7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Recent improvements in the microscale theory
of nonisothermal liquidvapor contact lines
P. Colinet and A.Ye. Rednikov
Laboratory TIPs (Transfers, Interfaces and Processes), Fluid Physics Unit, Universit6 Libre de Bruxelles,
C.P 165/67, Av. F.D. Roosevelt 50, B1050 Brussels, Belgium
pcolinet@ulb.ac.be and aredniko@ulb.ac.be
Keywords: liquidvapor phase change, heat transfer, apparent contact angle, micromacro coupling
Abstract
The microscale theory of evaporating contact lines is revisited (for the case of a onecomponent liquid and its pure
vapor) in the framework of continuum lubricationtype models allowing the prediction of the apparent contact angle
and of the microscale contribution to the heat flux. The analysis is restricted to perfectly flat and homogeneous
substrates maintained at constant temperature (at least on the small length scales considered here). Additional physical
effects not previously accounted for in the classical theory are discussed, such as the role of the spreading coefficient
S* often used in studies of wetting of nonvolatile liquids, the influence of pressure variations induced by intense
flows in the vapor phase, and the modifications brought about by considering small curvatures of the macroscopic
meniscus and of the substrate. It is shown that depending on the situation considered, some of these effects (and
others, briefly quoted as open questions in the conclusions) might need to be taken into account when numerically
modeling macroscopic liquidvapor flows involving contact lines.
Introduction
The mathematical and numerical modeling of multi
phase flows involving solid boundaries generally faces
important difficulties, due to the presence of unsteady
contact lines, where the classical macroscopic descrip
tion of interfaces fails. On the one hand, at a moving
contact line, there is a wellknown divergence of the vis
cous stress, which results from the explicit assumption
that the distance between the moving interface and the
motionless solid boundary vanishes at a welldefined lo
cation. On the other hand, for the nonisothermal liquid
vapor systems considered in this presentation, a similar
singularity of the heat flux (and of the evaporation rate)
occurs when assuming the liquidvapor interface to be
at saturation temperature T*t (/,,) (at a supposedly con
stant vapor pressure /.,, while the solid wall is main
tained at a different uniform temperature T*.
The present contribution will review some of the clas
sical and more recent models proposed to avoid these
divergences, focusing on physical approaches taking in
termolecular forces with the substrate into account, in
the form of a given disjoining pressure isotherm H* (*),
where $* is the local film thickness (distance between
the interface and the solid wall). If HI ($*) is such that
an adsorbed film exists ahead of the apparent contact
line, both viscous and thermal divergences are typically
smoothed out because the film thickness does not van
ish anymore. If such film does not exist, such as may
be the case e.g. in the partial wetting regime, it remains
to be seen whether these divergences can nonetheless be
avoided, possibly without introducing any other physical
ingredient (such as e.g. a slip length). Even though it is
apparently similar to the socalled precursor film model
(which assumes some computationally convenient con
stant residual film thickness, by p. ri,,h,,i a suitable
form of the disjoining pressure), the presented physical
approach has conceptual advantages over the latter, be
cause the actual existence of the film here depends on
characteristics among which the disjoining pressure
isotherm which can be measured from independent ex
periments.
The classical microscale theory of evaporating con
tact lines, pioneered by Potash & Wayner (1972) in the
case considered here (i.e. a pure liquid and its pure
vapor in contact with a superheated substrate), has led
to a series of works also based on the lubrication ap
proximation (small apparent contact angle between the
interface and the substrate), and which have in partic
ular revealed the existence of an important steady so
lution which we here call "microstructure", i.e. a re
gion of transition between an extended adsorbed film
(typically nanometersized) and a constant slope portion
(defining the apparent contact angle as seen on a macro
scopic scale). In particular, Stephan & Busse (1992)
have shown how to use such a microscale theory in the
context of heat pipes, by coupling the steady microstruc
ture with macroscopic numerical simulations of the heat
transfer in both solid and liquid phases. An extended
parametric analysis of the microstructure, emphasizing
its relevant microscales, and extending previous works
to the influence of Marangoni and vapor recoil effects,
has recently been performed by Rednikov et al. (2009).
In the present paper, after reviewing the classical the
ory and defining relevant scales, more recent theoretical
results of the authors are reviewed, focusing on effects
which could have an influence on the microscopic han
dling of contact lines, which is often a crucial issue for
the accurate simulation of multiphase flow phenomena
such as evaporation in capillaries or porous media, nu
cleate boiling, sessile droplet evaporation, ... First, in
the case of a disjoining pressure form H* (*) ~ 3,
which is usually associated with perfect wetting, it is
shown that the behavior of the microstructure may also
depend on the spreading coefficient S*. Actually, it is
clear that this dependency could already be expected
on the basis on studies of equilibrium shapes of liquid
films (see e.g. BrochartWyart et al. (1991) and Yeh et
al. (1999)). In this respect, the analysis described here
just extends such theories to the case of nonisothermal
situations involving evaporation, and shows in particu
lar that the abovementioned microfilm may be replaced
by another regime (a "truncated" microfilm, see Fig.
1) depending on the values of S* and of the superheat
AT* T T* > 0, where T* Tjt (,,) is the sat
uration temperature corresponding to the farfield pres
sure / ,,
Figure 1: Sketch of the contact line microstructure.
