If we suppose a normal ν less than ε to be drawn from the surface S into the liquid, we may divide the shell into elementary shells whose thickness is dν, in each of which the density and other properties of the liquid will be constant.
If we suppose a normal ν less than ε to be drawn from the surface S into the liquid, we may divide the shell into elementary shells whose thickness is dν, in each of which the density and other properties of the liquid will be constant.
The volume of one of these shells will be Sdν. Its mass will be Sρdν. The mass of the whole shell will therefore be S∫ε0ρdν, and that of the interior part of the liquid (V − Sε)ρ0. We thus find for the whole mass of the liquid
M = V ρ0− S∫ε0(ρ0− ρ) dν. (2)
To find the potential energy we have to integrate
∫∫∫χρ dxdydz. (3)
Substituting χρ for ρ in the process we have just gone through, we find
E = Vχ0ρ0− S∫ε0(χ0ρ0− χρ) dν. (4)
Multiplying equation (2) by χ0, and subtracting it from (4),
E − Mχ0= S∫ε0(χ − χ0) ρdν (5)
In this expression M and χ0are 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 χ expresses the energy of unit of mass of the liquid at a depth ν within the bounding surface. When the liquid is in contact with a rare medium, such as its own vapour or any other gas, χ is greater than χ0, 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.
Dupré 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,
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
T =∫ε0(χ − χ0) ρ dν, (7)
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 χ 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
E1− M1χ01= S∫ε10(χ1− χ01) ρ1dν1, (8)
E2− M2χ02= S∫ε20(χ2− χ02) ρ2dν2. (9)
Adding these expressions, and dividing the second member by S, we obtain for the tension of the surface of contact of the two liquids
T1·2=∫ε10(χ1− χ01) ρ1dν1+∫ε20(χ2− χ02) ρ2dν2. (10)
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 ∫ε0(χ − χ0)ρdν 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; σ the mass, and e the energy of unit of area; then
M = Sσ, (11)
E = Se. (12)
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 T dS, and the energy of the film will be increased by this amount. Hence
TdS = dE = Sde + edS. (13)
But since M is constant,
dM = Sdσ + σdS = 0. (14)
Eliminating dS from equations (13) and (14), and dividing by S, we find
In this expression σ 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 ρ is the density, and χ the energy of unit of mass at depth z, then
σ =∫c0ρdz, (16)
and
e =∫c0χρdz. (17)
Both ρ and χ 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 2 ε, there will be a stratum of thickness c − 2ε in the middle of the film, within which the values of ρ and χ will be ρ0and χ0. In the two strata on either side of this the law, according to which ρ and χ depend on the depth, will be the same as in a liquid mass of large dimensions. Hence in this case
σ = (c − 2ε) ρ0+ 2∫ε0ρdν, (18)
e = (c − 2ε) χ0ρ0+ 2∫ε0χρdν, (19)
T = 2∫ε0χρ dν − 2χ0∫ε0ρdν = 2∫ε0(χ − χ0) ρdν. (20)
Hence the tension of a thick film is equal 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 ε.
The symbol χ 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 theform χ − χ0, 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 ε of the bounding surface.
When a liquid is in thermal and dynamical equilibrium with its vapour, then if ρ′ and χ′ are the values of ρ and χ for the vapour, and ρ0and χ0those for the liquid,
χ′ − χ0= JL − p(1/ρ′ − 1/ρ0), (21)
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 χ is greater than χ0, and at points in the gas very near the surface of the liquid it is probable that χ is less than χ′, 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 χ 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′ (φ(ƒ) + Cƒ-2), (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′ Cƒ-2. 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 φ(ƒ), where φ(ƒ) is a function of ƒ which is insensible for all sensible values of ƒ, but which becomes sensible and even enormously great when ƒ is exceedingly small.
If we next introduce a new function of f and write
∫∞ƒφ(ƒ) dƒ = Π (ƒ), (23)
thenm m′Π(ƒ) 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 ƒ 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 Π(ƒ) is also insensible for sensible values of ƒ, but for insensible values of ƒ it may become sensible and even very great.
If we next write
∫∞ƒƒΠ(ƒ) dƒ = ψ(z), (24)
then 2πmσψ(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 σ; (2) The attraction of a particle m placed at a distance z from the plane surface of an infinite solid whose density is σ.
