Minerals Engineering 15 (2002) 1027–1041
This article is also available online at:
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Fracture toughness and surface energies of minerals:
theoretical estimates for oxides, sulphides, silicates and halides
D. Tromans
a,*
, J.A. Meech
b,1
a
b
Department of Metals and Materials Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1W5
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 singlephase 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
0.131 to 2.774 MPa m1=2 . The critical energy release rates for intragranular cracking, GIC , ranged from 0.676 to 20.75 J m2 . 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
The size reduction of minerals by comminution and
crushing technologies involves particle fracture and the
creation of new surface area. Usually, fracture occurs
because particles obtained from naturally occurring
minerals contain preexisting cracks (flaws) which, during the comminution process, propagate in response to
local tensile stress components acting normal to the
crack plane. Tensile stresses are generated even when the
external loading on the particle is predominantly compressive (Hu et al., 2001; Tromans and Meech, 2001).
During crack propagation, strain energy is released as
new surface area is generated. Resistance to fracture
under crack opening (mode I) conditions is termed the
fracture toughness, GIC . It is defined as the critical en-
*
Corresponding author. Tel.: +1-604-822-2378; fax: +1-604-8223619.
E-mail addresses: [email protected] (D. Tromans), jam@mining.
ubc.ca (J.A. Meech).
1
Tel.: +1-604-822-3984; fax: +1-604-822-5599.
ergy release rate per unit area of crack plane (J m2 ) that
is necessary for crack propagation and is related to the
mode I stress intensity factor for crack propagation
(KIC ) via Eq. (1) (Broek, 1982; Tromans and Meech,
2001):
1=2
KIC ð1 m2 Þ
¼ ðEGIC Þ
1=2
KIC Pa m1=2
ð1Þ
where E is the tensile elastic modulus (Pa), m is PoissonÕs
ratio and KIC (Pa m1=2 ) is given by Eq. (2):
KIC ¼ Y rc ðaÞ
1=2
Pa m1=2
ð2Þ
where rc is the critical tensile stress (Pa) for crack
propagation, a is the flaw size (m) and Y is a shape
factor related to the crack geometry, e.g. Y has the value
ðpÞ1=2 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 m2 Þ term in Eq. (1) implies plane strain
conditions, which is the usual situation for brittle fracture. For ideal brittle fracture (negligible plastic deformation at the crack tip), GIC is equivalent to 2c, where c
is the surface energy per unit area (J m2 ). Consequently,
0892-6875/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 8 9 2 - 6 8 7 5 ( 0 2 ) 0 0 2 1 3 - 3
1028
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
Nomenclature
flaw size (m)
stoichiometric number of atoms/molecule
bulk elastic modulus (Pa)
average particle diameter (m)
initial value of D (m)
final value of D (m)
average effective value of Df (m)
elementary charge (1:602177 1017 C)
equilibrium tensile elastic modulus (Pa)
tensile elastic modulus, R ¼ Rx (Pa)
permittivity in vacuum (8:854188 1012
C V1 m1 )
f
area fraction of grain boundary coincident
sites
Fr
surface roughness factor (>1)
GIC
critical crack energy release rate (J m2 )
ðGIC ÞGb GIC for grain boundary fracture (J m2 )
ðGIC ÞIF GIC for interfacial fracture (J m2 )
ðGIC ÞIP GIC for interphase fracture (J m2 )
Hþ
enthalpy of cation in gas phase (J mol1 )
H
enthalpy of anion in gas phase (J mol1 )
Hcr
crystal enthalpy (J mol1 )
k
a fraction (0.25 to 0.3)
KI
stress intensity (Pa m1=2 )
KIC
critical KI for crack propagation (Pa m1=2 )
ðKIC ÞGb KIC for grain boundary crack (Pa m1=2 )
ðKIC ÞIF KIC for interfacial cracking (Pa m1=2 )
ðKIC ÞIP KIC for interphase cracking (Pa m1=2 )
L
a crystal dimension, L0 , under strain (m)
L0
equilibrium crystal dimension (m)
m
multiplying factor >1
M
Madelung constant
Ma
combined Madelung constant, a2 =Man
MV
molar volume (m3 mol1 )
n
a number >1 related to B
N
atoms/m3 of unstrained crystal
NA
Avogadro number (6:023 1023 mol1 )
P
loading force (N)
a
an
B
D
Di
Df
Daef
e
E
ERx
E0
a higher c should lead to increased toughness of brittle
materials (e.g. minerals). Frequently, KIC and GIC
are used interchangeably as the measure of toughness,
because (1) they are directly related via Eq. (1) and (2)
experimental measurement of KIC is less difficult than
GIC . In this manner, earlier studies based on KIC measurements indicate that fracture toughness is one of the
parameters affecting power consumption during rock
breakage (Bearman et al., 1991; Napier-Munn et al.,
1999). Also, previous modelling studies by the authors
showed that the limiting particle size of finely milled
minerals is dependent upon KIC values (Tromans and
Meech, 2001).
R
R0
Rx
RLimit
Umolecule
UR
Ue
V
Wi
ðWi ÞSI
x
Y
a
DHf
DSA
DSEn
ex
U
c
cGb
ls
p
h
q
r
rc
rh
rmax
rP
rx
non-equilibrium average distance between
atoms (m)
average distance between atoms in unstrained
(equilibrium) crystal (m)
average distance between atoms in x-direction
due to rx (m)
limiting value of Rx ðR0 þ 2 109 m)
crystal energy per molecule (J)
crystal energy for N atoms at R (J)
equilibrium crystal binding energy for N
atoms at R0 (J m3 )
volume of N atoms (V ¼ NR3 m3 )
bond work index for crushing and grinding
(kWh/short ton)
Wi in SI units (J kg1 )
length of edge of unit cube containing N
atoms at R0 (m)
a shape factor for flaws
largest common valence (charge) on ions
enthalpy of crystal formation (J mol1 )
increase in surface area/unit volume (m1 )
increase in surface energy/unit mass (J kg1 )
tensile strain in x-direction
a fraction 2a=D (<0.5)
surface energy (J m2 )
grain boundary energy, cGb < c (J m2 )
shear modulus (Pa)
circumference/diameter ratio of a circle
angle between loading axis and plane of flaw
(deg)
density (kg m3 )
tensile stress (Pa)
critical tensile stress for cracking (Pa)
hydrostatic compression stress (Pa)
maximum theoretical tensile stress (Pa)
stress due to P (Pa)
tensile stress in x-direction (Pa)
It is evident that continued development of quantitative models of the comminution process, for purposes
of power consumption and particle fracture, should include the fracture toughness of the minerals involved.
An examination of the published literature indicates a
dearth of information on mineral toughness, with the
review by Rummel (1982) providing a useful but limited
set of data. The purpose of this study is to use the basic
physics and fundamental models developed for bonding
in ionic crystals, particularly the Born model (Sherman,
1932; Seitz, 1940) to develop theoretical relationships
from which quantitative estimates of GIC , KIC and c may
be obtained for over 40 crystalline minerals. These in-
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
clude rock forming minerals (e.g. silicates) and those of
relevance to mineral processing and hydrometallurgy
(e.g. oxides and sulphides).
2. Ionic crystal model
Bonding in some mineral crystals may be treated as
essentially ionic, where the atoms behave as ions with
charges in accordance with their normal chemical valence state(s). This concept works well for simple halide
crystals such as NaCl (halite) and CaF2 (fluorite). To a
first approximation, many oxides may be treated as
ionic crystals composed of metal cations and oxygen
anions O2 , such as ZnO (zincite) and spinel-type
structures related to MgAl2 O4 (Sherman, 1932; Verwey
and Heilman, 1947). Ionic bonding is less common in
sulphide minerals, where covalent bonding plays a larger
role (Vaughan and Craig, 1978). Consequently, any
ionic similarity between oxides and sulphides is restricted primarily to those of the alkali and Group II
metals (Sherman, 1932; Wells, 1962). In these cases the
sulphur is treated as the S2 anion, except for pyrite
(FeS2 ) where the sulphur is treated as the S anion. For
more complex cases, crystal anions may be treated as
negatively charged clusters of covalently bound atoms,
2
e.g. SiO2
4 in NaAlSiO4 (nepheline) and SO4 in CaSO4
(anhydrite).
The classical theory of ionic crystals, based on
the Born model (Sherman, 1932; Seitz, 1940) assumes
that the crystal energy is composed of an electrostatic
interaction between oppositely charged ions and a
shorter-range repulsive term. Using this model, the average non-equilibrium crystal energy per molecule
Umolcule (J) may be expressed to a first approximation in
the general form:
#
2 "
e
ðL0 Þn1 1
2
Umolecule ¼ a M
J
ð3Þ
L
4pE0
nLn
where a is the largest common factor in the valences
(number of charges) of all the ions, e is the elementary charge (1:602177 1019 C), E0 is the vacuum
permittivity (8:854188 1012 C V1 m1 ), p is the circumference/diameter ratio of a circle, L0 (m) is a characteristic 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
choice of L0 , and n is a number >1 related to the compressibility.
