Molecular random tilings as glasses

Juan P. Garrahan, Andrew Stannard, Matthew O. Blunt, and Peter H. Beton
PNAS September 8, 2009 106 (36) 15209-15213; https://doi.org/10.1073/pnas.0902443106

  1. Edited by Hans C. Andersen, Stanford University, Stanford, CA, and approved July 17, 2009 (received for review March 9, 2009)

Abstract

We have recently shown that p-terphenyl-3,5,3′,5′-tetracarboxylic acid adsorbed on graphite self-assembles into a two-dimensional rhombus random tiling. This tiling is close to ideal, displaying long-range correlations punctuated by sparse localized tiling defects. In this article we explore the analogy between dynamic arrest in this type of random tilings and that of structural glasses. We show that the structural relaxation of these systems is via the propagation–reaction of tiling defects, giving rise to dynamic heterogeneity. We study the scaling properties of the dynamics and discuss connections with kinetically constrained models of glasses.

Dimer coverings of lattices and random polygon tilings are problems of great interest both in physics and mathematics (1–5). One of the reasons is that they correspond to the idealized description of systems that display entropically stabilized critical phases and fractional excitations, being therefore relevant to condensed-matter systems such as quasicrystals (6, 7) and frustrated antiferromagnets (8–10).

However, actual experimental realizations of dimer covering/random tiling systems are very rare. Fig. 1 shows one of them: a molecular network formed by organic molecules adsorbed from solution onto a graphite substrate (11). The molecule p-terphenyl-3,5,3′,5′-tetracarboxylic acid (TPTC) (see Fig. 1A) binds to other TPTC molecules on the substrate adopting one of three possible orientations. Each molecule can then be mapped to a rhombus tile (see Fig. 1B), where the colors red, green, and blue indicate the molecular orientation. Neighboring molecules (tiles) can bind to neighbors in a parallel or “arrowhead” configuration, equivalent to junctions between tiles of the same or different color, respectively. Fig. 1C shows a scanning tunneling microscope (STM) image of the resulting molecular network of adsorbed TPTC, and Fig. 1D shows the corresponding rhombus tiling (11), where each molecule is represented by a tile.

Fig. 1.
Fig. 1.

Mapping of molecular network to a random rhombus tiling. (A) Molecular ball and stick diagram of TPTC. (B) Example of an arrangement of TPTC molecules linked via hydrogen bonds when adsorbed on substrate, and rhombus tile representation; the tiles are colored according to the three possible orientations of the molecule. (C and D) Mapping to a rhombus tiling. (C) An STM image of a typical area of TPTC network adsorbed on graphite; the backbones of the TPTC molecules appear as bright rods in the image. (D) The corresponding rhombus tiling. The molecular network is a rhombus tiling of the plane or, equivalently, a dimer covering of the honeycomb lattice. The white circle shows the position of a tiling defect. (E–L) Example of motion of tiling defects: The leftmost defect (upward pointing triangle) effectively makes two hopping steps, between E and G and between G and I. This motion is mediated by the desorption (F and H) and readsorption (G and I) of a tile. Between I and K, the rightmost defect (downward pointing triangle) makes a step to the left, which brings it into contact with the leftmost defect. (L) The two annihilate with the adsorption of the last tile (see ref. 11 for details.)

The molecular networks studied in ref. 11 are close to “perfect” rhombus tilings (or dimer coverings of the honeycomb lattice) (1–3, 8) in the sense that they contain rather few tiling defects, typically less than one defect per 300 adsorbed molecules (11). In Fig. 1C and D, one such tiling defect is identified. They are also entropically stabilized “random tilings” displaying algebraic spatial correlations (11), characteristic of a critical, or Coulomb, phase (1–3, 8). The structures, such as those in Fig. 1, are close to dynamically arrested at room temperature (11). The interaction energy between two neighboring molecules is several times kBT (11), so once a tiling is formed, tile removal is highly suppressed and structural relaxation is slow. Tile rearrangements mediated by propagation of defects have been observed experimentally, but so far on timescales of seconds for each event (11). This combination of an amorphous structure, albeit with critical spatial correlations, and very slow relaxation suggests an analogy between this kind of random tiling and structural glass formers (for reviews on the glass transition, see refs. 12–14).

The aim of this article is to discuss this analogy. For simplicity, we focus on the case where all tile–tile interactions are equal, because the dynamics for small bias* is qualitatively the same in the long-time dynamical regime; the initial growth dynamics will be discussed elsewhere. We show by means of numerical simulations that the low temperature dynamics of a rhombus tiling where the number of tiles is not conserved displays some of the features observed in the dynamics of structural glass formers, in particular dynamic heterogeneity. Relaxation in these random tilings is facilitated by tiling defects, a mechanism similar to that of kinetically constrained models of glasses. We will also discuss this connection.

