Sorption :

absorption of fluids (gas or liquid) by solids materials.

Permeation :

transport of fluids (gas or liquid) through solids materials.


Porous materials : Absorption and transport of fluids take place in pores inside the bulk of the material (substrate) and involve capillarity properties.
Dense materials : Absorption and transport of fluids take place in the material (sunstrate) and involve a dissolution process and a diffusion process.

This part will expose absorption and transport through dense materials.


3.1) General description :

The absorption process involves thermodynamic and kinetic phenomena. It could be described as follows :

external fluid --------> interface ------> bulk of the material
....................................and kinetic...............kinetic

3.2) Sorption thermodynamics

The interface is the surface of the solid material. It tears the fluid middle and the solid middle. At the interface takes place the adsorption equilibrium which is characterized by a thermodynamic equilibrium constant called the solubilty. The solubility S is defined as follows :

    S = as / af.
where :
    as is the activity of the fluid in the solid, usually assumed to be equal to the fluid concentration in the solid.
    af is the activity of the fluid in its proper middle :

      For a perfect gas af is assumed to be equal to the gas pressure in bars.
      For a pure liquid, af = 1.
      For a homogeneous blend of liquids or a salt solution, af is taken equal to the molar fraction.

The consideration of the solubility S in these terms is global. The temperature is an important factor for varying the solubility S. But characteristics of the substrate bulk such as chemical structure and physical structure, and behaviour of the absorbed fluid in the substrate are essential parameters to interprete the variations of the solubility.

The Unity of S is the cm3/cmHg but it may be expressed in m3/Pa.

For polymers, fluid solubilities range from 0.01 (for light gases such as hydrogen) to 1 cm3/cmHg (for water vapor and others).

3.2.1) For polymers, the following considerations are retained :

a) The existence of intermolecular free volumes : The chemical structure of macromolecular chains of polymers leads to the existence of elementary intermolecular free volumes. The size of the free volumes may influence the ability for a absorbed molecule to be located inside.
a.1) If the free volumes are not large enough, the fluid molecules cannot take place inside them and the solubility may be lowered.
a.2) If there is a lot of elementary free volumes large enough, The solubiliy of the fluid molecules may be then increased.

b) The existence of strong interaction sites : The chemical structure of macromolecular chains or impurities may lead to strong interactions (for example, polar interactions or hydrogen bonds) with the absorbed fluid molecules. The fluid molecules are then preferentially trapped on these parts of the polymer bulk until these are saturated.
b.1) If the concentration of strong interaction sites is high, the fluid molecules may be more absorbed.
b.2) If the concentration of strong interaction sites is low, the fluid molecules may be less absorbed.

3.2.2) For the absorbed fluids, the following considerations are retained :

a) The pressure of the fluid as gas : The more the pressure of the gas is high, the more the gas may be absorbed.

b) The condensability of the fluid as gas : If the gas is condensable like solvent or water vapor, the absorption of the gas may be increased because the gas may condensate in the polymer.

c) The volume of the fluid molecule : If the volume of the fluid molecule is large, it may have some difficulty to be absorbed inside the elementary free volume of the polymer bulk. Small molecules may be easily absorbed.

d) The chemical interactions between the fluid and the polymer :
d.1) Plasticization interactions : if they are strong enough, fluid molecules (such as solvents) can increase the mobility of the macromolecular chains and increase the size of elementary free volumes (relaxation or plasticization effect). The fluid molecules may be then more absorbed.
d.2) Strong interaction sites : If chemical interactions between absorbed molecules and some strong interaction sites are high, solubility may be increased.

In fact, the study of thermodynamical sorption properties requires the consideration of the whole of these factors and their influences because there can be some additional or compensatory effects between them.

