Capillary phenomenon

By a theorem of Euler, the sum (Ri -I-R ) is equal to the sum of the reciprocals of die radii of curvature of the surface along any two orthogonal tangents. 

Since K and H are positive, and R is positive for a convex surface, it follows from (1.13) that die internal pressure in a drop is h ier dian that in a liquid with a plane surface. Conversely the internal pressure widnn a liquid bounded by a concave spherical stuf ace is lower than that in die liquid with a plane surfoce since R is now ne tive. These results are the foundation of Laplace s theory of capfllarhy. Hie equation for the difference of pressure between p that of the liquid inside a spherical drop of radius R. and p , that the gas outside, is now called Laplace s equation  

It is quite general and not restricted to this molecular model. It is re-derived by purely thermodynamic arguments in Chapter 2 and used repeatedly in later chapters. 

It is interesting and useful to extend these results to a three-phase system of a solid in contact with a liquid and a vapour (again, of negligible density). The solid is of different chemical constitution from the liquid, and quite insoluble in it. We assume moreover, that it is a perfectly rigid molecularly uniform array of density pa. The intermolecular potential between two molecules of the species in the liquid is denoted Un, between two of those in the solid U22, and that between a molecule of each U 2- By the argument of the last section, the force per unit area between a slab of liquid and one of solid at separation I is 

This work is equal to the sum of the surface energies of the two new surfaces formed, liquid-gas and solid-gas, less that of the surface destroyed, liquid-solid  

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Modern experiment has proved beyond doubt that the so-called colloidal solutions are systems composed of two or more phases, i.c., heterogeneous, characterised by an enormously great extent of division, in which the surface of contact has, so to speak, been spread out throughout the whole mass. Capillary phenomena are therefore predominant here (cf. Ostwald, Kolloidchemic, Leipzig, 1909 Freundlich, Kapillarchemie, Leipzig, 1909). 

Studies of Wetting and Capillary Phenomena at Nanometer Scale with Scanning Polarization Force Microscopy... 

Capillary phenomena are due to the curvature of liquid surfaces. To maintain a curved surface, a force is needed. In Eq. (8), this force is related to the second term in the integral. The so-called Laplace pressure due to the force to maintain the curved surface can be expressed as, Pl = 2-yIr, where r is the curvature radius. The combination of the capillary effects and disjoining pressure can make a liquid film climb a wall. 

Nanometric Solid Deformation of Soft Materials in Capillary Phenomena... 

Capillary phenomena are also essential in tribology and in many biological systems, such as blood circulation and eye irrigation, involving the formation and persistence of the lachrymal film. 

Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

Haines, W. B. J. Agric. Science 17 (1927) 264. Studies in the physical properties of soils. IV. A further contribution to the theory of capillary phenomena in soil. 

As mentioned above, this approach treats each phase as a constituent to a mixture. Thus, all parameters are mixture parameters and must be averaged, usually by the saturation. Unlike the models mentioned at the end of the previous section, the models here use capillary phenomena. Furthermore, although the mixture moves at a mass-average velocity, interfacial drag between the phases and other conditions allow each separate phase velocity to be determined. The liquid-phase velocity is found by 9... 

In eq 51, the first term represents a convection term, and the second comes from a mass flux of water that can be broken down as flow due to capillary phenomena and flow due to interfacial drag between the phases. The velocity of the mixture is basically determined from Darcy s law using the properties of the mixture. The appearance of the mixture velocity is a big difference between this approach and the others, and it could be a reason the permeability is higher for simulations based on the multiphase mixture model. 

The liquid-solid or liquidj-solid-liquidz system is both a contact angle (Young s equation) and capillary phenomena (Laplace equation). These two parameters are... 

Yu.V. Naidich, N.F. Grigorenko and V.M. Perevertailo, Interphase and Capillary Phenomena in Crystal... 

Alvarellos, J. (2003) Fundamental Study of Capillary Phenomena in Porous Media. Ph.D. Dissertation, Georgia Tech, Atlanta, Georgia... 

Chambers, K.T. Radke, C.J. Capillary Phenomena in Foam Flow Through Porous Media in Interfadal Phenomena in Petroleum Recovery, Morrow, N.R. (Ed.), Dekker New York, 1991, pp. 191-256. 

The reasons for the difficulties encountered in the study of capillary phenomena in packings are not hard to find. We may cite the following ... 

The possibilities of the Poiseuille equation for explaining capillary phenomena in soils have had a certain amount of appeal to soil physicists. The theory of capillarity is so simple and so well established that if suitable soil parameters to express a bundle of capillaries could be found, the problem would be solved in the most direct fashion. Unfortunately, however, there has as yet been no completely satisfactory formula for the capillary bundle theory, although it has often given results of the proper magnitude in certain instances. We shall here develop the Poiseuille equation for special capillary conditions and point out the fallacies of this approach. 

Some contributions to the knowledge of capillary phenomena in connection with the heterogeneity of the soil. Soil Research, 1 239-301. 

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