Model

We simulate a dimer covering of the honeycomb lattice, which is equivalent to a rhombus tiling of the plane (1–3). That is, each rhombus tile is composed of an upward- and downward-pointing triangle face-to-face; these triangles are centered at the sites of a honeycomb lattice, each in a different sublattice. The dynamics we consider is one where the only possible moves are (i) the adsorption of a tile on the lattice if the two sites it would occupy are empty or (ii) the desorption of a tile. The number or tiles (and of tiling defects) is therefore not conserved. This resembles the experimental situation where molecules are exchanged between substrate and solution (11). We consider only single-tile moves.† We set the binding energy J between neighboring tiles to J = 1. For low temperatures, T < 1 (in units of J, and where kB = 1), we simulate the dynamics using a version of Borz–Kalos–Lebowitz, or continuous-time, Monte Carlo (20, 21), which is particularly efficient for this problem. We simulate systems of sizes N = 200 × 200 to N = 103 × 103 at all temperatures.

At low temperatures, when the density of tiles is high, desorption of tiles is rare. The energy barrier to remove a tile surrounded by four neighboring tiles is ΔE = 4, and the rate for that transition is suppressed by a factor of e−4/T. A more likely transition is the removal of a tile neighboring a defect, because the rate for this process scales as e−3/T. This will give rise to the effective propagation of tiling defects, as sketched in Fig. 1E–L. In Fig. 1E there are two tiling defects of opposite “charge” (3). Fig. 1F shows the desorption of a tile next to the leftmost defect (a process of rate ∝ e−3/T) and Fig. 1G the subsequent adsorption of another tile [a process of rate O(1), because it is energetically favorable]. The net effect is the hopping of the defect by one (sub)lattice site. The sequence represented in Fig. 1G–I shows a second such step. The effective hopping rate is therefore Γ ∝ e−3/T.

Defects of opposite charge can effectively react with each other. This is sketched in Fig. 1I–L. In this case, the rightmost defect hops one step to the left. Upon meeting the opposite defect, a gap large enough for a tile is formed, and the two defects are “annihilated” by the adsorption of a tile, at rate O(1) because this is energetically favorable. Of course, this process is reversible, and two opposing defects can be “created” by desorption of a molecular tile, at a rate ∝ e−4/T. The effective dynamics of defects, therefore, resembles a reversible A + B → 0 reaction–diffusion process (22); although, as we will see below, it is not clear that defect propagation is actually diffusive or that defect interaction can be approximated as being local (see also refs. 23 and 24).

Results

Fig. 2A shows the evolution in time of the concentration of defects, c(t), starting from an empty lattice at time 0, c(0) = 1, at various temperatures, T. After a short initial transient of fast, temperature-independent, tile adsorption, the system enters a regime of activated dynamics: Most defects are isolated, and energy barriers need to be crossed for the dynamics to progress. The dynamics becomes increasingly sluggish with decreasing temperature, and once times are long enough for defect motion to take place relaxation enters a scaling regime. Fig. 2B shows that the rate-limiting step is defect hopping: The long time data collapses if time is rescaled by the defect hopping rate, t → Γt. The defect concentration decays as c(t) ∼ (Γt)−α, with α ≈ 3/4. This exponent is somewhat different from the exponent α = 1/2 of two-species diffusion–annihilation, A + B → 0, in dimension two (22, 25). This could be an indication that defect propagation is nondiffusive, although an exponent of α = 3/4 can also be explained by initial state fluctuations in the tiling case that differ from those of the standard A + B → 0 problem.‡ Eventually, for times t ≫ e4/T, the reverse process 0 → A + B becomes accessible, and the concentration relaxes to its equilibrium value c(t) → ceq.§ The equilibrium properties of the tilings are shown in Fig. 3. The temperature dependence of the equilibrium defect concentration is given by ceq ≈ e−3/T (see Fig. 3A). This is what one would obtain for a noninteracting gas of defects on the lattice with an energy cost of E = 3 per defect. Fig. 3B shows the spatial correlations. A rhombus tiling can be mapped to a height field on the triangular lattice (8): The height h changes by ±1 unit when traversing the edges between tiles, according to the prescription of Fig. 3C. The main panel of Fig. 3B shows the height–height correlation function, 〈[h(r) − h(0)]2〉, as a function of distance r (along lattice directions), for various temperatures. At low temperatures this correlation approaches the ideal tiling limit 〈[h(r) − h(0)]2〉 = 9/π2 ln(r), corresponding to a Gaussian free-energy F = ∫d2x⃗(K/2)∣∂h(x⃗)∣2 for a continuous height field (3), with elastic constant K = π/9 (8). For finite T the logarithmic behavior is over a finite distance due to the presence of tiling defects (30). An alternative correlation function, 〈eik[h(r)−h(0)]〉 (27), is shown in Fig. 3B Inset, for the specific value k = π/5 of the “height space” reciprocal vector (the behavior is similar for other choices of k) (27). At low temperatures the function becomes algebraic indicating long-range correlations. The decay exponent is close to 9k2/2π2, as expected from the Gaussian form of the free-energy (27).

