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Evaporation of sessile liquid droplets
Colloids and Surfaces A: Physicochem. Eng. Aspects 291 (2006) 191–196
Evaporation of sessile liquid droplets G. Gu´ena a , C. Poulard a , M. Vou´e b , J. De Coninck b , A.M. Cazabat a,∗ a
b
Laboratoire de Physique Statistique de l’ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France UMH, Laboratoire de Mod´elisation Mol´eculaire, Materia Nova, 1 avenue Copernic, B-7000 Mons, Belgique Received 9 March 2006; received in revised form 10 July 2006; accepted 12 July 2006 Available online 15 July 2006
Abstract The evaporation of sessile liquid droplets on a completely wetted substrate is a complex dynamical process. Available models assume that the evaporation flux is controlled by the stationary diffusion of the liquid molecules in the atmosphere, and that the interface is in equilibrium with the gas phase just above it. This assumption must be reconsidered at the moving contact line, but the description of the thin edge of the drop is still an open question. The present paper reports experiments performed with alkanes on bare and grafted silicon wafers. It is shown that the short range part of the interaction plays no role on the dynamics, but provides an indirect evidence of the presence of a mesoscopic evaporating film left on the substrate ahead of the receding contact line. © 2006 Elsevier B.V. All rights reserved. JEL classification: 68.03.-g; 68.08.-p; 68.15.+e Keywords: Moving contact line; Evaporating drops
1. Introduction Films evaporating on heated substrates play a major role in heat exchangers and have been the subject of extensive, theoretical as well as experimental [1–10] studies. The evaporation flux is controlled by the imposed temperature difference T = TS − TG between the substrate and the atmosphere far away. The temperature at the free interface TI differs both from the one of the substrate TS and the one of the surrounding gas TG . A similar situation would be when a gas flow is imposed parallel to the free liquid interface [11]. Things are less clear if no temperature difference (or gas flow) is imposed [12–17]. Although thermal effects are present during evaporation, they do not necessarily play a role. A well known example is the slow evaporation of “coffee drops” [15–17]. Here, the evaporation flux is controlled by the stationary diffusion of the water molecules in the gas phase, the process is to first approximation isothermal, and the liquid interface is in equilibrium with the gas phase just above it [17,18].
∗
Corresponding author. Tel.: +33 1 42 34 69 52; fax: +33 1 42 34 69 50. E-mail address: [email protected] (A.M. Cazabat).
0927-7757/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2006.07.021
A theoretical analysis taking into account both thermal effects and diffusion in the gas phase has been recently provided for a flat film, evaporating on a horizontal substrate in a quiescent atmosphere without imposed temperature difference [19]. In that case, diffusion is the dominant process as soon as the film thickness is larger than the mean free path of the evaporating molecules in the gas phase (say, a few tens of nm). Then the interface is in equilibrium with the gas phase just above it (“diffusion model”). For thinner films, evaporation is controlled by the transfer of molecules across the interface, which is no longer at equilibrium with the contacting gas phase (“thermal model”). In the specific case of short alkanes (hexane to nonane) and water at room temperature, the validity of the “diffusion model” is wider, especially for nonane and octane, where diffusion should be dominant at any thickness. Hexane is somewhat at the verge. Heptane and water are intermediate. In the case of moving interfaces, the problem is largely open, noticeably because air entrainment may take place in the gas phase. From the experimental point of view, the “thermal model” is reported to describe satisfactorily the stationary behaviour of dip-coated films of volatile liquids in the complete wetting case, for withdrawing speeds 0.1–2.5 mm/s [20]. On the other hand, the “diffusion model” is customarily used to describe the
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evaporation of macroscopic sessile droplets [15–18,21], even in complete wetting, where thin precursor wetting films might play a role [22–25]. Here, the receding velocities are rather in the range 0–0.2 mm/s. The present paper reports two sets of experiments: (1) a first series of experiments on sessile drops of alkanes evaporating on bare oxidized silicon wafers. This duplicates our previous work, but ensures identical surroundings and therefore relevant comparison with the second series; (2) a second series, where hydrophobic layers have been grafted on the silica surface. While theoretical models are still missing, the mere experiment might suggest in which way the thin edge of wetting drops plays a role in the dynamic processes. 2. Materials and methods The liquids used are alkanes, from nonane to hexane (Aldrich, >99% pure). The substrates are silicon wafers, bearing typically 2 nm of natural oxide at the surface, supplied by Siltronix. As received, and if no specific chemical treatment is used, the silica surface is only partially hydroxylated and barely water-wettable. As it appears to be quite reproducible, we did not try to alter the ratio between silanol groups and siloxane bridges. Any reasonable cleaning of these “bare” wafers gives a complete wetting and reproducible results with the four alkanes. We also used hydroxylated surfaces incubated during 24 h in hexamethyldisilazane (“HMDZ wafers”), thereafter bearing a dense grafted layer of trimethyl groups. The critical surface tension of these grafted surfaces for the series of alkanes is 22.2 × 10−3 N/m [26], i.e., they are wetted by octane (surface tension 21.8 × 10−3 N/m at 20 ◦ C) and lighter alkanes, but not by nonane (surface tension 24.2 × 10−3 N/m at 20 ◦ C). Adsorption isotherms of octane on bare and grafted wafers have been recorded previously and provide relevant information for the present study [26,27]. The thickness z of the octane film is measured by ellipsometry as a function of the relative vapour pressure PV /Psat . The disjoining pressure is Π(z) =
kT PV log Psat
where the volume of the octane molecule is 270 × 10−30 m3 . The initial spreading parameter S0 : ∞ S0 = Π(z)dz 0
