A method to simulate crystallographic texture development using a small computer is presented. The growth behavior of a specified grain embedded in previously prepared matrix grains, for which the energy and mobility of grain boundaries as well as the mobility of triple lines depended on the relation between the orientations of the embedded and matrix grains, was first observed by a modified Potts-Monte Carlo-type three-dimensional grain growth simulation model. A specimen investigated using a single simulation corresponded to part of the mother specimen, and hence, was sufficiently small to be handled by a personal computer. The texture development in the mother specimen was then statistically estimated from results obtained by varying the size of embedded grains. The orientation-relation-dependent mobility of triple lines was found to substantially influence the texture change developed by the orientation-relation-dependent energy of grain boundaries.

Crystallographic textures of metallic materials change through recrystallization and grain growth after an annealing process.^1,2,3) Various researchers have noted that the texture evolution process is influenced by orientation-relation-dependent properties of grain boundaries⁴⁾ as well as by multiple junctions such as triple lines.^5,6,7,8) Stored energy^1,2) is, of course, an important factor in the recrystallization process. It is, however, not considered in the present work. Annealing experiments cannot actually be carried out with control of orientation- relation-dependent factors and, hence, computer simulations^2,3,6,7,9) are helpful in understanding how various factors influence the texture evolution.

Mapping a three-dimensional structure with a sufficient number of grains to statistically trace texture development requires a huge computer memory capacity. A method of grain embedding has been examined in this report to reduce the necessary memory capacity. Small partial specimens containing characteristic grain configurations are here sampled from a large virtual mother specimen representing the statistical population. The grain coarsening process of each partial specimen is simulated using a personal computer, and the structural development expected to occur in the mother specimen is statistically estimated from the results obtained for the small samples.

The used simulation algorithm was developed from the two-dimensional algorithm¹⁰⁾ and explained in the previous report.¹¹⁾ It belongs to varieties of the Potts-MC model.^8,12) In this paper, we present the results of textural developments with attention to the orientation-relation-dependent grain boundary energy and mobility as well as the triple line mobility.

A small specimen contained 256 × 320 × 256 cubic cells and a crystallographic texture of grains in a starting specimen was composed of two components: matrix component M and background component B. Details of the notations used in the following explanation were given in the previous paper.¹¹⁾

The origin of the boundary as well as the triple line energy was given by Eq. (1);   

When cell i was occupied by cell j, or when the orientation number of cell i was altered from O i before to O i after (=O_j), we calculated the energy change by Eq. (2);   

When, in the case of boundary migration, the combination of two texture components corresponding to O i before and O_j belongs to one of the higher boundary mobility combinations, R_permission is equal to 1, otherwise R_permission is equal to R (≥1.0). Similarly, in the case of triple line migration, R_permission was set to 1 or to R after the combination of relating three texture components. Permission ratio R controls the orientation relation dependence of both boundary and triple line mobility. The effect of a boundary energy on the migration rate also was represented by R.

Thus, for simplification of the system, only two levels were considered for boundary energy and mobility of boundaries and triple lines, and a characteristic dependence on the orientation relations among associated grains was defined for the combination of texture components to which they belonged.

The coarsening process was executed by iteration of the systematic MC steps,¹⁰⁾ where occupying cell j was selected from cells in the Von Neumann neighborhood of cell i. Initial structures were, as described in the previous report,¹¹⁾ obtained from starting structures by grain coarsening process with R = 1 and F = 100. They contained approximately 7200 grains. The grains were homogeneously distributed with respect to the texture components M and B.

The effects reported in the previous paper¹¹⁾ are summarized as follows (here R and F represent generally R_permission in Eq. (3) and f_in in Eq. (1), respectively).

When F was kept to 100, the boundary migration rate observed as shrinking rate of a spherical grain decreased roughly in inverse proportion to R² (Fig. 3 in the previous paper). This is a characteristic of the systematic MC steps.

Fig. 3.

Examples of the dependence of the relative growth ratio and the fraction of surviving embedded grains on the number of faces of a grain in the reference structure, as influenced by the energy and mobility of boundaries. (Online version in color.)

When R was kept to 1, the relative shrinking rate of a spherical grain decreased to 1.0, 1.0, 0.96, 0.83 corresponding to F = 99, 95, 80, 65, respectively. The migration rates were proportion to factor F, when 1/R was set to 94/100 and 91/100, respectively corresponding to F = 80 and 65 (see p = 0.94p(ΔE) and p = 0.91p(ΔE) in Fig. 5 in the previous paper).

Fig. 5.

Grain volume ratio, as influenced by the boundary energy factor F and the permission ratio R for boundaries. (Online version in color.)

The boundary energy, or energy originated from Eq. (1) and included in a certain finite boundary region per by its area, depends actually on the orientation of the boundary region with respect to the coordinates of array of cubic cells. The boundary energies were roughly estimated for F = 99, 95, 80, 65 to be, relatively to the value for F = 100, between 0.97 and 0.89, between 0.97 and 0.88, between 0.83 and 0.74, between 0.61 and 0.59, respectively (Fig. 8 in the previous paper).

