Phase field modeling with large driving forces 

Jin Zhang, Alexander F. Chadwick, David L. Chopp & Peter W. Voorhees

npj Computational Materials 9: 166 (2023)

(原文链接：https://doi.org/10.1038/s41524-023-01118-0)

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编辑概述：具有大驱动力的稳定相场模拟

相场法作为一种很有前途的材料设计工具，在建模材料的微观结构演化和计算设计方面取得了显著的成功，但在实现工程问题的定量预测方面仍存在障碍。较大的驱动力在工程问题和相场模拟中很常见。较大的驱动力意味着较大的表观驱动力，从而对界面宽度施加了一个很大的上限。这个上限将空间网格尺寸限制在纳米甚至亚纳米，禁止将相场模型应用于从微米到毫米的实际系统尺寸。尤其是考虑化学计量线化合物时，这种限制可能是显著的，通常使用的抛物线自由能在数值上可能是不稳定的，因此需要精细的空间和时间分辨率。这需要更多的计算资源，限制了以合适的大小模拟系统的能力，并降低了定量预测的能力。在本工作中，来自美国西北大学材料科学与工程系的Jin Zhang教授小组，提出了一种策略，以保持一个无条件稳定的相场分布独立于驱动力的大小。他们发现，稳定性问题是由扩散界面内的驱动力的非线性变化引起的，而不是驱动力的大小。使用三阶插值和恒定驱动力，可以在没有表面能项的情况下保持行波结构，他们提出了一种驱动力扩展方法，将驱动力投影到垂直于界面的恒定方向。这种方法缓解了驱动力对网格大小的限制。本研究提出的方法正确地捕捉了吉布斯-汤姆逊效应，并直接应用于高维，其提出的驱动力扩展方法只需要对相场方程进行简单的修改，可以直接应用于现有的相场模型，并易于适应许多数值技术。该方法可以用于大多数现有的相场模型，以处理由于大的表观驱动力导致的稳定性问题，从而在使用相场方法作为材料设计的定量工具方面迈出了重要的一步。

Editorial Summary: Stable phase field modeling with large driving force

As a promising material design tool, the phase field method has achieved remarkable success in modeling microstructure evolution and computational design of materials, but there are still barriers to achieving quantitative prediction for engineering problems. Large driving forces are common in engineering problems and phase field simulations. A larger driving force means a larger apparent driving force, thus imposing a strong upper bound on the interface width. This upper bound limits the spatial grid size to nanometers or even sub-nanometers, prohibiting the application of phase field models to practical system sizes ranging from micrometers to millimeters. This limitation can be dramatic especially when considering stoichiometric line compounds, and the commonly employed parabolic free energies can be numerically unstable, thus requiring fine spatial and temporal resolution. This demands more computational resources, limits the ability to simulate systems with a suitable size, and deteriorates the capability of quantitative prediction. In this work, a group led by Prof. Jin Zhang from the Department of Materials Science and Engineering, Northwestern University, proposed a strategy to maintain an unconditionally stable phase field profile independent of the magnitude of the driving forces. They found that the stability issue is caused by a nonlinear variation of the driving force within the diffuse interface instead of the magnitude of the driving force. By noticing that the traveling wave structure could be maintained without the surface energy term using the third-order interpolation and a constant driving force, they proposed a driving force extension method to project the driving force to a constant perpendicular to the interface. This method relieved the driving force constraint on grid size. The method proposed in this study correctly captured the Gibbs-Thomson effect and was directly applied to higher dimensions. The proposed driving force extension method only required a simple modification of the phase field equation, which could be directly applied to existing phase field models and was easy to adapt to many numerical techniques. The method could be applied to most existing phase field models to handle the stability problem due to large apparent driving forces, thus initiating a significant step forward in using the phase field method as a quantitative tool in materials design.