Structure of ionic liquids and concentrated electrolytes from a mesoscopic theory

Recently, underscreening in concentrated electrolytes was discovered in experiments and confirmed in simulations and theory. It was found that the correlation length of the charge-charge correlations, $\lambda_s$, satisfies the scaling relation $\lambda_s/\lambda_D\sim (a/\lambda_D)^n$, where $\lambda_D$ is the Debye screening length and $a$ is the ionic diameter. However, different values of n were found in different studies. In this work we solve this puzzle within the mesocopic theory that yielded n=3 in agreement with experiments, but only very high densities of ions were considered [A. Ciach A. and O. Patsahan, {\it J.Phys.: Condens. Matter} {\textbf 33}, 37LT01 (2021)]. Here we apply the theory to a broader range of density of ions and find that different values of n in the above scaling can yield a fair approximation for $\lambda_s/\lambda_D$ for different ranges of $a/\lambda_D$. The experimentally found scaling holds for $2 <a/\lambda_D<4$, and we find n=3 for the same range of the reduced Debye length. For smaller $a/\lambda_D$, we find n=2 obtained earlier in several simulation and theoretical studies, and still closer to the Kirkwood line we obtain n=1.5 that was also predicted in different works. It follows from our theory that n=3 (i.e. $\lambda_s$ is proportional to the density of ions) when the variance of the local charge density is large, and $\lambda_s$ is proportional to this variance times the Bjerrum length. Detailed derivation of the theory is presented.