3.2.3) Most common laws used to describe thermodynamical sorption properties :

The study of sorption properties is leaded by the drawing of sorption isotherms. The solubility of the fluid in the substrate is measured at a constant temperature and at different fluid activities (pressure for a gas, relative vapor pressure for vaporized water or solvent, and molar fraction for a liquid solution). The substrate my have different shapes : more or less thin membranes, powders, gels ...

a) Henry's law : The solubility is a constant k which only depends of the temperature. It may be the case of perfect gases in polymers when no particular strong interactions take place. The sorption isotherm obeys to the following equation :

    C = k P.
where :
    P is the applied gas pressure.
    C is the gas concentration in the substrate.

b) Double sorption mode : The solubility is for one part due to a Henry's process and for another part to a Langmu´r saturation process on strong interaction sites. The sorption isotherm obeys then to the following equation :

    C = k P + Co K P / (1 + K P).
where :
    P is the applied gas pressure.
    k illustrates the solubility of gas molecules which are not trapped on strong interaction sites.
    Co is the concentration of strong interaction sites in the substrate.
    K is the thermodynamic equilibrium constant of fluid molecules trapping by strong interaction sites inside the susbstrate.

c) BET sorption mode : It is used to study sorption isotherms of vapors when a processus of condensation is supposed to take place in the polymer. One have the following law :

    (1/C)*(Pr/(1-Pr)) = Pr*(a-1)/(a*Co) + 1/(a*Co)
where :
    Pr is the relative pressure of the vapor : Pr = P/Po were P is the absolute pressure and Po is the saturation pressure.
    Co is the concentration of strong interaction sites of condensation.
    a is a thermodynamic constant.

3.3) Sorption kinetics :

The study of absorption kinetics is complementary to the thermodynamic considerations. The equilibrium of sorption in a volumic material is reached step by step more or less rapidly because of the existence of kinetic parameters. These ones should be considered when the study of sorption thermodynamic equilibrium is difficult to be performed. These factors are influenced by the temperature and by both the substrate and the fluid. When a fluid molecule is absorbed in the substrate, it is first adsorbed at the substrate surface and it diffuses inside the material and may interact with some parts of the material bulk. For polymers, the following points are considered :

3.3.1) The speed of adsorption at the surface of the polymer : It depends on :

a) The size of the particles of substrate : The more the substrate is divided, the more its apparent specific surface is high and the more the fluid is rapidly absorbed.

b) The topology of the surface : Surfacic porosity of the substrate increases its surface area and the speed of adsorption.

c) The surfacic chemical structure of the substrate : It is possible to modify the surface chemistry and then modify its Wettability by the fluid.

3.3.2) The diffusion coefficient of the fluid molecules in the substrate : It depends on the fluid molecules accessibility to the internal parts of the substrate and is governed by :

a)) Concentration of large enough free volumes in regard to the size of the molecule. The size of the elementary free volumes can be modified by the chemical structure of macromolecular chains.
a.1) If the concentration of large enough elementary free volumes is low, fluid molecules have a low probability to find large enough free volumes and they may displace themselves slowly.
a.2) If the concentration of large enough elementary free volumes is high, fluid molecules have a high probability to find a large enough free volumes and they may displace themselves rapidly.

b) Mobility of macromolecular chains in the polymer :
b.1) In a vitreous polymer, the mobility of the molecular chains is low, (the polymer behaves like a glass) and the fluid molecules displace themselves slowly.
b.2) In a rubbery polymer, the mobilty of the molecular chains is high, the free volumes can be easily recombinated and offer much possibilities for the fluid molecule to displace themselves in the polymer.

c) The speed of trapping by strong interaction sites inside the bulk : Polar interactions or hydrogen bonding may take some time to be effective on strong interactions sites as a chemical interaction and then the saturation of the strong interaction sites may take time to be reached.

d) The speed of relaxation of the local environment by the fluid molecule :
d.1)If the relaxation occurs rapidly, it may be not a limiting factor and an absorbed solvent molecule will progress in a relaxed environment with high chain mobility and large elementary free volumes.
d.2)If the relaxation occurs slowly, an absorbed solvent molecule may displace itself slowly in the substrate until it succeeds to relax the polymer chains. When the solvent molecule finaly succeeds to relax the molecular chains, it displaces itself more rapidly only in the relaxed part.

As same as previously, the study of the sorption kinetic properties requires the consideration of the whole of thes factors and their influence because there can be some additional or compensatory effects between them.

