Liquid Crystalline Phase Behavior of Well-Defined ... - ACS Publications

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Liquid Crystalline Phase Behavior of Well-Defined Cylindrical Block Copolymer Micelles Using Synchrotron Small-Angle X‑ray Scattering Dominic W. Hayward,†,‡,§ Joe B. Gilroy,‡ Paul A. Rupar,‡ Laurent Chabanne,‡ Claire Pizzey,∥ Mitchell A. Winnik,⊥ George R. Whittell,*,‡ Ian Manners,‡ and Robert M. Richardson*,† †

H.H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, U.K. School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, U.K. § Bristol Centre for Functional Nanomaterials, University of Bristol, Bristol BS8 1TH, U.K. ∥ Diamond Light Source, Harwell Science & Innovation Campus, Didcot, Oxfordshire OX11 0DE, U.K. ⊥ Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada ‡

S Supporting Information *

ABSTRACT: The structure and phase behavior of colloidal solutions of monodisperse rod-shaped micelles, of different lengths (ca. 300−2100 nm) and formed from poly(ferrocenylsilane)-block-polyisoprene (PFS-b-PI) diblock copolymers, have been investigated using synchrotron smallangle X-ray scattering. The dimensions of the crystalline PFS core, solvated PI corona, and the overall radial polydispersity were measured, and relationships between the characteristics of the constituent copolymers and the internal structure of the self-assembled micelles have been established. In addition, the effects of micelle length, length distribution, concentration, composition, and block length on the liquid crystalline phase behavior of the micelles have been determined. It was found that micelle dispersions exist in three distinct phases: isotropic, nematic, and hexagonally packed, depending predominantly on their concentration and aspect ratio. The results have also highlighted the importance of the coronal composition and structure in determining the high-concentration behavior of micelle dispersions.

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solutions of anisotropic biological particles.12−17 Using genetically modified, polymer coated, filamentous viruses, progress has been made on testing the validity of Onsager’s theory at different lengths and with different interparticle interactions.18−20 The results showed that Onsager’s theory is not robust beyond the hard-rod limit and fails when particles exhibit moderate repulsion or even weak attraction. Theoretical studies and simulations have been conducted to generalize Onsager’s work for more complex systems;21 however, extensive, high-quality experimental data are required to inform and validate these models. The field of controllable self-assembly provides a new pathway to anisotropic nanostructures, and there have been a number of reports of self-assembled rod- and worm-like micelles forming lyotropic liquid-crystalline phases above particular volume fractions.22,23 However, there have been relatively few systematic studies detailing the phase behavior at different aspect ratios and concentrations. To date, the majority of work in this field has focused instead on rheological studies and the phase behavior under shear flow conditions.24−26 A

INTRODUCTION The ability to fabricate monodisperse, one-dimensional (1D) nanostructures is essential for the development of many technological applications at the nanoscale.1,2 Increasingly, these anisotropic building blocks are also being used to construct large-scale functional ordered assemblies for use in optoelectronic devices and metamaterials.3−5 In order for either of these applications to be suitable for widespread implementation, however, a number of challenges must first be addressed; the synthetic methods must give high yields and provide sufficient selectivity over the size, shape, and polydispersity of the particles produced. Furthermore, a quantitative understanding is required of how anisotropic structures behave and interact at the nanoscale to generate complex structures. Although significant progress has been made with regards to the synthetic methods,6 much work remains to be done to provide the accurate theoretical framework needed to begin exploiting anisotropic nanostructures to their full potential. The first description of ordering in assemblies of rod-like nanoparticles, in the limit of very high aspect ratios, was provided by Onsager in his work on the liquid-crystalline phase behavior of rod-like tobacco mosaic virus (TMV) solutions.7 Since then, many similar systems have been reported, generally based on surfactant stabilized inorganic nanorods8−11 and © XXXX American Chemical Society

Received: October 31, 2014 Revised: January 22, 2015

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DOI: 10.1021/ma502222f Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Formation of monodisperse cylindrical micelles via living CDSA. The polymers used in this study are shown in orange (PFS), red (PDMS), and blue (PI).

Figure 2. Block copolymers used in this study, where n, m, x, and y represent the number-average degree of polymerization of the corresponding blocks.

precursors, enabling access to the phase space where lyotropic liquid crystalline mesophases can be observed.35 This alignment behavior was used in recent synchrotron WAXS studies on PFS cylinders to reveal a highly crystalline core (diameter = 9 nm) with the polymer backbone perpendicular to the long axis of the micelle and with order reminiscent of a single crystal.36 Although the living CDSA method affords unprecedented control over cylinder lengths, comparatively little is known about the cross-sectional structures and thickness of the various sections, particularly in suspensions. Previously, micelle diameters have been measured using TEM,35,36 atomic force microscopy (AFM),36 and light scattering techniques,37,38 leading to estimates for the total diameter ranging from 9 to 50 nm. The broad range of values is due to differences in the block copolymer composition, the environment in which the measurements were conducted (deposited and dried versus in solution) and the sensitivity of the different techniques to different regions of the micelle cross-section. For example, TEM is very sensitive to the electron-rich cores but not to the relatively electron-poor coronae. Conversely, for AFM imaging, although the coronae may be more easily visible, their structure is strongly influenced by surface interactions with the hydrophobic substrate. In this work, a systematic investigation of micelle structure was conducted using synchrotron small-angle X-ray scattering (SAXS). This enabled nondestructive measurements of both the core and corona dimensions in situ and with high precision, for the first time allowing a detailed, self-consistent characterization of the micelle structure. The results were then used in conjunction with measurements of the field-responsive alignment of the micelles to map the liquid-crystalline phase behavior. The results shed light both on the factors influencing