is the integral of the disjoining pressure (z) over the whole range of thickness z. The adsorption isotherms z (PV /Psat ) are quite linear for very thin films (PV /Psat ≤ 0.7) with an initial slope of the order of 1 nm on bare wafers and less than 0.5 nm on grafted ones, where the values are scattered. S0 is found to be 21 × 10−3 N/m on the bare substrates, and 7 × 10−3 N/m on the grafted ones, a difference which is due to the short range part of the disjoining pressure [26,27]. In the long range part, the isotherms agree well
with a van der waals shape, calculated for the 4-layers system air–liquid octane–silica–silicon [28,29]. The thickness of the trimethyl layer is of the order of 0.3 nm and does not play for film thickness larger than 1 nm, typically. Drops with controlled volume are deposited on the solid substrate. The radius R(t) of the wetted spot increases first, because the liquid wets the substrate, then goes to a maximum, and later on decreases to zero, because of the evaporation going on. The drop disappears at a time t0 which is carefully recorded. The dynamics is followed under microscope in normal atmosphere, with mere protections against air draft, and special attention is given to the receding motion. A low magnification is used for the measurement of the radius, which is done with the entire drop in the field of the microscope. The measurement of the contact angle θ(t) from equal thickness interference fringes at the drop’s edge requires a higher magnification. This is a potential cause of artefacts. Anticaloric filters are required, to avoid heating due to the focussed microscope beam. We also use “extra long working distance” objectives to keep the surface of the objective far from the drop, and reduce the confinement of the atmosphere on top of it. However, shifting the substrate to keep the moving drop’s edge in the field of the microscope has to be done smoothly, not to induce convection in the gas phase. From that point of view, the case of alkanes is favourable, because they are heavier than air, therefore, no spontaneous buoyancy driven convection is expected (in contrast with water [30]). It is worth noting that the “contact angle” obtained from interference fringes corresponds to the range of thickness 0.2–1 m. 3. Data analysis, available models and open questions The evaporation of sessile drops can be usefully compared with the one of aerosols. The slow, isothermal evaporation of millimetric or micrometric aerosols droplets in normal atmosphere is controlled by the stationary diffusion of the liquid molecules in the gas phase [31–35]. The concentration c of these molecules obeys c = 0 in the atmosphere, with a constant value c = csat at the droplet surface, and another constant value at infinity. The evaporation rate per unit area of the drop is proportional to the inverse of the drop radius R(t), the rate of evaporation of the entire drop scales like R(t), and therefore R(t) is found to decrease with time as [31–35]: R(t) ∝ (t0 − t)0.5 For sessile droplets, the solution of the stationary diffusion equation c = 0 is more complex. Lot of mathematical functions with zero Laplacian and various boundary conditions can be found in textbooks of electromagnetism [36], as the Poisson equation for the electrostatic potential Φ writes Φ = 0. Then the available electrostatic formulae can be used for the diffusion problem by merely replacing the electrostatic potential Φ by the concentration c. If specific problems with moving interfaces and thin films are ignored, an evaporating sessile droplet is equivalent to a biconvex conducting lens put at constant potential, with another constant value of the potential at infinity. The corresponding formulae are rather intricate [15–18]. However, in
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the case of complete wetting, the dynamic contact angle is very small, of the order of one degree, and the electrostatic analogy is a flat disc, for which the formulae are much simpler. The evaporation flux per unit area at the drop surface is proportional to the concentration gradient which corresponds to the electric field. It can be written as: J(r) = √
j0 R2 − r 2
where R(t) is the disc radius and r the distance to disc centre [11,15,36]. The evaporation constant j0 depends on liquid and temperature. Integrated over the (flat) drop, this gives the rate of change of the volume V of the drop: dV = −2πj0 R(t) dt For a spherical cap with constant (small) contact angle, V∼ = (πR3 θ)/4, and the result is the same as for aerosols: R(t) ∝ (t0 − t)0.5 . This explains why the experimental data R(t) and θ(t) will be plotted in the following in a log–log representation with the interval t0 − t as the time variable. The first problem with wetting sessile drops is how to treat hydrodynamically the receding motion of the contact line. The second one in the “diffusion model” is the divergence of the evaporation flux at the macroscopic moving edge r → R(t). One sees immediately that the sensible parameter will be the “contact angle”, or more rigorously the interface slope close to the macroscopic contact line. 3.1. The receding contact line Let us address the first problem. If the liquid were nonvolatile, a flat film would be left on the substrate during the receding motion, which thickness would scale as Ca2/3 , as shown in a stationary analysis due to Landau and Levich [37]. Here, Ca = (η|U|)/γ is the capillary number, U = dR/dt the contact line velocity, η the viscosity of the liquid, and γ the surface tension. In this case, the dynamic contact angle is zero, and the relevant parameter which controls the crossover between the film and the bulk liquid is the curvature of the macroscopic interface. For a volatile liquid, the dynamic contact angle is finite [3,6,13,22–25]. Then the question is to know if the macroscopic wedge, where the angle θ is defined and measured, recedes on an extended film, just like in the Landau–Levich problem, or if evaporation truncates the film at a negligible distance of the macroscopic contact line. An extended film of micrometric thickness is present in the experiment by Garoff and co-workers [20], where a solid cylinder is pulled from a bath at a constant velocity. The characteristic velocities of our receding droplets are not much less than the ones used by Garoff and co-workers. However, the boundary conditions for the gas phase are different, and the evaporation dynamics may change significantly in case of air entrainment. No such film was previously observed with droplets [22–25].
