Chapter III: Part 3

then mm' [Pi](f) will represent--(1) The work done by the attractive
force on the particle m, while it is brought from an infinite distance
from m' to the distance f from m'; or (2) The attraction of a particle
m on a narrow straight rod resolved in the direction of the length of
the rod, one extremity of the rod being at a distance f from m, and
the other at an infinite distance, the mass of unit of length of the
rod being m'. The function [Pi](f) is also insensible for sensible
values of f, but for insensible values of f it may become sensible and
even very great.

If we next write
_
/ [oo]
| f[Pi](f)df = [psi](z), (24)
_/z

then 2[pi]m[sigma][psi](z) will represent--(1) The work done by the
attractive force while a particle m is brought from an infinite
distance to a distance z from an infinitely thin stratum of the
substance whose mass per unit of area is [sigma]; (2) The attraction
of a particle m placed at a distance z from the plane surface of an
infinite solid whose density is [sigma].

Let us examine the case in which the particle m is placed at a
distance z from a curved stratum of the substance, whose principal
radii of curvature are R1 and R2. Let P (fig. 2) be the particle and
PB a normal to the surface. Let the plane of the paper be a normal
section of the surface of the stratum at the point B, making an angle
[omega] with the section whose radius of curvature is R1. Then if O is
the centre of curvature in the plane of the paper, and BO = u,

1 cos^2[omega] sin^2[omega]
-- = ------------ + -----------. (25)
u R1 R2

Let POQ = [theta], PO = r, PQ = f, BP = z,

f^2 = u^2 + r^2 - 2ur cos[theta]. (26)

The element of the stratum at Q may be expressed by

[sigma]u^2 sin[theta] d[theta] d[omega],

or expressing d[Greek: th] in terms of df by (26),

[sigma]ur^-1 f df d[omega].

Multiplying this by m and by [pi](f), we obtain for the work done by
the attraction of this element when m is brought from an infinite
distance to P1,

m[sigma]ur^-1 f[Pi](f)dfd[omega].

Integrating with respect to f from f = z to f = a, where a is a line
very great compared with the extreme range of the molecular force, but
very small compared with either of the radii of curvature, we obtain
for the work
_
/
| m[sigma]ur^-1 ([psi](z) - [psi](a))d[omega],
_/

and since [psi](a) is an insensible quantity we may omit it. We may
also write

ur^-1 = 1 + zu^-1 + &c.,

since z is very small compared with u, and expressing u in terms of
[omega] by (25), we find
_ _ _
/ 2[pi] | /cos^2[omega] sin^2[omega]\ |
| m[sigma][psi](z) | 1 + z ( ------------ + ------------ ) | d[omega] =
_/0 |_ \ R1 R2 / _|
_ _
| 1 /1 1 \ |
2[pi]m[sigma][psi](z) | 1 + --z ( -- + -- ) |.
|_ 2 \R1 R2/ _|

This then expresses the work done by the attractive forces when a
particle m is brought from an infinite distance to the point P at a
distance z from a stratum whose surface-density is [sigma], and whose
principal radii of curvature are R1 and R2.

To find the work done when m is brought to the point P in the
neighbourhood of a solid body, the density of which is a function of
the depth [nu] below the surface, we have only to write instead of
[sigma][rho]dz, and to integrate
_ _
/ [oo] /1 1 \ / [oo]
2[pi]m | [rho][psi](z)dz + [pi]m ( -- + -- ) | [rho]z[psi](z)dz,
_/z \R1 R2/ _/z

where, in general, we must suppose [rho] a function of z. This
expression, when integrated, gives (1) the work done on a particle m
while it is brought from an infinite distance to the point P, or (2)
the attraction on a long slender column normal to the surface and
terminating at P, the mass of unit of length of the column being m. In
the form of the theory given by Laplace, the density of the liquid was
supposed to be uniform. Hence if we write
_ _
/ [oo] / [oo]
K = 2[pi] | [psi](z)dz, H = 2[pi] | z[psi](z)dz,
_/0 _/0

the pressure of a column _of the fluid itself_ terminating at the
surface will be

[rho]^2 {K + 1/2H(1/R1 + 1/R2)},

and the work done by the attractive forces when a particle m is
brought to the surface of the fluid from an infinite distance will be

m[rho] {K + 1/2H(1/R1 + 1/R2)},

If we write
_
/ oo
| [psi](z)dz = [theta](z),
_/z

then 2[pi]m[rho][theta](z) will express the work done by-the
attractive forces, while a particle m is brought from an infinite
distance to a distance z from the plane surface of a mass of the
substance of density [rho] and infinitely thick. The function
[theta](z) is insensible for all sensible values of z. For insensible
values it may become sensible, but it must remain finite even when z =
0, in which case [theta](0) = K.

If [chi]' is the potential energy of unit of mass of the substance in
vapour, then at a distance z from the plane surface of the liquid

[chi] = [chi]' - 2[pi][rho][theta](z).

At the surface

[chi] = [chi]' - 2[pi][rho][theta](0).

At a distance z within the surface

[chi] = [chi]' - 4[pi][rho][theta](0) + 2[pi][rho][theta](z).

If the liquid forms a stratum of thickness c, then

[chi] = [chi]' - 4[pi][rho][theta](0) + 2[pi][rho][theta](z) +

+ 2[pi][rho][theta](z - c).

