Path-Dependent Preparation of Complex Micelle Packings of a Hydrated Diblock Oligomer

Abstract

Small-angle X-ray scattering analyses reveal that the hydrated diblock oligomer n-C₁₆H₂₃(OCH₂CH₂)₂₀-OH (C₁₆E₂₀ or Brij 58) forms lyotropic liquid crystals (LLCs) exhibiting face-centered cubic (FCC), body-centered cubic (BCC), Frank–Kasper (FK) A15, and cylindrical (H_I) morphologies over the concentration range 30–65 wt % amphiphile. Heating LLCs comprising 54–59 wt % C₁₆E₂₀ drives the temperature-dependent phase transition sequence: A15 → BCC → H_I. However, rapidly quenching the resulting H_I phase from 70 to 25 °C initially forms a BCC phase that isothermally transforms into a complex, tetragonal FK σ phase comprising 30 quasispherical micelles. The metastability of this micellar σ phase is shown to depend on the sample cooling rate, thermal quench depth, and isothermal annealing temperature. We rationalize the preference for the A15 structure at 25 °C in terms of minimizing unfavorable water/hydrophobic contacts, while maximizing local particle sphericity. The symmetry breaking transition kinetics in these micellar LLCs apparently stem from the temperature-dependent activation barriers for phase nucleation and growth, which are intimately coupled to the time scales for micelle reconfiguration by amphiphile chain exchange and their spatial rearrangement. These findings highlight how thermal processing influences nucleation and growth of the self-assembled morphologies of intrinsically reconfigurable, soft spherical particles.

Short abstract

Thermal processing of lyotropic liquid crystals of the hydrated diblock oligomer Brij 58 in a manner reminiscent of metallurgy unexpectedly produces a complex, micellar Frank−Kasper σ phase.

Introduction

The structures and properties of diverse classes of materials arise from their path-dependent solidification transitions from the liquid state. Crystallization of a liquid of spherical particles typically results in the formation of high packing-fraction structures such as body-centered cubic (BCC), face-centered cubic (FCC), and hexagonally closest-packed (HCP) crystals, exemplified by the ubiquity of such ordered phases in elemental metals and their alloys,¹ solidified noble gases,² and hard sphere colloids.³ In some exceptional cases, pure metals and their alloys also crystallize into low-symmetry Frank–Kasper (FK) phases, wherein multiple classes of symmetry-equivalent particles pack into giant unit cells containing ≥7 particles.^4,5 One such exemplary packing is the FK σ phase formed by β-U,⁶ β-Ta,⁷ and FeCr alloys,¹ which comprises 30 particles of five distinct, symmetry-equivalent classes arranged in a large tetragonal unit cell. While ligand-decorated nanoparticles,^8,9 sphere-forming block polymers,¹⁰⁻¹³ and hydrated surfactant micelles form FCC and HCP structures,^14,15 the less dense BCC packing is also commonly observed.^16,17 Until recently, soft materials were long thought to adopt only these high-symmetry packings. However, recent experimental reports describe thermotropic liquid crystals,¹⁸⁻²⁰ block polymers,²¹⁻²⁴ giant shape amphiphiles,²⁵ and functional nanoparticles²⁶ comprising charge-neutral building blocks that form micellar FK phases and related dodecagonal quasicrystals.^{21,23,27−31} The appearance of these complex phases apparently originates from a frustrated balance between preserving the preferred local spherical particle symmetry, and filling space at constant density in order to maximize interparticle van der Waals cohesion.^32,33

FK A15 and σ phases have been recently recapitulated in lyotropic liquid crystals (LLCs) of minimally hydrated ionic surfactants.^34,35 In contrast to the aforementioned solvent-free self-assembling systems, low symmetry ionic LLC sphere packings arise from maximization of counterion-mediated electrostatic cohesion among the micelles while minimizing energetically expensive variations in counterion-headgroup solvation.³⁴⁻³⁶ The disparate natures of the noncovalent forces and the space filling constraints underlying complex phase formation in solventless nonionic and solvated ionic amphiphilic materials naturally beg the question: do solvated nonionic amphiphiles form FK structures? In this connection, earlier studies of the water concentration-dependent lyotropic mesophases formed by model poly(ethylene-block-ethylene oxide) diblock oligomers (“C_iE_j” surfactants) documented the formation of BCC, FCC, and HCP spherical micelle packings, as well as FK A15 phases.³⁷⁻⁴⁰ However, the low symmetry σ phase and other FK phases are strikingly absent in these nonionic lyotropic systems.

Herein, we describe the water concentration-dependent LLC self-assembly behavior of the nonionic amphiphile n-C₁₆H₃₃(OCH₂CH₂)₂₀-OH (C₁₆E₂₀, commercially available as Brij 58). Temperature-dependent synchrotron small-angle X-ray scattering (SAXS) studies between T = 25–100 °C reveal that C₁₆E₂₀ self-assembles into FK A15 phases at LLC compositions comprising ∼40 wt % water. Upon heating such samples, we observe an order-to-order transition (OOT) to a BCC phase, which gives way to a hexagonally-packed cylinders (H_I) morphology at higher temperatures. However, rapidly quenching this sample to 25 °C surprisingly leads to the development of an exceptionally well-ordered FK σ phase, which only slowly transforms into the original A15 state upon extended ambient temperature annealing. Thus, we demonstrate a previously unrecognized dependence of LLC mesophase morphology on sample thermal history for low molecular weight amphiphiles. These findings highlight analogies between the path-dependent processing of sphere-forming soft materials and metal alloys, establishing the importance of nucleation and growth kinetics in the self-assembly of spherical particles across disparate materials classes in accordance with Ostwald’s rule for the crystallization of solids.

