Phase field models frequently provide insight to phase transitions, and are
robust numerical tools to solve free boundary problems corresponding to the
motion of interfaces. A body of prior literature suggests that interface motion
via surface diffusion is the long-time, sharp interface limit of microscopic
phase field models such as the Cahn-Hilliard equation with a degenerate
mobility function. Contrary to this conventional wisdom, we show that the
long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free
energy undergoes coarsening, reflecting the presence of bulk diffusion, rather
than pure surface diffusion. This reveals an important limitation of phase
field models that are frequently used to model surface diffusion

Lee, Alpha A

Münch, Andreas

Süli, Endre

Applied Physics Letters

English

arXiv

Phase field models frequently provide insight into phase transitions and are robust numerical tools to solve free boundary problems corresponding to the motion of interfaces. A body of prior literature suggests that interface motion via surface diffusion is the long-time, sharp interface limit of microscopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function. Contrary to this conventional wisdom, we show that the long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free energy undergoes coarsening, reflecting the presence of bulk diffusion, rather than pure surface diffusion. This reveals an important limitation of phase field models that are frequently used to model surface diffusion

Lee, Alpha A.

ORA - Oxford University Research Archive

Degenerate mobilities in phase field models are insufficient to capture surface diffusion

Oxford University Research Archive

Degenerate mobilities in phase field models are insufficient to capture surface diffusionAlpha A. Lee, Andreas Münch, and Endre Süli  Citation: Applied Physics Letters 107, 081603 (2015); doi: 10.1063/1.4929696 View online: http://dx.doi.org/10.1063/1.4929696 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/107/8?ver=pdfcov Published by the AIP Publishing  Articles you may be interested in Response to “Comment on ‘Degenerate mobilities in phase field models are insufficient to capture surfacediffusion’” [Appl. Phys. Lett. 108, 036101 (2016)] Appl. Phys. Lett. 108, 036102 (2016); 10.1063/1.4939931  Comment on “Degenerate mobilities in phase field models are insufficient to capture surface diffusion” [Appl.Phys. Lett. 107, 081603 (2015)] Appl. Phys. Lett. 108, 036101 (2016); 10.1063/1.4939930  Control morphology of nanostructures with electric field Appl. Phys. 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Lee, Andreas M€unch, and Endre S€uliMathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom(Received 23 May 2015; accepted 16 August 2015; published online 27 August 2015)Phase field models frequently provide insight into phase transitions and are robust numerical tools tosolve free boundary problems corresponding to the motion of interfaces. A body of prior literaturesuggests that interface motion via surface diffusion is the long-time, sharp interface limit of micro-scopic phase field models such as the Cahn-Hilliard equation with a degenerate mobility function.Contrary to this conventional wisdom, we show that the long-time behaviour of degenerate Cahn-Hilliard equation with a polynomial free energy undergoes coarsening, reflecting the presence of bulkdiffusion, rather than pure surface diffusion. This reveals an important limitation of phase field mod-els that are frequently used to model surface diffusion. VC 2015 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4929696]A key problem in modelling phase transitions in materi-als lies in linking macroscopic interfacial motion and meso-scopic dynamics. A common approach is phase field models,which replace a sharp interface with order