Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 133

Figure 1. Sketch representing a pinned evaporative regime (left) and a depinning one (right).
Theoretical models currently available in the literature treat evaporating pinned
droplets having either spherical or ellipsoidal geometries [33]. These models give
accurate predictions of evaporative rates in isothermal conditions (i.e., conditions
where the droplet, the air and substrate are all at the same temperature). Droplet
evaporation is then essentially controlled by the diffusion of molecules from the liq-
uid phase into the gaseous one. Although giving good predictions, isothermal mod-
els do not always give a satisfactory agreement with the experiments [27, 29, 34] in
particular when trying to adapt them for droplets on moderately heated substrates.
From this perspective, full understanding of droplet evaporation phenomena is still
incomplete. The hydrodynamics that might develop in heated substrate conditions
is still, for example, in its exploratory phase. If heat dissipation by convective cells
is well known for large volumes, only a very limited number of studies have been
devoted to the hydrodynamics of small droplets [22-28, 38]. For such systems, the
main difficulty is due to the fact that evaporation processes strongly depend on
interfacial phenomena such as Marangoni driven convection that becomes funda-
mental when energy supply from wetting substrates have to be accounted for. The
interfacial temperature difference between the contact line of a deposited droplet
and its apex is generally less than a fraction of a degree Celcius, but such a small
temperature difference is sufficient to lead to the development of thermocapillary in-
stabilities [25] having their origin in the surface tension differences between 'warm'
and 'cold' areas of the interface. These instabilities generate a Marangoni flow on
the surface of the droplet that creates not only a shear on the adjacent liquid but also
modifies heat transfers as well as liquid circulation. All the above discussions show
that transport mechanisms of heat in small droplets are more complex than what
is usually expected for millimeter size objects. Droplets involve not only diffusive
based heat transfers but also convective phenomena. In droplet evaporation, we are
hence primarily concerned with the Marangoni effect caused by temperature gradi-
ents within the droplet. They can be the result of direct heat transfer processes or of
mass transfer processes involving enthalpy changes or of any other type of natural
or forced flow. There have been several attempts to produce numerical simulations
of droplet evaporation including or not the role of Marangoni effects. Most notable
are the models proposed by Liao [35], Di Marzo et al. [36], and Lorenz and Mic-
kic [17], which assume conduction into the droplet from the heated surface is the
dominant mode of heat transfer. Each model produces results with varying degrees
of success, but as far back as 1958, Zuiderweg and Harmens [37] reported the in-
fluence that internal fluid motion, and in particular the fluid circulation caused by
surface tension gradients, had on the rate of mass transfer in unsupported distilla-