Low-symmetry sphere packings of simple surfactantmicelles induced by ionic sphericitySung 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 andTheoretical 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-rials that typically exhibit high-symmetry packing arrangements,which optimize the interactions between their mesoscopic constitu-ents over multiple length scales. We report the discovery of an ionicsmall molecule surfactant that undergoes water-induced self-assembly into spherical micelles, which pack into a previouslyunknown, low-symmetry lyotropic liquid crystalline Frank–Kasper σphase. Small-angle X-ray scattering studies reveal that this complexphase is characterized by a gigantic tetragonal unit cell, in which30 sub-2-nm quasispherical micelles of five discrete sizes are arrangedinto a tetrahedral close packing, with exceptional translational orderover length scales exceeding 100 nm. Varying the relative concentra-tions of water and surfactant in these lyotropic phases also triggersformation of the related Frank–Kasper A15 sphere packing as well asa common body-centered cubic structure. Molecular dynamics simu-lations reveal that the symmetry breaking that drives the formationof the σ and A15 phases arises from minimization of local deviationsin surfactant headgroup and counterion solvation to maintain anearly spherical counterion atmosphere around each micelle, whilemaximizing counterion-mediated electrostatic cohesion among theensemble of charged particles.

self-assembly | liquid crystals | surfactants | Frank–Kasper phases | lyotropicphase

Molecular self-assembly provides a facile means of con-structing a plethora of multifunctional soft materials, with

mesoscopic structures that dictate their tailored properties andperformance applications. Driven by noncovalent interactionsbetween constituents, block polymers (1), giant shape amphiphiles(2), thermotropic liquid crystals (LCs) (3), lyotropic liquid crystals(LLCs) (4), and colloids (5) exemplify soft matter systems thatspontaneously form periodic 1D lamellar phases, 2D columnarstructures, and 3D packings of spherical particles. Columnar andspherical phases are useful as templates for mesoporous hetero-geneous catalysts (6) and as microscale photonic bandgap mate-rials (7). Manipulating supramolecular self-assembly to achievespecific materials morphologies and functions requires a funda-mental understanding of the interplay between the structure andsymmetry of the constituents and their multibody interactions.Although the packing of spherical objects (e.g., oranges and

billiard balls) seems intuitively simple, point particles form a diz-zying array of periodic crystals, quasicrystals (QCs), and structurallydisordered glasses. Metallic elements typically form high-symmetrybody-centered cubic (BCC), hexagonally closest-packed, and face-centered cubic (FCC) structures, due to the isotropy of metalliccohesion mediated by itinerant electrons (8). A few pure elements(e.g., Mn and U) form low-symmetry crystals with large andcomplex unit cells that maximize metallic cohesion against localconstraints, such as maximization of Fermi surface sphericity (9).Sphere-forming soft materials tend to prefer different packing

symmetries from those of metallic solids (10). Although squishyspheres do form BCC and FCC crystals, they also form tetrahe-drally closest-packed Frank–Kasper (FK) phases that containcombinations of 12-, 14-, 15-, and 16-coordinate lattice sites (11–14). The first FK A15 (15) and C15 (16) phases in ionic surfactant

and lipidic LLCs were identified over 30 y ago, yet the principlesgoverning their formation remain poorly understood (17). Morerecently, FK A15 and σ phases were documented in thermotropicLCs of wedge-shaped dendrons (18–20), linear diblock and mul-tiblock polymers (21, 22), and giant shape amphiphiles (2, 23).These studies culminated in the discovery of soft, dodecagonalQCs (23–26), for which the A15 and σ phases are 3D periodicapproximants.Many of the previously reported soft matter FK phases optimize

the van der Waals packing of hairy, uncharged particles that fillspace at constant density, while minimizing unfavorable interfacialinteractions between the particle cores and coronae (2, 17, 19, 21).In this paper, we describe the spontaneous formation of a new,direct LLC FK σ phase by simple ionic surfactant micelles in water.Complementary molecular dynamics (MD) simulations reveal apreviously unrecognized mechanism for forming low-symmetry,periodic materials from charged self-assembled particles.

