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. Dissipative particle dynamics constitutes a valuable tool for mesoscopic simulations of complex fluids. It was introduced by Hoogerbrugge and Koelman [1], [2] and has received growing interest in view of its potentiality in the study of complex flow problems as those arising in porous flow [1], colloidal suspensions [2]-[4], polymer suspensions [5], or multicomponent flows [6]. The technique has received a considerable theoretical backup [7]-[11] that provides solid ground for its use. DPD faces, however, a conceptual problem in that energy is not conserved: the dissipative particles interact with dissipative forces that depend on the relative velocities between particles. For this reason, a DPD system cannot sustain a temperature gradient, as has been pointed out by Marsh et al. [10]. Nevertheless, in the picture where the dissipative particles are understood as droplets or mesoscopic clusters of atoms [1], [12] it is apparent that the dissipated energy due to friction must be invested into increasing the internal energy of the cluster. Therefore, we propose in this letter to introduce an additional variable i which is interpreted as the internal energy of each particle. Along this new variable we introduce an entropy variable si = s( i) and a temperature Ti = [∂si/∂ i] −1. The variation of the internal energy must involve two different processes. On the one hand, temperature differences between particles produce variations in the internal energies through “heat conduction”. On the other hand, the friction forces dissipate energy which is transformed into internal energy through “viscous heating”. Let us analyze each process separately. () E-mail: [email protected] c © Les Editions de Physique 632 EUROPHYSICS LETTERS Conduction. – Let us assume for a while that the N dissipative particles of the system are at rest at arbitrary positions ri. We formulate the following equation of motion for the internal energy of each particle: