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Chapter 13
SOLUBLE SUBSTRATE
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Christophe Pirat, Jean Colombani and Alexandra Mailleur
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The studies of drop evaporation focus on the behavior of the liquid and on the influence of the environment. The substrate always plays a role on the evaporation, through heat exchange and via its wetting properties, but most of the time it is chosen as inert. Generally two model situations can be envisaged with regard to the evaporation of a sessile drop on an inert substrate. Either the contact line is free to move, so that the contact angle remains constant as the radius of the drop decreases, or the contact line is to be pinned, and then the contact angle decreases over time. In both situations the loss of mass obeys a power law with different exponents. In addition, mass loss due to evaporation, mainly at the contact line, is compensated by an outward flow. When colloidal particles are initially present in the liquid, the latter are deposited at the contact line, thus promoting the pinning.
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13.1 Interplay between evaporation and dissolution Yet, many situations exist in natural or industrial configurations, where liquid and solid react during the evaporation of a sessile liquid drop. In the case of mineral Droplet Wetting and Evaporation. DOI: http://dx.doi.org/10.1016/B978-0-12-800722-8.00013-8 © 2015 Elsevier Inc. All rights reserved.
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174 Droplet Wetting and Evaporation 20 2 1
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FIGURE 13.1 (Left) Atomic force microscopy picture of the cleavage surface of a gypsum (CaSO4, 2H2O) single crystal in water. The dissolution of the solid has led to the digging of parallelepiped (due to the monoclinic lattice of gypsum) etch pits. These pits are enclosed by atomic steps. (Right) The solvation of surface ions leads to the migration of the atomic steps and to the enlargement and deepening of the pits.
solids, the reaction at stake is mainly dissolution, and it may lead to a substantial modification for the solid surface as well as for the evaporation dynamics. This topic is quite new, and no coherent view exists at the moment. We present in this chapter the state-of-the-art of our knowledge on the evaporation of drops on a dissolving substrate.
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13.1.1 DISSOLUTION Dissolution is a heterogeneous reaction of the form solid 1 liquid-liquid. Its driving force is the difference in chemical potential between the solid and the initial liquid. When the two chemical potentials equalize, the dissolved species reaches the so-called solubility limit, or saturation concentration, of the solid in the liquid, and the reaction stops. The microscopic mechanism of dissolution is the detachment and solvation of units (mostly ions) from the solid. This detachment proceeds through the migration of atomic steps, the digging of etch pits, and so on (Figure 13.1). Many expressions of the dissolution flux have been observed, depending on the material. Its simplest shape is Jdiss 5 k (1 2 c/s), where k is the dissolution rate constant, c is the concentration of dissolved matter, and s is its solubility in the liquid (Rickard and Sjo¨berg, 1983; Colombani and Bert, 2007). The value of the dissolution rate constant varies in many orders of magnitude from one solid to another. Available values span from B10214 mol m22 s21—for instance, for kaolinite—to B1022 mol m22 s21—for instance, for rock salt. Whereas the studies on dissolution were originally limited to macroscopic investigations, the use since the early 2000s of recent far-field (vertical scanning interferometry) and near-field (atomic force microscopy) tools and the
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Soluble Substrate 175 AU:1 comparison with kinetic Monte Carlo molecular simulations have led to a better understanding of the microscopic mechanisms of dissolution. Questions such as “Why is the dissolution flux independent of the density of etch pits at the surface? Why is the dissolution flux dependent on the history of the surface? How do the atomic mechanisms combine to build a macroscopic matter flux?” have thereby begun to find answers (Lasaga and Luttge, 2001; Arvidson and Luttge, 2010).
