Phase Diagram of Rod–Coil Diblock Copolymer Melts

Mimicking effects of cholesterol in lipid bilayer membranes by self-assembled amphiphilic block copolymers

The effect of cholesterol on biological membranes is important in biochemistry. In this study, a polymer system is used to simulate the consequences of varying cholesterol content in membranes. The system consists of an AB-diblock copolymer, a hydrophilic homopolymer hA, and a hydrophobic rigid homopolymer C, corresponding to phospholipid, water, and cholesterol, respectively. The effect of the C-polymer content on the membrane is studied within the framework of a self-consistent field model. The results show that the liquid-crystal behavior of B and C has a great influence on the chemical potential of cholesterol in bilayer membranes. The effects of the interaction strength between components, characterized by the Flory-Huggins parameters and the Maier-Saupe parameter, were studied. Some consequences of adding a coil headgroup to the C-rod are presented. Results of our model are compared to experimental findings for cholesterol-containing lipid bilayer membranes.

Dynamic Processes and Mechanical Properties of Lipid-Nanoparticle Mixtures

In this study, we investigate the dynamic processes and mechanical properties of lipid nanoparticle mixtures in a melt via dissipation particle dynamic simulation. By investigating the distribution of nanoparticles in lamellar and hexagonal lipid matrices in equilibrium state and dynamic processes, we observe that the morphology of such composites depends not only on the geometric features of the lipid matrix but also on the concentration of nanoparticles. The dynamic processes are also demonstrated by calculating the average radius of gyration, which indicates the isotropic conformation of lipid molecules in the x-y plane and that the lipid chains are stretched in the z direction with the addition of nanoparticles. Meanwhile, we predict the mechanical properties of lipid-nanoparticle mixtures in lamellar structures by analyzing the interfacial tensions. Results show that the interfacial tension decreased with the increase in nanoparticle concentration. These results provide molecular-level information for the rational and a priori design of new lipid nanocomposites with ad hoc tailored properties.

General weak segregation theory with an application to monodisperse semi-flexible diblock copolymers

A general theory has been developed for a polydisperse semi-flexible multi-block copolymer melt. Using the Bawendi-Freed approach to model semi-flexible chains, an expression for the Landau free energy is derived in the weak segregation regime, which includes density and orientation order-parameters. The orientation order-parameter is described in the smectic phase and in more complicated structures, such as the hexagonal phase. The Landau free energy contains contributions of two kinds of interactions. The first kind is the Flory-Huggins interaction, which describes the incompatibility of chemically different blocks and may induce microphase separation. The second kind is the Maier-Saupe interaction, which may induce nematic ordering. In the framework of the weak segregation limit, the Landau theory allows us to predict phase structures in the melt as a function of the composition, persistence length, and the strength of the Flory-Huggins and Maier-Saupe interaction. The general theory is applied to a simple system of monodisperse semi-flexible diblock copolymers. In several phase diagrams, a number of possible phase structures are predicted, such as the bcc, hexagonal, smectic-A, smectic-C, and nematic phase. The influence of the Maier-Saupe interaction on the microphase structure is thoroughly discussed.

The stress deformation response influenced by the chain rigidity for mesostructures in diblock copolymers

A self-consistent field theory formalism based on the wormlike chain model is developed to investigate the stress-strain relation for mesostructures in diblock copolymers under the influence of chain rigidity, involving the adjustable simulation cell in the non-orthogonal coordinates by means of optimization of free energy. We elucidate the effect of the chain persistency broadly spanning from the Gaussian chain to the rigid rodlike chain on the elastic response of mesophases that deviate from the initial equilibrium structures. We analytically and numerically demonstrate that our current approach in the long chain limit recovers to the Gaussian-chain-based theory. Being ascribed to the distinct conformational behaviors for flexible chains and rigid rodlike chains, the tensile and compressive stresses applied to lamellae exhibit asymmetric deformation behaviors and the shear stress applied to the initial equilibrium hexagonal cylinders results in noticeable deviations in the shape and spatial arrangement of cylindroids for various chain rigidity values. For the zero stress, in addition, our approach can be straightforwardly utilized to explore the optimal size and shape of the simulation cell in order to achieve a stress free configuration of systems.

