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Final Project Influence of water microfilm on the sliding of water droplet Author Ignacio Jaime Domingo Director Zbigniew Zapałowicz Faculty of Mechanical Engineering and Mechatronics 2012 Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 2 1. Nomenclature ......................................................................................................................... 3 2. Aim and introduction ............................................................................................................. 4 3. Intermolecular Forces in Phase-Change Heat Transfer .......................................................... 5 3.1 Introduction ...................................................................................................................... 6 3.2 Basic Concepts ............................................................................................................... 10 3.2.1 Clapeyron effect ...................................................................................................... 10 3.2.2 Kelvin effect ............................................................................................................ 12 3.3 Interfacial free energy and Hamaker constant ................................................................ 15 4. Influence of dispersion forces on phase equilibria between thin liquid films and their vapour ............................................... 18 4.1 Introduction .................................................................................................................... 19 4.2 The nature of dispersion forces ...................................................................................... 23 4.3 The chemical potential of system dispersion forces ....................................................... 24 4.4 Dispersion forces and phase equilibria ........................................................................... 26 4.4.1 The Young - Laplace equation ................................................................................ 29 4.4.2 The Kelvin equation ................................................................................................ 33 4.5 Gas and liquid pressures apart from the interfaces ........................................................ 36 5. Experiments .......................................................................................................................... 38 5.1 Experiment 1 .................................................................................................................. 39 5.2 Experiment 2 .................................................................................................................. 41 6. Experimental data ................................................................................................................. 43 6.1 Experiment 1 .................................................................................................................. 44 6.1.1 Evaluation of the data .............................................................................................. 44 6.2 Experiment 2 .................................................................................................................. 59 7. Evaluated of microfilm thickness (Vladimir S. Ajaev) ........................................................ 64 7.1 Introduction .................................................................................................................... 65 7.2 Formulation .................................................................................................................... 69 7.3 Calculations .................................................................................................................... 73 8. Comparison with experimental results ................................................................................ 75 8.1 Model ............................................................................................................................. 76 9. Conclusions .......................................................................................................................... 83 10. Bibliography ....................................................................................................................... 85 Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 3 1. Nomenclature Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 4 2. Aim and introduction The aim of the project is investigate the phenomena of sliding of single water droplet on the flat surface when the solid body is covered by thin water microfilm. Based in a theory explanation and in the data of next experiments, the dynamic microscopic advanced angle has to be calculate and estimate with the most precision as it is possible. Then, after comparing with previous data experiments, we have to find which parameters have influence in this phenomena and in that way can change the evolution and the results of this phenomena. Moreover, the investigation of this phenomena have to be a reference for investigators in the future. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 5 3. Intermolecular Forces in Phase-Change Heat Transfer Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 6 3.1 Introduction Experimental observations and theoretical simulations of the effects of long-range intermolecular forces have demonstrated that the properties of small liquid systems Ž such as an evaporating drop . deviate from those of a bulk liquid. Deviations also occur near the junction of a liquid film with a substrate. Still additional deviations occur as the thickness of a thin liquid film decreases (such as near the ‘‘contact line’’ where the liquid-vapor interface intercepts the ‘‘substrate’’ ). These regions have received extensive study because of their fundamental importance in nature. Herein, this broad literature is reviewed and connected for the purpose of demonstrating its efficacy in modeling change-of-phase heat transfer (such as boiling, heat pipes, grooved evaporators and condensers, rewetting of heated surfaces, and so on ). An evaluation of the literature leads us to the conclusion that it is possible to start with two of the four fundamental forces of nature (gravity and electromagnetic (intermolecular )) and obtain the effects of shape (thickness and curvature ) , concentration, and temperature on the interfacial heat-transfer coefficient and stability of a thin liquid film. This basic understanding has led to important developments in heat-transfer systems. There is an extremely rich literature concerning equilibrium and no equilibrium curved liquid films. Therefore, judicious choices are made so that a coherent and consistent understanding of the phenomena of particular use to change-of-phase heat transfer can be discussed. For example, although computational molecular dynamics is extremely