| 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. |
