nextnano³ - Tutorial

next generation 3D nano device simulator

1D Tutorial

Poisson-Boltzmann equation: The Gouy-Chapman solution

Author: Stefan Birner
Please send comments to nextnano3 (-at-) wsi.tum.de.

If you want to obtain the input file that is used within this tutorial, please contact stefan.birner@nextnano.de.
-> 1DGouyChapman.in

Note: The electrolyte model is not implemented yet into the nextnano³ version of 2004_08_24. If you are interested in the executable, please contact stefan.birner@nextnano.de.

Poisson-Boltzmann equation: The Gouy-Chapman solution

• We solve the Poisson-Boltzmann equation for a monovalent salt, i.e. NaCl (Na⁺ Cl^-).
  For this particular case, our numerical solution of the Poisson-Boltzmann equation can be compared to the analytical one-dimensional Gouy-Chapman solution for a monovalent and symmetric salt (see PhD thesis of Sebastian Luber).
   
• The temperature is set to 300 K.
   
• Thus the electrolyte region (100 nm - 199 nm) contains the following ions:
   !---------------------------------------------------------------------------!
 ! The electrolyte (NaCl) contains two types of ions:
 !   1)  10 mM singly charged cations (Na+)    <- 10 mM NaCl
 !   2)  10 mM singly charged anions  (Cl-)    <- 10 mM NaCl
 !---------------------------------------------------------------------------!
 $electrolyte-ion-content

  ion-number        = 1            ! singly charged cations
  ion-valency       =  1d0         ! charge of the ion: Na+
  ion-concentration = 10d-3        ! Input in units of: [M] = [mol/l] = 1d-3 [mol/cm³]
  ion-region        = 100d0  199d0 ! refers to region where the electrolyte has to be applied to

  ion-number        = 2            ! singly charged anions
  ion-valency       = -1d0         ! charge of the ion: Cl-
  ion-concentration = 10d-3        ! Input in units of: [M] = [mol/l] = 1d-3 [mol/cm³]
  ion-region        = 100d0  199d0 ! 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 ion-concentration = 10d-3 ! [M].
   
• We assume an interface charge between the oxide and the electrolyte of -0.2 C/m² = -124.83 x 10¹² |e|/cm².
  
  $interface-states
  ...
  state-number      = 1               ! between SiO2 / Electrolyte at 100 nm
  state-type        = fixed-charge    ! sigma
  interface-density = -124.8301896d12 ! -0.2 [C/m^2] = -124.8301896 x 10^12 [|e|/cm^2]
 
• The pH value is 7, i.e. neutral.
  
  $interface-states
  ...
  state-number      = 2               ! between SiO2 / Electrolyte at 100 nm
  state-type        = electrolyte     !

$electrolyte
  ...
  pH-value          = 7d0             ! pH = -lg(concentration) = 7 -> concentration in [M]=[mol/l]
 
• The following figure shows the electrostatic potential for different salt concentrations at a fixed surface charge of -0.2 C/m².
  The potential at the surface at 100 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 spatial electrostatic potential.
  
  
  
  The Debye screening lengths (DebyeScreeningLength1D.dat) are indicated by the squares and the values are:
  - 1        M NaCl:   0.308 nm
- 0.1      M NaCl:   0.974 nm
-    10   mM NaCl:   3.080 nm
-     1   mM NaCl:   9.741 nm
-     0.1 mM NaCl:  30.79  nm

For a definiton of the Debye screening length, have a look here: $electrolyte
  
  The following figure shows the Debye screening length as a function of the NaCl concentration.
  
  
  
  In this simple tutorial where only monovalent salt is present, the nominal value of the NaCl concentration is equal to the ionic strength.
   
• 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    -0.117077979868865         -124.830189600000
 
• The following figure shows the ion distribution for a 0.1 M NaCl electrolyte. The multiples of the Debye screening lengths are indicated by the blue lines.
  The negative surface charge is screened by the positive Na⁺ ions whereas the negatively charged Cl^- ions are repelled from the surface.
  
  
   
• Linearization of the Gouy-Chapman model
  
  In this approximation (only for high salt concentrations or large surface charges!), the surface charge and the surface potential can be related through the basic capacitor equation:
  
        rho_s = phi_s C
  
  The following figure shows the surface potential at the SiO₂/electrolyte interface as a function of SiO₂/electrolyte interface charge for the monovalent salt NaCl at different salt concentrations calculated with the Poisson-Boltzmann equation.
  
  It can be clearly seen, that only for high salt concentrations or large surface charges the linearization of
     surface potential ( surface charge ) = phi_s(rho_s)
  is valid.
  
  
  
   

• Please help us to improve our tutorial. Send comments to nextnano3 (-at-) wsi.tum.de.