Electrostatic Suppression of the “Coffee Stain Effect” Alexander W. Wray,* ,† Demetrios T. Papageorgiou, ‡ Richard V. Craster, ‡ Khellil Sefiane, § and Omar K. Matar † † Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, U.K. ‡ Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2BZ, U.K. § Institute for Materials and Processes, School of Engineering, The University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, U.K. ABSTRACT: The dynamics of a slender, evaporating, particle-laden droplet under the effect of electric fields are examined. Lubrication theory is used to reduce the governing equations to a coupled system of evolution equations for the interfacial position and the local, depth- averaged particle concentration. The model incorporates the effects of capillarity, viscous stress, Marangoni stress, elecrostatically induced Maxwell stress, van der Waals forces, concentration-dependent rheology, and evaporation. Via a parametric numerical study, the one-dimensional model is shown to recover the expected inhomogeneous ring-like structures in appropriate parameter ranges due to a combination of enhanced evaporation close to the contact line, and resultant capillarity-induced flow. It is then demonstrated that this effect can be significantly suppressed via the use of carefully chosen electric fields. Finally, the three-dimensional behavior of the film and the particle concentration field is briefly examined. 1. INTRODUCTION It is a commonly experienced phenomenon that the evaporation of a nanoparticle-laden droplet gives rise to a distinctly inhomogeneous residue, even when the original particle distribution was entirely uniform. This results in what has come to be known as the “coffee stain” or “coffee ring” effect. It is of course commonly seen in situations such as drying coffee drops, or indeed in the drying of watercolor paint. 1 The effect is of significant interest in a wide array of practical contexts. While there are many contexts in which the ring-like inhomogeneity is a useful patterning technique, 2−4 it is also found to be an undesirable effect. This is the case, for instance, in DNA microarrays, 5,6 chemical recovery, 7 matrix- assisted laser desorption/ionization mass spectrometry, 8 and nanofabrication. 9 This phenomenon has been the subject of significant investigation, 10,11 with Deegan et al. explaining that the most important factors are an increased flux near the (pinned) contact line and a resultant capillarity-induced restoring flow. This results in an almost-complete mass flux of the particles in the bulk toward the contact line. In particular, Deegan et al. 1 gave an explicit ODE based on a lubrication approximation assuming a spherical cap. Similarly, Hu and Larson 12 have also given modeling results and demonstrate good agreement with their corresponding experiments. Hu and Larson 13 have used a lubrication-like analysis. However, as pointed out by Maki and Kumar, 14 this and other similar analyses suffer two flaws. First, the posited ansä tze do not themselves satisfy the lubrication form of the momentum equations. Second, the resultant velocity profiles are singular at the contact line, due to a singularity in the evaporative flux there. It is possible to artificially remove this singularity by having a flux that decays exponentially near the contact line, 15,16 or by relaxing the no-slip criterion there. 17 Hu and Larson 18 have shown that the introduction of Marangoni flow can in fact suppress the ring effect. However, they do note that the recirculation effect is significantly weaker in experiments than as predicted by theoretical calculations, at least for the case of water. This effect has since been extensively investigated. 19,20 There have since been studies of consistent lubrication-theory models both in the absence 21 and presence 22 of thermal effects. These used a depth-averaged concentration, an assumption relaxed by Maki and Kumar, 14 with resultant skin formation. There is a very large body of existing literature on this topic. An extensive review has been given recently by Larson, 23 to which we refer the interested reader for additional details. As mentioned above, the suppression of the coffee stain effect holds significant potential interest in practical and industrial settings. Thus, recent, predominantly experimental, results on the use of electric fields to suppress the ring stain effect hold a lot of interest for potential modeling. These results indicate that the “coffee stain” effect can be suppressed using either AC or DC electric fields. In the case of AC potential, the alternation and cycles of voltage lead to the oscillations of the contact line which inhibit the deposition of the particles in the same position. 24 Mampallil et al. 25 performed additional experiments to demonstrate the same mechanism over a range of parameters. In the case of DC fields, on the other hand, there are no oscillations of the contact line. Instead, the Received: February 28, 2014 Revised: May 2, 2014 Published: May 2, 2014 Article pubs.acs.org/Langmuir © 2014 American Chemical Society 5849 dx.doi.org/10.1021/la500805d | Langmuir 2014, 30, 5849−5858