  In the technique of microrheology, macroscopic rheological parameters as well
as information about local structure are deduced from the behavior of
microscopic probe particles under thermal or active forcing. Microrheology
requires knowledge of the relation between macroscopic parameters and the force
felt by a particle in response to displacements. We investigate this response
function for a spherical particle using the two-fluid model, in which the gel
is represented by a polymer network coupled to a surrounding solvent via a drag
force. We obtain an analytic solution for the response function in the limit of
small volume fraction of the polymer network, and neglecting inertial effects.
We use no-slip boundary conditions for the solvent at the surface of the
sphere. The boundary condition for the network at the surface of the sphere is
a kinetic friction law, for which the tangential stress of the network is
proportional to relative velocity of the network and the sphere. This boundary
condition encompasses both no-slip and frictionless boundary conditions as
limits. Far from the sphere there is no relative motion between the solvent and
network due to the coupling between them. However, the different boundary
conditions on the solvent and network tend to produce different far-field
motions. We show that the far field motion and the force on the sphere are
controlled by the solvent boundary conditions at high frequency and by the
network boundary conditions at low frequency. At low frequencies compression of
the network can also affect the force on the sphere. We find the crossover
frequencies at which the effects of sliding of the sphere past the polymer
network and compression of the gel become important.
