8828
Langmuir 2003, 19, 8828-8834
Freely Receding Evaporating Droplets C. Poulard,† O. Be´nichou, and A. M. Cazabat* Laboratoire de Physique de la Matie` re Condense´ e, UMR 7125 du CNRS, Colle` ge de France, 11 Place Marcelin Berthelot, 75231 Paris Cedex 05, France Received April 15, 2003. In Final Form: June 19, 2003 The evaporation rate of drops of pure, completely wetting liquids is investigated experimentally and theoretically in the situation where no pinning of the contact line is occurring. It is shown that the observed dynamics is controlled by the volatility of the liquid, as expected, but also by the properties of the wetting film left on the substrate. Contact line instabilities, and nonmonotonic behavior of the receding contact angle may be observed and are accounted for. However, the water still needs further investigations.
I. Introduction For practical reasons, processes occurring at free liquid interfaces under evaporation are widely studied. The stationary shape of menisci or of drops kept at constant volume on heated substrates has been investigated experimentally and theoretically by many authors.1,16,22,26,30,34 In the same way, the dynamics of †
E-mail address:
[email protected]. * Corresponding author. E-mail address: anne-marie.cazabat@ college-de-france.fr. (1) Anderson, D. M.; Davis, S. H. The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 1995, 7, 248. (2) Benamar, M.; Boudaoud, A.; Sultan, E. In preparation. (3) Be´nichou, O.; Cachile, M.; Cazabat, A. M.; Poulard, C.; Valignat, M. P.; Vandenbrouck, F.; Van Effenterre, D. Thin films in wetting and spreading. Adv. Colloid Interface Sci., in press. (4) Betterton, M.; Brenner, M.; Stone, H. Preprint. (5) Birdi, K. S.; Vu, D. T.; Winter, A. The study of the evaporation rates of small water drops placed on a solid surface. J. Phys. Chem. 1989, 93, 3702. (6) Blake, T. D. AIChe Spring meeting, New Orleans, LA, 1988; Paper I.a. (7) Blossey, R.; Bosio, A. Contact Line Deposits on cDNA Microarrays: A “Twin Spot Effect”. Langmuir 2002, 18, 2952. (8) Bourge`s-Monnier, C.; Shanahan, M. E. R. Influence of Evaporation on Contact Angle. Langmuir 1995, 11, 2820. (9) Cachile, M.; Be´nichou, O.; Cazabat, A. M. Evaporating droplets of completely wetting liquids. Langmuir 2002, 18, 7985. (10) Cachile, M.; Be´nichou, O.; Poulard, C.; Cazabat, A. M. Evaporating droplets. Langmuir 2002, 18, 8070. (11) Crank, F. The Mathematics of Diffusion, 2nd ed.; Oxford Science Publications: 1975. (12) Deegan, D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Capillarity flow as the cause of ring stains from dried liquid drops. Nature 1997, 389, 827. (13) Deegan, D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Contact line deposits in an evaporating drop. Phys. Rev. E 2000, 62, 757. (14) Deegan, R. D. Pattern formation in drying drops. Phys. Rev E 2000, 61, 475. (15) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface forces; Consultant bureau: New York and London, 1987. (16) Ehrard, P.; Davis, S. H. J. Fluid Mech. 1991, 229, 365. (17) Elbaum, M.; Lipson, S. G.; Wettlaufer, J. S. Evaporation preempts complete wetting. Europhys. Lett. 1995, 29 (6), 457. (18) Fanton, X.; Cazabat, A. M. Spreading and Instabilities Induced by a Solutal Marangoni Effect. Langmuir 1998, 14, 2554. (19) Fischer, B. J. Particle convection in an evaporating colloidal droplet. Langmuir 2002, 18, 60. (20) Hegseth, J. J.; Rashidnia, N.; Chai, A. Natural convection in droplet evaporation. Phys. Rev. E 1996, 54 (2),1640. (21) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. Wetting at the microscopic scale; influence of surface energy. Phys. Rev. Lett. 1990, 65, 599. (22) Hocking, L. M. On contact angles in evaporating liquids. Phys. Fluids 1995, 7, 2950. (23) Hosoi, A. E.; Bush, J. W. M. Evaporative instabilities in climbing films. J. Fluid Mech. 2001, 442, 217.
