Fracture Toughness and Surface Energies of Minerals: Theoretical Estimates for Oxides, Sulphides, Silicates and Halides D

Minerals Engineering 15 (2002) 1027–1041 This article is also available online at: www.elsevier.com/locate/mineng Fracture toughness and surface energies of minerals: theoretical estimates for oxides, sulphides, silicates and halides D. Tromans a,*, J.A. Meech b,1 a Department of Metals and Materials Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1W5 b Department of Mining and Mineral Process Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1W5 Received 6 August 2002; accepted 13 September 2002

Abstract Theoretical estimates of the ideal fracture toughness and surface energies of 48 minerals have been modelled by treating them as ionic solids, using the Born model of bonding. Development of the toughness model required calculation of the crystal binding enthalpy from thermodynamic data and the use of published elastic constants for single crystals. The principal minerals studied were oxides, sulphides and silicates, plus a few halides and sulphates. The study showed grain boundary fracture is most likely in single- phase polycrystalline minerals. However, the fracture toughness for grain boundary cracking in pure minerals is not signiﬁcantly lower than that for intragranular cracking. The computed critical stress intensity values for intragranular cracking, KIC, ranged from 1=2 2 0.131 to 2.774 MPa m . The critical energy release rates for intragranular cracking, GIC, ranged from 0.676 to 20.75 J m . The results are discussed with relevance to mineral comminution, including energy considerations, particle impact eﬃciency, and lower limiting particle size. Ó 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Comminution; Crushing; Grinding; Particle size

1. Introduction ergy release rate per unit area of crack plane (J m2) that is necessary for crack propagation and is related to the The size reduction of minerals by comminution and mode I stress intensity factor for crack propagation crushing technologies involves particle fracture and the (KIC) via Eq. (1) (Broek, 1982; Tromans and Meech, creation of new surface area. Usually, fracture occurs 2001): because particles obtained from naturally occurring 2 1=2 1=2 1=2 minerals contain preexisting cracks (ﬂaws) which, dur- KICð1 m Þ ¼ðEGICÞ KIC Pam ð1Þ ing the comminution process, propagate in response to local tensile stress components acting normal to the where E is the tensile elastic modulus (Pa), m is PoissonÕs 1=2 crack plane. Tensile stresses are generated even when the ratio and KIC (Pa m ) is given by Eq. (2): external loading on the particle is predominantly com- 1=2 1=2 pressive (Hu et al., 2001; Tromans and Meech, 2001). KIC ¼ Y rcðaÞ Pam ð2Þ

During crack propagation, strain energy is released as where rc is the critical tensile stress (Pa) for crack new surface area is generated. Resistance to fracture propagation, a is the ﬂaw size (m) and Y is a shape under crack opening (mode I) conditions is termed the factor related to the crack geometry, e.g. Y has the value 1=2 fracture toughness, GIC. It is deﬁned as the critical en- ðpÞ for a straight through internal crack of length 2a and the value 2ðpÞ1=2 for an internal penny-shaped (disc-shaped) crack of radius a (Broek, 1982). The ð1 m2Þ term in Eq. (1) implies plane strain * Corresponding author. Tel.: +1-604-822-2378; fax: +1-604-822- conditions, which is the usual situation for brittle frac- 3619. E-mail addresses: [email protected] (D. Tromans), jam@mining. ture. For ideal brittle fracture (negligible plastic defor- ubc.ca (J.A. Meech). mation at the crack tip), GIC is equivalent to 2c, where c 1 Tel.: +1-604-822-3984; fax: +1-604-822-5599. is the surface energy per unit area (J m2). Consequently,

Nomenclature

a flaw size (m) R non-equilibrium average distance between an stoichiometric number of atoms/molecule atoms (m) B bulk elastic modulus (Pa) R0 average distance between atoms in unstrained D average particle diameter (m) (equilibrium) crystal (m) Di initial value of D (m) Rx average distance between atoms in x-direction Df final value of D (m) due to rx (m) 9 Daef average effective value of Df (m) RLimit limiting value of RxðR0 þ 2 Â 10 m) 17 e elementary charge (1:602177 Â 10 C) Umolecule crystal energy per molecule (J) E equilibrium tensile elastic modulus (Pa) UR crystal energy for N atoms at R (J)

