Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys
Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys
Multiscale Modeling of Theta ' Precipitation in Al-Cu Binary Alloys
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2980 V. Vaithyanathan et al. / Acta Materialia 52 (2004) 2973–2987<br />
3.3. Elastic properties<br />
In addition to bulk and <strong>in</strong>terfacial energies, there is a<br />
significant elastic energetic contribution to the h 0 /<strong>Al</strong><br />
system, due to the fact that the h 0 precipitates are coherently<br />
stra<strong>in</strong>ed by the <strong>Al</strong> matrix. The h 0 plates are<br />
generally acknowledged to be coherent with the matrix<br />
along their broad faces. In addition, Doherty [7] has<br />
shown that the <strong>in</strong>terfacial rim <strong>of</strong> plate-shaped precipitates<br />
are not necessarily <strong>in</strong>coherent, but could also be<br />
coherent or semi-coherent. Observations <strong>of</strong> the presence<br />
<strong>of</strong> facets on the rims <strong>of</strong> h 0 precipitates (when viewed<br />
perpendicular to the coherent face <strong>of</strong> the plate) from<br />
TEM micrographs [4,29] prove that this <strong>in</strong>terface is not<br />
completely <strong>in</strong>coherent. Thus, equilibrium morphology<br />
calculations, such as <strong>in</strong> the present work, should <strong>in</strong>clude<br />
the contributions from the elastic energy anisotropy <strong>in</strong><br />
addition to the <strong>in</strong>terfacial energy anisotropy.<br />
To <strong>in</strong>clude the elastic energy contribution <strong>in</strong> the phasefield<br />
model, we require the follow<strong>in</strong>g: the stress-free misfit<br />
stra<strong>in</strong>s along the coherent and semi-coherent <strong>in</strong>terfaces<br />
and the result<strong>in</strong>g elastic stra<strong>in</strong> energy contributions from<br />
the coherency constra<strong>in</strong>ts. These quantities are all readily<br />
calculated from our first-pr<strong>in</strong>ciples methods.<br />
3.3.1. Stress-free misfit stra<strong>in</strong>s<br />
The equilibrium or stress-free T ¼ 0 K lattice parameters<br />
<strong>of</strong> pure, bulk fcc <strong>Al</strong> and h 0 -<strong>Al</strong> 2 <strong>Cu</strong> (CaF 2<br />
structure) calculated us<strong>in</strong>g first-pr<strong>in</strong>ciples (FLAPW-<br />
LDA) are<br />
a <strong>Al</strong> ¼ 0:3989 nm; a h<br />
0 ¼ 0:4016 nm;<br />
p<br />
c h<br />
0 ¼ ffiffi<br />
2 ah 0 ¼ 0:5679 nm:<br />
We note that very large k-po<strong>in</strong>t sets have been used to<br />
ensure the convergence <strong>of</strong> these structural parameters to<br />
several significant digits. From these lattice parameters,<br />
the calculated misfit d c along the coherent, broad face <strong>of</strong><br />
h 0 plates is<br />
d c ¼ a h 0 a <strong>Al</strong><br />
¼þ0:7%: ð19Þ<br />
a <strong>Al</strong><br />
<strong>Al</strong>though the h 0 plates are completely coherent across<br />
the broad face for lower ag<strong>in</strong>g temperatures, they may<br />
conta<strong>in</strong> widely spaced misfit dislocations at higher ag<strong>in</strong>g<br />
temperatures [30]. By measur<strong>in</strong>g the dislocation spac<strong>in</strong>g<br />
across h 0 broad faces, Weatherly and Nicholson [30]<br />
have <strong>in</strong>ferred a misfit value <strong>of</strong> jd c j¼0:57%, which<br />
agrees well with our first-pr<strong>in</strong>ciples calculated misfit. We<br />
also note that Weatherly and Nicholson could not ascerta<strong>in</strong><br />
the sign <strong>of</strong> the misfit from their observations, as<br />
they could not unambiguously determ<strong>in</strong>e the nature <strong>of</strong><br />
the Burgers vectors <strong>of</strong> the dislocations; our calculations<br />
clearly show that the h 0 plates are under a compressive<br />
stra<strong>in</strong> along the broad faces.<br />
Around the precipitate rim, the simple configuration<br />
<strong>of</strong> c h<br />
0 : a <strong>Al</strong> has an extremely large misfit stra<strong>in</strong> <strong>of</strong> 42%.<br />
Therefore, we have considered the misfit for a few <strong>in</strong>teger<br />
comb<strong>in</strong>ations <strong>of</strong> c h<br />
