sublattice. Each system is shown with a possible choice of composition axes
$\vec{q}^{\mathsf{T}}_{1}$ and $\vec{q}^{\mathsf{T}}_{2}$ and origin
$\vec{n}^{\mathsf{T}}_{0}$.

The geometric meaning of parametric composition is illustrated in Figure 20(a)
for the simple example of a binary A-B alloy with one sublattice per unit
cell. In this example, $n_{A}$ and $n_{B}$ track the number of A and B atoms
per unit cell. They are not independent, however, since A and B share the same
sites of the crystal, leading to the constraint that $n_{A}+n_{B}=1$. There
are two extremes in the subspace of allowed compositions: a crystal containing
only A atoms at $n_{A}=1$ and $n_{B}=0$ and a crystal containing only B atoms
with $n_{A}=0$ and $n_{B}=1$. These two extremes are illustrated in Figure
20(a) and are labeled as A and B. Any intermediate concentration of the alloy
will reside on the line connecting points A and B. The introduction of
independent parametric compositions requires the choice of an origin, for
example, pure A. In $n_{A}$ and $n_{B}$ space, this origin is represented with
the vector $\vec{n}_{0}$ as illustrated in Figure 20(a). The space of allowed
concentrations can be spanned with a second vector, $\vec{q}_{1}$, as
illustrated in Figure 20(a). Any allowed composition of A and B in this
crystal can then be expressed in vector form as

$\vec{n}=\vec{n}_{0}+x\vec{q}_{1}$ (105)

where $\vec{n}^{\mathsf{T}}=[n_{A},n_{B}]$ and $x$ is a parametric composition
within the subspace of allowed compositions. If the length of $\vec{q}_{1}$ is
equal to the distance between points 1 and 2 (the two extreme compositions) in
$n_{A}$ and $n_{B}$ space, then the parametric composition will take on any
value between 0 and 1.

Another example is shown in Figure 20(b) for a ternary A-B-C alloy having a
primitive cell containing one sublattice (i.e. $n=1$). There are now three
composition axes with variables $n_{A}$, $n_{B}$ and $n_{C}$. The constraint
that $n_{A}+n_{B}+n_{C}=1$ restricts the space of allowed compositions to a
two-dimensional subspace. In Figure 20(b) the origin is chosen to be pure A.
Two vectors are needed to span the two-dimensional subspace of allowed
compositions. These are denoted as $\vec{q}_{1}$ and $\vec{q}_{2}$ in Figure
20(b). These vectors connect the chosen origin,
$\vec{n}^{\mathsf{T}}_{0}=[1,0,0]$ to $\vec{n}^{\mathsf{T}}=[0,1,0]$ and
$\vec{n}^{T}=[0,0,1]$, respectively. Any concentration
$\vec{n}^{\mathsf{T}}=[n_{A},n_{B},n_{C}]$ can then be expressed as

$\vec{n}=\vec{n}_{0}+x_{1}\vec{q}_{1}+x_{2}\vec{q}_{2}$ (106)

where $x_{1}$ and $x_{2}$ are parametric compositions.

Many crystals are more complex than the two examples above. For example, a
crystal with two sites per primitive cell may host A and B atoms on the first
sublattice and B and C on the second sublattice. The number of each species
per primitive cell can again be tracked in a three dimensional space spanned
by the variables $n_{A}$, $n_{B}$ and $n_{C}$. However, not all values of
$n_{A}$, $n_{B}$ and $n_{C}$ are allowed due to the constraint of a fixed
number of sites per primitive cell and the additional constraint that $A$ and
$C$ atoms can only occupy one of the two sublattice, while $B$ can occupy both
sublattices.

The subspace of allowed values of $n_{A}$, $n_{B}$ and $n_{C}$ can be
identified by enumerating the extreme compositions. One extremum in the
subspace of allowed compositions corresponds to AC, with
$\vec{n}^{\mathsf{T}}=[1,0,1]$. Two other extrema in the subspace of allowed
compositions include the chemical formula AB with
$\vec{n}^{\mathsf{T}}=[1,1,0]$ and BC with $\vec{n}^{\mathsf{T}}=[0,1,1]$. A
fourth extremum corresponds to the chemical formula BB in which both
sublattices are occupied by B atoms, with $\vec{n}^{\mathsf{T}}=[0,2,0]$.
Figure 20(c) illustrates the extrema of allowed compositions for this
particular crystal and highlights the two-dimensional subspace of allowed
values of $n_{A}$, $n_{B}$ and $n_{C}$.

Parametric compositions can again be introduced to navigate within the two-
dimensional subspace of allowed compositions. First an origin needs to be
chosen. For example, $\vec{n}_{0}^{\mathsf{T}}=[1,0,1]$ corresponding to the
chemistry AC could be a possible origin. Two spanning vectors for the two-
dimensional subspace are $\vec{q}_{1}$ and $\vec{q}_{2}$ as illustrated in
Figure 20(c). The concentration of the compound in terms of parametric
compositions is again of the form

$\vec{n}=\vec{n}_{0}+x_{1}\vec{q}_{1}+x_{2}\vec{q}_{2}$ (107)

where $x_{1}$ and $x_{2}$ are parametric compositions.

In general, for a solid containing $s$ species, the number of each species per
primitive cell as collected in the $s$-dimensional vector, $\vec{n}$, can be
expressed in terms of parametric compositions $\vec{x}$ according to

$\vec{n}=\vec{n}_{0}+\boldsymbol{Q}\vec{x}$ (108)

where as before, $\vec{n}_{0}$ points to a chosen origin and the matrix
$\boldsymbol{Q}=[\vec{q}_{1},...,\vec{q}_{k}]$ collects as columns $k$
independent vectors that span the subspace of allowed values of $\vec{n}$,
consistent with the constraint of a fixed number of sites per primitive cell
and any additional sublattice constraints. The matrix $\boldsymbol{Q}$,
therefore, has dimensions $s\times k$, leading to $k<s$ parametric
compositions, $\vec{x}^{\mathsf{T}}=[x_{1},...,x_{k}]$.

The $k$ vectors $\vec{q}_{i}$ that span the subspace of allowed values of
$\vec{n}$ do not necessarily form an orthogonal set. It therefore becomes
useful to introduce a second set of vectors,
$[\vec{r}_{1},\dots,\vec{r}_{k}]$, that span the same subspace and that
satisfy $\vec{r}^{\mathsf{T}}_{i}\vec{q}_{j}=\delta_{i,j}$ for
$i,j=1,\dots,k$, with $\delta_{i,j}$ being the Kronecker delta. The matrix

$\boldsymbol{R}=[\vec{r}_{1},\dots,\vec{r}_{k}],$ (109)

whose transpose is the left pseudoinverse of $\boldsymbol{Q}$ (i.e.
${\boldsymbol{R}}^{\mathsf{T}}\boldsymbol{Q}=I$ where $I$ is a $k\times k$
identity matrix) makes it possible to determine the parametric compositions
$\vec{x}$ given the concentration variables per unit cell, $\vec{n}$,
according to

$\displaystyle\vec{x}={\boldsymbol{R}}^{\mathsf{T}}(\vec{n}-\vec{n}_{0})$
(110)

Since $\boldsymbol{Q}$ describes the allowed composition space, it has full
column rank and $\boldsymbol{R}^{\mathsf{T}}$ can be calculated using:

$\displaystyle{\boldsymbol{R}}^{\mathsf{T}}$
$\displaystyle=({\boldsymbol{Q}}^{\mathsf{T}}{\boldsymbol{Q}})^{-1}{\boldsymbol{Q}}^{\mathsf{T}}.$
(111)

The set of $\vec{r}_{i}$ vectors become equal to the set of $\vec{q}_{i}$
vectors when the latter form an orthonormal set.

When imposing the constraints of the crystal on diffusional flux expressions
in Appendix B, it will be convenient to utilize the orthogonal projection
operator defined as

$\mathbf{P}=\sum_{i=1}^{k}\vec{r}_{i}\vec{q}_{i}^{\mathsf{T}}=\boldsymbol{R}{\boldsymbol{Q}}^{\mathsf{T}},$
(112)

which is a $s\times s$ matrix with rank $k$. Any vector
$\vec{v}=\vec{n}-\vec{n}_{0}$ in the full $s$ dimensional composition space
will be projected onto the subspace of allowed compositions spanned by the
sets $\vec{q}_{i}$ or $\vec{r}_{i}$ when multiplied by $\mathbf{P}$. Any
vector $\vec{v}$ that is already in the space spanned by the column vectors of
$\boldsymbol{Q}$, (i.e. $\vec{v}=\boldsymbol{Q}\vec{x}$) is unaffected by
$\mathbf{P}$ (i.e. $\mathbf{P}\vec{v}=\vec{v}$), which can be verified by
substituting Eq. 111 into Eq. 112 and using the fact that
$({\boldsymbol{Q}}^{\mathsf{T}}{\boldsymbol{Q}})^{-1}$ is a symmetric matrix
and is equal to its transpose.

In general, it is most convenient to choose the length of each $\vec{q}_{i}$
such that it corresponds to one unit exchange of atoms per unit cell, i.e. the
resulting vector is of the form $[0,...,1,...,-1,...0]$. In the examples
treated so far, this has always been the case for the chosen $\vec{q}$
vectors. For crystals with more than one sublattice per primitive unit cell,
this will generally lead to parametric compositions that are not restricted to
the interval [0,1]. To illustrate this, consider the example of a crystal with
two sublattices in which A and B can occupy the first sublattice and A, B and
C can occupy the second sublattice. The composition space for this example is
illustrated in Figure 21.

