## Abstract

A novel continuum theory of incoherent interfaces with triple junctions is applied to study coupled grain boundary (GB) motion in three-dimensional polycrystalline materials. The kinetic relations for grain dynamics, relative sliding and migration of the boundary and junction evolution are developed. In doing so, a vectorial form of the geometrical coupling factor, which relates the tangential motion at the GB to the migration, is also obtained. Diffusion along the GBs and the junctions is allowed so as to prevent nucleation of voids and overlapping of material near the GBs. The coupled dynamics has been studied in detail for two bicrystalline and one tricrystalline arrangements. The first bicrystal consists of two cubic grains separated by a planar GB, whereas the second is composed of a spherical grain embedded inside a larger grain. The tricrystal has an arbitrary-shaped grain embedded inside a much larger bicrystal made of two cubic grains. In all these cases, analytical solutions are obtained wherever possible while emphasizing the role of various kinetic coefficients during the coupled motion.

## 1. Introduction

We develop a thermodynamically consistent continuum framework to study three-dimensional coupled grain boundary (GB) motion in the presence of triple junctions. A GB is modelled as a sharp incoherent interface connected to other GBs at junction curves. The irreversible dynamics at a GB is governed by its normal motion (GB migration) and a relative tangential sliding of the adjacent grains. The latter can arise owing to the intergranular viscous sliding, possibly as a result of the twist component of the GB, or/and as a result of coupling with GB migration [1]. In polycrystalline materials with rigidly deforming grains, as will be assumed presently, the sliding can be decomposed into a relative translation and a relative rotation between the adjacent grains. On the other hand, the irreversible dynamics at a junction is governed by the motion of the non-splitting junction curve. The presence of junctions can significantly influence the overall dynamics of all the GBs and the grains in their neighbourhood, for instance by inducing drag or altering diffusive flux [2]. The coupled motion, which requires sliding to be necessarily coupled with GB migration, is the dominant mechanism for both grain coarsening and plastic deformation in nanocrystalline (NC) materials with average grain size of the order of few tens of nanometres (hence a large volume fraction of GBs and triple junctions) [3,4]. This is unlike coarse-grained materials where GB migration and dislocation dynamics dominate grain coarsening and plastic deformation, respectively. The coupled motion has recently been studied theoretically [1,2,5,6], experimentally [7] and with molecular simulations [8]. Although some of these studies have included the effect of junction dynamics [2,8], all of them are restricted to two-dimensional grains and hence applicable only to polycrystals where each grain is columnar and identical in cross section along the length direction; such a restriction requires the GB to have only tilt, and no twist, character.

The main contributions of this paper include:

(1) A three-dimensional thermodynamic formalism including diffusion to deal with incoherent interfaces with junctions (§3). Junctions have been previously studied in the context of continuum thermodynamics but only with coherent interfaces and without diffusion [9,10]. On the other hand, thermodynamics of incoherent interfaces has been explored earlier without considering junctions [11]. All of these works were based on the framework of configurational mechanics. Our treatment, while extending to junctions with incoherent interfaces, takes an alternate viewpoint where we do not regard the configurational forces to be fundamentally on the same footing as standard forces (with their own balance laws, etc.). We introduce configurational forces in our formalism as mechanisms of internal power generation so as to ensure that the excess entropy production is restricted to interfaces and junctions. A two-dimensional version of this formalism was recently presented by the authors [2].

(2) Deriving kinetic relations for coupled GB motion in three dimensions (§4). The phenomenological kinetic equations are motivated from the dissipation inequalities derived from the second law of thermodynamics in confirmation with other standard balance laws of continuum physics. Our results extend earlier models of coupled GB motion to a three-dimensional setting. The first kinetic relations for the coupled motion were proposed by Cahn & Taylor [1,5] which were restricted to two-dimensions and only bicrystalline arrangements (hence no junctions). They also ignored the possibility of relative translation of grains while considering sliding at the GB only owing to the relative rotation. More recently, Basak & Gupta [2] have extended the model to include junctions and relative translation but still restricting themselves to two dimensions.

(3) Formulation of a vectorial geometrical coupling factor (appendix A). The coupling between the tangential and the normal motion of the GB is purely geometric and depends on the measure of incoherency at the boundary [1,12,13]. The incoherency is quantified by the net Burgers vector (given by Frank–Bilby relation) or equivalently by the interfacial dislocation density. The GBs in the present three-dimensional framework generally have a mixed character with both tilt and twist components. The coupling factor for a high angle planar symmetric tilt boundary, derived previously by Cahn

*et al.*[12,13], therefore, needs to be extended to include multiple sets of edge and screw dislocation arrays. The coupling factor now derived is a vectorial quantity rather than a scalar as has been the case in the earlier studies.

Our derivations are based on the following assumptions: (i) the individual grains experience only rigid deformations (i.e. translations and rotations); the deformations are, however, allowed to remain finite, (ii) the shape accommodation required for preventing void nucleation and interpenetration of the material in proximity of the GBs, during relative tangential motion between the grains, is accomplished by diffusion across as well as along the GBs and also along the junction curves; (iii) the velocities associated with various GBs, grains and junctions remain much smaller than the speed of sound in the material. The inertial effects are therefore ignored; (iv) the grains are free of defects and all the lattice imperfections are concentrated at the GBs and the junctions (this is reasonable for NC materials with their small grain size); and (v) no additional stress fields are present at the interface and the junction. The GBs are considered to be orientable surfaces (of arbitrary shapes) with five macroscopic degrees of freedom which include three misorientation angles and two independent variables describing the orientation of the GB. The junctions are arbitrary three-dimensional space curves with varying curvature, normal, binormal, torsion, etc. The excess energy density of a GB is assumed to depend on the five parameters (and additionally the curvature) mentioned above, whereas the excess energy density of a junction is assumed to depend only on the unit tangent associated with the junction curve.

