Mechanism of the Transition between Lamellar and Gyroid Phases

Nov 4, 2004 - Leiden Institute of Chemistry, Leiden University, P.O. Box 9502,. 2300 RA Leiden ... University of Central Lancashire, Preston, PR1 2HE,...
1 downloads 0 Views 364KB Size
Download PDF
Langmuir 2004, 20, 10785-10790

10785

Articles Mechanism of the Transition between Lamellar and Gyroid Phases Formed by a Diblock Copolymer in Aqueous Solution Ian W. Hamley,* Valeria Castelletto, Oleksandr O. Mykhaylyk, and Zhuo Yang Department of Chemistry, University of Leeds, Leeds LS2 9JT, U.K.

Roland P. May Institut Laue Langevin, F-38042 Grenoble, France

Kateryna S. Lyakhova and G. J. Agur Sevink Leiden Institute of Chemistry, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands

Andrei V. Zvelindovsky Centre for Materials Science, Department of Physics, Astronomy & Mathematics, University of Central Lancashire, Preston, PR1 2HE, U.K. Received June 18, 2004. In Final Form: September 17, 2004 The mechanism of the transition from a lamellar phase to a gyroid phase in an aqueous solution of a diblock copolymer has been studied by time-resolved synchrotron small-angle X-ray scattering. The transition occurs via a metastable perforated lamellar structure. The perforations initially have liquidlike ordering before developing hexagonal packing. The transient phase of irregularly perforated layers is revealed by the development of diffuse scattering peaks, just below the Bragg peaks of the lamellar structure. The diffuse scattering is modeled by Monte Carlo simulations of perforated layers. Following the formation of perforations, Bragg peaks characteristic of a hexagonal structure signal an ordering into a hexagonal lattice (with the concomitant loss of diffuse scattering). Computer simulations based on a dynamic density functional model reproduce these features. The hexagonal perforated lamellar phase is rapidly replaced by the gyroid phase. The domain spacing of the gyroid phase is larger than that of the perforated lamellar structure. The perforated lamellar and gyroid phases coexist for a defined period. The reverse transition from gyroid to lamellae occurs directly, with no transient or metastable intermediates.

1. Introduction The pathway associated with transitions between ordered phases in soft materials can often be intricate, reflecting the multitude of distinct structures that may be accessible. In block copolymer melts, it is now firmly established that lamellar (L), hexagonal-packed cylindrical (H), cubic spherical (BCC and HCP/FCC), and bicontinuous cubic gyroid (G, space group Ia3 h d) phases are the only stable equilibrium structures.1-3 A hexagonal perforated lamellar (HPL) structure has been observed as a metastable phase between L and G phases4-9 but is not stable * Author to whom correspondence should be addressed. (1) Matsen, M. W.; Schick, M. Phys. Rev. Lett. 1994, 72, 2660. (2) Hamley, I. W. The Physics of Block Copolymers; Oxford University Press: Oxford, 1998. (3) Hamley, I. W. J. Phys., Condens. Matt. 2001, 13, R643. (4) Hamley, I. W.; Koppi, K. A.; Rosedale, J. H.; Bates, F. S.; Almdal, K.; Mortensen, K. Macromolecules 1993, 26, 5959. (5) Hamley, I. W.; Gehlsen, M. D.; Khandpur, A. K.; Koppi, K. A.; Rosedale, J. H.; Schulz, M. F.; Bates, F. S.; Almdal, K.; Mortensen, K. J. Phys. France II 1994, 4, 2161. (6) Hajduk, D. A.; Takenouchi, H.; Hillmyer, M. A.; Bates, F. S.; Vigild, M. E.; Almdal, K. Macromolecules 1997, 30, 3788.

in equilibrium, at least in neat diblocks (a perforated lamellar structure has also been observed in blends10). This was predicted first using Semenov’s model for strongly segregated diblocks11 and in the random-phase approximation mean field theory in the weak segregation limit,12 as well as self-consistent mean field theory,1,13 and later confirmed experimentally. A perforated lamellar (PL) structure was first observed for a poly(ethylene propylene)-poly(ethylethylene) diblock copolymer melt.14 Later work using SANS on a shear-oriented specimen of the same sample identified both hexagonal modulated (7) Vigild, M. E.; Almdal, K.; Mortensen, K.; Hamley, I. W.; Fairclough, J. P. A.; Ryan, A. J. Macromolecules 1998, 31, 5702. (8) Hajduk, D. A.; Ho, R.-M.; Hillmyer, M. A.; Bates, F. S.; Almdal, K. J. Phys. Chem. B 1998, 102, 1356. (9) Hamley, I. W.; Fairclough, J. P. A.; Ryan, A. J.; Mai, S.-M.; Booth, C. Phys. Chem., Chem. Phys. 1999, 1, 2097. (10) Disko, M. M.; Liang, K. S.; Behal, S. K.; Roe, R. J.; Jeon, K. J. Macromolecules 1993, 26, 2983. (11) Fredrickson, G. H. Macromolecules 1991, 24, 3456. (12) Hamley, I. W.; Bates, F. S. J. Chem. Phys. 1994, 100, 6813. (13) Matsen, M. W.; Bates, F. S. Macromolecules 1996, 29, 7641. (14) Almdal, K.; Bates, F. S.; Mortensen, K. J. Chem. Phys. 1992, 96, 9122.

