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A new Phase-Field Model for Anisotropic Surface Diffusion: Anisotropic Cahn-Hilliard Equation with Improved Conservation (ACH-IC)

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Pith's one-line read An anisotropic Cahn-Hilliard model with a modified conserved quantity achieves second-order volume conservation while still recovering the anisotropic surface-diffusion sharp-interface limit.

desk verdict Genuinely improved phase-field model with a real second-order volume-conservation proof, but the headline theorem's regularity assumption is too weak for the pinch-off case the paper advertises. read the letter →

arxiv 2507.18048 v1 pith:SFO6J6P2 submitted 2025-07-24 math-ph math.MP

Zeyu Zhou , Wei Jiang , Tiezheng Qian , Zhen Zhang This is my paper · ORCID
classification math-phmath.MP MSC 35K5535B4074N20
keywords anisotropicsurfacediffusionphase-fieldmodelCahn-Hilliardequationvolumeconservationsecond-orderconvergenceOnsagervariationalprincipledegeneratemobilitymatchedasymptoticexpansions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a new phase-field model for anisotropic surface diffusion, called ACH-IC, in which the conserved quantity is not the phase variable itself but a modified function $Q(u)$ that more sharply approximates the two-phase step function. The paper argues that this replacement raises the accuracy of the volume enclosed by the interface from first order to second order in the interface thickness $\varepsilon$, while keeping energy dissipation and recovering the sharp-interface anisotropic surface diffusion law. The motivation is practical: standard Cahn-Hilliard models lose volume at first order and exhibit spurious shrinkage of small drops, and the paper shows numerically that the ACH-IC mitigates both defects, including in pinch-off events.

What carries the argument

The load-bearing object is the modified conserved quantity $Q(u)$, a normalized integral of $(1-z^2)^k$, together with its reciprocal $N(u)=1/Q'(u)$. In the evolution equation $u_t = \varepsilon^{-1} N(u) \nabla \cdot (M(u)\nabla(N(u)\mu))$, the factor $N(u)$ forces the flux to vanish in the bulk phases because it blows up at $u = \pm 1$, confining diffusion to the interface; the mobility $M(u)=(1-u^2)^l$ controls the rate there. The parameter condition $1 \le k \le l \le 2k+1$ is what the asymptotic analysis requires for the sharp-interface limit and the $O(\varepsilon^2)$ volume bound.

What would settle it

For an initially smooth anisotropic shape that develops a cusp or self-intersection in finite time, compute $|\,|\Omega_\varepsilon^+(T)| - |\Omega_\varepsilon^+(0)|\,|$ for a sequence of decreasing $\varepsilon$ before and after the singularity; if the error after the singularity scales like $\varepsilon$ rather than $\varepsilon^2$, the $O(\varepsilon^2)$ bound does not extend to the pinch-off regime. A direct check is to repeat the thin-tube pinch-off test of Section 4.6 with $\varepsilon = 0.01$ and $0.005$ and measure whether second-order convergence persists after the drops detach.

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Extended reading notes

Core claim

The central claim is that for the ACH-IC solution the volume of the domain enclosed by the zero level set satisfies $|\Omega_\varepsilon^+(t)| = |\Omega_\varepsilon^+(0)| + O(\varepsilon^2)$, and that the interface normal velocity obeys $V_\varepsilon = C_l \Delta_s(\nabla_s \cdot \xi(\nu)) + O(\varepsilon)$. The model is derived from Onsager's variational principle with the conservation law $\partial_t Q + \nabla \cdot J = 0$, where $Q(u) = \int_0^u (1-z^2)^k \, dz \; / \; \int_0^1 (1-z^2)^k \, dz$ approximates the step function of the two phases more closely than $u$ itself. Matched asymptotic expansions show that the leading-order profile remains a tanh interface, the first-order correction is uniformly bounded, and the degeneracy condition $1 \le k \le l \le 2k+1$ suffices for the sharp-interface limit. Numerical tests in two and three dimensions show second-order real-volume convergence and agreement with sharp-interface reference solutions, whereas the classical ACH shows first-order drift and spontaneous shrinkage.

Load-bearing premise

The proof of second-order volume conservation assumes the interface $\partial\Omega_\varepsilon^+(t)$ is finite and $C^2$-smooth at every time, an assumption that fails exactly at the pinch-off and topological-change events the model is advertised to simulate.