useful in analyzing the dynamics of molecules in small systems, additional development of this field to emphasize interfacial mass transfer is still needed to have an impact on change-of-phase heat-transfer systems. The gradient theory of fluid microstructures is not covered for the same reason translation of van der Waals, Davis and Scriven (1981 ) , and Evans et al. (1986 ) . Instead, models based on classical interfacial kinetic theory, quasi-thermodynamics, and continuum concepts are used to simulate the area averaged results that are experimentally observed. However, additional simulation at the molecular level will be needed as systems become smaller and optimized. In essence, the literature on the effect of a no uniform thin liquid film with shape dependent properties on phase-change heat transfer is reviewed and Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 7 connected. There have been experimental evaluations of the applicability of the Kelvin equation to highly curved interfaces. For example, Fisher and Israelachvili (1981 ) demonstrated the validity of the Kelvin equation with systems as small as 4 nm. Use of the bulk viscosity in extremely small thicknesses is still open to question. For example, conflicting results have been presented by Klein and Kumacheva (1995 ) and Knudstrup et al. (1995 ) . Although roughness can have an additional effect in some cases, roughness greater than that at approximately the molecular level is not addressed. Concerns about using continuum models with ultra-thin films are partially alleviated by averaging over a sufficiently large surface area and time so that the theoretical results can be experimentally evaluated for confirmation. Confirmed results can then be used for engineering designs. Since a liquid is deformable, the shape of a liquid film is a function of the three-dimensional (3-D ) intermolecular force field. Therefore, the transport processes in a thin film are a function of the liquid-solid system, temperature, concentration, and the shape which is a measure of the varying internal pressure (intermolecular force ) field. The optical measurement of shape gives the otherwise difficult measurement of relative pressure in small systems. Two powerful optical techniques based on reflectivity to measure the change in the thin film profile (intermolecular force profile ) with heat transfer are discussed herein: image analyzing interferometer (IAI ) ( DasGupta et al., 1995 ) and ellipsometry (Kim and Wayner, 1996 ). Other optical techniques are discussed by Oron et al. (1997 ). As outlined in many recent texts, intermolecular and surfaces forces are the result of the electronic structure of atoms and molecules . At equilibrium, these forces cause the adhesion of one substance to another, the cohesion in bulk liquids, the free energy associated with interfaces, and the liquid-vapour phase distribution in a closed system. An early example of the effect of dielectric properties on change-of-phase heat transfer is given by Wayner (1978 ). The many no covalent intermolecular interactions can be broadly classified as follows: Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 8 1 . Electrodynamics Lifshitz-van der Waals (LW ) forces (comprising the sum of the London dispersion interaction between two apolar molecules or atoms; the Debye induction interaction and the Keesom orientation interaction ). 2 . Polar forces (hydrogen bonding and Lewis acid-base interactions ) 3 . Purely electrostatic forces. The London dispersion forces are always present and control transport processes in very thin films with apolar systems. These dispersion forces are longrange forces, vary inversely with the film thickness raised to a power, and can be effective over large distances: from approximately 100 nm, at which an extremely small change from the bulk vapor pressure over a thin liquid film occurs, down to interatomic spacing ( 0.2 nm ) where the vapor pressure of an adsorbed monolayer can be extremely small. As described by the adsorption isotherm, the thickness of an adsorbed film is a function of the surrounding vapor pressure and substrate temperature. Herein, completely wetting apolar systems will be used as examples because, with simple systems, the van der Waals intermolecular force field can be easily modeled and experimentally studied. Using the insights thereby gained, these concepts can be extended to include more complicated systems. For a specific example, we start with the description of a constrained vapor bubble (CVB ) heat exchanger with transparent quartz walls which embodies many of the concepts that will be subsequently discussed. The CVB, which is presented in Figure 1 for a no isothermal microgravity environment, has various thin film regions that are of both basic and applied interest. It is formed by underfilling an evacuated small container with liquid. For a completely wetting system at equilibrium, liquid will coat all the walls of the container. For a finite contact angle system, some of the walls can have only an extremely small amount of adsorbed vapor, which changes the substrate properties. Liquid will fill a portion of the corners in both cases. Since the solid walls constrain the shape of the vapor bubble and, therefore, the equilibrium liquid film, the intermolecular force field in the thin liquid film is different from that in a bulk liquid. Both equilibrium, with a uniform temperature field, and no equilibrium T2 ≥ T1 studies have been made using the CVB. Although the thermal conductivity of quartz is small, transparent walls remove the uncertainty concerning liquid Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 9 shape and are highly desirable in basic research. When heat is supplied at one end and removed at the other end, the effective thermal conductivity due to the evaporation / condensation process can be orders of magnitude greater than that of copper. Vapor flows to the cold end and liquid flows to the hot end due to the shape-dependent intermolecular force gradient. As presented for a relatively large width (±3 mm ) , the cross-sectional area for vapor flow is much larger than that for liquid flow. As a result, the pressure in the vapor space is nearly Fig. 1. Constrained vapor bubble concept for a completely wetting system in a microgravity field with heat in at end 2 and heat out at end 1. constant even with high imposed heat fluxes. In this case, the high interfacial heat-transfer coefficient keeps the liquid-vapor interface nearly isothermal, except where the film is extremely thin. In the ultrathin film region where deviations in the intermolecular force field control the transport processes, large interfacial temperature gradients are sustainable. When sufficiently stressed, dry regions with contact lines occur. At the other extreme, the isothermal system is well suited to the study of interfacial thermodynamics. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 16 The picture of the completely wetting case conceptually represents, for an extremely thin case, a monolayer or, at the other extreme, a much thicker film. For the partially wetting case, a deceptively simple quantitative measure of the complex intermolecular force field in the contact line region is the apparent contact angle θ. The contact angle is defined as the angle between the tangents to the liquid-vapor and liquidsolid interfaces at a point where the film is sufficiently thick so that the transition regions in the interfaces do not overlap. For some cases, like a small drop on a substrate, this definition is still inexact since the observable tangent angle constantly changes. For the nonwetting case, θ > 90o . At this point, it is becoming obvious that mathematical interfaces and apparent contact angles are relatively crude but useful (at times) descriptions of the complex dynamic molecular world at interfaces and contact lines. For example, try to picture the real contact angle in the contact line region of a moving contact line using a molecular dynamic simulation model. The interfacial free energy can be composed of dispersion (London) d, polar (Keesom) p, induced dipole (Debye) i, electrostatic e, and acid-base interactions AB. The first three terms in this equation are the Lifshitz-van der Waals interactions σLW. Simple fluids like the alkanes have only LW interactions. Some representative values for the terms in Eq. 9 for polar fluids are given by van Oss (1994). For apolar liquids like the alkanes emphasized below, σ = σl = σLW. To connect these concepts and the Hamaker constant with experimental observations, we assume (at times) that there is no practical difference between these processes of interfacial formation occurring in a vacuum and an environment saturated with vapor or gas (σl ≡ σlv , σs ≡ σsg or σsv). However it is also important to realize that the interfacial free energy values can, in some cases, be substantially different in laboratory air because of the adsorption of foreign vapor molecules like water and hydrocarbons. At liquid-vapor interfaces, impurities may or may not concentrate at the surface and thereby affect the value of the interfacial free energy. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 17 A further complication can arise if the environment has a foreign gas which can adsorb on the liquid substrate and change σl to σlg. Therefore, there are many practical complications when these concepts are used in modeling transport processes. On the other hand, all these additional effects can be experimentally measured and theoretically modeled. As examples, practical experiments on the effect on the environment on the surface tension of stainless steel are discussed by Mantel and Wightman (1994) and a theoretical analysis of the wetting of gold by water is given by Parsegian (1977). The reader will find it necessary to use, as the situation dictates, both practical and ideal concepts to understand and apply the state of the art in surface transport processes. The current trend towards smaller closed devices is a trend towards more ideal systems. Equation 8 is the Young equation for the contact angle of a liquid on a substrate in its equilibrium vapor environment. This can also be written as where πe = σs - σsv is the equilibrium film pressure which accounts for adsorption on the solid substrate. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 18 4. Influence of dispersion forces on phase equilibria between thin liquid films and their vapour Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 19 4.1 Introduction Micro scale heat and mass transfer in nucleate boiling strongly depends on vapour–liquid phase equilibria of a thin liquid film adsorbed between the heated wall and the vapour bubble as first pointed out by Stephan and Hammer [1]. As an example, in Fig. 1 the curved line represents the interface between a vapour bubble and the liquid layer at a heated wall. In the so-called macro region the interface has an almost constant curvature K corresponding to the bubble radius rB, K = 2 / rB. Fig. 1. Liquid film between vapour bubble and heated wall: (—) region of strong influence and (- - -) region of weak influence In the micro-region the curvature of the interface sharply turns and ends in a non-evaporating liquid film adsorbed at the wall, the curvature being there K = 0. In this adsorbed film, in the so-called adhesion zone between wall and liquid film, attractive forces inhibit evaporation. In the micro-region between macro-region and the adsorbed film the curvature of the liquid– vapour interface undergoes a steep maximum, which for example for refrigerant R114 of 0.247 MPa boiling at a wall superheated by 5 K, amounts to Kmax ≈107 m-1 . Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 20 The curvature varies over a small distance of about 0.1 µm between that of the adsorbed film K = 0, the maximum value Kmax ≈ 107 m-1 in the micro-region and an almost constant value K ≈ 103 m-1 in the macro-region, thus leading to extremely high capillary forces, a strong liquid flow towards the interface, and hence in the thin liquid film to extreme high heat fluxes of up to 107 W/m2 [1–6]. Phase equilibria at the liquid–vapour interface are different from those at plane interfaces free of dispersion forces for three reasons: 1) The capillary forces due to the strong change in curvature in the micro-region also enhance the interface temperature TPH. The superheat over saturation temperature of a plane interface necessary for evaporation of the curved interface increases with capillary pressure. 2) Because of the thin liquid film in the micro-region the temperature drop TW - TPh between wall and gas–liquid interface can be as small or even smaller than the temperature drop TPh - TG between gas– liquid interface and gas-core. The molecular kinetic resistance at the liquid–vapour interface has to be taken into account. 3) In the adhesion zone and the micro-region long range interaction forces, so called dispersion forces, between solid wall and the molecules of the thin liquid film rise the temperature TPh at the interface. Because of the variation of curvature K(ζ) and film thickness δ(ζ) the interface temperature TPH is not constant but a function of the radial co-ordinate ζ. We can therefore state, as shown in Fig. 2, that the temperature drop TPh - Tsat is considerable, and thus the driving temperature difference TW - TPh for heat transfer is smaller than TW - Tsat of a plane film, when the dispersion forces and intermolecular resistance are negligible. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 21 The smaller temperature difference TW - TPh compared to TW - Tsat reduces the heat flux. This reduction in the micro-region, however, is partly compensated by the much smaller thermal resistance of the thin liquid film. Basic and pioneering theoretical and experimental studies of evaporating thin liquid films were mainly performed by Wayner and coworkers. A concise review of their research is given by Wayner. As shown by them the assumption of a constant interface temperature TPh and also a constant curvature K = const in the microregion does not hold. It is obvious that the liquid–vapour phase change is of fundamental interest and of practical significance in nucleate boiling heat transfer. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 22 A useful tool for the study of phase equilibria is the Kelvin equation, which shall be presented here for pure substances. Its derivation is usually based on the assumption of a fluid with ideal vapour and an incompressible liquid phase. Intermolecular forces between solid and liquid are not taken into account. In our case we also assume an ideal vapour and an incompressible liquid. However, because of the presence of dispersion forces the intermolecular forces between solid and liquid must be taken into account. They add to the capillary forces. Furthermore the temperature rises at the liquid–vapour interface because of the molecular kinetic resistance. Both effects cause a shift in the chemical potential, which shall be discussed in this paper. Then, with the aid of the Kelvin equation, deduced from the chemical potential, the variation of pressures in the liquid and vapour phase, as they occur in nucleate boiling when vapour bubbles exist at solid walls, will be studied. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 23 4.2 The nature of dispersion forces The adhesion of dispersion forces between the solid wall and the liquid film can be attractive or repulsive. As long range forces they can be effective from distances of interatomic spacing in the liquid film of about 0.2 nm to distances of some 10 nm. They are quantum mechanical in origin, and are present even if molecules are non-polar. Though the time average dipole of these molecules vanishes there exists a finite dipole moment given by the instantaneous position of the electron around the protons. These instantaneous dipoles generate an electric field inducing a dipole moment in any nearly neutral atoms giving rise to an instantaneous attractive force between the atoms. The time average of this force is finite. Attractive forces mainly determine the boiling point and the heat of vaporization of a substance, hence their importance when studying boiling processes. Dispersion forces were intensively studied by Israelachvili and in earlier papers by Derjaguin, who developed a model to convert these forces into a liquid pressure. The dispersion forces lead to an additional pressure pdisp in the liquid film, which under the assumption of an interaction pair potential w = -C / rn with coefficient C describing the interaction between two molecules and for example n = 6 for van der Waals forces becomes where η is the distance between a certain position within the liquid and an infinitely extended solid surface. The dispersion constant describing the interaction between two bodies 1 and 2 is defined as ρ1 and ρ2 are the number densities of molecules in the interacting bodies, and coefficient C describes the interaction between the atoms in the bodies. Instead of the dispersion constant Adisp the Hamaker constant A, after Hamaker , who did earlier studies to explore the forces between macroscopic bodies, is often used in the literature, Hamaker constants of van der Waals forces and hence dispersion constants can be calculated from bulk properties such as dielectric constants and refractive indices. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 24 4.3 The chemical potential of system dispersion forces With the aid of dispersion forces we can now proceed to determine the chemical potential of a homogeneous bulk phase, for instance the liquid phase in Fig. 3. Fig. 3. Dispersion forces acting on a liquid volume of unit area In all the literature known to the author, it is taken from the Gibbs–Duhem equation if gravity is not negligible. Quantities with an overbar represent molar quantities: S( p, T) are the molar entropy and V( p, T) the molar volume. If the above equations for the chemical potential were correct then according to the thermal equation of state V( p, T) or p = p( ρ, T) where ρ is the density, for an incompressible fluid the pressure inside the fluid would only be a function of temperature. This is correct when dispersion forces are absent, but does not hold when they are not negligible. As we saw before, Eq. (1) then the dispersion pressure becomes important, depending on the distance between fluid and the solid surface. We should therefore expect p = p( ρ, T,r), where r stands for the distance, and hence a chemical potential µ = µ( ρ, T ,r) and not µ = µ( ρ, T) as in the above equation. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 25 In order to derive the correct chemical potential of liquid phase we start from the Gibbs function with entropy S, volume V, co-ordinate r in the liquid and the mole number n. From Eq. (4) we obtain with Temperature T, pressure p, and chemical potential µ are defined as partial derivatives of E, T = δE / δS, p = -δE / δV, µ = δE / δn, but now depend on S, V, r, n. Strictly speaking, Fdisp stands here for all forces acting on the system. For reasons of simplicity we disregard all other forces except for the dispersion forces as the most important forces for evaporation of thin liquid films. Because the Gibbs function E, Eq. (4), is a homogeneous function of first order in the variables S, V, n its Euler-equation reads From this we have Together with Eq. (5) we obtain the Gibbs-Duhem equation The chemical potential µ = µ( ρ, T ,r) of the pure liquid therefore is given by where S, V and Fdisp again represent molar quantities. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 32 and Herein the Hamaker constants Aii are those of two dielectric or electrically non-conducting media i interacting across vacuum. They are given by the Lifshitz theory where εi is the dielectric constant of medium i, ni its refractive index in the visible, k is the Boltzmann constant, h the Planck constant and ve the Plasma frequency of the free electron. In nucleate boiling processes the solid wall is mostly a metal. For two metals interacting across vacuum the Hamaker constant is ASS ≈ 4*10-19 J. The Hamaker constants AS,L,σG and AS,Lσ,G are both negative, because of ASS > ALσ,Lσ > ALL, ALL > AσG,σG and AGG = 0. The negative sign is in agreement with the usually adopted sign convention according to which attractive forces are negative and repulsive forces positive. As follows then from the definition Eq. (11), dispersion pressures due to attractive forces are positive and those due to repulsive forces negative. The interaction forces between a solid wall and a wetting liquid surpass those between solid wall and gas- or σ-phase. The gas and σ-phase are kept away from the solid wall through the liquid, an effect which is reflected in negative values of the Hamaker constants AS,L,σG and AS,Lσ,G. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 33 4.4.2 The Kelvin equation With Eq. (15) the chemical potential Eq. (8) of a system with dispersion forces can be written Hence the chemical potential is µ = µ(T,p,pdisp). Strictly speaking a gravity term Mgdr should also appear in Eq. (24). As we are interested in phase equilibria this term appears, however, in the chemical potential of all phases in equilibrium and therefore cancels. We can leave it out here. Fig. 5. Local phase equilibrium between liquid L with gas G and σ-phase. We consider now (Fig. 5) an element dξ of a curved liquid film L of thickness δ(ξ) in phase equilibrium with its vapour G and the σ-phase between liquid and vapour. The temperature at the interfaces and in the σ-phase be TPh(ξ), the liquid-side pressure pPh,L and the gas-side pressure pPh,G. Phase equilibrium requires The condition for phase equilibrium at temperature TPh of the same substance if it had a plane interface and if dispersion forces were negligible would be Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 34 From that follows psat = psat (TPh). From Eqs. (25) and (26) we obtain and hence from Eq. (24) for a given interface temperature TPh: Assuming the liquid and the gas to be incompressible, we obtain The pressures pPh,L at the liquid interface and pPh,G at the vapour interface are connected by Eq. (19), in which the interface pressures pPh,L and pPh,G now replace the pressures pL and pG of the homogeneous phases. Elimination of p pPh,L in Eq. (27) and introducing the density ρ instead of the molar volume V = M / ρ with molar mass M, we obtain the Kelvin equation for the gasside pressure Corresponding, elimination of pPh,G in Eq. (27) yields the kelvin equation for the liquid-side pressure Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 35 It should be noted that the dispersion pressures pdisp,Ph,G and pdisp,Ph,L on both sides of the σ-interface are negative for a wetting liquid, as stated before. The Kelvin equations, Eqs. (28) and (29), hold for local equilibrium of any evaporating system independently if it is open or closed. For a spherical bubble (closed system) of constant curvature K in a liquid faroff from the solid wall we obtain the well known relations and The liquid pressure pPh,L and the gas pressure pPh,G at the temperature TPh are lower than the saturation pressure Tsat ( TPh ), Fig. 6. For a given saturation temperature Tsat the liquid must be superheated by ΔT = TPh - Tsat for vapour bubbles to exist. Fig. 6. Liquid and vapour pressure for spherical bubble, curvature K. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 36 4.5 Gas and liquid pressures apart from the interfaces If evaporation occurs, the gas pressure pG and the liquid pressure pL apart from the curved interface δ, Fig.1, differ from the interface pressures pPh,L and pPh,G. In the bulk of the gas, because of the intermolecular forces, the gas pressure pG is slightly lower than the interfacial pressure pPh,G , the difference pPh,G - pG being the driving force for evaporation. As follows from the kinetic theory or gases we have: with the heat flux q the condensation coefficient f and the enthalpy of evaporation Δhv. Only for vanishing heat fluxes q = 0, we have pPh,G = pG. In the micro region, where heat fluxes are very high and of the order 107 W/m2 the pressure difference pPh,G - pG is not negligible as the following example. With the aid of the Kelvin equation (28), together with Eq. (24) we also obtain the pressure pL (ξ, η) in the liquid film. Proceeding from the interface η = δ(ξ) where we have pPh,L = pPh,L (ξ, η = δ) in the direction η and neglecting gravity in the very thin liquid film, we have With Eq. (11) we find Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 37 The difference of the two dispersion pressures describes the pressure increase inside the liquid because of the action of dispersion forces on the liquid column of height δ - η. Adisp,SL is the dispersion constant for the solid–liquid interaction upon this liquid column. As follows from the combining rules we have which is positive for the wetting liquids considered here. Combining Eq. (33) with Eqs. (20) and (29) yields the pressure distribution in the liquid film whereas the dispersion constant Adisp,SL is positive, Adisp,Ph,L is negative as follows from Eq. (21). Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 38 5. Experiments Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 39 5.1 Experiment 1 N = Number of measurement V = Volume of water droplet (µl) ϕ = Humidity of air (%) T = Temperature of air (ᵒC) γ = Remark of the slope angle of plate ( ᵒ ) N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 5 37 20,77 59,75 2 5 37,2 20,77 73,92 3 5 36,5 20,45 52,25 4 5 36,5 20,48 45,50 5 5 36,4 20,5 27,50 1 10 36,4 20,6 32 2 10 36,3 20,63 28,08 3 10 36,1 20,58 28,25 4 10 36,4 20,55 24,83 5 10 36,3 20,6 24,33 1 15 36,3 20,65 22 2 15 36,3 20,7 21,75 3 15 41 20,79 32,33 4 15 40,7 20,93 28,66 5 15 41 20,97 41 1 20 40,6 21 25,33 2 20 41,5 21 33,33 3 20 40,5 21,02 35,92 4 20 40,5 21,03 39,25 5 20 40,6 21,03 35,25 Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 40 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 25 40,5 21,11 26,66 2 25 41,6 21,16 23,33 3 25 40,5 21,16 21,16 4 25 40,6 21,16 19 5 25 40,8 21,13 22,58 1 30 40,8 21,1 16,25 2 30 41,1 21,08 17,5 3 30 40,8 21,09 20,42 4 30 40,7 21,1 19,42 5 30 40,8 21,09 21,67 1 35 40,9 21,09 15,25 2 35 41,2 21,06 15,66 3 35 40,9 21,08 16,08 4 35 40,8 21,07 15,5 5 35 40,9 21,07 16,5 1 40 51,2 22,09 14,08 2 40 49,7 22,17 17,33 3 40 49,6 22,21 12,33 4 40 49,6 22,26 11,75 5 40 49,4 22,37 16,83 1 45 49,5 22,35 18,08 2 45 49,4 22,39 15,33 3 45 49,5 22,39 20,92 4 45 49,4 22,39 22,33 5 45 49,5 22,4 18,25 1 50 49,4 22,4 17,25 2 50 49,3 22,4 23,08 3 50 49,3 22,39 15,92 4 50 49,4 22,39 19,83 5 50 49,1 22,41 17,08 Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 41 5.2 Experiment 2 N V ϕ T Remarks (γ) µl % ᵒC ᵒ τ = 2 min 1 5 38,7 21,8 56,08 2 10 38,6 22,02 29,17 3 15 38,5 22,13 33,08 4 20 38,4 22,23 31,33 5 25 38,5 22,35 35,17 6 30 38,4 22,44 21,67 7 35 38,2 22,5 19,75 8 40 38,2 22,55 15,33 9 45 38,2 22,57 10,5 10 50 38,4 22,63 11,33 τ = 5 min 1 5 35,4 22,33 64,25 2 10 35,8 22,38 48,17 3 15 35 22,45 58,08 4 20 35,5 22,55 32,33 5 25 35,1 22,57 30,17 6 30 35,2 22,65 20 7 35 34,9 22,67 29,5 8 40 34,9 22,71 17,58 9 45 34,9 22,76 15,17 10 50 34,7 22,81 12,25 τ = 10 min 1 5 34,8 22,73 61,33 2 10 34,6 22,93 52,08 3 15 34,6 23,01 21,33 4 20 31,7 22,13 18,17 5 25 32 22,15 28,58 6 30 31,9 22,19 46,17 7 35 31,9 22,13 28,25 8 40 32,1 22,14 25,25 9 45 32,2 22,1 32,92 10 50 32,3 22,08 25,33 Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 48 µl = 15 γaverage 29,15 σ 2,74 γ + σ 31,89 γ - σ 26,41 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 15 