On the basic mechanism of grain growth: the role of texture on grain boundary migration rate

In this paper we will review some of the many achievements of Statistical Theory of Grain Growth (which is 40 years old, this year!), based on the classic law of grain growth. For instance, we will show that grain growth in presence of texture (which is the usual feature in a real microstructure) is not exclusively depending on the curvature of the grain boundary, as in this case the effect of curvature at the vertexes (in 2-D case) contributes as well (which, depending on the texture pattern, sometimes may be more relevant than curvature along the boundary). Such phenomenon requires a reformulation of Von Neuman’s equation in 2-D by involving, beside the different boundary mobilities, the different energies of the “third” grain boundary, which is pulling the vertex and inducing an extra curvature which is not balanced along the grain perimeter (for simplicity’s sake we are referring to a 2-D case, but the analysis can also be done in 3-D). We can then observe that boundary movement is not only depending on a couple of grains in contact (with the boundary curvature acting like in a bicrystal) but also includes the network of “third” grains operating at the vertexes. The overall consequence is that the growth kinetics and the grain size distribution shape are unique and variable for each texture component which contradicts classic growth laws as well as the expected results for the whole grain size distribution.


Introduction
Over 50 years ago, Hillert presented a new statistical model for Grain Growth (GG) in his seminal work [4], following initial , more naïve, theoretical considerations about the subject [1][2][3].He adapted certain aspects of Lifshitz's and Slyozov's method [5], which was originally used for particle coarsening (Ostwald ripening), to address normal grain growth.In this model, each grain is represented by a single parameter, its size R, allowing the entire microstructure to be characterized by a grain size distribution function (SDF) through a one-parametric theory.Hillert made heuristic assumptions that grains with a size smaller than a critical value, Rc, would shrink, while those larger than Rc would grow (Figure 1).By incorporating heuristic expressions for the rates of shrinkage and growth and utilizing a continuity equation, he derived an expression for Rc in the context of 2-D GG.Moreover, Hillert obtained a self-similar SDF, referred to as the Hillert distribution, which represents the long-time asymptote for the coarsening microstructure.
(1) Furthermore, Hillert defined a corresponding asymptotic time law for the growth of the mean radius  ̅ as  ̅ ∝ t 0.5 , which aligns with earlier, less sophisticated approaches to the problem.
Nevertheless, the validity of this growth law has rarely been confirmed through experiments, and the Hillert distribution has never been empirically verified.One of the main reasons for this is the challenging transition from the initial grain size distribution (typically obtained through processes like primary recrystallization, phase transformation, etc.) to reaching steady-state conditions for grain growth.During this transition, there may be a significant increase in the average grain size, sometimes by 1 or 2 orders of magnitude higher than the initial size.Additionally, the time exponent during this transient phase can significantly deviate from 0.5.For further information, refer to [6].
In general, such a model is unable to fully describe grain growth phenomena that encompass all the aspects of GG typically found in real microstructures, such as transient kinetics phenomena, as well as texture effects (non-uniform grain boundaries).Additionally, the model does not account for Zener and atom drag effects and their associated kinetics.Furthermore, the model fails to capture the topological features associated with grain growth microstructures, such as those forming the foundation of the Aboav-Wearie equation.The limitation in Hillert's theory stems from its assumption of a uniform grain boundary and its pursuit of an asymptotic solution in the space u = R / Rc, where the explicit expression for the critical radius Rc (LSW approach) is not necessary.In general, statistical approaches, such as Hillert's model, replace individual real grains in a microstructure with size classes, where each size class represents grains with the same grain size.By doing so, the evolution of the grain size distribution (GSD) formed by these various grain classes corresponds to the evolution of each size class, which is representative of the microstructure.The GSD may also be treated as a continuous function.
Figure 1.The statistical theory describes Grain Growth (GG) in the presence of uniform Grain Boundaries (GBs) using only one continuity equation: Where N(t) represents the total number of grains, φ(R) represents the grain size distribution (GSD), and dR/dt represents the mean growth rate of grains with a size of R. The implicit requirement for using the continuity equation is that of homogeneity assumption.
In this approach, the underlying hypothesis is known as the "homogeneity assumption."This assumption states that grains within the same size class, regardless of their spatial coordinates within the material, experience an identical environment of neighboring grains, leading to the same growth kinetics.In other words, all grains in a particular size class grow or shrink at the same rate.This homogeneity assumption is a kind of definition of a statistical approach since, instead of tracking the development of individual grains, it enables the system of grains to be grouped into classes based on their size (and orientation).As a result, the focus is on considering these distinct classes and their respective frequency  .
This assumption shares similarities with the approach used in Metallography for microstructure characterization of materials, in which measurements are taken from a sample, often a small section of the material.The underlying assumption here is that the material is "homogeneous," meaning that the microstructure is the same throughout the entire material, independent of its spatial coordinates.
Moreover, Equation ( 1) illustrates how the model describes the effect of the average curvature on the growth of grain R.This effect is achieved through the superposition of the boundary curvature of grain R and that of an average grain  , which represents all the surrounding grains in contact with grain R.
However, the limitations of Hillert's model can be overcome by the Statistical Theory of Grain Growth (STGG) or A-L Theory [7][8][9].The latter is formulated within the same framework of curvaturedriven boundary movement, incorporating the corners effect and utilizing a continuity equation, allowed by the 'homogeneous assumption.'The theory begins by considering a single event: a grain boundary shared by individual grains of sizes i and j.This is equivalent to considering a pair of grains i and j in contact, forming an i,j boundary with an average curvature C, obtained through the superposition of the grain boundary curvature of the individual grains,  = (1/ − 1/ ).To construct the microstructure as a statistical system, we must determine the probability of contact between grains.We derive this probability function using only fundamental principles such as volume conservation and topological laws, without relying on arbitrary assumptions.This approach allows us to explicitly derive the expression of the critical radius and handle any transient state, a common occurrence in nature.Furthermore, by considering the fundamental mechanism related to a pair of grains (i,j) sharing a boundary ij, it becomes possible to assign special properties to the boundary in terms of energy and mobility, which are consequences of the grains belonging to different crystallographic orientations.Furthermore, the theory comprehensively considers the impact of second-phase particles (Zener drag) on grain boundary energy and solute atoms on grain boundary mobility (atom drag) [10][11][12][13].In the following paragraph, a brief outline of the STGG will be presented, incorporating the Von Neumann equation [14][15][16] in a formulation applicable to pairs of grains.Notably, it will be demonstrated that Hillert's theory (implicit solution for the asymptotic case) is merely a specific instance of the broader STGG.For the sake of simplicity, our discussion will focus on a 2D microstructure, but it can readily be extended to the 3D case [16].

