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3.3.1 Strain due to Vacancy Migration
When an atom is exchanged for a vacancy, the neighboring atoms relax, leading to a total volume change given by
![$\displaystyle \Delta V = \symVacVol - \symAtomVol = -(1-\symVacRelFactor)\symAtomVol.$](img356.png) |
(3.30) |
Given a test volume
, the relative volume change associated with a change in vacancy concentration
is [147]
![$\displaystyle \frac{\Delta V}{V} = -(1-\symVacRelFactor)\symAtomVol\Delta\CV,$](img357.png) |
(3.31) |
so that the volumetric strain has the form
![$\displaystyle \frac{\Delta V}{V}=\symStrain_{11}^m + \symStrain_{22}^m + \symStrain_{33}^m = 3\symStrain^m = -(1-\symVacRelFactor)\symAtomVol\Delta\CV,$](img358.png) |
(3.32) |
where
refers to the migration strain.
Taking the time derivative of the above equation one gets
![$\displaystyle \ensuremath{\ensuremath{\frac{\partial \symStrain^m}{\partial t}}...
...RelFactor)\symAtomVol\ensuremath{\ensuremath{\frac{\partial \CV}{\partial t}}},$](img360.png) |
(3.33) |
and, since for the test volume the atom-vacancy exchange is governed by the continuity equation
![$\displaystyle \ensuremath{\ensuremath{\frac{\partial \CV}{\partial t}}} = -\ensuremath{\nabla\cdot{\vec\JV}},$](img361.png) |
(3.34) |
the components of the migration strain rate is given by
![$\displaystyle \ensuremath{\ensuremath{\frac{\partial \symVacMigStrain}{\partial...
...mVacRelFactor)\symAtomVol\ensuremath{\nabla\cdot{\vec\JV}}\right]\symKronecker.$](img362.png) |
(3.35) |
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Up: 3.3 Electromigration Induced Stress
Previous: 3.3 Electromigration Induced Stress
R. L. de Orio: Electromigration Modeling and Simulation