Indice di coerenza energetica nei cristalli di neve

Energetic Coherence Interpretation of Snow Crystal Morphology

The Energetic Coherence Interpretation of Snow Crystal Morphology is a conceptual framework proposed by C.J. Pérez Pulido (2026) within the ISHEA Institute research program. It presents an interpretive layer for understanding transitions in snow crystal morphology as mapped in the Nakaya diagram.

The framework does not replace established crystallographic or surface-kinetics models but proposes an integrative parameter intended to organize energetic contributions influencing crystal growth.

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Background

Snow crystal morphology has been systematically studied since the work of Ukichiro Nakaya (1954), who mapped crystal habit as a function of temperature and water vapor supersaturation. Subsequent theoretical advances, particularly by Kenneth G. Libbrecht, explained morphological transitions using surface attachment kinetics and diffusion-limited growth theory.

Ice in its most common atmospheric form (Ice Ih) exhibits hexagonal symmetry due to the tetrahedral hydrogen-bond network of water molecules. Under varying thermodynamic conditions, crystals form plates, columns, needles, or dendritic structures.

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Conceptual Framework

The coherence interpretation introduces a qualitative parameter K intended to integrate multiple energetic contributions:

Molecular cohesion energy of the hydrogen-bond lattice

Surface free energy

Thermal energy

Environmental perturbations

Dipolar interactions

Directional vapor flux associated with supersaturation

The parameter is dimensionless when normalized as energy densities and is proposed as an organizational index rather than an independent predictive model.

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Interpretation of Morphological Transitions

Within this framework:

High K regimes correspond to stable dendritic growth

Intermediate K regimes correspond to plate or column formation

Low K regimes correspond to irregular or unstable growth

The persistence of sixfold symmetry is interpreted as reflecting a high-coherence configuration of the hydrogen-bond network in Ice Ih.

The model is presented as consistent with diffusion-limited aggregation theory and classical thermodynamics.

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Relationship to Established Models

The coherence framework is explicitly described as complementary to:

Nakaya’s morphology diagram

Surface-kinetics models of crystal growth

Classical crystallography of ice Ih

Thermodynamic nucleation theory

It does not modify established physical laws but attempts to provide an integrative interpretive layer across scales.

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Proposed Quantitative Development

The framework suggests that formalization would require:

Measurable lattice cohesion energy per unit volume

Surface free energy values for basal and prism faces

Boltzmann thermal energy scaling

Supersaturation gradient quantification

Environmental perturbation indexing

Experimental validation would involve controlled laboratory crystal growth with correlation between calculated coherence index values and observed morphology.

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See Also

Snow crystal

Nakaya diagram

Ice Ih

Diffusion-limited aggregation

Fractal geometry

Surface kinetics

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References

Nakaya, U. (1954). Snow Crystals: Natural and Artificial.
Libbrecht, K. G. (2005, 2017). Surface kinetics and snow crystal growth.
Petrenko & Whitworth (1999). Physics of Ice.
Debenedetti (1996). Metastable Liquids.
Mandelbrot (1983). The Fractal Geometry of Nature.

SUPPLEMENTARY MATERIAL

Coherence in Snow Crystal Growth: An Energetic Interpretive Layer on the Nakaya Morphology Diagram

Author: C.J. Pérez Pulido
Institution: ISHEA Institute
Date: March 2026

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S1. Purpose and Scope

This Supplementary Document provides:

Full dimensional formalization of the coherence parameter K

Energy-density normalization framework

Replicable laboratory protocol

Quantitative worked examples

Threshold modeling proposal

Simulation pathway integration

Statistical validation methodology

Physical constants and data sources

This work does not replace classical crystallography or surface-kinetics theory. It introduces a structured energetic interpretive layer consistent with established thermodynamics.

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S2. Dimensional Formalization of the Coherence Index

We define a dimensionless coherence index:

K^* = \frac{E_c + E_m + E_{\rightarrow}}{E_s + E_t + E_e}

All terms are expressed as energy densities (J·m⁻³) to ensure dimensional consistency.

