When a liquid is raised against gravity due to the capillary action of, say, a paper towel or glass pipette, where does the energy come from to raise the liquid?
Does it actually come from the heat of the water, thus cooling it off? If so, does this go against the 2nd law of thermodynamics? I doubt it, but I really have no idea where the energy could come from.
Capillary action is an interesting, counterintuitive, and often misunderstood phenomena. For reference, here is the Wiki on capillary action. (The explanation appears to be correct–it’s always wise to check with Wikipedia–but I don’t have my physics text with me here at the office so I’m working from Mark’s.)
Capillarity (which causes the “action” of water rising in a tube) is a balance between the cohesion of a fluid (attraction to itself due to interatomic or intermolecular forces) and adhesion (attaction to another material via electrostatic forces). Water beading on your windshield, rather than rolling down as a purely Newtonian fluid must do, is exhibiting cohesion. The result of this is to crease a meniscus, a cross section of which looks like a suspension bridge, and for the same reason–balance of forces.
Okay, you knew all this; why does water appear to climb up the tube without any apparent applied force? And, more importantly, how can you utilize this to create an over-unity heat engine? Sadly, you can’t, and even if you could, the Patent Office would deny you proprietary rights to it. Water is naturally electrically polarized; even distilled (pure) water with no metallic or charge carrying impurities will have localized charge polarity (though it is a very poor conductor of electricity). Most pipette materials are some kind of ceramic, which holds a static charge as well; as a result, water molecules tend to form bonds at the boundary, and like English soccer fans, will climb over each other in order to get better access to the game. This makes them more ordered and is (locally) a lower energy configuration, even though they’ve (slightly) increased the gravitational potential energy of the entire mass. Those in the middle of the tube have no such incentive, and thus just sit there like a bunch of sleepy electrochemists in colloquium. So, forces are balanced, energy is conserved, and entropy is not diminished.
Because the behavior and fundamental characteristics of hydrogen bonds are not completely understood on the quantum level, the capillary effect is still a bit mysterious as far as the electrochemical interactions go; we’re not always certain why some materials exhibit such a adhesive affinity for others, and why some liquids are so empathetically cohesive. The mechanical and thermodynamic effects of capillarity are well understood, though, and do not serve to keep scientists and engineers after the shift whistle blows.
Stranger
Liquid is raised in a capillary tuve against the force of gravity because the attractive (adhesive) force of, say, water to the glass in the tube is stronger than the cohesive force of water-to-water. The same effect explains why there is a concave meniscus in a test tube holding water (note the same meniscus is convex is a mercury-filled tube because the cohesive force of mercury is stronger than it’s adhesive force to glass).
The energy required to draw up the liquid comed from the introduction of the capillary tube; there is no change in the entropy of the liquid (remember, you need to consider the entropy of the system as a whole–including the introduction of the glass tube in the system–before making entropy calculations.
The physics of capillary action work against any method for “drawing off” just the energy associated with the water being raised in the tube, e.g. poking a hole in the side of the tube and letting the liquid flow back into the container, which would represent a perpetual motion machine where capillary action raises water and gravity lets it fall perpetually. This site discusses these ideas, which perhaps will give you better details.
Stranger, that is one of the most entertaining scientific explanations fpr the uninitiated I’ve read in a long time
Heee! 
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Others have explained capillary action very well, so let me comment on this. The second law doesn’t state that entropy always increases. It states that in a closed system (i.e., one with no energy coming in), entropy increases on a global scale.
In a closed system like the universe, this holds true, but it doesn’t preclude entropy from decreasing on a local scale, especially in an open system like liquid in a tube. It’s a somewhat subtle distinction if you’re not used to this sort of thing, but keeping it in mind should help you to avoid confusion.
Forgive my skepticism, but I don’t think this is the answer. It’s incidental to what I’m really asking anyway. Let’s say you’ve already introduced the tube or whatever. Now we’re at a moment where a lot of water molecules are going to accelerate upward, climbing the walls of a tube or fibers of a paper towel. I know that the water is attracted to the material, but what is the “equal and opposite reaction” to the water’s acceleration?
Is this any different from when two iron magnets are set next to each other and being accelerating?
Thanks for the replies so far.