Bernal sphere

If large-scale space habitats for humans are ever built, it is likely that they will be based at least somewhat on designs such as the O NeiU cylinder and the Bernal sphere (similar to ships featured in the 1968 film 2001 A Space Odyssey and 1994-1998 television series Bafiyfow 5, respectively). The designs for these habitats promise living space for thousands of people and enough resources to have an agricultural industry in space. 

Figure 9.24 A radial distribution function (RDF) for a DRP of monospheres (right) and a scheme for its evaluation (left), Nmeenl4nR2 is an equivalent of IA(R), where A/mean is the mean number of spheres in the intervals of 0.2R. The solid curve illustrates the data obtained from neutron diffraction in liquid argon 1,2 are the experimental data by Scott [128] and Bernal [127] obtained for the models of steel spheres (cited in [127]).

Fig. 12.14 Stereoviews <4 the inner coordination spheres about the central ruthenium atom in the Orange (top) and violet (bottom) isomers of [(QH,> J,RiiftCF,bCiS2 -(CD). [From Bernal, I. Clearfield. A Rica. 1. S.. Jr J. Cryst. Mvt. Struct. 1974.4, 43-54. Reproduced with perrmssion.J...

If elements A and B are completely miscible in all compositions in their liquid state, it is reasonable to assume nearly equal interactions among the three possible cases A-A, B-B and A-B in their liquid state. Let us assume the atomic radii of A and B are about equal to one another (i.e., RA Rb)- With this assumption, the system is essentially a hard sphere or a Bernal model [9]. In the absence of gravitational or other external effects, the liquid solution will reach a dynamic equilibrium where the system is random-dense-packed and homogeneous throughout. Under these conditions we ask the following question pertaining to the atomic configuration. 

The theoretical model proposed herein for describing the liquid state has two parts, Pab (P) and exp[(At/Atc) -l]. The first part is based on the hard-sphere or Bernal s [9]... 

In 1954 Weiss32 used Bernal and Fowler s simplified solvation model,16 with an Inner Sphere of ionic coordination, i.e., a small spherical double layer around the ion of charge ze, followed by a sharp discontinuity at radius q, the edge of the Outer Sphere or Dielectric Continuum. He used a simple electrostatic argument to determine the energy to remove an electron at optical frequency from the Inner Sphere ... 

Since Bernal and Fowler,16 the charging radius r0 in the Born equation has been put equal to the Inner Sphere radius, or approximately the ion to water molecule center distance plus 1.4 A. At least for 1+ ions, this gives a fairly good approximation to the Gibbs energy of interaction of the ion with the outer Dielectric Continuum if aT and s are constant throughout the medium. High-valency ions are discussed in Section IV. 

We therefore arrive at Eq. (53), the classical expression used by Bernal and Fowler16 for the inner sphere, with the addition of a small additional term, which may be regarded as a +TAS term for the breakup of the solvent structure, which can be replaced by an estimate of the energy to break the structure of water. 

Bernal and Fowler postulated a fourfold tetrahedral corordination about each water molecule (Fig. 5. VIIIC). The hydrogens (which practically lie inside the large oxygen spheres) are centres of positive charge, the other two apices... 

We consider first the simulation of the atomic structure of vitreous silica because the majority of the simulations of amorphous oxides were done for this material. Some of these have simulated the formation of the vitreous silica surface in a very detailed fashion. Furthermore, the methods developed for the simulation of vitreous silica and its surface may be used with some modifications for other amorphous oxides. Subsequently, we consider less detailed methods of simulation of amorphous oxide surfaces which are not limited to Si02 but can be applied to various oxides. Finally the least detailed but the most general model - the Bernal surface (BS) - represents the atomic arrangement at the surface of any amorphous oxide (most important for physical adsorption) by the dense random packing of hard spheres. 

An even more general and correspondingly less detailed atomic model of amorphous oxide surfaces has been called the Bernal surface (BS)[3, 21]. It is based upon the fact that many oxides and halides can be regarded as close-packed arrays of large anions with much smaller cations occupying interstitial (usually tetrahedral or octahedral) positions (see., e.g. Ref. [4]). In line with this point of view, the BS is a surface of a collection of dense randomly packed hard spheres, a sphere representing an oxide anion. The cations in interstitial positions between hard spheres are excluded from the simulation since they do not attract adsorbed molecules due to their small polarizability. Thus only the atomic structure of the oxide ions is considered. This is called the Bernal structure and has been used for modelling simple liquids and amorphous metals [15]. 

The computer simulation of the Bernal atomic structure with a flat (on average) surface was carried out with the help of algorithm described in Refs. [3, 22]. The coordination number of hard spheres in this structure is about 8 which may be compared with the 0-0... 

Whereas Bernal had to rely primarily on mechanically produced random sphere packings in his work on liquid structure, the molecular dynamics and Monte Carlo computer simulation techniques have, during the past three decades, provided researchers with new and powerful experimental tools enabling a much closer look into the structure of the liquid—one has available the trajectory of every atom. Despite this, computer simulation has been used principally to calculate liquid-state correlation functions. This situation, to quote Lumsden and Wilson [6], ... appears to stem in part from a peculiar and fundamental relation that has always existed between experiment and theory in science the importance of experimental data is judged by the theory to which it is applied. As the physicist Arthur Eddington said half seriously, no fact should be accepted as true until it has been confirmed by theory. Unless an attractive theory exists that decrees certain kinds of information to be important, few scientists will set out to acquire the information. Thus, it is only infrequently that computer simulations have been used to characterize liquid structure in ways other than those dictated by the prevailing liquid theory. 

Our polygon description of the structure of 2D dense random packings of hard disks parallels Bernal s description of three-dimensional (3D) dense random packings of hard spheres as space-filling arrays of elementary polyhedral units ( Bernal holes, or canonical polyhedra ) [2-5]. Bernal s approach to 3D liquid structure is discussed in more detail in Section IV.A. 

Although Bernal found no evidence of crystalline regions in DRPs, he did find evidence of so-called pseudonuclei, which are exceptionally dense local arrangements of spheres consisting of helical chains of... 

In studying the structure of liquids, the model of dense random packing (DRP-model) suggested by Bernal [6.7] played an important role. In this model determined by the algorithm of the structure formation from spheres interacting according to some law), the correlations in the mutual location of atoms rapidly decrease with distance and the cybotaxis or crystallite orderings are absent. 

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