A single-parameter method for the determination of surface tension and contact angle

Colloids and Surfuces, 59 (1991) 129-148 Elsevier Science Publishers B.V., Amsterdam

A single-parameter method fcr the determination of suurf~cetension and contact angle H.K. Kuiken PhXps

Research

(Received

Laborafories,

P.O. Box 80.000,5600

31 August 1990; accepted 21 February

JA Eindhoven,

The Netherlands

1991)

Abstract The classical problem of determination of liquid surface tension from the shape of a sessile drop is reinterpreted. A non-dimensionalisation, first suggested by SmoIders, is chosen in which the shape of the drop is characterized solely by the value of the Bond number in which the observed maximum radius of the drop appears as a characteristic length. The Bond number is defined as Bo =pgr&/y, where p is the density, y is the surface tension of the liquid, g is the acceleration due to gravity and r_, is the maximum half width of the drop. A graphical presentation of the results allows an immediate comparison with experimental drops, eliminating the USC of iterative procedures. Graphs from which the contact angle can be read are also presented. A computer code is given for those cases where a refined approach is necessary. The power of the method is to be found in its simplicity and its diagnostic capabilities. Comparisons with experimental results are given.

INTRODUCTION

The determination of the shape of a sessile drop with a stationary inberior, i.e. a drop resting on a flat horizontal surface, is a classical problem. En the latter part of the last century Bashforth and Adams [ I] and Kelvin [ 2 ] studied this particular problem. Bashforth and Adams were the first to produce extensive tables covering all possible shapes. They found that by rendering the problem dimensionless, all these shapes were related to a one-parameter family of normalized shapes. This parameter is 8=,

pgb2

(1)

where p is the density of the liquid, g the acceleration due to gravity, y the surface tension and b the radius of curvature at the apex of the drop. Bashforth and Adams were able to achieve this parametric reduction by rendering the space variables dimensionless by means of b. Nowadays, j? is known as the Bashforth and Adams parameter.

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130

In more recent years the theory has been used as a tool to determine the surface tension of a liquid from the observed shapes of sessile drops. Basically, the idea is to measure the drop accurately and then to invoke some kind of curve-fit procedure. Originally, the tables of Bashforth and Adams [I] were used. Later authors integrated the equations themselves using modern computing facilities [ 3-8 1. What all these papers have in common is that the spatial coordinates are rendered dimensionless by means of an unknown scaling factor. Some authors [3-5,7] folIow Bashforth and Adams in selecting b, the radius of curvature at the apex, as the reference length. Other authors [6,8] select c= r y pg for a reference length, leaving a single parameter in the governing equation (Eqn (1) ). Since the values of both the length scaling parameter and the Bashforth and Adams parameter are unknown initially, the curve-fit procedures these authors propose involve a search in a two-parameter space. Some authors show that good initial estimates of either b or /3 can be given, but the fact remains that these methods involve searches in a two-parameter space. As far as the curve-fit aspects of these methods are concerned, most authors opt for a least-squares criterion in which the sum of the squares of the normal distances of the measured points to the object curve is minimized. This method may favour certain measurements, or even a single measurement, in such a way that the final values of the parameters are determined almost exclusively by these few measured points. This fact is obscured, and has therefore gone unnoticed, by the two-parameter searches described above. Another approach to the problem was proposed by Andreas et al. 191. Contrary to the curve-fit procedures described ab nye, these authors used actual length parameters of the measured drop in their attempt to determine surface tension. Their idea was to determine the value of the ratio

S=$

(2)

e

where d, is the equatorial (maximum) diameter of the drop and c& is the diameter measured a distance & away from the apex. They proceeded to show that surface tension could be obtained from (3) where H is a dimensionless function of S. This function can be calculated numerically for every value of S and is independent of a particular experiment, i.e. it is universal. On first consideration the method of Ref. [9 f seems a perfect one. It is simple and depends upon a single parameter only. Once S has been determined accurately, a correspondingly precise value of N follows, leading to a similarly

accurate value of y. However, it is here that we have to be careful in distinguishing between mathematical and experimental drops. There are two general effects causing the two to be different. First, there are the obvious difficulties associated with determining exact locations in space. These will cause a statistical scatter around the ideal shape and certain parts of the drop surface may be obscured, particularly near the base. Secondly, other physical effects which are not part of the Bashforth and Adams theory, such as convection within the liquid or a slight tilt of the supporting surface, may cause the experimental drop to deviate from any of the mathematically possible shapes resulting from the analysis of Bashforth and Adams. It is precisely a parameter such as S which is most sensitive to these latter effects, since its value is determined by their compounded influences along the drop surface. In the form in which it was originally put forward by Andreas et al. the method cannot be applied to sessile drops. Indeed, except in the limit where surface tension becomes infinite, or, which has the same effect, where gravity goes down to zero, d, cannot be measured at a distance & away from the apex. This is why Andreas et al. proposed their method for pendant drops, which are usually more elongated. Other authors [ 10,11] put forward modified versions of the methird. Although these authors also discuss their methods mostly in the context of pendant drops, their approach can be used in principle for sessile drops. These modified methods are sometimes called selected-plane metho--2s. Roe et al. [IO] were the first to discuss these. Apart from rl, they introduce a number of smaller diameters & and then measure the corresponding diameters & at a distance & from the apex. Next they define ratios Son the basis of these measurements and these in turn lead to corresponding values of 1;1’.The average of these If values is substituted in Eqn (2), resulting in a more accurate estimate of y. Again, even in its modified form, the selected-plane method is most suitable for pendant drops and less so for sessile drops. The authors of Refs [10,1X ] realized this, as their examples were for pendant drops only. In the case of a sessile drop one is severely limited in choice of reference diameters, as these have to be considerably smaller than the equatorial one. This leaves an important part of the drop out of consideration. Also, a given diameter may appear twice, both below and above the equator, which may be a source of ability. Roe et al. did not publish their modified 11 vs. S tables in the open literature but deposited them with the Library of Congress, Washington, IX. Also, since the drop is usually given as a set of measured coordinates, the various diameters can only be found by means of interpolation. This may be a source cf error. Nevertheless, the selected-planes method has one considerable advantage over the curve-fit approaches. Whereas the latter yield a single value of j based on an overall analysis of the measured shape, the selected-planes method produces a series of ordered H values, In tabulating the values of this parameter the experimenter will discern whether or not there is a deviating trend when

