COMPUTATIONAL MATERIALS ENGINEERING Episode 9 doc

• Solidification of liquids on cooling. When a liquid starts to solidify, new crystals are predominantly 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 solidification are crystals also nucleated within the liquid phase. • Precipitation of second phases in multicomponent solid or liquid matter. Due to equilibrium partitioning of individual components in the different phases of the alloy, the new phases grow with a chemical composition which is in general different from that of the parent phase. Therefore, solid-state nucleation frequently involves the (long-range) transport of atoms by diffusion. 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 nucleation barrier is overcome. This process is commonly treated in terms of probabilities, which makes nucleation a stochastic process. Once the nucleus has reached overcritical size, it can grow in a deterministic manner. Concepts to describe the nucleation and growth process are discussed in the 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 is some driving force for formation of clusters of pure B atoms. Let the initial configuration be a homogeneous solution of B in A and let the Gibbs free energy of unit volume of atoms in this configuration be G AB . If, by compositional fluctuations in the matrix, 1 a 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 as G BB . If it is further assumed that the reservoir of atoms in the initial configuration is sufficiently large such that the mean chemical composition of the atoms surrounding the cluster is unchanged by the nucleation process, the difference in bulk energy for unit volume of atoms transformed from the initial alloy into the cluster can be written as ∆G AB→BB = G BB − G AB =∆G 0 bulk (6.11) When looking at a single cluster and assuming that the cluster has spherical shape with a radius ρ, the bulk energy difference ∆G bulk between the initial configuration and the configuration after the cluster has formed is ∆G bulk = 4 3 πρ 3 · ∆G 0 bulk (6.12) Since the cluster now has a distinct shape and chemical composition other than the composition of the matrix, an interfacial area can be defined. Generally, the atomic binding in the interface between the atoms in the cluster and the atoms in the matrix is weaker than the binding between the like atoms on both sides of the interface and, consequently, this new interfacial region must be taken into account in the analysis of the cluster formation energy. A detailed quantification of the binding energies across interfaces is given later in Section 6.2.5 on interfacial energies. The contribution ∆G surf of the interfacial region to the total free energy of cluster formation can be expressed in terms of the specific interfacial energy γ and the geometrical surface area with ∆G surf =4πρ 2 · γ (6.13) 1 Although 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. 186 COMPUTATIONAL MATERIALS ENGINEERING The total energy change due to formation of this cluster is then ∆G =∆G bulk +∆G surf = 4 3 πρ 3 ∆G 0 bulk +4πρ 2 γ (6.14) Equation (6.14) manifests the basic concept behind CNT, which treats the total free energy change during cluster formation as the the sum of a term ∆G bulk , which is proportional to the volume of the new cluster, and a term ∆G surf , which is proportional to the surface area created during nucleation. Figure 6-4 displays these two terms as function of the cluster size together with the total free energy change. According to equation (6.14) and Figure 6-4, the early stages of cluster formation are characterized by an increase in total free energy with increasing cluster size. This means that, 20 Number of Atoms in Cluster Cluster Energy 40 60 100800 Interface Bulk Total 0.2 Cluster Radius Cluster Energy 0.4 0.6 1.21.00.8 0 Interface Bulk Total 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). Modeling Precipitation as a Sharp-Interface Transformation 187 not until a critical cluster size is reached, energy must be invested for each individual atom that attaches to the cluster. Even though the like B atoms are attracted by each other, small clusters of B atoms are energetically unfavorable and they are always more likely to dissolve than to grow. However, nucleation is a stochastic process and, with some probability, the random compositional fluctuations create clusters, which are large enough to grow. Once, a cluster has reached critical size, addition of extra atoms is a process where energy is gained rather than spent and cluster growth becomes more likely than cluster dissolution. At this point, the stochastic regime of precipitate nucleation switches