Marangoni flows during drying of colloidal films Stergios G. Yiantsios Department of Chemical Engineering, Aristotle University of Thessaloniki and Chemical Process Engineering Research Institute, P. O. Box 361, GR 570 01, Thermi, Thessaloniki, Greece Brian G. Higgins Department of Chemical Engineering & Materials Science, University of California, Davis, California 95616 Received 4 May 2006; accepted 10 July 2006; published online 21 August 2006 In this study, we consider the drying of a thin film that contains a stable dispersion of colloidal particles so that a coating of these particles is formed after the liquid is driven off by evaporation. For sufficiently thin films, we show that evaporative cooling can drive a Marangoni flow that results in surface deformation of the drying film. A thin-film approximation is used to describe the velocity and temperature fields, and the particle transport equation with convective terms retained is used to describe the concentration field. A coupled finite difference/spectral element scheme is implemented numerically to solve the particle transport equation, where high accuracy is required to resolve sharp gradients within the film and to ensure particle conservation during drying. The model employed is capable of describing the evolution of film thickness and concentration field up to the time when maximum packing is nearly reached at some point in the domain. Three types of film structures are observed, all characterized by a final nonuniform thickness. In the first type, observed at low Peclet numbers, the maximum concentration is reached at the thinnest points in the film, which surround elevations with lower particle concentrations. This mode of instability suggests that dried coatings will have pronounced nonuniformities, resulting in the formation of craters or pinholes. In the second type, observed at high Peclet numbers, a closely packed skin of nonuniform thickness is formed, with low concentration fluid remaining beneath the elevations. In the final stages of drying one would expect capillary pressure to pull particles in the underlying fluid toward the skin, thus creating voids under a seemingly homogeneously applied coating. Finally, still at relatively large particle Peclet numbers and when the destabilizing Marangoni stresses are sufficiently strong, floating lumps of closely packed particles may form in the vicinity of film elevations. © 2006 American Institute of Physics. DOI: 10.1063/1.2336262 I. INTRODUCTION The solution coating of colloidal particles is frequently used in the production of displays and other optical films. A crucial step in these manufacturing processes is the drying of the coated film by evaporation to yield a particulate film of uniform thickness. The main motivation behind the present work and the question attempted to be touched upon is whether Marangoni instabilities due to evaporation can have an effect on coated film quality and integrity. Nonuniformi- ties in colloidal particle deposits have been extensively ana- lyzed in the context of evaporating droplets because of en- hanced evaporation at the contact lines. 1–4 However, in coated products that are essentially two-dimensional 2-D in lateral extent, other factors rather than edge effects may be important. Routh and Russel 5 and Tirumkudulu and Russel 6 study theoretically and report experimental observations on drying colloidal dispersions in the form of thin films of finite lateral extent. In a thorough and insightful analysis they take into account several effects, such as the formation of a closely packed particle front at the periphery of the film, the motion of that front towards the film center, the effects of capillary pressure, which may result in a second front of dried par- ticles following the former, as well as in deforming the par- ticles and creating dry films of very small porosity. In their analysis, the Brownian diffusion of the particles is assumed large enough so that it effectively homogenizes the particle concentration across the liquid film. In a simpler setting, Routh and Zimmerman 7 consider a film of infinite lateral extent, evaporating at a constant rate and analyze the effect of a nonzero particle Peclet number by solving a one- dimensional 1-D diffusion equation. Thus, the assumption of uniform concentration is relaxed and even at relatively low Peclet numbers a concentration gradient is predicted to appear near the interface and advance towards the substrate. In their analysis, film deformation and convective motion are assumed to be absent. The present work focuses precisely on those two aspects, which may be driven by thermocapillary phenomena or other effects giving rise to surface tension gradients. Thus, the simple setting of an unbounded thin film of a colloidal dispersion, as in Routh and Zimmermann is retained, but the temperature field and the convective motion in the film due to Marangoni effects are analyzed. A thorough review on thermocapillary phenomena is not attempted here, but only some points relevant to the subse- quent discussion are highlighted. The interested reader may consult Davis, 8 Oron et al., 9 Van Hook et al., 10 and refer- PHYSICS OF FLUIDS 18, 082103 2006 1070-6631/2006/188/082103/11/$23.00 © 2006 American Institute of Physics 18, 082103-1