A Variational derivation of the continuum model for grain boundary motion

Motion of grain boundaries incorporating dislocation structure

Abstract

In this paper, we present a continuum model for the dynamics of low angle grain boundaries in two dimensions incorporating both the motion of grain boundaries and the dislocation structure evolution on the grain boundaries. This model is derived from the discrete dislocation dynamics model. The long-range elastic interaction between dislocations is included in the continuum model, which ensures that the dislocation structure on a grain boundary is consistent with the Frank’s formula. These evolutions of the grain boundary and its dislocation structure are able to describe both normal motion and tangential translation of the grain boundary and grain rotation due to both coupling and sliding. Since the continuum model is based upon dislocation structure, it naturally accounts for the grain boundary shape change during the motion and rotation of the grain boundary by motion and reaction of the constituent dislocations. Using the derived continuum grain boundary dynamic model, simulations are performed for the dynamics of circular and non-circular two dimensional grain boundaries, and the results are validated by discrete dislocation dynamics simulations.

keywords:
Grain boundary dynamics; dislocation dynamics; long-range elastic interaction; grain rotation; coupling and sliding
1\cortext

[mycorrespondingauthor]Corresponding author

1 Introduction

Grain boundaries are the interfaces of grains with different orientations and play essential roles in the polycrystalline materials [Sutton and Balluffi, 1995]. Grain boundaries migrate under various driving forces such as the capillarity force, the bulk energy difference, the concentration gradients across the boundary, and the applied stress field. The motion of grain boundaries crucially determines the mechanical and plastic behaviors of materials. The classical grain boundary dynamics models are based upon the motion by mean curvature to reduce the total interfacial energy [Herring, 1951, Mullins, 1956, Sutton and Balluffi, 1995] using the misorientation-dependent grain boundary energy [Read and Shockley, 1950]. There are extensive studies in the literature on such motion of grain boundaries by using molecular dynamics or continuum simulations, e.g. [Chen and Yang, 1994, Upmanyu et al., 1998, Kobayashi et al., 2000, Kazaryan et al., 2000, Upmanyu et al., 2002, Zhang et al., 2005, Upmanyu et al., 2006, Kirch et al., 2006, Elsey et al., 2009, Lazar et al., 2010, Esedoglu, 2016].

It has been shown that the grain boundary normal motion can induce a coupled tangential motion which is proportional to the normal motion, as a result of the geometric constraint that the lattice planes must be continuous across the grain boundary [Li et al., 1953, Srinivasan and Cahn, 2002, Cahn and Taylor, 2004]. Besides the tangential motion coupled with normal motion, there is another type of tangential motion that is the relative rigid-body translation of the grains along the boundary by sliding to reduce the grain boundary energy [Li, 1962, Harris et al., 1998, Kobayashi et al., 2000, Upmanyu et al., 2006, Esedoglu, 2016]. When a grain is embedded in another one, the tangential motions along a grain boundary give rise to a relative rotation between the two grains, leading to change of the misorientation of the grain boundary. In the grain rotation due to sliding, the misorientation angle goes to the nearby local energy minimum state (decreases for a low angle grain boundary), whereas in the grain rotation due to coupling, the misorientation angle increases.

Cahn and Taylor [Cahn and Taylor, 2004, Taylor and Cahn, 2007] formulated the phenomena of the coupling and sliding associated with the grain boundary motion. The tangential velocity induced by coupling effect is proportional the normal velocity with the coupling parameter , and the tangential velocity produced by sliding effect is . Accordingly, the total tangential velocity is the superposition of these two effect: . They discussed different cases for the rotation of a circular cylindrical grain embedded in another one [Cahn and Taylor, 2004]. When the grain does not have such symmetry, they proposed a generalized theory based on mass transfer by diffusion confined on the grain boundary [Taylor and Cahn, 2007].

