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Phase-field mobility for crystal growth rates in undercooled silicates, SiO$$_2$$ and GeO$$_2$$ liquids

過冷却中の珪酸塩、SiO$$_2$$, GeO$$_2$$融液中の結晶成長に対する界面易動度

河口 宗道  ; 宇埜 正美*

Kawaguchi, Munemichi; Uno, Masayoshi*

過冷却ケイ酸塩,SiO$$_2$$,GeO$$_2$$融液中の11種類の酸化物または混合酸化物の結晶化におけるフェーズフィールド易動度$$L$$と結晶成長速度をフェーズフィールドモデル(PFM)を用いて計算し、$$L$$の物質依存性を議論した。実験の結晶成長速度と$$L=1$$のPFMシミュレーションから得られた結晶成長速度の比は、両対数プロットで結晶成長における固液界面プロセスの$$frac{TDelta T}{eta}$$のべき乗に比例した。また$$L=A(frac{k_{B}TDelta T}{6pi^{2}lambda^{3}eta T_{m} })^{B}$$のパラメータ$$A$$$$B$$$$A=6.7times 10^{-6}-2.6$$m$$^4$$J$$^{-1}$$s$$^{-1}$$,$$B=0.65-1.3$$であり、材料に固有の値であることが分かった。決定された$$L$$を用いたPFMシミュレーションにより、実験の結晶成長速度を定量的に再現することができた。$$A$$$$T_{m}$$における単位酸素モル質量あたりの陽イオンモル質量の平均の拡散係数と両対数グラフで比例関係にある。$$B$$は化合物中の酸素モル質量あたりの陽イオンのモル質量の総和$$frac{Sigma_{i}M_{i}}{M_{O}}$$に依存する。$$frac{Sigma_{i}M_{i}}{M_{O}}leq 25$$では、陽イオンのモル質量が大きくなるにつれて$$B$$は小さくなる。陽イオンのモル質量は陽イオンの移動の慣性抵抗に比例するため、$$B$$は陽イオンのモル質量の逆数で減少する。$$frac{Sigma_{i}M_{i}}{M_{O}}geq 25$$の重い陽イオンのケイ酸塩の結晶化では、$$B$$は約$$0.67$$で飽和し、$$T_{p}approx 0.9T_{m}$$となる。

Phase-field mobility, $$L$$, and crystal growth rates in crystallization of 11 oxides or mixed oxides in undercooled silicates, SiO$$_2$$ and GeO$$_2$$ liquids were calculated with a simple phase-field model (PFM), and material dependence of the $$L$$ was discussed. Ratios between experimental crystal growth rates and the PFM simulation with $$L=1$$ were confirmed to be proportional to a power of $$frac{TDelta T}{eta}$$ on the solid/liquid interface process during the crystal growth in a log-log plot. We determined that parameters, $$A$$ and $$B$$, of the $$L=A(frac{k_{B}TDelta T}{6pi^{2}lambda^{3}eta T_{m} })^{B}$$ were $$A=6.7times 10^{-6}$$ to $$2.6$$m$$^4$$J$$^{-1}$$s$$^{-1}$$ and $$B=0.65$$ to $$1.3$$, which were unique for the materials. It was confirmed that our PFM simulation with the determined $$L$$ reproduced quantitively the experimental crystal growth rates. The $$A$$ has a proportional relationship with the diffusion coefficient of a cation molar mass average per unit an oxygen molar mass at $$T_{m}$$ in a log-log graph. The $$B$$ depends on the sum of the cation molar mass per the oxygen molar mass, $$frac{Sigma_{i}M_{i}}{M_{O}}$$, in a compound. In $$frac{Sigma_{i}M_{i}}{M_{O}}leq 25$$, the $$B$$ decreases with the cation molar mass increasing. The assumed cause is that the B represents the degree of the temperature dependence of the $$L$$. Since the cation molar mass is proportional to an inertial resistance of the cation transfer, the $$B$$ decreases with inverse of the cation molar mass. In crystallization of the silicates of heavy cation in $$frac{Sigma_{i}M_{i}}{M_{O}}geq 25$$, the $$B$$ saturates at approximately 0.67, which leads to $$T_{p}approx 0.9T_{m}$$.

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