Crosslinked polymer chains with excluded volume: A new class of branched polymers?

In this note microgels with and without excluded volume interactions are considered. Based on earlier exact computations on Gaussian mircogels, which are formed by self-crosslinking (with $M$ crosslinks) of polymer chains with chain length $N$ Flory type approximations are used to get first insight to their behavior in solution. It is shown that two different types of microgels exist: A special type of branched polymer whose size scales as $R \propto N^{2/5} M^{-1/5}$, instead of $R \propto N^{1/2}$. The second type are $c^*$ - microgels whose average mesh sizes $r$ are swollen and form self avoiding walks with a scaling law of the form $r = a (N/M)^{3/5}$.Comment: 5 pages, 2 figures, accepted in Macromol. Theory Simu

Elasticity in strongly interacting soft solids: polyelectrolyte network

This paper discusses the elastic behavior of a very long crosslinked polyelectrolyte chain (Debye-H\"uckel chain), which is weakly charged. Therefore the response of the crosslinked chain (network) on an external constant force $f$ acting on the ends of the chain is considered. A selfconsistent variational computation of an effective field theory is employed. It is shown, that the modulus of the polyelectrolyte network has two parts: the first term represents the usual entropy elasticity of connected flexible chains and the second term takes into account the electrostatic interaction of the monomers. It is proportional to the squared crosslink density and the Debye - screening parameter.Comment: submitted for publication to PR

Polyelectrolyte Networks: Elasticity, Swelling, and the Violation of the Flory - Rehner Hypothesis

This paper discusses the elastic behavior of polyelectrolyte networks. The deformation behavior of single polyelectrolyte chains is discussed. It is shown that a strong coupling between interactions and chain elasticity exists. The theory of the complete crosslinked networks shows that the Flory - Rehner - Hypothesis (FRH) does not hold. The modulus contains contributions from the classical rubber elasticity and from the electrostatic interactions. The equilibrium degree of swelling is estimated by the assumption of a $c^{*}$-network.Comment: submitted to Computational and Theoretical Polymer Scienc

How to break the replica symmetry in structural glasses

The variational principle (VP) has been used to capture the metastable states of a glass-forming molecular system without quenched disorder. It has been shown that VP naturally leads to a self-consistent random field Ginzburg-Landau model (RFGLM). In the framework of one-step replica symmetry breaking (1-RSB) the general solution of RFGLM is discussed in the vicinity of the spinodal temperature T_{A} in terms of ``hidden'' formfactors $\tilde g(k)$, g_{0}(k) and $\Delta(k)$. The self-generated disorder spontaneously arises. It is argued that at T < T_{A} the activated dynamics is dominant.Comment: 11 pages, no figures, accepted by Europhys. Let

Langevin Dynamics of a Polymer with Internal Distance Constraints

We present a novel and rigorous approach to the Langevin dynamics of ideal polymer chains subject to internal distance constraints. The permanent constraints are modelled by harmonic potentials in the limit when the strength of the potential approaches infinity (hard crosslinks). The crosslinks are assumed to exist between arbitrary pairs of monomers. Formally exact expressions for the resolvent and spectral density matrix of the system are derived. To illustrate the method we study the diffusional behavior of monomers in the vicinity of a single crosslink within the framework of the Rouse model. The same problem has been studied previously by Warner (J. Phys. C: Solid State Phys. {\bf 14}, 4985, (1981)) on the basis of Lagrangian multipliers. Here we derive the full, hence exact, solution to the problem.Comment: To appear in PRE, Figures on reques

Size and scaling in ideal polymer networks

The scattering function and radius of gyration of an ideal polymer network are calculated depending on the strength of the bonds that form the crosslinks. Our calculations are based on an {\it exact} theorem for the characteristic function of a polydisperse phantom network that allows for treating the crosslinks between pairs of randomly selected monomers as quenched variables without resorting to replica methods. From this new approach it is found that the scattering function of an ideal network obeys a master curve which depends on one single parameter $x= (ak)^2 N/M$, where $ak$ is the product of the persistence length times the scattering wavevector, $N$ the total number of monomers and $M$ the crosslinks in the system. By varying the crosslinking potential from infinity (hard $\delta$-constraints) to zero (free chain), we have also studied the crossover of the radius of gyration from the collapsed regime where R_{\mbox{\tiny g}}\simeq {\cal O}(1) to the extended regime R_{\mbox{\tiny g}}\simeq {\cal O}(\sqrt{N}). In the crossover regime the network size R_{\mbox{\tiny g}} is found to be proportional to $(N/M)^{1/4}$.Comment: latex, figures available on request, to be published: J. Phys I Franc

Collective Dynamics of Random Polyampholytes

We consider the Langevin dynamics of a semi-dilute system of chains which are random polyampholytes of average monomer charge $q$ and with a fluctuations in this charge of the size $Q^{-1}$ and with freely floating counter-ions in the surrounding. We cast the dynamics into the functional integral formalism and average over the quenched charge distribution in order to compute the dynamic structure factor and the effective collective potential matrix. The results are given for small charge fluctuations. In the limit of finite $q$ we then find that the scattering approaches the limit of polyelectrolyte solutions.Comment: 13 pages including 6 figures, submitted J. Chem. Phy

Langevin dynamics of the glass forming polymer melt: fluctuations around the random phase approximation

In this paper the Martin-Siggia-Rose (MSR) functional integral representation is used for the study of the Langevin dynamics of a polymer melt in terms of collective variables: mass density and response field density. The resulting generating functional (GF) takes into account fluctuations around the random phase approximation (RPA) up to an arbitrary order. The set of equations for the correlation and response functions is derived. It is generally shown that for cases whenever the fluctuation-dissipation theorem (FDT) holds we arrive at equations similar to those derived by Mori-Zwanzig. The case when FDT in the glassy phase is violated is also qualitatively considered and it is shown that this results in a smearing out of the ideal glass transition. The memory kernel is specified for the ideal glass transition as a sum of all water-melon diagrams. For the Gaussian chain model the explicit expression for the memory kernel was obtained and discussed in a qualitative link to the mode-coupling equation.Comment: 30 pages, 5 figure

Weak violation of universality for Polyelectrolyte Chains: Variational Theory and Simulations

A variational approach is considered to calculate the free energy and the conformational properties of a polyelectrolyte chain in $d$ dimensions. We consider in detail the case of pure Coulombic interactions between the monomers, when screening is not present, in order to compute the end-to-end distance and the asymptotic properties of the chain as a function of the polymer chain length $N$. We find $R \simeq N^{\nu}(\log N)^{\gamma}$ where $\nu = \frac{3}{\lambda+2}$ and $\lambda$ is the exponent which characterize the long-range interaction $U \propto 1/r^{\lambda}$. The exponent $\gamma$ is shown to be non-universal, depending on the strength of the Coulomb interaction. We check our findings, by a direct numerical minimization of the variational energy for chains of increasing size $2^4<N<2^{15}$. The electrostatic blob picture, expected for small enough values of the interaction strength, is quantitatively described by the variational approach. We perform a Monte Carlo simulation for chains of length $2^4<N<2^{10}$. The non universal behavior of the exponent $\gamma$ previously derived within the variational method, is also confirmed by the simulation results. Non-universal behavior is found for a polyelectrolyte chain in $d=3$ dimension. Particular attention is devoted to the homopolymer chain problem, when short range contact interactions are present.Comment: to appear in European Phys. Journal E (soft matter