Use of Cartesian Rotation Vectors in Brownian Dynamics Algorithms: Theory and Simulation Results

Summary: We have shown that the components of Cartesian rotation vectors can be used successfully as generalized coordinates describing angular orientation in Brownian dynamics simulations of non‐spherical nanoparticles. For this particular choice of generalized coordinates, we rigorously derived the conformation‐space diffusion equations from kinetic theory for both free nanoparticles and nanoparticles interconnected by springs or holonomic constraints into polymer chains. The equivalent stochastic differential equations were used as a foundation for the Brownian dynamics algorithms. These new algorithms contain singularities only for points in the conformation‐space where both the probability density and its first coordinate derivative equal zero (weak singularities). In addition, the coordinate values after a single Brownian dynamics time step are throughout the conformation‐space equal to the old coordinate values plus the respective increments. For some parts of the conformation‐space these features represent a major improvement compared to the situation when Eulerian angles describe rotational dynamics. The presented simulation results of the equilibrium probability density for free nanoparticles are in perfect agreement with the results from kinetic theory.

Simulation of p^(eq)(Φ) for free nanoparticles.     

Curvature effects upon interactions of polymer-grafted nanoparticles in chemically identical polymer matrices

We study the interactions between polymer-grafted nanoparticles immersed in a chemically identical polymer melt using a numerical implementation of polymer mean-field theory. We focus on the interpenetration width between the grafted and free chains and its relationship to the polymer-mediated interparticle interactions. To this end, we quantify the interpenetration width as a function of particle curvature, grafting density, and the relative molecular weights of the grafted and free chains. We show the onset of wetting and dewetting as a function of these quantities and explain our results through simple scaling arguments to include the effects of curvature. Subsequently, we show that the interparticle potentials correlate quantitatively with the trends displayed by the interpenetration widths.     

Fractional separation of polymers in nanochannels: Combined influence of wettability and structure

Trapping macromolecules in nanopits finds diverse applications in polymer separation, filtering biomolecules etc. However, tuning the locomotion of polymers in channels of nanoscopic dimensions is greatly restricted by the comparative advective and diffusive components of velocities. Using the polymer affinity toward the solvent and the wall, and the polymer structure, a mechanism is proposed to induce selective trapping of polymers. Similar to fractional distillation of hydrocarbons based on molecular weight, a technique of fractional segregation, depending on the channel wettability of polymeric chains at different depths in a pit that is located perpendicular to the flow is suggested. Depending on the properties of the polymeric chains and the surface chemistry, the segregation of the polymer at a particular level in the pit can be predicted. This behavior stems from the difference in polymer structure leading to a competition between wettability based enthalpic trapping and structure based entropic trapping. The results of this study suggest a novel way of separating biopolymers based on their structure without relying on the channel geometry. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 2118–2125     

Investigating the Dynamic Viscoelasticity of Single Polymer Chains using Atomic Force Microscopy

In single‐molecule force spectroscopy (SMFS), many studies have focused on the elasticity and conformation of polymer chains, but little attention has been devoted to the dynamic properties of single polymer chains. In this study, we measured the energy dissipation and elastic properties of single polystyrene (PS) chains in toluene, methanol, and N,N‐dimethylformamide using a homemade piezo‐control and data acquisition system externally coupled to a commercial atomic force microscope (AFM), which provided more accurate information regarding the dynamic properties of the PS chains. We quantitatively measured the chain length‐dependent changes in the stiffness and viscosity of a single chain using a phenomenological model consistent with the theory of viscoelasticity for polymer chains in dilute solution. The effective viscosity of a polymer chain can be determined using the Kirkwood model, which is independent of the intrinsic viscosity of the solvent and dependent on the interaction between the polymer and solvent. The results indicated that the viscosity of a single PS chain is dominated by the interaction between the polymer and solvent. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1736–1743     

Similarities between polyelectrolyte gels and biopolymer solutions

Small angle neutron scattering (SANS) measurements and osmotic swelling pressure measurements are reported for polyelectrolyte gels and solutions under nearly physiological conditions. A synthetic polymer (sodium-polyacrylate) and three biopolymers (DNA, hyaluronic acid, and polyaspartic acid) are studied. The neutron scattering response of these anionic polyelectrolytes is closely similar, indicating that at larger length scales the organization of the polymer molecules is not significantly affected by the fine details of the molecular architecture (e.g., size and chemical structure of the monomer unit, type of polymer backbone). The results suggest that specific interactions between the polyelectrolyte chains and the surrounding monovalent cations are negligible. It is found that the osmotic compression modulus of these biopolymer solutions determined from the analysis of the SANS response decreases with increasing chain persistence length. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3679–3686, 2006     

Deformation,scattering, and birefringence of flexible polymer chains under external forces or electric fields

