THE DETERMINATION O F CONTACT ANGLES FROM MEASUREMENTS OF T H E DIMENSIONS OF SMALL BUBBLES AND DROPS. I
THESPHEROIDAL SEGMENT METHODFOR ACUTEANGLES~ GUILFORD L. MACK Division of Chemistry, New York State Agricultural Experiment Station, Geneva, New York Received January 11, 193e INTRODUCTION
The present methods of measuring contact angles all require that the solid material be obtainable in some special shape, such as a flat plate or capillary tube. Many surfaces, for example, those of plant materials, occur in irregular forms and must be dealt with in situ, because of the inhomogeneity of the body. The chief value of the method herein described is that its applicability is largely independent of the form of the solid surface. Some of the earliest determinations of contact angles were made from measurements of the dimensions of bubbles and drops. The work has been confined to large drops, but the use of very small drops may be shown to possess several advantages: (1) Small drops may be applied successfully to small plane areas in an irregular surface or even to surfaces with an approximately constant degree of curvature. (2) Small drops show a much greater variation in height for small variations in magnitude of contact angle than do large drops. Accordingly, this dimension does not have to be measured with such extreme accuracy as is necessary with large drops. (3) Small drops assume the advancing angle of contact, while larger drops exhibit a fluctuating angle of contact varying roughly between the advancing and receding angles. Other generally applicable means of measuring contact angles are the various modifications of the tilting plate method (I), and methods depending upon visual observation of the image of a drop, bubble, or meniscus projected upon a screen or photographic plate ( 7 , 8, 3). It may be noted that in each of these methods the contact angle is measured at a single point on the surface. Such a procedure is somewhat unsatisfactory for Approved by the Director of the New York State Agricultural Station for publication as Journal Paper No. 44. 159 TEE JOURNAL OF PKYSICAL CAEMIBTRY, VOL. 40, NO. 2
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GUILFORD L. MACK
working with naturally occurring objects having variable surface properties. The proposed method has the advantage of yielding a value for the contact angle which is the integral of the sum of all the separate angles along the circumference of the drop. THEORY O F THE SPHEROIDAL SEGXCNT XETHOD
The principle of the spheroidal segment method is based upon the fact that the shape of a small drop of liquid having an acute angle of contact is only slightly affected by the influence of gravity. Consequently, the surface will be nearly spherical, and as a first approximation the drop may be considered to be a segment of a sphere. The angle between the horizontal plane through the base of the drop and the tangent to the spherical surface a t the point of contact is 0 = 2 tan-' ( h / z )
(1)
where h is the greatest height of the drop and z is the radius of the base of segment. The distance z is easily measured on a scale in the eyepiece of a low power microscope, but the value of h is usually much smaller and more difficult to measure. This is especially true in the case of small angles on a surface which is not perfectly plane. Another difficulty is the fact that small drops evaporate very rapidly if the surrounding vapor is not in equilibrium, a condition which is difficult to accomplish and even more difficult to maintain. The problem may be solved by substituting the Volume for the height as a measurable dimension of the drop. The volume of a number of equally formed drops may be measured in an auxiliary reservoir and such measurements mill be entirely independent of the shape of the solid surface. Furthermore, the volume measurement is made a t the time the drop is first formed and is unaffected by subsequent changes caused by evaporatior,. It was observed experimentally that an evaporating drop maintained its original radius long after its other dimensions, such as height, volume, and contact angle, had been considerably reduced. Thus from an experimental point of view, it appeared highly desirable to determine the contact angle from measurements of the radius and volume of the drop. This may be done by expressing the value of h in equation l as a function of the radius and volume of a spherical segment. The equation relating these quantities is
h3 + 3hs2 = 6 ( V / r ) No simple algebraic expression for h in terms of z and V can be obtained, however. The solution of the cubic equation leads to a result in the form of two slowly convergent infinite series.
DETERMINATION O F CONTACT ANGLES.
