Correlations in Disordered Melts of Asymmetric Diblock Copolymers

Simulation of diblock copolymer surfactants. III. Equilibrium interfacial adsorption

Monte Carlo simulations are used to study adsorption of highly asymmetric diblock copolymers to a polymer-polymer interface, and the results compared to self-consistent field theory (SCFT) predictions. The simulation model used here is a bead-spring model that has been used previously to study equilibrium and kinetic properties of spherical micelles [J. A. Mysona et al., Phys. Rev. E 100, 012602 (2019)2470-004510.1103/PhysRevE.100.012602; Phys. Rev. E 100, 012603 (2019)10.1103/PhysRevE.100.012603; Phys. Rev. Lett. 123, 038003 (2019)10.1103/PhysRevLett.123.038003]. Interfacial copolymer concentration Γ and interfacial tension γ are measured as functions of bulk copolymer concentration at concentrations up to the critical micelle concentration over a range of values of the Flory-Huggins χ parameter. The dependence of interfacial pressure Π = γ_{0}-γ on Γ (where γ_{0} is the interfacial tension in the absence of copolymer) is found to be almost independent of χ and to be accurately predicted by SCFT. The bare interfacial tension γ_{0} and total interfacial tension γ(Γ) can also be accurately predicted by SCFT using an estimate of χ obtained from independent analysis of properties of symmetric diblock copolymer melts. SCFT predictions obtained with this estimate of χ do not, however, adequately describe the thermodynamics of the coexisting bulk copolymer solution, as a result of contraction of the strongly interacting core block of dissolved copolymers. Accurate predictions of the relationship between bulk and interfacial properties can thus only be obtained for this system by combining SCFT predictions of the interfacial equation of state with a fit to the measured equation of state for the bulk solution.

Field theoretic approach for block polymer melts: SCFT and FTS

This perspective addresses the development of polymer field theory for predicting the equilibrium phase behavior of block polymer melts. The approach is tailored to the high-molecular-weight limit, where universality reduces all systems to the standard Gaussian chain model, an incompressible melt of elastic threads interacting by contact forces. Using mathematical identities, this particle-based version of the model is converted to an equivalent field-based version that depends on fields rather than particle coordinates. The statistical mechanics of the field-based model is typically solved using the saddle-point approximation of self-consistent field theory (SCFT), which equates to mean field theory, but it can also be evaluated using field theoretic simulations (FTS). While SCFT has matured into one of the most successful theories in soft condensed matter, FTS are still in its infancy. The two main obstacles of FTS are the high computational cost and the occurrence of an ultraviolet divergence, but fortunately there has been recent groundbreaking progress on both fronts. As such, FTS are now well poised to become the method of choice for predicting fluctuation corrections to mean field theory.

Mechanism of Micelle Birth and Death

In micellar surfactant solutions, changes in the total number of micelles are rare events that can occur by either of two mechanisms-by stepwise association and dissociation via insertion and expulsion of individual molecules or by fission and fusion of entire micelles. Molecular dynamics simulations are used here to estimate rates of these competing mechanisms in a simple model of block copolymer micelles in homopolymer solvent. This model exhibits a crossover with increasing degree of repulsion between solvent and micelle core components, from a regime dominated by association and dissociation to a regime dominated by fission and fusion.

Simulation of diblock copolymer surfactants. I. Micelle free energies

Semigrand hybrid Monte Carlo simulations are used to measure equilibrium properties of micelles formed in a simple bead-spring model of asymmetric A-B diblock copolymer surfactant molecules in an A homopolymer solvent, over a range of values of surfactant solubility. Simulations are used to accurately measure the free energy of formation of micellar clusters as a function of aggregation number over a wide range of values, and to characterize the crossover from spherical to rodlike micelle shape with increasing aggregation number. Dynamical properties of the same model are considered in an accompanying paper [Phys. Rev. E 100, 012603 (2019)10.1103/PhysRevE.100.012603].

Calibration of a lattice model for high-molecular-weight block copolymer melts

The Morse calibration is applied to a lattice model designed for efficient simulations of two-component polymer melts of high molecular weight. The model allows multiple occupancy per site, which results in high invariant polymerization indices, and interactions are limited to monomers within the same site, which enhances the computational speed. The calibration maps the interaction parameter of the lattice model, α, onto the Flory-Huggins χ parameter of the standard Gaussian-chain model, by matching the disordered-state structure function, S(k), of symmetric diblock copolymers to renormalized one-loop predictions. The quantitative accuracy of the calibration is tested by comparing the order-disorder transition of symmetric diblock copolymer melts to the universal prediction obtained from previous simulations. The model is then used to confirm the universality of fluctuation corrections to the critical point of symmetric binary homopolymer blends.

Calibration of the Flory-Huggins interaction parameter in field-theoretic simulations

Field-theoretic simulations (FTS) offer a versatile method of dealing with complicated block copolymer systems, but unfortunately they struggle to cope with the level of fluctuations typical of experiments. Although the main obstacle, an ultraviolet divergence, can be removed by renormalizing the Flory-Huggins χ parameter, this only works for unrealistically large invariant polymerization indexes, N¯. Here, we circumvent the problem by applying the Morse calibration, where a nonlinear relationship between the bare χ_b used in FTS and the effective χ corresponding to the standard Gaussian-chain model is obtained by matching the disordered-state structure function, S(k), of symmetric diblock copolymers to renormalized one-loop predictions. This calibration brings the order-disorder transition obtained from FTS into agreement with the universal results of particle-based simulations for values of N¯ characteristic of the experiment. In the limit of weak interactions, the calibration reduces to a linear approximation, χ ≈ z_∞χ_b, consistent with the previous renormalization of χ for large N¯.

Role of Chain Length in the Formation of Frank-Kasper Phases in Diblock Copolymers

The phase behavior of poly(styrene)-b-poly(1,4-butadiene) diblock copolymers with a polymer block invariant degree of polymerization N[over ¯]_{b}≈800 shows no evidence of Frank-Kasper phases, in contrast to low molar mass diblock copolymers (N[over ¯]_{b}<100) with the same conformational asymmetry. A universal self-concentration crossover parameter N[over ¯]_{x}≈400 is identified, directly related to the crossover to entanglement dynamics in polymer melts. Mean-field behavior is recovered when N[over ¯]_{b}>N[over ¯]_{x}, while complex low symmetry phase formation is attributed to fluctuations and space-filling constraints, which dominate when N[over ¯]_{b}<N[over ¯]_{x}.