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

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Chapter 2. <strong>Processes</strong> to Manufacture Foams and to Functionalize <strong>the</strong> Surface Period IV: The growth <strong>of</strong> <strong>the</strong> pores (<strong>the</strong> swelling <strong>of</strong> <strong>the</strong> polymer) continues until <strong>the</strong> vitrification (IV) where <strong>the</strong> amount <strong>of</strong> CO 2 inside polymer is not sufficient to maintain <strong>the</strong> plasticized state. In <strong>the</strong> first moments, <strong>the</strong> pore growth is controlled by <strong>the</strong> diffusion and <strong>the</strong>n <strong>the</strong> viscosity comes more significant and finally controls <strong>the</strong> end <strong>of</strong> <strong>the</strong> foaming process. Moreover, <strong>the</strong> growing pores can coalesce and reduce <strong>the</strong> global pore density (Part IV <strong>of</strong> <strong>the</strong> Figure 2.20). Figure 2.20: Evolution <strong>of</strong> process parameters and <strong>the</strong> occurring phenomena during <strong>the</strong> foaming with time. (Continuous and dotted lines correspond to P and T variations respectively). 4 Theoretical Background <strong>of</strong> Gas Foaming The modelling <strong>of</strong> scCO 2 foaming <strong>of</strong> polymers by pressure quench method requires <strong>the</strong> resolution <strong>of</strong> <strong>the</strong> diffusion equation as well as <strong>the</strong> degree <strong>of</strong> depression <strong>of</strong> <strong>the</strong> glass transition <strong>of</strong> polymer as a function <strong>of</strong> <strong>the</strong> amount <strong>of</strong> CO 2 sorbed. Indeed, <strong>the</strong> iso<strong>the</strong>rmal sorption data and its modelling are required for such study. 4.1 Diffusion The aim <strong>of</strong> studying <strong>the</strong> diffusion phenomenon is to calculate diffusion coefficients for <strong>the</strong> sorption-diffusion and <strong>the</strong> desorption-diffusion <strong>of</strong> CO 2 . These coefficients provide information <strong>of</strong> diffusion behaviour <strong>of</strong> CO 2 into or from <strong>the</strong> polymer with different conditions. The diffusion is vital to understand CO 2 -polymer interactions. The diffusion <strong>of</strong> CO 2 into polymers results in several changes in <strong>the</strong> polymer such as lowering <strong>the</strong> glass transition point, manipulating <strong>the</strong> chain mobility, swelling etc. Rubbery polymers (above T g ) obey simply <strong>the</strong> Fickian diffusion since <strong>the</strong>y have a homogenous liquid-like behaviour. Non- Fickian diffusion occurs in <strong>the</strong> glassy polymer and will be modelled by <strong>the</strong> rules <strong>of</strong> Fickian diffusion. For Fickian diffusion, unsteady state one dimensional equation is given by: C t C D x x Where C is <strong>the</strong> concentration <strong>of</strong> <strong>the</strong> gas, x is <strong>the</strong> distance that <strong>the</strong> gas diffuses, D is <strong>the</strong> diffusion constant and t is <strong>the</strong> time. This equation has been solved by Crank [1975] for constant diffusion coefficient inside a plane sheet: (2.1) - 48 -
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: - 49 -
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Institut National Polytechnique de
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pleased I was to work with all memb
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Publications and Conferences The wo
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ρ p Pellet density Splitting (Bra
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