evaporation of drops of pure liquids4,5,8,910,20,24,33,34,36,38 or colloidal dispersions7,12-14,19,31 on isothermal solid substrates is a very vivid field of investigation. Most studies deal with nonwetting liquids, which usually leads to an irreversible pinning of the contact line, a case where the hydrodynamic flow inside the drop is the most important parameter, controlling, for example, the deposition pattern of the colloidal particles. We focused on the opposite case of completely wetting liquids, where the contact line recedes freely during evaporation.9,10 This case is specific in two ways. First, the contact angle is very small, ∼1°, and therefore the drop is very thin. Second, the hydrodynamic problem involves a moving contact line, receding on a thin wetting film. In previous papers,9,10 hydrodynamic effects, and more precisely any flow parallel to the substrate, were ignored because of the thinness of the drops. We merely investigated the consequences of the structure of the evaporation term on the drop dynamics. The agreement with experiment was excellent in the main range of times for simple liquids such as alkanes, but with an adjustable parameter left: in fact, the contact angle at the beginning of the receding motion. Moreover, the anomalous behavior (24) Hu, H.; Larson, R. G. Evaporation of a sessile droplet on a substrate. J. Phys. Chem. 2002, 106, 1334. (25) Jackson, J. D. Classical Electrodynamics, 2nd ed.; Wiley: New York, 1975. (26) Kim, I. Y.; Wayner, P. C. Shape of an evaporating completely wetting extended meniscus. J. Thermophys. Heat Transfer 1996, 10, 320. (27) Lipson, S. G.; Samid-Merzel, N.; Tannhauser, D. S. Patterns in drying water films. Europhys. News 1998, 29, 116. (28) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: New York, 1991; Vol. I. (29) Lyushnin, A. V.; Golovin, A. A.; Pismen, L. M. Fingering instability of thin evaporating liquid films. Phys. Rev. E 2002, 65, 1602. (30) Morris, S. J. S. Contact angle for evaporating liquids predicted and compared with existing experiments. J. Fluid. Mech. 2001, 432, 1. (31) Parisse, F.; Allain, C. Drying of colloidal suspension droplets: experimental study and profile renormalization. Langmuir 1997, 13, 3598. (32) Picknett, R. G.; Bexon, R. The evaporation of sessile or pendant drops in still air. J. Colloid Interface Sci. 1977, 61, 336. (33) Pomeau, Y. C. R. Acad. Sci. Paris 2000, 328 (IIb), 411. (34) Qu, D. E.; Ram, E.; Garoff, S. Dip-coated films of volatile liquids. Phys. Fluids 2002, 14, 1154. (35) Quilliet, C.; Berge, B. Electrowetting: a revent outbreak. Curr. Opin. Colloid Interface Sci. 2001, 6 (1), 34. (36) Rafaı, S.; Bonn, D. Preprint. (37) Redon, C.; Brochart-Wyart, F.; Rondelez, F. Feestoon instabilities of slightly volatile liquids during spreading. J. Phys. II 1998, 2, 1671. (38) Shanahan, M. E. R. Simple theory of “stick-slip” wetting hysteresis. Langmuir 1995, 11, 1041.
10.1021/la030162j CCC: $25.00 © 2003 American Chemical Society Published on Web 09/18/2003
Freely Receding Evaporating Droplets
Langmuir, Vol. 19, No. 21, 2003 8829
Table 1. Values of x and y as a Function of the Parameter m of the Smoothing Function m
y
x
3 5 8 ∞
0.438 0.464 0.479 0.500
0.124 0.072 0.042 0.000
R(t)
of water droplets, first observed by Deegan13 and later by other groups,36 was not accounted for. The present paper provides a systematic study of alkanes and water in a wider range of times. Radii, contact angles, drop shapes, and contact line behavior are investigated. A coherent picture emerges; however, the complete quantitative understanding requires a numerical approach, now in the hands of theoreticians.2 A detailed report of the experimental setup and techniques has been presented elsewhere.9 Let us first summarize briefly our main previous results and analyses. II. Brief Summary of our Previous Work9,10 The drops are put on flat, smooth, isothermal, horizontal substrates and are studied using interference microscopy. The drops first spread, keep a constant radius for some time while the contact angle decreases, and then recede and finally disappear completely. Let t0 be the vanishing time. The radius R(t) of each drop is measured as a function of time t and plotted versus the time (t0 - t) to vanishing. The data for different drops of the same liquid receding on a given substrate collapse on a single curve, and a log-log plot is a straight line, indicating that R ∝ (t0 t)y. The slope y is found to be slightly less than 0.5 for alkanes on mica or oxidized silicon wafers9,10 but larger than 0.6 for water on mica.13,36 The general trend is that the more volatile the alkane, the larger the contact angle θ(t). The angle is found to decrease with (t0 - t) for all alkanes and can be measured with some accuracy in nonane and octane during the receding motion. The log-log plots, expected to be straight lines if θ ∝ (t0 - t)x, are not very linear, the slope x being less than 0.15 for the main part of the drop’s life but increasing significantly in the last second.3,9,10 A. Mathematical and Numerical Analysis without Hydrodynamic Flow. Let R(t) be the radius of the wetted spot, r the distance to the drop axis, h(r,t) the local drop thickness, h(r,t) U(h) the radial flow, averaged over the local velocity profile, and J(R,r,t) the local evaporation rate. The local conservation equation is
∂h + ∇⊥[hU] ) -J(R,r,t) ∂t
(1)
As the drops are thin, a model ignoring hydrodynamic flow parallel to the substrate was used first, to shed light on the evaporation term.
