The PF equations introduced in Section 1.15.2, that

is, eqns [2] and [3], are phenomenological, and one particular consequence is that they lack an absolute length scale. All scales observed in PF simulations are expressed in units of the interfacial width we of the appropriate field variable. As discussed in the previous section, for the case of one scalar conserved order parameter, this width we and the excess inter­facial free energy a are directly related to the gradi­ent energy coefficient к and the energy barrier between the two stable compositions Dfmax (see eqns [15] and [16]).

Beyond the difficulty of parameterizing к and Afmax to accurately reflect the properties of a given alloy system, the phenomenological nature of these coefficients creates additional problems. In particular, as the number of mesh points used in a simulation increases, the interfacial width, expressed in units of mesh point spacing, remains constant if no other parameter is changed. Increasing the number of mesh points thus increases the physical volume that is simulated but does not increase the spatial resolution of the simulations. If the intent is to increase the spatial resolution, one would have to increase к so that the equilibrium interface is spread over more mesh points. Equilibrium interfacial widths in alloy systems typically range from a few nanometers at high temperatures to a few angstroms at low tem­peratures. In the latter case, if the interface is spread over several mesh points, it implies that the volume assigned to each mesh point may not even contain one atom. This raises fundamental questions about the physical meaning of the continuous field vari­ables, and practical questions about the merits of PF modeling over atomistic simulations.

 

Another important problem related to the lack of absolute length scale in conventional PF modeling concerns the treatment of fluctuations. Fluctuations arise owing to the discrete nature of the microscopic (atomistic) models underlying PFMs. Furthermore, fluctuations are necessary for a microstructure to escape a metastable state and evolve toward its global equilibrium state, such as during nucleation. Fluctua­tions, or numerical noise, will also determine the initial kinetic path of a system prepared in an unstable state. The standard approach for adding fluctuations to the PF kinetic equations is to transform them into Langevin equations, and then to use the fluctuation- dissipation theorem to determine the structure and amplitude of these fluctuations. For instance, in the case of one conserved order parameter, the Cahn-Hilliard diffusion equation, that is, eqn [2], is transformed into the Langevin equation:

 

dC(r, t) dt

 

V

 

x(rF)

 

[19]

 

where X(r, t) is a thermal noise term. The structure of the noise term can be derived using fluctuation — dissipation105,106:

 

(x(r;t )) = 0 (X(r, t)X(r’/)) = -2кв TMV2d(r — r’)8(t — t’) [20;

 

where the brackets () indicate statistical averaging over an ensemble of equivalent systems. However, eqn [20] does not include a dependence of the noise amplitude with the cell size, which is not physical. Even if this dependence is added a posteriori, it is observed practically that this noise amplitude gives rise to unphysical evolution, as reported by Dobretsov eta/.107 While these authors have proposed an empirical solution to this problem by filtering out the short-length-scale noise in the calculation of the chemical potentials, a physically sound treatment of fluctuations requires a derivation of the PF equations starting from a discrete description. Recently, Bronchart et a/.100 have clearly demon­strated how to rigorously derive the PF equations from a microscopic model through a series of con­trolled approximations. We outline here the main steps of this derivation. The interested reader is referred to Bronchart et a/.100 for the full derivation. These authors consider the case of a binary alloy system in which atoms migrate by exchanging their position with atoms that are first nearest neighbors on a simple cubic lattice. A microscopic configuration is defined by the ensemble of occupation variables, or

spin values, for all lattice sites, C = {s}, where S = ±1 when the site i is occupied by an A or a B atom, respectively. The evolution of the probability distribution of the microscopic states is given by the following microscopic Master Equation (ME):

@PC: = -£ W (C! CiJ )P(C)

i; J

+ W(CiJ! C)P(CiJ) [21]

where the * symbol in the summation indicates that it is restricted to microscopic states that are connected to C through one exchange of the i and J nearest neighbor atoms, resulting in the configuration CiJ. The next step is to coarse-grain the atomic lattice into cells, each cell containing Nd lattice sites. It is then assumed that local equilibrium within the cells is achieved much faster than evolution across cells. The composition of the cell n, cn, is given by the average occupation of its lattice site by B atoms, and thus cn = 0, 1/Nj,, Nj/Nj. A mesoscopic config­uration is fully defined on this coarse-grained system by C = {cn}. A chemical potential can be defined within each cell and, if this chemical potential varies smoothly from cell to cell, the microscopic ME, eqn [21], can be coarse-grained into a mesoscopic ME: 2 *

lmn(C)exp

n, m

+ gain term

where a is the lattice parameter and d the number of lattice planes per cell (i. e., Nd = (d/a)3), в is the attempt frequency of atom exchanges, lmn(C) is a mobility function that is directly related to the microscopic jump frequency, b = (B T) 1, and mn(C) is the chemical potential in cell n. The * symbol over the summation sign indicates that the summation over m is only performed over cells that are adjacent to the cell n; the first term on the right — hand side of eqn [22] represents a loss term, and there is a similar gain term, which is not detailed.

The mesoscopic ME eqn [22] can be expanded to the second order using 1/Nd as the small parameter for the expansion. The resulting Fokker-Planck equa­tion is then transformed into a Langevin equation for the evolution of the composition in each cell n:

dc a2 в (n) ~ ~ ~

~@t = d2 к T^ ^l”m(C)[mm(C) — mm(C)] + Cn(t) [23]

B m

where the noise term Zn (t) is a Gaussian noise with first and second moments given by

<Cn(0> = 0

2 a2 (n)

<Cn(t)Cn(t’)> = — J2 lnf (C )8(t — t0)

d p 2 a2 ~

<Cn(t)Zm(t0)> = —- ^2 lnm(C )d(t — О [24]

Nd d2

While the structure of eqns [23] and [24] is quite similar to that of the phenomenological eqns [19] and [20], there are several key differences in these two descriptions. First, thermodynamic quantities such as the homogeneous free-energy density and the gradient energy coefficient are now cell-size dependent. These quantities can be evaluated sep­arately using standard Monte Carlo techniques.1 Second, the mobility coefficients, and thus the correlations in the Langevin noise, are func­tions of the local concentration, as well as of the cell size.

Bronchart eta/.100 applied their model to the study of nucleation and growth in a cubic A1-cBc system for various cell sizes, d = 6a, d = 8a, and d = 10a. The supersaturation is chosen to be small so that the critical nucleus size is large enough to be resolved by these cell sizes. As seen in Figure 4, for a given supersaturation, the evolution of the volume fraction of precipitates is independent of the cell size and in very good agreement with fully atomistic kinetic Monte Carlo (KMC) simulations (not shown in Figure 4).

The above results are important because they show that it is possible to derive and use PF equations that retain an absolute length scale defined at the atomistic level. The point will be shown to be very important for alloys under irra­diation. On the other hand, the work by Bronchart et a/.10 clearly highlights the difficulty in using quantitative PF modeling when the physical length scales of the alloy under study are small, as for instance in the case of precipitation with large supersaturation, which results in a small critical nucleus size, or in the case of precipitation growth and coarsening at relatively moderate temperature, which results in a small interfacial width. In these cases, one would have to reduce the cell size down to a few atoms, thus degrading the validity of the microscopically based PF equations since they are derived by relying on an expansion with respect to the parameter 1/Nd.