Classical Nucleation Theory Applied to Homogeneous Bubble

Jul 24, 2009 - Simulations were run for the flow of a polymer−PFA solution through an extrusion foaming die with an abrupt axisymmetric contraction...
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Ind. Eng. Chem. Res. 2009, 48, 7603–7615

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Classical Nucleation Theory Applied to Homogeneous Bubble Nucleation in the Continuous Microcellular Foaming of the Polystyrene-CO2 System Shunahshep Shukla* and Kurt W. Koelling The Ohio State UniVersity, Columbus, Ohio 43210

In the continuous production of microcellular thermoplastic foam, a polymer-physical foaming agent (PFA) solution is subjected to a rapid pressure drop through an extrusion foaming die. Simulations were run for the flow of a polymer-PFA solution through an extrusion foaming die with an abrupt axisymmetric contraction. The pressure drops across the die obtained through the simulations showed good qualitative agreement with experimental pressure drop measurements on the foaming extrusion die obtained in our laboratory. Field values of pressure, temperature, and velocity were obtained at each point in the foaming die. Once the values of pressure and temperature were obtained along each point in the foaming die, classical nucleation theory for bubble nucleation was invoked to predict the local bubble nucleation rate downstream of the saturation surface. The hydrodynamic constraints to the nucleation rate were calculated by using a modified form of the classical nucleation theory that accounted for the diffusional and viscosity constraints to the rate of homogeneous nucleation. The capillarity approximation was found not to be valid for bubble nucleation of CO2 in polymers; a correction accounting for the curvature dependence of surface tension was applied to get nonzero nucleation rates for the system to reconcile theoretically predicted rates with experimental observations. Introduction In the microcellular foaming of thermoplastic polymers, nucleation refers to the process in which a critical amount of a secondary component dissolved in a primary phase comes together to form a second stable phase once the initially stable primary phase is induced to cross the binodal and enter a metastable state. In the microcellular foaming of the polystyrene (PS)-carbon dioxide (CO2) system, bubbles of CO2 can, in principle, continue to nucleate and grow until the time that the two phases are completely partitioned into a stable liquid (PS) and a stable gas (CO2) at ambient conditions. However, in practice, vitrification of the matrix phase usually results in freezing of the foam morphology while the system is still in a metastable state, wherein minor external disturbances can trigger the sudden appearance of one or more new phases.1 Various mechanisms of nucleation have been proposed in the literature for nucleation of gas bubbles in thermoplastics, including homogeneous, heterogeneous, mixed-mode, shearinduced, and void nucleation theories.2 However, none of these theories give quantitative predictions for even the simplest cases of bubble nucleation in polymeric liquids. Notable work on predicting bubble nucleation in thermoplastic polymers using classical nucleation theory (CNT) has been carried out by Colton and Suh,3,4 Han and Han,5 and Goel and Beckman.6 Shafi and Flumerfelt,7 Ohshima and his co-workers,2 and Sirupurapu et al.8 have also attempted to predict homogeneous bubble nucleation rates in their experiments using CNT or modifications thereof. Their investigations are outlined briefly in the following sections. Colton and Suh3,4 were the first to develop a theoretical model based on CNT to predict the bubble nucleation density in amorphous thermoplastic microcellular foams. They chose the PS-zinc stearate system for their corroborative experimental work and found that their theoretical predictions did not agree well with their experimental observations. They acknowledged making some oversimplifying assumptions in their theoretical * To whom correspondence should be addressed. E-mail: [email protected].

