New Challenges in Aggregation Kinetics

聚合，即单体聚合成大的、有结构的簇，是科学和工业中许多现象的核心。在一个简化的模型中，簇生长是一个离散的随机过程，每次增加或减少一个单体。贝克-多林（BD）常微分方程通过表面反应控制小簇的生长。由于周围单体的扩散而形成的大簇可以用lifshitz - slyozoequation来描述。小的团簇成核并形成大的团簇是由自由能垒上的波动引起的缓慢过程。研究者Joseph Farjoun研究了成核的以下几个方面：(a) Zeldovich成核公式与实验结果不一致。它对超饱和度的值非常敏感，因此对簇大小的分布也非常敏感。研究者通过将超饱和分解到更高阶来寻求更精确的公式。此外，基于重整蒙特卡罗方法的模拟提供了更多关于成核速率的数据。(b)由Zeldovich成核率和LS增长率决定的团簇大小的长期分布是一个经典问题。研究者研究了与平滑相似解决方案的联系：对他博士论文中部分答案的深入分析。(c)二维成核分析是研究者博士论文工作的自然延伸。单体浓度在二维上对核的对数行为迫使成核过程在空间上是非均匀的。研究者通过将之前关于二维扩散的工作联系起来来研究这个问题。(d)研究者研究聚合过程对其他物理参数的动态依赖性。具体来说，他分析了温度变化的系统中的聚合。成核速率和形成的簇的总量是由温度降低和单体耗竭之间的平衡决定的，它们对超饱和有相互竞争的影响。通过聚集建模的现象——凝固、沉淀和蒸汽凝结成液滴——出现在许多生物物理和工业环境中：玻璃到晶体的转变、过冷液体和聚合物中的晶体成核、胶体结晶、超过临界胶束浓度（CMC）的球形聚集体的生长、通过淬火进入混相间隙的二元合金的粗化而产生的偏析，仅举几个具体例子。微芯片制造领域将从一种能够预测二维团簇的短期和长期行为的理论中受益匪浅。解决Zeldovich公式（预测集群形成的速度）和实验之间长期存在的分歧，找到聚集过程中各个阶段的时间尺度，以及二维聚集的理论（所有这些都是本项目的研究主题）允许更准确的预测和更好的设计。一个更现实的模型，可以解释温度的变化，可以用于蒸汽形成工业（如发电厂）的设计和天气预报。目前的聚合技术是极端的：非常短和非常长时间的行为，非常小和非常大的集群的行为已经被广泛研究。在这个项目中，研究员约瑟夫·法琼的工作弥补了这些差距。

Rosales0703937 Aggregation, the coalescence of monomers into large,structured clusters, is found at the heart of many phenomena inscience and industry. In a simplified model, cluster growth is adiscrete stochastic process of adding or shedding monomers, oneat a time. The Becker-Doring (BD) ordinary differentialequations govern the growth of small clusters by surfacereactions. Large clusters, growing by the diffusion of thesurrounding monomers, are described by the Lifshitz-Slyozovequations. Small clusters nucleate and form large ones by a slowprocess caused by fluctuations over a free-energy barrier. Theinvestigator, Joseph Farjoun, studies the following aspects ofnucleation: (a) The Zeldovich nucleation formula disagrees with theexperimental results. It is highly sensitive to thesuper-saturation's value, and hence on the distribution ofcluster-sizes. The investigator seeks a more accurate formula byresolving the super-saturation to higher order. In addition, asimulation based on renormalized Monte-Carlo methods is providingmore data on the nucleation rate. (b) The long-term distribution of clusters sizes, as determinedby the Zeldovich nucleation rate and LS growth rate, is aclassical problem. The investigator studies a connection to thesmooth similarity-solution: A deeper analysis of the partialanswer in his PhD thesis. (c) The analysis of 2-dimensional nucleation is a naturalextension of the work in the investigator's PhD thesis. Thelogarithmic behavior of monomer concentration about nuclei in2-dimensions forces the nucleation process to be spatiallynon-homogeneous. The investigator studies this by connecting toprevious work on 2-dimensional diffusion. (d) The investigator studies the dynamic dependency of theaggregation process on other physical parameters. Specifically,he analyzes aggregation in a system with a varying temperature. The nucleation rate and the total amount of clusters formed aredetermined by a balance between the decreasing temperature andthe depletion of monomers, which have competing effects on thesuper-saturation. Phenomena that are modeled by aggregation -- solidification,precipitation, and the condensation of vapor into droplets --appear in many biophysical and industrial contexts:Glass-to-crystal transformations, crystal nucleation inunder-cooled liquids, and in polymers, colloidal crystallization,growth of spherical aggregates beyond the critical micelleconcentration (CMC), and the segregation by coarsening of binaryalloys quenched into the miscibility gap, to name but a fewspecific examples. The field of micro-chip fabrication stands tobenefit greatly from a theory capable of predicting the short-and long-term behavior of 2-dimensional clusters. Resolving thelong-standing disagreement between the Zeldovich formula (whichpredicts the rate at which clusters are created) and theexperiments, finding the time-scales of the various phases of theaggregation process, and a theory for 2-dimensional aggregation(all of which are topics of research in this project) allows formore accurate predictions and better design. A more realisticmodel, which accounts for the varying temperature, can be used inthe design of vapor-forming industries (e.g. power plants), andin weather prediction. The current state of the art inaggregation is that of extremes: the behavior at very short andvery long times, the behavior of very small or very largeclusters have been studied extensively. In this project, thework of the investigator, Joseph Farjoun, bridges these gaps.