Modified Wulff Shape: Generalized Equilibrium

• Modified Wulff shape is a family of equilibrium constructions derived by perturbing the classical Wulff variational problem with altered support functions and additional curvature or external-field terms.
• It plays a crucial role in modeling anisotropic interfaces in crystallization, phase transitions, and liquid-drop energies across both continuum and discrete settings.
• Modifications include perturbative adjustments, added bending energies, and discrete or thermodynamic constraints that yield a variety of equilibrium geometries.

Searching arXiv for relevant papers on modified Wulff shapes, anisotropic Wulff constructions, and related variational/crystallization models. Modified-Wulff shape denotes a family of equilibrium-shape constructions obtained by modifying the classical Wulff variational problem. The common reference point is the classical Wulff body, defined by an orientation-dependent support function and characterized as the unique convex minimizer of anisotropic surface free energy at fixed volume. What is “modified” varies across the literature: the support function may be perturbed, higher-order curvature or external-field terms may be added, the effective support function may be extracted from a finite-$N$ statistical ensemble rather than prescribed a priori, or additional nonlocal and discrete constraints may alter the macroscopic minimizer. Taken together, these works indicate that the term does not designate a single universal object, but a class of generalized Wulff constructions adapted to anisotropic curvature problems, phase-equilibrium models, liquid-drop energies, and discrete crystallization limits (Miracle-Sole, 2013, Radin, 2016).

1. Classical baseline and support-function geometry

The classical starting point is Gibbs’ variational principle for a crystal region $B\subset \mathbb R^d$ of fixed volume. If $\tau(n)$ is the orientation-dependent surface tension, the total surface free energy is

$F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$

and equilibrium shapes minimize this functional under the constraint $|B|=V$. The corresponding Wulff shape is

$$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$

equivalently the convex body whose support function is $\tau$. In the formulation of Miracle-Solé, $W$ is the unique minimizer, up to translation, and $\tau$ coincides with the support function of $W$ (Miracle-Sole, 2013).

The smooth Euler–Lagrange condition for this classical problem is that the anisotropic mean curvature be constant: $B\subset \mathbb R^d$0 Thus the Wulff construction is not only a convex-geometric intersection of half-spaces, but also the geometric solution of a constant-anisotropic-mean-curvature problem.

A closely related smooth anisotropic formulation appears in Roth’s setting. For a smooth positive function $B\subset \mathbb R^d$1, the Wulff shape is

$B\subset \mathbb R^d$2

If

$B\subset \mathbb R^d$3

as a quadratic form on each tangent space, then $B\subset \mathbb R^d$4 is a smooth, strictly convex body. This formulation is the anisotropic differential-geometric analogue of the classical support-function construction (Roth, 2015).

2. Perturbative modifications and generalized equilibrium laws

The most direct modified-Wulff construction replaces the classical support function by a perturbed one. In the purely anisotropic case,

$B\subset \mathbb R^d$5

and the modified Wulff shape is defined by

$B\subset \mathbb R^d$6

In this setting the support function is simply

$B\subset \mathbb R^d$7

and the usual convex-geometric argument carries over: $B\subset \mathbb R^d$8 is convex and minimizes the perturbed anisotropic surface energy under fixed volume. Its anisotropic curvature satisfies

$B\subset \mathbb R^d$9

(Miracle-Sole, 2013).

More elaborate modifications leave the support-function framework only partially intact. If one adds a bending term and an external field, the functional becomes, schematically,

$\tau(n)$0

and the Euler–Lagrange equation becomes the generalized Young–Laplace law

$\tau(n)$1

In this regime the minimizer can no longer, in general, be represented purely as an intersection of linear half-spaces. The expression “modified Wulff shape” then refers to the unique solution of the full variational problem, convex if bending is small and perturbatively close to the classical Wulff body in weak-coupling limits (Miracle-Sole, 2013).

A standard illustrative case is the two-dimensional perturbation

$\tau(n)$2

The corresponding modified Wulff curve satisfies

$\tau(n)$3

and to first order

$\tau(n)$4

This yields the familiar four-petal perturbation of the circle (Miracle-Sole, 2013).

3. Anisotropic curvature functionals and stability of the Wulff shape

In Roth’s anisotropic curvature framework, modified-Wulff geometry is encoded through the $\tau(n)$5-Weingarten map

$\tau(n)$6

whose eigenvalues $\tau(n)$7 are the anisotropic principal curvatures. Their elementary symmetric polynomials define the higher-order anisotropic mean curvatures: $\tau(n)$8 For $\tau(n)$9, $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$0 reduces to the classical mean curvature (Roth, 2015).

The associated variational problem is built from the anisotropic $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$1-area functionals

$F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$2

and their positive linear combination

$F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$3

for $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$4 and nonnegative constants $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$5, not all zero. Under a volume constraint, the Euler–Lagrange equation is

$F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$6

Thus the critical hypersurfaces are exactly those for which a linear combination of anisotropic higher-order mean curvatures is constant (Roth, 2015).

