review analysis of solid state

8/7/2019 review analysis of solid state

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Analysis of solid state phase transformationkinetics: models and recipes

F. Liu1,2, F. Sommer2, C. Bos2 and E. J. Mittemeijer*2

The progress of solid-state phase transformations can generally be subdivided into three

overlapping mechanisms: nucleation, growth and impingement. These can be modelled

separately if hard impingement prevails. On that basis, an overview has been given of recent

numerical and analytical methods for determination of the kinetic parameters of a transformation.

The treatment focuses on both isothermally and isochronally conducted transformations. To

extend the range of transformations that can be described analytically, a number of more or less

empirical submodels, which are compatible with experimental results, has been included in the

discussion. It has been shown that powerful, flexible, analytical models are possible, once the

concept of time or temperature dependent growth exponent and effective activation energy, in

agreement with the existing experimental observations, has been adopted. An explicit (numerical)

procedure to deduce the operating kinetic processes from experimental transformation-rate data,

on the basis of different nucleation, growth and hard impingement mechanisms, has been

demonstrated. Without recourse to any specific kinetic model, simple recipes have been given for

the determination of the growth exponent and the effective activation energy from the

experimental transformation-rate data.

Keywords: Phase transformation, Kinetics, Nucleation, Growth, Impingement, Isothermal, Isochronal, Numerical and analytical kinetic analyses

IntroductionSolid-state phase transformations are important means

for the adjustment of the microstructure and thus the

tuning of the properties of materials. To exploit this tool

to full extent, much effort is spent on the modelling of

phase transformations, e.g. Refs. 113. The required

models should not be in particular of atomistic nature,

but pertain to larger, mesoscopic and even macroscopic

scales. However, atomistic simulations can be very

useful for the interpretation of the values obtained for

the kinetic parameters.1416

In the classical treatment,6 the JohnsonMehl

Avrami (JMA) approach15 plays a central role in

studies of transformations where nucleation and growthmechanisms operate. Very many experimental results ofphase transformation kinetics have been fitted with a

JMA model. The original JMA equation can only be

validated under certain conditions, in particular for

nucleation (see the following section).15 Under these

conditions, the growth exponent n and the effective

activation energy Q should be constant during the

course of the transformation. The JMA equation can

also be applied to transformations that do not satisfy

these conditions, but then n and Q are generally notconstant.6 When, in practice, n and Q are found not tobe constant, a traditional explanation is the suggestionof corresponding changes in the nucleation and

growth mechanisms during the course of the transfor-mation.1722 As shown in Ref. 6 and as will be shown

more explicitly in the present work, this is notnecessarily the only explanation (i.e. even with constantnucleation and growth mechanisms during the entire

transformation, Q and n do not have to be constant).

A general modular, transformation model is possiblethat recognises the three mechanisms, nucleation,growth and impingement of the growing new phase

particles, as entities that can be modelled separately ifhard impingement is adopted.23,24 In a treatment dealing

with the analysis of phase transformation kinetics, bothisothermally and non-isothermally (in particular iso-chronally, i.e. constant heating/cooling rate) conducted

transformations have to be considered. It will be shownhere that also for isochronally conducted transforma-tions, explicit equations for the effective activation

energy and the growth exponent can be given.

Two routes of this modular approach23,24 have been

shown to be feasible. First, a numerical description of

the transformation can be given. Fitting on that basis toexperimental data in principle leads to determination ofphysically meaningful parameters that are characteristic

for the process. This numerical method has not beenapplied often.25,26 Normally, special recipes arefollowed for the determination of kinetic parameters,

1State Key Laboratory of Solidification Processing, NorthwesternPolytechnical University, Xian 710072, China2

Max Planck Institute for Metals Research, Heisenbergstrasse 3, D70569Stuttgart, Germany

*Corresponding author, email [email protected]

2007 Institute of Materials, Minerals and Mining and ASM InternationalPublished by Maney for the Institute and ASM InternationalDOI 10.1179/174328007X160308 International Materials Reviews 2007 VOL 52 NO 4 19 3


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as the (effective) activation energy, on the basis of ananalytical description of the transformation kinetics,subject to assumptions. Furthermore, if both a numer-ical and an analytical description are available, one

favours the analytical description of the transformation,as the use of analytical expressions has an advantageover numerical calculations because the influence of,for example, the different nucleation, growth and

impingement models can be more easily identified andinvestigated. Hence, second, against this background,considerable effort has been made recently to arrive at

accurate, flexible analytical descriptions of phase trans-formation kinetics.2729 The analytical descriptions giveexplicit expressions for the (possible) change in Q and nas the transformation progresses.

The present paper has been devised to review theserecent developments and to provide a practical guide fortransformation kinetics analysis. Both approaches, thenumerical (the section on The numerical approach) andthe analytical (the section on The analytical approach),will be presented and discussed. An emphasis lies onthe analytical approach and therefore, in particular

those nucleation and growth modules that are suitablefor analytical analysis will be discussed (the sectionson Modes of nucleation and Modes of growth).Guidelines for a fit procedure to determine the modelparameters are given (the section Determination ofmodel parameters) and examples of applications toexperimental data are included (the section onApplications of kinetic model fitting to real transforma-tions). Furthermore, avoiding full fitting procedures asin the numerical and analytical approaches, recipes willbe given for the determination of the kinetic parameters,without recourse to a specific kinetic model (the sectionon Recipes to determine the effective activation energyand the growth exponent).

The number of different phase transformations thathas been described with JMA-like models is very large.The present review is restricted to transformations wherethe nucleation and growth mechanisms remain constant

during the course of the transformation and whichcomply with hard impingement.

Modes of nucleationIn the following, an overview is given of a number ofdifferent nucleation modes that have been presented inthe literature. Their differences are illustrated by Fig. 1,for the case of isothermal annealing. The nucleationmodes dealt with here and as presented in the literaturegenerally pertain to large undercooling or overheating ofthe transformation system. At this stage of the develop-ment, the equations presented below for the nucleationrate apply to a virtual, infinite volume of untransformedmaterial where the nuclei are not affected by the growthof other nuclei (see the discussion on extended volumebelow).

Upon a phase transformation, interfaces developbetween the old and the new phases, and (possibly)misfit strain is introduced in the system. Whereas theproduction of the new phase releases chemical Gibbsenergy, the creation of the interfaces and the introduc-tion of misfit strain cost Gibbs energy. According to the

classical nucleation theory,6,30

a critical particle size ofthe new phase can be defined such that if the particle isof subcritical size, it costs energy to increase the size of

the particle, whereas if the particle (nucleus) is ofsupercritical size, energy is released if the particle grows

further. The formation of particles of supercritical sizefrom particles of subcritical size is called nucleation.

The nucleation rate is determined by the number ofparticles of critical size and the rate of the jumping of

atoms through the interface between the parent phaseand the particles of critical size. The frequency ofjumping through the interface is given by an Arrheniusterm. The number of particles of critical size depends onthe critical free energy of nucleus formation DG*, whichaccording to the above description, depends on thedecrease of the chemical Gibbs free energy per unitvolume, the interface energy per unit area and the misfitstrain energy per unit volume. The classical theory ofnucleation gives the nucleation rate per unit volume:N(T(t)), i.e. the rate of formation of particles ofsupercritical size (i.e. nuclei), as6,30

:N(T(t))~Cv exp {

DG(T(t))zQN

RT(t) (1)

where R is the gas constant, T the temperature, C thenumber density of suitable nucleation sites, v thecharacteristic frequency factor and QN the activationenergy for the jumping of atoms through the interface*.

Continuous nucleationIf the undercooling or the overheating is very large, DG*can be considered to be very small as compared to RT.

a pre-existing nuclei (site saturation): N*; b continuous

nucleation: N0, a51; c continuous nucleation: N0, a51?5; d

mixed nucleation: N*, N0, a51; e mixed nucleation: N*, N0,

a51?5; f Avrami nucleation: N9, l05861018 s21 m23; g

Avrami nucleation: N9, l05661019 s21 m23; h Avraminucleation: N9, l053610

20 s21 m23

1 Evolution of the number of nuclei per unit volume with

time at constant temperature (T5600 K) for different

nucleation modes with N*5N95261021 m23, N05

1040 s2a m23

*Equation (1) is also often written including the Zeldovic non-equilibriumfactor (e.g. Ref. 31) and the characteristic frequency factor is often writtenas kT/h, with kas Boltzmanns constant and has Plancks constant. Here,

in accordance with Ref. 6, the Zeldovic factor is left out and a constantcharacteristic frequency is used (see also footnote on page 197), allowingthe development of analytical expressions for the (extended) fractiontransformed material (seesection on this subject below).

Liu et al. Solid state phase transformation kinetics

19 4 International Materials Reviews 2007 VOL 52 NO 4


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The nucleation rate per unit volume is then only deter-

mined by the atomic mobility for transport through the

interface, which for isothermally and non-isothermally

conducted transformations, gives23,24,27

:N(T(t))~N0 exp {

QN

RT t

(2)

where Cv has been combined into N0, the

temperature-independent nucleation rate. QN, the acti-vation energy for the jumping of atoms through

the interface between the particle of critical size and

the matrix, is defined for the remainder of this text as the

temperature-independent activation energy for nuclea-

tion. This type of nucleation is called continuous

nucleation characterised by a constant nucleation rate

at constant temperature; the number of nuclei equals 0

at t50. This type of nucleation often operates in the

crystallisation of amorphous alloys.32

It should be noted that for smaller undercooling or

overheating, DG* is not very small as compared to RT.

