Microcellular sheet extrusion system process design models for shaping and cell growth control.
INTRODUCTION
Microcellular plastics are innovative polymeric foams based on inert gas blowing agents such as carbon dioxide and nitrogen that have been demonstrated on thermoplastic, thermoset, elastomers, and liquid crystal polymers. Industrial application of these materials hinges on the development of processIng platforms capable of high volume production rates and compatible with existing infrastructure. Recent advances in microcellular polymer processing have demonstrated the feasibility of microcellular extrusion processes (1, 2). One of the critical steps in microcellular extrusion is maintaining the nucleation cell density of the polymer melt during shapIng operations through cell growth control. To prevent cell coalescence and deterioration of the nucleated cell density, one of these extrusion processes utilizes staged pressure drops to elimInate premature cell growth and maintain sufficiently small cell sizes (1, 3). Proper design of the extrusion system requires thoughtful design of the melt flow channel to maintain the accurate staged pressure drop and prevent loss of nucleated cells through cell coalescence. The complicated nature of the two phase polymer/gas solution flow precludes design of the flow channel using standard extrusion die design equations. In this paper, a series of design models are presented that approximate the pressure drop and flow rate experienced by a twophase polymer/gas solution flow through a die channel. The intent of these design models is to capture the major physical attributes of the complicated flow and provide a means of quickly estimating the required flow channel geometry and/or evaluating, to first order, the feasibility of a flow channel design. These models have been implemented in spreadsheet form for rapid design turnaround. Microcellular processing consists of three main processing elements: forming a polymer/gas solution, inducing a rapid thermodynamic instability that simultaneously nucleates a very large number of microvoids, and controlling/stabilizing the expanding microcells (413). A scanning electron microscope (SEM) micrograph of a typical microcellular polymer crosssection is shown in Fig. 1. Microcellular polymers are characterized by cells sizes on the order of 10 [[micro]meter] and cell densities on the order of [10.sup.9] cells/[cm.sup.3]. in addition to superior performance in some applications (1416), microcellular processing has the attractive feature of independent control over cell size and cell density allowing the materials to be designed for specific applications. Detailed descriptions of microcellular polymers and microcellular processing can be found elsewhere (5, 9, 10, 17). In order to develop design models for rapid evaluation and feedback process control, the models have been based on first order approximations capturing the critical physics of the microcellular growth process. The primary application of these models is for use in initial concept design of microcellular foaming dies providing designers a rapid method for comparing various design strategies. The utility of such first order design models allow for rapid evaluation of design concepts and enable designers to approach the foaming die design from a "what if" prospective. Once the foaming die design concept is selected and the basic layout identified, detailed design analysis should be carried out using more robust and physically accurate models as presented by Amon and Denson (29), Arefmanesh et al. (23), Ramesh et al. (24), and Hah and Yoo (28). The literature is rich with theoretical, and to a lesser extent, experimental treatments of cell growth in viscous, viscoeleastic, and elastic media. Epstein and Plesset (18) present one of the first theoretical treatments of cell growth in a supersaturated liquidgas solution using an approximate solution. Gent and Tompkins (19, 20) and Hobbs (21) present theoretical and experimental treatments of cell growth in elastomeric materials. Amon and Denson (22) present a cell model to predict cell growth where neighboring cells compete for available gas. Arefmanesh et al. (23) have extended this analysis by relaxing the assumption that a single cell is representative of all cells. Recently, Ramesh et al. (24) have presented a numerical and experimental study of cell growth in microcellular processing comparing Newtonian, powerlaw, and viscoelastic models. An additional class of models that have been developed focus on the injection molding process. Throne (25, 26) presents one of the earliest models predicting cell growth during injection of a gassaturated polymer in structural foam molding. Many of the concepts introduced by Throne have been integrated into the design models presented in this work. Villamizar and Han (27) present one of the first empirical studies of foam injection molding, while Han and Yoo (28) present empirical data for structural foam molding as well as a theoretical cell growth model of a single spherical gas bubble in a viscoelastic matrix during isothermal mold filling. The focus of this model is to predict mold filling caused by expansion of the single cell when a short shot of a polymer/blowing agent system is injected into the mold. Using the DeWitt model, they are able to capture important viscoeleastic phenomena such as sheardependent viscosity, normal stress effects, and stress relaxation. Amon and Denson (29) present a comprehensive study of cell growth during structural foam molding based on the cell model as a microscopic analysis for a short shot mold filling process. Their experimental data is based on bulk density measurements during mold filling. Upadhyay (30) developed a viseoelestic model based on the Leonov constitutive equation for the growth of a single bubble in an infinite medium comparing his model predictions with the data of Han and Yoo (28). MICROCELLULAR EXTRUSION SYSTEM OVERVIEW The continuous processing of microcellular polymers presented in previous work (1) is based on three subprocesses that include the processing of the polymer matrix, the processing of the microcellular structure, and the processing the net shape.