An improved dissipative coupling scheme for a system of Molecular Dynamics particles interacting with a Lattice Boltzmann fluid

We consider the dissipative coupling between a stochastic Lattice Boltzmann (LB) fluid and a particle-based Molecular Dynamics (MD) system, as it was first introduced by Ahlrichs and D\"unweg (J. Chem. Phys. 111 (1999) 8225). The fluid velocity at the position of a particle is determined by interpolation, such that a Stokes friction force gives rise to an exchange of momentum between the particle and the surrounding fluid nodes. For efficiency reasons, the LB time step is chosen as a multiple of the MD time step, such that the MD system is updated more frequently than the LB fluid. In this situation, there are different ways to implement the coupling: Either the fluid velocity at the surrounding nodes is only updated every LB time step, or it is updated every MD step. It is demonstrated that the latter choice, which enforces momentum conservation on a significantly shorter time scale, is clearly superior in terms of stability and accuracy, and nevertheless only marginally slower in terms of execution speed. The second variant is therefore the recommended implementation.Comment: 16 pages, 6 figure

Translational Diffusion of Polymer Chains with Excluded Volume and Hydrodynamic Interactions by Brownian Dynamics Simulation

Within Kirkwood theory, we study the translational diffusion coefficient of a single polymer chain in dilute solution, and focus on the small difference between the short--time Kirkwood value $D^{(K)}$ and the asymptotic long--time value $D$. We calculate this correction term by highly accurate large--scale Brownian Dynamics simulations, and show that it is in perfect agreement with the rigorous variational result $D < D^{(K)}$, and with Fixman's Green--Kubo formula, which is re--derived. This resolves the puzzle posed by earlier numerical results (Rey {\em et al.}, Macromolecules 24, 4666 (1991)), which rather seemed to indicate $D > D^{(K)}$; the older data are shown to have insufficient statistical accuracy to resolve this question. We then discuss the Green--Kubo integrand in some detail. This function behaves very differently for pre--averaged vs. fluctuating hydrodynamics, as shown for the initial value by analytical considerations corroborated by numerical results. We also present further numerical data on the chain's statics and dynamics.Comment: submitted to Journal of Chemical Physic

Simulation of a Single Polymer Chain in Solution by Combining Lattice Boltzmann and Molecular Dynamics

In this paper we establish a new efficient method for simulating polymer-solvent systems which combines a lattice Boltzmann approach for the fluid with a continuum molecular dynamics (MD) model for the polymer chain. The two parts are coupled by a simple dissipative force while the system is driven by stochastic forces added to both the fluid and the polymer. Extensive tests of the new method for the case of a single polymer chain in a solvent are performed. The dynamic and static scaling properties predicted by analytical theory are validated. In this context, the influence of the finite size of the simulation box is discussed. While usually the finite size corrections scale as L^{-1} (L denoting the linear dimension of the box), the decay rate of the Rouse modes is only subject to an L^{-3} finite size effect. Furthermore, the mapping to an existing MD simulation of the same system is done so that all physical input values for the new method can be derived from pure MD simulation. Both methods can thus be compared quantitatively, showing that the new method allows for much larger time steps. Comparison of the results for both methods indicates systematic deviations due to non-perfect match of the static chain conformations.Comment: 17 pages, 12 figures, submitted to J. Chem. Phy

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

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

Energy-stable linear schemes for polymer-solvent phase field models

We present new linear energy-stable numerical schemes for numerical simulation of complex polymer-solvent mixtures. The mathematical model proposed by Zhou, Zhang and E (Physical Review E 73, 2006) consists of the Cahn-Hilliard equation which describes dynamics of the interface that separates polymer and solvent and the Oldroyd-B equations for the hydrodynamics of polymeric mixtures. The model is thermodynamically consistent and dissipates free energy. Our main goal in this paper is to derive numerical schemes for the polymer-solvent mixture model that are energy dissipative and efficient in time. To this end we will propose several problem-suited time discretizations yielding linear schemes and discuss their properties

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

Progress in the Understanding of the Fluctuating Lattice Boltzmann Equation

We give a brief account of the development of methods to include thermal fluctuations into lattice Boltzmann algorithms. Emphasis is put on our recent work (Phys. Rev. E 76, 036704 (2007)) which provides a clear understanding in terms of statistical mechanics.Comment: Conference paper for CCP 2008, submitted to Computer Physics Communication

The Cassie-Wenzel transition of fluids on nanostructured substrates: Macroscopic force balance versus microscopic density-functional theory

Classical density functional theory is applied to investigate the validity of a phenomenological force-balance description of the stability of the Cassie state of liquids on substrates with nanoscale corrugation. A bulk free-energy functional of third order in local density is combined with a square-gradient term, describing the liquid-vapor interface. The bulk free energy is parameterized to reproduce the liquid density and the compressibility of water. The square-gradient term is adjusted to model the width of the water-vapor interface. The substrate is modeled by an external potential, based upon Lennard-Jones interactions. The three-dimensional calculation focuses on substrates patterned with nanostripes and square-shaped nanopillars. Using both the force-balance relation and density-functional theory, we locate the Cassie-to-Wenzel transition as a function of the corrugation parameters. We demonstrate that the force-balance relation gives a qualitatively reasonable description of the transition even on the nanoscale. The force balance utilizes an effective contact angle between the fluid and the vertical wall of the corrugation to parameterize the impalement pressure. This effective angle is found to have values smaller than the Young contact angle. This observation corresponds to an impalement pressure that is smaller than the value predicted by macroscopic theory. Therefore, this effective angle embodies effects specific to nanoscopically corrugated surfaces, including the finite range of the liquid-solid potential (which has both repulsive and attractive parts), line tension, and the finite interface thickness. Consistently with this picture, both patterns (stripes and pillars) yield the same effective contact angles for large periods of corrugation.Comment: 13 pages 9 figure

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