Trying to find out more about the increase in pressure on water droplets as their size decrease I googled ["surface tension" parachor] and stumbled on a rather interesting review of Einstein's first paper. The bit relating to the Parachor comes at the end.
======================================================== http://www.journals.royalsoc.ac.uk/(a0ntxpzxcdx5vcaqwtrf hdvs)/app/home/content.asp?referrer=contribution&format= 3&page=1&pagecount=6 -------------------------------------------------------- CONNECTION TO THE PARACHOR The paper is very difficult to understand, not least because of the large number of obvious misprints; from its lack of clarity we can only assume that it had not been independently refereed. We could well imagine Ostwald depositing it in the bin when the paper was received from Einsteins father. Yet it was an extraordinarily advanced paper for a recent graduate who was receiving no independent scientific advice. The idea behind a stoichiometric analysis of surface tension goes back to the work of R. Schiff, and this was well discussed in Ostwalds book, but to suppose that one could obtain some information about inter- molecular potentials by such an analysis is probably Einsteins own idea. It is interesting that the stoichiometric analysis of surface tension and its interpretation through molecular structure became very popular from the work on the parachor, a quantity introduced by Sugden[7] and others in the 1920s. The parachor is equal to the fourth root of the surface tension 94 John N. Murrell and Nicole Grobert divided by the difference between the liquid and gas densities, and multiplied by the molecular mass, and is a quantity that is largely independent of temperature. The parachor was subjected to a large number of analyses and interpretations in subsequent years, but the subject died a rapid death in the 1940s. ========================================================= (Mmm...They dropped the baton in favour of the "nuculus"). Now these fourth power laws keep cropping up. So far we have Temperature, Casimir, Vapour Pressure and now the Parachor. We also have a rather more subtle one, Jerk, if Jerk is looked at not from the velocity datum but from the more fundamental static datum, displacement. So what's with all these fourth power laws? Where do they all come from? The big hint is given by the fact that they turn up in the context of surface tension. Surfaces involve a square law. So two surfaces involve a fourth power law. But what are these two surfaces? They are the Solid Phase and the Fluid Phase. One is expanding (negative strain energy) whilst the other is contracting (positive strain energy). In effect, the Fluid Phase is the datum for measuring the change in the Solid Phase. It's a bit like the Tardis. If you're a Doctor Who fan you'll soon get the idea. Fortunately, I tackled the problem of the Nature of Spaces many years ago in: NOTE NO IN 55/70 A HIERARCHICAL DIPHASE MODEL OF MATERIAL BEHAVIOUR so I was able to recognise the answer when I saw it. For the benefit of Vortexians who don't have access the Files Section of the Beta-atmosphere Group I have copied the relevant excerpt below. ======================================================= MULTIPLICITY OF SPACES Just as there is a multiplicity of hierarchical fluids or particle fields, so there is a multiplicity of corresponding spaces. Any real object, since it is a hierarchical body, exists in a multitude of different spaces. Consequently it can be moving at one speed in a particular direction in one hierarchical space and quite another speed in the opposite direction in a complementary hierarchical space . A good example of this is provided by a cork floating on water. The two complementary spaces in which the cork exists are the air and the water. These spaces are particularly useful for explanatory purposes because they are separated and not intermingled as are hierarchical spaces in general. Usually the two hierarchical spaces, the air and the water, will be moving relative to one another. If we use one space as a frame of reference, the water say, the cork will be moving at a speed Vw in a direction theta deg. through that space. If on the other hand we use the air as a frame of reference then the cork will be moving at a speed v[a] in a direction (180 + theta)deg through the air space. The numerical sum of the relative speeds in the air and water spaces will equal the speed of the two spaces relative to each other. The movement of the cork through the two spaces in opposite directions will set up two drag forces, fw in direction (180 + theta deg.) in the water space and fa acting in direction theta deg. in the air space. The ratio va/vw will be governed by the requirement fw = fa. In the above example two particular hierarchical spaces were considered and these spaces were complementary, i.e. they were taken as comprising the totality of spaces in which the object existed. The spaces were defined as air and water space for ease of explanation but they could have been defined more precisely as water and not water space; this definition makes the totality of the space more apparent. The general principles which apply to two complementary hierarchical spaces can be extended to n complementary hierarchical spaces which can be separate as in the case of air and water or co-extensive as in the case of air and a cloud of gnats, say. In general a body will be moving at a different speed and in a different direction in every hierarchical space in which it exists. The relative speeds will be those for which the algebraic sum of the drag vectors is zero. At this point it is pertinent to ask the question, what is the relationship between a hierarchical space and Cartesian space? This question is best answered by drawing a comparison between Cartesian space and the interval between zero and one. In a unit of Cartesian space the number of points is unlimited. In a hierarchical space however the number of points is limited and can be specified numerically. Thus a particular hierarchical space corresponds to a particular class of fractions, 1/1000ths say. A different hierarchical space will correspond to a different set of fractions, 1/999 ths say. It can be seen therefore that we can carve an unlimited number of hierarchical spaces out of Cartesian space. The objection to the Cartesian concept of continuous space is that it involves unlimited numbers and these cannot be specified