model, viz. modeling the interactions between polymer chains assuming them to act as points with spherically symmetric potential fields, assuming that the polymer chains interact via a Lennard-Jones 6-12 potential and assuming hexagonal packing of nearest neighbor quasi-lattice sites with a coordination number of 12. These assumptions could possibly explain the large disparity in their theoretical predictions and experimentally observed results. Han and Han5 studied the PS-toluene system and realized that CNT incorporating macroscopic values of surface tension could not predict the experimentally observed bubble nucleation density for this system. They modified the CNT by considering two additional factors that might contribute to the formation of a critical nucleus in a polymer solution, namely change of free energy of the solvent due to the dissolution of macromolecules in it and modifying the free energy of formation of a critical bubble in a polymer solution to account for the fact that bubble nucleation in a polymer solution occurs under supersaturated conditions and not under thermodynamic equilibrium conditions for which CNT was originally developed. The macroscopic surface tension for the PS-toluene system was calculated using the parachor model. The value of the pre-exponential factor in the nucleation bubble density equation was determined using experimental nucleation rate data for the PS-toluene system. A key modification in their theory was for the free energy for critical nucleus formation in order to obtain calculated nucleation rates, which conformed to their experimental observations within 4 orders of magnitude. Goel and Beckman6 studied the batch microcellular foaming of the poly(methyl methacrylate) (PMMA)-CO2 system. They employed the correlation for the surface tension of mixtures given by Reid et al.9 to calculate the surface tension for the PS-CO2 mixture. They found that the agreement between their data and their CNT model calculations was very good at higher saturation pressures (above ∼15 MPa); however, CNT far underpredicted the nucleation rates at lower pressures (below ∼10.5 MPa). They attributed the higher experimental nucleation bubble densities to heterogeneous nucleation at lower pressures,

10.1021/ie8011243 CCC: $40.75  2009 American Chemical Society Published on Web 07/24/2009

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triggered by the presence of trace contaminants in the system. Goel and Beckman6 used an empirical model for the surface tension of the PMMA-CO2 mixture.10 Also, in the expression to evaluate the reversible work of critical nucleus formation and to compute the radius of the critical nucleus, the authors assume ∆P to be the quench pressure, which is different from the appropriate ∆P that should be used in the nucleation equation.11 Shafi and Flumerfelt7 used a perturbation solution (in terms of Peclet number) to determine the state of the critical cluster at the upper bound of the critical nucleation region with a view to using this information as a starting point for bubble growth calculations. They compared their analytical solution with the numerical solution of Patel12 and found excellent agreement between the two sets of results. However, Shafi and Flumerfelt7 have used results from vapor to liquid nucleation (condensation phenomena in homogeneous vapors) as discussed in the work of Feder et al.13 with Zeldovich’s model14 for gas bubble nucleation despite the fact that there are fundamental differences in the mathematical description of the two phenomena. Siripurapu et al.8 carried out microcellular foaming of the PMMA-CO2 system and used a modified form of CNT that accounted for the compressible nature of supercritical CO2 to describe the pore cell growth as a function of foaming temperature and supercritical CO2 saturation pressure. The surface tension for the PMMA-CO2 system was computed using density gradient theory based on the Sanchez-Lacombe equation of state (SLEOS).15 However, the authors have not mentioned the value of surface tension that they used anywhere in their work. They have cited the work of Harrison et al.16 with reference to their gradient theory calculations but have not mentioned the specific values of several parameters that they used in their gradient model to compute their results. As Li et al.17 showed earlier in their work, the density gradient theory could predict the macroscopic surface tension for the PS-CO2 system fairly well for higher saturation pressures (above ∼7 MPa); however, the theory severely underpredicted the interfacial tension of the PS-CO2 system at lower pressures (below ∼6 MPa). Siripurapu et al.8 have not discussed how the density gradient theory performed for their system or whether it was valid to use it over their experimental conditions for the PMMA-CO2 system. Furthermore, the authors did not give an explicit account of the diffusion and viscosity constraints on the rate of homogeneous nucleationsin fact, they treated the pre-exponential factor in the nucleation density equation as a “one time fixed parameter” to match their experimental observations and theoretical predictions. In this work, the relevance of applying classical nucleation theory in predicting bubble nucleation rates in thermoplastic polymer foamssparticularly the PS-CO2 systemshas been investigated. First, some simulation results that have been obtained earlier on pressure, temperature, and velocity profiles for the extrusion foaming of the PS-CO2 system are summarized. Then, the theoretical equations necessary to describe bubble nucleation in liquids using CNT are delineated. Finally, the consequences of applying CNT to predicting bubble number density in the PS-CO2 system are investigated. Also, the significance of the “viscosity” and “diffusional” constraints to nucleation in the case of highly viscous systems like the PS-CO2 system are explored. This work is new in at least two different ways from the work carried out previously in the field. Researchers working in the field have not worked out explicitly the magnitude and distribution of the viscosity or diffusional constraints on the

