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. 2023 Jan 11;8(3):3329–3338. doi: 10.1021/acsomega.2c06996

Phase-Field Modeling of Crystal Growth on the Surface and in the Volume of Na2O–2CaO–3SiO2 Glass under Periodic Temperature Conditions

Munemichi Kawaguchi †,*, Masayoshi Uno †
PMCID: PMC9878635  PMID: 36713699

Abstract

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We performed modeling of two- and three-dimensional phase-field methods (2D- and 3D-PFM) for crystal growth on the surface and in the volume of Na2O–2CaO–3SiO2 glass to investigate crystal growth behaviors under periodic temperature conditions. In this study, the periodic temperature conditions were set to 993 K for 180 s and 873 K for 252 s repeatedly. Phase-field mobilities, LS and LB, were determined to compare with the experimental surface crystal growth rate, uS, and the volume crystal growth rate, uB, at 873–1023 K. 2D-PFM with LS and LB reproduced quantitatively the temperature dependence of uS and uB. The parameters of LS and LB were consistent with those of 11 kinds of silicate melts, considering the surface and bulk diffusion coefficients. 3D-PFM simulated the single- and multinucleated crystal growth behaviors: the single-nucleated crystal simulation revealed that a ring was formed around the pre-existing crystal by heterogeneous nucleation. These radii obtained by 3D-PFM were comparable to the experimental values. The multinucleated crystal simulation revealed the contact and interaction between the crystals, e.g., new crystal rings could not be formed at the contacting region. In random nucleation, the 3D-PFM simulation demonstrated the crystal shape of the multinucleated crystals under periodic temperature conditions. It was comparable to the experimental photographs obtained by Yuritsyn et al.

1. Introduction

Under a heating process, glass-forming liquids are known to reveal predominant surface crystallization. In crystallization, several interesting phenomena occur such as crystal growth and heterogeneous and homogeneous nucleation on and in the glass. Especially, heterogeneous nucleation plays a more important role in the early stage of crystallization than homogeneous nucleation due to fewer driving forces.1−4 In the crystal growth, it is often more remarkable for the growth rates along the polished surface. The crystal growth rates5−8 in the volume of the melts and at the polished surface could be characterized by uB and uS, respectively, which have been discussed using the bulk and surface diffusion coefficients, DB and DS, respectively. In this study, uB and uS are called the “bulk crystal growth rate” and the “surface crystal growth rate,” respectively. In general, the difference between uB and uS is mainly due to the difference between DB and DS through the surface pathways because the crystal growth rates have a strong relationship with the diffusion coefficient. Therefore, uS could imply larger than uB because DS was much larger than DB, e.g., DS ∼ 1000DB.9

In the crystallization of Na2O–2CaO–3SiO2 glass,5,10−21 Yuritsyn et al. reported the demonstration experiments, which revealed the difference between uS and uB. In the experiments, they measured the uS and uB of the nucleated crystals on the polished surface under the periodic heating process and took beautiful photographs of the crystal shapes. Their experiments successfully applied the following properties of crystallization: at a low temperature, heterogeneous nucleation occurs extremely faster than the crystal growth on the polished surface. On the other hand, at a high temperature, bulk crystal growth rates become remarkable due to diffusion coefficients. Therefore, under the periodic heating treatment, the phenomena of only surface crystallization and the whole bulk crystallization occurred alternately. Each periodic heating treatment resulted in the formation of a new ring around the pre-existing crystals. The rings were a result of the increase in surface crystallization due to frequent heterogeneous nucleation. By applying the periodic heating treatment, uS and uB were found to be evaluated quantitatively from the radius of any given crystal versus heating time. Furthermore, Yuritsyn et al. showed distinguished results such as the temperature dependence of DB and DS from uS and uB obtained by the experiments. However, no studies have ever tried the simulation of the crystallization behaviors under periodic temperature conditions.