132

going from one part of the drop to another or alternatively, if there is only a statistical scatter around a m e a n value. Only in the latter case can the experimenter be confident that the shape of the drop is the result of a balance of only two forces: gravity vs surface tension. This diagnostic quality renders the method, in principle, superior to the other approaches put forward to date. W h a t w e intend to offer in this paper is a m e t h o d which is s o m e h o w intermediate between the methods based on the Bashforth and A d a m s approach combined with curve fitting a n d the method put forward by R o e et al, [10 ], but does not have the less desirable features of either. It is strongly diagnostic with respect to the determination of the influence of effects other than gravity and surface tension, it refers to measurements taken at the most sensitive positions on the surface of the drop and it does not rely on interpolated values between these measurements. Instead of curve fitting we strongly advocate some kind of averaging, circumventing a bias with respect to a subset of the measured points. T h e basic idea for the m e t h o d is given in a dissertation by Smolders [12 ]. Although the idea is old, it seems never to have been worked out using the computer capabilities n o w available as some of the other methods have been. Our m e t h o d assumes that the drop reaches its largest width somewhere above the surface on which it rests, not directly on it. Therefore, the material of the supporting surface will have to be selected carefully in order to ensure a contact angle larger than 90 °. Since the surface tension and contact angle can be measured independently of one another, a suitable surface can always be selected. Once the surface tension is known, an independent experiment involving another surface can be carried out for the determination of a particular contact angle. T h e angle m a y then be smaller than 90 °. T h e method is illustratedboth graphically and by means of a computer code. T h e graphical method is the first presented to date which offers the experireenter a quick way of assessing the quality of his measurements, without having to m a k e iterative use of t a b l e s or c o m p u t e r software. T h e c o m p u t e r code, on the o t h e r hand, is e x t r e m e l y diagnostic in n a t u r e and can easily be used for a sensitivity analysis with respect to the m e a s u r e d values. BASIC E Q U A T I O N S

If t h e surface tension is t h e s a m e everywhere on the drop surface, t h e liquid inside t h e drop will be a t rest. In t h a t case t h e s h a p e of the drop will be the result of a balance b e t w e e n gravitational a n d surface-tension forces. If tl ~~ pressure rise immediately below the surface of t h e drop is d e n o t e d b y p a n d ~. is the surface ~ension, t h e n t h e shape of t h e d r o p is governed b y t h e equation die N - p -

3~

(4)

133

where N is a field of unit vectors in three-dimensional space coinciding with the field of outward unit normals on the drop surface. As an alternative to Eqn (4) we could have chosen the more familiar formula of Laplace [ 131 P --

‘+’

Y-RI

(5)

R2

where R, and R2 are the principal radii of curvature of the drop surface. Equation (4 ) is rarely seen in the literature on drops but two of the few recent references are Refs [ 14,151. This is surprizing, since it is much simpler to proceed mathematically from Eqn (4) than from Eqn (5). A possible explanation is that Eqn (4) is mainly mathematical whereas Eqn (5) is more physical. Research on drops is usually carried out by people of a physical bias. Describing the surface of the drop, which is rad.ialIy symmetric, by

z=fW

(6)

and using a prime to denote differentiation with respect to r, we obtain the field of unit normals 1 N=

[1+

(fyy

L If’

1 1

,,2+;2)1/2f~~

(x*+yy)‘,2ft.-lj

(

where the factor f’ 1f’ 1-’ has been added to ensure that the outward normal is always obtained at the drop surface. This is a necessary addition frequently omitted in the literature since f has two values for certain values of r. The pressure inside the drop is a simple hydrostatic one P’PO

-+I%~

(8)

where p0 is the as yet unknown pressure at z= 0, p is the density of the drop material and g is the acceleration due to gravity. Strictly speaking, dp, the density difference between the liquid of the drop and the ambient fluid, should be taken not p. If the surrounding fluid is a gas, this amounts to only a small correction. Our coordinate system has been chosen such that the drop appears upside down in it. All points of the drop are in the upper z plane, except the highest which is at z= 0. Therefore, the pressure increases with z. Substituting Eqns (7 ) and (8) in Eqn (4 ) we obtain a differential equation for f

