Downloaded by guest on September 26, 2021 t iud eoti ausfrtooinain fteinterface the of orientations two for values and obtain crystal We BCC a liquid. sharp, between its a interface produces position, equilibrium specific planar an a vertical in at which them gradient, creates observing density dielectrophoresis a device, and this This particles In system. (16–18). the BCC–liquid bottle” “electric colloidal charging a by to achieved it is apply we paper, this In FCC–liquid colloidal Results to only interfaces far simulated (13–15). so numerically interfaces experimentally to but applied (8–12) been energy, has interface the method in of variation direction a the causes anisotropy, in normal interface the (θ where a liquid Hwang its Hyerim and crystal colloidal cubic a body-centered between interface the of Stiffness www.pnas.org/cgi/doi/10.1073/pnas.2005664117 8) (7, (CFM) model fluctuation gives capillary averaging, the time to upon according which spectrum, power a yields displace- ments these along of transformation interface Fourier plane. interface the average of fluctuations normal, positional its the of inter- surements the quantity, related stiffness a facial observed, be can interface dynamic modeling structural some that indicate and 6) experiments (4, solidification of ses and atom interface the in fol- atom per or them, fusion of normalize heat the to inter- by useful (5), Turnbull crystal–liquid is lowing it the energies, BCC, interfacial for compare than energy, greater then facial is nucleation the FCC for force of driving prefer- the that phase Since BCC (1–4). metastable nucleates the entially is it one, stable (FCC) the at is cubic phase alloys face-centered the some where of temperatures solidification undercooled the during Remarkably, melting. T kinetics preferential alloys. metallic the in stiffness for FCC over offered Its BCC explanations of liquid. nucleation the its to support and 0.26 mental crystal larger: significantly (FCC) was cubic between interface comparison, face-centered an For create interface. to a used the were of colloids plane charged the similar in isotropic were 0.18 and stiffnesses the and interfaces, (114) 0.15 the were and fluctua- (100) capillary the For the method. to tion according determined interfacial inter- was of their stiffness series spectrum, facial power time average a the measuring computing parti- By and colloidal positions bottle. charged electric using an 2020) liquid 26, in March their cles review and for body-centered crystals (received between 2020 (BCC) 23, cubic established August approved were and interfaces IL, Evanston, Equilibrium University, Northwestern Cruz, la de Olvera Monica by Edited 02138 MA colo niern n ple cecs avr nvriy abig,M 23;and 02138; MA Cambridge, University, Harvard Sciences, Applied and Engineering of School )/∂θ ietmaueet of measurements Direct α n los fe stehgettmeauepae ro to prior phase, highest-temperature the as metals often many in alloys, found and is phase (BCC) cubic body-centered he = 2 k | γ crystallization steitraestiffness; interface the is savco aallt h nefc and interface the to parallel vector a is /(∆ ρ h steaoi ubrdniyi h rsa.Analy- crystal. the in density number atomic the is (y H γ f γ ˜ , ρ o C utb mle hnfrFC To FCC. for than smaller be must BCC for , a edtrie rmatm eiso mea- of series time a from determined be can , z a 2/3 ai .Weitz A. David , , t α where ), ,where ), k | side mle o C hnfrFCC. for than BCC for smaller indeed is B melting T h|A /σ (k)| 2 γ (σ y | ∆ oee,aedfcl,bti a if but difficult, are however, , k 2 colloids and atcedaee) respectively, diameter), particle : H i B T = f /σ θ steetap ffso per fusion of enthalpy the is a,b k z γ ˜ B steaglrvrainof variation angular the is 2 k | n rn Spaepen Frans and , r h oriae nthe in coordinates the are hsrsl ie experi- gives result This . T 2 interfaces , hc u ocrystal to due which k, γ ˜ = γ γ (θ + .This ). a,1 ∂ [1] 2 γ n hi est oaodcemn rsdmnain Here, sedimentation. or creaming microscopy avoid confocal to in density scattering index minimize refractive their the to and both particles matches vol/vol the closely of 