The effect of a tensile stress or an electric field on the conformation of a flexible polymer chain has been studied by combining theory with numerical simulation. In the presence of such external agents, the macromolecule experiences the action of two opposite forces at the chain ends. Two models are considered: the Gaussian bead-and-spring chain, and the freely jointed chain with segments of fixed length. From simulated Brownian trajectories we calculate steady-state properties of the polymer under the continuing action of the external forces. Thus, we compute the chain deformation and expansion, measured by the square radius of gyration, and analyze the influence of the external force on low-angle scattering of radiation. The effect of the link orientation in the optical anisotropy or birefringence is also analyzed. From existing theories, we predict very simple relationships between expansion, low-angle scattering, and birefringence, valid for Gaussian chains of any length, and for long freely jointed chains. The simulation results confirm those relationships. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35: 689–697, 1997     

Selective adsorption behavior of polymer at the polymer–nanoparticle interface

Coarse‐grained molecular dynamics simulations are used to investigate the adsorption behavior of monodisperse and bidisperse polymer chains on the nanoparticle (NP) surface at various polymer–NP interactions, chain lengths, and stiffness. At a strong polymer–NP interaction, long chains preferentially occupy interfacial region and squeeze short chains out of the interfacial region. Semiflexible chains with proper stiffness wrap NPs dominantly in a helical fashion, whereas fully flexible chains constitute the surrounding matrix. As chain stiffness increases, the results of the preferential adsorption are the opposite. The chain‐length or chain‐stiffness‐induced selective adsorption behavior of polymer chains in the polymer–NP interfacial region relies on a delicate competition between entropic and enthalpic contributions to the total free energy. These results could provide insights into polymer–NP interfacial adsorption behavior and guide the design of high‐performance nanocomposites. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 1829–1837     

Nanotube confinement denatures protein helices

In striking contrast to simple polymer physics theory, which does not account for solvent effects, we find that physical confinement of solvated biopolymers decreases solvent entropy, which in turn leads to a reduction in the organized structural content of the polymer. Since our theory is based on a fundamental property of water-protein statistical mechanics, we expect it to have broad implications in many biological and material science contexts.     

A physical model for the longitudinal polarizabilities of polymer chains

The aim of this work is to provide a physical model to relate the polarizability per unit cell of oligomers to that of their corresponding infinite polymer chains. For this we propose an extrapolation method for the polarizability per unit cell of oligomers by fitting them to a physical model describing the dielectric properties of polymer chains. This physical model is based on the concept of a dielectric needle in which we assume a polymer chain to be well described by a cylindrically shaped nonconducting rod with a radius much smaller than its length. With this model we study in which way the polarizability per unit cell approaches the limit of the infinite chain. We show that within this model the macroscopic contribution of the induced electric field to the macroscopic electric field vanishes in the limit of an infinite polymer chain, i.e., there is no macroscopic screening. The macroscopic electric field becomes equal to the external electric field in this limit. We show that this identification leads to a relation between the polarizability per unit cell and the electric susceptibility of the infinite polymer chain. We test our dielectric needle model on the polarizability per unit cell of oligomers of the hydrogen chain and polyacetylene obtained earlier using time-dependent current-density-functional theory in the adiabatic local-density approximation and with the Vignale-Kohn functional. We also perform calculations using the same theory on truly infinite polymer chains by employing periodic boundary conditions. We show that by extrapolating the oligomer results according to our dielectric needle model we get good agreement with our results from calculations on the corresponding infinite polymer chains.     

Phase behavior predictions for polymer blends containing reversibly associating endgroups

A mean field model is developed to predict how polymer–polymer miscibility changes if polymers are functionalized with noncovalent, reversibly binding endgroups. The free-energy model is based on the Flory–Huggins mixing theory and has been modified using Painter's association model to account for equilibrium self-association of endgroups. Model input parameters include the length of polymer chains, a temperature-dependent interaction parameter, and a temperature-dependent equilibrium constant for each type of associating endgroup. The analysis is applied to 12 possible blend combinations involving self-complementary interactions and seven combinations involving hetero-complementary [i.e. donor–acceptor (DA)] interactions. Combinations involve both monofunctional and telechelic associating chains. Predicted phase diagrams illustrate how self-complementary interactions can stabilize two-phase regions and how DA interactions can stabilize single phase regions. The model is a useful tool in understanding the delicate balance between the combinatorial entropy of mixing polymer chains, the repulsive interactions between dissimilar polymers, and the additional enthalpic and entropic changes due to end-group association of chain ends. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3285–3299, 2007     

