Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
1
Non-Iso he mal App oach o Isokine ic C ys alliza ion P ocesses:
Applica ion o he Nanoc ys alliza ion o HITPERM Alloys.
J. S. Blázquez, C. F. Conde, A. Conde*
Depa amen o de Física de la Ma e ia Condensada. Ins i u o de Ciencia de Ma e iales, CSIC
Uni e sidad de Se illa. Apa ado 1065, 41080-Se illa, Spain.
Keywo ds: c ys alliza ion, phase ans o ma ion kine ics, nanoc ys alline mic os uc u e.
Abs ac
A new app oach is p oposed o he s udy o c ys alliza ion p ocesses unde he hypo hesis o
isokine ic beha io . F om a di ec ex ension o he A ami heo y o non-iso he mal egimes, an
exp ession o he local A ami exponen as a unc ion o he c ys alline olume ac ion is
ob ained om a single DSC un pe o med a a cons an hea ing a e and a aw es ima ion o he
ac i a ion ene gy (a a ia ion o he ac i a ion ene gy alue be ween 3 and 5 eV p oduces a
change on he A ami exponen smalle han ± 0.1). This app oxima ion was applied o he
nanoc ys alliza ion p ocess o a FeCoNbB(Cu) alloy se ies and he esul s ound a e in good
ag eemen wi h p e ious iso he mal analysis on he same alloys. The kine ics is cha ac e ized by
an ini ial A ami exponen close o uni y, which dec eases down o alues below 0.5 as he
c ys alline olume ac ion inc eases.
*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 protected]
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
2
1. In oduc ion
Nowadays, an impo an e o is de o ed o he esea ch o me as able s uc u es. A
s ong enhancemen o di e en physical p ope ies is ound o se e al me as able s uc u es
wi h espec o hei co esponding one in he he modynamical equilib ium s a e. Fo example,
in Fe-based alloys amo phous and nanoc ys alline mic os uc u es imp o e se e al o de s o
magni ude some p ope ies o bo h so and ha d magne ic ma e ials depending on he
composi ion [1-4] and quasic ys alline and nanoc ys alline s uc u es s ongly imp o e he
mechanical p ope ies o Al-base alloys [5].
Me as able s uc u es a e no in he modynamic equilib ium and, he e o e, hey can
spon aneously e ol e o new mo e s able s uc u es, al hough some imes s ill me as able ones.
A e hese ans o ma ion p ocesses, he ma e ials usually modi y hei p ope ies. Besides,
in e es ing me as able s uc u es can be p oduced in a con olled way om he e olu ion o a
p e ious me as able bu less s able s uc u e: o example, he con olled hea ea men o
ce ain amo phous alloys can p oduce nanoc ys alline o quasic ys alline s uc u es o jus
s uc u ally elaxed amo phous alloys in which he desi ed physical p ope ies a e s ongly
enhanced wi h espec o he amo phous p ecu so [1-5]. The e o e, kine ics o ans o ma ion
mus be conside ed as a e y impo an ask, which gi es in o ma ion ela i e o he s abili y
and hus he applicabili y o hese ma e ials, as well as suppo s he pa ame e s necessa y o
con ol he p oduc ion o a desi ed mic os uc u e.
The e olu ion o he me as able s uc u es can be s udied in bo h iso he mal and
non-iso he mal p ocesses. Non-iso he mal expe imen s in ela ion o iso he mal ones ha e he
ad an age o an easie and as e pe o mance, as well as a smalle noise o signal a io o
kine ic expe imen s. Howe e , he ange o applicabili y o he di e en non-iso he mal
me hods o analysis mus be conside ed, because mos o hem we e ob ained as app oxima ions
o heo ies which heo e ical applicabili y is on iso he mal egimes. In his wo k, a new non-
iso he mal app oach is p oposed o s udy he p ima y c ys alliza ion o amo phous alloys as an
ex ension o he Johnson-Mehl-A ami-Kolmogo o heo y o c ys alliza ion [6-10]. A
compa ison is pe o med be ween his new me hod and se e al non-iso he mal app oaches,
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
3
widely used o he s udy o non-iso he mal c ys alliza ion kine ics. The limi s o applicabili y
o di e en me hods as well as he ad an ages o his new me hod wi h espec o o he s ound
in he li e a u e a e emphasized. Finally, an applica ion o he p oposed me hod o expe imen al
da a is pe o med s udying he kine ics o he nanoc ys alliza ion p ocess o a FeCoNbB(Cu)
alloy se ies by di e en non-iso he mal me hods and compa ing he ob ained esul s wi h
p e ious ones om iso he mal kine ics [11].
2. Non-iso he mal kine ic models.
C ys alliza ion kine ics o me allic glasses is usually s udied in he ame o he
Johnson-Mehl-A ami-Kolmogo o (JMAK) heo y [6-10], which can be exp essed by he so-
called JMAK equa ion:
X = 1-exp[-{k( - 0)}n] (1)
whe e k is he equency ac o , he ime, 0 he induc ion ime, n he A ami exponen and X
he e ec i e c ys alline olume ac ion, no malized o he alue a he end o he p ocess. The
JMAK heo y was de eloped o iso he mal p ocesses, howe e , some app oxima ions ha e
been done o ex ending i o non-iso he mal ans o ma ions.
Nakamu a e al. [12,13] gene alized JMAK equa ion o non-iso he mal p ocesses unde
he app oxima ion o isokine ic beha io . This app oxima ion implies ha he c ys alliza ion
p ocess is independen o he he mal his o y o he sample. In Nakamu a equa ion, he
dependence o X wi h empe a u e, T, and ime is gi en by:
















