Anisotropic Spinodal Infills

• Anisotropic spinodal infills are phase-separation-derived internal networks with directional organization and distinct shell or interface features.
• They emerge from spinodal instabilities governed by elastic mismatch, anisotropic diffusion, and advanced computational frameworks.
• These microarchitectures offer tunable mechanical, electronic, and magnetic properties for architected materials and topological systems.

Searching arXiv for recent and relevant papers on anisotropic spinodal structures, spinodoids, and related phase-field/mechanics work. Anisotropic spinodal infills are spinodal-decomposition-derived internal structures in which phase-separated, shell-based, or compositionally modulated regions are directionally organized. In one formulation, they are “non-uniform, directionally structured networks” of amorphous polytetrahedral aggregates and bcc crystallites generated during liquid–solid spinodal decomposition (Gomez et al., 2010). In another, they are “self-assembled, compositionally modulated channels and interfaces” in spinodal superlattices of topological insulators (Usanmaz et al., 2018). In architected materials, they appear as stochastic, aperiodic, bicontinuous “spinodoid” microarchitectures whose directional stiffness, orientation, continuity, and anisotropy are explicitly controlled or optimized (Zheng et al., 2020, Deng et al., 29 Jun 2025). Across these settings, the common theme is that a spinodal instability sets an internal length scale, while kinetics, elasticity, diffusion, geometry, or optimization determines whether the resulting infill is isotropic only in the statistical sense or strongly anisotropic in local structure and effective response.

1. Thermodynamic and kinetic basis

A central description of liquid–solid spinodal decomposition uses a Landau free energy functional for a scalar order parameter $\psi(\mathbf{r})$, the deviation of particle density from the uniform liquid density,

$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$

with

$$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$

The dynamics is purely relaxational, conserving the order parameter, and is modeled by a Cahn–Hilliard equation,

$$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$

Linearization around the uniform liquid gives the amplification factor

$\lambda(k) = -D k^4 + \tau k^2 - b,$

so there is a finite band of unstable wavevectors and the most unstable mode occurs near

$k_0 \sim \sqrt{\frac{\tau}{D}}$

for shallow quenches near the spinodal line $\tau_S = 2\sqrt{bD}$ (Gomez et al., 2010). This finite-wavelength instability is the basic source of characteristic spinodal length scales.

In intercalation materials, anisotropy enters directly through elasticity. A phase-field model for Li concentration $c(\mathbf{x},t)$ uses

$$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$

with eigenstrain

$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$

For FePO$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$0/LiFePO$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$1, the mismatch values are

$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$2

so insertion produces expansion along two axes and contraction along the third. The resulting growth rate $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$3 is direction dependent, and the neutral curve $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$4 is anisotropic in $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$5 (Stanton et al., 2012).

Anisotropy can also arise from transport. In dynamical density functional theory with a constant diffusion tensor $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$6, the Fourier-space evolution of density fluctuations obeys

$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$7

The linear growth rate is

$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$8

In this setting, the instability band in $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$9 is unchanged, but the rate at which each direction’s mode grows depends on orientation through $$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$0, and anisotropic diffusion strongly reduces the coupling between different Fourier modes (Vuijk et al., 2018).

2. Morphological classes and the emergence of anisotropy

In liquid–solid spinodal decomposition with an isotropic scalar free energy, the early-stage density field is a superposition of random density waves with wavevectors near $$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$1. For near-critical quenches, the band is very narrow, so the pattern is a strongly correlated filamentary network of density fluctuations, and the two-point correlation function behaves like a Bessel function decaying as

$$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$2

The paper explicitly notes that there is “no built-in directional anisotropy in the free energy,” yet “anisotropic spatial organization” emerges through a filamentary random-wave morphology, surface-driven anisotropy of crystallization, and the network topology of amorphous infills (Gomez et al., 2010). This directly addresses a common misconception: explicit anisotropy in the free energy is not required for locally anisotropic spinodal infills.

The same model produces a sequence of structural motifs. Strong length scale selectivity induces the appearance of small precursors for crystallization with icosahedral order. These precursors grow in densely packed clusters of tetrahedra; about $$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$3 of particles participate in at least one regular tetrahedron, and the local tetrahedral order parameter distribution peaks at $$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$4. As the average size of the amorphous nuclei becomes large enough to reduce geometric frustration, crystalline particles with a body center cubic symmetry heterogeneously nucleate on the growing clusters. Crystalline particles begin to appear at radii $$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$5, with $$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$6 the mean interparticle spacing, and the crystalline volume fraction obeys

$$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$7

in the studied regime (Gomez et al., 2010). The resulting infill is composite: amorphous cores, bcc shells or patches, and dense polytetrahedral aggregates.