Results

LLC samples were produced by mixing the block oligomer C₁₆E₂₀ with variable amounts of ultrapure water (18 MΩ·cm), followed by sample homogenization by at least three cycles of centrifugation (4950g for 10 min each) and hand-mixing at 22 °C. The commercial sample of C₁₆E₂₀ (Brij 58), which derives from ethoxylation of n-C₁₆H₃₃OH, was used as received without additional purification. ¹H NMR end group analyses reveal that M_n,NMR = 1074 g/mol, while size-exclusion chromatography indicates that M_n,SEC = 1000 g/mol with Đ = M_w/M_n = 1.17 (against poly(styrene) standards). (See Figures S1 and S2 for ¹H NMR and size-exclusion chromatography traces, respectively.) The resulting LLC samples were sealed in hermetic, alodined differential scanning calorimetry (DSC) pans for synchrotron SAXS analysis at the Advanced Photon Source at Argonne National Lab. Detailed sample preparation and data acquisition protocols are provided in the Supporting Information. Previous studies of the phase behavior of C₁₆E₂₀ relied solely on optical microscopy and low resolution SAXS to identify the LLC mesophases formed, thus precluding definitive morphology assignments.^39,41,42

Figure 1 presents representative SAXS data for a series of aqueous LLCs formed by centrifugation in the concentration range 30–65 wt % C₁₆E₂₀, from which a phase map was constructed using data obtained during the first heating cycle for each composition. Figure 1A depicts a representative SAXS intensity profile for a sample with 32.1 wt % C₁₆E₂₀ at 25 °C, which exhibits five scattering maxima corresponding to (111), (200), (220), (311), and (222) peaks of a micellar face-centered cubic (FCC) lattice with a unit cell parameter a = 13.0 nm. While FCC packing persists in the range 30 ≤ [C₁₆E₂₀] ≤ 38 wt % at ambient temperature, we find that samples with [C₁₆E₂₀] = 30–32 wt % undergo a thermoreversible melting transition to a disordered (isotropic) micellar solution above 60 °C. In the range [C₁₆E₂₀] = 33–38 wt %, heating the as-formed FCC structure instead drives an order-to-order transition (OOT) to a pure body-centered cubic (BCC) morphology at T ≥ 60 °C. In some samples, we also observe the expected intervening window of FCC/BCC two-phase coexistence anticipated by Gibbs’ Phase Rule for two-component systems (Figure S3).⁴³ Further heating to T > 70 °C leads to a BCC → disorder transition, again with an expected window of two-phase coexistence in some cases.

Figure 1.

Aqueous LLC phase behavior of C₁₆E₂₀: (A) Synchrotron SAXS patterns exhibiting the formation of H_I, A15 + H_I, A15, BCC, and FCC phases at diblock oligomer concentrations ranging 30–65 wt % C₁₆E₂₀. (B) Temperature versus wt % C₁₆E₂₀ phase portrait depicting the composition-dependent LLC phases that form, based on synchrotron SAXS analyses of as-prepared samples at 22 °C on the first heating. The solid lines delineate the approximate locations of the phase boundaries.

On increasing [C₁₆E₂₀] to 40–54 wt %, the as-formed LLCs yield SAXS patterns with six peaks at q/q* = √2, √4, √6, √8, √12, √14 (q* = 0.0953 Å^–1 at 52.4 wt %) corresponding to a BCC micelle packing with a (310) scattering peak extinction. LLCs with [C₁₆E₂₀] = 40–45 wt % exhibit an order–disorder transition (ODT) near T = 80 °C, whereas samples with [C₁₆E₂₀] = 45–54 wt % instead exhibit an OOT to a two-dimensional (2D) hexagonally-packed cylinders (H_I) morphology that remains stable up to 100 °C. In the weight fraction range 46–50% C₁₆E₂₀ at 100 °C, the H_I phase coexists with an isotropic micellar solution as evidenced by the superposition of a broad correlation scattering peak on the sharp Bragg scattering (Figure S4). SAXS patterns for samples comprising [C₁₆E₂₀] = 54–59 wt % display at least 18 sharp SAXS maxima at q/q* = √2, √4, √5, √6, etc. consistent with the FK A15 phase with Pm3(−)n symmetry (space group #223). Note that A15 phases have been previously observed in aqueous LLCs of structurally homologous long-chain alcohol ethoxylates.³⁷⁻³⁹ The SAXS pattern in Figure 1A obtained at 57.9 wt % C₁₆E₂₀ is characteristic of an FK A15 phase with a unit cell parameter, a = 14.7 nm at 25 °C. At all compositions that form the FK A15 phase, we observe the same sequence of thermally induced OOTs: A15 → BCC → H_I (Figure 1B), where the OOT temperature decreases with increasing [C₁₆E₂₀]. When [C₁₆E₂₀] > 62 wt %, only a thermally stable H_I phase forms over the entire accessible range T = 25–100 °C with intercylinder distances d ≈ 7.29 nm. Generally, the ordered LLC unit cell parameters only weakly depend on hydration and temperature within a given pure phase window (<5% variation). A complete aqueous LLC phase diagram upon first heating for C₁₆E₂₀ is given in Figure 1B. Although the Gibbs phase rule anticipates two-phase coexistence windows between each pure LLC phase,⁴³ the composition resolution of our phase map (∼3 wt % increments) revealed only one such occurrence as noted.