The rate constant of polymer reversal inside a pore

Translocation of biopolymers through pores is implicated in many biological phenomena. Confinement within a pore often breaks ergodicity on experimental and/or biological time scales by creating large entropic barriers to conformational rearrangements of the chain. Here, we study one example of such hindered rearrangement, in which the chain reverses its direction inside a long pore. Our goal is twofold. First, we study the dependence of the time scale of polymer reversal on the pore size and on the polymer length. Second, we examine the ability of simple one-dimensional theories to quantitatively describe a transition in a system with a complex energy landscape by comparing them with the exact rate constant obtained using brute-force simulations and the forward flux sampling method. We find that one-dimensional transition state theory (TST) using the polymer extension along the pore axis as the reaction coordinate adequately accounts for the exponentially strong dependence of the reversal rate constant on the pore radius r and the polymer length N, while the transmission factor, i.e., the ratio of the exact rate and the TST approximation, has a much weaker power law r and N dependence. We have further attempted to estimate the transmission factor from Kramer's theory, which assumes the reaction coordinate dynamics to be governed by a Langevin equation. However, such an approximation was found to be inadequate. Finally, we examine the scaling behavior of the reversal rate constant with N and r and show that finite size effects are important even for chains with N up to several hundreds.     

Transport properties in concentrated polymer solutions

Transport properties of polymer solutions at finite concentration are derived in the partial draining case by formulating a static version of the theory given by Freed and Edwards (FE) for unentangled concentrated polymer solution. The method follows the Kirkwood—Riseman theory for infinitely dilute solutions: the dynamics of the polymer are ignored apart from the overall rotation or translation of the chain and the solvent velocity is given by the Navier—Stokes equations perturbed by point friction forces. The concentration dependence of viscosity and translational friction coefficient of finite chains obtained by numerical calculations are compared with the results of the FE closed-form solution. It is shown that the screening of the hydrodynamic interaction approximately follows Debye-like behavior in the entire range of concentration. The progressive balancing of the increasing intramolecular hydrodynamic interaction with its reduction due to the screening effects, as the molecular weight increases, is well evidenced by comparing results obtained at constant number concentration for different chain lengths.     

高分子动力学的单链模型

高分子单链模型是高分子稀溶液理论研究的基本模型.对其进行深入地分析,不仅有助于解决高分子稀溶液体系中溶液黏度和分子链扩散等基本问题,而且能够增进人们对高分子链结构与溶液性质间关联性的理解.虽然基于经典连续性介质力学的流体动力学理论可以定性,甚至半定量地获得稀溶液的一些重要性质,但是,随着科学技术的发展,人们从分子水平上建立了许多描述高分子稀溶液性质的模型和理论,期望能够定量地描述高分子稀溶液的性质.本文以高分子稀溶液中3个典型的单链模型为例(包括:不含流体力学相互作用的Rouse模型、含二体流体力学相互作用的Zimm模型和含多体流体力学相互作用的部分穿透球模型),综述高分子稀溶液的重要性质,并详细地给出其动力学方程的推导过程及其重要的研究进展.特别是,对于Rouse模型,本文还将其预言结果拓展到了短链高分子流体体系;此外,还介绍了这一领域的关键科学问题、发展前景和研究方向.     

Monomer density profiles for polymer chains in confined geometries: massive field theory approach

Taking into account the well known correspondence between the field theoretical ?(4) O(n)-vector model in the limit n → 0 and the behavior of long flexible polymer chains in a good solvent, the universal density-force relation is analyzed and the corresponding universal amplitude ratio B(real) is obtained using the massive field theory approach in fixed space dimensions d < 4. The monomer density profiles of ideal chains and real polymer chains with excluded volume interaction in a good solvent between two parallel repulsive walls, one repulsive and one inert wall, are obtained in the framework of the massive field theory approach up to one-loop order. Besides, the monomer density profiles for the dilute polymer solution confined in semi-infinite space containing mesoscopic spherical particle of big radius are calculated. The obtained results are in qualitative agreement with previous theoretical investigations and with the results of Monte Carlo simulations.     

Ergodicity breaking and conformational hysteresis in the dynamics of a polymer tethered at a surface stagnation point

We study the dynamics of long chain polymer molecules tethered to a plane wall and subjected to a stagnation point flow. Using a combination of theory and numerical techniques, including Brownian dynamics (BD), we demonstrate that a chain conformation hysteresis exists even for freely draining (FD) chains. Hydrodynamic interactions (HI) between the polymer and the wall are included in the BD simulations. We find qualitative agreement between the FD and HI simulations, with both exhibiting simultaneous coiled and stretched states for a wide range of fixed flow strengths. The range of state coexistence is understood by considering an equivalent projected equilibrium problem of a two state reaction. Using this formalism, we construct Kramers rate theory (from the inverse mean first passage time for a Markov process) for the hopping transition from coil to stretch and stretch to coil. The activation energy for this rate is found to scale proportionally to chain length or Kuhn step number. Thus, in the limit of infinite chain size the hopping rates at a fixed value of the suitably defined Deborah number approach zero and the states are "frozen." We present the results that demonstrate this "ergodicity breaking."     

