### Article

## Nonlocal statistical field theory of dipolar particles in electrolyte solutions

We present a nonlocal statistical field theory of a dilute electrolyte solution with a small

additive of dipolar particles. We postulate that every dipolar particle is associated with an

arbitrary probability distribution function (PDF) of distance between its charge centers.

Using the standard Hubbard–Stratonovich transformation, we represent the configuration

integral of the system in the functional integral form. We show that in the limit of a small

permanent dipole moment, the functional in integrand exponent takes the well known form

of the Poisson–Boltzmann–Langevin (PBL) functional. In the mean-field approximation we

obtain a non-linear integro-differential equation with respect to the mean-field electrostatic

potential, generalizing the PBL equation for the point-like dipoles obtained first by Abrashkin

et al. We apply the obtained equation in its linearized form to derivation of the expressions

for the mean-field electrostatic potential of the point-like test ion and its solvation free energy

in salt-free solution, as well as in solution with salt ions. For the ‘Yukawa’-type PDF we

obtain analytic relations for both the electrostatic potential and the solvation free energy of

the point-like test ion. We obtain a general expression for the bulk electrostatic free energy of

the solution within the Random phase approximation (RPA). For the salt-free solution of the

dipolar particles for the Yukawa-type PDF we obtain an analytic relation for the electrostatic

free energy, resulting in two limiting regimes. Finally, we analyze the limiting laws, following

from the general relation for the electrostatic free energy of solution in presence of both the

ions and the dipolar particles for the case of Yukawa-type PDF.