4
Freely Receding Volatile Drops of Wetting
Liquids
The effect of evaporation in the vicinity of the contact line has been studied both experimentally and theoretically. It seems to be generally recognized that evaporation leads to larger apparent contact angles. For instance, experiments with evaporating sessile droplets in air revealed that the contact angles during the drying of drops were finite although the liquids used were completely wetting (Cachile et al. [2002a,b], Poulard et al. [2003]). The measured apparent contact angles indicated an increase with evaporation rate whereas a complete wetting situation was achieved only in the absence of evaporation. In this chapter we present a similar experimental study as far as the motivation goes, yet different as far as the objectives and their realization are concerned.
In particular, the widely used interferometric techniques are here optimized to be applicable to visu-alize evaporating droplets of millimetric sizes whose apparent contact angle extends far beyond the ones reported in the aforementioned studies. We track the behavior of the apparent contact angle as well as the evolution of the radius in time. Moreover, the latter allows us to address the question of whether or not the speed of the receding contact line can be considered negligible as far as the appar-ent contact angle is concerned. Evappar-entually, we are able to calculate the evaporation-induced contact angles of the tested droplets and, by measuring the volume loss rate, correlate them with the intensity of evaporation.
P
vap@T= P
sat@TP
vap@T<< P
sat@TP
vap@T0 < P
sat@T(a) (b) (c)
Figure 4.1: Three schematic representations of liquid evaporation taking place in a cell. When the cell is sealed and no vapor is present in the gas phase (a) liquid molecules evaporate filling up the gas phase until (b) we are at an equilibrium where both evaporation and condensation occur at the same rate; (c) As soon as the seal is removed, evaporation continues until all the liquid is vanished (for a practically low ambient humidity).
4.1
Evaporating Droplets
The term evaporation is used to denote the transition of a substance from a liquid to a vapor phase. It takes place at the liquid-vapor interface where the liquid molecules with sufficient kinetic energy have more chances to overcome the intermolecular forces that hold them together. When this happens, the liquid molecules escape into the gas phase to form a vapor component therein (Fig. 4.1a). The opposite process, known as condensation, also occurs. At a fixed temperature, if the rate at which molecules escape the liquid phase is equal to the rate at which molecules return to it, we are at equilibrium (Fig. 4.1b). The vapor pressure at that moment (or the partial vapor pressure in case air molecules are present in the cell) is said to be equal to the saturation pressure of the liquid at this particular temperature. If the vapor pressure is lower than the saturation pressure, then evaporation prevails over condensation until equilibrium is achieved. In case that equilibrium cannot be reached, as in an open system where the atmosphere is far from being saturated with the vapor, all the molecules eventually move to the gas phase and the liquid disappears (Fig. 4.1c). In general, the saturation pressure is a good indicator of the level of evaporation at a certain temperature. In particular, liquids that are characterized by a high saturation pressure are expected to evaporate faster, or in other words be more volatile, than those with a lower value, when these are at the same temperature.
4.1. EVAPORATING DROPLETS
4.1.1
Local and Global Evaporation Rate
Let us describe the particular case of sessile droplet evaporation in an inert gas atmosphere, such as air, where ordinary diffusion is the dominant driving factor (Fig. 4.2a). Namely, the molecules that escape the surface of the droplet will be eventually transferred to those areas of the gas phase that are characterized by a low vapor concentration. Larger gradients of vapor concentration in the gas phase result in a faster diffusion of vapor into the air and consequently in higher evaporation rates, and vice versa.
For sufficiently small concentrations, the concentration field of the vapor around the drop at each moment of the process is governed by the diffusion equation
∂ c ∂ t = D∇
2c (4.1)
where c is the concentration of the vapor, t the time and D the diffusion coefficient.
Assuming a quasi-steady evaporation, as Deegan et al. [1997] did in an attempt to quantify the loss of mass that a pinned droplet experiences during such process, the vapor cloud has enough time to adjust in the atmosphere as compared with the total evaporation time. Consequently, Eq. (4.1) becomes
∇2c= 0 (4.2)
The boundary conditions are that the vapor cannot penetrate the substrate and that the air at the surface of the drop is saturated with vapor, whereas far from the drop the concentration is equal to the ambient vapor concentration, or equal to zero if the ambient atmosphere consists only of air. Once the vapor concentration around the drop has been defined, the local evaporation rate (mass loss per unit surface area per unit time) is given by the Fick’s 1stlaw of diffusion as,
j= −D∂ c
∂ n (4.3)
evaluated at the droplet surface.
It turns out that the local evaporation rate in a diffusion-limited regime is not uniform along the surface of the droplet, moreover diverging near the contact line of the drop, as it results in the framework of the just posed boundary-value problem. Particularly, close to the contact line one has (Deegan et al. [1997])
j(r,t) ∼ (R − r)−λ (4.4)
where R is the contact radius of the droplet, r is the radial distance measured from the symmetry axis, λ = (π − 2θ )/(2π − 2θ ) and θ is the contact angle of drop.
P
vap@T
= P
sat@T
P
vap@T
<< P
sat@T
P
vap@T
0 < P
sat@T
(a) (b)
Figure 4.2: (a) Contour plot of the vapor concentration distribution above a droplet of 1mm radius and 0.364mm height during the diffusion-controlled evaporation process. From Hu and Larson [2002]; (b) part of the top view of an evaporating water droplet. The induced flow inside the drop drags suspended particles towards the pinned contact line. From Yunkeret al. [2011].
