Partially reversible lithium plating

pybamm/CITATIONS.txt

@@ -296,6 +296,20 @@
   journal = {Journal of The Electrochemical Society}
 }
 
+@article{OKane2022,
+  author ="O{'}Kane, Simon E. J. and Ai, Weilong and Madabattula, Ganesh and Alonso-Alvarez, Diego and Timms, Robert and Sulzer, Valentin and Edge, Jacqueline Sophie and Wu, Billy and Offer, Gregory J. and Marinescu, Monica",
+  title  ="Lithium-ion battery degradation: how to model it",
+  journal  ="Phys. Chem. Chem. Phys.",
+  year  ="2022",
+  volume  ="24",
+  issue  ="13",
+  pages  ="7909-7922",
+  publisher  ="The Royal Society of Chemistry",
+  doi  ="10.1039/D2CP00417H",
+  url  ="http://dx.doi.org/10.1039/D2CP00417H",
+}
+
+
 @article{ORegan2021,
   author = {O'Regan, Kieran and Brosa Planella, Ferran and Widanage, W. Dhammika and Kendrick, Emma},
   title = {{Thermal-electrochemical parametrisation of a lithium-ion battery: mapping Li concentration and temperature dependencies}},

pybamm/input/parameters/lithium_ion/lithium_platings/okane2022_Li_plating/README.md

@@ -0,0 +1,7 @@
+# Lithium plating parameters
+
+Some example parameters for lithium plating from the paper:
+
+> O’Kane, S. E. J., Ai, W., Madabattula, G., Alonso-Alvarez, D., Timms, R, Sulzer, V., Edge, J. S., Wu, B., Offer, G. J., & Marinescu, M. (2022) Lithium-ion battery degradation: how to model it. Physical Chemistry: Chemical Physics, accepted.
+
+Note: this parameter set does not claim to be representative of the true parameter values. These are merely the parameter values that were used in the referenced papers.

pybamm/input/parameters/lithium_ion/lithium_platings/okane2022_Li_plating/SEI_limited_dead_lithium_OKane2022.py

@@ -0,0 +1,36 @@
+from pybamm import FullBroadcast, Parameter, Scalar
+
+
+def SEI_limited_dead_lithium_OKane2022(L_sei):
+    """
+    Decay rate for dead lithium formation [s-1].
+    References
+    ----------
+    .. [1] Simon E. J. O'Kane, Weilong Ai, Ganesh Madabattula, Diega Alonso-Alvarez,
+    Robert Timms, Valentin Sulzer, Jaqueline Sophie Edge, Billy Wu, Gregory J. Offer
+    and Monica Marinescu. "Lithium-ion battery degradation: how to model it."
+    Physical Chemistry: Chemical Physics 24, no. 13 (2022): 7909-7922.
+    Parameters
+    ----------
+    L_sei : :class:`pybamm.Symbol`
+        Total SEI thickness [m]
+    Returns
+    -------
+    :class:`pybamm.Symbol`
+        Dead lithium decay rate [s-1]
+    """
+
+    gamma_0 = Parameter("Dead lithium decay constant [s-1]")
+    L_inner_0 = Parameter("Initial inner SEI thickness [m]")
+    L_outer_0 = Parameter("Initial outer SEI thickness [m]")
+    L_sei_0 = L_inner_0 + L_outer_0
+
+    if L_sei < L_sei_0:
+        gamma = gamma_0
+    else:
+        L_inner_0 = Parameter("Initial inner SEI thickness [m]")
+        L_outer_0 = Parameter("Initial outer SEI thickness [m]")
+        L_sei_0 = L_inner_0 + L_outer_0
+        gamma = gamma_0 * L_sei_0 / L_sei
+
+    return gamma

pybamm/input/parameters/lithium_ion/lithium_platings/okane2022_Li_plating/parameters.csv