Fig. 8.

Volume increasing ratio, as influenced by the boundary energy factor F and the permission ratio R for a triple line. (Online version in color.)

When R_permission was set to R and 1 respectively to triple line and boundary migration, the ratio of mobility of triple line m_tl to that of boundary m_b observed by a three-grains-configuration was found to be described by a theoretical equation under the correspondence of M/(R−1) to m_tl/m_b. M was a kind of adjustable parameter that represents the difference between the geometrical shape of boundaries used by the theory and actual faceting boundary structure of Potts-MC model, and found to be 0.965 (Fig. 9 in the previous paper).

Fig. 9.

Grain volume ratio, as influenced by the boundary energy factor F and the permission ratio R for a triple line. (Online version in color.)

A lenticular spheroid grain belonging to texture component S was embedded in an initial structure (a red grain in Fig. 1(a)). The size of the spheroid grain was controlled by its volume and axial ratio. A reference structure was obtained using coarsening steps with given F and R until the number of the matrix grains decreased by 10% (Fig. 1(b)). The reference structure corresponds to a partial sample. The volume of the embedded grain and the mean volume of the matrix grains in the reference structure are denoted by V₀ and <V>₀, respectively, and those after coarsening are indicated by V and <V>, respectively. Ratios <V>/<V>₀ and (V/V₀)/(<V>/<V>₀) are hereafter referred to as the coarsening ratio of the matrix and the relative growth ratio of the embedded grain, respectively.

Fig. 1.

Sections observed by the grain embedding method. Grains colored by , , , and belong to texture component M, and those painted by , , , and belong to component B. (c) (boundary mobility effect) and (d) (boundary energy effect) correspond to Figs. 4(b) and 4(a), respectively. The matrix structures in (c) and (d) are very similar because they started from the same initial structure shown in (a). Note that the centers of curvature of boundaries between embedded grain and grains of component M, and those of boundaries between embedded grain and grains of component B are differently located in the two figures.

Boundaries were classified into two types. Boundaries between the embedded and matrix grains belonging to texture component M were defined as “special”; the others, specifically boundaries between the embedded and matrix grains belonging to texture component B and boundaries between matrix grains, were defined as “general.” Matrix composition parameter M/B in the figures that follow indicates the ratio between the total volume of grains belonging to texture component M and the total volume of grains belonging to texture component B. On the basis of the boundaries and triple line energy calculated using Eq. (1), combinations of neighboring cells were distinguished only by consideration of whether they corresponded to a special or general boundary. Unless specially described, the permission ratio was similarly given only after the combination of passive cell i and active cell j.

Reference structures were actually constructed from a common group of initial structures prepared at the beginning of the experiments. The effects of the factors controlling boundary migration were thus observed under almost the same matrix constructions (cf. Figs. 1(c) with 1(d)).

The following two parameters were calculated to represent the structural evolution. A parameter of textural development—specifically, the “volume increasing ratio”—was, as a function of the coarsening ratio of the matrix <V>/<V>₀, defined as the volume fraction of texture component S after grain coarsening divided by the volume fraction of texture component S in the reference structure (Eq. (5)). A parameter describing the relative grain size change—specifically, the “grain volume ratio”—was, as a function of the coarsening ratio of the matrix <V>/<V>₀, defined as (mean size of grains belonging to texture component S after grain coarsening/mean size of grains belonging to texture component S in the reference structure) divided by <V>/<V>₀ (Eq. (7)). Hereafter, <V>/<V>₀ is represented by j and the number of faces of an embedded grain in the reference structure is represented by i.

Initially, no dragging of the triple line migration on boundary migration was considered. The same R_permission was substituted into Eq. (3) independently from the triple line or boundary migration. Experiments were repeated by varying combinations of the initial structures and size of embedded grains (Fig. 2) until approximately 15 examples were obtained for each number of faces of the embedded grain in the reference structure. The mean of the relative growth ratio of the examples, r(j,i), is plotted in Fig. 3(a). The surviving fraction, s(j,i), was defined as the number of surviving cases divided by the number of examples (Figs. 2 and 3(b)). Parameter f_N(i) in Fig. 3(c) is the same as the previously reported histogram of the number of faces in the initial structure shown in Fig. 10 at coarsening ratio = 1.¹¹⁾ Parameter f_V(i) corresponds to the volume fraction of grains whose number of faces is specified by i. The data in Figs. 3(a) and 3(b) were collected when j reached 8. The labels in Fig. 3 representing combinations of parameters for special and general boundaries are shown in Table 1.

Fig. 2.

Twenty-one examples of coarsening processes observed for the case where the number of faces of embedded grains in the reference structure coincided with 19. The surviving fraction at <V>/<V>₀ = j = 8 is here equal to 16/21. The volume of embedded grains increased in case 1 and decreased in case 2 at j = 32. (Online version in color.)

