THEORY OF CAPILLARY ACTION
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 twentythousandth part of a millimetre.
We shall use the symbol e to denote this extreme range, beyond which the action of these forces may be regarded as insensible. If x denotes the potential energy of unit of mass of the substance, we may treat x as sensibly constant except within a distance e of the bounding surface of the fluid. In the interior of the fluid it has the uniform value Xo. In like manner the density, p, is sensibly equal to the constant quantity po, which is its value in the interior of the liquid, except within a distance e 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= f f fpdxdydz, (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 e, 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 a 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 a will be SE, and that of the interior space will
be V–SE.
If we suppose a normal v less than f to be drawn from the surface S into the liquid, we may divide the shell into elementary shells whose thickness is dv, in each of which the density and other properties of the liquid will be constant.
The volume of one of these shells will be Sde. Its mass will be
Spdv. The mass of the whole shell will therefore be S f f pdv, and
that of the interior part of the liquid (V–SE)po. We thus find for the whole mass of the liquid
m =Vpo—Sf:(po—P)dv. (2)
To find the potential energy we have to integrate
E =JJf xp dx dy dz (3)
Substituting xp for p in the process we have just gone through, we find
F =VxopoSf f(xopo–xi)dv (4)
Multiplying equation (2) by xo, and subtracting it from (4), E
E–Mxo=Sf 0(x–xo)Pdv (5)
In this expression M and xo are both constant, so that the variation of the righthand 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 x expresses the energy of unit of mass of the liquid at a depth v within the bounding surface. When the liquid is in contact with a rare medium, such as its own vapour or any other gas, x is greater than xo, 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 surfacetension of liquids.
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. r). 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
8 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,
F=dbTab=Ta, . . (6)
or the force arising from the surfacetension acting on a length a of the strip is Ta, so that T represents the surfacetension acting transversely on every unit of length of the periphery of the liquid surface. Hence if we write
T =f E(x—Xo)adv, . (7)
we may define T either as the surfaceenergy per unit of area, or as the surfacetension 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 x 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
E, — Mixoi = SP fl (xi — xoi) pid vi,
0
Ez—MYXoz =S5i( Xz—Xoz)Izdvz.
Adding these expressions, and dividing the second member by S, we obtain for the tension of the surface of contact of the two liquids
T1.s=f Si (xi – xo1)pldvi+f0 (xz— Xo2)p2dv2. . . (1 o)
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 tende 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 surfacetension. Thus water spreads itself out on a clean surface of glass. This shows
E
that f (x—xo)Pdv must be negative for water in contact with glass.
On the Tension of Liquid Films.—The method already given for the investigation of the surfacetension of a liquid, all whose dimensions are sensible,. fails in the case of a liquid film such as a soapbubble. 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; o the mass, and e the energy of unit of area; then
M =So., . (1I)
E = Se. (I2) Let us now suppose that by some change in the form of the boundary of the film its area is changed from S to S+dS. If its tension is T the work required to effect this increase of surface will be TdS, and the energy of the film will be increased by this amount. Hence
TdS=dE =Sde+edS (13) But since M is constant,
dM=Sde+edS=o (14) Eliminating dS from equations (13) and (14), and dividing by S, we find
(8) (9)
T (15)
In this expression o denotes the mass of unit of area of the film, and e the energy of unit of area.
If we take the axis of z normal to either surface of the film, the radius of curvature of which we suppose to be very great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth z, then
jo dz, (16)
e=f scads, (17)
Both p and x are functions of z, the value of, which remains the same when z—c is substituted for z. If the thickness of the film is greater than 2e, there will be a stratum of thickness C—2E in the middle of the film, within which the values of p and x will be po and xo. In the two strata on either side of this the law, according to which p and x depend on the depth, will be the same as in a liquid mass of large dimensions. Hence in this case
u=(C—2e)po+2ffPdv, t„ (18)
e = (c — 2f) xoPo+2 f fxPd v,
do de • de
–Cc= P°' dC=xOPO, ••dam=xo,
T=2ffxpdv—2x fepdv=2ff(x—xo)pdv. . . (20) 0 0 0
Hence the tension of a thick film is egbal to the sum of the tensions of its two surfaces as already calculated (equation 7). On the hypothesis of uniform density we shall find that this is true for films whose thickness exceeds f.
The symbol x is defined as the energy of unit of mass of the substance. A knowledge of the absolute value of this energy is not required, since in every expression in which it occurs it is under the
and
• (19)
form x—xo, that is to say, the difference between the energy in two different states. The only cases, however, in which we have experimental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas. It is impossible to make direct measurements of the properties of particles of the substance within the insensible distance e of the bounding surface.
