Drying-induced stresses in poroelastic drops on rigid substrates

We develop a theory for drying-induced stresses in sessile, poroelastic drops undergoing evaporation on rigid surfaces. Using a lubrication-like approximation, the governing equations of three-dimensional nonlinear poroelasticity are reduced to a single thin-film equation for the drop thickness. We find that thin drops experience compressive elastic stresses but the total in-plane stresses are tensile. The mechanical response of the drop is dictated by the initial profile of the solid skeleton, which controls the in-plane deformation, the dominant components of elastic stress, and sets a limit on the depth of delamination that can potentially occur. Our theory suggests that the alignment of desiccation fractures in colloidal drops is selected by the shape of the drop at the point of gelation. We propose that the emergence of three distinct fracture patterns in dried blood drops is a consequence of a nonmonotonic drop profile at gelation. We also show that depletion fronts, which separate wet and dry solid, can invade the drop from the contact line and localize the generation of mechanical stress during drying. Finally, the finite element method is used to explore the stress profiles in drops with large contact angles.

Figure 1

The alignment of fractures is driven by the deposit profile. (a) The steady-state profile of a dried blood drop. Circles represent experimental data from Sobac and Brutin [9] and the solid line is a polynomial fit. The superimposed heat map illustrates the difference between the radial and hoop stresses $\mathcal{S}={\epsilon }^{-2}(1+\nu )({\sigma }_{rr}-{\sigma }_{\theta \theta })$. The black dashed line is the $\mathcal{S}=0$ level set. The three regions correspond to (I) a dominant radial stress (orthoradial fractures), (II) a dominant orthoradial stress (radial fractures), and (III) similar stresses (nonoriented fractures). (b) The deposit associated with the profile in panel (a), showing the three fracture patterns suggested by the asymptotic theory. This figure has been adapted from Sobac and Brutin [9].

Figure 2

The dynamics of a drying poroelastic drop at a moderate evaporation rate ($\mathcal{Q}=1$). The spatio-temporal evolution of the (a) fluid fraction (porosity), (b) drop thickness, (c) pressure, (d) total in-plane stress, (e) radial and (f) vertical traction on the substrate. The parameter values are $\nu =0.45$, ${\varphi }_0^f=0.80$, and $q\left({\varphi }^f\right)\equiv 1$. The solutions are shown at times $t=0.01$, 0.2, 0.5, 1, and 2. Arrows show the direction of time; the dashed black lines denote the steady states.

Figure 3

Moderate evaporation rates can lead to the hoop stress exceeding the radial stress in initially parabolic drops. The rescaled difference between the radial and hoop stress $\mathcal{S}={\epsilon }^{-2}(1+\nu )({\sigma }_{rr}-{\sigma }_{\theta \theta })$ defined by (40) is shown as a heat map at times (a) $t=0.01$, (b) $t=0.5$, and (c) $t=2$. The drop profile $h(r,t)$ is shown as the solid black line. The white circle in (b) depicts the position of the depletion front $r_f\left(t\right)$, defined by ${\varphi }^f(r_f\left(t\right),t)=0.01$. The parameter values are $\mathcal{Q}=1$, $\nu =0.45$, ${\varphi }_0^f=0.80$, and $q\left({\varphi }^f\right)=1$.

Figure 4

Drop dynamics with a small evaporation rate ($\mathcal{Q}=0.1$). In panels (a)–(c), the initial fluid fraction is ${\varphi }_0^f=0.80$. In panels (d)–(f), the initial fluid fraction is ${\varphi }_0^f=0.50$. In all panels, $\nu =0.45$, $q\left({\varphi }^f\right)\equiv 1$, and the solutions are shown at times $t=0.1$, 2, 5, 10, and 20. Arrows show the direction of time; the dashed black lines denote the steady states.

Figure 5

The (a) equilibrium drop thickness $h_{\infty }$ and (b) radial elastic stress along the substrate for different drop aspect ratios $\epsilon =H/R$. Symbols denote quantities computed using the finite element method. The solid black lines denote asymptotic solutions. The legend applies to both panels. Here, $\nu =0.3$ and $J=0.5$.

Figure 6

The (a) radial elastic stress, (b) orthoradial elastic stress, and (c) pressure in a dried poroelastic drop with large initial contact angle. The drop has shed half of its volume due to fluid loss ($J=0.50$). We set $\epsilon =0.80$ (${\phi }_0\simeq {58}^{\circ }$) and $\nu =0.30$.