numerically nor grasped conceptually whereas the concept of hierarchical space can because of its discontinuous nature. This problem of continuity has occurred many times before. The classic example is of course the conflict between the wave and particle behaviour of electromagnetic radiation; a more revealing but less well known example is the one given by communication theory where it is only by treating a varying signal as a discontinuous process that problems of information content and coding become tractable(6) (12). EXTERNAL AND INTERNAL SPACES For every identifiable level of an object there will be an external and an internal hierarchical space. The internal space comprises the external space of the object at the next identifiable level down. The existence of the object at any level is the manifestation of the difference between the external and the internal space. For a hierarchical space where the difference between these two spaces is negligible, the existence of this space may be neglected in considering the properties of the object. If existence at any level is defined as the manifestation of a difference between the object and its environment at that level then for hierarchies where the difference between the internal and the external environments are negligibly small the existence in these hierarchies can in practice be neglected. The surface of an object is the boundary between its internal and external environment. Where there is a sharp discontinuity between internal and external environment there will be a sharp boundary; where the discontinuity between internal and external environment is more gradual the boundary surface will be correspondingly diffuse. Since an object exists in a hierarchical set of spaces it should have a corresponding hierarchical set of surfaces. It will be readily recognised that this is indeed so. Moreover, an object is only a single object at its highest hierarchical level of existence. At lower hierarchical levels it is a collection of objects. This raises the problem of the one and the many, the simple and the complex which will be dealt with in a later section. Just as an object is in equilibrium in relation to all the forces generated by existence in and movement through the external hierarchical spaces, so an element of the surface is in equilibrium between the external and internal environments. HIERARCHICAL SURFACES Consideration of internal and external spaces has focussed attention on the fact that a body has many different hierarchical surfaces. We normally consider the surface of an object to be the surface detectable by sight. In general this surface will also coincide with the surface detectable by touching the object with another object but many exceptions to this rule occur. An obvious example is that of a plate glass door under the right lighting conditions. A less obvious example is a pair of magnets the like poles of which are brought together; in this case the magnetic surface is exterior and much less sharp than the visual surface and we do not recognise its surface characteristic. Recognition of the existence of different hierarchical surfaces leads to the realisation that all repulsive forces between objects are manifestations of the deformation of a hierarchical surface. In other words, repulsive forces commence when the two hierarchical surfaces touch. This is shown in Fig 3 in relation to the Condon-Morse curve by dividing the force distance diagram into three separate regions. When the atoms are separated then the distance of the atomic surface from the atomic centre is d2. When the atoms are at their lowest potential energy position, i.e. total force F = 0 then the distance of the surface from the atomic centre is d1. The zone between d1 and d2 constitutes the surface zone, in other words the surface. This example draws attention to the importance of time and volume in relation to the concept of surface. The importance of time can be illustrated by a simple example. Suppose we have an aeroplane propeller rotating at high speed. If a goose flies into the propeller it will soon find that the surface of the propeller is defined by the swept volume of the static propeller. We may describe this surface as the dynamic propeller surface (an interesting example of dynamic volumes of this type is given by the Bènard cells formed by convection currents in a thin layer of fluid uniformly heated; see Fig 4). However for a bullet fired at the propeller the surface of the propeller is essentially the same as the surface of the static propeller. It can be seen that just as different hierarchical surfaces of an object become manifest as the spacial scale of scrutiny is altered so also different hierarchical surfaces become manifest as the temporal scale of scrutiny is altered. The importance of volume in relation to the concept of surface is that it gives it reality. A surface must not be though of as a geometric abstraction, as a two dimensional concept with no physical reality, but as a real entity, dependent on the antecedent existence of the object and the environment but quite distinct from them. Examples of this interaction term can be given for every system. The child is an example of the interaction term in the sociological field. It is dependent on the antecedent existence of its parents but it is quite distinct from them. In a biological context the surface or interaction term constitutes a skin or membrane which is a physical entity in its own right having a definite though usually small thickness. In terms of political geography the surface or interaction term comprises a border zone which will generally be found to constitute a small but distinct region in comparison with the two parent countries. In communications the surface term corresponds to the communication channel between the sender and the receiver, a physical entity distinct from either the sender or the receiver. In sentence structure the surface term corresponds to the verb which relates the subject to the object. In mathematics the surface or interaction term is the combinatorial sign between the two numbers. This combinatorial sign has a conceptual reality equal to the conceptual reality of the number concepts. ======================================================= You see how easy the answer can be when you adopt a top down strategy rather that a bottom up strategy. 8-) The whole is more than the sum of its parts. Cheers, Frank
