In this work, a phase-field model for recrystallization is developed based on the conservation laws. There has been no attempt to develop a phase-field model of recrystallization based on the conservation laws, even though various phase-field simulation models to reproduce the recrystallization phenomenon have been proposed. However, it is unclear what conservation laws are required for such a model. In the previous paper, toward solving this problem, we developed conservation laws of mass, momentum, angular momentum, and energy and a law of entropy at the lattice scale for the process of recrystallization. In this paper, first, two continuous variables, i.e., the order parameter and crystal orientation, are introduced into the balance equation of mass for a single phase and that of angular momentum for the lattice, respectively. Next, the fluxes of the order parameter and crystal orientation are derived from the law of entropy by the use of rational thermodynamics. Moreover, the diffusion coefficient and mass source are modeled to derive the evolution equations, i.e., phase-field equations of the order parameter and crystal orientation. Finally, for the phase-field equation of the crystal orientation, neglecting the conservative part and integrating the equation with respect to time under the first-order approximation, a phase-field model that is used for stable calculations is developed. This work aims to develop a phase-field theory on the basis of the change in crystal lattice during recrystallization. This paper gives a physical background to the methodological phase-field approach in the case of recrystallization.

The grain size and texture of a metal affect its mechanical and material properties and, thus, have attracted considerable attention. Recrystallization is a process by which texture is created, and many studies on recrystallization have been reported.1–9 In the recrystallization process, the recrystallization area and matrix area switch with each other (Fig. 1). At the same time, owing to the major advances in computational ability, recrystallization simulations10–13 have been carried out using the phase-field models that are continuum models.14–16

There are many well-known phase-field models that enable us to analyze material dynamics.17 The complicated morphology of dendrites can be reproduced by a phase-field model with an anisotropic diffusion term.14,18–23 The martensitic transformation can also be expressed by a phase-field model that considers elastic energy based on micromechanics.24–27 There are phase-field models which reproduce the phase transition in a binary alloy,28,29 as well as eutectic and peritectic growth and a magnetic phase.30–34 The phase-field crystal model is useful for expressing the periodic structure of atoms in terms of phase-field parameters.35,36 In addition, phenomena during solidification and grain growth and around dislocations have been studied using phase-field models37–42

These phase-field models are usually derived through the energy principle43 rather than being based on conservation laws in which the physical quantities are defined clearly. While energy functionals usually have equivalent balance equations, the balance laws of the phase-field models have scarcely been discussed. There are some important formulations of phase-field models based on mixture theory,44 and models of the phase transformation phenomenon are modeled with an interface within the thermodynamical balance laws.45–48 Referring to those literature studies, the physical aspect of the order parameter of the phase-field model can be discussed based on mixture theory.49,50 Specifically, it is still unclear what conservation law is equivalent to the governing equation of the crystal orientation of the Kobayashi–Warren–Carter (KWC)-type phase-field model.51–54 Here, the KWC-type phase-field model has two order parameters. One is the order parameter and the other is the crystal orientation. However, it is quite important to understand what conservation law the phase-field model is based on.

In the phase-field crystal model,35,36 there is no parameter related to crystal orientation. It only has the order parameter indicating the probability of the existence of atoms, and its energy functional includes higher-order gradients in order to achieve the profile of the order parameter.

If models are based on the energy principle, we cannot introduce material properties through fluxes. Although there are some reports in which materials with phase transformation are thermodynamically discussed,55 they do not treat the continuum evolution of crystal orientation, which is observed in static recrystallization. Considering the above, formulating conservation laws for recrystallization and redeveloping the phase-field model based on conservation laws are meaningful.

Generally, the basic conservation laws of the continuum are not introduced assuming that the material is homogeneous; thus, nm-scale-phenomena, such as atoms or molecules, or mm-scale-phenomena, such as crystal grain behavior, are not considered. Modeling of recrystallization in which microscale crystal lattice scale behavior affects macroscopic phenomena requires making the behaviors coarse in the μm or nm scale. The framework for such a continuum has been developed as the generalized continuum mechanics was developed.56 Well-known material models include strain gradient theory,57 n-order theory,58 micromorphic theory,59–61 micropolar theory,62–64 director theory,64 and multipolar theory.65,66 Specifically, the micropolar theory has been used for various materials so far.67–72 However, these models have a problem that the symmetry of the Cauchy stress disappears and asymmetric stress appears, which has never been observed in experiments. On the basis of these theories, a lot of studies attempt to bridge the discrete quantities to the continuum ones73,74 or to bridge microscale quantities with microstructures to the macroscale ones75,76 for various phenomena. As same as the micropolar theory, mixture theory has also been discussed for various phenomena.77–81 Additionally, the aforementioned theories have not been extended to mixture forms.

On the other hand, the balance equations for a continuum are derived assuming that a material including an internal structure can be seen as a continuum; thus, the microscopic physical quantities need not be considered. Microscopic quantities should be considered in the case that the behavior of atoms affects macroscopic phenomena such as recrystallization. Therefore, generalized continuum mechanics considering the microscopic structure in continuum theory has been formulated by Cosserat and others.

The authors previously modeled the behavior of materials during recrystallization by considering generalized continuum mechanics.82 On the basis of this model, summing and averaging the discrete conservation laws all over the Representative Volume Elements (RVEs) making up a mesoscale domain, the RVE-averaged conservation laws of mass, momentum, angular momentum, and energy were formulated in which, using the characteristic quantities at the bulk and lattice scales, the conservation law of angular momentum was divided into bulk and lattice parts. By converging the RVEs to a material point of the continuum model for the averaged conservation laws for phase p, the continuum conservation laws of mass, momentum, angular momentum, and energy for phase p were formulated. The obtained global forms of conservation laws were localized in the usual manner. Furthermore, by summing the localized conservation laws all over the phases and averaging them, the conservation laws of a mixture were derived. In addition, based on the conservation laws, the second law of thermodynamics was formulated. However, an order parameter that expresses the phases was not introduced and the constitutive equations of fluxes were not derived. The governing equations that numerically reproduce recrystallization must be developed from the basic equation.

