# Controlling the Fiber Diameter during electrospinning

Zach Wissner-Gross (April 21, 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.