Rouse Chains with Excluded Volume Interactions: Linear Viscoelasticity

Linear viscoelastic properties for a dilute polymer solution are predicted by modeling the solution as a suspension of non-interacting bead-spring chains. The present model, unlike the Rouse model, can describe the solution's rheological behavior even when the solvent quality is good, since excluded volume effects are explicitly taken into account through a narrow Gaussian repulsive potential between pairs of beads in a bead-spring chain. The use of the narrow Gaussian potential, which tends to the more commonly used delta-function repulsive potential in the limit of a width parameter "d" going to zero, enables the performance of Brownian dynamics simulations. The simulations results, which describe the exact behavior of the model, indicate that for chains of arbitrary but finite length, a delta-function potential leads to equilibrium and zero shear rate properties which are identical to the predictions of the Rouse model. On the other hand, a non-zero value of "d" gives rise to a prediction of swelling at equilibrium, and an increase in zero shear rate properties relative to their Rouse model values. The use of a delta-function potential appears to be justified in the limit of infinite chain length. The exact simulation results are compared with those obtained with an approximate solution which is based on the assumption that the non-equilibrium configurational distribution function is Gaussian. The Gaussian approximation is shown to be exact to first order in the strength of excluded volume interaction, and is found to be accurate above a threshold value of "d", for given values of chain length and strength of excluded volume interaction.Comment: Revised version. Long chain limit analysis has been deleted. An improved and corrected examination of the long chain limit will appear as a separate posting. 32 pages, 9 postscript figures, LaTe

A universal constitutive model for the interfacial layer between a polymer melt and a solid wall

In a preceeding report we derived the evolution equation for the bond vector probability distribution function (BVPDF) of tethered molecules. It describes the behavior of polymer molecules attached to a solid wall interacting with an adjacent flowing melt of bulk polymer molecules and includes all the major relaxation mechanisms such as constraint release, retraction and convection. The derived equation is quite universal and valid for all flow regimes. In the present paper the developed formalism is further analyzed. We begin our analysis with the simple case of slow flows. Then, as expected, a remarkable reduction of the theory is possible. Later on the more general case is considered. \u

Applying matrix product operators to model systems with long-range interactions

An algorithm is presented which computes a translationally invariant matrix product state approximation of the ground state of an infinite 1D system; it does this by embedding sites into an approximation of the infinite ``environment'' of the chain, allowing the sites to relax, and then merging them with the environment in order to refine the approximation. By making use of matrix product operators, our approach is able to directly model any long-range interaction that can be systematically approximated by a series of decaying exponentials. We apply our techniques to compute the ground state of the Haldane-Shastry model and present results.Comment: 7 pages, 3 figures; manuscript has been expanded and restructured in order to improve presentation of the algorith

A matrix product state based algorithm for determining dispersion relations of quantum spin chains with periodic boundary conditions

We study a matrix product state (MPS) algorithm to approximate excited states of translationally invariant quantum spin systems with periodic boundary conditions. By means of a momentum eigenstate ansatz generalizing the one of \"Ostlund and Rommer [1], we separate the Hilbert space of the system into subspaces with different momentum. This gives rise to a direct sum of effective Hamiltonians, each one corresponding to a different momentum, and we determine their spectrum by solving a generalized eigenvalue equation. Surprisingly, many branches of the dispersion relation are approximated to a very good precision. We benchmark the accuracy of the algorithm by comparison with the exact solutions of the quantum Ising and the antiferromagnetic Heisenberg spin-1/2 model.Comment: 13 pages, 11 figures, 5 table

Data augmentation in Rician noise model and Bayesian Diffusion Tensor Imaging

Mapping white matter tracts is an essential step towards understanding brain function. Diffusion Magnetic Resonance Imaging (dMRI) is the only noninvasive technique which can detect in vivo anisotropies in the 3-dimensional diffusion of water molecules, which correspond to nervous fibers in the living brain. In this process, spectral data from the displacement distribution of water molecules is collected by a magnetic resonance scanner. From the statistical point of view, inverting the Fourier transform from such sparse and noisy spectral measurements leads to a non-linear regression problem. Diffusion tensor imaging (DTI) is the simplest modeling approach postulating a Gaussian displacement distribution at each volume element (voxel). Typically the inference is based on a linearized log-normal regression model that can fit the spectral data at low frequencies. However such approximation fails to fit the high frequency measurements which contain information about the details of the displacement distribution but have a low signal to noise ratio. In this paper, we directly work with the Rice noise model and cover the full range of $b$-values. Using data augmentation to represent the likelihood, we reduce the non-linear regression problem to the framework of generalized linear models. Then we construct a Bayesian hierarchical model in order to perform simultaneously estimation and regularization of the tensor field. Finally the Bayesian paradigm is implemented by using Markov chain Monte Carlo.Comment: 37 pages, 3 figure

Optimisation of a Brownian dynamics algorithm for semidilute polymer solutions

Simulating the static and dynamic properties of semidilute polymer solutions with Brownian dynamics (BD) requires the computation of a large system of polymer chains coupled to one another through excluded-volume and hydrodynamic interactions. In the presence of periodic boundary conditions, long-ranged hydrodynamic interactions are frequently summed with the Ewald summation technique. By performing detailed simulations that shed light on the influence of several tuning parameters involved both in the Ewald summation method, and in the efficient treatment of Brownian forces, we develop a BD algorithm in which the computational cost scales as O(N^{1.8}), where N is the number of monomers in the simulation box. We show that Beenakker's original implementation of the Ewald sum, which is only valid for systems without bead overlap, can be modified so that \theta-solutions can be simulated by switching off excluded-volume interactions. A comparison of the predictions of the radius of gyration, the end-to-end vector, and the self-diffusion coefficient by BD, at a range of concentrations, with the hybrid Lattice Boltzmann/Molecular Dynamics (LB/MD) method shows excellent agreement between the two methods. In contrast to the situation for dilute solutions, the LB/MD method is shown to be significantly more computationally efficient than the current implementation of BD for simulating semidilute solutions. We argue however that further optimisations should be possible.Comment: 17 pages, 8 figures, revised version to appear in Physical Review E (2012