Numerical Investigation of Nucleating-Agent-Enhanced

Ind. Eng. Chem. Res. 2010, 49, 12783–12792

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Numerical Investigation of Nucleating-Agent-Enhanced Heterogeneous Nucleation C. Wang, S. N. Leung, M. Bussmann,* W. T. Zhai, and C. B. Park Department of Mechanical & Industrial Engineering, UniVersity of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8

Nucleating agents have long been employed in polymeric foaming processes to promote cell nucleation, increase cell density, and improve cell uniformity. This improvement in foam morphology is usually considered to result from the enhanced heterogeneous nucleation caused by the lower free energy barrier for cell nucleation. However, less is known about the underlying mechanisms of nucleating-agent-enhanced nucleation. In the polymer foaming process, pressure is a critical parameter that affects the degree of supersaturation of gas within a polymer-gas solution. In most previous theoretical studies on cell nucleation, a uniform pressure was assumed throughout the solution. Although this assumption may be acceptable when no particles have been added, its validity is questionable when nucleating agents are present. It has been speculated that growing cells that have already been nucleated generate local flow fields that induce tensile stresses around nearby particles, resulting in local pressure fluctuations. The discontinuity at the interface between a nucleating agent particle and the surrounding polymer melt yields local pressure and stress fields around the particle that are different from those in the bulk, which may enhance it as a potential heterogeneous nucleation site. This paper presents a numerical analysis to investigate the pressure profile in the vicinity of nucleating agents and provides new information about the underlying mechanism that promotes cell nucleation in the presence of nucleating agents. Introduction In polymer foam processing, cell nucleation refers to the process that leads to bubble formation in a polymer matrix or polymer composite matrix. There are two types of cell nucleation: homogeneous and heterogeneous. Homogeneous nucleation refers to spontaneous nucleation in a supersaturated solution that takes place in the bulk of the polymer matrix. Homogeneous nucleation is not considered the primary mechanism by which most cells form in plastic foams, because of too high an activation energy for cell nucleation.1,2 Heterogeneous nucleation involves the addition of a foreign phase, referred to as a nucleating agent, which presents a new surface on which the formation of bubbles can occur. Nucleating agents are commonly used in polymeric foaming processes to promote cell nucleation in order to increase cell density, reduce cell size, and improve cell uniformity.2 These desirable characteristics translate into notable advantages in a wide range of applications, from household products to engineering processes. Nucleating agents for polymer processing can be classified into organic agents, such as metal aromatic carboxylates, sorbitol derivatives, and organic phosphates; inorganic agents, such as talc and silica; and nanoparticles, such as nanoclay and carbon nanotubes. In heterogeneous nucleation, nucleating agents are critical for controlling the morphology of the cells being formed. Organic nucleating agents have been used to improve stiffness and transparency of a foam, while inorganic ones are applied to improve its mechanical properties. Talc or silica are usually solid particles and cannot be deformed; additives such as mineral oil (i.e., a liquid petroleum derivative or extract) are also used, as these droplets can deform and so improve the flexibility of the foam and make it less brittle. Solid particles and deformable additives will affect cell nucleation in different ways; the mechanism of cell nucleation around these materials is the subject of this paper. * To whom correspondence should be addressed.

Nucleation normally occurs at nucleation sites on the solid walls that confine a melt solution, but suspended particles and minute bubbles also provide nucleation sites. From classical thermodynamics, it is known that the presence of heterogeneous nucleating sites of various shapes reduce the free energy barrier to initiate cell nucleation3-6 and thereby aid in generating more cells. This model is the basis of many theoretical studies (e.g., see refs 7-12) of polymeric foaming processes published over several decades. According to classical nucleation theory,13,14 both the free energy barrier for heterogeneous nucleation (Whet) and the critical radius for bubble nucleation (Rcr) depend on the local pressure in the polymer matrix: Whet )

16πγlg3F(θc, β) 3(Pbub,cr - Plocal)2

Rcr )

2γlg Pbub,cr - Plocal

(1)

