Liquid Metal Dealloying (LMD)

• Liquid Metal Dealloying (LMD) is the corrosion of a precursor alloy by a molten metal, leading to selective dissolution and the formation of a nano‐porous network.
• The process is driven by diffusion, local interfacial equilibrium, and complex phase-field formulations that capture gradient penalties and solute leakage.
• Variations in melt chemistry and grain-boundary diffusion critically control front kinetics, connectivity, and mechanical scaling in LMD-derived nanoporous metals.

Liquid metal dealloying (LMD) is the corrosion of a two-component solid alloy by contact with a molten third component, leading to the selective dissolution of one species into the liquid and the self-organization of the retained species into a nano-porous or bicontinuous network with ultra-high interfacial area. In current theoretical and computational treatments, LMD is a diffusion-coupled moving-boundary problem in which local interfacial equilibrium, finite solubility of the retained element in the melt, interfacial instability, grain-boundary transport, and solvent chemistry jointly determine front kinetics, morphology, topology, and the resulting mechanical response (Lai et al., 2022, Lai et al., 2022, Bieberdorf et al., 2022, Bonneville et al., 25 Sep 2025, Ramallo et al., 24 Apr 2026).

1. Thermodynamic basis and governing fields

A standard LMD setting consists of a precursor solid alloy $A_xB_{1-x}$, where $A$ is the immiscible element to be retained and $B$ is the dissolvable element, immersed in a liquid metal melt $M$. At the solid–liquid interface, local equilibrium imposes chemical potentials $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$. The small but finite solubility of $A$ in $M$, measured from ternary phase diagrams, enables leakage of the nominally retained species into the melt; values such as $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$ are explicitly invoked in LMD analyses (Lai et al., 2022).

Phase-field descriptions introduce a non-conserved field $\phi(x,t)$, with $\phi\approx 1$ in the solid and $A$0 in the liquid, together with conserved composition fields $A$1 and $A$2, with $A$3. One representative free-energy functional is

$A$4

with $A$5 and

$A$6

The corresponding dynamics combine a modified Allen–Cahn evolution for $A$7 and coupled Cahn–Hilliard transport for the conserved species,

$A$8

or, in shorthand,

$A$9

These formulations are central in both mechanistic studies and surrogate-model development for LMD (Bonneville et al., 25 Sep 2025).

Alternative phase-field formulations enrich the chemical and interfacial terms. In the treatment summarized by Lai–Geslin–Karma, the free energy contains regular-solution contributions, latent-heat terms, and composition-gradient penalties, while the mobility matrix takes the form $B$0. Across these formulations, the same structural point recurs: selective dissolution is thermodynamically driven by the difference in chemical free energies of the species in the solid and liquid, while the retained network is stabilized and reshaped by gradient penalties and interfacial energy (Lai et al., 2022, Bonneville et al., 2024).

A persistent computational consequence is stiffness. Because the conserved-species equations contain fourth-order spatial derivatives through terms such as $B$1, explicit solvers require $B$2 for stability. This stiffness is one of the defining features of LMD phase-field computation and motivates reduced-order or learned surrogates (Bonneville et al., 25 Sep 2025).

2. Front propagation and instability selection

The kinetics of LMD have been analyzed through one-dimensional dissolution models and higher-dimensional stability analyses. In diffusion-limited form, the dealloying front position obeys

$B$3

where $B$4 is the diffusivity of the dissolving species in the melt and $B$5 is a Péclet number depending on interface reaction rates and diffusivities; at long times, the front velocity scales as $B$6. This $B$7 law is a central scaling for LMD (Lai et al., 2022).

A more detailed one-dimensional treatment distinguishes two regimes. In the early stage, the low solubility of the retained element in the melt causes accumulation of that element at the interface, generating a passivation layer. The peak retained-species fraction satisfies $B$8, and a simple phenomenology for the local front velocity is $B$9, with $M$0. This yields

$M$1

Once the interfacial composition reaches its tiny equilibrium solubility in the melt, the system enters a stationary regime in which both elements dissolve at fixed interfacial compositions and the classical diffusion-controlled $M$2 law emerges (Lai et al., 2022).

Pattern formation originates in instability of the dealloying front. Linear stability analysis of the interfacial alloy layer gives a Cahn–Hilliard-type dispersion relation,

$M$3

and, in the ternary-reduced binary formulation,

$M$4

Spinodal instability occurs if

$M$5

and the fastest-growing wavenumber $M$6 determines the initial ligament spacing $M$7, reported as $M$8 in phase-field units. The instability criterion also predicts a transition between planar dissolution and spinodal-driven ligamentation as the precursor and melt compositions vary (Lai et al., 2022).

