effect of anisotropic growth on the deviations from JMA kinetics

7/26/2019 effect of anisotropic growth on the deviations from JMA kinetics

1/11

Effects of anisotropic growth on the deviations fromJohnsonMehlAvrami kinetics

Feng Liu *, Gencang Yang

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xian, Shanxi 710072, China

Received 19 June 2006; received in revised form 19 October 2006; accepted 19 October 2006Available online 26 December 2006

This article is for the purpose of congratulating the 80th birthday of Chinese academician of Science, Prof. Yaohe Zhou.

Abstract

The effects of nucleation, growth and impingement on deviations from classical JohnsonMehlAvrami (JMA) kinetics are described.On the basis of Monte Carlo (MC) simulations of isothermal phase transformations, the deviations from the JMA kinetics due to theeffects of anisotropic growth have been investigated further. An analytical approach has been adopted to describe phase transformationssubjected to the effects of anisotropic growth. In combination with MC simulations, the effect of the memory time has been shownwhen the transition from JMA behavior to blocking behavior occurs during the transformation. On this basis, a physically realisticAvrami exponent has been deduced. A comparative study between the analytical description and a gammaalpha transformation occur-ring in an FeMn alloy under an applied load has been carried out. 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: JMA kinetics; Anisotropic growth; Avrami exponent

1. Introduction

The JohnsonMehlAvrami (JMA) equation [110] isgenerally applied to describe phase transformations involv-ing nucleation and growth. This equation provides anexpression for the volume fraction of material transformedas a function of time,f(t). The JMA description is precise ifthe assumptions imposed in their original derivations arenot violated. However, for many important physical situa-

tions, these conditions are not satisfied, and generalizationsof the JMA theory are required. Although some extensionsof the JMA theory have been made [1115], additionalwork needs to be completed, especially in the case of grainformation through anisotropic growth.

Monte Carlo (MC) simulations are well suited to testJMA kinetics because they fully account for the statisticalnature of nucleation and subsequent anisotropic growth

(mutual hindrance of growing grains). The effects of aniso-tropic growth on deviations from ideal JMA kinetics havebeen numerically quantified previously [16]. It wasobserved that the deviations become significant as theeffects of anisotropic growth increase.

MC calculations incorporating deviations from JMAkinetics due to anisotropic growth have recently beenextended [17], assuming a time(/temperature)-dependentnucleation rate and a time-independent and temperature-

dependent growth rate in combination with differentgrowth modes. A concept of memory time (i.e. the firsttime steps when transformation is purely JMA-like beforethe blocking of growing grains) has been introduced inthe transformation kinetics to evaluate the effect of aniso-tropic growth. A novel analytical description (see Section2.2) has been developed which incorporates classical JMAkinetics, but also accounts for the blocking if the anisotrop-ically growing grains have perpendicular anisotropy axes.All the MC results could be reproduced and are thus wellexplained by the novel analytical description[17].

1359-6454/$30.00 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.actamat.2006.10.022

* Corresponding author. Tel.: +86 029 88460374.E-mail address:[email protected](F. Liu).

www.actamat-journals.com

Acta Materialia 55 (2007) 16291639
mailto:[email protected]:[email protected]


2/11

However, the MC simulations described above andthe analytical descriptions involve some inherent flaws,with respect to both the Avrami exponent and its generalapplication (see Section 2.2.4). In the present work, ana-lytical expressions for the Avrami exponent of transfor-mations subjected to the effects of anisotropic growth

have been obtained by applying the analytical approachpresented in Ref. [8]. On this basis, an analytical descrip-tion of MC simulations is given (see Sections 3 and 4.1).Moreover, a comparative study between the analyticaldescription and the gammaalpha transformation occur-ring in an FeMn alloy under applied loads has beencarried out (see Section 4.2). This study not only showspossible applications of the current analytical phasetransformation model, but also provides new insightand understanding of phase transformations subjectedto the effects of anisotropic growth in real materials.

2. Theoretical background

2.1. Classical JMA kinetics

JMA kinetics hold only for extreme conditions, i.e.either pure continuous nucleation or pure site saturationas nucleation mechanisms [58]. This phenomenon hasbeen demonstrated using the analytical phase transforma-tion model [8]. If the explicit effects of transient nucle-ation are excluded, the nucleation rate per unit ofvolume (in three dimensions) or per unit of area (intwo dimensions) of untransformed material, I, as a func-tion of time, t, and temperature, T, can be expressed as

[7,8]

It; T I0 exp QnRT

1

withR as the gas constant andQnthe activation energy fornucleation. The termsite saturationis used where the num-ber of (supercritical) nuclei does not change during thetransformation, i.e. all nuclei of number N* per unit vol-ume are already present at t= 0[7,8]:

IT Ndt 0 2with d(t 0) denoting the Dirac function.