Editorial extensions

If this is right

  • If Eq. (18) is correct, phase-field simulations of anisotropic surface diffusion can use larger interface thickness $\varepsilon$ and coarser meshes while matching the volume accuracy of classical models run at much smaller $\varepsilon$.
  • The spontaneous shrinkage of small drops is mitigated and the critical radius below which drops vanish is drastically reduced, allowing stable simulation of thin tubes and pinch-off where the classical ACH collapses.
  • The model preserves energy dissipation for the classical anisotropic surface energy, so the improved conservation does not come at the cost of thermodynamic consistency.
  • For isotropic energy with $k=1$ and $l=2$, the ACH-IC recovers the second-order variational Cahn-Hilliard model of Ref. [7], placing that earlier model inside the new variational framework.
  • With a Willmore-type regularization for strongly anisotropic energies, the volume-conservation and approximation properties persist, with only the normal-velocity relation modified.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same device, conserving a steeper approximation of the step function instead of the phase variable, could be applied to other volume-preserving phase-field dynamics such as two-phase flow or solid-state dewetting, where first-order volume drift is a known limitation.
  • Because the $O(\varepsilon^2)$ theorem assumes $C^2$ regularity for all times, the good pinch-off behavior shown numerically is not covered by the proof; whether it holds uniformly through topological change is an open question that can be settled by the numerical convergence test proposed above.
  • The paper does not locate the ACH-IC critical radius analytically; measuring it as a function of $k$ and $l$ would give a practical design rule for choosing the model parameters.
  • The volume bound is proved for the difference $|\Omega_\varepsilon^+(t)| - |\Omega_\varepsilon^+(0)|$; a stronger statement about monotone volume loss at intermediate times, or about the volume of each connected component after pinch-off, is not established.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a variational phase-field model (ACH-IC) for anisotropic surface diffusion in which a generalized conserved quantity Q(u)=∫_0^u(1−z^2)^k dz / ∫_0^1(1−z^2)^k dz replaces u in the conservation constraint. The model is derived from Onsager's variational principle, and matched asymptotic expansions are used to show that the zero level set evolves according to V = C_l Δ_s(∇_s·ξ(ν)) + O(ε) and that the enclosed volume satisfies |Ω_ε^+(t)| = |Ω_ε^+(0)| + O(ε^2), i.e., second-order rather than first-order volume conservation. The paper supports these claims with a coarea-formula proof and with numerical experiments in two and three dimensions, including spontaneous-shrinkage, flower-shape, strongly anisotropic, and pinch-off examples.

Significance. If the main theorem holds on the claimed range of applicability, the ACH-IC model is a substantial improvement over classical anisotropic Cahn-Hilliard models: it preserves the variational/energy-dissipation structure while reducing volume drift from O(ε) to O(ε^2), and the numerics suggest that it permits larger ε for similar accuracy. The variational derivation via an improved conserved quantity is general and could be transferred to other phase-field settings. The matched asymptotic analysis is detailed and internally coherent, and the mobility constant C_l is derived rather than fitted. The paper also includes careful convergence tables and comparisons against a sharp-interface parametric FEM. However, the second-order conservation theorem is proved under regularity and uniformity assumptions that are not met in the pinch-off regime advertised in Section 4.6, and the strong-anisotropy case is asserted rather than proved; these gaps do not undermine the core model but require a substantial restatement of the theorem.