Results and DiscussionSynchrotron small-angle X-ray scattering (SAXS) was used to in-vestigate the aqueous LLC phase diagram of bis(tetramethy-lammonium) decylphosphonate (DPA-TMA2) between 25–100 °C,with water contents w0 = (moles H2O)/(moles DPA-TMA2) =0–44 (Fig. 1A). Aqueous LLCs were produced by thoroughly mixingmeasured amounts of DPA-TMA2 with ultrapure water (Materialsand Methods and SI Appendix). Samples with w0 ≥ 44 are freelyflowing fluids, indicative of disordered micellar solutions. Whenw0 = 31–42, we observe SAXS peaks at q/q* = √2, √4, √6, and

Significance

Surfactants (“soaps”) spontaneously self-assemble into spher-ical micelles in water, which pack into ordered crystallinestates. Such soft particles have long been assumed to adoptthe same closest-packed configurations observed with hardspheres (e.g., billiard balls). Here, we show that surfactantmicelles also form complex, tetrahedrally closest-packed Frank–Kasper (FK) phases. Surprisingly, the low-symmetry unit cells ofthese structures comprise multiple particle types with discretesize distributions. We demonstrate that these unexpectedstructures arise from simultaneous optimization of interparticleelectrostatic interactions and the spherical symmetry of thecharged ion clouds around each micelle. This discovery bridgesprevious reports of FK phases in neutral soft materials such asblock polymers, dendrimers, and giant shape amphiphiles andin metal alloys.

Author contributions: S.K., K.-J.J., A.Y., and M.K.M. designed research; S.K., K.-J.J., andM.K.M. performed research; S.K., K.-J.J., A.Y., and M.K.M. analyzed data; and S.K., K.-J.J.,A.Y., and M.K.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: maheshkm@UMN.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1701608114/-/DCSupplemental.

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sometimes √8, consistent with ionic micelles packed into BCC unitcells with edge lengths a ∼ 4.4 nm (Fig. 1B). These BCC LLCs

reversibly melt into disordered micellar solutions at elevatedtemperatures.Decreasing the surfactant hydration to w0 = 20–31 yields an

LLC that exhibits at least 50 instrument resolution-limited SAXSpeaks (Fig. 1B and SI Appendix, Fig. S1), inconsistent with anyknown lyotropic phase. Crystallographic analyses of the w0 =24.0 LLC reveal a tetragonal unit cell with P42/mnm symmetryand 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-diametermicelles in a water matrix on length scales ≥ 100 nm. The re-semblance between this remarkable diffraction pattern and thoseof thermotropic LC and block polymer FK σ phases (19, 21),coupled with the lattice symmetry and characteristic unit cellparameter ratio c/a = 0.526, strongly imply the formation of thefirst LLC σ phase.Le Bail SAXS data refinement combined with charge-flipping

algorithms (27, 28) enabled electron density map reconstructionfor this LLC σ phase (Fig. 2 A–C). The water-filled unit cellcontains 30 quasispherical micelles arranged into alternatingsparsely and densely populated layers, consistent with other σphase structures (19, 21). The micelles apparently have differentvolumes and exhibit soft facets, with the facets of neighboringmicelles facing one another (Figs. 2 B and C and 3A). In contrastto previously reported soft matter σ phases wherein the particlesmake van der Waals contacts, the ionic micelles in this LLC σphase 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, and so on (Fig. 1B), whichconform to cubic Pm3(–)n symmetry with unit cell parameters a ∼6.95 nm. The electron density reconstruction for this phase inFig. 2D is consistent with known soft matter A15 phases (15,18). The corner and center micelles are somewhat facetted inthe 90% isosurface plots, and the pairs of larger particles ineach unit cell face are severely distorted despite their spatialseparation by water (Fig. 2 E and F). At the lowest hydrationsstudied (w0 = 6), we observed a hexagonally packed cylindersmorphology (SI Appendix, Fig. S2).The σ and A15 LLC electron density maps reveal that they

comprise squashed micelles with different volumes, instead of theuniform spherical particles intuitively expected for a simple sur-factant/water mixture. These volume differences imply that eachclass 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 chainsas expected based on configurational entropy maximization.We conducted MD simulations to understand these surprisingresults. The decyl chain of DPA-TMA2 was coarse-grained in aGROMOS45a3 (29) united atom force field as a chain of 10beads with a phosphonate headgroup (see SI Appendix for MDsimulation details). Attempts to self-assemble the σ phase denovo in silico by combining SPC/E water (30) with the surfactantfailed. We instead seeded a 13.34 × 13.34 × 7.01-nm3 tetragonalsimulation cell with 30 identical micelles, each containing 32surfactant molecules located at the expected positions for the σ phase,along with 13 unaggregated surfactant molecules and 23,352 ex-plicit SPC/E waters corresponding to w0 = 24. The ensemble freeenergy was minimized at 298 K, by relaxing the water moleculeconfigurations and by allowing surfactant chain exchange be-tween the micelles.After 500 ns, the electron density map for the MD simulated σ

