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1911 Encyclopedia Britannica
Capillary Action
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CAPILLARY ACTION. 1 A tube, the bore of which is so small that it will only admit a hair (Lat. capilla ), is called a capillary tube. When such a tube of glass, open at both ends, is placed vertically with its lower end immersed in water, the water is observed to rise in the tube, and to stand within the tube at a higher level than the water outside. The action between the capillary tube and the water has been called capillary action, and the name has been extended to many other phenomena which have been found to depend on properties of liquids and solids similar to those which cause water to rise in capillary tubes.
The forces which are concerned in these phenomena are those which act between neighbouring parts of the same substance, and which are called forces of cohesion, and those which act between portions of matter of different kinds, which are called forces of adhesion. These forces are quite insensible between two portions of matter separated by any distance which we can directly measure. It is only when the distance becomes exceedingly small that these forces become perceptible. G. H. Quincke (Pogg. Ann. cxxxvii. p. 402) made experiments to determine the greatest distance at which the effect of these forces is sensible, and he found for various substances distances about the twentythousandth part of a millimetre.
Historical
According to J. C. Poggendorff (Pogg. Ann. ci. p. 551), Leonardo da Vinci must be considered as the discoverer of capillary phenomena, but the first accurate observations of the capillary action of tubes and glass plates were made by Francis Hawksbee ( PhysicoMechanical Experiments, London, 1709, pp. 13916 9; and Phil. Trans., 1711 and 1712), who ascribed the action to an attraction between the glass and the liquid. He observed that the effect was the same in thick tubes as in thin, and concluded that only those particles of the glass which are very near the surface have any influence on the phenomenon. Dr James Jurin ( Phil. Trans., 1718, p. 739, and 1719, p. 1083) showed that the height at which the liquid is suspended depends on the section of the tube at the surface of the liquid, and is independent of the form of the lower part of the tube. He considered that the suspension of the liquid is due 1 In this revision of James Clerk Maxwell's classical article in the ninth edition of the Encyclopaedia Britannica, additions are marked by square brackets.
to " the attraction of the periphery or section of the surface of the tube to which the upper surface of the water is contiguous and coheres." From this he showed that the rise of the liquid in tubes of the same substance is inversely proportional to their radii. Sir Isaac Newton devoted the 31st query in the last edition of his Opticks to molecular forces, and instanced several examples of the cohesion of liquids, such as the suspension of mercury in a barometer tube at more than double the height at which it usually stands. This arises from its adhesion to the tube, and the upper part of the mercury sustains a considerable tension, or negative pressure, without the separation of its parts. He considered the capillary phenomena to be of the same kind, but his explanation is not sufficiently explicit with respect to the nature and the limits of the action of the attractive force.
It is to be observed that, while these early speculators ascribe the phenomena to attraction, they do not distinctly assert that this attraction is sensible only at insensible distances, and that for all distances which we can directly measure the force is altogether insensible. The idea of such forces, however, had been distinctly formed by Newton, who gave the first example of the calculation of the effect of such forces in his theorem on the alteration of the path of a lightcorpuscle when it enters or leaves a dense body.
Alexis Claude Clairault (Theorie de la figure de la terre, Paris, 1808, pp. 105, 128) appears to have been the first to show the necessity of taking account of the attraction between the parts of the fluid itself in order to explain the phenomena. He did not, however, recognize the fact that the distance at which the attraction is sensible is not only small but altogether insensible. J. A. von Segner ( Comment. Soc. Reg. Getting. i. (1751) p. 301) introduced the very important idea of the surfacetension of liquids, which he ascribed to attractive forces, the sphere of whose action is so small " ut nullo adhuc sensu percipi potuerit." In attempting to calculate the effect of this surfacetension in determining the form of a drop of the liquid, Segner took account of the curvature of a meridian section of the drop, but neglected the effect of the curvature in a plane at right angles to this section.
