Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 140

approaches are indeed suitable when time scales can be separated in a way to fi-
nally focus on a system presenting a time evolution dominated by a single and well
defined phenomenon. Both previous conditions are to be satisfied when consider-
ing the evaporation of small pinned droplets on moderately heated substrates. In
the case of water droplets of few micro-liters deposited on copper rough substrates,
evaporation shows a pinned regime where droplets keep simple spherical cap ge-
ometries and where the relevant time scale is given by the time evolution of θ (or
equivalently h ). The latter turns out to be relatively slow and lasts from few tens of
seconds up to several tens of minutes for moderate substrate superheats. All other
time scales, usually which are much faster like the ones associated with contact line
fluctuations and molecular diffusion, can be averaged and further used to justify
quasi-steadiness hypotheses used for model simplifications.
D. Theoretical Model and Parameters
In this Paragraph we present the approximations and the basic equations that are im-
plemented numerically to illustrate this chapter with simulations of water droplet
evaporation on heated copper substrates. The volume of the droplets that we will
consider in the following is of a few micro-liters. This makes negligible the hy-
drostatic pressure contribution when compared to surface tension one and explains
why such small droplets keep a spherical cap geometry in their first evaporative
regime [8, 33, 38]. This system reduces to a problem with an axial symmetry
that presents the important advantage to allow a full description of the droplets
by limiting its study to only one of its section (i.e., to a single plane containing the
symmetry axis). This will simplify the equations of the theoretical model, ease their
numerical handling and significantly speed up simulations that will be restricted
to a two dimensional space instead of a full three dimensional one. The wetting
properties of water droplets on substrates such as aluminum or copper show the
existence of pinned evaporation regimes where the contact angle θ decreases with
time while contact radius R remains constant. As discussed previously, the param-
eters θ and R will be two of the basic variables used for droplet description and
are illustrated in Fig. 2. A third parameter denoted by L s is also necessary here
and presented in Fig. 2 and Fig. 3. It corresponds to the radius of the heated zone
of the substrate (substrate heater) that we assume to be a disk located in such a
way to preserve the axial symmetry of the droplet/substrate system as well as the
planar shape of the substrate (see Fig. 3). When L s /R < 1 the substrate heater
is smaller than R as illustrated in Fig. 2(a) and it is only the central part of the
droplet basis that is heated up. Conversely, when L s /R > 1 (Fig. 2(b)), the sub-
strate heater is greater than the contact radius and it is not only the complete
droplet/substrate interface that is heated but also the air nearby the contact line.
L s is a fundamental parameter in evaporation studies in particular for the descrip-
tion of evaporative mass flows [34, 48] and further opens interesting perspectives