The volume increasing ratio and the grain volume ratio were statistically estimated using Eqs. (5), (6), (7) on the assumption that the size distribution of grains belonging to component S in the reference structure was the same as those of matrix grains in the initial structure and that the grains belonging to component S did not collide with each other. Values of i ranging from 10 to 40 were summed; the value for r(j,i) is practically zero when i<10, whereas f_N(i) and f_V(i) are negligible when i>40.   

Fig. 4.

Volume increasing ratio, as influenced by the boundary energy factor F and the permission ratio R for boundaries. (Online version in color.)

The results labeled R = 1 in Figs. 4(a) and 5(a) show that, with decreasing boundary energy between grains belonging to texture components S and M, the total and mean volumes of grains (relative to that of matrix grains) in texture component S increase. Similarly, results labeled F = 100 in Figs. 4(b) and 5(b) show that the increase in mobility of boundaries between grains belonging to texture components S and M yields the same effects. The decrease in boundary energy and the increase in boundary mobility work multiplicatively rather than additively (R = 1.41 and R = 2 in Fig. 4(a)). The decrease in the migration rate of boundaries as a result of the decrease in boundary energy (Fig. 5(a) in the previous report¹¹⁾) insignificantly affects the boundary energy effect shown in Fig. 4(a). Legends R = 1/1.19 (= 0.84) and R = 1/1.41 (= 0.71) in Fig. 4(a) correspond to notations p = 0.84p(ΔE) and p = 0.71p(ΔE) in Fig. 5(a) in the previous report¹¹⁾ (see section 2.2). The increase in boundary energy, by contrast, substantially diminishes the effect of increasing mobility of boundaries (Fig. 4(b)).

The texture evolution through orientation-relation-dependent boundary energy has been suggested to be controlled more by changes in the migration direction of the triple line than by different migration rates of boundaries corresponding to differences in boundary energy.

Figures 6(a) and 6(b) correspond to Figs. 3(a) and 3(b) and show r(j,i) and s(j,i) for different matrix compositions. The effects of matrix composition on the volume increasing ratio and on the grain volume ratio are demonstrated in Fig. 7. The effects of increasing boundary mobility decrease more sensitively than those of decreasing boundary energy with decreasing fraction of texture component M, whose grains construct, with grains of texture component S, boundaries of higher mobility and lower energy.

Fig. 6.

Examples showing influence of the matrix composition on the relative growth ratio and the probability of survival. (Online version in color.)

Fig. 7.

Summary of the influence of matrix composition on the volume increasing ratio and the grain volume ratio. (Online version in color.)

There are generally four equivalent orientations in an ideal orientation in a rolled sheet presented as {hkl}<uvw>. In this sense, texture component M comprises four equivalent subcomponents. The special triple line migration is defined as an occupation event where the occupying cell belongs to the embedded grain, the cell to be occupied belongs to a grain of one of the subcomponents of M, and another cell in the von Neumann neighborhood of the cell to be occupied belongs to a grain of one of the other three subcomponents of M. Any triple line migration out of this definition was classified as general triple line migration. Permission ratio R_ij in Eq. (3) was controlled for the general and special triple line migration scenarios.

The results obtained under the condition of dragging of boundaries by triple lines are shown in Figs. 8 and 9. The legends are explained in Table 4. The curves shown by legend R = 1 in Figs. 8 and 9—specifically, the curves presenting data of non-triple-line dragging—are the same as those in Figs. 4 and 5.

The aforementioned figures demonstrate that the effects of the orientation-relation-dependent boundary energy and orientation-relation-dependent triple line mobility function multiplicatively rather than additively for both the volume increasing and grain volume ratios.

A modified Potts–MC-type grain coarsening model reported in the previous paper was applied to a hybridization method for estimating crystallographic texture evolution through the grain coarsening process. Small partial specimens, whose grain coarsening processes could be simulated using a personal computer, were through the grain embedding method sampled from a mother specimen, and the expected structural development in the mother specimen was statistically estimated from the results obtained by simulation. A grain that belonged to an object texture component was embedded in the center of the simulated specimen. Statistical estimation is, in principle, possible only while the embedded grain does not contact the specimen surfaces. When the object texture component is a minor component, the mother specimen should be large. The hybridization method is then expected to be especially useful.

The crystallographic texture of matrix grains was modeled by two components: M and B. Grain boundaries were classified into two types with respect to both energy and mobility: higher-energy or lower-energy boundaries and higher-mobility or lower-mobility boundaries. Boundaries between an embedded grain and grains belonging to texture component M were defined as special boundaries, and other boundaries were defined as general boundaries. Triple lines composed of a growing embedded grain and two grains belonging to texture component M were similarly defined as a special triple line, whereas others were defined as a general triple line.

The influences of the orientation-relation-dependent boundary energy and mobility and the triple line mobility on the structural evolution were examined. When the special boundaries had lower energy, the object texture component, in particular, was observed to increase. The special triple lines were observed to reduce this increase when they were low-mobility triple lines. The effects of grain boundary energy and triple line mobility work multiplicatively rather than additively. The results presented in this work suggest that, under certain conditions, the triple line mobility can influence the texture evolution during grain coarsening.

Any simulation modelling and grain size distribution could be used in this hybridization method.

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