Semi-Theoretical Modeling and Experimental Validation of the Extrusion Swell Ratio of Highly Concentrated Silver Paste in Micro-Extrusion

In micro-extrusion and direct ink writing, the nozzle outlet diameter is often used to estimate the deposited line width or free-filament diameter. However, highly loaded conductive silver pastes may exhibit pronounced extrusion swell after leaving the nozzle, resulting in a filament diameter larger than the nozzle inner diameter. To quantify this deviation, this study proposes a single-parameter semi-theoretical correction model based on radial force balance at the nozzle exit, integrating Herschel–Bulkley yield stress–shear-thinning rheology with a finite-deformation description. The exit radial stress is derived from pressure-driven circular tube flow, while the post-exit radial expansion is balanced against atmospheric constraint. A comprehensive correction force constant, C, is introduced to account for wall-induced energy dissipation, particle-structure rearrangement, residual elastic recovery, and model simplifications. After calibration using a transition-swelling nozzle, C was determined as 1.03 × 10⁻² N. The model was applied to six nozzle diameters and four nozzle length–pressure conditions. For Nozzles 1–4 with significant swelling, the mean absolute percentage error was 5.31%, while the overall error for all six nozzles was 11.84%, mainly due to overestimation for the nearly non-swelling Nozzle 6. For varying nozzle lengths, the error was 5.20%, and both experimental and predicted swell ratios decreased with increasing effective nozzle length. The model provides a semi-theoretical tool for estimating free-filament dimensions and analyzing nozzle-length effects, primarily under pronounced-swell conditions. Its predictive capability becomes limited as the swell ratio approaches unity, where additional corrections for wall slip, relaxation, and the zero-swell boundary are required.