The paper is organized in the following manner. In §2, we briefly introduce various geometrical preliminaries, including integral relations, required for our study. We derive the essential balance laws and dissipation inequalities in §3. In §4, we apply our theory to derive the kinetic relations for GB motion, grain dynamics and junction motion for two bicrystalline and one tricrystalline arrangements. A generalized derivation of the vectorial geometrical coupling factor is presented in appendix A. We conclude our study with a discussion on some open directions in §5.

## 2. Geometrical preliminaries

Let *b*, taken out from a polycrystalline arrangement depicted in figure 1*a*. It contains three subgrains *P*_{1}, *P*_{2} and *P*_{3}, three smooth GBs *Γ*_{1}, *Γ*_{2} and *Γ*_{3}, and a smooth junction curve *J*. The normal *n*_{i} to *Γ*_{i} is chosen such that it points inside *P*_{i}, where *i*=1,2,3. We denote the position vector of a point in *P* by ** x** and the time by

*t*. The grains are oriented differently with respect to a fixed coordinate.

Let ** A** and

**be second-order tensors. The derivative of a scalar-valued differentiable function of tensors, say**

*B**G*(

**), is a tensor ∂**

*A*_{A}

*G*defined by

*G*(

**+**

*A***)=**

*B**G*(

**)+∂**

*A*_{A}

*G*⋅

**+**

*B**o*(|

**|), where**

*B***|**

*B*^{2}=

**⋅**

*B***, where (⋅) represents the Euclidean inner product. Similar definitions can be made for vector and tensor-valued differentiable functions (of scalars, vectors and tensors). ∇ indicates the gradient of a differentiable field defined over**

*B**P*.

### (a) Bulk fields

Let *f* be a piecewise-smooth bulk field which is discontinuous across *Γ*_{i} and singular at *J*. We denote the jump of *f* across *Γ*_{i} as 〚 *f*]]=*f*^{+}−*f*^{−}, where *f*^{+} is the limiting value of *f* at ** x**∈

*Γ*

_{i}from the grain into which

*n*_{i}points and

*f*

^{−}is the limiting value from the other grain. If

*f*

_{1}and

*f*

_{2}are two piecewise continuous functions across

*Γ*

_{i}, and the product

*f*

_{1}

*f*

_{2}is distributive with respect to the sum, then 〚

*f*

_{1}

*f*

_{2}]]=〚

*f*

_{1}]]〈

*f*

_{2}〉+〈

*f*

_{1}〉〚

*f*

_{2}]], where 〈

*f*〉=(

*f*

^{+}+

*f*

^{−})/2 is the average of

*f*

^{+}and

*f*

^{−}. To deal with the singularity of the field at the junction, we carry out our analysis in a punctured region

*P*

_{ϵ}obtained by excluding a small tube

*T*

_{ϵ}of radius

*ϵ*from

*P*in the neighbourhood of the junction, cf. [9]. The outward normal to the boundary of the tube ∂

*T*

_{ϵ}is denoted by

**. The boundary ∂**

*m**T*

_{ϵ}moves with a velocity

**.**

*u*We assume that *f* satisfies the limit
*dv* is the volume measure on *P*. Using the standard transport relations for the bulk quantities, it can be shown that [9]
** v** is the particle velocity,

*V*

_{i}is the interfacial normal velocity,

*U*

_{i}=

*V*

_{i}−

**⋅**

*v*

*n*_{i}is the relative normal velocity of the interface and

*da*is the area measure on a surface. Let

**and**

*a***denote a piecewise-smooth vector and tensor field, respectively, defined in**

*A**P*but allowed to be singular at the junction. The divergence theorem requires [9]

### (b) Interfacial fields

Consider an orientable surface *Γ* (subscript *i* is presently dropped), with boundary a closed piecewise-smooth curve ∂*Γ*, and let ** n** and

*V*be the associated unit normal field and normal velocity field, respectively. The surface gradient of a scalar field

*g*, vector field

**and tensor field**

*g***, all smoothly defined over**

*G**Γ*, are defined as

**=**

*P***−**

*I***⊗**

*n***is the projection tensor (**

*n***is the identity tensor and ⊗ denotes the dyadic product); while calculating ∇**

*I**g*(etc.), one has to use a smooth extension of

*g*in a small neighbourhood of

*Γ*. The surface divergence of these fields is defined by

**, where tr represents the trace operator and the superscript ‘T’ stands for the transpose. The surface Laplacian of**

*k**g*is given by Δ

^{S}

*g*=div

^{S}(∇

^{S}

*g*). The curvature tensor field

**and the total curvature**

*L**κ*associated with

*Γ*are defined as

**=−∇**

*L*^{S}

**and**

*n**κ*=tr

**, respectively.**

*L*Let ** t** be the outward unit normal to the closed curve ∂

*Γ*such that

**⋅**

*n***=0. When**

*t***and**

*G***satisfy**

*g*

*G***=**

*n***0**and

**⋅**

*g***=0, respectively, the surface divergence theorem yields [14]**

*n**dl*is the length measure of a curve.

The normal time derivative of *g* following *Γ* is given by *g* as experienced by an observer sitting on the moving surface *Γ* [14]. The first term indicates the local rate of change of *g* at a fixed material position, whereas the second term represents the rate of change of *g* owing to influx of particles along ** n** as the interface moves with velocity

*V*. The following identities can be readily verified [15]:

On the other hand, the intrinsic time derivative of *g* following ∂*Γ* is given by ** w** is the intrinsic velocity of ∂

*Γ*, such that

**=**

*w**V*

**+**

*n**W*

**, and**

*t**W*is the velocity of ∂

*Γ*along

**[15]. Thus,**

*t**g*following

*Γ*and a term representing the change in

*g*owing to the incoming particles from the neighbourhood along the tangential direction

**.**

*t*We will need the following transport theorem for *Γ* such that a part of ∂*Γ* intersects with ∂*P* and the rest with *J* [9]:
*q*_{p} is the intrinsic (independent of the parametrization) velocity of the junction.

