How faceted liquid droplets grow tails - BIU

How faceted liquid droplets grow tailsShani Guttmana, Zvi Sapira,1, Moty Schultza, Alexander V. Butenkoa, Benjamin M. Ockob, Moshe Deutscha,and Eli Sloutskina,2

aDepartment of Physics and Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel; and bCondensed MatterPhysics & Materials Sciences, Brookhaven National Laboratory, Upton, NY 11973

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved December 11, 2015 (received for review August 8, 2015)

Liquid droplets, widely encountered in everyday life, have no flatfacets. Here we show that water-dispersed oil droplets can bereversibly temperature-tuned to icosahedral and other faceted shapes,hitherto unreported for liquid droplets. These shape changes areshown to originate in the interplay between interfacial tension andthe elasticity of the droplet’s 2-nm-thick interfacial monolayer, whichcrystallizes at some T = Ts above the oil’s melting point, with thedroplet’s bulk remaining liquid. Strikingly, at still-lower tempera-tures, this interfacial freezing (IF) effect also causes droplets to de-form, split, and grow tails. Our findings provide deep insights intomolecular-scale elasticity and allow formation of emulsions of tunablestability for directed self-assembly of complex-shaped particles andother future technologies.

emulsions | membranes’ buckling | topological defects |two-dimensional crystals | spontaneous emulsification

Of all same-volume shapes, a sphere has the smallest surfacearea A. Microscopic liquid droplets are, therefore, spheri-

cal, because this shape minimizes their interfacial energy γA for asurface tension γ > 0. Spontaneous transitions to a flat-facetedshape, which increases the surface area, have never been reportedfor droplets of simple liquids. Here we demonstrate that surfactant-stabilized droplets of oil in water, of sizes ranging from 1 to 100 μm,known as “emulsions” or “macroemulsions” (1), can be tuned tosharp-edged, faceted, polyhedral shapes, dictated by the molecu-lar-level topology of the closed surface. Furthermore, the physicalmechanism which drives the faceting transition allows the sign of γto be switched in a controllable manner, leading to a spontaneousincrease in surface area of the droplets, akin to the spontaneousemulsification (SE) (1, 2), yet driven by a completely different, andreversible, process.At room temperature, the spherical shape of our emulsions’

surfactant-stabilized oil droplets indicates shape domination byγ > 0 (oil: 16-carbon alkane, C16; surfactant: trimethyloctadecy-lammonium bromide, C18TAB, see SI Appendix, Fig. S1). How-ever, the observed shape change to an icosahedron at someT =Td, below the interfacial freezing temperature Ts (Fig. 1A),demonstrates that γ has become anomalously low and no longerdominates the shape. This γ-decrease upon cooling starkly con-trasts with the behavior of most other liquids, where γ increasesupon cooling (1). Direct in situ γ-measurements in our emulsions(SI Appendix), as well as pendant drop tensiometry of millimeter-sized droplets, confirm the positive dγðTÞ=dT here (Fig. 2). Wil-helmy plate method γðTÞmeasurements (3, 4) on planar interfacesbetween bulk alkanes and aqueous C18TAB solutions (blue circlesin Fig. 2A) also demonstrate the same dγðTÞ=dT > 0 at T <Ts.Thus, the anomalous positive dγ=dT below Ts is confirmed for theC16/C18TAB system by three independent methodologies.To elucidate the implications of the positive dγ=dT, we note

that thermodynamics equates an interface’s γ with its free-energychange upon a unity increase in area: γ =ΔE−TΔS. ΔE and ΔSare the concomitant internal energy and entropy changes of themolecules (alkanes in our case) transferred from the bulk to theexpanded interface. Because an interface is typically less orderedthan a bulk, ΔS is positive, yielding dγ=dT < 0. Conversely, ourdγ=dT > 0 yields ΔS< 0, suggesting that at T <Ts our droplet’s

interface is more ordered than its (liquid) bulk. Indeed, recentX-ray measurements (4) on planar C18TAB-decorated C16–waterinterfaces showed that the interfacial region freezes at T =Ts,yielding a crystalline monolayer of mixed, fully extended, surface-normal–aligned, C16 and C18TAB molecules. No further struc-tural changes were found upon cooling to C16 bulk freezing.Systematic studies with various combinations of alkane and sur-factant chain lengths (4) indicated that interfacial freezing (IF)occurs due to the unique match in molecular geometry betweenthe alkane and the surfactant. The surfactant’s bulky hydratedtrimethylammonium headgroups yield a surfactant molecule’sspacing that leaves just sufficient room for alkane molecule in-terdigitation in between the surfactant tails, thus maximizing thevan der Waals contacts. Similar IF has also been observed in al-kane monolayers at the planar liquid/air interface of aqueousC18TAB solutions (4, 5), where grazing-incidence X-ray diffractionmanifests that the frozen monolayer’s molecules form a 2D quasi–long-range hexagonal crystalline order.Because no changes occur in the liquid bulk phases at T =Ts,

the γðTÞ slope change at T =Ts is a direct measure of ΔS asso-ciated with the monolayer freezing (3–6). Furthermore, the slopeof our γ (Fig. 2) matches closely that of the frozen C16 monolayerat the planar surface of its own melt (6) (purple dashed line),where jΔSj≈ 0.9× 10−3Jm−2K−1 was measured. The near-equaldγðT <TsÞ=dT in emulsions and at the planar interfaces discussedabove strongly indicates that the IF observed in our dropletsindeed forms a hexagonally packed monolayer of fully extendedsurfactants and alkanes, aligned normal to the interface (Fig. 3 Aand B). However, although similar in forming an interfacially frozenmonolayer, the planar and spherical interfaces greatly differ.

Significance

Rounded oil-in-water emulsion droplets are ubiquitous in lifeand technology. We demonstrate that crystallization of themonomolecular nanolayer at the interface of these dropletsprovides a novel way to control the shape of liquid droplets. Inparticular, the droplets undergo a spontaneous faceting tran-sition, split, and grow tails. All these transitions are fully tem-perature-controllable and reversible. The observed phenomenamimic shape selection in virus capsids, virus-associated pyra-mid formation in lipid membranes, and the growth of rod-likebacteria, yet the underlying mechanism is completely new. Theobserved effects allow formation of emulsions with control-lable attributes, probe the fundamentals of molecular-scaleelasticity, and open new routes for self-assembly of complex-shape colloids.