(*) The creation of a microcellular structure is achieved by dissolving large gas concentrations into a polymer matrix and subjecting the saturated system to a rapid thermodynamic state change. This creates an unstable or supersaturated matrix that drives the nucleation of billions of microcells. Stable cells then grow as gas diffuses into the cells and reduces the bulk density of the material. based on these fundamental processing requirements, a hierarchical design strategy is used to synthesize the overall production system such that each of the processing functions is independently satisfied by a unique design parameter or process variable. A schematic of the microcellular sheet extrusion system is shown in Fig. 2. The basic concept of this design is the use of a single screw plasticating extruder to process the polymer matrix, a staged pressure loss to produce a microcellular structure, and a foaming/shaping die to produce the net shape. The plasticating extruder is used to melt and pump the pellets so the polymer is suitable for downstream processing. The polymer processing is accomplished in the first stage of a twostage extruder. At the beginning of the second extruder stage, the polymer/gas solution formation system begins. Here, a metered amount of gas or supercritical fluid is injected into the polymer melt and mixed to form a single phase solution using a technique presented previously (2, 31). The solution formation system supplies a singlephase solution at the exit of the breaker plate/flow stabilizer. Next, the singlephase solution flows into the foaming die system. The foaming die accomplishes the nucleation and cell growth functions of the microcellular processing system and the shaping function of the sheet processing system. A microcellular sheet then exits the foaming die and can be postprocessed to achieve an appropriate molecular orientation, surface finish, etc. A comprehensive presentation of the microcellular extrusion system and the microcellular extrusion process is presented by Baldwin et al. (1). The three critical processing functions of the foaming die are continuous nucleation, cell growth control, and shaping. In general, cell nucleation must be accomplished independently of cell growth and shaping, and cell growth must be accomplished independently of sheet shaping. To this end, the concept of shaping a nucleated polymer/gas solution flow is presented and experimentally verified. As an integral part of this development, firstorder process models were developed to aid in the design process. Preliminary results indicate these models adequately quantify the limited experimental results supporting their relevance in microcellular foaming die design. Moreover, the results of the critical experiments verify the overall performance of the microcellular extrusion system. DESIGN MODELS To satisfy the preshaping cell growth control requirement, it is necessary to design a shaping and cell growth control die that can maintain the required shaping pressures in order to minimize cell coalescence and loss of nucleated cell density. The design of a shaping and cell growth control die requires knowledge of the pressure loss effects of nucleated polymer/gas solutions. in general, the flow of a nucleated polymer/gas solution is complex and involves a heterogeneous system consisting of a polymer melt with dispersed cells having relatively large centertocenter spacing compared with the average cell diameters [ILLUSTRATION FOR FIGURE 3 OMITTED]. The pressure loss in the flow results from frictional effects at the die walls. As the pressure decreases along the die channel, the nucleated cells grow resulting in an increase in the specific volume (or the decrease of density) of the twophase system and the volumetric flow rate. Cell growth is driven by the pressure within the cells and diffusion of solution gas into the cells. Gas diffusion into the cells results from the decrease in solubility as the solution pressure decreases along the die channel. As the heterogeneous system expands, there is less polymer across any given area of the flow channel to support the shear stresses generated by the wall friction. This implies that the expanding solution has a lower apparent viscosity. The apparent viscosity also tends to decrease as a result of shear thinning as the volumetric flow rate increases. The flow of a nucleated polymer/gas solution in a slit of height 2[center dot]B is sketched in Fig. 3 where [P.sub.s] is the nucleated solution pressure entering the die slit, [P.sub.exit] is the system pressure at the die exit, and L is the die slit length. Figure 3 also illustrates the influence of the high shear regions near the die walls that tend to elongate the expanding cells near the walls. The shear thinning nature of the polymer flow leads to relatively fiat velocity profiles in the center characteristic of plug flow. Similar twophase flow morphologies are presented by Han and Villamizar (32) based on visualization studies for HDPE and PS flows blown with chemical agents. For the purposes of designing foaming dies to satisfy the shaping and cell growth control requirements, it is necessary to develop models to estimate the pressure loss and flow rate of a nucleated polymer/gas solution during die flow. The problem can be divided into four limiting cases that span the range of possible flow configurations. The first case accounts for the fact that gas diffuses into the cells as the solution pressure decreases such that cell growth during the die flow is driven by both the gas pressure in the cells and the diffusion of gas into the cells. The second case assumes that negligible gas diffuses into the cells after nucleation such that cell growth during the die flow is driven by the gas pressure in the cells postnucleation. This case represents an upper limit on the pressure loss during die flow. The third case assumes that all of the available solution gas diffuses into the cells prior to flowing into the die slit such that cell growth is driven by the gas pressure in the cells. This case represents a lower limit on the pressure loss in the die flow. The fourth case is similar to case two but accounts for the diffusion of gas into the cells prior to entering the foaming die and neglects gas diffusion during die flow. In the following discussion, each of the four modeling cases will be presented. While the following analysis incorporates a number of simplifying assumptions, it is important to keep in mind the intent of the analysis, which is to develop and approximate design model for estimating the pressure loss and flow rates of nucleated polymer/gas solutions so that foaming dies can be sized appropriately. In