Low-symmetry sphere packings of simple surfactant induced by ionic sphericity

Sung A Kima, Kyeong-Jun Jeongb, Arun Yethirajb, and Mahesh K. Mahanthappaa,1

aDepartment of Chemical Engineering & Materials Science, University of Minnesota, Minneapolis, MN 55455; and bDepartment of Chemistry and Theoretical Chemistry Institute, University of Wisconsin–Madison, Madison, WI 53706

Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved March 9, 2017 (received for review January 29, 2017) Supramolecular self-assembly enables access to designer soft mate- and lipidic LLCs were identified over 30 y ago, yet the principles rials that typically exhibit high-symmetry packing arrangements, governing their formation remain poorly understood (17). More which optimize the interactions between their mesoscopic constitu- recently, FK A15 and σ phases were documented in thermotropic ents over multiple length scales. We report the discovery of an ionic LCs of wedge-shaped dendrons (18–20), linear diblock and mul- small molecule surfactant that undergoes water-induced self- tiblock polymers (21, 22), and giant shape amphiphiles (2, 23). assembly into spherical micelles, which pack into a previously These studies culminated in the discovery of soft, dodecagonal unknown, low-symmetry lyotropic liquid crystalline Frank–Kasper σ QCs (23–26), for which the A15 and σ phases are 3D periodic phase. Small-angle X-ray scattering studies reveal that this complex approximants. phase is characterized by a gigantic tetragonal unit cell, in which Many of the previously reported soft matter FK phases optimize 30 sub-2-nm quasispherical micelles of five discrete sizes are arranged the van der Waals packing of hairy, uncharged particles that fill into a tetrahedral close packing, with exceptional translational order space at constant density, while minimizing unfavorable interfacial over length scales exceeding 100 nm. Varying the relative concentra- interactions between the particle cores and coronae (2, 17, 19, 21). tions of water and surfactant in these lyotropic phases also triggers In this paper, we describe the spontaneous formation of a new, – formation of the related Frank Kasper A15 sphere packing as well as direct LLC FK σ phase by simple ionic surfactant micelles in water. a common body-centered cubic structure. Molecular dynamics simu- Complementary molecular dynamics (MD) simulations reveal a lations reveal that the symmetry breaking that drives the formation σ previously unrecognized mechanism for forming low-symmetry, of the andA15phasesarisesfromminimization of local deviations periodic materials from charged self-assembled particles. in surfactant headgroup and counterion solvation to maintain a nearly spherical counterion atmosphere around each , while Results and Discussion maximizing counterion-mediated electrostatic cohesion among the Synchrotron small-angle X-ray scattering (SAXS) was used to in- ensemble of charged particles. vestigate the aqueous LLC phase diagram of bis(tetramethy- lammonium) decylphosphonate (DPA-TMA )between25–100 °C, self-assembly | liquid crystals | surfactants | Frank–Kasper phases | lyotropic 2 with water contents w = (moles H O)/(moles DPA-TMA ) = phase 0 2 2 0–44 (Fig. 1A). Aqueous LLCs were produced by thoroughly mixing measured amounts of DPA-TMA2 with ultrapure water (Materials olecular self-assembly provides a facile means of con- and Methods and SI Appendix). Samples with w0 ≥ 44 are freely Mstructing a plethora of multifunctional soft materials, with flowing fluids, indicative of disordered micellar solutions. When mesoscopic structures that dictate their tailored properties and w0 = 31–42, we observe SAXS peaks at q/q* = √2, √4, √6, and performance applications. Driven by noncovalent interactions between constituents, block polymers (1), giant shape amphiphiles Significance (2), thermotropic liquid crystals (LCs) (3), lyotropic liquid crystals (LLCs) (4), and colloids (5) exemplify soft matter systems that “ ” spontaneously form periodic 1D lamellar phases, 2D columnar Surfactants ( soaps ) spontaneously self-assemble into spher- structures, and 3D packings of spherical particles. Columnar and ical micelles in water, which pack into ordered crystalline spherical phases are useful as templates for mesoporous hetero- states. Such soft particles have long been assumed to adopt geneous catalysts (6) and as microscale photonic bandgap mate- the same closest-packed configurations observed with hard rials (7). Manipulating supramolecular self-assembly to achieve spheres (e.g., billiard balls). Here, we show that surfactant micelles also form complex, tetrahedrally closest-packed Frank– specific materials morphologies and functions requires a funda- Kasper (FK) phases. Surprisingly, the low-symmetry unit cells of mental understanding of the interplay between the structure and these structures comprise multiple particle types with discrete symmetry of the constituents and their multibody interactions. size distributions. We demonstrate that these unexpected Although the packing of spherical objects (e.g., oranges and structures arise from simultaneous optimization of interparticle billiard balls) seems intuitively simple, point particles form a diz- electrostatic interactions and the spherical symmetry of the zying array of periodic crystals, quasicrystals (QCs), and structurally charged ion clouds around each micelle. This discovery bridges disordered glasses. Metallic elements typically form high-symmetry previous reports of FK phases in neutral soft materials such as body-centered cubic (BCC), hexagonally closest-packed, and face- block polymers, dendrimers, and giant shape amphiphiles and centered cubic (FCC) structures, due to the isotropy of metallic in metal alloys. cohesion mediated by itinerant electrons (8). A few pure elements (e.g., Mn and U) form low-symmetry crystals with large and Author contributions: S.K., K.-J.J., A.Y., and M.K.M. designed research; S.K., K.-J.J., and complex unit cells that maximize metallic cohesion against local M.K.M. performed research; S.K., K.-J.J., A.Y., and M.K.M. analyzed data; and S.K., K.-J.J., constraints, such as maximization of Fermi surface sphericity (9). A.Y., and M.K.M. wrote the paper. Sphere-forming soft materials tend to prefer different packing The authors declare no conflict of interest. symmetries from those of metallic solids (10). Although squishy This article is a PNAS Direct Submission. spheres do form BCC and FCC crystals, they also form tetrahe- Freely available online through the PNAS open access option. drally closest-packed Frank–Kasper (FK) phases that contain 1To whom correspondence should be addressed. Email: [email protected]. – combinations of 12-, 14-, 15-, and 16-coordinate lattice sites (11 This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 14). The first FK A15 (15) and C15 (16) phases in ionic surfactant 1073/pnas.1701608114/-/DCSupplemental.

