Particle Dynamics in a Diblock-Copolymer-Based Dodecagonal Quasicrystal and Its Periodic Approximant by X-Ray Photon Correlation Spectroscopy.

Temperature-dependent x-ray photon correlation spectroscopy (XPCS) measurements are reported for a binary diblock-copolymer blend that self-assembles into an aperiodic dodecagonal quasicrystal and a periodic Frank-Kasper σ phase approximant. The measured structural relaxation times are Bragg scattering wavevector independent and are 5 times faster in the dodecagonal quasicrystal than the σ phase, with minimal temperature dependence. The underlying dynamical relaxations are ascribed to differences in particle motion at the grain boundaries within each of these tetrahedrally close-packed assemblies. These results identify unprecedented particle dynamics measurements of tetrahedrally coordinated micellar block polymers, thus expanding the application of XPCS to ordered soft materials.

Self-Assembly of Unusually Stable Thermotropic Network Phases by Cellobiose-Based Guerbet Glycolipids.

Bicontinuous thermotropic liquid crystal (LC) materials, e.g., double gyroid (DG) phases, have garnered significant attention due to the potential utility of their 3D network structures in wide-ranging applications. However, the utility of these materials is significantly constrained by the lack of robust molecular design rules for shape-filling amphiphiles that spontaneously adopt the saddle curvatures required to access these useful supramolecular assemblies. Toward this aim, we synthesized anomerically pure Guerbet-type glycolipids bearing cellobiose head groups and branched alkyl tails and studied their thermotropic LC self-assembly. Using a combination of differential scanning calorimetry, polarized optical microscopy, and small-angle X-ray scattering, our studies demonstrate that Guerbet cellobiosides exhibit a strong propensity to self-assemble into DG morphologies over wide thermotropic phase windows. The stabilities of these assemblies sensitively depend on the branched alkyl tail structure and the anomeric configuration of the glycolipid in a previously unrecognized manner. Complementary molecular simulations furnish detailed insights into the observed self-assembly characteristics, thus unveiling molecular motifs that foster network phase self-assembly that will enable future designs and applications of network LC materials.

Stabilizing a Double Gyroid Network Phase with 2 nm Feature Size by Blending of Lamellar and Cylindrical Forming Block Oligomers.

Molecular dynamics simulations are used to study binary blends of an AB-type diblock and an AB2-type miktoarm triblock amphiphiles (also known as high-χ block oligomers) consisting of sugar-based (A) and hydrocarbon (B) blocks. In their pure form, the AB diblock and AB2 triblock amphiphiles self-assemble into ordered lamellar (LAM) and cylindrical (CYL) structures, respectively. At intermediate compositions, however, the AB2-rich blend (0.2 ≤ x AB ≤ 0.4) forms a double gyroid (DG) network, whereas perforated lamellae (PL) are observed in the AB-rich blend (0.5 ≤ x AB ≤ 0.8). All of the ordered mesophases present domain pitches under 3 nm, with 1 nm feature sizes for the polar domains. Structural analyses reveal that the nonuniform interfacial curvatures of DG and PL structures are supported by local composition variations of the LAM- and CYL-forming amphiphiles. Self-consistent mean field theory calculations for blends of related AB and AB2 block polymers also show the DG network at intermediate compositions, when A is the minority block, but PL is not stable. This work provides molecular-level insights into how blending of shape-filling molecular architectures enables network phase formation with extremely small feature sizes over a wide composition range.

Flexible Nanoporous Materials by Matrix Removal from Cylinder-Forming Diblock Copolymers.

We describe a straightforward self-assembly route to nanoporous materials derived from a hexagonally-packed cylinder (HEX) morphology of a polyisoprene-block-polylactide (PI-b-PLA) diblock copolymer, by thermal cross-linking of the minority PI domains followed by selective chemical etching of the PLA matrix. The resulting mechanically stable and porous samples defy the expectation that the remaining cylinders cannot yield a robust, integrated material upon matrix removal. Scanning electron microscopy imaging reveals that this unexpected structural integrity stems from the interconnected nanofibrils therein, reflecting topological defects at the grain boundaries of the parent polydomain HEX nanostructure. Hydrodynamic radius-dependent poly(ethylene oxide) (Mn = 0.4-35 kg/mol) permeation behavior through these monoliths directly demonstrated the continuity and size selectivity of the nanoporous material. The ready accessibility of block copolymer HEX morphologies of varied chemistries suggests that this matrix etching strategy will enable the future design of functional, size-selective nanofiltration membrane materials.

