# Controlling the Fiber Diameter during electrospinning

Original entry: Zach Wissner-Gross, APPHY 226, Spring 2009

## Information

Controlling the fiber diameter during electrospinning

Sergey V. Fridrikh, Jian H. Yu, Michael P. Brenner, and Gregory C. Rutledge

Physical Review Letters, **2003**, 90, 144502

## Soft matter keywords

Electrospinning, Surface tension, Scaling

## Summary

Fridrikh et al. derive an analytical model of the forces involved in electrospinning [1]. In the process of electrospinning, a polymer fiber is created by ejecting the polymer in a solvent between two plates of opposite electric charge. When the plate voltages are high enough, electrostatic forces dominate forces due to surface tension, so that the polymer develops a very high aspect ratio (i.e., becoming a thin fiber) as it streams toward the opposing plate.

Electrospinning is further complicated by the "whipping instability," in which the electrospun fiber coils around itself. In their model, the authors are able to calculate the minimum possible fiber radius (for a thin viscous jet) before the whipping instability breaks the jet up into droplets (as in electrospraying [2]).

Their main result is that this terminal radius <math>h_t</math> scales as:

where <math>Q</math> is the flow rate of the polymer solution and <math>I</math> is current (due to the net downward motion of the fiber, which is simultaneously whipping around in circles). Figure 1 shows a log-log plot of <math>Q/I</math> vs. <math>h_t</math>, which we would expect to have a slope of 2/3. The fitted line, with a slope of 0.639, corresponds well with their theory, and the offset in the log-log plot suggests that the authors' electrospun fibers could be reduced in size by a factor of two but no more.

## Soft matter discussion

The physics in this article is all derived from the authors' Equation 1 (the equation of motion for an electrospun fiber), which I reproduce here in Figure 2. The lone term on the left side of the equation is simply the force per unit length (one can simply guess what each of the variables represents).

The first term on the right side is the force from the electric field acting on the surface charge of the fiber (<math>\sigma_0</math>), whereas everything inside the parentheses on the right side represents bending stress, to which the authors attribute the whipping instability. If this whole term (i.e., inside the parentheses) is negative, then the fiber can reduce its total energy with increased bending, and so the fiber is deemed unstable.

Inside the parentheses, we first see a surface tension term (which is positive and counteracts bending, since bending increases the surface area of the fiber), and then a Hookean normal stress term associated with stretching that occurs when the fiber is bent. The authors point on that this force is similarly stabilizing.

So then what is this third term, which, as the only term left, must be responsible for inducing any instabilities? According to the authors, "The third term is due to surface charge repulsion and is destabilizing." As the fiber stretches (as opposed to simply growing due to the continuing influx of polymer solution), the surface charge density drops with increasing area. In short, stretching is promoted by surface charge repulsion, but must counteract the stabilizing forces associated with stretching the material and increasing the area of the polymer-air interface.

The authors go on to neglect the stress term for thin fibers, since the stress term is scaled down by additional factor of <math>h</math>, the position-dependent radius of the fiber, as compared to the other terms. After throwing in volume conservation and a few identities involving flow rate and current, the authors obtain the final relationship between the lower bound of stable fiber radius and the ratio between flow rate and current.