For the same reason the concentration of the surface ofa solution is, in general, different from that of the solution in bulk. ... It is merely a ques...

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MEASUREMENT OF SURFACE TENSION By N.

Ernest Dorsey

ABSTRACT This paper (a) presents a brief survey of the more important of the methods which have been employed in the measurement of surface tension, (&) calls attention to some of the more important facts vrhich must be kept in mind by one wishing to succeed in such measurements, (c) indicates certain errors which are frequently made and shows how they may be avoided, and (d) gives A the working equations that are applicable to the methods considered. bibliography of more than 100 selected papers is appended. In each instance the purpose for which the reference was selected is indicated.

CONTENTS Page I.

II.

[II.

Introduction

General remarks 1. Surface tension 2. Negligible quantities— 3. Purity angle 4. Contact 5. Mathematical 6. Notation correction— 7. Reservoir

Methods of measurement— 1.

2.

Capillary tubes Virtual capillary

tube (method of Sentis)-

3.

Vertical

4.

Plate and cylinder

5.

Small pendant drops.. Large pendant drops—

6.

plates

I.

Every

set of

563 565

Pago III.

M Methods of measurement Continued.

565 566 566 568 570 571 572 573

13.

574

14.

7.

and bub-

bles 8.

Jaeger's method

9.

Drop weight

580 581

10. Pull of vertical film_.

on vertical plate. on ring Adhesion plate Pull on sphere Oscillating drops and

11. Pull 12. Pull

15.

576 577 578 579 579

Sessile drops

bubbles Vibrating jets 17. Ripples

589 590 592

16.

IV.

583 586 586 586 587 588

B

INTRODUCTION

measurements connecting the curvature of a liquid

surface at a given point with the difference in the hydrostatic pressure on the two sides of the surface at that point; every set of

observations connecting the velocity of propagation of a deforma563

Scientific

564

Papers of the Bureau of Standards

[Voi.zt

tion over the surface of a liquid with the nature of the deformation,

the size of the liquid mass, the undeformed shape of the surface, the density of the liquid, and the forces that enter into the phenomenon and every set of observations connecting the frequency of oscil;

lation of a liquid

mass

v^ith its size,

forces that enter into the problem,

undisturbed form, and the be used to determine the

may

value of the surface tension of the surface concerned.

The number

ahnost unlimited; and in point of fact the number that are more or less workable in practice, and even the number which have been actually used, is great, and the latter is being increased of methods

is

continually.

Such a wealth of available methods

is

ideal for the detection

the elimination of constant experimental errors, but

of great discordance when due attention

is

it is also

and

fruitful

not paid to such errors.

In the past, surface-tension measurements have frequently been marked by excessive care to secure a high reproducibility, accompanied by scant attention to the possible presence of constant errors and to the real significance of the quantity which is reproduced. There is a great need for a careful study of the various methods in order to determine {w) the proper interpretation of the quantity which is determined by each, (fi) the magnitude of the errors which are introduced by known small departures from the ideal conditions upon which the interpretation rests, and {c) the limits within which each of the several methods are trustworthy. Ferguson (21)^ has started upon such a program. His work will be followed with much interest. Unfortunately, the mathematical discussions in some of his earlier papers have been marred by errors (66, 78, 79) it is hoped that all such errors will be corrected as the work progresses. The object of the present paper is much less ambitious. Its aim is to present a brief survey of the more important of the methods which have been employed, to call attention to some of the more important facts which must be kept in mind by one wishing to succeed in such measurements, to indicate certain errors which are frequently made and to show how they may be avoided, and to give the working equations that are applicable to the methods which are here considered. The derivations of the equations and directions for the construction and use of the necessar^^ apparatus will have to be sought in other places, to the more important of which references will be given. The list of references does not pretend to be complete but is intended merely to direct the reader to one or more of the sources from which the required information can be obtained ;

most

satisfactorily.

iThe figures given in parentheses here and throughout the text numbers in the bibliography given at the end of this paper.

relate to the refereuce

Measurement of Surface Tensioii

Darsey]

II.

GENERAL REMARKS 1.

The term

565

SURFACE TENSION

" surface tension " conveys a fictitious, thougli very valu-

phenomenon under consideration. In the body of a liquid the molecular attractions acting upon an element of tlie liquid are, on the whole, completely balanced the element is pulled as strongly in one direction as in another. As the surface is approached this balance is destroyed. Near the surface the element is able, picture of the

;

pulled more strongly toward the liquid than toward the other medium from which the surface separates the liquid, or else the reverse is This lack of balance in the molecular forces acting upon the true. is what gives rise to the property that we call The lack of balance, and consequently the surface depends upon the properties of both of the media of which

elements of the liquid .surface tension.

tension,

common boundary.

the surface

is

erty of one

medium, but

the

is

Surface tension

is

not a prop-

the joint property of two media.

When

one speaks of the surface tension of a particular liquid, what is meant, or at least what should be meant, is the surface tension of the liquid in contact with its own pure vapor at the same temperature. Actually the surface tension of the interface separating a liquid from a gas or vapor depends but little upon the nature of the latter, provided that it does not react chemically with the liquid, and that certain exceptional conditions are excluded.

When we enlarge a clean liquid surface we do not stretch it in any proper sense of this term we merely bring into being an additional area of the same kind of surface as we had before. When we speak of a surface tending to contract under the action of its surface tension we do not mean that the particles in the surface tend to crowd nearer together while remaining in the surface, but that they tend ;

withdraw from the surface into the interior of the liquid. As we are concerned with a condition of stable equilibrium it is obvious that they can not so withdraw unless provision is made for a cor-

to

responding decrease in the area of the surface. The range through which the forces are unbalanced is negligibly small as compared with the smallest difference in length which we can experimentally measure under the conditions with which we are here concerned. All the equations employed in the reduction of surface-tension measurements are based upon the assumption that the transition layer in which the forces are unbalanced is of negligible thickness. Should conditions ever arise in which this assumption is not justified, then either

we must completely

reconsider the

equations, taking into account the variation in the forces as

through the surface layer, or we must change the

scale of

we pass

our opera-

566

Papers of the Bureau of Standards

Scientific

tions in such a

maimer that the thickness of the

iVoi.it

transition layer be-

comes negligible in comparison with the smallest length which we can measure upon the revised scale. Indeed, unless our scale is so coarse that the transition layer is negligibly thin, the concept of a "surface" tension loses its significance. 2.

NEGLIGIBLE QUANTITIES

Although frequently forgotten, it should be remembered that the terms " negligible " and " small " are purely relative. To say that something is negligible or small, without indicating either directly or by implication anything with which it is thus compared, is to talk nonsense. It can be negligible or small only in comparison with something else of the same kind. It is especially necessary to keep this clearly in

mind when considering the equations applicable to With few exceptions the working equa-

surface-tension phenomena.

tions are approximate only

and are validly applicable only under

the condition that certain lengths are negligible with reference to certain other lengths involved in the phenomenon. It makes no difference

how

small the

first

length

unless the comparison length in comparison with

it.

is

is,

the equation

is

not applicable

so great that the first is negligible

By this we mean that

in any

sum or

difference

of the two lengths the first may be neglected without changing the value of the sum or difference by an amount that is experimentally It does not mean that when the first length occurs as the numerator of a fraction of which the second is the denominator that this fraction can always be ignored; that depends upon the connection in which the fraction occurs. Will the value of the sum or other function in which it enters be changed by a significant

significant.

amount

if this fraction is

made

zero

?

If so, then the fraction

is

not

negligible with reference to the other terms that enter into the

function. 3.

PURITY

Although the purity of the materials used is of importance, it should be borne in mind that it is the purity of the surface separating the juxtaposed fluids that must primarily be considered. This is quite a different thing from the chemical purity of the materials Exceedingly small amounts of physical contamination may most seriously. Whatever tends to lower the surface tension will tend to accumulate in the surface. For this reason, other things being the same, the higher values of the surface tension

in bulk.

affect the surface

are worthy of the greater confidence.

is,

For the same reason the concentration of the surface of a solution in general, different from that of the solution in bulk. When

—

MeasxLremcnt of Surface Tension

Dorsey]

567

formed, diffusion sets in tending to bring about Consequently, the properties of a surface bounding a sohition should be expected to depend upon the age of the surface. It is merely a question of whether the change takes place slowly enough for us to be able to detect it with the means at our disposal. In some cases it does take place so slowly. Thus, Rayleigh (59) found that the tension of a freshly formed surface separating a soap solution from air differs very little from that of an air-water surface, but decreases markedly within a few seconds. The explanation is to be sought in the progressive accumulation of soap in the superficial layers. More recently du Noiiy (50a, 50b) and Johlin (35a) have shown that the surface of many colloidal solutions attain equilibrium very slowly. The tension of the surface may continue to decrease for hours after the surface is first formed, and the final value of the tension may be as much as 50 per This is explained by the procent lower than the initial value. gressive change in the concentration of the surface layer. In the case of pure fluids no such diffusion phenomena can exist, and it is very probable that the rearrangement of the molecules attendant upon the formation of a fresh surface takes place too rapidly to be detected by any means that has yet been proposed. We might likewise anticipate that the time required for the interface of two pure fluids to become saturated with each would be too short to be detected by any of our usual procedures. This also seems to agree with experience. Nevertheless, it is well to take pains to insure such saturation before making measurements. When a surface is exposed to the air it gradually becomes contaminated and its properties progressively change. Thus Schmidt (65) observed that the superficial viscosity of mercury increased manyf old in the course of a few hours after it was first exposed to a fresh surface

is

this difference in concentration.

the

air.

The surface

tion of the surface

tamination

is

it is difficult

—

tension it is

is

by the contaminaThe progressive effect of con-

likewise affected

reduced.

of the same sign as that of changing concentration

to separate the

two

effects.

