A New Perspective on the Poisson-Boltzmann Equation and some Related Results

Ping ShengDepartment of Physics

Hong Kong University of Science and Technology

1Phys. Rev. X 4 (1), 011042

Controlled Structural Formation of Soft MatterBeijing, China

August 12, 2015

Collaborators

Li WAN Shixin XU Maijia LIAO Chun LIU

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Introduction to electrokinetics

Distance from the interface

ζ : Zeta potentialλD: Debye length

0

• 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.

3

Si

Si

Si

Si

Electrical double layer

Si

O

HSi

Si

Si

Si

+-

+

-

-

+

Poisson-Boltzmann equation

.const

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.

• 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?

Boundary of the PB equation

22

1sinh

D

Surface charge

4

Treating electrical double layer as an integral entity

0

a

Fluid Surface charge

.const Boundary of PB equation

Present Boundary• The computational domain must be

overall neutral.

• A constant potential boundary condition would only yields a constant potential solution.

• From the Gauss Theorem, the only compatible Neumann boundary condition is zero normal electric field.

• Uniform boundary condition is not possible to describe the physical situation!

--- Go back to the charge separation process!--- Go back to the charge separation process!5

Surface potential trap model

Introduces an asymmetry to the positive and negative ions--Attributes an energy cost to interfacial charge separation process.

6

a r a for

( ) 0.f r 0 .r a for

Height of the potential trap is approximated by the hydrogen bond strength

( )( ) 1 cos ,

2

r af r

• Functional form of f(r) can not be arbitrary.

• the integration of over the whole solution domain

should yield zero(no net charge brought by the potential trap).

• The potential trap is only for and ions.

1 ( )f r crr r r

C

+H OH

Outline of the mathematical approach

• Derive a charge-conserved PB(CCPB) equation whose right hand side must integrate to zero.

• Total ion density must be written as:

---- and are denoted surface dissociated charge densities. For a positive surface potential trap, resides mostly inside the trap, and (= ) resides mostly in the diffuse layer.

• 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.

2On na

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Poisson-Nernst-Planck equations and their static limit

PB equation CCPB equation

0n nJ J

0n nJ J

0,

0.

B

B

en n

k T

en n

k T

exp /

exp /

B

B

n e k T

n e k T

exp ( ) /

exp ( ) /

B

B

n e f k T

n e f k T

n Assume

22

1sinh

D

1 2

22B

D

k T

e n

Be k T

exp[ ( ) / ] exp[ ( ) / ]oB Bdr e f k T n V dr e f k T

0

0

exp ( ) /

exp ( ) /21.

2 exp ( ) /

exp ( ) /

a

a

e f k TB

r e f k T drBoea nr

r r r e f k TB

r e f k T drB

( )2 ,e n n

,ne

D n nk TB

J

/ ( ) 0,nn t n u J

,e

D n nn k TB

J

/ ( ) 0,n t n n u J

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Global chemical potential

2On na

Reformulate the CCPB with a definition of chemical potential (Salt addition considered)

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1{ exp[ ( ) / ] exp[ ( ) / ]

exp[ ( ) / ] exp[ ( ) / ]}

OH Cl

H Na

er n e f k T n e k TB Br r r

n e f k T n e k TB B

0

0

{ exp[ ( ) / ] exp[ / ]}

ln2

{ exp[ ( ) / ] exp[ / ]}

a

B BOH ClB

a

B BH Na

n e f k T n e k T rdrk T

en e f k T n e k T rdr

OH Cl Na Hn n n n n

R.H.S.+ and – iondensities separately integratesto zero:

Solution process (I)

1{ exp[ ( ) / ] exp[ ( ) / ]

exp[ ( ) / ] exp[ ( ) / ]}

OH Cl

H Na

er n e f k T n e k TB Br r r

n e f k T n e k TB B

0

0

{ exp[ ( ) / ] exp[ / ]}

ln2

{ exp[ ( ) / ] exp[ / ]}

a

B BOH Cl

B

a

B BH Na

n e f k T n e k T rdrk T

en e f k T n e k T rdr

(1)

(2)

2

2

2{ exp[ ( ) / ] exp[ ( ) / ]}

0

2{ exp[ ( ) / ] exp[ ( ) / ]}

0

2

OH Cl

H Na

O

an e f k T n e k T rdrB Baan e f k T n e k T rdrB Ba

n na

10

(1) Solve (1) and (2) simultaneously, with as the inputs,

n

(3) From one can obtain the surface dissociation charge density as:

(4) Surface charge densities s can be obtained by integrating the net charge densities inside the surface potential trap.

