Conadian MinerologistVol. 23, pp. 273-285 (1985)

THE RELATIONSHIP BETWEEN MINERAL HARDNESS AND COMPRESSIBILITY(OR BULK MODULUS)

RONALD J. GOBLEDepartment of Geolog/, University of Nebroska-Lincoln, 433 Morrill Holl, Lincoln, Nebraska 68588-0340, U.S,A.

STEVEN D. SCOTTDeportrnent of Geologlt, University of Toronto, Toronto, Ontario MSS IAI

AssrRAcr nir Ia duret6 H comme deuxidme d6riv6e de U par rap-port i la distance interatomique r, H: (&U/M)ro.

Hardness has been defined as "the resistance of amineral to scratching." Compressibility is the isothermal (Traduit par la R6daction)change in volume of a compound with pressure and isdependent upon many of the same factorJthat determine Mots'cl4s: duret6, compressibilitd, module de compres-hardness. Bulk modulus K, the reciprocal of com- sion, 6nergie du r6seau.pressibility, has been defined as the second derivativeof lattice elergy U with change in volume, INTR'DUCTI.NK = V(d2UldV2)u^. Several authors have demonstratedan empirical corr'elation between hardness and com-pressibility for specific groups of minerals. Examination Numerous attempts have been made to derive an

bt data roi st soiidr tepiesenting I I structure tlpes shows empirical relationship between hardness and variousthiscorrelariontobeindirect.Thehardnessofiiubstance physigal parameters (e.9., Friedrich 1926,isdependentuponthesquareof theinteratomicdistance Goldschmidt 1926, Plendl&Gielisse 1963, Povaren-whereascompressibilityandbulkmodulusaredependent nykh 1964, Julg 1978). Most of these equationsupon the cube of the interatomic distance. Dependence incorporate several coefficients that are difficult toupon valence is more complex, with the data examined determine for a particular substance. Similarfalling into two groups, the Alkali Halide Group and the attempts at empiricai correlations between compres-Sulfide-Oxide Group. Empirical equations developed for sibility or bulk modulus and crystal parameters havethe two groups suggest that hardness should be definedas the second derivative of lattice energy with change in been more successful -(e'g', Bridgman ,1923, Ander-

interatomic distance, H = aatur;t-' son & Nafe 1965' D'L' Anderson 1967' Anderson& Soga t967, O.L. Anderson 1970, 1972, Plendl &

Keywords:hardness, compressibility, bulk modulus, lat- Gielisse 1969, 1970, Anderson & Anderson 1970).tice energy. Several investigators have demonstrated empirical

correlations between hardness and compressibility

S.MMATRE (e.g.,Plendlet al. 1965, Beckmann l9l, Scott 1973),suggesting that it might be possible to derive an

La duret6 d,un mineral mesure sa rdsistance d l'6ra- empirical equation relating hardness to compressi-

flure. La compressibilite d'un compos6, le changement bility and' therefore' to other more readily deter-

isotherme de son volume lorsqu,il est comprim6, d6pend 3i:Llp:q^parameters' The present study was

de plusieurs des facteurs qui ddterminent la duret6. Le undertaken to test the empirical hardness-module de compression K, qui est I'inverse de la com- compressibility correlation of earlier workers, topressibilit6, correspond au taux de changement de l,6ner- determine its theoretical basis and the possibility ofgie U du r6seau avec le changement en volume: usingitinapredictivefashion,andtodeterrnineanK : V(d2uldv2)yn, Plusieurs auteurs ont 6tabli une cor- empirical equation for hardness.r6lation empiriqud entre duret€ et compressibilit6 pourcertains groupes de mindraur. Un examen des donndes CnySfaf-CfmnfcAl FACTORS AFFECItr{c }IApONSSSpouf 8l composds solides qui representent ll types destructure montre que cette corrdlation est indirecte, La. Hardness is defined as "the resistance of a mineralourele q'une suDsBnce oepenq ou c:lrre de la custance rn-teratomique, tandis que compressibilite et module de com- to scratching" @ates & Jackson 1980) or aspression d6pendent-du cubi de cette distance. La rela- "resistance to plastic deformation, usually beingtion avec la valence est plus compliquee; les donndes indentation" (Thrush 1968). It is a complex butddfinissent deux groupes, celui des halog6nures alcalins poorly defined property dependent upon such fac-et celui des sulfures et oxydes. Des relations empiriques tors as the nature of the chemical bond, interatomicpour les deux groupes font penser que I'on dewait d6fi- distance, valence, atomic density and co-ordination

273

274 THE CANADIAN MINERALOGIST

nurrber. Several methods for approximating thehardness of solids are in common use. The Mohshardness scale ranks minerals in the order of theirability to scratch one another; numerical values ofhardness, r€mgrng from I to 10, have been assignedto certain index-minerals. The technical scale is simi-lar and uses many of the same index-minerals,although the range is increased to I to 15. TheVickers and Knoop techniques of measuring hard-ness €ue more precise; one measures the size of anindentation produced by a diamond stylus under aset load. Other methods include Shore hardness,based upon measurement of the height of reboundof a steel ball or hammer dropped upon the speci-men, polishing hardness and Brinnell hardness.Because the Mohs hardness and Vickers hardness arethe most commonly used by mineralogists, we havenot considered the others.

