On Power Law Scaling Dynamics for Time-fractional Phase Field Models during Coarsening
OOn Power Law Scaling Dynamics for Time-fractional PhaseField Models during Coarsening
Lizhen Chen ∗ , Jia Zhao † and Hong Wang ‡ March 13, 2018
Abstract
In this paper, we study the phase field models with fractional-order in time. Thephase field models have been widely used to study coarsening dynamics of materialsystems with microstructures. It is known that phase field models are usually derivedfrom energy variation, so that they obey some energy dissipation laws intrinsically.Recently, many works have been published on investigating fractional-order phase fieldmodels, but little is known of the corresponding energy dissipation laws. In this paper,we focus on the time-fractional phase field models and report that the effective freeenergy and roughness obey a universal power law scaling dynamics during coarsening.Mainly, the effective free energy and roughness in the time-fractional phase field modelsscale by following a similar power law as the integer phase field models, where the poweris linearly proportional to the fractional order. This universal scaling law is verifiednumerically against several phase field models, including the Cahn-Hilliard equationswith different variable mobilities and molecular beam epitaxy models. This new findingsheds light on potential applications of time fractional phase field models in studyingcoarsening dynamics and crystal growths.
Phase field models have been fully exploited to study many material, physical andbiology systems [11,22,25,30,30,32–34,36]. One important application of phase field modelsis to study coarsening dynamics [8, 9, 12, 14, 35], which is a widely observed phenomenonin material systems involving micro-structures. It is characterized by the dissipation ofexcessive Helmholtz free energy and the growth of a characteristic length scale [8]. Physicallyspeaking, the coarsening dynamics could often be characterized by a power law scaling,mainly if we introduce a characteristic length scale R ( t ), it grows like a power in time,i.e. R ( t ) ≈ ct /α , where α indicates the order, i.e., R ( t ) α − R ( t ) α ≈ c ( t − t ) for some ∗ Beijing Computational Science Research Center, Beijing, China; Email: [email protected]. † Utah State University, Logan, UT, USA; email: [email protected]. ‡ University of South Carolina, Columbia, SC, USA; email: [email protected]. a r X i v : . [ m a t h . NA ] M a r ositive constant c . The coarsening dynamics of phase field models have already been widelyinvestigated both theoretically and numerically [8]. For instance the Cahn-Hilliard equationwith smooth double-well potential and phase-dependent diffusion mobility is studied in [8],where the authors show that the coarsening dynamics of the Cahn-Hilliard equation isrelative to the mobility coefficient: for CH equation with constant mobility, the free energyscales as O ( t − ); for the two-sided degenerate mobility, the free energy scales O ( t − ), whichis independent of the volume fraction of each phase [8]; for the one-sided degenerate mobility,the free energy scales depending on the volume fraction of the phases. The coarsening ofthe molecular beam epitaxy (MBE) model has also been investigated [14, 15, 29]: for theMBE model with slope selection, the free energy scales as O ( t − ) and the roughness scalesas O ( t ); for the MBE model without slope selection, the roughness growth scales as O ( t ).Fractional phase field models have been studied in the last few years and have quicklyattracted significantly increased attentions ever since. The study of fractional phase-fieldmodels is motivated by the significantly increased applications of fractional partial differ-ential equations (FPDEs) to a wide variety of areas [19, 20, 23] and the rapid developedmathematical, numerical and computational analysis of FPDEs over the past few decades.The concept of FPDEs can be explained in the context of anomalously diffusive transport.The classical