Kinetic Arrest, Not Solidification: What Happens at the Glass Transition — Epoche C2
Medieval window panes are often thicker at the bottom, and the explanation usually offered is that glass is a liquid still creeping downwards under its own weight. That explanation is testable, and it fails. Testing it is also the shortest route into the real problem, because the quantity that decides it — the time a material needs to rearrange its own molecules — is the quantity that defines the glass transition itself. The clock inside the material Viscosity $\eta$ measures resistance to shear; in SI it is quoted in pascal seconds, Pa·s. (Older glass literature uses the CGS poise; $10^{13}$ poise $= 10^{12}$ Pa·s.) Viscosity converts into a time through the Maxwell relation, $$\tau = \frac{\eta}{G_\infty},$$ where $\tau$ is the structural relaxation time — the time for the molecules to forget an imposed arrangement — and $G_\infty$ is the instantaneous shear modulus, the stiffness the material shows when pushed faster than it can flow. The form follows from dimensions alone: Pa·s divided by Pa is seconds. The physics in it is that a solid stores shear stress and a liquid dissipates it, and $\tau$ is the crossover between the two behaviours. The laboratory convention puts the glass transition temperature $T_g$ where $\eta = 10^{12}$ Pa·s. For a molecular glass former with $G_\infty \approx 10^{9}$ Pa this gives $\tau = 10^{12}/10^{9} = 10^{3}$ s, about 17 minutes. Stiffer inorganic networks have $G_\infty \approx 10^{10}$ Pa and yield $\tau \approx 10^{2}$ s, the other common convention. The two definitions differ by one decade in $\tau$, and we shall see shortly that one decade is worth only a few kelvin. Now the window. Soda-lime glass has $T_g \approx 820$ K. Extrapolating its relaxation time down to room temperature along the steepest slope the data allow gives $\log_{10}(\tau/\mathrm{s}) \approx 2 + 40\,(820/300 - 1) \approx 71$, using the fragility index $m \approx 40$ defined below. The age of the universe is $1.38\times10^{10}$ yr $= 4.4\times10^{17}$ s, or $10^{17.6}$ s. The ratio is $10^{71-17.6} \approx 10^{53}$. The extrapolation is crude — below $T_g$ the material is out of equilibrium and ages faster than any equilibrium estimate — but no correction of that kind rescues the story by fifty decades. Zanotto's explicit calculation reaches the same verdict. The panes are thicker at the bottom because the crown-glass spinning process made them uneven, and glaziers set the heavy edge down. Fourteen decades over eighty kelvin Take o-terphenyl, the standard organic glass former. Near its melting point $T_m = 329$ K its viscosity is of order $10^{-2}$ Pa·s, about ten times that of water at room temperature. At $T_g = 246$ K it has reached $10^{12}$ Pa·s. That is $12 - (-2) = 14$ orders of magnitude, over $329 - 246 = 83$ K, which is a fall of only $83/329 = 25\%$ in absolute temperature. Nothing in the structure changes correspondingly: the static structure factor of the glass is nearly indistinguishable from that of the liquid, and there is no latent heat. What appears at $T_g$ is a step in the heat capacity, not the delta function a first-order transition would give. The decisive point is that $T_g$ depends on how fast you cool. Fragility is defined as the steepness of the Angell plot at $T_g$, $m = \mathrm{d}\log_{10}\tau / \mathrm{d}(T_g/T)$ evaluated at $T = T_g$. Differentiating with respect to $T$ gives $|\mathrm{d}\log_{10}\tau/\mathrm{d}T| = m/T_g$ there, so changing the cooling rate by one decade moves $T_g$ by $T_g/m$. $T_g$ (K) $m$ shift per decade, $T_g/m$ (K) silica (strong) 1450 20 73 o-terphenyl (fragile) 246 81 3.0 A phase transition temperature does not depend on how quickly one passes through it; this one does. Note also the curvature: 14 decades spread over 83 K is an average of $83/14 = 5.9$ K per decade, while the tangent at $T_g$ gives 3.0 K per decade. The slope has steepened by a factor of $5.9/3.0 = 2.0$ across the window. That steepening is exactly what the word "fragile" names. The coincidence that keeps the thermodynamic reading alive If the arrest were purely kinetic, nothing special would sit below $T_g$. Two extrapolations suggest something does. The excess entropy of the supercooled liquid over the crystal falls steeply and, extended below the data, vanishes at the Kauzmann temperature $T_K \approx 204$ K for o-terphenyl — 42 K below $T_g$, a ratio $T_g/T_K = 1.21$. Independently, fitting $\tau(T)$ to the Vogel-Fulcher-Tammann form $\tau = \tau_0\exp[DT_0/(T-T_0)]$ gives a divergence temperature $T_0 \approx 202$ K. The Adam-Gibbs argument ties the two together: if relaxation requires the cooperative rearrangement of a region whose size is set by the configurational entropy $s_c$, then $\tau = \tau_0\exp[A/(Ts_c)]$, which diverges precisely where $s_c$ vanishes. A thermodynamic extrapolation and a dynamic one land within a few kelvin of each other. That coincidence is the strongest evidence for an ideal glass transition — and it is entirely an extrapolation, over 40 K in which nobody has equilibrated anything. What the correlation length rules out Suppose the slowdown were ordinary critical slowing, $\tau \sim \xi^{z}$, with $\xi$ a correlation length and $z$ a dynamic exponent. Four-point correlation functions, which detect regions relaxing together rather than static order, give $\xi$ growing from about one molecular diameter to about four across the whole window. Then $z = 14/\log_{10}4 = 14/0.60 = 23$. Ordinary critical dynamics has $z \approx 2$ to $4$. An exponent of 23 is not a measurement of an exponent; it is evidence that the power law is the wrong functional form. Random first-order transition theory replaces it with activated scaling, $\ln(\tau/\tau_0) = \Upsilon\xi^{\psi}/(k_BT)$: the barrier to rearranging a region grows as a power of its size, and the time is exponential in the barrier. Check whether the numbers fit. With $\psi = 3/2$, a fourfold growth in $\xi$ multiplies the barrier by $4^{3/2} = 8$; cooling from 329 K to 246 K mult