Dynamics and unsteady morphologies at ice interfaces driven by D₂O–H₂O exchange

Significance

Freezing and melting of ice are one of the most common events on Earth. The dynamics of ice crystallization are relevant to climate research, mitigating frost damage in agriculture and construction, glacier dynamics, tissue and food preservation, and transportation. We describe the use of microfluidic devices, accompanied by precise temperature control, to examine the effect of H/D isotope exchange between liquid light water and solid heavy water on ice growth dynamics. These studies revealed unusual morphologies at the ice surface in contact with the liquid, including curious unsteady morphological features that give the appearance of oscillation due to complex interplay of H/D exchange, thermal gradients, and local surface curvature.

Abstract

The growth dynamics of D₂O ice in liquid H₂O in a microfluidic device were investigated between the melting points of D₂O ice (3.8 °C) and H₂O ice (0 °C). As the temperature was decreased at rates between 0.002 °C/s and 0.1 °C/s, the ice front advanced but retreated immediately upon cessation of cooling, regardless of the temperature. This is a consequence of the competition between diffusion of H₂O into the D₂O ice, which favors melting of the interface, and the driving force for growth supplied by cooling. Raman microscopy tracked H/D exchange across the solid H₂O–solid D₂O interface, with diffusion coefficients consistent with transport of intact H₂O molecules at the D₂O ice interface. At fixed temperatures below 3 °C, the D₂O ice front melted continuously, but at temperatures near 0 °C a scalloped interface morphology appeared with convex and concave sections that cycled between growth and retreat. This behavior, not observed for D₂O ice in contact with D₂O liquid or H₂O ice in contact with H₂O liquid, reflects a complex set of cooperative phenomena, including H/D exchange across the solid–liquid interface, latent heat exchange, local thermal gradients, and the Gibbs–Thomson effect on the melting points of the convex and concave features.

Ice formation is arguably the most common crystallization process on Earth, ranging from the formation of atmospheric snow and ice to annual cycles of freezing and thawing of ground water. Ice has more solid phases than any other substance, with 17 polymorphs reported (1) and another predicted (2). The growth dynamics of common, hexagonal (Ih) ice are essential in climate modeling, mitigating structural damage due to freeze–thaw cycles (3, 4), glacier dynamics (5), cryobiology and cryopreservation (6), and the design of new materials through ice templating (7). Heavy water (D₂O) also adopts the same Ih structure and crystal habit as light water (H₂O) (8), but melts at higher temperatures T_m=3.8°C vs. T_m=0°C, respectively. D₂O has been used as a solvent in the cold preservation of organs (9). The higher melting temperature and lower vapor pressure of D₂O are essential for estimating millennial global temperatures through analysis of deuterium composition in ice cores (10). The fractionation of water isotopomers is a manifestation of preferential freezing of D₂O with respect to H₂O, which affects the distribution of environmental deuterium (11, 12). Isotopic fractionation also can occur through H/D exchange, a process that has been investigated at temperatures ranging from −178 °C to −2 °C (13–20). H/D exchange at lower temperatures (up to −103 °C) has been investigated by measurement of changes in isotopic composition in thin stratified films of D₂O–H₂O–HDO (1–100 nm in thickness) (13, 14, 17, 19). The diffusion of tritium into bulk H₂O ice has been measured over the range −35 °C to −2 °C (18, 20), with diffusion coefficients spanning seven orders of magnitude. Comparison of diffusion coefficients for ice films and bulk crystals from measurements performed over such a large range of temperatures may be problematic because (i) extrapolation of Arrhenius behavior may not be reliable over the temperature range; (ii) different diffusion mechanisms may be operative at lower temperatures compared with those at higher temperatures (interstitial ion migration vs. vacancy migration); (iii) thin ice films may be polycrystalline or even amorphous, compared with the single crystallinity of bulk ice; and (iv) thin ice films tend to have more defects than single crystals, affording larger-than-actual diffusion rates in films (17). Herein, we describe ice crystallization on the surface of a D₂O crystal in liquid water within a microfluidic device for which temperature can be controlled with a precision of 0.001 °C and a stability of ±0.002 °C. At a constant temperature below the melting point of D₂O, 3.8 °C, the crystal interface melted slowly. As the system was cooled continuously at a sufficiently high rate, the ice front advanced, but retreated immediately after cooling ceased, signaling exchange of H for D in the solid D₂O that reduced the ice melting temperature. At a constant system temperature (the temperature that was measured by the thermistor), the liquid–solid interface exhibited cyclical and unsteady morphologies owing to the combined effects of H/D exchange, latent heat dissipation, local thermal gradients, and the Gibbs–Thomson effect.

D₂O–H₂O Phase Diagram

Several investigations of ice-containing mixtures of H₂O and D₂O (21–23), and implicitly HOD, have been reported, relying on vibrational spectroscopic assays of HOD through the appearance of separate O-H and O-D vibrational bands. These reports, however, have not addressed the melting (or freezing) habits of H₂O:D₂O mixtures. Measurement of the melting points across a range of H₂O:D₂O compositions can permit construction of a phase diagram, which in turn can be used to deduce the equilibrium composition of a H₂O–D₂O liquid–solid interface during growth or melting. This was achieved by insertion of a small amount of premixed solutions (0.5 $\mu$L total volume) with various H₂O:D₂O compositions into an oil droplet resting on a sapphire disc placed on a custom-built cold stage (Fig. S1). The oil droplet was then topped with a glass coverslip, extruding the oil and leaving water trapped between the coverslip and the sapphire disc (24). The assembly was cooled to −20 °C and then heated slowly to melt the ice partially, affording a disk-shaped single crystal of ice. Optical microscopy was used to determine the melting temperature of the single crystal, enabling construction of a simple binary phase diagram, which displayed a linear relationship between the melting temperature and isotopic composition (Fig. 1).

Fig. S1.

(Top) Custom-built cold-stage setting. (Bottom) Stability of the cold-stage temperature, measured with a thermistor, at 3.0 °C and after cooling to 2.0 °C at 0.05 °C/5 s. Insets depict the temperature stability before and after cooling.

Fig. 1.

Binary phase diagram for H₂O:D₂O mixtures determined from visual observation of freezing/melting (red circles) and the predicted melting points (black line). The phase diagram follows a simple linear dependence between melting temperature and isotopic composition. Inset shows an ice crystal formed in a 75%-H₂O–25%-D₂O mixture at 0.996 °C. The disc-shaped morphology is typical of crystals grown across the isotopic range. The upper face of the disc corresponds to the 0001 plane. (Scale bar, 10 $\mu$m.)