In this case, the full nucleation equation (equation (1))

must be used. Note that DG* in equation (1) depends on

temperature. An analytical treatment on the basis of

equation (1) is only possible for isothermal transforma-

tions (cf. the section on Extended volume and extended

transformed fraction); an isochronal process could be

subdivided in isothermal steps to perform an analytical

treatment per step.

The nucleation index a

With the use of equation (2), the nucleation rate is

constant at constant temperature during the entire

course of the transformation. To allow for dependence

of the nucleation rate on the degree of transformation, it

was proposed to introduce the empirical nucleation

index a.33,34 The number of particles (for isothermaltransformation) in that case can be expressed as

N(t)5n9ta, with n9 as a constant (a50 for zero nucleation

rate, a51 for constant nucleation rate, a.1 for

nucleation rate accelerating with progress of transfor-

mation). Values for a in the range of 0,a,1 imply a

nucleation rate decelerating with ongoing transforma-

tion. Such a case can better be described with the

Avrami nucleation mechanism (see below) and there-

fore, the range 0,a,1 is not considered further.

Anticipating the analysis of non-isothermal

(isochronal) kinetics, the nucleation index should be

introduced such that both time and temperature

dependences of the nucleation density can be considered.Accordingly, the relationship between N and t for

continuous nucleation in isothermal transformation

becomes

N(T,t)~N0ta exp {

QN

RT

(3)

and it follows (cf. equation (2))

:N(T,t)~aN0t

a{1 exp {QN

RT

(4)

For isochronal transformation, the number of nuclei

per unit volume of untransformed material can be

expressed as an integration of the continuousnucleation rate from T0 to T(t), i.e. as a function of

T(t) (Ref. 35)

N~

T t T0

N0 exp {QN

RT t0

!d

T t0

W

%RN0

QNWT t 2exp {

QN

RT t

(5)

where W is the constant heating rate dT(t)/dt. The

temperature integral has been approximated asdescribed in Ref. 7, T(t).T0 and QN=RTww1, implyingthat equation (5) holds for the case of isochronal heating

(see Appendix).

Now, adopting the philosophy as described above in

equation (3) for the case of isothermal annealing, it

appears appropriate for isochronal annealing to incor-

porate the nucleation index a as follows

N T t ~RN0T t

QN

T t

W

aexp {

QN

RT t

(6)

and the corresponding nucleation rate is given by

:N T t %N0

T(t)

W

a{1exp {

QN

RT t

(7)

For application of the transformation model incorpor-

ating the nucleation index, see the section on Appli-

cations of kinetic model fitting to real transformations.

It should be noted here that all equations involving

the nucleation index a only provide an empirical

description of the nucleation rate. When available,

models with a stronger theoretical justification should

be used. For a.1, the constant N0 can no longer be

interpreted as a temperature independent nucleation

rate; the unit of N0 should be given as s2a m23.

Saturation by pre-existing nucleiIn the section on Continuous nucleation, it was

assumed that the number of nuclei at the beginning of

the transformation is zero. Here, the case is considered

where there already is a number of pre-existing nuclei

(supercritical particles of the new phase) at t50 and the

further nucleation rate is zero. This implies for the

nucleation rate23,27

:N(T)~Nd t{0 (8)

for isothermal transformation and

:N T t ~Nd

T t {T0W (9)

for isochronal transformation, where N* is the number

of (pre-existing) nuclei per unit volume, d(t20) and

d{[T(t)2T0]/W} denote Dirac functions and W is the

constant heating/cooling rate with T05T (t50) and

T(t)5T0zWt with W5dT(t)/dt.

A typical situation where the nucleation rate can

be described by equation (8) or equation (9) is the

case of saturation caused by preferential

nucleation at grain boundaries, edges or corners as

discussed by Cahn.6,36 Depending predominantly on

the degree of supercooling (or superheating),

saturation of the (grain boundary) nucleation sites

can occur very early in the transformation leadingto a zero nucleation rate for the remainder of the

transformation.


International Materials Reviews 2007 VOL 52 NO 4 19 5


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The activation energy for supercritical particle formation QS

Upon rapid cooling/quenching of a phase stable at

elevated temperature, this phase can become metastable

at lower temperatures, e.g. an amorphous alloy or a

supersaturated crystalline solid solution may occur,

which strives for crystallisation or decomposition

respectively. Depending on the precise thermal history

of such metastable phases, more or less particles of a

new, stable phase may have been frozen in. If a heattreatment is applied subsequently to such a metastable

phase with frozen-in particles of the new, stable phase,

then those particles larger than the critical size (which

are called the nuclei of the new, stable phase24) can

grow, implying occurrence of initial pre-existing nuclei

with no formation of further nuclei (cf. equations (8)

and (9)). The critical size depends on temperature. Then,

given a certain size distribution for the frozen-in

particles of the new, stable phase, it is evident that the

number of nuclei (i.e. supercritical particles) acting in

the pre-existing nuclei nucleation mechanism is tem-

perature dependent. Hence, an activation energy QScan be introduced that controls the number of frozen-in

nuclei acting in the pre-existing nuclei mechanism upon

progressing transformation.

As a first approximation, the critical size of the frozen-in

new phase particles can be assumed to depend on

temperature as A/DT, with DT as the undercooling.30

Hence, the critical size increases with decreasing under-

cooling, i.e. with increasing annealing temperature.

Assuming that the size distribution of the frozen-in particles

of the new, stable phase exhibits an exponential tail towards

larger particle size, the temperature dependence of the

number of supercritical particles operating in the pre-

existing nuclei nucleation mode may be given as exp(QS/

RT). (Note that the number of active sites decreases with

increasing annealing temperature: thus QS is not anactivation energy in the usual sense, which would have

required a minus sign in the argument of the exponent.)

On this basis, an analytical treatment can be given but only

for the case of isothermal annealing, as explained in

Ref. 35.

For isothermal transformations conducted at different

annealing temperatures, different values for the number of

the supercritical particles thus occur for each annealing

temperature T. Equation (8) can then be rewritten as35

:N(T)~NS exp

QS

RT

d t{0 (10)

with NS* as a constant.

Avrami nucleationAccording to this nucleation mechanism, the particles of

supercritical size (nuclei) are formed from the particles

of subcritical size of number Nsub (:

N~{:

Nsub), such that

the total number of particles, of sub- and supercritical

size N9 is constant.24 The change of the number of

particles of supercritical size is thus equal to the product

of the number of particles of subcritical size Nsub and

the rate l, at which an individual subcritical particle

becomes supercritical

:N~{

:Nsub~lNsub (11)

It is supposed that l obeys Arrhenius-type temperature

dependence

l~l0 exp {QN

RT

(12)

with l0 as a temperature-independent rate. Uponintegration of equation (11), after separation of vari-

ables, using equation (12) and the boundary condition

that the number of subcritical particles equals N9 at t50,

it is obtained for the rate of formation of supercritical

particles at t

:N T(t) ~{

:Nsub~lN

0 exp {

t0

ldt

0@

1A (13)

By variation of l0, the mode of nucleation can be

varied from only pre-existing nuclei (l0 infinitely

large) to continuous nucleation (l0 infinitely small).This behaviour is shown for isothermal annealing in

Fig. 1.

For isothermal annealing, l is constant and equa-tion (13) becomes23

:N T,t ~Nl exp {lt (14)

For isochronal annealing, with T(t).T0 and QN/RT&1(usually QN/RT>25), equation (13) can be rewritten

using the approximation for the temperature integral as

given in Appendix7,27

:N T(t) %Nl exp {

Rl

QNWT t 2

(15)

Mixture of nucleation mechanismsIn practice, mixed types of nucleation may occur. The

specific name mixed nucleation represents a combination

of the pre-existing nuclei and continuous nucleation

modes: the nucleation rate is equal to some weightedsum of the nucleation rates according to continuousnucleation (equations (2), (4) and (7)) and pre-existing

nuclei (equations (8) and (9)). Hence, for isothermal

transformation27

:N(T,t)~Nd t{0 zN0 exp {

QN

RT

(16)

which after introduction of the nucleation index a (cf.equation (4)), becomes

:N(T,t)~Nd t{0 zaN0t

a{1 exp {QN

RT

(17)

and for isochronal transformation

27

:N T t ~Nd

T t {T0W

zN0 exp {

QN

RT t

(18)

which after introduction of the nucleation index a (cf.

equation (7)), becomes

:N T t ~Nd

T t {T0W

z

N0 exp {QN

RT(t)

T t

W

a{1" #(19)

where N* and N0 represent the relative contributions of

the two modes of nucleation.Unambiguous preference for this type of mixed

nucleation has been observed for the crystallisation of




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different amorphous alloys, indicating a contribution ofquenched in nuclei and of subsequent nucleation uponannealing.25,28,29,37

According to equations (1315), Avrami nucleation

approaches the continuous nucleation mode and thepre-existing nuclei mode, at the start and at the endof the transformation respectively (see Fig. 1).Therefore, a combination of pre-existing nuclei and

Avrami nucleation also offers a description of inter-mediate cases of nucleation. Hence, for isothermaltransformation28

:N(T,t)~Nd t{0 zN0l exp {lt (20)

and for isochronal transformation

:N T t ~Nd

T t {T0W

z

N0l exp {Rl

QNWT(t)2

(21)

where N* and N9 represent the relative contributionsof the two modes of transformation. For example,

the isothermal crystallisation of amorphous MgNialloys was shown to exhibit nucleation according toequation (20).28

A combination of continuous nucleation with

Avrami nucleation is physically less meaningful toconsider: according to both nucleation modes, thenumber of nuclei continuously increases with progres-sing transformation.

