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diff --git a/examples/Data/DoubleLayerCapacity/Lambda.npz b/examples/Data/DoubleLayerCapacity/Lambda.npz
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<g>
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<!-- $C_{dl}$ [μ$F/cm^2]$ -->
<g style="fill: #d62728" transform="translate(2093.210813 553) rotate(-90) scale(0.3 -0.3)">
+ <defs>
+ <path id="DejaVuSans-Oblique-46" d="M 1081 4666
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+L 1606 4134
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+L 3188 2759
+L 3084 2228
+L 1234 2228
+L 800 0
+L 172 0
+L 1081 4666
+z
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L 1962.392222 681.063415
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L 1962.392222 696.753948
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L 1962.392222 704.265874
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diff --git a/examples/ReproducableCode/DoubleLayerCapacity/Compressibility.py b/examples/ReproducableCode/DoubleLayerCapacity/Compressibility.py
index 153c3c35fd8af2ef65f0171a34ba591e83a4087b..88afaa1ab5b5181fc16627308e29d0433591df4e 100644
--- a/examples/ReproducableCode/DoubleLayerCapacity/Compressibility.py
+++ b/examples/ReproducableCode/DoubleLayerCapacity/Compressibility.py
@@ -16,7 +16,7 @@ src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../../..',
sys.path.insert(0, src_path)
from Eq02 import solve_System_2eq
-from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana
+from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana, C_DL_dimless_ana, C_DL_dim_ana
# Remove the src directory from sys.path after import
@@ -66,6 +66,7 @@ Phi_Pot_Diff_dimless = Phi_Pot_Diff_dim * e0/(k*T)
# K
K_vec = ['incompressible', 50_000, 40_000, 30_000, 20_000]
+# Charge
Q_DL_dim_ = []
Q_DL_dimless_ = []
for K in K_vec:
@@ -75,11 +76,11 @@ for K in K_vec:
Q_DL_dim_.append(Q_DL_dim_ana(y_A_R, y_C_R, y_N_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, nR_m, e0, LR, kappa))
# Double layer capacity
-C_DL_dim = [C_dl(q_dl, Phi_Pot_Diff_dim) for q_dl in Q_DL_dim_]
-C_DL_dimless = [C_dl(q_dl, Phi_Pot_Diff_dimless) for q_dl in Q_DL_dimless_]
-
-Phi_pot_center_array_dimless = Phi_pot_center(Phi_Pot_Diff_dimless)
-Phi_pot_center_array_dim = Phi_pot_center(Phi_Pot_Diff_dim)
+C_DL_dim_ = []
+C_DL_dimless_ = []
+for K in K_vec:
+ C_DL_dim_.append(C_DL_dim_ana(y_A_R, y_C_R, 1-y_A_R-y_C_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, nR_m, e0, LR, k, T, kappa))
+ C_DL_dimless_.append(C_DL_dimless_ana(y_A_R, y_C_R, 1-y_A_R-y_C_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, kappa))
# Plotting
@@ -117,8 +118,6 @@ fig.show()
# Double Layer Capacity
-Phi_pot_center_array_dimless = Phi_pot_center(Phi_Pot_Diff_dimless)
-Phi_pot_center_array_dim = Phi_pot_center(Phi_Pot_Diff_dim)
fig = plt.figure()
color_dimless = 'tab:purple'
color_dim = 'tab:red'
@@ -126,16 +125,16 @@ ax = fig.add_subplot(111, label="1")
ax2 = fig.add_subplot(111, label="2", frame_on=False)
# Plot dimensional data
-for i, c_dl in enumerate(C_DL_dim):
- ax.plot(Phi_pot_center_array_dim, c_dl, color=colors[i], label=f'$K$: {K_vec[i]}')
+for i, c_dl in enumerate(C_DL_dim_):
+ ax.plot(Phi_Pot_Diff_dim, c_dl, color=colors[i], label=f'$K$: {K_vec[i]}')
ax.grid()
ax.set_xlabel('$\delta \\varphi [nm]$', color=color_dim)
ax.set_ylabel('$C_{dl}$ [\u03bc$F/cm^2]$', color=color_dim)
ax.tick_params(axis='x', colors=color_dim)
ax.tick_params(axis='y', colors=color_dim)
-for i, c_dl in enumerate(C_DL_dimless):
- ax2.plot(Phi_pot_center_array_dimless, c_dl, color=colors[i])
+for i, c_dl in enumerate(C_DL_dimless_):
+ ax2.plot(Phi_Pot_Diff_dimless, c_dl, color=colors[i])
# ax2.grid()
ax2.xaxis.tick_top()
ax2.yaxis.tick_right()
@@ -150,49 +149,6 @@ fig.legend()
fig.tight_layout()
fig.show()
-# # y_alpha_L
-# y_A_np, y_C_np, y_S_np = np.array(y_A), np.array(y_C), np.array(y_S)
-# markers = ['--', '-', ':']
-# colors = ['tab:blue', 'tab:orange', 'tab:green', 'tab:red', 'tab:purple']
-# plt.figure()
-# for i in range(len(K_vec)):
-# clr = colors[i]
-# plt.plot(Phi_Pot_Diff_dimless, y_A_np[i,:,0], '--', color=clr)
-# plt.plot(Phi_Pot_Diff_dimless, y_C_np[i,:,0], '-', color=clr)
-# plt.plot(Phi_Pot_Diff_dimless, y_S_np[i,:,0], ':', color=clr)
-# dummy, = plt.plot(0, 0, color='grey', linestyle='--', label='$y_A$')
-# dummy, = plt.plot(0, 0, color='grey', linestyle='-', label='$y_C$')
-# dummy, = plt.plot(0, 0, color='grey', linestyle=':', label='$y_S$')
-# for index, K_ in enumerate(K_vec):
-# dummy, = plt.plot(0, 0, color=colors[index], linestyle='-', label=f'K = {K_}')
-# plt.grid()
-# plt.legend()
-# plt.xlabel('$\delta \\varphi$ [-]')
-# plt.ylabel('$y_\\alpha^L$')
-# plt.tight_layout()
-# plt.show()
-
-# n_sol_np = np.array(n_sol)
-# n_A_np, n_C_np, n_S_np = y_A_np * n_sol_np, y_C_np * n_sol_np, y_S_np * n_sol_np
-# plt.figure()
-# clr = colors[0]
-# for i in range(len(K_vec)):
-# clr = colors[i]
-# plt.plot(Phi_Pot_Diff_dimless, n_A_np[i,:,0], '--', color=clr)
-# plt.plot(Phi_Pot_Diff_dimless, n_C_np[i,:,0], '-', color=clr)
-# plt.plot(Phi_Pot_Diff_dimless, n_S_np[i,:,0], ':', color=clr)