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3.2. Divergence of the evaporation flux It is clear that the way to treat the divergence of the evaporation flux depends on whether or not there is a film left on the substrate. – If such a film does exist, then the evaporation flux takes a finite value at the crossover towards the film, i.e., at the apparent contact line r = R(t). The (non-physical) divergence has to be corrected farther, i.e., at the very edge of the film, and the regularization procedure there has no incidence on the measured contact angle at the crossover, nor on the observed dynamics for radius R(t) and contact angle θ(t). – On the contrary, if the film is so short that it does not play the role of an “intermediate length scale”, the regularization procedure takes place at the contact line, and controls the value of the contact angle, and therefore the observed dynamics for radius R(t) and contact angle θ(t). (a) One may consider that the free moving interface is not at equilibrium, in which case evaporation might be controlled by the transfer of molecules across the interface [19]. An interesting feature of the “thermal model” is that the evaporation flux does not diverge. Ignoring vapour recoil and Marangoni flow, the evaporation flux reads [1,4,19]: J(z) =
JT z + HT
where z is the local thickness, and JT scales as the imposed temperature difference T. The thickness HT depends only on physical parameters of the liquid and temperature [1,4,19], and is of the order of a micrometer for the alkanes considered. Even if the model does not hold if the temperature difference is a local, self consistent parameter, it contains a characteristic thickness which is a plausible threshold allowing the evaporation to level off, when the interface is far from equilibrium. (b) The opposite approach is to retain the assumption of local equilibrium at the free interface, in which case the “diffusion model” should be valid for the alkanes considered. Then, the argument is that thin wetting films evaporate more slowly than thick ones, due to the interactions with the substrate [13,14,23–25]. This also will lead to a levelling of the evaporation flux, because j0 scales like PV . However, even if the range of molecular forces is a few tens of nm, their incidence on the evaporation flux is significant only for very thin films, less than 1 nm, as evidenced by adsorption isotherms [23]. In that case, the thickness where the evaporation levels off is rather of the order of nm. 3.3. Available models for evaporating droplets As previously mentioned, the models used to describe the evaporation of sessile droplets assume a diffusion controlled process. In the non-wetting case, the contact line is pinned and the
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divergence of the evaporation flux at the drop edge is not critical [15–17]. In the complete wetting case, regularization is needed. A model has been recently proposed by Poulard et al. [25], where the evaporation flux is assumed to saturate at a distance ≈ a/θ 2 before the contact line, therefore at a√thickness hT ≈ a/θ. The molecular length a is given by a ≈ A/6πγ, where A is the modulus of the effective Hamaker constant. The thickness hT is of the order of 10 nm. The characteristic distance is obtained by balancing curvature and disjoining pressure, which is the way to calculate the interface shape at the very edge of a film [38,39]. It does not correspond to a decrease of evaporation due to disjoining pressure, nor to a crossover towards a film with negligible curvature, but provides a plausible thickness where for any reason the flux will level off. The predictions of the model are excellent [25,30], which means a saturation value J(r → R(t)) ≈ √ √ that introducing j0 / 2R ≈ θj0 / 2Ra in the vicinity of the macroscopic contact line is a correct procedure. However, this agreement does not prove that the model describes correctly the drop’s edge. Noticeably, the contact angle θ 0 at the maximum extension of the drop is predicted to be√related to the maximum value R0 of the radius by θ03 ≈ 3ηj0 /γ 2R0 a. The formula gives correctly the order of magnitude for the angle, however the predicted dependence between θ 0 and R0 is not obeyed [25]. As a matter of fact, the way used to calculate [38,39] excludes the presence of a film left on the substrate ahead of the contact line. As we shall see below, this is against experimental evidence. 4. Experimental results and discussion 4.1. Experimental data The measured radiuses of octane drops have been reported in a Log-Log representation on Fig. 1 for bare wafers, and Fig. 2 for grafted ones. The initial value of the drop volume is varied by a factor 100, which gives enough information while avoiding
Fig. 2. Log–log plot of recorded radius R(t0 − t) vs. time before vanishing t0 − t for octane drops deposited on HMDZ coated wafers. Range of deposited volumes: V = 0.02 L − 3 L. The continuous line has a slope: y = 0.48. The measured slopes for the receding part of the dynamics are in the range 0.48 ± 0.005.
gravity effects (for drops much larger than the capillary length) or loss of information for very small drops. It is clear that a power law R(t) ∝ (t0 − t)y is well obeyed during the main part of the retraction. The exponent y is found to be 0.48 ± 0.01 both on the bare wafers and on the grafted ones. Drop profiles agree well with spherical caps when the drop radius is of the order of the capillary length or less (1.8 mm for alkanes). For heptane, the exponent is the same, within experimental accuracy. The contact angles measured on small drops (V = 1 L) are reported on Fig. 3 for the two substrates. They are small, of the order of 10−2 rd. There is a limited range in the figure where a power law: θ(t) ∝ (t0 − t)x is obeyed. This is not surprising, because the volume of a thin spherical cap can be written as: πR3 θ V ∼ = 4 If the diffusion model is obeyed, then dV/dt ∝ R(t), which means that the angle has to follow a power law, which exponent x is such that 2y + x = 1. Actually, the relation is well obeyed for the drops investigated, considering the scattering of the experimental data (x = 0.03 ± 0.01 for the grafted wafer, x = 0.06 ± 0.01 for the bare wafer). Note that the value of the contact angle depends on the initial volume of the drop [25], and that a comparison can be done only on drops of same initial volume. 4.2. Discussion
Fig. 1. Log–log plot of recorded radius R(t0 − t) vs. time before vanishing t0 − t for octane drops deposited on a bare wafer. Range of deposited volumes: V = 0.04 L − 4 L. The continuous line has a slope: y = 0.48. The measured slopes for the receding part of the dynamics are in the range 0.48 ± 0.005.
It is clear that even for octane, which is close to a wetting transition on grafted surfaces, the behaviour is quite similar on both substrates. This means that the short range part of the interaction
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Fig. 3. Log–log plot of the contact angle θ(t0 − t) measured for two octane drops of same volume V = 1 L on bare and coated wafer, respectively. The slope x is ∼0.03 ± 0.01 on the grafted wafer and ∼0.06 ± 0.01 on the bare wafer, but considering the scattering of the points, the difference is too small to be significant. In both cases, the relation 2y + x = 1 is obeyed within experimental accuracy. The vertical lines correspond to the maximum extension of the drops.
