Glass Viscosity Curvature from Constraint-Driven Actualization: A Physical Parity with the Vogel-Fulcher-Tammann Relation

TL;DR

The paper tackles the problem of super-Arrhenius viscosity in glass-forming liquids by proposing a physically interpretable CPA-based mechanism. It introduces a CPA + C model where a temperature-dependent constraint load $\hat{C}(T)$ modulates the CPA rate, yielding $\log_{10}\eta(T)=\log_{10}\eta_{0}+\frac{\kappa}{T-T_{g}}[1+\lambda\hat{C}(T)]$, with $T_{g}$ as a finite CPA Lock-In temperature. Across three diverse datasets (OTP from Laughlin & Uhlmann 1972, OTP from Plazek 1994, and glycerol–water mixtures from Kumar 1994), the CPA + C model achieves $R^{2}$ values > 0.99 and reproduces viscosity over ~14 orders of magnitude, matching or slightly outperforming VFT. This work provides a physically grounded alternative to the VFT divergence, suggesting that the sensitivity to configurational constraint drives the observed curvature and enabling principled extrapolation and design of glass-forming materials.

Abstract

The Vogel-Fulcher-Tammann (VFT) equation empirically describes the super-Arrhenius viscosity of glass-forming liquids; however, its divergence at a finite temperature $T_{0}$ lacks a clear physical basis. Here, a formulation derived from Dynamic Present Theory (DP$Φ$) and its Continuous Present Actualization (CPA) framework is tested: a CPA Rate modulated by a temperature-dependent Constraint Load $C(T)$, the CPA + Constraint (CPA + C) model. This formulation was evaluated across three canonical datasets: ortho-terphenyl (OTP) measurements from Laughlin and Uhlmann (1972) and Plazek et al. (1994), and glycerol-water mixtures from Kumar et al. (1994). Across all evaluated systems, the CPA + C formulation demonstrated statistical parity with the VFT model ($R^{2} > 0.99$), reproducing the viscosity curve by more than 14 orders of magnitude. Unlike the VFT equation, this approach derives the nonlinear curvature from a physically interpretable mechanism: the increase in configurational constraint as the system approaches a CPA Lock-In threshold. These findings indicate that the residual noise observed in simpler models represents an actual physical signal, which the CPA + C model effectively isolates. This suggests that the VFT's empirical success originates from its implicit capture of an underlying constraint-driven dynamic, providing a physically interpretable foundation for one of the most enduring phenomenological equations in materials science.

Figures (3)

• Figure 1: Viscosity of ortho-terphenyl versus temperature (Laughlin & Uhlmann dataset). The CPA + Constraint (CPA + C) model (Eq. 3) achieves statistical parity with the empirical VFT equation (Eq. 1), successfully capturing the super-Arrhenius curvature over 14 orders of magnitude. Viscosity $\eta$ is in $\text{Pa}\cdot\text{s}$. Both formulations explain 99.67% of the variance ($R^{2} = 0.9967$) with a nearly identical RMSE of $\approx 0.235$.
• Figure 2: Validation on independent OTP dataset (Plazek et al., 1994). The CPA + C model demonstrates robustness across independent experimental trials. Notably, in this dataset, the CPA + C model slightly outperforms the VFT reference ($\text{AIC} = -33.81$ vs $-33.42$) with $R^{2} = 0.9932$, indicating that the constraint-based mechanism effectively captures the viscosity evolution without requiring the VFT singularity.
• Figure 3: Extension to hydrogen-bonded networks (Glycerol-Water mixtures, Kumar et al., 1994). The CPA + C formulation maintains statistical parity ($R^{2} \approx 0.9981$) across three distinct mixture concentrations. This suggests the constraint-driven actualization mechanism is generalizable beyond van der Waals liquids to systems with different bonding topologies.