1) DEFINITION OF TERMS:

Osmose :

Solvent transfer from a diluate solution to a concentrate solution through a permselective membrane.

Reverse osmosis

Separation process which involves a reverse transfer of osmosis.

2) OSMOSIS PNENOMENON

2.1) Macroscopic description :

A diluate solution or a pure solvent is put on one side of a membrane. A concentrate (with for instance salt) solution is put on the other side. The membrane is permselective : it means that has got high permeability to the solvent and low permeability to the salt. The solvent will cross the membrane from the diluate solution to the concentrate solution. Therefore, the salt will not cross the membrane or will cross it very lightly.

If the salt concentration is not too high so that one can consider that the solvent is the mean constituant in the concentrate solution (it means if one can assume that the solutions are diluate). The flux F of solvent is then proportional to the difference of concentration between the two sides of the membrane and can be expressed as follows :

    F = s Pe DeltaC
where :
    s is the membrane surface in m2
    Pe is the membrane permeability to the solvent in international unities
    DeltaC is the difference of salt concentration in mol/l.

2.2) Microscopic view :

The solvent molecules go through the membrane by a solution-diffusion process. The membrane is swelled by the solvent and the solubility is then high. The diffusion coefficient is high too but lower than in the liquid phase. Therefore the salt entities (ions in case of salts or molecules in case of other components such as sugars) may be stopped near the membrane face because they have no affinity with the membrane material ( their solubility in the membrane is then low as it may be for instance the case of ionic salts in non ionic membranes), or may diffuse very slowly in the membrane because of their big molecule size (case of sugars in regard to water molecules).

One can then explain the osmotic flux by the following assumptions :

On one hand, the salt entities in the concentrate solution may obstruct partially the intermolecular holes of the membrane at its surface so that only a few solvent molecules can go in the membrane. On the other face of the membrane where the diluate solution takes place, the holes are less obstructed and more solvent molecules can go in the membrane.

On the other hand, when a solvent molecule goes out of the membrane, it can easily push the salt intities in the solution and one can consider that the desorption of a solvent molecule is fast.

The dissymetry of these microscopic phenomena between the to faces of the membrane induice the global macroscopic flux observed from the diluate solution to the concentrate solution.

2.3) Calculation of the equilibrium state in case of an intermittent process :

In an intermittent process, on suppose that the volumes of the involved solutions are finite :

    - If the concentration of the more diluate solution is not nil, There is the existence of an equilibrium state in which the concentrations on both side of the membranes are equal.

    - If the diluate solution is the pure solvent, the solvent will go completly in the concentrate solution.

When the membrane selectivity is not infinite, a low flux of salt may be observed from the concentrate solution to the diluate solution. The total flux of solvent will be lower than in the case of an infinite selectivity and it also may be partially compensated by the flux of salt. The apparent measured membrane permeability to the solvent will be then lower than the intrinsic permeability.

If the membrane selectivity is equal to 1, the flux of salt and the flux of solvent may compensate themselves.

We have (Conservation of the salt) :

    Ceq V1' = C1 V1
    Ceq V2' = C2 V2
Where :
    V1 is the volume and C1 is the concentration of the diluate solution
    V2 is the volume and C2 is the concentration of the concentrate solution
    Ceq is the equilibrium concentration
    V1' and V2' are the equilibrium volumes
Then :
    Ceq = (V1 * C1 + V2 * C2) / (V1 + V2)
and :
    V1' = V1 * C1 / Ceq
    V2' = V2 * C2 / Ceq
The trowed quantity Q of solvent is then defined as :
    Q = V1 - V1' = V2' - V2
Remarks :
    V1' is lower than V1 and V2' is greater than V2.
    C1 is greater than Ceq and C2 is lower than Ceq.
    If C1 = 0 then V1' = 0. All the pure solvent goes through the membrane.

