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THEORY OF CAPILLARY
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THEORY OF CAPILLARY See also:ACTION When two different fluids are placed in contact, they may either diffuse into each other or remain See also:separate. In some cases See also:diffusion takes See also: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 See also:water and its saturated vapour are contained in the same See also:vessel. The conditions under which the thermal and See also:mechanical See also:equilibrium of two fluids, two mixtures, or the same substance in two See also:physical states in contact with each other, is possible belong to See also:thermodynamics. All that we have to observe at See also:present is that, in the cases in which the fluids do not mix of themselves, the potential See also:energy of the See also:system must be greater when the fluids are mixed than when they are separate. It is found by experiment that it is only very See also:close to the bounding See also:surface of a liquid that the forces arising from the mutual action of its parts have any resultant effect on one of its particles. The experiments of Quincke and others seem to show that the extreme range of the forces which produce capillary action lies between a thousandth and a twenty-thousandth See also:part of a millimetre. We shall use the See also: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 See also:mass of the substance, we may treat x as sensibly See also:constant except within a distance e of the bounding surface of the fluid. In the interior of the fluid it has the See also:uniform value Xo. In like manner the See also: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 See also:volume of a mass M of liquid bounded by a surface whose See also:area is S, the integral M= f f fpdxdydz, (1) where the integration is to be extended throughout the volume V, may be divided into two parts by considering separately the thin See also:shell or skin extending from the See also:outer surface to a See also:depth e, within which the density and other properties of the liquid vary with the depth, and the interior portion of the liquid within which its properties are constant. Since a is a See also:line of insensible magnitude compared with the dimensions of the mass of liquid and the See also:principal radii of curvature of its surface, the volume of the shell whose surface is S and thickness a will be SE, and that of the interior space will be V–SE. If we suppose a normal v less than f to be See also:drawn from the surface S into the liquid, we may See also:divide the shell into elementary shells whose thickness is dv, in each of which the density and other properties of the liquid will be constant. The volume of one of these shells will be Sde. Its mass will be Spdv. The mass of the whole shell will therefore be S f f pdv, and that of the interior part of the liquid (V–SE)po. We thus find for the whole mass of the liquid m =Vpo—Sf:(po—P)dv. (2) To find the potential energy we have to integrate E =JJf xp dx dy dz (3) Substituting xp for p in the See also:process we have just gone through, we find F =Vxopo-Sf f(xopo–xi)dv (4) Multiplying See also:equation (2) by xo, and subtracting it from (4), E E–Mxo=Sf 0(x–xo)Pdv (5) In this expression M and xo are both constant, so that the variation of the right-See also:hand See also: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 See also: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 See also:medium, such as its own vapour or any other See also:gas, x is greater than xo, and the surface energy is See also: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 See also:contract , itself is called the surface-tension of liquids. See also:Dupre has described an arrangement by which the surface- tension of a liquid film may be illustrated. A piece of See also:sheet See also:metal is cut out in the See also:form AA (fig. r). A very See also:fine slip of metal is laid on it in the position BB, and the whole is dipped into a See also:solution of See also:soap, or M. See also:Plateau's glycerine mixture. When it is taken out the 8 rectangle AACC if filled up by a liquid film. This film, however, tends to contract on itself, and the loose See also:strip of metal BB will, if it is let go, be drawn up towards AA, provided it is sufficiently See also:light and smooth. Let T be the surface energy per unit of area; then the energy of a surface of area S will be ST. If, in the rectangle AACC, AA=a, and AC = b, its area is S = ab, and its energy Tab. Hence if F is the force by which the slip BB is pulled towards AA, F=dbTab=Ta, . . (6) or the force arising from the 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 =f E(x—Xo)adv, . (7) we may define T either as the surface-energy per unit of area, or as the surface-tension per unit of See also: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 See also:free surface. If the liquid is in contact with another liquid, let us distinguish quantities belonging to the two liquids by suffixes. We shall then have E, — Mixoi = SP fl (xi — xoi) pid vi, 0 Ez—MYXoz =S5i( Xz—Xoz)Izdvz. Adding these expressions, and dividing the second member by S, we obtain for the tension of the surface of contact of the two liquids T1.s=f Si (xi – xo1)pldvi+f0 (xz— Xo2)p2dv2. . . (1 o) If this quantity is positive, the surface of contact will tend to contract, and the liquids will remain distinct. If, however, it were negative, the displacement of the liquids which tends to enlarge the surface