over to the deterministic regime of precipitate growth. In a first approach, the particular size where this transition occurs is given by the maximum of the nucleation 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 ρ to zero, that is, ∂∆G ∂ρ =4πρ 2 ∆G 0 bulk +8πργ =0 (6.15) The critical nucleation radius ρ ∗ follows with ρ ∗ = − 2γ ∆G 0 bulk = 2γ D (6.16) In equation (6.16), an effective driving force D has been introduced with D = −∆G 0 bulk . Note that D is closely related to the chemical driving force D ch , which has been introduced in equation (2.79) of Section 2.2.6. In the absence of mechanical stresses or other driving forces, the relation D =ΩD ch holds, where Ω is the molar volume. Back substitution of equation (6.16) into (6.14) yields the critical nucleation energy G ∗ with G ∗ = 16π 3 γ 3 D 2 (6.17) For initiation of a single nucleation event, the critical nucleation energy G ∗ must be overcome and the probability P nucl that this process occurs can be expressed as P nucl = exp  − G ∗ kT  (6.18) where k is the Boltzmann constant and T is the absolute temperature. From the probability of an individual nucleation event, the frequency of nucleation events in unit volume and unit time can be deduced. The respective quantity J is denoted as the nucleation rate and it quantifies the number of nuclei that are created in unit volume per unit time. The unity of J is [events/(m 3 s)]. Under steady state conditions, the nucleation rate J SS is proportional to the probability P nucl of a single nucleation event multiplied by the total number of possible nucleation sites N 0 . To obtain the exact expression for the steady state nucleation rate, further thermodynamic and kinetic aspects have to be taken into consideration. These will not be eluciated here and the interested reader is referred to, for example, the textbook by Khashchiev [Kha00] or the review by Russell [Rus80]. 188 COMPUTATIONAL MATERIALS ENGINEERING The rigorous treatment of nucleation in the framework of CNT delivers that the steady state nucleation rate can be interpreted as the flux of clusters in cluster size space, which grow from critical to overcritical size and, in condensed systems, J SS can be written as J SS = N 0 Zβ ∗ exp  − G ∗ kT  (6.19) In equation (6.19), the additional quantities Z and β ∗ have been introduced. The Zeldovich factor Z is related to the fact that the critical size of a nucleus is not exactly given by the maximum of the cluster formation energy. An additional energy contribution from thermal activation kT has to be taken into account because the thermal vibrations destabilize the nucleus as compared to the unactivated state. Z is often of the order of 1/40 to 1/100 and thus decreases the effective nucleation rate. The atomic attachment rate β ∗ 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 fluctuations and that the steady state nucleation rate J SS is determined by the flux of clusters in cluster size space, which grow from critical to overcritical size. In the derivation of the steady state nucleation rate, it has been assumed—without explicitly mentioning it—that the distribution 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 in the 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 from homogenization temperature into a supersaturated state, the sharp cluster size distribution, which initially consists of mainly monomers and dimers, becomes wider, because larger clusters are stabilized by the increasing influence of favorable B–B bonding over thermally induced mixing. 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 driving force for cluster formation is present, that is, clusters are thermodynamically unstable. If a positive 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 for situations, where the largest clusters have undercritical or supercritical size, respectively. In the first case, the cluster distributions are stationary and time invariant (equilibrium cluster distribution). 