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 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 assembly of a wide variety of networks, and suggest that kinetic trapping can be a useful way of making functional assemblies.


I. INTRODUCTION
Molecular self-assembly is a promising strategy for making useful materials, and has already produced many remarkable structures in the laboratory [1,2]. But it remains largely an empirical science, in the sense that we do not know in advance which components and which conditions will give rise to successful assembly. If we could go beyond empiricism, by identifying the physical concepts and rules that underpin molecular self-assembly, then presumably we could build materials with functionalities approaching those of biological materials. The pursuit of the underlying physical principles of self-assembly motivates a large body of ongoing theoretical work -Refs. [3][4][5] being three examples -and is the motivation for this paper.
Here we take the view that in pursuit of the physical principles that underpin self-assembly there is value in identifying physical mechanisms common to apparently unlike systems. We shall show that the self-assembly of a diverse collection of building blocks, one example of which comes from our own work, can indeed be understood within a common physical framework. These building blocks range in size from atoms to micron-scale polymers made of DNA, and interact through mechanisms as different as hydrogen bonds and covalent forces. We show that in a qualitative sense the self-assembly of these building blocks, which results in a range of phenomena that include periodic and nonperiodic networks in thermal equilibrium, and nonperiodic supercooled and glassy networks away from equilibrium, can be repro- * swhitelam@lbl.gov † Isaac.Tamblyn@uoit.ca ‡ Peter.Beton@nottingham.ac.uk duced by a statistical mechanical 'patchy particle' simulation model. The model accounts only for the geometry and strength of building block interactions, indicating that these two physical factors control assembly of the real networks. Furthermore, we use quantum mechanics and analytic statistical mechanics techniques to show why we think this is so: the thermodynamics of association of model building blocks and real building blocks into isolated polygons, which one might regard as the basic constituents of self-assembled networks, is in a qualitative sense the same. This similarity reveals that the model, despite containing none of the molecular or chemical detail of the real systems, nonetheless captures a key microscopic physical feature of the self-assembly of these systems, and explains why -or at least suggests why we should not be surprised when -the model and real building blocks, undergoing Brownian motion, give rise to similar equilibrium and dynamic phenomena.
In what follows we introduce the set of experimental examples we will focus on (Section II). We do a quantum mechanical (density functional theory, or DFT) analysis of one of these examples (Section III), to calculate the free energy cost of arranging molecules into isolated polygons. This calculation allows us to show that the experimental network is trapped far from equilibrium, but it also quantifies a key microscopic feature of this system, namely the thermodynamics of association of molecules into the basic polygon constituents of the network. We then introduce (Section IV) a statistical mechanical patchy particle model able to form networks. We show within a simple analytic approximation that the thermodynamics of association of model particles into polygons is similar to that of the real system studied in Section III. This similarity then provides a partial explanation for why equilibrium (Section V) and dynamic (Section VI) simulations of the model reproduce the range of behavior seen experimentally. We conclude in Section VII.  1. Spanning a lengthscale of three 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. Three-foldcoordinated building blocks can, in equilibrium, form the periodic honeycomb network (A) [6,7] or a nonperiodic polygon network (E) [8]. Dynamically, they can self-assemble as honeycomb polycrystals (B) [9], a polygon network that evolves to the honeycomb (C) [10], or a kinetically trapped polygon network glass (D). Model building blocks whose interactions (parameterized by strength 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). For image permissions, please see end of paper.
In isolation, each of the techniques we have used in this paper -self-assembly experiments, DFT calculations of assembled molecules, analytic statistical mechanical treatments of networks, and equilibrium and dynamic simulations of patchy particle models -has been used extensively by other authors; references are given in the text. The focus of this paper is not the use of these methods individually, but the chain of connections we have drawn between experiment, the quantum mechanics of molecular interactions, and the behavior of a statistical mechanical model. We have therefore chosen to consign much of the technical detail of the individual methods to Appendices, referenced from the relevant section of text, and have focused the narrative on developing this chain of connections. Our hope is that by doing so we have written a paper that appeals to a broad readership, par-ticularly those who are not expert with one or other of the techniques we have used.