Figure 21: The space of allowed compositions for a crystal with two
sublattices in which A and B can occupy the first sublattice and A, B and C
can occupy the second sublattice. A choice of composition axes
$\vec{q}^{\mathsf{T}}_{1}=[-1,1,0]$ and $\vec{q}^{\mathsf{T}}_{2}=[-1,0,1]$
with origin $\vec{n}^{\mathsf{T}}_{0}=[2,0,0]$ is shown along with the
stoichiometic compositions at the extrema of the allowed composition space:
AA, AC, BB and BC.

As the origin of the parametric composition space, one possible choice is pure
A, in which both sublattices are completely occupied by A atoms, i.e.
$\vec{n}^{\mathsf{T}}_{0}=[2,0,0]$. The subspace of allowed compositions can
be spanned by $\vec{q}^{\mathsf{T}}_{1}=[-1,1,0]$ and
$\vec{q}^{\mathsf{T}}_{2}=[-1,0,1]$ as illustrated in Figure 21. Notice that
$\vec{q}_{1}$ does not extend all the way from the AA composition to the BB
composition, but has a length that is half that distance. This is to ensure
that moving along that axis by a unit distance of $\vec{q}_{1}$ corresponds to
an exchange of one A atom with one B atom. By using this particular vector,
the parametric composition corresponding to this axis, $x_{1}$, ranges from 0
to 2, as two A atoms must be replaced by 2 B atoms to go from the AA
composition to the BB composition. This convention ensures that the exchange
chemical potentials that are conjugate to the resulting parametric
compositions are quantities of an atom as opposed to that of multiple atoms.

###  A.2 Examples of semi-grand canonical potentials

It is instructive to inspect the form of semi-grand canonical potentials and
the exchange chemical potentials $\vec{\tilde{\mu}}$ for specific examples. As
defined in Section 2.2, the semi-grand canonical potential takes the form

$\phi=g-\vec{\tilde{\mu}}^{\mathsf{T}}\vec{x}=\vec{\mu}^{\mathsf{T}}\vec{n}_{0}$
(113)

where $\phi=\Phi/N_{u}$ and $g=G/N_{u}$ and where
$\vec{\tilde{\mu}}=\boldsymbol{Q}^{\mathsf{T}}\vec{\mu}$. Another important
thermodynamic quantity is the semi-grand canonical enthalpy defined as

$\omega=e-\vec{\tilde{\mu}}^{\mathsf{T}}\vec{x}$ (114)

where $e$ is the average energy per primitive cell. The next sections
illustrate these potentials for concrete examples.

####  A.2.1 Binary and ternary alloys on a single sublattice

For the simple binary alloy with the choice of origin as shown in Figure
20(a),

$\displaystyle\vec{n}_{0}$ $\displaystyle=\left[\begin{array}[]{c}1\\\
0\end{array}\right]$ $\displaystyle\vec{q}_{1}$
$\displaystyle=\left[\begin{array}[]{c}-1\\\ 1\end{array}\right]$ (119)

Since the subspace of allowed compositions is one dimensional,
$\boldsymbol{Q}$ is a $2\times 1$ matrix

$\displaystyle\boldsymbol{Q}$ $\displaystyle=\left[\begin{array}[]{c}-1\\\
1\end{array}\right],$ (122)

According to Equation 10, the exchange chemical potential is the equal to

$\displaystyle\tilde{\mu}_{1}$ $\displaystyle=\mu_{B}-\mu_{A},$ (123)

and is the intensive variable that is conjugate to the parametric composition
$x_{1}$. The semi-grand canonical free energy becomes equal to the chemical
potential of A since, using Equation 12

$\displaystyle\phi=\vec{\mu}^{\mathsf{T}}\vec{n}_{0}=[\mu_{A},\mu_{B}]\left[\begin{array}[]{c}1\\\
0\end{array}\right]=\mu_{A}$ (126)

In semi-grand canonical Monte Carlo simulations, $\tilde{\mu}_{1}$ together
with temperature T are the input variables, while quantities such as the
average parametric composition $\langle x_{1}\rangle$ and the average semi-
grand canonical enthalpy, $\langle\omega\rangle=\langle
e-\tilde{\mu}_{1}x_{1}\rangle$ are outputs. Since $\tilde{\mu}_{1}$ is a
difference in the chemical potentials of B and A, it is necessary to know both
$\tilde{\mu}_{1}$ and the semi-grand canonical potential $\phi=\mu_{A}$ in
order to determine the individual chemical potentials $\mu_{A}$ and $\mu_{B}$.
This requires a free energy integration step described in Section 6.

Similar results are obtained for the ternary A-B-C alloy with origin and
parametric composition axes as chosen in Figure 20(b). Now there are two
exchange chemical potentials, with

$\displaystyle\tilde{\mu}_{1}$ $\displaystyle=\mu_{B}-\mu_{A},$ (127)
$\displaystyle\tilde{\mu}_{2}$ $\displaystyle=\mu_{C}-\mu_{A}$ (128)

while the semi-grand canonical potential becomes

$\phi=g-\tilde{\mu}_{1}x_{1}-\tilde{\mu}_{2}x_{2}=\mu_{A}$ (129)

and the semi-grand canonical enthalpy (at zero pressure) per primitive cell
becomes

$\omega=e-\tilde{\mu}_{1}x_{1}-\tilde{\mu}_{2}x_{2}$ (130)

####  A.2.2 Two binary sublattices sharing a common species

Consider the example of Figure 20(c) again for a crystal with two sites in the
primitive cell, with the first sublattice hosting A and B and the second
sublattice hosting B and C. For the choice of origin in Figure 20(c) and the
choice of parametric composition axes

$\displaystyle\vec{n}_{0}$ $\displaystyle=\left[\begin{array}[]{c}1\\\ 0\\\
1\end{array}\right]$ $\displaystyle\vec{q}_{1}$
$\displaystyle=\left[\begin{array}[]{c}0\\\ 1\\\ -1\end{array}\right]$
$\displaystyle\vec{q}_{2}$ $\displaystyle=\left[\begin{array}[]{c}-1\\\ 1\\\
0\end{array}\right]$ (140)

The two-dimensional composition subspace leads to a $3\times 2$
$\boldsymbol{Q}$ matrix

$\displaystyle\boldsymbol{Q}$ $\displaystyle=\left[\begin{array}[]{cc}0&-1\\\
1&1\\\ -1&0\end{array}\right],$ (144)

This matrix yields for the exchange chemical potentials

$\displaystyle\tilde{\mu}_{1}$ $\displaystyle=\mu_{B}-\mu_{C},$ (145)
$\displaystyle\tilde{\mu}_{2}$ $\displaystyle=\mu_{B}-\mu_{A}$ (146)

while the semi-grand canonical potential becomes

$\phi=g-\tilde{\mu}_{1}x_{1}-\tilde{\mu}_{2}x_{2}=\mu_{A}+\mu_{C}$ (147)

####  A.2.3 Binary and ternary sublattices sharing multiple common species

As a final example, consider the more complex crystal with two sublattices per
primitive cell with the first sublattice hosting A and B and the second
sublattice hosting A, B and C. The composition space for this example is shown
in Figure 21. In Figure 21, the origin and parametriccomposition axes are
chosen as

$\displaystyle\vec{n}_{0}$ $\displaystyle=\left[\begin{array}[]{c}2\\\ 0\\\
0\end{array}\right]$ $\displaystyle\vec{q}_{1}$
$\displaystyle=\left[\begin{array}[]{c}-1\\\ 1\\\ 0\end{array}\right]$
$\displaystyle\vec{q}_{2}$ $\displaystyle=\left[\begin{array}[]{c}-1\\\ 0\\\
1\end{array}\right]$ (157)

This leads to

$\displaystyle\boldsymbol{Q}$ $\displaystyle=\left[\begin{array}[]{cc}-1&-1\\\
1&0\\\ 0&1\end{array}\right],$ (161)

The exchange chemical potentials then become

$\displaystyle\tilde{\mu}_{1}$ $\displaystyle=\mu_{B}-\mu_{A},$ (162)
$\displaystyle\tilde{\mu}_{2}$ $\displaystyle=\mu_{C}-\mu_{A}$ (163)

The semi-grand canonical potential becomes

$\phi=g-\tilde{\mu}_{1}x_{1}-\tilde{\mu}_{2}x_{2}=2\mu_{A}$ (164)

The factor of two in front of the chemical potential of A emerges since the
primitive unit cell contains two sites, which are both occupied by A atoms at
the origin of the parametric composition space.

##  Appendix B Fluxes and driving forces for diffusion in crystals

###  B.1 Crystallographic constraints on flux expressions

Diffusion within a crystal that occurs far away from extended defects proceeds
under crystallographic constraints that affect the form of the flux
expressions, thermodynamic driving forces and the Onsager transport
coefficients. Indeed, diffusion within single crystal regions of a solid can
only lead to a spatial redistribution of different chemical species, but
cannot result in the extension of the crystal. It, therefore, occurs at
constant $N_{u}$, the number of unit cells. This imposes constraints on
diffusional fluxes and on the form of diffusional potentials (driving forces).
To establish these constraints, it is necessary to revisit the composition
variables of the previous section. For a crystal having a fixed number of unit
cells, $N_{u}$, the concentrations of $s$ species per unit cell,
$\vec{n}^{\mathsf{T}}$, reside in a lower dimensional subspace spanned by a
set of vectors $[\vec{q}_{1},\dots,\vec{q}_{k}]$. Parametric compositions
$\vec{x}$ serve as coordinates in this subspace. Diffusion can only result in
composition changes of $\vec{n}$ that stay in the subspace spanned by
$[\vec{q}_{1},\dots,\vec{q}_{k}]$. This means that any changes in $\vec{n}$
that leave this subspace are forbidden.