10.1021/la0484927 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/04/2004

10786

Langmuir, Vol. 20, No. 25, 2004

lamellar (HML) and HPL structures,4,5 although it is now believed that the HML phase is a transient structure and the HPL phase is metastable.15-17 Following the identification of the bicontinuous cubic structure as G,18,19 a more detailed discussion of the symmetry of the HPL phase was possiblesand it was concluded that it consists of hexagonal close-packed layers with preodominantly ABC packing (spacegroup R3 h m) but with a contribution from ABAB stacking.20 The relationship between the topology of the HPL and G phases was also noted. The HPL structure consists of planar tripods, which are distorted by rotation through 70.53° in the G phase, which is generated by the appropriate symmetry operations.20 Zhu et al. have modeled SAXS patterns from the HPL phase in detail on the basis of high-quality SAXS patterns (in three perpendicular planes) from PEO-PS diblocks sheared in the melt.21,22 They model the structure as a combination of two stackings of hexagonally perforated lamellaesABAB (hexagonal, spacegroup P63/mmc) and ABC (trigonal, spacegroup R3 h m) twinssthe latter being the predominant structure.21,22 Defects in the stacking sequence were imaged directly by transmission electron microscopy on stained samples.22 The transition from lowtemperature metastable HPL phase to G and from hightemperature hexagonal phase to G was examined for a blend of poly(ethylene propylene)-poly(dimethylsiloxane) diblock copolymers via SANS and SAXS.7 The growth of the G phase during isothermal annealing following an increase of temperature from the HPL phase was found to be extremely slow (taking up to 10 h), as was the complete development of G during an isothermal anneal following a quench from H. The G phase was bypassed completely during a slow cool from H to HPL, although presumably on thermodynamic grounds, the low temperature HPL phase would eventually transform into G. The kinetics for this process at low temperature appear to be very slow, i.e., the HPL phase is highly metastable in this system. For other diblocks where the PL phase is observed at higher temperature, the transformation into G can be monitored.8,20,23,24 Hajduk et al. undertook a particularly thorough examination of the transition from L to G in several diblock copolymers.8 They report that direct L-G transitions are suppressed by the high surface tension associated with grain boundaries between L and G. In contrast, the transition L to PL occurs readily. That from PL to G depends on the mismatch between spacings of the PL lamellae and the dense G {211} planes, the transition slowing as the mismatch increases. They note that the reverse G-to-L transition can occur directly, although it is slow. They propose that all transformations (15) Qi, S.; Wang, Z. G. Phys. Rev. E 1997, 55, 1682. (16) Laradji, M.; Shi, A.-C.; Noolandi, J.; Desai, R. C. Phys. Rev. Lett. 1997, 78, 2577. (17) Laradji, M.; Shi, A.-C.; Noolandi, J.; Desai, R. C. Macromolecules 1997, 30, 3242. (18) Schulz, M. F.; Bates, F. S.; Almdal, K.; Mortensen, K. Phys. Rev. Lett. 1994, 73, 86. (19) Hajduk, D. A.; Harper, P. E.; Gruner, S. M.; Honeker, C. C.; Kim, G.; Thomas, E. L.; Fetters, L. J. Macromolecules 1994, 27, 4063. (20) Fo¨rster, S.; Khandpur, A. K.; Zhao, J.; Bates, F. S.; Hamley, I. W.; Ryan, A. J.; Bras, W. Macromolecules 1994, 27, 6922. (21) Zhu, L.; Huang, P.; Cheng, S. Z. D.; Ge, Q.; Quirk, R. P.; Thomas, E. L.; Lotz, B.; Wittmann, J.-C.; Hsiao, B. S.; Yeh, F.; Liu, L. Phys. Rev. Lett. 2001, 86, 6030. (22) Zhu, L.; Huang, P.; Chen, W. Y.; Weng, X.; Cheng, S. Z. D.; Ge, Q.; Quirk, R. P.; Senador, T.; Shaw, M. T.; Thomas, E. L.; Lotz, B.; Hsiao, B. S.; Yeh, F.; Liu, L. Macromolecules 2003, 36, 3180. (23) Zhao, J.; Majumdar, B.; Schulz, M. F.; Bates, F. S.; Almdal, K.; Mortensen, K.; Hajduk, D. A.; Gruner, S. M. Macromolecules 1996, 29, 1204. (24) Khandpur, A. K.; Fo¨rster, S.; Bates, F. S.; Hamley, I. W.; Ryan, A. J.; Bras, W.; Almdal, K.; Mortensen, K. Macromolecules 1995, 28, 8796.