The idea of surfacetension introduced by Segner had a most important effect on the subsequent development of the theory. We may regard it as a physical fact established by experiment in the same way as the laws of the elasticity of solid bodies. We may investigate the forces which act between finite portions of a liquid in the same way as we investigate the forces which act between finite portions of a solid. The experiments on solids lead to certain laws of elasticity expressed in terms of coefficients, the values of which can be determined only by experiments on each particular substance. Various attempts have also been made to deduce these laws from particular hypotheses as to the action between the molecules of the elastic substance. We may therefore regard the theory of elasticity as consisting of two parts. The first part establishes the laws of the elasticity of a finite portion of the solid subjected to a homogeneous strain, and deduces from these laws the equations of the equilibrium and motion of a body subjected to any forces and displacements. The second part endeavours to deduce the facts of the elasticity of a finite portion of the substance from hypotheses as to the motion of its constituent molecules and the forces acting between them. In like manner we may by experiment ascertain the general fact that the surface of a liquid is in a state of tension similar to that of a membrane stretched equally in all directions, and prove that this tension depends only on the nature and temperature of the liquid and not on its form, and from this as a secondary physical principle we may deduce all the phenomena of capillary action. This is one step of the investigation. The next step is to deduce this surfacetension from a hypothesis as to the molecular constitution of the liquid and of the bodies that surround it. The scientific importance of this step is to be measured by the degree of insight which it affords or promises into the molecular constitution of real bodies by the suggestion of experiments by which we may discriminate between rival molecular theories.
In 1756 J. G. Leidenfrost (De aquae communis nonnullis qualitatibus tractatus, Duisburg) showed that a soapbubble tends to contract, so that if the tube with which it was blown is left open the bubble will diminish in size and will expel through the tube the air which it contains. He attributed this force, however, not to any general property of the surfaces of liquids, but to the fatty part of the soap which he supposed to separate itself from the other constituents of the solution, and to form a thin skin on the outer face of the bubble.
In 1787 Gaspard Monge (Memoires de l'Acad. des Sciences, 1787, p. 506) asserted that " by supposing the adherence of the particles of a fluid to have a sensible effect only at the surface itself and in the direction of the surface it would be easy to determine the curvature of the surfaces of fluids in the neighbourhood of the solid boundaries which contain them; that these surfaces would be linteariae of which the tension, constant in all directions, would be everywhere equal to the adherence of two particles, and the phenomena of capillary tubes would then present nothing which could not be determined by analysis." He applied this principle of surfacetension to the explanation of the apparent attractions and repulsions between bodies floating on a liquid.
In 1802 John Leslie (Phil. Mag., 1802, vol. xiv. p. 193) gave the first correct explanation of the rise of a liquid in a tube by considering the effect of the attraction of the solid on the very thin stratum of the liquid in contact with it. He did not, like the earlier speculators, suppose this attraction to act in an upward direction so as to support the fluid directly. He showed that the attraction is everywhere normal to the surface of the solid. The direct effect of the attraction is to increase the pressure of the stratum of the fluid in contact with the solid, so as to make it greater than the pressure in the interior of the fluid. The result of this pressure if unopposed is to cause this stratum to spread itself over the surface of the solid as a drop of water is observed to do when placed on a clean horizontal glass plate, and this even when gravity opposes the action, as when the drop is placed on the under surface of the plate. Hence a glass tube plunged into water would become wet all over were it not that the ascending liquid film carries up a quantity of other liquid which coheres to it, so that when it has ascended to a certain height the weight of the column balances the force by which the film spreads itself over the glass. This explanation of the action of the solid is equivalent to that by which Gauss afterwards supplied the defect of the theory of Laplace, except that, not being expressed in terms of mathematical symbols, it does not indicate the mathematical relation between the attraction of individual particles and the final result. Leslie's theory was afterwards treated according to Laplace's mathematical methods by James Ivory in the article on capillary action, under "Fluids, Elevation of," in the supplement to the fourth edition of the Encyclopaedia Britannica, published in 1819.
In 1804 Thomas Young (Essay on the " Cohesion of Fluids, " Phil. Trans., 1805, p. 65) founded the theory of capillary phenomena on the principle of surfacetension. He also observed the constancy of the angle of contact of a liquid surface with a solid, and showed how from these two principles to deduce the phenomena of capillary action. His essay contains the solution of a great number of cases, including most of those afterwards solved by Laplace, but his methods of demonstration, though always correct, and often extremely elegant, are sometimes rendered obscure by his scrupulous avoidance of mathematical symbols. Having applied the secondary principle of surfacetension to the various particular cases of capillary action, Young proceeded to deduce this surfacetension from ulterior principles. He supposed the particles to act on one another with two different kinds of forces, one of which, the attractive force of cohesion, extends to particles at a greater distance than those to which the repulsive force is confined. He further supposed that the attractive force is constant throughout the minute distance to which it extends, but that the repulsive force increases rapidly as the distance diminishes. He thus showed that at a curved part of the surface, a superficial particle would be urged towards the centre of curvature of the surface, and he gave reasons for concluding that this force is proportional to the sum of the curvatures of the surface in two normal planes at right angles to each other.