Previous work [1820] proposed to introduce thenucleation index to allow for the dependence of thenucleation rate on the degree of transformation. The con-tinuous nucleation rate equation (for isothermal transfor-mation) in that case can be expressed as I(t) = an 0ta1,with n0 as a constant (a= 0 for zero nucleation rate(i.e. site saturation), a= 1 for constant nucleation rateand a> 1 for increasing nucleation rate with the progressof transformation). The effect of the nucleation index onthe transformation is not the main focus in the presentwork; however, detailed descriptions are available inRefs. [1821]. For the case of interface-controlled growth,the textbook equation for the interface velocity Gis given

by [5]:

G G0 exp DGa

RT

1 exp DG

RT

3a

with G0 as the pre-exponential factor for growth, DGa the

activation energy for growth that equals the interface en-ergy barrier and DGthe driving force, which is the energydifference between the new phase and the parent phase.

For large undercooling or superheating, i.e. DG (>0) issufficiently large compared with RT, Eq. (3a) becomes[5,7,8]

G G0 exp QGRT

3b

withQG(=DGa) as the activation energy for growth andG0

the temperature-independent interface velocity. For thecase of diffusion-controlled growth, QG equals the activa-tion energy for diffusion, QD, and G0 equals the pre-exponential factor for diffusionD0.

The overall kinetics of transformations involving nucle-

ation and growth, i.e. the volume fraction transformed, f,as a function of time t, for isothermal conditions is gener-ally described by:

f 1 expktn 4

kT k0 exp QRT

5

The expressions for n, Qand k0are given in Table 1.

2.2. Deviations from JMA kinetics

The necessary associated size-dependent growth [9,10]

and transient nucleation[11,12]or a mixture of nucleationmodes [7,8] lead to deviations from JMA kinetics.Anisotropic growth (possibly leading to the blocking ofgrowing grains by the neighboring ones) is another impor-tant origin for deviations from JMA kinetics [13]. There-fore, deviations from JMA kinetics may be caused by (i)nucleation or (ii) growth and impingement.

Table 1Expressions for the (time and temperature independent growth exponent,n, the overall activation energy, Q, and the rate constant, k0, to be insertedinto Eqs.(4) and (5)for isothermal and isochronal annealing, respectively

Isothermal Isochronal

Continuous nucleation

n d/m+1 d/m+ 1

Qdm

QGQNn

dm

QGQNn

kn0gN0m

d=m0

d=m1gN0m

d=m0

CC

d=m1Qd=mG

Site saturation

n d/m d/mQ QG QG

kn0 gNmd=m0

gNmd=m0

Qd=m

G

For Cc see Ref.[8]. These values are valid for the JMA kinetics based oncontinuous nucleation (a= 1) or site saturation, with m as growth modeparameter (m = 1 for interface-controlled growth; m= 2 for diffusion

controlled growth), and das the dimensionality of the growth (d= 1,2,3).

1630 F. Liu, G. Yang / Acta Materialia 55 (2007) 16291639


3/11

2.2.1. Influence of nucleation

In principle, JMA kinetics are not applicable if a mix-ture of nucleation mechanisms are active. A situation ofmixed nucleation occurs when pre-existing nuclei are pres-ent at the onset of the transformation, and a constantnucleation rate holds upon transformation [7,8]. There-

fore, the Avrami exponent has an initial value of 1, 2or 3, but as the transformation proceeds it changes to avalue of 2, 3 or 4 that holds for continuous nucleationand linear growth (one-, two- or three-dimensional (1-D,2-D or 3-D)), respectively.

Furthermore, nucleation has to occur randomlythroughout the infinite space, and transient nucleation can-not be considered[15]for JMA kinetics to hold.

2.2.2. Influence of growth and impingement

As shown in Ref. [17], isotropic growth and parallelgrowth of anisotropically growing grains with identicalconvex shapes, in combination with continuous nucle-ation, does obey JMA kinetics, which is compatible withimpingement due to randomly dispersed nuclei. It isassumed here that the nuclei are dispersed randomlythroughout the whole volume, but suppose that at timet the actually transformed volume is Vt. If the time isincreased by dt, the extended and the actual transformedvolumes will increase by dVe and dVt, respectively. Fromthe change in the extended volume, dVe, only a part willcontribute to the change in the actually transformed vol-ume, dVt, namely, a part as large as the untransformedvolume fraction[15]. Hence

dVt V Vt

V

dVe ; dfdxe

1 f 6

Fig. 2(b) and (c) in Ref. [17] shows that the s (square), r(rectangular), rNRp (parallel anisotropic) and np (parallelneedle) growth modes, to a large extent, obey JMAkinetics, i.e. the data in an ln(ln(1 f)) vs. ln(t) plotfall on a straight line, and the Avrami exponent, derivedfrom the slope in Fig. 2(b) of Ref. [17], is constant forthese four modes almost throughout the entire intervalf= 0 to f= 1.

In the case of anisotropically growing particles, i.e.anisotropic orthogonal growth rNRo and 1-D orthogonal

growth no [17], the average time interval for the ran-domly dispersed particles to grow before blocking byother particles is smaller than the average time intervalfor isotropic growth, leading to strong deviations fromJMA kinetics. One phenomenological approach regardingthis blocking effect has been proposed by extendingEq. (6) to [17,21,22]:

df

dxe 1 fe 7

where eP 1. Impingement due to Eq. (7) is more severe(i.e. the difference between fand xe is larger than the dif-

ference due to Eq. (6)) and increases with e (see Fig. 1).

