On the Thermodynamics of the Swift-Hohenberg Theory
OOn the Thermodynamics of the Swift–Hohenberg Theory
LFR Espath ∗ , AF Sarmiento , L Dalcin , VM Calo King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia, andCurtin University, Bentley, Perth, Western Australia, Australia
Abstract
We present the microbalance including the microforces, the first- and second-order microstresses for theSwift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with thefree-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functionalendowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via aproper constitutive process. Finally, we present one highly-resolved three-dimensional numerical simulationto demonstrate the particular form of the resulting microstresses and their interactions in the evolution ofthe Swift–Hohenberg equation.
Keywords:
Swift–Hohenberg Theory, Thermodynamics of Continua
1. Introduction
The work of J. Swift and P. C. Hohenberg sought to explain the Rayleigh–Benard instabilities and theirpatterns (cf. Swift and Hohenberg (1977)). These well-ordered structures resemble crystalline structuresfound in material sciences. In their work, entitled “Hydrodynamic fluctuations at the convective instability”in 1977, Swift and Hohenberg proposed a new free-energy (cf. equation (21) therein) and a governingequation (cf. equation (19) therein) that motivate our study on the thermodynamics and the microbalancebetween the microforces and microstresses for this phenomenon as described by the Swift–Hohenberg theory.Phase-field theories based on first-order gradients were formalized throughout a balance of microforcesby Fried and Gurtin (1993, 1994); Fried (1996); Gurtin (1996). They sought to segregate the constitutiveequations from the balance laws. This dissociation is natural since the balance laws are related to differentphenomena whereas the constitutive equations are concerned with particular material behaviors.Another way to tackle the problem is throughout the virtual power machinery. Some authors investigatedits use in other contexts, particularly in solid mechanics, such as Fried and Gurtin (2006) (cf. equations(2-3) therein) and Dell’Isola et al. (2011), (cf. equations (55-57) therein). The virtual power machinery wasoriginally proposed by Toupin (1962, 1964) to develop a second-order gradient theory.The Swift–Hohenberg equation can be interpreted as a second-order gradient phase-field model, whichbroadens its applicability. In the context of solid mechanics, Miehe et al. (2016) used a phase-field crystal ∗ Corresponding author.
Email address: [email protected] (LFR Espath) Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology,Thuwal 23955-6900, Saudi Arabia Extreme Computing Research Center, King Abdullah University of Science and Technology, Thuwal 23955-6900, SaudiArabia National Scientific and Technical Research Council (CONICET), Sta Fe, Argentina Applied Geology, Western Australian School of Mines, Faculty of Science and Engineering, Curtin University, Perth, WA,Australia 6845 Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Kensington, WA, Australia6152
Preprint submitted to arXiv February 7, 2017 a r X i v : . [ phy s i c s . f l u - dyn ] F e b odel of ductile fracture in elasto-plastic solids under large strains, where the phase-field approximatessharp cracks. Whereas in the context of fluid mechanics, Praetorius and Voigt (2015) coupled phase-fieldcrystal with Navier–Stokes to model colloidal suspensions.In this work, we present a phase-field theory based on second-order gradients. Particularly, we presentthe thermodynamics and the derivation of the Swift–Hohenberg theory based on a microbalance between themicroforce and the first- and second-order microstresses. Moreover, we build a constitutive process for thegeneralization of the Swift–Hohenberg theory. Finally, to study the interplay between the first- and second-order microstresses, we use highly-resolved simulation and focus on the relevant details of the microstressinteractions.
2. Governing Equation
Consider the following evolutionary partial differential equation˙ φ = − κη, (1)where φ = φ ( x , t ) is the order-parameter with x and t being, respectively, the spatial and temporal coordi-nates, κ > η is the chemical potential. Notation : the differential operators grad ( · ), div ( · ), and lap ( · ) represent the gradient, divergence,and Laplacian, respectively. Also, grad ( · ) = grad grad ( · ), div ( · ) = div div ( · ), and lap ( · ) = lap lap ( · ).The symbol ⊗ denotes the tensor (dyadic) product. The number of underlines indicates the order of thetensor. The material (or total) time derivative is denoted by ()˙. Finally, ∂ ma is ∂ m ∂a m .
