COMPUTATIONAL MATERIALS ENGINEERING Episode 9 doc

Let us assume that there issome driving force for formation of clusters of pure B atoms.. If, by compositional fluctuations in the matrix,1a cluster of pure B atoms forms in the alloy at

• Solidification of liquids on cooling When a liquid starts to solidify, new crystals arepredominantly created on the walls of the liquid container Growth of the crystals occurs

in columnar or dendritic mode into the liquid core Only in the later stages of solidificationare crystals also nucleated within the liquid phase

• Precipitation of second phases in multicomponent solid or liquid matter Due to rium partitioning of individual components in the different phases of the alloy, the newphases grow with a chemical composition which is in general different from that of theparent phase Therefore, solid-state nucleation frequently involves the (long-range) trans-port of atoms by diffusion

equilib-In many cases, nucleation is not an easy process and does not happen without cost or effort.Usually, formation of a new phase needs some activation such that the classical nucleationbarrier is overcome This process is commonly treated in terms of probabilities, which makesnucleation a stochastic process Once the nucleus has reached overcritical size, it can grow in adeterministic manner Concepts to describe the nucleation and growth process are discussed inthe following sections

6.2.2 Macroscopic Treatment of Nucleation—Classical Nucleation Theory

Consider a homogeneous binary alloy with components A and B Let us assume that there issome driving force for formation of clusters of pure B atoms Let the initial configuration be ahomogeneous solution of B in A and let the Gibbs free energy of unit volume of atoms in thisconfiguration beGAB.

If, by compositional fluctuations in the matrix,1a cluster of pure B atoms forms in the alloy

at some arbitrary location, the Gibbs free energy of unit volume of this cluster can be defined

asGBB If it is further assumed that the reservoir of atoms in the initial configuration is suciently large such that the mean chemical composition of the atoms surrounding the cluster isunchanged by the nucleation process, the difference in bulk energy for unit volume of atomstransformed from the initial alloy into the cluster can be written as

When looking at a single cluster and assuming that the cluster has spherical shape with aradiusρ , the bulk energy difference∆Gbulkbetween the initial configuration and the configu-ration after the cluster has formed is

Since the cluster now has a distinct shape and chemical composition other than the position of the matrix, an interfacial area can be defined Generally, the atomic binding in theinterface between the atoms in the cluster and the atoms in the matrix is weaker than the bind-ing between the like atoms on both sides of the interface and, consequently, this new interfacialregion must be taken into account in the analysis of the cluster formation energy A detailedquantification of the binding energies across interfaces is given later in Section 6.2.5 on interfa-cial energies

com-The contribution∆Gsurfof the interfacial region to the total free energy of cluster formationcan be expressed in terms of the specific interfacial energyγand the geometrical surface areawith

1Although a solution can be homogeneous on a macroscopic level, that is, the solution contains no gradients in

concentration, there are always local, microscopical variations in chemical composition observable, which are

caused by the random walk of vacancies.

The total energy change due to formation of this cluster is then

∆G = ∆Gbulk+ ∆Gsurf= 4

3πρ3∆G0bulk+ 4πρ2γ (6.14)Equation (6.14) manifests the basic concept behind CNT, which treats the total free energychange during cluster formation as the the sum of a term∆Gbulk, which is proportional to thevolume of the new cluster, and a term∆Gsurf, which is proportional to the surface area createdduring nucleation Figure 6-4 displays these two terms as function of the cluster size togetherwith the total free energy change

According to equation (6.14) and Figure 6-4, the early stages of cluster formation arecharacterized by an increase in total free energy with increasing cluster size This means that,

FIGURE 6-4 Bulk and surface free energy of a spherical nucleus as a function of the number of

atoms N in the cluster (top) and the cluster radius R (bottom).

not until a critical cluster size is reached, energy must be invested for each individual atomthat attaches to the cluster Even though the like B atoms are attracted by each other, smallclusters of B atoms are energetically unfavorable and they are always more likely to dissolvethan to grow.