Solution process (II)

NOTE:

14 2( ) ( ) 10 ( )H OHn M n M M

(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.

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(a) In the inputs , the law of mass action must be obeyed. That is,

n

(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.

+H OH

| |

o f

foa ax x

Na+, H+,Cl-, OH-

Potential trap

H+, OH-

Surface-specific potential trap model

( )( ) 1 cos ,

2

r af r

r

a r a for

( ) 0.f r 0 .r a for

Potential trapheight

o f

Interfacial Potential trap intended to model the interfacial charge separation energy

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• Charge conserved PB equation with a definition of chemical potential (salt addition considered)

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1{ exp[ ( / ] exp[ ( ) / ]

exp[ ( ) / ] exp[ ( ) / ]}

1{ exp[ ( ) / ] exp[ ( ) /

)o

o o

o o

f

f B fOH H

Cl OH

Na H

er n e k T n e k TB Br r r

n e k T n e k TB B

er n e f k T n e f k

r r r

]}BT

OH Cl Na Hn n n n n

0

0

{ exp( / ) exp( / )} exp[ ( ) / ]

ln2

{ exp( / ) exp( / )} exp[ ( ) / ]

.

a a

B B BOH Cl OH

a

a a

B B BH Na H

a

o o fB

o o f

n e k T n e k T rdr n e f k T rdrk T

en e k T n e k T rdr n e f k T rdr

--Arising from charge neutrality condition:

Re-derivation of the PB equation from the CCPB equation

14 2( ) ( ) 10 ( )H OHn M n M M

( )

( )

1{( exp[ / ]

( exp[ / ]}

)

)

o

Cl OH

Na H

PB

PB

er n n e k TBr r r

n n e k TB

Outside potential trap: ( )PB

1 ( ) ( )exp[ / ] exp[ / ]

2 ( )( / )

sinh

oe PB PB

r n e k T e k TB Br r r

e PBn e k T

B

( ) ( )22

( ) ( )

1sinhPB PB

D

PB PBBe k T

Charge-regulation phenomenon

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.

• In the large channel width limit, our reformulation yields the

same results as the traditional PB equation.

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 . .

D

Zeta potential

-- 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.

( )2 20 0

2 2[( ) ] ( )a aPB rdr r rdr

a a

• Zeta potential --the basis of the EK effects that arises from charge separation and its associated potential variation

No Salt, pH=8.2 NaOH

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1 ( )

0 0

212 ( )

2a PBE Er rz zu u rdr dz z a aa

2 ( ) ( )

2

1PB PB

r r r

2

2

1 1u u EPz z zr r r z

2 2 ( ) ( )

2 2

1 1 1PB PBu u EPz z zr r r z r r r

Average velocity:

Outside Potential Trap

Isoelectric point and dipole inversion

Potential trap height γ=510mV

• Consistent with experiment: Net polar orientation of interfacial water molecules was observed to flip close to pH4.

• Isoelectric point is pH2.5, in good agreement with experimental data.

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Donnan potential

Net Charge Distribution:y

SiO2

FluidpH=7 No Salt

Nanochannel

ha

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h/2=0.2μm

a=0.2μm

0 2 4 × 34 /10 m

R=0.7μmNet electronic charge per μm

a: channel radius

2| |2

Na OH ClH

Na H OH Cl

k T n n nn efNBF d n n n n n n n no o o o k T k TA n n n n BB

x

Mixing entropyMixing entropy Electrical interactionElectrical interactionenergyenergy

Comparison with force measurements

• CCPB predicts both the range and the magnitude of the force as a function of pH and salt concentration.

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Thanks for your attention!

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