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13.1.2 INTERACTION BETWEEN DISSOLUTION AND EVAPORATION KINETICS Interaction between dissolution and evaporation will take place if the kinetics of both phenomena are comparable. The evaporation flux of liquid water in normal conditions is JeB1027 m3 m22 s21. The overall dissolution flux is not necessarily the above-mentioned flux Jdiss. Indeed, this flux characterizes the transfer of matter from solid to liquid at the surface. But if the diffusion flux Jdiff of the dissolved species from the surface to the bulk liquid is small compared to the dissolution flux, these species will accumulate close to the surface, the concentration will increase, the driving force will decrease, and the reaction will slow down. The overall flux will then be a combination of Jdiss and Jdiff (Colombani, 2008). Two limiting cases are usually defined. For slow-dissolving solids (like oxides), dissolution is the slowest step and therefore drives the whole kinetics, called reaction-driven or surface-driven, kinetics. On the contrary, for fastdissolving solids (like salts or sugars), diffusion is the slowest step, and it controls the kinetics, so-called diffusion-driven or transport-driven, kinetics. For the study of drop evaporation, fast-dissolving solids are preferred to guarantee a significant release of matter in the liquid. In this case, diffusion is expected to drive the dynamics. The diffusion flux writes Jdiff B D Δc/h, where D is the diffusion coefficient of the dissolved species, and Δc is the concentration difference inside the drop of height h. If we consider the exemplifying case of a water drop evaporating on NaCl, D is 1029 m2 s21, and Δc amounts at most to the solubility of NaCl in water—that is, 17%. For a drop of height h B 1 mm, this leads to a diffusion flux Jdiff B 2 3 1027 m3 m22 s21. Two conclusions can be drawn from this order of magnitude computation:
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The dissolution kinetics of all fast-dissolving solids ($1029 m3 m22 s21) should be faster than the mass transport flux Jdiff in the drop. Therefore, the overall dynamics of dissolution should be governed by diffusion and be therefore quite identical for all these solids, D varying little from one material to another. The evaporation and overall dissolution fluxes are comparable for fast-dissolving solids, and both mechanisms should therefore interfere during drop evaporation on a soluble substrate.
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176 Droplet Wetting and Evaporation
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13.1.3 COUPLING MECHANISMS BETWEEN DISSOLUTION
AND
EVAPORATION
Dissolution is expected to have an influence on the behavior of the liquid and of the solid. Dissolution will first induce the buildup of vertical concentration gradients inside the drop. It is also likely to anchor the triple line. The divergence of the evaporation flux at the pinned triple line will then lead to outward flows inside the drop, as in the case of the evaporation of colloidal suspensions, therefore creating horizontal concentration gradients. These matter gradients may create hydrodynamic instabilities, leading either to gravitational convection (solutal RayleighBe´nard) or capillary convection (solutal MarangoniBe´nard convection). These flows will in turn modify the evaporation and dissolution kinetics. In addition, these flows of matter inside the drop are expected to lead to a transfer of matter from the center to the periphery of the drop, thereby inducing the deposit of a ring at the end of the evaporation (like in the coffee stain situation; see Chapter 4). Because the dissolution proceeds via etch pitches deepening, the roughness of the substrate is also likely to be largely increased. In the following, we provide a short overview of recent works that deal with dissolution and/or solute concentration effects during the evaporation of a sessile drop.
13.2 Evaporation in presence of solute concentration with or without dissolution A study of the effect of saline Rayleigh convection on an evaporating sessile drop deposited on a nonsoluble substrate was recently carried out by Kang et al. (2013). The authors investigated the evaporation dynamics using NaCl aqueous drops on a hydrophobic smooth substrate and showed that the evaporation-induced density gradient affects the evaporation dynamics. Starting with a homogeneous concentration for the solute, concentration and temperature gradients progressively build up due to the fluid motion initiated by the evaporation at the drop surface. The resulting surface tension gradient will be the cause of a Marangoni flow, while another type of flow, due to the evaporation-induced density gradient (so-called evaporationinduced Rayleigh convective flow) and the resulting spatially nonuniform buoyancy force, will be generated as well. The internal flow in an almost hemispherical evaporating drop was visualized by a particle image velocimetry technique. It was shown that a stable toroidal vortex takes place, the flow being directed upward at the center, whatever the initial concentration in the range 0.0110.00 wt% mixture (see Figure 13.2 (Left)). Assuming that the problem is axially symmetric, a set of nondimensionalized governing equations was derived for the mass, momentum, and solute conservation
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Soluble Substrate 177 AU:1
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FIGURE 13.2 (Top) Flow visualization with fluorescent tracers for NaCl concentrations of (a) 0.01 wt%; (b) 0.1 wt%; (c) 1 wt%, with an exposure time of 20 s; and (d) 10 wt%, with an exposure time of 2 s. (Bottom) Effect of Rayleigh number on the concentration distribution and flow field in the pseudo-steady regime for Pe 5 0.197: (a) Ra 5 5 3 103; (b) Ra 5 104; (c) Ra 5 5 3 104; and (d) Ra 5 105. The left half of the plane is for the concentration distribution, and the right half of the plane is for the velocity field. Reproduced with permission from Kang et al. (2013). Copyright 2013, AIP Publishing LLC.