(2)

where γlg is the surface tension at the polymer-gas interface, Pbub,cr is the pressure within a critical bubble, Plocal is the local pressure in the polymer matrix, F is the ratio of the volume of a heterogeneously nucleated bubble to the volume of a spherical bubble having the same radius, and the parameters θc and β are related to the interfacial properties of an additive particle and the geometry of the particle, respectively. A number of studies have examined the nucleation enhancement mechanism during polymeric foaming processes.15-18 In most of these, researchers viewed the role of nucleating agents (e.g., talc) as providing more heterogeneous nucleating sites, thus promoting a more energetically favorable route for cell nucleation. More recently, the importance of shear stress upon bubble nucleation has been addressed. The shear stress can be induced by expanding gas cavities1 or by a shear flow field17,18

10.1021/ie1017207  2010 American Chemical Society Published on Web 11/15/2010

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Ind. Eng. Chem. Res., Vol. 49, No. 24, 2010

Figure 1. Micrographs of PS foaming with 2.1 wt % CO2 at 180 °C: (a) pure PS at 2.20 s and (b) PS + 5 wt % talc (CIMPACT 710) at 1.56 s.

induced during foaming. Han and Han reached a similar conclusion by observing continuous foaming in situ through transparent slit dies, leading them to suggest that cell nucleation

could be induced by flow or shear stress due to the motion of gas clusters, even at thermodynamically unsaturated conditions.16 Although the aforementioned studies provide valuable insights into heterogeneous nucleation, no systematic analysis has been presented to elucidate the mechanism of nucleation enhancement. In addition, in most of these studies, the system pressure in the foaming equipment was assumed to be the pressure inside the polymer-gas solution during foaming; i.e., Plocal ) Psystem in eqs 1 and 2, so any local pressure fluctuations were simply ignored. A recent study by Leung et al.19 demonstrates that the melt flow induced by the expansion of nucleated bubbles triggers the formation of new cells around them (see Figure 1). The authors speculated that the expanding cells generate tensile stress fields around nearby filler particles, resulting in local pressure fluctuations, above and below the nominal system pressure. The local decrease in pressure reduces the critical

Figure 2. Secondary bubble formation induced by bubble expansion.

Figure 3. Polymer-gas solution encloses an expanding bubble and a nearby rectangular clay particle.


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Figure 4. Spring analogy method. Each edge of an element is modeled as a fictitious spring. Table 1. Numerical Simulation Cases

radius for cell nucleation in these nearby regions. This, in turn, promotes the growth of preexisting gas cavities hidden in the crevices of the nucleating agent particles and leads to the preferential formation of new cells. Figure 2 presents a schematic of the phenomenon of secondary bubble formation induced by bubble expansion. This paper follows the study of ref 19 and documents a first attempt to investigate the local pressure variations around heterogeneous nucleation sites during polymeric foaming processes. The local movement of the polymer-gas solution caused by the expansion of a nucleated bubble can induce a varying or fluctuating stress field within the polymer matrix that can affect the local pressure at the interface between a nucleating agent and the surrounding polymer. The results are from a numerical model that predicts pressure changes in response to specified flow conditions. In the following sections, a model of a nucleation site is introduced. Several cases are studied to consider the possible dynamics of the nucleating agent particle in the site. A mathematical model based on physical laws and assumptions is presented; the numerical algorithm for solving the pressure

field around a heterogeneous nucleation site is explained; and finally, simulation results are presented. Geometry and Theoretical Considerations Model of a Nucleation Site and of Particle Dynamics. In this study, the finite element method is used to spatially discretize the physical domain. A nucleation site, at the interface between a nucleating agent particle and a polymer-gas solution, can be modeled within a 3D finite element volume to study the pressure change near the site. A hollow sphere can represent an already nucleated bubble within the 3D volume, and we allow the bubble to expand. A rectangular solid clay particle, which cannot be deformed, is considered a typical nucleating agent particle; an additive such as mineral oil, which can be deformed, is modeled as a spherical droplet. A schematic of an already nucleated spherical bubble near a rectangular clay particle, all enclosed within a volume of polymer gas solution, is illustrated in Figure 3. The dynamics of a solid particle in a polymer gas solution are complicated due to the nucleation-induced flow field and the small size of the particle. We speculate that the shear/extensional flow associated with the expansion of nucleated bubbles near these

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Figure 5. Discretized domain: a growing bubble and a rectangular particle within a polymer gas solution.

