Phase-field gradient theory

We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a microhypertraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a 'generalized Swift-Hohenberg equation'-a second-grade phase-field equation-and its conserved version, the 'generalized phase-field crystal equation'-a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

In this work, we propose a phase-field theory for enriched continua. This continuum framework extends and complements the work by Fried & Gurtin [1,2] and Gurtin [3]. However, we base our theory on Fosdick's approach [4,5] to derive it. We generalize the Swift-Hohenberg equation [6]-the second-grade phase-field equation-and its conserved version-the phase-field crystal equation. We build our theory on balances of microforces and microtorques while accounting for 'rough' arbitrary parts. Additionally, we present the configurational fields arising in this theory with its balance.
Brazovskiǐ [7] introduced the free-energy functional that delivers the equations of Swift-Hohenberg and phase-field-crystal. In the literature, these equations are typically found in the following form of gradient flows where ϕ is the phase field (left: nonconserved case, right: conserved case) and Ψ a free-energy functional depending on (ϕ, grad ϕ, grad 2 ϕ).
The outline of the work is as follows. In §2, to derive this continuum theory, we allow for different types of interactions between adjacent arbitrary parts P occurring inside the body B to characterize the primitive and fundamental contact microforce fields. Parts P of the body B are arbitrary and may exhibit a lack of smoothness on their boundaries ∂P along a curve C ⊂ ∂P. Thus, by balancing these traction fields, we arrive at the field equations of the phase-field gradient theory. In §3, we present the virtual power theorem. In §4, we derive the thermodynamic laws, in the form of the energy balance and the free-energy imbalance to derive suitable constitutive equations. In §5, we introduce the nonconserved second-grade phase-field equation, its specialization to the Swif-Hohenberg equation, and the configurational fields with its balance equation. In §6, we aim at the conserved second-grade phase-field equation by augmenting the nonconserved version with a balance of mass and emulate the developments of §5. In §7, we derive thermodynamically consistent boundary conditions for both, the nonconserved and conserved cases. In §8, we summarize this work. In appendix §A, we present the mathematical identities used to derive this theory.

Fundamental fields & field equations
The continuum theory by Fried & Gurtin [1,2] and Gurtin [3] represents a turning point in phase-field theories. In this collection of papers, Fried & Gurtin describe these phenomena from a mechanistic standpoint and derived the 'generalized Allen-Cahn (Ginzburg-Landau)' and the 'generalized Cahn-Hilliard' equations by introducing a balance of microforces. Their work makes explicit the underlying forces' that dictate the evolution of phase fields. Herein, we denote these equations as first-grade phase-field descriptions. In this section, we characterize the fundamental fields, based on Fosdick's approach [5], and derive the field equations that yield the phase-field gradient theory.

