Mathematical Modelling of High-Energy

Mathematical Modelling of High-Energy

Ball Milling of Powders

A. S. Kurlov*andA. I. Gusev

Institute of Solid State Chemistry, Ural Division of the RussianAcademy 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 timetand powder charge Mhas 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 particlesize 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 powderM. 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 weconsiderthe 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 and , respectively, where fv and fs are the form factors of the volume and the surface area (i.e. the proportionality 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/fvis constant.

Let the initial powder have the density d and the mass M. The number of particles in the initial powder is M/dVin=. 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)

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 smaller particles with the average linear sizeD,volume fvD3and surface area fsD2of one particle. The surface area of n particles is and increasing surface area caused by grinding is equal to S= S–Sin = .

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)

Let us determine the energy Esurfof 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)

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):

. (4)

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/D2and 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

VD/3 = [(3/D2)(C2/b2)]1/2D/3 = ,(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 Esurfexpended for the formation of an additional surface upon the cleavage of a crystalline particle:

.(7)

Substituting (2) and (7) into (1), we obtain a formula relating the milling energyЕmill to the mean particle size D after milling:

.(8)

For a particular powdered material the values of d, fs, fv,q, u, C, , G, , b, and sf are fixed, the milling energydepends onthe 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)

It follows from the relationship (9) that

, (10)

where A=(fs/fv)qu/sfd, 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 timetthe dependence of the size Dof ball-milled powder particles on the massM of a substance is described by the function

, (11)

with K=.Thus, the smaller the mass of ground substance at the same milling time is, the smaller the particle size of prepared powder is.

The value of the microstrains =l/ld/dcharacterizes the uniform deformation averaged over the crystal bulk, i.e. the relative change dof the interplanar spacing d as compared to its change in a perfect crystal. AccordingtotheHooke’slaw, l/l=/EwhereisthestressandEisthemodulusofelasticity.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/Ewhereсis 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 isdestroyed. 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 tand the 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))], wherepisafitting parameterandc0. In this case the function (10) can be written as

. (12)

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. Ithasbeenshown [11, 12] earlierthattheenergyexpendedformillingofthepowderisproportionaltothecubeoftheangular speed of rotation of the bearing disk, 3, andthemillingtimet, i.e. Еmill = 3t, where is a parameter typical for this mill, Rс is the radius of the circle described by the bowl axis, r isthe internal radius of the bowl, Nbis a number of grinding balls, m is the mass of each ball, and ak1 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 3-5% of the total kinetic energy is expended for milling of the powder [13]. In the case of the PM-200 Retsch planetary ball mill Rcis 0.075m, r is 0.0225 m and the total mass of the grinding balls is Nbm = 0.1 kg. Letak= 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:

. (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 coarse-grained 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 5s between runs. The total mass of the grinding ball charge is about 100 g, the number of balls in a charge Nbis ~450. Isopropyl alcohol (from 5 to 15 ml) is added to the powder and after milling the powder preparedis dried. The mass of the powder for ball milling is 10, 20, 25, and 33g.

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) method from broadening of diffraction reflections. XRD measurements are carried out in a ShimadzuXRD-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 2s at each point. The diffraction reflections were described by the pseudo-Voigt function.

Inthediffractionexperiment, themeansize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)= , 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).

Figure1. 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 20g 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. / Figure2. Determination of mean particle size Dand microstrains  in nanocrystalline WC powder prepared by ball milling of 20 g of the initial coarse-grained 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)/. 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 saccording to the formula ={[*(2)]/2s} [(tg)/2]. Note that we wrongly determined the quantity of microstrainsas  ={[*(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, andfor 20g of the WC powder ground for the same time D = 375nm 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 tleads to the smaller particle sizeD 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)

where aD=A/3=(fs/fv)qu/(sfd3)and bD=[Bln(Din/2b)]/3=(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=-0.004247ms/kgandbD=9.52ms/kgoraD= 4.247106nms/kg andbD = 9.52109nm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.

Figure3. 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 plane-taryball mill, the ball charge is 0.1kg and the rotation speed is 8.33 rps). Experi-

mental data on the particle size Dexp are approximated by the function (14) with the

fitted parameters aD=-4.247106nms/kg andbD=9.52109nms/kg;themicrostrain 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 aDand bDcan 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=720GPa,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 0.084nm2and 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 energyEat. 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50kJmol-1, whence qu= 4.431019J. Considering these values and taking fs/fv= 6, C=18 and =100, we shall have the constants A= 0.002 иB=0.85 Jmkg-1 for hexagonal tungsten carbide WC. At  0.0015kgm2, the angular rotation speed =8.33rps, and the initial particle size Din=610-6 m, we shall obtain aD=0.00232and bD=9.0899nms/kg (or aD= 2.3106 andbD=9.1109 nms/kg). The calculated and experimental values of aDand bD agree well.

Figure4. Theoreticalthree-dimensionaldependenceof 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 aDand bDis shown inFig. 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 .