Fig. 4. Picture of the receding contact line of an octane drop at high magnification. The droplets left on the substrate evaporate more slowly.
plays no role, at least for alkanes. Therefore, what is relevant for the dynamics of evaporation is the shape of the interface for thicknesses of the order of a hundred of nanometres, and not at the very edge of the drop. The grafted substrates provide the answer: films are visible at the drop’s edge on these surfaces, see Fig. 4. The films are longer with hexane, shorter with octane, and significantly thinner than 90 nm, which corresponds to the first black fringe. They are not seen directly, but indirectly revealed because they dewet on the grafted substrates, at the place where they become thin enough to feel the unfavourable short range interaction due to the trimethyl layer. Thicker rims and strings of small drops become observable. We do not see these films on bare wafers, because the film wets the substrate whatever its thickness [22–25]. It is not easy to measure the thickness of these films at a moving edge. As expected, they are much thinner than the ones
Fig. 5. Late time behaviour of a heptane drop. Log–log plot of: (a) radius R(t0 − t) in mm, (b) contact angle θ(t0 − t) in rd, versus time interval t0 − t in sec. While the behaviour of the radius follows a power law (y = 0.47) over the whole range of times, this is not the case for the contact angle. In the last tenth of second, the slope x increases from 0.08 (2y + x ≈ 1) to a value close to 0.5. In the last hundredth of second, there are not enough fringes to define properly a contact angle. This is why the measurements stop earlier for the angle than for the radius.
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observed by Garoff and co-workers [20], first because the velocities (more rigorously the capillary numbers) are less, but also because there is no hydrodynamic flow in the surrounding atmosphere. Note that there is a change in the dynamics of the contact angle at the very end of the drop’s life, see Fig. 5 for heptane. As there is no change in the dynamics of the radius, this means that the relation 2y + x = 1 is no longer obeyed, therefore that the “diffusion model” is no longer valid. Further experiments are needed to check the relative role of the increase of the contact line velocity, and the possible convection in the atmosphere. In any case, the measured contact angle is defined at the crossover towards a hydrodynamically controlled mesoscopic film, and not at a sharp edge [25]. An intermediate length scale must therefore be introduced in the forthcoming models. This is still an open problem. 5. Conclusion The study of evaporating sessile droplets on grafted surfaces, where the short range part of the interaction opposes wetting, allows us to answer at least some of the open questions concerning the dynamics of the process. The dynamic contact angle, measured at a typical thickness of the order of a fraction of m, is not controlled by the decrease in evaporation flux in ultra thin films. As a matter of fact, the apparent contact line recedes on a mesoscopic film, which thickness is probably a few tens of nanometres. The film is too thin to be seen directly under microscope, but indirect evidence is provided on grafted surfaces, where the thin part of the film dewets. The thicker rims or droplets become visible and delineate the edge of the continuous film ahead of the macroscopic contact line. Forthcoming models should therefore focus on the crossover between the receding wedge and the short, evaporating film left on the substrate. Acknowledgements This work has been partially supported by the European Community (Marie Curie and Objective 1 Phasing Out programs) and by the Minist`ere de la R´egion Wallonne. We gratefully acknowledge Josephine Conti for the preparation of the grafted substrates. The Laboratory of Statistical Physics of the Ecole Normale Sup´erieure is the UMR 8550 of CNRS, associated to the Universities Pierre et Marie Curie (Paris 6) and Denis Diderot (Paris 7). References [1] J.P. Burelbach, S.G. Bankoff, S.H. Davis, J. Fluid Mech. 195 (1988) 463.
