Polymers as compressible soft spheres

We consider a coarse-grained model in which polymers under good-solvent conditions are represented by soft spheres whose radii, which should be identified with the polymer radii of gyrations, are allowed to fluctuate. The corresponding pair potential depends on the sphere radii. This model is a single-sphere version of the one proposed in Vettorel et al., Soft Matter 6, 2282 (2010), and it is sufficiently simple to allow us to determine all potentials accurately from full-monomer simulations of two isolated polymers (zero-density potentials). We find that in the dilute regime (which is the expected validity range of single-sphere coarse-grained models based on zero-density potentials) this model correctly reproduces the density dependence of the radius of gyration. However, for the thermodynamics and the intermolecular structure, the model is largely equivalent to the simpler one in which the sphere radii are fixed to the average value of the radius of gyration and radiiindependent potentials are used: for the thermodynamics there is no advantage in considering a fluctuating sphere size.Comment: 21 pages, 7 figure

Elastic Lattice Polymers

We study a model of "elastic" lattice polymer in which a fixed number of monomers $m$ is hosted by a self-avoiding walk with fluctuating length $l$. We show that the stored length density $\rho_m = 1 - /m$ scales asymptotically for large $m$ as $\rho_m=\rho_\infty(1-\theta/m + ...)$, where $\theta$ is the polymer entropic exponent, so that $\theta$ can be determined from the analysis of $\rho_m$. We perform simulations for elastic lattice polymer loops with various sizes and knots, in which we measure $\rho_m$. The resulting estimates support the hypothesis that the exponent $\theta$ is determined only by the number of prime knots and not by their type. However, if knots are present, we observe strong corrections to scaling, which help to understand how an entropic competition between knots is affected by the finite length of the chain.Comment: 10 page

Individual Entanglements in a Simulated Polymer Melt

We examine entanglements using monomer contacts between pairs of chains in a Brownian-dynamics simulation of a polymer melt. A map of contact positions with respect to the contacting monomer numbers (i,j) shows clustering in small regions of (i,j) which persists in time, as expected for entanglements. Using the ``space''-time correlation function of the aforementioned contacts, we show that a pair of entangled chains exhibits a qualitatively different behavior than a pair of distant chains when brought together. Quantitatively, about 50% of the contacts between entangled chains are persistent contacts not present in independently moving chains. In addition, we account for several observed scaling properties of the contact correlation function.Comment: latex, 12 pages, 7 figures, postscript file available at http://arnold.uchicago.edu/~ebn

Identification of a polymer growth process with an equilibrium multi-critical collapse phase transition: the meeting point of swollen, collapsed and crystalline polymers

We have investigated a polymer growth process on the triangular lattice where the configurations produced are self-avoiding trails. We show that the scaling behaviour of this process is similar to the analogous process on the square lattice. However, while the square lattice process maps to the collapse transition of the canonical interacting self-avoiding trail model (ISAT) on that lattice, the process on the triangular lattice model does not map to the canonical equilibrium model. On the other hand, we show that the collapse transition of the canonical ISAT model on the triangular lattice behaves in a way reminiscent of the $\theta$-point of the interacting self-avoiding walk model (ISAW), which is the standard model of polymer collapse. This implies an unusual lattice dependency of the ISAT collapse transition in two dimensions. By studying an extended ISAT model, we demonstrate that the growth process maps to a multi-critical point in a larger parameter space. In this extended parameter space the collapse phase transition may be either $\theta$-point-like (second-order) or first-order, and these two are separated by a multi-critical point. It is this multi-critical point to which the growth process maps. Furthermore, we provide evidence that in addition to the high-temperature gas-like swollen polymer phase (coil) and the low-temperature liquid drop-like collapse phase (globule) there is also a maximally dense crystal-like phase (crystal) at low temperatures dependent on the parameter values. The multi-critical point is the meeting point of these three phases. Our hypothesised phase diagram resolves the mystery of the seemingly differing behaviours of the ISAW and ISAT models in two dimensions as well as the behaviour of the trail growth process

Metal-Insulator transition in the Generalized Hubbard model

We present the exact ground-state wave function and energy of the generalized Hubbard model, subjected to the condition that the number of double occupied sites is conserved, for a wide, physically relevant range of parameters. For one hole and one double occupied site the existence of the ferromagnetic ground-state is proved which allow one to determine the critical value of the on-site repulsion corresponding to the point of metal-insulator transition. For the one dimensional model the exact solution for special values of the parameters is obtained.Comment: 20 pages, LaTex. Mod.Phys.Lett.B 7 (1993) 1397; Journal of Physics: Condensed Matter (to appear