∂h ≈ -J(R,r,t) ∂t
j0
J(R,r,t) ) J(R(t),r) )
(2)
If the concentration of the evaporating compound in the gas phase is controlled by diffusion and is stationary, J does not depend explicitly on time and the problem has an electrostatic equivalent, which is, for thin drops, a charged disk at constant potential.11-13 The corresponding expression for J is11,25
x
(3)
r2 1- 2 R (t)
where j0 is the evaporation rate parameter. The divergence at the edge is nonphysical and was removed in our analysis9,10 by using a smoothing function:
Φ
x
r 1 - e-m 1 - r /R ) R 1 - e-m
()
2
2
(4)
Then
J(r) )
j0
x
r2 1- 2 R
R
j0
(Rr ) ) Rf(Rr )
Φ
(5)
A mathematical analysis is performed in the limit (t0 - t) f 0. It gives y values slightly less than 0.5 and shows that the exponents x and y are linked by 2y + x ) 1. A numerical study confirms the value of y. The mathematical values of y and x are given in Table 1 as a function of the parameter m of the smoothing function. These values are in good agreement with the ones measured on alkanes, where y is in the range 0.45-0.48. For nonane and octane, the measured values of x are satisfactory except in the last tenths of a second of the drop’s life. B. General Properties of Thin Evaporating Drops with Spherical Cap Shape. The integral of the conservation equation
∂h + ∇⊥[hU] ) -J(R,r,t) ∂t
(6)
over an almost flat drop is
∫0R(t)J(r,t)2πr dr
dV )dt
(7)
This can be written as
dV ) -j0Ψ(R(t),t) dt
(8)
where j0 depends only on the volatility of the liquid under the conditions of the experiment (temperature and relative humidity in the case of water). If J does not depend explicitly on time, the integral on the right hand side of eq 7 is a function of R only.
dV ) -j0Ψ(R(t)) dt
(9)
Instead of neglecting the hydrodynamic term, let us now assume the drop to be a spherical cap with small contact angle. This is well supported by experiments and is also a way to take into account at least partially the influence of the curvature in the hydrodynamic term. The volume of the drop is
π V = R3(t) θ(t) 4 If J has the general shape (eqs 3 and 5)
(10)
8830
Langmuir, Vol. 19, No. 21, 2003
J)
j0
( )
r f R(t) R(t)
Poulard et al.
(11)
then the integral in eq 7 with J given by eq 11 becomes
dV j0 ) dt R
∫0Rf(Rr )2πr dr ) -2πRj0∫01f(x)x dx
(12)
that is, the rate of change of the drop’s volume is proportional to the drop’s radius, as already noticed by Deegan.13 If, as suggested by experiments, we assume R ∝ (t0 t)y and θ ∝ (t0 - t)x, the exponents x and y are linked by the relation
2y + x ) 1
Figure 1. Contact angles versus evaporation rate for alkanes in corresponding states, from left to right: nonane, octane, heptane, hexane.