3.3.3) Diffusion coefficient and fick laws :

a) Thirst Fick law : In a homogeneous substrate and if one does not take in acount the interactions between the absorbed fluid molecules (essentially bumps between the absorbed molecules), the local flux density J of the diffusing molecules and their local concentration gradient dC/dx are related together by the first fick law :

    J = -D dC/dx (monodimensional equation here).
where : D is called the diffusion coefficient.

b) Second Fick law : It is the consequence of the first Fick law and it allows to do some integration with time t and abscisse x in the sample thickness. It is written as follows :

    dC/dt = -D d2C/dx2 (monodimensional equation here).

c) Application to the study of sorption kinetics : The Fickian sorption kinetics on a membrane is linear with the square root of time scale at the beginning of the sorption and the diffusion coefficient D can be calculated by measuring the initial slope Beta of the kinetic curve drawed with a square root of time scale :

    D = Pi Beta2 e2 / 16 / Coo2
where :
    e is the thickness of the membrane
    Coo is the equilibrium absorbed concentration.
A unity of D is the cm2/s but it may be given in m2/s.

For polymers, Diffusion values range from 10e-10cm2/s or less (for methan and heavier molecules to 10e-5 cm2/s (for hydrogzen or helium gases), depending on fluid and polymer.

Usually one measures the mass increase of the sample and not directly the concentration. The equilibrium concentration Coo is calculated like this :

    Coo = Moo / V
where :
    Moo is the equilibrium increase of mass
    V is the sample volume.

Then one may draw the following mass curve (Fickian case here) :

One have : Beta = A / V

    where A is the slope of the mass curve.

3.4)Sorption measurements

Sorption measurements may be performed in a sorption balance which is a balance with a high accuracy and able to measure very low weights. The balance is confined in a pressurized oven with gas or vapor being absorbed. A compensation of Archimedes shoving should be made to increase the accuracy of the measurements if this one is high as in the case of measurements under high gas pressures. Thus for exemple, a equivalent volume as sample one of some material which is known to not absorb studied gas or vapor can be put on the opposite nacelle than sample one. This material should have got roughly the same density as the sample one so that the mass compensation will be low. Then the mass compensation in one or the other nacelle can be made with lead balls for which ones Archimedes shoving can be neglegted in comparizon of polymer sample and compensatory material ones. Thus, one measures the evolution of sample mass increase with time.


4.1) General description :

The permeation process, performed through a dense membrane, involves thermodynamic and kinetic phenomena. It could be described as follows :

upstream external fluid --------> interface ------> bulk of the material----------> interface----------> downstream external fluid
..................................................and kinetic......................kinetic............................and kinetic

The study of permeation requires the drawing of a kinetic curve which represents the quantity Q(t) which has crossed the membrane.

Parameters and factors are the same as in sorption process but one should notice that adsorption equilibrium and sorption kinetics interfer on both face of the membrane : the upstream face where the inlet fluid pressure is applied and the downstream face where the outlet fluid pressure (inferior to the upstream pressure) is applied too.
For transport of the fluid molecules inside the sample bulk, in the case of polymeric membranes, size of elementary free volumes, macromolecular chain mobility, plasticization effects and existence of strong interaction sites may influence the permeability of the membrane in the same way as sorption equilibrium constant and diffusion.
One may find to principal cases of permeation kinetics :
a) The adsorption process is the limiting factor on upstream or downstream face of the membrane and diffusion inside the membrane is fast.
b) The diffusion inside the membrane is the limiting factor and the adsorption process on upstream and downstream faces is fast.

4.2) Fondamental relations :

The permeability Pe of a membrane is defined as follows :

    J = -Pe (P2 - P1) / e
where :
    P1 is the upstream pressure and P2 is the downstream pressure.
    e is the thickness of the membrane
    J is the flux density of fluid :
    J = F / s where F is the flux of fluid and s is the surface of the membrane.

Remark : If the diffusion is the limiting factor, the permeability is independant of the membrane thickness.

The kinetic curve Q=f(t) where Q is the quantity of throwed molecules allows to calculate the permeability by measuring the slope of the curve, which gives the flux F = dQ/dt.

