evaporation of pinned sessile microdroplets of water on a highly heat-conductive substrate: computer...

Eur. Phys. J. Special Topics 219, 143–154 (2013)© EDP Sciences, Springer-Verlag 2013DOI: 10.1140/epjst/e2013-01789-y

THE EUROPEANPHYSICAL JOURNALSPECIAL TOPICS

Regular Article

Evaporation of pinned sessile microdroplets ofwater on a highly heat-conductive substrate:Computer simulations

Sergey Semenov1,a, Victor M. Starov1,b, and Ramon G. Rubio2,c

1 Dept. of Chemical Engineering, Loughborough University, LE11 3TU Loughborough, UK2 Dept. of Quimica Fisica I, Universidad Complutense, 28040 Madrid, Spain

Received 1 July 2012 / Received in final form 18 January 2013

Published online 19 March 2013

Abstract. The aim of the current numerical study is to investigate theinfluence of individual effects (kinetic effects, latent heat of vaporiza-tion, Marangoni convection, Stefan flow, droplet’s surface curvature) onthe rate of evaporation of a water droplet placed on a highly heat con-ductive substrate for different sizes of the droplet (down to submicronsizes). We performed simulations for one particular set of parameters:the ambient relative air humidity is set to 70%, the ambient temper-ature is 20 ◦C, the contact angle is 90◦, and the substrate material iscopper. The Suggested model combines both diffusive and kinetic mod-els of evaporation. The obtained results allow estimation of the char-acteristic droplet sizes where each of the mentioned above phenomenabecomes important or can be neglected.

1 Introduction

The evaporation of sessile liquid droplets is important in a variety of industrial appli-cations such as spray cooling [1,2], ink-jet printing [3], tissue engineering [4], printingof microelectromechanical systems (MEMS) [5], surface modification [6,7], variouscoating processes [8], as well as biological applications [9]. A number of theoreticaland experimental investigations have been focussed on this phenomenon [10–18].Studying the evaporation of microdroplets can help understanding the influence of

the Derjaguin’s (disjoining/conjoining) pressure acting in the vicinity of the apparentthree-phase contact line [19–21].The aim of the presented computer simulations is to show how the evaporation of

pinned sessile submicron droplets of water on a solid surface differs from the evapora-tion of millimetre size droplets. The obtained results prove the importance of kineticeffects whose influence becomes more pronounced for submicron droplets.

a e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]

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http://dx.doi.org/10.1140/epjst/e2013-01789-y

144 The European Physical Journal Special Topics

Fig. 1. An axisymmetric sessile droplet on a solid substrate. θ and L are the contact angleand the radius of the droplet base, respectively. In the present studies, the contact angleis 90◦.

The model used below includes both diffusive and kinetic models of evaporationsimultaneously. the Hertz-Knudsen-Langmuir equation [22,23] is used as a boundarycondition at the liquid-gas interface instead of the condition of a saturated vapour.The overall evaporation rate is limited either by the rate of vapour diffusion intothe ambient gas or by the rate of molecule transfer across the liquid-gas interface.As a result, the vapour concentration at the liquid-gas interface falls in between itssaturated value and its value in the ambient gas. This intermediate value of vapourconcentration at the liquid-gas interface drives both transition of molecules fromliquid to gas (kinetic flux) and vapour diffusion into the ambient gas (diffusive flux).Thus the rate of evaporation is limited by the slower of these two processes.Computer simulations are performed using the software COMSOL Multiphysics.

The dependence of the total molar evaporation flux, Jc, on the radius of the contactline, L, of pinned droplets is studied below using a quasi-steady state approximation.

