Complex order-parameter phase-field models derived from structural phase-field-crystal models
Nana Ofori-Opoku, Jonathan Stolle, Zhi-Feng Huang, Nikolas Provatas
CComplex order-parameter phase-field models derived from structuralphase-field-crystal models
Nana Ofori-Opoku,
1, 2
Jonathan Stolle, Zhi-Feng Huang, and Nikolas Provatas Department of Materials Science and Engineering and Brockhouse Institute for Materials Research,McMaster University, Hamilton, Canada L8S-4L7 Department of Physics and Centre for the Physics of Materials,Rutherford Building, McGill University, Montreal, Canada, H3A-2T8 Department of Physics and Astronomy and Brockhouse Institute for Materials Research,McMaster University, Hamilton, Canada L8S-4M1 Department of Physics and Astronomy, Wayne State University, Detroit, USA, 48201
The phase-field-crystal (PFC) modeling paradigm is rapidly emerging as the model of choice wheninvestigating materials phenomena with atomistic scale effects over diffusive time scales. Recentvariants of the PFC model, so-called structural PFC (XPFC) models introduced by Greenwood et al. , have further increased the capability of the method by allowing for easy access to variousstructural transformations in pure materials [Phys. Rev. Lett. , 045702 (2010)] and binaryalloys [Phys. Rev. B. , 064104, (2011)]. We present an amplitude expansion of these XPFCmodels, leading to a mesoscale complex order-parameter (amplitude), i.e., phase-field representation,model for two dimensional square-triangular structures. Amplitude models retain the salient atomicscale features of the underlying PFC models, while resolving microstructures on mesoscales as intraditional phase-field models. The applicability and capability of this complex amplitude model isdemonstrated with simulations of peritectic solidification and grain growth exhibiting the emergenceof secondary phase structures. I. INTRODUCTION
Understanding complex phenomena during mi-crostructural and phase evolution in materials andcondensed matter systems, particularly those associatedwith system elasticity and plasticity, is at the heart ofmaterials science research. In situ investigation of thesephenomena is difficult by experimental means and ourtheoretical understanding of some of the underlyingmechanisms at work is often incomplete, mainly dueto the non-equilibrium nature and multiple scales onwhich these physical mechanisms operate. The designof engineering materials can thus benefit from tractable,yet fundamental, models that capture the full spectrumof microstructural phenomena.To date, the most successful microstructural model-ing approach has come from the use of phenomenologiesthat have their origins in Ginzburg-Landau and Cahn-Hilliard theories. These models intrinsically operate onthe length and time scales relevant to most microstruc-tural processes, i.e., mesoscopic, where information fromshorter time and length scales is introduced through ef-fective parameters. The most popular approache is thephase-field (PF) method. This method, notably, has seengreat success in the area of solidification [1–7].Over the last decade, another class of phase-fieldmodels has emerged, i.e., the phase-field-crystal (PFC)model [8, 9]. Unlike its traditional counterpart, the PFCmethod is an atomic-scale modeling formalism, operat-ing on atomistic length scales and diffusive time scales.The free energy of PFC models is minimized by peri-odic fields. As such, the method self-consistently incor-porates elasticity, multiple crystal orientations and topo-logical defects. It is rapidly becoming the methodology of choice when investigating atomistic scale effects overdiffusive time scales. It has been formally shown, by El-der and coworkers [9] and Jin and Khachaturyan [10],that PFC and PFC-type models, respectively, can be de-rived from classical density functional theory (CDFT).The PFC method has been successfully applied in the de-scription of solidification [11], spinodal decomposition [9],elasto-plasticity [12], thin film growth and island forma-tion [13], crystal nucleation and polymorphism [14, 15],amorphous or glassy states [16, 17], among many others.Most recently, an improved variant of the PFC modelhas emerged that allows one to control complex crystalstructures and their equilibrium coexistence with bulkliquid, i.e., the so-called structural PFC (
XPFC ) mod-els. Greenwood et al. [18, 19] accomplished this by intro-ducing a class of multi-peaked, two-point direct correla-tion functions in the free energy functional that containedsome of the salient features of CDFT, yet were simplifiedto be numerically efficient. This XPFC formalism waslater extended to binary [20] and N -component [21] alloy-ing systems, and applied to phenomena such as dendriticand eutectic solidification [21], elastic anisotropy [20],solute drag [22], quasi-crystal formation [23], solute clus-tering and precipitation mechanisms in Al-Cu [24] andAl-Cu-Mg [21, 25] alloys, and 3D stacking fault struc-tures in fcc crystals [26].Coarse-graining approaches have recently shown thatPFC-type models can be used to derive the form of tra-ditional PF models, expressed, however, in the form ofcomplex order-parameters, which makes it possible tosimulate different crystal orientations and defect struc-tures on mesoscopic length and time scales. These am-plitude models, remarkably, retain many salient atomisticlevel phenomena, making them prime candidates for mul- a r X i v : . [ c ond - m a t . m t r l - s c i ] J u l tiple scale modeling of microstructure phenomena. Re-cent amplitude descriptions have been used to describeanisotropic surface energy of crystal-melt interfaces inpure materials and alloys [27–29], solidification of multi-ple crystallites using an adaptive mesh [30], island andquantum dot formation [13], segregation and alloy so-lidification [31, 32], grain boundary premelting [33, 34],and lattice pinning effect on solid-liquid interfaces [35].However, these for the most part, have involved purematerials or binary alloys where both elements had thesame crystal structure, with calculations based on a sin-gle mode approximation of the system free energy func-tional, i.e., the correlation function was approximated bysingle peak function.The purpose of this work is to apply a new coarse-graining approach to the recent XPFC formalism. Recentstudies involving coarse-grained PFC models suggest thatan amplitude model capable of describing multiple crys-tal structures and elasto-plastic effects will be valuable inelucidating atomistic scale interactions, material proper-ties and dynamic processes at the mesoscale, as well asmotivating more consistent derivations of mesoscale con-tinuum models, such as PF models. Here, we present theamplitude expansion of the XPFC model of the singlecomponent system used in [18, 19], for two-dimensional(2D) square-triangular structures. At the core of our ap-proach is a Fourier method applied to the excess partof the free energy functional, coupled to the volume-averaging technique described in Refs. [28, 29, 36]. Afterderivation of the corresponding coarse-grained free en-ergy functional, we perform dynamic simulations illus-trating solidification and subsequent coarsening, peritec-tic growth and solid-solid interactions between differentcrystal structures.The remainder of this paper is organized as follows.We begin by reviewing the free energy functional of theXPFC model in Sec. II. Section III goes through the var-ious steps of generating a complex amplitude free energyfunctional, including the construction of an appropriatedensity expansion, then a brief remark on the volume-averaging technique and finally the coarse-graining of theXPFC free energy functional. The dynamics of the setof amplitude equations are discussed in Sec. V, followedby numerical illustrations of the model in Sec. VI. II. FREE ENERGY FUNCTIONAL FROM CDFT
This section reviews the free energy functional used inGreenwood et al. [18, 19]. Particularly, we highlight theexcess term in the free energy and examine its correlationkernel, which plays a central role in obtaining differentcrystal structures in the XPFC model. This is followedby a discussion of the equilibrium properties of the model.