When

one of the fluids separated by the surface is a gas, one would anticipate, from the great disparity in the density of a gas and a liquid, that under ordinary conditions the tension of the surface would be very slightly affected by the nature of the overlying gas

and would be

still less affected by the solution of the gas in the This appears to be borne out by experiment (100). On the other hand, when a liquid is near its critical point and is in contact with an inert gas under high pressure, it seems possible that the solution of the gas in the liquid might produce a considerable effect.

liquid.

;

Scientific

568

Papers of the Bureau of Standards 4.

ivoi. ti

CONTACT ANGLE

In many surface-tension problems it is necessary to know the angle at which the liquid surface meets a given solid wall. But the intersection of two surfaces defines two plane angles; which shall we take as the angle of contact of the liquid with the wall? The acute angle between the tangent to the surface at the point of contact and the tangent to the wall at the same point, each drawn in the plane that

is

normal to the

line of contact,

might be taken

as defining the angle of contact; but physical considerations reveal

not satisfactory. There are two fluids to be considered One is to be regarded as the at our disposal. angle of contact of the surface of fluids with the wall, and the other as the angle of contact of fluid2 with the wall. Either angle might be chosen as the angle of contact of fluids, the other being that of fluidg. But the more logical choice appears to be that which is universally accepted; namely, the angle of contact of

that this

is

and two angles are

is the angle of the limiting wedge of fluids bounded by the fluidi-fluidg surface and the fluids-wall and 180°, each included. It may have any value between surface. If the wall has an absolutely sharp edge at the line of contact, the concept of a tangent to the wall at and across the line of contact loses its significance, and consequently the significance of an angle of contact is likewise lost. But if the edge is rounded, no matter

fluidi

with the wall

that

is

how

short its radii of curvature, provided that they are long as

compared with the range of molecular forces, the concept holds. In practice, an edge is never infinitely sharp. If the surface were free from all other constraints, probably it would adjust itself in such a way that the angle of contact would be always the same for the same surface and wall, temperature and other pertinent conditions being the same. This equilibrium angle of contact is called the " contact angle " of the fluid with the wall under the

But all who have worked with mercury manometers know that a liquid surface may be subjected to constraints that appear closely akin to frictional resistance of the kind characteristic of the sliding of solids over solids. Under the action of such constraints the surface can not adjust itself as it otherwise would, and the angle of contact may take any one of a range of values. It is desirable to distinguish between the existing angle existing conditions.

of contact and that particular angle of contact which

is characequilibrium in the absence of such constraints as we have considered. For the latter, which is probably a characteristic of the fluids and the wall, we shall reserve the term " contact angle " the others will be designated as angles of contact.

teristic of

Measurement of Surface Tension

Doi-aoi/]

The

existence of constraints

line of contact

is

is

An

to be expected.

569 advance of the

necessarily accompanied by the replacing of

an

intimate contact of the solid and one fluid by an intimate contact of the solid and the other fluid. If there is any adhesion between the

and the fluids, the replacement will involve irreversible effects, and constraints of the kind considered will result and there is always solid

;

adhesion.

There is, however, one case in which the constraint will be absent. This is when one of the fluids adheres to the solid so strongly that it is not replaced by the second, but the second merely flows over a thin laj^er of the first which continues to be attached to the solid. We thus have in reality merely one surface that of the fluid-fluid boundary the solid merely serving as a support to the thin layer. If the thickness of the adhering layer exceeds the range of the molecular forces, then, except for viscous effects, which merely delay the attainment of equilibrium, there is perfect freedom of motion, and the apparent contact angle is zero. If the thickness is less than the range of molecular forces, then the apparent contact angle may be different from zero but will be definite if sufficient time is given for the layer to drain to its equilibrium thickness. Before such drainage is complete the angle may be expected to vary as the layer thins. In this case, also, the motion of the boundary surface over the solid will be quite free, but the freedom will decrease as the time required for drainage increases. The drainage will be affected not only by the viscosity of the fluid forming the layer, but also by that of the other fluid which presses it against the solid. In all these cases the actual angle of contact of the fluid-fluid surface with the solid is to be found at the free end of the layer far beyond the region with which we are primarily concerned. Any change in the surfaces, whether due to contamination or other cause, may be expected to cause a change in the angle of contact, except in the single case in which it is zero as the result of the solid being covered by a layer of fluid so thick that the molecular forces of the solid can not penetrate it. The main facts brought out by the large amount of experimental work that has been done in the study of angles of contact justify these conclusions. It has been found that in many cases the solid surface can be so prepared that the liquid will truly wet it (81, 64), will slide freely over the surface under these conditions the effective angle of contact is actually zero, the free surface passing by imperceptible degrees into the exceedingly thin film which remains coating the solid and extending to points far beyond the apparent line of contact. But when surfaces so prepared are allowed to dry, the liquid may then meet them at a finite angle and may no longer slide

—

—

;

10712°—26

2

570

Scientific

freely over

them

Papers of the Bureau of Standards In other cases

(64).

a given surface in such a

way

it

[voi. ti

seems impossible to prepare

that the liquid can truly wet

Generally, in such cases the liquid will not

move

it

(12).

freely over the solid,

will be both finite and variable; but occabe free, and then the contact angle though For any solid and any liquid an inconstant finite finite is constant. angle appears to be possible, but the insuring of a zero or other constant angle is in many cases very difficult if not impossible.

and the angle of contact

may

sionally the motion

For these reasons it is desirable to choose for the measurement of surface tension such methods as do not require a knowledge of the value of the angle of contact. Methods involving a knowledge of the angle of contact may, however, be safely used if care is taken to is perfectly wetted by the liquid at the time that

insure that the solid the measurement

is

made.

over the wall, although of contact

is

certainly

it

Or,

if

the liquid moves perfectly freely

does not wet

known to

it, provided that the angle the necessary precision. measure-

A

ment of the angle of contact under the conditions of observation in surface-tension measurement involves many difficulties, not the least of which is the fact that the angle required is that made by the last

To measure the angle under one under another involves the unsound assumption that it is necessarily the same in both cases. (Quincke (55, 56), Volkmann (82, 83, 84, 85, 86, 87), Anderson and Bowen (4), Bosanquet and Hartley (T), Ablett (1), Ferguson (23), Dorsey (12),Devaux (95,96).)

infinitesimal element of the surface. set of conditions

and to use

5.

Of

the

MATHEMATICAL

many mathematical

in connection with the

satisfactory 80).

it

discussions of the problems that arise measurement of surface tension, the most

and complete

is

probably that of Verschaffelt

(78, 79,

from those most deduces general equations which can be readily

It should be noted that his notation differs

commonly

used.

He

adapted to the various particular cases that arise in practice, but in only a few instances does he give an equation in the final form required for the reduction of observations. As the derivations of the equations applicable

methods which we

to

the

several

shall consider are readily available elsewhere,

it

and references to where The equations given will not always

will suffice to give here the final equations

their derivations

may

be found.

be of the same form as those given in the corresponding citations but will be derivable therefrom. The change in form has been made for the purpose of securing greater uniformity.

Measurement of Surface Tension

Dorsou]

6.

571

NOTATION

Throiigliout the remainder of this j^aper the following notation be used; when additional symbols are required they will be

•will

explained as introduced. Subscripts. Whenever it is necessary to distinguish between the two fluids of which the surface is the interface it will be done by means of the subscripts 1 and 2. When the problem involves a reference to a portion of the surface which is horizontal and flat, the subscript 1 will be used to denote that liquid Avhich lies below that

—

portion of the surface.

When

a surface of revolution

is

under con-

sideration the subscript 1 will be used to denote that liquid which

is

below the vertex of the surface. (See figs. 1, 2.) It will be noticed in Figure 1 that the fluid which is denoted by 1 when the meniscus in the tube is being considered is the same as that denoted by 2 when the suspended drop is being discussed. Quantities pertaining to the vertex of a s'urface of revolution will be distinguished, as necessary,

by means of the subscript

A^^±-,

T

0.

X— the choice of sign ,

is

determined by the fact that

A^ is essentially positive. A^ is half the product of the radius of a tube multiplied by the height above a flat surface that the liquid would rise in the tube if the meniscus were hemispherical; it is numerically equal to half the height that the liquid would rise under such conditions in a tube of unit radius. In the literature the symis* used to denote sometimes A^ and sometimes 2^.^; the latter is the usual custom of German and the former of English authors. ^=:The acceleration of gravity. h^p/(p^—P2)g', by definition. It will be noticed that p is the difference in the pressures exerted by two columns of liquid of height h, the density of one liquid being /o^ and that of the oth^- being p^/>=:The amount by which the pressure at a point infinitely near the surface and in fluids exceeds that at a neighboring point in fluidi. It measures the discontinuity in the pressure on crossing the surface. P=The total vertical pull exerted by a mass of liquid that is

bol a^

raised as a result of surface tension effects.

2r=r:The diameter of a tube, or the distance between

two

parallel

plates.

/?=Radius of curvature of the surface;

i?'

principal radii of curvature at the point where

and

p

^"

is

denote the

measured; Eq

the radius of curvature of the vertex of a surface of revolution. radius of curvature is to be considered positive if in the section to which it applies the surface is concave toward fluidg. In Figure is

A

2,

Bq is negatiA'C for the drop 2^= The surface tension.

(Z>)

and positive for the bubble {B).

572

Scientific

Papers of the Bureau of Standards

\_vol ti

^=The

angle of contact of the surface of a fluid with another surconsidered in this paper the second surface will be that of a solid wall. In every case it is necessary to indicate to which of the two fluids 6 applies. face.

In

all cases

P=The (See

density; p^

is

the density of fluids and

p^

that of fluidj.

figs. 1, 2.)