3. Free-Energy Functional
Consider the following class of free-energy functionalsΨ[ φ, grad φ, grad φ ] = Z Ω ψ [ φ, grad φ, grad φ ] dΩ= Z Ω "X a ς a φ a + γ grad φ · grad φ + β ( : grad φ ) dΩ , (2)where ς a , β , and γ are coefficients which control the relative contribution of each term to the energy grantingthese terms physical meaning. Ω is a body that occupies a fixed region in a three-dimensional Euclidean spaceand Γ its boundary. Here, grad φ = grad grad φ is the Hessian of the order-parameter φ and = e i ⊗ e j is the second-order identity tensor, where the set e i (with i = 1 , , and 3) forms an orthonormal Cartesianbasis.The equations that define the free energy (2) together with (1) yield the classical Swift–Hohenbergequation (cf. Swift and Hohenberg (1977)). Remark 1 (Functional derivative) . Consider the functional in the form (2), i.e., Ψ[ φ, grad φ, grad φ ] . Thus,its first variation is δ Ψ = Z Ω [ ∂ φ ψδφ + ∂ grad φ ψ · δ grad φ + ∂ grad φ ψ : δ grad φ ] dΩ . (3) Taking into account that the differential and variational operators are commutative, thus, the followingidentities hold div ( ∂ grad φ ψδφ ) = div ( ∂ grad φ ψ ) δφ + ∂ grad φ ψ · δ grad φ, (4a) div ( ∂ grad φ ψ · δ grad φ ) = div ∂ grad φ ψ · δ grad φ + ∂ grad φ ψ : δ grad φ, (4b) div ( div ( ∂ grad φ ψ ) δφ ) = div ( div ∂ grad φ ψ ) δφ + div ∂ grad φ ψ · δ grad φ, (4c)2 rom (3), we obtain δ Ψ = Z Ω (cid:2) ∂ φ ψ − div ∂ grad φ ψ + div ∂ grad φ ψ (cid:3) δφ dΩ+ Z Γ (cid:2) ( ∂ grad φ ψ − div ∂ grad φ ψ ) δφ + ∂ grad φ ψ · δ grad φ (cid:3) · n dΓ , (5) where n is the unit, outward normal to the domain Ω . Finally, the chemical potential η is η = δ Ψ δφ = ∂ φ ψ − div ∂ grad φ ψ + div ∂ grad φ ψ. (6)We use different expressions of this identity lap φ = tr grad φ = : grad φ according to what simplifiesthe derivations most. Remark 2 (Chemical potential for the free energy (2)) . The components of the functional derivative are ∂ φ ψ = P a aςφ ( a − , (7a) div ∂ grad φ ψ = div [2 γ · grad φ ] = 2 γ lap φ, (7b) div ∂ grad φ ψ = div [2 β ( : grad φ ) ] = 2 β lap φ. (7c) Then, the chemical potential reads η = X a aςφ ( a − − γ lap φ + 2 β lap φ. (8)
4. Microbalance and Free-Energy Imbalance
To start, we account for the rate of work done by external agencies on each kinematic process – i.e., external agencies × kinematic processes – to build the free-energy imbalance since there exist energies in thisphysical law governed by the Swift–Hohenberg equation. In phase-field theories, the kinematic processes arerelated to the order-parameter φ . We thus must account for rate of work stated on the temporal changes inthe order-parameter and its gradients. Finally, considering that the free-energy (2) is endowed with up tosecond-order gradients, the external chemical power expenditure ˙ w extc has the following form˙ w extc = div (cid:2) ˙ φ ( ξ − div Ξ ) + grad ˙ φ · Ξ (cid:3) + ˙ φ$, (9)being ξ and Ξ the first- and second-order microstresses – stresses-like objects – respectively, and $ a scalarbody microforce that represents the external force. The choice of an external chemical power expenditure inthe form of (9) is supported by an internal chemical power expenditure – detailed in remark 3, cf. (12) – asexpected in second-order theories. Additionally, to obtain a proper set of conjugate pairs stated on remark3, the local microbalance assumes the following new form div ( ξ − div Ξ ) + π + $ = 0 , (10)where π is scalar body microforce that represents the internal microforce. Remark 3 (Conjugate Pairs) . Consider the external chemical power expenditure (9) in its integral form Z Ω ˙ φ$ dΩ + Z Γ [ ˙ φ ( ξ − div Ξ )] · n dΓ + Z Γ [ grad ˙ φ · Ξ ] · n dΓ , (11)3 hich states that there are two traction-like objects, a zeroth-order effective microtraction ( ξ − div Ξ ) · n and first-order microtraction Ξ · n . Although ˙ φ and grad ˙ φ are not independent kinematic