Contact line arrest in solidifying spreading drops

When does a drop, deposited on a cold substrate, stop spreading? Despite the practical relevance of this question, for example, in airplane icing and three-dimensional metal printing, the detailed mechanism of arrest in solidifying spreading drops has remained debated. Here we consider the spreading and arrest of hexadecane drops of constant volume on two smooth wettable substrates: copper with a high thermal conductivity and glass with a low thermal conductivity. We record the spreading radius and contact angle in time for a range of substrate temperatures. The experiments are complemented by a detailed analysis of the temperature field near the rapidly moving contact line, by means of similarity solutions of the thermohydrodynamic problem. Our combined experimental and theoretical results provide strong evidence that the spreading of solidifying drops is arrested when the liquid at the contact line reaches a critical temperature, which is determined by the effect of kinetic undercooling.

Figure 1

Sketches of the different hypotheses for contact line arrest. Arrest occurs when (a) the contact angle of the spreading drop equals the angle of the freezing front at the contact line [14, 15], (b) a critical volume at the contact line is solidified [26], and (c) the liquid at the contact line reaches a critical temperature (present work).

Figure 2

(a) Sketch of the experimental setup (not to scale). (b) Snapshots at the moment of contact line arrest of (the lower part of) hexadecane drops spreading on a copper substrate for various substrate temperatures $T_0$. The arrest spreading radius $R_a$ and arrest contact angle ${\theta }_a$ are indicated in red.

Figure 3

Spreading radius versus time for a range of substrate temperatures (a) on a copper substrate and (b) on a glass substrate ($T_i=28{}^{\circ }\mathrm{C}$). The inset in (b) shows a close-up of the stick-slip behavior on glass for $T_0=8{}^{\circ }\mathrm{C}$ (left and right contact lines). The black line is the spreading law in the inertiocapillary regime [Eq. (1)].

Figure 4

Contact angle versus (a) time or (b) capillary number for a range of substrate temperatures on a copper substrate ($T_i=28{}^{\circ }\mathrm{C}$). The contact angle is plotted up to the moment of contact line arrest. The black line in (b) is the Cox-Voinov law [Eq. (3)] as fitted in the inertiocapillary regime with $ln(R_0/l)=25.7$.

Figure 5

(a) Arrest spreading radius and (b) arrest contact angle versus the substrate undercooling $\mathrm{\Delta }T=T_f-T_0$. Black indicates the copper substrate (properties at arrest) and red the glass substrate (properties at first pinning event). Closed symbols indicate drops that get arrested in the inertiocapillary regime, while open symbols indicate drops that get arrested in the viscocapillary regime or during the transition between both regimes. Dashed lines show the model with kinetic undercooling (see Sec. 3). A kinetic coefficient of $\kappa =0.011\mathrm{m}/\left(\mathrm{s}\mathrm{K}\right)$ is used.

Figure 6

(a) Geometry in the contact line region for calculation of the temperature field. (b)–(d) Isotherms (solid lines) and streamline patterns (dashed lines) for hexadecane drops on a glass substrate with a contact angle of (b) $\theta ={120}^{\circ }$, (c) $\theta ={90}^{\circ }$, and (d) $\theta ={60}^{\circ }$. The rescaled contact line temperatures are indicated.