The subject was next taken up by Pierre Simon Laplace (Mecanique celeste, supplement to the tenth book, pub. in 1806). His results are in many respects identical with those of Young, but his methods of arriving at them are very different, being conducted entirely by mathematical calculations. The form into which he threw his investigation seems to have deterred many able physicists from the inquiry into the ulterior cause of capillary phenomena, and induced them to rest content with deriving them from the fact of surfacetension. But for those who wish to study the molecular constitution of bodies it is necessary to study the effect of forces which are sensible only at insensible distances; and Laplace has furnished us with an example of the method of this study which has never been surpassed. Laplace investigated the force acting on the fluid contained in an infinitely slender canal normal to the surface of the fluid arising from the attraction of the parts of the fluid outside the canal. He thus found for the pressure at a point in the interior of the fluid an expression of the form p =K+ZH(1/R+i/R'), where K is a constant pressure, probably very large, which, however, does not influence capillary phenomena, and therefore cannot be determined from observation of such phenomena; H is another constant on which all capillary phenomena depend; and R and R' are the radii of curvature of any two normal sections of the surface at right angles to each other.
In the first part of our own investigation we shall adhere to the symbols used by Laplace, as we shall find that an accurate knowledge of the physical interpretation of these symbols is necessary for the further investigation of the subject. In the Supplement to the Theory of Capillary Action, Laplace deduced the equation of the surface of the fluid from the condition that the resultant force on a particle at the surface must be normal to the surface. His explanation, however, of the rise of a liquid in a tube is based on the assumption of the constancy of the angle ofjcontact for the same solid and fluid, and of this he has nowhere given a satisfactory proof. In this supplement Laplace gave many important applications of the theory, and compared the results with the experiments of Louis Joseph Gay Lussac.
The next great step in the treatment of the subject was made by C. F. Gauss (Principia generalia Theoriae Figurae Fluidorum in statu Aequilibrii, Göttingen, 1830, or Werke, v. 29, Göttingen, 1867). The principle which he adopted is that of virtual velocities, a principle which under his hands was gradually transforming itself into what is now known as the principle of the conservation of energy. Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles. This expression has been called the forcefunction. With its sign reversed it is now called the potential energy of the system. It consists of three parts, the first depending on the action of gravity, the second on the mutual action between the particles of the fluid, and the third on the action between the particles of the fluid and the particles of a solid or fluid in contact with it.
The condition of equilibrium is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum. This condition when worked out gives not only the equation of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid.
Gauss thus supplied the principal defect in the great work of Laplace. He also pointed out more distinctly the nature of the assumptions which we must make with respect to the law of action of the particles in order to be consistent with observed phenomena. He did not, however, enter into the explanation of particular phenomena, as this had been done already by Laplace, but he pointed out to physicists the advantages of the method of Segner and Gay Lussac, afterwards carried out by Quincke, of measuring the dimensions of large drops of mercury on a horizontal or slightly concave surface, and those of large bubbles of air in transparent liquids resting against the under side of a horizontal plate of a substance wetted by the liquid.
In 1831 Simeon Denis Poisson published his Nouvelle Theorie de action capillaire. He maintained that there is a rapid variation of density near the surface of a liquid, and he gave very strong reasons, which have been only strengthened by subsequent discoveries, for believing that this is the case. He proceeded to an investigation of the equilibrium of a fluid on the hypothesis of uniform density, and arrived at the conclusion that on this hypothesis none of the observed capillary phenomena would take place, and that, therefore, Laplace's theory, in which the density is supposed uniform, is not only insufficient but erroneous. In particular he maintained that the constant pressure K, which occurs in Laplace's theory, and which on that theory is very large, must be in point of fact very small, but the equation of equilibrium from which he concluded this is itself defective. Laplace assumed that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance. On these assumptions his results are certainly right, and are confirmed by the independent method of Gauss, so that the objections raised against them by Poisson fall to the ground. But whether the assumption of uniform density be physically correct is a very different question, and Poisson rendered good service to science in showing how to carry on the investigation on the hypothesis that the density very near the surface is different from that in the interior of the fluid.