prominent exception is the work by the Bates group, which details the aqueous phase behavior of poly(ethylene oxide)block-poly(butadiene) worm-like micelles as a function of concentration and temperature.27,28 Because of the difficulties in controlling the length and polydispersity of such selfassembled structures, however, it was not possible to investigate the effects of length, aspect ratio, and polydispersity on the phase behavior. Recent advances in the synthesis of crystalline-coil block copolymer micelles offer a promising solution. Significantly, unlike micelles derived from low molecular weight surfactants, block copolymer micelles are intrinsically much more robust due to the absence of exchange phenomena.29 It has been shown that block copolymers containing a poly(ferrocenyldimethylsilane) (PFS) core-forming metalloblock self-assemble in a selective solvent for the coblock to form 1-D cylindrical micelles and that this morphology is a direct consequence of the semicrystalline nature of the PFS.30 Furthermore, using core or coronal cross-linking chemistry, these block copolymer micelles can be transformed into permanent and mechanically tough nanostructures that retain their morphology even in good solvents for both blocks.31−33 A key feature of self-assembled PFS-containing cylindrical micelles is the “living” nature of the growth mechanism, which allows the ends to remain active to the addition of further amounts of block copolymer unimer.34 This leads to an increase in length by the epitaxial crystallization of the coreforming metalloblock (Figure 1) that is proportional to the amount of block copolymer added, in a process now known as living crystallization-driven self-assembly (CDSA).35 Further improvement in control over the micellar dimensions was achieved by preparing very well-defined seed micelle B


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Figure 3. SAXS patterns for (a) 25, (b, c) 50, and (d) 75 mg/mL decane dispersions of 640 nm long PFS53-b-PI637 micelles. The micelles in (c) are in a horizontal 4 V μm−1 electric field; those in (a), (b), and (d) are not subjected to an external field. The vertical scale bars represent log(intensity) of the scattered X-rays.

of nematic and hexagonal phases present in the same sample. The evolution of the phases with respect to concentration can also be clearly seen in the corresponding interparticle structure factors. These are proportional to the scattered intensity divided by the intensity from a dilute sample, shown in Figure 4. At low concentrations, the interparticle contribution is negligible so the structure factor is independent of Q. Increasing the concentration gives rise first to broad liquidlike peaks and eventually to sharper peaks corresponding to the (100), (110), (200), and (210) Bragg reflections from a hexagonal lattice. This sequence of phases, from isotropic through nematic to a hexagonal phase, is representative for almost all samples that exhibit alignment behavior, although the onset of nematic and hexagonal behavior varies from sample to sample and is discussed in more detail below. Micelle Structure. For samples exhibiting isotropic scattering with no interparticle interference peaks, the intensity was analyzed using a model for the scattering from long, core− shell cylindrical rods with a homogeneous core and a shell of decaying density. Geometrical considerations of chains grafted to particles lead to the following approximation of the polymer density profile within a micelle:39

the internal structure of the micelles and how the structure and morphology in turn affect the interparticle interactions and the macroscopic phase behavior.

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RESULTS AND DISCUSSION Overview of Phases Observed. In order to build up a clear picture of the micelle structure and interactions at the nanoscale, a broad range of variables were considered. To this end, 72 separate samples were produced, encompassing a range of corona chemical compositions (PFS-b-PDMS, PFS-b-PI, and cross-linked PFS-b-PI, Figure 2), micelle lengths (350, 640, 1160, and 2130 nm) and length polydispersities (0.13 ≤ σ/Ln ≤ 0.88), degree of polymerization of the corona (637, 890, 1250, and 1424), block ratios (∼1:10, ∼1:20), and micelle concentrations (10, 25, 50, 75, 100, and 150 mg/mL). The SAXS and field alignment study yielded three principal types of behavior: an isotropic phase, characterized by isotropic scattering monotonically decreasing from the center with no alignment in an electric field, a nematic phase, characterized by broad peaks in intensity at low Q and strong anisotropic scattering in an electric field, and a hexagonally packed phase, characterized by sharper Debye−Scherrer rings with no alignment in an electric field (Figure 3). In addition, some samples exhibited both Debye−Scherrer rings and an alignment response to an electric field; this was ascribed to the coexistence C


