Imbibition and evaporation of droplets of colloidal suspensions on permeable substrates

When evaporated on a substrate, a droplet containing solutes usually deposits the solutes onto the substrate in a coffee-ring pattern. Recent experiments have shown that substrate permeability can suppress the coffee-ring pattern and promote more uniform solute deposition. Motivated by these observations, we have developed a lubrication-theory-based model to describe imbibition and evaporation of droplets of colloidal suspensions on permeable substrates. The model consists of a system of one-dimensional partial differential equations accounting for the changing droplet shape and depth-averaged concentration of colloidal particles. We also incorporate a precursor film, disjoining pressure, and substrate topography to control contact-line motion of the droplet. Solvent evaporation is described using the well-known one-sided model, and imbibition of solvent by the substrate is assumed to only depend on the excess pressure on the liquid side. The governing equations are solved with finite-difference methods. Our results reveal that solvent evaporation and solvent imbibition have the same qualitative effect on the final particle deposition pattern. For the case where the substrate is smooth, we find that increasing imbibition or evaporation leads to a transition from a cone-shaped deposition pattern to a ring-shaped deposition pattern. For the case where the substrate is rough, the droplet contact line is pinned at a defect on the substrate, and the pinning-depinning transition leads to the bull's-eye deposition pattern often observed in experiments. Finally, we also find that particle adsorption onto the substrate can promote more uniform particle deposition patterns for both smooth and rough substrates, and solvent imbibition can indirectly suppress the coffee-ring pattern by inducing more particle adsorption.

Figure 1

(a) Droplet on top of a permeable substrate. A precursor film is present near the droplet edge and the substrate may have topographical features. (b) Cross section of the system. Pores inside the substrate are saturated with liquid.

Figure 2

(a) Height profile and evaporation flux profile of a pure-solvent droplet on an impermeable substrate. The profiles are recorded at $t=100$. The evaporation flux is highest near the contact line. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $E={10}^{-4}$, $I=0$, $b=8.18\times {10}^{-4}$, $K=2\times {10}^{-1}$, and $\delta ={10}^{-3}$. (b) Height profile and imbibition flux profile of a nonvolatile pure-solvent droplet. The profiles are recorded at $t=100$. The imbibition flux is relatively uniform inside the droplet. All parameters are the same as in (a) except for $E=0$, $I={10}^{-4}$, and $b=1.0\times {10}^{-3}$.

Figure 3

(a) Time evolution of the droplet radius of a pure-solvent droplet on a smooth substrate for ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$). For $E={10}^{-4}$ (evaporation only), $b=8.18\times {10}^{-4}$, $K=2\times {10}^{-1}$, and $\delta ={10}^{-3}$. For $I={10}^{-4}$ (imbibition only), $b={10}^{-3}$. (b) Time evolution of the droplet radius of a pure-solvent droplet on a rough substrate. The parameters are the same as in (a), except now the substrate has a Gaussian defect centered at $r=1.1$.

Figure 4

Droplet containing colloidal particles imbibed on a smooth substrate. (a) Time evolution of the droplet shape. (b) Time evolution of the particle concentration. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, $\text{Pe}={10}^{-4}$, and $c_0={10}^{-3}$. The profiles are recorded at $t_1=469$, $t_2=939$, $t_3=1409$, and $t_4=2338$.

Figure 5

Final profiles for particle concentration (triangles) and area density (squares). The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, $\text{Pe}={10}^{-4}$, and $c_0={10}^{-3}$.

Figure 6

Final particle area densities when (a) the evaporation rate is increased while imbibition is suppressed and (b) the imbibition rate is increased while evaporation is suppressed. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$) and (a) $b=8.18\times {10}^{-4}$, $K=2\times {10}^{-1}$, $\delta ={10}^{-3}$, $\text{Pe}={10}^{-4}$, and $c_0={10}^{-3}$ and (b) $b={10}^{-3}$.

Figure 7

Phase diagram of the final particle deposition pattern.

Figure 8

Early time evolution of a droplet laden with particles imbibed on a rough substrate with a Gaussian defect centered at $r=1.1$. (a) Time evolution of the droplet height $h$. (b) Time evolution of the particle concentration $c$. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, and $c_0=3\times {10}^{-3}$. The profiles are recorded at $t_1=469$, $t_2=1409$, $t_3=2349$, and $t_4=3289$.

Figure 9

Late time evolution of a droplet carrying particles being imbibed on a rough substrate with a Gaussian defect centered at $r=1.1$. (a) Time evolution of the droplet height $h$. (b) Time evolution of the particle concentration $c$. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, $\text{Pe}={10}^{-4}$, and $c_0=3\times {10}^{-3}$. The profiles are recorded at $t_1=3486$, $t_2=3505$, and $t_3=3523$ (labeled 1, 2, and 3, respectively).

Figure 10

Depinning of the contact line results in nonuniform bull's-eye patterns. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $\text{Pe}={10}^{-4}$, and $c_0=3\times {10}^{-3}$. For the case with only evaporation (solid lines), $b=8.18\times {10}^{-4}$, $K=2\times {10}^{-1}$, and $\delta ={10}^{-3}$. For the case with only imbibition (dashed lines), $b={10}^{-3}$.

Figure 11

Time evolution of (a) the bulk particle area density and (b) the adsorbed particle area density for a droplet imbibed by a smooth substrate. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, $\text{Pe}={10}^{-4}$, $c_0={10}^{-3}$, and $\alpha ={10}^{-3}$. The profiles are recorded at $t_1=469$, $t_2=939$, $t_3=1409$, $t_4=1879$, and $t_5=2379$.

Figure 12

Particle area densities resulting from imbibition of a droplet of a colloidal suspension on a smooth substrate. The final particle area density becomes more uniform as particle adsorption increases. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, $\text{Pe}={10}^{-4}$, and $c_0={10}^{-3}$.

Figure 13

Particle area densities resulting from evaporating a droplet of a colloidal suspension on a rough substrate with a defect centered at $r=1.1$. The final particle area density becomes more uniform as particle adsorption increases. The values of the parameters are ${\theta }_e=9.{42}^{\circ }$ ($A_1=3\times {10}^3$ and $A_2={10}^{-3}$), $I={10}^{-4}$, $b={10}^{-3}$, $\text{Pe}={10}^{-4}$, and $c_0=3\times {10}^{-3}$.

Figure 14

Contact-line region of a droplet (a) near a topographical defect on a substrate and (b) on a perfectly smooth substrate.