Dimensionality and design of isotropic interactions that stabilize honeycomb, square, simple cubic, and diamond lattices

We use inverse methods of statistical mechanics and computer simulations to investigate whether an isotropic interaction designed to stabilize a given two-dimensional (2D) lattice will also favor an analogous three-dimensional (3D) structure, and vice versa. Specifically, we determine the 3D ordered lattices favored by isotropic potentials optimized to exhibit stable 2D honeycomb (or square) periodic structures, as well as the 2D ordered structures favored by isotropic interactions designed to stabilize 3D diamond (or simple cubic) lattices. We find a remarkable `transferability' of isotropic potentials designed to stabilize analogous morphologies in 2D and 3D, irrespective of the exact interaction form, and we discuss the basis of this cross-dimensional behavior. Our results suggest that the discovery of interactions that drive assembly into certain 3D periodic structures of interest can be assisted by less computationally intensive optimizations targeting the analogous 2D lattices.

spatial dimension affects the design rules for assembly. For example, to what extent will an interaction designed to stabilize a given 2D lattice also favor an analogous 3D structure, and vice versa [47]? The answer is of fundamental interest and may also have important practical implications because finding interactions that stabilize lattices in 2D is a simpler and less computationally demanding material design problem than in 3D. Here, we study this question using computer simulations and model potentials designed by inverse statistical mechanical optimization [48,49].
In particular, we determine the 3D ordered lattices favored by models with isotropic potentials ϕ hc (or ϕ squ ) optimized to exhibit stable 2D honeycomb (or square) periodic structures, as well as the 2D ordered structures favored by isotropic interactions ϕ dia (or ϕ sc ) designed to stabilize 3D diamond (or simple cubic) lattices [50]. As we show, the isotropic potentials optimized for either 2D or 3D target structures also do surprisingly well at stabilizing the analogous lattices in the other dimension.
A specified target lattice is the ground state for a given pair potential ϕ and pressure p if, and only if, it is mechanically stable at this condition and its zero-temperature chemical potential (i.e., molar enthalpy) is lower than that of all other mechanically stable competing structures. Here, we use a stochastic optimization approach (described in detail elsewhere [30]) to discover new model pairwise interactions ϕ target that maximize the range of density ρ for which a 2D target lattice is the ground state. In our optimizations, we consider isotropic, convex-repulsive pair potentials that qualitatively mimic the soft, effective interactions of sterically-stabilized colloids or nanoparticles [51]. The form we adopt can be expressed [30] ϕ Here, x = r/σ is a dimensionless interparticle separation; and σ are characteristic energy and length scales; x cut is the dimensionless potential range; H is the Heaviside step function; and f shift (x) = P x 2 + Qx + R is a shifting function with fitting constants P, Q, R chosen to ensure ϕ(x cut ) = ϕ (x cut ) = ϕ (x cut ) = 0. All together, there are nine dimensionless parameters that can be varied in the optimization algorithm (x cut , A, n, λ 1 , k 1 , δ 1 , λ 2 , k 2 , δ 2 ); however, one is not independent of the others because we also require ϕ(1)/ = 1. From here forward, we report quantities implicitly nondimensionalized by appropriate combinations of and σ.
To identify the ground-state phase diagram for a given pair potential ϕ, we compare the p-dependent, zero-temperature chemical potentials of a wide variety of Bravais and non-Bravais lattices in a 'forward' calculation. Several methods for identifying candidate ground states are available, including evolutionary optimization [52,53] as well as shape matching and machine learning algorithms [54]. In this study, we use simulated annealing optimization [30] to determine free lattice parameters which minimize the chemical potentials of the structures subject to the constraint of mechanical stability, as determined by phonon spectra analysis [55]. In 2D, the Bravais lattices consist of oblique, rhombic, square, rectangular, and triangular symmetries; here, we limit our consideration of non-Bravais lattices to honeycomb, kagome, and other five-vertex semi-regular tilings, namely snub-hexagonal, snub-square, and elongated-triangular. For 3D, we consider the following Bravais and non-Bravais lattices identified in a previous study on closely related model interactions [56]: facecentred cubic (FCC), body-centred cubic (BCC), simple cubic (SC), diamond, pyrochlore, body-centred orthogonal (BCO), hexagonal (H), rhombohedral (hR), cI16, oC8, βSn, A7, A20, and B10. While the methods employed both to determine the interaction potentials optimal for a target lattice and to compute the corresponding ground states are identical in 2D and 3D, we note that calculations are significantly faster in 2D than in 3D due to the smaller number of competing structures to consider in 2D and the reduced dimensionality of the lattice sum and the phonon spectra evaluations.
