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Change in the critical nucleation radius and its impact on cell stability during polymeric foaming processes
Chemical Engineering Science 64 (2009) 4899 -- 4907
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Change in the critical nucleation radius and its impact on cell stability during polymeric foaming processes Siu N. Leung a, ∗ , Anson Wong a , Qingping Guo a , Chul B. Park a , Jin H. Zong b a b
Microcellular Plastics Manufacturing Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8 Department of Materials, Mechanical and Automation Engineering, Yanbian University of Science and Technology, Yanji, Jilin, People's Republic of China
A R T I C L E
I N F O
Article history: Received 22 August 2008 Received in revised form 9 June 2009 Accepted 28 July 2009 Available online 4 August 2009 Keywords: Bubble growth and collapse Diffusion Polymer foam Simulation Stability Visualization
A B S T R A C T
The critical radius of cell nucleation is a function of the thermodynamic state that is uniquely determined by the system temperature, system pressure, and the dissolved gas concentration in the polymer/gas solution. Because these state variables change continuously during the foaming process, the critical radius varies simultaneously despite the traditional concept that it is a fixed thermodynamic property for a given initial state. According to classical nucleation theory, the critical radius determines the fate of the bubbles. Therefore, the change in the critical radius during foaming has a strong impact on the stability of foamed cells, especially in the production of microcellular or nanocellular foams. In this study, the continuous change in the critical radius is theoretically demonstrated under atmospheric pressure while bubbles are generated and expanded by the decomposition of a chemical blowing agent. The experimental results observed from the visualization cell are used to support the theoretically derived concept. Sustainability of the nucleated bubbles is also discussed by comparing the bubble size to the critical radius. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Microcellular plastics are characterized by foams with cell densities ranging from 109 to 1015 cells/cm3 , cell sizes ranging from 0.1 to 10 m, and a specific density reduction of 5–95% (Park et al., 1995). Due to their high cell density, tiny cell size, and uniform cell size distributions, these foams have superior mechanical (Matuana et al., 1998), thermal (Glicksman, 1992), and acoustical properties (Suh et al., 2000) when compared with their solid counterparts. In this context, intense research efforts have been made to develop foams with higher cell density and smaller cell size. Successful implementations of nanocellular plastics (i.e., foam with cell density higher than 1015 cells/cm3 and cell sizes less than 0.1 m) have been reported using various batch foaming techniques (Krause et al., 2002; Fujimoto et al., 2003; Li et al., 2004; Yokoyama et al., 2004). But the high cost and slow production rates associated with these foaming techniques have limited commercial applications. Therefore, large-scale production of nanocellular plastics is still technologically challenging and economically unviable. In these circumstances, various researchers have attempted to understand the fate of nano-bubbles. Ljunggren and Eriksson (1997) analyzed the lifetime of a colloid-sized gas bubble in water. Tyrrell and Attard (2001) used atomic force microscopy
∗ Corresponding author. E-mail address: [email protected] (S.N. Leung). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.07.025
to produce images of nano-bubbles at hydrophobic surfaces in water. Taleyarkhan et al. (2002) investigated the intensive implosive collapse of gas and vapor bubbles experimentally. Despite the valuable insights on bubble growth and collapse processes offered by past researchers (Street, 1968; Fogler and Goddard, 1970; Venerus et al., 1998), only a limited number of publications have focused on identifying the fate and stability of bubbles in polymeric foaming processes. Guo et al. (2005) studied the effects of high-pressure gas on the lifespan of a CBA-blown bubble. Xu et al. (2005) investigated bubble growth and collapse phenomenon in lowdensity polyethylene (LDPE) foaming with chemical blowing agents (CBA) using computer simulation and the empirically observed data in a batch foaming process. Zhu and Park (2006) studied the stability of nano-bubbles and the notion of cell ripening in the polymeric foaming process using a computer simulation system. Their studies applied diffusion phenomena to explain the cell growth and collapse processes. At the same time, these processes can also be explained by classical nucleation theory (CNT) (Gibbs, 1961). According to CNT, a state parameter, called the critical radius (Rcr ), governs the bubble growth and collapse. Theoretically, (Rcr ) is a function of the thermodynamic state that is uniquely determined by the system pressure (Psys ), system temperature (Tsys ), and the dissolved gas concentration (C). In particular, nucleated bubbles larger than Rcr grow spontaneously, whereas those smaller than Rcr collapse. Tucker and Ward (1975) verified this concept experimentally and examined the stability of bubbles in an under-saturated water–oxygen solution.