processes – i.e., ˙ φ and grad ˙ φ cannot be prescribed independently – the external power may be rewritten, which is rigorouslyequivalent to (11), as two new zeroth-order microtractions that are power-conjugate to two independentkinematic processes, ˙ φ and ∂ n ˙ φ (cf. Fried and Gurtin (2006)).Now, using the divergence theorem in (11) and the microbalance equation (10), we obtain the internalchemical power expenditure in its integral form, − Z Ω ˙ φπ dΩ + Z Ω grad ˙ φ · ξ dΩ + Z Ω grad ˙ φ : Ξ dΩ , (12) depicting three power-conjugate pairs: ( ˙ φ, − π ) , ( grad ˙ φ, ξ ) , and ( grad ˙ φ, Ξ ) . Since grad ˙ φ is symmetric, we assume that Ξ is as well, without loss of generality.The free-energy imbalance states that the rate at which the free-energy changes in time has an upperbound given by the external rate of work, i.e.,˙ ψ (cid:54) div (cid:2) ˙ φ ( ξ − div Ξ ) + grad ˙ φ · Ξ (cid:3) + ˙ φ$. (13)Here, we apply the Coleman–Noll procedure (cf. Coleman and Noll (1963)) considering the following listof variables and their dependencies { ψ, ξ , Ξ , π } = f ( φ, grad φ, grad φ ) . (14)The set of functions { ψ, ξ , Ξ , π } is called thermodynamic or constitutive process if the conservation laws,microbalance and energy balance in our case, are satisfied (cf. §2, Coleman and Noll (1963)). This process iscalled admissible if it obeys the local free-energy imbalance (or Clausius–Duhem inequality) and is endowedwith a positive-definite finite absolute temperature.Now, considering the explicit dependencies listed in (14) and substituting them into the free-energyimbalance (13), we obtain ∂ φ ψ ˙ φ + ∂ grad φ ψ · ( grad φ )˙+ ∂ grad φ ψ : ( grad φ )˙ (cid:54) grad ˙ φ · ξ + ˙ φ div ξ − grad ˙ φ · div Ξ − ˙ φ div Ξ + grad ˙ φ : Ξ + grad ˙ φ · div Ξ + ˙ φ$. (15)Without constraints, the constitutive relations (14) might violate the free-energy imbalance (15). Rearrang-ing equation (15) and considering that there is no advection, the spatial and temporal derivatives commute.We thus build a constitutive process for ψ , π , ξ , and Ξ which implies that( ∂ φ ψ − $ − div ξ + div Ξ | (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) {z (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) } π ) ˙ φ + ( ∂ grad φ ψ − ξ ) · grad ˙ φ + ( ∂ grad φ ψ − Ξ ) : grad ˙ φ (cid:54) . (16)At some chosen point ( x , t ) we can set a field φ such that ˙ φ , grad φ , grad ˙ φ , grad φ , and grad ˙ φ havearbitrary values. Thus, one has to find the proper constitutive relations for { ψ, ξ , Ξ , π } to guarantee theinequality direction. Thus, taking into account that (16) is linear on ˙ φ , grad ˙ φ , and grad ˙ φ , only the trivialsolutions exist. Therefore, the constitutive relations for ξ , Ξ , and π are π = − ∂ φ ψ = − P a aςφ ( a − , (17a) ξ = ∂ grad φ ψ = 2 γ grad φ, (17b) Ξ = ∂ grad φ ψ = 2 β lap φ , (17c)where the identities listed in (7) where used to express these relations explicitly. Here, we observe thatour microbalance (10) with its constitutive relations (17) recovers the steady state of the Swift–Hohenbergequation, i.e., δ Ψ δφ = 0. 4o recover the Swift–Hohenberg equation as it was conceived, we have to include ˙ φ in the list of variables.Thus, the new list is { ψ, ξ , Ξ , π } = f ( φ, grad φ, grad φ, ˙ φ ) , (18)and its free-energy imbalance( ∂ φ ψ + π ) ˙ φ + ( ∂ grad φ ψ − ξ ) · grad ˙ φ + ( ∂ grad φ ψ − Ξ ) : grad ˙ φ + ∂ ˙ φ ψ ¨ φ (cid:54) . (19)Now, the inequality (19) is no longer linear on ˙ φ , admitting nontrivial solutions in its first term. Thus, wedefine π dis = π + ∂ φ ψ . The last terms admit only trivial solutions, therefore ∂ ˙ φ ψ = 0.The thermodynamic constraints yield the consistent constitutive relations π dis ˙ φ = ( π + ∂ φ ψ ) ˙ φ (cid:54) , (20a) ξ = ∂ grad φ ψ, (20b) Ξ = ∂ grad φ ψ. (20c)From (20a), one has several choices; however, to recover the Swift–Hohenberg equation we choose π dis = − ˙ φκ ,being κ > $ = 0.