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to the micelle axis. Applying the density profile in eq 1, the electron density of the shells, for 1 ≤ i ≤ N becomes ⎛ R i ⎞−α ρi = ρs + (ρp − ρs )⎜ ⎟ ⎝ R core ⎠

(4)

where R i = R core +

i T N

(5)

ρN+1 = ρs, the electron density of the solvent, and ρp is the electron density of the corona adjacent to the core. Since the length L of the micelles is significantly greater than 1/Qmin in this experiment, the orientational average of |F(Q⊥)|2 simply introduces a Q−1 factor, and the scattering intensity per unit volume becomes I(Q ) = nae

2πL |F(Q )|2 Q

(6)

where n is the number of micelles per unit volume and ae is the scattering length of one electron. Radial polydispersity is accounted for by averaging all radii over a Gaussian distribution with standard deviation σR. Nonlinear least-squares fits were conducted by allowing the scaling factor, background, core radius, corona thickness, radial polydispersity, and electron density of the corona near the core to refine. The electron density of the core was fixed at the electron density of PFS throughout. Because of the number of parameters to be fitted, a covariance analysis was conducted on the core radius, corona thickness, and radial polydispersity (see Figure S2). The extent of the coupling was found to be minimal and less than the variation between samples of the same composition (from which the error bars in subsequent plots are derived). Representative fits from all materials investigated are shown in Figure 5, and a summary of the results is given in Table 1. In order to gain an insight into the internal structure of the micelles, it is instructive to compare the experimentally determined geometrical parameters with the polymer characterization data. The materials used in this study are all based

Figure 4. Plots of effective structure factor vs Q from the SAXS results of the 640 nm long PFS 53-b-PI 637 micelles with increasing concentration. The effective structure factor is defined as the azimuthally averaged intensity relative to that of the 10 mg/mL sample with no external field.

⎧ϕ for r < R core ⎪ core ⎪ ⎛ r ⎞−α ϕ(r ) = ⎨ ϕcorona⎜ for R core ≤ r ≤ R core + T ⎟ ⎪ ⎝ R core ⎠ ⎪ ⎪0 for r > R core + T ⎩ (1)

where ϕcore and ϕcorona are the volume fractions of the core and corona chains, respectively, Rcore is the radius of the core, T is the thickness of the corona, and α is given by α=

(D − 1)(3ν − 1) 2ν

(2)

where D is the dimension of curvature of the core surface and ν is the Flory exponent. For cylindrical micelles in a good solvent for corona-forming block, α = 0.66. The structure factor for the cross section was calculated (assuming circular symmetry) using the cylindrical Bessel function J1 and by dividing the corona into N equally spaced cylindrical shells of radius Ri, such that N

F(Q⊥) = π ∑ (ρi + 1 − ρi )2 R i 2 i=0

2J1(Q⊥R i) Q⊥R i

Figure 5. Plot of radially averaged intensity vs Q from the SAXS results for the 10 mg/mL dispersions of representative, monodisperse micelles. The lines are fits of the model described in the text. All plots other than the 1160 nm PFS53-b-PI637 cross-linked data have been arbitrarily shifted in intensity for clarity.

(3)

where R0 is the radius of the core and ρ0 is its electron density and Q⊥ is the component of the scattering vector perpendicular D


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micelle lengths, particularly for the PFS53-b-PI637 micelles (Figures 6b and 6c), there is no correlation between these variables. The electron density of diffuse polyisoprene chains is both significantly less than that of the PFS core and relatively close to that of the decane solvent, resulting in much weaker scattering from the micelle coronae than from the cores. Nevertheless, it was possible to determine estimates of the radial extension of the corona based on the power-law decay model in eq 4. Figure 7a shows how the thickness of the corona varies with the number of repeat units in the corona chain. It is immediately clear that there is no direct linear relationship between the corona thickness and the length of the corona chains. As the extent of the corona will also depend on how stretched the chains are and therefore how densely the chains are packed at the core/corona boundary, this information must also be taken into account. From Figure 6a, it can be seen that the volume fraction of corona near the core is approximately inversely proportional to the core chain length. The extent of the corona is therefore better correlated with the block ratio as shown in Figure 7b. Micelles with similar block ratios tend to have similar corona thicknesses whereas the micelles with a much larger block ratio tend to have more extended coronae. In order to investigate the effects of cross-linking the corona on the overall micelle structure, an additional sample of monodisperse PFS53-b-PI637 micelles of length Ln = 1160 nm was prepared and subsequently cross-linked via metal-catalyzed hydrosilylation.33,43−45 In contrast to previous experiments showing little difference between the hydrodynamic radii of cross-linked and un-cross-linked micelles measured using dynamic light scattering at a scattering angle of 90°,33,43,45 the small-angle scattering data show some clear differences which can be observed both directly, in the intensity distribution (Figure 5), and indirectly via the fitted geometrical parameters (Table 1). First, the extent of the cross-linked coronae (17.5 nm) is substantially lower than that of the uncross-linked coronae (28.0−29.7 nm). Although this is the first time this effect has been observed with crystalline-coil block copolymers, it is not necessarily unexpected; cross-linking binds individual corona chains together, thereby restricting their freedom to extend fully. A similar “densification” phenomenon was observed in the cross-linked cores of poly(ethylene oxide)b-poly(butadiene) diblock copolymer giant worm-like micelles.27 More surprising, however, is the apparent substantial reduction in core radius from 4.2−4.6 to 3.0 nm on crosslinking. It is possible that this phenomenon is due to a rearrangement of the core chains as a result of the stresses imposed by the cross-linked periphery. Preliminary wide-angle X-ray scattering (WAXS) measurements made in the laboratory (Figure S3) show that the cores in the cross-linked micelles appear less crystalline than those in their non-cross-linked counterparts. Further investigation is required to determine exactly how this reorganization occurs and what effect, if any, it has on the physical properties of the micelles. Nematic Behavior. For a dispersion of perfectly aligned, rigid rods, the small-angle scattering pattern will be highly anisotropic. If the rods are rigid and the alignment is perfect, then the extent of the scattering will be inversely proportional to the dimensions of the particle; therefore, horizontally aligned rods will give rise to a narrow vertical streak in the diffraction pattern and vice versa. If the alignment is not perfect, however, the streak will broaden. By measuring the azimuthal intensity distribution at constant magnitude of Q, it is possible to