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However, no study has used the critical radius concept to explain bubble growth and collapse processes in polymeric foaming. It was previously believed that Rcr is constant during a polymeric foaming process. However, Psys or C or even both change continuously and effect a continuous change in Rcr . As Rcr governs a nucleated cell's stability, it is interesting to study its evolution during the plastic foaming process and to examine the sustainability of its bubbles under different conditions. In our previous work, a numerical simulation system to model the bubble growth process in polymeric foaming was developed and verified using in situ plastic-foaming visualization data (Leung et al., 2006). As an extension of the investigations conducted by Guo et al. (2005) and Xu et al. (2005), in the current study, we have modified our numerical simulation system to examine bubble growth and collapse processes in LDPE foaming with a CBA. Our goal is to elucidate the mechanisms that govern the following processes: bubble growth and collapse behaviors during plastic foaming; the relationship between the dynamic change of Rcr and the fate of the generated bubble; and the dependences of bubble lifespan on various thermo-physical parameters. These processes were captured in situ in batch foaming experiments to qualitatively compare them with those obtained in the numerically simulated results. 2. Theoretical framework In this study, bubble growth and collapse phenomena were theoretically investigated in a CBA-based, pressure-free foaming process. The continuous change of Rcr was simulated to demonstrate its role in governing the sustainability of nucleated bubbles. In general, a large number of bubbles grew in close proximity to each other in the polymer/gas solution during plastic foaming. Our previous work (Leung et al., 2006) verified that the well-known cell model (Amon and Denson, 1986) is appropriate for the simulation of bubble expansion in a physical blowing-agent-based foaming process. To simulate bubble expansion and collapse behaviors of CBA-blown bubbles, we modified the cell model by adjusting its mathematical models and simulation algorithm. Instead of simulating an isothermal process, the modified simulation program examines temperature increases during the heating process and their effect on the various thermophysical parameters. 2.1. Cell model Based on the cell model, Amon and Denson (1986) studied the diffusion-induced growth of a gas bubble surrounded by a thin film of Newtonian liquid to simulate cases where a large number of bubbles grow in close proximity during foaming. Polymer melts, however, are viscoelastic in nature. Therefore, instead of considering that bubbles grow in a Newtonian fluid, we adopted the strategy used in our previous work (Leung et al., 2006) and assumed that each bubble was surrounded by a shell of a viscoelastic plastic fluid with a finite volume. A schematic diagram of the cell model is illustrated in Fig. 1. In Fig. 1, Patm is the atmospheric pressure; Pbub (t, t ) is the bubble pressure at time t for the bubble nucleated at time t; Rbub (t, t ) is the bubble radius at time t for the bubble nucleated at time t; Rshell (t, t ) is the radius of the corresponding polymer/gas solution shell; C(r, t ,t) is the dissolved gas concentration at the radial position r; and CR (t, t) is the dissolved gas concentration at the bubble surface. To implement the cell model in the simulation algorithm of the CBA-based bubble growth and collapse processes, the following assumptions were made: (1) The bubble is spherically symmetric throughout the bubble growth and collapse processes. (2) The polymer/gas solution is incompressible.
Fig. 1. A schematic diagram of the cell model.
