62 RANDOM CLOSE-PACKED HARD-SPHERE MODEL11. Geometry of Random Packing of Hard SpheresBY J. D. BERNAL AND J. L. FINNEYA preliminary Voronoi polyhedron analysis of a random model has enabled an accurate fixing ofthe density of random close-packing. A wide variation (-15 % of the density of closest regularpacking) in local density is found, correlated with various topological types of polyhedra. The sus-pected predominance of 5-sided faces is verified, while the average number of faces per polyhedron,and the average number of sides per face, are a few per cent greater than the theoretical predictions ofCoxeter. Much data remain still to be analyzed for both physical and computer models.The radial distribution function of a random close-packed array of hard spheresshows a striking similarity with simple liquids, as does the difference in densitybetween this packing and the corresponding crysta1.Z 3 Ball-bearing models havebeen studied by Bernal 1 and Scott,49 5 and the synthesis of such models attemptedusing computers.Complete descriptions in terms of the coordinates of the packedcentres have been obtained, providing data for a thorough examination of the arrays.Among other reasons, the random packing is of special interest because of itssimilarities with the liquid state, and the geometry of the arrangement itself. Amanageable overall description is necessary if the thermodynamics of such an array isto be computed, though some progress (e.g., in the heat of fusion) can be made usingcomputer computations directly on the coordinates, and further advances may bepossible by exploring Monte Carlo techniques.The development of some kind of“ statistical geometry ” appears to be one of the most promising lines of attack.From our first measured ball-bearing array, a large ball and spoke model wasconstructed. This showed many interesting features.1 For example, consideringvectors between neighbours and near neighbours, the whole model was seen to consistof only five different kinds of polyhedra (“ canonical holes ”) as the basic units,suggesting that perhaps these might be treated as the building bricks of a statisticalgeometry. Our present work, however, concentrates on describing the model as anetwork of Voronoi polyhedra. If we take an array of points, and perpendicularlybisect the vectors between them, we obtain a large number of intersecting planes.To define the Voronoi polyhedron of a point, we select the smallest polyhedron SOformed about that point, ensuring no further possible planes can cut this chosen set.This polyhedron contains all those points closer to the chosen centre than to any other :hence, a network of such polyhedra completely fills space.The construction of thetwo dimensional “Voronoi polygon” is shown in fig. 3. Now each polyhedroncontains information sufficient to describe completely the neighbourhood of a point J. D . BERNAL AND J . L. FINNEY 63we are trying to develop some statistical theory to describe an extended random modelin these terms.In a pilot experiment,l a number of plasticene spheres were compressed together,and the distribution of faces on each resulting polyhedron recorded.More recently,we have developed a computer programme to elucidate the polyhedra exactly usingthe coordinates of a measured hard-sphere array. Before we could use the data ofScott,5, 6 it was necessary to correct the coordinates of a few centres apparentlyseparated by much less than a sphere diameter. This was done by building a largescale ball-and-spoke model-which incidentally exhibited the same structural featuresof our original model ; no rigorous structural analysis on the basis of the five canonicalholes has, however, been attempted. All the errors thus found were simply ones oftranscription. This large model also proved a help in choosing those centres whoseVoronoi polyhedra would not intersect the outer boundary : in this way 407 of the1006 centres were chosen.FIG.3.-The construction of a Voronoi polygon.The Voronoi computer programme starts by calculatingpolyhedron. As every vertex is equidistant from four centres,the vertices of eachthis basically reducesto solving four siinultaneous equations. All possible combinations of four centreswithin a restricting combination radius rc are taken, and the resulting solution acceptedas a vertex if no other centre is closer than the initial four. The value of rc was fixedat 1.6 diameters by trial runs on a limited number of polyhedra : if this proved in-sufficient for any particular polyhedron (as it did in two cases where rc was con-sequently increased to 1.61d) an inconsistency would arise in the form of a missingvertex.The reference numbers of the four centres associated with each vertex arestored : this provides sufficient data for the second half of the programme to sort outthe topology. Each polyhedron takes about two seconds on the University ofLondon’s Atlas computer.The programme produces the following data for each polyhedron : (1) the centrenumber and its coordinates ; (2) the coordinates of the vertices, with their associatedcentres ; (3) for each face : (a) the number of edges, (b) the area of the face, (c) thevolume subtended by each face at the centre (d) the centre-face perpendicular distance,(e) the number of the centre with which the face is shared; (4) the total number offaces, total area, total volume, and percentage density ; (5) the number of faces with3, 4, .. . etc. edges. e.g., 03651000 denotes three quadrangular, six pentagonal, fivehexagonal, and