Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota 55455, USA.
Langmuir. 2011 Sep 20;27(18):11347-63. doi: 10.1021/la202088s. Epub 2011 Aug 26.
When a coffee droplet dries on a countertop, a dark ring of coffee solute is left behind, a phenomenon often referred to as the coffee-ring effect. A closely related yet less-well-explored phenomenon is the formation of a layer of particles, or skin, at the surface of the droplet during drying. In this work, we explore the behavior of a mathematical model that can qualitatively describe both phenomena. We consider a thin axisymmetric droplet of a colloidal suspension on a horizontal substrate undergoing spreading and evaporation. In contrast to prior work, precursor films (rather than pinned contact lines) are present at the droplet edge, and evaporation is assumed to be limited by how quickly molecules can transfer out of the liquid phase (rather than by how quickly they can diffuse through the gas phase). The lubrication approximation is applied to simplify the mass and momentum conservation equations, and the colloidal particles are allowed to influence the droplet rheology through their effect on the viscosity. By describing the transport of the colloidal particles with the full convection-diffusion equation, we are able to capture depthwise gradients in particle concentration and thus describe skin formation, a feature neglected in prior models of droplet evaporation. The highly coupled model equations are solved for a range of problem parameters using a finite-difference scheme based on a moving overset grid. The presence of evaporation and a large particle Peclet number leads to the accumulation of particles at the liquid-air interface. Whereas capillarity creates a flow that drives particles to the droplet edge to produce a coffee ring, Marangoni flows can compete with this and promote skin formation. Increases in viscosity due to particle concentration slow down droplet dynamics and can lead to a reduction in the spreading rate.
当咖啡液滴在台面上干燥时，会留下一圈深色的咖啡溶质，这种现象通常被称为咖啡环效应。与之密切相关但研究较少的现象是，在干燥过程中，液滴表面会形成一层颗粒（或表皮）。在这项工作中，我们研究了一个可以定性描述这两种现象的数学模型的行为。我们考虑了一个在水平基底上展开和蒸发的胶体悬浮液的薄轴对称液滴。与之前的工作相比，在液滴边缘存在前导膜（而不是固定的接触线），并且假设蒸发受到限制的原因是分子从液相中转移的速度（而不是它们通过气相扩散的速度）。应用润滑近似简化了质量和动量守恒方程，并且通过颗粒对粘度的影响允许胶体颗粒影响液滴的流变学。通过用完整的对流扩散方程描述胶体颗粒的输运，我们能够捕获颗粒浓度的深度梯度，从而描述表皮形成，这是先前液滴蒸发模型中忽略的特征。使用基于移动重叠网格的有限差分方案，针对一系列问题参数求解了高度耦合的模型方程。蒸发和大颗粒 Peclet 数的存在导致颗粒在液-气界面处积聚。虽然毛细作用产生的流动将颗粒驱动到液滴边缘以产生咖啡环，但 Marangoni 流动可以与该流动竞争并促进表皮形成。由于颗粒浓度导致的粘度增加会降低液滴动力学速度，并导致扩展速率降低。