In dealing with surfaces of revolution, axes of coordinates will be laid so that the origin is at the vertex, the axis of is in the

X

tangent plane, and the positive direction of Y is upward. cases, of which there are but few, will be considered as they 7.

RESERVOIR CORRECTION

The fundamental equation

in capillarity is

v-t[1,+-^\ or, if the surface is

whence

T may

Other arise.

(1)

one of revolution,

be obtained at once

if

p^ and Rq^ or ^, ^', and R'^

are known.

In many cases the pressures po and p are derived from the observed height of a column of liquid. In practice, the vertical height of the column is usually measured from the vertex of the surface ofa liquid contained in a relatively large reservoir. Unless the sectional area of the reservoir exceeds a certain value, depending upon the liquid, the surface at the vertex will be appreciably curved. This curvature will introduce a pressure (equation 2). The total pressure corresponding to the observed height will be the sum of the pressure due to the curvature, taken with the proper sign, and the hydrostatic pressure due to the observed column. It is customary to express the pressure due to the curvature in terms of the equivalent hydrostatic column and to regard it as a correction to be added to the observed height. This correction may be denoted by the symbol he. Richards and Carver (64) have experimentally investigated the

amount of

this correction in the case of cylindrical reservoirs of cir-

cular section of radius r and have prepared a curve giving the values

r/A^ lying between A^. An optical

of h^/A-^^ corresponding to various values of 1.25 and 4.30. They use the symbol a to denote

method was used for measuring the curvature of the surface at the vertex. Rayleigh (62), and also Verschaffelt (79), have deduced expressions by means of which the value of h^/A may be computed when r/A is known, provided that r/A amounts to at least several

—

Measurement of Surface Tension

Dorsey]

573

Rayleigh gives a table of values of he/ A corresponding to r/A lying between G and 10. Values may also be computed from the exact tables prepared by Bashforth and Adams (5), either directly or from the table which was prepared from tliese tables by Sugden (70). The values given in Table 1 were obtained from Sugden 's table they essentially agree with the values given by Rayleigh and b}' Richards and Carver. units.

integral values of

;

Table [The capillary

rise

1.

Reservoir correction

^o in a cylindrical tube of circular section, radius

T r,

the capillary constant,

/

v

»

being A^]

T

/ic

r

h^

r

fto

A

A

A

A

A

A

1.75 2.00 2.25 2.50 2.75 3.00

0.76 .59 .46 .37 .30 .24

3.25 3.50 4.00 4.50 5.00 5.50

0.19 .15 .09 .054 .035 .022

6.0 6.5 7.0 8.0 9.0 10.0

0.015 .009 .006 .0023 .0009 .00034

In the same paper, Rayleigh shows that the equation of the surface near the vertex is 2/

where Jo

is

= /tc}/o(jj-lj

the Bessel function of the

first

(3)

kind and zeroth order,

i

denotes y — i, and he is the correction required. From this it can be shown that at a point 1.8J. distant from the vertex the surface will lie Ac higher, or lower, than it does at the vertex. This is Rayleigh's criterion for the flatness of a surface.

If the difference in

between the vertex and a point 1.8^ distant from it can not be detected, then the surface is flat within the limits of error in the In order that h^/A for a water-air surface vertical measurements. shall be as small as 0.001, it is necessary for the reservoir to have a radius of about 2.4 cm. That the curvature of a water-air surface may be apparent at 2.5 to 3 cm from a plane vertical wall was pointed out by the author (12) many years ago. In the further course of this paper we shall assume that this effect, arising from the curvature of the surface in the reservoir, has been considered and the proper correction applied. Where it seems desirable to emphasize the fact, we may speak of the height measured level

from a

flat surface.

III.

METHODS OF MEASUREMENT

If ^o is sufficiently small, po will be of such magnitude that it can be measured without serious difficulty. The determination of T by

equation (2) then reduces to the problem of finding Rq.

This can be

574

Scienti-fic

Papers of the Bureau of Standards

{voi. tt

done in some cases by means of optical methods (64) in other cases may be determined by the measurement of a photograph or of an enlarged projection of the surface (8, 15, 91) but in most cases it is more satisfactory to deduce the value of Ro from some other length that is more amenable to direct measurement. This is the usual procedure in the method of capillary tubes. ;

it

;

1.

CAPILLARY TUBES

For capillary tubes the value of Eq is deduced from the radius of the tube and the properties of the fluids. In this way equation (4) is obtained. In this equation ^^ denotes the angle of contact of the lower fluid (fluidi) with the wall.

^cos

^i=^ri+^ (^) sec^ +-i ("^Ysec^ ^ijsec*

di (1

6^ (1

-sin e,y

-sin

(1

^i)^ (1

+2 1ogi±f^j+...]

+2

sin

+sin

^i

d^)

+2

sin^-^i)

(4)

or, if the angle of contact is zero,

The

first expression, Verschaffelt (79), holds provided that the cube of r/ho is negligible in comparison with unity the second, Rayleigh (62), holds if the fourth power of r/h is negligible. Even the first ;

of these approximations is ample for most practical purposes. From its definition (see Notation) it is obvious that ho is the vertical distance of the vertex of the meniscus from the flat liquid surface in

a connecting reservoir, the same body of overlying fluid being in In designing the apparatus, an unnecessarily small r should not be chosen, as this limits the precision; T can not be determined to a higher precision than that to which r is measured. The disadvantages inherent in this method are: (a) The contact angle must be known; (h) there is difficulty in measuring r to the necessar}^ precision; (c) as the surface in the tube is small, a very small amount of contamination may produce a marked effect; (d) the cleaning of a small tube is not easy; and (e) the meniscus lies in an exposed position so that its temperature has to be inferred rather than measured. Conditions can probably be adjusted so that the last-named disadvantage is more apparent contact with each of these surfaces.

than

real.

Numerous modifications

of the simple capillary-tube method have been proposed and occasionally used. One of the most recent was

Measurement of Surface Tension

Dorsc!/]

575

proposed by Ferguson (20). He suggests applying a pressure to the capillary column so as to drive the meniscus down until its vertex By is just in the plane of the end of the tube, placed vertically. this procedure the meniscus is always in the same section of the tube, thus obviating the necessity of calibrating a long section of tubing, and the cleaning of the tube and the control of the temperature are facilitated. Obviously, this method is not applicable to the measurement of the surface tension of a liquid in contact with only its own pure vapor. For liquids that are obtainable in only very small quantities Kiplinger (38) proposed that a short column of liquid suspended in a capillary tube be used, the tube being tilted until the liquid surface at the lower end becomes plane. Then, if the contact angle is zero,

T =0.5 {p^-p^)grl

cos

A

(6)

approximately, where I is the length of the short column of liquid and A is the angle it makes with the vertical. His results run low by from 1 to 8 per cent. But, as Ferguson (22) has pointed out, the lower surface is never plane when the tube is tilted. Ferguson has proposed that the tube be left vertical and the pressure upon the suspended column be adjusted until the lower surface becomes plane, In this case the usual equation for capillary as optically tested. tubes applies. He found that individual determinations made by this method differed among themselves by as much as 7 per cent, but the probable error of a set of 28 observations amounted to only 0.25 per cent. Only a few cubic millimeters of liquid are required for the measurement.

Sugden (70) endeavored to eliminate the uncertainty of the effect due to the curvature of the surface in the reservoir by employing two capillaries and basing the computation upon the difference in the elevations of the vertices of the two menisci. He employed a method of approximation, using the tables computed by Bashforth and Adams (5). The increased difficulties introduced by the use of two capillaries would seem to offset by far any apparent advantage of the method.

Anderson and Bowen (3) proposed an optical method for measuring the curvature at the vertices of the surface in the capillar}^ and in the reservoir. These values inserted in the obvious manner into equation (2) permit the surface tension to be evaluated at once. The realization of the method requires a knowledge of the index of refraction of the liquid and the ability to estimate accurately the difference in elevation of

by

two images, one formed by refraction and

The determination is independent of the angle of contact but would seem to be difficult to employ with any the other

reflection.

576

ScientifiG

Papers of the Bureau of Standards

considerable precision.

For additional

\joi. tt

details the original

paper

should be consulted.

Bigelow and Hunter (5a) and Carver and Hovorka (10a) have used an interesting modification which is attributed to Oersted. In this the capillary consists of a perforated plate which rests on the top of a vertical tube of large diameter. The whole is immersed in a reservoir of liquid until the latter floods the plate; then the tube is raised slowly, or the level of the liquid in the reservoir is slowly lowered, until the meniscus which forms at the perforation breaks. This occurs when the distance from the surface of the liquid in the reservoir to the bottom of the meniscus at the perforation is equal to the height the liquid would rise in a tube of the diameter of the perforation. The method is said to be quite accurate and rapid. In this method the diameter is measured at the exact section at which the meniscus forms, and this section is the same for all liquids; the cleaning of the capillary is relatively easy. These are distinct advantages. The measurement, however, depends upon the determination of the incidence of instability, and this may introduce difficulties when exact determinations are desired; also, the perforating of the plate may present difficulties. problem which is of no little interest in connection with the determination of the surface tension by means of capillary tubes is the determination of the shape of the surface between two concentric cylinders, one of which is much smaller than the other. Apparently this problem has not been very completely solved even by approximations, but Verschaffelt (7T) has shown that in the region approximately midway between the tubes the curvature of the radial section of the surface differs but little from that of the corresponding region (that at the extremity of its minor axis) of the ellipse of which the

A

major axis is equal to the distance between the walls of the two tubes, the minor axis is equal to twice the height of the meniscus that is, to twice the distance between horizontal tangent to the surface and the line of contact of the liquid with the outer tube. He tested this relation by measuring the changes produced in the height to which a liquid rises in the inner tube when the inner radius of the outer ;

tube was changed from 1.04

cm

to 0.745, 0.505, 0.325,

and 0.295 cm,

respectively, the external radius of the inner tube being 0.0464 cm.