one septagonal faces.Cumulative histograms are output of: (1) the number of M-sided faces ; (2) thenumber of N-faced polyhedra ; (3) polyhedron density64 RANDOM CLOSE-PACKED HARD-SPHERE MODELAll these data are stored on tape, and are easily accessible for further computationif required.SURVEY OF INITIAL RESULTSDENSITY OF RANDOM PACKINGScott 4 has measured the density directly, allowing for surface effects by extra-polation. We have previously attempted a measurement by calculating the actualvolume of material within spheres of increasing radius from the centre of the array :this also involved extrapolation as the oscillations had not quite died out when thesphere cut the mass boundary.In contrast, the Voronoi polyhedron provides, within the limits of the data, anexact measurement of average density, as well as the local variations.The resultsfor 407 polyhedra are shown in fig. 4, which is a histogram of polyhedron density(percentage occupied by the sphere) in intervals of 0.25 % : the large spread from57 to 70 % is obvious. The overall average is 63.42 %, which occurs at a minimum% densityFIG. 4.-Polyhedron densities.in the histogram. If we plot separate histograms for polyhedra with the same numberN of faces ( N = 12, 13, . . ., 19), we find, as expected, that the mean density p increasesas the number of faces N falls, with considerable overlap between them. The resultsof these separate histograms are plotted in fig.5. The lengths of the ordinates indicatestandard deviations from the mean (except for polyhedra with 18 and 19 faces),and the figure by each point is the number of such N-faced polyhedra. The averagefor the whole 407 centres is also plotted for comparison.Not only the average density, but the peaks also appear to have no obviousstructural significance (e.g., in terms of polyhedron type). The first implication is thatthe average density is a purely statistical figure. But this does not invalidate thepossibility of there being a statistical upper limit to the overall density of randompacking, as we cannot consider the array solely in terms of isdated polyhedra.Forexample, although some very high density polyhedra occur, we cannot necessarilyincrease the proportion of these at the expense of those of lower density, as thegeometry of the packing may not allow this. Thus, although there may be localregions of density greater than the average they may occur only in conjunction withbalancing polyhedra of lower density such as to keep the average density below J . D . BERNAL A N D J . L . FINNEY 65maximum value. A study of the packing of polyhedra around a central polyhedronis an obvious next step. We are also planning to examine the changing local densitypattern in packings of different average densities, and see how the distribution of+I201 i-Ii LI II 12 13 14 I\ Ik 17 I$ I91 1 I 1number of faces per polyhedronFIG.5.-Polyhedron density against number of faces.polyhedron types, as well as their sizes, changes. A detailed comparison of ourresults with the density curve of Kiang,7 obtained from a completely random fragmenta-tion of space, may also be useful, although our system is much more restricted.FACES AND EDGES OF POLYHEDRABoth the average number N of faces per polyhedron, and the average number @of edges per face may be important in the overall statistical picture. The distributionof M and N are shown in figs. 6 and 7, which are histograms of number of polyhedrawith a given number N of faces, and number of faces with a given number M ofedges respectively.The mean N of 14.28 is considerably higher than the 13.6 obtainedfrom the plasticene spheres 1 and the 13.56 from Coxeter’s theoretical model.8 Thehistogram of the number M of edges per face verifies the predominance of five-fold,and shows traces of eight-, nine-, and even ten-fold faces. is 5-160, higher thanCoxeter’s 5.1 15.8, 9These higher values immediately suggest two points. (a) In the plasticene model,very small faces are likely to be difficult to observe. Taking a sphere diameter about*”, and faces less than 1 mm2 in area to be unobserved, the Voronoi analysis of theScott model shows that about 130 faces come below this limit. for the plasticenemodel is thus increased to about 13-9. This factor does nothing to remove thediscrepancy with Coxeter’s theoretical model.(b) If the packing density were lowerthan the maximum limit, J7 would be greater than the Rmin expected at that limit.Moreover, as fig. 5 suggests, a decrease in the spread of the density histogram keepingthe same average would decrease N, and a change in polyhedron distribution of a more66 RANDOM CLOSE-PACKED HARD-SPHERE MODELgeneral nature could have a similar effect. However, with regard to changes indistribution, we must bear in mind the possible restrictions imposed by the geometrvof the packing, as discussed above.Analysis of different density packings will provide some idea of how NI (and hence a) vary with density : although it may be that several different values of NI may occur-I214012cI008 06 04020,I13 - 14number of facesFIG.6.-Histogram of number of faces per polyhedron.2 8 0 02400.2000.VJ1600.44lw t3 '200.d800.400.0' 3 b 1number of edgesFIG. 7.