Author contributions: S.G., Z.S., B.M.O., M.D., and E.S. designed research; S.G., Z.S., M.S.,A.V.B., M.D., and E.S. performed research; S.G., B.M.O., M.D., and E.S. analyzed data; andS.G., Z.S., M.S., B.M.O., M.D., and E.S. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1Present address: Intel (Israel) Ltd., Kiryat Gat, Israel.2To whom correspondence should be addressed. Email: [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1515614113/-/DCSupplemental.

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A planar interface can be tiled perfectly by all 2D crystal symmetries,and thus imposes no constraints on the monolayer’s lateral crystal-line packing. By contrast, a sphere cannot be perfectly tiled by aplanar 2D lattice. Thus, packing defects are necessarily imposed, thenumber, nature, and symmetry of which depend on the monolayer’s2D crystalline order. As this study demonstrates, these geometry-imposed defects lead to a rich array of temperature-dependent ef-fects, which have no counterparts in the temperature evolution of aplanar interface, where no further changes occur upon cooling oncethe frozen monolayer is formed.

Faceting Dictated by TopologyWhen sufficiently high, γ dominates the droplets’ shape, theelastic energy of the frozen interfacial monolayer is negligible,and the droplets are spherical. However, below Ts, γ drops dra-matically on cooling, and if ΔT =Ts −Tm (where Tm is the oil’sbulk melting point) exceeds γðTsÞ=jΔSj, γ crosses zero above Tm.Thus, at some Td, while the droplet is still liquid (Tm <Td <Ts), γbecomes small enough for the interfacially frozen layer’s elas-ticity to become dominant (Fig. 2). However, the frozen mono-layer’s hexagonal molecular packing is incompatible with aspherical geometry: Euler’s formula (7) shows that hexagons donot perfectly tile a spherical surface (8). At least 12 fivefoldlattice defects must form to overcome this topological constraint(9); this is why soccer balls and C60 fullerenes have 12 pentagonalfacets (7). When strain relaxation mechanisms are absent (7, 10,11), the defects-induced strain energy leads to a lattice-mediatedeffective repulsion between the defects (10). Occupying thevertex positions of an inscribed icosahedron maximizes the de-fects’ separation, like in the classical problem of 12 electrons ona sphere, in Thomson’s atom model (12). Upon cooling into theelasticity-dominated regime, the 12 defects buckle, changing thespherical droplet into an icosahedron (Fig. 1 A, C, and D). The-oretical models have proposed a similar buckling of elastic spher-ical shells (13) to account for the icosahedral shapes of virus andbacteriophage capsids, onion carbon nanoparticles (14), and Cir-cogonia icosahedra, submillimeter-size unicellular organisms.* The

hexagonally packed shells in these systems possess 12 topologicaldefects. The buckling of a fivefold defect reduces its local radius ofcurvature (Fig. 3C). The associated lattice stretching energy nowscales only as logðR0Þ, where R0 is the droplet radius, instead of thestronger R2

0 scaling of unbuckled defects (13). This buckling re-laxation of stretching energy is limited by the penalty in bendingenergy. The balance is described by the dimensionless Föppl–vonKármán number ΓvK =YR2

0=κ (where Y and κ are the 2D Young’smodulus and the bending rigidity, respectively). For buckling (13)ΓvKb ≈ 150. Given Y=κ, this condition yields a minimal radius

allowing buckling: Rb0 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΓvKb κ=Y

q.

The driving force for the onset of icosahedral symmetry isuniversal for many systems (12, 16, 17); similar effects may (18, 19)also play a dominant role in forming icosahedral mixed-amphi-philes vesicles (18, 20–23). However, our emulsions fundamentallydiffer from those systems by the interfacial alkanes’ ability to freelyexchange with bulk alkanes, as in a grand-canonical ensemble.Thus, the droplet’s interfacial area A is not conserved and con-tributes γA to the energy balance, promoting the minimization ofA. This contribution scales as R2

0, becoming more dominant for thelargest droplets. Remarkably, large droplets (R0 J 40 μm) preserveupon buckling their overall-spherical shape, with only 12 tinyprotrusions observable, indicative of a local buckling at the defects’locations (Fig. 1E). In smaller droplets (R0 K 40 μm), γA is negli-gible, and a well-defined icosahedron emerges (Fig. 1 A, C, and D).We reproduced both behaviors in computer simulations (Fig. 1 Band F), which neglect the droplet’s buoyancy-induced distortion.Whereas buoyancy-squashing (SI Appendix) is negligible for vesi-cles (18, 20) and virus capsids (13), it sets a low limit on γ for thelargest emulsion droplets, precluding conversion into an icosahe-dron. Importantly, most previous studies of self-assembled icosa-hedra (13, 14, 16, 19, 21, 22) addressed submicrometer objects,which cannot be visualized by conventional optical microscopy.Our droplets preserve the icosahedral symmetry for much largerR0, thanks to the large lateral size of their interfacially frozenmonocrystals, mediating the interdefect repulsion. This size mayreach several millimeters (5, 6) owing to the possibility of ex-changing alkane molecules between liquid bulk and the frozeninterfacial monolayer, a very efficient mechanism for healing ofextra pairs of topological defects, beyond the ones required by

Fig. 1. Buckling in liquid emulsion droplets. (A–D) For small γ, the frozeninterface’s elasticity dominates. Small droplets become icosahedra (brightfield: A, simulated: B), exhibiting five vertex-emanating edges (lines). Con-focal microscopy reveals regular-icosahedron-identifying hexagonal (C) andpentagonal (D) cross-sections. Large droplets, having significant surface-areaA and -energy γA, remain spherical (bright field: E, simulated: F), but showprotrusions formed by defect buckling (arrows in E).

