On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit
On the van der Waals–Cahn–Hilliard phase-field model and its equilibria conditions in the sharp interface limit
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锐界面极限下的范德瓦尔斯-卡恩-希利亚德相场模型及其平衡条件
DOI：
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
C. Kraus
中科院分区：
文献类型：
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作者：
W. Dreyer;C. Kraus
我们研究相场模型的热力学一致性，其中包括自由能泛函中密度 ρ 的梯度项，例如范德华-卡恩-希利亚德模型。众所周知，只有当能量通量或熵通量由非经典形式表示时，熵不等式才允许自由能中的 ρ 的梯度项和高阶梯度项。我们确定了一种非经典熵通量，它不限于等温过程，因此梯度贡献是可能的。然后，我们研究了尖锐界面极限下范德华-卡恩-希利亚德相场模型的平衡条件。对于单一物质，热力学在尖锐界面处提供了两个跳跃条件，即相邻相吉布斯自由能的连续性和相应压力的不连续性，这通过平均曲率来平衡。我们证明这些条件也可以从尖锐界面极限的范德华-卡恩-希利亚德相场模型中提取。为此，我们证明密度渐近展开至一阶。结果基于局部能量估计和密度的均匀收敛结果。
We study the thermodynamic consistency of phase-field models, which include gradient terms of the density ρ in the free-energy functional such as the van der Waals–Cahn–Hilliard model. It is well known that the entropy inequality admits gradient and higher-order gradient terms of ρ in the free energy only if either the energy flux or the entropy flux is represented by a non-classical form. We identify a non-classical entropy flux, which is not restricted to isothermal processes, so that gradient contributions are possible. We then investigate equilibrium conditions for the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. For a single substance thermodynamics provides two jump conditions at the sharp interface, namely the continuity of the Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. We show that these conditions can be also extracted from the van der Waals–Cahn–Hilliard phase-field model in the sharp interface limit. To this end we prove an asymptotic expansion of the density up to the first order. The results are based on local energy estimates and uniform convergence results for the density.