9234
J. Phys. Chem. B 1998, 102, 9234-9238
Determination of the Peripheral Contact Angle of Sessile Drops on Solids from the Rate of Drop Evaporation H. Yildirim Erbil TUBITAK, Marmara Research Center, MKTAE, Department of Chemistry, P.O. Box 21, Gebze 41470, Kocaeli, Turkey ReceiVed: February 24, 1998; In Final Form: September 2, 1998
A method was developed to determine the initial peripheral contact angle of sessile drops on solid surfaces from the rate of drop evaporation by using the published precise data of Rowan et al. The constant drop contact radius throughout this stage of evaporation, the initial weight (or volume), the weight decrease by time, and the experiment temperature should be measured for this purpose. When water drops are considered, the relative humidity should also be known. The drop profile data are not required in the calculations. The peripheral contact angle so obtained may be regarded as the mean of all the various contact angles existing along the circumference of the drop. Thus, each determination yields an average result not unduly influenced by irregularities at a given point on the surface. In addition, the error in personal judgment involved in drawing the tangent to the curved drop profile at the point of contact is eliminated. The application of this method requires the use of the product of the vapor diffusion coefficient of the evaporating liquid with the vapor pressure at the drop surface which can be found directly by experiment by following the evaporation of fully spherical liquid drops. The other alternative method is the measurement of the drop surface temperature in order to find ∆pv and then to estimate the product D∆pv.
Introduction The surface tensions of solids cannot be measured directly because of the elastic and viscous restrains of the bulk phase, which necessitate the use of indirect methods. The only general method is the rather emprical one of estimating solid surface tension from that of the contacting liquid drop. In principle, a given pure liquid drop 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.1-3 However, in practice, a whole range of angles between advancing and receding values exists depending on the previous history of the triple line. Contact angle hysteresis became a valuable analytical tool providing information about surface heterogeneity, roughness, and polymer surface dynamics.1-7 The use of surface-sensitive techniques such as XPS (X-ray photoelectron spectroscopy or ESCA) and SSIMS (static secondary ion mass spectrometry) improved the correllation between contact angles and surface composition.1,8 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. An initial advancing contact angle will diminish toward a receding angle when the liquid constituting the meniscus starts to evaporate. 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 surface characterization processes. Birdi et al.9,10 reported the change of the mass and contact diameter of liquid drops placed on solids by time. They observed that the initial rate of evaporation was dependent on the radius of the liquid-solid interface, rb, by assuming a spherical cap geometry.9 A model based on the diffusion of vapor across the boundary of a spherical cap drop was considered to explain their data.
Shanahan and Bourges-Monnier pointed out that the liquid evaporation effect on the contact angle measurements would seem to have been largely neglected11 and they examined the influence of evaporation on contact angles of the liquid drops placed on polymer and glass surfaces.11,12 They have shown the existence of three stages in the evaporation process in open air conditions: In the first stage, the contact radius, rb, remains constant while θ and the drop height, h, decrease. In the second stage, h and rb diminish concominantly, thus maintaining θ more or less constant for smooth surfaces. This stage does not exist on rough surfaces.12 In the final stage, the drop disappears in an irregular fashion with h, rb, and θ tending to zero. They have also showed that when the surrounding atmosphere is saturated in the vapor of the given liquid, the contact angle remains constant.12 They proposed a theory to calculate the diffusion coefficient of the liquid vapor in air by using a drop model from saturation vapor concentration to zero over a stagnation layer thickness covering the drop meniscus.11,12 Recently, Rowan, Newton, and McHale13 examined the change in mass and the geometry of small water droplets on poly(methyl methacrylate) (PMMA) due to evaporation in open air. They examined only the first stage of drop evaporation where rb was constant, with θ and h decreasing with time. Measurements of θ, rb, and h with time in the regime of constant contact radius, valid for θ < 90°, were reported. It was shown that the rate of mass loss was proportional to h and rb, but not to the spherical radius, R. The results were explained by a vapor diffusion model based on a two-parameter spherical cap geometry and the observed constant value of the contact radius.13 Later, they applied this model to the cases where θ > 90°.14 There were some conceptual differences between the vapor diffusion model of Shanahan, Bourges-Monnier,11,12 and Rowan, Newton, and McHale13,14 which will be discussed in our Theory section.
