Droplet motion in one-component fluids on solid substrates with wettability gradients

Abstract

CitationDroplet motion in one-component fluids on solid substrates with wettability gradients 2012, 85 (5) Physical Review E Droplet motion on solid substrates has been widely studied not only because of its importance in fundamental research but also because of its promising potentials in droplet-based devices developed for various applications in chemistry, biology, and industry. In this paper, we investigate the motion of an evaporating droplet in onecomponent fluids on a solid substrate with a wettability gradient. As is well known, there are two major difficulties in the continuum description of fluid flows and heat fluxes near the contact line of droplets on solid substrates, namely, the hydrodynamic (stress) singularity and thermal singularity. To model the droplet motion, we use the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)] for the hydrodynamic equations in the bulk region, supplemented with the boundary conditions at the fluid-solid interface. In this continuum hydrodynamic model, various physical processes involved in the droplet motion can be taken into account simultaneously, e.g., phase transitions (evaporation or condensation), capillary flows, fluid velocity slip, and substrate cooling or heating. Due to the use of the phase field method (diffuse interface method), the hydrodynamic and thermal singularities are resolved automatically. Furthermore, in the dynamic van der Waals theory, the evaporation or condensation rate at the liquid-gas interface is an outcome of the calculation rather than a prerequisite as in most of the other models proposed for evaporating droplets. Numerical results show that the droplet migrates in the direction of increasing wettability on the solid substrates. The migration velocity of the droplet is found to be proportional to the wettability gradients as predicted by Brochard [Langmuir 5, 432 (1989)]. The proportionality coefficient is found to be linearly dependent on the ratio of slip length to initial droplet radius. These results indicate that the steady migration of the droplets results from the balance between the (conservative) driving force due to the wettability gradient and the (dissipative) viscous drag force. In addition, we study the motion of droplets on cooled or heated solid substrates with wettability gradients. The fast temperature variations from the solid to the fluid can be accurately described in the present approach. It is observed that accompanying the droplet migration, the contact lines move through phase transition and boundary velocity slip with their relative contributions mostly determined by the slip length. The results presented in this paper may lead to a more complete understanding of the droplet motion driven by wettability gradients with a detailed picture of the fluid flows and phase transitions in the vicinity of the moving contact line.

Thermal singularity and droplet motion in one-component fluids on solid substrates with thermal gradients

“…Through this study, the thermal singularity at small scales will be coupled with the droplet motion at large scales. We carry out numerical simulations by employing the DVDWT [26,27] supplemented with the boundary conditions at the fluid-solid interface [22,25]. Our numerical results show that through strong evaporation or condensation in the vicinity of the contact line, the thermal singularity makes the contact angle increase with the increasing substrate temperature.…”

“…Though there has been extensive work on the subject, many aspects of this seemingly simple phenomenon remain issues of interest to fundamental research. In the present work, we focus on the hydrodynamics of liquid droplets in one-component liquid-gas systems on heated or cooled substrates [16][17][18][19][20][21][22] and substrates with temperature gradients [5,23]. Here the major challenges lie in the complicated dynamics at the intersection of the free (liquid-gas) interface with the solid substrate, i.e., the threephase contact line.…”

“…Therefore, a temperature discontinuity is inevitable if the two isothermal interfaces are of different temperatures and intersect at the contact line [22,26,27]. To resolve the stress and thermal singularities, numerous models have been proposed for various systems [16,[19][20][21][22][25][26][27][29][30][31][32][33][34][35][36][37][38][39][40]. In particular, quite a few diffuse-interface models have been developed to describe the contact line motion using mechanisms missing from the sharp-interface treatments in fluid mechanics.…”

“…Physically, this continuum hydrodynamic model is closely related to the so-called model "H" which was originally devised to describe the critical dynamics of thermal fluctuations [41,42]. Supplemented with the hydrodynamic boundary conditions derived in our previous works [22,25], this model can be used to fully take into account the various physical processes involved in the contact line dynamics, including phase transition (evaporation or condensation) and capillary flow in the bulk fluid region [26,27], boundary slip of fluid [25,39,40,43], temperature slip across the fluid-solid interface [44,45], and mechanical-thermal cross coupling at the fluid-solid interface [46]. Due to the use of diffuse-interface method, the stress and thermal singularities are resolved automatically.…”

“…In one-component two-phase (liquid-gas) systems, the contact line can move through the mass transport across the liquid-gas interface [16,22,25,[32][33][34][35], while in two-component (binary) fluids, the contact line can move through the diffusive transport across the fluid-fluid interface [36][37][38][39][40]. Besides these mechanisms that are inherent in the phase field modeling, boundary slip of fluid velocity can be incorporated in diffuse-interface models as well [22,25,39,40]. In addition, the stress singularity can be removed if the (nominal) contact line is preceded by a thin precursor film [1,2].…”