For modelling of fracture toughness, it is necessary to
consider a large collection of atoms and express the nonequilibrium crystal energy UR in terms of the number of
atoms N contained in a unit volume (1 m3 ) of crystal at
equilibrium (unstrained crystal). Also, the average
1029
equilibrium inter-atomic distance R0 (m) and the nonequilibrium average inter-atomic distance R (m) replace
L0 and L in Eq. (1), respectively, leading to:
#
2 "
n1
N a2 M
e
ðR0 Þ
1
UR ¼
an
R
4pE0
nRn
#
2 "
n1
e
ðR0 Þ
1
¼ NMa
J
ð4Þ
R
4pE0
nRn
where Ma is a combined Madelung constant equal to
a2 M=an , and an is the stoichiometric number of atoms
per molecule. (Note that theoretical estimates of M
have been reported for several basic crystal structures
(Sherman, 1932; Moliere, 1955) and spinel oxides (Verwey and Heilman, 1947). In the current study, Ma is
calculated from elastic constants and thermodynamic
data.)
The value of N is obtained from:
N ¼ an ðNA =MV Þ
ð5Þ
where MV is the molar volume (m3 ), obtained by dividing the molecular weight by the crystal density, and NA
is the Avogadro number (6:023 1023 mol1 ).
At equilibrium, R ¼ R0 and Eq. (4) becomes:
2 NMa
e
1
Ue ¼
ð6Þ
1
J m3
n
R0
4pE0
where Ue is the crystal binding energy (J m3 ) at equilibrium (i.e. unstrained), and R0 ¼ N 1=3 .
Fig. 1 depicts the general shape of the resulting R–UR
curve obtained from Eq. (4), and shows the relative
positions of Ue and R0 .
2.1. Crystal binding energy
Values of Ue for selected mineral crystals at 298 K
were calculated from the molar enthalpies (J mol1 ) of
the individual cations (Hþ ) and anions (H ) in the gas
phase and the molar enthalpy (J mol1 ) of the crystal
(Hcr ) at 298 K via Eq. (7):
UR (Jm-3 )
+
0
RO
Ue
0
R (m)
Fig. 1. Schematic diagram showing the influence of average atomic
spacing, R, on the crystal energy per unit volume UR .
1030
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
Ue ¼
Hcr P
Hþ MV
P
H
¼
DHf
MV
J m3
ð7Þ
B ¼ V
where DHf is the molar enthalpy of formation
of the
P
crystal from the gaseous ions, and
indicates the
summation of enthalpies of the different
P cation and
anion components
(e.g.
in
MgAl
O
,
Hþ ¼ HMg2þ þ
2
4
P
2HAl3þ and
H ¼ 4HO2 ).
Values of Hþ , H and Hcr for the ionic and crystal
species considered are listed in Table 9 in the Appendix.
Values of DHf , MV , and the resulting value of Ue are
listed in Tables 1 and 2. Each MV was obtained from
published crystallographic data (PDF, 1995). Table 1
presents data for oxides and sulphides. Table 2 contains
data for three halides and minerals whose cations are
treated as groups of covalently bonded atoms contain2
ing oxygen (e.g. SO2
4 and SiO4 ).
orh
oV
¼V
o2 UR
oV 2
ð8Þ
Pa
R¼R0
where rh is hydrostatic compression stress (pressure)
and rh ¼ oUR =oV .
Now oUR =oV ¼ ðoUR =oRÞðoR=oV Þ, from which the
second derivative is obtained (Sherman, 1932):
oUR oR
2
o
o2 UR
oR o2 UR oR
o2 R oUR
oR oV
¼
¼
þ 2
2
2
oR
oV
oV
oV oR
oV
oR
"
#
2
o2 UR oR
¼
ð9Þ
oR2 oV
R¼R0
2.2. Bulk modulus of elasticity
where oUR =oR ¼ 0 when R ¼ R0 (see Eq. (4) and Fig. 1).
Thus from Eqs. (8) and (9):
"
2 #
o2 UR oR
ð10Þ
B¼V
oR2 oV
Consider a volume of crystalline mineral V , where
V ¼ NR3 (V ! 1 m3 as R ! R0 ). The bulk modulus B
(Pa) is given by the usual definition
From Eq. (4), o2 UR =oR2 may be obtained and inserted in
1
Eq. (10), together with V ¼ NR3 , oR=oV ¼ ð3R2 N Þ and
R ¼ R0 to yield the final equation for B:
R¼R0
Table 1
Values of DHf , MV and Ue for selected oxide and sulphide minerals at 298 K
Mineral
Cuprite
Formula
Cu2 O
Structurea
c Pn3m
DHf (kJ mol1 )
Ions
MV (105 m3 )
Ue (1011 J m3 )
2Cu , O
)3301.12
2.34382
)1.4084
þ
2
þ
2
Periclase
Lime
Barium oxide
Wustite
Cobalt oxide
Nickel oxide
MgO
CaO
BaO
FeO
CoO
NiO
c
c
c
c
c
c
Fm3m
Fm3m
Fm3m
Fm3m
Fm3m
Fm3m
Mg , O
Ca2þ , O2
Ba2þ , O2
Fe2þ , O2
Co2þ , O2
Ni2þ , O2
)3915.74
)3527.59
)3181.05
)3985.77
)4046.45
)4136.21
1.12413
1.67603
2.55891
1.20291
1.16387
1.09708
)3.4834
)2.1047
)1.2431
)3.3134
)3.4767
)3.7702
Bromellite
Zincite
BeO
ZnO
h P63 mc
h P63 mc
Be2þ , O2
Zn2þ , O2
)4568.86
)4124.53
8.31173
1.4340
)5.4969
)2.87622
Rutile
Cassiterite
TiO2
SnO2
tet P42 /mnm
tet P42 /mnm
Ti4þ , 2O2
Sn4þ , 2O2
)12491.16
)11769.59
1.8800
2.1546
)6.6442
)5.4626
Corundum
Hematite
Eskolaite
Titanium oxide
Al2 O3
Fe2 O3
Cr2 O3
Ti2 O3
trig
trig
trig
trig
2Al3þ , 3O2
2Fe3þ , 3O2
2Cr3þ , 3O2
2Ti3þ , 3O2
)15547.20
)15153.76
)15336.01
)15279.20
2.55603
3.0302
2.90556
3.13563
)6.0826
)5.0010
)5.2782
)4.8728
Spinel
Hercynite
Chromite
Nickel chromite
Zinc ferrite
Magnetite
Chrysoberyl
MgAl2 O4
FeAl2 O4
FeCr2 O4
NiCr2 O4
ZnFe2 O4
Fe3 O4
BeAl2 O4
c Fd3m
c Fd3m
c Fd3m
c Fd3m
c Fd3m
c Fd3m
o Pbnm
Mg2þ , 2Al3þ , 4O2
Fe2þ , 2Al3þ , 4O2
Fe2þ , 2Cr3þ , 4O2
Ni2þ , 2Cr3þ , 4O2
Zn2þ , 2Fe3þ , 4O2
Fe2þ , 2Fe3þ , 4O2
Be2þ , 2Al3þ , 4O2
)19485.91
)19585.31
)19372.94
)19474.37
)19275.17
)19166.40
)20133.21
3.975135
4.08005
4.42809
4.32945
4.52714
4.45526
3.43906
)4.9020
)4.8003
)4.3750
)4.4981
)4.2577
)4.3020
)5.8543
Galena
Sphalerite
Metacinnabar
Greenockite
Wurtzite
PbS
ZnS
b-HgS
CdS
ZnS
c Fm3m
c F43m
c F43m
h P63 mc
h P63 mc
Pb2þ , S2
Zn2þ , S2
Hg2þ , S2
Cd2þ , S2
Zn2þ , S2
)3082.62
)3621.36
)3551.49
)3385.35
)3611.53
3.1494
2.3783
3.0167
3.00478
2.3824
)0.97880
)1.5227
)1.1773
)1.1267
)1.5159
Pyrite
FeS2
c Pa3
Fe2þ , 2S
)3063.90
2.39318
)1.2803
a
R3c
R3c
R3c
R3c
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
Mineral
Formula
Structurea
DHf (kJ mol1 )
Ions
)2641.76
)2393.63
)2619.54
2.45457
5.21892
4.6009
)1.0763
)0.45865
)0.56936
Naþ , Al3þ , SiO2
4
2Co2þ , SiO2
4
2Ni2þ , SiO2
4
2Fe2þ , SiO2
4
Ca2þ , Mg2þ , SiO2
4
2Mg2þ , SiO2
4
2Mg2þ , SiO2
4
2Mg2þ , SiO2
4
)9654.40
)8557.23
)8721.64
)8448.30
)7988.42
)8337.33
)8302.39
)8293.6
5.4073
4.45093
4.25254
4.63347
5.1184
4.36654
4.04747
4.18098
)1.7854
)1.9226
)2.0509
)1.8233
)1.5607
)1.9094
)2.0513
)1.9836
2Al3þ , O2 , SiO42
)15993.21
5.15262
)3.1039
NaCl
KCl
CsCl
c Fm3m
c Fm3m
c Pm3m
Na , Cl
Kþ , Cl
Csþ , Cl
Fluorite
Barite
Anhydrite
CaF2
BaSO4
CaSO4
c Fm3m
o Pbnm
o Bmmb
Ca2þ , 2F
Ba2þ , SO2
4
Ca2þ , SO2
4
Nepheline
Cobalt olivine
Liebenbergite
Fayalite
Monticellite
Forsterite
Wadsleyite
Ringwoodite
NaAlSiO4
Co2 SiO4
Ni2 SiO4
Fe2 SiO4
CaMgSiO4
Mg2 SiO4
b-Mg2 SiO4
c-Mg2 SiO4
h P63
o Pmnb
o Pmnb
o Pmnb
o Pmnb
o Pmnb
o Ibmn
c Fd3m
Andalusite
Al2 SiO5
o Pnnm
CaAl2 Si2 O8
Ca3 Al2 Si3 O12
Mg3 Al2 Si3 O12
Fe3 Al2 Si3 O12
Ca3 Fe2 Si3 O12
Grossularite
Pyrope
Almandine
Andradite
a
2SiO2
4
tr P1
Ca2þ , 2Al3þ ,
c
c
c
c
3Ca2þ , 2Al3þ , 3SiO2
4
3Mg2þ , 2Al3þ , 3SiO2
4
2
3Fe2þ , 2Al3þ , 3SiO4
3Ca2þ , 2Fe3þ , 3SiO2
4
Ia3d
Ia3d
Pa3
Ia3d
Ue (1011 J m3 )
)0.29110
)0.19113
)0.15810
Halite
Sylvite
Cesium chloride
Anorthite
)786.51
)716.74
)667.285
MV (105 m3 )
2.7018
3.7500
4.22061
þ
)20069.79
10.0764
)1.9918
)27776.06
)28689.34
)28886.43
)27371.19
12.5226
11.3143
11.3590
13.1995
)2.2181
)2.5357
)2.5431
)2.0737
c ¼ cubic, h ¼ hexagonal, o ¼ orthorhombic, tr ¼ triclinic (quasi-monoclinic).