rate of homogeneous nucleation for the PS-CO2 system. This work throws some light on that. Also, previous researchers have not investigated the implications of using a curvature dependent surface tension approach to reconcile experimental results with results predicted by theory. In this work, we discuss these implications. Microcellular Extrusion Foaming of the PS-CO2 System. Continuous microcellular foaming of PS-CO2 mixtures for various processing conditions and for different weight percentages of CO2 dissolved in PS was carried out earlier in our laboratory. Polystyrene (685D) supplied by Dow Chemical Company was used as the matrix polymer, and bone-dry grade CO2 of 99% purity supplied by Praxair was used as the foaming agent. The PS pellets were fed into the extruder hopper, and once the PS phase was melted and homogenized in the extruder, CO2 at high pressure was injected into the molten polymer through an injection port in the extruder barrel wall. The shearing motion of the screw and the kneading motion of the mixing elements downstream of the CO2 injection port and the passage of the PS-CO2 mixture through a static mixer downstream of the extruder barrel aided in the solubilization of CO2 in the matrix phase. The PS-CO2 mixture was then subjected to a rapid pressure drop by forcing it through a cylindrical capillary die of circular cross-section. Thermodynamic instability was induced by the rapid pressure drop resulting in phase separation of the PS-CO2 solution, and since the polymer matrix was frozen by contact with ambient air before coalescence of the nucleated and growing gas bubbles became significant, a large number of submicrometer size gas bubbles were frozen in the polymer matrix and could be quantitatively analyzed under a scanning electron microscope (SEM). Han et al.18,19 have described a schematic of the experimental setup and discussed the apparatus used and the experimental findings in detail. Simulation results for 1 wt % CO2 dissolved in PS for different inlet melt temperatures (140, 160, 180, 200, 220, 240 °C) into the capillary die for two different screw rotation rates (10 and 30 rpm) have been described in our earlier work.20 In the present work, except where stated otherwise, a fixed inlet melt temperature (200 °C) and a fixed screw rotation rate (10 rpm) were arbitrarily chosen with a view to investigate the conditions under which the predictions of CNT and the observed experimental results could be reconciled. Of relevance to this work from our previous work20 are the die geometry, pressure, temperature, and velocity profiles in the extrusion foaming die, and the shear viscosity of the PS-CO2 mixture. A brief summary of these is provided in the discussion that follows. The polymer-CO2 solution exiting the static mixer, located immediately downstream of the extruder, enters the die block assembly. The flow passage through the die block comprises a die insert having an internal diameter of 7.8 mm and a length of 56.12 mm followed by the foaming capillary die, which is essentially a cylindrical hollow metal tube having an internal diameter of 0.5 mm and a length of 10 mm. The die block, due to its large thermal capacity, provides thermal homogenization of the PS-CO2 solution flowing through it. The slow flow of the PS-CO2 solution through the die insert serves to smoothen the velocity profile of the fluid across the flow cross-section after the tortuous passage of the fluid through the static mixer. For computational efficiency, only 23.4 mm of the die insert (along with 10 mm of the capillary die) were modeled,20 and a length of 33.4 mm thus marks the end of the capillary die. The die geometry modeled is reproduced here as Figure 1.

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Figure 1. (a) Geometry of the contraction. Since the die is axisymmetric, only the symmetric part of the 2D geometry needs to be meshed. (b) Enlarged view of the grid near the contraction. Grid adaption based on temperature gradients has been carried out in the region near the wall and near the entrance corner.

Figure 2. Pressure profiles along the die for different inlet PS-CO2 solution temperatures (10 rpm screw rotation rate).

The three parameter Cross model was found to give an excellent fit of the PS-CO2 solution viscosity for reasonably small amounts of CO2 dissolved in it. A detailed viscosity scaling for the PS-CO2 system was carried out,20 and the pressure, temperature, and velocity fields for the flow of the PS-CO2 solution through the foaming die incorporate an internally consistent viscosity model. Furthermore, the point (or local) values of viscosity used later on in this work in the nucleation bubble density equation are derived from this model. As expected from the geometry of the foaming die, most of the pressure drop in the die was seen to occur in the constricted 10 mm long end section for all temperatures studied. Due to the pressure dependence of viscosity and the associated nonisothermal effects, the pressure profile along the die was found to be highly nonlinear. Figure 2 shows pressure profiles along the

