def _in_facet(self, facet, entry):
"""
Checks if a Pourbaix Entry is in a facet.
Args:
facet: facet to test.
entry: Pourbaix Entry to test.
"""
dim = len(self._keys)
if dim > 1:
coords = [np.array(self._pd.qhull_data[facet[i]][0:dim - 1])
for i in range(len(facet))]
simplex = Simplex(coords)
comp_point = [entry.npH, entry.nPhi]
return simplex.in_simplex(comp_point,
PourbaixAnalyzer.numerical_tol)
else:
return True
def _get_3d_domain_simplexes_and_ann_loc(
points_3d: np.ndarray, ) -> Tuple[List[Simplex], np.ndarray]:
"""Returns a list of Simplex objects and coordinates of annotation for one
domain in a 3-d chemical potential diagram. Uses PCA to project domain
into 2-dimensional space so that ConvexHull can be used to identify the
bounding polygon"""
points_2d, v, w = simple_pca(points_3d, k=2)
domain = ConvexHull(points_2d)
centroid_2d = get_centroid_2d(points_2d[domain.vertices])
ann_loc = centroid_2d @ w.T + np.mean(points_3d.T, axis=1)
simplexes = [
Simplex(points_3d[indices]) for indices in domain.simplices
]
return simplexes, ann_loc
def _get_3d_formula_lines(
draw_domains: Dict[str, np.ndarray],
formula_colors: Optional[List[str]],
) -> List[Scatter3d]:
"""Returns a list of Scatter3d objects defining the bounding polyhedra"""
if formula_colors is None:
formula_colors = px.colors.qualitative.Dark2
lines = []
for idx, (formula, coords) in enumerate(draw_domains.items()):
points_3d = coords[:, :3]
domain = ConvexHull(points_3d[:, :-1])
simplexes = [
Simplex(points_3d[indices]) for indices in domain.simplices
]
x, y, z = [], [], []
for s in simplexes:
x.extend(s.coords[:, 0].tolist() + [None])
y.extend(s.coords[:, 1].tolist() + [None])
z.extend(s.coords[:, 2].tolist() + [None])
line = Scatter3d(
x=x,
y=y,
z=z,
mode="lines",
line={
"width": 8,
"color": formula_colors[idx]
},
opacity=1.0,
name=f"{formula} (lines)",
)
lines.append(line)
return lines
def get_pourbaix_domains(pourbaix_entries, limits=None):
"""
Returns a set of pourbaix stable domains (i. e. polygons) in
pH-V space from a list of pourbaix_entries
This function works by using scipy's HalfspaceIntersection
function to construct all of the 2-D polygons that form the
boundaries of the planes corresponding to individual entry
gibbs free energies as a function of pH and V. Hyperplanes
of the form a*pH + b*V + 1 - g(0, 0) are constructed and
supplied to HalfspaceIntersection, which then finds the
boundaries of each pourbaix region using the intersection
points.
Args:
pourbaix_entries ([PourbaixEntry]): Pourbaix entries
with which to construct stable pourbaix domains
limits ([[float]]): limits in which to do the pourbaix
analysis
Returns:
Returns a dict of the form {entry: [boundary_points]}.
The list of boundary points are the sides of the N-1
dim polytope bounding the allowable ph-V range of each entry.
"""
if limits is None:
limits = [[-2, 16], [-4, 4]]
# Get hyperplanes
hyperplanes = [
np.array([-PREFAC * entry.npH, -entry.nPhi, 0, -entry.energy]) *
entry.normalization_factor for entry in pourbaix_entries
]
hyperplanes = np.array(hyperplanes)
hyperplanes[:, 2] = 1
max_contribs = np.max(np.abs(hyperplanes), axis=0)
g_max = np.dot(-max_contribs, [limits[0][1], limits[1][1], 0, 1])
# Add border hyperplanes and generate HalfspaceIntersection
border_hyperplanes = [
[-1, 0, 0, limits[0][0]],
[1, 0, 0, -limits[0][1]],
[0, -1, 0, limits[1][0]],
[0, 1, 0, -limits[1][1]],
[0, 0, -1, 2 * g_max],
]
hs_hyperplanes = np.vstack([hyperplanes, border_hyperplanes])
interior_point = np.average(limits, axis=1).tolist() + [g_max]
hs_int = HalfspaceIntersection(hs_hyperplanes,
np.array(interior_point))
# organize the boundary points by entry
pourbaix_domains = {entry: [] for entry in pourbaix_entries}
for intersection, facet in zip(hs_int.intersections,
hs_int.dual_facets):
for v in facet:
if v < len(pourbaix_entries):
this_entry = pourbaix_entries[v]
pourbaix_domains[this_entry].append(intersection)
# Remove entries with no pourbaix region
pourbaix_domains = {k: v for k, v in pourbaix_domains.items() if v}
pourbaix_domain_vertices = {}
for entry, points in pourbaix_domains.items():
points = np.array(points)[:, :2]
# Initial sort to ensure consistency
points = points[np.lexsort(np.transpose(points))]
center = np.average(points, axis=0)
points_centered = points - center
# Sort points by cross product of centered points,
# isn't strictly necessary but useful for plotting tools
points_centered = sorted(
points_centered,
key=cmp_to_key(lambda x, y: x[0] * y[1] - x[1] * y[0]))
points = points_centered + center
# Create simplices corresponding to pourbaix boundary
simplices = [
Simplex(points[indices])
for indices in ConvexHull(points).simplices
]
pourbaix_domains[entry] = simplices
pourbaix_domain_vertices[entry] = points
return pourbaix_domains, pourbaix_domain_vertices
def get_chempot_range_map(self, limits=[[-2, 16], [-4, 4]]):
"""
Returns a chemical potential range map for each stable entry.