4.1. EVAPORATING DROPLETS
Several other studies (Hu and Larson [2002], Popov [2005]) presented a more complete formulation of the same problem, with the results confirming that of Deegan (Fig. 4.3). In particular, Popov [2005] provides an analytical expression for the local evaporation rate as a function of the contact angle, the radial coordinate and the drop contact radius. Particularly, in the limit of small contact angles (θ 1),
j(r) = 2 π
D(cs− c∞) √
R2− r2 (4.5)
where cs and c∞are the vapor concentrations at the surface, i.e., the saturation concentration, and far from the droplet, respectively.
Although the local evaporation rate introduces a singularity at the three-phase contact line, this sin-gularity is integrable. Thus, by integrating Eq. (4.5) over the surface of the drop one can obtain the global evaporation rate, i.e., the evaporative mass loss per unit time,
−dm
dt ≡ J = 1.3πRD(cs− c∞) (4.6) According to Eq. (4.6), for small contact angles the total evaporation rate in the diffusion-limited regime is proportional to the radius of the drop and not to its surface area as one might expect. Since during our experiments no vapor HFE is present in the atmosphere, it is c∞= 0. Thus, the above expression can be written as
J= 1.3πRDPsatM RgT
(4.7)
where Psat is the saturation pressure, M is the molar mass, Rg is the universal gas constant and T is the temperature. We should note here that the experimental evaporation rate is given in volume per unit time and it is related to the theoretical one through the expression Jexp= J/ρl, where ρl is the density of the liquid.
4.1.2
Evolution of the Droplet Radius During Evaporation
From the work of Deegan et al. [1997, 2000b] one can also extract information related to the evo-lution of the radius while the drop is evaporating. We already indicated that the global mass loss is proportional to the base radius of the droplet which in terms of volume loss can be expressed as
dV dt = −
J ρl
∼ −R (4.8)
If we now rearrange Eq. (2.9) we can have an expression for the volume of the droplet, assuming that it adopts a spherical cap shape and that the (small) apparent contact angle does not change in time,
V(t) =π θapR(t) 3
4 (4.9)
dV dt = 3πθapR(t)2 4 dR dt (4.10)
and combining Eqs. (4.8) and (4.10) we eventually get
R(t)2dR
dt = −β R(t) (4.11)
where β is a proportionality constant. Integrating from t = 0, where the initial radius is R0, Eq. (4.11) yields
R(t) = (R20− 2βt)1/2 (4.12) If we now define the extinction time of the droplet, i.e., when R(to) = 0, as
to= R02
2β (4.13)
then Eq. (4.12) gives the scaling law
R(t) ∼ (to− t)1/2 (4.14)
A study published in Shahidzadeh-Bonn et al. [2006] suggested that when natural convection can come into play, as in the case of a sessile water droplet whose vapor is lighter than air, the reported exponent can deviate from 0.5. Experiments performed by Cachile et al. [2002b] and Poulard et al. [2003] however, consider deviation from 0.5 to be not due to natural convection, but rather due to a varying contact angle. In particular, tests with freely receding evaporating drops of various alkane liquids showed that the exponent of the above similarity solution is slightly smaller than 0.5 with the apparent contact angle not being constant in time. Thus, they assumed that both the radius and the contact angle vary in time as
R∼ (t0− t)y (4.15)
θap∼ (t0− t)x (4.16)
Then relations (4.8) and (4.9) yield, respectively dV
dt ∼ −(t0− t)
y (4.17)
V ∼ (t0− t)3y+x (4.18)
By differentiating the latter expression and equating its exponent with the former’s, one obtains a more general form as far as the exponent of the radius is concerned,
4.1. EVAPORATING DROPLETS
or
y= 0.5(1 − x) (4.20)
This relation would now agree reasonably well with their findings. Moreover, alkanes produce a vapor that is heavier than air, therefore conducting experiments with hanging (pendant) drops should approach the situation of an evaporating sessile water droplet. Nevertheless, same exponents as in the case of sessile alkane droplets were reported. Thus, according to Guéna et al. [2006, 2007b], an exponent lower than 0.5 cannot be explained in terms of natural convection in this case. In addition to these tests, they also examined sessile and pendant water droplets with the exponent for the radius being 0.6 in both cases. Yet, the exponent x was found to be equal to −0.2 satisfying nicely the above equation. This also shows that in order for Eq. (4.19) to be obeyed the apparent contact angle should behave accordingly, namely either decrease (x > 0) or increase (x < 0) in time depending on the situation. Clearly, for a time-independent apparent contact angle, one should retrieve the exponent 0.5 for the radius evolution.
In the following sections we are going to examine whether this picture is in agreement with our experimental findings.
4.1.3
Evaporation-Induced Contact Angles
As we already noted above, the intense evaporation at the three-phase contact line region triggers a strong hydrodynamic flow inside the droplet, which will eventually transfer fluid to the contact line in order to replenish the evaporated mass. For this flow to take place, there must be a significant pressure drop towards a small vicinity of the contact line, which is also associated with a local curvature change. A side effect of this curvature change is the generation of a non-equilibrium contact angle in a microscopic region at the contact line, the consequences of which are apparent at a macroscopic scale as well, although the liquid undergoing evaporation is highly wetting (Cachile et al. [2002a,b], Poulard et al. [2003], Lee et al. [2008], Bonn et al. [2009]). Such evaporation-induced contributions can also be formed in the case of partial wetting droplets, however their relative importance in the total macroscopic apparent contact angle is decreasing with an increasing Young’s angle.