major comments (4)
  1. [Section 3.4, Eqs. (42)–(46)] The proof of Property 3 uses the first-variation expansion h(ζ)=h(0)+ζh'(0)+O(ζ^2) and, uniformly for ζ∈[0,|log ε|], writes h(εζ)−h(−εζ)=2ζεh'(0)+O(ε^2ζ^2) with a constant C1 independent of ζ. For a C^2 interface whose minimum radius of curvature is O(ε), the remainder in the first-variation formula is O(εζ^2), not O(ε^2ζ^2), because the constant in the O(ζ^2) term depends on curvature. Since C^2 regularity alone does not exclude curvature blow-up as ε→0, the estimate (43) and hence Eq. (18) are not uniform in interface geometry. The hypothesis in Property 3 must be strengthened (e.g., a uniform curvature bound) or the theorem must be restricted to time intervals on which curvature is O(1).
  2. [Section 3.2 and Section 4.6] The matched asymptotic expansion (16) relies on local normal coordinates in which ερ κ_i is treated as small, as seen in the expansions following Eq. (25). When a tube neck reaches radius O(ε), the principal curvature is O(1/ε) and ερ κ_i is not uniformly small for ρ up to O(|log ε|); hence Eq. (16) and the subsequent volume estimate are not valid in the pinch-off regime. Section 4.6 uses exactly this regime to advertise the model's conservation properties, so the theoretical guarantee stated in Property 3 does not cover the motivating application; a separate near-pinch analysis or an explicit exclusion of this regime is needed.
  3. [Section 2.2 and Appendix B, Remark B.3] The asymptotic analysis in Section 3 uses the unregularized N(u)=1/Q'(u) and assumes u≠±1 throughout, while the numerical scheme does not enforce |u|≤1 and no maximum principle or a priori bound is proved. The regularized Ñ(u) is introduced in Section 2.2 to avoid the singularity, but the matched asymptotics are not redone for the regularized model. Consequently Properties 1–3 are conditional on a solution regularity that is not established as part of the theorem; this should be stated explicitly among the hypotheses of Property 3.
  4. [Section 2.3 and Section 4.2] The proof in Section 3.4 is carried out for the unregularized model, whereas the paper states in Section 2.3 that in the strongly anisotropic case with Willmore regularization all properties except the normal velocity are retained 'by similar arguments.' No such analysis is supplied. Since Table 2 reports second-order volume conservation for α=0.2 with β=ε^2, the numerical claim for strong anisotropy is not covered by the theorem as proved and should be either proved or explicitly labeled as a numerical observation.
minor comments (6)
  1. [Eq. (5) and Eq. (7)] The phase field u is scalar, but the text writes u ∈ R^d in the display of the energy functional; this should be corrected to u ∈ R or u: Ω→[−1,1].
  2. [References] Reference [57] is listed as J. Comput. Phys. 223 (2017) 1–9; the actual publication year is 2007, so the citation should be corrected.
  3. [Section 2.2] The sentence 'The function 1/M(u) plays a role of friction coefficient' should read 'plays the role of a friction coefficient.'
  4. [Figure 9 caption] The caption contains the typo 'Sharp-inerface model'; it should be 'Sharp-interface model.'
  5. [Section 4.6] The statement that 'the thickness of the thin tube must be greater than 2ε when the ACH model is employed' appears to be an observation from a single simulation; it should be qualified or supported by a systematic study.
  6. [Section 3.3] The notation (N μ̄)_k is introduced in a parenthetical remark and is easy to miss; it would be clearer as a displayed definition before Eq. (29).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the O(ε²) volume-conservation result is proved from the exact conservation law for Q and matched-asymptotic expansions, not assumed or fitted.

full rationale

The paper's central claims—uniform expansion (16), sharp-interface velocity (17), and second-order volume conservation (18)—are derived in Section 3 via matched asymptotic expansions from the model (15). The conserved quantity Q is chosen in Section 2.2 so that Q(u) approximates the step function faster than u, but the O(ε²) bound in Section 3.4 is then established by a coarea-formula argument using the exact identity ∫Q(uε)dx = const, the first-variation expansion h(ζ) = h(0) + ζh'(0) + O(ζ²), and exponential decay of 1−Q(σ(ζ)); the proof is not an assumption of the conclusion. The mobility coefficient Cl is computed from an integral of M(U0), not fitted. The recovery of Bretin's model in the isotropic quadratic-mobility case is explicitly disclosed and does not smuggle in the result. Self-citations appear only as background or numerical-benchmark references and are not load-bearing for the asymptotic derivation. Concerns about the C2 regularity hypothesis in Property 3 (e.g., possible curvature blow-up near pinch-off) are legitimate rigor/completeness issues but are not circularity: the theorem is conditional on a regularity assumption and does not reduce to its inputs by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The ledger lists the hand-chosen parameters (k, l, α, β) and the regularity, parameter-range, and expansion assumptions used in the derivation. There are no fitted constants in the usual sense; the model parameters are selected by hand and not calibrated to data.

