Evaporation of Sessile Droplets of Liquidon Solid Substrates
S. Semenov, V.M. Starov, M.G. Velarde, and R.G. Rubio
1 Introduction
The process of evaporation of sessile droplets is of interest for both industryand academia. It is often difficult to study the evaporation process and relatedeffects experimentally [1], as this phenomenon is controlled by a number of exter-nal conditions. Thus computer simulations are used alongside with experimentalinvestigations to study this complex phenomenon. There are also a number ofother general problems related to evaporating droplets. Some of them require aconsideration of a droplet as an integral part of a wider problem, such as spraydynamics [2]. Sometimes it is required to predict the solidification in the courseof evaporation including formation of hollow shells [3]. Also the residue fromdried drops [4–6] has implications for a number of applications, including painting,coating processes, ink-jet printing, DNA chip manufacturing [7], formation of arraysof organic materials for video displays and photo sensors, fabrication a variety ofmicro-electro-mechanical (MEMS) devices [8]. The analysis of the influence ofproperties of solid substrates on the evaporation rate of sessile droplets can be usedfor production of various materials providing an optimal regime of work for airconditioners, dryers and cooling systems [9].
S. Semenov � V.M. Starov (�)Department of Chemical Engineering, Loughborough University, Loughborough, UKe-mail: [email protected]; [email protected]
M.G. VelardeInstituto Pluridisciplinar, UCM, Paseo Juan XXIII, 1, Madrid 28040, Spain
R.G. RubioDepartment of Quı́mica Fı́sica I, Facultad de CC. Quı́micas, UCM, Madrid 28040, Spain
Evaporation starts immediately after a deposition of a liquid droplet onto a solidsubstrate in non-saturated vapour atmosphere. In the case of complete wetting(equilibrium contact angle � is zero) the evaporation process is accompanied byspreading. The whole process can be subdivided into two stages: (a) an increase ofthe solid–liquid contact area cause by spreading, and (b) a decrease of the solid–liquid contact area caused by evaporation. Stage (a) is usually proceeds much fasterthan stage (b).
In the case of partial wetting (contact angle 0ı < � < 90ı) after a deposition of adroplet onto a solid surface, the whole process can be subdivided into the followingfour stages. Stage (0): a short spreading stage, when the solid–liquid contact areaincreases until contact angle, � , reaches its static advancing value, �ad . This stageis accompanied by the evaporation, but due to different time scales of spreadingand evaporation processes (spreading is much faster) the influence of evaporationduring the spreading stage is negligible. Evaporation cannot be neglected duringnext stages. In the presence of contact angle hysteresis the latter process occurs inthree stages [10, 11]. (i) During the first stage evaporation proceeds with a constantradius of the contact line, L, and decreasing contact angle, � , until the contact anglereaches the static receding value, �r . (ii) The next stage of evaporation developswith a constant contact angle, �r , and decreasing radius of the contact line, L.(iii) During the third stage both the radius of the contact line, L, and the contactangle, � , decrease until the droplet disappears. Stages (i) and (ii) are usually thelongest ones, while the last third stage is the shortest one and is the most difficultfor experimental investigations.
3 Dependence of Evaporation Flux on the Droplet Size
Most of the theoretical and computer simulation studies [12–17] assume thespherical shape of the cap of a sessile droplet. This is true only for small enough(L < 1 mm) droplets, when the capillary forces dominate over the body forces (e.g.gravity forces). In the case of diffusion controlled evaporation the latter model givethe following equation for the evaporation rate of a sessile droplet:
.0:00008957 C 0:6333 � � C 0:116 � �2 � 0:08878 � �3
C 0:01033 � �4/=sin�; � > �=18
(2)
Evaporation of Sessile Droplets of Liquid on Solid Substrates 287
where V is the droplet volume; t is time; F is a function of contact angle,derived by Picknett and Bexon [13], which equals 1 at � D �=2 (� is in radiansin (2));ˇ D 2� DM
�
�csat .Tsurf / � c1
�; D is the diffusion coefficient of vapour in
air; M is the molecular weight of the evaporating substance; � is the density of theliquid; csat is the saturated value of the molar concentration of vapour, consideredas a function of uniform temperature, Tsurf , of the droplet’s surface; c1 is themolar vapour concentration in the ambient air far away from the droplet. Note that(1) is deduced using the model which takes into account only vapour diffusion inthe surrounding air and ignores the temperature distribution along the droplet–airinterface. If contact angle, � , is constant (evaporation stage (ii) in case of contactangle hysteresis), then (1) gives the evaporation rate proportional to the first powerof the contact line radius, L.
