1998, 31, 6958. (30) Kelarakis, A.; Havredaki, V.; Derici, L.; Yu, G.-E.; Booth, C.; Hamley, I. W. J. Chem. Soc., Faraday Trans. 1998, 94, 3639. (31) Derici, L.; Ledger, S.; Mai, S.-M.; Booth, C.; Hamley, I. W.; Pedersen, J. S. Phys. Chem., Chem. Phys. 1999, 1, 2773.

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using SAXS, and transitions between different types of shear flows were elucidated.11 A twinned fcc structure with a high density of stacking faults due to the flow of sliding layers was observed to transform into large homogeneous single crystals of either twin, separated on a millimeter scale, on application of large amplitude oscillatory shear.12 The same system has recently been investigated in more detail using SAXS and SANS on samples subjected to steady shear in a Couette cell.14 Different mechanisms of flow were identified, depending on the shear rate. At low shear rates, the fcc structure was locally preserved, and the flow was mediated by defects between crystallites. However, at high shear rates, the melting of the structure was observed as characterized by a liquidlike structure factor. Intermediate shear rates (γ˘ ) 100 s-1) led to layer sliding, where hexagonal closepacked planes were aligned parallel to the Couette cell walls. We have previously investigated the effect of large amplitude shearing on cubic phases formed in concentrated solutions of EB diblock copolymers, using SAXS or SANS.13,15,27-38 In this paper, we report on the structure of the cubic gel phases formed by diblocks E96B18, E184B18, E315B17, and E398B19 in water. These copolymers are highly asymmetric, with long hydrophilic blocks attached to short hydrophobic ones. Consequently they form micelles with large coronas swollen by water, and thus the critical gel concentration is low. In fact, at a fixed temperature and nearly constant B block length, the critical gel concentration generally decreases with increasing E block length, down to 4 wt % for E398B19. Such low copolymer content gels are of potential technological interest, for example in drug delivery and oil recovery, and work is in progress to further extend the length of the water-soluble chains with end-attached hydrophobes. Here, we identify the structure of the gels using SAXS. Large-amplitude oscillatory shear is used to prepare highly aligned “crystals” of the gels, and this turns out to be essential to identify gels based on the symmetry of the diffraction patterns. 2. Experimental Section 2.1. Synthesis. The diblock copolymers E96B18, E184B18, E315B17, and E398B19 were prepared by sequential anionic polymerization of ethylene oxide followed by 1,2-butylene oxide.39 The monofunctional initiator was 2(2-methoxyethoxy)ethanol activated by reaction with potassium metal (mole ratio OH/K ≈ 5). Vacuum line and ampule techniques were used. Characterization of the copolymers by gel permeation chromatography [GPC, calibrated with poly(oxyethylene) standards] indicated narrow chain length distributions, i.e., Mw/Mn ≈ 1.05, where Mw and Mn are the mass-average and number-average molar mass, respectively. Molecular characteristics are listed in Table 1. Further details of characterization of the copolymer composition are provided elsewhere.39 The uncertainty in the number of B repeats is ( 2%, and in the number of E repeats is ( 1%. 2.2. Sol-Gel Boundaries. Samples of solution (0.5 g) were enclosed in small tubes (internal diameter ca. 10 mm), and (32) Hamley, I. W.; Pople, J. A.; Booth, C.; Yang; Y.-W.; King, S. M. Langmuir 1998, 14, 3182. (33) Hamley, I. W.; Pople, J. A.; Booth, C.; Derici, L.; Impe´ror-Clerc, M.; Davidson, P. Phys. Rev. E 1998, 58, 7620. (34) Pople, J. A.; Hamley, I. W.; Terrill, N. J.; Fairclough, J. P. A.; Ryan, A. J.; Yu, G.-E.; Booth, C. Polymer 1998, 39, 4891. (35) Fairclough, J. P. A.; Ryan, A. J.; Hamley, I. W.; Li, H.; Yu, G.-E.; Booth, C. Macromolecules 1999, 32, 2058. (36) Alexandridis, P.; Olsson, U.; Lindman, B. Langmuir 1997, 13, 23. (37) Pople, J. A.; Hamley, I. W.; Diakun, G. P. Rev. Sci. Instrum. 1998, 69, 3015. (38) Pople, J. A.; Hamley, I. W.; Fairclough, J. P. A.; Ryan, A. J.; Booth, C. Macromolecules 1998, 31, 2952. (39) Mingvanish, W.; Mai, S.-M.; Heatley, F.; Booth, C.; Attwood, D. J. Phys. Chem. B, accepted.