Fundamental fields.
Throughout what follows, B denotes a fixed region of a three-dimensional point space E. P ⊆ B is an arbitrarily fixed subregion of B with a closed boundary surface ∂P oriented by an outward unit normal n. The surface ∂P can lose its smoothness along a curve, namely an internal-edge C. Analyzing a neighborhood of an internal-edge C, two smooth surfaces ∂P ± are defined. Thus, the limiting unit normals of ∂P ± at C are denoted by the pair {n + , n − }, which characterizes the internal-edge C. Equivalently, the limiting outward unit tangent-normal of ∂P ± at C are {ν + , ν − }. As a notational agreement, C is oriented by the unit tangent t := t + such that ν + := t + × n + . Irrespectively of the surface S being a boundary of P the internal-edge remains a feature of the surface and not part of P. Furthermore, the body B and all its parts are open sets in E. Figure 1 depicts the part under discussion. To unfold the implications of considering arbitrary parts P that lack of smoothness at an internal-edge C arising in this theory, we begin by discussing the interaction of a smooth open surface S ⊆ P with a boundary-edge ∂S and its adjacent parts of P, that is P \ S. Here, S is oriented by an unit normal n with a boundary-edge ∂S oriented by the unit tangent t. Boundary-edges ∂S are equipped with an intrinsic Darboux frame, composed by the unit tangent t, unit normal n, and outward unit tangent-normal ν, where ν := t×n. Understanding the underlying mechanical interactions of a boundary-edge ∂S allows us to further the understanding of an internal-edge C. Figure 2 depicts a smooth open surface (left) and a nonsmooth open surface (right), which will serve to study boundary-and internal-edge interactions.   Before developing the field equations of the phase-field gradient theory, we describe the set of interactions of adjacent parts of B and the resulting fundamental fields as follows.
(i) Surface microtraction: The surface microtraction ξ S := ξ S (x, t; n, L) represents a microforce per unit area, acting on an oriented surface S ⊂ B at x ∈ S. It depends on S through the pair {n, L}, its outward unit normal n and its curvature tensor L (or the negative surface gradient of the unit normal, −grad S n). On the opposite side of S, S * , the surface microtraction ξ * S := ξ * S (x, t; n, L) is defined. We say that ξ * S is the intrinsic counterpart of ξ S . Each of these surface microtractions are developed by the contact of each side of a surface S with the adjacent parts of B.
(ii) Surface-couple microtraction: The surface-couple microtraction S := S (x, t; n) represents a microtorque per unit area, acting on an oriented surface S ⊂ B at x ∈ S. It depends on S through its outward unit normal n. On the opposite side of S, S * , the surface-couple microtraction * S := * S (x, t; n) is defined. We say that * S is the intrinsic counterpart of S . Each of these surface-couple microtractions are also developed by the contact of each side of a surface S with the adjacent parts of B.
(ii) Boundary-edge microtraction: The boundary-edge microtraction τ ∂S := τ ∂S (x, t; ν, n) represents a microforce per unit length, acting on an boundary-edge ∂S of an open oriented surface S ⊂ B at x ∈ ∂S. It depends on S through the pair {ν, n}, its outward unit tangent-normal ν and its unit normal n. The boundary-edge microtraction is developed by the contact of a boundary-edge ∂S with the adjacent parts of B \ S. The analysis of the boundary-edge microtraction and its definition is intrinsic to the study of open oriented surfaces. Thus, the boundary-edge microtraction does not contribute to balances that are reckoned on arbitrary parts P. However, the boundary-edge microtraction characterizes the internal-edge microtraction and more importantly, it supports the existence of a hypermicrostress field.
(iv) Internal-edge microtraction: The internal-edge microtraction τ C := τ C (x, t; n + , n − ) represents a microforce per unit length, acting on an internal-edge C of a nonsmooth oriented surface S ⊂ B at x ∈ C ⊂ S. It depends on S through the pair {n + , n − }, its unit normals defined at each smooth part of S. The internaledge microtraction is developed by the contact of an internal-edge C with the adjacent parts of B \ S.
(v) External microforce: The external microforce γ := γ(x, t) represents a body microforce per unit mass, acting on the body B. It is developed by external causes, outside of B.
In accounting for the surface-couple, the boundary-edge, and the internal-edge microtractions, in addition to the conventional surface microtraction and the external microforce, we obtain a general second-grade phase-field theory. Conversely, first-grade phase-field theories introduced by Fried & Gurtin [1,2] and Gurtin [3] are recovered when these additional microtraction fields are neglected.
2.2. Differential relations on evolving surfaces. We began our theory by assuming the body B to be fixed, and so any other part P and surface S. However, let the surfaces depicted in Figure 2 undergo deformation from S to y(S) =: S . This motion is somewhat arbitrary but smooth and needed to characterize variationally the contact microforce and microtorque fields arising from the interactions between adjacent parts. We then parameterize the motion from S to S , with a one-parameter family of smooth invertible mappings, such that (2) y 1], and the displacement u sufficiently smooth. Quantities with the subscript live on S . Defining the parameterized deformation gradient of this motion by (3) F := grad y | =1 and F := grad y , the unit tangent and unit normal at y on S , respectively, are where (·) represents the transposition. The curvature tensor L is (5) L := −grad S n = −(grad n)P and P := 1 − n ⊗ n, and the determinant of the deformation gradient J := det(F ). In a Darboux frame, the following relations hold, In appendix A, we detail the derivations leading to the relations (6), which we use in this section. With these identities, we have suitable machinery to characterize variationally all types of microtractions we introduce in our description.
Remark 2 (Scenarios of a nonsmooth edge ∂S). There are two particular scenarios to be considered when it comes to analyzing a nonsmooth edge ∂S. Consider a unit tangent t discontinuous at some x of ∂S. Then, in the first scenario, the unit normal n is continuous and the unit tangent-normal ν is discontinuous, while alternatively in the second scenario, n is discontinuous but the unit tangent-normal ν is continuous.
To illustrate both scenarios, one can imagine two particular cases. In the first scenario, imagine a square surface. The unit normal n is continuous everywhere, but at the corners the unit tangent t and the unit tangent-normal ν are discontinuous. In the second scenario, imagine half of a cone. The unit tangent-normal ν is continuous everywhere, but at its vertex the unit normal n and the unit tangent t are discontinuous.
Bearing in mind Remarks 1 and 2, we are led to speculate about the existence of a hypermicrostress tensor field as follows.
Theorem 1 (Existence of a hypermicrostress tensor field ). The interaction between an edge ∂S of a smooth open oriented surface S and the adjacent parts of B\S invokes the existence of a linear transformation Σ (x, t) ∈ Lin 2 , denoted as the hypermicrostress tensor field in B for all (x, t).
Proof. Consider the two particular scenarios on nonsmooth edge ∂S described in Remark 2: (i) First Scenario: t and ν are discontinuous at some x of ∂S while n is continuous; (ii) Second Scenario: t and n are discontinuous at some x of ∂S while ν is continuous.
The inner product of the jump (17) with t + yields, in the first scenario, and considering that t + · ν − = t − · ν + and t + · t − = ν − · ν + , we arrive at whereas, in the second scenario, and considering that t + · n − = t − · n + and t + · t − = n − · n + , we arrive at In the first scenario, by keeping t − fixed and consequently ν − fixed as well, the boundary-edge microtraction τ ∂S is linear in ν, that is, (22) τ ∂S (x, t; ν, n) := a · ν, a := a(x, t; n), and a · n = 0, ∀ n, ν ∈ Unit, n · ν = 0, alternatively, in the second scenario, by keeping t − fixed and consequently n − fixed as well, the boundaryedge microtraction τ ∂S is linear on n, that is, To encompass conditions (22) and (23) into a single one, we let {e 1 , e 2 , e 3 := n} be a orthonormal basis and drop the dependency on x and t. Combining (22) 1 and (23) 1 , we have that (a(n) · e α )e α = (b(e α ) · n)e α , where the index α goes from 1 to 2, leaving out the unit normal e 3 = n from the set e α . Now, noting that (24) is a(n), the boundary-edge microtraction τ ∂S (x, t; ν, n) = a(n)ν can be specified as Next, we express b(e α ) in a fixed orthonormal basis e i , such that b(e α ) = b i (e α )e i . By using this generic but fixed orthonormal basis, the the boundary-edge traction τ ∂S (x, t; ν, n) assumes the forms Therefore, based on the jump condition (17) there exists a linear transformation Σ (x, t) ∈ Lin, denoted as the hypermicrostress tensor field in B for all (x, t), such that Next, for the surface microtraction, we characterize a jump condition throughout the surface based on a surface balance of microforces and Theorem 1 as follows.
Proposition 1 (Surface microtraction: jump condition throughout the surface). Consider a smooth open surface S oriented by an unit normal n with boundary-edge ∂S. Let ξ S := ξ S (x, t; n, L) and ξ * S := (x, t; n, L) be the surface microtractions defined on opposite sides of S and τ ∂S := τ ∂S (x, t; ν, n) the boundaryedge microtraction. In balancing these microforces on S while accounting for (27) from Theorem 1, the following jump condition is obtained.
Proof. We postulate the surface balance of microforces on a smooth surface S as follows Next, consider the surface divergence theorem on a smooth open surface S, for any vector field g on S. Thus, with the expression for the boundary-edge microtraction (27) in the surface balance of microforces (29) and applying the surface divergence theorem for smooth open surfaces, we are led to While by localizing it, we arrive at the statement of this proposition, where div S (PΣ n) represents a jump condition throughout the surface S from one side to the other. Now, recalling the variational setting of the proof of the Cauchy theorem [4], we establish that there exists a microstress-like field ζ such that Note that, in the absence of higher-order effects, such as the hypermicrostress Σ , we recover the microtraction presented by Fried & Gurtin [1], that is, ξ S (x, t; n) = ζ(x, t) · n.