A phase-field method (PFM) is one of the strong simulation tools for the time evolution microstructure in a material due to thermodynamics, mass diffusion, and microelasticity.22−24 The PFM is characterized by a phase-field parameter, ϕ, in the total calculation region, and a diffuse interface is defined at a different phase interface. By treating free energy minimization in the total calculation system, the PFM solves the time evolution for ϕ. Furthermore, to address wetting and heterogeneous crystal nucleation on a wall using the PFM, Gránásy et al. suggested boundary conditions at the wall, including the free energy of nuclei formation and the contact angle.25

Our previous papers26,27 showed that the classical PFM with newly defined phase-field mobilities reproduced quantitatively the crystal growth rates in undercooled silicates, SiO2 and GeO2 liquids. Furthermore, it found that the phase-field mobilities were independent of the calculation dimension such as two or three dimensions. Their mobility parameters were described by the diffusion coefficients and by the sum of the ratio between the cation i and the oxygen molar mass, ∑Mi/MO.

In this study, we determined the surface and bulk phase-field mobilities, LS and LB, to compare the 2D-PFM simulation results using LS and LB with experimental uS and uB. Furthermore, the 3D-PFM simulation with LS and LB revealed the contact and interaction effects between the crystals on the polished surface of Na2O–2CaO–3SiO2 glass under periodic temperature conditions.

2. Model

2.1. PFM

In this paper, the crystal growth under periodic temperature conditions is modeled using a classical PFM. However, the temperature distribution is assumed to be constant in the total calculation region. The temperature is given as the input condition by the time function.

The phase-field variable, ϕ, is defined at the total calculation region. ϕ changes continuously from 1 to 0, corresponding to the solid and liquid phases, respectively. The classical PFM equation23,28,29 is derived using the total free energy density, Inline graphic, which includes the homogeneous and heterogeneous energy densities. The former is represented by bulk free energy densities, fS(T) and fL(T), in which the superscripts “S” and “L” denote the solid and liquid phases, respectively. The latter is represented by the double-well potential and the gradient energy density due to the overlap and gradient of ϕ at the interface.

2.1. 1
2.1. 2
2.1. 3

The kinetics of ϕ is derived from the time-depending Ginzburg–Landau equation, ∂ϕ/∂t = −LδF/δϕ.

2.1. 4

Here, fS(T) – fL(T) is the thermodynamics driving force for crystallization, which is represented by −ΔHmΔT/Tm. ΔHm, ΔT = Tm – T, and Tm are the melting enthalpy, the undercooling degree, and the melting temperature, respectively. W and κ denote the degrees of the double-well potential and the gradient energy, respectively.

2.1. 5
2.1. 6

where γ, δ, and b denote the interface energy, the interface thickness, and a constant number of ∼2.2, respectively.

2.2. Phase-Field Mobility

According to the screw dislocation growth model,8 a jump frequency over the phase interface corrects crystal growth rates, which depend on the effective diffusion coefficient. In Na2O–2CaO–3SiO2, the diffusion coefficient is represented by the Stokes–Einstein–Eyring (SEE) equation, Dη.

2.2. 7

where η is the viscosity, which is described by the Vogel–Fulcher–Tammann (VFT) equation, log η = −5.46 + 5196/(T – 547). The other parameters of eq 7 are α = 2.856 and d = 7.5 × 10–10 m. T and kB are the temperature and the Boltzmann constant, respectively. According to our previous papers,26,27 the phase-field mobility, LS,B, was defined as follows

2.2. 8

where A and B are the mobility parameters, Deff is the effective diffusion coefficient, and λ is the jump distance, which is usually taken as the molecular diameter. Here, Deff values for uS and uB are given by DS and DB, respectively. By comparing experimental uS and uB with the 2D-PFM results using LS,B = 1, we determined A and B in LS,B. In the procedure for determination of LS,B, first, we solved eq 4 with constant LS,B = 1. The growth rate, u0, includes only the thermodynamics driving force except for the interface process. Then, A and B of LS,B were calculated with the least square method from the ratios of experimental uS and uB to u0, respectively. We ran the 2D-PFM simulation using new LS,B to confirm the validity.

2.3. Computational Conditions

The widths of the grid spacing, dx, dy, and dz, were fixed at 0.1 μm. The interfacial width, δ, was set as 10 grids. 2D-PFM was used for the determination of LS,B. 3D-PFM was used to investigate the contact and interaction between the crystals under periodic temperature conditions and to demonstrate the crystal shapes. Figure 1 depicts the schematic computational system of (a) 2D-PFM and (b, c) 3D-PFM simulations.

