【作者】 宋文迪；

【导师】 杨玉良； 唐萍；

【作者基本信息】 复旦大学 ， 高分子化学与物理， 2011， 博士

【摘要】 经过50多年的不断发展和完善,自洽平均场理论(Self-Consistent Field Theory, SCFT)已经成为高分子凝聚态物理理论中十分重要且有效的理论方法,由于可以考虑不同的链拓扑结构和给出链构象信息,因此被广泛应用在非均相多组分高分子体系的热力学平衡态结构研究中。然而,关于蠕虫状链模型的自洽场理论,其数值算法远远没有高斯链自洽场理论那样完善。一方面,由于蠕虫状链模型中引入单位矢量u描述链的弯曲刚性,高分子链的构象概率不仅与空间位置r有关,还与链节取向方向u有关。对应的链节概率函数(传播因子)q(r,u,s)的扩散方程,在三维空间中是一个高达六维的偏微分方程；另一方面,蠕虫状链扩散方程中包含对取向角度u(θ,φ)的二阶梯度算符(?)u2,与平面Laplace算符(?)r2相比,其数值计算方法还不完善。除此之外,在半刚性链高分子体系中,由于链的刚性效应往往存在各向异性取向相互作用,其描述形式比各向同性相互作用更加复杂。因此,探索运用蠕虫状链自洽场理论及其有效的数值求解算法研究半刚性高分子的相行为,一直是高分子物理领域理论研究者长期以来致力于解决的难题。至今为止,关于蠕虫状链模型的自洽场理论,其数值算法还处于起步和探索阶段。其中较为具有代表性的是基于球谐函数Yl,m(u)展开的思想,对体系中函数的取向依赖性(u-dependence)进行变量分离,在球谐空间处理取向问题和完成角度Laplace算符的求解,再通过“合成”方法获得实空间下关于取向依赖u(θ,φ)的信息。由于数值计算量较大,球谐函数展开法在研究半刚性链体系时大多对取向进行简化处理,仅考虑对角度θ的依赖性,即令m=0,因此只得到nematic、smectic-A等液晶结构,却无法观察到smectic-C相。虽然借助免费的SPHEREPACK软件包,可以完成在实空间(θ,φ)和球谐空间(,,m)之间的快速变换,在一定程度上提高了计算效率,同时也可以考虑链节取向对角度φ的依赖性(m≠0)。然而,由于目前借助于SPHEREPACK软件包的球谐变换只适合于描述在球面上分布连续光滑的函数,因此该法不适用于强取向相互作用的体系。因此,本论文的思路是探索有效的数值算法,求解基于蠕虫状链模型的自洽场理论,并初步探讨其在实际半刚性链高分子体系中的应用。本论文的第一章首先从统计物理的基本原理出发,介绍不同柔顺性高分子链的相关模型(从高斯链到蠕虫状链直至完全刚性链),并以rod-coil嵌段共聚物为代表,介绍近四十多年来关于半刚性链高分子体系的实验和理论模拟研究进展。第二章将介绍基于蠕虫状链模型自洽场理论的实空间算法。其基本思想是：通过三维笛卡尔直角坐标系下,三角网格化的单位球面描述体系的取向变量u,并结合曲面Laplace-Beltrami算子的有限体积算法求解角度Laplace算符。该实空间算法的特点是,不需要对取向变量的描述空间进行变换(球谐分解与合成),表达具有直观性并且操作简单。更重要的是,由于对取向变量u的描述是真实的三维空间表达,不需要对体系的取向对称性进行预先假定,因此能够得到液晶取向与界面法向不在同一方向的有序结构(如smectic-C),这是目前大多研究中采用球谐函数展开法无法得到的。本章模拟了rod-coil两嵌段共聚物的相分离,计算了体系在Onsager排斥体积作用下,由于熵效应可以形成具有部分双层结构的smectic-A液晶相。在完全相同的参数条件下,与球谐函数展开法得到结果完全相同。Rod-coil嵌段共聚物是一类典型的半刚性链高分子体系,近年来在功能材料领域的应用前景就得到了人们越来越广泛的关注。尤其是以共轭高分子(conjugated polymer)作为刚性嵌段的一类聚合物,由于共轭结构的半导体和光学特性,其在有机光电器件领域的开发得到了深入研究和长足发展。然而,rod-coil的自组装行为与传统coil-coil柔性两嵌段共聚物有显著不同,究其原因主要包括以下两点：一方面,体系中不仅由于化学组成的差异性产生各向同性相互作用,而且由于链节的刚性效应产生各向异性取向相互作用；另一方面,rod和coil在分子链尺寸的描述上也有显著不同：对于柔性高斯链,通常使用无扰均方回转半径标度,Rg～N1/2；而对于半刚性或刚性链,则通常使用链的围线长度描述,L～N。因此,rod-coil体系的相图具有明显的不对称性,而且层状相稳定区域明显变大。特别的是,在有序层状结构中,rod-coil又可以排列成各种不同smectic微观结构：根据rod液晶取向与层状法线方向之间的夹角θt,可以区分为smectic-A (θt=0)和smectic-C (θt≠0),根据rod的排列方式可以区分为bilayer smectic、monolayer smectic和folding smectic。综上所述,rod-coil嵌段共聚物体系具有更为复杂的参数空间,既存在微相分离与取向相互作用的耦合,又存在rod与coil之间的尺寸差异效应。这些因素会造成rod-coil体系中具有特殊的分子链排列规律,从而表现出特殊的液晶相行为和微观聚集形态。因此,从理论角度研究rod-coil嵌段共聚物的相行为和组装规律,对指导实验合成、自组装和材料性能优化,具有重要的指导意义和实际应用价值。鉴于此目的,本章基于蠕虫状链模型的自洽场理论,系统研究了rod-coil嵌段共聚物在稀溶液和本体中的相行为和有序微观结构。第三章使用蠕虫状链模型描述刚性rod和柔性coil嵌段,忽略两种嵌段之间的各向同性Flory-Huggins相互作用,仅通过弯曲刚性模量((?)R=10和(?)C=0.1)来区分rod与coil。使用溶致液晶理论中的Onsagei排斥体积相互作用描述所有链节(包括rod-rod、rod-coil、coil-coil)之间的取向相互作用,以聚合物浓度G和rod嵌段体积分数fR为变量,计算得到了体系在一维空间下的相图(G-fR)。其中包括典型的isotropic、nematic及smectic相,它们对应的相区分布与球谐函数展开法得到的结果基本一致。特别的是,由于实空间算法对取向描述具有直观和完整性,无需对体系的液晶取向进行假定和简化,因此在一维空间相图中进一　步区分了smectic-C与smectic-A结构。第四章对rod-coil理论模型和相互作用描述进一步完善,同时也改进了数值算法,将体系中函数的空间依赖性使用基函数展开(Matsen谱方法),从而提高了对空间算符(?)r求解的精确度和稳定性。该章节重点讨论微相分离与取向相互作用比值μ/χ,和rod与coil间的尺寸不对称效应β=L/Rg,对rod-coil嵌段共聚物本体相行为和smectic微观结构的影响。结果发现,在相同的μ/χ参数下,β较小有利于双层结构的稳定；β较大则倾向于形成单层smectic,进一步增大β有利于smectic-C的稳定。另外,当β确定而体系微相分离作用明显时,rod-coil倾向于发生微相分离,同时nematic相区显著减小；而当取向相互作用占主导时,体系具有明显的液晶相行为,nematic稳定相区增加。以上结论与已有的实验结果和理论分析基本一致。另外还发现,体系仅在Maier-Saupe取向作用下就能形成isotropic、nematic与smectic液晶结构,这与第三章中使用Onsager排斥体积模型得到的结果基本类似。特别的,基于Maier-Saupe平均场理论的取向相互作用(μN)对rod/coil两相界面面积也有显著影响：随着μN增大,刚性rod间的取向程度增大,排列更加紧密,导致界面面积减小。此时,由于coil的构象熵效应,体系会向界面面积增大的微观结构转变。因此,在相图中观察到了有趣的folding smectic-A(fA)和folding smectic-C(fC)结构。