In coherent crystalline solids, anisotropy is more explicit. For LiFePO$$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$8/FePO$$W(\psi) = -\tau \psi^2 + \nu \psi^3 + \lambda \psi^4.$$9, simultaneous expansion along $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$0 and contraction along $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$1 leads to “tilted, striped phase boundaries in equilibrium.” Three-dimensional simulations reveal skewed, striped phases whose boundaries align along $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$2 or $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$3 planes, in qualitative agreement with experiments by Laffont et al., Ramana et al., and Chen et al. (Stanton et al., 2012). This is not merely a geometric detail: the preferred instability direction is selected by the tensorial mismatch $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$4 and the elastic kernel in Fourier space.

In fluids with anisotropic diffusion, the morphology becomes directionally focused. With sufficiently strong anisotropy, only modes along the fast direction show significant growth, the spectrum becomes almost one-dimensional, and real-space domains become almost filamentary or layered along the fast axis (Vuijk et al., 2018). In topological-insulator alloys, the same broad logic appears in a crystalline setting: the TlBiX$$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$5 systems are modeled with composition modulation along rhombohedral $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$6, so interfaces lie in $$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$7 planes and form lamellar or stripe-like spinodal superlattices. These interfaces act as quasi-two-dimensional channels embedded in a three-dimensional matrix (Usanmaz et al., 2018).

3. Spinodoids and architected anisotropic infills

Spinodoid microstructures are synthetic, spinodal-inspired architectures generated not by direct solution of the Cahn–Hilliard equations but by compact stochastic parametrizations. In one framework, a Gaussian random field is written as

$$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$8

and thresholded through

$$\frac{\partial \psi}{\partial t} = M \nabla^2 \left( \frac{\delta \Phi}{\delta \psi} \right).$$9

The design parameters are

$\lambda(k) = -D k^4 + \tau k^2 - b,$0

with $\lambda(k) = -D k^4 + \tau k^2 - b,$1 the average relative density and $\lambda(k) = -D k^4 + \tau k^2 - b,$2 cone angles controlling the anisotropic orientation distribution function. Adding a rotation angle $\lambda(k) = -D k^4 + \tau k^2 - b,$3 produces

$\lambda(k) = -D k^4 + \tau k^2 - b,$4

and the rotated homogenized stiffness is

$\lambda(k) = -D k^4 + \tau k^2 - b,$5

A deep neural network $\lambda(k) = -D k^4 + \tau k^2 - b,$6 is trained on $\lambda(k) = -D k^4 + \tau k^2 - b,$7 homogenized samples, achieves $\lambda(k) = -D k^4 + \tau k^2 - b,$8 on test data for each stiffness component, and reduces single predictions to approximately $\lambda(k) = -D k^4 + \tau k^2 - b,$9 instead of minutes for finite-element homogenization (Zheng et al., 2020).

A second framework uses Spectral Density Functions. A target phase field $k_0 \sim \sqrt{\frac{\tau}{D}}$0 with SDF $k_0 \sim \sqrt{\frac{\tau}{D}}$1 is combined with white noise $k_0 \sim \sqrt{\frac{\tau}{D}}$2 to reconstruct

$k_0 \sim \sqrt{\frac{\tau}{D}}$3

Thresholding then gives

$k_0 \sim \sqrt{\frac{\tau}{D}}$4

Within this representation, anisotropy is explicit. Isotropic spinodoids are described by two Lamé constants. Monoclinic spinodoids have one plane of symmetry and six independent stiffness components $k_0 \sim \sqrt{\frac{\tau}{D}}$5. Orthotropic spinodoids have two orthogonal symmetry planes and four independent components $k_0 \sim \sqrt{\frac{\tau}{D}}$6. The anisotropy controls are

$k_0 \sim \sqrt{\frac{\tau}{D}}$7

together with a frequency rotational angle $k_0 \sim \sqrt{\frac{\tau}{D}}$8. A Gaussian Process surrogate $k_0 \sim \sqrt{\frac{\tau}{D}}$9 maps descriptors to effective moduli, with prediction errors $\tau_S = 2\sqrt{bD}$0 for each spinodoid type (Deng et al., 29 Jun 2025).