II. SELF-ASSEMBLY ACROSS SCALES
Let us now introduce the experimental examples on which we will focus. Panels A to E of Fig. 1 summarize a range of phase behavior and dynamics exhibited by a diverse collection of building blocks. These building blocks self-assemble into planar networks by making three pairwise bonds. When bonds are distributed regularly around the building block, the network formed is the periodic honeycomb: consider carbon atoms [6] or a DNA star polymer [7] (panel A), as well as a host of other systems [11]. Three-fold coor-dination also permits the formation of nonperiodic variants of the honeycomb. Zachariasen showed in a sketch in 1932 [12] that irregular 3-fold coordination results in a network of polygons of different sizes. Such a network is seen in the case of silica [8] (panel E) on a surface. Furthermore, a range of dynamics is associated with network self-assembly. The covalently-associating molecule cyclohexa-m-phenylene forms polycrystals, sections of honeycomb network punctuated by grain boundaries [9] (panel B). Certain hydrogen-bonding molecules self-assemble initially as a nonperiodic polygon network that subsequently relaxes to the honeycomb [10] (panel C). A distinct dynamics is seen in the case of the trigonal molecule tris(4-bromophenyl)benzene (TBPB) [13] (panel D): this molecule forms a polygon network that does not evolve to the honeycomb. Preparation of this network is described in Appendix A.

III. MICROSCOPIC UNDERPINNING OF ONE PARTICULAR EXPERIMENT
The spectrum of behavior seen within this class of building blocks can be reproduced within a simple physical framework that resolves only coarse details of the geometry and energetics of building block interactions ( Fig. 1, simulation snapshots and lower panel). This framework was inspired by resolving, for the particular case of TBPB, the collective microscopic mechanisms that determine the basic polygon units of the network. In Fig. 2(a) we show a portion of the polygon network generated during TBPB self-assembly at 410 K on a gold surface (see Supplemental Information (SI)). As described in Appendix B, we used density functional theory (DFT), using functionals with (vdW-DF2) and without (B3LYP) van der Waals interactions, to calculate the relative energy cost, per molecule, for arranging molecules into isolated, regular n-gons. These n-gons approximate the basic elements of the network. This energy cost captures the essence of the thermodynamics of molecules' polygon-forming tendencies [14,15]. It is shown in Fig. 2(b). Three features are apparent: molecules favor the hexagon, whose geometry is commensurate with the symmetry of the molecule; molecules may form other polygons, at an energy cost on a scale approaching eV (calculations done on interacting loops give similar numbers; Appendix B); and the shape of the potential is not symmetric in n, as is sometimes assumed in idealized foam models [16].
Simple estimates based on the energy cost of forming isolated polygons of TBPB molecules suggest that the experimental network is trapped far from equilibrium. To a first approximation we see that the energy cost to turn a pair of hexagons into a heptagon and a pentagon is of order eV/2, indicating that in equilibrium at experimental temperatures the network should be a tiling of hexagons with characteristic linear distance between defects of order microns. As seen in Fig. 2(a), this is not the case. At one further level of refinement, a 'topological gas' calculation [16] (see Appendix C), a mean-field thermodynamic estimate that assumes the network to be composed of isolated polygons whose average size is 6, indicates that the network in thermal equilibrium should be the honeycomb up to a temperature of at least 500 K ( Fig. 2(c)). We therefore conclude that the polygon network seen in experiments is probably a nonequilibrium, glassy one (at this level of approximation we are not considering irregular polygons or interactions between polygons, and so we cannot prove conclusively that the network seen is a nonequilibrium one). Note that inclusion of van der Waals forces in our DFT calculations changes considerably our numerical estimate of the network ordering temperature, but not this qualitative conclusion (inset to Fig. 2(b)).

IV. A STATISTICAL MECHANICAL MODEL OF NETWORK FORMATION.
Motivated by our microscopic insight into this particular system, and by the ability of coarse-grained models to capture key physical features of complicated systems [3,5,[17][18][19][20][21], we next built a simple physical model of interacting 'building blocks' in an attempt to capture the essence of TBPB's self-assembly. The model accounts only for the geometry and strength of interactions between building blocks, and pays no attention to the atomic or chemical detail through which these features arise in the real system. Although our original focus was TBPB, we found that by varying two parameters of the model -binding strength and flexibility -we could reproduce the behavior of all the systems described in Fig. 1. This finding suggests that the same two factors control the self-assembly of those systems, independent of their molecular details.
Following work on 'patchy particle' simulation models [22][23][24][25], we consider striped discs living on a smooth, two-dimensional substrate ( Fig. 3(a)). Three stripes, each of angular width 2w, are placed regularly around the disc. Discs bind in a pairwise fashion, stripe-tostripe [26], with energy of interaction − . Full details of the interaction potential are described in Appendix D. In figures, stripes are green when bound in this fashion. The parameter w determines the flexibility of disc interactions: the broader the stripe (the larger is w), the less precisely need two discs align in order to bind.