5. First and Second Laws of Thermodynamics
Here, we detail the derivation of the first and second laws of thermodynamics in the usual manner, i.e.,directly from the governing equation and the free-energy (cf. Gurtin et al. (2010)), to further validate ourmodel for the microbalance.
First Law
The product between the material derivative of the phase-field and the chemical potential ( ˙ φη ) yieldsthe balance between the external and internal chemical rate of work.Consider the free energy in the form (2). In the general case,˙ ψ = ∂ φ ψ ˙ φ + ∂ grad φ ψ · ( grad φ )˙+ ∂ grad φ ψ : ( grad φ )˙+ ∂ θ ψ ˙ θ, (21)where θ is the absolute temperature, whereas an isothermal process implies˙ ψ = ∂ φ ψ ˙ φ + ∂ grad φ ψ · ( grad φ )˙+ ∂ grad φ ψ : ( grad φ )˙ . (22) Remark 4 (Identities) . The identities used in this section are div ( ˙ φ ∂ grad φ ψ ) = grad ˙ φ · ∂ grad φ ψ + ˙ φ div ∂ grad φ ψ, (23a) div ( ˙ φ div ∂ grad φ ψ ) = grad ˙ φ · div ∂ grad φ ψ + ˙ φ div ∂ grad φ ψ, (23b) div ( grad ˙ φ · ∂ grad φ ψ ) = grad ˙ φ : ∂ grad φ ψ + grad ˙ φ · div ∂ grad φ ψ. (23c)Here, we analyze the chemical powers using ˙ φη , i.e.,˙ φη = ˙ φ ( ∂ φ ψ − div ∂ grad φ ψ + div ∂ grad φ ψ ) . (24)Rewriting equation (24) using the identities (23), we obtain˙ φη = ∂ φ ψ ˙ φ + ∂ grad φ ψ · grad ˙ φ + ∂ grad φ ψ : grad ˙ φ − div (cid:0) ˙ φ ∂ grad φ ψ − ˙ φ div ∂ grad φ ψ + grad ˙ φ · ∂ grad φ ψ (cid:1) . (25)5onsidering that there is no advection, the spatial and temporal derivatives commute, and using equa-tions (1) and (25), we obtain ∂ φ ψ ˙ φ + ∂ grad φ ψ · ( grad φ )˙+ ∂ grad φ ψ : ( grad φ )˙+ κη = div (cid:0) ˙ φ ∂ grad φ ψ − ˙ φ div ∂ grad φ ψ + grad ˙ φ · ∂ grad φ ψ (cid:1) , (26)alternatively, if no external power is spent across the boundaries˙ ψ = − κη . (27)The internal and external powers are˙ w intc = κη + ∂ φ ψ ˙ φ + ∂ grad φ ψ · ( grad φ )˙+ ∂ grad φ ψ : ( grad φ )˙ , (28a)˙ w extc = div (cid:0) ˙ φ ∂ grad φ ψ − ˙ φ div ∂ grad φ ψ + grad ˙ φ · ∂ grad φ ψ (cid:1) . (28b)Note that κη + ∂ φ ψ ˙ φ = − ˙ φπ .The first law of thermodynamics represents the energy balance in the system and states explicitly theinterplay between the kinetic energy e k , the internal energy e i , the rate at which (mechanical and chemical)power is spent, and the rate at which energy in the form of heat is transferred, i.e.,˙ e T = ˙ e k + ˙ e i = ˙ w extm + ˙ w extc − div q + q, (29)where q is the heat flux, and q may either be a heat sink or source.Finally, we obtain the first law of thermodynamics˙ e i = ∂ φ ψ ˙ φ + ∂ grad φ ψ · ( grad φ )˙+ ∂ grad φ ψ : ( grad φ )˙+ κη − div q + q, (30)In the absence of heat transfer, the first law of thermodynamics (i.e., the balance of energy) reads˙ e i = ˙ ψ + κη , (31)where we used equation (28). Second Law
The second law of thermodynamics (in the form of the Clausius–Duhem inequality or entropy imbalance)states that the entropy s should grow at least at the rate given by the entropy flux q /θ added to the entropysupply q/θ , i.e., ˙ s (cid:62) − div (cid:16) q θ (cid:17) + qθ = 1 θ (cid:18) − div q + 1 θ grad θ · q + q (cid:19) . (32)By definition, the free energy is ψ = e i − θs. (33)Taking the material time derivative, we obtain˙ ψ = ˙ e i − ˙ θs − θ ˙ s. (34)Substituting the first law (30) and