2 Problem statement

Below, only droplets of the size less than 1mm are under consideration, that is theinfluence of gravity is neglected. It is assumed that an axisymmetric sessile dropletforms a sharp three-phase contact line and maintains a spherical-cap shape of theliquid-gas interface due to the action of liquid-gas interfacial tension. The geometryof the problem is schematically presented in Fig. 1.Due to a small size of the evaporating droplets under consideration, the diffusion

of vapour in the gas phase dominates its convective transport. The latter is confirmedby small values of both thermal and diffusive Peclet numbers: Peκ = Lu/κ < 0.05;PeD = Lu/D < 0.04, where L is the radius of the contact line, u (<1mm/s) isthe characteristic velocity of vapour convection due to evaporation (Stefan flow) andMarangoni convection, κ is the thermal diffusivity of the surrounding air, and D isthe diffusion coefficient of vapour in air at standard conditions.As in Ref. [24], the problem is solved under a quasi-steady state approximation.

It yields a simultaneous distribution of heat and mass fluxes in the system.The volume of the droplet diminishes because of evaporation. Therefore the liquid-

air interface moves with some velocity which can be calculated based on the knowledgeof the evaporation rate and two particular assumptions: the cap of the droplet pre-serves its spherical shape, and the contact line is pinned (L = const). The regime ofthe moving contact line is not considered in the present study.The parameters of the following materials are used in the present computer simu-

lations: copper as the substrate, water as the liquid in a droplet, and a humid air as asurrounding medium. The pressure in the surrounding gas is equal to the atmospheric

IMA6 – Interfacial Fluid Dynamics 145

pressure, the ambient temperature is 20 ◦C, the ambient air humidity is 70%, and thecontact angle is 90◦.

2.1 Governing equations in the bulk phases

This section presents the problem statement which is based on that proposed by Krahlet al. [25]. The following governing equations describe the heat and mass transfer inthe bulk phases:heat transfer in the solid phase:

ΔT = 0,

where Δ is the Laplace operator, and T is the temperature;heat transfer inside of fluids (liquid or gas):

u · ∇T = κΔT,where u is the fluid velocity,∇ is the gradient operator, and κ is the thermal diffusivityof the fluid;incompressible Navier-Strokes equations are used to model hydrodynamic flows in

both fluids (liquid or gas):

ρu · ∇u = ∇ · T,where ρ is the fluid density, ∇u is the gradient of the velocity vector, T is the fullstress tensor, and ∇ · T is the dot-product of the nabla operator and the full stresstensor. The full stress tensor is expressed via hydrodynamic pressure, p, and theviscous stress tensor, π, as:

T = −pI+ π,where I is the identity tensor. The Continuity equation for the fluids is also used:

∇ · u = 0.The diffusion-convection equation for vapour in the gas phase:

u · ∇c = DΔc, (1)

where c is the molar concentration of the liquid vapour.

2.2 Boundary conditions

The no-slip and no-penetration boundary conditions are used for the Navier-Stokesequations at the liquid-solid and gas-solid interfaces, resulting in a zero velocity atthese interfaces:

u = 0.

Let Γ be the liquid-gas interface. Let also jc be the density of the molar vapour fluxacross the liquid-gas interface, then the density of the mass vapour flux, jm, acrossthis interface is jm = jcM , where M is the molar mass of an evaporating substance(water). Let uΓ be the normal velocity of the liquid-gas interface itself, see Fig. 2. Then


Fig. 2. Notation: Γ is the liquid-vapour interface; uΓ is the normal velocity of the interfaceΓ in the direction of the normal unit vector n (from the liquid phase to the gaseous one); τis the unit vector tangential to the interface Γ; jm is the mass flux across the interface Γ.

the boundary condition for the normal velocity of liquid at the liquid-gas interfacereads:

ρl (ul · n− uΓ) = jm, (2)

where ρl is the liquid density, the subscript l stands for liquid. The expressions forthe evaporation flux, jm, and the normal interfacial velocity, uΓ, are specified below.The condition of the stress balance at the liquid-gas interface is used to obtain bound-ary conditions for the pressure and the tangential velocity:

(T · n)l − (T · n)g = −γ (∇ · n)Γ n+ γ′T (∇ΓT ) , (3)

where subscripts l and g stand for liquid and gas, respectively; T is the full stresstensor; γ is the interfacial tension of the liquid-gas interface; γ′T is the derivativeof the interfacial tension with the temperature; ∇ΓT is the surface gradient of thetemperature; (∇ · n)Γ is the divergence of the normal vector at the liquid-gas interfacewhich is equal to the curvature of the interface. The boundary condition (3) is a vectorone. The boundary conditions in the normal and tangential directions are deducedas follows: (i) the boundary condition for the thermal Marangoni convection whichdetermines the tangential component of the velocity vector is obtained by multiplyingEq. (3) by the tangential vector τ (see Fig. 2) and neglecting the viscous stress inthe gas phase (due to a small gas viscosity compared to the liquid viscosity); (ii)the similar procedure with the normal vector n (see Fig. 2) results in a boundarycondition for pressure in the liquid at the liquid-gas interface.The normal flux of vapour at the gas-solid interface is zero because there is no

penetration into the solid surface:

∇c · n = 0,where c is the molar concentration of the vapour in the air; and n is the unit vectorperpendicular to the solid-gas interface.Below, we consider the gas phase as the mixture of vapour and dry air. Note that

due to the mass conservation law, the mass flux of vapour, jm, perpendicular to theliquid-gas interface in the vapour phase should be equal to the mass flux of liquidperpendicular to the interface in the liquid phase:

jm = ρl (ul · n− uΓ) = ρv (ug · n− uΓ)− ρgDvapour in air∇ (ρv/ρg) · n. (4)

The density of the mass flux of a dry air, jm,air, across the liquid-gas interface isassumed to be zero:

jm,air = ρair (ug · n− uΓ)− ρgDair in vapour∇ (ρair/ρg) · n, (5)


where ρg, ρv and ρair are the densities of gas, vapour and dry air, respectively;D is the diffusion coefficient; ul and ug are the velocity vectors of liquid and gas,respectively; uΓ and n are shown in Fig. 2. Let us adopt the following assumption:ρg = ρair+ρv = const. The latter assumption means incompressibility of the gaseousphase. This assumption results in

∇ρair = −∇ρv. (6)

The following equality is valid for a binary mixture:

Dair in vapour = Dvapour in air = D. (7)

Substituting Eqs. (6) and (7) into Eqs. (4) and (5) and preforming simple algebraicmanipulations (note: it is assumed that ρg = const) yields an expression for thedensity of the mass flux across the liquid-gas interface, jm, as a function of the molarvapour concentration, c, in the gas phase:

jm =−D∇ρv · n1− ρv/ρg =

−D∇c · n1/M − c/ρg , (8)

where the following relation has been used: ρv = cM , M is the molar mass of anevaporating substance (water). On the other hand, the rate of mass transfer acrossthe liquid-gas interface is given by the Hertz-Knudsen-Langmuir equation [22,23]:

jm = αm

√MRT

2π(csat (T )− c) , (9)

where R is the universal gas constant; T and c are the local temperature in ◦K andthe molar vapour concentration at the liquid-gas interface, respectively; csat is themolar concentration of saturated vapour; αm is the mass accommodation coefficient.The kinetic model of gases (the Hertz-Knudsen-Langmuir equation with αm = 1)yields the maximal value of the molecular flux through the imaginary plane in thegaseous phase. In reality, when vapour molecules collide with the real liquid surface,not all of them go into the liquid phase, some of them are reflected back to thegaseous phase. Thus, the mass accommodation coefficient, αm, is the probabilitythat uptake of vapour molecules occurs upon collision of these molecules with theliquid surface. The value of αm depends on contact surfaces and can be determinedexperimentally [22]. Thus, according to Ref. [22], the minimal and average values ofαm measured for water are 0.04 and 0.5, respectively. In our computer simulationswe used αm = 0.5. The molar concentration of saturated vapour (csat = psat/RT )is taken as a function of a local temperature and a local curvature of the liquid-gasinterface according to the Clausius-Clapeyron [24] and Kelvin [26] equations:

ln

(psat,0

pref

)= −ΛR

(1

T− 1

Tref

)(Clausius− Clapeyron equation),

ln

(psat

psat,0

)=γKM

ρlRT(Kelvin equation),

where psat,0 and psat are saturated vapour pressures above the flat and curved liq-uid surfaces, respectively; Λ is the latent heat of vaporization; R is the universalgas constant; pref and Tref are tabulated values of the saturated vapour pressureand corresponding temperature; γ is the liquid-gas interfacial tension; M and ρl are


the molecular weight and the mass density of the evaporating liquid; and K is thecurvature of the liquid-gas interface, determined as K = (∇ · n)Γ, where n is theunit vector perpendicular to the liquid-gas interface and pointing into the gaseousphase.Combining Eqs. (8) and (9) we obtain a boundary condition for the diffusion

equation (1) at the liquid-gas interface:

−D∇c · n1/M − c/ρg = αm

√MRT

2π(csat (T )− c) .

Adding Eqs. (4) and (5), and taking into account the assumption (ρg = ρair + ρv =const), we arrive to the boundary condition for the normal velocity of gas at theliquid-gas interface:

ρg (ug · n− uΓ) = jm. (10)

The tangential velocity of the gas at the liquid-gas interface is determined by theno-slip condition:

ug · τ = ul · τ .The boundary conditions of temperature continuity are applied at all interfaces. Thecontinuity of the heat flux is applied on the solid-liquid and solid-gas interfaces. Atthe liquid-gas interface, the heat flux experiences discontinuity caused by the latentheat of vaporization:

kg (∇T )g · n− kl (∇T )l · n = jcΛ, (11)

where k is the thermal conductivity of the corresponding phase; Λ is the latent heatof vaporization (or enthalpy of vaporization [27], units: J/mol); n is the unit vector,normal to the liquid-gas interface, and pointing into the gas phase; jc is the surfacedensity of the molar flux of evaporation (mol · s−1 ·m−2) at the liquid-gas interface.The conditions of axial symmetry are applied at the axis r = 0. At the outer

boundary of the computational domain, the values of temperature, T∞, and vapourconcentration, c∞, are imposed.In our computer simulations, we assume that the droplet under consideration

retains a spherical-cap shape in the course of evaporation, and the contact line ispinned (L = const), then knowing the total mass evaporation flux, Jm =

∫ΓjmdA

(dA is the element of area of the interface Γ), we can calculate the normal velocity ofthe liquid-gas interface, uΓ, at any point of the interface:

uΓ =−JmπρlL2

(ω

√1−( rLsin θ)2− cos θ

)1 + cos θ

1− cos θ , ω ={1, z/L > − cot θ−1, z/L ≤ − cot θ ·

The origin of cylindrical coordinates (r, z) is supposed to be at the point of intersectionof the axis of the droplet symmetry with the liquid-solid interface (Fig. 2).

3 Computer simulations

Computer simulations are performed using the commercial software COMSOL Mul-tiphysics. The numerical method used in COMSOL is a finite element method withquadratic Lagrangian elements. The software transforms all equations into their weakform before discretization. The method of Lagrange multipliers is used to applyboundary conditions as constraints.


The computational domain is selected as a circle in (r, z) coordinates of the cylin-drical frame of reference. The centre of this circle is located at the origin of thecoordinates system. The radius of the computational domain is 100 times bigger thanthe radius of the contact line, L, which prevents the influence of the proximity ofouter boundaries on the solution inside the droplet.The choice of the contact angle of 90◦ allows avoiding the problem of singularity

of the diffusive evaporation flux at the three-phase contact line; but the problem ofthe viscous stress singularity remains. The current model does not employ any phys-ical mechanisms to overcome this problem. Instead, the mesh is refined around thecontact line to reduce the area of singularity influence. Thus the size of computationalmesh elements around the three-phase contact line is chosen to be 100 times smallerthan the droplet size. The growth rate for mesh elements is less than 1.1 in the wholecomputational domain.