A second effect discussed in this paper is the influ
ence of viscosityinduced pressure fluctuations in the va
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
por phase, which turn out to be important in some situ
ations and especially at large superheat. Finally, the im
portant issue of the coupling between microscopic and
macroscopic scales is addressed, along with the effect of
the small (on the microscale) macroscopic curvatures of
the liquidvapor meniscus and of the substrate. Rather
than accounting for the influences these macroscopic
curvatures have on the microstructure, the goal here is
merely to show that the micromacro coupling can often
be achieved by neglecting these influences, i.e. using the
apparent contact angle and microscale heat flux calcu
lated in the limit of vanishing curvatures. To conclude, a
functional form of possible generalized boundary condi
tions to be used in macroscopic codes is conjectured, and
open questions are pointed out. Note also that through
out this paper, we will use the notation
f* [f f
where f (without asterisk) is the dimensionless version
of a dimensional quantity f* (with asterisk), [f] being
its scale.
Classical theory and relevant microscales
A first essential feature of the classical theory is the use
of a kinetic law for the evaporative mass flux j*, which,
for small deviations with respect to local phase equilib
rium, can be written in the form
L* r*
3 2 (T Tat,loc) (1)
where L*, is a positive phenomenological coefficient,
* is the latent heat, T5 is the interfacial temperature,
and the local saturation temperature is defined by
Tsat,loc O (1
fn*(*) + 2y*H*
P *,C*
(2)
where 7* is the surface tension, H* is the mean inter
facial curvature, /' is a possible vapor pressure fluc
tuation (around the farfield value /,, taken at the in
terface, p* is the density (with indices 1 and v for the
liquid and the vapor, respectively). The deviations with
respect to T* accounted for in Eq. (2) are due to the so
called Kelvin's effects of disjoining and capillary pres
sures, the importance of which is generally noticeable at
very small scales. In addition, pressure fluctuations ob
viously affect the saturation temperature as well, which
is expressed in linearized form by the last contribu
tion in Eq. (2). Note that the classical theory, rely
ing on the socalled onesided assumption (Burelbach et
al. (1988)), assumes the vapor pressure to be constant,
hence i. 0. In a later section, we will relax this
hypothesis, however.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The "conductance" coefficient L*, originates from
the linear theory of irreversible processes (see e.g. Kjel
strup & Bedeaux (2008), though the possibility of an
interfacial temperature jump is neglected here), and in
particular illustrations hereafter, we will make use of its
kinetic theory (HertzKnudsenSchragelike) expression
Lw, , Pv (3)
S 2 a V R*
where fa is the accommodation coefficient, M* is the
molar mass and R* the universal gas constant.
Another key feature of the classical theory is the pres
ence of an extended adsorbed microfilm of constant
thickness ahead of the contact line (see Fig. 1), which
relies on the assumption that the disjoining pressure
isotherm I* = I* (*), at least within its sufficiently
high$* tail, is a positive decreasing function of the film
thickness (* (H* 0 as (* oo). In particular
computations, we shall be taking here this tail in the
form H* (*) A*/*3 with A* > 0 (A* is a modified
Hamaker constant), even though some of the arguments
developed later on do not qualitatively depend on such
a specific choice. Note that for nonpolar liquids, this
form is often used down to rather small $*. Then, the
thickness * of the microfilm in thermodynamic equilib
rium with the vapor at the temperature T* = To + AT*
of the substrate and at the vapor pressure *,, is found by
setting j= 0 in Eq. (1), which leads to
/ T 1/3
\ A* *T* (4)
As detailed by Rednikov et al. (2009), a natural set
of scales suitable for studying thin film shapes are [z]
[ ] = f for the vertical coordinate, [X] = /2/V a*
for the horizontal coordinate (where a* (A*/y*)1/2
is a molecular length scale), and [t] = 3 /Ey*e4 for
the time scale. In the latter scale, tr* is the dynamic vis
cosity, and
v[ ] a ( 5 )
Then, under the lubrication hypothesis c < 1, the dif
ferential equation governing the thin film evolution can
be written in dimensionless form as
ia3 (3 0 a2 )1 + j 0 (6)
at 3 O [ ax ax2 j
with
1 1/3 3 2 X2 ()/
j= (7)
+K +
The dimensionless numbers appearing in Eqs (6) and (7)
are the "evaporation number"
F ATJT
E *2 r)(8)
3 a*2 (G* p*Y)2
and the "kinetic resistance" number
K= A1*2
L*, L *2
2 fo R* T* \ 7T
_}a A* Ti (9)
fa 2 M* Cp* *2 *
Ja ~~ f "'
where A* is the heat conductivity. The quantity j de
fined by Eq. (7) represents the dimensionless local evap
oration flux, with the scale [j] =A A T*/ ** for the
mass flux and [j] )= A T*/Q} for the associated heat
flux. The numerator different from unity is due to the
Kelvin's effects of disjoining pressure and capillarity al
ready present in Eq. (2). The denominator can be inter
preted as the sum of mass transfer resistances due to heat
conduction though the liquid (just equal to in the se
lected scales) and to kinetic effects quantified by K. In
this sense, K < corresponds to an evaporation regime
limited by heat conduction (with local phase equilibrium
prevailing at the interface), while K > corresponds to
a "reaction"limited regime (where the liquid could be
treated as isothermal). Generally however, E, K and (
are considered to be 0(1) in the present analysis.
