Dynamics of polymers: A mean-field theory

Stochastic kinetic theory applied to coarse-grained polymer model

A stochastic field theory approach is applied to a coarse-grained polymer model that will enable studies of polymer behavior under non-equilibrium conditions. This article is focused on the validation of the new model in comparison with explicit Langevin equation simulations under conditions with analytical solutions. The polymers are modeled as Hookean dumbbells in one dimension, without including hydrodynamic interactions and polymer-polymer interactions. Stochastic moment equations are derived from full field theory. The accuracy of the field theory and moment equations is quantified using autocorrelation functions. The full field theory is only accurate for a large number of polymers due to keeping track of rare occurrences of polymers with a large stretch. The moment equations do not have this error because they do not explicitly track these configurations. The accuracy of both methods depends on the spatial degree of discretization. The timescale of decorrelation over length scales bigger than the spatial discretization is accurate, while there is an error over the scale of single mesh points.

Combining particle and field-theoretic polymer models with multi-representation simulations

Particle-based and field-theoretic simulations are both widely used methods to predict the properties of polymeric materials. In general, the advantages of each method are complementary. Field-theoretic simulations are preferred for polymers with high molecular weights and can provide direct access to chemical potentials and free energies, which makes them the method-of-choice for calculating phase diagrams. The trade-off is that field-theoretic simulations sacrifice the molecular details present in particle-based simulations, such as the configurations of individual molecules and their dynamics. In this work, we describe a new approach to conduct "multi-representation" simulations that efficiently map between particle-based and field-theoretic simulations. Our approach involves the construction of formally equivalent particle-based and field-based models, which are then simulated subject to the constraint that their spatial density profiles are equal. This constraint provides the ability to directly link particle-based and field-based simulations and enables calculations that can switch between one representation to the other. By switching between particle/field representations during a simulation, we demonstrate that our approach can leverage many of the advantages of each representation while avoiding their respective limitations. Although our method is illustrated in the context of complex sphere phases in linear diblock copolymers, we anticipate that it will be useful whenever free energies, rapid equilibration, molecular configurations, and dynamic information are all simultaneously desired.

Effects of Local Order Parameter Dependent Transport Coefficient in Diblock Copolymers Under Applied Electric Fields

We present an approach for constructing thermodynamically consistent time-dependent models relevant to thin films of diblock copolymers in applied electric fields. The approach is based on the principles of linear irreversible thermodynamics and in this work, it is applied to study the effects of electric  fields on thin  films of incompressible diblock copolymers. Enforcement of local incompressibility constraint at all times leads to a local order parameter dependent transport coefficient in the model for the diblock copolymers. The dependence of transport coefficient on the local order parameter is used to relate it with diffusion constant of Rouse chains and leads to sensitivity of the model to initial conditions. Also, transient behavior is found to be a affected when compared with an ad hoc model assuming a constant transport coefficient. Numerical results such as electric  field induced alignment of lamellae domains due to the  field are found to be in qualitative agreements with experiments.This approach opens up a systematic way of developing kinetic models for simulating effects of electrolytes added to thin  films containing diblock copolymers in the presence of applied electric  fields.

Dynamics of a single polyampholyte chain

Polymers that feature both positive and negative charges along chains, known as polyampholytes, represent a class of materials that hold promise for a new generation of energy storage devices, the design of which will require knowledge of the underlying structure and dynamics. Here, we develop a theory based on the Rouse model for the dynamic structure factor of a single polyampholyte chain in the weak coupling regime (negligible intramolecular electrostatics) or subjected to weak external electric fields (governed by linear response). Neglecting effects of small ions, we find deviations in scaling from the classic Rouse theory and make predictions for scattering experiments performed on polyampholytes. We find that, under weak coupling with arbitrarily strong fields, the dynamics are highly dependent on the charge distribution and consequently look at two representative examples-random charge densities and periodic charge densities-with different scaling properties. Under weak fields, the dynamics are largely independent of charge distribution. Finally, we investigate the influence of hydrodynamic effects and the implications of including inertial effects in the model.

PEDOT:PSS nano-particles in aqueous media: A comparative experimental and molecular dynamics study of particle size, morphology and z-potential

PEDOT

PSS is the most widely used conducting polymer in organic and printed electronics.