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Table 1. Molecular Characteristics of the Copolymers, and Micellar Thermodynamic and Hydrodynamic Radii at 20 °C39 copolymer

Mn/103 g mol-1 (NMR)

Mw/Mn (GPC)

rt/nma

rh/nma

E96B18 E184B18 E315B17 E398B19

5.5 9.4 15.1 18.9

1.03 1.03 1.04 1.06

14.9 18.0 17.2 21.5

15.9 19.5 21.8 26.6

a

By extrapolation of values at 25, 40, and 50 °C.

observed while slowly heating the tube in a water bath within the range 0-100 °C. The heating/cooling rate was 0.5 °C min-1 or less. Additional data points in the range 0 to -10 °C were obtained by cooling the tubes in an acetone/dry ice bath, care being taken to avoid freezing the solution at the lowest temperatures. The change from a mobile to an immobile system (or vice versa) was determined by inverting the tube. The method served to define a sol-gel transition temperature to (1 °C. The hard gels were found to be immobile in the inverted tubes over time periods of days to several months. In favorable cases, this simple method of detecting gelation, which is sensitive to the yield stress of the gel, has been shown to define the same hardgel phase boundary as rheometry and differential scanning calorimetry.4 The mobile and immobile solutions were clear at all temperatures. In particular, no clouding was observed at the hard gel (immobile-mobile) boundary, indicating that any biphasic regions are too narrow to be detected. 2.3. Rheology. Experiments were conducted using a Rheometrics RSAII rheometer, with a shear sandwich geometry. All measurements were conducted in the linear viscoelastic regime, identified via strain amplitude sweeps (typically A ) 0.3%). Data discussed in this paper were obtained at room temperature, where frequency sweeps served to confirm the cubic nature of the gels. 2.4. Small-Angle X-ray Scattering. SAXS experiments were conducted at the Synchrotron Radiation Source, Daresbury Laboratory, UK, on beamline 2.1, which is configured with an X-ray wavelength λ ) 1.5 Å. Details of this beamline and data collection electronics have been given elsewhere.40 Experiments were conducted with a 6 m camera length. Scattered photons were collected on a multiwire gas-filled area detector. A scattering pattern from a specimen of wet collagen (rat tail tendon) was used for calibration of the q scale (q ) 4πsinθ/λ, where the scattering angle is 2θ). Solutions were subjected to either steady shear or oscillatory shear, using a Couette cell described in detail elsewhere.37 Briefly, it comprises two concentric polycarbonate cylinders, with an inner stator (radius 25 mm) and an outer rotor, with a 0.5 mm gap for the sample. A cylindrical tube is incorporated in the inner stator, for the X-ray beam to pass through. The accessible shear rate range is 0.05 < γ˘ /s-1 < 540 for continuous shear and 0.05 104 Pa at a frequency ω ) 1 rad s-1. Frequency sweeps for the same gels show that G′ is a weak function of frequency in the range 10-2 < ω/rad s-1 < 102, while G′′(ω) has a broad minimum. This behavior is characteristic of a cubic phase.28,41,42 Further details on rheology will be the subject of a future paper. 3.2. SAXS. Experiments were conducted on each of E96B18, E184B18, E315B17, and E398B19 at four concentrations spanning the low concentration gel transition boundary. First, we consider the symmetry of the gel phase, bodycentered cubic versus face-centered cubic. We observe that micelles with the relatively shorter corona blocks tend to form a fcc structure, whereas micelles formed from diblocks with longer corona blocks form a bcc structure. This is confirmed by the results in Figure 2, which shows SAXS patterns obtained for gels of all four copolymers at approximately 10 wt % copolymer, following oscillatory shear at γ˘ ) 50 s-1, A ) 17 500%, apart from E96B18, the pattern for which shown in Figure 2 was obtained during shear under these conditions, for reasons discussed shortly. The SAXS patterns for the two shorter copolymers indicate a fcc structure for gels containing approximately 10 wt % copolymer. This is confirmed from the sequence of observed reflections and their azimuthal orientation. Figure 3 shows two-dimensional data from an unoriented gel of E96B18 (9 wt %) reduced to a plot of intensity versus q. The sequence of observed reflections matches closely those expected for a structure with fcc (spacegroup Fm3 h m) symmetry, with peaks in the positional ratio 1:x4/3:x8/3:x11/3. The SAXS pattern for E96B18 in Figure 2a is similar to that previously reported for a fcc gel of E40B10, although in that case orientation was achieved under steady shear.15,16 As shown in Figure 4a, the six inner peaks result from the hexagonal-close packed layers, while at least two sets of higher order reflections are also apparent. The layers of hexagonally packed micelles align parallel to the walls of the Couette cell, i.e.,