### (c) Junction fields

Let *δ* and *ι* denote the terminal points of the junction curve *J*, and let ** l** be the unit tangent to the curve such that it is directed towards

*ι*. The normal and binormal vectors associated with

*J*are denoted by

**and**

*ν***, respectively. The projection tensor**

*b***=**

*Q***−**

*I***⊗**

*ν***−**

*ν***⊗**

*b***=**

*b***⊗**

*l***maps any vector on to the tangential direction of**

*l**J*. The intrinsic velocity field of the junction, which can be decomposed as

*q*_{p}=

*q*

_{ν}

**+**

*ν**q*

_{b}

**, is such that (**

*b***−**

*I***)**

*Q***→**

*u*

*q*_{p}as

*ϵ*→0 [9]. The velocity of terminal points is denoted as

*χ*, is given by

*χ*is given by [9]

*κ*

_{J}is the scalar curvature of the junction curve. The gradient of

*χ*along the junction curve is defined as ∇

^{J}

*χ*=

**∇**

*Q**χ*. Similarly, for a vector field defined on

*J*, we introduce ∇

^{J}

*q*_{p}=(∇

*q*_{p})

**. We note the identity**

*Q*It is useful to decompose an integral over the tube surface around the junction as [9]
*T*_{ϵ} is the envelope of the circles *C*_{ϵ} of radius *ϵ*.

## 3. Balance laws and dissipation

We now obtain the consequences of balance of mass and momentum, as well as obtain local dissipation inequalities in the bulk, at the interface, and at the junction, while restricting to the assumptions enlisted in §1. In particular, we allow the individual grains to deform only rigidly, although without restricting the magnitude of the deformation so as to accommodate large intergranular slip and grain rotation.

### (a) Balance of mass

The rate of change of total mass in *P* is balanced by the mass transport into the region via bulk diffusion across ∂*P*, GB diffusion at the edge *Γ*_{i}∩∂*P*, and diffusion at *ι* and *δ*. Neglecting excess mass densities of the GBs and the junction, the mass balance can be written as
*ρ* is the mass density of the bulk grain, ** j** is the bulk diffusional flux,

*h*_{i}is the tangential diffusional flux on

*Γ*

_{i}, and

*h*

_{J}is the diffusional flux along the junction. Using transport theorem (2.2) and divergence theorems (2.3) and (2.7)

_{2}, and localizing the result owing to the arbitrariness of

*P*, we can obtain the following local equations

Here, and in rest of this paper, we have considered contact interactions to be concentrated over edges and points. These are to be understood in terms of the physical concentration of the relevant quantity over the respective domain. In contrast, if these interactions were to take place at edges and points which represent geometrical singularities on the boundary of *P*, then we would require a higher gradient theory to justify their existence [16].

### (b) Balance of linear momentum

Neglecting inertia and body forces, and assuming absence of interfacial and junction stress fields, the balance of linear momentum is given by
** σ** is the symmetric Cauchy stress tensor. Using (2.4), the following local equations are readily obtained [9]

*C*

_{ϵ}is zero in the limit

*ϵ*→0, although the stress field can still be singular at the junction (weak singularity).

### (c) Dissipation inequality

Let ** Ψ** be the bulk free energy density,

*γ*

_{i}the interfacial free energy per unit area of

*Γ*

_{i}, and

*η*the free energy per unit length of

*J*. For an isothermal environment, the mechanical version of the second law requires the rate of change of the total free energy pertaining to

*P*to be less than or equal to the total power input into

*P*[14,17], i.e.

*μ*is the chemical potential. We assume that the chemical potential is continuous across the interface and at the junction (i.e. local chemical equilibrium [17]). The first integral on the right-hand side of (3.9) is the power input through the tractions acting on ∂

*P*; the next three terms are contribution to power input owing to mass flux at ∂

*P*, ∂

*P*∩

*Γ*

_{i}, and the endpoints of

*J*, respectively. The last two terms are non-standard, and the significance of their appearance in the dissipation inequality needs to be discussed in detail. It is clear that as interfaces evolve, their domain of intersection with

*P*changes; there would be portions of the interface which, previously present outside

*P*, are now inside it, and vice versa. In other words, the set of points belonging to an interface and lying within

*P*changes continuously as the interface evolves. The change in the configuration of interfaces and junction curve within

*P*produces additional entropy at the edges ∂

*P*∩

*Γ*

_{i}and at the terminal points of

*J*. However, in the spirit of Gibbs thermodynamics, we require the excess entropy generation to be necessarily restricted to the GBs and the junction. The non-standard terms are therefore included in the inequality to ensure that there is no excess entropy production at the arbitrary edges and terminal points. These additional power input terms are to be considered in (3.9) only when the edges and the terminal points lie on the surface of an interior part of a body. The precise form of

*c*_{i},

*τ*_{i}and

**will depend on the constitutive prescriptions for free energies and stress, as well on the nature of allowable dissipative fluxes. At this point, these are to be understood as agents of power input, in conjugation with the respective intrinsic velocities, so as to ensure that the net entropy generation meets the above-mentioned requirement. Such terms also appear in the framework of configurational mechanics [9,11,15], where the existence of**

*ω*

*c*_{i},

*τ*_{i}and

**is assumed**

*ω**a priori*as fundamental forces which satisfy certain balance relations. Our treatment (see also [2,6,14]) is motivated purely from the viewpoint of quantifying excess entropy generation. We note that it may be possible to derive the balance laws as well as the global dissipation inequality starting with a more general framework, where the two-dimensional interface is obtained as a limit of a three-dimensional non-material domain, as is done in [18,19]. In doing so, it might be possible to have a better understanding of the non-standard power terms included in (3.9).