@@ -0,0 +1,14 @@
+Name [units],Value,Reference,Notes
+,,,
+,,,
+# Lithium plating parameters,,,
+Lithium metal partial molar volume [m3.mol-1],1.30E-05,Yang2017,
+Lithium plating kinetic rate constant [m.s-1],1E-9,,
+Exchange-current density for plating [A.m-2],[function]plating_exchange_current_density_OKane2020,,
+Exchange-current density for stripping [A.m-2],[function]stripping_exchange_current_density_OKane2020,,
+Initial plated lithium concentration [mol.m-3],0.00E+00,,
+Typical plated lithium concentration [mol.m-3],1000,1 molar,
+Lithium stripping transfer coefficient,0.35,,
+Lithium plating transfer coefficient,0.65,,
+Dead lithium decay constant [s-1],1E-6,,
+Dead lithium decay rate [s-1],[function]SEI_limited_dead_lithium_OKane2022,,

pybamm/input/parameters/lithium_ion/lithium_platings/okane2022_Li_plating/plating_exchange_current_density_OKane2020.py

@@ -0,0 +1,29 @@
+from pybamm import constants, Parameter
+
+
+def plating_exchange_current_density_OKane2020(c_e, c_Li, T):
+    """
+    Exchange-current density for Li plating reaction [A.m-2].
+    References
+    ----------
+    .. [1] O’Kane, Simon EJ, Ian D. Campbell, Mohamed WJ Marzook, Gregory J. Offer, and
+    Monica Marinescu. "Physical origin of the differential voltage minimum associated
+    with lithium plating in Li-ion batteries." Journal of The Electrochemical Society
+    167, no. 9 (2020): 090540.
+    Parameters
+    ----------
+    c_e : :class:`pybamm.Symbol`
+        Electrolyte concentration [mol.m-3]
+    c_Li : :class:`pybamm.Symbol`
+        Plated lithium concentration [mol.m-3]
+    T : :class:`pybamm.Symbol`
+        Temperature [K]
+    Returns
+    -------
+    :class:`pybamm.Symbol`
+        Exchange-current density [A.m-2]
+    """
+
+    k_plating = Parameter("Lithium plating kinetic rate constant [m.s-1]")
+
+    return constants.F * k_plating * c_e

pybamm/input/parameters/lithium_ion/lithium_platings/okane2022_Li_plating/stripping_exchange_current_density_OKane2020.py

@@ -0,0 +1,35 @@
+from pybamm import constants, Parameter
+
+
+def stripping_exchange_current_density_OKane2020(c_e, c_Li, T):
+    """
+    Exchange-current density for Li stripping reaction [A.m-2].
+
+    References
+    ----------
+
+    .. [1] O’Kane, Simon EJ, Ian D. Campbell, Mohamed WJ Marzook, Gregory J. Offer, and
+    Monica Marinescu. "Physical origin of the differential voltage minimum associated
+    with lithium plating in Li-ion batteries." Journal of The Electrochemical Society
+    167, no. 9 (2020): 090540.
+
+    Parameters
+    ----------
+
+    c_e : :class:`pybamm.Symbol`
+        Electrolyte concentration [mol.m-3]
+    c_Li : :class:`pybamm.Symbol`
+        Plated lithium concentration [mol.m-3]
+    T : :class:`pybamm.Symbol`
+        Temperature [K]
+
+    Returns
+    -------
+
+    :class:`pybamm.Symbol`
+        Exchange-current density [A.m-2]
+    """
+
+    k_plating = Parameter("Lithium plating kinetic rate constant [m.s-1]")
+
+    return constants.F * k_plating * c_Li