Fig. 2. Anomalous γðTÞ dependence. (A) Alkane–water interfaces, deco-rated by C18TAB surfactant, undergo IF at T = Ts, yielding a positive tem-perature slope of the interfacial tension γ at T < Ts. The γ-values of theemulsion droplets (green squares) agree with those measured for a planarinterface (blue circles) and for pendant millimeter-sized drops (red triangles).The γ-dominated, the elasticity-dominated, and the SE regimes are sche-matically marked. γ=γðTsÞ derived from ΔS of surface-frozen C16 alkane melt(5) is shown in purple. A magnified plot at ultralow γ is shown in B.

*The shape transformation reported by Jeong et al. (15) in lyotropic liquid crystal dropletsis driven by a completely different process: a bulk transition to a columnar phase. Thedroplet assumes a rotationally symmetric 3D shape of an object of revolution of ahexagon about its long axis, but exhibits no true planar facets as found here.

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Euler’s formula (7, 11). Also, the droplets’ large size allows one todirectly confirm by optical tweezing and polarized microscopy thatthe droplets’ bulk indeed remains liquid throughout these shapetransformations (SI Appendix and Movie S1).The delicate balance between γ, bending, and stretching of the

interface-frozen monolayer allows estimating the upper limit onits bending modulus as κ< 104 kBT (SI Appendix). The lowerlimit, κ> 103 kBT, is obtained by optical tweezing experiments (SIAppendix and Movie S2), where pN forces prove insufficient tobend an 8-μm-wide facet of a droplet, even for γ→ 0. These valuesare unusually high for soft condensed matter, yet some theoreticalmodels predict very high κ for crystalline monolayers (24, 25), andvalues up to 4× 103 kBT were measured in frozen mono- andbilayers of similar surfactants (17, 26–28). The estimated κ yieldsRb0 > 0.4 μm (SI Appendix). Droplets of R0 ≈ 1 μm are observed to

buckle, indicating Rb0 < 1 μm. The limits above on κ and Rb

0, to-gether with the estimated ΓvK

b , yield 6× 10−4 <Y < 0.04 N/m and avery low 3D Young’s modulus: 0.3<Y3D < 20 MPa; here Y3D =Y=dand d≈ 2 nm is the interfacial monolayer’s thickness (4, 5). In theclassical thin elastic plate theory (26, 28), Y3D = 12κð1− ν2Þd−3,where ν is the Poisson ratio, and the elastic properties are assumedto be isotropic. Even adopting a very large ν∼ 0.5, that of rubber,this Y3D exceeds 20 MPa by several orders of magnitude, epito-mizing the breakdown of the isotropic continuum elasticity con-cept at the single-molecule level.

Transient Negative γ Drives Tail Growth and SplittingThe anomalous positive slope of γðT <TsÞ (Fig. 2) has an addi-tional dramatic consequence: at TSE =Ts − γðTsÞ=jΔSj, γ becomestransiently negative, driving a spontaneous increase of the drop-lets’ interfacial area (2, 29–32) for T <TSE by, e.g., dropletsplitting (SI Appendix and Movie S3), and rod-like droplet for-mation (Fig. 4A, SI Appendix, and Movie S4).† Conserving vol-ume, the rods extend and thin, turning eventually intosuboptical-resolution nanorods; nanocoil tails also develop (Fig.4B, SI Appendix, and Movie S4). Other highly dynamic shapesare observed as well (Fig. 4C; see also SI Appendix). These

effects, reminiscent of SE en route to a microemulsion (1, 2), aredriven here by IF. Thus, unlike conventional SE, we do not ob-serve diffuse or budding interfaces (33, 34); rather, sharp facetsoccur. Moreover, SE occurs here on cooling and is tunable:Reheating the sample to T >Td increases γ, and all dropletsbecome spherical again (SI Appendix). No such controllability isexhibited by any previously reported spontaneously emulsifyingsystems. At T <TSE, SE goes on until adsorption to the in-creasing interface area depletes the bulk C18TAB concentration,increasing γ to zero. Another remarkable effect occurs when anoptical-tweezers–trapped droplet splits at T <TSE (SI Appendixand Movie S5), exhibiting a strong, directed flow from the (trap-ped) mother to the (untrapped) daughter. The daughter grows whilethe mother shrinks and eventually vanishes. Because the optical

Fig. 3. Molecular structure cartoons of the droplets’ interface. (A) The interfacial monolayer at T > Ts, comprising mixed C16 alkane and C18TAB surfactantmolecules. Yellow, blue, green, and red denote C, H, N+, and Br−, respectively. C18TAB headgroups are partially water-ionized and hydrated. (B) Only theinterface freezes at T = Ts, forming a crystalline, hexagonally packed, monolayer of extended, interface-normal, molecules (4) (Inset). Because a sphericalsurface cannot be tiled by hexagons, the frozen monolayer includes 12 fivefold defects. (C) At low γ elasticity dominates. The defect-induced strain is partlyrelieved by buckling (Inset).

Fig. 4. Complex shapes of liquid emulsion droplets. (A–C) For (transient)γ < 0, γA is minimized by increasing A through formation of constant-volumerods (A). Their lengths increase and widths decrease below optical resolu-tion. Nanocoil tails also occur (B), as do faceted shapes, possibly reflectinglocal free-energy minima (C and D). An observed hexagram (D) is reproducedin computer simulations (E), which neglect buoyancy squashing, of a low-γdroplet under nonequilibrium conditions.

†Note that the interfacial tension γ is defined between two phases in thermodynamicequilibrium, which is not the case when SE occurs. Thus, the term “negative γ” used hereshould be understood as a transient state. For a full discussion, outside the scope of thepresent article, see refs. 2, 29–31.

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trap’s temperature is slightly higher than the surroundings, only ananomalous dγ=dT > 0 can yield an energy reduction by the ob-served high-T to low-T flow, confirming again the droplet’s IF.Finally, cyclic temperature variation across TSE yields even moreunique droplet shapes, e.g., a hexagram (Fig. 4D), stronglysquashed by the buoyancy at its low γ. Interestingly, preliminarynonequilibrium computer simulations yield a similar symmetry(Fig. 4E), implying that these shapes correspond to the dominantlocal minimum of the free-energy landscape when exactly 12fivefold defects are present, as for the icosahedral droplets.Models allowing the motion of defects, as also the creation andthe annihilation of pairs of five- and sevenfold defects, have ingeneral a different energy landscape; in some of these models, thehexagram is the global energy minimum. Such minima and de-liberate seeding of lattice defects may be exploited to allowcontrolled formation of complex shapes (11, 16).

