Direct Observation of Crystallization and Melting with Colloids Hyerim Hwanga, David A

Direct observation of crystallization and melting with colloids Hyerim Hwanga, David A. Weitza,b, and Frans Spaepena,1

aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; and bDepartment of Physics, Harvard University, Cambridge, MA 02138

Edited by William L. Johnson, California Institute of Technology, Pasadena, CA, and approved December 6, 2018 (received for review August 17, 2018)

We study the kinetics of crystal growth and melting of two types position. A powerful tool to this end is the control of the particle of colloidal crystals: body-centered cubic (BCC) crystals and face- density through dielectrophoresis. Sullivan and coworkers (12, centered cubic (FCC) crystals. A dielectrophoretic “electric bottle” 13) demonstrated how dielectrophoresis can be used to manipu- confines colloids, enabling precise control of the motion of the late particles in a confined system in a device that they labeled an interface. We track the particle motion, and by introducing a “electric bottle.” This technique provides better control of con- structural order parameter, we measure the jump frequencies centration than other body forces, such as gravity or temperature of particles to and from the crystal and determine from these gradients (14, 15). the free-energy difference between the phases and the interface The electric bottle, shown in Fig. 1A with details in SI mobility. We find that the interface is rough in both BCC and Appendix, is a capacitor that contains the colloidal suspension as FCC cases. Moreover, the jump frequencies correspond to those a dielectric spacer. In an electric field, particles with a dielectric expected from the random walk of the particles, which trans- constant εp in a medium with a dielectric constant εm acquire an lates to collision-limited growth in metallic systems. The mobility induced dipole moment ~p. Dielectrophoresis occurs, because the of the BCC interface is greater than that of the FCC interface. In electric field gradient ∇E, shown in Fig. 1B, exerts a force on the addition, contrary to the prediction of some early computer simu- 2 induced dipole moment FDEP = (~p · ∇)E~ = −Vp εeff ε0∇E /2, lations, we show that there is no significant asymmetry between 2 with εeff = 3βεm /(1 − βφ) , β = (εp − εm )/(εp + 2εm ), and φ the mobilities for crystallization and melting. the volume fraction of the particles (12, 16, 17). In a sealed electric bottle system, the system reaches equilibrium when all kinetics | crystallization | melting | colloids | phase transformation dielectric forces are balanced by the osmotic pressure forces from the density gradients. This corresponds to the chemical rystallization (solidification) and melting are among the potential, including both electrical and chemical contributions, Cmost studied phenomena for both fundamental and techno- being uniform throughout the system (18). logical reasons. Although there is an impressive body of macro- The colloidal suspension consists of 1.8-µm diameter scopic and thermodynamic data on these phenomena, direct poly(methyl methacrylate) (PMMA) particles sterically stabi- observation of the transformations at the atomic level remains lized with poly(hydroxystearic acid) brushes. The particles are elusive, because the presence of two condensed phases inhibits suspended in a nonpolar solvent that is a mixture of 60 vol/vol % any probe of individual atomic movements at the interface. One decahydronaphtalene (cis-decalin) and 40 vol/vol % tetra- of the questions that could be addressed by such observations, chloroethylene. We chose this solvent to closely match both the for example, is that of the ultimate growth velocity. It has been refractive index of the particles (to minimize scattering in confo- proposed (1), based on measurements of velocity vs. undercool- cal microscopy) and their density (to avoid sedimentation). The ing (2), that the attachment rate of liquid atoms to a growing interaction between the particles is repulsive as a result of the pure metallic crystal is collision limited (i.e., close to the inter- addition of dioctyl sodium sulfosuccinate (AOT) (SI Appendix). atomic vibration frequency). In that case, the ultimate growth By adjusting the AOT concentration, the particles can be made velocity would approach the velocity of sound. Should the growth to crystallize in the face-centered cubic (FCC) or body-centered be, in part, diffusion limited (i.e., involve major switches of nearest neighbors), the ultimate velocity would be much lower. The answer to this question is of interest, for example, in evaluat- Significance ing the possibility of forming pure metallic glasses (3). Computer simulations have shed some light on this (4, 5), but a direct inves- This paper reports on colloid experiments that provide a tigation on the atomic level is still lacking. Another question unique 3D experimental view at the particle level of the fun- that would benefit from direct investigation is the possibility of damental mechanism by which liquids crystallize and crystals the mobility for melting being different from that for crystal- melt. Since these mechanisms cannot be observed directly in lization, which has been claimed in some simulations (6, 7) and atomic systems, the colloids serve as models to identify, for dismissed by later ones (8). There is, however, a way to probe example, the equivalent of the often-invoked collision-limited the melting and crystallization behavior on an actual physical growth of pure crystals. Other observations include the mea- system: dense colloidal suspensions form liquid and crystalline surement of equal mobilities for growth and melting and the phases that are structurally identical to the simple atomic ones, mobility of the body-centered cubic interface being greater such as metals, but the size (∼ 1 µm) and slowness of the par- than that of the face-centered cubic one. ticles allow them to be tracked individually in the confocal microscope. Author contributions: D.A.W. and F.S. designed research; H.H. performed research; H.H. and F.S. analyzed data; and H.H., D.A.W., and F.S. wrote the paper.y Results The authors declare no conflict of interest.y Colloidal suspensions and atomic systems exhibit similar phases, This article is a PNAS Direct Submission.y in which, for colloids, the particle concentration is a key param- Published under the PNAS license.y eter in the phase behavior (9). The growth of colloidal crystals 1 To whom correspondence should be addressed. Email: [email protected] from their liquid has been studied under conditions of nucleation This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. (10) or sedimentation (11), but a detailed study of the crystalliza- 1073/pnas.1813885116/-/DCSupplemental.y tion and melting kinetics requires closer control of the interface Published online January 7, 2019.