Von Neumann equation: a base for a statistical approach in 2-D
The equation for the grain growth kinetics of a single grain with area A was proposed by Von Neumann, assuming uniform boundary energy and mechanical equilibrium at the vertices (where there is a 120° angle between each pair of grain boundaries meeting at the common vertex, as shown in Figure 2) [14][15].The resulting equation is as follows: where n represents the number of corners, M=m•γ is the boundary diffusivity, and m and γ stand for the boundary mobility and energy, respectively.This result is derived directly, without any approximation, by summing the total curvature of the closed loop formed by the tensioned boundaries, resulting in a shrinking pressure or tension on the grain (-2π).Additionally, there is outward curvature induced by the "third" boundary at the vertices, generating a pulling tension (nπ/3) [15].The equilibrium of tensions at the vertex leads to the factor of π/3.Within the framework of the STGG [16][17], where grains of equivalent size are grouped into classes i, we can derive an alternative formulation of the Von Neumann (VN) equation.In the context of the STGG, which is based on pairs of grains i and j (boundaries ij), we can establish an equivalent VN equation for an ideal simplified microstructure consisting of grains j of equal size surrounding a specific grain i (figure 2).In this simplified scenario, the VN equation is obtained by summing up the various contributions generated by the boundaries ij surrounding the grain i.The formulation is as follows: where dAi j/dt represents the area change of a specific grain i induced by its contact with only grains of size j.Here, ni j denotes the maximum number of grains j that can remain in contact with grain i [16][17][18].Notably, the equal size of grains j results in an equal fraction of 2π angle coverage around the inner grain i ( ).Importantly, this formulation does not assume anything about the shape of the grains but relies solely on the packing capability of the surrounding grains, namely, the angle they cover while touching the perimeter of the inner grain i.The expression (4), pertaining to the case of equal-sized grains j surrounding grain i, can be formulated as follows for any pair of grains i and j, representing the contribution of a single boundary ij to the area change.
=  − =  −  (5) this can be regarded as a "reduced" Von Neumann equation since it specifically represents the contribution of the grain pair ij to the change in area of grain i.In this context, ij represents the inner angle subtended to the boundary ij and, along with the inclination angle, serves as the only relevant parameter in the kinetics expression.This equation must adhere to the antisymmetric relationship that holds for any pair of grains i and j.This relationship is linked to the fundamental physical constraint of volume conservation in the elementary process of boundary displacement, namely: 3. The number of corners nij and the boundary length lij From equations ( 5) and ( 6), the following relationship holds between nij and nji The harmonic mean between nij and nji is equal to 3 for any grains i and j.From a geometrical standpoint, another valid condition is : (and   =  ) (8) where lij = lji represents the boundary length shared between grain i and grain j and Pi is the perimeter of grain i.Here, the perimeter Pi is considered the fundamental geometrical parameter describing the grain, as it is directly proportional to the free boundary energy, which drives grain growth.By substituting the expression nij from equation ( 8) into equation (6b), we get: This equation provides the relationship between the number of grain sides (corners) nij and the perimeters of grains i and j .This relationship holds fundamental significance and can be utilized to derive various outcomes.By substituting equation ( 9) into equation (8), we obtain the expression for lij in terms of perimeters of grains i and j: The expressions nij and lij are derived based on fundamental principles, but they can also be interpreted geometrically in a simple approximated model consisting of circular grains [17].