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S3. Component Definitions and Calculations

S3.1 Cohesion Energy Density (Ec)

Hydrogen bond energy ≈ 20 kJ/mol
Effective lattice energy ≈ 40 kJ/mol

Conversion:

\frac{40,000 \, J/mol}{6.022\times10^{23}} = 6.64\times10^{-20} J/molecule

Ice Ih unit cell volume:

a = 4.52 Å
c = 7.35 Å

V_{cell} \approx 1.3\times10^{-28} m^3

Energy density:

E_c = \frac{6.64\times10^{-20}}{1.3\times10^{-28}} \approx 5.1\times10^{8} J/m^3

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S3.2 Surface Free Energy Density (Es)

Surface free energy (basal face at −10°C):

\sigma \approx 0.109 J/m^2

Volumetric approximation:

E_s = \frac{\sigma}{r}

For r = 10⁻⁵ m:

E_s \approx 1.09\times10^4 J/m^3

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S3.3 Thermal Energy Density (Et)

E_t = \frac{kT}{V_{cell}}

Boltzmann constant:

k = 1.38\times10^{-23} J/K

At T = 263 K:

kT = 3.63\times10^{-21} J

E_t \approx 2.8\times10^7 J/m^3

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S3.4 Dipolar Interaction Energy Density (Em)

Water dipole moment:

\mu = 1.85 D

Dipole–dipole interaction estimate:

U_{dd} \sim \frac{\mu^2}{4\pi\epsilon_0 r^3}

Using r ≈ 2.8 Å yields energy scale ≈ 10⁻²⁰ J

Converted to volumetric density:

E_m \sim 10^7 J/m^3

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S3.5 Supersaturation Flux Energy (E→)

Supersaturation:

\sigma_v = \frac{\rho_v - \rho_{eq}}{\rho_{eq}}

Diffusion flux:

J = -D \nabla \rho

D ≈ 2×10⁻⁵ m²/s

Energy contribution:

E_{\rightarrow} \sim \frac{L_s J}{v_{growth}}

Latent heat sublimation:

L_s = 2.83\times10^6 J/kg

This term increases sharply with supersaturation gradient.

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S3.6 Environmental Perturbation Term (Ee)

Modeled as:

E_e = \alpha ( \Delta P + \Delta C_{impurity} + \text{turbulence factor} )

Calibrated experimentally.

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S4. Numerical Example (−15°C, Moderate Supersaturation)

Numerator:

Ec ≈ 5.1×10⁸
Em ≈ 1×10⁷
E→ ≈ 5×10⁷

Denominator:

Es ≈ 1×10⁴
Et ≈ 2.8×10⁷
Ee ≈ 1×10⁷

K^* \approx 15

Predicted regime: dendritic growth stability.

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S5. Proposed Morphological Thresholds

K* Range Predicted Morphology

< 5 Irregular growth
5 – 10 Plates / Columns
≥ 10 Dendritic branching

Empirical calibration required.

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S6. Laboratory Replication Protocol

S6.1 Equipment

Controlled growth chamber (±0.1°C)

Supersaturation control (±1%)

Laminar airflow system

Optical microscope ≥1000×

Time-lapse imaging

Hygrometer and pressure control

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S6.2 Procedure

1. Set target temperature.

2. Establish defined supersaturation.

3. Introduce nucleation seed.

4. Record growth evolution.

5. Measure:

Tip velocity

Branch density

Fractal dimension

6. Calculate K*.

7. Correlate morphology vs K* regime.

Replicates: minimum n ≥ 30 per condition.

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S7. Simulation Integration

Modify surface-kinetics model:

\alpha_{eff} = \alpha_0 \cdot g(K^*)

Where:

g(K) > 1 in high-coherence regime
g(K) ≈ 1 in intermediate regime
g(K*) < 1 in low regime

Monte Carlo DLA simulations can integrate this weighting.

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S8. Statistical Validation

Measurements:

Fractal dimension (box-counting)

Branch density per mm

Growth velocity

Tip instability frequency

Analysis:

Pearson correlation between K* and fractal dimension

ANOVA across morphological classes

Significance threshold p < 0.05

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S9. Physical Constants

Boltzmann constant: 1.38×10⁻²³ J/K
Avogadro number: 6.022×10²³
Latent heat sublimation: 2.83×10⁶ J/kg
Ice surface energy: 0.109 J/m²
Diffusion coefficient (water vapor in air): 2×10⁻⁵ m²/s

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S10. Limitations

K* currently semi-phenomenological.

Threshold values require empirical calibration.

Environmental perturbation term is simplified.

Surface anisotropy not fully decomposed per crystallographic plane.

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S11. Reproducibility Statement

All equations are dimensionally consistent.
All constants are publicly available.
Laboratory setup follows established ice-growth methodologies.
Framework is compatible with existing surface-kinetics theory.

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S12. Future Work

Plane-specific energy density decomposition.

Time-dependent coherence evolution modeling.

Coupling with full Navier–Stokes vapor transport simulations.

Controlled microgravity validation experiments.

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References

Nakaya (1954)
Libbrecht (2005, 2017)
Petrenko & Whitworth (1999)
Debenedetti (1996)
Mandelbrot (1983)