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