1. Two ions case

(1)

1. Two ions case

A battery consists of a group of connected electrochemical cells. Here we will consider a typical lithium- ion cell which constitutes a basic component rechargeable Li batteries. On discharge, electrons leave one electrode (the anode) and travel through an external load where they can perform useful work and finally they enter second electrode (the cathode). Simultaneously Li

⁺

cations are released from the anode and enter the electrolyte where they carry positive charge to the cathode where they combined with electrons which travelled through the external circuit. The process is reversed on charging the cell.

Practical electrodes in Li-ion cells have host structures where Li atoms can be inserted/extracted reversibly. The process of insertion is also named the intercalation. Examples of such materials include specially prepared carbon/graphite, Li

x

TiS

2

(lithium titanium disulfide), and Li

x

CoO

2

(lithium cobalt oxide) which all exhibit layered structures. In what follows we assumed that the anode material is based on graphite while the cathode on lithium cobalt oxide.

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

discharging

1 2 2

charging

Li CoO

__x

 x Li

^

 xe

^

LiCoO (1)

and

discharging

6 6

charging

Li C

_x

C  x Li

^

 xe

^

(2)

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 (LiPF

6

), dissolved in organic polar

aprotic solvents composed mainly of various mixtures of carbonate esters.

(2)

The concentrations of Li

^

and PF

₆^

ions as functions of position (x) and time (t) will be denoted by

Li

( , ) c



x t and

PF6

( , )

c



x t , respectively. The NernstPlanck expression will be used for the fluxes

Li PF6

J



, J



which requires also introducing the electric potential ( , ).  x t This constitutive relation for the ionic flux splits it into two parts: diffusion and migration caused by the electric field. The relations between fluxes and electric current density at both electrodes serve as the boundary conditions for lithium ions and are in fact a modern version of Faraday’s law. The hexafluorophosphate ion ( PF

₆^

) is blocked at both electrodes so the corresponding boundary fluxes are zero. Thus, the electro-diffusion processes during charging of the cell are governed by the following set of mass balance equations, flux constitutive relations, and boundary conditions:

Li Li Li

Li Li Li Li

Li Li

J 0, where J ,

( ) ( )

J (0, ) , J ( , ) ,

c c F

D D c

t x x RT x

i t i t

t L t

F F



  

   

 

  

 

    

    

     



(3)

and

6 6 6

6 6 6 6

6 6

PF PF PF

PF PF PF PF

PF PF

J

0, where J ,

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

c c F

D D c

t x x RT x

t L t



  

   

 

  

 

    

    

   



(4)

Here, ( ) 0 i t  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.

The set of equations (3) and (4) is not complete because there are three unknown functions (two concentrations and electric potential), but the number of equations is two. To complete the description one can use Poisson’s equation which relates directly the electric potential to the volume charge distribution. However, for macroscopic problems on the order of millimeters or larger an approximation

x = 0 x = L

LiCoO

2

LiC

6

LiPF

6

in PC+EC+DEC

Li

⁺

Li

⁺

charging

discharging

(3)

obtain addition equation

Li

( , )

PF6

( , ).

c



x t  c



x t (5)

Denoting the common concentration in (5) by ( , ) c x t we can rewrite equations (3) and (4) as follows

6 6

2

Li Li

2

PF 2 PF

,

c c F

D D c

t x RT x x

c c F

D D c

t x RT x x



 

 

               

               

(6)

with boundary conditions

6 6

Li Li PF PF

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

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

c F i t c F

D t D c t t D t D c t t

x RT x F x RT x

c F i t c F

D L t D c L t L t D L t D c L t L t

x RT x F x RT x

 

   

   

      

   

      

   

(7)

Multiplying in (6) the first equation by

PF6

D



, the second by D

_Li

, and next adding gives

6

2

LiPF 2

,

c c

t D x

 

   (8)

where

⁶

6

Li PF LiPF

Li PF

2D D

D D D

 

  is known as the binary salt diffusion coefficient. Moreover, the boundary conditions can also be expressed in terms of only the common concentration ( , ). c x t The third and fourth equations of (6) give

Li Li

( ) ( )

(0, ) , ( , ) .

