Synergy between the crack pattern and substrate elasticity in colloidal deposits

Desiccation cracks in colloidal deposits occur to release the excess strain energy arising from the competition between the drying induced shrinkage of the deposit and its adhesion to the substrate. Here we report remarkably different morphology of desiccation cracks in the dried patterns formed by the evaporation of sessile drops containing colloids on elastomer (soft) or glass (stiff) substrates. The change in the crack pattern, i.e., from radial cracks on stiff substrates to circular cracks on soft substrates, is shown to arise solely due to the variation in elasticity of the underlying substrates. Our experiments and calculations reveal an intricate correlation between the desiccation crack patterns and the substrate's elasticity. The mismatch in modulus of elasticity between the substrate and that of the particulate deposit significantly alters the energy release rate during the nucleation and propagation of cracks. The stark variation in crack morphology is attributed to the tensile or compressive nature of the drying-induced in-plane stresses.

Figure 1

Top: Schematic displaying an aqueous sessile drop placed on a solid substrate. The nonuniform evaporative flux $J_{\mathrm{flux}}$, the solid substrate, and the three-phase contact line are marked. Bottom: Schematic of the annular particulate deposit with circular cracks formed after complete evaporation of the solvent.

Figure 2

Load vs. displacement plots measured for (a) glass substrate and (b) glass substrate coated with a PDMS layer. The slope S of the unloading part of the load-displacement curve, maximum penetration depth $h_{\max }$, contact depth $h_c$, residual depth after unloading $h_f$, and the maximum load $P_{\max }$ are indicated. The red dotted line, guide to the eye, shows the tangent to the unloading part of the curve drawn at ($P_{\max }$, $h_{\max }$). The estimated values of $E_s$ for glass and glass coated with PDMS elastomer are around 70 GPa and 6 MPa, respectively.

Figure 3

Optical microscopy images of the annular colloidal deposits exhibiting (a) circular cracks on soft substrate, (b) radial cracks on silane-coated stiff substrate, and (c) radial cracks on cleaned glass substrate. The crack patterns are consistently reproducible for at least 10 different trial experiments. The contact angle, ${\theta }_c$, subtended by a drop of water and the modulus of elasticity $E_s$ of various substrates are marked. The width, $w$, of the cracks in particulate deposits and the crumpled region of soft substrate are also marked.

Figure 4

(a) The optical microscopy image of the dried annular particulate deposit on a soft substrate with an accompanying crumpled layer (highlighted inside the dotted circle). (b) The corresponding height profile of the crumpled layer measured with an optical surface profile meter. The color bar indicates the height along the out-of-plane direction.

Figure 5

[(a)–(h)] Snapshots recorded during the last stages of the drying process. The snapshots are extracted from the video microscopy movie (see supplemental movie S1) recorded during the drying of a colloidal drop deposited on the soft substrate ($E_s\sim 6\mathrm{MPa}$). The morphological evolution of the crack patterns formed in the particulate deposit at $t_c$, the time at which colloidal dispersion transforms into a gel-like film and at various instant thereafter are shown. The width $w$ of the deposit, and the crack nucleating point A are marked. The direction of propagation of cracks is highlighted with arrows. The crumpling of the soft substrate is also marked by a dotted circle at the central region of the ringlike deposit.

Figure 6

[(a)–(h)] Snapshots show the propagation of cracks when colloidal silica dispersion's are dried on stiff substrates ($E_s\sim 70\mathrm{GPa}$). These snapshots at different drying times are extracted from the a movie recorded by video-microscopy movie (see supplemental movie S2). The morphological evolution of the crack patterns formed in the particulate deposit at various instant of time starting from $t_c$ at which colloids solidifies into a gel-like can be seen. The width $w$ of the deposit and the nucleating point of cracks are marked. The direction of crack propagation is indicated by arrows.

Figure 7

(a) Three-dimensional schematic representation of the annular particulate deposit on soft or stiff substrates. The colloidal deposit is considered as a coaxial cylinder of inner radius $r_1$ and outer radius $r_2$. The direction of in-plane stresses ${\sigma }_r$ and ${\sigma }_{\theta }$ are marked. The in-plane stresses ${\sigma }_r$, ${\sigma }_{\theta }$ as a function of scaled radial distance $r/r_2$ developed in the particulate deposited on the soft substrate and on the stiff substrate are shown respectively in (b) and (c). The arrows point to the inner boundary of the annular colloidal deposit.

Figure 8

(a) Schematic of an annular particulate deposit on a substrate. The atmospheric pressure is $P_o$ while the pressure in the pores of the particulate deposit is represented by $\mathrm{\Pi }$. (b) A section of the annular particulate deposit displaying evaporation along radial direction is depicted. The expression for the pressure ($\mathrm{\Pi }$) inside the pores in the particulate deposit is also displayed. The pressure is estimated at an instant when the particulate deposit is semisolid (gel like) and without cracks. The inner ($r_1$) and outer ($r_2$) radius of the particulate deposit, the width of the deposit ($w$) and the evaporative flux ($J_{\mathrm{flux}}$) are marked. (c) The variation of pressure $\mathrm{\Pi }$ with the scaled radial distance ($r/r_2$) at a given instant is depicted. It shows that the pressure at $r=r_2$ is maximum and increases nonlinearly as $r\to r_2$.