Theory of specific heat in glass-forming systems

Experimental measurements of the specific heat in glass-forming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called “specific heat peak” in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same $\alpha$-relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formers—unfortunately, one needs to understand each case by itself.

Figure 1

(Color online) A snapshot of the system at $T=0.44$. In colors we highlight the clusters of large particles in local hexagonal order. The colors have no meaning.

Figure 2

(Color online) The energy distribution functions for the binary mixture model, computed at constant volume, such that the volume agrees with the pressure $P=13.5$ [14] at each temperature.

Figure 3

(Color online) Dots: the temperature dependence of the average potential energy per in the binary mixture model, computed at constant volume, such that the volume agrees with the pressure $P=13.5$ [14] at each temperature. The continuous line represents the approximation furnished by the virial expansion, which obviously fails for $T<0.5$.

Figure 4

(Color online) Dots: the temperature dependence of the specific heat in the binary mixture model, computed at constant volume, such that the volume agrees with the pressure $P=13.5$ [14] at each temperature. The data indicate the existence of two specific heat peaks, one prominent at about $T=0.5$ and a smaller one at about $T=0.1$; see the inset for finer detail. The continuous line represents the approximation furnished by the virial expansion, which obviously fails for $T<0.5$.

Figure 5

(Color online) A test of Eq. (34). In points we present simulation results whereas the continuous line is given by Eq. (34).

Figure 6

(Color online) Specific heat at constant volume as predicted by the simple theory which is based on the mechanical equation of state supplied by Eqs. (55, 33, 34, 35). Note that the theory predicts the two peaks which are associated with the micromelting or microfreezing of the clusters of large and small particles, respectively. The magnitude of the peaks is too high, reflecting terms missing in the simple approach, like the effect of anharmonicity at the lowest temperatures which are negative, tending to decrease the height of the low-temperature peak.

Figure 7

(Color online) Specific heat at constant volume and at constant pressure as predicted by the simple theory which is based on the caloric equation of state supplied by Eqs. (56, 57, 58, 59). In agreeement with the simulations and the theory based on the mechanical equation of state (cf. Fig. 6) the theory predicts the two peaks to both $C_V$ and $C_P$ which are associated with the micromelting or microfreezing of the clusters of large and small particles, respectively. Note that $C_P⩾C_V$ as is expected from thermodynamics.

Figure 8

(Color online) The theoretical real part of the specific heat $\mathrm{Re}{C}_p\left(\omega \right)$ multiplied by the thermal conductivity for glycerol.

Figure 9

(Color online) The theoretical imaginary part of the specific heat $\mathrm{Im}{C}_p\left(\omega \right)$ multiplied by the thermal conductivity for glycerol.

Figure 10

(Color online) Evaluated values of the constant $c_M$ from the simulation results.