The second virial coefficient of bounded Mie potentials

The second virial coefficient (SVC) of bounded generalizations of the Mie m:n potential $ϕ(r)=λ[1/(aq+rq)m/q−1/(aq+rq)n/q]$⁠, where λ, a, q, m, and n are constants (a ≥ 0), is explored. The particle separation distance is r. This potential could be used as an effective interaction between polymeric dispersed colloidal particles of various degrees of interpenetrability. The SVC is negative for all temperatures for a, greater than a critical value, a_c, which coincides with the range of a, where the system is thermodynamically unstable. The Boyle temperature and the temperature at which the SVC is a maximum diverge to +∞ as a → a_c from below. Various series expansion expressions for the SVC are derived following on from those derived for the Mie potential itself (i.e., a = 0) in the study of Heyes et al. [J. Chem. Phys. 145, 084505 (2016)]. Formulas based on an expansion of the exponential in the Mayer function definition of the SVC are formally convergent, but pose numerical problems for the useful range of a < 1. High temperature expansion (HTE) formulas extending those in the previous publication are derived, which in contrast converge rapidly for the full a range. The HTE formulas derived in this work could be useful in guiding the choice of nucleation and growth experimental conditions for dispersed soft polymeric particles. Inter alia, the SVC of the inverse power special case of the Bounded Mie potential, $i.e.,ϕ(r)=1/(aq+rq)m/q$⁠, are also derived.