ConclusionsTo conclude, we demonstrated that IF in oil-in-water macro-emulsions allows γ to be dramatically tuned over a significanttemperature range. Consequently, an elasticity-dominated regimeemerges, where liquid droplets assume faceted icosahedral shapes.

IF-driven SE occurs at a lower temperature, giving rise to fasci-nating shape transformations, splitting, and tail growing of theliquid droplets. These novel effects may have applications in par-ticle synthesis, microencapsulation, and many other technolo-gies. Our ongoing experiments reveal identical effects for othersurfactants, alkanes, and concentrations. Moreover, the observedshape transition temperatures can be tuned for particular appli-cations by varying the lengths of the constituent molecules (4, 5).Loading the emulsion with spherical or anisotropic nanoparticlesshould allow the number and the energy of defects to be tuned,providing further control of the observed geometries. Future studiesof the full phase diagram of shape transitions should provide in-roads into the fundamental mechanisms of molecular-scale elas-ticity, now poorly understood, with far-reaching consequences fornanomedicine, low-dimensional physics, and nanotechnology.

ACKNOWLEDGMENTS. We thank T. Zemb, D. C. Rapaport, S. A. Safran, Y. Rabin,D. A. Weitz, and A. B. Schofield for discussions, A. Be’er and H. Taitelbaum forassistance in early microscopy measurements, M. Schneeberg and D. Friedmanfor technical assistance in cell construction, and I. Tkatz for assistance withgraphics. Acknowledgment is made to the Donors of the American ChemicalSociety Petroleum Research Fund for support of this research and to the KahnFoundation for the purchase of equipment.

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Supporting InformationGuttman et al. 10.1073/pnas.1515614113

Movie S1. Pulling a faceted liquid droplet and a bulk-frozen droplet by an optical trap. (A) A liquid C16 droplet in a C18TAB aqueous solution is opticallytrapped at T < TSE, where the droplet has a rectangular cross-section. The sample stage is laterally moved in opposite directions and the direction is reversed att∼10 s (see label in the bottom-left corner). “X” marks the trap position. The blue arrow indicates the hydrodynamic drag force acting on the droplet. The redarrow indicates the force applied by the optical trap. This force is proportional to the velocity of the trapped droplet, relative to the sample stage. At t = 13.5 s,the droplet is observed to escape the trap. Note that the trap is able to move the droplet with respect to the surrounding liquid only when the trap holds thedroplet’s boundary. When the trap holds a point inside the droplet’s bulk, the droplet is carried along with the surrounding liquid, indicating that the droplet’sbulk is liquid. (B) A similar trapping experiment, carried out with a bulk-frozen droplet of alkane (C22) in water as a control measurement. When the stagemoves, the hydrodynamic drag displaces the center of the droplet from the center of the trap by ∼2 μm, to have the optical force (red vector) balance thehydrodynamic drag (blue vector). Once the stage stops, the droplet returns to the center of the optical potential well. The observed response and the unevenappearance in bright-field microscopy are typical for a polycrystalline object, contrasting with the response and the appearance of the liquid droplet in A. Thetwo videos appear side-by-side to allow a more direct comparison.

Movie S1

Movie S2. Estimating the bending modulus of the interfacially frozen monolayer. The same droplet as in Movie S1A is dragged rapidly by its boundary, atT < TSE, to probe its elastic modulus. At T < TSE, the rapid growth of interfacial area leads to depletion of bulk surfactant, so that γ eventually reaches zero;the present experiment is started only after this transient γ-variation is over. As in Movie S1, the blue vector is the hydrodynamic drag force and the redvector is the force applied on the droplet by the optical trap. The sample stage is moved consecutively in opposite directions, reaching a velocity of 15 ± 1μm/s. The movie demonstrates that the droplet boundaries remain flat (unperturbed) upon dragging. This observation sets a lower limit on the bendingmodulus κ of the interfacially frozen monolayer (see SI Appendix).

Movie S2

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Movie S3. Droplet splitting event. A faceted droplet is demonstrated to spontaneously split at T < TSE, forming three smaller droplets. The total interfacialarea of the daughter droplets is larger than that of the initial droplets. Therefore, this splitting is energetically favorable in the SE regime where, tran-siently, γ < 0.

Movie S3

Movie S4. Optical microscopy: a typical temperature scan, demonstrating the shape transitions. Initially, T > Td, therefore the droplets are spherical. Thetemperature at a given time t is represented by the red circle, which is moving along the T(t) curve (Inset). The largest droplet, initially located below the centerof the frame, allows the shape transitions to be observed most clearly. This droplet distorts into an icosahedral shape on cooling below Td, at 00:15s. Rods andcoils are visible after 00:20 s, where T < TSE. At 00:22–01:14 s, the largest droplet springs a leak from its two sharp corners and then continuously loses materialover time. The green dotted line in the inset represents Td; the horizontal magenta dashes denote TSE. The movie is sped up by a factor of 38. The sample stagewas moved back and forth several times, to position the droplet at the center of the frame.

Movie S4

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Movie S5. A droplet in an optical trap, away from the walls of the capillary, at T < TSE. A liquid C16 droplet is optically trapped at T < TSE ∼ 20 °C and po-sitioned away from the capillary walls. “X” marks the trap position. The SE proceeds as usual, showing a spontaneous splitting and a daughter droplet for-mation, indicating that wall effects are not significant in our study. The slightly higher temperature at the center of the trap and the positive dγ/dT causematerial to be sucked out of the mother droplet into the daughter, making it grow. The reduced-size mother then ejects coiled tails, and eventually vanishesdue to material loss. Note the scale bar in the right bottom corner; the movie was sped up by a factor of 18.5.