parameters thatare continuous across the interface. Phase field models canbe constructed from systematic coarse-graining of the micro-scopic Hamiltonian and are often written as a gradient mini-misation of a certain microscopic free energy functional.1–3As such, they provide the crucial link between microscopicinteractions and the kinetics of phase separation and pattern-formation.Mesoscopic phase field descriptions have been widelyused in the literature as their numerical approximation is lesscomplicated than the approximation of macroscopic descrip-tions based on sharp interfaces. By using a continuous orderparameter field, phase field approaches are versatile enoughto capture topological changes and replace the numericallychallenging task of interface tracking with integration of atime-dependent partial differential equation. Therefore, thephase field formalism is increasingly used as a numericalapproximation for a wider class of free boundary problemsthan what the free energy describes microscopically.4–6A particularly noteworthy class of free boundary prob-lem is when the velocity of the interface vn is proportional tothe surface Laplacian of the mean curvature j andvn ¼MDsj; (1)where M is the mobility. Equation (1) is known as the sur-face diffusion flow and has been used as a model for manycomplex processes such as electromigration in metals,7 het-eroepitaxial growth,8 and more recently solid-soliddewetting.9The Cahn-Hilliard equation with degenerate mobility isthe commonly used phase field model to approximate surfacediffusion (e.g., Refs. 7–14), where the order parameter u isconserved and satisfies (in dimensionless units)ut ¼ r  j; j ¼ MðuÞrl; (2a)l ¼ 2r2uþ f 0ðuÞ; (2b)f uð Þ ¼ 141 u2ð Þ2; M uð Þ ¼ 1 u2; (2c)where M(u) is the mobility function; j is the flux; l is thechemical potential;  is the interfacial tension which deter-mines the width of the interface; and f(u) is the bulk freeenergy. Throughout the paper, we will assume the no-fluxcondition n  j ¼ 0, and the variational Neumann conditionn  ru ¼ 0 on the boundaries of the solution domain. Thetwo pure phases are denoted by u ¼ 61, and the interface islocated at the contour u¼ 0.The precise form of the mobility is usually chosen onthermodynamic grounds.15,16 For the lattice-gas entropysðuÞ ¼ u log uþ ð1 uÞ logð1 uÞ, the Einstein relationstipulates that the mobility is related to the entropy functions(u) via MðuÞ ¼ ð@2s=@u2Þ1 ¼ 1 u2. This motivates thechoice of mobility in (2).More elaborate multiphase-field models have beenconstructed in the literature to describe phase transitionwhen multiple phases coexist17,18 as is the case for manytechnologically important alloys. In addition, the inverseproblem of relating physical parameters of the phases tothe form of the free energy and the interfacial tension isnon-trivial.19,20 Here, for simplicity, we restrict ourselvesto considering a binary system with only two phases andconsider a simple quartic free energy; as is seen in the anal-ysis below, the fact the coarsening occurs in the sharp inter-face limit is a generic feature of the system (2) independentof its parameterisation.We are interested in the long-time behaviour when theinitial mixture has separated into regions where u is eitherclose to 1 or to 1, except for regions of width OðÞ close tothe interface over which u transitions between these tworegions. Heuristically, the width of this interface layerdecreases as ! 0. One may think that if the mobility func-tion is degenerate and vanishes at the pure phases, the fluxnormal to the interface is suppressed and therefore only sur-face diffusion via mass flux tangential to the interface canoccur. However, this heuristic argument neglects the fact thatthe gradient of the interface diverges as ! 