LeBail refinement of the one-dimensional (1D)-synchrotron SAXS profile of an A15 LLC using the JANA2006(44) software package enabled quantitative extraction of the structure factor intensities, which were used as inputs for the SUPERFLIP(45) charge-flipping algorithm to reconstruct a real-space electron density map for this phase. The electron density map for the A15 phase formed at 58.9 wt % C₁₆E₂₀ shown in Figure 2A represents the 90% negative electron density isosurface obtained by averaging 11 charge-flipping algorithm convergences (see Supporting Information for the SUPERFLIP input file). On the basis of the LLC composition and by assuming equal mass densities of the aqueous and hydrophobic domains, we estimate that the average aggregation number for the A15 phase micelles is N_agg = 127 diblock chains. However, the electron density map clearly indicates that the A15 phase consists of two symmetry-equivalent classes of quasispherical micelles, which appear to differ in size and shape. From the 2D electron density map of the z = 0 and 1 planes of the A15 phase (Figure 2B), the micelles appear to be polyhedral, with facets and apparent sizes that mirror the shapes and volumes of the Wigner–Seitz (W–S) cells for each occupied lattice position.

Figure 2.

(A) Electron density map (90% isosurface) of an A15 phase reconstructed from a 1D-synchrotron SAXS pattern derived from an LLC with 58.9 wt % C₁₆E₂₀, comprising six coordination number CN = 14 micelles (blue) and two CN = 12 micelles (red). (B) 2D electron density map of the z = 0 and 1 planes of the A15 phase showing faceted micelles, which face each other and mimic the underlying Wigner–Seitz cell structure of each corresponding lattice position.

Similarly faceted micelles in self-assembled FK A15 and σ phases have been previously noted in X-ray data-derived electron density reconstructions for wedge-shaped dendritic liquid crystals^18,19,46 and for lyotropic mesophases of ionic alkylphosphonate LLCs.³⁴ In the latter case, molecular dynamics (MD) simulations by Kim et al. revealed that the negatively charged alkylphosphonate micelle facets mirror the W–S cell structures of these aqueous FK LLC phases.³⁴ These MD simulations also demonstrated that the ionic micelle aggregation numbers (N_agg) in the σ phase vary by as much as ± 9% around the mean volume, whereas related self-consistent mean-field theory (SCMFT) calculations for solvent-free diblock polymer A15 phases indicate a ± 3% variation in the micelle sizes within a unit cell.²³ The limited resolution of the density map in Figure 2A precludes quantitative analyses of such differences in micelle volumes. Nonetheless, the micelles at different symmetry-equivalent positions appear to have different volumes and thus different N_agg values.

Thermally cycling LLCs comprising [C₁₆E₂₀] = 54–59 wt % between 25 and 70 °C unexpectedly leads to the formation of complex morphologies not observed on heating the as-prepared samples. Heating a sample comprising 58.7 wt % C₁₆E₂₀ resulted in the sequence of phase transitions A15 (25 °C) → A15 + BCC (37 °C) → H_I (60 °C) with increasing T. However, rapidly quenching this sample from 70 to 25 °C by immersion in a thermostated water bath induces its transformation from a H_I phase to a BCC morphology (a = 9.17 nm) with a series of low intensity SAXS peaks near the base of the (110) peak after 5 min (Figure 3). Isothermal annealing at 25 °C causes the latter peaks to grow in intensity at the expense of the BCC scattering maxima. After 50 min, the sample develops a remarkably complex scattering pattern with >50 sharp SAXS peaks with the most intense peaks occurring at intermediate q-values (Figure 3). This distinctive scattering signature corresponds to that of a pure FK σ phase (P4₂/mnm symmetry) with tetragonal unit cell parameters a = 28.4 nm and c = 14.9 nm (c/a = 0.525) (see Figure S5 for fully indexed pattern and Table S1 for complete peak listing with residuals). While similar scattering signatures have been obtained for wedge-shaped dendrons,¹⁹ block polymers,²¹⁻²³ and ionic surfactant LLCs,^34,35 this unexpected discovery constitutes an unrecognized example of a lyotropic FK σ phase in nonionic diblock oligomer LLCs.

Figure 3.

SAXS patterns obtained from an LLC having 58.7 wt % C₁₆E₂₀, upon quenching from 70 to 25 °C. The H_I phase rapidly transforms into BCC, evidenced by the seven sharp SAXS maxima recorded 2 min after quenching. The BCC phase then evolves into a well-developed FK σ phase on further isothermal annealing at 25 °C for 50 min.