Excepting for the smallest tube, the observed changes from those calculated by not over 0.004 cm, which was con-

in every case. differed

sidered satisfactory for his purpose. 2.

VIRTUAL CAPILLARY TUBE (METHOD OF SENTIS)

employed a very ingenious modification of the A modification which entirely eliminates the question of the angle of contact, provided that it is the same for each Sentis

(67, 68)

capillary tube method.

—

Measurement of Surface Tension

Dorsey]

577

of two consecutive observations. His method has very appropriately been called the method of " virtual capillary tube." He introduced into a vertical capillary tube such a column of liquid that a drop formed on the lower end of the tube, and, with the column, was supported by the surface forces. (Fig. 1.) He measured the maximum

horizontal diameter, 2r, of the drop.

Then

a vessel of the liquid

was

brought up from beneath until the surface of the liquid just touched the bottom of the drop. At this instant the column in the The position (A) of tube falls. the surface is noted, and the vessel then raised higher until the is meniscus of the column in the tube restored to the position (P) is which it occupied when the drop was measured; the surface is then tance

The vertical dis(AB) between these two posi-

tions,

corrected for the curvature

in

position B.

of the liquid in the vessel,

quantity

by

^0

is

the

which we have denoted Notation)

(See

it

;

3

meas-

ures the hydrostatic pressure which

the column exerted upon the vertex is,

of

the

drop.

The problem

therefore, exactly the

same as

that of a capillary tube with the

denser liquid on the upper side of the surface at the vertex, the angle contact being zero, and the diameter of the tube being equal

of

to that of the

maximum

section of the drop.

horizontal

Equation

applies directly, but

Fig. 1.

(5)

it should be noticed that in this case the quan-

Drop of liquid suspended from a tube

m

—

the Illustrating Sentis's et hod method of the virtual capillary tube

The approximations employed by Sentis were lower than that corresponding to equation (5).

tity Aq is negative.

3.

VERTICAL PLATES

Closely related to the rise in tubes is the rise between vertical plates. When the plates are parallel and the angle of contact of fluid-L is

zero,

r=p„.[l +0.2146(0-0.052(0+ 10712°—26

3

.

.

.]

(7)

578

Scientific

Papers of the Bureau of Standards

\yoj. ti

(Volkmann (81)) where the distance between the plates is 2r, the other quantities having the same meaning as before. If the plates are inclined to one another at the angle <^, the line of contact of the plates with one another being vertical,

and

if

the axes of coordinates

coincide, respectively, with the line of contact of the plates

the intersection of the

flat

and with

surface of the liquid in the reservoir with

the plane bisecting the angle between the plates, then to a first approximation the equation of the median section of the meniscus is

T cos (Pi

- P2)

^1

g tan

2

In order to facilitate the use engraved on one of the plates a (26) series of hyperbolas, for each of which the value of the product xy was known. Then, to make a measurement, the angle between the plates is changed by means of a micrometer screw until the liquid surface coincides with one of the hyperbolas, the plates being so placed as to make this possible. The hyperbola determines the value hence of xy, the reading of the micrometer fixes the value of if the value of ^1, the angle of contact of the lower fluid (fluidi) is known, T can be found at once. Quite recently Ferguson and Vogel (23a) have suggested a procedure for determining the surface tension from the positions of the points of the hyperbola with reference to any set of axes which are parallel to those of the hyperbola. This eliminates the difficulty of determining the exact position of the latter axes. This

is

the equation of an hyperbola.

of this method,

Grunmach

,

4.

PLATE AND CYLINDER

Wagstaff (88) used a plate pressed against the inside of a cylinder.

The

coordinates of a series of points upon the projection of the menisupon the plate were measured, and from these and the radius of the cylinder the value of T was calculated. The main advantages claimed for this method over the use of capillary tubes is the facility with which the apparatus can be cleaned and the fact that no very small distances have to be measured. The volume of liquid required cus

is large.

Obviously in all these cases the surface in the reservoir must be plane or due allowance must be made for its curvature. In the case of plates, it is also necessary that observations be confined to those portions that are so far from the edges that the shape of the meniscus is the same as if the plates were of infinite extent. For these reasons 13lates

are not so suitable as tubes.

In the use of the capillary rise methods for determining the tensions of liquid-liquid surfaces

it is

necessary that care be taken to super-

Measurement of Surface Tension

Dorsei/]

579

way as to facilitate the drainage of the trapped between the wall and the advancing liquid. On account of difficulties that may be produced by such trapping, it is

pose the liquids in such a film that

is

frequently impossible to test satisfactorily the freedom of motion

of the surface over the wall. 5.

Very similar

to the

SMALL PENDANT DROPS method employed by Sentis

is

that generally

knoAvn as the method of small pendant drops. In this method a drop is formed, the pressure in it is measured, and its form determined, either by a direct measurement of the drop or by the measurement of an enlarged photograph or projection of it. The results, being obtained from a direct measurement of the curvature of the drop at the point where p is measured, are independent of the contact angle. Obviously the method is equally applicable to bubbles or drops, as of oil in water, that are pendant upward, due account being taken of the sign of h (8, 15). 6.

LARGE PENDANT DROPS

In the case of large pendant drops the pressure is too small to be measured manometrically with precision. In this case measurements at two or more selected levels give the data required for computing the curvature at each level.

This determines the difference in pres-

sure from level to level, which, equated to the corresponding differ-

ence in hydrostatic pressure, permits the tension to be determined.

Measurements are usually made upon enlarged photographs or prodrop or bubble. The method employed for reducing the observations is, in part, determined by the manner in which the measurements have been distributed over the surface (15, 91, 93). In some cases the tables of Bashforth and Adams (5) should be of much assistance. No knowledge of the contact angle is required. Likewise, from any two suitable sets of measurements made at different levels upon the same surface, whatever its form, the tension can be computed. Such computation does not involve any angle of contact. Thus, Pekar (52) has studied the form of the surface in a cylindrical tube of circular section, and Eotvos (13) has studied a It should be remembered that all such section of a large drop. determinations are based upon the departure of the surface from a spherical form, thus resting upon just those factors that give rise jections of the

to the correction terms in the equations applicable to the rise in a capillary tube.

upon the

precise

This indicates the limitation of the method it rests measurement of quantities which in other methods

enter only as correction terras.

;

—

:

Scientific

580

7.

An

Papers of the Bureau of Standards

SESSILE DROPS

Woi.

tt

AND BUBBLES

problem is involved in the method of sessile In this method the form of a drop resting upon a flat horizontal surface, or of a bubble formed under such a surface, The quantities that are usually measured are the maxiis studied. mum horizontal diameter (2Z) of the drop or bubble, the distance {h) from the plane of maximum diameter to the vertex, and the distance {K) of the vertex from the plane upon which drop rests essentially similar

drops and bubbles.

P.

D '21-

-21

B p.

Fig. (fig.

2).

(99, 48)

Meridional section of a sessile drop (D) and hnhble (B)

2.

From the first two the surface tension can be determined by means of the equation A' = ^\[1

in which

^0

''"§*'S*

the absolute value of the radius of curvature at the

is

vertex of the drop or bubble. (13)

it

From

Verschaffelt's

(80)

equation

follows that

i_=M5 /I A \A

IZA

(10)

Bn approximately.

Whence

it is

omit

it.

evident that this term in equation (9) the preceding term is small; we

when

will be very small indeed

may

(9)

]

The equation

relating

J.,

K, and

6,

the angle of con-

tact of the outer fluid with the wall is (48)

2A

A=

(11)

2 cos^

SlKcos-

It should be especially noticed that here 6 is in every case the angle of contact of the outer fluid with the wall; when bubbles are con-

Measv/rement of Surface Tension

Doi'seyi

581

sidered, B is the angle of contact of the surrounding liquid with the

when drops are considered, 9 is, as before, the angle of contact of the surrounding fluid with the wall. The angle of contact of the

wall

;

is w, the supplement of B (see fig. 2). combining (9) and (11) the angle of contact can be determined. Equations equivalent to (9) and (11) are attributed (48) to Poisson. The approximation of neither of these equations will be satisfactory unless A /I is small as compared with unity. In much of the work that has been done by this method the drops or bubbles have been entirely too small for the approximations used to be satisfactory; in nearly all of the earlier work only the first term of each of these equations was used, thus making matters far worse. Quincke (54) determined experimentally, in both water and alcohol, the ratios of corresponding to air bubbles of various diameters to the k and of corresponding quantities for bubbles 10 cm in diameter and attempted to correct his values by means of them. Siedentopf (69) used drops about 1 cm in diameter and measured the maximum horizontal diameter and the curvature at the vertex, using an ophthalmometer. His observations were reduced by graph-

liquid forming the drop

By

K

ical

methods.

Although the method of sessile drops is very convenient for' determining the surface tension of fused materials at the instant of solidiprobably right in saying that " a really practical importance can not be ascribed to them." Others who have used and discussed the method are Heydweiller (32, 33, 33a), Lohnstein (43, 44), and Worthington (94). Heydweiller (33a) gives a

fication, Verschaffelt (80) is

table

which

facilitates the reduction of observations. 8.

