In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
h p://dx.doi.o g/10.1016/j.in e me .2007.12.011
1
An equi alen ime app oach o scaling he mechanical alloying
p ocesses
J. J. Ipus1, J. S. Blázquez1, V. F anco1, M. Millán1, A. Conde1, D. Oleszak2, T. Kulik2
1 Dp o. Física de la Ma e ia Condensada, ICMSE-CSIC, Uni e sidad de Se illa, P.O.
Box 1065, 41080, Se illa, Spain.
2 Facul y o Ma e ials Science and Enginee ing, Wa saw Uni e si y o Technology, ul.
Woloska 141, 02-507, Wa saw, Poland.
ABSTRACT. Dynamics o a single ball in o a plane a y ball mill is analyzed leading o
a cubic dependence o he powe ans e ed du ing milling wi h he o a ional speed, .
This leads o he de ini ion o an equi alen ime o desc ibe he s a e o ball milled
powde s independen ly o . Mechanical alloying o Fe75Ge20Nb5 composi ion is
s udied by a combina ion o expe imen al echniques (di e en ial scanning calo ime y,
scanning elec on mic oscopy, ene gy dispe si e x- ay spec ome y, x- ay di ac ion,
Mössbaue spec ome y and ib a ing sample magne ome y) and esul s e idence a
good ag eemen wi h he p edic ions o he equi alen ime app oach.
Keywo ds: C. Mechanical alloying and milling; C. Nanoc ys als
Co esponding au ho : P o . A. Conde
Depa amen o de Física de la Ma e ia Condensada. Uni e sidad de Se illa.
Apa ado 1065, 41080 Se illa (Spain).
Phone: (34) 95 455 28 85
Fax: (34) 95 461 20 97
E-mail: [email p o ec ed]
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2
1. In oduc ion
Ball milling has been shown as a e y e sa ile echnique o p oduc ion o
me as able sys ems: nanoc ys alline, amo phous, supe sa u a ed solid solu ions,
quasic ys als, e c [
1
]. Unlike ul a as cooling echniques, which can eeze high
empe a u e mic os uc u es a low empe a u es in a na ow composi ional ange
a ound he eu ec ic, ball milling is applied o a wide composi ional ange. Mechanical
e olu ion induced by ball milling is due o he ene gy ans e ed om he milling
media o he powde pa icles, con inuously submi ed o ac u e and cold welding
p ocesses which will de ine hei inal mo phology. Among he di e en ypes o ball
mills a ailable, plane a y ball mills a e widely used o p oduce such me as able
ma e ials.
In o de o unde s and he dynamics o plane a y ball mills, se e al au ho s ha e
used he app oach o single ball dynamics and ha e ex apola ed hei esul s o ac ual
milling p ocesses using se e al balls [
2
,
3
,
4
,
5
,
6
,
7
,
8
]. These s udies analyze he ene gy
ans e ed pe collision be ween he ball and he ial wall and he dependence o he
ball ajec o y on di e en pa ame e s such as a io be ween he equencies o main
disk and ials, ball o powde mass a io and o he s [1,2,3,4,5,6,7,8,
9
,
10
]. Those esul s
yield a pa abolic dependence o he in ensi y o a single ball-powde in e ac ion wi h he
o a ional speed. Mo eo e , he ajec o y desc ibed by his single ball was ound
independen o he o a ional speed [2]. These wo ea u es, as i will be shown in he
ollowing sec ion o his pape , migh lead o a cubic law o he es ima ed powe
ans e ed o he powde unde a single ball app oach. The eliabili y o his p edic ion
is es ed in he hi d sec ion o he pape by s udying he e olu ion o a mechanically
alloyed Fe75Nb5Ge20 powde a di e en milling in ensi ies du ing di e en imes.
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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3
2. App oach o he dynamics o he plane a y ball mill
2.1 Mo emen o a single ball in o he ial
Single ball app oxima ion has been used in he li e a u e o p edic he ene gy
ans e o powde du ing milling. In his sec ion, a b ie desc ip ion o he o a ional
speed dependence o bo h he ene gy in ol ed in di e en ball-powde in e ac ions and
he equency o hese e en s is gi en. Figu e 1 shows a scheme o he plane a y ball
mill, whe e

