A novel macrokinetic approach for mechanochemical reactions.pdf

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Chemical Engineering Science 58 (2003) 815–821

www.elsevier.com/locate/ces

A novel macrokinetic approach for mechanochemical reactions

F. Delogua, R. Orr Âu b, G. Caoa ;∗

aDipartimento di Ingegneria Chimica e Materiali, UnitÂa di Ricerca del Consorzio Interuniversitario Nazionale di Scienza e Tecnologia dei Materiali

(INSTM), UniversitÂa degli Studi di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy bConsorzio PROMEA Scarl, c/o Dipartimento di Fisica, Cittadella Universitaria di Monserrato, S.S. 554 bivio per Sestu, 09042 Monserrato (CA),

Italy

Abstract

Mechanical treatment by ball milling (BM) is an extremely versatile technique allowing for the synthesis of non-equilibrium structuresand the production of nanostructured alloys for advanced materials science technology as well as the degradation of toxic compounds and

the conventional mineral processing. In the present paper, we ÿrst review the recent developments in basic research on the kinetics of

phase transformation under milling. Secondly, we propose a novel phenomenological macrokinetic approach and we test its capability of

reproducing experimental data available in the literature. It was shown that the degree of conversion during BM can be correlated with

the milling time and the ball to powder mass ratio. The proposed approach may provide a useful framework for the rationalisation of

experimental results as well as the opportunity for further progress in the ÿeld of mechanochemical processing.

? 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Powders technology; Attrition; Ball milling; Reaction engineering; Kinetics

1. Introduction

Modern era of scientiÿc investigation on mechanochemi-

cal reactions, and mechanical alloying (MA) in particular, is

usually dated back to the pioneer work of Benjamin at the In-

ternational Nickel Company’s Paul D. Merica Research Lab-

oratory in the late 1960s. Mechanical “milling/mixing” of

powders, only later referred to as “MA” in a patent attorney,

was originally employed to combine dispersion strength-

ening with precipitation for the industrial production of

oxide-dispersion-strengthened nickel-based superalloys for

gas turbine engine components (Benjamin & Volin, 1974).

However, the subsequent discovery of metastable phase for-mation by ball milling (BM) opened the door to important

applications in Materials Science (Koch, Cavin, McKamey,

& Scarbrough, 1983).

Due to its actual commercial exploitability and to the un-

usual physical phenomena involved, BM is attracting inter-

est from both the technological and the scientiÿc point of

view. Applications currently range from Materials Science

∗ Corresponding author. Tel.: +39-70-675-5068; fax: +39-70-675-

5067.

E-mail address: [email protected] (G. Cao).

and advanced technology to traditional mineral processingand alternative wastes degradation (CMPS& F-Environment

Australia, 1997; Mulas, Loiselle, Schini, & Cocco, 1997;

Ito, 2000; Aksani & Sonmez, 2000; Suryanarayana, 2001).

Mechanochemical processing can be performed with a

variety of devices taking advantage of two basic mechan-

ical actions such as impact and attrition. In what follows,

our attention will be mainly devoted to impact grinders

since they are widely adopted to carry out mechanochemical

processes.

Usually carried out inside batch reactors, the grinding

of powders by BM consists in a series of impacts during

which powder particles become trapped between the externalsurfaces of two colliding bodies, as schematically shown in

Fig. 1, so that high deformation rates are obtained.

Cold-welding and fracturing phenomena are induced

together with comminution, continuous defect generation

and interface renewal processes which allow for structural

and chemical transformations to occur. Among the oth-

ers, the nanometric reÿnement of microstructure, as shown

in Fig. 2 for the case of milling of Ti, the extension of

solid solubility limits, the synthesis of novel crystalline

and amorphous phases and the ignition of self-sustaining

high-temperature reactions are some of the observed phe-

nomena (Suryanarayana, 2001).

0009-2509/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0009-2509(02)00612-7

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816 F. Delogu et al./ Chemical Engineering Science 58 (2003) 815– 821

Fig. 1. Sketch of powder behaviour under the mechanical action of theimpinging ball.

5 10 15 20 25 30 35 400

10

20

30

40

50

60

70

80

90

T i a v e r a g e

c r y s t a l l i t e s i z e , < L > ,

n m

Milling time, t , h

0

Fig. 2. Decrease of the average size L of the Ti coherent diraction

domains as a function of the processing time t . Milling was performed

at a speciÿc intensity I m = 0:24 W g−1.

