Generalized Swift–Hohenberg and phase-field-crystal equations based on a second-gradient phase-field theory

The Swift–Hohenberg and phase-field-crystal equations describe a multitude of processes involving spatiotemporal pattern formation. Swift and Hohenberg [1] derived the first of these equations through a weakly nonlinear approximation of the infinite Prandtl number version of the Boussinesq equations about the Rayleigh number corresponding to the onset of convective instabilitiies. Their equation can be formulated as a relaxation, or “gradient flow”, law for an energy functional that depends on a scalar order parameter along with its gradient and Laplacian. The resulting evolution equation is parabolic and involves a fourth-order spatial operator. As Burke and Knobloch [2] explain, a distinguishing feature of the Swift–Hohenberg equation that distinguishes it from other pattern forming equations, like the Kuramoto–Sivashinsky equation, but which stem from small wave-number expansions is that it exhibits a finite wavenumber instability at onset and, thus, admits stable spatially periodic equilibria. For this reason, it has been used to study disclinations (Golovin and Nepomnyashchy [3]), defects (Galla and Moro [4]), grain boundaries (Boyer and Viñals [5]), oscillons and other patterns in vibrated granular media (Crawford and Riecke [6]), and radiation-induced microstructural features in immiscible alloys (Simone et al. [7]).

Starting from the classical density functional theory for crystallization, as described by Ramakrishnan and Yussouff [8], Elder and Grant [9] derived the phase-field-crystal theory by approximation for the two-particle correlation function of the melt that enters the free-energy density of the crystal relative (measured relative to the homogeneous melt). The resulting energy functional is identical to that underpinning the Swift–Hohenberg equation. The salient order parameter is, however, a conserved quantity—the atomic number density. The phase-field-crystal equation is thus the conserved counterpart of the Swift–Hohenberg equation. This relationship is completely analogous to that between the Cahn–Hilliard equation and the Allen–Cahn equation. The spatial operator entering this parabolic equation is consequently of sixth-order. Apart from modeling crystal growth and dissolution, the phase-field-crystal equation has been used to study a broad spectrum of phenomena, including plasticity (Stefanovic et al. [10]), pattern formation in liquid crystals (Löwen [11]), epitaxial growth (Gránásy, et al. [12], Yua et al. [13]), glass formation (Berry and Grant [14]), two-dimensional materials like graphene (Seymour and Provatas [15], grain boundary motion, grain rotation, and dislocation reactions in body-centered-cubic bicrystals (Yamanaka et al. [16]), and plastic flow due to dislocation motion (Skaugen et al. [17]). Additionally, as Provatas et al. [18] explain, the phase-field-crystal equation also provides a natural basis for developing multiscale simulation methods.

Our goal in this paper is to develop a broadly applicable thermodynamically consistent framework that subsumes the Swift–Hohenberg and phase-field-crystal equations, accompanied by suitable boundary conditions and associated relations that guarantee that appropriate measures of total energy be nonincreasing along solution paths. To achieve this, we exploit and extend an approach used by Fried and Gurtin [19] to derive generalizations of the scalar Ginzburg–Landau equation and later applied by Gurtin [20] to derive generalized versions of the Cahn–Hilliard equation. In that approach, the phase field is a basic kinematical descriptor and power expenditures associated with its temporal variations, along with those of its spatial gradient, are ascribed to a system of microforces subject to their own balance. However, because the spatial operators entering the Swift–Hohenberg and phase-field-crystal equations are respectively two orders higher than those entering Ginzberg–Landau and Cahn–Hilliard equations, we must also account for power expenditures associated with temporal variations of the second spatial gradient of the phase field. This leads to a second-gradient phase-field theory.