second-order diffusion PDE was first presented by Fick in his study on how wa-ter and nutrients travel through cell membranes. Einstein and Pearson derived the diffusionPDE independently from first principles and random walk under the common assumptionsof (i) the existence of a mean free path and (ii) the existence of a mean time taken toperform a jump in the particle movements in the underlying processes. Under these as-sumptions, the variance of a particle excursion distance is finite. The central limit theoremconcludes that the probability of finding a particle somewhere in space satisfies a Gaussiandistribution, which gives a probabilistic description of the Fickian diffusion. In the lastfew decades more and more diffusion processes were found to be non-Fickian, ranging fromthe signaling of biological cells [21], anomalous electrodiffusion in nerve cells [13], foragingbehavior of animals [24] to viscoelastic and viscoplastic flow [23] and solute transport ingroundwater [4]. In these cases either the particle movements have many long jumps orhave experienced longer waiting times, namely, the assumptions (i) or (ii) is violated. Con-sequently, the processes can have large deviations from the stochastic process of Brownianmotion, leading to anomalously diffusive transport that exhibits heavy tails either in spaceor in time. Consequently, classical integer-order PDE models fail to provide an accurate de-scription of these problems. In contrast, it has been shown that these anomalously diffusivetransport can be better modeled by space-fractional PDE (for superdiffusive transport) ortime-fractional PDE (for subdiffusive transport) [19, 20].The same principle applies to phase-field models. If the coarsening dynamic processesexperience some long-range spatial interaction and so the assumption (i) is violated, thecorresponding phase-field models may be better described by space-fractional phase-fieldmodels [28]. As a matter of fact, the original phase-field model was expressed in terms ofan integration of Helmholtz free energy that has a nonlocal spatial interaction [5]. It wasapproximated by a Laplacian operator just for the modeling, mathematical and numerical2implicity. On the other hand, when the material during the coarsening process exhibitsmemory effect, the assumption (ii) is violated. The corresponding phase-field model may bedescribed by a time-fractional phase-field model [17]. In this paper we study time-fractionalphase-field models.In this paper, we lay out a foundation for time fractional phase field models by discover-ing a universal power laws scaling property. The scaling of effective free energy/roughnessin the time-fractional phase field models during coarsening follow a similar power law asthe integer phase field models, where the power is linearly proportional to the fractional or-der α . By far, there is the first report in literature. Several examples of fractional phasefield models with specific choices of free energies are presented to verify this new finding.A rigorous asymptotic analysis to this interesting correlation will be pursed in our futurework.In the rest of the paper, we will introduce the classical and fractional phase field modelsin Section 2. And in Section 3, we will provide the numerical approximations for thefractional phase field models. Afterward, we will present the power scaling laws for theeffective free energy and roughness of the time-fractional phase field models. In the end, wedraw a brief conclusion and point out possible future research directions. In this section, we first briefly introduce the classical (integer) phase field models andthe analogical time-fractional phase field models, subsequently. Interested readers couldrecommended to read [1, 2, 6, 23, 28] and the references therein for more details.
