How heavy is the curtain and how are you supporting the pipe? If you need more strength go with 1", 1.5" or even 2" pipes and forget about filling it. There are a few types of copper pipes: Type-L, Type-M and Type-K and Type-DWV which I don't know anything about but one is more rigid than the others.
Don't use cement, use concrete and rebar - much, much stronger. Copper will react with the concrete and develop pin holes but may take 20 or so years. Shouldn't be hard to fill at all, just use a vibrator and a shop vac at one end and let gravity work for you.
Rebar comes in 20-ft sections but isn't very stiff and 16-ft section small enough to go through a 1/2" pipe won't add much stiffness although it will prevent a buckle. 3/4" rod is awfully heavy. I think the black iron would be better.
However, if you're expecting an unsupported 16-ft section of such a small diameter to be stiff enough, I think you're going to be disappointed.
What is the maximum span between supports and number of supports?
You're not adding a material for it's strength - you're really just looking to keep the tube from buckling - so I don't know that a "stronger" material would add strength. For a given diameter the heavier the pipe/rod/tube the more it will deflect under its own weight, so lighter is better.
I'm not sure if it it would work, but it might be possible to use a length of plastic tubing attached to an expanding foam can. If the tubing doesn't prove to be have too much friction (it might), and the spray can pressure can push the foam through the tube for half the length of the pipe (inserted from both ends), then it could work. Insert the tube fully, spray a dollop, withdraw the tubing and see how far the foam comes out. Or, if you're feeling lucky, use minimally expanding foam and spray while continuously pulling it out.
Another option would be to use another material and paint/patinate it to look like copper and not have to worry about the strength.
The bigger the diameter the stiffer the tube, and the relative stiffness goes up quickly, so you'll have to decide how much sag is acceptable and how big of a rod is the esthetic maximum size.
The standard rebar is 20'. They cut it up at the home store.
I think mortar is the best choice. Buy a cheap plastic funnel and cut it down to a good fit on the pipe. Wear plastic gloves and shove it down the funnel until it comes out the other end. Tape them off and let it sit flat for a week or two. The water will come out eventually but the longer it takes the better. Makes the concrete harder if it slow cures. You could still try to work a #3 rebar in there with the mortar.
I am looking to use a length of 1/2" or 3/4" copper pipe as a rail for holding a curtain.
I would like to strengthen the rod by filling it with a liquid that solidifies. What would you recommend using?
Note I have considered using cement (regular or hydraulic) but am concerned that it will be hard to fill the pipe and that in any case the water won't have anywhere to go. Also, I have considered epoxy but again, I fear that it won't fill the pipe, plus it is very expensive.
Are there any other alternatives such as some type of liquid fiberglass?
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