Critical Cooling Rates Database

• The Critical Cooling Rates Database is a comprehensive resource that defines threshold cooling rates marking transitions between regimes like crystallization and vitrification.
• It integrates experimental measurements, simulations, and machine learning predictions with quantified uncertainties to ensure reproducibility and method transparency.
• The database supports diverse applications—from metallic glass formation to astrophysical plasmas—providing actionable insights into phase transitions and stability criteria.

A Critical Cooling Rates Database is a rigorously structured collection of quantitatively defined thresholds, rate coefficients, and their dependencies that mark transitions between distinct physical regimes (e.g., crystallization vs. vitrification, fragmentation vs. turbulence, stability vs. instability) in diverse physical, astrophysical, and materials systems. The database aggregates experimental measurements, theoretical models, simulation-derived rates, and machine-learning predictions, emphasizing reproducibility, uncertainty quantification, and parameter provenance for each entry. Its scope covers laboratory, computational, and cosmic environments, with direct application to fields such as metallic glass formation, astrophysical plasma evolution, nanomaterials, galaxy formation, and quantum optomechanics.

1. Definition and Scope

A Critical Cooling Rate (CCR), denoted $R_c$ (units: K/s), is the minimum (or threshold) time-averaged cooling rate required to prevent a specified phase or structural transition, such as the avoidance of crystallization (glass formation), the suppression of disk fragmentation, the maintenance of a meta-stable amorphous complexion, or the stability against thermal/catastrophic instabilities in plasma. For water, the directly measured value is $6.4\times 10^{6}$ K/s with an uncertainty of $\pm 0.5\times 10^6\, \mathrm{K/s}$ (Mowry et al., 2024).

The Critical Cooling Rates Database extends this concept to systems characterized by complex cooling functions $\Lambda(T, n, Z, ...)$, where "critical rate" may refer to the locus of instability criteria or the boundary in multidimensional physical parameter space.

2. Database Structure and Primary Measurement Protocols

Entries consist of the following principal components:

• Substance or System: Unambiguous naming (e.g., "Pure Water (H₂O)", "Cu–3 at.% Zr", "collisionally cooled ISM").
• Critical Cooling Rate (CCR), $R_c$: Measured or computed value with explicit units and uncertainty.
• Determination Methodology: Experimental setup (e.g., shaped microsecond laser-pulse melting and flash freezing with in situ electron diffraction for vitrification) (Mowry et al., 2024), simulation protocol (e.g., time-resolved molecular dynamics with precise rate ramping (Zhang et al., 18 Mar 2025)), or theoretical derivation (e.g., rate-equation solutions, threshold criteria).
• Supporting Parameters: Sample geometry/thickness, environmental conditions (e.g., vacuum, ambient pressure, redshift), auxiliary fields (e.g., composition, disorder, metallicity).
• Key Equations: Cooling rate definition $R=-dT/dt$; rate equations for energy evolution; analytic fits or tabulated critical curves.
• Uncertainty and Reproducibility: Statistical or systematic uncertainty quantitative bounds; database includes parameter provenance and method annotations.

Representative example from vitrification of water (Mowry et al., 2024):

Substance	Critical Cooling Rate	Measurement Method
Pure Water (H₂O)	$6.4\times 10^{6}$ K/s	Microsecond laser-pulse, flash freeze

3. Data Sources and Computational Protocols

Data are incorporated from diverse methodologies:

• Direct Experiment: Microsecond-resolved laser melting/freezing (cryo-EM vitrification), quench-gradient techniques for alloys, Compton cooling rates through resonance cross-sections for electrons in strong magnetic fields (Baring et al., 2011).
• Tabulated and Simulation Data: Multidimensional grids from codes such as Cloudy (ion-by-ion, element-by-element, non-equilibrium plasma cooling (Gnat et al., 2011, Lykins et al., 2012, Ploeckinger et al., 2020)), large-scale hydrodynamical simulations (critical rates for disk fragmentation (Baehr et al., 2015)), or time-dependent ODE solvers for non-equilibrium metal-enriched gas (Vasiliev, 2013).
• Machine Learning Models: Random forest regression trained on an expanded set of metallic glass CCRs, with database organization reflecting original feature provenance, error metrics, and predictive performance (Afflerbach et al., 2023).
• Analytic Fits/Instability Criteria: Threshold curves from Balbus’ TI analysis, extended with thermal conduction and generalized to map critical cooling loci in $(T, n, \lambda)$ parameter space (Stricklan et al., 19 May 2025), or critical curves in photochemistry (e.g., $k_{\mathrm{LW}}$ vs. $k_{\mathrm{H^{-}}}$ for H₂-cooling suppression (Wolcott-Green et al., 2016)).

Access modes include ASCII tables, HDF5 multidimensional arrays, online repositories, and programmatic interfaces (Python or C routines).