phase remarkably resembled the experimental one (Fig. 3 and SIAppendix, Fig. S3). Surfactant molecule exchange between theparticles yielded five statistically different micelle populationswith 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-coordinate lattice sites, respectively (SI Appendix, Table S2).After total free energy minimization, surfactant chains continue

A

B

Fig. 1. Aqueous lyotropic liquid crystalline phase behavior for DPA-TMA2.(A) Temperature versus surfactant hydration number w0 = (moles of H2O)/(moles DPA-TMA2) phase diagram, illustrating the composition-dependentformation of normal cylindrical (HI), and normal micellar A15, σ, and BCC or-dered phases, and disordered micelle solutions based on synchrotron SAXSanalyses. Narrow windows of two-phase coexistence typically occur in betweenthe pure phases (e.g., HI/A15 coexistence at w0 = 8). (B) Synchrotron SAXSpowder patterns for DPA-TMA2 LLCs corresponding to the A15 (red), σ (blue),and BCC (purple) phases labeled with the Miller indices for each reflection; seeSI Appendix, Fig. S1 for complete indexing of the FK σ phase SAXS pattern withmore than 50 peaks. a.u., arbitrary units.

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to exchange on timescales less than 30 ns with equal numbers ofsurfactant chain acceptance and expulsion events at each latticesite (SI Appendix, Table S2), consistent with a dynamic equi-librium. Our simulation results concur with experiments by Leeet al. (31) on a diblock polymer FK σ phase, which demon-strated that interparticle chain exchange enables formation of

quasispherical micelles with different aggregation numbers andvolumes that fill the various σ lattice sites. Micelle aggregationnumbers derived from our MD simulations quantitatively agree withthe geometrically calculated volume variations of the five different σphase Wigner–Seitz cells, which deviate by 91–106.5% around theaverage volume.

z

xy

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σ phaseA

D

B C

z = 1/2top view

y

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y

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z = 1/2

F

top view

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Fig. 2. Electron density reconstructions (90% isosurfaces) derived from synchrotron SAXS powder patterns for the tetragonal σ and cubic A15 FK phases at25 °C, withw0 = 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) aslice 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 thatoccupy the 2a (red) and 6c (blue) Wyckoff positions with respective coordination numbers Z = 12 and 14.

A B C

TMA mass density

120160

200240

y

x z = 0, 1

MD total mass density

800900

10001100

1200Experimental electron

density

(kg/m3)

(kg/m3)

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 thetetragonal unit cell of the experimentally derived electron density map (90% isosurface) wherein the micelles exhibit soft facets. (B) Slices of the total massdensity map from MD simulations through the same z = 0, 1 planes of the σ phase, and (C) the simulated TMA counterion density map that illustratescounterion localization (yellow) between the micelles to maximize interparticle electrostatic cohesion.

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An analogous MD simulation of the A15 LLC conducted atw0 = 12.1 at 353 K, in which a cubic cell with a = 6.79 nm wasseeded with eight identical micelles each containing 40 surfac-tants, equilibrates in 400 ns. This simulated electron density mapis again consonant with experiments, including the formation ofpairs of platelet micelles in the unit cell faces (Fig. 4 and SIAppendix, Fig. S4). Intermicellar surfactant chain exchange leadsto 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 simulationequilibration 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-mission EM analyses of the giant shape amphiphile A15 phaserecently reported by Cheng and coworkers (2).MD simulations of the σ and A15 phases reveal that the micelles