The subspace that is orthogonal to $[\vec{q}_{1},\dots,\vec{q}_{k}]$ has a
dimension $s-k$ and can be spanned by a set of orthogonal vectors
$[\vec{t}_{k+1},\dots,\vec{t}_{s}]$. Such a set of vectors can always be found
by finding the null space of $\boldsymbol{Q}=[\vec{q}_{1},\dots,\vec{q}_{k}]$.
The spanning vectors of the subspace orthogonal to $\boldsymbol{Q}$ can be
collected in the $s\times(s-k)$ matrix

$\boldsymbol{T}=[\vec{t}_{k+1},\dots,\vec{t}_{s}].$ (165)

The constraint of a constant number of unit cells $N_{u}$ on the composition
variable $\vec{n}$ can be expressed as

$\boldsymbol{T}^{\mathsf{T}}(\vec{n}-\vec{n}_{0})=\vec{0}$ (166)

which ensures that variations in the concentration $\vec{n}$ do not stray into
the subspace spanned by $[\vec{t}_{k+1},\dots,\vec{t}_{s}]$. Equation 166 form
a set of $s-k$ linear constraints on the concentration variables
$\vec{n}^{\mathsf{T}}=[n_{A},n_{B},\dots]$.

Constraints similar to Eq. 166 also apply to the diffusional fluxes of the
different species within a perfect crystal having a fixed number of unit cells
$N_{u}$. This is a consequence of the conservation of particles equation, that
relates each concentration variable $n_{m}$ for species $m$ to a corresponding
flux $J_{m}$ according to

$\frac{\partial(n_{m}/v_{u})}{\partial t}=-\nabla{J_{m}}$ (167)

where $v_{u}$ represents the volume of the unit cell of the crystal. Due to
the linearity of time and space derivatives, the constraints of Eq. 166 that
apply to $\vec{n}$ transfer to the fluxes $\vec{J}$ and can be expressed as

$\boldsymbol{T}^{\mathsf{T}}\vec{J}=\vec{0}$ (168)

where $\vec{J}$ collects the fluxes of the $s$ species of the crystal. There
are therefore $s-k$ linear constraints on the fluxes of each element within
the crystal when holding the number of unit cells constant.

The constraints on the fluxes also affect the Onsager transport coefficients
that appear in the flux expressions

$\displaystyle\vec{J}=-{\boldsymbol{L}}\nabla\vec{\mu},$ (169)

Since the constraints of Eq. 168 hold independent of the values of the driving
forces, $\nabla\vec{\mu}$ appearing in Eq. 169, the following must hold for
the Onsager matrix

${\boldsymbol{T}}^{\mathsf{T}}{\boldsymbol{L}}=\mathbf{0}$ (170)

where the $\mathbf{0}$ represents a $(s-k)\times s$ matrix of zeros. These
equations, therefore, constitute a set of $(s-k)\times s$ linear relationships
on the Onsager transport coefficients that emerge due to the constraint of a
fixed number of unit cells. Due to the Onsager reciprocity relationships
(${\boldsymbol{L}}={\boldsymbol{L}}^{T}$), the above equation can also be
rewritten as

${\boldsymbol{L}}{\boldsymbol{T}}=\mathbf{0}$ (171)

where the $\mathbf{0}$ now represents a $s\times(s-k)$ matrix of zeros.

The linear relationships between the Onsager transport coefficients can be
accounted for by rewriting the flux expressions 169. To this end, it is useful
to rely on projection operators for the subspace of allowed compositions when
fixing the number of unit cells, which was introduced in Appendix A, Eq. 112,
as

$\mathbf{P}_{c}=\boldsymbol{R}{\boldsymbol{Q}}^{\mathsf{T}}$ (172)

and for the subspace that is orthogonal to it, which can be expressed in terms
of the matrix $\boldsymbol{T}$ as

$\mathbf{P}_{o}=\sum_{i=k+1}^{s}\vec{t}_{i}{\vec{t}_{i}}^{\mathsf{T}}=\boldsymbol{T}{\boldsymbol{T}}^{\mathsf{T}}$
(173)

The projection operators, $\mathbf{P}_{c}$ projecting onto the allowed
composition space and $\mathbf{P}_{o}$ projecting onto its kernel, are
complementary, and sum to the identity matrix

$\mathbf{I}=\mathbf{P}_{c}+\mathbf{P}_{o}=\boldsymbol{R}{\boldsymbol{Q}}^{\mathsf{T}}+\boldsymbol{T}{\boldsymbol{T}}^{\mathsf{T}}.$
(174)

The identity operator, Eq. 174, can be inserted between the Onsager matrix and
the gradients in chemical potentials in the flux expressions, Eq. 169, to
yield

$\vec{J}=-{\boldsymbol{L}}\boldsymbol{R}\nabla({\boldsymbol{Q}}^{\mathsf{T}}\vec{\mu})$
(175)

where Eq. 171 was used. By further multiplying both sides by
${\boldsymbol{R}}^{\mathsf{T}}$, the flux expressions can be rewritten as

$\vec{\tilde{J}}=-\tilde{{\boldsymbol{L}}}\nabla\vec{\tilde{\mu}}$ (176)

where the $k$ independent exchange fluxes are defined as

$\vec{\tilde{J}}={\boldsymbol{R}}^{\mathsf{T}}\vec{J}$ (177)

and where

$\tilde{{\boldsymbol{L}}}={\boldsymbol{R}}^{\mathsf{T}}{\boldsymbol{L}}\boldsymbol{R}$
(178)

is a $k\times k$ Onsager transport coefficient matrix that is consistent with
the constraints of a constant number of unit cells. The
$\vec{\tilde{\mu}}={\boldsymbol{Q}}^{\mathsf{T}}\vec{\mu}$ are the exchange
chemical potentials defined in Appendix A. These expressions describe the
fluxes, transport coefficients and thermodynamic driving forces for diffusion
in a crystal with a fixed number of unit cells $N_{u}$.

The transformed fluxes, $\vec{\tilde{J}}$ reside in the dual space of the
exchange chemical potentials, $\vec{\tilde{\mu}}$.

###  B.2 Examples of flux expressions

####  B.2.1 Diffusion in a binary alloy with a single sublattice by direct
exchange

Consider the simple binary alloy in which diffusion occurs through direct
exchanges between A and B atoms on a single sublattice. As detailed in A.2.1,
for such a system

$\displaystyle\boldsymbol{Q}$ $\displaystyle=\left[\begin{array}[]{c}-1\\\
1\end{array}\right],$ (181)

is consistent with the composition formula A${}_{1-{x_{1}}}$B${}_{x_{1}}$. The
transpose of the left pseudo inverse of $\boldsymbol{Q}$ is

$\displaystyle\boldsymbol{R}$
$\displaystyle=\frac{1}{2}\left[\begin{array}[]{c}-1\\\ 1\end{array}\right].$
(184)

The space that is orthogonal to the space of allowed compositions is the
column vector space

$\displaystyle\boldsymbol{T}$
$\displaystyle=\frac{1}{\sqrt{2}}\left[\begin{array}[]{c}1\\\
1\end{array}\right].$ (187)

Hence according to Eq. 168, the crystallographic constraints require that

$\frac{1}{\sqrt{2}}[1,1]\left[\begin{array}[]{c}J_{A}\\\
J_{B}\end{array}\right]=\frac{1}{\sqrt{2}}(J_{A}+J_{B})=0,$ (188)

which imposes the constraint that a flux in A atoms must be compensated by an
opposite flux of B atoms (i.e. $J_{A}=-J_{B}$) in order to preserve crystal
sites.

For a binary system with direct exchange hops, the flux expressions take the
form

$\displaystyle\left[\begin{array}[]{c}J_{A}\\\ J_{B}\end{array}\right]$
$\displaystyle=-\left[\begin{array}[]{cc}L_{AA}&L_{AB}\\\
L_{AB}&L_{BB}\end{array}\right]\left[\begin{array}[]{c}\nabla\mu_{A}\\\
\nabla\mu_{B}\end{array}\right],$ (195)

Projecting the flux expressions to the subspace of allowed concentrations
consistent with a fixed number of unit cells yields

$\tilde{J}_{1}=\tilde{L}\nabla\tilde{\mu}_{1}$ (196)

where according to Eq. 178,

$\displaystyle\tilde{L}=\left[\begin{array}[]{cc}-1/2&1/2\end{array}\right]\left[\begin{array}[]{cc}L_{AA}&L_{AB}\\\
L_{AB}&L_{BB}\end{array}\right]\left[\begin{array}[]{c}-1/2\\\
1/2\end{array}\right]$ (202)

which yields

$\tilde{L}=\frac{1}{4}(L_{AA}+L_{BB}-2L_{AB})$ (203)

This expression can be further simplified by exploiting the constraints on the
${\boldsymbol{L}}$ matrix as codified by Eq. 171, which for this example can
be expressed explicitly as

$\displaystyle\left[\begin{array}[]{cc}L_{AA}&L_{AB}\\\
L_{AB}&L_{BB}\end{array}\right]\left[\begin{array}[]{c}1/\sqrt{2}\\\
1/\sqrt{2}\end{array}\right]=\left[\begin{array}[]{c}0\\\ 0\end{array}\right]$
(210)

These two linear equations make it possible to eliminate two of the three
Onsager coefficients such that Eq. 203 can be rewritten as

$\tilde{L}=L_{BB}$ (211)

