
EquationsSimulation: electrode modelVariables & Constantes
ChargesThere are 4 charge species to consider a priori in an electrochemical device: a) Positive and Negative ions in the electrolyte in the pores and in the separator.
b) Electrons and holes in the electrode (solid electronic conductors). Electrode sketchResistance is much smaller in carbon => the carrier path will be maximized in it. At high frequency only the pores close to the electrode surface will be reach
Electrode modelIn each electrode volume element there is a mixing of two homogeneous phases: the electrode matrix and the pores. Φ_{1}(x,t) is the electric potential in the matrix. Φ_{2}(x,t) is the electric potential in the electrolyte “far enough” from the pore surface. We aren’t studying the potential in the double layer…. Φ_{1}(x,t) and Φ_{2}(x,t) depend on the position in the depth of the electrode. i(x) = i_{1}(x) + i_{2}(x) where i_{1} is the current in the electrode matrix and i_{2} is the current in the electrolyte Current in the electrode matrixOnly electronic conduction in the electrode matrix (carbon), No diffusion. Ohm law => i_{1}: Gradient definition => Φ_{}
Liquid phase physical lawsTwo laws: Electrical field Coulombic force => Ohm law Charge migration Concentration gradient => Fick law Material diffusion (multiplied by the charge) to get charges diffusion Opposed to the migration DiffusionEinstein relation μ mobility [m^{2}/Vs] Brownian μ_{p} = v_{d}/ F. Electrical μ_{e} = v_{d}/ E.k_{B} Boltzmann’s constant
T temperature
Drude model σ conductivity n density of carriers z_{j} Nr of charge of specie j c charge molar concentration Faraday F = A e c_{+} = c_{} = c ρ = 0 Single charge carrierIn most of the case the electrolyte salt is made of single charge ions z_{+} = z_{} = 1 J_{+} = F N_{+} J_{} = F N_{} i_{2} = J_{+} + J_{} = F (N_{+}  N_{}) Total current in the electrolyte
Charge storage at the interfaceAt the interface between the pores and the electrolyte there is a uniform capacitance C = C(x) per unit of pore surface. The interface surface per unit of volume is given by a The current involved per unit of volume is J(x,t) The current of specie j J_{j}(x,t)
Interface neutralitydq change of interface electrode charge dq_{+ } change of positive ion charge dq_{+} /dq = dq_{} /dq = 1/2 Bring 4 electrons on the electrode surface => bring 2 negative ions and remove 2 positive ions in the electrolyte neighborhood. Current of chargeJ = (J_{+} + J_{}) => C capacity per pore surface unit Continuity equationConsider an electrode volume unit, Its capacitance is given by aC, where a is the pore surface per electrode volume unit and C is the capacitance per surface unit The current which flow through the unit surface (cross section of the unit volume) is i_{2} div i_{2} = [i_{2}(x+δx)i_{2}(x)] / δx dρ/dt = charge trapped in the double layer = aC dΦ_{2}/dt Continuity equation at the interface=> and => Equations in the electrodeElectronic conduction : Ionic conduction: Equations in the separatorNo electronic conduction (except selfdischarge through the separator) i_{1}(x,t) = 0 Field equations + BoundariesContinuity equations G : charge separation (generation) R : charge recombination
