Supercapacitor - ultracapacitor - Double layer capacitor theory
This chapter presents the different theories for the determination of the spatial charges distribution at a metal-electrolyte interface:
Double layer charges and potential
At the interface between a metal and an electrolyte (liquid or gel containing ionic conductors, for example solvent with a salt) appears a charged layer. In the electrolyte the charge is made of ions; in the conductor it may be either electrons or holes. To maintain the electrical neutrality of the system, a second layer charged with the opposite charges is necessary.
In the core of the electrolyte the potential drop is due to the ionic current in the resistance of the viscous solvent. The ionic resistance in the electrolyte may be understood as the friction of the ions in the solvent. The ionic resistance increases with the vicosity. In static, when there is no more current, the potential would be flat in the electrolyte.
The separator is a porous membrane which contains about 50% of electrolyte. The membrane provides the electronic insulation between the electrodes. A high porosity, a small percolation length and a small thickness provides a good ionic conductivity..
The different theories presented below assume different types of charge distribution at the interface.
Double layer model I: Helmholtz model
This is the simplest model for the spatial charge distribution at a metal-electrolyte interface. All the unpaired ions are at a distance d from the metal surface. The ion diffusion rate is infinite.
The model has been proposed in 1879 by Helmholtz.
In the negative electrode the potential U = ΦMetal.
Far from the negative electrode, in the electrolyte the potential U = ΦSolution. The same number of negative ions have left the solvent to collect on the positive electrode which isn't shown in the picture. In the electrolyte remains "paires" negative and positive ions which are not used at the interface to insure the potential difference ΦMetal - ΦSolution) = Q / C .
The ions form a compact mono-layer on the surface at a distance of about 1 nm. This distance is considered as the layer thickness. It depends on the ion size and on the voltage.
Double layer model II: Gouy- Chapman diffuse layer point-charge model (1913)
This more sophisticated model takes into account more physical phenomena. The positive and negative ions distance distributions from the interface are calculated with the following laws and assumptions:
Ni is the i ion concentration in the electrolyte far from the interface, zi the number of charge of the ion, e the electron charge.
Overestimation of the capacitance
Double layer model III: Stern model (1924)
This model combines Helmholtz compact layer model with a diffuse ion layer.
Ions have a finite size and include their solvated shell.
Solvation (hydration in the case of water)
The smaller the ions are, the stronger is the solvation
Cations are generally more solvated than anions
Requires a correction because of the capacitance overestimation.
Double layer model IV: Grahame model (1940)
Resulting capacitance of one layer
CGC is the Goüy - Chapman capacitance of the diffuse layer
Capacitance calculation methodology
Estimation of the maximum capacitance of a layer:
Atom sizes are given in pm (10-12m)
Ion size given in pm
Negative ions can go closer from the electrode => higher capacitance.
This is confirmed by cyclic voltammetry
Graphite basal plane:
Distance between the carbon atoms: 110 pm
Graphite interlaminar distance d002 = 350 to 400 pm
Carbon atomic radius: 70 pm
Carbon covalent radius: 77 pm
Carbon Van der Waals radius: 170 pm
Surface charge estimation
Carbon atomic mass: 12 g/mol
Carbon density: = 2.2 g/cm3 => 0.18 mol/cm3
Avogadro constant: NA = 6 1023 atoms/mol = > 1.1 1023 atoms/cm3
=> Carbon interatomic distance of 0.2 nm (actually d002 = 0.35 to 0.4 nm; the discrepancy is due to the hexagonal packing in the basal plane (0.11 nm between the carbon atoms)
=> Number of carbon atoms on a unit surface 2.3 1015 atoms/cm2
Same estimation with the ions
BF4-: 0.46 nm => 5 1014 molecules/cm2
Et4N+: 0.7 nm => 2 1014 molecules/cm2 or 3.3 10-6 moles/m2
With a charge 1+= 1.6 1019 C/charge => 3.2 10-5 Coulomb/cm2
Capacitance density estimation
Assuming a potential drop of 1 V on the interface layer, with Q = C U.
The maximum expected capacitance could be C = 32 mF/cm2
1 g of carbon powder has an estimated surface of about 1'000 m2
=> the electrode capacitance density has 320 F/g carbon
Today the best « organic » supercapacitor has an electrode capacitance of 100 F/g of carbon (25 F/g if estimated for the double layer).
Aqueous electrolytes bring higher capacitance (but do not forget the voltage limitation....) than organic electrolytes. A factor of 2 is generally found.
The surface charge density is in the range of 10-5 Coulomb/cm2
1 M Et4NBF4 in AN or PC => 6 1023 moles per litre of solution
The electric field in the interface is huge. Let assume a Helmoltz layer of 1 nm width:
The electric field E = U/d = 109 V/m (1'000 V/μm).
In metallized polypropylene film capacitor the electric strength is about 200 V/mm to get a 30 years lifetime of the component.
The pressure P = qE/S = 1000 kg/cm2
300 g of carbon which have a surface 1'000 m2/g need 1 mole of electrolyte, in other words, 1 litre of 1 M solution, to get a compact mono-layer of ions on the surface.
In most of the case we have to face a ion starving (not enough ions to get the maximum capacitance). Consequently the capacitance will depend on the electrolyte molarity.