Instead of the pivotal microforce balance of Fried and Gurtin [19] and Gurtin [20], we emulate Gurtin [21] and base our theory on a power balance. This fundamental postulate requires that the instantaneous power expended by all agencies external to any part of the region within which the phase field is defined be equal to the instantaneous power expended by all internal actions within that part. In line with the presence of the second gradient of the phase field, our theory includes a hypermicrostress and a hypermicrotraction above and beyond the microstress, its associated microtraction, and the internal and external microforce densities that enter the conventional first-gradient phase-field theory. We also relax the standard requirement that the microtraction distributed on a surface separating two parts depends only on the orientation of that surface. As consequences of the principle of virtual-power, we obtain a generalization of the pointwise microforce balance from the conventional first-gradient phase-field theory together with representations for the microtraction and the hypermicrotraction. The microstress and hypermicrostress both appear in the generalized microforce balance. Whereas the microtraction depends on surface orientation and surface curvature, the hypermicrotraction depends only on surface orientiation, albeit quadratically. As corollaries of the generalized pointwise microforce balance and the representations for the microtraction and hypermicrotraction, we derive partwise statements of microforce balance and hypermicroforce balance.

In our treatment of energetics, we follow the lead of Fried and Gurtin [19] and Gurtin [20] and confine attention to isothermal processes, with the consequence that the first and second laws of thermodynamics combine to yield a free-energy imbalance. Using the pointwise version of that imbalance, we find the largest class of smooth, thermodynamically admissible constitutive relations in which the free-energy density, internal microforce density, microstress, and hypermicrostress serve as dependent constitutive variables and the phase field along with its first and second spatial gradients and its temporal rate serve as independent constitutive variables. Augmenting our microforce balance with those constitutive relations delivers a broad generalization of the Swift–Hohenberg equation. For equations of phase-field-crystal type, the phase field is identified as the volume fraction of a conserved constituent governed by a transport equation expressing the balance of constituent content. Using a suitably modified version of the free-energy inequality, we find the largest class of smooth, thermodynamically admissible constitutive relations in which the list of dependent variables considered previously is augmented to include the constituent flux while the corresponding list of independent constitutive variables is augmented to include the chemical potential and its spatial gradient but no longer includes the temporal rate of the phase field. The constitutively augmented microforce balance then determines the chemical potential in terms of the phase field and its first, second, third, and fourth spatial gradients. Augmenting the constituent content balance with the constitutive relation for the constituent flux and the expression for the chemical potential then yields a broad generalization of the phase-field-crystal equation.

To derive thermodynamically consistent boundary conditions for our equations, we adapt an approach introduced by Fried and Gurtin [27] and subsequently extended by Duda et al. [28]. We restrict attention to situations where the boundary is unstructured and cannot support constituent transport. As a starting point, we formulate versions of microforce and hypermicroforce balances and the free-energy imbalance for an arbitrary boundary pillbox. For simplicity, we rule out coupling effects and dissipative interactions with the environment. This leads to what Fried and Gurtin [27] call ‘zero-dissipation conditions’. In the absence of constituent transport, we find that it is generally permissible to assign the microtraction and hypermicrotraction on the same portion of the boundary. When species transport is present, we find that it is additionally permissible to assign either the constituent flow or the chemical potential across a portion of the boundary. The thermodynamic structure of our theory ensures the existence of energy decay relations for our generalized Swift–Hohenberg and phase-field-crystal equations for this class of boundary conditions.

