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Polymer models describing the statistics of biomolecules under confinement have applications to a wide range of single-molecule experimental techniques and give insight into biologically relevant processes in vivo. In this paper, we determine the transverse position and bending correlation functions for a wormlike chain confined within slits and cylinders (with one and two confined dimensions, respectively) using a mean-field approach that enforces rigid constraints on average. We show the theoretical predictions accurately capture the statistics of a wormlike chain from Monte Carlo simulations in both confining geometries for both weak and strong confinement. We also show that the longitudinal correlation function is accurately computed for a chain confined to a slit and leverages the accuracy of the model to suggest an experimental technique to infer the (often unobservable) transverse statistics from the (directly observable) longitudinal end-to-end distance. [ABSTRACT FROM AUTHOR]

Morphology of symmetric ABCD tetrablock quaterpolymers in melt was studied by the Monte Carlo (MC) simulation, where the volume fractions of the block chains, f, kept the relationships of fA = fD and fB = fC, and the volume fraction of the two mid-blocks φ was defined as φ = fB + fC. Previous self-consistent field theory for ABCD reported morphological change including several structures; however, the scope was limited within a two-dimensional system. To the contrary, in this paper, MC simulations were carried out in three dimensions with changing the φ value finely, which resulted in finding a tetracontinuous structure in the range of 0.625 ⩽ φ ⩽ 0.75. Moreover the tetracontinuous structure has been found to be the gyroid structure, and the mean curvature of the B/C interface is nearly zero. We concluded that the B/C interface must be the Schoen gyroid surface, one of three-dimensional periodic minimal surfaces. The geometrical nature of the A/B interface should be equivalent to that of the C/D interface, and they stand as level surfaces to the Schoen gyroid surface. [ABSTRACT FROM AUTHOR]

The nonlinear viscoelastic behavior of the Fraenkel-chain model is studied with respect to the constitutive equation of the Rouse model. Distinctly different from the results of the Rouse model, the Fraenkel-chain model gives the following characteristic nonlinear behavior: (a) The two distinct dynamic modes in the relaxation modulus GS(t,λ)—as observed in the linear region reported in Paper I [Y.-H. Lin and A. K. Das, J. Chem. Phys. 126, 074902 (2007), preceding paper]—or in the first normal-stress difference function GΨ1(t,λ) are shown to have different strain dependences: strain hardening for the fast mode and strain softening for the slow mode. (b) The Lodge–Meissner relation GS(t,λ)=GΨ1(t,λ) holds over the whole time range, which has been shown both analytically and by simulation. (c) The second normal-stress difference is nonzero, being positive in the fast-mode region and negative in the slow-mode region. The comparisons between orientation and stress for all tensor components consistently confirm the strong correlation of the slow mode as well as its entropic nature with the segmental-orientation anisotropy as shown in the linear region studied in Paper I. A consequence of this correlation is the applicability of the stress-optical rule in the slow-mode region. This also leads to the expectation that the damping function h(λ)=GS(t,λ)/GS(t,λ→0) and the ratio between the first and second normal-stress differences, N2(t,λ)/N1(t,λ), are described by the orientation tensor which has the same form as that given by Doi and Edwards [J. Chem. Soc. Faraday Trans. 2 74, 1789 (1978); 74, 1802 (1978)] with independent-alignment approximation for an entangled system. The similarity between the slow mode of an entanglement-free Fraenkel-chain system and the terminal mode of an entangled polymer system as observed in the comparison of theory, simulation, and experiment suggests that the close correlation of the entropic nature of the mode with the orientation anisotropy—as of the Fraenkel segment or the primitive step in the Doi–Edwards theory—is a generally valid physical concept in polymer viscoelasticity. [ABSTRACT FROM AUTHOR]

This paper investigates the conformational and scaling properties of long linear polymer chains. These investigations are done with the aid of Monte Carlo (MC) and molecular dynamics (MD) simulations. Chain lengths that comprise several orders of magnitude to reduce errors of finite size scaling, including the effect of solvent quality, ranging from the athermal limit over the θ-transition to the collapsed state of chains are investigated. Also the effect of polydispersity on linear chains is included which is an important issue in the real fabrication of polymers. A detailed account of the hybrid MD and MC simulation model and the exploited numerical methods is given. Many results of chain properties in the extrapolated limit of infinite chain lengths are documented and universal properties of the chains within their universality class are given. An example of the difference between scaling exponents observed in actual solvents and those observed in the extremes of “good solvents” and “θ-solvents” in simulations is provided by comparing simulation results with experimental data on low density polyethylene. This paper is concluded with an outlook on the extension of this study to branched chain systems of many different branching types. [ABSTRACT FROM AUTHOR]