die for different inlet PS-CO2 solution temperatures at a 10 rpm screw rotation rate. For 1 wt % CO2 dissolved in PS and a 10 rpm screw rotation rate, a significant temperature variation was observed along and across the die cross-section due to viscous heat dissipation near the wall of the capillary die for the different inlet melt temperatures (at the inlet of the capillary die). The thermal conductivity of the PS-CO2 solution is low, and the residence time of the fluid in the capillary is short, so the heat generated near the wall does not have sufficient time to travel to the center of the capillary die. This results in a relatively large temperature increase of the polymer-CO2 solution near the die wall. Figure 3a shows the temperature distribution of the PS-CO2 solution for different radial distances from the die axis and along different lengths of the die for 180 °C inlet melt temperature in the die insert. Figure 3b shows the velocity variation of the PS-CO2 solution in the die. A typical fluid “particle” spends a relatively short time in the capillary die due to its relatively high rate of passage through the die. In fact, the average residence time of a fluid “particle” in the die is only ∼0.025 s. The temperature and velocity distributions for the other inlet melt temperatures follow qualitatively similar trends and so are not shown here. The cross-sectional morphology of the polymer foam exiting the extruder was examined using SEM. The bubble number density for a unit volume of foam was computed using the following well-known expression21 Nf )

( ) nM2 A

3/2

(1)

where n is the number of bubbles in the micrograph, A is the area of the micrograph (cm2), M is the magnification factor of the micrograph, and Nf is the number of bubbles per cubic centimeter of the final foam. Bubble Nucleation in Liquids as Described by CNT According to Gibbs,22 the critical nucleus would correspond to an aggregate of the new phase of a size for which the work

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Figure 3. (a) Temperature profiles along the die length (180 °C, 1.0 wt % CO2, 10 rpm screw rotation rate). (b) Velocity profiles along the die length (180 °C, 1.0 wt % CO2, 10 rpm screw rotation rate).

of formation (or free energy of formation) is a maximum. An increase in the size of such an aggregate or nucleus will lead to its further growth, while a decrease in the size of the nucleus will lead to its annihilation. Zeldovich,14 in his seminal work on phase transitions, described the evolution of spontaneously arising bubbles from a metastable liquid, considering explicitly both thermodynamically determined transitions and random effects. Kagan23 extended Zeldovich’s theory of bubble nucleation in homogeneous liquids to account for viscosity and inertial constraints on the rate of bubble nucleation in simple liquids. Katz and his co-workers24-26 further extended it to account for diffusional constraints in multicomponent liquid mixtures.

Kagan cast the equation of motion for a nucleating bubble in the following dimensionless form: 8 27 ω′ 2 2 3-b 1 ω′z3 + ω 1 + z + ω+ z- )0 9 64 ω 3 b 2

(

)

(

)

(2) where, the dimensionless quantities in eq 2 are defined as z)

rc

( drdr˙ ) βV rc

t

(3)

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ω)

3 βVtγµ 8 σ

(4)

2 2 2 β Vt Frc ω′ ) 3 σ

(5)

2σ rcpc

b)

(6)

Here, r is the radius of the nucleating bubble, rc is the radius of the critical nucleus, F is the density of the liquid, µ is the viscosity of the liquid, γ is a numerical coefficient that has a value of ∼4, σ is the interfacial tension, Vt denotes the average thermal velocity of the vapor molecules impinging on the nucleus and is given by Vt )

8kT πm

(7)

Here, m is the mass of a molecule, k is the Boltzmann constant, and T is the temperature. β is the condensation coefficient and for approximate calculations its value is taken to be 1.27 The dot over a quantity denotes its derivative with respect to time. We mention this equation explicitly, since the coefficient of the inertial term in Kagan’s paper has been incorrectly reported as 1/3 in the literature (instead of 2/3 as in eq 5 above). Inertia is not important in the majority of situations under which eq 2 has been applied; a particular solution of eq 2 for the case when viscosity of the fluid is the only factor limiting the rate of growth of precursor embryos to critical nuclei sizes and beyond was obtained by Kagan, and the expression for the nucleation rate, J, for the viscosity-limiting case can be expressed as J) 3N0