When

METHOD

JAEGER'S

the column of liquid that rise& in a tube dipping vertically

is slowly forced down by applied gas pressure, it is observed that the pressure steadily increases to a well-marked maximum just before a bubble escapes from the end of the tube. The value of this maximum depends upon the surface tension and the density of the liquid and upon the configuration of the apparatus. Jaeger (34, 35) first employed this phenomenon for measuring the surface tension, and the method is generally called by his name. He first employed it for the determination of the way in which the tension varies with the temperature and as at that time the equation relating the maximum pressure to the radius of the tube and to other factors had not been developed, he attempted only relative measurements. He used two tubes of different diameters and adjusted the difference in the depths of their immersion until the maximum pressure was the same for each. Until the line of contact reaches the end of the tube, the condition is exactly the same as in the case of capil-

into a liquid

;

582

Scientific

Papers of the Bureau of Standards

[Vol.ti

lary rise; whence it is evident that, to a first approximation, the pressure will be proportional to the surface tension multiplied by a

function of the radius of the tube. Hence, in the case of two tubes to different depths, the hydrostatic pressure corresponding

immersed

to the difference in

immersion will likewise be proportional

tension multiplied by a function of the radii of the two tubes.

to the

Fur-

evident that the next degree of approximation will depend upon the density of the liquid. Jaeger assumed that it would be obtained by simply multiplying the first order approximation by

thermore,

it is

a function of the density alone. Under the conditions of that work he found that this assumption fitted his observations within the limits of his experimental error, the density function being linear. Actually, the exact expression is not so simple. He emphasized the importance of the end of the tube being broken off perfectly smooth and flat and at right angles to the axis of the tube, and he comments on the advantage of using a broken, rather than a ground end. The advantages of this method are: {a) The continual renewal of the surface reduces the troubles arising from surface contamination, (h) the temperature control is facilitated by having the surface surrounded by a large volume of liquid, {c) the use of a light manometric liquid, or of an inclined manometer, facilitates the measurement of the pressure, {d) the radius that is to be measured, in the case of absolute measurements, lies exactly at the end of the tube. While the use of a light manometric liquid is of advantage in the case of relative measurements, either after the manner of Jaeger or in the case of a single tube when provision is made for adjusting the depth to exactly the same amount in every case, its advantage is rather illusory in other cases, as the precision is limited by the accuracy with which the depth of immersion can be determined. Since this early work, the equation relating the tension to the diameter of the tube, the maximum pressure, and the density of the liquid has been deduced by methods of approximation to a precision which is ample for experimental work. Unfortunately, several incorrect equations have been deduced and used, as pointed out by Schrodinger (66), and later by Verschaffelt (79). Four incorrect equations have been found in the literature (9, 24, IT, 10), viz:

^

-^

2

2

I-ttI

(9 3U)

(12)

(13) (14)

•^~ 2 (15)

Measv/rement of Surface Tensiori

Dofsey]

The

583

correct equation (66) is:

^-l['-i(0-KO"--] is the maximum excess of pressure in the bubble at the level of the end of the tube over that at the same level in the surrounding

where p liquid

and h=-.

tions exist,

—_

.

.

The equation assumes that

static condi-

and that the internal circumference of the tube, on which Apparently no attempt is circular and horizontal.

the bubble forms,

has been made to determine how far these conditions may be departed from without introducing appreciable errors, although Martin (49) states that in purely relative measurements the horizontality and planeness of the contour of the end of the tube are of little imporIt is very important that all such questions be carefully tance. studied.

For computing the tension from observations made with larger Sugden (71) has computed a table based upon those of Bashforth and Adams (5). Jaeger (34) has also compiled a table de-

tubes,

signed to aid in the reduction of observations. This table seems to be based upon the equations given by Fuestel (24), and consequently requires critical examination before it can be accepted with full (See Verschaffelt (79).) is equally applicable to the case of drops of one liquid formed in another, provided that the drop forms on the inner circumference of the tube. The equation is the same in all cases,

confidence.

The method

due attention being given to the sign of h. Verschaffelt's remarks (79) regarding the formation of drops and bubbles should be read by those interested in this method. A modification of this method has been proposed by "VYhatmough (89). It has been criticized by Cantor (10) and appears to have little to

recommend

it. 9.

DROP WEIGHT

In 1864 Tate (73) announced as the result of his observations that, Other things being the same, the weight of a drop of liquid is proportional to the diameter of the tube in which it is formed." Four years later Quincke (53) undertook to estimate the value of the surface tension from the weight of a fallen drop of the liquid. He stated, quite correctly, that the weight of the suspended drop just before falling is 27rr(r, provided that the inflow of liquid into the drop is such that at the end of the tube the pressure in the drop is the same as if the surface were plane. But this condition is never fulfilled in practice. He also clearly recognized both that a portion of the drop remains adhering to the tube, causing an error which "

584

Scientiftc

Papers of the Bureau of Standards

{Voi. 21

he thought would be negligible when the radius of the tube is very small, and that the conditions attending the breaking away of the drop are very complex; consequently, that the assumption that the weight of the fallen drop is equal to 2TrrT can at best lead only to approximate values of the tension, and that the method should be used only when others are either not applicable or involve excessive experimental difficulties. Unfortunately, his followers have quite generally failed to recognize the limitations he pointed out and have endeavored to elevate to the dignity of an exact law the approximate numerical relation which he employed.

Worthington (91) in 1881 pointed out that the relation generally assumed

W=z2^rT

(17)

could not be correct in any case, and that, in general, the relation between the weight (W) and the surface tension (T) would not be one of simple proportionality. Several writers have commented on the fact that the weight must depend upon how rapidly the drops form and must be sensitive to slight disturbances occurring at the instant in which instability is setting in; and Vaillant (76) Abonnenc (2), and Guye and Perrot (28b) have endeavored to determine experimentally the relation between the weight of the drop and the rate at which it forms. Even though the speed of formation were excessively slow, the effect of disturbances would persist, and, although the average of a number of drops formed under the same nominal conditions might under certain conditions exhibit a high degree of reproducibility, this would be no proof that the average is the same as would be obtained under other conditions, nor, in particular, would it be a proof that the average is the same as would be obtained in the absence of all extraneous mechanical jars, vibrations, or other disturbance.

Rayleigh (61), Kohlrausch (39), Harkins and Humphrey (31), and Harkins and Brown (1(M) have studied experimentally the variation of the ratio W/{2'7rrT) with the value of r/A; and Lohnstein (45) has computed the same relation on the assumption that the residue of the drop left pendant to the tube was the stable drop for which the surface at the line of contact with the tube had the same inclination to the vertical as that of the unstable one which fell. The various tables thus prepared exhibit marked differences, due doubtless to the effect of slight mechanical and other disturbances upon the drop while passing through its condition of unstable equilibrium. These differences emphasize the weakness of the method as a means for obtaining reliable measurements of surface tension. The tables are, nevertheless, in agreement in showing that the ratio 'W/{2TrrT) varies markedly with the value of r/A and passes through a minimum when r/A is about equal to ^2

"

Measwrement of Surface Tension

Dorsey]

Although these conclusions appear not

to be accepted

585 by Morgan

(50), Lohnstein (47) has pointed out that his observations are actually concordant with them. In the progress of his work Morgan has been forced to limit the size of the tips he uses to those having radii

that differ but

little

from

A -^2

for the liquids used for standardiza-

tion; their use for liquids for

which

mately

He

this relation is not approxievades the point by demanding that the drop shall have a certain singular shape, described as bagThis limitation is closely equivalent to the former but it is a like. criterion that can not be applied with rigor and leaves the observer uncertain whether the proper conditions have been satisfactorily met. The numerical relation leaves no such uncertainty but in every case, whatever be the criterion, there remains the uncertainty arising from the effects of slight disturbances occurring while the droj^ is in its critical condition just preceding its fall. The technique of this method, generally called the " drop-weight method, is as simple as anyone could wish. For rapid work and within the limits of its reliability it would be the method of choice, especially when only small amounts of liquid are available, but it is not, and probably can not be made suitable for use where really accurate determinations are required. The reproducibility characteristic of averages is likely to lead to a very false estimate of the accuracy that has been attained. For example, Eayleigh (61) says: fulfilled is unjustified.

;

;

Successive collections, made without disturbance, gave indeed closely accordant weights (often to one-thousandth part), but repetitions after cleaning and remounting indicated discrepancies amounting to one-half per cent, or even to 1 per cent.

Although the method is not promising for accurate determinations, be hoped that in Ferguson's proposed comparison of methods will be compared with others under all the stringent requirements

it is to it

characteristic of precise physical work, so that its advantages as well its disadvantages may be brought to light and that the conditions under which and the limits within which it can be trusted may be

as

finally established.

Papers treating of this method and published prior to 1917 have been critically reviewed by Guye and Perrot (28a) and by Perr6t (52a). Appended to their reviews are extensive bibliographies. In addition to the articles already referred to, the following should be consulted by those interested in this method: Davies (11), Ferguson (19), Guthrie (27, 28), Harkins and Brown (30), Lohnstein (46). The methods so far considered have been based upon the measurement of the pressure exerted by a curved liquid surface. Obviously, it is possible to compute the tension from a direct measurement of the pull exerted by the surface under certain conditions. This is the basis of the numerous balance methods that have been proposed and used.

:

586

Scientific

Papers of the Bureau of Standards

10.

{Voi. ti

PULL OF VERTICAL FILM

Hall (29), Fahrenwald (14), Foley (24a), and Lenard, v. DallwitzWegener, and Zachmann (42a) measured the pull exerted by the film formed upon a vertical rectangular frame when its uppermost side was raised above the flat surface of the liquid in the reservoir. The tension may be computed either from the maximum pull exerted by the film as the frame is raised or from the pull of the film after it has drained. Hall used both methods; the corrections to be applied are uncertain and troublesome. II.

Wilhelmy

PULL ON VERTICAL PLATE

weighed the pull upon a vertical plate dipping into is very thin in comparison to its horizontal length, the pull is approximately equal to Tcosd^ times the horizontal perimeter of the plate. Correction must be made for the effect of the liquid displaced by the submergence of the plate below the general level of the liquid. If the lower edge lies above the general level, there is likewise a correction for the liquid raised by adhesion to the bottom of the plate. Hall eliminated these hydrostatic corrections by making the weighing while the lower edge of the plate lay (90)

If the plate

the liquid.

in the level of the undisturbed surface.

