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Langmuir 2002, 18, 2636-2641
Drop Evaporation on Solid Surfaces: Constant Contact Angle Mode H. Yildirim Erbil,*,† G. McHale,‡ and M. I. Newton‡ Kocaeli University, Faculty of Sciences & Arts, Department of Chemistry, 41430, Izmit, Kocaeli, Turkey, and Department of Chemistry and Physics, The Nottingham Trent University, Clifton Lane, Nottingham NG11 8NS, United Kingdom Received September 24, 2001. In Final Form: January 17, 2002 There are two pure modes of evaporation of liquid drops on surfaces: one at constant contact area and one at constant contact angle. Constant contact area mode is the dominating evaporation mode for water and many other drops on solids when the initial contact angle is less than 90°. However, the constant contact angle mode has been reported in a few instances, such as water drop evaporation on poly(tetrafluoroethylene) where the initial contact angle is greater than 90°. In this work, we report the evaporation of n-butanol, toluene, n-nonane, and n-octane drops on a poly(tetrafluoroethylene) surface, which occurs with constant contact angle mode and an initial angle of less than 90°. Video microscopy and digital image analysis techniques were applied to monitor the drop evaporation. The decrease of the square of contact radius of these drops was found to be linear with time for most of the cases. This paper discusses the theoretical background and compares the experimental data with results from the previous models.
Introduction The evaluation of contact angle, θ, of test liquids on polymer surfaces is used to determine the surface tension of these polymers.1,2 In principle, a given pure liquid on an ideal (flat, homogeneous, smooth, rigid, and isotropic) solid should give a unique value for the equilibrium contact angle, θe, as determined by Young’s equation. In the small volume limit, the macroscopic shape of the fluid is independent of gravitational forces and on a flat surface gives a spherical cap. However, in practice, a whole range of angles between advancing and receding values exists depending on the previous history of the triple line. The contact angle variability (or wetting hysteresis) is attributed to surface roughness and chemical heterogeneity.3-5 On the other hand, the occurrence of liquid evaporation is inevitable unless the atmosphere in the immediate vicinity of the drop is saturated with the vapor of the liquid. Drop evaporation usually results in a decrease of θ. Thus, a more complete understanding of how evaporation influences the contact angle of the drop on polymer surfaces in still air or in controlled atmospheric conditions is very important in the wetting and surface characterization processes. In 1977, Picknett and Bexon6 reported on the mass and profile evolution of a slowly evaporating liquid (methyl acetoacetate) drop on a poly(tetrafluoroethylene) (Teflon) surface in still air. They distinguished two pure modes of evaporation: at constant contact angle with diminishing contact area and at constant contact area with diminishing contact angle. They also observed a mixed type where the * Corresponding author. E-mail:
[email protected]. Fax: 90 (262) 331 39 06. † Kocaeli University. ‡ The Nottingham Trent University. (1) Erbil, H. Y. Surface Tension of Polymers. In Handbook of Surface and Colloid Chemistry; Birdi, K. S., Ed.; CRC Press; Boca Raton, FL, 1997; Chapter 9, pp 259-306. (2) Erbil, H. Y. Langmuir 1994, 10, 2006. (3) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68 (7), 1744. (4) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (5) Chibowski, E.; Gonzales-Cabellero, F. J. Adhes. Sci. Technol. 1993, 7, 1195. (6) Picknett, R. G.; Bexon, R. J. Colloid Interface Sci. 1977, 61, 336.
