pan>14B alloy at various
temperatures and initial hydrogen pressures are shown in
Figure 1. As follows from Figure 1 if the reverse phase
transformation starts from higher initial hydrogen pres-
sure it leads to increase of phase transformation evolu-
tion for all transformation temperatures. In general, a
(a)
(b)
(c)
Figure 1. The isothermal kinetic diagrams for hydrogen induced reverse phase transformation in R2Fe14B alloy, where T is
the isothermal exposure temperature, t is the transformation time and 10, 30, 50, 70, 90 100% is the degrees of the reverse
transformation at different initial hydrogen pressure: (a) – 0.10 MPa; (b) – 0.15 MPa; (c) – 0.20 MPa.
Copyright © 2011 SciRes. MSA
Growth Kinetics of NdFe B Phase during Hydrogen-Induced Reverse Phase Transformation in NdFe B Type 1111
2 142 14
Nanocrystalline Magnetic Alloy
twofold increase of the initial hydrogen pressure (from
0.1 MPa up to 0.2 MPa) results in an acceleration of the
reverse phase transformation evolution in the R2Fe14B
alloy in ~1.52 times.
As can be seen from Equation (2) for transformation of
this type needs diffusion of alloy’s components and in
fact earlier on a base of kinetic, TEM and X-ray diffrac-
tion studies during reverse phase transformation in
Nd2Fe14B type alloys has been showed that transforma-
tions of this type maybe classified as diffusive phase
transformation in solid state and that the reverse trans-
formation process proceeds by the nucleation and growth
mechanism [11,12]. Because of this, according to de-
tailed TEM investigations [12] at first NdH2 phase disso-
ciates into Nd due to desorption of H2 and then -Fe dif-
fuses into Nd. The Fe2B grains act as boron carriers and
due to their small size and random distribution there are
short transfer-lengths for the boron atoms during recom-
bination. Finally the disproportionated mixture recom-
bines to the thermodynamically more stable Nd2Fe14B
phase.
Therefore, it is obviously that from viewpoint of clas-
sical kinetic theory of phase transformations in con-
densed state above-described hydrogen-induced reverse
phase transformation in Nd2Fe14B type alloys proceeds
by the nucleation and growth mechanism. Further, it is
possible to find an effective activation energy of phase
transformation process in accordance with Becker-Döe-
ring model of nucleation kinetics [13,14] if plots de-
pendence lnt  on 1/T where t  is the time, which is
needed for reaching of some degree of transformation
and T is the transformation temperature. With this goal
experimental data from Figure 1 were re-plotted in
co-ordinates lnt  versus which are shown in Fig-
ure 2. The obtained values of effective energy of phase
transformation for various hydrogen pressures and de-
grees of transformation are given in Tables 1.
T/1
As can be seen the from Table 1 all obtained values of
effective activation energy have good agreement by order
of magnitude with activation energy for iron atoms diffu-
sion in
-phase of Fe (QFe = 259.54 284.2 kJ/mol
[15,16]). Therefore, it is really possible to consider that
evolution process of reverse phase transformation is con-
trolled by iron atoms diffusion to new centres of
Nd2Fe14B phase. On the other hand, it is obviously that
the driving force of this transformation is dissociation of
NdH2 phase due to desorption of H2. Thus, in accordance
with above-described model transformation let’s obtain
then kinetic equation for hydrogen induced reverse phase
transformation in Nd2Fe14B type alloy taking into ac-
count influence of hydrogen pressure at direct phase
transformation stage on reverse transformation kinetics.
In accordance with well known Kolmogorov kinetic
theory of phase transformation in condensed state [17]
the volume of transformed area
 
during reverse phase
transformation can be written as follows:
34
() π
1exp ,
3
o
Vt
I
t
V



(2)
where V(t) is the transformed area volume at time mo-
ment t, Vo is the initial untransformed volume, I is the
nucleation rate of centres of new Nd2Fe14B phases,
 
is
the rate of growth of a new Nd2Fe14B phase.
On the other hand, rate of nucleation I of new
Nd2Fe14B phases centers in condensed systems in accor-
dance with Turnbull-Fisher model [18,19] in case of dif-
fusion-controlled growth is
,
GQ
RT
RT
Ie
h

(3)
where  G is the energy necessary for formation of
Nd2Fe14B critical nucleus, Q is the activation energy
diffusion of Fe atoms to centres of new Nd2Fe14B phases,

= 10–4 mol/m3 [20], R is the gas constant, h is the
Planck constant, T is transformation temperature.
Then, substitute equation (3) into (2) we can obtain
kinetic equation for volume of the transformed area

in
dependence on t transformation time and transformation
temperature T:
34
π
()1 exp3
GQ
RT
RT
te
h