interact through their tetramethylammonium (TMA) surfactantcounterion atmospheres. The soft and relatively hydrophobicTMA counterions localize along the center-to-center vectorsconnecting neighboring particles, with higher densities near themidpoints between micelle surfaces as shown in Figs. 3C and 4C.In other words, the counterions outline the boundaries of theWigner–Seitz cells for each lattice site. Counterion localizationalong these boundaries drives faceting of the soft micelles, whichdeform 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 decreasingw0 depends on packing charged particles in a manner thatmaximizes both interparticle cohesion and ionic sphericity. Ionicsphericity refers to the thermodynamic preference for a sphericalcounterion 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 counterionatmospheres at high w0, due to repulsions between the negativelycharged headgroups in each aggregate. Removing water fromthis micellar solution initially triggers a BCC packing, becausethe configurational entropy loss upon crystallization is minimizedin this periodic lattice of symmetry-equivalent micelles (32).Decreasing w0 in the BCC LLC reduces the average distancebetween micelles while increasing the solvated ion concentrationin the aqueous domain. This increased ion concentration

enhances charge screening between the surfactant headgroupswithin each aggregate, while also driving hydrophobic surfactanttail stretching to minimize unfavorable ion/alkane interactions(Fig. 5). Thus, the surfactant molecules pack more tightly intolarger micelles with diminished interfacial curvatures. Main-taining the BCC structure at smaller w0, with increased micellesize and smaller aqueous domain volume, severely distorts thecounterion atmospheres and particle cores away from theirpreferred spherical symmetries. Thus, a phase transition occursin which the nearly identical self-assembled, micellar particlesreconfigure.The sphericity of a polyhedron may be quantified by the iso-

perimetric quotient IQ = 36πV2/S3, where V and S are its re-spective volume and surface area (10). Objects with higher IQvalues are more spherical by this definition, with IQ = 1 being thelimiting case of a perfect sphere. Lee et al. (31) calculated the

y

x z = 0, 1

A B C

MD total mass density TMA mass densityExperimental electron

density

200250

300350

800900

10001100

1200

(kg/m3)

(kg/m3)

Fig. 4. Comparison of an experimental electron density map and MD sim-ulation results for the FK A15 phase. (A) Two-dimensional top view of the z =0, 1 planes of the experimentally derived A15 phase electron density map(90% isosurface), (B) slices of the total mass density map for the A15 phasefrom MD simulations, and (C) the simulated TMA counterion density map,showing counterion localization (yellow) between the deformed plateletmicelles in each unit cell face.

BCCNagg 23

FK σNagg 32

FK A15Nagg 42

w0 = 36

w0 = 24

w0 = 12

Fig. 5. As the water concentration (w0) decreases in the ionic surfactantLLCs, the increased ion concentration leads to greater chemical incompati-bility between the ion-rich aqueous domains and the hydrophobic regions,as well as enhanced charge screening between the negatively charged sur-factant headgroups. Thus, the surfactant tails stretch to drive formation oflarger and more deformable aggregates with lower interfacial curvatures,which pack into low-symmetry FK σ and A15 phases that maximize the ionicsphericities of the counterion atmospheres around the micelles.

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number-averaged isoperimetric quotients for the polyhedralWigner–Seitz cells of various low-symmetry sphere packings, andthey found that they decrease along the phase progression σ >A15 > BCC. Thus, we reason that reducing w0 beyond a criticalvalue in the BCC LLC forces a phase transition to the σ phase thatis facilitated by surfactant chain exchange between micelles tomaximize both ionic sphericity and the electrostatic cohesion be-tween micelles (Fig. 5). Further reduction in w0 triggers a σ →A15 phase transition to maintain the highest possible ionic sphe-ricity at even lower hydrations. From the MD simulations, the IQvalues for each of the micelles in σ and A15 phases were calcu-lated by defining the micelle surfaces using the locations of thesurfactant headgroup phosphorous atoms (see Materials andMethods and SI Appendix, Table S3 for calculation details). Thecalculated IQ values follow the trend anticipated by Lee et al. (31)based on the geometries of the Wigner–Seitz cells of the σ andA15 lattices. Because the increased ion concentration in theaqueous domains of the A15 phase better screens repulsions be-tween the charged surfactant headgroups within each micelle,these larger particles are more able to deform into the observedplatelet micellar aggregates.Ionic sphericity-induced FK σ phase LLC formation by coupled

mass and charge exchange through surfactant redistribution ex-tends the provocative analogy described by Lee et al. (31) betweenlow-symmetry phases of metal alloys and of soft spheres. MetallicFK σ phase alloys (e.g., Fe–Cr) maximize both metallic bondingand Fermi surface sphericity in reciprocal space, through chargeexchange between atomic sites. Lee et al. (31) showed that blockpolymer σ phases similarly optimize both short-range van derWaals interactions to fill space at constant density and real spaceparticle sphericity, enabled by interparticle block polymer chain(mass) exchange. The phenomenological similarity between LLCs,block polymers, and thermotropic LCs extends to the fact that theionic LLC phase sequence with decreasing w0 parallels that ofnonionic soft systems with decreasing temperature T (19, 31). Thisapparent equivalence between w0 in LLCs and T in thermotropicLCs and block polymers stems from the increased chemical in-compatibility between the particle cores and coronae as w0 (orT) decreases.