The surface-density of this stratum is [sigma] = c[rho]. The energy
per unit of area is
_
/ c
e = | [chi][rho]dz = c[rho]([chi]' - 4[pi][rho][theta](0)) +
_/0
_ _
/ c / c
+ 2[pi][rho]^2 | [theta](z)dz + 2[pi][rho]^2 | [theta](z - c)dz.
_/0 _/0

Since the two sides of the stratum are similar the last two terms are
equal, and
_
/ c
e = c[rho]([chi]' - 4[pi][rho][theta](0)) + 4[pi][rho]^2 | [theta](z)dz.
_/0

Differentiating with respect to c, we find

d[sigma] de
-------- = [rho], -- = [rho]([chi]' - 4[pi][rho][theta](0)) + 4[pi][rho]^2[theta](c).
dc dc

Hence the surface-tension
_
de / / c \
T = e - [sigma] -------- = 4[pi][rho]^2 ( | [theta](z)dz - c[theta](c) ).
d[sigma] \ _/0 /

Integrating the first term within brackets by parts, it becomes
_
/ c d[theta]
c[theta](c) - 0[theta](0) - | z -------- dz.
_/0 dz

Remembering that c(0) is a finite quantity, and that

d[theta]
-------- = -[psi](z),
dz

we find
_
/ c
T = 4[pi][rho]^2 | z[psi](z)dz. (27)
_/0

When c is greater than [epsilon] this is equivalent to 2H in the
equation of Laplace. Hence the tension is the same for all films
thicker than [epsilon], the range of the molecular forces. For thinner
films

dT
-- = 4[pi][rho]^2c[psi](c).
dc

Hence if [psi](c) is positive, the tension and the thickness will
increase together. Now 2[pi]m[rho][psi](c) represents the attraction
between a particle m and the plane surface of an infinite mass of the
liquid, when the distance of the particle outside the surface is c.
Now, the force between the particle and the liquid is certainly, on
the whole, attractive; but if between any two small values of c it
should be repulsive, then for films whose thickness lies between these
values the tension will increase as the thickness diminishes, but for
all other cases the tension will diminish as the thickness diminishes.

We have given several examples in which the density is assumed to be
uniform, because Poisson has asserted that capillary phenomena would
not take place unless the density varied rapidly near the surface. In
this assertion we think he was mathematically wrong, though in his own
hypothesis that the density does actually vary, he was probably right.
In fact, the quantity 4[pi][rho]^2K, which we may call with van der
Waals the molecular pressure, is so great for most liquids (5000
atmospheres for water), that in the parts near the surface, where the
molecular pressure varies rapidly, we may expect considerable
variation of density, even when we take into account the smallness of
the compressibility of liquids.

The pressure at any point of the liquid arises from two causes, the
external pressure P to which the liquid is subjected, and the pressure
arising from the mutual attraction of its molecules. If we suppose
that the number of molecules within the range of the attraction of a
given molecule is very large, the part of the pressure arising from
attraction will be proportional to the square of the number of
molecules in unit of volume, that is, to the square of the density.
Hence we may write

p = P + A[rho]^2,

where A is a constant [equal to Laplace's intrinsic pressure K. But
this equation is applicable only at points in the interior, where
[rho] is not varying.]

[The intrinsic pressure and the surface-tension of a uniform mass are
perhaps more easily found by the following process. The former can be
found at once by calculating the mutual attraction of the parts of a
large mass which lie on opposite sides of an imaginary plane
interface. If the density be [sigma], the attraction between the whole
of one side and a layer upon the other distant z from the plane and of
thickness dz is 2[pi][sigma]^2[psi](z)dz, reckoned per unit of area.
The expression for the intrinsic pressure is thus simply
_
/ [oo]
K = 2[pi][sigma]^2 | [psi](z)dz. (28)
_/0

In Laplace's investigation [sigma] is supposed to be unity. We may
call the value which (28) then assumes K0, so that as above
_
/ [oo]
K0 = 2[pi] | [psi](z)dz. (29)
_/0

The expression for the superficial tension is most readily found with
the aid of the idea of superficial energy, introduced into the subject
by Gauss. Since the tension is constant, the work that must be done to
extend the surface by one unit of area measures the tension, and the
work required for the generation of any surface is the product of the
tension and the area. From this consideration we may derive Laplace's
expression, as has been done by Dupre (_Theorie mecanique de la
chaleur_, Paris, 1869), and Kelvin ("Capillary Attraction," _Proc.
Roy. Inst._, January 1886. Reprinted, _Popular Lectures and
Addresses_, 1889). For imagine a small cavity to be formed in the
interior of the mass and to be gradually expanded in such a shape that
the walls consist almost entirely of two parallel planes. The distance
between the planes is supposed to be very small compared with their
ultimate diameters, but at the same time large enough to exceed the
range of the attractive forces. The work required to produce this
crevasse is twice the product of the tension and the area of one of
the faces. If we now suppose the crevasse produced by direct
separation of its walls, the work necessary must be the same as
before, the initial and final configurations being identical; and we
recognize that the tension may be measured by half the work that must
be done per unit of area against the mutual attraction in order to
separate the two portions which lie upon opposite sides of an ideal
plane to a distance from one another which is outside the range of the
forces. It only remains to calculate this work.

If [sigma]1, [sigma]2 represent the densities of the two infinite
solids, their mutual attraction at distance z is per unit of area
_
/ [oo]
2[pi][sigma]1[sigma]2 | [psi](z)dz, (30)
_/z

or 2[pi][sigma]1[sigma]2[theta](z), if we write
_
/ [oo]
| [psi](z)dz = [theta](z). (31)
_/z

The work required to produce the separation in question is thus
_
/ [oo]
2[pi][sigma]1[sigma]2 | [theta](z)dz; (32)
_/ 0

and for the tension of a liquid of density [sigma] we have
_
/ [oo]
T = [pi][sigma]^2 | [theta](z)dz. (33)
_/0

The form of this expression may be modified by integration by parts.
For
_ _ _
/ / d[theta](z) /
| [theta](z)dz = [theta](z).z - | z -----------dz = [theta](z).z + | z[psi](z)dz.
_/ _/ dz _/

Since theta(0) is finite, proportional to K, the integrated term
vanishes at both limits, and we have simply
_ _
/ [oo] / [oo]
| [theta](z)dz = | z[psi](z)dz, (34)
_/0 _/0

and
_
/ [oo]
T = [pi][sigma]^2 | z[psi](z)dz. (35)
_/0

In Laplace's notation the second member of (34), multiplied by 2[pi],
is represented by H.

