3 On the Spectrum of Relaxation Times in Coupled Diffusion …
53
At β/η = 0 i kT /GV m 1, the asymptotics of the second relaxation time at
λ → 0 corresponds to the mechanism of interdiffusion with a coefficient
D
+
0 =
4G
3
(M A V A (1 − φ 0 ) + M B V B φ 0 ),
(3.43)
which in the framework of the asymptotics under consideration significantly
exceeds the coefficient of conventional thermal diffusion: D
+
0 D
−
0 . In this case,
(3.43) corresponds to a sequental connection of diffusing bodies. If there is a considerable difference in mobility between the components of a binary system, for example,
M A M B , interdiffusion is determined by diffusion of the fast component (A). As
for the first branch, the diffusion mechanism due to the mean stress gradients is an
important part of the relaxation process.
Similar asymptotics of the branches of the function τ (λ) at β/η = 0 and λ → 0
for GV m /kT 1 can be interpreted as follows. The second branch corresponds to
thermal diffusion with the interdiffusion coefficient
D
+
0 = kT (M A ξ 0 + M B (1 − ξ 0 ))
(3.44)
corresponding to the Darken mechanism. In this case, the first branch should be
associated with the mechanism of slow diffusion with the interdiffusion coefficient
D
−
0 =
4G
3
M A M B V A V B
M A V A φ 0 + M B V B (1 − φ 0 )
,
(3.45)
with D
−
0 D
+
0 denoting a parallel connection of diffusing bodies. If there is an
essential difference in the mobility between the components of a binary system, for
example, M A M B , interdiffusion is determined by diffusion of the slow component (B). As before, the barodiffusion process is an important part of the relaxation
mechanism.
3.4 Conclusion
In this study, we have demonstrated the efficiency of the method for qualitative
study of rather slow (non-dynamic)-coupled diffusion and rheological processes.
The approach is based on the analysis of the relaxation times of a spatially perturbed
homogeneous stationary solution of the field equations for a one-dimensional model
problem, in which only the diffusion and rheological relations remain nontrivial.
Mathematically, the method is reduced to an eigenvalue problem and can be easily
studied using computer algebra systems. It has been used by Stephenson (1988) and
Brassart et al. (2018) and proved to be an effective tool for qualitative study of the
complex, coupled diffusion-rheological processes, accompanied by chemical reactions and changes in the microstructure of a deformable metal alloy. The asymptotics
53
At β/η = 0 i kT /GV m 1, the asymptotics of the second relaxation time at
λ → 0 corresponds to the mechanism of interdiffusion with a coefficient
D
+
0 =
4G
3
(M A V A (1 − φ 0 ) + M B V B φ 0 ),
(3.43)
which in the framework of the asymptotics under consideration significantly
exceeds the coefficient of conventional thermal diffusion: D
+
0 D
−
0 . In this case,
(3.43) corresponds to a sequental connection of diffusing bodies. If there is a considerable difference in mobility between the components of a binary system, for example,
M A M B , interdiffusion is determined by diffusion of the fast component (A). As
for the first branch, the diffusion mechanism due to the mean stress gradients is an
important part of the relaxation process.
Similar asymptotics of the branches of the function τ (λ) at β/η = 0 and λ → 0
for GV m /kT 1 can be interpreted as follows. The second branch corresponds to
thermal diffusion with the interdiffusion coefficient
D
+
0 = kT (M A ξ 0 + M B (1 − ξ 0 ))
(3.44)
corresponding to the Darken mechanism. In this case, the first branch should be
associated with the mechanism of slow diffusion with the interdiffusion coefficient
D
−
0 =
4G
3
M A M B V A V B
M A V A φ 0 + M B V B (1 − φ 0 )
,
(3.45)
with D
−
0 D
+
0 denoting a parallel connection of diffusing bodies. If there is an
essential difference in the mobility between the components of a binary system, for
example, M A M B , interdiffusion is determined by diffusion of the slow component (B). As before, the barodiffusion process is an important part of the relaxation
mechanism.
3.4 Conclusion
In this study, we have demonstrated the efficiency of the method for qualitative
study of rather slow (non-dynamic)-coupled diffusion and rheological processes.
The approach is based on the analysis of the relaxation times of a spatially perturbed
homogeneous stationary solution of the field equations for a one-dimensional model
problem, in which only the diffusion and rheological relations remain nontrivial.
Mathematically, the method is reduced to an eigenvalue problem and can be easily
studied using computer algebra systems. It has been used by Stephenson (1988) and
Brassart et al. (2018) and proved to be an effective tool for qualitative study of the
complex, coupled diffusion-rheological processes, accompanied by chemical reactions and changes in the microstructure of a deformable metal alloy. The asymptotics