B¼
Ma ðn 1Þ
9ðR0 Þ
4
e2
4pE0
ð11Þ
The isotropic value of B, equivalent to the averaged
modulus for the polycrystalline mineral, is obtained
from the corresponding isotropic tensile elastic modulus
E and PoissonÕs ratio m via Eq. (12) (Wachtman, 1996):
B¼
E
3ð1 2mÞ
ð12Þ
After inserting Eq. (12) in (11), and rearranging, n is
obtained:
4 3EðR0 Þ
4pE0
n¼1þ
ð13Þ
ð1 2mÞMa
e2
The PoissonÕs ratio is obtained from E and the isotropic
elastic shear modulus ls via the relationship m ¼
ðE=2ls Þ 1 (Wachtman, 1996). Values of E and ls were
computed from anisotropic stiffness and compliance
constants for single crystals compiled mainly by
Hearmon (1979, 1984), together with some constants
compiled by Bass (1995). Computations were conducted
in two ways: (1) from stiffness constants assuming uniform stress in the polycrystalline aggregate and (2) from
compliance constants assuming uniform strain in the
aggregate. The two resulting values for each modulus, E
and ls , were then averaged. The matrix and analytical
procedures required for the computations are outlined
by Gerbrande (1982) and Wachtman (1996). Resulting
E and m for the minerals studied are listed in Tables 3
and 4.
Note that reported measurements of E for some porefree, polycrystalline oxides are 310.9 GPa for MgO,
123.5 GPa for ZnO, 284.2 GPa for TiO2 and 402.8 GPa
for Al2 O3 (Wachtman, 1996). These are very close to the
values listed in Table 3 and lend confidence to the
computation methods used to obtain E.
2.3. Evaluation of Ma , n and tensile behaviour
At this stage, only two parameters, Ma and n, in Eqs.
(6) and (13) remain unknown. All others are either established constants (e), or obtainable from crystallographic data (N and R0 ), thermodynamic data (Ue ), and
stiffness/compliance data (E and m). Consequently, Eqs.
(6) and (13) are two simultaneous equations, which may
be solved by numerical analysis to obtain Ma and n.
Resulting computed values are listed in Tables 3 and 4,
together with the corresponding N , R0 , E and m.
Tensile behaviour may be analysed by subjecting a
unit cube of material containing N atoms to a uniaxial
stress. The initial length of the cube has the length
x ¼ N 1=3 R0 ¼ 1 m. When a uniaxial tensile stress is applied in the x-direction, x ! x þ ox and R in the xdirection (Rx ) goes from R0 ! R0 þ oR and ox ¼ N 1=3 oR.
The variation in uniaxial stress rx as the polycrystalline
mineral aggregate is extended in the x-direction is obtained from UR in Eq. (4) via the differentiation:
1 2m
rx ¼
3
oUR
ox
1 2m
¼
3N 1=3
oUR
oR
ð14Þ
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
N (1028 m3 )
R0 (1010 m)
Mineral
Formula
Cuprite
Cu2 O
7.7092
2.3496
29.91a
0.4542
2.1728
6.9606
0.0790
Periclase
Lime
Barium oxide
Wustite
Cobalt oxide
Nickel oxide
MgO
CaO
BaO
FeO
CoO
NiO
10.7159
7.1872
4.7075
10.0141
10.34990
10.9801
2.1054
2.4052
2.7695
2.1534
2.1299
2.0883
307.18b
196.96b
92.25a
127.53a
189.40a
232.46b
0.1794
0.2145
0.2805
0.3830
0.3282
0.2834
3.9154
3.8322
3.9485
3.873
3.9269
4.0585
4.1259
4.9177
5.0718
4.9364
4.7553
4.2704
0.1068
0.0971
0.0955
0.0969
0.0990
0.1050
Bromellite
Zincite
BeO
ZnO
14.4928
8.4002
1.9038
2.2833
396.07b
126.74b
0.2201
0.3527
4.2239
4.3604
3.8612
4.4875
0.1106
0.1022
Rutile
Cassiterite
TiO2
SnO2
9.6112
8.3864
2.1831
2.2846
285.10a
262.49b
0.2755
0.2915
10.046
9.075
2.86650
3.4575
0.1276
0.11682
Corundum
Hematite
Eskolaite
Titanium oxide
Al2 O3
Fe2 O3
Cr2 O3
Ti2 O3
11.7819
9.9384
10.3646
9.60413
2.0398
2.1589
2.1289
2.1836
399.76b
212.35b
314.56b
244.45b
0.2335
0.1379
0.2761
0.3003
6.256
10.914
6.268
6.537
3.6989
1.7588
3.9935
3.7679
0.1130
0.1553
0.1087
0.11201
Spinel
Hercynite
Chromite
Nickel chromite
Zinc ferrite
Magnetite
Chrysoberyl
MgAl2 O4
FeAl2 O4
FeCr2 O4
NiCr2 O4
ZnFe2 O4
Fe3 O4
BeAl2 O4
10.6062
10.3335
9.5213
9.7382
9.3130
9.4632
12.2595
2.1126
2.1310
2.1900
2.1736
2.2062
2.1944
2.0130
273.83b
222.10b
268.65b
106.62b
241.16a
230.33b
389.79a
0.2660
0.3199
0.2804
0.4466
0.2899
0.2616
0.2289
5.872
5.795
5.727
5.121
5.8082
6.1495
5.7187
3.5810
3.8527
4.1945
6.6564
4.0437
3.3686
3.6845
0.1149
0.1107
0.1059
0.0813
0.10800
0.1183
0.1132
Galena
Sphalerite
Metacinnabar
Greenockite
Wurtzite
PbS
ZnS
b-HgS
CdS
ZnS
3.8249
5.0649
3.9931
4.0089
5.0563
2.9680
2.7028
2.9257
2.9218
2.7043
80.04a
82.73a
48.51b
46.61a
86.85b
0.2706
0.3202
0.3802
0.3759
0.3037
4.049
4.519
4.638
4.449
4.5545
5.3476
4.5320
5.1584
5.0011
4.37870
0.0926
0.1016
0.0945
0.0962
0.1035
Pyrite
FeS2
7.5502
2.3660
296.05b
0.1553
1.931
E (GPa)
m
Ma
n
10.0622
rmax =E
0.0620
a
Hearmon (1984).
b
Hearmon (1979).