Modes of growthTwo (extreme) growth models are considered: one forvolume diffusion controlled growth and one for inter-face-controlled growth. Volume diffusion controlled

growth can occur for phase transformations where long-range compositional changes take place. The case ofinterface-controlled growth can occur if the growth isdetermined by atomic processes in the immediatevicinity of the interface, as the massive austenite (c)Rferrite (a) transformation in substitutional binary Fe-based alloys3840 and also in some cases of crystallisationreactions of amorphous alloys.28,29

The diffusion-controlled and interface-controlledgrowth modes can be given in a compact form. At timet, the volume Yof a particle nucleated at time t is givenby24

Y t,t ~gtt

n dt0@ 1A

d=m

(22)

with g as a particle-geometry factor (m3 m2d) and n asthe growth velocity, m as the growth mode parameter(interface-controlled growth: m51; volume diffusioncontrolled growth: m52; particle thickening obeys aparabolic growth law) and das the dimensionality of thegrowth.

For the case of interface-controlled growth (then

equation (22) is applied with m51), the textbookequation for the interface velocity v is given by6

v(T(t))~v0 exp {

DGa

RT(t)

1{ exp

DG

RT(t)

(23)

where n0 is the pre-exponential factor for growth, DGa is

the activation energy for the transfer of atoms through

the parent phase/new phase interface and DG is theenergy difference between the new phase and the parentphase.

For large undercooling or overheating, DGj j is largecompared to RT and equation (23) becomes

v(T(t))~v0 exp {QG

RT(t) (24)with QG (5DG

a) as the activation energy for growth andn0 as the temperature-independent interface velocity.For interpretation ofDGa, see Ref. 41.

For small undercooling or overheating, the drivingforce DGj j is small as compared to RTand equation (23)reduces to

v(T(t))~M({DG)

~M0 exp {QG

RT(t)

({DG(T(t))) (25)

where QG5DGa and M is the temperature dependent

interface mobility*. Note that DG in equation (25)

depends on temperature. For isothermal transforma-tions, Y can still be calculated analytically according toequation (22), after substitution of equation (25) ifDGisconstant for the integration. For isochronally conductedmeasurements, Y can only be calculated by numericalintegration. This has led to limited application ofequation (25), as compared to equation (24).

In general, growth can exhibit a mixed-mode char-acter: the transformation can start with interface-controlled growth and then a transition to diffusion-controlled growth can occur, as obtained from modelconsideration, e.g. for the isothermal austenite (c)Rferrite (a) transformation in FeC alloys.42 The transi-tion from interface-controlled growth to diffusioncontrolled growth has been observed during nano-crystallisation of amorphous Al-based alloys.32

The activation energy introduced for nucleation in thesection on Continuous nucleation QN (cf. equation (2))has been conceived as an activation energy for thetransfer from the matrix (cf. an atom) through the inter-face between the particle of critical size and the matrix;thereby the particle considered becomes a nucleus. Inthe current section, the activation energy introducedfor interface controlled growth QG (5DG

a) (cf. equa-tion (23)) has been conceived as an activation energy fortransfer from the matrix (of an atom) through theinterface between the growing particle (much larger than

a nucleus) and the matrix. Both activation energiesdepend on elementary atomic jumps. Yet, they can haveconsiderably different values. This may be due toconsiderably different structures for the interface withthe matrix for the minute embryos (particles smallerthan and just equal to critical size) and for the muchlarger (up to orders of magnitude) growing particles.

*Equation (24) contains v0 as the temperature-independent interfacevelocity. This means that v0 includes a temperature independentfrequency factor. In the derivation of equation (25) from equation (23) apre-exponential factor M05v0/RT appears. To arrive at a temperatureindependent factor M0 (equation (25)) the atomic vibration frequency isusually assumed to be proportional to kT/h(with kas Boltzmanns constant

andh

as Plancks constant). Because the influence of the temperaturedependence of the frequency factor is small in comparison to theexponential term the error that is made by adopting either v0 astemperature independent or M0 as temperature independent is small.




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For example, in initial stages, coherent interfaces mayoccur, whereas a growing particle may exhibit anincoherent interface. Then in the initial stage of growth,the activation energy may change due to the occurring

changes in the interface structure. Activation energiesfor interface mobilities can be determined by groups ofatomic jumps leading to effective activation energiesconsiderably larger than the activation energy for a

single atomic jump.41 Such processes may have sig-nificantly different net effects for minute embryos andlarge growing particles, in view of the different interface

structures.For the case of diffusion-controlled growth (then

equation (22) is applied with m52), n in equation (22)can generally be substituted by n according to equa-tion (24), where QG has to be replaced by the activationenergy for diffusion QD and n0 has to be replaced bythe pre-exponential factor for diffusion D0, i.e.n5D(T(t))5D0 exp(2QD/RT).

Unambiguous results for the activation energies ofboth the nucleation and growth mechanisms for thesame solid-solid state transformation are rare. The

methodology presented in the present review allowsthe separate determination of QN and QG in one kineticanalysis. First results with this approach demonstratethat QN can be larger than QG (Ref. 28) and that QN canbe smaller than QG or QD but exhibiting the same orderof magnitude.29 Clearly, much more experimental dataare necessary to arrive, possibly with the aid ofdedicated computer simulations,4345 at detailed inter-pretation of values determined for QN and QG or QD.

Extended volume and extendedtransformed fractionThe number of supercritical nuclei formed in a unit

volume, at time t during a time lapse dt, is given by:N(T(t))dt, with

:N(T(t)) according to equations (121).

The volume of each of these nuclei grows from t until taccording to equation (22) where it is supposed thatevery particle grows into an infinitely large parent phase,in the absence of other growing particles. In thishypothetical case, the volume of all particles at time t,called the extended volume Ve, is given by

Ve~

t0

V:

N t Y t,t dt (26)

with V as the sample volume, which is supposed to be

constant throughout the transformation. To evaluateequation (26) for non-isothermal transformation, it isnecessary to apply explicit time dependences for the

temperature Toccurring in the expressions for:

N and Y.The extended transformed fraction xe is defined as

xe:Ve

V~

t0

:N t Y t,t dt (27)

To arrive at explicit expressions for xe and as discussedin the section on Modes of growth, in the following,equation (24) will be applied for the growth functionboth for interface- and diffusion-controlled growth. For

the cases of only pre-existing nuclei and continuousnucleation, it now follows straightforwardly fromequation (27); first, for isothermal transformations

(equations (2) and (8))

xe~Kn0 t

n exp {nQ

RT

~bn (28)

with b~K0t exp{(Q=RT) and second, for isochronaltransformations (equations (7) and (9))

xe~Kn

0t

0

exp {Q

RT t dt24 35

n

~bn (29)

with

b%RT2

WK0 exp {

Q

RT

and for a.1 (continuous nucleation) with b>

T2K0 exp({Q=RT). For both isothermal and isochro-nal transformations, n is the constant growth exponentand Q is the constant effective activation energy (see

Table 3 for Kn0 , n and Q).

For all the other (mixed) nucleation modes considered(equations (13) and (1621)), the extended volume can

be shown to be given always by the addition of twoparts27,28 that can be conceived as due to pre-existingnuclei and to continuous-like nucleation (a>1) respec-

tively. Then, by extensive calculation, the followingexplicit analytical expressions for the extended trans-formed fraction can be obtained27

for isothermal transformation

xe~K0 t n t

tn t exp {n t Q t

RT

(30)

for isochronal transformation

xe~K0 T n T RT

2

W

n T

exp {n T Q T

RT (31)

for a51

xe~K0 T n T

T2 n T

exp {n T Q T

RT

(31a)

for a.1, which can be generally written as

xe~Kn0 a

nexp {

nQ

RT

(32)

where a is identified with either the annealing time t for

isothermal transformation or with RT2/W (a51) and T2

(a.1) for isochronal transformation.

In general, the kinetic parameters n, Q and K0 are

functions of time t (isothermal transformation) ortemperature T (isochronal transformation) and dependon model parameters as N*, N0 and a (mixed nucleation

incorporating a.1) or N9 and l0 (Avrami nucleation),QN, QS (pre-existing nuclei incorporating QS) and QG(see the section on Isochronal crystallisation of amor-

phous Zr50Al10Ni40). Explicit expressions for n, Q andKn0 , in terms of the general nucleation and growthmechanisms, for both isothermal and isochronal anneal-

ing (heating), have been listed in Tables 13.

