Surface tension measurements by means of the “microcone tensiometer”

JOURNAL OF COLLOID AND INTERFACE SCIENCE 2 3 , 179--194

(1966)

Surface Tension Measurements by Means •1-

of

the

"Microcone

°

.

~'J

.ens,ometer

W I L F R I E D H E L L E R , M I E N - H S I U N G CHENG, A~D B E T T Y E W. G R E E N E ~

Department of Chemistry, Wayne State University, Detroit, Michigan ~8202 Received September 20, 1965 I. BRIEF REVIEW OF THE

INTRODUCTION An observation undoubtedly made by virtually every experimental chemist is the following: if, after emptying a pipet, a drop remains halfway inside the conical end section, this drop will move to the narrow tip no matter whether the tip is pointing downward (case 1) or is horizontal (case 2). If it is pointing upward, the drop will move to a wider section of the conical end until it assumes an equilibrium position (case 3). In the most striking case 2, the movement is due exclusively to the difference in surface pressure at the two ends of the conical drop. In case 1 this pressure difference cooperates with the pressure resulting from gravity; in case 3--the only one where an equilibrium position of the drop inside the pipet is possible with well wetting liquids--the position of the drop is dictated by the competition between excess surface pressure and gravitational pressure. It is clear that this last phenomenon can be used for determining surface tensions of liquids or solutions using as little as 10-~ ml. The detailed theory of this almost completely ignored phenomenon has been developed recently (1). The present paper is concerned with its practical application. There exists no literature on pertinent experimentation. This work was supported in part by a grant from the National Institute of Health. 2 During the early experimental phase of this project, Dr. N. Tanaka contributed to it with a series of helpful semiquantitative measurements not considered here.

THEORY

Figure 1 gives a schematic picture of a conical capillary (fully drawn quadrangle) the axis of which forms an angle, ~, with the vertical. I t contains a drop, of axial length H, of a liquid which has a contact angle fl < 90 °. The drop assumes an equilibrium position, defined by hi and h2, in that section of the capillary where the pressure exerted by the weight of the drop upon the larger liquid-a~r surface $2

PH = Ve Apg cos co/S~

[1]

is just balanced by the excess surface pressure acting upon the smaller liquid-air surface $1 P s = 2 ~ , {[R1 cos(~ -

a)/S1]

[2]

JR: cos(~ + ~)/S~]}.

Here, Ve is the effective volume of the drop, Ap is the density difference between the drop and the surrounding gas phase, R1 and R2 are the capillary radii at the level of the respective liquid-glass-air junctions, g is the gravitational constant, and ~ defines the cone to the extent needed here. From the preceding equations, one obtains at once the surface tension

V~ Apg cos ~/2~$2 [RI cos (~ - ~)/$I] - [R2 cos (~ + ~)/S~] if ~ < 90 ° and T=

,y--_

179

V~ Apg cos w/2~$1 [R2 cos (~ + ~)/S~] - JR1 cos (~ - ~)/S,]

[3]

[4]

180

ttELLEI~, C H E N G , AND G R E E N E

G~ = (2 + cos 2 a ) / ( 1 - cos 2~2); [9a]

A

where

U! \ jf~

I

I~ /

I I

~

~2

I

_ ~

I

t

ha

I

2:

•4

2

(~ -- ~);

[10]

~r 4

1 (~ + a). 2

[10a]

1

5 ....

If (a q- ¢/) < 3 °, one will, as explained elsewhere (1), commit only a small error in ~, on replacing A in Eq. [6] by

=--~-t,.,

A' = H -/- ~/~R2 -t- RI(1 -

2/~R12/R~2) [6a]

and B in Eq. [7] by B' = 2{[cos(~

-

~)/RI] -

Fro. 1. Schematic illustration of method and of quantities contained in equations

if/3 > 90 °, it being necessary, in the latter case, that the tip of the cone faces downward (e > 90°). In order to arrive at a practically convenient expression, V~ has to be replaced by easily measurable quantities. Furthermore, both menisci have to be taken into account. The latter is done assuming the menisci to represent segments on a spherical surface. This assumption is valid in sufficiently narrow capillaries. One thus obtains, eventually, for ~ < 90 °, the explicit exact expression "/ = AApg cos co~B,

[5]

where A = R22(H + h~2 -t- h~l)

[6]

3

and B = 2(R:/R1)[R2 cos(¢~ - a) -

[Ta] ~)/R~]}.

If, in addition, B = 0 ° (aqueous solutions) B' simplifies to

g

1

[eos(~ +

R1 c o s ( ~ +

~)1.

[7]

B" = 2 c o s a ( ~

~-~).

[7b]

Finally, if (a -4- ~) << 3 ° and H >= i cm., and if no more than three significant figures are needed for % one may simplify Eq. [6a] further to A" = H - t - ~/~(R~ + R2).

[6b]

In the ease of well wetting liquids, including H20 and aqueous solutions, (a ~- ~) will always be far smaller than required by the simpler equations since fl is t h e n always small or zero and since, in addition, a must necessarily be kept very small if the capillaries are to have a convenient length. (Consider, for instance, a capillary as short as 20 cm. with the two terminal inner radii of 0.05 and 0.15 ram., a < 0°2'.) It will therefore, in general, be possible, to operate with the simple equations [5], [6a], and [7aJ and, in the case of aqueous solutions, with Eqs. [5], [6a], and [7b]. The comparative performance of the various equations will be tested later on a few examples. II. T H E A P P A R A T U S

Here, h~l = R~ tan (t ;

[8]

h~2 = R2 tan ~2 ;

[Sa]

G1 = (2 -~- cos 2~1)/(1 - - cos 2~1);

[9]

Figure 2 shows the "microcone tensiometer." The capillary is inserted in a conventional reflux condenser and is held in place by means of two perforated corks or rubber stoppers D~ and D2. The purpose of the