A. XPFC free energy functional of a singlecomponent system
The free energy functional for the XPFC model is de-rived from the classical density functional theory of Ra-makrishnan and Yussouff [37], containing two contribu-tions. The first is an ideal energy which drives the systemto constant homogenous fields, e.g. liquid. The secondcontribution is an excess term in particle interactions,truncated at the two-particle interaction, which drivesthe system to be minimized by periodic fields, i.e., solid.In dimensionless form, the resulting XPFC free energyfunctional can be written as [18, 19] Fk B T ρ o = (cid:90) d r (cid:26) F id k B T ρ o + F ex k B T ρ o (cid:27) . (1)where, F id k B T ρ o = n − η n χ n F ex k B T ρ o = − n (cid:90) d r (cid:48) C ( | r − r (cid:48) | ) n ( r (cid:48) ) . (2)Here, n is the dimensionless number density, k B is theBoltzmann constant, T is the temperature and ρ o is areference liquid density of the system. η and χ are con-stants, formally equal to unity, however as discussed in[19], deviations from unity allow for better tuning tothe full ideal energy, can aid in the mapping to ther-modynamic parameters, and can physically be motivatedfrom the contributions of the lowest-order component ofhigher-order particle correlation functions [38]. Finally, C ( | r − r (cid:48) | ) is the direct two-point correlation function atthe reference density ρ o . The construction of this latterexpression is what differentiates the XPFC from otherPFC variants. We briefly review this next. B. Correlation function, C ( | r − r (cid:48) | ) The correlation kernel for the XPFC model is con-structed in Fourier space, since real space convolutionsare simply multiplicative in Fourier space. This alsomakes the XPFC formalism better equipped for simu-lations using spectral methods. The correlation function C ( | r − r (cid:48) | ) defined at the reference density ρ o , is denotedas ˆ C ( | k | ) in Fourier space. A reciprocal space peak ofˆ C ( | k | ) [19], for a given mode, j , i.e., a peak correspond-ing to a family of planes of a desired crystal structure, isdenoted by ˆ C j = e − σ k jρjβj e − ( k − kj )22 α j . (3)The first exponential in Eq. (3) sets the temperature scalevia a Debye-Waller prefactor that employs an effectivetemperature parameter, σ . ρ j and β j represent the pla-nar and atomic densities, respectively, associated withthe family of planes corresponding to mode j . These pa-rameters are formally properties of the crystal structure,but can be exploited as constants for convenience and fit-ting purposes [20, 21]. The second exponential sets thespectral peak position at k j , where k j is the inverse of theinterplanar spacing for the j th family of planes in the unitcell of the crystal structure. Unlike spectral Bragg peaksresulting from diffraction experiments for single crystals,here, each peak is represented by a Gaussian function,with α j being the width of peak j . The { α j } have beenshown in Ref. [19] to set the elastic and surface energiesand their anisotropic properties. Including only thosepeaks of the most dominant family of planes, the totalcorrelation function for the crystal structure of interest,ˆ C , is then defined by the numerical envelope of all peaksˆ C j .Finally, a comment about the k = 0 mode of the cor-relation function. This mode is the infinite wavelengthmode and sets the bulk compressibilities of the system.For simplicity, in Refs. [18, 19], the value of the k = 0was set to zero. A nonzero amplitude at k = 0, however,merely shifts the local free energy at densities away fromthe reference density ( ρ o ), thereby causing a compressionof the phase diagram about the reference density [19].This, however, does not alter the stability of the equi-librium crystal structure, since the correlation kernel isconstructed about the reference density. It is noteworthythat in addition to setting some bulk properties of thesystem, the k = 0 will also have an effect on surfacesseparating bulk phases, e.g. surface energy. Therefore,in the following, our coarse-graining procedure is donein a general manner that considers a nonzero k = 0,admitting another degree of freedom in mapping to ther-modynamic properties of the XPFC model. C. Equilibrium properties
The free energy of Eq. (1) can be shown to yieldcoexistence of varying crystal structures in equilibriumwith liquid [18, 19]. In 2D, square-liquid and triangle-liquid phases have been studied. In three-dimensions(3D), face-centered cubic (fcc) and liquid, Hexagonal-close packed (hcp) and liquid and body-centered cubic(bcc) and liquid have been demonstrated with singleand two-peaked kernels. Furthermore, the free energyof Eq. (1) can also yield peritectic systems in both 2Dand 3D, where multiple solid phases can coexist with liq-uid. These peritectic systems are comprised of square-triangle-liquid and fcc-bcc-liquid in 2D and 3D, respec-tively.Figure 1, shows a sample phase diagram resulting fromminimization of the free energy in Eq. (1), here for 2Dstructures. The phase diagram is a result of an inputcorrelation kernel corresponding to a square crystal struc-ture. To stabilize a square crystal structure, the corre-lation function requires two peaks, k and k , corre-sponding to the first two primary family of planes for FIG. 1. Phase diagram resulting from the minimization of thefree energy of Eq. (1) for 2D structures expanded in a two-mode approximation. Parameters: k = 2 π and k = 2 π √ ρ = 1, β = 4, α = 1 and ρ = 1 / √ β = 4, α = 1. The emergent square and triangle structures havedimensionless lattice spacings of a sq = 1 and a tri = 2 / √ a square crystal structure. We choose, k = 2 π and k = 2 π √ ρ = 1, β = 1, α = 1 and ρ = 4, β = 4, α = 1. To stabilize the triangu-lar structure, a single primary peak is sufficient, sinceadditional peaks have a negligible effect on the total en-ergy [19]. We re-scale the position of that peak to becommensurate with the k peak of the square. In do-ing so, the two-peaked square correlation function cansimultaneously permit square and triangular structures,where the structure with the minimum energy can beparameterized by the average density, n o , and tempera-ture parameter, σ . After re-scaling, the emergent crys-tal structures will have dimensionless lattice spacings of a sq = 1 and a tri = 2 / √
3, respectively. To constructa phase diagram, a density mode approximation is in-troduced for each of the crystal structures of interest,inserted into the free energy and after following standardminimization techniques (see Appendix of Ref. [20]), thephase diagram shown in Fig. 1 is attained.Next we shall use the 2D system just discussed to con-struct a complex order-parameter model via a coarse-graining technique.