Table 1. Summary of Micelle Dimensions As Determined from TEM and SAXS micelle length polymer monodisperse samples PFS53-b-PI637 PFS53-b-PI637 PFS53-b-PI637 PFS53-b-PI637 cross-linked PFS53-b-PI637 PFS90-b-PI890 PFS133-b-PI1250 PFS63-b-PI1424 PFS60-b-PDMS670 polydisperse samples PFS53-b-PI637 PFS90-b-PI890 PFS63-b-PI1424

Lna

core radiusc

corona thicknessc

(nm)

σ/Lna,b

Rcore (nm)

T (nm)

σR/Rcorec,d

350 640 1160 1150

0.25 0.18 0.20 0.13

4.2 4.0 4.3 3.0

27 26 29 18

0.24 0.25 0.18 0.21

2130 805 715 495 915

0.15 0.18 0.21 0.17 0.15

4.2 4.1 3.5 4.3 3.9

27 29 28 47 24

0.20 0.35 0.56 0.21 0.26

700 690 530

0.70 0.65 0.88

4.6 3.9 4.4

29 26 50

0.25 0.51 0.19

a

As determined by manually tracing 500 individual micelles for each sample in TEM micrographs. bStandard deviation σ calculated assuming Gaussian distribution of lengths. cAs determined by SAXS data analysis. dStandard deviation σR from fitted nonlinear leastsquares fits as described above.

around highly asymmetric block ratios (∼1:10 and ∼1:20) and had been demonstrated to form rod-like micelles by TEM. Initial evidence from a study on the effects of increasing the chain lengths independently of the block ratio, conducted on a similar PFS-based block copolymer system, suggested that an increase in molecular mass results in a decrease in crystallinity of the core and eventually to a change in morphology.40 Although a comparison of the TEM images of micelles composed of block copolymers with different block ratios does not appear to show any major structural differences in this case, Figure 6a shows that the volume fraction of corona chains near the core/corona boundary decreases as a function of core chain length. This leads to the conclusion that the longer core chains are undergoing more folding per unit length. From the previously reported WAXS study36 it was determined that the core chains pack perpendicular to the long axis of the micelle; therefore, an increase in the number of chain folds in the core may, in turn, lead to an increase in the anisotropy (i.e., ellipticity) of the core cross section. This is supported by Figures 6b and 6c, showing the average core radii and relative polydispersities as functions of the core-block chain length for the monodisperse samples, respectively. As the core chain length increases, the apparent core radius decreases and the relative polydispersity increases. Although this may initially appear somewhat counterintuitive, it is thought to be a direct consequence of the degeneracy of polydispersity and ellipticity in the calculation of form factors;41,42 assuming there is no substantial change in the cross-sectional area, an increase in the ellipticity would appear in the form factor for a cylindrical rod, as a reduction in the characteristic radius R0 (i.e., the minor axis of the ellipse) and an increase in the apparent polydispersity. The existence of elliptical cross sections may also explain the previously reported observations of discrepancies between the micelle cross sections determined by TEM, AFM, and X-ray scattering.36 It should also be noted that, although there is a small spread in core radius and polydispersity for different E


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Figure 6. Plot of (a) volume fraction of corona chains adjacent to the core, (b) average core radius, (c) relative polydispersity vs number-average degree of polymerization of the core (Core DPn) for micelle dispersions exhibiting isotropic scattering. Multiple points of the same type represent different micelle lengths with identical chemical compositions.