ERx tensile elastic modulus, R ¼ Rx (Pa) Ue equilibrium crystal binding energy for N 12 3 E0 permittivity in vacuum (8:854188 Â 10 atoms at R0 (J m ) CV1 m1) V volume of N atoms (V ¼ NR3 m3) f area fraction of grain boundary coincident Wi bond work index for crushing and grinding sites (kWh/short ton) 1 Fr surface roughness factor (>1) ðWi ÞSI Wi in SI units (J kg ) 2 GIC critical crack energy release rate (J m ) x length of edge of unit cube containing N 2 ðGICÞGb GIC for grain boundary fracture (J m ) atoms at R0 (m) 2 ðGICÞIF GIC for interfacial fracture (J m ) Y a shape factor for flaws 2 ðGICÞIP GIC for interphase fracture (J m ) a largest common valence (charge) on ions 1 1 Hþ enthalpy of cation in gas phase (J mol ) DHf enthalpy of crystal formation (J mol ) 1 1 H enthalpy of anion in gas phase (J mol ) DSA increase in surface area/unit volume (m ) 1 1 Hcr crystal enthalpy (J mol ) DSEn increase in surface energy/unit mass (J kg ) k a fraction (0.25 to 0.3) ex tensile strain in x-direction 1=2 KI stress intensity (Pa m ) U a fraction 2a=D (<0.5) 1=2 2 KIC critical KI for crack propagation (Pa m ) c surface energy (J m ) 1=2 2 ðKICÞGb KIC for grain boundary crack (Pa m ) cGb grain boundary energy, cGb < c (J m ) 1=2 ðKICÞIF KIC for interfacial cracking (Pa m ) ls shear modulus (Pa) 1=2 ðKICÞIP KIC for interphase cracking (Pa m ) p circumference/diameter ratio of a circle L a crystal dimension, L0, under strain (m) h angle between loading axis and plane of flaw L0 equilibrium crystal dimension (m) (deg) m multiplying factor >1 q density (kg m3) M Madelung constant r tensile stress (Pa) 2 Ma combined Madelung constant, a =Man rc critical tensile stress for cracking (Pa) 3 1 MV molar volume (m mol ) rh hydrostatic compression stress (Pa) n a number >1 related to B rmax maximum theoretical tensile stress (Pa) 3 N atoms/m of unstrained crystal rP stress due to P (Pa) 23 1 NA Avogadro number (6:023 Â 10 mol ) rx tensile stress in x-direction (Pa) P loading force (N) a higher c should lead to increased toughness of brittle It is evident that continued development of quanti- materials (e.g. minerals). Frequently, KIC and GIC tative models of the comminution process, for purposes are used interchangeably as the measure of toughness, of power consumption and particle fracture, should in- because (1) they are directly related via Eq. (1) and (2) clude the fracture toughness of the minerals involved. experimental measurement of KIC is less difficult than An examination of the published literature indicates a GIC. In this manner, earlier studies based on KIC mea- dearth of information on mineral toughness, with the surements indicate that fracture toughness is one of the review by Rummel (1982) providing a useful but limited parameters affecting power consumption during rock set of data. The purpose of this study is to use the basic breakage (Bearman et al., 1991; Napier-Munn et al., physics and fundamental models developed for bonding 1999). Also, previous modelling studies by the authors in ionic crystals, particularly the Born model (Sherman, showed that the limiting particle size of finely milled 1932; Seitz, 1940) to develop theoretical relationships minerals is dependent upon KIC values (Tromans and from which quantitative estimates of GIC, KIC and c may Meech, 2001). be obtained for over 40 crystalline minerals. These in- D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1029 clude rock forming minerals (e.g. silicates) and those of equilibrium inter-atomic distance R0 (m) and the non- relevance to mineral processing and hydrometallurgy equilibrium average inter-atomic distance R (m) replace (e.g. oxides and sulphides). L0 and L in Eq. (1), respectively, leading to: " # Na2M e2 ðR Þn1 1 U 0 R ¼ n 2. Ionic crystal model an 4pE0 nR R " # e2 ðR Þn1 1 Bonding in some mineral crystals may be treated as NM 0 ¼ a n J ð4Þ essentially ionic, where the atoms behave as ions with 4pE0 nR R charges in accordance with their normal chemical valence state(s). This concept works well for simple halide where Ma is a combined Madelung constant equal to a2M=a , and a is the stoichiometric number of atoms crystals such as NaCl (halite) and CaF2 (fluorite). To a n n first approximation, many oxides may be treated as per molecule. (Note that theoretical estimates of M ionic crystals composed of metal cations and oxygen have been reported for several basic crystal structures anions O2, such as ZnO (zincite) and spinel-type (Sherman, 1932; Molieere, 1955) and spinel oxides (Ver- wey and Heilman, 1947). In the current study, M is structures related to MgAl2O4 (Sherman, 1932; Verwey a and Heilman, 1947). Ionic bonding is less common in calculated from elastic constants and thermodynamic sulphide minerals, where covalent bonding plays a larger data.) role (Vaughan and Craig, 1978). Consequently, any The value of N is obtained from: ionic similarity between oxides and sulphides is re- N ¼ anðNA=MVÞð5Þ stricted primarily to those of the alkali and Group II 3 where MV is the molar volume (m ), obtained by divi- metals (Sherman, 1932; Wells, 1962). In these cases the ding the molecular weight by the crystal density, and NA 2 sulphur is treated as the S anion, except for pyrite is the Avogadro number (6:023 Â 1023 mol1). (FeS2) where the sulphur is treated as the S anion. For At equilibrium, R ¼ R0 and Eq. (4) becomes: more complex cases, crystal anions may be treated as NM e2 1 negatively charged clusters of covalently bound atoms, U ¼ a 1 Jm3 ð6Þ 2 2 e R0 4pE0 n e.g. SiO4 in NaAlSiO4 (nepheline) and SO4 in CaSO4 (anhydrite). 3 where Ue is the crystal binding energy (J m ) at equili- The classical theory of ionic crystals, based on 1=3 brium (i.e. unstrained), and R0 ¼ N . the Born model (Sherman, 1932; Seitz, 1940) assumes Fig. 1 depicts the general shape of the resulting R–UR that the crystal energy is composed of an electrostatic curve obtained from Eq. (4), and shows the relative interaction between oppositely charged ions and a positions of Ue and R0. shorter-range repulsive term. Using this model, the average non-equilibrium crystal energy per molecule 2.1. Crystal binding energy Umolcule (J) may be expressed to a first approximation in the general form: Values of Ue for selected mineral crystals at 298 K " # were calculated from the molar enthalpies (J mol1)of 2 n1 e ðL0Þ 1 U 2M the individual cations (Hþ) and anions (H) in the gas molecule ¼ a n J ð3Þ 4pE0 nL L phase and the molar enthalpy (J mol1) of the crystal (H ) at 298 K via Eq. (7): where a is the largest common factor in the valences cr (number of charges) of all the ions, e is the elemen- 19 U (Jm-3) tary charge (1:602177 Â 10 C), E0 is the vacuum R permittivity (8:854188 Â 1012 CV1 m1), p is the cir- + cumference/diameter ratio of a circle, L0 (m) is a char- acteristic crystal dimension when the crystal is at equilibrium (i.e. unstrained) and L (m) is a non-equilibrium value, M is a constant per molecule (Madelung constant) dependent upon the crystal structure and 0 choice of L0, and n is a number >1 related to the com- RO pressibility. Ue For modelling of fracture toughness, it is necessary to - consider a large collection of atoms and express the non- 0 R (m) equilibrium crystal energy UR in terms of the number of 3 atoms N contained in a unit volume (1 m ) of crystal at Fig. 1. Schematic diagram showing the influence of average atomic equilibrium (unstrained crystal). Also, the average spacing, R, on the crystal energy per unit volume UR. 1030 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 P P 2 Hcr Hþ H DHf 3 orh o UR Ue ¼ ¼ Jm ð7Þ B ¼V ¼ V Pa ð8Þ MV MV oV oV 2 R¼R0 where DH is the molar enthalpy of formation of the f P where rh is hydrostatic compression stress (pressure) crystal from the gaseous ions, and indicates the and rh ¼oUR=oV . summation of enthalpies of the different cation and P Now oUR=oV ¼ðoUR=oRÞðoR=oV Þ, from which the anion componentsP (e.g. in MgAl2O4, Hþ ¼ HMg2þ þ second derivative is obtained (Sherman, 1932): 2HAl3þ and H ¼ 4HO2 ). oUR oR Values of Hþ, H and Hcr for the ionic and crystal o o2U oR oV oR o2U oR 2 o2R oU species considered are listed in Table 9 in the Appendix. R ¼ ¼ R þ R Values of DH , M , and the resulting value of U are oV 2 oR oV oR2 oV oV 2 oR f V e "# listed in Tables 1 and 2. Each MV was obtained from o2U oR 2 published crystallographic data (PDF, 1995). Table 1 ¼ R ð9Þ oR2 oV presents data for oxides and sulphides. Table 2 contains R¼R0 data for three halides and minerals whose cations are where oUR=oR ¼ 0 when R ¼ R0 (see Eq. (4) and Fig. 1). treated as groups of covalently bonded atoms contain- 2 2 Thus from Eqs. (8) and (9): ing oxygen (e.g. SO4 and SiO4 ). "# o2U oR 2 B ¼ V R ð10Þ 2.2. Bulk modulus of elasticity oR2 oV R¼R0 2 2 Consider a volume of crystalline mineral V , where From Eq. (4), o UR=oR may be obtained and inserted in 3 3 3 2 1 V ¼ NR (V ! 1m as R ! R0). The bulk modulus B Eq. (10), together with V ¼ NR , oR=oV ¼ð3R NÞ and (Pa) is given by the usual definition R ¼ R0 to yield the final equation for B:

Table 1

Values of DHf , MV and Ue for selected oxide and sulphide minerals at 298 K a 1 5 3 11 3 Mineral Formula Structure Ions DHf (kJ mol ) MV (10 m ) Ue (10 Jm ) þ 2 Cuprite Cu2O c Pn3m 2Cu ,O )3301.12 2.34382 )1.4084 Periclase MgO c Fm3m Mgþ,O2 )3915.74 1.12413 )3.4834 Lime CaO c Fm3m Ca2þ,O2 )3527.59 1.67603 )2.1047 Barium oxide BaO c Fm3m Ba2þ,O2 )3181.05 2.55891 )1.2431 Wustite FeO c Fm3m Fe2þ,O2 )3985.77 1.20291 )3.3134 Cobalt oxide CoO c Fm3m Co2þ,O2 )4046.45 1.16387 )3.4767 Nickel oxide NiO c Fm3m Ni2þ,O2 )4136.21 1.09708 )3.7702

2þ 2 Bromellite BeO h P63 mc Be ,O )4568.86 8.31173 )5.4969 2þ 2 Zincite ZnO h P63 mc Zn ,O )4124.53 1.4340 )2.87622

4þ 2 Rutile TiO2 tet P42/mnm Ti ,2O )12491.16 1.8800 )6.6442 4þ 2 Cassiterite SnO2 tet P42/mnm Sn ,2O )11769.59 2.1546 )5.4626

3þ 2 Corundum Al2O3 trig R3c 2Al ,3O )15547.20 2.55603 )6.0826 3þ 2 Hematite Fe2O3 trig R3c 2Fe ,3O )15153.76 3.0302 )5.0010 3þ 2 Eskolaite Cr2O3 trig R3c 2Cr ,3O )15336.01 2.90556 )5.2782 3þ 2 Titanium oxide Ti2O3 trig R3c 2Ti ,3O )15279.20 3.13563 )4.8728

