Nanomaterials, HEPA Filtration & Tissue Scaffolds, Taylor-Cone Electrospinning
Electrospinning Taylor-Cone Critical Voltage & GSM Formula
Calculate G. I. Taylor Critical Cone-Jet Onset Voltage Vc (kV), Mean Inter-Electrode Field Vc/H (kV/cm), Dry Nanofiber Production Rate (g/h), and Nonwoven Mat GSM (g/m²).
Why neat aqueous PEO/PVA solutions spark corona discharges before spinning—until surfactant cuts 72 dyn/cm surface tension below the Taylor threshold!
**Electrospinning** charges a polymer solution meniscus (`Surface Tension gamma` dyn/cm = mN/m, `Solids Concentration C` g/mL) pumped at `Q` (`mL/h`) through a metallic Spinneret Needle of Outer Radius `R` (`cm`) and Protrusion Length `L` (`cm`) spaced `H` (`cm`) from a grounded collector drum! Below a sharp electrostatic threshold, surface tension holds a rounded droplet; at **Sir Geoffrey Ingram Taylor’s (`1969`) Critical Voltage `V_c = SQRT(((4 * H^2) / L^2) * (LN((2 * L) / R) - 1.5) * (0.117 * PI() * R * gamma))`** (`kV`), Maxwell electrostatic traction snaps the meniscus into the `49.3°` half-angle **Taylor Cone**, launching a whipping polymer jet that flashes off solvent into `100–400` nm nanofibers at **Dry Fiber Mass Rate `dot_m_{dry} = Q * C`** (`g/h`) and **Roll-to-Roll Nonwoven Basis Weight `GSM = dot_m_{dry} / dot_A_{web}`** (`g/m²`)!
Verify on a `21`-Gauge (`R = 0.041` cm) PVDF / DMF Battery-Separator Electrospinner—Tip-to-Collector_Gap `H` = 15.0 cm (B2), Needle_Length `L` = 2.50 cm (B3), Outer_Radius `R` = 0.041 cm (B4), Solution_Surface_Tension `gamma` = 38.0 dyn/cm (B5), Syringe_Flow `Q` = 1.80 mL/h (B6), Polymer_Solids `C` = 0.120 g/mL (B7), and Substrate_Web_Rate `dot_A` = 0.24 m²/h (B8): Taylor Critical Voltage `V_c` equals `16.51` kV (E2 = `1.100` kV/cm average field in E3), Dry Nanofiber Output equals `0.216` g/h (E4), depositing a `0.90` g/m² (`GSM`) nanofiber veil (E5)!
| Taylor (1969) Electrospinning KPI (H=15 cm, L=2.5 cm, R=0.041 cm, gamma=38 vs 72 dyn/cm) | Formula | Verified Output |
|---|---|---|
| Taylor Critical Voltage Vc (kV) | SQRT((4*H^2/L^2)*(LN(2*L/R)-1.5)*(0.117*PI()*R*gam)) | 16.51 kV (22.72 kV corona risk @ 72 dyn/cm!) |
| Mean Onset Field Vc / H (kV/cm) | Critical_Voltage_Vc_kV / Gap_H_cm | 1.100 kV/cm (1.515 kV/cm @ neat water) |
| Dry Nanofiber Mass Rate (g/h) | Syringe_Flow_mL_h * Polymer_Conc_g_mL | 0.216 g/h / nozzle |
| Nanofiber Veil Basis Wt (g/m²) | Dry_Nanofiber_g_h / Web_Area_Rate_m2_h | 0.90 g/m² (HEPA slip-flow veil) |
G. I. Taylor (1969) Electrostatic Meniscus Instability & Nanofiber Web Formula
Validate `Gap H > L >= 0.3` cm, slender needle `2 * L / R > EXP(1.8)` (`LN(2*L/R) > 1.8`), `Surface_Tension gamma` in `[10, 90]` dyn/cm, `Flow Q > 0` mL/h, `Concentration C` in `(0.005, 0.60]` g/mL, and `Web_Area > 0` m²/h!
=LET(hg_c, B2, ln_c, B3, rn_c, B4, gm_d, B5, qf_m, B6, cp_g, B7, aw_m, B8, lg_t, LN((2 * ln_c) / rn_c) - 1.5, IF(OR(ln_c < 0.3, hg_c <= ln_c, rn_c <= 0.003, lg_t <= 0.3, gm_d < 10, gm_d > 90, qf_m <= 0, cp_g <= 0.005, cp_g > 0.6, aw_m <= 0), "Invalid Electrospin Input", ROUND((qf_m * cp_g) / aw_m, 2)))Un-Surfactanted Neat Water Solvent (gamma = 72 dyn/cm) Corona Breakdown Check (Set B5 = 72 dyn/cm)
Switch from organic DMF (`38` dyn/cm) to pure water (`gamma` B5 = 72 dyn/cm) for green PVA spinning without non-ionic Triton/Tween surfactant: Taylor Onset Voltage E2 jumps to **`22.72` kV** (`1.515` kV/cm in E3)—approaching needle-tip air ionization where corona discharges sputter beads instead of stable nanofibers!
=IF(ISTEXT(E5), E5, ROUND(SQRT(((4 * (B2 ^ 2)) / (B3 ^ 2)) * (LN((2 * B3) / B4) - 1.5) * (0.117 * PI() * B4 * B5)), 2))Try the example
Use the original values or change them to test your understanding.