The exchange chemical potential is equal to

$\displaystyle\tilde{\mu}_{1}$
$\displaystyle=\left[\begin{array}[]{cc}-1&1\end{array}\right]\left[\begin{array}[]{c}\mu_{A}\\\
\mu_{B}\end{array}\right]=\mu_{B}-\mu_{A},$ (215)

while the exchange flux is

$\tilde{J}_{1}=\left[\begin{array}[]{cc}-1/2&1/2\end{array}\right]\left[\begin{array}[]{c}J_{A}\\\
J_{B}\end{array}\right]=\frac{1}{2}(J_{B}-J_{A})=J_{B}$ (216)

where the third equality follows by using Eq. 188. The flux expression
projected into the constant number of crystal unit cells subspace can
therefore be expressed as

$J_{B}=-L_{BB}\nabla(\mu_{B}-\mu_{A})$ (217)

####  B.2.2 Diffusion in a binary alloy with a single sublattice mediated by a
vacancy mechanism

Figure 22: The space of allowed compositions for a crystal with a single
sublattice that can be occupied by A atoms, B atoms and vacancies (Va). A
choice of composition axes $\vec{q}^{\mathsf{T}}_{1}=[1,0,-1]$ and
$\vec{q}^{\mathsf{T}}_{2}=[1,0,-1]$ with origin
$\vec{n}^{\mathsf{T}}_{0}=[0,0,1]$ is shown along with the stoichiometic
compositions at the extrema of the allowed composition space: A, B and Va.
This is consistent with the composition formula
A${}_{x_{1}}$B${}_{x_{2}}$Va${}_{1-{x_{1}}-{x_{2}}}$.

A common scenario is diffusion in a binary A-B alloy that is mediated by a
vacancy. The crystal contains three species, A, B and vacancies denoted Va and
the concentration per unit cell $\vec{n}^{\mathsf{T}}=[n_{A},n_{B},n_{Va}]$.
For the purpose of setting up flux expressions, it is convenient to choose as
the origin for the parametric composition variables the crystal in which all
sites are fully vacant, i.e.

$\displaystyle\vec{n}_{0}$ $\displaystyle=\left[\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right]$ $\displaystyle\vec{q}_{1}$
$\displaystyle=\left[\begin{array}[]{c}1\\\ 0\\\ -1\end{array}\right]$
$\displaystyle\vec{q}_{2}$ $\displaystyle=\left[\begin{array}[]{c}0\\\ 1\\\
-1\end{array}\right].$ (227)

The space of allowed compositions consistent with the constraints of the
crystal, as illustrated in Figure 22, is

$\displaystyle\boldsymbol{Q}$ $\displaystyle=\left[\begin{array}[]{cc}1&0\\\
0&1\\\ -1&-1\end{array}\right],$ $\displaystyle\boldsymbol{R}$
$\displaystyle=\left[\begin{array}[]{cc}2/3&-1/3\\\ -1/3&2/3\\\
-1/3&-1/3\end{array}\right]$ (234)

The space that is orthogonal to space of allowed compositions is
$\vec{t}^{\mathsf{T}}_{3}=1/\sqrt{3}[1,1,1]$, such that

$\displaystyle\boldsymbol{T}$
$\displaystyle=\left[\begin{array}[]{c}1/\sqrt{3}\\\ 1/\sqrt{3}\\\
1/\sqrt{3}\end{array}\right].$ (238)

According to Eq. 168, the crystallographic constraints require that

$\frac{1}{\sqrt{3}}\left[\begin{array}[]{ccc}1&1&1\end{array}\right]\left[\begin{array}[]{c}J_{A}\\\
J_{B}\\\ J_{Va}\end{array}\right]=\frac{1}{\sqrt{3}}(J_{A}+J_{B}+J_{Va})=0,$
(239)

The flux expressions take the form

$\displaystyle\left[\begin{array}[]{c}J_{A}\\\ J_{B}\\\
J_{Va}\end{array}\right]$
$\displaystyle=-\left[\begin{array}[]{ccc}L_{AA}&L_{AB}&L_{AVa}\\\
L_{AB}&L_{BB}&L_{BVa}\\\
L_{AVa}&L_{BVa}&L_{VaVa}\end{array}\right]\left[\begin{array}[]{c}\nabla\mu_{A}\\\
\nabla\mu_{B}\\\ \nabla\mu_{Va}\end{array}\right],$ (249)

Linear constraints on the Onsager coefficients due to Eq. 171 makes it
possible to eliminate the Onsager coefficients involving vacancies (i.e.
$L_{AVa}=-(L_{AA}+L_{AB})$, $L_{BVa}=-(L_{AB}+L_{BB})$ and
$L_{VV}=-(L_{AVa}+L_{BVa})$). Projecting the Onsager coefficient matrix to the
fixed crystal subspace according to Eq. 178 leads to

$\displaystyle\tilde{{\boldsymbol{L}}}$
$\displaystyle=\left[\begin{array}[]{cc}L_{AA}&L_{AB}\\\
L_{AB}&L_{BB}\end{array}\right],$ (252)

where the above linear relationships between the Onsager coefficients were
used.

Projecting the fluxes according to Eq. 177 yields

$\displaystyle\left[\begin{array}[]{c}\tilde{J}_{1}\\\
\tilde{J}_{2}\end{array}\right]$
$\displaystyle=\frac{1}{3}\left[\begin{array}[]{ccc}2&-1&-1\\\
-1&2&-1\end{array}\right]\left[\begin{array}[]{c}J_{A}\\\ J_{B}\\\
J_{Va}\end{array}\right]=\left[\begin{array}[]{c}J_{A}\\\
J_{B}\end{array}\right]$ (262)

where the constraint on the fluxes, Eq. 239, was used to eliminate the vacancy
flux. The diffusion potentials, which are the exchange chemical potentials,
become

$\displaystyle\left[\begin{array}[]{c}\tilde{\mu}_{1}\\\
\tilde{\mu}_{2}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{ccc}1&0&-1\\\
0&1&-1\end{array}\right]\left[\begin{array}[]{c}\mu_{A}\\\ \mu_{B}\\\
\mu_{Va}\end{array}\right]=\left[\begin{array}[]{c}\mu_{A}-\mu_{Va}\\\
\mu_{B}-\mu_{Va}\end{array}\right]$ (272)

The resulting flux expressions in the fixed crystal frame becomes

$\displaystyle\left[\begin{array}[]{c}J_{A}\\\ J_{B}\end{array}\right]$
$\displaystyle=-\left[\begin{array}[]{cc}L_{AA}&L_{AB}\\\
L_{AB}&L_{BB}\end{array}\right]\left[\begin{array}[]{c}\nabla(\mu_{A}-\mu_{Va})\\\
\nabla(\mu_{B}-\mu_{Va})\end{array}\right],$ (279)

####  B.2.3 Interstitial diffusion with a fixed host lattice

Figure 23: The space of allowed compositions for a crystal with one fixed host
sublattice occupied by A atoms, and an interstitial sublattice occupied by B
atoms and vacancies (Va). A choice of composition axis
$\vec{q}^{\mathsf{T}}_{1}=[0,1,-1]$ with origin
$\vec{n}^{\mathsf{T}}_{0}=[1,0,1]$ is shown along with the stoichiometic
compositions at the extrema of the allowed composition space: AVa and AB,
along with compositions A, B and Va as guides for the eye. This is consistent
with the composition formula AB${}_{x_{1}}$Va${}_{1-{x_{1}}}$.

Another common system is one in which there is a fixed sublattice of A atoms
and an interstitial sublattice that allows vacancies and B atoms. This crystal
also contains three species, A, B and Va, and the concentration per unit cell
can be expressed as the vector $\vec{n}^{\mathsf{T}}=[n_{A},n_{B},n_{Va}]$.
Clearly, this system behaves like a binary alloy on the interstitial
sublattice, but with a fixed concentration of A atoms. This makes for a useful
system to use as an exercise because the results can be easily checked against
intuition.

The choice of composition origin and axis vectors consistent with the
composition formula AB${}_{{x_{1}}}$Va${}_{1-{x_{1}}}$ is

$\displaystyle\vec{n}_{0}$ $\displaystyle=\left[\begin{array}[]{c}1\\\ 0\\\
1\end{array}\right]$ $\displaystyle\vec{q}_{1}$
$\displaystyle=\left[\begin{array}[]{c}0\\\ 1\\\ -1\end{array}\right],$ (286)

as illustrated in Figure 23.

