Mechanisms of nucleation

10
Modelling precipitation kinetics: Evaluation of the thermodynamics of nucleation and growth Bastian Rheingans a,n , Eric J. Mittemeijer a,b a Institute for Materials Science, University of Stuttgart, Heisenbergstrasse 3, 70569 Stuttgart, Germany b Max Planck Institute for Intelligent Systems (formerly Max Planck Institute for Metals Research), Heisenbergstrasse 3, 70569 Stuttgart, Germany article info Article history: Received 29 January 2015 Received in revised form 9 April 2015 Accepted 30 April 2015 Available online 4 May 2015 Keywords: GibbsThomson effect Thermodynamics CALPHAD Precipitation kinetics Modelling abstract Modelling of (solid-state) precipitation kinetics in terms of particle nucleation and particle growth re- quires evaluation of the thermodynamic relations pertaining to these mechanisms, i.e. evaluation of the nucleation barrier and of the GibbsThomson effect. In the present work, frequently occurring problems and misconceptions of the thermodynamic evaluation are identied and a practical approach with regard to kinetic modelling is proposed for combined and unied analysis of the thermodynamics of nucleation and growth, based on the fundamental thermodynamic equilibrium consideration in a particlematrix system. A computationally efcient method for numerical determination of the thermodynamic relations is presented which allows an easy and exible implementation into kinetic modelling. & 2015 Elsevier Ltd. All rights reserved. 1. Introduction The dispersion of small second-phase particles within a parent- phase matrix, e.g. as resulting from a solid-state precipitation re- action, strongly inuences the properties of the two-phase system. In materials science, precipitation reactions are therefore widely used as a method to enhance materials performance in numerous elds of application [1]. Precise control of the reaction kinetics allows to tailor the microstructure evolving upon precipitation and thus to tune the material properties. Upon precipitation, particles of a solute(s)-rich β phase are formed within an α-phase matrix initially supersaturated in solute (s), leaving behind a solute(s)-depleted α-phase matrix. The kinetics of the precipitation reaction, typically described in terms of nucleation and growth of precipitate particles, strongly vary with the degree of solute supersaturation, i.e., at constant tem- perature, with phase composition. In order to account for this ef- fect in a model for precipitation kinetics the kinetics must be coupled to the thermodynamics of the alloy system. The numerical efciency of the kinetic model and the quality of its results are therefore directly linked to the evaluation of the system's ther- modynamics. Typical examples are models of KampmannWagn- er-numerical (KWN) type [2] (see e.g. [36] and Section 4): in this frequently applied type of modelling approach, the evolution of the particle size distribution is computed on the basis of numerical integration of a composition-dependent nucleation rate and a size- and composition-dependent growth rate for discrete time steps and discrete particle-size classes. Such models thus require numerous evaluations of thermodynamic relations. Unfortunately, up to now the current corresponding modelling practice often involves usage of incompatible thermodynamic models for nu- cleation and growth and redundant thermodynamic evaluations (see below). The present work proposes a practical route for the thermodynamically correct and numerically efcient coupling of kinetic model and thermodynamic description (for KWN-type modelling). In terms of thermodynamics, formation and stability of a pre- cipitate-phase particle are (in the simplest case) dened by two counteracting factors (see e.g. [7]): (i) The release of energy due to the decomposition of the supersaturated matrix phase into solute- depleted matrix phase and solute-rich precipitate phase. This re- lease of energy can be described as a difference of chemical Gibbs energies G x j j c ( ) of the (homogeneous) phases j , =αβ, dened by their respective compositions x j . (ii) The increase in energy due to the development of a particlematrix interface. 1 In the rate equations for nucleation and growth as typically used in KWN-type kinetic models, this stability consideration is represented by two different concepts: the energy barrier for Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/calphad CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry http://dx.doi.org/10.1016/j.calphad.2015.04.013 0364-5916/& 2015 Elsevier Ltd. All rights reserved. n Corresponding author. Fax: þ49 711 689 3312. E-mail addresses: [email protected] (B. Rheingans), [email protected] (E.J. Mittemeijer). 1 Within the scope of this work, only the case of a, in the Gibbsian sense, sharpinterface, i.e. an interface with a width small compared to the size of the particle will be considered. CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 4958

description

Rheingans and MIttemeijer (2015)

Transcript of Mechanisms of nucleation

Page 1: Mechanisms of nucleation

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–58

Contents lists available at ScienceDirect

CALPHAD: Computer Coupling of Phase Diagrams andThermochemistry

http://d0364-59

n CorrE-m

e.j.mitte

journal homepage: www.elsevier.com/locate/calphad

Modelling precipitation kinetics: Evaluation of the thermodynamics ofnucleation and growth

Bastian Rheingans a,n, Eric J. Mittemeijer a,b

a Institute for Materials Science, University of Stuttgart, Heisenbergstrasse 3, 70569 Stuttgart, Germanyb Max Planck Institute for Intelligent Systems (formerly Max Planck Institute for Metals Research), Heisenbergstrasse 3, 70569 Stuttgart, Germany

a r t i c l e i n f o

Article history:Received 29 January 2015Received in revised form9 April 2015Accepted 30 April 2015Available online 4 May 2015

Keywords:Gibbs–Thomson effectThermodynamicsCALPHADPrecipitation kineticsModelling

x.doi.org/10.1016/j.calphad.2015.04.01316/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Fax: þ49 711 689 3312.ail addresses: [email protected] (B. [email protected] (E.J. Mittemeijer).

a b s t r a c t

Modelling of (solid-state) precipitation kinetics in terms of particle nucleation and particle growth re-quires evaluation of the thermodynamic relations pertaining to these mechanisms, i.e. evaluation of thenucleation barrier and of the Gibbs–Thomson effect. In the present work, frequently occurring problemsand misconceptions of the thermodynamic evaluation are identified and a practical approach with regardto kinetic modelling is proposed for combined and unified analysis of the thermodynamics of nucleationand growth, based on the fundamental thermodynamic equilibrium consideration in a particle–matrixsystem. A computationally efficient method for numerical determination of the thermodynamic relationsis presented which allows an easy and flexible implementation into kinetic modelling.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The dispersion of small second-phase particles within a parent-phase matrix, e.g. as resulting from a solid-state precipitation re-action, strongly influences the properties of the two-phase system.In materials science, precipitation reactions are therefore widelyused as a method to enhance materials performance in numerousfields of application [1]. Precise control of the reaction kineticsallows to tailor the microstructure evolving upon precipitation andthus to tune the material properties.

Upon precipitation, particles of a solute(s)-rich β phase areformed within an α-phase matrix initially supersaturated in solute(s), leaving behind a solute(s)-depleted α-phase matrix. Thekinetics of the precipitation reaction, typically described in termsof nucleation and growth of precipitate particles, strongly varywith the degree of solute supersaturation, i.e., at constant tem-perature, with phase composition. In order to account for this ef-fect in a model for precipitation kinetics the kinetics must becoupled to the thermodynamics of the alloy system. The numericalefficiency of the kinetic model and the quality of its results aretherefore directly linked to the evaluation of the system's ther-modynamics. Typical examples are models of Kampmann–Wagn-er-numerical (KWN) type [2] (see e.g. [3–6] and Section 4): in thisfrequently applied type of modelling approach, the evolution of

gans),

the particle size distribution is computed on the basis of numericalintegration of a composition-dependent nucleation rate and asize- and composition-dependent growth rate for discrete timesteps and discrete particle-size classes. Such models thus requirenumerous evaluations of thermodynamic relations. Unfortunately,up to now the current corresponding modelling practice ofteninvolves usage of incompatible thermodynamic models for nu-cleation and growth and redundant thermodynamic evaluations(see below). The present work proposes a practical route for thethermodynamically correct and numerically efficient coupling ofkinetic model and thermodynamic description (for KWN-typemodelling).

