Revisiting the fiber gel point concentration model: from the perspective of flexibility and external fibrillation

Abstract

The gel point concentration model has seen significant success in industrial applications such as headbox optimization and predicting the length of high aspect ratio micro and nanoscale fibers. However, traditional models based on crowding number theory or effective medium theory exhibit significant inaccuracies when applied to highly refined pulps, primarily due to their neglect of key morphological properties: external fibrillation degree (D) and fiber flexibility (F). This study systematically investigates how these properties, alongside aspect ratio (A), modulate the gel point concentration (C_g) in bleached softwood kraft pulp and bleached bamboo kraft pulp. Experimental results reveal that C_g decreases with increasing external fibrillation degree or fiber flexibility, driven by distinct mechanisms: external fibrillation enhances the specific surface area of fibers and the number of fiber contacts, promoting the formation of fiber network; increased flexibility reduces fiber elastic modulus, enabling more deformable and extensive fiber–fiber interactions. To address these gaps, we developed an improved predictive model incorporating A, D, and F: \(C_{g} = 7.488 \times 10^{14} A^{ - 2.109} F^{ - 1.004} D^{ - 0.053}\) (30 < A < 70, 0 < D < 20%, 5 × 10¹⁰ N⁻¹ m⁻² < F < 3 × 10¹¹ N⁻¹ m⁻², R² = 0.954) which significantly outperforms traditional models. Notably, Sensitivity analysis further showed that the effect of external fibrillation is negligible (exponent = −0.053), justifying its exclusion in a simplified model yet robust model: \(C_{g} = 1.617 \times 10^{16} A^{ - 2.036} F^{ - 1.141}\) (R² = 0.950). These findings establish fiber flexibility as a critical determinant of C_g, on par with aspect ratio—a key insight absent in conventional frameworks. The proposed models provide a mechanistic basis for predicting gel point concentration in refined fiber suspensions, enabling more accurate optimization of papermaking and nanomaterial processing applications.