2þ 3þ 2 Spinel MgAl2O4 c Fd3m Mg , 2Al ,4O )19485.91 3.975135 )4.9020 2þ 3þ 2 Hercynite FeAl2O4 c Fd3m Fe , 2Al ,4O )19585.31 4.08005 )4.8003 2þ 3þ 2 Chromite FeCr2O4 c Fd3m Fe , 2Cr ,4O )19372.94 4.42809 )4.3750 2þ 3þ 2 Nickel chromite NiCr2O4 c Fd3m Ni , 2Cr ,4O )19474.37 4.32945 )4.4981 2þ 3þ 2 Zinc ferrite ZnFe2O4 c Fd3m Zn , 2Fe ,4O )19275.17 4.52714 )4.2577 2þ 3þ 2 Magnetite Fe3O4 c Fd3m Fe , 2Fe ,4O )19166.40 4.45526 )4.3020 2þ 3þ 2 Chrysoberyl BeAl2O4 o Pbnm Be , 2Al ,4O )20133.21 3.43906 )5.8543 Galena PbS c Fm3m Pb2þ,S2 )3082.62 3.1494 )0.97880 Sphalerite ZnS c F43m Zn2þ,S2 )3621.36 2.3783 )1.5227 Metacinnabar b-HgS c F43m Hg2þ,S2 )3551.49 3.0167 )1.1773 2þ 2 Greenockite CdS h P63mc Cd ,S )3385.35 3.00478 )1.1267 2þ 2 Wurtzite ZnS h P63mc Zn ,S )3611.53 2.3824 )1.5159

2þ Pyrite FeS2 c Pa3 Fe ,2S )3063.90 2.39318 )1.2803 a c ¼ cubic, h ¼ hexagonal, trig ¼ trigonal, tet ¼ tetragonal, o ¼ orthorhombic. D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1031

Table 2

Values of DHf , MV and Ue for selected halide, sulphate and silicate minerals at 298 K a 1 5 3 11 3 Mineral Formula Structure Ions DHf (kJ mol ) MV (10 m ) Ue (10 Jm ) Halite NaCl c Fm3m Naþ,Cl )786.51 2.7018 )0.29110 Sylvite KCl c Fm3m Kþ,Cl )716.74 3.7500 )0.19113 Cesium chloride CsCl c Pm3m Csþ,Cl )667.285 4.22061 )0.15810

2þ Fluorite CaF2 c Fm3m Ca ,2F )2641.76 2.45457 )1.0763 2þ 2 ) ) Barite BaSO4 o Pbnm Ba ,SO4 2393.63 5.21892 0.45865 2þ 2 ) ) Anhydrite CaSO4 o Bmmb Ca ,SO4 2619.54 4.6009 0.56936 þ 3þ 2 ) ) Nepheline NaAlSiO4 hP63 Na ,Al , SiO4 9654.40 5.4073 1.7854 2þ 2 ) ) Cobalt olivine Co2SiO4 o Pmnb 2Co , SiO4 8557.23 4.45093 1.9226 2þ 2 ) ) Liebenbergite Ni2SiO4 o Pmnb 2Ni , SiO4 8721.64 4.25254 2.0509 2þ 2 ) ) Fayalite Fe2SiO4 o Pmnb 2Fe , SiO4 8448.30 4.63347 1.8233 2þ 2þ 2 ) ) Monticellite CaMgSiO4 o Pmnb Ca ,Mg , SiO4 7988.42 5.1184 1.5607 2þ 2 ) ) Forsterite Mg2SiO4 o Pmnb 2Mg , SiO4 8337.33 4.36654 1.9094 2þ 2 ) ) Wadsleyite b-Mg2SiO4 o Ibmn 2Mg , SiO4 8302.39 4.04747 2.0513 2þ 2 ) ) Ringwoodite c-Mg2SiO4 c Fd3m 2Mg , SiO4 8293.6 4.18098 1.9836

3þ 2 2 Andalusite Al2SiO5 o Pnnm 2Al ,O , SiO4 )15993.21 5.15262 )3.1039

2þ 3þ 2 ) ) Anorthite CaAl2Si2O8 tr P1Ca, 2Al , 2SiO4 20069.79 10.0764 1.9918 2þ 3þ 2 ) ) Grossularite Ca3Al2Si3O12 c Ia3d 3Ca , 2Al , 3SiO4 27776.06 12.5226 2.2181 2þ 3þ 2 ) ) Pyrope Mg3Al2Si3O12 c Ia3d 3Mg , 2Al , 3SiO4 28689.34 11.3143 2.5357 2þ 3þ 2 ) ) Almandine Fe3Al2Si3O12 c Pa3 3Fe , 2Al , 3SiO4 28886.43 11.3590 2.5431 2þ 3þ 2 ) ) Andradite Ca3Fe2Si3O12 c Ia3d 3Ca , 2Fe , 3SiO4 27371.19 13.1995 2.0737 a c ¼ cubic, h ¼ hexagonal, o ¼ orthorhombic, tr ¼ triclinic (quasi-monoclinic).

M ðn 1Þ e2 Note that reported measurements of E for some pore- B ¼ a ð11Þ 4 free, polycrystalline oxides are 310.9 GPa for MgO, 9ðR0Þ 4pE0 123.5 GPa for ZnO, 284.2 GPa for TiO2 and 402.8 GPa The isotropic value of B, equivalent to the averaged for Al2O3 (Wachtman, 1996). These are very close to the modulus for the polycrystalline mineral, is obtained values listed in Table 3 and lend confidence to the from the corresponding isotropic tensile elastic modulus computation methods used to obtain E. E and PoissonÕs ratio m via Eq. (12) (Wachtman, 1996): E B ¼ ð12Þ 2.3. Evaluation of Ma, n and tensile behaviour 3ð1 2mÞ After inserting Eq. (12) in (11), and rearranging, n is At this stage, only two parameters, Ma and n, in Eqs. obtained: (6) and (13) remain unknown. All others are either es- tablished constants (e), or obtainable from crystallo- 4 3EðR0Þ 4pE0 n graphic data (N and R0), thermodynamic data (Ue), and ¼ 1 þ 2 ð13Þ ð1 2mÞMa e stiffness/compliance data (E and m). Consequently, Eqs. The PoissonÕs ratio is obtained from E and the isotropic (6) and (13) are two simultaneous equations, which may elastic shear modulus l via the relationship m ¼ be solved by numerical analysis to obtain Ma and n. s Resulting computed values are listed in Tables 3 and 4, ðE=2lsÞ1 (Wachtman, 1996). Values of E and ls were computed from anisotropic stiffness and compliance together with the corresponding N, R0, E and m. constants for single crystals compiled mainly by Tensile behaviour may be analysed by subjecting a Hearmon (1979, 1984), together with some constants unit cube of material containing N atoms to a uniaxial compiled by Bass (1995). Computations were conducted stress. The initial length of the cube has the length 1=3 in two ways: (1) from stiffness constants assuming uni- x ¼ N R0 ¼ 1 m. When a uniaxial tensile stress is ap- o form stress in the polycrystalline aggregate and (2) from plied in the x-direction, x ! x þ x and R in the x- o o 1=3o compliance constants assuming uniform strain in the direction (Rx) goes from R0 ! R0 þ R and x ¼ N R. aggregate. The two resulting values for each modulus, E The variation in uniaxial stress rx as the polycrystalline mineral aggregate is extended in the x-direction is ob- and ls, were then averaged. The matrix and analytical procedures required for the computations are outlined tained from UR in Eq. (4) via the differentiation: by Gerbrande (1982) and Wachtman (1996). Resulting o o E and m for the minerals studied are listed in Tables 3 1 2m UR 1 2m UR r ¼ ¼ ð14Þ and 4. x 3 ox 3N 1=3 oR R¼Rx 1032 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041

