Towards a theory of viscosity for glass-forming liquids

A theoretical relationship has been obtained between viscosity and the PVT properties of a glass-forming liquid. First an equation of state is determined for the liquid by using a hole model of the liquid state. This equation of state is then combined in a unique way with the Cohen-Turnbull theory of self-diffusion and an important result from Brownian motion theory to obtain what is essentially an equation of state for viscosity. The principal results of the theory are as follow: (i) The change in the thermal expansion coefficient Δα that occurs at the glass transition temperature Tg can be predicted from the WLF C2 constant (T∞ ≡ Tg − C2), Tg Δα = 2(T∞/Tg) × [exp (1 + 2T∞/Tg)−1]−1. Except for the polyakylmethacrylates, good agreement is obtained between observed and theoretical Δα values for many polymers. (ii) A theoretical upper bound for Tg Δα has been found and is equal to 0.159. To within the experimental uncertainties in measured Δα values, there are no known violations of this bound. (iii) The hole energy Eh required to form a hole in a glass-forming liquid is equal to 2RT∞. (iv) A glass-forming liquid tends to an isofree volume state at T∞. The calculated hole fraction equals e−3 = 0.0498. (v) A correlation has been found between the hole energy and a viscosity parameter. This correlation is sensitive to the chemical structure of a polymer chain backbone. (vi) Both WLF constants, C1 and C2, can be predicted from the observed Δα and viscosity-temperature behavior. Excellent agreement has been obtained for polyvinyl acetate and a nonpolymeric glass-forming liquid, tri-α-naphtyhl benzene. (vii) Even though the theory is a free-volume theory, the temperature derivative of the viscosity at constant volume is nonzero when we identify T∞ with the second-order phase transition temperature of the Gibbs-DiMarzio theory.