The probability factor wij
In a statistical system of grains with sizes i=1, 2, …, nc (where nc is the number of size classes in the system) a simple contact symmetry law must be valid.It states that the number of boundaries that grains of size i share with grains of size j must be equal to the number of boundaries that grains of size j have with grains of size i.This law can be expressed as follows: Here i and j represent the frequency of grains i and j in the system, respectively, while wij and wji are the probabilities of grains j being in contact with grains i and grains i being in contact with grains j, respectively.nij and nji denote the maximum number of contacts between grains i and grains j and between grains j and grains i, respectively.Namely, winij represents the fraction (number) of grain j in contact with a single grain i. Equation ( 11) is the basis for formulating the volume conservation condition valid in any grain growth system: This formulation assumes the validity of the antisymmetric condition (6).From equation ( 8), we can also derive: We can substitute the expression for nij derived from equation ( 13) into equation (11) to obtain: Hence, the probability wij must be a function of j and Pj as follows:  =    (15) Due to the condition ( 14), the function kij must satisfy the symmetry condition:  =  (16) Additionally, considering the definition of probability, the following must be valid: ∑  = 1 (17) (Where  is the number of grain size classes in the system); we can then derive: As the summation on index j results in a constant, this demonstrates that there is no dependency from i , meaning that ki j = kj.Due to the symmetry condition (16) it follows that kj=k which is a constant for all i and j.Consequently wij=wj; therefore, the sum on j leads to: The expression for wj in equation ( 15) can be simplified as follows: Where P represents the average value of Pj .Equation ( 11) can be simplified as: (21) Equation (20) holds paramount importance in the STGG.It is directly derived from simple geometrical definitions (see equations 8 and 9) and a basic statistical rule (equation 11).While previous papers introduced this fundamental expression as an assumption, referred to as the "randomness assumption," implying that the probability of contact between grains j and i depends only on the relative perimeter of grains j in the system [8,16], this paper demonstrates it, as it represents an important basic principle inherent to the space-filling condition and the symmetry requirements for a statistical set of grains Moreover, the probability function wij and its three-dimensional counterpart [16], becomes a fundamental criterion for averaging all microstructural properties in any statistical system, such as metallographic patterns and metallurgical microstructures.

The special linear relationship
The expression obtained for wij and nij (equations ( 20) and ( 9) respectively) from very basic assumptions allow us to calculate another significant equation, representing the average number of sides of grains belonging to size class i: This equation establishes what is known as the special linear relationship, which holds in a generic 2-D mosaic of grain structures [23,24].The key aspect of this relationship is that the smallest grain in the system tends to have 3 sides (Pi  0) whereas the grain of mean radius has 6 sides.This relationship has been demonstrated in numerous artificial 2-D microstructures and real 2-D cuts of 3-D microstructures in grain growth experiments [18].Moreover, in a 2-D microstructure of grains with only triple joints at the vertices, a general topological constraint must be valid.Specifically, the total average of the number of grain sides on the system must always be equal to 6.This is evident when summing over index i to average ni in equation (20): This further illustrates the intrinsic coherence of the expressions obtained in the previous derivation.Other important laws, such as the average grain boundary length of a grain i and the Aboav-Wearie equation, can be easily derived by using the  function for a similar averaging procedure [17].

