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

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 one-component 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.

Figure 1

A schematic for the two-dimensional system in numerical simulations. Bottom: A semicircular liquid droplet of radius $R_0$ is placed on the flat rigid solid surface at $z=0$ and is surrounded by gas. Here, the two solid surfaces are parallel to the $xy$ plane. The system is closed by applying the periodic boundary condition in the $x$ direction.

Figure 2

Cosine of the static contact angle, $\cos {\theta }_s$, is plotted as a function of the wettability number $\mathcal{W}.$ The ${}^{*}$ symbols are measured by fitting the level curve of $n=(n_l+n_g)/2$ with a circular arc. Each symbol corresponds to a nearly equilibrium state of a droplet on a homogeneous solid substrate with the fixed temperature $T_w=0.875T_c$ and a constant value of $\mathcal{W}.$ Dashed line: A linear least squares fitting gives $\cos {\theta }_s=-6.23\mathcal{W}.$

Figure 3

Time variation in the migration velocity $V_{\mathrm{mig}}.$ Three different temperatures are used for the bottom substrate with the cooled substrate: dots for $T_w=0.870T_c$, the ordinary substrate: stars for $T_w=0.875T_c$, and the heated substrate: crosses for $T_w=0.890T_c$.

Figure 4

The $+x$ direction: level curves of $n=(n_l+n_g)/2$ for droplets moving on solid substrates in the direction of increasing wettability. (a) The hydrophilic case with ${\theta }_s<{90}^{\circ }.$ (b) The hydrophobic case with ${\theta }_s>{90}^{\circ }.$ In both cases, $R_0=25\ell$, $\frac{d\cos {\theta }_s}{dx}\approx 1.5\times {10}^{-3}/\ell$, $L{s}_l=2.0$, and the ordinary substrate: $T_w=0.875T_c$ are used. Here, the droplet height $h_0$ is defined along the middle line, and $R_c$ is designated as the contact radius.

Figure 5

$V_{\mathrm{mig}}$ is plotted as a function of $h_0\frac{\gamma }{\eta }\frac{d}{dx}\cos {\theta }_s$ for the circles: hydrophilic and for the squares: hydrophobic cases. In both cases, $R_0=25\ell$, $L{s}_l=2.0$, and the ordinary substrate $T_w=0.875T_c$ are used. A linear least squares fitting gives the proportionality coefficient ${\alpha }_V\approx 0.16,$dashed line: for the hydrophobic case and ${\alpha }_V\approx 0.33,$solid line: for the hydrophilic case.

Figure 6

The tangential velocity $v_x$, measured on the middle line of the droplet, plotted as a function of $z$ at $t=2000{\tau }_0$. The dashed line represents the hydrophobic case (${\theta }_s>{90}^{\circ }$) and the solid line represents the hydrophilic case (${\theta }_s<{90}^{\circ }$)with $R_0=25\ell$, $\frac{d\cos {\theta }_s}{dx}\approx 1.5\times {10}^{-3}/\ell$, $L{s}_l=2.0$, and the ordinary substrate: $T_w=0.875T_c$.

Figure 7

Dependence of the coefficient ${\alpha }_V$ on (a) the nondimensionalized initial droplet radius $R_0/\ell$ for $l{s}_l/\ell =2.0$, (b) the nondimensionalized slip length of the liquid $L{s}_l=l{s}_l/\ell$ for $R_0/\ell =25.0$, and (c) the ratio $l{s}_l/R_0,$ evaluated for the ordinary substrate: $T_w=0.875T_c$. A linear least squares fitting gives the solid line for the hydrophilic case and the dashed line for the hydrophobic case in (c). Here, $l{s}_l$ is the slip length in the liquid phase, given by $l{s}_l=\nu m{n}_l/{\beta }_l.$

Figure 8

Color: density and arrow: velocity fields for droplets on solid substrates with and without wettability gradients. Each figure is labeled by a letter (a), (b), or (c) for the substrate temperature at $z=0$ and a number (1) or (2) indicating whether the wettability gradient is 1 for a stationary case: off or 2 for a moving case: on. The substrate temperature at $z=0$ is set to be the cooled substrate: (a) $0.870T_c$, (b) the ordinary substrate: $0.875T_c$, and (c) the heated substrate: $0.890T_c.$ The wettability number $\mathcal{W}$ at the bottom fluid-solid interface $z=0$ is set to be (1) a homogeneous constant $\mathcal{W}=0.1$ for the three figures in the left column and (2) a function linear in $x$, varying from $\mathcal{W}=0.1$ at $x=0$ to $\mathcal{W}=0.04$ at $x=L_x=250\ell$ for the three figures in the right column. Here, the dimensionless slip length $L{s}_l=2.0$ is used.

Figure 9

$\nabla ·\mathbf{v}$ at $z=0$ plotted as a function of $x$ along the solid substrate. Each figure is labeled by a letter (a), (b), or (c) for the substrate temperature at $z=0$ and a number (1) or (2) indicating whether the slip length is (1) large or (2) small. The substrate temperature at $z=0$ is set to be (a) the cooled substrate: $0.870T_c$, (b) the ordinary substrate: $0.875T_c$, and (c) the heated substrate: $0.890T_c$. In each figure, the dashed line represents the wettability gradient off: stationary case and the solid line represents the wettability gradient on: moving cases.

Figure 10

$V_{\mathrm{slip}}/V_0$ plotted as a function of $x$ along the solid substrate. Each figure is labeled by a letter (a), (b), or (c) for the substrate temperature at $z=0$ and a number (1) or (2) indicating whether the slip length is (1) large or (2) small. The substrate temperature at $z=0$ is set to be (a) the cooled substrate: $0.870T_c$, (b) the ordinary substrate: $0.875T_c$, and (c) the heated substrate $0.890T_c.$ In each figure, the dashed line represents the wettability gradient off: stationary case and the solid line represents the wettability gradient on: moving cases. Figures here have a one-to-one correspondence with those in Fig. 9.