60 vol/vol mixture of solvent 40 This mixture and (cis-decalin) solvent aphtalene a and acid) brushes poly(hydroxystearic diameter with stabilized electrosterically with are that particles (PMMA) methacrylate) chemical particle uniform system. the a the throughout by establishes potential caused which forces gradients, pressure concentration osmotic are forces the dielectric by when is equilibrium balanced bottle reaches electric it the system, As sealed respectively. a particles, the of volume and where β moment, gradi- dipole field induced the electric moment an dipole when induced ent an appear acquire forces particles Dielectrophoretic The field. tric h atce n h upnigmedium suspending the the of and midplane particles constants the The dielectric is in 1B. software, have Fig. profile Multiphysics frequency in COMSOL field a with plotted at electric calculated electrodes The slides, the two MHz. across applied 1 is of V Cr/Au 80 1-mm-wide of alternating kinet- have An voltage crystallization slides regions. electrode-free on bottom with and work alternated top earlier electrodes the in both used (18): we ics cell the of that in BCC–liquid details the of stiff- that than the higher well, be as to found interface interface. it is FCC–liquid particles, which an the of create ness of length to screening possible and (19). is charge potential of the Yukawa simulations changing screened numerical By a of using results interfaces the BCC–liquid with them compare and doi:10.1073/pnas.2005664117/-/DCSupplemental at online information supporting contains article This 1 the under Published Submission.y Direct PNAS a is article This F.S. interest.y and competing no H.H. declare paper.y authors the research; The wrote F.S. performed and D.A.W., H.H. H.H., and research; data; designed analyzed F.S. contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To oycnee ui rsasi h oiicto fmetal of solidification the in observa- of alloys. crystals nucleation long-standing preferential cubic the a lower body-centered transformations: is for phase in value invoked tion phenomeno- measured the explanation the supports dielectrophore- interfaces That logical by alternative bottle. of controlled that electric than colloids an crystal using in cubic sis liquid body-centered a its between and interface the stiff- of the of ness determination experimental the reports paper This Significance (ε = h olia upnincnit fcagdpoly(methyl charged of consists suspension colloidal The with 1A, Fig. in depicted schematically is bottle electric The ∇ E p − xrsaforce a exerts t eindfessihl from slightly differs design Its Methods. and Materials ε m b eateto hsc,HradUiest,Cambridge, University, Harvard Physics, of Department )/(ε NSlicense.y PNAS p 2 + ε F m DEP and ), ε p ( = and φ ~ p and ∇ · ε . y m ε ) epciey nteelec- the in respectively, , eff V https://www.pnas.org/lookup/suppl/ E ~ p 3 = NSLts Articles Latest PNAS = % r h ouefraction volume the are −V βε tetrachloroethylene. m p ε /(1 % eff σ ε decahydron- − 0 ∇E 1.8 = ) βφ 2 | 2 /2 f5 of 1 with µm on ~ p .

APPLIED PHYSICAL SCIENCES in nearest and next-nearest neighbor particles is used to distin- guish the crystal from liquid (27). A perfect BCC crystal has an order parameter 7, four from nearest neighbor pairs and three from next-nearest neighbor pairs. We classify particles with ψi > 3.5 as crystalline and all others as liquid. A rendering of crystalline and liquid particles is shown in Fig. 1E. In addition, we use the number of crystal bonds Z, which is the number of crystal neighbors of a particle (14, 18). In an ideal BCC crystal, Z is 14. Next, we categorize the crystalline particles (ψi > 3.5) into two groups: crystal (Z ≥ 13) and interface (12 ≥ Z ≥ 7). The inter- face particles lie between the two phases and form a continuous two-dimensional surface with an area of 45 × 90 µm2. In most cases, the interface is close to parallel to the yz plane, indicated in Fig. 1A. The height variations from the time-averaged position are given by h(y, z, t) = x(y, z, t) − hx(y, z)it . [2] Fig. 1. (A) Schematic figure of the electric bottle. (B) Calculated electric field profile at 80 V. (C and D) Confocal xy images of interfaces between The Fourier transform is calculated according to liquid and BCC crystal. The yz interfaces have orientations (C) (100) and (D)   (114). (Scale bar: 20 µm.) (E) Reconstructed images of crystal (yellow) and 2 ∆yz X −i( 2π )mj −i( 2π )nk liquid (blue) particles. Projections of the particle positions in the crystals are A(k) = h(m, n) = √  h(j , k)e L e L , [3] found in SI Appendix, Fig. S4. S jk

