Glasses constitute one of the most intriguing materials in condensed-matter1. Having a liquid-like disordered structure they behave mechanically as solids. Glasses are typically formed by cooling the liquid at a rate that overcomes the crystallization threat. One of the main features of glasses is the glass transition temperature, Tg, which characterizes the reversible transformation between the metastable supercooled liquid (SCL) state and the non-equilibrium amorphous solid-like material. The heat capacity of the glass is lower than that of the liquid and, upon heating, the jump in heat capacity marks the onset temperature of devitrification, Ton. Its value strongly depends on the previous thermal history of the glass and on the heating rate that follows a predefined cooling procedure. Another characteristic feature of glasses is that they age if stored below the glass transition temperature, Tg, for long periods of time2. Aging produces glasses with enhanced stability. The stability of a glass can be established by means of its limiting fictive temperature (Tf’), i.e. the temperature at which the glass would be in equilibrium with its own liquid. While the glass transition temperature can only be accessed by cooling from the liquid state, the limiting fictive temperature is a property of the glass. The enthalpic Tf’ is obtained by integration of the normalized heat capacity curve that is measured during a calorimetric cooling scan or during a calorimetric heating scan starting from a given glassy state. Improving the glass stability by aging is a rather inefficient process due to the exponential increase of the relaxation time (or viscosity) below the glass transition temperature. A breakthrough in the field was the recent discovery that vapour-deposition can produce glasses that rival in stability with ambers naturally aged for millions of years3. Those glasses, dubbed ultrastable glasses (UG), are typically grown at temperatures around 0.85 Tg, where Tg stands for the conventional glass (CG) transition temperature measured when the liquid is cooled at 10 K/min4,5,6,7,8. Besides the enhanced kinetic and thermodynamic stability, vapour-deposited stable glasses have been shown to exhibit striking properties with respect to a conventional glass obtained from the liquid. Among them, higher densities and higher sound velocities which imply higher modulus9,10, surface-initiated transformation mechanism into the supercooled liquid in thin films11,12,13,14, absence of TLS (tunnelling two-level systems) in Indomethacin (IMC) at cryogenic temperatures15 and lower heat capacities and thermal expansion coefficients16. In particular, ultrastable IMC glasses, one of the archetypical UG’s3,5, have a higher density by about 1.2% and a lower heat capacity of the glass by about 4%17.
While the properties of the glass transition temperature have been deeply studied as a function of temperature by calorimetry in many different glasses, the pressure dependence of the calorimetric glass transition is a subject relatively little explored18. The main reason can be attributed to experimental difficulties, in relation to applying high pressures in calorimetric experiments. On the contrary, dielectric or Pressure-Volume-Temperature (PVT) measurements are more abundant and permit to broadly infer several tendencies with respect to molecular interactions19,20. For instance, it has been found that glasses with strong molecular interaction of hydrogen bonding type, systematically show lower values of dTg/dP compared to glasses dominated by van der Waals forces18,21,22,23. Another universal feature of glasses is that over a sufficiently large pressure range the pressure dependence of Tg is non-linear, i.e. the effect of pressure on temperature weakens when pressure increases and can be adjusted with the empirical Andersson-Andersson equation24,
where κ1, κ2 and κ3 are empirical constants. Davies and Jones derived, based on the Ehrenfest equations, two expressions for dT/dP in the liquid state evaluated at Tg25. One of these expressions has been found to describe a large range of materials23:
where ∆α and ∆Cp refer to the difference in isobaric expansivity and heat capacity at Tg, between the liquid and its corresponding glass, and v is the molar volume at Tg. We are not aware of previous studies that analyse the pressure dependence of aged glasses over a wide range of stabilities.
In a previous work, we developed an empirical model that could simultaneously describe the relaxation time of the liquid and of glasses of different stability26. The model was built with data taken at ambient pressure and therefore only depends on temperature and density. What would be the effect of pressure on glasses of different stability? Can we explain the new data measured as a function of pressure introducing a density dependence on pressure? We present in this work measurements of the devitrification temperature of ultrastable and conventional IMC glasses as a function of pressure. We also propose a tentative extension of our previous empirical model that aims to describe the relaxation dynamics of the system as a function of temperature and pressure by considering the dependence of density on these variables.