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

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: (Formula: see text) where Q = K'Km - delta K' delta MA3, Q' = K' delta K - delta K'KA3, Km = 3 kappa m + 4 microns - 2 gamma/a, delta K = 3 kappa m - 3 kappa - 2 gamma/a, delta M = 4 microns + 2 gamma/a, K = 4 microns + 3 kappa + 2 gamma/a, K' = 3 kappa' + 4 microns - 2 gamma'/a', delta K' = 3 kappa m- 3 kappa' + 2 gamma'/a', A = a'/a, a and a' are the outer and the inner radii of the shell, gamma and gamma' 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, kappa and mu the bulk modulus and the rigidity of the medium, in which the prime and the subscripts m denote the quantities for the inside (r less than a') and the shell (a' less than r less than a) regions, respectively, and c the volume concentration of shell structures.