We showed earlier [18] that proportionality of the total evaporation flux, J, tothe droplet perimeter has nothing to do with a distribution of the local evaporationflux, j, over the droplet surface. Let us reproduce the derivation of that statement byconsidering a stationary diffusion equation for vapour in air:
1
r
@
@r
�
r@c
@r
�
C @2c
@z2D 0; (3)
where, r and z are radial and vertical coordinates, respectively; c is the molar vapourconcentration. The local flux, j, in normal direction to the droplet’s surface is
j D �D@c
@n
ˇˇzDh.r/ D �D
@c
@r
ˇˇˇˇzDh.r/nr C @c
@z
ˇˇˇˇzDh.r/
nz
!
; (4)
where D is the diffusion coefficient of vapour in the air; n, nr and nz are unit vectornormal to the liquid–air interface (pointing into the air), and its radial and verticalcomponents, respectively; h.r/ is the height of the droplet surface. Let us introducedimensionless variables using the same symbols as the original dimensional ones butwith an over-bar: Nz D z=L, Nr D r=L, Nh D h=L, Nc D c=�c, �c D csat .Tsurf /�c1.Then (4) can be rewritten as:
j D �D�c
L
@ Nc@NrˇˇˇˇzD Nh.Nr/nr C @ Nc
@NzˇˇˇˇzD Nh.Nr/
nz
!
D D�c
LA.Nr; Nz/; (5)
where A.Nr; Nz/ D � @ Nc@NrˇˇˇzD Nh.Nr/nz . Hence, the total flux is
J D 2�
Z L
0
rj
s
1 C�
@h
@r
�2
dr D 2�LD�c
Z 1
0
NrA.Nr; Nz/vuut1 C
@ Nh@Nr
!2
d Nr: (6)
288 S. Semenov et al.
The latter equations show that the total flux, J � L, and the local flux j � 1=L.Note, those properties do not depend on the distribution of the local evaporationflux, j, over the droplet surface. The latter conclusions agree with the previousconsideration by Guena et al. [19]. Note, that the latter properties are valid onlyin case of diffusion controlled evaporation.
4 Distribution of Evaporation Flux at the Droplet Surface
It was shown above that the proportionality of total evaporation flux, J, to theradius of the contact line, L, and accordingly its proportionality to the perimeterof the droplet does not necessary mean that evaporation occurs mostly at thedroplet perimeter. However, a number of researches showed that in case of contactangles � < 90ı the evaporation indeed is more intensive in a vicinity of the threephase contact line. Several different principles were utilised in order to explain thisphenomenon: (a) non-uniform distribution of vapour flux over the droplet surfacedue to the diffusion controlled process of vapour transfer to the ambient air [4–6];(b) action of Derjaguin’s (disjoining/conjoining) pressure at the three phase contactline [20–23]; (c) evaporative cooling of the liquid–gas interface (due to latent heatof vaporization) and formation of the temperature field leading to a comparativelymore intensive evaporation at the three phase contact line [24].
Deegan et al. [4, 5] studied the distribution of density of vapour flux over thespherical cap of a sessile droplet, solving the diffusion equation and neglecting thelatent heat of vaporization and thermocapillary flow inside the droplet. The obtainedsolution for the droplets with contact angles � < 90ı shows an infinite increase ofthe vapour flux in a vicinity of the three phase contact line. Such distribution of theflux, according to the authors, generates the flow inside the droplet, which transportsparticles to the edge of the droplet and results in a ring-like stain formation (coffeerings).