The ð1 2mÞ=3 term is included in Eq. (14) in recognition that extension in the x-direction is accompanied by
a Poisson contraction in the other two orthogonal directions. Completing the differentiation in Eq. (14) leads
to:
#
2 "
n1
1 2m
e
1
ðR
Þ
0
rx ¼
ðN 2=3 Ma Þ
Pa
2
nþ1
3
4pE0
ðRx Þ
ðRx Þ
of Tables 3 and 4. The ratio ranged from 0.0606 to
0.141 with an average and standard deviation of 0:095 0:02.
The tensile elastic modulus ERx at any value of Rx is
obtained from Eq. (15) by differentiation:
orx
orx
orx
orx
ERx ¼
¼x
¼ N 1=3 Rx
¼ Rx
oex
ox
ox
oRx
ð15Þ
ð16Þ
where N , R0 , Ma , n and m are obtained from Tables 3 and
4. (Note that rx ! 0 as Rx ! R0 , consistent with reality.)
Eq. (15) indicates that rx rises rapidly to a peak value,
followed by an asymptotic decrease as Rx increases from
R0 . Representative curves showing this behaviour for
several minerals are presented in Figs. 2 and 3. Each
peak stress corresponds to the maximum theoretical
tensile stress, rmax , of the flaw-free mineral crystal in the
absence of any plasticity effects. The value of Rx at rmax
is readily obtained from Eq. (15) via the condition
orx =oRx ¼ 0, from which rmax is obtained after insertion
in Eq. (15). The rmax =E ratio is listed in the last column
where ex is the tensile strain in the x-direction (i.e. ox=x).
Thus, from Eqs. (15) and (16):
ERx ¼
2 1 2m
e
Rx ðN 2=3 Ma Þ
3
4pE0
"
#
n1
ðn þ 1ÞðR0 Þ
2
Pa
ðRx Þnþ2
ðRx Þ3
ð17Þ
Importantly, the equilibrium value of the tensile modulus E, as Rx ! R0 , is obtained from Eq. (17) by substituting 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
R0 (1010 m)
E (GPa)
m
Ma
n
rmax =E
4.4585
3.2123
2.8541
7.3614
2.8201
3.1458
3.2722
2.3861
36.87a
24.09a
25.40a
109.26b
0.254
0.276
0.268
0.288
0.9164
0.92
0.869
1.7565
7.7245
8.4573
10.3857
7.1862
0.0740
0.0697
0.0606
0.0775
6.9244
7.8546
2.4352
2.3350
60.03c
74.36a
0.316
0.269
0.771
0.8317
10.694
8.4808
0.0594
0.0696
7.7970
9.4724
9.9143
9.0992
8.2371
9.6555
10.4166
10.0840
2.3408
2.1937
2.1606
2.2233
2.2983
2.1798
2.1253
2.1484
76.014a
163.48b
205.37d
135.94b
141.58d
201.39b
278.07d
292.12d
0.218
0.316
0.292
0.335
0.278
0.242
0.237
0.236
4.155
2.2547
2.249
2.264
2.2565
2.233
2.083
2.081
2.2681
6.9411
7.2194
6.7980
6.1154
6.1216
7.7418
8.3534
0.1410
0.0792
0.0772
0.0802
0.0856
0.0855
0.0739
0.0703
Al2 SiO5
9.3514
2.2031
246.54b
0.246
4.03
4.6854
0.0997
Anorthite
CaAl2 Si2 O8
7.7705
2.3434
103.28a
0.295
3.537
3.7892
0.1116
Grossularite
Pyrope
Almandine
Andradite
Ca3 Al2 Si3 O12
Mg3 Al2 Si3 O12
Fe3 Al2 Si3 O12
Ca3 Fe2 Si3 O12
9.6194
10.6467
10.6048
9.1261
2.1825
2.1099
2.1127
2.2211
263.35a
232.08b
241.73a
219.04b
0.242
0.267
0.276
0.235
2.55
2.6228
2.6044
2.6265
6.9153
5.8979
6.3754
5.9838
0.0793
0.0874
0.0834
0.0867
Mineral
Formula
Halite
Sylvite
Cesium chloride
Fluorite
NaCl
KCl
CsCl
CaF2
Barite
Anhydrite
BaSO4
CaSO4
Nepheline
Cobalt olivine
Liebenbergite
Fayalite
Monticellite
Forsterite
Wadsleyite
Ringwoodite
NaAlSiO4
Co2 SiO4
Ni2 SiO4
Fe2 SiO4
CaMgSiO4
Mg2 SiO4
b-Mg2 SiO4
c-Mg2 SiO4
Andalusite
N (1028 m3 )
a
Hearmon (1979).
Hearmon (1984).
c
Gerbrande (1982).
d
Bass (1995).
b
σx (GPa)
σx (GPa)
50
30
8
40
6
30
5
20
4
7
25
1. Cuprite
2. Galena
3. Zincite
4. Pyrite
5. Hercynite
6. Periclase
7. Rutile
8. Corundum
7
5
15
4
10
3
3
10
1. Halite
2. Anhydrite
3. Fluorite
4. Anorthite
5. Forsterite
6. Pyrope
7. Andalusite
6
20
2
5
2
1
1
0
0
5E-10
1E-9
1.5E-9
2E-9
2.5E-9
0
0
5E-10
Fig. 2. Computed uniaxial tensile stress behaviour of defect-free oxide
and sulphide minerals as the average distance between atoms in the xdirection, Rx , is increased from Rx ¼ R0 when rx ¼ 0.
E¼
#
2 "
1 2m
e
ðn
1Þ
ðN 2=3 Ma Þ
Pa
2
3
4pE0
ðR0 Þ
1E-9
1.5E-9
2E-9
2.5E-9
Rx (m)
Rx (m)
ð18Þ
Recognising that ðN 1=3 R0 Þ2 ¼ 1 m2 , multiplication of Eq.
2
(18) by 1=ðN 1=3 R0 Þ produces:
Fig. 3. Computed uniaxial tensile stress-extension behaviour of defectfree halide, sulphate and silicate minerals as the average distance
between atoms in the x-direction, Rx , is increased from Rx ¼ R0 when
rx ¼ 0.
E¼
#
2 "
1 2m
e
ðn 1Þ
ðMa Þ
4
3
4pE0
ðR0 Þ
Pa
ð19Þ
Upon rearrangement, Eq. (19) is seen to be identical to
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
3.1. Intragranular fracture
The area beneath each Rx –rx curve in Figs. 2 and 3
represent the average work (energy) per unit area of
crack plane that is required for ideal intragranular
brittle fracture (i.e. bond breakage with no plastic deformation). This work is equivalent to the brittle fracture toughness GIC , which may be expressed formally:
Z Rx ¼RLimit
rx oRx J m2
ð20Þ
GIC ¼
Rx ¼R0
where RLimit is an upper value of Rx beyond which the
ionic model becomes invalid and rx becomes negligible.
After substituting Eq. (15) for rx , integration of Eq.
(20) gives GIC :
"
2 1 2m
e
GIC ¼
ðN 2=3 Ma Þ
3
4pE0
!#Rx ¼RLimit
n1
ðR0 Þ
1
J m2
ð21Þ
n Rx
nðRx Þ
Rx ¼R0
Inspection of Figs. 2 and 3 indicates RLimit is reached
when the average separation between atoms in the xdirection increases by 2 nm, i.e. RLimit ¼ ðR0 þ 2 109 Þ
m. Resulting values of GIC , based on this limit and Eq.
(21), are listed in Tables 5 and 6. The necessary N , R0 ,
Ma , n and m were obtained from Tables 3 and 4.
The critical stress intensity for intragranular brittle
crack propagation KIC was obtained from Eq. (1), and
the average surface energy of each mineral was obtained
from the condition c ¼ GIC =2 for ideal brittle fracture.
These are listed in Tables 5 and 6.
