VOLUME 101, NUMBER 8, FEBRUARY 20, 1997
© Copyright 1997 by the American Chemical Society
LETTERS Evaporation of Microdroplets of Three Alcohols S. M. Rowan, G. McHale,* M. I. Newton, and M. Toorneman Department of Chemistry and Physics, The Nottingham Trent UniVersity, Clifton Lane, Nottingham NG11 8NS, U.K. ReceiVed: September 11, 1996; In Final Form: NoVember 19, 1996X
The evaporation of small droplets of water resting on poly(methyl methacrylate) (PMMA) has previously been shown to be dominated by an initial stage with constant contact radius and an almost linear decrease in contact angle and height. The time and contact radii dependencies were found to be in good agreement with a model based on diffusion. This letter reports the evaporation of small droplets of n-decanol, n-hexanol, and n-propanol resting on Teflon tape (polytetrafluoroethylene). Such systems provide much greater evaporation rates with some droplets completing the process in 1-2 s compared to minutes for the water on PMMA system.
Introduction It has previously been demonstrated that a small droplet of water resting on a poly(methyl methacrylate) PMMA surface has an initial contact angle less than 90° which then decreases linearly whilst retaining a constant contact radius until eventually the radius begins to shrink;1 the regime of constant contact area holds during the majority of the evaporation time. A diffusion model due to Birdi et al.2,3 was extended to the spherical cap geometry, valid for small droplets resting on solid surfaces, and applied to the observed data subject to an assumption of constant contact diameter. A similar study of the rate of change of mass, contact angle, and height for a sessile droplets of water on polymer surfaces has also been published by Bourge`s and Shanahan.4,5 Their study did not examine in detail the dependence of the evaporation process on contact diameter which was observed to be constant during a large portion of the experiment. The regime they examined related to time scales of minutes rather than seconds due to the choice of water as the volatile fluid. The constant contact diameter mode of evaporation for droplets with initial contact angles of less than 90° has also been observed for droplets of water on a flat and heated surface of Nickel.6 In this case the constant wetted area X
Abstract published in AdVance ACS Abstracts, February 1, 1997.
S1089-5647(96)02795-2 CCC: $14.00
was reported to cover 80% of the total time of evaporation. One early study by Picknett and Bexon7 in 1976 did report observing a nonconstant contact diameter mode of evaporation as well as the constant contact diameter mode for drops with contact angles less than 90°. However, this was for pendant, rather than sessile, drops of methyl acetoacetate on a PTFE surface, and the time scale involved was a comparatively long period of around 1 h. In this Letter we report observations of the evaporation of small droplets of n-decanol, n-hexanol, and n-propanol of contact diameters between 0.296 and 0.542 mm resting on Teflon tape. In these systems, the volatility of the liquids is much higher and the time scales for evaporation are relatively short, ranging from a few minutes to a few seconds. Experimental Details Experiments were performed using >98% pure n-decanol, n-hexanol, and n-propanol deposited onto a flat piece of Teflon tape on a supporting glass microscope slide. This choice of system was made because of the volatility of the alcohols, the initial contact angle of between 40° and 65°, and the good reflectivity of Teflon tape. Deposition was from a syringe brought lightly into contact with the substrate and this enabled contact radii between 0.296 and 0.542 mm to be obtained. The temperature for all experiments was constant at around 21.5 © 1997 American Chemical Society
1266 J. Phys. Chem. B, Vol. 101, No. 8, 1997
Letters droplets of n-propanol is repeated for both n-hexanol and n-decanol. The initial angles are around 51° and 42°, and the time scales for these less volatile alcohols increase to between 10 s and 1 min depending on initial droplet size. The initial linear change with time in contact angle can be explained using a theory of diffusion-controlled evaporation of small droplets that was developed for the water on PMMA system.1 For a constant contact radius r0 and a spherical cap shape, the equation describing the contact angle, θ, is
λ sin3 θ dθ )2 dt πr0 (1 - cos θ) Figure 1. Contact angles for droplets of n-propanol on Teflon tape with initial base diameters of 0.399 (×), 0.460 (+), 0.515 (O), and 0.542 mm (]) are initially linear in time. The smaller symbols in the upper part of the figure are the corresponding contact diameters, scaled by their initial values. The function (F(θ) + 1.592) × 180/1.632π is predicted to be proportional to time providing the contact radius remains constant, and this is shown (4) for the droplet with initial contact diameter 0.460 mm.
