W. C. Duer, J. R. Greensteln, G. B. Oglesby, and F. J. Miller0 The Rosentiel School of Marine and Atmospheric Science University of Miami Miami, Florida 33149
Qualitative Observations Concerning Packing Densities for Liquids, Solutions, and Random Assemblies of Spheres
Much discussion in the literature has heen devoted to the subject of the random packing of spheres in containers. This topic pervades many areas of science and engineering. For example, the subject relates to the packing of: (1) reactor beds in nuclear engineering (I); (2) columns in chemical engineering (2) and petroleum engineering (3); (3) assemblies having maximum theoretical density in mathematics (4); (4) mortar and concrete (5): . ., (5) . . ion-exchange resins (6): (6) various mixtures in metallurgy (7), ceramic engineering (8), and geology (9); and (7) atoms and molecules in the liquid state (in) The type of random packing which has been found most reproducible is that of random close packing (10) or dense random packing (11-13). A number of studies have been directed toward obtaining an understanding of the unbiased preparation (1,12) and the spatial description (14,15) of this system as well as more loosely randomly packed systems (13, 14). J. D. Bernal. in his earlv career (16) and in his later years (10,11,17-26)contributed significantly to the elucidation of a relationship between the packing of spheres and the liquid state. Some results for a number of workers, including Bernal, have been summarized (27) and indicate a correspondence between the radial distribution function peak positions for the liquid noble gases - and the dense randomly packed sphere asbem.hly. Finney (28)has extended Bernal's geometrical analyses and has arrived at the significant conclusion that the dense random packed assembly of spheres is a first-order approximation to the liquid state. However, Finney goes on to imply that this liquid model must await a theory of statistical geometry before its full predictive utility may he realized. Bernal (16) made a similar suggestion in 1937. Limited progress has been made alone these lines (29-32). significant studies have been carried out in order to relate the random packing of spheres to heats of fusion (22). entropies of fuiion(33), and the statistical thermodynamics of liquids (34,35). We shall discuss only those portions of these important investigations which have a bearing on packing density. The purpose of the present communication is twofold: (1) to discuss comparisons of packing densities derived from known molar volume data of liquids and solutions with packing densities which have, in the past, been observed for (2) to suggest assemblies of randomlv oacked s ~ h e r e sand ; further studies which should prove usefui in determinigthe utilitv of these assemblies of spheres as models for simple liquids and solutions. ~~~~~~
Variety of Names and Symbolsfor
Symbol Packing Density
Packing Densities for Spheres of One Size For random assemblies of soheres. the most freauentlv determined quantity has been ;he fraction of space dhich is filled: (number of soheres) (volume per sphere)/(volume of wntairkr). The variety of namesand .&b(hs which have heen used tor thisauantitv areciven in the table. Followinr! Bernal (21), we have EhosenL u&p, the packingdensity, to represent this quantity. Careful experimental studies (13, 28. 36), notably by Scott (12-14,37), and one theoretical study (29), have indicated that p = 0.637 f0.001 for the assembly of dense randomly packed spheres. If the monatomic elements in the liquid state are viewed as
Filled
Literature Cited
P
d
*
Pn P.
Y
~
\--I.
Fraction of Space
Drcp fP
Packing Fraction
@U
m I k Mean Particle v o l u m e Fraction
Cv
d
Mean Density
f
Solid Fraction Volume Packing Density Volume Fraction 1 V o i d Fraction Reduced Density
P" 95
1-e P
z Packingdensities, p, for the liquid elemenh at their melting temperatures, -us atomic number, I ( 0 , metals; m, noble gases: ', nonmetals: 0, metalloids).