Note that the computed GIC , KIC and c, in Tables 5
and 6 are average values for intragranular cracking on a
randomly oriented plane in polycrystalline material. It is
recognised that c may be crystallographically anisotropic in the same manner that elastic constants of single
crystals are anisotropic. For example, with crystals
having the NaCl structure (e.g. the Fm3m oxides in
Table 1), theoretical estimates indicate that for the
(1 1 0) plane the surface energy is 2.7 times that of the
(1 0 0) plane (Seitz, 1940). Also, recent first-principles
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
Mineral
Formula
Intragranular crack
GIC (J m2 )
c (J m2 )
Grain boundary crack
ðGIC ÞGb (J m2 )
KIC (MPa m1=2 )
cGb (J m2 )
ðKIC ÞGb (MPa m1=2 )
Cuprite
Cu2 O
0.886
0.163
0.4428
0.769
0.152
0.117
Periclase
Lime
Barium oxide
Wustite
Cobalt oxide
Nickel oxide
MgO
CaO
BaO
FeO
CoO
NiO
13.704
8.335
4.274
4.885
7.449
9.964
2.052
1.281
0.623
0.789
1.188
1.522
6.852
4.1680
2.137
2.443
3.725
4.982
12.293
7.382
3.768
4.333
6.624
8.921
1.943
1.206
0.590
0.743
1.120
1.440
1.411
0.953
0.506
0.552
0.825
1.043
Bromellite
Zincite
BeO
ZnO
17.238
5.599
2.613
0.842
8.619
2.799
15.545
4.990
2.481
0.795
1.693
0.609
Rutile
Cassiterite
TiO2
SnO2
18.445
14.845
2.293
1.974
9.223
7.423
16.885
13.438
2.194
1.878
1.560
1.407
Corundum
Hematite
Eskolaite
Titanium oxide
Al2 O3
Fe2 O3
Cr2 O3
Ti2 O3
19.250
20.750
14.617
12.269
2.774
2.099
2.144
1.732
9.625
10.375
7.309
6.135
17.387
19.438
13.135
11.059
2.636
2.032
2.033
1.644
1.863
1.312
1.482
1.210
Spinel
Hercynite
Chromite
Nickel chromite
Zinc ferrite
Magnetite
Chrysoberyl
MgAl2 O4
FeAl2 O4
FeCr2 O4
NiCr2 O4
ZnFe2 O4
Fe3 O4
BeAl2 O4
14.014
10.687
12.209
3.080
11.420
12.897
18.624
1.959
1.541
1.811
0.573
1.660
1.724
2.694
7.007
5.344
6.104
1.540
5.710
6.449
9.312
12.675
9.624
10.934
2.684
10.249
11.700
16.828
1.863
1.462
1.714
0.535
1.572
1.642
2.561
1.339
1.063
1.275
0.396
1.171
1.197
1.796
Galena
Sphalerite
Metacinnabar
Greenockite
Wurtzite
PbS
ZnS
b-HgS
CdS
ZnS
3.736
4.180
2.316
2.289
4.536
0.547
0.588
0.335
0.327
0.628
1.868
2.090
1.158
1.145
2.268
3.278
3.712
2.037
2.017
4.037
0.512
0.554
0.314
0.307
0.592
0.458
0.468
0.279
0.272
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
Mineral
Formula
Intragranular crack
GIC (J m2 )
c (J m2 )
KIC (MPa m1=2 )
cGb (J m2 )
Grain boundary crack
ðGIC ÞGb (J m2 )
ðKIC ÞGb (MPa m1=2 )
Halite
Sylvite
Cesium chloride
Fluorite
NaCl
KCl
CsCl
CaF2
1.155
0.758
0.676
3.179
0.206
0.135
0.131
0.589
0.577
0.379
0.338
1.589
0.993
0.647
0.570
2.754
0.191
0.125
0.120
0.548
0.162
0.111
0.106
0.425
Barite
Anhydrite
BaSO4
CaSO4
1.203
1.805
0.269
0.366
0.602
0.902
1.021
1.550
0.248
0.340
0.182
0.255
Nepheline
Cobalt olivine
Liebenbergite
Fayalite
Monticellite
Forsterite
Wadsleyite
Ringwoodite
NaAlSiO4
Co2 SiO4
Ni2 SiO4
Fe2 SiO4
CaMgSiO4
Mg2 SiO4
b-Mg2 SiO4
c-Mg2 SiO4
6.412
4.570
5.450
3.924
4.665
6.329
6.792
6.685
0.698
0.864
1.058
0.730
0.813
1.129
1.374
1.397
3.206
2.285
2.725
1.962
2.332
3.164
3.396
3.343
5.933
3.973
4.729
3.414
4.081
5.542
5.872
5.756
0.672
0.806
0.985
0.681
0.760
1.056
1.278
1.297
0.479
0.597
0.721
0.510
0.584
0.787
0.920
0.929
Andalusite
Anorthite
Al2 SiO5
CaAl2 Si2 O8
10.130
5.478
1.580
0.752
5.065
2.739
9.011
4.930
1.491
0.714
1.119
0.548
Grossularite
Pyrope
Almandine
Andradite
Ca3 Al2 Si3 O12
Mg3 Al2 Si3 O12
Fe3 Al2 Si3 O12
Ca3 Fe2 Si3 O12
7.356
7.348
7.102
7.154
1.392
1.306
1.310
1.252
3.678
3.674
3.551
3.577
6.396
6.452
6.208
6.271
1.298
1.224
1.225
1.172
0.960
0.896
0.894
0.883
rundum has a c-value ranging from 2.13 to 3.5 J m2 ,
depending on the proportion of oxygen to aluminium
atoms on the exposed surface (Tepesch and Quong,
2000).
The modelling and computational procedures in the
present study have not incorporated any crystallographic effects on GIC and c. This is not a serious
drawback. During crack propagation in a polycrystalline material, the macroscopic crack plane tends to remain normal to the applied tensile stress component,
even though the crystallographic orientation of the grain
with respect to the tensile component changes as the
crack propagates from grain to grain. The result is
usually a stepped fracture surface at the microscopic
level (see Tromans and Meech, 2002) that is composed
of crystalline facets of lower GIC (lower c) with higher
energy step edges. The overall result for many brittle
minerals will be the production of a macroscopic intragranular crack plane whose average GIC (average c) is
likely to be near that obtained from Eq. (21).
3.2. Grain boundary fracture
The fracture toughness for grain boundary cracking
ðGIC ÞGb is expected to be lower than that for random
plane intragranular cracking, because atoms are arranged irregularly in the grain boundary region. Such
irregularities are readily seen in the floating bubble raft
experiments of Bragg and Nye (1947), where surface
tension effects between bubbles (nearest neighbour in-
teractions) simulate behaviour of close-packed atom
structures. These bubble raft studies indicate that the
width of the boundary region is of the order of two atom
diameters, consistent with high resolution transmission
electron microscopy images of grain boundaries in Ni
(Benedictus et al., 1994) and Si (Shen et al., 1995). A
geometric construction of a simple 25° tilt boundary,
AB, between two idealised close-packed grains of the
same crystalline phase is shown in Fig. 4. Examination
of this figure indicates that the boundary exhibits regions of atom coincidence and non-coincidence, consistent with the appearance of boundaries in bubble rafts
(Bragg and Nye, 1947). In the coincident regions, the
atom separation across the boundary is the same as that
in the crystal (i.e. R0 ). In the non-coincident regions, the
separation across the boundary is > R0 . Further analysis
is simplified by assigning a fractional area (f ) of grain
boundary to co-incident sites and a fraction ð1 f Þ to
non-coincident sites. The separation of atoms across the
boundary in the ð1 f Þ fraction is treated as an averaged value mR0 , where m is a multiplying factor >1.
Consequently, ðGIC ÞGb is obtained by modifying Eq. (20)
to reflect the two fractional areas and m:
Z Rx ¼RLimit
ðGIC ÞGb ¼ ðf Þ
rx oRx
Rx ¼R0
Z
þ ð1 f Þ
Rx ¼RLimit
rx oRx
Rx ¼mR0
where RLimit ¼ ðR0 þ 2 109 Þ m.
J m2
ð22Þ
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 ðKIC ÞGb in the pure minerals
are also listed in Table 4, being obtained from the
analogous equation to that of Eq. (1):
1=2
ðKIC ÞGb ð1 m2 Þ
1=2
¼ EðGIC ÞGb
ðKIC ÞGb Pa m1=2
ð24Þ
The grain boundary energy, cGb , corresponding to the
boundary condition f ¼ 0:5 and m ¼ 1:5 is also listed in
Tables 5 and 6, being obtained from the difference in
toughness between intragranular and grain boundary
cracking:
cGb ¼ GIC ðGIC ÞGb
Fig. 4. Schematic diagram of a 25° tilt grain boundary, AB, showing
regions of coincidence and non-coincidence between atoms in the
neighbouring grains.
After substituting Eq. (15) for rx , integration of
the two terms in Eq. (22) gives the general form of
ðGIC ÞGb :
ðGIC ÞGb
2 1 2m
e
2=3
¼
ðN Ma Þ
3
4pE0
8"
!#Rx ¼RLimit
n1
<
ðR0 Þ
1
f
:
nðRx Þn Rx
Rx ¼R0
!#Rx ¼RLimit 9
=
ðR0 Þ
1
J m2
þ ð1 f Þ
;
nðRx Þn Rx
"
n1
Rx ¼mR0
ð23Þ
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
mixed). Consequently, even in the same polycrystalline
mineral phase, no two boundaries will be identical.