[2] L.M. Hocking, Phys. Fluids 7 (1995) 2950. [3] D.M. Anderson, S.H. Davis, Phys. Fluids A7 (1995) 248. [4] A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys. 69 (1997) 932. [5] S.J.S. Morris, J. Fluid Mech. 411 (2000) 59. [6] S.J.S. Morris, J. Fluid Mech. 432 (2001) 1. [7] S.J.S. Morris, J. Fluid Mech. 494 (2003) 297. [8] P.C. Wayner, C.Y. Tung, M. Tirumala, J.H. Yang, J. Heat Transfer 107 (1985) 182. [9] I.Y. Kim, P.C. Wayner, J. Thermophys. Heat Transfer 10 (1996) 320. [10] S. Miladinova, S. Slatchev, G. Lebon, J.C. Legros, J. Fluid Mech. 453 (2002) 153. [11] J. Crank, The Mathematics of Diffusion, second ed., Oxford University Press, Oxford, 1975. [12] J.C. Truong, P.C. Wayner, J. Chem. Phys. 87 (1987) 4180. [13] M. Sujanani, P.C. Wayner, Chem. Eng. Commun. 118 (1992) 89. [14] P.C. Wayner, Langmuir 9 (1993) 294. [15] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Nature (London) 389 (1997) 827. [16] R.D. Deegan, Phys. Rev. E 61 (2000) 475. [17] R.D. Deegan, O. Bakajin, T.F. Dupont, G. Huber, S.R. Nagel, T.A. Witten, Phys. Rev. E 62 (2000) 757. [18] Y.O. Popov, T.A. Witten, Eur. Phys. J. E. 6 (2001) 211. [19] E. Sultan, A. Boudaoud, M. Ben Amar, J. Fluid Mech. 543 (2005) 183. [20] D. Qu, E. Ram´e, S. Garoff, Phys. Fluids 14 (2002) 1154. [21] H. Hu, R.G. Larson, J. Phys. Chem. B 106 (2002) 1334. [22] M. Cachile, O. B´enichou, A.M. Cazabat, Langmuir 18 (2002) 7985. [23] M. Cachile, O. B´enichou, C. Poulard, A.M. Cazabat, Langmuir 18 (2002) 8070. [24] C. Poulard, O. B´enichou, A.M. Cazabat, Langmuir 19 (2003) 8828. [25] C. Poulard, G. Gu´ena, A.M. Cazabat, A. Boudaoud, M. Ben Amar, Langmuir 21 (2005) 8226. [26] N. Fraysse, PhD thesis, University Pierre et Marie Curie (Paris 6), 1991. [27] P. Levinson, M.P. Valignat, N. Fraysse, A.M. Cazabat, F. Heslot, Thin Solid Films 234 (1993) 482. [28] J.N. Israelachvili, Proc. R. Soc. Lond. A 331 (1972) 39. [29] G.J. Hirasaki, in: N.R. Morrow (Ed.), Interfacial Phenomena in Petroleum Recovery, Dekker, New York, 1991, pp. 23–99. [30] C. Poulard, G. Gu´ena, A.M. Cazabat, J. Phys. Cond. Mater. 17 (2005) S4213. [31] J.C. Maxwell, “Diffusion” collected scientific papers, in: Encyclopedia Britannica, Cambridge, 1877. [32] I. Langmuir, Evaporation of small spheres, Phys. Rev. 12 (1918) 368. [33] N.A. Fuchs, in: R.S. Bradley (Ed.), Evaporation and Droplet Growth in Gaseous Media, Pergamon Press, London, New York, 1959. [34] A. Frohn, N. Roth, Dynamics of Droplets, Springer, Berlin, New York, 2000. [35] N. Roth, K. Anders, A. Frohn, Size and evaporation rate measurements of optically levitated droplets, in: Proc. 3rd Int. Congress on Optical Particle Sizing, 1993, p. 371. [36] J.D. Jackson, Classical Electrodynamics, second ed., Wiley, New-York, 1975. [37] L. Landau, B. Levich, Acta Physicochim. URSS 17 (1942) 42. [38] P.G. de Gennes, Wetting: statics and dynamics, Rev. Mod. Phys. 57 (V3) (1985) 827. [39] P.G. de Gennes, H. Hervet, Dynamique du mouillage: films pr´ecurseurs sur solide sec, C. R. Acad. Sci. Paris 299 (1984) 499.
Evaporation of sessile liquid droplets G. Gu´ena a , C. Poulard a , M. Vou´e b , J. De Coninck b , A.M. Cazabat a,∗ a
b
Laboratoire de Physique Statistique de l’ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France UMH, Laboratoire de Mod´elisation Mol´eculaire, Materia Nova, 1 avenue Copernic, B-7000 Mons, Belgique Received 9 March 2006; received in revised form 10 July 2006; accepted 12 July 2006 Available online 15 July 2006
Abstract The evaporation of sessile liquid droplets on a completely wetted substrate is a complex dynamical process. Available models assume that the evaporation flux is controlled by the stationary diffusion of the liquid molecules in the atmosphere, and that the interface is in equilibrium with the gas phase just above it. This assumption must be reconsidered at the moving contact line, but the description of the thin edge of the drop is still an open question. The present paper reports experiments performed with alkanes on bare and grafted silicon wafers. It is shown that the short range part of the interaction plays no role on the dynamics, but provides an indirect evidence of the presence of a mesoscopic evaporating film left on the substrate ahead of the receding contact line. © 2006 Elsevier B.V. All rights reserved. JEL classification: 68.03.-g; 68.08.-p; 68.15.+e Keywords: Moving contact line; Evaporating drops
1. Introduction Films evaporating on heated substrates play a major role in heat exchangers and have been the subject of extensive, theoretical as well as experimental [1–10] studies. The evaporation flux is controlled by the imposed temperature difference T = TS − TG between the substrate and the atmosphere far away. The temperature at the free interface TI differs both from the one of the substrate TS and the one of the surrounding gas TG . A similar situation would be when a gas flow is imposed parallel to the free liquid interface [11]. Things are less clear if no temperature difference (or gas flow) is imposed [12–17]. Although thermal effects are present during evaporation, they do not necessarily play a role. A well known example is the slow evaporation of “coffee drops” [15–17]. Here, the evaporation flux is controlled by the stationary diffusion of the water molecules in the gas phase, the process is to first approximation isothermal, and the liquid interface is in equilibrium with the gas phase just above it [17,18].
∗
Corresponding author. Tel.: +33 1 42 34 69 52; fax: +33 1 42 34 69 50. E-mail address: [email protected] (A.M. Cazabat).