Worthington (92) rolled the increasing

the

pull.

He

strip into a cylindrical spiral, thus

called

this

modification

a

"capillary

multiplier." 12.

PULL ON RING

Timberg (75) measured the maximum pull exerted by the surface as a thin horizontal ring of platinum was detached from the liquid surface.

In reducing

his observations he

assumed that this pull

is

equal to the product of the surface tension times the sum of the outer and the inner perimeters of the ring. This approximation is of

Cantor (9), and more recently Verschaffelt (80), have deduced a more exact expression for thin rings of rectangular cross section. Verschaffelt's equation, which is more exact than Cantor's,

a low order.

is

equivalent to the following

(18)

Pr 4 [l-J2.8284

+ 0.6095y|}-i= + [3+2.585^| + 0.371^^j|!...]

T=^

7t=

,

^ ,

(19)

Measurement of Surface Tension

Dorsoy]

where

P

is

the maximiiin pull, r

is

the

mean

587

radius of the ring, 28

p^—pz is the amount the density of the liquid exceeds that of the overlying fluid.- By the use of an indicating instrument, such as a torsion balance, for the measurement of the maximum pull, the value of the surface tension can be very quickly determined. The method has been used for the study of the progressive changes which occur in the surface tension of colloidal and certain other solutions (50a). It should be noticed that the equation (18) is derived on the assumption that the shortest distance between the ring and the wall of the vessel containing the liquid is so great that at a point midway between the two the surface is truly flat. If the vessel is cylindrical, the surface will not be flat unless this minimum distance amounts to some 2 or 3 cm, depending upon the surface tension and the density of the liquid. If the vessel is so small that the liquid surface between the ring and the wall is nowhere flat, then, in order to obtain the value of the surface tension, a correction will have to be applied to the quantity T computed in accordance with equation (18). It is believed that the mathematical expression for this correction has not been derived, but it appears evident that the amount of the correction will depend upon the properties of the liquid under study. When the vessel is small, it is not generally allowable to assume that the surface tension is proportional to the maximum pull, although such the thickness of the

is

proportionality

may

rin<^,

practically exist for small variations in the

In much of the work done by

this method the vessel has been too small. The theory of various detachment methods has been discussed by Tichanowsky (T4a, 74b), and a recent paper by Lenard, v. DallwitzWegener, and Zachmann (42a) should be read by those expecting to use such methods.

tension.

13.

ADHESION PLATE

In the method known as that of the adhesion plate, the force required to lift a horizontal circular plate from the surface of the liquid is measured. Two modifications of the method have been used. In one the maximum pull exerted as the disk is raised is measured. Ferguson's expression (16), slightly modified in form, for this was written there has appeared an important paper (31a) by HarYoung, and Cheng treating of the maxinrum pull upon a circular ring of circular They have determined experimentally the value of the factor F in the cross section. equation T=PF/4 TriJ for various values of A/R and of r/R, R and r being, respectively, the i-adius of the ring and of its cross section. They denote by V the total volume of liquid (P/ptg) lifted by the tension and give four curves, each showing the variation of F with R^/V for a different fixed value of r/R. Their F corresponds formally to the expressions in the square brackets in equations (18) and (19) but is not comparable with them because they are derived on the assumption tha.t the axial width, of the ring is -

Since this paper

kins,

effectively infinite.

588

Scientific

maximum

force

{P)

Papers of the Bureau of Staruiards

when

the disk

is

perfectly

[voi. tt

wetted by the

liquid is

Verschaffelt carries the expression only to the second term.

The

assumed great. Ferguson says that in order to neglect the third term, r must be over 6 cm; he also calls attenradius of the disk

is

is not exactly the radius of the disk, but a slightly smaller one the difference is negligible if r is sufficiently great to make the equation applicable.

tion to the fact that the r in this equation ;

In the other modification, the pull when the film is truly vertical where it meets the edge of the disk is measured. In this case the value of the angle of contact does not enter into the equation. This method was used by Gallenkamp (25), The expression for the pull (Pi) as deduced

from the equation of Vershaffelt

1--E.

L=.v.[..i4-(^))

iPi-P2)g

(There

is

an

(80) is

,.,

7rr2(P:

error, arising

from an incorrect

sign, in Gallenkamp's

expression, p. 483.) 14.

PULL ON SPHERE

Ferguson (16, 18, 21) has suggested the use of a sphere, the pull being measured when the lowest point of the sphere is in the plane of the undisturbed surface of the liquid. His expression, slightly modified, for the pull (P) in this case when the angle of contact is zero, is

(22)

=4r|i-irAf_i3/A\ where

B

is

the radius of the sphere, and the other quantities have

the same significance as formerly. 15.

Owing

OSCILLATING DROPS AND BUBBLES

to the capillary forces, a drop or bubble that

from a spherical shape and then

is

deformed

left to itself will execute periodic

vibrations about its figure of equilibrium. These oscillations have been studied both mathematically and experimentally by Rayleigh

(57,58,59,63). From a study of such vibrations the surface tension can be determined. For success a rather elaborate technique is required. The amplitude of vibration must be very small. The method is especially

Measitremeni of Surface Tension

Dorsey]

589

permits the tension to be observed within a very is formed. Lenard (42) employed the method as applied to freely falling oscillating drops. He found that the tension decreased as the age of the drop increased; this he attributed, probably correctly, to the contamination of the surface. Recently Kutter (40) devised a very simple, but probably not very valuable, in that

it

short interval after the surface

method for determining the period of the drops. He made phenomenon described by Thomson and Newall (74) in 1885. When drops of one liquid fall into another with which it mixes, they form vortex rings that travel a greater or lesser distance into the second liquid. The depth to which they penetrate depends upon the form of the drop at the instant of impact as the distance precise,

use of the

;

the drops fall before striking the liquid surface

is

progressively in-

creased the depths reached pass through a succession of

and minima.

The

distance

from one maximum

to the distance that the drop falls while

it is

maxima

to the next is equal

executing one complete

His observations indicate that he can determine this distance with considerable precision but it would appear to be necessary for the vibrations to have quite an appreciable amplitude, as compared with the radius of the drop, if the phenomenon is to be sharply defined, and the mathematical relation connecting the period with the surface tension and other quantities assumes that the amplitude is small in comparison with the radius. For this reason confidence can not be placed in the results, although those displayed appear vibration.

;

to lead to nearly the correct values. i6.

VIBRATING JETS

If at any point fixed with reference to

its

source a jet of liquid

is

thrown into oscillations, which will exhibit themselves as stationary waves in the jet. Such vibrations have been studied both mathematically and experimentally by Rayleigh. His observations (57, 58) did not yield very satisfactory values for the tension. The subject was taken up again by Pedersen (51) with a very greatly improved technique. He showed that the method was capable of a precision that is comparable with those of other methods. The following year it was employed by Bohr (6). He elaborated Rayleigh's mathematical treatment and subjected to a periodic change in form,

it

will be

developed expressions that take account of the amplitude of the damping of the waves. He obtained for the tension a value somewhat lower than that found by Pedersen and more nearly in agreement with the best of those found by other methods. He thinks that Pedersen's values are too high because he worked too near the orifice, obtaining measurements before the vibrations and of the

vibration

had

settled

down

to its

normal condition.

Scientific

590

Papers of the Bureau of Standards 17.

[Voi. et

RIPPLES

The velocity with which waves travel over the surface of a liquid depends upon the surface tension as well as upon the acceleration of gravity. The relative importance of these two factors depends upon the length of the wave.

surface tension

is

the

When the wave length is so short that the more important, the waves are commonly

Under suitable conditions the surface tension can be determined from the wave length and the period of ripples. This is known as the ripple method. The effect of the surface tension upon the velocity of surface waves was investigated by Kelvin in 1871 (37). The ripple method was used first by Eayleigh (60) in 1890 in his study of the contamination of water surfaces. (See also Lamb (41), Tait (72), Rayleigh (63).) If the amplitude of the waves is so small that the square of the slope of the surface is at every point negligible in comparison with unity, if the waves are plane, and if the depth of the liquid is great in comparison with the length of the waves, then Kelvin's equation called ripples.

(23) is applicable ,^(Pi (PI + PP! 7^= coth^^ ^''''^ 27rr2 X A,

Is

the wave length, t

gravity,

T

the surface tension, and

where Pi is

is

- iBi:zpl9]^ Alt'

(23)

the period, g the acceleration of I is the depth of the liquid;

the density of the lower liquid.

Since

T

varies as the cube of

wave length and inversely as the square of the period, these quantities must be measured with extreme accuracy. This, however, can be done without great difficulty. The method obviously requires a large surface of liquid in order (a) that the waves reflected from the

the walls shall die out before reaching the portion under observation,

and

(h)

that the undisturbed surface shall be truly plane.

This

large surface invites contamination, but has the advantage of dis-

amount of contamination over a large surface, thus reducing its effect. In addition, it permits of a mechanical cleaning of the surface, thus enabling the observer to speak with confidence regarding the possible effect of the contamination that

tributing any small

The last is very desirable. The method is thoroughly trustworthy, but measurements must not be made too near the source, because, owing to the small amplitude of the waves, a very slight curvature of the surface will cause an appreciable relative displacement of the crests. The necessity for considering this source of error, which has been quite generally ignored, was pointed out by the author a number of years ago. In that work (12) it was found that the curvature as far as 3 cm from the plate used for generating the waves was sufficient exists.

