Measurement of small contact angles for sessile drops

Measurement of Small Contact Angles for Sessile Drops LEONARD R. FISHER CS1RO Division of Food Research, P.O. Box 52, North Ryde, N.S.W. 2113, Australia Received N o v e m b e r 7, 1978; accepted February 14, 1979 The contact angle (0) of a sessile liquid drop on a horizontal solid surface can be calculated from the drop volume and the radius o f the contact circle at the liquid/solid interface. A simple apparatus which allows simultaneous estimation of these two parameters is described, and tests of the method for two systems are reported. The first system is a 3.25 mole liter -1 solution of 1-propanol in water on paraffin wax. The advancing (0a) and receding (0r) contact angles at 20°C are found to be (59.5 -+ 1.0) ° and (54.3 +_ 0.3) °, respectively, in good agreement with the literature values and those found by direct measurement. The second system chosen is cyclohexane (a volatile liquid) on cleaved mica at 20°C. Two mica sheets were used. Mean contact angles of cyclohexane on the first mica sheet are 0~ = (7.45 _+ 0.10) °, 0r = (6.99 +-0.12) °, For cyclohexane on the second sheet the mean contact angles are 0a = (6.48--+ 0.31) °, Or = (5.56 +- 0.06) °, The difference b e t w e e n advancing and receding contact angles is statistically significant ( P < 0.01) for both sheets. Other methods of comparable accuracy exist for 0 ~> 30 °, but the accuracy of most o f these methods diminishes rapidly if 0 ~ 30 °. If calculation of the contact angle from the spacing between interference fringes is not appropriate, then estimation from drop volume and contact circle radius becomes the method of choice if 0 ~< 30 °. INTRODUCTION

the Young-Laplace equation (5-7). A number of approximate solutions have also been proposed (2, 5). No adequate solutions exist for sessile drops which do not have circular symmetry. 1 If it is known whether 0 is less than or more than 90° , and if the surface tension and density of the liquid are known, only two further parameters are needed for the calculation of the contact angle from the integrated Young-Laplace equation. For 0 < 90°, the combination most often used has been the maximum drop height and the radius of the contact circle (3). The combination of drop volume and contact circle radius, although suggested by Bikerman (4) and used for 0 > 90° (Ref. (2)), seems not actually to have been used for 0 < 90°.

The contact angle (0) of a sessile liquid drop on a horizontal solid surface can be obtained directly by measurement of the slope of a tangent to the profile (Fig. 1) at a point where the liquid and solid meet (1). The inevitable subjective quality of such measurements can lead to significant errors, particularly for small contact angles. Methods have therefore been developed (2-5) to derive the contact angles of drops with circular symmetry from parameters which can be measured more accurately. The parameters most often used are the maximum height, the drop volume, and the radius of the contact circle at the liquid/solid interface (Fig. 1). To calculate the contact angle, an expression is needed for the shape of the drop profile in terms of the measured parameters. For drops with circular symmetry, the drop profile (taken through the axis of symmetry) can be obtained by numerical integration of

i As Pujado (8) has pointed out, Larkin's solution (9) for a sessile drop on an inclined plane, even if verified, would be only one o f a family of possible solutions.

200 0021-9797/79/140200-06502.00/0 Copyright © 1979 by Academic Press, Inc. All rights of reproduction in any form reserved.

Journal of Colloid and Interface Science, Vol. 72, No. 2, November 1979

SESSILE DROP CONTACT ANGLES

(o,o)

(R,O)

+x

FIG. 1. Sessile drop profile (through the axis of symmetry) for 0 < 90° in Cartesian coordinates. Drop parameters most often used in the calculation of 0 are height (z), maximum height (z0), radius (x), radius of the contact zone (R), and volume. The aim of the present w o r k is to investigate the merits of this alternative method, especially in the m e a s u r e m e n t of small contact angles. M e a s u r e m e n t s are r e p o r t e d on two systems; a 3.25 mole liter -~ solution of 1-propanol in water, with paraffin wax as the solid substrate, and c y c l o h e x a n e with mica as the solid substrate.