The remainder of this paper is organized as follows: In Sect. 2, we develop expressions for internal and external expenditures that account explicitly for second gradient effects and state the associated power balance. In Sect. 3, we introduce and obtain the consequences of the principle of virtual-power. In Sect. 4, we describe interactions between the system with state characterized by the phase field and its surroundings. In Sect. 5, we pose the free-energy imbalance for situations where the phase field is not a conserved order parameter and use that imbalance to determine thermodynamically admissible constitutive relations for the microstress, hypermicrostress, and internal microforce density in terms of the phase field, its first and second spatial gradients, and its temporal rate. In Sect. 6, we augment the pointwise microforce balance with the constitutive relations obtained in Sect. 5 to arrive at our generalization of the Swift–Hohenberg equation. In Sects. 7 and 8, we consider the consequences of identifying the phase field with the volume fraction of a single, independent, mobile constituent, augment our theory to account for the transport effects, use the free-energy imbalance to determine thermodynamically admissible constitutive relations for the microstress, hypermicrostress, internal microforce density, and constituent flux, and arrive our generalization of the phase-field-crystal equation. In Sect. 9, we develop thermodynamically consistent boundary conditions that are compatible with our framework. In Sect. 10, we establish energy-decay relations for our generalized versions of the Swift–Hohenberg and phase-field-crystal equations. Finally, in Sect. 11, we summarize and discuss our primary findings.

We consider a medium described by a phase field \(\varphi \) defined on a fixed open region \({\mathcal {B}}\) of three-dimensional point space and some open time-interval. Our aim in this section is to generalize the first-gradient phase-field theory developed by Fried and Gurtin [19] to account explicitly for effects associated with the second gradient \(\hbox {grad}\,^2\varphi \) of \(\varphi \).

To formulate the principle of virtual-power, we assume that, at some arbitrarily chosen, but fixed, time, the internal microforce density \(\pi \), microstress \(\varvec{\xi }\), hypermicrostress \(\varvec{{\varSigma }}\), external microforce density \(\gamma \), surface microtraction \(\xi _{\scriptscriptstyle {\mathcal {S}}}\), and surface hypermicrotraction \({\sigma _{\scriptscriptstyle {\mathcal {S}}}}\) are known, but not necessarily independent, fields and consider \({\dot{\varphi }}\) as a virtual velocity \(\chi \) that can be specified independently of any actual process. Then, defining associated reckonings,andof the virtual internal and external power expenditures of a part \({\mathcal {P}}\) of \({\mathcal {B}}\), we emulate Gurtin [21] by imposing the requirement that \(\pi \), \(\varvec{\xi }\), \(\varvec{{\varSigma }}\), \(\gamma \), \(\xi _{\scriptscriptstyle {\mathcal {S}}}\), and \({\sigma _{\scriptscriptstyle {\mathcal {S}}}}\) be consistent with the principle of virtual-powerfor all choices of \({\mathcal {P}}\) and \(\chi \). Granted that the collection of all virtual velocities contains all actual, or realizable, velocities, since \({\mathcal {V}}_{\text {int}}({\mathcal {P}};{\dot{\varphi }})={\mathcal {W}}_{\text {int}}({\mathcal {P}})\) and \({\mathcal {V}}_{\text {ext}}({\mathcal {P}};{\dot{\varphi }})={\mathcal {W}}_{\text {ext}}({\mathcal {P}})\) the balance of power (7) is a consequence of the principle (10) of virtual-power for \(\chi ={\dot{\varphi }}\). Although it is certainly possible to build a theory on the basis of (10) without having first introduced \({\mathcal {W}}_{\text {int}}({\mathcal {P}})\) and \({\mathcal {W}}_{\text {ext}}({\mathcal {P}})\), the approach taken in Sect. 2 affords the opportunity to describe precisely the power-conjugate pairings that enter the internal and external power expenditures (2) and (6).

The seminal treatments of the principle of virtual power for elastic materials with couple stress and concomitant constitutive dependence on the second gradient of the deformation were advanced by Toupin [22, 23] and Germain [24]. For additional perspective on the principle, see Antman and Osborn [25] and Del Piero [26].