Equation (6) itself has the usual structure of a
lubricationtype equation, expressing that changes of the
film thickness are due to both the mass loss due to evap
oration (last term), and to the divergence of an hydro
dynamic flux driven by capillarity and disjoining pres
sure (second term). Note that, interestingly, c < 1 is
required both for the lubrication approximation to hold
([E] < [x]) and for the macroscopic approach to be ap
plicable (a* < *). Up to notations and scaling factors,
and excluding additional physical effects sometimes in
corporated into the model, Eq. (6) with (7) is of course
the same as elsewhere (cf Stephan& Busse (1992); Mor
ris (2001); Ajaev (2005), while for a more detailed pre
sentation in the same terms as here see Rednikov et al.
(2009), who show in particular that the Marangoni and
vapor recoil effects may be neglected in general).
We now consider some particular steady solutions of
Eq. (6) satisfying boundary conditions
 1 for x: oo (10)
i.e. the equilibrium adsorbed film to the left, and
 bx for x  +oo (11)
i.e. a vanishing curvature, constantslope region to the
right. The overall problem appears to be wellposed and
was parametrically studied by Rednikov et al. (2009).
In particular, the apparent rescaledd) contact angle b
b(E, K) is a found as a nonlinear eigenvalue of the prob
lem (note that the true contact angle is given by 0 = b,
see Fig. 1). As a key example in what follows, for the
parameters of Stephan & Busse (1992), i.e. the case
of ammonia in contact with aluminum at T, = 300 K
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
i.e. the integral flux calculated by approximating the film
profile in the microregion as a wedge with angle b (as is
indeed seen on a macroscale) and formally neglecting
Kelvin's effects in Eq. (7), but still keeping K in the
denominator in order to avoid a singular integral (even
though kinetic effects are typically significant at small
scale as well). A microscale "correction" is also defined
as
Jmicro(E, K)
lim [J(x) Jmacr(X)]
1 l(K 3]
b bk 2
0 100 200 300 400 X which can be calculated from tabulated values of
b(E, K) and k(E, K) given in Rednikov et al. (2009),
ure 2: Contact line microstructure and associated and is to be added locally at the apparent contact point
orative flux (E 0.124 and K 5.74). at the macroscopic level, to compensate for the approxi
mations made in calculating (15).
After having reviewed the classical theory and intro
a wall superheat AT = 1 K, we have e = 0.58,
duced scales and results useful in what follows, let us
0.124 and K = 5.74 (for f, = 1). The solution
consider additional physical effects which could have an
ined is represented in Fig. 2 in this case, and cor
onds to a value b 0.573 for the apparent contact i fence not only on the apparent contact angle b, but
le (0 0.332, i.e. about 19). Note that as is not also on the microscale contribution Jmicro of the contact
e ( 0.332, i.e. about 19. Notethe evaporation flux.that as is not
,,11 P_ +114 1 1 1 1A t i, line to the evaporation flux.
su small r l s iexamp Cn SCe IrSUtLS S UIIU mIereIy
taken as qualitatively valid, even though the lubrication
approximation itself turns out to be reasonably accurate
even for such moderate slopes.
As already studied by several authors, the evapora
tion flux strongly peaks in the transition region between
the adsorbed film (where j tends to 0 exponentially) and
the constant slope region (where j ~ x1). Such peak
may generally represent a substantial contribution to the
global evaporation flux, as emphasized by Stephan &
Busse (1992). In order to quantify more precisely this
microscale contribution to the flux, it is useful to intro
duce the integral flux
J(x) j (x) dx
J 00
1 1 93
E 80 d:
0IE 0 (12)
) 0x)
where the second equality is easily obtained from Eq.
(6), at steady state. Rednikov et al. (2009) have shown
that the leadingorder asymptotic behaviors of and J
for x  oc are
bx x 1g 2( X log 3 X (13)
4V k x
1J
bi(l e k
) o(lg X)
where k, just as b, can be found as part of the solution.
Then, we define a macroscopicc" contributionto the heat
flux as
S(15)
Jm.ar(X) Jx K + b(x o) (15)
Truncated microfilm regimes and the
spreading coefficient
First, we consider a possible effect of the spreading co
efficient S* 7= (7"* + y*), i.e. the difference of
energies per unit area for a "bare" substrate' and for a
substrate covered by a macroscopic liquid film in contact
with its vapor. A description in terms of both S* and A*
has indeed been used by de Gennes and coworkers (de
Gennes (1985); BrochartWyart et al. (1991)) for study
ing wetting of nonvolatile liquids, and the extension to
the volatile case presented here has been developed by
Rednikov & Colinet (2010a). An important concept in
troduced by de Gennes and coworkers is the "pancake",
i.e. the puddlelike equilibrium state of a macroscopic
volume of liquid which perfectly wets the substrate (i.e.
S* > 0, but yet the liquid does not spread to infinity
because of thin film forces with A* > 0, preventing the
film thickness to decrease down to molecular level). In
troducing the excess free energy of thinfilm forces as
P*(*) If (c*) d(* (17)
it can be shown that the thickness of the pancake is given
by the equation
s* =P*($) + n**($) (18)
'Note that Gibbs adsorption of vapor molecules can never be ex
cluded, and the value of should actually depend on such sur
face excesses, through the Gibbs adsorption equation.
Figi
evap
and
E
obta
resp
angle
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
and in the particular case I* ($*) = A* /*3, this yields
3 )* 1/2
2 S*
* ( 32* )
sp 2 S*
where a* is the molecular length scale introduced earlier.