In terms of thermodynamics, formation and stability of a pre-cipitate-phase particle are (in the simplest case) defined by twocounteracting factors (see e.g. [7]): (i) The release of energy due tothe decomposition of the supersaturated matrix phase into solute-depleted matrix phase and solute-rich precipitate phase. This re-lease of energy can be described as a difference of chemical Gibbsenergies G xj j

c ( ) of the (homogeneous) phases j ,= α β, defined bytheir respective compositions xj. (ii) The increase in energy due tothe development of a particle–matrix interface.1

In the rate equations for nucleation and growth as typicallyused in KWN-type kinetic models, this stability consideration isrepresented by two different concepts: the energy barrier for

1 Within the scope of this work, only the case of a, in the Gibbsian sense,“sharp” interface, i.e. an interface with a width small compared to the size of theparticle will be considered.

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nucleation and the Gibbs–Thomson effect, affecting the growth(rate) of a particle. In the classical theory of nucleation [8,9], the

rate of nucleation N is dominated by an energy barrier GΔ * forformation of a particle of critical size rn above which the particle isstable:

⎛⎝⎜⎜

⎞⎠⎟⎟N

GkT

exp ,1

∝ − Δ *

( )

where k and T denote the Boltzmann constant and the absolutetemperature, respectively.2 For the case of a precipitation reaction,

GΔ * and rn are functions of the change in chemical Gibbs energyg x x,c

,m ,pΔ ( )α β upon nucleation (with g x x,c,m ,p−Δ ( )α β being the

chemical driving force for nucleation) for given compositionsx ,mα and x ,pβ of the α-phase matrix and the β-phase precipitate,respectively, and of the interface energy γ per unit area, i.e.

G G g x x, ,c,m ,p γΔ * = Δ *(Δ ( ) )α β and r r g x x, ,c

,m ,p γ* = *(Δ ( ) )α β , thus re-flecting the two competing energy contributions. Growth of asolute-rich particle leads to solute depletion of the surroundingmatrix; particle growth can then (in any case eventually) becomerate-controlled by solute diffusion through the solute-depletedmatrix towards the particle. The growth rate of a spherical particleof radius r in a binary3 system A–B is then often described by[12,13]

rt

x x

k x x

Dr

dd

,2

,m ,int

,int ,int= −

′ − ( )

α α

β α

with the diffusion coefficient D of the solute component in thematrix and the atom fractions4 of solute x ,mα in the α-phase matrixremote from the particle, and x ,intα and x ,intβ in the α-phase matrixand in the β-phase particle at the particle–matrix interface,respectively; the factor k′ accounts for the difference in molarvolume of the α phase and the β phase. For x x,m ,int>α α , i.e. for apositive growth rate (considering precipitation of a solute-rich βphase, k x x,int ,int′ −β α is generally positive), the particle is stable andgrows; for x x,m ,int<α α , the particle is unstable and shrinks. x ,intα

and x ,intβ are often taken according to local establishment ofthermodynamic equilibrium at the interface.5 For a small particlesize, i.e. for a large ratio of interface area to particle volume, thestate of equilibrium between the α-phase matrix and the β-phaseprecipitate can strongly deviate from the state of equilibriumbetween the α- and β-bulk phases, i.e. the α phase and the βphase in the absence of the interface. This is the so-called Gibbs–Thomson effect, which, in compliance with the two counteractingcontributions of composition-dependent chemical Gibbs energyand interface energy, can be expressed by functions x ,intα and x ,intβ

depending on particle size (i.e. interface area) and interface energyγ per unit area, i.e. x x r,,int ,int ( )γ=α α and x x r,,int ,int ( )γ=β β .

Kinetic modelling of nucleation and growth thus requiresevaluation of the thermodynamics of the system defined by che-mical energy and interface energy. Usually, the interface energy γper area is taken as being constant within a certain range ofcomposition, particle size and morphology. This assumption ef-fectively allows to reduce the evaluation of the thermodynamicsfor the kinetic modelling to determination of (i) the nucleationbarrier as a function of the composition-dependent chemicaldriving force for nucleation g x x,c

,m ,p−Δ ( )α β and (ii) the composi-

tions x r,int ( )α and x r,int ( )β as a function of particle size r (cf. Section

2 For a full expression of N according to classical nucleation theory, see e.g.[10].

3 For multinary systems, see e.g. [11].4 For binary systems, the convention x xj j

B= will be used.5 cf. footnote at the end of Section 2.3.

2). Analytical expressions for g x x,c,m ,pΔ ( )α β , x r,int ( )α and x r,int ( )β ,

based on simple thermodynamic solution models for the chemicalGibbs energies of the α and the β phase, are an often used, nu-merically efficient way to implement thermodynamic data into thenumerical kinetic modelling. For instance, the Gibbs–Thomsoneffect in a binary system is often accounted for by application ofthe equation [14]

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟x r x r

V

RT rexp

2 1

3,int molγ

( ) = → ∞( )

α αβ

for the composition of the matrix at the particle–matrix interface,where x r( → ∞)α is the solute concentration of the α phase in thereference state of equilibrium between the α phase and the βphase with r → ∞, i.e. between the bulk phases in the absence ofthe interface. Vmol

β is the mean molar volume of the β-phase and Rdenotes the gas constant. Eq. (3) is based on the assumption thatthe thermodynamic behaviour of the α-matrix phase can bedescribed with the regular solution model and that the β-pre-cipitate phase is a pure phase, i.e. x r x r 1,int ( )( ) = → ∞ =β β . The

applicability of such analytical expressions can thus be severelylimited by the limited capability of the underlying simple solutionmodels to adequately describe the actual thermodynamic beha-viour of the α phase and the β phase.

Hence, in recent years, direct numerical derivation of the re-lations g x x,c

,m ,pΔ ( )α β , x r,int ( )α and x r,int ( )β from a comprehensivethermodynamic assessment of the alloy system (which is typicallybased on more complex solid solution models for the chemicalGibbs energies) has become more frequently applied, especially formulti-component systems (see, e.g., [5,6]). This trend is facilitatedby the increasing availability of such thermodynamic assessments,e.g. in form of CALPHAD data, and commercial software for ther-modynamic analysis (e.g. [15]). On the one hand, the numericaldetermination of the chemical driving force for nucleation,

g x x,c,m ,p−Δ ( )α β , for a given composition x ,mα of the matrix phase,

can be performed straightforwardly, for instance by application ofthe parallel tangent/maximum chemical driving force approach[16] (see Section 2). On the other hand, the numerical evaluationof the Gibbs–Thomson effect, i.e. the determination of the com-positions x r,int ( )α and x r,int ( )β , is much more elaborate [17–19],since it requires the evaluation of a thermodynamic equilibriumstate including the energy contribution of the interface, e.g. byminimisation of the total Gibbs energy [15]. In view of the corre-spondingly larger complexity and computational effort, directnumerical evaluation of the Gibbs–Thomson effect is in practiceoften avoided and simple analytical expressions such as Eq. (3),based on generally invalid solid solution models, are adopted in-stead. Obviously, problems of inconsistency arise when the rela-tions g x x,c