Consider an arbitrary subsurface \({\mathcal {A}}\) of the boundary \(\partial {\mathcal {B}}\) of the region \({\mathcal {B}}\) occupied by the system and let \({\varvec{n}}\) denote the outward unit normal on \(\partial {\mathcal {B}}\). Following Fried and Gurtin [27], we view \({\mathcal {A}}\) as a boundary pillbox, of infinitesimal thickness, that straddles a portion of the boundary. This allows us to isolate the physical processes within on both sides of \({\mathcal {A}}\). The geometric boundary of the surface \({\mathcal {A}}\) is simply its edge, the closed curve \(\partial {\mathcal {A}}\). Viewed, in contrast, as a boundary pillbox, \({\mathcal {A}}\) has parallel faces \(\pm {\mathcal {A}}\) with corresponding unit normals \(\pm {\varvec{n}}\) and a lateral face \(\partial {\mathcal {A}}\), as described by Fried and Gurtin [29].

Guided by the partwise microforce and hypermicroforce balances (30) and (32), we introduce environmental tractions \(\xi _{\text {env}}\) and \(\sigma _{\text {env}}\) and assume that, given any subsurface \({\mathcal {A}}\) of \(\partial {\mathcal {B}}\),represent the net microforce and hypermicroforce exerted by the environment on the portion \({\mathcal {A}}\) of the pillbox surface. To derive microforce and hypermicroforce balances for the pillbox, we first note that, by (28), the net microforce and net hypermicroforce exerted by the system on the portion \(-{\mathcal {A}}\) of the pillbox surface areReferring to (41)\(_1\) and (42)\(_1\), we infer that the microforce balance for \({\mathcal {A}}\) takes the formAnalogously, referring to (41)\(_2\) and (42)\(_2\), we see that the hypermicroforce for \({\mathcal {A}}\) takes the formSince \({\mathcal {A}}\) is arbitrary, (43) and (44) localize to yield balances valid pointwise on \(\partial {\mathcal {B}}\):Finally, invoking the representations (24) for the microstress \(\varvec{\xi }\) and hypermicrostress \(\varvec{{\varSigma }}=\varvec{{\varSigma }}^{\scriptscriptstyle \top }\), we find that the pointwise balances \(\partial {\mathcal {B}}\) can be written asand

We restrict attention to purely mechanical processes governed by the purely mechanical partwise free-energy imbalanceholds for all \({\mathcal {P}}\), where \(\psi \) denotes the free-energy density and \({\mathcal {W}}_{\text {ext}}({\mathcal {P}})\) is given by (6) with \(\xi _{\scriptscriptstyle {\mathcal {S}}}\) and \(\sigma_{\scriptscriptstyle\mathcal{S}}\) is determined according to the representations in (24). Using the power balance (7) in (48), we arrive at the pointwise free-energy imbalanceGuided by the presence of the power conjugate pairings \(\pi {\dot{\varphi }}\), \(\varvec{\xi }\cdot \hbox {grad}\,{\dot{\varphi }}\), and \(\varvec{{\varSigma }}\cdot \hbox {grad}\,^2{\dot{\varphi }}\) in (49), we consider a class of constitutive relations that determine \(\psi \), \(\pi \), \(\varvec{\xi }\), and \(\varvec{{\varSigma }}=\varvec{{\varSigma }}^{\scriptscriptstyle \top }\) at each point \({\varvec{x}}\) in \({\mathcal {B}}\) and each instant t of time as smooth functions of \(\varphi \), \(\hbox {grad}\,\varphi \), \(\hbox {grad}\,^2\varphi \), and \({\dot{\varphi }}\) at \(({\varvec{x}},t)\). Following Coleman and Noll [30], we allow the external microforce density \(\gamma \) to take any value necessary to ensure satisfaction of the microforce balance (23). It can then be shown that the dissipation inequality (49) is satisfied in all processes it is necessary and sufficient to require that:Although the response function \({\hat{\psi }}\) for the free-energy density \(\psi \) serves as a potential for the microstress \(\varvec{\xi }\), the hypermicrostress \(\varvec{{\varSigma }}\), and the equilibrium contribution to the internal microforce density \(\pi \), the dissipative contribution to \(\pi \) is specified through the nonnegative kinetic modulus B. In Sect. 3.3 we showed that if \(\varvec{{\varSigma }}\) includes a skew, reactive component \(\varvec{{\varSigma }}_r\), then \(\varvec{{\varSigma }}_r\) does not enter the local microforce balance (23) or the representations (24) for the microtraction \(\xi _{\scriptscriptstyle {\mathcal {S}}}\) and hypermicrotraction \({\sigma _{\scriptscriptstyle {\mathcal {S}}}}\). Thus, \(\varvec{{\varSigma }}_r\) is completely indeterminate and, consistent with this observation, no generality is incurred by omitting a reactive component from (51)\(_2\).