Given a domain Ω with smooth boundary ∂ Ω, we consider a material system with twocomponents. We introduce a phase variable φ to represent the volume fraction of onecomponent, such that 1 − φ represents the volume fraction of the other component. Forthe situations of material system with multiple components, more phase variables shouldbe introduced. We also introduce the notation F to represent the total free energy. Giventhe explicit expression of F , there are usually two types of phase field equations, namelythe Allen-Cahn type (AC) equation and the Cahn-Hilliard type (CH) equation.The Allen-Cahn equation could be considered as a gradient flow equation, i.e. the rateof change ∂ t φ is in the direction of decreasing gradient of the free energy functional F ( φ ),which reads as ∂ t φ = − λ δFδφ , in Ω T = Ω × (0 , T ] , ∇ φ · n = 0 , on ∂ Ω × (0 , T ] ,φ = φ , in Ω = Ω × { } , (2.1)where λ is the motility parameter, and δFδφ is the chemical potential ( the functional deriva-3ive of F with respect to φ ). Its energy dissipation law is dFdt = Z Ω δFδφ δφδt d x = Z Ω − λ (cid:16) δFδφ (cid:17) d x ≤ . (2.2)One other property of Allen-Cahn equation is its lacking of mass conservation, i.e. ddt Z Ω φd x = Z Ω − λ δFδφ d x = 0 . (2.3)The Cahn-Hilliard equation could be interpreted as derived by following the Fick’s firstlaw, where the flux goes from regions of high chemical potential to low chemical potential,which reads as ∂ t φ = ∇ · ( λ ∇ δFδφ ) , in Ω T = Ω × (0 , T ] , ∇ φ · n = 0 , ∇ δFδφ · n = 0 , on ∂ Ω × (0 , T ] ,φ = φ , in Ω = Ω × { } , (2.4)where λ is the motility parameter. For the Cahn-Hilliard equation, it possesses energydissipation property as follows dFdt = Z Ω δFδφ δφδt d x = − Z Ω λ |∇ δFδφ | d x + Z ∂ Ω λ δFδφ ∇ δFδφ · n dS ≤ , (2.5)and mass conservation property as ddt Z Ω φd x = Z Ω ∇ · ( λ ∇ δFδφ ) d x = Z ∂ Ω λ ∇ δFδφ · n ds = 0 . (2.6) Next, we introduce the corresponding time-fractional phase field models, i.e., the time-fractional Allen-Cahn (FAC) type equations and the time-fractional Cahn-Hilliard (FCH)type equations. Mainly, we simply replace the integer time derivatives by the fractionaltime derivatives.For simplicity, we consider periodic boundary conditions for the FPDEs in this paper.Thus, the fractional Allen-Cahn equation reads C D αt φ = − λ δFδφ , in Ω T = Ω × (0 , T ] ,φ = φ , in Ω = Ω × { } . (2.7)4nd the fractional Cahn-Hilliard equation reads C D αt φ = ∇ · ( λ ∇ δFδφ ) , in Ω T = Ω × (0 , T ] ,φ = φ , in Ω = Ω × { } . (2.8)Here C D αt (with α ∈ (0 , α defined by [16, 23] C D αt φ ( t ) := − α ) t Z φ η ( η )( t − η ) α dη, ≤ α < ,∂φ∂t , α = 1 , (2.9)with Γ( · ) the Gamma function. The Caputo type time-fractional differential operator pro-vides a means to model the sub-diffusive or long time memory behavior. Remark . There usually exist two ways to express the operator D αt : Caputo andRiemann-Liouville definitions. In the Caputo definition, the fractional derivative of order α , denoted by C D αt u ( x, t ) is defined as C D αt u ( x, t ) = 1Γ(1 − α ) Z t ∂u ( x, τ ) ∂τ dτ ( t − τ ) α , < α < , (2.10)where Γ( · ) denotes Gamma function. By using integration by parts, we have the alternativeformula C D αt u ( x, t ) = 1Γ(1 − α ) (cid:16) u ( x, t ) − u ( x, t α + α Z t u ( x, t ) − u ( x, s )( t − s ) α − ds (cid:17) . (2.11)The Riemann-Liouville derivative R D αt has the form R D αt u ( x, t ) = 1Γ(1 − α ) ∂∂t Z t u ( x, τ )( t − τ ) α dτ, < α ≤ . (2.12)The two definitions are linked by R D αt u ( x, t ) = u ( x, − α ) t α + C D αt u ( x, t ) . (2.13)In this paper, we stick to the Caputo’s definition for simplicity. In order to solve the time-fractional phase field models, accurate, efficient and stablenumerical approximations are necessary. There are mainly two major difficulties in dis-5retizing the models in time, mainly dealing with the time-fractional operator, and dealingwith the nonlinear terms on the right-hand-side of the models. Here we provide generalnumerical techniques to deal with these issues as follows.Recall the Caputo fractional derivative of order α (0 < α <
1) [3, 18] C D αt u ( x, t ) = 1Γ(1 − α ) Z t ∂u ( x, η ) ∂η ( t − η ) − α dη. (3.1)We consider the fast evaluation of the Caputo fractional derivative proposed by [10, 31].Before introduce the scheme, we approximated the kernel t − β (0 < β <
