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

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

Dynamics of polymeric manifolds in melts: Hartree approximation

The Martin-Siggia-Rose functional technique and the self-consistent Hartree approximation is applied to the dynamics of a D-dimensional manifold in a melt of similar manifolds.The generalized Rouse equation is derived and its static and dynamic properties are studied. The static upper critical dimension discriminate between Gaussian and non-Gaussian regimes, whereas its dynamic counterpart discriminates between Rouse- and renormalized-Rouse behavior. The dynamic exponents are calculated explicitly. The special case of linear chains shows agreement with MD- and MC-simulations.Comment: 4 pages,1 figures, accepted by EPJB as a Rapid Not

Dynamics of a polymer test chain in a glass forming matrix: The Hartree Approximation

In this paper the Martin-Siggia-Rose formalism is used to derive a generalized Rouse equation for a test chain in a matrix which can undergo the glass transition. It is shown that the surrounding matrix renormalizes the static properties of the test chain. Furthermore the freezing of the different Rouse modes is investigated. This yields freezing temperatures which depend from the Rouse mode index.Comment: to be published in Journal de Physique I

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

The USA: Challenges of the Superpower

Since the collapse of the Soviet Union function and mission of the United States in the contemporary world system is one of the most debatable problems of academic litera-ture. This article is an attempt to analyze most recent socioeconomic and political tendencies of the USA for better understanding the scale of ongoing transformation of the society. As the level of integration of contemporary world is very high, transfor-mation of the USA provokes tectonic changes and transformation of the world system, its structure and nature. This study argues that, although the US primacy in the world is significantly challenged and shaken by external and internal factors, the USA still preserves its traditional function of economic, financial, military and political superpow-er, but in a quite different environment. The article predominantly uses materials drawn from the Central Intelligence Agency (CIA, The World Fact book 2012); U.S. Census Bureau, Statistical Abstract of the USA - 2012, the US Federal Budgets 2010-2012, and other valuable literature and sources

Detachment of semiflexible polymer chains from a substrate - a Molecular Dynamics investigation

Using Molecular Dynamics simulations, we study the force-induced detachment of a coarse-grained model polymer chain from an adhesive substrate. One of the chain ends is thereby pulled at constant speed off the attractive substrate and the resulting saw-tooth profile of the measured mean force vs height $D$ of the end-segment over the plane is analyzed for a broad variety of parameters. It is shown that the observed characteristic oscillations in the $< f >$-$D$ profile depend on the bending and not on the torsional stiffness of the detached chains. Allowing for the presence of hydrodynamic interactions (HI) in a setup with explicit solvent and DPD-thermostat, rather than the case of Langevin thermostat, one finds that HI have little effect on the -$D$ profile. Also the change of substrate affinity with respect to the solvent from solvophilic to solvophobic is found to play negligible role in the desorption process. In contrast, a changing ratio $\epsilon_s^A / \epsilon_s^B$ of the binding energies of $A$- and $B$-segments in the detachment of an $AB$-copolymer from adhesive surface strongly changes the $$-$D$ profile whereby the $B$-spikes vanish when $\epsilon_s^A / \epsilon_s^B < 0.15$. Eventually, performing an atomistic simulation of a (bio)-polymer {\it polyglycine}, we demonstrate that the simulation results, derived from our coarse-grained model, comply favorably with those from the all-atom simulation.Comment: Latex, 12 pages, 8 figures, to appear in JC

Polyelectrolyte chains in poor solvent. A variational description of necklace formation

We study the properties of polyelectrolyte chains under different solvent conditions, using a variational technique. The free energy and the conformational properties of a polyelectrolyte chain are studied minimizing the free energy $F_N$, depending on $N(N-1)/2$ trial probabilities that characterize the conformation of the chain. The Gaussian approximation is considered for a ring of length $2^4<N<2^{16}$ and for an open chain of length $2^4<N<2^9$ in poor and theta solvent conditions, including a Coulomb repulsion between the monomers. In theta solvent conditions the blob size is measured and found in agreement with scaling theory, including charge depletion effects, expected for the case of an open chain. In poor solvent conditions, a globule instability, driven by electrostatic repulsion, is observed. We notice also inhomogeneous behavior of the monomer--monomer correlation function, reminiscence of necklace formation in poor solvent polyelectrolyte solutions. A global phase diagram in terms of solvent quality and inverse Bjerrum length is presented.Comment: submitted to EPJE (soft matter

The Hartree approximation in dynamics of polymeric manifolds in the melt

The Martin-Siggia-Rose (MSR) functional integral technique is applied to the dynamics of a D - dimensional manifold in a melt of similar manifolds. The integration over the collective variables of the melt can be simply implemented in the framework of the dynamical random phase approximation (RPA). The resulting effective action functional of the test manifold is treated by making use of the selfconsistent Hartree approximation. As an outcome the generalized Rouse equation (GRE) of the test manifold is derived and its static and dynamic properties are studied. It was found that the static upper critical dimension, $d_{\rm uc}=2D/(2-D)$, discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, ${\tilde d}_{uc}=2d_{\rm uc}$, distinguishes between the simple Rouse and the renormalized Rouse behavior. We have argued that the Rouse mode correlation function has a stretched exponential form. The subdiffusional exponents for this regime are calculated explicitly. The special case of linear chains, D=1, shows good agreement with MD- and MC-simulations.Comment: 35 pages,3 figures, accepted by J.Chem.Phy