0. Therefore,0003-6951/2015/107(8)/081603/4/$30.00 VC 2015 AIP Publishing LLC107, 081603-1APPLIED PHYSICS LETTERS 107, 081603 (2015) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  129.67.246.57 On: Sun, 28 Feb 2016 16:09:38whether the degenerate mobility function is sufficient to sup-press the normal mass flux at leading order is unclear.21,22The key result of this paper is the presence of a nonlin-ear bulk diffusion term at leading order for the interface ve-locity, and the correct sharp interface limit that describes thequasistationary evolution of the interface f, located at u¼ 0,is given byr  ðlrlÞ ¼ 0; in X; (3a)l ¼ 23j; on f; (3b)vn ¼23Dsjþ14lrnl on f; (3c)rnl ¼ 0; on @Xext; (3d)where vn is the normal velocity, and the definitions of Xand f are given in Figure 1. The possibility of bulk diffusionwas noted in earlier works21,23 and analysed recently by Daiand Du.24 However, in Ref. 24, the analysis was done on the(unphysical) solution branch with juj > 1 in some region.Our analysis below considers the physical branch of solutionwhere juj < 1 everywhere and derives the limiting model as! 0. It can be shown using rigorous mathematical analysisthat the physical branch with juj < 1 everywhere exists forall parameter values.25To begin our analysis, we drop the time derivative from(2). Rewriting the Laplacian in polar coordinates, and resolv-ing the boundary layer near the interface by noting that theinterface is OðÞ thick, we obtain the leading order solutionu0 rð Þ ¼ tanhr  r0 ; l ¼ ljf: (4)Here, r is the radial coordinate with respect to the centre ofthe osculating circle to the interface, r0 is the position of theinterface in these coordinates, and ljf is a constant. Equation(4) reveals how the interface is being represented as a contin-uous, albeit thin, order parameter profile of width  aroundr¼ r0. A key physical intuition is that the relaxation of thelocal order parameter profile to (4) is rapid (as it is driven bylocal rearrangement of particles), and the late stage dynamicsis quasi-static and determined by the movement of the inter-face, i.e., change in r0.To put this intuition onto firmer footing, we consider thechemical potential l away from the interface. There, the sys-tem is almost a pure phase (say u ¼ 1þ ~u; l ¼ ~l, with~u; ~l ¼ Oð1Þ), and the spatial variation of u is negligible sothat the Laplacian can be neglected. Expanding Equation(2b) in powers , the leading order term is given by~l ¼ f 00ð1Þ~u; (5)and substituting (5) into (2a), we obtain~ut ¼ M0 1ð Þr  ~ur~lð Þ ¼M0 1ð Þf 00 1ð Þr  ~lr~lð Þ: (6)Further, we use the aforementioned quasi-static approxima-tion and assume that ~ut is small. Thus, to leading order,r  ð~lr~lÞ ¼ 0: (7)The chemical potential at the interface can be obtainedby multiplying (2a) by @ru, and integrating both sides.Assuming that the azimuthal variations in u are asymptoti-cally smaller than the radial variation, we haveðr0þgr0gl@rudr ¼ðr0þgr0g 2r@r r@ruð Þ þ f 0 uð Þ @rudr; (8)where g is an intermediate lengthscale bridging the interfaceof width  and the macroscopic lengthscale; Kaplun’s exten-sion theorem in asymptotic theory provides a rigorous justifi-cation of the existence of such a lengthscale under mildassumptions.26 Close to the interface, u and l can beapproximated by (4). Substituting (4) and g   logð1=Þ into(8) and taking ! 0þ, we obtainljf ¼23j; (9)where j ¼ 1=r0 is the curvature of the interface. The abovechoice of g satisfies both the requirement that (4) is validnear the interface (we have assumed l to be constant overshort radial distances r; over O(1) length scales, this is not,in general, the case) and the integrals converge to (9) (forg ). From u0 in (4), it can be readily seen that we haveu0  61þ OðaÞ with some positive a for the above choiceof g, and this gives (9) in the limit ! 