H₂O Crystal Growth on D₂O Ice

Crystal growth resulting from ice deposition on a D₂O crystal in liquid H₂O was examined in a microfluidic device that initially contained neat D₂O. The temperature of the cold stage was controlled to within ±0.001 °C. After lowering the temperature to −20 °C to freeze the liquid in the microfluidic channel, the resulting D₂O ice was melted partially at 3.6 °C until a well-formed single crystal remained in the channel. The length of these crystals ranged from 80 $\mu$m to 800 $\mu$m. The channel dimensions of the primary compartment prescribed their width and height (150 $\mu$m and 20 $\mu$m, respectively). The liquid D₂O surrounding the crystal was then replaced with H₂O, and the temperature was lowered immediately to a value below 3 °C to limit melting of the crystal. The reliability of this protocol was assessed using an indicator dye that enabled visualization of the displacement of D₂O by H₂O. Injection of an H₂O solution containing sunset yellow dye FCF (for coloring food; 1% wt/wt; CAS no. 2783-94-0) into the microfluidic channel confirmed plug flow displacement of the liquid D₂O. All subsequent experiments were performed without sunset yellow, however. At constant system temperature below the D₂O melting point, the D₂O crystal interface melted, as observed by the retreat of the interface, signaling a continuous reduction in the melting point due to H/D exchange across the interface. For example, at 1 °C the solid–liquid interface retreated at a rate of ca. 0.5 $\mu$m/s. This exchange, which melts the interface, adds H to the D₂O ice while enriching the solution near the solid interface with D.

When the temperature was decreased at a rate of 0.002–0.1 °C/s from an initial temperature between 2 °C and 0.2 °C, the growth front of a D₂O crystal in contact with the added liquid H₂O advanced, which can be attributed to freezing of liquid near the interface that contained D₂O from prior melting of the interface. This new layer must have a composition governed by the phase diagram in Fig. 1. If the temperature was held constant after continuous cooling, this newly crystallized region melted over a few seconds. For example, lowering the temperature from 0.786 °C to 0.682 °C at 0.02 °C/s resulted in rapid growth of ice on the D₂O crystal, accompanied by the formation of a distinct interface between the new H₂O-rich ice and the D₂O ice (Fig. 2 A and B). After the cooling was stopped and the cold stage stabilized at 0.699 °C, the H₂O-rich ice melted rapidly at 2.2 $\mu$m/s to the original D₂O crystal boundary, after which the remaining D₂O crystal melted more slowly at 0.25 $\mu$m/s, limited by H/D exchange across the interface (Fig. 2 C and D). Above 0.002 °C/s, the growth rate of the ice front increased linearly with increasing cooling rate (Fig. S2). Raman microscopy of the newly H₂O-rich ice performed on an area ∼10 $\mu$m from the D₂O ice front revealed compositions near those expected from the phase diagram in Fig. 1 (Fig. S3).

Fig. 2.

(A) A D₂O crystal in H₂O liquid. (B) The crystal grew after lowering the temperature, revealing a distinct interface between the oval D₂O crystal in the center and the surrounding H₂O crystal. The dashed oval is a visual aid. (C) After raising the temperature, the newly formed ice retreated. (D) Raising the temperature melted the entire H₂O crystal, leaving the D₂O crystal with the original shape intact. (Scale bar, 20 $\mu$m.)

Fig. S2.

Growth rates of D₂O ice fronts at various cooling rates.

Fig. S3.

(A and B) Comparison of Raman spectra of ice crystals with a known composition (black, data from Fig. S7) with spectra acquired after quenching new-formed ice (red) grown at (A) 1 °C and (B) 1.7 °C.

In another experiment in which the temperature was lowered from 0.745 °C at 0.01 °C/s, the D₂O ice interface exhibited an unsteady scalloped morphology (Fig. 3 A–E and Movie S1). The cellular features of this interface merged to form a flat ice front upon further cooling (Fig. 3 F and G). Like the crystal in Fig. 2, the solid–solid interface between the D₂O crystal and the added H₂O ice was readily apparent (Fig. 3G, dashed box). Upon cessation of cooling the ice growth stopped, and the ice front retreated again to the position of the former D₂O crystal interface (Fig. 3H). The unsteady cellular features were not observed at cooling rates exceeding 0.02 °C/s (Fig. S4). Collectively, the cellular features in Fig. 3 suggest nonlinear behavior that results from a competition between the timescales for cooling and H/D exchange, the former promoting growth and the latter promoting melting of the ice interface. Both processes require exchange of latent heat. When the driving force (supercooling) is large (Fig. 3 D–G), growth becomes more competitive than H/D exchange, and the advancing growth front is flat, absent of cellular features. Notably, these features were not observed for D₂O ice in contact with D₂O liquid or H₂O ice in contact with H₂O liquid; thus any explanation must involve H/D exchange and diffusion as well as thermal effects.

Fig. 3.

(A) A single crystal of D₂O ice after replacement of liquid D₂O with H₂O. (B–E) The D₂O ice crystal revealed a cellular growth front as the temperature was decreased stepwise by 0.05 °C every 5 s (corresponding to 0.01 °C/s) with the number of cells increasing at first and then merging. (F and G) The ice front flattens and a faceted plane emerged, indicating the basal plane of the crystal (solid arrow, G). The original solid–solid interface is evident within the dashed rectangle. (H) After 36 s at near-constant temperature the newly formed ice completely retreated to the original position of the D₂O ice interface. (Scale bar in A, 25 $\mu$m.) (Movie S1.)

Fig. S4.

(A) The liquid H₂O–D₂O interface at 1.264 °C. (B) Advancement of the ice growth front after cooling for 6 s at 0.02 °C/s, revealing a flat advancing interface. (C) An advancing ice front, exhibiting a linear facet, which is commonly assigned to the basal (0001) plane (solid arrow), observed at a later stage of growth. (D) Upon cessation of cooling, the newly formed ice melted and the solid–liquid interface retreated to the original location. (Scale bar, 25 $\mu$m.)

The interface between D₂O and H₂O crystals was examined during rapid lowering of the temperature of a D₂O crystal in contact with liquid H₂O from 0.998 °C to −0.211 °C. The liquid in the channel froze, creating a noticeable discontinuity between the D₂O crystal and the new H₂O ice (Fig. S5). The interface between the D₂O and H₂O crystals gradually blurred, becoming imperceptible after ∼20 min (Fig. S5 C–E). The discontinuity suggests a grain boundary between the old (D₂O-rich) and the new (H₂O-rich) ice, which may reflect a disordered liquid layer that persisted during freezing, possibly akin to quasi-liquid layers invoked for grain boundary melting (25). It is reasonable to suggest that the optical contrast across the width of the discontinuity stems from a different refractive index ($\eta =1.24$ if amorphous) compared with hexagonal ice $(\eta =1.3)$ (26). The disappearance of this region likely reflects restructuring of the disordered region to match the lattice of the solid ice on either side. This process would require molecular motion, and it is reasonable to expect that the timescale would be comparable to that for diffusion. The blurring of the H₂O–D₂O discontinuity at the solid–solid interface formed as described above suggests a change in structure or composition due to H/D exchange, widening a mixed interface accompanied by the formation of HOD.

Fig. S5.

(A) A D₂O ice crystal in liquid H₂O. (B) After the temperature was lowered rapidly (ca. 0.5 °C over 3 s) a distinct solid–solid interface was formed. (C–E) At constant temperature the boundary remains distinct but begins to blur. (F) After raising the temperature to 0.943 °C, the H₂O retreats to the former interface, consistent with the behavior observed in Figs. 2 and 3. (Scale bar in A, 20 $\mu$m.)

