Pattern Formation in Thin-Film Ferromagnets and in Spinodal Decomposition of Polymer Solutions

这项计划中的工作调查了凝聚态物理中两个简单的连续介质模型的预测能力。模型是否以无序参数来预测特定空间图案的形成，这些图案是在特定区域内实验观察到的？所考虑的两个模型是a)微磁模型和b)两种不同粘度的不可压缩流体混合物的准定常流动模型，这种流动是由每个组分上的缓慢变化的力驱动的。在模型a)的情况下，目标是识别物理参数的制度，其中该模型再现了在具有薄膜几何结构的铁磁体中观察到的磁化的特征磁区和磁壁图案。对于模型b)，目的是询问它是否解释了海绵状图案的产生和自相似粗化，海绵状图案是在聚合物和水的混合物的调幅分解过程中聚合物相的浓度形成的。从数学上讲，模型a)以磁化中的非凸、非局部最小化问题的形式出现。模型b)以梯度通量类型的偏微分方程式的形式出现，它随时间演化聚合物浓度。对于模型a)，数学目标是刻画目标参数极限中极小值的变分极限。将会给出严格的证明。模型b)实际上是对实验者提出的模型的修正。我们的目标是证明修改是必要的，以至少在质量上重现他们的观测。将采用渐近分析和数值模拟相结合的方法。对某些现象进行建模是控制它们-并利用它们的关键。一个好的模型既简单又预测能力强。科学和工程中的模型通常是用数学语言建立的，用模型进行预测就等于解决了一个数学问题。本工作的目的是利用现代数学对两个具体的模型进行评估，以评估模型的预测质量并改进模型。第一个模型描述了磁性薄膜的行为，如磁记录介质中的涂层。目标是更好地理解微磁化图案的大小和形状。这是此类介质存储容量的一个主要因素。第二个模型描述了聚合物-水混合物随时间变化的行为。这种混合物用于聚合物加工，在生物化学中无处不在。我们的目标是了解聚合物浓缩过程中经常形成的模式。我们可以使用的新工具是一台功能强大的计算机，它可以用数字方法解决模型背后的数学问题。在计算机上实现数学问题是微妙的，需要对问题有很好的理论理解。数学分析中的现代工具已成为这项任务不可或缺的工具，并将与计划中的工作的数值计算结合使用。

The planned work investigates the predictive power of two simple continuum models in condensed matter physics. Do themodels predict the formation of certain spatial pattern in anorder parameter, patterns which are experimentally observed in special regimes? The two considered models are a) the micromagnetic model and b) a model for quasi-stationary flow of a mixture of two incompressible fluids ofdifferent viscosity, a flow driven by slowly changing forces on eachcomponent. In case of model a), the objective is to identify aregime of the physical parameters in which the model reproduces the characteristic domain and wall pattern of the magnetization which is observed in a ferromagnet with a thin film geometry. In case of model b), the objective is to inquire if it explains the creation and self-similar coarsening of sponge-like pattern, which are formed by the concentration of the polymer phase during spinodal decompositionof a mixture of polymer and water. Mathematically speaking, model a) comes in the form of a non-convex, non-local minimization problem in the magnetization. Model b) comes in the form of a partial differential equation of gradient flux type, which evolves the polymer concentration in time. For model a), the mathematical goal is to characterize variational limits of minimizers in the targeted parameter limit. Rigorous proofs will be given. Model b) actually is a modification of the model proposed bythe experimentalists. The goal is to show that the modificationis necessary to reproduce their observations at least qualitatively.A combination of asymptotic analysis and numerical simulation willbe employed.Modeling certain phenomena is the key to control them - and to makeuse of them. A good model is both simple and of high predictive power.Models in sciences and engineering usually are cast in mathematical language,and making prediction with the model then amounts to solving a mathematical problem.The goal of this work is to use modern mathematics for two specificmodels in order to assess the quality of a model's predictions andto improve the models. The first model describes the behavior of magnetic thin films such as coatings in magnetic recording media.The goal is a better understanding of the size and shape of micromagnetization patterns. This is a major factor in the storage capacity of such media. The second model describes the behavior of polymer-water mixtures over time. Such mixtures are used in polymer processing and are ubiquitous in biochemistry. The goal is to understand the patterns that often form during the polymer concentration process. The new tool at our disposal is a powerful computer, which allows to solve the mathematical problem behind the model numerically. Implementing a mathematical problem on the computer is delicate and requires a good theoretical understanding of the problem. Modern tools in mathematical analysis have become indispensible for this task and will be used in conjunction with numerical calculations for the planned work.