When is large enough, discs can form 3-fold coordinated polygon networks. We can gain microscopic insight into the network-forming tendencies of discs by calculating the thermodynamics of isolated bound polygons of discs (the basic elements of networks), just as we did for TBPB. We calculated this thermodynamics within a simple approximation that considers only the rotational freedom discs' possess when bound in this fashion. Details of this calculation are given in Appendix D; the resulting free energy per disc as a function of polygon edge number    (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 polygon number from experiment and as predicted in equilibrium (using the topological gas model) at two temperatures indicates that the network seen in experiment is not in equilibrium, and so is a kinetically trapped polygon glass.
n is where z 1 (n) ≡ max (0, 2w − π|n − 6|/3n) is the angle a disc can rotate without its stripes breaking contact with either of its two neighbors. Eq. (1) is plotted in Fig. 3(a). This rotational entropy is largest for the hexagon, because discs may rotate the full angular width of the stripe without breaking energetic contact. In other polygons, bound discs have less rotational freedom (as can be seen by looking at sketches of e.g. the pentagon vertex shown next to the free energy plot in Fig. 3(a)), and so the free energy per disc is larger than in the hexagon. Rotational entropy therefore favors network order [27]. The microscopic origin of this thermodynamics (rotational entropy) is therefore different than for the TPBP molecules of Fig. 2 (the energy cost of irregular bond angles). Despite this microscopic difference, the essence of both systems' polygon-forming tendencies is the same: they favor hexagons, and they can achieve, with some free energy cost, other polygons. Within the model, this cost is controlled by w, the binding flexibility. This similarity suggests that the model, although simple, captures the physics essential to TBPB polygon formation, and, by extension, network formation (because polygons are the key constituent of the latter).
Note that the strategy of considering the free energy cost of arranging building blocks into important microscopic elements of a larger structure was used with success in [27] (compare Fig. 2(b) of that paper with our Fig. 3(a)): here the same strategy allows us to compare model building blocks and real molecules in order to develop the connection between the two. The similarity of model building blocks and TBPB molecules with respect to their thermodynamics of polygon formation leads to similar behavior in the nonequilibrium regime in which TBPB is prepared; this is described below. Moreover, by varying model parameters controlling building block binding flexibility (w) and strength ( ), the model also reproduces the behavior of the other systems shown in Fig. 1. Thermodynamically, a meanfield topological gas estimate applied to the model (details given in Appendix E) predicts a crossover from a honeycomb network at small w (favored by discs' rotational entropy) to a polygon network at large w (favored by configurational entropy). The latter is a 2D analog of a 3D patchy colloid liquid shown to be stable with respect to its crystal at zero temperature [28]: that reference therefore identified the physics (the entropy associated with bond flexibility) that permits the fully-connected polygon network to be stable with respect to the honeycomb one.
Turning to standard equilibrium MC simulations of the discs themselves (see Appendix F), which account for interactions and fluctuations absent from the topological gas mean-field estimate, we show in Fig. 3(b) that the essence of the mean-field estimate, the change from an ordered network to a disordered one as a function of bond flexibility w, is confirmed by thermodynamic simulations [29]. (Note that in snapshots we draw polygons atop discs, but we simulated the discs themselves). In simulations, however, the transition from order to disorder is not a smooth crossover but a true phase transition. Temperature-concentration phase diagrams are shown in Fig. S4, demonstrating that in some regions of phase space there exists coexistence between ordered and disordered networks. The thermodynamics of the patchy 100 ns/sec (7) 100 ns/sec (7) (c) (c) disc model therefore interpolates between the examples of network order given in panel A of Fig. 1 (graphene and the DNA star [7]), and the order-disorder coexistence shown in panel E of Fig. 1 (silica). This finding, combined with our analysis of the DFT results of Ref. [14] ( Fig. S2), leads us to interpret the silica patterns de-scribed in Refs. [8,14] as thermodynamic phase coexistence between honeycomb and polygon networks [30], albeit frozen because of the low temperatures at which images were taken.

VI. MODEL ALSO REPRODUCES DYNAMICS SEEN IN DIFFERENT EXPERIMENTS.
A range of nonequilibrium behavior also emerges upon variation of binding energy and flexibility. In Figs. 4 and 5 we report the results of dynamical simulations [31], described in Appendix G, in which discs were allowed to exchange with and diffuse on an initially empty substrate. When interactions are inflexible (i.e. when w is small), only hexagons may form. Dynamically-generated networks in this regime are polycrystalline, having few grain boundaries in the weak bond (nucleation) regime, and many grain boundaries in the strong bond (spinodal) regime (Fig. S5). This behavior is like that of the covalent polycrystalline networks shown in panel B of Fig. 1 [9].