Comparing Eqs (4) and (19), it is then seen that the
thickness of the adsorbed microfilm is smaller than the
pancake thickness when S* < S'r, or equivalently when
AT* > ATer, where these critical values are defined by
S* 3a AT*
3 a*2 *2
s7r 2T' a
* 'T (S* 3/2
* p*a* 3 y*)
In order to determine whether these critical conditions
play any particular role, we will consider in what follows
that AT* (hence also f) is fixed, while S* will be con
sidered as a free parameter (keeping in mind that both
these parameters are here considered small enough for
* and* to fall in the range where I ($*) A* /*3 is
indeed applicable). Then, Rednikov & Colinet (2010a)
have shown by purely thermodynamic (equilibrium) ar
guments that for S* < S* (i.e. when < *) the ad
sorbed film turns out to be metastable in favor of a bare
surface state. Indeed, considering .,, as a constant exter
nally imposed pressure, and introducing the relevant free
energy Z* F* + i.,,V* (where F* is the Helmholtz
free energy and V* is the total volume of the system),
the difference of free energies between these two possi
ble equilibrium states can be written as
z Z*
P*(~( f ) + ( n*(*) S* (21)
where A* denotes the total substrate area, and indicesf
and b refer to the microfilm state and to the bare sub
strate state, respectively. Then, from Eq. (18) and if
P($*) + $*n($*) is monotonically decreasing with $*
(which is the case in the range of thicknesses considered
here), it is seen that Z* > Z* if * < *, and the micro
film state is indeed metastable.
In order to study the microstructure of the contact line
in this case, it is again necessary to find steady solutions
of Eqs (6) and (7), though now with the boundary con
dition
S 0 forx = 0 (22)
while the boundary condition (11) is unchanged. It
turns out that there exists a oneparameter family of such
steady solutions satisfying this boundaryvalue problem,
and that the parameter of this family can be associated
with S* through the additional hypothesis
( x)
S*
Sfor x  0
*cr
10
8 
6 S*/S=0.99997
2x
10 20 30 40 50 60 70
10
8 S*/S*=0.462
6 cr
4 1
2 x
5 10 15 20 25 30
8
6 S*/S*=0.924
cr
2 X
5 10 15 20 25 30 35 40
108
6
2 S*/S*=1.15
2 4 6 8
Figure 3: Microstructures with a truncated film profile
for E = 0.124, K 5.74, and various values of the
spreading coefficient S*.
which actually expresses the fact that the singularity at
x 0 is resolved in the same way as in the equilib
rium case, where the behavior (23) can be obtained by
minimizing the relevant free energy (Rednikov & Col
inet (2010a)), quite similarly to the calculation e.g. by
Yeh et al. (1999) in a nonsingular situation yielding a
microcontact angle at a bare surface (the "augmented
Young equation").
Interestingly, the family of such "truncated" film so
lutions, represented in Fig. 3, starts exactly at S* Sr
and formally exists for any value of S* lower that S*r
(including negative values). In itself, this also confirms
the conjecture made in posing Eq. (23). It is also seen
that the microfilm gets longer when approaching the
critical value S7r, while nothing particular happens at
S* 0 within this family of solutions. Note that the
case S* < 0 corresponds to a partial wetting situation at
equilibrium, hence to a finite contact angle even without
evaporation, while for S* > 0 (and A* > 0 as assumed
throughout this paper), the contact angle observed in
Fig. 3 (just as in Fig. 2) is intrinsically dynamic, i.e.
induced by intense evaporationinduced flows in the mi
croregion.
The apparent contact angle corresponding to mi
crostructures shown in Fig. 3 is represented in Fig. 4
as a function of S* /S ,. In addition to the case consid
ered by Stephan & Busse (1992), namely E 0.124
and K 5.74, other values of E and K are also rep
resented in order to appreciate the effect of these two
dimensionless parameters. In each case, in addition
with the constant angle solution (corresponding to mi
crostructures with an extended adsorbed microfilm, in
dependent of S*), a branch of truncated microfilm so
lutions emerges at S*/Sr 1, for which the contact
angle is weakly affected for 0 < S* < Sr, but becomes
significantly different for S* < 0. In particular, the
curves all tend asymptotically to the classical Young's
1.75
1.5
N5
. . 175
. . . 6 
. . .. 

 44
2 1 1 2
Young's law S*/S,*
Values with an extended microfilm
Solutions with a truncated microfilm
Branching points at S*=S,*r
Figure 4: Rescaled apparent contact angle b as a func
tion of the spreading coefficient S*. Case I: E = 0.124,
K = 5.74; case II : E = 7, K = 50; case III:
E = 0.124, K = 0.1; case IV : E = 0.01, K = 5.74.
law cos 0 1 + S*/7*, which reads in our scales (and
in the smallangle limit)
bYoung = S*/Scr (24)
This clearly shows that the contact angle is weakly
affected by evaporation in the partial wetting regime,
for S* sufficiently negative (though its absolute value
should not be too large in the framework of hypotheses
made above), while evaporation has a dominant influ
ence in the complete wetting regime where the equilib
rium contact angle obviously vanishes. For small values
of E in particular, evaporation has a weaker influence in
shaping the film (compared to the effect of flows induced
by liquid pressure gradients), and the apparent contact
angle gets close to Eq. (24) even for slightly negative
values of S, while it is almost constant (but small) as
soon as S* > 0. Note also that Rednikov & Colinet
(2010a) have also analyzed the dynamics of kinklike
fronts separating the bare surface (say, to the left) and
the extended microfilm (say, to the right), and calculated
their velocity as a function of the spreading coefficient
S*. The results obtained confirm the overall picture,
namely the front advances to the left when S* > S* (i.e.
the microfilm invades the bare portions of the substrate),
while it recedes to the right when S* < Sr (leaving a
bare surface behind it).
