METASTABLE POLYTYPES · SURFACE THERMODYNAMICS · PANDAT ASSESSMENT

Thermodynamic description of metastable 4H gold nanowires

A quantitative framework for relating the fcc–4H lattice-stability difference to nanowire dimensions, surface free energy, ligand adsorption, strain and interphase boundaries. The study combines published first-principles data, a reduced nano-CALPHAD screening model, a dedicated Pandat description and controlled experimental validation.

Research planJuly 2026Prepared byDENG JinxuanProposed platformPandat / PanPython

Study question

Under which combinations of temperature, diameter and surface condition can the 4H polytype compete thermodynamically with fcc Au?

STRUCTURAL AND THERMODYNAMIC BASIS

From stacking sequence to a lattice-stability function

Bulk gold adopts the face-centred cubic structure. Experimentally isolated 4H Au has a four-layer close-packed sequence, commonly written as ABCB. For thermodynamic modelling, it can be represented as the double-hexagonal close-packed polytype: ABCB and ABAC are symmetry-equivalent after relabelling layer positions.

This relationship permits hcp/dhcp axial-Ising results to provide an initial quantitative constraint. It does not remove the need for direct 4H phonon calculations and experimental parameter optimisation.

Close-packed layer sequence

Four-layer hexagonal repeat. Treated as the explicit metastable phase in the custom description.

AIM-DERIVED INPUT

Reproducible 0 K derivation

The published ANNI and ANNNI stacking-fault energies are converted to per-atom polytype energies using the close-packed fcc(111) area per atom.

A_atom = √3 a² / 4F_hcp − F_fcc = 2Aγ_ANNIF_4H − F_fcc = F_hcp − F_fcc − Aγ_ANNNI

Published and derived parameters used in the reduced model
Quantity	Value	Role in the model
fcc equilibrium lattice parameter	4.051 Å	Atomic area and atomic volume
γ_ANNI / γ_ANNNI	37.0 / 35.6 mJ m⁻²	hcp and 4H energy constraints
F_hcp − F_fcc	8.2051 meV atom⁻¹	Axial-Ising consistency check
F_4H − F_fcc	3.7921 meV atom⁻¹	Preliminary 0 K lattice-stability offset
Au surface energies, (111)/(100)/(110)	1.17 / 1.24 / 1.47 J m⁻²	fcc reference; SCAN+rVV10

REDUCED NANO-CALPHAD MODEL

A free-energy balance for a cylindrical nanowire

ΔG_4H−fcc^wire(T,d) = ΔG_4H−fcc^bulk(T) + 4V_mdΔγ + ΔG_ligand + ΔG_strain + ΔG_interface

Δγ is defined as γ_4H − γ_fcc. A negative differential surface energy favours 4H. The current model scans Δγ because facet-resolved 4H-Au surface energies are not yet available.

G_bulk

Lattice stability

Direct DFT energies and phonon free energies for fcc, hcp, dhcp and 4H Au.

G_surface

Facet contribution

Initially 4V_mγ/d; later replaced by Σγ_iA_i/V from HRTEM facets.

G_ligand

Adsorption correction

Coverage and facet-specific adsorption free energies for amine and thiol surface states.

G_int

Interfaces and strain

Added only for supported wires or identified 4H/fcc heterophase regions.

INTERACTIVE SCREENING RESULTS

Size–surface competition at finite temperature

Parameter sweep

Temperature 298 KWire diameter 12.6 nmDifferential surface energy -0.10 J m⁻²

Temperature treatmentThe screening code scales the 0 K offset with the reported 39% stacking-fault-energy reduction at 890 K. A final assessment should use tabulated phonon free energies or an optimised Gibbs function.

Calculated ΔG_4H−fcc+0.004 meV atom⁻¹

fcc has the lower free energy within this reduced surface scenario.