10.1021/jp9812785 CCC: $15.00 © 1998 American Chemical Society Published on Web 10/27/1998
Peripheral Contact Angle of Sessile Drops In all the above evaporation studies the spherical cap geometry was used. It was assumed that when a drop of fluid was sufficiently small, the influence of gravity becomes negligible and the drop assumes a spherical cap shape. However, although gravity effects are negligible, shape distortion of liquid drops often occurs when the drop is placed on solid surfaces.1,2,5,6 More recently, the ellipsoidal cap geometry was applied15 for sessile drops on solid surfaces in order to compare the differences between the surface area, drop volume, and evaporation rates between the spherical and ellipsoidal cap geometry approaches. Rowan et al.’s13 precise θ, rb, and h data were used for the calculations. A vapor diffusion model of the ellipsoidal cap geometry was also developed similar to Rowan et al.’s, and some success over the classical spherical cap geometry was obtained.15 In addition, it was realized that when the three different equation sets of the classical two-parameter spherical cap geometry were used to calculate the volume and the surface area of the drops, three different results were obtained, a point which has always been missed before in the existing literature.16 To overcome this deficiency, a new pseudo-spherical cap model was offered and mathematical expressions for a drop having all three measurable quantities as parameters were derived. The proposed new model was termed as the three-parameter spherical cap geometry.16 To find the correllation between the drop evaporation rate and the diffusion coefficient and vapor pressure of the drop liquids, the evaporation experiments for fully spherical drops were performed in still air for 11 liquids and the results were expressed by using the drop half-lifetime method.17 Weight (m2/3) versus time plots were drawn and clearly linear plots over the drop life were obtained. The product of the vapor diffusion coefficient and the drop surface vapor pressure was calculated directly from the experiments.17 On the other hand, it is a known fact that instead of measuring the contact angle directly, it may be calculated from the profile of the drop.18,19 With this method, the error in personal judgment involved in drawing the tangent to the curved drop surface at the point of contact is eliminated. Mack and Lee showed that the contact angle depends not only upon the size of the drop given by drop height and drop contact radius but also upon its shape given by the radius of curvature at the apex.18 They calculated the radius of curvature from the capillary constant of the drop liquid and rearranged the tables of Bashforth and Adams20 to find the average peripheral contact angle from the dimensions of the drop. They used the contact radius, rb, and the height, h, of the drop in their calculations, and the angle so obtained was regarded as the average of all the various contact angles existing along the circumference of the drop.18 Since a sessile drop may show varying horizontal profiles having different rb and h according to the direction of photographing or when one looks for a revolving drop along its axis, then the validity of the Mack and Lee paper18 from a single profile is questionable. Recently, Neumann and co-workers developed a new method (ADSA-CD) to determine the horizontal profile of the drop simultaneously with the top view which was specifically used to calculate the average contact diameter.21 Then they calculated the mean contact angle by using predetermined drop volumes, and this method resulted in contact angles approximately 10° greater than that of the angles calculated by applying spherical cap geometry, especially for low contact angles.21 In this paper, an attempt was made to determine the mean peripheral contact angle from the evaporation rate of water sessile drops on a PMMA surface by using the precise data of