Liquid-Crystal Ordering and Microphase Separation in the Lamellar Phase of Rod-Coil-Rod Triblock Copolymers. Molecular Theory and Computer Simulations

A molecular model of the orientationally ordered lamellar phase exhibited by asymmetric rod-coil-rod triblock copolymers has been developed using the density-functional approach and generalizing the molecular-statistical theory of rod-coil diblock copolymers. An approximate expression for the free energy of the lamellar phase has been obtained in terms of the direct correlation functions of the system, the Flory-Huggins parameter and the Maier-Saupe orientational interaction potential between rods. A detailed derivation of several rod-rod and rod-coil density-density correlation functions required to evaluate the free energy is presented. The orientational and translational order parameters of rod and coil segments depending on the temperature and triblock asymmetry have been calculated numerically by direct minimization of the free energy. Different structure and ordering of the lamellar phase at high and low values of the triblock asymmetry is revealed and analyzed in detail. Asymmetric rod-coil-rod triblock copolymers have been simulated using the method of dissipative particle dynamics in the broad range of the Flory-Huggins parameter and for several values of the triblock asymmetry. It has been found that the lamellar phase appears to be the most stable one at strong segregation. The density distribution of the coil segments and the segments of the two different rods have been determined for different values of the segregation strength. The simulations confirm the existence of a weakly ordered lamellar phase predicted by the density-functional theory, in which the short rods separate from the long ones and are characterized by weak positional ordering.

Different Mechanisms of Translational Symmetry Breaking in Liquid-Crystal Coil–Rod–Coil Triblock Copolymers

A molecular-statistical theory of coil-rod–coil triblock copolymers with orientationally ordered rod-like fragments has been developed using the density functional approach. An explicit expression for the free energy has been obtained in terms of the direct correlation functions of the reference disordered phase, the Flory–Huggins parameter and the potential of anisotropic interaction between rigid rods. The theory has been used to derive several phase diagrams and to calculate numerically orientational and translational order parameter profiles for different polymer architecture as a function of the Flory–Huggins parameter, which specifies the short-range repulsion and as functions of temperature. In triblock copolymers, the nematic–lamellar transition is accompanied by the translational symmetry breaking, which can be caused by two different microscopic mechanisms. The first mechanism resembles a low dimensional crystallization and is typical for conventional smectic liquid crystals. The second mechanism is related to the repulsion between rod and coil segments and is typical for block copolymers. Both mechanisms are analyzed in detail as well as the effects of temperature, coil fraction and the triblock asymmetry on the transition into the lamellar phase.

Microphase separation in helix-coil block copolymer melts: computer simulation

By means of molecular dynamics simulation, the process of the microphase separation in the melts of diblock helix-coil copolymers comprising a flexible and a helical block was studied. The resulting microstructures were examined, and the spatial distribution of the blocks and molecular packing were investigated. The phase diagram was built in terms of the fraction of the helical block and the incompatibility parameter of the blocks. The comparison of the diagrams for helix-coil and the classic coil-coil copolymer blends was carried out. It was shown that the total region where the ordering into distinctive microstructures takes place is similar for both diagrams. But for the helix-coil copolymers the area of the cylinders splits into the region of those with circular and elliptical cross-sections; the bicontinuous phase area is much wider; in the lamellar phases, the helical blocks were oriented precisely perpendicular to the lamellar interface, forming a cohesive interlocked structure of densely packed helices.

Self-assembly of rod-coil diblock copolymer-nanoparticle composites in thin films: dissipative particle dynamics

In this study, the assembled structures of rod-coil diblock copolymer and nanoparticle blends were studied via dissipative particle dynamics (DPD). Thin films were composed of soft confinement DPD fluid beads and the fluctuating film structure was maintained during the simulation process. Analysis of the position of nanoparticles was done in the smectic lamellar phase of the rod-coil polymer matrix, and density distributions of rods, coils, and nanoparticles were obtained as functions of the size of the nanoparticle and the DPD repulsion constant between the rod and the nanoparticle. The distribution of nanoparticles was explained by using the concept of translational entropy of nanoparticles, stretching energy of the polymer chain, relative repulsion enthalpy of nanoparticles to rods or coils, and the effect of the liquid crystalline rod.

Liquid Crystal Ordering in the Hexagonal Phase of Rod-Coil Diblock Copolymers

Density functional theory of rod-coil diblock copolymers, developed recently by the authors, has been generalised and used to study the liquid crystal ordering and microphase separation effects in the hexagonal, lamellar and nematic phases. The translational order parameters of rod and coil monomers and the orientational order parameters of rod-like fragments of the copolymer chains have been determined numerically by direct minimization of the free energy. The phase diagram has been derived containing the isotropic, the lamellar and the hexagonal phases which is consistent with typical experimental data. The order parameter profiles as functions of temperature and the copolymer composition have also been determined in different anisotropic phases. Finally, the spatial distributions of the density of rigid rod fragments and of the corresponding orientational order parameter in the hexagonal phase have been calculated.