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 R1and 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 ω 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,
Let
POQ = θ, PO = r, PQ = ƒ, BP = z,
ƒ² = u² + r² − 2ur cos θ. (26)
The element of the stratum at Q may be expressed by
σu² sin θ dθdω,
or expressing dθ in terms of dƒ by (26),
σur-1ƒ dƒ dθ.
Multiplying this by m and by π(ƒ), we obtain for the work done by the attraction of this element when m is brought from an infinite distance to P1,
mσur-1ƒΠ(ƒ) dƒdω.
Integrating with respect to ƒ from ƒ = z to ƒ = 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
∫mσur-1(ψ(z) − ψ(a)) dω,
and since ψ(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 ω by (25), we find
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 σ, and whose principal radii of curvature are R1and 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 ν below the surface, we have only to write instead of σ ρdz, and to integrate
where, in general, we must suppose ρ 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
K = 2π∫∞0ψ(z) dz, H = 2π∫∞0zψ(z) dz,
the pressure of a columnof the fluid itselfterminating at the surface will be
ρ² {K + ½H (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
mρ {K + ½H (1/R1+ 1/R2) }.
If we write
∫∞zψ(z) dz = θ(z),
then 2πmρθ(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 ρ and infinitely thick. The function θ(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 θ(0) = K.
If χ′ 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
χ = χ′ − 2πρθ(z).
At the surface
χ = χ′ − 2πρθ(0).
At a distance z within the surface
χ = χ′ − 4πρθ(0) + 2πρθ(z).
If the liquid forms a stratum of thickness c, then
χ = χ′ − 4πρθ(0) + 2πρθ(z) + 2πρθ(z − c).
The surface-density of this stratum is σ = cρ. The energy per unit of area is
e =∫c0χρ dz = cρ (χ′ − 4πρθ(0)) + 2πρ²∫c0θ(z) dz + 2πρ²∫c0θ(z − c) dz.
Since the two sides of the stratum are similar the last two terms are equal, and
e = cρ (χ′ − 4πρθ(0)) + 4πρ²∫c0θ(z) dz.
Differentiating with respect to c, we find
Hence the surface-tension
Integrating the first term within brackets by parts, it becomes
Remembering that θ(0) is a finite quantity, and that dθ/dz = − ψ(z), we find
T = 4πρ²∫c0zψ(z) dz. (27)
When c is greater than ε this is equivalent to 2H in the equation of Laplace. Hence the tension is the same for all films thicker than ε, the range of the molecular forces. For thinner films
Hence if ψ(c) is positive, the tension and the thickness will increase together. Now 2πmρψ(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 capillaryphenomena 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 4πρ2K, 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 + Aρ²,
where A is a constant [equal to Laplace’s intrinsic pressure K. But this equation is applicable only at points in the interior, where ρ 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 σ, 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 π σ² ψ(z) dz, reckoned per unit of area. The expression for the intrinsic pressure is thus simply
K = 2πσ²∫∞0ψ(z) dz. (28)
In Laplace’s investigation σ is supposed to be unity. We may call the value which (28) then assumes K0, so that as above
K0= 2π∫∞0ψ(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 (Théorie mécanique 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 σ1, σ2represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area
2πσ1σ2∫∞0ψ(z) dz, (30)
or 2πσ1σ2θ(z), if we write
∫∞0ψ(z) dz = θ(z) (31)
The work required to produce the separation in question is thus
2πσ1σ2∫∞0θ(z) dz; (32)
and for the tension of a liquid of density σ we have
T = πσ²∫∞0θ(z) dz. (33)
The form of this expression may be modified by integration by parts. For
Since θ(0) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply
∫∞0θ(z) dz =∫∞0zψ(z) dz, (34)
and
T = πσ²∫∞0zψ(z) dz. (35)
In Laplace’s notation the second member of (34), multiplied by 2π, is represented by H.
As Laplace has shown, the values for K and T may also be expressed in terms of the function φ, with which we started. Integrating by parts, we get
∫ψ(z) dz = z ψ(z) +1⁄3z3Π(z) +1⁄3∫z3φ(z) dz,
∫zψ(z) dz = ½ z2ψ(z) +1⁄8z4Π(z) +1⁄8∫z4φ(z) dz.
