Common Physical Framework Explains Phase Behavior and Dynamics of Atomic, Molecular, and Polymeric Network Formers

We show that the self-assembly of a diverse collection of building blocks can be understood within a common physical framework. These building blocks, which form periodic honeycomb networks and nonperiodic variants thereof, range in size from atoms to micron-scale polymers and interact through mechanisms as different as hydrogen bonds and covalent forces. A combination of statistical mechanics and quantum mechanics shows that one can capture the physics that governs the assembly of these networks by resolving only the geometry and strength of building-block interactions. The resulting framework reproduces a broad range of phenomena seen experimentally, including periodic and nonperiodic networks in thermal equilibrium, and nonperiodic supercooled and glassy networks away from equilibrium. Our results show how simple “design criteria” control the assembly of a wide variety of networks and suggest that kinetic trapping can be a useful way of making functional assemblies.

Popular Summary

Carbon atoms, microscopic stars made of polymerized DNA, and organic molecules can all form polygon networks, despite the fact that these building blocks differ in size by orders of magnitude and possess interactions as different as hydrogen bonds and covalent forces. Are there, then, “rules” that govern network assembly that transcend the apparent differences between building blocks? If so, mastery of such rules may one day allow us to “program” building blocks in order to self-assemble materials on demand. In this paper, we show that these networks and their self-assembly dynamics can indeed be reproduced by a model that resolves only the geometry of building blocks and the strength of their interactions, suggesting that these two parameters are more important than the atomic and chemical details of the building blocks themselves.

The central element of our work is the demonstration, using quantum mechanics and statistical mechanics, that the thermodynamics of molecular polygon formation—polygons being the key microscopic element of self-assembled networks—can be captured by a “patchy-particle” model containing no explicit chemical detail. Each material system can be related to a patchy particle with a particular patch size and binding energy. Large-scale self-assembly of patchy particles proceeds through their diffusion-driven association, and, because particles’ polygon-forming tendencies are similar to those of the real building blocks, this self-assembly reproduces the phenomena seen in experiments.

These phenomena include the formation of periodic and nonperiodic polygon networks in thermal equilibrium, similar to what is seen in the case of graphene and a silica bilayer, respectively. As nonequilibrium structures, we have observed honeycomb polycrystals, similar to those formed by a certain covalently associating molecule; a polygon network that evolves as time progresses to a honeycomb one, similar to what is seen within a hydrogen-bonding bimolecular system; and nonperiodic glassy networks, similar to those seen in experiments involving covalently associating organic molecules.

Our results suggest a unifying and practically useful framework for understanding and engineering large-scale structural assemblies.

Figure 1

Spanning a length scale of 3 orders of magnitude, the networks formed by a diverse collection of building blocks can be reproduced in simulation by accounting only for the geometry and strength of building-block interactions. Threefold-coordinated building blocks can, in equilibrium, form (A) the periodic honeycomb network [6, 7] or (E) a nonperiodic polygon network [8]. Dynamically, they can self-assemble as (B) honeycomb polycrystals [9], (C) a polygon network that evolves to the honeycomb [10], or (D) a kinetically trapped polygon-network glass. Model building blocks whose interactions (parametrized by strength $\epsilon$ and flexibility $w$) are motivated by quantum mechanical calculations (Fig. 2) can reproduce this spectrum of behavior. In equilibrium (grey lettering), such building blocks form the honeycomb network when their interactions are inflexible and a polygon network when their interactions are flexible (Fig. 3). Dynamically (blue lettering), within the regime of equilibrium-network order, building blocks self-assemble as honeycomb polycrystals when their interactions are inflexible (few polygons are generated dynamically) and as a polygon network when their interactions are flexible (many polygons are generated dynamically). If their interactions are weak, then the network evolves to the honeycomb; if their interactions are strong, then the network formed is a polygon glass (Figs. 4 and 5). Image permissions for Fig. 1: Panel A, top, reprinted (adapted) with permission from Ref. [7], copyright 2005, American Chemical Society. Panel B (experimental image) reproduced with permission from Ref. [9], copyright 2009, The Royal Society of Chemistry. Panel C (experimental image) reprinted (adapted) with permission from Ref. [10], copyright 2010, American Chemical Society. Panel E (experimental image) reprinted from Ref. [8], copyright 2012, The American Physical Society.

Figure 2

Analysis of one example from Fig. 1 reveals the microscopics of polygon formation. (a) Image of TBPB acquired using a scanning tunneling microscope fading to polygon representation (Fig. S1). (b) DFT calculations with (vdW-DF2) and without (B3LYP) van der Waals forces show the relative energy per TBPB molecule when bound in isolated, regular $n$-gons. Using this estimate of polygon thermodynamics in a topological-gas estimate (inset) shows the equilibrium network to a perfect honeycomb up to about 500 K. (Crystallinity $C$ is the fraction of the polygon network made up of hexagons [8].) (c) Histogram of the polygon number from the experiment and as predicted in equilibrium (using the topological-gas model) at two temperatures indicates that the network seen in the experiment is not in equilibrium, and so is a kinetically trapped polygon glass.

Figure 3

Model capturing the microscopics of Fig. 2 captures the range of phase behavior seen in experiments. (a) The rotational free energy per disk within bound $n$-gons is narrow when interaction flexibility $w$ is small and broad when $w$ is large. Although different in origin and functional form to the thermodynamics governing TBPB polygon formation, shown in Fig. 2, its essential features—the hexagon is favored, and other polygons are allowed with some geometrical strain—are similar. Top: Model geometry. Right: Sketches demonstrating the geometric strain felt by disks in nonhexagonal polygons; $\theta (n)=(n-2)\pi /n$ is the internal angle of a regular $n$-gon. (b) Thermodynamic simulations ($T/\epsilon =0.16$) show that the stable network undergoes a thermodynamic order-disorder transition as a function of stripe width $w$. This thermodynamics interpolates between examples of network order (panel A) and order-disorder coexistence (panel E) shown in Fig. 1. Network order $C$ is the number of hexagons divided by the total number of all polygons. Inset: Snapshot (Fig. S3) at thermodynamic order-disorder coexistence with $w=25°$ (Fig. S4).

Figure 4

Snapshots of self-assembled model networks: Compare the behavior of real systems in Fig. 1. Disks with inflexible bonds (small $w$) form polycrystals (left). Crystal grains are small if the binding strength is large (bottom), similar to cyclohexa-$m$-phenylene [9]. Disks with flexible bonds (right, large $w$) form evolving polygon networks if their bonds are weak (top), similar to the hydrogen-bonding molecules of Ref. [10], and form glasses if their bonds are strong (bottom), similar to TBPB. (A side-by-side comparison of theory and simulation is shown in Fig. S9.) Lower-case letters a–h match phase points in Fig. 5.

Figure 5

Model capturing the microscopics of Fig. 2 captures the range of nonequilibrium behavior seen in experiments. We report network order $C$ (the number of hexagons divided by the total number of all polygons) in a space of inverse bond strength $T/\epsilon$ and stripe width $w$, from dynamical simulations. When $w$ is small, polycrystals assemble (see also Fig. 4, left). For larger $w$, disordered polygon networks at early times (left panels) evolve into the stable honeycomb at later times (right panel; see also Fig. 4, upper right), as long as bonds are weak enough to break frequently as the network assembles. Otherwise, glasses are formed. Disks with unbreakable bonds (bottom) self-assemble into structures that interpolate between polycrystals (small $w$) and glasses (large $w$). Time $t$ is measured in millions of Monte Carlo cycles. Lower-case letters a–h match snapshots in Fig. 4.