4 Results and discussion

4.1 Isothermal evaporation

The model described here is an extension of the previous one [18,24] which wasdeveloped for a diffusion-limited evaporation of water droplets. In distinction to theprevious model, the present one takes into account additional phenomena: Stefan flowin the gas, the effect of curvature of the droplet’s surface on the saturated vapourpressure (the Kelvin’s equation), and the kinetic effects (also known as Knudseneffects). A numerical model allows switching individual phenomena on/off in orderto understand their contribution to the overall process of heat and mass transfer inthe course of evaporation. The effect of Stefan flow is switched off in such a way thatthe convective part of heat and mass fluxes is neglected (excluded from equations)comparing to the conductive one in the gaseous phase only.The value of the mass accommodation coefficient αm is taken as 0.5 which is the

average experimentally measured value of αm for water according to Ref. [22].When Kelvin’s and kinetic effects are switched off, then the model represents the

case of a diffusion-limited evaporation. This allows us to validate the present modelagainst the previous one [18,24] which in turn was validated against the availableexperimental data [18] for the diffusion-limited case. Computer simulations showedthat there is agreement with the previous results [24] (see Fig. 3).In Fig. 3, the total molar fluxes of droplet’s evaporation, Jc,i, are presented for

various isothermal cases (index i stands for the “isothermal”). The flux was com-puted according to the presented model in which the Stefan flow, heat transfer andthe thermal Marangoni convection were omitted. Figure 3 shows that the kineticeffects change the slope of curves (see triangles and circles) for submicron droplets(L < 10−6m) only. The influence of the curvature of the droplet’s surface (Kelvin’sequation) becomes significant only for droplet sizes less than 10−7m. However, atsuch low sizes the surface forces action (disjoining/conjoining pressure) has to betaken into account (not included in the present model).Note once more that the presented model is valid only for the droplet size bigger

than the radius of surface forces action, which is around 10−7m = 0.1μm. That is,the data in Fig. 3 for the droplet size smaller than 10−7m are presented only toshow the trend. Figure 3 shows that if the radius of the droplet base is bigger than10−7m, then (i) the deviation of the saturated vapour pressure caused by the dropletcurvature (Kelvin’s equation) can be neglected, (ii) the deviation from the pure dif-fusion model of evaporation can be neglected for the droplet size bigger than 10−6m,(iii) this deviation becomes noticeable only if the droplet size is less than 10−6m.


Fig. 3. Dependence of the droplet’s molar evaporation flux, Jc,i, on the droplet size, L,for the isothermal model of evaporation. The parameters used: αm = 0.5; θ = 90

◦; rela-tive air humidity is 70%. Note: the results for L < 10−7m do not have physical meaning,as additional surface forces must be included into the model. These points are shown todemonstrate the trends of the curves.

This deviation is caused by an increasing influence of the kinetic effects at the liquid-gas interface (Hertz-Knudsen-Langmuir equation) and this theory should be appliedtogether with the diffusion equation of vapour in the air if the droplet size is less than10−6m.The latter conclusions show that a consideration of evaporation of microdroplets,

completely covered by the surface forces action (that is less than 10−7m), shouldtake into account both the deviation of the saturated vapour pressure caused by thedroplet curvature and the kinetic effects.According to the model of diffusion-limited evaporation, the evaporation flux, Jc,i,

must be linearly proportional to the droplet size, L, that is Jc,i ∼ L. The latter is inagreement with the data presented in Fig. 3 for droplets bigger than 10−6m. However,for a pure kinetic model of evaporation (no vapour diffusion, uniform vapour pressurein the gas) flux Jc,i is supposed to be proportional to the area of the droplet’s surface,that is in the case of pinned droplets (constant contact area) Jc,i ∼ L2 should be sat-isfied. To check the validity of the latter models at various droplet sizes let us assumethat the dependency of the evaporation flux on the droplet radius has the followingform Jc,i = A(θ)L

