Mathematical Modelling of High-Energy
Ball Milling of Powders
A. S. Kurlov* and A. I. Gusev
Institute of Solid State Chemistry, Ural Division of the Russian Academy of Sciences,
Ekaterinburg 620990, Russia
ABSTRACT
A mathematical model of high-energy ball milling of powders has been proposed. An
analytical expression describing the size of nanocrystalline powder particles as a
function of the milling time t and powder charge M has been deduced. It is shown that
part of the energy is expended for initiation of microstrains during milling and, hence,
the process of the powder grinding is decelerated. It is established that both
decreasing of the initial powder charge and increasing of milling time lead to
reduction of mean particle size of ball-milled powder. The model and the experiment
have been compared using a powder of tungsten carbide. The mean size of the particles and the value of the microstrains in the ball-milled powder are determined by an
XRD-method from broadening of diffraction reflections. The particle size is evaluated
also using the scanning electron microscopy. The proposed model of ball milling of
powders allows substituting the empirical selection of milling conditions for the
theoretical determination of milling parameters.
INTRODUCTION
In the recent decade, the producing of substances and materials in the nanocrystalline state is one of the advancing directions in materials science [1-3]. Ball
milling is a simple and efficient method for making various nanocrystalline powders
[1, 4, 5] with particle size up to 20 nm or less. High-energy planetary, ball and vibratory mills are used for grinding.
The strength and the mechanical failure of solids are the objects in a number of
original and generalizing studies (see, e.g., [6, 7]) focusing on the analysis of the theory of strength and the failure mechanism (the interatomic bond breakage kinetics under plastic deformation, the formation and the growth of cracks, stress relaxation,
etc.). As to ball milling of powders and its final result, i.e. the particle size, the studies
of this kind are still performed at the empirical level since models relating the size of
particles in prepared nanocrystalline powders to the milling energy are unavailable.
In this paper we propose a physical and mathematical model of ball milling of
powders and discuss its applicability to estimating the particle size as a function of a
milling time t and a mass of the initial coarse-grained powder M. The model and the
experiment are compared using tungsten carbide WC powder as an example.
MILLING MODEL
While grinding the decrease of the particle size is accompanied by the initiation of microstrains in the particles. Below we consider the initial powder with particles having the average linear size Din. The volume of a particle in the initial powder
and its surface area are equal to Vin  f v Din3 and Sin  f s Din2 , respectively, where fv
and fs are the form factors of the volume and the surface area (i.e. the proportionality
1-105
factors controlled by the body shape). For spherical particles of the diameter Din the
form factors are fv = /6 and fs = , whence fs/fv = 6; for cubic particles with the edge
length Din the form factors fv = 1 and fs = 6, whence fs/fv = 6 too. If particles of the initial and ball-milled powders have the same shape in the first approximation (or the
particle shape distribution in the initial and ball-milled powders is the same), the ratio
between the form factors of the volume and the surface area fs/fv is constant.
Let the initial powder have the density d and the mass M. The number of particles in the initial powder is M/dVin = M /( df v Din3 ) . According to [7], when a powder
is being milled, the energy is expended for the rupture of interatomic bonds in crystals
and formation of the additional surface by cleavage of crystal particles. Therefore, the
milling energy Еmill can be written in the form
(1)
Emill  M ( Erupt  Esurf ) /( df v Din3 ) ,
where Erupt is the energy expended for the rupture of interatomic bonds in one particle
of the initial powder and Esurf is the formation energy of the additional surface under
the cleavage of one particle of the initial powder.
Grinding of one particle of the initial powder gives n  Din3 / D 3 smaller particles with the average linear size D, volume fvD3 and surface area fsD2 of one particle.
The surface area of n particles is S  nf s D 2  f s Din3 / D and increasing surface area
caused by grinding is equal to S = S – Sin = f s Din2 ( Din  D) / D .
Assume that the surface area of a face of a unit cell of the crystal is sf. In this
case the number of faces of unit cells, by which the cleavage occurred, is S/sf. If q
interatomic bonds with the binding energy u pass in each face of a unit cell, the energy of the rupture of bonds during grinding of one particle of the initial powder can be
defined as
(2)
Erupt  quS / sf  quf s Din2 ( Din  D) / sf D .
