1. 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),

2. Collaborators
Li WAN
Shixin XU
Maijia LIAO
Chun LIU

3. Introduction to electrokinetics
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 Si H Si
Potential - + + -
Distance from the interface -
ζ : Zeta potential
λD: Debye length +
Electrical double layer 3

4. Poisson-Boltzmann equation
Boundary of the PB equation a
Fluid
Surface charge
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?

5. Treating electrical double layer as an integral entity
Boundary of PB equation
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!
Present Boundary a
Fluid
Surface charge
--- Go back to the charge separation process!

6. Surface potential trap model
for
Height of the potential trap is approximated by the hydrogen bond strength
for
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.
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.

7. 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.

8. Poisson-Nernst-Planck equations and their
static limit
PB equation
CCPB equation
Assume

9. Global chemical potential
Reformulate the CCPB with a definition of chemical potential (Salt addition considered)
R.H.S.
+ and – ion
densities separately integrates
to zero:

10. Solution process (I) (1) (2)
(1) Solve (1) and (2) simultaneously, with as the inputs,
(3) From one can obtain the surface dissociation charge density as:

11. Solution process (II)
(4) Surface charge densities s can be obtained by integrating the net
charge densities inside the surface potential trap.
NOTE:
In the inputs , the law of mass action must be obeyed. That is,
(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.

12. Surface-specific potential trap model
Na+, H+,
Cl-, OH-
Potential trap
H+, OH-
Interfacial Potential trap intended to model the interfacial charge separation energy r
for
Potential trap
height

13. Re-derivation of the PB equation from the
CCPB equation
Charge conserved PB equation with a definition of chemical potential (salt addition considered)
Outside potential trap:
--Arising from charge neutrality condition:

14. 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.
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.
No Salt, pH=8.2 NaOH

15. Zeta potential Zeta potential
No Salt, pH=8.2 NaOH
Zeta potential
--the basis of the EK effects that arises
from charge separation and its
associated potential variation
Average velocity:
-- 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.

16. Isoelectric point and dipole inversion
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.

17. Donnan potential y 2 4 × Fluid Net Charge Distribution:
pH=7 No Salt
SiO2
Fluid
Net Charge Distribution: y
h/2=0.2μm
a=0.2μm 2 4 ×
R=0.7μm
Net electronic charge per μm a h
Nanochannel
a: channel radius

18. Comparison with force measurements Electrical interaction
energy
Mixing entropy
CCPB predicts both the range and the magnitude of the force as a function of
pH and salt concentration.

19. Thanks for your attention!