Molecular dynamics simulations have been performed to validate the theory of Cahn and Taylor on the coupling grain boundary motion to shear deformation for planar grain boundary [Cahn et al., 2006b, a], and grain boundary migration and grain rotation for closed circle cylindrical grain boundaries [Srinivasan and Cahn, 2002, Trautt and Mishin, 2012]. Experimental observations have also been reported on the migration of low angle planar tilt boundaries coupled to shear deformation in Al bicrystal with stress [Molodov et al., 2007, Gorkaya et al., 2009]. The ratios of the normal to the lateral motion that they measured are complied with the coupling factors in the theory and atomistic simulations by Cahn et al [Cahn and Taylor, 2004, Cahn et al., 2006b, a]. Phase field crystal model (an atomistic-level model) was employed to simulate the dynamics of a two-dimensional circular grain, and grain rotation and translation by motion and reaction of the constituent dislocations were observed [Wu and Voorhees, 2012]. Phase field crystal simulations also showed that the coupling of grain boundary motion in polycrystalline systems can give rise to a rigid body translation of the lattice as a grain shrinks and that this process is mediated by dislocation climb and dislocation reactions [McReynolds et al., 2016]. Three-dimensional phase field crystal simulations were further performed to investigate the motion, rotation and dislocation reactions on a spherical grain in a BCC bicrystal [Yamanaka et al., 2017]. Numerical simulations based upon the generalization of the Cahn-Taylor theory to noncircular grains [Taylor and Cahn, 2007] were performed using the level set method [Basak and Gupta, 2014].

In this paper, we present a continuum model for the dynamics of low angle grain boundaries in two dimensions incorporating both the motion of grain boundaries and the dislocation structure evolution on the grain boundaries (Eqs. (16) and (17) in Sec. 3). The long-range elastic interaction between dislocations is included in the continuum model, which ensures that the dislocation structures on the grain boundaries are consistent with the Frank’s formula for grain boundaries (the condition of cancellation of the far-field elastic fields). These evolutions of the grain boundary and its dislocation structure are able to describe both normal motion and tangential translation of grain boundaries and grain rotation due to both coupling and sliding effects. Since the continuum model is based upon dislocation structure, it naturally accounts for the grain boundary shape change during the motion and rotation of the grain boundary by motion and reaction of the constituent dislocations without explicit mass transfer. Our model can be considered as a generalization of the Cahn-Taylor theory [Cahn and Taylor, 2004] by incorporating detailed formulas of the driving forces for the normal and tangential grain boundary velocities that depend on the constituent dislocations, their Burgers vectors, and the grain boundary shape, as well as the shape change of the grain boundaries, and is different from their earlier generalization based on mass transfer via surface diffusion [Taylor and Cahn, 2007]. Note that in some existing continuum models for the motion of grain boundaries and grain rotation [Li, 1962, Kobayashi et al., 2000, Upmanyu et al., 2006, Esedoglu, 2016], evolution of misorientation angle was included to reduce the grain boundary energy density, which are able to capture the grain boundary sliding but not coupling. In our continuum model, we use dislocation densities on the grain boundary as variables instead of the misorientation angle, which enables the incorporation both the grain boundary coupling and sliding motions. Using the derived continuum grain boundary dynamic model, simulations are performed for the dynamics of circular and non-circular two dimensional grain boundaries. We also perform discrete dislocation dynamics simulations for the dynamics of these grain boundaries and the simulation results using the two models agree excellently with each other. In particular, both our continuum and discrete dislocation dynamics simulations show that without dislocation reaction, a non-circular grain boundary shrinks in a shape-preserving way due to the coupling effect, which is consistent with the prediction of the continuum model in Taylor and Cahn [2007] based on mass transfer via surface diffusion.

Our continuum grain boundary dynamics model is based upon the continuum framework for grain boundaries in Zhu and Xiang [2014] derived rigorously from the discrete dislocation dynamics model. Previously, a continuum model for the energy and dislocation structures on static grain boundaries has been developed [Zhang et al., 2017a] using this framework. In fact, the energetic and dynamic properties of grain boundaries were understood based on the underlying dislocation mechanisms in many of the available theories, simulations and experiments [Read and Shockley, 1950, Li et al., 1953, Srinivasan and Cahn, 2002, Cahn and Taylor, 2004, Cahn et al., 2006b, a, Molodov et al., 2007, Gorkaya et al., 2009, Trautt and Mishin, 2012, Wu and Voorhees, 2012, McReynolds et al., 2016, Yamanaka et al., 2017]. Although direct discrete dislocation dynamics simulations are able to provide detailed information of grain boundary or interface structures and dynamics [Lim et al., 2009, Quek et al., 2011, Lim et al., 2012], continuum models of the dynamics of grain boundaries incorporating their dislocation structures are still desired for larger scale simulations without tracking individual dislocations.

