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A two micrometer thick layer of water would evaporate nearly instantaneously, I’d think.
Without surface tension, water would evaporate much faster in general. Probably could set out a cup of water in the morning and it would be gone by bedtime.
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Your own body would basically be a fog cloud of evaporation all day until you were dehydrated like a skeleton in a salt flat, which is exactly what you would be, since no life anywhere could exist.
Especially if there were weaker intermolecular forces (which normally give it surface tension).
You could even mop it up by placing the mop in the middle of the room and just wait.
If there were no surface tension there would almost definitely be no wicking action either. If it didn't evaporate super quickly, water without surface tension would probably be a giant pain in the ass to clean up.
Surface tension and surface energy are related but different. The surface energy of the interface between water and air causes surface tension. The surface energy of the interface between water and mop fibers causes wicking.