J. Phys. Chem. B, Vol. 102, No. 46, 1998 9235 Rowan et al.13 No drop profile data were used in the calculations and only the knowledge of the drop contact radius was required. The initial volumes of the drops were calculated by using the three-parameter spherical geometric model and were taken from ref 16, and the product of the diffusion coefficient with the vapor pressure for water was taken from ref 17. Theory By geometry, when the parameters rb and θi are used, the initial volume of the spherical cap drop is given as
Vi )
πrb3(2 - 3 cos θi + cos3 θi) 3 sin3 θi
(1)
where rb is the contact radius between the solid-liquid interface and θi is the initial contact angle at time t ) 0 of the evaporation period. The initial surface area, Ai, of the liquid-air interface for this sessile drop is known to be
Ai )
2πrb2 1 + cos θi
(2)
Then, the initial Vi/Ai ratio is given as
Vi rb(2 + cos θi)(sin θi) ) Ai 6(1 + cos θi)
(3)
When the vapor concentration of the substance is considered, the initial evaporation rate is given as13-16
-FL
(dVdt ) ) D∫∇Bc dA ) D∫∂n∂c dA
(4)
where D is the diffusion coefficient (cm2 s-1) and FL is the density of the liquid (g cm-3); c is its vapor concentration (g cm-3) and ∂n denotes the element of the outward normal (cm); the integral of the concentration gradient is taken over the entire surface of the spherical cap drop. To perform the integration in eq 4, the concentration gradient is assumed to be radially outward, and using the boundary conditions c ) c∞ as R f ∞ and c ) c0 as R ) Rsphere, it is assumed that
∂c (c∞ - c0) ) ∂n R
(5)
where c0 is the vapor concentration at the surface of the drop, c∞ is the vapor concentration at infinite distance, and R is the radius of the drop. Equation 5 is the main assumption of the vapor diffusion model of Rowan et al.13 and Birdi et al.9 There were some conceptual differences between the vapor diffusion models of Shanahan et al.12 and Rowan et al.13 Shanahan et al.12 introduced a stagnation layer having a thickness and they used a drop model from saturation to zero over this stagnation layer thickness. Since they used a dish of silica gel within their experimental contact angle measurement box and allowed a “drying time” for the absorption of the water vapor present in this box by the silica gel, they assumed that c ) 0 as Rs f ∞ and ∂c/∂n ) c0R/Rs, where Rs is the radius of the spherical surrounding atmosphere (stagnation layer). Instead, Birdi et al.9 and Rowan et al.13 experimented only in the open air conditions where the water vapor was present in a definite amount in the ambient atmosphere which can be calculated if the relative humidity and the ambient temperature are known and they
9236 J. Phys. Chem. B, Vol. 102, No. 46, 1998
Erbil
assumed that eq 5 was valid in these circumstances. It is difficult to imagine that the concentration gradient goes to zero over a stagnation length for water vapor diffusion in open air conditions since it would then not be possible to match to the ambient environment. (However, for an organic liquid the vapor concentration in ambient air would be zero and there is no difference between both approaches). Equation 5 can be derived so that22 if we assume that the concentration gradient ) ∂c/∂R, and using a perfect sphere not supported by a surface where dV/dt ∼ ∂c/∂R and assuming that dV/dt is independent of R which enables this to be integrated with respect to R within limits of c∞ and csurface, then it is possible to obtain dV/dt ∼ ∆c/R, thus giving the assumption ∂c/∂R ) ∆c/R. Since rb ) R sin θi by geometry, and ∆c ) c∞ - c0, then by combining eqs 4 and 5, one obtains
-FL
∆c sin θi dA rb
(dVdt ) ) D ∫
(6)
Langmuir23 assumed that D is practically independent of the partial density of the vapor of the evaporating substance for the low concentrations of the evaporating vapors. By using the ideal gas laws, he replaced
∆c )
M∆pv RT
(7)