mode would change from one to the other at some point in the course of evaporation. In this mixed mode, the drop shape would vary suddenly resulting in an increase in the contact angle with a decrease in the contact circle diameter or, sometimes, a decrease in both quantities. They developed a theory to predict the evaporation rate and residual mass at any time in the life of the drop based on the spherical cap geometry. In 1989 Birdi, Vu, and Winter7 reported the mass and contact diameter of water and n-octane drops placed on glass and Teflon. They observed that the evaporation occurred in constant contact radius mode with diminishing contact angle and the rate of evaporation for different sized drops was dependent on the initial radius of the liquid-solid interface, rb, by assuming a spherical cap geometry. The experimental data supporting this were obtained by direct measurement of the variation of the mass of droplets with time rather than the observation of contact angles. A model based on the diffusion of vapor across the boundary of a spherical cap drop was considered to explain their data.7 In a subsequent paper, Birdi and Vu8 reported that two regimes can be observed during drop evaporation on substrates. In the first case, with contact angle θ < 90°, the evaporation rates were found to be linear and the contact radius constant. In the second case, with θ > 90°, the evaporation rate was nonlinear, the contact radius decreased, and the contact angle remained constant. However, they reported neither the θ value nor the mode of evaporation of n-octane drops placed on Teflon in this publication. In 1995, Rowan, Newton, and McHale9 examined the change in the profile of small water droplets on poly(methyl methacrylate) (PMMA) due to evaporation in open air. Experimentally, the drops maintained a constant contact radius over much of the evaporation time. Measurements of θ and the drop height, h, with time in the regime of constant contact radius, rb, valid for initial contact angle < 90° were reported. No attempt was made to provide (7) Birdi, K. S.; Vu, D. T.; Winter, A. J. Phys. Chem. 1989, 93, 3702. (8) Birdi, K. S.; Vu, D. T. J. Adhes. Sci. Technol. 1993, 7, 485. (9) Rowan, S. M.; Newton, M. I.; McHale, G. J. Phys. Chem. 1995, 99, 13268.
10.1021/la011470p CCC: $22.00 © 2002 American Chemical Society Published on Web 03/08/2002
Drop Evaporation on Solid Surfaces
measurements for the final, rapid stages of evaporation where the mixed mode of evaporation occurred. Rowan et al. noted that the model of Birdi and Vu7 did not distinguish between the two principal radii of curvature occurring at the contact line: these two radii do not have the same values. They therefore extended the model to a twoparameter spherical cap geometry. By incorporating the experimentally observed constant value of the contact radius, they were able to explain why the experimentally observed change in contact angle should appear linear in time.8 Shanahan and Bourges-Monnier10,11 in 1994 and 1995 also published a similar drop evaporation study. They used large drops of water and n-decane (3-5 mm3) on polyethylene, epoxy resin, and Teflon, both in a saturated vapor atmosphere and in open air. They obtained direct measurements of the change of θ, h, and rb with time. They have shown the existence of four distinct stages in the evaporation process in open air conditions: In the initial stage, which corresponds to a saturated vapor atmosphere, the contact radius, rb, remains constant while θ and drop height, h, decrease. In the second stage, which corresponds to a nonsaturated atmosphere, the same things happen but the evaporation rate increases. In the third stage, which follows for smooth surfaces, h and rb diminish concomitantly, thus maintaining θ more or less constant. This stage does not exist on rough surfaces. In the final stage, the drop disappears in an irregular fashion with h, rb, and θ tending to zero. This step is difficult to follow experimentally, and it is probably related to the triple line anchoring on local heterogeneous zones. For the drop evaporation corresponding to the second stage, they derived a diffusive evaporation model based on spherical cap geometry to explain the process and to calculate the diffusion coefficient of the liquid vapor in air.11 This model is based on similar assumptions to that of Rowan et al.9 but uses a self-consistent approach to obtain a radial diffusion gradient. The main difference physically is that using this form of concentration gradient allows for the restriction of space for vapor to diffuse into caused by the presence of the solid substrate. Shape distortion of liquid drops often occurs when the drops are placed on solid surfaces, although no gravity effect is present. Sessile drops are flattened and oblate ellipsoid shapes are obtained when such distortions happen. In 1997, Erbil and Meric12 applied an ellipsoidal cap geometry in order to compare the differences between surface area, drop volume, and evaporation rates between the spherical and ellipsoidal cap geometry approaches. In this paper, Rowan et al.’s9 data for the variation of θ, rb, and h with time for small water droplets on PMMA polymer were used for the calculations. An elliptical profile in the x-y plane is revolved around the y-axis to form an ellipsoidal cap geometry for the drop, and the corresponding surface area and the volume of this body of revolution are found in terms of rb, h, and θ. A vapor diffusion model of a drop based on a three-parameter ellipsoidal cap geometry was also developed similar to ref 9. When the three-parameter ellipsoidal cap model was used, the surface area and volume of the drop showed almost complete linear time dependence for most of the data.12 The same approach was later applied as a three-parameter pseudo-spherical cap geometry model.13 These three(10) Shanahan, M. E. R.; Bourges, C. Int. J. Adhes. Adhes. 1994, 14 (3), 201. (11) Bourges-Monnier, C.; Shanahan, M. E. R. Langmuir 1995, 11, 2820. (12) Erbil, H. Y.; Meric, R. A. J. Phys. Chem. B 1997, 101 (35), 6867. (13) Meric, R. A.; Erbil, H. Y. Langmuir 1998, 14, 1915.