To the above problems, in this study, continuous phase-field functions, i.e., the order parameter and crystal orientation, are introduced. The constitutive equations of fluxes are derived from the second law of thermodynamics developed in the previous paper through a thermodynamic discussion. Furthermore, by modeling the basic equations, phase-field equations are developed, with which stable calculations are possible. In addition, the result of simple simulations using the present model is shown. Additionally, the body force and radiant heat are not considered as conditions, unlike in the previous work.

Originally, the KWC-type model is a methodological model for practical applications such as recrystallization. Traditionally, such a methodological model is developed by the energy principle. Hence, the meaning of one of the order parameters of the KWC-type phase-field model, i.e., crystal orientation, has not been discussed. In order to reconsider what the order parameter is, it is important to develop the model from the viewpoint of a balance law and thermodynamic discussion. If we reformulate the phase-field model based on the balance law and thermodynamic discussion, these parameters should be driven from the four conservation laws and we can clarify the meaning of the parameters. Actually, one of the order parameters of the KWC-type phase-field model, i.e., crystal orientation, can be explained as the change in crystal lattice during recrystallization. The emphasis of this paper is on giving a physical background to the methodological phase-field approach in the case of recrystallization.

During the static recrystallization process, the atomic force is in unstable equilibrium owing to the thermal activation of atoms and stored dislocations on the boundary between the recrystallized phase and the matrix, as previously shown. Hence, crystal lattices rotate and undergo a transition to another stable state, as shown in Fig. 2. In this study, a crystal lattice is modeled as a rigid bar, i.e., a lattice element (LE), with the equivalent atomic mass shown in Fig. 3. The balance equations are as follows:

$ρ‘p+ρp div vp=b⌣p,$
(1)
$ρṡ=div M.$
(2)

Equations (16) and (23) are the local forms of the variables, and there is no expression for the global form in this manuscript because it has already been formulated in a previous report.82 Configuration is assumed to be the current one.

Equations (1) and (2) were derived from the conservation laws of mass for a single phase and angular momentum for the lattice, respectively, in the previous work. Here, ρp, vp, and $b⌣p$ are the mass density, velocity, and mass source of phase p, and ρ, s, and M are the mass density, angular momentum per unit mass, and couple stress of the mixture, respectively. On the other hand, the energy equation can be written as

$ρε̇=−div q+T⋅D+M⋅Λ̇,$
(3)

where ε, q, T, and D are the energy per unit mass, heat flux, Cauchy stress, and deformation rate of the mixture, respectively, and Λ ≡ grad θ. The quantity of phase p is expressed as

$Ap≡A+A⌢p,$
(4)

with its averaged quantity A and relative quantity $A⌢p$. In addition, $Àp$ is the material time derivative, which is defined as

$Àp≡∂Ap∂t+vpgrad Ap.$
(5)

Using Eqs. (4) and (5), Eq. (1) can be rewritten in the form of the usual material time derivative as

$ρ̇p+ρp div v+div(ρpv⌢p)−b⌣p=0.$
(6)

Furthermore, the usual continuous equation is derived as follows from the mass conservation for a mixture:

$ρ̇+ρ div v=0.$
(7)

A microscopic volume element dv can be expressed as the sum of the microscopic volume elements of the recrystallized phase and matrix as

$dv=dvr+dvm.$
(8)

Setting the volume fraction of the recrystallized phase as ϕ, the microvolume elements of the recrystallized phase and matrix are written as dvr = ϕdv and dvm = (1 − ϕ)dv, respectively. Moreover, considering the constant mass density all over the phases, a microscopic mass element can be calculated as

$dm=ρdvr+ρdvm=ρϕdv+ρ(1−ϕ)dv=ρrdv+ρmdv.$
(9)

From the above, the following relations are obtained:

$ρr=ϕρ, ρm=(1−ϕ)ρ.$
(10)

Defining the relative mass flux for phase p as

$j⌢p≡ρpv⌢p$
(11)

and changing the index to r, the following equation is obtained:

$ρ̇r+ρr div v+div j⌢r−b⌣r=0.$
(12)

Using Eq. (10) so as to introduce the order parameter into Eq. (12), the next equation is derived as follows:

$(ϕρ)+(ϕρ)div v+divj⌢r−b⌣r=0.$
(13)

Substituting Eq. (7) into Eq. (13),

$ϕ̇ρ=−div j⌢r+b⌣r$
(14)

is obtained. Writing ρ = const. and defining the order flux $j⌢r$ and mass source $b⌣r$ as

$jϕ≡j⌢r/ρ, bϕ≡b⌣r/ρ,$
(15)

the balance equation of the order parameter is obtained as

$ϕ̇=−div jϕ+bϕ.$
(16)

In the previous paper, the relations between the angular momentum per unit mass sp(k), the crystal orientation θp(k), and the angular velocity ωp(k) were formulated as

$sp(k)=J(k)ωp(k), ωp(k)≡θ̇p(k),$
(17)

where J(k) is the inertia moment tensor of a lattice element. In the same manner as in the previous work, by averaging Eq. (17), the corresponding relations for a mixture are obtained as

$s=Jω, ω≡θ̇,$
(18)

where J is the average inertia moment tensor, which was expressed as follows in the previous paper:

$J≡IΩ−ΩI,$
(19)

where $I≡ξ(k)⊗ξ(k)m$ and $Ω$ denotes the averaged spin angular velocity tensor of the lattice. Substituting Eq. (18)1 into Eq. (2) and dividing both sides by ρ, the following equation is obtained:

$(Jω)=div Iθ,$
(20)

where the orientation flux is defined as

$Iθ≡M/ρ.$
(21)

Substituting Eq. (19) into Eq. (20) and applying the previously obtained relations $J̇ω=IΩω−ΩIω=−ΩIω$, we obtain

$J̇ω−Jω̇=−ΩIω+Jω̇=div Iθ.$
(22)

Using Eq. (18), Eq. (22) is rewritten as

$Jθ̇−ΩIθ̈=div Iθ.$
(23)

Equation (23) is the balance equation of the crystal orientation.

The derivation procedure of the KWC-type phase-field model through a thermodynamic discussion is the same as that of the classical Clausius–Duhem relation, but the consideration of the rotation of the crystal lattice is different.