By applying the SUPERFLIP charge-flipping algorithm to SAXS data for the FK σ phase obtained by quenching an LLC with 58.7 wt % C₁₆E₂₀ from 60 to 25 °C and isothermally annealing it for 2 days, we obtained the electron density reconstruction shown in Figure 4A (see Supporting Information for the SUPERFLIP input file). This structure exhibits five classes of symmetry-equivalent micelles located in different environments having coordination numbers CN = 12, 14, and 15, consistent with prior reports of σ phases.^4,5,32,34 In a manner analogous to the A15 phase, the micelles in the σ phase exhibit facets corresponding to the W–S cell (Voronoi) lattice partition. The degree of micellar faceting in the σ phase (Figure 4B) is apparently less pronounced than in the A15 phase (Figure 2B). MD simulations of ionic micelles forming the σ phase,³⁴ and SCMFT calculations for σ phase-forming diblock polymers show that the micelle volumes vary by ± 9% around the mean value,^23,32,34 which mimics the volume the variation in the W–S cells of the lattice. Thus, the formation of the σ phase suggests the adoption of a discrete N_agg distribution among the micelles to enable the formation of the specific micelle sizes required by this structure.

Figure 4.

(A) Electron density reconstruction (90% isosurface) depicting the FK σ phase formed by a sample with 58.7 wt % C₁₆E₂₀ on quenching from 60 to 25 °C, followed by isothermally annealing at 25 °C for 2 days. The 30 particles within the tetragonal unit cell belong to five symmetry-equivalent classes exhibiting coordination numbers CN = 12 (green and yellow), 14 (blue and purple), 15 (red). (B) 2D-electron density map of the z = 0 and 1 planes of the σ phase, illustrating the more modest faceting of the micelles as compared to the A15 phase in Figure 2B.

Further isothermal annealing of this FK σ phase at 25 °C for 60 days leads to the development of new peaks coincident with an A15 phase (Figure 5). While the (002) reflection for the A15 phase is distinct from the (002) peak for the σ phase, the relatively close values are consistent with the observed epitaxial relationship between these tetrahedrally close packed FK phases.^19,29,30,35 Over a period of 150 days of isothermal annealing at 25 °C, the sample continues to evolve and eventually transforms into a pure A15 phase (Figure 5). Note that the unit cell parameter for this A15 phase is identical to that of the initially prepared LLC (a = 14.7 nm), indicating the apparent metastability of the σ phase with respect to the thermodynamically preferred A15 phase at 25 °C.

Figure 5.

SAXS patterns of the σ phase obtained from a quenched LLC with 58.7 wt % C₁₆E₂₀ upon ambient temperature annealing for 60 days show the development of peaks which index to A15 (red markers). Extended annealing for 150 days reveals that the σ phase completely reverts to the original A15 phase of the as-prepared sample, indicating that the σ phase is a long-lived metastable liquid crystalline state at 25 °C.

In view of the appearance of the FK σ phase as a long-lived, metastable liquid crystalline state, we carefully reassessed the potential accessibility of the σ phase on heating. Figure S6 furnishes temperature-dependent SAXS results for an LLC formed at 58.7 wt % C₁₆E₂₀ on heating in 2 °C increments, with a 5 min sample equilibration time at each temperature, in the window T = 25–60 °C. This in-depth analysis established that the phase sequence on heating is

Within the confidence of these high-resolution and high-sensitivity synchrotron SAXS analyses, we observe no evidence for even weak peaks corresponding to any other phases. Thus, we conclude that the σ phase is only accessible upon cooling from H_I via a transient BCC phase within the temperature and time resolution of our experiments.

We subsequently examined the impact of the sample cooling rate on the formation of the C₁₆E₂₀ LLC FK σ phase (Figure 6). Rapidly quenching a 58.7 wt % C₁₆E₂₀ LLC from 70 to 25 °C reproducibly forms a FK σ phase, which remains unchanged even after isothermal annealing for up to 3 days. On the other hand, cooling a sample with [C₁₆E₂₀] = 57.9 wt % from 70 to 25 °C at 0.1 °C/min (over 550 min) followed by annealing at 25 °C for 2 h yields a SAXS pattern indicative of an FK σ phase with at least three additional peaks that correspond to a coexisting A15 phase. More explicitly, the additional reflections correspond to the (200), (210), (211), and (400) planes of the A15 lattice.

Figure 6.

SAXS traces obtained by two different LLC sample cooling protocols from 70 to 25 °C: (1) quenching a sample formed at 58.7 wt % C₁₆E₂₀ at >60 °C/min and isothermally annealing for >3 days at 25 °C (blue), and (2) cooling a sample comprising 57.9 wt % C₁₆E₂₀ at 0.1 °C/min followed by isothermal annealing for 2 h at 25 °C (black). The rapid quench yields a pure σ phase, whereas slow cooling yields coexisting A15 and σ phases. The red lines correspond to (200), (210), (211), and (400) peak positions of the A15 structure, which are present in the black trace and are absent in the blue trace.