SURFACE TENSION MEASUREMENTS

181

FIG. 2. The apparatus reflux condenser is to thermostat the capillary, water from a thermostat being pumped through it continuously. The liquid drop, of nearly cylindrical shape, inside the capillary is seen at E. This section of the apparatus is shown, in magnified reproduction, in the rectangular inset. The position of the drop in the capillary can be changed at will by changing the inclination of the capillary with respect to the vertical (the angle ~o). To that effect, the reflux condenser and, with it, the capillary can be rotated about a horizontal axis at R. If solutions in common solvents are investigated, a small amount of solvent is placed in F in order to speed up equilibration between the drop and the surrounding gas phase. In Fig. 2 it is assumed that fl < 90. If fl > ~0, the condenser would have to be turned so that the narrow end of the capillary, facing G, faces downward. A large protractor to which a level, L, is attached gives a prelhninary, rough value of co. Next to the apparatus is

the traveling microscope needed for the measurements of H and of the quantity hT (see Section III). I t is provided with a filar eyepiece micrometer, a special attachment, which is needed for exact w determinations (see Section V). III. CALIBRATION OF THE CAPILLARY FOR ABSOLUTE v DETERMINATIONS WITHOUT USE OF AN INSTRUMENT CONSTANT The calibration of the conical capillary has to be carried out with great care if the method is to yield absolute surface tension data of high accuracy. The calibration experiments are, in principle, identical with those needed with cylindrical capillaries (such as used in connection with the capillary height method). T h e y differ, in detail, from the latter in two respects: (a) several sets of calibration data are necessary instead of the single set which suffices with cylinders; (b) the treatment of the calibration data is somewhat more time consuming.

182

HELLER, CHENG, AND GREENE

Working with a carefully cleaned ca!cilIary and triple distilled Hg, the axial length H of a drop of Hg of exactly known weight is determined with a traveling microscope for systematically varied values of h~. The latter quantity is, as shown in Fig. 1,3 the distance from an arbitrary reference mark, M, engraved near the wide end of the capillary, to the center, C, of the conical drop of Hg in the capillary. (In actual work, several auxiliary reference marks, spaced at about equal intervals between M and the tip, were used for the sake of convenience. At least two such marks can be detected easily in the inset to Fig. 2.) Not only the variation of the capillary radius, r, with h~ but also c~ and its possible variation with h~ are a priori unknown. An immediate direct determination of both quantities by a single series of calibrations is not icessible. It is therefore necessary that the mean radius 5 be calculated first, at any h~, by considering the I-Ig drop of volume, Vt~g, in a first approximation as a cylinder which contains a hemispheric meniscus at each end, viz., a In the present instance, the menisci have, of course, a curvature opposite to that given in Fig. 1.

VH~ = ~%rH -- 2/~37r.

(Since ~ > 90 °, H is here the distance between the summits of the two menisci in contradistinction to the situation illustrated by Fig. 1.) For the sake of convenience, one will, in general, prefer not to calculate 5 from Fq. [11], but rather in two steps, ~rst by evaluating the quantity 502 which neglects the menisci -

2

r0 -

\

z

i

Vttg

[12]

7rH

and subsequently by calculating i2 itself from r =f0

1 - t - 3 V H j = f°2 1 + ~ ] .

J

/

f'%

\ \

-~ ~ - - ~ - ~ ~

,o j /

\

[13]

(The correction factor in Eq. [13] clearly will be consequential only at stuN1 H.) The 5 values thus obtained as a function of h~ give a calibration curve which is sufficient if used in connection with an instrument constant (Section IV). If absolute ~/measurements are to be made, the cMibration just described must be repeated with systematically varied Hg volumes in order to find the true radius, r, and its variation

/

I'-

[11]

/ z 1 !

FIG. 3. Illustration and definition of some quantities used in Eqs. [14]-[I9], [23], and [271

SURFACE TENSION MEASUREMENTS

183

1.8

PARAMETER: Hg-WEIGHT (mg); [Hg-VOLUME {mlx104 ) ; 25 oC]

1,7

\ '\

\\

\ \\\

\

I : 62.4 [46. I] ]r: 40.1 [29.6] v r : 14.5 [10.7] %rn-: 0

•

,.7

ta.I

, 6 . a:

~,. \\ 1:3 1.6-3

D'\\NXNN ` ~ .l .

..

\',

•\ •

,t~,~- N

•

Z ,,n

\ \

" "

0~(

\

rr

\

\ \ _ Ill N% \

0x --

\~\ ~-\-

-

•

1.5

\

\ \ \

0 ""\Ta \

\

x•\\

N.\\i

E o \

\ . >~

vlr

~

x

I:i.

\

\

\>...

\

1.4

1.4

1.3

1

. la.o

,a.5

3 ,a.o

1 ,3.5

14.o ~ 14.5

15.o

i5.5

,6.o

hr(Cm}

Fro. 4. Detailed calibration curves for ~ limited section of the conical capillary used. For details see text. w i t h h, . T h e p r o c e d u r e to be followed m a y be e x p l a i n e d b y m e a n s of s c h e m a t i c Fig. 3. I t is clear t h a t t h e t r u e radius, r, 'at C (i.e., a t t h e d i s t a n c e h~ f r o m M ) is s m a l l e r t h a n ~. I n view of t h e simple relations H rl = r -- 9 ~ t a n a [14] and r2 = r q- H t a n a, 2

[15]

a n d since ~ = ~(r~

+ w ~ + r~),

[16]

t h e r e l a t i o n b e t w e e n ~(H) a n d r follows as _2

r

~

= r +

H 2 t a n 2 c~ 12

!171

F u r t h e r m o r e , in v i e w of

. . . . . 2

'

[1<

t h e error in t h e h~ v a l u e x = H 2 t a n a / 1 2 ( r l q- r2).

[19]

E q u a t i o n [17] allows one to o b t a i n the t r u e r vs. hr curve if several ? vs. h, calibration curves, o b t a i n e d w i t h s y s t e m a t i c a l l y v a r i e d VH~, are ~vailable. One s i m p l y constructs, for s y s t e m a t i c a l l y v a r i e d h~ v a l u e s as p a r a m e t e r , ~ vs. H 2 plots (or e q u i v a l e n t o t h e r plots). T h e i n t e r c e p t of such p l o t s yields t h e t r u e r ~ for a given h,.. T h e v a r i a tion of r w i t h hr thus o b t a i n e d , b y e x t r a p olation, is shown, for a l i m i t e d s e c t i o n of

ttELLER, CHENG, AND GREENE

184

p r e s e n t work. Here, a v a r i e d w i t h hr (see Fig. 8) and, c o n s e q u e n t l y , t h e m e a n &, a t a g i v e n h~, v a r i e d w i t h H . Therefore, t h e ~ vs. H 2 plots are, in r e a l i t y , s t r a i g h t lines o n l y as long as H is small enough so t h a t t h e v a r i a t i o n of a b e t w e e n R~ a n d R1 can be disregarded. W i t h capillaries of t h e t y p e used in this work, ~ vs. H 2 plots h a v e therefore no a d v a n t a g e over p r a c t i c a l l y m o r e c o n v e n i e n t ? vs. H 2 plots. T h e s e a l t e r n a t e p l o t s used p r i m a r i l y in the p r e s e n t w o r k give r d i r e c t l y as t h e i n t e r c e p t . T h e p e r t i n e n t equation, w h i c h follows f r o m E q . [17] b y expansion, is

2.2 Hg -WEIGHT

I : o 6 2 . 4 mg "r'r: e 40. I m g "~:= 19.0 mg 3/11": 0 . 0 mg

2.0

1.8 e4 O x

E o

1,6

ii.