2.4) Kinetics aspects :

The parameters which influence the osmotic flux are :
2.4.1) The difference of concentration : the more the difference between the concentrate solution and the diluate solution is high, the more the osmotic flux is high.
2.4.2) The permeability of the membrane : the more the permeability of the membrane is high, the more the osmotic flux is high and faster the equilibrium state is reached in the intermittent process.
2.4.3) The membrane thickness : the more the membrane is thick, the more the osmotic flux will be low.
2.4.4) The membrane selectivity : the less the membrane is selective, the more the apparent osmotic flux will be low because the flux of salt and the flux of solvent may compensate themselves.
2.4.5) The mobility of salt entities : This parameter may have some influence in the case of perfectly quiet solutions where the diffusion is the only one process of species mobility : The more the salt diffusion coefficient is law, the more the osmotic flux may be low. In stirred solutions this problem may disappear because the mobility of salt entities is then increased.
2.4.6) The temperature may have an influence because the membrane permeability generally increases with the temperature.

3) REVERSE OSMOSIS PHENOMENON

3.1) Macroscopic description :

When an increasing hydrostatic pressure P1 is applied on the membrane side where the concentrate solution is placed, the osmotic flux is slowed, stopped and then reversed. The threshold of reverse pressure is given by Van't Hoff law :

    DeltaP = DeltaC * R * T
where :
    DeltaP = P1 - P2. P2 is the hydrostatic pressure in the less concentrate solution, in Pascals
    DeltaC = C1 - C2. C1 is the higher concentration and C2 is the lower concentration, in mol/m3
    T is the temperature in Kelvins

    DeltaP is called the osmotic pressure.
    R = 8.314 J/K

3.2.1) Remarks :

The values of the osmotic pressure are very high even at moderate concentrations : for instance, at 273K and for a concentration of 1 mol/l, the osmotic pressure is about 22.7 Bars and the equivalent water column is then about 230 m high!

In a closed membranar cell compartment filled with a concentrate solution and plunged in a diluate solution, the osmotic flux will take place from the outside diluate solution to the concentrate enclosed concentrate solution. The hydrostatic pressure inside the cell will increase and will be greater than the outside value. The difference of pression will be given in fact by Vant'Hoff law. This phenomenon has got its application in case of biologic cells : Blood cells will swell and may explode when the blood is diluated in cool water because the salt concentration inside the cells is so high that the equilibrium osmotic pressure is sufficient to brake their membranes.

The osmotic pressure increases or decreases with the temperature.

Reverse osmosis is involved for exemple to purify water in some laboratory or industrial applications. The commercial devices for laboratories generally perform a good purification and the obtained water has the got a high resisivity. The problem is that the initial water must not be too concentrate with salts and that the obtained fluxes are usually low. The involved membrane surfaces are then great.

3.2.2) Simple experimental device in order to evidence the osmotic pressure :

The following experimental device allows to give a simple evidence of the osmotic pressure :

in order ti give good quantitative results, this device must be performed on the following points : The membrane must be perfectly embeded on its support in order to prevent weaks and should be supported under its bottom face in order to have no deformation. The tube must be not to thin in order to neglegt capillarity effets. The equilibrium state is long to be reached because permeabilities are low. Salt concentration must not be to high too. Thus one should prevent the evaporation of the concentrate solution during the experiment time. Therefore it can give good qualitative results after less than half an hour or one hour in most of cases and it may then be a good pedagogical device in order to evidence the osmotic flux.

3.2) Microscopic view :

One may suppose that the compressibility of the solvent could explain the origin of Van't Hoff law : The volumic concentration of solvent molecules in the concentrate solution will increase under the applied pressure because of the compressibility properties, this until it will be equal to the volumic concentration of solvent molecules in the diluate solution. (The compressibility properties are in fact induiced by the presence of intermolecular free volumes inside the liquid). In this point of view, the liquid solution is considered to behave as same as a gaseous blend. The problem is that the numerical calculation do not agree this assumption. For example in the case of water, the compressibility coefficient is ten times lower than the matching value which is attempted to validate this model. Another point of view may be the consideration of the intermolecular free volumes while taking in account that a liquid has got an important fraction of excluded volume which is occupied by the liquid molecules and that this excluded volume is constant.