These formulations also embed anisotropy into topology optimization. In the two-scale framework, the objective is compliance minimization subject to equilibrium and a volume constraint,

$\tau_S = 2\sqrt{bD}$1

while in the data-driven multiscale framework the design variables per element are

$\tau_S = 2\sqrt{bD}$2

A shallow neural network with 2 input neurons, 20 hidden neurons, and 10 output neurons maps element centroid coordinates to these descriptors, and gradients are computed through JAX automatic differentiation (Deng et al., 29 Jun 2025). A recurring result is that monoclinic spinodoids are selected where stress trajectories are predominantly unidirectional, orthotropic spinodoids where two dominant orthogonal load paths coexist, and isotropic spinodoids where stress directions are highly multi-directional (Deng et al., 29 Jun 2025).

4. Mechanical response, curvature, and robustness

The mechanical literature distinguishes solid spinodal models from shell spinodal models. In numerical and experimental studies of cellular materials with spinodal topologies, solid spinodal models in the density range of $\tau_S = 2\sqrt{bD}$3–$\tau_S = 2\sqrt{bD}$4 are relatively inefficient, with

$\tau_S = 2\sqrt{bD}$5

Shell spinodal models in the density range of $\tau_S = 2\sqrt{bD}$6–$\tau_S = 2\sqrt{bD}$7 are exceptionally stiff and strong, with

$\tau_S = 2\sqrt{bD}$8

and they are remarkably imperfection insensitive (Hsieh et al., 2019). This distinction is fundamental for the notion of infill: shell-based spinodal infills are not a minor variation on solid bicontinuous networks but a different mechanical regime.

Interpenetrating phase composites with spinodal shell reinforcements show the same pattern. For spinodal shell IPCs,

$\tau_S = 2\sqrt{bD}$9

whereas spinodal solid IPCs follow

$c(\mathbf{x},t)$0

Spinodal shell IPCs have comparable compressive strength and stiffness to IPCs reinforced by the Schwarz P TPMS and the octet truss-lattice, but exhibit far less catastrophic failure and greater damage resistance, particularly at high volume fraction of reinforcing phase (Zhang et al., 2021). The stochastic topology produces stochastically distributed cracks rather than shear-band-like localization, and the large surface area of the shell promotes matrix constraint and redistribution of load.

A more recent shell-based theory makes the curvature dependence explicit. In shell-based spinodal architected materials, each local patch is described by principal curvatures $c(\mathbf{x},t)$1, mean curvature

$c(\mathbf{x},t)$2

and Gaussian curvature

$c(\mathbf{x},t)$3

The areal stretching energy density under an affine macroscopic deformation in direction $c(\mathbf{x},t)$4 is

$c(\mathbf{x},t)$5

and the geometric proxy for stretch-to-bend ratio is

$c(\mathbf{x},t)$6

A corresponding strength proxy is

$c(\mathbf{x},t)$7

The paper shows that directional effective stiffness scales approximately as $c(\mathbf{x},t)$8, and that normalized stiffness and yield strength correlate strongly with the geometric proxies (Dhulipala et al., 15 May 2025). This provides a direct route from curvature distributions to anisotropic infill design.

5. Functional material realizations

One realization of anisotropic spinodal infills is electronic rather than structural. In TlBiX$c(\mathbf{x},t)$9 ($$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$0 S, Se, Te), all three binary alloys considered have positive formation enthalpy over the entire composition range, are immiscible at $$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$1, and display miscibility gaps at finite temperature. Reported consolute temperatures and critical compositions are approximately $$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$2 and $$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$3 for TlBiS$$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$4–TlBiSe$$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$5, $$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$6 and $$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$7 for TlBiSe$$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$8–TlBiTe$$F[c] = \int_\Omega\left\{f(c) + \frac{\rho}{2}\nabla c \cdot ({\bf K}\nabla c) + \frac{1}{2}{\bf E}:{\bf T}\right\}\mathrm{d}{\bf x},$$9, and $$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$0 and $$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$1 for TlBiS$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$2–TlBiTe$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$3 (Usanmaz et al., 2018). In the TlBiS$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$4–TlBiTe$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$5 system, the eutectic regime around $$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$6 TlBiTe$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$7 at $$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$8 allows spinodal superlattices and two-dimensional eutectic microstructures. The resulting interfaces host gapless or quasi-gapless states, and the work explicitly frames them as “quasi-2D metallic channels inside a 3D insulating or weakly conducting matrix” without vacuum interfaces (Usanmaz et al., 2018).