Impingement due to Eq. (7) implies that the effects ofanisotropic growth change only the relationship betweenfand xe (seeFig. 1), but does not change xe itself, whichis not compatible with MC simulations and leads to poorerfits[17].

2.2.3. Analytical description based on MC simulations[17]

In order to describe the effects of anisotropic growth onthe deviations from JMA kinetics, a relatively simple and

transparent analytical description has been developed,extending from and incorporating the JMA theory. Thisanalytical description accurately reproduces the numericalresults of MC simulations[17]. The process starts with for-mulations that closely follow MC simulations with discretetime steps and the corresponding increment of the fractiontransformed within every single period but ends with tradi-tional continuum equations, as in the usual JMA kinetics[17].

Generally, impingement leads to the blocking of thegrowing grains and thus reduces the time interval for grainsto grow, i.e. to contribute to the fraction transformed. Instandard JMA kinetics, this time interval (i.e. the mem-ory) always goes back to t= 0, so that all the precedingtime steps do contribute [17]. Upon transformation, theircontributions become less important due only to theimpingement factor (1 f); see Eq.(6).

Following this philosophy, an exact agreement betweenthe analytical description and MC simulations wasachieved for the np growth mode. Accordingly, a standardJMA equation for this parallel 1-D growth with continuousnucleation in 2-D space results [17]:

ln1 f wIGt2 8wherew is the width of the growing 1-D grain and Gis the

growth rate (unit length per unit time). Ifw= 1/2, Eq.(8)is

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

extended fraction, xe

realfractio

n,

f

=3

=2

=1.5

=1.2

=1

Fig. 1. The transformed fraction,f, as a function of the extended fraction,

xe, for the case of impingement by anisotropically growing grains,

corresponding to values ofe P 1.

F. Liu, G. Yang / Acta Materialia 55 (2007) 16291639 1631


4/11

compatible with Eq.(4)for n = 2, and k= (IG/2)1/2. In thecase of blocking events originating from the no growthmode, the memory does not revert to t = 0 but is limitedto a certain number of previous time steps [17]. Thus anequation different from Eq.(8)results[17]:

ln

1

f

2Cw ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI t t

p wIGt

2

9

where t*, the time when the transition from JMA behaviorto blocking behavior occurs, is given by:

tffiffiffiffiffiffiffi

C2

IG23

s 10

withCas a constant with a value close to 2[17]. Fort < t*,JMA kinetics, i.e. the last term in Eq. (9), holds. Analo-gously, an equation for the rNRo growth mode was ob-tained as[17]:

ln1 f 2Cw ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIt tp 1 2

NR 4C

3NR

ffiffi

Ip

Gt t3=2 23NR

IG2t3 11

wheret*, the time after which Eq.(11)becomes valid, is stillgiven by Eq.(10). A detailed description for derivations ofEqs.(8), (9) and (11) is available in Ref. [17].

2.2.4. Inherent flaws in the analytical description based on

MC simulationsThe Avrami method of plotting ln(ln(1 f)) vs. ln(t)

is commonly adopted to deduce the Avrami exponent, n,for transformations following JMA or non-JMA kinetics.

Namely, whether the data in an ln(ln(1 f)) vs. ln(t)plot fall on a straight line or not determines whether atransformation follows JMA or non-JMA kinetics [5,17].Actually, the ln(ln(1 f)) vs. ln(t) plot is not applicablefor non-JMA transformations. The Avrami exponent fornon-JMA transformations on the basis of Eqs. (9) and(11) cannot be deduced reasonably by using an Avramiplot, even if they fit MC simulations well [17].

In practice, an analytical description of transforma-tions is favored, as numerical calculations can be verycumbersome (one complication might be the distinctionbetween independent and dependent fit parameters[21,23]). Analytical expressions have an advantage overnumerical calculations because the influence of, for exam-ple, the different nucleation, growth and impingementmodels can be easily identified and investigated[21,23,25]. On this basis, an accurate, flexible analyticalphase transformation model has been developed thatincorporates a choice of nucleation, growth and impinge-ment mechanisms [8,21] which has been successfullyapplied [21,23,25].

The analytical description (i.e. Eqs. (8), (9) and (11))aims solely at interpreting MC simulations[17]. However,its application to general transformations is still an openquestion. Applying the analytical approach[8], reinterpre-

tation of the analytical description on the basis of MC

simulations is carried out, and physically realistic solu-tions for the Avrami exponent are analytically obtained(see Section 3).

3. An analytical phase transformation model based on MC

simulations

For thenpgrowth mode with continuous nucleation in a2-D space, combining Eq. (8) (with w= 1/2) and (1)(4)gives

ln ln1 x ln1=2I0G0 QN QGRT

2 ln t 12

Following JMA kinetics, a constant Avrami exponent,n= 2, and a constant effective activation energy Q=(QN+QG)/2, are obtained (seeTable 1).