Starov and Sefiane [24] suggested a physical mechanism of redistribution ofevaporation flux which is controlled by the temperature field rather than by theprocess of vapour diffusion into air. According to their model, there is convectionin the ambient air, so that vapour diffusion occurs only in a boundary layer. If thethickness of the boundary layer, ı, is constant then the vapour diffusion across thelayer is controlled by the difference of vapour concentrations in the ambient air andat the droplet surface. The latter is a function of the local temperature of the droplet’ssurface. In the model under consideration [24] the surface of a droplet is cooled bythe evaporation; meanwhile due to the high heat conductivity of the substrate thetemperature of the contact line is stayed equal to the ambient one. As a result thehigher temperature at the three phase contact line gives higher vapour concentrationand more intensive evaporation flux at the droplet’s perimeter (see Fig. 1).
Evaporation of Sessile Droplets of Liquid on Solid Substrates 289
Fig. 1 Temperature distribution over the droplet surface, Ts.r/. r is the radial coordinate; T0 is thetemperature of the substrate; Ts1 is the temperature of the droplet surface at which the evaporationflux vanishes; l is the contact line radius; � is a tiny area within the vicinity of the three phasecontact line, where evaporation mostly takes place [24]
5 Thermal Marangoni Convection
Studies of evaporation of droplets with contact angles � < 120ı were performed byGirard et al. [12, 14–17] and Hu and Larson [25–28]. Girard et al. investigated theinfluence of substrate heating [16, 17], air humidity [16] and Marangoni convection[12]. They concluded that contribution of Marangoni convection to the total vapourflux is negligible, whereas heating of the substrate is important. Hu and Larsoninvestigated the process of particles deposition and ring-like stain formation duringthe droplet evaporation [25–27]. They concluded that the density profile of theparticles deposit substantially depends on the Marangoni convection within thesessile droplet [28]. If Marangoni convection is present, then it results in a particledeposition at the droplet centre rather than at the edge. According to authors, thesuppression of Marangoni convection is the one of the important conditions for ring-like deposit formation.
Ristenpart et al. [29] investigated the influence of the substrate conductivity onreversal of Marangoni circulation within evaporating sessile droplet. They neglectedthe thermal conductivity of the surrounding air. The authors used predefineddistribution of evaporation flux over the droplet surface:
j.r/ D j0
�1 � .r=L/2
��1=2C�=�; (7)
where j0 is a constant determined by the diffusivity of vapour in the ambient airand the ambient humidity, r is the radial coordinate. This expression for j.r/ (a) isnot applicable for contact angles � > �=2, and (b) introduces the singularity at thethree phase contact line: j.L/
ˇˇ�<�=2 D 1. Despite of these assumptions made by
authors, their quantitative criteria for the circulation direction was experimentallyconfirmed.
290 S. Semenov et al.
Earlier [18, 30] we used numerical simulations to investigate instantaneousdistribution of heat and mass fluxes in the system consisting of a single sessiledroplet of a pure liquid, substrate, and surrounding air. Self-consistent systemof equations was solved including Navier–Stokes equations inside the droplet;heat transfer equations in substrate, droplet and gas; and vapour diffusion in thesurrounding air. Latent heat of vaporization and thermal Marangoni convection weretaken into account. It is shown in [18, 30] that presence of Marangoni convectionresults in deviations from the earlier deduced laws for both local evaporation fluxj L�1 and total evaporation flux, J L. Note, the latter dependences were deducedfor isothermal evaporation. If the mean temperature of the droplet surface is usedinstead of the temperature of the surrounding air for the vapour concentration onthe droplet surface then the calculated dependences for the total evaporation fluxcoincide with those calculated for the isothermal case [18, 30].
6 Influence of Heat Conductivity
It is well known that evaporation process consumes heat due to the latent heat ofvaporization. Because of that the droplet’s surface cools down and a heat flux fromthe surface is generated to compensate for heat losses. As the heat conductivities ofthe droplet liquid and substrate material are much higher than that of air then themajor part of the heat flux goes through the droplet and substrate. Thus the heatconductivities of liquid and substrate define the temperature drop at the dropletsurface. From the other hand the temperature of the droplet surface defines thevalue of saturated vapour concentration (in case of diffusive model of evaporation)and hence the evaporation rate. Dunn et al. [31, 32] solved the coupled problem ofvapour diffusion and heat transfer for the evaporation of sessile droplets of differentliquids on substrates with different thermal properties. They demonstrated bothexperimentally and numerically that the heat conductivity of the substrate stronglyinfluences the evaporation rate. Decreasing the heat conductivity of the substratecauses a decrease of the evaporation rate.
