Abstract
This paper constitutes a new contribution on the resolution of Mullins problem in the case of the evaporation-condensation and gives an exact and explicit solution of the second partial differential equation relative to the geometric profile of the grain boundary grooving. New analytical expressions of the solution, the groove profile, the derivative and the groove deep were obtained:begin{array}{c}y{boldsymbol{(}}x{boldsymbol{,}}t{boldsymbol{)}}{boldsymbol{=}}{boldsymbol{-}},sqrt{pi Ct},sin,theta ,{boldsymbol{[}}erfc{boldsymbol{(}}frac{x}{{bf{2}}sqrt{Ct}}{boldsymbol{)}}{boldsymbol{+}}{boldsymbol{sum }}_{n{boldsymbol{=}}{bf{1}}}^{infty },frac{{boldsymbol{(}}{bf{2}}n{boldsymbol{)}}{boldsymbol{!}}}{{{boldsymbol{(}}n{boldsymbol{!}}{boldsymbol{)}}}^{{bf{2}}}{{bf{2}}}^{{bf{2}}n},sqrt{{bf{3}}n}},si{n}^{{bf{2}}n}theta ,{boldsymbol{(}}erfc{boldsymbol{(}}frac{xsqrt{{bf{3}}n}}{{bf{2}}sqrt{ct}}{boldsymbol{)}}{boldsymbol{)}}{boldsymbol{]}} y{boldsymbol{{prime} }}{boldsymbol{(}}x{boldsymbol{,}}t{boldsymbol{)}}{boldsymbol{=}}{boldsymbol{+}},frac{sin ,theta }{sqrt{{e}^{{x}^{{bf{2}}}{boldsymbol{/}}{boldsymbol{(}}{bf{2}}ct{boldsymbol{)}}}{boldsymbol{-}}si{n}^{{bf{2}}}theta }},{rm{and}},{varepsilon }_{{bf{0}}}{boldsymbol{(}}theta {boldsymbol{)}}{boldsymbol{=}}sqrt{pi ct},sin ,theta ,{boldsymbol{[}}1{boldsymbol{+}}{boldsymbol{sum }}_{n{boldsymbol{=}}1}^{infty }frac{{boldsymbol{(}}{bf{2}}n{boldsymbol{)}}{boldsymbol{!}}}{{{boldsymbol{(}}n{boldsymbol{!}}{boldsymbol{)}}}^{{bf{2}}}{{bf{2}}}^{2n}sqrt{3n}}si{n}^{2n}theta {boldsymbol{]}}end{array}It was proved that the found solution gave more accurate results relative to those obtained by Mullins that neglected the first derivative (|y′| ≪ 1) relative to 1. The results obtained by this new solution can be advantageously used to give more precise solution of the general problem when combining the two phenomena relative to the evaporation-condensation and the surface diffusion in thin polycrystalline films.
Highlights
It was proved that the found solution gave more accurate results relative to those obtained by Mullins that neglected the first derivative (|y′| ≪ 1) relative to 1
We propose in this paper to study the grain boundary groove profiles in polycrystalline metal and to give an analytical solution relative to the only case of evaporation/condensation, more precise than of the solution found by Mullins[10] that supposed very small slops for all x values
The approximation |y′| ≪ 1 used by Mullins in the case of evaporation and condensation in case of the grain boundary grooving in polycrystalline thin films is well described by equations (17)
Summary
It was proved that the found solution gave more accurate results relative to those obtained by Mullins that neglected the first derivative (|y′| ≪ 1) relative to 1. The application areas include aerospace, aviation, railway, electrical distribution, automotive, home automation, oil industry These modules constitute an assembly of various materials (Fig. 1). The power chips are carried on a ceramic substrate which must ensure good electrical insulation and good thermal conduction. The most common topside interconnections in power semiconductor devices, consisting of the metallization and the wire bonds, are subjected in operation to high functional stresses. This is the result of an important difference between the coefficients of thermal expansion (CTE) of the materials in contact: metallization and wire bonds (aluminum) and dies (silicon). The end of life is rather characterized by bond-wire heel-cracks and lift-off[9]
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