36,3 20,65 22 2 15 36,3 20,7 21,75 3 15 41 20,79 32,33 4 15 40,7 20,93 28,66 5 15 41 20,97 41 Average 38,58 20,77 26,19 22 21,75 32,33 28,66 41 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 49 µl = 20 γaverage 33,82 σ 2,12 γ + σ 35,93 γ - σ 31,70 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 20 40,6 21 25,33 2 20 41,5 21 33,33 3 20 40,5 21,02 35,92 4 20 40,5 21,03 39,25 5 20 40,6 21,03 35,25 Average 40,775 21,02 35,9375 25,33 33,33 35,92 39,25 35,25 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 50 µl = 25 γaverage 22,55 σ 1,57 γ + σ 24,12 γ - σ 20,98 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 25 40,5 21,11 26,66 2 25 41,6 21,16 23,33 3 25 40,5 21,16 21,16 4 25 40,6 21,16 19 5 25 40,8 21,13 22,58 Average 40,8 21,14 22,55 26,66 23,33 21,16 19 22,58 0 5 10 15 20 25 30 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 51 µl = 30 γaverage 19,05 σ 1,48 γ + σ 20,53 γ - σ 17,58 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 30 40,8 21,1 16,25 2 30 41,1 21,08 17,5 3 30 40,8 21,09 20,42 4 30 40,7 21,1 19,42 5 30 40,8 21,09 21,67 Average 40,84 21,09 19,05 16,25 17,5 20,42 19,42 21,67 0 5 10 15 20 25 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 52 µl = 35 γaverage 15,80 σ 0,70 γ + σ 16,50 γ - σ 15,10 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 35 40,9 21,09 15,25 2 35 41,2 21,06 15,66 3 35 40,9 21,08 16,08 4 35 40,8 21,07 15,5 5 35 40,9 21,07 16,5 Average 40,94 21,07 15,80 15,25 15,66 16,08 15,5 16,5 15 15,2 15,4 15,6 15,8 16 16,2 16,4 16,6 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 53 µl = 40 γaverage 14,46 σ 1,62 γ + σ 16,08 γ - σ 12,85 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 40 51,2 22,09 14,08 2 40 49,7 22,17 17,33 3 40 49,6 22,21 12,33 4 40 49,6 22,26 11,75 5 40 49,4 22,37 16,83 Average 49,9 22,22 14,46 14,08 17,33 12,33 11,75 16,83 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 54 µl = 45 γaverage 18,98 σ 1,63 γ + σ 20,61 γ - σ 17,36 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 45 49,5 22,35 18,08 2 45 49,4 22,39 15,33 3 45 49,5 22,39 20,92 4 45 49,4 22,39 22,33 5 45 49,5 22,4 18,25 Average 49,46 22,38 18,98 18,08 15,33 20,92 22,33 18,25 0 5 10 15 20 25 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 55 µl = 50 γaverage 18,63 σ 1,68 γ + σ 20,31 γ - σ 16,95 N V ϕ T Remarks (γ) µl % ᵒC ᵒ 1 50 49,4 22,4 17,25 2 50 49,3 22,4 23,08 3 50 49,3 22,39 15,92 4 50 49,4 22,39 19,83 5 50 49,1 22,41 17,08 Average 49,3 22,40 17,52 17,25 23,08 15,92 19,83 17,08 0 5 10 15 20 25 0 1 2 3 4 5 6 Angles N Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 56 First graph: γaverage - ϕaverage γ AVERAGE ( ᵒ ) ϕ AVERAGE (%) 52,5 36,67 26,37 36,28 26,19 38,58 35,94 40,775 22,55 40,8 19,05 40,84 15,8 40,94 14,46 49,9 18,98 49,46 17,52 49,3  The graphic shows that the slope angle oh plate depends on the humidity of the air. When the humidity increases, the slope angle of plate decreases. y = 198,7e-0,051x 0 10 20 30 40 50 60 0 10 20 30 40 50 60 γ average ( ᵒ ) ϕ average (%) Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 57 Second graph: γaverage - Taverage γ AVERAGE ( ᵒ ) T AVERAGE (ᵒC) 52,5 20,57 26,37 20,59 26,19 20,77 35,94 21,02 22,55 21,14 19,05 21,09 15,8 21,07 14,46 22,22 18,98 22,38 17,52 22,4  This graphic shows that the slope angle of plate depends on the temperature of air. When temperature increases, the slope angle of plate decreases. But the range of temperature is very small ( about 2 ᵒC) and this conclusion can be wrong, it is only a supposition. y = 48442e-0,359x 0 10 20 30 40 50 60 20 20,5 21 21,5 22 22,5 γ average ( ᵒ ) T average (ᵒC) Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 64 7. Evaluated of microfilm thickness (Vladimir S. Ajaev) Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 65 7.1 Introduction The evolution of a liquid droplet that spreads on a solid surface is known to depend on the local conditions near the contact line where the surface of the droplet touches the solid. Incorporating such local conditions into the standard description of viscous flow in the liquid results in a non-physical shear-stress singularity which can be removed by relaxing the no-slip condition for the viscous flow or introducing a microscopic precursor film. Adding the effect of evaporation at the droplet surface may be expected to further complicate the problem and thus require some additional modeling assumptions. As we show below, this is not necessarily the case since evaporation alters the flow structure and thus allows us to employ mathematical models which are not appropriate for the isothermal case. The goal of this paper is to investigate moving contact lines in the presence of evaporation in the context of spreading and develop a mathematical model that does not involve any ad hoc assumptions about the value of the apparent contact angle. Developing such a model is especially important since moving contact lines in applications often appear when evaporation is also significant. Spreading can be analysed using a lubrication-type approach if droplet thickness is much smaller than its radius, as shown by Lopez, Miller & Ruckenstein (1976) and Greenspan (1978) for isothermal spreading. They took into account viscous and capillary effects and reduced the problem to a single partial differential equation for droplet thickness. A detailed investigation of the effect of different contact line models on droplet spreading was carried out by Haley & Miksis (1991). Additional physical effects, such as Marangoni stresses at the droplet surface and chemical reaction at the solid–liquid interface, were investigated later for non-volatile droplets in the framework of the same approach by Ehrhard & Davis (1991) and Braun et al. (1995). Experimental results for spreading of silicon oil on glass (Ehrhard 1993) are in agreement with the lubrication-type models. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 66 The effect of evaporation in droplet spreading was considered by Anderson & Davis (1995) in the framework of the lubrication theory for a two-dimensional model. They used the onesided model of evaporation which implies that all dynamical processes in the vapour are negligible. Corrections to the equilibrium value of the contact angle owing to contact-line motion and evaporation are both assumed relatively small and therefore their linear superposition is used to determine the contact angle. The dynamic contribution to the contact angle (owing to the flow near the moving contact line) is approximated by a linear relation between the speed and the cube of the contact-angle departure from the equilibrium value. Later, Hocking (1995) suggested that a different model for the dynamic contribution to the contact angle is more appropriate, but used the same superposition principle to investigate the combined effect of evaporation and fluid flow on the contact-line motion. Formulae for the corrections to the equilibrium value of the contact angle due to contact line motion in both papers are motivated by experiments with no evaporation at the droplet surface. In the present work, we use a different approach to contact-line modelling suggested by Potash & Wayner (1972) and Moosman & Homsy (1980) in their studies of steady contact lines on heated surfaces. It relies on the description of dry areas on heated surfaces by microscopic adsorbed films which are in thermodynamic equilibrium with both solid and vapour phases. Such equilibrium can be achieved for non-zero film thickness owing to action of London–van der Waals forces. The pressure in the film due to these forces is inversely proportional to the cube of film thickness. We note that the adsorbed film is introduced here, not as an artificial tool needed to remove the singularity at the contact line, but rather as a physical effect with experimental verification. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 67 The adsorbed film is always formed on the solid surface if the droplet is surrounded by vapour. A macroscopic interfacial shape, such as liquid film or constant-curvature meniscus, has to approach the adsorbed-film solution when the macroscopically dry region is approached. The contact line is then defined as the region of rapid change of interfacial curvature where the transition between the macroscopic shape and the adsorbed film takes place. The approach has been used successfully by DasGupta et al. (1993) and Morris (2001) for finding local solutions near the contact lines on heated surfaces and by Ajaev & Homsy (2001) for finding global shapes of steady vapour bubbles in microchannels. It has not been applied to spreading of volatile liquids on heated surfaces even though microscopic films are often used in models of isothermal spreading (de Gennes 1985; Glasner 2003). Fig. 1. A sketch of a thin volatile droplet spreading on a uniformly heated surface. Cartesian coordinates are shown. This approach was originally developed for liquids which are perfectly wetting under isothermal conditions. One of the main challenges in modelling contact lines with evaporation is to generalize this approach to the case of partial wetting. The assumption that the disjoining pressure is an inverse power of the film thickness is no longer applicable for this case. Two different approaches can be taken. We can use an experimentally motivated disjoining pressure curve that accounts for both attractive and repulsive interactions; the contact angle is then related to the area under this curve. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 68 Alternatively, we can derive a slope-dependent expression for the disjoining pressure near the contact line by integrating over all intermolecular interactions with a simple model potential and a cut off length, following Miller & Ruckenstein (1974), Hocking (1993) and Wu & Wong (2004). The latter approach is taken in the present work. We believe that both approaches are capable of describing experimental results, but we do not attempt a detailed comparison here. In the present work, we develop a model of spreading of a droplet of either perfectly or partially wetting volatile liquid on a uniformly heated surface. The droplet is in direct contact with a large reservoir of vapour. The model incorporates the effects of surface tension, gravity, evaporation, thermocapillarity and disjoining pressure in the framework of a lubrication-type approach. We note that many previous theoretical investigations of spreading have been carried out in the framework of two-dimensional models, while axisymmetric shapes are clearly more relevant experimentally. Therefore, we first present a complete formulation of the problem and discuss the several important results in the framework of a two-dimensional model to facilitate an easy comparison with the previous work. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 69 7.2 Formulation We consider a two-dimensional droplet of a volatile liquid of density ρ and viscosity µ on a uniformly heated rigid surface, as shown in figure 1. The fluid flow in the vapour phase directly above the liquid is, in general, coupled to the liquid flow in the droplet. However, in this study, we use the one-sided model of evaporation of Burelbach, Bankoff & Davis (1988). It implies that the density, dynamic viscosity and thermal conductivity of the vapour phase are very small compared to those of the liquid. Therefore, we take the limit when the corresponding non-dimensional ratios approach zero. However, the vapour density is retained in the boundary conditions where it multiplies the vapour velocity, which can be large. We define the capillary number according to where σo is the surface tension at the equilibrium saturation temperature, TS*, and the characteristic velocity is determined from the interfacial mass balance as Here, k is the thermal conductivity of the liquid, L is the latent heat of vaporization per unit mass, and R0 is the initial radius of the droplet. Let us consider the limit of small capillary numbers. In order to obtain physically meaningful solutions, we consider distinguished limits when physical quantities, as well as parameters of the problem, scale as certain powers of the capillary number. Solutions are obtained from the leading-order terms of an asymptotic expansion in powers of C1/3. We note that it is often convenient to consider an asymptotic expansion in terms of an aspect ratio of the droplet. However, the cube of the aspect ratio has to scale as the capillary number in order for this approach to result in experimentally relevant solutions, so it is essentially equivalent to our asymptotic expansion in powers of C1/3. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 70 For simplicity we carry out the derivation for a perfectly wetting liquid with disjoining pressure being inversely proportional to the cube of the thickness of the liquid layer on the solid surface. A more general case of the slope-dependent disjoining pressure is discussed briefly in the end of the next section. Let us choose the length scales in the horizontal and vertical directions as R0 and C1/3R0, respectively; the resulting non-dimensional Cartesian coordinate system, (x, y), is shown in figure 1. The vapour–liquid interface in our formulation is represented by a function y =h(x, t), where t is the time variable scaled by R0/U. We choose C1/3U as the velocity scale in the y-direction and C1/3σ0 / R0 as the pressure scale. The governing equations at leading order take the usual lubrication-type form: Here, B = ρgR02/σ0 is the Bond number, g is the acceleration due to gravity; the velocity in the x-direction is scaled by U, the non-dimensional temperature T is defined in terms of the dimensional one, T *, according to Let us now turn to the interfacial boundary conditions. In order to include the nondimensional evaporative mass flux J into the leading-order