The stability statement is particularly rigid. A critical immersion is $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$7-stable if the second variation of $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$8 is nonnegative on all volume-preserving normal variations. Roth proves that, under the convexity assumption $F_0(B)=\int_{\partial B}\tau(n(s))\,dS(s),$9, any closed, $|B|=V$0-stable critical immersion in $|B|=V$1 is, up to translation and homothety, the Wulff shape $|B|=V$2. The proof uses the special test function

$|B|=V$3

anisotropic Hsiung–Minkowski identities, and Newton-type inequalities. Stability forces equality in the anisotropic Newton inequalities, hence equality of the anisotropic principal curvatures at every point, so the hypersurface is umbilic in the $|B|=V$4-sense and therefore a homothetic image of $|B|=V$5 (Roth, 2015).

This result also clarifies a frequent source of confusion. A modified anisotropic curvature functional does not generically produce a large family of stable closed critical hypersurfaces. In the convex setting treated by Roth, stability collapses the moduli to the Wulff shape itself, up to translations and homotheties (Roth, 2015).

4. Thermodynamic and phase-equilibrium interpretations

Radin’s “modified” Wulff shape shifts the emphasis from prescribing a macroscopic surface tension to deriving an effective directional surface cost from a microscopic pressure–temperature ensemble. The microscopic model uses $|B|=V$6 point particles in $|B|=V$7 with pair potential

$|B|=V$8

For a fixed boundary shape $|B|=V$9, one considers the usual pressure–temperature ensemble with partition function

$$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$0

and the average total energy $$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$1. The shape $$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$2 is allowed to vary over a compact quotient space of star-shaped regions of fixed volume, and finite-$$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$3 minimizers exist by continuity and compactness. Accumulation points of these minimizers as $$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$4 are called the Wulff shapes for the thermodynamic parameters $$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$5 (Radin, 2016).

In this construction the classical support function $$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$6 is replaced by a directional excess surface-energy function

$$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$7

recovered implicitly from the finite-$$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$8 ensemble. The paper proves existence of finite-$$W=\bigcap_{n\in S^{d-1}}\{x\in\mathbb R^d:x\cdot n\le \tau(n)\},$$9 minimizers and of thermodynamic accumulation points, while uniqueness away from coexistence, the sphere in the pure fluid regime, the polyhedral crystal in the high-pressure regime, and facet regularity are described as expectations or open problems. In $\tau$0, whether the high-pressure Wulff shape is the regular hexagon remains open in this model, apart from the degenerate $\tau$1 “soft disks” limit (Radin, 2016).

A computationally explicit realization of modified Wulff behavior appears in the phase-field crystal treatment of the bcc–liquid interface. There the interface free energy $\tau$2 is obtained from the Euler–Lagrange equation of the PFC free-energy functional, evaluated for 18 orientations and fitted by an eight-term Kubic-harmonic expansion. The resulting Wulff construction is

$\tau$3

and the equilibrium shape evolves with reduced temperature $\tau$4: as $\tau$5, $\tau$6 becomes isotropic and the shape is a sphere; at small $\tau$7 the anisotropy is weak with maximum energy at $\tau$8 and minimum at $\tau$9; near $W$0 the minimum switches from $W$1 to $W$2; and at larger $W$3 the Wulff shape becomes a strongly faceted polyhedron dominated by $W$4 and $W$5 faces. The paper emphasizes that this equilibrium polyhedron differs from the rhombo-dodecahedron obtained in earlier finite-seed growth simulations (Podmaniczky et al., 2014).

5. Nonlocal energies and anisotropic liquid drops

In anisotropic liquid-drop models, the modified-Wulff problem couples an anisotropic perimeter to a repulsive nonlocal term. For a one-homogeneous convex surface tension $W$6, the anisotropic perimeter is

$W$7

and with the Riesz interaction

$W$8

the unit-volume energy is

$W$9

The Euler–Lagrange equation for a critical set is

$\tau$0

where $\tau$1 is the anisotropic mean curvature (Misiats et al., 2019).

Two distinct regimes are important. For smooth elliptic anisotropies, Choksi, Neumayer, and Topaloglu show that Wulff shapes are minimizers in the anisotropic liquid-drop problem if and only if the surface energy is isotropic; in particular, for $\tau$2, the Wulff shape of $\tau$3 satisfies the Euler–Lagrange equation only when $\tau$4. In sharp contrast, for certain crystalline surface tensions the Wulff shape is the unique minimizer, and for several anisotropic-repulsion variants it is again the minimizer in the small-mass regime (Choksi et al., 2018).