4072–4077 | PNAS | April 18, 2017 | vol. 114 | no. 16 www.pnas.org/cgi/doi/10.1073/pnas.1701608114 Downloaded by guest on September 28, 2021 reversibly melt into disordered micellar solutions at elevated A temperatures. Decreasing the surfactant hydration to w0 = 20–31 yields an LLC that exhibits at least 50 instrument resolution-limited SAXS peaks (Fig. 1B and SI Appendix, Fig. S1), inconsistent with any known lyotropic phase. Crystallographic analyses of the w0 = 24.0 LLC reveal a tetragonal unit cell with P42/mnm symmetry and lattice parameters a = 13.36 nm and c = 7.02 nm (SI Ap- pendix, Table S1). The numerous sharp SAXS reflections in- dicate exceptional translational ordering of sub-2-nm-diameter micelles in a water matrix on length scales ≥ 100 nm. The re- semblance between this remarkable diffraction pattern and those of thermotropic LC and block polymer FK σ phases (19, 21), coupled with the lattice symmetry and characteristic unit cell parameter ratio c/a = 0.526, strongly imply the formation of the first LLC σ phase. Le Bail SAXS data refinement combined with charge-flipping algorithms (27, 28) enabled electron density map reconstruction forthisLLCσ phase (Fig. 2 A–C). The water-filled unit cell contains 30 quasispherical micelles arranged into alternating sparsely and densely populated layers, consistent with other σ phase structures (19, 21). The micelles apparently have different volumes and exhibit soft facets, with the facets of neighboring micelles facing one another (Figs. 2 B and C and 3A). In contrast to previously reported soft matter σ phases wherein the particles make van der Waals contacts, the ionic micelles in this LLC σ phase sit in a water matrix and make no apparent physical contacts. LLCs formed at w0 = 10–18 typically display at least 18 SAXS peaks at q/q* = √2, √4, √5, √6, √8,andsoon(Fig.1B), which B conform to cubic Pm3(–)n symmetry with unit cell parameters a ∼ 6.95 nm. The electron density reconstruction for this phase in Fig. 2D is consistent with known soft matter A15 phases (15, 18). The corner and center micelles are somewhat facetted in the 90% isosurface plots, and the pairs of larger particles in