For computational efficiency of inverse optimizations in 2D or 3D, only a limited set of competing structures can be considered for a specific target lattice, ideally consisting of the lattices which have the lowest chemical potentials for the interaction type over the density range of interest. Here, we use a simple iterative process for determining the competitive lattice pools. Specifically, we (1) begin with a trial set of competitive structures; (2) carry out an inverse optimization calculation using this competitive pool to obtain parameters for a trial optimal potential; (3) perform an extensive forward calculation to determine the ground-state phase diagram of the trial potential; (4) as necessary, refine the competitive pool based on the lattices that appear in the forward calculation in (3) and return to step (2). The final pools determined from this method contained a diverse array of structures in 2D and 3D [57].
To obtain information about the thermal stability of the target lattices, we also perform Monte Carlo quench simulations in which a high-temperature fluid is instantaneously cooled down to a much lower temperature to observe assembly of the target structure. Our simulation sizes were chosen such that larger systems did not affect the results (for more details, see Table S1 and discussion in Supplementary Information). We note that interactions previously optimized to stabilize 3D target ground states of diamond (ϕ dia ) and simple cubic (ϕ sc ) lattices over a wide range of density-using methods identical to those employed here-lead to target crystalline phases with good thermal stability [58].
The interaction potentials we obtain for maximizing the density range of 2D honeycomband square-lattice ground states [59] together with previously optimized interactions for diamond-and simple cubic-lattice ground states [30], are shown in Fig. 1. Notice that interactions ϕ hc and ϕ dia are remarkably similar to one another, despite the fact that they were obtained from optimizations favoring different (albeit analogous) structures in different spatial dimensions. As is shown in the inset to Fig. 1, significant discrepancies between these potentials (i.e., the steeper repulsions of ϕ hc ) are only present for interparticle separations x < 0.6 that, as we confirm below, are closer than the nearest neighbor distance for the honeycomb or diamond lattices in the density range where the structures are stable for either model. Based on the similarity of these interactions, one might already expect that ϕ hc and ϕ dia would stabilize similar lattices in 2D and 3D. On the other hand, we see appreciable differences between the potentials ϕ squ and ϕ sc optimized to stabilize 2D square and 3D simple cubic lattices, respectively. Of the four interactions studied here, ϕ squ has the softest repulsive core and the longest range, while ϕ sc has the steepest core repulsion and the shortest range.
In Fig. 2, we show the results of our forward calculations, i.e., the 2D ground states for the four optimized potentials as a function of density [60]. Shaded regions represent densities where the ground state comprises two neighboring lattices in coexistence. First, we note that the 2D inverse optimization calculations succeed in their goal: stable honeycomb-and square-lattice ground states appear for ϕ hc and ϕ squ , respectively, over very wide density ranges, especially when compared to those of other repulsive, isotropic interaction models [36,39,61] known to form these phases. Perhaps more noticeable is not only that the 2D honeycomb lattice is stabilized over a similar density range by the 3D-optimized ϕ dia (a result now expected based on the similarity to ϕ hc shown in Fig. 1), but also that the square lattice is stabilized over a wide density range by ϕ sc (despite significant differences compared to ϕ squ ). In other words, for both cases, stable 2D ground states of interest were obtainable by optimizing interactions for a corresponding analogous target lattice in 3D.