Figure 7. Plot of corona thickness (measured from center of the core) vs (a) number-average degree of polymerization of the corona (Corona DPn) and (b) block ratio for all monodisperse PFS-b-PI micelle dispersions. Multiple points of the same type represent different micelle lengths with identical chemical compositions.

determine the orientational order parameter, Sa measure of the extent to which the rods in the dispersion are aligned. Thus,

the order parameter is equal to one if the rods are perfectly aligned with their long axes parallel to the applied field, zero for F


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Figure 8. (a) Plot of orientational order parameter vs Q, extracted from SAXS results of the 640 nm long PFS53-b-PI637 micelles. Letters in parentheses indicate the liquid crystalline phase of the micelles: (I)sotropic, (N)ematic, and (H)exagonally ordered. (b) Plot of order parameter vs number-average degree of polymerization of the corona (Corona DPn) for all dispersions that showed field-alignment behavior.

Figure 9. (a) SAXS patterns for 100 mg/mL decane dispersion of 530 nm long PFS63-b-PI1424 micelles. (b) Corresponding plot of azimuthally averaged intensity vs Q, labeled with Miller indices. The vertical scale bar represents log(intensity) of the scattered X-rays.

further detail in the Liquid Crystal Phase Diagram section. The values of the order parameter were in the range 0.65−0.75, which is slightly less than, but comparable to, those observed in previous studies of PFS50-b-PI550 seeded micelles of length 731 nm (S = 0.73)35 and in magnetically aligned dispersions of high-aspect-ratio tobacco mosaic virus (TMV) (S = 0.77).15 The values are also less than that predicted by Onsager’s theory for rigid rods (S = 0.79).47 In general, there are no significant trends in the order parameter with concentration, but it was found that the order parameters tend to decrease with increasing corona chain length (Figure 8b). Although Onsager’s theory predicts a strong concentration dependence for the order parameter of rigid rods, it has been shown that this effect is mitigated by flexibility, as the entropy penalty for aligning a flexible rod is much more severe than for a rigid rod.48 The dependence of the order parameter on corona chain length also suggests that the liquid-crystalline behavior is influenced by intermicelle interactions. At higher concentrations, the Debye−Scherrer rings seen in Figure 3d are indicative of long-range crystal-like order. As with many powder diffraction experiments, there may be a preferred orientation of the crystalline domains due to anchoring on the capillary walls for example. This has a significant influence on the azimuthal intensity distribution and results in the large negative values, exhibited by the hexagonally packed samples.

randomly aligned rods, and minus one-half for alignment perpendicular to the field. Scattering patterns were recorded twice for all micelle dispersions: first without the presence of any external fields and subsequently in the presence of an alternating (1 kHz) electric field with a peak-to-peak amplitude of 4.0 V μm−1. The order parameters were determined for all samples by measuring the azimuthal intensity distribution at scattering variables 0.01− 0.05 Å−1 using a standard method.35,36 Strictly speaking, the calculated values only represent a lower bound on the order parameters as it is not possible to conclusively disprove the existence of multiple domains. Nevertheless, the complete absence of subsidiary azimuthal peaks in any well-aligned sample is strongly indicative for the existence of monodomains. Figure 8a shows the variation in order parameter with Q for a representative sample. It can be seen that at isotropic and nematic concentrations the order parameter remains approximately constant with increasing Q, as expected for rods with relatively monodisperse radii.46 In principle, a strong field could induce observable orientational order in an isotropic phase, but this was not the case in this system. Samples with order parameters equal to zero (within error) were identified as isotropic phases. Samples with significant positive order parameters were identified as (well-aligned) nematic phases. This was confirmed by birefringence observations, discussed in G


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Macromolecules The micelles with the longest coronae (PFS63-b-PI1424) are not shown as they did not exhibit alignment at any concentration, proceeding directly from an isotropic phase to an immobile hexagonally packed one. Furthermore, the micelles with crosslinked polyisoprene coronae and PDMS coronae also showed no signs of field alignment at any concentrations (although the PFS-b-PDMS micelles did show structure factors similar to the samples identified as nematic at concentrations above 50 mg/ mL). From the dilute cross-linked samples, it was observed that the extent of the corona is dramatically reduced with respect to their un-cross-linked counterparts (Table 1). It is thought that without the extended and highly solvated corona chains, there will be more points of contact where the micelles are close enough to interact strongly and stick, at least temporarily. This, in turn, causes an open gel of cross-linked micellar rods to form, precluding the formation of any ordered phases. Such behavior would result in isotropic scattering patterns even at very high concentrations, as was observed experimentally. The lack of alignment or hexagonal ordering for the samples with PDMS coronae must be a direct result of the nature of the PDMS chains. Both examples of anomalous high-concentration behavior suggest that the chemical composition of the corona and, consequently, also the type of solvent have a strong influence on the interaction, aggregation, and alignment behavior of the micelles. Hexagonal Packing Behavior. As can be seen in Figures 3d and 4, at higher concentrations, diffuse peaks of the nematic give way to sharper Bragg peaks, demonstrating a translationally ordered phase. These scattering vectors are in the ratio of 1:√3:√4:√7, indicating that the micelles are packing with a 2D hexagonal lattice. This behavior is particularly striking for the PFS63-b-PI1424 micelles where the nematic phase is bypassed completely and higher order Bragg reflections can be readily indexed up to the (430) plane regardless of the polydispersity (Figure 9). For all other samples with non-cross-linked polyisoprene coronae, however, it was observed that the diffraction peaks were broader than the instrumental resolution (Figure 10) and the intensity of higher order reflections decreases rapidly with Q. This means that the phase does not have the long-range translational order of a true crystal, so the possibility of columnar or hexatic ordering must be considered.