Considerable eorts are currently devoted to a deeper

understanding of fundamentals of mechanically driven processes and to the optimisation of BM devices, which

are intimately connected. Indeed, the nature and the rate

of structural and chemical transformations depend on both

the powder properties and the ultimate limit of structural

and thermodynamic stability reached during the mechanical

processing. The latter one, in turn, should be related to the

severity of the mechanical deformation at impact and then

to the force acting on, or to the energy transferred to, the

powder trapped during a collision.

These aspects have been investigated theoretically

through the development of remarkable mathematical

models supported by numerous experimental results

(Suryanarayana, 2001). Modelling activities have devel-

oped along dierent lines of inquiry and the large number

of contributions prevents any detailed discussion. For this

reasons we briey recall here only the most outstanding

attempts to create a coherent framework for the rationali-

sation of fundamentals of BM, while the interested reader

may consider the speciÿc references (Khina & Froes, 1996;Suryanarayana, 2001).

The mechanics and the dynamics of mechanical alloy-

ing have been discussed in terms of so-called “local” and

“global” models. The former ones deal with the deforma-

tion response of powder particles involved in a collision

event, while the latter ones describe the speciÿc operative

features of the milling device (Courtney, 1995; Maurice &

Courtney, 1996). The eects of particle deformation, coa-

lescence and fracture on particle size, shape and hardness

are reasonably accounted for, while the approach does not

account for chemical interaction mechanisms and structural

transformations.

Such aspects are incorporated into a recent semi-empirical

model developed to suitably describe the structural reÿne-

ment processes (Tian & Atzmon, 1999). Supported by de-

tailed X-ray diraction analyses on powders milled at dif-

ferent intensities and temperatures, the model is based on a

previous work on the so-called “driven alloys” synthesised

by ion beam irradiation (Bellon & Martin, 1989). Accord-

ingly, the mechanism of structural reÿnement is basically

described in terms of so-called ballistic and thermal jumps.

The ÿnal microstructure then arises from the competition

between disordering and reordering phenomena which de-

pend, respectively, on the milling intensity and the thermal

migration of defects. The high degree of sophistication of the model shed new light on the possible microscopic mech-

anisms of defect generation, combination and annihilation.

At the same time, however, it also hinders any direct ex-

perimental evaluation of the critical parameters related to

microscopic quantities, such as the density and mobility of

lattice point defects. Therefore, it becomes dicult to iden-

tify speciÿc operating parameters and relate the reactive be-

haviour observed to the properties of the various chemical

systems investigated.

This consideration provides an a posteriori support to

the suggestions of the East European scientists involved in

BM research (Boldyrev, 1981; Butyagin, 1989). Follow-ing their basic observations, an alternative approach to the

local models which has been recently developed does not

take into explicit consideration the physics of the elemen-

tary reactive events occurring on the microscopic scale un-

der impact loading (Delogu, Monagheddu, Mulas, Schini,

& Cocco, 2000). Such events are recognised to be too com-

plex and co-related to be described by accurate models. In

addition, the large gap existing between theoretical and ex-

perimental results would not allow any validation of as-

sumptions related to the microscopic behaviour of matter

under mechanical processing conditions (Khina & Froes,

1996). Thus, the novel approach we propose in this work



F. Delogu et al./ Chemical Engineering Science 58 (2003) 815– 821 817

is a phenomenological one. Based on accurate experimen-

tal methodologies (Delogu et al., 2000; Delogu, Schini, &

Cocco, 2001), the proposed macrokinetic model relates the

total mechanical work performed on powders in the course

of the milling treatment to their degree of structural evo-

lution, with the ÿnal goal of highlighting possible relation-

ships among the processing parameters, the structural trans-formation observed and the characteristic properties of the

system investigated.

2. Model development

The evaluation of the mechanical work performed on

powders requires the reliable deÿnition of the main param-

eters of the mechanical treatment, such as the impact ve-

locity and the collision frequency, and the detailed charac-

terisation of the dynamics of milling regimes as a function

of the powder charge. Carried out in the past for the sim-

plest grinders only (Suryanarayana, 2001), such a charac-terisation has been made recently possible also for the more

complex shaker mills by the development of a suitable ex-

perimental methodology (Delogu et al., 2000).