Table 3

Values of N, R0, E, m, Ma, n and rmax=E ratio for oxide and sulphide minerals at 298 K 28 3 10 Mineral Formula N (10 m ) R0 (10 m) E (GPa) m Ma n rmax=E a Cuprite Cu2O 7.7092 2.3496 29.91 0.4542 2.1728 6.9606 0.0790 Periclase MgO 10.7159 2.1054 307.18b 0.1794 3.9154 4.1259 0.1068 Lime CaO 7.1872 2.4052 196.96b 0.2145 3.8322 4.9177 0.0971 Barium oxide BaO 4.7075 2.7695 92.25a 0.2805 3.9485 5.0718 0.0955 Wustite FeO 10.0141 2.1534 127.53a 0.3830 3.873 4.9364 0.0969 Cobalt oxide CoO 10.34990 2.1299 189.40a 0.3282 3.9269 4.7553 0.0990 Nickel oxide NiO 10.9801 2.0883 232.46b 0.2834 4.0585 4.2704 0.1050 Bromellite BeO 14.4928 1.9038 396.07b 0.2201 4.2239 3.8612 0.1106 Zincite ZnO 8.4002 2.2833 126.74b 0.3527 4.3604 4.4875 0.1022

a Rutile TiO2 9.6112 2.1831 285.10 0.2755 10.046 2.86650 0.1276 b Cassiterite SnO2 8.3864 2.2846 262.49 0.2915 9.075 3.4575 0.11682

b Corundum Al2O3 11.7819 2.0398 399.76 0.2335 6.256 3.6989 0.1130 b Hematite Fe2O3 9.9384 2.1589 212.35 0.1379 10.914 1.7588 0.1553 b Eskolaite Cr2O3 10.3646 2.1289 314.56 0.2761 6.268 3.9935 0.1087 b Titanium oxide Ti2O3 9.60413 2.1836 244.45 0.3003 6.537 3.7679 0.11201

b Spinel MgAl2O4 10.6062 2.1126 273.83 0.2660 5.872 3.5810 0.1149 b Hercynite FeAl2O4 10.3335 2.1310 222.10 0.3199 5.795 3.8527 0.1107 b Chromite FeCr2O4 9.5213 2.1900 268.65 0.2804 5.727 4.1945 0.1059 b Nickel chromite NiCr2O4 9.7382 2.1736 106.62 0.4466 5.121 6.6564 0.0813 a Zinc ferrite ZnFe2O4 9.3130 2.2062 241.16 0.2899 5.8082 4.0437 0.10800 b Magnetite Fe3O4 9.4632 2.1944 230.33 0.2616 6.1495 3.3686 0.1183 a Chrysoberyl BeAl2O4 12.2595 2.0130 389.79 0.2289 5.7187 3.6845 0.1132 Galena PbS 3.8249 2.9680 80.04a 0.2706 4.049 5.3476 0.0926 Sphalerite ZnS 5.0649 2.7028 82.73a 0.3202 4.519 4.5320 0.1016 Metacinnabar b-HgS 3.9931 2.9257 48.51b 0.3802 4.638 5.1584 0.0945 Greenockite CdS 4.0089 2.9218 46.61a 0.3759 4.449 5.0011 0.0962 Wurtzite ZnS 5.0563 2.7043 86.85b 0.3037 4.5545 4.37870 0.1035

b Pyrite FeS2 7.5502 2.3660 296.05 0.1553 1.931 10.0622 0.0620 a Hearmon (1984). b Hearmon (1979).

The ð1 2mÞ=3 term is included in Eq. (14) in recogni- of Tables 3 and 4. The ratio ranged from 0.0606 to tion that extension in the x-direction is accompanied by 0.141 with an average and standard deviation of 0:095 a Poisson contraction in the other two orthogonal di- 0:02. rections. Completing the differentiation in Eq. (14) leads The tensile elastic modulus ERx at any value of Rx is to: obtained from Eq. (15) by differentiation: " # 2 n1 1 2m e 1 ðR0Þ or or or or r N 2=3M Pa x x 1=3 x x x ¼ ð aÞ 2 nþ1 ERx ¼ ¼ x ¼ N Rx ¼ Rx 3 4pE0 ðRxÞ ðRxÞ oex ox ox oRx ð15Þ ð16Þ N R M n where , 0, a, and m are obtained from Tables 3 and where ex is the tensile strain in the x-direction (i.e. ox=x). 4. (Note that rx ! 0asRx ! R0, consistent with reality.) Thus, from Eqs. (15) and (16): Eq. (15) indicates that rx rises rapidly to a peak value, 2 followed by an asymptotic decrease as Rx increases from 1 2m e E R N 2=3M R . Representative curves showing this behaviour for Rx ¼ xð aÞ 0 " 3 4p#E0 several minerals are presented in Figs. 2 and 3. Each n1 peak stress corresponds to the maximum theoretical ðn þ 1ÞðR0Þ 2 Â nþ2 3 Pa ð17Þ tensile stress, rmax, of the flaw-free mineral crystal in the ðRxÞ ðRxÞ absence of any plasticity effects. The value of Rx at rmax is readily obtained from Eq. (15) via the condition Importantly, the equilibrium value of the tensile mod- orx=oRx ¼ 0, from which rmax is obtained after insertion ulus E,asRx ! R0, is obtained from Eq. (17) by sub- in Eq. (15). The rmax=E ratio is listed in the last column stituting R0 for Rx in Eq. (17): D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1033

Table 4

Values of N, R0, E, m, Ma, n and rmax=E ratio for selected halide, sulphate and silicate minerals at 298 K 28 3 10 Mineral Formula N (10 m ) R0 (10 m) E (GPa) m Ma n rmax=E Halite NaCl 4.4585 2.8201 36.87a 0.254 0.9164 7.7245 0.0740 Sylvite KCl 3.2123 3.1458 24.09a 0.276 0.92 8.4573 0.0697 Cesium chloride CsCl 2.8541 3.2722 25.40a 0.268 0.869 10.3857 0.0606 b Fluorite CaF2 7.3614 2.3861 109.26 0.288 1.7565 7.1862 0.0775

c Barite BaSO4 6.9244 2.4352 60.03 0.316 0.771 10.694 0.0594 a Anhydrite CaSO4 7.8546 2.3350 74.36 0.269 0.8317 8.4808 0.0696

a Nepheline NaAlSiO4 7.7970 2.3408 76.014 0.218 4.155 2.2681 0.1410 b Cobalt olivine Co2SiO4 9.4724 2.1937 163.48 0.316 2.2547 6.9411 0.0792 d Liebenbergite Ni2SiO4 9.9143 2.1606 205.37 0.292 2.249 7.2194 0.0772 b Fayalite Fe2SiO4 9.0992 2.2233 135.94 0.335 2.264 6.7980 0.0802 d Monticellite CaMgSiO4 8.2371 2.2983 141.58 0.278 2.2565 6.1154 0.0856 b Forsterite Mg2SiO4 9.6555 2.1798 201.39 0.242 2.233 6.1216 0.0855 d Wadsleyite b-Mg2SiO4 10.4166 2.1253 278.07 0.237 2.083 7.7418 0.0739 d Ringwoodite c-Mg2SiO4 10.0840 2.1484 292.12 0.236 2.081 8.3534 0.0703

b Andalusite Al2SiO5 9.3514 2.2031 246.54 0.246 4.03 4.6854 0.0997

a Anorthite CaAl2Si2O8 7.7705 2.3434 103.28 0.295 3.537 3.7892 0.1116

a Grossularite Ca3Al2Si3O12 9.6194 2.1825 263.35 0.242 2.55 6.9153 0.0793 b Pyrope Mg3Al2Si3O12 10.6467 2.1099 232.08 0.267 2.6228 5.8979 0.0874 a Almandine Fe3Al2Si3O12 10.6048 2.1127 241.73 0.276 2.6044 6.3754 0.0834 b Andradite Ca3Fe2Si3O12 9.1261 2.2211 219.04 0.235 2.6265 5.9838 0.0867 a Hearmon (1979). b Hearmon (1984). c Gerbrande (1982). d Bass (1995).

σ x (GPa) σx (GPa) 50 30 8

25 7 40 1. Cuprite 7 2. Galena 1. Halite 6 6 3. Zincite 20 2. Anhydrite 4. Pyrite 30 5 3. Fluorite 5. Hercynite 4. Anorthite 5 6. Periclase 15 5. Forsterite 7. Rutile 6. Pyrope 20 4 8. Corundum 4 7. Andalusite 10 3 3 10 2 2 5 1 1 0 0 0 5E-10 1E-9 1.5E-9 2E-9 2.5E-9 0 5E-10 1E-9 1.5E-9 2E-9 2.5E-9 Rx (m) Rx (m)

Fig. 2. Computed uniaxial tensile stress behaviour of defect-free oxide Fig. 3. Computed uniaxial tensile stress-extension behaviour of defect- and sulphide minerals as the average distance between atoms in the x- free halide, sulphate and silicate minerals as the average distance direction, Rx, is increased from Rx ¼ R0 when rx ¼ 0. between atoms in the x-direction, Rx, is increased from Rx ¼ R0 when rx ¼ 0.