The second law of thermodynamics was formulated as follows in the previous work:

$ρη̇+div ŝ≥0,$
(24)

where η is the entropy density and s is the local entropy flux considering the entropy flux due to the phase transition, which is expressed as

$ŝ=qθK−∑pμpj⌢pθK.$
(25)

Writing ρ = const. and v = −vm in Eq. (11), we obtain

$j⌢r=−j⌢m.$
(26)

Therefore, from Eqs. (15) and (26), the following equation is obtained:

$jϕ=j⌢r/ρ=−j⌢m/ρ.$
(27)

Using Eq. (27), the second term on the right-hand side of Eq. (25) is calculated as

$∑pμpj⌢pθK=μrj⌢r+μmj⌢mθK=ρ(μr−μm)j⌢ϕθK=ρμj⌢ϕθK,$
(28)

where μ is the chemical potential, which is defined as

$μ≡μr−μm.$
(29)

Substituting Eqs. (25) and (28) into inequality (24) and performing the divergence operation, the following inequality is derived:

$ρη̇+div qθK−P⋅jϕθK−ρμdiv j⌢ϕθK≥0,$
(30)

where P = ρpρ grad μ. Substituting Eqs. (3) and (16) into inequality (30) and multiplying θK/ρ to both sides, the Clausius–Duhem inequality is derived as

$φ=θKη̇−ε̇+Iθ⋅Λ̇−p⋅jϕ+μϕ̇−μbϕ≥0,$
(31)

where φ is the dissipation fraction per unit mass. In addition, Eq. (21) is incorporated into inequality (31). On the other hand, Helmholtz’s free energy per unit mass ψ is written as

$ψ=ε−θKη.$
(32)

Taking the material time derivative of both sides of Eq. (32), we obtain

$ψ̇=ε̇−θKη̇,$
(33)

where $θ̇K=0$ is considered owing to the assumption of an isothermal field. Therefore, by applying the Legendre transformation with Helmholtz’s free energy to inequality (31), the following inequality is derived:

$φ=−ψ̇+Iθ⋅Λ̇−p⋅jϕ+μϕ̇−μbϕ≥0.$
(34)

Dividing the orientation flux into conservative and dissipative parts using the relation $Iθ=Iθ(c)+Iθ(d)$, inequality (34) can be rewritten as

$φ=−ψ̇+Iθ(c)⋅Λ̇+Iθ(d)⋅Λ̇−p⋅jϕ+μϕ̇−μbϕ≥0.$
(35)

From inequality (35), the arguments of ψ can be chosen as

$ψ=ψ(ϕ, ϕ̇, Λ, Λ̇, p).$
(36)

In addition, on the basis of the set of axioms, the arguments of the constitutive quantities can be chosen as

$Iθ(c)=Iθ(c)ϕ,ϕ̇,Λ,Λ̇,p,μ =μϕ,ϕ̇,Λ,Λ̇,p,Iθ(d)=Iθ(d)ϕ,ϕ̇,Λ,Λ̇,p,jϕ =jϕϕ,ϕ̇,Λ,Λ̇,p.$
(37)

From Eq. (36), expanding $ψ̇$ using the chain rule, we obtain

$ψ̇=∂ψ∂ϕϕ̇+∂ψ∂ϕ̇ϕ̈+∂ψ∂Λ⋅Λ̇+∂ψ∂Λ̇⋅Λ̈+∂ψ∂p⋅ṗ.$
(38)

Substituting Eq. (38) into inequality (35), the following inequality is derived:

$φ=Iθ(c)−∂ψ∂Λ⋅Λ̇+μ−∂ψ∂ϕϕ̇+Iθ(d)⋅Λ̇−p⋅jϕ−μbϕ− ∂ψ∂ϕ̇ϕ̈−∂ψ∂Λ̇⋅Λ̈−∂ψ∂p⋅ṗ≥0.$
(39)

The sixth, seventh, and eighth terms of the middle part of inequality (39) are linear for each rate. Considering the possible arguments of ψ, the conditions in the case that inequality (39) is true are given as

$∂ψ̇∂ϕ̇=0, ∂ψ̇∂Λ̇=∂ψ̇∂p=O.$
(40)

Therefore, the arguments of ψ can be reduced to

$ψ=ψ(ϕ, Λ).$
(41)

Note that the case of strong nonequilibrium is neglected and that local balance is assumed in which the state quantity at the material point is dependent on the thermodynamic parameters. The first and second terms of the middle part in inequality (39) are also linear in each velocity; thus, the following equations are derived:

$Iθ(c)=∂ψ∂Λ, μ=∂ψ∂ϕ.$
(42)

From Eqs. (41) and (42), the arguments of the conservative part of the orientation flux and chemical potential are reduced to

$Iθ(c)=Iθ(c)(ϕ,Λ), μ=μ(ϕ,Λ),$
(43)

where it is assumed that the chemical potential is a state quantity. In this study, ψ is to be expressed as a polynomial of the scalar invariants that are constituted by its arguments. Here, the invariants of ϕ and $Λ$ are given as

$ϕ, Iθ1=tr Λ, Iθ2=tr Λ2, Iθ3=tr(ΛΛT),Iϕ=tr(∇ϕ⊗∇ϕ), tr{∇ϕ⊗(Λ∇ϕ)},$
(44)

where the scalar invariant of ∇ϕ is included so as to consider the effect of the gradient of the order parameter. On the other hand, the vector potential of the angular velocity does not exist. Therefore, $div θ̇≠0$ can be assumed, whereas $div ω=div θ̇=0$ in a usual solenoidal field. Because $div θ̇≠0$, div θ ≠ 0 can be assumed; thus, Iθ1 = tr Λ = tr(grad θ) = div θ is included in the arguments of ψ in this theory. On the other hand, the invariants whose order is higher than three are not considered so as not to generate high-order nonlinear terms. From the above, the fraction of free energy is given as

$ψ=ψ(ϕ, Iθ1, Iθ2, Iθ3, Iϕ).$
(45)