Movie S5

Other Supporting Information Files

SI Appendix (PDF)

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How faceted liquid droplets grow tailsSupplementary MaterialShani Guttman ∗, Zvi Sapir ∗ †, Moty Schultz ∗ , Alexander V. Butenko ∗ , Benjamin M. Ocko ‡, Moshe Deutsch ∗

and Eli Sloutskin ∗

∗Physics Department & Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 5290002, Israel,†Current address: Intel (Israel) Ltd., Kiryat Gat,

Israel, and ‡Condensed Matter Physics & Materials Sciences, Brookhaven National Laboratory, Upton, NY 11973, USA

Submitted to Proceedings of the National Academy of Sciences of the United States of America

Supplementary Materials and MethodsChemicals.C18TAB (Aldrich, ≥ 99 % and ≥ 98 % pure) wasrecrystallized 1-3 times from a methanol - acetone solution.C16 alkane, hexadecane, (Aldrich, 99 % pure) was percolated2-3 times through fresh activated basic alumina columns toremove polar components. Millipore Ultrapure 18.2 MΩ-cmwater was used throughout.

Fig. S1. Molecular structure. (A) Hexadecane (C16) normal-alkane,

CH3(CH2)14CH3. (B), Trimethyloctadecylammonium bromide (C18TAB or STAB)

surfactant, CH3-(CH2)17N(CH3)3Br. All chemical bonds are covalent except for

the bromine, which has an ionic bond with the nitrogen. Color code: carbon - grey,

hydrogen - white, nitrogen - blue, bromine - brown.

In planar interfaces measurements, as also in some of theemulsion measurements, all parts of the equipment coming incontact with the sample have been soaked in hot Piranha so-lution, a widely-used method for removing organic impurities.However, the large interfacial area of the emulsions signifi-cantly reduces the sensitivity of the measurements to impu-rities. Moreover, the ultralow γ in these emulsions rendersinterfacial adsorption of impurities energetically unfavorable.Therefore, cleaning the equipment with common organic sol-vents was found to yield identical results to Piranha cleaning.

Emulsions are prepared by mixing 0.05% of C16 in 0.3 mMto 1 mM water solution of C18TAB, and stirring at 50C for∼5 minutes for ∼ 100 µm droplets and for ∼ 12 hours for∼ 10 µm droplets, employing a hotplate/magnetic stirrer. Theinterface/bulk mass ratio in these emulsions is high; therefore,the concentration of C18TAB in the bulk aqueous phase dropsduring stirring as the C16 droplets get smaller and the total oil-water interfacial area grows. Thus, we disperse a small amountof the prepared emulsion in a large volume of a C18TAB aque-ous solution, where the concentration was 0.6, 0.7, or 0.8 mM;all results stay unchanged in this concentration range. Thisre-dispersion procedure, significantly reduces the sensitivity ofthe shape transition temperatures to the details of the emul-sion preparation process. No crystallization of the surfactantis observed in the relevant temperature range.

Optical microscopy.The emulsions are loaded by capillaryforces into Vitrocom R© 0.1× 2× 50 mm rectangular glass cap-illaries and sealed by an instant Epoxy glue. 0.6×0.6×50 mmcapillaries were used for the largest droplets. Unsealed sam-ples, contained between two glass slides, yield similar resultsexcept for having a much-reduced stability and not allowingmeasurements of large droplets. The capillary is glued withits wide face down onto an aluminum slide, having a narrowmachined opening for optical observations. The slide is in-serted horizontally into a home made temperature-controlledcell, mounted on the translation stage of an inverted micro-scope. The sample cell employed a pair of Peltier elements forbaseplate cooling, and a Lakeshore model 330 for PID tem-perature control via thin-film resistive heaters. Where oil-immersed objectives were used for imaging, their temperaturewas controlled by the Okolab R© temperature control system,based on a water circulator. Our setup allows the tempera-ture of the sample to be regulated to 0.01 C, in the rangefrom 10 to 50 C. At low temperatures, where water conden-sation from air interfered with the imaging, the experimentalsetup was covered by a plastic bag, inside which a dry nitrogenatmosphere was maintained.

Bright-field microscopy was carried out employing an in-verted Nikon Ti-E microscope, with a Nikon DS-Fi1 CCDinstalled for video acquisition. The microscope is part of aNikon A1R scanning confocal setup, allowing either bright-field, phase contrast, DIC, or confocal measurements to becarried out. Most measurements have been carried out inthe bright-field mode. For confocal images, the oil was flu-orescently labeled with the Bodipy 505/515 hydrophobic dye,which was then excited by the 488nm line of the Ar laser. Thedroplets float up in water by buoyancy, so that an uprightor horizontal optical axis microscopy proved optimal in somecases. In these cases, we rotated the optical axis employing anInverterScope R© objective inverter. The images were taken byPlan Apo 100x (NA=1.4), Plan Apo λ 100x (NA=1.45), and60x (NA=1.4) oil-immersed objectives. Where more carefultemperature determination was needed, we employed a dryPlan Fluor 100x (NA=0.9) objective, which does not touchthe sample capillary.

Three-dimensional confocal imaging was carried out in a res-onant scanning mode, with the 512×512 pixel frames collected

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at ∼7 fps. A piezo device was employed for rapid scanning ofthe sample along the optical axis. The voxel size in confocalmeasurements was typically chosen as 0.06 × 0.06 × 0.4 µm3,slightly oversampling the optical resolution with the confocalpinhole size set to 0.8 − 1 AU. In general, interfacial phe-nomena are highly sensitive to contaminations, such that thefluorescent staining for confocal microscopy may significantlydisturb the behavior of the system[1]. The non-polar Bodipy-505/515 dye, used in these studies, is among the very fewfluorescent dyes which are strongly hydrophobic and do notadsorb to charged oil-water interfaces. However, even withthis dye, strong irradiation by the laser source results in itsphotoinduced ionization, after which the dye adsorbs to theinterface, enhancing the spontaneous emulsification phenom-ena. To avoid these artifacts as much as possible, all but afew of the results in this work are based on the bright fieldmicroscopy, where fluorescent dye was not used.

For the preparation of the current manuscript, we measuredmany hundreds of droplet shapes. Still, within the availablestatistics, we could not see any significant variation of shapesacross the statistical ensemble. Thus, the free energy minimaof the system must be narrow and much deeper than 1 kBT.Importantly, the buckling temperature Td depends on the sizeof the droplets, as demonstrated in Fig. S2. This dependencewas used to obtain the data in Fig. S4, as described below.

