Coalescence of Printed Yield Stress Filaments in Direct Ink Writing

TL;DR

This work analyzes the arrested coalescence of two neighboring yield-stress filaments in direct ink writing by combining scaling theory, elasto-viscoplastic (EVP) simulations based on the Saramito model, and optical coherence tomography (OCT) experiments on Carbopol gels. A central finding is that in the viscoplastic limit the final bridge height scales nearly linearly with the plastocapillary number $\mathcal{J}$, with a geometry-dependent critical value where coalescence ceases. Elasticity can modify this outcome, enabling larger arrested bridges and producing transient oscillations that reflect a balance among capillary, elastic, and yield stresses; in the Kelvin–Voigt limit, the system exhibits damped oscillations with a frequency set by $\sqrt{\mathrm{Oh}_p/\mathrm{De}}$. Overall, the study provides a framework to predict deposition profiles and mitigate residual ridges in DIW, while highlighting the sensitivity to initial filament geometry and suggesting extensions to embedded printing and alternative constitutive models.

Abstract

In direct ink writing (DIW), neighbouring filaments of yield-stress inks are deposited side-by-side and are expected to merge into smooth, mechanically robust structures. Unlike Newtonian filaments, coalescence can arrest in finite time, leaving a permanent, non-flat ridge set by the competition between capillarity and rheology. Here we study the coalescence of two printed yield-stress filaments, combining scaling theory for the arrested state, direct numerical simulations, and DIW experiments on Carbopol gels imaged by optical coherence tomography. In the viscoplastic limit, we predict and observe an approximately linear decrease of the final bridge height with plastocapillary number and a critical yield stress above which coalescence does not initiate. Simulations further show that elasticity becomes important at high plastocapillary number, enabling larger final bridge heights via a crossover from a rigid Herschel--Bulkley solid to a deformable Kelvin--Voigt response. Our findings provide a framework for predicting deposition profiles and, ultimately, for mitigating residual topography in DIW.

Figures (9)

• Figure 1: Coalescence of two printed yield stress filaments: A. Experimental setup: two lines of yield–stress fluid are deposited onto a pre-wetted substrate using a custom direct-ink-writing system. A syringe pump extrudes material through a nozzle at a constant flow rate $Q$, while the substrate translates beneath it. The resulting filaments are imaged using Optical Coherence Tomography (OCT). B. OCT cross-sections of the coalesced filaments for materials with low and high yield stress, showing the final arrested shapes. The overlaid lines denote the tracked surface profiles. C. The coalescence geometry and parameters. Two filaments with combined cross-sectional area $\mathcal{L}^2$, initially separated by a gap, spread and merge to form a central bridge. The bridge height $\mathcal{H}(t)$ evolves until flow ceases due to yielding, producing a well-defined final shape as $t \to \infty$.
• Figure 2: Merging dynamics of quasi-viscoplastic filaments ($\mathrm{De} = 10^{-3}$). Panel A shows a time evolution of the shear rate for filaments with $\mathcal{J} = 0.4$: large plug regions form over time and coalescence stops eventually. Panel B illustrates the time evolution of filaments for three values of $\mathcal{J}$, illustrating the pronounced arrested coalescence for large $\mathcal{J}$. Panel C contains the time evolution of kinetic energy and the bridge height, showing how the bridge reaches an arrested shape at late-time in the presence of yield-stress. Panel C also shows the final bridge height as a function of $\mathcal{J}$, compared with the linear scaling predicted in equation \ref{['eq:theory_scaling_1']}. The prefactor $\Omega_2 = -0.4$ was obtained by fitting the slope of the numerical data. In all simulations we keep fixed $\mathrm{Oh}_s = 1/90$ and $\mathrm{Oh}_p = 8/90$.
• Figure 3: Merging dynamics of elastoviscoplastic filaments. Panel A shows time evolution of the flow type $(\xi)$ and solidification $(\mathcal{S})$ parameters for filaments with $\mathcal{J} = 0.5$ and $\mathrm{De} = 0.2$. We note that specific flow regimes can be overlapped with yielded and unyielded areas. Panel B illustrates the interface evolution of filaments for three values of $\mathrm{De}$ and fixed $\mathcal{J} = 0.5$, illustrating that going from a rigid to an elastic solid allows for additional deformation and stronger coalescence. Panel C presents the bridge height over time for various $\mathrm{De}$ and $\mathcal{J}$, showing that elasticity plays a greater role in the solid regime than in fluidized one. In all simulations we keep fixed $\mathrm{Oh}_s = 1/90$ and $\mathrm{Oh}_p = 8/90$ fixed.
• Figure 4: Oscillatory dynamics of Kelvin-Voigt filaments during coalescence. We use $\mathcal{J}=10^4$ in all simulations, guaranteeing that the material is always unyielded. Panel A contains a sweep of $\mathrm{De}$ for fixed $\mathrm{Oh}_s = 1/90$ and $\mathrm{Oh}_p = 8/90$, illustrating the harmonic oscillator behaviour with a frequency determined by $\mathrm{De}$. Panel B presents a sweep of $\mathrm{Oh}_s$ for fixed $\mathrm{De} = 0.01$ and $\mathrm{Oh}_p = 8/90$, showing that the oscillator decay depends on $\mathrm{Oh}_s$, but the frequency does not. The filament shapes during these oscillations can be seen in supplementary videos II and III.
• Figure 5: Steady-state bridge height as a function of $\mathcal{J}$ and $\mathrm{De}$. Panel A shows results for small Ohnesorge numbers, where fast and inertial coalescence takes place. These results illustrate that elasticity is more influential to the steady-state solution when $\mathcal{J}$ is high. Panel B shows results from Carbopol experiments and simulations with comparable (high) Ohnesorge numbers. The elasticity of Carbopol has little influence on the final shape of filaments. The theory line is the same calculated for the purely viscoplastic section in figure \ref{['fig:vp_dynamics']}.