Force among magnetic nanocylinders trapped in triangular arrays

We study the effect of the forces among magnetic nanocylinders trapped in a membrane such as those used to produce them. The interaction force between two parallel and identical magnetic nanocylinders is revisited to obtain expressions that can later be used to add them both in a discrete and a continuum approximation. At this point a particular geometry has to be assumed and we use a particular configuration reported in the literature, namely, a bundle of parallel magnetic cylinders trapped in a circular membrane. When a strong enough external magnetic field is imposed along the axis of the membrane (also along the axes of the cylinders) all magnetizations point along this direction and cylinders repel among themselves. In a first approximation we will consider a soft enough membrane so energy is mostly relaxed through a deformation of the membrane leaving the magnetization of the cylinders basically as it was in the absence of external field. Then we obtain the forces among these cylinders by two methods: one summing the contributions of a discrete number of objects and another one in which we consider a continuum distribution of them to reach larger systems. Numerical evaluation of these forces can reach 50 mdyn approximately. Such forces will act on the membrane at the positions of the magnetic cylinders; in the case of a circular silicon membrane of radius 1.0 mm the radial expansion of the membrane can be of the order of 1 nm. This effect could be larger for softer membranes. A discussion of experimental techniques to detect this phenomenon is also done followed by the proposal of a possible application.