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

R

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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