,m ,pΔ ( )α β for nucleation and x r,int ( )α and x r,int ( )β forgrowth are derived, analytically or numerically, by adoption ofdiffering, incompatible thermodynamic solution models for nu-cleation and for growth. This is a common shortcoming in kineticmodels of precipitation kinetics based on the KWN-approach (e.g.[2,3]).6 The problem becomes even more aggravated when anelastic strain energy contribution due to a precipitate/matrix misfitis taken into consideration only for nucleation but not for growth(or vice versa) without more ado (e.g. [22,23]). As a consequence,the kinetic model predictions may be strongly biased or

6 Naturally, this problem does not appear in kinetic models without dis-crimination of nucleation kinetics and growth kinetics, as in cluster dynamicsmodels, cf. e.g. [20]. Also, in kinetic models involving a consideration of the total(Gibbs) energy of the system [21], such problems are more readily avoided.

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B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–58 51

completely corrupted by the incongruent thermodynamic de-scriptions for nucleation and for growth.7

Against the above background, in the present work a generallyvalid method for combined, inherently consistent, numericalevaluation of nucleation barrier and Gibbs–Thomson effect isproposed which is founded on the Gibbsian treatment of nuclea-tion and growth by fundamental consideration of thermodynamicequilibrium in a particle–matrix system [24,25]: starting from ty-pically applied approaches for separate thermodynamic evalua-tions of nucleation and growth, it is first shown how these ap-proaches can be retraced to the same evaluation of thermo-dynamic equilibrium. Introducing assumptions typically madeupon coupling thermodynamics to kinetics (e.g. a constant inter-face energy per unit area), a computationally efficient method fornumerical evaluation of nucleation barrier, critical radius andGibbs–Thomson effect is presented, which can be easily adapted topractical cases of modelling transformation kinetics.

2. Theoretical background

2.1. Nucleation barrier

Evaluation of nucleation thermodynamics, i.e. determination of

the nucleation barrier GΔ * and the critical radius rn, classicallydeparts from a consideration of the change of Gibbs energy G rΔ ( )of the system with size of the precipitate particle formed, i.e. incase of formation of a spherical particle with radius r

G r r r43

4 , 43 2π Φ π γΔ ( ) = × + × ( )

adopting the sharp interface model (see footnote in Section 1) with aconstant (i.e. independent of r and composition), isotropic interfaceenergy γ per unit area. Φ denotes the contributions to the Gibbsenergy change which scale with the volume of the particle. Forprecipitation in the solid state, two contributions to Φ (per unitvolume of particle) are typically taken into consideration: the (nega-tive) change in chemical Gibbs energy gcΔ upon formation of a particleand an additional (positive) contribution of strain energy gelΔ , arisingdue to elastic accommodation of the particle/matrix misfit (with

g g 0c elΦ = Δ + Δ < ). gcΔ and gelΔ are usually treated as constantsfor a givenmatrix composition. The Gibbs energy change then shows amaximum for G r rd /d 0Δ ( ) = for r equal to the critical radius:

rg g

2 2,

5c el

γΦ

γ* = − = −Δ + Δ ( )

and with the nucleation barrier:

G G r rg g

43

163

.6

2 3

c el2

π γ γΔ * = Δ ( *) = * =(Δ + Δ ) ( )

For evaluating the change gcΔ in chemical Gibbs energy uponparticle formation, it is assumed that particle formation occurswithin an infinitely large matrix phase (or that the particle isnegligibly small), implying a constant matrix composition x ,mα

(and thus constant gcΔ for certain composition x ,pβ of the particle).The change in chemical Gibbs energy per unit volume of theparticle is defined as the difference g g gc c cΔ = −β α of the chemicalGibbs energy with gc

β being the chemical Gibbs energy of theparticle-forming components in the β phase of composition x ,pβ

7 Such type of inconsistencies can in an extreme case lead to the unphysicalscenario that a particle of certain size generated by nucleation immediately ex-periences a negative growth rate due to x x 0,m ,int( − ) <α α (cf. Eq. (2)), i.e. it wouldbe instantaneously, intrinsically unstable.

and gcα being the chemical Gibbs energy of the same components

in the supersaturated α-phase matrix of composition x ,mα eval-uated at the composition x ,pβ of the nucleus (see Fig. 1, note that alldata presented for the binary case in Figs. 1(b) and 3 pertain to theexample given in Section 4).

Introducing the chemical potentials xij j

c, ( )μ for the components

i in the (bulk) phase j ,= α β, the chemical Gibbs energy change permol, g g gc,mol c,mol c,molΔ = −β α , can be expressed as

g g g x x

x x x x

,

.7ai

i ii

i i

c,mol c,mol c,mol,m ,p

,pc,

,p ,pc,

,m∑ ∑μ μ

− = Δ ( )

= ( ) − ( )( )

β α α β

β β β β α α

For the case of nucleation, it thus follows with Eq. (5):

⎛⎝⎜⎜

⎞⎠⎟⎟

g x x x x x x

rg V

,

2

7b

ii i

ii ic,mol

,m ,p ,pc,

,p ,pc,

,m

el mol

∑ ∑μ μ

γ

Δ ( ) = ( ) − ( )

= −*

+ Δ( )

α β β β β β α α

β

For a binary system, Eq. (7b) can be visualised in a diagram ofmolar chemical Gibbs energy G xj j

c,mol ( ) vs. composition xj (see

Fig. 1). g x x,c,mol,m ,pΔ ( )α β is represented by the difference of the

ordinate value of the tangent of the G xc,mol ( )α α -curve drawn at x ,mα

(the given composition of the matrix phase) and evaluated at x ,pβ

(the composition of the nucleus), and the G xc,mol ( )β β -curve at

x x ,p=β β . For a given matrix composition x ,mα , Eq. (7b) does notuniquely define g x x,c,mol

,m ,pΔ ( )α β for nucleation, since the compo-

sition of the nucleus x ,pβ is not known a priori. In principle, forgiven x ,mα , the composition x ,pβ of the precipitate nucleus mayassume any value for which g x x,c,mol

,m ,pΔ ( )α β is negative. If x ,pβ is

chosen such that the tangent of the G xc,mol ( )β β -curve drawn at x ,pβ is

parallel to the tangent of the G xc,mol ( )α α -curve (fixed by the givenmatrix composition x ,mα ), the chemical driving force

g x x,c,mol,m ,p−Δ ( )α β assumes a maximum value and is solely defined

by the composition of the matrix phase, i.e.g x x g xmax ,c,mol

,m ,pc,molmax ,m( )−Δ ( ) = − Δ ( )α β α and x x x,p ,p ,m( )=β β α . This

is the so-called maximum chemical driving force approach orparallel tangent approach [16,7], an originally graphical method todetermine the maximum chemical driving force for a binarysystem which can be straightforwardly generalised to the multin-ary case to numerically determine g xc,mol

max ,m−Δ ( )α and

x x x,p ,p ,m( )=β β α (see Fig. 2). Other methods for describing nuclea-

tion depart from setting x ,pβ equal to the composition of the βphase corresponding with equilibrium between the α- and β-bulkphases, thus inherently employing a lower chemical driving force;in the following, x ,pβ will always be identified with the composi-tion of the β phase pertaining to the maximum chemical drivingforce. Use of the maximum chemical driving force is justifiedrecognising the strong dependency of the nucleation rate N on

g x x,c,mol,m ,pΔ ( )α β (Eqs. (1) and (6)), allowing to neglect nucleation

of particles of composition different from that corresponding withthe maximum chemical driving force [10].