Let us determine the energy Esurf of formation of an additional surface by
cleavage of crystal particles. According to [7, 8], this energy is hundreds of times
higher than the change of the surface energy Es = S, i.e. Esurf = Es, where  is the
proportionality coefficient. Then
(3)
Esurf  Es  S   f s Din2 ( Din  D) / D ,
where  is the specific (per unit surface area of the interface) excess energy arising
from a disordered network of edge dislocations. The authors [9] proposed a grain
boundary model taking into account the presence of dislocations in grains and derived
the following expression for the specific excess interface energy  arising from the
chaotic network of edge dislocations with the Burgers vector b = (b, 0, 0):
Gb 2  ln( Din / 2b)
.
(4)

4 (1   )
Here G and  denote the shear modulus and the Poisson ratio, b = |b| is the Burgers
vector magnitude, and   VD/3 and V are the linear and the volume density of dislocations, respectively.
According to [10], the volume density of dislocations, which are randomly distributed in the grain bulk, equals to the geometric mean of the densities of dislocations
D = 3/D2 and s = C2/b2. The former is connected with the grain size D, and the letter is determined by the microstrains , i.e.
V = (Ds)1/2.
(5)
Considering (5), the linear density of dislocations  is
1-106
  VD/3 = [(3/D2)(C2/b2)]1/2D/3 =  3C / 3b ,
(6)
where C is a constant for a given material and takes value from 2 to 25 [10].
Writing the specific excess energy  (4) with account of (6) and substituting it
into the formula (3) gives the energy Esurf expended for the formation of an additional
surface upon the cleavage of a crystalline particle:
3C  f s GbDin2 ( Din  D) ln( Din / 2b) 
2
. (7)
Esurf   f s Din ( Din  D) / D 
12
(1   )
D
Substituting (2) and (7) into (1), we obtain a formula relating the milling energy Еmill to the mean particle size D after milling:
E mill 
Mf s ( Din  D) 12 (1   )qu  3C Gbsf ln( Din / 2b) 

.
df v DDin 
12 (1   ) s f

(8)
For a particular powdered material the values of d, fs, fv, q, u, C, , G, , b, and
sf are fixed, the milling energy depends on the milling time t, while the particle size
and the microstrains are the functions of a milling time t and a powder mass M. Therefore the expression (8) can be rearranged to the form
(9)
Emill (t )  M [ Din  D(t, M )][ A  B (t, M ) ln( Din / 2b)] /[ D(t, M ) Din ] ,
It follows from the relationship (9) that
D(t ) 
M [ A  B ln( Din / 2b) (t )]
,
Emill (t )  M [ A  B ln( Din / 2b) (t )] / Din
(10)
where A = (fs/fv)qu/sfd, B  ( f s / f v ) 3C Gb /[12 (1  )d ] are some constants typical
for a given substance. One can easily see that the formula (10) satisfies the edge condition D(0,M) = Din, since at the initial moment of time t = 0 the milling energy
Emill(0) and the microstrains (0,M) are equal to 0.
For a given initial size Din of particles and the fixed milling time t the dependence of the size D of ball-milled powder particles on the mass M of a substance is described by the function
(11)
D(t  const , M )  M/( K  M / Din ) ,
E (t  const )
with K =
. Thus, the smaller the mass of ground
A  B[ln( Din / 2b)] (t  const , M )
substance at the same milling time is, the smaller the particle size of prepared powder
is.
The value of the microstrains  = l/l  d/d characterizes the uniform deformation averaged over the crystal bulk, i.e. the relative change d of the interplanar
spacing d as compared to its change in a perfect crystal. According to the Hooke’s
law, l/l = /E where  is the stress and E is the modulus of elasticity. The fracture
begins when a critical value of stress, max, equal to strength of substance at given deformation form is attained. The main deformation forms during high-energy ball milling are the compression and shear. Therefore max = с/E where с is a compressive or
bending strength and E is a bulk or shear modulus, respectively. The microstrains 
change from zero at t = 0 to some limiting value max, above which the crystal lattice
of the ball-milled substance is destroyed. When the milling time t is constant, the value of microstrains is inversely proportional to the mass M of ball-milled substance.
With this in mind, the dependence of the microstrains  on the milling time t and the
1-107
mass M of substance can be described by the empirical function
(t,M) = max[1  exp(ct/(M + p))] = (с/E)[1 - exp(ct/(M + p))], where p is a fitting
parameter and c < 0. In this case the function (10) can be written as
D(t , M ) 
M { A  B[ln( Din / 2b)] max [1  exp( ct /( M  p))]}
. (12)
Emill (t )  M { A  B[ln( Din / 2b)] max [1  exp( ct /( M  p))]} / Din
The formula (12) is the basic expression of the milling model, which defines
the mean size of powder particles as a function of the milling energy Emill(t).
