Asymptotic Study of a Smoluchowski Equation Arising in Modeling of Nematic Polymers

向列聚合物建模中产生的 Smoluchowski 方程的渐近研究

• 批准号: 0733126
• 负责人: Jesenko Vukadinovic
• 金额: $ 5.08万
• 依托单位: CUNY College of Staten Island
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-10-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0733126&HistoricalAwards=false
• 关键词: Asymptotic; Study; Smoluchowski; Equation; Arising

The investigator and his collaborators study the Smoluchowskiequation arising in the kinetic molecular theory of nematic liquidcrystalline polymers. The intermolecular interaction of excludedvolume effects, which is responsible for the isotropic-nematicphase transition, is modeled by a mean-field potential. Both theMayer-Saupe and the Onsager potentials are used. The main purposeof the project is a rigorous study of the dynamics and rheologicalproperties of such systems. The problems to be addressed includebifurcation analysis of both the equilibrium and the shear flowcase, and questions arising in the treatment of the equations bymeans of the classical theory of infinite dimensional systems. One of the central problems is the question of the existence ofinertial manifolds on which the partial differential equationreduces to a finite system of ordinary differential equations, theso-called inertial form. The starting point of the project isrecent results in which the isotropic-nematic phase transition wasresolved for the Smoluchowski equation using the Mayer-Saupepotential in the equilibrium case (no external flow), as well asthe existence of absorbing balls (cones) in Gevrey spaces ofanalytic functions. These questions remain open for the Onsagerpotential, and are addressed in the study. In the absence of theflow, the Smoluchowski equation is a gradient system thatpreserves uniaxial symmetry, and as such it exhibits fairly simpledynamical behavior. However, even a passive interaction with asymmetry-breaking shear flow introduces complicated dynamicalfeatures, such as different kinds of periodic solutions (tumbling,kayaking, log-rolling), and even chaos. These are characteristicfor intermediate shear rates. At low and high shear rates flowaligning takes place. A rigorous investigation of these dynamicalfeatures is conducted in the project. The main goal of the project is a mathematical understandingof the transition that occurs in a class of polymeric materialscontaining anisotropic molecules known as mesogens to a so-calledliquid crystal phase. Due to their anisotropy, the moleculesinteract with each other, resulting in alignment. This long-rangeorientational order resembles the one found in solid crystals,while, unlike solid crystals, the material retains fluidity due tothe absence of the long-range positional order of molecules. Thetransition to such a phase is induced by changes in thetemperature and the concentration of the mesogens in the solution. The distribution of the molecules can be described by a partialdifferential equation known as the Smoluchowski equation, which isthe main subject of this project. It was already employedsuccessfully to explain the transition to the liquid crystal phaseunder some very special conditions, with many unanswered questionsstill remaining. The significance of these liquid crystallinepolymers lies in the fact that crystal-like properties, combinedwith many of the useful and versatile properties of polymers, makethese materials suitable for a wide range of importantapplications. For example, liquid crystalline polymers areabundant in living systems such as DNA, polypeptides, and cellmembranes. Accordingly, they attract particular attention in thefield of biomimetic chemistry. An application of liquidcrystalline polymers that has been successfully developed forindustry is the area of high strength fibers. Kevlar, which isused to make such things as helmets and bullet-proof vests, isjust one example of the use of polymer liquid crystals inapplications calling for strong, light-weight materials. Theoptical properties of liquid crystalline polymers lead toapplications in optical imaging and the display industry. Liquidcrystalline polymers can be used to coat drugs in thepharmaceutical industry. However, as very complex materials,liquid crystalline polymers exhibit very complex behavior, andmany of the actual and potential applications of these materialsdepend on multidisciplinary research. A better mathematicalunderstanding of the process of aligning is essential for betterunderstanding of the behavior of liquid crystalline polymers. This project addresses some of the central questions regardingthis process.

研究者及其合作者研究了列液晶体聚合物的动力学分子理论中产生的Smoluchowskiequation。 造成各向同性 - 近代型转变的排除值效应的分子间相互作用是由平均场电位建模的。 都使用了Themayer-Saupe和Onsager电位。 该项目的主要目的是对此类系统的动力学和流变学的严格研究。 要解决的问题包括对平衡和剪切流箱的分析分析，以及在处理无限尺寸系统经典理论方程式中引起的问题。中心问题之一是存在的问题的问题，即偏微分方程将其偏向于普通微分方程的有限系统，即被称为惯性形式的问题。 该项目的起点是，在平衡情况下，使用Mayer-Saupepomitential（无外部流动）以及存在吸收球（锥体）在Gevrey空间中存在的均衡函数的存在。 这些问题对于OnSagerPotitional仍然是开放的，并在研究中得到了解决。 在没有潮流的情况下，Smoluchowski方程是一种梯度系统，可以表现出单轴对称性，因此表现出相当简单的动力学行为。 然而，即使是与不对称剪切流的被动互动也会引入复杂的动力特征，例如不同种类的周期性解决方案（翻滚，皮划艇，对数），甚至混乱。 这些是中间剪切速率的特征。 在低和高剪切速率下进行流式定位。 对这些动态表进行了严格的调查。 该项目的主要目标是在一类聚合物材料范围内的各向异性分子中发生的过渡的数学理解，称为中成蛋白到所谓的液化晶体相。 由于它们的各向异性，分子互相互动，导致对齐。 这种远程定向的顺序类似于在固体晶体中发现的阶，而与固体晶体不同，由于没有分子的远距离位置阶，材料会保持流动性。 在溶液中的thetemature和介体的浓度的变化引起了这种相对于这种相处的趋势。分子的分布可以通过称为Smoluchowski方程的partialDifferential方程来描述，这是该项目的主要主题。 它已经有效地解释了某些非常特殊的条件向液晶阶段的过渡，其中剩下许多未解决的问题。 这些液晶聚合物的重要性在于，类似晶体的特性与聚合物的许多有用和多功能特性相结合，Makethese材料适用于广泛的重要应用。 例如，在DNA，多肽和细胞膜等生物系统中，液晶聚合物在生物中繁殖。 因此，它们引起了仿生化学领域的特别关注。 成功开发的液态晶聚合物的应用是高强度纤维的领域。 凯夫拉尔（Kevlar）是制作头盔和防弹背心之类的东西，它是使用聚合物液晶贴贴的一个例子，要求使用强大的轻量级材料。 液晶聚合物的定性特性导致光学成像和展示行业的应用。 液态晶体聚合物可用于在制药行业涂上药物。 但是，作为非常复杂的材料，液晶聚合物表现出非常复杂的行为，并且对这些材料的实际和潜在应用依赖于多学科研究。 对对齐过程的更好数学理解对于更好地理解液晶聚合物的行为至关重要。该项目解决了有关此过程的一些主要问题。