Molecular theory of the tilting transition and computer simulations of the tilted lamellar phase of rod-coil diblock copolymers

Symmetric rod-coil diblock copolymers have been simulated using the method of dissipative particle dynamics in the broad range of the Flory-Huggins parameter. It has been found that the tilted lamellar phase appears to be the most stable one at strong segregation. The rod-coil copolymer tilt angle and orientational order parameters have been determined as functions of the segregation strength. The density functional theory of rod-coil diblock copolymers has been generalized to the case of the tilted lamellar phase and used to study the stability of the orthogonal lamellar phase with respect to tilt. The orthogonal phase indeed appears to be unstable in the broad region of the parameter space in the case of relatively strong segregation. It has also been shown that the transition into the tilted lamellar phase is determined by a strong coupling between two independent tilt order parameters.

Vertical Lamellae Formed by Two-Step Annealing of a Rod-Coil Liquid Crystalline Block Copolymer Thin Film

Silicon-containing block copolymer thin films with high interaction parameter and etch contrast are ideal candidates to generate robust nanotemplates for advanced nanofabrication, but they typically form in-plane oriented microdomains as a result of the dissimilar surface energies of the blocks. Here, we describe a two-step annealing method to produce vertically aligned lamellar structures in thin film of a silicon-containing rod-coil thermotropic liquid crystalline block copolymer. The rod-coil block copolymer with the volume fraction of the Si-containing block of 0.22 presents an asymmetrical lamellar structure in which the rod block forms a hexatic columnar nematic liquid crystalline phase. A solvent vapor annealing step first produces well-ordered in-plane cylinders of the Si-containing block, then a subsequent thermal annealing promotes the phase transition from in-plane cylinders to vertical lamellae. The pathways of the order-order transition were examined by microscopy and in situ using grazing incidence small-angle X-ray scattering and wide-angle X-ray scattering.

Target-Directed Design of Phase Transition Path for Complex Structures of Rod-Coil Block Copolymers

We apply the string method to the self-consistent mean-field theory framework of the rod-coil block copolymer system to calculate the minimum energy pathways in the rearrangement transitions of lamellae and cylinders with different orientations under certain epitaxial growth relationship. Metastable phases appearing in the reordering transition pathway tend to form the structure at low χN side of the order-order transition boundary compared with the initial phase. In particular, for complex network, metastable phases, such as single gyroid and perforated lamellae, need to select a rearrangement transition between lamellae or cylinders near the order-disorder transition boundary with the same epitaxial growth relationship but different orientations. It is confirmed that this strategy for obtaining complex metastable phases by rational design of rearrangement transition between specific phases in the phase diagram can be applied to a wide range of χN as well as the coil-coil block copolymer system. We further investigate the rearrangement transition behavior combining with the analysis of contribution from the free energy, entropy, degree of mixing between different blocks, and the average orientation degree of rods during the phase transitions. Based on this mechanism, we have developed a target-directed design strategy for constructing self-assembled metastable structures of rod-coil block copolymers.

Molecular theory of liquid-crystal ordering in rod-coil diblock copolymers

Molecular-statistical theory of rod-coil diblock copolymers is proposed using the general density functional approach which enables one to consider the cases of both weak and strong segregation. The free energy of the system is expressed as a functional of the phase-space densities of rod and coil monomers, which depend on the orientational and translational order parameters. Temperature-concentration phase diagrams are obtained and the profiles of all order parameters are calculated numerically by minimizing the polymer free energy. The lamellar phase is shown to possess strong orientational order which is partially induced by the phase structural anisotropy. The enhanced stability of the lamellar phase is determined by a combination of the microphase separation effects and the emergence of long-range smectic order.

Core-Shell and Zigzag Nanostructures from a Thin Film Silicon-Containing Conformationally Asymmetric Triblock Terpolymer

The self-assembly of multiblock copolymers generates diverse hierarchical nanostructures and greatly extends the range of microdomain geometries beyond those produced by diblock copolymers. We report the synthesis of a conformationally asymmetric ABC triblock terpolymer in which the end blocks are a mesogen-jacketed liquid crystalline polymer and poly(dimethylsiloxane), respectively, and its self-assembly under mixed solvent vapor annealing forms a range of sphere, cylinder, and perforated lamellar core-shell morphologies, as well as stacked multilevel structures. Sub-7 nm diameter SiO_x nanopatterns were generated by selective plasma etching of the small volume fraction Si-containing core block giving a line/space ratio of ∼1:4. Moreover, the conformational asymmetry of this terpolymer leads to zigzag cylinders on a flat substrate and stable cylinder alignment transverse to template sidewalls within lithographically patterned trenches.