Dodecagonal quasicrystals of oil-swollen ionic surfactant micelles.

A delicate balance of noncovalent interactions directs the hierarchical self-assembly of molecular amphiphiles into spherical micelles that pack into three-dimensional periodic arrays, which mimic intermetallic crystals. Herein, we report the discovery that adding water to a mixture of an ionic surfactant and n-decane induces aperiodic ordering of oil-swollen spherical micelles into previously unrecognized, aqueous lyotropic dodecagonal quasicrystals (DDQCs), which exhibit local 12-fold rotational symmetry and no long-range translational order. The emergence of these DDQCs at the nexus of dynamically arrested micellar glasses and a periodic Frank-Kasper (FK) σ phase approximant sensitively depends on the mixing order of molecular constituents in the assembly process and on sample thermal history. Addition of n-decane to mixtures of surfactant and water instead leads only to periodic FK A15 and σ approximants with no evidence for aperiodic order, while extended ambient temperature annealing of the DDQC also reveals its transformation into a σ phase. Thus, these lyotropic DDQCs are long-lived metastable morphologies, which nucleate and grow from a stochastic distribution of micelle sizes formed by abrupt segregation of varied amounts of oil into surfactant micelles on hydration. These findings indicate that molecular building block complexity is not a prerequisite for the formation of aperiodic supramolecular order, while also establishing the generic nature of quasicrystalline states across metal alloys and self-assembled micellar materials.

Reevaluation of Poly(ethylene-alt-propylene)-block-Polydimethylsiloxane Phase Behavior Uncovers Topological Close-Packing and Epitaxial Quasicrystal Growth.

Reanalysis of an asymmetric poly(ethylene-alt-propylene)-block-polydimethylsiloxane (PEP-PDMS) diblock copolymer first investigated in 1999 has revealed a rich phase behavior including a dodecagonal quasicrystal (DDQC), a Frank-Kasper σ phase, and a body-centered cubic (BCC) packing at high temperature adjacent to the disordered state. On subjecting the sample to large amplitude oscillatory shear well below the σ-BCC order-order transition temperature (TOOT), small-angle X-ray scattering evidenced the emergence of a twinned BCC phase that, on heating, underwent a phase transition to an unusually anisotropic DDQC state. Surprisingly, we observe no evidence of this apparent epitaxy on heating or cooling through the equilibrium σ-BCC transition. We rationalize these results in terms of a shear-induced order-order transition and an apparent BCC-DDQC epitaxy favored by micelle translation-mediated ordering dynamics far below TOOT.

Porous Fibers Templated by Melt Blowing Cocontinuous Immiscible Polymer Blends.

We report a scalable melt blowing method for producing porous nonwoven fibers from model cocontinuous polystyrene/high-density polyethylene polymer blends. While conventional melt compounding of cocontinuous blends typically produces domain sizes ∼1-10 µm, melt blowing these blends into fibers reduces those dimensions up to 35-fold and generates an interpenetrating domain structure. Inclusion of ≤1 wt % of a block copolymer compatibilizer in these blends crucially enables access to smaller domain sizes in the fibers by minimizing thermodynamically-driven blend coarsening inherent to cocontinuous blends. Selective solvent extraction of the sacrificial polymer phase yielded a network of porous channels within the fibers. Fiber surfaces also exhibited pores that percolate into the fiber interior, signifying the continuous and interconnected nature of the final structure. Pore sizes as small as ∼100 nm were obtained, suggesting potential applications of these porous nonwovens that rely on their high surface areas, including various filtration modules.

Shear-Modulated Rates of Phase Transitions in Sphere-Forming Diblock Oligomer Lyotropic Liquid Crystals.