 



t
(4)
As a rule, phase transformation kinetics for practical
application describes by curves showing time transfor-
mation t needs for reaching some degree of transforma-
tion

in dependence on transformation temperature T
Table 1. The effective activation energy of hydrogen induced reverse phase transformation in R2Fe14B alloy for various initial
hydrogen pressur e s P and degree s of phase transformation
.
, degree of transformation P = 0.1 MPa, initial hydrogen
pressure
P = 0.15 MPa, initial hydrogen
pressure
P = 0.2 MPa, initial hydrogen
pressure
0.5 198.31 7.51 kJ/mol 181.24 6.78 kJ/mol 214.23 8.05 kJ/mol
0.7 225.29 7.95 kJ/mol 177.83 7.96 kJ/mol 216.62 8.23 kJ/mol
0.9 224.46 8.14 kJ/mol 224.99 7.96 kJ/mol 198.96 8.21 kJ/mol
Copyright © 2011 SciRes. MSA
Growth Kinetics of NdFe B Phase during Hydrogen-Induced Reverse Phase Transformation in NdFe B Type
1112 2 142 14
Nanocrystalline Magnetic Alloy
(a)
(b)
Figure 2. The dependence of lnt versus 1/T for hydrogen induced reverse phase transformation in R2Fe14B alloy for degrees
of transformation 50% (a), 70% (b) and 90% (c) and following initial hydrogen pressure: 1 - 0.20 MPa; 2 - 0.15 MPa; 3 -
0.10 MPa.
(c)
[22]. Thus, in this case Equation (4) has been written in
following form:
 

1
1
4
4
4
3
4
ln 1
3
,.
π
U
R
T
h
tT e
RT



 (5)
Further, in isothermal conditions the rate of nucleation
I describing by Equation (3) is constant in accordance to
classical theory of nucleation rate in condensed systems
because and as a rule are constants for same
ore, we can believe that the rate of
nucleation of Nd2Fe14B phase is not depends on initial
W
0,22]
Q
falloy [2. There
Copyright © 2011 SciRes. MSA
Growth Kinetics of NdFe B Phase during Hydrogen-Induced Reverse Phase Transformation in NdFe B Type 1113
2 142 14
Nanocrystalline Magnetic Alloy
hydrogen pressure. Let’s determine further rate of nu-
cleation I and rate of growth of a new Nd2Fe14B phase
believed that the activation energy at transition of atoms
through interface of phases Q is equal in our case to ac-
tivation energy of diffusion of Fe atoms in -Fe phase Q
= 259.54 kJ/mol [16] and energy necessary for formation
of Nd2Fe14B critical nucleus G = 314.1 kJ/mol in ac-
cordance with data from [23]. Thus, using Equation (3)
and kinetic data from Figure 1 were calculated values of
rate of nucleation I and rate of growth of Nd2Fe14B phase
during phase transformation for various hydrogen
pressures and temperatures for degree of transformation

= 0.9 which are given in Tables 2.
In Figure 3 is presented dependence growth rate of
Nd2Fe14B phase versus hydrogen pressures
P
plotted
on the base of data from Table 2. As follows from Fig-
ure 3, the rate of growth
increase with increase of
hydrogen pressure at all temperatures.
Further, in accordance with Lyubov and Hillert kinetic
approach [20,21] for diffusion-controlled growth we as-
sume that dependence rate of growth
on hydrogen
pressure and transformation temperature T can be
approximated by following type function:
P
()UP
s
T
FF
RT
Me

,
where M is proportional to the mobility coefficient of the
interface of growing Nd2Fe14B phase, F is the molar
difference of the free energies of initials and final phases,
FS is the driving force of phase transformation which in
our case is proportional to the dissociation energy of
NdH2 phase, U(P) is the activation energy transition of
Fe atoms across the interface of Nd2Fe14B phase de-
pending on hydrogen pressure.
Then, let’s denote
()
s
FF
RT
M
AP

Further, activation energy transition of Fe atoms across
interface of phases U(P) in dependence on initial hydro-
gen pressure P has been approximated using data from
Table 2 by following function:
33
( )113.08 1078.77 10UP P
 
Finally, the rate of growth
depends on initial hydrogen
pressure P in R2Fe14B alloy by following type equation:
33
113.08 1078.77 10
() ,
P
RT
APe
 
(6)
where A(P) is the preexponential function that depends
on hydrogen pressures P, which are presented in Table 3
for various hydrogen pressure.
As can be seen from Table 3, A(P) function increase
with increase of initial hydrogen pressure that is possible
if the mobility coefficient of interface M will increase. In
our case hydrogen atoms can be considered as interstitial
atoms and increase of their concentration lead to increase
of diffusion of matrix atoms (Fe, Nd, B atoms in our case)
as was shown in [24-26]. Moreover, the hydrogen atoms
concentration was larger in case if reverse transformation
was started when hydrogen pressure was larger too and
in this case it lead to increase of diffusion of matrix at-
oms and it correspondingly lead to increase of mobility
of interface M. On the other hand, U(P) decrease with
hydrogen pressure increase. Hence, the rate growth of
new Nd2Fe14B phase
depends on two main factors:
increasing of mobility coefficient of interface M with
hydrogen pressure increase and decreasing of activation
energy transition of Fe atoms through interface of
Nd2Fe14B phase U(P) with hydrogen pressure increase.
Finally, substitute Equation (6) into Equation (5) we can
written the final equation descraibing dependence
transformation time t needs for reaching some degree of
transformation
on hydrogen pressures
P
and trans-
formation temperature
T
:
33
1
1
4
4
4
3
4
113.08 1078.77 10
,,
3(ln(1)) ,
π
()
GQ
RT
P
RT
tTP
he
RT
APe