However, if m can be assumed to be relatively constant,
and independent of f for high angle boundaries, then
Eq. (23) shows that ðGIC ÞGb has a maximum value when
f ¼ 0 and a minimum when f ¼ 1, with f ¼ 0:5 representing an average condition. Furthermore, examination of the boundary AB in Fig. 4 indicates m 1:5,
consistent with the general appearance of boundaries in
the bubble rafts of Bragg and Nye (1947), and f ¼ 0:54
(close to the average condition of 0.5). Consequently,
it is possible to make useful relative comparisons of
ðGIC ÞGb for different minerals by inserting f ¼ 0:5 and
m ¼ 1:5 in Eq. (23), together with RLimit ¼ ðR0 þ 2 109 Þ m. The resulting average ðGIC ÞGb values are listed
in Tables 5 and 6. These values should be taken as es-
J m2
ð25Þ
3.3. Interfacial fracture
For purposes of the current analysis, interfacial
fracture is defined as the propagation of cracks along the
interface between poorly bonded subparticles within a
larger mineral particle. Such interfaces are related to the
geological history of the mineral and are likely to be
present in minerals formed via deposition processes,
such as sedimentary rocks and conglomerates. It is not
possible to analyse this situation precisely, due to the
many variations of ‘‘poor bonding.’’ However, in principle, the fracture toughness for interfacial cracking
ðGIC ÞIF may be treated in a similar manner to Eq. (23) by
replacing ðGIC ÞGb with ðGIC ÞIF and recognising that, due
to poor interfacial bonding, f ! 0 and m 1. The result is that ðGIC ÞIF ðGIC ÞGb and ðGIC ÞIF GIC . Consequently, cracking along these interfaces will occur
preferentially whenever they are present.
3.4. Interphase fracture
In the present analysis, interphase fracture is defined
as cracking along the boundary between two different
crystalline phases. Many natural mineral bodies are
multiphase composites, being composed of one (or
more) mineral phase(s) dispersed in the matrix of a
different mineral phase. Bonding across the boundary between the different phases is stronger than that
for interfacial boundaries but not as strong as that
across grain boundaries in the pure, single-phase mineral. Modelling of interphase boundaries is difficult, due
to the variety of different mineral/mineral phase boundary combinations. However, interphase cracking toughness ðGIC ÞIP may be assessed in a semi-formal manner by
replacing ðGIC ÞGb in Eq. (23) by ðGIC ÞIP and then recognising that it is very likely that 0:5 > f > 0 and m > 1
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
but not 1. The overall result in terms of relative
toughness values becomes:
GIC > ðGIC ÞGb > ðGIC ÞIP ðGIC ÞIF
ð26Þ
Following from Eq. (1), the corresponding KIC values
for the different crack paths should follow the same
decreasing order as that shown in Eq. (26) (i.e. KIC >
ðKIC ÞGb > ðKIC ÞIP ðKIC ÞIF ). The relative fracture tough
ness for interphase cracking is likely to be an important
consideration in the liberation of different mineral
phases during milling. The relatively low value of
ðGIC ÞIF implies that interfacial cracking will be the first
particle fracture process to occur during milling, whenever such interfaces are present.
4. Discussion
4.1. Comments on toughness estimates
To the authorsÕ knowledge, the estimates in Tables 5
and 6 are the first time that theoretical toughness and
surface energy computations have been attempted on
such a large number of minerals. The computed toughness values are the lowest values possible, based on ideal
brittle fracture in pure, single-phase polycrystalline
minerals. They are associated with bond breakage in the
absence of any accompanying plastic deformation. With
reference to Table 5, GIC for intragranular fracture
ranges from 0.886 J m2 for cuprite to 20.75 J m2 for
hematite, a factor of 23.4 times. Similarly for Table 6,
GIC ranges from 0.676 J m2 for CsCl to 10.13 J m2 for
andalusite, a factor of 15 times. A lower toughness is
predicted for grain boundary cracking, indicating that
failure along suitably oriented high angle grain boundaries is a likely occurrence in polycrystals. However, the
average boundary toughness is not markedly lower than
that for intragranular cracking, For example, ðGIC ÞGb is
10–14% lower than GIC , and ðKIC ÞGb is 5–7% lower than
KIC .
There are scant experimental studies of KIC or GIC to
compare with the estimates in Tables 5 and 6. Reported
KIC values for fine-grained polycrystalline Al2 O3 (corundum) are in the range 2.5–4.5 MPa m1=2 (Wachtman,
1996). The measured GIC (or ðGIC ÞGb Þ for polycrystalline
MgO (periclase) is 8–10 J m2 (Davidge, 1974). Data
from single crystal experiments (McColm, 1990) suggest
that KIC is >1.2 MPa m1=2 for polycrystalline MgO
(periclase) and >2.5 MPa m1=2 for polycrystalline TiO2
(rutile). An experimental value of c for the (1 1 0) plane
in halite (NaCl) is 0.33 J m2 (Gilman, 1959). All of
these experimental data are sufficiently close to the
corresponding data in Tables 5 and 6 to indicate with
fair confidence that the estimates in Tables 5 and 6 are
of the right order of magnitude in both value and with
respect to each other.
1037
4.2. Relevance to comminution
4.2.1. Energy to create new surface
For purposes of estimating the total energy required
to generate new surface area during comminution of
polycrystalline minerals (i.e. the fracture energy), either
ðGIC ÞGb or GIC is adequate. Consider an initial spherical
particle of diameter Di (m) that is fractured (milled) into
very small particles with an average final diameter of Df
(m). The number of particles produced is ðDi =Df Þ3 and
the resulting increase in surface area per unit volume
DSA is obtained:
1
1
ð27Þ
DSA ¼ 6Fr
m1
Df Di
where Fr is a surface roughness factor (>1) introduced in
recognition of the fact that milled particles will not be
perfect spheres, but will exhibit roughened (stepped and
faceted) surfaces with a surface area/diameter ratio that
is Fr times that of spheres.
From Eq. (27), the increase in surface energy per unit
mass DSEn (J kg1 ) is obtained
DSA c 6Fr c 1
1
¼
DSEn ¼
q
q Df Di
6Fr c
J kg1 ; when Di Df
ð28Þ
qDf
where q is the mineral density (kg m3 ). (Note that c is
GIC =2 for intragranular cracking and ðGIC ÞGb =2 for
grain boundary cracking.)
Eq. (28) indicates that DSEn is likely to be proportional to c=q, assuming Fr does not vary widely between
different populations of particles. Using computed values of c from Table 5, and q from crystal data (PDF,
1995), the c=q ratio for, galena, sphalerite and corundum were calculated and normalised with respect to the
highest c=q ratio. The results are listed in Table 7 and
show that for the same Df , fracture of corundum requires approximately 10 times as much fracture energy
as galena and five times as much fracture energy as
sphalerite. King et al. (1997) measured the fracture
energy (J kg1 ) via single particle fracture tests on a
population of particles of the same minerals as those in
Table 7 by the drop weight technique. Their graphed
energy values at the 97% cumulative distribution were
normalised and listed for comparison with the normalised c=q values in Table 7. The agreement between their
experimental data and the computed c=q ratios is encouragingly good.
4.2.2. Comminution efficiency
Bond (1961) has determined and listed the average
standard work index Wi for the crushing and grinding of
numerous mineral bodies, where Wi is defined as the
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 J m kg1 )
Normalised c=q
Normalised fracture energya
Galena (PbS)
Sphalerite (ZnS)
Corundum (Al2 O3 )
1.868
2.090
9.625
7.597
4.097
3.989
2.459
5.1
24.13
0.102
0.21
1.0
0.1
0.2
1
a
Data from King et al. (1997).
is required to reduce the feed from an infinitely large
particle size to 80% passing 100 lm. (N.B. conversion of
Wi to SI units, ðWi ÞSI , requires that ðWi ÞSI ¼ Wi ð3:968 103 Þ J kg1 .) Bond (1961) cautioned the use of his Wi
values, because large variations may be encountered
between the same materials. This probably occurs because the ‘‘80% passing 100 lm’’ criterion provides no
information on the actual distribution of particle sizes,
which may be expected to vary between >100 and
<20 lm, with an average effective finished diameter of
Daef . Within these limitations, Eq. (28) may be used to
estimate the energy efficiency of particle fracture during
comminution via the ratio DSEn =ðWi ÞSI :
ð29Þ
4.2.3. Impact efficiency
Factors affecting impact efficiency may be understood
qualitatively in terms of fracture mechanics concepts,
based on the presence of inherent flaws in the particles.
Fig. 5a is a schematic diagram of an approximately
spherical mineral particle, of average diameter D, containing a penny-shaped flaw (crack) of radius a (viewed
along the plane of the flaw). The particle is subjected to
opposing impact forces P , such as occur when particles
are compressed between two balls in a ball mill, or two
rods in a rod mill. These forces generate a compressive
stress rP in the particle along the axis of impact and a
tensile stress krP (k < 1) within the particle that is normal to the impact axis. This is a well-recognised behaviour of spherical and cylindrical particles, which has
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
of the inefficiency arises from particles receiving numerous impacts before an impact of sufficient force is
received to cause particle fracture (e.g. King et al., 1997).
Fig. 5. Schematic diagram of small particle, average diameter D,
containing a flaw (crack) of radius a subjected to compressive force P .