"# "# 2 2 1 2m 2=3 e ðn 1Þ 1 2m e ðn 1Þ E ¼ ðN MaÞ 2 Pa ð18Þ E ¼ ðMaÞ 4 Pa ð19Þ 3 4pE0 ðR0Þ 3 4pE0 ðR0Þ

1=3 2 2 Recognising that ðN R0Þ ¼ 1m , multiplication of Eq. Upon rearrangement, Eq. (19) is seen to be identical to 1=3 2 (18) by 1=ðN R0Þ produces: Eq. (13) and conﬁrms the validity of Eq. (15) for rx. 1034 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041

3. Application to fracture toughness Inspection of Figs. 2 and 3 indicates RLimit is reached when the average separation between atoms in the x- 9 3.1. Intragranular fracture direction increases by 2 nm, i.e. RLimit ¼ðR0 þ 2 Â 10 Þ m. Resulting values of GIC, based on this limit and Eq. The area beneath each Rx–rx curve in Figs. 2 and 3 (21), are listed in Tables 5 and 6. The necessary N, R0, represent the average work (energy) per unit area of Ma, n and m were obtained from Tables 3 and 4. crack plane that is required for ideal intragranular The critical stress intensity for intragranular brittle brittle fracture (i.e. bond breakage with no plastic de- crack propagation KIC was obtained from Eq. (1), and formation). This work is equivalent to the brittle frac- the average surface energy of each mineral was obtained ture toughnessZ GIC, which may be expressed formally: from the condition c ¼ GIC=2 for ideal brittle fracture. Rx¼RLimit 2 These are listed in Tables 5 and 6. GIC ¼ rxoRx Jm ð20Þ Note that the computed GIC, KIC and c, in Tables 5 Rx¼R 0 and 6 are average values for intragranular cracking on a where RLimit is an upper value of Rx beyond which the randomly oriented plane in polycrystalline material. It is ionic model becomes invalid and rx becomes negligible. recognised that c may be crystallographically aniso- After substituting Eq. (15) for rx, integration of Eq. (20) gives G : tropic in the same manner that elastic constants of single "IC 2 crystals are anisotropic. For example, with crystals 1 2m 2=3 e GIC ¼ ðN MaÞ having the NaCl structure (e.g. the Fm3m oxides in 3 4pE0 !# Table 1), theoretical estimates indicate that for the R R n1 x¼ Limit (1 1 0) plane the surface energy is 2.7 times that of the ðR0Þ 1 2 Â n Jm ð21Þ (1 0 0) plane (Seitz, 1940). Also, recent ﬁrst-principles nðRxÞ Rx Rx¼R0 calculations have suggested that the basal plane in co-

Table 5 Computed toughness values for intragranular and grain boundary cracking, plus surface and grain boundary energies of oxide and sulphide minerals at 298 K 2 2 Mineral Formula Intragranular crack c (J m ) Grain boundary crack cGb (J m ) 2 1=2 2 1=2 GIC (J m ) KIC (MPa m ) ðGICÞGb (J m ) ðKICÞGb (MPa m )

Cuprite Cu2O 0.886 0.163 0.4428 0.769 0.152 0.117 Periclase MgO 13.704 2.052 6.852 12.293 1.943 1.411 Lime CaO 8.335 1.281 4.1680 7.382 1.206 0.953 Barium oxide BaO 4.274 0.623 2.137 3.768 0.590 0.506 Wustite FeO 4.885 0.789 2.443 4.333 0.743 0.552 Cobalt oxide CoO 7.449 1.188 3.725 6.624 1.120 0.825 Nickel oxide NiO 9.964 1.522 4.982 8.921 1.440 1.043 Bromellite BeO 17.238 2.613 8.619 15.545 2.481 1.693 Zincite ZnO 5.599 0.842 2.799 4.990 0.795 0.609

Rutile TiO2 18.445 2.293 9.223 16.885 2.194 1.560 Cassiterite SnO2 14.845 1.974 7.423 13.438 1.878 1.407

Corundum Al2O3 19.250 2.774 9.625 17.387 2.636 1.863 Hematite Fe2O3 20.750 2.099 10.375 19.438 2.032 1.312 Eskolaite Cr2O3 14.617 2.144 7.309 13.135 2.033 1.482 Titanium oxide Ti2O3 12.269 1.732 6.135 11.059 1.644 1.210

Spinel MgAl2O4 14.014 1.959 7.007 12.675 1.863 1.339 Hercynite FeAl2O4 10.687 1.541 5.344 9.624 1.462 1.063 Chromite FeCr2O4 12.209 1.811 6.104 10.934 1.714 1.275 Nickel chromite NiCr2O4 3.080 0.573 1.540 2.684 0.535 0.396 Zinc ferrite ZnFe2O4 11.420 1.660 5.710 10.249 1.572 1.171 Magnetite Fe3O4 12.897 1.724 6.449 11.700 1.642 1.197 Chrysoberyl BeAl2O4 18.624 2.694 9.312 16.828 2.561 1.796 Galena PbS 3.736 0.547 1.868 3.278 0.512 0.458 Sphalerite ZnS 4.180 0.588 2.090 3.712 0.554 0.468 Metacinnabar b-HgS 2.316 0.335 1.158 2.037 0.314 0.279 Greenockite CdS 2.289 0.327 1.145 2.017 0.307 0.272 Wurtzite ZnS 4.536 0.628 2.268 4.037 0.592 0.499

Pyrite FeS2 6.143 1.349 3.072 5.233 1.245 0.910 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1035

Table 6 Computed toughness values for intragranular and grain boundary cracking, plus surface and grain boundary energies of selected halide, sulphate and silicate minerals at 298 K 2 2 Mineral Formula Intragranular crack c (J m ) Grain boundary crack cGb (J m ) 2 1=2 2 1=2 GIC (J m ) KIC (MPa m ) ðGICÞGb (J m ) ðKICÞGb (MPa m ) Halite NaCl 1.155 0.206 0.577 0.993 0.191 0.162 Sylvite KCl 0.758 0.135 0.379 0.647 0.125 0.111 Cesium chloride CsCl 0.676 0.131 0.338 0.570 0.120 0.106

Fluorite CaF2 3.179 0.589 1.589 2.754 0.548 0.425

Barite BaSO4 1.203 0.269 0.602 1.021 0.248 0.182 Anhydrite CaSO4 1.805 0.366 0.902 1.550 0.340 0.255

Nepheline NaAlSiO4 6.412 0.698 3.206 5.933 0.672 0.479 Cobalt olivine Co2SiO4 4.570 0.864 2.285 3.973 0.806 0.597 Liebenbergite Ni2SiO4 5.450 1.058 2.725 4.729 0.985 0.721 Fayalite Fe2SiO4 3.924 0.730 1.962 3.414 0.681 0.510 Monticellite CaMgSiO4 4.665 0.813 2.332 4.081 0.760 0.584 Forsterite Mg2SiO4 6.329 1.129 3.164 5.542 1.056 0.787 Wadsleyite b-Mg2SiO4 6.792 1.374 3.396 5.872 1.278 0.920 Ringwoodite c-Mg2SiO4 6.685 1.397 3.343 5.756 1.297 0.929

Andalusite Al2SiO5 10.130 1.580 5.065 9.011 1.491 1.119 Anorthite CaAl2Si2O8 5.478 0.752 2.739 4.930 0.714 0.548