As discussed in detail in a previous work (Delogu et al.,

2000), the methodology has been ÿrst applied to the most

diuse commercial mill, the Spex Mixer/Mill mod. 8000,

and to a prototype vibrating grinder. In both cases, in or-

der to ensure inelastic conditions at impact the use of a sin-

gle milling ball and of relatively high powder charges was

a necessary condition to experimentally measure the main

milling parameters, namely the average energy at impact

and the frequency of collisions. The measurement of boththe impact energy, E , and the collision frequency, N , allows

one to deÿne the so-called intensity of the mechanochemical

processing, I , as follows:

I = NE = 12Nmbv

2; (1)

where I represents the rate of energy transfer to the milled

powders, mb is the mass of the single milling ball and v the

impact velocity. The total mechanical energy D transferred

to the powders can be easily obtained by multiplying the

milling intensity with the milling time t . It is however useful

to refer this parameter to the powder charge, mp, subject

to mechanical processing, in order to obtain the so-called

speciÿc milling dose, Dm, as follows:

Dm =It

mp

=NEt

mp

: (2)

Systematic investigation on the amorphisation of transition

metal binary mixtures proved that the milling energy dose

required to attain a given degree of structural transformation

is a characteristic, invariant quantity of the system under

examination (Delogu et al., 2001). Indeed, as shown in Figs.

3 and 4, the kinetic data determined by X-ray diraction

(XRD) analysis, concerning the evolution of the amorphous

mass fraction, , and the reduction of the crystallite size, L,

respectively, superpose when reported as a function of the

0 5 10 15 20 25 30 350.0

0.2

0.4

0.6

0.8

1.0

I m

= 0.03 W g-1

I m

= 0.12 W g-1

I m

= 0.38 W g-1

I m

= 2.67 W g-1

A m o r p h o u s f r

a c t i o n , α

Specific milling dose, Dm, kJ g-1

Fig. 3. Experimentally determined amorphous fraction in Ni40Ti60 samples

reported as a function of the speciÿc milling dose Dm. The experimental

data superpose despite the dierences in the speciÿc milling intensity I m.

0 5 10 15 20 250

10

20

30

40

50

60

70

Specific milling dose, Dm, kJ g-1

A v e r a g

e g r a i n s i z e , < L > , n m

I m

= 0.03 W g-1

I m

= 0.12 W g-1

I m

= 0.38 W g-1

I m

= 2.67 W g-1

Fig. 4. The average size of the crystallite domains, L, reported as

a function of the speciÿc milling dose Dm. Data refer to Ni powders

processed at dierent milling intensities.

speciÿc milling dose (Delogu et al., 2001), independently

of the speciÿc milling intensity, i.e. I m = I=mp, adopted.A simpliÿed kinetic model has been also developed on

the basis of a statistical approach to describe the structural

evolution of powders (Delogu et al., 2001). In particular, the

degree of amorphisation, for example, has been expressed

as follows:

= 1 − (1 + KI mt ) exp(− KI mt ): (3)

While the latter expression has been used to ÿt the exper-

imental data, as shown in Fig. 3, K represents the kinetic

constant of the transformation process, which takes, for each

binary system, a characteristic value that is experimentally




found to be qualitatively related to the mechanical properties

of the starting elemental powders.

The comparison among the energy doses required to reach

a certain degree of transformation for dierent systems, or,

alternatively, among the kinetic constants, has pointed out

an interesting relationship between milling dose and me-

chanical properties characteristic of the processed powders(Delogu et al., 2001). The relationship is, however, of qual-

itative nature and further work is needed to gain a deeper

insight.

As for the macrokinetic approach to mechanochemical

reactions, it is possible to generalise the above mentioned

mathematical model to dierent mills and milling conditions

including the use of a number of milling balls greater than

one. To this aim, it is useful to refer the milling intensity I

to the charge ratio C R. By recalling the deÿnitions of milling

intensity and milling dose in Eqs. (1) and (2), it is possible

to express the speciÿc milling intensity for the case of n

balls of equal mass mb as

I m =mb(

N iv2i )

2mp

; (4)

where N i and vi are the average collision frequency and

the average impact velocity of the ith ball, respectively.

