Computer Physics Communications 182 (2011) 2192–2200 Contents lists available at ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc Boundary condition considerations in lattice Boltzmann formulations of wetting binary fluids H.S. Wiklund a,∗ , S.B. Lindström b , T. Uesaka a a Fibre Science and Communication Network (FSCN), Mid Sweden University, SE-851 70 Sundsvall, Sweden b Department of Fibre and Polymer Technology, Royal Institute of Technology, SE-100 44 Stockholm, Sweden article info abstract Article history: Received 12 August 2010 Received in revised form 12 May 2011 Accepted 27 May 2011 Available online 14 June 2011 Keywords: Lattice Boltzmann Binary fluid Wetting surface Boundary condition We propose a new lattice Boltzmann numerical scheme for binary-fluid surface interactions. The new scheme combines the existing binary free energy lattice Boltzmann method [Swift et al., Phys. Rev. E 54 (1996)] and a new wetting boundary condition for diffuse interface methods in order to eliminate spurious variations in the order parameter at solid surfaces. We use a cubic form for the surface free energy density and also take into account the contribution from free energy in the volume when discretizing the wetting boundary condition. This allows us to eliminate the spurious variation in the order parameter seen in previous implementations. With the new scheme a larger range of equilibrium contact angles are possible to reproduce and capillary intrusion can be simulated at higher accuracy at lower resolution.  2011 Elsevier B.V. All rights reserved. 1. Introduction Numerical simulations of wetting binary fluids require, not only capturing the hydrodynamics of the fluid, but also tracking in time the interfaces between the different fluids. If a solid surface is in- troduced one must define the wetting properties between the fluid and the surface. In recent years the lattice Boltzmann method has been developed into an increasingly mature numerical scheme for studying non-ideal gases [1–4] and binary fluids [2,5,6]. An eval- uation of the three most common models has been presented by Huang et al. [7]. When simulating contact line motion it is necessary to imple- ment a boundary condition that defines the wetting properties between the two fluid phases and the solid. Depending on which lattice Boltzmann formulation is used to describe the fluid, im- plementations of such boundary conditions vary. The multi-phase model developed by Shan and Chen [2] has been extended to in- clude wetting boundaries and used to study capillary intrusion by Raiskinmäki [8], Chibbaro [9] and Diotallevi et al. [10,11] and droplets on super hydrophobic surfaces by Hyväluoma et al. [12]. Harting et al. [13] have developed a new approach based on the Shan–Chen model to investigate boundary slip in hydropho- bic micro-channels. One drawback with the Shan–Chen method is that the equilibrium contact angle cannot be known beforehand as a simulation input parameter, but instead must be obtained * Corresponding author. E-mail addresses: hanna.wiklund@miun.se (H.S. Wiklund), stefan.lindstroem@gmail.com (S.B. Lindström), tetsu.uesaka@miun.se (T. Uesaka). from the simulation itself. Recently, an approach, which is able to give an estimate of the static contact angle from the simulation parameters, has been proposed [14]. Swift et al. [15] developed the so-called free energy lattice Boltzmann method that gives the thermodynamically consistent equilibrium state. Based on the free energy lattice Boltzmann method, Desplat et al. [16] and Briant et al. [17] further developed the approach to simulate partial wetting and contact line motion in non-ideal gases and binary fluids. In their approach, a wetting boundary condition is incorporated us- ing the Cahn model [18], in which the surface free energy density function is expanded into a power series. For the representation of the surface free energy density function, Briant et al. [17] used a linear approximation whereas Desplat et al. [16] used a quadratic function. These models have been successfully employed in appli- cations, such as droplet spreading on heterogeneous surfaces [19] and capillary intrusion [20]. A linear or quadratic form for the surface free energy density function has originally been considered sufficient when studying partial wetting [16,17,21]. However, when assuming a linear form of the surface free energy density function, the order parameter has been found to vary exponentially from the surface to the bulk [21], creating a spurious variation of the order parameter at the solid surface. Although such a “film” can form on a surface at a molecular level [22], when modeling macroscopic flow with the lattice Boltzmann method, the interface width is already many or- ders of magnitude thicker than the real physical interface and any additional “film” is of non-physical nature. For the quadratic sur- face free energy, such a variation of the order parameter at the surface has not been reported in the literature, to the authors’ knowledge. Avoiding such unphysical variations is of particular 0010-4655/$ – see front matter  2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2011.05.019