Open-source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units.

Self-consistent field theory (SCFT) is a powerful approach for computing the phase behavior of block polymers. We describe a fast version of the open-source Polymer Self-Consistent Field (PSCF) code that takes advantage of the massive parallelization provided by a graphical processing unit (GPU). Benchmarking double-precision calculations indicate up to 30× reduction in time to converge SCFT calculations of various diblock copolymer phases when compared to the Fortran CPU version of PSCF using the same algorithms, with the speed-up increasing with increasing unit cell size for the diblock polymer problems examined here. Where double-precision accuracy is not needed, single-precision calculations can provide speed-up of up to 60× in convergence time. These improvements in speed within an open-source format open up new vistas for SCFT-driven block polymer materials discovery by the community at large.

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.

Influence of charge sequence on the adsorption of polyelectrolytes to oppositely-charged polyelectrolyte brushes.

When a solution of polyanionic chains is placed in contact with a polycationic brush, the polyanions adsorb into the brush. We investigate the influence of the charge sequences of the free and bound species on the thermodynamics of polyelectrolyte adsorption. As model systems, we consider free and brush polyelectrolytes with either block or alternating charge sequences, and study the adsorption process using coarse-grained Langevin dynamics with implicit solvent, explicit counterions, and excess salt. Free energy, internal energy, and entropy of adsorption are computed using umbrella sampling methods. When the number of polyanions exceed the number of polycations, the brush becomes overcharged. Free chains adsorb most strongly when both free and tethered chains have a block charge sequence, and most weakly when both species have an alternating sequence. Adsorption is stronger when the free polyanion has a block sequence and the tethered polycation is alternating than in the reverse case of an alternating free polymer and a tethered block copolymer. Sequence-dependent effects are shown to be largely energetic, rather than entropic, in origin.

Network Model of the Disordered Phase in Symmetric Diblock Copolymer Melts.

We present a model for the order-disorder transition of symmetric A-B diblock copolymer melts in which the disordered phase is treated as a bicontinuous network, and in which self-consistent field predictions of properties of an analogous ordered network are used to estimate some properties. Such a model is shown to accurately predict the latent heat of this transition. The dependence of the location of the transition upon the invariant degree of polymerization N[over ¯] is shown to be consistent with a simple hypothesis that the disordered bicontinuous structure is stabilized relative to an analogous ordered network by a nearly constant entropy per network junction.

Accelerating self-consistent field theory of block polymers in a variable unit cell.

Self-consistent field theory (SCFT) is one of the most widely used tools to study the equilibrium phase behavior of block polymers. We have extended an existing version of the Anderson-mixing iteration scheme to solve the highly nonlinear SCFT equations while simultaneously optimizing the unit-cell dimensions. This improved scheme substantially increases the computational efficiency compared to existing schemes.

Particle-directed assembly of semiflexible polymer chains.

We use Langevin dynamics simulations to study aggregation of semiflexible polymers driven by attractions between polymers and spherical particles. We consider a simple model with purely repulsive polymer/polymer and particle/particle interactions but attractive polymer/particle interactions. We find a rich "phase diagram" that contains several different types of globular and rod-like aggregates with either liquid-like or crystalline structure for the particle positions. Systems that exhibit rod-like aggregates with crystalline internal order exhibit a discontinuous rod-globule transition, while systems with liquid-like internal order exhibit a smooth crossover between isotropic and elongated aggregates with increasing chain stiffness. Polymers in elongated liquid-like aggregates often adopt helical configurations that wind around the axis of the aggregate.

Commensurability and finite size effects in lattice simulations of diblock copolymers.

Lattice Monte Carlo (MC) simulations provide an efficient method for exploring the structure and phase behavior of block polymer melts. However, the results of such simulations may differ from the equilibrium behavior of a hypothetical infinite system as a consequence of the finite size of the simulation box. Standard finite-size scaling techniques cannot be employed to remove the effects of a small system size due to incommensurability between the ordered structure domain spacing and the periodicity of the simulation box. This work describes a systematic approach to estimating the equilibrium domain spacing in lattice MC simulations of symmetric diblock copolymers, and thereby minimize the effects of incommensurability. Results for simulations with commensurate simulation boxes, which are designed to be commensurate with the preferred lattice periodicity but contain different numbers of unit cells, show that once the effects of incommensurability are removed, the effects of finite size alone are relatively small.

