  We consider an emulsion whose droplets contain a trapped species (insoluble
in the continuous phase), and study the emulsion's stability against coarsening
via Lifshitz-Slyozov dynamics (Ostwald Ripening). Extending an earlier
treatment by Kabalnov et al(Colloids and Surfaces, 24,(1987), 19-32), we derive
a general condition on the mean initial droplet volume which ensures stability,
even when arbitrary polydispersity is present in both size and composition of
the initial droplets. We distinguish "nucleated" coarsening, which requires
either fluctuations about the mean field equations or a tail in the initial
droplet size distribution, from "spinodal" coarsening in which a typical
droplet is locally unstable. A weaker condition for stability, previously
suggested by Kabalnov et al., is sufficient only to prevent "spinodal"
coarsening and is best viewed as a condition for metastability. The coarsening
of unstable emulsions is considered, and shown at long times to resemble that
of ordinary emulsions (with no trapped species), but with a reduced value of
the initial volume fraction of dispersed phase. We discuss the physical
principles relevant to the stability of emulsions with trapped species,
describing how these may be exploited to restabilise partially coarsened
emulsions and to "shrink" previously formed emulsion droplets to form
"miniemulsions".