However, nucleation is a stochastic process and, with some probability, the random sitional fluctuations create clusters, which are large enough to grow Once, a cluster has reachedcritical size, addition of extra atoms is a process where energy is gained rather than spent andcluster growth becomes more likely than cluster dissolution At this point, the stochastic regime

compo-of precipitate nucleation switches over to the deterministic regime compo-of precipitate growth In afirst approach, the particular size where this transition occurs is given by the maximum of thenucleation free energy∆G The radius of a sphere containing exactly the number of atoms at this

point is called critical nucleation radius ρ ∗ and the value of the free energy at the maximum

is the nucleation barrier or the critical nucleation energy G ∗ The position of the maximum

nucleation free energy can be found by setting the derivative of equation (6.14) with respect to

ρto zero, that is,

= 2γ

In equation (6.16), an effective driving force D has been introduced with D = −∆G0bulk.Note thatDis closely related to the chemical driving forceDch, which has been introduced inequation (2.79) of Section 2.2.6 In the absence of mechanical stresses or other driving forces,the relationD = ΩDchholds, whereΩis the molar volume

Back substitution of equation (6.16) into (6.14) yields the critical nucleation energyG ∗with

G ∗= 163π γ3

For initiation of a single nucleation event, the critical nucleation energyG ∗must be

over-come and the probabilityPnuclthat this process occurs can be expressed as

Pnucl= exp



− G ∗ kT



(6.18)

wherekis the Boltzmann constant andT is the absolute temperature From the probability of

an individual nucleation event, the frequency of nucleation events in unit volume and unit timecan be deduced The respective quantityJ is denoted as the nucleation rate and it quantifies the

number of nuclei that are created in unit volume per unit time The unity ofJis[events/(m3s)].Under steady state conditions, the nucleation rateJSSis proportional to the probabilityPnucl

of a single nucleation event multiplied by the total number of possible nucleation sitesN0.

To obtain the exact expression for the steady state nucleation rate, further thermodynamic andkinetic aspects have to be taken into consideration These will not be eluciated here and theinterested reader is referred to, for example, the textbook by Khashchiev [Kha00] or the review

by Russell [Rus80]

The rigorous treatment of nucleation in the framework of CNT delivers that the steady statenucleation rate can be interpreted as the flux of clusters in cluster size space, which grow fromcritical to overcritical size and, in condensed systems,JSScan be written as

In equation (6.19), the additional quantitiesZandβ ∗have been introduced The Zeldovich

factorZis related to the fact that the critical size of a nucleus is not exactly given by the mum of the cluster formation energy An additional energy contribution from thermal activation

maxi-kT has to be taken into account because the thermal vibrations destabilize the nucleus as pared to the unactivated state.Z is often of the order of 1/40 to 1/100 and thus decreases theeffective nucleation rate

com-The atomic attachment rateβ ∗takes into account the long-range diffusive transport of atoms,which is necessary for nucleus formation if the chemical composition of matrix and precipitate

differs Quantitative expressions for these quantities are given in Section 6.2.4

6.2.3 Transient Nucleation

In the previous section, we have found that clusters are created by random compositional tuations and that the steady state nucleation rateJSS is determined by the flux of clusters incluster size space, which grow from critical to overcritical size In the derivation of the steadystate nucleation rate, it has been assumed—without explicitly mentioning it—that the distribu-tion of clusters is in a stationary state, namely, the size distribution of clusters is time invariant(see ref [Rus80]) This is rarely the case, however, in practical heat treatment situations at least

fluc-in the fluc-initial stages

Consider a homogeneous solution of B atoms in an A-rich matrix, which has been

homog-enized at a temperature above the solution limit of the B clusters.2 After quenching fromhomogenization temperature into a supersaturated state, the sharp cluster size distribution,which initially consists of mainly monomers and dimers, becomes wider, because larger clus-ters are stabilized by the increasing influence of favorable B–B bonding over thermally inducedmixing Only after a characteristic time, which is determined by factors such as atomic mobil-ity, driving force, and interfacial energy, a stable distribution of clusters can be established,

which is denoted as the equilibrium cluster distribution The characteristic period until the equilibrium cluster distribution is reached is denoted as the incubation time τ