By contrast, a regime in which polygons can be generated dynamically is found when building block interactions are more flexible (i.e. when w is larger), still within the regime in which the network is ordered thermodynamically. Here, the initial pieces of self-assembling networks are made of a distribution of polygons, because collective microscopic motions lead to rapid formation of loops of particles that need not be six in number. When bonds are weak (i.e. when is small), this polygon network evolves to the thermodynamically stable honeycomb one. This two-step dynamics is like that seen in the H-bonded molecules shown in panel C of Fig. 1 [10]; simulations of model clathrin honeycomb self-assembly display a similar dynamics [32]. When bonds are strong (i.e. when is large), the polygon network is instead kinetically trapped, resulting in a glass. Slow relaxation of polygon defects in the face of strong bonds has been extensively discussed: see e.g. graphene [33], clathrins [32] and foams [34]. This dynamics is similar to that displayed by TBPB, the inspiration for the model.
Glasses' polygon distributions are sensitive to rates of particle deposition, indicating that they are not simply frozen versions of the disordered network stable in equilibrium at larger w (Fig. S6, Fig. S7). Instead, they are nonequilibrium structures whose polygon statistics is determined by collective microscopic motions (Fig. S8). The strong visual similarity between our simulations and experiments (Fig. S9) indicates that the model captures the physics that determines experimental patterns: molecules' substantial binding flexibility allows the formation, via a diffusive dynamics, of a range of polygons. These polygons are then 'frozen in' because bonds are too strong to be broken: we calculated from DFT the bond strength of TBPB be 5 eV, an effectively unbreakable 150 k B T at experimental temperatures. Our simulations also provide an explicit demonstration of the nonequilibrium T = (✏ s ✏ d ) 1 (11) min ⇣ 1, e ( E±µ) (c) T = (✏ s ✏ d ) 1 (11) 0.9 0.8 0.7 0.6 0.5 0.4 (1) (c) (c) (c)    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 / 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. Discs 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 on Fig. 4. origin suggested for isolated polygons made from the covalently-associating molecule 1,3,5-triiodobenzene [15].

VII. CONCLUSIONS.
We have shown that the thermodynamic and dynamic properties of self-assembled networks whose basic length-scales span three orders of magnitude can be reproduced within a common physical framework. This framework, developed using a combination of quantum mechanics and statistical mechanics, resolves only the geometry and strength of binding of network-forming building blocks, not their chemical and atomic details. This finding indicates that there exist basic 'design criteria' -here geometry and strength of binding -that control the assembly of the building blocks of Fig. 1. Our results also indicate that structure formation driven by irreversible bonds, sometimes not classed as 'self-assembly' [2], can nonetheless be considered within the same physical framework as assembly driven by reversible bonds: the behavior of covalently-associating molecules and those interacting via reversible bonds can be reproduced in different parameter regimes of the same model. The key limitation of our work is that it is of course qualitative, in respect of the comparison between experiments and statistical mechanical model. Nonetheless, quantum mechanics allows one to quantify the microscopic interactions between molecules, and so to make our approach quantitative with respect to a particular system, one could consider a statistical mechanical model with an interaction potential just complicated enough to permit exact reproduction of real molecules' free energy cost of polygon formation. We also note that we see no impediment to doing a similar study of other geometries in 2D [35], or in 3D: indeed, recent work has shown that simplified model particles that again focus only on geometry and energy scales of binding [36,37] (the latter being a 3D equivalent of the model studied here) can in 3D capture important structural and thermodynamic features seen in experiments done on water, and atomistic simulations of water and silica.
Our results also suggest ways of making functional materials by using kinetic trapping to generate defined nonequilibrium assemblies. Kinetic trapping, the failure of a set of self-assembling components to achieve the structure lowest in free energy, is often regarded as a nuisance, not a virtue. But the nonperiodic polygon networks studied here are generated by kinetic trapping. They have microscopic environments similar to the honeycomb, but mesoscopic environments substantially different, and so have properties not attainable to their periodic, equilibrium counterparts. Atomic-scale polygon network graphene has recently been predicted in simulations [38]; this material would have novel conductance properties [39]. Given that 'patchy particle' models like the one use here first appeared as models of colloids, we predict that colloids -perhaps 3-patch 'lock-and-key' ones [40] -could self-assemble as a nonperiodic polygon network, provided that their interactions are made sufficiently strong and flexible (Fig. 5). Such a material would have novel photonic properties [41].
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 from Ref. [9] with permission from 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) by The American Physical Society.