2.3.2.
Boundary-edge microtraction characterization and its implications: part 2. In analyzing the first integral in (8), for the point we wish to make, it is enough to analyze ξ S instead of ξ S + ξ * S . So, let be a functional on S . Additionally, let ξ S x , ξ S n , and ξ S L be the intrinsic partial derivatives of ξ S with respect to x, n, and L, respectively. The partial derivative ξ S x need not any additional characterization. The remainder derivatives, ξ S n and ξ S L need further understanding as follows.
Remark 3 (Characterization of the partial intrinsic derivatives of the surface microtraction). Let β( ) and B( ) be smooth parametric orthonormal-vector-and symmetric-tensor-valued functions of We then characterize the partial intrinsic derivatives of the surface microtraction as Analyzing each component, we conclude that Thus, the first variation of G(S ) is Consider the surface divergence theorem for any vector field g on S, and the identity with K := − 1 2 div S n. Using (38) and (39), and the partial intrinsic derivatives from remark 3, we arrive at Although this integral does have a contribution on ∂S, it does not characterize a jump condition. Thus, the surface microtraction does not affect the proof of (27).
Next, with the surface divergence theorem for smooth open surfaces (30) and the identity the surface balance of microtorques (42) specializes further to and localizing it, to arrive at the statement of this proposition, where PΣ n represents a jump condition across the surface S.