each unit cell face are severely distorted despite their spatial CHEMISTRY separation by water (Fig. 2 E and F). At the lowest hydrations studied (w0 = 6), we observed a hexagonally packed cylinders morphology (SI Appendix,Fig.S2). The σ and A15 LLC electron density maps reveal that they comprise squashed micelles with different volumes, instead of the uniform spherical particles intuitively expected for a simple sur- factant/water mixture. These volume differences imply that each class of symmetry-equivalent micelles in each FK structure con- tains a different and specific average number of surfactant chains, instead of every micelle having the same average number of chains as expected based on configurational entropy maximization. We conducted MD simulations to understand these surprising results. The decyl chain of DPA-TMA2 was coarse-grained in a GROMOS45a3 (29) united atom force field as a chain of 10 beads with a phosphonate headgroup (see SI Appendix for MD simulation details). Attempts to self-assemble the σ phase de novo in silico by combining SPC/E water (30) with the surfactant failed. We instead seeded a 13.34 × 13.34 × 7.01-nm3 tetragonal simulation cell with 30 identical micelles, each containing 32 Fig. 1. Aqueous lyotropic liquid crystalline phase behavior for DPA-TMA2. surfactant molecules located at the expected positions for the σ phase, = (A) Temperature versus surfactant hydration number w0 (moles of H2O)/ along with 13 unaggregated surfactant molecules and 23,352 ex- (moles DPA-TMA2) phase diagram, illustrating the composition-dependent w = σ plicit SPC/E waters corresponding to 0 24. The ensemble free formation of normal cylindrical (HI), and normal micellar A15, , and BCC or- energy was minimized at 298 K, by relaxing the water molecule dered phases, and disordered micelle solutions based on synchrotron SAXS analyses. Narrow windows of two-phase coexistence typically occur in between configurations and by allowing surfactant chain exchange be- the pure phases (e.g., H /A15 coexistence at w = 8). (B) Synchrotron SAXS tween the micelles. I 0 σ powder patterns for DPA-TMA2 LLCs corresponding to the A15 (red), σ (blue), After 500 ns, the electron density map for the MD simulated and BCC (purple) phases labeled with the Miller indices for each reflection; see phase remarkably resembled the experimental one (Fig. 3 and SI SI Appendix,Fig.S1for complete indexing of the FK σ phase SAXS pattern with Appendix, Fig. S3). Surfactant molecule exchange between the more than 50 peaks. a.u., arbitrary units. particles yielded five statistically different micelle populations with aggregation numbers Nagg = 29.8 ± 0.3 and 30.7 ± 0.1, 32.3 ± 0.2 and 33.1 ± 0.1, and 34.6 ± 0.6 for the 12-, 14-, and 15- sometimes √8, consistent with ionic micelles packed into BCC unit coordinate lattice sites, respectively (SI Appendix, Table S2). cells with edge lengths a ∼ 4.4 nm (Fig. 1B). These BCC LLCs After total free energy minimization, surfactant chains continue

Kim et al. PNAS | April 18, 2017 | vol. 114 | no. 16 | 4073 Downloaded by guest on September 28, 2021 A σ phase BC

z y y y x x x top view z = 1/2 D A15 phase E F

z y y y x x x top view z = 1/2

Fig. 2. Electron density reconstructions (90% isosurfaces) derived from synchrotron SAXS powder patterns for the tetragonal σ and cubic A15 FK phases at

25 °C, with w0 = 24.0 and 12.1, respectively. For the σ phase, images include (A) a 3D rendering of the unit cell, (B) a view through the top of the cell, and (C)a slice through the z = 1/2 plane. Colors indicate the five different symmetry-equivalent micelle types that occupy the 2b (red) and 8i (blue), 8i′ (green) and 8j (purple), and 4f (gold) Wyckoff positions with coordination numbers Z = 12, 14, and 15, respectively. Images of the A15 phase depict (D) the cubic unit cell, (E) a view through the top of the cell, and (F) a slice through the z = 1/2 plane. Colors indicate the two different symmetry-equivalent micelle types that occupy the 2a (red) and 6c (blue) Wyckoff positions with respective coordination numbers Z = 12 and 14.