R
is he posi ion o he ial cen e and i s modulus he adius o he main
disk;

, he posi ion o he ball espec o he cen e o he ial, which modulus is he
adius o he ial minus ha o he ball;
  Rp
is he posi ion o he cen e o he
ball; = +

0 is he angle o a ed by he main disk a ime and  i s angula speed;
= +

0 is he angle o a ed by he ial a ime and  i s angula speed. In his
scheme, he o a ions o he ial and ha o he main disk a e opposi e. Th ee
o hogonal e e ence sys ems ha e been de ined o posi ion he ball a poin P; a
Ca esian ine ial sys em
 
,,i j k
and wo pola and non-ine ial sys ems:
,,
R
ee







,
which ollows he o a ion o he main disk, and
,,
uu






, which ollows he
o a ion o he ial.
In he simpli ied s udied sys em, he pa icle mo emen mus ul ill some
equi emen s, also conside ed by o he au ho s [3,4]:
 Only he mo emen o one ball is conside ed.
 The mo emen occu s in a plane.
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4
 Only wo o ces a e conside ed; he no mal eac ion o he ial wall o e he
ball,

N
, and he ic ion o ce,


F
.
 The pa icle keeps ixed a a poin o he ial wall since N>0.
 When N=0, he ball de aches om he ial wall and mo es eely un il i impac s
again wi h he ial wall.
 A e he collision, he ball mo es again s uck o he wall.
A e analyzing he dynamics o he sys em and conside ing only he o ce
componen pa allel o

u
an exp ession o N pe mass uni o ball is ound,
)cos(
22

 R
m
N
(1)
This exp ession is meaning ul only o N>0, i N=0 he ball will de ach om he ial
wall, hus a de achmen ime, d, can be ob ained as:
 

















00
2
2
a ccos
1



R
d
(2)
Analogously o exp ession (1), conside ing he componen pa allel o


u
, an
exp ession o F

pe mass uni o ball is ob ained,
)sin(
2


 R
m
F
(3)
This solu ion will apply i he ball is assumed o be ixed a a poin o he ial wall.
Once he ball de aches om he ial wall i mo es eely wi h cons an eloci y un il i
collides again wi h he ial wall. In o de o calcula e he ime o ligh , , he sepa a ion
dis ance be ween he ball and he ial wall has been analyzed. In ac , he ball will
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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5
collide when
pR
equals he adius o he ial, . Figu e 2 shows he cu e o he
di e ence be ween he squa e o
pR
and he squa e adius o he ial as a unc ion o
ime om he de achmen o he plane a y mill used in sec ion 3, wi h R=0.125 m,
=0.017 m (i has been conside ed ha he ball has a adius o 0.005 m), a =150, 250
and 350 pm, being he alues o d and o he same o de o magni ude. By using
di e en alues o , keeping cons an he a io /, i can be obse ed ha bo h d and
a e p opo ional o 1/ (see exp ession (2) and inse o igu e 2, espec i ely) and,
he e o e, he numbe o de achmen and collision e en s emains cons an along a
pe iod o o a ion o he main disk.
Fo > d, N=0 (as well as F

) and he ball is no in con ac wi h he ial wall.
A e he ball collides wi h he wall, N will eco e a non ze o alue. The ajec o y o
he ball, o bo h ine ial and non-ine ial e e ence sys ems a e shown in igu e 3. I can
be obse ed ha he poin o collision, PC, is di e en o he posi ion o he de achmen
poin a he ime o collision, PD’. The e o e, a phase angle,

, mus be included in he
a gumen o he cosine unc ion o equa ion (1). A e he collision, he p ocess is
epea ed con inuously and igu e 4 shows he alue o N, solid line, du ing a o a ional
pe iod o he main disk along wi h he analy ical solu ion o equa ion (1), do ed line,
o compa ison.
2.2 Es ima ion o he dependence o he ene gy ans e on he equency
The simple model desc ibed abo e can be used o p opose a dependency o he
ene gy ans e ed du ing milling om he balls and ial o he powde . Se e al