The nature of this phase will be discussed in detail in a future publication. In order to better understand the nature of the intermicelle interactions, it is instructive to identify the important factors that determine the lattice parameter. For this, the lattice parameter determined from the Q value of the (100) peak was used because these peaks are the strongest and their positions could be determined with the least error. The lattice parameter is then given by a=

4π 3 Q 100

(7)

From Figure 11a, it can be seen that the intermicelle spacing decreases as the micelle dispersions change from the nematic to the hexagonally packed phase. Once the micelles are in the hexagonal phase, however, there is very little variation in micelle separation with increasing concentration. This again suggests a well-defined, noncompressible micelle cross section with a swollen corona. In Figure 11b,c it can be seen that the lattice parameter (i.e., the average distance between neighboring micelles) is not well correlated with the corona diameter; rather, it is much more strongly correlated to the number of monomer units in the corona chains. It is considered likely that this phenomenon is a result of the model used to fit the micelle cross section. The total diameter of the micelle was observed to depend on both the length of the chains in the corona and the chain folding in the core (and hence the corona density near the core) (Figure 6). High corona chain density near the core will result in stretched chains and an extended corona. Conversely, low chain density at the core will reduce the steric constraints on the corona chains, allowing them to “relax” to random-coil-like conformations; as a result, the majority of the corona bulk will be found closer to the core. However, as the polyisoprene corona is well solvated, some chains are expected to be stretched beyond the cutoff at R + T, in the model of the corona. Because of the minimal scattering contrast between the corona and the solvent, these extended chains do not significantly influence the scattering, and so the model does not need to capture them in order to fit well. However, they are likely to play a role in the spacing between the micelles, giving rise to the behavior observed in Figure 11. Finally, Figure 11d shows that the lattice parameter is also affected by the length of the micelles. Taken in conjunction with Figure S4, showing that the Bragg peaks become broader and of lower intensity with increasing length and previous work placing the persistence length of the micelles at ∼1 μm,38 this supports the proposition, demonstrated mathematically in the Supporting Information, that the longer micelles are more bent in thermal equilibrium and therefore do not pack as efficiently as the shorter ones. The fact that the polydisperse, self-seeded micelles do not follow this trend, despite having a similar nominal corona diameter, suggests that the flexibility contribution to the lattice parameter is determined by the longest micelles in the dispersion rather than the length average. Liquid Crystal Phase Diagram. To investigate the phase behavior of the micelles in greater detail, polarizing optical microscopy (POM) observations were performed on the concentrated micelle dispersions. The curved surface and thickness of the glass capillaries hindered observations at high magnifications, but it was possible to ascertain the phase of each dispersion and representative textures for the isotropic,

Figure 10. Peak widths from azimuthally averaged SAXS data, determined using nonlinear fitting with a Gaussian profile. Filled symbols represent translationally ordered micelle phases, and empty symbols represent the nematic phase. H


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Figure 11. Plot of lattice parameter vs (a) concentration, (b) corona diameter, (c) number-average degree of polymerization of the corona (Corona DPn), and (d) micelle length. Plots (a) and (c) include both the lattice parameter with and without an external applied electric field; these are not differentiated as the effects are small and there are no consistent differences. Plots (b) and (d) show only the average lattice parameter at concentrations of 100 and 150 mg/mL for clarity. In (a), solid squares represent dispersions exhibiting peaks characteristic of hexagonal packing and open triangles, dispersions exhibiting broad peaks. In (c), the 100 and 150 mg/mL data include only dispersions exhibiting hexagonal peaks, and the 50 mg/mL data include only dispersions exhibiting broad peaks.

Figure 12. Representative POM micrographs obtained from (a) 25 mg/mL (isotropic), (b) 50 mg/mL (nematic), and (c) 75 mg/mL (hexagonally packed) decane dispersions of 640 nm long PFS53-b-PI637 micelles. The micrographs were obtained from the same capillary tube employed in the SAXS studies reported in this paper. Scale bars: 100 μm.