It is interesting to note that a time-invariant cluster distribution can only exist when no drivingforce for cluster formation is present, that is, clusters are thermodynamically unstable If apositive driving force for precipitation exists, overcritical clusters will immediately grow in

a deterministic manner and will thus escape the stochastic distribution of clusters produced

by random compositional fluctuations Figure 6-5 schematically shows cluster distributions forsituations, where the largest clusters have undercritical or supercritical size, respectively Inthe first case, the cluster distributions are stationary and time invariant (equilibrium clusterdistribution) The shape of the distributions only depends on driving force and temperature

In the second case, precipitates are continuously nucleated and the shape of the size distributiondepends on time

2A homogeneous solution can be achieved by annealing for a sufficiently long time at a sufficiently high temperature above the solution temperature of the precipitate phase In this case, the vast majority of B atoms is present in the form of monomers and dimers and only a negligible number of larger clusters exists The supersaturated state is established by rapid quenching from homogenization temperature to reaction temperature.

FIGURE 6-5 Typical equilibrium cluster distributions without driving force for precipitation, (top)

and with driving force, (bottom) The top distributions are time invariant Under steady state tions, the bottom distributions will continuously create stable precipitates.

condi-When taking the incubation time τ for nucleation into account in the expression for thenucleation rate, a most pragmatic approach is to multiply the steady state nucleation rateJSS

by a smooth function, which is zero at timet= 0, and which approaches unity at timest > τ

In CNT, the traditional expression is

(6.21)

The transient nucleation rateJdescribes the rate at which nuclei are created per unit volumeand unit time taking into account the incubation timeτ It should be noted, finally, that the expo-nential function in equation (6.20) has received some criticism due to physical inconsistencies.Nonetheless, this approach is widely used because the error in practical calculation, which isintroduced by this weakness, is small compared to the uncertainties of other input quantities,such as the interfacial energy

6.2.4 Multicomponent Nucleation

The transient nucelation rate given in equation (6.21) can be rigorously derived for binary alloysystems However, already in ternary systems, the applied methodology becomes involved andtreatments of higher-order systems are more or less lacking In a first approximation, equation(6.21) can nevertheless be applied to multicomponent systems, provided that extended expres-sions for some of the quantities that appear in this relation are used

When investigating equation (6.21) closer, we find that some of the quantities are alreadyapplicable to multicomponent systems “as they are.” For instance, the number of potentialnucleation sitesN0is independent of the number of components and the Zeldovich factorZaswell as the critical nucleation energyG ∗already contain full multicomponent thermodynamic

information The critical quantity, which contains kinetic quantities describing multicomponent

diffusive fluxes, is the atomic attachment rateβ ∗ An approximate multicomponent expression

has been derived in ref [SFFK04] in the modeling of multicomponent multiphase precipitationkinetics based on the thermodynamic extremal principle The corresponding expression is pre-sented in Table 6-2 together with expressions for the other necessary quantities for evaluation

of multicomponent nucleation rates

Finally, an important note shall be placed on practical evaluation of multicomponent

nucle-ation rates It has not yet been emphasized that all quantities in Table 6-2 rely on the a priori knowledge of the chemical composition of the nucleus The term “a priori” means that we have

to input the nucleus composition in all formulas without really knowing what this compositionshould be

Luckily, there are some concepts that help us in making reasonable guesses of what a cessful” and realistic nucleus composition might be In a first step, it is assumed that the wide

