Alternative mechanism for coffee-ring deposition based on active role of free surface

When a colloidal sessile droplet dries on a substrate, the particles suspended in it usually deposit in a ringlike pattern. This phenomenon is commonly referred to as the “coffee-ring” effect. One paradigm for why this occurs is as a consequence of the solutes being transported towards the pinned contact line by the flow inside the drop, which is induced by surface evaporation. From this perspective, the role of the liquid-gas interface in shaping the deposition pattern is somewhat minimized. Here, we propose an alternative mechanism for the coffee-ring deposition. It is based on the bulk flow within the drop transporting particles to the interface where they are captured by the receding free surface and subsequently transported along the interface until they are deposited near the contact line. That the interface captures the solutes as the evaporation proceeds is supported by a Lagrangian tracing of particles advected by the flow field within the droplet. We model the interfacial adsorption and transport of particles as a one-dimensional advection-generation process in toroidal coordinates and show that the theory reproduces ring-shaped depositions. Using this model, deposition patterns on both hydrophilic and hydrophobic surfaces are examined in which the evaporation is modeled as being either diffusive or uniform over the surface.

Figure 1

(a) An evaporating sessile droplet containing a dilute suspension of nonvolatile particles. (b) The lines of constant $\alpha$ and $\theta$ and positive direction of velocity components in a (left-handed) toroidal coordinate system that fits the boundaries of the drop on the meridian plane $A-A$.

Figure 2

The cumulative fraction of particles captured by the liquid-gas interface $N_s/N_{\mathrm{tot}}$ and the fraction of droplet's lost volume $({V–}_0-V–)/{V–}_0$ versus ${\theta }_c/{\theta }_{c_0}$ for droplets of different initial contact angles. The symbols and solid lines represent $N_s/N_{\mathrm{tot}}$ and $({V–}_0-V–)/{V–}_0$, respectively.

Figure 3

The final distribution of particles in the form of bar charts for droplets with initial contact angles (a) ${\theta }_{c_0}={30}^{\circ }$, (b) ${\theta }_{c_0}={60}^{\circ }$, (c) ${\theta }_{c_0}={90}^{\circ }$, (d) ${\theta }_{c_0}={120}^{\circ }$, and (e) ${\theta }_{c_0}={150}^{\circ }$. The distance from the axis of symmetry to the contact line is divided into 20 bins, and each bar represents the fraction of particles located in the corresponding radial interval. The blue and red bars illustrate the results of the one-dimensional continuum model and particle tracing simulations, respectively. Also, the insets are magnified replots for $0\le r/R\le 0.95$. (f) The fraction of particles in the last interval $0.95<r/R\le 1$ (denoted by $N_{\mathrm{edge}}/N_{\mathrm{tot}}$) as a function of the initial contact angle ${\theta }_{c_0}$. The circles recap the results of (a)–(e) for the deposition due to the Stokes flow inside the drop induced by a diffusive evaporation profile. The diamonds and squares show particle accumulations at the edge due to, respectively, potential flow induced by the same diffusive evaporation and Stokes flow generated by a uniform evaporative flux. Due to the similarity, only the results of the continuum calculations are shown.

Figure 4

Particle deposition during evaporation of a colloidal droplet with ${\theta }_{c_0}={15}^{\circ }$. (a) The cumulative fraction of particles arriving at the contact line $N_{\mathrm{edge}}/N_{\mathrm{tot}}$ versus time $t/t_f$, where $t_f$ is the total drying time whose value for the experiment of Deegan et al. [1] is 1300 s. Also, in their experiment the maximum number of particles deposited at the contact line is $N_{{\mathrm{edge}}_{\max }}=0.9N_{\mathrm{tot}}$. To be consistent with the experiments, in this plot, $N_{\mathrm{edge}}$ represents the number of particles in the interval of $0.99<r/R\le 1$. (b) and (c) Radial distribution of height-averaged concentration of particles at $t/t_f=0.12,0.51,0.9$. Here, $\overline{C}$ denotes the total (bulk and surface) concentration of solutes averaged over the height of the drop $z_b$ at each radial position, and $z_b\overline{C}$ is normalized by its corresponding value at $r/R=0$ and $t/t_f=0.12$. Symbols represent the measurements of Deegan et al. [3], whereas the solid lines denote the predictions of the continuum model. The simulations are carried out down to ${\theta }_c=1.5^{\circ }$ to obtain the results for $t/t_f=0.9$.