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 distribution depends on time. 2 A 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. Modeling Precipitation as a Sharp-Interface Transformation 189 Log(Cluster Size) Log(Cluster Density) No Supersaturation n * Log(Cluster Size) Log(Cluster Density) Stable Precipitate Growth n * 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 conditions, the bottom distributions will continuously create stable precipitates. When taking the incubation time τ for nucleation into account in the expression for the nucleation rate, a most pragmatic approach is to multiply the steady state nucleation rate J SS by a smooth function, which is zero at time t =0, and which approaches unity at times t>τ. In CNT, the traditional expression is 190 COMPUTATIONAL MATERIALS ENGINEERING J = J SS · exp  − τ t  (6.20) and we finally obtain J = N 0 Zβ ∗ exp  − G ∗ kT  exp  − τ t  (6.21) The transient nucleation rate J describes the rate at which nuclei are created per unit volume and unit time taking into account the incubation time τ. It should be noted, finally, that the exponential 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 is introduced 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 alloy systems. However, already in ternary systems, the applied methodology becomes involved and treatments 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 expressions 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 already applicable to multicomponent systems “as they are.” For instance, the number of potential nucleation sites N 0 is independent of the number of components and the Zeldovich factor Z as well as the critical nucleation energy G ∗ already contain full multicomponent thermodynamic information. The critical quantity, which contains kinetic quantities describing multicomponent diffusive fluxes, is the atomic attachment rate β ∗ . An approximate multicomponent expression has been derived in ref. [SFFK04] in the modeling of multicomponent multiphase precipitation kinetics based on the thermodynamic extremal principle. The corresponding expression is presented 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 nucleation 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 composition should be. Luckily, there are some concepts that help us in making reasonable guesses of what a “successful” and realistic nucleus composition might be. In a first step, it is assumed that the wide 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 compositions, precipitates which appear first and/or precipitates which appear in the highest number density will be the most successful ones in the direct growth competition of an entire precipitate population. In this sense it should be sufficient to consider this most successful representative precipitate composition only. In the second step, a chemical composition is chosen for the representative nucleus, which most closely represents the situation under which the nucleus is formed. For instance, in a system with only substitutional elements, the composition which gives the highest chemical Modeling Precipitation as a Sharp-Interface Transformation 191 TABLE 6-2 Expressions for Evaluation of Multicomponent Nucleation Kinetics Based on Equation (6.21) Quantity Value Comment Z (dim. less) Zeldovich factor  −1 2πkT ∂ 2 ∆G ∂n 2  1 2 n number of atoms in the nucleus β ∗ s −1 Atomic attachment rate 4πρ ∗2 a 4 Ω  n  i=1 (c ki − c 0i ) 2 c 0i D 0i  −1 ρ ∗ crit. nucl. radius a atomic distance Ω molar volume c i concentrations D 0i diffusion coeff. ∆G ∗ (J) Critical nucleation energy 16π 3 γ 3 k F 2 F effective driving force γ interfacial energy ρ ∗ (m) Critical nucleation radius 2γ k F τ (s) Incubation time 1 2β ∗ Z 2 driving force 3 could be a reasonable choice because maximum driving force D leads to (i) approximately maximum thermodynamic stability and often also to (ii) maximum nucleation rates (the nucleation barrier G ∗ is minimum in the exponential term of the nucleation rate J ). However, the second statement is not always true. If substantial long-range diffusional transport of atoms toward the nucleus is necessary to grow the nucleus, this process can be very costly in terms of time. Much higher nucleation rates and, thus, a higher nucleus density could be achieved with compositions, which are somewhere in between the maximum driving force composition and a composition with minimum necessary solute transport. The parameter deter- mining the amount of necessary diffusive transport is the atomic attachment rate β ∗ (see Table 6-2). This quantity is a maximum, if the chemical composition of the nucleus is most closely the composition 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 and fast diffusing interstitial elements, such as carbon and nitrogen. Under specific conditions, the growth 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 thermodynamic stability and high content of carbide forming elements, but slow nucleation kinetics due to slow diffusion of these elements. Figure 6-6 shows the theoretical nucleation rates for cementite precipitates (FeCr) 3 C in the ternary system Fe–3wt%Cr–C as evaluated with the CNT relations given in Table 6-2 for varying Cr content of the cementite nuclei. The different curves are related 3 For practical evaluation of the composition