7 Ring-Like Stain Formation
The formation of ring-like stains during the droplet evaporation has been studiedby a number of scientists. Deegan et al. [5] studied deposition of particles in avicinity of the three phase contact line contact line and reasons for this phenomenon.The authors concluded that formation of ring-like stains requires “a weakly pinningsubstrate and evaporation”. Hu and Larson [28] reported that formation of suchdeposits requires not only a pinned contact line but also suppression of Marangoniflow. They demonstrated of deposit in the centre of the droplet.
Evaporation of Sessile Droplets of Liquid on Solid Substrates 291
Bhardwaj et al. [6] solved numerically a complex problem of drying of dropletsof colloidal solutions and deposits formation. Their model takes into accountthe Navier–Stokes equations, convection and conduction heat transfer equations,Marangoni convection and receding of the three phase contact line. The interactionof the free surface with the peripheral deposit and eventual depinning were alsosimulated. The diffusion of vapour in the atmosphere was solved numerically,providing an exact boundary condition for the evaporative flux at the droplet–air interface. The formation of different deposit patterns both theoretically andexperimentally the possibility of Marangoni flow reverse and formation obtainedexperimentally is explained by their simulations.
8 Complete Wetting
In the case of complete wetting droplets spread out completely over a solidsubstrate, and contact angle decreases down to zero value. Lee et al. [33] consideredprocess of simultaneous spreading and evaporation of sessile droplets in the caseof complete wetting. In order to model the spreading they [33] considered Stokesequations under a low slope approximation. For the modelling of evaporationthe proportionality of the total evaporation flux, J, to the contact line radius, L,was assumed. The whole process of spreading/evaporation was divided into twostages: (a) a first fast but short stage spreading stage, when the evaporation can beneglected, and the droplet volume, V, is approximately constant; (b) a second slowerstage, when the spreading process is almost over, contact angle approximatelyconstant, and evolution is determined by the evaporation. On the basis of thisanalysis the contact line radius, L, is considered as a function of the droplet volume,V, and contact angle, � . Time derivative of L.V; �/ gives two velocities of thecontact line:
dV.V; �/
dtD @L.V; �/
@�
d�
dtC @L.V; �/
@V
dV
dtD vC � V�; (8)
where vC is the spreading velocity, and v� is the “shrinkage” velocity due to theevaporation:
vC D dL.V; �/
dtjV �const D @L.V; �/
@�
d�
dt; (9)
v� D �dL.V; �/
dtj��const D �@L.V; �/
@V
dV
dt: (10)
The spreading velocity of the contact line, vC, is obtained by Starov et al. in [34]:
vC D 0:1
�4V
�
�0:3 �10�!
�
�0:11
.t C t0/0:9; (11)
292 S. Semenov et al.
where � is the surface tension of the liquid; � is the dynamic viscosity of the liquid;! is the effective lubrication parameter [35]; t0 is the duration of the initial stage ofspreading when the capillary regime of spreading is not applicable [34]. Equation(11) is derived from (9) using the formula for L.t/ obtained by Starov et al. in [35]:
L.t/ D L0 .1 C t=�/0:1 ; (12)
where L0 is the droplet base radius after the very fast initial stage is over; � D3�L0
10�
���L3
0
4V
�3
; is the dimensionless constant [35] connected to the effective
lubrication parameter ! [35]. The velocity v� is obtained from (10) using (1):
v� D ˇF.�/L2
3V: (13)
Substituting (11) and (13) into (8) leads to the following equation:
dL
dtD 0:1
�4V
�
�0:3 �10�!
�
�0:11
.t C t0/0:9
� ˇF.�/L2
3V: (14)
The latter gives a system of two differential (1) and (14) with following boundaryconditions [33]:
V.0/ D V0; (15)
L.0/ D L0 D"
10�!
�
�4V0
�
�3#0:1
t0:10 ; (16)
where V0 is the initial droplet volume and L0 is the contact line radius after thevery fast initial stage is over. Solution of this system of equation in non-dimensionalform gives a universal law of process of simultaneous spreading and evaporation forthe case of complete wetting, which was validated against experimental data fromvarious literature sources [33] (see Figs. 2 and 3).