“suc-variety of possible compositions can be substituted by a single characteristic composition This

step is rationalized by the reasoning that, from the variety of different possible chemical positions, precipitates which appear first and/or precipitates which appear in the highest numberdensity will be the most successful ones in the direct growth competition of an entire precipitatepopulation In this sense it should be sufficient to consider this most successful representativeprecipitate composition only

com-In the second step, a chemical composition is chosen for the representative nucleus, whichmost closely represents the situation under which the nucleus is formed For instance, in asystem with only substitutional elements, the composition which gives the highest chemical

TABLE 6-2 Expressions for Evaluation of Multicomponent Nucleation Kinetics Based on



−1 2πkT

2γ k F

τ

(s)

Incubationtime

1

2β ∗ Z2

driving force3 could be a reasonable choice because maximum driving forceD leads to (i)approximately maximum thermodynamic stability and often also to (ii) maximum nucleationrates (the nucleation barrierG ∗is minimum in the exponential term of the nucleation rateJ).

However, the second statement is not always true If substantial long-range diffusional port of atoms toward the nucleus is necessary to grow the nucleus, this process can be verycostly in terms of time Much higher nucleation rates and, thus, a higher nucleus density could

trans-be achieved with compositions, which are somewhere in trans-between the maximum driving forcecomposition and a composition with minimum necessary solute transport The parameter deter-mining the amount of necessary diffusive transport is the atomic attachment rateβ ∗(see Table

6-2) This quantity is a maximum, if the chemical composition of the nucleus is most closely thecomposition of the matrix, namely, minimum transport of atoms is necessary to form a nucleus

A typical example for this latter situation is given in the precipitation of carbides and nitrides

in steels, where the precipitates are composed of slow diffusing substitutional elements andfast diffusing interstitial elements, such as carbon and nitrogen Under specific conditions, thegrowth of carbides with a composition close to the matrix composition and only transport of fast

diffusing carbon and nitrogen is more favorable than forming precipitates with high namic stability and high content of carbide forming elements, but slow nucleation kinetics due toslow diffusion of these elements Figure 6-6 shows the theoretical nucleation rates for cementiteprecipitates (FeCr)3C in the ternary system Fe–3wt%Cr–C as evaluated with the CNT relationsgiven in Table 6-2 for varying Cr content of the cementite nuclei The different curves are related

thermody-3For practical evaluation of the composition with highest driving force, see Section 2.2.6

0 0 1e + 004

1e + 006 1e + 004

1e + 026 1e + 024 1e + 022 1e + 020 1e + 018 1e + 016

b c

c

d

d f

e

b c d e f

nucle-ation rate JSas a function of the Cr content of a cementite precipitate in the Fe–Cr–C system (from ref [KSF05b]).

to different carbon content of the supersaturated matrix, which is equivalent to different drivingforces for precipitation

The analysis demonstrates that, under situation of high supersaturation, the highest

nucle-ation rates are achieved for the so-called paraequilibrium composition, which is the particular

chemical composition where matrix and precipitate have the same amount of substitutional ments and only the amount of interstitial elements differs.4At the paraequilibrium composition,

ele-β ∗is a maximum, because only fast diffusing carbon atoms are needed to grow the precipitate,

4The term “paraequilibrum” composition is related to a specific type of constrained equilibrium, in which

equilibration of chemical potentials is only achieved for interstitial elements, whereas substitutional elements are not allowed to partition between the phases The term “orthoequilibrium” composition denotes the full, unconstrained thermodynamic equilibrium for substitutional and interstitial elements.

and the Fe to Cr ratio is identical in precipitate and matrix With decreasing supersaturation, thechemical driving force decreases and, at some point, the nucleation rate for paracompositioncementite goes to zero, whereas the driving force for higher-chromium nuclei is still sufficient

to support significant nucleation

Finally a note is dropped on practical evaluation of the optimum nucleus composition From

a physical point of view, the particular composition, which yields the highest nucleation rate,

is often a most reasonable choice However, computation of this composition is not alwayseasy because equation (6.19) must be scanned in the entire composition space In practicalsimulation, orthoequilibrium and paraequilibrium composition are popular choices due to thefact that they are often available from thermodynamic equilibrium calculation without additionalcomputational cost