Finally, it is noted that within the theoretical frame-work of this section, equation (30) holds exactly,whereas equation (31) is subject to the approximation

made for the temperature integral in nucleation andgrowth functions and thus pertains to the case ofisochronal heating.




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Table1

Expressions

forthe

(time

and

temp

erature

dependencies

ofthe)growthe

xponentn,the

overallactivation

energ

y

Q

and

the

rate

constantK0,to

be

inserted

into

equation(32)

forisothermalannealingandisochr

onalannealingrespectively*

Isothermal

Isochronal

Mixednucleation

27

n

n~

d mz

1

1z(r2=r1){

1

n~

d mz

1

1z(r

2=r1){

1

Q

d mQGz

n{

d m

QN

!0n

d mQGz

n{

d m

QN

!0n

Kn 0

gv

d=m

0

(d=m)z11

=1z(r

2=r1)

1

N

1z

r2 r1

!1=1zr2=r1

N0

1z

r2 r1{1

"

#

(

)1=1z(r

2=r1)

1

gv

d=m

0

(d=m)z11

=1z(r

2=r

1)

1

N

Qd=m

G

1z

r2 r1

"

#1=1zr

2=r1

CcN

0

1z

r2 r1{1

"

#

(

)1=1z(r

2=r1)

1

r2/r1

N0texp

{QN=RT

(d=m)z1N

CcQ

d=m

G

N0exp

{QN=RT

(d=m)z1N

RT

2

W

Avrami

nucleation

27

n

d mz

1

1z(r

2=r1)

d mz

1

1z(r

2=r1)

Q

d mQGz

n{

d m

QN

!0n

d mQGz

n{

d m

QN

!0n

Kn 0

gN

0f(lT)v

d=m

0

(d=m)z1

l1=1zr2=r1

0

(lt)

1=1z(r2=r1)

1

gv

d=m

0

N0f(lRT

2=QNW)

(d=m)z1

l1=1

zr2=r1

0

l

RT

2

W

1=1z(r

2=r1)

1

Cc

r2/r1

lt

(d=m)z1

CcQ

d=m

G

(d=m)z1

RT

2

W

l

Avrami

nucleationpluspre-existingnuclei

n

d mz

1

1z(r

2=r1)

d mz

1

1z(r

2=r1)

Q

d mQGz

n{

d m

QN

!0n

d mQGz

n{

d m

QN

!0n

Kn 0

gv

d=m

0

(d=m)z11

=1z(r

2=r1)

Nz

N0ltf(lt)

2(d

=m)z1

1z

r2 r1{

1

lt

(d=m)z1

!1=1z(r

2=r1)

1|

N0l

0

f(lt)

2

1z

r2 r1

lt

(d=m)z1

{1

!1=1z(r

2=r

1)

2 6664

3 7775

gv

d=m

0

d=m

z1

1=1z

r2=r1

1

Qd=

m

G

Nz

N0Q

d=m

G

Ccl

RT

2

=W

fl

RT

2

=QNW

2

d=m

z1

|

1z

1

d=m

z1

CcQ

d=m

G

RT

2

W

l

r2 r1

{1

!

8>>>< >>>:

9>>>= >>>;1=

1z

r2=r1

{1

|

N0l

0Ccfl(RT

2=QNW)

2

1

z

r2 r1

1

(d=m)z1

CcQ

d=m

G

RT

2

W

l

{1

"

#1=1z(r

2=r1)

r2/r1

N

N0z

lt

2

(d=m)z1

flt

!01 2f(lt)

N

N0z

CcQ

d=m

G

l

2

(d=m)z1

RT

2

W

f

l

RT

2

QNW

"

#,1 2

f

l

RT

2

QNW

*ForCc

,f(lt)andf(lRT2

/QNW),seeRef.27.Thes

evaluesarevalidfortheanalyticalmode

lbasedonmixednucleation(a51)andA

vraminucleationpluspre-existingnuclei,

aspresentedinthepresent

paper.

n0

andQG

havetobesubstitutedbyD0a

ndQD

inthecaseofvolumediffusion-controlledgrowth.




8/20

Table2

Expressions

forthe

(time

and

temp

erature

dependencies

ofthe)growthe

xponentn,the

overallactivation

energ

y

Q

and

the

rate

constantK0

,to

be

inserted

into

equation(32)

forisothermalannealingandisochr

onalannealingrespectively*

Mixed

nucleation

Isothermal

Isochronal

n

d mz

a

1z

r2=r1

{1

d mz(

1za

2

)

1

1z

r2=r1

{1

Q

d mQGz

n{(d=m

)

a

QN

!=n

d mQGz

n{(d=m)

(1za)=2

QN

!=n

Kn 0

Interfacecontrolled

(m51)

Kn 0

gv

d 0=

Pd i~

1

iazi

{1=1z(r2=r1){

1

"

#N

1z

r2 r1

!1=1zr2=r1

N0

1z

r2 r1{1

"

#1=1z(r2=r1){

1

8< :

9= ;

gv

d=m

0

(d=mz1)1

=1z(r2=r1){

1

N

RQGW

d=m

1z

r2 r1

"

#1=1zr2=r1

CcN

0

Rd=mz1

Wd=mza

1z

r2 r1{1

"

#

(

)1=1zr2=r1){

1

r2/r1

Interfacecontrolled

(m51)

r2/r1

Pd i~

1

iazi

h

iN0taex

p({QN=RT)

N

CcQ

d=m

G

N0exp

{QN=RT

(d=mz1)N

RT

1za

Wa

Kn 0

Volumediffusionco

ntrolled(m52)

gv

d=2

0

=

Pa i~

1

i

(d=2)zi

{1=1z(r2=r1){

1

"

#N

1z

r2 r1

!1=1zr2=r1

N0

1z

r2 r1{1

"

#1=1z(r2=r1){

1

8< :

9= ;

r2/r1

Volumediffusionco

ntrolled(m52)

Pa i~

1

i

(d=2)zi

h

iN0t

aexp({QN=RT)

N

*ForCc

,seeRef.27.Thesevaluesarevalidforth

eanalyticalmodelbasedonmixednuclea

tionincorporatingthenucleationindexa,aspresentedinthepresentpaper.Equationspresentedforisothermal

transformationsandvolumediffusioncontrolledgrowthareonlyvalidforintegervaluesof

a.n0

andQG

havetobesubstitutedbyD

0

andQD

inthecaseofvolumediffusion

controlledgrowth.




9/20

Because the interpretation of values determined for

the activation energy of a phase transformation often

led and leads to confusion in the literature,4650 the

following remarks are made.

For all combinations of nucleation (a51) and growth

modes, with Arrhenius temperature dependences, as

considered in the present review, the effective, overall

activation energy of the transformation can be given as

(cf. Tables 13)

Q~(d=m)QGz n{d=m QN

n(33)

As noted above, Q (through n) is time and temperature

dependent, whereas the activation energies for nuclea-

tion and growth QN and QG are constants. Therefore, an

observation of change of Q with time or temperature,

i.e. during the course of a transformation, needs not be

considered as an experimental artefact or as a con-

sequence of change of transformation mechanism.

Modes of impingementA relationship between the actually transformed volume

Vt and the extended transformed volume Ve or betweenthe real transformed fraction f5Vt/V and the extendedtransformed fraction xe5V

e/V is required. The expres-

sions for the extended transformed volume/fraction donot account for the overlap of growing particles (hardimpingement). Furthermore, in diffusion-controlled

transformations, as can pertain to nano-crystallisationof amorphous alloys32,51,52 and the gammaalpha trans-formation in (carbon containing) alloyed steels,53,54 theoverlap of diffusion fields surrounding the growingparticles has to be considered (soft impingement). Somespecific analytical and numerical approaches to accountfor the diffusion fields surrounding the product-phaseparticles have been proposed,5154 which however, areunsuited for general applications. Within the context ofthe general modular transformation model,23,24 modelsfor (hard) impingement are discussed below.

Randomly dispersed nuclei

Suppose that the nuclei are dispersed randomlythroughout the total volume. If the time is increasedby dt, the extended and the actual transformed volumeswill increase by dVe and dVt respectively. From thechange of the extended volume dVe, only a part willcontribute to the change of the actually transformedvolume dVt, namely a part as large as the untransformedvolume fraction.15 Hence

dVt~V{Vt

VdVe;

df

dxe~1{f (34)

and thus

f~1{ exp {xe (35)

Anisotropic growthIn the case of anisotropically growing particles, the timeinterval that particles, after their randomly dispersednucleation, can grow before blocking by other particlesoccurs is, on average, smaller than for isotropicgrowth.13,15,16 This blocking effect due to anisotropicgrowth leads to hard impingement that results in strong

Table 3 Expressions for the (time and temperature dependencies of the) growth exponent n, the overall activationenergy Q and the rate constant K0, to be inserted into equation (32) for isothermtal annealing and isochronalannealing respectively*

Isothermtal (a51) Isothermtal ( a.1) Isochronal ( a51) Isochronal ( a.1)

Continuous nucleationn d=mz1 d/mza d=mz1 d=mz 1za =2Q (d=m)QGzQN

n

(d=m)QGzaQNn

(d=m)QGzQNn

dm

QGzn{d=m1za =2 QN

nK0

n

gN0nd=m0

n

gN0nd=m0

n

gN0nd=m0 Cc

n

gnd=m0

d=mz1CcN0

Rd=mz1

Wd=mza

Isothermal (QS50) Isothermal (QS.0) Isochronal (QS50) Isochronal (QS.0)Pre-existing nuclein d/m d /m d/m Q QG (d=m)QG{QS

n

QG

K0n

gNnd=m0 gN

nd=m0 gN

nd=m0

*For Cc, seeRef. 27. These values are valid for the analytical model based on continuous nucleation incorporating the nucleation indexor pre-existing nuclei incorporating QS, as presented in the present paper. n0 and QG have to be substituted by D0 and QD in the caseof volume diffusion controlled growth.

