Motion By Curvature In Phase Transitions

相变中的曲率运动

• 批准号: 9801337
• 负责人: Gieri Simonett
• 金额: $ 6.32万
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9801337&HistoricalAwards=false
• 关键词: Motion; Curvature; Phase; Transitions

PI: Gieri Simonett DMS-9801337 ABSTRACT: Over the last years the PI has worked on various free boundary problems and has studied existence, uniqueness, regularity, and qualitative properties of classical solutions for such models as the gravitational flow of a fluid in a porous medium, the multi-dimensional one-phase Hele-Shaw problem, the one and two-phase Mullins-Sekerka model,the quasi-stationary Stefan problem with surface tension, and the surface diffusion flow.This research led to the solution of some long-standing open problems. In this project the PI will continue to study geometric evolution problems for surfaces driven by mean curvature. These models are widely used in material sciences, physics, and chemistry to model phase changes, domain growth,and interface controlled crystal growth. Progress on the mathematical front will necessarily have an impact in material sciences. The Mullins-Sekerka model is a nonlocal geometric evolution law in which the normal velocity of a propagating interface depends on the jump across the interface of the normal derivative of a function which is harmonic on either side and which equals the mean curvature on the propagating interface. It was introduced to study solidification and liquidation of materials of zero specific heat and has attracted considerable attention since then.Important contributions by Alikakos, Bates and Chen have tied this model to a singular limit for the Cahn-Hilliard equation, a fourth order parabolic equation which is widely used as a model for phase separation and coarsening phenomena in a melted binary alloy. This model has also been proposed to account for aging or Ostwald ripening in phase transitions. In general, the kinetics of a first order phase transition is characterized by a first stage where small droplets of a new phase are created out of the old phase, e.g., solid formation in an undercooled liquid. The first stage, called nucleation,yields a large number of small particles .During the next stage the nuclei grow rapidly at the expense of the old phase.When the phase regions are formed,the mass of the new phase is close to equilibrium and the amount of undercooling is small,but large surface area is present.At the next stage, the configuration of phase regions is coarsened, and the geometric shape of the phase regions become simpler and simpler, eventually tending to regions of minimum surface area with given volume. The driving force of this process comes from the need to decrease the interfacial energy. There have been considerable effortsin finding a theory which describes Ostwald ripening, and the Mullins-Sekerka model is a prominent candidate.The surface diffusion flow and the intermediate surface diffusion flow are geometric evolution problems which model morphological changes where surface diffusion and interface kinetics are the transport mechanisms. These laws constitute a class of dynamic problems where the volume is conserved and the driving force is surface energy reduction.

PI：Gieri Simonett DMS-9801337摘要： 在过去的几年里，PI致力于各种自由边界问题，并研究了存在性，唯一性，正则性和定性性质 流体重力流动等模型的经典解 在多孔介质中，多维单相Hele-Shaw问题， 单相和两相Mullins-Sekerka模型，准稳态Stefan模型， 问题的表面张力，和表面扩散流。这项研究导致解决了一些长期悬而未决的问题。在这个项目中 PI将继续研究曲面的几何演化问题 由平均曲率驱动。这些模型广泛应用于材料科学、物理学和化学中，以模拟相变、畴生长和界面控制的晶体生长。数学前沿的进展将 必然会对材料科学产生影响。 Mullins-Sekerka模型是一个非局部几何演化律，其中 传播界面的法向速度取决于跳跃 在函数的法向导数的界面上， 在两边都是调和的，它等于 传播界面。它被用来研究零比热物质的凝固和液化，并引起了人们的广泛关注。Alikakos，Bates和Chen的重要贡献 已经将该模型与Cahn-Hilliard方程的奇异极限联系起来，Cahn-Hilliard方程是一种四阶抛物方程，广泛用作熔融二元合金中相分离和粗化现象的模型。 这个模型也被提出来解释老化或奥斯特瓦尔德熟化 在相变中。通常，一级相变的动力学的特征在于第一阶段，其中新相的小液滴从旧相产生，例如，过冷固态成形 液体第一个阶段称为形核，产生大量的小颗粒。在下一个阶段，形核以旧相为代价迅速生长。当相区形成时，新相的质量接近平衡，过冷量很小，但存在大的表面积。在下一个阶段，相区的结构粗化，并且相位区域的几何形状变得越来越简单， 最终趋向于具有给定体积的最小表面积的区域。 该过程的驱动力来自于降低界面能的需要。有相当大的努力找到一个理论，描述Ostwald熟化，和Mullins-Sekerka模型是一个突出的候选人。表面扩散流和中间表面扩散流的几何演化问题，模型的形态变化，表面扩散和界面动力学的传输机制。 这些定律构成了一类动态问题， 是守恒的，驱动力是表面能减少。