Figure 1.3 Specific viscosity as a function of polymer concentration to determine entanglement concentration for PEO of various molecular weights. Scanning electron microscopy (SEM) (PEO 600 kDa) showing the transition from beaded fibers to uniforms as the polymer concentration increases above the entanglement concentration. For neutral polymers in a good solvent, e.g. aqueous PEO, concentrations above ~2.5× the entanglement concentration form uniform fibers.

      Source: Image of beaded fibers is reprinted from Fong et al. (1999). Copyright (1999), with permission from Elsevier.

      To quantify the degree of entanglement required to achieve uniform fibers, semiempirical relationships have been used (Shenoy et al. 2005; McKee et al. 2004, 2006). The entanglement concentration can be determined by measuring the viscosity (zero‐shear) as a function of polymer concentration and examining the scaling relationship between the specific viscosity and concentration. Note that the specific viscosity (η_sp) accounts for the viscosity of the solvent

      (1.9)

where η ₀ is the zero‐shear viscosity of the polymer solution and η_s is the viscosity of the solvent. Upon the onset of polymer entanglement, there is a sharp increase in the scaling relationship based on the theory for semidilute solutions, below the entanglement concentration η_sp α [C]^1.25 and above the entanglement concentration η_sp α [C]^4.8. For linear, neutral polymers in a good solvent, the transition from beaded fibers to uniform fibers generally occurs at a polymer concentration ~2–2.5× the entanglement concentration (Figure 1.3). This approach has worked well for a number of polymer systems, e.g. polyvinyl alcohol (aqueous), PEO (aqueous), polystyrene in tetrahydrofuran, poly(D‐lactic acid) in DMF, and poly(L‐lactic acid) in dichloromethane (Shenoy et al. 2005; McKee et al. 2004). Analogous approaches have been developed for polyelectrolyte solutions. Polyelectrolytes transition from droplets to fibers at ~8 × C_e. The viscosity scaling relationships for polyelectrolytes to determine the entanglement concentration are an increase in the scaling relationship from η_sp α [C]^0.5 to η_sp α [C]^1.5 (McKee et al. 2006).

      The entanglement concentration can also be used to predict nanofiber diameter based on polymer concentration. A master curve for fiber diameter (d_f) as a function of concentration φ can be constructed as follows:

      (1.10)

      where d_f,e is the diameter of the fibers electrospun at the entanglement concentration φ_e. This result agrees well with the theoretical scaling of 2.3 (Wang et al. 2016). Long and coworkers showed comparable results with multiple polymers including linear, randomly branched, highly branched, and star polymers (McKee et al. 2004). This approach, which considers polymer concentration, viscosity, and polymer molecular weight (because the entanglement concentration decreases as polymer molecular weight increases), is convenient (Andrady 2008). Due to the high deformation rates, the entangled polymer solutions behave like elastic swollen gels. The rapid stretching of the gel has recently been considered the main mechanism of fiber formation. These results imply that the elasticity of the entangled polymer solution rather than the viscosity influences the final fiber diameter (Wang et al. 2016).

Rutledge and coworkers attribute ability to form uniform fibers to elasticity of the polymer solution noting that the presence of entanglements is a sufficient but not necessary condition for the fluid polymer to demonstrate strong elastic properties. A lack of sufficient elasticity leads to droplets or beaded fibers. To prevent beaded fibers, Rayleigh's breakup instability must be suppressed. The instability can be slowed down or suppressed by the viscoelasticity of the polymer solution. The timescales for instability and the viscoelasticity can be quantified with the Deborah number (De), i.e. the ratio of the fluid relaxation time and instability growth time. Using blends of poly(ethylene oxide) and poly(ethylene glycol) of various molecular weights to tune the elasticity of the solution, they show a transition from beaded fibers to uniform fibers with increasing Deborah number. At high Deborah number (≫̸1), the capillary forces that lead to the Rayleigh instability activate the elastic response and delay jet breakup. Deborah numbers above 6 results in uniform fibers because the instability is completely suppressed by elastic forces or arrested at a very early stage of instability growth. There was no observed correlation between the Newtonian viscosity/Ohnesorge number of the fluid and the fiber morphology indicating the elasticity measured by relaxation time is critical for governing the fiber morphology (Yu et al. 2006). The elastic properties were measured using a capillary breakup extensional rheometer, whereas measuring the entanglement concentration is commonly done with a dynamic shear rheometer. Thus, entanglement concentrations are a popular practical approach.

      1.4.2 Solvent Selection

Solvent selection is also an important consideration. Currently, solvent selection is based on trial and error. Further, the solvent characteristics cannot be independently varied to tune desired fiber properties. The key properties of commonly used electrospinning solvents are provided in Table 1.1. Practically, solvent volatility affecting evaporation rate is an important consideration. If solvent remains when the fiber is deposited, the wet fibers can fuse together or result in a ribbon‐like fiber. To avoid such issues, more volatile solvents can be used. However, highly volatile solvents can result in needle clogging. Further, the average fiber diameter of polystyrene fibers has been observed to be inversely proportional to solvent boiling point. Fibers between ~5 and 0.2 μm were achieved by varying the boiling point of the solvent (Wannatong et al. 2004).

In addition to the rate of evaporation, the solvent affects the conformation of the dissolved polymer chains as well as the solution conductivity, surface tension, and viscosity. Generally, good solvents that yield open conformations of polymer chains (to promote entanglement) and sufficient polymer solubility to achieve high solids contents are preferred for electrospinning (Andrady 2008). Luo and coworkers considered 49 solvents for electrospinning polycaprolactone. Solubility alone is not sufficient for electrospinning. They note that solvents with partial (Luo et al. 2010; Shenoy et al. 2005) to high solubility with moderate dispersion forces tended to form uniform fibers. Solvents with strong van der Waals forces/hydrogen bonding and weak polar force formed large droplets (micron to millimeter). Solvents with high molecular weight, low dielectric constant, low boiling point, and/or strong dispersion forces could not be used for electrospinning (Luo et al. 2012).

Table 1.1 Key solvent properties for common electrospinning solvents.