3D Printing

Checked against published tests. PEEK: 252 tensile tests, typical error 11 MPa. PEEK with carbon fibre: 154 tests, typical error 14 MPa.

Process inputs and their ranges follow the PEEK printing literature compiled by Fu and Zobeiry (2026). PEEK 450G: T_m 343 °C, T_g 143 °C, 1.30 g/cm³, about 100 MPa and 3.7–4 GPa injection molded (same source, Table 1).

Heat transfer. Bead width w = 1.1 × nozzle diameter. Each bead keeps one temperature: heat goes to the air (h = 30 W/m²K, natural convection with no part fan; Lee et al. 2022), to the coupon below (warmed by the platform, more in thin coupons) and to the beads it touches. Printing builds heat up: the part and the air next to it run ΔT = ρc_p(T_nozzle − 90 °C)·h / (t_layer·h_air) above the 90 °C chamber, so fast printing and short layers keep it hotter. Beads leave the nozzle at its temperature, up to 100 °C cooler at high flow w·h·v (the hot end cannot melt it fully). Colour scale as in Lee et al. (2022).

Contact width (viscous flow). η = 245 Pa·s at 400 °C, E_a = 193 kJ/mol, fitted to the PEEK 450G viscosity curve of Wang et al. (2019). While a bond is above 245 °C (supercooled melt) the nozzle squeezes melt into it: contact width D_ic (the neck between two beads, as a share of the bead) = stadium contact + (1 − stadium) Φ/(1 + Φ), Φ = (0.2 mm/h)^2.5 ∫ p/η dt with p = 25 kPa. Necks also grow by surface tension (Pokluda et al. 1997; Bellehumeur et al. 2004). The open corners between beads are the voids; infill below 100% adds gaps.

Autohesion. Where the beads touch, chains interdiffuse across the interface by reptation: degree of autohesion D_au = (∫dt/t_r)^1/4 (Wool and O'Connor 1981; Lee and Springer 1987), with the reptation time t_r following the same E_a (Yang and Pitchumani 2002). Bond quality = D_ic × D_au.

Crystallization (nucleation and growth). As a bead cools, nuclei appear: N(T) = 1.5×10⁻³ e^{0.025(310 − T)} per µm³ (cooled deeper, more and smaller spherulites). Every spherulite front grows at the Lauritzen–Hoffman rate G(T) = G₀ e^{−U*/R(T − T_∞)} e^{−K_g/(T ΔT f)}, fitted to the spherulite growth rates measured by Wynn and Zobeiry (2023): G₀ = 7.2×10⁵ µm/min, K_g = 4.6×10⁵ K², U* = 2800 J/mol, T_∞ = T_g − 30 °C, T°_m = 380.5 °C; no growth below T_g. Fraction transformed F = 1 − e^−E, E = Σ (4π/3) N_k (∫G dt since nuclei k formed)³ + (4φ/d)∫G_tc dt (Kolmogorov–Avrami). Each newly transformed volume crystallizes only up to X_∞(T), the most reachable where it forms (Hamid et al. 2026): about 35% at 260–280 °C, 25% at 200 °C, 10% at 150 °C, so fast-cooled beads end less crystalline. Crystallizing releases 130 J/g of crystal, which warms the bead by about 20 °C at 30%. Cooled at a constant rate, this model leaves PEEK nearly amorphous above about 3000 °C/min, close to the 2700 °C/min measured by fast-scan calorimetry (Hamid et al. 2026). Carbon fibre surfaces nucleate below 320 °C, at small undercooling, so a transcrystalline layer starts growing outward before most bulk nuclei appear; it grows at its own rate G_tc (K_g = 5.4×10⁵ K², G₀ = 3.2×10⁶ µm/min, fitted to the same measurements) and meets the spherulites where the fronts collide. Each new layer reheats the ones below (Lee et al. 2022). The spherulite view is the same calculation in a slice along the bead above: each point belongs to the front that reaches it first. Its fibres lie in the slice; at ±45° the fibres of the bead below cross it and show end-on. Measured crystallinity: 11–20% (Lee et al. 2022), 17% (Zhao et al. 2018), 22–32% (Zhang et al. 2023).

Strength. Bulk (72 + 6X MPa) × e^{−0.63·voids}. Across the printed tests crystallinity changes strength little: bonds and voids decide it. Moulded PEEK does gain strength with crystallinity, from 72 to 100 MPa between 16 and 39% (Talbott et al. 1987). The weakest of: beads breaking, bonds pulled apart (σ₂/sin²θ), shear between beads (τ/(sinθ cosθ)). The section is two perimeter walls, three solid layers at the top and bottom, and the infill core.

Test. The walls, the solid layers and each simulated bead are separate elements, each with its own strength from its bonds, voids and crystallinity (±5% scatter). They stretch together and break one by one; the curve is their sum.

Calibration. Fitted to the 252 PEEK tests, from 28 studies in the database that extends Fu and Zobeiry (2026); within one study the typical error is 6 MPa. Raster angle matters more here than in the tests, and thin layers gain less than measured.

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