Many hardness data are expressed as either Mohsor Vickers hardness but not both, so that conver-sion from one scale to the other is necessarv. Two

2 4 6 8 t OrcHS HARDNESS

Ftc. 1. Comparison of Mohs and Vickers hardness data( + ) of Tables l, 2 and 3. The correlations of Beckmann(1971), VHN: 86.3 - 90.9 Hn + 34.6H;2, and West-brook & Conrad (1973), VHN = 5.25 g.z.r3, for theMohs index minerals (O) are showu as solid and dashedlines, respectively.

schemes of conversion, by Beckmann (1971) and byWestbrook & Conrad (1973), are shown in Figurel, together with published examples for which bothVickers and Mohs hardness are known. The curveof Westbrook & Conrad fits the data better and wasused in our study. Whereas there is good agreementin Figure I for the Mohs index-minerals upon whichthe empirical curves are based, there is considerablescatter of the data points (up to 3 H. units) fornon-index minerals. This is at least partly due to thefact that the Viskers-hardnes data that we have usedare average in ranges of hardness listed in the sourcefiterature and, overall, are for different loads (inmany cases not included in the source data). For thisreason we have used one type of hardness data (FI-,Mohs) where possible.

Several formulae have been proposed for estimat-ing hardness from basic crystal-chemical parameters.Most contain terms involving valence and meaninteratomic distance, as well as various constantsrelated to such factors as the crystal structure, thecovalency of the bonds, the co-ordination number,and atomic number. None of the formulae proposedto date are particularly useful in predicting hardnessbecause most contain terms that are variable withstructure or bulk composition. In an attempt to cir-cumvent this problem, Plendl & Gielisse (1963) pro-posed that hardness is equal to the volumetric cohe-sive energy, U/V. In the case of Mohs hardness, theyfound that:

U/Y =24(Hn-2.7) for "hard-core" solids,4 < H - < 9 ,

U/Y = 48(H--3.3) for "soft-core" solids,H- > 4, and

U/V : % H^3 for all solids, H* ( 4.

(la)

(lb)(1c)

U/V is given in kcal cm-'. They defined as "hard-core" those binary solids either of the AB type inwhich A or B (or A and B) consist of elements ofatomic number less than 10 or of any other type inwhich one element has an atomic number less thanl0 and the other element has an atomic number lessthan 18. "Soft-core" solids include all others.

Cnysr^q,L-CHEMTcAL FACTons AFrsc"uNcCoMpnrssnrr,rrY AND Bun Monulus

Compressibility is the isothermal change in volumeof a compound with pressure, B = -(l/VXdV/dP)r(Birch 1966), and is dependent upon the same fac-tors that determine hardness. Compressibility can bemeasured directly from the change in volume withconfining pressure or can be computed from elasticconstants. The bulk modulus is the reciprocal of thecompressibility, K = t/9, and is more useful thancompressibility in the development of predictivemodels.

at,attUJz.oOEsU'TElrlYc)t

fEsTBnooK 6co{MD (|t7t)

The systernatics of elastic constants were developedby Birch (1960), who demonstrated that the densityand the mean atomic weight are the only variablesof importance in describing the compressional veloc-ity of sound of rocks and minerals. Several empiri-cal formulae have been developed, based upon therelationship between bulk modulus and the inversefourth power of the interatomic distance (e.g., Bridg-man 1923, Anderson & Nafe 1965, D.L. Anderson1967, Anderson & Soga 1967, O.L. Anderson 1970,1972, Anderson & Anderson 1970, Plendl & Gielisse1969, 1970). A summary paper by Wang (1978)shows that:

K : V(d'zUld\p)v^:Y(dzU/df)(dr/dVE,^ Q-a): (AZJ,"*(n-r))/(9You\ " eb)

where,4 is the Madelung constant, Z. the anionicvalence, Z" the cationic valence, e the electroniccharge, n the repulsive force parameter, r the meanatomic radius and Vo the specific volume. IfAZuZ,"*(n-l),/ro in equation (2b) is assumed to beconstant for isostructural compounds, then:

KVm: constant Qc)

where V- is the molar volume and 16 is the intera-tomic distance at equilibrium. Anderson & Nafe

275

(1965), D.L. Anderson (1967), Anderson & Soga(1967), Anderson & Anderson (1970), and O.L.Anderson (1972)have evaluated equations (2b) and(2c) for a number of different elements and typesof compounds.

RELATToNSHIp BET'$T,EN H-en-uwess anpCoMpREssrBrlrrt (oR But- Monulus)

Beckmann (1971) demonstrated empirical corre-lations between hardness and compressibility forspecific groups of minerals. Scott (1973) used a simi-lar empirical correlation for sulfide minerals topredict the compressibility of troilite. In contrast toBeckmann, who fitted straight lines to the data, Scottfitted a curve. Plendl et al. (1965) expressed theempirical correlation between hardness and compres-sibility as:

l/0 = p/V(Zm/q):H/(Zm/q) (3)

where B is the compressibility, U/V the volumetriccohesive energy, H the hardness, Z the maximumvalence, m the number of component atoms, and qthe number of atoms per molecule. If the Mohs hard-ness from equations (la), (lb), (1c) is substituted inequation (3), bulk modulus is used in place of inverse

THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY

rng le i . DATA usED rN DETER!4 rN tNG THE coNs rANT_EMprR tcAL RELAT loNsHrps AMoNo BULK MoDULus ,MOHS HARDNESS, MOLAR VOLUME AND VALENCE FOR THE SULFIDE-OXIDE GROUP

STRUCTURE TYPE IZ I Vm U K HgA N D C o M P o U N D

' a r c ' ( c n 3 / m o t e ) ( k c a l , / m o l e ) ( M b a r )vHN , rur/23/4 yrft l t tr ' l t

( k g l n m ' ,Hn

( o b - p r )

NaC I

Z n S ( c )

NaC I

Z n S ( c )

Z n S ( h )

Z n S ( c )NaC I

Z n S ( c )

24 .92 5 . 74 1 . t?7 .721.8t t . 41 6 . 8l 1 . 524.51 1 . 2l t . 2, 1 . 44 0 . 420.' l4 1 . O4 0 . 02 3 . 82 7 . 43 4 . 0

