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Electrospinning Model
Scott McNamara Martin Mee Department of Materials Engineering Drexel University
Abstract
Electrospinning is a rising technology capable of producing nanofibers from polymeric solutions. As the fluid emerges from a hypodermic needle, interesting changes in the jet diameter occur. In this project, we have attempted to model these changes.
Background
Electrospinning is a simple, quick, inexpensive, non-mechanical, electrostatic method in which a polymer solution in many possible solvents is placed into a hypodermic syringe at a fixed distance of about 5 to 30 cm from a metal cathode such as copper or aluminum foil. The negative terminal of a variable high voltage transformer, capable of delivering 30,000 V, is grounded to the aluminum foil target cathode. The
Figure 1: Electrospinning schematic.
positive terminal is either attached to the metal capillary tip (1.2mm) of the syringe, which is filed to produce a flat edge, or connected to a copper wire and placed into the polymer solution. The syringe is then placed directly over the aluminum foil or at a convenient angle (5 degrees was found in many experimental procedures). As the
voltage is increased between the anode and cathode, the charge over comes the surface tension of the suspended polymer drop at a critical value. This value is a function of distance and varies with the polymer solution being spun. When the surface tension is released, a jet stream of polymer is produced. The polymer molecules in the trajectory repel each other due to the positive charge induced on each molecule. This separation allows for rapid evaporation of the solvent, not only due to repelling polymer molecules, but also due to repelling solvent molecules, all being positive. In the jet, the polymer undergoes electrically induced bending instabilities, which hyper stretches the extruding jet. This stretching, along with the solvent evaporation, reduces the diameter of the jet in a volume called the Taylor cone. Fiber diameter is controlled by polymer solution
Figure 2: Region attempting to be modeled. The Taylor cone is also apparent.
concentration and the charge density. With ideal conditions, a non-woven mat consisting of dry, multiple meter fibers can accumulate on the target cathode of aluminum foil. Note that some polymers are susceptible to degradation following a 25,000V electrospinning process. It has been ascertained that the variables affecting the onset of the polymer jet expelled from the capillary tip can be narrowed down to two types, operating parameters and fluid properties. Of the fluid properties, viscosity and conductivity play the most important role, with density, surface tension, permeativity, dielectric constant, and viscoelasticity following. As far as the process variables are concerned, the electric field, which is a function of the applied voltage and the distance over which the voltage drop to ground occurs, is the most important. And as stated earlier the volumetric flow rate is an important parameter, however fibers with a fairly uniform diameter differential can still be produced without regulating this parameter. So long as the fluid is viscous enough to sustain suspended drop immediately outside of the capillary tip, and not reaching the solid thresh hold, the volumetric flow rate will be determined by the electric field intensity. The total electric current between anodes is composed of convection and conduction currents and translates to an indirect measure of surface charge density. It is this surface charge density that is used to predict the bending instabilities. This is the lowest portion of the jet, which rapidly whips and then splits and spiders before collecting on the cathode. Most of the current research is focused here in this area due to its intricacy and hence lack of complete understanding. It has been found that the
so the differentials with respect to time in the first, second and third equations were set equal to zero. The dimensionless equations were as follows:
E
v v r
g g r
K K r
πβ
ρ γ
ρ γ
ρ γβ
0
0
2
2 0
3 0
The constants were for PEO in water and were set equal to these values:
η = 1.67 Pa s ε = 78. γ = 0.0606 N/m r 0 = 0.000045 m Einf = 40,000 V/m ρ = 1,000 kg/m 3 K = 0.012 S/m έ = 1 g = 9.8 m/s
Numerical Simulation
The initial equations and constants were all entered into Maple. Equations were then entered to convert the differential equations to linear equations (see Appendix 1). These equations were then substituted into the previous equations where necessary. Equation 1 was solved once for the differential of the radius and a second time for the velocity to produce two equations. Equation 2 was solved for the differential of the surface charge density, while equation 3 was solved for the third differential of the jet radius and the second differential of the jet velocity and the fourth equation was solved for the electric field. These equations were used in the Euler forward method for solving differential equations. The equation used in this method is:
y(n+1)=y(n)+dx*f(x(n),y(n))
This equation is stating that the next value for a variable can be determined by adding the change in distance times the differential value to the previous value for the variable, where the change in distance is much less than zero. To properly carryout this method, several initial conditions need to be known. These act as starting values for the variables, which can be added to the step times differential values. The initial conditions were:
z = 0 v = 0.015 m/s
σ = 0 h = 4.5 10 -5^ m E = 50 V/m
At time zero, the position z is at its initial position, or a distance of zero. The surface charge density is taken to be zero because at time zero, the surface is much larger than after the Taylor cone. The larger surface spreads the charge out and is effectively equal to zero. The remaining values were determined by previous experiments. Graphs were then produced of the jet radius versus position with varying steps in the position.
Results and Discussion
Two graphs with different steps, 10-6^ and 10-8^ in the positioning, neither of which correctly model the Taylor cone of electrospinning. In both graphs, the jet radius increases as the jet gets farther away from its initial position. Photographs and other numerical models show that the jet radius decreases as it gets farther from its origin.
Figure 3: Graph with a 10-6^ increment in the position.
Possition vs. Radius
0.00E+
5.00E+
1.00E+
1.50E+
2.00E+
2.50E+
3.00E+
3.50E+
4.00E+
0 0.0005 0.001 0.0015 0. z (meters)
h (meter)
Appendix 1 Important Maple Data
Relations
= ∂
z h( z ) wh( z )
z v( z ) w( z )
z
σ( z ) α ( z )
z
w( z ) u( z )
z
wh( z ) whh( z )
z
whh( z ) whhh( z )
Differential Equations
z h( z ) −
h( z )w( z ) v ( z )
z v( z ) − 2
v( z )wh( z ) h ( z )
z σ( z )=−.1000000000 10 -15^ (.1000000000 10 17 σ( z )wh( z )v( z )
- .1000000000 10 17 σ( z )h( z )w( z )+ .1058038507 10 10 h( z ) E wh( z ))/( h( z )v( z ))
z whh( z )=−.1000000000 10 -12^ ( −.1000000000 10 14 w( z )v( z )h( z )^2
- .1000000000 10 14 wh( z )+.2203856748 10 17 v( z )^2 h( z )w( z )wh( z )
- .1256637061 10 15 σ( z )α( z )h( z )^2 +.2271847336 10 13 σ( z ) E h( z )
- .3280165289 10 10 h( z )^2 + .1101928374 10 17 v( z )^2 h( z )^2 u( z )) h( z )^2
z w( z )=−.9075000005 10 -16^ ( −.1000000000 10 14 w( z )v( z )h( z )^2
- .1000000000 10 14 wh( z )+.1000000000 10 14 whhh( z )h( z )^2
- .1256637061 10 15 σ( z )α( z )h( z )^2 +.2271847336 10 13 σ( z ) E h( z )
- .3280165289 10 10 h ( z )^2 + .2203856748 10 17 v( z )^2 h ( z )w( z )wh( z )) (v( z )^2 h( z )^2 )
E :=.5000000000 ( −.5531344650 10 18 wh( z )σ( z )+.2434734306 10 19
- .4286792104 10 20 h( z )whh( z )wh( z )σ( z )+.4286792104 10 20 h( z )^2 whh( z )α( z )
- .4286792104 10 20 wh( z )^2 h( z )α( z )−.5531344650 10 18 h( z )α( z )
- .4286792104 10 20 wh( z )^3 σ( z )−1. sqrt (−.2693470916 10 37 wh( z )σ ( z ) − .2693470916 10 37 h ( z )α( z )+.2087439960 10 39 wh( z )^3 σ( z )
- .2087439960 10 39 h ( z )^2 whh( z )α( z )+.2087439960 10 39 wh( z )^2 h( z )α ( z )
- .2087439960 10 39 h ( z )whh( z )wh( z )σ( z )+ .5927931141 10 37 ) ) ( .3003125000 10 20 wh( z )^2 h( z )whh( z ) +.2500000000 10 16 − .3875000000 10 18 wh( z )^2 +.1501562500 10 20 wh( z )^4 − .3875000000 10 18 h ( z )whh( z )+ .1501562500 10 20 h( z )^2 whh( z )^2 )
z wh( z )=(.5000000000 ( −.3675317309 10 22 h ( z )^3 α( z )^2 wh( z )^2
− .7350634617 10 22 h( z )^2 wh( z )^3 σ( z )α( z )
- .9484689828 10 20 wh( z )σ( z )h( z )^2 α ( z )+.3402560352 10 11 h( z )wh( z )σ( z )
- .3402560352 10 11 h ( z )^2 α( z )−.3675317309 10 22 h( z )wh( z )^4 σ( z )^2
- .4742344914 10 20 wh( z )^2 σ( z )^2 h( z )+ .4742344914 10 20 h( z )^3 α( z )^2 + .1600000 10 7 sqrt .1072149577 10( 19 h( z )^3 wh( z )σ( z )α( z )
- .5360747885 10 18 h ( z )^4 α( z )^2 −.8053359481 10 17 h( z )^2 wh( z )^3 σ( z )^3 − .2416007844 10 18 wh( z )^2 σ( z )^2 h( z )^3 α( z ) − .2416007844 10 18 wh( z )σ( z )h( z )^4 α ( z )^2 −.8053359481 10 17 h( z )^5 α( z )^3
- .5360747885 10 18 h ( z )^2 wh( z )^2 σ( z )^2 ) ) (.1837658654 10 22 h( z )^2 wh( z )^2 σ ( z )^2
- .3675317309 10 22 h ( z )^3 wh( z )σ( z )α ( z )+ .1837658654 10 22 h ( z )^4 α( z )^2 ), .5000000000 ( −.3675317309 10 22 h( z )^3 α ( z )^2 wh( z )^2 −.7350634617 10 22 h( z )^2 wh( z )^3 σ( z )α( z )
- .9484689828 10 20 wh( z )σ( z )h( z )^2 α ( z )+.3402560352 10 11 h( z )wh( z )σ( z )
- .3402560352 10 11 h( z )^2 α( z )−.3675317309 10 22 h( z )wh( z )^4 σ( z )^2
- .4742344914 10 20 wh( z )^2 σ( z )^2 h( z )+ .4742344914 10 20 h( z )^3 α( z )^2 − .1600000 10 7 sqrt .1072149577 10( 19 h( z )^3 wh( z )σ( z )α( z )
- .5360747885 10 18 h ( z )^4 α( z )^2 −.8053359481 10 17 h( z )^2 wh( z )^3 σ( z )^3 − .2416007844 10 18 wh( z )^2 σ( z )^2 h( z )^3 α( z ) − .2416007844 10 18 wh( z )σ( z )h( z )^4 α ( z )^2 −.8053359481 10 17 h( z )^5 α( z )^3
- .5360747885 10 18 h ( z )^2 wh( z )^2 σ( z )^2 ) ) (.1837658654 10 22 h( z )^2 wh( z )^2 σ ( z )^2
- .3675317309 10 22 h ( z )^3 wh( z )σ( z )α ( z )+ .1837658654 10 22 h ( z )^4 α( z )^2 ) )