Given the recent recognition that thermal processing sensitively impacts the observed spherical micelle morphologies of neat diblock polymers,²³ we probed the influence of temperature quench depth on the formation of the LLC σ phase formed at [C₁₆E₂₀] = 58.7 wt %. A relatively shallow quench of an H_I phase at 70 °C (q* = 0.0984 Å^–1) to 50 °C led to the formation of a BCC LLC (q* = 0.0977 Å^–1) that remains stable upon 50 °C annealing for ∼10 h (Figure S7). We note that the SAXS peak intensities for the BCC LLC evolve even after phase nucleation during the annealing period, suggesting a relatively long time scale for micelle size equilibration even at this relatively high temperature. This size equilibration, which presumably occurs through intermicellar chain exchange of C₁₆E₂₀ amphiphiles, highlights the dynamic nature of these reconfigurable soft particle assemblies. Rapidly cooling a sample with the same composition from 70 to 40 °C (into the A15/BCC coexistence window observed on heating) initially drives formation of a BCC structure, which is followed by σ phase nucleation evidenced by the evolution of a second set of weaker intensity SAXS peaks 10 min after the quench (Figure S8). Annealing this LLC at 40 °C for ∼18.5 h leads to reduced σ phase SAXS peak intensities, suggesting a preference for the BCC structure at 40 °C. Finally, quenching the sample from 70 to 0 °C drives a transition from the H_I to a BCC morphology exhibiting seven somewhat broad SAXS peaks with (q/q*)²= 1, 2, 3, ..., etc. (Figure S9). Upon isothermal annealing at 0 °C over 4 days, this BCC morphology remains unchanged (i.e., the SAXS peak intensities do not change) with no evidence for σ or A15 phase nucleation or growth. Since the BCC morphology comprises only one class of symmetry equivalent micelles, the formation of either of the FK A15 (two micelle types) or σ (five micelle types) phases requires both significant micelle reconfiguration through interparticle unimer exchange and nucleation of a denser, tetrahedral close packing.^32,34

The fact that quenching of the above LLC to 0 °C leads to a stable BCC phase is consistent with cessation of chain exchange, which retards mass exchange-dependent phase transitions. Note that SAXS patterns for these quenched samples do not exhibit any signs of crystallization of the alkyl tails, the PEO segments, or the water in the LLC. Therefore, the cessation of chain exchange must stem from an increased penalty for extraction of a hydrocarbon chain into the aqueous domain due to the increased chemical incompatibility of the alkyl core and the aqueous corona at low temperatures.⁴⁷ A similar temperature-dependent frustration of interparticle chain exchange that delays σ phase nucleation and growth was previously reported by Gillard et al., upon quenching neat diblock polymers from the disordered state to below the ergodicity temperature (T_erg). Below T_erg, long-lived nonequilibrium states comprising disordered yet nonergodic, liquid-like micelle packings become trapped.^22,24,27,32 The authors argued that the jammed, glassy state forms due to increased core–corona segregation at temperatures below T_erg, which leads to the loss of relaxation modes required for equilibrium structure formation.

Rapidly quenching the 58.7 wt % C₁₆E₂₀ LLC from 70 to 30 °C followed by isothermal annealing leads to the formation of coexisting BCC and σ phases after 15 min (Figure 7). After 20 min of annealing at 30 °C, we also observe the splitting of the σ-phase (002) SAXS peak (red markers) and the appearance of a new, lower intensity peak at q = 0.0606 Å^–1. After further annealing this sample for 19.5 h postquench, we observe the appearance of the (110), (200), (210), and (211) peaks of the A15 phase. However, these SAXS features do not develop upon sample quenching to 25 °C and subsequent annealing for 3 days. These findings collectively suggest that the thermally activated process of interparticle chain exchange accelerates the reversion of the metastable σ phase to the A15 phase.

Figure 7.

SAXS traces depicting the nucleation of the FK σ and A15 phases from a BCC morphology, upon quenching an aqueous LLC comprising 58.7 wt % C₁₆E₂₀ from 70 to 30 °C with subsequent isothermal annealing. BCC/σ phase coexistence is observed after 20 min of isothermal annealing at 30 °C, after which the BCC peak intensities (dashed lines) decrease with concomitant growth of an A15 LLC morphology indicated by the appearance of the (110), (200), (210), and (211) SAXS peaks (red triangles).

The results of these various thermal processing experiments for the 58.7 wt % C₁₆E₂₀ aqueous LLCs are graphically summarized in Figure 8. The as-made sample produced by centrifugation of water and the diblock oligomer followed by hand-mixing at 22 °C leads to the formation of an A15 phase with a = 14.75 nm. Heating the A15 phase to 70 °C drives the formation of an H_I LLC, which undergoes the depicted time-dependent transformations upon quenching to temperatures T ≤ 50 °C. At all quench temperatures, we initially observe the nucleation of a BCC phase with slightly different unit cell parameters. At 50 °C, the BCC phase remains stable upon isothermal annealing at the quench temperature. Quenching the LLC sample to 40 °C initially forms a BCC phase, followed by nucleation of a σ phase that reverts to BCC upon extended annealing at 40 °C for 18.5 h. At T = 25 or 30 °C, the initially formed BCC phase transforms into a σ phase that slowly reverts to the original A15 phase (same unit cell parameter).

