Phase-field model for crystallization in alkali disilicate glasses: Li₂O–2SiO₂, Na₂O–2SiO₂ and K₂O–2SiO₂

Abstract

This study developed phase-field method (PFM) technique in oxide melt system by using a new mobility coefficient (L). The crystal growth rates (v 0) obtained by the PFM calculation with the constant L were comparable to the thermodynamic driving force in normal growth model. The temperature dependence of the L was determined from the experimental crystal growth rates and the v 0. Using the determined L, the crystal growth rates (v) in alkali disilicate glasses, Li 2 O2SiO 2 , Na 2 O2SiO 2 and K 2 O2SiO 2 were simulated. The temperature dependence of the v was qualitatively and quantitatively so similar that the PFM calculation results demonstrated the validity of the L. Especially, the v obtained by the PFM calculation appeared the rapid increase just below the thermodynamic melting point (T m) and the steep peak at around T m 100K. Additionally, as the temperature decreased, the v apparently approached zero ms ¹1 , which is limited by the L representing the interface jump process. Furthermore, we implemented the PFM calculation for the variation of the parameter B in the L. As the B increased from zero to two, the peak of the v became steeper and the peak temperature of the v shifted to the high temperature side. The parameters A and B in the L increased exponentially and decreased linearly as the atomic number of the alkali metal increased due to the ionic potential, respectively. This calculation revealed that the A and B in the L were close and reasonable for each other.

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

“…∑ i M i / M O was the sum of cation molar mass per oxygen molar mass. The triangles show the A and B of the other silicates, SiO 2 and GeO 2 , according to our previous paper. , The straight lines in Figures and show the approximation curves for the triangles in the other silicates, SiO 2 and GeO 2 . , Figures and revealed that the A and B of Na 2 O–2CaO–3SiO 2 depended on their diffusion coefficients ( D η , D S , and D B ). Compared with the solid lines determined by the other silicates, the A and B determined by D S and D B are more reasonable values than those by D η .…”

“…Furthermore, to address wetting and heterogeneous crystal nucleation on a wall using the PFM, Grańaśy et al suggested boundary conditions at the wall, including the free energy of nuclei formation and the contact angle. 25 Our previous papers 26,27 showed that the classical PFM with newly defined phase-field mobilities reproduced quantitatively the crystal growth rates in undercooled silicates, SiO 2 and GeO 2 liquids. Furthermore, it found that the phase-field mobilities were independent of the calculation dimension such as two or three dimensions.…”

“…26,27 The straight lines in Figures 10 and 11 show the approximation curves for the triangles in the other silicates, SiO 2 and GeO 2 . 26,27 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 D S and D B at dx = 0.1, 0.05, 0.02, and 0.01 μm.…”

“…The other parameters of eq are α = 2.856 and d = 7.5 × 10 –10 m. T and k B are the temperature and the Boltzmann constant, respectively. According to our previous papers, , the phase-field mobility, L S,B , was defined as follows L normalS , normalB = A true( D eff normalΔ T 2 π T normalm λ 2 true) B where A and B are the mobility parameters, D eff is the effective diffusion coefficient, and λ is the jump distance, which is usually taken as the molecular diameter. Here, D eff values for u S and u B are given by D S and D B , respectively.…”

“…The other parameters of eq 7 are α = 2.856 and d = 7.5 × 10 −10 m. T and k B are the temperature and the Boltzmann constant, respectively. According to our previous papers, 26,27 the phase-field mobility, L S,B , was defined as follows i k j j j j j y…”