When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x fop the vapour, and po and xo those for the liquid,
x'—xo=JL—p(I/P —I/po), . . . (2I)
where J is the dynamical equivalent of heat, L is the latent heat of unit of mass of the vapour, and p is the pressure. At points in the liquid very near its surface it is probable that x is greater than xo, and at points in the gas very near the surface of the liquid it is probable that x is less than x', but this has not as yet been ascertained experimentally. We shall therefore endeavour to apply to this subject the methods used in Thermodynamics, and where these fail us we shall have recourse to the hypotheses of molecular physics.
We have next to determine the value of x in terms of the action between one particle and another. Let us suppose that the force between two particles m and m' at the distance f is
F=mm'(4'(f)+Cf2), (22)
being reckoned positive when the force is attractive. The actual force between the particles arises in part from their mutual gravitation, which is inversely as the square of the distance. This force is expressed by m m' Cf2. It is easy to show that a force subject to this law would not account for capillary action. We shall, therefore, in what follows, consider only that part of the force which depends on O(f), where ¢(f) is a function off which is insensible for all sensible values of f, but which becomes sensible and even enormously great when f is exceedingly small.
If we next introduce a new function of f and write
f fo(f)df=lI(f), (23)
then m m' II(f) will represent—(I) 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 (24 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 II(f) is also insensible for sensible values of f, but for insensible values off it may become sensible and even very great.
If we next write
fz fII(f)df='G(z), (24)
then 27rmo /i(z) will represent—(I) The work done by the attractive force while a particle m is brought from an infinite distance to a 6 distance z from an infinitely thin stratum of
the substance whose mass per unit of area is
a; (2) The attraction of a particle m placed p= at a distance z from the plane surface of an
infinite solid whose density is a.
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 RI 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 w with the section whose radius of curvature is R. Then if 0 is the centre of curvature in the
plane of the paper, and BO =u,
I cos2w sinew
u = Rl +— 2 (25)
Let POQ=9, PO=r, PQ=f, BP=z,
fr=u2+r2—2ur cos o (26)
The element of the stratum at Q may be expressed by
au2 sin a do dw,
or expressing do in terms of df by (26),
our I f dw.
Multiplying this by m and by r(f), we obtain for the work done by the attraction of this element when m is brought from an infinite distance to Pi,
maurIfII(f)dfdw.
Integrating with respect to f from f =s 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
fmaur' (>L (z) —+P(a))dw,
and since +/i(a) is an insensible quantity we may omit it. We may also write
ur 1= I +zu '+ &c.,since z is very small compared with u, and expressing u in terms of co by (25), we find
fe 7ma(z) I+z(cos I
Rw+siRn 2w) dw =2armo+,&(z) I+2 Iz(II I
+I/
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 surfacedensity is v, and whose principal radii of curvature are Rl 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 v below the surface, we have only to write instead of a pdz, and to integrate f
27rmf z pII'(Z)dz+7rm (RI+R ll 2/ J zpz,/,(z)dz,
where, in general, we must suppose p a function of z. This expression, when integrated, gives (I) 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
K=27rfo¢(z)dz, H=27rf ~z¢(z)dz, 0
the pressure of a column of the fluid itself terminating at the surface will be
p2(K+IH(I/Ri+I/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
mp(K+2H(I/Ri+I/R2)},
If we write
f z /'(z)dz=9(z),
then 27rmpe(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 p and infinitely thick. The function 0(z) is insensible for all sensible values of z. For insensible values it may become sensible, but it must remain finite even when z=o, in which case 9(o) =K.
If x' 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 x=x'—27rpo(z)•
x = x' 27rpe(o).
At a distance z within the surface
x= x'—47rpe(o) +2arpe(z).
If the liquid forms a stratum of thickness c, then
x = x' 47rpo(0) +27rp0(z) +27rpo(z —c).
The surfacedensity of this stratum is a = cp. The energy per unit of area is
e =f xpdz=cp(X 47rpe(o))+27rp2f 00(z)dz+27rp f 0e(c—z)dz. 0 0 0
Since the two sides of the stratum are similar the last two terms are equal, and
e = cp(x' 4xpe(0)) +411p2f00(z) dz. 0
Differentiating with respect to c, we find
dcr do= p, cc=p(x'—47pe(0))+47rp2o(c)•
Hence the surfacetension
T =e— d =47rp2(f 06(z)(iz — ce(c)).
Integrating the first term within brackets by parts, it becomes
cO(c) —oO(o) —f o dedz.
