T H E DETERMINATION OF CONTACT ANGLES FROM MEASUREMENTS OF T H E DIMENSIONS OF SMALL BUBBLES AND DROPS. I1
THESESSILEDROPMETHODFOROBTUSEANGLES‘ GUILFORD L. MACK AND DOROTHY A. LEE Division of Chemistry, New York State Agricultural Ezperiment Station, Geneva, New York
Received June $0, 1935 INTRODUCTION
It has been suggested in a previous communication (3) that widely variant surface energies may exist a t closely adjoining points on a surface. Well-substantiated theory as to the surface structure of solid catalytic materials is in accord with this view (7). The “active patches” on the catalytic surfaces are an extreme example of irregularity in the surface energy, but it seems reasonable to suppose that such irregularities may exist to a lesser degree in nearly all ordinary surfaces. Photographic evidence in support of this proposition appears in the work of Wark and Cox (9), who found that the same air bubble under a mineral surface wet with water might have an angle of contact on the right side different from that on the left. Instead of measuring the contact angle directly, it may be calculated from the dimensions of the drop. The angle so obtained may be regarded as the integral of the sum 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 a t a given point on the surface. For precise determinations the method should have an especial advantage over the usual procedure of direct measurement, because the error in personal judgment involved in drawing the tangent to the curved drop surface a t the point of contact is eliminated. This error becomes increasingly important as the contact angle approaches 180°,while the dimensions of the drop may be measured with the same degree of accuracy as before. If the image of the drop is recorded on a photographic plate, the capillary Presented a t the Twelfth Colloid Symposium, held a t Ithaca, New York, June 20-22, 1935. Approved by the Director of the New York State Experiment Station for publication as Journal Paper No. 94, J u n e 19, 1935. I69
170
GUILFORD L. MACK AND DOROTHY A, LEE
constant of the liquid may be determined, without any additional experimental data, by the method of Dorsey (2). This serves as a valuable checli upon the purity of the liquid-air interface and upon the reliability of the contact angle determination. THEORETICAL
The tables of Bashforth and Adams (1) give the necessary information for calculating the contact angle from the dimensions of the drop. For thc present purpose a considerable rearrangement of their data was necessary, because the units tabulated by them cannot be readily calculated from experimental data. In choosing other units, those have been selected which may be measured precisely or which may be readily calculated from other experimental data. The most easily measured dimensions of the drop represented by figure 1 are the total height z and the radius r. These two lengths determine the I I(
A
I
2r
i
__ ___x - __ ____ - - - -__B -- - - -
- - ----
FIQ.1. Outline of a drop AOB resting upon the horizontal plate A B
size of the drop. Unfortunately, the contact angle depends not only upon the size of the drop but also upon its shape. In a drop of given size, the shape may be related to the capillary constant a of the liquid, or to the radius of curvature a t the apex of the drop ( b in figure 1). The contact angle is thus a function of two independent variables. Hence the contact angles contained in the body of table 1 must be related to both the size factor x/r listed in the vertical column at the left and to the shape factor r / b arranged in the horizontal column a t the top of the table. The procedure employed in making up table 1 was as follows: The original Bashforth and Adams tables contain, for each 5' interval of 0, values of z / b corresponding to given values of r / b and /3 = 2b2/a2. The values of z / r were found by dividing z / b by the corresponding value of r/b. Then for each value of 8, the values of z / r corresponding to equal increments of r/b were obtained by numerical interpolation. For each even value of r / b so obtained, the values of e corresponding to equal incre-
DETERMINATION OF CONTACT ANGLES
171
ments of z / r were determined. This could be done with sufficient accuracy by means of graphical interpolation. The values of e are correct to the nearest 0.1'. This error is well within the accuracy of most experimental data. The calculations for the factor r / b have been extended only to r / b = 0.90. This is far enough to include a drop of water nearly 4 mm. in diameter. TABLE 1 V a l u e s of 0 corresponding to related values of z / r and r/b T