with highest driving force, see Section 2.2.6 192 COMPUTATIONAL MATERIALS ENGINEERING 0 0 1e + 004 Cr cem (Site-Fraction) Cr cem (Site-Fraction) dfm (J/mol) 5000 0.2 0.4 0.6 0.8 1 0 0.01 1 1e + 006 1e + 004 Beta* (s -1 ) 100 0.2 0.4 0.6 0.8 1 Cr cem (Site-Fraction) 0 1e + 026 1e + 024 1e + 022 1e + 020 1e + 018 1e + 016 J S (m -3 s -1 ) 0.2 0.4 0.6 0.8 1 a a a b b c c d d f f Matrix C-Content (wt-%): a 0.1 b 0.05 c 0.03 d 0.015 e 0.005 f 0.001 e e b c d e f FIGURE 6-6 Chemical driving force df m , atomic attachment rate beta ∗ , and steady state nucleation rate J S as 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 driving forces for precipitation. The analysis demonstrates that, under situation of high supersaturation, the highest nucleation 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 elements and only the amount of interstitial elements differs. 4 At the paraequilibrium composition, β ∗ is a maximum, because only fast diffusing carbon atoms are needed to grow the precipitate, 4 The 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. Modeling Precipitation as a Sharp-Interface Transformation 193 and the Fe to Cr ratio is identical in precipitate and matrix. With decreasing supersaturation, the chemical driving force decreases and, at some point, the nucleation rate for paracomposition cementite 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 always easy because equation (6.19) must be scanned in the entire composition space. In practical simulation, orthoequilibrium and paraequilibrium composition are popular choices due to the fact that they are often available from thermodynamic equilibrium calculation without additional computational cost. 6.2.5 Treatment of Interfacial Energies In the previous sections, we have introduced the interfacial energy γ as a convenient physical quantity, which describes the energy of the narrow region between precipitate and matrix. In reality, however, this quantity is most delicate and, only in rare cases, reliable values of γ are known. 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 the interfacial energy as a parameter. A most popular method in this respect is to compare experiments on phase transformation and precipitation kinetics to corresponding theoretical models and 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 that γ is (strongly) dependent on a number of parameters, such as crystallographic misorientation, elastic misfit strains, degree of coherency, and solute segregation. All this makes interfacial 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 a coherent phase boundary to interatomic bonding and finally to the thermodynamic quantity enthalpy. This approach allows for an estimation of effective 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 consideration of atomic bonding was laid by W. L. Bragg and E. J. Williams [BW34] in 1934. In this work, the concept of nearest-neighbor bonds was introduced and applied to estimate the total energy of a crystal based on the sum of binding energies of neighboring atoms. This idea was 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 interfacial energy 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 2 consists 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 broken bonds in the original blocks 1 and 2. 5 Although 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. 194 COMPUTATIONAL MATERIALS ENGINEERING Block 1 Block 2 Block 3 Block 4 AA A ABB B B FIGURE 6-7 Calculation of interfacial energies. Interface Matrix Precipitate FIGURE 6-8 Two-dimensional coherent interface with nearest-neighbor broken bonds. According to this thought experiment, the specific interfacial energy γ is evaluated as the difference in bond energies between the two separate blocks and the energy of the interchanged blocks per unit interfacial area. Thus, we can write γ = E new AB − E broken AA − E broken BB , (6.22) where the energy E refers to unit area of interface. The energies in equation (6.22) are easily obtained by counting the broken bonds in the blocks. Figure 6-8 schematically illustrates broken bonds across a two-dimensional interface. In a general crystal structure, let z S be the number of bonds across the interface counted per atom and let n S be the number of surface atoms per unit area within the surface plane. Accordingly, we have E broken AA = n S z S 2 ·  AA E broken BB = n S z S 2 ·  BB E new AB = n S z S ·  AB (6.23) The factor 1/2 for the like A–A and B–B bonds avoids counting bonds twice. This factor is missing in the expression for the AB bonds because we have two identical A–B interfaces (see Figure 6-7). For the interfacial energy we finally have γ = n S z S   AB − 1 2 ( AA +  BB )  (6.24) Modeling Precipitation as a Sharp-Interface Transformation 195 [...]