EXPERIMENTAL
To compare results of the proposed model and the experiment the initial
coarse-grained WC powder with the mean particle size Din  6 m was ground to the
nanocrystalline state with the particle size up to 10 nm.
The powder was milled in a PM-200 Retsch planetary ball mill. It has been
shown [11, 12] earlier that the energy expended for milling of the powder is proportional to the cube of the angular speed of rotation of the bearing disk, 3, and the mill64  3(r / Rc ) 4
3
2
2 1/ 2
3
ing time t, i.e. Еmill =  t, where   8 ak N b m( Rc  r ) Rc
is a
64  16(r / Rc ) 2
parameter typical for this mill, Rс is the radius of the circle described by the bowl axis,
r is the internal radius of the bowl, Nb is a number of grinding balls, m is the mass of
each ball, and ak << 1 is a coefficient showing how much energy is expended for the
ball-milling of the powder. Indeed, in the course of crushing and milling most of the
energy is expended for elastic deformation of the milling system, i.e. for the interaction of the grinding bodies with the walls of the grinding chamber, while less than 35% of the total kinetic energy is expended for milling of the powder [13]. In the case
of the PM-200 Retsch planetary ball mill Rc is 0.075 m, r is 0.0225 m and the total
mass of the grinding balls is Nbm = 0.1 kg. Let ak = 0.01, then the coefficient  is
equal to ~0.0015 kgm2.
Taking into account Еmill = 3t, the formula (12) can be written as an expression relating the post-milling particle size D to the milling time:
D(t , M ) 
M { A  B[ln( Din / 2b)] max [1  exp( ct /( M  p))]}
.
 t  M {A  B[ln( Din / 2b)] max [1  exp( ct /( M  p))]} / Din
3
(13)
It follows from (13) that the increasing of angular rotation speed  and milling
time t, decreasing of mass M of the powder and the size Din of particles in the initial
powder provide decreasing of post-milling particle size D. It can be seen from (13)
that generation of the microstrains  retards powdering.
Let us consider the experimental results obtained from milling of a coarsegrained WC powder (with the mean size of particles Din  6 m) in a PM-200 Retsch
planetary ball mill. The milling process is run in an automatic mode at a rotation
speed  = 8.33 rps, with the grinding direction reversed every 15 min with an interval
of 5 s between runs. The total mass of the grinding ball charge is about 100 g, the
number of balls in a charge Nb is ~450. Isopropyl alcohol (from 5 to 15 ml) is added
to the powder and after milling the powder prepared is dried. The mass of the powder
for ball milling is 10, 20, 25, and 33 g.
The mean size D of the coherent scattering regions and the value of the microstrains  in the ground WC powders are determined by an X-ray diffraction (XRD)
1-108
method from broadening of diffraction reflections. XRD measurements are carried out
in a Shimadzu XRD-7000 diffractometer using CuK1, 2 radiation at the angles 2 of
10 to 140 with the scan step (2) = 0.03 and the exposure time of 2 s at each
point. The diffraction reflections were described by the pseudo-Voigt function.
In the diffraction experiment, the mean size D of the coherent scattering region is D = /[cosd(2 )] where d(2 ) is a diffraction reflection broadening
caused by the small particle size,  is the radiation wavelength, and  is the scattering
angle [1, 14]. The diffraction reflection broadening (2) is defined as (2) =
(FWHM exp ) 2  (FWHM R ) 2 , where FWHMexp is the full width at half-maximum
of an experimental diffraction reflection and FWHMR is the instrumental angular
resolution function of the diffractometer. The angular resolution function
FWHMR(2) of a diffractometer is determined in a special diffraction experiment
with the cubic lanthanum hexaboride LaB6 (NIST Standard Reference Powder 660a).
The particle size distribution in the WC powder is determined using a Laser
Scattering Particle Size Distribution Analyzer a HORIBA-Laser LA-920.
Figure 1 presents X-ray diffraction patterns of the initial coarse-grained WC
powder and nanocrystalline powders of tungsten carbide made by milling for 10 h. All
the diffraction reflections become much wider after milling (Fig. 1b). The quantitative
analysis of broadening of the reflections of the nanocrystalline powder showed that it
is due to both the small size of the particles and the microstrains (Fig. 2).
