Metal Whiskers: Failure Modes & Mitigation Strategies - NEPP - NASA

u = c ⋅∆L
⋅ A⋅T
du
dt
T o
Whisker
C
u = C ⋅T
c =
V
⎛ C ⎞
u = ⎜ ⎟⋅V
⋅T
= c ⋅V
⋅T
⎝ V ⎠
∂T
= c ⋅ ∆L
⋅ A⋅
∂t
Derivation of Melting Current of a
Metal Whisker in Vacuum
∆ L
L
2a T o
2
⎡ ∂T
⎤ ⎡ Lz
⋅T
⎛ ∂ T ⎞ ⎤ 2
⎢c
⋅∆L ⋅ A⋅
− ⎢
⋅∆L
⋅ A⎥
= J ⋅ ⋅∆L
⋅ A
t ⎥ ⎜
⎣ x ⎟
ρ
2
⎣ ∂ ⎦ ρ ⎝ ∂ ⎠ ⎦
melt,
vac
December 5, 2007 Metal Whiskers:
Failure Modes and Mitigation Strategies
I
du
dt
⎡2
= ⎢
⎣
+ Φ
L
R
0
Z
=
source = I ⋅ R
2
I = Je
⋅
T
0
source
du/dt + Φ = source
2
⎡ ∂T
⎤ ⎡ Lz ⋅T
⎛ ∂ T ⎞⎤
2
⎢
c ⋅
⎥
− ⎢
⎥ = ⋅ ρ
⎣ ρ ⎜
⎟ J
2
⎣ ∂t
⎦ ⎝ ∂x
⎠⎦
Assume both ends of
Whisker are thermally
grounded to T o
Convection loss = 0 for vacuum
Neglect radiation loss
⎛ ∂J
⎞
Φ = ⎜ ⎟⋅
∆L
⋅ A
⎝ ∂x
⎠
2
∂T
∂J
∂ T
J = −kT
⋅
= −kT
⋅ 2
∂x
∂x
∂x
⎛ ∂ T ⎞
Φ = −kT
⋅ ⎜ ⋅∆L
⋅ A
x ⎟ 2
⎝ ∂ ⎠
2 2 2
LZ
⋅T
source = ( Je
⋅ A ) ⋅⎜
⎟
kT
=
⎝ A ⎠
2
LzT
⎛ ∂ T ⎞
Φ = − ⎜ ⋅ ∆L
⋅ A
x ⎟ 2
ρ ⎝ ∂ ⎠
ρ
⎤
⎥
⎦
A
cos
⎛ ρ ⋅ ∆L
⎞
−1
⋅ ∆L
R =
A
ρ
2 ( J ⋅ A)
⋅ ⋅ L
source = e ρ ∆
⎛ T
⎜
⎝ T
0
melt
⎞
⎟
⎠
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