Figure 1.

Figure 1

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Schematic of computational domains of (a) 2D-PFM, (b) Single and two nucleation in 3D-PFM. (c) Multinucleation in 3D-PFM surrounded by a dashed square. dx, dy, and dz mean the widths of the grid spacing.

The computational domain used 200 × 200 nodes in the 2D-PFM simulation. The calculation temperature was set to constant at T = 800–1100 K. Since the experimental temperature ranged from 873 to 1023 K, the experimental temperature was within the calculation temperature. The SEE equation (Dη) was used as the effective diffusion coefficient in eq 8. However, we will discuss the effects of the surface and bulk diffusion coefficients (DS and DB, respectively), as suggested in the previous experimental paper.5

LS,B was used in 3D-PFM. LS was set only for the first layer on the surface (at z = dz) and LB was set elsewhere. The periodic temperature conditions were set to 993 K for 180 s and 873 K for 252 s repeatedly, as referred to in experimental conditions.5 Under periodic temperature conditions, nucleation would occur frequently at 993 K. Therefore, the nucleation conditions for 3D-PFM simulations follow three kinds of nucleation: In (i) single nucleation, one nucleated crystal was assumed at (dx, dy, dz) as a 1/4 partial model to investigate the crystal growth behavior. In (ii) two nucleation, two nucleated crystals were assumed at (dx, dy, dz) and (L, L, dz) as a 1/4 partial model. Here,L is 43 μm. In (iii) random nucleation, several nucleated crystals were assumed randomly on the x–y plane (at z = dz) to investigate the contact and interaction between the crystals. In all of the cases, the initial nucleated crystals were set to a hemisphere with a radius of 0.55 μm. Table 1 summarizes the nucleation conditions in (i), (ii), and (iii). The 3D-FPM computational domain used 430 × 430 × 200 nodes in (i) and 700 × 700 × 140 nodes in (ii)–(iii).

Table 1. Nucleation Conditions in 3D-PFM Simulations: (i) Single Nucleation, (ii) Two Nucleation, and (iii) Random Nucleationa.

  number of nucleated crystals (N), nucleation positions time of ith nucleation (ti)
(i) single nucleation N = 1, (0, 0, 0) t1 = 0 s
(ii) two nucleation N = 1, (−0.3 L, −0.3 L, 0) t1 = 0 s
  N = 1, (0.3 L, 0.3 L, 0) case 1: t2 = 0 s, case 2: t2 = 432 s, case 3: t2 = 864 s, case 4: t2 = 1296 s
(iii) random nucleation N = 3, at random t1 = 0 s
  N = 2, at random t2 = 432 s
  N = 1, at random t3 = 864 s
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a

L is 43 μm.

3. Results

3.1. Determination of Phase-Field Mobility

Figure 2 depicts the ratios between experimental crystal growth rates provided in the previous papers5,10 and u0 obtained by the PFM simulation with LS,B = 1 in Na2O–2CaO–3SiO2 glass. To calculate the ratios, we used both the Dη of the SEE equation and DS and DB. In a log–log plot, the ratios appear to be approximately proportional to the power of 6πDeffΔTλ/kB. Here, 6πDeffΔTλ/kB implies TΔT/η in the assumption of the Stokes–Einstein diffusion coefficient for the comparison with other silicates.26,27 Note that the solid black lines of uS and uB using Dη show different slopes, but the solid red lines of uS and uB using DS and DB show the almost same slopes, respectively. This is because DS and DB are reasonable as effective diffusion coefficients. Table 2 shows the parameters, A and B, of LS and LB in Na2O–2CaO–3SiO2. The dashed line in Figure 3 depicts the surface crystal growth rate (uS) obtained by the 2D-PFM simulation with LS. Similarly, the solid line in Figure 3 depicts the bulk crystal growth rate (uB) obtained by the 2D-PFM simulation with LB. Open and closed circles in Figure 3 depict experimental uS and uB provided in Yuritsyn’s paper.5 Since they show good agreement at the temperature range of 830–1020 K, the A and B of LS and LB could be determined enough accurately. In the 2D-PFM simulation, uS was larger than uB at 830–1020 K. Especially, uS/uB∼ 129 was remarkable at 873 K, which was comparable to the experimental value. As the temperature increased, the difference between uS and uB decreased, and uS and uB became equal at 1028 K. We discuss LS and LB using DS and DB in Section 4.

