12.5 Change of Phase: Evaporation and Condensation12.5.1 Interfacial Conditions We now consider the case of an evaporating condensing thin film of a simple liquid lying on a heatedcooled
Trang 112.5 Change of Phase: Evaporation and Condensation
12.5.1 Interfacial Conditions
We now consider the case of an evaporating (condensing) thin film of a simple liquid lying on a heated(cooled) plane surface held at constant temperature ϑ0which is higher (lower) than the saturation tem-perature at the given vapor pressure It is assumed that the speed of vapor particles is sufficiently low, sothat the vapor can be considered an incompressible fluid
The boundary conditions appropriate for phase transformation at the film interface z h are now
for-mulated The mass conservation equation at the interface is given by the balance between the liquid andvapor fluxes through the interface
j ρυ(vυ vi) n ρ
where j is the mass flux due to evaporation;ρυandρ
f are, respectively, the densities of the vapor and the
liquid; vυand vf are the vapor and liquid velocities at z h; and v iis the velocity of the interface Equation(12.79a) provides the relationship between the normal components of the vapor and liquid velocities atthe interface The tangential components of both of the velocity fields are equal at the interface:
where L is the latent heat of vaporization per unit mass; k th,υ,µυ,ϑυare, respectively, the thermal
con-ductivity, viscosity, and the temperature of the vapor; vυ,r vυ vi, vf,r vf viare the vapor and uid velocities relative to the interface, respectively;υυ,
liq-n vυ,r n, vf,n vf,r n are the normal components
of the latter; and ef, eυare the rate-of-deformation tensors in the liquid and the vapor, respectively InEquation (12.80b) the first term represents the contribution of the latent heat, the combination of thesecond and the third terms represents the interfacial jump in the momentum flux, the combination of thefourth and the fifth terms represents the jump in the conductive heat flux at both sides of the interface,while the combination of the last two terms is associated with the viscous dissipation of energy at bothsides of the interface
Since ρυ/ρ
f 1, typically of order 103, it follows from Equation (12.79a) that the magnitude of thenormal velocity of the vapor relative to the interface is much greater than that of the liquid Hence, thephase transformation causes large accelerations of the vapor at the interface where the back reaction,called the vapor recoil, represents a force exerted on the interface During evaporation (condensation) thetroughs of the deformed interface are closer to the hot (cold) plate than the crests, so they have greater
evaporation (condensation) rates j The dynamic pressure at the vapor side of the interface is much larger
than that at the liquid side,
1
2
Trang 2Momentum fluxes are thus greater in the troughs than at the crests of surface waves Vapor recoil is a
destabilizing factor for the interface dynamics for both evaporation (j 0) and condensation (j 0) [Burelbach et al., 1988] Scaled with j2, see Equation (12.84), the vapor recoil is only important for appli-cations where very high mass fluxes are involved
Vapor recoil generally exerts a reactive downward pressure on a horizontal evaporating film Bankoff(1961) introduced the effect of vapor recoil in the analysis of the film boiling In this analysis the liquidoverlays the vapor layer generated by boiling and leads to the Rayleigh–Taylor instability of an evaporatingliquid–vapor interface above a hot horizontal wall In this case the vapor recoil stabilizes the film boilingbecause the reactive force is greater for the wave crests approaching the wall than for the troughs
To obtain a closure for the system of governing equations and boundary conditions, an equation relatingthe dependence of the interfacial temperature ϑ
iand the local pressure in the vapor phase is added[Plesset and Prosperetti, 1976; Palmer, 1976; Sadhal and Plesset, 1979] Its linearized form is
sis the absolute saturation temperature, ˆαis the accommodation coefficient, Rυ is the universal gas constant,
and M wis the molecular weight of the vapor [Palmer, 1976; Plesset and Prosperetti, 1976; Burelbach et al.,1988] Note that the absolute saturation temperature ϑsserves now as the reference temperature instead of
ϑ∞in the normalization, Equation (12.49) When ∆ϑi 0, the phases are in thermal equilibrium with eachother, and in order for net mass transport to take place, a vapor pressure driving force must exist, given forideal gases by kinetic theory [Schrage, 1953] The latter is represented in the linear approximation by theparameter ~ K [Burelbach et al., 1988] Departure from ideal behavior is addressed in the parameter K by ~
the presence of an accommodation coefficient ˆαdepending on interface/molecule orientation and stericeffects which represents the probability of a vapor molecule sticking upon hitting the liquid–vapor interface.The set of the boundary conditions Equations (12.80) can be simplified to what is known as a “one-sided” model for evaporation or condensation [Burelbach et al., 1988] in which the dynamics of the liq-uid are decoupled from those of the vapor This simplification is possible because of the assumption ofsmallness of density, viscosity, and thermal conductivity of the vapor with respect to the respective prop-erties of the liquid The vapor dynamics are ignored in the one-sided model, and only the mass conser-vation and the effect of vapor recoil stand for the presence of the vapor phase
The energy balance Equation (12.80b) becomes
αρυL
Trang 3The procedure of asymptotic expansions outlined in the beginning of this chapter is used again toderive the pertinent evolution equation The dimensionless mass balance Equation (12.13) is modified by
the presence of the non-dimensional evaporative mass flux, J jdL/k th(ϑ0 ϑs)
is substituted into Equation (12.85b) to obtain the sought evolution equation The general dimensionless
evolution Equation (12.21) will then contain an additional term EJ, which arises from the mass flux because of evaporation and condensation now expressed via the local film thickness H.
