PHYSICS OF GLASSES, AMORPHOUS SOLIDS AND DISORDERED CRYSTALS
Name: .........................................................................
Date: .........................................................................
PROBLEM:
The viscosity of many supercooled liquids obey the empirical Vogel-TammannFulcher (VTF) equation:

a 

 T  T0 
   exp 
where it is often written a = D T0, with D being an adimensional parameter that measures the
“strength” of the liquid in Angell´s classification.
Another well-known empirical equation for the viscosity of many glass-forming
liquids (though not exhibiting any divergence) is the Williams-Landel-Ferry (WLF) equation:
g
ln 

  A(T  Tg ) 

  


  B  (T  Tg ) 
where A and B are constants.
On the other hand, the free-volume theory basically assumes that molecular volume
can be divided into the part occupied by the molecules (vo) and that part in which the
molecules are free to move, the “free volume” (vf). It further supposes that a supercooled
liquid can be described by a distribution of liquid-like and solid-like cells, with or without
enough free volume for molecular diffusion. The free volume is considered to be partitioned
randomly among the cells and hence the viscosity is found to be given by
  C exp( b
vo
)
vf
which is the so-called Doolittle equation, C and b being empirical constants.
(a) Show that the Doolittle equation is equivalent to the VTF law, if the assumption is made
that the thermal expansion is linear in temperature above the glass-transition temperature.
(b) Show further that the insertion of the relation for the free-volume temperature dependence
into the Doolittle equation gives an equation of the WLF type.