22.08.2015   Comprehensive nuclear materials

Numerical integration of the kinetics equations

The main idea of the grouping methods for numerical evaluation of the ME is to replace a group of equations described by the ME with an ‘averaged’ equation. Such a procedure was proposed by Kiritani75 for describing the evolution ofvacancy loops during aging of quenched metals. Koiwa84 was the first to examine the Kiritani method by com­paring numerical results with the results of an analytical solution for a simple problem. Serious disagreement was found between the numerical and analytical results, raising strong doubts regard­ing the applicability of the method. The main objec­tion to the method75 in Koiwa84 is the assumption used by Kiritani75 that the SDF within a group does not depend on the size of clusters. However, Koiwa did not provide an explanation of where the inaccuracy comes from. The Validity of the Kiritani method was examined thoroughly by Golubov et a/.85 The general conclusion of the analysis is that the grouping method proposed by Kiritani is not accurate. The origin of the error is the approxi­mation that the SDF within a group is constant as was predicted by Koiwa.84 Thus, the disagree­ment found in Koiwa84 is fundamental and cannot be circumvented. Because it is important for under­standing the accuracy of the other methods sug­gested for numerical calculations of cluster evolution,
the analysis performed in Golubov et a/.85 is briefly highlighted below.

It follows from eqn [18] that the total number of clusters, N(t) = ^2T=2 f (x, t) and total number of defects in the clusters, S(t) = ^Ц=2 xf (x, t), are described by the following equations:

f = J 0.t) N

@ S 1

— = J(1, t)+ J(x, t) [37]

x=1

Подпись: Figure 3 Size distribution function of voids calculated in copper irradiated at 523 K with the damage rate of 10—7dpas—1 for doses of 10—2-10—1 dpa. The dashed and solid lines correspond to the Kiritani method and the new grouping method, respectively. The thick line corresponds to the steady-state function, g(x). Reproduced from Golubov, S. I.; Ovcharenko, A. M.; Barashev, A. V.; Singh, B. N. Philos. Mag. A 2001, 81, 643-658. where the generation term in eqn [18] is dropped for simplicity. Equations [36] and [37] are the con­servation laws which can be satisfied when one uses a numerical evaluation of the ME. When a group method is used, the conservation laws can be satisfied for reactions taking place within each group.69 How­ever, this is not possible within the approximation used by Kiritani75 because a single constant can be used to satisfy only one of the eqns [36] and [37]. To resolve the issue, Kiritani75 used an ad hoc modifica­tion of the flux J(x,); therefore, the final set of equations for the density of clusters within a group, Fj, are as follows

dF,

dt

 
image783

Jj ]

 

[38]

 

It is worth noting that this comparison also sheds light on the relative accuracy of other numerical solutions of the F—P equation such as in Bondarenko and Konobeev,76 Ghoniem and Sharafat,77 Stoller and Odette,78 and Hardouin Duparc et a/.79

Equations [36] and [37] provide a way of getting a simple but still reasonably correct grouping method for numerical integration of the ME. Indeed, the two conservation laws, eqns [36] and [37], require two parameters within a group at least. The simplest approximation of the SDF within a group of clusters (sizes from x,—1 to x, = x,—1 + Ax, — 1) can be achieved using a linear function

 
image784 image785

[39]

 

Jj

 

where Ax, is the width of the ‘j’ group. Equations [38] and [39] indeed satisfy both the conservation laws. However, they do not provide a correct description of cluster evolution described by the ME because the flux Jj in eqn [39] depends on the widths of groups and these widths have no physical meaning. An example of a comparison of the calcu­lation results obtained using the Kiritani method with the analytical and numerical calculations based on a more precise grouping method is pre­sented in Figure 3. Note that in the limiting case where the widths of group are equal, Ax, = Ax, +1, the flux Jj is equal to the original one, J(x, t). In this limiting case, eqns [38] and [39] correspond to those that can be obtained by a summation of the ME within a group and therefore they provide con­servation of the total number of clusters, N(t), only. This limiting case is probably the simplest way to demonstrate the inaccuracy of the Kiritani method.

