Multidimensional equilibria and their stability in copolymer–solvent mixtures

Evolution and Competition of Block Copolymer Nanoparticles

“…The free boundary approximation of (5) is used as a more tractable model for our analysis. This results in the usual way (see, e.g., [27,34,45,46]) from the limit \epsilon \rightar 0, wherein (\Phi , \Psi ) \rightar (\Phi 0 , \Psi 0 ) = \bfitu A,B,S almost everywhere. The limiting configuration may therefore be associated with corresponding domains \Omega A,B,S .…”

“…Theoretical models have been successful at the replication and prediction of copolymer nanoparticle assembly [9,23,24,25,26,27,28,29]. The primary focus of past work has been to reproduce equilibrium morphologies seen in experiments.…”

“…As common with phase-field approaches, there is a natural limiting case where sharply defined domains of different composition form. This allows passage to a free boundary problem which describes the evolution of domain boundaries, rather than the composition itself [27,34,35]. This type of free boundary problem is employed here to study domain interface configurations.…”

“…Localized or confined multidomain structures have been studied previously in the context of density functional theory and its free boundary limit. This includes lamellar configurations and their stability [36,37,38], concentric geometries [27,39], double bubbles [40], and dynamic phenomena [41,42]. Numerical work using Cahn--Hilliard-type systems has also revealed a wide range of equilibrium morphologies [9,26].…”

“…The fundamental objects in this study are nanoparticle equilibria, which are stationary solutions to the free boundary problem. A wide variety of such configurations have been observed [9,27]. These structures are in equilibrium with their local environment, in the sense that in isolation the chemical potential \mu which drives polymer mass diffusion is constant everywhere.…”