Remembering that 0(o) is a finite quantity, and that de= —#(z), we find
T =47rp2f z+~(z)dz (27)
When c is greater than e this is equivalent to 2H in the equation of Laplace. Hence the tension is the same for all films thicker than e, the range of the molecular forces. For thinner films
dT
dc =4'rp2c~(c).
Hence if >G(c) is positive, the tension and the thickness will increase together. Now 27rmp+,(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
At the surface
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 47rp2K, 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+Ap2,
where A is a constant [equal to Laplace's intrinsic pressure K. But this equation is applicable only at points in the interior, where p is not varying.]
The intrinsic pressure and the surfacetension 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 a, the attraction between the whole of one side and a layer upon the other distant z from the plane and of thickness dz is 27ra2,G(z)dz, reckoned per unit of area. The expression for the intrinsic pressure is thus simply
K=27ra fo ¢(z)dz (28)
In Laplace's investigation a is supposed to be unity. We may call the value which (28) then assumes Ko, so that as above
Ko =27fo ¢(z)dz (29)
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 02 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area
27rala2f :+G(z)dz, (30)In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write
fu I, (z)dz=sf y z3¢(z)dz, fa z4(z)dz=ifo z44(z)dz; (46) Ko = 3 f o z3~(z)dz, To = $ f o z44'(z)dz (37)
A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that
¢(.l) =e t3f (38)
From this we obtain
11(z)=01e 0z, 'G(z)=t3(ISz+1)e tsz, . . ~
Ko =47rf3 , To =37r$ (40
The range of the attractive force is mathematically infinite, but practically of the order 15', 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 4,(z) =z". We get
zn+1
n+4.n+3.n+Il o
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=o, Ko 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.
4,(z) =I fromz=otoz=a, 0(z) =o from z=a to z= co; and corresponding therewith,
= 2a(a2 —z2) — 3 (a3 —z3)
from z=o to z=a, +G(z) =o from z=a to z=oo, Equations (37) now give
9
Ko = 3 u z3dz = 6 , (45)
To 8 f o z4dz =— 06 (46) The numerical results differ from those of Young, who finds that
" the contractile force is onethird 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=*aK; whereas according to the above calculation T = AaK. 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. 189o, or Scientific Papers, vol. iii. P. 397.]
ON SURFACETENSION
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 SurfaceTension.—1. For any given liquid surface, as the surface which separates water from air, or oil from water, the surfacetension 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 surfacetension 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 surfacetension to vary as the square of the
or 27raar28(Z), if we write
f Z ¢(z)dz=o(z) (31)
The work required to produce the separation in question is thus 27ra,a fo 9(z)dz; (32) and for the tension of a liquid of density a we have
T=7raJ o B(z)dz (33)
The form of this expression may be modified by integration by parts. For
fO(z)dz =0(z).z—fzded  dz =0(z).z+fz,G(z)dz.
Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply
fo 0(z)dz=f:zkz)dz, (34)
T=7ra?fo z¢(z)dz (35)
In Laplace's notation the second member of (34), multiplied by 27r, is represented by H.
As Laplace has shown, the values for K and T may also be expressed in terms of the function di, with which we started. Integrating by parts, we get l
f1G(z)dz = z,G(z) +ae3II (z) +y fz34(z)dz, fz'G(z)dz = 2z2,G(z) kiz4II (z) + a fe44(z)dz.
and
so that
zn+3
Ko =
. . . . (41)
(42)
(43) (44)
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 a 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 surfacetension is to make the pressure on the concave side exceed the pressure on the convex side by T (1/Rl +1/R2), where T is the intensity of the surfacetension and RI, 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 0 (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
b to the three surfaces of separation
at the point 0 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
4 the other sides be and ca are taken
If four fluids, a, b, c, d, meet in a point 0, 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 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 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 0 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 0 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—i. 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 surfacetensions 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 hotwater, soup or gravy. But when the surfacetension 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 surfacetension 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 surfacetension 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 surfacetension 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 surfacetension 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 surfacetensions. If a and b are the two fluids and c the solid then the equilibrium of the tensions at the point 0 depends only P
on that of thin components parallel to the surface, because the surfacetensions 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 a,
Tb,—T,a—Tab cos a=o,
Whence
cos a=(Tba—T,a)/Tab.
As an experiment on the angle of contact only gives Flo. 4. us the difference of the surfacetensions at the solid
surface, we cannot determine their actual value. It is theoretically probable that they are often negative, and may be called surfacepressures.
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 1400. Quincke makes it 128° 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 18o° 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 surfacepressure 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
a
Q
R
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+2T24T'12, and this, as we have seen, may be equated to 2T12. Hence
End of Article: THEORY OF CAPILLARY 

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