9 Partial Wetting and Contact Angle Hysteresis
Earlier in [36] we discussed the evaporation of sessile water droplets in the presenceof contact angle hysteresis. Model presented in [36] describes non-isothermaldiffusion limited evaporation. It takes into account latent heat of vaporization, heattransfer in solid, liquid and gas phases, and thermal Marangoni convection in waterdroplet. The rate of diffusion limited evaporation depends on vapour concentrationin the ambient air and on concentration of the saturated vapour over the droplet’ssurface. The latter depends on surface temperature. Thus we decided to compareevaporation rate, J, from our computer simulations [18] with the one calculatedusing (1), but substituting the average temperature, Tav, of the droplet’s surface
Evaporation of Sessile Droplets of Liquid on Solid Substrates 293
Fig. 2 Spreading/evaporation in the case of complete wetting: dimensionless radius againstdimensionless time comparing different liquid droplets spreading/evaporating on solid substrates.Experimental points from various literature sources and the solid line according to the theoreticalprediction [33]
obtained from computer simulations, instead of Tsurf . These two evaporation ratescoincide within the simulation error bar [18]. Thus (1) with new parameter ˇ
ˇ2�DM
�Œcsat .Tav/ � c1 ; (17)
where Tav is the average temperature of the droplet’s surface, provides a very goodapproximation of the droplet’s evaporation rate for non-isothermal case.
The average temperature, Tav, of the droplet’s surface depends on thermalconductivities of all phases, on ambient temperature, T1, air humidity, H , andcontact angle, � , formed by the droplet in contact with a substrate. In absenceof external heating or cooling of the substrate, Tav can be approximated with thefollowing expression (when T1 is close to 20ıC) [36]:
Tav D Tsurf
ˇˇksD0�D�=2 C T1 � Tsurf
ˇˇksD0�D�=2
1 C F.�/ Œ.ka=kw/ .sin� � 0:75� C 4:61/; (18)
where function F.�/ is defined by (2); ka, kw and ks are thermal conductivities ofair, water droplet and substrate, respectively; Tsurf
ˇˇksD0�D�=2 is the root of (19)
with respect to Tsurf :
�D�csat
�Tsurf
� Hcsat .T1/� D ka
�T1 � Tsurf
; (19)
294 S. Semenov et al.
Fig. 3 Spreading/evaporation in the case of complete wetting: dimensionless contact angle againstdimensionless time curve for the behaviour of the droplet radius (see Fig. 2) comparing theoretical(solid curve) and experimental data (symbols) [33]
where D is the diffusion coefficient of vapour in air, � is the latent heat ofvaporization (in J/mol). Equation (19) represents the relation between diffusiveevaporation rate and conductive heat flux through the air (air convection is neglectedin [36]), in case when thermal conductivity of the substrate, ks , is zero and contactangle is 90ı, which results in absence of heat transfer through the substrate and thedroplet.
Using (1) with the new parameter ˇ, see (17), and assuming that ˇ is almostconstant during the process of droplet evaporation [36], allowed us to develop atheory describing stages of the process of sessile droplet evaporation in case ofcontact angle hysteresis (i), constant radius of the droplet base, L D L0, but thecontact angle decreases over time from the static advancing contact angle down tostatic receding contact angle; and (ii), constant contact angle, � , equals to the staticreceding contact angle but radius of the droplet base decreases over time. Resultshave been compared with the experimental data from literature for evaporation ofwater droplets on different substrates. First, experimental data for stage (i) havebeen fitted with the theoretical curve by adjusting unknown parameter ˇ. Then thesame values of ˇ have been used for plotting theoretical curves of stage (ii), whichshowed very good agreement with experimental data (see Fig. 4).