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I
A preferable procedure is to transform equation 2 into the dimensionless form (*/6)(h/z)* (./2) (Vz)= V / x s
+
and tabulate numerical solutions of h / z in terms of V/za. Table 1contains the values of h/z corresponding to values of V/z3 from 0 to 2.0944, this being the range within which the angle remains acute. TABLE 1 Related values of V/xa and h / x h/z
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0ooo 0.0636 0.1267 0.1887 0.2495 0.3085 0.3657 0.4208 0.4738 0.5248 0.5737
1.1 1.2 1.3 1.4 1.5 1.6 1.7
1.8 1.9 2.0 2.1
0.6206 0.6656 0.7089 0.7504 0.7904 0.8288
0.8659 0.9016 0.9361 0.9695 1.0018
TECHNIQUE
The solid to be investigated was imbedded on a paraffined glass plate. The liquid was placed in a short length of glass tubing, one end of which had been drawn out into a very fine capillary. A spiral in the capillary tube facilitated the manipulation of the tip. Drops of the liquid were formed slowly on the tip by gravitational force. By touching the glass tip to the solid surface at definite time intervals, drops of a constant and reproducible size were detached. Ten or more drops were measured a t a time so that each determination yielded a truly average result. To minimize evaporation the solid and supporting plate were enclosed in a low form weighing bottle. The cover of this bottle was fitted with a window of optically plane glass. The diameters of the drops were observed through this window in the ocular scale of a low power microscope. The volume of liquid used for a given number of drops was obtained by measuring the difference in height of the liquid in the upper part of the glass tube before and after the drops were removed. Since the diameter of the tube was known, the cylindrical volume could be calculated. I n this investigation a tilting plate method and a direct observation method were used in order to check the measurements made by the spheroidal segment method. The procedure used in the tilting plate method was that of Adam and Jessop (1). In the visual observation method the apparatus was arranged so that the drop could be viewed directly in a
162
GUILFORD L. hIACK
microscope. The eyepiece mas fitted with an accurately centered erossline micrometer disc and a pointer which indicated the angular measure on a scale fastened around the draw tube of the microscope. The glass tip on which the drop formed was allowed to remain on the surface under examination, so that expanding and contracting drops could be produced by changing the air pressure a t the other end of the glass tube. TABLE 2 E f e c t of d r o p size u p o n the contact angle Distilled water on apple wax R A D I U S O F DROP
cm
1
1
VOLUME OF DROP
.
CONTACT ANGLE CORRECTED
cm.8
5g0 50" 53" 66" 74 90"20' 88"30' 89"30' S8"50'
0,00365 0.00142 o.oon39 0 . no038 0 00041 0.00046 0,00035 0 , ooni5 0,00012
0.1962 0,1212 0.0773 0.0687 0.0663 0,0601 0.0560 0.0418 0.0390
TABLE 3 C o m p a r i s o n of contact angles obtained b y different methods
I
SYSTEM
Distilled water
011
A J G L E O F COBTACT METHOD
Spheroidal segment
azobenzene. . . . . , , . .
i( 1
Ldvancing
Receding
89" 90 92"
62 64"
0.05 per cent soap solution on paraffin ...
Spheroidal segment Tilting plate Direct observation
48" 48" 49"
27' 30"
Distilled water on apple wax.. . . . . . .
Spheroidal segment Tilting plate
89 O 87 O
62'
DISCUSSION O F RESULTS
Bartell and Hatch (2) found that a series of fairly large drops detached from a capillary tip formed widely different angles of contact on the same surface. This observation is confirmed by the results contained in the upper portion of table 2. But it is apparent that if the drop is made sniall enough, the angle attains a constant maximum value. In table 3 it is shown that this maximum is the advancing contact angle.
t
DETERMINATION O F CONTACT ANGLES.