For thenogrowth mode with continuous nucleation in a2-D space, Eq. (9) can be used to describe the non-JMAtransformation subjected to the effect of anisotropicgrowth. From Eq.(9), the extended fraction can be consid-ered to be composed of two parts, i.e. xe= xe1+xe2, onepart withn1= 0.5, the other withn2= 2. The ratio betweenthe two extended fractions is given by

wI1G1t2

2Cwffiffiffiffiffiffiffiffiffiffi

I1teffp r2

r1 xe2

xe113

where teff=t t*. Analogous to the analytical approachadopted in Ref.[8], different values ofI2andG2can be cho-sen in such a way that xe is due only to the part withn1= 0.5

x0e1 2Cwffiffiffiffiffiffiffiffiffiffi

I1teffp wI1G1t2 2Cw ffiffiffiffiffiffiffiffiffiffiI2teffp 14a

or that xe is only due to the part with n2= 2,

x0e2 2Cwffiffiffiffiffiffiffiffiffiffi

I1teffp

wI1G1t2 wI2G2t2 14bIntegrating Eqs.(13), (14a) and (14b)gives

xe 1r1 r2

Xr1i1

x0e1i Xr2i1

x0e2i !

and

xe 2Cw ffiffiffiffiffiffiffiffiffiffiI2teffp 11r2r1wI2G2t21

1 r

2r1 1" # 15

From Eqs.(1), (3), (7) and (17), one can obtain

ln1x 2Cwffiffiffiffiffiffi

I02p 1

1r2r1wI02G021

1 r2r1 1"

exp 1

1 r2=r1 1QG QN 11r2=r1

1

2QN

RT

0@

1A

t

t

1

2 1r2r1

t

1

1 r2r1 1# 16



5/11

where

I02 wffiffiffiffiffiffi

I01p

t t 1=2 2Cw N01G01 exp 1=2QNQGRT

t2

wffiffi

tp

0@

1A

24

35

2

17a

G02 12Ct

exp

1=2QN

QG

RT ffiffiffiffiffiffi

I01p

t t 1=2 2CI01G01 exp 1=2QNQGRT

t2

24 3517b

With reference to Ref. [8], the kinetic parameters for thecurrent transformation can be given as

n 12 1 r2

r1

21 r2

r1

1 2 3=21 r2=r1 18a(seeFig. 2(b))

Q

11

r2r1

n1Q1 11

r2r1 1n2Q2

n ; Q1 QG QN2 ;

Q2 QN 18b

kn0 2Cwffiffiffiffiffiffi

I02p

1

1r2r1wI02G021

1 r2r1 1" #

18c

For the rNRogrowth mode with continuous nucleation ina 2-D space, Eq.(11)is obtained (see Section2.2.3). Anal-ogously, the whole extended fraction on the right side ofEq.(11) can be considered to be composed of two parts:one part with n= 3, the other with a mixture of n= 0.5and n= 1.5. Finally, an analytical expression analogous

to Eq.(16)is obtained as

where

G02 t t n1ffiffiffiffiffiffiffiffiffi

Ut3p

t t n11exp

QGRT

For n1seeAppendix.

I02 UG22t

3 exp QNRT

U 2Cwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I t t p

1 2NR

4C

3NR

ffiffiI

p Gt t3=2

2

3NR

IG2

t

3

With reference to Ref. [8], the kinetic parameters for thecurrent transformation can be given as

n 3 32 1

1r2=r11 r4=r3 20a

(seeFig. 3(b))

Q

11r4r3

1

1 r2r1

1 QG QN20B@ 1CA 11 r4r3

1 2QG QN n

20b

kn0 2Cwffiffiffiffiffiffi

I02p

ffiffiffiffiffiffi

I02p

G1t t

1 2NR

11 r2r1 1

1r4r3 "

4C3NR

ffiffiffiffiffiffiI02

p exp QN

RT

ffiffiffiffiffiffiI02p G02 t t t t

! 11 r2r1 1

1r4r3

23NR

I02G202

11 r4r3 1

# 20c

where

2G1teff3w NR 2

r2

r1;

23NR

I1G21 t

3

2Cw ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI1 t t p 1 2NR 4C3NR ffiffiIp

1G1

t

t

3=2h i r4

r3

ln1x

266666664

2Cwffiffiffiffiffiffi

I02p

ffiffiffiffiffiffi

I02p

G2t t

1 2NR

11 r2r1 1

1r4r3 4C3NR


p exp QN

RT

ffiffiffiffiffiffiI02p G02t ttt

! 11 r2r1 1

1r4r3 :

2

3NRI02G202

11 r4r3 1 exp 1

1r4r31

1 r2r1 1 QG QN2

0

B@

1

CA 1

1 r4r3 12QG QN

RT

0

BBBBBBBB@

1

CCCCCCCCAt t

32

1

1

r2

r1 112 11

r2r

1

11

r4r

3 t

3

1 r4r3 1

3

77777777519

http://-/?-http://-/?-


6/11

The detailed derivation of Eq. (11) is given in theAppendix.