As Laplace has shown, the values for K and T may also be expressed in
terms of the function [phi], with which we started. Integrating by
parts, we get
_ _
/ /
| [psi](z)dz = z[psi](z) + (1/3)z^3[PI](z) + 1/3 | z^3[phi](z)dz,
_/ _/
_ _
/ /
| z[psi](z)dz = 1/2z^2[psi](z) + (1/8)z^4[PI](z) + 1/8 | z^4 [phi](z)dz.
_/ _/

In all cases to which it is necessary to have regard the integrated
terms vanish at both limits, and we may write
_ _ _ _
/ [oo] 1 / [oo] / [oo] 1 / [oo]
| [psi](z)dz = -- | z^3 [phi](z)dz, | z[psi](z)dz = -- | z^4 [phi](z)dz; (36)
_/0 3 _/0 _/0 8 _/0

so that
_ _
2[pi] / [oo] [pi] / [oo]
K0 = ----- | z^3[phi](z)dz, T0 = ---- | z^4 [phi](z)dz. (37)
3 _/0 8 _/0

A few examples of these formulae will promote an intelligent
comprehension of the subject. One of the simplest suppositions open to
us is that

[phi](f) = e^([beta] f). (38)

From this we obtain

[Pi](z) = [beta]^(-1) e^([beta] z), [psi](z) = [beta]^(-3)([beta]z + 1) e^(-[beta] z), (39)

K0 = 4[pi][beta]^(-4), T0 = 3[pi][beta]^(-5). (40)

The range of the attractive force is mathematically infinite, but
practically of the order [beta]^(-1), and we see that T is of higher
order in this small quantity than K. That K is in all cases of the
fourth order and T of the fifth order in the range of the forces is
obvious from (37) without integration.

An apparently simple example would be to suppose [phi](z) = z^n. We
get

z^(n+1) z^(n+3)
[PI](z) = - -------, [psi](z) = ---------,
n+1 n+3 . n+1

2[pi]z^(n+4) |[oo]
K0 = --------------- | (41)
n+4 . n+3 . n+1 |0

The intrinsic pressure will thus be infinite whatever n may be. If n +
4 be positive, the attraction of infinitely distant parts contributes
to the result; while if n + 4 be negative, the parts in immediate
contiguity act with infinite power. For the transition case, discussed
by William Sutherland (_Phil. Mag._ xxiv. p. 113, 1887), of n + 4 = 0,
K0 is also infinite. It seems therefore that nothing satisfactory can
be arrived at under this head.

As a third example, we will take the law proposed by Young, viz.

[phi](z) = 1 from z = 0 to z = a, \
[phi](z) = 0 from z = a to z = [oo]; / (42)

and corresponding therewith,

[Pi](z) = a - z from z = 0 to z = a, \
[Pi](z) = 0 from z = a to z = [oo], / (43)

[psi](z) = 1/2a(a^2 - z^2) = 1/3(a^3 - z^3) from z = 0 to z = a, \
[psi](z) = 0 from z = a to z = [oo], / (44)

Equations (37) now give
_
2[pi] / [oo] [pi]a^4
K0 = ----- | z^3dz = -------, (45)
3 _/0 6
_
[pi] / a [pi]a^5
T0 = ---- | z^4 dz = -------. (46)
8 _/0 40

The numerical results differ from those of Young, who finds that "_the
contractile force is one-third of the whole cohesive force of a
stratum of particles, equal in thickness to the interval to which the
primitive equable cohesion extends_," viz. T = (1/3)aK; whereas
according to the above calculation T = (3/20)aK. The discrepancy seems
to depend upon Young having treated the attractive force as operative
in one direction only. For further calculations on Laplace's
principles, see Rayleigh, _Phil. Mag._, Oct. Dec. 1890, or _Scientific
Papers_, vol. iii. p. 397.]

ON SURFACE-TENSION

Definition.--_The tension of a liquid surface across any line drawn on the surface is normal to the line, and is the same for all directions of the line, and is measured by the force across an element of the line divided by the length of that element._

_Experimental Laws of Surface-Tension._--1. For any given liquid surface, as the surface which separates water from air, or oil from water, the surface-tension is the same at every point of the surface and in every direction. It is also practically independent of the curvature of the surface, although it appears from the mathematical theory that there is a slight increase of tension where the mean curvature of the surface is concave, and a slight diminution where it is convex. The amount of this increase and diminution is too small to be directly measured, though it has a certain theoretical importance in the explanation of the equilibrium of the superficial layer of the liquid where it is inclined to the horizon.

2. The surface-tension diminishes as the temperature rises, and when the temperature reaches that of the critical point at which the distinction between the liquid and its vapour ceases, it has been observed by Andrews that the capillary action also vanishes. The early writers on capillary action supposed that the diminution of capillary action was due simply to the change of density corresponding to the rise of temperature, and, therefore, assuming the surface-tension to vary as the square of the density, they deduced its variations from the observed dilatation of the liquid by heat. This assumption, however, does not appear to be verified by the experiments of Brunner and Wolff on the rise of water in tubes at different temperatures.

3. The tension of the surface separating two liquids which do not mix cannot be deduced by any known method from the tensions of the surfaces of the liquids when separately in contact with air.

