Model of spontaneous droplet transport on a soft viscoelastic substrate with nonuniform thickness

Dynamic wetting of droplets on soft solids has many industrial and biological applications which require an understanding of the underlying fluid transport mechanism. Here we study the case of a droplet on a viscoelastic substrate of variable thickness which is known to give rise to a spontaneous droplet transport. This phenomenon is known as droplet durotaxis and has been observed experimentally. Here we develop a model assuming a small linear gradient in substrate thickness to reveal the physical mechanism behind this transport phenomena. We show the variable thickness causes an asymmetric deformation along the drop contact line, which causes a variation in the contact angle. This generates a net driving force on the drop, causing it to move in the direction of higher thickness. The resulting drop velocity is determined by balancing the work done by the moving drop with the viscoelastic dissipation of the substrate (viscoelastic braking) and computed from a self-consistent model. We find our results to be in qualitative agreement to previously reported experimental findings.

Figure 1

Orientation of the triangular cusp formed at the tip of the wetting ridge. Macroscopic solid-liquid contact angle $\alpha$ changes as the cusp rotates through a tilt angle $\varphi$ which depends upon the elastic response of the solid substrate. The solid angle formed inside the cusp is denoted as ${\theta }_s$.

Figure 2

Definition sketch showing a two-dimensional liquid droplet of radius $R$ in contact with the free surface of a solid substrate with a variable thickness $-h+\kappa x$. The drop applies surface tension forces on the solid at the contact line $x=R$ and bulk fluid pressure along the length of the droplet

Figure 3

Vertical deformation $u_z$ of a soft substrate at the liquid-solid interface $z=0$ as it depends upon (a) thickness gradient $\kappa$ ($\mathrm{\Lambda }=1,{\mathrm{\Gamma }}_s=1,{\mathrm{\Gamma }}_l=2,v=0$) and (b) dimensionless velocity $v$ ($\mathrm{\Lambda }=1,{\mathrm{\Gamma }}_s=1,{\mathrm{\Gamma }}_l=2,\kappa =0.3,n=0.5$).

Figure 4

(a) Advancing (${\alpha }_a$) and receding (${\alpha }_r$) receding contact angles of the drop in degrees plotted against the aspect ratio $\mathrm{\Lambda }$. Here $\kappa =0.25$. (b) Dimensionless driving force $\mathrm{\Delta }cos\alpha$ against $\mathrm{\Lambda }$ for different thickness gradient $\kappa$. In both figures, $v=0,{\mathrm{\Gamma }}_s=0.5,{\mathrm{\Gamma }}_l=2$.

Figure 5

Driving force $\mathrm{\Delta }cos\alpha$ plotted against the elastocapillary number ${\mathrm{\Gamma }}_l$ with different aspect ratios $\mathrm{\Lambda }$. Here, $\nu =0.5,{\mathrm{\Gamma }}_s=0.5{\mathrm{\Gamma }}_l,\kappa =0.3,v=0$.

Figure 6

Driving force $\mathrm{\Delta }cos\alpha$ against scaled velocity $v$ for different power-law exponent $n$. Here, $\mathrm{\Lambda }=1,{\mathrm{\Gamma }}_s=2,{\mathrm{\Gamma }}_l=5,\kappa =0.4$.

Figure 7

(a) Dimensionless velocity $v$ plotted against aspect ratio $\mathrm{\Lambda }$ with different thickness gradient $\kappa$. Here, ${\mathrm{\Gamma }}_s=1,{\mathrm{\Gamma }}_l=2,n=0.6$. (b) Velocity in $\mu \mathrm{m}/\mathrm{s}$ plotted against relaxation timescale $\tau$ with different power-law exponent $n$. Here, $\mathrm{\Lambda }=1,{\mathrm{\Gamma }}_s=3,{\mathrm{\Gamma }}_l=6,\kappa =0.4,R=10\mu \text{m}$.

Figure 8

Solid angle ${\theta }_s$ plotted against the scaled velocity $v$ with different elastocapillary numbers ${\mathrm{\Gamma }}_s$. Here, $\mathrm{\Lambda }=1,{\mathrm{\Gamma }}_s=0.5{\mathrm{\Gamma }}_l,n=0.5,\kappa =0.3$.