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PHYSICS Table 1. Average rates of attachment and detachment at The jump frequency k of an atom in a singly activated pro- ∗ the interfaces cess can be written as k = Γexp(−∆G /kB T ), in which Γ is an ∗ −1 −1 attempt frequency and ∆G is an activation energy. In some Jump rate, s · site interface BCC FCC cases, such as pure metals or noble gases, the mechanism is Equilibrium thought to be collision limited, which corresponds to the absence + k 0.1025 ± 0.0014 0.0731 ± 0.0010 of an activation barrier ∆G∗ ≈ 0 and hence, k = Γ. Evidence − k 0.1026 ± 0.0014 0.0731 ± 0.0010 for this type of growth includes measurements of the growth Growth speed as a function of temperature (1, 25, 26). To establish + k 0.0885 ± 0.0013 0.0679 ± 0.0010 the attempt frequency Γ in our experiments, we investigated − k 0.0859 ± 0.0013 0.0645 ± 0.0010 the Brownian motion of the particles in the crystal, interface, Melting and liquid at equilibrium. We determined the mean square dis- + k 0.0893 ± 0.0013 0.0696 ± 0.0010 placements (MSDs) of the particles, which are plotted as a − k 0.0932 ± 0.0013 0.0732 ± 0.0010 function of time in Fig. 5. The MSD of the crystal particles sat- 2 2 2 urates at h∆r iBCC = 0.72 and h∆r iFCC = 0.44 µm due to confinement by their lattice cages, while particles at the interface † potential jump sites Nsite at the interface. We estimate the lat- and in the liquid have diffusive motion, which is evident from ter as Nsite = Al/Vp , where Vp is the average volume of the the linearly increasing MSD. That the MSD of interfacial par- particles in the crystal (equal to the average Voronoi cells in ticles is less than that in the liquid is the result of their partial SI Appendix, Table S1), A = 1,200 µm2 is the cross-sectional constraint by the crystal. A least squares linear fit to the liquid 2 1/3 data gives diffusion coefficients Dliquid = h∆r i/6t of 0.0264 ± interface area, and l = Vp is the particle dimension along the ± µm2 growth direction. The imaging interval of 4 s is sufficiently short 0.0010 and 0.0239 0.0011 /s for BCC and FCC, respec- Γ for each observation of a jump to represent a single event. The tively. The attempt frequency, , is the inverse of the average λ results are listed in Table 1. At equilibrium, the attachment and time that it takes a particle to travel the jump distance, j . The + − attempt frequency for Brownian motion in three dimensions is detachment rates are equal (k = k ), while for crystallization, 2 + − − + Γ = 6D /λ k > k , and for melting, k > k . The average jump distances 3D liquid j , which using the average measured values for λ ± ± −1 λj for attachment and detachment at equilibrium were mea- j mentioned above, gives 0.23 0.01 and 0.25 0.01 s for sured to be 0.83 ± 0.01 and 0.76 ± 0.01 µm for BCC and FCC, BCC and FCC, respectively; these values are factors of two and respectively. Note that these jump distances λj are significantly three, respectively, greater than the jump frequencies for attach- shorter than the interparticle spacings (SI Appendix, Table S2). ment and detachment listed in Table 1. Given that the attach- In other words, a small adjustment λj in the position of the parti- ment and detachment jumps have a pronounced directionality cle changes its nature between crystalline and liquid and thereby, 1/3 moves the interface locally by a much larger distance l = Vp . A 2D projection of the trajectory of a single particle near the interface in equilibrium is shown in Fig. 3. It shows the particle first jumping back and forth between crystal and liquid, making a quasirandom walk. After the particle attaches itself to the crystal, it is trapped for a time. The changing positions of BCC– and FCC–liquid interfaces during growth and melting are shown in Fig. 4. The interface velocities v = ∆xinterface /t were determined by measuring the total length of the crystal xc directly on the cell once every 8 s. The resulting “macroscopic” velocities are listed in Fig. 4. The “microscopic” interface velocities are obtained from the measured net jump rates. When liquid particles join the crystal, each attaching particle adds a volume Vp to crystal. + − When ∆Njump = N − N particles join, the interface moves by 1/3 l∆Njump /Nsite with l = Vp . The microscopic interface velocity is then l∆N jump = l(k + − k −). [1] Nsite ∆t