The equations relating parametric composition, $\vec{x}$, and concentration
per unit cell, $\vec{n}$, (Eqs. 2, 110, and 111) are

$\displaystyle\vec{n}$ $\displaystyle=\vec{n}_{0}+\boldsymbol{Q}\vec{x},$
$\displaystyle\vec{x}$
$\displaystyle={\boldsymbol{R}}^{\mathsf{T}}(\vec{n}-\vec{n}_{0}),$
$\displaystyle{\boldsymbol{R}}^{\mathsf{T}}$
$\displaystyle=({\boldsymbol{Q}}^{\mathsf{T}}{\boldsymbol{Q}})^{-1}{\boldsymbol{Q}}^{\mathsf{T}},$

which for this system and choice of composition axes take the values

$\displaystyle\boldsymbol{Q}$ $\displaystyle=\left[\begin{array}[]{c}0\\\ 1\\\
-1\end{array}\right]$ $\displaystyle\boldsymbol{R}$
$\displaystyle=\frac{1}{2}\left[\begin{array}[]{c}0\\\ 1\\\ -1\\\
\end{array}\right].$ (293)

The null space of Q is the space of disallowed compositions,

$\displaystyle\boldsymbol{T}$ $\displaystyle=\left[\begin{array}[]{cc}1&0\\\
0&\frac{1}{\sqrt{2}}\\\ 0&\frac{1}{\sqrt{2}}\end{array}\right],$ (297)

which can be used to define the constraints on the the concentration per unit
cell (Eq. 166), flux (Eq. 168), and Onsager coefficients (Eq. 171) according
to

$\displaystyle\boldsymbol{T}^{\mathsf{T}}(\vec{n}-\vec{n}_{0})$
$\displaystyle=\vec{0},$ $\displaystyle\boldsymbol{T}^{\mathsf{T}}\vec{J}$
$\displaystyle=\vec{0},$ $\displaystyle{\boldsymbol{L}}{\boldsymbol{T}}$
$\displaystyle=\mathbf{0}.$

For this system and choice of composition axes, the concentration constraints
are

$\displaystyle\left[\begin{array}[]{ccc}1&0&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right]\left(\left[\begin{array}[]{c}n_{A}\\\
n_{B}\\\ n_{Va}\end{array}\right]-\left[\begin{array}[]{c}1\\\ 0\\\
1\end{array}\right]\right)$ $\displaystyle=\left[\begin{array}[]{c}0\\\
0\end{array}\right],$ (308)

which gives the expected result

$\displaystyle n_{A}$ $\displaystyle=1,$ (309) $\displaystyle n_{B}+n_{Va}$
$\displaystyle=1,$

The flux constraints are

$\displaystyle\left[\begin{array}[]{ccc}1&0&0\\\
0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array}\right]\left[\begin{array}[]{c}J_{A}\\\
J_{B}\\\ J_{Va}\end{array}\right]$ $\displaystyle=\left[\begin{array}[]{c}0\\\
0\end{array}\right],$ (317)

which gives the expected result

$\displaystyle J_{A}$ $\displaystyle=0,$ (318) $\displaystyle J_{B}$
$\displaystyle=-J_{Va},$

The Onsager coefficients constraints are

$\displaystyle\left[\begin{array}[]{ccc}L_{AA}&L_{AB}&L_{AVa}\\\
L_{AB}&L_{BB}&L_{BVa}\\\
L_{AVa}&L_{BVa}&L_{VaVa}\end{array}\right]\left[\begin{array}[]{cc}1&0\\\
0&\frac{1}{\sqrt{2}}\\\ 0&\frac{1}{\sqrt{2}}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}0&0\\\ 0&0\\\ 0&0\end{array}\right],$
(328)

which gives the obvious result

$\displaystyle L_{AA}=L_{AB}=L_{AVa}=L_{AB}$ $\displaystyle=0,$ (329)

and also the expected result

$\displaystyle L_{BB}=L_{VaVa}=-L_{BVa}.$ (330)

The choice of composition space results in exchange chemical potentials (Eq.
10) defined as

$\displaystyle\vec{\tilde{\mu}}=\boldsymbol{Q}^{T}\vec{\mu}$

which take the values

$\displaystyle\left[\begin{array}[]{c}\tilde{\mu}_{1}\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{c}0\\\ 1\\\
-1\end{array}\right]\left[\begin{array}[]{c}\tilde{\mu}_{A}\\\
\tilde{\mu}_{B}\\\ \tilde{\mu}_{Va}\end{array}\right],$ (338)

or equivalently

$\displaystyle\tilde{\mu}_{1}$ $\displaystyle=\mu_{B}-\mu_{Va}.$ (339)

The flux expressions and Onsager coefficients can be projected onto the
allowed composition space (Eqs. 176, 177 and 178),

$\displaystyle\vec{\tilde{J}}$
$\displaystyle=-\tilde{{\boldsymbol{L}}}\nabla\vec{\tilde{\mu}},$
$\displaystyle\vec{\tilde{J}}$
$\displaystyle={\boldsymbol{R}}^{\mathsf{T}}\vec{J},$
$\displaystyle\tilde{{\boldsymbol{L}}}$
$\displaystyle={\boldsymbol{R}}^{\mathsf{T}}{\boldsymbol{L}}\boldsymbol{R}.$

which for this system and choice of composition axes gives

$\displaystyle\left[\begin{array}[]{c}\tilde{J}_{1}\end{array}\right]$
$\displaystyle=\frac{1}{2}\left[\begin{array}[]{ccc}0&1&-1\end{array}\right]\left[\begin{array}[]{c}J_{A}\\\
J_{B}\\\ J_{Va}\end{array}\right],$ (345)

or equivalently

$\displaystyle\tilde{J}_{1}$
$\displaystyle=\frac{1}{2}\left(J_{B}-J_{Va}\right),$ (346)

and

$\displaystyle\left[\begin{array}[]{c}\tilde{L}_{11}\end{array}\right]$
$\displaystyle=\frac{1}{4}\left[\begin{array}[]{ccc}0&1&-1\end{array}\right]\left[\begin{array}[]{ccc}L_{AA}&L_{AB}&L_{AVa}\\\
L_{AB}&L_{BB}&L_{BVa}\\\
L_{AVa}&L_{BVa}&L_{VaVa}\end{array}\right]\left[\begin{array}[]{c}0\\\ 1\\\
-1\end{array}\right],$ (355)

or equivalently

$\displaystyle\tilde{L}_{11}$
$\displaystyle=\frac{1}{4}\left(L_{BB}+L_{VaVa}-2L_{BVa}\right).$ (356)

The exchange chemical potentials, projected fluxes, and projected Onsager
coefficients can be combined with the constraints found previously to yield
the simplified one-dimensional results

$\displaystyle\tilde{J}$ $\displaystyle=-\tilde{L}\nabla\tilde{\mu}$ (357)
$\displaystyle\tilde{J}$ $\displaystyle=\tilde{J}_{1}=J_{B},$
$\displaystyle\tilde{L}$ $\displaystyle=\tilde{L}_{11}=L_{BB}.$
$\displaystyle\tilde{\mu}$ $\displaystyle=\tilde{\mu}_{1}=\mu_{B}-\mu_{Va}.$

## References

  * [1] J. W. Cahn, On spinodal decomposition, Acta Metallurgica 9 (9) (1961) 795–801.
  * [2] J. E. Hilliard, Spinodal decomposition, Phase Transformation 497 (1970).
  * [3] S. M. Allen, J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (6) (1979) 1085–1095.
  * [4] P. W. Voorhees, Ostwald ripening of two-phase mixtures, Annual Review of Materials Science 22 (1) (1992) 197–215.
  * [5] S. M. Allen, R. W. Balluffi, W. C. Carter, Kinetics of materials, John Wiley & Sons, 2005.
  * [6] L.-Q. Chen, Phase-field models for microstructure evolution, Annual review of materials research 32 (1) (2002) 113–140.
  * [7] A. Van der Ven, J. C. Thomas, B. Puchala, A. R. Natarajan, First-principles statistical mechanics of multicomponent crystals, Annual Review of Materials Research 48 (2018) 27–55.
  * [8] G. H. Teichert, A. R. Natarajan, A. Van der Ven, K. Garikipati, Machine learning materials physics: Integrable deep neural networks enable scale bridging by learning free energy functions, Computer Methods in Applied Mechanics and Engineering 353 (2019) 201–216.
  * [9] D. De Fontaine, Cluster approach to order-disorder transformations in alloys, in: Solid state physics, Vol. 47, Elsevier, 1994, pp. 33–176.
  * [10] G. Ceder, A derivation of the Ising model for the computation of phase diagrams, Computational Materials Science 1 (2) (1993) 144–150.
  * [11] A. Van De Walle, G. Ceder, The effect of lattice vibrations on substitutional alloy thermodynamics, Reviews of Modern Physics 74 (1) (2002) 11.
  * [12] A. Van der Ven, Z. Deng, S. Banerjee, S. P. Ong, Rechargeable alkali-ion battery materials: theory and computation, Chemical Reviews 120 (14) (2020) 6977–7019.
  * [13] D. Landau, K. Binder, A guide to Monte Carlo simulations in statistical physics, Cambridge University Press, 2021.
  * [14] C. M. Bishop, N. M. Nasrabadi, Pattern recognition and machine learning, Vol. 4, Springer, 2006.
  * [15] C. Wolverton, A. Zunger, Ni-Au: A testing ground for theories of phase stability, Computational materials science 8 (1-2) (1997) 107–121.
  * [16] V. Ozoliņš, C. Wolverton, A. Zunger, Cu-Au, Ag-Au, Cu-Ag, and Ni-Au intermetallics: First-principles study of temperature-composition phase diagrams and structures, Physical Review B 57 (11) (1998) 6427.
  * [17] C. Wolverton, A. Zunger, Cation and vacancy ordering in LixCoO2, Physical Review B 57 (4) (1998) 2242.
  * [18] A. Kohan, P. Tepesch, G. Ceder, C. Wolverton, Computation of alloy phase diagrams at low temperatures, Computational materials science 9 (3-4) (1998) 389–396.
  * [19] C. Wolverton, V. Ozolins, A. Zunger, Short-range-order types in binary alloys: a reflection of coherent phase stability, Journal of Physics: Condensed Matter 12 (12) (2000) 2749.
  * [20] C. Wolverton, First-principles theory of 250 000-atom coherent alloy microstructure, Modelling and Simulation in Materials Science and Engineering 8 (3) (2000) 323.
  * [21] S. Barabash, V. Ozolins, C. Wolverton, First-principles theory of competing order types, phase separation, and phonon spectra in thermoelectric AgPbmSbTem+2 alloys, Physical Review Letters 101 (15) (2008) 155704.
  * [22] J. Chan, J. Z. Liu, H. Raebiger, S. Lany, A. Zunger, Relative stability, electronic structure, and magnetism of MnN and (Ga, Mn)N alloys, Physical Review B 78 (18) (2008) 184109.
  * [23] S. Hao, Z. Lu, C. Wolverton, Quaternary phase diagrams of spinel Li${}_{y}\square_{1-y}$MnxNi2-xO4 and composite cathode voltages for concentration gradient materials, Physical Review B 94 (1) (2016) 014114.
  * [24] P. D. Tepesch, A. F. Kohan, G. D. Garbulsky, G. Ceder, C. Coley, H. T. Stokes, L. L. Boyer, M. J. Mehl, B. P. Burton, K. Cho, et al., A model to compute phase diagrams in oxides with empirical or first-principles energy methods and application to the solubility limits in the CaO–MgO system, Journal of the American Ceramic Society 79 (8) (1996) 2033–2040.
  * [25] P. Tepesch, G. Garbulsky, G. Ceder, Model for configurational thermodynamics in ionic systems, Physical Review Letters 74 (12) (1995) 2272.
  * [26] A. Van der Ven, M. Aydinol, G. Ceder, G. Kresse, J. Hafner, First-principles investigation of phase stability in LixCoO2, Physical Review B 58 (6) (1998) 2975.
  * [27] B. Han, A. Van der Ven, G. Ceder, B.-J. Hwang, Surface segregation and ordering of alloy surfaces in the presence of adsorbates, Physical Review B 72 (20) (2005) 205409.
  * [28] F. Zhou, T. Maxisch, G. Ceder, Configurational electronic entropy and the phase diagram of mixed-valence oxides: the case of LixFePO4, Physical Review Letters 97 (15) (2006) 155704.
  * [29] Y. Hinuma, Y. S. Meng, K. Kang, G. Ceder, Phase transitions in the LiNi0.5Mn0.5O2 system with temperature, Chemistry of Materials 19 (7) (2007) 1790–1800.
  * [30] H. Das, A. Urban, W. Huang, G. Ceder, First-principles simulation of the (Li–Ni–vacancy)O phase diagram and its relevance for the surface phases in Ni-rich Li-ion cathode materials, Chemistry of Materials 29 (18) (2017) 7840–7851.
  * [31] A. Seko, K. Yuge, F. Oba, A. Kuwabara, I. Tanaka, T. Yamamoto, First-principles study of cation disordering in MgAl2O4 spinel with cluster expansion and Monte Carlo simulation, Physical Review B 73 (9) (2006) 094116\.
  * [32] X. Zhang, M. H. Sluiter, Cluster expansions for thermodynamics and kinetics of multicomponent alloys, Journal of Phase Equilibria and Diffusion 37 (2016) 44–52.
  * [33] V. Blum, A. Zunger, Mixed-basis cluster expansion for thermodynamics of bcc alloys, Physical Review B 70 (15) (2004) 155108.
  * [34] L. Cao, C. Li, T. Mueller, The use of cluster expansions to predict the structures and properties of surfaces and nanostructured materials, Journal of Chemical Information and Modeling 58 (12) (2018) 2401–2413.
  * [35] D. Wen, M. S. Titus, First-principles study of suzuki segregation at stacking faults in disordered face-centered cubic co-ni alloys, Acta Materialia 221 (2021) 117358.
  * [36] J. J. Cordell, J. Pan, A. C. Tamboli, G. J. Tucker, S. Lany, Probing configurational disorder in ZnGeN2 using cluster-based Monte Carlo, Physical Review Materials 5 (2) (2021) 024604.
  * [37] J. J. Cordell, L. Pucurimay, R. R. Schnepf, B. L. Levy-Wendt, M. F. Toney, G. J. Tucker, S. Lany, A. C. Tamboli, Simulation and characterization of cation disorder in ZnGeP2, Journal of Materials Research 37 (12) (2022) 1986–1996.
  * [38] B. Burton, M. Sluiter, First principles phase diagram calculation for the 2D TMD system ${W}{S}_{2}$–${W}{T}e_{2}$, Calphad 63 (2018) 142–147.
  * [39] X. Zhang, M. H. Sluiter, Kinetically driven ordering in phase separating alloys, Physical Review Materials 3 (9) (2019) 095601.
  * [40] F. Körmann, A. V. Ruban, M. H. Sluiter, Long-ranged interactions in bcc NbMoTaW high-entropy alloys, Materials Research Letters 5 (1) (2017) 35–40.
  * [41] N. A. Zarkevich, T. L. Tan, D. D. Johnson, First-principles prediction of phase-segregating alloy phase diagrams and a rapid design estimate of their transition temperatures, Physical Review B 75 (10) (2007) 104203.
  * [42] A. Van der Ven, G. Ceder, Vacancies in ordered and disordered binary alloys treated with the cluster expansion, Physical Review B 71 (5) (2005) 054102.
  * [43] A. Van der Ven, J. C. Thomas, Q. Xu, B. Swoboda, D. Morgan, Nondilute diffusion from first principles: Li diffusion in LixTiS2, Physical Review B 78 (10) (2008) 104306.
  * [44] B. Puchala, A. Van der Ven, Thermodynamics of the Zr-O system from first-principles calculations, Physical Review B 88 (2013) 094108.
  * [45] M.-H. Chen, B. Puchala, A. Van der Ven, High-temperature stability of $\delta^{\prime}$-ZrO, Calphad 51 (2015) 292–298.
  * [46] J. C. Thomas, A. Van der Ven, Finite-temperature properties of strongly anharmonic and mechanically unstable crystal phases from first principles, Physical Review B 88 (21) (2013) 214111.
  * [47] A. A. Belak, A. Van der Ven, Effect of disorder on the dilute equilibrium vacancy concentrations of multicomponent crystalline solids, Physical Review B 91 (22) (2015) 224109.
  * [48] J. G. Goiri, A. Van der Ven, Phase and structural stability in Ni-Al systems from first principles, Physical Review B 94 (9) (2016) 094111.
  * [49] A. R. Natarajan, E. L. Solomon, B. Puchala, E. A. Marquis, A. Van der Ven, On the early stages of precipitation in dilute Mg–Nd alloys, Acta Materialia 108 (2016) 367–379.
  * [50] M. S. Titus, R. K. Rhein, P. B. Wells, P. C. Dodge, G. B. Viswanathan, M. J. Mills, A. Van der Ven, T. M. Pollock, Solute segregation and deviation from bulk thermodynamics at nanoscale crystalline defects, Science Advances 2 (12) (2016) e1601796.
  * [51] N. H. Gunda, A. Van der Ven, First-principles insights on phase stability of titanium interstitial alloys, Physical Review Materials 2 (8) (2018) 083602.
  * [52] N. H. Gunda, B. Puchala, A. Van der Ven, Resolving phase stability in the Ti-O binary with first-principles statistical mechanics methods, Physical Review Materials 2 (3) (2018) 033604.
  * [53] S. K. Kolli, A. Van der Ven, First-principles study of spinel MgTiS2 as a cathode material, Chemistry of Materials 30 (7) (2018) 2436–2442.
  * [54] S. K. Kolli, A. Van der Ven, Controlling the electrochemical properties of spinel intercalation compounds, ACS Applied Energy Materials 1 (12) (2018) 6833–6839.
  * [55] A. R. Natarajan, P. Dolin, A. Van der Ven, Crystallography, thermodynamics and phase transitions in refractory binary alloys, Acta Materialia 200 (2020) 171–186.