Figure 8.

Summary of the results of various thermal processing protocols applied to an LLC with 58.7 wt % C₁₆E₂₀. The progression of phases upon quenching to 50, 40, 30, and 20 °C is depicted as a function of log(time). Dotted arrows represent time intervals where no SAXS data were obtained. Numerical values in the parentheses correspond to the unit-cell parameters (in nm) of the respective phases. The unit cell parameters of the BCC phase were obtained immediately following the quench (t = 0 s). The unit cell parameter of the A15 phase as-formed by centrifugation is equal to that of samples that were quenched and subsequently annealed at 25 °C and 30 °C for 150 days and 19.5 h, respectively.

No unexpected or unusually high safety hazards were encountered during sample handling or preparation. X-ray scattering analyses do pose significant safety hazards, and all institutional radiation safety protocols were followed to minimize exposure.

Discussion

The water concentration-dependent self-assembly behavior of C₁₆E₂₀ into lyotropic sphere packings at 25 °C originates from a frustrated balance of minimizing water-hydrophobic contacts, while maximizing poly(ethylene oxide) (PEO) corona chain hydration and minimizing repulsive micellar packing (steric) interactions. In dilute aqueous solution (high H₂O contents), theoretical and experimental studies have shown that the PEO coronae of C₁₂E₈ micelles partially dehydrate and accumulate near the interface between the hydrocarbon core and the water to mitigate energetically unfavorable contacts.^48,49 The micelles consequently exhibit a core-shell-brush structure, wherein the hydrocarbon core is surrounded by a dehydrated PEO layer from which hydrated PEO chains extend. Thus, the core appears larger than anticipated by the amphiphile composition, and the spatial extent of the hydrated corona is smaller than expected. In a manner akin to hard spheres, these so-called “crew-cut micelles” pack into a closest-packed FCC lattice.^10,50 However, maintaining this crew-cut structure as the LLC hydration decreases would drive further accumulation of ethylene oxide units at the water/hydrocarbon interface and result in unfavorable intramicellar corona chain overlap. To obviate these unfavorable steric interactions, the PEO segments of the amphiphiles become more uniformly hydrated and stretched so that the interfacial area per chain decreases.^10,51−53 The resulting “hairy” micelles, comprising hydrocarbon cores decorated with stretched PEO arms, now prefer a lower packing fraction BCC structure to prevent unfavorable intermicellar corona chain overlap. Further decreasing the H₂O content in the LLC drives coronae chain stretching (reduced interfacial area per surfactant) and transforms the BCC structure into the lower packing fraction A15 phase to further alleviate interparticle coronal repulsions.^{1,10,11,53−55} At a fixed temperature, note that N_agg increases with decreasing water content along the lyotropic sequence: FCC → BCC → A15 → H_I.

The complex temperature-dependent phase behavior of the C₁₆E₂₀ LLCs on heating may be rationalized in terms of the temperature-dependent water solubility of PEO.⁵² On heating the A15 LLC formed at 54–59 wt % C₁₆E₂₀, the PEO coronae of the hairy micelles partially dehydrate and retract toward the hydrophilic/hydrophobic interface due to reduced PEO solubility with increasing temperature. Thus, heating effectively transforms the hairy micelles into partially crew-cut micelles that prefer to pack into a BCC lattice. Further heating drives greater PEO dehydration and brings the micelles into close proximity, so that they fuse along the <111>-direction of BCC to form the cylindrical micelles of the H_I phase.^37,56 In a manner consistent with previous reports on solvated block polymer and small molecule amphiphile dispersions,^37,56,57 the last transition is thermoreversible. This same molecular-level mechanism also explains the somewhat counterintuitive thermal transition from a dense FCC packing to sparsely packed BCC for LLCs with [C₁₆E₂₀] < 39 wt %. Heating the crew-cut micelles of the FCC drives further accumulation of ethylene oxide segments near the water/hydrocarbon interface, as a consequence of reduced PEO solubility with increasing temperature. The resulting “dry” PEO segments near the hydrophobic/hydrophilic interface may penetrate into the hydrophobic particle core. Thus, the particle cores swell, and a more loosely packed BCC phase forms in order to mitigate intermicellar corona repulsions.

The observed formation of the C₁₆E₂₀ LLC FK σ phase from a BCC phase is reminiscent of previous reports from Bates and co-workers on di- and tetra-block polymer melts.^22,32 Since the FK σ phase comprises five distinct classes of symmetry-equivalent micelles (Figure 4) with a specific N_agg distribution that exhibits a ± 9% variation around the mean value, the transition from a BCC phase (in which all micelles are symmetry-equivalent) must stem from micelle reconfiguration by interparticle amphiphile exchange.²⁴ Bates and co-workers have rationalized σ phase formation in terms of maximizing the van der Waals cohesion of the particle ensemble while maximizing local spherical particle symmetry. In other words, the block polymers fill space at constant density while minimizing both the interfacial area between the core and corona and energetically expensive variations in corona segment stretching.^32,58 Similar arguments have been advanced by Rappolt et al. in the case of inverse micellar LLCs.³³ In order to maximize local particle sphericity, the micelles reconfigure into a lattice with a higher average coordination number to introduce more facets to the W–S cells to render them more spherical. According to a metric developed by Polya,⁵⁹ the sphericity of an object with volume V and surface area S may be quantified using the dimensionless isoperimetric quotient (IQ) = 36πV²/S³, such that IQ_sphere = 1_. The number-averaged values of the IQs for the constituent W–S cells for various phases ascend in the order: BCC (0.7534) < A15 (0.7617) < σ (0.7623). Consequently, one expects that a frustrated BCC phase may transform into an FK σ phase to maximize local particle sphericity and global lattice cohesion.

