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Register a new userDynamics of polymers: classic results and recent developments
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In this chapter we review concepts and theories of polymer dynamics. We think of it as an introduction to the topic for scientists specializing in other subfields of statistical mechanics and condensed matter theory, so, for the readers reference, we start with a short review of the equilibrium static properties of polymer systems. Most attention is paid to the dynamics of unentangled polymer systems, where apart from classical Rouse and Zimm models we review some recent scaling and analytical generalizations. The dynamics of systems with entanglements is also briefly reviewed. Special attention is paid to the discussion of comparatively weakly understood topological states of polymer systems and possible approaches to the description of their dynamics.
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We present the results of analytic calculations and numerical simulations of the behaviour of a new class of chain molecules which we call thick polymers. The concept of the thickness of such a polymer, viewed as a tube, is encapsulated by a special three body interaction and impacts on the behaviour both locally and non-locally. When thick polymers undergo compaction due to an attractive self-interaction, we find a new type of phase transition between a compact phase and a swollen phase at zero temperature on increasing the thickness. In the vicinity of this transition, short tubes form space filling helices and sheets as observed in protein native state structures. Upon increasing the chain length, or the number of chains, we numerically find a crossover from secondary structure motifs to a quite distinct class of structures akin to the semi-crystalline phase of polymers or amyloid fibers in polypeptides.
We explore the effect of an attractive interaction between parallel-aligned polymers, which are perpendicularly grafted on a substrate. Such an attractive interaction could be due to, e.g., reversible cross-links. The competition between permanent grafting favoring a homogeneous state of the polymer brush and the attraction, which tends to induce in-plane collapse of the aligned polymers, gives rise to an instability of the homogeneous phase to a bundled state. In this latter state the in-plane translational symmetry is spontaneously broken and the density is modulated with a finite wavelength, which is set by the length scale of transverse fluctuations of the grafted polymers. We analyze the instability for two models of aligned polymers: directed polymers with a line tension and weakly bending chains with a bending stiffness.
We investigate the ejection dynamics of a ring polymer out of a cylindrical nanochannel using both theoretical analysis and three dimensional Langevin dynamics simulations. The ejection dynamics for ring polymers shows two regimes like for linear polymers, depending on the relative length of the chain compared with the channel. For long chains with length $N$ larger than the critical chain length $N_{c}$, at which the chain just fully occupies the nanochannel, the ejection for ring polymers is faster compared with linear chains of identical length due to a larger entropic pulling force; while for short chains ($N<N_c$), it takes longer time for ring polymers to eject out of the channel due to a longer distance to be diffused to reach the exit of the channel before experiencing the entropic pulling force. These results can help understand many biological processes, such as bacterial chromosome segregation.
We present a model for semiflexible polymers in Hamiltonian formulation which interpolates between a Rouse chain and worm-like chain. Both models are realized as limits for the parameters. The model parameters can also be chosen to match the experimental force-extension curve for double-stranded DNA. Near the ground state of the Hamiltonian, the eigenvalues for the longitudinal (stretching) and the transversal (bending) modes of a chain with N springs, indexed by p, scale as lambda_lp ~ (p/N)^2 and lambda_tp ~ p^2(p-1)^2/N^4 respectively for small p. We also show that the associated decay times tau_p ~ (N/p)^4 will not be observed if they exceed the orientational time scale tau_r ~ N^3 for an equally-long rigid rod, as the driven decay is then washed out by diffusive motion.
Using analytical techniques and Langevin dynamics simulations, we investigate the dynamics of polymer translocation into a narrow channel of width $R$ embedded in two dimensions, driven by a force proportional to the number of monomers in the channel. Such a setup mimics typical experimental situations in nano/micro-fluidics. During the the translocation process if the monomers in the channel can sufficiently quickly assume steady state motion, we observe the scaling $tausim N/F$ of the translocation time $tau$ with the driving force $F$ per bead and the number $N$ of monomers per chain. With smaller channel width $R$, steady state motion cannot be achieved, effecting a non-universal dependence of $tau$ on $N$ and $F$. From the simulations we also deduce the waiting time distributions under various conditions for the single segment passage through the channel entrance. For different chain lengths but the same driving force, the curves of the waiting time as a function of the translocation coordinate $s$ feature a maximum located at identical $s_{mathrm{max}}$, while with increasing the driving force or the channel width the value of $s_{mathrm{max}}$ decreases.
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