2 The transformed fraction f as a function of the

extended fraction xe for the case of impingement by

anisotropically growing particles corresponding to j>1

and for the case by impingement of non-random nuclei

distributions corresponding to e>1




10/20

deviations from classical JMA kinetics.16 Considering

this blocking effect, one phenomenological approach*

accounting for impingement in this case has been

proposed13,16 by extending equation (34) to

df

dxe~(1{f)j (36)

where j>1. Impingement due to equation (36) is more

severe, i.e. the difference between f and xe is larger thandue to equation (34), and increases with j (see Fig. 2).

For j.1, integrating equation (36) gives

f~1{ 1z j{1 xe {1=(j{1)

(37)

For j51, equation (36) reduces to equation (34) (see

Fig. 2).

Non-random nuclei distributionsA more regular dispersion of the growing particles, as

compared to the randomly dispersed case, would lead to

a smaller influence of the impingement correction, i.e.

the difference between xe and f is smaller. If the initial

material is polycrystalline, the corners of the grains ofthe parent phase can act as preferred nucleation sites.

This implies that the dispersion of the growing particles

cannot be considered as truly random and thus, the

impingement correction as indicated by equation (34)

does not hold exactly. The dispersion of the growing

particles will also be not exactly periodic, corresponding

to an inexactly homogeneous parent crystallite size and

shape, and hence the impingement correction for ideally

periodically dispersed growing particles is also incor-

rect.38 A general approach to impingement for non-

randomly distributed nuclei can be proposed

df

dxe~1{fe (38)

with e>1. The impingement according to equation (38)

with e.1 is less distinct than the one according to

equation (34), because the difference between f and xe is

smaller, and decreases with increasing e (see Fig. 2).

Departing from equation (38), it follows that an

analytical expression for f cannot be obtained for all

values of e. However, a simple recursive procedure is

possible. Provided the differences between the consecu-

tive f values, fiz1 and fi, and the corresponding xevalues, xe(fiz1) and xe(fi), are sufficiently small, equa-

tion (38) can be written in difference form and it follows

fiz1~ 1{fe

i

xe fiz1 {xe fi zfi (39)Given the initial condition f150 and xe(f1)50, xe(f2) is

calculated for t2 (isothermal annealing) or T2 (isochronal

annealing) and f2 follows from equation (39), etc.

For special values of e, analytical expressions result.

Obviously for e51, equation (35) results. For e52, it

follows

f~ tanh xe (40)

This approach was first used to describe the kinetics of

the massive austenite (c)Rferrite (a) transformation in

iron and iron-based alloys.3840

The transformed fractionThe general recipe for deriving an explicit analytical

formulation or numerically calculating values for the

degree of transformation in terms of the modular

transformation model is as follows (see Fig. 3). The

extended volume is calculated according to equa-

tion (26) using the appropriate nucleation mode (see

above) and the appropriate growth mode (see above).

The expression for the extended volume (see the section

above) is then substituted into the appropriate impinge-

ment correction (see the previous section) to give the

degree of transformation. Note that values of a, e and j

different from unity imply that a phenomenological

description of nucleation and/or impingement in the

fraction transformed is given.

The numerical approachThe cornerstone of the numerical calculation is the

evaluation of the extended fraction xe (equation (27))

that implies the execution of a nested, double integration

(see equations (22) and (27)). Even with modern

computers, the full numerical evaluation of equa-tion (27) is not trivial. In a typical fit of the model to

experimental data, equation (27) must be calculated very

many times. Not only must equation (27) be evaluated

for many different times/temperatures (typically y100

points per xe-curve (or f-curve)), but also for a number

of different temperatures/heating rates. Thus, a fit to

experimental data may require very many function

iterations before the optimal set of parameters is found.

To directly use equation (27) in a fit of the model to

experimental data, the numerical evaluation of the

double integral represented by equation (27) must be

done as efficiently as possible. Two different approaches

can be followed to solve equation (27) in a numericalway fast enough for application in a fit procedure that

can be applied in practice:

*Equation (36) is one of several phenomenological approaches possible.

16

For isothermal transformations with continuous nucleation an analyticaltheory is available,16,55 which, however, cannot be expressed as a relationbetween f and xe.

3 Schematic diagram of the modular transformation

model: the specific models for nucleation, growth and

impingement have to be substituted into the expres-

sion for the extended transformation fraction xe to cal-

culate the real transformed fraction f (see the sections

on Modes of nucleation, Modes of growth and

Modes of impingement)




11/20

(i) because the integrands are smooth functions ofthe variables, high order algorithms as describedin Ref. 56 can calculate the integrals with arelatively small number of function evaluations,

or an even more sophisticated algorithm asdescribed in Ref. 57 can be applied

(ii) a different approach involves development of analgorithm optimised for the specific problem. The

inner integral of equation (27) concerns theevaluation of Y. As follows from equation (22),Y is obtained by integrating the growth velocity

from time (of nucleation) t to time t. Forequation (27), this integral has to be evaluatedfor all times within the considered range of t.However, once Yhas been calculated for t50, allwork has in fact already been done for the othertimes t and by storing the result for t50, Y doesnot have to be calculated for the other values oftanymore. The same principle can be used for theouter integral of equation (27): once it has beenevaluated for the end time of t, most of thecalculation work for the intermediate times has in

fact already been performed and by storing theresults for t5tend, xe can be calculated with aminimal amount of work for all other values of t.

The analytical approachThe recipe described above can also be adopted for thederivation of analytical formulations of the degree oftransformation f. The analytical procedures depart from

equations (30) and (31) for xe (with parameters fromTables 13). Analytical descriptions of f provide moredirect insight into functional dependences and are oftenused in practice. This explains the large, also recent,interest in analytical descriptions of transformationkinetics712,2729 and their application (see, out of an

enormous body of such literature, a few very recentexamples2022,28,29,35,37).

In contrast with the original JMA equation,15 whichis still at present often but in many cases inappropriatelyused,17,18,2022 the present overview provides a summaryof possible, flexible analytical approaches. It may be saidthat the variety of single and combined nucleation andimpingement models considered offers the possibility todescribe real phase transformations in many cases.

The main limitation in the application of theconsidered nucleation and growth modes to arrive atanalytical expressions for f is that the undercooling or

overheating of the transforming system is relativelylarge, in order that Arrhenius temperature dependencesfor the nucleation and growth rates are assured (cf. thediscussions in the sections on Modes of nucleation and

Modes of growth). For small undercooling or over-heating, the nucleation and growth modes cannot bedescribed using an Arrhenius temperature dependencewith a constant activation energy (cf. for continuous

nucleation, equation (1) and for growth, equation (25)).Another limitation of the analytical expressions for f

is caused by the use of the temperature integral for

isochronal transformations (see Appendix). As noted inthe section on Extended volume and extended trans-formed fraction, this means that equation (31) is only avalid approximation for isochronal heating. The accu-racy of this approximation is determined by themagnitude of RT/Q (where Q can be either QG orQN), i.e. the first neglected term of the series expansion(see Appendix). Especially, for high values ofn, the errormade by this approximation can be significant eventhough RT/Q is small. To assess the total error made bythe approximation, a comparison with a direct numer-

ical calculation can be made (as described above in thesection on The numerical approach). Such a compar-ison shows that typically, the activation energies QN andQG are relatively insensitive to the error made by theapproximation but, for example, pre-exponential factorsof the nucleation rate, as N* and N0, can showconsiderable deviations from their true values.

Numerical v. analytical approachThe analytical formulations for xe as given by equa-tions (30) and (31) have some important advantagesover a direct numerical solution. The analytical solu-tions can show which parameters can be determinedindependently from a fit of the model to experimental

results (see also the section on Determination of modelparameters). The analytical equations are also easierand much faster to evaluate. The last point becomesimportant for a fit of the model to experimental results(see above).

On the other hand, the numerical evaluation of xe(equation (27)) allows to consider cases not amenable toanalytical treatment in general, as the case of smallundercooling or overheating for interface-controlledgrowth (i.e. v obeys equation (25)) and the case ofisochronal cooling for which the analytical approxima-tion given by equations (29) and (31) does not hold.

Table 4 Values of the model parameters used for the numerical calculations of isothermally and isochronallyconducted phase transformations for the case of only pre-existing nuclei, continuous nucleation (incorporatingnucleation index a) and mixed nucleation as nucleation mode, in combination with interface controlled growthas growth mode*

Parameters d/m N*, m23 N0, m23 s2a QN, kJ mol

21 QG, kJ mol21 n0, m s

21 a

Isothermal 3 0 161028 200 300 161010 13 561016 161028 200 300 161010 13 161018 161028 200 300 161010 13 161019 161028 200 300 161010 13 161019 0 200 300 161010 13 161019 161028 200 300 161010 2

Isochronal 3 0 161028 200 300 161010 13 161020 0 200 300 161010 1

3 16

10

18

16

10

28

200 300 16

10

10

13 161018 161028 200 300 161010 1.5

*cf. the section on Effect of model parameters and impingement and Figs. 4ad and 5ad.