4. Theoretical Framework and Key Equations

Each database entry is underpinned by rigorous theoretical models that relate cooling rates to identifiable physical transitions. Representative equations include:

• Definition of cooling rate (experiment/theory):

$$R = -\frac{dT}{dt} = \frac{T(t_1) - T(t_2)}{t_2 - t_1}$$

• Plasma and astrophysical cooling:

$$\dot{E}_{\rm cool} = n_e n_H \sum_E A_E \Lambda_E(T)$$

with $$\Lambda_E(T) = \sum_i x_{E,i}(T) \Lambda_{e,E\,i}(T)$$.

• Thermal instability thresholds (Balbus):

$$\Lambda(T) > \Lambda_c^{\rm isochoric}(n,T) \equiv \Gamma(n) + T \frac{d\Lambda}{dT}$$

$$\Lambda(T) > \Lambda_c^{\rm isobaric}(n,T) \equiv \Gamma(n) + \frac{1}{2}\left[T \frac{d\Lambda}{dT} + n \frac{d\Gamma}{dn}\right]$$

• Glass formation and metallic systems:

$$R_c = A / D_c^B \quad\text{(for fit to critical casting diameter)}$$

$\ln \tau(T) = \ln \tau_0 + \nu + y T$

associating kinetics of nucleation with entropic regime bifurcations (Zhang et al., 18 Mar 2025).

5. Applications and Representative Physical Regimes

Critical Cooling Rates Databases provide the reference and discrimination thresholds for:

• Glass Formation and Vitrification: $R_c$ for water ($6.4\times 10^6\,\mathrm{K/s}$ (Mowry et al., 2024)), metallic glasses (data-driven $R_c$ for thousands of alloy compositions (Afflerbach et al., 2023)), and classified dynamical regimes in supercooled liquids (GFR vs CFR (Zhang et al., 18 Mar 2025)).
• Phase, Complexion, and Metastability Transitions: Amorphous-to-ordered intergranular transitions in nanocrystalline alloys with $R_c$ varying by orders of magnitude on chemical complexity (Grigorian et al., 2020).
• Astrophysical Plasmas and ISM Models: Ion-by-ion and element-by-element critical cooling rates for shock, radiative, or photoionized plasmas (Gnat et al., 2011, Lykins et al., 2012); cooling suppression in early protogalaxies (critical radiation backgrounds for $H_2$-chemistry (Wolcott-Green et al., 2016)).
• Thermal and Catastrophic Instabilities: Mapped critical cooling curves to delineate stable and unstable (TI/CC) regions under local and global conditions (Stricklan et al., 19 May 2025).
• Quantum-Level Optomechanics and Particle Trapping: Threshold ambient pressure and feedback cooling rates required for nanoparticle stability and ground-state cooling (Jazayeri, 2019).
• Disk Fragmentation and Gravitational Instability: Critical cooling parameter $\beta_{\rm crit}$ for the onset of self-gravitating disk fragmentation, exhibiting strong dependence on numerical model and cooling law (Baehr et al., 2015).
• White Dwarf Cooling Sequences: Cooling timescales and their critical inflection points as a function of mass, composition, and boundary conditions (Salaris et al., 2010).

6. Practical Database Implementation and Usage

The standard schema includes explicit fields for physical quantities ($R_c$, composition, method, uncertainty), environmental/control parameters (density, temperature, metallicity, pressure, field strength), solution or interpolation routines (gridded tables, analytic expressions, or machine learning models), and reproducibility meta-information.

Key recommendations for users:

• Interpolate logarithmically for all quantities spanning multiple orders of magnitude (e.g., $\log_{10} R_c$ or $\log_{10} \Lambda(T)$).
• For multi-parameter dependencies, discretize on a fine $(T,n,Z,...)$ grid and include all variables affecting criticality (e.g., system size, wavelength, conduction).
• Where cooling and heating processes are tabulated separately, the critical rate for stability is typically at the sign change (zero-crossing) of $$\Lambda_{\rm net}(T) = \Lambda_{\rm cool}(T) - \Gamma_{\rm heat}(T)$$.
• Tag each entry with both the physical transition it delineates (e.g., vitrification, fragmentation onset, TI/CC stability, glass formation) and the reliability/validation class (experiment, simulation, analytic, machine-learned).

7. Uncertainties, Validation, and Limitations

All database entries include explicit or recommended uncertainty bounds (e.g., $\pm 0.5\times 10^6\,\mathrm{K/s}$ for water $R_c$ (Mowry et al., 2024); $\pm15\%$ or $\pm25\%$ for plasma/atomic data (Lykins et al., 2012); RMSE in $\log_{10} R_c$ for machine-learning models (Afflerbach et al., 2023)). Systematic limitations are flagged: model-dependent thresholds (e.g., resolution dependence in disk fragmentation), incomplete atomic/molecular data in cooling curves, limited transferability of simulation-trained models, and caveats regarding environmental parameters (e.g., UV field, shielding, thermal conduction).

The database is an evolving infrastructure designed to enable fast, robust stability or transition queries, support data-driven model calibration, and ensure that cross-comparisons, meta-analyses, and simulation embedding are performed on a consistent quantitative basis.