A different approach to theoretically describe the rate of evaporative flux j in the isothermal case is
known in the literature [Sharma, 1998; Padmakar et al., 1999] This approach is based on the extendedKelvin equation that accounts for the local interfacial curvature and the disjoining and conjoining pressures,
both entering the resulting expression for the evaporative mass flux j It was shown by Padmakar et al (1999)
that their evaporation model admits the emergence of a flat adsorbed layer remaining in equilibrium withthe ambient vapor phase, and thus in this state the evaporation rate from the film vanishes This adsorbedlayer, however, is usually several molecular spacings thick, which is beyond the resolution of continuumtheory
12.5.2 Evaporation/Condensation Only
We first consider the case of an evaporating or condensing thin liquid layer lying on a rigid plane held atconstant temperature Mass loss or gain is retained, while all other effects are neglected
Solving first Equation (12.53) along with boundary conditions Equations (12.51a) and (12.86) and
eliminating the mass flux J from the latter yields the dimensionless temperature field and the evaporative
mass flux through the interface
Trang 4An initially flat interface will remain flat as evaporation or condensation proceeds If surface tension,
thermocapillary, and convective thermal effects are negligible (i.e., M S εRP 0), it will give rise to
a scaled evolution equation of the form
so that H/K 1 at this point However, H/K 101at d 30 nanometers, so that the resistance to
con-duction is small compared to the interfacial transport resistance Shortly after, van der Waals forcesbecome appreciable
12.5.3 Evaporation/Condensation, Vapor Recoil, Capillarity, and
eva-1
3
3
2
Trang 5Taking into account van der Waals forces and thermocapillarity, the complete evolution equation for athin heated or cooled film on a horizontal plane surface was given by Burelbach et al (1988) in the form
KM P1
2
Hξ ξ
S(H3Hξξξ)ξ 0 (12.91)
with M εM Here the first term represents the rate of volumetric change; the second one the mass
loss/gain; the third, fourth, and fifth ones the attractive van der Waals, vapor recoil, and thermocapillaryterms, all destabilizing; while the sixth term describes the stabilizing capillary force This was the first fullstatement of the possible competition among various stabilizing and destabilizing effects on a horizontalplate, with scaling making them present at the same order Other effects such as gravity may be included
in Equation (12.91) Joo et al (1991) extended the work to an evaporating (condensing) liquid film ing down a heated (cooled) inclined plate
drain-Oron and Bankoff (1999) studied the two-dimensional dynamics of an evaporating ultrathin film on
a coated solid surface when the potential Equation (12.31d) was used Three different types of the tion of a volatile film were identified One type is related to low evaporation rates associated with rela-
evolu-tively small E 0 when holes covered by a liquid microlayer emerge, and the expansion of such holes isgoverned mainly by the action of the attractive molecular forces These forces impart the squeeze effect
to the film and, as a result of this, the liquid flows away from the hole In this stage the role of tion is secondary.Figure 12.9 displays such an evolution of a volatile liquid film Following the nucleation
evapora-of the hole and during the process evapora-of surface dewetting, one can identify the formation evapora-of a large ridge,
or drop, on either side of the trough The former grows during the evolution of the film until the drops
at both ends of the periodic domain collide A further recession of the walls of the dry spot leads to theformation of a single large drop that flattens and ultimately disappears, according to Equation (12.89).The stages of the film evolution shown in Figure 12.9(a) are very similar to that sketched in Figure 3 ofElbaum and Lipson (1995) This type of evolution also resembles the results obtained by Padmakar et al.(1998) for the isothermal film subject to hydrophobic interactions and to evaporation driven by the dif-ference between the equilibrium vapor pressure and the pressure in the vapor phase Such films thin uni-formly to a critical thickness and then spontaneously to dewet the solid substrate by the formation ofgrowing dry spots when the solid was partially wetted In the completely wetted case, thin liquid films evolved
to an array of islands that disappeared by evaporation to a thin equilibrium flat film Two other regimescorresponding to intermediate and high evaporation rates were discussed in Oron and Bankoff (1999)