Universality of block copolymer melts.

Simulations of five different coarse-grained models of symmetric diblock copolymers are compared to demonstrate a universal (i.e., model-independent) dependence of the free energy and order-disorder transition (ODT) on the invariant degree of polymerization N̄. The actual values of χN at the ODT approach predictions of the Fredrickson-Helfand (FH) theory for N̄ ≳ 10(4) but significantly exceed FH predictions at lower values characteristic of most experiments. The FH theory fails for modest N̄ because the competing phases become strongly segregated near the ODT, violating an underlying assumption of weak segregation.

Boundary Frustration in Double-Gyroid Thin Films.

Self-consistent field theory for thin films of AB diblock polymers in the double-gyroid phase reveals that in the absence of preferential wetting of monomer species at the film boundaries, films with the (211) plane oriented parallel to the boundaries are more stable than other orientations, consistent with experimental results. This preferred orientation is explained in the context of boundary frustration. Specifically, the angle of intersection between the A/B interface and the film boundary, the wetting angle, is thermodynamically restricted to a narrow range of values. Most termination planes in the double gyroid cannot accommodate this narrow range of wetting angles without significant local distortion relative to the bulk morphology; the (211)-oriented termination plane with the "double-wave" pattern produces relatively minimal distortion, making it the least frustrated boundary. The principle of boundary frustration provides a framework to understand the relative stability of termination planes for complex ordered block polymer phases confined between flat, nonpreferential boundaries.

Diffusion of surfactant from a micellar solution to a bare interface. 1. Absorbing boundary.

We analyze dynamic adsorption of surfactant from a micellar solution to a rapidly created surface that acts as an absorbing boundary for surfactant monomers (single molecules), along which the monomer concentration vanishes, with no direct micelle adsorption. This somewhat idealized situation is analyzed as a prototype for situations in which strong suppression of monomer concentration accelerates micelle dissociation, and will be used as a starting point for analysis of more realistic boundary conditions in subsequent work. We present scaling arguments and approximate models for particular time and parameter regimes and compare the resulting predictions to numerical simulations of the reaction-diffusion equations for a polydisperse system containing surfactant monomers and clusters of arbitrary aggregation number. The model considered here exhibits an initial period of rapid shrinkage and ultimate dissociation of micelles within a narrow region near the interface. This opens a micelle-free region near the interface after some time τe, the width of which increases as t1/2 at times t≫τe. In systems that exhibit disparate fast and slow bulk relaxation times τ1 and τ2 in response to small perturbations, τe is usually comparable to or greater than τ1 but much less than τ2. Such systems exhibit a wide intermediate time regime τe

Fluctuations in symmetric diblock copolymers: testing theories old and new.

Computer simulations are used to study composition fluctuations in disordered diblock copolymer melts over a range of values of the chain length N, and test several theories for the structure factor S(q). Specifically, we test the random-phase approximation (RPA), which is based on a self-consistent field treatment of fluctuations, the Fredrickson-Helfand theory, which was designed to describe fluctuations near the order-disorder transition, and the relatively new renormalized one-loop (ROL) theory. The results confirm claims that the RPA is exact in the limit N→∞ and that the ROL theory yields the dominant corrections to the RPA within a systematic expansion in powers of N(-1/2), and show that the ROL theory is much more accurate than either older theory.

Translationally invariant slip-spring model for entangled polymer dynamics.

The topological effect of noncrossability of long flexible macromolecules is effectively described by a slip-spring model, which represents entanglements by local, pairwise, translationally invariant interactions that do not alter any equilibrium properties. We demonstrate that the model correctly describes many aspects of the dynamical and rheological behavior of entangled polymer liquids, such as segmental mean-square displacements and shear thinning, in a computationally efficient manner. Furthermore, the model can account for the reduction of entanglements under shear.

Nonlinear dynamics in micellar surfactant solutions. II. Diffusion.

We discuss diffusion in micellar surfactant solutions in a form appropriate for analyzing experiments that involve large deviations from equilibrium. A general nonlinear dynamical model for inhomogeneous systems is developed that describes the effects of diffusion and micelle kinetics as a set of coupled partial differential equations for unimer concentration, micelle number concentration, average micelle aggregation number, and, optionally, the variance of the micelle aggregation number. More specialized models are developed to describe slow dynamics in situations in which the system stays in a state of partial local equilibrium or full local equilibrium. As an illustrative example of a nonlinear transport phenomenon, we discuss a simple model of diffusion from an initially homogeneous micellar solution to a rapidly created absorbing interface with fast unimer adsorption.