In efforts to produce polymeric materials with tailored physical properties, significant interest has grown around the ability to control the spatial organization of nanoparticles in polymer nanocomposites. One way to achieve controlled particle arrangement is by grafting the nanoparticle surface with polymers that are compatible with the matrix, thus manipulating the interfacial interactions between the nanoparticles and the polymer matrix. Previous work has shown that the molecular weight of the grafted polymer, both at high grafting density and low grafting density, plays a key role in dictating the effective inter-particle interactions in a polymer matrix. At high grafting density nanoparticles disperse (aggregate) if the graft molecular weight is higher (lower) than the matrix molecular weight. At low grafting density the longer grafts can better shield the nanoparticle surface from direct particle-particle contacts than the shorter grafts and lead to the dispersion of the grafted particles in the matrix. Despite the importance of graft molecular weight, and evidence of non-trivial effects of polydispersity of chains grafted on flat surfaces, most theoretical work on polymer grafted nanoparticles has only focused on monodisperse grafted chains. In this paper, we focus on how bidispersity in grafted chain lengths affects the grafted chain conformations and inter-particle interactions in an implicit solvent and in a dense homopolymer polymer matrix. We first present the effects of bidispersity on grafted chain conformations in a single polymer grafted particle using purely Monte Carlo (MC) simulations. This is followed by calculations of the potential of mean force (PMF) between two grafted particles in a polymer matrix using a self-consistent Polymer Reference Interaction Site Model theory-Monte Carlo simulation approach. Monte Carlo simulations of a single polymer grafted particle in an implicit solvent show that in the bidisperse polymer grafted particles with an equal number of short and long grafts at low to medium grafting density, the short grafts are in a more coiled up conformation (lower radius of gyration) than their monodisperse counterparts to provide a larger free volume to the longer grafts so they can gain conformational entropy. The longer grafts do not show much difference in conformation from their monodisperse counterparts at low grafting density, but at medium grafting density the longer grafts exhibit less stretched conformations (lower radius of gyration) as compared to their monodisperse counterparts. In the presence of an explicit homopolymer matrix, the longer grafts are more compressed by the matrix homopolymer chains than the short grafts. We observe that the potential of mean force between bidisperse grafted particles has features of the PMF of monodisperse grafted particles with short grafts and monodisperse grafted particles with long grafts. The value of the PMF at contact is governed by the short grafts and values at large inter-particle distances are governed by the longer grafts. Further comparison of the PMF for bidisperse and monodisperse polymer grafted particles in a homopolymer matrix at varying parameters shows that the effects of matrix chain length, matrix packing fraction, grafting density, and particle curvature on the PMF between bidisperse polymer grafted particles are similar to those seen between monodisperse polymer grafted particles. [ABSTRACT FROM AUTHOR]

The classical bond-fluctuation model (BFM) is an efficient lattice Monte Carlo algorithm for coarse-grained polymer chains where each monomer occupies exclusively a certain number of lattice sites. In this paper we propose a generalization of the BFM where we relax this constraint and allow the overlap of monomers subject to a finite energy penalty [variant_greek_epsilon]. This is done to vary systematically the dimensionless compressibility g of the solution in order to investigate the influence of density fluctuations in dense polymer melts on various static properties at constant overall monomer density. The compressibility is obtained directly from the low-wave vector limit of the static structure factor. We consider, e.g., the intrachain bond-bond correlation function P(s) of two bonds separated by s monomers along the chain. It is shown that the excluded volume interactions are never fully screened for very long chains. If distances smaller than the thermal blob size are probed (s≤g) the chains are swollen according to the classical Fixman expansion where, e.g., P(s)∼g-1s-1/2. More importantly, the polymers behave on larger distances (s>g) like swollen chains of incompressible blobs with P(s)∼g0s-3/2. [ABSTRACT FROM AUTHOR]

