Influence of contact angle on slow evaporation in two-dimensional porous media

We study numerically the influence of contact angle on slow evaporation in two-dimensional model porous media. For sufficiently low contact angles, the drying pattern is fractal and can be predicted by a simple model combining the invasion percolation model with the computation of the diffusive transport in the gas phase. The overall drying time is minimum in this regime and is independent of contact angle over a large range of contact angles up to the beginning of a transition zone. As the contact angle increases in the transition region, the cooperative smoothing mechanisms of the interface become important and the width of the liquid gas interface fingers that form during the evaporation process increases. The mean overall drying time increases in the transition region up to an upper bound which is reached at a critical contact angle ${\theta }_c$. The increase in the drying time in the transition region is explained in relation with the diffusional screening phenomenon associated with the Laplace equation governing the vapor transport in the gas phase. Above ${\theta }_c$ the drying pattern is characterized by a flat traveling front and the mean overall drying time becomes independent of the contact angle. Drying time fluctuations are studied and are found to be important below ${\theta }_c$, i.e., when the pattern is fractal. The fluctuations are of the same order of magnitude regardless of the value of contact angle in this range. The fluctuations are found to die out abruptly at ${\theta }_c$ as the liquid gas interface becomes a flat front.

Figure 1

Example of phase distribution during drying in a model porous medium made of a monolayer of $1\mathrm{mm}$ glass beads randomly distributed between two glass plates. Vapor escapes from the four sides of model porous medium: (a) $\theta \sim 30°–40°$, (b) $\theta \sim 105°–107°$. The black areas correspond to liquid saturated regions. Glass beads forming the porous medium are visible in the regions (in gray) invaded by the gas phase.

Figure 2

Model porous medium formed by a regular array of disks of random diameter. Distribution of fluids at $t=0$ (invading fluid in gray, displaced fluid in white). The interface is visible as a series of arcs joining the bottom row of disks. Vapor escapes from bottom edge of system.

Figure 3

(Color online) Schematic example of arcs location before and after invasion of one pore and definition of contact angle.

Figure 4

(Color online) Probability (in percent) of interface local growth mechanisms as a function of contact angle (in degrees) in the displaced fluid.

Figure 5

(Color online) Growth of interfacial arcs during the invasion. The arrows indicate the meniscus displacement direction. Local invasion events: (a) burst, (b) touch, (c) and (d) arc coalescence (overlap).

Figure 6

(Color online) Mean overall drying time $⟨t_0⟩$ as a function of contact angle; $t_{\mathit{ref}}=t_{oD}\left({\theta }_c\right)$, i.e., the overall drying for system $D$ for $\theta ⩾{\theta }_c$. The inset shows the evolution of $⟨t_0⟩∕t_o\left({\theta }_c\right)$, where $t_o\left({\theta }_c\right)$ is the overall drying for the considered system for $\theta ⩾{\theta }_c$.

Figure 7

Example of drying patterns (liquid phase in dark gray, gas phase in light gray) in the transition region for one realization of a $50\times 50$ disk array (system $D$). Value of contact angle is indicated at top of each vertical series of patterns; the numbers of invaded pores from top row to bottom row are 1250, 1500, 1750, and 2000, respectively. The five vapor isoconcentration lines shown together with each pattern correspond to $P_v∕P_{vs}=0.2$ (the closest to the porous medium surface), 0.4, 0.6, 0.8, and 0.9995 (the farthest from the porous medium surface), respectively.

Figure 8

(Color online) Equivalent flat front position as a function of mean overall saturation for various values of contact angle.

Figure 9

(Color online) Evolution of $\frac{t_S\left({\theta }_c\right)-{⟨t_S⟩}_{\theta }}{t_o\left({\theta }_c\right)}$ as a function of average overall saturation. $t_0\left({\theta }_c\right)$ is the overall drying time for $\theta ={\theta }_c$. ${⟨t_S⟩}_{\theta }$ and $t_S\left({\theta }_c\right)$ are the mean drying time needed to reach a given average saturation $S$ for $\theta ={\theta }_c$ and the drying time to reach $S$ when $\theta ={\theta }_c$, respectively.

Figure 10

(Color online) Standard deviation $\sigma$ of overall drying time as a function of contact angle in the transition region for different porosity.

Figure 11

Evolution of overall drying time standard deviation near ${\theta }_c$ $(\sim 101°)$. The inset shows the variation of $\sigma$ over the transition zone.

Figure 12

(Color online) Probability density function (PDF) of overall drying time for four values of contact angle (corresponding to symbols on the curve in the inset, which shows the evolution of mean overall drying time as a function of contact angle in the transition region for $15\times 15$ disk systems).

Figure 13

(Color online) Evolution of standard deviation of overall liquid saturation ${\sigma }_S$ as a function of average saturation $⟨S⟩$. The inset shows the evolution of $⟨S⟩$ as a function of time.