在有机光电高分子材料的应用研究中发现,激子(电子-空穴对)的有效扩散距离仅为10nm,因此寻取合理途径来控制纳米结构的聚集尺度,并获得优越的两相界面和微区取向,成为优化材料性能的一项关键技术。实验研究发现,向rod-coil嵌段共聚物中共混化学组分相同的rod或coil均聚物,即能获得rod与coil组分的合适比例,是一种高效且简便的调控体系自组装形态的手段。有鉴于此,第五章沿用第四章中的理论模型和混合数值算法,针对od-coil嵌段共聚物与rod或coil均聚物共混体系的相行为展开研究。结果发现,rod-coil/rod和rod-coil/coil共混物在相同的作用条件(χN和μN)和coil体积分数(fC)下具有更宽的近晶相(smectic)稳定区间。另外,根据自洽场理论得到的嵌段浓度分布和链节分布信息,还重点考察了有序结构中微区尺寸(包括层状周期、rod富集区尺寸、coil富集区尺寸等)和界面形态的变化规律,发现rod和coil均聚物在rod-coil嵌段共聚物本体中的相容机理有显著不同。Rod均聚物会与rod嵌段相互穿插、并排列成有序的液晶取向微区,此时rod/coil的界而面积变化会诱导体系在单层smectic和双层smectic之间发生转变。Coil均聚物则由于分子量不同,与coil嵌段具有不同的相容性。当分子量较大时, coil均聚物倾向于形成独立的浓度富集区,并显著增大coil组分的微区尺寸,但rod/coil界面和rod微区形态基本不发生变化；当分子量较小时,coil均聚物会溶胀在coil嵌段的微区内,并向界面附近渗透,造成rod/coil两相界面变宽。更多还原

【Abstract】 During the past more than fifty years, the self-consistent field theory (SCFT) has been highly improved as one of the most important and successful theories in polymer science. In principle, SCFT can be used to tackle various thermodynamic problems in equilibrium multi-component systems by taking account of the architectures of macromolecules and providing the information of chain configuration. However, in the framework of SCFT based on the semiflexible or wormlike chain model, the numerical methods are quite limited compared to that of Gaussian chain model. On one hand, the state of a segment is specified by its position and orientation, and the propagator q(r,u,s) satisfies a diffusion-like equation in the six dimensional spaces (6D) including two additional variables to describe the chain orientation. On the other hand, the angular Laplace▽u2 in the diffusion equation of q(r,u,s) presents as a Laplacian on a unit sphere, which is more complicated than the spatial Laplace▽r2 for Gaussian chain propagator q(r,s). Additionally, the characterization of orientational interaction due to the chain rigidity also incorporates an external difficult problem. As a result, the numerical solution for wormlike chain SCFT addresses a challenge to the polymer physics community and efficient methods are desirable.Until now, the numerical methods for the semiflexible SCFT are quite limited, among which one of the representative strategies is the spherical harmonic method. It expands the orientational dependence (u-dependence) by spherical harmonics Yl,m(u) to tackle with the angular Laplace operator▽u2,and then synthesizes to the expression in real space u(θ,φ). Because the computation is exceedingly costly in general case of m≠0 in the spherical harmonics Yl,m (u), the most applications of this method have been restricted to the absence ofφ-dependency. In this case, m=0 is assumed to reduce the computational demand, but only finds axial-symmetric structures like nematic and smectic-A, without the observation of axial-asymmetry structures such as smectic-C. Recently, a software named as SPHEREPACK has been applied to the transformation between real space (θ,φ) and spherical harmonics space (l,m) for the orientational description. This technique improves the computational speed to some extent and goes straightforward to take the consideration ofφ-dependency (m≠0) of chain orientation. However, the subroutines in SPHEREPACK using the "triangular truncated" expression for the spherical harmonic expansion allow us to approximate a smooth function to arbitrary precision for some integer value of the index l. Therefore the spherical