this analysis, the model is kept as simple as possible while capturing the major physics of the complex twophase flow. Such models have great utility in the design of prototype shaping and cell growth control devices. More important, they provide reasonable guidelines to the basic interactions of the major processing variables involved in continuous shaping and cell growth control operations. Nucleated Flow With Concurrent Gas Diffusion For the modeling case where the gas diffuses into the cells as the pressure decreases (i.e., the concurrent diffusion case), the pressure loss and volumetric flow rate of a nucleated polymer/gas solution can be estimated based on the frictional losses at the die walls. This is accomplished by treating the nucleated solution as a nonNewtonian fluid having a bulk viscosity and flow rate that are dependent on the volume fraction of cells (voids) in the polymer matrix (given by Eqs 1 and 2, respectively, where [[Phi].sub.v] is the volume fraction of voids). [[Eta].sub.f] [equivalent to] [Eta]([[Phi].sub.v]) (1) Q [equivalent to] Q([[Phi].sub.v]) (2) Based on the apparent viscosity measurements of Oyanagi and White (33), the rheological behavior of polymeric foams follows similar shearthinning behavior to the neat polymer. Moreover, Oyanagi and White's results indicate that the presence of a blowing agent tends to decrease the apparent viscosity by a multiplicative constant (i.e., showing equivalent power law factors, n). The developments of Draynik (34) in polymer foam theology also indicate decreases in apparent viscosity for many thermoplastic foam flow fields. Therefore, as a first order approximation, the nonNewtonian viscosity of the nucleated solution, [[Eta].sub.f], will be given by the modified power law relation of Eq 3 where [m.sub.f] is the power law constant of the foam. [Mathematical Expression Omitted] (3) The ability of the nucleated solution to resist shear forces is a strong function of the solution void fraction where the presence of cells or voids reduces the effective area of polymer resisting shear. Moreover, the viscosity of the gas within the voids is negligible compared with the viscosity of the polymer matrix. Therefore, the nonNewtonian viscosity of the nucleated solution can be approximated by Eq 4. For the case where the mass fraction of gas is relatively small, Eq 4 simplifies to Eq 5 where [[Rho].sub.p] is the polymer density and [v.sub.g/p] is the specific volume of the gas phase relative to the polymer mass. In this analysis, the expected decrease in the apparent viscosity due to the presence of dissolved gas in the polymer matrix has been neglected since reliable estimates for polystyrene/C[O.sub.2] systems are not available. Estimates of the apparent viscosity of polymer/gas solutions have been presented in two studies. Blyler and Kwei (35) have reported viscosity decreases of 20% and 22% in low and high density polyethylene containing 0.32% dissolved gas from the decomposition of a chemical blowing agent. Under the assumption that the gases produced from a decomposed chemical blowing agent are completely dissolved in a polymer melt, Han and Villamizar (32) estimated from experimental measurements an apparent viscosity decrease of 10 to 37% for polystyrene containing dissolved C[O.sub.2]. The impact of the apparent viscosity is discussed further in the model comparison section, where predicted results account for both reduced base polymer power law factors m due to dissolved gas and base polymer power law factors m for the neat polymer. [m.sub.f] [equivalent to] m(1  [[Phi].sub.v]) (4) [m.sub.f] [equivalent to] m (1  1/1 + 1/[[Rho].sub.g][v.sub.g/p]) (5) Using the power law approximation of Eqs 3 and 5, the bulk flow of a nucleated polymer/gas solution can be described by Eq 6 for steady, fully developed, nonNewtonian flow through a constant crosssection slit of height 2[center dot]B and width W (36). It should be noted that using the lubrication approximation, this analysis can be modified to accommodate gradual tapers in crosssectional area where W = W(z) and B = B(z). in addition, as a first order approximation, the flow will be assumed isothermal and entrance effects will be neglected. The former assumption neglects any viscous work that can raise the flow temperature. For capillary flows, the viscous dissipation localized near the capillary walls can increase the melt temperature by 10 to 15 K (36). The later assumption can lead to large pressure errors in some flow fields, particularly at higher flow rates (37, 38).  dp/dz = [[2 + 1/n/2W[B.sup.2]Q].sup.n] [m.sub.f]/B (6) The total pressure loss can then be obtained by integrating Eq 6 over the slit length, L, where [P.sub.s] is the solution pressure at the slit entrance and [P.sub.exit] is the solution pressure at the slit exit. In Eq 7, the volumetric flow rate, Q, and the power law coefficient, [m.sub.f], are both functions of the local pressure, p. [Mathematical Expression Omitted] (7) In order to integrate Eq 7, the pressure dependence of the volumetric flow rate and power law factor must be determined. The volumetric flow rate of the nucleated solution can be determined by accounting for the specific volume of the polymer/gas system, [v.sub.m], which is a strong function of the local pressure. In the case of relatively low gas concentrations, the volumetric flow rate is given by Eq 8, where [Mathematical Expression Omitted] is the polymer mass flow rate. [Mathematical Expression Omitted] (8) Again using the approximation that the gas concentration is relatively low, the specific volume of the polymer/gas system is given by the sum of the specific volume of the gas phase relative to the polymer mass, [v.sub.g/p], plus the specific volume of the polymer matrix. Neglecting any swelling of the polymer matrix caused by the dissolved gas, the specific volume of the polymer is given by the inverse of the neat polymer density, [[Rho].sub.p]. In this case, the volumetric flow rate is given by [Mathematical Expression Omitted]. (9) To determine [v.sub.g/p], the gas within the cells can be treated as being in quasistatic equilibrium and approximated as a perfect gas such that the total volume of the gas phase, [V.sub.g], is given by [V.sub.g] = [m.sub.g] RT/[P.sub.g] (10) where [m.sub.g] is the mass of gas phase, R is the gas constant (equal to the universal gas constant divided by the molecular weight of the gas), T is the absolute temperature, and [P.sub.g] is the pressure of the gas phase. Here the gas phase in each cell is treated uniformly throughout the polymer/gas system (i.e., the cells are treated as having an equivalent average cell size). Mass conservation of the gas maintains that the mass of gas phase must equal the mass of the total gas initially dissolved into the polymer, [m.sub.T], less the mass of gas remaining in solution, [m.sub.s]. Equation 10 then becomes [V.sub.g] = ([m.sub.T]  [m.sub.s]) RT/[P.sub.g]. (11) The initial mass of dissolved gas is given by the initial gas concentration times the polymer mass. The mass of the remaining dissolved gas can be approximated usIng Henry's law assuming at any given instant the solution gas concentration is uniform and the system is in quasistatic equilibrium. Normalizing Eq 11 relative to the polymer mass then yields [v.sub.g/p] = [V.sub.g]/[m.sub.p] = ([c.sub.[infinity]]  [K.sub.s](T)[P.sub.g])RT/[P.sub.g] (12) where [c.sub.[infinity]] is the Initial gas concentration and [K.sub.s] is Henry's law constant. Inherent to Eq 12 is the assumption that at any given instant, a sufficient amount of gas has diffused Into the cells to establish a quasistatic equilibrium. Such a state is typified by uniform solution gas concentrations (i.e., no gas depleted regions around the cells). Strictly speaking, this assumption does not hold, although it provides a useful approximation in obtaining an estimate of the pressure loss and flow rates In the nucleation solution flows. Based on the quasistatic approximation, the pressure of the gas phase can be related to the local solution pressure using Laplace's Eq 13 where the cells are treated as having a uniform radius, R, P is the solution pressure, and [[Gamma].sub.bp] is the surface tension of the gas/polymer interface. Note that the uniform cell radius is typically a reasonable assumption in microcellular processing because of the narrow cell size distributions achieved with this technology. Inherent to Eq 13 is the assumption that the cell growth dynamics are sufficiently rapid to allow nearequilibrium conditions to be achieved at any given instant. In addition, it should be noted that Eq 13 treats the flow field as a viscous medium and neglects the elastic energy of the polymer matrix. At higher melt temperatures, this is a reasonable approximation for many polymers. An alternate to Eq 13 would be to treat the expanding matrix as neoHookean and use the developments of Denecour and Gent (39) and Gent and Tompkins (19, 20).(**) [P.sub.g] = 2[[Gamma].sub.bp]/R + P (13) Next, it is necessary to relate the radius of the cells back to the specific volume of the gas phase. This can be achieved using Eq 14 for a known density of nucleated cells, [[Rho].sub.c] (remember that the cell density is determined by an independent processing step). Rearranging Eq 14 yields the average cell radius given by Eq 15. [V.sub.g] = [[Rho].sub.c] [m.sub.p]/[[Rho].sub.p](4/3 [Pi][R.sup.3]) (14) R [(3[[Rho].sub.p]/4[Pi][[Rho].sub.c] [v.sub.g/p]).sup.1/3] (15) By combining Eqs 12, 13, and 15, an expression relating the gas phase pressure and the solution pressure is derived and given by Eq 16. [Mathematical Expression Omitted] (16) Calculating the gas phase pressure for a given solution pressure, requires Eq 16 to be solved iteratively, precluding a closed form solution of Eq 7. Therefore, a finite difference technique can be used to solve Eq 7. Using Fig. 3 as a reference, the slit flow can be divided into N elements of length [Delta]z. The incremental pressure drop [Delta]P over element i + 1 is then given by [Delta][P.sub.i+1] [approximately equal to] [[2 + 1/n/2W[B.sup.2] [Q.sub.i]].sup.n] [m.sub.fi]/B [Delta]z (17) and the total pressure at i + 1 is given by [P.sub.i+1] [approximately equal to] [P.sub.i] + [Delta][P.sub.i+1] (18) Equations 17 and 18 can be solved using the boundary condition that at the slit exit the solution pressure is given by P(x = L) = [P.sub.exit] = P(i = N). The solution to Eqs 17 and 18 is accomplished by determining the local gas phase pressure, [P.sub.g], at the solution pressure, [P.sub.i], via Eq 16. Equations 5, 9, and 12 can then be used to calculate the local power law coefficient, [m.sub.fi], and the local flow rate, [Q.sub.i]. The values of [m.sub.fi] and [Q.sub.i] are then used to calculate the local pressure drop, [Delta][P.sub.i+1], and the local pressure, [P.sub.i+1], using Eqs 17 and 18. This iterative solution proceeds until i equals N1. Pressure loss and flow rate estimates typical of nucleated solution flows during shaping and cell growth operations are presented in Fig. 4. The model parameters used for the calculations were for a polystyrene/C[O.sub.2] system having T = 150 [degrees] C, [[Rho].sub.p] = 1.04 g/[cm.sup.3], [[Gamma].sub.bp] = 0.0314 N/m (40), [[Rho].sub.c] = 1.5 x [10.sup.9] cells/[cm.sup.3], [K.sub.s] = 0.0039 kg(C[O.sub.2])/kg(PS)MPa (31), [c.sub.[infinity]] = 0.05, [Mathematical Expression Omitted], n = 0.29, and m = 40750 Pa [s.sup.n]. The slit dimensions used were B = 0.32 mm, W = 25.4 mm, L = 20.3 mm, and N = 20 (i.e., [Delta]z = 1.02 mm). The boundary condition used was [P.sub.exit] = 0.965 MPa (140 psi). This boundary condition is based on the findings of Han et al. (41) and Han and Villamizar (32). Hah and coworkers measured pressure profiles of polymer foam flows through various die geometries. These measurements revealed exit pressure losses equal to nearly 10% of the die entrance pressures. Figure 4 demonstrates some of the important aspects of nucleated solution flows during shaping and cell growth operations. Notice that the volumetric flow rate increases parabolically along the slit length, and the solution pressure decreases nonlinearly along the slit length. In contrast, a neat polymer flow would have a constant flow rate and a linear decrease in pressure over the slit length. The pressure loss trends of Fig. 4 can be understood by looking back at Eq 6. The decrease in pressure gradient along the slit length results from the decrease in the effective viscosity of the heterogeneous polymer/gas matrix along the slit length. The viscosity decrease follows from the expansion of the nucleated solution that reduces the effective area supporting the shear stresses. In this case, the viscosity decrease is sufficiently large to overcome the contributions of the increasing flow rate along the slit length. Notice from Eq 6 that an increasing flow rate tends to increase the pressure loss. The nonlinear pressure