Hydration of the amphiphilic diblock oligomer C16H33(CH2CH2O)20OH (C16E20) leads to concentration-dependent formation of micellar body-centered cubic (BCC) and Frank-Kasper A15 lyotropic liquid crystals (LLCs). Quiescent thermal annealing of aqueous LLCs comprising 56-59 wt % C16E20 at 25 °C after quenching from high temperatures established their ability to form short-lived BCC phases, which transform into long-lived, transient Frank-Kasper σ phases en route to equilibrium A15 morphologies on a time scale of months. Here, the frequency and magnitude of applied oscillatory shear show the potential to either dynamically stabilize the metastable BCC phase at low frequencies or increase the rate of formation of the A15 to minutes at high frequencies. Time-resolved synchrotron small-angle X-ray scattering (TR-SAXS) provides in situ characterization of the structures during shear and thermal processing. This work shows that the LLC morphology and order-order phase transformation rates can be controlled by tuning the shear strain amplitude and frequency.

Protonation-Driven Aqueous Lyotropic Self-Assembly of Synthetic Six-Tail Lipidoids.

We report the aqueous lyotropic mesophase behaviors of protonated amine-based "lipidoids," a class of synthetic lipid-like molecules that mirrors essential structural features of the multitail bacterial amphiphile lipid A. Small-angle X-ray scattering (SAXS) studies demonstrate that the protonation of the tetra(amine) headgroups of six-tail lipidoids in aqueous HCl, HNO3, H2SO4, and H3PO4 solutions variably drives their self-assembly into lamellar (Lα) and inverse micellar (III) lyotropic liquid crystals (LLCs), depending on acid identity and concentration, amphiphile tail length, and temperature. Lipidoid assemblies formed in H2SO4(aq) exhibit rare inverse body-centered cubic (BCC) and inverse face-centered cubic (FCC) micellar morphologies, the latter of which unexpectedly coexists with zero mean curvature Lα phases. Complementary atomistic molecular dynamics (MD) simulations furnish detailed insights into this unusual self-assembly behavior. The unique aqueous lyotropic mesophase behaviors of ammonium lipidoids originate in their dichotomous ability to adopt both inverse conical and chain-extended molecular conformations depending on the number of counterions and their identity, which lead to coexisting supramolecular assemblies with remarkably different mean interfacial curvatures.

From Order to Disorder: Computational Design of Triblock Amphiphiles with 1 nm Domains.

Using molecular dynamics simulations and transferable force fields, we designed a series of symmetric triblock amphiphiles (or high-χ block oligomers) comprising incompatible sugar-based (A) and hydrocarbon (B) blocks that can self-assemble into ordered nanostructures with sub-1 nm domains and full domain pitches as small as 1.2 nm. Depending on the chain length and block sequence, the ordered morphologies include lamellae, perforated lamellae, and hexagonally perforated lamellae. The self-assembly of these amphiphiles bears some similarities, but also some differences, to those formed by symmetric triblock polymers. In lamellae formed by ABA amphiphiles, the fraction of B blocks "bridging" adjacent polar domains is nearly unity, much higher than that found for symmetric triblock polymers, and the bridging molecules adopt elongated conformations. In contrast, "looping" conformations are prevalent for A blocks of BAB amphiphiles. Above the order-disorder transition temperature, the disordered states are locally well-segregated yet the B blocks of ABA amphiphiles are significantly less stretched than in the lamellar phases. Analysis of both hydrogen-bonded and nonpolar clusters reveals the bicontinuous nature of these network phases. This simulation study furnishes detailed insights into structure-property relationships for mesophase formation on the 1 nm length scale that will aid further miniaturization for numerous applications.

Reduction of Punch-Sticking Propensity of Celecoxib by Spherical Crystallization via Polymer Assisted Quasi-Emulsion Solvent Diffusion.

