= 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.]
[Illustration: FIG. 6.]
_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.
[Illustration: FIG. 7.]
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.