12/20

Numerical evaluation also allows use of the full

nucleation rate equation (equation (1)). Thereby, the

range of different transformation mechanisms that can

be described with the modular transformation model is

increased considerably.

Effect of model parameters andimpingementTo illustrate the role of various kinetic model para-

meters, the dependences of the growth exponent n, the

effective, overall activation energy Q and the degree of

transformation f on transformation time/temperatureand the development of the transformation rate df/dt on

the degree of transformation have been illustrated for

the cases of isothermal and isochronal transformations

in Figs. 4ad and 5ad respectively. Here, mixed

nucleation (a>1), interface-controlled growth and var-

ious modes of impingement have been considered. Thevalues for the model parameters have been gathered in

Table 4.

Growth exponent and effective activationenergyFor both isothermal and isochronal transformations, ifN*50, continuous nucleation prevails and the values ofn (54) and Q (5(QNz3QG)/4) are constant throughoutthe transformation (see the horizontal bold dotted linesin Fig. 4a and b, and the open circles in Fig. 5a and b)and if N050, the only nuclei are the pre-existing nucleiand the values ofn (53) and Q (5QG) are also constantthroughout the transformation (see the horizontal boldfull lines in Fig. 4a and b, and the open triangles inFig. 5a and b). Upon increasing N*, the contribution ofthe pre-existing nuclei becomes of increasing importanceand thus, the value of n decreases and the value of Qincreases (see Figs. 4a, b 5a and b). n and Q change as afunction of transformation time (Fig. 4a and b) ortemperature (Fig. 5a and b): n changes from 3 to 4 andQ changes from QG to (QNz3QG)/4. Incorporation of

the nucleation index (a.1) leads to values of n and Qbeyond these ranges (see the dash-dot lines in Fig. 4aand b, and the solid lines in Fig. 5a and b).

4 a the growth exponent n and b the effective activation energy Q at T5800 K: solid line only pre-existing nuclei,

N*51610

19

m23

, N050 s21

m23

, dashed line pure continuous nucleation, N*50 m

23

, N05161028

s21

m23

, $ mixednucleation (a51), N*5561016 s21 m23, N051610

28 s21 m23, & mixed nucleation (a51), N*5161018 s21 m23,

N05161028 s21 m23, . mixed nucleation (a51), N*5161019 s21 m23, N051610

28 s21 m23, dotted line mixed

nucleation (a52), N*5161019 s22 m23, N05161028 s21 m23; c the transformed fraction f at T5800 K, for different

nucleation and impingement modes: random nucleation, anisotropic growth (j52) and non-random nucleation (e52),

as functions of time t, with N*5161018 m23 and N05161028 s2a m23; d the transformation rate df/dt, for mixed

nucleation with N*5161018 m23 and N05161028 s2a m23, at T5780 and 800 K, as a function of f




13/20

Transformed fraction and transformation rateThe effect of impingement on the transformationbehaviour is illustrated for both isothermal and iso-chronal annealing in Figs. 4c, d, 5c and d respectively.Clearly, for any nucleation mode considered, the time

(isothermal transformation) or temperature (isochronal

transformation) to attain the same fraction transformedis reduced with changing the mode of impingement from

equation (38) to (34) and then to equation (36).

For any kind of nucleation and growth modesconsidered, the height of the df/dt (df/dT51/W df/dt)peak maximum occurs at the same f value applying thesame impingement model and irrespective of the appliedannealing temperature or applied heating rate, moves to

larger f values with changing the mode of impingementfrom equation (34) to (36) (j52) and then to equa-tion (38) (e52) (Figs. 4d and 5d).

If impingement according to random nucleation (seeequation (34)) is taken as a reference, impingement in

the case of anisotropic growth (equations (36) and (37))retards the transformation (i.e. the height of the df/dtmaximum decreases) while the df/dt maximum moves to

smaller values of f with increasing j (Figs. 4d and 5d),

whereas impingement according to non-random nuclea-tion (equations (38) and (39)) accelerates the transfor-

mation (i.e. the height of the df/dt maximum increases)

while the df/dt maximum can appear at larger f values

with increasing e.

Determination of model parametersAccording to the modular model,23,24 for extreme

conditions such as pure continuous nucleation (a>1)

or where growth occurs only at pre-existing nuclei,equations (30) and (31) reduce to equations (28) and

(29): an analytical solution characterised by the follow-

ing set of model parameters: n0 (or D0), QN and QG (or

QD) together with either N* or N0 and a, with constant

values for n, Q and K0 (see Table 3). For these extreme

conditions, n, Q and K0 can be chosen as the fittingparameters.

For all mixed nucleation modes considered in thisoverview, n, Q and K0 are time or temperature dependent.

Accordingly, the parameters to be fitted aren0 (or D0), QN

5 a the growth exponent n and b the effective activation energy Q as a function of temperature T at different heating

rates as indicated: # pure continuous nucleation, N05161028 s21 m23, N*50 m23, e only pre-existing nuclei,

N050 s21 m23, N*5161020 m23, dashed line mixed nucleation (a51), N05161028 s21 m23, N05161018 m23, solidline mixed nucleation (a51?5), N051610

28 s21?5 m23, N*5161018 m23; c the transformed fraction f as a function of

temperature T for various impingement modes and for different nucleation modes, at W51 K s21: N*5161018 m23,

N05161028 s2a m23; d the transformation rate df/dT as a function of f for various impingement modes and for mixed

nucleation at heating rates as indicated: N*5161018 m23, N05161028 s2a m23




14/20

and QG (or QD) together with either N*, N0 and possibly

a (mixed nucleation with possibly a>1), or N9 and l0(Avrami nucleation), or N*, N9 and l0 (Avrami nuclea-

tion plus pre-existing nuclei) (see the sections on Modes

of nucleation and Modes of growth).

As follows from the equations gathered in Tables 13,

n0 (or D0) is not an independent parameter. In the case of

mixed nucleation (with a>1), it is multiplied with N0 and

N*

, in the case of Avrami nucleation withl0 or in the caseof Avrami nucleation plus pre-existing nuclei with l0, N*

and N9. This means that a change in n0 can be fully

compensated by a change in the other parameters.

Parameter determinationThe discussion above implies that individual values for

n0, N0 and N* (mixed nucleation with a>1), or n0 and l0(Avrami nucleation), or n0, l0, N* and N9 (Avrami

nucleation plus pre-existing nuclei), cannot be deter-

mined independently by fitting to experimental kinetic

data. Hence for a fit to kinetic data, one of these

parameters must be given an estimated, fixed value. In

the recipe given below, a fixed value for n0 will be used.

It must be realised that the value assigned to n0 directlyinfluences the absolute value of the other parameters.

For example, in the case of mixed nucleation (with a51),

ifn0 is taken as 106 m s21, but in reality is (found to be)

1010 m s21, then the values obtained for N0 and N* by

the fit procedure will be much too high (yet, because

both N0 and N* deviate with the same factor, the ratio

N0/N* is found correctly).

On this basis, the following procedure for determina-

tion of the model parameters by fitting to the experi-

mental data is appropriate:

(i) choose values for a and for n0 (or D0). Physically

reasonable initial values for these parameters can

be chosen, for example within the range of valuesgiven in Tables 4 and 5

(ii) fit the model to the experimental data which leads

to optimal values for QN and QG, and for N0 and

N* (mixed nucleation with a>1), or l0 (Avrami

nucleation), or l0, N* and N9 (Avrami nucleation

plus pre-existing nuclei).

If the optimal value for a is unknown, this procedure

should be repeated for different values of a.

With extra experimental information, it is possible toarrive at absolute values for the fit parameters. Thenucleation rate as described in the section on Modes ofnucleation is the nucleation rate for a hypothetical

infinite volume. The real nucleation rate is given by:

NR(t)~:

N(t)(1{f) (41)

i.e. the real nucleation rate decreases (with respect to:

N(t) as given in the section on Modes of nucleation)as the transformation progresses. Integrating equa-tion (41) with respect to time gives the real number ofnew particles (per unit volume) formed during thetransformation5

NR~

t0

:N(t)(1{f(t))dt (42)

Equation (42) can be evaluated numerically. To thisend, experimental or calculated (as the model has

already been fitted) f values can be used. Fittingequation (42) to the experimentally determined numberof product grains will give absolute values for N0 and N*(mixed nucleation with a>1), or l

0(Avrami nucleation),

or l0, N* and N9 (Avrami nucleation plus pre-existingnuclei), as their ratios (as N0/N* in the case of mixednucleation) are already known (see above).

Sensitivity to experimental errorsFor a correct interpretation of the values obtained forthe parameters of a fit of the modular model toexperimental data, it is important to know how different

experimental errors affect the fit results. Experimentalerrors are made in temperature and in transformedfraction measurements. These errors can be simulatedseparately and combined departing from ideal, calcu-

lated f-curves.