 
n
d TkX
0
)(exp1 (2)
Fo iso he mal p ocesses, a empe a u e TISO, his equa ion yields di ec ly he JMAK
equa ion (1), being cons an k(TISO) = k. On he o he hand, in he case o a cons an hea ing a e
dT/d =

, i is possible o w i e:
 
















 
n
T
To
ndTTkX )(/1exp1

(3)
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
4
which can be w i en as:





 n
TZ
X

)(
exp1 (4)
whe e
n
T
To
dTTkTZ 





)()( (5)
being Z(T) he c ys alliza ion unc ion. Exp ession (4), known as he Ozawa equa ion [14],
allows o ob ain he A ami exponen as he slope o he plo o Ln[-Ln[1-X(T)]] s Ln[1/

] o
a selec ed empe a u e T and di e en

alues. Unde his app oxima ion, i is wo h no icing
ha , a a gi en empe a u e o di e en scanning a es he p ocess migh be in a di e en s age
o de elopmen . The e o e, in his app oach, kine ic pa ame e s, as A ami exponen and
ac i a ion ene gy (Q), a e conside ed independen o X.
F om Ozawa heo y i is also possible o ob ain a simple exp ession o Q i an
A henius law is used o desc ibe he e olu ion o Z(T) [15]. Thus, exp ession (4) can be
ew i en as:
Ln[-Ln[1-X]] = Ln[Z0] - Q/RT – nLn[

] (6)
and o a cons an alue o X, achie ed a di e en empe a u es, T(X0), depending on

, yields:
d(Ln[

])/d(1/T(X0)) = - Q/(nR) (7)
Besides hese app oaches which gi e a local in o ma ion o kine ic pa ame e s n(X) and
Q(X), he e a e di e en app oxima ions o he JMAK equa ion o non-iso he mal p ocesses
om which i is possible o ob ain a e aged alues o he kine ic pa ame e s. Mos o hem use
he hypo hesis o maximum ans o ma ion a e a he peak empe a u e (Tp), being
(d2X/d 2)p = 0. Thus, by di e en ep esen a ions o he pa ame e s a Tp i is possible o ob ain
a e age alues o Q and in some cases o n. Among hese me hods he Kissinge [16,17], Augis-
Benne [18] and Gao-Wang [19] me hods a e ound. Al hough he o me is he mos widely
used, i s heo e ical alidi y o desc ibing c ys alliza ion o amo phous alloys has been
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
5
ques ioned by some au ho s [20,21]. The Kissinge me hod can be summa ized in he
exp ession:




 
R
Q
Td
TLnd
p
p
/1
/2

(8)
On he o he hand, he Augis-Benne me hod conside s he in luence o he onse
empe a u e and yields he exp ession:






 
R
Q
Td
TTLnd
p
op 

/1
/

(9)
Finally, he Gao-Wang me hod gi es wo exp essions, which allow calcula ing bo h Q:






 
R
Q
Td
d dXLnd
p
p
/1
/ (10)
and he A ami exponen :
(dX/d )p = 0.37n

Q/(RTp2) (11)
The e o e, om plo s o Ln[

/Tp2], Ln[

/(Tp-To)] and Ln[(dX/d )p] e sus 1/Tp o
di e en alues o

(Kissinge , Augis-Benne and Gao-Wang plo s, espec i ely), i is possible
o es ima e Q alues. I is wo hy o no e ha he in o ma ion ob ained om hese h ee me hods
is somehow a e aged in empe a u e, due o he change o Tp and To wi h

. Howe e , he alue
o X a Tp is, in a good app oxima ion, independen o

and, he e o e, i can be conside ed ha
he alues o he kine ic pa ame e s ob ained by hese me hods co espond o he alue o X a
he peak empe a u e, X(Tp).
An impo an di e ence be ween he Ozawa me hod and he p e ious ones is ha he
Ozawa me hod allows o selec di e en alues o X, and no only ha o Tp. I mus be aken
in o accoun ha , in Ozawa heo y, kine ic pa ame e s a e supposed o be cons an du ing he
whole c ys alliza ion p ocess. Howe e , we could in e p e exp ession (7) as a dependence on T
o he kine ic pa ame e s only h ough X. Tha means he e is no dependence on he he mal
his o y o he sample bu on he c ys alliza ion s age, i is said, in he ame o isokine ic
app oxima ion used by he Nakamu a heo y.

Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
6
3. Di ec ex ension o JMAK equa ion o non-iso he mal kine ic egimes.
The me hod p oposed in his wo k consis s on a di ec ex ension o he JMAK equa ion
o non-iso he mal p ocesses. Al hough he idea seems o be qui e simple and aw, i s alidi y
and limi a ions will be demons a ed. I Eq.(1) is di ec ly used o non-iso he mal ea men s
wi h a cons an hea ing a e, i should be aken in o accoun ha - 0 = (T-To)/

, whe e To is he
empe a u e a he c ys alliza ion onse . The e o e, i can be w i en:
X = 1-exp[-{k’(T-To)/

}n] (12)
whe e k’ is a new equency ac o .
The ela ionship be ween he di e en equa ions can be obse ed h ough he Nakamu a
equa ion. Assuming an A henius dependence o k’ wi h T:
k’ = k’0exp[-Q/RT] (13)
whe e k’0 is a cons an , Q is he ac i a ion ene gy and R is he gas cons an and compa ing
equa ion (3), (4) and (12), i is possible o w i e:
 
n
T
To
oTZdTTkTTRTQk /1
0)()()(/exp'   (14)
k(T) = k’0{1+(T-To)Q/RT2}exp[-Q/RT] (15)
Exp ession (15) is a mo e complica ed o m han he simple A henius law:
k(T) = k0exp[-Q/RT] (16)
no mally used o desc ibe he empe a u e dependence o some kine ic pa ame e s, as he
equency ac o and he induc ion ime. In ou case, o simpli y he exp essions, A henius law
was used o desc ibe he beha io o he pa ame e k’.
In he case o JMAK equa ion (1), o iso he mal p ocesses he local A ami exponen
a a selec ed alue o X, n(X), can be ob ained om he slope o Ln[-Ln[1-X]] s Ln[ - 0]:




 
 
0
1
( ) d Ln Ln X
n X d Ln
 
 
 
 (17)
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
7
bu , in he case o non-iso he mal ans o ma ions he dependency o k’ wi h T mus be
conside ed. The e o e, he exp ession (17) would be:






 
 
 


 
 
 





















T
T
RT
Q
n
TTLnd
kLnd
n
TTLnd
XLnLnd o
oo
11
/
)'(
1
/
1

(18)
I is clea ha (18) educes o (17) o iso he mal ea men s, whe e T = To is cons an .
F om (18) i will be easy o ob ain n(X).
The p oposed app oxima ion shows wo main ad an ages wi h espec o he Ozawa
app oach:
Fi s , in Ozawa app oxima ion, he local A ami exponen , n(T), is calcula ed as a
unc ion o T o di e en alues o

and, he e o e, i is implici ly assumed ha he p ocess is
independen o X, which alue depends on

o a gi en empe a u e. Howe e , in he di ec
ex ension o JMAK o non-iso he mal p ocesses, he local A ami exponen , n(X), is calcula ed
as a unc ion o X, independen ly o

.
Second, an impo an cha ac e is ic o he di ec ex ension o JMAK o non-iso he mal
p ocesses is ha , as o iso he mal cases, n can be ob ained di ec ly om (18) using only one
un, a a single alue o he cons an hea ing a e,