-# dummy, = plt.plot(0, 0, color='grey', linestyle='--', label='$n_A$')
-# dummy, = plt.plot(0, 0, color='grey', linestyle='-', label='$n_C$')
-# dummy, = plt.plot(0, 0, color='grey', linestyle=':', label='$n_S$')
-# for index, K_ in enumerate(K_vec):
-# dummy, = plt.plot(0, 0, color=colors[index], linestyle='-', label=f'K = {K_}')
-# plt.grid()
-# plt.legend()
-# plt.xlabel('$\delta \\varphi$ [-]')
-# plt.ylabel('$n_\\alpha^L [-]$')
-# plt.tight_layout()
-# plt.show()
-
# Store data
-np.savez('../../Data/DoubleLayerCapacity/DLKap_Compressibility.npz', Lambda2=Lambda2, a2=a2, K_vec=K_vec, kappa=kappa, Molarity=Molarity, y_A_R=y_A_R, y_C_R=y_C_R, z_A=z_A, z_C=z_C, p_R=p_R, phi_R=phi_R, Volt_start=Volt_start, Volt_end=Volt_end, n_Volts=n_Volts, Phi_Pot_Diff_dimless=Phi_Pot_Diff_dimless, Phi_Pot_Diff_dim=Phi_Pot_Diff_dim, Phi_pot_center_array_dim=Phi_pot_center_array_dim, Phi_pot_center_array_dimless=Phi_pot_center_array_dimless, Q_DL_dimless_=Q_DL_dimless_, Q_DL_dim_ =Q_DL_dim_, C_DL_dimless=C_DL_dimless, C_DL_dim=C_DL_dim)
+np.savez('../../Data/DoubleLayerCapacity/DLKap_Compressibility.npz', Lambda2=Lambda2, a2=a2, K_vec=K_vec, kappa=kappa, Molarity=Molarity, y_A_R=y_A_R, y_C_R=y_C_R, z_A=z_A, z_C=z_C, p_R=p_R, phi_R=phi_R, Volt_start=Volt_start, Volt_end=Volt_end, n_Volts=n_Volts, Phi_Pot_Diff_dimless=Phi_Pot_Diff_dimless, Phi_Pot_Diff_dim=Phi_Pot_Diff_dim, Q_DL_dimless_=Q_DL_dimless_, Q_DL_dim_ =Q_DL_dim_, C_DL_dimless=C_DL_dimless_, C_DL_dim=C_DL_dim_)
diff --git a/examples/ReproducableCode/DoubleLayerCapacity/CompressibleValidationAnalyticalMethod.py b/examples/ReproducableCode/DoubleLayerCapacity/CompressibleValidationAnalyticalMethod.py
index 992b84d9b4db907449a8c256d29ef76bf5c30c97..75c7f6343dabb7182320753bde1c2a6d8467b508 100644
--- a/examples/ReproducableCode/DoubleLayerCapacity/CompressibleValidationAnalyticalMethod.py
+++ b/examples/ReproducableCode/DoubleLayerCapacity/CompressibleValidationAnalyticalMethod.py
@@ -17,7 +17,7 @@ sys.path.insert(0, src_path)
from Eq04 import solve_System_4eq
from Eq02 import solve_System_2eq
-from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana
+from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_num_dim, Q_DL_dimless_ana, Q_DL_dim_ana, C_DL_dimless_ana, C_DL_dim_ana
# Remove the src directory from sys.path after import
@@ -27,42 +27,6 @@ del sys.path[0]
import matplotlib.pyplot as plt
import numpy as np
-# # Define Parameter
-# e0 = 1.602e-19 # [As]
-# k = 1.381e-23 # [J/K]
-# T = 293.75 # [K]
-
-# # Load numerical data
-# data_numerical = np.load('../../Data/DoubleLayerCapacity/Compressibility.npz')
-# Lambda2 = data_numerical['Lambda2']
-# a2 = data_numerical['a2']
-# K_vec = data_numerical['K_vec']
-# temp = K_vec[1:].astype(np.float64)
-# K_vec_converted = [K_vec[0]]
-# [K_vec_converted.append(temp_) for temp_ in temp]
-# K_vec = K_vec_converted
-# kappa = data_numerical['kappa']
-# Molarity = data_numerical['Molarity']
-# y_R = data_numerical['y_R']
-# z_A = data_numerical['z_A']
-# z_C = data_numerical['z_C']
-# p_right = data_numerical['p_right']
-# Volt_start = data_numerical['Volt_start']
-# Volt_end = data_numerical['Volt_end']
-# n_Volts = data_numerical['n_Volts']
-# phi_left_dim = data_numerical['phi_left_dim']
-# phi_left_dimless = data_numerical['phi_left_dimless']
-# phi_right = data_numerical['phi_right']
-# y_A_num = data_numerical['y_A']
-# y_C_num = data_numerical['y_C']
-# y_S_num = data_numerical['y_S']
-# phi_num = data_numerical['phi']
-# p_num = data_numerical['p']
-# x_num = data_numerical['x']
-# Q_num = data_numerical['Q']
-# C_DL_num = data_numerical['C_DL']
-# phi_left_center = data_numerical['phi_left_center']
-
# Define Parameter
e0 = 1.602e-19 # [As]
k = 1.381e-23 # [J/K]
@@ -97,9 +61,10 @@ p_right = 0
# phi^L domain
Vol_start = 0
Volt_end = 0.75
-n_Volts = 10#0
+n_Volts = 25#0
-phi_left_dimless = np.linspace(Vol_start, Volt_end, n_Volts) * e0/(k*T)
+phi_left_dim = np.linspace(Vol_start, Volt_end, n_Volts)
+phi_left_dimless = phi_left_dim * e0/(k*T)
# Numerical calculations
@@ -116,46 +81,75 @@ for i, phi_bcs in enumerate(phi_left_dimless):
x_num.append(x_)
Q_num = []
+Q_num_dim_ = []
for j in range(len(phi_left_dimless)):
Q_num.append(Q_num_(y_A_num[j], y_C_num[j], n(p_num[j], K), x_num[j]))
+ Q_num_dim_.append(Q_num_dim(y_A_num[j], y_C_num[j], n(p_num[j], K), x_num[j], z_A, z_C, nR_m, e0, LR))
Q_num = np.array(Q_num)
+Q_num_dim_ = np.array(Q_num_dim_)
dx_ = phi_left_dimless[1] - phi_left_dimless[0] # [1/V], Assumption: phi^L is uniformly distributed
-C_DL_num = (Q_num[1:] - Q_num[:-1])/dx_ # [µAs/cm³]
-C_DL_num = np.array(C_DL_num)
C_dl_num = C_dl(Q_num, phi_left_dimless)
+dx_dim = phi_left_dim[1] - phi_left_dim[0]
+C_dl_num_dim_ = C_dl(Q_num_dim_, phi_left_dim)
+
# Analytical calculations
-# K = K_vec[0]
-# print(K)
-# print(phi_left_dimless)
Q_ana = Q_DL_dimless_ana(y_R, y_R, 1-2*y_R, z_A, z_C, phi_left_dimless, phi_right, p_right, K, Lambda2, a2, kappa)
-C_DL_ana = C_dl(Q_ana, phi_left_dimless)
+Q_ana_dim_ = Q_DL_dim_ana(y_R, y_R, 1-2*y_R, z_A, z_C, phi_left_dimless, phi_right, p_right, K, Lambda2, a2, nR_m, e0, LR, kappa)