Langmuir, Vol. 18, No. 7, 2002 2637
parameter models do not allow for the presence of the substrate which restricts the space for vapor diffusion. In 1997, McHale et al. examined the much more rapid evaporation of three different alcohol microdroplets on PMMA (θ < 90°) where the evaporation is dominated by an initial stage with constant contact radius and decreasing θ.14 Later, in 1998, they monitored the evaporation of water droplets on Teflon (θ > 90°) where the evaporation is dominated by an initial stage with constant contact angle and decreasing rb.15 In this publication, they reworked their earlier model based on diffusion across the liquid-vapor interface,9 but with an assumption of constant contact angle rather than constant base radius. This model predicted a linear dependence on time for the square of the contact radius and so fitted the experimental data.15 Erbil16,17 attempted to determine the initial peripheral contact angle of sessile water drops on PMMA and methylacetoacetate drops on Teflon substrates from the rate of evaporation by using the experimental data of refs 9 and 6. The peripheral contact angle so obtained was regarded as the average of all the various θ’s existing along the circumference of the drop. For this purpose, a required parameter, the product of the diffusion coefficient and vapor pressure, DPv, was found experimentally from the rate of evaporation of the fully spherical liquid droplets in the same medium. The value of the peripheral θ was found to deviate by only 3.5% from the reported tangential θ, but the proposed method is very sensitive to the evaporation rate which needs to be very slow for good results.17 Erbil et al.18 examined the reasons for the flattening of evaporating water drops (θ < 90°) where the evaporation is dominated by constant contact radius and decreasing θ. They developed a single image sequencing (SIS) method to determine whether droplets of fluids are spherical or ellipsoidal. Internal flows in the drop during evaporation were suggested as one possible cause of the flattening of the drop profile as the drop size reduces, in contradiction to expectations based on equilibrium ideas. In a subsequent paper, they applied video microscopy to follow the time-dependent evaporation of sessile drops in order to determine the receding contact angles of water drops on PMMA and poly(ethylene terephthalate) surfaces.19 In this work, the evaporation of n-butanol, toluene, n-nonane, and n-octane drops on Teflon has been investigated by applying video microscopy and digital image analysis techniques. Contact angles were found to be approximately constant and to be less than 90° during the first stage of drop evaporation. The decrease of the square of the drop contact radius and the square of the drop height was found to be linear with time for most of the cases. Previous diffusion models are discussed and compared with the experimental data for this special case. Theory Spherical Cap Drops. When a liquid drop is sufficiently small and surface tension dominates over gravity, the drop forms a spherical cap shape. A spherical cap shape can be characterized by four different parameters, (14) Rowan, S. M.; McHale, G.; Newton, M. I.; Toorneman, M. J. Phys. Chem. 1997, 101, 1265. (15) McHale, G.; Rowan, S. M.; Newton, M. I.; Banerjee, M. K. J. Phys. Chem. B 1998, 102, 1964. (16) Erbil, H. Y. J. Phys. Chem. B 1998, 102, 9234. (17) Erbil, H. Y. J. Adhes. Sci. Technol. 1999, 13, 1405. (18) Erbil, H. Y.; McHale, G.; Rowan, S. M.; Newton, M. I. J. Adhes. Sci. Technol. 1999, 13, 1375. (19) Erbil, H. Y.; McHale, G.; Rowan, S. M.; Newton, M. I. Langmuir 1999, 15, 7378.
