Bulk modulus of dispersions of spherical shell structures taking into account interfacial tensions

Abstract

The bulk modulus of a dilutely dispersed system of spherical shell structures is obtained taking into account interfacial tensions at both the interfaces of the shell as follows: $${\mathit{\kappa }}^{\ast }={\displaystyle \frac{\mathit{\kappa }\mathrm{Q}+\frac{4}{3}\mathrm{c}\mathit{\mu }{\mathrm{Q}}^{\prime }}{\mathrm{Q}-\mathrm{c}{\mathrm{Q}}^{\prime }}}$$ where $\mathrm{Q}={\mathrm{K}}^{\prime }{\mathrm{K}}_{\mathrm{m}}-\mathrm{\Delta }{\mathrm{K}}^{\prime }\mathrm{\Delta }{\mathrm{MA}}^3$ , ${\mathrm{Q}}^{\prime }={\mathrm{K}}^{\prime }\mathrm{\Delta }\mathrm{K}-\mathrm{\Delta }{\mathrm{K}}^{\prime }{\mathrm{KA}}^3$ , ${\mathrm{K}}_{\mathrm{m}}=3{\mathit{\kappa }}_{\mathrm{m}}+4\mathit{\mu }-2\mathit{\gamma }\mathrm{/}\mathrm{a}$ , $\mathrm{\Delta }\mathrm{K}=3{\mathit{\kappa }}_{\mathrm{m}}-3\mathit{\kappa }-2\mathit{\gamma }\mathrm{/}\mathrm{a}$ , $\mathrm{\Delta }\mathrm{M}=4{\mathit{\mu }}_{\mathrm{m}}-4\mathit{\mu }+2\mathit{\gamma }\mathrm{/}\mathrm{a}$ , $\mathrm{K}=4{\mathit{\mu }}_{\mathrm{m}}+3\mathit{\kappa }+2\mathit{\gamma }\mathrm{/}\mathrm{a}$ , ${\mathrm{K}}^{\prime }=3{\mathit{\kappa }}^{\prime }+4{\mathit{\mu }}_{\mathrm{m}}-2{\mathit{\gamma }}^{\prime }\mathrm{/}{\mathrm{a}}^{\prime }$ , $\mathrm{\Delta }{\mathrm{K}}^{\prime }=3{\mathit{\kappa }}_{\mathrm{m}}-3{\mathit{\kappa }}^{\prime }+2{\mathit{\gamma }}^{\prime }\mathrm{/}{\mathrm{a}}^{\prime }$ , A = a′/a, a and a′ are the outer and the inner radii of the shell, γ and ${\mathit{\gamma }}^{\prime }$ the interfacial tensions at r = a and r = a′, in which r is the radial distance from the origin at the center of the shell structure, κ and μ the bulk modulus and the rigidity of the medium, in which the prime and the subscripts m denote the quantities for the inside (r<a′) and the shell (a′<r<a) regions, respectively, and c the volume concentration of shell structures.