2.2. Gibbs–Thomson effect

Calculation of the growth rate requires evaluation of the Gibbs–Thomson effect, i.e. determination of the compositions x r,int ( )α andx r,int ( )β as a function of particle size (cf. Eq. (2)). This involvesanalysis of the equilibrium state for the β-phase particle em-bedded in the α-phase matrix, accounting for the contributions ofthe chemical energies, of the interface energy and of the elasticmisfit-strain energy. Thermodynamic equilibrium, here at constant

Page 4: Mechanisms of nucleation

Fig. 1. Chemical Gibbs energy curves G xc,mol ( )α α and G xc,mol ( )β β for the α phase (matrix) and the β phase (particle) in a binary system A–B as a function of composition,respectively: (a) schematic representation; (b) chemical Gibbs energy curves for the system Cu–Co at T 763 K= and p 105= Pa [36], employed for the case study in Section 4(A: Cu, B: Co, α: fcc (Cu), β: fcc (Co)). The chemical driving force gc,mol−Δ assumes a maximum value gc,mol

max−Δ if the tangent on the G x curvec,mol ( )−β β is parallel to the tangentof G xc,mol ( )α α in x ,mα , the given composition of the matrix (as shown here; parallel tangent/maximum driving force approach). This condition uniquely defines the com-position x ,pβ of the β-phase particle.

Fig. 2. Chemical Gibbs energy surfaces G xc,mol ( )α and G xc,mol ( )β for the α phase andthe β phase in a ternary system A–B–C, respectively. Here, the parallel tangents ofthe binary system correspond to parallel tangent planes (depicted in gray). Ap-plying the parallel tangent/maximum chemical driving force method yields

g xc,mol,mΔ ( )α (vertical red line) and the composition x x,p ,m( )β α of the β-phase par-

ticle (for better visualisation, the G x surfacec,mol ( )−β and the tangent planes wereplotted semi-transparent). (For interpretation of the references to colour in thisfigure caption, the reader is referred to the web version of this paper.)

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–5852

p and T, corresponds to equality of the (total8) chemical potentialsμi

j for each component i in all phases of the system j, i.e.

. 8i i,m ,pμ μ= ( )α β

Elaboration of these equilibrium conditions for the system parti-cle–matrix in terms of size and shape of the second-phase particle,composition (field), elastic strain (field) etc. can become extremelycumbersome due to the interdependence of these parameters.Therefore, without restricting the generality of the following dis-cussion, several simplifying assumptions and boundary conditionsare introduced here:

8 Derived from the total energy function, i.e. in the present case the total Gibbsenergy incorporating the contributions of chemical Gibbs energy, interface energyand elastic energy, see e.g. [26].

(i)

9

contrenerg

the matrix phase is taken infinitely large,

(ii) particle and matrix are separated by a sharp interface (see

above), with a size- and composition-independent, isotropicinterface energy per unit area γ,

(iii)

the elastic misfit-strain energy gelΔ per unit volume can bereasonably well described by assumption of phases of con-stant compositions and of isotropic elasticity; the presence ofthe interface has no effect on the state of stress (i.e. the effectof interface stress is neglected, cf. [27,28,25]),

(iv)

the particle is of spherical shape (a consequence of the elasticisotropy and the isotropy of γ; see (ii) and (iii) above),

(vi)

the molar volume of the precipitate phase is independent ofcomposition (i.e. the partial molar volumes V imol,

β of thecomponents of the β phase have the same value).

These assumptions, all more or less implicitly already introducedin the above discussion on the thermodynamics of nucleation,allow expression of the equilibrium condition in terms of phasecomposition and particle size on the basis of equality of the totalchemical potential of the components:

x x r x x r, , , , , 9i i,m ,m ,p ,p ,m ,pμ μ( ) = ( ) ( )α α β β α β

and also allow a separation of the contributions of interface en-ergy, elastic energy and chemical energy to the (total) chemicalpotential (cf. e.g. [7]), the latter then being the only contributiondepending on (variable) phase composition:

⎜ ⎟⎛⎝

⎞⎠x x

rg V

2,

10i i ic,,m

c,,p

el mol,μ μ γ( ) = ( ) + + Δ( )

α α β β β

with xij j

c,μ ( ) being the chemical potential of component i in thebulk phase j in the unstressed state, i.e. the chemical potentialpertaining to the composition-dependent chemical Gibbs energyonly (cf. Eq. (7)).9

Eq. (10) represents a system of equations (one equation foreach component i) which defines thermodynamic equilibrium forthe system particle–matrix. For given values of the interface en-ergy γ, the elastic strain energy gelΔ and the partial molar volume

V imol,β , Eq. (10) allows to derive expressions for the compositions

x r x r,m ,int( ) = ( )α α and x r x r,p ,int( ) = ( )β β as a function of particle ra-dius r employing suitable functions for the chemical potentials

xij j

c,μ ( ) (i.e. employing suitable thermodynamic solution models

For r → ∞, the elastic energy contribution, in contrast to the interface energyibution, does not vanish, since both the chemical Gibbs energy and the elasticy contribution scale with the volume of the particle.

Page 5: Mechanisms of nucleation

10 Eq. (13b) is only a necessary condition, while Eq. (10) is a sufficientcondition.

11 Eq. (7) results from a maximisation of the system energy change as a functionof the radius r. Combined with a maximisation of the chemical driving force as afunction of the composition xβ at given x ,mα , this corresponds, under the presentassumptions, to the determination of equilibrium as a function of r and xβ (thefunction G r g x x x, ,c,mol

,mΔ ( Δ ( = ))α α β shows a saddle-point at r r x x, ,p*= =β β ). In-deed, the maximum chemical driving force/parallel tangent method is derived froma consideration of thermodynamic equilibrium [16].

12 In case these assumptions do not hold, e.g. in the (very) early stages ofparticle growth before particle-growth kinetics become rate-controlled by long-range volume diffusion of solute only (i.e. for an, in comparison, low interfacemobility), an approach must be chosen which does not use this point of departure;see here, e.g., [21] and, more generally, [30] and [31].

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–58 53

G xc,mol ( )α α and G xc,mol ( )β β for the α and β phases).In case of binary systems A–B, closed, analytical expressions for

x r,int ( )α and x r,int ( )β can be obtained if very simple mathematicalexpressions hold for the chemical potentials xi

j jc,μ ( ), i.e. if very

simple solid solution models are applicable (cf. Eq. (3)). For ex-ample, adopting the ideal solution model for the α phase andassuming a pure β phase of the solute component B, the chemicalpotential xj j

c,Bμ ( ) is given by x RT xlnc,B c,B,0μ μ( ) = + ( )α α α α , and

xc,B c,B,0μ μ( ) =β β β , since x x 1,p= =β β (the corresponding chemical po-

tential c,Aμ β of component A in the β phase is not defined). Then,

RT x rlnc,B,0

c,B,0( ( ))μ μ+ → ∞ =α α β holds for equilibrium between the

bulk phases, i.e. in the absence of interface and misfit strain. Ap-plying Eq. (10) for the equilibrium between the α-phase matrixand the β-phase precipitate in this case then leads to a Gibbs–Thomson-type equation:

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

⎞⎠⎟⎟x r x r x r

rg

V

RTexp

2.