To investigate the effect of an error in the tem-perature measurement, four isothermal f-curves weregenerated with QN5200 kJ mol

21, QG5300 kJ mol21,

N051028 m23 s21, N*51018 m23, n0510

10 m s21 and

d/m53 (mixed nucleation, a51; see also Table 4 and

Fig. 4) for T5790, 800, 810 and 820 K. If a systematicerror of 2 K is made in the temperature, then a fit of themodel gives values for QN and QG within 1% of theoriginal, true QN and QG values with which the f-curves

Table 5 Kinetic parameters as determined by fitting the analytical phase transformation model to the isochronalcrystallisation of amorphous Pd40Cu30P20Ni10 after different pre-annealings for 600 s at different temperaturesTpre (Ref. 25)

Analytical model based on Tpre, K N*, m23 l0, s

21 N0, s21 m23 N9, m23

QN,kJ mol21

QG,kJ mol21 n

Q,kJ mol21 Error, %

Mixed nucleation and volumediffusion controlled growth

620 1.161019 4.261041 256 330 2.5 301 17622 1.361020 3.561041 255 325 14623 2.361020 4.161041 254 321 12625 6.161020 5.561041 255 315 11626 7.461020 7.461040 255 315 10628 2.261021 2.261041 250 315 12629 8.161021 2.161040 253 320 1.5 320 11

Avrami nucleation and volumediffusion controlled growth

620 4.661017 1.261024 252 313 2.5 299 17622 1.061018 8.161023 247 310 13623 9.861018 9.861023 259 309 14625 8.261019 1.361024 254 320 14626 5.061019 1.161022 249 319 15

628 8.4610

20

2.0610

24

256 315 15629 1.061021 1.061022 253 318 1.5 318 14

Analytical models based on interface-controlled growth and mixed nucleation or Avrami nucleation .100




15/20

were generated. The values found for N* and N0 alsoonly deviate minimally from their original values.However, if a statistical error of 1 K is imposed

(some temperatures are too high, others too low), then adirect fit ofQN, QG, N* and N0 is rather sensitive to theinitial values chosen for the fit parameters; different

initial values lead to rather different results for theparameters determined by fitting (QN and QG candeviate up to 30% of the original, true values) althoughthe goodness of fit is nearly equal (for the different sets

of final values for the parameters).

The influence of an error in the measurement of f hasbeen assessed for the same data set as above. An error infis simulated by shifting an entire curve 1% down or up.No matter whether all curves are shifted in the samedirection (systematic error) or whether some curves areshifted up and others down (statistical error), the fitresults are similar: QG, N0 and N* are close to theiroriginal, true values, but QN can deviate up to 35% fromits original value.

In the section on Recipes to determine the effectiveactivation energy and the growth exponent, recipes for

the direct determination of the effective activation energyQ and the growth exponent n from experimental data arepresented. Once both Q and n are known as a function off, QG and QN can be determined by fitting equation (33)to the (Q, n) data set. The values found for QG and QN bythis two-step procedure (applied to simulated data witherrors in T and/or f as described above) are closer (seebelow) to the original, true values, then as determinedby a direct fit (as above) of the model, here with thefour parameters QN, QG, N* and N0. Therefore, it isrecommended to first determine QG and QN (via Q and n)by the recipes described in the section on Recipes to

determine the effective activation energy and the growthexponent. Values for parameters as N* and N0 can thenbe found by fitting the model to the experimental

transformation curves with fixed QG and QN (firstdetermined by the two-step procedure). Applying thistwo-step fit procedure to the simulated data containingstatistical errors in both Tand f, QG and QN were found

to be within 15% of their original values ( N* and N0 candeviate up to a factor of 104 of their original, true values).

Applications of kinetic model fitting toreal transformations

Crystallisation of bulk amorphousPd40Ni10P20Cu30The kinetics of the crystallisation of amorphousPd40Ni10P20Cu30 was investigated on the basis of dif-ferential scanning calorimetry (DSC) measurements as

presented in Ref. 25. Isothermal pre-annealing in thesupercooled liquid temperature range has significantinfluence on the kinetics of the transformation (see Fig. 6).

The kinetic parameters were determined by fitting theanalytical modular phase transformation model to,simultaneously, all DSC runs recorded isochronallyemploying five different heating rates.

Of all nucleation, growth and impingement modesconsidered, it was found that mixed nucleation (a51) orAvrami nucleation, volume diffusion controlled growth

and impingement for randomly dispersed nuclei providethe best description of the isochronal crystallisationbehaviour of amorphous Pd40Cu30P20Ni10 (see Fig. 6).

A gradual change of nucleation mode can be realisedby increase of the pre-annealing performed before the

kinetic analysis. As indicated by the observed increase of

N*

upon pre-annealing (mixed nucleation with a51; cf.equation (18)) and by the increase of l0 upon pre-annealing (Avrami nucleation; cf. equation (15)), it

6 Rate of enthalpy change divided by the heating rate,

dDH/Wdt5dDH/dT, due to isochronal crystallisation of

amorphous Pd40Cu30P20Ni10, at the heating rates indi-

cated, as measured (symbols), and as fitted (lines), by

adopting mixed nucleation and volume diffusion con-

trolled growth, after pre-annealing for 600 s at 623, 626

and 629 K




16/20

follows that the character of the nucleation mode

changes from largely continuous nucleation to only

pre-existing nuclei for increasing pre-annealing. Also,

depending on the degree of pre-annealing performed, the

obtained value of n varies between d/m (3/2) pertaining

to only pre-existing nuclei and d/mz1 (5/2) pertaining to

continuous nucleation (see Tables 3 and 5).

In particular for the cases of intermediate pre-

annealing, n and Q are not constant during thetransformation (see Fig. 7). Values for QN and QD are

a direct result of the fitting of the kinetic model. They

can also be determined from the dependence of Q on n

by fitting of equation (33); this makes sense if Q and n

have been determined without full fitting of a kinetic

model (see the section on Recipes to determine the

effective activation energy and the growth exponent).

The values for QN and QD determined here by full fitting

of the kinetic model (see Table 5) indeed do not depend

on the degree of pre-annealing, as it should be.

Isochronal crystallisation of amorphous

Zr50Al10Ni40The crystallisation kinetics of the amorphousZr50Al10Ni40 alloy was measured by means of

isochronally conducted DSC scans. For details about

the experiments performed, see Ref. 37. The measured

heat release due to crystallisation is proportional to df/

dT, i.e.

dDH

dT~DHt

df

dT

with DHt as the total enthalpy of crystallisation. DHt(df/

dT) has been calculated on the basis of equations (31)

and (35) and has been fitted to the experimental DSCdata (dDH/dT).

Of all nucleation, growth and impingement modes

considered, it was found that the isochronal crystal-

lisation of Zr50Al10Ni40 can best be described by mixed

nucleation with a nucleation index a54?6, interface-

controlled growth and impingement according to ran-

dom nucleation (see Fig. 8a). The importance of

incorporating the nucleation index (a.1) in the

model is illustrated by Fig. 8: evidently, the results

obtained by fitting using an imposed value of a51

(Fig. 8b) give a bad fit to the experimental data. A value

of the nucleation index distinctly larger than one

indicates that the nucleation rate is substantiallyincreased during the transformation. This is reflected

in the occurrence of a relatively high transformation-rate

7 a the growth exponent n and b the overall effective

activation energy Q as a function of temperature for

isochronal annealing, at the heating rates indicated,

for the crystallisation of amorphous Pd40Cu30P20Ni10after pre-annealing for 600 s at 626 K

8 Rate of enthalpy change, dDH/W dt5dDH/dT, due to

isochronal crystallisation of amorphous Zr50Al10Ni40, at

the heating rates indicated, as measured (symbols)and as fitted (lines), by adopting mixed nucleation as

nucleation mode and interface-controlled growth as

growth mode




17/20

maximum, which in no way can be described by classicalnucleation modes.

Recipes to determine the effectiveactivation energy and the growthexponentWithout recourse to any specific kinetic model, it ispossible to determine values for an important kineticparameter as the overall activation energy Q fromexperimental data of the degree of transformation independence on time and temperature.7,35 Thereby, a

strong practical need is satisfied. Determination of thegrowth exponent n requires adoption of an impingementmodel.

In this section, simple recipes are given for determina-

tion of the time and/or temperature dependent growthexponent n and effective activation energy Q. Sub-sequently, the constant QN and QG values can be

determined from the dependence of Q on n usingequation (33).23,35

Effective activation energy Q(f)(i) Isothermal transformationWithout recourse to any kinetic model, values for theactivation energy can be obtained from the lengths oftime tf needed at different isothermal annealing tem-

peratures, to attain a certain fixed value of f. Withrespect to that given f value, the values obtained byplotting ln tf versus 1/T are usually approximated by a

straight line, from the slope of which a value for Q(f) isobtained.7 However, this procedure is only strictly validif the effective activation energy Q is constant during thetransformation.