(40) Towns-Andrews, E.; Berry, A.; Bordas, J.; Mant, G. R. Murray, P. K.; Roberts, K.; Sumner, I.; Worgan, J. S.; Lewis, E.; Gabriel, A. Rev. Sci. Instrum. 1989, 60, 2346.

(41) Jones, J. L.; McLeish, T. C. B. Langmuir 1995, 11, 785. (42) Kossuth, M. B.; Morse, D. C.; Bates, F. S. J. Rheol. 1999, 43, 167.

3. Results and Discussion

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Figure 2. SAXS patterns obtained at 20 °C for 10 wt % gels under shear at γ˘ ) 50 s-1, A ) 17 500%, except E96B18 (9 wt % gel, during shear at these conditions). (a) E96B18, (b) E184B18, (c) E315B17, (d) E398B19. The shear direction is horizontal.

Figure 3. Profile of intensity versus q for a 9 wt % gel of E96B18 at 20 °C, obtained by radial integration of an isotropic SAXS pattern. The arrows denote the positions of expected reflections for a fcc structure.

in the (v,e) plane, with a close-packed direction parallel to v. As discussed in detail elsewhere,10,15,16,18-22 the presence of six inner reflections indicates that the stacking sequence of layers of hexagonally packed micelles cannot be purely ABCABC... as in a fcc structure, but must include sequences of ABAB... stacking as in a hexagonal closepacked structure. Thus, we choose to index the patterns following shear based on a hexagonal close-packed structure, using Miller indices for the two-dimensional lattice. The fraction of the different stacking arrangements is shear-rate dependent, although a full analysis of this is beyond the scope of the present paper. Related shearinduced effects on the relative stability of fcc and hcp structures in gels of E55B8 are the subject of a forthcoming publication.43 The SAXS pattern for E184B18 in Figure 2b is also consistent with a hexagonal close-packed structure,

although there are differences compared to the SAXS pattern for E96B18. First, a much higher degree of orientation is apparent for the latter copolymer, despite the fact that this pattern was obtained at rest after shearing, whereas the SAXS pattern for E96B18 was obtained during shear (at the same shear rate). Second, four weak additional off-meridional reflections can be discerned close to the first-order reflections (Figure 4b). Since these are oriented at (55° with respect to the meridian, they appear to be 110-type reflections from a small fraction of bcc crystals in the gel, because as will be discussed shortly this arrangement of reflections is observed for bcc gels. Previously, we have observed 200 type reflections close to the inner hexagon of peaks, but these are at (45° (a similar observation was made by others14) and this does not correspond to the location of the peaks in the pattern for E184B18. We turn now to the copolymers with longer corona blocks. The SAXS patterns for E315B17 and E398B19 gels (10 wt %) shown in Figure 2c and Figure 2d respectively are quite distinct from those for the fcc structures in Figure 2a and Figure 2b, in that there are no higher order reflections. Despite this, we are confident that these patterns for E315B17 and E398B19 are from a bcc structure, for the following reasons. First, there is no evidence of any reflections at x4/3q* characteristic of a fcc structure. Second, the azimuthal positions of the reflections are identical to those of the 110 reflections in bcc phases of EmBn diblock gels previously investigated by us. Specifically, the two pairs of off-meridional peaks in each pattern are oriented at (55° with respect to the meridian (Figure 4c). This is the signature of a twinned bcc structure, as detailed previously.28,32 The patterns can be indexed on (43) Daniel, C.; Hamley, I. W.; Mingvanish, W.; Booth, C.; Messe, L. In preparation.