+C_dl_ana = C_DL_dimless_ana(y_R, y_R, 1-2*y_R, z_A, z_C, Phi_pot_center(phi_left_dimless), phi_right, p_right, K, Lambda2, a2, kappa)
+C_dl_ana_dim = C_DL_dim_ana(y_R, y_R, 1-2*y_R, z_A, z_C, Phi_pot_center(phi_left_dimless), phi_right, p_right, K, Lambda2, a2, nR_m, e0, LR, k, T, kappa)
# Plotting
plt.figure()
-# plt.plot(phi_left, Q_num[1] - Q_ana, label='Difference')
-# plt.plot(phi_left_dimless, Q_num - Q_ana, label='Difference')
-# plt.plot(phi_left_dimless, Q_num[1], label='Numerical')
+# plt.plot(phi_left, Q_num - Q_ana, label='Difference')
+plt.title('Charge (dimensionless)')
plt.plot(phi_left_dimless, Q_num, label='Numerical')
plt.plot(phi_left_dimless, Q_ana, label='Analytical')
plt.grid()
plt.legend()
plt.xlabel('$\delta \\varphi$ [-]')
-plt.ylabel('$Q_{num} - Q_{ana} [-]$')
+# plt.ylabel('$Q_{num} - Q_{ana} [-]$')
+plt.ylabel('$Q[-]$')
plt.tight_layout()
plt.show()
plt.figure()
-# plt.plot(Phi_pot_center(phi_left), C_DL_num[1] - C_DL_ana, label='Difference')
-# plt.plot(Phi_pot_center(phi_left_dimless), C_DL_num - C_DL_ana, label='Difference')
-# plt.plot(Phi_pot_center(phi_left_dimless), C_DL_num[1], label='Numerical')
-plt.plot(Phi_pot_center(phi_left_dimless), C_DL_num, label='Numerical')
-plt.plot(Phi_pot_center(phi_left_dimless), C_DL_ana, label='Analytical')
+# plt.plot(phi_left, Q_num - Q_ana, label='Difference')
+plt.title('Charge (diomensions)')
+plt.plot(phi_left_dim, Q_num_dim_, label='Numerical')
+plt.plot(phi_left_dim, Q_ana_dim_, label='Analytical')
+plt.grid()
+plt.legend()
+plt.xlabel('$\delta \\varphi$ [-]')
+# plt.ylabel('$Q_{num} - Q_{ana} [-]$')
+plt.ylabel('$Q[µAs/cm³]$')
+plt.tight_layout()
+plt.show()
+
+plt.figure()
+# Use Center points to evaluate on same x-points
+plt.title('Charge (dimensionless)')
+plt.plot(Phi_pot_center(phi_left_dimless), C_dl_num, label='Numerical')
+plt.plot(Phi_pot_center(phi_left_dimless), C_dl_ana, label='Analytical')
+plt.grid()
+plt.legend()
+plt.xlabel('$\delta \\varphi$ [-]')
+# plt.ylabel('$C_{dl,num} - C_{dl,ana} [-]$')
+plt.ylabel('$C_{dl}[-]$')
+plt.tight_layout()
+plt.show()
+
+plt.figure()
+# Use Center points to evaluate on same x-points
+plt.title('Charge (dimensionless)')
+plt.plot(Phi_pot_center(phi_left_dim), C_dl_num_dim_, label='Numerical')
+plt.plot(Phi_pot_center(phi_left_dim), C_dl_ana_dim, label='Analytical')
plt.grid()
plt.legend()
plt.xlabel('$\delta \\varphi$ [-]')
-plt.ylabel('$C_{dl,num} - C_{dl,ana} [-]$')
+# plt.ylabel('$C_{dl,num} - C_{dl,ana} [-]$')
+plt.ylabel('$C_{dl}[µAs/cm²]$')
plt.tight_layout()
plt.show()
\ No newline at end of file
diff --git a/examples/ReproducableCode/DoubleLayerCapacity/IncompressibleValidationAnalyticalMethod.py b/examples/ReproducableCode/DoubleLayerCapacity/IncompressibleValidationAnalyticalMethod.py
index cd0ec8a03b34e1bf99594e9e21812474160977a9..06f249e3b45b210fae0bd770dd78c41c07db5606 100644
--- a/examples/ReproducableCode/DoubleLayerCapacity/IncompressibleValidationAnalyticalMethod.py
+++ b/examples/ReproducableCode/DoubleLayerCapacity/IncompressibleValidationAnalyticalMethod.py
@@ -17,7 +17,7 @@ sys.path.insert(0, src_path)
from Eq04 import solve_System_4eq
from Eq02 import solve_System_2eq
-from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana
+from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_num_dim, Q_DL_dimless_ana, Q_DL_dim_ana, C_DL_dimless_ana, C_DL_dim_ana
# Remove the src directory from sys.path after import
@@ -61,9 +61,10 @@ rtol = 1e-4 # ! Change back to 1e-8
# phi^L domain
Vol_start = 0.1 # ! Change back to 0
Volt_end = 0.75
-n_Volts = 10#0
+n_Volts = 30#0
-phi_left = np.linspace(Vol_start, Volt_end, n_Volts) * e0/(k*T)
+phi_left_dim = np.linspace(Vol_start, Volt_end, n_Volts)
+phi_left = phi_left_dim * e0/(k*T)
# Numerical calculations
@@ -80,45 +81,75 @@ for i, phi_bcs in enumerate(phi_left):
x_num.append(x_)
Q_num = []
+Q_num_dim_ = []
for j in range(len(phi_left)):
Q_num.append(Q_num_(y_A_num[j], y_C_num[j], n(p_num[j], K), x_num[j]))
+ Q_num_dim_.append(Q_num_dim(y_A_num[j], y_C_num[j], n(p_num[j], K), x_num[j], z_A, z_C, nR_m, e0, LR))
Q_num = np.array(Q_num)
+Q_num_dim_ = np.array(Q_num_dim_)
dx_ = phi_left[1] - phi_left[0] # [1/V], Assumption: phi^L is uniformly distributed
-C_DL_num = (Q_num[1:] - Q_num[:-1])/dx_ # [µAs/cm³]
-C_DL_num = np.array(C_DL_num)
C_dl_num = C_dl(Q_num, phi_left)
-
+dx_dim = phi_left_dim[1] - phi_left_dim[0]
+C_dl_num_dim_ = C_dl(Q_num_dim_, phi_left_dim)
# Analytical calculations
Q_ana = Q_DL_dimless_ana(y_R, y_R, 1-2*y_R, z_A, z_C, phi_left, phi_right, p_right, K, Lambda2, a2, kappa)
-C_DL_ana = C_dl(Q_ana, phi_left)
+Q_ana_dim_ = Q_DL_dim_ana(y_R, y_R, 1-2*y_R, z_A, z_C, phi_left, phi_right, p_right, K, Lambda2, a2, nR_m, e0, LR, kappa)
+C_dl_ana = C_DL_dimless_ana(y_R, y_R, 1-2*y_R, z_A, z_C, Phi_pot_center(phi_left), phi_right, p_right, K, Lambda2, a2, kappa)
+C_dl_ana_dim = C_DL_dim_ana(y_R, y_R, 1-2*y_R, z_A, z_C, Phi_pot_center(phi_left), phi_right, p_right, K, Lambda2, a2, nR_m, e0, LR, k, T, kappa)
# Plotting
plt.figure()
# plt.plot(phi_left, Q_num - Q_ana, label='Difference')
+plt.title('Charge (dimensionless)')
plt.plot(phi_left, Q_num, label='Numerical')