11

,m ,intel

mol,Bγ( ) = ( ) = ( → ∞) + Δ( )

α α αβ

As compared to Eq. (3), this expression includes an additionalcorrection for the elastic energy contribution. Similar analyticalexpressions for x r,int ( )α and x r,int ( )β , based on other (simplifying)assumptions, can be found in numerous works (see e.g. [29,4] forthe case of g 0elΔ = , and [28] including a consideration of elasticmisfit-strain and interface stress for the special case of ideal solidsolutions).

For describing the Gibbs–Thomson effect in the general case,i.e. without resorting to specific simple solution models for the αand β phases and without restriction to binary systems, Eq. (10)has in principle to be evaluated numerically to obtain the (nu-merical) relationships x r,int ( )α and x r,int ( )β , see e.g. [17,18] or [19].

2.3. The equivalence of nucleation thermodynamics and growththermodynamics

Now considering the thermodynamics of both nucleation andgrowth, as in case of kinetic modelling, separate analysis of nu-cleation thermodynamics (Eq. (7b)) and of growth thermo-dynamics (the Gibbs–Thomson effect, Eq. (10)) is unnecessary:both nucleation and growth thermodynamics are based on thesame thermodynamic stability consideration. In case of nucleation,

the critical size r r g x x x g, , ,c,mol,m ,p ,m

elγ* = (Δ ( ( )) Δ )α β α for stability ofa nucleus is determined, which, for given γ and gelΔ , is defined by adifference in chemical Gibbs energy (Eq. (7b)) and thus by thecompositions x ,mα and x x x,p ,p ,m= ( ))β β α of the α-matrix phase andthe β-precipitate phase, respectively. In case of growth, the com-

positions x r x r, ,,int ,m( ) ( )γ γ=α α of the α-phase matrix and

x r x r, ,,int ,p( ) ( )γ γ=β β of the β-phase precipitate particle, as defined

by a difference in chemical Gibbs energy (c.f. Eq. (10)), are eval-uated for a particle of certain size r stable in an α-phase matrix.Thermodynamic analysis in both cases thus breaks down to anevaluation of the equilibrium state for an α-phase matrix of certaincomposition containing a β-phase particle of certain compositionand certain size r, associated with a certain interface energy andelastic strain energy. As such, this is a consideration of thermo-dynamic equilibrium in a heterogeneous system with counter-acting contributions to the total energy scaling differently withsize of the second-phase domain (see [24], p. 252 ff., for a dis-cussion of particle formation/nucleation and stability in a one-component system ensuing from a consideration of thermo-dynamic equilibrium, there in terms of the variables size andpressure).

The equivalence of the nucleation and growth thermodynamics

as presented above may be elucidated in the following way: fromEqs. (9) and (10), the total molar Gibbs energyg x r x,i i imol

,p ,p ,p ,pμ= ∑ ( )β β β β of a precipitate particle with radius r and

composition x ,pβ is given by (cf. Eq. (9))

x r x x r x, ,12i i i

ii i

,p ,p ,p ,m ,m ,p∑ ∑μ μ( ) = ( )( )

β β β α α β

and (cf. Eq. (10))

⎜ ⎟⎡⎣⎢⎢

⎛⎝

⎞⎠

⎤⎦⎥⎥x

rg V x x x

2

13ai i i ii

i ic,,p

el mol,,p

c,,m ,p∑ ∑μ γ μ( ) + + Δ = ( )

( )β β β β α α β

⎡⎣⎢⎢

⎤⎦⎥⎥

⎛⎝⎜⎜

⎞⎠⎟⎟x x

rg V x x

2,

13bi i ii

i ic,,p ,p

el mol c,,m ,p∑ ∑μ γ μ( ) + + Δ = ( )

( )β β β β α α β

with V x Vi i imol,,p

mol∑ =β β β and x 1i i,p∑ =β . In a diagram of molar

chemical Gibbs energies vs. composition of a binary system, Eq.(13b), under the constraint of satisfying Eq. (10),10 for eachcomponent i, represents a construction of two parallel tangents,one of the G x curvec,mol ( )−α α and one of the G x curvec,mol ( )−β β , with a

vertical offset of r g V2 / el mol( )γ + Δ β , since this additional term to

each ic,μ β is independent of composition. Eq. (13b), under theconstraint of satisfying Eq. (10) for each component, thus isidentical with Eq. (7) derived specifically for nucleation, providedthe latter is evaluated according to the parallel tangent/maximumchemical driving force approach (see discussion below Eq. (7)).Hence, evaluation of nucleation thermodynamics by combinationof Eq. (7) with the maximum chemical driving force approachrepresents an evaluation of the thermodynamic equilibrium of thesystem particle–matrix as a function of the variables x ,mα , x ,pβ and rand thus comprises both nucleation thermodynamics and growththermodynamics.11

Irrespective of the simplifying assumptions introduced above,equivalence of nucleation thermodynamics and growth thermo-dynamics generally holds as long as nucleation is treated as oc-curring via a critical state for which the particle is in an unstableequilibrium with the matrix and growth is treated as involvinglocal establishment of thermodynamic equilibrium at the particle–matrix interface (see Section 1).12 Separate evaluation of nuclea-tion thermodynamics and growth thermodynamics is then gen-erally unnecessary, as will be illustrated in the next section.

Despite the fundamental nature of the equivalence of nuclea-tion thermodynamics and growth thermodynamics [24], it is fre-quently disregarded for analysis of the thermodynamics of nu-cleation and growth with respect to kinetic modelling (but, cf. e.g.,[27,28,16,7,4]) – a possible reason may lie in typical textbooktreatments of nucleation thermodynamics in terms of an energychange (cf. Eq. (4)), and of growth in terms of compositions only(instead of, e.g., activities; cf. Eq. (2)) [14]. Moreover, its implica-tions for thermodynamic evaluation upon kinetic modelling basedon a modelling approach as outlined in Section 1 have, to the

Page 6: Mechanisms of nucleation

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–5854

authors’ knowledge, up to now not been utilised at all.