= r + 1.4

1.2

LOT

I

9

_

III

I

13

I

15

17

19

211

hr(cm) FIG. 5. Calibration curves for entire practically useful length of conical capillary. (For meaning of two arrows, see Section VII, 1.) t h e c a p i l l a r y , b y curve V I I in Fig. 4. C u r v e s I, I I , a n d V I are ~ vs. h~ curves, t h e w e i g h t (W) or v o l u m e of H g used for t h e c a l i b r a t i o n a t 25°C, ( V ~ ) being t h e p a r a m eter. T h r e e i m p o r t a n t a d d i t i o n a l ~ vs. h~ curves, for V ~ = 19.0 X 10 -4, 17.6 X 10 -4, a n d 14.0 X 10 -4 ml., are n o t shown in o r d e r n o t to o v e r l o a d t h e figures. ( T h e curves Ia, I I a , a n d V i a will be discussed later.) T h r e e of t h e e x p e r i m e n t a l curves and t h e e x t r a p o l a t e d curve are shown in Fig. 5 for t h e entire m o s t useful section of t h e s a m e capillary. ( R a d i i l a r g e r t h a n 0.20-0.22 a n d smaller t h a n 0.10-0.12 ram. are p r a c t i c a l l y less useful t h a n i n t e r m e d i a t e radii.) I t is i m p o r t a n t to describe in m o r e d e t a i l t h e m e t h o d s a v a i l a b l e for o b t a i n i n g , b y e x t r a p o l a t i o n , t h e t r u e r vs. h, curve. T h e ~2 vs. H ~ plots a l r e a d y referred to should r e p r e s e n t s t r a i g h t lines for a n y H r a n g e if a is c o n s t a n t . T h i s was n o t t h e case w i t h t h e h o m e m a d e capillaries used in t h e

H 2 tan ~ a

H 4 tan 4 a

24r

1152r 3

~- . - . .

[20]

T h e l i m i t i n g slope extends over a sufficiently wide r a n g e of H v a l u e s to allow secure e x t r a p o l a t i o n to r p a r t i c u l a r l y if t h e v a r i a t i o n of a b e t w e e n R1 a n d R2 is t a k e n into account. T h i s is a p p a r e n t f r o m Fig. 6, where t h e l i m i t i n g slope is seen to be cons t a n t for t h e four lowest H g v o l u m e s (four lowest H values) used, a f t e r t h e v a r i a t i o n of a w i t h h T has b e e n c o r r e c t e d for (full circles). I t should be n o t e d t h a t t h e correct i o n does n o t a l t e r t h e two lowest p o i n t s of t h e plots. H e r e c o r r e c t e d a n d u n c o r r e c t e d (open circles) d a t a coincide. The open circles represer~t the primary uncorrected data based upon considering the drop as equivalent to a cylinder of radius ~ having a volume identical to that of a truncated cone of constaat a (Eq. [16] (Fig. 3)). This simple truncated cone is subsequently replaced by two truncated cones, A and B, of differing a but equal length, which are contiguous at h~ . To that effect, a preliminary tracing of ~ vs. h~ is made by means of the ~ vs. h,. curve obtained with the smallest ttg volume (curve VI in Fig. 4). (For the final a vs. h~ curve see Fig. 8.) With H and h~ known, the a values, a2 pertinent to h~, a~ pertinent to [h~ -(H/2)], and a8 , pertinent to [hr + (H/2)], are available at once. For the calculation of the volume of one of the two cones, the a value aA = (a~ + a2)/2 is used; for the calculation of the volume of the contiguous cone, aB = (a2 + a3)/2 is used. The corrected ~ value obtained from such a double cone (full circles) represents therefore the radius of a cylinder having a volume identical with that of a cone of stepwise varied a. An unnecessary minor further improvement in the ~ values could be obtained by replacing the

185

SURFACE TENSION MEASUREMENTS 1.70

~

t

\

1.65

~

,.,o

\,

///

W:mgHg USED e PRIMARY DATA -,CORRECTEDDATA

\

,

x

I I_ 1

~ ~

ro

15.6

._____

_

'-.

IL,.

,,o

-

1.40

[.55

I

I

I

I

I

I

I

I

1

f

I

0

4

8

12

16

20

24

28

:32

36

40

,

I

I

44

48

Ha(eraa) FIG. 6. Graphical evaluation of true capillary radius according to Eq. [20] t r u n c a t e d cone by 3 or 4 contiguous differential cones of varied a. In order to get plots of the t y p e shown in Fig. 6 for h~-values which have 3 significant figures. To t h a t effect all points p e r t i n e n t to a given h~ were read off a set of large m a s t e r calibration curves.