When the surface is curved, the effect of the surface-tension is to make the pressure on the concave side exceed the pressure on the convex side by T (1/R1 + 1/R2), where T is the intensity of the surface-tension and R1, R2 are the radii of curvature of any two sections normal to the surface and to each other.

If three fluids which do not mix are in contact with each other, the three surfaces of separation meet in a line, straight or curved. Let O (fig. 3) be a point in this line, and let the plane of the paper be supposed to be normal to the line at the point O. The three angles between the tangent planes to the three surfaces of separation at the point O are completely determined by the tensions of the three surfaces. For if in the triangle abc the side ab is taken so as to represent on a given scale the tension of the surface of contact of the fluids a and b, and if the other sides bc and ca are taken so as to represent on the same scale the tensions of the surfaces between b and c and between c and a respectively, then the condition of equilibrium at O for the corresponding tensions R, P and Q is that the angle ROP shall be the supplement of abc, POQ of bca, and, therefore, QOR of cab. Thus the angles at which the surfaces of separation meet are the same at all parts of the line of concourse of the three fluids. When three films of the same liquid meet, their tensions are equal, and, therefore, they make angles of 120 deg. with each other. The froth of soap-suds or beaten-up eggs consists of a multitude of small films which meet each other at angles of 120 deg.

If four fluids, a, b, c, d, meet in a point O, and if a tetrahedron ABCD is formed so that its edge AB represents the tension of the surface of contact of the liquids a and b, BC that of b and c, and so on; then if we place this tetrahedron so that the face ABC is normal to the tangent at O to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at O to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at O to the other three lines of concourse of the liquids, and the other five edges of the tetrahedron will be normal to the tangent planes at O to the other five surfaces of contact.

If six films of the same liquid meet in a point the corresponding tetrahedron is a regular tetrahedron, and each film, where it meets the others, has an angle whose cosine is -1/3. Hence if we take two nets of wire with hexagonal meshes, and place one on the other so that the point of concourse of three hexagons of one net coincides with the middle of a hexagon of the other, and if we then, after dipping them in Plateau's liquid, place them horizontally, and gently raise the upper one, we shall develop a system of plane laminae arranged as the walls and floors of the cells are arranged in a honeycomb. We must not, however, raise the upper net too much, or the system of films will become unstable.

When a drop of one liquid, B, is placed on the surface of another, A, the phenomena which take place depend on the relative magnitude of the three surface-tensions corresponding to the surface between A and air, between B and air, and between A and B. If no one of these tensions is greater than the sum of the other two, the drop will assume the form of a lens, the angles which the upper and lower surfaces of the lens make with the free surface of A and with each other being equal to the external angles of the triangle of forces. Such lenses are often seen formed by drops of fat floating on the surface of hot water, soup or gravy. But when the surface-tension of A exceeds the sum of the tensions of the surfaces of contact of B with air and with A, it is impossible to construct the triangle of forces; so that equilibrium becomes impossible. The edge of the drop is drawn out by the surface-tension of A with a force greater than the sum of the tensions of the two surfaces of the drop. The drop, therefore, spreads itself out, with great velocity, over the surface of A till it covers an enormous area, and is reduced to such extreme tenuity that it is not probable that it retains the same properties of surface-tension which it has in a large mass. Thus a drop of train oil will spread itself over the surface of the sea till it shows the colours of thin plates. These rapidly descend in Newton's scale and at last disappear, showing that the thickness of the film is less than the tenth part of the length of a wave of light. But even when thus attenuated, the film may be proved to be present, since the surface-tension of the liquid is considerably less than that of pure water. This may be shown by placing another drop of oil on the surface. This drop will not spread out like the first drop, but will take the form of a flat lens with a distinct circular edge, showing that the surface-tension of what is still apparently pure water is now less than the sum of the tensions of the surfaces separating oil from air and water.

The spreading of drops on the surface of a liquid has formed the subject of a very extensive series of experiments by Charles Tomlinson; van der Mensbrugghe has also written a very complete memoir on this subject (_Sur la tension superficielle des liquides_, Bruxelles, 1873).

When a solid body is in contact with two fluids, the surface of the solid cannot alter its form, but the angle at which the surface of contact of the two fluids meets the surface of the solid depends on the values of the three surface-tensions. If a and b are the two fluids and c the solid then the equilibrium of the tensions at the point O depends only on that of thin components parallel to the surface, because the surface-tensions normal to the surface are balanced by the resistance of the solid. Hence if the angle ROQ (fig. 4) at which the surface of contact OP meets the solid is denoted by [alpha],

T_(bc) - T_(ca) - T_(ab) cos[alpha] = 0,

Whence

cos[alpha] = (T_(bc) - T_(ca))/T_(ab).

As an experiment on the angle of contact only gives us the difference of the surface-tensions at the solid surface, we cannot determine their actual value. It is theoretically probable that they are often negative, and may be called surface-pressures.

The constancy of the angle of contact between the surface of a fluid and a solid was first pointed out by Dr Young, who states that the angle of contact between mercury and glass is about 140 deg. Quincke makes it 128 deg. 52'.

If the tension of the surface between the solid and one of the fluids exceeds the sum of the other two tensions, the point of contact will not be in equilibrium, but will be dragged towards the side on which the tension is greatest. If the quantity of the first fluid is small it will stand in a drop on the surface of the solid without wetting it. If the quantity of the second fluid is small it will spread itself over the surface and wet the solid. The angle of contact of the first fluid is 180 deg. and that of the second is zero.

If a drop of alcohol be made to touch one side of a drop of oil on a glass plate, the alcohol will appear to chase the oil over the plate, and if a drop of water and a drop of bisulphide of carbon be placed in contact in a horizontal capillary tube, the bisulphide of carbon will chase the water along the tube. In both cases the liquids move in the direction in which the surface-pressure at the solid is least.