4. Discussion

4.1. Effects of anisotropic growth, extended fraction and

Avrami exponent

For all the nucleation modes considered, the extendedvolume can be shown to be always given by the additionof two parts[8]: one part that can be conceived as due topure site saturation and the other that can be conceivedas due to pure continuous nucleation. By extensive calcula-tions, the following explicit analytical expressions for the

extended transformed fraction can be obtained[21,23]:

(i) for isothermal transformation,

xe k0tnttnt exp ntQtRT

21

(ii) for isochronal transformation,

xe k0TnT RT2

U

nTexp nTQT

RT

! 22

Explicit expressions for n, Q and k0, in terms of generalnucleation and growth mechanisms, for both isothermaland isochronal annealings have been given in Refs.[8,21,23]. The kinetic parameters (n, Q and K0) are deter-

mined by the extended fraction, as has been demonstrated

0 100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

transformation time, s

transformedfraction,

f

t*

3=180s

t*

2=140s

t*

1=100s

a

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

transformed fraction, f

Avramiexponent,n

t*

1t*

2t*

3b

Fig. 3. The transformed fraction,fas a function of transformation time, t(a) and the Avrami exponent, n, as a function off(b) for pure continuousnucleation and rRNo growth mode: N0= 1 10

5 (s1 m3), QG=1.1 105 (kJ mol1), QN= 1 10

5 (kJ mol1), d/m= 2, m0= 1.8 105

(m s1),T= 740 K. In (a), the thick (overlapping) solid lines represent thefraction transformed within (0! t1), (0! t2) and (0! t3), and the thinsolid, thick dashed and thick dashdotted lines represents the fractiontransformed within (t1! t), (t2! t) and (t3! t), respectively.

0 2000 4000 60000

0.2

0.4

0.6

0.8

1


transformedfraction,

f

t*

3=1000s

t*

2=500s

t*

1=100s

a

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

transformed fraction, f

Avramiexponent,n

t*

1t*

2t*

3 b

Fig. 2. The transformed fraction,f, as a function of transformation time,t(a) and the Avrami exponent,n, as a function off(b) for pure continuousnucleation and no growth mode: N0= 4 10

7 (s1 m3), QG= 1.2 105

(kJ mol1), QN= 2 105 (kJ mol1), d/m= 1, m0= 1 10

5 (m s1),

T= 900 K. In (a), the thick (overlapping) solid lines represent the fractiontransformed within (0! t1), (0! t2) and (0! t3), and the thin solid, thickdashed and thick dashdotted lines represents the fraction transformedwithin (t1! t), (t2! t) and (t3! t), respectively.

http://-/?-http://-/?-


7/11

for mixed nucleation using the analytical phase transfor-mation model [8,21,23,25]. The cases of site saturationand continuous nucleation follow directly from Eqs. (21)and (22):

(i) for isothermal transformation[6,24]

xe kn0tn exp nQRT

bn 23

with b k0texp QRT [6], and(ii) for isochronal transformation,

xe kn0Z t

0

exp QRTt

dt

n bn 24

with

b ffi RT2

UQk0exp Q

RT

from Ref.[24].

For both isothermal and isochronal transformations, nis the constant Avrami exponent and Q is the constanteffective activation energy (see Table 1). Combining Eq.(23)with Eq.(4)gives

ln ln1 f n lnk0 nQRT

n ln t 25

In a classical analysis (e.g. as in Eq.(8) where n, Qand k0are constant upon isothermal transformation)ln(ln(1 f)) is plotted as a function of ln(t), resulting instraight lines with slopes ofn = 2, 3 or 4 in the case of con-

tinuous nucleation and linear growth (1-, 2- or 3-D) (seeTable 1).

Since Eqs. (9) and (11) follow non-JMA kinetics, theAvrami method cannot be used to deduce the Avramiexponent (see Section2.2.4). Numerical calculations basedon I0, G0,QG, QN and Tcan be performed to constructphase transformations under the effects of anisotropicgrowth. Note that, for numerical calculations, terms suchas 2Cw and w in Eq. (9) and 2Cw(1-2/NR), 4C/3NR and2/3NR in Eq. (11) are incorporated with I0 and G0. Inthe present work, combining Eq. (16) and (19) with Eqs.(18a) and (20a) gives the relationship of ft and nf, asshown inFigs. 2 and 3, respectively.

With reference to[17], the effects of anisotropic growthare characterized by the value oft*, i.e. the time step whenthe transition from JMA to blocking behavior occurs. Lar-ger t* implies smaller effects of anisotropic growth and inturn, higher transformation rates, in accordance with MCsimulations[17].

The Avrami exponent from Eq. (18a) is determinedmainly by r2/r1, the ratio between the two extended frac-tions originating from JMA and blocking behavior. Ift t*, the extended fraction origi-nating from blocking behavior increases with transforma-tion, thus decreasing n. Particularly if t t*, the value ofr2/r1tends to be zero, and n is close to 0.5 (seeFig. 2(b)).For sufficiently large t*, the value of r2/r1 (within t* ! t)is limited, and thus n decreases almost linearly with f(seethe nf curve with t3 1000 s in Fig. 2(b)). In contrast,for large effects of anisotropic growth (e.g. t* = 100 s), aninitial strong decrease of n occurs after which a near-plateau is reached where n hardly varies with f (see thenfcurve with t1 100 s inFig. 2(b)).

