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)