Fig. 3. (a) Composition x x,p ,m( )β α of the β-phase particle and (b) maximum change inmolar chemical Gibbs energy g xc,mol

max ,mΔ ( )α , both as a function of the α-matrix com-position x ,mα , as derived from the chemical Gibbs energy curves of the α and β phases(see Fig. 1(b)) by application of the maximum chemical driving force method. Thecontribution of the elastic energy is represented by the dotted line in (b). For

g x 0c,molmax ,mΔ ( ) =α , i.e. zero chemical driving force, x ,mα and x ,pβ assume the composition

values for equilibrium between bulk phases in the absence of interface energy andelastic energy (indicated by the dashed vertical lines in (a), (b) and (c)). For

g x g 0c,mol,m

elΔ ( ) + Δ =α , x ,mα and x ,pβ assume the values for the so-called coherentequilibrium of the bulk phases (indicated by the solid vertical lines in (a), (b) and (c)).(c) The relation r g x g, ,c,mol

max ,melγ(Δ ( ) Δ )α as a function of the matrix composition x ,mα

(black curve), calculated by use of Eq. (5) (for γ and gelΔ see Section 4). For nucleation,this relation can be evaluated for given matrix composition x ,mα in order to obtain thecritical radius r r g x g, ,c,mol

max ,melγ= (Δ ( ) Δ )⁎ α . For growth, the same relation can be eval-

uated inversely for given radius r of the precipitate-phase particle in order to obtain thecorresponding equilibrium composition of the matrix x x r,int ,m= ( )α α as a function of

3. Usage of the common stability consideration upon numer-ical evaluation of nucleation and growth thermodynamics

Modelling of nucleation and growth kinetics requires numer-ous evaluations of the corresponding thermodynamic relations, i.e.nucleation barrier and Gibbs–Thomson effect (cf. Section 1). In thissituation, the equivalence of nucleation thermodynamics andgrowth thermodynamics as presented in the previous section al-lows, for the typically made assumption of a constant interfaceenergy γ and a constant elastic energy contribution gelΔ , to devisea convenient method for computationally efficient analysis ofthermodynamic data.

For nucleation, numerical evaluation of the chemical drivingforce can be performed straightforwardly by application of themaximum chemical driving force approach: for given matrixcomposition x ,mα , g xc,mol

max ,mΔ ( )α can be directly evaluated from the

chemical Gibbs energy functions G xc,mol ( )α and G xc,mol ( )β , for exampleby first calculating the tangent (plane) T α of G xc,mol ( )α in x ,mα , i.e.T x x x x, i i i

,mc,

,mμ( ) = ∑ ( )α α α α , and then finding the maximum of

G x T x x g x,c,mol,m

c,molmax ,m( ) − ( ) = − Δ ( )β α α α and the corresponding

composition x x x,p ,m= ( )β α of the β-phase particle (Figs. 3 (a,b) and4). Upon kinetic modelling, for given values of γ and gelΔ , the

critical radius r r g x g r x g, , , ,c,molmax ,m

el,m

elγ γ* = (Δ ( ) Δ ) = ( Δ )α α can becomputed using Eq. (5) (see Figs. 3(c) and 5). During the actual

computation process, the numerical relation r r x g, ,,melγ* = ( Δ )α

can then be accessed rapidly for certain x ,mα , e.g. via a table look-up.For growth, the equilibrium of the particle–matrix system (i.e.

the Gibbs–Thomson effect) has to be evaluated [17–19] for givenvalues of radius r, interface energy γ and elastic energy gelΔ basedon Eq. (10) (see Section 2; or in Eq. (13), see e.g. [18], the ad-ditionally required constraint (see below Eq. (13)) can then e.g. beobtained by introduction of the Gibbs–Duhem equation). Now,instead of either (numerically) evaluating Eq. (10) (see, e.g., [17–19]) or adopting some approximate description of the Gibbs–Thomson effect (as Eq. (11); see, e.g.[2,3,23]), the equivalence ofevaluating the thermodynamics of nucleation and of the treatmentof thermodynamic equilibrium of the particle–matrix system canbe utilised: values for the matrix compositionx x r g, ,,int ,m

elγ= ( Δ )α α as a function of the radius can simply be

obtained by inverse evaluation of the relation r r x g, ,,melγ* = ( Δ )α

as already determined for nucleation, e.g. via an inverse table look-up. The corresponding composition x ,intβ of the precipitate phasecan then be derived from the relation x x,p ,m( )β α (see Figs. 3(a),(c) and 5).With the thermodynamic relations for nucleation andfor growth thus based on the same thermodynamic analysis,(i) any inconsistencies of the thermodynamic descriptions fornucleation and growth are inherently excluded and (ii) separateevaluation of the Gibbs–Thomson effect [17–19] is renderedobsolete.13

In binary systems, for a given radius r, the inverse evaluationmethod provides a unique set of compositionsx x r g, ,,int ,m

elγ= ( Δ )α α and x x r g x x, ,,int ,pel

,p ,m( )γ= Δ = ( )β β β α (this

the particle radius. The composition x x r,int ,p= ( )β β of the β phase can then be derivedvia the relation x x,p ,m( )β α shown in (a). The gray curve in (c) represents the relationx r,int ( )α as derived from the classical Gibbs–Thomson equation (3), i.e. neglecting theelastic energy contribution gelΔ and using a highly simplified thermodynamic solutionmodel (note that when using the classical Gibbs–Thomson equation, x 1,p =β is as-sumed). For r → ∞, the composition of the matrix in the first case (exact treatment)approaches the composition for coherent equilibrium (i.e. the solid vertical line), in thesecond case (the approximate Gibbs–Thomson equation) the composition for in-coherent equilibrium (i.e. the dashed vertical line). All numerical results shown in thisfigure pertain to the example case considered in detail in Section 4.

13 The critical radius and the so-called no-growth radius rc, the radius forwhich x r,int ( )α equals the current composition x ,mα of the matrix and the growthrate (Eq. (2)) then equals zero, are then inherently identical (compare e.g. [22]).Arguably, the thermodynamics of nucleation and of growth may be different: forexample, after nucleation of a particle, the misfit strain generated by the particlecan be relaxed in the further course of its growth, e.g. via vacancy condensation orintroduction of misfit dislocations. The kinetics of such additional mechanismsshould then be explicitly introduced into the kinetic model.

Page 7: Mechanisms of nucleation

Fig. 4. Maximum change in chemical Gibbs energy g xc,molmax ,mΔ ( )α as a function of

the matrix concentration x ,mα for a ternary system A–B–C with phases α and β (asurface in this figure; cf. Fig. 2). The corresponding composition relation x x,p ,m( )β α

between the β-precipitate phase and the α-matrix phase is here represented as aplot of the surface g x xc,mol

,p ,mΔ ( ( ))β α . Each composition pair x x,,p ,m( )β α is related bya tie-line (the dashed red line represents the tie-line for the case depicted in Fig. 2).For g x 0c,mol

,mΔ ( ) =α , i.e. zero chemical driving force, the ternary equilibrium be-tween the α- and β-bulk phases is obtained (cf. Fig. 3). (For interpretation of thereferences to colour in this figure caption, the reader is referred to the web versionof this paper.)