. Howe e , i will be necessa y o ha e an
es ima ion o he ac i a ion ene gy alue which, o example, could be ob ained om li e a u e
da a. In ac , as i will be shown below, he main esul s ob ained using he di ec ex ension o
JMAK equa ion o non-iso he mal egimes a e no signi ican ly a ec ed e en by an impo an
change o Q alue.
4. Applica ion o nanoc ys alliza ion p ocess o FeCoNbB(Cu) alloys.
The me hod p oposed abo e has been applied o he nanoc ys alliza ion p ocess o a
FeCoNbB(Cu) alloy se ies. Amo phous ibbons, abou 20 m in hickness, o nominal
composi ion Fe78-xCoxNb6B16-yCuy (x = 18, 39, 60; y = 0, 1) we e p epa ed by a single wheel
mel spinning echnique. Di e en ial scanning calo ime y (DSC) scans o as-cas samples we e
pe o med in a Pe kin-Elme DSC7 a di e en hea ing a es (2.5, 5, 10, 20, 40 and 80 K/min).
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
8
T ansi ion empe a u es o Pb and C O3K s anda ds we e used o he calib a ion o he
calo ime e and co ec ion o he mal lag in he DSC scans. Values o X we e calcula ed om
he no malized in eg a ion o he dH/d signal a e sub ac ion o he baseline. In he case o he
alloy wi h 18 a . % o Co and wi h Cu hea ed a 80 K/min, he nanoc ys alliza ion p ocess was
no comple ed in he empe a u e ange a ailable o he DSC7 (up o 1000 K). The e o e, his
a e was disca ded in he analyses ha in ol e a good knowledge o he baseline.
Figu e 1 shows, as an example, DSC plo s o Fe60Co18Nb6B16 amo phous alloy o he
nanoc ys alliza ion p ocess a di e en hea ing a es. As

inc eases, he wid h o he
nanoc ys alliza ion peak dec eases and Tp shi s o highe alues (as i is expec ed o a
he mally ac i a ed p ocess).
As i was poin ed abo e, isokine ic beha io was a equi emen o applying Nakamu a
equa ion. Ins an aneous nuclea ion would ul ill his equi emen . Fo he nanoc ys alliza ion o
FeCoNbB(Cu) alloys, iso he mal s udies showed ha nuclea ion is mainly es ic ed o he
ini ial s ages o he p ocess [11]. In ac , in he case o he alloys wi h Cu, clus e s o his
elemen , o med p e ious o he onse o nanoc ys alliza ion, ac as he e ogeneous nuclea ion
cen e s o he -FeCo nanoc ys als [22], suppo ing he ins an aneous nuclea ion
app oxima ion. On he o he hand, in he case o Cu ee alloys, he p esence o quenched in
nuclei migh be in oked.
In he aim o a sys ema ic compa ison be ween he di ec ex ension o JMAK and he
o he me hods, Q alues we e ob ained by di e en me hods. Figu e 2 shows he Kissinge
(abo e), Augis-Benne (cen e ) and Gao-Wang (below) plo s o he s udied alloys. The linea
egime is well ep oduced in all he h ee me hods o all he s udied samples and, as can be
obse ed in Table I, Q alues om he h ee me hods a e in a good ag eemen o e e y alloy.
The ob ained alues o Q a e simila o hose ound in o he NANOPERM and HITPERM
alloys [23-25].
Once he Q alue has been calcula ed, i is possible o use exp ession (18) o ob ain he
local A ami exponen , o he nanoc ys alliza ion p ocess. Figu e 3 p esen s plo s o he local
A ami exponen , n(X), o he Fe60Co18Nb6B16 alloy ob ained om DSC scans a di e en
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
9
scanning a es using he Q alue ob ained om he Kissinge me hod. As can be obse ed, all
he cu es a e nea ly collapsing in a single one, in ag eemen wi h he expec ed independen
beha io o n(X) wi h

de i ed om he isokine ic assump ion. The obse ed di e ences a e
qui e small conside ing he unce ain y o he baseline, an inhe en p oblem o he DSC analyses
o ans o ma ions ex ended o a b oad empe a u e ange (~ 100-150 K in ou case). Figu e 4
p esen s he n(X) cu es ob ained o he di e en alloys hea ed a