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178 Droplet Wetting and Evaporation
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equations, introducing the nondimensional Peclet (Pe), salinity Rayleigh (Ra), and Prandtl (Pr) numbers, respectively. Pe 5 a0u0/D, Ra 5 gβc0a03/νD, and Pr 5 ν/D, where a0 5 1023 m stands for the initial base radius of the drop, u0 5 1026 m s21 is the constant contracting velocity of the drop surface, g 5 9.81 m s22 is the acceleration of gravity, β 5 7.1023 wt%21 is the solutal expansion coefficient, c0 5 1 wt% is the initial concentration, ν 5 1026 m s22 is the kinematic viscosity, and D 5 1029 m s22 is the molecular diffusivity of the solute. The main findings are as follows. Because almost no fluid motion was observed during the evaporation of a drop of pure water, it was assumed that the thermal contribution to the flow was negligible. Interestingly, it was observed that the boundary condition at the drop surface was no slip instead of the free surface condition. It is likely due to either the presence of a surfactant in the liquid or a contaminant in the air, even if the experiments were carefully conducted in a closed chamber at a temperature of 25 C and a relative humidity of 45%. This effect has been recently debated for other systems where the slip condition at a liquidgas interface plays a crucial role (Bolognesi et al., 2014; Berkelaar et al., 2014; Ybert and Di Meglio, 1998). Because the convection time is greater than the molecular diffusion time, the low flow field (of creeping-flow type) can be decoupled from the concentration field. It follows that the body force (Ra(a/a0)2(c 2 c0 )/c0), where a is the base radius, c is the concentration, and c0 is the concentration at the origin), proportional to the effective concentration, is rapidly time-invariant and that this pseudo-steady state is rapidly reached, even in the presence of convection, as checked by the authors experimentally and numerically (Figure 13.2 (Right)). Thus, in this configuration of Pr .. 1, the flow field can be considered by neglecting its time history, and it is only governed by the Rayleigh number, while the concentration field is driven by the Peclet number. As the body force determines the fluid motion, a scaling analysis shows that the flow velocity is proportional to (RaPe) in the low Rayleigh-number-limit, while it scales approximately as (Ra1/2) in the high Rayleigh-number-limit, probably because of the mixing enhancement (see Figure 17 in Kang et al., 2013, for a comparison of computed and measured velocity versus evaporation time for various Rayleigh numbers). It is well known that a polymeric coating could affect the wettability of a substrate. It is particularly the case for water drops deposited on substrates coated with a hydrophilic soluble polymer (i.e., a good solvent for the drop). Tay et al. (2010) have extensively studied the wetting dynamics and the drying of water drops deposited onto soluble-polymer-coated silicon substrates, using either a neutral polymer (PDMA) or two charged polymers—namely, a cationic poyelectrolyte (PDADMAC) or a zwitterionic polymer (PZ). In all cases, around a 200 nm-thick coating is deposited by spin coating. The authors conclude that both the spreading and the pinning dynamics depend on the type of polymer used. Basically, they observe that after an initially
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Soluble Substrate 179 AU:1
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FIGURE 13.3 (Top Left) Radius evolution for a water drop with the tree polymers during spreading, pinning, and receding. (Top Right) Schematic drawing of the mechanisms at stake in the vicinity of the contact line during evaporation. (Bottom) After complete evaporation, top view and schematic height profile of the deposit from left to right PDMA, PDADMAC, and PZ polymers. Reproduced in part from Tay et al. (2010) with permission of The Royal Society of Chemistry.
