Chemistry Reference
In-Depth Information
where
k
B
is the Boltzmann constant. From these equations we can see that at constant
molecular size, the rate of such diffusive motions will be critically controlled by both
temperature and viscosity.
Another more general way to express molecular mobility quantitatively is to de
ne the
relaxation time
, a parameter that directly indicates the timescale over which a single
rotation takes place or the time over which a molecule undergoes translation across a given
reference distance; the greater the viscosity and hence the smaller the diffusion coef
τ
cient,
τ
the greater the translational or rotational relaxation time. To determine the value of
under a
given set of conditions, one can experimentally perturb the system out of equilibrium
mechanically, electrically, or magnetically and observe the rate
(
t
) at which the property
returns toward the equilibrium state. Dynamic mechanical analysis (DMA) is generally
used to measure the rate of relaxation and viscosity; however, primary relaxations can also
be measured by electrical perturbation of dipoles in the molecule and measurement of the
dipole relaxation back toward an equilibriumstate using dielectric spectroscopy (DES). For
a system exhibiting a single mode of relaxation that follows
ϕ
first-order kinetics, such as a
pure liquid above its melting temperature, one can write
ϕ
t
exp
t
=
τ
;
(1.5)
where
τ
is the reciprocal of the
first-order rate constant. In such a case, one would expect
the temperature dependence of
τ
to follow the Arrhenius equation:
τ
T
τ
0
exp
E
a
=
RT
;
(1.6)
τ
0
is the relaxation time at the high temperature limit, on the order of 10
12
s, and
E
a
is the activational energy associated with the process. Indeed, equilibrium liquids
generally exhibit Arrhenius kinetics. Supercooled liquids at temperatures approaching
T
g
, however, appear to exhibit more than a single relaxation time generally described by a
“
where
exponential form of Equation 1.5 that takes into account this distribution of
relaxation modes. For example, a widely applicable empirical equation in such situations
is the Kohlrausch-Williams-Watts (KWW) equation:
stretch
”
=
τ
β
;
ϕ
t
exp
t
(1.7)
where
represents the distribution of different
relaxation times, with values falling between 0 and 1; a value of 1 indicates a single mode
of relaxation and smaller values represent an increasing number of relaxation modes.
Most amorphous systems of pharmaceutical interest yield values of
τ
is the average relaxation time and
β
β
in the range of
0.3
-
0.6 [16]. When such an equation is applicable, the temperature dependence of
τ
can be
expressed by the nonexponential empirical Vogel
-
Tammen
-
Fulcher (VTF) equation:
τ
T
τ
0
exp
DT
0
=
T
T
0
;
(1.8)
where
D
and
T
0
are constants.
D
is called the
an indication of the
activation energy of the diffusive relaxation process, and
T
0
is the temperature at which
“
strength parameter,
”
