Glassy Spin Dynamics in Geometrically Frustrated Buckled Colloidal Crystals

Geometrical frustration arises when the lattice geometry prevents local interaction energies from minimizing simultaneously. Whether and how geometrically frustrated spins or charges in clean crystals exhibit glassy dynamics remain elusive due to the lack of measurements on microscopic dynamics. Here, we employ buckled monolayer colloidal crystals to mimic frustrated antiferromagnetic Ising spins on triangular lattices and measure single-spin dynamics using video microscopy. Both attractive and repulsive colloidal crystals buckled into zigzag stripes with glassy dynamics at low effective temperatures in experiment and simulation. The simple local spin configurations enable uncovering correlations among structure, dynamics, and soft vibrational modes. Machine learning analysis further reveals facilitated dynamics to be an important mechanism of structural relaxation. Moreover, our simulation reveals a similar structure and dynamics in lattice Coulomb liquids. Hence, spin-lattice coupling and long-range interaction can similarly lift degeneracy, induce a rugged landscape, and, thus, produce glassy dynamics.

Popular Summary

At the molecular level, glass and crystal could not be more different. In a crystal, the molecules are arranged in an orderly periodic pattern; in a glass, the molecular layout is disordered. These different structures lead to very different molecular behaviors. Surprisingly, some crystalline materials in recent experiments have shown indirect evidence of behaving like glass. These crystals are “geometrically frustrated”—the atoms are arranged in such a way as to prevent the system from reaching its lowest energy state, leading to a range of exotic behavior. A lack of microscopic observations, however, means these behaviors are poorly understood. We directly observed glassy dynamics in a geometrically frustrated crystal at the single-particle level for the first time, with an experimental setup that shows promise for understanding other important aspects of glass behavior.

Our setup uses buckled colloidal crystals composed of micrometer-sized spheres dispersed in water. The spheres can mimic the simplest form of geometric frustration: Three atoms, each with either a spin-up or spin-down state, arranged on a triangular lattice cannot minimize their interaction energy. By tuning the temperature of the crystals and recording the motion of the spheres with a camera, we were able to identify glassy behavior of single particles. The crystalline structure setup and ability to measure particle trajectories make this a novel method for identifying correlations between structure and dynamics, as well as testing spin-based glass theories.

Our soft-matter system bridges different types of materials and behaviors, and we expect that our findings will motivate future studies of glassy dynamics in geometrically frustrated crystals.

Figure 1

Correlations among structures, soft vibrational modes, and dynamics in the buckled attractive colloidal crystal. The simulation counterpart is Fig. S4 of SM [14]. (a) Three spins on a triangular plaquette cannot simultaneously satisfy all antiferromagnetic interactions. (b) Schematic of the 1.5-layer colloid confined between two walls with spheres buckled up (white) and down (yellow). (c) All 13 possible local spin configurations, i.e., motifs, labeled with the number of frustrated bonds (blue lines) attached to the central spin. The motif’s energy $E$ is the number of frustrated bonds. The weight of a bond at the edge is $1/2$ because it is shared between two motifs; hence, $E=6/2$ for motif 0, $E=1+4/2$ for motif 1, and so on. Panels (d)–(f) are the same sample region. Scale bar: $10\mu \mathrm{m}$. (d) The raw image with bright spheres buckled up and dark spheres buckled down. (e) The color-coded mean-square displacement (MSD) of each particle normalized by the ensemble-averaged value. (f) The color-coded total magnitude of soft low-frequency modes whose participation ratio $p<0.2$ [15]. The dashed ellipses mark an ordered region in (d) which has weak motions in (e) and fewer soft modes in (f); the solid-line ellipses mark a disordered region in (d) which has stronger motions in (e) and more soft modes in (f).

Figure 2

Typical normal modes. (a) The nonlocalized plane wave with the lowest frequency $\omega$. (b) A quasilocalized mode with a low $\omega$ and a low participation ratio $p$. (c) A mode with a high $\omega$ and low $p$. The polarization vectors in (a)–(c) show the vibration amplitude and direction of individual particles. (d) Participation ratios of all modes.

Figure 3

The dynamics of the repulsive buckled crystal in (a)–(c) and the Coulomb liquid in (d)–(f). The simulation counterpart of (a)–(c) is Fig. S5 of SM [14]. (a),(d) Typical real-space structures. Coulomb charges (white spots) fill half of the triangular lattice. (b),(e) Spin or charge autocorrelation function $C(t)$ ensemble averaged over all motifs. Panel (b) is reproduced from Ref. [10] with permission, copyright 2008, Nature Publishing Group. (c),(f) Persistence function of each type of motif. This function describes the fraction of motifs that have not changed state during time $t$. Solid curves in (b),(c),(e),(f) are stretched exponential fits, $\exp [-(t/\tau {)}^{\beta }]$, with the relaxation time $\tau$ shown in the insets. Temperature $T=25.3°\mathrm{C}$ in (a),(c) and $T=0.04$ in (d),(f).

Figure 4

Glassy dynamics in the repulsive buckled crystal. The simulation counterpart is Fig. S6 of SM [14]. (a) Four-point spin correlation function $C_4(r,t=\tau )$. (b) Dynamical four-point susceptibility ${\chi }_4(t)$. (c) The probability distributions of the cluster size $N_c$ of the 10% most flipped spins at different temperatures. The fitting line $P(N_c)\propto N_c^{-\mu }$, where $\mu$ is shown in the inset. (d)–(f) Statistics of the transition events among 13 motifs shown by circular flow plots at (d) $24.7°\mathrm{C}$, (e) $25.3°\mathrm{C}$, and (f) $27.1°\mathrm{C}$. The length of the outer thick arc is proportional to the total number of spin flips from or towards the labeled motif number. The flips from the motif are labeled by the color of the motif arc. Some inflows and outflows between motifs 2c and 3b are labeled with arrows as examples. The inner thin arc length is proportional to the number of flips from the motif, and is half the length of its corresponding outer thick arc. Hence, the incoming and outgoing flows are equal, as required by the detailed balance under equilibrium.