Fig. 5. The relation r x g, ,,melγ( Δ )α as a function of the matrix composition x ,mα in a

ternary A–B–C system with phases α and β, calculated from the maximum changein chemical Gibbs energy g xc,mol

max ,mΔ ( )α as shown in Fig. 4 by use of Eq. (5), witharbitrarily chosen values for γ and gelΔ (for g 0elΔ = , this figure corresponds to theso-called “Gibbs–Thomson phase diagram” as presented in [18]). The correspond-ing composition relation x x,p ,m( )β α is represented by a plot of the surfacer x x g, ,,p ,m

elγ( ( ) Δ )β α . Corresponding relations pertaining to different values of γ andgelΔ can be easily re-obtained from the g xc,mol

max ,mΔ ( )α -data via Eq. (5). When con-sidering nucleation, the critical radius r r x g, ,,m

elγ= ( Δ )⁎ α and the composition ofthe nucleus x x,p ,m( )β α can directly be obtained from r x g, ,,m

elγ( Δ )α , equivalent tothe binary case (cf. Fig. 3). In case of growth, direct inverse evaluation ofr x g, ,,m

elγ( Δ )α in order to determine x r,int ( )α and x r,int ( )β is not possible (in contrastto the binary case, Fig. 3 (c)): For a given value of r, an infinite number of com-position pairs x x,,p ,m( )β α exists (equivalent to equilibrium between bulk phases, i.e.for r → ∞ and g 0elΔ = ). For modelling of particle growth in a multi-componentsystem under assumption of equilibrium at the particle–matrix interface, particleradius and as well as compositions must therefore be tracked. Size and composi-tions then follow coupled trajectories on the r x g, ,,m

elγ( Δ )−α andr x x g, ,,p ,m

elγ( ( ) Δ )β α -surfaces defined by the diffusion kinetics of the differentcomponents in the matrix phase.

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–58 55

corresponds to the existence of a single tie-line at given p and Trelating one pair of compositions x x,,m ,p( )α β , analogous to equili-brium between bulk phases in a binary system, cf. Fig. 3). Startingfrom a composition x x x,p ,m ,int( ) =β α β of the particle defined by thecomposition of the matrix x x,m ,int=α α at the moment of its nu-cleation, the compositions x ,intα and x ,intβ then evolve upon particlegrowth (see, e.g., Eq. (2)) according to x x r g, ,,int ,m

elγ= ( Δ )α α and

x x r g, ,,int ,pel( )γ= Δβ β .

In a multinary system, however, equilibrium is not uniquelydefined by the value of the particle radius r (the system of Eqs. (10)would then be an underdetermined system), i.e. the compositionof the phases cannot be determined directly via the inverse eva-luation method (this corresponds to the existence of multiple tie-lines at given p and T relating multiple pairs of compositionsx x,,m ,p( )α β , analogous to phase equilibrium between bulk phases ina multinary system; cf. Figs. 4 and 5). Starting from the, also for themultinary case, known compositions x x r g, ,,int ,m

elγ= ( Δ )α α and

x x r g x x, ,,int ,pel

,p ,m( )γ= Δ = ( )β β β α of matrix and particle at the

moment of nucleation, upon particle growth the generally differ-ing diffusivities of the components in the matrix can then lead to acontinuous change of the compositions in addition to the size ef-fect. Modelling of the growth kinetics in a multi-component sys-tem considering the Gibbs–Thomson effect therefore requirestreatment of the coupled evolutions of particle radius and of phasecompositions under the equilibrium constraint provided by Eq.(10) (see e.g. [11]). This constraint corresponds to subjecting thesize-composition trajectory of the particle to the relationsr x g, ,,m

elγ( Δ )α and x x,p ,m( )β α obtained via the inverse evaluationmethod (Fig. 5): particle size and the compositions at the interface

then follow a path, which is governed by the particle-growth ki-netics in the multi-component system [11], on the size-composi-tion surfaces determined by thermodynamics (i.e. the r x ,m( )−α andr x surfaces,p( )−β in Fig. 5; cf. [5,6]).

The equivalence of nucleation and growth thermodynamics inprinciple also allows application of a reverse procedure to obtainthe thermodynamic relations between particle size and phasecompositions: in a first step, the evaluation of the growth ther-modynamics via Eq. (10) could be performed for chosen values of rto determine x x r g, ,,int ,m

elγ= ( Δ )α α and x x r g, ,,int ,pelγ= ( Δ )β β , for

given interface energy γ and elastic energy contribution gelΔ . Then,

in a second step, the corresponding relation r r x g, ,,melγ* = ( Δ )α ,

required for description of the nucleation kinetics, could be ob-tained by inverse evaluation. However, upon assumption of con-stant γ and gelΔ , determination of the thermodynamic relations aspresented above is much more convenient that this reverse pro-cedure: evaluation of the thermodynamics of growth as a first stepwould require a complete evaluation of thermodynamic equili-brium via Eq. (10) for each value of γ, gelΔ and r [17–19]. The ap-proach via nucleation thermodynamics, by contrast, allows to split

Page 8: Mechanisms of nucleation

Table 1Data employed for simulation of the precipitation kinetics in a binary model systemA–B mimicking a Cu–0.5 at% Co alloy at T 763 K= . γ: interfacial energy; Q, D0: ac-tivation energy for diffusion, corresponding pre-exponential factor (used for bothnucleation and growth); Vmol: molar volume; E: Young's modulus (elastic isotropyapproximation), G: shear modulus.

Parameter Value Source, remark

γ 220 mJ m2 (assessed)

Q 214 kJ mol 1− [40]

D0 4.3 10 m s5 2 2× − [40]

A (fcc Cu)Vmol 7.3 10 m mol6 3 1× − − [37]

E 18.1 10 Pa10× [41], T 750 K=G 6.6 10 Pa10× [41], T 750 K=B (fcc Co)Vmol 6.8 10 m mol6 3 1× − − [37]

E 31.1 10 Pa10× [42], T 710 K=G 11.0 10 Pa10× [42], T 710 K=

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–5856

the thermodynamic evaluation into two independent steps: oncethe chemical driving force for nucleation g xc,mol

max ,m−Δ ( )α and the

relation x x,p ,m( )β α have been derived from the thermodynamicassessment of the system (determination of critical, equilibriumphase compositions), the numerically simple Eq. (5) can be easily(re-)evaluated for given values of γ and gelΔ to obtain the nu-merical relation r x g, ,,m

elγ( Δ )α pertaining to these values (de-termination of critical, equilibrium particle size). The methodproposed in this paper, with evaluation of the nucleation ther-modynamics as a first step, thus represents a potent and nu-merically efficient way of handling the numerical thermodynamicdata and allows for instance to use the interface energy γ as a fitparameter in a computational algorithm (see [32]) or to efficientlyincorporate a size dependency of the interface energy14 rγ γ= ( ).

The proposed treatment builds on the separability of the che-mical Gibbs energy contribution, the interface energy contributionand elastic energy contribution (Eqs. (9)→ (10)), which allowsapplication of the maximum chemical driving force method toindependently determine the chemical contribution. Accordingly,the presented approach breaks down as soon as interface energyand elastic energy (and the molar volume of the precipitate phase)show a composition dependency which cannot be neglected, thusprohibiting application of the maximum chemical driving forcemethod [16]. Analysis of the thermodynamics of nucleation andgrowth would then require determination of equilibrium on thebasis of a more general consideration of the total energy (cf. Eq.(8)). However, the fundamental equivalence of the thermo-dynamics of nucleation and of growth, as two aspects of the samefundamental thermodynamic equilibrium principle, still remainsintact and allows inverse evaluation of the thermodynamic rela-tions, thus avoiding inconsistent, unnecessarily separate thermo-dynamic evaluations of nucleation and growth.

4. Application example

To visualise the usage of the inverse evaluation method uponmodelling the kinetics of precipitation reactions, thus avoidinginconsistent thermodynamic descriptions for nucleation andgrowth, the precipitation kinetics were simulated for a binary A–Bmodel system (using for the kinetics a KWN-type approach asfrequently employed for practical applications; for details on thekinetic modelling, see [32]).