(13)
The expression of the evaporation rate proposed by Deegan (eq 3) or a regularized form of it with the general shape (eq 11) is valid only if the process is controlled by diffusion and stationary. If convection takes place, which can be the case at late times because the receding velocity is large, or if the evaporation rate is limited by any other process, different shapes for J will result, and the relation in eq 13 will not be obeyed. For example, if J is given by
J ) j0g
(Rr )
(14)
then
∫0Rg(Rr )2πr dr ) -2πR2j0∫01g(x)x dx
dV ) -j0 dt
(15)
Now, the rate of change of the volume scales such as the square of the radius, and the relation becomes
y+x)1
(16)
Note that if J is constant one obtains x ) y ) 0.5, which was the trend observed9,10 at the end of the life of octane drops. However, in the general case, only relations between x and y can be obtained by the present analysis, but no information on x and y separately can be obtained. III. New Experiments The aim of the new experiments was to provide bases for answering the still open questions: (i) Which are the parameters controlling the value of the dynamic contact angle? Evaporation surely plays a role (the contact angle for nonvolatile liquids would be zero), but which one, and are there other parameters? Does the thickness of the wetting film around the drop play a role? (ii) An increase of x at late times has been observed with octane. Is it a general feature, and if so, does it bear any relation to a change in the evaporation rate J(R,r,t)? (iii) Is there any specificity with the behavior of water drops which could explain the anomalous value of the exponent y? Therefore, the experiments were developed in the following directions: (i) a study of the dependence of contact angle on evaporation rate in alkanes; (ii) a study of drop shape and wetting films around the drops; (iii) a study of the alkane drops just before vanishing; (iv) a systematic study of water on oxidized silicon wafers. For thin enough silica layers (∼2 nm) water wets the wafers, and the optical contrast is better than that on mica,
Figure 2. Some profiles of an evaporating heptane droplet from (t0 - t) ) 0.012 s (bottom) to 0.056 s (top), recorded and well fitted by spherical caps (continuous lines).
allowing us to measure the contact angle and to have good pictures of the contact line. A. Dependence of Contact Angle on Evaporation Rate. To compare contact angles for different liquids, one has to define “corresponding states”. Assuming that J does not depend explicitly on time, eq 9 suggests that “corresponding states” could be obtained for a given value of j0(t0 - t), at least for liquids belonging to a given chemical series, like the alkanes. The value of j0(t0 - t) will be chosen in order to have all the drops in the range where x is small, that is, a situation where the assumption of stationarity is plausible and where the models work well. Obviously, the late time behavior just before drop vanishing must be excluded. The values of the contact angle at a time (t0 - t) such that j0(t0 - t) ) 3 × 10-6 cm2, which corresponds to (t0 t) ) 3 s for nonane (j0 ) 10-6 cm2 s-1), are plotted in Figure 1 as a function of j0. The substrates were silicon wafers bearing a thick silica layer, cleaned with detergent, rinsed several times with distilled water, and dried on the spin coater with a jet of methanol. Reproducible data were obtained with that procedure. The contact angle vanishes for j0 ) 0 and increases with j0. The best fit with a power law might suggest an exponent close to 0.5. B. Drop Shape and Wetting Films with Alkanes. A systematic study of drop shape and wetting films has been performed on the same substrates as the ones above. The thickness profiles of the drops are always satisfactorily fitted with spherical caps, till close to the end of the drop’s life (see Figures 2 and 3 for heptane). A film is visible around receding hexane drops (see Figure 4), which means that it is relatively thick, 20 nm or more. The film is not seen with heptane (see Figure 3) nor with heavier alkanes; therefore, it is thinner, 10 nm or less, which is more or less the limit of detection on our wafer. The structure of the film left in Figure 4 provides an indirect evidence of a significant hydrodynamic flow inside
Freely Receding Evaporating Droplets
Figure 3. Heptane droplet on an oxidized silicon wafer at (t0 - t) ) 0.296 s. No film is visible around the drop.
Langmuir, Vol. 19, No. 21, 2003 8831
Figure 5. log-log plot of the drop radius (in millimeters) versus the time interval (t0 - t) (in seconds) for hexane (diamonds, y ∼ 0.48) and heptane (squares, y ∼ 0.50) on an oxidized silicon wafer.