The result, however, of Poisson's investigation is practically equivalent to that already obtained by Laplace. In both theories the equation of the liquid surface is the same, involving a constant H, which can be determined only by experiment. The only difference is in the manner in which this quantity H depends on the law of the molecular forces and the law of density near the surface of the fluid, and as these laws are unknown to us we cannot obtain any test to discriminate between the two theories.
We have now described the principal forms of the theory of capillary action during its earlier development. In more recent times the method of Gauss has been modified so as to take account of the variation of density near the surface, and its language has been translated in terms of the modern doctrine of the conservation of energy.' J. A. F. Plateau (Statique experimentale et theorique des liquides ), who made elaborate study of the phenomena of surfacetension, adopted the following method of getting rid of the effects of gravity. He formed a mixture of alcohol and water of the same density as olive oil, and then introduced a quantity of oil into the mixture. It assumes the form of a sphere under the action of surfacetension alone. He then, by means of rings of ironwire, disks and other contrivances, altered the form of certain parts of the,surface of the oil. The free portions of the surface then assume new forms depending on the equilibrium of surfacetension. In this way he produced a great many of the forms of equilibrium of a liquid under the action of surfacetension alone, and compared them with the results of mathematical investigation. He also greatly facilitated the study of liquid films by showing how to form a liquid, the films of which will last for twelve or even for twentyfour hours. The debt which science owes to Plateau is not diminished by the fact that, while investigating these beautiful phenomena, he never himself saw them, having lost his sight in about 1840.
G. L. van der Mensbrugghe (Mem. de l'Acad. Roy. de Belgique, xxxvii., 1873) devised a great number of beautiful illustrations 1 See Enrico Betti, Teoria della Capillarita: Nuovo Cimento (1867); a memoir by M. Stahl, " Ueber einige Punckte in der Theorie der Capillarerscheinungen," Pogg. Ann. cxxxix. p. 239 (1870); and J. D. Van der Waal's Over de Continuiteit van den Gasen Vloeistoftoestand. A good account of the subject from a mathematical point of view will be found in James Challis's " Report on the Theory of Capillary Attraction," Brit. Ass. Report, iv. p. 2 35 (1834).
of the phenomena of surfacetension, and showed their connexion with the experiments of Charles Tomlinson on the figures formed by oils dropped on the clean surface of water.
Athanase Dupre in his 5th, 6th and 7th Memoirs on the Mechanical Theory of Heat (Ann. de Chimie et de Physique, 18661868) applied the principles of thermodynamics to capillary phenomena, and the experiments of his son Paul were exceedingly ingenious and well devised, tracing the influence of surfacetension in a great number of very different circumstances, and deducing from independent methods the numerical value of the surfacetension. The experimental evidence which Dupre obtained bearing on the molecular structure of liquids must be very valuable, even if our present opinions on this subject should turn out to be erroneous.
F. H. R. Liidtge (Pogg. Ann. cxxxix. p. 620) experimented on liquid films, and showed how a film of a liquid of high surfacetension is replaced by a film of lower surfacetension. He also experimented on the effects of the thickness of the film, and came to the conclusion that the thinner a film is, the greater is its tension. This result, however, was tested by Van der Mensbrugghe, who found that the tension is the same for the same liquid whatever be the thickness, as long as the film does not burst. [The continued coexistence of various thicknesses, as evidenced by the colours in the same film, affords an instantaneous proof of this conclusion.] The phenomena of very thin liquid films deserve the most careful study, for it is in this way that we are most likely to obtain evidence by which we may test the theories of the molecular structure of liquids.
Sir W. Thomson (afterwards Lord Kelvin) investigated the effect of the curvature of the surface of a liquid on the thermal equilibrium between the liquid and the vapour in contact with it. He also calculated the effect of surfacetension on the propagation of waves on the surface of a liquid, and determined the minimum velocity of a wave, and the velocity of the wind when it is just sufficient to disturb the surface of still water.
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 X i. 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 f pdx dydz, (I) 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 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 e 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 will be and that of the interior space will be V  SE. If we suppose a normal v less than E 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 Sdv. Its mass will be Spdv. The mass of the whole shell will therefore be S f E 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  S E ( Po  P)dv.. . (2) 0 To find the potential energ we have to integrate E=f xpdxdydz.. .. Substituting xp for p in the process we have just gone through, we find E = p p x p)d v.