2 2

c i t c i t

t L t

x D F



x D F



 

 

  (9)

Therefore, the description is reduced to the following PDE problem

6

2

LiPF 2

0

Li Li

, (0 , 0),

( , 0) ,

( ) ( )

(0, ) , ( , ) .

2 2

c c

D x L t

t x

c x c

c i t c i t

t L t

x D F



x D F



 

   

  

 

  

  

 



(10)

It is convenient to rewrite the above problem in terms of the flux

(4)

0

Li Li

0, ( , 0) ,

( ) ( )

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

c J

t x

c x c

i t i t

t t L t t

F F

 

 

  

  

 

 

      



(11)

where

LiPF6

J c

D x

  

 and transference numbers are

6 6

Li PF

Li PF Li PF

, D .

t D t

D D D D

 

   

 

  ⁽¹²⁾

After solving the above system for ( , ) c x t we can use it to obtain the electric potential ( , ).  x t Subtract the second equation form the first one in (6) to get

6

2

PF Li

2

PF Li

,

D D

F c

RT x c x D D x



^ ^

 

        

      (13)

and now integrate this equality from 0 to x

6 6

PF Li PF Li

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

D D D D

F F c c

c x t c t x t t

RT x RT x D D x D D x

 

^ ^ ^ ^

   

 

   

      

         

    (14)

Using the boundary conditions displayed in (7) and (10) we can eliminate ( c )(0, ) t x





 and c (0, ) x t



 from

the above equation

6 6

6

6 6

PF Li PF Li

PF Li Li

PF Li PF Li

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

1 (0, ) ( ) ( , ) 2 (0, ) ( ) ( , )

2 D D D D

F c c c

c x t t x t t

RT x x D D x D D x

D D c c D c c

t t t x t t t t x t

D D x x D D x x



^ ^ ^ ^

   

  

   

 

   

     

       

 

      

        

     

 

 

6 6

Li

PF Li PF Li

PF Li Li PF Li

( ) ( )

( ) ( , ) ( ) ( , ),

2 ( )

D i t c i t c

t t x t t t x t

D D D F x D D F x



   

    

 

    

   

(15)

thus

6 6

PF Li

( ) 1 ln

( , ) ( ) ( , )

( ) ( , )

F i t c

x t t t x t

RT x D D F c x t x



 

 

  

   (16)

and integrating over x from 0 to L gives the potential drop between electrodes

(5)

6 6

PF Li

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

( ) (0, )

V

cell

t L t t t t

F D D F c t

 

 

 

    

  

  (17)

where

_{( , )}

( )

0^L_{c x t}^dx

. G t  

2. Three species (two ions and one neutral) with dissociation/association reaction

Dissociation of the LiPF

6

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 consider the following dissociation/association reaction in the solvent

6 6

LiPF Li PF

f

b

k k





(18)

where k

_f

and k

_b

are forward and backward reaction rate constants, respectively. Assuming the first order dissociation reaction

LiPF6

( 

_f

 k c

_f

) and second order association reaction

Li PF6

( 

_b

 k c c

_b  

) we can postulate the reaction term:

6 6

LiPF Li PF

.

f b

R  k c  k c c

 

(19)

The physical meaning of R is as follows: the number of Li

^

moles produced/consumed in unit volume (m

³

) per unit time (s). From the mass balance expresses by reaction (18) we have R

_LiPF₆

  R ,

PF6

.