Figure 4. Hexane droplet on an oxidized silicon wafer at (t0 - t) ) 0.380 s. The film visible ahead of the contact line keeps the fingerprint of a hydrodynamic flow inside the evaporating drop. To enhance the visibility of the film, hexane with traces of hexene (0.5%) has been used for the present picture.
the drop itself. Such hydrodynamic motions inside evaporating drops are well-known for drops where the contact line is pinned and the contact angle fairly large.12-14,19,20,32,40 In the present case, the drops are very thin and therefore the hydrodynamic flow is reduced. The main effect of that flow is to keep the spherical cap shape till the end (see Figure 2). In fact, this is why the model9,10 summarized in section 2.1 works so well. In the limit (t0 - t) f 0 for a very flat drop, the quadratic term is dominant in the drop’s shape, and this is the condition which mathematically leads to the relation 2y + x ) 1. Therefore, the asymptotic behavior of the mathematical model is just a spherical cap. A more complex effect of the hydrodynamic flow is to control the value of the contact angle at the edge of the drop. That point will be discussed more precisely in the last part of the paper. C. Late Time Experiments with Alkanes. For values of j0(t0 - t) < 10-6 cm2, which corresponds to (t0 - t) ) 1 s for nonane, a significant increase of the slope x is observed. With some dexterity and luck, and with the help of an ultrafast camera (FASTCAM-ultima 1024 model 1K with 250 frames per second), the late behavior of radius and angle was recorded on the same systems. It is confirmed that no significant change of the exponent y for the radius is observed for alkanes until the drop disappears; one example is given in Figure 5. The values of y are well defined and slightly less than 0.5. The behavior of the angle is more complex and is reported in Figure 6. There are clearly two regimes, (39) Vallet, M.; Berge, B. Electrowetting of water and aqueous solutions on poly(ethylene terephthalate) insulating films. Polymer 1996, 37 (12), 2465. (40) Wang, G. AIP Conference proceeding 197, Drops and bubbles; American Institute of Physics: 1988. (41) Xu, L.; Lio, A.; Ogletree, D. F.; Salmeron, M. Wetting and capillary phenomena of water on mica. J. Phys. Chem. B 1998, 102, 540. (42) Xu, L.; Salmeron, M. Scanning polarization force microscopy study of the condensation and wetting properties of glycerol on mica. J. Phys. Chem. B 1998, 102, 7210.
Figure 6. log-log plot of the contact angle (in radians) versus j0(t0 - t) (unit: 10-6 cm2) for hexane (squares), heptane (triangles), and nonane (diamonds). The straight lines have the slope 0.5.
separated by a threshold for the value j0(t0 - tthre) ∼ 2 × 10-6 cm2. Above the threshold, x is small, and the relation 2y + x ) 1 is obeyed. Below the threshold, x increases significantly, and the trend is that ultimately x ∼ y ∼ 0.5 and therefore y + x ) 1. It is remarkable that the crossover takes place approximately at the same value of j0(t0 - t) for all the alkanes investigated, while j0 changes by a factor of 34 from nonane to hexane (which would correspond to a shift of 1.5 on the x-axis of the log-log plot). The idea of “corresponding states” appears to have a real physical meaning, and the fact that j0 is a convenient scaling factor for the time dependence, as suggested by eq 9 in the case where J does not depend explicitly on time, is of importance for the data analysis. An increase of x means a faster decrease of the contact angle, which may have various origins. A plausible one was proposed previously10 in the framework of the molecular theory of wetting proposed by Blake.6 The additional decrease of angle at high receding velocities was related to the finite frequency of the molecular jumps at the contact line. The orders of magnitude were acceptable. A better assumption is that the shape of the evaporation term differs from eq 11, which can be due to any nonstationary process, like convection in the gas phase, or specific properties of the thin parts of drops and wetting films ahead, as discussed elsewhere.3,10 Whatever the process, it will lead to a flatter dependence of the evaporation rate J with r, and the rate of evaporation of the volume will no longer be proportional to the drop’s radius R. We have seen above that, in the limit where J is constant, and if the drop is a spherical cap, one expects
8832
Langmuir, Vol. 19, No. 21, 2003
Poulard et al.
Figure 7. Festoon like instabilities of the contact line of a water droplet evaporating on an oxidized silicon wafer, with a thin oxide layer. Relative humidity, RH ∼ 0.6.
Figure 9. Linear plot of the contact angle versus elapsed time for a water droplet on an oxidized silicon wafer. The origin of times is taken when the drop starts to recede. The inset represents the end of the drop’s life.
Figure 8. Last seconds of the life of a water drop on an oxidized silicon wafer. The first picture is taken when the contact angle is maximum (time 257.88 s on the next figure). Relative humidity, RH ∼ 0.6.