In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write
∫∞0ψ(z) dz =1⁄3∫∞2z3φ(z) dz,∫∞0zψ(z) dz =1⁄8∫∞0z4φ(z) dz; (36)
so that
A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that
φ(ƒ) = eβƒ. (38)
From this we obtain
Π(z) = β-1e-βz, ψ(z) = β-3(βz + 1)e-βz, (39)
K0= 4πβ-4, T0= 3πβ-5. (40)
The range of the attractive force is mathematically infinite, but practically of the order β-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 φ(z) = zn. We get
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, K0is 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.
and corresponding therewith,
Equations (37) now give
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 =1⁄3aK; whereas according to the above calculation T =3⁄20aK. 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, orScientific 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 thedensity, 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, R2are 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° 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°.
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 α,
Tbc− Tca− Tabcos α = 0,
Whence
cos α = (Tbc− Tca) / Tab.
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°. 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 180° 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 Dupré. If T12denote the interfacial tension, the energy corresponding to unit of area of the interfaceis 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′12for 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, T12would 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′12must exceed the mean of T1and T2; otherwise work would have to beexpendedto effect a close alternate stratification of the two bodies, such as we may suppose to constitute a first step in the process of mixture (Dupré,Théorie mécanique de la chaleur, p. 372; Kelvin,Popular Lectures, p. 53).
The value of T′12has already been calculated (32). We may write
T′12= πσ1σ2∫∞0θ(z) dz =1⁄8πσ1σ2∫∞0z4φ(z) dz; (48)
and in general the functions θ, or φ, 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.
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 Oberfläche zweier Flüssigkeiten hat eine geringere Oberflächenspannung als die Differenz der Oberflächenspannung der Flüssigkeiten 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 = K0σ², (49)
T = T0σ², (50)
where K0, T0are 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= σ1σ2T0; (51)
so that by (47), (50),
T12= (σ1− σ2)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 σ1, σ2, σ3being in descending order of magnitude, we may write
T31= (σ1− σ2+ σ2− σ3)² T0= T12+ T23+ 2(σ1− σ2) (σ2− σ3)T0;
so that T31necessarily 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 σ is made in two equal steps, the thickness of the intermediate layer of density ½σ 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 σ, 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 σn+1, contiguous to σnand σn+2. Before the change we have (σn− σn+1)² + (σn+1− σn+2)², and afterwards (σn− σn+2)². The secondminusthe first, or the increase in the sum of tensions, is thus
2(σn− σn+1) (σn+1− σn+2).
Hence, if σn+1be intermediate in magnitude between σnand σn+2, the sum of tensions is increased by the abolition of the stratum; but, if σn+1be 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 T12is 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 2πr, so that the resultant of this tension is a force 2πrT cos α 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 πr²h, and its weight is πρgr²h, when ρ is its density and g the intensity of gravity. Equating this force with the resultant of the tension
πρgr²h = 2πrT cos α,
or
h = 2T cos α/ρgr.
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 / ρgr.
For mercury in a glass tube the angle of contact is 128° 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 α to the vertical. Hence the resultant of the surface-tension is 2lT cos α. 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 ρghla. Equating the forces—
ρghla = 2lT cos α,
whence
h = 2T cos α/ρga.
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; θ1, θ2the inclination of the surface to the horizon at P1and P2; y1, y2the heights of P1and P2above 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 = −ρgy,
where ρ 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 P1P2A2A1are—first, the horizontal pressures, −½ρgy1² and ½ρgy2²; second, the surface-tension T acting at P1and P2in directions inclined θ1and θ2to the horizon. Resolving horizontally we find—
T(cosθ2− cosθ1) + ½gρ(y2² − y1²) = 0,
whence
or if we suppose P1fixed and P2variable, we may write
cosθ = constant − ½gρy² / 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(sinθ2− sinθ1), and this is therefore equal to the area P1P2A2A1multiplied by gρ. 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’sNatural 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 α1and α2, 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 + Bx², h2= h1+ Aa + Ba²,
whence
cot α1= −A, cot α2= A + 2Ba
T(cos α1+ cos α2) = ρga(h1+ ½Aa +1⁄3Ba²),
whence we obtain