Spatial correlations in the dynamics of glassforming liquids: Experimental determination of their temperature dependence

We use recently introduced three-point dynamic susceptibilities to obtain an experimental determination of the temperature evolution of the number of molecules $N_{\mathrm{corr}}$ that are dynamically correlated during the structural relaxation of supercooled liquids. We first discuss in detail the physical content of three-point functions that relate the sensitivity of the averaged two-time dynamics to external control parameters (such as temperature or density), as well as their connection to the more standard four-point dynamic susceptibility associated with dynamical heterogeneities. We then demonstrate that these functions can be experimentally determined with good precision. We gather available data to obtain the temperature dependence of $N_{\mathrm{corr}}$ for a large number of supercooled liquids over a wide range of relaxation time scales from the glass transition up to the onset of slow dynamics. We find that $N_{\mathrm{corr}}$ systematically grows when approaching the glass transition. It does so in a modest manner close to the glass transition, which is consistent with an activation-based picture of the dynamics in glassforming materials. For higher temperatures, there appears to be a regime where $N_{\mathrm{corr}}$ behaves as a power-law of the relaxation time. Finally, we find that the dynamic response to density, while being smaller than the dynamic response to temperature, behaves similarly, in agreement with theoretical expectations.

Figure 1

(Color online) Three different estimates (i)–(iii) detailed in Sec. 3B for the full curve $T\partial \chi (\omega ,T)∕\partial T$ for glycerol at temperatures $T=196.6$, 201.6, 207.1, 211.1, 217.2, and $225.6\mathrm{K}$ (from left to right). Full lines are for method (i), dashed lines for method (ii), and points for method (iii). The three methods give consistent results, with small deviations that are well understood.

Figure 2

(Color online) $N_{\mathrm{corr},T}$ versus ${\tau }_{\alpha }∕{\tau }_0$ on a log-log plot for glycerol (top) and $o$-terphenyl (bottom). $N_{\mathrm{corr},T}$ is calculated by using $c_p\left(T_g\right)$ (circles), $c_p\left(T\right)$ (squares), $\Delta c_p\left(T_g\right)$ (diamonds), and $\Delta c_p\left(T\right)$ (triangles). The two upper (lower) curves correspond to using $\Delta c_p$ $\left(c_p\right)$; ${\tau }_0$ is arbitrarily set to $1\mathrm{ps}$.

Figure 3

(Color online) $N_{\mathrm{corr},T}$ versus $T_g∕T$ for $m$-toluidine evaluated from dielectric (triangles) and photon correlation (circles) spectroscopies close to $T_g$, and by coherent quasielastic neutron scattering (squares) at high temperature. (We have not considered the two lowest temperature data points for which the determination of the relaxation profile is too uncertain.) The inset shows the corresponding relaxation times (determined from the maximum of the imaginary part of the dielectric susceptibility, from a KWW fit of the correlation functions obtained from photon correlation spectroscopy and neutron scattering at $q=1.3{\mathrm{\AA }}^{-1}$).

Figure 4

(Color online) Top: Evolution of $N_{\mathrm{corr},T}$ for the different materials indicated when the glass transition is approached. The results are shown on a logarithmic scale as a function of ${\tau }_{\alpha }∕{\tau }_0$, where ${\tau }_0$ is a microscopic time scale (see the text). The full line is from Eq. (19) and describes a crossover from a power law scaling at high temperatures (small ${\tau }_{\alpha }$) to a logarithmic growth close to the glass transition, Eq. (19). Bottom: Using the freedom left by unknown normalizations of order unity we obtain a better collapse of the data onto the fit.

Figure 5

(Color online) $N_{\mathrm{corr},T}∕N_{\mathrm{corr},T}\left(T_g\right)$ (top) and ${\tau }_{\alpha }∕{\tau }_0$ (bottom) versus $T_g∕T$ for different glass-forming liquids (indicated in the figure) on a logarithmic scale; $T_g$ is defined by ${\tau }_{\alpha }\left(T_g\right)=10\mathrm{s}$.

Figure 6

(Color online) $N_{\mathrm{corr},4}$ versus $T_g∕T$ for glycerol, $m$-toluidine, and $o$-terphenyl (top to bottom) calculated via the lower bounds given in Eqs. (11) (squares) and 12 (diamonds). Also displayed is the contribution due to energy fluctuations (circles). The inset shows the ratio R versus $T_g∕T$ [see Eq. (22)].