= 10 K/min.
To apply he app oxima ion p oposed in his wo k and summa ized in equa ion (18),
i s , i is necessa y o es ima e Q. This es ima ion can be ob ained om li e a u e da a. Fo
example, in he case o nanoc ys alliza ion p ocesses o -Fe phase, Q anges om 3.3 o
4.5 eV [1]. Figu e 5 shows, o he Fe60Co18Nb6B16 alloy hea ed a 20 K/min, plo s o exp ession
(18) in which he e ec o his inde e mina ion in he Q alues can be obse ed. E en using ha
wide ange o ac i a ion ene gie alues (3 o 5 eV), he A ami exponen shows a maximum
dispe sion o ± 0.1, which dec eases as he c ys alline olume ac ion inc eases.
Ozawa heo y also allows o ob ain a local alue o n, bu as a unc ion o T (Figu e 6).
As poin ed be o e, his me hod implici ly assumes ha kine ic pa ame e s, bo h n and Q, a e
independen o he c ys alliza ion s age (i is said independen o X), which is a s ong
es ic ion. Howe e , o all he s udied alloys a ange o empe a u e a which n(T) is nea ly
cons an is ound. This ange co esponds o he long ails obse ed a high X alues, a e he
peak empe a u e, in nanoc ys alliza ion DSC cu es. A hese ails, whe e he e olu ion o
en halpy, and hus o X, is slowe is whe e he equi emen s o he Ozawa me hod a e be e
ul illed: he sys em being in a simila si ua ion o s ongly hinde ed, slow g ow h p ocess a
his empe a u e ange, independen ly o he scanning a e. Fo a be e unde s anding o he
meaning o he empe a u e dependence o he A ami exponen , n(T), in he Ozawa app oach,
we can use he esul s ob ained om exp ession (18). F om each n i(X), ob ained o e e y

i
alue, i is easy o ob ain a ni(T) h ough he ela ion X(T) ob ained by he no malized
in eg a ion o he co esponding DSC signal. Fo he Fe60Co18Nb6B16 alloy, igu e 7 shows n(T)
plo s ob ained o di e en hea ing a es h ough exp ession (18) and om he Ozawa me hod.
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
16
Table I
Ac i a ion ene gies ob ained by di e en me hods.
Alloy
Kissinge [26]
(eV)
 0.2
Augis-Benne
(eV)
 0.2
Gao-Wang
(eV)
 0.2
Fe60Co18Nb6B15Cu1 3.6 3.8 3.7
Fe39Co39Nb6B15Cu1 3.6 3.6 3.6
Fe18Co60Nb6B15Cu1 3.1 3.2 3.1
Fe60Co18Nb6B16 3.9 4.4 4.4
Fe39Co39Nb6B16 3.6 3.3 3.3
Fe18Co60Nb6B16 3.5 3.4 3.4

Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
17
Figu e 1
700 750 800 850 900 950
80 K/min
40
20
10
2.5
5
(1/

)·dH/d (a.u.)
T (K)
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
18
Figu e 2
1.25 1.30 1.35 1.40
Ln((dX/d )
p
)
1/TP (10-3*K-1)
Ln(

/T
P
2
)
Ln(

/(T
p
-T
o
))
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
19
Figu e 3
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
80 K/min
40 K/min
20 K/min
10 K/min
5 K/min
2.5 K/min
X =

H(T)/

H o al
n(X)
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
20
Figu e 4
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
n(X)
X
60 % Co 1 % Cu
39 % Co 1 % Cu
18 % Co 1 % Cu
60 % Co 0 % Cu
39 % Co 0 % Cu
18 % Co 0 % Cu
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
21
Figu e 5
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
Q = 3 eV
Q = 4 eV
Q = 5 eV
n(X)
X

Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
22
Figu e 6
750 800 850 900 950
0.0
0.5
1.0
T (K)
n (T)
60 % Co 1 % Cu
39 % Co 1 % Cu
18 % Co 1 % Cu
60 % Co 0 % Cu
39 % Co 0 % Cu
18 % Co 0 % Cu
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
23
Figu e 7
750 800 850 900 950
0.0
0.5
1.0 2.5 510 20 40
80 K/min
n(T)
T (K)
Ac a Ma e ialia. Vol. 53. Núm. 8. 2005. Pag. 2305-2311
h p://dx.doi.o g/10.1016/j.ac ama .2005.01.037
24
Figu e 8
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0 Q(X)
746 K
716 K
Q=c e
n(X)
X