2.4.
Analysis on a nonsmooth open oriented surface S. Applying Fosdick's procedure [5], we proved that the edge microtraction invokes the existence of a hypermicrostress field. Moreover, the edge microtraction is linear on ν and n through the hypermicrostress. Next, we study the surface balance of microforces and microtorques on a nonsmooth open oriented surface S with boundary ∂S while considering the lack of smoothness of S along C.
2.4.1. Surface balance of microforces. Having characterized the boundary-edge microtraction (27) and the surface microtraction jump (28), we determine the internal-edge microtraction on a nonsmooth open oriented surface S as follows.
Proposition 3 (Internal-edge microtraction). Consider a nonsmooth open surface S with boundary ∂S and an internal-edge C. Let ξ S := ξ S (x, t; n, L) and ξ * S := (x, t; n, L) be the surface microtractions defined on opposite sides of S, τ ∂S := τ ∂S (x, t; ν, n) and τ C := τ C (x, t; n + , n − ) the boundary-and internal-edge microtractions, respectively. In balancing these microforces on S while accounting for (27) from Theorem 1 and (28) from Proposition 1, the following representation for the internal-edge microtraction is obtained, Proof. We postulate the surface balance of microforces on S as follows Owing to the lack of smoothness at an internal-edge C, the surface divergence theorem over a closed surface exhibits a surplus, that is, where { {g · ν} } := g + · ν + + g − · ν − , for any smooth vector field g on S with limiting values g + and g − on C. For open surfaces, the surface divergence theorem (48) reads for any vector field g on S.
By the surface divergence theorem (49) with the boundary-edge microtraction (27) and the jump condition for the surface microtraction (28) between opposite sides of S, the surface balance of microforces (47) specializes to and localizing it, we arrive at the statement of this proposition.
2.5. Analysis on an arbitrary part P. On an arbitrary part P, we postulate the partwise balance of microforces as follows where π and γ are the internal and external microforces densities. We also postulate the partwise balance of microtorques on an arbitrary part P, which reads where ξ is a microstress and r = x − o a vector for a fixed point o.
2.5.2. Pointwise microtorque balance. Substituting the internal-edge microtraction (46) and the surface microtration representation (33) into the partwise balance of microtorques (52), we obtain while considering the identity, and applying the surface divergence theorem for nonsmooth closed surfaces (48), we arrive at Now, applying the volume divergence theorem, we obtain the following representation where the first integral is zero by the pointwise balance of microforces (54). Thus, the partwise balance of microtorques yields To annihilate the second integral, the surface-couple microtraction is assumed to have the form (60) S = (n ⊗ n)Σ n, yielding the partwise balance of microtorques Localizing this expression, we arrive at the pointwise balance of microtorques We can now rewrite the surface microtraction (33) as Furthermore, accounting for the pointwise microtorque balance (62) into the pointwise balance of microforces (54), we arrive at the field equation of the phase-field gradient theory, which reads Espath et al. [8,9] also derived this equation. For convenience, we define the hypermicrotraction and its relation with the surface-couple microtraction (65) σ S := n · Σ n, S := σ S n. Remark 4 (On the symmetry of the hypermicrostress). There are some evidences that suggest that the hypermicrostress should be symmetric, and we list them as follows. evidence (i) Recalling that the internal-edge microtraction has the representation τ C = { {ν · Σ n} } and since n ± and ν ± live in the same plane, (n + ⊗ ν + + n − ⊗ ν − ) ∈ Sym, only the symmetric part of the hypermicrostress contributes to the internal-edge microtraction, that is, evidence (ii) Recalling that the surface-couple microtraction has the representation S = (n ⊗ n)Σ n, only the symmetric part of the hypermicrostress contributes to it, that is, (67) (n ⊗ n)Σ n = (n ⊗ n) sym (Σ )n; evidence (iii) Given that P skw (Σ )n = skw (Σ )n and with the following identity which holds for any tensor field A defined on S, then (69) div S (skw (Σ )n) = n · div S (skw (Σ ) ) = −n · div S (skw (Σ )) and (70) n · (n ⊗ n : (grad (skw (Σ ))) = 0, we obtain (71) − n · div (skw (Σ )) = div S (P (skw (Σ ))n).
Thus, recalling that the surface microtraction has the representation ξ S = (ξ − div Σ ) · n − div S (PΣ n), only the symmetric part of the hypermicrostress contributes to it, that is, evidence (iv) Recalling that the field equation of the phase-field gradient theory reads div (ξ − div Σ ) + π + γ = 0, only the symmetry part of the hypermicrostress contributes to it, since div 2 Σ = div 2 (sym (Σ )).

2.5.3.
Action-reaction principle. An important consequence of the representations of the microtractions is that the surface microtraction and the surface-couple microtraction are local at any point x on S and any time t and that the internal-edge microtraction is local at any point x on C and any time t. Moreover, we state that ξ S depends on S through the unit normal n and the curvature tensor L of S at x, and S depends on S through n (cubically), whereas τ C depends on C through the unit normals {n + , n − } (or equivalently through the unit tangent-normals {ν + , ν − }). Next, letting −S (−C) denote the surface S (internal-edge C) oriented by −n ({−n + , −n − }), with reference to (5), −S has curvature tensor −L, we see that The relations (73) 1,2 make explicit the action-reaction principle in terms of microtractions between two smooth surfaces endowed with opposite unit normals and opposite curvature tensors at a point. Conversely, the relation (73) 3 presents the action-reaction principle between two parts of the same oriented nonsmooth surface divided by an internal-edge, which in turn is defined by a pair of discontinuous unit normals at a point.