In terms of Eq. (20a), the Avrami exponent is deter-mined by both r4/r3, the ratio between the two extendedfractions contributed from JMA and blocking behavior,and r2/r1, the ratio between the two kinds of blockingbehavior. Ift< t*, then the value of r4/r3 tends to be infi-nite, and n is equal to 3 (see Fig. 3(a) and (b)). For t> t*,the increase inr2/r1and the decrease inr4/r3result in a spe-

cific evolution ofn withf. As shown inFig. 3(b), ift > t*, acontinuous decrease of n with f occurs, and after a mini-mum value is reached, a continuous increase of n occursuntil the end of the transformation. Note that the mini-mum value ofn shifts to higher values offwith increasingt*, i.e. with the decreasing effects of anisotropic growth. Ift t*, then the value ofr4/r3tends to zero, and nis equalto 3/2-1/(1+ r2/r1) (see the nfcurve with t

3 inFig. 3(b)).

Only if the value of r2/r1 becomes infinite with t* ! t is afinal value ofn= 1.5 achieved. Otherwise, a final value ofn< 1.5 results.

In combination with Section3, it is concluded that the

variations of the Avrami exponent are consistent with theextended fraction. The effects of anisotropic growth leadsto an evolution of the extended fraction from JMA (Eq.(8)) to non-JMA behaviors (Eqs.(9) and (11)), changing naccordingly. These phenomena were also observed by MCsimulations [17,26,27]. Therefore, the Avrami exponentsdeduced from the analytical description on the basis ofMC simulations (i.e. Eqs. (8), (9) and (11)) provide physi-cally realistic explanations for deviations from JMA kinet-ics due to the effects of anisotropic growth.

4.2. Future application of the analytical description

4.2.1. Dilatometric experiments and analysis

The preparation of alloy samples (Fe2.13 at.% Mn)and the calibration procedure for the raw experimentaldata are described in Ref.[28]. The sample was heated fromroom temperature up to 1223 K at a rate of 20 K min1

and kept at this temperature for 30 min. Then it was cooleddown (at a rate of 20 K min1) to room temperature. Uni-axial compressive loads were applied using a servohydrau-lic system at 1173 K and were cancelled at 873 K during thecontinuous cooling of the alloy samples.

The length change recorded for samples under appliedloads during cooling is shown in Fig. 4. From the experi-

mental data, the fractions ofc and a phases (e.g. fa) and



8/11

the transformation rate, dfa/dt, during c ! a transforma-tion can be calculated using a lever rule [29].

4.2.2. JMA and non-JMA transformation behaviors

On the basis of the obtained dilatometric experimentalresults, the alloy samples can be categorized into two types(i.e. A and B), depending on their initial grain size and ini-tial grain size distribution [28]. In the present work, we

focus on the transformation behavior under the effects ofanisotropic growth and illustrate the possible future appli-cation of the analytical description (see Sections 2.2.3 and3). The c ! a transformation mechanism under appliedloads, including the influence of the initial grain size andinitial grain size distribution, will be published elsewhere[28].

From the dilatometric measurements performed for typeA samples under an 150N applied load, the evolution offawith t and the evolution of dfa/dt with fa, are shown inFig. 5(a) and (b), respectively. Analogous results per-formed for type B samples under 50, 150 and 400 N appliedloads are shown in Fig. 6(a)(d).

As shown inFig. 5(a) and (b), a usual S-shaped curve isobserved for fa,and dfa/dt exhibits only one maximumpeak. The optimization procedure (i.e. for Fig. 5(b)) isexplained in Ref.[28]. Given the same impingement mode,the peak maximum of dfa/dt occurs at the same fvalue,irrespective of the nucleation and growth modes consid-ered, and the applied annealing temperature or appliedheating rate[21]. FromFig. 5(b), it is shown that the peakmaximum in dfa/dt vs. fa corresponds to fa= 0.540.58.Within the experimental uncertainty, the gammaalphatransformation for type A can be concluded to followJMA kinetics, i.e. random nucleation and isotropic growth

[21,23,25, 28,30].

For type B samples subjected to 50, 150 and 400 Napplied loads, unusual S-shaped curves are observed forthe evolution of fa in c ! a transformations (seeFig. 6(a)(c)). From Fig. 6(a), (b) and (d), a single peakmaximum of dfa/dt corresponds to fa 0.540.58,whereas double or multi-peaks in dfa/dt vs. fa can beinferred fromFig. 6(c), illustrating that c

!atransforma-

tions of type B do follow non-JMA kinetics.From Eq.(7), it is implied that a high value of e gives

high anisotropy, corresponding to the peak maximum indfa/dt vs. fa at small fa reflected in the relation between fand xe in Fig. 1and by a comparison between Figs. 5(b)and 6(d). However, Eq.(7), as a phenomenological descrip-tion, cannot provide a physically realistic explanation forthe deviation due to the effects of anisotropic growth.

Actually, the analytical phase transformation model(Section3) based on MC simulations deals only with iso-thermal transformations (see Figs. 2 and 3). The c ! atransformations as shown in Figs. 5(a) and 6(a)(c) are

conducted isochronally. Therefore, the current analytical

0 20 40 60 80 100

0

0.2

0.4

0.6

0.8

1

f


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

0.05

f

df /dt

experimentaloptimization

a

b

Fig. 5. The ferrite fraction, fa as a function of time, t (a), and thetransformation rate, dfa/dt, as a function offa(b) as determined from thelength change measured for Fe2.13 at.% Mn (A) samples under 150 Napplied load with a cooling rate of 20 K min1.