8 , 25 0 . 13 3 . 8

2 1 . 72 2 . 0

1 2 . 11 t . 51 4 . 41 5 . 41 2 . 4

( 2 1 7 '( 2 1 9 '( 2 1 t '

(25't '

( 9 0 0 )( 9 8 7 )a7 54 '( 9 5 8 )( 9 5 J )( 8 1 0 )t763 '( 8 6 6 )(655 '(662 ' , '( 7 8 8 )1727 '( 6 8 6 )

( 1 0 7 4 )(7 46)( 7 0 5 '( 9 6 t )( 1 A 4 t

( 2 7 6 7 t(2500)

( t 1 9 2 )(2100,

7 4

6 t 55 8 0

7 6| 080

225a749

1 6 46 02 8

195

9 0r 1 5 0

7 2f i 02 5 4'17 8214

1 4971200

1 1 . 7 6i l . 1 61 0 . 0 4

9 . 5 59 . 4 7

1 1 . 1 21 1 . J 51 2 . 6 6

, . 9 0 r1 0 . 2 41 2 . 1 11 t . 2 8

9 . 8 01 0 . 1 4| 0 . f , 4t 0 . 3 9t 0 . 8 0

9 . 6 9' 1 0 . t 1

1 2 . 1 61 0 , 9 7' | 0 . 7 9. l . l . 7 9

1 0 . 8 11 0 . t 41 0 . 2 01 0 , 2 7I 1 . 9 61 0 . 6 31 2 . ' t 5t 0 . E 0

90, 2

7 2 427892480

AgBrAgC IA s lCuBrCUC IC u lCa0Co0HgsMsoMn0P b sPbTa

CdT€HgTeZnSZ n S eZnT6B€0cdsCdSsZnOZ n S1 /zctFes2TaCT l cT I NucZrCs l c

2222222

0 . 4 0 7 2 . 50 . 4 4 2 . 1 . 7 50 . 2 4 t ! . 5o . t 4 5 2 . 40 . 3 9 8 2 . 2 50 . t 5 5 2 . 51 . t J 6 4 . 51 . 4 5 2 ( 4 . 8 )0 . 2 3 0 * 2 . 5f ; 5 1 8 5 . t t1 . 5 4 1 5 . 50 . 6 0 9 2 . 6 50 . 4 0 8 5 . 00 . 4 2 4 3 . 5o . 4 2 4 2 . 4o . 4 t 7 2 , 50 . 7 6 t t , 7 5o , 5 9 5 1 , 7 50 . 5 1 0 ( 2 . 6 ' , )2 . 4 9 4 8 . 50 . 6 1 3 3 . 20 . 5 1 7 3 . O1 . 4 3 7 4 . 5o . 7 6 7 t . 7 5o . 7 7 5 t . 7 52 . 1 6 9 9 . 02 . 4 0 1 9 . 02 . 9 4 1 9 . 5' t . 6 t 4 ( 6 . 1 )2 .23 ' , t 8 . '? . 4 6 4 9 . 5

2 3 . 5 4 0 . 51 5 . 2 4 - 0 . 51 7 . 9 2 - 0 . 12 ' t . 9 7 0 . 51 I . 6 t - 0 . |2 5 . 9 2 0 . 71 9 . 5 9 - 0 . I1 5 . 4 0 - 1 . 2t 4 . 6 9 - 0 . 8I 6 . 8 1 - 0 . 71 9 . t 5 0 . I' | 6 . 6 1 - O . 42 2 . 2 4 0 . 41 6 . 6 2 - 0 . 52 0 . 9 7 0 . 21 8 . 4 2 - 0 . 11 9 . 5 4 0 . I2 1 . 4 7 0 . 41 8 . 7 t - 0 . 12 1 , 7 7 | . 01 9 . 5 0 0 . I' t 9 . 7 5 0 . 1' 1 6 . 3 | - 0 . 8' t 9 . 4 9 - 0 . 11 8 . 5 4 0 . 12 0 . 0 5 0 . 4r 8 . 8 2 - O , 21 9 . 2 1 0 . 01 6 . 4 2 - 0 . 82 0 . 8 8 0 . 72 0 . 2 0 0 . 5

V a l r e s l n b r a c k e t s ( ) a r e o s t l m a t s d .H m ( o b - p r ) l s t h e d l f l e r e n c a b o t r s e n

A n o m a l o u s v a l u e s a r E l n d l c a t e dt h e o b s e r v s d M o h s h a r d n e e s € n d

b y a s t e r I s k s r .f h 6 t p r s d l c f s d u s l n g e q u a t l o n ( 1 0 ) .

276 THE CANADIAN MINERALOGIST

z:tr

clrI

-o.5l og K

Ftc. 2. Bulk modulus and hardness of solids. The curves show correlations for sul-fide data by Beckmann (1971) and Scott (1973) as well as a solution to equations(6b) and (6c) with tr = ozos.

T A B T E 2 . D A T A U S E D I N D E T E R M I N I N G T H E C O N S T A N T E M P T R I C A L R E L A T I O N S H I P S A M O N G B U L K M O D U L U S ,M O H S H A R D N E S S , M O L A R V O L U M E A N D V A L E N C E F O R T H E A L K A L I H A L I D E O R O U P

aoaA

+

STRUoTURE TypE l z - - l s t l z e l ^ vm u K H6A N D C o M P o U N D ( c m t m o l e ) ( k c a l . / m o t e ) ( M b a r )

vHN ̂ Kv. , /z 314 p-u?/3 tz2l3 *(kg/ nnzt " ' 1 o6'1p r )

