Generalized Einstein Equation for Ceramics Suspension Rheology

This work introduces the Generalized Einstein-type Equation Rheological Gaussian Model (E²RG) for suspension rheology, extending the linear Einstein-type form to concentrated systems while preserving the correct Einstein limit at very low volume fractions. The formulation enables accurate prediction of relative viscosity across the entire packing interval from φ = 0 to φ_max. Unlike empirical power-law or exponential approaches, E²RG does not apply an exponent to particle concentration; instead, it modulates the entire Einstein expression through a correction based on the Gaussian error function, providing a smooth, physically consistent transition from ideal to nonideal regimes without introducing arbitrary constants. A key advantage of E²RG is its intrinsic verifiability: the model parameters retain clear physical meaning, reflecting the intensity of particle interactions rather than serving as free-fitting constants. After applying the correction term, linearization enables verification of whether the fit has correctly separated the ideal contribution from interaction effects, providing an internal self-consistency check of the model decomposition, confirming that the two parameters fulfill their intended roles rather than acting as compensating free variables. Beyond its mathematical robustness, the E²RG formulation is consistent with the Central Limit Theorem (CLT), since the Gaussian correction is consistent with the Gaussian limit expected from the cumulative effect of multiple weak constraints of many weak, multiplicative interaction constraints acting at the particle scale. The transition at φ_max. aligns with the jamming framework, offering a physically coherent interpretation of flow cessation without invoking a true viscosity divergence. Together, these features distinguish E²RG from conventional empirical models and make it a conceptually transparent and practically versatile tool for describing suspension flow behavior.