of contact would be aided by the molecular forces, so that the liquids, if not kept separate by gravity, would at length become thoroughly mixed. No instance, however, of a phenomenon of this See also:kind has been discovered, for those liquids which mix of themselves do so by the process of diffusion, which is a molecular See also:motion, and not by the spontaneous puckering and replication of the bounding surface as would be the See also: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 See also:body be solid we can often demonstrate this; for the liquid tende to spread itself over the surface of the solid, so as to increase the area of the surface of contact, even although in so doing it is obliged to increase the free surface in opposition to the surface-tension. Thus water spreads itself out on a clean surface of See also:glass. This shows E that f (x—xo)Pdv must be negative for water in contact with glass. On the Tension of Liquid Films.—The method already given for the investigation of the 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 See also:lead to an estimate of the range of the molecular forces, or at least of the depth within a liquid mass, at which its properties become sensibly uniform. We shall therefore indicate a method of investigating the tension of such films. Let S be the area of the film, M its mass, and E its energy; o the mass, and e the energy of unit of area; then M =So., . (1I) E = Se. (I2) Let us now suppose that by some See also: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 See also:work required to effect this increase of surface will be TdS, and the energy of the film will be increased by this amount. Hence TdS=dE =Sde+edS (13) But since M is constant, dM=Sde+edS=o (14) Eliminating dS from equations (13) and (14), and dividing by S, we find (8) (9) T (15) In this expression o denotes the mass of unit of area of the film, and e the energy of unit of area. If we take the See also:axis of z normal to either surface of the film, the See also:radius of curvature of which we suppose to be very See also:great compared with its thickness c, and if p is the density, and x the energy of unit of mass at depth z, then jo dz, (16) e=f scads, (17) Both p and x are functions of z, the value of, which remains the same when z—c is substituted for z. If the thickness of the film is greater than 2e, there will be a stratum of thickness C—2E in the See also:middle of the film, within which the values of p and x will be po and xo. In the two strata on either side of this the See also:law, according to which p and x depend on the depth, will be the same as in a liquid mass of large dimensions. Hence in this case u=(C—2e)po+2ffPdv, t„ (18) e = (c — 2f) xoPo+2 f fxPd v, do- de • de –Cc= P°' dC=xOPO, ••See also:dam=xo, T=2ffxpdv—2x fepdv=2ff(x—xo)pdv. . . (20) 0 0 0 Hence the tension of a thick film is egbal to the sum of the tensions of its two surfaces as already calculated (equation 7). On the See also:hypothesis of uniform density we shall find that this is true for films whose thickness exceeds f. The symbol x is defined as the energy of unit of mass of the substance. A knowledge of the See also:absolute value of this energy is not required, since in every expression in which it occurs it is under the and • (19) form x—xo, that is to say, the difference between the energy in two different states. The only cases, however, in which we have experimental values of this quantity are when the substance is either liquid and surrounded by similar liquid, or gaseous and surrounded by similar gas. It is impossible to make See also:direct measurements of the properties of particles of the substance within the insensible distance e of the bounding surface. When a liquid is in thermal and dynamical equilibrium with its vapour, then if p' and x' are the values of p and x fop the vapour, and po and xo those for the liquid, x'—xo=JL—p(I/P —I/po), . . . (2I) where J is the dynamical See also:equivalent of See also:heat, L is the latent heat of unit of mass of the vapour, and p is the pressure. At points in the liquid very near its surface it is probable that x is greater than xo, and at points in the gas very near the surface of the liquid it is probable that x is less than x', but this has not as yet been ascertained experimentally. We shall therefore endeavour to apply to this subject the methods used in Thermodynamics, and where these fail us we shall have recourse to the hypotheses of molecular physics. We have next to determine the value of x in terms of the action between one particle and another. Let us suppose that the force between two particles m and m' at the distance f is F=mm'(4'(f)+Cf-2), (22) being reckoned positive when the force is attractive. The actual force between the particles arises in part from their mutual See also:gravitation, which is inversely as the square of the distance. This force is expressed by m m' Cf-2. It is easy to show that a force subject to this law would not See also:account for capillary action. We shall, therefore, in what follows, consider only that part of the force which depends on O(f), where ¢(f) is a See also:function off which is insensible for all sensible values of f, but which becomes sensible and even enormously great when f is exceedingly small. If we next introduce a new function of f and write f fo(f)df=lI(f), (23) then m m' II(f) will represent—(I) The work done by the attractive force on the particle m, while it is brought from an See also:infinite distance from m' to the distance f from m' ; or (24 