In this paper, we construct an effective model for the dynamics of an excluded-volume chain under confinement by extending the formalism of Rouse modes. We make specific predictions about the behavior of the modes for a single polymer confined to a tube. The results are tested against Monte Carlo simulations using the bond-fluctuation algorithm which uses a lattice representation of the polymer chain with excluded-volume effects. [ABSTRACT FROM AUTHOR]

In the first paper of this series, we developed a new one-dimensional Monte Carlo approach for the study of flexible chains that are translocating through a small channel. We also presented a numerical scheme that can be used to obtain exact values for both the escape times and the escape probabilities given an initial pore-polymer configuration. We now present and discuss the fundamental scaling behaviors predicted by this Monte Carlo method. Our most important result is the fact that, in the presence of an external bias E, we observe a change in the scaling law for the translocation time τ as function of the polymer length N: In the general expression τ∼Nβ/E, the exponent changes from β=1 for moderately long chains to β=1+ν or β=2ν for very large values of N (for Rouse and Zimm dynamics, respectively). We also observe an increase in the effective diffusion coefficient due to the presence of entropic pulling on unbiased polymer chains. [ABSTRACT FROM AUTHOR]

Monte Carlo simulations of a model fluid in which the particles interact via a continuous potential that has a logarithmic divergence at a pair separation of σ, which we introduced in J. G. Powles et al., Proc. R. Soc. London, Ser. A 455, 3725 (1999), have been carried out. The potential has the form, [lowercase_phi_synonym](r)=-ε ln(f(r)), where ε sets the energy scale and f(r)=1-(σ/r)m. The value of m chosen was 12 but the qualitative trends depend only weakly on the value of m, providing it is greater than 3. The potential is entirely repulsive and has a logarithmic divergence as ∼-ln(r/σ-1) in the r→σ limit. Predictions of the previous paper that the internal energy can be computed at all temperatures using the standard statistical mechanics formula for continuous potentials are verified here. The pressure can be calculated using the usual virial expression for continuous potentials, although there are practical limitations in resolving the increasingly important contribution from the r→σ limit at reduced temperatures greater than ∼5. The mean square force ≤F2≥ and infinite frequency shear G∞ and bulk K∞ moduli are only finite for T*=kBT/ε<1. The logarithmic fluid’s physical properties become increasingly more like that of the hard sphere fluid with increasing temperature, showing a sharp transition in the behavior of the mean square force and infinite frequency elastic constants at T*=1. The logarithmic fluid is shown to exhibit a solid-fluid phase transition. [ABSTRACT FROM AUTHOR]

Proteins contain many voids, which are unfilled spaces enclosed in the interior. A few of them have shapes compatible to ligands and substrates and are important for protein functions. An important general question is how the need for maintaining functional voids is influenced by, and affects other aspects of proteins structures and properties (e.g., protein folding stability, kinetic accessibility, and evolution selection pressure). In this paper, we examine in detail the effects of maintaining voids of different shapes and sizes using two-dimensional lattice models. We study the propensity for conformations to form a void of specific shape, which is related to the entropic cost of void maintenance. We also study the location that voids of a specific shape and size tend to form, and the influence of compactness on the formation of such voids. As enumeration is infeasible for long chain polymer, a key development in this work is the design of a novel sequential Monte Carlo strategy for generating large number of sample conformations under very constraining restrictions. Our method is validated by comparing results obtained from sampling and from enumeration for short polymer chains. We succeeded in accurate estimation of entropic cost of void maintenance, with and without an increasing number of restrictive conditions, such as loops forming the wall of void with fixed length, with additionally fixed starting position in the sequence. Additionally, we have identified the key structural properties of voids that are important in determining the entropic cost of void formation. We have further developed a parametric model to predict quantitatively void entropy. Our model is highly effective, and these results indicate that voids representing functional sites can be used as an improved model for studying the evolution of protein functions and how protein function relates to protein stability. [ABSTRACT FROM AUTHOR]