equation (34) into equation (32), and using standard argumentsof Coleman and Noll (cf. Coleman and Noll (1963)) the entropy is s = − ∂ θ ψ , and the second law ofthermodynamics is obtained in the form of the entropy imbalance, i.e.,˙ s = 1 θ (cid:18) κη − θ grad θ · q (cid:19) (cid:62) , (35)In the absence of heat transfer, the entropy imbalance yields κη (cid:62) . (36)Finally, we conclude from (36) and (27) that ˙ ψ (cid:54) . A Constitutive Process for the Generalization of the Swift–Hohenberg Theory
Here, we generalize the dependencies in the list of variables (14) and define { ψ, ξ , Ξ , π } = f ( φ, grad φ, grad φ, ˙ φ, grad ˙ φ, grad ˙ φ ) . (37)The free-energy imbalance is now given by( ∂ φ ψ + π ) ˙ φ + ( ∂ grad φ ψ − ξ ) · grad ˙ φ + ( ∂ grad φ ψ − Ξ ) : grad ˙ φ + ∂ ˙ φ ψ ¨ φ + ∂ grad ˙ φ ψ · grad ¨ φ + ∂ grad ˙ φ ψ : grad ¨ φ (cid:54) . (38)While the inequality is no longer linear on ˙ φ , grad ˙ φ , and grad ˙ φ , admitting nontrivial solutions for its firstthree terms, the last three terms admit only trivial solutions, i.e., ∂ ˙ φ ψ = ∂ grad ˙ φ ψ = ∂ grad ˙ φ ψ = 0. Thus,we define π dis = π + ∂ φ ψ, (39a) ξ dis = ξ − ∂ grad φ ψ, (39b) Ξ dis = Ξ − ∂ grad φ ψ. (39c)Now, the inequality to be enforced is π dis ˙ φ − ξ dis · grad ˙ φ − Ξ dis : grad ˙ φ (cid:54) . (40)For simplicity, we assume that π dis , grad ˙ φ , and Ξ dis are defined as a linear combination of ˙ φ , grad ˙ φ ,and grad ˙ φ . Thus, π dis = − α ˙ φ − a · grad ˙ φ − A : grad ˙ φ, (41a) ξ dis = σ ˙ φ + S · grad ˙ φ + Σ : grad ˙ φ, (41b) Ξ dis = U ˙ φ + grad ˙ φ · Υ + G : grad ˙ φ. (41c)Rearranging (40) and considering (41), we obtain − α ˙ φ − S : grad ˙ φ ⊗ grad ˙ φ − G :: grad ˙ φ ⊗ grad ˙ φ − ( a + σ ) · ( ˙ φ grad ˙ φ ) − ( A + U ) : ( ˙ φ grad ˙ φ ) − ( Υ + Σ ) ... grad ˙ φ ⊗ grad ˙ φ (cid:54) . (42)We assume that we cannot assert the direction of the inequality (42) for all terms, but we can state that α , S , and G must be positive definite. Additionally, assuming that { α, a , A , σ , S , Σ , U , Υ , G } are constantcoefficients, we can look for the trivial solution for the remaining terms. Thus, assuming that a + σ = 0, A + U = 0, and Υ + Σ = 0, we obtain π = − ∂ φ ψ − α ˙ φ − a · grad ˙ φ − A : grad ˙ φ, (43a) ξ = ∂ grad φ ψ − ˙ φ a + S · grad ˙ φ + Σ : grad ˙ φ, (43b) Ξ = ∂ grad φ ψ − A ˙ φ − grad ˙ φ · Σ + G : grad ˙ φ. (43c)Finally, using the constitutive relations (43) in the new form of the microbalance (10), we obtain thegeneralized Swift–Hohenberg equation div (cid:20) ∂ grad φ ψ − ˙ φ a + S · grad ˙ φ + Σ : grad ˙ φ − div (cid:18) ∂ grad φ ψ − A ˙ φ − grad ˙ φ · Σ + G : grad ˙ φ (cid:19)(cid:21) − ∂ φ ψ − α ˙ φ − a · grad ˙ φ − A : grad ˙ φ + $ = 0 , (44)7nd considering the free-energy (2), we obtain − P a aςφ ( a − + 2 γ lap φ − β lap φ − α ˙ φ − a · grad ˙ φ + S : grad ˙ φ + 2 Σ ... grad ˙ φ − G :: grad ˙ φ + $ = 0 . (45)
7. Numerical Experiment
