Updated Ecker2015 diffusivity to better match the original paper

pybamm/CITATIONS.bib

@@ -783,3 +783,11 @@ @article{Wycisk2022
   author = {Dominik Wycisk and Marc Oldenburger and Marc Gerry Stoye and Toni Mrkonjic and Arnulf Latz},
   keywords = {Lithium-ion battery, Voltage hysteresis, Plett-model, Silicon–graphite anode},
 }
+@article{Yuan2023,
+  title = {A electrochemical-electro-thermal coupled computational framework to simulate the performance of Li-ion batteries at cell-level: Analysis on the thermal effects},
+  journal = {arXiv preprint},
+  year = {2023},
+  doi = {https://doi.org/10.48550/arXiv.2303.09838},
+  url = {https://arxiv.org/abs/2303.09838},
+  author = {Han Yuan and Shen Li and Tao Zhu and Simon O'Kane and Carlos Garcia and Gregory Offer and Monica Marinescu},
+}

pybamm/input/parameters/lithium_ion/Ecker2015.py

@@ -31,7 +31,15 @@ def graphite_diffusivity_Ecker2015(sto, T):
         Solid diffusivity
     """
 
-    D_ref = 8.4e-13 * np.exp(-11.3 * sto) + 8.2e-15
+    log_D_ref = (
+        -15.06
+        + 2.722 * np.exp(-((sto / 0.0777) ** 2))
+        + 1.503 * np.exp(-(((sto - 0.17) / 0.1153) ** 2))
+        + 1.055 * np.exp(-(((sto - 0.3118) / 0.04089) ** 2))
+        + 1.555 * np.exp(-(((sto - 0.6061) / 0.04545) ** 2))
+        + 1.215 * np.exp(-(((sto - 1) / 0.02759) ** 2))
+    )
+    D_ref = 10**log_D_ref
     E_D_s = 3.03e4
     arrhenius = np.exp(-E_D_s / (pybamm.constants.R * T)) * np.exp(
         E_D_s / (pybamm.constants.R * 296)
@@ -179,7 +187,12 @@ def nco_diffusivity_Ecker2015(sto, T):
         Solid diffusivity
     """
 
-    D_ref = 3.7e-13 - 3.4e-13 * np.exp(-12 * (sto - 0.62) * (sto - 0.62))
+    log_D_ref = (
+        -12.67
+        - 1.597 * np.exp(-(((sto - 0.6169) / 0.07859) ** 2))
+        - 1.608 * np.exp(-(((sto - 1) / 0.0931) ** 2))
+    )
+    D_ref = 10**log_D_ref
     E_D_s = 8.06e4
     arrhenius = np.exp(-E_D_s / (pybamm.constants.R * T)) * np.exp(
         E_D_s / (pybamm.constants.R * 296.15)
@@ -634,5 +647,6 @@ def get_parameter_values():
             "Hales2019",
             "Richardson2020",
             "OKane2020",
+            "Yuan2023",
         ],
     }

pybamm/input/parameters/lithium_ion/Ecker2015_graphite_halfcell.py

@@ -1,4 +1,5 @@
 import pybamm
+import numpy as np
 
 
 def li_metal_electrolyte_exchange_current_density_Xu2019(c_e, c_Li, T):
@@ -61,9 +62,17 @@ def graphite_diffusivity_Ecker2015(sto, T):
         Solid diffusivity
     """
 
-    D_ref = 8.4e-13 * pybamm.exp(-11.3 * sto) + 8.2e-15
+    log_D_ref = (
+        -15.06
+        + 2.722 * np.exp(-((sto / 0.0777) ** 2))
+        + 1.503 * np.exp(-(((sto - 0.17) / 0.1153) ** 2))
+        + 1.055 * np.exp(-(((sto - 0.3118) / 0.04089) ** 2))
+        + 1.555 * np.exp(-(((sto - 0.6061) / 0.04545) ** 2))
+        + 1.215 * np.exp(-(((sto - 1) / 0.02759) ** 2))
+    )
+    D_ref = 10**log_D_ref
     E_D_s = 3.03e4
-    arrhenius = pybamm.exp(-E_D_s / (pybamm.constants.R * T)) * pybamm.exp(
+    arrhenius = np.exp(-E_D_s / (pybamm.constants.R * T)) * np.exp(
         E_D_s / (pybamm.constants.R * 296)
     )
 
@@ -119,12 +128,12 @@ def graphite_ocp_Ecker2015(sto):
     t = 0.196176
 
     u_eq = (
-        a * pybamm.exp(-b * sto)
-        + c * pybamm.exp(-d * (sto - e))
-        - r * pybamm.tanh(s * (sto - t))
-        - g * pybamm.tanh(h * (sto - i))
-        - j * pybamm.tanh(k * (sto - m))
-        - n * pybamm.exp(o * (sto - p))
+        a * np.exp(-b * sto)
+        + c * np.exp(-d * (sto - e))
+        - r * np.tanh(s * (sto - t))
+        - g * np.tanh(h * (sto - i))
+        - j * np.tanh(k * (sto - m))
+        - n * np.exp(o * (sto - p))
         + q
     )
 
@@ -173,7 +182,7 @@ def graphite_electrolyte_exchange_current_density_Ecker2015(c_e, c_s_surf, c_s_m
     )  # (A/m2)(m3/mol)**1.5 - includes ref concentrations
     E_r = 53400
 
-    arrhenius = pybamm.exp(-E_r / (pybamm.constants.R * T)) * pybamm.exp(
+    arrhenius = np.exp(-E_r / (pybamm.constants.R * T)) * np.exp(
         E_r / (pybamm.constants.R * 296.15)
     )
 
@@ -351,8 +360,8 @@ def electrolyte_conductivity_Ecker2015(c_e, T):
 
     # add temperature dependence
     E_k_e = 1.71e4
-    C = 296 * pybamm.exp(E_k_e / (pybamm.constants.R * 296))
-    sigma_e = C * sigma_e_296 * pybamm.exp(-E_k_e / (pybamm.constants.R * T)) / T
+    C = 296 * np.exp(E_k_e / (pybamm.constants.R * 296))
+    sigma_e = C * sigma_e_296 * np.exp(-E_k_e / (pybamm.constants.R * T)) / T
 
     return sigma_e
 
@@ -538,5 +547,6 @@ def get_parameter_values():
             "Xu2019",
             "Richardson2020",
             "OKane2020",
+            "Yuan2023",
         ],
     }