2) via a sum-of-exponentials approximation efficiently on the interval [ δ, T ] with δ = min ≤ K ≤ N ∆ t n and theabsolute error (cid:15) . That is to say, there exits positive real numbers s i and ω i ( i = 1 , . . . , K exp )such that (cid:12)(cid:12) t β − K exp X i =1 ω i e − s i t (cid:12)(cid:12) ≤ (cid:15), t ∈ [ δ, T ] . Then we split the fractional-in-time derivative term into a sum of local part and historypart as following: C D αt φ ( x , t k +1 ) = 1Γ(1 − α ) t Z φ s ( x , s )( t − s ) α ds, = 1Γ(1 − α ) t k +1 Z t k φ s ( x , s )( t − s ) α ds + 1Γ(1 − α ) t k Z φ s ( x , s )( t − s ) α ds := C l ( t k +1 ) + C h ( t k +1 ) , (3.2)The local part C l ( t k +1 ) and the history part C h ( t k +1 ) are approximated respectively asfollows: C l ( t k +1 ) ≈ φ ( x , t k +1 ) − φ ( x , t k )∆ t αk +1 Γ(2 − α ) , and C h ( t k +1 ) = 1Γ(1 − α ) h φ ( x , t k )∆ t αk +1 − φ ( x , t ) t αk +1 − α t k Z φ ( x , s )( t − s ) α ds i ≈ − α ) h φ ( x , t k )∆ t αk +1 − φ ( x , t ) t αk +1 − α K exp X i =1 ω i t k Z e − ( t k +1 − τ ) s i φ ( x , τ ) dτ i = 1Γ(1 − α ) h φ ( x , t k )∆ t αk +1 − φ ( x , t ) t αk +1 − α K exp X i =1 ω i U hist,i ( t k +1 ) i , U hist,i ( t k +1 ) = t k Z e − ( t k +1 − τ ) s i φ ( x , τ ) dτ = e − s i ∆ t k +1 U hist,i ( t k ) + t k Z t k − e − ( t k +1 − τ ) s i φ ( x , τ ) dτ ≈ e − s i ∆ t k +1 U hist,i ( t k ) + e − s i ∆ t k +1 s i ∆ t k h ( e − s i ∆ t k − s i ∆ t k ) φ ( x , t k )+(1 − e − s i ∆ t k − e − s i ∆ t k s i ∆ t k ) φ ( x , t k − ) i . Denote C D αt φ n +1 = φ n +1 − φ n ∆ t αn +1 Γ(2 − α ) + 1Γ(1 − α ) h φ n ∆ t αn +1 − φ t αn +1 − α K exp X i =1 ω i U n +1 hist,i i , where U n +1 hist,i = e − s i ∆ t n +1 U nhist,i + e − s i ∆ t n +1 s i ∆ t n h ( e − s i ∆ t n − s i ∆ t n ) φ n +(1 − e − s i ∆ t n − e − s i ∆ t n s i ∆ t n ) φ n − i . We further utilize the stabilized semi-implicit strategy [26, 27] to discretetize the non-linear terms on the right hand side to end up with linear schemes. Given the free energy F , if we rewrite it as F = ( 12 Lφ, φ ) + E, (3.3)where L is the linear operators that are separated from F , and E is the rest term.The scheme for the fractional Allen-Cahn equation is given as follows. Scheme . Give the initial condition φ = φ . After we obtain φ n , n ≥
1, we can get φ n +1 via C l ( t n +1 ) + C h ( t n +1 ) = λ h Lφ n +1 + S ∆( φ n +1 − φ n +1 ) − (cid:16) δEδφ (cid:17) n +1 i , (3.4)where ( • ) n +1 = 2( • ) n − ( • ) n − is a second-order extrapolation, and S is stabilizing constant.The operators C l , C h are defined in (3.2).The scheme for the fractional Cahn-Hilliard equation is given as follows. Scheme . Give the initial condition φ = φ . After we obtain φ n , n ≥
1, we can get φ n +1 via C l ( t n +1 ) + C h ( t n +1 ) = ∇ · h λ ( φ n +1 ) ∇ (cid:16) Lφ n +1 + δEδφ n +1 (cid:17)i + λ h − S ∆ ( φ n +1 − φ n +1 ) + S ∆( φ n +1 − φ n +1 ) i , (3.5)7here ( • ) n +1 = 2( • ) n − ( • ) n − is a second-order extrapolation, and S , S are stabilizingconstant. The operators C l , C h are defined in (3.2). In this section, we present our numerical findings on the fractional-order phase fieldmodels, that is the scaling of effective free energy/roughness in the time-fractional phasefield models during coarsening follow a similar power law as the integer phase field models,where the power is linearly proportional to the fractional order α .Here we chose periodic boundary conditions in the square domain. Define the roughnessmeasure function W ( t ) as follows: W ( t ) = s | Ω | Z Ω (cid:16) φ ( x, y, t ) − φ ( x, y, t ) (cid:17) d Ω , (4.1)where φ ( x, y, t ) = R Ω φ ( x, y, t ) d Ω. By the linear least square fitting, we get the absolutevalue of slope for each linear line of energy and roughness, β ( α ) , R ( α ), defined aslog E ( α, t ) = β ( α ) − β ( α ) log t, log W ( α, t ) = R ( α ) + R ( α ) log t. (4.2) Fractional Cahn-Hilliard equations with the double-well potential.