  * [56] N. H. Gunda, A. Van der Ven, Understanding the interactions between interstitial and substitutional solutes in refractory alloys: The case of Ti-Al-O, Acta Materialia 191 (2020) 149–157.
  * [57] J. S. Bechtel, J. C. Thomas, A. Van der Ven, Finite-temperature simulation of anharmonicity and octahedral tilting transitions in halide perovskites, Physical Review Materials 3 (11) (2019) 113605.
  * [58] M. D. Radin, J. C. Thomas, A. Van der Ven, Order-disorder versus displacive transitions in Jahn-Teller active layered materials, Physical Review Materials 4 (4) (2020) 043601.
  * [59] Z. Deng, G. Sai Gautam, S. K. Kolli, J.-N. Chotard, A. K. Cheetham, C. Masquelier, P. Canepa, Phase behavior in rhombohedral nasicon electrolytes and electrodes, Chemistry of Materials 32 (18) (2020) 7908–7920.
  * [60] J. Bhattacharya, A. Van der Ven, First-principles study of competing mechanisms of nondilute Li diffusion in spinel LixTiS2, Physical Review B 83 (14) (2011) 144302.
  * [61] J. G. Goiri, S. K. Kolli, A. Van der Ven, Role of short-and long-range ordering on diffusion in Ni-Al alloys, Physical Review Materials 3 (9) (2019) 093402\.
  * [62] S. K. Kolli, A. Van der Ven, Elucidating the factors that cause cation diffusion shutdown in spinel-based electrodes, Chemistry of Materials 33 (16) (2021) 6421–6432.
  * [63] A. van de Walle, M. Asta, Self-driven lattice-model Monte Carlo simulations of alloy thermodynamic properties and phase diagrams, Modelling and Simulation in Materials Science and Engineering 10 (5) (2002) 521.
  * [64] A. Van De Walle, M. Asta, G. Ceder, The alloy theoretic automated toolkit: A user guide, Calphad 26 (4) (2002) 539–553.
  * [65] A. Van de Walle, M. Asta, First-principles investigation of perfect and diffuse antiphase boundaries in HCP-based Ti-Al alloys, Metallurgical and Materials Transactions A 33 (2002) 735–741.
  * [66] O. Adjaoud, G. Steinle-Neumann, B. Burton, A. Van de Walle, First-principles phase diagram calculations for the HfC–TiC, ZrC–TiC, and HfC–ZrC solid solutions, Physical Review B 80 (13) (2009) 134112.
  * [67] C. Ravi, H. Sahu, M. Valsakumar, A. van de Walle, Cluster expansion Monte Carlo study of phase stability of vanadium nitrides, Physical Review B 81 (10) (2010) 104111.
  * [68] G. Ghosh, A. Van de Walle, M. Asta, First-principles calculations of the structural and thermodynamic properties of bcc, fcc and hcp solid solutions in the Al–TM (TM= Ti, Zr and Hf) systems: a comparison of cluster expansion and supercell methods, Acta Materialia 56 (13) (2008) 3202–3221.
  * [69] C. B. Gopal, A. van de Walle, Ab initio thermodynamics of intrinsic oxygen vacancies in ceria, Physical Review B 86 (13) (2012) 134117.
  * [70] C. Nataraj, E. J. L. Borda, A. van de Walle, A. Samanta, A systematic analysis of phase stability in refractory high entropy alloys utilizing linear and non-linear cluster expansion models, Acta Materialia 220 (2021) 117269.
  * [71] S. Zhu, J. Shittu, A. Perron, C. Nataraj, J. Berry, J. T. McKeown, A. van de Walle, A. Samanta, Probing phase stability in crmonbv using cluster expansion method, calphad calculations and experiments, Acta Materialia (2023) 119062.
  * [72] D. A. Kitchaev, Z. Lun, W. D. Richards, H. Ji, R. J. Clément, M. Balasubramanian, D.-H. Kwon, K. Dai, J. K. Papp, T. Lei, et al., Design principles for high transition metal capacity in disordered rocksalt Li-ion cathodes, Energy & Environmental Science 11 (8) (2018) 2159–2171.
  * [73] E. Decolvenaere, E. Levin, R. Seshadri, A. Van der Ven, Modeling magnetic evolution and exchange hardening in disordered magnets: The example of Mn1-xFexRu2Sn Heusler alloys, Physical Review Materials 3 (10) (2019) 104411.
  * [74] D. A. Kitchaev, E. C. Schueller, A. Van der Ven, Mapping skyrmion stability in uniaxial lacunar spinel magnets from first principles, Physical Review B 101 (5) (2020) 054409.
  * [75] J. L. Zuo, D. Kitchaev, E. C. Schueller, J. D. Bocarsly, R. Seshadri, A. Van der Ven, S. D. Wilson, Magnetoentropic mapping and computational modeling of cycloids and skyrmions in the lacunar spinels GaV4S8 and GaV4Se8, Physical Review Materials 5 (5) (2021) 054410.
  * [76] G. Pilania, V. Kocevski, J. A. Valdez, C. R. Kreller, B. P. Uberuaga, Prediction of structure and cation ordering in an ordered normal-inverse double spinel, Communications Materials 1 (1) (2020).
  * [77] G. Pilania, B. Puchala, B. P. Uberuaga, Distortion-stabilized ordered structures in A2BB′O7 mixed pyrochlores, npj Computational Materials 5 (1) (2019) 7.
  * [78] M. Ångqvist, P. Erhart, Understanding chemical ordering in intermetallic clathrates from atomic scale simulations, Chemistry of Materials 29 (17) (2017) 7554–7562.
  * [79] C.-H. Huang, L. Gharaee, Y. Zhao, P. Erhart, J. Marian, Mechanism of nucleation and incipient growth of Re clusters in irradiated W-Re alloys from kinetic Monte Carlo simulations, Physical Review B 96 (9) (2017) 094108.
  * [80] M. Ångqvist, J. M. Rahm, L. Gharaee, P. Erhart, Structurally driven asymmetric miscibility in the phase diagram of W-Ti, Physical Review Materials 3 (7) (2019) 073605.
  * [81] J. Brorsson, Y. Zhang, A. E. Palmqvist, P. Erhart, Order–disorder transition in inorganic clathrates controls electrical transport properties, Chemistry of Materials 33 (12) (2021) 4500–4509.
  * [82] K. Wang, D. Cheng, C.-L. Fu, B.-C. Zhou, First-principles investigation of the phase stability and early stages of precipitation in Mg-Sn alloys, Physical Review Materials 4 (1) (2020) 013606.
  * [83] D. Cheng, E. R. Hoglund, K. Wang, J. M. Howe, S. R. Agnew, B.-C. Zhou, Atomic structures of ordered monolayer GP zones in Mg-Zn-X (X= Ca, Nd) systems, Scripta Materialia 216 (2022) 114744.
  * [84] K. Wang, D. Cheng, B.-C. Zhou, Generalization of the mixed-space cluster expansion method for arbitrary lattices, npj Computational Materials 9 (1) (2023) 75.
  * [85] D. Cheng, K. Wang, B.-C. Zhou, Crystal structure and stability of phases in Mg-Zn alloys: A comprehensive first-principles study, Acta Materialia 242 (2023) 118443.
  * [86] A. R. Natarajan, J. C. Thomas, B. Puchala, A. Van der Ven, Symmetry-adapted order parameters and free energies for solids undergoing order-disorder phase transitions, Physical Review B 96 (13) (2017) 134204.
  * [87] A. B. Bortz, M. H. Kalos, J. L. Lebowitz, A new algorithm for Monte Carlo simulation of Ising spin systems, Journal of Computational Physics 17 (1) (1975) 10–18.
  * [88] C. Uebing, R. Gomer, A Monte Carlo study of surface diffusion coefficients in the presence of adsorbate–adsorbate interactions. I. Repulsive interactions, The Journal of Chemical Physics 95 (10) (1991) 7626–7635.
  * [89] K. A. Fichthorn, W. H. Weinberg, Theoretical foundations of dynamical Monte Carlo simulations, The Journal of Chemical Physics 95 (2) (1991) 1090–1096.
  * [90] F. Soisson, A. Barbu, G. Martin, Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation, Acta Materialia 44 (9) (1996) 3789–3800.
  * [91] F. Bulnes, V. Pereyra, J. Riccardo, Collective surface diffusion: n-fold way kinetic Monte Carlo simulation, Physical Review E 58 (1) (1998) 86.
  * [92] I. Belova, G. Murch, Collective diffusion in the binary random alloy, Philosophical Magazine A 80 (3) (2000) 599–607.
  * [93] F. Soisson, G. Martin, Monte Carlo simulations of the decomposition of metastable solid solutions: Transient and steady-state nucleation kinetics, Physical Review B 62 (1) (2000) 203.
  * [94] K. A. Fichthorn, M. Scheffler, Island nucleation in thin-film epitaxy: a first-principles investigation, Physical Review Letters 84 (23) (2000) 5371.
  * [95] A. Van der Ven, G. Ceder, M. Asta, P. Tepesch, First-principles theory of ionic diffusion with nondilute carriers, Physical Review B 64 (18) (2001) 184307.
  * [96] I. Belova, G. Murch, Behaviour of the diffusion vacancy-wind factors in the concentrated random alloy, Philosophical Magazine A 81 (7) (2001) 1749–1758.
  * [97] A. Tarasenko, F. Nieto, L. Jastrabik, C. Uebing, Collective surface diffusion of repulsively interacting particles on a triangular lattice: Comparison of real-space renormalization group and Monte Carlo approaches, Physical Review B 64 (7) (2001) 075413.
  * [98] A. Van der Ven, G. Ceder, First principles calculation of the interdiffusion coefficient in binary alloys, Physical Review Letters 94 (4) (2005) 045901.
  * [99] M. A. Hartmann, R. Weinkamer, P. Fratzl, J. Svoboda, F. D. Fischer, Onsager’s coefficients and diffusion laws—a Monte Carlo study, Philosophical Magazine 85 (12) (2005) 1243–1260.
  * [100] E. Clouet, C. Hin, D. Gendt, M. Nastar, F. Soisson, Kinetic Monte Carlo simulations of precipitation, Advanced Engineering Materials 8 (12) (2006) 1210–1214.
  * [101] F. Soisson, Kinetic Monte Carlo simulations of radiation induced segregation and precipitation, Journal of nuclear materials 349 (3) (2006) 235–250.
  * [102] B. Uberuaga, D. Bacorisen, R. Smith, J. Ball, R. Grimes, A. Voter, K. Sickafus, Defect kinetics in spinels: Long-time simulations of MgAl2O4, MgGa2O4, and MgIn2O4, Physical Review B 75 (10) (2007) 104116.
  * [103] D. Andersson, T. Watanabe, C. Deo, B. Uberuaga, Role of di-interstitial clusters in oxygen transport in UO2+x from first principles, Physical Review B 80 (6) (2009) 060101.
  * [104] A. Van der Ven, H.-C. Yu, G. Ceder, K. Thornton, Vacancy mediated substitutional diffusion in binary crystalline solids, Progress in Materials Science 55 (2) (2010) 61–105.