The changes in the composition of the D₂O single ice crystal were measured by Raman microscopy at subfreezing temperatures in the range −0.2 °C to −0.5 °C, which were performed with a precision of ±0.001 °C and spatial resolution of ±0.5 $\mu$m. The Raman microscope was focused on the D₂O side of the conjoined H₂O–D₂O ice crystal, 10 $\mu$m from the original D₂O–H₂O interface. The reduction in D₂O content and the increase in HOD due to H/D exchange were evident from changes in the intensity and the position of the ν_O-D bands between 2200 cm⁻¹ and 2600 cm⁻¹ (Fig. S6). Initially, the spectrum revealed ν_OD stretching modes typical for pure D₂O at 2337 cm⁻¹ and 2483 cm⁻¹ (23, 27). Over time, the intensity of the D₂O peaks decreased concomitant with a shift of the 2337-cm⁻¹ peak toward 2440 cm⁻¹, the ${\nu }_{\text{OD}}$ stretching vibration for HOD (28). The amount of H/D was determined by measuring the reduction in the integrated intensity in the region of interest and by using a calibration curve of predetermined H₂O:D₂O compositions (Materials and Methods and Figs. S6 and S7). The diffusion coefficient of H⁺ (or H₂O) into the D₂O ice was determined using a standard semiinfinite slab model (Eq. 1), where $C_s$ is the H₂O fraction at the D₂O–H₂O interface (determined from the phase diagram), C is the H₂O fraction at a distance x from the D₂O–H₂O interface, x is the distance from the D₂O–H₂O interface where the Raman measurement is positioned (x = 10 $\pm$ 0.5 $\mu$m), D is the diffusion coefficient, and t is time. A reasonable fit of Eq. 1 to the data afforded D = 0.59 $\pm$ 0.1 $\times$ 10⁻¹⁰ cm²$\cdot$s⁻¹ (Fig. 4). Notably, the timescale for diffusion of H₂O through a typical 3-$\mu$m-thick layer using this value of D is ∼25 min, comparable to the time for near disappearance of the interface region in Fig. S5E.

$$C=C_s\left[1-erf\left(\frac{x}{2\sqrt{Dt}}\right)\right]$$ [1]

The aforementioned diffusion coefficient compares favorably to the values of D = 1.0 $\pm$ 0.2 $\times$ 10⁻¹⁰ cm²$\cdot$s⁻¹ reported for D₂O diffusion in polycrystalline ice at −1.8 °C and D = 0.1 $\pm$ 0.1 $\times$10⁻¹⁰ cm²$\cdot$s⁻¹ for T₂O diffusion in single-crystal ice at −10 °C (18, 29). Using the reported value of $D_0=10.63$ cm²/s measured for T₂O diffusion in a single crystal of H₂O ice (18), the data in Fig. 4 were consistent with an activation energy E_a = 58.7 ±0.2 kJ/mol. This value is comparable to that reported for T₂O in ice, E_a = 59.8 ± 6.7 kJ/mol (18); diffusion of T₂O into natural single ice crystals, E_a = 65.7 ± 8.3 kJ/mol (22); and the activation energy for desorption of H₂O from 6-$\mu$m-thick ice films (58.1 ± 0.8 kJ/mol) (30). In contrast, the diffusion coefficient for HF, which may diffuse as an intact molecule or as ions, in single ice crystals at −10 °C was reported as D = 0.8 ± 0.5 $\times$ 10⁻⁶ cm²$\cdot$s⁻¹, substantially larger that the values above (31). Collectively, these data implicate diffusion of intact water molecules through a vacancy mechanism for H/D exchange in the D₂O crystal (8, 18, 32).

Fig. S6.

(A) Representative time-dependent Raman spectra of D₂O ice acquired at 10 $\pm$ 0.5 $\mu$m from the D₂O–H₂O interface, collected at the time intervals indicated (minutes). The data were background subtracted and baseline corrected, and the integrated intensities between 2200 cm⁻¹ and 2500 cm⁻¹ (dashed lines) were normalized to the first spectrum collected at $t=0$ min and calibrated using the calibration curve (Fig. S7B). The first and last spectra acquired at 0 min and 540 min are shown as thick solid lines for clarity. (B) The O-D concentration calculated from the spectra in A and four additional Raman measurements. These data points were averaged and binned (Fig. 4).

Fig. S7.

(A) Raman spectra of single crystals of ice at various D₂O–H₂O compositions (%D₂O in H₂O), measured just below their melting points. The data were background subtracted and baseline corrected, after which the spectra of pure H₂O were subtracted from each spectrum. (B) Dependence of the integrated intensity for the various D₂O–H₂O compositions in A. Each intensity value was normalized to the integrated intensity at 100% D₂O. The slope of this plot was used to calibrate the integrated intensity measured during H/D exchange. The difference in the values of the diffusion coefficient with and without this calibration curve was only 5%.

Fig. 4.

The decrease of D₂O concentration in a D₂O crystal due to H/D exchange between the D₂O crystal and an adjoining H₂O crystal, measured at x = 10 $\pm$ 0.5 $\mu$m from the H₂O–D₂O interface. The data were fitted to a semiinfinite model (Eq. 1). Data points represent an average of at least three measurements. Error bars correspond to the SE. See Fig. S6 for details.

Unsteady Interface Morphologies

As described above, when the temperature of the microfluidic channel containing a D₂O crystal in contact with liquid H₂O was lowered quickly from ca. 2.6 °C to a constant temperature in the range 2.6–0.6 °C, the flat ice front grew, but then retreated, with lower rates of retreat at lower temperatures. Cooling the device from 2.6 °C to below 0.6 °C also resulted in an initial period of growth followed by retreat of the H₂O-rich ice front. Once the ice front retreated to the original D₂O crystal interface, unsteady morphologies like those in Fig. 3 were observed, growing and melting in an unsteady, almost oscillatory, manner across the interface (Fig. 5 and Movie S2). These unsteady morphologies were not observed for D₂O ice in contact with D₂O liquid or H₂O ice in contact with H₂O liquid, arguing against impurities as a major factor in their formation. The interface was decorated with convex features having radii of curvature ranging from 5 $\mu$m to 15 $\mu$m, alternating with deep concave features having large negative curvature. Under constant system temperature, the convex and concave features cycled every 20–120 s, persisting during a slow retreat of the average position of the D₂O crystal interface, which typically required 8 h to traverse the length of the microfluidic chamber (3.6 mm, Movie S3). These unsteady morphologies vanished if the temperature was raised by >0.025 °C (Movie S2). In a few trials, the convex and concave features of the interface coincide in successive cycles to give the appearance of wavelike patterns, denoted by the vertical dashed lines in Fig. 5. Most experiments, however, reveal an unsteady cycling of the interface morphology, usually with new convex features emerging out of regions of sharp negative curvature where shrinking convex features intersect the interface. The evolution of the curved features was not a consequence of crystal anisotropy (Fig. S8).

Fig. 5.

Unsteady morphologies oscillating between positive (convex) and negative (concave) curvature at the solid–liquid interface during melting of a D₂O ice crystal in liquid H₂O under isothermal conditions (0.506 °C). In this example, the unsteady morphologies oscillated between convex and concave at fixed positions throughout, as denoted by the dashed lines in the corresponding panels on the right. The dashed lines intersect regions of negative curvature in A and E, and regions of positive curvature in C and G. Bottom Center panel illustrates the relationship between the interface temperatures and the bulk melting temperature as well as the gradients of D and H in the solid and liquid phases. The solid–liquid interface here is defined by x = 0. Local thermal gradients due to local nonisothermal conditions accompanying melting and freezing of the convex and concave features also are expected (not shown).

