Influence of the Processes Parameters on the Properties of The ...

Chapter 2.
<strong>Processes</strong> to Manufacture Foams and to Functionalize <strong>the</strong> Surface
M
M
t
1
n0
2 2
8 D(2n
1)
t
exp
2 2
2
(2n
1)
4a
(2.2)
where a is <strong>the</strong> semi-thickness <strong>of</strong> <strong>the</strong> polymer pellet, M t denotes <strong>the</strong> total amount <strong>of</strong> diffusing substance
which has entered <strong>the</strong> polymer during time t, and M ∞ <strong>the</strong> corresponding quantity after infinite time. On <strong>the</strong>
o<strong>the</strong>r hand, in <strong>the</strong> case <strong>of</strong> <strong>the</strong> desorption-diffusion, M ∞ denotes <strong>the</strong> amount <strong>of</strong> CO 2 at zero time, M t , <strong>the</strong>
quantity <strong>of</strong> CO 2 which remains in <strong>the</strong> polymer at time t and n is gas molecules per cm 3 .
The reduced version <strong>of</strong> Equation (2.2) for short times is also given by [Crank, 1975].
M
M
t
Dt
2
2
a
1
2
1
1
2
n na
2 ( 1)
ierfc
(2.3)
n
( Dt)
Literature and experimental work revealed that, during <strong>the</strong> sorption and desorption <strong>of</strong> CO 2 into <strong>the</strong>
polymer, <strong>the</strong> diffusion coefficient is not constant. It depends on <strong>the</strong> concentration <strong>of</strong> CO 2 sorbed by polymer.
One can say that <strong>the</strong> more CO 2 absorbed, <strong>the</strong> more CO 2 can diffuse easily into or from <strong>the</strong> polymer. The
dependency <strong>of</strong> <strong>the</strong> diffusion coefficient on <strong>the</strong> concentration is due to <strong>the</strong> several effects caused by <strong>the</strong>
sorption <strong>of</strong> <strong>the</strong> CO 2 into <strong>the</strong> polymer matrix such as manipulation <strong>of</strong> <strong>the</strong> polymer matrix, swelling
(decreasing <strong>the</strong> bulk polymer density), lowering <strong>the</strong> glass transition point, lowering <strong>the</strong> interfacial tension
and <strong>the</strong> viscosity <strong>of</strong> <strong>the</strong> polymer. Also, by activated state <strong>the</strong>ory <strong>the</strong> diffusion coefficient is also temperature
dependent and it increases with <strong>the</strong> increasing temperature [Koros and Madden, 2004].
As proposed by Crank, <strong>the</strong> average diffusivity <strong>of</strong> CO 2 in polymers can be measured in a
desorption experiment. For early stages <strong>of</strong> diffusion (sorption or desorption), <strong>the</strong> amount <strong>of</strong> gas remaining in
a plane sample at any time is related to <strong>the</strong> diffusion coefficient. This procedure has been applied to analyze
<strong>the</strong> sorption <strong>of</strong> CO 2 into <strong>the</strong> polymers [Kumar and Weller, 1994b; Berens and Huvard, 1989b], and it can be
used when <strong>the</strong> sorption curve is plotted against
approximately linear as far as M t /M ∞ = 0.5.
2
t / l
, where l is <strong>the</strong> thickness <strong>of</strong> sample, if it is
The Sanchez-Lacombe’s equation <strong>of</strong> state (SL-EOS) has been used to predict <strong>the</strong> behaviours <strong>of</strong>
polymer-gas mixtures [Sanchez and Lacombe, 1976]. SL-EOS is a well defined statistical mechanical model
which is not a physical model <strong>of</strong> sorption <strong>of</strong> a gas into a polymer but an equation <strong>of</strong> state which defines <strong>the</strong>
capacity <strong>of</strong> sorption as well as <strong>the</strong> swelling <strong>of</strong> <strong>the</strong> polymer. The SL-EOS is known as a lattice-gas model
since <strong>the</strong> P-V-T properties <strong>of</strong> a pure component are calculated assuming that <strong>the</strong> component is broken into
parts or “mers” that are placed into a lattice and are allowed to interact with a mean-field-type
intermolecular potential. To obtain <strong>the</strong> correct system density, an appropriate number <strong>of</strong> holes is also put
into specific lattice sites, hence <strong>the</strong> name lattice-gas model [McHugh and Krukonis, 1994]. We have to
underline that this equation can be used for rubbery and glassy states <strong>of</strong> <strong>the</strong> amorphous or liquid-like
polymers. In its basic form, <strong>the</strong> SL-EOS is given by:
2
1
R
PR
TR
ln(1
R
) (1 )
R 0
(2.4)
r
where T R , P R , ρ R are reduced temperature, pressure and density respectively and r represents <strong>the</strong> number <strong>of</strong>
lattice sites occupied by one molecule.
The reduced parameters can be calculated by:
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