Power balance
Once the theory is built upon balances of fundamental fields, the 'principle of virtual powers' becomes a theorem. We thus state this theorem as follows.
Theorem 2 (The virtual power theorem). Assuming that the external virtual power is expended by the following conjugate pairs (i) Surface microtraction power per unit area: {γ, χ} on P; (ii) Surface microtraction power per unit area: {ξ S , χ} on ∂P; (iii) Surface-couple microtraction power per unit area: { S , ω} on ∂P, with ω := grad χ; (iv) Internal-edge microtraction power per unit length: {τ C , χ} on C ⊂ ∂P, together with the assumption that the field equation (64) is satisfied, the virtual power balance (74) holds for any scalar smooth and admissible virtual field χ.
Proof. By using the hypermicrotraction representation (65)  The virtual power balance, that is, the balance between the external and internal powers, holds for any arbitrary part P and any choice of the virtual field χ.
where ε and η represent the internal-energy density and entropy density, q is the heat flux, r is the heat supply, and ϑ > 0 is the absolute temperature.
Since, by (78), W ext (P) = W int (P), we may substitute the expression (76) defining the power expended on P by external agencies on the right-hand side of the energy balance (81) 1 by the expression (77) defining the internal power of P. The result of localizing both of (81) is Important to what follows is the free-energy density (83) ψ := ε − ϑη.
Then, since (82) 2 may be written as if we multiply this equation by ϑ and subtract it from (82) 1 , we arrive at the pointwise free-energy imbalance

4.2.
Pointwise and partwise free-energy imbalances for isothermal processes. Applications in which thermal changes are negligible are encompassed by the present framework when attention is restricted to isothermal processes, namely to processes in which For such processes, the expression (83) for the free-energy density specializes to (87) ψ = ε − ϑ 0 η and the pointwise free-energy imbalance (85) has the simple form Further, if we multiply (82) 2 by ϑ 0 and subtract it from (82) 1 we arrive at a partwise free-energy imbalance requiring that the temporal increase in free energy of P be less than or equal to the power expended by external agencies on P.
The imbalance (89) is often the implicit starting point of most purely mechanical theories. Of course, the pointwise imbalance (88) may be derived as a direct consequence of (89), the power balance (78), and the expression (77) for the internal power without introducing the notion of temperature; in that sense, the imbalance stands on its own as a starting point for the development of purely mechanical theories. 5 The idea of using a virtual-power principle to generate an appropriate form of the external power expenditure in the energy balance was originated by Gurtin [11, §6].

Nonconserved second-grade phase field equation
We hereafter restrict attention to purely mechanical processes governed by the isothermal version (88) of the pointwise free-energy imbalance. Guided by the presence of the power conjugate pairings πφ, ξ · gradφ, and Σ : grad 2φ in that inequality, we consider a class of constitutive equations that delivers the free-energy density ψ, internal microforce π, microstress ξ, and hypermicrostress Σ at each point x in B and each instant t of time in terms of the values of the phase field ϕ, its first and second gradients grad ϕ and grad 2 ϕ, and its time rateφ at that point and time.
We do not prescribe a constitutive equation for the external microforce γ but instead allow it to be chosen in any way that is needed to ensure the satisfaction of the field equation (64). Arguments introduced by Coleman & Noll [12] can then be adapted to show that for the dissipation inequality (88) to be satisfied in all processes it is necessary and sufficient to require that: • The free-energy density ψ is given by a constitutive response functionψ that is independent ofφ: (90) ψ =ψ(ϕ, grad ϕ, grad 2 ϕ).
• The internal microforce π is given by a constitutive response functionπ that splits additively into a contribution derived from the response functionψ and a dissipative contribution that, in contrast toψ,ξ, andΣ , depends onφ and must be consistent with a residual dissipation inequality: π dis (ϕ, grad ϕ, grad 2 ϕ,φ)φ ≤ 0.
In view of the constitutive restrictions (90)-(92), the response function for the free-energy density serves as a thermodynamic potential for the microstress, the hypermicrostress, and the equilibrium contribution to the internal microforce. A complete description of the response of a material belonging to the class in question thus consists of providing scalar-valued response functionsψ and π dis . Whereasψ depends only on ϕ, grad ϕ, and grad 2 ϕ, π dis depends also onφ. Moreover, π dis must satisfy the residual dissipation inequality (92) 2 for all choices of ϕ, grad ϕ, grad 2 ϕ, andφ.