201

adjusted to m a k e the stainless steel plate accurately horizontal. All e x p e r i m e n t s were p e r f o r m e d at ambient t e m p e r a t u r e [(20 + 3)°C]. E a c h experiment was begun by placing the solid substrate on the middle of the stainless steel plate. In the experiments reported here the solid substrate was either a step-free sheet of cleaved mica, about 3 c m square, cut from a larger sheet, or a layer of paraffin wax, p r e p a r e d as a thin film by melting and spreading onto the stainless steel plate. The apparatus, with the solid substrate in position, is then leveled [while suspended from the loop (F)] to within 0.1 ° of the horizontal with the aid of a cathetometer. The apparatus (plus solid substrate) is first tared in the balance. F o r the measurement of advancing contact angles, liquid is

MATERIALS AND METHODS

Materials Ruby Muscovite mica from Bihar, India, was chosen for its excellent cleaving properties. The paraffin wax was histological grade and was not subjected to further purification. 1-Propanol was Reagent grade, redistilled via a short reflux column and a small middle fraction taken (bp = 97.2°C; lit. 97.2°C (10)). C y c l o h e x a n e (Analytical Reagent grade) was similarly redistilled (bp = 80.7°C; lit. 80.74°C (10)). The w a t e r was tripleglass-distilled, and was used immediately after r e m o v a l f r o m the still.

Experimental Method The apparatus (Fig. 2) replaces the balance pan in a standard analytical balance capable of reading to 10 -5 g (a Mettler H160T). The working surface is a flat stainless steel plate (A) with a scale in millimeter divisions m a r k e d on it. The plate rests on three ball bearings (B) which are in turn glued to supporting bars (C). Leveling screws (D) attached to the supporting f r a m e (E) can be

D FIG. 2. Apparatus for measurements of sessile drop mass and diameter. Journal o f Colloid and Interface Science, Vol. 72, No. 2, November 1979

202

LEONARD R. FISHER

fed f r o m a microsyringe onto the substrate surface. P h o t o g r a p h s of the drop are taken, the c a m e r a being directed horizontally at the 45 ° mirror (G), so that the resulting p h o t o g r a p h gives a plan view of the drop. The m a s s is read at the time of each photograph. Receding contact angles are measured similarly, liquid being r e m o v e d f r o m the drop b y syringe or, if the liquid is volatile, by evaporation. Drop diameters are calculated f r o m measu r e m e n t s of enlarged p h o t o g r a p h s , with the scale on the stainless steel plate used for calibration. The radius of a drop can be found in this w a y with a reproducibility o f ___0.02 m m . Most drops are not circular in outline to within this margin of error. The average ratio o f m a x i m u m to m i n i m u m radius o f the contact circle (Rmax/Rmin) is 1.09 -- 0.03 for c y c l o h e x a n e on mica and 1.13 _ 0.05 for 3.5 mole liter -1 1-propanol in w a t e r on paraffin wax. Possible reasons for such noncircularity h a v e been discussed by B i k e r m a n (11). CALCULATIONS The m e t h o d depends on the relationship b e t w e e n the contact angle (0), the drop volume (v), and the radius (R) of the contact circle s at the liquid/solid interface (Fig. 1). This relationship cannot be e x p r e s s e d in closed analytical form, but a semiempirical formula can be derived which agrees well with the results o f accurate numerical calculations p e r f o r m e d by other w o r k e r s (5). 2 For noncircular drops, the contact angle is an average value of the contact angles about the drop perimeter. Two possible ways to calculate an average contact angle are (i) to take R = ( Rma x + Rmin)/2, or (ii) to calculate maximum and minimum values of the contact angle (from Rmia and Rmax, respectively) and to take the arithmetic mean of these contact angles. The difference between values of the contact angle calculated by these two methods depends on the value of Rmax/Rmin. For Rmax/Rr.in = 1.10, contact angles calculated by the second method are about 1% higher than those calculated by the first method. For Rmax/Rmin= 1.20, the difference is about 2%. All values reported in this paper have been calculated by the first method. Journal of Colloid and Interface Science, Vol. 72, No. 2, November 1979

F o r a liquid of density p and liquid/vapor interfacial tension 3', we first define a dimensionless radius ( 0 of the contact circle at the liquid/solid interface by ( = R(3"lpg) -1/2,

[1]

where g is the acceleration due to gravity. An analytical expression (12) which yields an a p p r o x i m a t e value o f 0 accurate to within 10% for 0 <- 30 ° and 1 <- ( _< 4.5 is then tan 0 ~ tan 0a~p = (4~'/zrR3){((/4)[Ii(O/12(O]},

[2]

where I 1 ( 0 and 12(0 are modified Bessel functions of the first kind of orders 1 and 2, respectively? More accurate expressions for 0, derived to c o v e r the experimental range o f values of (, are 0

:

0app -[- [42 + 20(] × 10-6~pp + [0.65 -- 0.33(] × 10-604a0,,

[3]

which is accurate to within 0.1% of the values found by numerical integration (5) for 0 °-< 0 - < 3 0 ° and l <- ( <- 4.5, and 0 = 0.729( + [1 - 0.051(] O~pp + 8.93

×

10-4(02app + [42 + 20(]