n, where n is the exponent to be extracted from our model and Ais a function of the contact angle, θ. In general, it is necessary to calculate the partialderivative

n = ∂ (lnJc,i)/∂ (lnL), (12)

in order to compute the exponent n using the present model. However, in the case ofa pinned droplet the only varying parameter for each individual curve in Fig. 3 wasthe contact line radius, L. Hence, the partial derivatives in Eq. (12) can be replacedby ordinary derivatives and in this way n was calculated using the data presented inFig. 3. The calculated values of n are presented in Fig. 4.One can see from Fig. 4 that the exponent n, as expected, is equal to 1 for a pure

diffusive isothermal model of evaporation within the whole studied range of L values(diamonds in Fig. 4). In the case when kinetics effects and Kelvin’s equation are bothtaken into account additional to the pure diffusion, Fig. 4 shows that the diffusion


Fig. 4. The exponent n for the dependence Jc,i = A(θ)Ln for the isothermal model of

evaporation. The parameters used: αm = 0.5; θ = 90◦; relative air humidity is 70%. Note:

the results for L < 10−7m do not have physical meaning, as the surface forces action mustbe included into the model here. These points are shown to demonstrate the trends of curves.

model of evaporation dominates for droplets with the size bigger than 10−5m, thatis for droplets bigger than 10μm.Taking into account kinetic effects only additional to the diffusion without the

Kelvin’s equation (triangles in Fig. 4) results in a smooth transition from the lin-ear dependence Jc,i ∼ L, that is n = 1 (the diffusive model) to the quadratic oneJc,i ∼ L2, that is n = 2 (the kinetic model) as the droplet size decreases down toL = 10−9m (see Fig. 4). The latter shows that Jc,i is tending to be proportional toL2 as the droplet size decreases which means that evaporation flux becomes propor-tional to the area of the liquid-gas interface. The influence of the curvature (Kelvin’sequation) on the saturated vapour pressure results in a substantially lower exponentn as compared with the kinetic theory (Fig. 4). However, the latter happens onlyfor a droplet completely in the range of the surface forces action, that is, less than10−7m [20]. Below this limit, the droplet does not have a spherical cap shape any-more even on the droplet’s top (microdroplets according to Ref. [20]). The evaporationprocess in the latter case should be substantially different from the considered above.Thus the range of sizes less than 10−7m is not covered by the presented theory.

4.2 Influence of thermal effects

Computer simulations were also performed including both Kelvin’s equation and thekinetic effects in the case when the thermal effects were taken into account. The latterwas made to show the influence of the latent heat of vaporization, Marangoni con-vection and Stefan flow on the evaporation process. Droplet’s evaporation rates, Jc,were normalized using those, Jc,i (circles in Fig. 3), from the isothermal model. Theresults are presented in Fig. 5.Figure 5 shows that the latent heat of vaporization reduces the evaporation flux

as compared to the isothermal case (that is Jc/Jc,i < 1) in all cases considered. Therelative reduction of the evaporation rate (caused by the latent heat of vaporization)reaches the maximum for droplets with L ∼ 10−5m. The latter size according toFigs. 3 and 4 is in the range of a diffusion-limited evaporation. For smaller droplets,when the kinetic effects come into play, the evaporation rate (circles and triangles in


Fig. 5. The influence of the latent heat of vaporization, Marangoni convection, and Stefanflow on the evaporation rate in the case when the kinetic effects and the Kelvin’s equationare included into the model. Jc and Jc,i are the total molar flux of evaporation and that inthe isothermal case, respectively.