The underlying theory developed by Colinet and Rednikov [2011] suggests an expression for the calculation of such angles. The theory concerns a microregion (with a length typically from tens to hundreds of nm) around an immobile contact line, which can be part of an evaporating, into an inert gas atmosphere, droplet or meniscus. Here we are considering the former case only and it is in this microregion that the evaporation induced angles are generated (Fig. 4.4).
The problem is formulated in the framework of the lubrication (thin-film) approximation according to which inertial forces are negligible with respect to viscous forces (creeping flow) and the vertical length scale, [z], is much smaller than the horizontal one, [x], indicating that we are dealing with thin drops. Here x, z are the Cartesian coordinates such that z = h(x) corresponds to the liquid-gas interface and z = 0 to the substrate surface as shown in Fig. 4.4. We have [h] = [z], and in accordance to with what was said
ε = [h]
[x] 1 (4.21)
macro-region micro-region jev θev solid liquid air q h(x) θap x
Figure 4.4: Sketch of the microstructure near the contact line where the evaporation-induced angles are lo-cated.
θev∼ ε (4.22)
with θevbeing the angle induced in the microregion due to evaporation.
Since we are in the microstructure of the contact line, no gravity effects are considered, while both the Laplace pressure and the disjoining pressure must be taken into account regarding the pressure difference, ∆P, across the liquid-vapor interface. As far as the disjoining pressure is concerned, only the long range stabilizing Van der Waals forces are chosen (Bonn et al. [2009], Colinet and Rednikov [2011]) in order to particularly focus on a perfectly wetting situation. The ambient pressure in the air is assumed to be constant.
Based upon these assumptions and considerations one can simplify significantly the Navier-Stokes equations and extract eventually an expression for the velocity field in the microregion, i.e., µl∂zzu− ∂xP= 0, from which it follows that
[u] ∼ [h] 2 [x]
[∆P]
µl (4.23)
where µl is the kinematic viscosity of the liquid.
For this microregion the scales are chosen assuming that the disjoining pressure and the Laplace pressure are of the same order of magnitude. Thus,
[∆P] ∼ γ [h] [x]2 ∼
A
[h]3 (4.24)
From Eqs. (4.21) and (4.24) we have [x] = α/ε2 and [h] = α/ε where α =pA/γ and with γ and A being the liquid surface tension and the Hamaker constant, respectively.
Finally, according to mass conservation, the mass flux, q, towards the contact line should balance the local mass flux due to evaporation, jev, such that
∂ q
4.1. EVAPORATING DROPLETS
From this equation we see that the two terms are of the same order of magnitude, therefore it is [q]
[x] ∼ [ jev] (4.26)
The local mass flux for a diffusion-limited evaporation into an inert gas is given, in compliance with Eq. (4.4), as follows (Colinet and Rednikov [2011])
jev=
Dgρvsat
l1/2x1/2 (4.27)
where Dg is the diffusion coefficient, ρvsat is the saturation vapor density, l is a macroscopic length scale and x is the distance from the contact line at which the local evaporation flux is calculated. As for the scale of the local mass flux [q], this can be expressed as
[q] ∼ ρl[u][h] (4.28)
From these developments, one in particular obtains
θev∼ ε ∼ νlDgρvsat α1/2l1/2γ 1/3 (4.29) where νlis the kinematic viscosity of the liquid.
If one assumes that the evaporation rate is limited by diffusion also globally, i.e., no convection effects are present in the gas, then the macroscopic length scale is proportional to the drop radius R and particularly it is l = π2
2R (Deegan et al. [1997, 2000b], Popov [2005]). In this case, Eq. (4.29) yields θev∼ νlDgρvsat α1/2R1/2γ 1/3 (4.30) From this relation, we can deduce that at a given temperature the evaporation-induced contact angle depends slightly on the radius. i.e., θev∼ R−1/6and specifically it increases as the radius decreases.
Effect of Receding Velocity
Since, the evaporation-induced contact angle is localized at a scale that is hardly accessible experi-mentally, it is worthy to relate it to the macroscopic apparent contact angle of the droplet. This can be done through the Cox-Voinov law, recalled below,
θ3ap= θev3+ 9Ca lnR/(2e2L)
(4.32) with Ca being negative for an evaporating drop with a receding contact line. From Eq. (4.32) it follows that, in case of a static contact line, the apparent contact angle is directly associated with the evaporation-induced angle. Once again, it is necessary to remark that the evaporation-modified Cox-Voinov law is valid in the limit of small velocities, i.e., Ca 1.
This expression allows us to isolate the evaporation-induced contact angle from any possible velocity induced contributions. For the purpose of this study, we are going to choose the value of the logarithm to be 15. It is a rough estimation assuming a macroscopic length scale of the order of 2mm and a microscopic scale of the order of 1nm. The modified Cox-Voinov is then re-written below as
θ3ap= θev3 + 9 · 15Ca (4.33)
4.2
Results
Experiments are performed on a 3mm thick polycarbonate plate (2x1.5cm) and are repeated 10 times for each tested liquid. From the liquids listed in Table 3.1, however, only three are going to be extensively used. In particular, we choose two liquids of which the evaporation rate is considerably different, i.e., HFE-7100 and HFE-7500 as well as one whose evaporation rate falls in between, i.e., HFE-7200. Performing preliminary tests with HFE-7000, whose saturation pressure is about 2.5 times larger that HFE-7100, resulted in a strong dew formation which distorts the obtained results. The injection of liquid using a syringe takes place manually and therefore the initial volume of the deposited drop is not accurately specified. Thus, drops of different initial radii, varying between 1.5 and 2mm, are generated per experiment. The deposited drop is then let to evaporate freely in open air. The evaporation takes place in normal ambient conditions which are noted down before each experiment but they are not controlled. In the section to follow we show and explain in detail the experimental results concerning each liquid.