Hamley et al.

occur through nucleation and growth of the final phase for shallow quenches. However, for fast quenches, they comment that thermodynamically unstable states could be accessed.8 Qi and Wang have modeled pathways between ordered structures in weakly segregated block copolymers via cell dynamics simulations,25,15 although they have not considered the case of the L-to-G transition, which is the focus of the current paper. The transition between H and G phases in a diblock copolymer blend was found to occur epitaxially, with well-defined relationships between the lattices of the shear-aligned structures.18 Matsen developed a model for this transition based on self-consistent mean-field theory and proposed that it occurs epitaxially, although it was not established that the domain spacing is conserved across the transition.26 In solution, block copolymers behave as surfactants. In the present work, we investigate the mechanism of the transition between lamellar and gyroid phases in an aqueous solution of a poly(ethylene oxide)-based oligomeric diblock copolymer that is closely related to the alcohol ethoxylate nonionic surfactants (CnEm, C ) methylene, E ) ethylene oxide). The transition between lamellar, gyroid, and hexagonal-packed cylinder phases has previously been investigated for a number of these systems.27-30 Ranc¸ on and Charvolin observed that L or H phases grow epitaxially from the G phase formed by the surfactant C12E6 (C denotes methylene) in water.27 Specifically, the {211} planes of the G phase were found to be epitaxially related to the {001} planes of the L phase and the {10} planes of the H phase. In addition, they observed growth of hexagonal-packed cylinders along the 〈111〉 directions of the G phase.27 Clerc et al. propose that the transition occurs via an intermediate hexagonal perforated layer structure which puckers to form the channels of the G structure.28 Intermediate phases have also been reported for C16E7 in D2O, including a defective lamellar phase containing uncorrelated water-filled defects31 and a rhombohedral mesh phase (space group R3 h m), which is also observed for C30E9 in D2O.32,33 Interestingly, the latter is only observed on cooling from the high-temperature L phase prior to the G phase. On heating, only G is observed. The transition from the high-temperature L phase to the lower-temperature G phase in C16E7/H2O on slow cooling has been studied in greater detail by SAXS and observed to occur via a transient perforated layer structure which transforms to a rhombohedral structure immediately prior to formation of the gyroid structure.29,30 Imai and co-workers have also examined the nature of the diffuse scattering in the PL phase. They find excess scattering beyond that expected for thermally fluctuating lamellae, described by the Caille´ structure factor.29,30,34 The excess scattering was compared to results from a cell dynamics simulation using the time-dependent Ginzburg-Landau equation to de(25) Qi, S.; Wang, Z. G. Phys. Rev. Lett. 1996, 76, 1679. (26) Matsen, M. W. Phys. Rev. Lett. 1998, 80, 4470. (27) Ranc¸ on, Y.; Charvolin, J. J. Phys. Chem. 1988, 92, 2646. (28) Clerc, M.; Levelut, A. M.; Sadoc, J. F. J. Phys. France II 1991, 1, 1263. (29) Imai, M.; Kawaguchi, A.; Saeki, A.; Nakaya, K.; Kato, T.; Ito, K.; Amemiya, Y. Phys. Rev. E 2000, 62, 6865. (30) Imai, M.; Saeki, A.; Teramoto, T.; Kawaguchi, A.; Nakaya, K.; Kato, T.; Ito, K. J. Chem. Phys. 2001, 115, 10525. (31) Funari, S. S.; Holmes, M. C.; Tiddy, G. J. T. J. Phys. Chem. 1994, 98, 3015. (32) Fairhurst, C. E.; Holmes, M. C.; Leaver, M. S. Langmuir 1996, 12, 6336. (33) Leaver, M.; Fogden, A.; Holmes, M.; Fairhurst, C. Langmuir 2001, 17, 35. (34) Imai, M.; Nakaya, K.; Kawakatsu, T.; Seto, H. J. Chem. Phys. 2003, 119, 8103.

