Face saturation at ρ * = 4 ln 2: a universal threshold separating cooperative from individual relaxation in hydrogen-bonded liquids
- 1Independent Researcher
Abstract
The non-Arrhenius crossover in the dielectric relaxation of liquid water near 35-45 • C has been documented for four decades. No existing model predicts the crossover temperature without a fitted parameter. We show that the crossover occurs at a topological threshold ρ * = 4 ln 2, derived from the 4-simplex (K5) and not adjusted. A periodic table of K5 hydrides (CH4, NH3, H2O, HF and their third-row analogues) reveals that water is the only simple hydride combining a permanent dipole with an effective hydrogen-bond density ρ above the threshold. The crossover at ∼ 40 • C is a direct consequence of this unique position. We test the universality of ρ * on a database of 11 hydrogen-bonding liquids with published temperature-dependent dielectric relaxation data. Using the measured number of hydrogen bonds per molecule nHB as the effective density ρ, the threshold classifies 10 of 11 liquids correctly: B19 • Regime II (ρ * < ρ < 2ρ *): cooperative. Water, formamide: 2/2. • Regime I (ρ < ρ *): individual. DMF, DMA, MeOH, EtOH, PrOH, DMSO, CH3CN: 7/7. • Regime III (ρ > 2ρ *): vitrification. Glycerol: 1/1. The single exception is N-methylformamide (NMF, nHB = 2.0 < ρ * , cooperative). Neutron diffraction shows NMF forms linear chains, not triangular clusters: its cooperativity arises from sequential bond breakage, a mechanism distinct from K5 face saturation. The gap between the lowest cooperative ρ (formamide, 3.08) and the highest individual ρ (ethanol, 1.5) straddles ρ * = 2.773 with no overlap. All nHB values are sourced from neutron diffraction or molecular dynamics; all τ (T) classifications are from the original authors. No existing model (Tanaka, Stanley, Poole) extends beyond water. D2O (nHB = 3.76) and ethylene glycol (nHB = 3.0-3.7) are predicted to show cooperative relaxation. Both are testable immediately. The threshold also predicts a critical concentration in water-alcohol mixtures: the non-Arrhenius crossover in τ (T) should disappear at x * MeOH ≈ 0.35 (25 • C) in water-methanol, and x * EtOH ≈ 0.39 in water-ethanol. B20 In the (T, nHB) plane, the cooperative-individual boundary is a horizontal line at nHB = ρ * , regardless of how nHB is varied (temperature, concentration, or pressure). Four K5 constants, together with one measured input (f coord), predict the relaxation time τ of 6 hydrides with τ free derived from geometry, and of 5 further molecules across three decades (3-1500 ps). All 11 predictions fall within a factor 3. B17 Companion script: 123 tests, 20 blocks, all PASS.
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BibTeX
@misc{blatiere2026face,
title = {{Face saturation at ρ * = 4 ln 2: a universal threshold separating cooperative from individual relaxation in hydrogen-bonded liquids}},
author = {Jean-Baptiste Blatiere and Jean-Baptiste BLATIERE},
howpublished = {ChemRxiv},
year = {2026},
month = aug,
doi = {10.26434/chemrxiv.15001225/v2},
url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15001225/v2},
note = {CC BY},
}
RIS
TY - UNPB
TI - Face saturation at ρ * = 4 ln 2: a universal threshold separating cooperative from individual relaxation in hydrogen-bonded liquids
AU - Jean-Baptiste Blatiere
AU - Jean-Baptiste BLATIERE
PY - 2026
DA - 2026/08/31
PB - ChemRxiv
DO - 10.26434/chemrxiv.15001225/v2
UR - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15001225/v2
AB - The non-Arrhenius crossover in the dielectric relaxation of liquid water near 35-45 • C has been documented for four decades. No existing model predicts the crossover temperature without a fitted parameter. We show that the crossover occurs at a topological threshold ρ * = 4 ln 2, derived from the 4-simplex (K5) and not adjusted. A periodic table of K5 hydrides (CH4, NH3, H2O, HF and their third-row analogues) reveals that water is the only simple hydride combining a permanent dipole with an effective hydrogen-bond density ρ above the threshold. The crossover at ∼ 40 • C is a direct consequence of this unique position. We test the universality of ρ * on a database of 11 hydrogen-bonding liquids with published temperature-dependent dielectric relaxation data. Using the measured number of hydrogen bonds per molecule nHB as the effective density ρ, the threshold classifies 10 of 11 liquids correctly: B19 • Regime II (ρ * < ρ < 2ρ *): cooperative. Water, formamide: 2/2. • Regime I (ρ < ρ *): individual. DMF, DMA, MeOH, EtOH, PrOH, DMSO, CH3CN: 7/7. • Regime III (ρ > 2ρ *): vitrification. Glycerol: 1/1. The single exception is N-methylformamide (NMF, nHB = 2.0 < ρ * , cooperative). Neutron diffraction shows NMF forms linear chains, not triangular clusters: its cooperativity arises from sequential bond breakage, a mechanism distinct from K5 face saturation. The gap between the lowest cooperative ρ (formamide, 3.08) and the highest individual ρ (ethanol, 1.5) straddles ρ * = 2.773 with no overlap. All nHB values are sourced from neutron diffraction or molecular dynamics; all τ (T) classifications are from the original authors. No existing model (Tanaka, Stanley, Poole) extends beyond water. D2O (nHB = 3.76) and ethylene glycol (nHB = 3.0-3.7) are predicted to show cooperative relaxation. Both are testable immediately. The threshold also predicts a critical concentration in water-alcohol mixtures: the non-Arrhenius crossover in τ (T) should disappear at x * MeOH ≈ 0.35 (25 • C) in water-methanol, and x * EtOH ≈ 0.39 in water-ethanol. B20 In the (T, nHB) plane, the cooperative-individual boundary is a horizontal line at nHB = ρ * , regardless of how nHB is varied (temperature, concentration, or pressure). Four K5 constants, together with one measured input (f coord), predict the relaxation time τ of 6 hydrides with τ free derived from geometry, and of 5 further molecules across three decades (3-1500 ps). All 11 predictions fall within a factor 3. B17 Companion script: 123 tests, 20 blocks, all PASS.
ER -