Grossularite Ca3Al2Si3O12 7.356 1.392 3.678 6.396 1.298 0.960 Pyrope Mg3Al2Si3O12 7.348 1.306 3.674 6.452 1.224 0.896 Almandine Fe3Al2Si3O12 7.102 1.310 3.551 6.208 1.225 0.894 Andradite Ca3Fe2Si3O12 7.154 1.252 3.577 6.271 1.172 0.883 rundum has a c-value ranging from 2.13 to 3.5 J m2, teractions) simulate behaviour of close-packed atom depending on the proportion of oxygen to aluminium structures. These bubble raft studies indicate that the atoms on the exposed surface (Tepesch and Quong, width of the boundary region is of the order of two atom 2000). diameters, consistent with high resolution transmission The modelling and computational procedures in the electron microscopy images of grain boundaries in Ni present study have not incorporated any crystallo- (Benedictus et al., 1994) and Si (Shen et al., 1995). A graphic effects on GIC and c. This is not a serious geometric construction of a simple 25° tilt boundary, drawback. During crack propagation in a polycrystal- AB, between two idealised close-packed grains of the line material, the macroscopic crack plane tends to re- same crystalline phase is shown in Fig. 4. Examination main normal to the applied tensile stress component, of this figure indicates that the boundary exhibits re- even though the crystallographic orientation of the grain gions of atom coincidence and non-coincidence, con- with respect to the tensile component changes as the sistent with the appearance of boundaries in bubble rafts crack propagates from grain to grain. The result is (Bragg and Nye, 1947). In the coincident regions, the usually a stepped fracture surface at the microscopic atom separation across the boundary is the same as that level (see Tromans and Meech, 2002) that is composed in the crystal (i.e. R0). In the non-coincident regions, the of crystalline facets of lower GIC (lower c) with higher separation across the boundary is > R0. Further analysis energy step edges. The overall result for many brittle is simplified by assigning a fractional area (f ) of grain minerals will be the production of a macroscopic in- boundary to co-incident sites and a fraction ð1 f Þ to tragranular crack plane whose average GIC (average c)is non-coincident sites. The separation of atoms across the likely to be near that obtained from Eq. (21). boundary in the ð1 f Þ fraction is treated as an averaged value mR0, where m is a multiplying factor >1.

3.2. Grain boundary fracture Consequently, ðGICÞGb is obtained by modifying Eq. (20) to reflect the two fractional areas and m: The fracture toughness for grain boundary cracking Z Rx¼RLimit ðGICÞ is expected to be lower than that for random o Gb ðGICÞGb ¼ðf Þ rx Rx plane intragranular cracking, because atoms are ar- Rx¼R0 Z ranged irregularly in the grain boundary region. Such Rx¼RLimit o 2 irregularities are readily seen in the floating bubble raft þð1 f Þ rx Rx Jm ð22Þ R ¼mR experiments of Bragg and Nye (1947), where surface x 0 9 tension effects between bubbles (nearest neighbour in- where RLimit ¼ðR0 þ 2 Â 10 Þ m. 1036 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041

timates for high angle boundaries in pure, single-phase minerals, with no segregation of impurity atoms at the boundary (i.e. ‘‘clean’’ boundaries). The corresponding critical stress intensity factors for

grain boundary cracking ðKICÞGb in the pure minerals are also listed in Table 4, being obtained from the analogous equation to that of Eq. (1): 2 1=2 1=2 1=2 ðKICÞGbð1 m Þ ¼ EðGICÞGb ðKICÞGb Pam ð24Þ

The grain boundary energy, cGb, corresponding to the boundary condition f ¼ 0:5andm ¼ 1:5 is also listed in Tables 5 and 6, being obtained from the diﬀerence in toughness between intragranular and grain boundary cracking:

2 cGb ¼ GIC ðGICÞGb Jm ð25Þ Fig. 4. Schematic diagram of a 25° tilt grain boundary, AB, showing regions of coincidence and non-coincidence between atoms in the 3.3. Interfacial fracture neighbouring grains. For purposes of the current analysis, interfacial After substituting Eq. (15) for r , integration of x fracture is defined as the propagation of cracks along the the two terms in Eq. (22) gives the general form of interface between poorly bonded subparticles within a : ðGICÞGb larger mineral particle. Such interfaces are related to the ðGICÞ geological history of the mineral and are likely to be Gb 2 present in minerals formed via deposition processes, 1 2m 2=3 e ¼ ðN MaÞ such as sedimentary rocks and conglomerates. It is not 3 4pE0 8" !# possible to analyse this situation precisely, due to the R R < n1 x¼ Limit ðR0Þ 1 many variations of ‘‘poor bonding.’’ However, in prin- Â : f n ciple, the fracture toughness for interfacial cracking nðRxÞ Rx Rx¼R0 may be treated in a similar manner to Eq. (23) by " !# 9 ðGICÞIF Rx¼RLimit = n1 replacing ðGICÞGb with ðGICÞIF and recognising that, due ðR0Þ 1 2 þð1 f Þ n ; Jm to poor interfacial bonding, f ! 0 and m 1. The re- nðRxÞ Rx Rx¼mR0 sult is that ðGICÞIF ðGICÞGb and ðGICÞIF GIC. Con- ð23Þ sequently, cracking along these interfaces will occur preferentially whenever they are present. In practice, f will depend on the characteristics of the crystal structure, the misorientation angle across the boundary and the type of boundary (e.g. twist, tilt or 3.4. Interphase fracture mixed). Consequently, even in the same polycrystalline mineral phase, no two boundaries will be identical. In the present analysis, interphase fracture is defined However, if m can be assumed to be relatively constant, as cracking along the boundary between two different and independent of f for high angle boundaries, then crystalline phases. Many natural mineral bodies are

Eq. (23) shows that ðGICÞGb has a maximum value when multiphase composites, being composed of one (or f ¼ 0 and a minimum when f ¼ 1, with f ¼ 0:5 rep- more) mineral phase(s) dispersed in the matrix of a resenting an average condition. Furthermore, examina- different mineral phase. Bonding across the bound- tion of the boundary AB in Fig. 4 indicates m 1:5, ary between the different phases is stronger than that consistent with the general appearance of boundaries in for interfacial boundaries but not as strong as that the bubble rafts of Bragg and Nye (1947), and f ¼ 0:54 across grain boundaries in the pure, single-phase min- (close to the average condition of 0.5). Consequently, eral. Modelling of interphase boundaries is difficult, due it is possible to make useful relative comparisons of to the variety of different mineral/mineral phase bound-

ðGICÞGb for different minerals by inserting f ¼ 0:5and ary combinations. However, interphase cracking tough- m ¼ 1:5 in Eq. (23), together with RLimit ¼ðR0 þ 2 Â ness ðGICÞIP may be assessed in a semi-formal manner by 9 10 Þ m. The resulting average ðGICÞGb values are listed replacing ðGICÞGb in Eq. (23) by ðGICÞIP and then rec- in Tables 5 and 6. These values should be taken as es- ognising that it is very likely that 0:5 > f > 0 and m > 1 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1037 but not 1. The overall result in terms of relative 4.2. Relevance to comminution toughness values becomes: 4.2.1. Energy to create new surface G > ðG Þ > ðG Þ ðG Þ ð26Þ IC IC Gb IC IP IC IF For purposes of estimating the total energy required Following from Eq. (1), the corresponding KIC values to generate new surface area during comminution of for the different crack paths should follow the same polycrystalline minerals (i.e. the fracture energy), either decreasing order as that shown in Eq. (26) (i.e. KIC > ðGICÞGb or GIC is adequate. Consider an initial spherical ðKICÞGb > ðKICÞIP ðKICÞIF). The relative fracture tough particle of diameter Di (m) that is fractured (milled) into ness for interphase cracking is likely to be an important very small particles with an average final diameter of Df 3 consideration in the liberation of different mineral (m). The number of particles produced is ðDi=Df Þ and phases during milling. The relatively low value of the resulting increase in surface area per unit volume

ðGICÞIF implies that interfacial cracking will be the ﬁrst DSA is obtained: particle fracture process to occur during milling, whenever such interfaces are present. 1 1 1 DSA ¼ 6Fr m ð27Þ Df Di

