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.

High-speed printing processes are a leading technology for the large-scale manufacture of a new generation of nanoscale and microscale devices. Central to all printing processes is the transfer of liquid from one surface to another, a seemingly simple operation that is still not well understood. A useful idealization of liquid transfer is a liquid bridge with moving contact lines being deformed between two separating surfaces. The fluid mechanics of such bridges are relevant not only to printing, but also to other important applications, such as adhesion, tribology, biology, oil recovery, and microfluidics.

In many technological applications, excess solvent must be removed from liquid droplets to deposit solutes onto substrates. Often, the substrates on which the droplets rest may possess some roughness, either intended or unintended. Motivated by these observations, we present a lubrication-theory-based model to study the drying of droplets of colloidal suspensions on a substrate containing a topographical defect. The model consists of a system of one-dimensional partial differential equations accounting for the shape of the droplet and depth-averaged concentration of colloidal particles. A precursor film and disjoining pressure are used to describe the contact-line region, and evaporation is included using the well-known one-sided model. Finite-difference solutions reveal that when colloidal particles are absent, the droplet contact line can pin to a defect for a significant portion of the drying time due to a balance between capillary-pressure gradients and disjoining-pressure gradients. The time-evolution of the droplet radius and contact angle exhibits the constant-radius and constant-contact-angle stages that have been observed in prior experiments. When colloidal particles are present and the defect is absent, the model predicts that particles will be deposited near the center of the droplet in a cone-like pattern. However, when a defect is present, pinning of the contact-line accelerates droplet solidification, leading to particle deposition near the droplet edge in a coffee-ring pattern. These predictions are consistent with prior experimental observations, and illustrate the critical role contact-line pinning plays in controlling the dynamics of drying droplets.

We develop a method which incorporates spring-spring repulsions into Brownian dynamics simulations of flexible polymers. The distance of closest approach between two springs is computed, and a repulsive force is then applied based on this distance. Repulsive potentials of the exponential and power-law forms are considered. We demonstrate that our method is capable of accounting for excluded-volume effects in start-up of extensional flow. Equilibrium simulations indicate that spring-spring repulsions can be used to prevent the passage of two springs through each other, and thus maintain the topological integrity of polymer molecules. The method developed here is expected to be useful for simulating entanglement phenomena in both single and multichain systems.

The behaviour of the interface between stratified thin liquid films bounded by parallel solid surfaces and subject to van der Waals forces which drive dewetting is studied in this work. Chemically homogeneous surfaces are considered first; this is followed by an investigation of chemically heterogeneous surfaces. The lubrication approximation is applied to obtain a single nonlinear evolution equation which describes the interfacial behaviour, and both the linear stability and nonlinear development of the interface are examined. The sensitivity of the interfacial rupture time to problem parameters such as the viscosity ratio, initial interfacial height, interfacial tension, and magnitude of the van der Waals forces is characterized in detail for the homogeneous case. This serves as a basis for a study of the heterogeneous case, where the strong dependence of the rupture time on the length scale of the heterogeneity is found to be relatively independent of changes in the remaining problem parameters. The mechanisms underlying the rupture-time behaviour are also explored in detail. The results suggest a route by which one liquid can become emulsified in the other, and may be beneficial to industrial processes such as lithographic printing which are based on wettability phenomena.