Bulk penalty3.2969 meV atom⁻¹

Critical diameter12.59 nm

4H-favoured sidefcc-favoured side

A negative total difference denotes a lower calculated 4H nanowire free energy. This is a sensitivity calculation, not evidence that each selected Δγ is physically realised.

Calculated crossover diameter, d_c (nm)
T (K)	−0.05	−0.10	−0.15	−0.20 J m⁻²
0	5.47	10.94	16.41	21.88
298	6.29	12.59	18.88	25.17
500	7.01	14.01	21.02	28.02
700	7.89	15.78	23.68	31.57
890	8.97	17.94	26.91	35.88

LIGAND SENSITIVITY · 298 K SCENARIO

Required fractional coverage

Example inputs: 0.10 eV adsorption advantage per ligand, 0.25 nm² per site and clean-surface Δγ = −0.02 J m⁻².

Wire diameter 10 nm

0.93

Within the stated coverage rangeThese are scenario inputs, not measured oleylamine parameters. The calculation defines adsorption-energy targets for subsequent DFT.

PANDAT AND EXPERIMENTAL PROGRAMME

Assessment, prediction and withheld-sample validation

Consistent first principles

Calculate fcc, hcp and 4H/dhcp energies, phonons, elastic stability, clean facets and ligand adsorption with matched functionals, spin–orbit treatment and convergence criteria.

Custom thermodynamic description

Define FCC_A1, hcp, dhcp and 4H as explicit phases; link metastable phases to fcc through physically interpretable lattice-stability functions.

Nano geometry and surface state

Introduce cylinder and faceted surface terms, adsorption coverage, strain and 4H/fcc interface energy with uncertainty bands.

Controlled validation

Measure diameter, length, ligand coverage, annealing time and temperature, morphology and initial/final 4H–fcc fractions.

Conservative parameter optimisation

Fit bulk terms first, then surface and ligand corrections. Retain one diameter range and one ligand condition outside calibration.

18-month work plan

Months 1–4Baseline

Reproduce 4H-rich samples, compile constraints, define phases and begin DFT.

Months 5–8Initial assessment

Complete relative-energy and surface calculations; construct the first T–r map.

Months 9–13Validation dataset

Annealing and ligand exchange; quantify phase fraction, morphology and surface coverage.

Months 14–18Model evaluation

Refit, perform withheld-sample validation, uncertainty analysis and database release.

INTERPRETIVE RULE

Thermodynamic preference is not kinetic accessibility.

A sample retained outside the calculated 4H stability region will be classified as metastable and kinetically retained, not forced into an artificial equilibrium region. Time-dependent transformation will be analysed separately, initially with a Johnson–Mehl–Avrami form, and compared with the Pandat chemical driving force.

SCOPE AND SOURCES

Evidence, uncertainty and model limits

Data limitation

No direct calorimetric description exists for 4H Au. Relative DFT energies and observed transformation boundaries therefore anchor the first assessment.

Surface limitation

Facet-resolved 4H surface energies and ligand free energies are not yet available. Results are reported as sensitivity ranges.

Morphology limitation

Rayleigh breakup may precede phase transformation. Morphological instability must be recorded independently during heating.

Fan et al.

Stabilization of 4H hexagonal phase in gold nanoribbons

Nature Communications 6, 7684 (2015)

DOI ↗02

Li & Schönecker

First-principles prediction of the stacking fault energy of gold at finite temperature

Acta Materialia 135, 88–95 (2017)

DOI ↗03

Patra et al.

Properties of real metallic surfaces: Effects of density functional semilocality and van der Waals nonlocality

PNAS 114, E9188–E9196 (2017)

DOI ↗04

Bajaj et al.

Phase stability in nanoscale material systems: extension from bulk phase diagrams

Nanoscale 7, 9868–9877 (2015)

DOI ↗05

Han et al.

Gas-assisted transformation of gold from fcc to the metastable 4H phase

Nature Communications 11, 552 (2020)

DOI ↗