To solve this equation a set of boundary conditions is required. First of all there is

f(O)=f’(O)=O

(10)

134

describing the condition at the top of the drop. One further condit;ion is needed to determine po. It is here that we shall make a reference to the experimental problem; the difficulty in accurately measuring the heigh% of t:he drop. It is even more difficult to measure the width of the drop at its base. It would seem to be easier to measure the maximum width of a .lrop which is bulging out sideways. So let us consider drops which reach their largest width somewhere between the top and the bottom and let us refer to this as r= r,,,. The required third boundary condition is then (11)

f’ (&ax) =m

showing that the surface of the drop is vertical ar ~=r,,,. (This condition holds if r= r,,, is approached from bekow. If approached from above, f’ kw )=-00.) A DIMENSIONLESS

FORMULATION

We shall now render the system introducing

r= rmoxR

of Eqns

(9)-(11)

dimensionless

(12)

f(r) =L,,~UO

We now have

z’ L

Vi

._!.

.‘. T’ RL

[l+!z’!‘])

I = (x+BoZ)

[l-r (2’ )‘I 3’2

(13)

i"

subject to Z(0) -2

(0) =o

(14)

.Z’=ar,atR=R,,,=l,

(15)

and In Eqn (13) Bo is the so-called Bond number Bo=

PgrLX Y

(16)

which is dimensionless. It is similar to the Bashfotih and Adnms parameter /3, but contrary to the parameter of Eqn (1) it contains one ( y) instead of two (b,y) unknown quantities. As can be seen, it is a measure of the relative magnitudes of gravitational and surface-tension forces taken in direct relation to the actual size of the drop under consideration. Further, x is a free parameter that will have to be determined from the boundary conditions. For given values of Bo a;ld 3~we can salve Eqn (131, subject to the boundary conditions of Eqn (14). In general, that solution will not satisfy Eqn (15). This will be the case only for fixed pairs (Bo,~). Since Bo is the given parameter, we have ~=x(Bo). This function is tabulated in Table 1. The numerical procedure to integrate the system is described in the Appendix. One might use apline interpolation or

135

TABLE 1 Values of the parameter x for various values of the Bond number in the range 0 -z Bo c 50. For each value of i the corresp~nd~ugvalue of 130 is given by Bo = i2/50 i

Xi

0

2.00000000

1

1.99335892

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1.97373454 1.94196863 1.89330241 IA4721368 1.78726203 ~.720984$7 1.64983098

1.57512593 X.49805791 1.4 1967562 1.34089278 1.26249584 1.18515324 1 . 10942507 1.03577273 .

i

Xi

i

Xi

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33

0.96456826 0.89610340 0.83059812 0.76820875 0.70903575 0.653 13089 O-60050403 0.55112946 0.50495168 0.46189070 0.42~8~689 0.38470526 0.35033935 0.31861456 0.28939110 0.26252651 0.23787777

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.21530307 0.19466330 0.17582313 0.15865199 0.14302471 0.12882203 0.~~533090 0.~0424466 0.09366316 0.08409266 ~*07544577 0.06764125 0.06060384 0.05426397 0.04855753 0.04342553 0.03881385

quadratic interpolation to find accurate intermediate values of I within the range 0 c Bo G 50. This range is wide enough to cover all practical cases. It is not expected that the sessile-drop method will be used for BOB 50 since the drops are then too flat. Although the difference between our approach and the earlier ones may seem only trivial at first, there is a vast difference when it comes to the application of the method. The scaling is now done beforehand by means of an observable of the system under consideration, namely r_. In the approaches based on the Bashforth and Adams theory the scaling factor can only be found iteratively. Although our sc;aled equations seem tc involve two parameters, namely Bo and K, one of these is a function of the other, so that the known normalized shape is a function of a single parameter, the Bond number, only. Therefore, ~=x(Bo) being a universal function which is independent of any particular set of measurements, the search for the best curve involves a one-parameter family of shapes only. MEASUR?NG

SURFACE TENSION

GRAPHICALLY

As we have seen, the shape of the drop is a finction of the Bond number Bo (Eqn (16) ) only. In Fig. 1 we present a series of shapes. Suppose one has measured the shape 02 a drop experimentally and war:ts to deduce the surface

136 1.0 I 2

1.6

1.4

l.O1.2

1.5

l.C

0.L

0-f

0.

0.

0.

Fig. 1. Normalized shape of a sessile dwp for Bo = 0.5 (0.5 ) 5.0. The squares rafe: to a meas-xement reported in Fig. 5 of Ref. [6].

tension y from this shape. The first thing to do is to measure the half width of the drop, which we have called r,,,,. Next one divides all the measured coordinates by r,,,. The resulting normalized shape is compared with the shapes presented in Fig. 1. The best fit will give an estimate of 30~ Since p, g an-3 r,,, are known, the value of y follows immediately from Eqn ( 16). Itis emphasized again that the method should only be applied to drops which reach their maximum width above the surfaces on which they rest, not on them. Therefore, to measure surface tension one should select surfaces for which the liquid is relatively non-wetting. CURVE FITTING

BY MEANS

OF LEAST-SQUARES

Inspection of the normalized shapes of Fig. 1 reveals that a method based on a least-squares Et, OF any related method which attempts to minimize a sum of absolute distances, is unsuitable here. Indeed, the Bond curves are closely

137

spaced in the region between the apex and the equator plane. They are spreading out in the region closer to the supporting surf;ace. Therefore, if a sum of unweighted squares of distances or unweighted absolute distances is minimized, points that are closest to the supporting surface will dominate the outcome. A simple textbook example will clarify this. Suppose we have a problem in which a set of measurements is to be fitted by a straight line Z= arR,where cy is the parameter which is to be determined. Suppose also that we have two measurements: (R,,Z,) = (10,lO) and (R2,Z2) = (O.&l).The normal distance of a point (Ri,Zi) to the curve is lzi

(17)

-CYR;I

Therefore, a ieast-squares routine will attempt to minimize 100(1-cVcu)“+ (l-0.5&?