1

Measurement of Surface Tension

Dorsey]

to displace the apparent crests

59

by an appreciable amount. Consemade closer to the generator

quently, no measurement should be

than

3.5 or

observations

Several experimenters have, however, employed a few millimeters of the generator. Some

4 cm.

made within

investigators, instead of using plane waves, have used the interference pattern produced by two sets of circular waves, assuming that the effects of the two sets of waves are directly additive; it is not evident that this is permissible, as here we have to do with a surface (For experimental curvature, not with a mere surface elevation.

and data

details

Bohr Of

(6),

see

Grunmach

Kayleigh (60), Dorsey (12), Kaliihne (36), (101, 102).)

the methods that have been considered a knowledge of the

angle of contact

is

not required in {a) Sentis's method of virtual

capillary tube, (b) pressure in a pendent drop,

(c)

measurement

of the form of the surface so as to get the curvature at two different levels (this includes the

method of Anderson and Bowen),

[d) ses-

drops or bubbles when the maximum horizontal diameter and either the distance from the vertex to the plane of the maximum diameter or the curvature at the vertex are used, {e) Jaeger's method, (/) pull of a true film, {g) maximum pull upon a thin ring, {h) pull on an adhesion plate when the surface is vertical at the line of sile

contact,

(^)

oscillation of

drops,

{j)

vibration of

jets,

and {k)

ripples.

Those that require a knowledge of the angle of contact are

{a) all

others involving capillary rise in tubes, between plates, or against

when the total thickness is upon a vertical plate (Wilhelmy's method) {d) maximum pull upon an adhesion plate, and {e) pull upon a sphere. The drop-weight method stands by itself as one involving undetermined factors. Before closing, reference should be made to a few other articles. Those interested in capillary phenomena themselves will wish to read the article on " capillarity " in the Encyclopedia Britannica, and along with that Kelvin's comments (105) should be read. Boys's article (97) on the drawing of curves by their curvature, Pockels's article (107) on capillarity, and Maxwell's chapter on capillarity (106) should also be read. These and a few others that have not solid walls, {h) sessile drops or bubbles

used, {c) pull

,

been referred to in the preceding portions of this article have been placed in a supplementary list at the end of the bibliography. The entries

which are unnumbered have not been recently

are believed to be correct.

verified; they

592

Scientific

Papers of the Bureau of Standards IV.

K,

Ablett,

Abonnenc, Anderson,

4.

u

BIBLIOGRAPHY

Mag. (6), 46, pp? 244-256; 1923. Contact angle. Comp. Rend., 168, pp. 556-557; 1919. Fallen drops. A., and Bowen, J. E., Phil. Mag. (6), 31, pp. 143-148; Capillary tube and curvature. Anderson, A., and Bowen, J. E., Phil. Mag. (6), 31, pp. 285-289;

1.

2. 3.

{Yql

Phil.

L.,

1916.

1916.

Contact angle. 5. Bashforth and Adams; An attempt to test the theory of capillary action. Cambr. "Univ. Press 1888. Shape of drops numerical tables computed. 5a. Bigelow, S. L., and Hunter, F. W., J. Physical Chem., 15, pp. 867-380; 1911. Capillary height by means of perforated plate. Jet vibration. 6. Bohr, N., Phil. Trans. (A), 309, pp. 281-317; 1909. 7. Bosanquet, C. H., and Hartley, H., Phil. Mag. (6), 42, pp. 456-462; 1921. Contact angle, measurement of. Measurement of small 8. Boys, v., Soc. Chem Ind. J., 39, pp. 58-60; 1920. drop interf acial tension. 9. Cantor, M., Wied. Ann., 47, pp. 399-423 1892. Detaching of ring Jaeger's method. 10. Cantor, M., Ann. Physik (4), 7, pp. 698-700; 1902. Jaeger's method. 10a. Carver, E. K., and Hovorka, F., J. Am. Chem. Soc, 47, pp. 1325-1328; 1925. Capillary height by means of perforated plate. 11. Davies, A. C. H., Am. Chem. Soc. J., 40, pp. 784-785; 1918. Tips for dropweight method. 12. Dorsey, N. E., Phil. Mag. (5), 44, pp. 369-396; 1897; also Phys. Rev.,, 5, 170-181, 213-230; 1897. Ripples. 13. Eotvos; Wied. Ann., 27, pp. 448-459; 1886. Curvature measured optically. 14. Fahrenwald, A. W., Opt. Soc. Am. J., 6, pp. 722-733; 1922. Pull by fihn. 15. Ferguson, A., Phil. Mag. (6), 33, pp. 417-430; 1912. Photographic measure;

;

;

;

ment of pendant 16.

17. 18. 19.

20.

;

drops.

Mag. (6), 26, pp. 925-934; 1913. Force on sphere, and on horizontal disk. Ferguson, A., Phil. Mag. (6), 28, pp. 128-138; 1914. CapiUary tube. Ferguson, A., Phil. Mag. (6), 28, pp. 149-153; 1914. Forces on sphere. Ferguson, A., Phil. Mag. (6), 30, pp. 632-637; 1915. Drop weight. Ferguson, A., and Dowson, P. E., Faraday Soc. Trans., 17, pp. 384-392; 1921-22 also Manchester Lit. Phil. Soc, Mem., 65, No. 5 1921. CapilFerguson,

A.,

Phil.

;

;

lary tube method.

Ferguson, A., Faraday Soc Trans., 17, pp. 370-383; 1921-22; also Manchester Lit. Phil. Soc, Mem., 65, No. 4; 1921. Studies in capillarity. Capillary tube, 22. Ferguson, A., Proc Phys. Soc (Lond.), 36, pp. 37-44; 1923. small volume of liquid. 23. Ferguson, A., Phil. Mag. (6), 47, pp. 91-94; 1924. Contact angle. 23a. Ferguson, A., and Vogel, I., Proc. Phys. Soc, 38, pp. 193-203 1926. The hyperbola method. Jaeger's method. 24. Feustel, R., Ann. Physik (4), 16, pp. 61-92; 1905. 24a. Foley, A. L., Physical Rev., 3, pp. 381-386 1896. Pull on mica frame. Adhesion plate. 25. Galleukamp, W., Ann. Physik (4), 9, pp. 475-494; 1902. Inclined 26. Grunmach, L., Verb. Deut. physik. Ges., 12, pp. 847-859; 1910. 21.

;

;

plates

;

hyperbolas.

Proc Roy. Soc (A), 14, pp. 22-33; 1865. Bubbles. Proc Roy. Soc (A), 13, pp. 444-483; 1864. Drops. Guye, Ph. A., and Perrot, F. L., Arch. Sci. Phys. et Nat, 11, pp. 225-345;

27. Guthrie, F.,

28. Guthrie, F.,

28a.

1901.

Critical study of drop-weight method.

Bibliography.

Measu/rement of Surface Tension

Dorset/]

593

28b. Giiye, Ph. A., and Perrot, F. L., Arcli. Sci. Phys. et Nat., 15, pp. 132; 1003. Experimental study of form and weight of drops, both static and

dynamic. 29. Hall, T.

Phil.

P.,

Mag.

(5),

36, pp. 385-413; 1893.

Weighing methods;

bibliography of various methods. 30. Harkins. W. D., and Brown, F. E., Am. Chem. Soc. J., 38, pp. 246-252 ; 1916. Drop weight. 31. Harkins, W. D., and Humphery, F. C, AM. Chem. Soc. J., 38, pp. 228-246;

Drop weight

capillary tube. Three articles. Young, T. F., and Cheng, L. H., Science, 64, pp. 333-336; Detaching of anchor ring. Experimental determination of ratio 1926. of tension to maximum pull and its variation with the capillary constant 1916.

31a. Harkins,

W.

and the

;

D.,

size of the ring,

Wied. Ann., 62, pp. 694-699; 1897. Sessile drops. Wied. Ann., 62, pp. 700-701; 1897. Sessile drops. 33a. Heydweiller, A., Wied. Ann., 65, pp. 311-319; 1898. Drops; table to

32. Heydvveiller, A.,

33. Heydweiller, A.,

facilitate reduction of observation.

Chem., 101, pp. 1-214 1917. Jaeger's method. Akad. Wiss. Wien (2a), 100, pp. 245-270; 1891.

34. Jaeger, F. M., Zeits. Anorg. 35. Jaeger,

F. M.,

Sitzb.

;

Jaeger's method. 35a. Johlin, J. M., Science,

64,

Detaching of

pp. 93-94; 1926.

ring.

Variation

of tension with age of surface. 36.

Ann. Physik (4), 7, pp. 440-476; 1902. Ripples. Mag. (4), 42, pp. 362-377; 1871. Ripples; mathematical. Inclined Kiplinger, O. C, Am. Chem. Soc. J., 42, pp. 472-476; 1920.

Kalahne,

A.,

37. Kelvin, Phil. 38.

capillary.

Kohlrausch, Fr., Ann. Physik (4), 22, pp. 191-194; 1907. Drop weight. Oscillating 40. Kutter, V., Physik, Zeits., 17, pp. 424-429, 573-579; 1916. drops; vortex rings. 41. Lamb, H., Hydrodynamics, Chap. IX; Cambridge Univ. Press. 42. Lenard, P., Wied. Ann., 30, pp. 209-243; 1887. Oscillating drops. 42a. Lenard, P., v. Dallwitz-Wegener, R., and Zachmann, E., Ann. der Physik, 74, pp. 381-404; 1924. Detachment methods. Frame and ring. Sessile drops. 43. Lohnstein, Th., Wied. Ann., 53, pp. 1062-1073 1894. Sessile drops. 44. Lohnstein, Th., Wied. Ann., 54, pp. 713-723; 1895. Drop weight. 45. Lohnstein, Th., Ann. Physik (4), 22, pp. 767-781; 1907. Drop weight. 46. Lohnstein, Th., Zeits. physik, Chem., 64. pp. 686-692; 1908. 47. Lohnstein, Th., Zeits. physik, Chem., 84, pp. 410-418; 1913. Drop weight. 48. Magie, W. F., Phil. Mag. (5), 26, pp. 162-183; 1888. Contact angle; sessile 39.

