If there is a meltdown at the Fukushima reactors, would the control rods melt also? According to the reports, the shutdown procedures at all the Fukushima reactors were successful, and all the control rods were fully inserted.
So - if there was a meltdown, would the control rods also melt and blend into the resulting material (corium)? If so, would that have the effect of "diluting" the corium in radioactive terms and stabilising it to some extent? (I guess it would depend on the relative melting points of the fuel and the control rods: if the rods are of boron, the melting point is a lot higher than that of uranium.)
 A: A nuclear reactor system is a very delicately balanced system for sustained chain reaction controlled by the control rods. 
As in fukushima reactors, which are light water reactor where you have enriched uranium, the criticality depends heavily on the core configuration and the position of control rods.
In a core meltdown, what it meant is the uranium core get melted and settles down and the core configuration gets damaged; for control rod mechanism to work the control rods should be surrounded by active core
Actually, that's how reactor power is reduced by inserting the control rods into the core, Now, when core meltdown occurs the core becomes inaccessible for control rods, since it settles down.

So, the question whether control rods melts is heavily dependent on the mature of accident. Control rods are generally made of stainless steel with boron or cadmium or hafnium, while core is Uranium which is somewhat like soft metals sodium having melting point near 1100 degree Celsius, while SS has melting point near 1500 degree Celsius. Again some reactor used $UO_2$ as fuel which has higher melting point than metallic uranium. I don't know fukushima used which one!!
So, due to the temperature difference, is seems unlikely that control rods melted in fukushima, rather it seems to me that control rod melting is a good thing for controlling the reactor in case of a core meltdown.
A: An accident which progressed in a somewhat similar manner is the TMI-2 Meltdown.
In-vessel inspection at TMI-2 has revealed that an upper cavity existed in which all structural materials had melted and relocated. The first reaction which occurs in the progression of core melt is the formation of Zirconium eutectics. Soon after, structural elements such as grid spacers are integrated, and after that, all core materials, including the control rod elements are melted.
The melting point of the fuel (>3100 K) is significantly higher than control rod materials (highest ~3000 K). The existence of control rod materials in the Corium mass will not mitigate the thermal damage the mass causes during migration, because the nuclear reactions occurring are the decay of fission products.
The chain reaction rates in the reactors at Fukushima were already decreasing at the maximum possible rate before and during the melt. This is because the fuel geometry and material distribution is such that the likelihood of a neutron born in the fuel being absorbed and causing fission, scaled by the number of neutrons emitted during fission, is much less than one.
The only radioactive dilution that occurred during the accident is the continuing release of fission products (and radioactive daughters) to the environment.
The key to mitigation of the radioactive consequences of a core meltdown is containment. The last line of defense against release is the reactor containment, each of which failed at Fukushima. Another important containment barrier is the confinement of the Corium mass to the reactor pressure vessel. This is achieved by maintaining sufficient cooling to the mass to prevent RPV failure and outflow. Outflow occurred at Chernobyl, but did not occur at TMI-2. The distribution of core material won't be known at Fukushima until detailed Muon tomography or physical inspection occurs.
A: A typical control rod will be stainless steel plus boron carbide, boron steel, a silver/cadmium containing metal rod or halfmium. If these were heated up then they would be likely to melt along with the fuel. It is likely that the molten pool of fuel would have mixed with it the control rod material if a "melt down" occurs.