Fig. S2. A typical bright field image of an emulsion demonstrates the depen-

dence of the buckling transition temperature Td on droplet radii. The temperature

of this image was set such that the largest droplet is on the verge of converting into

an icosahedron, while the smaller droplets already have well-developed icosahedral

shapes. The smallest droplets in this image are below optical resolution, so that their

shapes cannot be resolved.

Macroscopic γ measurements.The Wilhelmy plate method,well documented in the literature[2, 3, 4, 5], was used tomeasure the interfacial tension of the planar buried oil-water interfaces, employing a glass plate of 41-mm circum-ference. The pendant drop tensiometry employed a home-built temperature-regulated cell, electronically controlled to0.01 C, mounted on a commercial Dataphysics R© OCA20 op-tical shape-measuring system operated in the pendant-dropmode. We re-scaled the drop tensiometry data slightly to makethe slope of γ(T ) at T > Ts match the Wilhelmy plate result;this accounts for the slow rate of C18TAB adsorption onto theinterface of a newly-created drop.

Microscopic γ measurements.For optical microscopy, theemulsion is loaded into a glass capillary (see Optical Mi-croscopy section, above). The liquid C16 droplets float to thecapillary’s top wall, under the joint action of buoyancy andgravity. At T > Ts, where γ ≈ 5 × 10−3 J/m2, the droplet’sdistortion from a sphere is negligible, as demonstrated bythree-dimensional (3D) confocal microscopy (Fig. S3A). Uponcooling, however, a flat circular region of a progressively largerradius rc appears at the droplet’s top (Fig. S2B), indicatingsquashing against the capillary wall, and thus - ultralow γvalues. Gravity and buoyancy are bulk forces scaling withdroplet radius R0 as R3

0; the Laplace pressure due to γ scalesas R−1

0. Thus, rc ∝ R2

0 is expected and, indeed, measured(Fig. S3C ), indicating that the flattening is fully describedby an interplay between gravity, buoyancy, and γ. Mostimportantly, the observed rc allows the temperature depen-dent ultralow γ of these microscopic droplets to be measuredin situ, which is highly challenging otherwise. Specifically,for small droplet distortions[6] γ = (2/3)R4

0∆ρgr−2

c , where∆ρ = 0.23 g/cm3 is the density mismatch between alkane andwater, and g = 9.8 m/s2. We plot the γ(T ) values, thus ob-tained, in Fig. 2 of the main text (green symbols); note thepositive slope dγ/dT > 0, indicative of an interfacial freezing.Also, we obtain the γ values at the buckling transition for arange of (relatively large) droplet radii. These values matchnicely the trend observed for smaller R0, based on Td(R0)measurements (see Fig. S4).

B

rc

A

40 m 40 m

rc

R0

C

R0

Fig. S3. Microdroplet shape analysis. Shapes of capillary-contained oil-in-water

emulsion droplets are determined by the balance between gravity, buoyancy and in-

terfacial tension γ , allowing ultralow γ values to be measured in situ. (A), At high

γ, all droplets are spherical (confocal microscopy reconstruction). (B), At low γ, thetopmost part of the droplets flattens as the droplet is squashed against the top wall of

the capillary (see cartoon in (C)). (C), The measured contact area radii rc of droplets

of radii R0 (symbols) follow the theoretical scaling rc ∝ R0

2(solid line). The inter-

facial tension γ, plotted in Fig. 2 (green symbols) of the main text, is extracted from

the prefactor of this relation (see Supplementary Materials and Methods).

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0 40 80 1200.0

0.1

0.2

0.3

Droplet shape analysis | S|(Td-TSE)

d (m

N/m

)

R0 ( m)

Fig. S4. The surface tension γd at the buckling transition temperature Td de-

creases with increasing droplet radius R0. For the largest droplets (R0 > 40µm),

the γd values (black squares) were obtained by optical microdroplet shape analysis (see

Supplementary Materials and Methods). For the small droplets, where the buoyancy-

induced distortions are too small to be reliably measured by optical microscopy, γd(red circles) were estimated from the buckling temperature Td of each droplet as

Td = |∆S|(Td − TSE). TSE is the temperature of spontaneous emulsification,

where γ(TSE) = 0. |∆S| is the entropy loss on freezing of a C16 monolayer[2, 5].

Note the smooth joining of the two sets of data, strongly supporting our results. The

fitted exponentially decreasing line is only a guide to the eye.

Optical tweezers.Home-built optical tweezing setup, based ona diode-pumped Nd:YAG laser (at 1064nm), was employed totest the mechanical properties of the droplets. To roughlycalibrate the force F applied on the emulsion droplet by thetweezing laser, we move the sample at different velocities, andfind the minimal velocity that drags the droplet out of theoptical trap. The force due to the viscous Stokes drag at ve-locity v on a sphere of radius R0, in a medium with a viscos-ity η is: Fv = 6πηR0v; this estimate yields Fmax ≈ 4 pN forthe maximal trapping force which we can apply on a dropletof R0 ≈ 8 µm under typical experimental conditions. Asan additional test for our tweezing setup, we prepared anheptane-in-water emulsion where the interfacial tension is ul-tralow. This emulsion, stabilized by 1.14mM AOT (dioctylsodium sulfosuccinate) in presence of salt (0.05M NaCl), wasstudied by Ward et al.[7]. No interfacial freezing phenomenaoccur in this emulsion, so that the droplets are readily de-formed at low γ. Pulling an emulsion droplet at a velocity of25± 1 µm/s, stretches the droplet, so that the aspect ratio ofa droplet of R0 ≈ 8 µm reaches ∼ 1.2 under steady state con-ditions. Balancing the Laplace pressure by the Stokes drag[7]:γ = FvbR0/[πa(2ab−bR0−aR0)], where a and b are the semi-axes of the stretched droplet, we obtain an ultralow value forthe interfacial tension of the droplet γ ≈ 8 × 10−6N/m, ofthe same order of magnitude as Ward et al.[7], confirming thevalidity of our analysis.