Sections of some of them were shown in Fig. 4. The lower series of curves (Roman numerals only) are the basis for the uncorrected values (open circles) in Fig. 6. The upper series of curves (Roman numerals followed by the letter a) represent the corrected values (identified by full circles in Fig. 6). Whereas ~ vs. H 2 plots as illustrated by Fig. 6 are very satisfactory, they are affected b y the slight inconvenience t h a t the location, with respect to the abscissa, of the points pertinent to the same H g weight, varies with h , . This is avoided on using as the independent variable the square of the volume or of the weight, W, of Hg. The slope of the ~2 vs. W ~ plots is according to Eq. [17] tan2a/12 ~%2p~g and the limiting slope of ~ vs. W 2 plots is, according to Eq. [20], t a n ~ a / 2 ~ r~47r2p~ ,

where peg is the density of ttg. At low and moderate W values, a practically straight line is obtained, allowing again a simple extrapolation to r. Examples of such plots are given in Fig. 7. H a v i n g established the r vs. h~ calibration curve, the angle a and its variation with h, must be determined. One m a y use the slope of either ¢2 vs. H 2 or ~ vs. H 2 plots for the evaluation of a and of its variation with hr. However, it was found simpler to obtain this information from a chord plot derived from the r vs. h r curve (curve V I I in Fig. 5). The resulting tan a vs. hr curve is givert in Fig. 8. 4 The latter data, in conjunction with the r vs. hr data discussed above, form the basis for absolute v determinations with the present method, as described in Section V I I . IV. CALIBRATION OF THE CAPILLARY FOg 7 DETERMINATIONS USING AN INSTRUMENT CONSTANT If it is not necessary to achieve the highest possible accuracy for v and, particularly, if it is sufficient to convert measured data The bars of length Ah~ represent the mean t a n a between the limits of hr indicated.

186

HELLER, CHENG, AND GREENE 1.70

PARAMETER : hr PRIMARY DATA e CORRECTED DATA "

1.65

19..0

__----------U

1.60

o ,.66

1~.6

E

(,) v "

1.50

1.45

i .0

1.40

1.35 0

I

I

I

I

I

I

I

I

I

4

8

12

16

20

24

28

32

36

I 40

W~Z( rngax I 0 -~'1

FIG. 7. Alternate graphical evaluation of true capillary radius into absolute ~, values by means of an "instrument constant," then the capillary calibrations may be simplified appreciably. I t will then suffice to compile just one calibration curve, for instance, curve II in Fig. 4 (5). One may bring this curve into coincidence with the extrapolated curve, VII, by means of a factor D = f(h~). Fortunately, the variation of D with h,. is only slight particularly at large h~(small a)-values. Thus the specific factors D~ and D2 at h~ = 12 and 15 era., respectively, indicated in Fig. 4 differ by only 5-6%. Therefore, a single conversion factor D will be justified as long as h~ is not varied within wide limits. Whenever a single/9 value is applicable, the apparent ~, values, ~,op., determined with the aid of a single ~ vs. h~ curve (e.g., curve II in Fig. 4) may therefore be corrected by means of a single correction factor (instrument constant) C, provided the variation of C with the pertinent variables does not exceed that of D. In order to check this point the quantitative relationship between C and D had to be determined. For the present purpose, it is sufficient to use the simplest

equations, viz., Eqs. [6b] and [7b], in conjunction with Eq. [5], in order to define this relationship. Representing ~/Apg cos ~o by F, one has

F = HR1 R~ -q- (~/~)R1 R~(R1 + R2) 2 cos a (R2 - R1)

[21]

On the other hand, the apparent F1 value, Far., derived by means of a single calibration curve, such as curve II in Fig. 4, Fo,. = H~?I R2 + ( ~ ) R 1 t?~(R1 + R~)

[221

2 cos ~(/~2 -/71) On omitting, as inconsequential, terms containing /5 2 and /3 a, one obtains F Fop.

--

"F 7a,.

--

1

J0[H(/~, +/~2) + (1/i)(R1 ~

[23]

+ 4~1 R~) ] HR~ R~ + (}~)~?t~ + (1A)~1/~ ~ The second term in Eq. [23] evidently represents the explicit value of the correc-

SURFACE TENSION MEASUREMENTS

187

12,6

ll.8

I1.0

10.2

9.4

8.6

0 x

7.8

7.0

6.2

\

5.4

4.6

3.8

3.0

I0

I

I

I

I

II

12

13

14

I 15

I 16

I 17

I 18

19

hr FIG. 8. G r a p h i c a l d e t e r m i n a t i o n of the (tan ~) vs. h~ f u n c t i o n

tion factor [24]

instrument constant) C in Eq. = ~

(1 -

C),

[241

where the "apparent" surface tension Yap is assumed to have been derived from a single r vs. h~ calibration curve. If R~ << H >> R2, and a < 1 ° (this will generally apply), the second t e r m in Eq. [23] m a y be simplified to [ / ~ + /~2 _~

2]

[25]

I t follows from Eq. [25] t h a t (C//)) is not a constant unless R~ and R2 are always the same. Within the h, range for which one m a y consider /) as a constant, C will, at constant h~, increase with H and, at constant H, decrease with h~. If the convenience of a single instrument constant is to be

taken advantage of, it is therefore advisable not to v a r y hr within more than modest limits and to use roughly the same liquid volumes. The former condition can fortunately be met easily by adjusting ~0 according to need and the latter condition represents no problem. The last remaining question pertains to the actual practical procedure to be used in order to find a value of C for a particular section of a capillary. This is done by using a single calibration curve for the calculation of the apparent surface tension v~,., of a liquid of known surface tension % For this purpose, one also has to use the apparent aap. vs. h,. curve (aa~. : the apparent a value). From Eq. [24] C follows at once. Fronl C, in turn, /) can be obtained immediately, if desired.

188

HELLER, CHENG, AND Gt~EENE

The value of C (and/or D) having been established for a given section of a capillary (for a given approximate hr range), it will be wise to determine the limits existing, beyond the h~ range selected, for the application of a single C value without risking significant errors in the results. For this purpose, the liquid drop may be shifted systematically in both directions until % cMculated from Eq. [24], ceases to be a constant on using the single value of C. One may, of course, if desired, establish the variation of C with ha and H if the entire length of the capillary is to be used with various liquid volumes. Actual numerical ~/ determinations by means of an instrument constant are ineluded and discussed in Section VII. All the experiments reported below were carried out with a single capillary. Its characteristies were as follows: Overall length: 38 cm. ; length of experimentally useful calibrated section: 13.5 em.; limiting diameters within calibrated section: largest: 5 × 10.2 era. ; smallest: 2.1 X 10.2 cm. ; liquid volume needed to give a cylindrical drop of 1 era. length at narrow end: 3.83 X 10.4 ml. V. DETERMINATION OF w The larger co, the more important is its accurate determination. In order to make accurate determinations relatively easy, the traveling microscope is equipped with an eyepiece micrometer (Fig. 2) so that distanee measurements in two orthogonal directions are possible. The character of the measurements follows from Fig. 3. Starting to the left of C at the upper wall of the capillary, an arbitrary distance, a, is covered by vertical displacement of the tubus of the traveling microscope. Next, the eyepiece micrometer is displaced horizontally over the distance, b, until the capillary wall is reached again. The measurement of a and b is followed by that of the distances c and d at the lower wall of the capillary. This completes one pair of measurements. The simple relationships to be used for the calculation of co are:

a/b = c/d=

torah(co -~- a);

[26]

tan(co -- a).

