User:Brian P. Josey/Notebook/2010/05/14

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Iron and Neodymium Yoke
I ran new simulations on the iron and neodymium yoke, and compared them side by side. A new simulation was needed to account for the tip width of the cone magnets, add smoothing of the data, increase the range of the data, and definitively prove the strength of the neodymium yoke. In my original model of the neodymium yoke, I had the tip of the cone a little too wide, at 0.01 inches. This was just an estimate, and after I remeasured it, I determined that it had a width of 0.006 inches, and comparing the two, there is a slight difference.



This is the magnetic field magnitude I simulated for the iron yoke, and it illustrates the issue better than the graph of the neodymium magnet. Near the tip of the cone, on the far left side of graph, the field experiences a leveling out and is not as sharp as elsewhere in the graph. This is the result of the increasing importance of the lateral direction of the magnetic field as the ferritin approaches the tip of the magnet, and the width of the cone's tip affects this. For my earlier simulations, this was not an issue, but I now have to take it into account as I look at the area nearest to the cone.

Force Difference
I compared the force on a single ferritin protein in both the neodymium magnetic yoke, and the iron electromagnet, and I found something a little surprising. This time, I calculated that there would be a stronger force with the iron electromagnet than with the neodymium yoke in certain places. While this is surprising, it is actually very small and insignificant. Here is the numerical difference between the force in both cases:



Here a positive value indicates that the iron electromagnet exerts a stronger force on the ferritin. The percent difference:



In the percent difference, a positive percentage indicates that the neodymium magnet exerts a stronger force on the ferritin than the iron magnet. The reason for the switch in the sign between the numerical and percentage difference is the result from the convention that an attractive force has a negative sign.

I was then able to measure the average force exerted on the ferritin in three cases: near the tip, out to 0.0025 in; within the flow cell; and the first 0.01 inch in the micro-centrifuge tube. The average forces that I calculated, in fN, are:

Neodymium Yoke
 * Near Tip: 0.975
 * Flow Cell: 0.186
 * Tube: 0.041

Iron Electromagnet
 * Near Tip: 0.451
 * Flow Cell: 0.264
 * Tube: 0.050

From this, I calculated that the neodymium yoke was 0.524 fN, or 53.78% on average stronger than the iron magnet, near the tip. The iron electromagnet, however, was 0.078 fN or 42.02% stronger in the flow cell, and 0.009 fN or 23.17% stronger in the area of the tube. This illustrates that neodymium yoke is a better choice for working near the ferritin, while and iron electromagnet might be the better choice away from the ferritin. I am still not a hundred percent sure why this is, and I want to find out why. However, the absolute difference in these areas are small, but the percent difference is significant. If it turns out that iron electromagnets are a better choice for a flow cell, then being able to tune the current, and in turn the force, including turning it off could be easier to use than the permanent magnets when working on a microscope.

Terminal Velocity
Using Stoke's Law I was able to determine the speed of ferritin going through water in different areas of the simulation. To do this, I balanced the force of the magnet with the force of drag from the liquid. Here is the equation I used:


 * $$F = 6 \pi\, \mu\, R\, V \,$$

where:
 * F is the force (in pN = 103 fN)
 * &mu; is the viscosity of water (on the order of 10-6pN*s/μm*nm)
 * R is the radius of the ferritin (6 nm)
 * V is the ferritin's velocity (in μm/s).

For the flow cell, the medium is water, but I am assuming that the viscosity in a cell is the same, but I need to double check that. The viscosity is in non-standard units to cancel out nicely with my data. Rearranging this equation, I solve for the velocity:


 * $$ V = \frac {F}{6 \pi\ \mu\ R }\, $$

Where the speed is the maximum speed possible, the terminal velocity, where the force of drag is equal to the force of attraction from the magnet. The final speeds I was able to calculate, in μm/s are:

Neodymium Yoke
 * Near Tip: 8.620
 * Flow Cell: 1.646
 * Tube: 0.3583

Iron Electromagnet
 * Near Tip: 3.984
 * Flow Cell: 2.337
 * Tube: 0.441

As indicated by the force from above, my simulation indicates that there will be a small increase in speed in the iron yoke. However, at this scale it is negligible.


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