The rest of the paper is organized as follows. In Sec. 2, we describe the two dimensional settings of the grain boundaries with their dislocation structures, and review the continuum framework in Zhu and Xiang [2014] based on which our continuum model will be developed. In Sec. 3, we present our continuum model for the motion of grain boundaries. The variational derivation method of the continuum model is presented in the Appendix. In Sec. 4, we discuss the calculation of the misorientation angle based on the Frank’s formula, which is maintained by the long-range dislocation interaction of the grain boundary during its motion. In Sec. 5, we present the formula for the change of misorientation angle (grain rotation) derived from our continuum dynamics model. We also derive the formulas for the tangential motions of grain boundaries due to the coupling and sliding effects based on our continuum model, and make comparisons with the formulas in the Cahn-Taylor theory [Cahn and Taylor, 2004, Trautt and Mishin, 2012]. Simulation results for the dynamics of circular and non-circular two dimensional grain boundaries using our continuum model and comparisons with discrete dislocation dynamics simulation results are presented in Sec. 6. Conclusions and discussion are made in Sec. 7.

2 Grain boundaries in two dimensions and review of the continuum framework in Zhu and Xiang [2014]

We consider the two dimensional problem that one cylindrical grain is embedded in another grain with arbitrary cross-section shape. The inner grain has a misorientation angle relative to the outer grain, and the rotation axis is parallel to the cylindrical axis. The grain boundary is then a pure tilt boundary.

Figure 1: The cross-section of a cylindrical grain boundary with geometric center . A point on the grain boundary can be written in polar coordinates as , where is the radius and is the polar angle. The normal and tangent directions on the grain boundary are denoted by and , respectively.

We assume that the cross-section curve of the grain boundary is in the plane and the rotation axis is in direction, and the geometric center (mass center) of the cross-section of the inner grain is the origin of the plane, see Fig. 1. Each point on the grain boundary can be written in the polar coordinates as , where is the radius and is the polar angle. We parameterize the two dimensional grain boundary by , and all the functions defined on , such as , are functions of the parameter . In particular, a point on the grain boundary has the coordinate . The derivative of a function defined on the grain boundary with respect to is denoted by , i.e., . The grain boundary tangent direction , the grain boundary normal direction , and the curvature of the grain boundary at a point on the grain boundary can be calculated as

(1)
(2)
(3)

The normal direction of the grain boundary in Eq. (2) is defined such that , where is the unit vector in the direction. We also have the relations , where is the arclength parameter of the grain boundary , and

(4)

Assume that on the grain boundary, there are dislocation arrays corresponding to different Burgers vectors with length , , respectively. All the dislocations are parallel to the axis, i.e., they are points in the plane.

Our continuum model is based on the continuum framework proposed in Zhu and Xiang [2014]. The dislocation densities on the grain boundaries are described by the dislocation density potential functions. A dislocation density potential function is a scalar function defined on the grain boundary such that the constituent dislocations of the same Burgers vector are given by the integer-valued contour lines of : , where is an integer. The dislocation structure can be described in terms of the surface gradient of on the grain boundary, which is . In particular, the local dislocation line direction is , and the inter-dislocation distance along the grain boundary is . Accordingly, the dislocation density per unit length on the grain boundary is

(5)

Here we allow the dislocation density to be negative to include dislocations with opposite line directions.

In our continuum model, it is more convenient to use the dislocation density per unit polar angle, which is

(6)

Here we have used Eqs. (4) and (5). The surface gradient can be expressed in terms of as

(7)

The dislocation line direction defined by becomes

(8)

When there are dislocation arrays on the grain boundary, these constituent dislocations are represented by , corresponding to different Burgers vectors , , respectively.