However, the geometrically predicted phase stability of σ over A15 based on the IQ formalism does not explain the observed metastability of the σ LLC with 54–59 wt % C₁₆E₂₀ in H₂O at 25–30 °C. From the electron density reconstructions given in Figures 2 and 4, we note that the polyhedral micelle cores in the C₁₆E₂₀ FK phases reflect the W–S cells (Voronoi partition) of their underlying crystal lattices. Therefore, the area of the W–S cells serves as a geometric proxy for the micellar interfacial area in each morphology. Upon calculating the W–S cell area per diblock chain for LLCs formed with 58.7 wt % C₁₆E₂₀ at 25 °C (A_int in units of nm²/diblock chain) assuming a uniform interfacial tension across all interfaces, one finds that A_int decreases in the order BCC (1.602) > σ (1.354) > A15 (1.353). This calculation demonstrates that the equilibrium A15 phase formed at 25 °C exhibits a slightly lower interfacial area per chain as compared to the metastable σ phase. This trend in A_int runs counter to that for IQ, suggesting that the observed morphology reflects a complex optimization of the interfacial contact area and the entropic penalty for chain stretching to completely fill space. Grason and co-workers recently reported a general approach for predicting the phases formed by soft spherical particles in the polyhedral interface limit (PIF), which highlights the subtle thermodynamic balance underlying phase selection in these systems.^58,60

These geometric arguments lead to a nuanced picture of the driving forces for FK σ and A15 phase formation in C₁₆E₂₀ LLCs upon thermal quenching to 25 °C. At 70 °C, the PEO corona chains of the H_I micelles are concentrated near the hydrocarbon/water interface due to decreased water solubility of PEO at higher temperatures. Quenching this state to 25 °C drives hydration of the PEO segments, causing the cylinders to pinch off and epitaxially form discontinuous spherical micelles packed onto a BCC lattice. Rapid cooling of the H_I phase likely leads to a nonequilibrium micelle size distribution in the BCC structure. Note that the rates at which the micellar coronae of these newly formed particles approach their equilibrium hydration state and equilibrium micelle size distribution are governed by the relatively slow rate of amphiphile chain exchange between the micellar particles (see Figure S7; vide supra). As the PEO segments drive toward their equilibrium hydration, the incompatibility between the hydrophobic particle core and the hydrated corona increases. Consequently, the increased enthalpic penalty of hydrocarbon–water contacts drives the transition from the BCC phase to σ followed by A15 via intermicellar amphiphile exchange so as to minimize A_int. The fact that the average N_agg are nearly the same in the BCC (123.2), σ (127.7), and A15 (128.2) phases formed with 58.7 wt % C₁₆E₂₀ at 25 °C suggests that micelle fission/fusion events are not operative, and that a combination of chain exchange and spatial redistribution of the micelles drives the overall transition from BCC to A15.

The observed phase sequence BCC → σ → A15 on quenching from 70 to 25 °C suggests that the barrier for BCC → σ is relatively lower than that for the direct BCC → A15 transition. Quenching the H_I phase at 70 to 25 °C drives a low energy, epitaxial transition to a nonequilibrium BCC phase.^37,56,57 We speculate that the distribution of N_agg for the spherical micelles in this BCC structure is somewhat broad due to the rapid quench conditions. The breadth of this N_agg distribution more readily allows the formation of the disperse yet discrete distribution of sphere sizes requisite for σ phase formation, wherein N_agg varies by ± 9% around a mean value. This transition also enables the system to lower its free energy by reducing A_int. As the chains approach their equilibrium hydration, the LLC undergoes a transition to the original A15 phase (formed directly from centrifugation of C₁₆E₂₀ with H₂O) to further reduce A_int while also narrowing the variation in N_agg to ± 3% around a mean value. Given the nearly identical average N_agg values for the σ and A15 phases, this transition must involve a subtle spatial rearrangement of the micelles in addition to chain exchange, whereby the slow kinetics implies a significant nucleation barrier. The fact that we observe a direct A15 → BCC transition on heating from 25 to 70 °C with no intervening phases also suggests that there is a significant barrier to nucleation and growth of the σ phase from A15, possibly due to the required broadening of the micellar N_agg distribution. Ha et al. have recently reported similarly slow nucleation of FK A15 phases of phosphonate block polymer LLCs in ionic liquids upon cooling from the disordered micellar state.⁶¹ These findings highlight the competing roles of nucleation and growth kinetics that underlie complex phase formation in amphiphilic soft materials. Note that the formation of a metastable σ phase upon quenching LLCs to T ≤ 30 °C is reminiscent of Ostwald’s step rule in solid crystallization: the least stable polymorph nucleates first upon cooling a melt or a supersaturated solution of the solid.⁶² In this present case, both the metastable BCC and σ structures nucleate first and only slowly convert to the equilibrium A15 phase.

