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

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.

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.

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.

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.

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].

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.

Equilibration of Micelle-Polyelectrolyte Complexes: Mechanistic Differences between Static and Annealed Charge Distributions.

The role of charge density and charge annealing in polyelectrolyte complexation was investigated through systematic comparison of two micelle-polyelectrolyte systems. First, poly(dimethylaminoethyl methacrylate)-block-poly(styrene) (PDMAEMA-b-PS) micelles were complexed with poly(styrenesulfonate) (PSS) at pH values above and below the pKa of PDMAEMA to investigate the role of charge annealing in the complexation process. Second, complexes of poly(DMAEMA-stat-oligo(ethylene glycol) methyl ether methacrylate)-block-poly(styrene) (P(DMAEMA-stat-OEGMA)-b-PS) micelles with the same PSS at low pH were used to investigate how the complexation process differs when the charged sites are in fixed positions along the polymer chains. Characterization by turbidimetric titration, dynamic light scattering, and cryogenic transmission electron microscopy reveals that whether or not the charge distribution can rearrange during the complexation process significantly affects the structure and stability of the complexes. In complexes of PDMAEMA-b-PS micelles at elevated pH, in which the charge distributions can anneal, the charge sites redistribute along the corona chains upon complexation to favor more fully ion-paired configurations. This promotes rapid rearrangement to single-micelle species when the micelles are in excess but traps complexes formed with PSS in excess. In complexes with static charge distributions introduced by copolymerization of DMAEMA with neutral OEGMA monomers, on the other hand, the opposite is true: in this case, reducing the charge density promotes rearrangement to single-micelle complexes only when the polyanion is in excess. Molecular dynamics simulations show that disruption of the charge density in the corona brush reduces the barrier to rearrangement of individual ion pairs, suggesting that the inability of the brush to rearrange to form fully ion-paired complexes fundamentally alters the kinetics of complex formation and equilibration.

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.

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.

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.

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.

Micellization kinetics of diblock copolymers in a homopolymer matrix: a self-consistent field study.

Self-consistent field theory is used to calculate free energy barriers and reaction rates for the spontaneous association and dissociation of micelles formed of block copolymers in a homopolymer matrix. The barriers are prohibitively large for copolymers of typical molecular weights when the unimer (free surfactant) concentration is near the equilibrium critical micelle concentration (CMC). As a result, polymeric micelles normally cannot reach true thermodynamic equilibrium. The rates of association and dissociation are, however, sensitive to unimer concentration, making it possible to form or destroy micelles at observable rates in sufficiently highly supersaturated or subsaturated solutions, respectively, even when both reactions are suppressed near the equilibrium CMC. The barrier to dissociation is particularly sensitive to unimer concentration, and vanishes when the unimer concentration is only slightly (for example, a percentage of a few tens) below the equilibrium CMC.