Using the values of the jump rates, from Table 1, we obtain microscopic velocities: vgrowth = 0.0081 ± 0.0056 µm/s and vmelting = −0.0120 ± 0.0058 µm/s for the BCC interface and vgrowth = 0.0089 ± 0.0037 µm/s and vmelting = −0.0099 ± 0.0039 µm/s for the FCC interface. The close agreement between the macroscopic and microscopic interface velocities validates the choices that we made in deﬁning the order parameters and the attachment/detachment criteria. To check if the magnitude of the electric ﬁeld has any effect on the jump frequencies and growth velocities, we established that identical results could be obtained at 60 and 80 V (SI Appendix, Figs. S2 Fig. 3. A 2D projection of the trajectory of a particle near the BCC–liquid and S3). interface at equilibrium. (A) The color gradient of the pixels encodes points along a particle trajectory (red star, starting point; blue star, end point). The average position of the crystal–liquid interface is shown by the dotted line; the crystalline and liquid areas are yellow and blue, respectively. (B) A map †Potential sites are those where the arrival or departure of a particle would turn it into of the particle identity (yellow, liquid; orange, interface; red, crystalline) a crystalline or a liquid one, respectively. along its trajectory.

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Melting −0.0428 −0.0504 | o 4 no. ± ± 0.0198 0.0196 | 1183 -

PHYSICS Table 3. Mobilities of the BCC and FCC crystal–liquid interface in chemical potential, ∆G, caused by the concentration gradient during growth and melting produced by the dielectrophoretic forces. The driving force is the gradient of the chemical potential, Fx = ∇G = ∆G/λ. The drift Interface mobility, M (107 m2/J · s) BCC crystals FCC crystals velocity in response to the driving force is, according to Newton’s Equilibrium 3.26 ± 0.04 1.91 ± 0.03 third law, Growth 2.80 ± 0.04 1.64 ± 0.02 Fx τ ∆Gλ vd = = . [4] Melting 2.90 ± 0.04 1.83 ± 0.04 2m 2kB T τ In a 1D random walk, a particle makes l steps to the right with a probability p in n trials and n − l steps to the left with a and the FCC interfaces. This is in line with the most recent probability q = 1 − p so that hli = np and hn − li = nq. The aver- simulation results (8), which contradicted earlier findings of age distance that the particle travels is the net number of steps asymmetry (6, 7). in the growth direction (suppose to the right): hx(n)i = hliλ − These measurements provide a direct view of the particle- hn − liλ. The drift velocity is then vd = hx(n)i/τ with n = t/τ, level dynamics and behavior of crystallization and melting at a which then gives vd = (p − q)λ/τ. Equating this to the drift crystal–fluid interface in a colloidal system. The observed behav- velocity in Eq. 4 then gives ∆Gλ/2kB T τ = (p − q)λ/τ. Thus, ior provides an experimental test of computer simulations and we find ∆G yields insight into the properties of atomic systems that are not p − q = . [5] otherwise accessible experimentally. 2kB T The ratio of these probabilities is also the ratio of the jump Materials and Methods frequencies in the transformation: The sample cell (Fig. 1A) consists of two coverslips (22 × 22 mm2) separated by a spacer (∼200 µm). The top plate has three 1-mm-wide gold electrodes + ∆G p k 1 + 2k T separated by 1 mm. The bottom plate is a continuous indium tin-oxide elec- = = B , [6] q k − 1 − ∆G trode. We apply an ac voltage across the electrodes at a frequency of 1 MHz. 2kB T The calculated electric field profiles in the center plane between the plates at 60 and 80 V are shown in Fig. 1B. which gives for the driving force Appendix: Driving force for the biased random walk (32) k + − k − A particle with a mass m has a Brownian velocity vB = λ/τ, ∆G = 2kB T . [7] k + + k − where λ and τ are jump distance and time, respectively. Application of the equipartition principle to one component 2 2 ACKNOWLEDGMENTS. We thank Winfield Hill (Rowland Institute at Har- of the kinetic energy, kB T /2 = mvB /2, gives m = kB T /vB = vard) for his assistance with the design and construction of the electronics. 2 2 kB T τ /λ . The phase transformation is driven by a difference This work was supported by NSF Contracts DMR-1206765 and DMR-1611089.

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