  Magnetophoretic force acting on a rigid spherical cluster of single-domain
nanoparticles in a constant-gradient weak magnetic field is investigated
numerically using the Langevin dynamics simulation method. Nanoparticles are
randomly and uniformly distributed within the cluster volume. They interact
with each other via long-range dipole-dipole interactions. Simulations reveal
that if the total amount of particles in the cluster is kept constant, the
force decreases with increasing nanoparticle concentration due to the
demagnetizing field arising inside the cluster. Numerically obtained force
values with great accuracy can be described by the modified mean-field theory,
which was previously successfully used for the description of various dipolar
media. Within this theory, a new expression is derived, which relates the
magnetophoretic mobility of the cluster with the concentration of nanoparticles
and their dipolar coupling parameter. The expression shows that if the number
of particles in the cluster is fixed, the mobility is a nonmonotonic function
of the concentration. The optimal concentration values that maximize the
mobility for a given amount of magnetic phase and a given dipolar coupling
parameter are determined.

Mesoscale and Nanoscale Physics