Electric energies of a charged sphere surrounded by electrolyte

István P Sugár

doi:10.3934/biophy.2021012

Abstract

<abstract> By using the recently generalized version of Newton's Shell Theorem <xref ref-type="bibr" rid="b6">[6]</xref> analytical equations are derived to calculate the electric potential energy needed to build up a charged sphere, and the field and polarization energy of the electrolyte inside and around the sphere. These electric energies are calculated as a function of the electrolyte's ion concentration and the radius of the charged sphere. The work needed to build up the charged sphere, <italic>ECC</italic> (i.e. the total charge-charge interaction energy) decreases with increasing ion concentration of the electrolyte because of the electrolyte ions' increasing screening effect on the charge-charge interaction. The work needed to build up the charged sphere appears as a sum of the field and polarization energy of the electrolyte. At zero ion concentration the electrolyte's field energy is equal with <italic>ECC</italic> while the polarization energy is zero. At high electrolyte ion concentrations (<italic>C</italic> > 10<italic>mol</italic>/<italic>m</italic>3) 50% of <italic>ECC</italic> appears as the polarization energy of the electrolyte, 25% as the electrolyte's field energy inside the sphere and 25% as the electrolyte's field energy around the sphere. </abstract>

Highlights

IntroductionThe head groups of membrane lipids have either single charge (e.g. tetraether lipids [1,2]) or electric dipole (e.g. phospholipids [1,3])
The head groups of membrane lipids have either single charge or electric dipole
Theoretical models of lipid membranes usually focus on short range (Van der Waals) lateral interactions between nearest neighbor lipids and ignore the long range charge-charge interactions [3,4]

Summary

Introduction

The head groups of membrane lipids have either single charge (e.g. tetraether lipids [1,2]) or electric dipole (e.g. phospholipids [1,3]). According to the Generalized Shell Theorem the potential around a charged sphere of radius R is (see Eq. in ref.). In this paper we consider a single charged sphere (as in ref.6) and calculate the charge-charge interaction energies and the electrolyte’s field energy inside and around the charged sphere. According to the Generalized Shell Theorem, Eqs., the potential on the surface of the charged sphere of radius R with charge q is: V(R, q) ke∙q∙λD ε∙R2. The energy of the electric field inside the charged sphere, Z ≤ R, is calculated by using Eqs. (see Appendix 1): Ein ε. The energy of the electric field around the charged sphere, Z > R, is calculated by using Eqs. (see Appendix 2): Eout

Results

Discussion

Conclusions