Conductivity
Low bulk membrane conductivity in a fuel cell results in a fuel cell with larger voltage losses which results in lower performance. Assuming all other components are equal, a fuel cell with high bulk membrane conductivity will have better performance due to smaller voltage losses.
Bulk Conductivity (σ) is a material property. Thus it is independent of a sample's dimensions. Bulk Resistivity (ρ) is also a material property. The two are inversely related:
σ = 1 / ρ
Resistance
Resistance (R) is a material quantity. Thus it is dependent on a sample's dimensions. The unit of resistance is Ohms (Ω). Resistance is related to conductivity and resistivity through the following equation:
R = (ρ * L) / A = L / (σ * A)
Where L is the Length parallel to charge flow and A is the cross sectional Area perpencicular to the charge flow.
Conductivity Calculation
Conductivity can be calculated by measuring a resistance for a given sample then using the samples geometry to calculate conductivity.
σ = 1/ρ = L / (R * A) = L / (R * T * W)
A sample calculation is made below
Conducivity in polymers is much lower than that of metals. So conductivity in the polymer industry is typically expressed in mS/cm. Where a S is a Siemen and has units equal to 1/Ohm
Area Specific Resistance
Area Specific Resistance is also material quantity, as it is dependent on a sample's dimensions. This quantity, when expressed in units of mΩ*cm2, can be multiplied by the current density in A/cm2 to get mV. When referring to Area Specific Resistance we must be clear as to what is causing the resistance. While the measured Area Specific Resistance is dominated by the membrane in an optimized fuel cell, it is possible to have an Area Specific Resistance much higher than the membrane's Area Specific Resistance in a fuel cell that is not yet optimized. This is the result of the resistance introduced by other fuel cell components and interfaces. Area Specific Resistance is calculated by multiplying the Area times the Resistance:
R * A = L / σ
Unit Analysis
To convert conductivity to Area Specific Resistance we must do some unit analysis. L is the distance parallel to charge travel. In a fuel cell, this is the membrane thickness which is typically expressed in microns (μm). Thus we can do the following unit analysis.
The result of the above calculations shows that if we use conductivity in mS/cm, L in microns, then we simply divide the L by conductivity and multiply by 100 to get an Area Specific Resistance with units of mΩ*cm2