Modeling of electrolyte processes in a typical Li-ion battery

Modeling of electrolyte processes in a typical Li-ion battery

Case 1: Fully dissociated salt

The processes occurring at the electrodes of a typical Li-ion battery can be summarized in schematic form as follows

At negative electrode

At positive electrode

x = 0 x = L

LiCoO₂ LiC₆ LiPF₆ in PC+EC+DEC

Li⁺

Li⁺ charging

discharging

Electrolyte: a lithium salt in a mixture of polar aprotic organic solvents, e.g. propylene carbonate (PC), ethylene carbonate (EC), diethyl carbonate (DEC) etc.

discharging

1 2 2

charging

Li CoO_x  xLi^  xe^  LiCoO

discharging

6 6

charging

Li C_x  C  xLi^  xe^

 Lithium ions (Li⁺) after crossing the electrode/electrolyte interface move in the electrolyte solution which is based on some Li-salt, such as lithium hexafluorophosphate (LiPF₆) or lithium perchlorate (LiClO₄), dissolved in a mixture of organic polar aprotic solvents composed mainly of various carbonate esters.

 The concentrations of Li⁺ and PF₆^- ions as functions of position (x) and time (t) will be denoted by c_Li+ and c_PF6- respectively.

 The NernstPlanck expression will be used for the fluxes of Li⁺ and PF₆^- ions. It requires introducing the electric potential (). This constitutive relation for the ionic flux splits it into two parts:

diffusion (caused by the concentration gradient) +

migration (caused by the electric field)

 The relations between fluxes and electric current density at both electrodes serve as the boundary conditions (Faraday’s laws). The hexafluorophosphate anion (PF₆^-) is blocked at both electrodes so the corresponding boundary fluxes are zero.

Model Development – grasping the physics and

chemistry of our system

Denote by i_app(t)>0 is the charging current density supplied to the cell from an external power source, F is the Faraday constant, R is the universal gas constant, and T is the temperature in K.

Equations for Li⁺ (in the form of PDEs):

Boundary conditions for Li⁺:

Equations for PF₆^ (in the form of PDEs):

Boundary conditions for PF₆^ :

Li Li Li

Li Li Li Li

J 0, where J ,

mass conservation for Li Nernst Planck flux for Li

diffusion migration

c c F

D D c

t x x RT x



 

  

   



        

   

 

 

Li Li

( ) ( )

J (0, ) i^app t , J ( , ) i^app t

t L t

F F

     

6 6

6 6 6

6 6 6 6

PF PF PF

PF PF PF PF

J 0, where J ,

mass conservation for PF Nernst Planck flux for PF

diffusion migration

c c F

D D c

t x x RT x



 

  

   



        

   

 

 

6 6

PF PF

J (0, ) 0, Jt  ( , ) 0L t 

The set of equations in the previous slide is not complete because there are five unknown functions (two concentrations, two fluxes, and electric potential), but the number of equations is four. To complete the description we will use an approximation such called the electroneutrality condition. In our case of two ions case Li+ and PF₆^- leads to a simple additional equaation

Denoting the common concentration by c(x,t) we can rewrite equations as follows

Li ( , ) PF6 ( , ) c  x t c  x t

6

6

6 6 6

Li

Li Li Li

PF

PF PF PF

J 0 , J ,

J 0 , J

mass conservation for Li Nernst Planck flux for Li

mass conservation for PF Nernst Planck flux f

c c F

D D c

t x x RT x

c c F

D D c

t x x RT x





 



  





  





        

   

   

    

   

   

 

,

or PF^



The boundary conditions with assumed electroneutrality condition take now the following explicite form

6 6

6 6

Li Li

PF PF

Li Li

PF PF

(0, ) (0, ) (0, ) ( ),

at 0 :

(0, ) (0, ) (0, ) 0,

( , ) ( , ) ( , ) ( ),

at :

( , ) ( , ) ( , ) 0.

app

app

i t

c F

D t D c t t

x RT x F

x c F

D t D c t t

x RT x

i t

c F

D L t D c L t L t

x RT x F

x L c F

D L t D c L t L t

x RT x









 

 

 

 

     

  

 

 

  

  



     

 

     

 





When we insert the electroneutrality condition (c = c_Li = c_PF6) into the mass conservation equations and fluxes we obtain the problem with only one concentration:

Equations:

Initial conditions:

Boundary conditions:

6

6 6 6

Li

Li Li Li

PF

PF PF PF

J 0, J ,

J 0, J ,

c c F

D D c

t x x RT x

c c F

D D c

t x x RT x







  



  

        

   

        

   

6

6 6

6

6 6

PF

Li PF

Li PF

PF

Li PF

Li PF

J (0, ) ( ), J (0, ) 0,

J ( , ) ( ), J ( , ) 0.

app

app

D i t

t t

D D F

D i t

L t L t

D D F



 

 



 

 

  



  

 ( ,0) 0

c x  c

with the flux

After some algebraic manipulations, the set of PDEs shown above can be

reformulated to the following problem (no new physical assumptions are involved!):

equation:

initial condition:

boundary conditions:

where is the salt diffusion coefficient and c₀ is a given concentration of salt in the cell at the time when a charging process starts.