Advances toward the effective use of block copolymers as organic photovoltaic active layers

Donor/acceptor block copolymers for organic photovoltaic active layers are discussed from first principles through the modern state-of-the-art and future perspectives.

Liquid crystalline bilayers self-assembled from rod-coil diblock copolymers

The structure and phase behaviour of bilayer membranes self-assembled from rod-coil diblock copolymers are studied using the self-consistent field theory, focusing on the occurrence and relative stability of liquid crystalline phases induced by the geometric shape and orientational interaction of the rod-blocks. A variety of liquid crystalline bilayers, corresponding to the smectic phases in bulk systems, are predicted to occur as equilibrium phases of the system. The ordered morphologies and phase behaviour of the system are analyzed. Phase diagrams of the self-assembled bilayers are constructed. The theoretical results provide an understanding of the formation mechanisms of these intricate phases.

Demixing by a Nematic Mean Field: Coarse-Grained Simulations of Liquid Crystalline Polymers

Liquid crystalline polymers exhibit a particular richness of behaviors that stems from their rigidity and their macromolecular nature. On the one hand, the orientational interaction between liquid-crystalline motifs promotes their alignment, thereby leading to the emergence of nematic phases. On the other hand, the large number of configurations associated with polymer chains favors formation of isotropic phases, with chain stiffness becoming the factor that tips the balance. In this work, a soft coarse-grained model is introduced to explore the interplay of chain stiffness, molecular weight and orientational coupling, and their role on the isotropic-nematic transition in homopolymer melts. We also study the structure of polymer mixtures composed of stiff and flexible polymeric molecules. We consider the effects of blend composition, persistence length, molecular weight and orientational coupling strength on the melt structure at the nano- and mesoscopic levels. Conditions are found where the systems separate into two phases, one isotropic and the other nematic. We confirm the existence of non-equilibrium states that exhibit sought-after percolating nematic domains, which are of interest for applications in organic photovoltaic and electronic devices.

Extremely asymmetric phase diagram of homopolymer-monotethered nanoparticles: Competition between chain conformational entropy and particle steric interaction

The phase behaviors of homopolymer-monotethered nanoparticles (HMNs) in melt are investigated via a theoretical method combining self-consistent field theory for polymers and density functional theory for hard spheres. An extremely asymmetric phase diagram is observed: (i) microphases are only possible for the volume fraction of the tethered polymer f_A > 0.35; (ii) in addition to lamellar phase, the system can only self-assemble into various morphologies with a polymer-rich matrix, including gyroid phase, cylindrical phase, and spherical phase. In the frame of this theory, the critical point for HMNs' microphase separation is significantly lower than that of linear diblock copolymers. Furthermore, the characteristic length of microphase-separated structures of HMNs is much smaller than that of linear diblock copolymers with the same molecular weight. Our calculation results on morphologies and characteristic length agree well with recent simulations and experimental observations.

Thermodynamics of a Compressible Maier-Saupe Model Based on the Self-Consistent Field Theory of Wormlike Polymer

This paper presents a theoretical formalism for describing systems of semiflexible polymers, which can have density variations due to finite compressibility and exhibit an isotropic-nematic transition. The molecular architecture of the semiflexible polymers is described by a continuum wormlike-chain model. The non-bonded interactions are described through a functional of two collective variables, the local density and local segmental orientation tensor. In particular, the functional depends quadratically on local density-variations and includes a Maier–Saupe-type term to deal with the orientational ordering. The specified density-dependence stems from a free energy expansion, where the free energy of an isotropic and homogeneous homopolymer melt at some fixed density serves as a reference state. Using this framework, a self-consistent field theory is developed, which produces a Helmholtz free energy that can be used for the calculation of the thermodynamics of the system. The thermodynamic properties are analysed as functions of the compressibility of the model, for values of the compressibility realizable in mesoscopic simulations with soft interactions and in actual polymeric materials.

Emergence of ordered network mesophases in kinetic pathways of order-order transition for linear ABC triblock terpolymers

Applying the string method to the self-consistent field theory (SCFT) of ABC linear triblock copolymers, we developed a new strategy to design kinetic pathways for the formation of stable or metastable network mesophases in order-order transition (OOT) processes. The design principle regarding the kinetic pathways between distinct mesophases is based on the matching relationships of both domain spacing and dominant Fourier components of the density distributions. The results suggest that complex ordered network mesophases, such as alternating diamond (D^A) and alternating plumber's nightmare (P^A) could be obtained in kinetic pathways between simple phases covering lamellae, cylinders and spheres. By virtue of the minimal free energy pathway (MEP) obtained, we could acquire the epitaxial relationship and phase transition mechanism. Furthermore, we managed to regulate the MEP by changing the block composition to adjust packing frustration. Two new metastable networks, core-shell five-pronged and six-pronged morphologies, were found in the kinetic pathways, further demonstrating the regulating mechanism. The results will contribute to a better understanding of the kinetic relationship between simple phases and complex networks, thus providing a platform for soft materials design via the OOT route and guiding experimental procedures to fabricate ordered network mesophases.