(13)

resulting in a=GY?!40! A s a result. the ‘optimal’ straight line goes almost exactly through the pain.. (R,,Z,). Suppose that we now have (R,,Z,) = (LO.5) instead. The routine now tries to minimize 100(3_-Ll!)2+

(0.5-c#

(1-9)

resulting in Q!= 201/202 which is almost the same as the earlier result. Apparently, the point closer to the origin, where the object curves are more closely spaced, has hardly any influence. Of ccurse, one answer to this problem might be to apply a weighted leastsquares method, giving more weight to measure_ .ents in the denser regions. Weights which are proportional to the spacing between the object curves might be appropriate. However, such a weighted least-squares method will be highly susceptible to errors in the measurement of points located in the more closely packed regions. As a remedy, the weights might be reduced accordingly. Clearly, this would introduce a great deal of arbitrariness. PARAMETER

-4VERAGING

AND RELATED

METHODS

A method which dces more justice to each individual measurement is to determine the values of the curve parameter for each point separately, and then to invoke some kind of averaging principle. Returning to the example of the previous section let us assume that we have six individual measurements: ( 1,l), (2,2.5), (3,5),(4,4),(5,3)and (6,4). The individual a! values are 1,1.25,1.67, 1,0.6and 0.67. The average of these values is a = 1.03. However, it is clear that the average can be influenced quite strongly by outlyers, particularly if only a few measured points are available. In the example given here the outlyers are distributed evenly around an acceptable average, so that the associated errors are evened out. This is not so when the errors in the outlyers have the same sign, adversely affecting the average. Therefore it may ‘be better to choose the

138

median value of the observed a! values. This would yield cy= 1 in our example. An unweighted least-squares fit would have produced cx=/3.84 showing a strong bias towards the po!nts that are furthest away from the origin. A method which gives the value of the curve parameter for each individual measured point is particularly suited to reveal trend-like deviations from the ideal gravity-surface-tension induced shapes. In such cases we expect the curve parameter to vary in a monotonic fashion. Referring again to our straight-line model, let us suppose that a physical effect which is not part of the ideal model yields a parabola Z= LY~R-aR2, where o( > 0) is small enough for the curve to be an almost straight line in the measurement region but not quite. Assuming that the measurements are now accurate, a search within the one-parameter ( CY) cnve collection will show a series of slowly decreasing va!ues of c1 in a direction away from. the origin. A calculation of cyfor each measured point will reveal this trend. An unweighted least-squares fit will be biased towards the lower values of a! that are associated with points furthest away from the origin. The m&hod will produce a straight line which undershoots most of the measured points and, in the absence of a diagnostic element in the methcd, this will go unnoticed. Applying this to the collection of Bond curves, one can draw similar conclusions. If other physical e:ffects, besides gravity and surface tension, have a marked influence on the shape of the drop, the drop will either be too flat or too high. Assuming, as art example, the first to be true, the measured points which are closest to the apex will yield Bond numbers that are too large, and points measured closer tc the supporting surface will also produce Bo values that are too large, but. only slightly so. To illustrate this we refer to Fig. 1 in which the measurements of a drop reported by EIuh end Reed [6] are plotted against the Bond curves. We note a trend from larger to lower Bo values as we go from the apex to the supporting surface. The range may be estimated to be from Bo =4 to Bo= 2.5. Clearly, unknown physical effects, be they evaporation-driven convective flows or temperature-induced natural convection, affected the shape of the drop of IIuh and Reed. In the absence of these detrimental effects the drop would probably have been less flat, pointi.ng to a Bond number br!low any of those of the measured points. A least-squares routine, when a;;::1 ied to this case, would probably give a value just below Bo = 3. In any case, should tb~zyhave been able to use the diagnostic tool we present in this paper, IIuh and Reed miglli have decided to rethink their experiment. It should be obvious that a two-parameter curve fit fails to brin-- to light the effects we discussed here. COMPUTERIZED

SELECTION

OF AN OPTIMAL

BOND

NUMBER

As has been shown, the method proposed here can be used to determine the Bond number graphically. Then the surface tension of the liquid follows diieCtly from

