Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 131

interface, that determines the droplets mass evolution and therefore the change in
time of their volume [17-19].
Evaporation and vaporization phenomena correspond both to a transition of a
substance from its liquid phase to its gaseous phase. But the difference between
these two transformations is their characteristic time. Evaporation is a slow process
when compared to vaporization. Quasi-equilibrium hypotheses can hence be in-
voked. This will allow important simplifications of the models used in particular for
the description of unsteadiness and to overcome difficulties in the implementation
of time evolution. Due to the complexity of evaporative processes, a broad range
of spatial and temporal scales are involved. This problem has led to the develop-
ment of specific numerical approaches to simulate either hydrodynamic phenomena
or the dynamics at the molecular scales. Despite well adapted modeling [20] and
numerical methods, the available computing power is still too limited to include si-
multaneously in the simulations both the full detail of microscopic phenomena and
hydrodynamics occurring at macroscopic scales. This is why we will propose in this
chapter to illustrate our purpose with a simplified model dealing with droplet evap-
oration on the basis of a quasi-steady hydrodynamics. Molecular phenomena like
liquid density changes due the anisotropy at liquid/air interfaces is not described
explicitly as in molecular dynamic simulations although they contribute indirectly
to the model through liquid/gas interfacial properties.
The evaporation of a volatile liquid in contact with unsaturated air is inevitable
because of the diffusivity of the molecules of this latter in the surrounding atmo-
sphere [21-23]. Generally it is a relatively slow process, as long as ambient and
substrate temperature, remain sufficiently smaller than the saturation one. Evap-
oration can even stop when the atmosphere becomes saturated with the liquid
vapor. The study of evaporation phenomena hence requires not only a precise de-
scription of the liquid but also needs to account for its vapor concentration when
studying mass transfers. These latter are governed by molecular mechanisms that
have a characteristic timescale of the order of 10 − 10 s [24], which is the typical
timescale for molecular evaporation mechanisms. Another typical time is deter-
mined by the diffusion of molecules in the gas phase and given by the ratio R 2 /D ,
where D is the diffusion coefficient of the vapor-liquid molecules in the air (typi-
cally D 0 . 2443 cm 2 s − 1 for water vapor in air) and R is the contact line radius

that corresponds to the coexistence line between the three phases: liquid, solid and
gas. R is about 1 mm for the droplets that will be considered in this chapter. These
two estimates give R 2 /D > 10 − 5 s and show that the time characteristics of molec-
ular mechanisms and diffusion phenomena differ by five orders of magnitude. The
slowest mechanism will control evaporation and here it is diffusion in the gaseous
phase. These estimates explain why millimeter droplets evaporate only within few
minutes for moderate substrate temperature [25] and why they rapidly reach a quasi-
steady state as evaporation takes place [26].
Besides the previous time scale estimations and quasi-steady properties, heated
droplets also raise the specific problem of their coupling with the heating substrate

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