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6
mechanisms can be conside ed as candida es o ene gy ans e om he balls and ial
o he powde .
Fi s , while he ball is in con ac wi h he ial wall, he powde which is in
be ween is comp essed and shea ed and he wo k done o e he powde du ing his
comp ession could be app oxima ed o:
2
1kdNdxNWN
(4)
whe e d is a dis ance o he o de o he powde pa icle size (mic ome e ) and k1 is a
cons an . Fo =150 pm, he ene gy ans e ed by uni mass o ball can be es ima ed
as ~10-4 J/kg o d~1 m. Analogously, i could be possible o es ima e a simila o de
o magni ude o he shea wo k done by he ic ion o ce.
On he o he hand, when he ball, a e being de ached, impac s agains he ial
wall, a change in i s eloci y is necessa y o allow he ball o mo e wi h he ial again.
This change implies a educ ion in he kine ic ene gy o he ball, which can be
calcula ed as:
 
2
2
)(
2
1impdC mE 
(5)
whe e ( d) is he speed o he ball a he de achmen ime and imp is he modulus o he
speed o he ial wall a PC:
 
 
 
2
2cossin)( 






 R
R ddddd


(6)
    
 
    
2
2cossin 






 R
R d d d dimp


(7)
The alues o EC pe uni mass o ball, as i occu s o he es ima ed wo k o
comp ession, and as expec ed by exp essions (6) and (7), ollows a 2 law. Fo =150
pm he o de o magni ude o he ans e ed ene gy is 1 J/kg pe collision e en .
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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Some au ho s ha e es ima ed he ene gy ans e ed o he powde , assuming a
He zian collision app oxima ion and a e es ima ing he amoun o powde in be ween
he ball and he ial wall [4,6,8]. Se e al pa ame e s a ec he inal esul s, as ball o
powde mass a io and elas ic p ope ies o he milling media as well as o he powde ,
e c, which we e cons an along he expe imen s pe o med in his wo k. The e o e, he
 dependence o he ene gy ans e ed du ing he milling p ocess is p ese ed.
As i has been shown in igu e 4, du ing a pe iod o o a ion o he main disk, a
cons an numbe o comp ession e en s and impac s occu . Assuming he wo k done
du ing each comp ession is desc ibed by equa ion (4) and ha o each impac is
desc ibed by equa ion (5), he a e age powe could be es ima ed, o long imes, as he
whole wo k done du ing a pe iod o o a ion o he main disk di ided by his ime:
 
3
2

k
T
EWn
PCN
(8)
whe e n is he cons an numbe o e en s pe pe iod and k2 is a cons an . The e o e, he
ene gy ans e ed a e a ime would be p opo ional o 3 and, i se e al alues o 
a e used (keeping cons an he a io /) an equi alen ime, eq, can be de ined as:
 