Before deriving the consequence of the global dissipation inequality, we briefly digress to discuss the status of configurational balance laws within our framework. In a purely conservative setting (e.g. nonlinear elasticity), these balance laws, which are independent of the standard balance laws, are derived from the horizontal variation of the energy functional [20]. In non-conservative setting such balances are usually postulated (rather than derived) as independent relations [21,22]; an exception is the recent work by Mariano [23,24] where the balance of configurational forces is derived from the invariance of the so-called relative power under roto-translating changes in observers. In our work (see also [2,6]), the configurational balance laws appear in the form of kinetic relations (see §4) which are motivated from the local dissipation inequalities obtained from (3.9). Under equilibrium, they reduce down to the familiar balance laws obtained using variational arguments; for instance, junction kinetic equation reduces to Herring's relation under equilibrium solution (see the end of this subsection). In the present formalism, non-trivial balances of configurational forces appear only at the interfaces and junctions.

Using transport theorems (2.2), (2.9) and (2.10), divergence theorems (2.3) and (2.7), and the decomposition (2.11), we can rewrite the inequality (3.9) as
*Ω*=** Ψ**−

*ρμ*(the grand canonical potential). To derive (3.12), on the other hand, we have used (3.3) and (3.7); here,

**=**

*E**Ω*

**−**

*I***is the projection of the bulk Eshelby tensor on the current configuration.**

*σ*To determine the precise form of *c*_{i}, *τ*_{i} and ** ω**, and also to obtain the local dissipation inequalities associated with the grains, GBs and junction, we will now prescribe the constitutive nature of the GB energy and the junction energy. Towards this end, we assume the GB energy to depend on the misorientation between the grains, the normal to the GB and curvature. The former two dependencies are standard in material science literature (cf. ch. 12 in [25]). The curvature dependence is primarily introduced to regularize the governing partial differential equations for capillary-driven GB motion, which otherwise become backward parabolic and hence unstable in certain ranges (GB spinodals) of the orientations. We follow Gurtin & Jabbour [15] in assuming the following quadratic dependence of GB energy on curvature:

**is the misorientation tensor given by (**

*Θ*

*R*^{+})

^{T}

*R*^{−};

*ϵ*

_{1}and

*ϵ*

_{2}are scalar constants such that

*ϵ*

_{1}>0 and

*ϵ*

_{2}+

*ϵ*

_{1}/2>0. The rotation

*R*^{+}is the orientation tensor of the grain into which

**points, and**

*n*

*R*^{−}is the orientation tensor of the other grain. We introduce

**is symmetric and satisfies**

*M***=**

*MP***. The junction energy density, on the other hand, is assumed to be a function of the unit tangent along the junction curve, i.e.**

*M**Γ*

_{i}, and per unit length of

*J*, respectively. The expressions for these rates are given in equations (3.21)–(3.23). In deriving the above inequality, we have also assumed the junctions to be non-splitting, i.e.

*V*

_{i}=

*q*_{p}⋅

*n*_{i}. The first term in the above inequality is the net entropy generation within the grains. The next two terms are excess entropy generation at the GBs and at the junction, respectively. The rest of terms in (3.18) are the entropy production rate at the edges

*Γ*

_{i}∩∂

*P*, and the terminal points

*δ*and

*ι*. We however require that the excess entropy production must not have any contribution from the edges of the GBs and the terminal points of

*J*, all of which are a part of ∂

*P*. This is reasonable, because the entropy generation in

*P*should only be within the grains, at the interfaces, and at the junction. Any additional source should vanish. Consequently,

Substituting (3.19) and (3.20) back into (3.18), and localizing the result, we obtain the following local dissipation inequalities

Equation (3.21) gives the entropy production rate per unit volume within the grain. For rigidly deforming grains ** σ**⋅∇

**=0 and div**

*v***=0. Additionally, if we assume that**

*v***⋅∇**

*j**μ*≤0. Equation (3.22) contains the dissipation rate per unit area of the GB, with contribution from boundary migration, relative translation of grains at the boundary, GB diffusion and misorientation change. The inequality therein forms a basis for postulating kinetic relations for coupled GB motion, as is done in §4. It can also be a starting point for motivating kinetic relations for motion of incoherent phase boundaries with diffusion- and curvature-dependent boundary energy [11,15], as well as for a variety of physical phenomena involving coherent interfaces [17]. An analogous inequality, valid for a one-dimensional interface in a two-dimensional grain, was derived recently by Basak & Gupta [6].

Equation (3.23) gives the net dissipation rate per unit length of the junction curve, with contribution owing to motion of the curve, diffusion along it, and evolution of orientation of the intersecting boundaries. A comment is in order regarding the contribution owing to the latter, represented by the last term on the LHS of the inequality. It is evident that this term is linear in scalar parameters (*ϵ*_{1} and *ϵ*_{2}) which appear in the curvature-dependent part of the GB energy. The scalar parameters are usually infinitesimally small, ensuring that curvature-dependent part of the energy is significant only at the corners [15]. The curvatures of intersecting boundaries, as they approach the junction, are finite and hence |*τ*_{i}| are small. We can therefore ignore the last term on the LHS of the inequality (3.23) within the present analysis. Second, in the context of GB dynamics, we assume the density field to remain bounded at the junction curve and the velocity in each grain to be resulting only a simple rigid body motion (hence no strains in the grain). With these assumptions, and keeping in mind the weak singularity condition (3.8), we can show that the closed integral terms in (3.23) vanish in the limiting sense. We can rewrite (3.23) and (3.25) under all these considerations as

## 4. Kinetic relations

The governing equations for coupled GB dynamics with junctions can be derived starting from inequalities (3.22) and (3.23) by first identifying various dissipative fluxes, and the associated driving forces and then assuming linear kinetics. Towards this end, we can use Onsager's relations [27] in an unambiguous manner, because fluxes are clearly defined as time-rate of change of the state variables. That this is not so in general, as in the case of heat conduction and viscous flow, has been pointed out by Truesdell [28]. We illustrate the derivation of kinetic relations using three crystalline arrangements as described in the following subsections.