Nonlinear dynamics in micellar surfactant solutions. I. Kinetics.

This is the first of a pair of articles that present the theory of kinetic and transport phenomena in micelle-forming surfactant solutions in a form that facilitates discussion of large deviations from equilibrium. Our goal is to construct approximate but robust reduced models for both homogeneous and inhomogeneous systems as differential equations for unimer concentration c_{1}, micelle number concentration c_{m}, average micelle aggregation number q and (optionally) aggregation number variance σ_{m}^{2}. This first article discusses kinetics in homogeneous solutions. We focus particularly on developing models that can describe both weakly perturbed states and states in which c_{1} is suppressed significantly below the critical micelle concentration, which leads to rapid shrinkage and dissociation of any remaining micelles. This focus is motivated by the strong local suppression of c_{1} that is predicted to occur near interfaces during some adsorption processes that are considered in the second article. Toward this end, we develop a general nonlinear theory of fast stepwise processes for systems that may be subjected to large changes in q and c_{1}. This is combined with the existing nonlinear theory of slow association and dissociation processes to construct a general model for systems governed by stepwise reaction kinetics. We also consider situations in which the slow process of micelle creation and destruction instead occurs primarily by micelle fission and fusion, and analyze the dependencies of micelle lifetime and the slow relaxation time upon surfactant concentration in systems controlled by either association-dissociation or fission-fusion mechanisms.

Relationships among coarse-grained field theories of fluctuations in polymer liquids.

Two closely related field-theoretic approaches have been used in previous work to construct coarse-grained theories of corrections to the random phase approximation for correlations in block copolymer melts and miscible polymer blends. The "auxiliary field" (AF) approach is based on a rigorous expression for the partition function Z of a coarse-grained model as a functional integral of an auxiliary chemical potential field. The "effective Hamiltonian" (EH) approach is instead based on an expression for Z as a functional integral of an observable order parameter field. The exact effective Hamiltonian H(eff) in the EH approach is defined as the free energy of a system with a constrained order parameter field. In practice, however, H(eff) has often been approximated by a mean-field free energy functional, yielding what we call a mean-field effective Hamiltonian (MFEH) approximation. This approximation was the starting point of both the Fredrickson-Helfand analysis of fluctuation effects in diblock copolymers and earlier work on the Ginzburg criterion in polymer blends. A more rigorous EH approach by Holyst and Vilgis used an auxiliary field representation of the exact H(eff) and allowed for Gaussian fluctuations of this field. All applications of both AF and EH approaches have thus far relied upon some form of Gaussian, or "one-loop" approximation for fluctuations of a chemical potential and/or order parameter field about a mean-field saddle-point. The one-loop EH approximation of Holyst and Vilgis and the one-loop AF theory are equivalent to one another, but not to the one-loop MFEH theory. The one-loop AF and MFEH theories are shown to yield predictions for the inverse structure factor S(-1)(q) that (in the absence of further approximations to either theory) differ by a function that is independent of the Flory-Huggins interaction parameter χ. As a result, these theories yield predictions for the peak scattering intensity that exhibit a similar χ-dependence near a spinodal. The Fredrickson-Helfand theory for the structure factor in disordered diblock copolymer melts is an asymptotic approximation to the MFEH one-loop theory that captures the dominant asymptotic behavior of very long, symmetric copolymers very near the order-disorder transition.

Renormalized one-loop theory of correlations in disordered diblock copolymers.

A renormalized one-loop (ROL) theory developed in previous work [P. Grzywacz, J. Qin, and D. C. Morse, Phys. Rev E. 76, 061802 (2007)] is used to calculate corrections to the random phase approximation (RPA) for the structure factor S(q) in disordered diblock copolymer melts. Predictions are given for the peak intensity S(q∗), peak position q∗, and single-chain statistics for symmetric and asymmetric copolymers as functions of χ(e)N, where χ(e) is an effective Flory-Huggins interaction parameter and N is the degree of polymerization. The ROL and Fredrickson-Helfand (FH) theories are found to yield asymptotically equivalent results for the dependence of the peak intensity S(q∗) upon χ(e)N for symmetric diblock copolymers in the limit of strong scattering, or large χ(e)N, but to yield qualitatively different predictions for symmetric copolymers far from the ODT and for asymmetric copolymers. The ROL theory predicts a suppression of S(q∗) and a decrease of q∗ for large values of χ(e)N, relative to the RPA predictions, but an enhancement of S(q∗) and an increase in q∗ for small χ(e)N. The decrease in q∗ near the ODT is shown to be unrelated to any change in single-chain statistics, and to be a result of inter-molecular correlations. Conversely, the predicted increase in q∗ at small values of χ(e)N is a direct result of non-Gaussian single-chain statistics.