139

y=

?&Lc

(2W

80

The right-hand side of Eqn (20) now contains values which are all known to the experimenter. If the measured curve deviates visibly from any curve in the Bond collection, as, for instance, in the case in Fig. 1,there is no point in trying to improve the estimate by means of a computer code. The experimenter may wish to redo his experiment taking proper precautions. For those cases where a further inquiry is in order we have written a computer code which produces the Bond number for each of the individual measurements. On the basis of these values one might select the arithmetic average or the median. The values of Table 1 are an integral par: of the routine. For each va1ue of Bo a spline interpolation yields a V&P of x which produr;es an R maxvalue of 12 10v4.Of course, we might have made this part of the routine more accurate, but there is no point. in doing this since most of the 80 values are quite sensitive to even small variations in the measurements. We sha11 illustrate this later. Rotenberg et al. [ 71 suggest a far greater accuracy can be achieved, but these authors cornpale their curve-fit routine with exact solutions of the Bashforth and Adams equations. This, obviously, yields an accuracy which is limited only by the truncation error of the computer. A slight scatter or a faint trend superimposed upon their ideal target curve would have revealed a considerable drop in accuracy. But, again, the bias of the least-squares fit with respect to regions closest to the supporting plane, where sensitivity is reduced, might have obscured this. As an example, suppose we had written a routine which selects one particular element from the collection of straight lines 2 = QR, and does so by invoking a least-squares formalism. Next, we proceed to illustrate the method by feeding it a collection of points which all lie on Z= R to within the accuracy allowed by the computer. Then we demonstrate the power of the code by producing a= l. Clearly, this would be a numerical exercise within a numerical model along the lines of Ref. [ 7 3. What one needs to know about a code, that is when the method is to be used to analyze real experimental data, are statements about its sensitivity and preferences. MEASURING

THE CONTACT

ANGLE

Non-wetting liquids (90”
180”)

This case refers to drops which are bulging outwards. To measure the contact ang1e we determine the maximum height zmaxof the drop. This value is then divided by r,,, to obtain the normalized height. This ratio, which is denoted by z,,,, fixes the value of the contact angle uniquely. Figure 2 can be used here. In this figure, # is defined as the angle between the outer surface of the drop

140

and the wetted part of the supporting plane. It is clear from Fig. 2 that in this $-range the method works best for the smaller Bond numbers. This can always be achieve d by taking drops which are small enough. ~e~~~ng ~~~~~~ (0 D< cp< 90 “1 We assume that the surface tension y has already been determined, possibly by means of an experiment in which the supporting surface and the drop were in a relative non-wettmg state. In the present experimental situation where the liquid is wetting the substrate, we measure the maximum radius rm,, which is now reached on the supporting plane. We can now calculate the value of a modified Bond number which is also defined by Eqn (16 j. Since our third boundary condition reads Z = Z,,, at R = 1, we can find a unique solution of

Zmax

/rmax

B

Fig. 2. Contact angle @ as a function of the ratio of maximum height and maximum half width of the drop for Bo = 0.5 (0.5 )5-O_This graph refers to drops that reach their maximum width above the supporting surface. 180. 180. 140. 120. 100. 0.5

0.7

0.9

1.1

1.3

1.5 ~maxf~max-

Fig. 3. Contact angle q&as a function of the ratio of maximum height and maximum half width of the drop for Bo = 0.5 (0.5 )5.0. This graph refers to drops that reach their maximum width on the supporting surface.

141

the basic Eqn (13) and hence a uuique value of the contact angle @ Some pertinent results are plotted in Fig. 3. In this #-range the method works best for Bond numbers that are not too small, i.e. for larger drops that are flattened out. Apart from using the graphs, it is also possible for us to determine contact angles by means of the computer code. APPLICATION

OF THE COMPUTER

CODE TO A REAL DROP

The purpose of this section is to show how the method can be applied in practice, and to discuss the effects which may influence the measurements. Sometimes these detrimental influences may seem quite trivial. It is not oltr purpose to put forward sophisticated measuring techniques here. Indeed, Lhe particular drop we study in this section is not ideal, at least not in the indirect way it has come tc us, so that the application of advanced measuring ‘techniques would be out of place. It is precisely through this less-than-perfect drop that we are able to demonstrate the diagnostic power of a method based on Bond numbers. In Ref. [16] (Fig. 3, p. 200) a drop is shown which has a very sharp image. Our Bond-number characterization of drops is independent of the size of the drop. It depends only on its shape. A phot megative of the above-mentioned drop having roughly the same size as the original photograph of Ref. [ 161 was produced. Then we made an enlarged print in which the drop had a maximum diameter of 174.8 mm and a height of 95.3 mm. First the drop was measured by slicing it horizontally with a thin shs.rp blade at various distances from a horizontal reference plane. This reference plane was thought to coincide with the plane on which the drop rested. As can be seen in Ref. 1161, the reference plane is very smooth and flat at one side of the drsp, and fairly rough at the other. Therefore, to define the reference plane, we took the part of the plane closest to the drop. To actually measure the lengths of the slits FIruler was used which allowed an accuracy of roughly ‘+O.l mm. Results of this exercise are shown in Table 2a. The first column of Table 2a gives the distances of th.e slits from the top of the drop. The second column shows the corresponding half widths. These were obtained by first measuring the total widths across the drop and then dividing these values by a factor of two. The last column presents the Bond number corresponding to each of the measured points If the drop had been perfect and our measurements infinitely accurate, all Bond numbe: would have been the same. However, looking at Table 2a we see a variation and possibly a slight trend from higher Bond numbers in the region near the top of the drop to slightly lower Bo values near the supporting pl.ane. Further, the first Bo value is noticeably lower than all the other valueF,. This can be attributed to the extreme sensitivity of the Bond curves to c?hanges of location in this region.

