Recently, Fedelich
et al.[
20] measured the increase in the microstructural periodicity
λ and the
γ channel width
ω of CMSX-4 commercial superalloys as a function of service time, temperature, and applied stress. An analytical expression was proposed, which seems to capture the findings from detailed observations on the transition of
γ′ morphology from cuboid to rafted shape. In their work, the processes of isotropic coarsening and unidirectional coarsening,
i.e., rafting, are considered separately and are assumed to be governed by different driving mechanisms. Based on geometrical considerations, the channel width in the two different procedures is given by
$$ \omega_{\text{cube}} (t) = \left( {1 - \sqrt[3]{{f_{{\gamma^{\prime}}} }}} \right)\lambda (t) $$
(1)
$$ \omega_{\text{raft}} (t) = \left( {1 - f_{{\gamma^{\prime}}} } \right)\lambda (t), $$
(2)
where
f
γ′
is the volume fraction of precipitates and
λ(
t) is the microstructural periodicity. The isotropic coarsening of
λ(
t) was captured by the fitted equation used for specimens crept at 1223 K (950 °C):
$$ \lambda (t) = \lambda_{0} \sqrt {1 + \beta t} . $$
(3)
A dimensionless parameter, termed rafting degree
ξ, is defined to quantify the microstructural changes. It presents the normalized channel width, which increases from 0 to 1 when the microstructure changes from cubes to plates:
$$ \xi = \frac{{\omega (t) - \omega_{\text{cube}} (t)}}{{\omega_{\text{raft}} (t) - \omega_{\text{cube}} (t)}}. $$
(4)
The analytical expression to describe the degree of rafting as a function of time and external stress based on experimental results is given by
$$ \xi = 1 - { \exp }( - b\sigma^{n} t). $$
(5)
Fundamentally, the aforementioned model is an experimentally based analytical expression, which can be applied only for CMSX-4 alloy at the temperature 1223 K (950 °C), where
β,
b, and
n are all fitting parameters not having any physical meaning. To extend the application scope of this model over a wider temperature range, Fedelich
et al.[
21] further collected the microstructural evolution data of CMSX-4 over a wide temperature range and built the following equation to describe the kinetics of evolving periodicity and channel width:
$$ \lambda_{{\left[ {001} \right]}} = \lambda_{[001]}^{0} \left[ {1 + D_{0} {\text{exp}}\left( { - \frac{{Q_{\lambda } }}{RT}} \right)t} \right]^{a} $$
(6)
$$ \omega_{[001]} = \omega_{\text{cube}} + \left( {\omega_{\text{raft}} - \omega_{\text{cube}} } \right)\times\left\{ {1 - { \exp }\left[ { - A_{0} {\text{exp}}\left( { - \frac{{Q_{\xi } }}{RT}} \right)\left( {1 + \frac{\sigma }{{\sigma_{0} }}} \right)^{p} t} \right]} \right\}. $$
(7)
In these two equations,
D
0,
Q
λ
,
a,
A
0,
Q
ξ
,
σ
0, and
p are also fitting parameters. This modified equation excellently describes the microstructural degradation of CMSX-4 at different stress and temperature levels with the correlation factors
r
2 > 0.97. However, all the parameter values in the equation remain fitted values, which means for other superalloy systems with slightly different chemical compositions, the applicability of the model should be reassessed and the parameter values need to be redetermined.