For each combination of \({\hat{\psi }}\) and B, the local microforce balance (23) and the constitutive relations (51) and (52) for \(\varvec{\xi }\), \(\varvec{{\varSigma }}\), and \(\pi \) yield an evolution equation for \(\varphi \):If, in particular, we choose the response function \({\hat{\psi }}\) and the kinetic modulus B such thatandwhere \(\triangle \) denotes the Laplacian and \(\lambda >0\), \(\ell >0\), and \(\beta >0\) are constant material parameters, then the evolution equation (53) specializes toIf, moreover, the potential f and the coefficients \(\lambda \), \(\ell \), and \(\beta \) are chosen suitably, then (56) can be made to coincide with the Swift–Hohenberg [1] equation, in which case \(\gamma \) is a source term that accounts for random fluctuations. For this reason, we refer to the progenitor (53) of (56) as the ‘generalized Swift–Hohenberg equation.’

For each choice of \({\hat{\psi }}\), the local microforce balance (23) and the constitutive relations (51) and (63) for \(\varvec{\xi }\), \(\varvec{{\varSigma }}\), and \(\pi \) determines the chemical potential \(\mu \) in the formwhere the external microforce density \(\gamma \) can embody, for instance, the potential energy, per unit volume, of an external gravitational or electromagetic field. The associated evolution equation for \(\varphi \) arises from using (66) in the consequenceof augmenting the constituent-content balance (58) with the constitutive relation (64) for the constituent flux \(\varvec{\jmath }\). Due to the dependence of the response function \({\hat{\psi }}\) on \(\hbox {grad}\,^2\varphi \), \(\mu \) as determined by (66) involves fourth-order spatial derivatives of \(\varphi \) and (67) thus includes sixth-order spatial derivatives of \(\varphi \).

Mimicking the assumptions leading from the generalized Swift–Hohenberg equation (53) to the more familiar specialization (56), we take \({\hat{\psi }}\) to be of the form (54) and assume that the mobility tensor has the formwhere \(M>0\) is a material parameter. Then, (66) reduces toand the evolution equation for \(\varphi \) that arises on using (68) and (69) takes the formwhere we have introduced an effective constituent supply rate \({\bar{s}}=s+M\triangle \gamma \). If, moreover, the potential f and the coefficients \(\lambda \), \(\ell \), and M are chosen suitably, then (56) can be made to coincide with the phase-field-crystal equation originated by Elder and Grant [9], in which case \({\bar{s}}\) is a source term that accounts for random fluctuations. For this reason, we refer to the evolution equation that arises from using (66) to eliminate \(\mu \) from (67) as the ‘generalized phase-field-crystal equation.’

We next establish energy-decay relations for the generalized Swift–Hohenberg Eq. (53) and the phase-field-crystal equation arising from combining (66) and (67). In so doing, we assume that the assigned microtraction and hypermicrotraction conditions (74) and (75) on all of \(\partial {\mathcal {B}}\) and for the generalized phase-field-crystal equation arising from combining (66) and (67) subject to the assigned microtraction and hypermicrotraction conditions (74) and (75) on all of \(\partial {\mathcal {B}}\). Additionally, we assume either that the assigned constituent flow condition (76) or that the assigned chemical potential condition (77) holds on all of \(\partial {\mathcal {B}}\). In the latter case, we restrict attention to situations in which the system is in contact with a reservoir with uniform chemical potential \(\mu _{\text {env}}={\text {constant}}\).