0927-7757/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2006.07.021
A theoretical analysis taking into account both thermal effects and diffusion in the gas phase has been recently provided for a flat film, evaporating on a horizontal substrate in a quiescent atmosphere without imposed temperature difference [19]. In that case, diffusion is the dominant process as soon as the film thickness is larger than the mean free path of the evaporating molecules in the gas phase (say, a few tens of nm). Then the interface is in equilibrium with the gas phase just above it (“diffusion model”). For thinner films, evaporation is controlled by the transfer of molecules across the interface, which is no longer at equilibrium with the contacting gas phase (“thermal model”). In the specific case of short alkanes (hexane to nonane) and water at room temperature, the validity of the “diffusion model” is wider, especially for nonane and octane, where diffusion should be dominant at any thickness. Hexane is somewhat at the verge. Heptane and water are intermediate. In the case of moving interfaces, the problem is largely open, noticeably because air entrainment may take place in the gas phase. From the experimental point of view, the “thermal model” is reported to describe satisfactorily the stationary behaviour of dip-coated films of volatile liquids in the complete wetting case, for withdrawing speeds 0.1–2.5 mm/s [20]. On the other hand, the “diffusion model” is customarily used to describe the
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evaporation of macroscopic sessile droplets [15–18,21], even in complete wetting, where thin precursor wetting films might play a role [22–25]. Here, the receding velocities are rather in the range 0–0.2 mm/s. The present paper reports two sets of experiments: (1) a first series of experiments on sessile drops of alkanes evaporating on bare oxidized silicon wafers. This duplicates our previous work, but ensures identical surroundings and therefore relevant comparison with the second series; (2) a second series, where hydrophobic layers have been grafted on the silica surface. While theoretical models are still missing, the mere experiment might suggest in which way the thin edge of wetting drops plays a role in the dynamic processes. 2. Materials and methods The liquids used are alkanes, from nonane to hexane (Aldrich, >99% pure). The substrates are silicon wafers, bearing typically 2 nm of natural oxide at the surface, supplied by Siltronix. As received, and if no specific chemical treatment is used, the silica surface is only partially hydroxylated and barely water-wettable. As it appears to be quite reproducible, we did not try to alter the ratio between silanol groups and siloxane bridges. Any reasonable cleaning of these “bare” wafers gives a complete wetting and reproducible results with the four alkanes. We also used hydroxylated surfaces incubated during 24 h in hexamethyldisilazane (“HMDZ wafers”), thereafter bearing a dense grafted layer of trimethyl groups. The critical surface tension of these grafted surfaces for the series of alkanes is 22.2 × 10−3 N/m [26], i.e., they are wetted by octane (surface tension 21.8 × 10−3 N/m at 20 ◦ C) and lighter alkanes, but not by nonane (surface tension 24.2 × 10−3 N/m at 20 ◦ C). Adsorption isotherms of octane on bare and grafted wafers have been recorded previously and provide relevant information for the present study [26,27]. The thickness z of the octane film is measured by ellipsometry as a function of the relative vapour pressure PV /Psat . The disjoining pressure is Π(z) =
kT PV log Psat
where the volume of the octane molecule is 270 × 10−30 m3 . The initial spreading parameter S0 : ∞ S0 = Π(z)dz 0
is the integral of the disjoining pressure (z) over the whole range of thickness z. The adsorption isotherms z (PV /Psat ) are quite linear for very thin films (PV /Psat ≤ 0.7) with an initial slope of the order of 1 nm on bare wafers and less than 0.5 nm on grafted ones, where the values are scattered. S0 is found to be 21 × 10−3 N/m on the bare substrates, and 7 × 10−3 N/m on the grafted ones, a difference which is due to the short range part of the disjoining pressure [26,27]. In the long range part, the isotherms agree well
with a van der waals shape, calculated for the 4-layers system air–liquid octane–silica–silicon [28,29]. The thickness of the trimethyl layer is of the order of 0.3 nm and does not play for film thickness larger than 1 nm, typically. Drops with controlled volume are deposited on the solid substrate. The radius R(t) of the wetted spot increases first, because the liquid wets the substrate, then goes to a maximum, and later on decreases to zero, because of the evaporation going on. The drop disappears at a time t0 which is carefully recorded. The dynamics is followed under microscope in normal atmosphere, with mere protections against air draft, and special attention is given to the receding motion. A low magnification is used for the measurement of the radius, which is done with the entire drop in the field of the microscope. The measurement of the contact angle θ(t) from equal thickness interference fringes at the drop’s edge requires a higher magnification. This is a potential cause of artefacts. Anticaloric filters are required, to avoid heating due to the focussed microscope beam. We also use “extra long working distance” objectives to keep the surface of the objective far from the drop, and reduce the confinement of the atmosphere on top of it. However, shifting the substrate to keep the moving drop’s edge in the field of the microscope has to be done smoothly, not to induce convection in the gas phase. From that point of view, the case of alkanes is favourable, because they are heavier than air, therefore, no spontaneous buoyancy driven convection is expected (in contrast with water [30]). It is worth noting that the “contact angle” obtained from interference fringes corresponds to the range of thickness 0.2–1 m. 3. Data analysis, available models and open questions The evaporation of sessile drops can be usefully compared with the one of aerosols. The slow, isothermal evaporation of millimetric or micrometric aerosols droplets in normal atmosphere is controlled by the stationary diffusion of the liquid molecules in the gas phase [31–35]. The concentration c of these molecules obeys c = 0 in the atmosphere, with a constant value c = csat at the droplet surface, and another constant value at infinity. The evaporation rate per unit area of the drop is proportional to the inverse of the drop radius R(t), the rate of evaporation of the entire drop scales like R(t), and therefore R(t) is found to decrease with time as [31–35]: R(t) ∝ (t0 − t)0.5 For sessile droplets, the solution of the stationary diffusion equation c = 0 is more complex. Lot of mathematical functions with zero Laplacian and various boundary conditions can be found in textbooks of electromagnetism [36], as the Poisson equation for the electrostatic potential Φ writes Φ = 0. Then the available electrostatic formulae can be used for the diffusion problem by merely replacing the electrostatic potential Φ by the concentration c. If specific problems with moving interfaces and thin films are ignored, an evaporating sessile droplet is equivalent to a biconvex conducting lens put at constant potential, with another constant value of the potential at infinity. The corresponding formulae are rather intricate [15–18]. However, in
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the case of complete wetting, the dynamic contact angle is very small, of the order of one degree, and the electrostatic analogy is a flat disc, for which the formulae are much simpler. The evaporation flux per unit area at the drop surface is proportional to the concentration gradient which corresponds to the electric field. It can be written as: J(r) = √
j0 R2 − r 2
where R(t) is the disc radius and r the distance to disc centre [11,15,36]. The evaporation constant j0 depends on liquid and temperature. Integrated over the (flat) drop, this gives the rate of change of the volume V of the drop: dV = −2πj0 R(t) dt For a spherical cap with constant (small) contact angle, V∼ = (πR3 θ)/4, and the result is the same as for aerosols: R(t) ∝ (t0 − t)0.5 . This explains why the experimental data R(t) and θ(t) will be plotted in the following in a log–log representation with the interval t0 − t as the time variable. The first problem with wetting sessile drops is how to treat hydrodynamically the receding motion of the contact line. The second one in the “diffusion model” is the divergence of the evaporation flux at the macroscopic moving edge r → R(t). One sees immediately that the sensible parameter will be the “contact angle”, or more rigorously the interface slope close to the macroscopic contact line. 3.1. The receding contact line Let us address the first problem. If the liquid were nonvolatile, a flat film would be left on the substrate during the receding motion, which thickness would scale as Ca2/3 , as shown in a stationary analysis due to Landau and Levich [37]. Here, Ca = (η|U|)/γ is the capillary number, U = dR/dt the contact line velocity, η the viscosity of the liquid, and γ the surface tension. In this case, the dynamic contact angle is zero, and the relevant parameter which controls the crossover between the film and the bulk liquid is the curvature of the macroscopic interface. For a volatile liquid, the dynamic contact angle is finite [3,6,13,22–25]. Then the question is to know if the macroscopic wedge, where the angle θ is defined and measured, recedes on an extended film, just like in the Landau–Levich problem, or if evaporation truncates the film at a negligible distance of the macroscopic contact line. An extended film of micrometric thickness is present in the experiment by Garoff and co-workers [20], where a solid cylinder is pulled from a bath at a constant velocity. The characteristic velocities of our receding droplets are not much less than the ones used by Garoff and co-workers. However, the boundary conditions for the gas phase are different, and the evaporation dynamics may change significantly in case of air entrainment. No such film was previously observed with droplets [22–25].