An important correction to earlier intuition is the role of solid-state diffusion. Although the solid diffusivity is four to five orders of magnitude smaller than the liquid diffusivity, it affects both dissolution kinetics and ligament morphologies. Finite $M$9 broadens the retained-species interfacial peak, increases the effective Péclet number toward the phase-diagram prediction, and can promote parallel laminæ rather than highly tortuous bicontinuous networks. This directly contradicts the common simplification that solid-state transport may be neglected without qualitative consequence (Lai et al., 2022).

3. Connectivity, topology, and solvent chemistry

Topology control in LMD is commonly discussed in terms of the solid volume fraction $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$0 in the dealloyed layer. In mass-conservation form, $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$1 in the absence of leakage of the retained element. With retained-element leakage at a rate determined by $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$2 and $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$3,

$\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$4

Genus number $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$5 correlates with $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$6 by topological invariants, and highly bicontinuous structures require $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$7 above a percolation threshold $\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$8. In this framework, bulk diffusive transport of the immiscible element in the liquid melt strongly influences the evolution of the solid fraction and topology during dealloying (Lai et al., 2022).

A compact topology criterion is the ratio

$\mu_i^s(\phi,c)=\mu_i^l(\phi,c)$9

For $A$0, corresponding to high leak or low front velocity, $A$1 at the front falls below threshold and connectivity is lost. For $A$2, connectivity is maintained. This parameterization clarifies why melt chemistry exerts such strong morphological control. Adding Ti to Cu increases $A$3 solubility in the melt, lowers front velocity, but increases $A$4 leak, which lowers $A$5 below the percolation threshold and yields structures that detach, fragment, or dissolve. Adding Ag to Cu lowers both $A$6 and $A$7, reduces $A$8 leakage, keeps $A$9 above $M$0, and preserves high-genus bicontinuous topology (Lai et al., 2022).

LMD is often conflated with electrochemical dealloying (ECD), but the two processes differ in their controlling transport and topology-selection mechanisms.

Aspect	ECD	LMD
Front rate	Constant interface-controlled $M$1	Diffusion-limited $M$2
Retained species in fluid	Negligible bulk diffusion, $M$3	Finite solubility and bulk diffusion of $M$4
Pattern formation	Surface-spinodal decomposition	Unstable diffusion-coupled growth
Solid fraction $M$5	Remains uniform	Spatially varying due to leakage

This distinction matters for both interpretation and design. In ECD, topologically connected bicontinuous structures occur only at low $M$6 content and low voltage. In LMD, the front velocity rapidly drops because of diffusion-limited kinetics, and bulk diffusion of the retained species generates spatially varying $M$7 and topology (Lai et al., 2022).

Experimental comparisons support the topology criterion. For Ta–Ti precursors dealloyed for $M$8 at $M$9, pure Cu produced rapid dealloying depth of $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$0, an $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$1-leak layer $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$2, a dealloyed layer $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$3, and a connected core $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$4, with a predominantly disconnected, fine porous layer near the interface. Under otherwise comparable conditions, $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$5 produced a slower depth of $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$6, no leak layer, $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$7, and a fully connected bicontinuous morphology with uniform $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$8 ligament size. The same study also reports that the classical coarsening law $c_{\mathrm{Ta}}^l\approx 10^{-3}\!-\!10^{-2}$9 with $\phi(x,t)$0 fails to capture LMD data; instead, bulk diffusive transport of the retained species and concurrent dissolution/redeposition dominate ligament evolution (Lai et al., 2022).

4. Grain-boundary-mediated dealloying

Grain boundaries are not passive defects in LMD. A multi-phase-field model extending the original approach of Geslin et al. treats two solid grains and one liquid using three non-conserved phase fields $\phi(x,t)$1 and three conserved concentrations $\phi(x,t)$2. In that formulation, the local diffusivity is

$\phi(x,t)$3

so diffusion is fast in the liquid, negligible in the bulk solid, and enhanced along the diffuse grain boundary. The model is implemented with explicit Euler for the phase fields and a semi-implicit spectral scheme, following Badalassi et al., for the concentrations, enabling large $\phi(x,t)$4 and $\phi(x,t)$5 simulations while maintaining a realistic interfacial thickness (Bieberdorf et al., 2022).