To test the same approach in the other direction, i.e., whether optimizing analogous 2D structures will stabilize 3D target lattices of interest, we also determine the 3D ground states for ϕ hc and ϕ squ . The results, presented in Table I, show that ϕ hc and ϕ squ indeed display wide stability regions for diamond and simple cubic lattice ground states, respectively. In fact, not only are the density ranges of the stable diamond lattice comparable for ϕ hc and ϕ dia , but the density range of the simple cubic lattice for ϕ squ is even slightly wider than that of ϕ sc [62]. The latter result likely reflects the fact that the faster optimizations targeting 2D ground states enables a more thorough exploration of parameter space during the calculation than is practical in the 3D optimizations.
That particles with isotropic interactions encoded to form 3D diamond (or simple cubic) lattices also display 2D honeycomb (or square) arrays, although nontrivial, is in some sense not surprising. The tetrahedrally-coordinated diamond lattice itself consists of undulating interconnected trivalent honeycomb networks, and the simple cubic structure comprises square arrays stacked in registry. However, the outcome that particles with interactions designed to stabilize 2D honeycomb (or square) lattices also favor diamond (or simple cubic) lattices and not other morphologies containing honeycomb (or square) motifs such as graphite (or body-centered cubic) structures is much more interesting.
To understand these results, it is helpful to recall that-for isotropic potentials-the zerotemperature chemical potential depends only on the pair interaction and properties of coordination shells located at distances closer than the interaction cut-off, x < x cut . In Fig. 3, we plot the interparticle separations corresponding to the first, second, and third coordination shells {x 1 , x 2 , x 3 } for the four lattices of interest here-honeycomb (hc), square (squ), diamond (dia) and simple cubic (sc)-considering densities where these lattices are the ground states for the models ϕ hc and ϕ squ . First, note that there is considerable overlap between the coordination-shell distances of the honeycomb and diamond structures. Thus, an isotropic potential which stabilizes a honeycomb structure in 2D is expected to be an excellent (if not necessarily optimal) candidate for forming a diamond lattice in 3D, and vice versa. This helps to explain the near identical potentials, ϕ hc and ϕ dia , despite their being obtained via optimization of different target structures in different spatial dimensions.
To gain further insights, we also compare the coordination-shell distances of the honeycomb lattice with another related 3D structure, graphite, which consists of stacks of 2D honeycomb (i.e., graphene) sheets. Note that only the nearest-neighbor distances of mechanically stable 3D graphite lattices align with the first coordination-shell separations of 2D honeycomb structures, and there is substantial mismatch of other relevant coordination distances (i ≥ 2) (see Fig. S1 in the Supplementary Information). In this important sense, graphite-while closely related to the honeycomb lattice in other ways-is not as analogous to honeycomb as the 3D diamond structure is in its relation between interaction and coordination-shell structure, and is thus, not favored as a ground-state by ϕ hc at any density.
In comparing the other case of square versus simple cubic lattices, we see that the first two coordination shells of these structures similarly overlap, but the third shell positions are not in alignment. This result-together with the ground-state calculations presented abovesuggests that, for short-range interactions, the common separation distances between the nearest and next-nearest neighbors for square and simple cubic structures in enough to allow for an optimal 2D square-forming potential to assemble into 3D simple cubic structures, and vice versa. However, the differences in the third-shell distances might help to explain the significant variations in the optimized potentials targeting 2D-square (ϕ squ ) versus 3D-simple cubic (ϕ sc ) lattices shown in Fig. 1.
In Fig 4a- Table S1). In Fig. 4g, To summarize, we have investigated the cross-dimensional phase behavior of specifically designed isotropic interactions with low coordination. In particular, we have determined the 3D ordered lattices favored by isotropic potentials ϕ hc (or ϕ squ ) optimized to exhibit stable 2D honeycomb (or square) lattice structures, as well as the 2D periodic structures favored by isotropic potentials ϕ dia (or ϕ sc ) optimized to assemble into 3D diamond (or simple cubic) morphologies. We find surprising transferability of interactions designed to stabilize analogous structures in 2D and 3D, and we gain insights into this behavior by studying the different ways in which information in the analogous target structures encodes itself in the optimal isotropic potentials through the coordination-shell geometry.