142 T A B L E 2a Values o f t h e B o n d n u m b e r o f s e l e c t e d poi:nts a l o n g a d r o p a p p e a r i n g o n p. 200 o f Ref. [ 16 ] m e a sured by t h e b l a d e - a n d - r u l e r m e t h o d using a large-size p h o t o g r a p h of t h e drop (zi distance f r o m t o p ( r a m ) r~, h a l f w i d t h a t z = z ~ ( r a m ) ) i

zi

ri

Bol

1 2 3 4 5 6 7 8 9 10 11 12 13 14

4.3 9.3 14.3 19.3 24.3 29.2 34.4 39.4 44.4 49.4 79.4 84.4 89.3 94.4

30.6 44.5 54.0 61.3 67.3 72.1 76.3 79.6 82.3 84.4 85.4 83.3 80.4 76.0

1.736 1.884 1.873 1.844 1.857 1.855 1.856 1.857 1.878 1.899 1.778 1.813 1.817 1.817

rm,x ----87.3; zc..x = 99.4; Bo.v---- 1.840; BOmed-----1.85,6; ~av = 154 ° 2 0 ' ; emma=-" 155 ° 2 0 ' .

T h i s is i m m e d i a t e l y o b v i o u s f r o m F i g . 1. H a d w e m e a s u r e d t h e f i r s t s l i t a t 4.2 m m i n s t e a d o f 4.3 m m f r o m t h e t o p o f t h e d r o p , l e a v i n g t h e v a ! u e o f r u n a l t e r e d , t h e n a B o v a l u e o f 1.92 w o u l d h a v e r e s u l t e d . I n s t e a d o f b e i n g l o w e r t h a n e a c h o f t h e o t h e r B o v a l u e s , it n o w o v e r s h o o t s a l l o f t h e m . T h i s c l e a r l y d e m o n s t r a t e s t h a t g r e a t c a r e h a s t o b e e x e r c i s e d in e s t i m a t i n g t h e a c c u r a c y o f t h e c a l c u l a t e d Bo values. T h e s e n s i t i v i t y of t h e Bo v a l u e s b e c o m e s less severe as t h e m e a s u r e m e n t s a r e t a k e n f u r t h e r a n d f u r t h e r a w a y f r o m t h e t o p o f t h e d r o p , b u t it increases again as we come closer to the maximum-width region of the drop. A f t e r t h a t t h e s e n s i t i v i t y d e c r e a s e s r a p i d l y ( s e e F i g . 1 ). T h e r e f o r e , w e r e j e c t m e a s u r e m e n t s w h i c h a r e o u t s i d e t h e i n t e r v a l 0.2-< R < 0 . 9 8 e x c e p t , o f c o u r s e , the measurement of the maximum radius. In order to investigate whether we can improve upon our simple hand meas u r e m e n t d e s c r i b e d a b o v e , we d i d a n o t h e r m e a s u r e m e n t u s i n g a p r o j e c t i o n m i c r o s c o p e w i t h a n x - y t a b l e m o u n t e d o n t o it. I n p r i n c i p l e , m e a s u r e m e n t s a s accurate as _ 1 micron can be carried out with such an instrument, but we h a v e f o u n d b y t r i a l a n d e r r o r t h a t ± 5 m i c r o n s is c l o s e r t o t h e t r u t h i n o u r c a s e . W e u s e d a p o s i t i v e i m a g e o f t h e d r o p p r i n t e d o n a s m a l l f i l m slide. T h e size o f t h e d r o p o n t h i s s l i d e w a s a b o u t 18.5 m m b y 10.5 m m . O u r m e a s u r e m e n t s , w h i c h a r e i n m i c r o n s , a r e s h o w n i n T a b l e 2b. S u r p r i s i n g l y , t h e s e a c c u r a t e m e a surements showed hardly any improvemen~ upon our earlier hand-obtained ones. The scatter seems to be reduced somewhat, but now there appears to be a d e f i n i t e t r e n d f r o m l o w e r t o h i g h e r B o v a l u e s W h i c h is t h e r e v e r s e o f t h e s l i g h t

143 TABLE

2b

Values of the Bond number of selected points along a drop appearing on p. 200 of Ref. [ 161 measured with a projection microscope usir.d a small film-slide image of the drop (z;, distance from top (microns); Fi, half width at z=Zi (microns) ) i

zi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

500 1000 15OG 2000 2500 3000 3500 4000 4500 5000 5500 8500 9000 9500 lOOQ0 10400

fi

Bo,

3365 4725 5685 6450 7045 7550 7985 8345 8630 8880 9060 9100 8895 8610 8200 7735

1.553 1.733 1.758 1.803 1.747 1.732 1.757 1.770 1.739 1.788 1.766 1.811 1.820 1.812 1.812 1.807

trend displayed by Table 2a. More importantly, the values of Table 2b seem to be lower than the corresponding values of Table 2a across the board. While being measured the slide was kept under a glass plate, to prevent it from bending. This would otherwise have caused a distortion of the width-height ratios. In explaining the discrepancy between the values of Tables 2a and 2b we are led to believe that the material of the slide is not of constant form in time. During its processing the slide will undergo heat treatment among other things. This may deform it preferentially. The reason that we discuss matters which seem as trivial as these is to focus the attention on the enormous sensitivity of drop measurements on all kinds of factors. It should be a sobering thought that a simple preferential shrinkage of a slide can have a noticeable effect on the determination of a Bond number and, therefore, on that of surface tension. A third measurement of the shape of the drop of Ref. 1161 was carried out on a projection microscope using the photonegative from which the photograph which led to Table 2a was made. The results are shown in Table 2c. In comparison with the previous two, this measurement shows very little scatter. Owing to this increased accuracy, we now discern a distinct, albeit limited, trend from lower I3o values near the top to larger ones near the base of the drop. This trend is much smaller than the huge one revealed by the drop of I-Iuh and Reed [ 6 3 which could be detected by simple graphical methods. The