### (a) Bicrystal-I

The first bicrystalline arrangement is as shown in figure 2*a* such that the traction on the outer boundaries perpendicular to *e*_{1} is *τ**e*_{2} and the traction on the outer boundaries perpendicular to *e*_{2} is *τ**e*_{1}. We neglect all kinds of atomic diffusion. The GB is considered to be of a mixed type, with both tilt and twist components, where the misorientation is given by ** Θ**=

**+**

*I**θ*

_{1}(

*e*_{1}×)+

*θ*

_{3}(

*e*_{3}×) for small

*θ*

_{1}and

*θ*

_{3}, where (

**×) represents a skew tensor with components given by (**

*e***×)**

*e*_{jk}=

*ε*

_{jlk}

*e*

_{l}(here

*ε*

_{jlk}is the permutation symbol); as discussed in appendix A,

*θ*

_{1}and

*θ*

_{3}determine the twist and the tilt characteristic, respectively, of the GB. Whereas the deformation of grains in the wake of a moving GB, under the external loading considered here, is simple shear for a tilt GB, it is more complicated if the GB is of mixed type [7]. The array of edge dislocations is driven by the Peach–Koehler force to move the GB in normal direction while translating the grains parallel to the GB. On the other hand, the simultaneous movement of two perpendicular sets of screw dislocation arrays results into a relative rotation of the adjacent grains about the GB normal. The GB motion, the relative tangential translation and the grain rotation are, in general, all coupled to each other.

The state of stress throughout the bicrystal is taken as ** σ**=

*τ*(

*e*_{2}⊗

*e*_{1}+

*e*_{1}⊗

*e*_{2}); this clearly satisfies both the equilibrium equations and the traction boundary conditions. Based on the experimental observations in [7,29], we assume the tilt angle to remain fixed while allowing the twist angle to evolve owing to the relative rotation between the grains. The axis of rotation is taken to coincide with

*e*_{1}. Without loss of generality, the grain

*G*

^{+}can be assumed to remain stationary, i.e.

*v*^{+}=

**0**, and

*G*

^{−}moving with a velocity

*G*

^{−}in the direction of

*e*_{2}and

**=**

*x**x*

_{1}

*e*_{1}+

*x*

_{2}

*e*_{2}+

*x*

_{3}

*e*_{3}is the position vector. Observing that 〚

*v*

_{n}]]=0, the dissipation inequality (3.22) reduces to

*f*

_{θ}=(

*τx*

_{3}−∂

*γ*/∂

*θ*

_{1}) and

*f*

_{c}=−

*τ*. Following Onsager, we postulate the following pair of coupled kinetic relations [6]:

*β*

_{2}=

*θ*

_{3}is the geometrical coupling factor as calculated in the previous section. Note that the geometrical coupling exists only with respect to the translational velocity, as the rotational part of

*v*^{−}amount to pure sliding. Accordingly, it is the direction of

*x*

_{1}and

*x*

_{2}, an initially planar GB will not necessarily remain planar (cf. [7]) and GB diffusion will be required to prevent void-formation/interpenetration at the GB.

### (b) Bicrystal-II

We consider a bicrystal with a spherical grain *G*^{−} (of radius *R*) embedded within a much larger grain *G*^{+}, as shown in figure 2*b*, with misorientation between the grains given by ** Θ**=

**+**

*I**θ*

_{1}(

*e*_{1}×)+

*θ*

_{2}(

*e*_{2}×)+

*θ*

_{3}(

*e*_{3}×); i.e. grain

*G*

^{−}was obtained by rotating it from

*G*

^{+}by small angles

*θ*

_{1},

*θ*

_{2}and

*θ*

_{3}about

*e*_{1},

*e*_{2}and

*e*_{3}, respectively, where the orthonormal basis vectors {

*e*_{1},

*e*_{2},

*e*_{3}} form a coordinate frame with origin at the centre of the sphere. The GB

*G*

^{+}to remain fixed and allow the inner grain

*G*

^{−}to rotate (without translating). We assume that the external stress is absent, the grains are rigid and free of defects, the free energy of the grains is vanishing, and the volumetric diffusion is absent. Let us consider an orthonormal spherical basis {

*e*_{R},

*e*_{ξ},

*e*_{ϕ}} with origin at the centre of the embedded grain (

*ξ*is the polar angle and

*ϕ*is the azimuthal angle). The GB normal

**points into**

*n**G*

^{+}, hence

**=**

*n*

*e*_{R}. The angular velocity of

*G*

^{−}(axial vector of

*v*^{−}=

**×**

*w***, where**

*x***is the position vector. Recalling that**

*x*

*v*^{+}=

**0**we obtain

*V*_{t}=

**(**

*P*

*v*^{+}−

*v*^{−})=−

*v*^{−}, where

*V*_{t}represents the relative tangential velocity of

*G*

^{−}with respect to

*G*

^{+}at the GB.

The spherical GB rotates and shrinks without any shape change. This does not require any shape accommodation mechanism such as GB diffusion. We will therefore consider ** h**=

**0**. Assuming isotropic GB energy and defining

**=**

*θ**θ*

_{1}

*e*_{1}+

*θ*

_{2}

*e*_{2}+

*θ*

_{3}

*e*_{3}(the axial vector of

**−**

*Θ***) the dissipation inequality, given by (3.22), reduces to**

*I*_{2}with

**to obtain**

*x*_{1}relates

**introduced in appendix A, whereas (4.5)**

*β*_{2}relates

_{3}is motivated from the two-dimensional counterpart of the present discussion [6]. Next, we specialize these kinetic equations for a spherical GB, under various additional assumptions, and present analytical solutions wherever possible. When reduced to a two-dimensional setting, the derived relations will be identical to those obtained previously for a circular GB [1,6].

*GB migration*: when both geometrical coupling and GB sliding are negligible, kinetic equations (4.3)_{1} and (4.3)_{2} simplify to the well-known equations for curvature driven GB migration: ** w**=

**0**. Using

*κ*=−1/

*R*(for a spherical GB), they can be solved to obtain

**(**

*θ**t*)=

*θ*_{0}, where

*R*

_{0}is the initial GB radius and

*θ*_{0}is the initial misorientation. Both

*γ*have been treated as constants.