Figure 10. Linear plot of the time dependence of the radius at the end of the previous drop.
x ∼ y ∼ 0.5. This prediction is in excellent agreement with experiment. D. Water Drops on Oxidized Silicon Wafers. The evaporation of water drops on mica substrates has been studied by Deegan13 and other groups,36 with similar but intriguing results. The exponent y is larger than 0.5, but apparently the angle decreases with time, which means that x should be positive and that the relation 2y + x ) 1 is not obeyed. However, the contrast on mica is not good enough to allow precise measurements of the contact angle. Oxidized silicon wafers provide a good optical contrast and are water wettable if the silica layer is thin.21 They were cleaned using a “piranha” solution (1/3H2O2, 2/3H2SO4), which ensures a good reproducibility of the data for (t0 - t) larger than, typically, 3 s. At later times, the general features are reproducible, but the behavior may be shifted from one drop to the other. In the first range, (t0 - t) g 3 s, we measured y ∼ 0.60 and x ∼ 0.1. The values of y are similar to the ones observed on mica, and the positive value of x is compatible with the qualitative trend of a decrease of the angle seen on that substrate. Our former experiments on mica had shown a slight festoon-like structure of the receding contact line, but we did not trust the substrate enough to analyze the data at that time. The same, now well reproducible instability, is present on the silicon substrates and observable typically for (t0 - t) between 50 and 250 s. A characteristic picture is shown in Figure 7. The last part of the drop’s life is significantly more complex. The contact angle first decreases, as said above, but suddenly increases again. This is quite spectacular because the circular fringes which usually disappear regularly at the top of the drop start to reappear for a short time. Then the contact angle decreases steeply, and the drop disappears.
Figure 11. Schematic disjoining pressure (a, left) and interaction free energy (b, right) in thin water films on hydrophilic substrates.
Sometimes, the drop literally “explodes”, as in Figure 8. Such a behavior indicates that the vanishing time is no longer a relevant parameter for the dynamics in that range of times and that a log-log plot with the interval (t0 - t) as a time variable will severely distort the data at the end of the drop’s life. This is why we chose a linear representation for both angle (Figure 9) and radius (Figure 10). Water films on silica or quartz have been extensively studied by Derjaguin and co-workers (ref 15, p 341). Let Psat be the saturated vapor pressure at the temperature of the experiment, and let Pv be the actual vapor pressure. A film of thickness z is in equilibrium with the gas phase on top of it if the disjoining pressure Π(z) is given by15,28
Π(z) )
kT Psat ln $ Pv
(17)
The disjoining pressure isotherms in these systems are not monotonic, and thick β films may exist at the same pressure as that for thin R films (see Figure 11a). This is
Freely Receding Evaporating Droplets
Figure 12. Image of a very small water droplet placed on the wafer and evaporated slowly. Around the droplet, a ring of microdrops is progressively formed, by recondensation of the water molecules on the R film. The contact angle of the droplet is approximately 7°.
the origin of the complex patterns observed on water films at equilibrium just below saturation.17,27,29 In the present case, we are far from equilibrium; therefore, the effective value of the vapor pressure above the liquid surface depends on dynamical factors. In the same way, the thickness of the film left depends both on the shape of the disjoining pressure and on the value of the receding velocity. However, the sudden increase of the contact angle is clearly due to a transition from some “dynamical β” state to a thinner “dynamical R” state of the film at the edge of the drop. Note that for these wafers the oxide layer is thin, and the presence of the underlying silicon increases the disjoining pressure at relatively large film thicknesses, that is, for β films. This increases the height of the energy barrier for the β w R transition and stabilizes the thicker film (see Figure 11b). Apparently, the “dynamical” β film is present during the main part of the evaporation of the water drops, and the transition is induced at large receding velocities, either by convective processes in the gas phase or by hydrodynamic effects at the contact line. The maximum value of the angle is on the order of 8.3 × 10-2 rad (or ∼5°). This is the same order of magnitude as the one measured on initially very small droplets (see Figure 12) where the liquid evaporates and recondenses on the R film left. This process was observed by Derjaguin and co-workers (ref 15, p 341), and our observations are reminiscent of the ones by Lipson17,27 and Salmeron on mica.42,41 