plt.plot(phi_left, Q_ana, label='Analytical')
plt.grid()
plt.legend()
plt.xlabel('$\delta \\varphi$ [-]')
-plt.ylabel('$Q_{num} - Q_{ana} [-]$')
+# plt.ylabel('$Q_{num} - Q_{ana} [-]$')
+plt.ylabel('$Q[-]$')
plt.tight_layout()
plt.show()
plt.figure()
-# plt.plot(Phi_pot_center(phi_left), C_DL_num - C_DL_ana, label='Difference')
+# plt.plot(phi_left, Q_num - Q_ana, label='Difference')
+plt.title('Charge (diomensions)')
+plt.plot(phi_left_dim, Q_num_dim_, label='Numerical')
+plt.plot(phi_left_dim, Q_ana_dim_, label='Analytical')
+plt.grid()
+plt.legend()
+plt.xlabel('$\delta \\varphi$ [-]')
+# plt.ylabel('$Q_{num} - Q_{ana} [-]$')
+plt.ylabel('$Q[µAs/cm³]$')
+plt.tight_layout()
+plt.show()
+
+plt.figure()
+# Use Center points to evaluate on same x-points
+plt.title('Charge (dimensionless)')
plt.plot(Phi_pot_center(phi_left), C_dl_num, label='Numerical')
-plt.plot(Phi_pot_center(phi_left), C_DL_ana, label='Analytical')
+plt.plot(Phi_pot_center(phi_left), C_dl_ana, label='Analytical')
plt.grid()
plt.legend()
plt.xlabel('$\delta \\varphi$ [-]')
-plt.ylabel('$C_{dl,num} - C_{dl,ana} [-]$')
+# plt.ylabel('$C_{dl,num} - C_{dl,ana} [-]$')
+plt.ylabel('$C_{dl}[-]$')
plt.tight_layout()
plt.show()
-print('phi_left:', phi_left)
-# print('Q_num:', Q_num)
-print('Q_ana:', Q_ana)
\ No newline at end of file
+plt.figure()
+# Use Center points to evaluate on same x-points
+plt.title('Charge (dimensionless)')
+plt.plot(Phi_pot_center(phi_left_dim), C_dl_num_dim_, label='Numerical')
+plt.plot(Phi_pot_center(phi_left_dim), C_dl_ana_dim, label='Analytical')
+plt.grid()
+plt.legend()
+plt.xlabel('$\delta \\varphi$ [-]')
+# plt.ylabel('$C_{dl,num} - C_{dl,ana} [-]$')
+plt.ylabel('$C_{dl}[µAs/cm²]$')
+plt.tight_layout()
+plt.show()
diff --git a/examples/ReproducableCode/DoubleLayerCapacity/Lambda.py b/examples/ReproducableCode/DoubleLayerCapacity/Lambda.py
index a3e06a941594cbc6c5a0d79cf81e31e68e3e2097..835c529d939628eb171e05a01f14c5c2f604b704 100644
--- a/examples/ReproducableCode/DoubleLayerCapacity/Lambda.py
+++ b/examples/ReproducableCode/DoubleLayerCapacity/Lambda.py
@@ -13,7 +13,7 @@ import os
src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../../..', 'src')
sys.path.insert(0, src_path)
-from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana
+from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana, C_DL_dimless_ana, C_DL_dim_ana
# Remove the src directory from sys.path after import
@@ -69,8 +69,12 @@ for Lambda2 in Lambda2_vec:
Q_DL_dimless_.append(Q_DL_dimless_ana(y_A_R, y_C_R, y_N_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, kappa))
Q_DL_dim_.append(Q_DL_dim_ana(y_A_R, y_C_R, y_N_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, nR_m, e0, LR, kappa))
-C_DL_dim = [C_dl(q_dl, Phi_Pot_Diff_dim) for q_dl in Q_DL_dim_]
-C_DL_dimless = [C_dl(q_dl, Phi_Pot_Diff_dimless) for q_dl in Q_DL_dimless_]
+# Double layer capacity
+C_DL_dim_ = []
+C_DL_dimless_ = []
+for Lambda2 in Lambda2_vec:
+ C_DL_dim_.append(C_DL_dim_ana(y_A_R, y_C_R, 1-y_A_R-y_C_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, nR_m, e0, LR, k, T, kappa))
+ C_DL_dimless_.append(C_DL_dimless_ana(y_A_R, y_C_R, 1-y_A_R-y_C_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, kappa))
@@ -109,8 +113,6 @@ fig.show()
# Double Layer Capacity
-Phi_pot_center_array_dimless = Phi_pot_center(Phi_Pot_Diff_dimless)
-Phi_pot_center_array_dim = Phi_pot_center(Phi_Pot_Diff_dim)
fig = plt.figure()
color_dimless = 'tab:purple'
color_dim = 'tab:red'
@@ -118,16 +120,16 @@ ax = fig.add_subplot(111, label="1")
ax2 = fig.add_subplot(111, label="2", frame_on=False)
# Plot dimensional data
-for i, c_dl in enumerate(C_DL_dim):
- ax.plot(Phi_pot_center_array_dim, c_dl, color=colors[i], label=f'$\lambda^2$: {Lambda2_vec[i]}')
+for i, c_dl in enumerate(C_DL_dim_):
+ ax.plot(Phi_Pot_Diff_dim, c_dl, color=colors[i], label=f'$\lambda^2$: {Lambda2_vec[i]}')
ax.grid()
ax.set_xlabel('$\delta \\varphi [nm]$', color=color_dim)
ax.set_ylabel('$C_{dl}$ [\u03bc$F/cm^2]$', color=color_dim)
ax.tick_params(axis='x', colors=color_dim)
ax.tick_params(axis='y', colors=color_dim)
-for i, c_dl in enumerate(C_DL_dimless):
- ax2.plot(Phi_pot_center_array_dimless, c_dl, color=colors[i])
+for i, c_dl in enumerate(C_DL_dimless_):
+ ax2.plot(Phi_Pot_Diff_dimless, c_dl, color=colors[i])
# ax2.grid()
ax2.xaxis.tick_top()
ax2.yaxis.tick_right()
@@ -143,4 +145,4 @@ fig.tight_layout()
fig.show()
# Save the results
-np.savez('../../Data/DoubleLayerCapacity/Lambda.npz', Lambda2_vec=Lambda2_vec, a2=a2, K=K, kappa=kappa, z_A=z_A, z_C=z_C, phi_R=phi_R, p_R=p_R, Vol_start=Vol_start, Volt_end=Volt_end, n_Volts=n_Volts, Phi_pot_center_array_dimless=Phi_pot_center_array_dimless, Phi_pot_center_array_dim=Phi_pot_center_array_dim, Phi_Pot_Diff_dim=Phi_Pot_Diff_dim, Phi_Pot_Diff_dimless=Phi_Pot_Diff_dimless, C_DL_dim=C_DL_dim, C_DL_dimless=C_DL_dimless, Q_DL_dim_=Q_DL_dim_, Q_DL_dimless_=Q_DL_dimless_, Molarity=Molarity)
+np.savez('../../Data/DoubleLayerCapacity/Lambda.npz', Lambda2_vec=Lambda2_vec, a2=a2, K=K, kappa=kappa, z_A=z_A, z_C=z_C, phi_R=phi_R, p_R=p_R, Vol_start=Vol_start, Volt_end=Volt_end, n_Volts=n_Volts, Phi_Pot_Diff_dim=Phi_Pot_Diff_dim, Phi_Pot_Diff_dimless=Phi_Pot_Diff_dimless, C_DL_dim=C_DL_dim_, C_DL_dimless=C_DL_dimless_, Q_DL_dim_=Q_DL_dim_, Q_DL_dimless_=Q_DL_dimless_, Molarity=Molarity)