(4) Multiplying equation (2) by xo, and subtracting it from (4), E E
(5) In this expression M and 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 surfacetension of a liquid film may be illustrated. A piece of sheet metal is cut out in the form AA (fig. I). 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 B 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 =d b T ab =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) p d v,. .
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 1  Mixoi S f El E  4 S 2 Adding these expressions, and dividing the second member by S, we obtain for the tension of the surface of contact of the two liquids T,. 2 (xi  xoi)Pidvi+f 2 (xi  xn2)P2dv2.. .
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 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 surfacetension. Thus water spreads itself out on a clean surface of glass. This shows that f E (x  xo) p dv 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 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; a the mass, and e the energy of unit of area; then M=So, (II) 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 TdS, and the energy of the film will be increased by this amount. Hence TdS = dE = Sde dedS. But since M is constant, dM = Sda+adS =o. Eliminating dS from equations (is) and (14), and dividing by S, we find T = e  v??, (15) In this expression a 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 o = f o dz, (16) and e = f a xpdz,. . (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 c2E in the middle of the film, within which the values of p and x will be pc and 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 a= (c2E)po+2fopdv, . (18) e= (c2E)xo p o+2 f oxp d v, . (19) dode, de o =Po, Clc = Xopo,
dv=Xo, d T=2 f xpdv2x f pdv=21 0 (x  xi) pd v. (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 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 (3) 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 for the vapour, and po and Xo those for the liquid, x'  xo=JL  p(I/p'  I/pc),. ... (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 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 '(0(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 OM, where 4)(f) is a function of f 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 .f 4(f) d f =Il (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 (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 II(f) is also insensible for sensible values of f, but for insensible values of f it may become sensible and even very great.
If we next write f II (f) d f =, P(z), (24) then 27rmagz) will represent  (I) 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 Z o; (2) The attraction of a particle m placed Q 2 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 R 1 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 FIG. 2. section whose radius of curvature is R1.
Then if 0 is the centre of curvature in the plane of the paper, and BO =u, I _ cos sinew u R 1 R2 Let POQ=o, PO=r, PQ=f, BP=z, f 2 = u 2 +r 2 2ur cos 0 (26) The element of the stratum at Q may be expressed by ou t sin o do dw, or expressing do in terms of df by (26), our 1fdfdw. Multiplying this by m and by 7( f ), we obtain for the work done by the attraction of this element when m is brought from an infinite distance to P1, maur1 fI I (f)dfdw.
Integrating with respect to f from f =z 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 (1,G (z)  111(a))dw, and since (a) is an insensible quantity we may omit it. We may also write ur 1 = I +zu 1+ &c., since z is very small compared with u, and expressing u in terms of w by (25), (we find l 21 mv i fi(z) i I +z(c R w + ' R 2 w) do ) = 27rmoti(z) I fZZ (Ki + R2/ 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 a, and whose principal radii of curvature are R 1 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 ?
27rmf .. p? +2r (z) dzm (+)f ? zpzL(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 =27rf o,/i(z)dz, H =27rf o zi/i(z)dz, the pressure of a column of the fluid itself terminating at the surface will be p2{K+1H(I/RidI/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+zH(I/Ri+I/Ro)}
If we write
(.0 J then 27rmpo(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 0(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'  22 7rp 7rpe e ((zo)) .. At the surface At a distance z within the X su=rxfa'ce  X = X'  47rpo(o ) +27rp0(z).
If the liquid forms a stratum of thickness c, then x = x'  47rpo(o) +27rpo(z) +27rpo(Z  c). The surfacedensity of this stratum is a = cp. The energy per unit of area is e = f xpdz=cp(X' 4lrpe(o))+27rp'f c 0(z ) dz+27rp fee(c  z)dz.
Since the two sides of the stratum are similar the last two terms are equal, and Differentiating with respect to c, we find dc = P' d p(x ' 41rpe (0))+47rp20(c). Hence the surfacetension =e  =47rp 2 (f 0(z)dz  ce(c)). Integrating the first term within brackets by parts, it becomes  fo de Remembering that 0(o ) is a finite quantity, and that Viz =  (z), we find T = 4 7rp f a ,/.(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 do =4'rp24(c).
Hence if il.(c ) is positive, the tension and the thickness will increase together. Now 2 7rmpi,t(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 B (25) e = c p (X'  4 7rpe ( 0 ) ) I474°o(z)dz.
