Polymer chain in a quenched random medium: slow dynamics and ergodicity breaking

The Langevin dynamics of a self - interacting chain embedded in a quenched random medium is investigated by making use of the generating functional method and one - loop (Hartree) approximation. We have shown how this intrinsic disorder causes different dynamical regimes. Namely, within the Rouse characteristic time interval the anomalous diffusion shows up. The corresponding subdiffusional dynamical exponents have been explicitly calculated and thoroughly discussed. For the larger time interval the disorder drives the center of mass of the chain to a trap or frozen state provided that the Harris parameter, $(\Delta/b^d) N^{2 - \nu d} \ge 1$, where $\Delta$ is a disorder strength, $b$ is a Kuhnian segment length, $N$ is a chain length and $\nu$ is the Flory exponent. We have derived the general equation for the non - ergodicity function $f(p)$ which characterizes the amplitude of frozen Rouse modes with an index $p = 2\pi j/N$. The numerical solution of this equation has been implemented and shown that the different Rouse modes freeze up at the same critical disorder strength $\Delta_c \sim N^{-\gamma}$ where the exponent $\gamma \approx 0.25$ and does not depend from the solvent quality.Comment: 17 pages, 6 figures, submitted to EPJB (condensed matter

Localization and freezing of a Gaussian chain in a quenched random potential

The Gaussian chain in a quenched random potential (which is characterized by the disorder strength $\Delta$) is investigated in the $d$ - dimensional space by the replicated variational method. The general expression for the free energy within so called one - step - replica symmetry breaking (1 - RSB) scenario has been systematically derived. We have shown that the replica symmetrical (RS) limit of this expression can describe the chain center of mass localization and collapse. The critical disorder when the chain becomes localized scales as $\Delta_c \simeq b^d N^{-2 + d/2}$ (where $b$ is the length of the Kuhn segment length and $N$ is the chain length) whereas the chain gyration radius $R_{\rm g} \simeq b (b^d/\Delta)^{1/(4 - d)}$. The freezing of the internal degrees of freedom follows to the 1-RSB - scenario and is characterized by the beads localization length $\bar{{\cal D}^2}$. It was demonstrated that the solution for $\bar{{\cal D}^2}$ appears as a metastable state at $\Delta = \Delta_A$ and behaves similarly to the corresponding frozen states in heteropolymers or in $p$ - spin random spherical model.Comment: 18 pages, 6 figures, submitted to J. Chem. Phy

Collapse or Swelling Dynamics of Homopolymer Rings: Self-consistent Hartree approach

We investigate by the use of the Martin - Siggia - Rose generating functional technique and the self - consistent Hartree approximation, the dynamics of the ring homopolymer collapse (swelling) following an instantaneous change into a poor (good) solvent conditions.The equation of motion for the time dependent monomer - to - monomer correlation function is systematically derived. It is argued that for describing of the coarse - graining process (which neglects the capillary instability and the coalescence of ``pearls'') the Rouse mode representation is very helpful, so that the resulting equations of motion can be simply solved numerically. In the case of the collapse this solution is analyzed in the framework of the hierarchically crumpled fractal picture, with crumples of successively growing scale along the chain. The presented numerical results are in line with the corresponding simple scaling argumentation which in particular shows that the characteristic collapse time of a segment of length $g$ scales as $t^* \sim \zeta_0 g/\tau$ (where $\zeta_0$ is a bare friction coefficient and $\tau$ is a depth of quench). In contrast to the collapse the globule swelling can be seen (in the case that topological effects are neglected) as a homogeneous expansion of the globule interior. The swelling of each Rouse mode as well as gyration radius $R_g$ is discussed.Comment: 20 pages, 7 figures, submitted to Phys. Rev.

Thermal Degradation of Adsorbed Bottle-Brush Macromolecules: Molecular Dynamics Simulation

The scission kinetics of bottle-brush molecules in solution and on an adhesive substrate is modeled by means of Molecular Dynamics simulation with Langevin thermostat. Our macromolecules comprise a long flexible polymer backbone with $L$ segments, consisting of breakable bonds, along with two side chains of length $N$, tethered to each segment of the backbone. In agreement with recent experiments and theoretical predictions, we find that bond cleavage is significantly enhanced on a strongly attractive substrate even though the chemical nature of the bonds remains thereby unchanged. We find that the mean bond life time decreases upon adsorption by more than an order of magnitude even for brush molecules with comparatively short side chains $N=1 \div 4$. The distribution of scission probability along the bonds of the backbone is found to be rather sensitive regarding the interplay between length and grafting density of side chains. The life time declines with growing contour length $L$ as $\propto L^{-0.17}$, and with side chain length as $\propto N^{-0.53}$. The probability distribution of fragment lengths at different times agrees well with experimental observations. The variation of the mean length $L(t)$ of the fragments with elapsed time confirms the notion of the thermal degradation process as a first order reaction.Comment: 15 pages, 7 figure

Directed Polymers with Constrained Winding Angle

In this article we study from a non-perturbative point of view the entanglement of two directed polymers subjected to repulsive interactions given by a Dirac $\delta-$function potential. An exact formula of the so-called second moment of the winding angle is derived. This result is used to provide a thorough analysis of entanglement phenomena in the classical system of two polymers subjected to repulsive interactions and related problems. No approximation is made in treating the constraint on the winding angle and the repulsive forces. In particular, we investigate how repulsive forces influence the entanglement degree of the two-polymer system. In the limit of ideal polymers, in which the interactions are switched off, we show that our results are in agreement with those of previous works.Comment: 34 pages, LaTeX + RevTeX 4, minor typos corrected, 2 figures added, a major revision of Section