5.1.
Nonconserved second-grade phase-field equation. We now detail the logic of the derivation that leads us to define the Swift-Hohenberg equation as a particular case of the nonconserved second-grade phasefield equation. Thus, we here generalize the Swift-Hohenberg equation and fit this description within the second-grade phase-field framework we built in this section. Using (91) and (92) in the field equation (64), we obtain an evolution equation for the phase field. We refer to (93) as a 'nonconserved second-grade phase-field equation. where f is a function of ϕ, and λ > 0, > 0, β > 0 are problem-specific-constants. Here, f and λ carry dimensions of energy per unit volume, carries dimensions of length, and β carries dimensions of (dynamic) viscosity. Granted thatψ and π dis are as defined in (94), the thermodynamic restrictions (91) and (92) yield where = div grad denotes the Laplacian and a superposed prime denote differentiation with respect to ϕ.
Using the particular constitutive relations (95) in the field equation (64), we obtain the Swift-Hohenberg [6] equation

Configurational fields.
Configurational forces are primitive entities that describe the motion of interfaces as well as the thermodynamics of their evolution. These forces are associated with the integrity of the material structure and the evolution of defects. Moreover, configurational forces expend power associated to the transfer of matter. Now, we recall the configurational balance presented by Fried [13], for a part P which renders after localization the following pointwise version where C is the configurational stress tensor (Eshelby stress tensor) while f and e are the internal and external forces. The transfer of matter defines kinematic processes which characterize the power expended by the configurational forces. Thus, we first establish how configurational forces expend power in an immaterial migrating arbitrary part P , where υ is the migrating boundary velocity defined on ∂P , letting n denote the outward unit normal. We consider that the migrating boundary ∂P may exhibit a lack of smoothness along a curve C with limiting normals {n + , n − }. Furthermore, the configurational traction Cn is assumed to be power conjugate to υ on ∂P .
We now use the external virtual power (76), where γ, ξ S , and τ C are conjugates to the virtual field χ, while σ S is conjugate to ∂χ/∂n . Thus, we set as virtual fields the advective terms to follow the motion of ∂P . Next, with the surface microtraction (63) and the hypermicrotraction (65) 1 , consider the following identities and Now, bearing in mind that grad ϕ · grad S υ = 0 together with the surface divergence theorem on a nonsmooth closed surface (48), we are led to the following external configurational power Since the nature of the motion of ∂P involves only the normal component υ·n , the power must be indifferent to the tangential component of υ, implying where α is a scalar field. Thus, we can express the second integral of (102) as Appealing to the free-energy imbalance (89) for a migrating arbitrary part P with a velocity υ, we can state that which implies that α = ψ. Having α and with equation (103), we determine the configurational stress. Invoking (91), (92) 1 , and (98), we determine the internal and external configurational forces. Thus, the explicit form of the configurational stress tensor is while the internal and external configurational forces (108) f = −π dis grad ϕ and e = −γ grad ϕ, respectively.

