A holistic model for melt electrowritten three-dimensional structured materials based on residual charge

Abstract

The printing accuracy of polymer melt electrowriting is adversely affected by the residual charge entrapped within the fibers, especially for three-dimensional (3D) structured materials or multilayered scaffolds with small interfiber distances. To clarify this effect, an analytical charge-based model is proposed herein. The electric potential energy of the jet segment is calculated considering the amount and distribution of the residual charge in the jet segment and the deposited fibers. As the jet deposition proceeds, the energy surface assumes different patterns, which constitute different modes of evolution. The manner in which the various identified parameters affect the mode of evolution are represented by three charge effects, including the global, local, and polarization effect. Based on these representations, typical modes of energy surface evolution are identified. Moreover, the lateral characteristic curve and characteristic surface are advanced to analyze the complex interplay between fiber morphologies and residual charge. Different parameters contribute to this interplay either by affecting residual charge, fiber morphologies, or the three charge effects. To validate this model, the effects of lateral location and grid number (i.e., number of fibers printed in each direction) on the fiber morphologies are investigated. Moreover, the "fiber bridging" phenomenon in parallel fiber printing is successfully explained. These results help to comprehensively understand the complex interplay between the fiber morphologies and the residual charge, thus furnishing a systematic workflow to improve printing accuracy.

Keywords: Charge polarization; Energy analysis; Fiber morphologies; Melt electrohydrodynamic printing; Residual charge.

Figure 1

Schematic of melt electrowriting system.

Figure 2

Schematic of jet deposition process. (A) Schematic of the incoming jet segment of interest (in blue) approaching the topmost two layers of the scaffold. (B) The incoming jet segment of interest is modeled as a pair of charge segments whose charge amount are qdl and Qdl, respectively. Each fiber in the topmost two layers of the scaffold is modeled as two infinitely long uniform charge lines, whose linear charge densities are q and Q₀, respectively. In this way, all positive (negative) charge lines in the topmost two layers form a positive (negative) charge grid. The grey plane in (A) denotes the midplane between the positive and negative charge grids, whose center defines the origin of the coordinates system. (C) is the side view of (A) and (B) with polymer material and charge superimposed together. The distance between the centroids of positive and negative charges is 2L. The interfiber distance is S_f. The fiber diameter is d_f. Fiber A is an arbitrary deposited fiber in the topmost two layers of the scaffold. ① (or ②) and ③ (or ④) denote the positive (or negative) charge segment in the incoming jet segment of interest and charge line in Fiber A, respectively.

Figure 3

Exemplification of three charge effects (global, local and polarization effect) in two modes. (A) and (F) schematize the equivalent physical scenarios for Mode 1 (B–D) and Mode 2 (G–I), respectively. Phase 1 in (A) schematizes a positively charged segment approaching a positively charged plate. Phase 2 in (A) schematizes a positively charged segment approaching a positively charged grid. Phase 3 in (A) schematizes a charged segment with polarization approaching a charge grid with polarization. (B) can be understood in a similar way. The dimensionless parameters enabling Mode 1 and Mode 2 include α = 3, β = 3 (for Mode 1) or 0.1(for Mode 2), ξ = 15, η = 1, and K = 1. Black curves in (B–D, G–I) denote the locations on the energy surface prescribed by the toolpath or briefly prescribed locations. (E) and (J) show the evolution of black curves in these two modes with z. As explained later, these curves are essentially the lateral characteristic curves for Mode 1 and Mode 2.

Figure 4

Characterization of energy surface. (A) Dependence of energy surface on lateral position when z=5. Curve A and B are the intersecting curves of the energy surface with the plane x=0 and y=0. Based on the mathematical analysis, is not a function of y, and is not a function of x in Equation VI; the energy surface can be generated by translating Curve A along the path defined by Curve B or vice versa. Curve B is defined as the lateral characteristic curve, which is shown in (B). Curve C and D are also shown in (B) when z is toggled at 30 and 1.1, respectively. The dash-dotted line denotes the prescribed locations. When z is continuously varied, lateral characteristic curves continuously evolve, and form a characteristic surface as shown in (C).

Figure 5

The interplay between the fiber wall morphology, residual charge, and three charge effects (global, local, and polarization). (A–C) Dependence of characteristic surface on β. α = 3; β = 0.1, 1.5 and 3 for (A), (B), and (C), respectively; ξ = 10, η = 1and K = 2. The dash-dotted lines denote the prescribed toolpath, and the red curves denote the actual deposition trajectory. Since there is currently no way to determine β, the characteristic surfaces and the actual deposition trajectories may not exactly match as shown, therefore, (A–C) are qualitatively showing the dependence of jet deposition trajectory on β. (D–F) show the dependence of lateral characteristic curves at different z on β. Α = 3, ξ = 10, η = 1 and K = 1. (A), (B), and (C) correspond to the blue, green, and red curves in (D), (E) and (F), respectively. The purple rectangle in (F) denotes the region dominated by the polarization effect. (G) shows the evolution of wall structure with layer number. (H) schematizes the process of the jet approaching the intersection point after the lateral deviation has been formed. Point P on the incoming jet will be deposited on point R on the preexisting fiber. (I) summarizes the interplay of fiber morphologies, residual charge, and three charge effects (global, local, and polarization). Moreover, the ways in which different parameters factor into this interplay are also shown.

Figure 6

Effect of lateral location on jet deposition. (A) Lateral characteristic curves at different z. Parts of the orange curve in (A) denoted by ①②③ are locally enlarged and superimposed at the peak values in (B) for comparative analysis of the slopes. The decreasing rate of the energy function at the outer side is lowest for ① and highest for ③. The dash-dotted line denotes the prescribed path. (C) shows the dependence of the “spindle” size on lateral location (layer printing time 57 s). The closer it is to the scaffold periphery, the more obvious the “spindle-like” structure is. Voltage: 16 kV; pressure: 70 kPa; material temperature: 95°C; stage speed: 16 mm/s; layer printing time 57 s. α = β = 3, ξ = 60, η = 1, K = 3. All the curves are translated to compare their difference.

Figure 7

Effect of grid number on jet deposition. (A) and (B) show the dependence of lateral characteristic curves on K at z=60 and 10, respectively. (C–E) show the variation of “spindle-like” structure when K increases from 2 to 5. Due to the scope of the camera, (E) shows only part of the scaffold. As the grid number increases, the “spindle-like” structure becomes less significant. Voltage: 16 kV; pressure: 70 kPa; material temperature: 95°C; translational stage speed: 15 mm/s; layer printing time 92 s. α = β = 3, ξ = 60, η = 1. All the curves are translated when necessary to compare their difference.

Figure 8

Energy surface and lateral characteristic curves for parallel walls. (A) Schematic of “fiber bridging” phenomenon observed in printing parallel fibrous walls. (B) Energy surface at z = 5. (C) Energy surface at z = 2. The black curves show the lateral characteristic curves. (D) and (E) show the dependence of the lateral characteristic curves on β. For (B–E), α = 3, ξ = 5, η = 1, and K = 3. β = 0.5 for (B) and (C).