N a C I K B rK C IKFK IL I B rL t c l

. L l IN a B rNaC INaFN a lRbBrR b c IRbFR b lBaSCaSPb Se5 r 5

N l A s F a SZ n S ( c , A l S b

0aAb' G a P

0€SbI nAsl n PI n S b

5 4 l . A 4t 6 l . 4 a2 8 1 . 3 27 2 2 , 2 23 8 1 . 5 22 0 1 . 1 61 2 1 , 0 05 6 1 . 8 84 6 | . 6 82 8 1 . 3 22 0 t . t 66 4 2 . O 47 2 2 . 2 05 4 1 . 8 44 6 t . 6 89 0 2 . 5 5

1 5 9 . 31 6 5 , 4' I 89 .7| 5 0 . 8I 8 8 . 51 9 9 . 22 4 0 , 11 7 4 . 1t 7 4 . 5t 8 r . 12 1 3 . 41 6 5 . 9' | 53 .5I 6 0 . 71 8 t . 6' t 45 . t

l 7 2 5 '( 7 6 5 )

i l l 7 t )t 1 2 0 1 '( 1 t 0 7 )( t 2 0 t )f i 1 7 0 )i l 1 9 6 )(1 l t2 )

0 . 1 5 1 2 , 00 . t 8 0 2 , 00 . t z t 2 , 80 . 1 2 o 2 , 20 . 2 5 7 2 . 5

0 . 6 6 3 5 , 7 50 . 1 8 8 ( 2 . 8 )0 . | 8 8 2 . 4 50 . 2 4 4 2 . O0 . 4 9 0 2 . 2 50 . t 6 t 2 . 3o , 1 3 4 ( 2 . 7 '0 . 1 6 0 ( 2 . 6 ,0 . 2 7 t ( 2 . 5 '0 . t l t ( 2 . 7 )o . 5 4 5 ' . 00 . 4 r t 4 . 00 .3 t9 2 .70 . 4 1 7 3 , 3o . 7 4 6 4 . 00 . 5 9 t ( 4 , 9 'o . 1 5 6 4 . 50 . 8 8 7 5 . 00 . 5 6 t 4 . 50 .579 (4 .7 ' , r ,0 . 7 2 5 t 5 . l '0 . 4 6 7 ' . 8

1 5 , 4 5 - 0 . 51 5 . 1 6 0 . 01 1 . 2 1 o . f ,1 4 . 1 7 - 0 . 21 1 . 1 2 - 0 . 2| 9 . 2 6 0 . 61 7 . 1 7 0 . 41 5 . 2 5 0 . 01 4 . 7 2 - 0 . 11 5 . 6 t - 0 , t1 1 . 7 0 - 0 . 71 5 . 5 8 - 0 . 5l a ? t n a

1 7 . 6 8 0 . 51 2 . 2 9 - 0 . 61 5 . 8 8 0 . , l1 7 . t 3 0 . J1 8 . 3 1 0 . 6t 4 . r0 -o .21 6 . 9 4 0 . '1 3 . 4 4 - 0 . 4' t 7 . 3 7 0 . 6t 5 . 5 6 - 0 . 61 4 . 0 2 - 0 . 51 5 . 8 9 0 . 1| 6 . 1 | 0 . 2t 6 . 5 9 0 . 4t 5 . 0 t - 0 . t

z2222

3

4 1 , t

2 t . 15 1 . 22 5 . 12 0 . t

9 . 85 2 . 5t 2 . 02 6 . 91 4 . 84 0 . 84 8 . 44 2 . 92 7 , 05 9 , 5t 9 . 32 7 . 73 4 , 53 2 , 91 a , 21 4 . 82 7 . 22 4 , 4t 4 . 53 t . ' lt 0 . 44 0 . 9

6 , 5 4

7 , 4 66 . 3 85 , 4 56 . r 96 . 5 06 . t l6 . 0 26 , 5 67 . 2 56 . 5 76 . 6 86 . 8 67 , 3 76 . 6 0

5 . 9 75 . 6 56 . 8 66 . 7 96 . 8 86 . 8 57 . 2 16 , 4 76 . t 97 . 3 56 . t 7

8382

7 81 0 51 2 8165

8 8A A

95t 3 080

76

52

2 t44007 1 7942420t57412220

V a l u o s I n b r a c k e t s ( ) a r e o s t l n a t e d .H m ( o b - p r ) l s t h o d l f f e r e n c e b o t r o e n t h e o b s e r v e d M o h s h a r d n e s s a n d t h a t p r e d l c t e d u s t n g e q u a f t o n ( 1 2 ) .

compressibility, and the units axe converted, we findthat:

K : [0.s74(H ^-2.1)]/ (Zm/ q)("hard-core", 4<H.<9) (4a)

K : [. 147(H ^- 3.3)]/ (Zm/ q)("soft-core", H. > 4) (4b)

K = [0.012(H ^)3)/ (Zm/ q)(all solids, H.<4) (4c)

K is given in Mbar-l. Similar relationships can bederived by cornbining Zhdanov's (1965) equation forthe cohesive energy U of an ionic crystal:

U: -(Z;ZceNA/r)(r-r/n) (5)

in which N is Avogadro's number, with equations(la,b,c) and (2b). Ndrops out because of differencesin definition of volume, giving:

(9Kln) = 0.574(H^-2.7)("hard-core", 4<Hm<9) (6a)

(9Kln): 1.147(H--3.3)("soft-core", H- > 4) (6b)

(9Kln) : 0.012(HJ3(all solids, H. < 4) (6c)

Equations (4) and (6) are identical if (9/n) equals(Zm/q).