spreading stage, for charged polymers, the drop undergoes pinning and then recedes at the very end of the evaporation, while it recedes immediately after the spreading for PDMA (with a constant contact angle; Figure 13.3 (Top Left)). Since the polymers are not chemically grafted onto the substrate, they are essentially dissolved into the drop. A detailed characterization of the residual deposit shows that, for charged polymers, it forms a ring-like accumulation at the periphery of the drop, while it forms a crater at the center of the drop for the neutral polymer (Figure 13.3 (Bottom)). The spreading dynamics is similar for the three configurations, but the difference in the evaporation dynamics between neutral and charged polymers can be explained as follows. Rapidly after the spreading, polymer molecules accumulate in the vicinity of the contact line, resulting in a glassy zone that will not contribute further to the evaporation. A concentration gradient builds up and makes a transition zone between the bulk and the glassy area, whose size will be essentially driven by the nature of the polymer. The dynamics of the contact line will depend on the relative strength of the evaporative flux (that will tend to make the contact line recede), and the osmotic pressure present in the transition zone (that will tend to promote a flux of water toward the edge; Figure 13.3 (Top Right)). For the charged polymers, these two contributions equilibrate, which causes the pinning of the contact line in conjunction with the formation of a deposit of
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180 Droplet Wetting and Evaporation
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polymer. For the neutral polymer, the osmotic pressure is negligible, because it originates from the high concentration of electric charges. As a consequence, the contact line starts to recede immediately after spreading, leaving a continuous glassy deposit on the surface. It must be noted that when salt is added to charged polymers, as expected, the pinning disappears because the osmotic pressure weakens due to the induced screening of the charges in the transition zone. For a detailed study of the spreading dynamics on soluble substrates, the reader can refer to Dupas et al. (2013) and the references herein. If we now focus on the evaporation of a drop deposited on an initially flat and smooth soluble substrate (i.e., not a soluble coating), it is expected that the solvent will quickly initiate a process of dissolution. The consequences are twofold. First, a concentration gradient will be established inside of the drop. On the other hand, the surface condition will be changed during evaporation. Cordeiro and Pakula (2005) investigated numerically the behavior of evaporating nanometer-sized drops at nonsoluble and soluble surfaces. They performed inlattice Monte Carlo simulations to investigate the evaporation dynamics, as well as the deposit formation. They used the cooperative motion algorithm method with periodic boundary conditions by considering liquid drops on solid in equilibrium with a vapor phase. The details about the method can be found in Cordeiro and Pakula (2005). Roughly, this method involves an ensemble of beads on a tridimensional lattice. There is no empty site. Three kinds of beads are considered in their approach: pure solvent particles, substrate particles, and vacuum. Each bead can replace one of its neighbors. At each time step, the probability of a jump for a bead is given by the boltzmann factor p 5 exp(2E/kT), the sum of all the displacements being zero. kT is the temperature at equilibrium, and E is the energy of interaction of the moving bead given by Ei 5 nijeij 1 nikeik for the site i, with e the interaction energy between two beads and n the number of interactions. The jump is considered if and only if p is greater than a random number. To account for evaporation, if a solvent bead is surrounded by vacuum beads, then it is transformed in a vacuum bead. Gravity is not taken into account. When the energy interactions are carefully set via an appropriate density function, it is possible to observe microwell and ring formation for a swelling substrate. The numerical results are obtained for a drop with an initial contact angle being roughly 80 , measured by fitting the shape of the generated drop as a spherical cap. It shows that if the evaporation takes place on a nonsoluble homogeneous and smooth substrate, as expected, the contact radius decreases and the contact angle remains constant (Figure 13.4 (Right a)). For a soluble substrate, first the system follows this regime, but a microwell is progressively dug, and a peripheral ring is formed underneath the drop that will cause a pinning of the contact line. Figure 13.4 (Left) shows a comparison of the evaporation dynamics for
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Soluble Substrate 181 AU:1
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FIGURE 13.4 (Left) Snapshots of evaporating drops. Left: nonsoluble substrate, Right: soluble substrate. For each snapshot, the radial average is also shown, together with the number of solvent beads remaining within the drop. (Right) Evolution during the drop evaporation, and for each profile, the corresponding fit with a spherical cap. (a) Nonsoluble substrate, (b) soluble substrate. Reprinted with permission from Cordeiro and Pakula (2005). Copyright (2005) American Chemical Society.
