Counter-ion fluctuations in the presence of a spherical macromolecule

The effective interactions of charged macromolecules (e.g. polyelectrolytes) are still not fully understood. The role of counter-ion fluctuations, in the presence of these macromolecules, seems to be crucial in understanding these effective interactions. We consider a single charged spherical macromolecule, enclosed in a finite volume, in the presence of point-like counter-ions in an electrically neutral solution. Writing the partition function as a functional integral and only keeping terms up to quadratic order, we calculate the free energy in this approximation exactly.