In the continuum framework presented in Zhu and Xiang [2014], the total energy associated with grain boundaries can be written as

(9)

where is the energy due to the long-range elastic interaction between the constituent dislocations of the grain boundaries, is the line energy of the constituent dislocations corresponding the commonly used grain boundary energy in the literature [Sutton and Balluffi, 1995, Read and Shockley, 1950] and is a generalization of the classical Read-Shockley energy formula [Read and Shockley, 1950], and includes the energy due to the interactions between the dislocation arrays and other stress fields such as the applied stress.

The variations of this total energy with respect to the change of the grain boundary and the change of dislocation structure on the grain boundary are

(10)
(11)

respectively, where

(12)

Here is the total force on the -th dislocation array, including the force due to the long-range elastic dislocation interaction , the force due to the local interaction between dislocations , and the force due to other stress fields (such as the applied stress field) , for . These forces are consistent with the Peach-Koehler force on a dislocation in the discrete dislocation dynamics model.

Following the discrete dislocation dynamics model [Xiang et al., 2003, Zhu and Xiang, 2010, 2014], the continuum dynamics of these dislocation arrays along the grain boundary is given by

(13)

where the dislocation velocity , and is the mobility associated with the total Peach-Koehler force of the -th dislocation array. If in the dynamics process, generation and removal of dislocations are critical, e.g. [Li et al., 1953, Srinivasan and Cahn, 2002, Upmanyu et al., 2006, Wu and Voorhees, 2012, Trautt and Mishin, 2012], using Eq. (11), the dislocation evolution equations are

(14)

where is the total force on the -th dislocation array in Eq. (12) or some of its contributions on the right-hand side, is some positive constant. The choice of Eqs. (13) and/or (14) depends on the physics of the dynamics process. The motion of the grain boundary in general is, following Eq. (10),

(15)

where is the mobility.

This continuum framework is general and applies to any dislocation arrays. Based on this framework, we will develop a two dimensional continuum model for the dynamics of grain boundaries including the motion and tangential translation of the grain boundaries and grain rotation that is consistent with the discrete dislocation dynamics model and atomistic simulations, see the next section. In the two dimensional problems, this new dislocation representation method based on dislocation density potential functions ’s and the classical method using the scalar dislocation densities are equivalent, see Eqs. (6) and (7). Continuum dynamics model for grain boundaries in three dimensions is being developed and will be presented elsewhere, in which it is more convenient to use dislocation density potential functions for the structure of the constituent dislocations.

3 The continuum model for grain boundary motion

In this section, we present our continuum model for grain boundary motion based on densities of the constituent dislocations. The grain boundary motion is coupled with the evolution of the constituent dislocations on the grain boundary, under both the long-range and local interactions of dislocations. The misorientation angle changes due to the coupling and sliding motions and the shape change of the grain boundary are naturally accommodated by the motion and reaction of the constituent dislocations. A critical idea in the continuum model is that when obtaining the driving force for the grain boundary motion, the variation of the total energy is calculated based on the conservation of the constituent dislocations [Zhu and Xiang, 2014], instead of the fixed grain boundary energy density as did in the literature. An example of such variational derivation is presented in the Appendix for the two dimensional problem in terms of the contribution of the (local) grain boundary energy.

We consider the dynamics of a closed curved grain boundary with radius function and dislocation arrays on with Burgers vector , , as described in the previous section. The motion of the grain boundary and the evolution of the dislocation structure on it are described by

(16)
(17)