Using Eq. (45), Eq. (42) can be expanded as

$Iθ(c)=∂ψ∂ϕ∂ϕ∂Λ+∂ψ∂Iθ1∂Iθ1∂Λ+∂ψ∂Iθ2∂Iθ2∂Λ+∂ψ∂Iθ3∂Iθ3∂Λ+∂ψ∂Iϕ∂Iϕ∂Λ.$
(46)

Calculating Eq. (46) assuming the independence of the arguments, the explicit constitutive equation of $Iθ(c)$ can be written as

$Iθ(c)=∂ψ∂Iθ1∂Iθ1∂Λ+∂ψ∂Iθ2∂Iθ2∂Λ+∂ψ∂Iθ3∂Iθ3∂Λ+∂ψ∂Iϕ∂Iϕ∂Λ=∂ψ∂Iθ11+∂ψ∂Iθ22Λ+∂ψ∂Iθ32ΛT=aθc01+aθcΛ+aθc′ΛT,$
(47)

where the coefficients are defined as

$aθc0(ϕ, Iθ1, Iθ2, Iθ3, Iϕ)≡∂ψ∂Iθ1,aθc(ϕ, Iθ1, Iθ2, Iθ3, Iϕ)≡2∂ψ∂Iθ2,aθc′(ϕ, Iθ1, Iθ2, Iθ3, Iϕ)≡2∂ψ∂Iθ3.$
(48)

Note that aθc0, aθc, and $aθc′$ in Eq. (48) are the functions of ϕ and Λ.

Substituting Eqs. (41) and (42) into inequality (35), the pure dissipation function is obtained as

$φ=Iθ(d)⋅Λ̇−p⋅jϕ−μbϕ≥0.$
(49)

Considering the quasi-compound process, φ can be divided into pure dissipation functions relating to fluxes φJ and the mass source φb as

$φ=φJ+φb, φJ=Iθ(d)⋅Λ̇−p⋅jϕ, φb=−μbϕ.$
(50)

Furthermore, from Eq. (49), we obtain

$φJ=Iθ(d)⋅Λ̇−p⋅jϕ≥0.$
(51)

Considering Eq. (43) and expanding p = grad μ using the chain rule, inequality (51) can be expressed as

$φJ=Iθ(d) ⋅ Λ̇−p ⋅ jϕ=Iθ(d) ⋅ Λ̇−∂μ∂ϕΛ∇ϕ+∂μ∂ΛϕΛ⊗∇← ⋅ jϕ.$
(52)

Using Eq. (42), the third term on the right-hand side of Eq. (52) is calculated as

$∂μ∂Λϕ=∂∂Λ∂ψ∂ϕΛϕ=∂∂ϕ∂ψ∂ΛϕΛ=∂Iθ(c)∂ϕΛ,$
(53)

where the region in which Λ is fixed is nearly equal to the region where Λ = O. The region where grad θ = O is the region inside phases, i.e., ϕ = 0 or ϕ = 1. Therefore, ψ is constant and the third term on the right-hand side of Eq. (52) takes a value of zero. From the above, inequality (51) is reduced to

$φJ=Iθ(d)⋅Λ̇−cπ∇ϕ⋅jϕ≥0,$
(54)

where cπ = ∂μ/∂ϕ. Moreover, applying the normality rule to the curved surface of the dissipation function, the constitutive equations of the fluxes are obtained as

$Iθ(d)=v∂φJ∂Λ̇, jϕ=−vcπ∂φJ∂(∇ϕ),$
(55)

where v is constant. From inequality (51), the arguments of φJ are reduced to

$φJ=φJ(ϕ,ϕ̇,Λ,Λ̇,p)=φJ(ϕ,ϕ̇,Λ,∇ϕ,Λ⊗∇←),$
(56)

where p is expanded with the arguments in Eq. (43)2. Applying Eq. (56) to Eq. (55), the arguments of the constitutive quantities are reduced to the same as those of φJ,

$Iθ(d)=Iθ(d)(ϕ,ϕ̇,Λ,∇ϕ,Λ⊗∇←),jϕ =jϕ (ϕ,ϕ̇,Λ,∇ϕ,Λ⊗∇←).$
(57)

Moreover, representing φJ as a polynomial of its scalar invariant, the invariants $Λ̇$ and ∇θ are given as

$ϕ, Iθ1=tr Λ, Iθ2=tr Λ2, Iθ3=tr(ΛΛT),Iν1=tr Λ̇, Iν2=tr Λ̇2, Iν3=tr(Λ̇Λ̇T),Iϕ=tr(∇ϕ⊗∇ϕ), tr{∇ϕ⊗(Λ∇ϕ)}.$
(58)

In the same way as for the conservative part of the dissipation function, neglecting the invariants whose order is higher than three, the dissipation function takes the following form:

$φJ=φJ(ϕ, Iθ1, Iθ2, Iθ3, Iν1, Iν2, Iν3, Iϕ).$
(59)

Substituting Eq. (59) into Eq. (55)2, the constitutive equation of the order flux is given as

$jϕ=−νcπ∂φJ∂ϕ∂ϕ∂(∇ϕ)+∂φJ∂Iϕ∂Iϕ∂(∇ϕ)+∂φJ∂Iθ1∂Iθ1∂(∇ϕ)+∂φJ∂Iθ2∂Iθ2∂(∇ϕ)+ ∂φJ∂Iθ3∂Iθ3∂(∇ϕ)+∂φJ∂Iν1∂Iν1∂(∇ϕ)+∂φJ∂Iν2∂Iν2∂(∇ϕ)+∂φJ∂Iν3∂Iν3∂(∇ϕ),$
(60)

where the arguments in Eq. (56) are independent of each other. Thus, Eq. (60) can be rewritten as

$jϕ=−νcπ∂φJ∂Iϕ∂Iϕ∂(∇ϕ).$
(61)

Simplifying Eq. (61) further, the following equation is obtained:

$jϕ=−νcπ∂φJ∂Iϕ∂Iϕ∂(∇ϕ)=−νcπ∂φJ∂Iϕ2∇ϕ=−aϕ∇ϕ,$
(62)

where aϕ is defined as

$aϕ(ϕ, Iθ1, Iθ2, Iθ3, Iν1, Iν2, Iν3, Iϕ)≡2νcπ∂φJ∂Iϕ.$
(63)

