Rheological Study of Transient Networks with Junctions of Limited Multiplicity II. Sol/Gel Transition and Rheology

Viscoelastic and thermodynamic properties of transient gels formed by telechelic polymers are studied on the basis of the transient network theory that takes account of the correlation among polymer chains via network junctions. The global information of the gel is incorporated into the theory by introducing the elastically effective chains according to the criterion by Scanlan and Case. We also consider effects of superbridges whose backbone is formed by several chains connected in series with several breakable junctions inside. Near the critical concentration for the sol/gel transition, superbridges becomes infinitely long along the backbone, thereby leading to the short relaxation time $\tau$ of the network. It is shown that $\tau$ is proportional to the concentration deviation $\Delta$ near the gelation point. The plateau modulus $G_{\infty}$ increases as the cube of $\Delta$ near the gelation point as a result of the mean-field treatment, and hence the zero-shear viscosity increases as $\eta_0\sim G_{\infty}\tau\sim\Delta^4$. The dynamic shear moduli are well described in terms of the Maxwell model, and it is shown that the present model can explain the concentration dependence of the dynamic moduli for aqueous solutions of telechelic poly(ethylene oxide).Comment: 18 pages, 12 figures, 2 table

Kinetics and thermodynamics of first-order Markov chain copolymerization

We report a theoretical study of stochastic processes modeling the growth of first-order Markov copolymers, as well as the reversed reaction of depolymerization. These processes are ruled by kinetic equations describing both the attachment and detachment of monomers. Exact solutions are obtained for these kinetic equations in the steady regimes of multicomponent copolymerization and depolymerization. Thermodynamic equilibrium is identified as the state at which the growth velocity is vanishing on average and where detailed balance is satisfied. Away from equilibrium, the analytical expression of the thermodynamic entropy production is deduced in terms of the Shannon disorder per monomer in the copolymer sequence. The Mayo-Lewis equation is recovered in the fully irreversible growth regime. The theory also applies to Bernoullian chains in the case where the attachment and detachment rates only depend on the reacting monomer

On the chain length dependence of local correlations in polymer melts and a perturbation theory of symmetric polymer blends

The self-consistent field (SCF) theory of dense polymer liquids assumes that short-range correlations are almost independent of how monomers are connected into polymers. Some limits of this idea are explored in the context of a perturbation theory for mixtures of structurally identical polymer species, A and B, in which the AB pair interaction differs slightly from the AA and BB interaction, and the difference is controlled by a parameter alpha Expanding the free energy to O(\alpha) yields an excess free energy of the form alpha $z(N)\phi_{A}\phi_{B}$, in both lattice and continuum models, where z(N) is a measure of the number of inter-molecular near neighbors of each monomer in a one-component liquid. This quantity decreases slightly with increasing N because the self-concentration of monomers from the same chain is slightly higher for longer chains, creating a deeper correlation hole for longer chains. We analyze the resulting $N$-dependence, and predict that $z(N) = z^{\infty}[1 + \beta \bar{N}^{-1/2}]$, where $\bar{N}$ is an invariant degree of polymerization, and $\beta=(6/\pi)^{3/2}$. This and other predictions are confirmed by comparison to simulations. We also propose a way to estimate the effective interaction parameter appropriate for comparisons of simulation data to SCF theory and to coarse-grained theories of corrections to SCF theory, which is based on an extrapolation of coefficients in this perturbation theory to the limit $N \to \infty$. We show that a renormalized one-loop theory contains a quantitatively correct description of the $N$-dependence of local structure studied here.Comment: submitted to J. Chem. Phy

Three-phase coexistence with sequence partitioning in symmetric random block copolymers

We inquire about the possible coexistence of macroscopic and microstructured phases in random Q-block copolymers built of incompatible monomer types A and B with equal average concentrations. In our microscopic model, one block comprises M identical monomers. The block-type sequence distribution is Markovian and characterized by the correlation \lambda. Upon increasing the incompatibility \chi\ (by decreasing temperature) in the disordered state, the known ordered phases form: for \lambda\ > \lambda_c, two coexisting macroscopic A- and B-rich phases, for \lambda\ < \lambda_c, a microstructured (lamellar) phase with wave number k(\lambda). In addition, we find a fourth region in the \lambda-\chi\ plane where these three phases coexist, with different, non-Markovian sequence distributions (fractionation). Fractionation is revealed by our analytically derived multiphase free energy, which explicitly accounts for the exchange of individual sequences between the coexisting phases. The three-phase region is reached, either, from the macroscopic phases, via a third lamellar phase that is rich in alternating sequences, or, starting from the lamellar state, via two additional homogeneous, homopolymer-enriched phases. These incipient phases emerge with zero volume fraction. The four regions of the phase diagram meet in a multicritical point (\lambda_c, \chi_c), at which A-B segregation vanishes. The analytical method, which for the lamellar phase assumes weak segregation, thus proves reliable particularly in the vicinity of (\lambda_c, \chi_c). For random triblock copolymers, Q=3, we find the character of this point and the critical exponents to change substantially with the number M of monomers per block. The results for Q=3 in the continuous-chain limit M -> \infty are compared to numerical self-consistent field theory (SCFT), which is accurate at larger segregation.Comment: 24 pages, 19 figures, version published in PRE, main changes: Sec. IIIA, Fig. 14, Discussio