mass-conservation condition, we scale the dimensional flux by ρUC1/3. With this choice and the above length and velocity scales, the non-dimensional leading-order conditions for conservation of mass and energy at the interface are written in the form: Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 71 Equation (9) can be interpreted as the balance between the heat conducted through the droplet and the latent heat of the phase change at the interface. Heat transfer from the liquid film to the vapour is assumed negligible here, but it can be easily accounted for in the framework of our approach. The normal stress condition at the interface includes contributions from capillarity and disjoining pressure: where pv is the non-dimensional vapour pressure. We assume that the disjoining pressure is inversely proportional to the cube of film thickness and introduce a nondimensional parameter, ε = |A| / (σ0R02C) which is assumed to be an order of one quantity in the asymptotic limit C → 0, although its numerical value may be small (~10-3) ; A is the Hamaker constant. We assume that the surface tension is a linear function of temperature, and introduce the modified Marangoni number M = γTS*/σ0. We note that this parameter is sometimes referred to as the capillary number, but in our case it is essentially equivalent to the modified Marangoni number introduced by Gramlich et al. (2002) for thin-film flows over topographic features. With this choice, the shear stress condition at the interface is written as The scaled interfacial temperature T i is related to the local mass flux and pressure jump at the interface through the non-equilibrium condition, which can be written in the following form: Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 72 where Here, R is the gas constant per unit mass, ρv is the vapour density. According to (13), the departures of local temperature at the interface from the equilibrium value are characterized by two non-dimensional parameters, K and δ. The kinetic parameter, K, measures the relative importance of kinetic effects at the interface. The parameter δ characterizes the effect of changes in liquid pressure on the local phase-change temperature at the interface. Derivation of (13) is based on a simple linear relation between the mass flux and the vapour pressure. We note that alternative approaches have been suggested in the literature (Rose 2000), but we do not attempt to review them here. At the solid–liquid interface, the liquid velocity is zero and the non-dimensional value of the temperature is fixed at T =T0 > 0. The non-dimensional temperature T0 is an important control parameter in experiments. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 73 7.3 Calculations Thickness of adsorbed film from equation given by Ajaev: δ = 9,4181 * 10-10 * (R0)-1/3 where R0 = β r0 β = B ( ) 0,5 For water droplet B = 0,87. If the water droplet is slowly deposition on the surface than the Weber number is equal near zero ( We ≈ 0 ). Spreading factor is equal: β ≈ 1,2304 r0 = ( )1/3 Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 80 The next graph show the relation between tangent of dynamic microscopic advanced angle in relation to slope angle of the plate and to thickness of the liquid film. This data of the experimental results must be compared with the next experimental data: V = 20 µl -5E-08 0 5E-08 0,0000001 1,5E-07 0,0000002 2,5E-07 0 10 20 30 40 50 tg θ γ Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 81 Average values of slope angle of plate with relationship to tg θ. V ( nl) γaverage ( ᵒ ) tg θ 5 52,5 -1,11089E-08 10 26,37 1,84269E-08 15 26,19 9,45329E-09 20 35,94 2,46196E-08 25 22,55 2,65703E-09 30 19,05 7,96059E-10 35 15,8 3,37688E-10 40 14,46 -1,03064E-08 45 18,98 4,29255E-10 50 17,52 -1,26908E-08 This graph shows that exists a lineal relation between slope angle of plate and tg θ. But the two negative values must be deleted because some error in the measurements can be happened. y = 2E-09x - 3E-08 -1,5E-08 -1E-08 -5E-09 0 5E-09 1E-08 1,5E-08 2E-08 2,5E-08 3E-08 0 5 10 15 20 25 30 35 40 tg θ γaverage ( ᵒ ) Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 82 Then the next graph, show the correct relation between this parameters: And the next graph shows the relation of volume of droplet between slope angle of plate in different times of measurements. y = 1E-17x6,2012 0 5E-09 1E-08 1,5E-08 2E-08 2,5E-08 3E-08 3,5E-08 4E-08 4,5E-08 5E-08 0 5 10 15 20 25 30 35 40 tg θ γ average ( ᵒ ) τ = 2 min τ = 5 min τ = 10 min τ = 15 min τ = 20 min 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 γ ( ᵒ ) V (µl) Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 83 9. Conclusions Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 84 Comparing the experimental data with the experimental results by Zapalowicz we can explain the next conclusions about the verification of the model of droplet´s spreading on the solid body surface coated with a water microfilm.  The values of the dynamic microscopic advanced angle have been estimated with a small range of error.  The process of evaluated of thickness microfilm by Vladimir S Ajaev for each volume of droplet, allows estimate the dynamic microscopic advanced angle with high grade of verification  The relationship between the slope angle of plate and dynamic microscopic advanced angle for equals and different values of thickness of microfilm, it is similar. For small values of slope angle of plate is almost lineal. For bigger values, the relationship is more difficult of evaluate. The humidity of air and volume of water droplets can be very important in this aspect. On the other hand, the influence of temperature is not important.  The slope angle oh plate depends directly on the humidity of the air. When the humidity increases, the slope angle of plate decreases.  The slope angle of plate depends on the temperature of air in according to the experiment. When temperature increases, the slope angle of plate decreases. But the range of temperature is very small ( about 2 ᵒC) and this conclusion can be wrong, it is only a supposition.  The slope angle of plate depends directly on the volume of water droplet. When volume increases, the slope angle of plate decreases.  The influence of the time in this phenomena it can't be evaluated. More experiments are needed for finding conclusions about this aspect. Influence of water microfilm on the sliding of water droplet Jaime Domingo, Ignacio 85 10. Bibliography  Peter C. Wayner, Jr., " Intermolecular forces in phase - change heat transfer: 1998 Kern award review " 1999.  Karl Stephan, " Influence of dispersion forces on phase equilibria between thin liquid films and their vapour" 2002.  Vladimir S. Ajaev, " Spreading of thin volatile liquid droplets on uniformly heated surfaces, 2003  Zbigniew Zapałowicz, " Initial verification of the model of droplets spreading on the solid body surface presence of liquid microfilm" 2003.