The kinetics equation
By multiplying equation ( 5) the fraction of grains j in contact with a grain i, (wjnij), and summing over all possible grains j considered we obtain the following expression for the area change of grain i: This equation is equivalent to VN equation (3) but formulated in the space of size classes i.Moreover, from the reduced VN equation ( 5), by substituting the value for nij obtained from equation ( 9) and using equation ( 10), we can transform the "reduced Von Neumann" equation as follows: By repeating the previous averaging procedure, we can derive the kinetics of area change:  / = ∑    /| = ( / − 1) (26) This expression is "exact" for a mosaic of 2-D grains with three sides in contact at each vertex.If we define the area and perimeter of a grain through a function of a unique parameter (as is typically done with a circular grain of equivalent radius R), we can easily obtain Hillert's expression for radius change: In 2D, the critical radius being  =  .
We can observe the significance of the approximation in Hillert's equation.Like many classical models of grain growth, it employs a single parameter to describe the grain size, often the equivalent radius R, as an approximation of the average curvature's radius.Both the area and the perimeter of a grain in a 2-D scenario (or grain volume and contact surface in 3-D) are then represented by a unique function of the equivalent radius (as an approximation to the more general equation (26) for easier calculations [18,19]).However, in general cases, the area and perimeter of a grain serve independent roles.Additionally, the fact that a grain does not have a circular shape demonstrates that both parameters must be considered.Even in roughly equiaxed, irregularly shaped grains, the two quantities cannot be reduced to a single parameter R.
Furthermore, the same statistical treatment can be extended to 3D grain growth (more details can be found elsewhere [16]), where grains are treated as spheres instead of circles (2D).By repeating the procedure and starting with a "reduced VN equation" in 3D, we obtain: Here,  represents the average number of faces of a grain of size i, and  is the average perimeter of the contact surfaces [16].
With the simplification of using a single parameter as the equivalent radius  , the kinetics equation in 3D can be written as: Here,  =  / = represents the "critical radius" [7,16].This crucial parameter now has an explicit analytical expression, linked to the entire grain size distribution through the standard deviation.Consequently, it becomes possible to calculate the 3D evolution of the grain size distribution and the grain growth kinetics during any transient phase using equation (30) and the related continuity equation (=specific grain number, () = grain size frequency distribution): (()/) = −(()/)/ (31)

Texture effects including boundary energy and mobility variation
In STGG, since it is based on basic principles, we are able to include a natural differentiation of boundary mobility.We explore a population of grains with different orientations: each grain is identified by its size i and orientation H while in contact with grains of other sizes j and orientations K.
(with i=1,2…Nc=number of size classes, and H=1, 2…Nt = number of orientation classes).By using the homogeneity assumption, we can replace the individual grains υ and μ with their corresponding classes i and H, and j and K, respectively.We then apply a continuity equation for each texture component H.Moreover, the equilibrium at the vertices of the forces leads to a deviation from the /3 equilibrium angle.As a consequence, an additional effect emerges, which contributes to the curvature effect of the grain boundary.This effect is attributed to the presence of a "third" grain at all the vertices in a 2D case [20], or to the arrangement of grains along the contact face perimeter in 3D [9].Detailed calculations in 2D, which lead to the rate of area change of grain i,H, were provided in references [20][21].However, the final relevant results can be expressed as follows: The two terms in brackets on the right side of equation ( 32) describe different causes for the GB movement.The first term represents the contribution of the pulling tension present at the grain corners to the moving GB.On the other hand, the second term accounts for the contribution of the curvature along the GB (see figure 3).In the special case of  =  = const;  =  = const for all GBs, it follows that  and  ̅ converge to .Consequently, only the term associated with GB curvature remains in equation ( 32), and the 2D growth rate is written as equation ( 27), which represents a Hillerttype equation (uniform grain boundary).For the application to a 3D case, in the simplest approach, which considers the effect of the surrounding grains of different orientations around the perimeter of contact [9], the equation of motion for grain i,H can be written as: The generic term  is the frequency of K oriented grains in the j-th size class.
The average tension  in equation ( 33) can be calculated by applying the homogeneity assumptions.In this case, each grain i,H is considered to have the same randomly distributed surrounding grains, as described previously ( expression in 3D) [9].The neighboring grains generate tension on the i,H grains along the common boundaries with grain j,K (3D drawing in Figure 3).These neighboring grains can be viewed as a random sample of the whole grain size and orientation distribution, and thus they have the same statistical parameters.As a result, the value of  can be calculated by a process of averaging, taking into account the probability of contact with surrounding "third" grains from the whole distribution: Here, each tension  is weighted by the total grain boundary area of the texture component L in the system, which represents the contribution of "third" grains around the couple of i,H and j,K grains.Like in equation (32, this preliminary average on "third grains" allows to reduce the basic equation to that of a two grains model involving (i,H) and (j,K) grains.
Therefore, for each texture component H, there is an average mobility  and a critical driving force  .The latter can also be interpreted by a critical radius  with the usual meaning for H grains.
Furthermore, since all grains of same orientation H and same size i have the same growth rate, it is possible to formulate a continuity equation (homogeneity assumption).This equation describes the change, as a function of time, of the number of grains per unit volume  (or the frequency φ ) due to the " flow " of the grains of class i -1 into i and of class i into i + 1 for the growing grains, as well as the flow from class i + 1 into i and from i into i -1 for shrinking grains.
With being the drift velocity given by equation (33) and Δ the width of the size classes, one obtains a separate equation for each orientation class H: The effect on the H grains' flux exerted by grains of other orientations influences only the drift velocity .The evolution over time of the grain size distribution  can be calculated by numerically integrating the set of equations ( 33) and (35), provided that the initial size distributions for each texture component H, the matrices of mobility  and boundary energy  are known.
More sophisticated equations for 3D can be provided [22], in line with equation (32) for 2D.However, for practical applications, the previous simplified approach is largely adequate, as will be shown hereon.