Three Bead Rotating Chain model shows universality in the stretching of proteins

We introduce a model of proteins in which all of the key atoms in the protein backbone are accounted for, thus extending the Freely Rotating Chain model. We use average bond lengths and average angles from the Protein Databank as input parameters, leaving the number of residues as a single variable. The model is used to study the stretching of proteins in the entropic regime. The results of our Monte Carlo simulations are found to agree well with experimental data, suggesting that the force extension plot is universal and does not depend on the side chains or primary structure of proteins

Equilibrium properties of charged microgels: a Poisson-Boltzmann-Flory approach

The equilibrium properties of ionic microgels are investigated using a combination of the Poisson-Boltzmann and Flory theories. Swelling behavior, density profiles, and effective charges are all calculated in a self-consistent way. Special attention is given to the effects of salinity on these quantities. It is found that the equilibrium microgel size is strongly influenced by the amount of added salt. Increasing the salt concentration leads to a considerable reduction of the microgel volume, which therefore releases its internal material -- solvent molecules and dissociated ions -- into the solution. Finally, the question of charge renormalization of ionic microgels in the context of the cell model is briefly addressed

Monte Carlo Algorithm for Simulating Reversible Aggregation of Multisite Particles

We present an efficient and exact Monte Carlo algorithm to simulate reversible aggregation of particles with dedicated binding sites. This method introduces a novel data structure of dynamic bond tree to record clusters and sequences of bond formations. The algorithm achieves a constant time cost for processing cluster association and a cost between $\mathcal{O}(\log M)$ and $\mathcal{O}(M)$ for processing bond dissociation in clusters with $M$ bonds. The algorithm is statistically exact and can reproduce results obtained by the standard method. We applied the method to simulate a trivalent ligand and a bivalent receptor clustering system and obtained an average scaling of $\mathcal{O}(M^{0.45})$ for processing bond dissociation in acyclic aggregation, compared to a linear scaling with the cluster size in standard methods. The algorithm also demands substantially less memory than the conventional method.Comment: 8 pages, 3 figure

Non-ideal behavior of intramolecular structure factor of dilute polymers in a theta solvent

We study the configurational properties of single polymers in a theta solvent by Monte Carlo simulation of the bond fluctuation model. The intramolecular structure factor at the theta point is found to be distinctively different from that of the ideal chain. The structure factor shows a hump around $q\sim 5/R_g$ and a dip around $q\sim 10/R_g$ in the Kratky plot with $R_g$ being the radius of gyration. This feature is apparently similar to that in a melt. The theoretical expression by the simple perturbation expansion to the first order in terms of the Mayer function can be fitted to the obtained structure factor quite well, but the second virial coefficient cannot be set to zero.Comment: 8 pages, 7figure

Collapse transition of a square-lattice polymer with next nearest-neighbor interaction

We study the collapse transition of a polymer on a square lattice with both nearest-neighbor and next nearest-neighbor interactions, by calculating the exact partition function zeros up to chain length 36. The transition behavior is much more pronounced than that of the model with nearest-neighbor interactions only. The crossover exponent and the transition temperature are estimated from the scaling behavior of the first zeros with increasing chain length. The results suggest that the model is of the same universality class as the usual theta point described by the model with only nearest-neighbor interaction.Comment: 14 pages, 5 figure

Simplified Onsager theory for isotropic-nematic phase equilibria of length polydisperse hard rods

Polydispersity is believed to have important effects on the formation of liquid crystal phases in suspensions of rod-like particles. To understand such effects, we analyse the phase behaviour of thin hard rods with length polydispersity. Our treatment is based on a simplified Onsager theory, obtained by truncating the series expansion of the angular dependence of the excluded volume. We describe the model and give the full phase equilibrium equations; these are then solved numerically using the moment free energy method which reduces the problem from one with an infinite number of conserved densities to one with a finite number of effective densities that are moments of the full density distribution. The method yields exactly the onset of nematic ordering. Beyond this, results are approximate but we show that they can be made essentially arbitrarily precise by adding adaptively chosen extra moments, while still avoiding the numerical complications of a direct solution of the full phase equilibrium conditions. We investigate in detail the phase behaviour of systems with three different length distributions: a (unimodal) Schulz distribution, a bidisperse distribution and a bimodal mixture of two Schulz distributions which interpolates between these two cases. A three-phase isotropic-nematic-nematic coexistence region is shown to exist for the bimodal and bidisperse length distributions if the ratio of long and short rod lengths is sufficiently large, but not for the unimodal one. We systematically explore the topology of the phase diagram as a function of the width of the length distribution and of the rod length ratio in the bidisperse and bimodal cases.Comment: 18 pages, 16 figure