Figure 2.

Figure 2

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Ratios of experimental crystal growth rates,5uS,B, and u0 obtained by PFM calculations with LS,B = 1 in crystallization of undercooled liquids. Experimental uS,B used the data in Table 1 in ref (5) by Yuritsyn et al.

Table 2. Phase-Field Mobilities (LS and LB) for the Crystal Growth Rates on the Surface and in the Volume of Na2O–2CaO–3SiO2.

    phase-field mobility parameters using Dη
 phase-field mobility parameters using DS and DB
mobility temperature (K) A B A B
LS 820–1050 36.2 0.332 10.4 1.11
LB 820–1050 16.9 0.734 16.7 1.06
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Figure 3.

Figure 3

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Temperature dependence of surface and bulk crystal growth rates (uS and uB, respectively) in Na2O–2CaO–3SiO2 obtained by the 2D-PFM calculations and the previous experiment.5 Experimental uS,B used the data in Table 1 in ref (5) by Yuritsyn et al.

3.2. Crystal Growth Behavior of a Single-Nucleated Crystal

To observe the crystal growth behavior under periodic temperature conditions, we performed the 3D-PFM simulation for the crystal growth of a single-nucleated crystal on a polished surface. LS and LB determined in Section 3.1 were used to set that the heterogeneous nucleation and crystal growth rates along the surface and in other directions were consistent with the experiment.

Figure 4 depicts the crystal growth behavior on the polished Na2O–2CaO–3SiO2 surface, such as the crystal radius, R, and crystal shapes at (a) 87 s, (b) 870 s, and (c) 1392 s. Open and closed circles in Figure 4 show R at z = dz (on the interface) and z = 2dz, respectively. The crystal growth rates changed drastically due to the periodic temperature: the rates decreased at a low temperature and increased at a high temperature. When the temperature changed periodically, the rates increased and decreased periodically with the temperature. However, the open circle showed a slight increase at low temperatures, e.g., at 180–432, 612–864, 1044–1296, and 1476–1728 s, because the surface crystal growth rate was not too low. It was caused by heterogeneous nucleation. Next, in crystal shapes, the formation of a ring seemed around the pre-existing crystal, as shown in Figure 4b. It was found that the ring was a trace formed by heterogeneous nucleation at low temperatures. At high temperatures, the crystal growth occurred in all directions, not just along the surface, but at a low temperature of 873 K, the crystal growth occurred remarkably only along the polished surface (uS). uS/uB was close to 129 times at 873 K. Namely, the ring was caused by the difference between the surface and bulk crystal growth rates, which increased remarkably at relatively low temperatures such as 873 K, as shown in Figure 3. Figure 5 compares the crystal radius at z = dz (open circle) with the radii of the 1st, 2nd, 3rd, and 4th rings (open square). All of the open squares were obtained from the radius of each ring at every 87 s in 0–1740 s. They showed different values because the radii of all rings grew slightly with time. The standard deviation to the mean value was less than 13% at 1749 s. It would be difficult in principle to reduce the error in the crystal growth rates obtained from the radius of the ring in the crystal under periodic temperature conditions. Figure 6 depicts the cross section of the growing crystal at 0, 174, 348, 522, 696, 870, 1044, 1218, 1392, and 1566 s. As this simulation was calculated with the 1/4 partial model, Figure 6 illustrates symmetric crystals at the left and right sides. The filled triangles show the traces of the rings, which resulted from the low-temperature condition. It was found that the filled triangles gradually moved outward with time. The increase of open squares in Figure 5 corresponds to the movement of the traces in Figure 6. If they try to measure the surface crystal growth rates from the radius of the crystal ring under periodic temperature conditions, they should calculate the difference of the radii of the rings, e.g., the difference in the radii between 1st and 2nd rings, 2nd and 3rd rings, and 3rd and 4th rings. The reason is that all of the rings grow gradually with time at the same speed at the same temperature condition. The slight crystal growth length can be offset by taking the difference in the radius of each ring.