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the drop height (h), the contact radius (rb), the radius of the sphere forming the spherical cap (RS), and the contact angle (θ). By geometry, the relationships between the two radii, the contact angle, and the volume of the spherical cap (Vc) at any instant in time are
rb ) RS sin θ
and
( )
3Vc RS ) πβ
1/3
(1)
where
β ) (1 - cos θ)2(2 + cos θ) ) 2 - 3 cos θ + cos3 θ (2) The height of the spherical cap above the supporting solid surface is related to the two radii and the contact angle by
h ) RS(1 - cosθ)
and
(2θ)
h ) rb tan
(3)
Thus, a spherical cap shaped drop can be characterized by using any two of the above four parameters. When the horizontal solid surface is taken into account, the rate of volume decrease by time is given as6,9-11
-
( )
4πRSD dVc ) (cS - c∞) f(θ) dt FL
(4) (cm2/
where t is the time (s), D is the diffusion coefficient s), cS is the concentration of vapor at the sphere surface (at RS distance) (g/cm3), c∞ is the concentration of the vapor at infinite distance (R∞ distance) (g/cm3), FL is the density of the drop substance (g/cm3), and f(θ) is a function of contact angle of the spherical cap. There are at least three different versions for the solutions of the above general eq 4 in the literature.6,9-11 Using the analogy between the diffusive flux and electrostatic potential, Picknett and Bexon6 derived the exact solution. Rowan et al.9 derived an approximate solution by assuming the vapor concentration gradient (dc/dR) to be radially outward and equal to (c∞ - cS)/RS. Although this solution is approximate, it is a closed-form solution which gives good results around θ ) 90°. Independently, Bourges-Monnier and Shanahan11 used a self-consistent approach to derive an alternative approximate but closed-form solution with a vapor concentration gradient (dc/dR) radially outward. The solution of Bourges-Monnier and Shanahan is remarkably consistent with the exact solution of Picknett and Bexon except for small angles where it has a singularity and diverges. Picknett and Bexon6 converted the problem of determining the evaporation rate of a sessile drop to a problem of evaluating the capacitance of an isolated conducting body of the same size and shape as the drop as a equiconvex lens. Their solution resulted in
f(θ)Picknett & Bexon )
( )
1 C 2 RS
(5)
where C is the capacitance of the equiconvex lens, which can be obtained by using stream functions.6 To facilitate easier calculations, they also provided two empirical polynomial fits to the capacitance factor in eq 5. For 0 e θ < 0.175 rad (0 e θ < 10°),
C ) 0.6366θ + 0.09591θ2 - 0.06144θ3 RS and for 0.175 e θ e π rad (10° e θ < 180°),
(6)
C ) 0.00008957 + 0.6333θ + 0.116θ2 RS 0.08878θ3 + 0.01033θ4 (7) Picknett and Bexon6 indicated that the differences between the values obtained from finite series and those from Snow’s formula are in all cases less than 0.2% and are generally less than 0.1%. Rowan et al.9 assumed that the vapor concentration gradient (dc/dR) is equal to (c∞ - cS)/RS and is radially outward, and after performing the integral they found
f(θ)Rowan et al. )
(1 - cos θ) 2
(8)
Bourges-Monnier and Shanahan11 also assumed that the diffusion of liquid vapor from the liquid drop into the surrounding atmosphere is purely radial, but they obtained a self-consistent solution for the concentration gradient. They did so by considering a spherical cap shell of surface area ALV ) f(R′,θ′) defined by a coordinate system (R′,θ′) based on the center of the drop, through which vapor diffuses. Their approximation to the concentration gradient therefore began with
