Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 147

there is a one to one correspondence between time and contact angle we will only
consider in the following contact angles. The smaller they will be the closer the
droplet will be to complete evaporation.
This choice of initial conditions implies that the droplet is initially at the tem-
perature of the substrate ( T s ) and that air is at the temperature of the far field ( T ∞ ).
The temperature across the water/air interface is hence initially discontinuous but
will be smoothed out right after the first time step of the simulations. Simulations
are coherent with the temperature continuity condition imposed in the model. Prac-
tically, such an initial condition corresponds to an experimental procedure where
water droplets would itself be initially at temperature T s before being deposited
on the substrate that would be rapidly heated up to T s right before this deposition.
This initial condition has been used for the validation of the model of Paragraph D.
Literature proposes numerous experimental data of evaporating droplets describing
more or less precisely working conditions [10, 30, 32, 34]. The large number of
parameters required for a complete experimental description of such a system and
especially the technical difficulty to properly control all of them is an important
limiting factor regarding to simulation validation. This difficulty is probably one of
the major obstacles to carry out reliable comparisons between experimental results
and simulations where all the parameters have to be known. Controlled experimen-
tal conditions coherent with the model presented here are available with T s = 60 ◦ C,

26 ◦ C, H T ∞ = 0 . 515 mm for micro-liter volume droplets [34]. FEM

numerical simulations carried out for these systems show a good agreement for the
time evolution of the droplets volume provided the radius of the substrate heater L s
is set to L s ≈

= 40%, R =

1 . 5 R [25, 48]. This indicates that, although very simplified, the model
of Paragraph D is sufficient to provide relevant insights for pinned evaporating wa-
ter droplets.
Figure 5 displays the dimensionless temperature T ∗

in air when L s = 1mm

60 ◦ C for different contact angles (i.e., different successive times). This
representation is restricted to the left half section of the droplet (the complete con-
figuration can be readily obtained by axi-symmetry). At a given contact angle (i.e.,
a given time) a fast decrease of T ∗ due to the poor heat conductivity of air is
observed when moving from the droplet interface to the far field in the normal
direction to the interface [26, 76]. The corresponding gradient is almost constant
everywhere when θ

and T s =

80 ◦ and increases at the contact line level as contact angle
decreases. This increasing trend is controlled by the value of L s and occurs only
for L s /R

=

1 while when L s /R > 1, air nearby the contact line is heated by the
substrate up to temperature T s , and temperature gradients become smaller. It results
here that when L s /R 1 (resp., L s /R > 1) mass flux is increased (respectively

decreased) in the vicinity of the contact line.
The rising trend of the mass flux nearby the contact line can generate impor-
tant contributions to the overall mass loss of the droplets. In the case L s /R

1
illustrated in Fig. 5, it is initially the same as everywhere else on the interface as
suggested by the uniform normal gradients of T ∗ of Fig. 5(a) and as discussed. But

=