Figure 4.

Figure 4

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Crystal growth behavior of single nucleation on the polished surface of Na2O–2CaO–3SiO2 under periodic temperature conditions: crystal radius, R, periodic temperature, and crystal shapes at (a) 87 s, (b) 870 s, and (c) 1392 s.

Figure 5.

Figure 5

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Time dependence on the crystal radius, R, under periodic temperature conditions (heat treatment time: 180 s and temperature: 993 K). Open squares denote the crystal radius, R, taken from the stacked rings at every 87 s in 0–1740 s.

Figure 6.

Figure 6

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Cross section of a crystal on the polished surface of Na2O–2CaO–3SiO2 under periodic temperature conditions at 0, 174, 348, 522, 696, 870, 1044, 1218, 1392, and 1566 s during crystal growth.

3.3. Contact and Interaction between Two Crystals

Figure 7 depicts the 3D-PFM simulation results for the contact and interaction between two crystals: (a), (c), (e), and (g) in Figure 6 show crystal shapes in the nucleation of one more crystal at different timings, (a) 0 s, (c) 435 s, (e) 867 s, and (g) 1305 s. We set that the nucleation occurred in reaching temperatures from 873 to 993 K because heterogeneous nucleation should occur frequently on the polished surface at nearly 860 K.10 Here, the timings and initial radius of nucleation are given by the user’s input as calculation conditions in Table 1. Figure 7b,d,f,h shows crystal shapes in the two crystals contacting and growing. When the two crystals grew up and contacted each other, they could not grow up further along the contacting direction. The crystal grew up in the directions without crystals. In the crystal growth rates, any anomalous behavior could not be observed such as a sudden rapid crystal growth rate. Their interaction was independent of the crystal sizes, as shown in Figure 7b,d,f,h: the large crystal did not grow up on top of the other small crystals. It spread to the periphery of the other crystal along the polished surface and then eventually integrated. Furthermore, each crystal ring was connected at the same height level as each ring, as shown in Figure 7. It was because the crystal growth rates including the heterogeneous nucleation were the same as all of the crystals under the same periodic temperature conditions. Their crystal growth behavior in interacting with another crystal can be observed commonly under periodic temperature conditions. However, it was the most important interaction that the contact of two crystals affected the heterogeneous nucleation. When one crystal grew up and covered the polished surface of the periphery of the other crystal, heterogeneous nucleation could not occur at the covered surface. As a result, the rings that were caused by heterogeneous nucleation at a low temperature did not occur on the covered surface. In Figure 7b,f, the traces as rings were observed at the only uncontacted surface. Figure 8 depicts the cross section of two crystal interactions on the polished surface of Na2O–2CaO–3SiO2 under periodic temperature conditions at 0, 174, 348, 522, 696, 870, 1044, 1218, 1392, and 1566 s. The filled triangles show the traces of the rings that were formed at low temperatures. In Figure 8a (nucleation timing: t2 = 0 s), the filled triangles could be observed between the two crystals. On the other hand, in Figure 8b–d, the filled triangles could not be observed between the two crystals because the crystals contacted each other. The reasons were simply the timing of the heating process and the crystal contact. The ring could be formed in only the early time prior to the crystal contact. In the photographs of the previous experiments,5 the ring seemed to vanish at the contacting regions, which was similar behavior to the present 3D-PFM results. Thus, the surface crystal growth including heterogeneous nucleation was revealed to play a role in the formation of crystal rings. The present 3D-PFM simulation succeeded to reproduce the surface and bulk crystal growth behaviors including heterogeneous nucleation.

Figure 7.

Figure 7

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Snapshots for nucleated crystals (a, c, e, g) and contact and interaction between two crystals during growing (b, d, f, h).

Figure 8.

Figure 8

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Cross section of two crystals interacting on the polished surface of Na2O–2CaO–3SiO2 under periodic temperature conditions at 0, 174, 348, 522, 696, 870, 1044, 1218, 1392, and 1566 s during crystal growth at different nucleated timings: (a) t2 = 0 s, (b) t2 = 432 s, (c) t2 = 864 s, and (d) t2 = 1296 s.