Conserved second-grade phase field
We here extend our theory to the case where the phase field represents the concentration of a conserved species with chemical potential µ, flux , and external rate of species production s, while continuing to restrict attention to isothermal processes. Following Gurtin's derivation of the Cahn-Hilliard equation [3, §3], we therefore supplement the field equation (64) by a partwise species balance After localizing it, we obtain the pointwise version of the species balance (110)φ = s − div  Moreover, we augment the partwise free-energy imbalance (89) to account for the rate at which energy is transferred to P due to species transport, yielding Localizing (111) and using the field equation (64) and the pointwise species balance (110) to eliminate the external microforce γ and rate of species production s, we arrive at the pointwise free-energy imbalance (112)ψ + (π − µ)φ − ξ · gradφ − Σ : grad 2φ +  · grad µ ≤ 0.
• Granted that the species flux  depends smoothly on the gradient grad µ of the chemical potential µ, it is given by a constitutive response function of the form (116)  =(ϕ, grad ϕ, grad 2 ϕ, µ, grad µ) = −M (ϕ, grad ϕ, grad 2 ϕ, µ, grad µ)grad µ, where the mobility tensor M must obey the residual dissipation inequality (117) grad µ · M (ϕ, grad ϕ, grad 2 ϕ, µ, grad µ)grad µ ≥ 0 for all choices of ϕ, grad ϕ, grad 2 ϕ, µ, and grad µ. In contrast to the theory previously developed for a phase field that is not a conserved species, a complete description of the response of a material belonging to the present class consists of providing a scalar-valued response functionψ and a tensor valued response function M . Whereasψ depends only on ϕ, grad ϕ, and grad 2 ϕ, M may depend also on µ and grad µ. Moreover, M must satisfy the residual dissipation inequality (117) for all choices of ϕ, grad ϕ, grad 2 ϕ, µ and grad µ.
6.1. Conserved second-grade phase-field equation. We now detail the logic of the derivation that leads us to define the phase-field crystal equation as a particular case of the conserved second-grade phasefield equation. Thus, we here generalize the phase-field crystal equation and fit this description within the conserved second-grade phase-field framework we built in this section.
In analogy to the comment immediately after the equation (93), this suggests the possibility of referring to (119) with µ given by (118) as a 'conserved second-grade phase-field equation.' 6.1.1. Phase-field crystal equation. Mimicking the assumptions leading from (93) to the Swift-Hohenberg equation (96), we choose the definition ofψ to be (120)ψ(ϕ, grad ϕ, grad 2 ϕ) = f (ϕ) + 1 2 λ(ϕ 2 − 2 2 |grad ϕ| 2 + 4 (tr (grad 2 ϕ)) 2 ), In addition, we stipulate that the mobility tensor depends at most on ϕ and is isotropic, so that Combining (122) 3 and (123) and using the resulting expression in the pointwise species balance (119), we obtain the phase-field crystal equation 6.2. Configurational fields. In accounting for species transport, species migration and the associated energy flow that occur in conjunction with the motion of the migrating part P , must be considered. The partwise species balance (109) is rewritten as where the free-energy imbalance for a migrating volume P (106) is specialized from (111), becoming Thus, the expressions (100) and (126) yield the explicit form of the configurational stress tensor, the internal and external configurational forces, for the generalized conserved second-grade phase-field equation, 7. Boundary conditions 7.1. Nonconserved second-grade phase field. We here extend Fried & Gurtin's [14,15] procedure, also expoited by Duda et al. [16] for the Cahn-Hilliard equation, to determine thermodynamically consistent boundary conditions by tailoring the surface balances of microforces and microtorques. In taking the surface S in expressions (29) and (42) (while including the terms relative to the internal-edge microtration) to the limit such that the surface coincides with the boundary, S ⊆ ∂B, the surface ξ S , surface-couple S , boundary-edge τ ∂S , and internal-edge τ C microtractions happen to represent external actions, ξ S env , S env , τ ∂S env , and τ C env , respectively, from the environment of B. That is, in this limit the surface balances of microforces (29) and the surface balances of microtorques (42) become ((ξ S env − (ξ − div Σ ) · n + div S (PΣ n)) + ( S env − (n ⊗ n)Σ n)) r da = 0, ∀ S ⊆ ∂B and t.
Localizing expressions (132) and (133), we obtain the explicit representations of τ ∂S env and τ C env . Replacing τ ∂S env and τ C env in the surface balance of microforces (128) and localizing it, we arrive at the representation of ξ S env . Finally, with ξ S env , τ ∂S env and τ C env in (134) and localizing it, we are led to S env . Thus, the explicit form of the environmental microtractions are The relations (135), represent the first set of suitable boundary conditions, where ξ S env , σ env , τ ∂S env and τ C env are given on S. In the variational context, these are natural boundary conditions on S nat . Next, we require that the temporal increase in free energy of S to be zero. This implies in particular that the power expended on S be great or equal than zero. We express the partwise surface free-energy imbalance as The power expended on S * by the surface ξ * S and surface-couple * S microtractions is given by Thus, with (137) expression (136) becomes For pasive environments, the quantities ξ S env , σ env , τ ∂S env and τ C env equal zero and so does the power expenditure exerted by the environment W env (S). However, we consider a more general setting and allow for non-homogeneous boundary conditions. Thus, we set the external power to be (139) W env (S) = S ξ S envφenv + σ env ∂φ env ∂n da + ∂S τ ∂S envφenv ds + C τ C envφenv ds, whereφ env and ∂φ env /∂n are the limits of the time derivative of the environmental phase field and its normal derivative at S. Accounting for expression (139), the partwise surface free-energy imbalance (138) renders (140) Since ξ S env = ξ S , σ env = σ S , τ ∂S env = τ ∂S , and τ C env = τ C , uncoupling the integral on different parts followed by localization, we obtain the following pointwise conditions Expressions in (141) will serve us to design suitable boundary conditions for this continuum mechanical theory. In what follows, we propose some classes of boundary conditions. We focus on the uncouple case where (142) which is sufficient but not necessary to guarantee the inequality direction of (141). The second possible set of boundary conditions is given by the trivial solution of (142) when used as an equality, that is, the assignment These are essential boundary conditions on S ess . Note that, S ess ∩ S nat = Ø. The third possible set of boundary conditions is given ifφ env and ∂φ env /∂n are prescribed, while ξ S env = ξ S , σ env = σ S , τ ∂S env = τ ∂S , and τ C env = τ C are given by with a, b, c, d > 0. Combining ξ S env = ξ S , σ env = σ S , τ ∂S env = τ ∂S , and τ C env = τ C from (135) with (144), we obtain the mixed boundary conditions on S mix .