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.

Renormalized one-loop theory of correlations in polymer blends.

The renormalized one-loop theory is a coarse-grained theory of corrections to the random phase approximation (RPA) theory of composition fluctuations. We present predictions of corrections to the RPA for the structure function S(k) and to the random walk model of single-chain statics in binary homopolymer blends. We consider an apparent interaction parameter chi(a) that is defined by applying the RPA to the small k limit of S(k). The predicted deviation of chi(a) from its long chain limit is proportional to N(-1/2), where N is the chain length. This deviation is positive (i.e., destabilizing) for weakly nonideal mixtures, with chi(a)N less than or approximately 1, but negative (stabilizing) near the critical point. The positive correction to chi(a) for low values of chi(a)N is a result of the fact that monomers in mixtures of shorter chains are slightly less strongly shielded from intermolecular contacts. The predicted depression in chi(a) near the critical point is a result of long-wavelength composition fluctuations. The one-loop theory predicts a shift in the critical temperature of O(N(-1/2)), which is much greater than the predicted O(N(-1)) width of the Ginzburg region. Chain dimensions are found to deviate slightly from those of a random walk even in a one-component melt and contract slightly as thermodynamic repulsion is increased. Predictions for S(k) and single-chain properties are compared to published lattice Monte Carlo simulations.

On the chain length dependence of local correlations in polymer melts and a perturbation theory of symmetric polymer blends.

The self-consistent field (SCF) approach to the thermodynamics of dense polymer liquids is based on the idea that short-range correlations in a polymer liquid are almost independent of how monomers are connected into polymers over larger scales. Some limits of this idea are explored in the context of a perturbation theory for symmetric polymer blends. We consider mixtures of two structurally identical polymers, A and B, in which the AB monomer pair interaction differs slightly from the AA and BB interactions by an amount proportional to a parameter alpha. An expansion of the free energy to first order in alpha yields an excess free energy of mixing per monomer of the form alphaz(N)phi(A)phi(B) in both lattice and continuum models, where z(N) is a measure of the number of intermolecular near neighbors per monomer in a one-component (alpha=0) reference liquid with chains of length N. The quantity z(N) decreases slightly with increasing N because the concentration of intramolecular near neighbors is slightly higher for longer chains, creating a slightly deeper intermolecular correlation hole. We predict that z(N)=z(infinity)[1+betaN(-1/2)], where N is an invariant degree of polymerization and beta=(6/pi)(3/2) is a universal coefficient. This and related predictions about the slight N dependence of local correlations are confirmed by comparison to simulations of a continuum bead-spring model and to published lattice Monte Carlo simulations. We show that a renormalized one-loop theory for blends correctly describes this N dependence of local liquid structure. We also propose a way to estimate the effective interaction parameter appropriate for comparisons of simulation data to SCF theory and to coarse-grained theories of corrections to SCF theory, which is based on an extrapolation of perturbation theory to the limit N-->infinity.

Simulation of diblock copolymer surfactants. II. Micelle kinetics.

Molecular dynamics (MD) simulations are used to measure dynamical properties of a simple bead-spring model of A-B diblock copolymer molecules, and to characterize rates and mechanisms of several dynamical processes. Dynamical properties are analyzed within the context of a kinetic population model that allows for both stepwise insertion and expulsion of individual free molecules and occasional fission and fusion of micelles. Kinetic coefficients for stepwise processes and micelle fission have been extracted from MD simulations of individual micelles. Insertion of a free surfactant molecule into a preexisting micelle is shown to be a completely diffusion-controlled process for the model studied here. Estimates are given for rates of rare events that create and destroy entire micelles by competing mechanisms involving stepwise association and dissociation or fission and fusion. Both mechanisms are shown to be relevant over the range of parameters studied here, with association and dissociation dominating in systems with more soluble surfactants and fission and fusion dominating in systems with less soluble surfactants.