(3) The inertial forces are negligible. (4) The accumulation of adsorbed gas molecules on the bubble surface is negligible. (5) The gas inside the bubble obeys the ideal gas law throughout the bubble growth and collapse processes. (6) The polymer/gas solution is a weak solution. (7) The dissolved gas concentration at the polymer/gas interface can be related to the Pbub using Henry's law: CR (t, t ) = KH Pbub (t, t )
(1)
where KH is Henry's law constant. (8) After the system has been heated up to the set point temperature, the bubble growth and collapse processes are isothermal. (9) The effects of gravity on the bubble growth and collapse processes are negligible. (10) The initial accumulated stress around the growing bubble is zero. (11) The initial bubble volume is the same as the volume of the decomposed CBA particle. 2.2. Mathematical formulations The bubble expansion and shrinking mechanisms can be described by a standard group of governing equations that include the following: (i) the momentum equation; (ii) the mass balance equation over the bubble; and (iii) the gas diffusion equation in the surrounding polymer melt. Using Assumption 3 and considering the surrounding pressure to be Patm , the dynamics of the aforementioned system are governed by the conservation of momentum in the radial direction. The corresponding momentum equation can be written as (Arefmanesh and Advani, 1991) Pbub − Patm −
2lg Rbub
+2
Rshell
Rbub
rr − r
dr = 0
(2)
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Laplace equation: Rcr =
where lg is the surface tension at the polymer/gas interface; and rr and are the stress components in the r and directions, respectively, of the spherical coordinates. The integration term in Eq. (2) can be evaluated by considering the quasi-linear, upperconvected Maxwell model with the Lagrangian coordinate transformation (Arefmanesh and Advani, 1991). Using Assumption 4, the change rate of gas content inside the expanding/collapsing bubble must balance with both the rate at which gas diffuses in and out of the bubble and the rate of gas generated through CBA decomposition. In this way, with the conditions of Assumption 5, the mass conservation equation can be obtained: 4 d 3 dt
Pbub R3bub Rg Tsys
2 *C = 4R D *r
+ r=Rbub
dngen dt
(3)
where Rg is the universal gas constant; ngen is the number of moles of gas being generated; and D is the gas diffusivity in the polymer. With a knowledge of the concentration gradient at the bubble surface, Eq. (3) can be solved with Eq. (2) to obtain Pbub and Rbub at a particular instant. Therefore, it is necessary to determine the concentration profile around the gas bubble. The diffusion equation can be written as (Arefmanesh and Advani, 1991) follows: •
*C dngen R2bub Rbub *C D *c 2 *C + = + r dt r2 *t *r r 2 *r *r
for r ⱖ R
(4)
Eq. (4) can be solved by imposing the following boundary and initial conditions: For r > Rbub ,
C(r, 0) = xKH Patm
(8)
where Pbub,cr is the pressure inside the critical bubble. Because a critical bubble is in an unstable equilibrium state with its surrounding polymer/gas solution, Pbub,cr can be determined by considering the equilibrium between the chemical potential of the gas inside the bubble (i.e., g ) and that of the gas in the polymer/gas solution outside the bubble (i.e., g,sol ). Using Assumptions 5 and 6, g and g,sol can be expressed as Eqs. (9) and (10), respectively (Ward and Tucker, 1975). P g (T, Pbub,cr ) = g (T, Psys ) + kT ln bub,cr (9) Psys
Fig. 2. Change in free energy vs. bubble radius.
2lg Pbub,cr − Psys
(5)
For t > 0,
C(Rshell , t) = KH Patm
(6)
For t ⱖ 0,
C(Rbub , t) = KH Pbub
(7)
where x is the degree of gas saturation in the polymer melt. Eqs. (2)–(7) constitute a complete set of equations that describes bubble growth and collapse behaviors in the polymer/gas solution. 2.3. Determination of critical radius According to classical nucleation theory (Gibbs, 1961), and as shown in Fig. 2, a critical bubble is in an unstable equilibrium state with its surroundings. Fig. 2 illustrates how nucleated cells larger than the critical radius (Rcr ) grow spontaneously, whereas those smaller than Rcr collapse. Theoretically, Rcr is a function of the thermodynamic state that is uniquely determined by the system pressure (Psys ), the system temperature (Tsys ), and the dissolved gas concentration (C). Past studies showed that Rcr can be estimated by the
g,sol (T, Psys , CR ) = g (T, Psys ) + kT ln
CR Csat
(10)
where Psys is the pressure inside the polymer/gas solution and Csat is the saturated gas concentration corresponding to Psys . As a result, the chemical equilibrium condition (i.e., g = g,sol ) can be solved to determine Pbub,cr : Pbub,cr = Psys
CR CR = Csat KH
(11)
Using Eq. (11), Eq. (8) can be rewritten as follows: Rcr =
2lg CR − Psys KH
(12)
Eq. (12) can be employed to determine the continuously changing Rcr during bubble expansion and collapse processes. The sizes of Rcr at different times can be compared with Rbub to investigate the relationship between Rcr and the sustainability of nucleated bubbles under different conditions. 3. Implementation of a computer simulation 3.1. Numerical simulation algorithm Due to the nonlinearity and coupling of the governing equations that describe bubble growth and collapse behaviors, an analytical solution was not possible. Hence, a numerical simulation algorithm, which integrates the explicit finite difference scheme and the fourthorder Runge–Kutta method, was employed to solve Eqs. (2)–(4), and thereby simulated the expanding and shrinking phenomena of the formed bubble. The simulation involved two major numerical difficulties: (i) a moving boundary and (ii) a large concentration gradient at the bubble–polymer interface. The free-moving boundaries for the governing equations were immobilized by the Lagrangian coordinate transformation (Arefmanesh and Advani, 1991). A variable mesh, which had grid points clustered near the interface, was used to overcome the numerical challenges caused by the steep concentration gradient at the interface. The convergence of the finite difference scheme was verified by simulating bubble growth and collapse profiles using a different number of mesh points. It was found that the simulated bubble-growth profiles converged when 100 or more mesh points were used. As a result, 100 mesh points were used throughout this study to simulate the bubble growth and collapse phenomena. In CBA-based plastic foaming, bubble generation is initiated by heating the system to a temperature that is higher than the CBA's decomposition temperature (Tdecomposition ). To reflect this heating process, the simulation program was modified to take into account the
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Fig. 4. TGA curve of Celogen䉸 OT at heating rates of 10 and 20 ◦ C/min.
