Strategies To Estimate the Pressure Drop Threshold of Nucleation for

Jan 23, 2009 - Two methods to predict the pressure drop threshold (ΔPthreshold) to initiate bubble nucleation in polymer foaming processes are develo...
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Ind. Eng. Chem. Res. 2009, 48, 1921–1927

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Strategies To Estimate the Pressure Drop Threshold of Nucleation for Polystyrene Foam with Carbon Dioxide Siu N. Leung, Anson Wong, Chul B. Park,* and Qingping Guo Microcellular Plastics Manufacturing Laboratory, Department of Mechanical & Industrial Engineering, UniVersity of Toronto, Toronto, Ontario, Canada M5S 3G8

Two methods to predict the pressure drop threshold (∆Pthreshold) to initiate bubble nucleation in polymer foaming processes are developed. One method uses the modified nucleation theory developed in our previous work, while the other utilizes computer simulations to model the growth profiles of the first observable bubbles in batch foaming experiments. These two approaches have shown reasonably good agreement qualitatively with each other in their ∆Pthreshold predictions. Moreover, the effects of the pressure drop rate, the gas content, and the processing temperature on ∆Pthreshold are demonstrated. It was found that the pressure drop rate does not have a significant effect on ∆Pthreshold, while increasing the gas content or the processing temperature leads to a decrease in ∆Pthreshold. Introduction Foamed plastics with a high cell density and narrow cell size distribution offer superior mechanical properties such as higher toughness and specific tensile stress, as well as better thermal and acoustic insulation properties when compared to their solid counterparts.1-3 Due to the substantial increase in the price of plastic resins in recent years, reduction of the raw material cost is one research area that is of major economical interest to plastics manufacturers. To this end, research has been undertaken to investigate the fundamental mechanism governing volume expansion of plastic foams.4-6 In a continuous plastic foam extruding process, cell nucleation usually occurs inside the die after the pressure of the polymer/ gas solution drops below the solubility pressure. Upon cell nucleation, cells start to grow before the polymer/gas solution exits the die. This cell growth phenomenon is termed “premature cell growth”. An excess amount of premature cell growth would lead to rapid cell growth upon die exit. This accelerated cell growth will promote gas loss during the foam cooling process, so the foam shrinks before it stabilizes, and subsequently a low volume expansion ratio can result.7 In order to accurately determine the amount of premature cell growth, it is first necessary to identify the onset point of cell nucleation. Much research in the past assumes cell nucleation occurs right after the system pressure drops below the solubility pressure.8,9 However, since cell nucleation is a kinetic process, a certain amount of pressure drop beyond the solubility pressure is needed to create a sufficient level of supersaturation to initiate cell nucleation. This pressure drop is termed a “pressure drop threshold” (i.e., ∆Pthreshold) in this study. A fundamental understanding of the mechanisms governing ∆Pthreshold will assist the development of design strategies in foaming systems to suppress premature cell growth and to better control cell morphology as well as the volume expansion ratio of foamed plastics. In particular, foamed products with a high volume expansion ratio can be achieved while maintaining a uniform cell morphology. At the other end of the spectrum, by knowing the onset point of cell nucleation, it is possible to develop innovative means to suppress cell growth to produce nanocellular foamed products. * To whom correspondence should be addressed. Telephone: 416978-3053. E-mail: [email protected].

In the past, little effort has been made to study the mechanisms that govern ∆Pthreshold due to the difficulty in gathering such empirical data. This study aims to fill this gap by investigating the effects of the pressure drop rate (-dP/dt), the gas content, and the processing temperature (Tsys) on ∆Pthreshold. To achieve this, a semiempirical and a theoretical approach were used to determine the onset time of cell nucleation at each experimental condition. ∆Pthreshold results from the two approaches were then compared. Methodology The overall research strategy, which includes a semiempirical approach and a theoretical approach, is illustrated in Figure 1. Implementation of the Semiempirical Method. In the semiempirical method, the batch foaming visualization system developed by Guo et al.10 was employed to determine the time at which the first bubble occurred in the plastic foaming process. However, due to the limited optical resolution of the system, the smallest bubble that could be captured by the equipment was about 3-5 µm in diameter. Since the critical bubble’s diameter is believed to be in the nanometer range in typical polymeric foaming processes, there is a time delay between the onset moment of nucleation and the time at which the first bubble appears in the visualization system. In this context, the

Figure 1. Overall research methodology.