Fig. 2.
Fig. 2.

Relaxation to equilibrium in the random rhombus tiling. (A) Relaxation of the concentration of defects c(t) as a function of time (in units of Monte Carlo sweeps), starting from an empty lattice, c(0) = 1, for various temperatures T. (B) Same as A, but time is rescaled by the defect effective hopping rate Γ ≈ e−3/T. The curves collapse at long times in this representation. The dotted line indicates the power-law decay (Γt)−1/2 expected from diffusion–pair-annihilation, A + B → 0, in d = 2. The observed behavior is closer to c(t) ∼ (Γt)−3/4, as indicated by the dashed line.

Fig. 3.
Fig. 3.

Equilibrium properties of random rhombus tiling. (A) Equilibrium concentration of defects, ceq, as a function of temperature T. The straight line corresponds to the fit ceq = e−3/T. (B) Equilibrium height correlations at various temperatures. The main graph shows 〈[h(r) − h(0)]2〉 as a function of distance r. As the defect concentration decreases with decreasing temperature, the curves approach the ideal tiling behavior 〈[h(r) − h(0)]2〉 = 9/π2 ln(r). (Inset) The correlation function 〈eikΔ/h(r)〉, where Δh(r) ≡ h(r) − h(0), for k = π/5; the correlation behaves in a similar way for other choices of k. The dashed line is the power-law behavior for the ideal tiling, r−9k2/2π2 = r−9/50 for this choice of k. Once again, the lower the temperature, the longer the algebraic regime. For one of the temperatures, T = 0.4, we show that 〈eikΔh(r)〉 = e−k2/2〈 [h(r)−h(0)]2〉, as expected from a Gaussian form of the free-energy for the height. (C) Scheme for obtaining the height representation of a tiling. A displacement along a tile edge leads to an increase in height by +1 or −1 as shown (cf. Fig. 1).

As described above, at low temperatures structural rearrangements are most likely in the neighborhood of tiling defects. This gives rise to heterogeneous relaxation, as illustrated in Fig. 4. Here we plot the local autocorrelation function Ci(t) ≡ δni(t),ni(0), where ni stands for the state of site i in the lattice, say ni = 0,1,2,3 for empty, or occupied by a red, green, or blue tile, respectively. More precisely, Fig. 4 shows the corresponding persistence field, Pi(t) = ∏t′=0t Ci(t): if site i has never relaxed up to time t then Pi(t) = 1, and Pi(t) = 0 otherwise. The different images show how relaxation is distributed in space at different times. Clearly, the system relaxes in a heterogeneous, spatially correlated manner. Fig. 4 also suggests that the size of these spatial dynamic correlations grows with decreasing temperature. This is very similar to dynamic heterogeneity in structural glass formers (for reviews on dynamic heterogeneity, see refs. 31–34).

Fig. 4.
Fig. 4.

Dynamic heterogeneity in random rhombus tilings. The images show the persistence field Pi(t) of the local autocorrelation function Ci(t) (see text) at various times t, for representative equilibrium trajectories, at two different temperatures T. Black indicates Pi(t) = 1, and white indicates Pi(t) = 0. The average relaxation time is τ (see Fig. 5). Relaxation is clearly heterogeneous. The size of dynamic heterogeneity grows with decreasing temperature.