144 TABLE

2c

Values of the Bond number of selected points aIong a drop appearing on p. 200 of Ref. [161 measured with a projection microscope using a photonegative image of the drop (q, distance from top (microns ); r,, halfwidth at t=Zi {microns) )

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 i3i)OO 21000 22000 23000 24000 25000

7555 10530 12695 14430 15870 17085 18140 19055 19835 20505 21080 21555 21950 22055 21620 21050 80270 19190

1.773 1.785 1.783 1.789 1.790 1.782 1.788 1.800 1.800 1.798 1.804 1.799 1.806 1.811 1.818 1.815 1.82:' 1.828

Gk>~==51"50'* r ,,,-_22695;~,,,,=25780;~ollV=1.800;Bomf~=~-6~O;'?,,,=151"50';

trend of the drop of Ref. [ 161, however, could be brought to light only by a relatively refined measurement using a proper indirect representation of the drop_ Still, we must realize that the actual original drop from which the photograph of Ref. [16] stems, may not have had this trend at all! As it is only through the photographical representation printed in Ref. [ 161 that we have access to it, we have no way of knowing how many distorting sources there may have been, It is not completely unthinkable that the slight trend we revealed in Table 2c may fmd its roots in a distortion of the paper of the book in which the photograph was actually printed. To conclude the discussion of the drop appearing in Ref. [16] we remark that if we multiply the values appearing in the second column (r values) of Table 2b, including the value of r,,,, by a factor of 1.002 and recalculate the corresponding So value-5, we obtain a set of results which is roughly in agreement with that of Table 2c. However, in comparison with the numbers listed in Table 2c there is a much greater scatter among these new results. This shows that a preferential shrinkage in the z-direction may indeed have affected the shape of the drop. We might even go so far as attributing the two dips in the list of Bond numbers as shown by the sixth and the ninth points (Table 2b) to the two holes appearing near the edge of the film slide roughly alongside the

145

locations where these measurements were taken. One can hardly point to a source of error more trivial than this one! Yet, its effect is noticeable. If there is shrinkage (or dilation) in a piece of plastic material, transport holes punched along its edge will most certainly be of influence. If instead of r, = 7550 microns we would have had r6= 7570 microns, which would mean a difference of only 20 microns on the slide, then Bo6 would have been I.79 which is roughly in line with the values of Table 2c. DISCUSSION

In this paper we have presented a method for the determination of surface tension from the shape of a sessile drop which appears to have some advantages over existing methods put forward in this field to date. Its strength derives ficlllyfrom the fact that all the shapes possible within the surface-tension vs gravity model are characterized by a single dimensionless parameter which, apart from the unknown surface tension, contains only observables of the system. Previous methods which are based on the scalings introduced by Bashforth and Adams [I J need two-parameter iterations, where one of the parameters is an unknown scaling factor and the other a shape factor. There exists a relationship between these two parameters, but in each instance this has to be determined again. Because of the two-parameter character of these approaches, the special nature of the collection of possible shapes, which shows both closely-spaced and widely-spaced regions, does not seem to have been discussed to date. Because of this it has gone unnoticed until ow that the points ale_. -g +Fr= surface of the drop are not equivalent and that their influence is varying. This. affects in particular curve-fit procedures based on unweighted least squares which favour measurements in the widely-spaced regions. In our method the iteration is carried out within the mathematical model once and for all. This allows a presentation of all possible shapes in terms of the most realistic system parameter. We feel that this presentation is both perspicuous and of direct practical use. The power of the method is to be found in its diagnostic features. For each of the measured points we can now calculate the corresponding curve parameter 130. If these values are all the same, the drop is perfect in a surface-tension vs gravity sense, and we may conclude that the measurements were carried out with extreme accuracy. However, we have shown by studying actual examples, that real drops have a strong tendency to deviate from the above-mentioned idcal mathematical drops. The slightest deviation in the shape of the object curve, sometir s of the order of a few microns only, may lead to much increased variations in the curve parameter. Therefore, statements which have appeared in the literature suggesting that accuracies of 0.1% or less can be reached, are somewhat misleading. A closer examination of the investigations which promoted such statements reveals that they concerned mock experi-

146

ments on drops which were represented by cne of the Bashforth and Adams curves. It has been our intention in this paper to show the limits of what can be achieved with the sessile-drop method in the way of determining surface tension. It is quite probable that if we redo one of the measurements reported in ‘I ..Mes 2a-c, for instance by repositioning the reference surface a little or defining in a slightly different manner when the crossing lines of the projection microscope are in a particular on-line position, we will come up with slightly different values. It has not been our purpose here to prpsenc the most accurate or sophisticated measurements, but only to show how they are to be zrpreted, and to what extent c c -elusions based upon t’hem are tc be trusted. For one thing, it would be advisable as advocated by Anastasiadis et al. [SJ to measure points in the view plane without giving reference to any supporting plane. A special piece of software will then decide, on the basis of these measurements, where the top of the drop is to be found and what its horizontal position is. Our software does not yet include this feature, but it, is relatively straightforward to include such extras.