[In order to express the dependence of the tension at the interface of two bodies in terms of the forces exercised by the bodies upon themselves and upon one another, we cannot do better than follow the method of Dupre. If T12 denote the interfacial tension, the energy corresponding to unit of area of the interface is also T12, as we see by considering the introduction (through a fine tube) of one body into the interior of the other. A comparison with another method of generating the interface, similar to that previously employed when but one body was in question, will now allow us to evaluate T12.

The work required to cleave asunder the parts of the first fluid which lie on the two sides of an ideal plane passing through the interior, is per unit of area 2T1, and the free surface produced is two units in area. So for the second fluid the corresponding work is 2T2. This having been effected, let us now suppose that each of the units of area of free surface of fluid (1) is allowed to approach normally a unit area of (2) until contact is established. In this process work is gained which we may denote by 4T'12, 2T'12 for each pair. On the whole, then, the work expended in producing two units of interface is 2T1 + 2T2 - 4T'12, and this, as we have seen, may be equated to 2T12. Hence

T12 = T1 + T2 - 2T'12 (47)

If the two bodies are similar,

T1 = T2 = T'12;

and T12 = 0, as it should do.

Laplace does not treat systematically the question of interfacial tension, but he gives incidentally in terms of his quantity H a relation analogous to (47).

If 2T'12 > T1 + T2, T12 would be negative, so that the interface would of itself tend to increase. In this case the fluids must mix. Conversely, if two fluids mix, it would seem that T'12 must exceed the mean of T1 and T2; otherwise work would have to be _expended_ to effect a close alternate stratification of the two bodies, such as we may suppose to constitute a first step in the process of mixture (Dupre, _Theorie mecanique de la chaleur_, p. 372; Kelvin, _Popular Lectures_, p. 53).

The value of T'12 has already been calculated (32). We may write

_
/ [oo]
T'12 = [pi][sigma]1[sigma]2 | [theta](z)dz =
_/0
_
1 / [oo]
= -- [pi][sigma]1[sigma]2 | z^4 [phi](z)dz; (48)
8 _/0

and in general the functions [theta], or [phi], must be regarded as capable of assuming different forms. Under these circumstances there is no limitation upon the values of the interfacial tensions for three fluids, which we may denote by T12, T23, T31. If the three fluids can remain in contact with one another, the sum of any two of the quantities must exceed the third, and by Neumann's rule the directions of the interfaces at the common edge must be parallel to the sides of a triangle, taken proportional to T12, T23, T31. If the above-mentioned condition be not satisfied, the triangle is imaginary, and the three fluids cannot rest in contact, the two weaker tensions, even if acting in full concert, being incapable of balancing the strongest. For instance, if T31 > T12 + T23, the second fluid spreads itself indefinitely upon the interface of the first and third fluids.

_
/| T23
/
3 /
<----------* 2
T31 1 \
\
_\| T12

The experimenters who have dealt with this question, C.G.M. Marangoni, van der Mensbrugghe, Quincke, have all arrived at results inconsistent with the reality of Neumann's triangle. Thus Marangoni says (_Pogg. Annalen_, cxliii. p. 348, 1871):--"Die gemeinschaftliche Oberflache zweier Flussigkeiten hat eine geringere Oberflachenspannung als die Differenz der Oberflachenspannung der Flussigkeiten selbst (mit Ausnahme des Quecksilbers)." Three pure bodies (of which one may be air) cannot accordingly remain in contact. If a drop of oil stands in lenticular form upon a surface of water, it is because the water-surface is already contaminated with a greasy film.

On the theoretical side the question is open until we introduce some limitation upon the generality of the functions. By far the simplest supposition open to us is that the functions are the same in all cases, the attractions differing merely by coefficients analogous to densities in the theory of gravitation. This hypothesis was suggested by Laplace, and may conveniently be named after him. It was also tacitly adopted by Young, in connexion with the still more special hypothesis which Young probably had in view, namely that the force in each case was constant within a limited range, the same in all cases, and vanished outside that range.

As an immediate consequence of this hypothesis we have from (28)

K = K0[sigma]^2, (49)

T = T0[sigma]^2, (50)

where K0, T0 are the same for all bodies.

But the most interesting results are those which Young (_Works_, vol. i. p. 463) deduced relative to the interfacial tensions of three bodies. By (37), (48),

T'12 = [sigma]1[sigma]2T0; (51)

so that by (47), (50),

T12 = ([sigma]1 - [sigma]2)^2 T0 (52)

According to (52), the interfacial tension between any two bodies is proportional to the square of the difference of their densities. The densities [sigma]1, [sigma]2, [sigma]3 being in descending order of magnitude, we may write

T31 = ([sigma]1 - [sigma]2 + [sigma]2 - [sigma]3)^2 T0

= T12 + T23 + 2([sigma]1 - [sigma]2)([sigma]2 - [sigma]3) T0;

so that T31 necessarily exceeds the sum of the other two interfacial tensions. We are thus led to the important conclusion that according to this hypothesis Neumann's triangle is necessarily imaginary, that one of three fluids will always spread upon the interface of the other two.

Another point of importance may be easily illustrated by this theory, viz. the dependency of capillarity upon abruptness of transition. "The reason why the capillary force should disappear when the transition between two liquids is sufficiently gradual will now be evident. Suppose that the transition from 0 to [sigma] is made in two equal steps, the thickness of the intermediate layer of density 1/2[sigma] being large compared to the range of the molecular forces, but small in comparison with the radius of curvature. At each step the difference of capillary pressure is only one-quarter of that due to the sudden transition from 0 to [sigma], and thus altogether half the effect is lost by the interposition of the layer. If there were three equal steps, the effect would be reduced to one-third, and so on. When the number of steps is infinite, the capillary pressure disappears altogether." ("Laplace's Theory of Capillarity," Rayleigh, _Phil. Mag._, 1883, p. 315.)