diff --git a/examples/ReproducableCode/DoubleLayerCapacity/Molarity.py b/examples/ReproducableCode/DoubleLayerCapacity/Molarity.py
index ab531e1bdd5f08a9b625b02f03cc41674987b4f7..20c453b107fe86d19ba66ae60e9fb9f5bdd9685b 100644
--- a/examples/ReproducableCode/DoubleLayerCapacity/Molarity.py
+++ b/examples/ReproducableCode/DoubleLayerCapacity/Molarity.py
@@ -13,7 +13,7 @@ import os
src_path = os.path.join(os.path.dirname(os.path.abspath(__file__)), '../../..', 'src')
sys.path.insert(0, src_path)
-from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana
+from Helper_DoubleLayerCapacity import Phi_pot_center, dx, C_dl, n, Q_num_, Q_DL_dimless_ana, Q_DL_dim_ana, C_DL_dimless_ana, C_DL_dim_ana
# Remove the src directory from sys.path after import
del sys.path[0]
@@ -66,8 +66,14 @@ for mol in Molarity:
Q_DL_dimless_.append(Q_DL_dimless_ana(y_A_R, y_C_R, y_N_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, kappa))
Q_DL_dim_.append(Q_DL_dim_ana(y_A_R, y_C_R, y_N_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, nR_m, e0, LR, kappa))
-C_DL_dim = [C_dl(q_dl, Phi_Pot_Diff_dim) for q_dl in Q_DL_dim_]
-C_DL_dimless = [C_dl(q_dl, Phi_Pot_Diff_dimless) for q_dl in Q_DL_dimless_]
+# Double layer capacity
+C_DL_dim_ = []
+C_DL_dimless_ = []
+for mol in Molarity:
+ y_A_R, y_C_R = mol / nR_mol, mol / nR_mol
+ y_N_R = 1 - y_A_R - y_C_R
+ C_DL_dim_.append(C_DL_dim_ana(y_A_R, y_C_R, 1-y_A_R-y_C_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, nR_m, e0, LR, k, T, kappa))
+ C_DL_dimless_.append(C_DL_dimless_ana(y_A_R, y_C_R, 1-y_A_R-y_C_R, z_A, z_C, Phi_Pot_Diff_dimless, phi_R, p_R, K, Lambda2, a2, kappa))
@@ -106,8 +112,6 @@ fig.show()
# Double Layer Capacity
-Phi_pot_center_array_dimless = Phi_pot_center(Phi_Pot_Diff_dimless)
-Phi_pot_center_array_dim = Phi_pot_center(Phi_Pot_Diff_dim)
fig = plt.figure()
color_dimless = 'tab:purple'
color_dim = 'tab:red'
@@ -115,16 +119,16 @@ ax = fig.add_subplot(111, label="1")
ax2 = fig.add_subplot(111, label="2", frame_on=False)
# Plot dimensional data
-for i, c_dl in enumerate(C_DL_dim):
- ax.plot(Phi_pot_center_array_dim, c_dl, color=colors[i], label=f'M: {Molarity[i]}')
+for i, c_dl in enumerate(C_DL_dim_):
+ ax.plot(Phi_Pot_Diff_dim, c_dl, color=colors[i], label=f'M: {Molarity[i]}')
ax.grid()
ax.set_xlabel('$\delta \\varphi [nm]$', color=color_dim)
ax.set_ylabel('$C_{dl}$ [\u03bc$F/cm^2]$', color=color_dim)
ax.tick_params(axis='x', colors=color_dim)
ax.tick_params(axis='y', colors=color_dim)
-for i, c_dl in enumerate(C_DL_dimless):
- ax2.plot(Phi_pot_center_array_dimless, c_dl, color=colors[i])
+for i, c_dl in enumerate(C_DL_dimless_):
+ ax2.plot(Phi_Pot_Diff_dimless, c_dl, color=colors[i])
# ax2.grid()
ax2.xaxis.tick_top()
ax2.yaxis.tick_right()
@@ -140,4 +144,4 @@ fig.tight_layout()
fig.show()
# Save the results
-np.savez('../../Data/DoubleLayerCapacity/Molarity.npz', Lambda2=Lambda2, a2=a2, K=K, kappa=kappa, z_A=z_A, z_C=z_C, phi_R=phi_R, p_R=p_R, Vol_start=Vol_start, Phi_pot_center_array_dimless=Phi_pot_center_array_dimless, Phi_pot_center_array_dim=Phi_pot_center_array_dim, Volt_end=Volt_end, n_Volts=n_Volts, Phi_Pot_Diff_dim=Phi_Pot_Diff_dim, Phi_Pot_Diff_dimless=Phi_Pot_Diff_dimless, C_DL_dim=C_DL_dim, C_DL_dimless=C_DL_dimless, Q_DL_dim_=Q_DL_dim_, Q_DL_dimless_=Q_DL_dimless_, Molarity=Molarity)
+np.savez('../../Data/DoubleLayerCapacity/Molarity.npz', Lambda2=Lambda2, a2=a2, K=K, kappa=kappa, z_A=z_A, z_C=z_C, phi_R=phi_R, p_R=p_R, Vol_start=Vol_start, Volt_end=Volt_end, n_Volts=n_Volts, Phi_Pot_Diff_dim=Phi_Pot_Diff_dim, Phi_Pot_Diff_dimless=Phi_Pot_Diff_dimless, C_DL_dim=C_DL_dim_, C_DL_dimless=C_DL_dimless_, Q_DL_dim_=Q_DL_dim_, Q_DL_dimless_=Q_DL_dimless_, Molarity=Molarity)
diff --git a/examples/Visualizations/DoubleLayerCapacity.py/VisCompressibility.py b/examples/Visualizations/DoubleLayerCapacity.py/VisCompressibility.py
index 1849af8c96e7c06af81b0c76167b040213691829..998a8d226b2dea813e6e74df4b420913646d5afd 100644
--- a/examples/Visualizations/DoubleLayerCapacity.py/VisCompressibility.py
+++ b/examples/Visualizations/DoubleLayerCapacity.py/VisCompressibility.py
@@ -20,8 +20,6 @@ C_DL_dim = data['C_DL_dim']
C_DL_dimless = data['C_DL_dimless']
Phi_Pot_Diff_dim = data['Phi_Pot_Diff_dim']
Phi_Pot_Diff_dimless = data['Phi_Pot_Diff_dimless']
-Phi_pot_center_array_dim = data['Phi_pot_center_array_dim']
-Phi_pot_center_array_dimless = data['Phi_pot_center_array_dimless']
K_vec = data['K_vec']
@@ -57,7 +55,7 @@ for i, q_dl in enumerate(Q_DL_dim_):
ax1_dim.xaxis.tick_top()
ax1_dim.yaxis.tick_right()
ax1_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
-ax1_dim.set_ylabel('$Q$ [\u03bc$F/cm^3]$', color=color_dim, fontsize=labelsize)
+ax1_dim.set_ylabel('$Q$ [\u03bc$C/cm^3]$', color=color_dim, fontsize=labelsize)
ax1_dim.xaxis.set_label_position('top')
ax1_dim.yaxis.set_label_position('right')
ax1_dim.tick_params(axis='x', colors=color_dim, labelsize=labelsize)
@@ -66,7 +64,7 @@ ax1_dim.tick_params(axis='y', colors=color_dim, labelsize=labelsize)