[27]

The co value actually used is the mean of the results obtained with Eqs. [26] and [27]. In case of major divergence of the two primary co values, one or more additional pairs of measurement are made. VI. EFFECT OF VISCOSITY AND SURFACE ROUGHNESS A change in co will cause the conical drop in the capillary to move until it assumes a new stationary position. With water and aqueous NaC1 solutions, the only systems studied thus far, the kinetics is practically the same. I t takes the conical drop close to 25 seconds to cover half the distance between its old and its new stationary position but nearly 10 minutes before the movement stops completely. The asymptotic hr vs. time curve is almost the same for movement towards either the wide or the narrow end of the capillary. Although the attainment of a new stationary position is slow, it is important to note that the very small changes found after 3 minutes do not lead to changes of v outside of the limits of the overall experimental error (see Tables I and II). 5 I t also should be noted that the rate at which the menisci approach their final position, increases with the capillary radius. The preliminary numerical data just given apply to a radius < 0.2 ram. The conclusion to be drawn from these observations is that it is advisable to watch the two menisci for a sufficient period of time so as to be sure that they have reached their final or practically final position for the particular co value selected. These phenomena are, of course, completely analogous to those observed on forcing a change in level in the capillary height method. In that method, possible errors due to slow creep are mostly disregarded. It is important to realize that the final position reached by the conical drop is a true equilibrium position only if the inner surface of the capillary is ideally smooth. If it is not ideally smooth (none of the capillaries tested by us turned out to have an ideally smooth inner surface), the true position of equilibrium between gravitational 5 In contradistinction to % the variation of the individual ~.i- values (see Section VII) is not negligible prior to about 10 minutes.

SURFACE TENSION MEASUREMENTS

189

T

~]l]: 74

66

70

6564

, °:

x

_" T ~ / / ~~ r

V~5o rn, Io4

6973 6s 7z

z:4a.2,

63

~:

o

26.'78

/:"/r'/

az

7,

~

66

4 //•

",.:.,o6 60

,,,'"59

~

~//"

~

3 _._//

64 68

l

636,

r.~ 57 z

70

61 65

56

~

•

6064

55

59 65

54

o

53

58

ml

I,

/
/ ~ /

64 I-

51

62

62

e,~

~

and

57 61

ml

54 58

50

/

49 48 7.0

/ I S.O

I 9.0

I 10.0

I I1.0

601/./,~ 13.0 I 13.0

I 14.0

I 15.0

.j_

,j.. 14.0

-~ 1 12.0

t ]6.0

15.0 1 I"~0

I IS,O

,r 16.0

53 57 52 56

]9,0

hr (cm)

FIG. 9. D e p e n d e n c e of h, on c0 and the direction of flow in the capillary. Inset: D a t a t a k e n f r o m Table I

and excess surface pressure will not be reached but must be found indirectly. The problem and its solution follow readily from Fig. 9. Open circles represent the final hr value attained after flow towards the wide end of the capillary subsequent to a reduction in ~0 to the value indicated on the ordinate. Full circles represent the final h, value reached ~fter flow, in the opposite direction, subsequent to an increase in ~o to the value indicated on the ordinate. I t is seen that a single curve c~rmot satisfy both open and full circles even on making allowance for the scattering of the data. Considering, for example, o~ = 63 ° and the curve p~ir II, it is seen that on flow from the narrower to the wider section of the capillary (initial o) < 63°), the flow can be expected to stop at hr = 14.8 cm. For the same co, but with the flow proceeding from the opposite direction (initial o~ > 63°), the flow can be expected to stop ~t 14.6 cm. The true equilibrium position, which is not attained, is therefore between 14.6 and 14.8 cm. Although it cannot be expected to be exactly halfway between these two values, the most

reasonable procedure is to use the intermediate value, here 14.7, as the likely equilibrium value of hr. Any departure of the ~rue-unattained-equilibrium position from this "halfway" hr value is not large enough to be of consequence. This follows from the fact (Section VII) that results obtained on this basis yield ~, values which agree with those reported in the literature. A further argument in favor of this "halfway" value will be given presently. It is of interest in this connection to compare "r values calculated from the actually reached two limiting h~ values. Using the mean of such values given in Table I, one finds t h a t the two values differ, on the average, by 1.03% and differ from their mean, on the average, by 0.51%-0.52%. The possible error committed on considering the " h a l f w a y " value and the equilibrium value of hT as identical, is therefore very small (<< 0.5%).

Figure 9 shows that the difference between the two final h , values, reached on trying to approach equilibrium for a given o~ from opposite directions, is apparently independent of h~ excepting possibly very

HELLER, CHENG, AND GREENE

190

TABLE I SURFACE TENSION OF DOUBLE DISTILLED W2~TER AT 25.00 ° 4- 0.05°C.

(Eq. [5] in conjunction with Eqs. [6a] and [7a]) Expt.

e(°)

H (cm.)

k~ (cm)

hr

H_ -

H

hr q- ~

n'i

A . V o l u m e Used: 1.667 N 10 -~ ml.

la lb 2a 2t) 3a 3b 4a 4b 5a 5b 6a 6b 7a 7b 8a 8b

60.10 59.82 61.26 61.26 61.63 61.68 61.68 61.85 63.00 62,73 63.90 64.28 64,00 64.28 64,46 64,45

2.2497 2.1728 2.5302 2.4759 2.5147 2.4722 2.5729 2.4837 2.7860 2.7099 2,9442 3.0086 2.9466 2.9393 3,0844 3.0403

9a 9b 10a 10b lla llb

60.47 60.75 60.87 61.13 61.30 61.32

4.1501 4.2338 4.3158 4. 3 2 0 5 4.4768 4.2395

12.9898 12.6163 14.1062 13.9057 14.1110 13.9155 14.3194 13.9243 15.1795 14.8881 15.8348 16.1525 15.9955 16.0412 16.5312 16. 2 9 5 1