pybamm/models/full_battery_models/base_battery_model.py

@@ -67,7 +67,7 @@ class BatteryModelOptions(pybamm.FuzzyDict):
                 Can be "full" (default), "constant", or "current-driven".
             * "lithium plating" : str
                 Sets the model for lithium plating. Can be "none" (default),
-                "reversible" or "irreversible".
+                "reversible", "partially reversible", or "irreversible".
             * "loss of active material" : str
                 Sets the model for loss of active material. Can be "none" (default),
                 "stress-driven", "reaction-driven", or "stress and reaction-driven".
@@ -198,7 +198,12 @@ def __init__(self, extra_options):
                 "Marcus-Hush-Chidsey",
             ],
             "interface utilisation": ["full", "constant", "current-driven"],
-            "lithium plating": ["none", "reversible", "irreversible"],
+            "lithium plating": [
+                "none",
+                "reversible",
+                "partially reversible",
+                "irreversible",
+            ],
             "lithium plating porosity change": ["false", "true"],
             "loss of active material": [
                 "none",

pybamm/models/submodels/interface/lithium_plating/base_plating.py

@@ -19,7 +19,10 @@ class BasePlating(BaseInterface):
     .. [1] SEJ O'Kane, ID Campbell, MWJ Marzook, GJ Offer and M Marinescu. "Physical
            Origin of the Differential Voltage Minimum Associated with Li Plating in
            Lithium-Ion Batteries". Journal of The Electrochemical Society,
-           167:090540, 2019
+           167:090540, 2020
+    .. [2] SEJ O'Kane, W Ai, G Madabattula, D Alonso-Alvarez, R Timms, V Sulzer,
+           JS Edge, B Wu, GJ Offer and M Marinescu. "Lithium-ion battery degradation:
+           how to model it". Physical Chemistry: Chemical Physics, 24:7909, 2022
 
     **Extends:** :class:`pybamm.interface.BaseInterface`
     """
@@ -53,7 +56,7 @@ def get_coupled_variables(self, variables):
 
         return variables
 
-    def _get_standard_concentration_variables(self, c_plated_Li):
+    def _get_standard_concentration_variables(self, c_plated_Li, c_dead_Li):
         """
         A private function to obtain the standard variables which
         can be derived from the local plated lithium concentration.
@@ -82,20 +85,32 @@ def _get_standard_concentration_variables(self, c_plated_Li):
         L_plated_Li_av = pybamm.x_average(L_plated_Li)
         Q_plated_Li = c_plated_Li_av * param.L_n * param.L_y * param.L_z
 
+        c_dead_Li_av = pybamm.x_average(c_dead_Li)
+        L_dead_Li = c_dead_Li  # dead Li "thickness", required by porosity submodel
+        L_dead_Li_av = pybamm.x_average(L_dead_Li)
+        Q_dead_Li = c_dead_Li_av * param.L_n * param.L_y * param.L_z
+
         variables = {
             "Lithium plating concentration": c_plated_Li,
             "Lithium plating concentration [mol.m-3]": c_plated_Li * c_scale,
             "X-averaged lithium plating concentration": c_plated_Li_av,
             "X-averaged lithium plating concentration"
             " [mol.m-3]": c_plated_Li_av * c_scale,
+            "Dead lithium concentration": c_dead_Li,
+            "Dead lithium concentration [mol.m-3]": c_dead_Li * c_scale,
+            "X-averaged dead lithium concentration": c_dead_Li_av,
+            "X-averaged dead lithium concentration"
+            " [mol.m-3]": c_dead_Li_av * c_scale,
             "Lithium plating thickness": L_plated_Li,
             "Lithium plating thickness [m]": L_plated_Li * L_scale,
             "X-averaged lithium plating thickness [m]": L_plated_Li_av * L_scale,
-            "Loss of lithium to lithium plating [mol]": Q_plated_Li * c_scale,
-            "Loss of capacity to lithium plating [A.h]": Q_plated_Li
-            * c_scale
-            * param.F
-            / 3600,
+            "Dead lithium thickness": L_dead_Li,
+            "Dead lithium thickness [m]": L_dead_Li * L_scale,
+            "X-averaged dead lithium thickness [m]": L_dead_Li_av * L_scale,
+            "Loss of lithium to lithium plating [mol]": (Q_plated_Li + Q_dead_Li)
+            * c_scale,
+            "Loss of capacity to lithium plating [A.h]":
+            (Q_plated_Li + Q_dead_Li) * c_scale * param.F / 3600,
         }
 