*Coupled motion without GB sliding*: at temperatures far below the melting point, GB viscous sliding becomes negligible and geometrical coupling plays the dominant role in grain rotation [12]. The kinetic relations (4.3)_{1} and (4.3)_{2} then take the form ** β**=(

**−**

*Θ***)**

*I*

*e*_{R}(see appendix A for the latter expression), (4.5)

_{1}and a cross-product of second kinetic relation with

**=**

*x**R*

*e*_{R}furnishes

*R*(

*t*)

**(**

*θ**t*)=

*R*

_{0}

*θ*_{0}for all

*ξ*and

*ϕ*. Note that we have used the same geometrical coupling factor associated with a planar GB for analysing the kinetics of a curved GB (see appendix A); this assumption is motivated from the two-dimensional atomistic studies where the geometrical coupling factor was observed to be same for planar and curved GBs (cf. [8] and the references therein). We use the isotropic energy of the form

*γ*

_{0}and

*A*

_{c}are constants depending on the materials. Restricting ourselves to a spherical GB (for which

*κ*=−1/

*R*, and

*Coupled motion without geometrical coupling*: at temperatures close to the melting point, viscous sliding at the GB dominates over the geometrical coupling to govern grain rotation [12]. The kinetic relations (4.3)_{1} and (4.3)_{2} then reduce down to _{2} requires *R* and ** θ**:

*Fully coupled motion*: we finally consider the situation when both geometrical coupling and GB sliding will contribute comparable to the evolution of grain rotation. Assuming _{1} and (4.3)_{2}, we derive
** w** can be obtained by substituting (4.3)

_{1}in (4.3)

_{2}

### (c) Tricrystal

We consider a tricrystal, as shown in figure 2*c*, where grain *G*_{1} is embedded inside a larger bicrystal made of two cubic grains *G*_{2} and *G*_{3}. The tricrystal is subjected to external stress. The configuration has three GBs , with unit normals denoted by *n*_{i} (*i*=1,2,3), and a closed junction curve *J*. The normals are chosen such that both *n*_{1} and *n*_{2} point into *G*_{1}, whereas *n*_{3} points into *G*_{2}. In deriving the kinetic laws, we assume both the volumetric diffusion and the diffusion along the junction to be negligible. In addition, the magnitude of applied stresses are taken to be small enough, so that elastic and plastic deformation of the grains can be neglected. Under the combined effects of GB capillary force and the applied stress field the GBs will migrate, grain *G*_{1} will rotate and translate (as a rigid body), grains *G*_{2} and *G*_{3} will translate rigidly relative to each other, and the junction *J* will move in space, all coupled to each other.

In the absence of defects within the grain, the orientation field associated with grain *G*_{i}, denoted by *R*_{i}, will be homogeneous throughout the grain. We define the misorientation tensor for the respective GBs as *G*_{2} and *G*_{3} to be non-rotating, thereby fixing their orientations once for all. As a result *v*_{3}=**0**, where ** w** is the angular velocity of

*G*

_{1}(the axial vector of

*C*_{i}is the rigid translation of

*G*

_{i}(grain

*G*

_{3}has been assumed to remain fixed). We define the relative translation velocity of adjacent grains at the respective GBs as

^{S}

*h*_{a}=−

*ρ*

**×**

*x*

*n*_{a}⋅

**for**

*w**a*=1,2, and

**=**

*j***0**,

*h*

_{J}=0, and

*ρ*,

**are non-singular at the junction) to obtain**

*v*

*h*_{a}=

*A*_{a}

**and**

*w*

*A*_{i}is a second-order tensor which depends on the geometry of (see [2] for a similar calculation for a two-dimensional tricrystal). We also relate these fluxes to the chemical potential by assuming the Fick's law for superficial diffusion

*h*_{i}=−

*D*_{i}∇

^{S}

*μ*for

*i*=1,2,3 [15] where

*D*_{i}is the (symmetric and tangential) diffusivity tensor along .

For the present case, the dissipation inequality in the bulk (3.21) is trivially satisfied. The dissipation inequalities at the GBs, given by (3.22), are however non-trivial and will be used in the following to derive the kinetic equations. Using Fick's law, and neglecting the terms of the order of *i*)
*D*_{i} which satisfies *Υ*_{a} is the axial vector associated with the skew part of *a*=1,2). In writing them, we assume the relative translational velocities *G*_{1}; a theory without this assumption can easily be constructed along similar lines. The kinetic relations are taken as
_{1}, (4.10)_{3} and (4.10)_{2} back into the dissipation inequality (4.9), we derive the following restrictions:

Furthermore, when sliding is active and _{1} with the help of (4.10)_{2} to obtain
*a*=1 and *a*=2, respectively. The two expressions can be added to obtain the following expression for ** w**:

*G*

_{1}in (4.12).

The kinetic relations for *β*_{3}, and *V*_{3} from (4.13)_{2} in (4.13)_{1} and integrating the final expression over *G*_{1} has been neglected.

On the other hand, the governing equation for average translation velocity of the embedded grain can be obtained by first integrating (4.10)_{3} for *a*=1 and 2, respectively, and then adding them to obtain
_{2} we have the governing equation for the normal motion of

Finally, we derive the kinetic relations which govern junction dynamics. For negligible diffusion along the junction curve, the dissipation inequality (3.27) simplifies to *V*_{i}=*q*_{p}⋅*n*_{i} (see [2,30] for a detailed calculation in two-dimension).

To conclude, the complete set of kinetic equations governing the coupled GB motion in the tricrystalline arrangement includes (4.11) for the motion of _{2} for the motion of *G*_{1} (the outer grains are non-rotating) (4.15) and (4.14) for the translation of grains *G*_{1} and *G*_{2}, respectively (whereas grain *G*_{3} is stationary), and (4.16) (in association with the compatibility condition) for the motion of the junction curve.