An important phenomenon was found in the last stage of the evolution of an evaporating film wherethe latter finally disappears by evaporation: prior to that the film equilibrates, so that its disappearance ispractically uniform in space The film equilibration is caused by the “reservoir effect,” which is driven bythe difference in disjoining pressures and manifests itself by feeding the liquid from the large drops intothe ultrathin film that bridges between them
Oron and Bankoff (2001) studied the dynamics of condensing thin films on a horizontal coated solidsurface In the case of a relatively fast condensation, where the initial depression of the interface rapidlyfills up because of the enhanced mass gain there, the film equilibrates and grows uniformly in space
according to Equation (12.89) Note that E 0 When condensation is relatively slow, the evolution ofthe film exhibits several distinct stages The first stage, dominated by attractive van der Waals forces, leads
to the opening of a hole covered by a microlayer, as shown in the first three snapshots ofFigure 12.10(a).This is accompanied with continuous condensation with the highest rate of mass gain attained in the
microlayer region corresponding to the smallest thickness H in Equation (12.87) However, opposite to
the evaporative case [Oron and Bankoff, 1999], where the “reservoir effect” arising from the differencebetween the disjoining pressures causes feeding of the liquid from the large drops into the microlayer andfilm equilibration, in the condensing case the excess liquid is driven from the microlayer into the largedrops This effect is referred to as the “reversed reservoir effect.” The thickness of the microlayer remainsnearly constant because of local mass gain by condensation compensating for the impact of the reversereservoir effect The first stage of the film evolution terminates in the situation where the size of the hole
1
3
Trang 6is the largest The receding of the drops stops due to the increase of the drop curvature and buildup ofthe capillary pressure that comes to balance with the squeeze effect of the attractive van der Waals forces.From this moment the hole closes driven by condensation, as shown in Figure 12.10(a, b) Once the holecloses, the depression fills up rapidly, the amplitude of the interfacial disturbance decreases, and the filmtends to flatten out The film then grows uniformly in space following the solution Equation (12.89) with
negative E
Oron (2000c) studied the three-dimensional evolution of an evaporating film on a coated solid surfacesubject to the potential Equation (12.31d) The main stages of the evolution repeat those mentioned pre-viously in the case of a non-volatile film in the section on isothermal films, except for the stage of disap-pearance accompanied by the reservoir effect Because of the reservoir effect, the minimal film thicknessdecreases very slowly during the stage of film equilibration
12.5.4 Flow on a Rotating Disc
Reisfeld et al (1991) considered the axisymmetric flow of an incompressible viscous volatile liquid on ahorizontal, rotating disk The liquid was assumed to evaporate because of the difference between the
FIGURE 12.9 The evolution of a slowly evaporating film: (a) the initial and intermediate stages of the film tion, and (b) the final stage of the evolution The curves in both graphs correspond to the interfacial shapes in con-secutive times (not necessarily equidistant) (Reprinted with permission from Oron and Bankoff (1999).)
Trang 7evolu-vapor pressures of the solvent species at the fluid–evolu-vapor interface and in the gas phase This situation isanalogous to the phase two of spin coating process.
The analysis is similar to what is done in the section on isothermal films, but now with an additional
parameter describing the process of evaporation, which for a prescribed evaporative mass flux j is defined as
2
3
FIGURE 12.10 The evolution of a slowly condensing film on the horizontal plane (a) The curves from the bottom
to the top correspond to consecutive times (not necessarily equidistant) (b) The curves from the left to the right respond to consecutive times (not necessarily equidistant) The flat curve corresponds to the interface at a certain timeafter which the film grows uniformly in space according to Equation (12.89) In the graph (a), the dashed lines rep-
cor-resent the location of the solid substrate H 0 (Reprinted with permission from Oron and Bankoff (2000).)