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Hamley et al. Table 2. Comparison of Micellar Dimensions at 20 °Ca copolymer

q*/Å-1 before shear

after shear

r/nm from SAXS

rt/nm from SLSb

E96B18 E184B18 E315B17 E398B19

0.0268 0.0248 0.0241 0.0199

0.0257 0.0230 0.0241 0.0199

14.3 15.5 16.0 19.3

14.9 18.0 17.2 21.5

a Values listed were obtained from SAXS on 10 wt % gels (9 wt % for E96B18). The thermodynamic radii were obtained via fits of the structure factor for hard spheres to static light scattering data from solutions of different concentrations.39 b Details as Table 1.

Figure 4. Indexation of SAXS patterns. (a) 9 wt % gel of E96B18. The superposed hexagons highlight the 6-fold symmetry that results from the stacked hexagonal close-packed layers. The presence of inner reflections indicates that the stacking sequence is not uniquely ABCABC... as in a fcc structure, but that there must be ABAB... sequences in addition. (b) Indexation of SAXS pattern for the 10 wt % gel of E184B18. In addition to the hexagonal arrangement of the primary reflections, weak side peaks are observed close to the inner peaks, at (55° with respect to the meridian. These are probably due to a small fraction of bcc phase, coexisting with the fcc structure (c) 10 wt % gel of E315B17 (the SAXS pattern for the E398B19 gel has the same symmetry). The sample is a multiply twinned, directionally oriented crystal. The labels A, B, C, and D correspond to different twins obtained by rotation around the [111] direction.

the basis of a directionally oriented crystal, as discussed elsewhere.30-33 Essentially, the bcc phase is oriented with a [111] direction along the shear direction, and flow is in {110}, {211}, and {321} planes, for which the [111] direction is a common zone axis. A similar shear-induced oriented SANS pattern has been observed for block copolymer melts.44,45 The positions of the first-order reflections, q* obtained for gels at 10 wt % (except E96B18, at 9 wt %) are compared to thermodynamic radii in Table 2. Interestingly, a (5 ( 1)% decrease in q* was observed when comparing values for the samples forming a fcc phase upon shearing at γ˘ ) 50 s-1, A ) 17 500%. Micellar radii from SAXS were obtained from the unit cell dimensions (before shear), using the appropriate packing fractions for bcc and fcc structures. SAXS thus yields effective total micellar radii. The thermodynamic radius, obtained via the concentration dependence of the inverse intensity in static light scattering experiments,2,3,30,31,39 provides a reliable indicator of the critical concentration for micellar packing. In particular, for the copolymers studied here, good correspondence has been observed between δt-1 and the critical gel concentration.39 Here δt ) vt/va is the thermodynamic expansion factor, where vt is the thermodynamic volume (vt ) 4/3πrt3) and va is the anhydrous volume of a micelle. Table 2 shows that the micellar radius from SAXS is similar to, and shows the same trend with E block length as, the thermodynamic radius. Assuming a uniform hard sphere structure, the radius of gyration in dilute solution can be obtained from dynamic light scattering (DLS) measurements of the hydrodynamic radius, rh, as rg ) x3/5rh. The radius of gyration in dilute solution shows the same trend with E block length as the thermodynamic radius (Table 1), and thus also as the micellar radius from SAXS. A correspondence between the radius of gyration (obtained in the zero concentration limit) and the micellar radius from SAXS on gels has been noted elsewhere,8,33 although a large increase in micellar mass in the gel has been noted.33 Comparison of Table 1 and Table 2 shows that for each copolymer the micellar radius from SAXS is closer to the thermodynamic radius than to the hydrodynamic radius, as expected, since the thermodynamic radius is directly related to the excluded volume of one micelle for another. We now consider the qualitative effects of shear conditions. For the bcc gels, oscillatory shear led to highly aligned twinned bcc crystals at a shear rate as low as γ˘ ) 5 s-1. Only a limited number of strain amplitudes were applied, a high degree of orientation being observed at A ) 17 500%, but not A ) 3500%. Structural transitions were not observed during shear. A further increase in the (44) Almdal, K.; Koppi, K. A.; Bates, F. S. Macromolecules 1993, 26, 4058. (45) Koppi, K. A.; Tirrell, M.; Bates, F. S.; Almdal, K.; Mortensen, K. J. Rheol. 1994, 38, 999. (46) Daniel, C.; Hamley, I. W.; Messe, L.; Mingvanish, W.; Mai, S.M.; Booth, C. unpublished results.