Evaporation of Sessile Droplets of Liquid on Solid Substrates 295
Fig. 4 Evaporation of sessiledroplets of water on differentsubstrates in case of partialwetting with contact anglehysteresis: comparison oftheoretical curves (solution of(1) with ˇ from (17)) withexperimental data. (a) stage(i) of the evaporation process,when contact line radius isconstant L D L0. (b) stage(ii) of the evaporationprocess, when contact angleis constant and equal its staticreceding value �r . ` D L=L0
is the non-dimensionalcontact line radius. Q� and N�are non-dimensional times forstages (i) and (ii) respectively.Redrawn from [36].Experimental data are takenfrom [10, 37, 38]
In Fig. 4 non-dimensional times are defined as follows:
Q� D � C B .�ad / ;
N� D 2F.�r/
3f .�r/.� � �r / ;
where � D tˇ=L20; L0 is the value of L during the first evaporation stage (i); B.�/ D
R �=2
� f 0.�/=F.�/d� ; f 0.�/ � df .�/=d� ; F.�/ is defined by (2); �r is the value of� at the beginning of stage (ii), when receding of the contact line starts; f .�/ DVL3 D �
3
.1�cos�/2.2Ccos�/
sin3�under assumption of a spherical cap shape of the droplet.
296 S. Semenov et al.
10 Kelvin’s and Kinetic Effects
Below we studied the influence of kinetic effects on evaporation of pinned sessilewater droplets of submicron size on copper substrate. The model takes into accountthe influence of curvature of the droplet’s surface (Kelvin’s equation for the pressureof saturated vapour above a curved liquid–gas interface) on total evaporation rate, J .
Kinetic effects are included into model using Hertz–Knudsen–Langmuirequation, (20), for local evaporation/condensation flux at the droplet–air interfaceas a boundary condition for the diffusion equation of vapour in air:
jm D ˛m
rMRT
2�Œcsat .T / � c ; (20)
where jm is the surface density of mass flux across the droplet–air interface indirection from liquid to air; ˛m is the mass accommodation coefficient (probabilitythat uptake of vapour molecules occurs upon collision of those molecules with theliquid surface); R is the universal gas constant; T and c are the local temperaturein ıK and molar vapour concentration at the liquid–gas interface, respectively; M isthe molecular weight of the evaporating substance; csat is the molar concentrationof saturated vapour at the droplet–air interface.
Equation (20) is based on the kinetic theory of gases and it links evaporation flux,jm, with local vapour concentration, c, and local temperature, T, at the liquid–airinterface.
The model also includes thermal effects: latent heat of vaporization and thermalMarangoni convection, whose presence reduced evaporation rate for less than 5%(water on copper substrate) as well as thermal conductivity in all phases. It is shownthat Stefan flow, generated in air by the evaporation process, reduced evaporationrate by less than 0.2%.
Results of computer simulations are presented in Fig. 5. The model used is validonly for droplet size bigger than the radius of surface forces action, which is around10�7 m D 0.1 �m. Results for contact line radius L < 10�7 m have no physicalmeaning, as additional surface forces have to be included into the model at thisscale (disjoining/conjoining Derjaguin’s pressure). The range of sizes L < 10�7 mis shown only for the demonstration of curves tendencies.
Figure 5 shows that kinetic effects become important only for submicron droplets(L < 10�6 m), therefore a deviation from the pure diffusion model of evaporationcan be neglected for the droplet size bigger than 10�6 m. Kelvin’s effect influencesthe evaporation rate, J, only for droplets of size L < 1:1 � 10�8 m, and can beneglected for droplets of size L > 10�7 m.
The latter shows that a consideration of evaporation of microdroplets completelycovered by the surface forces action (that is less than 10�7 m) should include bothdeviation of the saturated vapour pressure caused by the droplet curvature (Kelvin’seffect) and the kinetic effects.
Evaporation of Sessile Droplets of Liquid on Solid Substrates 297
Fig. 5 Influence of Kelvin’s equation and kinetic effects on evaporation rate, J, of pinned sessiledroplets of water on copper substrate. L is the contact line radius
Though the influence of above mentioned effects on total evaporation flux, J, isnegligible for droplets of size L > 10�6 m, they may be important for the calculationof a local evaporation flux at the three-phase contact line.
11 Derjaguin’s (Disjoining/Conjoining) Pressure
From theoretical point of view two singularities have to be coped with simulta-neously [39] at the three-phase contact line. The first problem is associated withthe well-known problem of a singularity at the moving three phase contact line: asingularity of the viscous stress caused by an incompatibility of no-slip boundarycondition at the solid–liquid interface with boundary condition of moving liquid–air interface. The second problem is associated with the specific behaviour of theevaporation flux at the perimeter of the droplet. The latter singularity is caused byan incompatibility of a boundary condition for vapour flux at liquid–air interfacewith the boundary condition of no vapour penetration at the solid–air interface.