(a) Flaw inclined at angle h with respect to the loading axis. (b) Plane
of flaw parallel to the loading axis (h ¼ 0).
% efficiency ¼
DSEn
6Fr c
100 100
qDaef ðWi ÞSI
ðWi ÞSI
Table 8
Estimated crushing and grinding efficiency based on c, q and the Bond work index
Mineral
b
Feldspar
Galena
Garnetc
Hematite
Magnetite
Pyrite
Rutile
Fluorite
a
c (J m2 )
q (103 kg m3 )
Wi (kWh ton1 )a
ðWi ÞSI (kJ kg1 )
Efficiency (%)
2.739
1.868
3.678
10.375
6.449
3.072
9.223
1.589
2.761
7.597
3.597
5.270
5.197
5.013
4.250
3.181
11.67
10.19
12.37
12.68
10.21
8.9
12.12
9.76
46.31
40.43
49.1
50.31
40.51
35.32
48.1
38.73
0.97
0.27
0.94
1.76
1.38
0.78
2.03
0.58
BondÕs data (1961).
Anorthite chosen as representative.
c
Grossularite chosen as representative.
b
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
been treated in detail by Hu et al. (2001). Previously, the
authors approximated k to PoissonÕs ratio m (Tromans
and Meech, 2001). This approximation is probably adequate for a qualitative assessment, because the detailed
stress analysis of Hu et al. on the horizontal diametric
plane indicates that k is 0.28 when m is 0.15 and 0.25
when m is 0.3.
The plane of the flaw in Fig. 5a is inclined at an angle
of h degrees with respect to the impact axis, so that the
component of induced tensile stress normal to the crack
plane is krP cos h. Also, the component of compressive
stress acting normal to the crack plane (tending to close
the crack) is rP sin h, so the resulting crack opening
stress is rP ðk cos h sin hÞ. The resulting stress intensity
KI acting on the flaw is obtained from Eq. (2):
KI ¼ Y rP ðk cos h sin hÞa1=2
a 1=2
¼ 2rP ðk cos h sin hÞ
p
Pa m1=2
ð30Þ
1=2
where Y is 2ðpÞ
(Broek, 1982).
For a particle to be fractured KI ¼ KIC . Hence, a
larger KIC (see Tables 5 and 6) requires a higher rP
(larger P ) for particle fracture. If P is insufficient, fracture will not occur despite repeated impacts (impact
inefficiency). However, if the orientation of the particle
changes during successive impacts, so that h ! 0 and
kðcos h sin hÞ ! k, KI will increase at constant P and
may reach KIC (particle fracture). Thus, KIC , P and flaw
orientation (h) determine impact efficiency. In ball mills
and rod mills a distribution of P takes place, due to the
random nature of the particle/ball interactions, leading
to inefficient particle fracture. An obvious way to narrow this distribution and increase P is by high compression roller mill grinding, as proposed and developed
by Sch€
onert (1988). In several instances, such mills
are reported to consume less energy (Sch€
onert, 1988;
McIvor, 1997) and exhibit improved interparticle separation (i.e. liberation via interphase cracking), particularly in the processing of diamond ores (McIvor,
1997).
4.2.4. Limiting fine particle size
Inspection of Fig. 5 shows the maximum flaw size
must be less than the particle diameter (i.e. 2a < D).
Using the condition KI ¼ KIC and h ¼ 0, Eq. (30) may be
reset in terms of D by expressing 2a as a fraction U of D,
and rearranging (Tromans and Meech, 2001)
UD ¼ p
KIC
2krP
2
m
ð31Þ
where U < 0:5 and h ¼ 0.
Hence, for a constant rP , the lower limiting average D
obtainable via particle fracture is expected to be strongly
2
dependent upon ðKIC Þ . For example, examination of
1039
the KIC values of zincite (0.842 MPa m1=2 ) and rutile
(2.293 MPa m1=2 ) in Table 5 suggests that the limiting D
of rutile is likely to be 7 times larger than zincite. This,
of course, only applies at very fine particle sizes when
only one or two flaws are present. It will not be applicable at larger particle sizes where a significant population of different flaw sizes is likely (a population
distribution that may be dependent on the type of
mineral and its previous history). The influence of KIC
on fine particle fracture indicated by Eq. (31), together
with the estimated KIC values in Tables 5 and 6, should
prove useful for guiding the production of ultra-fine
mineral powders via techniques such as stirred ball mills
(Wang and Forssberg, 1997).
Note that with ultra-fine particles ðD 1 lm or less),
a brittle-ductile transition may be obtained at high compressive loading stresses (rP ). This occurs when high induced tensile components (high krP ) remain insufficient
to produce particle fracture, whereas the shear stress
components (rP =2) due to rP become sufficient to
promote significant dislocation motion in the particle
(plastic deformation). This has been described in detail
previously (Tromans and Meech, 2001). It contributes
to increased impact inefficiency (i.e. large energy consumption with minimal particle fracture).
5. Conclusions
1. The theoretical modelling of ideal fracture toughness
gave fracture estimates of KIC , GIC and c for 48 minerals that appear to be of the right order of magnitude in both value and with respect to each other.
It is the first time modelled estimates have been reported for such a large number of minerals.
2. Fracture toughness for high angle grain boundary
cracking ðGIC ÞGb in pure single-phase minerals is less
than GIC for intragranular cracking, the difference being of the order of 10–14%.
3. Reported differences in fracture energy (increase
in surface energy per unit mass) between galena,
sphalerite and corundum tested via single particle,
drop-weight fracture tests correlate well with relative
differences in their computed GIC =q ratios (toughness/
density).
4. Comparisons between the standard Bond work index
for different minerals and the ideal surface energy required for generating new surface indicate the energy
efficiency of crushing and grinding operations to be
very low, of the order of 1%.
5. The impact efficiency of particle fracture is dependent
upon the loading force, the size and orientation of inherent particle flaws with respect to the loading axis,
and KIC .
6. The average limiting particle size in ultra-fine grinding is strongly influenced by KIC .
1040
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
Acknowledgements
The authors wish to thank the Natural Sciences and
Engineering Council for financial support of the research study.
Appendix. Enthalpies of ions and crystals
Molar enthalpies of cations (Hþ ), anions (H ) and
crystals (Hcr ) at 298 K are listed in Table 9. All enthalpies were obtained from a thermodynamic database
compiled by Roine (1999), except for a few ionic species
whose enthalpies were estimated. The enthalpy of Sn4þ
was estimated via Eq. (7) using the experimental value of
DHf 11769.592 (kJ mol1 ) for SnO2 listed by Sherman
(1932). The Hþ of Ti4þ was estimated from the average
enthalpies of Sn4þ , Pb4þ (9556.256 kJ mol1 ) and Mo4þ
(10025.002 kJ mol1 ), the latter two being obtained from
Roine (1999). The Hþ for Ti3þ was assumed to be approximately 1000 kcal (4184 kJ) lower than that for
Ti4þ , because the reported Hþ for Mo3þ is 1017.5 kcal
(4257.22 kJ) lower than that for Mo4þ (Roine, 1999).
The molar enthalpy of Zn2þ was obtained from the
averaged Hþ of Co2þ , Cd2þ , Fe2þ , Hg2þ and Ni2þ .