III. COMPLEX ORDER-PARAMETER MODEL:2D SQUARE-TRIANGLE STRUCTURES
Recently, numerous works have been published thatperform amplitude expansions, particularly of PFC-typemodels. The main approaches that have been usedare: the multiple scale analysis [31, 35, 38, 39], volume-averaging method [28, 29, 36] and the renomarlizationgroup (RG) approach [30, 40–42], with the multiple scalemethod being the most widely applied across disciplines.Older works where these expansions have been performeddirectly on CDFT models, like the work of Haymet andOxtoby [43, 44] and Lakshmi et al. [45] fall under thevolume-averaging method. Others still, e.g. Kubstrup etal. [46], fall under the multiple scale analysis. The cen-tral theme in all these techniques is that the density canbe separated into so-called “fast” length scales, wherethe density oscillates rapidly, and “slow” length scales,where the amplitudes of the oscillations vary slowly withrespect to the rapidly varying oscillation of the density.Beyond this, each method has its own additional under-lying assumptions and approximations.A noteworthy consideration is the validity or accuracyof the various methods in arriving at the same self con-sistent system of equations. Namely, the multiple scaleanalysis and RG methods operate on the PFC equationsof motion, after which the coarse-grained free energyfunctional is derived. The volume-averaging method canoperate on both the PFC free energy functional and thedynamical equations, however it has been implementedfor the most part at the energy functional level of thePFC or CDFT free energy functionals. A point of criti-cism against the volume-averaging method, has been thelack of a covariant gradient operator [47] in the amplitudeequations. However, this problem can be circumventedby expanding to higher order in the amplitude expan-sion. In previous implementations, only a second orderexpansion was taken of the “slow” variables (i.e., the am-plitudes) [28, 29]. While to second order, surface energycalculations can be performed quite quantitatively, dy-namic simulations become fixed to certain orientations.It has been shown [48], that an expansion of amplitudesappearing in the excess term to at least fourth order isnecessary in order to recover the lowest order covariantgradient operator in the volume-averaging approach.This work will use the volume-averaging technique toperform calculations, in conjunction with a novel methodto handle the excess term in the PFC free energy func-tional. To begin, we first discuss the separation of scalesvia an expansion of the density that describes two crys-tal lattices. After discussing the density expansion, webriefly outline the basic features of the volume-averagingmethod. As the method has been published elsewhere,the outline given will highlight the important concepts ofthe method, after which it is applied to the ideal portionof the free energy. Finally we introduce the method ofhandling the excess term, which completes the amplitudederivation for the 2D XPFC model.
A. Density expansion in two lattices
The PFC suite of models for a pure material con-tain only a single dimensionless density field, n . A self-consistent method of putting forth a density expansionwhich incorporates multiple crystal structures is nontriv- ial. For the XPFC, in 2D these crystal structures arecrystals of triangular and square symmetry. Kubstrup et al. [46] in a study of pinning effects between frontsof hexagonal (i.e., triangular) and square phases, haveproposed a construction through which variable phasescan be described by a single expansion definition. Thisdensity expansion, for the XPFC model, can be writtenas, n ( r ) = n o ( r ) + (cid:88) j A j ( r ) e i k j · r + (cid:88) m B m ( r ) e i q m · r + c.c., (4)where n o ( r ) is the dimensionless average density and isa “slow” variable, “ c.c. ” denotes the complex conjugate, { A j } represent the amplitudes describing the first modeof our structures, while all { B m } represent the ampli-tudes for the second mode. Like the dimensionless aver-age density, the amplitudes are also “slow” variables. FIG. 2. (color online) Schematic representation of, mis-oriented by 30 degrees, the vector sets { k , k , k } and { k , k , k } , respectively, which form a resonant set andcompromise two triangular lattices. Vectors k i and k i +3 ( i = 1 , ,
3) are orthogonal to each other, forming the firstmode of the square correlation kernel. The other set of vec-tors, { q m } , correspond to different orientations of the sec-ond mode of the correlation kernel necessary to stabilize thesquare structure in the XPFC formalism, and are formed froma linear combination of the orthogonal pairs from the two tri-angular sets. Note that only the first mode { k j } was consideredby Kubstrup et al. [46] in the study of pattern forma-tion, while in Eq. (4) we have included both the zerothmode n o (as a result of density conservation) and alsothe second mode { q m } , which is needed for stabilizingthe square structure in the XPFC formalism. The den-sity expansion we construct can be schematically inferredfrom Fig. 2 in terms of the required set of reciprocal lat-tice vectors. Figure 2 represents the reciprocal latticevectors that enter the density expansion in Eq. (4), hav-ing two interlaced triangular structures mis-oriented by30 ◦ , i.e., vectors k , k , k and k , k , k each forming atriangular lattice, respectively. It will be useful in whatfollows that the property of resonance is satisfied by thesetwo vector sets. Resonance between density waves is sat-isfied when k + k + k = 0 and k + k + k = 0. Thesquare structure can be partly constructed from combina-tions of the reciprocal lattice vectors of the two triangularsets. For example, k and k (which are orthogonal, i.e., k · k = 0) represent the first mode of a square lattice,while the second mode of the square can be constructedfrom a linear combination, such as q = k + k and q = k − k . Analogous associations can be made forthe second and third set of square lattices which arisefrom the two interlaced triangular lattices. In total, thedensity expansion for a system described by the vectorsof Fig. 2 amount to 12 vectors and therefore 12 complexamplitudes. FIG. 3. (color online) Second schematic representation ofthe reciprocal set of basis vectors which comprise a densitysimultaneously describing crystals with square and triangularsymmetry respectively. Vectors k , k , k form a resonantset and compromise a single triangular lattice. Vectors k and k are orthogonal forming the first mode of the squarecorrelation kernel. The other set of vectors, { q m } dashed-dotted, comprise the second mode of the square correlationkernel. The expansion described by Eq. (4) and the vectors ofFig. 