I'm looking for a type of plastic membrane that would let water through it from the outside, but wouldn't allow water to escape once it's inside the membrane.
Ideally available in a roll.
Does such a plastic membrane exist?
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The comments above have gently hinted that it would be a violation of thermodynamics for a passive membrane to force a substance in a single direction. Molecules have an equal probability of moving through the membrane in either direction, unless you apply a force to make them move in a specific direction, such as electro-osmosis or active water transport in plants.
Consider that if such a membrane as you posit existed, you could use it to pump water from a lower level to a higher, and then power a water-wheel: perpetual motion. Sadly, Maxwell's demon is getting a bit arthritic to shuttle water molecules, and the dampness only aggravates its condition.
No passive membrane can achieve this goal but...
A passive structure that permits one-way passage of water vapour can't exist as it would violate the laws of thermodynamics (effectively it would be able to pump water continuously with no input of energy which is obviously impossible).
But practical membranes and active devices can do a useful job of something similar. Gore-Tex, for example, allows the passage of water vapour but not liquid water. So it is widely used in clothing to create breathable but waterproof fabrics. This is a useful goal but not magic: the flow of vapour is not directional but depends on the relative concentration on the two sides of the membrane (but keeping out liquid water is very useful in clothing so even this non-miraculous property is very beneficial compared to impervious membranes that would trap all the sweat and water vapour inside the clothing until it condenses and makes you wet even though the rain has been kept out).
Consider also osmotic membranes. They can often allow selective passage of small molecules like water but prevent the passage of salts or larger molecules. But, operated passively, what happens is that water passes from the side with a higher water concentration to the one with lower concentration (ie from pure water to sea water). They can be used to purify water by "pushing" the impure side driving the reaction in the opposite direction from the thermodynamically favoured one so water travels from the impure side to the pure side.
Some polymers and minerals (eg some zeolytic molecular sieves) can very selectively absorb water. This might look like a potential route to a pump. But they eventually saturate and can't absorb any more (in effect the laws of thermodynamics reassert themselves by equalising the internal "pressure" of water in the pump with the external "pressure" of water outside the pump. There is no way around this that doesn't break the rules of thermodynamics.
The only way to drive one-way passage of water (or anything else) is to provide energy to the system. An amazing example of this is the ionic membrane technology described here which uses a very low current input to continuously "pump" water vapour to dehumidify the inside of electrical junction boxes with no mechanical parts. So, selective pumping is possible, but requires energy.