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

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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 À2 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 mÀ2) 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 À 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 2ðpÞÀ1=2 for an internal penny-shaped (disc-shaped) crack of radius a (Broek, 1982). The ð1 À 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 mÀ2). Consequently, 0892-6875/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S0892-6875(02)00213-3 1028 D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041 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) À9 Daef average effective value of Df (m) RLimit limiting value of RxðR0 þ 2  10 m) À17 e elementary charge (1:602177  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 À12 À3 E0 permittivity in vacuum (8:854188  10 atoms at R0 (J m ) CVÀ1 mÀ1) 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) À1 Fr surface roughness factor (>1) ðWi ÞSI Wi in SI units (J kg ) À2 GIC critical crack energy release rate (J m ) x length of edge of unit cube containing N À2 ðGICÞGb GIC for grain boundary fracture (J m ) atoms at R0 (m) À2 ðGICÞIF GIC for interfacial fracture (J m ) Y a shape factor for flaws À2 ðGICÞIP GIC for interphase fracture (J m ) a largest common valence (charge) on ions À1 À1 Hþ enthalpy of cation in gas phase (J mol ) DHf enthalpy of crystal formation (J mol ) À1 À1 HÀ enthalpy of anion in gas phase (J mol ) DSA increase in surface area/unit volume (m ) À1 À1 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 À2 KIC critical KI for crack propagation (Pa m ) c surface energy (J m ) 1=2 À2 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 mÀ3) M Madelung constant r tensile stress (Pa) 2 Ma combined Madelung constant, a =Man rc critical tensile stress for cracking (Pa) 3 À1 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 À1 NA Avogadro number (6:023  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 ÞnÀ1 1 U 0 R ¼ n À 2. Ionic crystal model an 4pE0 nR R " # e2 ðR ÞnÀ1 1 Bonding in some mineral crystals may be treated as NM 0 ¼ a n À J ð4Þ essentially ionic, where the atoms behave as ions with 4pE0 nR R charges in accordance with their normal chemical va- lence 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 O2À, 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).
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