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nextnano^{3}  Tutorial
PoissonBoltzmann equation: The GouyChapman solution
Author:
Stefan Birner
If you want to obtain the input file that is used within this tutorial, please
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> 1DGouyChapman_template.in
2DGouyChapman_template.in
3DGouyChapman_template.in
PoissonBoltzmann equation: The GouyChapman solution
 We solve the PoissonBoltzmann equation for a monovalent salt, i.e. NaCl
(Na^{+} Cl^{}).
For this particular case, our numerical solution of the PoissonBoltzmann
equation can be compared to the analytical onedimensional GouyChapman
solution for a monovalent and symmetric salt.
 The temperature is set to 298.15 K = 25°C, the static dielectric constant
of water is set to 78.
 The electrolyte region (0 nm  200 nm) contains the following ions:
!! ! The electrolyte (NaCl) contains
two types of ions: ! 1) 100 mM singly charged cations (Na^{+})
< 100 mM NaCl ! 2) 100 mM singly charged anions (Cl^{})
< 100 mM NaCl !!
$electrolyteioncontent
ionnumber =
1
! singly charged cations ionvalency =
1d0 !
charge of the ion: Na^{+} ionconcentration =
100d3 ! Input in units of: [M] = [mol/l] = 1d3 [mol/cm³] ionregion =
0d0 200d0 ! refers to region where
the electrolyte has to be applied to
ionnumber =
2
! singly charged anions ionvalency =
1d0 !
charge of the ion:
Cl^{} ionconcentration = 100d3
! Input in units of: [M] = [mol/l] = 1d3 [mol/cm³] ionregion =
0d0 200d0 ! refers to region where
the electrolyte has to be applied to
We vary the NaCl concentration from
 1 M NaCl
 0.1 M NaCl
 10 mM NaCl
 1 mM NaCl
 0.1 mM NaCl
Consequently, we have to vary ionconcentration =
100d3 ! [M] .
 We assume an interface charge between the oxide and the electrolyte of
 0.2 C/m^{2} = 124.83 x 10^{12} e/cm^{2}.
$interfacestates
...
statenumber = 1
! between contact / electrolyte at 0 nm statetype =
fixedcharge !
sigma interfacedensity =
124.830193d12 ! 0.2 [C/m^2] = 124.830193 x
10^12 [e/cm^2]
 The pH value is 7, i.e. neutral.
$interfacestates
...
statenumber = 2
! between contact / electrolyte at 0 nm statetype =
electrolyte !
$electrolyte
...
pHvalue =
7d0
!
pH = lg(concentration) = 7 > concentration in [M]=[mol/l]
 The following figure shows the electrostatic potential for different salt
concentrations (0.1 mM, 1 mM, 10 mM, 0.1 M and 1 M) at a fixed surface charge of
 0.2 C/m^{2}.
The potential at the surface at 0 nm, that arises due to the fixed surface
charge density that is in contact with the electrolyte, is screened by the
ions in the solution and the resulting distribution of the ions depends on the
value of the spatially varying electrostatic potential.
The Debye screening lengths (DebyeScreeningLength.txt ) are
indicated by the squares and the values are:
 1 M NaCl: 0.303
nm
 0.1 M NaCl: 0.959 nm
 10 mM NaCl: 3.032 nm
 1 mM NaCl: 9.589 nm
 0.1 mM NaCl: 30.308 nm
(Here, the concentrations of the H3O^{+} and OH^{}
ions slighly influence the Debye screening length (last two digits) because
in the nextnano³ simulations these ions are always present and their
concentrations depend on the pH value.)
For a definiton of the Debye screening length, have a look here:
$electrolyte
The following figure shows the Debye screening length for a monovalent salt
such as NaCl as a function of the salt concentration.
For a monovalent salt the nominal value of the salt concentration is equal
to the ionic strength which is a measure for the screening of charges in a
solution.
 The surface potential can be found in this file:
InterfacePotentialDensity_vs_pH1D.dat It reads for a salt concentration of 0.1 M NaCl:
pH value interface potential [V]
interface density (1*10^12 [e/cm^2]) 7.000000
0.123478240580906
124.830193000000
 The following figure shows the ion distribution for a 0.1 M NaCl
electrolyte. The multiples of the Debye screening lengths (kappa^{1} = 0.959 nm) are indicated by the
vertical lines.
The negative surface charge is screened by the
positive Na^{+} ions
whereas the negatively charged Cl^{}
ions are repelled from the surface.
At about 5 nm both ions reach their equilibrium concentration of 0.1 M.
 Linearization of the GouyChapman model
In this approximation (only for high salt concentrations or
small surface
charges), the surface charge and the surface potential can be related through
the basic capacitor equation:
sigma_{s} = phi_{s} C_{DL}
where C_{DL} is the capacitance per unit area of the electric double
layer. The following figure shows the surface potential at the
solid/electrolyte interface as a function of interface charge for a monovalent salt
such as NaCl at different salt concentrations calculated with the
PoissonBoltzmann equation (symbols).
The solid lines are the solutions of the analytical Grahame equation for a
monovalent salt which relates surface potential to surface charge.
The dotted lines are the solutions of the DebyeHückel approximation for a
monovalent salt which relates surface potential to surface charge.
It can be clearly seen, that only for high salt concentrations or
small surface charges the linearization is valid.
The linearization of the PoissonBoltzmann equation is called "DebyeHückel
approximation".
It can be switched on using:
! electrolyteequation =
PoissonBoltzmann
! PoissonBoltzmann equation (default)
electrolyteequation = DebyeHueckelapproximation ! DebyeHückel approximation
 Capacitance
The numerical PoissonBoltzmann calculations for the capacitance (using the
same data as in the previous figure are shown next.
The capacitance increases rapidly for higher potentials but at very small
surface potentials, the capacitance is equal to the approximation of
parallel plate capacitor model of the electric double layer.
This is expected because in the limit of low potentials, the solution of the
PoissonBoltzmann equation must converge to the solution of the DebyeHückel
equation.