-Histogram showing distribution of number of edges per faceJ . D. BERNAL AND J . L . FINNEY 67for the same density, it may be possible to deduce a relationship between overalldensity and N. As the values of N and density for the Scott model fit reasonablywell on the N against density curve of fig. 5, an investigation of local regions ofdifferent densities may be of interest in this connection.TOPOLOGICAL CHARACTERISTICSIf the examination of some characteristics of a set of polyhedra showed theoccurrence of certain " types ", in the simplest case an array could be described interms of the frequency of occurrence M i of type Ai.This system, however, gives usno information about how different polyhedra are spatially arranged : an extensiontherefore is to formulate a matrix showing the frequency of occurrence nil of type Acnext to type Aj :A further extension adds a third dimension to the matrix, showing the type of faceBk by which Ai and Aj are joined. The general case occurs when all polyhedra are ofdifferent types, SO that nijk = 0 or 1.One possible characteristic by which similarity between polyhedra can be assessedis the topology. In this connection, it might be worth elucidating all the possibletopologically different types of polyhedra with N faces, and see whether any of thesetypes are specificially excluded or frequently occurring in a Voronoi network. Thistype elucidation has been done for N = 4 to 12 inclusive, after which the task of furtherextension for large N becomes unmanageable.If interesting results come out ofcomparing our 24 twelve-faced polyhedra with the possible 77 types, it may be worthconsidering further work along these lines.PRELIMINARY TOPOLOGICAL RESULTSA predominance of certain types is evident. Table 1 shows those polyhedrontypes with five or more examples present, together with the full plasticene sphereanalysis for comparison. 255 of the 407 polyhedra (63 %) are thus accounted for :the total number of plasticene polyhedra was 65. The number of exact correspond-ences between the two models, especially in the six and eight pentagonal face groups,is worth noting.And there are other near correspondences between polyhedra differ-ing only slightly, e.g., 04460 and 13451, where one quadrangular face has becometriangular, and one hexagonal has become septagonal ; similarly with 13532 and0454 1.What this representation fails to do is to allow for the differences in relativearrangements of faces; for example, there are five distinct polyhedra of type 03630(the 6 referring to the pentagonal faces). However, we now have a programme tosort out these sub-types. So far we have found that even where two polyhedra hav68 RANDOM CLOSE-PACKED HARD-SPHERE MODELdeintical arrangements of the same types of faces, they are not quantitatively similar.A case in point is polyhedra 1 and 322 in table 2, which have only four faces (1,9, 10, 13) corresponding vaguely in size and distance from the centre.Can two suchTABLE 1A B(Scott model) (plasticcne)1141 50( 1)4B04420(1)04440(1)04450(7) 0#50( 1)04460( 1 1) 04460(1)04470(5)1 343 1 (1)13441(2)1345 l(l0) 1345 l(3)13461 (2)6A B03620( 1)03630(?) 03630(5)03640(30) 03640(5)03650(17) 03650(3)03660( 17) 03660(5)12641 (5)04642( 1)8A B028 1 O(2)02820(6) 02820(3)02830( 13) 028 30( 3)02840( 15) 02840(4)02 8 50( 1 0)02860( 1)11831(8)10A B01”10”20(16) 01“10”20(4)01”10”30(11)01 ”1 0”40(2)A B30533 l(1)5A B13532(6)04541 (2)0453 1 (1 )12530( 1)7A B03 72 1 (3)0373 1 (1)0374 l(8)12722(7)12732(11)9A B10930( 1)02951 (1)12A B00” 1 2”00( 1 )00” 12”20( 1)01 ”1 2”00(5)polyhedra be considered to be similar for the purposes of our required geometricaldescription, and if not, how can the differences be represented? Much remains to bestudied merely with respect to this one analysis, which may throw further light onproblems such as these.FUTURE WORKThe polyhedron analysis of this and other models will be pursued, as variouslymentioned above, using in addition auxiliary tools such as computer programmes toproduce stereographic projections and stereoscopic pictures of the polyhedra. WJ .D . BERNAL AND J . L . FINNEY 69intend to measure a large (-3000 centres) random close-packed model on our co-ordinate measuring machine, which will give a much greater accuracy of measurementthan anything at present available : the increased size will give greater flexibility andstatistical reliability, and more scope for investigating local density fluctuations.Analyses of models of different densities may give some indication of how to deal withTABLE 2faceno.1234567891011121314no.ofedges75455664645555face area1 3225.84.60.34-33-65.04.90.95.00-53.74.24-20.55.72-72.73.31 *75.85-62-34-80.22.75.14.22.8distance betweencentre and face1 3221.581 -632.131-661.68.1-601.592.051 -592.171 -601.601-642.181.611.921 *891 *782.051.591.621 -921 6 02.281 -781-591.591-81temperature variations (such as by polyhedron dilatations or a change in the distribu-tion of types). The use of the Voronoi polyhedron as a basis for thermodynamiccalculations is to be investigated, and we are also intending an approach to the thermo-dynamics via Monte Carlo computer techniques. Along these lines we see thedevelopment of further consequences of a random-packed liquid, and thus hope toprovide more points for experimental comparison.1 J. D. Bernal, Proc. Roy. SOC. A , 1954, 280, 299.2 J. D. Bernal, Nature, 1960, 185, 68.3 J. D. Bernal and J. Mason, Nature, 1960, 188,910.4 G. D. Scott, Nature, 1960,188,908.5 G. D. Scott, Nature, 1962, 194,956.6 J. D. Bernal, J. Mason and K. R. Knight, Nature, 1962, 194,958.7T. Kiang, 2. Astrophysik, 1966, 64, 433.8 H. S. M. Coxeter, Introduction to Geometry, (John Wiley, 1961), p. 411.9 J. D. Bernal, in Liquids ; Structure, Properties, Solid Interactions, ed. T. J. Hughel. (ElsevierPub. Co. Amsterdam, 1965), p. 25