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Figure 5. SAXS patterns from a 9 wt % gel of E96B18 at 20 °C under the following conditions. (a) Shearing at γ˘ ) 5 s-1, A ) 17 500%, (b) Shear stopped, (c) Shearing at γ˘ ) 50 s-1, 17 500%, (d) Shear stopped.

sharpness of Bragg reflections was observed for a shear rate γ˘ ) 50 s-1; however the effect of shear rate was also not examined in detail. The effect of shear on the fcc gels led to structural transformations, specifically rearrangements in stacking sequence, as exemplified by the shifts in q* discussed above. Again, only a limited number of shear rates and strain amplitudes were applied, precluding a full study of critical conditions for macroscopic alignment of crystals. We observed that generally oscillatory shear produced better orientation of the samples than steady shear. Foaming of the sample was particularly a problem for steady shear rates γ˘ > 50 s-1. As mentioned above, data for E96B18 are presented for gels under shear. This is because following shear, a loss of orientation of the hexagonal close-packed crystal was observed. This is illustrated in Figure 5. Figure 5a shows the same SAXS pattern as Figure 2a. Upon cessation of shear, orientation of the gel is completely lost, as shown by the SAXS pattern containing rings in Figure 5b, which is produced by an unoriented fcc structure. Figure 5c shows the SAXS pattern obtained at a higher shear rate, obtained immediately after Figure 5b and shows the recovery of orientation (to a higher degree than in Figure 5a). Upon cessation of shear, partial relaxation of the orientation occurred, as indicated by the appearance of grainy Bragg peaks in Figure 5d. Such relaxation effects were not observed for the fcc gels formed by E184B18, nor the bcc gels. These results suggest that the fcc gel of E96B18 has a lower susceptibility to shear. A similar difference in the susceptibility to shear was observed for fcc gels of E40B10 at different concentrations.15,16 4. Effect of Copolymer Composition on the Symmetry of the Gel Structure Table 3 summarizes the symmetry of gel structures for micellar solutions of EmBn block copolymers observed in

Table 3. Symmetry of Gel Structures (at 20 °C, Except Where Stated)a copolymer (m/n ratio) E22B7 (3.1) E131B37 (3.5) E40B10 (4) E41B8 (5.1) E96B18 (5.3) E106B16 (6.6) E55B8 (6.9) E90B10 (9) E184B18 (10.2) E210B16 (13.1) E103B15E103 (13.7) E315B17 (18.5) E398B19 (20.9)

concentration (wt %) (in H2O, unless stated)

structure

35 20-30 25b 34-38b 26 40 9-15 30-45 20 35 20-40b,c 30-70 10 30-60 30-45 10 10

bcc46 fcc46 fcc13,15,16 bcc13,15,16 fcc35 bcc35 fcc (this work) bcc30 fcc43 bcc46 bcc32,33 bcc28,31 fcc (this work) bcc30 bcc30 bcc (this work) bcc (this work)

a All parameters refer to solutions in water, except salt solutions where stated. b In 0.2 M K2SO4. c Values obtained for E86B10.

this, and previous, work from our group. McConnell et al. have previously presented a phase diagram for PS-PI diblocks in decane in terms of the ratio of the micellar coronal radius (from the hydrodynamic radius) to core radius (from SANS) for each copolymer.8 In the absence of measured core radii for all copolymers, we have simply used the ratio m/n as a measure of the corona to core size. In addition, we use concentration as an ordinate rather than the volume fraction of the core block. Figure 6 shows the phase diagram for the two lattice types, fcc versus bcc. As expected, the fcc phase is favored for “crew cut” micelles, i.e., those with short coronas compared to the core radius, at low concentrations, and the bcc phase is favored for softer spheres, i.e., those with relatively large coronas. A

