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 />
4.3 Nucleati<strong>on</strong><br />
As we have explained earlier, when <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure is reduced from <str<strong>on</strong>g>the</str<strong>on</strong>g> equilibrium state, <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> nuclei occurs as a result <str<strong>on</strong>g>of</str<strong>on</strong>g> supersaturati<strong>on</strong>. This number <str<strong>on</strong>g>of</str<strong>on</strong>g> nuclei can be calculated by using<br />
<str<strong>on</strong>g>the</str<strong>on</strong>g> classical nucleati<strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g>ory <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> physical chemistry. <strong>The</strong> number <str<strong>on</strong>g>of</str<strong>on</strong>g> pores generated can be modeled by<br />
calculating <str<strong>on</strong>g>the</str<strong>on</strong>g> number <str<strong>on</strong>g>of</str<strong>on</strong>g> nuclei generated.<br />
In <str<strong>on</strong>g>the</str<strong>on</strong>g> classical nucleati<strong>on</strong> <str<strong>on</strong>g>the</str<strong>on</strong>g>ory, <str<strong>on</strong>g>the</str<strong>on</strong>g> free energy <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> formati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a cluster <str<strong>on</strong>g>of</str<strong>on</strong>g> radius r in a<br />
closed iso<str<strong>on</strong>g>the</str<strong>on</strong>g>rmal system in chemical equilibrium is given by Tsivintzelis et al. [2007a] and Adams<strong>on</strong> and<br />
Gast [1997]:<br />
3<br />
4 r<br />
G P<br />
4r<br />
2 <br />
(2.17)<br />
3<br />
where r is <str<strong>on</strong>g>the</str<strong>on</strong>g> radius <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> spherical cluster, γ is <str<strong>on</strong>g>the</str<strong>on</strong>g> interfacial tensi<strong>on</strong> and ΔP is <str<strong>on</strong>g>the</str<strong>on</strong>g> pressure between <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
two sides <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g> interface.<br />
radius r c :<br />
When ΔG is plotted against cluster size, <strong>on</strong>e obtains a curve that shows a maximum at a critical<br />
dG<br />
dr<br />
2<br />
0 rc<br />
<br />
(2.18)<br />
P<br />
Equati<strong>on</strong> 2.18 is <str<strong>on</strong>g>the</str<strong>on</strong>g> Young-Laplace equati<strong>on</strong>. <strong>The</strong> maximum value <str<strong>on</strong>g>of</str<strong>on</strong>g> ΔG for homogenous<br />
nucleati<strong>on</strong> is obtained by substituting equati<strong>on</strong> 2.17 into equati<strong>on</strong> 2.19):<br />
G<br />
*<br />
hom<br />
3<br />
16<br />
<br />
2<br />
3P<br />
(2.19)<br />
<strong>The</strong> interfacial tensi<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> mixture is calculated by <str<strong>on</strong>g>the</str<strong>on</strong>g> following equati<strong>on</strong> given by Goel and<br />
Beckman [1994a]:<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
4<br />
mix<br />
4<br />
<br />
mix<br />
<br />
polymer<br />
( 1 wCO2<br />
)<br />
(2.20)<br />
<br />
polymer<br />
where ρ are <str<strong>on</strong>g>the</str<strong>on</strong>g> densities and w CO2s is <str<strong>on</strong>g>the</str<strong>on</strong>g> weight fracti<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> CO 2 absorbed by <str<strong>on</strong>g>the</str<strong>on</strong>g> polymer.<br />
<strong>The</strong> surface tensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> PLGA 50:50 can be taken as 35.3 N/m as reported by Vargha-Butler et al.<br />
[2001]. <strong>The</strong> interfacial tensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> CO 2 is essentially zero at supercritical state, thus it will not be introduced<br />
in <str<strong>on</strong>g>the</str<strong>on</strong>g> equati<strong>on</strong>. A more complicated model which predicts <str<strong>on</strong>g>the</str<strong>on</strong>g> depressi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> interfacial tensi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> a polymer<br />
as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> CO 2 sorpti<strong>on</strong> is proposed by Harris<strong>on</strong> et al. [1996]. Unfortunately, this model requires<br />
experimental data for <str<strong>on</strong>g>the</str<strong>on</strong>g> depressi<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> interfacial tensi<strong>on</strong> as a functi<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> CO 2 and will not be<br />
c<strong>on</strong>sidered here.<br />
1994a]:<br />
Steady state rate <str<strong>on</strong>g>of</str<strong>on</strong>g> nucleati<strong>on</strong> can be described by <str<strong>on</strong>g>the</str<strong>on</strong>g> following equati<strong>on</strong> [Goel and Beckman,<br />
*<br />
G<br />
<br />
hom<br />
N <br />
0<br />
Cf<br />
0<br />
exp (2.21)<br />
kT <br />
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