°C to within 1 °C, although the ambient relative humidity did vary between experiments. The profile of the droplets was observed using a Kru¨ss contact angle meter with the image also recorded onto a video recorder operating at 25 frames per second. The droplet images could also be captured by a personal computer and digitized for analysis. All reported measurements of contact angles are the average of three measurements from a single frame and are for the left-hand side of the droplets. A difference of several degrees was observed between the contact angles on the left and right sides of some droplets although the time dependences of the angles were similar. Results and Discussion In Figure 1 we plot results for the evaporation of four droplets of n-propanol with initial base diameters of 0.399, 0.460, 0.515 and 0.542 mm. The contact diameters remain at their initial values for much of the evaporation time, and this is shown in the upper part of the figure by the smaller symbols which represent the contact diameter scaled by the initial value. When the contact diameter does begin to change in the final stages of evaporation it does so very rapidly. A systematic trend can be observed with the length of time the contact diameter is constant increasing as the initial contact diameter increases due to a greater initial volume of fluid. In the early stages of evaporation, during which the contact diameter is constant, the change in contact angle can be approximated by a linear fit, and this is shown by the solid lines in Figure 1. The contact angles of these droplets are initially 59.6°, 43.0°, 44.9°, and 45.5° and then decrease systematically; the smallest of these droplets appears to have an anomalously large initial angle. Figure 1 also shows that the scatter in the contact angle for any one droplet increases significantly as the contact diameter begins to change. Closer examination of the data suggests the variation in measured angle is actually a precursor indication of the subsequent reduction in the contact diameter. The scatter in the measured value of contact angle is not a consequence of difficulties in measuring the angle, but is physical and appears to indicate the onset of an instability in the contact line. Qualitatively, the edge profile appears to undulate at this stage of the evaporation. The linearity in the initial stages of the evaporation of the n-propanol is particularly impressive given the time scale of around 1-4 s that is involved for the smallest of the droplets. The pattern of evaporation followed by the
(1)
where λ ) 2πD(c∞ - c0)/F, F is the density of liquid, D is the diffusion coefficient of the vapor, and c is its concentration; it should be emphasized that the contact radius is different from the spherical radius of the droplet. Equation 1 can be solved and gives a function of the contact angle, F(θ), proportional to time
F(θ) ) ln[tan(θ/2)] + (1 - cos θ)/sin2 θ ) -2λ(t - t0)/(πr02) (2) where t0 is a constant of integration. It was shown in ref 1 that numerically F(θ) is essentially linear in angle between 30° and 90° and an extremely good fit is given by
F(θ) ≈ -1.592 + 1.632θ
(3)
where θ is in radians. The experimental data for some of the droplets of alcohol on Teflon tape shows a constant contact radius for angles below 30°, and so it is useful to consider deviations from the linear approximation. The low-angle approximation to F(θ) is
1 F(θ) ≈ - ln 2 + ln θ 2
(4)
These two approximate formula (eqs 3 and 4) give an almost perfect fit to F(θ) over the range 0-90° with the transition between the low and high angle approximations occurring at around 28°. Thus, it is expected that above this value the contact angle will change linearly in time with a slope decreasing with increasing initial contact radius and below it the contact angle should take on an exponential dependence. Figure 1 shows that the data initially follows a linear change, but toward 28° the experimental data lie systematically above the linear fit as expected from the transition to the low-angle exponential form of F(θ); this behavior is also confirmed by the evaporation of n-hexanol and n-decanol. In most cases the change from the linear to exponential curves occurs during the time the contact diameter remains constant. To examine the applicability of eq 2, the function (F(θ) + 1.592) × 180/1.632π for the droplet of n-propanol with base diameter 0.399 mm has been included in Figure 1 (4 symbols). The function F(θ) should change linearly with time over the whole of the constant contact diameter regime and has been scaled to correspond to the linear fit in the early stages. As the contact diameter begins to reduce an increasing scatter is observed in the contact angle. Theory predicts that droplets with larger contact diameters will have a lower rate of change of contact angle, and this is consistent with the data. However, as the droplet size increases, gravity will become important, and the shape will no longer be a spherical cap. When the contact radius is much less than the capillary length, κ-1 ) (γLV/Fg)1/2, the influence of gravity can be neglected. For a water-air system at 20 °C the capillary