10.1021/ie800079x CCC: $40.75  2009 American Chemical Society Published on Web 01/23/2009

1922 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 Table 1. Experimental Conditions for Foaming Experiments and Computer Simulations cases 1 2 3 4 5 6 7 8 9 10

gas content (wt %) (Psat (MPa)) 4.0%(9.71) 5.0%(12.1) 6.0%(14.7) 7.0%(16.8) 5.0%(12.5) 5.0% (12.9) 5.0% (13.4) 5.0% (12.1) 5.0% (12.1) 5.0% (12.1)

max -dP/dt (MPa/s)

processing temp (°C)

22 22 22 22 47 47 47 32 40 47

140 140 140 140 160 180 200 140 140 140

bubble growth simulation software developed in our previous work11 was utilized to estimate this time delay to minimize the error in the determination of ∆Pthreshold. In this study, foaming experiments of different processing conditions (summarized in Table 1) were conducted to elucidate the effects of -dP/dt, gas content, and Tsys on ∆Pthreshold. Each experiment was carried out three times, and the average ∆Pthreshold was determined. It should be noted that when studying the effect of Tsys on ∆Pthreshold, Psat was varied in order to maintain a constant CO2 content (i.e., 5.0 wt %). The polymer and the physical blowing agent used for the foaming experiments were polystyrene, PS (Styron PS685D, Dow Chemical Ltd.), and carbon dioxide, CO2 (99% pure, BOC Canada Ltd.), respectively. The MFI of the PS is 1.5 g/10 min, and its specific gravity is 1.04 g/cm3. The setup of the batch foaming visualization system (refer to Figure 2) and the experimental procedures are detailed in ref 10. The in situ visualization data were analyzed to obtain the time at which the first bubble occurred and its growth profile. Using the simulation algorithm in our previous work,11 the growth profile of the first bubble being observed in the experiment was simulated for each experimental condition to depict the onset time, tonset, of the bubble nucleation. This process involves simultaneously solving the continuity equation, the momentum equation, the constitutive equations, and the diffusion equation in a spherical coordinate system.12,13 In this work, the bubble growth simulation software11 was employed to find the onset moment of cell nucleation that will lead to a leastsquares fit between the simulated and the experimentally measured bubble growth profiles. Consequently, the ∆Pthreshold was determined by subtracting the system pressure at tonset (i.e., Psys(tonset)) from the saturation pressure, Psat. Implementation of the Theoretical Method. The theoretical method is based on an integrated model that combines the modified nucleation theory14 and the aforementioned bubble growth simulation model. The onset time of nucleation was

Figure 3. Schematic of a heterogeneously nucleated bubble.

determined to be the time at which the cell densities exceeded 10,000 bubbles/cm3. This metric was used because one bubble observed in the batch foaming visualization system (i.e., a circular viewing area of ∼500 µm in diameter) is equivalent to a cell density of ∼10,000 bubbles/cm3. Since the onset times of nucleation determined in the semiempirical approach depended on the bubble growth profiles of the first observable bubble, this metric on cell density was used to compare the onset times determined from the two approaches. The overall algorithm of this foaming simulation is discussed in our previous work.14 For the cell nucleation simulation, the cell-population density with respect to the unfoamed polymer volume, Nb,unfoam, was determined by integrating the total cell nucleation rate over time using eq 1: Nb,unfoam(t) )

∫ [J t

dt

(1)

where Jhom(t’) and Jhet(t’) are the homogeneous nucleation rate per unit volume of polymer/gas solution and the heterogeneous nucleation rate per unit surface area of nucleating agents, respectively, that can be computed using eqs 2-9; Ahet is the total surface area of the heterogeneous nucleating sites per unit volume of the polymer/gas solution; and t and t′ are the time of the current simulation step and the time at which bubbles are nucleated, respectively.