REFERENCES 1

2

9 10 11 12a 12b 13 14 75 16 17

F. Bashforth and J.C. Adams, An Attempt to Test the Theories of Capillary Attraction, Cambridge University Press, Cambridge, 1883. Lord Kelvin, Popular Lectures and Addresses, Vol. I, The Royal Institution, London, 1886, pp. l-72. D.N. Staicopolus, J. Colloid Sci., 17 (1962) 439. J.N. Butler and B.H. Bloom, Surf. Sci., 4 (1966) 1. H. Vos and J.M. Los, J. Colloid Interface Sci., 74 (1980) 360. C. Huh and R.L. Reed, J. Colloid Interface Sci., 91 (1983) 472. Y. Rotenberg, L. Buruvka and A.W. Neumann, J. Colloid Interface Sci., 93 (1983) 169. S.H. Anastasiadis, J.-K. Chen, J.T. Koberstein, A-F. Siegel, J.E. Sohn and J-A. Emerson, J. Colloid Interface Sci., 119 (1987) 55. J.M. Andreas, E.A. Hauser and W.B. Tucker, J. Phys. Chem., 42 (1938) 1001. R.-J. Roe, V.L. Bacchetta and P.M.G. Wong, J. Phys. Chem., 71 (1967) 4190. C-J. Lyons, E. Elbing and 1.R. Wilson, J. Chem. Sot. Faraday Trans., 81 (1985) 327. C.A. Smolders, Contact angIes, wetting and de-wetting, Ph.D. Thesis, University of Utrecht, 1961. C.A. Smolders and E.M. Duyvis, Rec. Trav Chim. Pays-Bas, 80 (1961) 635. L.D. Landau ,?nd E.M. Lifshitz, Fluid Mechanics, Fergamon, London, 1963, p. 231. P. Concus and R. Finn, Philos. Trans. R. Sot. London, 292 (1979) 307. R. Finn, Equilibrium Capillary Surfaces, Springer-Verlag, Berlin, 1985. A. Passerone, in B.N. Chapman and J.C. Anderson (Eds), Science and Technology of Surface Coating, Academii. ‘?ess, London, 1974, p. 200. SW. RienF’ r v ‘h.. 24 (1990) 293.

147 APPENDIX

Because of the nature of Zqn (15 ), it is difficult to numerically solve Eqn (13)) particularly since we hax?e to proceed beyond that point into a region where the surface curves backwards. Clearly, this difficulty arises only because of the special choice of coordinate system. We can remove it by selecting the arc length sI rather than the radial coordinate R as the independent variable. Using a dot t.o denote differentiation with respect to s, we have P+_i”=1

(Al)

lt can be shown that Eqn (13) is now transformed into i’=

(x+BoZ)k&&-1

(A2)

Differentiating Eqn (A1 ) with respect to s we can also derive a second-order differential equation for R &-

(~+B02)i+i~R-~

(A3)

The boundary conditions follow immediately from Eqns ( 14) and ( 15 ) .Z(O)=i(O)=R(O)=O

l?(O)=1

(A4)

and R mnx=1

(at s=ss,)

(A5)

Since Eqn (18) implies that the curve must be parallel to the 2 axis there, we must have A:=1 at s=s,. It can be seen that these conditions are consistent with Eqn (Al ) . There is still a minor complication left in the system of equations (A2)(A5). The right-hand sides of Eqns (A2) and (A.3) contain a factor R -’ which according to Eqn (A4) becomes infinite at s ~0. It is true that this behaviour is offset by multiphcative terms that are equal to zero at Z=O. Nevertheless, this condition is awkward from a numerical point of view. A remedy for this is afforded by a series-expansion solution of the system valid for values of s which are close to zero. Some trial and error will show that we may write R=s

5

ic0

u$?

Z=S’

~ i=O

his”

iA6)

where

Subsituting Eqn (A6) in Eqns (Al) and (AZ) we may derive a set of recursive formulae for the coefficients ai and his Omitting the details of the derivation, we have

148

Z(Zi+l)Uj=-

i-l

i-l

C fZj+l>(zi-2j*1)izjczi_j-4x j=t

j=0

G+l)[I-j)bjbi_j_l

(A8)

and (itl)2bi=i

i

Ci_jdj-(i+l)ig

G+l.)ai_jbj

(A9)

j=O

iQ

where CO=X

Ci=BObi_,(i=

1,2,***.)

(AW

and

It is very easy to program the system (A?)-(API). It ensbles us to find a soIution of (A2)-(A4) for given values of Bo andx. From there onwards one can integrate Eqns (A2 ) and (A3 ) numerically without any difficulty. Finally, we mention that Rienstra [ 171 gives an asymptotic solution for thq shape of the drop which is valid for Bo-+ W. His method yields good numerical results for values of I30 as low as 3.