We have collected published data for 8l solids inorder to test the derived relationships between hard-ness and bulk modulus or compressibility (Tables l,2, 3). Values of hardness are from Palache et ol.(1944, l95l), Deer et al. (1962), Beckmann (1971),Ivan'ko (1971) and Uytenbogaardt & Burke (1971).Bulk moduli and compressibilities are taken largelyfrom the compilation of Simmons & Wang (1971),supplemented by data from Birch (1966), Anderson& Anderson (1970), Beckmann (1971), Anderson(1972), Scott (1973) and King (1977). Bulk modulusand compressibilify data are averaged values at roomtemperature and one atmosphere pressure. The dataset is plotted in Figure 2 together with Beckmann's(1971) empirical correlation-curve for sulfides anddisulfides, Scott's (1973) curve for sulfides, and equa-tions (6b) and (6c) with the value of n as calculatedfor sphalerite, rzns:5.1. Although Scott's curvegives a better "average" fit for the sulfides than doesBeckmann's, as single curve satisfies all of the datain Figure 2. On the other hand, there do seem to betrends in the bulk modulus - hardness plots fordifferent types of materials. In order to explore thesetrends, we have concentrated on the relationshipsshown for the NaCl and ZnS structures beforeattempting to expand these relationships to otherstructural types.

The volumetric cohesive energies U/V formaterials having the NaCl and ZnS structures are

277

listed in Tables I , 2 and 3 , and plotted against hard-ness in Figure 3. The empirical correlations of Plendl& Gielisse (1963) (equations la, b, c) are shown andfit the average data but, for specific structure-typesand ionic charges within these structure types, moreprecise correlations are evident. For example, thealkali halides conform to the expressionU/Y:2.2(Iln-2.5) whereas according to Plendl &Gielisse the relationship should be U/V =H^t".Thus, the empirical correlations of Plendl & Gielisseseem to be oversimplified.

Equation 3 demonstrates.a correlation betweenvolumetric cohesive energy U/V and bulk modulusK. Equation 3 is shown in Figure 4 (solid line) for

log VHNf . o r . 5 2 .o 2 .5 3 .O 3 .5

2 . 5

o.o o,5 l .olog HM

Frc. 3. Hardness and volumetric cohesive energy for theNaCl and ZnS structures. Correlations predicted byPlendl & Gielisse (1963) with equations (la, b, c) areshown by solid lines; a curve fitted to the alkali halidedata is shown as a dashed line.

TI{E RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY

:)I

C'II

f t.to_ct-stt""tu;A

+ z " l0

s z- :e / l

+ Z = lf

. Z = 2

urv- qa(Ht3 gJ +^

o O I u /V=24(Hn-2 .71o.. I

u\/v.H:N/z .[

uru = urHM-2.s]

llidt-siro;mEl

+ z " I

O z " z

0 z " st?;s sficr!@]. z . Ia z - 2L 2 " 3

l lno t l t tod

l?,ou no,*--.-?'/

278 TI{E CANADIAN MINERALOGIST

:1". .

o . 5

- o .5 o . o o 5l o g K

Ftc. 4. Bulk modulus and volumetric cohesive energy,compensated for valence, for the NaCl and ZnS struc-tures. The solid line represents the fit of plendl et al.(1965). The dashed lines represent least-squares fits foralkali halide, Z = 1, and non-alkali halide, Z : 2, data.

materials with the NaCl and ZnS structures. Aspredicted, there is a good correlation (R : 0.99, usinga log scale) that is largely independent of structuretype. However, for the alkali halides and thosematerials with divalent cations, a more precise corre-lation would be (U/V) or Kr.1et0.03 (R = 0.997, logscale used), suggesting that the factor n in equation(6) lor 9q/Zm in equation (3)] is approximatelyinversely proportional to K%, that is:

n=9q/Zmqr/(ZuZa4)%. A)

Equation (7) suggests that in order to correct for theeffect of ionic charge, K should be multiplied by Z%rather than by Z as was done by Plendl et ol. (1965).In order to evaluate these expressions properly, wemust first know the exact correspondence betweenn and both r andZ. This relationship is also impor-tant in empirical expressions used for deriving thebulk modulus (e.9., equation 2) and has been studiedby Anderson (1970) and Anderson & Anderson(1970).

AB-type compounds

Tables I and 2 and Figures 5 and 6 illustrate the,relationship between bulk modulus and volume forAB-type compounds in the halite, sphalerite, wurt-zite and niccolite structures. The relationships are

oro

* l .

=tt!o- t .

T A B L E f . D A T A U S E D I N D E T E R M T N T N O T H E C O N S T A N T E U P I R I C A L R E L A T I O N S H T P S A I , I O N 6 B U L K M O D U L U S ,MoHS HARDNESS, MoLAR VoLUME AND VALENCE FOR AxBy_TIpE COMPOUNDS

sTRUcruRE rypE lz" lAND COMPOUND

Y m t *( c n 3 7 n o t e ) ,.n)llr,

rv'rz! n,ufti3 rz$ ,o!!o"r

K H .( M b a r )

CaQ. BaFZ l .4 lC a F t l . 4 lc d F , l . 4 lP b F 2 1 . 4 1S r F : l . 4 l

r l r * . C a C i 2 l . 4 l- M g F r - l . 4 t

SULF I DE-OX I DE GROUPC u 2 O C u 2 0 1 . 4 1A s i s A s i s r . 4 rM g A t 2 o F e l 0 4 2 . 3 1