Here, we present a highly-resolved three-dimensional simulation to depict the interplay between the first-and second-order microstresses. We use a high-order NURBS-based finite element solver, PetIGA: Dalcinet al. (2016), which has been used extensibly in the modeling of multiphyscis processes including phase-fieldapplications, Thiele et al. (2013); Sagiyama et al. (2015); Vignal et al. (2015b,a, 2016); Sarmiento et al.(2016). We solve the resulting equation in its weak primal version, the regular version Swift–Hohenbergequation, composed by equations (1) and (6). We employ a tensor-product B-spline approximation with 64 elements of polynomial degree 4 with C continuity at element interfaces.The free-energy coefficients in equation (2) are given by ς = 0; ς = −
12 ; ς = 0; ς = 14 ; ς a> = 0; γ = − β = 12 , (46)while the constitutive modulus is κ = 1 in (1) and the initial condition is defined as φ ( x ,
0) = tanh (cid:18) x − − r e cos (cid:18) x π [ L (cid:19)(cid:19) + (cid:18) x − − r e sin (cid:18) x π [ L (cid:19)(cid:19) − r c t h , (47)where π [ = 3 . . . . , L = 40, r e = 0 . L , r c = 0 . L , and t h = 0 . L . The domain is a cube withdimensions ( L x , L x , L x = L ), with periodic boundary conditions.The term P a ς a φ a in the free-energy (2) is a double-well potential function with the coefficients listed in(46). This function is defined in [ − √ , √
2] with minima at − φ , ξ , and Ξ at t = 10. Here, one can observe the different structuresof ξ and Ξ at early stages, represented by isosurfaces. While ξ captures the effects along a lattice nearthe interface, Ξ accounts for nonlocal effects in the neighborhood of the lattice. Roughly speaking, − div Ξ provides, in an averaged sense, the interaction among neighboring lattices.Figure 2 shows the evolution of φ colored by its Laplacian from t = 1 to t = 145. Here, we identifysome symmetries in the structure of the phase-field that are preserved in time. For instance, at x = 10and x = 30, both ξ and div Ξ do not have an out-of-plane component, meaning that the wavelength in x Figure 1: (Color online) From left to right: order-parameter, first-order microstress, and second-ordermicrostress (colored by the order-parameter) at t = 10.8 a) t = 1 (b) t = 10 (c) t = 20(d) t = 30 (e) t = 50 (f) t = 145 Figure 2: (Color online) Evolution of the order-parameter φ colored by its Laplacian. t = 11 t = 15 t = 7 Figure 3: (Color online) Detail of the microstresses evolution in the plane x = 10direction is 20. There is also a π rad rotational symmetry around x . Additionally, the simulation is freeof numerical oscillations (as shown by Vignal (2016) numerical oscilations can yield nonphysical solutions)and no numerical dissipation is used.Figure 3 depicts a slice at x = 10 at early stages to show first- and second-order microstresses over theisovalue surface of φ = − .
2. As shown by Espath et al. (2016), ξ and − div Ξ have a positive inner producton their overall behavior if the coefficients γ and β have the same sign, and they are parallel if the isosurfacesof φ have a constant curvature. In the detail of this figure, we show the microstresses evolution from t = 7to t = 15; there is a deviation of − div Ξ from − ξ , where the larger the curvature changes the larger the9 a) Plane x x (b) Plane x x Figure 4: (Color online) On the left a slice x = 20. On the right a slice x = 10 at t = 50.deviation from one another.Figure 4 shows two slices of the domain. On the left, a slice at x = 20 depicts several smaller orderedstructures, whereas on the right, a slice at x = 10 depicts two large structures at t = 50.