Counts (arb. units)
1,2
radiation
(a)
(b)
*(2 ) = [ (2 )cos ]/ (nm-1)
0.25
CuK
t = 36000 sec, M = 20 g
D = 37 + 5 nm
= 0.75 + 0.02 %
0.20
0.15
0.10
0.05
0.00
40
60
80
100
2 (degrees)
0
120
Figure 1. X-ray diffraction patterns of the hexagonal WC powders with different mean size
of particles: (a) initial coarse-grained (Din  6
m) powder and (b) nanocrystalline powders
prepared by high-energy ball-milling of 20 g of
the initial WC powder in a PM-200 Retsch
planetary ball mill during 10 h. Diffraction reflections of the nanocrystalline WC powder are
broadened considerably. Weak reflections observed at angles 2 = 34.6, 38.0, and 39.6º
corresponds to impurity W2C phase.
2
4
6
s = (2sin )/
8
(nm-1)
10
12
Figure 2. Determination of mean particle size D and microstrains  in nanocrystalline WC powder prepared by ball
milling of 20 g of the initial coarsegrained WC powder during 10 h:
D = 375 nm,  = 0.750.02 %.
Values of the size and strain broadening are separated and the size D of the
coherent scattering regions and the value of the microstrains  are determined by the
Williamson-Hall method [1, 14] using the dependence of the reduced broadening
*(2 ) = [(2 )cos ]/ of the (hkl) reflections on the scattering vector s = (2sin )/.
1-109
In this case, the mean size D is calculated by extrapolating the dependence of the
reduced broadening *(2 ) on the scattering vector s to s = 0, i.e. D = 1/*(2 ) =
/[cos(2 )] at  = 0 because (2 )| = 0  d(2 ). The value of the microstrains 
characterized the relative change of the interplanar spacing. The value of the microstrains in relative units is found from the slope  of the straight line approximating
the * dependence on s according to the formula  = {[*(2 )]/2s}  [(tg)/2]. Note
that we wrongly determined the quantity of microstrains as  = {[*(2)]/4s} 
[(tg)/4] in works [11, 12, 15] and it is underestimated in 2 times. According to the
estimates, the mean size D of the coherent scattering regions and the value of the
microstrains  depend on the mass of the ground powder. For example, for 10 g of the
WC powder ground for 10 h the values of D and  are 165 nm and 0.00700.0002
(or 0.700.02 %), respectively, and for 20 g of the WC powder ground for the same
time D = 375 nm and  = 0.00750.0002 (or 0.750.02 %).
The particle size distribution in the ball-milled WC powder is determined also
with using a laser analyzer HORIBA-Laser LA-920. According to the results obtained
the smallest particles in WC nanocrystalline powder are ~80 nm in size, half of all the
particles are less than 170 nm in size, and 95% of all the particles has the size of not
over 500 nm. This means that the nanopowder particles are agglomerated. Observations in a scanning electron microscope (SEM) also pointed to easy agglomeration of
the ball-milled powder: the agglomerates are 100 to 400 nm in size.
The mean particle size estimated from broadening of the diffraction reflections
of WC nanopowder is in qualitative agreement with the relevant results of the particle
size distribution and the SEM examination. The particle size determined from the
XRD analysis is smaller because the size of agglomerates rather than the size of separate particles is estimated by the SEM. It should be noted also that the XRD method is
a volume technique and, hence, the size of particles is volume-averaged in this method.
Experimental dependences of the mean size D of the WC particles and the microstrains  on the milling time t and the mass M are shown in Fig. 3: at the given
milling parameters, the particle size decreased quickly and the microstrains were
building up during the first 100-150 min of the milling process. As the milling time
increased further, the dependences D(t) and (t) asymptotically approached some limiting values. Decreasing powder mass M at the constant milling time t leads to the
smaller particle size D and increasing of microstrains .
Given the same angular rotation speed  and an equal powder charge M of one
and the same initial powder, the relationship (13) takes the form
D(t,M) = M[aD + bD(t,M)]/{t + M[aD + bD(t,M)]/Din},
(14)
3
3
where
aD = A/ = (fs/fv)qu/(sfd )
and
bD = [Bln(Din/2b)]/3 =
3C (fs/fv)Gbln(Din/2b)/[12(1 - )d3].