of the nematic phase occurs at lower concentrations for longer micelles, and the onset of hexagonal packing behavior occurs at lower concentrations for shorter micelles. In Figure 13b all uncross-linked PFS-b-PI samples are shown, together with the linear phase boundaries from Figure 13a. The agreement is generally very good with the exception of both monodisperse and polydisperse PFS63-b-PI1424 samples at 50 mg/mL and the monodisperse PFS63-b-PI1424 sample at 25 mg/mL. In Figure 13c, the thickness of the corona (calculated from the dilute

nematic, and hexagonally packed phases are shown in Figure 12. These results were used, in conjunction with the alignment behavior and analysis of the structure factor, to uniquely determine the phase of the micelle dispersions at each concentration. The phase diagrams are shown in Figure 13. In Figure 13a, only the seeded PFS53-b-PI637 micelles are shown. The micelles are all chemically identical and differ only in average length. Here, we see the emergence of two important trends: the onset I


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Figure 13. Phase diagrams of PFS-b-PI micelles in decane. (a) Shows how the phase of monodisperse, un-cross-linked PFS63-b-PI1424 micelles varies with length and concentration. (b) Shows how the phase of all un-cross-linked PFS-b-PI micelles varies with length and concentration (same phase boundaries as (a)). (c) Shows how the phase of all un-cross-linked PFS-b-PI micelles varies with aspect ratio and concentration. Different symbol shapes differentiate between different compositions. Closed symbols represent monodisperse samples and open symbols polydisperse samples.

Figure 14. POM micrographs obtained from decane dispersions of 350 nm long PFS53-b-PI637 micelles at increasing concentrations: (a) ∼20 mg/ mL, showing tactoid formation, (b) ∼40 mg/mL, showing formation of extended nematic, and (c) ∼60 mg/mL, showing coexistence of nematic and hexagonally packed phases. The 15 mg/mL sample showed no birefringence. Scale bars: 100 μm.

dispersion data) is also taken into account to give the aspect ratio of the micelles. This improves the agreement with the experimental data and leaves only one anomalous point: the PFS63-b-PI1424 sample at 25 mg/mL. As predicted by Onsager’s theory,7 the onset of nematic behavior depends on both aspect ratio and volume fraction, where the relative concentration required to attain the nematic phases decreases with increasing aspect ratio. The theory predicts that as the concentration increases, the micelles will begin to interact, leading first to local nematic ordering and subsequently to a homogeneous nematic phase. For long rigid

rods, the onset of the biphasic and nematic regions is predicted to occur at volume fractions of φI−N = 3.289

φN = 4.192

D L

D L

(8) (9)

Assuming the diameter of the rigid rods to be equivalent to the diameter of the corona as determined from the isotropic scattering data, the onset of nematic behavior is predicted to occur between 80 and 590 mg/mL (8−60 vol %). The observed behavior shows that liquid crystalline phases occur at J


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sufficient number of aggregates are present in the sample, they exhibit peaks in the scattered intensity (at higher Q than the diffuse peak of the nematic) but are small enough to be reoriented by the surrounding rods which remain in the nematic phase. Increasing the concentration beyond this point gives rise to sharper peaks in the scattering pattern, reminiscent of powder diffraction from a crystal. This suggests that the hexagonal aggregates increase in size until they can no longer be oriented by the field, resulting in a granular structure where individual grains each have a highly ordered internal structure but are randomly oriented with respect to each other. With respect to the immobile, nematic samples, the degree of alignment for particles in an external field is governed by the interplay between a number of different contributions, notably concentration 50 and length,51 in a nontrivial manner. Importantly, the coupling to the aligning field is expected to increase with both concentration and rod length. This is a possible explanation for why the shortest and most dilute of the nematic samples do not show alignment behavior; however, further alignment experiments at intermediate concentrations would be necessary to confirm this. Furthermore, mechanical probing of the 2130 nm micelle dispersions show that the viscosity of the high-concentration samples that do not field align is much higher than the lower concentrations samples that do align. It is therefore reasonable to assume that the increase in viscosity inhibits the alignment behavior of the long micelles. Finally, from Figure 13 it can be seen that the nematic− hexagonal phase boundary is not only more strongly dependent on the micelle length than the isotropic−nematic boundary, but it also shows the reverse trend; the shorter micelles show a stronger tendency to “crystallize” at lower concentrations than the longer micelles. This is thought to be a consequence of the fact that the longer micelles will flex more than shorter micelles resulting in less efficient packing, as discussed in the Hexagonal Packing Behavior section and in the Supporting Information.