Jhom ) N

Whom )

(

Whom 2γlg exp πm kBTsys

)

(2)

16πγ3lg

(3)

3(Pbub,cr - Psys)2

where N is the number of gas molecules per unit volume; γlg is the surface tension at the liquid-gas interface; m is the molecular mass of the gas; Whom is the free energy barrier to initiate homogeneous nucleation; kB is the Boltzmann constant; Tsys is the processing temperature; Pbub,cr is the pressure in a critical bubble; and Psys is the system pressure. Jhet )

∫ F (β)N

2/3

β

β



Q(θc, β)

Whet )

Figure 2. Schematic of the batch foaming visualization system.10

hom(t ′ ) + Ahet Jhet(t ′ )]

0

(

)

Whet 2γlg exp dβ πmF(θc, β) kBTsys (4)

16πγ3lgF(θc, β) 3(Pbub,cr - Psys)2

(5)

where Whet is the free energy barrier to initiate heterogeneous nucleation; F is the ratio of the volume of the heterogeneously nucleated bubble to the volume of a spherical bubble having the same radius; and Q is the ratio of the surface area of the liquid-gas interface of the heterogeneously nucleated bubble to the surface area of a spherical bubble with the same radius.

Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1923

Figure 4. Visualized batch foaming data taken from PS/CO2 foaming experiments.

Using simple geometry, F and Q were determined to be functions of θc and β, where θc is the contact angle (i.e., the angle between the bubble surface and the solid surface measured in the liquid phase) and β is the semiconical angle. Both θc and β are shown in Figure 3. The volume of a heterogeneously nucleated bubble, Vbub, and the surface area of its liquid-gas interface, Alg, can be determined by eqs 6 and 7. Vbub )

{

πR3bub cos θc cos2(θc - β) 2 - 2 sin(θc - β) + 3 sin β

}

Alg ) 2πR2bub(1 - sin(θc - β))

(6) (7)

where Rbub is the radius of curvature of the nucleated bubble. Hence, the expressions of F and Q can be derived: F(θc, β) )

{

Vbub 1 ) 2 - 2 sin(θc - β) + 4 3 4 πR 3 bub cos θc cos2(θc - β) sin β Q(θc, β) )

Ahet 4πR2bub

)

1 - sin(θc - β) 2

}

(8) (9)

Ruijter et al.15 observed a very weak dependence of θc on temperature for squalane on polyethylene terephthalate (PET). Due to the unavailability of θc data for a PS/CO2 solution on a sapphire surface, the dependence of θc on temperature was assumed to follow that for the squalane on PET and θc was being considered as constant in this study. The value of θc (i.e., 85.7°) was determined by choosing the value that describes the experimental data of a particular run. The same θc was then used for all other simulations. It should be emphasized that the assumptions being made on θc need to be re-evaluated in the future if the data becomes available. Since the foaming experiments were carried out using a thin plastic sample placed on a sapphire window inside a high pressure chamber, the sapphire surface would act as heterogeneous nucleating sites