I
163
It has been shown ( 6 , 9 ) that both expanding and contracting drops or bubbles may be caused to assume an equilibrium contact angle by tapping the support. This is true only for large drops, for vibration had no effect upon the advancing angles formed by small drops. Recent work has seemed to indicate that advancing and receding angles are themselves equilibrium angles for the surface under each of two different conditions (4) *
From the foregoing it appears that the ease with which a drop may be made to spread further over a solid surface after having formed the advancing contact angle depends largely upon the drop size. The hydrostatic pressure due to the weight of the drop is exerted in the direction of further spreading. The capillary pressure within the drop is opposed to any extension of the liquid air surface. Since this pressure is inversely proportional to the radius of curvature, it will be greater in small drops. Hence with increasing drop size both the hydrostatic and capillary forces favor the probability of irregular extension of the drop. CORRECTION FOR T H E EFFECT O F GRAVITATIONAL FORCE ON T H E FORM O F T H E DROP
In the past considerable discredit has been cast upon sessile drop methods because unwarranted assumptions were made in the development of the mathematical equations to describe the form of the capillary surface. Therefore, in order that the proposed method may be employed with confidence, it is necessary to examine critically the effect of gravitational force upon the shape of the drop. The formulation of a correction term to compensate for this deviation from the spherical form would serve not only to evaluate the error due to this source, but also to determine the range of conditions within which the error is sufficiently small so that’it may be ignored. Consider a drop of liquid resting upon a solid surface OP under the influence of gravity and surface tension. I n figure 1 the solid lines PR and PR’ are the actual forms of two drops making contact angles of e and 0’ with OP. The dotted lines PQ and PQ’ represent spherical surfaces and are the forms which the drops would assume if the gravitational force were removed. The surfaces PQ and PR have a common tangent P M , and PQ’ and PR’ also have a common tangent PM’. The measured quantities are 5 and V , where V is the volume generated by rotating the area OPR about the OR axis. The equation for the contact angle requires that the volume be enclosed by a spherical surface. A spherical surface PQ’ can be constructed such that the volume under PQ‘ is equal to that under PR (the true volume of the drop). This volume is used to calculate h, and the value of h / z thus obtained yields a value of 0’/2 or 8’ which is somewhat less than the true angle of contact 0.
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GUILFORD L. MACK
By evaluating the small distance e, the true contact angle may be found from the equation tan (e/2) = (h
+ e)/z
(4)
In order to calculate e we need to have given the approximate angle of contact e’, the radius z of the drop, and the capillary constant a of the liquid. From 0’ and z the volume V’ of the spherical segment PO&’ may be calculated by geometrical methods. For the same angle e’, the volume
I
P
FIG.1. Outline of a drop of liquid resting upon a solid surface under the influence of surface tension and gravity.
of the actual drop POR’ may be calculated from the tables of Bashforth and Adams ( 5 ) . Let the difference between the two volumes be AV’ = VrlQ,- VpO,,. If (e - e’) is small, AV’ will be very nearly equal to = VPOQ~, i V = V POQ- V P O H . Since by construction VPOR
AV = =
VPOQ
-
VPOQ’
( ~ / 6 )[ ( h
+ e)3 + 3z2(h + e ) - (I?
+ 32’h)I
Since e is very small, the term e3 may be ignored and the equation reduces to e%
+ e(z2 + h Z )- 2AV/n
=0
(5)
105
I
DETERMINATION O F CONTACT ANGLES.
The general solution of the quadratic equation for e may be expressed in the form
The terms within the brackets may be expanded in series according to the binomial theorem. After simplifying and expressing in numeric form, the equation becomes
_e --
5
2AV/nz3 [I + ( h / 4 2 1 - 1
~ - ( 2 A V / ~ z ~ ) ~ ( h / z )2 ( 2 A V / ~ z ~ ) ~ ( h-/ z )... [I + (h/4211-3 11 + (hi421-5
(6)
+
TABLE 4 Related values of e’, x/a, and AV/x3 Values of A V / x 3 are given in the body of the table for the corresponding values of 0’ and x/a along the margins 2
la
8’
90 85 SO 75 70 65 60 50 40 30 20 10
0.1
0.2
0.3
0.017 0.011 0.008 0.006 0.004 0.003 0.003 0.002 0.001 0.000 0.000 0.000