On the other hand, considering Eq. (59), Eq. (55) is expanded as

$Iθ(d)=ν∂φJ∂ϕ∂ϕ∂Λ̇+∂φJ∂Iθ1∂Iθ1∂Λ̇+∂φJ∂Iθ2∂Iθ2∂Λ̇+∂φJ∂Iθ3∂Iθ3∂Λ̇+∂φJ∂Iν1∂Iν1∂Λ̇+ ∂φJ∂Iν2∂Iν2∂Λ̇+∂φJ∂Iν3∂Iν3∂Λ̇+∂φJ∂Iϕ∂Iϕ∂Λ̇.$
(64)

Considering the independence of the arguments in Eq. (56), Eq. (64) can be rewritten as

$Iθ(d)=ν∂φJ∂Iν1∂Iν1∂Λ̇+∂φJ∂Iν2∂Iν2∂Λ̇+∂φJ∂Iν3∂Iν3∂Λ̇=ν∂φJ∂Iν11+∂φJ∂Iν22Λ̇+∂φJ∂Iν32Λ̇T=aθd01+aθdΛ̇+aθd′Λ̇T,$
(65)

where the constants in Eq. (65) are defined as

$aθd0(ϕ, Iθ1, Iθ2, Iθ3, Iν1, Iν2, Iν3, Iϕ)≡ν∂φJ∂Iν1,aθd(ϕ, Iθ1, Iθ2, Iθ3, Iν1, Iν2, Iν3, Iϕ)≡2∂φJ∂Iν2,aθd′(ϕ, Iθ1, Iθ2, Iθ3, Iν1, Iν2, Iν3, Iϕ)≡2∂φJ∂Iν3.$
(66)

Note that aθd0, aθd, and $aθd′$ in Eq. (66) are the functions of ϕ, Λ, and $Λ̇$, respectively.

It is noted that by substituting the constitutive equations of the fluxes expressed as Eqs. (62) and (65) into inequality (51) and assuming the constants to be positive, we find that the right-hand side remains positive.

Substituting Eq. (16) into Eq. (62), the basic equation of the order parameter is obtained as

$ϕ̇=−div(−aϕ grad ϕ)+bϕ.$
(67)

Assuming that the diffusion coefficient aϕ (m2/s) is constant and bϕ is a function of ϕ, i.e., bϕ = bϕ(ϕ), Eq. (67) is rewritten as

$ϕ̇=aϕ div(grad ϕ)+bϕ(ϕ),$
(68)

where the term in the parentheses of the second term on the right-hand side of Eq. (68) denotes the argument. Next, bϕ(ϕ) is formulated as the mass source term due to the phase transition. First, bϕ(ϕ) is represented as the sum of the mass sources due to the autocatalytic reaction bca and grain boundary migration bgb,

$bϕ(ϕ)≡bca+bgb.$
(69)

The equation of the autocatalytic reaction can be written as

$R+M→2R,$
(70)

where R and M denote the recrystallized phase and matrix, respectively. From Eq. (70), bgb can be represented as [see Figs. 4(a) and 4(b)]

$bca=kaϕ(1−ϕ)/τ,$
(71)

where ka is a coefficient and τ (s) is a representative interval of the autocatalytic reaction. Moreover, the autocatalytic reaction during recrystallization is due to the energy barrier and the stored dislocation energy. Therefore, ka can be represented as the sum of the effects of both energies as

$ka=kag+kab,$
(72)

where kag is a coefficient that represents the effect of the stored dislocation energy Es and kab is a coefficient that represents the effect of the energy barrier Wb. Furthermore, Es is the driving force of nucleus growth, i.e., the matrix region undergoes a transition to the recrystallized phase owing to this driving force. According to the above, kag can be modeled as

$kag=aagEsϕ(1−ϕ),$
(73)

where aag (m3/J) is a constant. In addition, assuming that the energy barrier exists in the region between the matrix and recrystallized phase, i.e., ϕ = 1/2 and ϕ is to transit to the recrystallized phase when ϕ ≥ 1/2 and the matrix when ϕ ≥ 1/2, kag is modeled as

$kag=aabWb(ϕ−1/2),$
(74)

where aab (m3/J) is a constant. Substituting Eqs. (72)–(74) into Eq. (71), bca is expressed as

$bca={aagEsϕ(1−ϕ)+aabWb(ϕ−1/2)}ϕ(1−ϕ)/τ.$
(75)

Here, Fig. 5 shows the illustration of the effect of Es and Wb. On the other hand, grain boundary migration without crystal lattice rotation is supposed as the nucleus shrinks to the curvature center of the nucleus [see Fig. 4(c)]. Therefore, the reaction equation is given as

$R→M.$
(76)

From Eq. (76), the velocity equation can be expressed as

$bgb=−kgbϕ/τ,$
(77)

where kgb is a coefficient. Moreover, the grain boundary energy is introduced for the modeling of kgb. On the basis of the Read–Shockley concept,83 Egb is dependent on the difference in the crystal orientation of grains. In a three-dimensional field, Egb is dependent on the invariants that are conjugate with the crystal orientation gradient grad θ. The invariants whose order is lower than two are written as

$tr Λ=div θ,tr ΛΛT=Λ⋅Λ=grad θ⋅grad θ=grad θ2,tr Λ2=Λ⋅ΛT=grad θ⋅(grad θ)T,tr ΛΛT=grad θ, tr Λ2=grad θ⋅(grad θ)T,$
(78)

where Λ = grad θ. In the invariants in Eq. (78), div θ does not form energy by itself; thus, the second-order quantities such as

$(div θ)2, grad θ, grad θ⋅(grad θ)T,grad θ, grad θ⋅(grad θ)T$
(79)

are adopted, where, considering that Egb = 0 when ϕ = 0, i.e., a matrix, Egb can be written as a polynomial of the invariants of Eq. (79) as

$Egb=Erϕ2212ε1(div θ)2+s2grad θ+12ε2grad θ2+ s3grad θ⋅(grad θ)T+12ε3 grad θ⋅(grad θ)T≡12Erϕ2Egb*.$
(80)

