The critical Plasto-Capillary number and minimum feature size in embedded 3D printing*

Embedded 3D printing enables freeform manufacturing of intricate components by suspending an extruded ink material in a non-Newtonian yield-stress fluid bath, enabling fabrication of small, delicate, and soft structures not achievable by other means. While it is known that interfacial capillary effects should limit the minimum feature size, there is persistent disagreement in the published literature between the theoretical plasto-capillary length d = 2Γ/σ_y, set by a balance between interfacial tension Γ and bath yield stress σ_y, and the experimentally observed minimum feature sizes. Although this has been rationalized by adjusting the apparent value of interfacial tension Γ, here we introduce and experimentally test the hypothesis that the critical diameter is set by the dimensionless Plasto-Capaillary number, Pl = σ_yd/(2Γ), having a non-trivial critical value different than one, and therefore there is no need to adjust Γ. We study several Newtonian inks (uncured polydimethylsiloxane (PDMS), highly refined mineral oil, silicone oil) extruded into a wide range of non-Newtonian viscoplastic bath materials (polyacrylic acid microgels, polysaccharide microgels, nanoclay gel, and micro-organogels). Across this wide parameter space, we observe a critical value of Pl = σ_yd/(2Γ) = 0.21±0.03.  We explain this being less than one by analogy to other critical dimensionless groups with yield stress fluids, such as the gravitational stability of a suspended sphere or bubble where the effective area is larger than the naïve estimate set only by diameter d. These results provide a new way to determine the minimum feature size based on the rheological properties involved in embedded 3D printing, as d = 0.42 (Γ/σ_y).