# Double Layer Capacity
# Plot dimensional data
for i, c_dl in enumerate(C_DL_dimless):
- ax2_dimless.plot(Phi_pot_center_array_dimless, c_dl, color=colors[i], lw=lw)
+ ax2_dimless.plot(Phi_Pot_Diff_dimless, c_dl, color=colors[i], lw=lw)
ax2_dimless.grid()
ax2_dimless.set_xlabel('$\delta \\varphi [-]$', color=color_dimless, fontsize=labelsize)
ax2_dimless.set_ylabel('$C_{dl} [-]$', color=color_dimless, fontsize=labelsize)
@@ -74,7 +72,7 @@ ax2_dimless.tick_params(axis='x', colors=color_dimless, labelsize=labelsize)
ax2_dimless.tick_params(axis='y', colors=color_dimless, labelsize=labelsize)
for i, c_dl in enumerate(C_DL_dim):
- ax2_dim.plot(Phi_pot_center_array_dim, c_dl, color=colors[i], lw=lw)
+ ax2_dim.plot(Phi_Pot_Diff_dim, c_dl, color=colors[i], lw=lw)
ax2_dim.xaxis.tick_top()
ax2_dim.yaxis.tick_right()
ax2_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
diff --git a/examples/Visualizations/DoubleLayerCapacity.py/VisIncompressible.py b/examples/Visualizations/DoubleLayerCapacity.py/VisIncompressible.py
index 1ce5786f8a7524c0c634aa2e9780ddc6cb66afdd..f003af58f34b45cb973aea4adc241b1ec3365157 100644
--- a/examples/Visualizations/DoubleLayerCapacity.py/VisIncompressible.py
+++ b/examples/Visualizations/DoubleLayerCapacity.py/VisIncompressible.py
@@ -20,8 +20,6 @@ Molarity = data_Molarity['Molarity']
Q_DL_dimless_ = data_Molarity['Q_DL_dimless_']
Phi_Pot_Diff_dimless = data_Molarity['Phi_Pot_Diff_dimless']
C_DL_dim = data_Molarity['C_DL_dim']
-Phi_pot_center_array_dimless = data_Molarity['Phi_pot_center_array_dimless']
-Phi_pot_center_array_dim = data_Molarity['Phi_pot_center_array_dim']
C_DL_dimless = data_Molarity['C_DL_dimless']
data_PB = np.load('../../Data/DoubleLayerCapacity/PB.npz')
@@ -68,7 +66,7 @@ for i, q_dl in enumerate(Q_DL_dim_):
ax1_dim.xaxis.tick_top()
ax1_dim.yaxis.tick_right()
ax1_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
-ax1_dim.set_ylabel('$Q$ [\u03bc$F/cm^3]$', color=color_dim, fontsize=labelsize)
+ax1_dim.set_ylabel('$Q$ [\u03bc$C/cm^3]$', color=color_dim, fontsize=labelsize)
ax1_dim.xaxis.set_label_position('top')
ax1_dim.yaxis.set_label_position('right')
ax1_dim.tick_params(axis='x', colors=color_dim, labelsize=labelsize)
@@ -79,7 +77,7 @@ ax1_dim.set_ylim(np.min(Q_DL_dim_)*10/9, np.max(Q_DL_dim_)*10/9)
# Double Layer Capacity
ax2_dimless.plot(Phi_pot_center_array_dimless_PB, C_DL_dimless_PB, '--', color=color_PB, lw=lw, alpha=0.5)
for i, c_dl in enumerate(C_DL_dimless):
- ax2_dimless.plot(Phi_pot_center_array_dimless, c_dl, color=colors[i], lw=lw)
+ ax2_dimless.plot(Phi_Pot_Diff_dimless, c_dl, color=colors[i], lw=lw)
ax2_dimless.grid()
ax2_dimless.set_xlabel('$\delta \\varphi [-]$', color=color_dimless, fontsize=labelsize)
ax2_dimless.set_ylabel('$C_{dl} [-]$', color=color_dimless, fontsize=labelsize)
@@ -88,7 +86,7 @@ ax2_dimless.tick_params(axis='y', colors=color_dimless, labelsize=labelsize)
ax2_dimless.set_ylim(np.min(C_DL_dimless)*(-10/9), np.max(C_DL_dimless)*10/9)
for i, c_dl in enumerate(C_DL_dim):
- ax2_dim.plot(Phi_pot_center_array_dim, c_dl, color=colors[i], lw=lw)
+ ax2_dim.plot(Phi_Pot_Diff_dim, c_dl, color=colors[i], lw=lw)
ax2_dim.xaxis.tick_top()
ax2_dim.yaxis.tick_right()
ax2_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
@@ -101,7 +99,7 @@ ax2_dim.set_ylim(np.min(C_DL_dim)*(-10/9), np.max(C_DL_dim)*10/9)
lines_labels = [ax.get_legend_handles_labels() for ax in fig.axes]
lines, labels = [sum(lol, []) for lol in zip(*lines_labels)]
-fig.legend(lines, labels, bbox_to_anchor=(0.735,1.03), ncol=6, fontsize=labelsize)
+fig.legend(lines, labels, bbox_to_anchor=(0.81,1.03), ncol=6, fontsize=labelsize)
fig.tight_layout()
fig.savefig('../../Figures/DoubleLayerCapacity/DLKap_Molarity.svg', bbox_inches='tight')
fig.show()
@@ -163,7 +161,7 @@ for i, q_dl in enumerate(Q_DL_dim_):
ax1_dim.xaxis.tick_top()
ax1_dim.yaxis.tick_right()
ax1_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
-ax1_dim.set_ylabel('$Q$ [\u03bc$F/cm^3]$', color=color_dim, fontsize=labelsize)
+ax1_dim.set_ylabel('$Q$ [\u03bc$C/cm^3]$', color=color_dim, fontsize=labelsize)
ax1_dim.xaxis.set_label_position('top')
ax1_dim.yaxis.set_label_position('right')
ax1_dim.tick_params(axis='x', colors=color_dim, labelsize=labelsize)
@@ -216,8 +214,6 @@ Lambda2_vec = data_Lambda['Lambda2_vec']
Q_DL_dimless_ = data_Lambda['Q_DL_dimless_']
Phi_Pot_Diff_dimless = data_Lambda['Phi_Pot_Diff_dimless']
C_DL_dim = data_Lambda['C_DL_dim']
-Phi_pot_center_array_dim = data_Lambda['Phi_pot_center_array_dim']
-Phi_pot_center_array_dimless = data_Lambda['Phi_pot_center_array_dimless']
C_DL_dimless = data_Lambda['C_DL_dimless']
# Plotting
@@ -249,7 +245,7 @@ for i, q_dl in enumerate(Q_DL_dim_):
ax1_dim.xaxis.tick_top()
ax1_dim.yaxis.tick_right()
ax1_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
-ax1_dim.set_ylabel('$Q$ [\u03bc$F/cm^3]$', color=color_dim, fontsize=labelsize)
+ax1_dim.set_ylabel('$Q$ [\u03bc$C/cm^3]$', color=color_dim, fontsize=labelsize)
ax1_dim.xaxis.set_label_position('top')
ax1_dim.yaxis.set_label_position('right')
ax1_dim.tick_params(axis='x', colors=color_dim, labelsize=labelsize)