k k dc ))2 dR ALV 2πR′ (1 - cos θ′)
(9)
where k is a constant. Since (RS cos θ) ) (R′ cos θ′) by geometry, then it is possible to integrate eq 9 and, using the boundary conditions for the concentration at the drop surface and far removed from the drop, to calculate the value of k in a self-consistent manner. Using the resulting concentration gradient with eq 4, the model of BourgesMonnier and Shanahan11 gives
f(θ)Shanahan et al. ) -
cos θ 2 ln(1 - cos θ)
(10)
The physical origin of the difference between eqs 8 and 10, for f(θ), can be most clearly seen for the limiting case of θ ) 180°, which corresponds to a fully spherical drop. Equation 8 then accurately reduces to the known result for the evaporation of a fully spherical drop surrounded by free space. However, for sessile drops while θ ) 180° corresponds to a fully spherical drop, that drop is in contact with a plane wall rather than being surrounded by free space into which vapor can diffuse. It is this case which the Picknett and Bexon solution (eq 5) describes and which is more accurately approximated by eq 10 than eq 8. At 180°, the exact solution, eq 5, gives fPicknett & Bexon(θ) ) 0.694 while eq 10 gives fShanahan et al.(θ) ) 0.721 and eq 8 gives fRowan et al.(θ) ) 1. The difference between eq 5 and the two approximate solutions, eqs 8 and 10, reduces as the angle reduces from 180°, and at 90° all three solutions are the same numerically. Below 90°, the difference between the exact and the approximate solutions increases with eq 10 being the more accurate approximation for much of the angular range. However, eq 10 diverges and is singular at 0° whereas eq 8 is analytically well-behaved in this limit. A comparison of the behavior of the equations for f(θ) is shown in Figure 1. Since we analyze only the constant contact angle mode of drop evaporation in this work, we will find constant f(θ) values with different numerical factors for each approach at the end of the analysis.
Drop Evaporation on Solid Surfaces
Langmuir, Vol. 18, No. 7, 2002 2639 Table 1. Drop Evaporation Experiment Results
liquid
Tair (°C)
n-nonane n-butanol n-octane toluene
mean contact angle (θo)
evaporation period (t, s)
35.5 ( 1.0 45.9 ( 1.2 36.6 ( 1.9 43.8 ( 2.0
24.9 29.5 23.6 22.1
initial drop volume (Vci, cm3)
expt [(K) f(θ)] (cm2/s)
10-4
10-5
0.89 × 2.07 × 10-4 1.90 × 10-4 4.80 × 10-4
120 120 45 90
1.420 × 1.785 × 10-5 4.740 × 10-5 5.790 × 10-5
expt (K) (cm2/s) 7.288 × 10-5 7.707 × 10-5 20.070 × 10-5 24.480 × 10-5
Table 2. Physical Properties of the Drop Liquids at Experiment Air Temperaturea liquid
Tair (°C)
molecular mass M (g/mol)
density FL (g/cm3)
diffusion coefficient in air, D (cm2/s)
vapor pressure (Pv)Air (mmHg)
(DPv)Air (mmHg cm2/s)
cS (g/cm3)
n-nonane n-butanol n-octane toluene
24.9 29.5 23.6 22.1
128.26 74.12 114.23 92.14
0.715 0.807 0.701 0.866
0.061 0.089 0.069 0.084
4.60 9.05 12.75 24.25
0.281 0.805 0.880 2.037
3.174 × 10-5 3.554 × 10-5 7.870 × 10-5 12.14 × 10-5
a
Reference 21.
the time dependence of the contact radius, 2
rb )
rbi2
-
4D(cS - c∞) sin2 θ
t
FL(1 - cos θ)(2 + cos θ)
(13)
which can be used as an indication of constant θ evaporation mode when fitted by the experimental data. Experimental Section
Figure 1. Comparison of the exact solution and the two approximate solutions for f(θ).