Punch-sticking during tablet compression is a common problem for many active pharmaceutical ingredients (APIs), which renders tablet formulation development challenging. Herein, we demonstrate that the punch-sticking propensity of a highly sticky API, celecoxib (CEL), can be effectively reduced by spherical crystallization enabled by a polymer assisted quasi-emulsion solvent diffusion (QESD) process. Among three commonly used pharmaceutical polymers, poly(vinylpyrrolidone) (PVP), hydroxypropyl cellulose (HPC), and hydroxypropyl methylcellulose (HPMC), HPMC was the most effective in stabilizing the transient emulsion during QESD and retarding the coalescence of emulsion droplets and the initiation of CEL crystallization. These observations may arise from stronger intermolecular interactions between HPMC and CEL, consistent with solution 1H NMR analyses. SEM and X-ray photoelectron spectroscopy confirmed the presence of a thin layer of HPMC on the surfaces of spherical particles. Thus, the sticking propensity was significantly reduced because the HPMC coating prevents direct contact between CEL and the punch tip during tablet compression.

Consequences of Convex Nanopore Chemistry on Confined Water Dynamics.

A fundamental understanding of confined water is crucial for developing selective ion transport and water purification membranes, yet the roles of nanopore geometry and functionality on confined water dynamics remain unresolved. We report the synthesis of perdeuterated ionic alkylsulfonate amphiphiles and their water-induced self-assembly into lyotropic liquid crystal (LLC) mesophases with well-defined, convex, sulfonate-lined nanopores. Quasielastic neutron scattering (QENS) measurements demonstrate that the water self-diffusion coefficients within these sulfonate-lined convex nanopores depend on the hydration level and amphiphile counterion identity (H+, K+, NMe4+). The consistency of the observed counterion-dependent water dynamics trends with those of carboxylate LLCs is rationalized on the basis of similarities in the counterion spatial distributions in the water-filled channels, which we deduce from electron density maps derived from small-angle X-ray scattering (SAXS) analyses. These findings indicate that water diffusion is systematically faster in sulfonate-lined nanopores as compared to carboxylate-lined pores due to weaker water interactions with the softer and more hydrophobic-SO3- functionalities. These molecular-level insights into the relationships between convex pore wall chemical functionalities, hydrated counterions, and confined water diffusion may inform future development of new nanoporous media.

Aqueous Lyotropic Mesophase Behavior of Gemini Dicarboxylate Surfactants Swollen with n-Decane.

We report detailed small-angle X-ray scattering (SAXS) studies of the impact of variable n-decane loadings on the lyotropic liquid crystalline (LLC) phase behaviors of homologous bis(tetramethylammonium) gemini didecanoate surfactants TMA-7x, which derive from dimerizing decanoic acid through its α-carbon with hydrocarbyl linkers -(CH2)x- where x = 3, 4, 5, and 6. TMA-7x amphiphiles with x = 3 or 5 exhibit a strong propensity to form normal double gyroid (G) LLC network mesophases over wide surfactant hydration ranges, as compared to homologues with x = 4 or 6. On swelling aqueous TMA-7x LLC mesophases with up to 35 wt % n-decane, we demonstrate that odd-carbon linked surfactants (x = 3 or 5) form G and normal double diamond (D) phases over wide water concentration windows with T = 22-100 °C. Complementary studies of decane-swollen TMA-7x (x = 4 or 6) aqueous LLCs instead demonstrate significantly diminished network phase stability, in favor of hexagonally-packed cylinder phases and a zoo of complex quasispherical micelle packings, which include micellar C14 and C15 Laves phases (P63/mmc and Fd3(-)m symmetries, respectively) and high-symmetry hexagonally close packed (HCP) and body-centered cubic (BCC) arrangements. These rich phase behaviors are rationalized in terms of linker length parity-dependent surfactant conformations and the delicate free energy balance that guides the packing of these geometrically anisotropic amphiphiles by minimizing unfavorable water-hydrophobic contacts, maximizing ionic surfactant-headgroup counterion solvation with minimal local variations, and maximizing electrostatic cohesion within these supramolecular assemblies.

Emergence of a C15 Laves Phase in Diblock Polymer/Homopolymer Blends.