6.2.5 Treatment of Interfacial Energies

In the previous sections, we have introduced the interfacial energyγas a convenient physicalquantity, which describes the energy of the narrow region between precipitate and matrix Inreality, however, this quantity is most delicate and, only in rare cases, reliable values ofγareknown One of the reasons for this is the fact that γ cannot be measured directly by experi-mental means Interfacial energies can only be obtained by indirect methods, that is by compar-ison of suitable experiments with the corresponding theoretical treatment, which includes theinterfacial energy as a parameter A most popular method in this respect is to compare exper-iments on phase transformation and precipitation kinetics to corresponding theoretical modelsand determine the interfacial energy by adjusting γ such that simulation and experiment are

in accordance Another problematic aspect in using interfacial energies in kinetic simulations

is the fact thatγ is (strongly) dependent on a number of parameters, such as crystallographicmisorientation, elastic misfit strains, degree of coherency, and solute segregation All this makesinterfacial energies a never-ending story of scientific interest, research, and also misconception

In this section, a popular approach is presented, which relates the interfacial energy of acoherent phase boundary to interatomic bonding and finally to the thermodynamic quantity

enthalpy This approach allows for an estimation of e ffective interfacial energies, which can be

used in computer simulations as first estimates.5

The Nearest-Neighbor Broken-Bond Model

The theoretical foundation for the first approach to calculating interfacial energies from eration of atomic bonding was laid by W L Bragg and E J Williams [BW34] in 1934 In thiswork, the concept of nearest-neighbor bonds was introduced and applied to estimate the totalenergy of a crystal based on the sum of binding energies of neighboring atoms This idea was

consid-shortly after (1938) applied by R Becker in his nearest-neighbour broken-bond model [Bec32].

Some years later, in 1955, D Turnbull [Tur55] made the connection between the interfacialenergy and the enthalpy of solution This concept is briefly reviewed now

Consider two blocks of material Block 1 is composed of pure A atoms, whereas block 2consists of pure B atoms Divide each block into two sections, and interchange the half blocks(see Figure 6-7) The energy of the newly formed interfaces in blocks 3 and 4 can be calculated

as the sum of the energies of the new bonds in blocks 3 and 4, minus the energy of the brokenbonds in the original blocks 1 and 2

5Although derivation of the following expressions is demonstrated rigorously only for coherent interfaces,

in many metallic systems, the values for γ obtained by this methodology can also be applied to incoherent

interfaces.

Block 1 Block 2 Block 3 Block 4

FIGURE 6-7 Calculation of interfacial energies.

Interface

FIGURE 6-8 Two-dimensional coherent interface with nearest-neighbor broken bonds.

According to this thought experiment, the specific interfacial energyγ is evaluated as the

difference in bond energies between the two separate blocks and the energy of the interchangedblocks per unit interfacial area Thus, we can write

γ = EnewAB − EAAbroken− EbrokenBB , (6.22)where the energyErefers to unit area of interface The energies in equation (6.22) are easilyobtained by counting the broken bonds in the blocks Figure 6-8 schematically illustrates brokenbonds across a two-dimensional interface

In a general crystal structure, letz S be the number of bonds across the interface counted

per atom and letn S be the number of surface atoms per unit area within the surface plane.Accordingly, we have

According to equation (6.24), the coherent interfacial energy is determined by the the tallographic neighborhood and the difference in bond energies between like and unlike atoms.Since AA, BB, and ABare not always readily available, these quantities can conveniently besubstituted by a quantity, which is nowadays easily accessible in the framework of computa-tional thermodynamics: the enthalpy of solution.

