Classical Models of the Interface between an Electrode and Electrolyte

M.Sc. Ekaterina GongadzeFaculty of Informatics and Electrical Engineering

Comsol ConferenceMilano 200913 10 2009

Interdisziplinäre Fakultät Fakultät für Informatik und Elektrotechnik Medizinische FakultätMathematisch-Naturwissenschaftliche Fakultät Fakultät für Maschinenbau und Schiffstechnik

13.10.2009

Presented at the COMSOL Conference 2009 Milan

Outline

MotivationElectrical Double Layer (EDL)y ( )EDL Models

• Classical ModelsClassical ModelsEDL Capacitance ComparisonF t G lFuture Goals

2

Motivation

Implant & Body Implant & Body FluidsFluidsFluidsFluids

Electrical Electrical

Double LayerDouble Layer

3

Source: Lüthen et al. (2005),Biomaterials 26

Electrical Double Layer (EDL)

• What?

The interface between a charged surface and an electrolyte

• How big?

Thickness: 0.1 - 100 nm (~ ionic strength of the solution)

• Where?

Implant´s surfaceElectrolyte batteryElectrolyte battery In microsystems, around charged nanoparticles (biomolecules, latex beads…)Aurora, solar corona, pulsars

4

Distribution of Ions

D t i d b th f tDetermined by three factors:

• electrostatic (Coulombic) forceselectrostatic (Coulombic) forces • diffusion• specific interactionsp

5

Helmholtz Model

Potential profile determined by Poisson’s equation: q

022

02

2

dxd

dxd

r

022

02

2

dxd

dxd

r

εr relative permittivity of medium

ε0 permittivity of free space

ρ space charge density

φ electric potentialφ electric potential

x distance tangential to the electrode‘s surface

d diameter of the solvated counter

6

d diameter of the solvated counter ion

Gouy-Chapman Model

Assumptions:

• ions as point-like charges• dilute ionic solution• continuum dielectric solvent• continuum dielectric solvent

7

Stern Model

Helmholtz layerHelmholtz layer+

Diffuse double layery

Total capacitance:

CH and Cdiffuse in series:

)(111

0diffuseHd CCC

8

Numerical Simulation

• Geometry and Subdomain Settings

Gouy-Chapman Model:Linearized Poisson BoltzmannSubdomain Settings Linearized Poisson-Boltzmann Equation:

zekTn sinh8

2/10

whereElectrolyte:

NaCl

Ele

ctro

de

Ø 2

0 nm

kTx r 2

sinh0

k Boltzmann constant

20 nm0.3 nm 0.3 nm

E k Boltzmann constante unit chargezi ion i charge numberc concentration of ion i

Helmholtz Model:Poisson Equation:

02d

ci concentration of ion iT temperatureni0 number of ion i in bulk

9

02 dx

Numerical Simulation (2)

Stern Model: • Boundary conditions

Subdomain 1:

02d

symmetry

Subdomain 2:

02 dx

continuity

pote

ntia

l

pote

ntia

l

dom

ain

1

dom

ain

2

kTzekTn

x 2sinh8

2/10

El.

p

El.

Sub

d

Sub

d

kTx r 20 symmetry

10

Postprocessing

• Capacitance computation

i

ndADQ 0QC

where Q – free electric charge dA i i fi i i l

H l h lt G St M d l E i t

QC dA - vector representing an infinitesimal element of area D0 - electric displacement field

Helmholtz Model

Gouy-Chapman Model

Stern Model Experiment

C [µF/cm2] 231 41 77 16 57 92 6Cdl [µF/cm2]analytical

231.41 77.16 57.92 6

Cdl [µF/cm2]i

231.69 77.03 57.61 6

11

numerical

Classical Models – an overview

Assumptions:• ions as point like charges

Disadvantages:

i t t d i t h• ions as point-like charges• dilute ionic solution• continuum dielectric solvent

• ions treated as point charges• only significant interactions -

Coulombic ones• constant electrical permittivity

Advantages:• Simple implementation • Analytical solution

• not time-dependent• infinite concentration of counter-

ions near the charged surface• inhomogeneity of the charge • Analytical solution o oge e y o e c a gedistribution not included• finite size & shape anisotropy of

molecules not incorporated• distorted structure by steric effectdistorted structure by steric effect

& hydration force not included• overlapping problem

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Summary & Outlook

• Importance of EDL

• Classical Models

• Need of a Sophisticated M d lModel

• Modelling of a non-planar electrode‘s surface

13

Source: Lange, R. et al (2009), Material and cell biological investigations on structured biomaterial surfaces with regular geometry, IBI 2009