3.4. Random Nucleation and Growth

Figure 9 depicts the 3D-PFM simulation results: the heterogeneous nucleation happened at random on the polished surface (Table 1), and the nucleated crystals grew up as they contacted and interacted with each other under periodic temperature conditions. In the 3D-PFM simulation, we set a similar nucleation ratio, in which three crystals nucleated in the initial at the calculation region and the number of nucleated crystals decreased with time. Therefore, the numbers of nucleated crystals were set to 3 at 0 s, 1 at 432 s, and 1 at 864 s, respectively. These conditions match approximately the experimental findings. Figure 9a represents the initial calculation system, in which three nucleated crystals were placed randomly on the surface. As time proceeded, the crystal grew up remarkably at 993 K, as shown in Figure 8b. Especially, the crystal growth along the surface is more conspicuous than the vertical orientation and their crystal shapes spread horizontally. In Figure 9c, only the surface crystal growth proceeded in the surroundings due to heterogeneous nucleation because of the low temperature of 873 K, which led to the formation of a ring around all of the pre-existing crystals. In Figure 9d–f, a new crystal was placed at 432 and 864 s. Then, their crystals contacted and interacted with the other crystals. As we mentioned in Section 3.3, the new rings, which resulted from a trace of heterogeneous nucleation at low temperatures, did not occur at the contacting region.

Figure 9.

Figure 9

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Crystal growth behaviors for random nucleated crystals under periodic temperature conditions. Three, one, and one nucleated crystals were assumed to be placed at 0, 432, and 864 s.

These present 3D-PFM simulation results correspond to the reduced version of the experimental photographs for the crystal nucleation and growth on the polished surface of Na2O–2CaO–3SiO2 under the periodic heating process.5 The 3D-PFM simulation reproduced the realistic crystal growth behavior and their crystal shapes on the polished surface such as the crystal radius, the rings caused by the periodic heating process, the interval between the rings, and the relative positions of the rings in the crystal and the other crystal. Thus, we could reproduce the complex crystal shapes on the polished surface quantitatively by just setting LS and LB in the classical PFM. It concluded that this simplicity and applicability were some of the characteristics and advantages of this PFM.

4. Discussion

4.1. Relationship of Crystal Growth Rates with DS, DB, and Dη

The previous experimental paper5 suggested the decoupling temperatures of Na2O–2CaO–3SiO2. They denoted temperatures at which the effective diffusion coefficients of surface and bulk are equal to Dη; therefore, DS and DB are equal to Dη over the decoupling temperatures. The surface and bulk decoupling temperatures, TS and TB, are close to 1015 and 929 K, respectively. The decoupling temperatures denote the switching diffusion mechanisms from the SEE equation (Dη) to an Arrhenius-type law (DS and DB); 1028 K in Figure 3 obtained by 2D-PFM was slightly higher than TS. Therefore, we should note the relationship of the crystal growth rates with DS, DB, and Dη at less than TS and TB, especially for DS.

Figures 10 and 11 depict the interrelationships of the scale factor of phase-field mobility, A, and the exponent number of phase-field mobility, B, respectively. The open circle and square show A and B determined by Dη, respectively. The closed circle and square show those determined by DS and DB, respectively.8∑iMi/MO was the sum of cation molar mass per oxygen molar mass. The triangles show the A and B of the other silicates, SiO2 and GeO2, according to our previous paper.26,27 The straight lines in Figures 10 and 11 show the approximation curves for the triangles in the other silicates, SiO2 and GeO2.26,27Figures 10 and 11 revealed that the A and B of Na2O–2CaO–3SiO2 depended on their diffusion coefficients (Dη, DS, and DB). Compared with the solid lines determined by the other silicates,26 the A and B determined by DS and DB are more reasonable values than those by Dη. The reason was that the effective diffusion coefficients changed from Dη to DS and DB as the temperature decreased from TS and TB. Therefore, DS and DB have the potential to work effectively for crystal nucleation and growth in undercooled Na2O–2CaO–3SiO2.

Figure 10.