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Table 4. Effect of Copolymer EmBn Composition, Concentration, and Temperature on Micellar Association Number, N, and Interaction Potential constant

increase

effect on micellar association number and potential

T, c, n T, c, m T, n, m c, n, m

m n c T

N decreases, hence a softer potential: fcc f bcc N increases, hence a harder potential: bcc f fcc easier penetration, hence a softer potential: fcc f bcc N increases to a plateau value. Initially a harder potential as N increases. Thereafter a softer potential as the solvent approaches the theta condition

mately 2rg. Assuming the same relative expansion factors, we can estimate the ratio of corona to core radii for micelles of the other copolymers. Since the repeat length in a fully extended B block is 0.36 nm, and the statistical segment length in a coiled E block is bE ) 0.56 nm, the ratio of corona/core radii is 2.1xm/n. The topology of the phase diagram, however, is unchanged if we use this variable for the abscissa instead of just m/n, and is not shown here. 5. Summary

Figure 6. Phase diagram showing regions of stability of fcc (O) and bcc (9) phases as a function of diblock EmBn concentration and m/n. At a fixed concentration, points corresponding to a given structure delineate the range of concentrations examined, i.e., in many cases the structure was also identified at intermediate concentrations.

particularly distinct feature of our phase diagram, independent of our choice of abscissa, is the observation of concentration-dependent transitions between fcc and bcc structures. This differs from the results of McConnell et al.8 Thermal transitions between fcc and bcc structures have also been observed for gels of E40B10 in aqueous salt solution, however Figure 6 is an isothermal phase diagram. The change in potential from that for interacting soft spheres to that for interacting hard spheres upon decreasing concentration is believed to be due to an increased density of coronal chains, which in turn means less chain penetration, and so a harder potential. The density of coronal chains will tend to increase with an increase in association number, N. It follows that a change in micellar composition, concentration, or solution temperature that favors an increase in N will lead to a harder potential, as summarized in Table 4. This is also confirmed by the ratio rh/rt obtained from Table 1. For a perfect hard sphere, rh/rt ) 1. For E96B18 and E184B18, rh/rt ≈ 1.07, close to the hard sphere limit. However for E315B17 rh/rt ) 1.27, and E398B19 rh/rt ) 1.24, consistent in both cases with softer spheres that are expected to pack in a bcc array. The sizes of the corona and core have been directly obtained from SANS intensity profiles of the micellar form factor for E90B10, which were modeled to provide the corresponding radii.31 The model indicated that the core blocks are highly extended, to approximately 60% of their fully extended length, whereas the corona blocks are best described as stretched coils, with dimensions approxi-

The structure of the cubic micellar gel phase formed by diblocks E96B18, E184B18, E315B17, and E398B19 in water was determined using SAXS on shear oriented specimens. Comparing gels at 10 wt % copolymer at 20 °C, those formed by the former two diblocks were fcc, whereas the latter were both bcc. For the bcc gels, although only a single order of reflections was observed, the extent of orientation was higher than for the fcc gels. The fcc gels show more subtle effects under shear, especially a shearinduced increase in domain spacing. In addition, the fcc gel of E96B18 was less susceptible to permanent alignment than gels formed by the other copolymers. These effects are presumably the result of flow of layers of hexagonalpacked micelles in the (v,e) plane, with flow along a closepacked direction. This is not the observed flow mechanism for bcc gels, which instead adopt a twinned structure, with flow along a close-packed [111] direction. The fcc and hcp arrangements have the same packing fraction and appear to have similar free energies so that flow can change the proportions of different stacking sequences. In addition, we have combined the results for the cubic structure for the copolymers examined in this paper, with those previously obtained by us, to produce a phase diagram in terms of copolymer concentration and asymmetry (block size ratio). This shows that fcc structures are observed at low concentrations for m/n e 10, whereas the bcc lattice is found for all copolymers at high concentration, and is the only lattice type observed for m/n g 10. Acknowledgment. We are grateful to the EPSRC for financial support of this work (grants GR/L22645 for synthesis and GR/L79854 for SAXS/rheology). The Thai Government provided a Research Studentship for W.M. Both C.D. and L.M. were supported by the European Union Training and Mobility Network “Complex Architectures in Diblock Copolymer Based Polymer Systems”. LA991035J