2, each corresponding to one of the 12 amplitudesmay prove to be intractable or at the least tedious andcumbersome to deal with. A simpler more intuitive ex-pansion, is also proposed here as a comparison. This is illustrated by the reciprocal lattice vectors of Fig. 3.Unlike the previous expansion, this expansion requires 6vectors and hence 6 amplitudes. At first glance, thereseems to be a limited number of degrees of freedom af-forded to us by an expansion of this kind, in particu-lar the pre-set orientation of the square structure thatis constrained to the { k , k } direction. This and othernuances that may exist between the two expansions maybe ascertained through numerical simulations. For con-venience, we will be using this latter expansion in ourderivation to follow. In Appendix B, we also report thecomplex amplitude model derived from the 12 amplitudeexpansion described by Eq. (4). The simpler density ex-pansion based on the vectors of Fig. 3 is written as n ( r ) = n o ( r ) + (cid:88) j A j ( r ) e i k j · r + (cid:88) m B m ( r ) e i q m · r + c.c. (5) B. Volume-averaging technique for coarse-graining
As mentioned in the previous section, the amplitudes { A j } and { B m } along with the dimensionless averagedensity n o , are all slowly varying on atomic scales. Afterinserting the density expansion of Eq. (5) into the XPFCfree energy terms of Eq. (2), to lowest order the termsthat will survive in the coarse-graining procedure arethose where the oscillating exponential phase factors van-ish. In particular, under coarse-graining, the free energyeffectively becomes a series of terms with “slow” vari-ables multiplying phase factors of the form e i ∆ Q l · r , where∆ Q l are sums or differences in the reciprocal lattice vec-tors. As in all coarse-graining approaches, the lowestorder approximation, i.e., so-called “quick and dirty” ap-proach [40], amounts to the situation where the only sur-viving coarse-grained terms result from all ∆ Q l ≡
0, i.e.,where a resonant condition is satisfied. This is the stan-dard condition from the symmetry requirement of trans-lational invariance of the total free energy [49].Formally, coarse-graining can be done by the volume-averaging method using a convolution operator [28],which can be defined by, (cid:104) f ( r ) (cid:105) V ≡ (cid:90) ∞−∞ d r (cid:48) f ( r (cid:48) ) ξ V ( r − r (cid:48) ) , (6)where f ( r (cid:48) ) is the function being coarse-grained, for ourpurposes collections of “slow” variables or “slow” vari-ables multiplied by phase factors and V is the coarse-graining volume, i.e., typically the volume of a unit cell.The function ξ V in the integrand of Eq. (6) is a smooth-ing function that is normalized to unity, i.e., (cid:90) ∞−∞ d r ξ V ( r − r (cid:48) ) ≡ . (7)In the long wavelength limit, L slow (cid:29) L (cid:29) a where L ∼ V /d , in d -dimensions, where L slow is the lengthscale of variation of the “slow” variables (i.e. microstruc-tural features), and a is the equilibrium lattice spacing.This condition implies that the function ξ V ( r ) varies ondimensions much larger than the lattice constant, e.g. a = 2 π/ | k j | , but much less than the length scale of varia-tion of the average density and amplitudes. Equation (6)is formally applied by changing the dependent variable inthe free energy functional from r to r (cid:48) , multiplying theresulting free energy by the left hand side of Eq. (7) (i.e.,1) and inverting the order of integration, thus arriving ata series of terms of the form of Eq. (6). Equation (6) de-fines a noninvertible limiting procedure that can be usedto average a function over some volume. The reader is referred to Refs. [21, 28, 36] for greater detail about theapplication of the volume-averaging convolution opera-tor. C. Coarse-graining the ideal term
Inserting the density expansion of Eq. (5) into the idealportion, F id , of the free energy, and coarse-graining asdescribed in the preceding section, yields to lowest orderin the average density and amplitudes, F cgid k B T ρ o V = (cid:90) d r (cid:40) n o − η n o χ n o
12 + (cid:0) − η n o + χ n o (cid:1) (cid:18) (cid:88) j | A j | + (cid:88) m | B m | (cid:19) − ( η − χn o ) [ A A A + A ∗ A ∗ B + A A ∗ B ∗ + c.c. ]+ χ (cid:88) j A j ( A ∗ j ) + (cid:88) m B m ( B ∗ m ) + 2 χ (cid:88) j (cid:88) m>j | A j | | A m | + (cid:88) j (cid:88) m | A j | | B m | + | B | | B | + χ (cid:2) A ∗ A ∗ A ∗ B ∗ + 2 A ∗ A ∗ A B ∗ + A B ∗ B ∗ + A B ∗ B + c.c. (cid:3) (cid:41) , (8)where ∗ denotes the complex conjugate and in the coarse-grained free energy the spatial variable, r , is scaled by thelattice constant a . As alluded to earlier, the correlation-containing excess term has received some attention in thecoarse-graining of PFC models. In the following section,we introduce a general Fourier method to coarse-grainthis term in the context of the present XPFC model. D. Coarse-graining the excess term
We first rewrite the correlation kernel in its Fourierseries representation, i.e., C ( | r − r (cid:48) | ) = (cid:90) d k ˆ C ( | k | ) e i k · r e − i k · r (cid:48) . (9)The convolution term, the integral over d r (cid:48) , in the freeenergy involving the excess term then becomes, G = (cid:90) d r (cid:48) C ( | r − r (cid:48) | ) n ( r (cid:48) )= (cid:90) d r (cid:48) (cid:90) d k ˆ C ( | k | ) e ik · r e − ik · r (cid:48) n ( r (cid:48) ) . (10)Next we take the Taylor series expansion of the correla-tion function around k = 0, i.e., the infinite wavelengthmode of the correlation, to all orders[50]. This expansion can be compactly written as,ˆ C ( | k | ) = ∞ (cid:88) l =0 l ! ( k ) l ∂ l ˆ C ∂ k l (cid:12)(cid:12)(cid:12)(cid:12) k =0 . (11)This functional Taylor series expansion is formally ex-act, as it goes to all orders. Substituting the densityexpansion, Eq. (5), into the convolution term of the freeenergy and employing the definition of the Fourier trans-form yields G = (cid:90) d k ∞ (cid:88) l =0 ϑ l ( k ) l ˆ n o ( k ) e ik · r (12)+ (cid:90) d k ∞ (cid:88) l =0 ϑ l ( k ) l (cid:88) j ˆ A j ( k − k j ) e ik · r + (cid:90) d k ∞ (cid:88) l =0 ϑ l ( k ) l (cid:88) m ˆ B m ( k − q m ) e ik · r + c.c., where we have made the following definition, ϑ l = 1 l ! ∂ l ˆ C ∂ k l (cid:12)(cid:12)(cid:12)(cid:12) k =0 , (13)and ˆ n o , ˆ A j and ˆ B m are the corresponding Fourier com-ponents of the average density and amplitudes respec-tively. Next we re-sum the correlation function for theaverage density part of the convolution term in Eq. (12),and make consecutive changes of variables, i.e., k (cid:48) = k − k j and then k (cid:48) = k − q m , for the second and thirdterms of Eq. (12), respectively. Following these steps, wearrive at G = (cid:104) ˆ C ( | k | )ˆ n o ( k ) (cid:105) r (14)+ (cid:88) j (cid:90) d k (cid:48) ∞ (cid:88) l =0 ϑ l ( k (cid:48) + k j ) l ˆ A j ( k (cid:48) ) e ik (cid:48) · r e ik j · r + (cid:88) m (cid:90) d k (cid:48) ∞ (cid:88) l =0 ϑ l ( k (cid:48) + q m ) l ˆ B m ( k (cid:48) ) e ik (cid:48) · r e iq m · r + c.c. Applying the definition of the Fourier transform to thesecond and third terms on the RHS of Eq. (14) yields, G = (cid:104) ˆ C ( | k | )ˆ n o ( k ) (cid:105) r + (cid:88) j e ik j · r (cid:104) ˆ C ( | k + k j | ) ˆ A j ( k ) (cid:105) r + (cid:88) m e iq m · r (cid:104) ˆ C ( | k + q m | ) ˆ B m ( k ) (cid:105) r + c.c., (15)where [ ] r denotes the inverse Fourier transform. Equa-tion (15) represents the total convolution term of theexcess free energy.To complete the coarse-graining of the excess term, wemultiply the convolution term in Eq. (15) by the expan-sion of the density field, i.e., n G , and apply the convolu- tion operator in Eq. (6), to obtain the lowest order result; F cgex k B T ρ o V = (cid:90) d r (cid:40) − n o (cid:104) ˆ ξ V ( k ) ˆ C ( | k | )ˆ n o ( k ) (cid:105) r (16) − (cid:88) j A ∗ j (cid:104) ˆ C ( | k + k j | ) ˆ A j ( k ) (cid:105) r − (cid:88) m B ∗ m (cid:104) ˆ C ( | k + q m | ) ˆ B m ( k ) (cid:105) r + c.c. (cid:41) , where ˆ ξ V is the convolution operator in Fourier space,which filters out ˆ C oscillations beyond its k = 0 peakwith some decay range in Fourier space. The explicitderivation of this term is discussed in Sec. IV.Several things are worth noting in Eq. (16). It be-comes evident that the rotational invariance nature ofa system, afforded through the covariant gradient op-erator in real space, is manifested here in the correla-tion kernel, which has as input a shifted wavenumberfor the respective modes being considered. This shiftedwavenumber samples low- k value deviations (long wave-length limit) around the peaks of the original correlationfunction. This essentially treats each reciprocal spacepeak of the original correlation kernel as a correspond-ing effective “ k = 0” mode. Like the microscopic XPFCmodel, the full correlation kernel, in this amplitude for-malism, is the numerical envelope of all reciprocal spacepeaks included to represent the crystal structural of in-terest.Combining Eq. (8) and (16), we arrive at a completecoarse-grained free energy for the structural PFC modelof the form, F cg = (cid:90) d r (cid:40) n o − η n o χ n o
12 + (cid:0) − η n o + χ n o (cid:1) (cid:18) (cid:88) j | A j | + (cid:88) m | B m | (cid:19) − ( η − χn o ) [ A A A + A ∗ A ∗ B + A A ∗ B ∗ + c.c. ]+ χ (cid:88) j A j ( A ∗ j ) + (cid:88) m B m ( B ∗ m ) + 2 χ (cid:88) j (cid:88) m>j | A j | | A m | + (cid:88) j (cid:88) m | A j | | B m | + | B | | B | + χ (cid:2) A ∗ A ∗ A ∗ B ∗ + 2 A ∗ A ∗ A B ∗ + A B ∗ B ∗ + A B ∗ B + c.c. (cid:3) − n o (cid:104) ˆ ξ V ( k ) ˆ C ( | k | )ˆ n o ( k ) (cid:105) r − (cid:88) j A ∗ j (cid:104) ˆ C ( | k + k j | ) ˆ A j ( k ) (cid:105) r − (cid:88) j A j (cid:104) ˆ C ( | k − k j | ) ˆ A j ( − k ) (cid:105) r − (cid:88) m B ∗ m (cid:104) ˆ C ( | k + q m | ) ˆ B m ( k ) (cid:105) r − (cid:88) m B m (cid:104) ˆ C ( | k − q m | ) ˆ B m ( − k ) (cid:105) r (cid:41) . (17) E. Recovering the amplitude representations ofother PFC models
Our Fourier method from the above section can alsobe used to recover the covariant gradient operators found in amplitude expansions of other PFC models. Here weconsider an expansion of the correlation around k = 0,in powers of k , similar to the standard PFC model ofElder and co-workers [8, 51] but generalized to all orders.This can be compactly written as,ˆ C ( | k | ) = ∞ (cid:88) l =0 l ! ( k ) l ∂ l ˆ C ∂ ( k ) l (cid:12)(cid:12)(cid:12)(cid:12) k =0 , (18)We note that this expansion can be generally valid pro-vided the correlation is some well-behaved function andexpressible to reasonable accuracy in even powers of k .This is true for most correlations derived from exper-iments or first principle calculations or those that canbe fit to such techniques. An appropriate example isthe eighth order fitting of Jaatinen and Ala-Nissila [52],which was found to be an accurate and efficient approxi-mation to CDFT. Applying the same arguments leadingup to Eq. (12) gives G = (cid:90) d k ∞ (cid:88) l =0 ε l ( k ) l ˆ n o ( k ) e ik · r (19)+ (cid:90) d k ∞ (cid:88) l =0 ε l ( k ) l (cid:88) j ˆ A j ( k − k j ) e ik · r + (cid:90) d k ∞ (cid:88) l =0 ε l ( k ) l (cid:88) m ˆ B m ( k − q m ) e ik · r + c.c., where we have made the following definition, ε l = 1 l ! ∂ l ˆ C ∂ ( k ) l (cid:12)(cid:12)(cid:12)(cid:12) k =0 . (20)Using the definition of the Fourier transform on the RHSof Eq. (19) leads to G = ∞ (cid:88) l =0 ε l ( −∇ ) l n o ( r ) + ∞ (cid:88) l =0 ε l ( −∇ ) l (cid:88) j A j ( r ) e ik j · r + ∞ (cid:88) l =0 ε l ( −∇ ) l (cid:88) m B m ( r ) e iq m · r + c.c. (21)Noting that, ∇ → ∇ + 2 ik j · ∇ − k j (the covariant gra-dient operator), when Laplacian operators act on termsof the form A j ( r ) e ik j · r , Eq. (21) becomes G = ∞ (cid:88) l =0 ε l ( −∇ ) l n o ( r ) (22)+ (cid:88) j e ik j · r ∞ (cid:88) l =0 ε l (cid:26) − (cid:0) ∇ + 2 ik j · ∇ − k j (cid:1) (cid:27) l A j ( r )+ (cid:88) m e iq m · r ∞ (cid:88) l =0 ε l (cid:26) − (cid:0) ∇ + 2 iq m · ∇ − q m (cid:1) (cid:27) l B m ( r )+ c.c. Equations (21) and (22) show that an infinite set of co-variant gradient operators (in the long wavelength limit)is needed to accurately capture the salient features, in real space, of a correlation kernel constructed in Fourierspace, reflecting that the latter would require an infiniteseries of square gradient terms to be represented in a tra-ditional PFC form. If we neglect all second mode contri-butions and truncate the series at l = 2 in Eq. (22), werecover the amplitude expansion of the standard PFCmodel [31], after the usual application of the coarse-graining operation. To make contact with the generalizedformalism of the previous section, the amplitude termsare rewritten in terms of an inverse Fourier transform viaa change of variable, and in Fourier space the resultingcorrelation kernel expansion is re-summed, resulting inthe same coarse-grained free energy form as Eq. (16). IV. PERIODIC INSTABILITY ARISING FROMTHE AVERAGE DENSITY