harmonic transformation based on the SPHEREPACK software is applicable in the situation where the orientational interaction is not so strong. Otherwise the reliability will decrease as the increase of orientational ordering. Under this consideration, we aim to propose a generic approach for solving the self-consistent field theory (SCFT) equations for semiflexible wormlike chains, and apply the new method to some representative semiflexible polymer systems such as rod-coil diblock copolymers.For this purpose, in Chapter One we firstly introduce two typical chain models, named as Gaussian chain and Wormlike chain, for polymers with different flexibility. Taking the rod-coil block copolymer as a representative system of semiflexible polymers, the recent forty-year progresses in both experiments and theoretic simulations are introduced briefly.In Chapter Two, we propose a new real-space numerical implementation of the SCFT for semiflexible polymers. The segment orientational vector u, is mapped to the surface of a unit sphere, which is discretized using an icosahedron triangular mesh. And then a finite volume algorithm is employed to evaluate the Laplacian on the unit sphere▽u2. The significant advantage of this approach is that the u-dependence of the system is described in true 3D Euclidean space, thus it does not restrict the nematic director of the ordered phase and can conveniently distinguish the smectic-C from smectic-A. To evaluate the capabilities of the generic real-space numerical implementation, we firstly applied this method to a simple system, rod-coil diblock copolymer dilute solution. As expected, we successfully obtained the partial bilayer smectic-A phase, which is in agreement with the prediction of the spherical harmonic expansion strategy under the same parametric condition. With this verification, we conclude that the newly improved real-space method for semiflexible SCFT is reliable and efficient.The rod-coil diblock copolymer is a representative semiflexible polymer system, which has attracted increasing recent attention as unique potential functional materials, among which the conjugated polymers are one of the most fascinating examples as economic and efficient organic optoelectronic devices. The self-assembly of rod-coil block copolymer exhibits quite different behaviors from that of traditional flexible system, which originates from two main reasons. On one hand, there exist not only the anisotropic Flory-Huggins interaction between rod and coil due to the difference of chemical components, but also the additional orientational interaction between rigid rods, which can promote the microphase separation and liquid crystal ordering respectively. On the other hand, the rod and coil blocks have different scaling behaviors as a function of molecular weight N:the unperturbed coil size scales as Rg～N1/ whereas the rod block has a characterize length that scales as L～N. This difference in size scaling creates a packing frustration, thus requires an additional parameter to describe the size mismatch between the rods and coils. Therefore, the phase diagram of rod-coil presents a significant asymmetry in comparison with the coil-coil diblock copolymer. In particular, the lamellar phase can further be divided into various different smectic configurations. According to the tilt angleθt