Figure 10

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Interrelationship between the scale factor of phase-field mobility (A) in bulk and surface crystal growth rates (uS and uB). A represented by open triangles were reproduced in Figure 4 in ref (26) by Kawaguchi et al.

Figure 11.

Figure 11

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Interrelationship between the exponent number of phase-field mobility (B) in bulk and surface crystal growth rates (uS and uB) and the sum of cation molar mass per oxygen molar mass (∑iMi/MO). B represented by open triangles were reproduced in Figure 5 in ref (26) by Kawaguchi et al.

Next, we investigated the dependence of A and B on the width of the grid spacing, dx, in 2D-PFM. Figure 12 depicts A determined by DS and DB at dx = 0.1, 0.05, 0.02, and 0.01 μm. A increased from 5.18 to 16.7 as dx increased from 0.01 μm to 0.1 μm. However, the increase of A was much smaller than the range of A in Figure 10. The range was close to 5 × 106. Figure 13 depicts B determined by DS and DB at dx = 0.1, 0.05, 0.02, and 0.01 μm. Although B at dx = 0.1 μm was larger than the others, the values of B determined by DS and DB were almost constant. Therefore, this concluded that the difference depending on dx could seem to be negligibly small in A and B.

Figure 12.

Figure 12

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Scale factor of phase-field mobility (A) in bulk and surface crystal growth rates (uS and uB) depending on dx = 0.1, 0.05, 0.02, and 0.01 μm.

Figure 13.

Figure 13

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Exponent number of phase-field mobility (B) in bulk and surface crystal growth rates (uS and uB) depending on dx = 0.1, 0.05, 0.02, and 0.01 μm.

4.2. Contact Angles Depending on uS

Figure 14 depicts the dependence of cross-sectional crystal shapes at 1392 s on the surface crystal growth rates, uS/x (x = 1.0, 1.1, 1.3, 1.5, 1.7, 2.0, 2.5, 3.0, 4.0, and 8.0). The scales at the y and z directions were standardized by z0 = 17.9 μm, which is the height of the crystal at 1392 s. The filled triangles show the traces of the rings resulting from the low-temperature condition in uS/1.0. The black line (uS/1.0) corresponds to that at 1392 s in Figure 6. Some lines (uS/4.0 – uS/2.5) were hidden by the gray line (uS/8.0) because of the almost same values everywhere. The crystal shapes show an almost flat slope at larger uS except for the traces of the rings because the crystallization occurs on the polished surface for a very short time. Their contact angles show small values, e.g., θ ∼ 29.5° in uS/1.0 and θ ∼ 32.4° in uS/1.1. On the other hand, at smaller uS, the cross-sectional crystal shape seemed like a sphere. The reason is that the crystallization proceeded at enough speed of uB over the second layer, even if small uS was set at the first layer on the polished surface. The pinning could not occur in the case of small uS defined at only the first layer, indicating that the crystal growth rate along the polished surface became nearly uB. The contact angle was close to 85.9°. As a result, the cross-sectional crystal shape became a sphere at small uS, but it became a flat slope at large uS. Open circles and filled circles shown in Figure 15 depict the contact angles at the triple line at 1392 and 1740 s, respectively. They depended on the surface crystal growth rate, which was standardized by uS. In less than 0.3 (uB > uS), the contact angle was constant at θ ∼ 85.9 or 86.7°, which denoted a nonwetting condition. The initial condition of the contact angle (θ ∼ 90°) strongly affected them further in uB > uS. On the other hand, as the surface crystal growth rate increased, the contact angle decreased in inverse proportion to the surface crystal growth rate. This behavior depended on the simulation time, as shown in Figure 15. As the simulation time proceeded, the contact angle approached a steady state. In more than 0.457 (uB < uS), the 1740 s simulation results (filled circles) decreased more rapidly than the 1392 s simulation results (open circles). The turning point was around uB/uS ∼ 0.457 (uB ∼ uS). Considering the crystal growth rates at the triple line (uB in the vertical direction and uS in the parallel direction of the polished surface), the contact angle (θ) should be associated with the ratio of uB and uS, i.e., θ ∼ arctan(uB/uS). The dashed line in Figure 15 depicts θ = arctan(uB/uS), which can represent quantitatively the contact angle in more than 0.457 (uB < uS) in 1740 s. Figure 16 depicts the interfacial energies at 1392 and 1740 s, which were calculated by γ cos θ from the θ of Figure 15. In the nonwetting region of less than 0.3, the interfacial energy was close to 0.06 J/m2. However, in more than 0.457, the interfacial energy increased with the surface crystal growth rate. It should meet the wetting condition qualitatively. Therefore, Figure 16 reveals that the high interfacial energy enabled the crystal to cover the polished surface at high surface crystal growth rates. According to the theory of the development of surface grooves at the grain boundaries of a heated polycrystal by Mullins,30 the surface current of atoms was proportional to the interfacial energy, which was derived from the C. Herring’s chemical potential and Nernst–Einstein relation. This denoted that the surface crystal growth rate was associated with the interfacial energy: the surface crystal growth rates increase as the interfacial energy increasing. The rings caused by heterogeneous nucleation at a low temperature were more remarkable in large uS, as shown in Figure 14, because the ratio of uS/uB was larger. In the previous PFM simulation for heterogenous nucleation focusing on the boundary condition at the walls,25 the free energy of formation and the contact angle were determined as a function of undercooling. The present 3D-PFM simulation revealed that the surface and bulk diffusion properties and interface energies strongly affected the contact angle.