free parameters (4)
  • k (exponent in conserved quantity Q) = 1 (numerics)
    Defines Q(u)=(∫_0^u (1−z^2)^k dz)/(∫_0^1 (1−z^2)^k dz); chosen by hand, central to the improved step-function approximation. The condition 1≤k≤l≤2k+1 is required.
  • l (mobility exponent) = 2 (numerics)
    Defines mobility M(u)=(1−u^2)^l; controls the mobility coefficient C_l=2/3∫(1−z^2)^{l−1}dz and the analysis condition (13).
  • α (anisotropy strength) = 0.05 (weak), 0.2 (strong)
    Parameter in the four-fold anisotropic energy (49); set for numerical tests, not fitted to data.
  • β (Willmore regularization strength) = ε^2 (strong anisotropy tests)
    Regularization coefficient for non-convex anisotropy; chosen by hand for numerical experiments.
assumptions (5)
  • domain assumption The interface Γ_ε(t) is a smooth (C^2) closed hypersurface for all times in the asymptotic analysis.
    Stated in Section 2.3 property 3; used in Section 3.4 via the expansion h(ζ)=h(0)+ζh'(0)+O(ζ^2) from the first variation of perimeter. Fails during pinch-off.
  • domain assumption The phase variable u stays away from ±1 in the asymptotic expansions, so N(u)=1/Q'(u) is nonsingular.
    Section 2.2: 'under the assumption that u≠−1,1 in the expansions'; the regularization Ñ is introduced but not used in the formal calculations.
  • ad hoc to paper The parameters satisfy 1 ≤ k ≤ l ≤ 2k+1 and l−2k < 1+1/w so the outer expansion closes.
    Condition (13) and surrounding text; restricts the family of mobilities to make the leading-order system elliptic.
  • standard math Standard matched asymptotic expansion machinery: outer/inner expansions, matching of flux, and solvability conditions are valid.
    Used throughout Section 3; no convergence proof is supplied for the formal expansions.
  • domain assumption The Torabi-type energy (5) with regularized normal vector n̂ is the correct phase-field approximation of the anisotropic surface energy (2).
    Section 1 and Eq. (5); the whole model is built on this energy, and results depend on its form.
invented entities (1)
  • Conserved quantity Q(u)
    purpose: Replaces u as the phase-field variable whose integral is conserved, so that ∫Q(u) approximates the volume of {u>0} more accurately.
    Q is constructed explicitly from (1−z^2)^k with normalization; it has no observable handle outside the model. Its benefit is internal, measured by the asymptotic and numerical analysis.

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Pith. "Pith review of A new Phase-Field Model for Anisotropic Surface Diffusion: Anisotropic Cahn-Hilliard Equation with Improved Conservation (ACH-IC)." pith.science (2026). https://pith.science/paper/SFO6J6P2

@misc{pith2026250718048,
  author       = {Pith},
  title        = {Pith review of: A new Phase-Field Model for Anisotropic Surface Diffusion: Anisotropic Cahn-Hilliard Equation with Improved Conservation (ACH-IC)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SFO6J6P2}},
  note         = {Machine review of arXiv:2507.18048}
}
read the original abstract

As popular approximations to sharp-interface models, the Cahn-Hilliard type phase-field models are usually used to simulate interface dynamics with volume conservation. However, the convergence rate of the volume enclosed by the interface to its sharp-interface limit is usually at first order of the interface thickness in the classical Cahn-Hilliard model with constant or degenerate mobilities. In this work, we propose a variational framework for developing new Cahn-Hilliard dynamics with enhanced volume conservation by introducing a more general conserved quantity. In particular, based on Onsager's variational principle (OVP) and a modified conservation law, we develop an anisotropic Cahn-Hilliard equation with improved conservation (ACH-IC) for approximating anisotropic surface diffusion. The ACH-IC model employs a new conserved quantity that approximates a step function more effectively, and yields second-order volume conservation while preserving energy dissipation for the classical anisotropic surface energy. The second-order volume conservation as well as the convergence to the sharp-interface surface diffusion dynamics is derived through comprehensive asymptotic analysis. Numerical evidence not only reveals the underlying physics of the proposed model in comparison with the classical one, but also demonstrates its exceptional performance in simulating anisotropic surface diffusion dynamics.

Figures

Figures reproduced from arXiv: 2507.18048 by the authors.

Figure 1
Figure 1. Q(u) for different k, when u = tanh  dist(x,∂Ω +) √ 2ε  with ε = 0.05. The function 1/M(u) plays a role of friction coefficient, measuring the hardness for atoms to move in the bulk phases (where u = ±1). For example, when we take M(u) = (1 − u 2 ) l with l > 1, the dissipation functional Φ is consistent with the ones in Refs. [35, 25]. High-order degeneracy in M(u) (i.e., l > 2) was typically required to achieve … view at source ↗
Figure 2
Figure 2. Equilibrium shapes obtained by the ACH-IC (solid line) and ACH [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Equilibrium shapes obtained by the ACH-IC (solid line) and ACH [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: (a) Comparisons of equilibrium shapes obtained by the ACH- [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: (a) Comparisons of the interface profiles obtained by the [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]
Figure 6
Figure 6. Figure 6: (a) and (b) show the flux J distribution for the ACH and ACH-IC models at an early stage of evolution (t = 2 × 10−5 ) in the isotropic case. The arrows indicate the direction and magnitude of the flux, while the black solid line represents the interface. (c) and (d) sh…
Figure 7
Figure 7. Figure 7: (a) and (b) show the flux J distribution for the ACH and ACH-IC models at a later stage of evolution (t = 2 × 10−4 ) in the isotropic case. The arrows indicate the direction and magnitude of the flux, while the black solid line represents the interface. (c) and (d) sho…
Figure 8
Figure 8. Figure 8: Comparisons about the evolution of an initially “flower” shap [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: comparison in energy evolution of an initially “flower” shape t [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: Several snapshots depicting the simulation of the pinch- [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · cited by 1 Pith paper