To perform the integration, the general eq 4 is written in terms of Vc and β,
( )
-
( )
dVc 4πD 3Vc ) dt FL πβ
1/3
(cS - c∞) f(θ) ) KVc1/3 f(θ) (11)
where
K)
4π2/331/3D(cS - c∞) FLβ1/3
and is a constant independent of time and volume. Equation 11 can be readily integrated between Vci (initial) when t f 0 and Vc when t f t,
2 Vc2/3 ) Vci2/3 - K f(θ) t 3
(12)
Then it is possible to construct a linear Vc2/3 - t plot for given Vci2/3 and θ values for each approach and compare it with experimental data. Time Dependence of Contact Radius. For the constant contact angle evaporation mode, the drop shape remains that of a spherical cap but with a diminishing drop height and a diminishing area of contact between the liquid and surface. This is the expected behavior for an ideal system with equilibrium between liquid, solid, and vapor, where there is no difference between advancing and receding contact angles. McHale et al.15 argued that a small variation of contact angle is expected but that the contact angle is the slow variable compared to the contact radius, and they therefore derived
A chamber housing both plan and side view video cameras having dimensions 75 × 75 × 100 cm was used in all the experiments. n-Butanol, toluene, n-octane, and n-nonane liquids were deposited as a sessile drop from a microliter syringe onto a flat, transparent, and smooth Teflon substrate. The substrate surface was cleaned of impurities and oily material by washing with ethyl alcohol, wiping with a piece of cloth containing dilute soap solution, washing with distilled water thoroughly, and finally drying at 50° C for 4 h under a 3-5 mmHg vacuum. Water drop contact angles were found to be 110 ( 2° on the cleaned Teflon surfaces. The evaporation of the drops was recorded in the closed chamber in still air. Drops were illuminated with both side and bottom lamps of low intensity. The temperature for all experiments was measured by a thermocouple probe located in the chamber. The relative humidity (RH) in the chamber was measured by both using a hair hygrometer (Polymeter, Casela, London Inc.) and a wet and dry bulb temperature hygrometer located inside the chamber. The relative humidity was found to be approximately constant at 53% for n-butanol and 54% for n-nonane, n-octane, and toluene drops during evaporation on Teflon. The polymer surface was sufficiently reflective to enable an optical profile of a droplet and its reflection to be obtained. The evolution in the side profile and plan view of the drops was then recorded onto videotape operating at 25 frames/s by simultaneously using two video cameras equipped with suitable objectives. The images were subsequently transferred onto a personal computer using a Data Translations DT-3152 scientific framegrabber card working at a resolution of 768 × 576 and with a one-to-one pixel aspect ratio. Digital image analysis was performed using a simple thresholding routine to obtain the profile. The UTHSCSA ImageTool program was used, together with in-house routines, to perform the image analysis.
Results and Discussion The repeatability of the drop evaporation rate experiments was checked by monitoring the evaporation of two different-sized liquid drops on Teflon, and only very small differences in the rate of evaporation were found. The air temperature in the evaporation chamber, the constant mean contact angle value, the initial drop volume, and the evaporation period in which the drop evaporates without altering this θmean value are given in Table 1. The evaporation mode was checked by plotting the time dependence of the square of the contact radius, rb, which
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Table 3. Comparison of Experimental Data with Theory Picknett & Bexon liquid n-nonane n-butanol n-octane toluene
[(K) f(θ)]
(cm2/s) 10-5
1.521 × 2.083 × 10-5 4.306 × 10-5 6.314 × 10-5
% deviation +7.1 +16.7 -9.2 +9.1
Shanahan et al. [(K) f(θ)]
% deviation
[(K) f(θ)] (cm2/s)
% deviation
10-5
+24.2 +26.2 +4.7 +19.3
0.677 × 10-5 1.172 × 10-5 1.979 × 10-5 3.406 × 10-5
-52.3 -34.3 -58.3 -41.2
1.763 × 2.253 × 10-5 4.962 × 10-5 6.906 × 10-5
Figure 2. Time dependence of the square of the drop contact radius, rb2.
Figure 3. Time dependence of the two-thirds power of the drop volume, Vc2/3.
is given in Figure 2. As seen in this figure, all the liquid drops gave linear plots obeying eq 13 and showing the constant contact angle evaporation mode behavior. Then, Vc2/3 - t plots were drawn in Figure 3 for every drop by using the spherical cap drop volumes, Vc, which were given directly by the image analysis program. The high linearity for all the drops is seen in Figure 3, as exhibited by the high regression coefficients R2 ) 0.99597-0.99923. Then, the experimental [(K) f(θ)] products were determined from the slopes of these lines by using eq 12. To find the
Rowan et al.