profiles predicted by this macroscopic treatment of nucleated solution flow agree in form with the experimental findings of Hah et al. (41) and Hah and Villamizar (32). Han and coworkers present pressure profile measurements in polymer foam flows indicating nonlinear trends similar to those of Fig. 4 with the exception of the pressure profiles near the exit reported by Han et al. (41), which show increased pressure gradients near the die exit. Nucleated Flow With Post Gas Diffusion In addition to the concurrent diffusion case presented above, it is important to understand the limiting cases for the flow of nucleation polymer/gas solutions. Predicting the limiting cases will provide a useful comparison for the intermediate models and will further illustrate the critical processIng characteristics that need to be addressed in the design of shaping and cell growth control devices. As a limiting case, consider the flow of a nucleated polymer/gas system in which only the gas present in the stable nuclei is available for expanding the cells (i.e., the postdiffusion case). In this case, gas diffusion into the cells is treated as being very slow compared with the flow rate during shaping operations. All of the appreciable gas diffusion occurs postshaping. The analysis follows the same lines and is restricted by the same assumptions of the previous section. Again, the flow field of interest is a planar slit flow of a nucleated polymer/gas solution as shown in Fig. 3. The gas phase present in the stable nuclei can be determined based on classical nucleation theory. For spherical nuclei, the critical radius size, [R.sub.o], is given by [R.sub.o] [r.sup.*] = 2[[Gamma].sub.bp]/[Delta][P.sub.nuc] (19) where [Delta][P.sub.nuc] is the pressure change used to instigate the thermodynamic instability for nucleation. The mass of the gas phase can then be determined by equating Eqs 10 and 14 and substituting Eq 19 for the solution conditions postnucleation. Immediately after nucleation, the solution temperature is [T.sub.n] and the pressure of the gas phase in the cells is [P.sub.go] [approximately equal to] [Delta][P.sub.nuc]. The mass of the gas phase immediately after nucleation is then given by [Mathematical Expression Omitted]. (20) Substituting Eq 20 into Eq 10 yields the specific volume of the gas phase relative to the polymer mass given by Eq 21. The specific gas volume applies at any point along the flow field under the assumption that the gas amount is fixed at the nucleation stage. [Mathematical Expression Omitted] (21) In Eq 21, T is the solution temperature during the slit flow that can be different from the solution temperature at nucleation. Finally, an expression for the gas phase pressure as a function of the solution pressure can be obtained by substituting Eq 15 and 21 into Eq 13. [P.sub.g] = P + [Delta][P.sub.nuc] [[T/[T.sub.n] [Delta][P.sub.nuc]/[P.sub.g]].sup.1/3] (22) Equation 22 can then be used to evaluate the pressure of the gas phase as a function of the local solution pressure. In order to determine the local pressure change and the local solution pressure via Eqs 17 and 18, Eq 22 can be used to evaluate the local gas phase pressure, [P.sub.g]. Equation 21 can then be used to calculate the local specific volume of the gas. The local volumetric flow rate and the local power law coefficient can then be determined from Eqs 9 and 5. Finally, the pressure loss and the volumetric flow rate profiles along the slit can be determined using a finite difference technique. Nucleated Solution Flow With Complete Gas Diffusion As a second limiting case, consider the flow of a nucleated polymer/gas system in which all of the available gas has diffused into the cells prior to shaping or cell growth operations (i.e., the prediffusion case). In this case, gas diffusion into the cells is treated as being very rapid compared with the flow rate such that all of the available gas diffuses into the cells prior to shaping. The analysis follows the same lines and restrictions as above. Again, the flow field of interest is a planar slit flow of a nucleated polymer/gas solution as shown in Fig. 3. By assumption, all of the available gas has diffused into the cells prior to the slit flow. Therefore, the mass of the gas phase is constant throughout the shaping process. In this case, the initial gas concentration is determined independently by the solution formation system such that the mass of the gas is given by [m.sub.g] = [c.sub.[infinity]][[m.sub.p]. (23) The specific volume of the gas phase relative to the polymer mass is obtained by substituting Eq 23 into Eq 10. [v.sub.g/p] = [c.sub.[infinity]] RT/[P.sub.g] (24) In the case of complete gas diffusion, the local gas phase pressure can be obtained by substituting Eqs 24 and 15 into Eq 13, yielding [P.sub.g] = P + 2[[Gamma].sub.bp] [[3[[Rho].sub.p]/4[Pi][[Rho].sub.c] RT/[P.sub.g] [c.sub.[infinity]].sup.1/3] (25) Equation 25 can then be used to evaluate the pressure of the gas phase as a function of the local solution pressure. In order to determine the local pressure change and the local solution pressure via Eqs 17 and 18, Eq 25 can be used to evaluate the local gas phase pressure, [P.sub.g]. Equation 24 can then be used to calculate the local specific volume of the gas. The local volumetric flow rate and the local power law coefficient can then be determined form Eqs 9 and 5. Finally, the pressure loss and the volumetric flow rate profiles along the slit can be determined using a finite difference technique. Nucleated Solution Flow With Partial Gas Diffusion A final case to consider for the flow of a nucleated polymer/gas solution is a variation of that presented above. In this case, the gas is assumed to diffuse into the cells prior to slit flow creating a quasistatic equilibrium state where the gas concentrations are uniform and the mass of gas in the cells is uniquely determined by the solution pressure at the inlet to the slit, [P.sub.s] (i.e., the partial prediffusion case). First, the gas mass available prior to entering the slit flow must be determined. To accomplish this, it is assumed that a quasistatic equilibrium condition is reached just upstream of the slit where equilibrium mass of gas has diffused into the cells and uniform gas concentrations have been achieved in the solution. Using Henry's law and an initial gas phase pressure, [P.sub.go], just upstream of the slit, the constant mass of the gas phase is given by [m.sub.g] = ([c.sub.