Complex liquid-crystal nanostructures in semiflexible ABC linear triblock copolymers: A self-consistent field theory

We show that two series of ABC linear triblock copolymers possess sequences of order-to-order phase transitions between microphase-separated states, as the degree of flexibility of the semiflexible middle B-blocks varies. The spatial and orientational symmetries of these phases, some of them containing liquid-crystal ordering, are analysed in comparison with related structures previously determined experimentally and theoretically. A theoretical framework based on the self-consistent field treatment of the wormlike-chain model, which incorporates the Flory-Huggins and Maier-Saupe interactions in the free energy, is used here as a basic foundation for numerical calculations. We suggest that tuning the flexibility parameter, which reduces to the concept of degree of polymerization in the coil-like limit and characterizes the chain-persistency in the rod-like limit, provides a promising approach that can be used to design the resulting microphase-separated structures in semiflexible copolymer melts.

Perspective: parameters in a self-consistent field theory of multicomponent wormlike-copolymer melts

We review a formalism that can be used to calculate the microphase-separated crystallographic structures of multi-component wormlike polymer melts. The approach is based on a self-consistent field theory of wormlike polymers where the persistence length of each component is an important parameter. We emphasize on an analysis of the number of independent parameters required to specify a problem in general, for a system that includes Flory-Huggins and Maier-Saupe energies. Examples of recent applications are also briefly demonstrated: AB homopolymer interface, AB diblock copolymers, and rod-coil copolymers.

Scattering and Gaussian Fluctuation Theory for Semiflexible Polymers

The worm-like chain is one of the best theoretical models of the semiflexible polymer. The structure factor, which can be obtained by scattering experiment, characterizes the density correlation in different length scales. In the present review, the numerical method to compute the static structure factor of the worm-like chain model and its general properties are demonstrated. Especially, the chain length and persistence length involved multi-scale nature of the worm-like chain model are well discussed. Using the numerical structure factor, Gaussian fluctuation theory of the worm-like chain model can be developed, which is a powerful tool to analyze the structure stability and to predict the spinodal line of the system. The microphase separation of the worm-like diblock copolymer is considered as an example to demonstrate the usage of Gaussian fluctuation theory.

Effect of Geometrical Asymmetry on the Phase Behavior of Rod-Coil Diblock Copolymers

The effect of geometrical asymmetry β (described by the length-diameter ratio of rods) on the rod-coil diblock copolymer phase behavior is studied by implementation of self-consistent field theory (SCFT) in three-dimensional (3D) position space while considering the rod orientation on the spherical surface. The phase diagrams at different geometrical asymmetry show that the aspect ratio of rods β influences not only the order-disorder transition (ODT) but also the order-order transition (OOT). By exploring the phase diagram with interactions between rods and coils plotted against β, the β effect on the phase diagram is similar to the copolymer composition f. This suggests that non-lamellae structures can be obtained by tuning β, besides f. When the rods are slim compared with the isotropic shape of the coil segment (β is relatively large), the phase behavior is quite different from that of coil-coil diblock copolymers. In this case, only hexagonal cylinders with the coil at the convex side of the interface and lamella phases are stable even in the absence of orientational interaction between rods. The phase diagram is no longer symmetrical about the symmetric copolymer composition and cylinder phases occupy the large area of the phase diagram. The ODT is much lower than that of the coil-coil diblock copolymer system and the triple point at which disordered, cylinder and lamella phases coexist in equilibrium is located at rod composition f_R = 0.66. In contrast, when the rods are short and stumpy (β is smaller), the stretching entropy cost of coils can be alleviated and the phase behavior is similar to coil-coil diblocks. Therefore, the hexagonal cylinder phase formed by coils is also found beside the former two structures. Moreover, the ODT may even become a little higher than that of the coil-coil diblock copolymers due to the large interfacial area per chain provided by the stumpy rods, thus compensating the stretching entropy loss of the coils.

Theoretical simulations of nanostructures self-assembled from copolymer systems

This article provides an overview of recent simulation investigations of the nanostructures and structure–property relationships in copolymer systems.