In the firstexample, we study the fractional Cahn-Hilliard (FCH) equation, equipped with the freeenergy F = Z Ω h ε |∇ φ | + 14 φ (1 − φ ) i d x , (4.3)where the first term represents the enthalpy, and the second term is the bulk energy func-tional. The fractional Cahn-Hilliard equation with a variable mobility reads C D αt φ = ∇ · ( λ ( φ ) ∇ µ ) , ( x , t ) ∈ Ω × (0 , T ] ,µ = − ε ∆ φ + φ (1 − φ )( − φ ) , ( x , t ) ∈ Ω × (0 , T ] ,φ ( x ,
0) = φ ( x ) , x ∈ Ω , (4.4)where λ ( φ ) is the motility parameter, which could be chosen as [8]( i ) λ ( φ ) = λ , ( ii ) λ ( φ ) = λ | − φ | ; ( iii ) λ ( φ ) = λ | φ | . (4.5)First of all, we focus on the FCH equation (4.4) with a constant mobility λ ( φ ) = λ .We follow the parameters used in [8], i.e. consider domain [0 , π ] , and ε = 0 . λ =0 .
02, and we choose random initial condition φ t =0 = 0 . × rand ( − , O ( t − α ),8 a) (b) Figure 4.1: Energy scaling laws for the FCH model with constant mobility λ = λ . (A) The log-log plot of the effective energy and time with differentfractional-order α . (b) The least-square fitted energy decay power β ( α )for different fractional order α .which is consistent with O ( t − ) as indicated in [8] for the integer Cahn-Hilliard equationwith constant mobility. In addition, the coarsening dynamics for the FCH model (2.8) ofconstant mobility with fractional-order α = 0 . λ ( φ ) = λ | − φ | . We usethe same initial values and parameters as the example above. The energy plot at differenttime slots with various fractional orders are summarized in Figure 4.3. We observe thatthe energy scales approximately like O ( t − α ), which is consistent with O ( t − ) as indicatedin [8] for the integer Cahn-Hilliard equation with variable mobility λ ( φ ) = λ | − φ | . Inaddition, the corresponding coarsening dynamics with α = 0 . Time-fractional molecular beam epitaxy models.
In the second example, weconsider the fractional molecular beam epitaxy (FMBE) model. Given a smooth domainΩ, and use φ ( x , t ) : Ω → R to denote the height function of MBE, and the effective freeenergy is given as E ( φ ) = Z Ω h ε | ∆ φ | + f ( ∇ φ ) i d Ω . (4.6)Here the first term represents the isotropic surface diffusion effect with ε a constant control-ling the surface diffusion strength, and the second term approximates the Enrlich-Schwoebeleffect that the adatoms stick to the boundary from an upper terrace, contributing to thesteepening of mounds in the film [7].If we choose f ( ∇ φ ) = − ln(1 + |∇ φ | ), the corresponding FMBE model with slope9 a) (b) (c)(d) (e) (f) Figure 4.2: Time snapshots of coarsening dynamics driven by the FCHmodel with constant mobility λ = λ . The profiles of φ at different timeslots t = 5 , , , , ,
145 are presented.selection reads C D αt φ ( x , t ) = − M (cid:16) ε ∆ φ − ∇ · (( |∇ φ | − ∇ φ ) (cid:17) , ( x , t ) ∈ Ω × (0 , T ] ,φ ( x ,
0) = φ ( x ) , x ∈ Ω . (4.7)For the FMBE model with slope selection, the energy scaling laws are summarized in 4.5. Itindicates that the energy scales approximately like O ( t − α ), which is consistent with O ( t − )as indicated in [29] for the integer MBE equation with slope selection. Similarly, as shownin Figure 4.6, the roughness scales approximately like O ( t α ), which is consistent with O ( t )as indicated in [29] for the integer MBE equation with slope selection. One time sequencefor the time-fractional MBE model with α = 0 . f ( ∇ φ ) = ( |∇ φ | − , the corresponding10 a) (b) Figure 4.3: Energy Scaling Laws for the FCH mdoel with variable mobil-ity λ = λ | − φ | . (A) The log-log plot of the effective energy and timewith different fractional-order α . (b) The least-square fitted energy decaypower β for different fractional order α . This figure illustrates that theenergy scales approximately like O ( t − α ), which is consistent with O ( t − )as indicated in [8] for the integer Cahn-Hilliard equation.time-fractional MBE model without slope selection reads C D αt φ ( x , t ) = − M (cid:16) ε ∆ φ + ∇ · ( ∇ φ |∇ φ | ) (cid:17) , ( x , t ) ∈ Ω × (0 , T ] ,φ ( x ,