nonsoluble and soluble substrates. As a result, the contact radius becomes constant, and the contact angle starts to decrease (Figure 13.4 (Right b)). This study supports that the evaporation/dissolution of a drop modifies the topography of the surface and in turn affects the evaporation dynamics, partly because of the induced pinning of the triple line. It must be noted that the authors claim that the results can be extended to micrometer-sized or even millimeter-sized drops. It should be pointed out that dissolution-induced watermarks can unexpectedly occur during evaporation. Such an example is presented in Belmiloud et al. (2012). The dynamics of the drying of a sessile drop of ultrapure water on silicon substrates have been studied. The silicon surfaces were made by dHF-etching (0.5% HF for 5 min, so-called hereafter “hydrophobic”), while more hydrophilic surfaces were obtained by exposing these surfaces to the air for 1 day (so-called hereafter hydrophilic), allowing for the formation of a hydrophilic layer of oxidized Si. The experiments have been conducted in a well-controlled ultra-clean environment (clean room class 1, relative humidity of 40%, temperature of 22 C) with deoxygenated water. Sessile drop evaporation was carried out in both hydrophobic (just after dHF) and hydrophilic configurations. On a hydrophobic Si surface, a pinned phase
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182 Droplet Wetting and Evaporation 0.7
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FIGURE 13.5 (Left) Evolution of the square diameter of a water drop deposited on a Si wafer (a) and a SiO2 wafer (b) at the end of the evaporation—that is, before and after the formation of a watermark (plateau). (Right) Topology taken at the center across a ring-shaped watermark for a hydrophilic Si wafer. Reprinted with permission from Belmiloud et al. (2012). Copyright 2012, The Electrochemical Society.
was first observed (constant contact radius, decreasing contact angle starting from 70 ), followed by a receding phase (contact angle of 32 , decreasing radius), as shown in Figure 2 of Belmiloud et al. (2012). By comparing the evaporation on Si and SiO2 surfaces in the receding phase, it is observed, as expected, that smaller evaporating drops dry slower with a higher receding contact angle. In both cases, the liquid-solid surface area decreases linearly with time. Focusing on the very last stage of evaporation, a similar behavior is observed for Si and SiO2 wafers. Interestingly, it is shown that just before complete evaporation, the contact line jumps due to the depinning on a residual ring, or so-called watermark (Figure 13.5 (Left)). After the depinning, the decrease in contact surface is faster than before the pinning begins, while the contact angle is measured almost constant around 2 for both Si and SiO2 surfaces. A high-resolution profilometry (HRP) of the surfaces clearly shows a so-called coffee ring made of thin residues, while a cluster of bigger residues is visible at the center (Figure 13.5 (Right)). The thickness of the ring is larger for SiO2 than for Si. In this specific case, the deposits are known to be a result of chemical reactions. First, the ambient oxygen that diffuses into the water drop will create silica and silicic acids. The resulting Si(OH)4 molecules will form hydrated polymeric chains due to their rising concentration throughout the evaporation. At the final stage, these silica-based residues of very low solubility in water will precipitate and will show dynamics similar to those observed with evaporating colloidal solutions. Beyond this specific case, which shows how complex the cleaning process of wafers for microelectronics applications can be, it must be pointed out that it is common for the process of evaporation of a sessile drop to involve some sort of reaction with the substrate that might significantly affect both the evaporation dynamics and the subtract integrity.
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Soluble Substrate 183 AU:1 500 μm
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FIGURE 13.6 (Left) Side view of a 3 μL pure water drop deposited on a NaCl substrate at initial time and after complete evaporation (t 5 88 s). The initial contact angle is about 15 . (Right) Top view of a 3 μL pure water drop on a NaCl substrate at different times. The deposit is clearly visible.
13.3 Water drop evaporation on a flat crystal salt substrate The evaporation of a pure water drop deposited on NaCl soluble single crystals of salt has been explored. After the crystals have been cleaved (10 3 10 mm, 2 mm in height), they are polished with sandpaper (grit size down to 3 μm). A water drop of a few microliters is gently deposited on the substrate. A contact angle measurement device is used to follow the evaporation dynamics. The drop evaporates at ambient pressure inside a transparent closed cell (100 3 100 3 100 mm3) to avoid any perturbations from the ambient. The side view is recorded throughout the evaporation via a CCD camera fitted with a macro lens positioned laterally. To complete this observation, an upright reflected light microscope fitted with a high-resolution camera is used to observe surface modifications after evaporation. These preliminary experiments showed unambiguously the strong influence of the substrate dissolution on the evaporation process for a water drop. After drop deposition, it is observed that the contact line is nearly circular for well-polished crystals substrates. First, it induces a pinning of the triple line during almost the whole evaporation (Figure 13.6 (Left)), instead of a receding triple-line typical of water drops on soft inert substrates. Second, the evaporation/dissolution interplay leads to an increase of the roughness of the solid surface. Finally, like in colloidal suspensions, the evaporation of a pure water drop on a soluble substrate results in the deposit of a ring at the periphery of the drop. Top view observations clearly reveal the accumulation of a ring-shaped deposit at the edge of the drop that seems to be homogeneous at the end of the evaporation (Figure 13.6 (Right)). A typical 3D profilometry of the surface is shown in Figure 13.7. It reveals a digging at the center as deep as 15 μm, while the deposit reaches around 15 μm
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184 Droplet Wetting and Evaporation μm
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−10.0
1.5 mm
−15.0 −20.6 f0040
p0170
FIGURE 13.7 Three-dimensional profile obtained by profilometry after complete evaporation.