where S is the total area in real space, L is the number of grid the PMMA particles have higher polarizability (εp = 2.6) than points, and ∆yz = 0.5 µm = 0.28 σ is the spacing of the grid ε the suspending medium ( m = 2.3); they move toward stronger at which the values of h are interpolated. Edge effects are min- electric fields. imized by imposing a circular mask with diameter equal to the Repulsive interactions between particles are established by edge length of the interface. adding dioctyl sodium sulfosuccinate (AOT). The AOT sur- The inverse averaged power spectra h|A(k)|2i−1, obtained factant molecules, which form inverse micelles above a critical from 200 configurations of the (100) and (114) interfaces, are micelle concentration (cmc), charge the particles (20–23). We shown as linear contour plots in Fig. 2. They appear to be chose 20 mM concentration to produce a balance between long azimuthally isotropic at low k, which corresponds to the contin- screening lengths (at low concentration but above the cmc) and uum limit of the CFM. The variation of the interfacial stiffness high surface charges (at high concentration), at which we antici- with angle in the plane of the interface is described by a second- κ−1 ≈ µ pate a screening length of 0.9 to 1 m (23); under these rank tensor (28). This implies that the stiffness of a fourfold conditions, these colloids form crystals at lower volume frac- symmetric (100) interface should be isotropic, as indeed it is. tions than hard-sphere colloids. These are known as colloidal The stiffness of a (114) interface, however, should be anisotropic, Wigner crystals, in analogy to the crystals created by electrons which it is not. A similar observation was made in hard-sphere on a semiconductor surface (24). FCC–liquid interfaces, where the (110) interfacial stiffness was The particles were located in a confocal microscope (25). found to be isotropic (14, 15). These discrepancies remain The sample cell was equilibrated for 3 d in the microscope unexplained. room, after which the electric field was applied for 3 d to reach The power spectra are angularly averaged to calculate the the dielectrophoretic equilibrium. This produced a crystal–liquid interfacial stiffness. Fig. 3A is a log–log plot of h|A(k)|2i vs. k for interface about 100 µm away from the electrode edge. Each par- −2 3 (100). At low k, the data follow the k behavior of the CFM. We ×100× µm −1 −1 ticle in a volume of 100 60 was imaged every 15 s. chose the range of 0.32 σ < k < 0.63 σ because a straight- The imaging frequency was chosen to sample a large interface line fit through the points in that range had a slope closest to area differently from the earlier measurements on crystallization 2 −1 −2(−2.03 ± 0.18). Fig. 3B shows the linear fit of h|A(k)| i vs. kinetics, where the image volume needed to be kept sufficiently k 2, which according to Eq. 1, gives an interfacial stiffness of γ˜ = small to capture the jump frequencies (18). We scanned 200 2 −4 0.149 ± 0.010 kB T /σ with an offset −0.003 ± 0.003 σ . For µ stacks, consisting of 240 xy slices every 0.25 m. the (114) interface, the power spectrum of Fig. 3C gives a range Fig. 1 C and D shows two confocal slices of the equilibrated BCC–liquid interfaces. We prepared the suspension with an ini- tial particle volume fraction φ ∼ 0.1 (particle number density ρ = 0.033 µm−3), less than required for crystallization. We chose this volume fraction based on our empirical observations of the phase behavior (26). The number density of the BCC phase near the interface is ρ = 0.039 µm−3. The corresponding conditions of packing fraction and AOT concentration are consistent with the