βVt b

 ( σ kT0

1 ω+

) (ω + 3 -b b )

e-1/3K

3+b + b

2

+ 4ω (8)

where 4πrc2σ K) kT0

16πσ3 3∆P2

the equality of the chemical potential of CO2 in the gas (vapor) phase and the liquid (polymer) phase. The construction for evaluation of ∆P is shown in Figure 4 following the approach suggested by McClurg.11 Blowing agent chemical potential is plotted vs pressure for the case of 1.0 wt % CO2 dissolved in PS. The point of intersection of the two curves corresponds to saturation conditions (the binodal). The term pE is the ambient pressure (i.e., the pressure of the surrounding liquid). Also, p* is the pressure inside the critical nucleus. The difference in the chemical potentials at pE provides the driving force for nucleation. Once the liquid phase becomes metastable in any vapor-liquid system undergoing transition in the direction of stable liquid (or liquid mixture) to stable vapor (or vapor mixture), the ambient pressure on the liquid, p0, is less than the pressure on it at equilibrium.25 The Poynting correction accounts for the fact that the metastable liquid is at an actual pressure p0 that is different from the saturation pressure ps of the liquid corresponding to vapor-liquid equilibrium. The Poynting correction factor can be expressed as28 exp

∫

p0

ps

( ) Vˆ1 dp RT

(11)

where Vˆ1 is the partial molar volume of the condensed phase. The Poynting correction factor can become significant for situations when ps is much different from p0 since it is an exponential function of pressure. If the condensed phase is not near its critical point, the volume of the liquid usually does not show a pronounced dependence on pressure and temperature, and in that case, the liquid can be considered as incompressible and the Poynting correction takes the following simple form:28

(

Vˆ1(p0 - ps) exp RT

)

(12)

For the PS-CO2 system, Vˆ1 can be evaluated using the following relation.15

[

dV˜ ∂V ) r10V*1 V˜ + φ2 Vˆ1 ) ∂N1 T,P,N2 dφ1

]

(13)

where V is the volume of the mixture. The term dV˜ /dφ1 can be evaluated as (9)

and N0 is the Avogadro number. Note that the criterion justifying neglect of the inertia terms has been reported as ω′/ω2 , 1 in the literature; however, the physical basis of making such an assumption is not clear. Blander26 suggests that the criterion justifying neglect of inertia terms should be ω′ ) 0, and that is the assumption we have made here. Evaluation of ∆P in the Expression for Reversible Work of Critical Nucleus Formation. The work required for critical nucleus formation can be expressed as W)

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dV˜ ) -ψT˜P*β dφ1

where β is the isothermal compressibility of the mixture. T˜P*β can be expressed by the following relation: T˜P*β )

1 V˜ [1/(V˜ - 1) + 1/r - 2/V˜ T˜]

(15)

ψ is a dimensionless function defined by

{[( ) ( ) F˜ ν

(10)

where σ is the interfacial tension between the stable gas in the bubble and the surrounding metastable liquid phase and ∆P is the pressure difference across the bubble interface. The composition of the gas phase can be assumed to be pure CO2 since the solubility of PS in CO2 is negligible. The composition of the polymer rich phase can be determined from

(14)

ν

ψ)

]

1 1 + (φ12 - νφ22)X1 T˜1 T˜2

1 1 P˜V˜ (ν - 1)(φ1 + νφ2) - 0 + 0 T˜ r1 r2 (φ1 + νφ2)2

}

(16)

X1, which characterizes the interaction between the solute and the solvent, is given as

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X1 ≡

∆P*V*1 kT

(17)

Figure 5 depicts the difference between psat and p* on the application of the Poynting correction. Although the absolute value of the Poynting correction is small, the effect of this correction on the nucleation rate is significant, since the nucleation rate, J, varies with ∆P as exp(-1/∆P2). Surface Tension of the PS-CO2 System. The surface tension of the PS-CO2 system was measured by Li et al.17 at 200 °C using axisymmetric drop shape analysis. For conditions for which the surface tension data is not available, researchers have used density gradient theory to compute the interfacial tension of the polymer-soluble gas interface. The method is described in detail by Harrison et al.16 and Li et. al.17 Figure 6 shows the variation of macroscopic surface tension with temperature for the PS-CO2 system as obtained from the work of Leung et al.29 The surface tension obtained by either experimental measurements on macroscopic bubbles/drops or by density gradient theory is too high for the bubble nucleation rate to have a nonzero realistic value. Using the macroscopic surface tension values of Figure 6 in the nucleation equation, for example, gives nucleation values that are