Fig. S8.

Convex and concave features (A and D) were observed regardless of the growth direction of the D₂O ice crystal, as indicated in C and B.

These observations suggest that the growth and dynamics of the unsteady morphologies depend on a complex set of cooperative phenomena as described below, including H/D exchange from the H₂O-rich liquid into the D₂O-rich ice crystal, latent heat exchange, local thermal gradients, and the Gibbs–Thomson effect on the melting points of the convex and concave features. The Gibbs–Thomson effect (33) predicts that the melting temperature of a curved interface, $T_i$, can be calculated using Eq. 2, where $T_m$ is the melting temperature of the bulk ice (which is nearly equal to the temperature measured by the thermistor mounted in a copper plate beneath the microfluidic device; Fig. S1), $\gamma$ is the energy of the solid–liquid interface, $\kappa$ is the curvature or reciprocal radius of the curved feature, and $L$ is the latent heat of ice). The $\kappa$ term is positive when the feature is convex (34). Convex features (positive curvature) are susceptible to melting because $T_i>T_m$, whereas concave features (negative curvature) can promote ice growth because $T_i<T_m$. Using the range of curvature radii observed here, a surface tension $\gamma =0.033$ J$\cdot$m⁻² and L = 3.34 × 10⁻⁸ J$\cdot$m⁻³, the magnitude of the $T_i$ values for the convex and concave features is anticipated to be in the range 0.001 ≤ $|T_i|$ ≤ 0.005 °C. This small difference is within the precision of temperature control and measurement in the microfluidic device (as measured at the thermistor), which is viewed as essential for the observation of the unsteady morphologies that otherwise would go unnoticed.

$$\frac{T_m}{T_i}=1-\left(\frac{\gamma \kappa }{L}\right)$$ [2]

Considering the above factors, a preliminary framework for understanding the growth and dynamics of the unsteady interface morphology was constructed. Ice grows initially upon cooling because of the presence of D₂O near the interface, which diffuses into the liquid phase, creating a supercooled condition. The ice then retreats when H₂O diffuses into the solid phase, inducing melting as $T_m$ decreases. During this growth and retreat, the cellular features form on the interface, possibly because of the nonequilibrium conditions created by the requirement for effective exchange of latent heat during freezing and melting following the temperature step. Once the unsteady morphologies appear, melting of the convex features, aided by the Gibbs–Thomson effect, enriches the concentration of D₂O above the concave region, creating again a supercooled condition that favors freezing in this region. Conversely, the melting point of the adjacent concave feature is lower than $T_m$, which would induce ice growth in this region. This growth would consume the D₂O molecules in the liquid above it due to preferential freezing of D₂O at the lower temperature. Heat transfer between the convex and concave regions may then occur, aided by the difference in thermal diffusivities. The thermal diffusivity of ice (1.50 $\times$ 10⁻² cm²$\cdot$s⁻¹) is 10 times greater than that of liquid water or poly(dimethylsiloxane) (PDMS) (1.50 $\times$ 10⁻³ cm²$\cdot$s⁻¹ and 1.00 $\times$ 10⁻³ cm²$\cdot$s⁻¹, respectively) and 3 times greater than the glass bottom of the microfluidic chamber (5.50 $\times$ 10⁻³ cm²$\cdot$s⁻¹).

Latent heat absorbed by melting of the convex features could therefore lower the temperature and hence supercool the concave interface. Considering the latent heat of fusion and the heat capacity of solid ice, and assuming an idealized adiabatic condition, a convex feature modeled as a hemicylinder with a 5-$\mu$m radius and a 20-$\mu$m height absorbs latent heat that would lower the temperature of an adjacent hemicylinder of equal volume in the adjacent solid region by 0.16 °C. Notably, local irradiation of a region of the unsteady ice interface at a constant system temperature with a 980-nm IR laser resulted in ice growth at neighboring (not irradiated) regions (Movie S4), which is consistent with the release of D₂O into the liquid near the ice interface upon melting and the supercooling effect. Moreover, the expected content of D₂O at x = 5 $\mu$m and t = 20 s, based on the diffusivity in the liquid (D = 10⁻⁵ cm²$\cdot$s⁻¹), is consistent with the composition of mixed ice expected at the constant system temperature (Fig. 1).

This heuristic framework demands a more quantitative understanding of the growth and dynamics of the unsteady morphologies. A typical starting point is a linear stability analysis of the ice front. It is well known that a flat interface may be linearly unstable under certain conditions, notably when subject to undercooling in the liquid, which gives rise to the well-known Mullins–Sekerka instability (34, 35). The observations here cannot be explained by a Mullins–Sekerka instability, which would predict a melting front to be stable. Moreover, nonplanar morphologies for H₂O ice contacting H₂O liquid or D₂O ice contacting D₂O liquid were never observed. Therefore, both temperature and concentration diffusion must be included in a linear stability analysis, but such an analysis appears to be analytically intractable here.

Consequently, numerical simulations were attempted to validate pieces of the heuristic framework and to explore the values of parameters that might give rise to unsteady morphologies. Phase-field simulations (36) that included thermal diffusion, concentration diffusion, and surface energy effects (Supporting Information) were performed, allowing for different diffusion constants in the solid and liquid phases and a concentration-dependent melting temperature (with respect to the amount of H in D₂O ice). Numerical limitations prevented simulations using the measured diffusivities or melting temperatures; therefore, simulations were aimed toward capturing the dynamics qualitatively. The initial configuration for the simulations included pure D₂O solid at one end of the domain separated by a sharp, possibly curved interface from a domain of pure liquid H₂O. The temperature at both ends of the domain was fixed at the same constant value (see Table S1 for the parameter values used in the phasefield simulations).

Table S1.

Parameter values used in the simulations shown in Figs. S9 and S10

Parameter	Fig. S10	Fig. S9
a0	1.2	1.38
b0	0.4a0 = 0.48	0.4a0 = 0.55
${ϵ}_{\mathrm{\Phi }}$	0.03	0.03
L0	0.2	0.2
J1	10	20
J2	1	4
DS(T)	0.2	0.05
DL(T)	0.02	0.02
DS(C)	0.02	0.01
DL(C)	0.2	0.4

Simulations of a flat interface upon imposing a constant temperature between the $T_m$ values for D₂O and H₂O usually exhibited either melting or freezing, but with a transition region at intermediate temperatures where the interface froze initially and then melted (Fig. S9). This change in direction was due to a high initial flux of D₂O into the liquid phase, creating a supercooled condition that causes the interface to freeze, followed by a slower flux of H₂O into the solid phase, lowering the melting temperature and causing melting. The timescales and temperature profile in the simulations differed from those in the experiment, which also exhibited freezing followed by melting. Nevertheless, the simulations support the premise that the direction of interface motion reflects the difference in the timescales of the fluxes of D₂O and H₂O between liquid and solid and the subsequent coupling to the melting temperature.

Fig. S9.