[infinity]]  [K.sub.s](T)[P.sub.go])[m.sub.p]. (26) Substituting Eq 26 into Eq 10 gives the specific volume of the gas phase. [v.sub.g/p] = [V.sub.g]/[m.sub.p] = ([c.sub.[infinity]]  [K.sub.s](T)[P.sub.go]) RT/[P.sub.g] (27) Using an analogous derivation of the gas phase pressure presented in the previous section, the initial gas phase pressure just upstream of the slit, [P.sub.go], is given by Eq 28 where [P.sub.s] is the solution pressure just upstream of the slit. Equation 28 can be used to determine [TABULAR DATA FOR TABLE 1 OMITTED] the gas phase pressure when the inlet solution pressure, [P.sub.s], is known. [P.sub.go] = [P.sub.s] + 2[[Gamma].sub.bp] [[3[[Rho].sub.p]/4[Pi][[Rho].sub.c] RT/[P.sub.go] ([c.sub.[infinity]]  [K.sub.s](T)[P.sub.go])].sup.1/3] (28) Similarly, the gas phase pressure at any point in the slit flow field is given by Eq 29. [P.sub.g] = P + 2[[Gamma].sub.bp] [[3[[Rho].sub.p]/4[Pi][[Rho].sub.c] RT/[P.sub.g] ([c.sub.[infinity]]  [K.sub.s](T)[P.sub.go])].sup.1/3] (29) Equation 28 can be used to evaluate the initial pressure of the gas phase at the slit inlet using an estimated solution pressure at the slit inlet. Equation 29 can then be used to evaluate the pressure of the gas phase as a function of the local solution pressure. In order to determine the local pressure change and the local solution pressure via Eqs 17 and 18, Eq 29 can be used to evaluate the local gas phase pressure, [P.sub.g]. Equation 27 can then be used to calculate the local specific volume of the gas. The local volumetric flow rate and the local power law coefficient can then be determined form Eqs 9 and 5. Finally, the pressure loss and the volumetric flow rate profiles along the die slit can be determined using the finite difference technique discussed above. Comparison of Nucleated Solution Flow Models In order to compare these four models, the parameters given in Table 1 for a polystyrene/C[O.sub.2] system typical of microcellular polymer sheet extrusion were used. The pressure profile results from the concurrent diffusion model, the postdiffusion model, the prediffusion model, and the partial prediffusion model are plotted in Fig. 5. Notice first that the postdiffusion case (i.e., where only the nucleated gas phase is present during slit flow) and the prediffusion case (i.e., where all of the solution gas is present in the gas phase) give the upper and lower pressure loss limits for the nucleated solution flow. The total pressure losses predicted by these limiting eases differ by 63%. The postdiffusion case is interesting in that it has a constant pressure gradient similar to the neat polymer flow. From closer inspection of this case, it is found that the postdiffusion case and the neat polymer flow are identical based on the approximation that the apparent viscosity of the polymer/gas system is equal to that of the neat polymer. This results from the assumption that only the nucleated gas phase is available for expanding the cells during slit flow. Since the nucleated gas phase has very little volume, the initial cell size is quite small, which leads to large surface tension contributions and negligible cell growth as the solution pressure decreases. A similar situation holds for the partial prediffusion case where it is assumed equilibrium concentrations exist prior to slit flow, and gas diffusion during slit flow is negligible. This case also demonstrates a nearly constant pressure gradient. In addition, notice that the pressure gradient is less than that of the postdiffusion ease (the neat polymer flow). The lower pressure gradient results from the finite cell size and specific volume of gas phase existing upstream of the slit. The presence of these cells tends to lower the bulk viscosity of the nucleated solution flow. The pressure loss results for the concurrent diffusion and prediffusion cases both show nonlinear behavior with decreasing pressure gradients along the slit length. The decreasing pressure gradients indicate that the viscosity changes dominate the pressure loss despite the fact that the flow rates increase along the slit length. Since the rationale of designing the die is to maintain small cell sizes during shaping operations, it is important to compare the estimated average cell sizes for the four models. The postdiffusion model estimates an average cell radius at the slit inlet of 0.00021 [[micro]meter]. At the other limit, the prediffusion case estimates a cell radius of 4.0 [[micro]meter]. The concurrent diffusion and partial prediffusion cases estimate entrance cell radii of 1.9 [[micro]meter] and 2.2 [[micro]meter], respectively. Since the final cell size range for microcellular polymers is 10 to 20 [[micro]meter], initial shaping cell sizes around 2 [[micro]meter] should be sufficient to prevent degradation of the cell density and therefore satisfy the preshaping cell growth control functional requirement. Indeed this is the ease based on the initial experimental results presented later in this section. In terms of the final cell sizes, the prediffusion, concurrent diffusion, partial prediffusion, and post diffusion models predict final cell radii of 8.5, 8.3, 5.2, and 0.002 [[micro]meter], respectively. The estimates of Figs. 5 and 6 have a number of implications on the design of shaping and cell growth control devices. in general, each of these eases can be associated with a particular type of die design and system flow depending on a dimensionless parameter consisting of the characteristic flow rate and the characteristic gas diffusion rate. The characteristic flow rate of a fluid particle is given by [Mathematical Expression Omitted] (30) where L is the channel length and A is the channel crosssectional area. The characteristic diffusion rate is given by [TABULAR DATA FOR TABLE 2 OMITTED] [Mathematical Expression Omitted] (31) where the characteristic diffusion distance is approximated by the centertocenter distance between cells, [Mathematical Expression Omitted]. In general, one can associate the different flow cases with a particular foaming die configuration as summarized in Table 2 based on the ratio of Eq 32. Notice, at very high processing rates, the nucleated solution flow would tend to behave more like the postdiffusion flow of Figs. 4 and 5. This follows since at very high processing rates the die flow occurs over a much shorter time frame then that of gas diffusion into the cells. For foaming die designs having large postnucleation channel