A lattice Boltzmann model for coupled diffusion

Diffusion coupling between different chemical components can have significant effects on the distribution of chemical species and can affect the physico-chemical properties of their supporting medium. The coupling can arise from local electric charge conservation for ions or from bound components forming compounds. We present a new lattice Boltzmann model to account for the diffusive coupling between different chemical species. In this model each coupling is added as an extra relaxation term in the collision operator. The model is tested on a simple diffusion problem with two coupled components and is in excellent agreement with the results obtained through a finite difference method. Our model is observed to be numerically very stable and unconditional stability is shown for a class of diffusion matrices. We further develop the model to account for advection and show an example of application to flow in porous media in two dimensions and an example of convection due to salinity differences. We show that our model with advection loses the unconditional stability, but offers a straightforward approach to complicated two-dimensional advection and coupled diffusion problems. Diffusion processes play a major role in the transport of heat and chemical components in nature. They have been shown to play an important role in the mass balance and element distribution in lakes and oceans [24,31] as well as the composition of minerals at the crystal scale. When multiple chemical components diffuse simultaneously, coupling of each component can arise because of local charge conservation for the case of ions or because the different components interact through bonds to form larger diffusing compounds. Physical and numerical models for multicomponent diffusion must account for these coupling effects. The multicomponent diffusion problem is usually described by a diffusion matrix where the non-diagonal terms represent the coupling interaction between the different diffusing species. The determination of the non-diagonal terms of the diffusion matrix is difficult as it depends on the collective behavior of all diffusing species present. The general approach used to determine these coefficients is semi-empirical, where the theoretical basis is described by Onsager's theory for non-equilibrium thermodynamics [28,29,25–27]. The non-diagonal terms of the diffusion matrix can be related to Onsager's phenomenological coefficients, however explicit relationships require experimental measurements [26,16,17,8,9]. Various multicomponent transport models have been developed for a macroscopic description of the transport equations using finite differences [8], finite volume and finite element methods [23]. However, these methods prove to be challenging …