BCC–liquid phase boundary identified by Kanai et al. (21). The crystal always forms with a (110) plane parallel to the cover- slip, presumably because this is the crystal plane with the largest areal density and hence, the favored one in heterogeneous crys- tal nucleation. Two vertical interfaces formed side by side at the same electrode edge, one with a (100) crystal boundary and the other with a (114) boundary. We define the interface by assigning a structure to each indi- vidual particle. An order parameter ψi , which we define as the Fig. 2. Linear contour plots of the inverse Fourier power spectrum number of in-line neighbor pairs (bond angles of 180 ± 19◦) h|A(k)|2i−1 of the (A) (100) and (B) (114) interfaces planes.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.2005664117 Hwang et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 pnigofescue ytedfeec ndniybtenthe negligible. between indeed density are in phases difference two the in by caused shown offsets displace- sponding As the horizontal experiments, the force. and by present stabilized vertical, dielectrophoretic is is interface the The interface horizontal. In the are ments role: (15). no negligible plays gravity were later offsets the particles, polymeric In the density-matched (14). more phases by the two for with the experiments accounted between In be difference quantitatively density horizontal. mea- the could were was which there interface colloids, offsets, silica the surable heavier the and with force experiments gravity, this the by 15), the (14, provided on experiments colloid effect was earlier differential the In negligible phases. a two having position interface D 3D offset Fig. an with in fit linear sponding 0.32 pro- height of region. interfacial function (114) (D) a and as (100) (B files (B) the CFM. of spectra with power averaged consistent lines) (dashed (C and wn tal. et Hwang directions of 3. Fig. hs ausaemc oe hntoeo h hard-sphere work earlier the in of studied those interface than (ρ crystal–liquid lower FCC much colloidal crys- are FCC values the These of density is number tal The vol- equilibrium. high 5A at at interface phase Fig. FCC repulsion fractions. the col- weaker produces causes charge-stabilized ume and AOT we of particles mM the interface, 5 system only between similar FCC–liquid of addition a the The in loids. of latter those the created with not interfaces does liquid hence, and interface (100) anisotropy. isotropic any the signify of uncer- that the than 10%. of of about variation The estimate is angular 2. which an apparent Fig. measurements, provides in the also observed in isotropy values tainty the the confirm in they variation and shown are 4, results Fig. The determined. in be could stiffness the of dence * ealddsuso fteucranylmti in is limit uncertainty the of discussion detailed A yfitting By ic ti fitrs ocmaetepoete fteBCC– the of properties the compare to interest of is it Since r ls ozr stersl ftefreta tblzsthe stabilizes that force the of result the is zero to close are .7and 0.27 = σ 14 nefcs h le oscrepn oa to correspond dots filled The interfaces. (114) ) ρ −1 (A 0.067 = < and h|A k k ˆ 0.002 C < ntepaeo h nefc,teaglrdepen- angular the interface, the of plane the in h oe spectra power The ) (k)| φ 0.65 µm 0 = k 2 2 ± ihalna ersint h ml aevector wave small the to regression linear a with , i −3 −1 1,1)adcoe otoeo our of those to closer and 15) (14, .53) 0.003 σ −1 n t ouefato is fraction volume its and , vs. sacnoa lc fa FCC–liquid an of slice confocal a is hr h F ple.Tecorre- The applies. CFM the where k σ 2 −4 o o ausof values low for , γ ˜ gives fte(1)itraei smaller is interface (114) the of htteofesi i.3 Fig. in offsets the That . h|A(k)| and γ ˜ IAppendix SI 0.179 = h corre- the Appendix, SI nes fteangularly the of Inverse D) 2 i −1 vs. . ± k k ∗ (100) (A) the for ln range a along , k 0.011 oeta the that Note −2 dependence φ k 0.20. = B B T and /σ 2 isaie rmteaiorp fteitrailenergy interfacial the of anisotropy the from arises ties normalized the of energies values relative interfacial the comparing for proxies valid FCC the in colloids range. charged