11.8649 11.5299 12.8411 12.6677 12.8536 12.6794 13.0329 12.6824 13.7865 13. 5 3 3 1 14.3627 14.6482 14.5222 14.5715 14.9890 14.7749

14.1146 13.7027 15.3713 15.1436 15.3683 15.1516 15.6058 15.1661 16.5725 16.2430 17.3069 17.6568 17.4688 17.5108 18.0734 17.8152 Average :

72.79 72.26 73.25 72.60 72.05 71.71 73.11 71.14 72.83 72.53 72.21 71.39 72.17 71.90 72.40 71.49 72.24

72.53 72.93 71.88 72.13 72.68 71.80 72.04 71.95

B. V o l u m e Used: 3.426 X 10 -~ m l .

13.0804 13.2610 13.4452 13.4136 13. 8 6 0 3 13.7639

15.1554 15.3779 15.6031 15.5738 16.0987 16. 2068 Average: Overall Average: Deviation from literature value (71.97 4- 0.1):

small drop v o l u m e s (curve pair I I I ) , for which points o n the ascending a n d descending curve seem to differ less t h e wider the capillary section i n v o l v e d is. M o r e syst e m a t i c e x p e r i m e n t s are needed to clarify this point. T h e same applies to the q u e s t i o n as to w h e t h e r or n o t the drop v o l u m e has a n effect o n t h e difference b e t w e e n the ascending a n d the descending curve of a given pair of curves. T h i s difference is practically the same for all v o l u m e s used (16.6742.21 X 10-4 ml.) except the smallest (11.06 X 10.4 ml.; curve pah" I I I ) . It is worth while mentioning here that the difference of ascending and descending curves cannot be accounted for by minor drop evaporation, which would, of course, be a trivial effect. To con-

11.0053 11.1441 11.2873 11.2533 11.6219 11.3209

72.18 71.85 72.37 71.57 72.63 71.30 71.98 72.17 0.28%

72.01 71.97 71.97

sider, for instance, curve pair II, the volume of the drop, as derived from H and ha, was, near the beginning of this series, 26.58 X 10-4 ml. (average of volumes calculated for points 1 and 2). Nearly 8 hours later, the volume averaged for points 3 and 4 (points obtained near tile conclusion of this series) was 26.74 ~4 10-4 ml. This difference is within the limits of experimental error. I t has b e e n t a k e n for g r a n t e d t h u s far t h a t the n o n a t t a i n m e n t of the t r u e equilibr i u m position of the drop is due to surface roughness. No other reasonable e x p l a n a t i o n can be found. Moreover, the following considerations m a k e this e x p l a n a t i o n also d y n a m i c a l l y plausible: Considering a g a i n the h~ v a l u e of 14.7 a t ~o -- 63 ° (curve pair I I i n Fig. 9), t h e p e r t i n e n t Ri a n d R~ values

SURFACE TENSION MEASUREMENTS are 1.295 X 10-2 a n d 1.557 X 10-2 era., respectively. Values of H a n d a follow as 4.15 cm. a n d 2 X 10-6 steradian, respectively. F r o m these data, one finds t h a t for TABLE II COMPARISON OF PERFORMANCE OF RIGOROUS AND APPROXIMATING EQUATIONS USED IN CONNECTION WITH EQ. [5]. 71 VALUES

(Using several experimental data of Table I)

3a 3b 8a 8b 9a 9b lla 11b

72.05 71.70 72.40 71.49 72.18 71.85 72.63 71.30

72.05 71.71 72.40 71.49 72.18 71.85 72.63 71,30

72.00 71.65 72.36 71.45 72.12 71.80 72.58 71.25

72.00 71.65 72.36 71.45 72.13 71.80 72.59 71.26

Equations in reference 1.

191

h~ = 14.7 the excess surface pressure balanced b y g r a v i t a t i o n a l pressure would be 1870.4 dynes cm.-% A t h~ = 14.6 a n d 14.8 the excess surface pressure would be 1861.8 a n d 1876.0 dynes cm. -2, respectively, a n d the g r a v i t a t i o n a l pressure would be the same. T h e residual pressure which is u n a b l e to lead to a f u r t h e r m o v e m e n t of the drop towards: its ideal e q u i l i b r i u m position is therefore 8.6 (h~ = 14.6) a n d 5.6 d y n e s cm. -2 (hr = 14.8 cm.), respectively. 6 Expressed in millimeters of Hg, this is 6.4 X 10-3 a n d 4.3 X 10-3, respectively. These figures show clearly t h a t the d y n a m i c differences b e t w e e n " a s c e n d i n g " a n d "descending" h~ vs. co curves are in fact so small t h a t t h e y m a y be a c c o u n t e d for b y flow i n h i b i t i o n r e s u l t i n g from v e r y m i n u t e irregu6 The similarity of these residual pressures is a further argument in favor of the "halfway" hr value as the most plausible equilibrium value.

TABLE III SURFACE TENSION OF AQuEous NACL SOLUTIONS AT 25.00° 4- 0.05°C. A: OUTLININGRECOMMENDED DETAILED PROCEDURE IN COLLECTION OF DATA (Eq. [5] in conjunction with Eqs. [6a] and [7a]) (c: moles of NaC1 per 1000 g. of solution) Expt.

e

(qul.V2~°C X 104)

~a (degrees)

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

0.5 0.5 0.5

24.63 24.63 24.63

1.0

26.95

1.0 1.0 1.5 1.5 1.5 1.5 2.0 2.0 2.0 2.5 2.5 2.5 2.5 3.0 3.0 3.5 3.5 3.5 3.5 4.0 4.0 4.0

26.95 26.95 27.12 27.12 27.12 27.12 29.06 29.06 29.06 23.62 23.62 23.62 23.62 25.84 25.84 23.31 23.31 23.31 23.31 25.71 25.71 25.71

60.87 62.02 63.13 61.62 62.28 63.80 60.70 61.78 62.23 63.57 60.33 60.93 62.77 61.15 62.15 63.32 63.73 63.20 64.41 59.95 61.71 62.00 63.55 63.21 64.00 64.69

16

17 18 19 20 21 22 23 24 25 26

COb (degrees)

60.93 61.93 63.05 61.56

62.33 63.66 60.59 61.93 60.28 63.50 60.31 61.01 63.26 61.15 62.14 63.26 63.66 63.10 64.32 60.02 61.67 61.78 63.48 63.40 62.21 64.92

Ha (oy£)

]f:~b (en~.)

