(362g) Extension of Glass Transition Model to Mixtures

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/τ_g)_isothermal =
[(y_g^P/Pg - y_g)/(y_g^Pf/Pg  - y_g)] log₁₀ (τ_f/τ_g) 

log₁₀ (τ/τ_g)_isobaric = [(F_g^Tg/T - F_g )/(F_g^Tg/Tf - F_g )] log₁₀ (τ_f/τ_g)

where τ is the dielectric
relaxation time, y_g  = exp(P_gv/k_BT_g),
F_g  = exp(h+P_gv/k_BT_g),
T is temperature, P is pressure, and k_B 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, h_mixand
v_mix. With these constants, y_g
 (for isothermal mixture data) or F_g
(for isobaric mixture data) can be calculated.  Given τ_g,
τ_f, P_g,and T_g,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/v_mix
= X_a/v_a + X_b/v_b  

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 T_g,
P_g, and τ_f 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).