  1. [1]

    Geometric evolution problems, distance function and viscosity solutions

    Luigi Ambrosio, Norman Dancer, and L Ambrosio. Geometric evolution problems, distance function and viscosity solutions . Springer, 2000

    work page 2000

  2. [2]

    Convexity splitting in a phase field model for surface diffusion

    Rainer Backofen, Steven M Wise, Marco Salvalaglio, and Axel Voigt . Convexity splitting in a phase field model for surface diffusion. Int. J. Numer. Anal. Model. , 16:192–209, 2019

    work page 2019

  3. [3]

    Srolovitz, and Yan Wang

    Weizhu Bao, Wei Jiang, David J. Srolovitz, and Yan Wang. Stable eq uilibria of anisotropic particles on substrates: A generalized Winterbottom construction. SIAM J. Appl. Math. , 77(6):2093–2118, 2017

    work page 2017

  4. [4]

    A parametric fin ite element method for solid-state dewetting problems with anisotropic surface energies

    Weizhu Bao, Wei Jiang, Yan Wang, and Quan Zhao. A parametric fin ite element method for solid-state dewetting problems with anisotropic surface energies. J. Comput. Phys. , 330:380–400, 2017

    work page 2017

  5. [5]

    Numer ical approximation of anisotropic geo- metric evolution equations in the plane

    John W Barrett, Harald Garcke, and Robert N¨ urnberg. Numer ical approximation of anisotropic geo- metric evolution equations in the plane. IMA J. Numer. Anal. , 28(2):292–330, 2008

    work page 2008

  6. [6]

    Stress-induced acceleration and ordering in solid-state dew etting

    Francesco Boccardo, Fabrizio Rovaris, Ashwani Tripathi, Fran cesco Montalenti, and Olivier Pierre- Louis. Stress-induced acceleration and ordering in solid-state dew etting. Phys. Rev. Lett., 128(2):026101, 2022. 29

    work page 2022

  7. [7]

    Appr oximation of surface diffusion flow: A second-order variational Cahn-Hilliard model with degenerate mo bilities

    Elie Bretin, Simon Masnou, Arnaud Sengers, and Garry Terii. Appr oximation of surface diffusion flow: A second-order variational Cahn-Hilliard model with degenerate mo bilities. Math. Mod. Meth. Appl. S., 32(04):793–829, 2022

    work page 2022

  8. [8]

    The Cahn–H illiard equation with a concen- tration dependent mobility: motion by minus the laplacian of the mean c urvature

    John W Cahn, Charles M Elliott, and Amy Novick-Cohen. The Cahn–H illiard equation with a concen- tration dependent mobility: motion by minus the laplacian of the mean c urvature. Eur. J. Appl Math. , 7(3):287–301, 1996

    work page 1996

Show all 60 references
  1. [9]

    Efficient energy stable schemes with spec tral discretization in space for anisotropic Cahn-Hilliard systems

    Feng Chen and Jie Shen. Efficient energy stable schemes with spec tral discretization in space for anisotropic Cahn-Hilliard systems. Commun. Comput. Phys. , 13(5):1189–1208, 2013

    2013

  2. [10]

    Coarsening mechanism for systems gove rned by the Cahn–Hilliard equation with degenerate diffusion mobility

    Shibin Dai and Qiang Du. Coarsening mechanism for systems gove rned by the Cahn–Hilliard equation with degenerate diffusion mobility. Multiscale Model. Sim. , 12(4):1870–1889, 2014

    2014

  3. [11]

    Computat ion of geometric partial differential equations and mean curvature flow

    Klaus Deckelnick, Gerhard Dziuk, and Charles M Elliott. Computat ion of geometric partial differential equations and mean curvature flow. Acta Numer. , 14:139–232, 2005

    2005

  4. [12]

    Onsager’s variational principle in soft matter

    Masao Doi. Onsager’s variational principle in soft matter. J. Phys. Condens. Matter , 23(28):284118, 2011

    2011

  5. [13]

    Application of the O nsager-Machlup integral in solving dynamic equations in nonequilibrium systems

    Masao Doi, Jiajia Zhou, Yana Di, and Xianmin Xu. Application of the O nsager-Machlup integral in solving dynamic equations in nonequilibrium systems. Phys. Rev. E , 99(6):063303, 2019