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3.2. Divergence of the evaporation flux It is clear that the way to treat the divergence of the evaporation flux depends on whether or not there is a film left on the substrate. – If such a film does exist, then the evaporation flux takes a finite value at the crossover towards the film, i.e., at the apparent contact line r = R(t). The (non-physical) divergence has to be corrected farther, i.e., at the very edge of the film, and the regularization procedure there has no incidence on the measured contact angle at the crossover, nor on the observed dynamics for radius R(t) and contact angle θ(t). – On the contrary, if the film is so short that it does not play the role of an “intermediate length scale”, the regularization procedure takes place at the contact line, and controls the value of the contact angle, and therefore the observed dynamics for radius R(t) and contact angle θ(t). (a) One may consider that the free moving interface is not at equilibrium, in which case evaporation might be controlled by the transfer of molecules across the interface [19]. An interesting feature of the “thermal model” is that the evaporation flux does not diverge. Ignoring vapour recoil and Marangoni flow, the evaporation flux reads [1,4,19]: J(z) =
JT z + HT
where z is the local thickness, and JT scales as the imposed temperature difference T. The thickness HT depends only on physical parameters of the liquid and temperature [1,4,19], and is of the order of a micrometer for the alkanes considered. Even if the model does not hold if the temperature difference is a local, self consistent parameter, it contains a characteristic thickness which is a plausible threshold allowing the evaporation to level off, when the interface is far from equilibrium. (b) The opposite approach is to retain the assumption of local equilibrium at the free interface, in which case the “diffusion model” should be valid for the alkanes considered. Then, the argument is that thin wetting films evaporate more slowly than thick ones, due to the interactions with the substrate [13,14,23–25]. This also will lead to a levelling of the evaporation flux, because j0 scales like PV . However, even if the range of molecular forces is a few tens of nm, their incidence on the evaporation flux is significant only for very thin films, less than 1 nm, as evidenced by adsorption isotherms [23]. In that case, the thickness where the evaporation levels off is rather of the order of nm. 3.3. Available models for evaporating droplets As previously mentioned, the models used to describe the evaporation of sessile droplets assume a diffusion controlled process. In the non-wetting case, the contact line is pinned and the
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divergence of the evaporation flux at the drop edge is not critical [15–17]. In the complete wetting case, regularization is needed. A model has been recently proposed by Poulard et al. [25], where the evaporation flux is assumed to saturate at a distance ≈ a/θ 2 before the contact line, therefore at a√thickness hT ≈ a/θ. The molecular length a is given by a ≈ A/6πγ, where A is the modulus of the effective Hamaker constant. The thickness hT is of the order of 10 nm. The characteristic distance is obtained by balancing curvature and disjoining pressure, which is the way to calculate the interface shape at the very edge of a film [38,39]. It does not correspond to a decrease of evaporation due to disjoining pressure, nor to a crossover towards a film with negligible curvature, but provides a plausible thickness where for any reason the flux will level off. The predictions of the model are excellent [25,30], which means a saturation value J(r → R(t)) ≈ √ √ that introducing j0 / 2R ≈ θj0 / 2Ra in the vicinity of the macroscopic contact line is a correct procedure. However, this agreement does not prove that the model describes correctly the drop’s edge. Noticeably, the contact angle θ 0 at the maximum extension of the drop is predicted to be√related to the maximum value R0 of the radius by θ03 ≈ 3ηj0 /γ 2R0 a. The formula gives correctly the order of magnitude for the angle, however the predicted dependence between θ 0 and R0 is not obeyed [25]. As a matter of fact, the way used to calculate [38,39] excludes the presence of a film left on the substrate ahead of the contact line. As we shall see below, this is against experimental evidence. 4. Experimental results and discussion 4.1. Experimental data The measured radiuses of octane drops have been reported in a Log-Log representation on Fig. 1 for bare wafers, and Fig. 2 for grafted ones. The initial value of the drop volume is varied by a factor 100, which gives enough information while avoiding
Fig. 2. Log–log plot of recorded radius R(t0 − t) vs. time before vanishing t0 − t for octane drops deposited on HMDZ coated wafers. Range of deposited volumes: V = 0.02 L − 3 L. The continuous line has a slope: y = 0.48. The measured slopes for the receding part of the dynamics are in the range 0.48 ± 0.005.