4.2.1
HFE - 7100
Here, the most volatile liquid of the three under examination is tested. When the drop is deposited on the polycarbonate plate, it tends to retract immediately and therefore we rarely observe any continuous spreading. As a result, no spreading regime is part of the treated data. We should highlight that the drop retains an almost perfectly axisymmetric shape as it retracts. During the retraction, waves travelling along the contact line are also often observed, making the latter look locally protruded and consequently breaking the radial symmetry of the drop. In most cases such waves start to fade out for radii smaller than 1 ± 0.1mm. Contact line oscillations have also been reported in past studies and linked to exchanges with the gas phase (Cachile et al. [2002a]) or to thermocapillary instabilities (Sobac and Brutin [2012], Carle et al. [2012]).
4.2. RESULTS
(a) (b)
Figure 4.5: Apparent contact angle versus the elapsed time for (a) a single experiment and (b) ten different experiments (HFE-7100).
(a) (b)
(a) (b)
Figure 4.7: (a) Apparent contact angle and (b) radius versus the time remaining for the drop to evaporate completely after averaging the results of different experiments (logarithmic scale) (HFE-7100).
The plot in Fig. 4.5a illustrates the evolution of the contact angle in time regarding a single experiment. The error bars show the variation in the measurement of the contact angle for different cross-sections of the drop with a 95% confidence interval, i.e., twice the standard deviation, which is about 1◦. The main source of this variation could be related to the determination of the contact line and to slight inclination of the substrate. Repeating the experiment a considerable number of times yields similar results (Fig. 4.5b). It was mentioned already, however, that the deposited volume of liquid is not controlled accurately, therefore each experiment is characterized by a different initial radius and consequently by a different lifetime.
Thus, to add a time-independent flavor in the representation of the results we define the vanishing time, to, of the drop and we plot the contact angle versus the time remaining till the drop evaporates completely, as well as versus the radius; in line with Cachile et al. [2002a]. This will also allow us to merge all the experimental runs (e.g. inset plot in Fig. 4.6a) into one curve (main graph of Fig. 4.6a). There one can see that the contact angle is found to be constant during most of the life of the drop, and equal to 8.8 ± 1◦. Yet, it decreases rapidly 2.5 seconds before the end of the experiment, as we approach radii smaller than 0.50mm (Fig. 4.6b).
The uncertainty related to the cross-sectional variation in the measurement of the contact angle is higher than the repeatability error before the radius has decreased significantly and lower at the latest stages of the evaporation. Thus, in the figures to follow we are going to show each time the maximum source of uncertainty.
We also plot the evolution of the contact angle and the radius with time in a logarithmic scale (Fig. 4.7). Both indicate a linear behavior at least up to one second before the drop vanishes. Fit-ting the data, which concern the main part of retraction while excluding the last moments, with a first order polynomial we obtain x = 0.003 and y = 0.57 (the uncertainty on the fittings is about 1%), with xand y being the exponents for the angle and radius as functions of time, according to Eqs. (4.15) and (4.16).
4.2. RESULTS
(a) (b)
Figure 4.8: Contact line velocity versus (a) the time remaining for the drop to evaporate completely and (b) the drop radius after averaging the results of different experiments (HFE-7100).
(a) (b)
Figure 4.9: (a) Ratio of the angle increment induced by the speed of the contact line to the experimentally determined contact angle; (b) Measured apparent contact angle and the evaporation-induced contact angle after extracting from the former the influence of the contact line speed (HFE-7100).
we mentioned already, to check whether this range of velocities is large enough to start considering hydrodynamic effects we need to calculate the capillary number and compare the second term of the right hand side of Eq. (4.33) with the obtained contact angles. In Fig. 4.9a we see the ratio between these two values. Apparently, the speed of the contact line becomes more important for smaller radii and it is more likely to make the macroscopic angle deviate from the evaporative angle in this regime. Hence, it is worth to calculate the evaporation-induced contact angle according to Eq. (4.33). The results indicate indeed that the difference is not significant for big radii while it becomes more pronounced at later stages (Fig. 4.9b). The evaporation induced constant angle has a value of about 9.2 ± 1◦, but it decreases near the end. The plot, however, cannot be continued to radii as small as in the case of the apparent angle due to the smoothing of the data.
(a) (b)
Figure 4.10: (a) Volume versus radius after averaging the results of different experiments. The inset shows the volume versus the time remaining for the drop to evaporate completely for all the experimental runs; (b) global evaporation rate versus the radius again after averaging the results of different experiments (HFE-7100).
0.7mm, contradicting this way the concept of a linear increase of the evaporation rate with the radius. However, the results tend to adopt a linear behavior at smaller radii. In the discussion in section 4.3 we are going to provide some possible explanations concerning why this can be the case.
