8.3.1 Vacancy Conservation
Based on mass/vacancy conservation law, the following derivation can be given
(Fig. 8.6):
q x À q xþΔx
À
Á ΔyΔz þ GΔxΔyΔz
Â
à Δt ¼ c þ Δc
ð
Þ V þ ΔV
ð
ÞÀcV
À∇q þ G
ð
Þ VΔt ¼ Δ cV
ð Þ
setΔt ! 0
) À∇q þ G
ð
Þ V ¼
∂ ccV
ð
Þ
∂t
¼ V
∂c
∂t
þ c
∂V
∂t
) À∇q þ G ¼
∂c
∂t
þ c
∂V=V
∂t
¼
∂c
∂t
þ c
∂ε V
∂t
)
∂c
∂t
¼ À∇q þ G À c
∂ε V
∂t
ð8:18Þ
where:
C V0 —Thermodynamic equilibrium vacancy concentration in the absence of stress
field.
c—Normalized vacancy concentration and c ¼
C V
C V0
.
C V —Instantaneous vacancy concentration.
ε V —Volumetric strain. It is assumed that a diffusing atom leaves behind a
vacancy or takes an interstitial position. Both cases lead to volumetric strain only.
t—Time.
Substituting Eqs. (8.6), (8.8), and (8.11) into Eq. (8.5), we obtain the q,vacancy
flux in the following form:
q x
q x+Δ
x
Δ
x
Δ
y
Δ
z
έ x ΔtΔ
x
Initial State
Δx, Δy,Δz, V= ΔxΔyΔz, c
After Δt
Δx+ έ x ΔtΔx
Δy+ έ y ΔtΔy
Δz+ έ z ΔtΔz
V+ ΔV
c+Δc
Fig. 8.6 Illustration of vacancy conservation law
8.3 Laws of Conservation
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