Fig. 5 quantifies equilibrium relaxation and dynamic heterogeneity. Fig. 5A shows the average (connected and normalized) autocorrelation function C(t) ≡ (〈Ci(t)〉 − A)/(1 − A), where A ≡ 〈Ci(∞)〉 = ceq2 + (1 − ceq)2/3. As expected, the autocorrelation function decays more slowly the lower the temperature. The characteristic timescale for relaxation, τ, obtained from these correlations is approximately τ = τ0eΔ/T, with τ ≈ 6.6 (see Fig. 5B). Relaxation times thus increase with decreasing temperature following an Arrhenius law. In the context of the glass transition, this is often termed “strong” glass-forming behavior (12–14). Moreover, the autocorrelations are close to exponential, rather than stretched exponential (12–14). This could mean that relaxation is not collective, but Fig. 4 suggests otherwise. The exponent Δ appears nontrivial: we have that Δ < 7, the energy barrier to remove two neighboring and parallel tiles (which would allow the adsorption of a distinct tile in the space created); this is the lowest energy barrier to purely local relaxation of the autocorrelation. This indicates that relaxation is achieved more effectively through defect propagation, a collective mechanism. Interestingly, we also have Δ > 6, which is the value one would expect for diffusing defects, although this may simply be due to logarithmic corrections to diffusion in dimension two.‖

Fig. 5.
Fig. 5.

Relaxation timescales and lengthscales in random rhombus lining. (A) Average equilibrium autocorrelation function C(t) at various temperatures T. (B) The same as A but as a function of scaled time t/τ, showing that the relaxation timescale for the autocorrelation function follows an Arrhenius law, τ ∝ eΔ/T, with Δ ≈ 6.6. (C) Four-point susceptibility χ4(t) at the same temperatures as in A. The four-point susceptibility has a maximum at t ≈ τ. The peak value appears to scale as χ4(peak) ∼ tλ, with λ ≈ 0.4 (see dotted line). (D) Scaled susceptibility, χ4(t)/τλ as a function of t/τ. The growth toward the peak would appear to follows the power law χ4 ≈ tμ, with μ ≈ 1.6, at low temperatures.

In Fig. 5C we show the “four-point” susceptibility, χ4(t) = N(1b− A)−2 [〈(1/N2) ΣijCi(t)Cj(t)〉 − C2(t)], which measures sample to sample fluctuations in the correlator C(t). This is an observable often used to quantify dynamic heterogeneity (31–37). It measures the degree of spatial fluctuation in the relaxational dynamics, allowing one to uncover spatial dynamical correlations, which are either absent or not apparent in static structural fluctuations. Our observed χ4 is (i) nonmonotonic in time, peaking at times close to the structural relaxation time, t ≈ τ, where dynamic heterogeneity is most prominent; and (ii) its peak value increases with decreasing temperature, indicating that dynamical fluctuations are larger at lower T. These are two of the central features of χ4 observed in glass formers (31–34). The four-point susceptibility also displays dynamic scaling at low temperatures, which again seems slightly different from what one expects from diffusing excitations in two dimensions (36): the peak value, χ4peak = χ4(τ), appears to scale as a power of the relaxation time, χpeak4 ∼ τλ, with λ ≈ 0.4 (rather than λ = 1); and the growth of χ4 toward the peak goes as χ4(t) ∼ tμ, with μ ≈ 1.6 (rather than μ = 2), see Fig. 5D. Better data would be required to confirm these scaling laws.

Discussion

We have shown that random tilings of the kind corresponding to the experimental system of ref. 11 display features commonly associated with glass forming systems, most notably a pronounced slowdown at low temperatures and accompanying heterogeneous relaxational dynamics. In these nearly dynamically arrested tilings, structural relaxation occurs through the propagation of rare localized tiling defects. This is an example of the mechanism of dynamic facilitation (38, 39), whereby defects “facilitate” molecular rearrangements in their immediate vicinity. The fact that defects are scarce at low T and that their motion is activated leads to the observed slowdown, and to fluctuation-dominated, heterogeneous dynamics (39). Relaxation by means of defect propagation-reaction is the hallmark of kinetically constrained models (KCMs) of glasses (40). The dynamics of the random tiling systems studied here is very close to that of these idealized models. For the simple rhombus tiling, when the number of tiles is not conserved, we have found Arrhenius timescales and exponential relaxation. This is similar to the simpler KCMs, such as the Fredrickson–Andersen model (40) in dimensions two or more (41). The observed dynamic scaling properties may suggest, however, that defect dynamics is not simply that of diffusion–annihilation–creation, but this requires further exploration.

It is possible that more complex random tiling systems, such as those giving rise to quasicrystals (3), display even richer slow dynamics, in particular, superArrhenius timescales at low temperatures and stretched relaxation functions.** Random tilings also offer a further testing ground for theories of the glass transition. Specifically, it would be of interest to see whether their slow dynamics can be explained using the “mosaic” perspective on glasses (45, 46). Here we have a system with a finite configurational entropy density given by all of the possible local tiling arrangements, the central tenet of the mosaic approach (45, 46). The mismatch between mosaics, however, does not produce extended interfaces but rather localized defects, which as shown above make the dynamics close to that of KCMs. Can one therefore explain the structural relaxation of these systems via the entropic droplet picture of refs 45 and 46? And could this highlight connections between the mosaic approach and that based on dynamic facilitation and KCMs (47)? Further study will be necessary to address these and other questions.