particles generate biaxial stretching or shear near the nucleation sites. When the nucleated bubble is expanding, an initially suspended clay particle nearby will experience a force due to the flow, that in turn generates a shear field in the vicinity. The moving melt due to the bubble expansion will then induce the simultaneous motion of the clay with the melt. At steady state, the asymmetrically distributed expanding bubbles due to nonuniform nucleation around a clay particle may also cause the clay particle to tumble or turn, because clay particles are asymmetric and have a large aspect ratio. The tumbling or turning will also generate local shear fields around the clay particle, superimposed on the translation of the particle away from the expanding bubble. The expansion of a bubble will also lead to extensional forces around the clay particle. The extensional force is a function of the distance between the particle and the bubble, which will impel the particle to move toward the expanding bubble. Therefore, the net movement of the particle will be equal to the melt flow due to the expanding bubbles. The extensional forces will induce a local pressure field around the clay particle. Okamoto et al. showed that solid particles in a polymer mixture eventually will align along a cell wall20 and thus help cells withstand the stretching forces that might otherwise break the thin cell walls, improving the modulus of the foam. In this study, several possible motions of a particle are considered. Interface Tracking Method. The mixture of polymer melt and blowing agent can be considered to be a single-phase fluid. However, the presence of 0.5-5 wt % solid clay and the prenucleated bubbles make the mixture a two-phase (liquid and solid) material. In this study, one particle was considered at a

time, and in each case the local pressure was determined in the vicinity; an interface tracking method was used to track the interface between the fluid and solid phases. The solid particle and the predeveloped bubble were modeled as hollow volumes within the polymer-gas solution; the particle movement and the bubble growth were imposed on these hollow volumes. The interface tracking method has been widely used21-23 in numerical modeling to treat a multiphase flow as a single phase. It is a suitable method for this study because we are only interested in the pressure field of the polymer gas solution (the liquid phase); the solid phase is simply disclosed by inducing possible particle dynamics. The modeling was carried out for the polymer melt mixture, and the interface between the polymer melt and the additive particle was tracked explicitly by an unstructured adaptive mesh.

Methodology Conservation Laws. The mixture of polymer melt and blowing agent is assumed to be a single-phase mixture, and the flow is assumed incompressible, isothermal, and steady, if the particle is fixed; the flow is unsteady if the particle moves with the flow field or deforms due to the shear stress of the flow. The flow is governed by conservation laws for mass and momentum, which are in the form of a set of partial differential equations: ∇·u ) 0

(3)


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Table 2. Material data and operating conditions material polystyrene

CO2 nucleating agent particle mineral oil operation conditions

F

parameters

quantity

MFI (g/min) Mn Mw/Mn specific gravity zero viscosity µ0 (Pa s) power law index, n weight percent (wt %) size: L × W × H (µm) radius (µm) melt temperature (°C) initial pressure (Pa)

0.15 120 000 2.6 1.04 4000.00 0.4 2.0 4×4×1 4 180 0

DU ) ∇·σ + f Dt

(4)

u is the velocity vector, σ is the stress tensor, F is fluid density, and f is an external force term. The stress tensor is required to obey the constitutive equations: σ ) -PI + τ

(5)

τ ) 2µ(γ˙ )d

(6)

where P is the fluid pressure, I is the identity tensor, µ is the dynamic viscosity, and d is the rate of deformation tensor given by 1 d ) [(∇u) + (∇u)T] 2

(7)

where γ˙ is the local shear rate defined by γ˙ ) √2tr(d · d)

(8)

The polymer melt is modeled as a purely viscous fluid, where the shear rate (γ˙ ) dependent viscosity of the melt is described by the Cross model:

Figure 7. Velocity contours and vectors as the bubble grows.

ALE Formulation. The arbitrary Lagrangian-Eulerian (ALE) formulation is used to treat the interface between the polymer melt mixture and a solid particle. The ALE formulation is a generalized kinematic description that allows arbitrary mesh movement, with the mesh velocity decoupled from the fluid velocity. The governing eqs 3 and 4 are nondimensionalized by introducing a nondimensional Reynolds number Re ) UL/µ (for molten polymer flow, the Reynolds number is very small; i.e., Re , 1). L is a characteristic length, U is a characteristic velocity, and µ is a characteristic dynamic viscosity. After the ALE formulation is applied, the nondimensional governing equations become ∇·u ) 0

(10)

(9)

∂u + Re(u - uG)∇u ) -∇P + ∇ · [µ · (∇u) + (∇u)T] ∂t (11)

where µ0 is the zero shear viscosity and τ* and n are fitted parameters.

where uG is the mesh velocity. If the nucleating clay does not deform and is not moving with the flow, uG ) 0; otherwise, uG accounts for the moving boundary.

