A hybrid Green-Kubo (hGK) framework for calculating viscosity from short MD simulations

Abstract

Viscosity calculation from equilibrium molecular dynamics (MD) simulations relies on the traditional Green-Kubo (GK) framework, which integrates the stress autocorrelation function (SACF) over time. While the formalism is exact in the linear response regime, the traditional approach often suffers from poor convergence and requires extensive phase space sampling, which is computationally demanding for soft matter and polymer systems. In this Letter, we introduce a hybrid Green-Kubo (hGK) framework that alleviates these limitations by partitioning the SACF into two physically meaningful regimes: (i) a short time ballistic component extracted directly from short MD simulations, and (ii) a long time relaxation tail represented using analytically motivated functions, $φ(τ)$, fitted only to short trajectories. This strategy bypasses the need for extensive sampling while preserving physical rigor. Benchmarking against SPC/E water confirms excellent agreement with established results, and we further demonstrate the efficacy of the method for challenging electrolyte systems (EC-LiTFSI and PEO-LiTFSI), for which the GK framework fails to converge. The computational savings are substantial, with reductions of several orders of magnitude in required sampling, achieved without compromising predictive accuracy. We also discuss the limitations of the hGK framework and outline clear avenues for refinement, including optimal tail selection and robust identification of relaxation regimes in noisy stress data. The hGK framework presented in this Letter provides a conceptually simple, broadly applicable, and computationally efficient route for viscosity prediction in molecular liquids, polymer melts, and ionically conducting soft materials.

Figures (3)

• Figure 1: Schematic illustration of the hGK framework. Left: A representative SACF showing an oscillatory short time region followed by a noisy long time tail. Only data up to $\tau_u$ are reliably sampled; beyond this window, the SACF is replaced by a fitted tail function $\phi(\tau)$ that captures the slow relaxation inaccessible to the traditional GK framework. Right: Running integral of viscosity $\eta(\tau)$ from the traditional GK framework, which fails to plateau, compared to the smooth convergence obtained using hGK. Tail reconstruction recovers the missing long time contribution and enables reliable viscosity calculation from short trajectories.
• Figure 2: Benchmarking the hGK framework proposed in this Letter to SPC/E water: (a) Normalized SACF, (b) Identification of the optimal boundaries for the fitting window with width $\tau_\Delta=\tau_u-\tau_l$, (c) Predicted viscosity as a function of $\tau_\Delta$ for different analytical tail models for $\phi(\tau)$. (d) Running integral of viscosity from the GK and hGK using different tail models. The hGK approach using stretched exponential yields the closest match with the GK and provides a smooth, noise free plateau.
• Figure 3: Demonstration of the hGK framework for liquid and polymer electrolytes: normalized SACF of (a) EC-LiTFSI and (b) PEO-LiTFSI electrolytes with slow tail highlighted in the respective insets. Running integral of viscosity for (c) EC-LiTFSI and (d) PEO-LiTFSI electrolytes. The insets to the figures show the convergence behavior as a function of $\tau_\Delta$ used to fit $C(\tau)$ (see Figure \ref{['fig:figure2']}). The hGK framework recovers stable plateaus for both systems despite their slow relaxation and noisy long time SACF tails.