4.2.2
HFE - 7200
Although this particular liquid is considered to be less volatile as compared to the HFE-7100, the spreading stage remains very short. Here, too, the drop retracts reasonably fast and axisymmetrically with the aforementioned waves still being present. However, these are much more intense in terms of speed giving the feeling that the drop is rotating as it recedes. The waves disappear as the radius decreases but this time the critical radius is about 0.5 ± 0.2mm. As far as the apparent contact angle is concerned, it follows the same trend as previously with the angle being constant during most of the experimental time, yet lower than before and equal to 6 ± 1◦. As soon as the drop has shrunk below a certain radius (< 0.35mm) the angle starts to decrease rapidly. This decrease occurs approximately 1.5 seconds before the drop vanishes.
Regarding possible hydrodynamic influences on the obtained angles, in Fig. 4.12 one can see that the receding velocity of the contact line has not decreased in absolute value as compared to HFE-7100, although a less volatile liquid is used now. Since, however, the apparent contact angle is lower than before, hydrodynamic contributions can have a larger impact on the angle induced by evaporation according to Eq. (4.33) as shown in Fig. 4.13. The calculated evaporation-induced angle is initially about 6.5 ± 1◦but it increases slightly as the drop becomes smaller.
As far as the evolution of the contact line and the radius with time is concerned, it is again linear when presented in a logarithmic scale and the corresponding fittings reveal results approximately the same as before, i.e., y = 0.56 and x = −0.007 (the uncertainty on the fittings is about 1%). Although the apparent angle seems to decrease as time passes, this decrease is really minor (Fig. 4.14).
4.2. RESULTS
(a) (b)
Figure 4.11: Apparent contact angle versus (a) the time remaining for the drop to evaporate completely and (b) the drop radius after averaging the results of different experiments (HFE-7200).
(a) (b)
Figure 4.12: Contact line speed versus (a) the time remaining for the drop to evaporate completely and (b) the drop radius after averaging the results of different experiments (HFE-7200).
(a) (b)
(a) (b)
Figure 4.14: (a) Apparent contact angle and (b) radius for HFE-7200 versus the time remaining for the drop to evaporate completely after averaging the results of different experiments (logarithmic scale) (HFE-7200).
(a) (b)
4.2. RESULTS
4.2.3
HFE - 7500
The last series of experiments concerns HFE-7500 which is the least volatile in comparison with HFE-7100 and 7200. Performing experiments with this liquid proved to be not an easy task for two main reasons. One concerns what was discussed already in section 3.4 with respect to static electricity. Yet, even after tackling that issue, we had then to deal with the mobility of the droplet due to which the drop would sometimes escape the field of view of the camera. Let us describe briefly the reason behind this. The weak evaporation induces lower apparent contact angles than before, so the initially deposited drop tends to spread in order to lower its apparent contact angle to the value of the evaporation induced one. In particular, for 1µl of HFE-7500, which is approximately the maximum volume used in the previous cases, the spreading drop covers now a much bigger area and the radius exceeds by far the capillary length. Gravity, therefore, becomes important and even a small inclination of the substrate may make the drop slide accordingly, driving it out of the field of view before the recession starts. To prevent this, we use a much thinner needle ensuring this way that the injected volume is less than 1µl. Moreover, as the measured angles are small in this case, we can afford lowering the magnification factor of the lens which in return offers a larger field of view. Hence, even if there is a slight movement of the drop this will still be under our observation.
Having overcome these experimental constrains, we can now perform tests that are at last character-ized by a good repeatability. During these tests we see that the shape of the drop, as it spreads reaching a maximum radius, can gradually deviate from a perfectly circular shape but its circumference still remains convex. Axisymmetry is established once again as soon as recession starts. In Fig. 4.16a we present the evolution of the contact angle for a single experiment that corresponds to a 0.5µl drop. There one can see that indeed the contact angle initially decreases while the radius (Fig. 4.16b) is in-creasing, indicating that spreading is visible in the processed data. As soon as the drop starts receding, the apparent contact angle increases again before it eventually reaches a plateau.
Note that this behavior actually contradicts the classical Cox-Voinov picture and the computations of Todorova et al. [2012] according to which the receding angle should be smaller than the angle at the moment when the contact line is momentarily immobile (minimum point in Fig. 4.16). Thus, it is likely that the angle at the minimum point does not correspond to the actual evaporation-induced contact angle but is rather a result of a transient regime, whose origin is not quite clear.
In addition, due to the smaller thickness the contact line is now more prone to get pinned on local defects or dust particles. Moreover, the pinning lasts longer since the contact line recedes at a small pace, owing to the low evaporation rate. Travelling waves are still traced during the particular exper-iments. The speed at which they propagate is much slower in this case, while occasionally they can even cause local spreading. As happened for the other two liquids, however, the wavy behavior disap-pears below a critical size of the drop radius, which has been found approximately to be 1.2 ± 0.3mm, and the recession continues intact from that point and on.
For the time being we will limit the results to the “smooth-receding” regime where the apparent contact angle appears to be constant and which we assume to represent the most the evaporation-induced contact angle.
(a) (b)
Figure 4.16: Apparent contact angle versus (a) the elapsed time and (b) the radius for a single experiment (HFE-7500).
(a) (b)
4.2. RESULTS
(a) (b)
Figure 4.18: (a) Apparent contact angle and (b) radius versus the time remaining for the drop to evaporate completely after averaging the results of different experiments (logarithmic scale) (HFE-7500).
