A New Perspective on the Poisson-Boltzmann
Equation and some Related Results
Ping Sheng
Department of Physics
Hong Kong University of Science and Technology
Controlled Structural Formation of Soft Matter
Beijing, China
August 12, 2015
Phys. Rev. X 4 (1), 011042
1

Li WAN Shixin XU Maijia LIAO Chun LIU
2

• Charge separation at the fluid-solid interface (e.g. silica-water interface) leads to an electrical double layer, which is the basis of electro-kinetics .
Si
O
H
-
+
+
-
-
+
Electrical double layer
0
Distance from the interface
ζ : Zeta potential
λ
D
: Debye length
3

const .
Boundary of the PB equation
 2
 
1
 2
D sinh
  
0 a
Fluid
•
Integral of the right hand side of the PB equation yields the net charge in the Debye layer.
---- Counter-balanced by the surface charge.
Surface charge
• Hence, the PB equation divides the electrical double layer into two halves.
---- They can be problems when the fluid channel width is smaller then twice the Debye length.
---- Is a holistic treatment possible?
4

Treating electrical double layer as an integral entity
Fluid const .
Boundary of PB equation
Present Boundary
• The computational domain must be overall neutral.
a
• A constant potential boundary condition would only yields a constant potential solution.
 
0
• From the Gauss Theorem, the only compatible Neumann boundary condition is zero normal electric field.
Surface charge
• Uniform boundary condition is not possible to describe the physical situation!
--- Go back to the charge separation process!
5



2
( )

0.

( r
 a )

, for a for r a
0 r a .
Height of the potential trap is approximated by the hydrogen bond strength
Introduces an asymmetry to the positive and negative ions
--Attributes an energy cost to interfacial charge separation process.
•
•
Functional form of f(r) can not be arbitrary.
1
 
 r
 f r
 r 
 c


C should yield zero(no net charge brought by the potential trap).
• The potential trap is only for H
+
 and ions.
6

•
Derive a charge-conserved PB(CCPB) equation whose right hand side must integrate to zero.
•
Total ion density must be written as: n
O

 n



2

 a
----



 

• Definition of a global chemical potential that is derived from the condition of overall charge neutrality.
---- Different from the local electrochemical potential component such as ion concentration and local electrical potential.
7

PB equation
J n


J n

 n
 n




Assume
 exp exp
0

 e

 e

/ k T
/

 n



J
/
/ n  n n t t 
 
 u u

D


(
(
B
 
B e e n n

 n
) n
 
 e

 n

J
 n  n
 


0,



0.
, k T n


 J n 
 
D


 n

 2    e
 n

(


 n
 dr
)
 
,


,
0,

0,

CCPB equation n
 n


J n


J n


0
 

   f ) / k T exp

 e (
  f ) / k T

 f ) / k T ]
 n V
   dr exp[
 e (
  f ) / k T ]
 2
 
1
 2
D sinh


D  2
 k T
B
2 e n



  e
  k T
1

 
 r


 r 

2
 o










 a
0
 a
0 exp e (
 r exp e (


 r exp
 e (
  f ) /
B f ) /
B exp
 e (
  f ) /
B f ) /
B
8











.


Reformulate the CCPB with a definition of chemical potential
(Salt addition considered)
1
  r r  r

 r 

 e
{ n
OH


 n
H

 exp[
 e (
  f ) /
B
]
 n
Cl


) /
B
] f ) /
B
]
 n
Na

 exp[
 e (
 
) /
B
]}
R.H.S.
+ and – ion densities separately integrates to zero:
  k T
B
2 e ln


 a
0
{ n

OH
 e



 a
0
{ n

H
 exp[
 e (
 
 f ) / k T
B
]
 n

Cl
 f ) / k T
B
]
 n


Na exp[ e

/ k T
B
]} rdr exp[
 e

/ k T
B
]} rdr



 n
    
OH

 n
Cl

 n
Na

 n
H

 n n
O

 n



2

 a
9

1
  r
 
 r

 e
{ n
OH


 n
H

 exp[
 e (
  f ) /
B
]
 n
Cl


) / f ) /
B
]
 n
Na

 exp[
 e (
 
) /
]
]}
  k T
B
2 e ln




 a
 a
0 n

{
OH
 e

{ n
H

 exp[
 e (
 

) / ]
B

 f k T n
Cl
 f ) / k T
B
]
 n

Na
 exp[ e

/ k T
B
]} rdr exp[
 e

/ k T
B
]} rdr
0 n

(1) Solve (1) and (2) simultaneously, with as the inputs,





(3) From one can obtain the a
2
2
 a

0
{ n
OH

 surface dissociation charge density f ) /
B
]
 n
Cl


) /
B
]} rdr

 as :

