Copolymers play an important role in a range of soft-materials applications and biological phenomena. Prevalent works on block copolymer phase behavior use flexible chain models and incorporate interactions using a mean-field approximation. However, when phase separation takes place on length scales comparable to a few monomers, the structural rigidity of the monomers becomes important. In addition, concentration fluctuations become significant at short length scales, rendering the mean-field approximation invalid. In this work, we use simulation to address the role of finite monomer rigidity and concentration fluctuations in microphase segregation of random copolymers. Using a field-theoretic Monte-Carlo simulation of semiflexible polymers with random chemical sequences, we generate phase diagrams for random copolymers. We find that the melt morphology of random copolymers strongly depends on chain flexibility and chemical sequence correlation. Chemically anti-correlated copolymers undergo first-order phase transitions to local lamellar structures. With increasing degree of chemical correlation, this first-order phase transition is softened, and melts form microphases with irregular shaped domains. Our simulations in the homogeneous phase exhibit agreement with the density-density correlation from mean-field theory. However, conditions near a phase transition result in deviations between simulation and mean-field theory for the density-density correlation and the critical wavemode. Chain rigidity and sequence randomness lead to frustration in the segregated phase, introducing heterogeneity in the resulting morphologies.

Random copolymers play an important role in a range of soft materials applications and biological phenomena. An individual monomer is typically a single chemical unit whose length is comparable to or less than a Kuhn length, resulting in a monomer segment that is structurally rigid at length scales of a segregated domain. Previous work on random copolymer phase segregation addresses the impact of correlations between the chemical identities along the chains for flexible polymers. In these works, a single monomer unit is effectively a large polymer block that behaves as a random walk without conformational correlation associated with semiflexibility. In our work, we develop a model of semiflexible random copolymers using the wormlike chain model to capture conformational correlation of the polymer chains. To address the thermodynamics of microphase segregation and the structure of the segregated domains, we develop a random phase approximation up to quartic order in density fluctuations that leverages our exact results for the statistical behavior of the wormlike chain model. In this work, we focus on the quadratic-order expansion of the free energy, which provides the mean-field spinodal of the homogeneous phase. We explore the impact of conformational and chemical correlations on the formation of inhomogeneous microphases at the spinodal point. We show that the onset of phase segregation and the correlation length of domains are extremely sensitive to chain rigidity. ■ INTRODUCTIONBlock copolymers have attracted attention in numerous applications because of their ability to self-assemble into nanostructured morphologies. 1−3 Extensive theoretical and experimental studies have predicted and characterized the ordered morphologies of block copolymers with well-defined chemical architecture such as diblock and triblock copolymers. 4−7 In contrast, the phase behavior of multiblock copolymers is less well understood. In particular, theoretical, computational, and experimental studies on random copolymers, whose monomer chemical identities are stochastically arranged along the polymers, are far less prevalent. 8−11 Commercial products such as high impact polystyrene (HIPS), styrene−butadiene rubber (SBR), and Nafion are blends of macromolecules with randomly distributed chemical repeat units. 12,13 In these materials, the mesoscale morphology greatly influences their mechanical and transport properties. Recently, there has been interest in making new soft materials by introducing chemical stochasticity into block copolymers, such as tapered diblock copolymers 14 and gradient copolymers, 15−17 due to their ability to relieve packing frustration during self-assembly. Biology also needs to overcome random variability to achieve self-assembly in macromolecular environments in scenarios such as protein folding and chromatin condensation. 18−20 Despite the abundance of random copolymers in various industrial applications and biological systems, the understanding of their phase behavior and structure− property relationship ...

The order-disorder phase transition and the associated phase diagrams of semiflexible diblock copolymers are investigated using the wormlike chain model, incorporating concentration fluctuations. The free energy up to quartic order in concentration fluctuations is developed with chain-rigidity-dependent coefficients, evaluated using our exact results for the wormlike chain model, and a one-loop renormalization treatment is used to account for fluctuation effects. The chain length N and the monomer aspect ratio α directly control the strength of immiscibility (defined by the Flory-Huggins parameter χ) at the order-disorder transition and the resulting microstructures at different chemical compositions f_{A}. When monomers are infinitely thin (i.e., large aspect ratio α), the finite chain length N lowers the χN at the phase transition. However, fluctuation effects become important when chains have a finite radius, and a decrease in the chain length N elevates the χN at the phase transition. Phase diagrams of diblock copolymers over a wide range of N and α are calculated based on our fluctuation theory. We find that both finite N and α enhance the stability of the lamellar phase above the order-disorder transition. Our results demonstrate that polymer semiflexibility plays a dramatic role in the phase behavior, even for large chain lengths (e.g., N≈100).

We present a simulation study of the equilibrium thermodynamic behavior of semiflexible diblock copolymer melts. Using discretized wormlike chains and field-theoretic Monte Carlo, we find that concentration fluctuations play a critical role in controlling phase transitions of semiflexible diblock copolymers. Polymer flexibility and aspect ratio control the order− disorder transition Flory−Huggins parameter χ ODT N. For polymers with low aspect ratios, fluctuations strongly elevate the phase transition χ ODT N at finite molecular weights. For high aspect-ratio polymers, chain semiflexibility decreases the phase transition χ ODT N. We find that the simulated phase behavior agrees well with our recently developed fluctuation theory based on wormlike chain configurations and a one-loop treatment of concentration fluctuations.

We present a field-theoretic model to predict the equilibrium thermodynamic behavior of semiflexible diblock copolymers and random copolymers in the presence of solvent. We find that in both systems polymer–solvent contacts dramatically influence the thermodynamic behavior with decreasing the copolymer segment length (i.e., molecular weight). When a copolymer has unequal monomer composition, both polymer length and solvent concentration have a strong influence on the phase transition spinodal and magnitude of the critical wave modes. Diblock copolymers exhibit an expanded region of the lamellar phase in the phase diagram with decreasing chain length and polymer concentration. Such effects suggest a breakdown of the dilute approximation for solutions of short diblock copolymers. Random copolymer solutions also exhibit changes in the phase-transition spinodal and critical wave mode at asymmetric chemical compositions. This effect is highly relevant to most random copolymer materials, since a monomer is typically a low-molecular-weight chemical unit.