Transition Mechanism between Lamellar and Gyroid Phases

scribe the dynamics of fluctuating lamellae.35 Additional evidence for perforated layers is provided by the nonideal swelling observed in the lamellar phase, on decreasing temperature toward the G phase.36 This was analyzed to estimate the volume fraction of perforations. The same method has been applied by us to analyze nonideal swelling observed for diblock E18B10 in water (which is similar to the system studied in the present paper).37 Defective lamellar phases have been observed for many other surfactants.38-41 There have been fewer reports on intermediate phases for lipids. For a ternary fatty acid/ lipid system in water, it has been proposed that the transformation from L to G occurs via a bicontinuous “double diamond” structure (D, space group Pm3 h n).42 A mechanism for the transformation was put forward based on the distortion of hexagonal channels (perforations) into interpenetrating diamond lattices prior to the rearrangement of this structure into G. It has to be emphasized, however, that the formation of bicontinuous cubic phases other than G is commonly observed for lipids,43 whereas none have been observed so far for diblock copolymers in the melt or in binary solution. Such structures are not expected due to packing frustration.13 Here, we examine in detail the nature of the transition between L and G phases in a diblock copolymer surfactant in aqueous solution and find that it proceeds via a transient perforated lamellar intermediate and with a discontinuous change in domain spacing. The signature of the appearance of perforations in the lamellae is the development of pronounced diffuse scattering just below the first Bragg reflection (with an additional higher-order peak). Ordering of the perforations into a hexagonal lattice occurs in a subsequent step as the precursor to development of the three-dimensional crystalline order of the G structure. The diffuse scattering is modeled on the basis of Monte Carlo simulations of packed holes in two dimensions, with initial random orientation but developing near closepacked order as the density of perforations increases. The L-to-PL transition is modeled in detail by a threedimensional simulation based on dynamic density functional theory. 2. Experimental Section 2.1. Synthesis and Characterization. The diblock copolymer was prepared by sequential anionic polymerization, as described elsewhere,44 using the monofunctional initiator, 2-(2-methoxyethoxy) ethanol and deuterated monomer (ethylene oxide). Deuteration was performed to enhance contrast for SANS; deuteration does not affect the results presented, as confirmed in separate experiments on the nondeuterated analogous oligomer, E18B10. Samples were characterized by 13C NMR and matrix-assisted laser-desorption ionization (MALDI) mass spectroscopy to determine the average composition and by gel permeation chromatography (GPC) to determine the width of the molar mass distribution. The average composition of the block copolymer was CH3OE2(dE)18B10. The distribution was relatively (35) Saeki, A.; Yonezawa, F. Prog. Theor. Phys. Suppl. 2000, 128, 418. (36) Minewaki, K.; Kato, T.; Yoshida, T.; Imai, M.; Ito, K. Langmuir 2001, 17, 1864. (37) Castelletto, V.; Fisher, J.; Hamley, I. W.; Yang, Z. Colloids Surf. A 2002, 211, 9. (38) Holmes, M. C.; Charvolin, J. J. Phys. Chem. 1984, 88, 810. (39) Ke´kicheff, P.; Cabane, B. J. Phys. France 1987, 48, 1571. (40) Ke´kicheff, P.; Cabane, B. Acta Crystallogr. B 1988, 44, 395. (41) Funari, S. S.; Holmes, M. C.; Tiddy, G. J. T. J. Phys. Chem. 1992, 96, 11029. (42) Squires, A. M.; Templer, R. H.; Seddon, J. M.; Woenckhaus, J.; Winter, R.; Finet, S.; Theyencheri, N. Langmuir 2002, 18, 7384. (43) Seddon, J. M. Biochim. Biophys. Acta 1990, 1031, 1. (44) Deng, N.-J.; Luo, Y.-Z.; Tanodekaew, S.; Bingham, N.; Attwood, D.; Booth, C. J. Polym. Sci. B: Polym. Phys. 1995, 33, 1085.