× 10-603app + [0.65 -- 0.33(] x 10-604a~,,

[4]

which is accurate to within 0.1% of the values found by numerical integration (5) for 30 ° < 0 -< 60 ° and 2.5 <- ( -< 4.5. The values of 0 and 0avv in Eqs. [3] and [4] are expressed in degrees. Equation [3] or Eq. [4] has b e e n used to calculate all of the contact angle values reported in the present work. 3 For convenience of numerical calculation, Eq. [2] can be approximated by tan

O~p =

-~

x I ~: + f3/8 + ~5/192 + ~Y9216 +_ ~:9,'737280 l (~: ~-~3/-i-~+ ~:5/384 + ~:V23040 + ~9/2211840J "

SESSILE DROP CONTACT ANGLES

TABLE 1 Contact Angles of 3.25 mole liter -1 1-Propanol/Water on Paraffin Wax ~

Run no.

Oa

No. of measurements

1 2 Overall

(58.7 _+ 0.9) ° (61.0 _+ 2.4) ° (59.5 _+ 1.1) °

5 3 8

Or

No. of measurements

(53.7 ___ 0.2) ° (54.9 _+ 0.6) ° (54.3 ~- 0.4) °

2 2 4

E r r o r s q u o t e d are s t a n d a r d e r r o r s .

RESULTS

(i) 1-Propanol/Water (3.25 moles liter -1) on Paraffin Wax This system was chosen because the relatively high contact angle ( 0 - ~ 60 °) allows comparison with other methods of measurement (2, 13). Two runs were performed, a run being defined as a series of measurements of drop mass and radius, first after the placement of a sessile drop, then after each addition of liquid to the drop, and after each removal of liquid from the drop by a syringe. Values of 0 are calculated from Eq. [4]. The values of 3/and p are 28.3 mN/m (calculated from Eq. [3] in (13)) and 962 kg/m 3 (measured by pycnometer), respectively, at 20°C. The results are given in Table I. The mean values found for the advancing (0a) and receding (0r) contact angles are 0a = (59.5 + 1.1) ° (mean of eight readings) and Or = (54.3 + 0.4) ° (mean of four readings). No significant d e p e n d e n c e o f the contact angle on contact circle radius was found. Measurement by protractor of the slope of the tangent to the drop profile gave 0a = (61 + 2) °, 0r = (52-+ 2) ° (mean o f 10 readings in both cases). Other workers (13), using a Wilhelmy balance technique, have found 0a = 63 ° and 0r = 53 °, in each case with a standard error of about -+3°, as judged from the scatter of their experimental points. No d e p e n d e n c e of contact angle on time was found. This rules out the possibility that significant dissolution o f the

203

paraffin wax occurred over the time of an experiment (<5 min in all cases). The difference between the advancing and receding contact angles for this system is almost certainly due to surface roughness of the paraffin wax (14, 15).

(ii) Cyclohexane on Mica A number of runs were performed on two different mica sheets. A run here is defined as a series of measurements of drop mass and radius, first after the placement of a sessile drop, in some cases after further addition of liquid to the drop, and then as the drop is allowed to evaporate. Complete evaporation of a drop took at least 10 min, since the atmosphere of the balance chamber was nearly saturated with cyclohexane vapor. Values of 0 were calculated from Eq. [3], with 3 / = 25.5 mN/m and 19 = 779 kg/m 3 at 20°C (10). The results are given in Table II. With the exception of Run 1 on mica sheet 1, the advancing contact angle during a run was greater than the receding contact angle. For both mica sheets, the overall difference between advancing and receding contact angles (0a and 0r, respectively) is statistically significant at a high level of confidence ( P < 0.01) (see Table II). No significant dependence of contact angle on contact circle radius was found. The results in Table II are all for mica which has been freshly cleaved in room air (16). A fresh mica surface exposed to room air quickly gains a monolayer of watersoluble material (17) which, while carbonaceous,4 does not appear to be adsorbed hydrocarbon (which would not be watersoluble). The contact angle of cyclohexane on a clean mica surface is expected to be zero (18). This was confirmed for the mica surfaces used in the present experiments 4 A n a l y s i s b y e l e c t r o n s p e c t r o s c o p y for c h e m i c a l analysis of the exposed surface of mica which has b e e n f r e s h l y c l e a r e d in air r e v e a l s the p r e s e n c e o f significant a m o u n t s o f c a r b o n ( - 5 % ) in the first f e w atomic layers. Journal of Colloid and Interface Science, Vol. 72, No. 2, November 1979

204

LEONARD R. FISHER TABLE II Contact Angles of Cyclohexane on Mica~

Mica sheet no.