This function works by using scipy's HalfspaceIntersection
function to construct all of the 2-D polygons that form the
boundaries of the planes corresponding to individual entry
gibbs free energies as a function of pH and V. Hyperplanes
of the form a*pH + b*V + 1 - g(0, 0) are constructed and
supplied to HalfspaceIntersection, which then finds the
boundaries of each pourbaix region using the intersection
points.
Args:
limits ([[float]]): limits in which to do the pourbaix
analysis
Returns:
Returns a dict of the form {entry: [boundary_points]}.
The list of boundary points are the sides of the N-1
dim polytope bounding the allowable ph-V range of each entry.
"""
tol = PourbaixAnalyzer.numerical_tol
all_chempots = []
facets = self._pd.facets
for facet in facets:
chempots = self.get_facet_chempots(facet)
chempots["H+"] /= -0.0591
chempots["V"] = -chempots["V"]
chempots["1"] = chempots["1"]
all_chempots.append([chempots[el] for el in self._keys])
# Get hyperplanes corresponding to G as function of pH and V
halfspaces = []
qhull_data = np.array(self._pd._qhull_data)
stable_entries = self._pd.stable_entries
stable_indices = [
self._pd.qhull_entries.index(e) for e in stable_entries
]
qhull_data = np.array(self._pd._qhull_data)
hyperplanes = np.vstack([
-0.0591 * qhull_data[:, 0], -qhull_data[:, 1],
np.ones(len(qhull_data)), -qhull_data[:, 2]
])
hyperplanes = np.transpose(hyperplanes)
max_contribs = np.max(np.abs(hyperplanes), axis=0)
g_max = np.dot(-max_contribs, [limits[0][1], limits[1][1], 0, 1])
# Add border hyperplanes and generate HalfspaceIntersection
border_hyperplanes = [[-1, 0, 0,
limits[0][0]], [1, 0, 0, -limits[0][1]],
[0, -1, 0, limits[1][0]],
[0, 1, 0, -limits[1][1]], [0, 0, -1, 2 * g_max]]
hs_hyperplanes = np.vstack(
[hyperplanes[stable_indices], border_hyperplanes])
interior_point = np.average(limits, axis=1).tolist() + [g_max]
hs_int = HalfspaceIntersection(hs_hyperplanes,
np.array(interior_point))
# organize the boundary points by entry
pourbaix_domains = {entry: [] for entry in stable_entries}
for intersection, facet in zip(hs_int.intersections,
hs_int.dual_facets):
for v in facet:
if v < len(stable_entries):
pourbaix_domains[stable_entries[v]].append(intersection)
# Remove entries with no pourbaix region
pourbaix_domains = {k: v for k, v in pourbaix_domains.items() if v}
pourbaix_domain_vertices = {}
# Post-process boundary points, sorting isn't strictly necessary
# but useful for some plotting tools (e.g. highcharts)
for entry, points in pourbaix_domains.items():
points = np.array(points)[:, :2]
center = np.average(points, axis=0)
points_centered = points - center
# Sort points by cross product of centered points,
# then roll by min sum to to ensure consistency
point_comparator = lambda x, y: x[0] * y[1] - x[1] * y[0]
points_centered = sorted(points_centered,
key=cmp_to_key(point_comparator))
points = points_centered + center
shift = -np.lexsort(np.transpose(points))[0]
points = np.roll(points, shift, axis=0)
# Create simplices corresponding to pourbaix boundary
simplices = [
Simplex(points[indices])
for indices in ConvexHull(points).simplices
]
pourbaix_domains[entry] = simplices
pourbaix_domain_vertices[entry] = points
self.pourbaix_domains = pourbaix_domains
self.pourbaix_domain_vertices = pourbaix_domain_vertices
return pourbaix_domains