0) = φ ( x ) , x ∈ Ω . (4.8)For the time-fractional MBE model without slope selection, the roughness scales approxi-mately like O ( t α ), which is consistent with O ( t ) as indicated in [29] for the integer MBEequation without slope selection. The results are summarized in 4.9. One time sequencefor the time-fractional MBE model without slope selection of time-fractional order α = 0 . In this paper, we propose time-fractional phase field models, develop their efficient,accurate, full discrete, linear numerical approximation. Correlations of scaling laws withfractional order is studied. The numerical scheme utilizes the fast algorithm for the Caputofractional derivative operator in time discretization and Fourier spectral method in spatialdiscretization. Our proposed scheme can handle long time simulation in high dimensions.The numerical approximation strategy proposed in this paper can be readily applied tostudy many classes of time-fractional and high dimensional phase field models.With the proposed time fractional Cahn-Hilliard equation, we study the power law11 a) (b) (c)(d) (e) (f)
Figure 4.4: Time snapshots of coarsening dynamics driven by the FCHmodel with variable mobility λ = λ | − φ | . The profiles of φ at differenttime slots t = 5 , , , , ,
145 are presented.scaling behavior of effective free energies with both constant mobility and variable mobilityin coarsening dynamics. For time-fractional Cahn-Hilliard equation with constant mobility,it scales as O( t − α ), which is consistent with the well known result ( t − ) for integer orderCahn-Hilliard model [8]. For time fractional Cahn-Hilliard equation with phase-variabledependent mobility λ = λ | − φ | , the effective energy scales as O( t − α ), which is consistentwith the result of O( t − ) for the integer Cahn-Hilliard equation with such variable mobility.With the proposed fractional MBE model, we observe that the scaling law for theenergy decays as O ( t − α ) and the roughness increases as O ( t α ) for the MBE model withslope selection, and the roughness increases as O ( t α ) for the MBE model without slopeselection. That is to say, the coarsening rate of MBE model could be manipulated by thefractional order α , and it is linearly proportional to α . This is the first time in literatureto report/discover such scaling correlation. It provides a potential application field forfractional differential equations. 12 a) (b) Figure 4.5: Energy scaling laws for the time-fractional molecular beamepitaxy equation with slope selection. (A) The log-log plot of the effectiveenergy and time with different fractional-order α . (b) The least-squarefitted energy decay power β for different fractional order α .Overall, we lay out a foundation for time fractional phase field models by discoveringa universal power laws scaling property. The scaling of effective free energy/roughness inthe time-fractional phase field models during coarsening follow a similar power law as theinteger phase field models, where the power is linearly proportional to the fractional order α . This universal scaling law is verified numerically against several phase field models,including the Cahn-Hilliard (CH) equations with different variable mobilities and molecularbeam epitaxy (MBE) models. This new finding sheds light on potential applications oftime fractional phase field models in studying coarsening dynamics and crystal growths.A rigorous asymptotic analysis to this interesting correlation will be pursed in our futurework. Acknowledgment
Lizhen Chen would like to acknowledge the support from National Science Foundation ofChina through Grant 11671166 and U1530401 , Postdoctoral Science Foundation of Chinathrough Grant 2015M580038. Jia Zhao is partially supported by a seed grant (ResearchCatalyst Grant) from Office of Research and Graduate Studies at Utah State University.Hong Wang is partially supported by the OSD/ARO MURI Grant W911NF-15-1-0562 andby the National Science Foundation under Grant DMS-1620194.13 a) (b)
Figure 4.6: Roughness scaling laws for the time-fractional molecular beamepitaxy equation with slope selection. (A) The log-log plot of the rough-ness and time with different fractional-order α . (b) The least-square fittedenergy decay power β for different fractional order α . References [1] M. Ainsworth and Z. Mao. Analysis and approximation of a fractional cahn-hilliardequation.
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