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Revised BiBTeX keys.
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tkphd committed Jan 6, 2025
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Expand Up @@ -13,7 +13,7 @@ Phase field models are, quite generally, extensions of classical
non-equilibrium thermodynamics. There are quite a few treatments of this in the
literature. Here we will follow formulations similar to those developed by
Sekerka and Bi in "Interfaces for the Twenty-First Century."
{cite}`bi_phase-field_1998` In that spirit, we will start with a very general
{cite}`Bi1998` In that spirit, we will start with a very general
formulation, including hydrodynamics, and then simplify the problem. For those
interested in starting with a simple model, you can skip over much of the
initial formulation and jump to the section on the further reduction of the
Expand Down Expand Up @@ -64,10 +64,10 @@ A complete discussion of how to use the above rules in this context is outside
the scope of this best-practice guide, but we offer a highly abbreviated
discussion of the ''flavor,'' following the ideas of irreversible
thermodynamics (we largely are following works like deGroot and Mazur
{cite}`groot_non-equilibrium_2013`, although the continuum mechanics community
{cite}`Groot2013`, although the continuum mechanics community
may be more comfortable with Noll, Coleman, and Truesdale
{cite}`malvern_introduction_1969`), as well as the aforementioned work by
Sekerka and Bi {cite}`bi_phase-field_1998`.
{cite}`Malvern1969`), as well as the aforementioned work by
Sekerka and Bi {cite}`Bi1998`.

### Mass

Expand Down Expand Up @@ -239,7 +239,7 @@ arise, and also, for careful readers, how to extend this approach to multiple
phases and additional gradient corrections, we will proceed with some more
simplifications for a less complex system. For those who are interested in
solid state systems that can creep, the work of Mishin, Warren, Sekerka, and
Boettinger (2013) {cite}`mishin_irreversible_2013` extends this framework.
Boettinger (2013) {cite}`Mishin2013` extends this framework.
Here we eliminate the ${\bf v}$ equations by fiat, assuming that only diffusion
controls the evolution of the system, which is often reasonable in microgravity
situations. We can also go further, and consider an isothermal system. Then we
Expand Down Expand Up @@ -299,7 +299,7 @@ modeling a liquid-solid binary alloy, although the details are less important
that understanding that a specific choice of state function has to come from
_somewhere_. Following the treatment in the Annual Reviews of Materials
Research (2001) by Boettinger, Warren, Beckerman and Karma
{cite}`boettinger_phase-field_2002` we note that the free energy can be
{cite}`Boettinger2002` we note that the free energy can be
determined through a multi-step process where the two components are called $A$
and $B$ respectively:

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