Melting Point Modeling (Ambient Pressure)

This repository evaluates how well classic melting-point theories explain metal melting points using public elemental data at ~1 atm. It is intended as a transparent baseline before building richer predictive models.

What This Repo Produces

• A metals-only dataset assembled from mendeleev.
• A theory-to-proxy evaluation report with metrics and plots.

Report

• reports/melting_point_model_report.md is the primary artifact. It documents the dataset, scope, model mapping, and fit metrics.
• Plots referenced by the report live in reports/figures/.

Theories and Relationships

This section lists the physical ideas considered and how they are mapped to available proxies in the current baseline. Each theory description includes a note on whether it is evaluated with the current dataset.

Theory-to-Feature Mapping (Current Baseline)

Theory	Proxy features in this repo	Status
Thermodynamic equilibrium / Richards rule	fusion_heat	Evaluated
Lindemann criterion (partial)	lattice_constant, atomic_weight	Evaluated (partial)
Born mechanical stability	None (elastic constants missing)	Not testable
Cohesive energy / bonding strength	evaporation_heat	Evaluated
Debye temperature relation	None (Debye temperature missing)	Not testable
Bonding type / coordination / structure	atomic_radius, density, atomic_volume	Evaluated
Electronic structure (metals)	en_pauling, en_allen, electron_affinity	Evaluated
Miedema model parameters	en_miedema, miedema_electron_density, miedema_molar_volume	Evaluated
Empirical melting curves (pressure)	None (pressure parameters missing)	Not testable

Thermodynamic equilibrium and Richards rule

Melting occurs when the Gibbs free energies of solid and liquid are equal. At equilibrium,

$$ G_s(T,p) = G_l(T,p). $$

At fixed pressure this gives

$$ T_m = \frac{\Delta H_{\mathrm{fus}}}{\Delta S_{\mathrm{fus}}}. $$

Richards rule assumes $\Delta S_{\mathrm{fus}}$ varies weakly across metals, so $T_m$ scales approximately with $\Delta H_{\mathrm{fus}}$. This turns the theory into a testable relationship by checking whether fusion enthalpy alone can explain variation in melting points. Status: evaluated. Proxy features used here: fusion_heat.

Lindemann criterion (vibrational instability)

Melting is triggered when RMS atomic vibration reaches a fraction of the interatomic spacing,

$$ \sqrt{\langle u^2 \rangle} = c,a. $$

Under typical approximations the melting temperature can be expressed as

$$ T_m = \frac{2\pi^2 m c^2 a^2 \Theta_D^2 k_B}{h^2}. $$

This shows how mass, lattice spacing, and phonon stiffness influence $T_m$. In this baseline we do not have phonon spectra or Debye temperature, so we only approximate the structure and mass terms. Status: partially evaluated. Proxy features used here: lattice_constant, atomic_weight.

Born mechanical stability (elastic softening)

Crystals become mechanically unstable when elastic constants violate stability criteria. For cubic crystals, a common set is

$$ C_{11} - C_{12} > 0,\quad C_{11} + 2C_{12} > 0,\quad C_{44} > 0. $$

In this picture, melting corresponds to a stability constant softening toward zero as temperature rises, for example

$$ C_{44}(T_m) = 0 $$

or $C_{11}(T_m) - C_{12}(T_m) = 0$. This connects $T_m$ to elastic constants and their temperature dependence. Status: not testable with current data because elastic constants are not present in the dataset.

Cohesive energy and bonding strength

Bonding strength feeds into $\Delta H_{\mathrm{fus}}$ and therefore $T_m$ via the equilibrium relationship above. Cohesive energy captures the energy required to separate a solid into isolated atoms and should correlate with the stability of the solid phase. If cohesive energy is high, melting should require higher temperatures. Status: evaluated. Proxy features used here: evaporation_heat as a cohesive energy proxy.

Debye temperature and phonon spectrum

Debye temperature $\Theta_D$ tracks lattice stiffness through the phonon spectrum,

$$ \Theta_D = \frac{\hbar \omega_D}{k_B}. $$

Under Lindemann-type arguments, $T_m \propto \Theta_D^2$ when other parameters are fixed. This makes $\Theta_D$ a direct physical predictor of melting stability. Status: not testable with current data because Debye temperature is not present in the dataset.

Bonding type, coordination, and crystal structure

Bonding and packing change both $\Delta H_{\mathrm{fus}}$ and $\Delta S_{\mathrm{fus}}$, affecting $T_m$ even at fixed chemistry. Close packed structures, for example, typically lead to stronger bonding and higher melting points than open structures. The current dataset does not encode full crystal structures, so we rely on coarse geometric proxies. Status: evaluated with coarse structural proxies. Proxy features used here: atomic_radius, density, atomic_volume.

Electronic structure (metals)

Valence electron count and electron density influence metallic bonding strength and thus $T_m$ via $\Delta H_{\mathrm{fus}}$. Trends in electronegativity and electron affinity correlate with how strongly atoms share electrons and how cohesive the metallic bond is. Status: evaluated. Proxy features used here: en_pauling, en_allen, electron_affinity.

Miedema model (metallic bonding parameters)

The Miedema model uses semi-empirical parameters to estimate cohesive behavior in metals. A common form for the enthalpy of formation of a binary alloy is

$$ \Delta H = \frac{2P}{n_{ws}^{1/3}}(\Delta \phi)^2 + Q(\Delta n_{ws}^{1/3})^2 - R, $$

where $\phi$ is a work-function-like electronegativity parameter, $n_{ws}$ is the electron density at the Wigner-Seitz boundary, and $P, Q, R$ are empirical constants. In this project we do not compute $\Delta H$ directly and we do not treat $\Delta H$ as the target. Instead, we use the elemental Miedema parameters as proxies for metallic bonding strength and relate them to the target melting point $T_m$ via regression. This clarifies the role of Miedema parameters: they are inputs, not the target. Status: evaluated. Proxy features used here: en_miedema, miedema_electron_density, miedema_molar_volume.

Empirical melting-curve relations (pressure dependence)

The Simon-Glatzel equation is often used for pressure-dependent melting:

$$ T_m = T_{\mathrm{ref}}\left(\frac{P - P_{\mathrm{ref}}}{a} + 1\right)^b. $$

At ambient pressure the parameters are not identifiable, so this is not evaluated in the current dataset. Status: not testable with current data.

How To Reproduce

python -m venv .venv
.venv/bin/python -m pip install -r requirements.txt
.venv/bin/python scripts/build_dataset.py
.venv/bin/python scripts/evaluate_theories.py

References

[1] https://en.wikipedia.org/wiki/Melting_point [2] https://en.wikipedia.org/wiki/Entropy_of_fusion [3] https://en.wikipedia.org/wiki/Enthalpy_of_fusion [4] https://en.wikipedia.org/wiki/Lindemann_criterion [5] https://en.wikipedia.org/wiki/Debye_temperature [6] https://en.wikipedia.org/wiki/Born_criteria [7] https://en.wikipedia.org/wiki/Simon%E2%80%93Glatzel_equation [8] https://en.wikipedia.org/wiki/Lattice_energy [9] https://en.wikipedia.org/wiki/Crystal_structure [10] https://en.wikipedia.org/wiki/Coordination_number [11] https://en.wikipedia.org/wiki/Metallic_bonding