PEDOT

PSS films have been extensively studied to understand the morphology, ionic and electronic conductivity of the polymer. However, the polymer dispersion, which is used to cast or spin coat the films, is not well characterized and not well understood theoretically. Here, we study in detail the particle morphology, size, charge density and zeta potential (z-potential) by coarse-grained MD simulations and dynamic light scattering (DLS) measurements, for different pH levels and ionic strengths. The PEDOT:PSS particles were found to be 12 nm-19 nm in diameter and had a z-potential of -30 mV to -50 mV when pH was changed from 1.7 to 9, at an added NaCl concentration of 1 mM, as measured by DLS. These values changed significantly with changing pH and ionic strength of the solution. The charge density of PEDOT:PSS particles was also found to be dependent on pH and ionic strength. Besides, the distribution of different ions (PSS^-, PEDOT⁺, Na⁺, Cl^-) present in the solution is simulated to understand the particle morphology and molecular origin of z-potential in PEDOT:PSS dispersion. The trend in change of particle size, charge density and z- potential with changing pH and ionic strength are in good agreement between the simulations and experiments. Our results show that the molecular model developed in this work represents very well the PEDOT:PSS nano-particles in aqueous dispersion. With this study, we hope to provide new insight and an in-depth understanding of the morphology and z-potential evolution in PEDOT:PSS dispersion.

Dynamic Self-Consistent Field Approach for Studying Kinetic Processes in Multiblock Copolymer Melts

The self-consistent field theory is a popular and highly successful theoretical framework for studying equilibrium (co)polymer systems at the mesoscopic level. Dynamic density functionals allow one to use this framework for studying dynamical processes in the diffusive, non-inertial regime. The central quantity in these approaches is the mobility function, which describes the effect of chain connectivity on the nonlocal response of monomers to thermodynamic driving fields. In a recent study, one of us and coworkers have developed a method to systematically construct mobility functions from reference fine-grained simulations. Here we focus on melts of linear chains in the Rouse regime and show how the mobility functions can be calculated semi-analytically for multiblock copolymers with arbitrary sequences without resorting to simulations. In this context, an accurate approximate expression for the single-chain dynamic structure factor is derived. Several limiting regimes are discussed. Then we apply the resulting density functional theory to study ordering processes in a two-length scale block copolymer system after instantaneous quenches into the ordered phase. Different dynamical regimes in the ordering process are identified: at early times, the ordering on short scales dominates; at late times, the ordering on larger scales takes over. For large quench depths, the system does not necessarily relax into the true equilibrium state. Our density functional approach could be used for the computer-assisted design of quenching protocols in order to create novel nonequilibrium materials.

Dynamical self-consistent field theory captures multi-scale physics during spinodal decomposition in a symmetric binary homopolymer blend

We study the spinodal decomposition in a symmetric, binary homopolymer blend using our recently developed dynamical self-consistent field theory. By taking the extremal solution of a dynamical functional integral, the theory reduces the interacting, multi-chain dynamics to a Smoluchowski equation describing the statistical dynamics of a single, unentangled chain in a self-consistent, time-dependent, mean force-field. We numerically solve this equation by evaluating averages over a large ensemble of replica chains, each one of which obeys single-chain Langevin dynamics, subject to the mean field. Following a quench from the disordered state, an early time spinodal instability in the blend composition develops, before even one Rouse time elapses. The dominant, unstable, growing wavelength is on the order of the coil size. The blend then enters a late-time, t, scaling regime with a growing domain size that follows the expected Lifshitz-Slyozov-Wagner t^1/3 power law, a characteristic of a diffusion-driven coarsening process. These results provide a satisfying test of this new method, which correctly captures both the early and late time physics in the blend. Our simulation spans five orders-of-magnitude in time as the domains coarsen to 20 times the coil size, while remaining faithful to the dynamics of the microscopic chain model.