Table 3 Numerical values of physical properties of LDPE and N2 system at 160–190 ◦ C (Arvanitoyannis et al., 1998; Lee and Flumerfelt, 1995, 1996; Mitsoulis et al., 2003; Pop-Iliev et al., 2007). Physical properties 2
D (m /s) KH (mol/m3 Pa) lg (N/m) 0 (N s/m2 ) (s)
Fig. 3. The overall simulation algorithm (Leung et al., 2006). Table 1 Properties of LDPE (Nova Chemicals, 2004/2005). Properties of LDPE
LDPE (LC0522A)
Melt index (g/10 min) Density (g/cm3 ) Melting temperature (◦ C)
4.50 0.922 110
Table 2 Properties of Celogen䉸 OT (Chemtura Corporation, 2004). Properties of CBA
Celogen䉸 OT
Gas, yield (cm3 /g) Specific gravity Decomposition temperature (◦ C)
N2 , 125 1.55 158–160
ramping up of temperature in the experiment's initial phase and the changes in the thermo-physical properties of the polymer/gas system caused by the temperature increase. The overall numerical simulation algorithm is illustrated in Fig. 3. 3.2. Materials and physical parameters The polymer and CBA considered in this study were low-density polyethylene (LDPE) supplied by Nova Chemicals (i.e., Novapol䉸 LC0522A) and Celogen䉸 OT supplied by Chemtura Corporation, respectively. Tables 1 and 2 show a summary of the material properties of the LDPE and CBA. A thermogravimetric analysis was done to study the decomposition behavior of Celogen䉸 OT using a thermogravimetric analyzer
Value 3.10×10−9 –6.04×10−9 4.18×10−5 –5.13×10−5 0.026–0.028 1431.99–2450.0 0.00634–0.00909
(TGA) (TA Instruments Q50). About 10 mg of pure CBA sample was loaded into the TGA and heated from 30 to 100 ◦ C at heating rates of 10 and 20 ◦ C/min. A stream of N2 was used for purging. The sample weight was recorded as a function of temperature. Fig. 4 illustrates the results. It can be observed that the onset temperature of CBA decomposition was measured as 155–160 ◦ C. Moreover, the observed decomposition rate of Celogen䉸 OT was extremely high regardless of the heating rate. Therefore, in the computer simulation, the CBA particle was assumed to decompose and generate the respective amount of gas instantaneously. Because the majority of gas released by decomposing Celogen䉸 OT was N2 , all the physical constants were estimated based on the system of LDPE and N2 . The diffusivity (D) and Henry's law constant (KH ) for N2 in LDPE were estimated based on the data and charts presented by Arvanitoyannis et al. (1998) and Lee and Flumerfelt (1995). The surface tension (lg ) values for LDPE melts were obtained from Lee and Flumerfelt (1996). The zero-shear viscosity (0 ) and the relaxation time () of LDPE melts were measured by our group and Mitsoulis et al. (2003). Table 3 provides a summary of the physical constants employed in this research. 3.3. Initial conditions For the decomposition of CBA particles, it was assumed that the amount of gas being released was in proportion to the CBA mass. For a single (i.e., 3 m×3 m×3 m) Celogen䉸 OT particle, the number of moles of N2 being generated, which were calculated from the specific gravity (i.e., 1.55) and the gas yield (i.e., 125 cm3 /g: 91% N2 and 9% H2 O) (Nova Chemicals, 2004/2005), was found to be 2.335×10−13 mol. Moreover, the average particle size of Celogen䉸 OT was about 3 m (Xu et al., 2005). By assuming that the volume of the initial bubble was equal to the free volume generated by the decomposing CBA particle, that the CBA particle had a cubic shape, and that the generated bubble was spherical and occupied the same volume of the CBA particle, the initial Rbub generated by a 3 m×3 m×3 m
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Fig. 6. Simulated lifespan of a CBA-generated bubble at various degrees of saturation (x).
