{"title":"斯威夫特-霍恩伯格和相场晶体模型中的梯度弹性","authors":"Lucas Benoit-Maréchal, Marco Salvalaglio","doi":"10.1088/1361-651x/ad42bb","DOIUrl":null,"url":null,"abstract":"The Swift–Hohenberg (SH) and phase-field crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.","PeriodicalId":18648,"journal":{"name":"Modelling and Simulation in Materials Science and Engineering","volume":"65 1","pages":""},"PeriodicalIF":1.9000,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Gradient elasticity in Swift–Hohenberg and phase-field crystal models\",\"authors\":\"Lucas Benoit-Maréchal, Marco Salvalaglio\",\"doi\":\"10.1088/1361-651x/ad42bb\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The Swift–Hohenberg (SH) and phase-field crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.\",\"PeriodicalId\":18648,\"journal\":{\"name\":\"Modelling and Simulation in Materials Science and Engineering\",\"volume\":\"65 1\",\"pages\":\"\"},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2024-05-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Modelling and Simulation in Materials Science and Engineering\",\"FirstCategoryId\":\"88\",\"ListUrlMain\":\"https://doi.org/10.1088/1361-651x/ad42bb\",\"RegionNum\":4,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATERIALS SCIENCE, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Modelling and Simulation in Materials Science and Engineering","FirstCategoryId":"88","ListUrlMain":"https://doi.org/10.1088/1361-651x/ad42bb","RegionNum":4,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
斯威夫特-霍恩伯格(SH)模型和相场晶体(PFC)模型是研究图案形成、集体有序和通过平滑有序参数产生缺陷等现象的最简便而又强大的方法。它们以自由能函数为基础,其中固有地包含了弹性效应。梯度弹性(GE)理论通过引入额外的特征长度来解释微观尺度上的弹性效应,本研究探讨了如何将梯度弹性纳入 SH 和 PFC 模型。在介绍了这些理论和模型的基本原理后,我们首先计算了各种晶格对称性的近似特征长度。然后,我们讨论了位错应力场的数值模拟以及与第一和第二应变梯度弹性分析解的比较。我们发现,位错诱导的弹性场的有效 GE 特性长度与相相关长度一样取决于自由能参数,从而揭示了它们如何随淬火深度而变化。本研究中的发现使我们能够利用 SH 和 PFC 模型对图案形成和晶体系统中的小尺度弹性效应进行全面的讨论和分析,重要的是完成了其中的弹性分析。此外,我们还为有序-无序相变背景下的 GE 提供了微观基础。
Gradient elasticity in Swift–Hohenberg and phase-field crystal models
The Swift–Hohenberg (SH) and phase-field crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.
期刊介绍:
Serving the multidisciplinary materials community, the journal aims to publish new research work that advances the understanding and prediction of material behaviour at scales from atomistic to macroscopic through modelling and simulation.
Subject coverage:
Modelling and/or simulation across materials science that emphasizes fundamental materials issues advancing the understanding and prediction of material behaviour. Interdisciplinary research that tackles challenging and complex materials problems where the governing phenomena may span different scales of materials behaviour, with an emphasis on the development of quantitative approaches to explain and predict experimental observations. Material processing that advances the fundamental materials science and engineering underpinning the connection between processing and properties. Covering all classes of materials, and mechanical, microstructural, electronic, chemical, biological, and optical properties.