Importantly, in relation with the divergence problem
mentioned in the introduction, here note that within the
present approach, the divergences, if any, turn out to
be rather soft and integrable. While this is obvious for
the extended microfilm solutions, Rednikov & Colinet
(2010a) have shown that for the steady truncated micro
film regime, not only the pressure p, but also the viscous
Sb
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
stress T =  Op/9x and the evaporation flux j all tend
to a constant for x  0 (i.e. at the point where the film
thickness formally vanishes). Even in the case of the
moving kink solutionjust mentioned, the corresponding
asymptotics arep ~ j ~ c log X, T ~ c 1/2, where c
is the kink velocity. Even though these quantities indeed
diverge, they remain integrable (finite evaporation flux
and total stress).
Finally, it is interesting to mention that when S* > 0
is considered to be fixed (though small enough as al
ready pointed out), extended microfilm solutions are ex
pected when the superheat AT* < AT*r, where this
critical value as been defined in Eq. (20). In contrast,
for AT* > AT*r, truncated microfilm solutions are ex
pected. While this has only weak consequences on the
apparent contact angle 0 = cb in general (see Fig. 4), a
larger influence is expected on the microscale heat flux,
hence on the microscopic correction Jmicro introduced
in the previous section. This has to be analyzed further,
however.
Influence of vapor pressure fluctuations
In the previous sections, the problem of the contact line
microstructure has been considered in the framework of
onesided models, where pressure is considered constant
and homogeneous in the vapor phase. In reality, pressure
fluctuations in the vapor can be expected, especially in
the microregion where the evaporation flux is quite in
tense (see Fig. 2). Yet, even though the corresponding
velocity in the gas is quite high (of the order of [j] /pt), it
can be estimated that the corresponding Reynolds num
ber Re [j] [x]/, remains small (because the length
scale [x] of the microstructure is quite small). Then, de
viations 1', '= j,, with respect to the farfield pres
sure /,, have the typical scale [P ] i ([j]/p*) [x] 1, as
estimated for an evaporationinduced Stokes flow on the
length scale [x].
Such fluctuations can potentially affect the liquid film
behavior both mechanically (through the normal mo
mentum balance at the interface) and thermodynami
cally (as already said in relation with Eq. (2), vapor
pressure fluctuations locally affect the saturation tem
perature). The mechanical effect will be neglected how
ever, as [p),] is generally much smaller than the scale
[pi] A*/q*3 = 7y*/3[z]2 of pressure fluctuations
in the liquid due to intermolecular forces and capillarity.
At the contrary, as can be appreciated from Eq. (2), the
impact of vapor pressure fluctuations on Tsat,ioc turns
out to be quantified by the dimensionless parameter
P* [Al] P* V
and in view of the fact that p* < p*, this parameter a
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
is not necessarily small (an example is given hereafter).
Note also that in Eq. (25), v* stands for the kinematic
viscosity. The expression for the local evaporation flux
now reads
1 1/Q3 3 2P/dx2 ap
3= Tr
i.e. generalizing Eq. (7).
In order to examine the impact of a nonzero value of
a, an expression for Pv, is needed. This is obtained by
solving the Stokes problem V2V , Vp, 0, V. 0
in the halfspace z > 0 (as for c 0, the film is flat and
its thickness canbe neglected on the large length scale
[x] characterizing flows in the vapor), with the boundary
conditions e.V j(x) and e.V 0 at z 0 while
V,  0 at infinity. This linear problem is conveniently
solved using Fourier transforms in the horizontal direc
tion, yielding the sought closure relationship
2 J
Pr (X) 
7r J 00
j"(x) log Ix x dx
As it could be expected, the effect of the vapor pres
sure nonuniformity is nonlocal with regard to the film,
resulting in an integrodifferential character of the prob
lem formed by Eqs (6), (26) and (27). As for boundary
conditions for its steady solution, we will keep on us
ing Eqs (10) and (11), where the apparent contact angle
should now depend on a as well, i.e. b b(E, K, a).
Note that the possibility of truncated microfilms is not
considered in this section. Importantly, it can be demon
strated (see Rednikov & Colinet (2009) for more details)
that the effect of pressure fluctuations remains localized
within the microstructure, i.e. the leadingorder asymp
totics for x +oo is found to be P5, ~ log xl/x2
for the pressure fluctuation, and this decay is sufficiently
fast for Eqs (13) and (14) to remain valid (though now k
depends on a as well). For x oo, oe also decays
as log xl/x2, but the film thickness and the heat flux be
have algebraically (contrary to the exponential behavior
obtained for a = 0), respectively as ( 1 ~ log x/x2
andj ~ log xl/4.
As for the important relationship b b(E, K, a), it
has presently been approximated by a perturbation anal
ysis around a = i.e. calculating the firstorder cor
rection to the value obtained earlier (for the classical
model), say bo(E, K). For small a, the pressure dis
tribution can be computed using the classical result, say
jo(x), in Eq. (27). For the ammonia/aluminum example
considered above, with Ty = 300 K and AT* 1 K
(for which jo(x) is presented in Fig. 2), this yields
the result represented in Fig. 5. This profile then ap
pears as an inhomogeneity in the linearized (around the
leadingorder solution) steady version of Eq. (6), used
20 40 60
Figure 5: Leadingorder vapor pressure deviation (with
respect to the farfield value) at the liquidvapor interface
for E 0.124, K 5.74.
to determine the first correction li(x) to the film pro
file = o(x) + a 1(x). In this way, the result for
the rescaled contact angle is in particular obtained as
b =bo(E,K)+abi(E, K).
