Pep Español

Statistical Mechanics of Dissipative Particle Dynamics.

The stochastic differential equations corresponding to the updating algorithm of Dissipative Particle Dynamics (DPD), and the corresponding Fokker-Planck equation are derived. It is shown that a slight modification to the algorithm is required before the Gibbs distribution is recovered as the stationary solution to the Fokker-Planck equation. The temperature of the system is then directly related to the noise amplitude by means of a fluctuation-dissipation theorem. However, the correspondingly modified, discrete DPD algorithm is only found to obey these predictions if the length of the time step is sufficiently reduced. This indicates the importance of time discretisation in DPD.

Dissipative particle dynamics with energy conservation

Dissipative particle dynamics (DPD) does not conserve energy and this precludes its use in the study of thermal processes in complex fluids. We present here a generalization of DPD that incorporates an internal energy and a temperature variable for each particle. The dissipation induced by the dissipative forces between particles is invested in raising the internal energy of the particles. Thermal conduction occurs by means of (inverse) temperature differences. The model can be viewed as a simplified solver of the fluctuating hydrodynamic equations and opens up the possibility of studying thermal processes in complex fluids with a mesoscopic simulation technique.

Mori–Zwanzig formalism as a practical computational tool

An operational procedure is presented to compute explicitly the different terms in the generalized Langevin equation (GLE) for a few relevant variables obtained within Mori-Zwanzig formalism. The procedure amounts to introducing an artificial controlled parameter which can be tuned in such a way that the so-called projected dynamics becomes explicit and the GLE reduces to a Markovian equation. The projected dynamics can be realised in practice by introducing constraints, and it is shown that the Green-Kubo formulae computed with these dynamics do not suffer from the plateau problem. The methodology is illustrated in the example of star polymer molecules in a melt using their center of mass as relevant variables. Through this example, we show that not only the effective potentials, but also the friction forces and the noise play a very important role in the dynamics.

Boundary conditions in dissipative particle dynamics

We study a continuum model for simulating solid boundaries of a DPD fluid. A layer of frozen DPD particles is placed on the wall boundary and a continuum limit is taken. This produces effective dissipative and random forces. In addition, a reflection at the wall must be specified in order to confine the fluid. Three different wall reflection laws are studied and their effectiveness in producing stick boundary conditions and correct temperature distributions is assessed by simulation of a planar Couette flow.

Hamiltonian Adaptive Resolution Simulation for Molecular Liquids

Adaptive resolution schemes allow the simulation of a molecular fluid treating simultaneously different subregions of the system at different levels of resolution. In this work we present a new scheme formulated in terms of a global Hamiltonian. Within this approach equilibrium states corresponding to well-defined statistical ensembles can be generated making use of all standard molecular dynamics or Monte Carlo methods. Models at different resolutions can thus be coupled, and thermodynamic equilibrium can be modulated keeping each region at desired pressure or density without disrupting the Hamiltonian framework.

Boundary Models in DPD

We present a model for treating solid boundaries of a DPD fluid. The basic idea is to model the stick boundary conditions by assuming that a layer of DPD particles is stuck on the boundary. By taking a continuum limit of this layer effective dissipative and stochastic forces on the fluid DPD particles are obtained. The boundary model is tested by a simulation of planar Couette flow which allows the performance of vicosimetric measurements. We analyze the conditions that ensure a proper stick boundary condition for an impenetrable wall, comparing with previous methods used.

Incompressible smoothed particle hydrodynamics

We present a smoothed particle hydrodynamic model for incompressible fluids. As opposed to solving a pressure Poisson equation in order to get a divergence-free velocity field, here incompressibility is achieved by requiring as a kinematic constraint that the volume of the fluid particles is constant. We use Lagrangian multipliers to enforce this restriction. These Lagrange multipliers play the role of non-thermodynamic pressures whose actual values are fixed through the kinematic restriction. We use the SHAKE methodology familiar in constrained molecular dynamics as an efficient method for finding the non-thermodynamic pressure satisfying the constraints. The model is tested for several flow configurations.

Consistent scaling of thermal fluctuations in smoothed dissipative particle dynamics

Dissipative particle dynamics (DPD) as a model of fluid particles suffers from the problem that it has no physical scale associated with the particles. Therefore, a DPD simulation requires an ambiguous fine-tuning of the model parameters with the physical parameters. A corrected version of DPD that does not suffer from this problem is smoothed dissipative particle dynamics (SDPD) [P. Espanol and M. Revenga, Phys. Rev. E 67, 026705 (2003)]. SDPD is, in fact, a version of the well-known smoothed particle hydrodynamics method, albeit with the proper inclusion of thermal fluctuations. Here, we show that SDPD produces the proper scaling of the fluctuations as the resolution of the simulation is varied. This is investigated in two problems: the Brownian motion of a spherical colloidal particle and a polymer molecule in suspension.

Force autocorrelation function in Brownian motion theory

The force autocorrelation function of an infinitely massive Brownian particle is studied with a molecular dynamics simulation. The plateau time problem, the calculation of the friction coefficient, and the relationship between the stochastic and real force are discussed.

Large scale and mesoscopic hydrodynamics for dissipative particle dynamics

Decay rates and related transport coefficients of hydrodynamic disturbances in the isothermal dissipative particle dynamics (DPD) fluid depend strongly on how the probing wave length (in simulations: sizes of colloidal particles, polymers, pores, etc.) compares to the dynamic correlation length and to the range of the DPD forces. In this article the wave number dependent transport properties (dispersion relations) of the DPD fluid are calculated analytically using methods of kinetic theory, as a natural generalization of the work by Marsh et al. [Phys. Rev. E 56, 1976 (1997)]. The Navier–Stokes transport coefficients are recovered in the hydrodynamic limit of long wavelength disturbances.