For the modeling of the BioNEMS-Drill we shall attempt to determine the intensity of the magnetic flux to move the device in a fluid without compromising the device integrity. We started by making the following suppositions.

• All the nanoparticles are monocrystalline and equal in size (spheres of r = 11nm for Fe₃O₄ and r = 21nm for Fe₃O₄@SiO₂).
• The amount of DNA BP between every nanoparticle is the same, this is same distance between nanoparticles.
• Though DNA is flexible, the force in a single particle is in the order of aN (attonewtons), while the extension force of the DNA is in the order of fN (femtonewtons), meaning the DNA is rather a rigid body.
• DNA and HU will not unbind from each other since the force needed for this to happen is about 6.3 pN.
• All drags are the same.
• The medium is homogeneous
• The system can be reduced to a single mass and a single drag.

First we considered the equation for the Lorentz Force (1) and the Drag equation (2). These two are the main forces acting on the device considering as an agglomerate of magnetic nanoparticles.

Where Q is the charge of the magnetic nanoparticle, v is the velocity of the magnetic nanoparticle, and B is the magnetic flux in Teslas. Also η is the dynamic viscosity of the medium, rd is the stokes radius (practically equal to the hydrodynamic radius), and Vd is the device’s velocity.

The charge Q can be obtained from the volume of the magnetite, the volume of the magnetite‘s unit cell, and the amount of Bohr magnetons (MB) per unite cell. Through these magnitudes we can use (4) to calculate the number of unpaired electrons (e-) and thus multiply it by the charge of the electron (q) to obtain the total charge of the particle.

The device’s total magnetic force (Fm) would be the summation of the individual forces from the Magnetite NP’s, but since every nanoparticle has the same size and shape, the total force will be the individual force multiplied by the number of nanoparticles in the device (n).

In this way we can calculate the total force in the device by subtracting the Drag Force from the Magnetic force.

Consider the particle velocity being 43.95 µm/s in a field of 1.4 T. The fluid is water. How many particles are needed in the device to move it. First we need to determine the charge of every particle.

Then we calculate the individual magnetic force and the drag force.

To cause movement the magnetic force should be higher than the drag force. By substituting (6) in (7) we can determine the amount of NP’s were the force is 0:

If n is greater than this number there should be movement:

So the device must contain at least 581 to move through water.