short the being phase of repulsion the with of consistent value lower the preferred to the close of to value corresponds higher which the FCC, that note We FCC. for diagram phase the fraction, packing gives measured the at boundary gives gives diagram concentration AOT the fusion diagram: agreement phase good the in AOT the is from obtained concentration length screening the phase, BCC the For stiffnesses normalization: the of Turnbull values their the with nucle- compare to crystal useful the is affect it kinetics, directly ation properties interfacial the Since Discussion between found was interfaces 29–31). hard-sphere discrepancy 15, (110) interface (14, the similar of (110) simulations A and unlike the experiments anisotropic. Furthermore, found be for clear. simulations to ones the immediately measured measurements, not the the are charged below that for substantially sim- reasons The stiffness are (19). the potential values interfacial Yukawa in ulation screened BCC a reported uses the also which colloids, of is quantity simulation latter only The dimen- order. from the pronounced arises compare therefore differences quantities We these sionless hard- densities. colloid of the different charged of very Part those the the than interfaces. smaller of FCC is that turn, sphere in than which smaller interface, (15, FCC are results interfaces simulation BCC of and units experiments In colloid 19). of data loidal 5D Fig. k in fit of hard- The (exponent colloidal 15). on but (14, observations 0.32 the symmetry interfaces earlier in twofold FCC–liquid the isotropic its sphere is with from FCC–liquid stiffness agreement expectation the interfacial in the on they its not f): 5B, against Fig. and are plane, (110). in c they is plane shown 3 but interfacial the As figure field, of orientation (21, FCC The al. boundary. the phase with et in consistent Kanai fall entirely of do not diagram also phase are the They system. BCC–liquid ftehr-peesse,w s h au acltdb Alder 1. fusion by Table of calculated in (33): value entropy the al. the use For we et system, structure. subtle frac- hard-sphere the micelle the packing of AOT of phase because the the differ the and in may (i.e., values liquid shifts other basis the the observational whereas and tion), direct FCC a between has boundary it range since the value in fall may depend by which systems, calculated Yukawa entropies for parameter the 5) the (Fig. on use (32) we al. et colloids, Robbins charged the For ∆s † e gr nrf 3adfiue3cadfi e.21. ref. in f and c 3 figure and 23 ref. in 4 figure See B ebleeta h omlzditrailstiffnesses, interfacial normalized the that believe We col- other with together 1, Table in summarized are data The eosreta h ausof values the that observe We T f /σ σ stedmninesetoyo uin(nt of (units fusion of entropy dimensionless the is κ −1 2 ∆ −1 iha offset an with s < f 0.18 = .9 o h C hs,tevlei o clear: not is value the phase, FCC the For 0.69. = α ˜ ∆ k −2.1 ausfrteFChr-peeclodinterfaces, colloid hard-sphere FCC the for values s < f κ k −1 0.80 .9 h eutn ausof values resulting The 1.09. = B λ T γρ .TeFCetoyo uintherefore fusion of entropy FCC The µm. ˜ = ± h ifrnebtentetoquanti- two the between difference The α. 0.81 = κ /σ −2/3 κρ −1 σ gives 5C, Fig. in 0.1) −0.008 ∆ 2 −1 −1/3 h tfnse ftecagdcolloid charged the of stiffnesses the , s 0.91 = f hc orsod obs ierfit linear best to corresponds which , /k .4t .2 euetelower the use We 0.72. to 0.54 = B .Frtecytllqi phase crystal–liquid the For µm. where , T ± κ hc hwasmlrbtless but similar a show which , α ˜ 0.005 −1 .Ti ie netoyof entropy an gives This µm. o C r oe hnthose than lower are BCC for α ˜ κ † 1.6 = = −1 σ ihta bandfrom obtained that with γρ ˜ −4 NSLts Articles Latest PNAS ∆H stesreiglength. screening the is −2/3 . α ˜ f ,bttephase the but µm, γ ˜ o hre colloid charged for = 0.264 = ∆s γρ ˜ α f −2/3 ˜ k B r listed are T γ k ± where , B which , ∆ | are α, ˜ (5). ) 0.015 s f5 of 3 f for is ,

APPLIED PHYSICAL SCIENCES Fig. 4. Interfacial stiffness as a function of in-plane angle for the (A) (100) and (B) (114) interfaces. The horizontal lines indicate the averaged stiffness.

gives rise to the term γ00 in Eq. 1. This means that the values experiments in which the nucleation of metastable BCC phase is of γ˜ at the high-symmetry orientations represent extrema in the favored. angular dependence and that they bracket the value of the only It would be of interest to apply our results directly to the weakly anisotropic γ, as is indeed seen in computer simulations interpretation of homogeneous crystal nucleation experiments (6). The experiments on the FCC hard-sphere colloids show that in similar colloidal systems. Tan et al. (34) have performed the spread of γ˜ and hence, α˜ is about 20%, which means that α such experiments and report the presence of hexagonal close is about 10% lower than the maximum value of α˜ at (110). Using packed–like precursor coordinations in the liquid leading to the this result on the preferred value of α˜ for the charged colloid formation of metastable BCC or FCC phases. To test the clas- FCC (110) interface gives α ≈ 0.8. The anisotropy of the BCC– sical nucleation interpretation, however, it would be necessary liquid interfacial energy is much weaker than that for FCC, with to perform the experiments under conditions that yield suf- the spread between the extreme values of γ˜ being less than 5%, ficiently large critical nuclei with a well-formed crystal–liquid so that α˜ is close to α (6). interface. We therefore feel justified, for the purposes of comparison, in using the measured interfacial stiffnesses as proxies for the Materials and Methods interfacial energy and that the inequality we have measured The electric bottle (Fig. 1A) consists of two glass slides (22 × 22 mm2) that between α˜ for BCC and FCC is the most direct experimental are separated by ∼200 µm and filled with a colloidal suspension. The top evidence to date for the classical nucleation interpretation of the and bottom plates have three 1-mm-wide Cr/Au electrodes separated by

Fig. 5. (A) Confocal xy image of an interface between liquid and FCC crystal, with yz interface orientation (110). (Scale bar: 20 µm.) (B) Linear contour plot of the inverse Fourier power spectrum. (C) The power spectrum h|A(k)|2i−1 vs. k.(D) Inverse of the angularly averaged power spectrum as a function of k2 with a linear regression to the small wave vector region.

4 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.2005664117 Hwang et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 0 .J ot .At,Aoitccmuaino iuddfuiiy oi-iudinterfacial solid-liquid diffusivity, liquid of computation Atomistic Asta, M. Hoyt, J. J. 10. 1 .R ors opeempigo h nstoi reeeg ftecrystal-melt the of energy free anisotropic the of mapping Complete Morris, R. J. 11. 2 .R ors .Sn,Teaiorpcfe nryo h enr-oe crystal-melt Lennard-Jones the of energy free anisotropic The Song, X. Morris, R. J. 12. 3 .HradzGza,E .Wes h qiiru nrni rsa-iudinterface crystal-liquid intrinsic equilibrium The Weeks, R. E. Hernandez-Guzman, J. 13. 4 .B asenr .A et,F pee,Sifeso h rsa-iuditraei a in interface crystal-liquid the of Stiffness Spaepen, F. Weitz, A. D. Ramsteiner, B. I. 14. 5 .Z a Loenen van Z. S. 15. 