Multicomponent and multiphase modeling and simulation of reactive wetting

A multicomponent and multiphase model with fluid motion is developed. The model is used to study reactive wetting in the case where concentration change of the spreading liquid and the substrate occurs. With the introduction of a Gibbs energy functional, the governing equations are derived, consisting of convective concentration and phase-field equations which are coupled to the Navier-Stokes equations with surface tension forces. The solid substrate is modeled hydrodynamically with a very high viscosity. Arbitrary phase diagrams, surface energies, and typical dimensionless numbers are some input parameters into the model. An axisymmetric model with an adaptive finite element method is utilized. Numerical simulations were done revealing two stages in the wetting process. First, the convection-dominated stage where rapid spreading occurs. The dynamics of the wetting is found to match with a known hydrodynamic theory for spreading liquids. Second, the diffusion-dominated stage where we observed depression of the substrate-liquid interface and elevation of the contact line region.

Figure 1

The axisymmetric model where each phase $\alpha$, $\beta$, and $\gamma$ contains a fraction of substitutional elements $A$, $B$, and $C$. The governing equations are expressed in cylindrical coordinates $\left(r,z\right)$. The angle ${\theta }_a$ is defined as the apparent contact angle and the $\sigma$’s are interfacial energies. The triple junction is not assumed or forced to remain at the initial substrate height although it stays there during the early stage of spreading.

Figure 2

The adaptive mesh at two different times with a superimposed contour plot of the phase-field variables.

Figure 3

(Color online) Calculated phase diagram of an arbitrary $A-B-C$ system at 1527 °C. The three crosses and dots represent the initial composition in the $\alpha ,\beta$, and $\gamma$ phases used for the EIC and OIC cases, respectively.

Figure 4

(Color online) The contour plots (level 0.5) of the phase-field variables for the EIC case with (a) SE1, (b) SE2, and (c) SE3, at time $t=0,8,15,100$ (marked 0,1,2,3), respectively.

Figure 5

Cutoff velocity profile of the EIC case with surface energy SE1 (a) at $t=1$ with $u_{\text{max}}=0.1135$ and (b) at $t=100$ with $u_{\text{max}}=0.0015$.

Figure 6

Apparent contact angle vs dimensionless time for EIC and OIC with surface energy SE1.

Figure 7

The evolution of the drop base radius in time for EIC and OIC with surface energies SE1, SE2, SE3.

Figure 8

Dynamics of the first stage of spreading with EIC and OIC compared to Cox’s theory with viscosity ratio ${\mu }_{\gamma }/{\mu }_{\alpha }=1$ [42].

Figure 9

(Color) Concentration profiles of the $A$ atoms for EIC [(a),(c),(e)] and OIC [(b),(d),(f)] at times $t=0,10$, and 175, respectively. Isolines are plotted in (d) and (f) with $A$ isoconcentrates 0.50, 0.55, 0.60, 0.65, and 0.70 (out to inwards).

Figure 10

(Color) Concentration profiles of the $B$ atoms for EIC [(a),(c),(e)] and OIC [(b),(d),(f)] at times $t=0,10$, and 175, respectively. Isolines are plotted in (d) and (f) with $B$ isoconcentrates 0.10, 0.15, and 0.20 (top to bottom).