    2019

  6. [14]

    Recent developments in the theory of sur face diffusion

    JD Doll and AF Voter. Recent developments in the theory of sur face diffusion. Annu. Rev. Phys. Chem. , 38(1):413–431, 1987

    1987

  7. [15]

    An aniso tropic phase-field model for solid-state dewetting and its sharp-interface limit

    Marion Dziwnik, Andreas M¨ unch, and Barbara Wagner. An aniso tropic phase-field model for solid-state dewetting and its sharp-interface limit. Nonlinearity, 30(4):1465, 2017

    2017

  8. [16]

    Surface diffusion

    Gert Ehrlich and Kaj Stolt. Surface diffusion. Annu. Rev. Phys. Chem. , 31(1):603–637, 1980

    1980

  9. [17]

    A diffuse-interface approach for solid-state dewetting with anisotropic surface energies

    Harald Garcke, Patrik Knopf, Robert N¨ urnberg, and Quan Zh ao. A diffuse-interface approach for solid-state dewetting with anisotropic surface energies. J. Nonlinear Sci. , 33(2):34, 2023

    2023

  10. [18]

    Phase-field investigation of the coarsening of porous structures by surface diffusion

    Pierre-Antoine Geslin, Mickael Buchet, Takeshi Wada, and Hide mi Kato. Phase-field investigation of the coarsening of porous structures by surface diffusion. Phys. Rev. Mater. , 3(8):083401, 2019

    2019

  11. [19]

    Comparison of phase-field models for surface diffusion

    Clemens Gugenberger, Robert Spatschek, and Klaus Kassner . Comparison of phase-field models for surface diffusion. Phys. Rev. E , 78(1):016703, 2008

    2008

  12. [20]

    A vector thermodynamics for a nisotropic surfaces: I

    David W Hoffman and John W Cahn. A vector thermodynamics for a nisotropic surfaces: I. fundamentals and application to plane surface junctions. Surf. Sci. , 31:368–388, 1972

    1972

  13. [21]

    Jiang, Q

    W. Jiang, Q. Zhao, and W. Bao. Sharp-interface model for simu lating solid-state dewetting in three dimensions. SIAM J. Appl. Math. , 80:1654–1677, 2020

    2020

  14. [22]

    P hase field approach for simulating solid-state dewetting problems

    Wei Jiang, Weizhu Bao, Carl V Thompson, and David J Srolovitz. P hase field approach for simulating solid-state dewetting problems. Acta Mater., 60(15):5578–5592, 2012

    2012

  15. [23]

    A perimeter-decreasing and area-con serving algorithm for surface diffusion flow of curves

    Wei Jiang and Buyang Li. A perimeter-decreasing and area-con serving algorithm for surface diffusion flow of curves. J. Comput. Phys. , 443:110531, 2021

    2021

  16. [24]

    Sharp-interface approach for simula ting solid-state dewetting in two dimen- sions: A Cahn–Hoffman ξ-vector formulation

    Wei Jiang and Quan Zhao. Sharp-interface approach for simula ting solid-state dewetting in two dimen- sions: A Cahn–Hoffman ξ-vector formulation. Phys. D , 390:69–83, 2019

    2019

  17. [25]

    Application of Onsager’s variational principle to the dynamics of a solid toroidal island on a subs trate

    Wei Jiang, Quan Zhao, Tiezheng Qian, David J Srolovitz, and Weizh u Bao. Application of Onsager’s variational principle to the dynamics of a solid toroidal island on a subs trate. Acta Mater., 163:154–160, 2019. 30

    2019

  18. [26]

    Degenerate mob ilities in phase field models are insuffi- cient to capture surface diffusion

    Alpha A Lee, Andreas M¨ unch, and Endre S¨ uli. Degenerate mob ilities in phase field models are insuffi- cient to capture surface diffusion. Appl. Phys. Lett. , 107(8):081603, 2015

    2015

  19. [27]

    A numerical study of ele ctro-migration voiding by evolving level set functions on a fixed cartesian grid

    Zhilin Li, Hongkai Zhao, and Huajian Gao. A numerical study of ele ctro-migration voiding by evolving level set functions on a fixed cartesian grid. J. Comput. Phys. , 152(1):281–304, 1999

    1999

  20. [28]

    A note on a phase-field model for anisotropic sy stems

    Luca Lussardi. A note on a phase-field model for anisotropic sy stems. Asymptotic Analysis, 94:241–254, 2015

    2015

  21. [29]

    Sets of finite perimeter and geometric variational problems : an introduction to Geo- metric Measure Theory