In order to overcome the problem of singularities at the three-phase contactline it is necessary to replace mathematically inconsistent boundary conditions byphysically correct ones. It can be done using Derjaguin’s (disjoining/conjoining)pressure concept. This approach does not introduce a three-phase contact line and,
298 S. Semenov et al.
therefore, rules out any singularity problems in a vicinity of the otherwise apparentthree-phase contact line [20].
Surface forces (Derjaguin’s pressure) act in a vicinity of the apparent three-phasecontact line so called transition zone [20]. The presence of the Derjaguin’s pressuredisturbs the initial special profile of the liquid droplet in a vicinity of the three phasecontact line. It is also known that a very thin adsorbed film forms on a solid surface,which is at the thermodynamic equilibrium with the vapour concentration in thesurrounding humidity. That is, in the humid air, water vapour forms a thin waterfilm on the surface of a solid substrate. Thus, a liquid–air interface of a sessileliquid droplet is actually in a contact with this adsorbed water film (or a film ofother substance); thus, in this approach there is no true three-phase contact line.
First models of evaporation in a vicinity of the three phase contact line based onconsideration of Derjaguin’s (disjoining/conjoining) pressure action were developedby Potash and Wayner [22], and Moosman and Homsy [21], who used theDerjaguin’s pressure to model the transport phenomena in an evaporating two-dimensional meniscus (both in the case of complete wetting). In [22] the authorscalculated the meniscus profile, heat flux profile, and pressure gradient profile. In[23] the authors deduced a meniscus profile changes relative to the static isothermalone, as well as an evaporation flux from the interface using a perturbation theory.Both [22] and [23] demonstrated that a large heat and evaporation fluxes occur inthe transition region between the capillary meniscus and the adsorbed layer. Stephanet al. [40] investigated experimentally evaporation in heat pipes with groovedwalls. They confirmed the theoretical conclusions by Moosman and Homsy [21] onprediction that significant part of the evaporation flux is localized at the three-phasecontact line.
Ajaev et al. [23] studied both static and dynamic values of the apparent contactangle for gravity-driven flow of a volatile liquid down of a heated inclined plane. Theauthors investigated macroscopic boundary conditions which could be used witha conventional continuum approach and agreed with the micro scale phenomenaat the contact line. They found the profile of the liquid–vapour interface in theregion of the apparent three-phase contact line and determined the dependence of themacroscopic contact angle on the temperature of the contact line and the velocityof its motion. The interface profile in the region was determined by a disjoiningpressure action and asymptotically approaches the adsorbed thin liquid film. It wasfound that the curvature of the interface at that transition region is very high. Authors[23] also investigated the effect of evaporation on moving contact line in the caseof partial wetting. They proposed a generalization of the approach of Moosman andHomsy [21] and Ajaev et al. [41, 42].
Diaz et al. [43] studied a static puddle taking into account capillarity, gravity anddisjoining pressure. They found an analytical solution for the shape of the vapour–liquid interface in the transition zone between adsorbed liquid layer and capillaryregion.
Eggers et al. [44] studied the evolution of a droplet of pure liquid on a solidsubstrate in case of complete wetting and intensive evaporation. They coupledviscous flow with evaporation from droplet’s surface and its precursor film. The
Evaporation of Sessile Droplets of Liquid on Solid Substrates 299
evaporation was limited by a diffusion of vapour into the surrounding atmosphere.Authors found that their model describes well the final stage of evaporation whendrop radius goes to zero like L .t0 � t/˛ , where ˛ has value close to 1/2, which is inagreement with experiments.
All the above examples of disjoining pressure action at the apparent three-phase contact line gave the evidence of possibility to construct physically consistentmacroscopic boundary conditions at the contact line taking into account microscopicphenomena.
References
1. Haschke, T., Lautenschlager, D., Wiechert, W., Bonaccurso, E., Butt, H.-J.: Simulation ofEvaporating Droplets on AFM-Cantilevers. In: Excerpt from the Proceedings of the COMSOLMultiphysics User’s Conference, Frankfurt, 2005