Presenting again the evolution of the contact angle (Fig. 4.18a) and the radius (Fig. 4.18b) with time in a logarithmic scale the linear fittings return a slope of x = −0.003 for the apparent contact angle and y = 0.58 for the radius in the framework of Eqs. (4.15) and (4.16) (the uncertainty on the fittings is about 1%).
Now, we are going to investigate the role of hydrodynamics. Again we begin by defining the reced-ing speed concernreced-ing each individual experiment before extractreced-ing the average trend. Overall, the obtained velocities are characterized by a good repeatability. Yet, contrary to what has been observed so far regarding the two other liquids, some of the drops now exhibit a fluctuating contact line speed. These fluctuations can be attributed partially to pinning effects, which as said above are more likely to occur due to the low apparent contact angle, but mostly to the waves travelling along the contact line. Both pinning and sudden spreading (associated with the mentioned waves) cannot be controlled and therefore they appear randomly during an experiment. Consequently, different experiments can exhibit different fluctuations at different moments or exhibit no fluctuations whatsoever. In Fig. 4.19 we present the contact line speed versus the vanishing time and the radius after combining all the data to a single plot. There one can see an averaged version of the fluctuations that have occurred during the tests. The fluctuations fade out for radii smaller than 1mm possibly because for the same radius the waves travelling along the periphery of the contact line are fading out as well. More details are mentioned in the following section.
As for the absolute values of the speed, these are now considerably lower than before. Yet, the fact that the apparent contact angle has also decreased makes the speed-induced angle the dominating term (Fig. 4.20a). Eventually, the actual angle induced by evaporation is increased by 21%, i.e., to 4.1 ± 0.5◦. Especially for small radii the increase is even up to 70% with respect to the experimentally measured one (Fig. 4.20b).
(a) (b)
Figure 4.19: Contact line speed versus (a) the time remaining for the drop to evaporate completely and (b) the drop radius after averaging the results of different experiments (HFE-7500).
(a) (b)
Figure 4.20: (a) Ratio of the angle induced by the speed of the contact line to the experimentally determined contact angle; (b) Measured apparent contact angle and the induced contact angle by evaporation after ex-tracting from the former one the influence of the contact line speed (HFE-7500).
(a) (b)
4.3. DISCUSSION
(a) (b)
(c) (d)
Figure 4.22: Comparing the experimental results of different liquids; (a) global evaporation rate with respect to a linear fit of the evaporation rate for small radii (up to 0.5mm) and (b) percentage difference between these two; (c) global evaporation rate with respect to the pure diffusion case and (d) percentage difference between these two. All quantities are plotted versus the radius of the drop.
4.3
Discussion
In this section we are going to discuss the results presented above and compare them with the theory introduced at the beginning of the chapter. In addition to that, we will also give some possible ex-planations on why the current experimental outcome might differ from some already published in the literature.
4.3.1
Global Evaporation Rate
both figures one can indeed see that although for larger drops the global evaporation rate deviates more from the anticipated linear behavior, this deviation is less pronounced in the case of HFE-7500 and that could lead us to the conclusion that as far as HFE-7500 is concerned, we are closer to the pure-diffusion regime for which the theories described in the introductory paragraphs are valid. We also compare in Fig. 4.22c the experimentally determined evaporation rate with the theoretically expected one given by Eq. (4.7). At a first glance, the experimental results are of the same order of magnitude as the theoretical ones. This means that the disagreement is mostly concerning the trend of the global evaporation rate decrease. The percentage difference between them, given in Fig. 4.22d, indicates a larger mismatch for HFE-7200 and 7500 as compared to the (small-radius) linear fit shown in Fig. 4.22b. Actually, the agreement between the theory and the experiment seems to be rather good for the latter two liquids at small radii. As for HFE-7100, the theoretical evaporation rate even overpredicts the experimental one for small drop radii and hence the negative difference. This could be possibly due to a progressive cooling-down of the substrate, whereas we compute the theoretical evaporation rate using the temperature of the substrate measured just before depositing the drop. Another source of error, could be related to the material properties used to calculate the global evaporation rate according to Eq. (4.7). Yet, the most crucial question to be addressed is why we do not obtain experimentally a linear evolution of the evaporation rate with the radius, which is especially manifest for more volatile liquids. According to observations of recent studies (Kelly-Zion et al. [2009, 2011, 2013]) the difference can be attributed to the role of buoyancy-induced convection which affects the evaporating process and leads to a deviation with respect to the simple proportionality predicted by Popov’s model, based on pure diffusion. This indeed could explain the initially large difference between the experiments and the theory.
We are now going to compare the experimental results particularly with the study of Kelly-Zion et al. [2011], in which the evaporation of droplets of rather heavy liquids, different mainly in the vapor pressure, of various sizes is studied considering that both diffusion Ed and natural convection Ec are important mechanisms that influence the global evaporation rate E, and therefore
E= Ed+ Ec (4.34)
or dividing by Ed
E∗= 1 + Ec∗ (4.35)
where E∗= E/Edis the dimensionless global evaporation rate and Ec∗ is a dimensionless convection term defined as Ec/Ed. The results suggested that the larger the radius of the droplet the larger the deviation from the diffusion-limited evaporation. In Fig. 4.23a we replot the data in the form of Eq. (4.35) for various radii and we see that global evaporation is enhanced for larger radii, which was also clear from Fig. 4.22c as well.
