Collaborative Research: Numerical Simulation of the Morphosynthesis of Polycrystalline Biominerals

多晶生物矿物是热力学上稳定的生物成因矿物的晶型，其特征是层叠的晶体，相邻层之间有矿物桥。独特的晶体织构产生了特殊的材料性能，如韧性、耐腐蚀性和耐温性，这使得这些晶体在光学纳米结构(光子带隙、衍射栅)和特殊涂层(例如在半导体器件技术中)具有极大的吸引力。因此，目前许多材料科学家都在努力实现这类生物矿物的合成。该项目的目的是提供结晶过程的数学模型和数值模拟的算法工具，以了解该过程的机理，使实验人员能够优化他们的实验室设置，从而为工业相关的生产线铺平道路。多晶生物矿物的形态合成遵循多阶段结晶过程，包括聚合物诱导的液体前体(PILP)相、由于成核而产生的球晶和马赛克中晶薄结构的再结晶。PILP相由生物矿物和与乙醇混合的阴离子聚合物的水溶液组成，通过液体混合物中富含聚合物的PILP液滴进行液-液相分离。混合是由表面声波(SAWS)操纵的流体流动处理的，其中SAW由两个锥形叉指换能器在双模式下工作产生。多晶开始形成球晶，球晶散布在衬底上，形成均匀的球晶薄膜。连续冷却导致球晶薄膜再结晶成马赛克多晶薄结构。表征PILP相的液液相分离可以用一个由不可压缩的Navier-Stokes方程和Cahn-Hilliard方程组成的耦合系统来描述。对于数值模拟，该项目将使用基于时间隐式离散和C0内部惩罚不连续Galerkin(C0-IPDG)的分裂格式在空间上相对于计算域的单纯三角剖分进行离散。该研究将研究分裂方法的收敛问题，并利用面向目标的双权方法实现时空自适应。作为多晶过程的数学模型，该项目研究了由测量局部结晶度的动力学方程、生物矿物的浓度场、定向场和冷却过程中温度演变的热方程组成的相场模型。浓度场方程为四阶Cahn-Hilliard方程。同样，在时间上隐式离散，在空间上用C0-IPDG方法离散，该项目将使用分裂方法和具有最终时间所需结晶度的时空自适应性的双重加权残差作为目标泛函。模型验证将基于合作实验室提供的实验数据，并将进行系统的参数研究，以调查各种工艺参数的影响。

Polycrystalline biominerals are thermodynamically stable crystal polymorphs of biogenic minerals featuring stacked layers of crystals with mineral bridges between adjacent layers. The unique crystal texture gives rise to specific material properties such as toughness, corrosion resistance, and temperature resistance, which makes these crystals highly attractive for optical nanostructures (photonic band gaps, diffraction gratings) and for special coatings (e.g., in semiconductor device technology). Therefore, many material scientists are currently trying to realize the synthesis of such biominerals. The aim of this project is to provide both a mathematical model for the crystallization process and algorithmic tools for numerical simulations in order to understand the mechanisms of the process, to enable the experimentalists to optimize their laboratory settings, and thus to pave the way for an industrially relevant production line.The morphosynthesis of polycrystalline biominerals follows a multistage crystallization process including a polymer-induced liquid-precursor (PILP) phase, the occurrence of spherulites due to nucleation, and the recrystallization of mosaic mesocrystal thin structures. The PILP phase consists of an aqueous solution of the biomineral and an anionic polymer mixed with ethanol and features a liquid-liquid phase separation in terms of polymer-rich PILP droplets in the liquid mixture. The mixing is taken care of by a surface acoustic waves (SAWs) manipulated fluid flow where the SAWs are generated by two tapered interdigital transducers operating in dual mode. The polycrystallization sets in with the formation of spherulites that spread across the substrate to form a uniform spherulitic thin film. Continuous cooling leads to a recrystallization of the spherulitic thin film into a mosaic polycrystalline thin structure. The liquid-liquid phase separation characterizing the PILP phase can be described by a coupled system consisting of the incompressible Navier-Stokes equations and a Cahn-Hilliard equation. For the numerical simulation, the project will use a splitting scheme based on an implicit discretization in time and C0 Interior Penalty Discontinuous Galerkin (C0-IPDG) methods for discretization in space with respect to simplicial triangulations of the computational domain. The research will study the convergence of the splitting method and realize space-time adaptivity by the goal oriented dual weighted approach. As a mathematical model for the polycrystallization the project investigates a phase field model consisting of the dynamic equations for the measure of local crystallinity, the concentration field for the biomineral, the orientation field, and a heat equation for the evolution of the temperature during the cooling process. The equation for the concentration field is a fourth order Cahn-Hilliard type equation. Again, discretizing implicitly in time and by C0-IPDG methods in space, the project will use a splitting method and dual weighted residuals for space-time adaptivity featuring a desired crystallinity at final time as the objective functional. A model validation will be based on experimental data provided by cooperating laboratories and a systematic parameter study will be performed to investigate the influence of various process parameters.