Boundary conditions for the classical Swift-Hohenberg equation.
As for the classical Swift-Hohenberg equation, although the essential boundary conditions (143) remain the same, the natural (135) and mixed (144) boundary conditions can be specialized. For the choice (94) 1 ofψ, (95) 1,2 yield ξ · n = −2λ 2 ∂ϕ/∂n, n · div Σ = λ 4 ∂( ϕ)/∂n, PΣ n = λ 4 ϕ P n = 0, n · Σ n = λ 4 ϕ, and ν · Σ n = λ 4 ϕ ν · n = 0. Thus, the natural boundary conditions (135) read while the mixed boundary conditions (144), when taking into account (145), become 7.3. Conserved second-grade phase field. In addition to the surface balances of microtractions (128) and microtorques (129), we supplement the system with the partwise surface species balance on S which by localization renders Here,  env represents the transfer of mass from the environment into S. As we augmented the partwise free-energy imbalance (89) with the energy transfer rate to P due to species transport to arrive at (111), we augment (136) with S µj · n da to obtain Analogously, (139) is augmented to become where µ env is the limit of the environmental chemical potential at S. The surface free-energy imbalance, under the assumptions that led to (140), from expression (149) while accountig for (150), we arrive at Following the same procedure that led to (141) 1 and taking into account the pointwise surface species balance (148), we obtain the following additional inequality, Note that, (141) 2,3 remain the same. For the sake of simplicity, we opt to split and study (152) term by term as we did in (142). Thus, the only additional condition is The first additional natural boundary condition arises from the pointwise surface species balance (148), where  env is prescribe and µ env is obtained from using (153) as an equality. Conversely, from (153), we obtain trivially the essential boundary condition, where µ env is prescribed and  env is computed from the pointwise surface species balance (148). A mixed boundary condition is obtained by prescribing µ env and evaluating the transfer of mass from the environment with on S mix , respectively.

Summary
The continuum mechanical theory we introduce in this paper allows us to describe the underlying mechanical interactions which yield popular phase-field models such as the Swift-Hohenberg and phase-field crystal equations. We also recognize and account for the lack of smoothness in arbitrary parts which renders additional interations between different parts. In considering these interactions, we generalize these phase-field models. We summarize the results as follows.
Remark 5 (Summary of results independent of constitutive relations). The fundamental fields and the field equations are (i) Surface microtraction on S: (ii) Surface microtraction on the opposite side of S: (vi) Internal-edge microtraction on C: σ S := n · Σ n =⇒ S = σ S n. (viii) Balance of microforces on a part P: div ζ + π + γ = 0.
(ix) Balance of microtroques on a part P: (xi) The virtual power balance on a part P: where χ is a smooth and admissible scalar virtual field and ∂χ/∂n its normal derivative. (xii) Configurational balance: div C + f + e = 0.

Acknowledgments
We are indebted to Professor Eliot Fried. We had many exhaustive discussions in which he gave us valuable ideas, constructive comments, and encouragement. The first author would also like to thank his Quantities with the subscript refer to the deformed configuration, whereas quantities without the subscript live on the reference or undeformed configuration. The point x represents a point in the undeformed configuration S, whereas y represents its mapping onto the deformed configuration S . The function y is a-one-parameter family-parameterization, which linearizes the mapping x → y along the displacement u. Thus, F is the total deformation gradient, while F is the parameterized deformation gradient. Finally, let J := det F . Gurtin et al. [17, (equations (6.15) and (8.4))] presents deformations laws for the unit tangent and unit normal vectors of a surface undergoing deformation from S to S . These vectors preserve their norm under the following transformations while the elements of length ds, area da, and volume dv, [17, (equations (7.14), (8. It is also convenient to express a vector a as a = (a · t)t + (a · n)n + (a · ν)ν. Finally, we denote the curvature tensor by L := −grad S n while the mean curvature by L := − 1 2 div S n. We here aim at computing the following derivatives The last three quantities (168) 5,6,7 represent, respectively, the rate of change of elements of length, area, and volume. In order to compute the inverse of the deformation gradient, by Neumann series [18], given the invertible second-order tensor A and the second-order tensor B, we define the following identity and letting A := (1 − )1 and B := F , we arrive at When it comes to analyzing A −1 , with A = (1 − )1 , we restrict our analysis to the range [0, 1[ without loss of generality. To determine the rate of change of the unit normal with respect to in (168) 1 when a surface undergoes deformation from S → S , we establish the following identities. The rate of change of the inverse of the deformation gradient with respect to is while the unit normal presents the following rate of change Although it does not preserve its norm. Thus, to determine the rate of change of inverse of the norm of the normal, |F − n| −1 , we first compute to compute the rate of change of the norm, |F − n| = F −1 F − : n ⊗ n, To determine the rate of change of the unit tangent with respect to in (168) 2 when a surface undergoes deformation from S → S , we establish the following identities. The unit tangent presents the following rate of change Again, this rate does not preserve its norm. Thus, to determine the rate of change of the inverse of the norm, |F t| −1 , we establish that to compute the rate of change of the norm, |F t| = F F : t ⊗ t, We can now compute the first variation of the curvature tensor as dL d = grad ((grad S u) n) (1 − n ⊗ n) + grad n grad u(1 − n ⊗ n) − grad n ((grad S u) n ⊗ n + n ⊗ (grad S u) n) = grad S ((grad S u) n) − L grad S u + L(grad S u) n ⊗ n, where we use the fact that n ⊗ (grad S u) n = (n ⊗ n)grad S u.