between the nematic director and lamellar normal, the smectic phases can be classified into smectic-A (θt=0) and smectic-C (θt≠0), as well as monolayer, bilayer and folded structures, according to the geometric arrangement of rods. In this regard, the parametric space of rod-coil block copolymer is more complicated, which consists of the coupling between microphase separation and orientational interaction, as well as the size asymmetry between rod and coil. These factors can induce extraordinary chain packing rules of rod-coil, to exhibit specific liquid crystal behaviors and various microphase structures. From this point of view, it is desirable to perform theoretic studies on the phase behavior of semiflexible rod-coil block copolymers, to provide good guidance for experimental synthesis, self-assembly and optimization of materials. For this purpose, based on the semiflexible SCFT method proposed in Chapter Two, we calculate the phase diagrams of rod-coil diblock copolymers in dilute solution and condensed melt under different interactional conditions, to focus on the investigation of liquid crystal behavior and smectic microstructures.In Chapter Three, we consider a rod-coil diblock copolymer model, in which both the rod and coil blocks are described by the worm like chain. And for simplicity, the anisotropic Flory-Huggins interaction between rod and coil is ignored, with only bending rigidity difference (ξR=10 andξC=0.1) to distinguish these two blocks. The Onsager exclude-volume interaction is employed to characterize the orientational interaction between all segments, including rod-rod, rod-coil and coil-coil in the system. The solutions of the different liquid crystalline phases including isotropic, nematic and smectic structures, allow us to construct a phase diagram as a function of polymer density and rod volume fraction (G-fR) in one dimensional space (1D), which is in qualitatively agreement with previously theoretical predictions. In particular, the chain orientation u is considered in 3D Euclidean space and thus the smectic-C can be conveniently distinguished from smectic-A, which presents a major problem in the spherical harmonic method.In Chapter Four, we propose a hybrid numerical approach to the SCFT of semiflexible-coil diblock copolymers. In this method, the spatial dependence of the SCFT functions is expanded in terms of a series of basis functions, to improve the numerical accuracy and stability. The self-assembly and liquid-crystalline ordering of rod-coil copolymers are governed by four parameters:the Flory-Huggins interactionχN, the Maier-Saupe interactionμN, the coil volume fraction fC,and the size asymmetry ratio between rods and coilsβ=L/ Rg. We focus on the effect of interplay between microphase separation and orientational interaction characterized by the ratioμ/χ, as well as size asymmetry ratioβ, on the phase behavior and various microstructures of the smectic phases. According to the numerical results of SCFT, when the system experiences the same interactional condition (μ/χ), the smallerβfavors the formation of bilayer smectic, while the largerβprefers to monolayer smectic and especially smectic-C. When the size asymmetry ratioβis fixed, the rod-coil diblock with smallμ/χpromotes the microphase separation and the nematic phase region is greatly compressed. As