         return variables

pybamm/models/submodels/interface/lithium_plating/no_plating.py

@@ -25,7 +25,7 @@ def get_fundamental_variables(self):
         zero = pybamm.FullBroadcast(
             pybamm.Scalar(0), "negative electrode", "current collector"
         )
-        variables = self._get_standard_concentration_variables(zero)
+        variables = self._get_standard_concentration_variables(zero, zero)
         variables.update(self._get_standard_overpotential_variables(zero))
         variables.update(self._get_standard_reaction_variables(zero))
         return variables

pybamm/models/submodels/interface/lithium_plating/plating.py

@@ -22,7 +22,10 @@ class Plating(BasePlating):
     .. [1] SEJ O'Kane, ID Campbell, MWJ Marzook, GJ Offer and M Marinescu. "Physical
            Origin of the Differential Voltage Minimum Associated with Li Plating in
            Lithium-Ion Batteries". Journal of The Electrochemical Society,
-           167:090540, 2019
+           167:090540, 2020
+    .. [2] SEJ O'Kane, W Ai, G Madabattula, D Alonso-Alvarez, R Timms, V Sulzer,
+           JS Edge, B Wu, GJ Offer and M Marinescu. "Lithium-ion battery degradation:
+           how to model it". Physical Chemistry: Chemical Physics, 24:7909, 2022
 
     **Extends:** :class:`pybamm.lithium_plating.BasePlating`
     """
@@ -31,6 +34,7 @@ def __init__(self, param, x_average, options):
         super().__init__(param, options)
         self.x_average = x_average
         pybamm.citations.register("OKane2020")
+        pybamm.citations.register("OKane2022")
 
     def get_fundamental_variables(self):
         if self.x_average is True:
@@ -39,14 +43,24 @@ def get_fundamental_variables(self):
                 domain="current collector",
             )
             c_plated_Li = pybamm.PrimaryBroadcast(c_plated_Li_av, "negative electrode")
+            c_dead_Li_av = pybamm.Variable(
+                "X-averaged dead lithium concentration",
+                domain="current collector",
+            )
+            c_dead_Li = pybamm.PrimaryBroadcast(c_dead_Li_av, "negative electrode")
         else:
             c_plated_Li = pybamm.Variable(
                 "Lithium plating concentration",
                 domain="negative electrode",
                 auxiliary_domains={"secondary": "current collector"},
             )
+            c_dead_Li = pybamm.Variable(
+                "Dead lithium concentration",
+                domain="negative electrode",
+                auxiliary_domains={"secondary": "current collector"},
+            )
 
-        variables = self._get_standard_concentration_variables(c_plated_Li)
+        variables = self._get_standard_concentration_variables(c_plated_Li, c_dead_Li)
 
         return variables
 
@@ -59,20 +73,26 @@ def get_coupled_variables(self, variables):
         c_plated_Li = variables["Lithium plating concentration"]
         j0_stripping = param.j0_stripping(c_e_n, c_plated_Li, T)
         j0_plating = param.j0_plating(c_e_n, c_plated_Li, T)
+        # phi_ref is part of the de-dimensionalization used in PyBaMM
         phi_ref = param.U_n_ref / param.potential_scale
 
         eta_stripping = delta_phi + phi_ref + eta_sei
         eta_plating = -eta_stripping
-        prefactor = 1 / (2 * (1 + self.param.Theta * T))
+        prefactor = 1 / (1 + self.param.Theta * T)
+        # NEW: transfer coefficients can be set by the user
+        alpha_stripping = self.param.alpha_stripping
+        alpha_plating = self.param.alpha_stripping
 
