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a simple analytical model focusing on the whipping of the jet treats the jet as a slender viscous object. Rutledge and coworkers assume that the final fiber diameter is dictated by an equilibrium between Coulombic charge repulsion on the surface of the jet and the surface tension of the liquid jet. The model predicts that terminal jet radius (r_j)

(1.1)

      where Q is the volumetric flow rate, I is the electric current, γ is the surface tension, is the dielectric constant of the outside medium (typically air), and χ is the dimensionless wavelength of the instability of the normal displacement. The fiber diameter (d) is related to the terminal jet radius and the polymer concentration, c. When compared to experimental results, the model accurately predicted the diameter of polyethylene oxide (PEO) fibers (within 10%) and polyacrylonitrile fibers (within 20%). The theory overpredicted stretching for polycaprolactone fibers, which had relatively low conductivity and high solvent volatility (Fridrikh et al. 2003).

      Recently, Stepanyan and coworkers developed an electrohydrodynamic model of the jet elongation in which kinetics of elongation and evaporation govern the nanofiber diameter. Using the timescale of elongation to nondimensionalize the force balance, the timescale of solvent evaporation, and concentration‐dependent material functions (e.g. relaxation time), they predict the scaling relationship for the final fiber radius (r_f) is

(1.2)

where k is the solvent evaporation rate, ρ_s is the solution density, η_o is the solution viscosity, Q is the volumetric flow rate, and I is the electric current. The result for Eq. (1.2) reduces to Eq. (1.1) in the limit of very slow evaporation. The viscosity dependence, η_o ^1/3, and (Q/I)^2/3 agree with experimental results. Using polyvinylpyrrolidone (PVP) in various alcohol solutions as well as polyamide/polyacrylonitrile in dimethylformamide (DMF) solutions, the fiber diameter was experimentally observed to scale with the evaporation rate, k, to the 1/3 power (Stepanyan et al. 2014, 2016).

      The current in electrospinning (I) and the volumetric charge density (Q/I) is commonly used in modeling and theoretical approaches. Experimental investigations by Yarin and coworkers determined that

      (1.3)

      (1.4)

      (1.5)

      where V is applied voltage, Q is flow rate, C polymer concentration, M molecular weight, and H tip to collector distance. Using PEO/water/ethanol mixtures, the solvent properties also slightly affect the current and volumetric charge density. From this work, it is evident that the fiber size is affected by applied electric field strength (applied voltage and time to collector distance), flow rate, and polymer solution (Thompson et al. 2007). Thus, these predictions for fiber size rely on several model parameters that cannot be easily related to measurable variables.

      In another work by Rutledge and coworkers, they experimentally determined that the total current in electrospinning, given by

      (1.6)

      varies electric field (E), volumetric flow rate (Q), and solution conductivity (K). In electrospray, the measured current scales linearly with Q. The authors attributed the observed Q ^0.5 dependence on combined jet and spray arising from secondary jetting. Secondary jetting was considered a mechanism for dynamic removal of charge from the surface of the jet which affects final fiber diameter and reducing jet stretching due to surface charge repulsion. The secondary jetting can be minimized by reducing the volumetric flow rate or solution conductivity (Bhattacharjee et al. 2010).

      Alternative scaling analyses based on the spinning solution properties and electrospinning operating conditions alone have been developed. Based on electrohydrodynamic theory, the Taylor–Melcher slender body theory relates jet kinematics to measurable fluid properties and process variables. Helgeson et al. validated the electrohydrodynamic model with measurements of the jet radius and velocity via in situ high‐speed photography and velocimetry in the straight portion of viscoelastic electrospinning jet and PEO/NaCl as a model system. Dimensional analysis of the validated model involves two important quantities the electroviscous number characterizing the electromechanical stress relative to shear stress and the Ohnesorge number (Oh). The relationship for the final fiber diameter was determined to be

      (1.7)

where w_p is the mass fraction of polymer in solution, ρ is solution density, γ is surface tension, η ₀ is the zero shear viscosity, Q is the volumetric flow rate, is the extensional viscosity, ε is the dielectric constant of the fluid, is the dielectric constant of the surrounding medium (typically atmosphere), and E ₀ is the strength of the applied electric field. This analysis explicitly contains the extensional viscosity of the fluid known to control fiber formation. The extensional viscosity can be estimated as 3η ₀ for a Newtonian fluid. This approach assumes the final fiber is directly proportional to jet radius in the straight portion of the jet and the actual relationship may be system‐dependent, i.e. influenced by solution conductivity and mechanics of the bending instability (Helgeson et al. 2008). The predictions agree with experimental observations over a broad range of polymer concentrations and voltages. Upon addition of a significant amount of NaCl, the trends are consistent, but the scaling factor changes and may be attributed to the differences in the bending instability.

      Recently, using a force balance on a bent jet considering electric field and surface charge for a Newtonian fluid, Gadkari predicted the final fiber diameter (d_f) scales as

      (1.8)

      where η_o is viscosity, K is conductivity, Q is flow rate, L is tip to collector distance, and V is applied voltage. When compared to experimental results, the scaling prediction qualitatively matches the observed trend for viscosity, flow rate, and applied voltage. The model overpredicts the dependence on volumetric flow rate (0.5 dependence predicted, whereas a 0.3 dependence has been observed experimentally). Experimental values for the scaling relationships for ΔV, L, and K were not available (Gadkari 2014).

      1.2.2 Experimental Results

      Collectively, theoretical considerations indicate that fiber size is affected by applied electric field strength (applied voltage and time to collector distance), volumetric flow rate, and polymer solution (viscosity and conductivity). However, experimental results have been system‐dependent. For example, experiments increasing the voltage has been observed to decrease fiber diameter for many systems such as polyacrylonitrile/DMF

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