µ(γ˙ ) )

µ0 1 + (µ0γ˙ /τ*)1-n

Figure 6. Mesh deformations with time as the bubble inside a polymer-gas solution expands.

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Figure 8. Pressure contours for case I, at (a) t ) 0.1 s and (b) t ) 7 s. Np

p)

∑pψ j

(13)

j

j)1

where there are N ) 10 degrees of freedom for velocity (in each coordinate direction) and Np ) 4 degrees of freedom for pressure. Following a Galerkin spatial discretization, the governing equations are written in semidiscrete form as [M]

d{u} + ([N] - [NG]) + [S]{u} + [L]T{p} ) dt ∂u -pn + dS Γ ∂n

∫(

[L]{u} ) 0

Figure 9. Pressure profiles along line AA in Figure 8, at different times.

Numerical Algorithm. The governing eqs 10 and 11 are spatially discretized using a Galerkin finite element approach in conjunction with P2-P1 tetrahedral Taylor-Hood elements. The unknown velocity and pressure fields are expressed in terms of the shape functions φj and ψj and the nodal velocity and pressure values uj and pj: N

u)

∑uφ

j j

)

(14) (15)

where {u} and {p} are the vectors of nodal velocity and pressure. [M], [S], and [L] are elemental matrices, S is the boundary of the elemental volume, and n is a normal vector. Mesh Deformation. In modeling flow problems with moving boundaries, a technique is required for updating the mesh as the domain deforms. A widely used approach is the spring analogy method, which treats a deforming spatial domain as a mass of elastic material, so that boundary displacements are spread into the mesh through elastic forces. In this work, the semitorsional spring analogy24 method, in which the edges of the mesh elements (see Figure 4) are considered to be fictitious springs, is used to describe the movement of the mesh, as the clay or mineral oil droplet moves. The semitorsional spring defines the stiffness of an edge i-j of an element as the sum of its lineal stiffness and its semitorsional stiffness:

(12)

j)1

Figure 10. Pressure contours for case II when the particle is fixed, at (a) t ) 0.1 s and (b) t ) 6 s.

kij ) klineal + ksemitorsional ij ij

(16)


Figure 11. Pressure profiles along line AA in Figure 10 at different times.

where klineal can be defined by the coordinates of the two nodes ij is defined as connected by an edge and ksemitorsional ij NEij

ksemitorsional ij

)

∑ sin1 θ 2

m)1

ij m

(17)

where NEij is the number of elements sharing edge i-j and θmij is the facing angle, defined as the angle that faces the edge i-j on the mth element attached to the edge. The mesh-updating algorithm was implemented in a finite element code25 for solving the 3D incompressible Navier-Stokes equations in an ALE formulation. The implementation was validated by solving simple flows with moving boundaries. Results and Discussion Three cases were modeled, summarized in Table 1. First, a solid rectangular plate, which represents a particle or a group of particles of the nucleating agent (e.g., talc) immersed in a polymer melt mixture, is considered. The plate may be considered stuck to a fixed surface during foaming, when it is a group of agglomerated particles and has too large a mass or moves with the mixture flow if it is a single particle. The movement can also be multiplex: it can simply move in the flow direction with the melt, spin due to shearing, or move randomly. Several cases were studied; simulation results are presented to illustrate how pressure changes around the particle.

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Figure 13. Pressure profiles along line AA in Figure 12, at different times.