to facilitate foaming. The irregular surface geometries at the microscopic level of the sapphire window were approximated as a series of conical cavities with random β. In order to account for the randomness of β at different locations, a uniform probability density function (i.e., Fβ) from 0 to π/2 was used to represent the distribution of β. Moreover, in eqs 3 and 5, the values of Pbub,cr were calculated using the approach detailed in our previous work.16 Results and Discussion Since the free energy barrier to initiate bubble nucleation (i.e., Whom and Whet) and the thermodynamic fluctuation (i.e., kBTsys) are inside the exponential term of eqs 2 and 4, they would be the dominant factors that govern the nucleation rates and ∆Pthreshold. Thus, our discussion about the dependence of ∆Pthreshold on -dP/dt, gas content, and Tsys focuses on investigating their effects on Whom, Whet, and kBTsys. Effect of -dP/dt on ∆Pthreshold. Figure 4 shows a sample of foaming visualization data of PS/CO2 foaming at different pressure drop rates. Figures 5a and 6a show that, in both approaches, ∆Pthreshold remained approximately the same while the maximum cell density was increased by raising -dP/dt from 22 MPa/s to 47 MPa/s. One-way analysis of variance (ANOVA)17,18 was applied to confirm the lack of effect of -dP/dt on ∆Pthreshold with the semiempirical results, and it was shown that the results were indeed statistically insignificant (refer to Table 2). Figure 7 indicates that the cell nucleation rate increased more rapidly at higher -dP/dt and led to an earlier tonset. However, in the beginning phase of the foaming processes (i.e., Nb,unfoam < 10,000 cells/cm3), it can be observed that the nucleation rates were the same with the same amount of pressure drop in each case. During this initial period, the gas content in the polymer/ gas solution was virtually unchanged. Therefore, γlg and Pbub,cr were identical in all cases when Psys was the same (i.e., the same amount of pressure drop). Moreover, Tsys was held constant and the surface geometry of the nucleating sites was independent of the processing conditions. Consequently, -dP/dt showed no effect on ∆Pthreshold.

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Figure 5. Effects of (a) -dP/dt, (b) CO2 gas content, and (c) Tsys on ∆Pthreshold (error bars: 3 × standard deviation).

Effect of Gas Content on ∆Pthreshold. The amount of dissolved gas content in a polymer was varied from 4% to 7% in 1% increments by adjusting the saturation pressure, which was determined from PS/CO2 solubility measurements carried out using the gravimetric method with a magnetic suspension balance by Li et al.19 The gravimetric method measures the mass of a polymeric sample during sorption of gas under high pressure in situ. The buoyancy effect caused by the swelling of the sample is then compensated by equations of states (EOS), such as the Sanchez-Lacombe EOS, to obtain accurate solubility data. The detailed theory and methodology of solubility measurements is outlined in ref 20. Figures 5b and 6b show that, in both approaches, ∆Pthreshold decreased and the maximum cell density was increased by raising the CO2 content from 4 to 7 wt %. In particular, the one-way ANOVA test about the significance of the gas content effect with the semiempirical results indicates that the results were significant with higher than 99% confidence (refer to Table 2). By simple inspection, Figure 5b seems to suggest that the semiempirical results exhibited a steeper decrease of ∆Pthreshold than that of the

Figure 6. Effects of (a) -dP/dt, (b) CO2 gas content, and (c) Tsys on maximum cell density (error bars: 3 × standard deviation). Table 2. One-Way ANOVA Results experimental parameter

P-value

significance (% probability)

max -dP/dt (MPa/s) CO2 gas content (wt %) processing temp (°C)

0.886 0.000 0.012

99% >98%

theoretical results with the same increase in CO2 gas content. However, the theoretical results still fell within three standard deviations of the semiempirical results, and hence, their differences were not statistically significant. According to the results from the simulation conducted with varying the CO2 content, the nucleation rates increased and tonset became earlier with increasing the CO2 content. Since all four cases had the same pressure drop rate, a higher CO2 content would lead to a lower ∆Pthreshold. On the other hand, when there is a higher gas concentration, γlg also decreases,21 which should lead to the reduction of Whom and Whet and an increase in the nucleating rate. Hence, this explains why the overall trend of ∆Pthreshold decreases with a higher CO2 content.

Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009 1925

Figure 8. Sensitivity analysis of the surface tension’s effect on bubble growth.

Figure 7. Pressure drop profiles and effects of -dP/dt on (a) nucleation rate (simulation) and (b) cell density (simulation).

