The critical plastocapillary number for a Newtonian liquid filament embedded into a viscoplastic fluid

Abstract

The yield stress of a viscoplastic material can stabilize an embedded fluid tunnel against capillarity-induced breakup, enabling remarkable technologies such as embedded 3D printing of intricate, freeform, and small components. However, there is persistent disagreement in the published literature between the observed minimum stable diameter, $d_{min}$, and the theoretical plastocapillary length $L_{pc}=2\Gamma /{\sigma }_y$, with interfacial tension $\Gamma$ and bath yield stress ${\sigma }_y$, leading to a prior hypothesis that the apparent surface tension $\Gamma$ is much smaller to enforce $d_{min}=L_{pc}$. Here we introduce and experimentally test a new hypothesis that the critical diameter is set by the dimensionless plastocapaillary number, $Y_{\Gamma }={\sigma }_{y}d/2\Gamma$, having a non-trivial critical value different than one, $Y_{\Gamma c}\ne 1$, and therefore the prior hypothesis of adjusting $\Gamma$ to enforce $L_{pc}=d_{min}$ is incorrect. 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 $Y_{\Gamma c}=0.21\pm 0.03$. 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 yield stress acts upon an effective area larger than the naïve estimate set only by embedded object diameter $d$. These results provide a new way to understand and predict the minimum stable diameter of embedded liquid filaments, as in embedded 3D printing, as $d_{min}=Y_{\Gamma c}(2\Gamma /{\sigma }_y)$.