M g A | 2 0 4 2 . t lM n F e 2 0 4 2 . 5 1N t F e z O a 2 . 3 1

A f e o r A t e o i 2 . 4 5crioi z.4sreio\ 2.45F e T t 0 r 2 , 4 5

s l q s t o 2 ' 2 , A 3caF2 Tho t 2 .83

uoz 2 .85' l l $ Sn02 2 ,8 t- r l o i 2 . 8 3

1 8 . 01 2 . 3' | t . 81 5 . 71 4 . 72 5 , 6

9 . 8

2 t . 45 4 . 4l l . l

9 . 9I t . 61 0 . 9

9 . 7l 0 . t' I 0 . 6I I . '1 3 . 21 2 . ,1 0 . 8

9 . 4

20542

5751 585

2265

95t6 J l

1 1 2 0t 0 7 87 7 7

9 8 t

'1 .22

7 . 5 0

5 . 7 8' t .27

4 . 4 5 r6 . 0 2

1 0 . 6 8I t . 0 9t 0 . 6 81 1 , 4 61 0 . 6 t1 0 . 5 1| 1 . 0 81 0 . 9 8

1 . 9 7 '' I 1 . 6 91 2 , 0 0l 0 . t l

9 . 0 6

1 4 . 4 7r 5 . 0 7t 4 . 6 61 4 . 1 91 4 . 8 51 8 . 4 2 rt 6 . 1 9

2 4 , 1 5t 9 . 0 3t 7 . 0 92 l . ' t It 6 . 8 61 4 . o 72 0 . 6 1z t . 2 81 5 . 4 3' t 5 ,9 t1 7 , 6 2l E . l 5l 5 . 9 91 7 , 1 01 4 , 4 8

-o ,2- 0 . I- 0 . 2- 0 . 5- 0 . I

0 . 80 . 0

- o . 7

- 0 . 8- l . 8

0 . 60 . 8

- l . 2- 0 . 6- 0 . 4

- 0 . 8- 2 . 1

0 . 5 6 7 3 . 00 . 8 6 2 4 . O0 . 8 9 ' 4 . O0 . 6 1 1 t . 20 . 6 9 9 t , 50 . 2 2 9 | 5 . 0 r0 . 8 6 9 5 . 0

0 . 5 1 5 3 . 7 50 . 1 9 8 2 . 2 51 . 8 7 2 6 . 02 . 0 2 0 8 . 01 . 8 5 1 5 . 7 51 . 8 2 t 5 . 02 . 4 2 1 9 . 02 . 2 3 7 8 . 52 , 1 2 4 6 . 0r . 7 8 6 6 . 00 . f , 8 0 r 7 . 01 . 9 t 1 6 . 5z , l 2 a 6 . 02 . O 4 1 7 . 02 . 1 0 1 6 . 5

1 2 7272

t 6 J3 1 7

A n o m a l o ! s v a l ! s s a r s l n d l c a t s d ! y a s l n g l o a s i 6 r l s k r . y m r r l n d l c a f 6 s m o l a r v o l u m e f o r a f o r m u l a , u n l tb a s s d u p o n o n e a n l o n ( e . o , l / 4 F e ? O a ) . b a s a z s u p e r s c i l p t h € s t h e v a l u s I f o r t h e A t k a i l H € i l d s G r o u pa n d 3 / 4 f o r f h e s u l f l d 6 - 6 x l d e g r o ' u p i c , i e a r n i l t s i u y h € s ' f h e v a t u e I f o r t h e A t k a l t H s l l d s o r o u p a n d2 / 3 l o r t h s S u l f l d e - O x l d s . G r o u p . i { m ( o b s - p r ) t s t h e d t f l e r e n c E b e t t e e n t h e o b s € r v e d U o h s h € r d n 6 s s a n dl h a t p r e d l c t e d u s t n g e q u s t l o n s i l 0 ) a n d f i 2 ) .

SULF I DE. OX IDE GROUP

+ Z . Io Z = 2E 2 " 4

THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY

2 A 3 0MOLAR VOLUME

FIo. 5. Bulk modulus - molar volume data for the Sulfide-Oxide Group' The curvesrepresent the equation K = 10.85 Z /V n wlth Z = | (+), Z = 2 (O)' Z = 3 (nosymbol), arrdZ:4 (tr). The equation determined by linear regression is in logarith-mic form: log (KV- ) = 1.025 + 0.781 log Z; R : 0.9765'

ALKALI HALIDE GROUP. l . t . l . t . t l

279

@fJDoo n s

YJDo

+ Z = |o 2 . 2L z = 3

3

2

I

l r l , l , l , lo to 20 30 40 50 60

MOI-AR VOLUME

Frc. 6. Bulk modulus - molar volume data for the Alkali Halide Group. The curvesrepresent the equation K = 6.65 Z/Y nwith Z : I ( + ), Z = 2 (O), Z = 3 (A), andZ = 4 (no symbol). The equation determined by linear regression is in logarith-mic form: log (KVr) = 0.821 + 1.010 log Z; R: 0.9919.

SULFIDE. OXI DE

+ Z o l

o 2 . 2E 2 . 4

280 THE CANADIAN MINERALOGIST

essentially those shown by Anderson & Anderson(1970), Anderson (1972), Plendl & Gielisse (1969,1970) and Plendl et al. (1965). We have chosen touse molar volume in place of the specific volume orcube of the interatomic distance used by some ofthese investigators. We have divided our data intotwo empirical groups, the Alkali Halide Group andthe Sulfide-Oxide Group. The group names simplyreflect the dominant compounds in each group andare not intended to be exclusive; other types of com-pounds are included in each group. Figures 5 and6 show the relationship between bulk modulus andmolar volume for these two groups, as well as theequation fitted to the data by linear and nonlinearregression; logarithmic scales were used for these fits.For the Alkali Halide Group the data are describedby the equation:

KVm = 6.65 Z" (8)

For the Sulfide-Oxide Group the appropriate equa-tion is:

KVm: 10.84 Z" (9)

In practice the proportionality of K and l,/V- wasdetermined separately, and a fit of log KV. againstlog Z" was then made. Correlation coefficients ofgreater than 0.97 confirm that there is a good corre-lation among bulk modulus, molar volume andvalence.