It is seen from Fig. 3 that the experimental dependence D(t, M=const) are approximated well by the function (14) with the parameters aD = -0.004247 ms/kg and
bD = 9.52 ms/kg or aD = 4.247106 nms/kg and bD = 9.52109 nms/kg ( measured
in relative units). The microstrains  (rel. units) are described by the empirical function (t,M) = max[1 - exp(ct/(M + p))], where max = 0.007802, c = -0.00000273 kg/s,
and p = 0.0126 kg.
1-110
D (t = const,M) (nm)
M = 0.01 kg
100
80
60
Dtheor(t,M=const)
40
20
0
0.008
(t,M=const) (rel. units)
D(t,M) = M[aD + bD(t,M)]/{t +M[aD + bD(t,M)]/Din}
0.006
0.004
0.002
0
50
40
30
20
t = 36000 sec
0.01
0.02
0.03
M (kg)
Dexp(t,M=const)
1
2
3
4
6 t x10 -4 (sec)
5
M = 0.01 kg
 (t = const,M) (rel. units)
D (t,M=const) (nm)
120
0.90
t = 36000 sec
0.85
0.80
0.75
0.70
0.65
0.01
0.02
0.03
M (kg)
Figure 3. Dependence of the mean particle size D (○) and the microstrains  (●) on the
milling time t and mass M of a coarse-grained WC powder (PM-200 Retsch planetary ball mill, the ball charge is 0.1 kg and the rotation speed is 8.33 rps). Experimental data on the particle size Dexp are approximated by the function (14) with the
fitted parameters aD = -4.247106 nms/kg and bD = 9.52109 nms/kg; the microstrain
variation is described by the empirical function (t,M) = max[1 - exp(ct/(M + p))].
The theoretical dependence Dtheor(t) is shown as dotted line
The parameters aD and bD can be evaluated theoretically. Hexagonal tungsten
carbide WC with the unit cell constants a = 0.29060 and c = 0.28375 nm has the density d = 15.8 gcm-3, the Young’s modulus E = 720 GPa, the shear modulus G = 274
GPa, and the Poisson ratio  = 0.31 [16]. For a given unit cell lattice constants a and c
value sf  0.084 nm2 and the magnitude of the Burgers vectors b = 0001, (⅓)11-23
and (⅓)2-1-10 is b  0.28-0.29 nm [17].
The quantity qu can be evaluated from the WC atomization energy Eat. A unit
cell of tungsten carbide includes one WC formula unit and has 6 faces. Therefore qu ~
Eat/6NA, where NA is the Avogadro number. The atomization energy of hexagonal
tungsten carbide WC, which is determined from thermodynamic data [18, 19], is
Eat = 160050 kJmol-1, whence qu = 4.431019 J. Considering these values and taking fs/fv = 6, C = 18 and  = 100, we shall have the constants A = 0.002 и B = 0.85
Jmkg-1 for hexagonal tungsten carbide WC. At   0.0015 kgm2, the angular rotation speed  = 8.33 rps, and the initial particle size Din = 610-6 m, we shall obtain
1-111
aD = 0.00232 and bD = 9.0899 nms/kg (or aD = 2.3106 and bD = 9.1109 nms/kg).
The calculated and experimental values of aD and bD agree well.
250
200
150
D (nm)
100
50
0.06
0
10000
20000
30000
40000
50000
t (sec)
60000
0.04
0.02
0
M (kg)
Figure 4. Theoretical three-dimensional dependence of the particle size D on the milling
time t and mass M of initial coarse grained WC powder with Din = 6 m
The theoretical three-dimensional dependence Dtheor(t,M) which is calculated
by the formula (14) using theoretical values of aD and bD is shown in Fig. 4. The small
discrepancy between the experimental and calculated results is explained primarily by
the rough estimation of qu value and the empirical coefficients a, fs/fv, C and .
CONCLUSION
The proposed high-energy ball milling model describes the particle size of
prepared nanocrystalline powder as a function of the milling time t and powder charge
M. The proposed model of ball milling of powders allows changing from the empirical selection of milling conditions to the theoretical determination of milling parameters taking into account physical characteristics of the initial powder. The model is
applicable to single-phase solids. The presence of many phases in powder mixtures
should be considered by special manner.
Acknowledgements
This work was supported by the Ural and Siberian Divisions of the Russian
Academy of Sciences (projects Nos. 09-С-3-1014 and OFI 00-3-11UT) and the Russian Foundation for Basic Research (grant No. 10-03-00023a).
1-112
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