-        if self.options["lithium plating"] == "reversible":
+        if self.options["lithium plating"] in ["reversible", "partially reversible"]:
             j_stripping = j0_stripping * pybamm.exp(
-                prefactor * eta_stripping
-            ) - j0_plating * pybamm.exp(prefactor * eta_plating)
+                prefactor * alpha_stripping * eta_stripping
+            ) - j0_plating * pybamm.exp(prefactor * alpha_plating * eta_plating)
         elif self.options["lithium plating"] == "irreversible":
             # j_stripping is always negative, because there is no stripping, only
             # plating
-            j_stripping = -j0_plating * pybamm.exp(prefactor * eta_plating)
+            j_stripping = -j0_plating * pybamm.exp(
+                prefactor * alpha_plating * eta_plating
+            )
 
         variables.update(self._get_standard_overpotential_variables(eta_stripping))
         variables.update(self._get_standard_reaction_variables(j_stripping))
@@ -85,26 +105,42 @@ def get_coupled_variables(self, variables):
     def set_rhs(self, variables):
         if self.x_average is True:
             c_plated_Li = variables["X-averaged lithium plating concentration"]
+            c_dead_Li = variables["X-averaged dead lithium concentration"]
             j_stripping = variables[
                 "X-averaged lithium plating interfacial current density"
             ]
-            # Note a is dimensionless (has a constant value of 1 if the surface
-            # area does not change)
             a = variables["X-averaged negative electrode surface area to volume ratio"]
+            L_sei = variables["X-averaged total SEI thickness"]
         else:
             c_plated_Li = variables["Lithium plating concentration"]
+            c_dead_Li = variables["Dead lithium concentration"]
             j_stripping = variables["Lithium plating interfacial current density"]
             a = variables["Negative electrode surface area to volume ratio"]
+            L_sei = variables["Total SEI thickness"]
 
         Gamma_plating = self.param.Gamma_plating
+        # In the partially reversible plating model, coupling term turns reversible
+        # lithium into dead lithium. In other plating models, it is zero.
+        if self.options["lithium plating"] == "partially reversible":
+            dead_lithium_decay_rate = self.param.dead_lithium_decay_rate(L_sei)
+            coupling_term = dead_lithium_decay_rate * c_plated_Li
+        else:
+            zero = pybamm.Scalar(0)
+            coupling_term = zero
 
-        self.rhs = {c_plated_Li: -Gamma_plating * a * j_stripping}
+        self.rhs = {
+            c_plated_Li: -Gamma_plating * a * j_stripping - coupling_term,
+            c_dead_Li: coupling_term
+        }
 
     def set_initial_conditions(self, variables):
         if self.x_average is True:
             c_plated_Li = variables["X-averaged lithium plating concentration"]
+            c_dead_Li = variables["X-averaged dead lithium concentration"]
         else:
             c_plated_Li = variables["Lithium plating concentration"]
+            c_dead_Li = variables["Dead lithium concentration"]
         c_plated_Li_0 = self.param.c_plated_Li_0
+        zero = pybamm.Scalar(0)
 
-        self.initial_conditions = {c_plated_Li: c_plated_Li_0}
+        self.initial_conditions = {c_plated_Li: c_plated_Li_0, c_dead_Li: zero}

pybamm/models/submodels/porosity/reaction_driven_porosity.py

@@ -26,9 +26,11 @@ def __init__(self, param, options, x_average):
     def get_coupled_variables(self, variables):
         L_sei_n = variables["Total SEI thickness [m]"]
         L_sei_0 = self.param.L_inner_0_dim + self.param.L_outer_0_dim
+        L_pl_0 = self.param.c_plated_Li_0_dim
         L_pl_n = variables["Lithium plating thickness [m]"]
+        L_dead_n = variables["Dead lithium thickness [m]"]
 
-        L_tot = (L_sei_n - L_sei_0) + L_pl_n
+        L_tot = (L_sei_n - L_sei_0) + (L_pl_n - L_pl_0) + L_dead_n
 
         a_n = variables["Negative electrode surface area to volume ratio [m-1]"]