For the ammonia example of Stephan & Busse (1992),
we find bl 0.001 and a = 9. As bo = 0.573, this
means that vapor pressure fluctuations lead to a decrease
of about _' of the apparent contact angle, which is in
deed hardly measurable. The dimensionless parameter a
can be seen to be proportional to the superheat however,
such that for larger superheats (say for AT* 10 K,
where a ~ 90), the effect is expected to be much
more significant, even though the perturbation approach
might not be applicable in this case. On the other hand,
note that the considered case of ammonia is character
ized by quite a large value of the saturation pressure,
/,, 10.5 bar. As a consequence, the vapor density is
rather large, while the vapor kinematic viscosity v* is
rather small, so that the property ratios entering the def
inition (25) of a are not so large. For smaller saturation
pressures (and accordingly smaller T*), one can expect
much greater values of a owing to the property ratios.
Yet another effect is that the number K also tends to in
crease with AT* and as .,, decreases, leading to smaller
j and larger effective scales along x (see Rednikov et
al. (2009)) and thus (as it can be conjectured) tending
to reduce the effect of vapor pressure nonuniformity. A
further parametric study is needed and other examples
should be considered in order to fully clarify its impor
tance. Our main conclusion here is that in certain cases
it may probably be quite significant.
Macroscopic curvature(s) and the micromacro
coupling
Finally, Rednikov & Colinet (2010b) have also con
sidered the possibility of small curvatures, both of the
meniscus and of the substrate, though otherwise remain
ing in the framework of the classical model (the micro
film is assumed to be extended, and the vapor pressure is
supposed to remain uniform). Only their main conclu
sions are reproduced in this section.
Assuming that a given curvature Km of the vapor
liquid interface is specified towards the macroscopic re
gion (e.g. corresponding to the curvature of the macro
scopic meniscus near the apparent contact line) and if
K, is the wall curvature (assumed to be constant on the
scales involved here), the boundaryvalue problem for
the film thickness is still composed of the steady ver
sion of Eq. (6), but the evaporation flux is now given
1 1/O3 3 32%/x2 3 K,,
J T
and the boundary conditions are
= (1 3 )1/3 as x oc (29)
and
K~ 2 as x +oc (30)
where
K = K_ Kw > 0 (31)
is assumed to be positive. Note that as written here, the
dimensional scale of all the curvature quantities (Km, Kw
and K) is given by
[ ] 2
[Kj [x]2 [1]
3 a*2
E;
Note also that the integral flux expression (12) re
mains valid here. As compared to Eq. (7), the change
in the numerator of j in Eq. (28) reflects the fact
that 02Q/lz2 is no longer the free surface curvature
to appear in the term representing the Laplace pres
sure. Now the free surface curvature is rather given by
02/Oxz2 + Kw. The thickness of the adsorbed microfilm
in (29) changes accordingly as compared to ( = 1.
Examining the x * oc asymptotics of the (steady
version of the) full problem (6, 2830), it turns out that
S~ K (X x0)2 + b (x Xo)
+% + 0 (74) as x +oo (33)
where b = b(E, K,K, Kw),C 3 C 3(E,K, K,K ),
and x0 can be arbitrary. We also note, using (12) with
(33), that
J(+oo) =15 K3 3/(2 E) (34)
so that with a nonzero curvature the integral flux
reaches saturation at infinity, unlike the constantslope
classical case characterized by the logarithmic diver
gence (14).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Even though according to Eq. (32) and for Kc~, Kw, =
0(1), the corresponding radiuses of curvature of the
meniscus and of the substrate are formally e 1 times
larger than the horizontal length scale [x] and e2 times
larger than the film thickness scale [i], they remain ex
tremely small in practice. For instance, in the ammonia
example of Stephan & Busse (1992), and for a superheat
AT 1 K, we get [] 1 = 2.83 nm (and moreover, ac
cording to Eqs (4) and (32), this curvature radius scale is
inversely proportional to AT). Hence, it is quite relevant
in most practical situations to consider the limit of small
values of the dimensionless curvatures Kc, Kw and K,
and hence the firstorder correction to the classical the
ory valid for K, = K = 0.
In what follows, we consider Kw = O(K) and K < 1.
In this limit, it seems rather clear that the steady solution
to the problem (6, 2830) should be close to the classical
microstructure (see e.g. Fig. 2), at least in some (inner)
region corresponding to 0(1) values of x. In particular,
the behavior (11) should be recovered for large x, but
not too large. For very large x, one rather expects to
reach the asymptotics (33). As shown by Rednikov &
Colinet (2010b), an intermediate zone exists in which
the relevant variables are
X KX = K!
and the solution has the form
bx+ x + Ki
2
where the "overlap function" 1 can be calculated ex
plicitly, which in particular requires its matching for
x * 0 with the asymptotics for x  oc of the in
ner solution (i.e. Eq. (13), which however needs to be
completed by firstorder corrections in K < 1 and in
Kw < 1). Then, considering the x  oc asymptotics
and rendering it in the form (33) yields the result
S b = K bll + b12 + (7 logk + 3 log2 k
6 log 2 log2 (37:
where bl (E, K) and b12(E, K) are known functions of
E and K, just as b(E, K) and k(E, K).
