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40 SMT Magazine • February 2015 trochemical potential F = m + ej that needs to be constant in space in order to minimize the sys- tem free energy. Here, m is the chemical poten- tial (change in the system energy in response to the change of number of particles in it by one; the difference in chemical potentials between two substances equals minus that of the corre- sponding work functions). The chemi cal poten- tial is sensitive to all kind of chemical features, deformations, defects, etc. Figure 12 shows how the concentrations of particles n are significantly different in the two regions with different chem- ical potentials. As a result, the particles move to the region of lower chemical potential. Because of the accumulated electric charge, there is now electric field E in the system. However, its cor- responding drift current nEb (b is the mobility) is totally balanced by the diffusion current Ddn/ dx, where D is the diffusion coefficient and x is the coordinate in the direction of concentration gradient. The important conclusion is that it is a metal where there is no current while the elec- tric field is not vanishing. It follows from the latter that some struc- tural or compositional inhomogeneities are needed for the electric charge redistribution triggering whisker growth. Grain structure can be one (but not the only) example of such inho- mogeneities. This is consistent with the general observation that whiskers are unlikely on metal surfaces built of very large grains, and that the presence of grain boundaries can be essential. Before pointing at more specific factors behind electric field variations, the following general examples (i–iii) are aimed at illustrating the un- derlying physics. i. Consider local stresses (due to grain bound- aries, dislocations, or external loads) modulat- ing interatomic distances in a metal, thus mak- ing some local regions denser than the others. Because of these local variations, some re- gions will present deeper potential wells for the electrons (a phenomenon known as the defor- mation potential D = dm / du 1 eV where u is the dilation). Should these regions be mutually independent, the Fermi level positions would vary between them. However, because they are connected, the free electrons will move to level out the Fermi level across the entire system thus minimizing the system free energy. As a result, the above regions become electrically charged. ii. Spatial variations in chemical composi- tion of alloys would similarly result in modula- tions of work function, which will be leveled out in the course of electron redistribution cre- ating local electric fields similar to the previous example. iii. The extreme case of the latter is present- ed by a contact of different metals with unequal Figure 12: a sketch of a spatial distribution of the electric potential including two regions with the relative- ly low (left) and relatively high (right) chemical potential m. To utilize that difference in chemical poten- tials, the electrons partially moved from right to left creating non-uniform electric charge distribution and its corresponding non-uniform electric potential j and electric field e. The quantity that remains constant throughout the region is the electro-chemical potential F = m + ej where e is the electron charge. eLeCTrOSTaTIC MeCHaNISM OF NuCLeaTION aNd GrOWTH OF MeTaL WHISKerS continues Feature