Acknowledgments

We thank Chris Henley for correspondence. This work was supported in part by Engineering and Physical Sciences Research Council Grant EP/D048761/1.

Footnotes

  • 1To whom correspondence should be addressed. E-mail: juan.garrahan{at}nottingham.ac.uk

  • Author contributions: J.P.G., A.S., M.O.B., and P.H.B. performed research and J.P.G. and P.H.B. wrote the paper.

  • The authors declare no conflict of interest.

  • This article is a PNAS Direct Submission.

  • ↵* Due to a slight mismatch in the distance between the two end phenyl rings in a TPTC molecule and that between phenyl rings across a carboxylic–carboxylic hydrogen bond, the tilings of ref. 11 are not “ideal” (3) because there is a small energetic penalty for the parallel tile arrangement. They are “interacting” random tilings (15–18). This difference in energy is however below the critical value at which such tilings would undergo a Kosterlitz–Thouless (KT) transition to an ordered phase (18).

  • ↵† In particular, we ignore moves where a cluster of tiles can be replaced without disturbing neighboring tiles. For example, in Fig. 1E–G there is a central cluster of three tiles, red–blue–green, forming a hexagon of side one. This triplet can be replaced by a green–red–blue combination without disturbing the surroundings. These kinds of moves are usually the ones considered for perfect tilings (see, e.g., ref. 19). In our case, they would be suppressed due to the energy cost of removing three tiles. Furthermore, we do not consider moves where tiles “slide” on the substrate, because this kind of process does not seem to apply to the experimental situation of ref. 11 either.

  • ↵‡ We can apply an argument similar to that used in ref. 26 for the XY model. If we think of tiling defects as vortices (27), due to Stokes law, the initial state fluctuations in the difference between upward-pointing and downward-pointing defects in a region of size L scales as L (26) rather than L (25). If the defects are diffusive, after a time Γt ∼ L2, that initial difference is all that will remain, all other defects having annihilated each other (25). The density will then scale as c(t) ∼ L/L2 ∼ (Γt)−3/4. (We are grateful to Alan Bray for an important discussion leading to this observation.)

  • ↵§ The structural relaxation of random tilings by means of the creation of an opposing pair of defects and their subsequent propagation has been considered before. See for example the “zipper” rearrangements in square–triangle tilings (28).

  • ↵ An isolated defect in the tiling is a dislocation in the height representation, with Burgers vector of magnitude +3 (−3) for an upward (downward) pointing defect (27). The fluctuations in height can be computed even in the presence of defects by removing the poles in the height field (29). That is, from the height difference h(x⃗) − h(x⃗′) between ⃗x and x⃗′, calculated according to the prescription of Fig. 3C, the contribution of all defects is subtracted, h(x⃗)− h(x⃗′) → h(x⃗) − h(x⃗′) − Σk qk θ(x⃗′ − y⃗k, x⃗ − y⃗k)/2π, where the sum is over all defects k of charge qk = ±3 at position y⃗k, and θ(a⃗,b⃗) indicates the angle subtended by vectors a⃗ and b⃗. This prescription ensures that the corrected height field is single-valued.

  • ↵‖ When relaxation is due to propagation of free defects, the persistence function is approximately given by P(t) ≈ e−c𝒩(t), where c is the concentration of defects, and 𝒩(t) is the average number of distinct sites visited by a defect in time t. For diffusing defects in two dimensions, 𝒩(t) ≈ (Γt)/ln(Γt). This, combined with the equilibrium concentration, ceq, gives, up to logarithmic corrections, P(t) ∼ e−t/τ, with τ ∼ ceq−1 Γ−1 ∝ exp(6/T).

  • ↵** The relation between quasicrystals and certain glasses has been discussed recently in ref. 42. The slow dynamics of other kind of tilings is studied also in ref. 43. Other random molecular networks with glassy characteristics have been reported in ref. 44.

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Molecular random tilings as glasses
Juan P. Garrahan, Andrew Stannard, Matthew O. Blunt, Peter H. Beton
Proceedings of the National Academy of Sciences Sep 2009, 106 (36) 15209-15213; DOI: 10.1073/pnas.0902443106
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