Liquid spreading on cold surfaces: Solidification-induced stick-slip dynamics

We study the contact line dynamics of a liquid drop on a substrate at temperature $T_s$ colder than the freezing temperature $T_m$. The purpose is to determine experimentally the criterion for a solidification-induced pinning of the liquid at its triple line. The sub- strate is driven at constant velocity, while the upper part of the drop is pinned to the injection pipe. Within a certain range of substrate velocity and temperature difference $T_m-T_s$, the solidification can induce contact-line pinning. This pinning occurs when the velocity of the substrate is slower than a critical value $V_c$ and leads to a discontinuous stick-slip dynamics of the contact line, while at high enough velocity the contact-line motion remains continuous. Stick-slip dynamics appear when the solid front, showing dendritic structure, catches up the advancing contact line. We study the dynamics of the stick-slip regime at different substrate velocities $V_s$, $T_m-T_s$, and injection flow rate $Q$. We relate the dependence of $V_c$ on $T_m-T_s$ to the velocity of the solidification front $V_f$ calculated and measured with an independent experiment. These results are consistent with a mechanism of kinetic undercooling where the front velocity follows a trend reminiscent to that measured and predicted in dendritic solidification front growth.

Figure 1

Sketch of the experimental setup.

Figure 2

Typical shape of the drop of hexadecane near the advancing line in the continuous regime (left: $\mathrm{\Delta }T=0.25{}^{\circ }\mathrm{C}$, $V_s=0.8$ mm/s, $Q=200\mu \mathrm{l}/\min$) and in the stick-slip regime (right: $\mathrm{\Delta }T=2{}^{\circ }\mathrm{C}$, $V_s=1\mathrm{mm}/\mathrm{s}$, $Q=100\mu \mathrm{l}/\min$). In the latter situation, the solid front catches up the advancing line. In both cases and in the next images, the substrate moves from right to left. The green line represents the substrate vertical position, below which the reflected image of the drop can be seen.

Figure 3

Sequence of transient behavior of a drop experiencing continuous advance on a cold moving substrate ($\mathrm{\Delta }T=0.25{}^{\circ }\mathrm{C}$, $V_s=0.8\mathrm{mm}/\mathrm{s}$, $Q=200\mu \mathrm{l}/\min$). The solid-liquid front visibly recedes away from the advancing line. The drop shape and contact-line location reach a steady state. The corresponding typical video is available in Ref. [24].

Figure 4

Sequence of transient behavior of a drop experiencing discontinuous (stick-slip) dynamics on a cold moving substrate ($\mathrm{\Delta }T=0.75{}^{\circ }\mathrm{C}$, $V_s=0.5\mathrm{mm}/\mathrm{s}$, $Q=500\mu \mathrm{l}/\min$). The solid-liquid front visibly catches up to the advancing line before pinning occurs, leading to an arrest of the contact line and an increase of the advancing angle. The corresponding typical video is available in Ref. [24].

Figure 5

Advancing dynamical contact angle ${\theta }_{\mathrm{ad}}$ and relative position of contact line ($X-X_0$) versus time, of hexadecane, over several stick-slip events ($\mathrm{\Delta }T=0.75{}^{\circ }\mathrm{C}$, $V_s=1\mathrm{mm}/\mathrm{s}$, $Q=1100\mu \mathrm{l}/\min$).

Figure 6

Critical velocity $V_c$ versus $\mathrm{\Delta }T$ for pentadecane (blue diamonds) and hexadecane (red diamonds). The blue and red circles stand for the velocity of the solid front directly measured in a bulk experiment (setup in Fig. 10 detailed in Sec. 5), respectively, for pentadecane and hexadecane. The red and blue plain lines represent the best fit by a power law $V_c\propto \mathrm{\Delta }{T}^{2.65}$. The power law is also emphasized in the insert where the data are plotted in log-log axes.

Figure 7

Relative velocity of the advancing contact line versus time $(t-t_0)$, with $t_0$ is the time at the onset of sliding. The velocity is averaged from seven successive stick-slip events, which are represented in the insert ($\mathrm{\Delta }T=0.75{}^{\circ }\mathrm{C}$, $V_s=1\mathrm{mm}/\mathrm{s}$, $Q=1100\mu \mathrm{l}/\min$).The arrow below shows the approximate time locations of the slip and stick phases, with the gradation of color indicating the relative uncertainty of the transition when considering the averaged velocity $\langle V\rangle$.

Figure 8

Frequency of stick-slip versus substrate velocity $V_s$ for various $\mathrm{\Delta }T=0.75$, 2.25, and $3.25{}^{\circ }\mathrm{C}$. Insert: Frequency versus velocity difference with respect to threshold $(V_c-V_s)$. The plain lines result from the best linear fit of the data.

Figure 9

Frequency of stick-slip close to the threshold of continuous dynamics versus $V_c\left(\mathrm{\Delta }T\right)$.

Figure 10

(Top) Sketch of the bulk solidification setup allowing for the direct observation of the dynamics of the front. (Bottom) A typical sequence of the solidification front progression, showing dendritic shape with needles ($\mathrm{\Delta }T=1.5{}^{\circ }\mathrm{C}$, hexadecane).