Fig. 3), becomes much smaller than the corresponding evaporation rate in case of apure diffusive model (diamonds in Fig. 3). Thus, due to this reduction of the evapora-tion rate, the latent heat of vaporization produces a much smaller thermal effect (seeEq. (11)), as compared to the pure diffusive model. This can be clearly seen in Fig. 5(squares): the ratio Jc/Jc,i in the kinetic regime of evaporation (when the droplet sizetends to zero) tends to one: limL→0 Jc/Jc,i = 1, which means a disappearance of theeffect of the latent heat of vaporization.Taking into account the kinetic effects, Kelvin’s equation, the latent heat of

vaporization, and thermal Marangoni convection (for water droplets) affects dropletsof size L > 10−5m (triangles in Fig. 5). For water droplets of size L < 10−5m theinfluence of Marangoni convection is negligible and the evaporation rate, Jc, coincideswith that for a model which includes only the latent heat of vaporization (squares inFig. 5).The effect of Stefan flow in the surrounding gas slightly changes the evaporation

rate in the present model (circles in Fig. 5) and makes it lower due to the appearanceof an outward convective heat flux in the gas above the droplet. In our particularcase, this effect appeared to be much weaker than other effects considered above. Theinfluence of the Stefan flow can be neglected. Note that the influence of the thermaleffects on the kinetics of evaporation is less than 5% (according to Fig. 5).

5 Conclusions

Computer simulations of evaporation of small sessile droplets of water are performed.The present model combines diffusive and kinetic models of evaporation. The effectof latent heat of vaporization, thermal Marangoni convection and Stefan flow in thesurrounding gas were investigated for a particular system: water droplet on a heatconductive substrate (copper) in air at standard conditions. The results of modellingallow an estimation of the characteristic droplet sizes when each of the mentionedabove phenomena becomes important or can be neglected.The presented model is valid only for the droplet size bigger than the radius of

surface forces action, which is around 10−7m = 0.1μm. That is, the data in Figs. 3


and 5 for the droplet size smaller than 10−7m are presented only to show the trend.When the radius of the droplet base, L, bigger than 10−7m then (i) the deviationof the saturated vapour pressure caused by the droplet curvature (the Kelvin’s equa-tion) can be neglected, (ii) the deviation from the pure diffusion model of evaporationcan be neglected for the droplet size bigger than 10−6m, (iii) this deviation becomesnoticeable only if the droplet size is less than 10−6m. This deviation is caused byan increasing influence of the kinetic effects at the liquid-gas interface (the Hertz-Knudsen-Langmuir equation) and this theory should be applied together with thediffusion equation of vapour in the air if the droplet size is less than 10−6m.The latter conclusions show that a consideration of evaporation of microdroplets

completely covered by the surface forces action (that is less than 10−7m) should in-clude both deviation of the saturated vapour pressure caused by the droplet curvatureand the kinetic effects.The latent heat of vaporization results in a temperature decrease at the surface

of the droplet. Due to that, the evaporation rate is reduced. This effect is more pro-nounced in the case of a diffusion-limited evaporation (L > 10−5m), when the vapourpressure at the droplet’s surface is saturated and determined by the local tempera-ture. The effect of Marangoni convection in water droplets is negligible for droplets ofsize L < 10−5m. For the system considered above, the Stefan flow effect appeared tobe weaker than that of thermal Marangoni convection for L > 10−4m, but strongerfor L < 10−4m. However, in all cases its influence is small and can be neglected.According to Fig. 5 the influence of the latent heat of vaporization on the kinetics ofevaporation is less than 5%.The presented model can be applied for evaporation of any other pure simple

liquids, not water only.

This work has been done under the umbrella of COST Action MP1106. Research was sup-ported by the European Union under Grant MULTIFLOW FP7-ITN-2008-214919, and EP-SRC UK grant EP/D077869/1. The work of R.G. Rubio was supported in part by theSpanish Ministerio de Ciencia e Innovacion through grant FIS2009-14008-C02-01, and byESA through project MAP-AO-00-052. Both V.M. Starov and R.G. Rubio acknowledge asupport from European Space Agency (PASTA project).

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evaporation of pinned sessile microdroplets of water on a highly heat-conductive substrate: computer...

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