One practical implication of the observed physics in this study is that the design of certain 3D lattices can greatly benefit from knowledge of potentials derived to maximize the stability of analogous 2D structures, information which can be obtained at relatively modest computational expense. The computational efficiency gained from this approach might be most valuable in multi-step optimization processes, where the goal to search for an interaction potential favoring a target structure is only one of several objectives within the design calculation. It will also be interesting in future studies to explore the effects of the interaction range on the cross-dimensional behavior of isotropic interactions obtained through inverse design, especially where one limits the potential range to encompass only two coordination shells. While we focus here on the dimensionality dependence of design rules pertaining to target structures formed by isotropic interactions, it will also be informative to study the effect of spatial dimension on other classes of interactions, e.g., short-ranged anisotropic interactions of patchy particles relevant to 2D and 3D assembly scenarios.
Finally, in the context of cross-dimensional freezing behavior, we note the differences between the soft repulsive interactions studied here-which enthalpically stabilize lowcoordinated periodic structures-and hard-sphere systems where entropy drives the particles to adopt close-packed periodic structures at high density. For the latter, crystallization from the fluid becomes increasingly more challenging in higher spatial dimensions due to correspondingly stronger geometric frustration [63,64]. The role that frustration plays in the dimensionality dependence of crystallization for particles with considerably softer repulsions remains a potentially rich area for future study.  Effective pair potential, Fig. 1. Isotropic, convex-repulsive potentials, ϕ hc and ϕ squ (described in the text), which maximize the density range of mechanically stable 2D honeycomb-and square-lattice ground states, respectively. Also shown are previously designed potentials, ϕ dia and ϕ sc , [30] that maximize the density range of mechanically stable 3D diamond-and simple cubic-lattice ground states, respectively. The inset highlights subtle differences between ϕ dia and ϕ hc .  Table S2 in Supplementary Information. [1] Junhu Zhang, Yunfeng Li, Xuemin Zhang, and Bai Yang, "Colloidal self-assembly meets nanofabrication: From two-dimensional colloidal crystals to nanostructure arrays," Adv. Mat.   is not quite defect-free but the configuration energy is only 0.09% higher than that of the perfect lattice. See Table S1 in Supplementary Information. used cuboid box shapes, and found no significant differences in the crystallization behavior and free energies. For ϕ dia , the fluid did not assemble into a perfect diamond crystal within the simulation time. We allowed the system to evolve for 7 × 10 5 MC steps, and no change was seen in the lowest-energy configuration during the final cycle of 10 5 MC steps. However, as can be seen in the table, the energy difference between the perfect diamond crystal and the crystal formed on quenching is about 0.09%, and from Fig. 4 in main text, there are only very subtle discrepancies between the pair distribution function of the quenched configuration and the perfect diamond lattice.  and graphite-II (c/a=1.67) ρ = [0.98, 1.05] lattices, the distances correspond to the density range at which each lattice is mechanically stable (i.e. the lowest phonon frequency has a positive value).
We also highlight the honeycomb motif in the graphite and the diamond structures.
In Fig. S1, we plot the interparticle separations corresponding to the coordination shells of honeycomb, diamond and graphite structures within the interaction range for ϕ hc . Graphite structures with axial ratios (c/a) in the ranges [1.64, 1.69] and [1. 25, 1.27] are found to be optimal and mechanically stable in the density range of interest. However, on comparison with a larger pool of structures, A7-II and diamond (see Table I) have lower molar enthalpy and are chemically stable. We clearly see that there is no coordination-shell overlap beyond the nearest neighbour distances for all the four lattices, and hence, ϕ hc is not able to stabilize any of the graphite structures.