Langmuir, Vol. 20, No. 25, 2004 10787 narrow (Mw/Mn ) 1.03). For SAXS, the polymer concentration was 63.1 wt% in H2O. For SANS, it was 59 wt% in D2O. 2.2. Small-Angle X-ray Scattering. SAXS experiments were conducted on station 8.2 at the Synchrotron Radiation Source, Daresbury Lab, UK. Samples were placed in DSC pans, modified by the insertion of mica windows to allow transmission of the X-ray beam. The pans were placed in a Linkam differential scanning calorimeter of single-pan design. The low thermal mass of the pans permitted rapid heating and cooling (using liquid nitrogen). Samples were heated to the L phase (at 55 or 65 °C) and quenched to the G phase at low temperature. Heating runs were also performed, confirming the equilibrium nature of hightemperature L and low-temperature G phases, and the reversibility of the transition between these two phases, although the mechanism of transformation is not the same in each direction. The X-ray wavelength was λ ) 1.5 Å. The scattering data were recorded with a 3.4 m sample-to-detector distance using a quadrant detector that produces a one-dimensional profile. SAXS patterns were corrected for detector response. This operation transforms the recorded diffraction patterns into profiles equivalent to those obtained with a Debye-Scherrer geometry. The wavevector scale (q ) (4π sin θ)/λ, where 2θ is the scattering angle) was calibrated using a specimen of wet collagen. 2.3. Small-Angle Neutron Scattering. SANS experiments were performed at D22 at the Institut Laue Langevin, Grenoble, France. Details of the beamline can be found elsewhere.45 Measurements were performed with a 3 m sample-to-detector distance, 14.4 m collimation, and neutrons with λ ) 6 Å. A gasfilled area detector was used to collect the scattering pattern. The q scale was calculated from the known sample-to-detector distance.

3. Results and Discussion 3.1. SAXS and SANS Experiments. SAXS was used to determine the equilibrium phase boundary between G and L phases. During repeated slow heating ramps (2 °C/min), this transition was located at (45 ( 1) °C. No intermediate phase was observed as a steady-state (equilibrium or metastable) structure, in contrast to prior work on block copolymer melts. The following discussion refers to truly transient intermediate structures observed between the L and G phases during rapid temperature changes. SAXS profiles obtained during a quench from the L phase at 65 °C to the G phase at 25 °C (the average cooling rate was measured to be 27 °C/min) are shown in Figure 1. Frames were collected every 10 s, and 25 °C was reached at frame 11. The sharp Bragg reflections at q* ) 0.078 Å-1 and q ) 0.157 Å-1 confirm an initial L structure (with period d ) 80.3 Å). A large increase in diffuse scattering is observed as a shoulder at the low q side of this peak (with an associated second-order peak) prior to the development of higher-order peaks at x3q* and 2q*, indicating a hexagonal structure (d ) 85 Å) which appears as the temperature reaches 25 °C. The diffuse scattering is clear in an individual SAXS intensity profile, shown in Figure 5. After approximately 1 min, additional reflections are observed at high q which may be indexed to spacegroup Ia3 h d, corresponding to the G structure. The transition from a HPL to the final G structure occurs with a distinct difference in q* for the HPL structure (which follows the temperature-dependent q* for the L phase) to that for the G phase, two sets of peaks coexisting for 140 s after 25 °C is reached (Figure 1, inset). In contrast to Imai et al., who studied slow cooled samples of a related nonionic surfactant, C16E7, in water,29,30 we find no evidence for a rhombohedral intermediate during a quench. The development of diffuse scattering during the quench and the subsequent transformation into a transient hexagonal (45) http://www.ill.fr/YellowBook/D22/

10788

Langmuir, Vol. 20, No. 25, 2004

Hamley et al.

Figure 2. SAXS intensity profiles recorded during a heating ramp from 25 to 55 °C (heating rate 40 °C/min).

Figure 1. SAXS intensity profiles recorded during a quench from 65 to 25 °C (cooling rate 22 °C/min). Inset: position and area of first-order peak as a function of time following temperature profile indicated. The squares refer to the L (HPL) structure and the circles to G.

structure provide valuable information on the mechanism of transformation from L to G. It can be noted that the development of diffuse scattering in the L phase prior to transformation to a HPL structure produces a scattering pattern very similar to that reported recently for a 55 wt% solution of C16E6 in water.34 As noted by Imai et al., the position of the diffuse scattering peak just below the first Bragg reflection (at approximately 0.9q*) is consistent with earlier calculations on the least-stable fluctuation mode of undulating lamellae.46 Qi and Wang anticipated diffuse scattering peaks at this location on the basis of an analysis of diblock copolymer melts at the spinodal of the lamellar phase. The instability led to a perforated lamellar structure, which was considered to have either hexagonal close-packed or body-centered cubic structure in three dimensions, the two being nearly degenerate in free energy (Laradji et al. considered earlier unstable modes of fluctuating block copolymer lamellae, but the analysis was restricted to in-plane fluctuations16,17). During the transformation, q* for the L (and subsequent PL) phase shifts continuously, as shown in Figure 1 (inset). However, just after 25 °C is reached, a peak appears at q* ) (0.082 ( 0.001) Å-1 signaling the formation of a G phase. The area of this peak then grows, while that of the PL phase decreases. The transformation is complete approximately 140 s after the quench temperature is reached. It is clear from the data that q* is not the same in L and G phases, i.e., the gyroid {211} planes do not form directly from the lamellar planes. The development of in-plane perforations occurs with a concomitant decrease in lamellar spacing in a symmetry-breaking transition that establishes local three-dimensional order based on hexagonally perforated layers as a precursor to the 3D crystalline order of the G phase. The development of perforations may occur via a nucleation and growth or “spinodal”-type mechanism. Our kinetic data unfortunately do not permit us to distinguish between these possibilities at present. The transformation from PL to G phase occurs very rapidly, and consistent with the interpretation of Hajduk et al., this corresponds to a close match in PL {001} and G {211} planes.8 (46) Qi, S.; Wang, Z.-G. Macromolecules 1997, 30, 4491.