A simulation that began with pure D₂O in the solid phase ($y<0.5$) and pure H₂O in the liquid phase ($y>0.5$) and evolved at constant temperature near the transition between overall melting and freezing. (A) Concentration of H₂O as a function of $y$ at various times. The location of the interface (contour $\varphi =0.5$) is shown as a red cross in B. (B) Speed of the interface as a function of time. The dynamics are as follows: D₂O initially spills rapidly into the liquid phase (B, orange curve), raising the melting temperature, creating a supercooled condition and causing the interface to freeze (A, positive speed). H₂O then diffuses into the solid phase (B, yellow and purple curves), lowering the melting temperature and causing the interface to melt (A, negative speed.) The initial condition was $T(x,y,0)=1$, $\varphi (x,y,0)=c(x,y,0)=1/(1+e^{16(y-L_y/2)})$ so the interface was initially flat and stayed flat thereafter. Other parameters for the simulations are shown in Table S1.

The evolution of an interface that began with a nonflat (sinusoidal) shape also was explored. Simulations clearly reveal that the temperature of a convex feature is higher than the temperature of the surroundings (i.e., the constant system temperature), which provokes its disappearance through melting. Conversely, the adjacent concave feature is cooler, which will provoke freezing in this region and a lower D₂O concentration above it due to preferential freezing of D₂O at the lower temperature (Fig. S10B). This preliminary modeling did not find evidence of a linear instability, however, and it could not replicate the proposed feedback between the convex and concave features. Instead, all interfaces eventually decayed to a flat morphology, and the locations of the convex and concave features never moved (Figs. S10C and S11B, for example.)

Fig. S10.

(A) Initial conditions for phase, temperature, and concentration. The solid phase is at $y=0$ and the liquid phase is at $y=L_y=3$. (B) The same fields at nondimensional time 0.4. The concentration of H₂O is lower in the concave region, indicating a higher concentration of D₂O there. The temperature is cooler on the liquid side of the interface and hottest in the solid where the convex feature is melting. (C) Location of the interface (contour $\varphi =0.5$) as a function of horizontal coordinate $x$, at different times shown in the key. A sinusoidal interface decays to nearly flat by time 0.9. Nondimensional parameters for this simulation are shown in Table S1.

Fig. S11.

This is similar to Fig. S10, but for the model in SI Numerical Phase-Field Model, Alternative Phase-Field Model. (A) Phase, temperature, and concentration fields at nondimensional time 0.4. Phase and concentration are indistinguishable from the original model, but temperature differs slightly; notably the convex region is colder than the concave region. (B) Location of the interface (contour $\varphi =0.5$) as a function of horizontal coordinate $x$, at different times shown in the key. The initial condition was the same as in Fig. S10, so is not shown. Nondimensional parameters for this simulation are shown in Table S1.

These preliminary numerical investigations are limited by numerous factors—most notably grid resolution, numerical stiffness that makes use of the experimental parameter values impractical, and a 10-dimensional parameter space that has not yet been examined exhaustively. The discrepancy between the numerical results and experimental observations also may reflect other, as yet unidentified physics. Perhaps defects that form during the continuous oscillatory-like cycling of melting and freezing at the interface play a role; or perhaps the morphologies are oscillating between nonplanar, quasi-steady ice fronts, which are known to exist in simpler systems (34). Although the modeling reported here was numerical, it would be useful to explore whether there is a parameter regime in which the planar interface is linearly unstable, and it still seems possible that improved theory could predict unsteady dynamics like those seen in the experiments (37).

Collectively, the observations described herein reveal that ice crystallization can still surprise, generating behavior that cannot be explained readily by nonlinear models. Liquid exchange around a single crystal of ice in a microfluidic device, combined with highly precise temperature control, created conditions in which the competing influences of heat transfer, surface curvature, and H/D exchange afforded unusual and persistent features that melted and froze cyclically on the ice interface as it retreated due to melting. Preliminary numerical simulations captured the freezing and melting of a flat interface and the relationship between thermal and concentration fluxes at the interface, as well as a dependence of the fluxes on surface curvature. Further modeling is required, however, to capture the sensitive coupling and feedback between temperature and concentration fluxes, as well as other physical factors, which produce the oscillatory behavior. Although oscillating growth fronts have been reported during condensation of water on thin metal films that imposed a temperature gradient (38), to our knowledge, the unsteady morphologies observed here have not been observed previously on ice interfaces, and they appear to be unique to D₂O solid-H₂O liquid interfaces. The behavior is somewhat reminiscent of oscillations observed during vapor–liquid–solid growth of sapphire nanowires, however (39). Moreover, the microfluidic device configuration enabled measurement of self-diffusion of water in single crystals of ice, using Raman microscopy. The methodology described here promises utility for investigations of other crystalline materials, and we anticipate it can lead to further discoveries in ice crystallization, including the effects of ice crystallization inhibitors.

Materials and Methods

Microfluidic Solution Exchange Experiments.

A temperature-controlled cold stage that was previously described in ref. 24 was placed on an inverted microscope (DMIRE2; Leica Microsystems Inc.). The temperature of the cold stage could be adjusted with a precision of ±0.001 °C and stability of ±0.002 °C (Fig. S1). A sCMOS (Zyla 5.5; Andor) camera was used to acquire images. A sapphire disc (2.5 cm diameter) was placed between the cold stage and the microfluidic device to minimize temperature gradients. Immersion oil was introduced to the surface of the sapphire disc, and the microfluidic chip was then placed on top of the sapphire disc. The microfluidic channels were filled with 99.96% D₂O (Cambridge Isotopic Laboratories) and the temperature decreased to ca. −20 °C to freeze the D₂O in the channel. The temperature was then increased to melt the ice blocking the inlet and outlet and to form a D₂O single ice crystal, with a distinct boundary. The D₂O liquid in the channels was then exchanged with doubly distilled H₂O to create a D₂O–H₂O solid–liquid interface. A 980-nm IR laser (Wuhan Laserlands Laser Equipment Co.) was used for local ice melting.

Raman Shift Measurements.

The H/D composition of ice crystals and H₂O/H⁺ diffusion into D₂O ice were measured with a Raman microscope (DXR Raman microscope; Thermo Fisher Scientific), using a 532-nm excitation laser operating at 10 mW, with a 2-cm⁻¹ resolution and slit width of 50 $\mu$m. Due to the upright configuration of the microscope, the cold stage was mounted upside down so that the channel was accessible to the Raman excitation beam. The exchange of D₂O with H₂O was performed in a manner similar to that described above. After the D₂O–H₂O solid–liquid interface was created, the temperature was decreased to −0.2 °C to −0.5 °C to freeze the liquid H₂O in the channel. Raman measurements were collected on a region of the D₂O ice crystal located 10 ± 0.5 $\mu \text{m}$ from the original D₂O–H₂O solid–solid interface. Data were collected at regular intervals using an exposure time of 15 s, and the final intensity for each interval was calculated from the sum of 15 measurements. The Raman spectra of the time-lapse measurements were background subtracted and baseline corrected before determination of the total intensity. The intensities used to evaluate the diffusion coefficient were calculated from the integral of the intensity between 2200 cm⁻¹ and 2500 cm⁻¹, the region that captures the bands attributable to ν_O-D for both D₂O and HOD (Fig. S6). The intensity data were corrected using a calibration curve for various compositions of D₂O:H₂O mixtures (Fig. S7), although this did not substantially affect the fit of the data or the calculated diffusion constant (Eq. 1).

