Self-Assembly at a Nonequilibrium Critical Point

Abstract

We use analytic theory and computer simulation to study patterns formed during the growth of two-component assemblies in two and three dimensions. We show that these patterns undergo a nonequilibrium phase transition, at a particular growth rate, between mixed and demixed arrangements of component types. This finding suggests that principles of nonequilibrium statistical mechanics can be used to predict the outcome of multicomponent self-assembly, and suggests an experimental route to the self-assembly of multicomponent structures of a qualitatively defined nature. DOI: 10.1103/PhysRevLett.112.155504 PACS numbers: 81.16.Dn, 05.20.-y, 81.16.Fg Nature builds its materials using multiple component types. The properties of these materials depend on the properties of their components, and on how components are distributed spatially: the exciton transfer properties of a light-harvesting complex, for instance, depend on the way that its constituent proteins cluster [1]. Achieving similar mesoscale spatial control with many synthetic selfassembled materials [2] is made difficult by the fact that the self-assembly of multicomponent systems generally happens "far" from equilibrium, where we possess few predictive theoretical tools. Although self-assembly is a nonequilibrium process, much of our understanding of it is based upon a physical picture that assumes dynamics to play no role except to convey a system along the "easiest" pathways on its free-energy landscape [3]. This "nearequilibrium" or "quasiequilibrium" assumption tends to hold, for instance, for simple one-component systems under mild nonequilibrium conditions [4][5][6][7][8]. It fails when there exist time scales within a given self-assembly process that exceed the time of the experiment or computer simulation. For one-component systems, long time scales can emerge if bonds between particles are strong, and so break infrequently, which can happen when subjected to "harsh" nonequilibrium conditions (e.g., conditions of deep supercooling). Binding errors made as components associate can then fail to anneal as structures grow, and the result is a kinetically trapped structure rather than an object corresponding to a favored position on the free-energy landscape [9][10][11][12].The self-assembly of multicomponent solid structures, however, is also affected by kinetic traps that emerge even under mild nonequilibrium conditions. The long time scale responsible for such trapping is the slow interchange of component types within solid structures [13][14][15][16][17][18][19][20][21][22][23]. Consider Fig. 1(a), which shows in cartoon form a fluid of two types of mutually attractive particles (call them "red" and "blue"), present in equal number, self-assembling into a solid structure. The equilibrium pattern of red and blue within this structure-shown in the figure as demixed red and blue domains-is determined only by the particles' mutual interactions. But achieving this equilibrium pattern via self-assembly requires that particle types interchange their positio...

Design principles for nonequilibrium self-assembly

“…24. The agreement between the two estimates confirms that the system has anomalously large fluctuations at the nonequilibrium critical driving force we identify.…”

“…We now demonstrate these ideas by considering a 2D analog of the fiber growth process described above (ref. 24 and Fig. 2A).…”

“…This model was first introduced in ref. 24. We again imagine two monomer types with interactions between like monomers set by the energy scale s and interactions between unlike monomers set by the energy scale to the intensive equilibrium free energy of the assembly, µcoex ≈ 2 s −kBT ln 2, the assembly does not grow on average.…”

“…In ref. 24,Whitelam et al demonstrated that by tuning the chemical potential driving force, this system can be positioned at a nonequilibrium critical point even though the coupling constant J is higher than Jc. Given these nontrivial thermodynamic properties, this nonequilibrium 2D model system is ideal for illustrating how our bounds can be effective even in conditions with large correlation lengths.…”

“…To confirm that the value of the critical driving force, δµc computed using the Onsager method above is indeed the same one as the critical driving force discovered in ref. 24, we also determine δµc following the method described in ref. 24.…”