We consider a fractional phase-field crystal (FPFC) model in which the classical Swift–Hohenberg equation (SHE) is replaced by a fractional order Swift–Hohenberg equation (FSHE) that reduces to the classical case when the fractional order |$\beta =1$|⁠. It is found that choosing the value of |$\beta $| appropriately leads to FSHE giving a markedly superior fit to experimental measurements of the structure factor than obtained using the SHE (⁠|$\beta =1$|⁠) for a number of crystalline materials. The improved fit to the data provided by the fractional partial differential equation prompts our investigation of a FPFC model based on the fractional free energy functional. It is shown that the FSHE is well-posed and exhibits the same type of pattern formation behaviour as the SHE, which is crucial for the success of the PFC model, independently of the fractional exponent |$\beta $|⁠. This means that the FPFC model inherits the early successes of the FPC model such as physically realistic predictions of the phase diagram etc. and, therefore, provides a viable alternative to the classical PFC model. While the salient features of PFC and FPFC are identical, we expect more subtle features to differ. The prediction of grain boundary energy arising from a mismatch in orientation across a material interface is another notable success of the PFC model. The grain boundary energy can be evaluated numerically from the PFC model and compared with experimental measurements. The grain boundary energy is a derived quantity and is more sensitive to the nuances of the model. We compare the predictions obtained using the PFC and FPFC models with experimental observations of the grain boundary energy for several materials. It is observed that the FPFC model gives superior agreement with the experimental observation than those obtained using the classical PFC model, especially when the mismatch in orientation becomes larger.

中文翻译：

---

分数相场晶体建模：分析，近似和图案形成

我们考虑分数相场晶体（FPFC）模型，其中经典的Swift–Hohenberg方程（SHE）被分数阶Swift–Hohenberg方程（FSHE）取代，当分数阶| $ \ beta时，分数阶Swift–Hohenberg方程（FSHE）简化为经典情况= 1 $ |⁠。发现选择| $ \ beta $ |的值 适当地导致FSHE比使用SHE获得的结构因子的实验测量结果具有明显优越的拟合（⁠| $ \ beta = 1 $ |⁠）用于多种晶体材料。对分数阶偏微分方程提供的数据的改进拟合促使我们研究基于分数自由能函数的FPFC模型。结果表明，FSHE具有良好的位置，并且表现出与SHE相同的图案形成行为，这对于PFC模型的成功至关重要，而与分数指数| $ \ beta $ |⁠无关。这意味着FPFC模型继承了FPC模型的早期成功，例如相图的物理逼真预测等，因此为经典PFC模型提供了可行的替代方法。尽管PFC和FPFC的显着特征相同，但我们希望更多细微的特征有所不同。PFC模型的另一个显着成功是对由于跨材料界面取向不匹配而产生的晶界能的预测。可以通过PFC模型对晶界能进行数值评估，并与实验测量值进行比较。晶界能是一个导出的量，并且对模型的细微差别更加敏感。我们将使用PFC和FPFC模型获得的预测与几种材料的晶界能的实验观察结果进行比较。