volumes ([L.sub.pn] [A.sub.pn]) and relatively small shaping channel volumes ([L.sub.s] [A.sub.s]), the nucleated solution flow would tend to behave like the partial prediffusion case. In this case, enough time is allowed for gas diffusion to equilibrate prior to slit flow. However, during slit flow, velocities are sufficiently large to inhibit appreciable gas diffusion. If the flow rates and diffusion rates are on the same order, then the concurrent diffusion case would tend to estimate the correct pressure loss and flow rates. Finally, under conditions where the flow rates are low and the flow channel volumes (L A) are high, the prediffusion case would tend to estimate the correct pressure losses and flow rates of nucleated polymer/gas solutions. [Mathematical Expression Omitted] (32) To verify the general applicability of the design models, a critical experiment was conducted to compare the flow rate results of a specific foaming die design, namely the die design of Table 1, with the analysis results of Figs. 4 through 6. The details of the experiment are presented elsewhere (1, 3). The plots in Figs. 4 through 6 correspond to the same flow parameters as the planar sheet extrusion experiments presented previously (1, 3). In this experiment, the microcellular sheet extrusion system of Fig. 2, without post sheet processing heaters or stretching, was used with the process parameters of Table I to produce microcellular sheets. Details of the die design and processing system are presented by Baldwin, Park, and Suh (13, 9, 31). For the experimental microcellular sheet extrusion system, Eq 32 yields: [Mathematical Expression Omitted] Therefore, the nucleated sheet die flow fell between the concurrent diffusion case and the prediffusion case based on Table 2. Furthermore, the shaping pressure achieved by the foaming die, [P.sub.s] = 9.65 MPa (1400 psi) (also plotted in [ILLUSTRATION FOR FIGURE 5 OMITTED]), was in reasonable agreement with the predicted values of 10 MPa and 11.5 MPa (1450 psi and 1670 psi) for the prediffusion and concurrent diffusion flow configurations. Moreover, if in Eq 4 one accounts for the lower apparent viscosity of polymer/gas solutions, the model results and experiment comparison improves. based on Han and Villamizar's (32) measurements, an estimated average apparent viscosity decrease for polystyrene/gas systems is 20% (where values reported range from a 10 to 37% reduction). Note that an average is used to approximate the increase in apparent viscosity expected as the dissolved gas diffuses from solution into the expanding cells thereby lowering the dissolved gas concentrations in solution during the growth process. If in Eq 4, m is decreased by 20% to reflect an average apparent viscosity decrease for a polystyrene/C[O.sub.2] solution, the prediffusion, concurrent diffusion, partial prediffusion, and postdiffusion models predict shaping pressures (at the foaming die inlet) of 8.8, 10.2, 12.7, and 14.4 MPa, respectively. This also illustrates the importance of accurate apparent viscosity data for polymer/gas solutions. While the results of Blyler and Kwei (35) and Han and Villamizar (32) provide important insights to this phenomena, it is clear that further research in this area is needed. Based on the cell morphology and geometry of the extruded microcellular PS for a nucleated cell density of 1.5 x [10.sup.9] cells/[cm.sup.3] with an average cell size of 11[[micro]meter], the feasibility of nucleating and postshaping a polymer/gas solution and extruding microcellular polymers has been demonstrated. Also, notice the predictions based on the neat polymer flow overpredicted the actual pressure loss by 66%. Thus, the pressure loss estimates from the nucleated solution flow models appeared to capture the major physics of the shaping and cell growth processes to provide reasonable estimates for foaming die designs. The volumetric flow rate results for the four nucleated polymer/gas solution flow models are shown in Fig. 6. Notice first the limiting flow rates given by the postdiffusion case and the prediffusion case. The postdiffusion case demonstrates a constant flow rate equal to the neat polymer flow that results from the negligible expansion of the nucleated gas phase during slit flow. in contrast, the prediffusion case shows a rapid increase in flow rate over the slit length resulting from the expanding cells that contain all the available gas. Notice further that the prediffusion case has an initial flow rate twice that of the neat polymer because of the larger initial cell size at the slit entrance. The intermediate case of concurrent diffusion shows flow rate trends similar to the prediffusion case with slightly larger gradients near the slit exit because of the larger expansion rates occurring for simultaneous gas diffusion. The other intermediate case of partial prediffusion, where equilibrium conditions are reached prior to entering the slit and no diffusion occurs in the slit, shows a more gradual change in the flow rate over the slit length since the gas available to expand the cells is limited. Finally, the analysis of the previous sections applies equally well for tubular filament flows. in this case, Eq 33 is substituted for Eq 6 where D is the diameter of the tube. The solution to Eq 33 proceeds in a similar manner to the slit flow case where a finite difference technique is used to locally solve the relative pressure loss and solution pressure. [Mathematical Expression Omitted] (33) Again, a critical experiment was conducted to compare the flow rate results of a specific foaming die design, namely the tubular filament die design of Table 1, with the analysis results of Fig. 7. The shaping pressure achieved by the foaming die, [P.sub.s] = 6.89 MPa (1000 psi) plotted in Fig. 7, was in reasonable agreement with the predicted values of 7.2 MPa and 8.1 MPa (1040 psi and 1170 psi) for the prediffusion and concurrent diffusion flow configuration plotted in Fig. 7. based on the cell morphology and geometry of the extruded microcellular PS for a nucleated cell density of 1.0 x [10.sup.9] cells/[cm.sup.3] with an average cell size of 10 [[micro]meter], the feasibility of nucleating and post shaping a polymer/gas solution and extruding microcellular polymers has also been demonstrated for filaments. This reconfirms that the nucleated solution flow models appeared to capture the major physics of the shaping and cell growth processes to provide reasonable