 

f (x) = L0(x — (x), ) + L1 N

where (x)j = x, — 1/2 (Ax, — 1) is the mean size of the group. Equations for L0, L] are as follows69

 

dI0

dt

 

[J (x,—1)—Jx (x,)]

 

[41]

 

di;

dt

—<xi—‘’+j <x’)—■J (x’)—0}|42]

 

image078

is the dispersion of the group. Equations [41] and [42] describe the evolution of the SDF within the group approximation. Note that the last term in the brackets on the right-hand side of eqn [42] follows from the corresponding term in eqn [38] in Golubov et a/.85 when the rates P(x, t), Q(x, t) are independent from x within the group. Note also that the factor ‘_ 1 /Ax1 ’ is missing in eqn [38] in Golubov eta/.85

As can be seen from eqns [41] and [42], in the case where Dx1 = 1, eqns [41] and [42] transform to eqn [18], that is f (x1) = L0 and L = 0 in contrast with Kiritani’s method, where the equation describing the interface number density of clusters between ungrouped and grouped ones has a special form (see, e. g., eqn [21] in Koiwa84). It has to be empha­sized that this grouping method is the only one that has demonstrated high accuracy in reprodu­cing well-known analytical results such as those by Lifshitz-Slezov-Wagner86,87 (LSW) and Greenwood and Speight88 describing the asymptotic behavior of SDF in the case of secondary phase particle evolu — tion89 and gas bubble evolution90 during aging.

A different approach for calculating the evolution of the defect cluster SDF is based on the use of the F-P equation. Note that the use of eqn [33] as an approximate method for treating cluster evolution is not new, for the work initiated by Becker and Doring64 has been brought into its modern form by Frenkel.66 An advantage of the F-P equation over the ME is based on the possibility of using the differential equation methods developed for the case of continu­ous space. Quite comprehensive applications of the analytical methods to solve the F-P have been done by Clement and Wood.83 It has been shown83 that convenient analytical solutions of the F-P equation cannot be obtained for the interesting practical cases. Thus, several methods have been suggested for an approximate numerical solution for it. The simplest method is based on discretization of the F-P equa — tion76-79 that transforms it to a set of equations for the clusters of specific sizes similar to the ME; in both the cases the matrix of coefficients of the equation set is trigonal. This method is convenient for numerical calculations and allows calculating cluster evolution up to very large cluster sizes (e. g., Ghoniem81). How­ever, this method is not accurate because it is identical to the approach used by Kiritani75 in

which SDF was approximated by a constant within a group. Thus, all the objections to Kiritani’s method discussed above are valid for this method as well. Also note that the method has a logic problem. Indeed a chain of mathematical transformations, namely ME to F-P and F-P to discretized F-P, results in a set of equations of the same type, which can be obtained by simple summation of ME within a group. More­over, the last equation is more accurate compared to the discretized F-P because it is a reduced form of the ME.

Another approach for numerical integration of the F-P equation was suggested by Wehner and Wolfer (see Wehner and Wolfer80). The method allows cal­culating cluster evolution on the basis of a numerical path-integral solution of the F-P equation which provides an exact solution in the limiting case where the time step of integration approaches zero. For a finite time step, the method provides an approximate solution with an accuracy that has not been verified. Moreover, there was an error in the calculation presented in Wehner and Wolfer80,91 and so the accu­racy of the method remains unclear. A modification of this method according to which the evolution of large clusters is calculated by employing a Langevin Monte Carlo scheme instead of the path integral was suggested by Surh eta/.82 The accuracy of this method has not been verified as an error was also made in obtaining the results presented in Surh et a/.82,91

The momentum method for the solution of the F-P equation used by Ghoniem81 (see also Clement and Wood83) is quite complicated and may provide only an approximate solution. So far, none of the methods suggested for numerical evaluation of the F-P equation has been developed and verified to a sufficient degree to allow effective and accurate calculations of defect cluster evolution during irradi­ation in the practical range of doses and temperatures.

Похожие посты: