41
Langmuir 1987, 3,41-45
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regime of high volume fraction, we have shown before' that
f(4) = [l- 1.892(1 - 4)1/2]2
(4
1)
(33)
and
Thus, a t 4 = 0.99, we find f = 0.66 and Ap/Apo = 2.92. As 4 tends to unity, f approaches one and Ap tends to infinity according to eq 33 and 34. As indicated above, the picture is less clear for intermediate and low volume fraction. Although we measured f(4) between $o and 4 = 0.975: the data are somewhat uncertain for reasons stated elsewhere.2 However, if we accept the previous empirical relationship6 0.70
0.90
0.80
1.00
$
Figure 9. Speculative curve of reduced capillary pressure, APIAP,, vs. 4.
then the derived A p ( 4 ) / A p oappears as in Figure 9. As discussed above, it may not be particularly disturbing that Ap($)/Ap0 drops below unity a t low volume fraction. Because of the uncertainties in the general validity of eq 35, however, it is prudent to consider the data in Figure 9 as being somewhat speculative.
Since submission of this paper, we have become aware of two recent publications by Kann,lo some aspects of which are related to, but clearly less advanced than, our work. (10) Kann, K. B. Colloid J. USSR (Engl. Transl.) 1984,46,397; 1985, 47, 744.
Investigation of Liquid Drop Evaporation by Laser Interferometry R. N. O'Brien* and Paul Saville Department of Chemistry, University of Victoria, Victoria, BC, V8W 2Y2 Canada Received April 24, 1986. In Final Form: September 16, 1986 The evaporation of a sessile drop of several liquids has been studied by two interferometers simultaneously. Liquid-phase interferograms which gave depth contours of the drop during its lifetime and some details of its final evaporation were supplemented by simultaneous vapor-phase interferograms. It appears that convection occurs in the liquid phase and also in the vapor phase; the vapor-phase physical structure appears to be a torus. An advancing foot, in this case a retreating foot, may have been detected.
Introduction Rates of evaporation from drops, sessile or suspended, have been carefully studied by LOU,^ Yang and Nouri,2 and Cammenga et a1.13 but the manner of evaporation has attracted much less interest. Rayleigh4 and others5 have established the conditions under which bouyancy-driven (density difference) convection should occur in pure liquid drops and Marangoni and Pearson6p7have derived the theory for surface-tension-driven (Marangoni) convection (1) Lou, Y. S. J. Appl. Phys. 1978, 49, 2350. (2) Yang, W.-J.; Nouri, A. Lett. Heat Mass Transfer 1981, 8, 115. (3) Cammenga, H. K.; Schreiber, D.; Barnes, G. T.; Hunter, D. S. J. Colloid Interface Sci. 1984, 98, 585. (4) Lord Rayleigh Phil. Mag. 1916, 32, 529. (5) (a) Pellew, A.; Southwell, R. V. Proc. R. SOC. London, A 1940,176, 312. (b) Law, A. R. Proc. R. SOC. London, A 1929,125,180. (c) Sparrow, E. M.; Goldstein, R. J.; Jonsson, V. K. Fluid Mech. 1964, 18, 513. (6) Marangoni, C. Nuouo Cimento 1871, 16, 239; 1878, (3) 3, 97. (7) Pearson, J. R. A. J.Fluid Mech. 1958,4, 489.
0743-7463/87/2403-0041$01.50/0
and there is experimental evidence for the general adequacy of theory.s Cammenga et aL3find Marangoni convection absent in water when theory predicts its presence. Yang and Nouri2 find three stages of drop evaporation by shadow graph studies in several polar liquids including polygonal cells as one stage. The relationship between increasing drop size on a solid substrate (or alternatively evaporation of a sessile drop) and dynamic contact angles is real enough though admittedly obscure a t this time as can be inferred from the work of Schwartz and Tejada? They extensively review (8)(a) Schmidt, R. J.; Milverton, S. W. Proc. R. SOC.London, A 1935, 152,586. (b) Silveston, P. L. Forsch. Ingenieurwes. 1958,24, 29,59. (c) Palmer, H. J.; Berg, J. C. J.Fluid Mech. 1971,47,779. (d) Berg, J. C.; Boudart, M.; Acrevos, A. J. Fluid Mech. 1966,24, 721. (9) Schwartz, A. M.; Tejada, S. B. J. Colloid Interface Sci. 1972,38, 359.
0 1987 American Chemical Society
42 Langmuir, Vol. 3, No. 1, 1987
O’Brien and Saville
Figure 2. Schematic diagram of the optical system used for producing vapor-phase interferograms where L = laser, D = drop, and C = camera.
Figure 1. Schematic diagram of the interferometer and optical system for producing fringe contours in the liquid drop where L = laser, C = camera, BS = beam splitter, M = first surface mirror, D = drop, and G = glass flat. the literature on the process of the attainment of an equilibrium for a sessile drop shortly after being placed on a solid. They also include the effects of roughness of the solid (on the outflow on drop settling) and of large contact angles. Their own work extensively tests the extant theory. They find two regions which can be described by different theories which apply at different velocities of flow. We have followed by laser interferometry the phenomenon of an evaporating drop, which is essentially a dynamic contact angle study, but in the opposite sense to the usual direction in the slow velocity regime which should conform to Blake and Haynes’ theorylO and give a Haynes plot (Schwartzg). Or if evaporation is too slow and the retreat of the liquid solid vapor (LSV) line is very slow the finding of Elliot and Riddiford may apply,l’ which is that the dynamic contact angle is equal to the equilibrium contact angle. We show some evidence for the “advancing foot” or in our case “retreating foot” as the drop leaves behind a layer of liquid of about one-quarter wavelength of light of the He-Ne laser (- 1500 A), in reasonable agreement with generally accepted literature values. We also obtain information on the final stages of drop evaporation. Using a vapor-phase and a liquid-phase interferometer simultaneously we obtain information on convection in the two phases.
Experimental Section The substrate for all experiments was clean glass. Glass flats (XI4 smooth over 2.5 cm, X = 632.8 nm)were immersed in chromic acid solution in an ultrasonic cleaner for 15 min or more then tested by immersion in water for any residual hydrophobicity. Two interferometerswere used: one previously described’* and shown schematically in Figure 1 and another used to follow the gas-phase concentration of the vapor evaporating from the drop. Figure 2 is a schematic of this interferometer. The interferometer giving the contours of the liquid drop operates by reflection of the laser light from the liquid-air (plus vapor) interface interfering with the light reflected from the liquid-glass interface, a bright fringe occurring at every X/2n increase in depth of the drop where n is the refractive index of the liquid. Those fringes form a contour map of the drop with a contour interval of 632.8/(2 X 1.33126) or 237.7-nm intervals. A further more intense fringe system of equispaced fringes was produced by the two sides of the glass flat not being exactly parallel. (10)Blake, T.D.;Haynes, J. M.J. Colloid Interface Sci. 1969,30,421. (11) (a) Elliot, G. E. P.; Riddiford, A. C. Nature (London) 1962,4843, 795. (b) Elliot, G. E. P.; Riddiford, A. C. J. Colloid Interface Sci. 1967, 23, 389. (12) O’Brien, R. N.; Saville, P. Can. J . Chem. 1985,63, 2339.
The second interferometerconsisted of two 5.0-cm square g h flats with 90% reflecting coatings held in an adjustable frame so that multiple-beam (Fizeau type) fringes were produced and these parallel, equispaced wedge fringes were perturbed by the changes in concentration of the vapor from the drop plus minor contributions from temperature effect (the cooling of the air associated with evaporation). The validity of the drop contour fringes was tested by weighing the 1-pL drop and by a rough calculation of each layer outlined by a fringe and summing the fringes. The drops were deposited with a microliter syringe. They weighed approximately 1 mg and the volume of the drop from the contour calculation and summation was 0.86 p L which should be multiplied by the refractive index of water (-1.33126) to give 1.14 MLor 0.8 X 1.33126 = 1.06 p L if the last fringe is considered to be the top of the drop. This value is expected to be too low since a flat contour rather than a rounded one was used. But countering this error is that the top contour is considered as one complete layer which it cannot be so that extra volume is thus added. We consider 1.0 & 0.1 is the likely limit of the volume calculation accuracy due in part to initial evaporation from the drop as it achieves the wet-bulb temperature and inaccuracies in delivery. Video cameras were arranged so that interferograms of the drop’s liquid contour and vapor concentration were made simultaneously and could be viewed (on rerun) on two monitors set side by side and in fast, normal, and slow times and by individual frames. A frame or two of a 0.5” grid was developed to give magnification. It was found that the dark, wide fringes from the sides (nonparallelsides) of the flat the drop was placed on were 1.04 mm apart, center to center, measured over five fringes (and averaged) where cooling or the drop itself has not affected them. The conditions for interferometry in the vapor phase are ideal as far as clarity of fringes to be expected is concerned. But the refractive index changes to be measured are extremely small and occur over a very small path length of light or a small part of the total path length. Fringe systems in the liquid drop are much more difficult to obtain. Because of the reflection rules, the total amount of light returned for interference purposes from the liquid-vapor interface of the drop cannot be expected to be more than about 5% of the original intensity. Given that in the optical system, which must have surfaces other than the ones of interest (the liquid-vapor interface), there will be other reflections of similar intensity and, of course, the more than 90% of the original admitted light (plus stray light from laboratory lighting etc.), the fringe system of interest will always be just a perturbing ripple on an intense background. A more sophisticated optical system could, of course, improve signal-to-noiseratios but probably not more than by a factor of 2 or 3. Besides water, drops of methanol, toluene, and diethyl ether were videotaped as they evaporated.
Results and Discussion The 1-pL drop of water when placed on the glass flat can be seen to distort seven fringes (Figure 3). It is therefore approximately 7 m m in diameter. The fringes that appear in Figure 3 are a combination of normal wedge fringes (fringes of equal path length) and we believe Fresnel fringes from the edge of the drop. Pure wedge fringes have been obtained and published12 but in the interest of clarity (the pure wedge fringes are not sharp and are difficult to obtain as pure fringes, Figure 4), Figure 3 fringes are used in the belief that they also represent thickness contours of the drop. Careful scrutiny of Figure 3 will show that part of the density (intensity) of the
Investigation of Liquid Drop Evaporation
Figure 3. Series of interferograms of an evaparating water droplet on a clean glass flat. Frames 1-4 are taken at 2-min intervals. Frames 5 and 6 are 15 s apart near the disappearance of the drop, approximately 10 min after frame 1. Force of gravity out of page.
Langmuir, Vol. 3, No. 1, 1987 43
Figure 5. Series of interferogramsshowing stages in the evaporation of spectroscopically pure toluene. Force of gravity out of page.
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