Figure 3. SANS intensity profiles obtained on heating from 21 to 55 °C and back to 31 °C (temperatures measured at sample). The temperature was increased in steps, with 20 min of equilibration at each temp prior to 5 min of data acquisition.

The reverse transition, from G to L, occurs rapidly and without transient intermediate structures. Figure 2 shows SAXS data obtained in a representative run. There is no evidence for the build-up of diffuse scattering during the transition, although there is a shift in domain spacing back to the initial q* for L. It appears that the G-to-L transition occurs spontaneously, without the need for intermediate structures. In contrast, for the L-to-G transition, there is a barrier to nucleation and growth which occurs at a greater rate initially in the PL phase before the thermodynamic stability of the G phase overwhelms the transient kinetics.8 Data from small-angle neutron scattering experiments are shown in Figure 3. Although the resolution (due to wavelength spread) is poorer than for SAXS, we are able to enhance the contrast between the partially deuterated E block and the B block. Figure 3 shows data obtained during a heat/cool cycle from 21 to 52 °C, then back to 31 °C. The sample was heated in set 5 °C increments (the temperature was recorded at the sample) and equilibrated at each temperature prior to acquisition of scattering data, as time-resolved measurements of the transformation process were not possible with SANS with the collimation conditions used. In the initial profile in Figure 3, the high q shoulder on the first peak is actually the poorly resolved second-order reflection at x8/6q* from the G structure seen by SAXS (Figures 1 and 2). On heating, this disappears and the second order reflection from L develops. At the same time, there is a large increase in diffuse scattering below the first Bragg reflection. This scattering is more pronounced than in the SAXS data, which is due to the enhanced contrast of the protonated B block against the partly deuterated E block in D2O. The scattering length density calculated for the E block, CH3OE2(dE)18 is 4.9 ×

Transition Mechanism between Lamellar and Gyroid Phases

1010 cm-2, that for (CH2CH2CHCH3O)10 is 0.15 × 1010cm-2, and that for D2O is 6.3 × 1010 cm-2. The relative contrast of B against the solvent is more than four times that for the E block, whereas in SAXS, the electron density contrast is ∼10%. The fact that the diffuse scattering is stronger when the contrast for B is enhanced is consistent with the diffuse scattering originating from D2O perforations in the hydrophobic B layers. The last four profiles in Figure 3 show that L transforms back into G on cooling. The volume fraction of the hydrophobic B layer, φB, can be obtained from the weight fraction of surfactant, together with the masses and volumes of the both blocks in the diblock.36,37,47 We obtained φB ) 0.29. The thickness of the hydrophobic bilayer, 2δB ) 31 Å has been determined previously from analysis of SAXS data for the L phase of E18B10.48 The water-filled perforations in the B lamellae take up a fractional area of the bilayer fw ) 1 - (dφB/ 2δB).49 Close to the transition to the G phase (d ) 77 Å), this takes the value fw ) 0.29. 3.2 Modeling of Diffuse Scattering. At the beginning of the transformation from the L phase into the HPL phase, a diffuse scattering peak is observed on the low q side of the first-order lamellar peak accompanied also by another broad peak at q ≈ 0.15 Å-1 (the SAXS data obtained by Imai et al. for C16E6 in H2O show a similar feature46). The first diffuse peak leads to a broadening of the first-order Bragg peak, but the second order diffuse peak is clearly separated from the second-order Bragg peak and has less effect on its width. For this reason, the diffuse scattering cannot be ascribed to simple long-range disorder of lamellae (Caille´ effect). We suggest that both the diffuse scattering peaks and sharp peaks of the HPL phase that appear later result from the same phenomenon, i.e., the perforation of the L phase layers. The process of perforation can be divided into two steps. The first step is a random perforation of the L phase layers causing the diffuse scattering peaks in the diffraction patterns. The second step is the ordering of the perforations into close-packed hexagonal arrays characterized by a continuous disappearance of the diffuse scattering in the diffraction patterns and simultaneous evolution of the sharp diffraction peaks indexed to a hexagonal lattice (Figure 1). These suggestions have been used in a structural model developed to interpret the diffraction patterns observed. A Monte Carlo algorithm has been performed to simulate the perforated layer structure evolving through an intermediate state from a disordered liquidlike packing of holes to a close-packed hexagonal arrays of holes. Initial holes were randomly positioned in-plane. These were used as “nucleation sites” for additional perforations, which were located radially according to close-packing but without fixed hexagonal orientation. Two parameters were used in the simulations. The first parameter (Rm) is the defined minimum distance between holes (Rm was set equal to the distance 85 Å between centers of the nearest holes in the HPL structure), which enables the distances between neighboring holes Ri to be controlled (Ri > Rm) in the process of generation of the structure. The second parameter is the angle between bond vectors of neighboring perforations, which is biased to nπ/3 ( ∆φ (n is a whole number), where ∆φ changes from π/6 for random orientation to 0 for a completely ordered structure. In the initial stage of the transformation, it is assumed that the holes (47) Luzzati, V. In Biological Membranes; Chapman, D., Ed.; Academic Press: London, 1968; p 71. (48) Alexandridis, P.; Olsson, U.; Lindman, B. Langmuir 1997, 13, 23. (49) Fairhurst, C. E.; Holmes, M. C.; Leaver, M. S. Langmuir 1997, 13, 4964.