Surface instabilities of minority chains in dense polymer brushes: A comparison of density functional theory and quasi-off-lattice self-consistent field theory

This work studies surface instabilities in switchable homopolymer brushes where the minority chain differs in length from the brush chains. Both off-lattice numerical self-consistent field theory and classical density functional theory are employed. It is found that the two methods agree well with each other as long as the same equation of state for the polymer chains is used.     

Ion transport and molecular organization are coupled in polyelectrolyte-modified nanopores

Chemically modified nanopores show a strong and nontrivial coupling between ion current and the structure of the immobilized species. In this work we study theoretically the conductance and structure in polymer modified nanopores and explicitly address the problem of the coupling between ion transport and molecular organization. Our approach is based on a nonequilibrium molecular theory that couples ion conductivity with the conformational degrees of freedom of the polymer and the electrostatic and nonelectrostatic interactions among polyelectrolyte chains, ions, and solvent. We apply the theory to study a cylindrical nanopore between two reservoirs as a function of pore diameter and length, the length of the polyelectrolyte chains, their grafting density, and whether they are present or not on the outer reservoir walls. In the very low applied potential regime, where the distribution of polyelectrolyte and ions is similar to that in equilibrium, we present a simple analytical model based on the combination of the different resistances in the system that describes the conductance in excellent agreement with the calculations of the full nonequilibrium molecular theory. On the other hand, for a large applied potential bias, the theory predicts a dramatic reorganization of the polyelectrolyte chains and the ions. This reorganization results from the global optimization of the different interactions in the system under nonequilibrium conditions. For nanopores modified with long chains, this reorganization leads to two interesting physical phenomena: (i) control of polyelectrolyte morphology by the direction and magnitude of ion-fluxes and (ii) an unexpected decrease in system resistance with the applied potential bias for long chains due to the coupling between polyelectrolyte segment distribution and ion currents.     

Microstructure and depletion forces in polymer-colloid mixtures from an interfacial statistical associating fluid theory

By using a classical density functional theory (interfacial statistical associating fluid theory), we investigate the structure and effective forces in nonadsorbing polymer-colloid mixtures. The theory is tested under a wide range of conditions and performs very well in comparison to simulation data. A comprehensive study is conducted characterizing the role of polymer concentration, particle/polymer-segment size ratio, and polymer chain length on the structure, polymer induced depletion forces, and the colloid-colloid osmotic second virial coefficient. The theory correctly captures a depletion layer on two different length scales, one on the order of the segment diameter (semidilute regime) and the other on the order of the polymer radius of gyration (dilute regime). The particle/polymer-segment size ratio is demonstrated to play a significant role on the polymer structure near the particle surface at low polymer concentrations, but this effect diminishes at higher polymer concentrations. Results for the polymer-mediated mean force between colloidal particles show that increasing the concentration of the polymer solution encourages particle-particle attraction, while decreasing the range of depletion attraction. At intermediate to high concentrations, depletion attraction can be coupled to a midrange repulsion, especially for colloids in solutions of short chains. Colloid-colloid second virial coefficient calculations indicate that the net repulsion between colloids at low polymer densities gives way to net attraction at higher densities, in agreement with available simulation data. Furthermore, the results indicate a higher tendency toward colloidal aggregation for larger colloids in solutions of longer chains.     

Transfer-matrix theory of the crystal–columnar transition in polymeric liquid crystals

The statistical–mechanical problem of the transition between crystalline and columnar phases in a main-chain liquid–crystalline polymer is treated in a simple model in which only longitudinal motions of the polymer chains are permitted. A mean-field approximation for the interchain potential is used to obtain a self-consistent equation for the crystalcolumnar transition temperature. When applied to typical homopolymers this theory correctly predicts transition temperatures above the degradation temperature; when applied to a crude model of a random copolymer a temperature in the observed range is predicted.     

Enthalpy of mixing for polymer solutions based on Gibbs excess function limit of hard spheres mixtures

Enthalpy of mixing of polymer with solvent has been evaluated by the new polymer/solvent theory proposed by the authors in a previous article. The new theory was based on the excess Gibbs function limit of hard sphere mixtures with infinite size difference. The calculated enthalpy of mixing for polymer/solvent mixtures by the new theory, agreed with experimental data with good accuracy and indicated that the theory is capable to produce the enthalpy of mixing in the whole concentration range of polymer compared with Flory–Huggins theory. Also the calculations provided information on the studied polymer chains and the molecular interaction effects which were consistent with the properties of polymers and solvents used in the mixtures.