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

Collaborators Li WAN Shixin XU Maijia LIAO Chun LIU

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

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?

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!

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.

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.

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

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:

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:

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.

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

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:

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

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.

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.

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

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.

Thanks for your attention!