;

drops. 49. Martin,

E.,

Sitzb.

Akad. Wiss. Wien

(2a),

123,

pp.

2491-2507;

1914.

Jaeger's method. 50.

Morgan,

J.

weight.

L. R., Am. Chem. Soc, J., 37, pp. 1461-1467; 1915. Drop Numerous other papers by same author and collaborators in

this journal.

50a.

du

Noiiy, P. L., Science,

59, pp. 580-582

;

1924.

Surface tension of solu-

tions of colloids. 50b.

du

Noiiy, P. L., J. de Physique (6), 6, pp. 145-153; 1925.

Surface tension

of colloids by ring method.

Pedersen, P. O., Phil. Trans. (A), 207, pp. 341-392; 1908. Jet vibration. Pekar, D., Zeits. physik. Chem., 39, pp. 433-452; 1902. Optical method. 52a. Perrot, F. L., J. chim. phys., 15, pp. 164-207; 1917. Drop-weight method. Review of work since 1900. Bibliography. 53. Quincke, G., Pogg. Ann., 135, pp. 621-646; 1868. Drop weight.

51. 52.

594 54.

Papers of the Bureau of Standards

Scientific

Quincke,

Pogg. Ann., 160, pp. 337-374; 1877.

G.,

57. 58. 59. 60.

61.

Sessile drops.

Wied. Ann., 2, pp. 145-194 1877. Contact angle. Quincke, G., Wied. Ann., 52, pp. 1-22; 1894. Contact angle. Rayleigh, Proc. Roy. Soc. (A), 29, pp. 71-97; 1879. Jet vibration. Rayleigb, Proc. Roy Soc. (A), 34, pp. 130-145; 1882. Jet vibration. Rayleigh, Proc. Roy. Soc. (A), 47, pp. 281-287; 1890. Jet vibration. Rayleigh, Phil. Mag. (5), 30. pp. 386-400; 1890. Ripples; contamination. Rayleigh, Phil. Mag. (5), 48, pp. 321-337; 1899. Drop weight; jets; con-

55. Quincl^e, G., 56.

ivoi. ti

;

tamination. Proc. Roy. Soc. mathematical.

62. Rayleigh,

(A), 92, pp. 184r-195; 1916.

Capillary tube,

Theory of sound, 2, pp. 351-375. Macmillan & Co. 1896. Richards, T. W., and Carver, E. K., Am. Chem. Soc. J., 43, pp. 827-S47; 1921. Contact angle curvature of surface capillary rise. 65. Schmidt, Th., Wied. Ann., 16, pp. 633-660; 1882 Surface viscosity. 66. Schrodinger, E., Ann. Physik (4), 46, pp. 413-418; 1915. Jaeger's method, 63. Rayleigh,

;

64.

;

;

mathematical.

de Phys. (2), 6, pp. 571-573; 1887. Virtual capillary. Journ. de Phys. (3), 6, pp. 183-187; 1897; also Annales, Univ. Grenoble. Virtual capillary; contact angle. Sessile drops. 69. Siedentopf, H., Wied. Ann., 61, pp. 235-266 1897. Capillary 70. Sugden, S., J. Chem. Soc. (Lond.), 119, pp. 1483-1492; 1921. 67. Sentis, H., Journ. 68. Sentis, H.,

;

tubes. 71.

Sugden, S., method.

Chem.

J.

Soc.

121,

(Lond.),

pp.

858-866;

1922.

Jaeger's

Roy. Soc. Edinb., 17, pp. 110-115; 1890. Ripples. Mag. (4), 27, pp. 176-180; 1864. Drop weight. 74. Thomson, J. J., and Newall, H. F., Proc. Roy. Soc. (Lond.), (A), 39, pp. 417-436; 1885. Vortex rings by falling drops. 74a. Tichanowsky, I. L, Physik Zeits., 25, pp. 299-302; 1924. Theory of 72. Tait, P. G., Proc. 73. Tate, T., Phil.

detaching ring. 74b. Tichanowsky,

I.

Correction for inclination. I.,

Physik

Zeits.,

26,

pp.

522-525;

1925.

Theory of

several detachment methods. 75.

Wied. Ann., 30, pp. 545-561; 1887. PuU on ring. Compt. Rend., 160, pp. 596-598; 1915. Drop weight. Verschaffelt, J., Comm. Leiden, No. 32 1896. Coaxial tubes. Verschaffelt, J., Comm. Leiden, Suppl. No. 42c 1918 also K. Akad. Amsterdam, Proc, 21, pp. 357-865; 1919. Small drops and bubbles, matheTimberg,

G.,

76. Vaillant, P.,

77. 78.

;

;

;

matical.

Comm.

79. Verschaffelt, J.,

1918 also K. Akad. AmsterApplications of small drops and

Leiden, Suppl. No. 42d

dam, Proc, 21, pp. 366-374; 1919.

;

;

bubbles.

81.

Comm. Leiden, Suppl. No. 42e 1918 also K. Akad. Amsterdam, Proc, 21, pp. 836-849; 1919. Large drops and bubbles; mathematical and experimental. Volkmann, P., Wied. Ann., 11, pp. 177-209; 1880. Capillary tubes; criti-

82.

Volkmann,

80. Verschaffelt, J.,

;

;

cally experimental. P.,

Wied. Ann., 53, pp. 633-663; 1894.

Capillary tubes;

criti-

cally experimental. 83. 84.

Volkmann, Volkmann,

P., P.,

Wied. Ann., 53, pp. 664-666; 1894. Remarks on preceding. Wied. Ann., 56, pp. 457-491; 1895. Surface tension of

water. 85.

Volkmann, P., Wied. Ann., 62, pp. 507-521 1894 and 1895.

;

1897.

Remarks on papers of

Measurement of Surface Tension

Doi-seyi

Volkmann,

8G.

P.,

Wied. Ann., 06, pp. 194-20G; 1898.

595

Capillary tubes, critically

experimental.

Volkmann,

87.

Wied. Ann., 28, pp. 135-144; 188G. Contact angle. Mag. (6), 45, pp. 88-94; 1923. Rise between plate

P.,

88. Wagstaff, J. E. P., Phil.

and cylinder.

Whatmough, W.

91.

Jaeger's H., Zeits. physik, Cbem., 39, pp. 129^193; 1902. method, modified. Wilbelmy, L., Pogg. Ann., 119, pp. 176-217; 1^63. Pull on vertical plate. Worthington, A. M., Proc. Roy. Soc. (A), 32, pp. 362-377; 1881. Pendant

92.

Worthington,

89.

90.

drops.

A.

Phil.

M.,

Mag.

(5),

19,

43^6;

pp.

Capillary

1885.

multiplier.

Worthington, A. M., Phil. Mag. (5), 19, pp. 46-48; 1885 Pendant drops. Worthington, A. M., Phil. ]\Iag. (5), 20, pp. 51-66; 1885. Sessile drops.

93. 94.

SUPPLEMENTAL BIBLIOGRAPHY Devaux, H., Journ. de Phys. et Rad.

95.

(6), 4, pp. 293-309; 1923.

Wetting of

surfaces. 96.

Devaux, H., Bull. soc. fr. physique, p. 184S 1923. Wetting of surfaces. Boys, v., Phil. Mag. (5), 36, pp. 75-82; 1893. Drawing curves. Ferguson, A., Phil. Mag. (6), 24, pp. 837-844; 1912. Large capillary surface, mathematical treatment. Ferguson, A., Phil. Mag. (6), 25, pp. 507-520; 1913. Large drops; mathe;

97. 98.

99.

matical treatment.

Mag.

100. Ferguson, A., Phil.

(6),

28,

pp. 403-412; 1914.

method;

Jaeger's

effect of overlying gas.

101. 102.

Grunmach, Grunmach,

L.,

Grunmach, L., and Bein, W., Wiss. Abh. K. Norm. Eichungs-Komm, Heft, 9, pp 1-18 1917 also Zeits. Instk., 39, p. 195 1919. Inclined plates. Harkins, W. D., and Brown, F. E., Am. Cbem. Soc. J., 41, pp. 499-524; 1919. Drop weight table of corrections. Kelvin, Popular lectures and addresses, 1, pp. 1-79. Maxwell, J. C, Theory of heat, chap. 20. Pockels, A., Winklemann Handbuch der Physik (ed. 2), 1. Quincke, G., Pogg. Ann. 105, pp. 1-48; 1858. Sessile drops; mercury. Quincke, G., Pogg. Ann., 139, pp. 1-89; 1870. Sessile drops; interfacial ;

104.

Ann. Physik. (4), 9, pp. 1261-1285; 1902. Ripples. Wlss. Abb. K. Norm. Eichungs-Komm, Heft, 7, pp. 45-129;

Ripples.

1908. 103.

L.,

;

;

;

105. 106. 107.

108. 109.

:

tension. 110. Quincke,

G.,

Wied. Ann., 61, pp. 267-280; 1897.

Critical discussion of

capillary measurements. Clairaut, Theorie de la figure de la terre. Chap. X. 110. Gauss,

Werke, 5, pp. 30-77.

in statu aequilibrii

;

;

1743.

Principia generalia theoriae figurae fluidorum

1830.

Laplace, Theorie de Taction capillaire, Mec. celeste, L. X. Suppl.

;

1806.

Mathieu, Theorie de la capillarite 1883. Poisson, Nouvelle theorie de Taction capillaire 1831. Young, Th. Phil. Trans. (A), 1, p. 65; 1805. Plateau, J., Statique exp^rimentale et theorique des liquides, Paris; 1873. ;

;

Washington, June

1,*1926.

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