4. Discussion where Fr is a surface roughness factor (>1) introduced in recognition of the fact that milled particles will not be 4.1. Comments on toughness estimates perfect spheres, but will exhibit roughened (stepped and faceted) surfaces with a surface area/diameter ratio that To the authorsÕ knowledge, the estimates in Tables 5 is Fr times that of spheres. and 6 are the first time that theoretical toughness and From Eq. (27), the increase in surface energy per unit mass DS (J kg1) is obtained surface energy computations have been attempted on En such a large number of minerals. The computed tough- DSAc 6Frc 1 1 DSEn ¼ ¼ ness values are the lowest values possible, based on ideal q q Df Di brittle fracture in pure, single-phase polycrystalline 6Frc 1 minerals. They are associated with bond breakage in the Jkg ; when Di Df ð28Þ qD absence of any accompanying plastic deformation. With f 3 reference to Table 5, GIC for intragranular fracture where q is the mineral density (kg m ). (Note that c is 2 2 ranges from 0.886 J m for cuprite to 20.75 J m for GIC=2 for intragranular cracking and ðGICÞGb=2 for hematite, a factor of 23.4 times. Similarly for Table 6, grain boundary cracking.) 2 2 GIC ranges from 0.676 J m for CsCl to 10.13 J m for Eq. (28) indicates that DSEn is likely to be propor- andalusite, a factor of 15 times. A lower toughness is tional to c=q, assuming Fr does not vary widely between predicted for grain boundary cracking, indicating that different populations of particles. Using computed val- failure along suitably oriented high angle grain bound- ues of c from Table 5, and q from crystal data (PDF, aries is a likely occurrence in polycrystals. However, the 1995), the c=q ratio for, galena, sphalerite and corun- average boundary toughness is not markedly lower than dum were calculated and normalised with respect to the highest c=q ratio. The results are listed in Table 7 and that for intragranular cracking, For example, ðGICÞGb is show that for the same Df , fracture of corundum re- 10–14% lower than GIC,andðKICÞGb is 5–7% lower than KIC. quires approximately 10 times as much fracture energy There are scant experimental studies of KIC or GIC to as galena and five times as much fracture energy as compare with the estimates in Tables 5 and 6. Reported sphalerite. King et al. (1997) measured the fracture 1 KIC values for fine-grained polycrystalline Al2O3 (co- energy (J kg ) via single particle fracture tests on a rundum) are in the range 2.5–4.5 MPa m1=2 (Wachtman, population of particles of the same minerals as those in Table 7 by the drop weight technique. Their graphed 1996). The measured GIC (or ðGICÞGbÞ for polycrystalline MgO (periclase) is 8–10 J m2 (Davidge, 1974). Data energy values at the 97% cumulative distribution were from single crystal experiments (McColm, 1990) suggest normalised and listed for comparison with the norma- 1=2 that KIC is >1.2 MPa m for polycrystalline MgO lised c=q values in Table 7. The agreement between their 1=2 (periclase) and >2.5 MPa m for polycrystalline TiO2 experimental data and the computed c=q ratios is en- (rutile). An experimental value of c for the (1 1 0) plane couragingly good. in halite (NaCl) is 0.33 J m2 (Gilman, 1959). All of these experimental data are sufficiently close to the 4.2.2. Comminution efficiency corresponding data in Tables 5 and 6 to indicate with Bond (1961) has determined and listed the average fair confidence that the estimates in Tables 5 and 6 are standard work index Wi for the crushing and grinding of of the right order of magnitude in both value and with numerous mineral bodies, where Wi is defined as the respect to each other. work input in kWh/short ton (short ton ¼ 2000 lb) that 1038 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041

Table 7 Normalised c=q for galena, sphalerite and corundum Mineral c (J m2) q (103 kg m3) c=q (104 Jmkg1) Normalised c=q Normalised fracture energya Galena (PbS) 1.868 7.597 2.459 0.102 0.1 Sphalerite (ZnS) 2.090 4.097 5.1 0.21 0.2 Corundum (Al2O3) 9.625 3.989 24.13 1.0 1 a Data from King et al. (1997). is required to reduce the feed from an infinitely large 4.2.3. Impact efficiency particle size to 80% passing 100 lm. (N.B. conversion of Factors affecting impact efficiency may be understood

Wi to SI units, ðWi ÞSI, requires that ðWi ÞSI ¼ Wi ð3:968 Â qualitatively in terms of fracture mechanics concepts, 3 1 10 Þ Jkg .) Bond (1961) cautioned the use of his Wi based on the presence of inherent flaws in the particles. values, because large variations may be encountered Fig. 5a is a schematic diagram of an approximately between the same materials. This probably occurs be- spherical mineral particle, of average diameter D, con- cause the ‘‘80% passing 100 lm’’ criterion provides no taining a penny-shaped flaw (crack) of radius a (viewed information on the actual distribution of particle sizes, along the plane of the flaw). The particle is subjected to which may be expected to vary between >100 and opposing impact forces P, such as occur when particles <20 lm, with an average effective finished diameter of are compressed between two balls in a ball mill, or two Daef . Within these limitations, Eq. (28) may be used to rods in a rod mill. These forces generate a compressive estimate the energy efficiency of particle fracture during stress rP in the particle along the axis of impact and a comminution via the ratio DSEn=ðWi ÞSI: tensile stress krP (k < 1) within the particle that is normal to the impact axis. This is a well-recognised be- DSEn 6Frc % efficiency ¼ Â 100 Â 100 ð29Þ haviour of spherical and cylindrical particles, which has ðÞWi SI qDaef ðWi ÞSI where Df is replaced by Daef (<100 lm). Table 8 shows estimated efficiencies for crushing and grinding of several mineral bodies, using average Wi values obtained by Bond (1961), c from Tables 5 and 6, q for the pure minerals (PDF, 1995), an approximate value of 3 for Fr and an assumed value of 40 lm for Daef . The resulting efficiencies should be taken as approximate values, but it is evident that they are all very low, of the order of 1%. The low efficiency of comminution processes has been long recognised (Bond, 1961; Austin, 1984; King et al., 1997). Apart from energy losses due to particle deformation, particularly at very small particle sizes (Tromans and Meech, 2001), much Fig. 5. Schematic diagram of small particle, average diameter D, of the inefficiency arises from particles receiving nu- containing a flaw (crack) of radius a subjected to compressive force P. merous impacts before an impact of sufficient force is (a) Flaw inclined at angle h with respect to the loading axis. (b) Plane received to cause particle fracture (e.g. King et al., 1997). of flaw parallel to the loading axis (h ¼ 0).

Table 8 Estimated crushing and grinding eﬃciency based on c, q and the Bond work index 2 3 3 1 a 1 Mineral c (J m ) q (10 kg m ) Wi (kWh ton ) ðWi ÞSI (kJ kg ) Eﬃciency (%) Feldsparb 2.739 2.761 11.67 46.31 0.97 Galena 1.868 7.597 10.19 40.43 0.27 Garnetc 3.678 3.597 12.37 49.1 0.94 Hematite 10.375 5.270 12.68 50.31 1.76 Magnetite 6.449 5.197 10.21 40.51 1.38 Pyrite 3.072 5.013 8.9 35.32 0.78 Rutile 9.223 4.250 12.12 48.1 2.03 Fluorite 1.589 3.181 9.76 38.73 0.58 a BondÕs data (1961). b Anorthite chosen as representative. c Grossularite chosen as representative. D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1039