400 600 800 1000 12004

2

0

2

4

6

8

10

12

l,*105

m

Temperature, K

Fig. 4. Measured length changes of Fe2.13 at.% Mn during continuouscooling (20 K min1) from 1223 K to room temperature. Uniaxial

compressive (e.g. 150 N) load was applied at 1173 K and cancelled at873 K.



9/11

description cannot be adopted to describe the deviationsdue to the effects of anisotropic growth. However, adetailed comparison between Figs. 6(a)(c), 2(a) and 3(a)demonstrates that all the fat curves are analogous, i.e.they are composed of two segments indicative of two trans-formation behaviours (JMA and blocking behaviors). Withincreased applied loads, the inflection point betweenthe two segments moves to higher t and fa values (see theevolution of t* inFig. 6(a)(c)). In combination with Sec-tion4.1, it is concluded that an increased load gives largermemory time, in accordance with lower effects of aniso-tropic growth[17].

Thus the current analytical description based on MCsimulations does create a wider possibility to investigatethe effects of anisotropic growth. By this means, the currentanalytical description can probably be applied to more gen-eral transformations, bringing new insight and understand-ing of phase transformations subjected to the effects of

anisotropic growth in real materials.

5. Conclusions

On the basis of MC simulations[17]of isothermal phasetransformations, the effects of anisotropic growth on devi-ations from the classical JMA kinetics have beeninvestigated.

(1) Adopting the analytical approach used to derive theanalytical phase transformation model[8], the effectsof anisotropic growth have been reinterpreted usingan analytical transformation model. In combinationwith MC simulations, the effect of the memory timewhen the transition from JMA to blocking behavioroccurs on the transformation has been shown, andphysically realistic Avrami exponents were deduced.

(2) In combination with MC simulations, it is concludedthat, the effects of anisotropic growth change theextended fraction from JMA (Eq. (8)) to non-JMA

(Eqs.(9) and (11)), thus leading to a change ofn.

0 20 40 60 800

0.2

0.4

0.6

0.8

1

f


t*

0 20 40 60 800

0.2

0.4

0.6

0.8

1

f


t*

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

f


t*

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

f

df/dt

experimentaloptimization

a

b

c

d

Fig. 6. The ferrite fraction,fa

, as a function of time, t, as determined from the length change measured for Fe2.13 at.% Mn (B) samples under 50 (a),150 (b) and 400 N (c) applied load with a cooling rate of 20 K min1. The transformation rate, dfa/dt, as a function offa (d) as determined from thelength change measured for Fe2.13 at.% Mn (B) samples under 50 N applied load with a cooling rate of 20 K min 1.



10/11

(3) A comparative study, performed between the analyt-ical description and the gammaalpha transforma-tion occurring in a FeMn alloy under appliedloads, shows that, based on MC simulations, thepresent analytical phase transformation model cre-ates a wider possibility to investigate the effects of

anisotropic growth. By this means, the current ana-lytical description can probably be applied to moregeneral transformations, bringing new insight andunderstanding to phase transformations subjectedto the effects of anisotropic growth in real materials.

Acknowledgements

The authors are grateful for the financial support of NewCentury Excellent Person Supporting Project (NCET-05-870), the Fundamental Research Project of National De-fense of China (A2720060295), the Project Sponsored bythe Scientific Research Foundation for the Returned Over-

seas Chinese Scholars, State Education Ministry(N6CJ0002), the Scientific and Technological CreativeFoundation of Youth in Northwestern Polytechnical Uni-versity, and the Natural Science Foundation of China(Grant No. 50501020, 50395103 and 50431030). F. Liu isalso grateful to Prof. E.J. Mittemeijer and Prof. F. Sommerfor their valuable and essential instruction and cooperation.

Appendix

First, the mixed part (in Eq.(11)) is considered. For thispart, the ratio of the extended fraction with n = 1.5 to that

with n= 0.5 is given as

2Gteff3wNR 2t

r2

r1A1

Analogous to the treatment performed for Eq.(9), Eq.(11)can be rewritten as

xe 2Cwffiffiffiffiffi

I20p

1 2NR

11r2r1

"

4C3NR ffiffiffiffiffi

I20p G20

11 r2r1 1 teff

32 11r2r1

# 2

3NRIG2 t 3

A2The whole extended fraction can be considered to be com-posed of two parts: xe=xe1+xe2, one part withn = 3, theother with n1 32 11r2r1. The ratio between the twoextended fractions is given as

23NR

I1G21t3

2Cwffiffiffiffiffi

I20p

1 2NR

11r2r1 4C

3NR

ffiffiffiffiffiI20

p G20

11 r2r1 1teff1

r2r12r2r1

" #

r4

r3 A3

whereffiffiffiffiffiffiffiI020

p 2Cw


p


p G1t t

1 2

NR

A4

ffiffiffiffiffiffiffiI020

p G020

4C

3NRffiffiffiffiffiffiI01p

exp QGRT ffiffiffiffiffiffi

I01p

G01

t

t

t t !" #A5

Analogously, different values ofI02and G02can be chosenin such a way that xe is only due to the part with n2= 3.Combining Eqs.(A2), (A4) and (A5)gives26664 2Cw


p


p G1t t

1 2

NR

11r2r1 :