SI Materials and Methods

The protocol for the microfluidic device fabrication was described previously (24). The microfluidic device was designed using AutoCAD2015 (Autodesk, Inc.), and transparency films (Artnetpro) were used as masks. Molds for fabricating the devices were created by spin-coating (Cost Effective Equipment, Model 100, Rolla, MO) SU8-2025 on a silicon wafer (3-inch diameter; University Wafer) at 500 rpm for 10 s and then 3,000 rpm for 60 s. The wafers were baked at 65 °C for 2 min and 95 °C for 5 min, followed by UV exposure for 25 s through a mask, using an MJB3 contact mask aligner (Karl Suss). After a postexposure bake at 65 °C for 1 min and then 95 °C for 3 min, the mold was treated with SU8 developer (Microchem) for 4.5 min and then washed with isopropanol and water. Microfluidic chips were fabricated by carefully pouring a degassed 10:1 mixture of PDMS elastomer (SYLGARD 184; Dow Corning) and curing agent on the mold, followed by baking at 80 °C for 60 min. After the PDMS layer solidified, it was removed from the mold and inlet and outlet holes were punched. A glass coverslip and PDMS microfluidic chip were cleaned for 50 s with an oxygen plasma cleaner (model PDC-001; Harrick Plasma Cleaner) and then bonded together by placing the PDMS chip on top of the coverslip.

SI Raman Measurements Near Ice Growth Fronts

The compositions of newly formed ice (upon cooling) near the liquid–solid interface were measured using Raman microscopy. A typical growth sequence lasted on the order of minutes, and H/D exchange caused retreat of the ice front within minutes if the cooling rate was slow or if cooling was stopped at T>0°C. This timescale is not sufficiently long for Raman measurements with sufficient signal-to-noise ratio. Therefore, Raman spectra were acquired after quenching the system to T = −0.1 °C immediately after the ice front retreated to the original D₂O boundary, freezing all liquid in the channel so that a Raman spectrum of the region of the previously frozen H₂O-rich ice could be acquired. As expected from the phase diagram (Fig. 1), the Raman spectra acquired after quenching from 1 °C compared favorably with that obtained for ice containing 25% D₂O whereas the Raman spectra acquired after quenching from 1.7 °C compared favorably with that obtained for ice containing 50% D₂O in H₂O (Fig. S3). These results substantiate that Raman spectra acquired near the interface accurately reflect the expected thermodynamic compositions.

SI Numerical Phase-Field Model

The direct numerical simulation of complex geometries created by phase transformations is a challenging problem with a long history. One flexible and widely used approach is the phase-field method, which uses a diffuse-interface approximation of the liquid–solid interface (36). Designing and implementing a phase-field model in the present setting requires making numerous modeling choices. A preliminary model is presented in this section.

Model.

The goal of this model is to simulate the joint evolution of temperature $T(x,y,t)$, concentration $c(x,y,t)$ of H₂O, and an order parameter (hereafter “phase”) $\varphi (x,y,t)$. Here $t\in [0,\mathrm{\infty })$ is time, and $(x,y)\in [0,L_x]\times [0,L_y]$ represents the spatial variables, with $x$ being the horizontal, along-interface direction, and $y$ varying from solid at $y=0$ to liquid at $y=L_y$. The phase is a continuous function that takes the value $\varphi (x,y)=1$ in the solid, $\varphi (x,y)=0$ in the liquid, and $\varphi (x,y)\in (\mathrm{0,1})$ in the transition region between solid and liquid. As the width of the transition region tends to zero, the model captures the well-known characteristics of phase interfaces, for example the importance of curvature through the Gibbs–Thomson effect. The free energy of the system is assumed to be of the form

$$F[\varphi ,c,T]=\int \left(\frac{1}{2}{|\sqrt{H}{W}_{\varphi }\nabla \varphi |}^2+f(\varphi ,c,T)\right)dx.$$ [S1]

The first term in the integrand, which is related to the surface energy of the interface, depends on parameters $W_{\varphi }$ governing the width of the interface and $H$ governing the height of the energy barrier between the phases. The second term in the integrand is the bulk free energy, which is modeled as

$$f(\varphi ,c,T)=Hg(\varphi )+\frac{c_p}{2T_{sys}}{\left(T-T_m(c)\right)}^2+R_0{(c-c_{\mathrm{\infty }})}^2.$$ [S2]

The first term in the bulk free energy models the free energy as a function of phase. Consistent with the literature on phase-field modeling (36) it chosen to be a double-well potential $g(\varphi )={\varphi }^2{(1-\varphi )}^2$, with a barrier of height $H$ between the wells. The second term in Eq. S2 is a quadratic approximation of the free energy about the melting temperature $T_m(c)$. It depends on the heat capacity $c_p$, assumed to be independent of concentration, and the system temperature $T_{sys}$. Although the factor in the denominator is usually $T_m$, the error in replacing it by $T_{sys}$ is presumed to be small enough to be neglected. The melting temperature is taken to be an affine function of concentration as suggested by the experimental measurements (Fig. 1)

$$T_m(c)=a-bc,$$ [S3]

where $a$ is the melting temperature of pure D₂O, and $a-b$ is the melting temperature of pure H₂O.

The final term in Eq. S2 measures the entropy of mixing. It depends on parameter $R_0=R{T}_{sys}$, where $R$ is the gas constant, and on the concentration $c_{\mathrm{\infty }}$ to which the system relaxes; the numerical value of $c_{\mathrm{\infty }}$ is immaterial in this model. Note that the true mixing entropy is $RT((1-c)\log (1-c)+c\log c)$; here again there is a quadratic approximation, ignoring any higher-order nonlinear effects.

Next consider the dynamical evolution of $\varphi$, $T$, and $c$. It is important to remember that there are two conserved quantities: mass and (thermal) energy. As mass is conserved, the evolution of $c$ should have the form $c_t+\nabla \cdot J_c=0$, where $J_c$ is a suitably defined vector of mass flux. The analogous calculation for energy involves the enthalpy, which is represented by

$$\gamma =T-\frac{L}{c_p}P(\varphi ),$$ [S4]

where $L$ is the latent heat (assumed to be independent of concentration) and $P(\varphi )=(3-2\varphi ){\varphi }^2$ is an interpolation function, which increases monotonically from $P(0)=0$ to $P(1)=1$. To ensure conservation of energy, the evolution of $\gamma$ should have the form ${\gamma }_t+\nabla \cdot J_{\gamma }=0$, where $J_{\gamma }$ is a suitably defined vector of energy flux. The phase $\varphi$ is of course not conserved; its evolution should have the property that the free energy decreases. With these considerations in mind, it is useful to view the free energy as a function of $\varphi$, $c$, and $\gamma$,

$$F[\varphi ,c,\gamma ]=\int \left(\frac{1}{2}{|\sqrt{H}{W}_{\varphi }\nabla \varphi |}^2+f(\varphi ,c,\gamma )\right)dx,$$ [S5]

where the bulk term is now

$$f(\varphi ,c,\gamma )=Hg(\varphi )+\frac{c_p}{2T_{sys}}{\left(\gamma -T_m(c)+\frac{L}{c_p}P(\varphi )\right)}^2+R_0{(c-c_{\mathrm{\infty }})}^2.$$ [S6]

Since $P\prime (0)=P\prime (1)=1$, the function $\varphi \mapsto f(\varphi ,c,\gamma )$ has local minima at $\varphi =0$ and $\varphi =1$ for all $c$ and $\gamma$.

