When two different fluids are placed in contact, they may either diffuse
into each other or remain separate. In some cases diffusion takes place
to a limited extent, after which the resulting mixtures do not mix with
each other. The same substance may be able to exist in two different
states at the same temperature and pressure, as when water and its
saturated vapour are contained in the same vessel. The conditions under
which the thermal and mechanical equilibrium of two fluids, two
mixtures, or the same substance in two physical states in contact with
each other, is possible belong to thermodynamics. All that we have to
observe at present is that, in the cases in which the fluids do not mix
of themselves, the potential energy of the system must be greater when
the fluids are mixed than when they are separate.
It is found by experiment that it is only very close to the bounding
surface of a liquid that the forces arising from the mutual action of
its parts have any resultant effect on one of its particles. The
experiments of Quincke and others seem to show that the extreme range of
the forces which produce capillary action lies between a thousandth and
a twenty-thousandth part of a millimetre.
We shall use the symbol [epsilon] to denote this extreme range, beyond
which the action of these forces may be regarded as insensible. If [chi]
denotes the potential energy of unit of mass of the substance, we may
treat [chi] as sensibly constant except within a distance [epsilon] of
the bounding surface of the fluid. In the interior of the fluid it has
the uniform value [chi]0. In like manner the density, [rho], is sensibly
equal to the constant quantity [rho]0, which is its value in the
interior of the liquid, except within a distance [epsilon] of the
bounding surface. Hence if V is the volume of a mass M of liquid bounded
by a surface whose area is S, the integral
_ _ _
/ / /
M = | | | [rho] dx dy dz, (1)
_/_/_/
where the integration is to be extended throughout the volume V, may be
divided into two parts by considering separately the thin shell or skin
extending from the outer surface to a depth [epsilon], within which the
density and other properties of the liquid vary with the depth, and the
interior portion of the liquid within which its properties are constant.
Since [epsilon] is a line of insensible magnitude compared with the
dimensions of the mass of liquid and the principal radii of curvature of
its surface, the volume of the shell whose surface is S and thickness
[epsilon] will be S[epsilon], and that of the interior space will be
V - S[epsilon].
If we suppose a normal [nu] less than [epsilon] to be drawn from the
surface S into the liquid, we may divide the shell into elementary
shells whose thickness is d[nu], in each of which the density and
other properties of the liquid will be constant.
The volume of one of these shells will be Sd[nu]. Its mass will be
S[rho]d[nu]. The mass of the whole shell will therefore be
_
/ [epsilon]
S | [ro]d[nu],
_/0
and that of the interior part of the liquid (V - S[epsilon])[rho]0. We
thus find for the whole mass of the liquid
_
/ [epsilon]
M = V [rho]0 - S | ([rho]0 - [rho]) d[nu]. (2)
_/0
To find the potential energy we have to integrate
_ _ _
/ / /
E = | | | [chi][rho] dx dy dz (3)
_/_/_/
Substituting [chi][rho] for [rho] in the process we have just gone
through, we find
_
/ [epsilon]
E = V[chi]0[rho]0 - S | ([chi]0[rho]0 - [chi][rho]) d[nu]. (4)
_/0
Multiplying equation (2) by [chi]0, and subtracting it from (4),
_
/ [epsilon]
E - M[chi]0 = S | ([chi] - [chi]0) d[nu]. (5)
_/0
In this expression M and [chi]0 are both constant, so that the
variation of the right-hand side of the equation is the same as that
of the energy E, and expresses that part of the energy which depends
on the area of the bounding surface of the liquid. We may call this
the surface energy.
The symbol [chi] expresses the energy of unit of mass of the liquid at
a depth [nu] within the bounding surface. When the liquid is in
contact with a rare medium, such as its own vapour or any other gas,
[chi] is greater than [chi]0, and the surface energy is positive. By
the principle of the conservation of energy, any displacement of the
liquid by which its energy is diminished will tend to take place of
itself. Hence if the energy is the greater, the greater the area of
the exposed surface, the liquid will tend to move in such a way as to
diminish the area of the exposed surface, or, in other words, the
exposed surface will tend to diminish if it can do so consistently
with the other conditions. This tendency of the surface to contract
itself is called the surface-tension of liquids.
[Illustration: FIG. 1.]
+-----------------------+
|///////////////////////|
|///////////////////////|
|//A+---------------+A//|
|///| |///|
|///| |///|
B =========================== B
|///|C C|///|
+---+ +---+
Dupre has described an arrangement by which the surface-tension of a
liquid film may be illustrated. A piece of sheet metal is cut out in
the form AA (fig. 1). A very fine slip of metal is laid on it in the
position BB, and the whole is dipped into a solution of soap, or M.
Plateau's glycerine mixture. When it is taken out the rectangle AACC
if filled up by a liquid film. This film, however, tends to contract
on itself, and the loose strip of metal BB will, if it is let go, be
drawn up towards AA, provided it is sufficiently light and smooth.
Let T be the surface energy per unit of area; then the energy of a
surface of area S will be ST. If, in the rectangle AACC, AA = a, and
AC = b, its area is S = ab, and its energy Tab. Hence if F is the
force by which the slip BB is pulled towards AA,
d
F = --- Tab = Ta, (6)
db
or the force arising from the surface-tension acting on a length a of
the strip is Ta, so that T represents the surface-tension acting
transversely on every unit of length of the periphery of the liquid
surface. Hence if we write
_
/ [epsilon]
T = | ([chi] - [chi]0) [rho] d[nu], (7)
_/0
we may define T either as the surface-energy per unit of area, or as
the surface-tension per unit of contour, for the numerical values of
these two quantities are equal.
If the liquid is bounded by a dense substance, whether liquid or
solid, the value of [chi] may be different from its value when the
liquid has a free surface. If the liquid is in contact with another
liquid, let us distinguish quantities belonging to the two liquids by
suffixes. We shall then have
_
/ [epsilon]1
E1 - M1[chi]01 = S | ([chi]1 - [chi]01) [rho]1 d[nu]1, (8)
_/ 0
_
/ [epsilon]2
E2 - M2[chi]02 = S | ([chi]2 - [chi]02) [rho]2 d[nu]2. (9)
_/ 0
Adding these expressions, and dividing the second member by S, we
obtain for the tension of the surface of contact of the two liquids
_ _
/ [epsilon]1 / [epsilon]2
T1.2 = | ([chi]1 - [chi]01) [rho]1 d[nu]1 + | ([chi]2 - [chi]02) [rho]2 d[nu]2. (10)
_/0 _/0
If this quantity is positive, the surface of contact will tend to
contract, and the liquids will remain distinct. If, however, it were
negative, the displacement of the liquids which tends to enlarge the
surface of contact would be aided by the molecular forces, so that the
liquids, if not kept separate by gravity, would at length become
thoroughly mixed. No instance, however, of a phenomenon of this kind
has been discovered, for those liquids which mix of themselves do so
by the process of diffusion, which is a molecular motion, and not by
the spontaneous puckering and replication of the bounding surface as
would be the case if T were negative.
It is probable, however, that there are many cases in which the
integral belonging to the less dense fluid is negative. If the denser
body be solid we can often demonstrate this; for the liquid tends to
spread itself over the surface of the solid, so as to increase the
area of the surface of contact, even although in so doing it is
obliged to increase the free surface in opposition to the
surface-tension. Thus water spreads itself out on a clean surface of
glass. This shows that
_
/ [epsilon]
| ([chi] - [chi]0) [rho] d[nu]
_/0
must be negative for water in contact with glass.
_On the Tension of Liquid Films._--The method already given for the
investigation of the surface-tension of a liquid, all whose dimensions
are sensible, fails in the case of a liquid film such as a soap-bubble.
In such a film it is possible that no part of the liquid may be so far
from the surface as to have the potential and density corresponding to
what we have called the interior of a liquid mass, and measurements of
the tension of the film when drawn out to different degrees of thinness
may possibly lead to an estimate of the range of the molecular forces,
or at least of the depth within a liquid mass, at which its properties
become sensibly uniform. We shall therefore indicate a method of
investigating the tension of such films.
Let S be the area of the film, M its mass, and E its energy; [sigma]
the mass, and e the energy of unit of area; then