In order to take into account the role of natural convection in the evaporation process, Kelly-Zion et al. [2011] suggested an empirical model based on the Grashof number, given as
E∗= 1 + 0.310Gr0.216 (4.36) with the Grashof number defined as
4.3. DISCUSSION
(a) (b)
Figure 4.23: The dimensionless evaporation rate versus (a) the droplet radius and (b) the Grashof number.
where ρm and ρa is the density of the air-vapor mixture and the air respectively, g is the gravitation acceleration, R is the contact radius of the drop and νa is the kinematic viscosity of air. In their case Eq. (4.36) was applicable to all the tested liquids and radii, indicating a certain level of universality. Nevertheless, when we compare it with our results (Fig. 4.23b) there is only a qualitative agreement. On top of that, our experimental results do not collapse on the same curve, which would possibly allow us to suggest different coefficients for Eq. (4.36) or even a different model. This could be due to the fact that our tested liquids have different molecular weights (M7500= 414g/mol ' 1.7M7100) whereas the liquids used in Kelly-Zion et al. [2011] all have a comparable molecular weight (M ∼ 90g/mol). It should also be noted here that the study of Kelly-Zion et al. [2011] extends to much larger radii (∼ 25mm) than considered here and consequently to higher Grashof numbers (∼ 105) too.
4.3.2
Apparent and Evaporation-Induced Contact Angles
In Fig. 4.24 we present in the same plot some of the results obtained for the different liquids tested. The measured apparent contact angle (Fig. 4.24a) is constant for all the cases except for small radii where a decrease of the angle can be seen. It is rather remarkable coincidence that the speed-induced and evaporation-induced angles combine to yield the typical constant apparent contact angle receding behavior, known from partially wetting evaporating droplets. It is only near the end of the lifetime of the drop that the combination does not yield a constant apparent angle. One could couple this contact angle decrease with the sharp increase of the receding velocity observed for small radii (Fig. 4.24b ). Focusing now on the evolution of the evaporation-induced contact angle (Fig. 4.24c), the most appar-ent and rather expected conclusion is that the more volatile a liquid is, the higher the angle triggered by evaporation also is. Moreover, we find that especially for HFE-7200 and HFE-7500 this angle appears to increase as the radius gets smaller. This is indeed what is expected theoretically as was shown in section (4.1.3) through the following equation
θev∼ νlDgρvsat α1/2R1/2γ 1/3 (4.38)
-(a) (b)
(c) (d)
Figure 4.24: Comparing the experimental results for different liquids; (a) apparent contact angle, (b) contact line speed and (c) evaporation-induced contact angle in the normal as well as in (d) logarithmic scale versus the drop radius.
0.14 (or -1/7) which differs from the theoretical prediction of -0.16 by 15%. For HFE-7200, a slope of -0.07 (or -1/14) is found and for HFE-7100 the angle is almost perfectly constant and it even slightly decreases at the end. An explanation regarding this decrease can be possibly found in the thermal properties of the substrate. Polycarbonate has a poor thermal conductivity (0.19–0.22 W/(m·K) at normal ambient conditions), therefore when HFE-7100 is tested, which is a highly volatile liquid, a cooling-down of the initial temperature of the surface might occur progressively. This, indeed, would imply a decrease of the evaporation rate and consequently a decrease in the evaporation induced angle, too. Other explanations could hold as well, related for instance to the validity of Cox-Voinov and to the fact that evaporation rate is not following the pure-diffusion case as mentioned above except maybe for HFE-7500, which could explain, partially at least, why for this liquid there is a better agreement between the theory and the experiment regarding the -1/6 power law.
4.3. DISCUSSION
Let us now relate the evaporation induced contact angle with the evaporation rate of the droplet. As was shown already, the evaporation rate does not follow the pure-diffusion prediction, therefore an ad-hoc modification is proposed assuming the same local evaporation profile but simply rescaled to yield a representative global evaporation rate. Particularly, from Eq. (4.7) it follows that
J
R∼ Dgρ sat
v (4.39)
Relation (4.38) can be then expressed in terms of the experimentally determined global evaporation rates Jexp to yield (4.40)
θev∼ νlJexpρl R3/2γ 1/3 (4.40) As it is also mentioned in section 4.1.1, the experimental evaporation rate is given in volume per unit time and therefore it is multiplied with the liquid density for unit consistency purposes.
We will now try to verify whether the experimentally determined evaporation induced angles scale with the evaporation rate extracted from the measurements as well as the radius in a manner similar to that described in relation (4.40). In Fig. 4.25a we plot the angles regarding not only each liquid but different radii, as well. We also perform a few experiments after heating the substrate to 50◦and 70◦. As polycarbonate is a poor heat conductor the temperature at the surface is measured to be 41◦ and 52◦, respectively. The induced contact angles as well as the evaporation rates are now larger than before. In addition to that, contact angles regarding a pendant case, namely the droplet is hanging from the bottom side of the polycarbonate substrate, are plotted, as well. Finally, we conducted a limited number of experiments with hexane and heptane droplets again in a sessile droplet configuration. The calculated evaporation induced contact angles, after subtracting the effect of the contact line velocity, are also shown in Fig. 4.25a. Particularly for heptane, if we take into account the receding speed of the droplet, the calculated evaporation-induced angle is found to be about 4◦, i.e., two times larger than the apparent one.
The solid line in Fig. 4.25a corresponds to a kx1/3 power law. For k = 2293, this was found to satisfy rather well all the plotted points.
