Random close-packed state

The assumption of a symmetric potential-well may not be too far off from the physical reality in that in a random-close-packed liquid state, atoms would tend to slid-by one another tangentially, instead of colliding head-on. 

What we have just defined, in a very simplistic maimer, is something called free volume. This is not the same as the empty or unoccupied volume found in the close-packed state (Figure 10-55). Consider the stacking of solid spheres. If we pack these in a regular or ordered fashion there will always be some gaps between the balls. If we allow these spheres to pack randomly, like peas that you throw into a colander to drain, there will be even more of this unoccupied volume, but the spheres will still be close packed, jammed up against their neighbors. However, molecules are not static, like macroscopic balls. Above the absolute zero of temperature they have motion, but in the close-packed state this will just involve vibrations around a mean position. 

The slight polydispersity in particle size allows the system to avoid the crystalline phase and reach the metastable glass state. Above ( ) = 0.58, the system is metastable with polydispersity, the random close-packing volume fraction shifts to higher values. 

Point 7 has very special properties. It occurs at random close-packing density and provides a definition of it. Point / is an isostatic point, where the number of particle contacts equals the number of force balance equations to describe them. As a result, it is a purely geometrical point the properties of the state at point / are independent of potential. Also, point / appears to be a zero-temperature mixed phase transition, with a discontinuity in the number of contacts, characteristic of first-order phase transitions, and diverging length scales, characteristic of second-order phase transitions. 

J.L. Finney, Random Packings and the Structure of Simple Liquids. I The Geometry of Random Close Packing, Proc. Roy. Soc., 319A (1970) 479-494 also, J.L. Finney, Random Packings and the Structure of the Liquid State, PhD Thesis, University of London, 1968. 

The liquid state of ionic compounds seems also to be rather simply analyzed. The change from crystal to liquid is much less dramatic in ionic solids than in covalent solids, since the ionic crystal is already close-packed. The liquid is presumably a somewhat random, but locally neutral, conglomeration of ions. Most of the studies of ionic compounds made in this part of the book have not depended greatly on details of structure and so remain appropriate in the liquid. It should be noted, however, that for studies of transport and diffusion, one should... 

We have already pointed out that the most stable forms of the solid state bonding of elemental boron and metals differ in an essential aspect. Hence, in the solidification of a melt containing a random mixture of metal and boron atoms the observed structure will be determined by a balance between the tendencies for boron to form a covalently bound network and the metal to form a close-packed lattice. Among other things, this competition will depend on relative metal and boron concentrations and one expects in proceeding from the metal-rich to the boron-rich borides that the B-B bonded network will become more extensive and dominant. 

Due to the need to use case-by-case analysis the Kister studies [136, 137] focused on item 1. The data evaluated came from published reports by Fractionation Research (FRI) and Separation Research Program (SRP) at the University of Texas, taken from commercial size equipment rather than laboratory research columns. The FRI data includes No. 2 and No. 2.5 Nutter random rings packing, aind Norton s Intalox 2T structured packing, each considered currently state-of-the-art or close to it, while the sieve and valve trays were of FRI s latest designs, plus Nutter s proprietary valve trays, all using 24-in. tray spacing. 

Crystallization Temperature - Temperature (or range of temperatures) at which polytetrafluoroethylene crystallizes. PTFE chains which were randomly distributed in the molten or gel state become aligned into a close-packed ordered arrangement during the crystallization process. 

The previous section showed how the van der Waals equation was extended to binary mixtures. However, much of the early theoretical treatment of binary mixtures ignored equation-of-state effects (i.e. the contributions of the expansion beyond the volume of a close-packed liquid) and implicitly avoided the distinction between constant pressure and constant volume by putting the molecules, assumed to be equal in size, into a kind of pseudo-lattice. Figure A2.514 shows schematically an equimolar mixture of A and B, at a high temperature where the distribution is essentially random, and at a low temperature where the mixture has separated into two virtually one-component phases. 

The best way to picture the solid state is in terms of closely packed, highly ordered particles in contrast to the widely spaced, randomly arranged particles of a gas. The liquid state lies in between, but its properties indicate that it much more closely resembles the solid than the gaseous state. It is useful to picture a liquid in terms of particles that are generally quite close together. 

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