Trang 8Equation (12.92) models the combined effect of local mass loss, capillary forces and centrifugal drainage,none of which describe any kind of instability.
For most spin coating applications S is very small, and the corresponding term may be neglected,
although it may be very important in planarization studies where the leveling of liquid films on roughsurfaces is investigated Therefore, Equation (12.92) can be simplified
This simplified equation can then be used for further analysis Looking for flat basic states H H(τ),Equation (12.93) is reduced to the ordinary differential equation which is to be solved with the initial
condition H(0) 1 In the case of E 0, both evaporation and drainage cause thinning of the layer.
Equation (12.93) describes the evolution in which the film thins monotonically to zero thickness in afinite time in contrast with an infinite thinning time by centrifugal drainage only Explicit expressions for
H(τ) and for the time of film disappearance are given in Reisfeld et al (1991) In the condensing case
E 0 drainage competes with condensation to thin the film Initially the film thins due to drainage until
the rate of mass gain because of condensation balances the rate of mass loss by drainage At this point the
film interface reaches its steady location H |E|1/3 The cases where inertia is taken into account are sidered in Reisfeld et al (1991), where linear stability analysis of flat base states is given
con-Experiments with volatile rotating liquid films [Stillwagon and Larson, 1990] showed that the finalstage of film leveling was affected by an evaporative shrinkage of the films Therefore, they suggested sep-arating the analysis of the evolution of evaporating spinning films into two stages with fluid flow domi-nating the first stage and solvent evaporation dominating the second one [Stillwagon and Larson, 1992]
In this chapter the physics of thin liquid films is reviewed and various examples of their dynamics relevantfor MEMS are presented, some of them with reference to the corresponding experimental results Theexamples discussed examine isothermal, non-isothermal with no phase changes, and evaporating andcondensing films under the influence of surface tension, gravity, van der Waals, and centrifugal forces Thelong-wave theory has been proven to be a powerful tool for the research of the dynamics of thin liquid films.However, there exist several optional approaches suitable for a study of the dynamics of thin liquidfilms Direct numerical simulation of the hydrodynamic equations (Navier–Stokes and continuity)[Scardovelli and Zaleski, 1999] mentioned briefly in the introduction represents one of these options Avariety of methods were developed to carry out such simulations: techniques based on Finite ElementsMethod (FEM) [Ho and Patera, 1990; Salamon et al., 1994; Krishnamoorthy et al., 1995; Tsai and Yue,1996; Ramaswamy et al., 1997], techniques based on the boundary-integral method [Pozrikidis, 1992,1997; Newhouse and Pozrikidis, 1992; Boos and Thess, 1999], surface tracking technique [Yiantsios andHiggins, 1989], and others Another optional approach is that of molecular dynamics (MD) simulations[Allen and Tildesley, 1987; Koplik and Banavar, 1995, 2000] Refer directly to these works for more detail
A new approach treating the film interface as a diffuse rather than a sharp one, as presented in thischapter, was recently developed [Pismen and Pomeau, 2000] and applied to various physical situations[Pomeau, 2001; Pismen, 2001; Bestehorn and Neuffer, 2001; Thiele et al., 2001a, b; 2002a, b; 2003].Lastly, new frontiers in the investigation of the dynamics of thin liquid films were recently discussed inthe special issue of “European Physical Journal E, Vol 12(3), 2003” An attempt was made to bridgebetween numerous theoretical and experimental results in order to explain the main mechanism(s) liable
to rupture of a film Open questions, controversial approaches, and contradictory conclusions were all inthe focus of the discussion [Ziherl and Zumer, 2003; van Effenterre and Valignat, 2003; Morariu et al.,2003; Kaya and Jérôme, 2003; Bollinne et al., 2003; Sharma, A., 2003; Thiele, 2003; Stöckelhuber, 2003;Richardson et al., 2003; Müller-Buschbaum, 2003; Green and Ganesan, 2003; Oron, 2003; Manghi andAubouy, 2003; Reiter, 2003]
1
3
2
3
Trang 9It is a pleasure to express my gratitude to A Sharma, G Reiter, and U Thiele for reading the manuscriptand sharing their comments and thoughts with me A Sharma, S Herminghaus, M F Schatz, and E.Zussman are acknowledged for providing me with their experimental results used in this chapter
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