In general, Q is a function of both transformationtime and temperature (cf. Tables 1 and 2). For iso-

thermal annealing, the ratio r2/r1 can be expressed asc9ta exp(2aQN/RT) (with c9 as a constant; cf. Tables 1and 2). From equations (35), (37) and (40), i.e. irres-pective of specific nucleation, growth and impingementmodes, it then follows

d ln tf

d 1=T ~

dm

QGz1

1z r2=r1 {1 QN

dmz a

1z r2=r1 {1

R~

Q f TR

with

Q f T~dm

QGz1a

n f T{dm

QN

n f T(43)

Plotting of ln tf versus 1/T will not yield a truly straightline. Thus, the value ofQ(f) depends on the temperaturewhere the slope of the plot of ln tfversus 1/Tis taken. Todetermine the slope in practice, several annealingtemperatures are required. The slope of the straightlines drawn through the data points of two of thesetemperatures can then be considered as an approxima-tion for Q(f) corresponding to a temperature betweenthese two annealing temperatures. Only for the limitingcases, r2/r1R0 or r2/r1R, the ln tf versus 1/T curvebecomes a truly straight line and Q then equals QG or[(d/m)QGzQN]/n, corresponding to n5d/m and n5d/mza respectively. The dependence of Q on f at the

temperature (range) considered can be determined byrepeating the above procedure for a chosen number of fvalues.

(ii) Isochronal transformation

Without recourse to any kinetic model, values for the

activation energy can be obtained upon isochronalannealing from the temperature Tf needed to attain a

certain fixed value off, as measured for different heating

rates. With respect to that given f value, the data points

in a plot of ln(Tf2/W) versus 1/Tf are usually approxi-

mated by a straight line, from the slope of which a value

for Q(f) is obtained.7

However, this procedure is strictlyvalid only if the effective activation energy Q is constant

during the transformation.

In general, Q is a function of both transformationtime and temperature (cf. Tables 1 and 2). From equa-

tions (35), (37) and (40), i.e. irrespective of specific

nucleation, growth and impingement models, it then

follows

d l n (T2f=W)

d(1=Tf)R~{

dm

QGz1

1z r2=r1 {1 QN

dmz 1za

21

1z r2=r1 {1

~{Q f W

with

Q f W~

dm QGz 21za n f W{ dm

QN

n f W

(44)

Plotting of ln(Tf2/W) versus 1/Tf will not yield a truly

straight line. Thus, the value of Q(f) depends on the

heating rate where the slope of the plot of ln(Tf2/W)

versus 1/Tf is taken. To determine the slope in practice,several heating rates are required. The slope of the

straight line drawn through the data points of two of

these heating rates can be considered as an approxima-

tion for Q(f) corresponding to a heating rate between

these two heating rates. Only for the limiting cases,

r2/r1R0 or r2/r1R, the ln(Tf2/W) versus 1/Tf curve

becomes a truly straight line and Q then equals QG or

(d=m)QGzQN=n, corresponding to n5d/m and n5d/mz(1za)/2 respectively. The dependence of Q on f at

different heating rates can be determined by repeating

the above procedure for a chosen number of f values.

For the case of a constant effective activation energy,

an equation like equation (44) was proposed originally

by Kissinger58 for a special case of reaction kinetics,

although instead of Tf, the temperature where the

transformation-rate maximum occurs was used. This

so-called Kissinger equation has been and is often used,

usually without justification. Mittemeijer7 was the first

to derive the general validity of the analysis provided

that the temperature of maximal transformation rate is

replaced by the temperature of the same degree oftransformation Tf.

Growth exponent n(f)(i) Isothermal transformation

Probably, the most popular method to determine n from

isothermal annealing data is based on the slope of the

straight line that should occur upon plotting

ln[2ln(12f)] as a function of lnt and thus6

d ln{ ln(1{f)

d ln t~n (45)

This result holds strictly only for classical JMA

kinetics,15

implying a constant value of n. In general,n is not constant (see the section on Effect of model

parameters and impingement).




18/20

In general, n is a function of both transformation time

and temperature (cf. Tables 1 and 2). The ratio r2/r1 can

be rewritten for isothermal transformation as c0ta (with

c0 as a constant; cf. Tables 1 and 2). From equa-

tions (35), (37) and (40), it then follows for the

corresponding modes of impingement

d ln { ln 1{f

d ln t~

d

mz

a

1z r2=r1 {1~n t T~n(f)T (46)

d ln1{f 1{j{1

j{1

d ln t~

d

mz

a

1z r2=r1 {1~n t T~n(f)T (47)

d ln H f

d ln t~

d

mz

a

1z r2=r1 {1~n t T~n(f)T (48)

with a51 for the Avrami nucleation and

Avrami nucleation plus pre-existing nuclei and a>1

for mixed nucleation. H(f) denotes the numerical

relationship between xe and f, expressed as

xe fiz1 ~ 1{fe

i {1 fiz1{fi zxe fi according toequation (39). For e52, an analytical form of equa-tion (48) results

d ln arctanh f

d ln t~

d

mz

a

1z r2=r1 {1~n t T~n(f)T(49)

The curves obtained by plotting ln[2ln(12f)] or

ln{[(12f)12j21]/(j21)} or ln[H(f)] or ln arctanh(f),

versus lnt, do not yield truly straight lines (r2/r15f(t)).

The slope at time t provides the value of n(t) at time t.

Only for the limiting cases, r2/r1R0 or r2/r1R, the

above curves become straight lines with their slopes

equal to d/m or d/mza respectively. Once the depen-

dence of n on t is known, the dependence of n on f can

be determined straightforwardly for each isothermalannealing experiment as the relationship between t and f

has been determined experimentally for each annealing

temperature.

(ii) Isochronal transformation

A value for the growth exponent can be obtained from the

transformed fraction fT attained at a certain fixed value of

T, as measured for different heating rates. With respect to

that given T value, the values obtained by plotting

ln[2ln(12fT)] versus ln W can be represented by a straight

line if the growth exponent n is constant during trans-

formation. The slope of this straight line then equals n

d ln{ ln 1{fT

d lnW~n (50)

This result holds strictly only for the extreme conditions

as indicated below equation (45).

In general, n is a function of both transformation time

and temperature (cf. Tables 1 and 2). For all the

nucleation and growth modes considered in the present

review, the value of the growth exponent n can be deter-

mined in correspondence with the prevailing impinge-

ment mechanisms (equations (35), (37) and (40)) according

to equations (35), (37) and (40) respectively

d ln{ ln 1{fT

d lnW~

d

mz

1za

2

1

1z r2=r1 {1

~n(T)W~n(f)W (51)

d ln1{fT

j{1{1

j{1

d lnW~

d

mz

1za

2

1

1z r2=r1 {1

~n(T)W~n(f)W (52)

d ln H fT

d lnW~

d

mz

1za

2

1

1z r2=r1 {1

~n(T)W~n(f)W (53)

For e52, an analytical form of equation (53) results

d ln arctanh f Td lnW

~d

mz

1za

2

1

1z r2=r1 {1

~n(T)W~n(f)W (54)

with a51 for the Avrami nucleation and Avraminucleation plus pre-existing nuclei and a>1 for

mixed nucleation. Plotting of ln[2ln(12fT)], orln{[(12fT)

12j21]/(j21)}, or ln[H(fT)] or ln arctanh(fT)

versus ln W does not yield a truly straight line (r2/r15f(W)). The value ofn(T)W depends on the heating rate

where the slopes of the above plots are taken. Severalheating rates are required to determine the slope in

practice. The slope of the straight line drawn throughthe data points of two of these heating rates can beconsidered as an approximation for n(T)W correspondingto a heating rate between these two heating rates. Only

for the limiting cases, r2/r1R0 or r2/r1R, the abovecurves become truly straight lines with their slopes equalto d/m or d/mz(1za)/2 respectively.

The dependence of n on T at a given heating rate(range) can be determined by repeating the aboveprocedure for a chosen number of T values. Once the

dependence of n on T is known, the dependence of n on

f, for the heating rate considered, can be determinedstraightforwardly as the relationship between T and fhas been determined experimentally for each heating

rate.

The recipes given here have been applied in analysingthe crystallisation of amorphous alloys.35,37 Once thevalues for n and Q have been determined as a function off, the constant values for QN and QG can be obtainedby fitting of equation (33) to the deduced (Q, n) datapoints.25

ConclusionsBy adopting a modular approach, nucleation, growthand impingement modes can be straightforwardlyincorporated in a quantitative description of phase

transformation kinetics. An efficient numerical analysisis possible by utilising the properties of the nesteddouble integral to be evaluated for determination of theextended transformed volume fraction (the section The

numerical approach).

The approach presented also provides a flexible,

analytical description of phase transformation kineticsduring both isothermal and isochronal anneals, i.e. alsofor cases where extreme kinetic conditions (continuousnucleation or growth only at pre-existing nuclei) cannot

be assumed. In particular, mixed nucleation, a mixture

of pre-existing nuclei and continuous nucleation(incorporating the nucleation index a), Avraminucleation, Avrami nucleation plus pre-existing nuclei,




19/20

interface-controlled growth and volume-diffusion con-

trolled growth and impingement for random and non-random nucleation and for anisotropic growth have

been dealt with.

No matter which of the nucleation and growth modes

govern the transformation, the extended volume fraction

of transformation xe can be given as (equations 30 and

31)

xe~K0 t n t

tn t exp { n t Q t RT

for

review analysis of solid state

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