Macroscopic equations for microscopic transport processes

通常，工程材料由大约10^23个原子或分子组成，而材料的行为是其所有粒子相互作用的结果。尽管有如此大量的单个粒子，但各种各样的流动和传热问题可以用相当少的偏微分方程组来描述，例如著名的流体和气体流动的纳维斯托克斯方程和傅立叶方程，或费克扩散定律。由于粒子之间的强烈相互作用(例如气体分子的碰撞)，扰动有效地分布在粒子之间，因此在许多应用中，粒子不是作为单个粒子，而是作为一个连续体。然而，连续介质的响应是微观尺度上的过程的结果，这为研究微观行为和宏观行为之间的关系提供了重要的理由。燃料电池用的PEM膜需要许多传输系数，而对微观行为的了解可以为后者的建模提供信息。在更极端的条件下可能会受到影响--例如稀薄气体流动、微流、非平衡蒸发和冷凝以及其他界面过程--标准的工程定律被打破，必须被高级模型取代，后者在原子尺度上解释了更多过程的细节。他说，现代工程设计依赖于对设备及其工艺的数值模拟。好的和准确的数学模型对于这项任务是不可或缺的。这项拟议的研究考虑了微观世界和宏观世界之间的关系，以推导和验证稀薄气体、用于燃料电池的聚合物膜以及蒸发和冷凝过程的数学传输模型。由此产生的模型将在宏观层面上描述过程，即作为一个连续体，但包含来自微观材料行为的知识。因此，它们将超越标准的工程定律，为这些复杂而重要的过程的高级模拟打开大门。

Typically, engineering materials consist of about 10^23 atoms or molecules, and the behavior of a material results from the interplay of all its particles.  Despite this huge number of individual particles, a wide variety of flow and heat transfer problems can be described by a rather low number of partial differential equations, e.g. the well-known equations of Navier-Stokes and Fourier for fluid and gas flows, or Fick's law of diffusion. Due to the intense interaction between the particles (e.g. collisions of gas molecules), disturbances are effectively distributed between the particles, so that in many applications the particles behave not as individuals, but as a continuum.      Nevertheless, the response of the continuum results from processes on the microscopic scale, and this gives an important reason to study the relation between the microscopic and the macroscopic behavior.  Macroscopic models for advanced materials--e.g. PEM membranes for fuel cells--require many transport coefficients, and understanding of the microscopic behavior can inform the modelling of the latter.      Under more extreme conditions--e.g. rarefied gas flows, microflows, non-equilibrium evaporation and condensation and other interface processes--the standard engineering laws break down, and must be replaced by advanced models, which account for more details of the processes on the atomic scale.      Modern engineering design relies on numerical simulations of devices, and the processes therein. Good and accurate mathematical models are indispensable for this task. The proposed research considers the relation between the micro- and the macroworld in order to derive, and validate, mathematical transport models for rarefied gases, polymer membranes for use in fuel cells, and evaporation and condensation processes. The resulting models will describe the processes on a macroscopic level, i.e. as a continuum, but incorporate knowledge from the microscopic material behavior. Thus, they will go beyond the standard engineering laws and open the door for advanced simulations of these complicated and important processes.