0.041 0.029 0.021 0.016 0.012 0,009 0,007 0.005 0.003 0.002 0.001 0.000
0.085 0.062 0,046
0.034 0.026 0.020 0.016 0.010 0.006
0.004 0.002 0.001
1
0.4
0.143 0.105 0.078 0.059 0.045 0.035 0.027 0.017 0.011 0.007 0.004
0.002
1
0.5
0.205 0.154 0.115 0.0%
0.068 0.053 0.041 0.026 0.016 0.011 0.006 0.003
Substituting this value of e / x in equation 4, one obtains tan (e/2)
=
tail (e’/2)
+ (2AV/d)
C O S ~ ( B ’ /~ ) ( ~ A v / & ) ~tan (O’j2) COS^ (ella) + . . . . . . . (71
The series is convergent if
[(~AV/TX tan ~ )@’/a) cos4 (e’/2)12 < 1
It is apparent that no error need be involved in the previous assumption that AV’ = AV, since a more exact value of AV may be obtained by successive approximations to the value of e. Practically, the firsf approximation is well within the experimental error. The relative magnitude of the correction terms in equation 7 may be
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GUILFORD L. MACK
shown by means of an example. Suppose a drop of water to have the following characteristics: surface tension, 72.8 dynes per centimeter; radius, 0.1 em.; contact angle, 90". Then V = 6.5 X ~ m . and ~ , the total correction due to the influence of gravity is +lolo'. In equation 7 the error in ignoring all correction terms beyond the first is -0'0'44''. If the angle of contact is reduced to 60", the radius and surface tension remaining the same, the total correction is only +0"29'. These calculations make it apparent that for the present purpose the gravitational effect is of little practical importance in measuring small contact angles. CORRECTION FOR T H E CURVATURE O F T H E SOLID SURFACE
The application of the method can be extended to include solid surfaces with an approximately constant degree of curvature. Two corrections to the determined angle of contact must be made. First the tangent to TABLE 5 Cowectzons f o r curvature of the solid surface R / x = 20 AXGULAR CORRECTIOK
+2"52'
e Volume correction
90
-0"41'
n-
-1"7' -1"35' -2"3' -227' -2"45' -252'
IC)
60 45 30 15
0
Net correction
+2"11' +1"45' +1"17' fO"49'
+0"25' +0°7' +O"O'
the solid surface at the point of contact is inclined a t an angle to the horizontal plane through the base of the drop. This additive correction is partially counterbalanced by that due to the small solid segment which projects above the base of the drop. If x is the radius of the drop, and R is the radius of curvature of the solid surface, the angular correction 4 is given by the equation tan6j2
=
R / x - d ( R / ~ )21
The change in contact angle caused by subtracting the volume of the solid segment from the volume of the drop may be calculated from @ and x according to the methods previously given (equations 1 and 3 and table 1). Khile the angular correction depends only upon R/x, the volume correction is also a function of the contact angle. Table 5 shows the relative values of these corrections when R / x = 20. The angular correction is aliiioqt exactly inversely proportional to R/x, and the volume correction
DETERMINATION O F CONTACT ANGLES.
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167
deviates only slightly more from the linear relationship. Hence for practical purposes, table 5 may be used for any value of R / x . It is only necessary to multiply any given correction by the appropriate value of 20
tz/z' SUMMARY
A method has been devised for the determination of acute angles of contact which is largely independent of the form of the solid surface. It was shown that the contact angle is a function of the radius and volume of a small spherical drop of liquid. An equation has been developed for evaluating the effect of gravitational force upon the form of the drop. The validity of the method has been checked by repeating the determination of the contact angles upon the same materials by two different methods. Satisfactory agreement waa obtained in all cases. The author wishes to express his appreciation to Professor F. E. Bartell for much helpful advice. REFERENCES
(1) ADAMAND JESSOP: J. Chem. SOC. 127, 1863 (1925). (2) BARTELLAND HATCH:Colloid Symposium Monogmph 11,ll (1934). (3) BARTELL AND MERRILL:J. Phys. Chem. 36, 1178 (1932). (4)BARTELLAND WOOLEY:J. Am. Chem. SOC.66, 3518 (1933). (6) BASHFORTH AND ADAMS:An Attempt to Test the Theory of Capillary Action. Cambridge (1883). (6) BOSANQUET AND HARTLEY: Phil. Mag. [6] 42, 456 (1921). (7) O'KANE, WESTGATE,GLOVER,AND LOWRY:N. H. Agr. Expt. Station Tech. Bull. 39 (1930). (8) TAGGART, TAYLOR,AND INCE:Am. Inst. Mining Met. Engrs., Tech. Pub. 204 (1929). (9) WARK:J. Phys. Chem. 37,636 (1933).