Finally, numerical simulations were run for a case involving a spherical particle, as it deforms when the polymer melt mixture is squeezed. Geometry of the Nucleating Site and Simulation Parameters. The numerical domain, which represents a volume of polymer-gas mixture with a nucleation site, is modeled as a spherical volume, Figure 3. The domain is much larger than the particle, to limit the influence of the boundary conditions specified at the domain surface. Inside the polymer-gas solution, a solid rectangular plate or a spherical droplet is submerged in a mixture of polymer melt and blowing agent. Schematics of the numerical geometries for all cases studied are shown in Table 1, in which case I represents a translation of the particle with the melt as a bubble expands; case II depicts a particle that is fixed, spinning, and moving randomly (spinning and translation); and case III illustrates a deformable droplet in a squeezing flow field. Figure 5 illustrates the spatial discretization, which is the finite element mesh for the geometry of case I. The material considered in this study is Styron 685D polystyrene with a weight-average molecular weight of 315 000 g/mol (The Dow Chemical Co.). The zero shear viscosity and the fitted parameters for calculating the viscosity of the PS and gas solution were adopted from ref 26. The physical blowing agent is 99% pure CO2. The nucleating agent is a talc particle, Stellar 410 (Luzenac), that has been idealized as a rectangular plate. An isothermal condition was assumed. A summary of

Figure 12. Particle and bubble positions for case II when the particle is spinning, at (a) t ) 0.1 s and (b) t ) 7 s.

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Figure 14. Particle and bubble position for case II, when the particle is moving randomly (translation and spinning), at (a) t ) 0.2 s and (b) t ) 7 s.


material data used for the calculations and the operating conditions considered are listed in Table 2. Case I, Rectangular Talc Particle: Translation. Consider a solid plate particle that moves as a nucleated bubble expands (Figure 3). The nucleated bubble is assumed to have an initial radius of 10 µm.19 The bubble radius expands linearly at an experimentally observed rate,19 and the solid plate moves with the melt radially outward from the bubble surface, at the same velocity, so that the distance between the bubble surface and the solid particle remains constant. Mesh deformations with time as the bubble expands are shown in Figure 6. The result of the flow field in vector form is shown in Figure 7. The flow field changes with time, and that affects the pressure profile around the particle. The pressure profiles at two different times are shown in Figure 8 and indicate that pressure around the particle changes with time, and pressure magnitudes around the particle increase with time. The pressure changes along the vertical lines in Figure 8 are plotted at different times in Figure 9. The pressure increases dramatically for the first few seconds and eventually reaches a steady state when the bubble has expanded for about 5 s and the bubble radius increases to 10.8 µm. The pressure field around the solid plate is different from the bulk pressure, especially at the top and bottom of the particle. Dramatic pressure variations at the bottom and the top of the particle represent potential nucleation sites.

Case II, Rectangular Talc Particle: Fixed, Spinning, or Moving Randomly (Translation and Spinning). First, consider a solid plate particle held motionless in a flow field induced by an expanding bubble (Figure 4). The bubble is growing at an experimentally observed expansion rate.19 The particle may be held fixed if it is part of a group of agglomerated particles attached to a spot of the processing wall. The pressure profiles at t ) 0.1 s and t ) 6 s are shown in Figure 10, and the pressure changes along AA are also plotted versus time in Figure 11. Figures 10 and 11 indicate that (a) the pressure around the nucleation site changes with time and eventually reaches a constant magnitude; (b) in the flow direction, there are extreme values of pressure at the top and the bottom of the particle, and the pressure around the solid plate is significantly different from that of the mixture; and (c) opposite to case I, the lower pressure appears at the bottom of the particle, which indicates a potential nucleation site around this area. Second, consider a solid plate spinning about an arbitrary fixed corner (in this case, the corner was assigned to be the one close to the bubble, Figure 12) in a shear flow due to the uneven growth of nonuniformly distributed bubbles. The solid plate is assumed to spin at a speed of τ, τ (m/s) ) ((bubble expansion rate) cos(t) dt)

(18)

where t is time and dt ) 0.001 s is the incremental time step. Figure 12 illustrates the domain at t ) 0.1 s, an initial state when the bubble is about to expand and the solid particle lies horizontally above the bubble, and at t ) 7 s when the bubble has expanded and the particle has spun a certain degree. Since the bubble is growing and the particle is spinning about a fixed point, the distance between the fixed point of the particle and the bubble (the value of ds in Figure 12) is decreasing as the bubble expands; this is different from the case when the particle is moving randomly (spinning and translation). The pressure changes with time along line AA in Figure 12 are plotted in Figure 13 and illustrate that the pressure changes more dramatically with time compared to the fixed particle case. Moreover, positive and negative pressure values are not symmetrically distributed as positive pressure values are greater than the negative values, compared to case I, due to the asymmetric orientation of the particle relative to the incoming flow. Finally, consider a solid particle spinning and moving with the melt at the same time, representing a random motion. The domain at two times is shown in Figure 14: at t ) 0.2 s this is an initial stage; at t ) 7 s the particle has spun and moved forward with the melt. For this case, the distance between the particle and the bubble, the value of ds in Figure 14, will not