The attraction of a particle m on a narrow straight See also:rod resolved in the direction of the length of the rod, one extremity of the rod being at a distance f from m, and the other at an infinite distance, the mass of unit of length of the rod being m'. The function II(f) is also insensible for sensible values of f, but for insensible values off it may become sensible and even very great. If we next write fz fII(f)df='G(z), (24) then 27rmo /i(z) will represent—(I) The work done by the attractive force while a particle m is brought from an infinite distance to a 6 distance z from an infinitely thin stratum of the substance whose mass per unit of area is a; (2) The attraction of a particle m placed p= at a distance z from the See also:plane surface of an infinite solid whose density is a. Let us examine the case in which the particle m is placed at a distance z from a curved stratum of the substance, whose principal radii of curvature are RI and See also:R2. Let P (fig. 2) be the particle and PB a normal to the surface. Let the plane of the See also:paper be a normal See also:section of the surface of the stratum at the point B, making an See also:angle w with the section whose radius of curvature is R. Then if 0 is the centre of curvature in the plane of the paper, and BO =u, I cos2w See also:sinew u = Rl +— 2 (25) Let POQ=9, PO=r, PQ=f, BP=z, fr=u2+r2—2ur See also:cos o (26) The See also:element of the stratum at Q may be expressed by au2 See also:sin a do See also:dw, or expressing do in terms of df by (26), our I f dw. Multiplying this by m and by r(f), we obtain for the work done by the attraction of this element when m is brought from an infinite distance to Pi, maur-IfII(f)dfdw. Integrating with respect to f from f =s to f =a, where a is a line very great compared with the extreme range of the molecular force, but very small compared with either of the radii of curvature, we obtain for the work fmaur' (>L (z) —+P(a))dw, and since +/i(a) is an insensible quantity we may omit it. We may also write ur 1= I +zu '+ &c.,since z is very small compared with u, and expressing u in terms of co by (25), we find fe 7ma(z) I+z(cos I Rw+siRn 2w) dw =2armo+,&(z) I+2 Iz(II I +I/ This then expresses the work done by the attractive forces when a particle m is brought from an infinite distance to the point P at a distance z from a stratum whose surface-density is v, and whose principal radii of curvature are Rl and R2. To find the work done when m is brought to the point P in the neighbourhood of a solid body, the density of which is a function of the depth v below the surface, we have only to write instead of -a pdz, and to integrate f 27rmf z pII'(Z)dz+7rm (RI+R ll 2/ J zpz,/,(z)dz, where, in See also: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 See also:long slender See also: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 See also:Laplace, the density of the liquid was supposed to be uniform. Hence if we write K=27rfo¢(z)dz, H=27rf ~z¢(z)dz, 0 the pressure of a column of the fluid itself terminating at the surface will be p2(K+IH(I/Ri+I/R2)}, and the work done by the attractive forces when a particle m is brought to the surface of the fluid from an infinite distance will be mp(K+2H(I/Ri+I/R2)}, If we write f z /'(z)dz=9(z), then 27rmpe(z) will See also:express the work done by°the attractive forces, while a particle m is brought from an infinite distance to a distance z from the plane surface of a mass of the substance of density p and infinitely thick. The function 0(z) is insensible for all sensible values of z. For insensible values it may become sensible, but it must remain finite even when z=o, in which case 9(o) =K. If x' is the potential energy of unit of mass of the substance in vapour, then at a distance z from the plane surface of the liquid x=x'—27rpo(z)• x = x' -27rpe(o). At a distance z within the surface x= x'—47rpe(o) +2arpe(z). If the liquid forms a stratum of thickness c, then x = x' -47rpo(0) +27rp0(z) +27rpo(z —c). The surface-density of this stratum is a = cp. The energy per unit of area is e =f xpdz=cp(X -47rpe(o))+27rp2f 00(z)dz+27rp f 0e(c—z)dz. 0 0 0 Since the two sides of the stratum are similar the last two terms are equal, and e = cp(x' -4xpe(0)) +411-p2f00(z) dz. 0 Differentiating with respect to c, we find dcr do= p, c-c=p(x'—47pe(0))+47rp2o(c)• Hence the surface-tension T =e— d =47rp2(f 06(z)(iz — ce(c)). Integrating the first See also:term within brackets by parts, it becomes cO(c) —oO(o) —f o dedz. Remembering that 0(o) is a finite quantity, and that de= —#(z), we find T =47rp2f z+~(z)dz (27) When c is greater than e this is equivalent to 2H in the equation of Laplace. Hence the tension is the same for all films thicker than e, the range of the molecular forces. For thinner films dT dc =4'rp2c~(c). Hence if >G(c) is positive, the tension and the thickness will increase together. Now 27rmp+,(c) represents the attraction between a particle m and the plane surface of an infinite mass of the liquid, when the distance of the particle outside the surface is c. Now, the force between the particle and the liquid is certainly, on the whole, attractive; but if between any two small values of c it should be repulsive, then for films whose thickness lies between these values the tension will increase as the thickness diminishes, but for all other cases the tension will diminish as the thickness diminishes. We have given several examples in which the density is assumed to be uniform, because See also:Poisson has asserted that capillary At the