Our results rely on identities for the partial time-derivative \({\dot{\psi }}\) of the free-energy density \(\psi \) which we now obtain. To derive those identities, we first observe that, from the relations (50), (51)\(_1\), and (51)\(_2\) for the free-energy density \(\psi \), microstress \(\varvec{\xi }\), and hypermicrostress \(\varvec{{\varSigma }}\) in the constitutive theories in the generalized Swift–Hohenberg equation (53) and the generalized phase-field-crystal equations (66)–(67),Using the local microforce balance (23) in (86), we next see thatThe relation (87) holds regardless of whether constituent transport is present. Referring, in particular, to (52), we see that in the absence of constituent transport (87) specializes towhere, for brevity, we have suppressed the argument \((\varphi ,\hbox {grad}\,\varphi ,\hbox {grad}\,^2\varphi ,{\dot{\varphi }})\) of the kinetic modulus B. Alternatively, referring to (63), we see that in the presence of constituent transport (87) specializes toFurthermore, using the pointwise constituent-content balance (58) and the constitutive relation (64) for the constituent flux \(\varvec{\jmath }\), we see that (90) is equivalent towhere, for brevity, we have suppressed the argument \((\varphi ,\hbox {grad}\,\varphi ,\hbox {grad}\,^2\varphi ,\mu ,\hbox {grad}\,\mu )\) of the mobility tensor \({\varvec{M}}\).

In this work, we used the principle of virtual power and a free-energy imbalance to develop a second-gradient phase-field theory. In addition to the microtraction, microstress, and internal and external microforce densities that enter in the first-gradient phase-field theory by Fried and Gurtin [19], this theory involves a hypermicrostress and a corresponding hypermicrotraction. In our theory, the hypermicrostress and hypermicrotraction enter to account for power expenditures associated with temporal variations of the second spatial gradient of the phase field. The microforce balance (23) that arises from the principle of virtual power (10) in conjunction with the thermodynamically compatible constitutive relations (51) and (52) yields a family of evolution equations that encompasses and broadly generalizes the classical Swift–Hohenberg equation. When the phase field represents the volume fraction of a conserved constituent, a suitably augmented version (62) of the free-energy imbalance yields constitutive relations (51) and (63) which, in conjunction with the microforce balance (23) determines the chemical potential as a functional of the phase field. Together with the constituent-content balance (58) and the constitutive relation (64) for the constituent flux yields a family of evolution equations that encompass and broadly generalizes the phase-field-crystal equation. The first-gradient phase-field theory of Fried and Gurtin [19] is recovered if higher-order terms, represented by the hypermicrotraction and hypermicrostress, arising from the second-gradient of the phase field are neglected. We also derived thermodynamically consistent boundary conditions (74)–(77) for our two families of evolution equations and associated energy-decay relations. For our generalization of the Swift–Hohenberg equation, the microtraction and hypermicrotraction are prescribed over the entire boundary of the region upon which the phase field is defined. For our generalization of the phase-field-crystal equation, the normal component of the constituent flux and the chemical potential are also prescribed on complementary portions of the boundary. In this case, we present two alternative energy decay relations, one corresponding to situations where only the normal component of the constituent flux is prescribed and the other corresponding to situations where the system is in contact with a reservoir of uniform chemical potential.

The framework presented here can be extended to incorporate other physical effects, such as deformation or thermal transport, and thus provides a platform for developing theories in which the phase field evolves in tandem with other independent kinematic or thermodynamic fields. Another possible enhancement, closely aligned with Gurtin’s [20] derivation of the viscous Allen–Cahn equation, would involve accounting for viscous contributions to the microstress and hypermicrostress, yielding further generalizations of the Swift–Hohenberg and phase-field-crystal equations. For brevity, we leave such developments for future consideration.