Equilibrium mechanisms of self-limiting assembly

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (macromolecules, particles, etc.) into ordered multiunit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits and the other consisting of a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving noncovalent interactions between identical subunits "measure" and select the size of assembled structures? This review begins with an introduction to the basic statistical mechanical framework for assembly thermodynamics that is used to highlight the key physical ingredients ensuring that equilibrium assembly will terminate at finite dimensions. Then examples of self-limiting assembly systems are introduced, and they are classified within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter (micellization of amphiphiles and shell and tubule formation of tapered subunits) as well as less widely known classes of assemblies, such as short-range attractive or long-range repulsive systems and geometrically frustrated assemblies. For each of these self-limiting mechanisms, the physical mechanisms that select equilibrium assembly size, as well as the potential limitations of finite-size selection, are described. Finally, alternative mechanisms for finite-size assemblies are discussed, and contrasts are drawn with the size control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.