Long-wavelength fluctuations and the glass transition in two dimensions and three dimensions

Abstract

Phase transitions significantly differ between 2D and 3D systems, but the influence of dimensionality on the glass transition is unresolved. We use microscopy to study colloidal systems as they approach their glass transitions at high concentrations and find differences between two dimensions and three dimensions. We find that, in two dimensions, particles can undergo large displacements without changing their position relative to their neighbors, in contrast with three dimensions. This is related to Mermin-Wagner long-wavelength fluctuations that influence phase transitions in two dimensions. However, when measuring particle motion only relative to their neighbors, two dimensions and three dimensions have similar behavior as the glass transition is approached, showing that the long-wavelength fluctuations do not cause a fundamental distinction between 2D and 3D glass transitions.

Keywords: colloidal glass transition; dimensionality; long-wavelength fluctuations; phase transition; two-dimensional physics.

Fig. 1.

Structural relaxation in two and three dimensions. (A–C) Self-intermediate scattering functions characterizing translational motion, using the wave vector $k$ corresponding to the peak of the structure factor (Materials and Methods). (D–F) Bond-orientational correlation functions. The columns correspond to 2DH, 2DS, and 3D experiments. The parameters for the experiments are as follows: ${\varphi }_{2DH}=0.55$, 0.65, 0.70, 0.74, 0.75,0.76, 0.78, and 0.78; 2DS (${\mathrm{\Gamma }}_{2DS}=60$, 100, 100, 140, 180, 310, 300, and 460); 3D ${\varphi }_{3D}=0.40$, 0.42, 0.52, 0.53, 0.54, 0.54, and 0.58. These parameters increase from Left to Right in each panel; or equivalently, from Bottom to Top.

Fig. 2.

Translational, bond-orientational, and bond-break correlation functions. (A–C) The solid curves are $F_S(\mathrm{\Delta }t)$ (translational correlations) and the dashed curves are $C(\mathrm{\Delta }t)$ (bond-orientational correlations) for the 2DH, 2DS, and 3D samples as labeled. The colors indicate different control parameters. For 2DH, the colors black, red, blue, and green denote ${\varphi }_{2DH}=0.55$, 0.75, 0.78, and 0.78, respectively. For 2DS, the colors black, red, blue, and green denote ${\mathrm{\Gamma }}_{2DS}=60$, 180, 310, and 460, respectively. For 3D, the colors black, red, blue, and green denote ${\varphi }_{3D}=0.42$, 0.52, 0.54, and 0.58, respectively. (D–F) The solid curves with circles are $F_{S-CR}(\mathrm{\Delta }t)$ (cage-relative translational correlations). The dashed curves are $C(\mathrm{\Delta }t)$, which are identical to those shown in (A–C). (G–I) The solid curves with circles are $F_{S-CR}(\mathrm{\Delta }t)$ (cage-relative translational correlations) and the dot-dashed curves are $B(\mathrm{\Delta }t)$ (bond-break correlations) for the 2DH, 2DS, and 3D samples.

Fig. 3.

Vector displacement correlations. The data are for 2DH (filled circles), 2DS (filled triangles), and 3D (open squares). The displacements are calculated using a timescale $\mathrm{\Delta }t$ such that $F_S(\mathrm{\Delta }t)=0.5$. These are measured for all pairs of particles separated by the nearest-neighbor spacing $d$. $d$ is determined from the large–large peak position in the pair correlation function $g(r)$ at the highest concentrations, and has values $d=3.38$, 6.5, and 3.10 $\mu$m for 2DH, 2DS, and 3D, respectively. [The location of the $g(r)$ peak depends slightly on $\varphi$ for 2DH and 3D experiments, and more strongly on $\mathrm{\Gamma }$ for the 2DS experiments; for consistency, we keep $d$ fixed to these specific values.] The lines are least-squares fits to the data. The data are plotted as a function of ${\tau }_{\alpha }/{\tau }_{\alpha 0}$ where ${\tau }_{\alpha 0}$ is the relaxation timescale for the large particles in a dilute sample. The 2DH (closed circles), 2DS (closed triangles), and 3D (open squares) samples have ${\tau }_{\alpha 0}=$ 5.4, 20, and 3.8 s, respectively.

Fig. 4.

Particle displacements. These images show displacement vectors of particles using a time interval $\mathrm{\Delta }t$ chosen such that $F_s(\mathrm{\Delta }t)=0.5$. For the 3D image, we use an $xy$ cut at fixed $z$. All scale ticks are at 10-$\mathrm{\mu }$m intervals, and all displacement vectors are multiplied by 2 for easier visualization. The circles denote particle positions and sizes. Samples are ${\varphi }_{2DH}=0.78$, ${\mathrm{\Gamma }}_{2DS}=300$, and ${\varphi }_{3D}=0.54$, from left to right, with corresponding $\mathrm{\Delta }t=4,290$, 1,720, and 3,540 s. ${\tau }_{\alpha }$ for these samples are 14,000, 3,800, and 7,600 s, respectively. Circles with no arrows are those with displacements less than 10% of symbol size.

Fig. 5.

MSDs and cage-relative MSDs. The data (A–C) are for the experiments as indicated. The solid curves are MSDs $⟨\mathrm{\Delta }{r}^2⟩$ calculated for all particles, normalized by d as described in the legend to Fig. 3. The solid curves with circles are cage-relative MSDs. The colors indicate different control parameters, as given in Fig. 2. For the 3D samples, the $z$ direction is neglected due to noise and also to facilitate the comparison with the 2D experiments.

Fig. 6.

Transient localization parameter. (A) ${\gamma }_{min,CR}$ is the minimum logarithmic slope of the cage-relative MSDs. (B) Difference ${\gamma }_{min,CR}-{\gamma }_{min}$ between the original MSD data and the cage-relative version. Negative values indicate the enhancement of measured transient localization using the cage-relative analysis.