It turns out that the use of the “quick and dirty” ormultiple scale method to coarse-grain the standard PFCmodel leads to a term of the form n o (1 + ∇ ) n o [38, 39],which can become unstable to periodic oscillations in theaverage density that replicates those of the original den-sity field, n , particularly around sharp solid-liquid inter-faces. Often a second long wavelength approximation ismade to suppress the associated terms responsible forthe instability [39]. In our approach this instability isself-consistently eliminated through the convolution op-erator. We qualify this statement by showing the explicitsteps required to coarse-grain the average density termin Eq. (15).We start with the form of the correlation contribu-tion of the average density term prior to introducing thevolume-averaging kernel of Eq. (7). From Eq. (15), wehave H = − (cid:90) d r (cid:48) n o ( r (cid:48) ) (cid:104) ˆ C ( | k | )ˆ n o ( k ) (cid:105) r (cid:48) . (23)After inserting the volume-averaging kernel, we have H cg = − (cid:90) d r (cid:48) (cid:90) d r (cid:20)(cid:90) d q ˆ ξ V ( q ) e iq · r e − iq · r (cid:48) (cid:21) × n o ( r (cid:48) ) (cid:90) d k ˆ C ( | k | ) ˆ n o ( k ) e ik · r (cid:48) . (24)Here, ˆ ξ V is the averaging (or convolution) kernel inFourier space, which restricts the wavenumber q to smallvalues, i.e., q < /L , approximately the same as thefirst Brillouin zone or similarly the first peak of the cor-relation function. Note that the average density vari-able, n o ( r (cid:48) ), is slowing varying, while the multiplicationof ˆ C ( | k | )ˆ n o ( k ) can lead to rapid oscillations due to in-stabilities caused by the correlation kernel. On scales ofthe rapidly oscillating term, it is reasonable to take an ex-pansion of n o ( r (cid:48) ), i.e., n o ( r (cid:48) ) → n o ( r ), which allows us toremove it from the integral over r (cid:48) . Next, the noninvert-ible procedure, described above, occurs by switching theorder of integration d r with d r (cid:48) , after which we integratethe equation with respect to r (cid:48) yielding, H cg = − (cid:90) d r n o ( r ) (cid:90) d q ˆ ξ V ( q ) ˆ C ( | q | )ˆ n o ( q ) e iq · r = − (cid:90) d r n o ( r ) (cid:104) ˆ ξ V ( k ) ˆ C ( | k | )ˆ n o ( k ) (cid:105) r , (25)where [ ] r denotes the inverse Fourier transform and inthe last line we have changed the wavenumber variablefor convenience, i.e., q → k . Equation (25), clearlydemonstrates that all the small wavelength modes in theoriginal correlation function associated with the periodicinstability of n o are suppressed by convolving with thevolume-averaging kernel. In other words, considering thevolume-averaging kernel as a filter (in this case a low-passfilter), it smooths or eliminates all the high-mode (smallwavelength) peaks resulting from the correlation func-tion. This effectively allows the system to only samplethe long wavelength information of the correlation func-tion around k = 0.Equivalently, this can also be motivated from the mul-tiple scale method of coarse-graining. In that method,a small parameter, (cid:15) [13, 38], is introduced in a pertur-bation type expansion which results in the wavenumberbeing described by k → k + (cid:15) K , where K representsthe large wavelength modes. Considering the long wave-length behavior of the average density, this results in thecorrelation function being evaluated at (cid:15) K , i.e., ˆ C ( | (cid:15) K | ),effectively shifting the modes sampled by the correlationto only those around k = 0. It is worth noting that if onesimply applies the so-called “quick and dirty” approach,of any of the coarse-graining methods when considering adensity jump, the average density term will not be coarse-grained, resulting in a term which still possesses the smallscale feature of the original free energy functional. V. DYNAMICS
Dynamics of the complex order-parameters comprisingthe coarse-grained free energy follow the usual variationalprinciple of traditional phase-field models. Particularly,the average density, n o , obeys conserved dissipative dy-namics, while the amplitudes A j and B m follow noncon-served dissipative dynamics [53]. Specifically we have ∂n o ∂t = ∇· (cid:18) M n o ∇ δF cg δ n o (cid:19) + ∇ · ζ n o , (26) ∂A j ∂t = − M A j δF cg δ A ∗ j + ζ A j , (27)and ∂B m ∂t = − M B m δF cg δ B ∗ m + ζ B m . (28) The reader is referred to Appendix A for the full set ofexplicitly written dynamic equations. The coefficients M n o , M A j and M B m denote the mobility parametersof the average density and each corresponding ampli-tude, respectively, and strictly speaking can be func-tions of the various fields in the free energy functional.We have appended to these equations of motion, thestochastic variables ζ n o , ζ A j and ζ B m , which model coarsegrained thermal fluctuations acting on the average den-sity and amplitudes, respectively. Formally, they satisfythe fluctuation-dissipation theorem, i.e., (cid:104) ζ ν ( r , t ) (cid:105) = 0and (cid:104) ζ ν ( r , t ) ζ ν ( r (cid:48) , t (cid:48) ) (cid:105) = Γ ν δ ( r − r (cid:48) ) δ ( t − t (cid:48) ), where ν de-notes the average density or one of the amplitude fields,with Γ ν ∝ M ν k B T . Huang et al. [38] have formallyshown how these coarse-grained stochastic variables arederived, in an amplitude equation formalism from dy-namic density functional theory through multiple scaleanalysis. Next we showcase the dynamics properties ofthe derived XPFC amplitude model through three typesof microstructure simulations. VI. APPLICATIONS
It is well known that most engineering materials con-tain multiple phases and components. While the latter isnot explored in this work, we can explore a system pos-sessing multiple phases with the amplitude formalism de-veloped above. In this section, we demonstrate that thecomplex amplitude model is capable of describing twodifferent crystal symmetries by exploring solidification,coarsening and peritectic growth. We then look at theemergence of a second phase, during grain growth, fromthe boundaries of a single phase polycrystalline system.In the sections to follow, simulations were performedusing Eqs. (A1)-(A7). A semi-implicit Fourier techniquewas used to solve the system of equations. Unless statedotherwise, numerical grid spacing of ∆ x = 0 . t = 1 have been used. Furthermore, all thermalfluctuations have been neglected, unless indicated other-wise. Following the original XPFC derivation, here wetake ˆ C ( | k = 0 | ) = 0. For simplicity, we also take all mo-bility coefficients to be equal to unity, i.e., M ν = 1, where ν is one of the corresponding fields ( n o , { A j } , or { B m } )in the free energy functional of Eq. (17). Finally, all sim-ulations were conducted in the phase space determinedby the equilibrium phase diagram in Fig. 1. A. Solidification and coarsening
As a first illustration of our amplitude model, we sim-ulate the solidification of a polycrystalline network ofgrains having triangular symmetry. Our simulation do-main was set to 4096 × ×
512 lattice spacings. Ini-tially, the system was seeded with ∼
100 triangular crys-tallites randomly distributed in a uniform liquid. Each0
FIG. 4. (color online) Solidification and coarsening images from a simulation run. The evolution of the system progresses intime from left to right, i.e., t = 1 , , t = 5 , , t = 30 ,
000 and t = 100 , A , where red regions denote large magnitudes and blue low magnitudes. crystallite had a radius of 30 grid spacings ( ∼ n o = 0 . σ = 0 .