7 .E ense,M .Slia,P .Cakn .VnBadrn ocnrtn colloids Concentrating Blaaderen, Van A. Chaikin, M. P. Sullivan, T. M. Leunissen, E. M. 17. Sullivan T. M. 16. m eapya Cvlaeo 0Vars h lcrdsa frequency a at Availability. electrodes Data the across V 80 of MHz. voltage 1 AC of an apply We mm. 1 wn tal. et Hwang .J .Hy,Aoitcadcniummdln fdnrtcsolidification. dendritic of modeling continuum and Atomistic Hoyt, solid-liquid the J. of J. anisotropy the solidification. computing 9. for Method Karma, A. in Asta, M. Hoyt, J. J. Fluctuations 8. in mobilities and Karma, energies free interfacial A. Crystal-melt Hoyt, 7. J. J. Asta, M. Sun, Y. D. metals. 6. liquid in nuclei crystal of Formation Turnbull, D. 5. Galenko, P. Holland-Moritz, D. Herlach, D. 4. droplets. Fe-Ni of Undercooling Cech, E. R. 1. .T okan .M elc,W L W. Herlach, M. D. Volkmann, T. 3. metallurgy. in structures Metastable Turnbull, D. 2. reeeg,adkntccefiin nA n Ag. and Au in coefficient kinetic and energy, free R Eng. energy. free interfacial (1993). Fe. BCC and FCC (1950). 2006). (Elsevier, nefc nAl. in interface experiments. solidification A Containerless II. Part melts. Fe-Cr-Ni (1981). interface. fcolloids. of adshr olia ytmmaue rmcplayfluctuations. capillary from measured system colloidal hard-sphere iudinterfaces. liquid (2010). 041603 iheeti edgains .Pril rnpr n rwhmcaimo hard- of mechanism growth and bottle. transport electric Particle an in I. crystals gradients. sphere-like field electric with 6–6 (1997). 461–469 28, 2–6 (2003). 121–163 41, .Ce.Phys. Chem. J. rc al cd c.U.S.A. Sci. Acad. Natl. Proc. neeti otefrcolloids. for bottle electric An al., et h atcenme density number particle the C 10 0.132 (110) BCC (110) BCC 0.26 (100) BCC 0.18 (110) FCC 0.15 (114) BCC (100) BCC stiffness, interface crystal–liquid parameter, the of values measured and Calculated 1. Table C 10 .305 .712 .7Hr-peeclod a onne l (15) al. et Loenen van (15) al. et Loenen colloids van Hard-sphere (15) al. et Loenen colloids van Hard-sphere colloids 1.17 Hard-sphere 0.90 1.04 1.27 0.98 1.13 0.27 0.27 0.53 0.27 0.53 0.53 0.53 0.41 0.47 (110) FCC (111) FCC (100) FCC C (111) BCC hs e.B Rev. Phys. hs e.B Rev. Phys. esrmn ftesifeso adshr olia crystal- colloidal hard-sphere of stiffness the of Measurement al., et hs e.Mater. Rev. Phys. l td aaaeicue nteatceand article the in included are data study All hs e.Lett. Rev. Phys. 9032 (2003). 3920–3925 119, y x 414(2002). 144104 66, 713(2004). 174103 69, α ˜ γ ˜ i h text) the (in sr ulainadpaeslcini undercooled in selection phase and Nucleation oser, ¨ (k 865(2019). 085605 3, B ± ± ± T /σ 59–50 (2009). 15198–15202 106, 50(2001). 5530 86, .102 .6 0.49 0.48 0.067 0.40 0.039 0.20 0.039 0.01 0.11 0.01 0.11 0.01 .Ce.Phys. Chem. J. 2 eatbeSld rmUdroldMelts Undercooled from Solids Metastable rn.Mtl.AIME Metall. Trans. ) eal ae.Tas B Trans. Mater. Metall. ρ φ hs e.B Rev. Phys. hs e.E Rev. Phys. hs e.Lett. Rev. Phys. ρ 658(2008). 164508 128, ftecytl iesols stiffness, dimensionless crystal; the of .Ap.Phys. Appl. J. (µm 116(2002). 214106 206, 8 (1956). 585 206, eal ae.Trans. Mater. Metall. −3 173(2006). 015703 96, ) 3441–3458 48, hs e.E Rev. Phys. IAppendix SI 1022–1028 21, γρ ˜ 695–708 12, ae.Sci. Mater. −2/3 0.120 0.138 0.136 ± ± ± /k .306 hre olisTi work This work work This This colloids Charged colloids colloids Charged Charged 0.67–0.92 0.68 0.03 0.58 0.03 0.03 82, B . T 4 .Tn .X,L u iulzn iei ahaso ooeeu ulainin nucleation homogeneous of pathways kinetic Visualizing Xu, L. Xu, N. Tan, P. Yukawa High-density of 34. V. dynamics. dynamics molecular in and Studies Young, A. diagram D. Hoover, Phase G. W. Grest, Alder, J. S. B. hard-sphere G. the 33. Kremer, of K. energies Robbins, free interfacial O. M. Anisotropic Song, 32. X. Houk, A. Mu, melt. the Y. from growth 31. crystal hard-sphere for coefficient Kinetic Laird, B. B. Amini, M. 30. 