For smooth anisotropies in the small-nonlocality regime, Figalli, Maggi, and collaborators prove that minimizers are nevertheless close to the Wulff body. If $\tau$5 is the Wulff shape scaled to $\tau$6, any minimizer $\tau$7 can be written as a normal graph

$\tau$8

with $\tau$9 as $W$0, and one obtains the explicit rate

$W$1

The energy deficit satisfies

$W$2

and a spectral-gap estimate gives

$W$3

for nearby competitors. A plausible implication is that, in nonlocal anisotropic problems, “modified Wulff shape” often means not exact coincidence with the classical Wulff body, but controlled convergence toward it under perturbative scaling (Misiats et al., 2019).

6. Discrete, atomistic, and charge-dependent modified Wulff shapes

In discrete-to-continuum settings, modified Wulff shapes arise as homogenized minimizers of missing-bonds energies rather than as direct inputs. For a Bravais lattice $W$4 and a finite-support interaction potential $W$5, the surface-type energy of a finite configuration $W$6 is

$W$7

After rescaling, the functionals $W$8 $W$9-converge to the anisotropic perimeter

$B\subset \mathbb R^d$00

If $B\subset \mathbb R^d$01 has the form $B\subset \mathbb R^d$02, then the Wulff shape is a zonotope. The same program extends to multigrid quasicrystal tilings, where non-crystallographic Wulff shapes such as the regular decagon arise from homogenized perimeter densities with quasiperiodic symmetry (Nin et al., 2021).

A three-dimensional atomistic realization is provided by hard spheres on the FCC and HCP lattices with sticky-disk potential

$B\subset \mathbb R^d$03

The continuum limit energies are again of perimeter type. The explicit Wulff set for FCC is the truncated octahedron, while for HCP it is a truncated elongated hexagonal bipyramid. The computed minimal ratios satisfy

$B\subset \mathbb R^d$04

so FCC crystallization is preferred for $B\subset \mathbb R^d$05 large enough (Cicalese et al., 2022).

A different type of modification comes from an additional global invariant. In the two-dimensional ionic square-lattice model with two atomic types, free minimization yields almost neutral ground states with

$B\subset \mathbb R^d$06

and neutral crystals converge, after rescaling by $B\subset \mathbb R^d$07, to a square Wulff shape. If instead one prescribes the net charge $B\subset \mathbb R^d$08, the energy undergoes a transition at the saturation charge $B\subset \mathbb R^d$09. At the saturation charge, the macroscopic shape changes from square to diamond; beyond $B\subset \mathbb R^d$10, square-lattice crystallization fails. In that context, the term modified Wulff shape refers explicitly to the change in limiting shape induced by supplementing the usual perimeter minimization with the extra charge constraint (Friedrich et al., 2019).

These discrete examples correct another common misconception. Modified-Wulff behavior need not come from altering a smooth surface tension by a small perturbation. It can also emerge from coarse graining, from the symmetry of an underlying lattice or quasicrystal tiling, or from extra constraints such as net charge that effectively modify the admissible macroscopic geometry (Nin et al., 2021, Friedrich et al., 2019).

7. Conceptual synthesis and open directions

Across these settings, the modified-Wulff shape remains tied to the same structural principle: equilibrium geometry is encoded by a positively homogeneous directional cost, whether introduced explicitly as $B\subset \mathbb R^d$11, generated by anisotropic curvature functionals, extracted from finite-$B\subset \mathbb R^d$12 ensemble energetics, or obtained as a homogenized surface tension from an atomistic model. The classical support-function construction survives exactly in some cases, perturbatively in others, and only implicitly in ensemble-based or strongly constrained problems (Miracle-Sole, 2013, Radin, 2016).

The main points of divergence concern uniqueness, regularity, and the role of extra interactions. In Roth’s convex anisotropic-curvature setting, stability singles out the Wulff shape itself. In smooth anisotropic liquid-drop models with isotropic repulsion, exact Wulff optimality generally fails unless the surface energy is isotropic, although minimizers remain close to the Wulff body for small nonlocality. In crystalline and discrete models, faceted polyhedra, zonotopes, squares, diamonds, truncated octahedra, hexagons, decagons, and related shapes arise as exact or asymptotic minimizers, reflecting the underlying anisotropy and any additional constraints (Roth, 2015, Choksi et al., 2018, Cicalese et al., 2022).

Open problems identified in the literature include rigorous proof of the solid–fluid phase transition in Radin’s ensemble-based model, characterization of the two-dimensional high-pressure Wulff shape, regularity theory for three-dimensional crystalline limit shapes with planar facets meeting at crystallographic angles, and extensions to coexistence problems involving two anisotropic surface tensions (Radin, 2016). In this sense, the modified-Wulff shape is best understood not as a single fixed object, but as a unifying label for equilibrium shapes produced when the classical Wulff variational principle is generalized without abandoning its core anisotropic-geometric logic.