A second realization is electro-chemo-mechanical. In LiFePO$$\mathbf{E} = \frac{1}{2}\left(\nabla\mathbf{u} + \nabla\mathbf{u}^{\mathrm{T}}\right) - c\,\mathbf{M}, \qquad \mathbf{M} = \mathrm{diag}(M_{11},M_{22},M_{33}).$$9/FePO$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$00, anisotropic coherency strain narrows the spinodal gap, selects preferred wavevectors, and yields tilted, striped phase boundaries in equilibrium (Stanton et al., 2012). The material-specific eigenstrain values

$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$01

produce direction-dependent suppression of instability and favor fluctuations along the direction of greatest expansion. The resulting striped or lamellar infills are therefore elasticity selected rather than merely diffusion limited.

A third realization is magnetic. In alnico, spinodal decomposition of a bcc solid solution into Fe–Co–rich $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$02 and Ni–Al–rich $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$03 is biased by magnetic annealing, which aligns $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$04 rods along the $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$05 direction closest to the applied field. The microstructure evolves from an initially isotropic, interpenetrating network to highly anisotropic rods, followed by Cu-rich spheres, rods, and blades and additional $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$06 rod precipitation during draw. The intrinsic coercivity rises from $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$07 in the solutionized condition to $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$08 after $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$09 magnetic annealing, to about $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$10 after $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$11 magnetic annealing, and to $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$12 after full draw (Zhou et al., 2018). In this case, “anisotropic spinodal infills” are the ensemble of rods, blades, and interfacial phases that control shape anisotropy and magnetic isolation.

6. Misconceptions, limitations, and open directions

A first misconception is that spinodal infills are necessarily periodic. The architected-material literature explicitly defines spinodoids as stochastic and aperiodic, and the mechanically efficient shell topologies generated from Cahn–Hilliard evolution are likewise non-periodic (Zheng et al., 2020, Hsieh et al., 2019). A second misconception is that anisotropy requires a built-in anisotropic free energy. In the liquid–solid Landau model, the gradient and nonlocal terms are isotropic, yet local anisotropy emerges through random-wave morphology, curvature-dependent heterogeneous nucleation, and the topology of the amorphous network (Gomez et al., 2010).

The present theories also have clear regime limits. The curvature-guided shell framework is developed for the thin-shell regime

$$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$13

and outside this range transverse shear and more complex three-dimensional stress states become important (Dhulipala et al., 15 May 2025). The two-scale topology-optimization framework assumes separation of scales and is restricted to linear elastic stiffness; nonlinear behavior, yielding, large strains, and damage are not part of the surrogate (Zheng et al., 2020). The topological-insulator work does not solve the full time-dependent Cahn–Hilliard equation but assumes sinusoidal composition modulation along $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$14 and periodic superlattice structures (Usanmaz et al., 2018).

Several open directions are already identified inside the literature. The isotropic liquid–solid Landau model states that anisotropic gradient coefficients $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$15 or anisotropic nonlocal kernels would produce direction-dependent correlation lengths and genuinely anisotropic spinodal morphological length scales (Gomez et al., 2010). The architected-material literature points to nonlinear and multi-physics surrogate models, full 3D rotations, adaptive resolution, and manufacturing-aware optimization as natural extensions (Zheng et al., 2020). In filled polymer blends, the current Sanchez–Lacombe framework treats fillers as isotropic spherical inclusions, but the paper explicitly argues that an anisotropic generalization would require orientation distribution functions, tensorial stress responses, and a direction-dependent instability analysis; its approximate spinodal formula already gives deviations in the spinodal temperature of less than $$\Phi[\psi] = \int dV \left[ W(\psi) + D (\nabla \psi)^2 + b \int dV' \frac{\psi(\mathbf{r}) \, \psi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \right],$$16 relative to the exact calculation in the low-compressibility regime (Chervanyov, 15 Jan 2026).

Taken together, these works suggest that “anisotropic spinodal infills” is not a single microstructure class but a family of spinodal-derived internal organizations. They range from amorphous polytetrahedral skeletons with bcc patches, to tilted striped phases selected by coherency strain, to lamellar topological channels, to shell-based spinodoid architectures whose anisotropy is optimized against local stress trajectories. The unifying principle is that a spinodal instability supplies a bicontinuous or finite-wavelength scaffold, while anisotropy emerges from elastic mismatch, diffusion tensors, magnetic or crystallographic bias, surface-energy design, or data-driven multiscale optimization (Gomez et al., 2010, Stanton et al., 2012, Usanmaz et al., 2018, Deng et al., 29 Jun 2025).