Grain Growth Experiments in the presence of Texture
GG experiments on single-phase Al 0.5% Mn and Al 3% Mg alloys are summarized in the following chapter, interpreted by the STGG [23].The samples were cold rolled and subsequently recrystallized.corresponding volume fractions of the involved texture components, while figure 4c shows the corresponding {111} pole figures.
After the primary recrystallization, the texture primarily consists of the Cube component, which sharpens significantly after short annealing times, at the expense of the high fraction of randomly oriented grains (phon) also present.During the fast GG period, two texture components, denoted as the P-component (77°, 45°, 0°) and the Q-component (55°, 0°, 0°), which were originally too small to be resolved, grow while the Cube component diminishes until it is entirely consumed.In the case of the Al-3% Mg alloy, it was cold rolled by 95%.For primary recrystallization, two different annealing temperatures were applied: one series of the samples were annealed for 10 minutes at 300 °C (series I), and another series for 0.5 minutes at 450 °C (series II).These different annealing temperatures resulted in different initial textures.This difference is evident from the {111} pole figures in Figure 5a, where the Cube component at 300 °C appears much sharper than at 450 °C, and from Figure 5c, where the initial volume fractions for the Cube orientation considerably differ (40% compared to 80%).All samples were subsequently annealed for GG at 450 °C. Figure 5b shows the development of the mean radius as function of time in a double logarithmic plot for both series.Qualitatively, they behave similarly: Initially, the average grain size grows slowly, followed by a period of fast GG, which eventually transitions into a very slow increase in grain size.However, in series I, the kinetic curve of the fast growth period is much steeper, resulting in a grain size that is more than two times higher than in series II samples.In both cases the sharp increase in grain sizes is again accompanied by a complete transition of texture (Figure 5c).Both series start with a strengthening of the cube component (up to nearly 100% volume fraction for series I), resulting in a texture which mainly consists of the Q-and Rcomponent (59°,37°,63°) for series I, and only of the Q-component for series II.