@@ -257,7 +253,7 @@ ax1_dim.tick_params(axis='y', colors=color_dim, labelsize=labelsize)
# Double Layer Capacity
for i, c_dl in enumerate(C_DL_dimless):
- ax2_dimless.plot(Phi_pot_center_array_dimless, c_dl, color=colors[i], lw=lw)
+ ax2_dimless.plot(Phi_Pot_Diff_dimless, c_dl, color=colors[i], lw=lw)
ax2_dimless.grid()
ax2_dimless.set_xlabel('$\delta \\varphi [-]$', color=color_dimless, fontsize=labelsize)
ax2_dimless.set_ylabel('$C_{dl} [-]$', color=color_dimless, fontsize=labelsize)
@@ -265,7 +261,7 @@ ax2_dimless.tick_params(axis='x', colors=color_dimless, labelsize=labelsize)
ax2_dimless.tick_params(axis='y', colors=color_dimless, labelsize=labelsize)
for i, c_dl in enumerate(C_DL_dim):
- ax2_dim.plot(Phi_pot_center_array_dim, c_dl, color=colors[i], lw=lw)
+ ax2_dim.plot(Phi_Pot_Diff_dim, c_dl, color=colors[i], lw=lw)
ax2_dim.xaxis.tick_top()
ax2_dim.yaxis.tick_right()
ax2_dim.set_xlabel('$\delta \\varphi [V]$', color=color_dim, fontsize=labelsize)
diff --git a/examples/Visualizations/VisElectrolyticDiode.py b/examples/Visualizations/VisElectrolyticDiode.py
index c078eedd6fa3b1bc641605225fa201803f18b3f0..708d8da4477ce71e4b0b3a8a9e5437f4356bb5a5 100644
--- a/examples/Visualizations/VisElectrolyticDiode.py
+++ b/examples/Visualizations/VisElectrolyticDiode.py
@@ -87,7 +87,7 @@ for bias, (y_A, y_C, y_S, phi, p, x, y) in enumerate(zip(y_A_packed, y_C_packed,
axs[bias,2].set_title(f'$y_C \in ({round(np.min(y_C),2)},{round(np.max(y_C), 2)})$', fontsize=titlesize)
c = axs[bias,3].tricontourf(x, y, y_S, cmap=color_theme_solvent, levels=vmap_solvent)
- c.set_clim(0, 1)
+ # c.set_clim(0, 1)
cbar = fig.colorbar(c, ax=axs[bias,3])
cbar.ax.tick_params(labelsize=labelsize)
axs[bias,3].tricontour(x, y, y_S, colors='black', levels=levels_contour)
diff --git a/src/Helper_DoubleLayerCapacity.py b/src/Helper_DoubleLayerCapacity.py
index bcc81905999e85a716c49b5e7d2241a3369afa87..e47b37efa121e9c1b5ae89707ed25366f1a01fcc 100644
--- a/src/Helper_DoubleLayerCapacity.py
+++ b/src/Helper_DoubleLayerCapacity.py
@@ -153,10 +153,7 @@ def Q_DL_dimless_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float
'''
Calculates charge of the system using the analytical method in dimensionless units
- Q = sgn(φᴸ −φᴿ)λ√(2/(κ+1))(√(Λᴸ)−√(Λᴿ))
-
- with
- Λ = ln(∑_α D_α exp(−z_α φ + Eₚ a²)
+ Q = λ^2(∂ₓ φᴸ − ∂ₓ φᴿ)
Parameters
----------
@@ -190,48 +187,22 @@ def Q_DL_dimless_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float
float
Charge of the system in dimensionless units
'''
+ if solvation != 0:
+ raise ValueError('Solvation number must be 0 for the analytical method')
z_N = 0
E_p = p_R
if K == 'incompressible':
# Assume E_p = p_R = 0
- D_A = y_A_R / (np.exp(-(solvation+1)*a2*p_R-z_A*phi_R))
- D_C = y_C_R / (np.exp(-(solvation+1)*a2*p_R-z_C*phi_R))
+ D_A = y_A_R / (np.exp(-a2*p_R-z_A*phi_R))
+ D_C = y_C_R / (np.exp(-a2*p_R-z_C*phi_R))
D_N = y_N_R / (np.exp(-a2*p_R))
- # fsolve method
- # Q_DL = []
- # dx_phi_L = 0
- # for phi_L_ in phi_L:
- # def func(p_L):
- # # A_Term = D_A * np.exp(-z_A*phi_L)
- # A_Term = D_A * np.exp(-(solvation+1)*a2*p_L-z_A*phi_L_)
- # C_Term = D_C * np.exp(-(solvation+1)*a2*p_L-z_C*phi_L_)
- # N_Term = D_N * np.exp(-a2*p_L)
-
- # return (A_Term + C_Term + N_Term) - 1
- # p_L = fsolve(func, dx_phi_L)
- # dx_phi_L = np.sqrt(p_L - E_p) * np.sqrt(2*a2/Lambda2)
- # Q_DL_ = Lambda2 * dx_phi_L
- # Q_DL.append(Q_DL_)
- # Q_DL = np.array(Q_DL)
-
- # Fixed Point Iteration
- # def func_L(p_L):
- # CLambda = D_A * np.exp(-z_A * phi_L - solvation*a2*p_L) + D_C * np.exp(-z_C * phi_L - solvation*a2*p_L) + D_N
- # return np.log(CLambda)/a2
-
- # p_L = fixed_point(func_L, 0, maxiter=5_000)
- # Lambda = np.sqrt(Lambda2)
- # a = np.sqrt(a2)
- # Q_DL = np.sqrt(2) * Lambda * a * np.sqrt(p_L - E_p)
-
- # ! ToDo: Implement full analytical for solvation == 0
# Analytical method
E_p = p_R
- CLambda_L = np.log(D_A * np.exp(-z_A * phi_L - (solvation+1) * E_p) + D_C * np.exp(-z_C * phi_L - (solvation+1) * E_p) + D_N * np.exp(-z_N * phi_L - (solvation+1) * E_p))
- # CLambda_R = np.log(D_A * np.exp(-z_A * phi_R + E_p * a2) + D_C * np.exp(-z_C * phi_R + E_p * a2) + D_N * np.exp(-z_N * phi_R + E_p * a2))
+ CLambda_L = np.log(D_A * np.exp(-z_A * phi_L - E_p) + D_C * np.exp(-z_C * phi_L - E_p) + D_N * np.exp(-z_N * phi_L - E_p))
+
Lambda = np.sqrt(Lambda2)
- Q_DL = (phi_L-phi_R) / np.abs(phi_L-phi_R) * Lambda * np.sqrt(2/(solvation+1)) * (np.sqrt(CLambda_L))# - np.sqrt(CLambda_R)) # dx_phi_R -> 0
+ Q_DL = (phi_L-phi_R) / np.abs(phi_L-phi_R) * Lambda * np.sqrt(2) * (np.sqrt(CLambda_L))
return Q_DL
else:
C_A = y_A_R / ((K + p_R - 1)**(-1*a2*K)*np.exp(-z_A*phi_R))
@@ -239,10 +210,8 @@ def Q_DL_dimless_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float
C_N = y_N_R / ((K + p_R - 1)**(-1*a2*K)*np.exp(-z_N*phi_R))
CLambda_L = C_A * np.exp(-z_A*phi_L) + C_C * np.exp(-z_C*phi_L) + C_N
- # CLambda_R = C_A * np.exp(-z_A*phi_R) + C_C * np.exp(-z_C*phi_R) + C_N
Left = np.sqrt((1/CLambda_L)**(-1/(a2*K)) + 1 - K - E_p)
- # Right = np.sqrt((1/CLambda_R)**(-1/(a2*K)) + 1 - K - E_p)
Lambda = np.sqrt(Lambda2)
a = np.sqrt(a2)
@@ -255,10 +224,7 @@ def Q_DL_dim_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float, ph