The governing picture is one of coupled grain-boundary migration and dealloying. Grain-boundary migration velocity obeys $\phi(x,t)$6, where the driving force $\phi(x,t)$7 is the local excess grand-potential gradient. As the liquid–solid triple line at the grain boundary dissolves $\phi(x,t)$8 more rapidly because of enhanced boundary diffusion, the local solid becomes enriched in $\phi(x,t)$9. Uphill diffusion of $\phi\approx 1$0 along the migrating boundary provides an extra evacuation path for the more noble element rejected by the advancing liquid channel. Two reinforcing mechanisms are reported: a solute-flux mechanism and a curvature mechanism. The triple junction maintains the Young’s-law force balance to within numerical tolerance (Bieberdorf et al., 2022).

The morphological consequences are pronounced. In $\phi\approx 1$1, for initial $\phi\approx 1$2, bicontinuous dealloying forms a uniform ligament network within each grain, but at the grain boundary a single deep channel penetrates $\phi\approx 1$3 further. In $\phi\approx 1$4, for $\phi\approx 1$5, regions away from the boundary develop $\phi\approx 1$6 nodules that grow into ligaments, whereas the migrating boundary develops $\phi\approx 1$7 ridges and linear corrosion channels. The boundary migrates in opposite $\phi\approx 1$8-directions at different $\phi\approx 1$9-locations, a result presented as a possible origin of experimentally observed waviness (Bieberdorf et al., 2022).

Grain-boundary migration also disrupts final connectivity. The $A$00-rich block formed by the migrating boundary, termed the GBMD region, attaches selectively to one grain’s ligament network and leaves a widening liquid gap to the other grain, asymmetrically disrupting connectivity. This behavior is reported to be in qualitative agreement with SEM observations of Ta–Ti LMD by McCue et al. and Fe–Ni LMD by Joo et al. A further result is that grain-boundary migration can locally bypass the parting-limit constraint: even for $A$01, where no bulk bicontinuous front forms in single-crystal runs, the boundary still nucleates a singular deep liquid channel via GBMD (Bieberdorf et al., 2022).

5. Phase-field computation and surrogate extrapolation

Direct numerical simulation of LMD phase-field models is computationally expensive because the stiffness of the conserved-species equations forces extremely small time steps and because the evolving microstructure must be resolved on fine grids. Representative DNS setups use $A$02 degrees of freedom such as $A$03 to resolve sub-nanometer features, with $A$04. Reaching $A$05 then requires $A$06 time steps, and a single $A$07, $A$08 run is reported to take $A$09 days on a large CPU cluster; extending to $A$10 or to times $A$11 longer becomes intractable, with cost scaling roughly as $A$12 (Bonneville et al., 25 Sep 2025).

One response has been operator-learning surrogates. The U-Shaped Adaptive Fourier Neural Operator (U-AFNO) couples a U-Net encoder–decoder to an Adaptive Fourier Neural Operator bottleneck implemented as a frequency-space vision transformer. In the reported configuration, the latent AFNO contains $A$13 blocks and $A$14 heads, and the model is trained on 87 high-fidelity runs with snapshots every $A$15. After 100 auto-regressive leaps, U-AFNO achieves $A$16, $A$17, and $A$18, on par with the intrinsic high-fidelity-to-high-fidelity discrepancy across different initial seeds, while the relative $A$19 error in total solid mass is $A$20. The solver cost is reported as $A$21 per $A$22 steps on 128 CPU cores, versus $A$23 per leap on one A100 GPU (Bonneville et al., 2024).

A later direction uses purely convolutional, conditionally parameterized U-Nets. In one version, the network has 5 encoder levels with $A$24 convolutions and stride 2, 5 decoder levels with transposed $A$25 convolutions, skip connections, convolutional self-attention in the bottleneck, and physics-aware padding: circular in the horizontal direction, zero at the top boundary, and value replication at the bottom. Parameter conditioning uses an MLP to inject the time-skipping interval $A$26 and a concentration parameter, enabling leaps of $A$27 solver steps and adaptation to diverse alloy systems. During inference, translational invariance allows the same weights to process larger domains such as $A$28, and auto-regressive rollout applies the learned map repeatedly in time. Within the training regime, quantities-of-interest errors are reported as $A$29; when extrapolating to domains twice as tall and time horizons $A$30 longer, the error remains $A$31. For a $A$32 reference simulation, the surrogate completes the same horizon in $A$33 on one A100 GPU, corresponding to a $A$34 speed-up (Bonneville et al., 25 Sep 2025).