where M is the molecular weight, ∆pv is the vapor pressure difference of the evaporating substance, R is the gas constant, and T is the absolute temperature in Kelvins. ∆pv is the pressure difference between the saturation vapor pressure on the surface of the drop at its surface temperature (drop surface cooling by evaporation should be considered17) with the vapor pressure of the same subtance at a considerable distance far from the drop at the ambient temperature. D is dependent on the ambient temperature. The liquid density, FL, is dependent on the bulk drop temperature. When water drops on solids are used
∆pv ) p0 - p∞
(8)
where p0 is the saturation water vapor pressure at the drop surface temperature and p∞ is the water vapor pressure in the ambient still air which can be calculated by multiplying the relative humidity, RH, and the water vapor pressure at the ambient temperature. By combining eqs 6 and 7 and integrating, one obtains
(D∆pv)M dV (Ai sin θi) )dt FLRTrb
(Ai sin θi)rb
)
(2 + cos θi) (1 + cos θi)
Results and Discussion The precise data of Rowan, Newton, and McHale13 were used in the calculations. They applied water drops on a PMMA polymer as this gave a static contact radius and an initially large contact angle of around 80° which progressively decreased over a period of a few minutes. The temperature for all experiments was constant to within 0.5 °C at 21.5 °C and the relative humidity was typically 55%. The contact radii were 0.293, 0.324, 0.381, 0.451, 0.491, and 0.585 mm and the approximate angular range was from around 86°-39°. The numerical values of the change of the drop height and the tangential contact angle by time for constant drop radii was obtained from Dr. Glen McHale and reported previously in Table 1 of ref 15. The average of the six experimental θi was found to be 80.3° ( 5.2° which deviated 6.5% from the mean. However, when the data giving the lowest θi ) 75.6° for rb ) 0.451 mm, and the highest θi ) 86.0° for rb ) 0.293 mm, were discarded, then the average of the four experimental θi was found to be 80.1° ( 1.2° which deviated only 1.5% from the mean and is within the range of any experimental error. The initial volumes of the sessile drops can be determined if the drops are weighed (or their volumes are precisely metered by means of a microsyringe) at the beginning of the evaporation experiment and if the density of the drop liquid is known precisely at the experiment temperature. However, for most of the literature relating contact angles, only drop contact radii and/ or drop heights were reported and no publication was found in the literature for the simultaneous report of the drop volume and contact angle during evaporation. To overcome this problem, we first calculate the drop volumes of Rowan et al.’s13 precise data by using a three-parameter spherical cap geometric model16 and we assumed that these volume figures were obtained by weighting experiments. This was the only possible procedure to test the applicability of our attempt to compare the calculated peripheral contact angles with the experimentally measured tangential contact angles. Volume Evaluation. When the three different equation sets of the classical two-parameter spherical cap geometry (which are given in Table 1 of ref 16) are used to calculate the volume and the surface area of the drops, three different results are obtained which effect the quality of any analysis applied. To minimize this problem, all the measurable three parameters, rb, θ, and h, are combined in a single equation to describe the volume of the spherical cap. One of the possible threeparameter spherical cap equations is given below:
(9) V3 )
Thus, it is possible to obtain the product of the initial area, Ai, and sinus of the initial contact angle, sin θi, if the constant contact radius, rb is measured and the decrease of the volume (or weight) of the sessile drop is monitored and the product of D∆pv is known. The product of D∆pv can be determined by following the evaporation of fully spherical drops17 or can be estimated if the drop surface cooling is measured. When the (Ai sin θi) value is obtained, it is possible to calculate the initial peripheral contact angle, θi, by using the rearranged form of eq 3:
6Vi
Since all the terms on the left side of eq 10 are known experimentally, then it is possible to calculate θi.
