Criterion for Fingering Instabilities in Colloidal Gels

We sandwich a colloidal gel between two parallel plates and induce a radial flow by lifting the upper plate at a constant velocity. Two distinct scenarios result from such a tensile test: (i) stable flows during which the gel undergoes a tensile deformation without yielding, and (ii) unstable flows characterized by the radial growth of air fingers into the gel. We show that the unstable regime occurs beyond a critical energy input, independent of the gel’s macroscopic yield stress. This implies a local fluidization of the gel at the tip of the growing fingers and results in the most unstable wavelength of the patterns exhibiting the characteristic scalings of the classical viscous fingering instability. Our work provides a quantitative criterion for the onset of fingering in colloidal gels based on a local shear-induced yielding in agreement with the delayed failure framework.

Figure 1

(a)–(d) Patterns obtained at a lift velocity $v_l=200\mu \mathrm{m}/\mathrm{s}$ for increasing initial gap thicknesses $h_0=100$, 300, 500, and $900\mu \mathrm{m}$ in tensile tests performed with a 8% wt. carbon black gel placed between two parallel plates of radius $R=20\mathrm{mm}$. (e) Stability diagram: radial velocity $v_r$ versus initial gap thickness $h_0$. Closed symbols denote stable conical patterns, open symbols denote unstable fingering patterns. The data points corresponding to the four images are indicated as triangles. The red line denotes the critical velocity $v_r^{*}$ separating the stable from the unstable regime. Inset: $v_r^{*}/h_0$ versus the inverse of the yield stress ${\sigma }_c^{-1}$ for carbon black gels at 4%, 6%, 8%, and 10% wt. ${\sigma }_c$ is determined as the crossover of $G^{\prime }$ and $G^{\prime \prime }$ during a stress sweep from 1 to 100 Pa (frequency $f=1\mathrm{Hz}$, waiting time of 5 s per point).

Figure 2

Energy input $E$ associated with the separation of two plates sandwiching a 8% wt. carbon black gel initially separated by $h_0$. Closed symbols denote stable patterns, open symbols denote unstable patterns. The red line defines the critical energy $E^{*}$ beyond which fingers grow. $E^{*}$ exhibits a power law with $h_0$, with an exponent $d_f/3$, where $d_f=2.1$. Inset: critical energy per particle $E^{*}/(h_0/a{)}^{d_f/3}$ versus yield stress for carbon black gels at 4%, 6%, 8%, and 10% wt.

Figure 3

(a) Normal force $F_N$ versus gap thickness $h$ during a tensile test ($v_l=200\mu \mathrm{m}/\text{s}$). The two curves correspond to $h_0=200\mu \mathrm{m}$ resulting in viscous fingering (open symbols) and $h_0=1000\mu \mathrm{m}$ resulting in a stable conical deposit (closed symbols). The blue lines denote the power-law exponents of the first relaxation step: $\alpha =5$ for $h_0=200\mu \mathrm{m}$ and $\alpha =2.5$ for $h_0=1000\mu \mathrm{m}$. The green lines denote the power-law exponents of the second relaxation step: $\beta \simeq 2$ for both experiments. (b) Exponents $\alpha$ and $\beta$ versus $h_0$. With increasing gap thickness, $\alpha$ transitions from a Newtonian-like response ($\alpha =5$—blue line) to a yield stress dominated behavior ($\alpha =2.5$—blue line). $\beta$ is roughly constant over the range of gap thickness explored ($\beta \simeq 2$—green line). (c) The relative extent $R_p/R$ of the pattern versus $h_0$. Inset: definition of $R$ and $R_p$. The vertical lines in (b) and (c) denote the boundary between the stable and the unstable regime.

Figure 4

Most unstable wavelength ${\lambda }_c$ of the fingering patterns versus (a) the initial gap thickness $h_0$ and (b) the radial velocity $v_r$. In (a), the solid line corresponds to a power-law exponent of $3/2$, the dashed line to ${\lambda }_c^{*}=1.76\mathrm{mm}$. In (b), the line corresponds to a power-law exponent of $-1/2$. Data obtained with a 8% wt. carbon black gel and two different plate radii $R=20\mathrm{mm}$ (circle) and 30 mm (triangle).