3.4167 3.3435 3.7065~:~13.6304~. 3.9811 ~ 3 . 8 9 6 0 3.7699 111~3.6776 3.9083 ,~:~ 3.8718 4.4134 4.2946 3.6280 3.5137 3.8797 3.8678 4.0467 3.6181 4.3802 4.3802 3.6688 3.5904 3.9224 3.8294 4.4557 4.4990 3.1593 3.0378 3.3272 3.3051 3.6336 3.5305 3.7161 3.5250 3.9570 3.7815 4.3149 4.1515 2.7568 2.6019 3.1871 3.0695 3.2445 3.0596 3.6233 3.4568 3.9174 3.7656 4.1079 3.3088 4.4052 4.2590

" "Y~a

.yib

,y

73.37 73.43 72.88

72.48 72.85 72.44 72.74 72.82 72.81 74.62 75.08 74.29 74.36 75.08 75.17 74.75 75.89 75.98 75.43 74.65 76.40 76.12 76.89 77.00 76.76 76.46 77.40 78.42 77.21

72.93 73.14 72.66 73.12 73.09 73.04 75.09 75.34 74.81 74.55 75.38 75.52 75.42 76.39 76.18 75.93 75.45 77.12 76.66 77.64 77.70 77.53 77.26 78.46 78.66 78.13

73.49

73.35 73.27 75.56 75.61 75.33 74.73 75.67 75.87 76.08 76.90 76.38 76.43 76.25 77.83 77.20 78.38 78.41 78.29 78.06 79.52 78.90 79.05

192

HELLER, CHENG, AND GREENE

79

78

77

~ 76 f.o w

Z 75 >,. ,.-,, F.

25 o_+0.05oc

t,

74

75

72

7'10

0.5

J[O

[I.5 21.0 21.5 ~.0 I CNaCl ( MOLES I lO00g)

3!5

1 4,0

Fro. 10. Surface tension of aqueous NaC1 solution as a function of molality. larities of the capillary surface (surface roughness). I n conclusion, it m a y be mentioned t h a t the preliminary results illustrated by Fig. 9 m a y possibly open the door for a quantitative determination of surface roughness of inner capillary walls. VII. RESULTS 1. THn SURFACE TENSIO~ OF DISTILLED WATER Table I gives the results obtained for two drops of double distilled water one having a volume of 16.67 X 10 -4, the other one of 34.26 X 10-4 ml. Experiments defined b y numbers followed by "a" refer to the final position of the conical drop of water after flowing from the smaller towards the larger

end of the capillary; experiments identified by numbers followed b y the letter "b" pertain to the inverse direction of flow. Each of these pairs of measurements differs from the others in the average ~ value. The individual values within a pair (2nd column) were made as similar as practically possible without lengthy trials. The over-all variation of was 4.6 °, which led to a change in hr by 3.9 cm. The extreme positions of the menisci-defined b y [h~ - ( H / 2 ) ] and [h~ + (H/2), respectively--were reached in experiments 9a and 8a, respectively, and are indicated by the arrows in Fig. 5. The entire range of the capillary utilized was therefore 7 cm., i.e., a little more t h a n half the useful calibrated section of the capillary. I t is defined by a minimum and m a x i m u m capillary radius of 0.124 and 0.171 ram., respectively. The individual ~/ values, v i , refer to a single measurement. T h e y are, as stated, affected b y surface roughness. The significant 7 values given in the subsequent column are those obtained on averaging the results of one pair of measurements. These 7 values do not show any systematic variation with or h , , nor can one detect any systematic variation of 7 with the volume of water used. The largest and smallest of the eleven 7 values obtained (72.93 and 71.80 dynes era. -1) differ by - 0 . 2 % and 1.3 % from the literature value of 71.97. T h e mean of the eleven 7 values (72.17) differs from the literature value b y only 0.28%. These results are a very gratifying proof of the high accuracy of absolute ~, values obtainable with the microcone tensiometer. E v e n more accurate 7 values can be expected with capillaries of constant c~, where uncertainties in calibrations would be eliminated almost completely.

T A B L E IV SURFACE TENSION OF AQUEOUS NACL SOLUTIONS AT 25.00 ° 4- 0.05°C. B: Final results and comparison with literature data.

c v,~. VL A(%)

0.5 72.91 i0.2 72.78 4-0.1 +0.17

1.0 73.08 4-0.1 73.59 =£:0.1 --0.71

1.5 74.95 ±0.4 74.40 ±0.1 +0.74

2.0 75.44 4-0.1 75.24 4-0.1 +0.27

2.5 76.02 ±0.4 76.04 ±0.2 --0.03

3.0 76.89 ±0.2 76.86 ±0.2 +0.04

3.5 77.55 4-0.3 77.67 4-0.2 --0.15

4.0 78.43 4-0.3 78.50 ±0.2 --0.10

SURFACE TENSION MEASUREMENTS

193

T h e results given in T a b l e I were o b t a i n e d w i t h E q . [5] in c o n j u n c t i o n w i t h t h e simple e q u a t i o n s [6a] a n d [7a]. T a b l e I I shows foa 8 of t h e 22 7~ v a l u e s t h e results on using, in a d d i t i o n , o t h e r equations. T h e first p e r t i n e n t e o l u n m shows t h e results o b t a i n e d w i t h t h e rigorous e q u a t i o n s [6] a n d [7]. I t is seen t h a t t h e results o b t a i n e d w i t h t h e simplified e q u a t i o n s [6a] a n d [7a] used in T a b l e I a n d r e p e a t e d in c o l u m n 2 are q u a n t i t a t i v e l y t h e same. T h i s could be e x p e c t e d for w a t e r (¢~ = 0) a n d t h e small a v a l u e of the capil-

l a r y used. T h e results o b t a i n e d w i t h t h e still simpler e q u a t i o n [6b], differ, as foreseen, in t h e f o u r t h figure. F i n a l l y , results a r e g i v e n on using e q u a t i o n s w h i c h differ