Effect of Processing Temperature on ∆Pthreshold. Similar to the effect of gas content, Figures 5c and 6c indicate both approaches suggested that ∆Pthreshold decreases when Tsys was increased from 140 to 200 °C. However, a higher Tsys would lead to a slight reduction in the maximum cell density. The theoretical results fell within three standard deviations of the semiempirical results, so their differences are not significant. The one-way ANOVA test about the significance of the temperature effect shows that the results were significant but with a lower confidence (i.e, 98%) than the previous case (refer to Table 2). This suggests that the effect of Tsys is not as strong as that of the gas content in the ranges that were considered in this study. This finding agrees with the theoretical results, which exhibit only a slight decreasing trend in ∆Pthreshold with increasing Tsys. In theory, an increase in Tsys increases the mobility of the gas molecules. The increased thermal fluctuation means there is a higher chance of the gas molecules forming clusters that are larger than the critical radius for cell nucleation. Therefore, according to the simulation results, a higher Tsys would increase the nucleation rate. Furthermore, it would reduce γlg, but the changes are very small at the pressures used in this study.21 This implies that there would only be a slight decrease in Whom and Whet. Hence, the decrease in ∆Pthreshold with increasing Tsys is not as significant as the case with increasing gas content. Sensitivity Analysis Although γlg of PS/CO2 has been measured as a function of gas content and temperature, the small radius of the critical nucleus may not validate the use of this data because of the curvature effect on surface tension.22 In addition, the data for relaxation times (λ) of PS/CO2 solutions is unavailable. Therefore, sensitivity analyses of these two parameters on bubble growth profiles were undertaken to estimate the impact of such

Figure 9. Sensitivity analysis of the relaxation time’s effect on bubble growth.

Figure 10. Sensitivity analysis of the contact angle’s effect on the simulated pressure drop threshold.

errors on the ∆Pthreshold results obtained in the semiempirical approach. It should be noted that sensitivity analysis results of γlg and λ obtained in this study agree with our previous work.11 Furthermore, because of the unavailability of the contact angle data for the PS/CO/sapphire system, a constant value was assumed (i.e., 85.7°) at different system temperatures. Effect of Surface Tension at the Liquid-Vapor Interface (γlg). The effect of γlg on bubble growth profiles was studied by varying its value over a range of 1.923-38.546 mJ/m2. The initial bubble radius in each case was assumed to be 1% larger than the critical radius. The sensitivity analysis is illustrated in Figure 8. Note that the base case is γlg ) 18.7 mJ/m2, which corresponds to experimental case 6 in Table 1. It was shown that the effect of γlg on the overall bubble growth profiles was minimal. This result was consistent with our previous work.11