to exchange on timescales less than 30 ns with equal numbers of quasispherical micelles with different aggregation numbers and surfactant chain acceptance and expulsion events at each lattice volumes that fill the various σ lattice sites. Micelle aggregation site (SI Appendix,TableS2), consistent with a dynamic equi- numbers derived from our MD simulations quantitatively agree with librium. Our simulation results concur with experiments by Lee the geometrically calculated volume variations of the five different σ et al. (31) on a diblock polymer FK σ phase, which demon- phase Wigner–Seitz cells, which deviate by 91–106.5% around the strated that interparticle chain exchange enables formation of average volume.

ABC

y

x z = 0, 1 (kg/m (kg/m 800 900 1000 1100 1200 120 160 200 240 3 3 ) ) Experimental electron MD total mass density TMA mass density density

Fig. 3. Comparison of an experimental electron density map and MD simulation results for the LLC FK σ phase. (A) A slice through the z = 0, 1 planes of the tetragonal unit cell of the experimentally derived electron density map (90% isosurface) wherein the micelles exhibit soft facets. (B) Slices of the total mass density map from MD simulations through the same z = 0, 1 planes of the σ phase, and (C) the simulated TMA counterion density map that illustrates counterion localization (yellow) between the micelles to maximize interparticle electrostatic cohesion.

4074 | www.pnas.org/cgi/doi/10.1073/pnas.1701608114 Kim et al. Downloaded by guest on September 28, 2021 A B C enhances charge screening between the surfactant headgroups within each aggregate, while also driving hydrophobic surfactant tail stretching to minimize unfavorable ion/alkane interactions (Fig. 5). Thus, the surfactant molecules pack more tightly into larger micelles with diminished interfacial curvatures. Main- taining the BCC structure at smaller w0, with increased micelle size and smaller aqueous domain volume, severely distorts the y counterion atmospheres and particle cores away from their preferred spherical symmetries. Thus, a phase transition occurs x z = 0, 1 (kg/m (kg/m 800 900 1000 1100 1200 200 250 300 350 3 in which the nearly identical self-assembled, micellar particles 3 ) ) Experimental electron reconfigure. MD total mass density TMA mass density density The sphericity of a polyhedron may be quantified by the iso- perimetric quotient IQ = 36πV2/S3,whereV and S are its re- Fig. 4. Comparison of an experimental electron density map and MD sim- IQ ulation results for the FK A15 phase. (A) Two-dimensional top view of the z = spective volume and surface area (10). Objects with higher 0, 1 planes of the experimentally derived A15 phase electron density map values are more spherical by this definition, with IQ = 1 being the (90% isosurface), (B) slices of the total mass density map for the A15 phase limiting case of a perfect sphere. Lee et al. (31) calculated the from MD simulations, and (C) the simulated TMA counterion density map, showing counterion localization (yellow) between the deformed platelet micelles in each unit cell face.

An analogous MD simulation of the A15 LLC conducted at w0 = 12.1 at 353 K, in which a cubic cell with a = 6.79 nm was seeded with eight identical micelles each containing 40 surfac- BCC tants, equilibrates in 400 ns. This simulated electron density map w = 36 is again consonant with experiments, including the formation of N 23 0 pairs of platelet micelles in the unit cell faces (Fig. 4 and SI agg Appendix, Fig. S4). Intermicellar surfactant chain exchange leads to statistically different micelle aggregation numbers of Nagg = 36.9 ± 0.2 and 41.0 ± 0.1 (SI Appendix, Table S2). Again, dy- namic surfactant chain exchange continues after MD simulation equilibration with an average time between chain entry or ex- pulsion events at each lattice site of less than 20 ns (SI Appendix, Table S2). These Nagg values quantitatively agree with the ∼15–