16 in equilibrium would give a finalsolid fraction of approximately 0 .
87 according to the leverrule. The amplitudes of the initial triangular nuclei werechosen to satisfy A θj = A j e iδ k j ( θ ) · r ( j = 1 , , A j is the corresponding amplitudes of the original referencebasis, δ k j ( θ ) = K j ( θ ) − k j , with θ being the randomlychosen orientation between the interval [ − π/ , π/
6] and K j ( θ ) the rotated triangular reciprocal lattice vectors.In Fig. 4, we show some snapshots of the solidificationand coarsening process. In descending order of rows fromtop to bottom, Fig. 4 displays the average density field,the reconstructed atomic density (from a portion of thesimulation domain) and the magnitude of A , respec-tively, with simulation times t = 1 , , t = 5 , , t =30 ,
000 and t = 100 , t = 5 , t = 30 ,
000 and t = 100 , B. Peritectic growth
Our second demonstration of the above amplitudemodel exploits the multi-phase nature of the XPFC mod-eling formalism. Here we illustrate peritectic growth,where the two solid structures have different crystallinesymmetries. The simulation cell was a rectangular do-main of size 768 × ∼ ×
128 latticespacings), where we initialized the system with alternat-ing square and triangular structures having length 200,and width 100 grid spacings respectively. The averagedensity was set to n o = 0 .
07, at the approximate peritec-tic temperature, σ = 0 . (a) (b)(c) (d)FIG. 5. (color online) Simulation snapshots, at t = 10 , A , which is nonzero in both solid structures; areas oflarger magnitudes are depicted in red and zero magnitudes areblue. (d) Magnitude of amplitude B , which is only nonzeroin the square phase. Color scheme is the same as in (c). area marked on the average density (top right), the mag-nitude of A (non-zero for both structures, bottom left),and the magnitude of B (non-zero for the square phase,bottom right). C. Grain growth and emergence of second phasestructures
To further illustrate the robust capability of the am-plitude model derived in this work, here we examine theemergence of a secondary phase (square), from the grainboundaries and triple junctions of a polycrystalline net-work of grains having triangular symmetry. The initialcondition was taken from the solidification simulation ofour triangular system, in Sec. VI A, at t = 5 , σ = 0 .
1. The system was left for athousand time steps to allow complete coalescence and merger of the grains having triangular symmetry. Af-ter merger, a nonzero noise amplitude of 0 .
001 for allstochastic variables was introduced for all dynamic equa-tions, thus activating thermal fluctuations in the systemfor one thousand time steps. Once nucleation of thesquare phase was apparent, the noise amplitude was setback to zero.Figure 6 shows several snapshots during the systemevolution, exhibiting the emergence of the secondarysquare phase from the boundaries of the triangular poly-crystalline network. From top to bottom, the plots dis-play the average density, n o , the reconstructed density, n , the magnitude of amplitude A and the magnitude of B (which is only nonzero for the square phase). Timeincreases from left to right in Fig. 6. Clearly evidentin the progression of the images in Fig. 6 is onset andsubsequent growth of the secondary phase. This illus-trates the further capability of our amplitude model indescribing the self-consistent nucleation and growth ofphases, a phenomena that cannot be captured currentlywith phase-field and other mean field type formalisms. VII. SUMMARY
In this paper, we reported on a new Fourier tech-nique for deriving complex amplitude models for PFCand PFC-type free energy functionals. Details of themethod were discussed in the context of the structuralPFC formalism for single component systems in 2D. Ourapproach was also shown to recover forms of previousapproaches, as well as address the issue of the periodicinstability of the average density. The dynamics of themodel were demonstrated with simulations of solidifica-tion and coarsening, peritectic solidification involving dif-ferent crystal structures, and grain growth exhibiting nu-cleation and growth of a secondary phase, phenomena ofrelevance in microstructural evolution, where the lattertwo cannot be captured with currently available meanfield formalisms such as the phase-field method.Complex amplitude models were introduced as a wayto provide a link between the standard phase-field ap-proach and the phase-field-crystal approach. Having de-veloped a complex amplitude model capable of describ-ing multiple crystal structures and elasto-plastic effects,this work has demonstrated the nature of such a bridgebetween the methodologies by directly incorporating theproperties of the microscopic correlation function. Oper-ating on larger scales, the model was shown to capturethe salient atomistic scale features inherent in several im-portant phase transformations currently outside the ca-pability of the standard phase-field approach. As a noveltechnique, our method can accept as input any derivedor experimentally calculated correlation function, whichmakes it applicable to a myriad of systems. It is expectedthat such a method, when combined with novel meshalgorithms, can truly represent a multi-scale modelingparadigm for investigating microstructural processes gov-2