(10)
πrb2h (2 + cos θ) 3 (1 + cos θ)
(11)
Surface Area Evaluation. With the same reasoning, one obtains the following for the surface area of the spherical cap drop:
A3 )
2πrbh sin θ
(12)
The change of the drop height and contact angle by time for a constant drop radius was taken from ref 13 and the volumes of the evaporating six water drops on a PMMA polymer having different sizes were calculated by using eq 11 and were given in Table 1. The volume decrease by time of the largest four
Peripheral Contact Angle of Sessile Drops
J. Phys. Chem. B, Vol. 102, No. 46, 1998 9237
TABLE 1: Decrease of the Volumes of the Evaporating Six Water Drops on a PMMA Polymer Having Different Sizes13 by Time, Which Were Calculated by Using Equation 11 time (s)
V3 (10-4 cm3) rb ) 0.0585 cm
V3 (10-4 cm3) rb ) 0.0491 cm
V3 (10-4 cm3) rb ) 0.0451 cm
V3 (10-4 cm3) rb ) 0.0381 cm
V3 (10-4 cm3) rb ) 0.0324 cm
V3 (10-4 cm3) rb ) 0.0293 cm
3.202
1.851
1.316
0.844
3.049
1.816
1.210
0.800
2.965
1.750
1.179
0.737
2.878
1.637
1.100
0.680
2.750
1.580
1.025
0.615
2.646
1.487
0.960
0.557
0.431 0.400 0.385 0.353 0.335 0.314 0.285 0.260 0.246 0.217 0.206 0.189
2.542 2.465 2.361 2.248 2.166 2.067 1.939
1.399 1.303 1.219 1.143 1.045 0.976
0.888 0.820 0.753 0.693 0.650 0.590
0.507 0.445 0.408
0.536 0.508 0.466 0.445 0.412 0.387 0.376 0.344 0.310 0.286 0.267 0.256 0.221
0 15 30 45 60 75 90 105 120 135 150 165 180 210 240 270 300 330 360
Figure 1. The volume decrease by time for the largest four drops.
TABLE 2: Evaporation Rates, Ai sin θi Products and the Results of Equation 10 for All the Water Drops contact radius rb (cm)
initial volume (10-4 cm3)
evaporation rate -(dV/dt) (10-7 cm3/s)
Ai sin θi
(2 + cos θi) / (1 + cos θi)
0.0585 0.0491 0.0451 0.0381 0.0324 0.0293
3.202 1.851 1.316 0.844 0.536 0.431
3.3943 2.7765 2.1970 1.8844 1.7106 1.4855
0.019 210 0.013 189 0.009 586 0.006 946 0.005 362 0.004 211
1.709 559 1.715 025 1.826 407 1.913 569 1.851 190 2.096 014
drops was given in Figure 1. The regression coefficients (RSq.) were found to vary between 0.9949 and 0.9984 for all the drops. The evaporation rates -(dV/dt) which were found from the slopes of these plots were given in Table 2. The product of the diffusion coefficient at 21.5 °C with the water vapor pressure at the drop surface temperature in 55% relative humidity (D∆pv) was found to be equal to 1.052 cm2 mmHg s-1 by graphical interpolation of the values given for the evaporation of fully spherical water drops in ref 17. The molecular weight of water was taken as 18.016 g mol-1, the density of water at 21.5° C was taken as 0.9979 g cm-3, and
the gas constant was taken as 62 360 cm3 mmHg mol-1 K-1. Then the product of the initial surface area and sinus of the initial contact angle (Ai sin θi) was calculated by using eq 9 and was given in Table 2. The terms on the left side of eq 10 were calculated correspondingly by using the values of initial volumes and contact radii and were also given in Table 2. The arithmetic average of these values was 1.851 961 which corresponds to an initial peripheral contact angle of 80.0°. This value deviated only 0.4% from the average of the tangential contact angles of 80.3°. It was noted that this procedure worked very well with the data for the drop having its rb ) 0.324 mm and also good with the drops having their rb ) 0.381 and 0.451 mm. Some discrepancy occurred for the other three drops. This may be due to the improper reporting of the relative humidity which is generally accepted as a fudge factor in the usual contact angle determination practice. When the (D∆pv) product was divided by the water diffusion coefficient at 21.5 °C which corresponded to 0.247 cm2 s-1, then a water vapor preesure difference of 4.26 mmHg was obtained. Since the water vapor