КОЛЛОИДНЫМ ЖУРНАЛ, 2007, том 69, № 1, с. 5-12
МАТЕРИАЛЫ XIII МЕЖДУНАРОДНОЙ КОНФЕРЕНЦИИ "ПОВЕРХНОСТНЫЕ СИЛЫ"
УДК 532.64
THE DYNAMICS OF EVAPORATING SESSILE DROPLETS © 2007 r. G. Guena, C. Poulard, A. M. Cazabat1
Laboratoire de Physique Statistique de I'Ecole Normale Superieure 24 rue Lhomond, 75231 Paris Cedex 05, France Поступила в редакцию 30.06.2006 г.
Experiments on sessile drops evaporating in normal atmosphere without applied thermal gradient are reported and compared with an available theoretical model. The liquids used are alkanes, water and, more recently, poly-dimethylsiloxane oligomers. The substrates are silicon wafers, completely wetted by the liquid. Experiments with hanging drops allow us first to discard any influence of convection in the gas phase on the drop dynamics. The model assumes the process to be controlled by the stationary diffusion of the evaporating molecules in the gas phase. For alkanes and water, and in a limited range of drop sizes, where gravity can be ignored, the model accounts very well for the dynamics of the drop radius, and rather well for the contact angle. This is no longer the case with the polydimethylsiloxane oligomers, where the very small contact angles require a more elaborated analysis of the drop edge.
1. INTRODUCTION
The dynamics of evaporation of spherical liquid droplets in ambient atmosphere has been studied for long. A remarkable experimental observation is that, for slow evaporation, the drop radius scales like the square root of the time interval t0 - t, where t0 is the time
where the drop disappears, R(t) ^ Jt0 - t. This means that, in a somewhat counter-intuitive way, the evaporation rate of a moderately volatile aerosol droplet
with instantaneous volume V(t) = 4( t - is proportional, not to the drop area, as it would be the case in vacuum, but to the drop radius ^^ x -R(t) [1-4]. As
a matter of fact, a slow enough evaporation is controlled by the stationary diffusion of the evaporating molecules in the gas phase, the liquid free interface being in local equilibrium with the atmosphere just above it. That simple model ignores the thermal effects induced by evaporation, and holds for droplets sizes significantly larger than the mean free path of the molecules in the gas phase [3-4].
The recent work by Deegan and co-workers [5] has attracted a renewed interest on the dynamics of evaporating sessile drops [5-8]. Most studies deal with the patterns left by solutions or colloidal dispersions, a case where the contact line is pinned. In contrast, we fo-cussed on pure liquids and complete wetting, where the contact line moves freely [9-11]. Although the liquid wets completely the substrate, the contact angle does not vanish because, even during drop retraction, there is
1 Corresponding author. E-mail address: anne-marie.cazabat@
upmc.fr.
a flow directed towards the edges to replace the evaporated liquid [5-8]. However, the angle is very small, a few degrees or less, especially during the receding motion. Therefore the drop is thin, which is of importance both for the hydrodynamic flow, which is mainly parallel to the substrate, in contrast with non-wetting cases where convection rolls can be observed [12-15], and for the smoothing of any vertical gradient.
The assumption of stationary, diffusion driven evaporation has been used by Deegan and co-workers to calculate the local evaporation flux J(r, (R(t)) at the free surface of an isothermal drop of radius R(t) deposited on a substrate [5-8, 16]. At the distance r from the drop axis, and for small contact angle, i.e., for a flat disc of radius R(t), an electrostatic analogy leads to:
J( r, t) =
J 0
Jr (t)2-¡
(1)
Here, the evaporation parameter j0 is known for a given liquid at a given temperature. The local evaporation flux diverges at the contact line, but the evaporation rate for the total drop volume V(t) is simply:
^ = -2 njo R (t).
(2)
Just like in aerosols, the evaporation rate is found proportional to the drop radius, and for the same reason, i.e., the assumption that evaporation is controlled by stationary diffusion in the gas phase.
Moreover, if the thin drop is a spherical cap of (small) contact angle 0(t), then another simple relation can be written
V (t ):
4 R3( t Ж t)
(3)
10 F
GUENA h ap.
10
0.01
10
100 300
to - t (sec)
Fig. 1. Two octane drops, volume 3 pL, on bare wafer, complete wetting. Squares: sessile drop, circles, hanging drop. Typical log-log plots of radius R(t) (mm, upper curves) and contact angle 0(t) (rad, lower curve) as a function of time interval t0 - t. Just after drop deposition (right), the radius increases and the contact angle decreases rapidly. Then the radius goes to a maximum and afterwards decreases according to a power law (straight line, upper curve). During drop retraction, the angle can also be fitted to a power law (straight line, lower curve). Within the last 0.5 s, the angle decreases more rapidly (not shown). Values of the exponents j = 0.47, x ~ 0.08. For smaller drops (volume 1pL), the exponent y is the same y = 0.47, but x ~ 0.04.
