Ring to Mountain Transition in Deposition Pattern of Drying Droplets

When a droplet containing a nonvolatile component is dried on a substrate, it leaves a ringlike deposit on the substrate. We propose a theory that predicts the deposit distribution based on a model of fluid flow and the contact line motion of the droplet. It is shown that the deposition pattern changes continuously from a coffee ring to volcanolike and to mountainlike depending on the mobility of the contact line and the evaporation rate. An analytical expression is given for the peak position of the distribution of the deposit left on the substrate.

Figure 1

Schematic of a droplet and three different deposition patterns with axisymmetry in a cylindrical coordinate system (side view). Relevant parameters are the radius of the contact line $R(t)$, the height of the droplet at the center $H(t)$, the contact angle $\theta (t)$, and the profile of liquid-vapor interface $h(r,t)$.

Figure 2

Profile of the deposit left on the substrate when the drying is completed. The coffee ring to the volcanolike and then to the mountainlike pattern transition induced by changing the value of $k_{\text{cl}}$ from 100 to 0. (a) The case of a fast evaporation rate characterized by $k_{\mathrm{ev}}=1$ and (b) the case that the evaporation rate is small with $k_{\mathrm{ev}}=10^{-3}$. For both cases, ${\theta }_e={\theta }_0=0.2$ and $\mathrm{\Delta }t/{\tau }_{\mathrm{ev}}=10^{-5}$.

Figure 3

The peak position $r_p$ of the deposit density distribution is plotted against $k_{\text{cl}}$. The results of the numerical calculation for various values of $k_{\mathrm{ev}}$ are denoted by symbols, while the analytical result (26) is denoted by the black-solid line. All other parameters are the same as in Fig. 2.

Figure 4

Evolution of the profile of the droplet’s interface between the liquid and vapor, $h(r,t)/H_0$, and solute density distribution $\mu (r,t)/(H_0{\varphi }_0)$ for (a), (b) a fast evaporation rate $k_{\mathrm{ev}}=1$, and for (c), (d) a slow evaporation rate $k_{\mathrm{ev}}=10^{-3}$. In all figures, the black dash-dot line represents the initial state, while the red solid line is the final state after drying up. The time steps in (b) and (d) are the same as in (a) and (c), accordingly. The arrow is the direction of the time evolution of the drying process. Other parameters are $k_{\text{cl}}=5$, ${\theta }_e={\theta }_0=0.2$, and $\mathrm{\Delta }t/{\tau }_{\mathrm{ev}}=10^{-5}$.