pressure in the ambient still air, p∞, at 21.5° C and 55% relative humidity was 19.231 × 0.55 ) 10.577 mmHg, then the saturation water vapor pressure at the drop surface (p0) was found to be 14.837 mmHg by using eq 8 which corresponds to a drop surface temperature of 17.3 °C. This means that a drop surface cooling of 4.2 °C took place during evaporation which agrees well with the reported values in the literature.17,24 Conclusion It was shown that the initial mean peripheral contact angle of sessile drops on solid surfaces can be successfully determined from the rate of drop evaporation by applying a simple method to the precise data of Rowan et al.13 Only the drop contact radius, initial weight (or volume), and the weight decrease by time and the temperature should be measured for this purpose. When water drops are considered, the relative humidity should also be known. The product of the diffusion coefficient with the vapor pressure at the drop surface for water is taken from ref 17 in which this product was found directly by experiment by following the evaporation of fully spherical water drops. The other alternative method is the measurement of the drop surface temperature in order to find ∆pv and then to estimate the D∆pv product. The peripheral contact angle so obtained may be
9238 J. Phys. Chem. B, Vol. 102, No. 46, 1998 regarded as the mean of all the various contact angles existing along the circumference of the drop. Thus, each determination yields an average result not unduly influenced by irregularities at a given point on the surface. In addition, the error in personal judgment involved in drawing the tangent to the curved drop surface at the point of contact is eliminated. References and Notes (1) Erbil, H. Y. Surface Tension of Polymers. In Handbook of Surface and Colloid Chemistry; Birdi, K. S., Ed.; CRC Press Inc.: Boca Raton, FL, 1997; Chapter 9, pp 259-306. (2) Morra, M.; Occhiello, E.; Garbassi, F. AdV. Colloid Interface Sci. 1990, 32, 79. (3) Good, R. J. Contact Angle, Wettability and Adhesion; K. L. Mittal, Ed.; VSP: Utrecht, 1993; pp 3-36. (4) Johnson, R. E.; Dettre, R. H. J. Phys. Chem. 1964, 68 (7), 1744. (5) De Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. (6) Shanahan, M. E. R. J. Phys. D: Appl. Phys. 1989, 22, 1128. (7) Andrade, J. D.; Gregonis, G. E.; Smith, L. M. Physicochem. Aspects Polym. Surf. 1985, 2, 911. (8) Erbil, H. Y.; Yasar, B.; Suzer, S.; Baysal, B. M. Langmuir 1997, 13, 5484.
Erbil (9) Birdi, K. S.; Vu, D. T.; Winter, A. J. Phys. Chem. 1989, 93, 3702. (10) Birdi, K. S.; Vu, D. T. J. Adhesion Sci. Technol. 1993, 7 (6), 485. (11) Shanahan, M. E. R.; Bourges, C. Int. J. Adhesion AdhesiVes 1994, 14 (3), 201. (12) Bourges-Monnier, C.; Shanahan, M. E. R. Langmuir 1995, 11, 2820. (13) Rowan, S. M.; Newton, M. I.; McHale, G. J. Phys. Chem. 1995, 99, 13268. (14) McHale, G.; Rowan, S. M.; Newton, M. I.; Banarjee, M. J. Phys. Chem. B 1998, 102, 1964. (15) Erbil, H. Y.; Meric, R. A. J. Phys. Chem. B 1997, 101 (35), 6867. (16) Meric, R. A.; Erbil, H. Y. Langmuir 1998, 14, 1915. (17) Erbil, H. Y.; Dogan, M., submitted to J. Phys. Chem. B, 1998. (18) Mack, G. L.; Lee, D. A. J. Phys. Chem. 1936, 40, 169. (19) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1983, 93, 169; Kwok, D. Y.; Hui, W.; Lin, R.; Neumann, A. W. Langmuir 1995, 11, 2669. (20) Bashforth, F.; Adams, J. C. An Attempt to Test the Theory of Capillary Action; Cambridge University Press: London, 1883. (21) Skinner, F. K.; Rotenberg, Y.; Neumann, A. W. J. Colloid Interface Sci. 1989, 130, 25; Li, D.; Cheng, P.; Neumann, A. W. AdV. Colloid Interface Sci. 1992, 39, 347. (22) McHale, G. Private communication, 1998. (23) Langmuir, I. Phys. ReV. 1918, 12, 368. (24) Peiss, N. C. J. Appl. Phys. 1989, 65, 5235.