4 0

The same groups were used in determining theempirical relationship anxong Mohs hardness, molarvolume and valence. The data for the Sulfide-OxideGroup are shown in Fiirure 7. The correlation canbe described by the equation:

H.V-% = 19,16 Z"% (10)

Again, the proportionality of H. and V--% wasdetermined separately, and a fit of log (H.V.%)against log (Z) then made. The correlation coeffi-cient, though not as high as for the KY^ versus Z"data, is still over 0.90. In the case of the Alkali HalideGroup, no immediate correlation of H.V-% viithZ" is apparent, the data for the alkali halides(2": l) showing considerable scatter. However, ifthese data are plotted separately against total num-ber of electrons T, in the formula, a correlation isapparent (Fie. 8), defined by the equation:

H.V.* : ll.4l + 0.30 T" (l la)

This relationship was used fo define an "effectivevalence" Z" for the alkali halides. This "effectivevalence" was based upon a minimum of 12 electronsin the alkali halides. Rearranging equation (lla)based on this l2-electron minimum leads to theequation:

50302 Ql oMOLAR VOLUME

Ftc' 7. Mohs hardness - molar volume data for the sulfide-oxide Group. The curvesrepresent the equation Hm = 18.60 Z% /V m% with Z = | (+ ), Z = 2 (O), Z = 3(no symbol), arrd Z:4 (tr). The equation determined by linear regression is inlogarithmic forml log (H.V.%):1.289+0.6261og Z; R --0.902t.

H-V.* : 14.99 Il - (T e- 12)/ 50.211 ( l lb)

ALKALI HALIDES

THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY

o 2 0 4 0 6 0 8 0 l o oTOTAL NTJMBER OF ELECTRONS

Ftc. 8. Dependence of H.V.% on total number of electrons in the structure for thealkali halides. The line determined by linear regression is:H-V.% : 11.41 + 0.299 (total number of electrons); R = 0.9096.

v'',(a42/3

Frc. 9, Mohs hardness - molar volume data for the Sulfide-Oxide Group. Molarvolume data for the alkali halides have been adjusted to compensate for the differ-ence between the effective valence, 2", alLd the true valence, Z.For the remain-ing compounds the effective valence is equal to the true valence, and Z/2" eqaalsl. The curves represent the equation Hn = 15,01 Z/Y ^% $/ith Ze : | (+), Z" = )(a), Z"= 3 (A), aad Ze= 4 (no symbol). The equation determined by linearregression is in logarithmic form: log (H*v.'z/; =1.176+ 1.023 log zriR :0 .9399.

281

4 0

q 3 0( \ l g

tEE

2 0

30

+ Z . lo 2 " 2d 2 " 3

282 THE CANADIAN MINERALOGIST

where:

l-(T"-L2)/50.21 = "effective valence" :2" (l lc)

Values of Z" for the alkali halides are tabulated inTable 2; LiF, with 12 electrons, has an effectivevalence of l. H- and Vm data for the Alkali HalideGroup are plotted in Figure 9; values^of V. havebeen adjusted by a quantity (ZJZ)% to permitinclusion of alkali halide data; for the non-alkalihalides, Z":2". The empirical relationship for alldata in the Alkali Halide Group is:

H.V.'/ : L5,38 Z" (r2)

The correlation coefficient, using logarithmic scales,is again greater than 0.90.

The empirically derived equations for both the,Sulfide-Oxide Group and the Alkali Halide Groupare summarized in Table 4. Equations (8), (9), (10)and (12) have been rearranged to put them into com-mon formats in equations (8a), (9a), (l0a) and (l2a).As can be seen, for the Alkali Halide Group the term(n 2",J3) is common to both the bulk modulusand hardness equations; "effective valence" mustbe used in place ofvdlence in the hardness equation.For the Sulfide-Oxide Grodp the term (24 Z) isconmon to both equations, but the exponent on thisterm differs for the bulk modulus and hardness equa-tions.

Combination of the empirical equations (8)through (12) and (8a) through (l2a) results in equa-tions (13), (l3a), (14) and (l4a) relating hardness to

bulk modulus for the Alkali Halide and Sulfide-Oxide Groups. The approximate term (9Vnv"/4) iscommon to the equations for both groups. The equa-tions differ in that the Alkali-Halide-Group equa-tions have a term compensating for the differentexponents on the valence term for the hardness andbulk-modulus equations. The latter term may or maynot have physical significance; regression analysis of(H. /KVnl6) against Z" shows a dependence onvalence, but the correlation coefficient is low. Equa-tions (13) through (l4a) show that, as noted by Beck-mann (1971) and Scott (L973), there is a relationshipbetween bulk modulus or compressibility and hard-ness. However, as is shown by these equations, thiscorrelation is not direct but involves other terms aswell.

Ap;type compounds

Initially, we restricted ourselves to a considerationof data for AB-type compounds. Table 3 shows therelationship between bulk modulus and molarvolume for A*B;type compounds in the fluorite,rutile, quartz, corundum, spinel, cuprite and argen-tite structures; the ratio x:y is not equal to l. Forthe volume term, we have chosen to use the molarvolume based upon one anion; for valence we haveused the square root of the product of the averagecation and anion valences, Zo. With these defini-tions, {Br-type compounds fall into two gxoupscorresponding to our original Alkali Halide andSulfide-Oxide Groups. Table 3 lists observed valuesof (KY^/Z) and (H.Vr%/Zr) for the Sulfide-

TABLE 4 . SUMMARY OF EMPIRICAL CORRELATIONS

ALKAL I HAL IDE

g q u a l l o n

OROUP

n u n D e r

S U L F I D E - O X I D E O R O U P

€ q u € i l o n

K . 6 . 6 5 Z " / V ,

a . ( 2 0 / 3 1 4 / l n

a^ . 15 .3s ze1y f i l3

z ( g l 4 , ( . 2 o l t , 2 e l \ n z / 3

H m - 2 . 3 r t u \ { , 3 t t z " l z " t x

= l g l q t r y L l 3 ) ( z / z , K_ m e c

K - t 0 . 84 t f ; l a t t ^ e

e ( 2 4 z c r 3 l 4 / v n

9 a

H m : 1 9 . 1 6 z l l 3 1 Y ^ 2 1 3 l 0

z'r 1\ (24 ,"r '/3 ttrt l ' I oaH m . 1 . 7 s n ! 1 3 t t t / z " t L / 1 2 * e + t o + 1 4