0þ as claimed.In the quasi-static approximation, we neglect time de-pendence except for the slow motion of the interface. Toobtain the interface velocity, we focus on the boundary layerregion and move into a Lagrangian frame by making thetransformation ut 7!ut  ðv  rÞu. Now, noting that the nor-mal velocity is much larger than the lateral velocity, we haveFIG. 1. Illustration of the asymptotic structure of the degenerate mobilitycase. The domain Xext is split into interfaces f where u¼ 0 which enclose thesolid domains Xþ where u> 0 (coloured red), and the region X with vapor(colored green) where u< 0. The normal n to the interfaces points out of Xþ.081603-2 Lee, M€unch, and S€uli Appl. Phys. Lett. 107, 081603 (2015) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  129.67.246.57 On: Sun, 28 Feb 2016 16:09:38ðv  rÞu  2vn@ru, where 2vn is the normal velocity, scaledin anticipation of the order of the right-hand side which willturn out to be Oð2Þ. Equation (2a) becomes2vn@ru ¼ @rðj  nÞ  @sðj  tÞ; (10)where s denotes the coordinate tangent to the interface, andn and t are the (outward pointing) unit normal and the unittangent vector, respectively. Close to the interface, the tan-gential flux is given by j  t ¼ MðuÞ@sl ¼ MðuÞ@sljf.Thus, again substituting (4) for u, and integrating over theinterface, we arrive at22vn ¼  j  n½ r0þgr0gþ432@ssj: (11)Now, for  g 1, we obtainj  njr0þg ¼ M uð Þ@rl ¼ 2 M0 1ð Þf 00 1ð Þl@rlþ O 3ð Þ;and j  njr0g ¼ 0 as the system reaches the pure phase u¼ 1deep in Xþ. Therefore, all in all, we obtain (3c), where wehave identified @ss with the surface Laplacian Ds. Equation(3c) shows that the interface velocity in the sharp interfacemodel has two contributions: one from surface diffusion,Equation (1), which is local to the interface, and another con-tribution from nonlinear bulk diffusion which satisfies aporous-medium Equation (3a). Unlike pure surface diffusion,the mass flux arising from bulk diffusion couples disjointsolid domains with each other. This results in coarseningwhere larger solid domains grow at the expense of smallerones, which cannot happen for pure surface diffusion.Moreover, for non-circular interfaces, the contribution fromsurface and bulk diffusion enters the interface evolution tothe same order, i.e., the effect of the latter does not becomenegligible compared to the former even when letting ! 0.To test this prediction of our analysis, we consider therelaxation of an azimuthal perturbation to a radially symmet-ric stationary state with radius r0 and hence curvaturej ¼ 1=r0. For azimuthal perturbations proportional tocos mh, the pure solid diffusion model (1) predicts an expo-nential decay rater ¼  23j4m2ðm2  1Þ: (12)In contrast, the decay rate in the porous medium model (3) isgiven byr ¼ j4 23m2ðm2  1Þ þ 19mðm2  1Þtanh m log jð Þ : (13)Table I shows the decay rate numerically obtained bysolving the phase field model, Equations (2). It shows howthe decay rate of the azimuthal perturbation to the axisym-metric base state tends to the linearised sharp interfacemodel with the contribution from nonlinear bulk diffusion,rather than to the one for pure surface diffusion.Our analysis only applies to free energy functions f forwhich f 0ð61Þ ¼ 0 and f 00ð61Þ > 0 at the minima u ¼ 61.Indeed, according to the asymptotic analysis in Ref. 27, thedouble obstacle free energyfdoðuÞ ¼1 u2 if juj < 11 if juj 	 1;gives rise to pure surface diffusion flow for MðuÞ ¼ 1 u2 inthe sharp interface limit (! 