According to Laplace's hypothesis the whole energy of any number of contiguous strata of liquids is least when they are arranged in order of density, so that this is the disposition favoured by the attractive forces. The problem is to make the sum of the interfacial tensions a minimum, each tension being proportional to the square of the difference of densities of the two contiguous liquids in question. If the order of stratification differ from that of densities, we can show that each step of approximation to this order lowers the sum of tensions. To this end consider the effect of the abolition of a stratum [sigma]_(n+1), contiguous to [sigma]_n and [sigma]_(n+2). Before the change we have ([sigma]_n - [sigma](n+1))^2 + ([sigma]_(n+1) - [sigma]_(n+2))^2, and afterwards ([sigma]_n - [sigma]_(n+2))^2. The second _minus_ the first, or the increase in the sum of tensions, is thus

2([sigma]_n - [sigma]_(n+1))([sigma]_(n+1) - [sigma]_(n+2)).

Hence, if [sigma]_(n+1) be intermediate in magnitude between [sigma]_n and [sigma]_(n+2), the sum of tensions is increased by the abolition of the stratum; but, if [sigma]_(n+1) be not intermediate, the sum is decreased. We see, then, that the removal of a stratum from between neighbours where it is out of order and its introduction between neighbours where it will be in order is doubly favourable to the reduction of the sum of tensions; and since by a succession of such steps we may arrive at the order of magnitude throughout, we conclude that this is the disposition of minimum tensions and energy.

So far the results of Laplace's hypothesis are in marked accordance with experiment; but if we follow it out further, discordances begin to manifest themselves. According to (52) ____ ____ ____ \/ T31 = \/ T12 + \/ T23, (53)

a relation not verified by experiment. What is more, (52) shows that according to the hypothesis T12 is necessarily positive; so that, if the preceding argument be correct, no such thing as mixture of two liquids could ever take place.

There are two apparent exceptions to Marangoni's rule which call for a word of explanation. According to the rule, water, which has the lower surface-tension, should spread upon the surface of mercury; whereas the universal experience of the laboratory is that drops of water standing upon mercury retain their compact form without the least tendency to spread. To Quincke belongs the credit of dissipating the apparent exception. He found that mercury specially prepared behaves quite differently from ordinary mercury, and that a drop of water deposited thereon spreads over the entire surface. The ordinary behaviour is evidently the result of a film of grease, which adheres with great obstinacy.

The process described by Quincke is somewhat elaborate; but there is little difficulty in repeating the experiment if the mistake be avoided of using a free surface already contaminated, as almost inevitably happens when the mercury is poured from an ordinary bottle. The mercury should be drawn from underneath, for which purpose an arrangement similar to a chemical wash bottle is suitable, and it may be poured into watch-glasses, previously dipped into strong sulphuric acid, rinsed in distilled water, and dried over a Bunsen flame. When the glasses are cool, they may be charged with mercury, of which the first part is rejected. Operating in this way there is no difficulty in obtaining surfaces upon which a drop of water spreads, although from causes that cannot always be traced, a certain proportion of failures is met with. As might be expected, the grease which produces these effects is largely volatile. In many cases a very moderate preliminary warming of the watch-glasses makes all the difference in the behaviour of the drop.

The behaviour of a drop of carbon bisulphide placed upon clean water is also, at first sight, an exception to Marangoni's rule. So far from spreading over the surface, as according to its lower surface-tension it ought to do, it remains suspended in the form of a lens. Any dust that may be lying upon the surface is not driven away to the edge of the drop, as would happen in the case of oil. A simple modification of the experiment suffices, however, to clear up the difficulty. If after the deposition of the drop, a little lycopodium be scattered over the surface, it is seen that a circular space surrounding the drop, of about the size of a shilling, remains bare, and this, however often the dusting be repeated, so long as any of the carbon bisulphide remains. The interpretation can hardly be doubtful. The carbon bisulphide is really spreading all the while, but on account of its volatility is unable to reach any considerable distance. Immediately surrounding the drop there is a film moving outwards at a high speed, and this carries away almost instantaneously any dust that may fall upon it. The phenomenon above described requires that the water-surface be clean. If a very little grease be present, there is no outward flow and dust remains undisturbed in the immediate neighbourhood of the drop.]

_On the Rise of a Liquid in a Tube_.--Let a tube (fig. 6) whose internal radius is r, made of a solid substance c, be dipped into a liquid a. Let us suppose that the angle of contact for this liquid with the solid c is an acute angle. This implies that the tension of the free surface of the solid c is greater than that of the surface of contact of the solid with the liquid a. Now consider the tension of the free surface of the liquid a. All round its edge there is a tension T acting at an angle a with the vertical. The circumference of the edge is 2[pi]r, so that the resultant of this tension is a force 2[pi]rT cos[alpha] acting vertically upwards on the liquid. Hence the liquid will rise in the tube till the weight of the vertical column between the free surface and the level of the liquid in the vessel balances the resultant of the surface-tension. The upper surface of this column is not level, so that the height of the column cannot be directly measured, but let us assume that h is the mean height of the column, that is to say, the height of a column of equal weight, but with a flat top. Then if r is the radius of the tube at the top of the column, the volume of the suspended column is [pi]r^2h, and its weight is [pi][rho]gr^2h, when [rho] is its density and g the intensity of gravity. Equating this force with the resultant of the tension

[pi][rho]gr^2h = 2[pi]rT cos[alpha],

or

h = 2T cos ([alpha]/[rho]gr).

