Authors: R. Bhardwaj, D. Attinger
Affilation: Columbia University, United States
Pages: 420 - 423
Colloidal drops evaporating on a solid substrate can be used as vehicles for dispensing or organizing small particles suspended within them. The pattern left after evaporation can be a ring-like structure such as the dried DNA drop  (Figure 1), a multi-dimensional organized array , a polygonal network pattern  or a uniform pattern  . Often, ring-like patterns appear, which consist of a mound of particles packed at the wetting line. This mechanism has been explained by Deegan et al  as follows: during the evaporation of a colloidal drop, the wetting line remains pinned and the evaporation rate is the fastest in the vicinity of the wetting line because of the geometry of the vapor diffusion problem. The particles are therefore transported by the radially outward fluid flow field and accumulate on the wetting line region. Other applications would rather require uniformly deposited spots, such as ink jet printing. While several analytical theories and numerical models (based on simplified fluid dynamics) exist [5, 6] , there is a need for an exhaustive numerical treatment able to explain and predict the deposition pattern from fundamental principles of heat transfer, mass transfer and fluid dynamics. In our study, we extend a mathematical modeling for drop deposition described in [7, 8] to account for solvent evaporation and particle transport in colloidal drops. The 2D axisymmetric finite element model solves the unsteady equations of mass, momentum and heat transfer. All governing equations are expressed in a Lagrangian framework, which provides accurate modeling of the free surface motion and the associated Laplace stresses. The boundary conditions at the free surface are the mass and energy jump conditions. The diffusion equation for vapor transport is numerically solved in the far field of the drop. The evaporative flux at the free surface is therefore obtained directly from the drop-substrate geometry and thermodynamic conditions, rather than through analytical correlations such as in . The transport of solute, i.e. particles Y [kg of solute/kg of solution] in a solution is simulated by solving the diffusion equation in a 2D axisymmetric form as : where DPL is the diffusion coefficient of solute in solvent (= 4.0e-13 m2/s for 1 micrometer particles). Initial condition is taken as Y = Initial concentration of the solute in solution (= 0.01), and the initial wetting angle is taken as a parameter. The boundary conditions are: At z = 0; (No penetration); At r = 0; (Axisymmetric). At the drop-air interface [10, 11]: where j is the evaporative flux [kg/m2-s] on the drop-air interface, rl is the density of the solvent. Results Figure 2 shows the buildup of particles (left) and streamlines (right) at t = 240 ms during the evaporation of colloidal 12 nL water drop on a fused silica substrate heated at 82oC. Initially, the droplet contains a uniform 1% concentration of particles and is at ambient temperature, with an initial wetted radius of 265 micrometer. The right side of the figure shows a radially outward flow pattern with velocities on the order of 10 micrometer/sec. The left-hand side of the figure shows that particles build up at the wetting line and close to the liquid-air interface, a feature that can be explained by the magnitude of the evaporation rate and which is confirmed by experiments . Ring formation is also shown in the second part (Figure 2).
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