R



 R

Further we will use the abbreviations:

₆

1 Li 2 PF6 3 LiPF

.

c  c



, c  c



, c  c Taking into the charge numbers

1

1,

2

1,

z   z   z

₃

 0 (LiPF

6

is electrically neutral) and the fact that PF

6

and LiPF

6

do not cross the electrode/electrolyte interface, we have the following mass conservation set of equations

2

1 1 1

1 2 1

2

2 2 2

2

3 3

3 2

( E) ,

,

c c F c

D D R

t x RT x

c c F c

D D R

t x RT x

c c

D R

t x

     

  

     

  

   

 

(20)

with reaction term R  k c

_f ₃

 k c c

_b _{1 2}

and boundary conditions

1 1

2 2

3 3

( ) ( )

J (0, ) , J ( , ) ,

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

i t i t

t L t

F F

t L t

   

 

(21)

(6)

Applying now the electroneutrality condition c

₁

  c

₂

c , the system (20), (21) is transformed to

2 2 2

3 3

1 2 1 2 2 2 3 2

( E) ( E)

, , c c ,

c c F c c c F c

D D R D D R D R

t x RT x t x RT x t x

 

     

       

        (22)

with boundary conditions

1 1 1 1

2 2 2 2

3 3

( ) ( )

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

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

(0, ) 0, ( , ) 0.

c i t c i t

D t D cE t D L t D cE L t

x F x F

c c

D t D cE t D t D cE L t

x x

c c

D t D L t

x x

 

       

 

     

 

   

 

(23)

From the first two equations in (22) we have

2

2 2 1 2 2 1 2

2

1 1 2 2 1 2 1

2

2 1 2 1 2 2 1

( E) ,

( E)

( ) 2 ( ) ,

c c F c

D D D D D D R

t x RT x

c c F c

D D D D D D R

t x RT x

c c

D D D D D D R

t x

     

  

     

  

 

   

 

(24)

that is

2

,

c c

t D x

 

   (25)

with

² ¹

1 2

2 D D . D  D D

 We see that elimination of the electric field has been possible as in the case of a full dissociation. The third equation in (20) does not contain the electric field, so we can convert the system of PDEs (20) to two equations

2

2 2 3

2

3 3 2

3 2 3

, ,

f b

c c

D k c k c

t x

c c

D k c k c

t x

    

 

  

   

  



(26)

or in the mass balance form with explicitly written fluxes

3 3

J ,

c R

t x

c R

t x

 

  

 

 

   

  



(27)

(7)

x x

 

Boundary conditions (23)

1 1

3 3

( ) ( )

(0, ) , ( , ) ,

2 2

(0, ) 0, ( , ) 0,

c i t c i t

D t D L t

x F x F

c c

t L t

x x

 

  

  

 

  

  



(28)

or expressed explicitly by fluxes

1 1

3 3

( ) ( )

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

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

i t i t

t t L t t

F F

t L t

      

 

  



(29)

For the initial condition we assume that the dissociation degree of the salt is 0    1. At equilibrium (t=0) this gives

0 3 0

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

c x   c c x    c (30)

where c

₀

is the analytical concentration of the salt (mol/m

³

).

The electric field and potential is obtained in the same manner as in the case without reaction:

1 2

2 1

1 2

( ) 1

( , ) ( ) ln ( , ),

( ) ( , )

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

( ) (0, )

F i t

E x t t t c x t

RT D D F c x t x

F i t c x t

x t t G x t t t

RT   D D F c t

    

 

   



(31)

where

0

( , )

x

d

G x t

c t



   and transference numbers are defined as usual, t

_i

 D

_i

/ ( D

₁

 D

₂

). In particular

2 1

1 2

( ) ( , )

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

( ) (0, )

cell

RT i t c L t

V t L t t G t t t

F D D F c t

  ^ ^

          (32)

where

0

( ) ( , ) .

( , )

L

d

G t G L t

c t



   

Table 1. Example of typical values of parameters for the case of homogeneous salt dissociation reactions in the cell (1=Li

⁺

, 2=PF

6

-

, 3=LiPF

6

).

L[mm] D

1

[m

²

/s] D

2

[m

²

/s] D

3

[m

²

/s] i

cell

[A/m

²

] k

f

[s

^-1

] k

b

[m

³

/(mols)]  c

0

[mol/dm

³

]

0.3 1.110

^-11

1.610

^-11

3.510

^-10

30 0.04 1.1210

^-5

0.5, 0.75, 0.95 1.3