surface phenomena would not take place unless the density varied rapidly near the surface. In this assertion we think he was mathematically wrong, though in his own hypothesis that the density does actually vary, he was probably right. In fact, the quantity 47rp2K, which we may call with See also: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 See also: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 See also: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 See also:intrinsic pressure K. But this equation is applicable only at points in the interior, where p 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 See also: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 27ra-2,G(z)dz, reckoned per unit of area. The expression for the intrinsic pressure is thus simply K=27ra fo ¢(z)dz (28) In Laplace's investigation a is supposed to be unity. We may call the value which (28) then assumes Ko, so that as above Ko =27fo ¢(z)dz (29) The expression for the superficial tension is most readily found with the aid of the See also:idea of superficial energy, introduced into the subject by See also:Gauss. Since the tension is constant, the work that must be done to extend the surface by one unit of area See also:measures the tension, and the work required for the See also:generation of any surface is the product of the tension and the area. From this See also:consideration we may derive Laplace's expression, as has been done by Dupre (Theorie mecanique de la chaleur, See also:Paris, 1869), and See also:Kelvin (" Capillary Attraction," Proc. See also:Roy. Inst., See also: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 See also: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 See also:time large enough to exceed the range of the attractive forces. The work required to produce this See also: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 See also:half the work that must be done per unit of area against the mutual attraction in See also: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 0-2 represent the densities of the two infinite solids, their mutual attraction at distance z is per unit of area 27rala2f :+G(z)dz, (30)In all cases to which it is necessary to have regard the integrated terms vanish at both limits, and we may write fu I, (z)dz=sf y z3¢(z)dz, fa z4(z)dz=ifo z44(z)dz; (46) Ko = 3 f o z3~(z)dz, To = $ f o z44'(z)dz (37) A few examples of these formulae will promote an intelligent comprehension of the subject. One of the simplest suppositions open to us is that
¢(.l) =e t3f (38)
From this we obtain
11(z)=0-1e 0z, 'G(z)=t-3(ISz+1)e tsz, . . ~
Ko =47rf3 , To =37r$ (40
The range of the attractive force is mathematically infinite, but practically of the order 15-', and we see that T is of higher order in this small quantity than K. That K is in all cases of the See also:fourth order and T of the fifth order in the range of the forces is obvious from (37) without integration.
An apparently See also:simple example would be to suppose 4,(z) =z". We get
zn+1
n+4.n+3.n+Il o
The intrinsic pressure will thus• be infinite whatever n may be. If n+4 be positive, the attraction of infinitely distant parts contributes to the result; while if n+4 be negative, the parts in immediate contiguity See also:act with infinite See also:power. For the transition case, discussed by See also: T=*aK; whereas according to the above calculation T = AaK. The discrepancy seems to depend upon Young having treated the attractive force as operative in one direction only. For further calculations on Laplace's principles, see See also:Rayleigh, Phil. Mag., Oct. Dec. 189o, or Scientific Papers, vol. iii. P. 397.] ON SURFACE-TENSION See also: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 See also:Laws of Surface-Tension.—1. For any given liquid surface, as the surface which separates water from See also: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 See also: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 See also:concave, and a slight diminution where it is See also: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 See also:horizon. 2. The surface-tension diminishes as the temperature rises, and when the temperature reaches that of the See also:critical point at which the distinction between the liquid and its vapour ceases, it has been observed by See also:Andrews that the capillary action also vanishes. The See also: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, there-fore, assuming the surface-tension to vary as the square of the or 27raar28(Z), if we write f Z ¢(z)dz=o(z) (31) The work required to produce the separation in question is thus 27ra,a fo 9(z)dz; (32) and for the tension of a liquid of density a we have T=7raJ o B(z)dz (33) The form of this expression may be modified by integration by parts. For fO(z)dz =0(z).z—fzded - dz =0(z).z+fz,G(z)dz. Since 0(o) is finite, proportional to K, the integrated term vanishes at both limits, and we have simply fo 0(z)dz=f:zkz)dz, (34) T=7ra?fo z¢(z)dz (35) In Laplace's notation the second member of (34), multiplied by 27r, is represented by H. As Laplace has shown, the values for K and T may also be ex-pressed in terms of the function di, with which we started. Integrating by parts, we get l See also:f1G(z)dz = z,G(z) +ae3II (z) +-y fz34(z)dz, fz'G(z)dz = 2z2,G(z) -kiz4II (z) + a fe44(z)dz. and so that zn+3 Ko = . . . . (41) (42) (43) (44) density, they deduced its See also:variations from the observed See also:dilatation of the liquid by heat. This See also:assumption, however, does not appear to be verified by the experiments of See also:Brunner and See also:Wolff on a rise of water in tubes at different temperatures. 