February 2015 • SMT Magazine 35
sion L, after which the growth rate remains on
average constant. The physical reason of that
acceleration is that the longer the whisker the
stronger the electrostatic energy gain and its
corresponding thermodynamic force pulling
the whisker. This process can be explosive (posi-
tive feedback) in nature: The stronger the force,
the faster the growth, the greater the length,
and the stronger the force, etc., as long as the
force increases with length (for h>L, the field appears strongly fluctu-
ating (Figure 2) which significantly suppresses
the pulling force. As a result, the pulling force
becomes independent of whisker length (Figure
5). That distant independent average force re-
sults in constant average growth rates.
The predicted time dependent growth rate is
sketched in Figure 6. The dormant time t
0
cor-
responds to whisker development over short
distances hL. For the
overgrown whiskers with h>>L, the growth rate
becomes time independent on average.
5. The latter statement of constant growth
rate for whisker lengths h>>L holds true on av-
erage. In reality, a whisker encounters a random
distribution of fields in the course of its growth
as can be realized from Figure 4. A whisker head
passes through multiple regions of various direc-
tions and strengths. Correspondingly, the whis-
ker length growth rate fluctuates between low
and high values. It can be shown that the most
significant electrostatic energy gain is achieved
via the increase of whisker length rather than
the diameter. Therefore, when the material sup-
ply is limited, the energy gain will dictate that
whiskers grow in length as much as possible.
However, when the length growth is suppressed
Figure 5: Top left: free energy vs. whisker length h where the downhill slope corresponds to the
growth stage of whiskers; W is the nucleation barrier, h0 is the critical nucleation dimension of the
needle-shaped embryo. The shape of that slope is determined by the charged patch model shown to
the right of the free energy plot where r is the distance from a metal surface and l is the characteristic
patch dimension. bottom: the free energy vs. whisker length with the marked regions of qualitatively
different dependencies.
eLeCTrOSTaTIC MeCHaNISM OF NuCLeaTION aNd GrOWTH OF MeTaL WHISKerS continues
Feature