Eq. (16) gives the normal velocity of the grain boundary , where , and are the forces on a dislocation with Burgers vector due to the long-range elastic interaction between dislocations, the local interaction between dislocations, and the applied stress field, respectively, and is the mobility of the dislocations. The velocity of a point on the grain boundary is the weighted average of the velocities of dislocations on the grain boundary with different Burgers vectors. This modification has been made from the variational normal velocity formula in Eq. (15) so that the velocity of the grain boundary is consistent with the dynamics of its constituent dislocations [Cahn and Taylor, 2004]. Here following the available phase field crystal simulations [Wu and Voorhees, 2012, McReynolds et al., 2016, Yamanaka et al., 2017], we assume that the temperature is high and the dislocation climb mobility is comparable with the glide mobility. Influences of the dislocation glide and climb mobilities on the grain boundary motion will be further explored in the future work. Another driving force for the grain boundary motion is the difference between the bulk energy densities of the two grains denoted by [Cahn and Taylor, 2004, Trautt and Mishin, 2012], and is the grain boundary mobility associated with this driving force. In the simulations of a finite grain embedded in another one, instead of using the grain boundary normal velocity in Eq. (16) directly, it is more convenient to use the inward radial velocity

(18)

where is the angle between the normal direction of the grain boundary and its inward radial direction (see Fig. 1).

Eq. (17) describes the evolution of the dislocation structure on the grain boundary. (Recall that for a function .) These equations are based on the dislocation motion along the grain boundary, driven by the forces due to the long-range elastic interaction between dislocations, the local interaction between dislocations, and the applied stress field. The first term in Eq. (17) is the motion of dislocations along the grain boundary following conservation law in Eq. (13) driven by the long-range elastic force. The importance of this term is to maintain the dislocation structure and the condition of cancellation of the far-field stress fields for the grain boundary (or the Frank’s formula [Frank, 1950, Bilby, 1955]) in a way that is consistent with the discrete dislocation dynamics. The quantity in the second term in Eq. (17) is the grain boundary energy density (when the long-range elastic interaction vanishes), which comes from the local dislocation interaction. The third term in Eq. (17) is due to the effect of the applied stress. Since the equilibrium dislocation structure on a grain boundary is stable except for the change of the misorientation angle (i.e. sliding along a fixed grain boundary) [Xiang and Yan, 2017], the major influences of the local energy and the applied stress are to change the dislocation structure by reactions [Li et al., 1953, Srinivasan and Cahn, 2002, Upmanyu et al., 2006, Wu and Voorhees, 2012, Trautt and Mishin, 2012], leading to grain boundary sliding. In these processes, the dislocations on the grain boundary are not conserved and the last two terms in Eq. (17) account for such changes of dislocation structure due to these two driving forces (see also Eq. (14)), in which and are the mobilities associated with the driving forces of the local energy and the applied stress, respectively.

In these evolution equations, the continuum long-range force on a dislocation of the -th dislocation arrays located at the point is

(19)

where is the line element of the integral along the grain boundary , the point varies along in the integral, and is the force acting on a dislocation of the -th dislocation array at the point generated by the -th dislocation array, with

(20)
(21)
(22)

Here is the shear modulus and is the Poisson ratio. These formulas can be derived from the continuum long-range elastic force on a low angle grain boundary in three dimensions in Ref. [Zhu and Xiang, 2014] (Eqs. (1) and (13) there) based on the discrete dislocation model [Hirth and Lothe, 1982].

The local force on a dislocation of the -th dislocation arrays in these evolution equations is

(23)

where is the curvature of the grain boundary. The energy density of the short-range interaction of all the dislocations on the grain boundary, i.e., the grain boundary energy density when the long-range elastic fields cancel out, is [Zhu and Xiang, 2014, Zhang et al., 2017a]

(24)

where is a dislocation core parameter and is a numerical cutoff parameter.

The Peach-Koehler force due to the applied stress is , where is the line direction of a dislocation in the -th dislocation array on the grain boundary. Using Eq. (8), the contribution of the applied stress the grain boundary velocity given in Eq. (16) can be written as

(25)

and the contribution of the applied stress to the dislocation structure evolution equation (17) can be written as

(26)

where the last expression holds for a constant applied stress , otherwise the expression in the middle should be used.

Remark: Essentially, it is equivalent to use the dislocation density per unit length , , as the variables in the continuum grain boundary dynamics model in Eqs. (16) and (17) based on the relation in Eq. (6). However, the evolution equations of ’s are no longer as simple as Eq. (17) because the arclength of the grain boundary also evolves as the grain boundary migrates.