Figure 5

Repulsive NIPA colloidal crystals at the three temperatures indicate stronger facilitated dynamics at lower $T$, i.e., higher $\varphi$. The first column shows the buckled crystals at different temperatures. The first 10%, 20%, and 30% of flipped spins are colored in green in the last three columns, respectively. They show that the newly flipped spins are mainly around the previously flipped spins, i.e., facilitation. Dashed-line and solid-line ellipses at $24.7°\mathrm{C}$ mark an ordered and a disordered region, which correspond to nonexcited and excited regions, respectively.

Figure 6

Dynamic facilitation in the repulsive buckled crystal. (a),(b) Actively flipping regions labeled in blue in the $(32\mu \mathrm{m}{)}^2\times 4\mathrm{s}$ spacetime at $27.1°\mathrm{C}$ and $(32\mu \mathrm{m}{)}^2\times 200\mathrm{s}$ spacetime at $24.7°\mathrm{C}$, respectively. The excitation coarse-graining time is 0.067 s in (a) and 0.33 s in (b). Different coarse-graining times yield similar spacetime structures, but shorter times make the structure too crowded to be clearly visualized. Excitations dispersed in spacetime at high $T$ in (a) and coalesced into clusters at low $T$ in (b). (c) Densities of the excitations, unstable, and stable motifs. (d) Probability distributions of ${\tau }_x$ and ${\tau }_p$. $⟨{\tau }_x⟩<⟨{\tau }_p⟩$ below $26.5°\mathrm{C}$, which signals the onset of DH.

Figure 7

Mobility propagation through facilitation. (a) The probability distributions $P_F(r)$ and $P_F^{*}(r)$. $P_F(r)$ and $P_F^{*}(r)$ for two time scales ($\mathrm{\Delta }t<\tau$ and $\mathrm{\Delta }t\approx \tau$) are shown for comparison. (b) Mobility transfer functions at different temperatures.

Figure 8

Softness distributions. The distributions of $S$ of all spins, mobile spins with facilitation, and local motifs at (a) $25.3°\mathrm{C}$ and (b) $27.1°\mathrm{C}$. The curves of local motifs are rescaled by 25% for clarity.

Figure 9

Principal component biplots. The first two principal components for (a) mobile and (b) nonmobile spins at $25.3°\mathrm{C}$. The percentages of explained variations for the first two principal components are 56.5% and 27.1%, respectively. They are higher than the mean value of 20%, reflecting a greater contribution to the spin dynamics. Symbols represent different motifs. The cluster distribution results from the discrete energy of motifs. (c) The percentage contribution of the five features $E_0,\dots ,E_4$. The percentages are calculated from $e_i^2$ for the first two principal components, where $e_i$ is the component corresponding to the $i$th feature of the first and second principal components.

Figure 10

(a)–(c) Examples of the ground states of AFIT. Panels (b) and (c) are ground states of deformable AFIT, but (a) is not. (a) Among $e^{0.323N}$ ground states, a typical one has disorder structures composed of motifs 0, 1, 2b, 2c, and 3c. (b) Random zigzag stripe patterns can be viewed as ordered 1D arrays with alternative Ising states along the $x$ direction stacking randomly in the $y$ direction; hence, there are $2^{\sqrt{N}}$ ways of stacking, i.e., $2^{\sqrt{N}}=e^{\ln 2\sqrt{N}}\sim e^{\sqrt{N}}$ patterns. The factor $\ln 2$ depends on the boundary condition, and thus is neglected in the main text. To jump to another ground state of the deformation AFIT model, one row of spin ($\sqrt{N}$ spins) needs to shift simultaneously, which corresponds to a high-energy barrier and results in glassiness. Similarly, in buckled crystals, when the effective temperature is lowered towards the ground state, there are less free spins (motif 3c) and, consequently, jumping from one low-temperature state to the other requires more spins to collectively flip; thus, it is more glassy. (c) More ordered zigzag stripes with longer persistence lengths due to the order-by-disorder effect illustrated in (d). (d) The order-by-disorder effect illustrated in phase space. We replot Fig. 3 of Ref. [5]. Ordered real-space structures correspond to the bottom left region in the phase space. In the top panel without the order-by-disorder effect, the system tends to be in a disordered state because the vast majority of ground states are disordered. The bottom panel depicts a phase space with the order-by-disorder effect: the system tends to be in an ordered state because thermal fluctuation can explore more phase space in the ordered region slightly above the ground state. As quenching towards the ground state (i.e., from the blue region to the black region), the system tends to inherit its ordered structure in a rugged free-energy landscape. (e) Schematic of two types of deformation in a solid-solid transition. When a crystalline domain of the parent phase transforms to a single crystalline domain of the product phase, the change in its shape induces a high strain energy. In real materials, the parent crystal usually transforms into small martensitic domains with alternative lattice orientations to cancel the strain energy. This mechanism is likely responsible for the formation of the medium-sized rather than the large-sized stripe domains in Fig. 1 herein and Fig. S4(a) of SM [14].