The H of S2 was obtained by subtracting the reported value for the electron affinity of sulphur,
)332.209 kJ mol1 (Sherman, 1932). from the molar
enthalpy of sulphur in the gas phase, 279.91 kJ mol1 at
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
3þ
Al
Ba2þ
Be2þ
Ca2þ
Co2þ
Cd2þ
Cr3þ
Csþ
Cuþ
Fe2þ
Fe3þ
Hþ
5485.998
1661.002
2993.999
1926.0
2841.999
2623.858
5648.4
458.403
1089.275
2752.001
5715.001
Cation
Hþ
Anion
H
Hg
Kþ
Mg2þ
Naþ
Ni2þ
Pb2þ
Sn4þ
Ti3þ
Ti4þ
Zn2þ
2890.002
514.009
2347.998
609.341
2930.001
2371.868
9258.954b
5429.404b
9613.404
2807.572b
Cl
F
O2
S
S2
SiO2
4
SO2
4
)233.953
)255.078
966.504
70.178
612.119b
1464.4
)740.568
2þ
Crystal enthalpies (kJ mol1 )a
Mineral
Formula
Hcr
Mineral
Formula
Hcr
Cuprite
Periclase
Lime
Barium oxide
Wustite
Cobalt oxide
Ni-oxide
Bromellite
Zincite
Rutile
Cassiterite
Corundum
Hematite
Eskolaite
Ti-oxide
Spinel
Hercynite
Chromite
Ni-chromite
Zinc ferrite
Magnetite
Chrysoberyl
Galena
Sphalerite
Cu2 O
MgO
CaO
BaO
FeO
CoO
NiO
BeO
ZnO
TiO2
SnO2
Al2 O3
Fe2 O3
Cr2 O3
Ti2 O3
MgAl2 O4
FeAl2 O4
FeCr2 O4
NiCr2 O4
ZnFe2 O4
Fe3 O4
BeAl2 O4
PbS
ZnS
)156.063
)601.241
)635.089
)553.543
)267.270
)237.944
)239.701
)608.354
)350.460
)944.747
)577.630
)1675.692
)824.248
)1139.701
)1520.884
)2299.903
)1995.299
)1458.124
)1381.557
)1171.579
)1118.383
)2301.2
)98.634
)201.669
Metacinnabar
Greenockite
Wurtzite
Pyrite
Halite
Sylvite
Cs-chloride
Fluorite
Barite
Anhydrite
Nepheline
Cobalt olivine
Liebenbergite
Fayalite
Monticellite
Forsterite
Wadsleyite
Ringwoodite
Andalusite
Anorthite
Grossularite
Pyrope
Almandine
Andradite
b-HgS
CdS
ZnS
FeS2
NaCl
KCl
CsCl
CaF2
BaSO4
CaSO4
NaAlSiO4
Co2 SiO4
Ni2 SiO4
Fe2 SiO4
CaMgSiO4
Mg2 SiO4
b-Mg2 SiO4
c-Mg2 SiO4
Al2 SiO5
CaAl2 Si2 O8
Ca3 Al2 Si3 O12
Mg3 Al2 Si3 O12
Fe3 Al2 Si3 O12
Ca3 Fe2 Si3 O12
)49.371
)149.369
)191.836
)171.544
)411.120
)436.684
)442.835
)1225.912
)1473.20
)1434.108
)2094.661
)1408.836
)1397.234
)1479.902
)2250.030
)2176.935
)2141.999
)2133.200
)2590.314
)4242.999
)6632.862
)6280.188
)5265.233
)5769.987
a
b
Values from database by Roine (1999).
Estimated by authors.
D. Tromans, J.A. Meech / Minerals Engineering 15 (2002) 1027–1041
References
Austin, L.G., 1984. Gaudin Lecture, Concepts in process design of
mills, Reprinted. In: Kawatra, S. (Ed.), Comminution Practices.
SME Inc., Littleton, CO, pp. 339–348.
Bass, J.D., 1995. Elasticity of minerals, glasses and melts. In: Ahrens,
T.J. (Ed.), Mineral Physics and Crystallography: A Handbook of
Physical Constants. American Geophysical Union, Washington,
DC, pp. 45–63.
Bearman, R.A., Barley, R.W., Hitchcock, A., 1991. Prediction of
power consumption and product size in cone crushing. Minerals
Engineering 4 (12), 1243–1256.
Benedictus, R., Han, K., B€
ottger, A., Zandbergen, H.W., Mittemeijer,
E.J., 1994. Solid state amorphisation reaction in Ni/Ti multilayers:
amorphisation along grain boundaries. In: Johnson, W.C., Howe,
J.H., Laughlin, D.E., Soffa, W.A. (Eds.), Solid ! Solid Phase
Transformations. TMS, Warrendale, PA, pp. 1027–1032.
Bond, F.C., 1961. Crushing and grinding calculations. Allis Chalmers
publication no. 07R9235B, pp. 1–14.
Bragg, L., Nye, J.F., 1947. A dynamical model of a crystal structure.
Proceedings of the Royal Society of London 190, 474–481.
Broek, D., 1982. Elementary Engineering Fracture Mechanics, third
ed. Martinus Nijhoff Publishers, The Hague, Netherlands.
Davidge, R.W., 1974. Effects of microstructure on the mechanical
properties of ceramics. In: Bradt, R.C., Hasselman, D.P.H., Lange,
F.F. (Eds.), Fracture Mechanics of Ceramics, vol. 2. Plenum Press,
New York, NY, pp. 447–468.
Gerbrande, H., 1982. Elastic wave velocities and constants of elasticity
of rocks and rock forming minerals. In: Angenheister, G. (Ed.),
Landolt-B€
ornstein Tables, Group V, vol. 1b. Springer-Verlag,
Berlin, pp. 1–98.
Gilman, J.J., 1959. Cleavage, ductility, and tenacity in crystals. In:
Averbach, B.L., Felbeck, D.K., Hahn, G.T., Thomas, D.A. (Eds.),
Fracture. John Wiley and Sons Inc., New York, NY, pp. 193–222.
Hearmon, R.F.S., 1979. The elastic constants of crystals and other
anisotropic materials. In: Hellwege, K.-H., Hellwege, A.M. (Eds.),
Landolt-B€
ornstein Tables, Group III, vol. 11. Springer-Verlag,
Berlin, pp. 1–244.
Hearmon, R.F.S., 1984. The elastic constants of crystals and other
anisotropic materials. In: Hellwege, K.-H., Hellwege, A.M. (Eds.),
Landolt-B€
ornstein Tables, Group III, vol. 18. Springer-Verlag,
Berlin, pp. 1–154.
Hu, G., Otaki, H., Lin, M., 2001. An index of the tensile strength of
brittle particles. Minerals Engineering 14 (10), 1199–1211.
King, R.P., Tavares, L.M., Middlemiss, 1997. Establishing the energy
efficiency of a ball mill. In: Kawatra, S. (Ed.), Comminution
Practices. SME Inc., Littleton, CO, pp. 311–316.
McColm, I.J., 1990. Ceramic Hardness. Plenum Press, New York, NY,
p. 269, 283.
McIvor, R.E., 1997. High pressure grinding rolls––a review. In:
Kawatra, S. (Ed.), Comminution Practices. SME Inc., Littleton,
CO, pp. 95–97.
1041
Moliere, K., 1955. Gitterenergien von Kristallen. In: Eucken, A. (Ed.),
Landolt-B€
ornstein Tables, Bd. I:4. Springer-Verlag, Berlin, pp.
534–545.
Napier-Munn, T.J., Morrell, S., Morrison, R.D., Kojovic, T., 1999.
Mineral comminution circuits: their operation and optimisation.
In: JKMRC Monograph Series in Mining and Mineral Processing
2. The University of Queensland, Australia, pp. 53–54, 80.
Powder Diffraction File (PDF), 1995. PCPDFWIN, JCPDS-International Center for Diffraction Data, Swarthmore, PA, USA.
Roine, A., 1999. Outokumpu HSC Chemistry for Windows, Version
4.1, Chemical Reaction and Equilibrium Software with Extensive
Thermochemical Database. Outokumpu Research Oy, Pori, Finland.
Rummel, F., 1982. Fracture and flow of rocks and minerals: strength
and deformability. In: Angenheister, G. (Ed.), Landolt-B€
ornstein
Tables, Group V, vol. 1b. Springer-Verlag, Berlin, pp. 141–194.
Sch€
onert, K., 1988. A first survey of grinding with high-compression
roller mills. International Journal of Mineral Processing 22, 401–
412.
Seitz, F., 1940. The Modern Theory of Solids. McGraw-Hill Inc., New
York, NY, pp. 76–98.
Shen, T.D., Koch, C.C., McCormick, T.L., Nemanich, R.J., Huang,
J.Y., Huang, J.G., 1995. The structure and property characteristics
of amorphous/nanocrystalline silicon produced by ball milling.
Journal of Materials Research 10 (1), 139–148.
Sherman, J., 1932. Crystal energies of ionic compounds and thermochemical applications. Chemical Reviews XI (August–December),
93–169.
Tepesch, P.D., Quong, A.A., 2000. First-principles calculation of
a-alumina (0 0 0 1) surfaces energies with and without hydrogen.
In: Deak, P., Frauenheim, T., Pederson, M.R. (Eds.), Computer
Simulation of Materials at Atomic Level. Wiley-VCH, Berlin,
FRG, pp. 377–387.
Tromans, D., Meech, J.A., 2001. Enhanced dissolution of minerals:
Stored energy, amorphism and mechanical activation. Minerals
Engineering 14 (11), 1359–1377.
Tromans, D., Meech, J.A., 2002. Enhanced dissolution of minerals:
conjoint effects of particle size and microtopography. Minerals
Engineering 15, 263–269.
Vaughan, D.J., Craig, J.R., 1978. Mineral Chemistry of Metal Sulfides.
Cambridge University Press, New York, NY (156–190).
Verwey, E.J., Heilman, E.L., 1947. Physical properties and cation
arrangement of oxides with spinel structures: I Cation arrangement in spinels. Journal of Chemical Physics 15 (4), 174–
180.
Wachtman, J.B., 1996. Mechanical Properties of Ceramics. John Wiley
and Sons Inc., New York, NY, pp. 24–35, 161.
Wang, Y., Forssberg, E., 1997. Ultra-fine grinding and classification of
minerals. In: Kawatra, S. (Ed.), Comminution Practices. SME,
Littleton, CO, pp. 203–214.
Wells, A.F., 1962. Structural Inorganic Chemistry, third ed. Oxford
University Press, London, UK.