Operator Product Expansion on a Fractal: The Short Chain Expansion for Polymer Networks

We prove to all orders of renormalized perturbative polymer field theory the existence of a short chain expansion applying to polymer solutions of long and short chains. For a general polymer network with long and short chains we show factorization of its partition sum by a short chain factor and a long chain factor in the short chain limit. This corresponds to an expansion for short distance along the fractal perimeter of the polymer chains connecting the vertices and is related to a large mass expansion of field theory. The scaling of the second virial coefficient for bimodal solutions is explained. Our method also applies to the correlations of the multifractal measure of harmonic diffusion onto an absorbing polymer. We give a result for expanding these correlations for short distance along the fractal carrier of the measure.Comment: 28 pages, revtex, 4 Postscript figures, 3 latex emlines pictures. Replacement eliminates conflict with a blob resul

Self-avoiding Tethered Membranes at the Tricritical Point

The scaling properties of self-avoiding tethered membranes at the tricritical point (theta-point) are studied by perturbative renormalization group methods. To treat the 3-body repulsive interaction (known to be relevant for polymers), new analytical and numerical tools are developped and applied to 1-loop calculations. These technics are a prerequisite to higher order calculations for self-avoiding membranes. The cross-over between the 3-body interaction and the modified 2-body interaction, attractive at long range, is studied through a new double epsilon-expansion. It is shown that the latter interaction is relevant for 2-dimensional membranes at the theta-point.Comment: 57 pages, gz-compressed ps-fil

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

Large Orders for Self-Avoiding Membranes

We derive the large order behavior of the perturbative expansion for the continuous model of tethered self-avoiding membranes. It is controlled by a classical configuration for an effective potential in bulk space, which is the analog of the Lipatov instanton, solution of a highly non-local equation. The n-th order is shown to have factorial growth as (-cst)^n (n!)^(1-epsilon/D), where D is the `internal' dimension of the membrane and epsilon the engineering dimension of the coupling constant for self-avoidance. The instanton is calculated within a variational approximation, which is shown to become exact in the limit of large dimension d of bulk space. This is the starting point of a systematic 1/d expansion. As a consequence, the epsilon-expansion of self-avoiding membranes has a factorial growth, like the epsilon-expansion of polymers and standard critical phenomena, suggesting Borel summability. Consequences for the applicability of the 2-loop calculations are examined.Comment: 40 pages Latex, 32 eps-files included in the tex

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

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

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

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)

Renormalization Theory for Interacting Crumpled Manifolds

We consider a continuous model of D-dimensional elastic (polymerized) manifold fluctuating in d-dimensional Euclidean space, interacting with a single impurity via an attractive or repulsive delta-potential (but without self-avoidance interactions). Except for D=1 (the polymer case), this model cannot be mapped onto a local field theory. We show that the use of intrinsic distance geometry allows for a rigorous construction of the high-temperature perturbative expansion and for analytic continuation in the manifold dimension D. We study the renormalization properties of the model for 0<D<2, and show that for d<d* where d*=2D/(2-D) is the upper critical dimension, the perturbative expansion is UV finite, while UV divergences occur as poles at d=d*. The standard proof of perturbative renormalizability for local field theories (the BPH theorem) does not apply to this model. We prove perturbative renormalizability to all orders by constructing a subtraction operator based on a generalization of the Zimmermann forests formalism, and which makes the theory finite at d=d*. This subtraction operation corresponds to a renormalization of the coupling constant of the model (strength of the interaction with the impurity). The existence of a Wilson function, of an epsilon-expansion around the critical dimension, of scaling laws for d<d* in the repulsive case, and of non-trivial critical exponents of the delocalization transition for d>d* in the attractive case is thus established. To our knowledge, this provides the first proof of renormalizability for a model of extended objects, and should be applicable to the study of self-avoidance interactions for random manifolds.Comment: 126 pages (+ 24 figures not included available upon request), harvmac, SPhT/92/12

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