3
0
/ eq
(9)
whe e 0 is a e e ence equency (in he ollowing s udy 0=150 pm, which is he
lowes equency used).
As milling ime inc eases, ene gy is ans e ed om milling media o powde
yielding mic os uc u e e olu ion and di e en ans o ma ions [1]. Quan i ying
expe imen ally his ene gy is a di icul ask as i s acquisi ion is no comple ely
e e sible. In o de o o e come hese obs acles, some au ho s ha e p oposed ene gy
maps o app oxima ely compa e mic os uc u es ob ained om di e en milling
condi ions [3,8]. In his wo k, i has been assumed ha he s a e o he powde is
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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8
uni ocally de e mined by he amoun o ene gy ans e ed o he powde s. The e o e,
powde o Fe75Nb5Ge20 composi ion has been cha ac e ized by using di e en s uc u al
and magne ic echniques a e milling a di e en equencies du ing di e en imes,
keeping cons an he / a io. Resul s om di e en expe imen al echniques ha e
been escaled using an equi alen ime de ined by equa ion (9) wi h 0=150 pm.
3. Applica ion o expe imen al esul s
3.1 Expe imen al echniques
Fe75Nb5Ge20 composi ion was p oduced om pu e powde s (pu i y99 %) by
ball milling in a F i sch Pul e ise e 5 plane a y ball mill using ha dened s eel balls (10
mm diame e ) and ials. The ini ial powde mass was 5 g and he ball o powde a io
10:1. The a io be ween he o a ional speed o he ial () and ha o he main disk ()
was ixed o /=-2. Th ee di e en alues o  we e used: 150, 250 and 350 pm.
Some powde was aken ou a e selec ed imes ( om 1 h o 150 h), opening and
closing he ials in a gon a mosphe e o a oid oxygen and humidi y con amina ion.
Size and mo phology o he powde pa icles we e s udied by scanning elec on
mic oscopy (SEM) in a Jeol JSM-6460 LV and ene gy dispe si e X- ay (EDX) analyses
we e pe o med using an Incax-sigh o Ox o d Ins umen s. The c ys alline s uc u e
was s udied by X- ay di ac ion (XRD) using Cu-K adia ion in a B uke D8I
di ac ome e and he local en i onmen o Fe a oms was analyzed by Mössbaue
spec ome y (MS). Mössbaue spec a we e eco ded a oom empe a u e in a
ansmission geome y using a 57Co(Rh) sou ce. The alues o he hype ine pa ame e s
we e ob ained by i ing wi h NORMOS p og am [
11
]. The mal s abili y o he samples
was s udied by di e en ial scanning calo ime y (DSC) using a Pe kin-Elme DSC7
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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9
unde a gon low. Speci ic sa u a ion magne iza ion,

S was measu ed in a Lakesho e
7407 ib a ing sample magne ome e (VSM), applying a maximum ield o 1.5 T.
3.2 Calo ime y
Figu e 5 shows he DSC scans o he alloy a e milling 150 h a di e en
o a ional speeds. Bo h samples milled a 250 and 350 pm exhibi simila cu es,
showing a b oad exo he mic maximum gene ally asc ibed o elaxa ion and c ys al
g ow h e ec s [1]. This ac implies ha he ene gy eleased a e hea ea men is no
p opo ional o he amoun o ene gy supplied by he milling media, as milling a 350
pm mus be mo e ene ge ic han milling a 250 pm. On he o he hand, he sample
milled a 150 pm exhibi s, along wi h he b oad exo he m, an endo he mic peak. This
endo he m is asc ibed o he e ogenei ies obse ed in he powde ; as i will be shown
la e by XRD and MS echniques, se e al c ys alline phases coexis in he powde
o med a e 150 h milling a 150 pm. A de ailed discussion on he he mal e olu ion o
milled samples as a unc ion o he milling ime will be epo ed elsewhe e [
12
] as i is
ou o he scope o he p esen pape .
3.3 Powde pa icle size and composi ion
SEM images we e used o measu e he a e age powde pa icle size, <d>,
(s a is ic o e an a e age o 200 pa icles pe sample) as a unc ion o ime and
o a ional speed. Figu e 6.a shows he alues o <d> as a unc ion o ime (uppe panel)
and he equi alen ime de ined by equa ion (9) (lowe panel). Fo samples milled a
150 pm, powde pa icle size keeps almos cons an (~15 m) o low milling imes o
dec ease o longe imes down o a cons an alue o ~5 m. Fo samples milled a 250
pm, <d> s a s dec easing a e y low milling imes and he same cons an alue, ~5
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16
Table 2
Milling ime anges o he h ee ypes o MS i ings pe o med: a) single si e a HF=33
T, b) si e a 33 T plus wo magne ic hype ine ield dis ibu ions and c) using a single
magne ic hype ine ield dis ibu ion.
 ( pm)
[/0]3
a) single HF=33 T
b) HF=33 T +HF
dis ibu ion
c) single HF
dis ibu ion
(h)
eq (h)
(h)
eq (h)
(h)
eq (h)
150
1
up o 10
20 o 150
>150
250
4.6
up o 1
4.6
5-20
23-92
50
230
350
12.7
none
2-5
25.4-63.5
10
127