Letters length is 2.7 mm, but for the three alcohols κ-1 reduces to around 1.8 mm. The range of contact radii in the data is from 0.148 to 0.271, mm and since the largest of these is greater than 10% of the capillary length, a slight deviation of the shape from a perfect spherical cap is anticipated. Nonetheless, we would expect the theory to be reasonably accurate with deviations due to gravity being a first-order correction for the droplets in this study. The solid-liquid interfacial forces in the alcohol-Teflon tape and water-PMMA systems are not directly comparable since in the latter hydrogen bonding occurs in addition to van der Waals forces. Nonetheless, the observed behavior of the contact angle and contact radius as evaporation proceeds is similar and poses some interesting questions. Firstly, for the water on PMMA system it seems surprising that the initial contact angle should be close to the accepted value for the equilibrium contact angle measured with a saturated vapor system. This also seems to be the case with the alcohols on Teflon tape which have initial angles of 45° for n-propanol, 51° for n-hexanol, and 42° for n-decanol; these are averages of four droplets for each alcohol. The equilibrium value of contact angle for n-propanol on polytetrafluoroethylene has been reported8 to be 43°, which is again similar to the initial value we record in the system with evaporation, although there is a larger drop to drop variation than in the water on PMMA case. This variation is probably an indication of the experimental difficulty presented by the higher evaporation rate in defining an initial contact angle although it may also be a consequence of the differences in interfacial forces between the two systems. A second unresolved question is why the contact radius should remain constant for a substantial fraction of the total evaporation time even though the contact angle is systematically reducing. This means the increasing component of the surface tension force at the contact line directed radially inward must be balanced by a force not present at the moment of droplet deposition. The suggestion that increasing scatter in the contact angle measurement as the end of the constant contact diameter period is reached is due to an instability would be consistent with the observations presented by Rymkiewicz and Zapacowicz6 for the evaporation of droplets from a heated surface. In their case the supply of heat generated flow patterns within the droplet that became unstable at the time when retraction of the contact diameter was about to occur. In our case the initial droplet may be in a local equilibrium with no flow within the droplet. This would allow the use of the usual argument balancing forces at the contact line and the initial contact angle would then be given by γSV ) γSL + γLVcos θ, where the γij’s are the interfacial tensions. An angle similar to the equilibrium system would then be expected; the difference would only be in the use of γij’s taken with respect to an unsaturated rather than saturated vapor. As the evaporation process establishes itself, the surface of the droplet will cool with the apex of the droplet cooling more than the contact line with the solid. We would expect a nonuniform temperature distribution to result, and this would cause the surface tension
J. Phys. Chem. B, Vol. 101, No. 8, 1997 1267 to vary, giving a tangential stress in the surface in the direction opposite to the surface temperature gradient and would have two consequences. Firstly, a Marangoni force would arise as the evaporation proceeded, and this may provide the extra force necessary to maintain the observed constant contact radius and reducing contact angle. Secondly, the change in surface energy due to the temperature gradient would result in a flow of fluid from the warmer part of the surface to the cooler part of the surface. A compensating flow would also occur from the interior of the spherical cap toward the contact line; this is not a Rayleigh-Bernard thermally induced convective flow, but a surface tension induced flow. The breakdown in such a flow as the contact radius started to reduce might then account for the scatter in the observed contact angles, although it is also possible that any surface roughness would increase the scatter in this stage of the evaporation. This hypothesized mechanism is not the only possibility, but it is one that may account for the initial value of contact angle and the constant contact radius observed in these systems. Conclusion Experimental data has been presented for the evaporation of small droplets of n-propanol, n-hexanol, and n-decanol on Teflon tape. The initial contact angles are less than 90°, and the evaporation is dominated by an initial stage with constant contact radius. This stage is characterized by a linear change in contact angle with time, which progressively becomes an exponential. The evaporation of these droplets is on the time scale of seconds and confirms previous observations for water on PMMA. The crossover from constant contact radius to reducing radius is indicated by an increase in the scatter in the contact angle, and this appears to indicate the onset of an instability. It is suggested that evaporative cooling establishes a surface tension driven flow within the droplet which is then responsible for the deviations from the initial contact angle. The Marangoni force created by the temperature variation across the droplet surface may also be responsible for the maintenance of a constant contact radius over much of the evaporation time. Acknowledgment. The authors thank Dr. S. D. Lubetkin for discussions on the topic of the Marangoni force. References and Notes (1) Rowan, S. M.; Newton, M. I.; McHale, G. J. Phys. Chem. 1995, 99, 13268. (2) Birdi, K. S.; Vu, D. T.; Winter, A. J. J. Phys. Chem. 1989, 93, 3702. (3) Birdi, K. S.; Vu, D. T. J. Adhes. Sci. Technol. 1993, 7, 485. (4) Shanahan, M. E. R.; Bourge`s, C. Int. J. Adhes. Adhes. 1994, 14, 201. (5) Bourge`s, C; Shanahan, M. E. R. C. R. Acad. Sci. Paris 1993, 316, 311. (6) Rymkiewicz, J; Zapacowicz, Z. Int. Commun. Heat Mass Transfer 1993, 20, 687. (7) Picknett, R. G.; Bexon, R. J. Colloid Interface Sci. 1977, 61, 336. (8) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; John Wiley & Sons: New York, 1982; Chapter 10.
