3.3.1 Strain due to Vacancy Migration

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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

$\displaystyle \Delta V = \symVacVol - \symAtomVol = -(1-\symVacRelFactor)\symAtomVol.$

(3.30)

Given a test volume , the relative volume change associated with a change in vacancy concentration $\Delta\CV$ is [147]

$\displaystyle \frac{\Delta V}{V} = -(1-\symVacRelFactor)\symAtomVol\Delta\CV,$

(3.31)

so that the volumetric strain has the form

$\displaystyle \frac{\Delta V}{V}=\symStrain_{11}^m + \symStrain_{22}^m + \symStrain_{33}^m = 3\symStrain^m = -(1-\symVacRelFactor)\symAtomVol\Delta\CV,$

(3.32)

where $\symStrain^m$ refers to the migration strain.

Taking the time derivative of the above equation one gets

$\displaystyle \ensuremath{\ensuremath{\frac{\partial \symStrain^m}{\partial t}}... ...RelFactor)\symAtomVol\ensuremath{\ensuremath{\frac{\partial \CV}{\partial t}}},$

(3.33)

and, since for the test volume the atom-vacancy exchange is governed by the continuity equation

$\displaystyle \ensuremath{\ensuremath{\frac{\partial \CV}{\partial t}}} = -\ensuremath{\nabla\cdot{\vec\JV}},$

(3.34)

the components of the migration strain rate is given by

$\displaystyle \ensuremath{\ensuremath{\frac{\partial \symVacMigStrain}{\partial... ...mVacRelFactor)\symAtomVol\ensuremath{\nabla\cdot{\vec\JV}}\right]\symKronecker.$

(3.35)

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R. L. de Orio: Electromigration Modeling and Simulation