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Figure 16. Pressure contours for case III, at (a) t ) 0.1 s and (b) t ) 1.0 s.


change as the particle moves with the melt as the bubble expands. The pressure profile along the line AA in Figure 14 was plotted versus time; the result is shown in Figure 15. The result indicates that the pressure difference around the particle increases with time and approaches a constant distribution. Moreover, the pressure around the particle is symmetrically distributed, as the translation motion is imposed onto the spinning particle. Case III: Deforming Spherical Droplet. Finally, consider a spherical mineral oil droplet that can be deformed in the center of a spherical mixture of polymer melt. The simulation was carried out to examine the pressure changes when the oil droplet is squeezed at a fixed rate, and as the polymer melt mixture is shrinking as the droplet is squeezed. The pressure profiles at t ) 0.2 s and t ) 3 s are shown in Figure 16. At t ) 0.2 s, the droplet is about to be squeezed, so the size of the droplet represents the original state, while, at t ) 3 s, the droplet has been squeezed to a smaller size. The pressure profile is plotted with time along line AA in Figure 16, and the pressure changes are illustrated in Figure 17. Figure 17 indicates that pressure changes with time as the deformable droplet is squeezed, and the pressure difference around the droplet eventually reaches a steady value at a certain time. Conclusions Pressure is the most critical parameter that affects cell nucleation. The presence of nucleating agents that promote cell

nucleation creates discontinuities in the foaming mixture. Knowledge of the pressure variation and distribution around a nucleation site is a key to understanding the underlying mechanism that promotes cell nucleation. To highlight this, numerical simulations of the pressure profiles around nucleation sites in a mixture of polymer melt and blowing agent have been calculated. Several cases were studied, and the issues that affect pressure distribution were investigated. Nucleation sites are surfaces of suspended particles or of minute bubbles contacting the melt solution. Nucleating agents in a polymer melt solution can behave diversely as they are dispersed in the solution and as the initially tiny nucleated bubbles expand. The dynamics of nucleating agents induce flow fields that have significant effect on the local pressure distribution. The simulation results indicate that the pressure profile around nucleating agents can vary significantly from the surroundings, which implies that the assumption of using one system pressure while ignoring any local pressure fluctuations is imprecise. In general, regions that experience significant variations in pressure depend on the configuration of the nucleation sites and on the dynamics of the nucleating agents. These pressure variations decrease the free energy barrier for heterogeneous nucleation and the critical radius for bubble nucleation, and therefore induce cell propagation and a high cell nucleation rate. This, on the other hand, elucidates the mechanism of heterogeneous nucleation in the presence of nucleating agents from a pressure distribution point of view. Literature Cited (1) Lee, S. T. Shear Effects on Thermoplastic Foam Nucleation. Polym. Eng. Sci. 1993, 33, 418. (2) Leung, S.; Park, C. B.; Li, H. Numerical Simulation of Polymeric Foaming Processes Using a Modified Nucleation Theory. Plast., Rubber Compos. 2006, 35, 93. (3) Han, J. H.; Han, C. D. A Study of Bubble Nucleation in a Mixture of Molten Polymer and Volatile Liquid in a Shear Flow Field. Polym. Eng. Sci. 1988, 28, 1616. (4) Turnbull, D.; Vonnegut, B. Nucleation Catalysis. Ind. Eng. Chem. 1952, 44, 1292. (5) Fletcher, N. H. Size Effect in Heterogeneous Nucleation. J. Chem. Phys. 1958, 29, 572. (6) Fletcher, N. H. J. Chem. Phys. 1959, 31, 1136. (7) Cole, R. Boiling Nucleation. AdV. Heat Trans. 1974, 10, 85. (8) Lee, J. G.; Flumerfelt, R. W. A Refined Approach to Bubble Nucleation and Polymer Foam Processing: Dissolved Gas and Cluster Size Effects. J. Colloid Interface Sci. 1996, 184, 335. (9) Shafi, M. A.; Joshi, K.; Flumerfelt, R. W. Bubble Size Distributions in Freely Expanded Polymer Foams. Chem. Eng. Sci. 1997, 52, 635. (10) Joshi, K.; Lee, J. G.; Shafi, M.; Flumerfelt, R. W. Prediction of Cellular Structure in Free Expansion of Viscoelastic Media. J. App. Polym. Sci. 1998, 67, 1353.

