Stability Criteria for Embedded Printing of Newtonian Inks: Yield Stress vs. Interfacial Tension

Embedded 3D printing (EM3D) enables freeform patterning of soft matters by extruding liquid inks into a yield-stress support matrix. While EM3D has been successfully applied to viscoplastic and shear-thinning inks, it remains difficult for Newtonian fluids-such as silicone oil or liquid metal-due to two major instabilities: (i) matrix yielding caused by nozzle motion and (ii) Rayleigh–Plateau (RP) breakup of the printed filament. To address the first, we performed flow visualization via particle image velocimetry and found that the yielded region around the needle scales as , where is needle speed and is needle diameter. For RP instability, we derived and validated a stability criterion balancing interfacial tension and matrix yield stress , given by . This criterion predicts the transition between the stable and unstable printing regimes for Newtonian inks with extremely low (35 mN/m) and high (345 mN/m) interfacial tension. Our results establish simple, predictive guidelines for EM3D with Newtonian fluids, enabling its extension to soft electronics, bioprinting, and multi-material manufacturing.