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I had a rod. I broke it into two. Now I wish to make it one i.e. to join those (not glue or any thing as such) as if the rod was not broken at all. This is our objective!

As I broke the rod apart, the area from where the rod was broken would be rough. So, If I were to join those, then one possible way would be to polish and make the area extremely smooth such that we would say that the coefficient of friction is 0. But Ironically, When I bring those rods together .. as the area is perfectly smooth, they will align so well that the friction force acting between those would tend to infinity when tried to apply force perpendicular to the length of the rod .. make them a single rod again.

Assumption: Friction force is due to the result of the alignment of various hills and valleys that exist on a surface on a microscopic level and due to interlocking of those. i.e. The Classical reason for existence of friction(and not due to the cohesive and adhesive and electromagnetic forces of interaction)

So this seems to be solution but, If we apply the force on both the rods in opposite direction taking them apart along the length of the rod, then we would definitely be able to separate it.

Is my argument right? Is there any other solution to this question that you might know of?

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Just having the surfaces perfectly smooth does not necessarily make for a strong bond which has to be between the molecules which make up the rod.
Richard Feynman put it rather well when he was writing about what is called cold welding between two flat and ultra-clean surfaces brought together in a vacuum:

The reason for this unexpected behaviour [cold welding of copper] is that when the atoms in contact are all of the same kind, there is no way for the atoms to “know” that they are in different pieces of copper. When there are other atoms, in the oxides and greases and more complicated thin surface layers of contaminants in between, the atoms “know” when they are not on the same part.

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