4 Misorientation angle and effect of long-range dislocation interaction

In our continuum model, the misorientation angle between the two grains can be calculated based on the Frank’s formula, which is maintained during the motion of the grain boundary by the long-range elastic interaction between the constituent dislocations of the grain boundary.

4.1 Frank’s formula and misorientation angle

With the continuum model in Eqs. (16) and (17) for the motion of the grain boundary and evolution of the dislocation structure on it, the misorientation angle between the two grains at any point on the grain boundary can be calculated based on the Frank’s formula [Frank, 1950, Bilby, 1955], which is a condition for an equilibrium grain boundary dislocation structure and is equivalent to the cancellation of the long-range elastic fields [Frank, 1950, Bilby, 1955, Zhu and Xiang, 2014]. Using the representation of dislocation density potential functions, the Frank’s formula is [Zhu and Xiang, 2014]

(27)

where is the rotation axis and is any vector tangent to the grain boundary. In the case of two dimensional tilt boundaries being considered in this paper, the rotation axis is in the direction, and the Frank’s formula can be written as

(28)

or

(29)

An equivalent form is

(30)

Using the Frank’s formula in Eq. (28) or (29), the misorientation angle at each point on the grain boundary can be calculated by

(31)

Note that an alternative formula to calculate the misorientation angle is . Integrating Eq. (31) over the grain boundary, we have

(32)

where is the circumference of the grain boundary. This formula can be used to evaluate during the evolution of the grain boundary in which the shape of the grain boundary and the dislocation densities on it change. In this case, the pointwise formula in Eq. (31) may not give a perfectly constant value over the grain boundary.

4.2 Frank’s formula maintained by the long-range dislocation interaction

It has been shown that the Frank’s formula is equivalent to the cancellation of the long-range elastic fields generated by the constituent dislocations of the grain boundary [Frank, 1950, Bilby, 1955, Zhu and Xiang, 2014]. In the continuum model in Eqs. (16) and (17), the Frank’s formula is maintained by the long-range dislocation interaction. This is because the long-range dislocation interaction is much stronger than the local dislocation interaction (grain boundary energy) except for very small size grain (comparable with the inter-dislocation distance on the grain boundary) [Zhu and Xiang, 2014]. As a result, approximately during the motion of the grain boundary, the long-range dislocation interaction vanishes and the Frank’s formula holds except when the grain size is very small.

A proof of the equivalence of the Frank’s formula and cancellation of the long-range elastic fields generated by the constituent dislocations for a curved grain boundary can be found in Zhu and Xiang [2014]. Here we present an alternative calculation in two dimensions to show their equivalence. In fact, the long-range stress field generated by the constituent dislocations of the grain boundary , in the infinite two dimensional space, can be written as

(33)

where the point varies along the grain boundary in the integrals, is the line element of the integrals, and

(34)
(35)

Using divergence theorem and the fact that , we have

(36)

where is recalled to be the unit normal vector of the grain boundary and is the inner grain. Using Eqs. (33) and (36), the stress field due to the long-range dislocation interaction can be written as

(37)

Therefore, the long-range stress field , and accordingly the long-range elastic interaction energy of the constituent dislocations, vanish when the Frank’s formula in Eq. (28) holds.

5 Grain rotation and coupling and sliding motions

In this section, we present formulas for the rate of change of the misorientation angle (grain rotation) and the tangential velocities of the coupling and sliding motions of the grain boundary. Derivation of these formulas are based on the Frank’s formula, which is maintained by the long-range dislocation interaction in our continuum model in Eqs. (16) and (17).

5.1 Grain rotation

Using the Frank’s formula in Eq. (29), it can be calculated that the rate of change of the misorientation angle is

(38)

where is the outward radial direction of the grain boundary.

