Embedded 3D Printing of Newtonian Fluids in Elasto-viscoplastic Matrix

Embedded 3D printing (EM3D) enables freeform patterning of soft materials by extruding ink into a yield-stress supporting matrix. While prior studies have focused on viscoplastic or shear-thinning inks, printing Newtonian fluids─such as silicone oil and liquid metal─remains challenging due to (i) matrix yielding induced by needle motion and (ii) Rayleigh-Plateau (RP) instability driven by interfacial tension. In this study, we investigate the EM3D of Newtonian inks with extremely low surface tension (silicone oil) and extremely high surface tension (Galinstan liquid metal), embedded in an elasto-viscoplastic Laponite matrix. Flow visualization with particle image velocimetry reveals that matrix yielding around the needle scales with (γ̇_c/γ̇_Y )^1/3, where γ̇_c = U/d is the characteristic shear rate and γ̇_Y = 2πfγ_Y is the yield threshold derived from amplitude sweep rheology. We demonstrate that printing orthogonal to a straight-needle aggravates matrix yielding and compromises print fidelity. To resolve this issue, we propose a bent-needle geometry, which reduces the yielded region and improves filament stability by minimizing stress propagation along the needle path. To address RP instability, we derive a theoretical stability criterion that balances interfacial tension Γ and yields stress τ_Y, given by τ_Y ∝ Γ/d. This prediction is experimentally validated using Newtonian inks with distinct interfacial tensions (35 mN/m for silicone oil and 345 mN/m for Galinstan). Our findings provide a unified design framework for Newtonian-ink EM3D, incorporating both rheological and geometric strategies to overcome flow-induced instability. This work expands the accessible material space for EM3D by providing fundamental insights into fluid-matrix interactions, offering practical guidelines for reliable printing of Newtonian inks in soft electronics and bioprinting applications.