3. JUSTIFICACIÓN TEÓRICA
3.6 LA FORMACIÓN EN EDUCACIÓN INTERCULTURAL DEL PROFESORADO
One particularly simple approach used by Rowan et al. [122] has been to approximate the concentration gradient of vapour to be radially outward and equal to
pcs´ c8q{R (4.8)
where cs is the vapour concentration at the liquid-vapour interface of the droplet
and is assumed to be its saturation value, c8 is the concentration of vapour far removed
the surface of the droplet which is assumed to be its ambient value. This approximation to the concentration gradient is intuitive and accurate for contact angles close to 90˝.
R
r
Solid
Liquid
Figure 4.13: Concentration profile as modelled by Rowan et al. (Solid line at the distance of r and Shanahan and coworkers (Dashed line and the solid line at r.)
However, it neglects the contact angle dependence that is introduced by the presence of the substrate which restricts the available space into which vapour can diffuse. With this approximation, Equation 4.7 gives the evaporation rate to be,
dV
dt “ ´λho (4.9)
where λ “ 2πDpcs´ c8q{ρ and ho “ hpz “0q
In the case of the droplet on a lubricant-impregnated surface, the bottom of the droplet is surrounding by a layer of the non-volatile lubricating liquid (Figure 4.11). The droplet’s liquid-vapour interfacial area is reduced due to the presence of a "skirt" of
lubricating liquid that rises to a height hs, above the surface features. The presence of
"skirt" modifies the liquid-vapour surface area, ALV “2πR2p1 ´ cos θpzqq “ 2πRhpzq,
through which evaporation occurs which in turn modifies the rate of evaporation of the droplet as follows, dV dt “ ´λho „ 1 ´ hs ho (4.10) Because of the existence of a skirt of lubricating liquid surrounding the base of the droplet, the evaporation rate can therefore be expected to be reduced by a factor of r1 ´ hs{hoscompared to a droplet of identical volume and having the equivalent contact
angle on a surface without a layer of lubricating liquid.
The evaporation rate can be re-written using the droplet volume and the contact angle, θs“ θpz “ hsq, deduced from the spherical cap profile, but measured at the top
of the lubricant skirt,
dV dt “ ´λRp1 ´ cos θsq “ ´λ „ 3V πβpθo 1{3 p1 ´ cos θsq (4.11)
where the spherical radius R has been replaced by the droplet volume and the contact angle, θo “ θpz “ 0q, deduced at the top of the solid surface structure from
the spherical cap profile, Assuming both the contact angle at the solid surface and the top of the lubricant skirt, as deduced from the spherical cap profile, are approximately constant, as can be expected on a SLIPS surface with no contact line pinning, this can be integrated in time, t, to give,
V ptq2{3 « Vi2{3´ 2λ 3 „ 3 πβpθoq 1{3 p1 ´ cos θsqt (4.12)
where Vi is the initial droplet volume and V ptq is the volume contained within the
spherical cap shape down to the top of the surface features at z “ 0 at any subsequent time t. The volume therefore reduces following a t3{2 power law and this gives a
linear time dependence for the deduced solid-liquid contact area ASL “ πr2o, where
r2o « r2i ´2λt sin 2θ op1 ´ cos θsq πβpθoq “ r2i ´ 2λt sin 2θ o πp1 ´ cos θoqp2 ` cos θoq ˆ 1 ´ cos θs 1 ´ cos θo ˙ (4.13) This is the same as previously reported by Erbil et al. for the constant contact angle mode of evaporation [129], but with a correction factor due to the lubricant skirt. The correction can be written using a Taylor series expansion as,
ˆ 1 ´ cos θs 1 ´ cos θo ˙ «1 ´ ∆θ sin θo 1 ´ cos θo (4.14) where ∆θ “ θs´ θo is the difference in contact angles deduced from the spherical
cap profile by measuring at the top of the lubricant skirt and at the top of the solid surface features.