0), and also if the logarithmicfree energy f ðuÞ ¼ u2 þ aðu log uþ ð1 uÞ logð1 uÞÞwith a > 0 is used instead.In conclusion, our analysis establishes that phase fieldmodels for pure surface diffusion cannot be realised usingthe Cahn-Hilliard equation with the degenerate mobility andGinzburg-Landau free energy as in (2), as was repeatedlyassumed in the literature.9,10,13,14 A nonlinear bulk diffusionterm appears to leading order of the sharp interface limit,hence affecting the coarsening behaviour on the same timescale as surface diffusion. In particular, it allows, on thistime scale, disjoint interfaces to coarsen and cannot be sup-pressed by reducing .We note that the derivation presented in this work couldbe made mathematically robust via matched asymptotic anal-ysis. Such an approach was applied to analyse the Cahn-Hilliard equation with constant mobility.28,29 Extending themethod of matched asymptotics to model (2) with a degener-ate mobility will be the subject of a subsequent publication.The heuristic approach presented here, however, revealsclearly the salient physics involved in the sharp interfacelimit.1D. Anderson, G. B. McFadden, and A. Wheeler, Annu. Rev. Fluid Mech.30, 139 (1998).2L.-Q. Chen, Annu. Rev. Mater. Res. 32, 113 (2002).3N. Provatas and K. Elder, Phase-field Methods in Materials Science andEngineering (John Wiley & Sons, 2011).4K. Elder, M. Grant, N. Provatas, and J. Kosterlitz, Phys. Rev. E 64,021604 (2001).5H. Emmerich, Adv. Phys. 57, 1 (2008).6I. Steinbach, Modell. Simul. Mater. Sci. Eng. 17, 073001 (2009).7M. Mahadevan and R. M. Bradley, Physica D 126, 201 (1999).8A. R€atz, A. Ribalta, and A. Voigt, J. Comput. Phys. 214, 187 (2006).9W. Jiang, W. Bao, C. V. Thompson, and D. J. Srolovitz, Acta Mater. 60,5578 (2012).10K. Kitahara and M. Imada, Prog. Theor. Phys. Suppl. 64, 65 (1978).11D. N. Bhate, A. Kumar, and A. F. Bower, J. Appl. Phys. 87, 1712 (2000).12D.-H. Yeon, P.-R. Cha, and M. Grant, Acta Mater. 54, 1623 (2006).13S. Wise, J. Kim, and J. Lowengrub, J. Comput. Phys. 226, 414 (2007).14S. Torabi, J. Lowengrub, A. Voigt, and S. Wise, Proc. R. Soc. A 465, 1337(2009).15G. Giacomin and J. L. Lebowitz, Phys. Rev. Lett. 76, 1094 (1996).16E. B. Nauman and D. Q. He, Chem. Eng. Sci. 56, 1999 (2001).17J. Tiaden, B. Nestler, H. Diepers, and I. Steinbach, Physica D 115, 73(1998).18J. Eiken, B. B€ottger, and I. Steinbach, Phys. Rev. E 73, 066122 (2006).19A. A. Wheeler, W. Boettinger, and G. McFadden, Phys. Rev. A 45, 7424(1992).20S. G. Kim, W. T. Kim, and T. Suzuki, Phys. Rev. E 60, 7186 (1999).TABLE I. Relaxation rates for m¼ 2 obtained from the linearised phasefield model are shown for different values of  in the first four columns andcompared to the eigenvalues obtained for linearised sharp interface modelsfor pure surface diffusion (12) and the porous medium type model (13) inthe next-to-last and the last column, respectively. 0.01 0.005 0.002 0.001 Eq. (13) Eq. (12)R 133.2 133.8 136.3 137.0 2137.4 2128081603-3 Lee, M€unch, and S€uli Appl. Phys. Lett. 107, 081603 (2015) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  129.67.246.57 On: Sun, 28 Feb 2016 16:09:3821A. J. Bray and C. L. Emmott, Phys. Rev. B 52, R685 (1995).22C. Gugenberger, R. Spatschek, and K. Kassner, Phys. Rev. E 78, 016703(2008).23J. Cahn and J. Taylor, Acta Metall. Mater. 42, 1045 (1994), ISSN 0956-7151.24S. Dai and Q. Du, Multiscale Model. Simul. 12, 1870 (2014).25C. M. Elliott and H. Garcke, SIAM J. Math. Anal. 27, 404 (1996).26M. H. Holmes, Introduction to Perturbation Methods (Springer Science &Business Media, 2012), Vol. 20.27J. W. Cahn, C. M. Elliott, and A. Novick-Cohen, Eur. J. Appl. Math. 7,287 (1996).28R. L. Pego, Proc. R. Soc. London, Ser. A 422, 261 (1989).29N. D. Alikakos, P. W. Bates, and X. Chen, Arch. Ration. Mech. Anal. 128,165 (1994).081603-4 Lee, M€unch, and S€uli Appl. Phys. Lett. 107, 081603 (2015) Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions.  IP:  129.67.246.57 On: Sun, 28 Feb 2016 16:09:38

Alpha A. Lee

Andreas Münch

Endre Süli

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Degenerate Mobilities in Phase Field Models are Insufficient to Capture
  Surface Diffusion

Degenerate Mobilities in Phase Field Models are Insufficient to Capture Surface Diffusion

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