Grafted polymer chains suppress nanoparticle diffusion in athermal polymer melts

We measure the center-of-mass diffusion of poly(methyl methacrylate) (PMMA)-grafted nanoparticles (NPs) in unentangled to slightly entangled PMMA melts using Rutherford backscattering spectrometry. These grafted NPs diffuse ∼100 times slower than predicted by the Stokes-Einstein relation assuming a viscosity equal to bulk PMMA and a hydrodynamic NP size equal to the NP core diameter, 2R_core = 4.3 nm. This slow NP diffusion is consistent with an increased effective NP size, 2R_eff ≈ 20 nm, nominally independent of the range of grafting density and matrix molecular weights explored in this study. Comparing these experimental results to a modified Daoud-Cotton scaling estimate for the brush thickness as well as dynamic mean field simulations of polymer-grafted NPs in athermal polymer melts, we find that 2R_eff is in quantitative agreement with the size of the NP core plus the extended grafted chains. Our results suggest that grafted polymer chains of moderate molecular weight and grafting density may alter the NP diffusion mechanism in polymer melts, primarily by increasing the NP effective size.

Fluctuation Effects on the Brush Structure of Mixed Brush Nanoparticles in Solution

A potentially attractive way to control nanoparticle assembly is to graft one or more polymers on the nanoparticle, to control the nanoparticle-nanoparticle interactions. When two immiscible polymers are grafted on the nanoparticle, they can microphase separate to form domains at the nanoparticle surface. Here, we computationally investigate the phase behavior of such binary mixed brush nanoparticles in solution, across a large and experimentally relevant parameter space. Specifically, we calculate the mean-field phase diagram, assuming uniform grafting of the two polymers, as a function of the nanoparticle size relative to the length of the grafted chains, the grafting density, the enthalpic repulsion between the grafted chains, and the solvent quality. We find a variety of phases including a Janus phase and phases with varying numbers of striped domains. Using a nonuniform, random distribution of grafting sites on the nanoparticle instead of the uniform distribution leads to the development of defects in the mixed brush structures. Introducing fluctuations as well leads to increasingly defective structures for the striped phases. However, we find that the simple Janus phase is preserved in all calculations, even with the introduction of nonuniform grafting and fluctuations. We conclude that the formation of the Janus phase is more realistic experimentally than is the formation of defect-free multivalent mixed brush nanoparticles.

Hybrid particle-continuum simulations coupling Brownian dynamics and local dynamic density functional theory

We present a multiscale hybrid particle-field scheme for the simulation of relaxation and diffusion behavior of soft condensed matter systems. It combines particle-based Brownian dynamics and field-based local dynamics in an adaptive sense such that particles can switch their level of resolution on the fly. The switching of resolution is controlled by a tuning function which can be chosen at will according to the geometry of the system. As an application, the hybrid scheme is used to study the kinetics of interfacial broadening of a polymer blend, and is validated by comparing the results to the predictions from pure Brownian dynamics and pure local dynamics calculations.

Solvent vapor annealing in block copolymer nanocomposite films: a dynamic mean field approach

Polymer nanocomposites are an important class of materials due to the nanoparticles' ability to impart functionality not commonly found in a polymer matrix, such as electrical conductivity or tunable optical properties. While the equilibrium properties of polymer nanocomposites can be treated using numerous theoretical and simulation approaches, in experiments the effects of processing and kinetic traps are significant and thus critical for understanding the structure and the functionality of polymer nanocomposites. However, simulation methods that can efficiently predict kinetically trapped and metastable structures of polymer nanocomposites are currently not common. This is particularly important in inhomogeneous polymers such as block copolymers, where techniques such as solvent vapor annealing are commonly employed to improve the long-range order. In this work, we introduce a dynamic mean field theory that is capable of predicting the result of processing the structure of polymer nanocomposites, and we demonstrate that our method accurately predicts the equilibrium properties of a model system more efficiently than a particle-based model. We subsequently use our method to predict the structure of block copolymer thin films with grafted nanoparticles after solvent annealing, where we find that the final distribution of the grafted nanoparticles can be controlled by varying the solvent evaporation rate. The extent to which the solvent evaporation rate can affect the final nanoparticle distribution in the film depends on the grafting density and the length of the grafted chains. Furthermore, the effects of the solvent evaporation rate can be anticipated from the equilibrium nanoparticle distribution in the swollen and dry states.