Our model

of a magnetic field around an iron nanoparticle is based on the model of the magnetic field around a magnet described in [18]. The electromagnetic potential in the point r near a permanent magnet of volume V is equal to (6) where M is the CP673451 magnetization vector at the point dV, the vector R is the difference between AZD5582 in vitro source of the magnetic field dV and the point r, R is the length of R. The intensity of the magnetic field H can be subsequently computed as (7) Finally, the magnetic force between the source of the intensity of magnetic field H and a permanent magnet of volume with a magnetization vector M 0 at the point r is equal to (8) In our previous work [19], the scalar potential of the magnetic field around one homogeneous spherical iron

nanoparticle with radius a located at the point (0,0,0) was derived as follows: (9) where a is the radius of the nanoparticle, and (x 1,x 2,x 3) are the coordinates of the point r. Here, the direction of selleck inhibitor the magnetization vector M is set towards x 3, and M is the magnitude of the vector M. From Equations 7 and 8, the analytical computation of the magnetic force between two iron nanoparticles can be obtained. Since nanoparticles aggregate, the magnetic force between aggregates must be derived. One aggregate can be composed of millions of nanoparticles. It would be time-consuming and Tolmetin very difficult to analytically compute all these forces. As a consequence, the forces are computed numerically, either as a sum of the magnetic forces between every nanoparticle in one aggregate with every nanoparticle in the second aggregate (10) or as one magnetic force between two averaged aggregates [20]. (11) where is the volume of a nanoparticle,

r 2j is the location of the centre of the j-th nanoparticle in the second aggregate, M 2j is the magnetization vector of the j-th nanoparticle in the second aggregate, M 1A and M 2A are the averaged magnetization vectors (Equation 12) of the first and the second aggregate respectively, and is the volume of the second aggregate. The averaged aggregate is a big homogeneous particle with its direction of magnetization vectors M A which is computed as a vector sum of the magnetization vectors of all nanoparticles in the aggregate M A and computed as an average of the sizes of all nanoparticles divided by the number of nanoparticles in the aggregate n. (12) The structure of aggregates When particles aggregate due to magnetic forces, the rate of aggregation depends on the magnetization vectors of the aggregating particles and on the distance between the particles. The rate of aggregation changes with the changing number of nanoparticles within the aggregates, that is, the changing scale of the structure by order.