b
1.00
0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 __
90.0 92.9 95.7 98.6 101.5 104.5 107.5 110.5 113.6 116.7 120.0 123.4 126.9 130.5 134.4 138.6 143.1 148.2 154.2 161.8 180.0
-
0.99
0.98
0.97
096
0.95
90.1 93.5 96.8 100.3 103.9 107.5 111.2 115.1 119.2 123.5 128.3 133.4 139.2 146.1 155.0 174.2
90.9 94.4 97.9 101.6 105.3 109.1 113.1 117.3 121.8 126.5 131.8 137.7 144.8 153.8 171.t
0.94
__ ___ _ _ _ _ _ _ _ _
90.8 93.8 96.8 99.9 102.9 106.0 109.2 112.5 115.8 119.3 123.0 126.5 130.8 135.1 139.8 145.1 151.4 159.8
91.7 94.8 97.9 101.1 104.3 107.6 111.0 114.6 118.2 122.1 126.1 130.4 135.1 140.3 146.4 153.9 165.9
92.5 95.7 99.1 102.4 105.9 109.4 113.0 116.8 120.8 125.1 129.8 134.6 140.1 146.7 155.1 171.5
~
I-I--
91.8 95.4 99.1 102.9 106.8 110.9 115.1 119.7 124.6 129.9 135.8 142.8 151.2 167.0
0.93
0.90
90.1 91.5 92.7 95.6 96.4 99.8 100.4 104.2 104.3 105.7 107.: 108.8 108.5 110.1 111.l 113.7 112.8 118.9 117.3 124.6 122.2 131.0 127.6 138.3 133.5 147.3 140.3 161.4 148.9 161.1
l l
It has been shown (3) that greater precision in contact angle measurements can be attained by the use of small drops. Hence there is nothing to be gained by working with larger drops of correspondingly flatter shape. The quantity b used in table 1 cannot be experimentally determined except by optical methods unsuited to the attainment of the requisite degree of accuracy. It may be readily calculated, however, from the value of the capillary constant a. The simplest relation between a and b is given by the equation
h = -a2 b
1i2
GUILFORD L. MACK AND DOROTHY A. LEE
where h is the height of rise of the liquid in a capillary tube of unit radius. By substituting this value of h in Rayleigh's equation (4) for a in terms of T and h, and transforming, a solution for b is obtained in the form of an infinite series, as follows: b =r
r3 2rb r7 +-(3log2 - 2) + - (78log2 - 53) - .. 3a2 9a4 27a6
S'erschaffelt (8) has developed equation 2 in this form as far as the third term on the right-hand side of the equation, and has pointed out its usefulness for calculating the value of b. By taking known values of b from the Bashforth and Adams tables, substituting the corresponding values of r and a in equation 2, and solving for b, the error involved by leaving off successive terms of the infinite series may be calculated. This procedure reveals an interesting fact which no one appears to have noticed heretofore. The last two terms in equation 2 add very little to the accuracy of the approximation even when r / a is small. For larger values of r / u , these latter termr actually increase the error beyond that existing after the calculation of the second term. The simplified equation
_b -- 1 + - 1- r2 r
3 a2
(3)
is amply sufficient for the present purpose. For r / a = 0.5, the error in calculating b from equation 3 is 0.06 per cent, from equation 2 it is 0.05 per cent. These amounts are insignificant in comparison with the experimental error in determining the contact angle. A further advantage in the use of equation 3 is that the value of a need be known only approximately. An error of 1per cent in the determination of a produces an error of only 0.06 per cent in the value of b when r/a = 0.3. The value of r / a must be determined with great precision only when '6 approaches 180". For smaller angles, the values of r / a and r / b have much less effect on the determination of e. Sugden (6) has published a table which may also be used for finding r/b when r / a is known. EXPERIMENTAL
The apparatus necessary to measure the dimensions of the small drops and bubbles consists of the following items. A low-power microscope is equipped with a filar micrometer eyepiece or a camera attachment. The solid surface requires a holder adjustable by rack and pinion movements in three directions. A microburet of the type used by Rehberg (5) is needed to form the very small bubbles and drop.. The solid holder and the tip of the microburet project downyard through the glass cover into an
DETERMINATION OF CONTACT ANGLES
173
absorption cell with plane glass sides. The cell holds the liquid into which air bubbles are blown, and, in the case of drops in air, it protects the surface from contamination by dust particles. Rapid evaporation of the very small drops is prevented by previously saturating the air within the cell
FIQ.2. Water drop on paraffin solidified in moist air. 8 (calculated) = 107.7'; ft'l = 107.5"; 8, = 102.8".