Langmuir, Vol. 20, No. 25, 2004 10789

Figure 4. Results from Monte Carlo computer simulations on perforated layer structures with increasing order (left to right, see text for details). Top row: structures (disks represent perforations in layer). Middle row: corresponding fast Fourier transforms (centers of the images correspond to q ) 0). Bottom: one-dimensional diffraction patterns obtained from radially integrated FFT images (to satisfy the Debye-Scherrer geometry, the obtained patterns were normalized by the radius of integration, Lorentz-polarization factors were not included).

have a liquidlike arrangement, i.e., no preferred packing (Figure 4, ∆φ ) π/6); however, as the transformation proceeds and the density of perforations increases, hexagonal close-packing will be favored (Figure 4, ∆φ ) 0.4 rad and Figure 4, ∆φ ) 0.2 rad) and ultimately perfected (Figure 4, ∆φ ) 0). The diffraction patterns for these simulated structures were computed by a radial integration of the Fourier transforms (FFT) of the images, produced using the software package ImageJ (author Wayne Rasband, http:// rsb.info.nih.gov/ij/). Figure 4 shows that two diffuse scattering peaks, corresponding to liquidlike disordered packing of holes. evolve into three sharp peaks, corresponding to close-packed hexagonal structure. This is similar to the evolution of scattering data observed experimentally. It is noted that the results obtained correspond to the two-dimensional structure of a single layer, whereas the real structure includes additional lamellar periodicity along the layer normal. Thus, to make the experimental data and the calculated data comparable, the calculated diffraction patterns (Figure 4) have to be superimposed onto a diffraction profile corresponding to the lamellar structure (Figure 5). It has to be also mentioned that the diffuse scattering observed in the SAXS patterns before formation of the ordered HPL structure cannot be due to stacking faults formed in HPL structures during the transformation from L to G.21,22 Although the stacking faults would lead to the presence of broad diffuse peaks at the position of hexagonal reflections hkl where h - k ) 3n ( 1, sharp peaks with h - k ) 3n corresponding to the hexagonal structure must also be present.50,51 In other words, there would be a sharp 110 peak in the diffraction pattern from the hexagonal structure in addition to the broadened diffuse scattering peaks. However, this is not observed experimentally. (50) Warren, B. E. X-ray Diffraction; Addison-Wesley: Reading, Massachusetts, 1969. (51) Dux, C.; Versmold, H. Phys. Rev. Lett. 1997, 78, 1811.

10790

Langmuir, Vol. 20, No. 25, 2004

Hamley et al.

Figure 5. SAXS profiles for the L and PL phases. (a) Initial SAXS data in L phase at 65 °C, (b) model. The simulated X-ray scattering profile is the superposition of sharp diffraction patterns from a lamellar structure and diffuse scattering originating from perforations of the layers, (c) SAXS data in PL phase (90 s after start of quench, Figure 1).