gravity effects (for drops much larger than the capillary length) or loss of information for very small drops. It is clear that a power law R(t) ∝ (t0 − t)y is well obeyed during the main part of the retraction. The exponent y is found to be 0.48 ± 0.01 both on the bare wafers and on the grafted ones. Drop profiles agree well with spherical caps when the drop radius is of the order of the capillary length or less (1.8 mm for alkanes). For heptane, the exponent is the same, within experimental accuracy. The contact angles measured on small drops (V = 1 L) are reported on Fig. 3 for the two substrates. They are small, of the order of 10−2 rd. There is a limited range in the figure where a power law: θ(t) ∝ (t0 − t)x is obeyed. This is not surprising, because the volume of a thin spherical cap can be written as: πR3 θ V ∼ = 4 If the diffusion model is obeyed, then dV/dt ∝ R(t), which means that the angle has to follow a power law, which exponent x is such that 2y + x = 1. Actually, the relation is well obeyed for the drops investigated, considering the scattering of the experimental data (x = 0.03 ± 0.01 for the grafted wafer, x = 0.06 ± 0.01 for the bare wafer). Note that the value of the contact angle depends on the initial volume of the drop [25], and that a comparison can be done only on drops of same initial volume. 4.2. Discussion
Fig. 1. Log–log plot of recorded radius R(t0 − t) vs. time before vanishing t0 − t for octane drops deposited on a bare wafer. Range of deposited volumes: V = 0.04 L − 4 L. The continuous line has a slope: y = 0.48. The measured slopes for the receding part of the dynamics are in the range 0.48 ± 0.005.
It is clear that even for octane, which is close to a wetting transition on grafted surfaces, the behaviour is quite similar on both substrates. This means that the short range part of the interaction
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Fig. 3. Log–log plot of the contact angle θ(t0 − t) measured for two octane drops of same volume V = 1 L on bare and coated wafer, respectively. The slope x is ∼0.03 ± 0.01 on the grafted wafer and ∼0.06 ± 0.01 on the bare wafer, but considering the scattering of the points, the difference is too small to be significant. In both cases, the relation 2y + x = 1 is obeyed within experimental accuracy. The vertical lines correspond to the maximum extension of the drops.
Fig. 4. Picture of the receding contact line of an octane drop at high magnification. The droplets left on the substrate evaporate more slowly.
plays no role, at least for alkanes. Therefore, what is relevant for the dynamics of evaporation is the shape of the interface for thicknesses of the order of a hundred of nanometres, and not at the very edge of the drop. The grafted substrates provide the answer: films are visible at the drop’s edge on these surfaces, see Fig. 4. The films are longer with hexane, shorter with octane, and significantly thinner than 90 nm, which corresponds to the first black fringe. They are not seen directly, but indirectly revealed because they dewet on the grafted substrates, at the place where they become thin enough to feel the unfavourable short range interaction due to the trimethyl layer. Thicker rims and strings of small drops become observable. We do not see these films on bare wafers, because the film wets the substrate whatever its thickness [22–25]. It is not easy to measure the thickness of these films at a moving edge. As expected, they are much thinner than the ones
Fig. 5. Late time behaviour of a heptane drop. Log–log plot of: (a) radius R(t0 − t) in mm, (b) contact angle θ(t0 − t) in rd, versus time interval t0 − t in sec. While the behaviour of the radius follows a power law (y = 0.47) over the whole range of times, this is not the case for the contact angle. In the last tenth of second, the slope x increases from 0.08 (2y + x ≈ 1) to a value close to 0.5. In the last hundredth of second, there are not enough fringes to define properly a contact angle. This is why the measurements stop earlier for the angle than for the radius.
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observed by Garoff and co-workers [20], first because the velocities (more rigorously the capillary numbers) are less, but also because there is no hydrodynamic flow in the surrounding atmosphere. Note that there is a change in the dynamics of the contact angle at the very end of the drop’s life, see Fig. 5 for heptane. As there is no change in the dynamics of the radius, this means that the relation 2y + x = 1 is no longer obeyed, therefore that the “diffusion model” is no longer valid. Further experiments are needed to check the relative role of the increase of the contact line velocity, and the possible convection in the atmosphere. In any case, the measured contact angle is defined at the crossover towards a hydrodynamically controlled mesoscopic film, and not at a sharp edge [25]. An intermediate length scale must therefore be introduced in the forthcoming models. This is still an open problem. 5. Conclusion The study of evaporating sessile droplets on grafted surfaces, where the short range part of the interaction opposes wetting, allows us to answer at least some of the open questions concerning the dynamics of the process. The dynamic contact angle, measured at a typical thickness of the order of a fraction of m, is not controlled by the decrease in evaporation flux in ultra thin films. As a matter of fact, the apparent contact line recedes on a mesoscopic film, which thickness is probably a few tens of nanometres. The film is too thin to be seen directly under microscope, but indirect evidence is provided on grafted surfaces, where the thin part of the film dewets. The thicker rims or droplets become visible and delineate the edge of the continuous film ahead of the macroscopic contact line. Forthcoming models should therefore focus on the crossover between the receding wedge and the short, evaporating film left on the substrate. Acknowledgements This work has been partially supported by the European Community (Marie Curie and Objective 1 Phasing Out programs) and by the Minist`ere de la R´egion Wallonne. We gratefully acknowledge Josephine Conti for the preparation of the grafted substrates. The Laboratory of Statistical Physics of the Ecole Normale Sup´erieure is the UMR 8550 of CNRS, associated to the Universities Pierre et Marie Curie (Paris 6) and Denis Diderot (Paris 7). References [1] J.P. Burelbach, S.G. Bankoff, S.H. Davis, J. Fluid Mech. 195 (1988) 463.
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