Grain Growth Simulation (3D) with 2 Texture Components and Comparison with experiments
Two simulations were conducted, considering two texture components, A and B [23,24].A log-normal grain size distribution was chosen as the initial state for both simulations, but with different initial volume fractions, VA and VB, for components A and B. In the first simulation, VA was set to 90% (VB = 10%), and in the second simulation, VA was set to 70% (VB = 30%).The energies and mobilities of GBs between grains of the same component were assumed to be equal ( =  ,  =  ), and relatively small compared to the values of the (large angle) A-B GBs, namely,  / =1.4 and  / =5.0 .Figures 6a and 6b show the results of the simulations.The time evolution of the grain size and textures in the simulations qualitatively matched experimental findings during GG in Al-3% Mg (figure 5), where different volume fractions also existed in the initial state.Initially, the majority component A sharpens, and later, the minority component B completely consumes the A-component.Additionally, confirmation of some phenomena was found, such as a retardation of the texture change in the case of a sharper initial texture.For Al-3% Mg (series I) a quantitative fitting of the simulation to the experimental results was also conducted.The 2D grain radii were transformed into 3D grain radii using the polyhedron model [25].The partial grain size distributions, separated for components C (Cube) and Q+R, were determined using the etch pit technique.Figures 7a and 7b show the results, and all stages of GG could be described very accurately.
The initial experimental grain size distributions were used as input for the simulations, and the normalized values of γ and m were chosen as (  = 3.9 • 10  /) Now, let's discuss the two different effects that anisotropic energies and mobilities have on GG kinetics.It's important to recognize that the B grains, having a lower initial frequency, are mainly surrounded by A grains, forming A-B boundaries with high energies and mobilities.On the other hand, the A grains, due to their higher initial frequency, are mostly surrounded by low-angle GBs with low energies and mobilities.
First, let's consider the effect of different mobilities while assuming constant GB energies.Due to their high mobility, small B grains will disappear at the beginning of GG, and large B grains will grow rapidly, causing the mean grain size of B grains to increase quickly.As a result, in the intermediate stage of GG, a few large B grains will be embedded in a fine-grained A matrix, which is a characteristic feature of abnormal GG.Eventually, B grains will occupy most of the volume, leading to a complete change in texture.Next, let's focus on the anisotropy of GB energies while assuming constant mobilities.The energy of the GBs surrounding B grains (A-B) is larger than that of their radial GBs (mostly A-A).
As a result, assuming equilibrium at the GB triple points, the boundaries of B grains must be curved more convexly compared to the case of isotropic γ (see Figures 8a, b).Consequently, the critical number of sides as well as the critical radius for B grains are shifted to higher values, and their probability to shrink is increased compared to the case of constant γ.This may cause an initial decrease in the volume fraction of B grains.Both effects, caused by the anisotropy of GBs, are predicted by the AL model.The comparison with experiments shows that the model can provide a quantitative description of texture-   controlled GG.Similar conclusion can be drawn for Cu-Zn alloys [24] and GOES.In the latter, in combination with texture, GG inhibition, provided by second phase particles, also plays an important role.This complexity is taken into account and rationalized by the STGG [13,26].

Conclusions
The STGG is inspired by and represents a natural evolution of Neuman's and Hillert's classic approaches.In a simplified version for practical applications, it serves as a generalization of the latter method.Due to the limited applicability of the traditional models, it was necessary to investigate the fundamentals of the statistical model.Starting from a basic contact between two grains and deriving the probability of contacts between grains, by utilizing only the fundamental constraints provided by the topological rules (such as volume conservation and symmetry requirements), an explicit formulation of the statistical model is obtained.As a result, the STGG is capable of addressing various typical aspects of grain growth and predicting their effects.These aspects include grain growth in transient phases (normal case), texture evolution, Zener and atom drags effects, predictions of new topological rules, and more.Moreover, the introduction of non-uniform grain boundaries (with varying energy and mobility) in this manner allows for a clearer understanding of the need to consider the effect of "third grains."The evaluation of the average curvature between two grains becomes influenced by the presence of additional grains.This effect becomes particularly noticeable at grain corners (edges), and depending on the level of texture intensity, it may even overshadow the effect of the boundary curvature along the 2D boundary (or 3D face).In this framework Texture evolution on Al-3% Mg has been quantitative measured in details and successfully compared with the grain growth kinetics predicted by AL model.

Figure 2 .
Figure 2. Ideal microstructure of equal sized grains j around a grain i. Inclination angle of π/3 due to equilibrium of forces at the vertices.
29)In the classical cases (Hillert/Statistical Model), an additional simplifying assumption of  =   is made, resulting in a one-parametric model:

Figure 3 .
Figure 3. Schematic for the effect of equilibrium at the vertexes (2D) and along the edges of the perimeter of contact surface (3D), in case of presence of different orientations (texture effect).

Figure 6 .
Figure 6.Model calculation data as function of annealing time for initial volume fractions of VA = 70% and of VA = 90%.a) Mean radius; b) Volume fractions VA and VB.

Figure 7 .
Figure 7. Experimental vs model data as function of annealing time for Al3%Mg (series I).a) Total mean radius and partial mean radii of Texture components; b) Volume fractions of Texture components.

Figure 8 .
Figure 8. Growth rate (a) for a grain in a microstructure with isotropic GB energies and (b) for a grain of the minority texture component in the anisotropic energy case.