3.3. Dynamic Density Functional Theory Modeling. To monitor the transition between L and PL in detail, we employed simulations based on dynamic self-consistentfield theory for polymers, also known as the dynamic density functional theory (DDFT).52,53 The polymers are modeled as Gaussian chains E4B6. The bead-bead interaction can be characterized by the well-known dimensionless Flory-Huggins parameter χEB.52 The timeevolution of the density fields for the two components E and B is given by a set of Langevin equations for diffusion dynamics, with appropriate noise terms. Applied shear gives rise to a convection term in the dynamic equation.53 The set of equations is solved numerically by the application of a Crank-Nicolson scheme, with a dimensionless time increment, ∆τ ) 0.45 (for more details see ref 52). The dimensionless time step k is defined as k ) τ/∆τ. By using this dynamic scheme, we follow an experimentally realistic pathway, including visits to long-living metastable states. The initial homogeneous mixture (all density fields are equal to the mean density) is quenched at time step 0 for χEB ) 2.91 and evolves into a highly defected structure. Shear with dimensionless shear rate 0.001 is applied between time steps 4000 and 8000 to remove defects and induce global orientation. After preshearing, we obtain a perfect lamellar phase (Figure 6a, left). At time step 35 000, the Flory-Huggins χ parameter is instantaneously changed to χEB ) 1.63. Following this change, a HPL phase develops (undulations with a wavelength smaller than the lamellar thickness are visible in the first image in Figure 6b). The perforations within a single layer initially develop with irregular order; however, hexagonal packing develops, as shown in Figure 6b. In the early stage, the development of perforations is accompanied by the formation of connections between lamellae in the originally kinked part of the lamellae, resulting in a global reorientation and flattening of the resulting HPL phase (details are given in Figure 6b). The resulting HPL phase has a reduced interlamellar distance compared to the initial L phase, in good agreement with the SAXS results in Figure 1. The HPL phase is remarkably stable; the HPL phase (52) Fraaije, J. G. E. M.; van Vlimmeren, B. A. C.; Maurits, N. M.; Postma, M.; Evers, O. A.; Hoffmann, C.; Altevogt, P.; Goldbeck-Wood, G. J. Chem. Phys. 1997, 106, 4260. (53) Zvelindovsky, A. V.; Sevink, G. J. A.; van Vlimmeren, B. A. C.; Maurits, N. M.; Fraaije, J. G. E. M. Phys. Rev. E 1998, 57, 4879.

Figure 6. Snapshots of the L-G transition from a mesoscopic computer simulation. The initial lamellar morphology (L) was obtained by quenching from a homogeneous mixture and consequent preshearing. After 35 000 time steps, the interblock interaction parameter was changed (reflecting a change of the temperature). Isodensity surfaces of E block are shown at the level 0.5. (a) From left to right: L structure at 35 000 time steps (χEB ) 2.91); hexagonal perforated lamellar (HPL) at 120 000 time steps (χEB ) 1.63); reference gyroid-like structure (G) starting from a homogeneous mixture and χEB ) 1.63. (b) Details of the L-to-HPL transition. The snapshots focus on one layer taken from the whole simulation box. From left to right and top to bottom: structures for 97 000, 97 400, 97 600, 98 000, 99 000, and 145 000 time steps.

at 120 000 time steps (see Figure 6a, middle) remains stable up to 330 000 time steps (at which time the simulation is stopped). As HPL is known to be an unstable bulk structure, we are stuck in a metastable state with a lifetime that exceeds our simulation range. For this reason, we also consider a quench to χEB ) 1.63 starting from a homogeneous mixture, resulting in the gyroid-like structure G (Figure 6a, right) after only 4000 time steps. 4. Summary In summary, we found that the transition between lamellar and gyroid phases occurs via a hexagonal perforated lamellar intermediate structure and is accompanied by a large reduction in domain spacing as the lamellar phase transforms into a hexagonal perforated lamellar intermediate prior to forming a stable gyroid structure. Modeling the diffuse scattering confirms that the spacing of the perforations is close to the lamellar spacing and this structure then acts as an intermediate prior to transformation into the gyroid phase. Mesoscopic computer simulations are able to capture the L-HPL transition and show a reduction in domain spacings as in the experiments. They also indicate that perforations within layers initially form with irregular ordering followed by the development of hexagonal packing. Acknowledgment. We thank EPSRC (UK) for grants GR/ N22052 and GR/N0853 that supported O.O.M. and V.C., respectively. We thank Gu¨nter Grossmann at the SRS, Daresbury lab for assistance with the SAXS experiments and the ILL for the award of beamtime (ref 9-11-953). LA0484927