Discontinuous Shear Modulus Determines the Glass Transition Temperature

A solid—amorphous or crystalline—is defined by a finite shear modulus while a fluid lacks such. We thus experimentally investigate the elastic properties of a colloidal glass former near the glass transition: Spectroscopy of vibrational excitations yields the dispersion relations of longitudinal and transverse phonons in the glassy state. From the long-wavelength limit of the dispersion relation, we extract the bulk and the shear modulus. As expected, the latter disappear in a fluid and we measure a clearly resolved discontinuous behavior of the elastic moduli at the glass transition. This not only determines the transition temperature $T_G$ of the system but also directly addresses recent discussions about elasticity during vitrification. We show that low-frequency excitations in our system are plane waves such that continuum elasticity theory can be used to describe the macroscopic behavior.

Popular Summary

Glassy materials such as obsidian have been used by humankind for millions of years. Even so, there is still no theoretical description of glassy materials that is able to explain all of the phenomena observed during the process of vitrification (i.e., the formation of glasses). Even worse, there is no consensus on how to exactly define the glassy state; an open question concerns the essential versus sufficient features of a glass. A supercooled fluid is commonly called a glass if the viscosity exceeds a certain value, irrespective of whether the viscosity increases but stays finite or diverges at the ideal glass transition. While viscosity refers to liquids, the shear elasticity contains complementary information from the solid state. Here, we demonstrate how an investigation of elastic excitations (phonons) provides further insights into the formation of amorphous solids.

We employ microscopy data of a two-dimensional colloidal system containing a few hundred thousand micrometer-sized colloidal particles. These colloidal particles are two different sizes, and they are all superparamagnetic such that their interactions can be controlled using an external magnetic field. Our data include video microscopy (obtained at two frames per second monitoring about 2300 particles with an optical resolution of 100 nanometers), and we are able to probe the particles’ dynamics after they relax. We measure bulk and shear moduli; these elastic moduli should become discontinuous at the glass transition temperature. We test different system temperatures, and we find, as expected, that the dynamics of the colloidal particles decrease at lower temperatures. Using acoustic spectroscopy, we show that shear elasticity develops discontinuously from zero during vitrification.

We expect that our results will pave the way for better understanding the diverse array of glasses currently used in industrial and research physics.

Figure 1

MSD for both species of particles at different temperatures. The kink at high temperatures slowly develops into a plateau. However, it is impossible to distinguish unambiguously between solid and fluid states from this dynamical data. The solid black line corresponds to free diffusion and the vertical dashed line to the sampling time $\mathrm{\Delta }t$ used in the following analysis to determine the elasticity.

Figure 2

Dispersion relations of longitudinal and transverse excitations rescaled with the linear dependence on $\mathrm{\Gamma }$ expected in harmonic systems (filled and empty symbols in the top and bottom plots, respectively). The curves for different temperatures do not collapse, which can only be explained if structural rearrangement occurs. Because of the better optical resolution, only trajectories of big particles are analyzed here and in the following plots.

Figure 3

$\mu$ and $2\mu +\lambda$ are given by the intercept of an extrapolation (indicated by green arrows) derived from the longitudinal and transverse dispersion bands (filled and empty black circles) for a solid (blue) and fluid (red) sample. The gray shaded region is used for fitting the data, and the inset shows the same plot with logarithmic scaling.

Figure 4

Temperature-dependent bulk and shear moduli. The data suggest a jump of moduli at the transition temperature, which is located at ${\mathrm{\Gamma }}_T=195\pm 5$. The lower plot zooms in on the ordinate. In the solid, the moduli grow linearly due to increased pressure indicated by dashed lines, and we extrapolate this linear behavior down to the fluid. The inset shows a zoom of ordinate and abscissa: The data are not compatible with a continuous increase of elasticity as, e.g., given by the dotted lines.

Figure 5

Finite time effects investigated by varying the time interval $\mathrm{\Delta }t$ used to determine the particle’s “equilibrium position.” On the glassy side, the shear modulus does not decay to zero even for the longest integration times. Data of the system are at $\mathrm{\Gamma }=287$ in the solid (blue) and $\mathrm{\Gamma }=103$ in the fluid (red) while the bulk and shear moduli are marked by filled and empty symbols. The inset shows the high-frequency region in the frozen state, which suggests a power-law behavior. Upper and lower black lines are functions with exponents $-\frac{1}{3}$ and $-\frac{3}{4}$.

Figure 6

(a) The spectrum of eigenvalues of the displacement field covariance matrix of big particles. By scaling with $\omega$, the inset reveals convincing Debye behavior. (b) shows $D(\omega )$ for various inverse temperatures. The red arrows in (a) labeled c–f indicate positions in the spectrum where the mode structure is shown in panels (c)–(f) for $\mathrm{\Gamma }=200$. The corresponding “frequencies” are given above the respective graphs. For details, see the text.