The observation of complex, Frank-Kasper (FK) particle packings in diblock polymer melts has until recently been limited to low molecular weight, conformationally asymmetric polymers. We report temperature-dependent small-angle X-ray scattering (SAXS) studies of blends of a sphere-forming poly(styrene-block-1,4-butadiene) (SB) diblock polymer (Mn = 33.3 kg/mol, D = Mw/Mn = 1.08, fB = 0.18) with two different poly(1,4-butadiene) (B) homopolymer additives. When the B additive Mn is the same as that of the diblock core-forming B segment, these blends remarkably form tetrahedrally close-packed FK σ and Laves C14 and C15 phases with increasing B content. However, binary blends in which the B additive Mn is 60% of that of the diblock B segment form only the canonical body-centered cubic (BCC) particle packing and hexagonally-packed cylinders (HEXc). The observed phase behavior is rationalized in terms of "wet" and "dry" brush blending, whereby higher B Mn drives stronger localization of the homopolymer in the particle cores while preserving the interfacial area per SB diblock chain. The consequent packing constraints in these blends destabilize the BCC packing, and FK phases emerge as optimal minimal surface solutions to filling space at constant density while maximizing local particle sphericity.

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

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 (HI) 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 → HI. However, rapidly quenching the resulting HI 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.

Order-to-Disorder Transitions in Lamellar Melt Self-Assembled Core-Shell Bottlebrush Polymers.

We report the synthesis and melt self-assembly behaviors of densely grafted, core-shell bottlebrush (csBB) polymers derived from covalently linking narrow dispersity, symmetric composition ABA-type triblock polymers through their chain midpoints. Derived from sequential ring-opening polymerizations of ε-decalactone and rac-lactide initiated from 5-norbornene-2-exo,3-exo-dimethanol, poly(lactide-block-ε-decalactone-block-lactide) macromonomers (Mn = 9.2-17.8 kg/mol; D = 1.19-1.25) were enchained by living ring-opening metathesis polymerization (ROMP) into csBBs with backbone degrees of polymerization Nbb = 8-43. Temperature-dependent small-angle X-ray scattering (SAXS) studies indicate that the critical triblock arm degree of polymerization (Narm) required for melt segregation decreases with increasing Nbb, leading to reductions in the accessible ordered lamellar microdomain (d) spacings. We derive a phenomenological relationship between the critical triblock arm segregation strength at the order-disorder transition (χNarm)ODT and Nbb to enable the future design of microphase separated core-shell bottlebrushes, which self-assemble at sub-10 nm length scales for nanolithography and nanotemplating applications.

Ion-Specific Confined Water Dynamics in Convex Nanopores of Gemini Surfactant Lyotropic Liquid Crystals.

The impact of pore geometry and functionality on the dynamics of water nanoconfined in porous media are the subject of some debate. We report the synthesis and small-angle X-ray scattering (SAXS) characterization of a series of perdeuterated gemini surfactant lyotropic liquid crystals (LLCs), in which convex, water-filled nanopores of well-defined dimensions are lined with carboxylate functionalities. Quasielastic neutron scattering (QENS) measurements of the translational water dynamics in these dicarboxylate LLC nanopores as functions of the surfactant hydration state and the charge compensating counterion (Na+, K+, NMe4+) reveal that the measured dynamics depend primarily on surfactant hydration, with an unexpected counterion dependence that varies with hydration number. We rationalize these trends in terms of a balance between counterion-water attractions and the nanopore volume excluded by the counterions. On the basis of electron density maps derived from SAXS analyses of these LLCs, we directly show that the volume excluded by the counterions depends on both their size and spatial distribution in the water-filled channels. The translational water dynamics in the convex pores of these LLCs are also slower than those reported in the concave pores of AOT reverse micelles, implying that water dynamics also depend on the nanopore curvature.

Micellar Mimicry of Intermetallic C14 and C15 Laves Phases by Aqueous Lyotropic Self-Assembly.