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17
Figu e cap ions
Figu e 1. Scheme o he h ee e e ence sys ems used.
Figu e 2. De achmen dis ance be ween ball and ial wall du ing ligh o 150, 250 and
350 pm, he c osses show he momen o collision. The inse shows he linea
ela ionship be ween he ime o ligh , , and he o a ional pe iod o he main disk, T.
Figu e 3. Le ; ine ial e e ence sys em: solid line, ball ajec o y, dashed line,
ajec o y o he cen e o mass o he ial, do ed ci cles show he ial posi ion a he
de achmen and collision e en s. Righ , non-ine ial sys em: Ci cle ep esen s he ial;
hick line, ball ajec o y; s aigh solid lines indica e he de achmen poin o he ball
om he wall a he de achmen ime, PD, and a he collision ime, PD’; PC indica es he
collision poin o he ball;  is he angle be ween PD’ and PC.
Figu e 4. No mal o ce exe ed o e he ball (con inuous line) and analy ical solu ion o
equa ion 6 (do ed line) du ing one pe iod o he main disk o a ion calcula ed o
=150 pm.
Figu e 5. DSC scans o Fe75Ge20Nb5 samples a e 150 h milling a di e en
equencies.
Figu e 6. a) Powde pa icle size as a unc ion o ac ual ime and equi alen ime. b)
A e age C con en as a unc ion o ac ual ime and equi alen ime.
Figu e 7. XRD pa e ns o samples a e di e en milling imes a di e en milling
in ensi ies. C osses, bcc-Fe; squa e, cc-Ge; ci cles, bcc-Nb; iangle, in e me allic.
Figu e 8. a) Angula posi ion, 2, and b) ull wid h a hal maximum, FWHM, o he
(110) di ac ion maximum o he -Fe(Ge,Nb) phase as a unc ion o ac ual ime
(abo e) and equi alen ime (below). c) A ea ac ion o cc Ge and bcc Fe phases
Figu e 9. Mössbaue spec a o samples a e di e en milling imes a di e en
equencies.
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18
Figu e 10. a)A e age hype ine magne ic ield, <HF>, and b) sa u a ion magne iza ion
o he di e en s udied samples as a unc ion o ac ual ime (abo e) and equi alen ime
(below).
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19
Figu e 1




e
eR
u
u
R
P
j
i
p
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20
Figu e 2
0.0 0.5 1.0 0.0
0.1
0.2
0.00 0.02 0.04 0.06
-4
-3
-2
-1
0
1
2
3
4
5
350 pm 250 pm
|R-p|2- 2 (10-4 m2)
ime (s)
150 pm
(s)
T (s)
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21
Figu e 3.
0.00 0.05 0.10 0.15
0.00
0.05
0.10
0.15
Y (m)
X (m)
-0,02 0,00 0,02
-0,02
0,00
0,02
PD'PC
PD
Y (m)
X (m)


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22
Figu e 4.
0.0 0.5 1.0
0.0 0.1 0.2 0.3 0.4
-10
0
10
20
30
40
50
60
N/m (N/kg)
ime (s)
ime/T
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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23
Figu e 5.
400 500 600 700 800 900
350 pm
Tempe a u e (K)
250 pm
Exo (2 W/g)
150 pm
dH/d
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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24
Figu e 6.
110 100 1000
0
5
10
15
110 100 1000
0
5
10
15
150 pm
250 pm
350 pm
<d> (m)
milling ime (h)
a)
<d> (m)
·[]3 (h)
110 100 1000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
110 100 1000
0.0
0.1
0.2
0.3
0.4
0.5
0.6
b)
C con en (%)
milling ime (h)
C con en (%)
·[]3 (h)
In e me allics. Vol. 16. Núm. 3. 2008. Pag. 470-478
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25
Figu e 7.
30 40 50 60 70
1 h
2 (deg ees)
5 h
10 h
20 h
50 h
100 h
150 pm
150 h
30 40 50 60 70
1 h
2 (deg ees)
5 h
10 h
20 h
50 h
100 h
250 pm
150 h
30 40 50 60 70
2 h
2 (deg ees)
5 h
10 h
20 h
50 h
100 h
350 pm
150 h