Kristian K. Müller-Nedebock

In situ X-ray structural studies of a flexible host responding to incremental gas loading

Crystallographic pressure-lapse snapshots of a porous material responding to gas loading were used to investigate the stepwise uptake of carbon dioxide and acetylene molecules into discrete confined spaces. Based on the data, a qualitative statistical mechanical model was devised that reproduces even subtle features in the experimental gas sorption isotherms.

Matrix models of discretely bending, stiff polymers

Polymer models which make use of the Ising model and transfer matrix techniques remind us, for example, of the work of Flory [Statistical mechanics of chain molecules, 1969] and Zimm and Bragg [J Chem Phys, 31 (1959) 526]. We investigate the properties of some such polymer models where the chain conformation can be described solely by an Ising-like parameterization and a set of independent, predetermined bond direction vectors or by a Potts-like model for directions of bond vectors on a lattice, with the specific aim of understanding more closely the connection of constraints and forces on the chain ends for polymers which, in general, are of arc length corresponding to their persistence lengths. Instances of these models are directed helical walks, random sequential walks, bimodally distributed in direction walks or relatively short, stiff chains fixed into a network. The behavior of this model under deformation in statistical mechanics and its dynamical properties under Glauber dynamics are discussed.

Entanglements in polymers: II. Networks

The effect of the preservation of the topology of the entanglement in systems of polymer loops which are part of a gel is investigated. A simple two-state invariant of this topology is implemented in the preservation of the state of linking of sets of loops. We compute the free energy of a network by making use of a variational principle and thereby to go beyond the limit of phantom chains. We compute the contribution due to entanglements to the reduced stress of the network.

Entropic competition in polymeric systems under geometrical confinement

Using molecular dynamics simulation, we investigate the effect of confinement on a system that comprises several stiff segmented polymer chains where each chain has similar segments, but length and stiffness of the segments vary among the chains which makes the system inhomogeneous. The translational and orientational entropy loss due to the confinement plays a crucial role in organizing the chains which can be considered as an entropy-driven segregation mechanism to differentiate the components of the system. Due to the inhomogeneity, both weak and strong confinement regimes show the competition in the system and we see segregation of chains as the confining volume is decreased. In the case of strong spherical confinement, a chain at the periphery shows higher angular mobility than other chains, despite being more radially constrained.

ENTANGLEMENTS IN POLYMERS : I. ANNEALED PROBABILITY FOR LOOPS

The effect of the preservation of the topology of the entanglement in systems of polymer loops is investigated. We define a simple two-state linking invariant for a pair of loop polymer chains and show its relationship to the familiar Gaussian linking number. The probability of linking two disjoint sets chosen from a dense melt of closed-loop chains is determined. We discuss the form of the resulting probability and the approximations necessary in obtaining it.

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 fluctuations in this charge of 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.

Operator Formalism for Topology-Conserving Crossing Dynamics in Planar Knot Diagrams

We address here the topological equivalence of knots through the so-called Reidemeister moves. These topology-conserving manipulations are recast into dynamical rules on the crossings of knot diagrams. This is presented in terms of a simple graphical representation related to the Gauss code of knots. Drawing on techniques for reaction–diffusion systems, we then develop didactically an operator formalism wherein these rules for crossing dynamics are encoded. The aim is to develop new tools for studying dynamical behaviour and regimes in the presence of topology conservation. This necessitates the introduction of composite paulionic operators. The formalism is applied to calculate some differential equations for the time evolution of densities and correlators of crossings, subject to topology-conserving stochastic dynamics. We consider here the simplified situation of two-dimensional knot projections. However, we hope that this is a first valuable step towards addressing a number of important questions regarding the role of topological constraints and specifically of topology conservation in dynamics through a variety of solution and approximation schemes. Further applicability arises in the context of the simulated annealing of knots. The methods presented here depart significantly from the invariant-based path integral descriptions often applied in polymer systems, and, in our view, offer a fresh perspective on the conservation of topological states and topological equivalence in knots.

Motor-driven dynamics of cytoskeletal filaments in motility assays

We model analytically the dynamics of a cytoskeletal filament in a motility assay. The filament is described as rigid rod free to slide in two dimensions. The motor proteins consist of polymeric tails tethered to the plane and modeled as linear springs and motor heads that bind to the filament. As in related models of rigid and soft two-state motors, the binding and unbinding dynamics of the motor heads and the dependence of the transition rates on the load exerted by the motor tails play a crucial role in controlling the filaments dynamics. Our work shows that the filament effectively behaves as a self-propelled rod at long times, but with non-Markovian noise sources arising from the coupling to the motor binding and unbinding dynamics. The effective propulsion force of the filament and the active renormalization of the various friction and diffusion constants are calculated in terms of microscopic motor and filament parameters. These quantities could be probed by optical force microscopy.

Counterion correlations and attractions in dense polyelectrolyte solutions

We study concentrated solutions of highly charged polyelectrolyte solutions beyond the mean-field approximation by explicitly taking into account the effects of both counterion and polymer chain fluctuations. One part of the fluctuating counterion densities moves freely throughout the bulk. The second counterion density component consists of a fraction of counterions fluctuating along the length of the chains. The degree of condensation and the effective intra-chain potential are determined self-consistently. We observe the effects of this interaction which is attractive over short distances and can lead to a collapse as a sufficiently high level of counterion condensation is achieved.

Stiff polymer in monomer ensemble

We employ an ordered monomer ensemble formalism in order to develop techniques to investigate a stiff polymer chain which is confined to a certain region. In particular, we calculate the segment density for a given location and segment orientation distribution within the confining geometry. With this method the role of the stiffness can be examined by means of differential equations, integral equations, or recursive relations for both continuum and lattice models. A suitable choice of lattice model permits an exact analytical solution for the segment location and orientation density for a chain between two parallel plates. For the stiff polymer in a spherical cavity we develop an integral equation formalism which is treated numerically, and in the same spherical geometry, a different model of the polymer displays a solution of a differential equation.