above the surface. These results are consistent with those obtained for the numerical study of the drop evaporation on the soluble substrate. On the contrary, no digging was measured for the evaporation with watermarks formation on a silicon wafer discussed above. In the long term, this work demonstrates the influence of the dissolution on the kinetics of evaporation of a drop and the formation of the deposit formed after the evaporation process.
References Arvidson, R.S., Luttge, A., 2010. Mineral dissolution kinetics as a function of distance from equilibrium— New experimental results. Chem. Geol. 269, 7988. Belmiloud, N., Tamaddon, A.H., Mertens, P.W., Struyf, H., Xu, X., 2012. Dynamics of the drying defects left by residual ultra-pure water droplets on silicon substrate. ECS J. Solid State Sci. Technol. 1 (1), 3439. Berkelaar, R.P., Dietrich, E., Kip, G.A., Kooij, E.S., Zandvliet, H.J., Lohse, D., 2014. Exposing nanobubble-like objects to a degassed environment. Soft Matter 10 (27), 49474955. Bolognesi, G., Cottin-Bizonne, C., Pirat, C., 2014. Evidence of slippage breakdown for a superhydrophobic microchannel. Phys. Fluids 26, 082004. Colombani, J., 2008. Measurement of the pure dissolution rate constant of a mineral in water. Geochim. Cosmochim. Acta 72, 56345640.
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Soluble Substrate 185 AU:1 Colombani, J., Bert, J., 2007. Holographic interferometry study of the dissolution and diffusion of gypsum in water. Geochim. Cosmochim. Acta 71, 19131920. Cordeiro, R.M., Pakula, T., 2005. Behavior of evaporating droplets at nonsoluble and soluble surfaces: modeling with molecular resolution. J. Phys. Chem. B 109, 41524161. Dupas, J., Verneuil, E., Talini, L., Lequeux, F., Ramaioli, M., Forny, L., 2013. Diffusion and evaporation control the spreading of volatile droplets onto soluble films. Interfacial Phenom. Heat Transf. 1 (3), 231243. Kang, K.H., Lim, H.C., Lee, H.W., Lee, S.J., 2013. Evaporation-induced saline Rayleigh convection inside a colloidal droplet. Phys. Fluids 25, 042001. Lasaga, A., Luttge, A., 2001. Variation in crystal dissolution rate based on a dissolution stepwave model. Science 291, 24002404. Rickard, D., Sjo¨berg, E.L., 1983. Mixed kinetic control of calcite dissolution rates. Am. J. Sci. 283, 815830. Tay, A., Lequeux, F., Bendejacq, D., Monteux, C., 2010. Wetting properties of charged and uncharged polymeric coating—effect of the osmotic pressure at the contact line. Soft Matter 7 (10), 47154722. Ybert, C., Di Meglio, J., 1998. Ascending air bubbles in protein solutions. Eur. Phys. J. B 4, 313319.
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To protect the rights of the author(s) and publisher we inform you that this PDF is an uncorrected proof for internal business use only by the author(s), editor(s), reviewer(s), Elsevier and typesetter MPS. It is not allowed to publish this proof online or in print. This proof copy is the copyright property of the publisher and is confidential until formal publication.
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NON-PRINT ITEM
Abstract The studies of drop evaporation focus on the behavior of the liquid and on the AU:2 influence of the environment. The substrate always plays a role on the evaporation, through heat exchange and via its wetting properties, but most of the time it is chosen as inert. Generally two model situations can be envisaged with regard to the evaporation of a sessile drop on an inert substrate. Either the contact line is free to move, so that the contact angle remains constant as the radius of the drop decreases, or the contact line is to be pinned, and then the contact angle decreases over time. In both situations the loss of mass obeys a power law with different exponents. In addition, mass loss due to evaporation, mainly at the contact line, is compensated by an outward flow. When colloidal particles are initially present in the liquid, the latter are deposited at the contact line, thus promoting the pinning. Keywords: Dissolution; hydrophobic; interferometry; osmotic pressure; polymer; AU:3 substrate
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