TABLE V

T a b l e I I I gives t h e results on t h e v a r i a t i o n of 7 w i t h t h e c o n c e n t r a t i o n of a q u e o u s NaC1 solutions. T h e l e t t e r a, used here as a s u b s c r i p t to 7i identifies a g a i n t h e results o b t a i n e d on flow f r o m t h e n a r r o w to t h e wide end of c a p i l l a r y ; t h e s u b s c r i p t b refers to t h e inverse d i r e c t i o n of flow p r i o r to e s t a b l i s h m e n t of t h e final d r o p position. A t l e a s t two such pairs a n d a t t h e m o s t four pairs of m e a s u r e m e n t s were m a d e w i t h each drop. T h e i n d i v i d u a l 7 i v a l u e s o b t a i n e d for each pair, 7~o a n d 7~b, were a v e r a g e d to yield 7. T h e averages, 7a~., of these 2 - 4 7 values are c o m p a r e d in T a b l e I V w i t h t h e l i t e r a t u r e values, 7~ (2). T h e y differ from t h e m a t t h e m o s t b y 0 . 7 % and, on t h e average, b y 0.3%. F i g u r e 10 i l l u s t r a t e s t h e results. T h e s t r a i g h t line r e p r e s e n t s t h e m e a n of t h e l i t e r a t u r e d a t a . T h e full circles are t h e 7a,. values as given in T a b l e I V a n d

from those given here in the treatment of the gravitational (hydrostatic) pressure but which incorporate approximations identical with those contained in Eqs. [6a] and [Ta].

INSTRUMENT CONSTANT, C , FOR THE hr RANGE

11.50-16.00 ASSUMINGUSE OF CALIBRATION CURVE II IN FIG. 4 Literature value of 7 used: 71.97 dynes cm. -1. Expt.*

D X 104 (cm)

Tap.

la lb 2a 2b 3a 3b

74.75 74.09 75.47 74.60 74.44 73.81 4a 74.41 4b 73.52 Average: 74.39 C: 3.253 X 10-2 D: 2.38 X 10-4

2.49 2.35 2.35 2.34

* The numbers in this column refer to the respective experiments in Table I.

2. THE SURFACE TENSION OF NACL SOLUTIONS A. Absolute 7 Values.

TABLE VI SURFACE TENSION OF AQUEOUS N A C L SOLUTIONS CALCULATED BY MEANS OF CALIBRATION CURVE I I IN FIG. 4 AND THE INSTRUMENT CONSTANT C GIVEN IN TABLE V Expt.

NaCI

(moles/lO00 g.)

hr -- H (cm.) hr -[- H (c¢n.) 2

I~r (cm.)

(3'i)ap.

la lb

0.5 0.5

11.6620 11.4655

15.0789 14.8090

3.4167 3.3435

74.55 74.21

2a 2b 8a

0.5 0.5 1.5

12.4052 12.1961 11.8045

16.1117 15.8265 15.6842

3.7065 3.6304 3.8797

75.12 74.31 76.90

8b 15a

1.5 2.5

11.7623 11.8530

15.6304 15.1802

3.8678 3.3272

76.36 78.36

15b 21a

2.5 3.5

11.7676 11.5689

15.0727 14.7560

3.3051 3.1871

78.10 80.09

21b

3.5

11.2431

14.3126

3.0695

78.88

(~ap.)av.

"/c

~/L

13,(%)

74.55

72.12

72.78

--0.91

76.63

74.14

74.40

--0.35

78.23

75.69

76.04

-- 0.46

79.49

76.90

77.67

-- 0.99

194

I-IELLER, CHENG, AND GREENE

the length of the bars represents the maximum spread of the individual ~/ values in Table III. B. "y Values Obtained by Means of an Instrument Constant

If one does not aim for the highest possible accuracy of the ~, values, one may simplify the calibration procedure and convert relative ~, values into absolute ~, values by means of an instrument constant, C. The practical usefulness of this possibility, outlined in Section IV, was tested on the NaC1 solutions. Since a single constant C can apply only to a limited section of the capillary, this section had to be picked. For the sake of convenience, the section encompassed by Fig. 4 (h~ :11.5-16.0 era.) was selected. Eight of the twenty-two experiments, with water, considered in Table I pertain to conical drops the entire length of which was within these limits. On the assumption that only the calibration curve II of Fig. 4 is available (obtained with 40.1 rag. Hg), the apparent 7 values, ~'a~., given in Table V are obtained for these eight experiments. These ~'ap. values are, of course, far too high. From their average, 74.39, and the literature value, 71.97, the instrument constant C is obtained at once by means of Eq. [24]. In addition, the individual D values, resulting from Eq. [25], and their mean, /), are calculated. Table VI gives the individual apparent values (~'i)ap. obtained for several of the NaC1 solutions considered in Table I I I using calibration curve II of Fig. 4. Experiments are excluded in which (hr -t- H / 2 ) and (h~ -- H / 2 ) were appreciably outside of the range of h~ values for which C was obtained in Table V. From these individual apparent ~/ values, the average apparent 1/ values,(%p.)~v, are obtained. If we now use

the instrument constant, the corrected ~/ value, ~'c, is obtained. The last column gives the per cent deviation of % from the literature value ~'L repeated in the preceding column. The % values are clearly less good than those obtained on operating the method as an absolute method (Table IV). For some purposes, however, the error of 0.4 %-1.0 % will not be considered serious enough to forego the convenience of the relative ~, determinations by means of an instrument constant. SUMMARY A brief survey is given on the theory of a new method for determining the surface tension of very small amounts of liquids or solutions (10-3-10 -4 ml.). The method is based on the fact that a liquid drop in a conical capillary will, in the absence of compensating factors, flow towards the tip owing to an imbalance of surface pressures at the two menisci. This imbalance can be relieved by proper inclination of the capillary, i.e., by proper hydrostatic pressure. A simple apparatus is described by means of which surface tensions of water and of aqueous NaC1 solutions were determined with an average error not in excess of 0.3 %. A detailed description of the rather elaborate calibration technique is given which is needed if the method is to give absolute surface tensions. An alternate much simpler, but less accurate, principle is also discussed which involves the use of an instrument constant. The relatively small but pronounced effect of surface roughness and a simple method for eliminating its effect from the results is also discussed. REFERENCES 1. ItELLEn, W., J. Chem. Phys. 40, 3292 (1964). 2. International Critical Tables, Vol. IV. McGraw-Hill, New York, 1928.