Colloidal electrophoresis: Scaling analysis, Green-Kubo relation, and numerical results

We consider electrophoresis of a single charged colloidal particle in a finite box with periodic boundary conditions, where added counterions and salt ions ensure charge neutrality. A systematic rescaling of the electrokinetic equations allows us to identify a minimum set of suitable dimensionless parameters, which, within this theoretical framework, determine the reduced electrophoretic mobility. It turns out that the salt-free case can, on the Mean Field level, be described in terms of just three parameters. A fourth parameter, which had previously been identified on the basis of straightforward dimensional analysis, can only be important beyond Mean Field. More complicated behavior is expected to arise when further ionic species are added. However, for a certain parameter regime, we can demonstrate that the salt-free case can be mapped onto a corresponding system containing additional salt. The Green-Kubo formula for the electrophoretic mobility is derived, and its usefulness demonstrated by simulation data. Finally, we report on finite-element solutions of the electrokinetic equations, using the commercial software package COMSOL.Comment: To appear in Journal of Physics: Condensed Matter - special issue on occasion of the CODEF 2008 conferenc

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

Numerical electrokinetics

A new lattice method is presented in order to efficiently solve the electrokinetic equations, which describe the structure and dynamics of the charge cloud and the flow field surrounding a single charged colloidal sphere, or a fixed array of such objects. We focus on calculating the electrophoretic mobility in the limit of small driving field, and systematically linearise the equations with respect to the latter. This gives rise to several subproblems, each of which is solved by a specialised numerical algorithm. For the total problem we combine these solvers in an iterative procedure. Applying this method, we study the effect of the screening mechanism (salt screening vs. counterion screening) on the electrophoretic mobility, and find a weak non-trivial dependence, as expected from scaling theory. Furthermore, we find that the orientation of the charge cloud (i. e. its dipole moment) depends on the value of the colloid charge, as a result of a competition between electrostatic and hydrodynamic effects.Comment: accepted for publication in Journal of Physics Condensed Matter (proceedings of the 2012 CODEF conference

Universal scaling and characterisation of gelation in associative polymer solutions

A Brownian dynamics algorithm is used to describe the static behaviour of associative polymer solutions. Predictions for the fractions of stickers bound by intra-chain and inter-chain association, as a function of system parameters, such as the number of stickers, the number of monomers between stickers, the solvent quality, and concentration are obtained. A systematic comparison with the scaling relations predicted by the mean-field theory of Dobrynin (Macromolecules, 37, 3881, 2004) is carried out. Different regimes of scaling behaviour are identified depending on the monomer concentration, the density of stickers on a chain, and the solvent quality for backbone monomers. Simulation results validate the predictions of the mean-field theory across a wide range of parameter values in all the scaling regimes. The value of the des Cloizeaux exponent proposed by Dobrynin for sticky polymer solutions, is shown to lead to a collapse of simulation data for all the scaling relations considered here. Three different signatures for the characterisation of gelation are identified, with each leading to a different value of the concentration at the sol-gel transition. The modified Flory-Stockmayer expression is found to be validated by simulations for all three gelation signatures. Simulation results confirm the prediction of scaling theory for the gelation line that separates sol and gel phases, when the modified Flory-Stockmayer expression is used. Phase separation is found to occur with increasing concentration for systems in which the backbone monomers are under theta-solvent conditions, and is shown to coincide with a breakdown in the predictions of scaling theory.Comment: 34 pages, 22 figures, includes Supplemental Material, accepted versio

Electrophoresis of colloidal dispersions in the low-salt regime

We study the electrophoretic mobility of spherical charged colloids in a low-salt suspension as a function of the colloidal concentration. Using an effective particle charge and a reduced screening parameter, we map the data for systems with different particle charges and sizes, including numerical simulation data with full electrostatics and hydrodynamics and experimental data for latex dispersions, on a single master curve. We observe two different volume fraction-dependent regimes for the electrophoretic mobility that can be explained in terms of the static properties of the ionic double layer.Comment: Substantially revised versio

Dynamics and Scaling of 2D Polymers in a Dilute Solution

The breakdown of dynamical scaling for a dilute polymer solution in 2D has been suggested by Shannon and Choy [Phys. Rev. Lett. {\bf 79}, 1455 (1997)]. However, we show here both numerically and analytically that dynamical scaling holds when the finite-size dependence of the relevant dynamical quantities is properly taken into account. We carry out large-scale simulations in 2D for a polymer chain in a good solvent with full hydrodynamic interactions to verify dynamical scaling. This is achieved by novel mesoscopic simulation techniques

Implicit and explicit solvent models for the simulation of a single polymer chain in solution: Lattice Boltzmann vs Brownian dynamics

We present a comparative study of two computer simulation methods to obtain static and dynamic properties of dilute polymer solutions. The first approach is a recently established hybrid algorithm based upon dissipative coupling between Molecular Dynamics and lattice Boltzmann (LB), while the second is standard Brownian Dynamics (BD) with fluctuating hydrodynamic interactions. Applying these methods to the same physical system (a single polymer chain in a good solvent in thermal equilibrium) allows us to draw a detailed and quantitative comparison in terms of both accuracy and efficiency. It is found that the static conformations of the LB model are distorted when the box length L is too small compared to the chain size. Furthermore, some dynamic properties of the LB model are subject to an $L^{-1}$ finite size effect, while the BD model directly reproduces the asymptotic $L \to \infty$ behavior. Apart from these finite size effects, it is also found that in order to obtain the correct dynamic properties for the LB simulations, it is crucial to properly thermalize all the kinetic modes. Only in this case, the results are in excellent agreement with each other, as expected. Moreover, Brownian Dynamics is found to be much more efficient than lattice Boltzmann as long as the degree of polymerization is not excessively large.Comment: 11 figures, submitted to J. Chem. Phy

Dynamic crossover scaling in polymer solutions

The crossover region in the phase diagram of polymer solutions, in the regime above the overlap concentration, is explored by Brownian Dynamics simulations, to map out the universal crossover scaling functions for the gyration radius and the single-chain diffusion constant. Scaling considerations, our simulation results, and recently reported data on the polymer contribution to the viscosity obtained from rheological measurements on DNA systems, support the assumption that there are simple relations between these functions, such that they can be inferred from one another.Comment: 4 pages, 6 figures, 1 Table. Revised version to appear in Physical Review Letters. Includes supplemental material

What is the Entanglement Length in a Polymer Melt ?

We present results of molecular dynamics simulations of very long model polymer chains analyzed by various experimentally relevant techniques. The segment motion of the chains is found to be in very good agreement with the repatation model. We also calculated the plateau-modulus G_N. The predicitions of the entanglement length N_e from G_N and from the mean square displacements of the chains segments disagree by a factor of about 2.2(2), indicating an error in the prefactor in the standard formula for G_N. We show that recent neutron spin echo measurements were carried out for chain lengths which are too small for a correct determination of N_e.Comment: 5 pages, 4 figures, RevTe