Fig. 5. A schematic diagram of the experimental setup.
CBA particle was estimated to be 1.86 m. Consequently, based on Assumption 5, the initial Pbub was determined. At the beginning of each experiment, the LDPE sample was equilibrated at 150 ◦ C and would be saturated with N2 in the atmosphere under Patm . During the rapid temperature increase, the solubility of N2 in LDPE became higher, resulting in an under-saturation of N2 . When the system temperature was increased to 190 ◦ C, the degree of saturation was about 80%. Finally, when the CBA decomposed, the generation of N2 might have led to saturation or over-saturation of N2 in the LDPE melt. It was impossible, however, to determine precisely the degrees of saturation of N2 in the LDPE melt. Consequently, the various degrees of saturation—(x), 80%, 100%, 105%, and 110%—were considered in the computer simulation to cover different possible scenarios.
4. Experimental verification In order to verify the theory describing bubble growth and collapse phenomena, the computer simulation results were compared with the experimental data of bubble growth and collapse as observed using a hot-stage optical microscope-based image processing system. Fig. 5 illustrates the schematic diagram of the experimental setup. The hot-stage (Linkam HFS 91) with a precision temperature controller (Linkam TP 93) was used to heat the system temperature to the desired level at a controlled rate. An optical system, which consists of a high speed CMOS camera coupled with a high magnification zoom lens and an optic fiber transmissive light source, was installed to allow for bright-field observation and video recording of the plastic sample during the foaming process.
4.1. Sample preparation Film samples of LDPE with 0.25 and 0.50 wt% Celogen䉸 OT were prepared using a compression molding machine equipped with a digital temperature controller (Fred S. Carver Inc.). LDPE powders were dry-blended with the specific amount of Celogen䉸 OT powders. The mixture was then molded into a 500 m thick film by using a hot press, which was pre-heated to a temperature above the LDPE's melting point and below the CBA's decomposition temperature.
4.2. Experimental procedure In the experiments, the sample was first heated up and equilibrated at 150 ◦ C on the hot stage. Then, the system temperature was rapidly ramped up to 190 ◦ C to initiate bubble generation. Experiments were conducted at two CBA contents (i.e., 0.25 and 0.50 wt%). 5. Results and discussion 5.1. Computer simulation Fig. 6 shows simulated bubble growth and/or collapse behaviors under different degrees of saturation. The results indicate that the bulk gas concentration, which was affected by the CBA content and changed with time, influenced the maximum bubble size and the bubble lifespan. When x was low (e.g., 80% and 100%), the simulated lifespan was extremely short (i.e., < 1 s). The rapid dissolution of these small gas bubbles would quickly lead to the saturation or oversaturation (i.e., increased x) of N2 in the LDPE melt, especially in the regions where the CBA particles were dispersed densely. Once the degree of saturation was high enough (e.g., x = 110%), the CBAblown bubbles would sustain and grow. Figs. 7(a)–(f) illustrate the proposed mechanisms of bubble growth and collapse in CBA-based foaming processes. Initially, during the heating process, the CBA decomposes, and the gas generated increases the dissolved gas content in the polymer matrix. Once the dissolved gas content is sufficiently high, the subsequent CBA decomposition will then form a bubble that can be sustained and grow. Due to its small size, this newly generated bubble has a high internal pressure; therefore, a high concentration gradient will develop around it and cause a rapid diffusion of gas from the bubble to its surroundings. As a result, a thin gas-rich layer forms around the bubble. Meanwhile, the high pressure inside the bubble will lead to rapid hydrodynamic-controlled bubble growth. When the bubble grows larger, its internal pressure reduces. Therefore, the gas concentration at the bubble surface will decrease dramatically. At that moment, the gas-rich layer surrounding the bubble will become the source of gas sustaining its continuous expansion. Finally, due to the gas loss from the sample surface to its surroundings, the bubble will shrink. To elucidate the relationship between the Rcr and bubble growth and collapse behaviors, the actual bubble radii (Rbub ) and Rcr were compared at different times. Fig. 8 shows the evolution of Rcr during bubble growth and collapse processes when the LDPE melt was oversaturated with N2 at a saturation level of 110%. The simulated result shows that Rcr was infinitely large initially because the LDPE melt was fully saturated with N2 . As T increased
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Fig. 7. The proposed mechanism of bubble growth and collapse in CBA-induced foaming is as follows: (a) heating, (b) bubble generation, (c) bubble expansion, (d) maximum bubble growth, (e) bubble collapse, and (f) bubble disappearance.
