# Mechanisms of nucleation

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Modelling

precermhe Gtherposefunefan

ase paa solidrties ofeactionials perntrol oolving.a solu

frequently applied type of modelling approach, the evolution of

dened by twoof energy due tohase into solute-e phase. This re-f chemical Gibbs, , dened by

in energy due to

represented by two different concepts: the energy barrier for

Contents lists available at ScienceDirect

els

CALPHAD: Computer Coupling of Phase Diagrams andThermochemistry

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 4958http://dx.doi.org/10.1016/j.calphad.2015.04.013

0364-5916/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author. Fax: 49 711 689 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,sharp interface, i.e. an interface with a width small compared to the size of theparticle will be considered.modynamics. Typical examples are models of KampmannWagn-er-numerical (KWN) type [2] (see e.g. [36] and Section 4): in this

In the rate equations for nucleation and growth as typicallyused in KWN-type kinetic models, this stability consideration iskinetics 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 numericalefciency of the kinetic model and the quality of its results aretherefore directly linked to the evaluation of the system's ther-

In terms of thermodynamics, formation and scipitate-phase particle are (in the simplest case)counteracting factors (see e.g. [7]): (i) The releasethe decomposition of the supersaturated matrix pdepleted matrix phase and solute-rich precipitatlease of energy can be described as a difference oenergies G xj jc ( ) of the (homogeneous) phases j =their respective compositions xj. (ii) The increasethe development of a particlematrix interface.1formed within an -phase matrix initially supersaturated in solute(s), leaving behind a solute(s)-depleted -phase matrix. The

modelling).tability of a pre-1. Introduction

The dispersion of small second-phphase matrix, e.g. as resulting fromaction, strongly inuences the propeIn materials science, precipitation rused as a method to enhance materelds of application [1]. Precise coallows to tailor the microstructure evthus to tune the material properties

Upon precipitation, particles ofrticles within a parent--state precipitation re-the two-phase system.s are therefore widelyformance in numerousf the reaction kineticsupon precipitation and

te(s)-rich phase are

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 efcient coupling ofkinetic model and thermodynamic description (for KWN-typeModelling precipitation kinetics: Evaluanucleation and growth

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

a Institute for Materials Science, University of Stuttgart, Heisenbergstrasse 3, 70569 Stub Max Planck Institute for Intelligent Systems (formerly Max Planck Institute for Metals

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:GibbsThomson effectThermodynamicsCALPHADPrecipitation kinetics

a b s t r a c t

Modelling of (solid-state)quires evaluation of the thnucleation barrier and of tand misconceptions of theto kinetic modelling is proand growth, based on thesystem. A computationallyis presented which allows

journal homepage: www.n of the thermodynamics of

rt, Germanyearch), Heisenbergstrasse 3, 70569 Stuttgart, Germany

ipitation kinetics in terms of particle nucleation and particle growth re-odynamic relations pertaining to these mechanisms, i.e. evaluation of theibbsThomson effect. In the present work, frequently occurring problemsmodynamic evaluation are identied and a practical approach with regardd for combined and unied analysis of the thermodynamics of nucleationdamental thermodynamic equilibrium consideration in a particlematrixcient method for numerical determination of the thermodynamic relationseasy and exible implementation into kinetic modelling.

& 2015 Elsevier Ltd. All rights reserved.

evier.com/locate/calphad

nucleation and the GibbsThomson 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 G * forformation of a particle of critical size rn above which the particle is

2). Analytical expressions for g x x,c ,m ,p ( ) , x r,int ( ) and x r,int ( ) ,based on simple thermodynamic solution models for the chemicalGibbs energies of the and the phase, are an often used, nu-

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 495850stable:

N GkTexp , 1

*( )

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

G * and rn are functions of the change in chemical Gibbs energyg x x,c ,m ,p ( ) upon nucleation (with g x x,c ,m ,p ( ) being the

chemical driving force for nucleation) for given compositionsx ,m and x ,p of the -phase matrix and the -phase precipitate,respectively, and of the interface energy per unit area, i.e.G G g x x, ,c ,m ,p * = *( ( ) ) and r r g x x, ,c ,m ,p * = *( ( ) ) , thus re-

ecting 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 AB is then often described by[12,13]

rt

x x

k x x

Dr

dd

,2

,m ,int

,int ,int=

( )

with the diffusion coefcient D of the solute component in thematrix and the atom fractions4 of solute x ,m in the -phase matrixremote from the particle, and x ,int and x ,int in the -phase matrixand in the -phase particle at the particlematrix interface,respectively; the factor k accounts for the difference in molarvolume of the phase and the phase. For x x,m ,int> , i.e. for apositive growth rate (considering precipitation of a solute-rich phase, k x x,int ,int is generally positive), the particle is stable andgrows; for x x,m ,int

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

Against the above background, in the present work a generally

g g 5c el + ( )

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 4958 51and with the nucleation barrier:

G G r rg g

43

163

.6

2 3

c el2

* = ( *) = * =( + ) ( )

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

(and thus constant gc for certain composition x ,p of the particle).The change in chemical Gibbs energy per unit volume of theparticle is dened as the difference g g gc c c = of the chemicalGibbs energy with gc

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

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( )

B. Rheingans, E.J. Mittemeijer / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 50 (2015) 495852 p and T, corresponds to equality of the (total8) chemical potentialsij for each component i in all phases of the system j, i.e.

. 8i i,m ,p = ( )

Elaboration of these equilibrium conditions for the system parti-clematrix in terms of size and shape of the second-phase particle,composition (eld), elastic strain (eld) 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:

(vi)

allowcomchem

and

Fig. 1. Chemical Gibbs energy curves G xc,mol ( ) and G xc,mol ( ) for the phase (matrix)respectively: (a) schematic representation; (b) chemical Gibbs energy curves for the syste(A: Cu, B: Co, : fcc (Cu), : fcc (Co)). The chemical driving force gc,mol assumes a maxof G xc,mol ( ) in x ,m , the given composition of the matrix (as shown here; parallel tanposition x ,p of the -phase particle.

Fig. 2. Chemical Gibbs energy surfaces G xc,mol ( ) and G xc,mol ( ) for the phase and

the phase in a ternary system ABC, re