The evolution law can now be specified: It is

$$\tau \frac{\partial \varphi }{\partial t}=-\frac{1}{H}\frac{\delta F}{\delta \varphi }$$ [S7]

$$\frac{\partial \gamma }{\partial t}=\nabla \cdot \left(M_T(\varphi )\nabla \frac{\delta F}{\delta \gamma }\right)$$ [S8]

$$\frac{\partial c}{\partial t}=\nabla \cdot \left(M_C(\varphi )\nabla \frac{\delta F}{\delta c}\right).$$ [S9]

Here $\tau$ is a timescale, which must ultimately be related to the small parameter $W_{\varphi }$ since it should be chosen so that the interface moves with a speed of order $1$. The functions $M_T(\varphi )$, $M_C(\varphi )$ are the temperature and concentration mobilities (diffusivities), respectively, which are assumed to depend on phase via $\varphi$. The logic here is familiar: $\varphi$ does steepest-descent dynamics for the free energy $F$, while for the conserved variables $\gamma$ and $c$ the associated fluxes are proportional to the gradients of the “chemical potentials” $\frac{\delta F}{\delta \gamma }$ and $\frac{\delta F}{\delta c}$.

To be more explicit, after absorbing some of the constants into the diffusivities, the evolution equations are

$$\tau \frac{\partial \varphi }{\partial t}=W_{\varphi }^2{\nabla }^2\varphi -g\prime (\varphi )-\frac{L}{H{T}_{sys}}\left(\gamma -T_m(c)+\frac{L}{c_p}P(\varphi )\right)P\prime (\varphi )$$ [S10]

$$\frac{\partial \gamma }{\partial t}=\nabla \cdot \left(M_T(\varphi )\nabla \left(\gamma -T_m(c)+\frac{L}{c_p}P(\varphi )\right)\right)$$ [S11]

$$\frac{\partial c}{\partial t}=\nabla \cdot \left(M_C(\varphi )\nabla \left(c+\frac{b{c}_p}{R_0{T}_{sys}}(\gamma -T_m(c)+\frac{L}{c_p}P(\varphi ))\right)\right).$$ [S12]

It is convenient to nondimensionalize the model S10–S12 above, using the following characteristic scales:

• •Length $\to$ $L_x$ = horizontal width of container
• •Temperature $\to$ $T_{sys}$
• •Time $\to$ $\tau$.

The choice of time nondimensionalization does not come from any physical characteristic timescale, but rather is used for convenience because time then enters only in the choice of mobilities. The nondimensional variables are

$$\stackrel{\sim }{\gamma }=\frac{\gamma }{T_{sys}},\stackrel{\sim }{t}=\frac{t}{\tau },\stackrel{\sim }{x}=\frac{x}{L_x},\stackrel{\sim }{a}=\frac{a}{T_{sys}},\stackrel{\sim }{b}=\frac{b}{T_{sys}},\stackrel{\sim }{\alpha }=\frac{\tau }{L_x^2}\alpha ,{\stackrel{\sim }{M}}_{T(c)}=\frac{\tau }{L_x^2}{M}_{T(c)}.$$

Substituting into [S10–S12] and removing all of the tildes affords the following equations:

$$\frac{\partial \varphi }{\partial t}=ϵ{}_{\varphi }\nabla ^2\varphi -g\prime (\varphi )-J_1(\gamma -a_0+b_{0}c+L_{0}P(\varphi ))P\prime (\varphi )$$ [S13]

$$\frac{\partial \gamma }{\partial t}=\nabla \cdot \left(M_T(\varphi )\nabla \left(\gamma +b_{0}c+L_{0}P(\varphi )\right)\right)$$ [S14]

$$\frac{\partial c}{\partial t}=\nabla \cdot \left(M_C(\varphi )\nabla \left(c+J_2(\gamma +b_{0}c+L_{0}P(\varphi ))\right)\right).$$ [S15]

Eqs. S13–S15 form the model that is simulated. There are six nondimensional parameters:

${ϵ}_{\varphi }$ = $\frac{W_{\varphi }^2}{L_x^2}$

$L_0$ = $\frac{L}{c_p{T}_{sys}}$

$J_1$ = $\frac{L}{H}$

$J_2$ = $\frac{b{c}_p}{R_0}$

$a_0$ = $a/T_{sys}$

$b_0$ = $b/T_{sys}$.

In addition, the modeler must prescribe the mobilities. Since the aim is to model the interplay between reduced concentration diffusivity and increased temperature diffusivity in the solid, the simulated model uses mobilities which interpolate continuously between constant diffusivities in the solid, $D_S^{(T)},D_S^{(C)}$ for temperature and concentration, respectively, and constant diffusivities in the liquid, $D_L^{(T)},D_L^{(C)}$, using $P(\varphi )$ as the interpolation function:

$$M_T(\varphi )=D_S^{(T)}P(\varphi )+D_L^{(T)}(1-P(\varphi )),M_C(\varphi )=D_S^{(C)}P(\varphi )+D_L^{(C)}(1-P(\varphi )).$$ [S16]

Including the mobilities, the simulated model contains a total of 10 nondimensional parameters. Finally, the model needs boundary conditions, which are chosen as follows:

• •In the $x$ direction: periodic in all variables.
• •Concentration: $c=0$ at $y=0$, $c=1$ at $y=L_y$.
• •Phase: $\varphi =1$ at $y=0$, $\varphi =0$ at $y=L_y$.
• •Temperature: $\gamma =1-L_0$ at $y=0$, $\gamma =1$ at $y=L_y$. This comes from the boundary condition $T=T_{sys}$ at $y=0,L_y$, combined with the boundary conditions for $\varphi$.

Model in the temperature variable.

The model S13–S15 expressed in terms of $T$ is

$$\frac{\partial \varphi }{\partial t}=ϵ{}_{\varphi }\nabla ^2\varphi -g\prime (\varphi )-J_1(T-T_m(c))P\prime (\varphi )$$ [S17]

$$\frac{\partial T}{\partial t}=\nabla \cdot \left(M_T(\varphi )\nabla \left(T-T_m(c)\right)\right)+L_{0}P\prime (\varphi )\frac{\partial \varphi }{\partial t}$$ [S18]

$$\frac{\partial c}{\partial t}=\nabla \cdot \left(M_C(\varphi )\nabla \left(c+J_2(T-T_m(c))\right)\right).$$ [S19]

A rough description of the dynamics is as follows: (i) The phase evolves (melts or freezes) in places near the front where the local temperature differs from the local melting temperature; (ii) when freezing (melting) happens, latent heat is released (absorbed), causing a local increase (decrease) of temperature on the side of the front that is changing phase; (iii) meanwhile both temperature and concentration diffuse to make the local temperature equal the melting temperature.

Alternative Phase-Field Model.