The properties of the A–B system were chosen to mimic thesystem Cu–Co, a classical model system for precipitation reactions(see e.g. [33–35]): upon annealing of supersaturated Cu-rich al-loys, spherical Co-rich precipitate particles of fcc structure areformed which are (initially) fully coherent with the fcc Cu-richmatrix phase. Thermodynamic data was taken from a recentCALPHAD assessment of the system Cu–Co [36] at T 763 K= (andp 10 Pa5= ): from the chemical Gibbs energy curves (Fig. 1(b)),employing the maximum chemical driving force method,

g xc,molmax ,m−Δ ( )α and the precipitate particle composition x x,p ,m( )β α

were derived as a function of the matrix composition x ,mα (Fig. 3).Due to the small lattice misfit between fcc Cu and fcc Co [37], theelastic energy contribution gelΔ is relatively small (but not negli-gible, cf. [38]): at the chosen alloy composition of Cu 0.5 at% Co,

gelΔ (here calculated using Eshelby's model [39]) amounts to about14 % of the initial maximum chemical driving force. Furtherparameters employed for kinetic modelling are given in Table 1.

The precipitation kinetics were simulated for two different

14 Note that in this case, Eq. (5) (and, correspondingly, also Eq. (10)) does nothold in the usual form as given in Section 2; it must be derived from Eq. (4) with

rγ γ= ( ).

scenarios: (i) Consistent scenario: use of the relation r x g, ,,melγ( Δ )α

as determined via Eq. (5) from the maximum chemical drivingforce g xc,mol

max ,m−Δ ( )α (Fig. 3(b), (c)) and of the relation x x,p ,m( )β α

(Fig. 3 (a)) for both nucleation and growth via the inverse eva-luation method (consistent thermodynamic model). (ii) Incon-sistent scenario: use of the relations r x g, ,,m

elγ( Δ )α and x x,p ,m( )β α

for particle nucleation only; description of the Gibbs–Thomsoneffect for growth, i.e. of x r,int ( )α , by use of Eq. (3). The compositionof the particle x r x r,int ,p( ) = ( )β β is then equal to 1, and the con-tribution of elastic energy is neglected for particle growth (in-consistent thermodynamic model).

The resulting precipitation kinetics according to both scenariosare shown in Fig. 6. Pronounced differences of the results for bothscenarios can be observed: only in the very early stages, the meanmatrix composition x t,m¯ ( )α , the mean radius r t¯ ( ), the critical radius

r t*( ) and the no-growth radius r tc ( ) (i.e. the particle radius forwhich x r x t,int ,m( ) = ¯ ( )α α and r td /d 0= , Eq. (2)), and the particlenumber density N(t) show a relatively similar evolution for bothcases. Note that for scenario (ii), due to employing an inconsistentthermodynamic model, the critical radius and the no-growth ra-dius are not identical: an existing particle of size equal to thecritical particle size, or even of somewhat smaller size, still ex-hibits a positive growth rate (see footnote in Section 1), whereas anucleating particle of critical size is in (unstable) equilibrium withthe matrix. Application of the classical Gibbs–Thomson equationin scenario (ii) thus results in an acceleration of the growth kineticsas compared to scenario (i). As a consequence, towards laterstages, the enhanced growth kinetics in scenario (ii) lead to amarkedly faster decrease of x t,m¯ ( )α and thus to a deceleration ofthe nucleation kinetics and a lower particle number density N(t).When the mean matrix composition x t,m¯ ( )α approaches the lim-iting composition of the so-called coherent equilibrium (i.e. ac-counting for the elastic energy contribution gelΔ , see Fig. 3), inscenario (i) the critical radius for nucleation rn and the no-growthradius rc (which are in this case identical) continuously increaseand eventually approach the mean particle radius r , correlatedwith the onset of distinct particle coarsening, i.e. dissolution of

particles with r r rc< = *. In scenario (ii), by contrast, x ,m¯α can fallbelow the matrix composition for coherent equilibrium, which isnow only limiting for particle nucleation. Consequently, the criticalradius for nucleation rn first strongly increases and is then nolonger defined when x ,m¯α falls below this composition value. Thekinetic model of inconsistently coupled nucleation and growththus degenerates to a model of particle growth (and coarsening)

Page 9: Mechanisms of nucleation

Fig. 6. Simulation of the precipitation kinetics in a binary A–B model system (Cu–0.5 at% Co, T 763 K= , p 10 Pa5= ) as a function of time for two different cases ofthermodynamic modelling of nucleation and growth (cf. Fig. 3 (c)): (i) consistentthermodynamic model based on the relations r x g, ,,m

el( )γ Δα and x x,p ,m( )β α for bothnucleation and growth (black curves); (ii) inconsistent thermodynamic model as-suming the classical Gibbs–Thomson equation (Eq. (3)) for growth (gray curves).(a) Evolution of the mean matrix composition x t,m¯ ( )α ; for case (i), x t,m¯ ( )α ap-proaches the composition of the matrix phase for coherent equilibrium, for case (ii)x t,m¯ ( )α approaches the composition for incoherent equilibrium (cf Fig. 3).(b) Evolution of the particle number density N(t). (c) Evolution of the mean radiusr t¯ ( ), critical radius r t( )⁎ and no-growth radius r tc ( ). For the consistent scenario (i),r⁎ and rc, both derived from the fundamental relation r x g, ,,m

el( )γ Δα , are identical.For the inconsistent scenario (ii), r t( )⁎

first strongly increases when x t,m¯ ( )α ap-proaches the matrix composition for coherent equilibrium and is then undefinedwhen x t,m¯ ( )α drops below this composition value. The model of initially combinednucleation and growth thus degenerates to a model of particle growth only.

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 49–58 57

only, with x ,m¯α slowly approaching the limiting matrix composi-tion for incoherent equilibrium.

Finally, above discussed discrepancies between the consistent

and the inconsistent scenario pertain to the real case Cu–Co, amodel system of only modest misfit strain development. Pro-nounced cases of large(r) misfit abound and can lead to muchlarger discrepancies.

5. Conclusions

Current practice of modelling precipitation kinetics frequentlyinvolves inconsistent and redundant evaluations of the ther-modynamics of nucleation and growth. As demonstrated in thiswork, inconsistent thermodynamic modelling (manifested by,e.g., the non-identity of rn and rc) results in inadvertent, arti-ficial enhancement or attenuation of either nucleation orgrowth, accompanied by a breakdown of the model of initiallycoupled nucleation and growth. It has been exemplary shownhere how typical approaches for separate evaluation of nu-cleation barrier and Gibbs–Thomson effect can in fact be re-traced to a single thermodynamic equilibrium considerationcommon to both mechanisms [24].

On this basis, a general approach for a single, consistent nu-merical evaluation of nucleation barrier and Gibbs–Thomsoneffect is proposed which (i) inherently excludes the corruptionof kinetic modelling by incongruent description of thermo-dynamics and (ii) renders separate evaluation of, e.g., theGibbs–Thomson effect obsolete.

For typical assumptions made upon kinetic modelling, a nu-merically efficient method for coupling thermodynamic data,as e.g. derived from a CALPHAD database, and kinetic model ispresented which reduces access to the thermodynamic data-base to a minimum and allows to easily incorporate a (com-position-independent) contribution of elastic energy for bothnucleation and growth, or a size-dependent interface energy.

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