Here, Er (J/m3) is the reference grain boundary energy and $Egb*$ is a nondimensional quantity expressed as

$Egb*=12ε1(div θ)2+s2grad θ+12ε2grad θ2+ s3grad θ⋅(grad θ)T+12ε3grad θ⋅(grad θ)T,$
(81)

where ε1 (m2), s2 (m), ε2 (m2), s3 (m), and ε3 (m2) are constants. In Eq. (81), in accordance with the general energy form, ϕ2/2 is employed. The constant kgb in Eq. (77) must be synchronized with the temporal change in grain boundary migration when grad θ is fixed, i.e., only due to the phase change. From Eq. (80), (∂Egb/∂t)$Λ$ is obtained as

$∂Egb∂tΛ=Egb*Erϕϕ̇=Egb*12Erϕ2,$
(82)

where, from Eq. (82), kgb can be expressed as

$kgb=Egb*=12ε1(div θ)2+s2grad θ+12ε2grad θ2+ s3grad θ⋅(grad θ)T+12ε3grad θ⋅(grad θ)T.$
(83)

Substituting Eq. (83) into Eq. (77), bgb is modeled as

$bgb=−12ε1(divθ)2+s2||gradθ||+12ε2||gradθ||2+ s3gradθ⋅(gradθ)T+12ε3gradθ⋅(gradθ)Tϕ/τ.$
(84)

Substituting Eqs. (75) and (84) into Eq. (69), the reaction term due to the mass source is written as

$bϕ=aagEsϕ(1−ϕ)+aabWbϕ−12ϕ(1−ϕ)/τ− 12ε1(div θ)2+s2grad θ+12ε2grad θ2+ s3grad θ⋅(grad θ)T+12ε3 grad θ⋅(grad θ)Tϕ/τ.$
(85)

Substituting Eq. (85) into Eq. (68) and defining the temporal diffusion α (m2) and mobility Mϕ (1/s) as α = aϕτ and Mϕ ≡ 1/τ, respectively, the equation of the order parameter is obtained as

$ϕ̇=Mϕα div(grad ϕ)+aagEsϕ(1−ϕ)+aabWbϕ−12ϕ(1−ϕ)− 12ε1(div θ)2+s2grad θ+12ε2grad θ2+ s3grad θ⋅(grad θ)T+12ε3 grad θ⋅(grad θ)Tϕ.$
(86)

Here, we can see the same form as the phase-field equation in Eq. (86).

FIG. 5.

Changes in the autocatalytic mass source due to various Es and Wb: (a) illustration of the mass source based on various values of Es, and (b) illustration of the mass source based on various values of Wb.

FIG. 5.

Changes in the autocatalytic mass source due to various Es and Wb: (a) illustration of the mass source based on various values of Es, and (b) illustration of the mass source based on various values of Wb.

Close modal

The orientation flux is expressed as the sum of the conservative and dissipative parts. Therefore, using Eqs. (47) and (65), the orientation flux Iθ is written as

$Iθ=Iθ(c)+Iθ(d)=aθc01+aθcΛ+aθc′ΛT+aθd01+aθdΛ̇+aθd′Λ̇T.$
(87)

Substituting Eq. (87) into Eq. (23), the basic equation of the crystal orientation is obtained as

$Jθ̈−ΩIθ̇=divaθc01+aθcΛ+aθc′ΛT+aθd01+aθdΛ̇+aθd′Λ̇T.$
(88)

Considering $Λ$ = grad θ, Eq. (88) is also expressed as

$Jθ̈−ΩIθ̇=divaθc01+aθc(grad θ)+aθc′(grad θ)T+ {aθd01+aθd(grad θ̇)+aθd′(grad θ̇)T}.$
(89)

From Eq. (89), it should be noted that the governing equation of the crystal orientation based on the conservation laws is distinctly different from that of the conventional KWC-type phase-field model based on the energy principle. The former takes the form of a sum of parabolic and hyperbolic functions, whereas the latter is expressed as a parabolic equation. Therefore, the proposed equation has the advantage that it can avoid the physical contradictions of a parabolic partial differential equation such as an infinite propagation speed of a material. However, Eq. (89) includes a second-order temporal partial differential term and the equation is given as a sum of parabolic and hyperbolic functions, making it difficult to derive a stable calculation. Therefore, we next use Eq. (89) to derive a model with which a relatively stable simulation can be carried out. Neglecting the conservative part of Eq. (89), only the dissipative part remains,

$Jθ̈−ΩIθ̇=divaθd01+aθdΛ̇+aθd′Λ̇T.$
(90)

Next, both sides of Eq. (90) are integrated with respect to time. First, we consider the following equation:

$θ̈=div(aθd grad θ̇).$
(91)

Integrating both sides of Eq. (91) with respect to time, we obtain

$θ̇=div(aθd grad θ)−∫div(ȧθd grad θ)dt,$
(92)

where c(x) = 0 is assumed as the initial condition. Consider Eq. (92). Approximating the average velocity as $ȧθd≈δaθd/δt$, the second term on the right-hand side of Eq. (92) is calculated as

$∫div(ȧθd grad θ)dt≈∫divδaθdδt grad θdt,$
(93)

where δt is an infinitesimal temporal interval. Moreover, according to Fig. 6, Eq. (93) can be expressed as the area of a trapezoid as

$∫divδaθdδt grad θdt=divδaθdδtgrad θt+grad θ2δt.$
(94)

$grad θt+grad θ2=12grad θ−δ(grad θ)+grad θ=grad θ−12δ(grad θ).$
(95)

Therefore, substituting Eq. (95) into Eq. (94) gives

$∫divδaθdδt grad θdt=divδaθdδtgrad θ−12δ(grad θ)δt.$
(96)

Substituting Eq. (96) into Eq. (93) gives

$∫div(ȧθd grad θ)dt=div(δaθd grad θ)−12 divδaθdδ(grad θ).$
(97)

