Calphad

c_iA

Equation

\[c^\alpha_{iA} = \sum_j e^\alpha_{ij} \frac{\partial M^\alpha_A}{\partial y^\alpha_j}\]

c_iG

Equation

\[c^\alpha_{iG} = -\sum_j e^\alpha_{ij} \frac{\partial G^\alpha_M}{\partial y^\alpha_j}\]

c_iPot

Equation

\[c^\alpha_{i\mathrm{Pot}} = -\sum_j e^\alpha_{ij} \frac{\partial^2 G^\alpha_M}{\partial \mathrm{Pot} \partial y^\alpha_j}\]

cond_row_rhs

This function takes a condition symbol and dispatches on the correct method that condition that returns the elements in the row and the right hand side.

The number of columns are determined by the fixed_free_terms - there's one column for each free chemical potential, free potential, and free phase amount.

get_Delta_y_mat

Notationally, Δy is a vector of length(phaserecord.sitefractions) that updates the site fractions. However in this case, we need to plug in the results from the solution vector that we do not have symbolic variables for. To resolve that, we'll design the site fractions to be a matrix that can be used to take the dot product with the solution vector (padded with a prefixed one to handle the c_iG term). The usage is therefore

Examples

delta_y_M = get_Delta_y_mat(prx, ["A", "B"], [T, P, N_A, N_B])
# ... compute solution
soln = [3271.04833019, 7271.04833015, 1e-16]
delta_y = delta_y_M * vcat(1, soln...)

Equation

\[\sum_i \Delta y_i^\alpha = \sum_i \left(c_{iG}^\alpha + \sum_A c_{iA}^\alpha \mu_A + \sum_\mathrm{Pot} c_{i\mathrm{Pot}}^\alpha \Delta \mathrm{Pot} \right)\]

get_N_A_row_rhs

Examples

using Symbolics
@variables T P N_A
@variables Y_BETA_A Y_BETA_B
G_BETA_A = 8000.0-10.0*T;
G_BETA_B = 12000.0-10.0*T;
G_BETA = Y_BETA_A*G_BETA_A + Y_BETA_B*G_BETA_B + R*T*(Y_BETA_A*log(Y_BETA_A) + Y_BETA_B*log(Y_BETA_B));
mass_BETA = [Y_BETA_A, Y_BETA_B];
state_variables = [P, T];
site_fractions = [Y_BETA_A, Y_BETA_B];
prx = PhaseRecord("BETA", G_BETA, mass_BETA, state_variables, site_fractions);

r, rrhs = get_N_A_row_rhs([prx], 1, N_A, [], [1, 2], [P, T], [], [], [1])

Equation

\[\sum_{B_{\mathrm{free}}} \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{iB} \mu_B + \sum_\mathrm{Pot} \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{i\mathrm{Pot}} \Delta \mathrm{Pot} + \sum_\beta M_A^\beta \Delta \aleph^\beta \\ = - \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{iG} - \sum_{B_{\mathrm{fixed}}} \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{iB} \mu_B + \left( \tilde{N}_A - \sum_\alpha \aleph^\alpha M^\alpha_A \right)\]

get_N_row_rhs

See the function get_N_A_row_rhs. This is conceptually the same as an N_A condition with an inner loop over all the elements in each column.

Equation

\[\sum_{B_{\mathrm{free}}} \sum_A \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{iB} \mu_B + \sum_\mathrm{Pot} \sum_A \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{i\mathrm{Pot}} \Delta \mathrm{Pot} + \sum_\beta \sum_A M_A^\beta \Delta \aleph^\beta \\ = \sum_A \left(- \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{iG} - \sum_{B_{\mathrm{fixed}}} \sum_\alpha \sum_i \aleph^\alpha \frac{\partial M_A^\alpha}{\partial y_i^\alpha} c_{iB} \mu_B + \right) + (\tilde{N} - N)\]

get_stable_phase_row_rhs

Examples

srow, srhs = get_stable_phase_row_rhs([prx], 1, [], [1, 2], [P, T], [], [], [1])

Equation

\[\sum_{B_{\mathrm{free}}} M_B^\alpha \mu_B + \sum_\mathrm{Pot} -\frac{\partial G_M^\alpha}{\partial \mathrm{Pot}} \Delta \mathrm{Pot} + \sum_\beta 0 = G_M^\alpha + \sum_{B_{\mathrm{fixed}}} -M_B^\alpha \mu_B\]

get_x_A_row_rhs

See the function get_N_A_row_rhs. This is the corresponding mole fraction condition.

Equation

\[\sum_{B_{\mathrm{free}}} \sum_\alpha \sum_i \frac{\aleph^\alpha c_{iB}}{N} \left( \frac{\partial M_A^\alpha}{\partial y_i^\alpha} - x_A \sum_C \frac{\partial M_C^\alpha}{\partial y_i^\alpha} \right) \mu_B \\ + \sum_\mathrm{Pot} \sum_\alpha \sum_i \frac{\aleph^\alpha c_{i\mathrm{Pot}}}{N} \left( \frac{\partial M_A^\alpha}{\partial y_i^\alpha} - x_A \sum_C \frac{\partial M_C^\alpha}{\partial y_i^\alpha} \right) \Delta \mathrm{Pot} \\ + \sum_{\beta_\mathrm{free}} \frac{1}{N} \left( M_A^\beta - x_A \sum_C M_C^\beta \right) \Delta \aleph^\beta \\ = - \sum_\alpha \sum_i \frac{\aleph^\alpha c_{iG}}{N} \left( \frac{\partial M_A^\alpha}{\partial y_i^\alpha} - x_A \sum_C \frac{\partial M_C^\alpha}{\partial y_i^\alpha} \right) \\ - \sum_{B_{\mathrm{fixed}}} \sum_\alpha \sum_i \frac{\aleph^\alpha c_{iB}}{N} \left( \frac{\partial M_A^\alpha}{\partial y_i^\alpha} - x_A \sum_C \frac{\partial M_C^\alpha}{\partial y_i^\alpha} \right) \mu_B \\ + \left( \tilde{x}_A - x_A \right)\]

Examples

using Symbolics
@variables N P T N_A MU_B NP_ALPHA N
elements = ["A", "B", "C"]
phases = ["ALPHA", "BETA", "GAMMA"]
condition_dict = Dict(
   N => 1.0,
   P => 101325.0,
   T => 300.0,
   N_A => 0.5,
   MU_B => -10000,
   NP_ALPHA => 0.8,
)
unpack_indices(elements, phases, condition_dict)