Recall that the coupling tangential motion of the grain boundary is associated with conservation of dislocations [Srinivasan and Cahn, 2002, Cahn and Taylor, 2004]. From the evolution of dislocation densities in our continuum model in Eq. (17), this means that the mobilities and due to the grain boundary local energy and the applied stress, respectively, are zero, because dislocations are not conserved under these motions. Further with the assumption of Frank’s formula, the first term in Eq. (17) due to the long-range dislocation interaction also vanishes. This gives for all under the pure coupling motion of the grain boundary. Therefore, the first term in Eq. (38) is associated with the coupling motion of the grain boundary. The second term in Eq. (38) is due to the change of dislocation densities. As shown above, when , the constituent dislocations are not conserved, leading to the sliding motion of the grain boundary [Srinivasan and Cahn, 2002, Cahn and Taylor, 2004]. Denoting these two contributions in Eq. (38) due to the coupling and sliding motions by and , respectively, we have

(39)
(40)

Note that in some existing continuum models for the motion of grain boundaries and grain rotation [Li, 1962, Kobayashi et al., 2000, Upmanyu et al., 2006, Esedoglu, 2016], evolution of misorientation angle was included to reduce the grain boundary energy density, which are able to capture the grain boundary sliding but not coupling. The grain rotation formula in Eq. (38) in our continuum model incorporates both the coupling and sliding motions and is consistent with the discrete dislocation dynamics model.

The grain rotation formula in Eq. (38) is derived as follows. Taking time derivative in Eq. (29), we have

(41)

It can be calculated that

(42)

Taking dot product with and , respectively, in Eq. (41) and using Eq. (42) and the Frank’s formula in Eq. (29), we have and . The second equation gives , and inserting it into the first equation, we have . This gives Eq. (38) by using the equation that , which can be obtained using the formulas of and in Eqs. (1) and (2).

5.2 Coupling and sliding velocities

The tangential velocity of the grain boundary is calculated based on the change of misorientation angle and the fixed center, which is

(43)

where is recalled to be the angle between the normal direction of the grain boundary and its inward radial direction, and , see Fig. 1. By Eqs. (43), (39), (40) and (18), we have

(44)
(45)
(46)

where and are the tangential velocities of the grain boundary due to the coupling and sliding motions, respectively.

By Eqs. (46) and (17) and the Frank’s formula (which means that the long-range dislocation interaction vanishes), we write the sliding velocity as

(47)
(48)
(49)

Here the sliding velocities and come from variation of the grain boundary local energy and the applied stress, respectively.

When the grain boundary is circular, is constant, , , and . At a point on the circular grain boundary where all the dislocations have the same Burgers vector that is in the direction , i.e., , we have and , and accordingly,

(50)

where is the shear component of the applied stress along the grain boundary. Note that at an arbitrary point on the circular grain boundary, dislocation densities of multiple Burgers vectors are in general nonzero, and the full formula in Eqs. (47)–(49) should be used instead of the simple formula in Eq. (50). As a result, the circular grain boundary may not remain circular during its evolution except for some special cases, e.g., without sliding ( for all ), see the simulations in the next section.

For this special case of circular grain boundary with single Burgers vector, the tangential velocity of the grain boundary given previously by the Cahn-Taylor model [Cahn and Taylor, 2004, Trautt and Mishin, 2012] is

(51)

where is the coupling factor, , is the grain boundary sliding mobility, and is the applied shear stress. Comparing Eqs. (50) and (51), we can see that our formula gives the same coupling term (the first term in Eq. (50)) as that in the Cahn-Taylor model. The last two terms in our formula in Eq. (50) that accounts for the grain boundary sliding are also the same as the corresponding terms in the Cahn-Taylor model in Eq. (51) based on the relations and if we further assume and .

Therefore, our continuum model generalizes the Cahn-Taylor model [Cahn and Taylor, 2004] by incorporating detailed formulas of the driving forces for the grain boundary coupling and sliding tangential motions that depend on the constituent dislocations, their Burgers vectors, and the grain boundary shape, as well as the shape change of the grain boundaries. For the special case of a circular grain boundary with a single Burgers vector, our tangential velocity formula reduces to the equation in the Cahn-Taylor model.

5.3 Shape-preserving grain boundary motion under pure coupling

Assume that the Frank’s formula holds during the evolution of the grain boundary. Taking time derivative in the Franks formula in Eq. (30) and using Eq. (42), we have