Random Close-Packed Hard-Sphere ModelI. Effect of Introducing HolesBY J. D. BERNAL AND S . V. KINGBirkbeck College, University of London, Malet Street,London, W.C. 1.Received 19th December, 1966The structures have been examined of lower density random models, obtained by removing atomsfrom the close-packed, random-model built here. Radial distribution functions and coordinationdistributions for models containing different proportions of holes have been calculated.Radial distribution functions calculated from X-ray or neutron diffraction patternsof liquids show that the effect of temperature is less to increase the interatomicdistances between atoms than to reduce the number of neighbours of each atom.That is, the position of the first peak in the radial distribution function changes littlebut the total area of the peak which is proportional to the number of first neighboursdecreases.In the random close-packed sphere model of a liquid the increase in volumecorresponding to increase in temperature 1 could be effected in two ways.First, theproportions of the basic polyhedra could be changed, i.e., there would be an increasein the proportions of the larger polyhedra so that there would be more archimedeanantiprisms and tetragonal dodecahedra and fewer of the smaller polyhedra, theoctahedra and tetrahedra. Secondly, atomic-sized holes could be introduced intothe body of the liquid.By using the first method alone it is only possible to achieve a maximum increasein volume of about 17 %. It is of interest therefore to investigate the effects of intro-ducing atomic sized holes at random into the hard sphere model (or rather to removeatoms at random from the model).Atoms were removed randomly from the model.Initially, the removal of atomswas restricted so that no two holes could be neighbours, a " neighbour " being definedto be within J2 diameters of another atom. There will be a limit to the number ofholes which can be introduced in this way. A preliminary calculation was carried outto ascertain the maximum possible number of holes which could be removed fromthe model so that no two holes were neighbours.This was done as follows: the central atom in the mass was removed, then,looking at the atoms in order of their distance from the centre of the mass, each atomwas tested if it was a neighbour of a previously chosen hole. The atom remainedin the assembly if it was not a neighbour of any previously chosen hole ; then it wasremoved from the assembly.This process continued until all atoms in the assemblyhad been visited. The maximum number of holes which could be introduced in thisway was 18-6 % (a neighbour being defined as within 1.414), roughly correspondingto an increase in volume of 18.6 %. If a neighbour was defined as being within 1-1atomic diameters (position of potential = 0 for Lennard-Jones potential) the number6J . D . BERNAL AND S . V . KING 61of holes which could be introduced was 26 %. For a neighbour defined as within1.05 diameters the number of holes which could be introduced was 28-3 %.Two sets of calculations were carried out on introducing holes at random into thehard sphere assembly : (A) atoms were removed only if they were not neighbours ofdistance from centre in atomic diametersFIG.1.-Radial distribution functions. Random model with no holes and 35 % holes.coordination number (for neighbours within 1-1 diam.)FIG. 2.-Coordination distribution functions. Random model with no holes and 35 % holes.holes already introduced (i.e.? within 1.414); (B) atoms were removed at random,holes could be neighbours. In the first (A) set of calculations after about 16.6 %of holes had been introduced it was impossible to introduce any further holes unlessholes were allowed to be neighbours.In both sets of calculations, holes were introduced in increasing proportions of5 %.First 5 % holes were introduced then 10, 15, 20, 25, 30, 35,40 and finally 45, 62 RANDOM CLOSE-PACKED HARD-SPHERE MODELthis being well above the critical volume. In the first (A) set of calculations after thesaturation point of 16.6 % had been reached all further holes were allowed to beneighbours. For each of these new assemblies the radial distribution function andcoordination distribution function was calculated.In general outline, the effect of introducing holes into a random close-packedhard-sphere assembly is analogous to the effect of temperature on a real liquid. Thepeaks of the radial distribution functions become lower and broader (fig. 1). In thecoordination distribution functions the mean of the curve moves to lower values ofcoordination, and the coordination distribution function itself broadens and becomeslower. This is clearly seen by comparing the coordination distribution histogramobtained from the random model with no holes with that for the random model with35 % holes. The mode falls from 9 in the first case to 6 in the latter (fig. 2)