The result (37) shows that the coefficient b appearing
in Eq. (33) remains close to the classical result b for
K < 1 and Kw = O(K). This implies that the clas
sical result (see Rednikov et al. (2009) for a full para
metric study) may indeed be used in most practical sit
uations where the curvatures of the meniscus and of the
substrate remain small compared to the typical curvature
scale (32) of the microstructure. When these macroscale
curvatures increase however, their lowestorder effect
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
may possibly be appreciated on the basis of Eq. (37),
but the most important result here is probably that Eq.
(33) may be used as a basis for achieving the coupling
between macroscopic and microscopic scales. Indeed,
even though this aspect should be studied in more details
in a future work, it seems natural at this stage to iden
tify xo with the apparent contact point at the macroscale,
where a "generalized boundary condition" applies in the
form of a relationship b(E, K, K, K,) between the ap
parent slope of the film profile at that point, and the lo
cal curvatures of the meniscus and of the substrate. Note
that b also depends on the local substrate temperature at
xo, through the value of the superheat AT* (in this re
spect, see Stephan & Busse (1992) and Morris (2000)
for an analysis of the assumption of a locally isother
mal substrate). As just said, in most situations of prac
tical interest, b b(E, K), and the micromacro cou
pling is then achieved by imposing the classical value of
the apparent contact angle on a locally flat and isother
mal substrate (as done by Stephan & Busse (1992) and
Rossomme et al. (2008) in the context of heat pipes, for
instance).
Another important aspect of this micromacro cou
pling is the microscale contribution to the mass (or heat)
flux. Rednikov & Colinet (2010b) have demonstrated
that the remaining coefficient in Eq. (33) is given by
2E 3 k + k
S 15b 2 38)
When used in (34), it yields the value of the integral flux
J(+00) 1 3 + log k (39)
b 2 2 b )
which, combined with the values of b = b(E, K) and
k = k(E, K) calculated for the classical microstructure
problem, allows to evaluate the effect of macroscopic
curvature upon J. Interestingly, the result (39), already
obtained by Morris (2003) in a different way, is also
found by adding the microscopic correction (16) to the
macroscopic flux calculated in the same way as for Eq.
(15), i.e.
macro
[ oX dx
So K+ b(x o) + ( xo)2/2
1 2 b2
Slog K ) for K < 1 (4(
b \K)
Hence, even when a (small) curvature of the menis
cus exists, the evaporation flux may be correctly evalu
ated by summing up a macroscopicc" contribution (eval
uated by approximating the meniscus near the wall as
a parabola, and neglecting Kelvin's effects in Tsat,ioc,
though still including the kinetic resistance) and a mi
croscale correction given by Eq. (16).
Conclusions and perspectives
In this paper, several recent results obtained by the au
thors have been summarized, focusing on effects not pre
viously accounted for in the classical microscale theory
of contact lines formed by a pure liquid, its vapor, and
a superheated smooth substrate. First, the possibility of
truncated microfilm regimes, instead of extended ones,
has been explored by introducing the spreading coeffi
cient S* into the picture and considering it as a (positive
or negative) parameter, at fixed superheat AT* > 0 and
Hamaker constant A* > 0. In particular, a critical pos
itive value of S* has been identified, below which trun
cated microfilms are preferred over extended ones, with
possible consequences on the apparent contact angle and
on the microscale contribution to the mass (or heat) flux.
For negative values of S*, the theory presented here ap
pears as a natural generalization of the classical theory
to the partial wetting case, for which it is shown that the
apparent contact angle is generally not much larger than
the equilibrium one given by Young's law.
In a second stage, the effect of pressure non
uniformities generated by intense (but still viscous)
evaporationinduced flows in the vapor was analyzed,
through the thermodynamicc) effect these may have on
the local saturation conditions at the interface. It has
been shown that this effect can be significant, depending
on the fluid used, on the operating pressure (or satura
tion temperature) and on the wall superheat AT* (and
especially at large values of the latter). A dimensionless
parameter has been identified to quantify its importance,
and a coherent framework useful for its full study has
been established.
Then, the influence of macroscopic curvatures of the
meniscus and of the substrate has been studied, under
the assumption that these remain much smaller than the
typical curvature of the thin film in the microstructure re
gion. In this case, it has been shown that both curvatures
have only a weak effect on microstructure characteristics
such as the apparent contact angle and the microscale
contribution to the heat flux. The integral heat flux it
self is more strongly affected, as the latter is found to
diverge logarithmically in the limit of vanishing macro
scopic curvature.
Finally, a strategy has been proposed to incorporate
the results of these models (including the classical one,
the validity of which remains guaranteed in many situa
tions) in macroscopic numerical simulations of a variety
of nonisothermal liquidvapor flows (e.g. heat pipes,
boiling, evaporating droplets, ...). This micromacro
coupling relies on a careful consideration of the asymp
totics of the microstructure at large distances from the
contact line, and shows in particular that in the situa
tions considered here, an effective macroscopic bound
ary condition can be applied in the form of a relationship
between the local slope and curvature of the meniscus
at the apparent contact point. Even though conceptu
ally interesting in itself, it must be emphasized that the
practical usefulness of this relationship remains limited
to small macroscopic curvatures (compared to the large
curvature in the microregion), and reduces in this case
to imposing a local slope obtained in the limit of van
ishing macroscopic curvatures. It is perhaps only in the
case of microfluidic devices that the effect of the menis
cus and substrate curvatures on the local slope can be
appreciated, although other corrections might have to be
incorporated as well in this case. Besides, quite remark
ably, another feature of this micromacro coupling is the
identification of a microscopic correction to the evapo
rative flux to be applied locally at the contact line, in ad
dition to the contribution evaluated by neglecting all mi
croscale effects but the kinetic resistance to evaporation
(deviation from local chemical potential equilibrium at
the interface).