assemblies of spheres, p is given by where N A is Avogadro's number, r is anatomic radius, v ~ ~ , , i d is the molar volume of the liquid, and Vo is (4/3)aNnr3. We have used eqn. (1) to c&ulate values of p for many of the elements in the liquid state at their melting temperatures For the metals andnonmetals, the and atmospheric V~i,,,jd values were obtained from tabulations of densities and atomic masses (38). The values of r were taken as one-half the observed minimum atomic separation in the elemental crystals. For the metals and nonmetals, the atomic sepgations as given by Weast (38) were used. The values of r and V~i,,,id for the noble gases were obtained from Moelwyn-Hughes (39). It is found that using a somewhat different set (40) of radii made no dramatic changes in the results. For illustrative purposes, the calculated values of p are plotted against atomic number. Z, in the fimre. Excluding Bi, ~ aU,, G:, Sn, Sh, He, and the nonmeta1s;the points form hand which encompasses the value of P = 0.637 for dense random packing ohtained from arrunl sphere packing studies. The median tor these elements is 0.63R; furthermnre. 90% of these values are within f0.04 of this median value. If 1%were
a
Volume 54. Number 3. March 1977 / 139
taken as an estimate for the standard deviation in the experimental r's and 1'1,,,,.,'a individually, an uncertainty, three standnrd deviations, in 0 romr~arableto 0.04 is obtained. The values of p for t h e excluded elements (mentioned above) are smaller than 0.637 by a t least 0.1. The nonmetal and metalloid behavior is understandable when it is recalled that these elements tend to exist as polyatomic moities. The low value of p for He is typical of its non-classicalproperties. In relation to these elements, Vold (41) found in a computer investigation concerned with sediments that cohesive forces between particles lead to very open packings and thus small oackine densities. ~ h e L e h a v i o rdepicted in the figure has, in the past, been mentioned (11, 42, 43) but not clearly demonstrated. This previous work has dealt primarily with the relative change in volume on fusion. The underlvina assumption of one study (42) relating to volume changes on fusion-has been that the radius of the atom changes during the phase change. I n his study on fusion, Miller (42), using p = 0.637 for the liquid state, a packing density for the solid based on unit cell type, and ohserved fractional chanees in volume on fusion. calculated the ratios for atomic r a g i in the liquid and solidstates. The ratios he found were in aualitative aereement with those ohtained from diffraction s'tudies on t i e liquid and solid states. Ry approximating the rndii by me-half thcdistance (44.43) corresoondine ru the first neak in the liauid radial distrihution functibn as obtained fro& diffraction studies a t the melting temoerature. the ranee of values of 0 for the liquid elements as c&nparedto that i f the figure may be somewhat reduced. Nevertheless, the scatter remains great due, a t least in part, to different values of radii ohtained by different workers. A further study by Ross and Miller (46) has related the relative volume change, the radius change, the packing density change, and a coordination number change upon fusion. Mikolaj and Pings (47) have demonstrated that coordination numbers from liquid state diffraction studies contain large inherent uncertainties due to the existence of several nossible calculation schemes. Ross and Miller (46)circumvenied this difficulty by employing the results of Gotoh (48,49). Using a cell approach with a binomial probability allocation for randonly distributing two regularly packed systems, Gotoh derived an equation relating packing density and coordination number. This equation was found to he in reasonahle aareement with resulk for real systems as well as those for dknse and loose randomly packed assemblies of spheres. Finney and Bernal (22), by using the coordinate data for one of Scott's random sphere assemblies (12) and a Lennnrd-Junes fi-12 mt~rniolesuiarpotential function, related the hrats of fusion lior t he rare gnsev excluding He) to the change in distance of closest annroaih durine the ~ h a s chanee. e " Their . results are in agreement with Miller's (42) volumestudy in as much as the distance of closest approach (twice the radius) was found to be less in the liquid than in the solid. Further, the derived distances for the liquid state qualitatively agreed with those from diffraction studies on the liquids. However, the ranee " of values for distances of closestanorosch ~ ~from diffraction studies precludes exacting comparisons. All of the above mentioned studies involving spheres or atoms of one type have contributed to a realization of the ultimate aoal: a useful and tractable model for the liauid state. little regard has been paid to the effect 0; temperature and pressure. Investiaations oertainine to all of the independent variables in thekquation of stateare needed. Le Fevre (50) observed, by plottina RTIPV (from Molecular Dynamics and Monte ~ a r ~ d s i u d iof e sspheres) against p , that as P m, p 0.637. Doubtless, his observation is fundamentally significant and may simply express the fact that real liquids are more compressible than hard spheres. Further, the question arises as to how the monumental work of Morrel and Hildebrand (51,521 in 1934 and 1936 on spheres in random motion (simulating temperature effects) may be used with the
.
-
~
- -
140 1 Journal of Chemical Education
Packing Densities for Binary Mixtures of Spheres For a binary mixture of spheres, the packing density, p, may be formulated as where ni is the number of moles of spheres of type i, Fi0is ( 4 / 3 ) * N ~ r iand ~ Vi = (aVlani),j is the partial molar volume for spheres of type "i" inthe mixture. Noting that the volume of the contaber V = n ~ hn2Vz and letting p:! = n z F z 0 l V and 01 = nl VI"/(V - n2V20), one may obtain the equation
+
P = 02
.~.