The previous equation can be easily simpliÿed by regarding

the n balls as indistinguishable milling bodies characterised

by equal, statistically averaged, collision frequencies and

impact velocities. It then follows that

I m ≈nmb N v

2

2mp= AC R; (5)

where n is the number of balls, C R = nmb=mp is the charge

ratio and A is a phenomenological constant.

By considering the degree of amorphisation deÿned in Eq.

(3) as a parameter related to the structural evolution of pow-

ders, we recall here that the constant K has to be regarded

as the characteristic kinetic constant of the transformation

process. Using a statistical approach, it has been recently

demonstrated (Delogu et al., 2001) that the transformation

rate constant is proportional to the mass fraction of powder

charge cold worked, on the average, at each impact. There-

fore

K ˙m∗

mp

; (6)

where m∗ is the mass of powder cold worked, on the aver-

age, at each impact. When n balls are employed, it can be

assumed that the mass of powder cold worked at impacts in

a cycle of the mill scales linearly with the number and the

mass of the balls. Indeed, as the number of balls increases,

the mass of powder trapped also increases. This is partic-

ularly true in the case of the Fritsch Pulverisette ball mill,

where milling balls give rise to impact sequences following

distinct trajectories along the base of the vial, as shown in

Fig. 5.

The volume of powder trapped also increases when the

balls size, and therefore their mass, increase. It then follows

Movement of

the supporting

disk

Centrifugal

force

R o t a t i o n o f t h e g r i n d i n g b o w l

Fig. 5. Horizontal section of the Fritsch Pulverisette 5 planetary mill.

A sketch of the ball and powder dynamics is shown together with the

motion components of both the supporting disk and the vial, rotating

along dierent axes in opposite directions.

that the kinetic constant appearing in Eq. (3) may be also

expressed as

K ≈Bnmb

mp= BC R; (7)

where B is a constant of proportionality.

At low conversion degrees, Eq. (3) can be safely approx-

imated as follows:

≈ ( KI mt )2: (8)

The milling time t necessary to observe the low conversion

degree is therefore

t ≈ 1= 2

KI m:

The replacement of K and I m with the approximate expres-

sions in Eqs. (5) and (7) leads to the following relationship:

t ≈1= 2

ABC 2 R=

C

C 2 R; (9)

where C =1= 2=AB is a phenomenological constant which de-

pends on the conversion degree considered, on the milling

parameters and the intrinsic physical and chemical proper-

ties of the system investigated.

3. Results and discussion

In what follows, the reliability of the macrokinetic model

derived above is tested by comparison with literature exper-

imental data. In particular, mixtures of commercially avail-

able, high purity, Mo and Si powders in stoichiometric ratio

1:2 were milled in a Fritsch Pulverisette 5 ball mill by us-

ing zirconia vial and 50 zirconia balls. A rotation speed of

250 rpm was used, while the C R was changed in the range

between 7 and 21 by changing the powder charge inside

the vial. In order to prevent oxygen, nitrogen and humid-

ity contamination, all the powder handling procedures were




Fig. 6. XRD patterns relative to Mo–Si powders milled 3 h at 250 rpm.

Each milling trial is characterised by theC R

value quoted. Diraction

lines pertaining to the elemental powders and the ÿnal product phases

are explicitly indicated.

performed inside a glove box under Ar atmosphere (Orr Âu,

Woolman, Cao, & Munir, 2001). Structural transformation

was followed by XRD. A Scintag XDS 2000 X-ray dirac-

tometer, equipped with Cu- K radiation tube, was used to

carry out XRD on powders milled under dierent C R and

milling time conditions.

It is worth noting that the ÿrst appearance of XRD peaks

products formed during milling corresponds to a MoSi2 massfraction in the mixture of about 0.05, i.e. suciently low

conversion degree to apply Eq. (8).

The eect of the C R on the rate of product phase forma-

tion is clearly shown in Fig. 6, where diraction patterns

relevant to powders milled 3 h at dierent C R are reported.

It is seen that diraction lines pertaining to the intermetallic

compound become more and more intense as the C R value

increases. Therefore, a larger weight fraction of the MoSi2

compound is formed in the mixture, in the same time inter-

val, when the C R is raised from 7 to 21. Correspondingly,

the highest transformation rate pertains to the milling trial

performed at the highest C R value.As shown in Fig. 7, Eq. (9) is able to satisfactorily re-

produce the experimentally determined trend of the milling

time required for the appearance of the MoSi2 phase as a

function of the charge ratio.