Mathematical reformulation to a simpler form

 the final form

0

Li Li

J 0, ( ,0) ,

( ) ( )

J(0, ) (1 ) ^app , J( , ) (1 ) ^app , c

t x

c x c

i t i t

t t L t t

F F

 

   

 



     

LiPF6

J c,

D x

  



6 6

6

Li PF

LiPF

Li PF

D 2D D

D D

 

 

 

The potential drop across the cell electrolyte can be obtained easily from the presented model. After some calculations and two integrations we get

where

The parameters are known as the transference numbers.

Electric potential drop in the electrolyte of the cell

6 6

2 PF Li 2

PF Li

( ) ( ) ( , )

( ) ( , ) (0, ) ( ) ln ,

(0, )

cell

diffusion part

migration part of potential

of potential

RT i t G t RT c L t

V t L t t t t

F D D F c t

 

 

    

   



6

6 6

Li Li

Li PF

0 Li PF Li PF

( ) , , .

( , )

L

dx D D

G t t t

c x t D D D D

 

 

   

  

 



Li , PF6

t  t 

Modeling of electrolyte processes in a typical Li-ion battery

Case 2: Partially dissociated salt

Extension of the model:

including the salt dissociation/association reaction

x = 0 x = L

LiCoO₂ LiC₆ LiPF₆ in PC+EC+DEC

Li⁺

Li⁺ charging

discharging

Boundary conditions (e.g. heterogeneous

reactions)

Boundary conditions (e.g. heterogeneous

reactions) Transport processes +

possible homogeneous reactions

Dissociation of the LiPF₆ salt in a mixture of organic solvents of a typical Li-ion battery is usually not complete, and it also can be reverted (association). Hence, we will extend our model by considering the following dissociation/association reaction in the solvent:

6 6

LiPF ^f Li PF

b

k k

  



We assume that for the dissociation/association reaction

the forward direction has first order kinetics, and the backward has the second order kinetics with homogeneous rate constants k_f [s^-1] and k_b [m³/(mols)] respectively, then the reaction term can be expressed as

Obviously the mass balans requires that:

To simplify the notation we shall use further:

Extension of the model:

Development of the reaction term

6 6

LiPF Li PF

f b

R  k c k c c 

6 6

LiPF ^f Li PF

b

k k

  



6 6

LiPF , Li , PF

R  R R   R R   R

6 6

1 Li , 2 PF , 3 LiPF .

c  c  c  c  c  c

The electroneutrality condition requires that c=c₁=c₂, so we obtain the following form of the PDEs in the conservative-flux form (suitable for a COMSOL project):

Extension of the model: Summary of equations

where

Mathematically when the expressions for fluxes and reaction term will be inserted into the mass balance equations, the above system takes the following form of a system of parabolic-type PDEs:

2 2

2 3

2 2

3 3

3 2 3

, ,

salt f b

f b

c c

D k c k c

t x

c c

D k c k c

t x

    

 

  

   

  



3 3

J ,

J ,

c R

t x

c R

t x

 

  

 

 

   

  



3 3 3

2 3

J , J ,

.

salt

f b

c c

D D

x x

R k c k c



    

 

 

6

1 2

3 LiPF

1 2

2 , .

salt

D D D D D

D D

 



Extension of the model: initial conditions and realtion between k

, k

, and 

If we denote the total salt concentration by c₀ [mol/m³] and the degree of salt dissociation by , then at equilibrium we have

or in a simplified notation

Under the electroneutrality condition (c₁=c₂=c) this gives

At equilibrium the forward and backward reactions are balanced so R = 0. Thus

so finally

6 6

0 0 LiPF 0

Li^eq

,

,

(1 )

c

  c c

  c c    c

1^eq 0

,

,

(1 )

c   c c   c c    c

1^eq 2^{eq eq} 0

,

(1 )

c  c  c   c c    c

3 1 2

2

0 0

0

0 (1 ) ( )

eq eq eq

f b

f b

k c k c c

k





 

  

2 0

1

f b

k k c 

 



Equations:

Fluxes and reaction:

Boundary conditions:

where is the salt diffusion coefficient,

Extended model: the final form

Initial conditions:

is the Li⁺ transference number, and c₀ is a given concentration of the LiPF₆ salt in the cell at the beginning, and the rate constant

Unknown quantities:

3 3

3 2

3 3 3

0 3 0

Li Li

3 3

J

J ,

J , J ,

( ,0) , ( ,0) (1 )

( ) ( )

J(0, ) (1 ) , J( , ) (1 ) ,

J (0, ) 0, J ( , ) 0

salt f b

app app

c

c R R

t x t x

c

D c D R k c k c

x x

c x c c x c

i t i t

t t L t t

F F

t L t

 

 

 

      

   



      

 

  

     

 

6

6

Li PF

Li PF

2

salt

D D D

D D

 

 

  ₆

Li Li

Li PF

t D

D D





 

 

2

0 / (1 ).

f b

k  k c  

6 3 3

Li :^ c c x t ( , ), LiPF : c c x t( , )