  * [105] F. Soisson, C. Becquart, N. Castin, C. Domain, L. Malerba, E. Vincent, Atomistic Kinetic Monte Carlo studies of microchemical evolutions driven by diffusion processes under irradiation, Journal of Nuclear Materials 406 (1) (2010) 55–67.
  * [106] A. Van der Ven, J. Bhattacharya, A. A. Belak, Understanding Li diffusion in Li-intercalation compounds, Accounts of chemical research 46 (5) (2013) 1216–1225.
  * [107] F. Soisson, T. Jourdan, Radiation-accelerated precipitation in Fe–Cr alloys, Acta Materialia 103 (2016) 870–881.
  * [108] K. Li, C.-C. Fu, M. Nastar, F. Soisson, Predicting atomic diffusion in concentrated magnetic alloys: The case of paramagnetic Fe-Ni, Physical Review B 107 (9) (2023) 094103.
  * [109] A. Abu-Odeh, B. P. Uberuaga, M. Asta, Barrier-free predictions of short-range ordering/clustering kinetics in binary FCC solid solutions, Acta Materialia 257 (2023) 119185.
  * [110] B. Puchala, J. C. Thomas, A. R. Natarajan, J. G. Goiri, S. S. Behara, J. L. Kaufman, A. Van der Ven, CASM—A software package for first-principles based study of multicomponent crystalline solids, Computational Materials Science 217 (2023) 111897.
  * [111] F. Larché, J. W. Cahn, Overview no. 41 the interactions of composition and stress in crystalline solids, Acta Metallurgica 33 (3) (1985) 331–357.
  * [112] J. Cahn, F. Larche, An invariant formulation of multicomponent diffusion in crystals, Scripta metallurgica 17 (7) (1983) 927–932.
  * [113] J. M. Sanchez, F. Ducastelle, D. Gratias, Generalized cluster description of multicomponent systems, Physica A: Statistical Mechanics and its Applications 128 (1-2) (1984) 334–350.
  * [114] R. Drautz, M. Fähnle, Spin-cluster expansion: Parametrization of the general adiabatic magnetic energy surface with ab initio accuracy, Physical Review B 69 (10) (2004) 104404.
  * [115] H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, John Wiley & Sons, Inc., 1985.
  * [116] B. Fultz, Vibrational thermodynamics of materials, Progress in Materials Science 55 (4) (2010) 247–352.
  * [117] T. L. Hill, Statistical mechanics: principles and selected applications, Courier Corporation, 2013.
  * [118] E. T. Jaynes, Information theory and statistical mechanics, Physical review 106 (4) (1957) 620.
  * [119] A. R. Allnatt, A. B. Lidiard, Atomic transport in solids, Cambridge University Press, 2003.
  * [120] A. R. Allnatt, E. L. Allnatt, Computer simulation of phenomenological coefficients for atom transport in a random alloy, Philosophical Magazine A 49 (5) (1984) 625–635.
  * [121] K. Kehr, K. Binder, S. Reulein, Mobility, interdiffusion, and tracer diffusion in lattice-gas models of two-component alloys, Physical Review B 39 (8) (1989) 4891.
  * [122] G. H. Vineyard, Frequency factors and isotope effects in solid state rate processes, Journal of Physics and Chemistry of Solids 3 (1-2) (1957) 121–127.
  * [123] H. Eyring, The activated complex in chemical reactions, The Journal of Chemical Physics 3 (2) (1935) 107–115.
  * [124] S. R. De Groot, P. Mazur, Non-equilibrium thermodynamics, Courier Corporation, 2013\.
  * [125] G. E. Murch, Diffusion Kinetics in Solids, John Wiley & Sons, Ltd, 2001, Ch. 3, pp. 171–238.
  * [126] Y. Haven, Rept. Conf. Defects Crystalline Solids, London, 1954.
  * [127] K. Lejaeghere, G. Bihlmayer, T. Björkman, P. Blaha, S. Blügel, V. Blum, D. Caliste, I. E. Castelli, S. J. Clark, A. Dal Corso, et al., Reproducibility in density functional theory calculations of solids, Science 351 (6280) (2016) aad3000.
  * [128] J. Sanchez, Cluster expansion and the configurational theory of alloys, Physical Review B 81 (22) (2010) 224202.
  * [129] J. C. Thomas, J. S. Bechtel, A. Van der Ven, Hamiltonians and order parameters for crystals of orientable molecules, Physical Review B 98 (9) (2018) 094105.
  * [130] R. Drautz, Atomic cluster expansion for accurate and transferable interatomic potentials, Physical Review B 99 (1) (2019) 014104.
  * [131] A. van de Walle, G. Ceder, Automating first-principles phase diagram calculations, Journal of Phase Equilibria 23 (4) (2002) 348–359.
  * [132] G. L. Hart, V. Blum, M. J. Walorski, A. Zunger, Evolutionary approach for determining first-principles hamiltonians, Nature materials 4 (5) (2005) 391–394.
  * [133] T. Mueller, G. Ceder, Bayesian approach to cluster expansions, Physical Review B 80 (2) (2009) 024103.
  * [134] L. J. Nelson, V. Ozoliņš, C. S. Reese, F. Zhou, G. L. Hart, Cluster expansion made easy with Bayesian compressive sensing, Physical Review B 88 (15) (2013) 155105.
  * [135] L. J. Nelson, G. L. Hart, F. Zhou, V. Ozoliņš, et al., Compressive sensing as a paradigm for building physics models, Physical Review B 87 (3) (2013) 035125.
  * [136] J. G. Goiri, A. Van der Ven, Recursive alloy Hamiltonian construction and its application to the Ni-Al-Cr system, Acta Materialia 159 (2018) 257–265.
  * [137] J. C. Thomas, B. Puchala, A. Van der Ven, Generalized cluster expansion Hamiltonians, To be submitted (2023).
  * [138] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO-the Open Visualization Tool, Modelling and Simulation in Materials Science and Engineering 18 (1) (2010).
  * [139] K. Binder, D. Heermann, Monte Carlo Simulation in Statistical Physics: An Introduction, Physics and astronomy online library, Springer, 2002.
  * [140] L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Physical Review 65 (1944) 117–149.
  * [141] F. Walsh, A. R. Natarajan, A. Van der Ven, Order parameters for antiferromagnetic structures: A first-principles study of iridium manganese, Physical Review Materials 6 (4) (2022) 044402.
  * [142] J. C. Thomas, A. Van der Ven, The exploration of nonlinear elasticity and its efficient parameterization for crystalline materials, Journal of the Mechanics and Physics of Solids 107 (2017) 76–95.
  * [143] R. Gomer, Diffusion of adsorbates on metal surfaces, Reports on progress in Physics 53 (7) (1990) 917.
  * [144] G. L. Montet, Integral methods in the calculation of correlation factors in diffusion, Physical Review B 7 (1973) 650–662.
  * [145] B. Sadigh, P. Erhart, Calculation of excess free energies of precipitates via direct thermodynamic integration across phase boundaries, Physical Review B 86 (2012) 134204.
  * [146] G. H. Teichert, A. R. Natarajan, A. Van der Ven, K. Garikipati, Scale bridging materials physics: Active learning workflows and integrable deep neural networks for free energy function representations in alloys, Computer Methods in Applied Mechanics and Engineering 371 (2020) 113281.
  * [147] J. M. Hammersley, D. C. Handscomb, Monte Carlo methods, Monographs on Statistics and Applied Probability, Chapman & Hall, 1964.
  * [148] A. Gelman, D. B. Rubin, Inference from iterative simulation using multiple sequences, Statistical Science 7 (4) (1992) 457–472.
  * [149] C. J. Geyer, Practical markov chain monte carlo, Statistical Science 7 (4) (1992) 473–483.
  * [150] C. J. Geyer, Introduction to Markov Chain Monte Carlo, in: S. Brooks, A. Gelman, G. Jones, X.-L. Meng (Eds.), Handbook of Markov Chain Monte Carlo, Chapman and Hall/CRC, 2011.
  * [151] D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 4th Edition, Cambridge University Press, 2014.
  * [152] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equation of state calculations by fast computing machines, The Journal of Chemical Physics 21 (6) (1953) 1087–1092.
  * [153] M. A. Novotny, Monte carlo algorithms with absorbing markov chains: Fast local algorithms for slow dynamics, Physical Review Letters 74 (1995) 1–5.
  * [154] B. Puchala, M. L. Falk, K. Garikipati, An energy basin finding algorithm for kinetic Monte Carlo acceleration, The Journal of Chemical Physics 132 (13), 134104 (04 2010).
  * [155] K. A. Fichthorn, Y. Lin, A local superbasin kinetic Monte Carlo method, The Journal of Chemical Physics 138 (16), 164104 (04 2013).
  * [156] N. Wang, T. Hammerschmidt, J. Rogal, R. Drautz, Accelerating spin-space sampling by auxiliary spin dynamics and temperature-dependent spin-cluster expansion, Physical Review B 99 (9) (2019) 094402.
  * [157] S. Duane, A. D. Kennedy, B. J. Pendleton, D. Roweth, Hybrid monte carlo, Physics letters B 195 (2) (1987) 216–222.
  * [158] M. Betancourt, A conceptual introduction to hamiltonian monte carlo, arXiv preprint arXiv:1701.02434 (2017).
  * [159] Ole Stauning, Fadbad++, (Accessed 1 September, 2023).
URL http://uning.dk/fadbad.html

  * [160] C. S. Adorf, P. M. Dodd, V. Ramasubramani, S. C. Glotzer, Simple data and workflow management with the signac framework, Computational Materials Science 146 (2018) 220–229.
  * [161] V. Ramasubramani, C. Adorf, P. Dodd, B. Dice, S. Glotzer, signac: A Python framework for data and workflow management, in: F. Akici, D. Lippa, D. Niederhut, M. Pacer (Eds.), Proceedings of the 17th Python in Science Conference (SciPy 2018), 2018, pp. 152–159.
  * [162] Bokeh Development Team, Bokeh: Python library for interactive visualization, (Accessed 1 September, 2023).
URL https://bokeh.pydata.org/en/latest/

  * [163] B. Puchala, G. Tarcea, E. A. Marquis, M. Hedstrom, H. V. Jagadish, J. E. Allison, The materials commons: A collaboration platform and information repository for the global materials community, JOM 68 (8) (2016) 2035–2044.
  * [164] CASM Website, (Accessed 1 September, 2023).
URL https://prisms-center.github.io/CASMcode_docs/