Concentration-dependent supramolecular self-assembly of amphiphilic molecules in water furnishes a variety of nanostructured lyotropic liquid crystals (LLCs), which typically display high symmetry bicontinuous network and discontinuous micellar morphologies. Aqueous dispersions of soft spherical micelles derived from small molecule amphiphile hydration typically pack into exemplary body-centered cubic and closest-packed LLCs. However, investigations of hydrated mixtures of the ionic surfactant tetramethylammonium decanoate loaded with 40 wt % n-decane (TMADec-40) revealed the formation of a high symmetry bicontinuous double diamond LLC, as well as cubic C15 and hexagonal C14 Laves LLC phases that mirror the MgCu2 and MgZn2 intermetallic structure types, respectively. Detailed small-angle X-ray scattering analyses demonstrate that the complex C15 and C14 LLCs exhibit large unit cells, in which 12 or more ∼3-4 nm diameter micelles of multiple discrete sizes arrange into tetrahedral close packing arrangements with exceptional long-range translational order. The symmetry breaking that drives self-assembly into these low-symmetry LLC phases is rationalized in terms of a frustrated balance between maximizing counterion-mediated micellar cohesion within the ensemble of oil-swollen particles, while simultaneously optimizing local spherical particle symmetry to minimize molecular-level variations in surfactant solvation.

Counterion-Regulated Dynamics of Water Confined in Lyotropic Liquid Crystalline Morphologies.

The dynamics of confined water is of fundamental and long-standing interest. In technologically important forms of confinement, such as proton-exchange membranes, electrostatic interactions with the confining matrix and counterions play significant roles on the properties of water. There has been recent interest on the dynamics of water confined to the lyotropic liquid crystalline (LLC) morphologies of Gemini dicarboxylate surfactants. These systems are exciting because the nature of confinement, for example, size and curvature of channels and surface functionality is dictated by the chemistry of the self-assembling surfactant molecules. Quasielastic neutron scattering experiments have shown an interesting dependence of the water self-diffusion constant, Dα, on the identity (denoted α) of the counterion: at high hydration, the magnitude of the water self-diffusion constant is in the order DTMA < DNa < DK, where TMA, Na, and K refer to tetramethyl ammonium, sodium, and potassium counterions, respectively. This sequence is similar to what is seen in bulk electrolyte solutions. At low hydrations, however, the order of water self-diffusion is different, that is, DNa < DTMA < DK. In this work, we present molecular dynamics simulations for the dynamics of water in the LLC phases of dicarboxylate Gemini surfactants. The simulations reproduce the trends seen in experiments. From an analysis of the trajectories, we hypothesize that two competing factors play a role: the volume accessible to the water molecules and the correlations between the water and the counterion. The excluded volume effect is the largest with TMA+, and the electrostatic correlation is the strongest with Na+. The observed trend is a result of which of these two effects is dominant at a given water to surfactant ratio.

Counterion-Dependent Access to Low-Symmetry Lyotropic Sphere Packings of Ionic Surfactant Micelles.

The water-driven self-assembly of homologous dianionic surfactants into lyotropic liquid crystals (LLCs) is investigated, with a focus on understanding how surfactant headgroup and counterion identities guide supramolecular spherical mesophase selection. Using temperature-dependent small-angle X-ray scattering (SAXS), we demonstrate that 2-alkylmalonate surfactants (CnMal-M2) with n = 8 (octyl) or 10 (decyl) and M = K+, Cs+, or (CH3)4N+ form both simple and complex micelle packings. Observed spherical morphologies include body-centered cubic (BCC), hexagonally closest-packed (HCP), and tetrahedrally closest-packed Frank-Kasper (FK) A15 and σ phases (Pm3(-)n and P42/mnm symmetries, respectively). Previously observed in only one other minimally hydrated surfactant, the σ phase is a rare LLC morphology comprising a low-symmetry unit cell containing 30 sub-2-nm quasispherical micelles, each of which belongs to one of five symmetry-equivalent classes with discrete aggregation numbers. Temperature versus water concentration phase maps for CnMal-M2 LLCs reveal that σ-phase formation depends sensitively on the size and polarizability of the surfactant counterion and the length of the surfactant alkyl tail. These observations are rationalized in terms of a delicate interplay between global packing symmetry and local particle symmetry, and the extent to which counterion-headgroup correlations enforce the latter structures in these LLC phases.