2 a
2 a

0
{ n
H

 exp[
 e (
 
 n
O  n



2

 a f ) /
B
]
 n
Na

 exp[
 e (
 
) /
B
]} rdr
(1)
(2)
10

(4) Surface charge densities s can be obtained by integrating the net charge densities inside the surface potential trap.
NOTE:
(a) In the inputs , the law of mass action must be obeyed. That is, n

H
 n


( M )
 n

OH

( M )

10

14 ( M 2 )
(b) In the presence of salt, acid, or alkaline additions, there will be ions
 other than and . These ions do not interact with the solid surface. That is, they should not be allowed to enter the surface potential trap. A mathematical implementation of this constraint is described below .
(c) Since the only interaction between the different types of ions is electrical, hence besides the interaction with the solid surface, all often positive and negative ions are treated equally.
11

 o
 
 
 x o | a


 
 x f f
| a

 o
Na + , H + ,
Cl , OH -
Potential trap
 f
H + , OH -
Interfacial
Potential trap intended to model the interfacial charge separation energy


2
( )

0.

( r
 a )

, for a for r a
0 r a .
Potential trap height

12 r

•
Charge conserved PB equation with a definition of chemical potential (salt addition considered)
1
1 n



OH
 r r

1

 n






Cl
 r r o
1 f

 r
 n





Na
 e r
 e r n o
{ n
Cl
 e
Na
{




 r n
 n


 o

OH

H


 e
2
 e


 n n
 e e




(
 e n


2
 exp[

 o
Na
 f
(
Cl
PB
(

(


)


)


H n
/
(

 k T
PB
1
OH

) f
B e
2

D

)
) sinh
)
( PB )
(
]
]
/
/
--Arising from charge neutrality condition :

)
B k T


 n e
B
 
OH
 e
H

( PB
B
]
( exp[
( PB  )
) n
H

/
/
 e (



B exp[ e
 o
 o
 
) /
( PB )
]

]} (

 f
) /
/
 k T
B
]


 f ) / k
B
T ]} n

H

( M )
 n

OH

M
 
14
M
2
( ) 10 ( )
 
2 e ln




0
0 a


{
 n
OH
 exp( e
 o
/ k T
B
)
 n

Cl
 a



{ n
H
 exp(
 e
 o
/ k T
B
)
 n

Na
 exp( e
 o
/ k T
B
)} rdr exp(
 e
 o
/ k T
B
)} rdr

 a
 a
 a
 a

 n
OH
 n
H


 f exp[
 e (
 f

 f ) / k T rdr
B
] f ) / k T rdr
B
13
]




.

Charge regulation phenomenon denotes the fact that the surface charge can not remain constant as the fluid channel width decreases.
Usually surface site density, equilibrium constants pK, Stern layer capacitance parameters are needed for explaining the data.
No Salt , pH=8.2 NaOH
• s contains some ions captured from the bulk. However, when the radius decreases below , s is seen to approach . .

•
In the large channel width limit, our reformulation yields the same results as the traditional
PB equation.

•
Zeta potential
-the basis of the EK effects that arises from charge separation and its associated potential variation
Average velocity: u z




2 r u a

1
2
2 z
 a
2
0 a
2 r



2
2 r

 a
 r
2 z
2

[(
(
2 z
PB

)

1
1 r

 

1

 r
2

  z

E
( PB ) r
 r

 1
E

( PB ) z 1
0


]

  z
P z
  rdr


2 
 r
(
 
(
E z a
)
2

( PB )
2
1
 
 r a   r
0
( PB ) a d r a


 

E z


No Salt , pH=8.2 NaOH
-Zeta potential has the same value as the negative of the chemical potential in large fluid channel down to 10 μm but the two deviate from each other as the fluid channel width diminishes. In particular, the zeta potential, or EK effect, diminishes in small channels.
15

Potential trap height γ=510mV
Outside
Potential Trap
•
Isoelectric point is pH2.5, in good agreement with experimental data.
• Consistent with experiment: Net polar orientation of interfacial water molecules was observed to flip close to pH4.
16

Net electronic charge per μm y
pH=7 No Salt
Fluid
Net Charge Distribution:
R=0.7μm a
SiO
2
Nanochannel h a=0.2μm h/2=0.2μm
0 2 4 × 10 4
/
 m
3 a: channel radius
17

F
 k T
B
A

 d x

 n
Na
 n n
Na
 n o
 n
H
 n n
H
 n o
 n
OH
 n n
OH
 n o
 n
Cl
 n n n
Cl
 
 o 2 k T
B
|
 
|
2  efN


Mixing entropy
Electrical interaction energy
•
CCPB predicts both the range and the magnitude of the force as a function of
18 pH and salt concentration.

19