crys-Interatomic Bonding and Enthalpy

Recalling the formalism of the regular solution model of Section 2.2.3, the total bond energy of

one mole of pure A atoms and one mole of pure B atoms is

γ=n S z S

∆His defined as the change in enthalpy on dissolving one mole of substance B in A In theframework of computational thermodynamics, the solution enthalpy can easily be calculated for

a given configuration Letfbe the phase fraction of a precipitate phase The change in enthalpy

on transferring one mole of atoms from the matrix phase into the precipitate is∆H = ∂H/∂f

and the interfacial energy can finally be written as

6.3 Diffusion-Controlled Precipitate Growth

Precipitation is a phenomenon where atoms agglomerate to form clusters of a new phase Thedriving force for the process comes from favorable atomic bonding of the cluster atoms, and

it is expressed in terms of the supersaturation of the original solution, that is, an undercoolingbelow the solution temperature of the precipitate phase A typical heat treatment for controlledprecipitation consists of (i) solution annealing above solution temperature, (ii) rapid quenching

to precipitation temperature, and (iii) annealing for a given period of time The first step isperformed to homogenize the material and to evenly distribute the solute atoms In the thirdstep, the precipitate-forming atoms cluster and grow

Common types of precipitates in solid-state materials are carbides, nitrides, carbonitrides,and oxides as well as special precipitate-forming intermetallic phases If the chemical compo-sition of the matrix and the precipitate differs, the precipitation process involves the long-rangetransport of atoms by diffusion In many cases, precipitate and matrix also differ in the type ofcrystal structure, and precipitation needs simultaneous short-range lattice rearrangement How-ever, diffusion is often the slower process and, therefore, the rate-controlling factor If the growthrate of a precipitate is governed by the diffusive transport of atoms, the reaction is denoted as

being di ffusion controlled If the growth rate of a precipitate is determined by the rearrangement process of atoms across the interface, the reaction is said to be interface controlled.

When looking at typical life cycles of precipitates, or entire precipitate populations, a tional categorization is often made into three stages of precipitation, which are

tradi-• Nucleation: The initial stages in precipitate life Stochastic process driven by microscopic

thermal and compositional fluctuations

• Growth: Controlled attachment of atoms to the precipitate Deterministic process driven

by chemical and/or mechanical driving forces

• Coarsening: Dissolution of small precipitates in favor of larger ones Driven by

curvature-induced pressure, that is, the Gibbs–Thomson effect (compare Section 2.2.7)

Figure 6-9 shows a typical sequence of snapshots during the lifetime of a precipitate tion Assume that the open circles represent precipitate-forming atoms Figure 6-9(a) displays astate, where the hypothetical system is held at very high temperature, such that the majority ofatoms occurs as monomers Formation of larger clusters is prohibited by the strong thermal acti-vation of the atoms In Figure 6-9(b), the temperature of the system has been decreased and theattractive bond energies between the precipitate-forming atoms leads to formation of small butunstable clusters Nevertheless, the system is still above solution temperature of the precipitatephase and no larger clusters occur

popula-Figure 6-9(c) represents a situation, where a first stable nucleus of the precipitate phase hasformed The vicinity of the stable precipitate is depleted from solute atoms, because these have

diffused toward the stable cluster In the Figure 6-9(d), the simultaneous nucleation and growth

of precipitates has proceeded Temperature has been further decreased and strong clustering

of even smaller precipitates is observed At this stage, temperature is held constant and theprecipitation reaction is continued in isothermal regime

The Figures 6-9(e) and (f) show two snapshots from the last stage of precipitation, that

is, coarsening or Ostwald ripening Many of the smallest precipitates or clusters have alreadydissolved and the corresponding atoms have attached to the larger precipitates, which con-tinue to grow at the expense of the smaller ones The driving force for this process comesfrom the Gibbs–Thomson effect (compare Section 2.2.7), which says that the concentration ofsolute atoms around curved interfaces is higher the smaller the radius of the curved surface is