Even though the effects studied here have been treated
separately, it can be conjectured that the mentioned re
lationship between the local slope and curvature would
in general simultaneously depend on all the relevant pa
rameters characterizing these effects. This relationship
is in general multivalued and transitions between dif
ferent regimes (e.g. extended and truncated microfilm
regimes) are possible. The generality of this micro
macro coupling strategy has to be studied further how
ever, in particular for unsteady situations involving con
tact line motions (although when the velocity of dis
placement of the apparent contact point remains weak
compared to the large microscale velocities, it is reason
able to assume that the results obtained may be used in
a quasistatic sense), negative meniscus curvatures (e.g.
for sessile droplets, by opposition with boiling bubbles
for instance), shortscale substrate curvature (i.e. rough
ness), ... Even though the lubrication theory usually ap
plies beyond the range of thin film slopes it is intended
for, it would certainly be important as well to study sit
uations characterized by large contact angles (and even
possibly larger than 90 ), which are excluded from the
present analysis. Finally, in a larger perspective, it would
also be interesting to consider other disjoining pressure
characteristics (negative values of the Hamaker constant,
Sshaped disjoining pressure isotherms, ...), and more
generally to compare microscale theories presented here
with hybrid models, using truly microscopic (or meso
scopic) theories to model the smallest scales.
Acknowledgements
The authors are grateful to Vadim Nikolayev, Peter
Stephan and Lounes Tadrist for numerous interesting
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
discussions in the frame of the Topical Team on Heat
Transfer sponsored by the European Space Agency. Part
of this research was funded by the ARCHIMEDES pro
gram granted by the Communaut6 Francaise de Bel
gique, and by the BOILING program funded by the Eu
ropean Space Agency and by the Belgian Science Pol
icy. PC also thankfully acknowledges financial support
of the Fonds de la Recherche Scientifique FNRS.
References
Ajaev VS., Spreading of thin volatile liquid droplets on
uniformly heated surfaces, J. Fluid Mech., Vol. 528, pp.
279296, 2005.
BrochardWyart F., di Meglio J.M., Qu6r6 D., and de
Gennes P.G., Spreading of nonvolatile liquids in a con
tinuum picture, Langmuir, Vol. 7, pp. 335338, 1991.
Burelbach J.P, Bankoff S.G., and Davis S.H., Nonlinear
stability of evaporating/condensing liquid films, J. Fluid
Mech., Vol. 195, pp. 463494, 1988.
de Gennes P.G., Wetting : statics and dynamics, Rev.
Mod. Phys., Vol. 57, pp. 827863, 1985.
Kjelstrup S. and Bedeaux D., Nonequilibrium thermo
dynamics of heterogeneous systems, World Scientific,
Singapore, 2008.
Morris S.J.S., A phenomenological model for the con
tact region of an evaporating meniscus on a superheated
slab, J. Fluid Mech., Vol. 411, pp. 5989, 2000.
Morris S.J.S., Contact angles for evaporating liquids
predicted and compared with existing experiments, J.
Fluid Mech., Vol. 432, pp. 130, 2001.
Morris S.J.S., The evaporating meniscus in a channel, J.
Fluid Mech. Vol. 494, pp. 297317, 2003.
Potash M. and Wayner PC., Evaporation from a two
dimensional extended meniscus, Int. J. Heat Mass Trans
fer, Vol. 15, pp. 18511863, 1972.
Rednikov A.Ye., Rossomme S. and Colinet P., Steady
microstructure of a contact line for a liquid on a heated
surface overlaid with its pure vapor: parametric study
for a classical model, Multiphase Sci. Tech., Vol. 21, pp.
213248, 2009.
Rednikov A.Ye. and Colinet P., Evaporationdriven con
tact angles on perfectly wettable heated substrates: ac
counting for local vapor pressure nonuniformity, Pro
ceedings of the Eurotherm Nr. 84 Seminar on Thermo
dynamics of Phase Changes, Namur, Belgium, May 24
27th, 2009.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Rednikov A.Ye. and Colinet P., Truncated versus ex
tended adsorbed microfilms at a vaporliquid contact
line on a heated substrate, submitted to Langmuir,
2010a.
Rednikov A.Ye. and Colinet P., Vaporliquid steady
meniscus at a superheated wall: asymptotics in an in
termediate zone near the contact line, to appear in Mi
crogravity Sci. Tech., 2010b.
Rossomme S., Goffaux C., Hillewaert K. and Colinet P.,
Multiscale Numerical Modeling of Radial Heat Trans
fer in Grooved Heat Pipe Evaporators, Microgravity Sci.
Tech., Vol. 20, pp. 293297, 2008.
Stephan PC. and Busse C.A., Analysis of the heat trans
fer coefficient of grooved heat pipe evaporator walls, Int.
J. Heat Mass Transfer, Vol. 35, pp. 383391, 1992.
Yeh E.K., Newman J., and Radke C.J., Equilibrium con
figurations of liquid droplets on solid surfaces under the
influence of thinfilm forces. Part I. Thermodynamics.
Colloids and Surfaces A, Vol. 156, pp. 137144, 1999;
Part II. Shape Calculations. Colloids and Surfaces A,
Vol. 156, pp. 525546, 1999.