estimates for foaming die designs. CONCLUSION In this paper, a new microcellular sheet extrusion system was presented based on the principle of shaping a nucleated polymer/gas solution flow under pressure and close temperature control. In this way, the initial cell growth was controlled to prevent degradation of the nucleated cell density during shaping. Cell growth control was achieved using carefully designed foaming dies. The die designs were accomplished using a series of design models that comprise the majority of this paper. The sheet extrusion systems and design models were verified form experiments validating the concept of shaping a nucleated polymer/gas solution. The feasibility of shaping a nucleated polymer/gas solution represents a significant advancement for microcellular plastics process technology. Through proper design of the foaming die, nucleated solution flows can be shaped to arbitrary dimensions while maintaining the functional independence of cell nucleation, cell growth, and shaping. To maintain functional independence, stringent pressure and temperature design specifications, which supersede those of conventional foam processing, must be met by the foaming die design. As a means of aiding the design process, a series of models were developed for predicting pressure losses and flow rates of nucleated polymer/gas solutions. A comparison of the model predictions and the actual foaming die design performance showed good agreement for limited data. Moreover, the relatively simple models capture the major physics of the complicated twophase flow field and provide a sound base from which scaleup of the foaming die concept to industrial levels can be achieved. In general, the concurrent diffusion and the prediffusion models showed the best agreement with the limited experimental data. Moreover, these models predicted a nonlinear pressure loss through the die that has been experimentally measured by Han and coworkers (29, 41). The decreasing pressure gradients predicted by the concurrent diffusion and prediffusion models indicate that viscosity changes dominate the pressure loss despite the fact that the flow rates increase along the die length. It is also found that predictions based on the neat polymer flow overpredict the actual pressure loss by 66%. The concurrent diffusion and prediffusion models predict highly nonlinear volumetric flow rates in contrast to the constant flow rate predicted for the neat polymer flow and the postdiffusion model. Finally, a convenient method for classifying nucleated polymer/gas solution flow was presented based on the dimensionless ratio of the characteristic flow rate to the characteristic gas diffusion rate. Flow classification allows for the rapid identification of the appropriate design model for microcellular foaming die design. ACKNOWLEDGMENTS The authors would like to thank Dr. Bruce M. Kramer and Dr. Maria Burka for their interest in this work and the National Science Foundations, Grant Number CTS9114738, for supporting this research. This work was performed at the MITIndustry Microcellular Plastics Research Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts. NOMENCLATURE B = Half slit height of channel [m]. c = Instantaneous or local equilibrium concentration of gas molecules [kg(gas)/kg(polymer)]. [c.sub.eq] = Equilibrium concentration of gas in a polymer [kg(gas) / kg(polymer)]. [c.sub.w] = Gas concentration at the cell wall surface [kg(gas)/kg(polymer)]. [c.sub.[infinity]] = Initial equilibrium gas concentration [kg(gas)/kg(polymer)]. D = Diameter of tubular flow channel [m]. [K.sub.s] = Henry's law constant [kg(gas)/kg(polymer)Pa]. L = Flow channel length [m]. m = factor in power law constitutive equation [Pa [s.sup.n]]. [m.sub.f] = Power law coefficient for the nucleated polymer/gas solution [Pa [s.sup.n]]. [m.sub.g] = Mass of the gas phase [kg]. [m.sub.p] = Mass of the polymer [kg]. [m.sub.s] = Mass of the gas remaining in solution [kg]. [m.sub.T] = Total mass of the gas [kg]. [Mathematical Expression Omitted] = Polymer mass flow rate [kg/s]. n = Exponent in power law constitutive equation. [p.sub.g] = Gas pressure in a cell or bubble [Pa]. [p.sub.[infinity]] = Pressure of the polymer far from the cells [Pa]. [P.sub.b] = Barrel pressure at the gas injection port [Pa]. [P.sub.exit] = Pressure at the foaming die exit [Pa]. [P.sub.g] = Pressure of the gas phase in the calls [Pa]. [P.sub.n] = Pressure at the nucleation device [Pa]. [P.sub.s] = Shaping pressure at the foaming die inlet [Pa]. [Delta]P = Pressure loss along the flow channel ([P.sub.inlet]  [P.sub.outlet]) [Pa]. [Delta][P.sub.nuc] = Effective pressure change driving nucleation [Pa]. Q = Volumetric flow rate [[m.sup.3]/s]. R = Average cell radius [m]. R = Universal gas constant/molecular weight of the gas [J/kg K]. T = Absolute temperature [K]. [T.sub.n] = Temperature at nucleation [K]. [T.sub.s] = Temperature at shaping [K]. [v.sub.g/p] = Specific volume of the gas phase relative to the polymer mass [[m.sup.3]/kg]. [v.sub.m] = Specific volume of the polymer/gas system [[m.sup.3]/kg]. [v.sub.r] = Radial velocity component of the polymer or polymer/gas interface [m/s]. W = Width of the flow channel slit [m]. [Delta]z = incremental distance along the flow channel (L/N) [m]. [[Phi].sub.v] = Volume fraction of gas phase. [[Gamma].sub.bp] = Surface energy of the polymer/bubble interface [N/m]. [Eta] = NonNewtonian apparent viscosity [Pa s]. [[Rho].sub.p] = density of polymer [kg/[m.sup.3]]. * A detailed presentation of the microcellular sheet extrusion system design is presented by Baldwin (1, 2). ** If the expanding cells are treated as having thin shells d thickness, [t.sub.o], supporting the extensional stresses, then the relationship between the cell pressure and radius is given by (where G is the shear modulus and [R.sub.o] is the initial cell radius) [P.sub.g]/G = (2[t.sub.o]/[R.sub.o])(([R.sub.o]/R)  [([R.sub.o]/R).sup.7]). If the expanding cells are approximated as having thick or infinite walls, then the cell pressure is given by [P.sub.g]/G = 5/2  2([R.sub.o]/R)  (([R.sub.o]/R).sup.4])/2. See Gent and Tompkins (19, 20) for similar expressions accounting for the pressure surrounding the cells. REFERENCES 1. D. F. Baldwin, C. B. Park, and N. P. Suh, Polym. Eng. Sci., 36, 1446 (1996). 2. C. B. Park, D. F. Baldwin, and N. P. Sub, Polym. Eng. Sci. 35, 432 (1995). 3. D. F. Baldwin, PhD thesis, Mech. 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