0.1
0.01
10
100
1000 2000 t0 - t (sec)
Fig. 2. Five different water drops on a bare wafer, complete wetting. One sessile drop with volume ~ 1.3 pL was used for the measurement of the radius (upper curve, squares, mm), three sessile drops with volumes between 0.5 and
3 pL were used for the measurements of the angle (lower curves, rad). One hanging drop with volume ~1 pL was used for the measurement of the radius (upper curve, circles, mm). Value of the exponents: y ~ 0.6 for the radius, both for sessile and hanging drops, x —0.2 for the angle of the sessile drops. UV-ozone cleaning.
6
1
1
making similarity and differences with the aerosol case more obvious. For example, if the contact angle is constant, the radius will scale as R(t) ^ Jt0 -1. And if the radius obeys a power law R(t) ^ (t0 - t)y, then the angle also will obey a power law 9(t) ^ (t0 - t)x and there will be a simple relation 2y + x = 1 between the exponents. As scaling laws and spherical cap shapes can be checked experimentally, these simple remarks will be helpful to guide more elaborated analyses.
2. EXPERIMENTS WITH ALKANES AND WATER [9-11, 17-19]
Drops with controlled volume are deposited on bare or grafted oxidized silicon wafers, made wettable by proper cleaning. With water, bare wafers are required, and several cleanings have been used, cleaning with sulfochromic acid, with piranha solution, or UV-ozone cleaning [18]. The three procedures ensure water wettability, but the two last ones were found to provide more reproducible substrates, slightly more hydrophilic with the piranha cleaning. With alkanes, the cleaning is not critical for bare wafers [9-11, 17]. Grafted wafers have been incubated with hexamethyldisilazane; therefore they bear an adsorbed layer of trimethyl groups. They are wetted by octane and lighter alkanes, but not by nonane [19].
The drop first spreads, because the liquid wets the substrate, then the radius goes to a maximum, referred to as R0 in the following. Finally the drop recedes and the radius vanishes at a time t0.
The radius R(t)of the wetted spot and the contact angle 9(t) are measured as a function of elapsed time t. The time t0 where the drop disappears is recorded with precision, the use of a fast camera being required for hexane and heptane [17].
During the receding motion, a power law R(t) ^ (t0 -- t)y is observed. The exponent y is slightly less than 0.5 for alkanes (between 0.44 for nonane and 0.49 for hexane), see Fig. 1, and close to 0.6 for water [6, 18], see Fig. 2. It does not depend significantly on the drop volume (see Fig. 3 taken from [19]). The behaviour of the angle is more complex. There is a fast decrease during the advancing motion, then a range of times where a power law 9(t) ^ (t0 - t)x is acceptably obeyed. There, the trend 2y + x = 1 is observed for small drops (maximum radius twice the capillary length or less), while larger drops have somewhat higher values of x, see the caption on Fig. 1.
At the very end of the drop's life, a fast decrease of the angle is always observed [11, 17], which means that the relation 2y + x = 1 is no longer obeyed. With alkanes, this corresponds to high receding velocities, and one may expect that the assumption of stationary, diffusion controlled evaporation is no longer relevant.
With water, the time where the angle starts to fall in depends on the cleaning and does not always correspond to large receding velocities [11]. It is probable that the specific shape of the disjoining pressure isotherm of water on a hydrophilic substrate, as extensively studied by Derjaguin and co-workers [20-22], plays also a role in such late-time behaviour.
Water is at present the only known liquid where y is significantly larger than 0.5, which is still a matter of debate [18, 23]. In fact, experiments with water require special care for at least two reasons [17-18].
First, the cleaning of the silica surface is critical with water. However, from one cleaning procedure to another, a mere shift of the log-log plots of radius and angle is observed (except at late times, as already mentioned). As such a shift does not change the values of the exponents x and y, there are enough data to define them with acceptable accuracy, see Fig. 2, even if the late-time behaviour is qualitatively dependent of the cleaning procedure.
Second, water vapour being lighter than air, convection in the atmosphere is easily generated. Large scale convection has been assumed to be at the origin of the large value of y [23]. We do not agree with that assumption considering the small size of our drops. What is clear however is that, when the substrate is shifted without care under a microscope objective, significant changes result for the angle (not for the radius). A specific setup without microscope has been required to obtain the first well controlled, reproducible results f
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