Computer simulations.Computer simulations have been car-ried out using the Surface Evolver software[8]. The dropletinterface was described by a set of triangulation vertices. Anicosadeltahedral triangulation lattice was used, including 12frozen 5-fold defects[9, 10] located at the icosahedral posi-tions. The edges are Hookean springs, with the stretchingenergy given by[11] Es = (ε/2)

∑〈ij〉 (|ri − rj | − a)2, where

ε =√3Y/2, a is the length of an unstrained edge, and the

summation is carried out over all nearest-neighbor vertices.a sets the length unit in our simulation. The bending en-

ergy is[11] Ed = (κ/2)∑

〈IJ〉 (nI − nJ )2, where κ = 2κ/

√3,

and the summation is over all nearest-neighbor plaquettes ofthe triangulated surface, with unit normals nI . As in theexperiment, the bulk volume of the droplets was kept con-stant in the simulations; importantly, bulk volume was notconserved in other theoretical studies[10] which model lipidvesicles[12, 13, 14, 15] and viruses[11] . A more complex model,where the number of defects is variable[9, 12, 16] and defectscan move, is currently under construction.

Supplementary DiscussionBright-field microscopy video of shape transitions.A typicalvideo of a temperature scan, demonstrating the shape transi-tions discussed in the main text, appears in Video4.mov. Thevideo starts with a cooling scan (00:00-00:24s), carried outat a rate of 2mK/s. Initially the central droplet is spherical.While the ∼ 2 nm-thick interfacial monolayer is below the op-tical resolution, so that no visible changes occur at T = Ts,this monolayer drives the buckling at T < Td (00:15s), withthe droplet adopting an icosahedral shape. On further cool-ing, γ switches sign from positive to (transiently) negative forT < TSE (00:20s), causing the onset of spontaneous emul-sification (SE). Note the sharp edges of the droplets (00:20-01:14s), which contrast conventional SE, where all interfacesare diffuse and disordered. Note the tails emanating from thelargest droplet, which become thinner as they grow, formingnano-coils. Significant active (non-Brownian) motion occursin the SE regime, driven by the (transiently) negative γ. Thedramatic increase in the interfacial area of the emulsion atT < TSE leads to surfactant depletion, so that the largestdroplet becomes rounded (00:37s - 01:14s). Upon reheatingto T > Td, the droplets turn spherical. The heating scan,carried out at a rate of 15 mK/s, starts at 01:12s. Note thatthe sizes of reheated droplets are smaller than at 00:00s, asthe reheating process breaks up the nano-coils, forming newdroplets.

In the SE regime, where γ < 0, droplet splitting events areoccasionally observed, since an increase in the total oil-waterinterfacial area is now thermodynamically favored. One suchevent is shown in Video3.mov, where a large triangular dropletsplits, forming three smaller droplets.

Confirming the liquid nature of the droplets’ bulk phase.Optical microscopy and optical tweezing measurements havebeen carried out to rule out the possibility that the shapechanges observed are due to crystallization of the droplet’sbulk. For comparison, fully-frozen droplets of alkane in wa-ter have been prepared, employing C22 (docosane) as an oiland C16TAB (cetyltrimethylammonium bromide) as a surfac-tant. The emulsion was prepared at an elevated temperature(T = 50 C); on cooling to room temperature, C22 dropletsfully solidified. The bulk of these droplets looks inhomoge-neous under either bright-field (panel b in Video1.mov) orpolarized microscopy, and can be readily distinguished fromour interfacially-frozen droplets, the liquid bulk of which ap-pears uniform. Nematic or smectic phases of alkanes are bire-fringent and would have been readily detected by polarizedlight microscopy; no optical activity could be detected for ourinterfacially-frozen emulsion droplets, indicating, again, thatthey are simple liquids.

We also compared the response to trapping forces of a fullyfrozen droplet of C22 (R0 ≈ 5 µm) to the response of a facetedliquid droplet of C16. In both cases, we fix the trap position(denoted by an × symbol in the movie clips) and move thesample stage. As we demonstrate in Video1.mov (panel b),when the solid droplet is displaced from the center of the trap

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by a hydrodynamic drag, an optical restoring force immedi-ately emerges, opposing this displacement; a balance betweenthese two forces is achieved shortly after. Similarly, when thestage motion stops, the optical force makes the droplet returnto the center of the trap. By contrast, displacing the liq-uid droplet from the center of the optical trap (panel a) doesnot give rise to a restoring optical force. The force emergesonly when the trap is right at the boundary of the droplet,as expected from a liquid having a zero shear modulus. Inconclusion, our experiments unequivocally show the bulk ofthe faceted C16 droplets to be liquid over the full temperaturerange employed in the present study.

Excluding wetting effects at the capillary walls.The bal-ance of buoyancy and gravity forces on oil droplets in wa-ter pushes them up, so that the phenomena discussed in thiswork typically occur next to the top wall of the borosilicateVitrocom R© capillary. To confirm that the borosilicate inter-face does not play any significant role in the observed effects,we employed an optical trap to move the droplets away fromthe glass wall. These optically-trapped droplets were demon-strated to follow the same behavior as the ones sitting next tothe glass wall. In particular, the droplet in Video5.mov splitsat T < TSE, forming a daughter droplet (t=50s, see the ex-periment time label, in the left bottom corner of the frame).The optically-trapped mother droplet, located far from thecapillary walls, grows tails much like the untrapped droplets,indicating that wall interactions play no significant role in theobserved phenomena. Interestingly, with dγ/dT being posi-tive for the interfacially-frozen emulsions (see main text), thedroplets gain energy by moving away from the vicinity of theoptical trap, where the temperature is slightly higher, and thusγ is higher. This ”Inognaram” (inverse Marangoni) effect[17]makes the daughter droplet suck the liquid from, and thusgrow at the expense of, the mother droplet (66 < t < 76s).Such behavior provides an additional confirmation that inter-facial freezing indeed takes place in our emulsions. Finally,as the mother droplet converts into a nano-coil, the trappingforce decreases, so that the nano-coil eventually leaves the trap(t = 245s).