    Francesco Maggi. Sets of finite perimeter and geometric variational problems : an introduction to Geo- metric Measure Theory . Cambridge University Press, 2012

    2012

  22. [30]

    Existence of solutions for anisot ropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions

    Ahmad Makki and Alain Miranville. Existence of solutions for anisot ropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete Cont. Dyn-S , 9(3):759, 2016

    2016

  23. [31]

    Un esempio di γ-convergenza

    Modica and Mortola. Un esempio di γ-convergenza. Boll. Un. Mat. Ital. A , 14:526, 1977

    1977

  24. [32]

    Theory of thermal grooving

    William W Mullins. Theory of thermal grooving. J. Appl. Phys. , 28(3):333–339, 1957

    1957

  25. [33]

    Reciprocal relations in irreversible processes

    Lars Onsager. Reciprocal relations in irreversible processes. I. Phys. Rev. , 37(4):405, 1931

    1931

  26. [34]

    Reciprocal relations in irreversible processes

    Lars Onsager. Reciprocal relations in irreversible processes. II. Phys. Rev. , 38(12):2265, 1931

    1931

  27. [35]

    A variational app roach to moving contact line hydrodynamics

    Tiezheng Qian, Xiao-Ping Wang, and Ping Sheng. A variational app roach to moving contact line hydrodynamics. J. Fluid Mech. , 564:333–360, 2006

    2006

  28. [36]

    Surface evolut ion of elastically stressed films under deposition by a diffuse interface model

    Andreas R¨ atz, Angel Ribalta, and Axel Voigt. Surface evolut ion of elastically stressed films under deposition by a diffuse interface model. J. Comput. Phys. , 214(1):187–208, 2006

    2006

  29. [37]

    Faceting of equilibrium and metastable nanostructures: a phase-fi eld model of surface diffusion tackling realistic shapes

    Marco Salvalaglio, Rainer Backofen, Roberto Bergamaschini, Fr ancesco Montalenti, and Axel Voigt. Faceting of equilibrium and metastable nanostructures: a phase-fi eld model of surface diffusion tackling realistic shapes. Cryst. Growth Des. , 15(6):2787–2794, 2015

    2015

  30. [38]

    Morphological evolution of pit-patterned si (001) substrates driven by surface-energy r eduction

    Marco Salvalaglio, Rainer Backofen, Axel Voigt, and Francesco Montalenti. Morphological evolution of pit-patterned si (001) substrates driven by surface-energy r eduction. Nanoscale Res. Lett., 12:1–8, 2017

    2017

  31. [39]

    Phase-field simulations of faceted Ge/Si-crystal arrays, me rging into a suspended film

    Marco Salvalaglio, Roberto Bergamaschini, Rainer Backofen, Ax el Voigt, Francesco Montalenti, and Leo Miglio. Phase-field simulations of faceted Ge/Si-crystal arrays, me rging into a suspended film. Appl. Surf. Sci. , 391:33–38, 2017

    2017

  32. [40]

    D oubly degenerate diffuse inter- face models of anisotropic surface diffusion

    Marco Salvalaglio, Maximilian Selch, Axel Voigt, and Steven M Wise. D oubly degenerate diffuse inter- face models of anisotropic surface diffusion. Math. Method. Appl. Sci. , 44(7):5406–5417, 2021

    2021

  33. [41]

    Doubly degener ate diffuse interface models of surface diffusion

    Marco Salvalaglio, Axel Voigt, and Steven M Wise. Doubly degener ate diffuse interface models of surface diffusion. Math. Method. Appl. Sci. , 44(7):5385–5405, 2021

    2021

  34. [42]

    Stabilized predictor-corrector schemes fo r gradient flows with strong anisotropic free energy

    Jie Shen and Jie Xu. Stabilized predictor-corrector schemes fo r gradient flows with strong anisotropic free energy. Commun. Comput. phys. , 2018

    2018

  35. [43]

    A new class of efficient and robus t energy stable schemes for gradient flows

    Jie Shen, Jie Xu, and Jiang Yang. A new class of efficient and robus t energy stable schemes for gradient flows. SIAM Rev. , 61(3):474–506, 2019

    2019

  36. [44]

    Interfaces in crystalline materials

    Adrian P Sutton. Interfaces in crystalline materials. Monographs on the Physice and Chemistry of Materials, pages 414–423, 1995

    1995

  37. [45]

    Linking anisotropic sharp and diffu se surface motion laws via gradient flows

    Jean E Taylor and John W Cahn. Linking anisotropic sharp and diffu se surface motion laws via gradient flows. J. Stat. Phys. , 77:183–197, 1994. 31

