A timescale-based framework for predicting filament stability in direct ink writing

Direct ink writing (DIW) often fails not because inks cannot be extruded, but because filaments lose continuity after nozzle exit due to insufficient resistance to extensional deformation. Although shear rheology governs extrusion, shape retention, and post-shear recovery, it does not fully capture this post-extrusion instability. Here, we isolate this extensional bottleneck using a rheology-decoupled model system in which shear properties are held nearly constant while filament extensibility is tuned by adding trace amounts of hydrolyzed polyacrylamide (HPAA) to a Pluronic F127 matrix. Despite nearly identical yield stress, shear-thinning behavior, viscoelastic moduli, and post-shear recovery, the formulations exhibit large differences in extensional response. By combining direct-writing experiments with capillary breakup extensional rheometry, we establish a predictive framework that links material response to processing conditions through two dimensionless parameters: the Deborah number, De=λ/τ_F, which characterizes the persistence of elastic stresses, and a process-imposed stretching parameter, Π=dvH/Q, which quantifies the kinematic deformation during deposition. The stability of the filament is organized within a stability map in the (De,Π) space. For practical implementation, we show that die swell at the exit of the nozzle correlates with De under fixed conditions, providing a simple proxy for locating a formulation within the stability map. Validation with independent formulations and benchmark printing tests demonstrates an accurate prediction of stable, transitional, and unstable regimes.