FIG. 6. (color online) Time evolution of grain growth exhibiting emergence of a secondary phase (square) at the boundariesand triple junctions of the primary solidified phase (triangular). System evolution progresses from left to right, i.e., t = 1 , t = 2 ,
000 and t = 9 , | A | and | B | , respectively. Red indicates areas of large magnitudes while blue represents a magnitude of zero. erned by elasticity and defects operating on diffusionaltime and length scales. ACKNOWLEDGMENTS
We thank Sami Majaneimi, Michael Greenwood andKen Elder for their insight and constructive discussions.N.P. acknowledges support from the National Science and Engineering Research Council of Canada (NSERC).Z.F.H. acknowledges support from the National ScienceFoundation under Grant No. DMR-0845264. We alsothank Compute Canada, particularly Clumeq and Shar-cnet, for computing resources.3
Appendix A: Dynamic Equations
In Sec. V, we introduced the variational principles ap-plied to the coarse-grained free energy functional, F cg ,in arriving at the set of dynamic equations. Here, weexplicitly apply the variational principles and write theresulting equations of motion.For the average density we have, ∂n o ∂t = ∇· (cid:32) M n o ∇ (cid:40) n o − η n o χ n o − (cid:104) ˆ ξ V ( k ) ˆ C ( k )ˆ n o ( k ) (cid:105) r + (2 χ n o − η ) (cid:18) (cid:88) j | A j | + (cid:88) m | B m | (cid:19) + 2 χ [ A A A + A ∗ A ∗ B + A A ∗ B ∗ + c.c. ] (cid:41)(cid:33) . (A1)Equations for the first mode of the amplitudes read, ∂A ∂t = − M A (cid:40) (cid:0) − η n o + χ n o (cid:1) A − ( η − χn o ) [ A ∗ A ∗ + A B + A ∗ B ]+ χA | A | + 2 (cid:88) j (cid:54) =1 | A j | + (cid:88) m | B m | + 2 χ A ∗ B B − (cid:104) ˆ C ( | k + k | ) ˆ A ( k ) (cid:105) r (cid:41) , (A2) ∂A ∂t = − M A (cid:40) (cid:0) − η n o + χ n o (cid:1) A − ( η − χn o ) A ∗ A ∗ + 2 χA ∗ [ A B ∗ + A ∗ B ∗ ]+ χA | A | + 2 (cid:88) j (cid:54) =2 | A j | + (cid:88) m | B m | − (cid:104) ˆ C ( | k + k | ) ˆ A ( k ) (cid:105) r (cid:41) , (A3) ∂A ∂t = − M A (cid:40) (cid:0) − η n o + χ n o (cid:1) A − ( η − χn o ) A ∗ A ∗ + 2 χA ∗ [ A B ∗ + A ∗ B ∗ ]+ χA | A | + 2 (cid:88) j (cid:54) =3 | A j | + (cid:88) m | B m | − (cid:104) ˆ C ( | k + k | ) ˆ A ( k ) (cid:105) r (cid:41) , (A4) ∂A ∂t = − M A (cid:40) (cid:0) − η n o + χ n o (cid:1) A − ( η − χn o ) [ A ∗ B + A B ∗ ]+ χA | A | + 2 (cid:88) j (cid:54) =4 | A j | + (cid:88) m | B m | + χ [2 A A B + 2 A ∗ B B ∗ + 2 A ∗ A ∗ B ∗ ] − (cid:104) ˆ C ( | k + k | ) ˆ A ( k ) (cid:105) r (cid:41) . (A5)Finally, for the second set of amplitudes, correspondingto the second set of reciprocal lattice vectors, we have ∂B ∂t = − M B (cid:40) (cid:0) − η n o + χ n o (cid:1) B − ( η − χn o ) A A + χB | B | + 2 (cid:88) j | A j | + | B | + χ (cid:2) A A ∗ A ∗ + A B + A B ∗ (cid:3) − (cid:104) ˆ C ( | k + q | ) ˆ B ( k ) (cid:105) r (cid:41) , (A6) ∂B ∂t = − M B (cid:40) (cid:0) − η n o + χ n o (cid:1) B − ( η − χn o ) A A ∗ + χB | B | + 2 (cid:88) j | A j | + | B | + χ (cid:2) A ∗ A ∗ A ∗ + ( A ∗ ) B + A B ∗ (cid:3) − (cid:104) ˆ C ( | k + q | ) ˆ B ( k ) (cid:105) r (cid:41) . (A7) Appendix B: Amplitude Equations for 12 VectorDensity Expansion
In Sec. III A, where we considered a density mode ex-pansion for our coarse-graining procedure, we arrived attwo expansions. While in the text, we opted to go withthe simpler of the expansions, it was not motivated fromany physical arguments or considerations, but rather forconvenience. In this appendix, we present the coarse-grained free energy functional associated with the den-sity mode expansion containing 12 complex amplitudes.Before proceeding, we recall the density expansion of theform n ( r ) = n o ( r ) + (cid:88) j A j ( r ) e i k j · r + (cid:88) m B m ( r ) e i q m · r + c.c. (B1)The derivation of the amplitude equation for 12 ampli-tudes is motivated and follows from the same arguments4and approximations that lead us to the coarse-grainedfree energy functional of the simpler 6 complex ampli- tude energy of Eq. (17). The coarse-grained free energyfunctional of the 12 complex amplitude expansion reads, F cg = (cid:90) d r (cid:40) n o − η n o χ n o
12 + (cid:0) − η n o + χ n o (cid:1) (cid:18) (cid:88) j | A j | + (cid:88) m | B m | (cid:19) − ( η − χn o ) (cid:104) A A A + A A A + B B B + A A ∗ B + A A ∗ B + A A B ∗ + A A ∗ B + A A B ∗ + A A B ∗ + B B B ∗ + c.c. (cid:105) + χ (cid:88) j A j ( A ∗ j ) + (cid:88) m B m ( B ∗ m ) + 2 χ (cid:88) j (cid:88) m>j | A j | | A m | + (cid:88) j (cid:88) m | A j | | B m | + (cid:88) j (cid:88) m>j | B j | | B m | + χ (cid:104) A B ∗ B ∗ + A B B ∗ + A B B ∗ + A B B ∗ + A B ∗ B ∗ + A B ∗ B ∗ + c.c. (cid:105) + 2 χ (cid:104) A A A B + A A A B + A A A B + A A A B ∗ + A A A B ∗ + A A A B ∗ + A A A ∗ B + A A A ∗ B + A A A ∗ B + A A A ∗ B + A A A ∗ B + A A A ∗ B + A A B B + A A B B + A A B B + A A ∗ B B + A A ∗ B B ∗ + A A ∗ B B ∗ + c.c. 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