20% variation in aggregation numbers estimated from trans- CHEMISTRY mission EM analyses of the giant shape amphiphile A15 phase recently reported by Cheng and coworkers (2). MD simulations of the σ and A15 phases reveal that the micelles σ interact through their tetramethylammonium (TMA) surfactant FK counterion atmospheres. The soft and relatively hydrophobic w0 = 24 TMA counterions localize along the center-to-center vectors Nagg 32 connecting neighboring particles, with higher densities near the midpoints between micelle surfaces as shown in Figs. 3C and 4C. In other words, the counterions outline the boundaries of the Wigner–Seitz cells for each lattice site. Counterion localization along these boundaries drives faceting of the soft micelles, which deform to minimize counterion–headgroup distance variations. Reminiscent of the electron probability density in covalent bonds, this counterion arrangement optimally screens Coulombic repul- sions between the negatively charged micelle surfaces while max- imizing micellar cohesion in the ordered liquid crystalline state. The free energy balance underlying the LLC phase progres- sion of disordered micelles → BCC → σ → A15 with decreasing FK A15 w 0 depends on packing charged particles in a manner that w0 = 12 maximizes both interparticle cohesion and ionic sphericity. Ionic Nagg 42 sphericity refers to the thermodynamic preference for a spherical counterion cloud around each micelle, which maximizes surfac- tant counterion–headgroup solvation and minimizes energeti- cally costly molecular-level variations. The surfactants form high- curvature, isolated spherical micelles with spherical counterion atmospheres at high w0, due to repulsions between the negatively charged headgroups in each aggregate. Removing water from Fig. 5. As the water concentration (w0) decreases in the ionic surfactant this micellar solution initially triggers a BCC packing, because LLCs, the increased ion concentration leads to greater chemical incompati- bility between the ion-rich aqueous domains and the hydrophobic regions, the configurational entropy loss upon crystallization is minimized as well as enhanced charge screening between the negatively charged sur- in this periodic lattice of symmetry-equivalent micelles (32). w factant headgroups. Thus, the surfactant tails stretch to drive formation of Decreasing 0 in the BCC LLC reduces the average distance larger and more deformable aggregates with lower interfacial curvatures, between micelles while increasing the solvated ion concentration which pack into low-symmetry FK σ and A15 phases that maximize the ionic in the aqueous domain. This increased ion concentration sphericities of the counterion atmospheres around the micelles.

Kim et al. PNAS | April 18, 2017 | vol. 114 | no. 16 | 4075 Downloaded by guest on September 28, 2021 number-averaged isoperimetric quotients for the polyhedral to dodecagonal QCs, our observations suggest design principles Wigner–Seitz cells of various low-symmetry sphere packings, and for driving colloidal materials to form soft QCs at various length they found that they decrease along the phase progression σ > scales. Reports of surfactant-templated mesoporous silicate FK > w A15 BCC. Thus, we reason that reducing 0 beyond a critical phases and 12- and 18-fold block polymer lyotropic QCs reflect value in the BCC LLC forces a phase transition to the σ phase that initial discoveries in this exciting direction (26, 33, 34). is facilitated by surfactant chain exchange between micelles to The formation of complex FK phases in ionic surfactant LLCs maximize both ionic sphericity and the electrostatic cohesion be- through the exchange of mass and charge is reminiscent of charge w σ → tween micelles (Fig. 5). Further reduction in 0 triggers a exchange in binary metallic alloys, comprising two elements with A15 phase transition to maintain the highest possible ionic sphe- different atomic masses. Thus, this discovery bridges previous re- ricity at even lower hydrations. From the MD simulations, the IQ ports of FK phases neutral soft materials (2, 18, 19, 21–23, 31, 35) values for each of the micelles in σ and A15 phases were calcu- and metal alloys. However, the ubiquity of icosahedral QCs in lated by defining the micelle surfaces using the locations of the Materials and multicomponent metal alloys (36) starkly contrasts the observation surfactant headgroup phosphorous atoms (see – Methods and SI Appendix,TableS3for calculation details). The of only dodecagonal QCs in soft materials to date (23 26). Ico- calculated IQ values follow the trend anticipated by Lee et al. (31) sahedral QCs, which are quasiperiodic in 3D, typically form in based on the geometries of the Wigner–Seitz cells of the σ and ternary alloys wherein the metal sites exhibit decoupled variations A15 lattices. Because the increased ion concentration in the in both particle mass (atomic number) and charge distribution. aqueous domains of the A15 phase better screens repulsions be- However, dodecagonal QCs are layered structures that are qua- tween the charged surfactant headgroups within each micelle, siperiodic in each 2D layer. Ionic LLCs, in which the σ and these larger particles are more able to deform into the observed A15 approximant phases form by coupled exchange of mass and platelet micellar aggregates. charge to maximize cohesive energy and ionic sphericity, possibly Ionic sphericity-induced FK σ phase LLC formation by coupled shed light onto this dichotomy in quasicrystallinity: icosahedral mass and charge exchange through surfactant redistribution ex- ordering may require the independent interparticle exchange of tends the provocative analogy described by Lee et al. (31) between both mass and charge to obtain a distribution of sphere sizes that low-symmetry phases of metal alloys and of soft spheres. Metallic adopt packings with 3D quasiperiodic order. FK σ phase alloys (e.g., Fe–Cr) maximize both metallic bonding and Fermi surface sphericity in reciprocal space, through charge Materials and Methods