A slightly different model was explored wherein the temperature and concentration diffuse independently. This model in nondimensional form is

$$\frac{\partial \varphi }{\partial t}=ϵ{}_{\varphi }\nabla ^2\varphi -g\prime (\varphi )-J_1(T-T_m(c))P\prime (\varphi )$$ [S20]

$$\frac{\partial T}{\partial t}=\nabla \cdot \left(M_T(\varphi )\nabla T\right)+L_{0}P\prime (\varphi )\frac{\partial \varphi }{\partial t}$$ [S21]

$$\frac{\partial c}{\partial t}=\nabla \cdot \left(M_C(\varphi )\nabla c\right).$$ [S22]

By defining $\gamma$ as in Eq. S11 and rewriting Eq. S21 as an equation for $\frac{\partial \gamma }{\partial t}$, one gets an equation equivalent to Eq. S21 with no time derivative on the right-hand side.

Preliminary simulations with this model produced results that differed in small ways from the first model, but with no significant qualitative differences. Notably, no oscillations, instabilities, or nonlinear growth patterns were observed.

Numerical Scheme.

Eqs. S13–S15 were solved semiimplicitly, using finite differences to calculate derivatives on the right-hand sides of Eqs. S14 and S15, and treating the Laplacian $ϵ{}_{\varphi }\nabla ^2\varphi$ in Eq. S13 implicitly in Fourier space.

We first construct an equally spaced grid as

$$x=(\mathrm{\Delta }x,2\mathrm{\Delta }x,\mathrm{\dots },N_x\mathrm{\Delta }x),y=(\mathrm{\Delta }y,2\mathrm{\Delta }y,\mathrm{\dots },N_y\mathrm{\Delta }y).$$

The grid spacings $\mathrm{\Delta }x$, $\mathrm{\Delta }y$ and number of points $N_x$, $N_y$ are related by $N_x\mathrm{\Delta }x=L_x$ (= 1), $(N_y+1)\mathrm{\Delta }y=L_y$. The relationship is different for the $x$ and $y$ variables because we have periodic boundary conditions in $x$, so we assume $u(0,y)=u(L_x,y)$ is assumed for any field $u$ (and similarly for any derivatives of $u$), but there are Dirichlet boundary conditions in $y$ so $u(x,0),u(x,L_y)$ are fixed, known values.

A fixed time step $\mathrm{\Delta }t$ was chosen as $\mathrm{\Delta }t=\mu \cdot \min ({(\mathrm{\Delta }x)}^2,{(\mathrm{\Delta }y)}^2)\cdot \min (D_S^{(C)},D_S^{(T)},D_L^{(C)},D_L^{(T)})$, where $\mu$ is the Courant number, set to $\mu =\mathrm{0.4.}$ The fields $\gamma ,c$ are updated as

$$u^{n+1}=u^n+\mathrm{\Delta }t(A_u^n{j}_u^n+h_u^n),$$ [S23]

where $u^n$ is either $\gamma$ or $c$ at the $n$th time step, $j_u^n$ is the quantity in the innermost brackets [either $\gamma +b_{0}c+L_{0}P(\varphi )$ for $\gamma$ or $c+J_2(\gamma +b_{0}c+L_{0}P(\varphi ))$ for $c$] at time-step $n$, $A_u^n$ is the numerical discretization of the operator $\nabla \cdot (M_u(\varphi )\nabla \cdot )$ and time-step $n$, and $h_u^n$ is any extra contribution from the boundary terms. In the numerical implementation, $u^n$, $j_u^n$, and $h_u^n$ are vectors of length $N_p=N_x\times N_y$, and $A_u^n$ is an $N_p\times N_p$ sparse matrix.

The operator $\nabla \cdot (M_u(\varphi )\nabla \cdot )$ was discretized using a second-order scheme based on finite differences (40). For a given field $v(x,y)$ evaluated at grid points $v_i=v(i\mathrm{\Delta }x,y)$, the $x$ component of the operator was discretized as

$${\partial }_x\left(M_u(\varphi ){\partial }_{x}v\right)\to M_u({\varphi }_{i-1/2})v_{i-1}-(M_u({\varphi }_{i-1/2})+M_u({\varphi }_{i+1/2}))v_i+M_u({\varphi }_{i+1/2})v_{i+1}$$ [S24]

and similarly for the $y$ component of the operator. Fields were evaluated at half grid points, using the usual averaging operator $u_{i+1/2}=(u_i+u_{i+1})/2$ and similarly for $u_{i-1/2}$. When a particular index extends beyond the set $\{\mathrm{1,2},\mathrm{\dots },N_{x(y)}\}$, a boundary condition is substituted: $u_{N_x+1}=u_1$ for the $x$ part of the operator and the known values of $u_0$, $u_{N_y+1}$ for the $y$ part of the operator. This implies including a vector of boundary values $h_u^n$ for the $y$ operator, since Eq. S24 will be an affine function of $v$, not a linear function.

The field $\varphi$ is updated as

$$(1-{ϵ}_{\varphi }\mathrm{\Delta }t{\nabla }^2){\varphi }^{n+1}={\varphi }^n-\mathrm{\Delta }t\left(g\prime ({\varphi }^n)+J_1\left({\gamma }^n-a_0+b_0{c}^n+L_{0}P({\varphi }^n)\right)\right).$$ [S25]

The right-hand side $r^n$ is calculated explicitly at each grid point. Its fast Fourier transform ${\widehat{r}}^n$ is then calculated, so ${\widehat{\varphi }}^{n+1}$ can be solved for in Fourier space, and finally the inverse Fourier transform gives ${\varphi }^{n+1}$.

The fast Fourier transform works best for periodic functions. To account for the Dirichlet boundary conditions in $y$, all functions were shifted by an affine function of $y$ so the boundary conditions are 0 on both sides of the domain. The resulting function was reflected antisymmetrically about $y=L_y$, and the final transformed function is denoted with a tilde. For example, the right-hand side is transformed as

$${\stackrel{\sim }{r}}^n(x,y)=\{\begin{array}{cc}{r}^n(x,y)-(1-\frac{y}{L_y})\hfill & y\in [0,L_y)\hfill \\ -r^n(x,L_y-y)+(1-\frac{L_y-y}{L_y})\hfill & y\in [L_y,2L_y)\hfill \end{array}.$$ [S26]

The transformed right-hand side ${\stackrel{\sim }{r}}^n(x,y)$ is used to solve for the transformed phase ${\stackrel{\sim }{\varphi }}^{n+1}(x,y)$ ($y\in [0,2L_y)$), and the transformations are then reversed to obtain ${\varphi }^{n+1}$. This transformation ensures the phase and its derivative are continuous functions on the interval $[0,2L_y)$ with periodic boundary conditions, and therefore derivatives calculated in Fourier space are better behaved. The transformation does not ensure the second derivative of ${\stackrel{\sim }{\varphi }}^n$ is continuous over the periodic interval, however, so calculating the Laplacian induces Gibbs phenomena (oscillations on the scale of the grid). These oscillations can be made negligible by choosing an initial condition whose second derivative is nearly zero for $y\in \{0,L_y\}$.

Examples.

A variety of parameter choices were explored, and results from two simulations are illustrated here. Both simulations had numerical parameters $L_x=1$, $L_y=3$, $N_y=61$, and $N_x=24$. The nondimensional parameters are provided in Table S1.