~~
~
+ (1 - P
(3)
~ P I
The subscripts "2" and "1" in the present paper represent large and small spheres, respectively. Equation (3) was used by Visscher and Bolsterli (53) in a computer simulation of the random oackine Althoueh .. of svheres. . .. eons. . (2) . . and (3) are tbrmally equivalent, we feel that eqn. 12) is preferable because it fits readilv into the framework ufthe methods of solution thermodyn&ics. Visscher and Bolsterli (53) as well as Yerazunis. Bartlett. and N~ssan(54) h a w presented an intuuive argument u,hich sets an upper twund to the pnrkinr denzits fur a h~narvmixture of spheres. The argument is h&ed on the assumpti& that the maximum p occurs when both the large and small spheres individually exhibit dense random packing, the small spheres being packed& thevoids determined by the large sphere assembly and V I ~ I Vapproaches ~ ~ zero. When the packing density for an assembly of spheres of one size is taken as 0.637, the argument leads to 0.8682 for the maximum packing density for the mixture. Both of the studies define a volume fraytion,) ,oflarge spheres as a measure of compusition 1) = n ? \ ' , " / ( n ~1'1" + n-\'lO)l. For the limiting case mentioned ahwe. \ approaches 0.7.337. At the same time.. X ..,. the mole fraction ofsmall spheres, approaches unity due to the fact that the small spheres are becomina infinitesimal hut vet must occupy a finite volume. This analysis has some ;tility for macroscopic problems (8)hut is of little use for mixtures of atoms and molecules since their radii do not differ by orders of magnitude. O this sitThe other extreme occurs when v2° = ~ I When uation prevails, p = 0.637 for dense random packina- and therefore p does not change with y. If eqn. (1) is presumed to hold for pure liquids composed of spherical atoms or molecules, it may be combined with eqn. (2) for a binary solution to obtain ~
(PO - P)/PO = (ti - v i d e a i ) P = A ~ ~(4) ~ where V = X1Vl X202 and v i d e a l = ( X I P I ' + X2v2°)l~o, in which po, 0.637, is the packing density for spheres of one type in dense random packing and Xi represents mole fraction of component "i". Since p 2 po for mixtures of spheres (2,5,8,36,53-591, eqn. (4) demands that AV, 5 0. This is observed to be the case for many mixtures of metals (7), mixtures of liquid argon and krypton (60), and mixtures of large molecules having roughly spherical shane such as carbon tetrachloride and octamethyl-cyclotktra-siloxane(61). For the mixtures of metals t h r Iwger the rndius ratio the more negative is l~,,,,,..This is in amerment umith the binarysphere parking studies. On the otherhand, binary mixtures bf elements which are known to form compounds or have stronainteractions (metal: metalloid, 2 0. This ma; meral: nonmetal) (7)are kno& to hare ~i',,,,., he rariunnlived by arain invnkine. Vold's tinding ( 4 1 ) , that cohesive forces lead t o small pack-ing densities; thus p 5 po, whence A V,,,,,, 2 0. With regards to volume considerations alone, AY,,,.,, is nonzero for spheres of different size. This suggests that the generalized Raoult's law used by Klotz (62) for example might not he the most logical choice for defining the volume for an
+
L
~~~
owev very
knowledge of the static sphere assemblies to achieve the ultimate goal.
~
"ideal" solution. Nevertheless, it remains the most convenient for expressing mixing data. The binary sphere packing study most relevant to mixtures of atoms is that of Mangelsdorf and Washington (56). They determined packing densities for mixtures of spheres having radius ratios in the range 1-1.6, while for the metals in the periodic table the largest radius ratio is about two. The results for the three mixtures which they investigated were expressible as
-
AV,,
=BXlXz
(5)
where B is a negative constant dependent upon the radius ratio. Real chemiral mixtures which &form to this funcrional form are termed quadratic mixtures by Rowlinson (63). One of the most seeminelv - " adverse binarv mixtures of spheres occurring in nature is that of molten salts. Pelton (43) examined the molar volume data for a select few molten salts. He chose salts having large anions relative to the cations. So that, at least in the regular solid state, the anions could be considered in contact and the cations considered to occupy lattice holes determined by the anions. Using anion radii estimated from molten salt radial distribution function data, he essentially calculated pz = n 2 V P / ( n l i i ,
+ nzi7d
(6)
