Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 132

on which they are deposited [27-29] and of the role of their wetting properties in
the overall evaporation process [30]. A water droplet of few micro-liters on a hy-
drophobic substrate will adopt an almost spherical geometry, whereas in the case
of wetting substrates (such as copper or aluminum) the same droplet will take a
spherical cap shape. From the point of view of thermodynamics, such a system
must exhibit a unique equilibrium contact angle θ given by Young's equation (in
the limit of validity of this latter). However for evaporating droplets in a nonsat-
urated atmosphere, the mass balance is in favor of the gaseous phase where the
chemical potential of liquid is smaller. The system is therefore out of equilibrium
and the contact angle deviates from the equilibrium one. It becomes an out of equi-
librium quantity and will exhibit a time evolution that will depend on the droplet
characteristics.
The temperature T s of the substrate plays key role in this process. If it is larger
than the saturation temperature, the droplets evolve too quickly for a simplified
model to be applicable for its description. For smaller temperatures however, the
water comes in close contact with the substrate and, if it is made of copper or
aluminum, the droplet will wet it. In this chapter, the surrounding air has atmo-
spheric pressure. Under these conditions, the evaporation dynamics shows several
well identified evolution regimes that do not depend solely on the hydrophobic or
hydrophilic nature of the substrate. Experimental conditions and in particular the
way the droplet is heated are also important inputs here. Extensive experiments
have been devoted to the description of droplets evaporation dynamics [31] and
their analysis shows three main evaporation regimes [32]. In the first one, evapo-
ration leads to a progressive reduction of the height ( h ) of the droplet and of its
contact angle. The droplet keeps a spherical cap shape and its contact line is almost
a circle. This regime is called pinned droplet regime since R remains constant. For
water on aluminum or copper substrates, this pinned regime is also the longest since
it corresponds to the evaporation of more than 80% of the initial droplet volume for
moderate values of T s . This first regime ends when the θ reaches a critical receding
value that depends on temperature, on the nature of the substrate and of course on
the liquid. For water droplets on copper or aluminum substrates, pinned regimes
can be observed for contact angles as small as θ 10 degrees. The second evapora-

tion regime is a receding contact angle regime. The contact line is no more pinned
and both h and R decrease. The droplet progressively shrinks but with values of θ
than can be constant or increase if there is recoil. Pinned and recoil regimes and are
illustrated in Fig. 1. In the last evaporation phase, h , θ and R finally all simulta-
neously vanish. Among these three regimes, the first one is of fundamental interest
not only due the important volume of evaporated liquid but also because of the pos-
sibility to use simplified hydrodynamic models for the droplet simulation [25, 26].
It allows indeed a step by step description of the hydrodynamic occurring within
the droplet including the properties at the liquid/gas interface governing the heat
and mass transfers.

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