Furthermore, substituting Eq. (97) into Eq. (92) gives

$θ̇=div(aθd grad θ̇)−div(δaθd grad θ)+12 divδaθdδ(grad θ)=div{(aθd−δaθd)grad θ̇}+O{δaθdδ(grad θ)}≈div{(aθd−δaθd)grad θ̇}≡div(āθd grad θ̇),$
(98)

where small quantities whose order is higher than two are neglected. In addition, $āθd=āθd(ϕ, Λ,∇ϕ, Λ̇)$ is assumed because $aθd=aθd(ϕ,Λ,∇ϕ,Λ̇)$. Selecting the arguments ϕ and ||Λ||, $āθd=āθd(ϕ, Λ)$ is obtained. In addition, $āθd=aθd−δaθd=aθdt+δaθd−δaθd=aθdt$ is equal to aθd at a known time. From the above, integrating both sides of Eq. (90) with respect to time, we obtain

$θ̇=J−1 divaθd01+aθdΛ+aθd′ΛT.$
(99)

Next, the diffusion coefficient of the orientation flux is modeled. Because Eq. (99) governs the change in orientation when grains impinge on each other, aθd0, aθd, and $aθd′$ must be synchronized with the temporal change in Egb when ϕ is fixed. Therefore, (∂Egb/∂t)ϕ can be expressed as

$∂Egb∂tϕ=Erϕ2ε1(div θ)div θ̇+s2grad θ⋅grad θ̇grad θ+ε2 grad θ⋅grad θ̇+ s3grad θ⋅(grad θ̇)Tgrad θ⋅(grad θ)T+ε3 grad θ⋅(grad θ̇)T=Erϕ2ε1(div θ)div θ̇+s2grad θ+ε2grad θ⋅grad θ̇+ s3grad θ⋅(grad θ)T+ε3grad θ⋅(grad θ̇)T,$
(100)

where $(grad θ)≈grad θ̇$ and $(div θ)≈div θ̇$ are assumed because of the extremely small time interval. In addition, ${grad θ ⋅ (grad θ)T}=2grad θ ⋅ (grad θ̇)T$ is assumed. Comparing Eq. (99) with Eq. (100), aθd0, aθd, and $aθd′$ can be modeled as

$aθd0=ad0ϕ2ε1, aθd=ad0ϕ2s2grad θ+ε2,aθd′=ad0ϕ2s3(grad θ)⋅(grad θ)T+ε3,$
(101)

where ad0 (m2/s) is a constant. Substituting Eq. (101) into Eq. (99), the three-dimensional phase-field equation of the crystal orientation is obtained as

$θ̇=Mθ divϕ2ε11+ϕ2s2grad θ+ε2Λ+ ϕ2s3(grad θ)⋅(grad θ)T+ε3ΛT,$
(102)

where MθJ−1ad0 (1/s) is the mobility tensor expressing the development of the crystal orientation. Moreover, considering that crystal lattice rotation occurs at the boundary region, the inertia moment per unit mass J can be expressed as a function of the order parameter J = J(ϕ) (m2). Therefore, M can also be written as M(ϕ). In addition, Eq. (102) can be regarded as the KWC-type-phase-field equation of the crystal orientation extended to three dimensions.

The three-dimensional quantities of the crystal orientation θ and inertia moment tensor J are given as

$θxθyθz, JxxJxyJxzJyxJyyJyzJzxJzyJzz.$
(103)

Now, reducing the problem to the two-dimensional problem in the xy plane, Eq. (103) is represented as

$00θz≡00θ→θ, 00Jxz00JyzJzxJzyJzz≡00Jxθ00JyθJθxJθyJθz→00Jθi,$
(104)

where θ and J−1 each have one component allowing them to be reduced to be written as the quantities θzθ and JziJθi, respectively. Considering that θi = θ3θ is the only component in the two-dimensional problem, the following calculation becomes possible:

$div θ→θi,i=θ3,3=θ,3=0,tr Λ2=Λ⋅ΛT=(grad)⋅(grad)T→θi,jθj,i=θ3,jθj,3=θ3,3θ3,3=0,grad θ→θi,j=θ3,j=θ,j→grad θ,1→δij=δ3j.$
(105)

Applying Eq. (105) to Eq. (83) gives

$kgb=s2||gradθ ||+12ε2||gradθ ||2.$
(106)

Substituting Eq. (106) into Eq. (77) and using Eqs. (68), (69), and (75), we obtain

$ϕ̇=Mϕaagdiv(grad)+{Esϕ(1−ϕ)+aabWb(ϕ−1/2)}ϕ(1−ϕ)− (s2||gradθ ||+12ε2||gradθ ||2)ϕ.$
(107)

Equation (107) is the conventional KWC-type phase-field equation of the order parameter. Equation (107) is represented with an index as

$θ̇i=Mθik(ϕ2ε1),k+Mθikϕ2s2||θl,m||+ε2θk,j,j+ Mθikϕ2s3θl,mθm,l+ε3θj,k,j.$
(108)

From Eq. (105), Eq. (108) can be expressed as

$θ̇3=Mθ3k(ϕ2ε1),k+Mθ3kϕ2s2||∇θ||+ε2θk,j,j+Mθ3k(ϕ2ε3θj,k),j=Mθ33(ϕ2ε1),3+Mθ33ϕ2s2||∇θ||+ε2θ3,j,j+Mθ33(ϕ2ε3θj,3),j.$
(109)

Since there is no rotation outside of the plane, the second and third terms on the right-hand side of Eq. (109) can be deleted. As a result, Eq. (109) is reduced to

$θ̇=Mθ∇⋅ϕ2s||∇θ||+ε∇θ,$
(110)

where MθMθ33, ss2, and εε2. Equation (110) coincides with the governing equation of the crystal orientation of the KWC-type phase-field model.

We would like to propose the use of the partial difference method for the final equations (107) and (110). However, in the case that Eq. (90) is solved instead of Eq. (110), the constrained interpolation profile scheme, which is one of the numerical schemes used to solve the hyperbolic partial differential equations, can be employed. The finite element method is generally used in the case that the deformation field is considered via the motion equation.