(cm2/s)
Figure 4. Comparison of the experimental data for toluene with the theory.
experimental K constant, the physical properties of the drop liquids at experiment air temperature were calculated by using the literature values from ref 21 and are given in Table 2. The concentration of vapor at the sphere surface, cS, was calculated by assuming the ideal gas behavior of the liquid vapors so that cS ) PvMw/RT. Then it is possible to find the experimental K constant by using eq 11, and the results are reported in Table 1. To compare the experimental results with the previously published models, f(θ) values of Picknett and Bexon, Rowan et al., and Bourges-Monnier and Shanahan were calculated by using eqs 5, 8, and 10, respectively. Then the products of these numerical contact angle functions with the experimental K constant, [(K) f(θ)] were found; these are reported in Table 3 with their deviations from the experimental [(K) f(θ)] values. An indicative Vc2/3 - t plot for the toluene drop is given in Figure 4 in order to show the deviation of the experimental results from these models. As seen in Table 3 and in Figure 4, the best fit at the experimental air temperature can be achieved when the Picknett and Bexon model is applied. Rowan et al.’s approximate model9 is a closed-form solution which gives compatible results with Picknett and Bexon’s exact results around θ ) 90°; however, it gives negative deviations of more than 50% once the contact angle falls below 40° as can be seen from the experimental results in Table 3. Bourges-Monnier and Shanahan’s11 approximate and closed-form solution is remarkably consistent with the exact solution of Picknett and Bexon being better than 4% accurate for contact angles in the range 57°-180°. However, this approximation deviates by more than 10% (20) Erbil, H. Y.; Dogan, M. Langmuir 2000, 16, 9267. (21) Chemical Engineer’s Handbook, 5th ed.; Perry, R. H., Chilton, C. H., Eds.; McGraw-Hill: New York, 1973.
Drop Evaporation on Solid Surfaces
for angles below 43° and for angles below 10° is a poorer estimate than that of Rowan et al., although both approximate models differ significantly from the exact solution in this low-angle range. The experimental data correspond to contact angles which are in the range 35°-46°, and this is reflected in Table 3 by the relatively poor consistency of the results using the Bourges-Monnier and Shanahan model with the experimental data. If we also consider the drop surface cooling during evaporation, it is possible to find the vapor pressure, Pv, which gives the exact fit with the [(K) f(θ)] values of the Picknett and Bexon model. Then it is possible to calculate the drop surface temperature from this Pv by using ln Pv - 1/T plots. A drop surface cooling of 1.1 °C for n-nonane, 2.2 °C for n-butanol, 0 °C for n-octane, and 1.2 °C for toluene was estimated by the Picknett and Bexon model. With the same reasoning, the Shanahan et al. model gives a drop surface cooling of 3.1 °C for n-nonane, 3.4 °C for n-butanol, 0.5 °C for n-octane, and 2.9 °C for toluene drops. The presence of water vapor in the chamber (RH ) 53%) may retard the evaporation of n-butanol, since this liquid absorbs water. However, it is a very difficult task to measure the experimental drop surface temperature and also to calculate it from an energy balance at the dropair interface in such a large chamber where small convection currents may be present. Consequently, these drop cooling figures are indicative only and the figures for
Langmuir, Vol. 18, No. 7, 2002 2641
n-nonane and n-butanol are very reasonable if we compare the results with the evaporation of fully spherical drop liquids.20 Conclusion In this work, we reported the constant contact angle evaporation mode of n-nonane, n-butanol, n-octane, and toluene drops on a poly(tetrafluoroethylene) surface. Video microscopy and digital image analysis techniques were applied to monitor the drop evaporation. The decrease of the square of contact radius of these drops was found to be linear with time. The high linearity for all the drops was found in all the experimental Vc2/3 - t plots. Then it is possible to compare these plots with the previously published models. The Picknett and Bexon model which was derived from the diffusion of the exact spherical shape analysis gave the best fit for all the four drops. In this model, the drop surface cooling during evaporation is estimated to lie in the range 0-2.2 °C for the drops studied. Acknowledgment. The authors gratefully acknowledge the support of The Royal Society, U.K., and TUBITAK, Turkey, for funding this work through a European collaborative exchange. LA011470P