ConclusionsIonic sphericity represents a previously unrecognized mechanismby which soft, charged particles assemble into both simple andcomplex 3D packings. The observation of ionic surfactant FK σand A15 LLC phases anticipates the provocative possibility thationic sphericity may induce similar self-assembly phenomena inlarger charged colloids at macroscopic length scales. Comparedwith the short-range forces that underlie FK phase formation innonionic LCs and block polymers, ionic sphericity combined withlong-range electrostatic forces between charged colloids may yieldmacroscale crystals with unusual photonic and phononic proper-ties. Because these FK phases are also 3D periodic approximants

to dodecagonal QCs, our observations suggest design principlesfor driving colloidal materials to form soft QCs at various lengthscales. Reports of surfactant-templated mesoporous silicate FKphases and 12- and 18-fold block polymer lyotropic QCs reflectinitial discoveries in this exciting direction (26, 33, 34).The formation of complex FK phases in ionic surfactant LLCs

through the exchange of mass and charge is reminiscent of chargeexchange in binary metallic alloys, comprising two elements withdifferent atomic masses. Thus, this discovery bridges previous re-ports of FK phases neutral soft materials (2, 18, 19, 21–23, 31, 35)and metal alloys. However, the ubiquity of icosahedral QCs inmulticomponent metal alloys (36) starkly contrasts the observationof only dodecagonal QCs in soft materials to date (23–26). Ico-sahedral QCs, which are quasiperiodic in 3D, typically form internary alloys wherein the metal sites exhibit decoupled variationsin both particle mass (atomic number) and charge distribution.However, dodecagonal QCs are layered structures that are qua-siperiodic in each 2D layer. Ionic LLCs, in which the σ andA15 approximant phases form by coupled exchange of mass andcharge to maximize cohesive energy and ionic sphericity, possiblyshed light onto this dichotomy in quasicrystallinity: icosahedralordering may require the independent interparticle exchange ofboth mass and charge to obtain a distribution of sphere sizes thatadopt packings with 3D quasiperiodic order.

Materials and MethodsFor details see SI Appendix. DPA-TMA2 was synthesized by stoichiometricdeprotonation of decylphosphonic acid with TMA hydroxide in methanol toproduce an analytically pure surfactant sample, which was characterized by1H, 13C, and 31P NMR spectroscopy and elemental analysis (C, H, N, and P).Lyotropic LC samples were prepared by combining measured amounts ofultrapure water (>18 MΩ resistance) with DPA-TMA2 in 1-dram vials, fol-lowed by three cycles of iterative high-speed centrifugation (4,996 × g for10 min) and hand mixing; all samples were allowed to rest at 22 °C for atleast 24 h before X-ray analysis. Temperature-dependent SAXS measure-ments were conducted on samples enclosed in alodined aluminum pans atthe 12-ID-B and DND-CAT 5-ID-D of the Advanced Photon Source. Sampleswere heated to the desired temperature using a thermostated sample stagewith a temperature accuracy of ± 0.1 °C and allowed to equilibrate for 5 minbefore data acquisition. SAXS data analysis details, including the electrondensity map reconstruction methodology, are described in detail inSI Appendix.

ACKNOWLEDGMENTS. We thank Frank S. Bates, Chris Leighton, and JesseMcDaniel for helpful discussions. This work was supported by US Depart-ment of Energy, Basic Energy Sciences Contract DE-SC0010328. SynchrotronSAXS data were acquired at the X-ray Sciences Division 12-ID-B and theDuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) 5-ID-Dbeamlines of the Advanced Photon Source (APS). DND-CAT is supported byE. I. DuPont de Nemours & Co., The Dow Chemical Company, and Northwest-ern University. The APS is an Office of Science User Facility operated for theUS Department of Energy (DOE) by Argonne National Laboratory and sup-ported by DOE Contract DE-AC02-06CH11357.

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