1926 Ind. Eng. Chem. Res., Vol. 48, No. 4, 2009

As a consequence, the value of γlg would not affect the fitting of the bubble growth simulation data to the empirical results and, hence, the estimation of tonset. This means that the simulations carried out in this study are valid despite the uncertainty of the validity of surface tension data at the molecular level. Effect of Relaxation Time (λ). λ is a characteristic parameter used to describe the viscoelastic nature of a polymer melt. In a physical sense, a larger λ implies a longer time is needed for the accumulated stress of the polymer melt to relax and also for stress to accumulate.23,24 The effect of λ on bubble growth profiles was studied by varying its value over a range of 0.1-1000 s, and the results are shown in Figure 9. Note that the base case is λ ) 27 s, which corresponds to pure PS at the experimental conditions outlined in case 6 of Table 1. Since the simulations in this study focused on initial bubble growth, it was expected that a lower λ would lead to a smaller bubble due to the higher rate of stress accumulation. But the effect of λ on the bubble growth profile was very minimal, which was consistent with the finding in our previous study on bubble growth.11 Since bubble growth processes are very insensitive even to a wide range of λ, the validity of the simulation results carried out in this study should not be undermined by the lack of available data on λ for PS/CO2 solutions. Effect of the Contant Angle (θc). θc is a parameter that relates to the wettability of the polymer on the nucleating agent’s surface (i.e., sapphire window). A larger θc means a worse wetting of the polymer on the sapphire window and thereby a better wetting of the gas on the sapphire window. This is beneficial to cell nucleation. Therefore, it is expected that a larger θc will lead to a lower energy barrier, a faster nucleation rate, and a lower ∆Pthreshold. This is also reflected in the classical nucleation theory as indicated in eq 5. In this study, because no data is available for the contact angle of the PS/CO/sapphire system, a constant value was assumed (i.e., 85.7°) for different system temperatures in the computer simulation. The effects of the size θc on ∆Pthreshold are illustrated in Figure 10. It can be observed that the theoretically simulated ∆Pthreshold was relatively sensitive to the change of θc. This means that an accurate measurement of θc is critical to verify the validity of the theoretical approach to predict ∆Pthreshold. Therefore, the assumptions being made on θc will need to be re-evaluated in the future if the data becomes available. Justification of Termination Points of Simulations The bubble growth simulation software used in this study assumed no interactions between bubbles. Therefore, the simulations must be terminated before the bubbles have grown to a point at which interaction between bubbles becomes significant. To this end, it is first noted that the foamed cells matrix can be approximately represented by tetrahedral structures, in which each bubble is centered at a vertex of a tetrahedron. Assuming that each of the sides of the tetrahedrons is lo and each bubble has an identical radius ra, then contact between adjacent bubbles takes place when ra g lo/2. Using a safety factor of 2, the termination point of simulations was chosen to be ra ) lo/4 to ensure that interaction between bubbles is minimal. Conclusion Using a semiempirical approach and a theoretical approach, analyses of the effect of the pressure drop rate, gas content, and processing temperature on pressure drop threshold

(∆Pthreshold) were conducted. The results from both approaches have shown a reasonably good agreement qualitatively, which is consistent with our theories. Using one-way ANOVA, it was demonstrated that pressure drop rate has no effect on ∆Pthreshold, while ∆Pthreshold decreases by increasing the gas content and processing temperature. The gas content showed a more significant effect than processing temperature in the ranges that were considered in this study. Acknowledgment The authors are grateful to the Consortium of Cellular and Micro-Cellular Plastics (CCMCP), AUTO21, and the Natural Sciences and Engineering Research Council of Canada (NSERC) for the financial support of this project. Nomenclature Alg ) surface area of the liquid-gas interface of a heterogeneously nucleated bubble, m2 Ahet ) surface area of nucleating agents per unit volume of polymer melt, m2/m3 C ) dissolved gas concentration, mol/m3 D ) diffusivity, m2/s F ) ratio of the volume of the nucleated bubble to the volume of a spherical bubble with the same radius, dimensionless Jhet ) heterogeneous nucleation rate (per unit area of nucleating agent), #/m2 · s Jhom ) homogeneous nucleation rate (per unit volume of polymer), #/m3 · s kB ) Boltzmann constant, m2kg/s2 · K m ) molecular mass of gas molecules, kg N ) number density of dissolved gas molecules, #/m3 Nb,unfoam ) cell density with respect to unfoamed volume, #/m3 Pbub,cr ) bubble pressure of a critical bubble, Pa Psat ) saturation pressure Psys ) system pressure, Pa ∆P ) pressure difference across the interface of gas and polymer, Pa Q ) ratio of the surface area of the nucleated bubble to the surface area of a spherical bubble with the same radius, dimensionless Rbub ) radius of curvature of a bubble, m t ) current time, s Tsys ) system temperature, K Vbub ) volume of a heterogeneously nucleated bubble, m3 Whet ) free energy barrier to heterogeneously nucleate a bubble, J Whom ) free energy barrier to homogeneously nucleate a bubble, J Greek Letters β ) semiconical angle of a heterogeneously nucleating site, rad γlg ) surface tension, N/m θc ) contact angle, rad λ ) relaxation time, s Fβ ) probability density function of β, dimensionless

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ReceiVed for reView January 17, 2008 ReVised manuscript receiVed November 28, 2008 Accepted December 8, 2008 IE800079X