Glass transition of glycerol in the volume-temperature plane

We assess the relative importance of spatial congestion and lowered temperature in the slowing dynamics of supercooled glycerol near the glass transition. We independently vary both volume $V$ and temperature $T$ by applying high pressure and monitor the dynamics by measuring the dielectric susceptibility. Our results demonstrate that both variables are control variables of comparable importance. However, a generalization of the concept of fragility of a glass-former shows that the dynamics are quantitatively more sensitive to fractional changes in $V$ than $T$. We identify a connection between the fragility and a recently proposed density-temperature scaling which indicates that this conclusion holds for other liquids and polymers.

Figure 1

Inset: ${\epsilon }^{″}$ vs $\nu$ at $217\mathrm{K}$ and $340\mathrm{MPa}$. The line is Davidson-Cole fit: $\epsilon ={\epsilon }_{\infty }+({\epsilon }_0-{\epsilon }_{\infty })∕(1-i\nu ∕{\nu }_{\alpha }{)}^{\beta }$. The relaxation frequency ${\nu }_p$ is defined to be the peak frequency. Main figure: Isobaric plot of ${\nu }_p$ vs $T$. The circles represent data at 83, 164, 212, 278, 358, 392, 527, 539, 560, 689, and 876 $\pm 4\mathrm{MPa}$. The crosses are $0.1\mathrm{MPa}$ data [1]. The solid lines are Cohen-Grest fits: ${\log }_{10}(T\xi ∕{\nu }_p)=b∕[T-T_c+\sqrt{(T-T_c{)}^2+CT}]$. $C$, $b$, and $\xi$ are fit parameters. $\xi$ and $b$ vary weakly with $P$ [11], while $C=3\mathrm{K}$ is held fixed at all $P$. $T_c=V_{a}P$, where $V_a$ is a volume parameter. The theory predicts that the molecular volume, $V_m=4.605(b∕C)V_a$, whence $V_m=1700{\mathrm{\AA }}^3$.

Figure 2

(a) Isochoric plot. The solid lines are Vogel-Fulcher-Tammann-Hesse (VFTH) fits [11]: ${\nu }_p={\nu }_{\infty }\exp [E∕(T_0-T\left)\right]$. The heavy line shows $0.1\mathrm{MPa}$ isobaric data [1]. $V\equiv 1$ at $273.15\mathrm{K}$ and $0.1\mathrm{MPa}$. Data are shown for $V=0.970$, 0.951, 0.942, 0.931, 0.921, 0.916, 0.900, 0.895, 0.895, 0.878, and 0.860 $\pm 0.3\%$. (b) Isothermal plot. These data are obtained from isothermal cuts to Fig. 1. The 15 temperatures from left to right are: 214 to $242\mathrm{K}$ in steps of $2\mathrm{K}$. The solid lines are Doolittle fits [11]: ${\nu }_p={\nu }_d\exp [D∕(V_0-V\left)\right]$. The error bars in $V$ are systematic errors introduced in transforming from the $P\text{−}T$ plane to the $V\text{−}T$ plane.

Figure 3

(a) Field of fragility vector $(M_T,M_V)$. The scale bar shows the size of the vectors. The dashed line is a 45° reference line showing that $M_V>M_T$ for all vectors. We can evaluate only one component for the gray vectors. The solid curves from top left to bottom right show the glass transition line of ${\nu }_p=0.01\mathrm{Hz}$; the horizontal and vertical gray vectors at this glass transition line are the isochoric and isothermal fragilities, respectively. (b) Test of $V\text{−}T$ scaling, ${\log }_{10}{\nu }_p=f\left(V^{\gamma }T\right)$. The best possible collapse of data is achieved with $\gamma =1.4$.

Figure 4

Glass transition in the $V\text{−}T$ plane. The solid lines are constant ${\nu }_p$ contours. The error bars are of comparable size to the symbols. The solid circles are $T_0$ from isochoric VFTH fits and the open circles $V_0$ from isothermal fits to the Doolittle equation to data sets that extend to $0.01\mathrm{Hz}$. The open squares are from Ref. 4. The dashed lines involve greater extrapolations of the data.