A dynamic Monte Carlo simulation of the collapse transition of polymer chains is presented. The chains are represented as self-avoiding walks on the simple cubic lattice with a nearest-neighbor contact potential to model the effect of solvent quality. The knot state of the chains is determined using the knot group procedure presented in the accompanying paper. The equilibrium knot spectrum and the equilibrium rms radius of gyration as functions of the chain length and the contact potential are reported. The collapse transition was studied following quenches from good-to poor-solvent conditions. Our results confirm the prediction that the newly formed globule is not yet at equilibrium, since it has not yet achieved its equilibrium knot spectrum. For our model system, the relaxation of the knot spectrum is about an order of magnitude slower than that of the radius of gyration. The collapse transition is also studied for a model in which both ends of the chain remain in good-solvent conditions. Over the time scale of these simulations, knot formation is frustrated in this inhomogeneous model, verifying that the mechanism of knotting is the tunneling of chain ends in and out of the globule. [ABSTRACT FROM AUTHOR]

This paper reports the first computational estimation of the comb polymers’ third virial coefficients. The number of the chains in the comb polymers range from 5 to 11. An algorithm that counts the contributing terms of the third virial coefficients in an accelerated manner is presented along with its efficiency dependence on the polymers’ size. In addition, the second virial coefficients are estimated for the comb polymers and compared to previously reported results. [ABSTRACT FROM AUTHOR]

We simulated the translocation process of a polymer chain from a source container to a drain container through a short nanochannel. We utilized the bond fluctuation model coupled with Monte Carlo dynamics in our simulations. The calculation results show that the excluded volume effect significantly affects the polymer’s translocation time τ. This time depends nonmonotonically on the polymer length N. For a fixed nanochannel length, τ decreases when the polymer length increases. τ, however, increases when the polymer length exceeds a certain threshold. This observation differs from those predicated for a Gaussian chain. In this paper, we will further present our findings to explain this phenomenon. The knowledge we gain from this research can enhance the understanding of complex transport processes in many biological systems. [ABSTRACT FROM AUTHOR]

Bounded potentials are good models to represent the effective two-body interaction in some colloidal systems, such as the dilute solutions of polymer chains in good solvents. The simplest bounded potential is that of penetrable spheres, which takes a positive finite value if the two spheres are overlapped, being 0 otherwise. Even in the one-dimensional case, the penetrable-rod model is far from trivial, since interactions are not restricted to nearest neighbors and so its exact solution is not known. In this paper the structural properties of one-dimensional penetrable rods are studied. We first derive the exact correlation functions of the penetrable-rod fluids to second order in density at any temperature, as well as in the high-temperature and zero-temperature limits at any density. It is seen that, in contrast to what is generally believed, the Percus-Yevick equation does not yield the exact cavity function in the hard-rod limit. Next, two simple analytic theories are constructed: a high-temperature approximation based on the exact asymptotic behavior in the limit T→∞ and a low-temperature approximation inspired by the exact result in the opposite limit T→0. Finally, we perform Monte Carlo simulations for a wide range of temperatures and densities to assess the validity of both theories. It is found that they complement each other quite well, exhibiting a good agreement with the simulation data within their respective domains of applicability and becoming practically equivalent on the borderline of those domains. A comparison with numerical solutions of the Percus-Yevick and the hypernetted-chain approximations is also carried out. Finally, a perspective on the extension of our two heuristic theories to the more realistic three-dimensional case is provided. [ABSTRACT FROM AUTHOR]

Hypothetical scanning Monte Carlo (HSMC) is a method for calculating the absolute entropy S and free energy F from a given MC trajectory developed recently and applied to liquid argon, TIP3P water, and peptides. In this paper HSMC is extended to random coil polymers by applying it to self-avoiding walks on a square lattice—a simple but difficult model due to strong excluded volume interactions. With HSMC the probability of a given chain is obtained as a product of transition probabilities calculated for each bond by MC simulations and a counting formula. This probability is exact in the sense that it is based on all the interactions of the system and the only approximation is due to finite sampling. The method provides rigorous upper and lower bounds for F, which can be obtained from a very small sample and even from a single chain conformation. HSMC is independent of existing techniques and thus constitutes an independent research tool. The HSMC results are compared to those obtained by other methods, and its application to complex lattice chain models is discussed; we emphasize its ability to treat any type of boundary conditions for which a reference state (with known free energy) might be difficult to define for a thermodynamic integration process. Finally, we stress that the capability of HSMC to extract the absolute entropy from a given sample is important for studying relaxation processes, such as protein folding. [ABSTRACT FROM AUTHOR]