3. The tension of the surface separating two liquids which do not mix cannot be deduced by any known method from the tensions of the surfaces of the liquids when separately in contact with air. When the surface is curved, the effect of the surface-tension is to make the pressure on the concave side exceed the pressure on the convex side by T (1/Rl +1/R2), where T is the intensity of the surface-tension and RI, R2 are the radii of curvature of any two sections normal to the surface and to each other. If three fluids which do not mix are in contact with each other, the three surfaces of separation meet in a line, straight or curved. Let 0 (fig. 3) be a point in this line, and let the plane of the paper be supposed to be normal to the line at the point O. The three angles between the tangent planes b to the three surfaces of separation at the point 0 are completely determined by the tensions of the three surfaces. For if in the triangle See also:abc the side ab is taken so as to represent on a given See also:scale the tension of the surface of contact of the fluids a and b, and if 4 the other sides be and ca are taken If four fluids, a, b, c, d, meet in a point 0, and if a See also:tetrahedron See also: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 See also:face ABC is normal to the tangent at 0 to the line of concourse of the fluids abc, and turn it so that the edge AB is normal to the tangent plane at 0 to the surface of contact of the fluids a and b, then the other three faces of the tetrahedron will be normal to the tangents at 0 to the other three lines of concourse of the liquids, and the other five edges of the tetrahedron will be normal to the tangent planes at 0 to the other five surfaces of contact. If six films of the same liquid meet in a point the corresponding tetrahedron is a See also:regular tetrahedron, and each film, where it meets the others, has an angle whose cosine is—i. Hence if we take two nets of See also:wire with hexagonal meshes, and place one ,on the other so that the point of concourse of three hexagons of one See also: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 See also: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 See also:lens, the angles which the upper and See also: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 See also:fat floating on the surface of hotwater, soup or See also: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 See also:train oil will spread itself over the surface of the See also:sea till it shows the See also:colours of thin plates. These rapidly descend in See also: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 See also: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 See also: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 See also:series of experiments by See also: Hence if the angle ROQ (fig. 4) at which the surface of contact OP meets the solid is denoted by a, Tb,—T,a—Tab cos a=o, Whence cos a=(Tba—T,a)/Tab. As an experiment on the angle of contact only gives Flo. 4. us the difference of the surface-tensions at the solid surface, we cannot determine their actual value. It is theoretic-ally 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 See also:mercury and glass is about 1400. Quincke makes it 128° 52'. If the tension of the surface between the solid and one of the fluids exceeds the sum of the other two tensions, the point of contact will not be in equilibrium, but will be dragged towards the side on which the tension is greatest. If the quantity of the first fluid is small it will stand in a drop on the surface of the solid without wetting it. If the quantity of the second fluid is small it will spread itself over the surface and wet the solid. The angle of contact of the first fluid is 18o° and that of the second is zero. If a drop of See also:alcohol be made to See also:touch one side of a drop of oil on a glass See also:plate, the alcohol will appear to See also:chase the oil over the plate, and if a drop of water and a drop of bisulphide of See also:carbon be placed in contact in a See also:horizontal capillary See also: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 inter-face 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 See also:T12 denote the interfacial tension, the energy corresponding to unit of area of the interface a Q R is also T12, as we see by considering the introduction (through a fine tube) of one body into the interior of the other. A comparison with another method of generating the interface, similar to that previously employed when but one body was in question, will now allow us to evaluate T12. The work required to cleave asunder the parts of the first fluid which lie on the two sides of an ideal plane passing through the interior, is per unit of area 2T1, and the free surface produced is two See also:units in area. So for the second fluid the corresponding work is 2T2. This having been effected, let us now suppose that each of the units of area of free surface of fluid (1) is allowed to approach normally a unit area of (2) until contact is established. In this process work is gained which we may denote by 4T'12, 2T'12 for each pair. On the whole, then, the work expended in producing two units of interface is 2T1+2T2-4T'12, and this, as we have seen, may be equated to 2T12. Additional information and CommentsThere are no comments yet for this article.
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