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Received Dec 19, 2017; Accepted Feb 12, 2018
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1. Introduction
The study of drying droplets, especially those containing colloidal particles or salts, has become a topic of wide interest in recent years. This is evident from the fact that international conferences on droplets are regularly organized (Droplets Conferences and EMN Droplets) and books exclusively devoted to this subject have been published [1–8]. There are, in addition, several excellent review articles on droplets [9–21]. Different features of this problem, such as the rate of drying, evolution of the drop geometry, and the final pattern formed, depend on a number of parameters, notably the composition of the drying fluid, ambient conditions during drying, and the substrate which supports the drop.
The widespread interest in the drying droplets with inclusions stems from important and innovative applications, mainly in medical science and in technology. When the fluid in the droplet evaporates, the solid material left behind can be distributed in a wide variety of patterns on the substrate. Inclusions may be salts, nanoparticles in the form of nanorods, nanotubes, or any other shape, starches, proteins, and so on. Patterns formed can range from a simple ring at the periphery of the droplet, the so-called coffee ring [22, 23], to multiple rings forming bands, fractal and multifractal aggregates of salt crystals, or nanoparticles [24–27]. In addition, the dried drop may develop crack patterns [8, 28–43], which can also be induced by external fields [35, 44]. It is important to realize that the shape of the droplet plays a crucial role in generating these patterns. Unless the drop is too large, its shape may be roughly approximated by a section of a sphere or a spherical cap. In some cases, when a crust or skin forms on the free surface [45], buckling instability may develop [46–49]. Nonuniform evaporation over different regions of the drop surface generates convection currents that determine mass transfer. Temperature and concentration gradients also develop [50–53], leading to surface tension gradients, resulting in thermal Marangoni flow [54–59] or/and solutal Marangoni flow [60]. Obviously, drying out of a large...