 
 






(7)
Table 2. The rate of nucleation I and rate of growth of a new Nd2Fe14B phase
of phase transformation determined for
various hydrogen pr e ssures P and transformation temperatures T for degree of transformation

= 0.9 in R2Fe14B alloy.
T, temperature (K) I, nucleation rate (m–3s–1)
, rate of growth at P =
0.1 MPa (ms–1)
, rate of growth at P =
0.15 MPa (ms–1)
, rate of growth at P =
0.2 MPa (ms–1)
983 0.392103 3.44810–6 5.12510–6 6.06410–6
1003 1.623103 4.33610–6 5.93910–6 7.11410–6
1023 6.357103 8.89510–6 10.29110–6 12.4210–6
Copyright © 2011 SciRes. MSA
Growth Kinetics of NdFe B Phase during Hydrogen-Induced Reverse Phase Transformation in NdFe B Type
1114 214 214
Nanocrystalline Magnetic Alloy
P, hydrogen pressure (MPa)
Figure 3. The growth rate of Nd2Fe14B phase
versus hy-
drogen pressure P for various transformation temperature:
1 - 710˚C; 2 - 730˚C; 3 - 750˚C for degree of transformation

= 0.9 in R2Fe14B alloy.
Table 3. The preexponential function A(P) at different ini-
tial hydrogen pressures P determined for degree of reverse
hydrogen induced transformation

= 0.9 in R2Fe14B alloy.
P, initial hydrogen
pressure (MPa)
A(P), the preexponential function
in Equation (6)
0.10 94.73 10–2
0.15 97.41 10–2
0.20 98.76 10–2
In Figure 4 is shown the isothermal kinetic diagram
for hydrogen induced reverse phase transformation in
R2Fe14B alloy calculated on the base of Equation (7) and
data from Table 3.
Thus, it is possible to conclude that obtained Equation
(7) well describes experimental results by influence of
value of initial hydrogen pressure on kinetic of reverse
phase transformation in R2Fe14B alloy in terms of two
main kinetic factors, i.e. increasing of mobility coeffi-
cient of interface M with hydrogen pressure increase and
decreasing of activation energy transition of Fe atoms
across interface of Nd2Fe14B phase U(P) with hydrogen
pressure increase.
4. Conclusions
The effect of initial hydrogen pressure on the kinetics of
growth of Nd2Fe14B phase during hydrogen induced re-
verse phase transformations in the industrial R2Fe14B
alloy has been studied.
It has been established that, as the temperature and the
initial hydrogen pressure increase it lead to considerable
acceleration of evolution of reverse phase transformation.
It has been obtained that kinetics of reverse phase trans-
formation process is controlled by Fe atoms diffusion
Figure 4. The isothermal kinetic diagram for hydrogen in-
duced reverse phase transformation in R2Fe14B alloy
calculated by Eq. (7) for degree of transformation

= 0.999
for various hydrogen pressure: 1 – 0.10 MPa; 2 – 0.15 MPa;
3 – 0.20 MPa. Points are experimental data from Figure 1
for following hydrogen pressur e s.
and the rate growth of Nd2Fe14B phase
increase with
increase of initial hydrogen pressure.
On the base of Kolmogorov kinetic theory of evolution
of phase transformation in condensed state the equation
describing the isothermal kinetic diagrams of such type
of transformations has been obtained. It has been shown
that effect of initial hydrogen pressure on kinetics of hy-
drogen induced reverse transformations in R2Fe14B alloy
the may be described by following type equation:
33
1
1
4
4
4
3
4
113.08 1078.77 10
,,
3(ln(1))
π
()
GQ
R
T
P
RT
tTP
he
RT
APe
 
 






where t(
,T,P) is the transformation time for various de-
gree of transformation
at different temperature T and
initial hydrogen pressure P, G is the energy necessary
for formation of Nd2Fe14B phase critical nucleus, Q is the
activation energy diffusion of Fe atoms to centers of
Nd2Fe14B phase,

= 10–4 mol/m3, R is the gas constant, h
is the Planck constant, T is transformation temperature,
A(P) is the preexponential function depending on hydro-
gen pressure, P is the initial hydrogen pressure.
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Growth Kinetics of Nd2Fe14B Phase during Hydrogen-Induced Reverse Phase Transformation in Nd2Fe14B Type
Nanocrystalline Magnetic Alloy
Copyright © 2011 SciRes. MSA
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