## 5. Concluding remarks

We have presented a thermodynamically consistent three-dimensional study of coupled GB motion in the presence of junctions, hitherto restricted to two-dimensional crystalline materials. Towards this end, we introduced a novel continuum mechanics-based theory of irreversible dynamics of incoherent interfaces with junctions, which allows for diffusion in the bulk, on the interface, and along the junction curve. The various local dissipation inequalities derived therein were used to motivate kinetic relations for the coupled GB motion in two bicrystals and one tricrystal. These relations were solved analytically whenever it was possible to do so, but were otherwise left in a form amenable to numerical computations. In any case, the results clearly demonstrated the effect of coupling on the grain dynamics. Consider for instance the shrinking of an isolated grain, embedded within a larger grain, under the action of capillary. Without coupling, the embedded grain can disappear only by shrinking to a vanishing size. However with coupling, the grain can disappear by aligning its orientation with the outer grain even before it has shrunk significantly [6]. The proposed kinetic relations also emphasize the coupling of junction dynamics with both grain and GB motion. Depending on the junction mobility, grain dynamics can experience a substantial drag compared with the case with no junctions [2].

This work can form a basis for research in several future directions. The theory of incoherent interfaces, which includes junctions and diffusion, is in fact applicable to a more general situation where the grains are allowed to deform plastically. The resulting framework would be useful for phenomena which involves coupling of plastically deforming bulk with moving incoherent interfaces and junctions [31]. Second, the kinetic relations derived for the tricrystal can be used to study the coupled motion in polycrystalline materials containing large number of grains and junctions. Of course, this will demand significant computational effort and hence efficient numerical algorithms. A related direction of work would be to develop numerical techniques (such as level set methods) for solving equations of coupled motion of an embedded grain with anisotropic constitutive properties. Third, the computation of the vectorial coupling factor, which has been restricted here to small angle GBs, should be extended to large angle GBs. Finally, three-dimensional atomistic simulations will be required to clarify the nature of various kinetic coefficients (including the coupling factor), in particular, regarding their dependence on three misorientation angles and two orientation angles of the GB.

## Data accessibility

This research involves no data.

## Funding statement

Neither author received funding to carry out this research.

## Author contributions

A.G. planned the research. A.B. conducted the research, and worked out all the derivations and examples. Both A.G. and A.B. analysed the results and contributed equally to the writing.

## Acknowledgements

We thank anonymous reviewers for their constructive comments which led to considerable improvement in the manuscript.

## Appendix A. Geometrical coupling factor

During the migration of a tilt or a mixed GB, the adjacent grains undergo a tangential motion giving rise to a coupled dynamics (ch. 14 in [25], and [7,12,29]). The deformation of the grain in the wake of a moving GB, during coupled motion, is essentially controlled by the intrinsic edge dislocation content at the GB. Screw dislocations, if present, just glide along the GB plane and contribute only to grain sliding without affecting the coupling process [7]. For a moving planar symmetric tilt GB, whose wake experiences a simple shear deformation, Cahn *et al.* [12] introduced geometrical coupling factor as the ratio of the relative tangential velocity (in the absence of viscous sliding) to the GB velocity. For a large misorientation range of a symmetric tilt boundary, containing single array of edge dislocations, the coupling factor (denoted by *β*) was calculated to be *θ* is the misorientation angle. This was later verified both in experiments and atomistic simulations [13,29]. In general, however, most of the GBs are mixed, containing multiple sets of edge and screw dislocation arrays. If the above definition of the geometrical coupling factor is generalized to an arbitrary GB, the result will be a vector given by
*a*.

*Symmetric tilt GB*: let ** F** be the total deformation gradient of the grains with respect to a fixed reference configuration. Compatibility at the boundary requires 〚

**]]=**

*F***⊗**

*a*

*n*_{r}and 〚

**]]=−**

*v**V*

_{r}

**, where**

*a***is an arbitrary vector, whereas**

*a*

*n*_{r}and

*V*

_{r}, respectively, are the normal vector and the normal velocity of the GB in the reference configuration. Without loss of generality, we can assume that

*F*^{+}=

**and**

*I*

*v*^{+}=

**0**. Consequently,

*n*_{r}=

**and**

*n**V*

_{r}=

*V*[14]. On the other hand, the multiplicative decomposition of

**, under the present assumption of elastically rigid grains, takes the form**

*F***=**

*F*

*R*

*F*^{p}[14], where

**is the lattice rotation tensor and**

*R*

*F*^{p}is the plastic deformation gradient. If we assume the plastic deformation to be isochoric (det

*F*^{p}=1), and that

*F*^{p+}=

**, then the above considerations lead to**

*I*

*F*^{−}=

**+**

*I***⊗**

*β***, where**

*n***=−**

*β*

*P***is a tangential vector (the superscript ‘minus sign’ will be suppressed hereafter). The total Burgers vector**

*a***of all the GB dislocations cut by a unit vector**

*B***lying on the GB plane is given by the Frank–Bilby equation**

*p***=(**

*B***−**

*I*

*R*^{T})

**=(**

*p***−**

*I*

*F*^{p})

**[14]. Assuming all the edge dislocations at the GB to glide in a single slip direction, we can write the resulting plastic distortion rate as**

*p***and**

*s***stand for slip rate, unit slip vector and unit normal to the slip plane, respectively (**

*m***and**

*s***are mutually perpendicular). With initial values of**

*m**ζ*and

*F*^{p}as 0 and

**respectively, time integration of the evolution equation yields**

*I*,*G*

^{−}is related to that of

*G*

^{+}by an anticlockwise rotation of angle

*θ*

_{3}about

*e*_{3}-axis, then we can write

**=**

*p*

*e*_{2}, and recalling that

**=|**

*B***|**

*B***, we obtain**

*s**ζ*=−|

**|/(**

*B***) and consequently,**

*m*⋅*e*_{2}*et al.*[12]. For small misorientation angle, the coupling factor takes a simple form

**=**

*β**θ*

_{3}

*e*_{2}(ch. 14 in [25]).

