Qualitative characterisation of effective interactions of charged spheres on different levels of organisation using Alexander’s renormalised charge as reference

Effective interactions are conveniently determined from experimental or numerical data by fitting a Debye–Hückel potential with an effective charge Z ∗ and an effective electrolyte concentration c ∗ as free parameters. In this contribution we numerically solved the Poisson–Boltzmann equation to obtain the so-called renormalised charge Z PBC ∗ . For sufficiently large bare charge Z one finds a saturation of Z ∗ which scales as Z ∗ = A a / λ B , where a is the particle radius, λ B the Bjerrum length and A a proportionality factor of order (8–10). The saturation value increases with increased total micro-ion concentration and shows a shallow minimum as a function of packing fraction. In addition, the bulk shear modulus G was measured along the melting line of a colloidal crystal to obtain Z G ∗ and molecular dynamics simulations were performed within the primitive model for a pair of particles at different added salt concentration to obtain Z MD ∗ . Z PBC ∗ was then used as reference for an extensive comparison to other effective charges as obtained in the present paper and taken from literature. We observe Z G ∗ to be somewhat smaller than Z PBC ∗ and other bulk experimental effective charges, while the simulation yields Z MD ∗ ≈ Z ≫ Z PBC ∗ . These differences are discussed in the light of charge renormalisation concepts and three and many body interactions.