9 .L aicak .R ors .B ar,Teaiorpchr-peecrystal-melt hard-sphere anisotropic The Laird, B. boundary B. in grain Morris, measures of R. J. Determination structure Mendelev, Davidchack, I. L. crystal M. R. Srolovitz, local 29. J. D. of Du, D. Applications Zhang, H. Jones, 28. P. A. Ackland, particle PhD J. bottle,” colloidal G. electric an improve 27. in colloids to with studied algorithm transitions “Crystal-liquid iterative Hwang, H. An 26. Nakamura, N. Jensen, E. K. 25. colloidal experimental of elasticity Anisotropic Weitz, A. D. Spaepen, F. Russell, R. E. 20. melting and Heinonen crystallization of V. observation 19. Direct Spaepen, F. Weitz, A. D. Hwang, H. 18. 4 .Wge,Efcso h lcrnitrcino h nrylvl feetosin electrons of levels energy the solvents. on nonpolar interaction in electron stabilization the of Charge Effects Weitz, Wigner, A. E. D. 24. Dufresne, R. E. Hsu, entropic F. of M. observation Direct 23. Weitz, A. D. Schall, P. Spaepen, F. Zaccone, A. Sprakel, J. 22. Kanai T. 21. avr o i sitnewt h einadcntuto fteelec- the and of DMR-1206765 Contracts construction NSF and by design supported been the DMR-1611089. has with work This assistance tronics. his for Harvard ACKNOWLEDGMENTS. inrcrystals. Wigner Phys. Chem. tblzto fBCcytl ermelting. near crystals BCC of stabilization solvents. olia crystallization. colloidal spheres. and disks hard for (1968). entropy 3696 and state of equation systems. interfaces†. crystal-melt Lett. Rev. Phys. nefca reeeg rmfluctuations. from energy free interfacial simulation. dynamics molecular from stiffness simulation. and experiment (2016). MA Cambridge, University, Harvard thesis, locating. colloids. with metals. Langmuir α ˜ rn.FrdySoc. Faraday Trans. rsalzto n enrn etn fcagdclod nnonpolar in colloids charged of melting reentrant and Crystallization al., et .Ce.Phys. Chem. J. hs e.E Rev. Phys. e.Si Instrum. Sci. Rev. 8148 (2005). 4881–4887 21, uaasmlto ennne l (19) al. et Heinonen (19) al. et Heinonen simulation Yukawa simulation Yukawa uaasmlto ennne l (19) al. et Heinonen simulation Yukawa uaasmlto ennne l (19) al. et Heinonen simulation Yukawa 475(2013). 044705 138, C rsa-uditrailfe nryi uaasystems. Yukawa in energy free interfacial crystal-fluid BCC al., et γρ rc al cd c.U.S.A. Sci. Acad. Natl. Proc. ˜ 112(2006). 216102 97, hs e.E Rev. Phys. −2/3 Sample 331(2015). 030301 91, 2631 (1988). 3286–3312 88, /k etakWnedHl fteRwadIsiueat Institute Rowland the of Hill Winfield thank We a.Phys. Nat. .Py.Ce.B Chem. Phys. J. 613(2016). 066103 87, B 7–8 (1938). 678–685 34, hs e.B Rev. Phys. 330(2015). 032310 91, T n h Turnbull the and ; γ ˜ h akn fraction, packing the ; 37 (2014). 73–79 10, 514(2006). 054104 73, 5060 (2005). 6500–6504 109, 1018 (2019). 1180–1184 116, .Ce.Phys. Chem. J. hs e.Lett. Rev. Phys. pl hs Lett. Phys. Appl. Source NSLts Articles Latest PNAS 970(2006). 094710 125, 803(2017). 088003 118, .Ce.Phys. Chem. J. 297(2006). 121927 88, φ; 3688– 49, | f5 of 5 J.

APPLIED PHYSICAL SCIENCES