Hence the mean height to which the fluid rises is inversely as the radius of the tube. For water in a clean glass tube the angle of contact is zero, and

h = 2T/[rho]gr.

For mercury in a glass tube the angle of contact is 128 deg. 52', the cosine of which is negative. Hence when a glass tube is dipped into a vessel of mercury, the mercury within the tube stands at a lower level than outside it.

_Rise of a Liquid between Two Plates_.--When two parallel plates are placed vertically in a liquid the liquid rises between them. If we now suppose fig. 6 to represent a vertical section perpendicular to the plates, we may calculate the rise of the liquid. Let l be the breadth of the plates measured perpendicularly to the plane of the paper, then the length of the line which bounds the wet and the dry parts of the plates inside is l for each surface, and on this the tension T acts at an angle [alpha] to the vertical. Hence the resultant of the surface-tension is 2lT cos[alpha]. If the distance between the inner surfaces of the plates is a, and if the mean height of the film of fluid which rises between them is h, the weight of fluid raised is [rho]ghla. Equating the forces--

[rho]ghla = 2lT cos[alpha],

whence

h = 2T cos ([alpha]/[rho]ga).

This expression is the same as that for the rise of a liquid in a tube, except that instead of r, the radius of the tube, we have a the distance of the plates.

_Form of the Capillary Surface_.--The form of the surface of a liquid acted on by gravity is easily determined if we assume that near the part considered the line of contact of the surface of the liquid with that of the solid bounding it is straight and horizontal, as it is when the solids which constrain the liquid are bounded by surfaces formed by horizontal and parallel generating lines. This will be the case, for instance, near a flat plate dipped into the liquid. If we suppose these generating lines to be normal to the plane of the paper, then all sections of the solids parallel to this plane will be equal and similar to each other, and the section of the surface of the liquid will be of the same form for all such sections.

Let us consider the portion of the liquid between two parallel sections distant one unit of length. Let P1, P2 (fig. 7) be two points of the surface; [theta]1, [theta]2 the inclination of the surface to the horizon at P1 and P2; y1, y2 the heights of P1 and P2 above the level of the liquid at a distance from all solid bodies. The pressure at any point of the liquid which is above this level is negative unless another fluid as, for instance, the air, presses on the upper surface, but it is only the difference of pressures with which we have to do, because two equal pressures on opposite sides of the surface produce no effect.

We may, therefore, write for the pressure at a height y

p = -[rho]gy,

where [rho] is the density of the liquid, or if there are two fluids the excess of the density of the lower fluid over that of the upper one.

The forces acting on the portion of liquid P1P2A2A1 are--first, the horizontal pressures, -1/2[rho]g y1^2 and 1/2[rho]g y2^2; second, the surface-tension T acting at P1 and P2 in directions inclined [theta]1 and [theta]2 to the horizon. Resolving horizontally we find--

T(cos[theta]2 - cos[theta]1) + 1/2g[rho](y2^2 - y1^2) = 0,

whence

g[rho]y1^2 g[rho]y2^2
cos[theta]2 = cos[theta]1 + ---------- - ----------,
2T 2T

or if we suppose P1 fixed and P2 variable, we may write

cos[theta] = constant - 1/2g[rho]y^2/T.

This equation gives a relation between the inclination of the curve to the horizon and the height above the level of the liquid.

Resolving vertically we find that the weight of the liquid raised above the level must be equal to T(sin[theta]2 - sin[theta]1), and this is therefore equal to the area P1P2A2A1 multiplied by g[rho]. The form of the capillary surface is identical with that of the "elastic curve," or the curve formed by a uniform spring originally straight, when its ends are acted on by equal and opposite forces applied either to the ends themselves or to solid pieces attached to them. Drawings of the different forms of the curve may be found in Thomson and Tait's _Natural Philosophy_, vol. i. p. 455.

We shall next consider the rise of a liquid between two plates of different materials for which the angles of contact are [alpha]1 and [alpha]2, the distance between the plates being a, a small quantity. Since the plates are very near one another we may use the following equation of the surface as an approximation:--

y = h1 + Ax + Bx^2, h2 = h1 + Aa + Ba^2,

whence

cot[alpha]1 = -A, cot[alpha]2 = A + 2Ba

T(cos[alpha]1 + cos[alpha]2) = [rho]ga(h1 + 1/2Aa + (1/3)Ba^2),

whence we obtain

T / \ a / \
h1 = ------- ( cos[alpha]1 + cos[alpha]2 ) + --( 2cot[alpha]1 - cot[alpha]2 )
[rho]ga \ / 6 \ /

T / \ a / \
h2 = ------- ( cos[alpha]1 + cos[alpha]2 ) + --( 2cot[alpha]2 - cot[alpha]1 ).
[rho]ga \ / 6 \ /

Let X be the force which must be applied in a horizontal direction to either plate to keep it from approaching the other, then the forces acting on the first plate are T + X in the negative direction, and T sin [alpha]1 + 1/2g[rho]h1^2 in the positive direction. Hence

X = 1/2g[rho]h1^2 - T(1 - sin[alpha]1).

For the second plate

X = 1/2g[rho]h2^2 - T (1 - sin[alpha]2).