much lower concentrations (25−50 mg/mL, 2−6 vol %), which implies that the micellar solutions cannot be fully described using a rigid hard rod model. As the relatively low aspect ratio and micelle flexibility would both act to suppress rather than promote the nematic and hexagonal phases, the reason for this discrepancy is most likely due to attractive forces between the micelle coronae. This may also explain why the phases cannot be completely determined by aspect ratio and concentration alone. Furthermore, the phase diagrams in Figure 13 show only monophasic regions and suggest that the boundaries between phases are linear. This is not necessarily the case, so in order to explore the nature of the phase boundaries more fully and establish whether the phases can coexist, POM observations were carried out at intermediate concentrations. At 15 mg/mL, there is no birefringence at all, indicating a wholly isotropic phase. Because of the small dimensions of the capillary and the rapid evaporation of solvent, the subsequent micrographs, shown in Figure 14, were taken at nonequilibrium conditions, resulting in large uncertainties on the instantaneous local concentrations. Nevertheless, the observed phenomena shed light on the nature of the phase boundaries and reveal the existence of biphasic regions. Above concentrations of approximately 20 mg/mL, small, weakly birefringent regions with a regular, spindle-like, pointed ellipsoid morphology became visible throughout the sample in an otherwise featureless isotropic dispersion (Figure 14a). This phenomenon was ascribed to the formation of nematic droplets or tactoids. The shape and internal structure of tactoids are governed by the competition between anchoring strength, surface tension, and the elastic properties of the nematic;49 as the component micelles are arranged predominately along the long axis of the tactoid, the surface tension has different values along the major and minor axes, giving rise to the distinct spindle-like shapes. Observations at higher concentrations (∼20−40 mg/mL) revealed increasing numbers of tactoids in the bulk solution. Eventually, shear forces generated by the evaporation of hexane at the meniscus caused the droplets to break apart and form an extended nematic phase between the meniscus and the capillary walls (Figure 14b). As the concentration increased further, the nematic phase appeared to aggregate and become more birefringent, with the Schlieren texture eventually giving way to the granular “crystalline” structure visible at the left-hand side of Figure 14c. These results may also provide an explanation for the observed discrepancies between the phase and alignment behavior. Although for most dispersions the structure factor, order parameter, and POM texture all indicated the same phase for the same conditions, a small number of samples showed discrepancies. This occurred both for samples that showed strong, broad peaks in the structure factor (similar to those shown by nematic samples) but did not align (350 nm PFS53-bPI637 50 mg/mL, 2130 nm PFS53-b-PI637 75−150 mg/mL, 715 nm PFS133-b-PI1250 25 mg/mL, 915 nm PFS60-b-PDMS670 50− 150 mg/mL) and samples that had a crystalline texture but did align (350 nm PFS53-b-PI637 75 mg/mL, 690 nm PFS90-b-PI890 75 mg/mL, 715 nm PFS133-b-PI1250 75−100 mg/mL). The alignment of micelle dispersions already exhibiting peaks characteristic of hexagonal packing is ascribed to the coexistence of nematic and hexagonal phases as seen in Figure 14c. It is thought that as the concentration of micelles in a homogeneous nematic increases, some regions will begin to pack closer together to form hexagonally packed aggregates (analogous to the nematic tactoids in an isotropic phase). If a

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SUMMARY A comprehensive, systematic study of the structure and liquid crystalline phase behavior of well-defined cylindrical micelles with a crystalline PFS core and a PI corona was conducted using synchrotron SAXS techniques. The study was made possible by the ability to control the micelle dimensions by the living CDSA method. Using the form factor for a homogeneous core and a corona of decaying density, the core and corona dimensions and associated polydispersities could be established and compared to the characteristics of the constituent copolymers. In this way it was possible to determine that the core cross sections remained approximately constant across all samples which, in turn, indicated that chain folding of the core block increased with increasing block length. From the accompanying increase in radial polydispersity, it is considered likely that this also gives rise to a corresponding increase in cross-sectional ellipticity. Furthermore, it was observed that the relationship between chain length and micelle diameter appears to depend more strongly on the block ratio than the length of the coronal chains. More data are required to fully characterize the exact relationship. The effects of micelle length, length distribution, concentration, composition, and block length on liquid crystalline phase behavior were also examined. This has provided a number of insights into the interparticle interactions in micelle dispersions, in particular the composition and radial extent of the corona have been shown to be critical parameters in K


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determining the isotropic−nematic phase behavior. Micelle length was also found to be important for the transition to a hexagonally packed phase. Further work is required to refine the phase diagrams presented in this work and expand the applicability to other semiflexible nanorods; however, it is hoped that this will eventually lead to the development of a robust theory governing the interactions and phase behavior of these tunable liquid crystal colloidal systems.

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ASSOCIATED CONTENT

S Supporting Information *

Experimental details for the polymer synthesis, polymer characterization data, an error analysis for the fits, and theory on the bending of cylindrical rods. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (R.M.R.). *E-mail [email protected] (G.R.W.). Present Addresses

J.B.G.: Department of Chemistry, The University of Western Ontario, 1151 Richmond Street N., London, ON, N6A 5B7, Canada. P.A.R.: Department of Chemistry, The University of Alabama, Tuscaloosa, AL 35487. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS D.W.H. was supported by EPSRC doctoral training centre grant [EP/G036780/1]. J.B.G. and P.A.R. are grateful to the NSERC of Canada and to the EU for postdoctoral fellowships. I.M. thanks the European Research Council for an Advanced Investigator Grant. M.A.W. thanks NSERC Canada for financial support. The authors gratefully acknowledge the Diamond Light Source synchrotron facility for a beamtime award (experiment SM6035). The authors also thank Dr. Torben Gädt for synthesizing PFS28-b-PDMS560 copolymer and Mr. Alexander J. Robertson for his role in analyzing SAXS data.

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REFERENCES

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