When a thin film of a colloidal suspension flows over a substrate, uneven distribution of the suspended particles can lead to an uneven coating. Motivated by this phenomenon, we analyse the flow of perfectly wetting films and droplets of colloidal suspensions down an inclined plane. Lubrication theory and the rapid-vertical-diffusion approximation are used to derive a coupled pair of one-dimensional partial differential equations describing the evolution of the interface height and particle concentration. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentrationdependent viscosity and diffusivity. We find that for sufficiently high Péclet numbers, even small initial concentration inhomogeneities produce viscosity gradients that cause the film or droplet front to evolve continuously in time instead of travelling without changing shape as happens in the absence of colloidal particles. At high enough particle concentrations, particle diffusion can lead to the formation of long-lived secondary flow fronts in films. Our results suggest that particle concentration gradients can have a dramatic influence on interface evolution in flowing films and droplets, a finding which may be relevant for understanding the onset of patterns that are observed experimentally.
Wetting of permeable substrates by liquids is an important phenomenon in many natural and industrial processes. Substrate heterogeneities may significantly alter liquid spreading and interface shapes, which in turn may alter liquid imbibition. A new lubrication-theory-based model for droplet spreading on permeable substrates that incorporates surface roughness is developed in this work. The substrate is assumed to be saturated with liquid, and the contact-line region is described by including a precursor film and disjoining pressure. A novel boundary condition for liquid imbibition is applied that eliminates the need for a droplet-thickness-dependent substrate permeability that has been employed in previous models. A nonlinear evolution equation describing droplet height as a function of time and the radial coordinate is derived and then numerically solved to characterize the influence of substrate permeability and roughness on axisymmetric droplet spreading. Because it incorporates surface roughness, the new model is able to describe the contact-line pinning that has been observed in experiments but not captured by previous models.
A droplet of a colloidal suspension placed on an inclined substrate may sag under the action of gravity. Solvent evaporation raises the concentration of the colloidal particles, and the resulting viscosity changes may influence the sag of the droplet. To investigate this phenomenon, we have developed a mathematical model for perfectly wetting droplets based on lubrication theory and the rapid-vertical-diffusion approximation. Precursor films are assumed to be present, the colloidal particles are taken to be hard spheres, and particle and liquid dynamics are coupled through a concentration-dependent viscosity and diffusivity. Evaporation is assumed to be limited by how rapidly solvent molecules can transfer from the liquid to the vapor phase. The resulting one-dimensional system of nonlinear partial differential equations describing the evolution of the droplet height and particle concentration is solved numerically for a range of initial particle concentrations and substrate temperatures. The solutions reveal that the interaction between evaporation and non-Newtonian suspension rheology gives rise to several distinct regimes of droplet shapes and particle concentration distributions. The results provide insight into how evaporation and suspension rheology can be tuned to minimize sagging as well as the well-known coffee-ring effect, an outcome which is important for industrial coating processes.
Water droplet spreading and imbibition on alkaline hydrolyzed melt-blown poly(butylene terephthalate) (PBT) fiber mats is studied with both experiments and numerical simulations to elucidate the influence of surface hydrolysis. Melt-blown PBT fiber mats were exposed to a NaOH-methanol solution for controlled periods of time, resulting in surface hydrolysis of the fiber mats and the transformation of their surface wetting properties from hydrophobic to superhydrophilic. Visualization experiments reveal the influence of the hydrolysis time on the rate of droplet absorption by superhydrophilic hydrolyzed PBT fiber mats, and are used to characterize droplet spreading and imbibition. A mathematical model was employed to understand the observed spreading and imbibition behavior and the lower effective permeability seen for long hydrolysis times.
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