... 0.0 0.1 0.3 0.5 0.7 0 .9 1.1 1.3 1.5 1.7 1 .9 Normalized Precipitate Radius 1.0 0.8 0.6 0.4 0.2 0.0 0.1 0.3 0.5 0.7 0 .9 1.1 1.3 1.5 1.7 1 .9 Normalized Precipitate Radius J N 20 C 15 Normalized Number Density -3 -1 Log(Nucleation Rate J [m s ]) Log(Number Density N [m-3]) 25 1 1 10 5 -4 -2 0 2 4 8 6 3.0 2.5 2.0 1.5 1.0 0.5 0.0 10 Log(Time[s]) 0.1 0.3 0.5 0.7 0 .9 1.1 1.3 1.5 1.7 1 .9 Normalized Precipitate... factor κ = zS /zL in fcc, bcc, and hcp structures is frequently in the order of 0.28 < κ < 0.33 With this approximation, equation (6. 29) can deliver reasonable first estimates of interfacial energies even for general matrix–precipitate interfaces 196 COMPUTATIONAL MATERIALS ENGINEERING 6.3 Diffusion-Controlled Precipitate Growth Precipitation is a phenomenon where atoms agglomerate to form clusters of a... (c0 − cαβ ) ∂ρ = ∂t ρ (cβ − cαβ ) (6. 49) Again, using the dimensionless supersaturation S , we finally arrive at S ∂ρ =D ∂t ρ ρρ = DS ˙ ρ= (6.50) (6.51) ρ2 + 2SDt 0 (6.52) ˙ where ρ is the position of the interface, that is, the radius of the precipitate and ρ = ∂ρ/∂t is the interface velocity, namely, the growth rate of the precipitate 204 COMPUTATIONAL MATERIALS ENGINEERING 6.3.4 Moving Boundary Solution... dissolution processes are described in a deterministic way Precipitate nucleation has already been dealt with in Section 6.2.2; modeling of the growth of precipitates is the topic of this section 198 COMPUTATIONAL MATERIALS ENGINEERING The third process, coarsening, can be treated explicitly in an analytical manner or implicitly by numerical integration of the evolution laws for precipitate growth and dissolution... Atoms cb v = ¶x/¶t Ab c0 A0 cab x Z Distance x FIGURE 6-12 Schematic drawing of real and linearized concentration profile of B atoms around the moving interface of a growing precipitate 202 COMPUTATIONAL MATERIALS ENGINEERING (cβ − cαβ ) ∂c ∂ξ = D (ξ) ∂t ∂x (6.34) Accordingly, the velocity of the planar interface ∂ξ/∂t can be directly evaluated if the concentration gradient of B atoms is known In Zener’s... precipitate–matrix interface, which causes the diffusional fluxes into and out of the interface In the following sections, three approaches will be presented that solve the problem with 200 COMPUTATIONAL MATERIALS ENGINEERING A Temperature a cb cab B a+b b Concentration of B FIGURE 6-11 Binary phase diagram for a hypothetical alloy A–B The alloy is solution treated in the α one-phase region at point... is reached This algorithm is commonly known as the numerical Kampmann–Wagner (NKW) model With this methodology, in every time interval, a new precipitate class is created (provided that 206 COMPUTATIONAL MATERIALS ENGINEERING For All Precipitates Next Time Step Pre-Proc.: Initialize and Set Up Parameters Nucleation? Add Precipitate Class Growth Evaluate Dissolution? Remove Prec Class Post-Proc.: Evaluate... Subsequently, three models for evaluation of the growth kinetics of individual spherical precipitates are briefly reviewed 6.3.2 Zener’s Approach for Planar Interfaces In 194 9, Clarence Zener proposed a simple and elegant approximate solution [Zen 49] to the diffusion-driven movement of planar, sharp interfaces The basic assumptions that he makes are that (i) the interfacial compositions are determined by local... asymptotic limit of coarsening, the precipitate distribution approaches a stationary state, where the shape of the normalized size distribution is time invariant In the representative 208 COMPUTATIONAL MATERIALS ENGINEERING size distribution for the coarsening regime (point D in Figure 6-14), the distribution obtained with the NKM model is compared to the theoretical distribution of the LSW theory... performed over all precipitates k Concentration of B Atoms cb v = ¶r/¶t Ab A0 c0 ca Jin ρ Distance r Z FIGURE 6-16 Mean-field concentrations around a single precipitate in the SFFK approach 210 COMPUTATIONAL MATERIALS ENGINEERING . approximation, equation (6. 29) can deliver reasonable first estimates of interfacial energies even for general matrix–precipitate interfaces. 196 COMPUTATIONAL MATERIALS ENGINEERING 6.3 Diffusion-Controlled. practical evaluation of the composition with highest driving force, see Section 2.2.6 192 COMPUTATIONAL MATERIALS ENGINEERING 0 0 1e + 004 Cr cem (Site-Fraction) Cr cem (Site-Fraction) dfm (J/mol) 5000 0.2. for γ obtained by this methodology can also be applied to incoherent interfaces. 194 COMPUTATIONAL MATERIALS ENGINEERING Block 1 Block 2 Block 3 Block 4 AA A ABB B B FIGURE 6-7 Calculation of