Glass is a molecularly disordered, dense, and highly viscous form of matter formed when a fluid is cooled or compressed in a manner that avoids nucleation and crystallization. Widespread in nature and technology, glassy materials are ordinarily formed when organic or inorganic liquids are cooled sufficiently rapidly that nucleation cannot occur. Such materials are characterized by a huge increase in viscosity as temperature decreases or pressure increases. Mixtures of glass-forming compounds exhibit similar behavior. Various models have been developed to describe the temperature and/or pressure dependence of viscosity or dielectric relaxation time for pure glassforming compounds. These models have sometimes been applied to mixtures. However, directly applying a pure compound model to a mixture simply results in parameters that are only valid for the mixture in question. A method to relate pure compound properties to mixture properties is needed.
A prior correlation model1for glass formation based on cluster-size distribution kinetics is here applied to binary mixtures of glassforming compounds. The model describes how rapidly cooling or compressing a liquid leads to structural arrest and a consequent sharp rise in viscosity or dielectric relaxation time. The model has two formulations, one for isothermal data and one for isobaric data. The isothermal and isobaric correlation models are, respectively:
log10 (τ/τg)isothermal = [(ygP/Pg - yg)/(ygPf/Pg - yg)] log10 (τf/τg)
log10 (τ/τg)isobaric = [(FgTg/T - Fg )/(FgTg/Tf - Fg )] log10 (τf/τg)
where τ is the dielectric relaxation time, yg = exp(Pgv/kBTg), Fg = exp(h+Pgv/kBTg), T is temperature, P is pressure, and kB is the Boltzmann constant. Subscript g indicates a value at the glass transition (defined as the point at which τ = 1 second) and subscript f indicates a value at a fluid condition.
The correlation model contains two constants, one related to heats of transformation (h) and one related to volumes of transformation (v). Using constants found at one set of temperature and pressure conditions for a pure compound, the correlation model has been shown capable of predicting dielectric relaxation at another set of pressure and temperature conditions for that compound1. These constants can be considered properties of the glassforming fluid. The constant h is found from pure component isobaric data, whereas the constant v is found from isothermal data. Considering these constants to be properties of the pure component, mixing rules may be applied to determine constants for a mixture. To apply the model to mixtures of glassforming compounds, constant pressure and constant temperature dielectric relaxation data for the pure components are required. Isothermal data are first fit by the isothermal correlation model to determine v. Once v is known, the isobaric correlation model can be used with pure component isobaric data to determine h. Simple mixing rules are then applied to determine constants to describe the mixture, hmix and vmix. With these constants, yg (for isothermal mixture data) or Fg (for isobaric mixture data) can be calculated. Given τg, τf, Pg, and Tg, for the mixture, the dielectric relaxation for the mixture can be predicted at various temperatures and pressures, using the correlation models previously developed.
Various binary mixtures of glassformers are examined, all of them at constant pressure and varying temperature. Once model constants are determined for the pure components, several simple mixing rules are used to determine mixture constants. All these mixing rules calculate mixture parameters using the fraction of the mixture that is due to each compound. Thus, the mixture parameters are different for each composition of a binary mixture. An example of mixing rule used is:
1/vmix = Xa/va + Xb/vb
where subscripts a and b indicate the two pure components in the mixture, and X is the mole fraction of a component in the mixture. The dielectric relaxation time is then predicted using the mixture constants in the correlation model, with Tg, Pg, and τf of the mixture. Certain mixing rules have proven superior to others in predicting dielectric relaxation for the mixture.
In summary, we have presented an application of a prior correlation model for pressure and temperature dependence of dielectric relaxation time to mixtures of binary glassforming compounds. Model constants for the components in the mixture are used to determine mixture constants, which are then used in the correlation model to predict dielectric relaxation time of the mixture.
1L.A. Brenskelle and B.J. McCoy, J. Chem. Phys. 124, Art. No. 084502 (2006).