We present an analytical theory of electrostatic interactions of two spherical dielectric particles of arbitrary radii and dielectric constants, immersed into a polarizable ionic solvent (assuming that the linearized Poisson–Boltzmann framework holds) and bearing arbitrary charge distributions expanded in multipolar terms. The presented development entails a novel two-center re-expansion analytical theory that expands upon and improves the existing ones, bypassing the conventional expansions in modified Bessel functions. On this basis, we develop a specific matrix formalism that facilitates the construction of asymptotic expansions in ascending order of Debye screening terms of potential coefficients, which are then employed to find exact closed-form expressions for the total electrostatic energy. In particular, this work allows us to explicitly and precisely quantify the k-screened terms of the potential coefficients and mutual interaction energy. Specific cases of monopolar and dipolar distributions are described in particular detail. Comprehensive numerical examples and tests of series convergence and the relative balance of leading and higher-order terms of the mutual interaction energy are presented depending on the inter-particle distance and particles’ radii. The results of this work find application in soft matter modeling and, in particular, in computational biophysics and colloid science, where the availability of increasingly larger experimental structures at the atomic-level resolution makes numerical treatment challenging and calls for more efficient expressions and an increased range of validity.

Zebrafish are powerful animal models for understanding biological processes and the molecular mechanisms involved in different human diseases. Advanced optical techniques based on fluorescence microscopy have become the main imaging method to characterize the development of these organisms at the microscopic level. However, the need for fluorescence probes and the consequent high light doses required to excite fluorophores can affect the biological process under observation including modification of metabolic function or phototoxicity. Here, without using any labels, we propose an implementation of a Mueller-matrix polarimeter into a commercial optical scanning microscope to characterize the polarimetric transformation of zebrafish preserved at different embryonic developmental stages. By combining the full polarimetric measurements with statistical analysis of the Lu and Chipman mathematical decomposition, we demonstrate that it is possible to quantify the structural changes of the biological organization of fixed zebrafish embryos and larvae at the cellular scale. This convenient implementation, with low light intensity requirement and cheap price, coupled with the quantitative nature of Mueller-matrix formalism, can pave the way for a better understanding of developmental biology, in which label-free techniques become a standard tool to study organisms.

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