the increase ofμ/χ, the orientational interaction becomes dominant and the rod-coil exhibits typical liquid crystal behavior with the expansion of nematic region. In particular, the system can experience isotropic, nematic and smectic structures only under the Maier-Saupe interaction (χN=0), which resembles the situation under the Onsager excluded-volume interaction. In addition, we find that the orientational interaction based on the Maier-Saupe mean field theory plays an important role in the rod/coil interfacial morphology. As the increase ofμN, the orientational ordering of rigid segments increases, which induces more close packing of rod blocks and decreases the rod/coil interfacial area. In this case, for the demanding of coil stretching entropy, the system will transform to microstructures with more interfacial area. Especially, the folding smectic-A (fA) and folding smectic-C (fC) are observed in this situation.A desirable feature for the conjugated materials to be useful is their self-assembly into well defined nanostructures. The domain size on the order of 10nm is a crucial requirement for optoelectronic applications, for the limit of efficient length scale of exciton diffusion. In this regard, it presents a necessary object to well control the nanoscale patterns of rod-coil segregation, the donor-acceptor interfacial morphology and domain orientation, to acquire good device performance. The addition of homopolymers which is chemically identical to one of the blocks to the rod-coil block copolymers is demonstrated to be an effective route for achieving the nanostructure optimization including domain spacing, rod orientation and rod-coil interfacial property without additional synthesis. In this regard, systematically theoretic study on the phase behavior of rod-coil diblock copolymers blended with rod or coil homopolymers presents a subject of much interest.Therefore in Chapter Five, we apply the SCFT method proposed in Chapter Four to blending systems of rod-coil/rod and rod-coil/coil, to examine the blending phase behavior as a function of the phase segregation strength and coil volume fractions. The stability of smectic phases significantly increases with the addition of homopolymers under the same parametric situation. According to SCFT results of block and segment density distributions, we explore the microstructure domain sizes including the lamellar period length, the rod domain size, the coil domain size and the interfacial width. The rod and coil homopolymers present different solubilization mechanics into the rod-coil matrix. The molecular weight matched rod homopolymers interdigitate with the rod blocks and align together to the nematic ordering, which increases the rod/coil interfacial and leads to the transformation between monolayer and bilayer smectic structures. The molecular weight of coil homopolymers is predicted to play an important role in the solubility. With small molecular weight, the coil homopolymers tend to be swollen in the coil-block area. In this case the lamellar period increases slightly and the rod/coil interfacial width is broadened. However, with high molecular weight, the coil homopolymers can segregate into independent layers, which can obviously increase the coil domain size without influencing the interfacial width and rod domain morphology.更多还原