exchange between atomic sites. Lee et al. (31) showed that block For details see SI Appendix. DPA-TMA2 was synthesized by stoichiometric polymer σ phases similarly optimize both short-range van der deprotonation of decylphosphonic acid with TMA hydroxide in methanol to Waals interactions to fill space at constant density and real space produce an analytically pure surfactant sample, which was characterized by particle sphericity, enabled by interparticle block polymer chain 1H, 13C, and 31P NMR spectroscopy and elemental analysis (C, H, N, and P). (mass) exchange. The phenomenological similarity between LLCs, Lyotropic LC samples were prepared by combining measured amounts of > Ω block polymers, and thermotropic LCs extends to the fact that the ultrapure water ( 18 M resistance) with DPA-TMA2 in 1-dram vials, fol- × ionic LLC phase sequence with decreasing w0 parallels that of lowed by three cycles of iterative high-speed centrifugation (4,996 g for nonionic soft systems with decreasing temperature T (19, 31). This 10 min) and hand mixing; all samples were allowed to rest at 22 °C for at least 24 h before X-ray analysis. Temperature-dependent SAXS measure- apparent equivalence between w0 in LLCs and T in thermotropic LCs and block polymers stems from the increased chemical in- ments were conducted on samples enclosed in alodined aluminum pans at the 12-ID-B and DND-CAT 5-ID-D of the Advanced Photon Source. Samples compatibility between the particle cores and coronae as w0 (or T) decreases. were heated to the desired temperature using a thermostated sample stage with a temperature accuracy of ± 0.1 °C and allowed to equilibrate for 5 min Conclusions before data acquisition. SAXS data analysis details, including the electron density map reconstruction methodology, are described in detail in Ionic sphericity represents a previously unrecognized mechanism SI Appendix. by which soft, charged particles assemble into both simple and σ complex 3D packings. The observation of ionic surfactant FK ACKNOWLEDGMENTS. We thank Frank S. Bates, Chris Leighton, and Jesse and A15 LLC phases anticipates the provocative possibility that McDaniel for helpful discussions. This work was supported by US Depart- ionic sphericity may induce similar self-assembly phenomena in ment of Energy, Basic Energy Sciences Contract DE-SC0010328. Synchrotron larger charged colloids at macroscopic length scales. Compared SAXS data were acquired at the X-ray Sciences Division 12-ID-B and the with the short-range forces that underlie FK phase formation in DuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) 5-ID-D nonionic LCs and block polymers, ionic sphericity combined with beamlines of the Advanced Photon Source (APS). DND-CAT is supported by E. I. DuPont de Nemours & Co., The Dow Chemical Company, and Northwest- long-range electrostatic forces between charged colloids may yield ern University. The APS is an Office of Science User Facility operated for the macroscale crystals with unusual photonic and phononic proper- US Department of Energy (DOE) by Argonne National Laboratory and sup- ties. Because these FK phases are also 3D periodic approximants ported by DOE Contract DE-AC02-06CH11357.

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