Run

no,

0a

No. of measurements

Or

No. of measurements

1

1 2 3 4 5 Overall

(7.86 _+0.22)° (7.54 -+ 0.13)° (7.15)° (7.23)° (7.15 -+ 0.13)° (7.35 +- 0.10)°

2 4 1 1 2 8

(8.06 _+ 0.26)° (6.79 _+ 0.04)° (6.97 _+ 0.10)° (6.58 _+ 0.16)° (6.83 +_ 0.07)° (6.77 _ 0.12)°

4 7 2 5 5 19

2

1 2 3 Overall

(6.81 _+ 0.34)° (5.52)° (6.08)° (6.48 -+ 0.31)°

4 1 1 6

(5.73 + 0.13)° (5.46 _+ 0.10)° (5.53 + 0.03)° (5.56 _+0.06)°

6 9 6 21

a Errors quoted are standard errors. after they were cleaned by exposure to strong ultraviolet radiation (19) for 24 hr. The difference between advancing and receding contact angles cannot be due to surface roughness, since the surface of the mica was step-free, and this has been found to be a sufficient criterion for the surface to be molecularly smooth (20). The difference may be due, however, to chemical or energetic heterogeneity of the mica surface. A n o t h e r possibility is that droplet cooling during evaporation may produce surface tension gradients and hence a Marangoni instability (21). DISCUSSION AND CONCLUSIONS For a perfectly circular liquid/solid contact region, the a c c u r a c y of the estimated contact angle depends on the a c c u r a c y with which both the contact circle radius and the drop volume can be measured. With the experimental method reported here, the contact circle radius could be measured with a reproducibility of +-0.02 m m and the drop volume with a reproducibility of +-(5 x 10-5) ml. F o r cyclohexane on mica, this corresponds to reproducibility in 0 ranging from ---1.5% (for R ~ 7 mm) to +-4.5% (for R--~ 3 ram). The corresponding ranges for 3.25 mole liter -1 1-propanol in water on paraffin wax are _ 1 and +_2%. Journal of Colloid and Interface Science, Vol. 72, No. 2, N o v e m b e r 1979

The standard errors actually found for the receding contact angles (_+ 1.7% overall for cyclohexane on mica sheet 1, - 1 % for cyclohexane on mica sheet 2, and +_1% for 3.25 mole liter -1 1-propanol/water on paraffin wax) are near the upper end of the expected range in each case. Noncircularity of the drop has surprisingly little effect on the scatter of the calculated contact angles. For example, the overall standard errors for the receding contact angle of cyclohexane on mica are +_0.12° for mica sheet 1 and +_0.06° for mica sheet 2, although the average values of Rmax/Rmin a r e 1.05 and 1.16, respectively. The standard errors for the advancing contact angles (+_1.4% overall for cyclohexane on mica sheet 1, _+4.8% for cyclohexane on mica sheet 2, and +- 1.7% for 3.25 mole liter -1 1-propanol/water on paraffin wax) are greater than those for the receding contact angles. This is in accordance with the experience of other workers (14), and is probably a real effect, due to surface roughness for the 3.25 mole liter -1 solution of 1-propanol in water on paraffin wax, and to surface heterogeneity or the presence of a Marangoni instability in the case o f mica. The only other method capable of such a c c u r a c y for contact angles in the range 5° ~< 0 ~< 30° is to calculate the contact angle

SESSILE DROP CONTACT ANGLES

from the separation of m o n o c h r o m a t i c interference fringes (22). This m e t h o d is not always suitable; for example, it is difficult to use with volatile liquids and the accuracy is greatly diminished if the substrate scatters or strongly absorbs light. The m e t h o d described in this paper thus fills a gap, being the only reasonably accurate m e t h o d in many cases for 0 ~< 30 ° and also for 30 ° ~< 0 ~< 60 ° in cases where measurements of drop height cannot be made.

10.

ACKNOWLEDGMENTS

11.

Thanks are due to Mr. W. Rushton (of the CSIRO Division of Food Research), who developed the photographic system which allowed the accurate resolution of the three-phase contact line upon which the accuracy of the final results depend. Thanks are also due to Dr. A. Buckley (of the CSIRO Division of Textile Physics), for performing the ESCA analyses, and to Ms. Elaine Smith and Ms. Mary Wiilcox (of the CSIRO Division of Mathematics and Statistics) for aid with the statistical analyses, and to Dr. G. E. Hibberd (Bread Research Institute of Australia) for a number of suggestions. The sample of mica used was kindly provided by Dr. J. N. Israelachvili, of the Australian National University.

12.

6.

7. 8. 9.

13. 14. 15. 16. 17. 18. 19.

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Journal of Colloid and Interface Science, Vol. 72, No. 2, November 1979