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(11) Shimoda, M.; Tsujimura, I.; Tanigaki, M.; Ohshima; Polymeric, M. Foaming Simulation for Extrusion Processes. J. Cell. Plast. 2001, 37, 517. (12) Mao, D.; Edwards, J. R.; Harvey, A. Prediction of Foam Growth and its. Nucleation in Free and Limited Expansion. Chem. Eng. Sci. 2006, 61, 1836. (13) Gibbs, J. W. On the Equiilibrium of Heterogeneous Substances. Sci. Pap. J. Willard Gibbs 1961, 1, 337. (14) Blander, M.; Katz, J. L. Bubble Nucleation in Liquids. AIChE J. 1975, 21, 833. (15) Chen, L.; Sheth, H.; Wang, X. Effects of Shear Stress and Pressure Drop Rate on Microcellular Foaming Process. J. Cell. Plast. 2001, 37, 353. (16) Han, J. H.; Han, C. D. Bubble Nucleation in Polymeric Liquids. II. Theoretical Considerations. J. Polym. Sci., Part B: Polym. Phys. 1990, 28, 743. (17) Gao, C. Y.; Zhou, N. Q.; Peng, X. F.; Zhang, P. Optimized Polystyrene Cell Morphology by Orthogonal Superposition of Oscillatory Shear. Polym. Plast. Tech. Eng. 2006, 45, 1025. (18) Guo, M. C.; Peng, Y. C. Study of Shear Nucleation Theory in Continuous Microcellular Foam Extrusion. Polym. Test. 2003, 22, 705. (19) Leung, S. N.; Wong, A.; Park, C. B. A New Interpretation to the Talc-Enhanced Polymeric Foaming Process. SPE, ANTEC, Tech. Pap. 2009, 0190. (20) Okamoto, M.; Nam, P. H.; Maiti, P.; Kotaka, T.; Nakayama, T.; Takada, M.; Ohshima, M.; Usuki, A.; Hasegawa, N.; Okamoto, H. Biaxial

Flow-Induced Alignment of Silicate Layers in Polypropylene/Clay Nanocomposite Foam. Nano Lett. 2001, 1, 503. (21) Sussman, M.; Fatemi, E. An Efficient Interface Preserving Level Set Redistancing Algorithm and its Application to Interfacial Incompressible Fluid Flow. Journal of Scientific Computing 1990, 20, 1165. (22) Lakehal, D.; Meier, M.; Fulgosi, M. Interface Tracking Towards the Direct Simulation of Heat and Mass Transfer in Multiphase Flows. Int. J. Heat Fluid Flow. 2002, 23, 242. (23) Salih, O. U.; Gretar, T. A Front-Tracking Method for Viscous, Incompressible, Multi-Fluid Flows. J. Comput. Phys. 1992, 100, 25. (24) Blom, F. J. Considerations on the Spring Analogy. Int. J. Numer. Methods Fluids 2000, 32, 647. (25) Minev, P.; Ethier, C. A Characteristic/Finite Element Algorithm for Time Dependent 3-D Advection-Dominated Transport Using Unstructured Grids. Journal of Computer Methods in Applied Mechanics and Engineering 2003, 192, 1281. (26) Wang, J.; Park, C. B.; James, D. Effect of Die Land Length on Die Pressure During Foam Extrusion-Part I Experimental Observations. SPE, ANTEC, Tech. Pap. 2007.

ReceiVed for reView August 14, 2010 ReVised manuscript receiVed October 11, 2010 Accepted October 19, 2010 IE1017207

Numerical Investigation of Nucleating-Agent-Enhanced

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