One may calculate the permeability Pe as follows :

    Pe = - F e / [(P2 - P1) s]

The unity of the permeability is the Barrer : according to the above formulae, 1 Barrer = 10e-10 cm3.cm /cmHg /cm2/s

P2 - P1 is given en cmHg : 1 cmHg = 1330 Pa

Permeabilty may be also expressed in international unities as : 1 IU = 1 m3.m/Pa/m2/s

One have 1 IU = 1.33e17 Barrer.

Permeabilities of polymers most currently range from 0.1 (methan and higher hydrocarbons) to 10 000 Barrers (water vapor), depending on the fluid and on the polymer.

4.3) Permeability measurements :

Permeation properties may be studied with a permeation cell :
The downstream volume in which the gas flows through the membrane is known with a high accuracy. The downstream manometer has got a high accuracy and measures low pressures. After the vacuum had be made in the whole cell through the gas outlet by closing the gas inlet and opening the bypass, the gas outlet and the by pass are closed. Then the gas inlet is opened so that a defined gas pressure is applied on the upstream face membrane. One records the variations of the downstrem pressure with time and draws the permeation kinetics. One may record the permeation kinetics while the gas outlet is opened too, if the low gas flow through the opened outlet can be directly measured. These two methods are equivalent if the downstream pressure can be neglegted in comparizon with the upstream pressure.

In the case of a closed downstream volume, one have the relation :

    Q = P V / Po
where :
    Q is the quantity of flowed gas in cm3
    P is the downstrem pressure in cmHg or in Pa
    V is the downstream volume in cm3
    Po = 76 cmHg or 101300 Pa
Thus one may easily calculate the membrane permeability.

4.3) Fickian case :

If Fick law is verified for diffusion in the membrane and if the sorption equilibria are instantaneously reached on each face of the membrane, one have the following relation :

    Pe = D * Sm
where :
    Sm is the mean solubility coeffcient.
    D is the diffusion coefficient.
    Pe is the permeability.

Remark : In general case, this equation cannot be used to relate parameters but it allows to describe roughly the relationship between solubility, diffusion coefficient and permeability if the diffusion is the limiting factor :
a) A high diffusion coefficient may compensate a low solubility.
b) A high solubility may compensate a low diffusion coefficient if the diffusing molecules are not too strongly trapped on some interaction sites.

In case of the membrane is initially empty of fluid molecules and the downstream pressure is nil, the kinetic curve has got a transient part at short times and a stationary part at long times. The diffusion coefficient D may be estimated by measuring the time-lag Teta at the intersection of the asymptotic slope of the stationary regime with the axis of time. One have the following relation :

    D = e * e / (6 Teta)

where e is the thickness of the membrane.


These simulation have been written in order to simulate some aspects of sorption and permeation in polymers. Each one is not exhaustive and just illustrates some particular case. The different involved thermodynamic and kinetic constants are represented by random values compared to parametrized numerical thresholds. A two dimensional visualization will allow to give a topological view of the sample in its profile. in these visualizations, only global informations are given on localization of the fluid molecules and on polymer structure. No chemical structure is represented and different interactions are globally represented. A geodesic draughtboard shape network is involved which allows to give an esthetic representation of bulk material and molecules. The molecules have got diagonal displacements. The complementary draugthboard network than molecules one is used to characterize the material matrix. The following example illustrates the case of double sorption mode simulation :

Overview of the simulations :

5.1) Gas sorption and gas permeation : Visualization of permeation cell and homogeneous membrane. No particular interactions are taken in acount. Langevin dynamics is involved.

5.2) Fickian sorption and Fickian permeation : An homogeneous membrane is visualized. No particular interactions are taken in acount. The aim of the program is the description of Fick laws. Langevin dynamics is involved.

5.3) Double sorption mode and permeation with a double sorption mode : The existence of strong interaction sites is taken in acount but there is no plasticization effect.

5.4) Sorption and permeation with plasticization effects : The local plasticization of the membrane by the molecule is taken in acount but the existence of strong interaction sites is not involved.

Introduction proposed by Dr Jean-Yves Dolveck
Fev 2008