While cathodes in magma electrolysis units might be made of carbon because only metals collect at the anode and a layer of fused ferrosilicon will form covered by molten ferrosilicon and “magma” there should be no problem.  Oxygen forms at the anode and will burn most metals up, even tungsten.  Platinum has bee tried but a platinum silicon alloy forms.  Also, platinum is very dense and expensive, and is not “Moon-makeable.”  We need other anode materials.  Oxides are a possibility because they are already oxidized and will not undergo any more reactions with oxygen. 

Calcium oxide can be heated to incandescence, the limelight. It melts at 2570 C. From Lime and Magnesia, 1924, E. Benn, we find that the electrical resistance of CaO decreses with temperature.
Temperature Ohms per cubic cm.
ordinary 7100 million
763 C. 70 million
930 C. 4175000
1105 C. 600000
1235 C. 104000
1370 C. 2045
1466 C. 91


The electrical resistivity of metal increases with temperature.  It decreases with temp. for semiconductors and semiconducting ceramics.

However, my calculations show that with 30 square meters of cylindrical electrode area with 2 cm thick steel cores surrounded by 2 cm of fused CaO, to flow 136,000 amps we need a voltage of  82.6 V to overcome the resistance of the ceramic and this will dissipate 11 megawatts!  We need a more conductive ceramic.  CaO could react to form slag and CaSi to that would melt off (mp 1500 C. and 1000 C. respectively) and erode the ceramic anode. 

Alumina is a possibility.  

http://hypertextbook.com/facts/2006/EuniceHuang.shtml

Al2O3  2.5E6 ohm meters (2.5 million ohms) at 900 C.  Compare to CaO at 930 C. 4.175 megaohms per cc

Could alumina’s resistance drop to lower levels at magma electrolysis temps of 1400 to 1600 C.? Alumina melts at 2000 C.

Calcium titanate (CaTiO3)  is another material that could be made on the Moon, m.p. 1980 C., that would be of interest. 

Titanium carbide is perhaps the most interesting  “Moon-makeable” choice for a cathode material.
From:
http://www.matweb.com/search/SpecificMaterial.asp?bassnum=BCTiCA we find that:

It’s [TiC] electrical resistivity is only 0.00018 to 0.00025 ohms-centimeter at 20 C.  If it acts like any other ceramic it’s resistance should drop even lower at high temps.  4.94 gr/cc  mp 3065 C solidus 3050 C. liquidus 3080 C.  tensile strength at 1200 C. 8560 psi   CTE linear 7.7 microns/m C. 

Will the carbon in the TiC oxidize to CO or CO2?  Or will a thin protective coating of TiO2 form that isn’t thick enough to reduce electrical conductivity?  Or will the TiC just resist oxidiation?

At:
http://www.eovak.com/industries/industries03.html#5 we read that carbides are inert in many chemical and corrosive enivronments.

From:
http://www.ceramics.mmat.ubc.ca/int/KCM.html#NTitanium%20Carbide
We find that titanium carbide is:
“  Very hard, refractory material, known for its high wear-resistance and thermal shock resistance. Used for bearings, nozzles, cutting tools, wear parts. Available as high-purity or technical grade, depending on carbon content. Used as cermet reinforcement, for components such as jet engine blades and cemented carbide tool bits. Electrical conductor especially at high temperatures.

Electrical conductor at high temps! It’s resistance is even low at 20 C. so it should be possible to flow enough electricity thru it to heat up the regolith and get the electrolysis going.  This could be the anode material of choice.  Will it react with silicon or other metals in the melt?  How long with a titanium carbide anode last?  How difficult will it be to cast this ceramic on the Moon?  Perhaps we could sinter powders together with focused solar rays. 

More research into this matter is called for.  As for the resistance of the motlen silicate or basalt melt, it is neglible. Just 10E-4 ohm meter

See:
http://www.islandone.org/MMSG/aasm/AASM5C.html

Table 5.9.- Properties Of Cast Basalt
Physical properties Average numerical value, MKS units
Density of magma @ 1473 K 2600-2700 kg/m3
Density of solid 2900-2960 kg/m3
Hygroscopicity 0.1%
Tensile strength 3.5X107 N/m2
Compressive strength 5.4X108 N/m2
Bending strength 4.5X107 N/m2
Modulus of elasticity (Young's modulus) 1.1X1011 N/m2
Moh's hardness 8.5
Grinding hardness 2.2X105 m2/m3
Specific heat 840 J/kg K
Melting point 1400-1600 K
Heat of fusion 4.2X105 J/kg(+/-30%)
Thermal conductivity 0.8 W/m K
Linear thermal expansion coefficient
... 273-373 K 7.7X10-6 m/m K
... 273-473 K 8.6X10-6 m/m K
Thermal shock resistance 150 K
Surface resistivity 1.0X1010 ohm-m
Internal resistivity 1.0X109 ohm-m
Basalt magma viscosity 102-105 N-sec/m2
Magma surface tension 0.27-0.35 N/m
Velocity of sound, in melt @ 1500 K 2300 m/sec (compression wave)
Velocity of sound, solid @ 1000 K 5700 m/sec (compression wave)
Resistivity of melt @ 1500 K 1.0X10-4 ohm-m
Thermal conductivity,
... melt @ 1500 K 0.4-1.3 W/m K
... solid @ STP 1.7-2.5 W/m K
Magnetic susceptibility 0.1-4.0X10-8 V/kg
Crystal growth rate 0.02-6X10-9 m/sec
Shear strength ~108 N/m2
Magma Electrolysis Electrodes
David Dietzler, 2007