Instabilities in Droplets Spreading on Gels

We report a novel surface-tension driven instability observed for droplets spreading on a compliant substrate. When a droplet is released on the surface of an agar gel, it forms arms or cracks when the ratio of surface-tension gradient to gel strength is sufficiently large. We explore a range of gel strengths and droplet surface tensions and find that the onset of the instability and the number of arms depend on the ratio of surface tension to gel strength. However, the arm length grows with an apparently universal law $L\propto t^{3/4}$.

Figure 1

Sample images of droplets spreading on gel agar substrates. (a)–(c) Shadowgraph images adapted from [16]; bright lines represent the outer boundaries of the arms. (a) Triton X-305 solution starburst: droplet concentration $\varphi =50\mathrm{ppm}$ ($\sigma =63\mathrm{mN}/\mathrm{m}$), $X=0.12\%\mathrm{w}$ ($G=16\mathrm{Pa}$), and $V=5\mu \mathrm{L}$ after 14 s. (b) Triton X-305 solution wispy drop: droplet concentration $\varphi =100\mathrm{ppm}$ ($\sigma =60\mathrm{mN}/\mathrm{m}$), $X=0.14\%\mathrm{w}$ ($G=28\mathrm{Pa}$), and $V=5\mu \mathrm{L}$ after 27 s. (c) PDMS starburst: $\sigma =20\mathrm{mN}/\mathrm{m}$, $X=0.08\%\mathrm{w}$ ($G=1.3\mathrm{Pa}$), $\nu =1000\mathrm{cS}$, and $V=5\mu \mathrm{L}$ after 5 s. (d) Scattering image of Triton X-305 starburst (inset) and cross section of arm.

Figure 2

Mean number of arms on individual droplets as a function of (a) droplet surface tension ${\sigma }_d$; (b) gel elastic modulus $G$; and (c) $M\lambda$ [see Eq. (1)], assuming ${\lambda }_{\mathrm{PDMS}}=12{\lambda }_{\mathrm{Triton}}$. Inset: Schematic of Marangoni/elastic instability. Open symbols are Triton X-305 solutions, with ${\sigma }_d=72\mathrm{mN}/\mathrm{m}$ (⊲), $71\mathrm{mN}/\mathrm{m}$ (◇), $70.6\mathrm{mN}/\mathrm{m}$ (★), $70\mathrm{mN}/\mathrm{m}$ (⊳), $63\mathrm{mN}/\mathrm{m}$ (□), $60\mathrm{mN}/\mathrm{m}$ (×), $59\mathrm{mN}/\mathrm{m}$ (▽), $57\mathrm{mN}/\mathrm{m}$ (○), $55\mathrm{mN}/\mathrm{m}$ (*), $53\mathrm{mN}/\mathrm{m}$ (△), $50\mathrm{mN}/\mathrm{m}$ (+). Solid symbols (here overlapping) are PDMS: ${\sigma }_d=20.1\mathrm{mN}/\mathrm{m}$ and $\nu =10\mathrm{cS}$ (■), ${\sigma }_d=20.8\mathrm{mN}/\mathrm{m}$ and $\nu =50\mathrm{cS}$ (◆), ${\sigma }_d=20.9\mathrm{mN}/\mathrm{m}$ and $\nu =100\mathrm{cS}$ (●), ${\sigma }_d=21.2\mathrm{mN}/\mathrm{m}$ and $\nu =1000\mathrm{cS}$ (★). Symbol colors represent gel substrate: 1.3 Pa (blue), 3.7 Pa (magenta), 5.6 Pa (cyan), 8.2 Pa (black), 11.7 Pa (yellow), 16 Pa (green), 28 Pa (red). Droplet volume is $2.5\mu \mathrm{L}$. Each point is averaged over at least ten drops at the same parameters. Horizontal error bars are due to $G$ and ${\sigma }_g$; vertical bars are the standard error for the measurements. No data at $\varphi =1000\mathrm{ppm}$ are included, due to the unreliable counting of numerous arms.

Figure 3

Length of the five arms (A, B, C, D, E) of a Triton X-305 starburst (inset) at a function of time. Time $t$ is measured from the start of each arm’s growth, not the drop release time. Solid lines are fits with a free exponent; dashed lines are fixed $\alpha =3/4$. Gel is 3.7 Pa and droplet has $V=5\mu \mathrm{L}$ and ${\sigma }_d=60\mathrm{mN}/\mathrm{m}$.

Figure 4

Fit value of spreading exponent $\alpha$ vs (a) shear modulus $G$, for constant $V=5\mu \mathrm{L}$, (b) surface tension ${\sigma }_d$, for constant $V=5\mu \mathrm{L}$, (c) arm width, for constant $V=5\mu \mathrm{L}$, and (d) droplet volume, $V$, for constant gel and droplet properties. Points are averaged over the arms of a single droplet. Legend as given in Fig. 2.

Figure 5

Mean spreading prefactor $C$, in $\mathrm{mm}/{\sec }^{3/4}$, as a function of mean arm width. Each point is averaged over the arms of a single drop, with max or min values shown as vertical bars. See the legend in Fig. 2 for parameters of each point. Size of symbol indicates droplet size, with the majority taken at $V=5\mu \mathrm{L}$.