4C

3NRffiffiffiffiffiffiI01p exp QGRT ffiffiffiffiffiffiI01

p G01

t

t

t t ! 1

1 r2r1

1

exp

QN2 1

1 r2r1

1QGRT

0BBB@

1CCCAteff

32 1

1r2r1

37775

23NR

I01G201 exp

2QG QNRT

t3

23NR

I02G202 exp

2QG QNRT

t3;

or that xe is only due to the part with n1,26664 2Cw


p


p G1t t

1 2

NR

11r2r1

4C3NR


p exp QG

RT

ffiffiffiffiffiffiI01p G01t tt t

! 11 r2r1 1

exp

QN2 1

1 r2r1

1QGRT

0BBB@

1CCCA

teff32 11r2r1

37775 23NRI01G201 exp 2QG QNRT t3

26664 2Cw


p


p G02t t

1 2

NR

11r2r1

4C3NR


p exp QG

RT


! 11 r2r1 1

exp

QN2 1

1 r2r1

1QGRT

0BBB@

1CCCA

teff32 1

1r2r1

37775



11/11

if,26664

2Cwffiffiffiffiffiffi

I01p

ffiffiffiffiffiffi

I01p

G1t t

1 2NR

11r2r1

4C3NR

ffiffiffiffiffiffiI01p exp QGRT


! 1

1 r2r1 1

exp

QN2 1

1 r2r1

1QGRT

0BBB@

1CCCAteff

32 1

1r2r1

37775

23NR

I01G201 exp

2QG QNRT

t3 U

then Eq.19can be obtained. Finally, the kinetic parame-ters are deduced, as given by Eqs. (20a), (20b) and (20c).

References

[1] Johnson WA, Mehl RF. Trans Am Inst Min (Metall) Engs 1939;135:1.

[2] Avrami M. J Chem Phys 1939;7:1109.[3] Avrami M. J Chem Phys 1940;8:212.[4] Avrami M. J Chem Phys 1941;9:177.

[5] Christian JW. The theory of transfomation in metals and alloys, Part1 equilibrium and general kinetics theory. Oxford: Pergamon Press;1975.

[6] Mittemeijer EJ. J Mater Sci 1992;27:3977.[7] Kempen ATW, Sommer F, Mittemeijer EJ. J Mater Sci 2002;37:1321.[8] Liu F, Sommer F, Mittemeijer EJ. J Mate Sci 2004;39:1621.[9] Weinberg MC, Kapral R. J Chem Phys 1989;91:7146.

[10] Bradley RM, Strenski PN. Phys Rev B 1989;40:8967.[11] Kashchiev D. Surf Sci 1969;14:209.[12] Kelton KF, Greer AL, Thompson CV. J Chem Phys 1983;79:6261.[13] Andrienko YA, Brilliantov NV, Krapivsky PL. Phys Rev A 1992;

45:2263.[14] Sekimoto K. Physica A 1986;135:328.[15] Sekimoto K. Phys Lett A 1984;105:390.[16] Pusztai T, Granasy L. Phys Rev B 1998;57:14110.[17] Kooi BJ. Phys Rev B 2004;70(22):224108.[18] Ruitenberg G, Petford-Long AK, Doole RC. J Appl Phys 2002;92:

3116.[19] Liu F, Sommer F, Mittemeijer EJ. J Mater Sci. Available from: .[20] Ranganathan S, Heimendahl M. J Mater Sci 1981;16:2401.[21] Liu F, Sommer F, Mittemeijer EJ. Int Mater Rev, in press.[22] Starink MJ. J Mater Sci 2001;36:4433.[23] Liu F, Sommer F, Mittemeijer EJ. J Mater Res 2004;19:2586.[24] Kempen ATW, Sommer F, Mittemeijer EJ. Acta Mater 2002;50:1319.[25] Liu F, Sommer F, Mittemeijer EJ. Acta Mater 2004;52:3207.[26] Shneidman VA, Weinberg MC. J Non-Cryst Sol 1993;160:89.[27] Weinberg MC, Birnie DP, Shneidman III VA. J Non-Cryst Sol 1997;

219:89.[28] Liu F, Sommer F, Mittemeijer EJ, unpublished work.[29] Liu YC, Sommer F, Mittemeijer EJ. Acta Mater 2003;51:507.[30] Malek J. Thermochimica Acta 2000;355:239.

http://-/?-http://dx.doi.org/10.1007/s10853-006-0802-4http://dx.doi.org/10.1007/s10853-006-0802-4http://dx.doi.org/10.1007/s10853-006-0802-4http://dx.doi.org/10.1007/s10853-006-0802-4http://-/?-

effect of anisotropic growth on the deviations from JMA kinetics

Documents

Transcript of effect of anisotropic growth on the deviations from JMA kinetics