'''
Calculates charge of the system using the analytical method in dimensionless units
- Q = sgn(φᴸ −φᴿ)λ√(2/(κ+1))(√(Λᴸ)−√(Λᴿ))
-
- with
- Λ = ln(∑_α D_α exp(−z_α φ + Eₚ a²)
+ Q = λ^2(∂ₓ φᴸ − ∂ₓ φᴿ)
Parameters
----------
@@ -273,7 +239,7 @@ def Q_DL_dim_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float, ph
z_C : float
Charge number of cations
phi_L : float
- Value of electric potential at the left boundary
+ Value of electric potential at the left boundary (dimensionless units)
phi_R : float
Value of electric potential at the right
p_R : float
@@ -302,4 +268,144 @@ def Q_DL_dim_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float, ph
Q_DL *= nR_m * e0 * LR
Q_DL *= 1e+6
Q_DL *= 1/(1e+4)
- return Q_DL
\ No newline at end of file
+ return Q_DL
+
+
+
+def C_DL_dimless_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float, phi_L:float, phi_R:float, p_R:float, K:str|float, Lambda2:float, a2:float, solvation:float) -> float:
+ '''
+ Calculates the double layer capacity of the system using the analytical method in dimensionless units
+
+ C_dl = ∂Q/∂ φᴸ
+
+ Parameters
+ ----------
+ y_A_R : float
+ Value of anion fraction at the right boundary
+ y_C_R : float
+ Value of cation fraction at the right boundary
+ y_N_R : float
+ Value of neutral fraction at the right boundary
+ z_A : float
+ Charge number of anions
+ z_C : float
+ Charge number of cations
+ phi_L : float
+ Electric potential at the left boundary [V]
+ phi_R : float
+ Electric potential at the right boundary
+ p_R : float
+ Pressure at the right boundary
+ K : str | float
+ Bulk modulus, use 'incompressible' for an incompressible mixture
+ Lambda2 : float
+ Dimensionless parameter
+ a2 : float
+ Dimensionless parameter
+ solvation : float
+ Solvation number
+
+ Returns
+ -------
+ float
+ Double layer capacity of the system in dimensionless units
+ '''
+ if solvation != 0:
+ raise ValueError('Solvation number must be 0 for the analytical method')
+ z_N = 0
+ E_p = p_R
+ if K == 'incompressible':
+ # Assume E_p = p_R = 0
+ D_A = y_A_R / (np.exp(-a2*p_R-z_A*phi_R))
+ D_C = y_C_R / (np.exp(-a2*p_R-z_C*phi_R))
+ D_N = y_N_R / (np.exp(-a2*p_R))
+
+ CLambda_L = np.log(D_A * np.exp(-z_A * phi_L - E_p * a2) + D_C * np.exp(-z_C * phi_L - E_p * a2) + D_N * np.exp(-z_N * phi_L - E_p * a2))
+
+ xi_CLambda_L = D_A * np.exp(-z_A * phi_L - E_p * a2) + D_C * np.exp(-z_C * phi_L - E_p * a2) + D_N * np.exp(-z_N * phi_L - E_p * a2)
+ dxi_CLambda_L = - D_A * z_A * np.exp(-z_A * phi_L - E_p * a2) - D_C * z_C * np.exp(-z_C * phi_L - E_p * a2) - D_N * z_N * np.exp(-z_N * phi_L - E_p * a2)
+
+ Lambda = np.sqrt(Lambda2)
+
+ PreTerm = (phi_L-phi_R) / np.abs(phi_L-phi_R) * Lambda * np.sqrt(2)
+ DerivedTerm = dxi_CLambda_L / (2 * xi_CLambda_L * np.sqrt(np.log(xi_CLambda_L)))
+
+ return PreTerm * DerivedTerm
+ else:
+ C_A = y_A_R / ((K + p_R - 1)**(-1*a2*K)*np.exp(-z_A*phi_R))
+ C_C = y_C_R / ((K + p_R - 1)**(-1*a2*K)*np.exp(-z_C*phi_R))
+ C_N = y_N_R / ((K + p_R - 1)**(-1*a2*K)*np.exp(-z_N*phi_R))
+
+ z = 1 - K - E_p
+ y = 1/(a2*K)
+
+ CLambda = C_A * np.exp(-z_A*phi_L) + C_C * np.exp(-z_C*phi_L) + C_N * np.exp(-z_N*phi_L)
+ dClambda = -C_A * z_A * np.exp(-z_A*phi_L) - C_C * z_C * np.exp(-z_C*phi_L) - C_N * z_N * np.exp(-z_N*phi_L)
+
+ Lambda = np.sqrt(Lambda2)
+ a = np.sqrt(a2)
+
+ DerivedTerm = y * CLambda**(y-1) * dClambda / (2 * np.sqrt(CLambda**y + z))
+
+ PreTerm = (phi_L-phi_R) / np.abs(phi_L-phi_R) * Lambda * a * np.sqrt(2)
+
+ return PreTerm * DerivedTerm
+ raise ValueError('Invalid input for K')
+
+
+
+def C_DL_dim_ana(y_A_R:float, y_C_R:float, y_N_R:float, z_A:float, z_C:float, phi_L:float, phi_R:float, p_R:float, K:str|float, Lambda2:float, a2:float, nR_m:float, e0:float, LR:float, k:float, T:float, solvation:float) -> float:
+ '''
+ Calculates the double layer capacity of the system using the analytical method in dimensionless units
+
+ C_dl = ∂Q/∂ φᴸ
+
+ Parameters
+ ----------
+ y_A_R : float
+ Value of anion fraction at the right boundary
+ y_C_R : float
+ Value of cation fraction at the right boundary
+ y_N_R : float
+ Value of neutral fraction at the right boundary
+ z_A : float
+ Charge number of anions
+ z_C : float
+ Charge number of cations
+ phi_L : float
+ Value of electric potential at the left boundary (dimensionless units)
+ phi_R : float
+ Value of electric potential at the right
+ p_R : float
+ Value of pressure at the right boundary
+ K : str | float
+ Bulk modulus, use 'incompressible' for an incompress
+ Lambda2 : float
+ Dimensionless parameter
+ a2 : float
+ Dimensionless parameter
+ nR_m : float
+ Reference number density in 1/m^3
+ e0 : float
+ Dielectric constant
+ LR : float
+ Reference length in m
+ k : float
+ Boltzmann constant
+ T : float
+ Temperature
+ solvation : float
+ Solvation number
+
+ Returns
+ -------
+ float
+ Double layer capacity of the system in µAs/cm²
+ '''
+ C_DL = C_DL_dimless_ana(y_A_R, y_C_R, y_N_R, z_A, z_C, phi_L, phi_R, p_R, K, Lambda2, a2, solvation)
+ C_DL *= nR_m * e0 * LR
+ # ! ToDo
+ C_DL *= 1e+6
+ C_DL *= 1/(1e+4)
+ C_DL *= 1 /(k*T/e0)
+ return C_DL
\ No newline at end of file
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