In previous work on sessile droplets evaporating in a constant contact angle mode, a function fpθq was introduced by Erbil et al. in order to take account in a common notational format of the dependence of the concentration gradient of vapour [129], between the surface of droplet and far from the droplet, on the contact angle arising from different models, this notation was also used in studies of droplet evaporating from superhydrophobic surfaces [127]. In this notation, Equation 4.9 becomes,
dV
dt “ ´2λRfpθq (4.15)
where the simple approximation pc8´ csq{R used by Rowan et al., to the concen-
tration gradient that ignores any variation with contact angle gives [122],
f pθqRowan et al. “
1 ´ cos θ
2 (4.16)
Bourgès-Monnier and Shanahan also assumed the diffusion of liquid vapour from the droplet to be radially outward (Figure 4.13) [121]. They considered a spherical cap shell of surface area ALV “ f pR1, θ1q using a coordinate system pR1, θ1q based on the
center of the drop, their approximation to the concentration gradient therefore began with
dc dR “ ´ k ALV “ k 2πR12p1 ´ cos θ1q (4.17)
where k is a constant. Using the resultant concentration gradient, they obtained an improved self-consistent model for evaporation of small sessile droplets and it is equivalent to,
f pθqShanahan et al. “
´cos θ 2 logep1 ´ cos θq
(4.18) This solution is remarkably consistent with the exact solution of Picknett and Bexon except for small angles where it has a singularity and diverges. For fpθq, the physical origin of the difference between Equation 4.16 and 4.18 can be most clearly seen for the limiting case of a completely spherical drop where θ “ 180˝. In this case, Equation
4.16 accurately reduces to the known result for the evaporation of a fully spherical drop in free space. However, in the case where a sessile droplet having a contact angle of 180˝ is in contact with a surface, Picknett and Bexon’s solution (Equation 4.21) is more
accurately approximated by Equation 4.18 than Equation 4.16. For θ “ 180˝, Equation
4.21 gives a value of 0.694 while Equation 4.18 and Equation 4.16 give 0.721 and 1, respectively. The difference between Picknett and Bexon’s exact solution and the two approximations decreases as the angle reduces from 180˝. All three solutions become
numerically identical at 90˝. Below this, the three solutions start to diverge once again
and Equation 4.18 becomes the more accurate approximation.
More recently, Stauber et al. studied the evaporation of droplets on strongly hydrophobic substrates with a focus on the constant contact angle and constant contact radius modes, their formulas are equivalent to [130],
f pθqStauber et al. “ sin θgpθq
4p1 ` cos θq2 (4.19) where θgpθq p1 ` cos θq2 “tan ˆ θ 2 ˙ ` ż8 0 cosh2 pθτ q sinhp2πτqtanhrτpπ ´ θqsdτ (4.20) Using the analogy between the diffusive flux and electrostatic potential, Picknett
and Bexon derived a exact solution for Equation 4.7 for determining the evaporation rate of sessile drop and they gave a numerically accurate polynomial interpolation for
f pθq [118],
f pθqPicknett and Bexon
$ ’ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ ’ % 1 2p0.6366θ ` 0.09591θ 2 ´0.06144θ3q 0˝ ă θ ă10˝ 1 2p0.00008957 ` 0.6333θ ` 0.116θ 2 10˝ ă θ ă180˝ ´0.08878θ3 `0.01033θ4q (4.21)
where θ in the series in in radians.
Following the earlier approach of using the apparent contact angle at height hs of
the lubricant skirt above the surface structure to define the droplet’s liquid-vapour interfacial area through which evaporation occurs, Equation 4.15 becomes
dV
dt “ ´2λRfpθsq (4.22)
Writing the spherical cap radius in terms of the drop volume and apparent contact angle θo at z “ 0 and assuming both θo and θs are approximately constant, the volume
dependence on time can be found,
V ptq2{3 « Vi2{3´ 4λ 3 „ 3 πβpθoq 1{3 f pθsqt (4.23)
where Vi is the droplet’s initial volume at t “ 0. In terms of the apparent contact
radius at z “ 0, this can be re-written as,
r20 « r2i ´ 4λt sin2θ of pθsq πβpθoq “ r2i ´ 2λt sin2θ o πp1 ´ cos θoqp2 ` cos θoq ˆ 2fpθsq 1 ´ cos θo ˙ (4.24)