The evaporation of these small droplets is abnormally slow compared to that of the ones with a larger initial radius, as noticed by previous authors.17,27,41,42 IV. Data Analysis We can now reconsider the questions asked above: (i) The parameters controlling the value of the contact angle are clearly the evaporation rate j0, as shown by the experiments on alkanes, but also the thickness of the wetting film ahead of the drop’s edge, as observed with water. That part needs further comments. (ii) The long time behavior of the angle is clearly related to a change in the structure of J(r), and therefore, it is well understood qualitatively. More precise conclusions would require complex numerical analyses. (iii) The water drops are quite specific. The evaporation is slow, and the contact angle is relatively large. Further discussion is needed to see which behaviors are specific for water on a hydrophilic substrate and which ones are simply the consequence of the low velocities and “large” receding angles observed during the main part of the drop’s life. A. Coherent Picture in the Case of Alkanes. The numerical model without hydrodynamic flow (section 2.1) accounts well for the observed values of x and y in the
Langmuir, Vol. 19, No. 21, 2003 8833
range where the evaporation rate is mainly controlled by diffusion (eqs 3 and 4), but it gives no information on the actual value of the contact angle. Many models have been proposed for the calculation of the contact angle of evaporating liquids in stationary situations, usually in the presence of applied thermal gradients.1,12-14,16,22,24,26,30,34 Streamlines inside drops have been drawn by some authors.1,19,22,29,32,40 The contact angle of these drops is controlled by the hydrodynamic flow toward the contact line to replenish the fluid at the stationary edge. Even if the present system is essentially isothermal and not stationary, it is clear that the parameter controlling the value of the contact angle is still the hydrodynamic flow toward the edges, needed because they evaporate faster than the main drop. More precisely, one expects ∇⊥J(r,t) to be the relevant parameter. Moreover, ahead of the edge, the hydrodynamic solution has to match with a thin wetting film, as noticed by Hocking.22 Here, the specific properties of the wetting film, that is, the disjoining pressure, come into play. We can therefore understand (i) why the contact angle increases with evaporating rate in alkanes (whatever the shape of J(r,t) is, J(r,t) is proportional to j0) and (ii) how it depends qualitatively on the wetting film ahead of the drop’s edge (the angle increases if the film is thinner).22 One would expect the thin film thickness to appear in a logarithmic term22 and to have a weak influence on the dynamics. However, the present situation is nonstationary and therefore more complex, because the thickness of the film left depends also on the receding velocity. More precise conclusions require numerical calculations.2 Afterward, a comparison with the data obtained with alkanes will allow us to estimate precisely the relative role of the film thickness and the evaporation rate in the value of the receding contact angle. Experiment suggests that in water the dynamics is especially sensitive to the thickness of the thin film. Although the evaporation rate is low, between nonane and octane, the contact angle is significantly larger. The β film left is clearly much thinner than the films left by alkanes, even on a substrate where the underlying silicon enhances the wettability. In conclusion, the parameters controlling the contact angle are well identified and their effect is qualitatively understood. Numerical calculations are expected to give precise answers in the case of alkanes. The situation is probably more complex with water. B. Contact Line Instability. The festoon-like instability present in water in a large range of times has been further studied. The picture is that, in some places, the contact line recedes more slowly and tends to leave a thicker, transitory fingerprint on the substrate. The instability disappears at larger velocities. The same pattern was seen furtively in heptane at the beginning of the receding motion, where the contact angle is comparable to the one of water and the receding velocity is in the same range of values. What is specific in water is merely the wide range where the instability is observed. In hexane, fingers were also observed, but they were much thinner and looked like filaments literally “ejected” on the substrate.9 (i) The picture in hexane is reminiscent of the instabilities observed in electrowetting,35,39 which are due to the large value of the electric field at the contact line. Here, the evaporation rate replaces the electric field, and a specific stability analysis is required to determine whether or not the contact line may be unstable in the evaporation case.