In Fig. 4.25b we present a similar plot regarding this time the apparent contact angles. This is to show that a 1/3 power law is retrieved even when the velocity contributions are not excluded, i.e., when Eq. (4.33) (in which the exact value of the logarithm is questionable) is not employed. The value of k in this case is 2158.
Having the value of the coefficient k one can now calculate the molecular scale α, based on the complete expression for θev, as given by Colinet and Rednikov [2011]
(a) (b)
Figure 4.25: Scaling of the (a) evaporation-induced and (b) apparent contact angle with the evaporation rate for the three HFEs at normal and heated conditions, as far as a sessile configuration is concerned. Pendant drops of HFE-7100 and 7200 as well as hexane and heptane sessile drops in normal ambient conditions are also shown.
Dgρvsat = J
4R (4.43)
Thus, Eq. (4.42) becomes
θev= 1.27 α1/6 νlJexpρl R3/2γ 1/3 (4.44) and therefore k = 1.27 α1/6 or α = 1.27 k· π 180 6 (4.45) For both cases shown in Fig. (4.25) the molecular scale is found to be approximately 1nm. According to de Gennes et al. [2004], the equivalent value for water is 10 times smaller. Nevertheless, the molecular weight of HFE is approximately 15 times larger than the one of water which, at a first stage, could explain why we retrieve a larger value.
4.3.3
Radius and Apparent Contact Angle Evolution in Time
4.3. DISCUSSION
(a) (b)
Figure 4.26: Comparing the experimental results of different liquids; (a) apparent contact angle and (b) radius versus the time remaining for the drop to evaporate completely; both plots are shown in a logarithmic scale.
derive a more general expression accounting for situations where the evaporation rate of the droplet depends on its base radius with a power n, that is
dV
dt ∼ −R(t)
n (4.46)
where n is slightly larger than one. Following a similar procedure as in section 4.1.2 we end up with the following expression
(3 − n)y + x = 1 (4.47)
The apparent contact angle according to Fig. 4.26a is constant in time, hence x is practically zero and then
y= 1
3 − n (4.48)
One should note here that in Eq. (4.19) it is the exponent based on the apparent contact angle and not the angle induced by evaporation that is used.
Then, one can see that for n > 1 Eq. (4.48) yields y > 0.5. Nevertheless, when the evaporation rate is plotted in logarithmic scale it cannot be fully approximated as a power of the drop radius therefore any effort of confirming or improving relation (4.19) might be questionable after all. One could isolate the data for which evaporation seems to decrease linearly with the radius in order to extract a power law but even when this was attempted Eq. (4.47) was not satisfied. Yet, the range of data is quite small and the accuracy of the fit might not be reliable enough to extract any conclusions at this point.
X Y 2Y+X
HFE-7100 +0.003 0.57 1.14 HFE-7200 -0.007 0.56 1.12 HFE-7500 -0.003 0.58 1.16
4.3.4
Travelling Waves and Fluctuations of the Contact Line
When we described the results regarding HFE-7500, and specifically the evolution of the receding speed, we referred to some fluctuations. In this section we dedicate some lines to this behavior, which could be also related to the wavy behavior of the contact line observed during the experiments with the other two liquids as well. We limit our description to the case of HFE-7500, which could naively be described as a “slow-motion” version of the other two liquids where effects that would possibly be damped in case of HFE-7100 and 7200, due to the strong evaporation, can now survive in time and in space.
The aforementioned fluctuating contact line speed is in fact related to a back and forth motion of the contact line itself. In particular, whenever the drop gets locally pinned, the receding motion is slightly hindered while the apparent angle decreases. As soon as the pinned part of the drop eventually escapes, an increase of the speed of the contact line is noticed and the local apparent contact angle catches up with the angles of the rest of the drop. Travelling waves also result in speed fluctuations since they can occasionally cause local spreading. In Fig. 4.27 we examine an individual experiment where such a behavior is intense. Initially, the drop retracts normally (Fig. 4.27a), however at some moment the receding motion is disturbed by a wave and local spreading is observed at the south-west part of the drop (Fig. 4.27b,c). The spreading delays the global retraction of the drop lowering simultaneously the apparent contact angle at this spot and consequently the average apparent contact angle that results from the various profiles of the drop. When spreading becomes faint the drop quickly tends to re-establish its axisymmetric shape (Fig. 4.27d). The latter stage is characterized by acceleration of the contact line and an increase of the apparent contact angle. In the graph shown in Fig. 4.27e we plot the contact line speed and the contact angle versus the vanishing time for the receding regime. There one can see the effect, i.e., the simultaneous decrease and then increase, that the local spreading has on both the apparent contact angle and the absolute value of the contact-line velocity that characterize the entire drop. We should highlight again at this point the difference between the local velocity at the point of spreading from the global (averaged) one. Specifically, while the spreading part of the drop indicates the existence of a positive velocity (advancing contact line) this is still depicted in Fig. 4.27e as a negative one (receding contact line), but with a smaller absolute value.
Immediately after the described fluctuation takes place another one begins which is triggered by a local pinning of the contact line as seen on the north-east part of the drop of Fig. 4.27d. Although this has a clear effect on the receding velocity of the drop, which again decreases before it increases once more, we cannot say the same for the apparent contact angle possibly because the disturbed part of the drop is now much smaller than before, and therefore the local decrease of the contact angle has a small effect on the calculated average of the contact angle.
4.3. DISCUSSION
(a) (b)
(c) (d)
(e)