It should be noted that the experimental trend observed

in the case of Mo–Si system is quite common for gradual

phase transformations in binary mixtures, as shown by the

experimental data quoted in Fig. 8 for a restricted number

of systems (Orr Âu et al., 2001; Delogu et al., 2001). Unfor-

tunately, a very limited number of experimental data sets is

available to test the model and to evaluate the characteristic

constant C , reported in Table 1 for the examined cases.

0 5 10 15 20 250

3

6

9

12

T i m e t o p r o d u c t

d e t e c t i o n , t , h

Charge ratio, C R

Fig. 7. Time elapsed from the beginning of the milling runs to the

detection by XRD of the MoSi2 phases as a function of the C R. XRD

detection limits amount to about 5% in weight. The best-ÿtted curve,

corresponding to Eq. (9), is also shown.

0.0 0.5 1.0 1.5 5 10 15 200

4

8

12

16

20

Cu40

Ti60

Ni40

Ti60

MoSi2

T i m e t o p r o d u c t d e t e c t i o n , t , h

Charge ratio, C R

Fig. 8. Milling time to product detection as a function of C R for the

systems Cu40Ti60, Ni40Ti60 and MoSi2.

Table 1

Values of the characteristic constant C as obtained by ÿtting of existing

experimental data

Chemical system Characteristic constant C , h

Ni40Ti60 0.9

Cu40Ti60 3.0

MoSi2 573.0

This is a clear consequence of the very small number of

experimental studies explicitly devoted to the quantitative

description of the kinetics of mechanochemical processes

under dierent conditions. A systematic investigation on




the behaviour of the dierent classes of chemical systems,

necessary to individuate correlations between kinetics,

milling parameters and materials properties, has yet to be

performed.

However, even from the small number of systems exam-

ined, it seems possible to predict the structural evolution of

powders under milling conditions by resorting to a relativelysimple approach which takes both material properties and

milling parameters into account, through a phenomenolog-

ical constant and the C R, respectively. However, it should

be noted that the C R itself is not able to suitably charac-

terise the milling conditions, as clearly demonstrated else-

where (Delogu et al., 2000), even if it may be regarded as

a useful operating parameter for the approximate deÿnition

of working conditions in industrial mills.

Further progress is, therefore, necessary in order to gain a

better description of the milling processes and, consequently,

a deeper understanding of the physico-chemical behaviour

of powders under impact loading.

4. Concluding remarks

Even if based on a phenomenological approach, the

macrokinetic model adopted in this work provides a satis-

factory rationalisation of structural evolution under milling

conditions. In particular, the model allows for the accu-

rate description of the structural transformations occurring

inside a mechanochemical reactor and the dependence

of the transformation rate on the intensity of mechani-

cal treatment. In addition, the present approach seems to be advantageous in view of the possible, easy extension

to mills of dierent geometry and characteristic milling

action.

The present contribution represents only a ÿrst step to-

wards the complete description of powder processing in high

energy ball mills. Further work is needed to mark a deÿ-

nite progress in the ÿeld of mechanochemical activation that

could allow a deeper understanding of the complex phe-

nomenology involved.

Nomenclature

A phenomenological constant, W g−1

B phenomenological constant, g J−1

C phenomenological constant, h

C R ball to powder mass ratio or charge ratio

Dm speciÿc milling dose, J g−1

E impact energy, J

K kinetic constant, g J−1

I intensity of the mechanochemical processing, W

I m speciÿc intensity of the mechanochemical process-

ing, W g−1

m∗ mass of powder cold worked, on the average, at

impact, g

mb mass of a single ball, g

mp mass of powder, g

n number of balls

N average collision frequency, Hz

N i collision frequency of the ith ball, Hzt milling time, h

v average impact velocity, m s−1

vi impact velocity of the ith ball, m s−1

Greek letters

degree of conversion

Acknowledgements

Prof. G. Cocco, Dipartimento di Chimica, UniversitÂa degli

Studi di Sassari, Italy, and Prof. Z. A. Munir, Department of

Chemical Engineering and Materials Science, University of

California at Davis, USA, are gratefully acknowledged for

useful discussions and valuable support.

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A novel macrokinetic approach for mechanochemical reactions.pdf