*General tilt boundary with small misorientation*: restricting ourselves to small misorientation, we consider a tilt GB such that ** R**=

**+**

*I**θ*

_{2}(

*e*_{2}×)+

*θ*

_{3}(

*e*_{3}×), where (

**×) represents a skew tensor with components given by (**

*e***×)**

*e*_{jk}=

*ε*

_{jlk}

*e*

_{l}(here

*ε*

_{jlk}is the permutation symbol). In other words, grain

*G*

^{−}is obtained by rotating the reference grain

*G*

^{+}anticlockwise about

*e*_{2}and

*e*_{3}by small angles

*θ*

_{2}and

*θ*

_{3}, respectively. The dislocation density tensor at the GB, defined as

**=(**

*α***−**

*I*

*F*^{p})(

**×)=(**

*n***−**

*I*

*R*^{T})(

**×) [14,33], takes the form**

*n***=**

*α**θ*

_{2}

*e*_{1}⊗

*e*_{2}+

*θ*

_{3}

*e*_{1}⊗

*e*_{3}. This represents two arrays of edge dislocations having line direction along

*e*_{2}and

*e*_{3}, and slip direction

*e*_{1}, with densities

*θ*

_{2}and

*θ*

_{3}, respectively. To calculate the geometrical coupling factor, we exploit linearity in extending the result for a symmetric tilt boundary to the present situation to obtain

**=**

*β**θ*

_{3}

*e*_{2}−

*θ*

_{2}

*e*_{3}. The coupling factor, therefore, has contributions from both arrays of edge dislocations.

*Twist GB with small misorientation*: we now consider a twist GB with small misorientation such that the grain *G*^{−} is rotated by an anticlockwise angle *θ*_{1}, about *e*_{1}-axis, with respect to grain *G*^{+}. For small angle, we can write ** R**=

**+**

*I**θ*

_{1}(

*e*_{1}×). As a result

**=−**

*α**θ*

_{1}(

*e*_{2}⊗

*e*_{2}+

*e*_{3}⊗

*e*_{3}), which represents two arrays of screw dislocations (both with density

*θ*

_{1}) with line directions parallel to

*e*_{2}and

*e*_{3}. With this in mind, we assume the plastic deformation gradient as

*F*^{p}=

**+**

*I**ζ*(

*s*_{1}⊗

*m*_{1}+

*s*_{2}⊗

*m*_{2}), where we have considered two mutually orthogonal slip systems with equal slip magnitude, such that |

*s*_{1}|=|

*s*_{2}|=|

*m*_{1}|=|

*m*_{2}|=1 and

*s*_{1}⋅

*s*_{2}=

*s*_{1}⋅

*m*_{1}=

*s*_{2}⋅

*m*_{2}=

*m*_{1}⋅

*m*_{2}=0. On the other hand, the total deformation gradient is of the form considered above, i.e.

**=**

*F***+**

*I***⊗**

*β*

*e*_{1}. With the assumption of small deformation and small misorientation, the multiplicative decomposition of the deformation gradient becomes an additive decomposition so as to yield

**⊗**

*β*

*e*_{1}=

*θ*

_{1}(

*e*_{1}×)+

*ζ*(

*s*_{1}⊗

*m*_{1}+

*s*_{2}⊗

*m*_{2}). Projecting this onto

*e*_{1},

*e*_{2}and

*e*_{3}, we obtain

**=**

*β**ζ*(

*s*_{1}(

*m*_{1}⋅

*e*_{1})+

*s*_{2}(

*m*_{2}⋅

*e*_{1})), and

*s*_{1}⋅

*e*_{1}=

*s*_{2}⋅

*e*_{1}=0,

*m*_{1}=

*s*_{2}and

*m*_{2}=

*s*_{1}. These results immediately furnish

**=**

*β***0**. We have therefore shown that the geometrical coupling factor for a twist boundary (with small misorientation) is zero, hence confirming the qualitative arguments provided in [1,7,12].

*A cubic grain embedded in a large grain*: as an application of the results obtained above, we now consider an example where a cubic grain (whose edges are aligned with directions *e*_{1}, *e*_{2} and *e*_{3}) is embedded inside another grain such that the (infinitesimal) misorientation between them is given by ** R**=

**+**

*I**θ*

_{1}(

*e*_{1}×)+

*θ*

_{2}(

*e*_{2}×)+

*θ*

_{3}(

*e*_{3}×). The surface dislocation density tensor for the GB with normal

*e*_{1}can be calculated as

**=−**

*α**θ*

_{1}(

*e*_{2}⊗

*e*_{2}+

*e*_{3}⊗

*e*_{3})+

*θ*

_{2}

*e*_{1}⊗

*e*_{2}+

*θ*

_{3}

*e*_{1}⊗

*e*_{3}; the GB is of a mixed type consisting of two mutually perpendicular sets of edge dislocations (with densities

*θ*

_{2}and

*θ*

_{3}) and two mutually perpendicular sets of screw dislocations (both with densities

*θ*

_{1}). The dislocation content at other boundaries can be obtained in a similar manner. In evaluating the geometrical coupling factor associated with the boundary with normal

*e*_{1}, we exploit linearity in our arguments (owing to small misorientation) to combine the results obtained above for tilt and twist boundaries to write

*β*_{e1}=

*θ*

_{3}

*e*_{2}−

*θ*

_{2}

*e*_{3}. We can similarly calculate the coupling factors for the GBs with normal

*e*_{2}and

*e*_{3}as

*β*_{e2}=−

*θ*

_{3}

*e*_{1}+

*θ*

_{1}

*e*_{3}, and

*β*_{e3}=

*θ*

_{2}

*e*_{1}−

*θ*

_{1}

*e*_{2}, respectively. It is easily verifiable that

*β*_{−e1}=−

*β*_{e1}, etc., where

*β*_{−e1}represents the coupling factor associated with the face with normal −

*e*_{1}.

- Received February 24, 2015.
- Accepted April 9, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.