    1994

  38. [46]

    A new phase-field model for strongly anisotropic systems

    Solmaz Torabi, John Lowengrub, Axel Voigt, and Steven Wise. A new phase-field model for strongly anisotropic systems. Proc. R. Soc. A , 465(2105):1337–1359, 2009

    2009

  39. [47]

    Voorhees

    Peter W. Voorhees. The theory of ostwald ripening. J. Stat. Phys. , 38(1):231–252, 1985

    1985

  40. [48]

    Onsager’s variationa l principle in active soft matter

    Haiqin Wang, Tiezheng Qian, and Xinpeng Xu. Onsager’s variationa l principle in active soft matter. Soft Matter , 17(13):3634–3653, 2021

    2021

  41. [49]

    Cahn–Hoffman ξ-vector and its relation to diffuse interface models of phase transit ions

    AA Wheeler. Cahn–Hoffman ξ-vector and its relation to diffuse interface models of phase transit ions. J. Stat. Phys. , 95:1245–1280, 1999

    1999

  42. [50]

    Solving the reg ularized, strongly anisotropic Cahn– Hilliard equation by an adaptive nonlinear multigrid method

    Steven Wise, Junseok Kim, and John Lowengrub. Solving the reg ularized, strongly anisotropic Cahn– Hilliard equation by an adaptive nonlinear multigrid method. J. Comput. Phys. , 226(1):414–446, 2007

    2007

  43. [51]

    G. Wulff. Zur frage der geschwindigkeit des wachstums und dera ufl¨ osung der krystallfl¨ achen.Z. Kristal- logr, 34(1):449–530, 1901

    1901

  44. [52]

    A finite element analysis of t he motion and evolution of voids due to strain and electromigration induced surface diffusion

    L Xia, Allan F Bower, Z Suo, and CF Shih. A finite element analysis of t he motion and evolution of voids due to strain and electromigration induced surface diffusion. J. Mech. Phys. Solids , 45(9):1473–1493, 1997

    1997

  45. [53]

    Anisotropic surfac e diffusion of CO on Ni (110)

    Xu-Dong Xiao, XD Zhu, W Daum, and YR Shen. Anisotropic surfac e diffusion of CO on Ni (110). Phys. Rev. Lett. , 66(18):2352, 1991

    1991

  46. [54]

    Efficient and lin ear schemes for anisotropic Cahn– Hilliard model using the stabilized-invariant energy quadratization (S -IEQ) approach

    Zhen Xu, Xiaofeng Yang, Hui Zhang, and Ziqing Xie. Efficient and lin ear schemes for anisotropic Cahn– Hilliard model using the stabilized-invariant energy quadratization (S -IEQ) approach. Comput. Phys. Commun., 238:36–49, 2019

    2019

  47. [55]

    Xiaofeng Yang. Efficient and energy stable scheme for the hydr odynamically coupled three components Cahn-Hilliard phase-field model using the stabilized-invariant energy quadratization (S-IEQ) approach. Journal of Computational Physics , 438:110342, 2021

    2021

  48. [56]

    Numerical approximation s for the molecular beam epitaxial growth model based on the invariant energy quadratization metho d

    Xiaofeng Yang, Jia Zhao, and Qi Wang. Numerical approximation s for the molecular beam epitaxial growth model based on the invariant energy quadratization metho d. J. Comput. Phys. , 333:104–127, 2017

    2017

  49. [57]

    Pengtao Yue, Chunfeng Zhou, and James J. Feng. Spontaneo us shrinkage of drops and mass conservation in phase-field simulations. J. Comput. Phys. , 223(1):1–9, 2017

    2017

  50. [58]

    Variational approach to droplet transport via bendotaxis: thin film dynamics and model reduction

    Zhen Zhang and Tiezheng Qian. Variational approach to droplet transport via bendotaxis: thin film dynamics and model reduction. Phys. Rev. Fluids , 7(4):044002, 2022

    2022

  51. [59]

    A parametric finite elemen t method for solid-state dewetting problems in three dimensions

    Quan Zhao, Wei Jiang, and Weizhu Bao. A parametric finite elemen t method for solid-state dewetting problems in three dimensions. SIAM J. Sci. Comput. , 42(1):B327–B352, 2020

    2020

  52. [60]

    Dynamics of small solid particles on substrates of arbitrary topography

    Quan Zhao, Wei Jiang, Yan Wang, David J Srolovitz, Tiezheng Qian , and Weizhu Bao. Dynamics of small solid particles on substrates of arbitrary topography. Acta Mater., 281:120407, 2024. 32

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.