The rheological behaviors of polymer brush under oscillatory shear flow were investigated by nonequilibrium Monte Carlo simulation. The grafted chain under oscillatory shear flow exhibited a waggling behavior like a flower, and the segments were found to have different oscillatory phases along the chain contour. Stress tensor was further obtained based on the statistics of sampled configuration distribution functions. The simulation reproduced the abrupt increase of the first normal stress difference N1 with the flow velocity over a critical value, as observed in the experiment of Klein et al. [Nature (London) 352, 143 (1991)]. However, our simulation did not reproduce the brush thickening with shear velocity increased, which was suggested to be responsible for the abrupt increase of N1 in the above-mentioned paper. This simulation demonstrates that the increase of normal stress might be an inherent behavior of polymer brush due to chain deformation under flow. [ABSTRACT FROM AUTHOR]

In this paper we present a comprehensive lattice Monte Carlo study of long chain conformations at solid–polymer melt interfaces. Segmental scale interfacial features, like the bond orientational distribution were found to be independent of surface–segment energetics, and statistically identical with Helfand’s predictions for the full-occupancy, infinite chain length limit. Conformational statistics of chains longer than 5–6 statistical segments followed the predictions of the Scheutjens–Fleer theory and the same power laws as a single ideal chain at the critical value of the surface–segment adsorption free-energy. Our simulations tested the predictions of random walk next to a ‘‘reflective’’ surface statistics for the spatial variations of chain dimensions and chain center of mass density. It was found that these statistics furnish the correct long chain limit, independently of surface–segment energetics. The random walk next to a ‘‘reflective’’ boundary predictions for the ‘‘adsorbed’’ amount and the distributions of tail, loop, and train number, as well as tail and loop size were in quantitative agreement with the simulation data. The correspondence between random walks and real (or simulated) chains required the knowledge of a single, microscopic parameter, the number of chemical segments per statistical segment, a∞. This quantity was very close to the average length of adsorbed sequences (trains), in the long chain limit. Our simulations tested thoroughly and established firmly the validity of ‘‘reflective’’ boundary statistics in the melt. The inevitability of these statistics has broad implications on ‘‘desorption’’ kinetics, chain mobility, and chain relaxation, which are currently under study. [ABSTRACT FROM AUTHOR]

In this paper, we present the results of Monte Carlo simulations of the static properties of polymer melts confined between hard walls. The simulations are conducted in the canonical ensemble with a method that is a combination of reptation and crankshaft motions. 1250 polymer chains each comprising of 100 connected beads are placed in a box which allows for the simulation of a typical polymer melt confined between two hard plates at a separation of 51 bead diameters. Noncovalently bonded beads are assumed to interact with an empirical 6-12 Lennard-Jones potential which has parameters chosen to simulate a polyethylene melt at 400 K. From the analysis of the simulation results we show the existence of two relevant length scales in the problem. Single-chain statistics are perturbed by the wall, and this effect is screened out only after one proceeds to a distance comparable to twice the unperturbed radius of gyration of the polymer chain. However, many-chain statistics, i.e., packing and orientation of chain segments, are screened out as soon as one proceeds about three times the bead size from the wall. The simulation also allows for the study of the conformations of chains near the wall, and we observe that chains near the surface are flattened into nearly two-dimensional structures. The interface therefore corresponds to a region where chain configurations gradually evolve from this nearly two-dimensional structure to the unperturbed, three-dimensional Gaussian configurations in the bulk. [ABSTRACT FROM AUTHOR]

The three-parameter model of excluded volume is employed to calculate both the moments {n=2,3,...} of the end-to-end vector distance of a linear homopolymer and the average square moment for a single branch of an f-armed star. These results complement the calculations of , the radius of gyration RG, the Kirkwood approximation to the hydrodynamic radius RH, and the second virial coefficient A2 for linear chains given in paper I. These calculations are intended to check the self-consistency of the ternary interaction theory against recent simulation data for lattice chains under theta conditions (A2=0) using the fact that higher moments and properties of branched polymers are more sensitive to the presence of ternary interactions. No free parameters are involved in the comparison, and the agreement between theory and numerical data is quite satisfactory. In addition, a general discussion is provided of differences in values of various polymer properties that can result from different methods of regularizing the continuum theory. [ABSTRACT FROM AUTHOR]