We presented the mass conservation law of single phase and the conservation law of spin angular momentum, which have been developed in the previous work, by the use of the order parameter and crystal orientation. The constitutive equations of fluxes of the order parameter and crystal orientation were thermodynamically derived. Additionally, we modeled the reaction term of the basic equation of the order parameter and diffusion coefficient of the basic equation of the crystal orientation by synchronizing with the temporal change in the grain boundary energy. The conclusions we obtained from this study are summarized.

1. The balance equation of the order parameter is the same as the mass conservation law in which the volume fraction of the recrystallized phase is expressed by the order parameter. The mass source of the governing equation of the order parameter can be modeled by the reaction rate equation of grain growth and grain boundary migration.

2. The balance law of the crystal orientation is derived by the conservation law of the spin angular momentum for the mixture of the lattice. The governing equation in this study was derived in the form of the sum of the hyperbolic equation and parabolic equation. If we assume that the recrystallization is limited to the dissipative process and integrating the above equation with respect to time, the three-dimensional parabolic-type equation of the crystal orientation is obtained.

3. When the phase-field model for recrystallization is derived based on the conservation laws, the conservation laws should be developed for the generalized continuum in which the lattice is modeled as a kind of directa.

Currently, we do not consider the elastic energy in the model. However, we can take the elastic energy into account. In that case, the elastic energy can be considered in the source term based on reaction velocity theory. If we develop the energy of a grain boundary not only with the gradient of the crystal orientation but also with the elastic energy, this can be achieved. This model is expected to be interesting since it is reasonable to consider the elastic energy around dislocations when simulating recrystallization phenomena. The development of this model is a future work.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

We derive the material derivative of the moment of inertia averaged in the RVE $J̇$. First, we define J(k). When a lattice element is modeled as a bar (Fig. 7) and its center of gravity is fixed, the spin angular momentum of the lattice element is expressed as follows:

$∫Vlξ′(k)×ρ′v′(k)dvl=∫Vlξ′(k)×ρ′Ω(k)ξ′(k)dvl,$
(A1)

where ξ(k) is the position vector to the arbitrary point from the center of the gravity of the lattice element, ρ′ is the mass density of the lattice element, Ω(k) is the spin angular velocity, defining Ω(k) ≡ dual ω(k) [ ], and vl is the volume of one lattice. Defining I(k)ξ(k)ξ(k), the integrated function of Eq. (A1) can be expanded as

$(ξ′(k)×ρ′Ω(k)ξ′(k))i=ρ′eijkξj′(k)Ωkl(k)ξl′(k)=ρ′eijkekmlξj′(k)ξl′(k)ωm(k)=ρ′δim′(k)δjl′(k)−δil′(k)δjm′(k)ξj′(k)ξl′(k)ωm(k)=ρ′(ξj′(k)ξj′(k))δim′(k)−ξi′(k)ξm′(k)ωm(k)=ρ′(ξk′(k)ξk′(k))δij′(k)−ξi′(k)ξj′(k)ωj(k)=ρ′J′(k)ω(k)i.$
(A2)

Here, J(k) is given as

$J′(k)≡(trI′(k))1−I′(k).$
(A3)

Substituting Eq. (A2) into Eq. (A1) and considering ω(k) is constant in the region of vl, the left-hand side of Eq. (A1) is reduced to

$∫Vlξ′(k)×ρ′v′(k)dvl=∫Vlρ′(k)J′(k)ω(k)dvl=m(k)J(k)ω(k).$
(A4)

Here, J(k) is expressed as

$J(k)≡1m(k)∫Vlρ′(k)J′(k)dvl.$
(A5)

Considering Eq. (A5) and writing J(k) using I(k),

$J(k)≡J′(k)ml=(trI′(k))1−I′(k)ml=trI′(k)ml1−I′(k)ml=(tr I(k))1−I(k),$
(A6)

where $ml$ means the mass averaging in the lattice element. Hence, taking a material derivation of both sides of Eq. (A6), we have

$J̇(k)=tr İ(k)1−İ(k),$
(A7)

where the material derivative of I(k), $İ(k)$, can be written as

$İ(k)=I′(k)ml=ξ′(k)⊗ξ′(k)ml=ξ′̇(k)⊗ξ′(k)ml+ξ′(k)⊗ξ′̇(k)ml.$
(A8)

Using orthogonal tensor R(k) and initial position vector $ξ′0(k)$, ξ(k) can be expressed as the follows:

$ξ′(k)=R(k)ξ′0(k).$
(A9)

Taking the material derivative of Eq. (A9), we obtain

$ξ̇′(k)=Ṙ(k)ξ0′(k)=Ṙ(k)R(k)Tξ′(k)=Ω(k)ξ′(k),$
(A10)

where we define $Ω(k)≡Ṙ(k)R(k)T$ [ ]. Substituting Eq. (A10) into Eq. (A8), the following equation is derived:

$J̇=J̇(k)m=I(k)Ω(k)−Ω(k)I(k)m=I(k)mΩ−ΩI(k)m+I(k)Ω̃(k)m−Ω̃(k)I(k)m.$
(A11)

Here, Ω and $Ω̃(k)$ are the mass average and fluctuation of Ω(k), respectively, and $m$ is the mass average in the RVE. Next, dividing I(k) into the average and fluctuation in the RVE as $I(k)=I(k)m+Ĩ(k)≡I+Ĩ(k)$, Eq. (A11) is reduced as follows:

$J̇=(IΩ−ΩI)+Ĩ(k)Ω̃(k)−Ω̃(k)Ĩ(k)m.$
(A12)

Generally, the correlation of the fluctuation is not always infinitesimal. In this case, I has the order of square of the length of the lattice element, whereas $Ĩ(k)$ has the order of the square of the amplitude of atom vibration around equilibrium position, which meaning that ⟨ ⟩ in right-hand side is negligible since sufficient small order to ( ) in right-hand side. Therefore, Eq. (A12) can be rewritten as follows:

$J̇=IΩ−ΩI.$
(A13)

Moreover, multiplying ω to Eq. (A13), and using the relationships (Ωω)i = Ωijωj = eikjωkωj = 0, we obtain the following equation:

$J̇ω=IΩω−ΩIω=−ΩIω.$
(A14)
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