Does there exist a membrane that has unbalanced concentration as equilibrium? In osmosis, there is flow of solvent particles from a region of lower solute concentration to a region of higher solute concentration, until such time that the concentration on each sides are equal.
But does there exist a membrane in which the equilibrium concentration is not necessarily equal (without applying unbalanced pressure)?
 A: 1) Osmosis is more commonly defined as the diffusion of water across a membrane. Unless specifically talking about water its best to just say diffusion. (just a small clarification)
2) The solutions on either side of the selectively permeable membrane will attempt to reach the same concentration (in units of molarity = moles of solute/liter of solution) of solutes by the movement of species across the membrane. The goal is for both sides to reach the same osmotic pressure. 
The membrane cannot affect the osmotic pressure of the solutions. It merely decides which species are allowed to move for the system to reach equilibrium. 
3) At equillibrium, the concentration of each species in the solutions may be different, as long as both systems have the overall same concentration of solutes. 
An example of this is if you have a semi-permeable membrane which allows the movement of ions but not large proteins. If the initial conditions are the same concentration of ions on both sides of the membrane, but an uneven concentration of protein across the membrane, the ions will move to balance out the uneven concentration of proteins. At equilibrium the osmotic pressure of both sides will be equal, but the individual concentration of species on each side will not be equal. 