Hence

X = 1/4g[rho](h1^2 + h2^2) - T{1 - 1/2(sin[alpha]1 + sin[alpha]2)},

or, substituting the values of h1 and h2,

1 T^2
X = -- --------- (cos[alpha]1 + cos [alpha]2)^2
2 [rho]ga^2

- T {1 - 1/2(sin [alpha]1 + sin[alpha]2)

- (1/12)(cos[alpha]1 + cos[alpha]2)(cot[alpha]1 + cot[alpha]2)},

the remaining terms being negligible when a is small. The force, therefore, with which the two plates are drawn together consists first of a positive part, or in other words an attraction, varying inversely as the square of the distance, and second, of a negative part of repulsion independent of the distance. Hence in all cases except that in which the angles [alpha]1 and [alpha]2 are supplementary to each other, the force is attractive when [alpha] is small enough, but when cos[alpha]1 and cos[alpha]2 are of different signs, as when the liquid is raised by one plate, and depressed by the other, the first term may be so small that the repulsion indicated by the second term comes into play. The fact that a pair of plates which repel one another at a certain distance may attract one another at a smaller distance was deduced by Laplace from theory, and verified by the observations of the abbe Hauy.

_A Drop between Two Plates._--If a small quantity of a liquid which wets glass be introduced between two glass plates slightly inclined to each other, it will run towards that part where the glass plates are nearest together. When the liquid is in equilibrium it forms a thin film, the outer edge of which is all of the same thickness. If d is the distance between the plates at the edge of the film and [PI] the atmospheric pressure, the pressure of the liquid in the film is [Pi] - (2T cos[alpha])/d, and if A is the area of the film between the plates and B its circumference, the plates will be pressed together with a force

2AT cos[alpha]
------------- + BT sin[alpha],
d

and this, whether the atmosphere exerts any pressure or not. The force thus produced by the introduction of a drop of water between two plates is enormous, and is often sufficient to press certain parts of the plates together so powerfully as to bruise them or break them. When two blocks of ice are placed loosely together so that the superfluous water which melts from them may drain away, the remaining water draws the blocks together with a force sufficient to cause the blocks to adhere by the process called _Regelation_.

[An effect of an opposite character may be observed when the fluid is mercury in place of water. When two pieces of flat glass are pressed together under mercury with moderate force they cohere, the mercury leaving the narrow crevasses, even although the alternative is a vacuum. The course of events is more easily followed if one of the pieces of glass constitutes the bottom, or a side, of the vessel containing the mercury.]

In many experiments bodies are floated on the surface of water in order that they may be free to move under the action of slight horizontal forces. Thus Sir Isaac Newton placed a magnet in a floating vessel and a piece of iron in another in order to observe their mutual action, and A.M. Ampere floated a voltaic battery with a coil of wire in its circuit in order to observe the effects of the earth's magnetism on the electric circuit. When such floating bodies come near the edge of the vessel they are drawn up to it, and are apt to stick fast to it. There are two ways of avoiding this inconvenience. One is to grease the float round its water-line so that the water is depressed round it. This, however, often produces a worse disturbing effect, because a thin film of grease spreads over the water and increases its surface-viscosity. The other method is to fill the vessel with water till the level of the water stands a little higher than the rim of the vessel. The float will then be repelled from the edge of the vessel. Such floats, however, should always be made so that the section taken at the level of the water is as small as possible.

[_The Size of Drops._--The relation between the diameter of a tube and the weight of the drop which it delivers appears to have been first investigated by Thomas Tate (_Phil. Mag._ vol. xxvii. p. 176, 1864), whose experiments led him to the conclusion 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." Sufficient time must of course be allowed for the formation of the drops; otherwise no simple results can be expected. In Tate's experiments the period was never less than 40 seconds.

The magnitude of a drop delivered from a tube, even when the formation up to the phase of instability is infinitely slow, cannot be calculated a priori. The weight is sometimes equated to the product of the capillary tension (T) and the circumference of the tube (2[pi]a), but with little justification. Even if the tension at the circumference of the tube acted vertically, and the whole of the liquid below this level passed into the drop, the calculation would still be vitiated by the assumption that the internal pressure at the level in question is atmospheric. It would be necessary to consider the curvatures of the fluid surface at the edge of attachment. If the surface could be treated as a cylindrical prolongation of the tube (radius a), the pressure would be T/a, and the resulting force acting downwards upon the drop would amount to one-half ([pi]aT) of the direct upward pull of the tension along the circumference. At this rate the drop would be but one-half of that above reckoned. But the truth is that a complete solution of the statical problem for all forms up to that at which instability sets in, would not suffice for the present purpose. The detachment of the drop is a _dynamical_ effect, and it is influenced by collateral circumstances. For example, the bore of the tube is no longer a matter of indifference, even though the attachment of the drop occurs entirely at the outer edge. It appears that when the external diameter exceeds a certain value, the weight of a drop of water is sensibly different in the two extreme cases of a very small and of a very large bore.

But although a complete solution of the dynamical problem is impracticable, much interesting information may be obtained from the principle of dynamical similarity. The argument has already been applied by Dupre (_Theorie mecanique de la chaleur_, Paris, 1869, p. 328), but his presentation of it is rather obscure. We will assume that when, as in most cases, viscosity may be neglected, the mass (M) of a drop depends only upon the density ([sigma]), the capillary tension (T), the acceleration of gravity (g), and the linear dimension of the tube (a). In order to justify this assumption, the formation of the drop must be sufficiently slow, and certain restrictions must be imposed upon the shape of the tube. For example, in the case of water delivered from a glass tube, which is cut off square and held vertically, a will be the external radius; and it will be necessary to suppose that the ratio of the internal radius to a is constant, the cases of a ratio infinitely small, or infinitely near unity, being included. But if the fluid be mercury, the flat end of the tube remains unwetted, and the formation of the drop depends upon the internal diameter only.

The "dimensions" of the quantities on which M depends are:--

[sigma] = (Mass)^1 (Length)^(-3),
T = (Force)^1 (Length)^(-1) = (Mass)^1 (Time)^(-2),
g = Acceleration = (Length)^1 (Time)^(-2),

of which M, a mass, is to be expressed as a function. If we assume

M [approximately equals] T^{x}.g^{y}.[sigma]^{z}.a^{u},