For the few salts which he investigated, pz was 0.64 f 0.01, remarkahlv close to the value of 0.637. This lends credence to Pelton's contention that these anions exhibit approximate dense random ~ a c k i n and e that the relativelv small cations are distributedin hol&. Packlng Densities and Aqueous Solutions Ideas relatine to the ~ a c k i n edensities for assemblies of spheres have aisisted in theorGica1 interpretations of the partial molt volume for ions, VjO's, a t infinite dilution in water. The Vjo's are considered to be composed of several terms, one of which is an intrinsic ionic volume (64). . . This term, in several instances, was assumed to be vij" = (413) rNnri3 which is approximately 2.5 X 10-Lri3, where ri represents the crystal ionic radius in nm and Viio is in m3mole-1. However, studies involving empirical curve fitting lead to coefficients of r i 3 near 4.5. Benson and Copeland (65) argued that the 4.5 as opposed to 2.5 arose from a packing density effect, 2.5 X 10-3 ri3/pi * 4.5 X 10-3 ri3. Considerable controversy has existed in the past regarding the correct method to be employed in arriving a t ionic partial molar volumes, Via's (64). King (66,67) used the concept of packing density to arrive a t Vio's for several ions, notably the proton, in aqueous solution. His results are in excellent aereement with the results from other techniaues One . (68). . unique aspect of King's method is that it involves no semitheoretical exoression for electrical contrihutions to the partial molar vhume of ions. King (66,67), following Bondi (fig), defined the packing density by p=
vvmio
(7)
whereTwlepresents the van der Wads molar volume. The ratio, V,,./Vio, essentially represents the spherical volume of the solute divided by the volume it occupies in the solution. In this manner, King determined p's not only for ions in water but also for many organic molecules in water a t 25OC. In essence, he found that as the solute size increased, p approached the value for the dense random packing of spheres, pO. The p for organic solutes and carboxvlate anions amroached 00 asym&tically from below and thep for the cations and hdid; ion approarhed p , ~asymptotically from ahove. This aualitative behavior occurred even though most of the solkes were nonspherical in shape. Stokes and Robinson (70) prompted by Alder's finding (59) (now known to be false) that the packing denoity for binary mixtures of seheres was independent of composition, found that plots of Vo,lt in aqueous solution versus (413)rN~ri"
were a-. ~ ~ r o x i m a t elinear. lv This can also he shown to be true for V",,I,, even for nonaqueous solvents, and is evidently due to the small change - in .D for the ranee - of ionic radii considered. Conclusions and Suggestions for Future Work I t has been shown that dense random assemblies of hard spheres have volume properties which are qualitatively similar to those of simule liauids and solutions comoosed of snherical atoms and moiecules. Further, although not directly related to thesubiect of the mesent discussion. literature results indicate that these assemblies also offer a hualitative picture for surface phenomena (20,26). Precise experimental data of several types are required in order to better compare these sphere assemblies to real chemical systems. Work needs to be directed at obtaining the precise equations of state (PVT surfaces) for many of the elements and binary elemental solutions. In conjunction with this, even more precise diffraction studies are needed on these same systems in order to obtain intermolecular separations. Very few diffraction studies of the required type (76) (as a function of P and T)have been performed. Finally, a more precise study of binary sphere packing needs to be made. This studv should cover a lareer radius ratio ranee than that of ~ a n i e l s d o rand f ~ n s h i & t o n( X i . 01'course it is realized that 3 hard snhereassemhlv cannnt correspond exactly to an atomic assembly. However, the ~ r e s e n qualitative t corres~ondencebetween the model and real systems suggests that there may exist renwnnl)le effectwe radii which would maximiw this corresu~~ndence. Thereigre. in addition to experimental work the question must be in: vestigated as to which radii should be used in computing packing densities.
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Acknowledgment One of us (J. R. Greenstein) from Miami Edison Senior High School was a research participant of the science program sponsored by the Dade County School System. This work was supported by the Office of Naval Research (N00014-75-C0173) and the National Science Foundation (GA-40532). Literature Clted Ill Tingafe,G.A.,Nucl. Enp. D&n. 24. 153 119791. 121 Ridway,K..andTabuck. K. J..Hri(. Chem E W . . 12.:181 119671. I31 0wen.A. O..and Watiun. K. M.,Naliunol Petroleum News. 36, R ~ 7 9 5119441. I41 Rogers. C. A . Pmc. London Moth. Srrc., 8,609 119SRI. I51 Furna8.C. C..Ind. nnd En#. Chem.. 23.1052 119:ilI. 161 Parrish. J.R.,Noture. 190.8W 1196ll. 17) Wilmn, J. R.. Mrl. Reu.. IO.381 11966). 18) McGeary R. K., J. Arne,. Cwom Silr, 44,613 1198l). (91 Vairnys, J. R..and Pi1heam.C. C.. "Annual Reviewof Earth and Planetary Sciences." Annual Reviews. Inc.. Palo All". Califl~mia.1971. Vol :1. p. 343. 110) Berna1.J D.,Natur#. lR1.111 11969). I111 Bernal, J. D.,snd Masm.J..Nmluir. 188,910 119601. (121 Sc0tt.G. O..Nalure. 188,908 119601. I131 Scott,G. L a n d Kilguur, D. M.,Hril. J. Appl. Phyr.. S P I 2. 2.863 11969). I141 Se0fl.G. D..Charlerwerth.A. M..and Mak. M. K.. J. Chem. Phyr.. 40,611 1196