Epitaxial transitions between micellar phases in aqueous LLCs of a homologous C₁₂E₁₂ amphiphile were previously investigated by Sakya et al.³⁷ The aqueous LLC phase behaviors of C₁₂E₁₂ and C₁₆E₂₀ are similar in that both amphiphiles form FCC, BCC, A15, and H_I phases with increasing diblock oligomer concentration. Additionally, C₁₂E₁₂ LLCs display the temperature-dependent OOT sequence A15 → BCC → H_I → disordered micelles in a narrow composition range. Cooling such LLCs from high temperature while applying shear triggers an epitaxial transition from an aligned H_I to a shear-oriented BCC phase as expected. However, further cooling of this oriented BCC phase led to a complete loss of alignment and the development of SAXS patterns that could not be indexed. On the basis of similarities between the LLC phase behaviors of C₁₂E₁₂ and C₁₆E₂₀ and the thermal processing protocols employed, we surmise that the unreported pattern may have been a C₁₂E₁₂ LLC σ phase or some other tetrahedrally closest-packed FK structure.

Conclusion

The aqueous LLC phase behavior of the model diblock oligomer C₁₆E₂₀ was investigated using temperature-dependent SAXS. In the composition range 54–59 wt % C₁₆E₂₀, we observe the formation of a FK A15 phase upon mixing the amphiphile with water. Upon heating this LLC, we observe temperature-dependent order–order transitions to a BCC structure, followed by formation of a H_I morphology. However, rapidly quenching the H_I phase from 70 to 25 °C leads to the nucleation and growth of a previously unrecognized FK σ phase, comprising 30 quasispherical micelles packed into a gigantic tetragonal unit cell. This unexpected finding establishes that the σ phase generically forms in systems comprising hydrated ionic surfactants, hydrated nonionic diblock oligomers, and solvent-free diblock polymers in spite of the disparate intermolecular forces stabilizing these self-assembling systems. Extended ambient temperature annealing of the LLC FK σ phase results in its remarkably slow reversion to the original A15 phase over the course of 150 days. These findings highlight the unappreciated dependence of LLC morphology on sample thermal history for low molecular weight amphiphiles, which can guide the nucleation and growth of metastable intermediate phases.

The accessibility of myriad particle packings in a given amphiphile system depends on a subtle competition between equilibrium self-assembly thermodynamics and the barriers to phase nucleation and growth. Since thermal processing of both hard and soft materials enables access to fundamentally interesting and commercially important metastable states, one may be tempted to draw parallels between their processing characteristics. For example, rapidly quenching low carbon-content austenite (FCC γ-Fe) drives its diffusionless transformation into a body-centered tetragonal martensitic steel that is metastable with respect to α-ferrite (BCC) and cementite (Fe₃C).⁶³ However, quasispherical particle packings observed in solvated block polymer and related amphiphile dispersions stem from diffusion-dependent, interparticle chain exchange between the micelles to reconfigure their sizes and consequent packings. The kinetics of these chain exchange events depend sensitively on the incompatibility between the core domain and the solvent and the core block mobility.^64,65 High amphiphile concentrations and lattice ordering of their micelles are known to decelerate chain exchange by over an order of magnitude, as compared to dilute solutions.⁴⁷ Therefore, slow unimer dynamics enable the nucleation of higher energy states which become kinetically trapped, because large activation barriers inhibit fast relaxation to the equilibrium structure.^13,23 Aside from thermal processing, we envision that unimer dynamics in the systems studied here may be tuned by variations in solution pH, addition of exogenous electrolytes, or other environmental stimuli. This conceptual framework for understanding the complex free energy landscape underlying solvated amphiphile self-assembly suggests that subtle changes in the conditions under which they aggregate and assemble may beget access to exciting, new, and undiscovered states.

Supporting Information Available

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acscentsci.8b00903.

• Experimental details, SAXS patterns obtained from LLCs with 32.9 and 46.5 wt % C₁₆E₂₀ (Figures S3 and S4), indexed SAXS pattern for the σ phase formed at 58.7 wt % C₁₆E₂₀ (Figure S5), and listing of peak positions and residuals for the σ phase (Table S1); SAXS patterns in 2 °C increments upon heating from 25–60 °C (Figure S6); time resolved 1D SAXS traces when sample with 58.7 wt % C₁₆E₂₀ when quenched to 50, 40, and 0 °C (Figures S7–S9); details about electron density reconstructions and SUPERFLIP input files for the A15 (58.9 wt % C₁₆E₂₀) and σ phase (58.7 wt %, quenched and annealed at 25 °C for 2 days) (PDF)

Author Contributions

A.J. and M.K.M. designed the research, D.Y.Z., B.L.D, and A.J. performed the research, A.J., D.Y.Z, B.LD., and M.K.M. analyzed the data, and A. J. and M.K.M. wrote the manuscript. All authors reviewed and agreed to the final manuscript.

The authors declare no competing financial interest.