3.3) Calculation of the equilibrium state in case of an intermittent process :

The quantity of solvent which goes from the concentrate solution to the diluate solution in the case of a perfectly selective membrane can be calculated as follows :

One has (conservation of the salt) :

    C1 V1 = C1'' V1''
    C2 V2 = C2'' V2''
where :
    V1 is the initial volume of the concentrate solution and V2 is the initial volume of the diluate solution.
    C1 is the initial concentration of the concentrate solution and C2 is the initial concentration of the diluate solution.
    V1'' is the final volume of the concentrate solution and V2'' is the final volume of the diluate solution.
    C1'' and C2'' are respectively the salt concentrations of concentrate and diluate solutions.

    C1'' - C2'' = DeltaP / (R * T)

Where DeltaP is the differential applied pressure.

The throwed quantity Q of solvent is defined as follows :

    Q = V1 - V1'' = V2'' - V2

If DeltaP is greater than R T (C1 - C2) then Q is positive and V1'' is greater than V1.
If DeltaP is lower than R T (C1 - C2) then Q is negative and V1'' is lower than V1.

Remark :

Changing the pressure will change the equilibrium state.
If the membrane is not perfectly selective, the salt will go in the less concentrate solution as same has the solvent. At the final state, the concentrate solution will completly cross the membrane.

3.4) Kinetics aspects :

Some kinetic factors are the followings :
3.4.1) The permeability of the membrane : the more the membrane permeability is high, the more the reverse osmotic flux is high, and faster the equilibrium state is reached in the case of the intermittent process.
3.4.2) The membrane thickness : the more the membrane is thick, the more the reverse osmotic flux will be low.
3.4.3) In a continuate process, increasing the pressure will increase the flux of solvent. The efficient differential pressure DeltaPapp which induices the flux of solvent may be calculated as follows :

    DeltaPeff = Papp - DeltaP
where Papp is the applied pressure and DeltaP = R T (C1 - C2).

Then the flux Q of throwed solvent is related to the permeability of the membrane as follows :

    Q = DeltaPeff * Pe' * s / e
Where :
    Pe' is the permeability of the membrane in regard to the applied pressures. Pe' is different of Pe which is defined with the consideration of concentrations.
    s is the membrane surface
    e is the membrane thickness

Remark : Through Van't Hoff low, on may write a relation between Pe' and Pe :

    Pe' = R T Pe

4) MONTE-CARLO SIMULATIONS

One has build two complementary Monte-Carlo simulations of intermittent osmosis process : one based on Langevin Dynamics and another one based one excluded volume dynamics, this in order to compare simulation results and then do some discussion.

4.1) General description :

The visualization area is two dimensional. The walls of osmosis cell and membrane are represented by colored lines. The liquid molecules are visualized by square blue pixels. a molecule can move in the horizontal or vertical directions. Each molecule and each pixel of membrane and cell walls are associated to an elementary energetic contribution defined as a number. The cohesion condition of the liquid which authorizes the displacement of a liquid molecule is that a sufficient total energetic contribution must always be found in a square zone of 7 * 7 pixels around the displaced molecule position. Salt molecules are represented by yellow colored pixels and are submitted to the same rules in their displacements. The salt is supposed to be composed of only one specie (like for instance a sugar solution) in order to simplify the simulation. The energetic contributions of cell and membrane are chosen in order to give flat interactions so that the capillarity forces can be neglegted.

In the simulation which involves the langevin dynamics, no restriction is made about the number of molecules on a host site and the cohesion condition is just suffucient in order to authorize the molecule displacements.

In the simulation which involves the excluded volume dynamics, no more than one molecule can occupy on a host site and the displacements are authorized both when the host site is empty and when the cohesion condition is satisfied.

The compressibility coefficient of the liquid will be higher in the case of excluded volume dynamics than in the case of Langevin dynamics. This will lead to two different results about the verification of Van't Hoff Law.

Because the number of involved molecules is relatively low (8000 molecules) and because the fluxes are low, one should repeat each simulation run at least three times in order to abtain a good accuracy of measurements. But a good qualitative exploration can be made even if the simulation runs are not repeated.

4.2) Parameters taken in account :

    - The salt concentration in the upstream compartment of the cell
    - The salt mobility in the solvent
    - The membrane thickness
    - The membrane permeability to the solvent
    - The membrane selectivity
    - The upstream applied pressure in case of reverse osmosis
 
 
   
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Applet developed by Dr Jean-Yves Dolveck
June 2006