Figure 14.

Figure 14

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Dependence of cross-sectional crystal shapes on the surface crystal growth rate, uS/1.0, uS/1.1, uS/1.3, uS/1.5, uS/1.7, uS/2.0, uS/2.5, uS/3.0, uS/4.0, and uS/8.0 at 1392 s. z0 is 17.9 μm.

Figure 15.

Figure 15

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Contact angles depending on the surface crystal growth rate standardized by uS at 1392 and 1740 s.

Figure 16.

Figure 16

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Interfacial energy at the triple line (γ cos θ) depending on the surface crystal growth rate standardized by uS at 1392 and 1740 s.

5. Conclusions

We performed phase-field modeling of crystal growth on the surface and in the volume of Na2O–2CaO–3SiO2 glass under periodic temperature conditions using 2D-PFM and 3D-PFM. The main conclusion remarks are the following three points:

  • (1)

    The phase-field mobilities, LS and LB, were determined with 2D-PFM simulations. The surface and bulk diffusion coefficients, DS and DB, have the potential to work effectively for crystal nucleation and growth rates in undercooled Na2O–2CaO–3SiO2. This revealed that the parameters, A and B, of LS and LB determined by DS and DB were consistent with other silicates. Furthermore, the crystal growth rates obtained by 2D-PFM using LS and LB were comparable to the experimental values.

  • (2)

    By defining LS at the first layer on the polished surface and LB elsewhere, we could perform quantitively the 3D-PFM simulation such as the crystal radius and formation of the rings around the pre-existing crystals under periodic temperature conditions. The advantage of this PFM modeling is that it is simple and can reproduce the surface and bulk crystal growth rates.

  • (3)

    In contact and interaction during the crystal growth, the contact of crystals affecting the heterogeneous nucleation is considered the most important. When one crystal grew up and covered the polished surface around another crystal, heterogeneous nucleation could not occur at the covered surface. As a result, new crystal rings that were caused by heterogeneous nucleation at a low temperature could not be formed at the contacting region.

To demonstrate the performance of this simple PFM modeling, we performed the 3D-PFM simulation considering the experimental conditions such as random nucleation and nucleation frequency. 3D-PFM revealed that complex shapes between several crystals were comparable to experimental photographs.5

Acknowledgments

One of the authors (M.K.) expresses gratitude to colleagues at the Research Institute of Nuclear Engineering, University of Fukui and colleagues at Japan Atomic Energy Agency’s Tsuruga Comprehensive Research and Development Centre for various discussions.

Glossary

Abbreviations

2D-PFM

two-dimensional phase-field method

3D-PFM

three-dimensional phase-field method

VFT equation

Vogel–Fulcher–Tammann equation

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. M.K.: conceptualization, methodology, software, validation, investigation, writing—original draft, and writing—review and editing. M.U.: supervision and reviewing—original draft.

The authors declare no competing financial interest.

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