8834
Langmuir, Vol. 19, No. 21, 2003
(ii) The festoon-like structure would rather indicate the presence of surface tension gradients in the vicinity of the contact line. Fingering instabilities are frequently observed during the spreading of volatile liquids, and they are due to thermal gradients developing at the drop’s edge.3,37 These gradients fade out with time, mainly due to hydrodynamic flow inside the drop, and they are not expected to survive in the receding motion. However, the larger value of the evaporation rate at the contact line could maintain locally a colder ring, not reached by the main hydrodynamic flow. In that case, an outwardly directed surface tension gradient exists in the vicinity of the contact line. If for any reason a transverse modulation of the drop thickness takes place, the thicker parts will recede more slowly, or even spread out, as in hexane. Such a modulation could be sustained or amplified by at least two mechanisms, which are not mutually exclusive. The common assumption is that the hydrodynamic flow does not reach the vicinity of the contact line and that the local temperature is controlled by diffusion. Then the situation is reminiscent of the one studied experimentally by Fanton18 and theoretically by Hosoi and Bush,23 where an alcohol-water mixture evaporates in a clock glass, except that here a thermal gradient replaces the concentration gradient. The starlike instability observed at the crossover between the main liquid in the glass and the thin film climbing on the walls was explained in the paper by Hosoi and Bush23 assuming the following: (i) The local concentration is controlled by diffusion and evaporation; therefore, the free interface is depleted in the more volatile compound, with the thicker parts being more depleted. (ii) The relaxation of the interface is much slower than the diffusive process. The conclusion is that if some perturbation in the film thickness shows up, thicker parts will have a larger surface tension. The surface tension gradient will cause a flow toward the thicker parts, amplifying the thickness perturbation. The same analysis can be used here. The main liquid corresponds to the drop, which is connected to a thin film through a wedge which is not reached by the hydrodynamic flow. Let us assume the following: (iii) The local temperature is controlled by diffusion and evaporation; therefore, the free interface is colder than the substrate, with the thickest parts being the coldest. (iv) The relaxation of the interface is much slower than the thermal diffusive process. Therefore, a transverse modulation of the drop thickness in the wedge will be amplified, either because the surface tension is larger in the thicker parts, which drives liquid to them, or because the evaporation rate j0(T) is lower there, with the same result. We do not know yet if the instability observed with hexane is the same as the one observed with heptane and water. The basic feature is always the large value of J at the edge of the drop. If it is enough to make the contact line unstable, or if a surface tension gradient is needed, is still an open question. Even if the possible mechanisms of the instability are identified, no information on the value of the wavelength or on the role of contact angle and possibly receding velocity is obtained for the moment. Contrary to the case of the alcohol-water mixture,18,23 the slope of the interface is determined by the hydrodynamic flow, that is, by the evaporative process itself.
Poulard et al.
Therefore, a more elaborate analysis is needed to choose between the possible mechanisms and to obtain a reliable prediction on the interplay between the various parameters.2 C. Specific Behavior of Water. A part of the strange behavior of water (well visible festoon-like instability and nonmonotonic change of the contact angle) is at least qualitatively understood as being related to the specific shape of the disjoining pressure isotherm, which leads to relatively large contact angles despite the low evaporation rate. The anomalous value of the exponent y is as yet not accounted for. Experiment shows that the drop’s shape is well fitted by a spherical cap. Therefore, one may write
∫0RJ(r,t)2πr dr
dV )dt
π V = R3 θ 4
(18) (19)
In water, where all the processes are slow (we do not address here the last seconds of the drops’ life), one may logically assume that J does not depend explicitly on time. Then, the integral on the right hand side of eq 18 is a function of R only.
dV ) -j0Ψ(R(t)) dt
(20)
(i) If Ψ(R) scales such as R, the relation 2y + x ) 1 results. (ii) If Ψ(R) scales such as R2, the relation becomes y + x ) 1. (iii) If Ψ(R) scales such as Rz, the relation becomes y(3 - z) + x ) 1. All our data with water are compatible with a value of z ∼ 1.5. One might say that giving a value for z does not provide any better understanding of the physical problem, which is absolutely true. The aim of the present discussion is to show that the problem to be solved is to find the correct expression for the local evaporation rate J(r,t). This is still to be done. V. Conclusion The evaporation rate of drops of pure, completely wetting liquids is investigated experimentally and theoretically in the situation where no pinning of the contact line is occurring. It is shown that the observed dynamics is controlled by the volatility of the liquid, as expected, but also by the properties of the wetting film left on the substrate. The receding contact angle increases with the volatility and decreases if the thickness of the wetting film increases. Just before the drops disappear, the evaporation rate flattens out and becomes asymptotically constant, which explains the stronger decrease of the contact angle in that range of times. Contact line instabilities may be observed at low receding velocities, and their possible origins are identified. Most of the specific behavior of water is simply the result of a low evaporation rate and a very thin wetting film. However, the anomalous value of the exponent y for the radius of evaporating water drops is still to be explained, even if it is clearly due to a specific structure of the evaporation rate. Acknowledgment. We thank Martine Benamar, Areski Boudaoud, and Eric Sultan for valuable discussions. LA030162J