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 41rp 2 K, 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 27r6 2 P(z)dz, reckoned per unit of area. The expression for the intrinsic pressure is thus simply K= 2 iro 2 f 1,G(z)dz (28) In Laplace's investigation o is supposed to be unity. We may call the value which (28) then assumes Ko, so that as above Ko =27rf (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 a i, a 2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area 21ra l a fZ '(z)dz, (30) or 27ra l 02 0(z), if we write f 4,(z)dz=0(z ) (31) The work required to produce the separation in question is thus 2 7ru l a o 0 (z)dz; (32) and for the tension of a liquid of density a we have T = a f o 0 (z)dz. (33) The form of this expression may be modified by integration by parts. For f0(z)dz =0(z).z  fz d dz) dz =0(z).z+fz4,(z) dz. Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply f 0(z)dz f: (z)dz, (34) and T= ref: z1,1,(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 cs, with which we started. Integrating by parts, we get J l'(z)d z = zI, G (z) + 3 z 3 I I (z) 3 f z3Cb(z)dz, fzqi(z)dz = J z21 '(z) + k z41 I (z) + a fz4(1)(z)dz.
In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write f o (z)dz = 3f2' z 3 4(z)dz, f o z(z)dz = 'a' z4 cb(z)d z; (36) so that Ko = 3 f o z3 ?(z) dz, To = $? o z 4 0(z)dz A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that From this we obtain II(z) = 1 31 e1 3z, '(z) =133 ((3z+I)e ? z , (39) K o = 4 7 ( 4, To = 3715
(40) The range of the attractive force is mathematically infinite, but practically of the order (31, 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 II (z) _  n +1, «z)=++, zn+1 zn+3 27rzn+4 (41) K0 = n+4 n+3 fl+I 0 The intrinsic pressure will thus b 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. ¢(z) =I from z =o to z =a, t(z) =o from z=a to z=oo;
'
and corresponding therewith, II(z)=a  z from z=o to z=a, II(z)=o from z=a to z=oo,
' ‘ G (z) = (a 2  z2)  3 (a3  z3) from z=o to z =a,. .. 4,(z ) =0 from z = a to z=co, Equations (37) now give 2 it " ira4 Ko =  z i dz = 6, 3 0 _ 7r fa 4 sra5 T o 8 fo zdz=40 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 = 3aK; whereas according to the above calculation T = 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, 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.  I. 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 (37) ?(f ) =eP f
(38) 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 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 surfacetension is to make the pressure on the concave side exceed the pressure on the convex side by T (1 /R I i /R 2), where T is the intensity of the surfacetension and R 1, 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 to the three surfaces of separation at the point 0 are completely determined by the tensions of the b o a 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 be 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 0 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 soapsuds or beatenup eggs consists of a multitude of small films which meet each other at angles of I 20°.
If four fluids, a, b, c, d, meet in a point 0, and if a tetrahedron AB CD 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, an 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 hot water, 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 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, Tbc  T ca  Tab cos a = o, Whence cos a = (Tbc  Tca)/Tab.
As an experiment on the angle of contact only gives 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 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 surfacepressure at the solid is least.
a [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 b Q FIG. 4. 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 2T 1, and the free surface produced is two units in area. So for the second fluid the corresponding work is 2 T2. This having been effected, let us now suppose that each of the units of area of free surface of fluid (I) 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 2T 12. Hence T12 = T1 +T2  2T'12 (47) If the two bodies are similar, T, = T2 =T',2; and T12 = o, 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, T12 would 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'12 must exceed the mean of T 1 and T2; otherwise work would have to be expended to effect a close alternate stratification of the two bodies, such as we may suppose to constitute a first step in the process of mixture (Dupre, Theorie mecanique de la chaleur, p. 372; Kelvin, Popular Lectures, p. 53). The value of T'12 has already been calculated (32). We may write T'12 = rQio f ° B(z) dz = bra i v 2 f 04 z 4 (z)dz;.. (48) ° and in general the functions 0, or 4), 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 3 quantities must exceed the third, and T 31 I 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 abovementioned 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 Oberflache zweier Fliissig keiten hat eine geringere Oberflachenspannung als die Differenz der Oberflachenspannung der Fliissigkeiten 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 watersurface 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 = Koo' 2, where Ko, To are 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 = 0102To; (51) so that by (47), (50), T,2 = (o,  (72) 2 To. .. .. .. (52) Acco Copyright Statement Bibliography Information 
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