A multi-blob representation of semi-dilute polymer solutions

A coarse-grained multi-blob description of polymer solutions is presented, based on soft, transferable effective interactions between bonded and non-bonded blobs. The number of blobs is chosen such that the blob density does not exceed their overlap threshold, allowing polymer concentrations to be explored deep into the semi-dilute regime. This quantitative multi-blob description is shown to preserve known scaling laws of polymer solutions and provides accurate estimates of amplitudes, while leading to orders of magnitude increase of simulation efficiency and allowing analytic calculations of structural and thermodynamic properties.Comment: 4 pages, 4 figure

Renormalization of the one-loop theory of fluctuations in polymer blends and diblock copolymer melts

Attempts to use coarse-grained molecular theories to calculate corrections to the random-phase approximation (RPA) for correlations in polymer mixtures have been plagued by an unwanted sensitivity to the value of an arbitrary cutoff length, {\it i.e.}, by an ultraviolet (UV) divergence. We analyze the UV divergence of the inverse structure factor $S^{-1}(k)$ predicted by a `one-loop' approximation similar to that used in several previous studies. We consider both miscible homopolymer blends and disordered diblock copolymer melts. We show, in both cases, that all UV divergent contributions can be absorbed into a renormalization of the values of the phenomenological parameters of a generalized self-consistent field theory (SCFT). This observation allows the construction of a UV convergent theory of corrections to SCFT phenomenology. The UV-divergent one-loop contribution to $S^{-1}(k)$ are shown to be the sum of: (i) a $k$-independent contribution that arises from a renormalization of the effective $\chi$ parameter, (ii) a $k$-dependent contribution that arises from a renormalization of monomer statistical segment lengths, (iii) a contribution proportional to $k^{2}$ that arises from a square-gradient contribution to the one-loop fluctuation free energy, and (iv) a $k$-dependent contribution that is inversely proportional to the degree of polymerization, which arises from local perturbations in fluid structure near chain ends and near junctions between blocks in block copolymers.Comment: 35 pages, 2 figure

Phase diagram of mixtures of colloids and polymers in the thermal crossover from good to θ\theta solvent

We determine the phase diagram of mixtures of spherical colloids and neutral nonadsorbing polymers in the thermal crossover region between the $\theta$ point and the good-solvent regime. We use the generalized free-volume theory (GFVT), which turns out to be quite accurate as long as $q = R_g/R_c\lesssim 1$ ($R_g$ is the radius of gyration of the polymer and $R_c$ is the colloid radius). Close to the $\theta$ point the phase diagram is not very sensitive to solvent quality, while, close to the good-solvent region, changes of the solvent quality modify significantly the position of the critical point and of the binodals. We also analyze the phase behavior of aqueous solutions of charged colloids and polymers, using the extension of GFVT proposed by Fortini et al., J. Chem. Phys. 128, 024904 (2008)

Renormalized one-loop theory of correlations in polymer blends

The renormalized one-loop theory is a coarse-grained theory of corrections to the self-consistent field theory (SCFT) of polymer liquids, and 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 chain length. This deviation is positive (i.e., destabilizing) for weakly non-ideal mixtures, with \chi_{a} N \alt 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 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 ${\cal O}(N^{-1/2})$, which is much greater than the predicted ${\cal O}(N^{-1})$ width of the Ginzburg region. Chain dimensions deviate slightly from those of a random walk even in a one-component melt, and contract slightly with increasing $\chi_{e}$. Predictions for $S(k)$ and single-chain properties are compared to published lattice Monte Carlo simulations.Comment: submitted to J. Chem. Phy

Surface segregation of conformationally asymmetric polymer blends

We have generalized the Edwards' method of collective description of dense polymer systems in terms of effective potentials to polymer blends in the presence of a surface. With this method we have studied conformationally asymmetric athermic polymer blends in the presence of a hard wall to the first order in effective potentials. For polymers with the same gyration radius $R_g$ but different statistical segment lengths $l_{A}$ and $l_{B}$ the excess concentration of stiffer polymers at the surface is derived as % \delta \rho _{A}(z=0)\sim (l_{B}^{-2}-l_{A}^{-2}){\ln (}R_{g}^{2}/l_{c}^{2}{)%}, where $l_{c}$ is a local length below of which the incompressibility of the polymer blend is violated. For polymer blends differing only in degrees of polymerization the shorter polymer enriches the wall.Comment: 11 pages, 7 figures, revtex