1=2 been treated in detail by Hu et al. (2001). Previously, the the KIC values of zincite (0.842 MPa m ) and rutile authors approximated k to PoissonÕs ratio m (Tromans (2.293 MPa m1=2) in Table 5 suggests that the limiting D and Meech, 2001). This approximation is probably ad- of rutile is likely to be 7 times larger than zincite. This, equate for a qualitative assessment, because the detailed of course, only applies at very fine particle sizes when stress analysis of Hu et al. on the horizontal diametric only one or two flaws are present. It will not be appli- plane indicates that k is 0.28 when m is 0.15 and 0.25 cable at larger particle sizes where a significant popu- when m is 0.3. lation of different flaw sizes is likely (a population The plane of the flaw in Fig. 5a is inclined at an angle distribution that may be dependent on the type of of h degrees with respect to the impact axis, so that the mineral and its previous history). The influence of KIC component of induced tensile stress normal to the crack on fine particle fracture indicated by Eq. (31), together plane is krP cos h. Also, the component of compressive with the estimated KIC values in Tables 5 and 6, should stress acting normal to the crack plane (tending to close prove useful for guiding the production of ultra-fine the crack) is rP sin h, so the resulting crack opening mineral powders via techniques such as stirred ball mills stress is rP ðk cos h sin hÞ. The resulting stress intensity (Wang and Forssberg, 1997). KI acting on the flaw is obtained from Eq. (2): Note that with ultra-fine particles ðD 1 lm or less), a brittle-ductile transition may be obtained at high com- 1=2 KI ¼ Y rP ðk cos h sin hÞa pressive loading stresses (r ). This occurs when high in- P a 1=2 duced tensile components (high kr ) remain insufficient ¼ 2r ðk cos h sin hÞ Pam1=2 ð30Þ P P p to produce particle fracture, whereas the shear stress components (r =2) due to r become sufficient to Y 1=2 P P where is 2ðpÞ (Broek, 1982). promote significant dislocation motion in the particle For a particle to be fractured K K . Hence, a I ¼ IC (plastic deformation). This has been described in detail larger K (see Tables 5 and 6) requires a higher r IC P previously (Tromans and Meech, 2001). It contributes P P (larger ) for particle fracture. If is insufficient, frac- to increased impact inefficiency (i.e. large energy con- ture will not occur despite repeated impacts (impact sumption with minimal particle fracture). inefficiency). However, if the orientation of the particle changes during successive impacts, so that h ! 0 and k cos h sin h k, K will increase at constant P and ð Þ! I 5. Conclusions may reach KIC (particle fracture). Thus, KIC, P and flaw orientation (h) determine impact efficiency. In ball mills 1. The theoretical modelling of ideal fracture toughness and rod mills a distribution of P takes place, due to the gave fracture estimates of KIC, GIC and c for 48 min- random nature of the particle/ball interactions, leading erals that appear to be of the right order of magni- to inefficient particle fracture. An obvious way to nar- tude in both value and with respect to each other. row this distribution and increase P is by high com- It is the first time modelled estimates have been re- pression roller mill grinding, as proposed and developed ported for such a large number of minerals. by Schoonert€ (1988). In several instances, such mills 2. Fracture toughness for high angle grain boundary are reported to consume less energy (Schoonert,€ 1988; cracking ðG Þ in pure single-phase minerals is less McIvor, 1997) and exhibit improved interparticle sepa- IC Gb than G for intragranular cracking, the difference be- ration (i.e. liberation via interphase cracking), parti- IC ing of the order of 10–14%. cularly in the processing of diamond ores (McIvor, 3. Reported differences in fracture energy (increase 1997). in surface energy per unit mass) between galena, sphalerite and corundum tested via single particle, 4.2.4. Limiting fine particle size drop-weight fracture tests correlate well with relative Inspection of Fig. 5 shows the maximum flaw size differences in their computed GIC=q ratios (toughness/ must be less than the particle diameter (i.e. 2a < D). density). Using the condition KI ¼ KIC and h ¼ 0, Eq. (30) may be 4. Comparisons between the standard Bond work index reset in terms of D by expressing 2a as a fraction U of D, for different minerals and the ideal surface energy re- and rearranging (Tromans and Meech, 2001) quired for generating new surface indicate the energy efficiency of crushing and grinding operations to be 2 KIC very low, of the order of 1%. UD ¼ p m ð31Þ 2krP 5. The impact efficiency of particle fracture is dependent upon the loading force, the size and orientation of in- where U < 0:5andh ¼ 0. herent particle flaws with respect to the loading axis, Hence, for a constant rP , the lower limiting average D and KIC. obtainable via particle fracture is expected to be strongly 6. The average limiting particle size in ultra-fine grind- 2 dependent upon ðKICÞ . For example, examination of ing is strongly influenced by KIC. 1040 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041

1 Acknowledgements DHf 11769.592 (kJ mol ) for SnO2 listed by Sherman 4þ (1932). The Hþ of Ti was estimated from the average The authors wish to thank the Natural Sciences and enthalpies of Sn4þ,Pb4þ (9556.256 kJ mol1) and Mo4þ Engineering Council for ﬁnancial support of the re- (10025.002 kJ mol1), the latter two being obtained from 3þ search study. Roine (1999). The Hþ for Ti was assumed to be approximately 1000 kcal (4184 kJ) lower than that for 4þ 3þ Ti , because the reported Hþ for Mo is 1017.5 kcal Appendix. Enthalpies of ions and crystals (4257.22 kJ) lower than that for Mo4þ (Roine, 1999). The molar enthalpy of Zn2þ was obtained from the 2þ 2þ 2þ 2þ 2þ Molar enthalpies of cations (Hþ), anions (H)and averaged Hþ of Co ,Cd ,Fe ,Hg and Ni . 2 crystals (Hcr) at 298 K are listed in Table 9. All enthal- The H of S was obtained by subtracting the re- pies were obtained from a thermodynamic database ported value for the electron aﬃnity of sulphur, compiled by Roine (1999), except for a few ionic species )332.209 kJ mol1 (Sherman, 1932). from the molar whose enthalpies were estimated. The enthalpy of Sn4þ enthalpy of sulphur in the gas phase, 279.91 kJ mol1 at was estimated via Eq. (7) using the experimental value of 298 K (Roine, 1999).

Table 9 Molar enthalpies of gaseous ions and crystalline minerals at 298 K Cation and anion enthalpies (kJ mol1)a

Cation Hþ Cation Hþ Anion H Al3þ 5485.998 Hg2þ 2890.002 Cl )233.953 Ba2þ 1661.002 Kþ 514.009 F )255.078 Be2þ 2993.999 Mg2þ 2347.998 O2 966.504 Ca2þ 1926.0 Naþ 609.341 S 70.178 Co2þ 2841.999 Ni2þ 2930.001 S2 612.119b 2þ 2þ 2 Cd 2623.858 Pb 2371.868 SiO4 1464.4 3þ 4þ b 2 ) Cr 5648.4 Sn 9258.954 SO4 740.568 Csþ 458.403 Ti3þ 5429.404b Cuþ 1089.275 Ti4þ 9613.404 Fe2þ 2752.001 Zn2þ 2807.572b Fe3þ 5715.001

Crystal enthalpies (kJ mol1)a

Mineral Formula Hcr Mineral Formula Hcr

Cuprite Cu2O )156.063 Metacinnabar b-HgS )49.371 Periclase MgO )601.241 Greenockite CdS )149.369 Lime CaO )635.089 Wurtzite ZnS )191.836

Barium oxide BaO )553.543 Pyrite FeS2 )171.544 Wustite FeO )267.270 Halite NaCl )411.120 Cobalt oxide CoO )237.944 Sylvite KCl )436.684 Ni-oxide NiO )239.701 Cs-chloride CsCl )442.835

Bromellite BeO )608.354 Fluorite CaF2 )1225.912 Zincite ZnO )350.460 Barite BaSO4 )1473.20 Rutile TiO2 )944.747 Anhydrite CaSO4 )1434.108 Cassiterite SnO2 )577.630 Nepheline NaAlSiO4 )2094.661 Corundum Al2O3 )1675.692 Cobalt olivine Co2SiO4 )1408.836 Hematite Fe2O3 )824.248 Liebenbergite Ni2SiO4 )1397.234 Eskolaite Cr2O3 )1139.701 Fayalite Fe2SiO4 )1479.902 Ti-oxide Ti2O3 )1520.884 Monticellite CaMgSiO4 )2250.030 Spinel MgAl2O4 )2299.903 Forsterite Mg2SiO4 )2176.935 Hercynite FeAl2O4 )1995.299 Wadsleyite b-Mg2SiO4 )2141.999 Chromite FeCr2O4 )1458.124 Ringwoodite c-Mg2SiO4 )2133.200 Ni-chromite NiCr2O4 )1381.557 Andalusite Al2SiO5 )2590.314 Zinc ferrite ZnFe2O4 )1171.579 Anorthite CaAl2Si2O8 )4242.999 Magnetite Fe3O4 )1118.383 Grossularite Ca3Al2Si3O12 )6632.862 Chrysoberyl BeAl2O4 )2301.2 Pyrope Mg3Al2Si3O12 )6280.188 Galena PbS )98.634 Almandine Fe3Al2Si3O12 )5265.233 Sphalerite ZnS )201.669 Andradite Ca3Fe2Si3O12 )5769.987 a Values from database by Roine (1999). b Estimated by authors. D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 1041

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