A related 2026 framework extends the convolution-only strategy with a flood-fill field corrector and a conditional diffusion model for initial conditions. The U-Net conditions on $A$35, enforces fields in $A$36 through a final sigmoid, and uses physically informed padding to match periodic side boundaries, a top Dirichlet condition, and a bottom Neumann condition. The flood-fill corrector resets predicted uncorroded regions to the exact initial concentration, while a DDPM generates synthetic deformed-interface initial conditions conditioned on $A$37. On the reported benchmarks, QoI relative errors are $A$38 in the training regime and $A$39 in large-scale, long-time extrapolation, with worst case $A$40 for high $A$41. One U-Net forward pass with $A$42 takes $A$43 on A100 versus 59 minutes for the direct solver, and end-to-end speed-ups reach $A$44. Synthetic initial conditions degrade QoI errors by only $A$45 in training and negligibly in extrapolation (Bonneville et al., 8 Jan 2026).

These surrogate studies also delimit current failure modes. Because LMD dynamics are chaotic, the surrogates do not reproduce pixel-wise microstructures exactly; instead they target quantities of interest such as interface curvature, penetration depth, ligament statistics, and spatial auto-correlations. The ligament height metric is repeatedly identified as the most challenging. Extreme auto-regressive extrapolation may lead to unphysical patterns or drift, and the models do not exactly satisfy PDE residuals away from the explicitly corrected regions (Bonneville et al., 25 Sep 2025, Bonneville et al., 8 Jan 2026).

6. Mechanical scaling and materials design

LMD is used not only to generate morphology but also to tune mechanical response through topology and connectivity. Ramallo et al. study single-crystalline nanoporous tantalum produced by liquid metal dealloying of Ti65Ta35 in a Cu40Bi60 melt at $A$46 for 5 minutes under Ar. The resulting network has ligament diameter $A$47, solid volume fraction $A$48, pure bcc Ta by XRD, and $A$49 at.% oxygen by SEM/EDS (Ramallo et al., 24 Apr 2026).

The mechanical behavior is expressed relative to Gibson–Ashby scaling for open-cell foams,

$A$50

For nanoporous Ta with $A$51, nanoindentation yields $A$52 and $A$53, with the data best fit by

$A$54

The paper explicitly contrasts this with nanoporous Au, where $A$55, and attributes the difference to enhanced ligament connectivity in the Cu–Bi LMD system (Ramallo et al., 24 Apr 2026).

The mechanistic basis of that enhanced connectivity is solvent chemistry. In a pure Cu bath, Ta has finite solubility at the solid/fluid interface, leading to broken ligaments and lower connectivity. Adding Bi lowers the activity of Cu by forming a Cu–Bi eutectic with $A$56, so that

$A$57

The reported interpretation is that reduced Ta leak results in a more bicontinuous network, with fewer dead-ends, higher genus, and more uniform load paths. This directly parallels the Ag-in-Cu topology-control strategy reported for Ta–Ti dealloying, where reduced leak of the retained species preserves connectivity and suppresses fragmentation (Lai et al., 2022, Ramallo et al., 24 Apr 2026).

Molecular-dynamics simulations indicate that the deformation mechanism is not anomalous. Using LAMMPS with an extended Finnis–Sinclair potential and an atomistic diamond Berkovich tip, the simulations show surface-nucleated glissile $A$58 dislocations, annihilation at free surfaces, occasional $A$59 sessile lines, and localized twinning below the contact. Densification under the indenter is minimal, $A$60, and the plastic zone extends only $A$61 the penetration depth, in agreement with post-indent SEM. The stiffness-density response is therefore attributed to connectivity rather than to unusual deformation mechanisms (Ramallo et al., 24 Apr 2026).

The resulting design rules are consistent across the literature. Melt compositions that minimize solubility of the retained element, such as Ag additions to Cu or Bi-containing Cu baths, preserve bicontinuity. Temperature and time control coarsening and volume fraction: higher $A$62 or longer $A$63 coarsen ligaments and reduce $A$64, whereas lower $A$65 or shorter $A$66 yield finer ligaments and higher $A$67, although incomplete dissolution becomes a competing risk. Precursor composition tunes the amount of dissolvable species and thus the final solid fraction. Taken together, these results establish solvent chemistry, front kinetics, and retained-species leakage as coupled levers for topology and mechanical scaling in LMD-derived nanoporous metals (Lai et al., 2022, Ramallo et al., 24 Apr 2026).