8. Conclusions
We analyze the thermodynamics of the Swift–Hohenberg theory. Our derivation is based on a microbal-ance between the microforce and the first- and second-order microstresses. In the Swift–Hohenberg theory,we obtain an effective microstress ξ − div Ξ depending on a first- ( ξ ) and second-order ( Ξ ) microstresses.After explicitly stating the first and second laws of thermodynamics for this model, we generalize the modeland detail some simple parameter choices. Finally, a highly-resolved simulation shows the interplay betweenthe first- and second-order microstresses.
9. Acknowledgments
This publication was made possible in part by the CSIRO Professorial Chair in Computational Geoscienceof Curtin University, the National Priorities Research Program grant 7-1482-1-278 from the Qatar NationalResearch Fund (a member of the Qatar Foundation), and by the European Union’s Horizon 2020 Researchand Innovation Program of the Marie Skłodowska-Curie grant agreement No. 644202, the J. Tinsley OdenFaculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES)of the University of Texas at Austin has partially supported the visits of VMC to ICES, the Spring 2016Trimester on “Numerical methods for PDEs”, organized with the collaboration of the Centre Emile Borelat the Institut Henri Poincare in Paris supported VMC’s visit to IHP in October, 2016.
References
Coleman, B. D., Noll, W., 1963. The thermodynamics of elastic materials with heat conduction and viscosity. Archive forRational Mechanics and Analysis 13 (1), 167–178.Dalcin, L., Collier, N., Vignal, P., Cˆortes, A. M. A., Calo, V. M., 2016. PetIGA: A framework for high-performance isogeometricanalysis. Computer Methods in Applied Mechanics and Engineering 308, 151–181.Dell’Isola, F., Seppecher, P., Madeo, A., 2011. Beyond euler-cauchy continua: The structure of contact actions in n-th gradientgeneralized continua: a generalization of the cauchy tetrahedron argument. In: Variational Models and Methods in Solidand Fluid Mechanics. Springer, pp. 17–106.Espath, L. F. R., Sarmiento, A. F., Vignal, P., Varga, B. O. N., Cortes, A. M. A., Dalcin, L., Calo, V. M., 2016. Energy exchangeanalysis in droplet dynamics via the Navier–Stokes–Cahn–Hilliard model. Journal of Fluid Mechanics 797, 389–430.Fried, E., 1996. Continua described by a microstructural field. Zeitschrift f¨ur angewandte Mathematik und Physik ZAMP47 (1), 168–175. ried, E., Gurtin, M. E., 1993. Continuum theory of thermally induced phase transitions based on an order parameter. PhysicaD: Nonlinear Phenomena 68 (3), 326–343.Fried, E., Gurtin, M. E., 1994. Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D:Nonlinear Phenomena 72 (4), 287–308.Fried, E., Gurtin, M. E., 2006. Tractions, balances, and boundary conditions for nonsimple materials with application to liquidflow at small-length scales. Archive for Rational Mechanics and Analysis 182 (3), 513–554.Gurtin, M. E., 1996. Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D:Nonlinear Phenomena 92 (3), 178–192.Gurtin, M. E., Fried, E., Anand, L., 2010. The mechanics and thermodynamics of continua. Cambridge University Press.Miehe, C., Aldakheel, F., Raina, A., 2016. Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory. International Journal of Plasticity.Praetorius, S., Voigt, A., 2015. A navier-stokes phase-field crystal model for colloidal suspensions. The Journal of chemicalphysics 142 (15), 154904.Sagiyama, K., Rudraraju, S., Garikipati, K., 2015. Unconditionally stable, second-order accurate schemes for solid state phasetransformations driven by mechano-chemical spinodal decomposition. arXiv preprint arXiv:1508.00277.Sarmiento, A., Cortes, A. M. A., Garcia, D., Dalcin, L., Collier, N., Calo, V. M., 2016. PetIGA-MF: a multi-field high-performance toolbox for structure-preserving B-splines spaces. Journal of Computational Science 18, 117–131.Swift, J., Hohenberg, P. C., 1977. Hydrodynamic fluctuations at the convective instability. Physical Review A 15 (1), 319.Thiele, U., Archer, A. J., Robbins, M. J., Gomez, H., Knobloch, E., 2013. Localized states in the conserved Swift–Hohenbergequation with cubic nonlinearity. Physical Review E 87 (4), 042915.Toupin, R. A., 1962. Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis 11 (1), 385–414.Toupin, R. A., 1964. Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis 17 (2), 85–112.Vignal, P., 2016. Thermodynamically consistent algorithms for the solution of phase-field models. Ph.D. thesis, King AbdullahUniversity of Science & Technology.Vignal, P., Collier, N., Dalcin, L., Brown, D., Calo, V. M., 2016. An energy-stable time-integrator for phase-field models.Computer Methods in Applied Mechanics and Engineering.Vignal, P., Dalcin, L., Brown, D. L., Collier, N., Calo, V. M., 2015a. An energy-stable convex splitting for the phase-fieldcrystal equation. Computers & Structures 158, 355–368.Vignal, P., Sarmiento, A., Cˆortes, A. M. A., Dalcin, L., Calo, V. M., 2015b. Coupling Navier–Stokes and Cahn–Hilliardequations in a two-dimensional annular flow configuration. Procedia Computer Science 51, 934–943.ried, E., Gurtin, M. E., 1993. Continuum theory of thermally induced phase transitions based on an order parameter. PhysicaD: Nonlinear Phenomena 68 (3), 326–343.Fried, E., Gurtin, M. E., 1994. Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D:Nonlinear Phenomena 72 (4), 287–308.Fried, E., Gurtin, M. E., 2006. Tractions, balances, and boundary conditions for nonsimple materials with application to liquidflow at small-length scales. Archive for Rational Mechanics and Analysis 182 (3), 513–554.Gurtin, M. E., 1996. Generalized Ginzburg–Landau and Cahn–Hilliard equations based on a microforce balance. Physica D:Nonlinear Phenomena 92 (3), 178–192.Gurtin, M. E., Fried, E., Anand, L., 2010. The mechanics and thermodynamics of continua. Cambridge University Press.Miehe, C., Aldakheel, F., Raina, A., 2016. Phase field modeling of ductile fracture at finite strains: A variational gradient-extended plasticity-damage theory. International Journal of Plasticity.Praetorius, S., Voigt, A., 2015. A navier-stokes phase-field crystal model for colloidal suspensions. The Journal of chemicalphysics 142 (15), 154904.Sagiyama, K., Rudraraju, S., Garikipati, K., 2015. Unconditionally stable, second-order accurate schemes for solid state phasetransformations driven by mechano-chemical spinodal decomposition. arXiv preprint arXiv:1508.00277.Sarmiento, A., Cortes, A. M. A., Garcia, D., Dalcin, L., Collier, N., Calo, V. M., 2016. PetIGA-MF: a multi-field high-performance toolbox for structure-preserving B-splines spaces. Journal of Computational Science 18, 117–131.Swift, J., Hohenberg, P. C., 1977. Hydrodynamic fluctuations at the convective instability. Physical Review A 15 (1), 319.Thiele, U., Archer, A. J., Robbins, M. J., Gomez, H., Knobloch, E., 2013. Localized states in the conserved Swift–Hohenbergequation with cubic nonlinearity. Physical Review E 87 (4), 042915.Toupin, R. A., 1962. Elastic materials with couple-stresses. Archive for Rational Mechanics and Analysis 11 (1), 385–414.Toupin, R. A., 1964. Theories of elasticity with couple-stress. Archive for Rational Mechanics and Analysis 17 (2), 85–112.Vignal, P., 2016. Thermodynamically consistent algorithms for the solution of phase-field models. Ph.D. thesis, King AbdullahUniversity of Science & Technology.Vignal, P., Collier, N., Dalcin, L., Brown, D., Calo, V. M., 2016. An energy-stable time-integrator for phase-field models.Computer Methods in Applied Mechanics and Engineering.Vignal, P., Dalcin, L., Brown, D. L., Collier, N., Calo, V. M., 2015a. An energy-stable convex splitting for the phase-fieldcrystal equation. Computers & Structures 158, 355–368.Vignal, P., Sarmiento, A., Cˆortes, A. M. A., Dalcin, L., Calo, V. M., 2015b. Coupling Navier–Stokes and Cahn–Hilliardequations in a two-dimensional annular flow configuration. Procedia Computer Science 51, 934–943.