III. Solid-Solid Nucleation: In lecture we discussed how solid precipitates forming within another solid phase would initially form coherent interfaces, due to the lower interfacial energy, Y coherent However, coherent interfaces are typically associated with some strain energy in the precipitate (ẞ), since it must be stretched or compressed so that its lattice parameter matches that of the host (α). Generally, the strain energy term may be expressed as Eε²V, where Е is the elastic modulus of the precipitate phase, & is the strain and can be determined from the lattice parameters aa and aß and VB is the volume of a single precipitate. Consequently, the strain energy term becomes large as the precipitates grow, and there will be a size at which it is energetically preferable for the precipitates to release their strain energy and form an incoherent interface with the host, even though the interfacial energy Y incoherent can be significantly larger than Y coherent Produce a graph of the critical size (cube edge length) at which a ẞ precipitate will undergo a coherent to incoherent transition for a lattice parameter mismatch equivalent to strains ranging from 1% to 10%. Assume the precipitates form in the shape of a cube with edge length d, and that: a = 4.00Å, Eß = 100 GPa, Y coherent = 0.1 J/m², Y incoherent = 1.0 J/m², and Ag=-1.0×101 J/atom. Furthermore, assume the crystal structure of each phase has one atom per unit cell.
III. Solid-Solid Nucleation: In lecture we discussed how solid precipitates forming within another solid phase would initially form coherent interfaces, due to the lower interfacial energy, Y coherent However, coherent interfaces are typically associated with some strain energy in the precipitate (ẞ), since it must be stretched or compressed so that its lattice parameter matches that of the host (α). Generally, the strain energy term may be expressed as Eε²V, where Е is the elastic modulus of the precipitate phase, & is the strain and can be determined from the lattice parameters aa and aß and VB is the volume of a single precipitate. Consequently, the strain energy term becomes large as the precipitates grow, and there will be a size at which it is energetically preferable for the precipitates to release their strain energy and form an incoherent interface with the host, even though the interfacial energy Y incoherent can be significantly larger than Y coherent Produce a graph of the critical size (cube edge length) at which a ẞ precipitate will undergo a coherent to incoherent transition for a lattice parameter mismatch equivalent to strains ranging from 1% to 10%. Assume the precipitates form in the shape of a cube with edge length d, and that: a = 4.00Å, Eß = 100 GPa, Y coherent = 0.1 J/m², Y incoherent = 1.0 J/m², and Ag=-1.0×101 J/atom. Furthermore, assume the crystal structure of each phase has one atom per unit cell.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
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Transcribed Image Text:III. Solid-Solid Nucleation:
In lecture we discussed how solid precipitates forming within another solid phase would initially
form coherent interfaces, due to the lower interfacial energy, Y coherent However, coherent
interfaces are typically associated with some strain energy in the precipitate (ẞ), since it must be
stretched or compressed so that its lattice parameter matches that of the host (α). Generally, the
strain energy term may be expressed as Eε²V, where Е is the elastic modulus of the
precipitate phase, & is the strain and can be determined from the lattice parameters aa and aß
and VB is the volume of a single precipitate. Consequently, the strain energy term becomes large
as the precipitates grow, and there will be a size at which it is energetically preferable for the
precipitates to release their strain energy and form an incoherent interface with the host, even
though the interfacial energy Y incoherent can be significantly larger than Y coherent
Produce a graph of the critical size (cube edge length) at which a ẞ precipitate will undergo a
coherent to incoherent transition for a lattice parameter mismatch equivalent to strains ranging
from 1% to 10%. Assume the precipitates form in the shape of a cube with edge length d, and
that: a = 4.00Å, Eß = 100 GPa, Y coherent = 0.1 J/m², Y incoherent = 1.0 J/m², and
Ag=-1.0×101 J/atom. Furthermore, assume the crystal structure of each phase has one atom
per unit cell.
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