Bending modulus: a low limit estimate based on the opticaltweezing.By optically trapping a droplet while moving thesample stage at a constant velocity, the droplet is subjectedto a significant hydrodynamic drag force Fv. For the dropletshown in Video2.mov, no elastic distortion is detectable bymicrocopy indicating that the distortion is below optical reso-lution. Since its past history (see Video captions) ensures thatγ = 0 N/m, the droplet shape is now elasticity-dominated and

any distortion observed while dragging should provide an es-timate for the now-dominant bending modulus κ.

x

L

r

Fig. S5. Distortion of an optically-trapped droplet by hydrodynamic drag. A

schematic representation of bending of the top flat facet of a droplet with a rectan-

gular cross-section. L and ∆x are defined in the magnified scheme on the right. ris the radius of curvature of the bent facet, if bending occurs.

With the viscosity of water, η ≈ 10−3 Pa s, and the lineardimension of a droplet, 2L ≈ 8 µm (see Fig. S5), we estimatethe Stokes drag force as Fv ≈ 1 pN, slightly lower than themaximal trapping force Fmax, calculated above; pN forces aretypical in optical tweezing experiments. To estimate the lowlimit on the bending rigidity κ of the frozen monolayer, we ap-proximate the bending work done by a force F on an initially-flat surface as F∆x, where ∆x is defined in Fig. S5. Whenγ = 0 N/m, the elastic energy cost of bending the droplet’s flatfacet is[18]: Ed = 0.5κr−2AF , where r is the curvature radiusin the distorted state and AF ≈ (2L)2 is the facet area. Forsimplicity, the distortion is assumed to be in one dimensiononly, as shown in Fig. S5. Relating the radius of curvature rto the distortion amplitude ∆x, (r +∆x)2 ≈ r2 + L2, we ob-tain: r ≈ (L2 −∆x2)/2∆x. Substituting this into Ed = F∆x,yields: κ ≈ F (L2−∆x2)2/(8L2∆x), so that κ has to be of theorder of 1000 kBT for the facet distortion to be resolvable bymicroscopy. A more accurate estimate has to account for thebuoyant squashing of the droplets at T < TSE; this correctionwould increase the estimated κ. Our calculation neglects theelastic contribution of other facets, adjacent to the one whichis being bent; this contribution should decrease the estimatedκ. Assuming that all these smaller corrections cancel out, κshould be larger than 1000 kBT.

Estimation of the elastic constants.To obtain the estimatesfor the elastic constants in the main text, we express theelastic energy of an unbuckled monolayer with 12 five-folddefects as[11] Es = D + 6BκΓvK/ΓvK

b , where D is a con-stant contribution and B ≈ 1.3. In our case, R0 is muchlarger than the lattice constant, ∼ 0.5 nm, of the interfacially-frozen monolayer[3] , so that pairs of seven-fold and five-folddefects may potentially form, organized into chains[16] , or‘grain boundary scars’. Taking the scars, coupled in our caseto the interfacial topology, explicitly into account is highlychallenging[9, 12] . However, for a spherical surface, both thenumber of defect pairs and the energy of an individual pair areproportional[16] to R0, so that the total energy scales as R2

0.

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Therefore, this contribution does not change the functionaldependence of Es on R0.

minimal

10-1 100101

102

103

104

105

max

imal

Rb 0

/ k B

T

Rb0 ( m)

Fig. S6. Limits on γd, as obtained from the theory of elasticity and optical tweez-

ing. The theoretical estimate (Eq. (1) in this document) for the bending modulus κis plotted (red curve) as a function of Rb

0, the smallest droplet radius where buckling

occurs. The curve corresponds to R0 = 5µm, where γb = 2.1 × 10−4 J/m2

(see Fig. S4). Droplets of R0 > 1µm, sufficiently large for optical microscopy, are

observed to buckle, setting the upper limit (blue dash-dotted line) on κ(Rb0). Thus,

the physically-relevant values of κ(Rb0) reside in the orange-lined region. Optical

tweezing experiments indicate that κ > 1000 kBT , limiting κ(Rb0) values to the

orange-turquoise shaded region. The intersection of the red curve with the borders

of this region yields an estimate for κ as 103 < κ < 104 kBT . Similar plots for

other R0, within the experimental range, yield the same order of magnitude for κ .

The elastic energy in the buckled state is[11] : Eico =D+6Bκ[1 + ln(ΓvK/ΓvK

b )]. The Gaussian rigidity is droppedin both Es and Eico, since by the Gauss-Bonnet theorem,the Gaussian curvature does not change on buckling[11] .When the interfacial elasticity is negligible, the interfacialenergy difference between a sphere and an icosahedron ofthe same volume is ∆Eint(γ) = γ(5

√3a2 − 4πR2

0), where

a = R0

[16π/(15 + 5

√5)]1/3

is the icosahedron’s edge length,

so that ∆Eint(γ) ≈ 0.8γR2

0. Denoting the interfacial tensionat the onset of the buckling by γd ≡ γ(Td), the balance be-tween the elastic energy and the interfacial tension is given by:Es = Eico + ∆Eint(γd). Substituting Eico and ∆Eint aboveyields:

κ = 0.8γdR2

0/6B[(R0/Rb0)

2 − 2 ln(R0/Rb0)− 1], [1]

Since dγ/dT = −∆S and, by the definition of TSE,γ(TSE) = 0, we obtain γ(Td) = |∆S|(Td−TSE) ( Fig. S4). Fordroplets of R0 ≈ 5 µm, we measure Td−TSE ≈ 0.25 C, whichyields γd ≈ 2.1×10−4 J/m2. Plugging these R0 and γd valuesinto Eq. 1, we obtain the κ(Rb

0) variation shown in Fig. S6.Having R0 ≈ 1 µm for the smallest droplets where buckling isclearly observed yields Rb

0 < 1 µm. Since κ increases with Rb0,

the κ(Rb0) relation yields an upper limit on κ, κ < 104 kBT

(see Fig. S6).Optical tweezing experiments yield κ > 103 kBT (see

above), so that according to Fig. S6, Rb0 > 0.4 µm.

Combining the estimated limits on κ and Rb0, as also the

simulated[11] ΓvKb ≈ 150, we obtain the limits on Y in the

main text.While the current analysis neglects possible spontaneous

curvature of the frozen monolayer, molecular geometryconsiderations[4] and the agreement of simulated icosahedralshapes with the experimental ones (Fig. 1 in the main text),justify this approximation.

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