' t s / 4 , t v { 3 t t t t zaz " t L /12 r t 4a

8

8a

1 Z

12a

8+ l 2+ l 5

' l5a

G E N E R A L R E L A T I O N S H I P S

e q u a i l o n

K-v(d2u/dv2tyo"r,r.:':ro"r'rrn_iirritiri"r^

, - l A Z u Z " e , . . , . , - , , .

mB m " ( A Z e 4 o ( n - l ) ) / ( 4 V m )

H n . ( A z a A " 2 { n - , , ) / t { 4 v r )

( z 4 z J l / 1 2 ,

H . ( d q J l d r z ) r o . H m

2a

2b

lt6+2b'156

I 4a+2b' | 5b

I t€+l 5br | 5

e sulfide-oxlde groupA olkoll holide group+ \ry compounds

THE RELATIONSHIP OF HARDNESS WITH COMPRESSIBILITY

MOHS HARDNESS

Frc. 10, The difference between observed and calculated Mohs hardness as a func-tion of observed Mohs hardness. Data are shown for the Sulfide-Oxide Group(e ), Alkali Halide Group (A), and for A*\-type compounds in both groups (+)'The curve was constructed through points determined by averaging data over one-half unit intervals in Mohs hardness.

283

goor l

4'3

3

3oUJzoE

r - loIo= -2

Oxide Group. As can be seen, these values are closeto the coefficients in equations (8), (9), (10) and (12)and are within the ranges for AB-type compoundsshown in Tables I and 2, with the exception of twoanomalous materials, CaCl2 and SiO2. The fact thatquartz, SiO2, is anomalous may indicate that thistype of correlation will not be applicable to silicates.

CoNcr.usroNs

Equations (8), (9), (10) and (12) demonstratethatempirical relationships exist by which bulk modu-lus and hardness can be modeled for certain groupsofminerals. Equations (8a), (9a), (10a) and (12a) fur-ther demonstrate that these empirical equations havea number of terms in common; combining equation(8a) with (l?a\ to derive (l3a) and equation (9a) with(l0a) to derive (l4a) demonstrates the basis forempirical correlations between hardness and bulkmodulus (or compressibility) established by previousinvestigators. Equations (2a) and (2b) show that forbulk modulus these empirical equations can berelated to a theoretical concept,.that is, bulk modu-lus is proportional to the second derivative of lat-tice energy with change of volume, (d2uldvlvEquation (2b) can be combined with equations (133)and (14a) to produce equations (l5a) and (l5b) asshown in Table 4; comparison with equation (2a)suggests that the Mohs hardness is proportional10 the second derivative of lattice energy withchange of interatomic distance, (&U/df),o. "Abso-

lute hardness" could therefore be defined by theequation:

H: @2U/df)ro a Hm (15)

rather than being defined as equal to the volunetriccohesive 0attice) energy, U/Y, as proposed by Plendl& Gielisse (1963). The dependence ofhardness upon16 rather than upon Ve is a reflection of the direc-tional nature of hardness; i.e., we are stretching andbreaking bonds in one direction rather than com-pressing the structure in three direclions.

One of the purposes of this study was to checkthe possibility of using the empirical Mohs hardness- bulk modulus correlations to estimate compressi-bilities. Examination of the scatter of data in the var-ious figures and tables shows that, in most cases,application of the empirical equations relating bulkmodulus to molar volume would be a more accuratemethod for estimating bulk modulus. However, theuse of the equations relating hardness to molarvolume may lie in determining in which group a com-pound belongs prior to applying the appropriateequation relating compressibility to molar volume.

It is possible that the equations derived could alsobe used to estimate hardness. Tables l, 2 and 3 andFigure 10 show the deviation of the values of Mohshardness estimated by use of equations (10) and (12)from the observed values. The spread is generallyabout one-half unit about the observed value, butthe predicted value is consistently low in the range

284 THE CANADIAN MINERALOGIST

6 to 7 and consistently high in the range 8 to 10. In len un&nichtmetallischen kristallinen Substanzen.the range 6 to 7 this is probably due to inaccuracies Kristall und Technik 6, 109-117.in the data (most of the data points are for A"B"compounds); in the range S to l0 the discrepaicj BIncH, F. (1960):'Thevelocityof compressionalwavesmay be real, since many iuthors have suggestA th; 1i.1.:H^.

l0 kilobars. J. Geophys' Res' 65'the Mohs hardness scale should be expandedbetween rudr-rruz'

9 and l0' (1966): compressibility: elastic constants. .rz

AcrNowr_rocEMENrs *T:A$.j,::)Hf "##'i;f;i;clarke, Jr.,

We gratefully acknowledge the work of Z.L,Mandziuk who, in a student project at the Univer-sity of Toronto, laid the groundwork for ourresearch. We also thank H.E. King Jr. and C.T.Prewitt of the State University of New York at StonyBrook for helpful discussions following an oralpresentation of our work at the 1978 Joint AnnualMeeting of GSA/GAC/MAC in Toronto (Scott &Goble 1978). Our research was supported by oper-ating grant A7069 to Scott from the Natural Sciencesand Engineering Research Council of Canada andby a Postdoctoral Fellowship to Goble from N.R.C.Negotiated Development Grant D56 to D.W. Strang-way on behalf of the Department of Geology,University of Toronto.

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