0 and a <strong>Al</strong> unit cells. For a<br />
configuration <strong>of</strong> m h 0 to n <strong>Al</strong> unit cells, respectively,<br />
where m and n are positive <strong>in</strong>tegers, the misfit along the<br />
semi-coherent <strong>in</strong>terface (d sc ) def<strong>in</strong>ed as<br />
d sc ¼ mðc h 0Þ nða <strong>Al</strong>Þ<br />
nða <strong>Al</strong> Þ<br />
ð20Þ<br />
is as follows for m 6 4: (i) 1c h<br />
0 : 1a <strong>Al</strong> – +42.4%, (ii)<br />
1c h<br />
0 : 2a <strong>Al</strong> – )28.8%, (iii) 2c h<br />
0 : 3a <strong>Al</strong> – )5.1%, (iv)<br />
3c h<br />
0 : 4a <strong>Al</strong> – +6.8%, (v) 3c h<br />
0 : 5a <strong>Al</strong> – )14.6%, and (vi)<br />
4c h<br />
0 : 5a <strong>Al</strong> – +13.9%.<br />
We verify that the 2c h<br />
0 : 3a <strong>Al</strong> configuration for the<br />
semi-coherent <strong>in</strong>terface, observed by Stobbs and Purdy<br />
[8], is the most favorable low-<strong>in</strong>teger comb<strong>in</strong>ation <strong>in</strong><br />
terms <strong>of</strong> lattice misfit with d sc ¼ 5:1%. Lattice parameter<br />
measurements <strong>of</strong> h 0 give a value <strong>of</strong> d sc ¼ 4:3% for<br />
this misfit [8,31]. Aga<strong>in</strong>, our calculations are <strong>in</strong> reasonable<br />
agreement with observations. We note that for<br />
larger-unit cell comb<strong>in</strong>ations (i.e., thicker h 0 plates), one<br />
can obta<strong>in</strong> even smaller misfit stra<strong>in</strong>s: For example, the<br />
5c h<br />
0 : 7a <strong>Al</strong> and 7c h<br />
0 : 10a <strong>Al</strong> comb<strong>in</strong>ations have d sc ¼þ<br />
1:7% and 0:3%, respectively. However, we conf<strong>in</strong>e our<br />
attention <strong>in</strong> this work to the 2c h<br />
0 : 3a <strong>Al</strong> configuration. For<br />
a more complete discussion <strong>of</strong> the <strong>in</strong>terfacial structure,<br />
and mechanisms for h 0 precipiate growth, we refer the<br />
reader to [4,8,31] and references there<strong>in</strong>.<br />
3.3.2. Coherency stra<strong>in</strong> energies:<br />
For the phase-field model, we also need to know the<br />
stra<strong>in</strong> energy penalty associated with the misfit stra<strong>in</strong>s<br />
calculated above. We have calculated the coherency<br />
stra<strong>in</strong> energy for a h 0 /<strong>Al</strong> system <strong>in</strong> an analogous manner<br />
to that used to construct the <strong>Al</strong>/<strong>Cu</strong> coherency stra<strong>in</strong><br />
enter<strong>in</strong>g the mixed-space cluster expansion [17,19]. This<br />
calculation <strong>in</strong>volves a direct computation <strong>of</strong> Eq. (16)<br />
whereby <strong>Al</strong> and h 0 -<strong>Al</strong> 2 <strong>Cu</strong> are each deformed <strong>in</strong> an<br />
‘‘epitaxial’’ geometry, stra<strong>in</strong>ed to a common lattice<br />
constant <strong>in</strong> a plane perpendicular to an orientation, ^G.<br />
Summ<strong>in</strong>g together these ‘‘epitaxial’’ energies and subsequently<br />
m<strong>in</strong>imiz<strong>in</strong>g with respect to the common lattice<br />
constant gives the coherency stra<strong>in</strong> energy <strong>of</strong> Eq. (16).<br />
Fig. 7 shows the calculated coherency stra<strong>in</strong> for the h 0 /<strong>Al</strong><br />
system for hydrostatic stra<strong>in</strong>, as well as epitaxial stra<strong>in</strong><br />
along [1 0 0] and [1 1 1] orientations.<br />
We note that due to the homogeneous modulus approximation<br />
used <strong>in</strong> the phase-field model, it is necessary<br />
that we extract a s<strong>in</strong>gle, averaged set <strong>of</strong> harmonic<br />
elastic constants from these calculations. We note that<br />
first-pr<strong>in</strong>ciples total energy calculations are not constra<strong>in</strong>ed<br />
by the homogeneous modulus approximation,<br />
nor by harmonic elasticity theory, and <strong>in</strong>clude both<br />
harmonic and anharmonic elastic effects. However, we<br />
analyze our results with<strong>in</strong> the context <strong>of</strong> harmonic<br />
elasticity, and use this theory to extract the average<br />
elastic modulii <strong>of</strong> the h 0 /<strong>Al</strong> system from the results <strong>of</strong>