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## Nonlocal statistical field theory of dipolar particles forming chain-like clusters

We present a nonlocal statistical field theory of a diluted solution of dipolar particles which are capable of

forming chain-like clusters in accordance with the ’head-to-tail’ mechanism. As in our previous study [Yu.A.

Budkov 2018 J. Phys.: Condens. Matter 30 344001], we model dipolar particles as dimers comprised of oppositely charged point-like groups, separated by fluctuating distance. For the special case of the Yukawa-type

distribution function of distance between the charged groups of dipolar particles we obtain an analytical

expression for the electrostatic free energy of solution within the random phase approximation. We show

that an increase in the association constant leads to a decrease in the absolute value of the electrostatic free

energy of solution, preventing its phase separation which is in agreement with the former computer simulations and theoretical results. We obtain a non-linear integro-differential equation for the self-consistent

field potential created by the fixed external charges in a solution medium, taking into account the association of dipolar particles. As a consequence of the derived self-consistent field equation, in regime of weak

electrostatic interactions, we obtain an analytical expression for the electrostatic potential of the pointlike test ion, surrounded by the chain-like clusters of the dipolar particles. We show that in the mean-field

approximation the association does not change the bulk dielectric permittivity of the solution, but increases

the solvation radius of the point-like charge, relative to the theory of non-associating dipolar particles.