z/r
FIQ.3. Water drop on paraffin solidified in moist air. z / r
= 1.281; r / b = 0.9919;
=
1.239; r / b = 0.9768;
0 (calculated) = 107.4"; et = 107.1"; 8, = 107.3".
with the vapor from other drops of the same liquid. A comparator or travelling microscope is convenient for measuring the small distances on the photographic plates, particularly if the capillary constant is to be determined by Dorsey's method (2).
174
GUILFORD L. MACK AND DOROTHY A. LEE
The following cases will serve to illustrate the use of the proposed method of determining contact angles. The paraffin surfaces shown in figures 2 and 3 were solidified in air of 60 per cent humidity and then kept for one hour in the cell containing saturated water vapor. The drop in figure 2
8
FIG.4. Water drop on paraffin solidified in dry air. z / r = 1.310; r / b = 0.9955; (ralculated) = 108.9O; 81 = 109.8"; 8, = 108.0".
FIG.5 . Air bubble under paraffin immersed in 0.01 per cent soap snlLtion. Z I T ..391; r j b = 0.9615; 0 (calculated) = 180 - 122.4 = 57.6";Oj = 57.8"; 4 = 57.0".
=
was selected as an extreme example of irregularity in a paraffin surface. It is apparent that the low value of the contact angle on the right side of the drop is not characteristic of the surface as a whole. The observed angle on the left side agrees well with the calculated value. Figure 3 shows
DETERMINATION OF CONTACT ANGLES
175
a much larger drop a little more than 2 mm. in diameter, while in figure 4 a fairly small drop of less than half that size is shown. In the latter case the paraffin was solidified in air dried over phosphorus pentoxide, and the photograph was taken immediately after the surface was placed in the absorption cell. The method may be applied to bubbles as well as drops (figures 5 and 6). The paraffin surface is immersed in aqueous solutions of a proprietary spreading agent, and air bubbles are formed on the under side of the plate. The disadvantage of direct measurement is evident in figure 6. As the contact angle approaches 180", it becomes increasingly difficult to draw the tangent to the curved surface a t the point of contact. The dimensions z , r, and b may still be determined with the same degree of accuracy, however.
FIG.6. Air bubble under paraffin imme'rsed in 0.1 per cent soap solution. 1.659; r / b = 0.3936; 0 (calculated) = 180 - 180 = 0 f 0.5".
Z/T =
SUMMARY
A method has been proposed for determining obtuse contact angles from measurements of three dimensions of sessile drops of bubbles under a plate. These dimensions are the vertical height z , the horizontal radius T , and the radius of curvature b a t the apex of the curved surface. A simplified equation is given for calculating the value of b from that of the capillary constant of the liquid. The proposed method is shown to have several advantages over the direct method of measurement with a protractor. REFERENCES (1) BASHFORTH AND ADAMS: An Attempt to Test the Theory of Capillary Action. Cambridge (1883).
176
GI.ILFORD L. MACK A N D
nonoTnY
A. LEE
(2) D O R ~ E Y J.: Wash. Acad. Sei. 18, 505 (1928). (3) MACK:J. Phys. Chem. 40, 159 (1936). (-1) RAYLEIGH: Proc. Roy. SOC.London 92A, 184 (1915). (5) REHBERG: Biochem. J. 19, 270 (1925). (6) SUGDEN: J. Chem. SOC.119, 1483 (1921). (7) TAYLOR: J. Phys. Chem. 30, 145 (1926). (8) VERSCHAFFELT: Proc. Acad. Sci. Amsterdam 21, 366 (1919). (9) WARKA K D C o x : Am. Inst. Mining Met. Engrs., Tech. Pub. No. 401,