increase due to the gas depletion around the bubble while it grew and the gas was lost to its surroundings. Finally, when Rcr became larger than Rbub (i.e., 6 s), the bubble started to collapse. As the bubble was collapsing, its internal pressure increased and led to the reduction of Rcr . Nevertheless, as Rcr remained larger than Rbub , the bubble continued to shrink and finally disappeared. This result can also be used to explain the pre-nucleation phase of CBA-based plastic foaming suggested by Pop-Iliev et al. (2007). Within the region where the CBA particles were dispersed densely, many tiny bubbles nucleated as CBA decomposed. Nevertheless, because these cells were extremely small, the time for these bubbles to collapse was very short. Eventually, only a small number of bubbles could survive, which helps to explain why the number of survived bubbles were much less than the number of CBA particles. Fig. 8. Simulated bubble size (Rbub ) and critical radius (Rcr ) (x = 110%).
and the CBA decomposed, a large amount of N2 was generated within a small volume, which dramatically increased the N2 concentration at the polymer/gas interface. Hence, the Rcr reduced rapidly and allowed for the generation of the bubble. After that, Rcr started to
5.2. Computer simulation vs. experimental simulation Figs. 9(a) and (b) show a series of micrographs taken in situ during the CBA-based foaming, with the addition of 0.25 and 0.50 wt% CBA, respectively. A bubble, which is indicated by an arrow, has been chosen from each set of experiments to demonstrate its growth and collapse phenomena. Based on observation, a typical curve describing
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Fig. 9. Bubble growth and collapse phenomena with different CBA contents: (a) 0.25 wt% Celogen䉸 OT and (b) 0.50 wt% Celogen䉸 OT.
Fig. 10. Simulated vs. experimentally observed lifespan of bubbles.
Fig. 11. Effect of diffusivity (D) on bubble's sustainability.
the lifespan of a bubble had two stages: bubble growth and bubble collapse. The bubble growth process was both hydrodynamic- and diffusion-controlled. The hydrodynamic-controlled bubble growth was predominant at the onset of the process, while the diffusioncontrolled process sustained the subsequent growth of the nucleated bubble. As gas was continuously lost to the surroundings through the sample surface, the bubble eventually shrank and completely dissolved into the polymer matrix. Fig. 10 shows how the computer-simulated bubble growth and collapse phenomenon and the experimental simulations both follow a similar trend and have good qualitative agreement. This confirms the validity of the computer simulation model and the theory that supports it. On the other hand, a quantitative discrepancy between the computer simulated results and those observed experimentally is noted. There are many possible reasons for the differences between the two sets of results. First, although the heating rate employed in the experiment was 1.5 ◦ C/s, the actual heating rate of the LDPE sample might have been lower due to the low thermo-conductivity of polymer. To reflect the possible effect of the lower heating rate on bubble lifespan, a simulation was run at a heating rate of 0.5 ◦ C/s for comparison. Fig. 10 illustrates the result. Second, the bubble-to-bubble interactions in the experiments were not considered in the computer simulations. Third, the initial thickness of the shell of LDPE melt around the bubble being studied in the computer simulation was estimated
based on half of the sample thickness (i.e., 500 m/2 = 250 m). However, when a bubble was formed in the LDPE film sample during the experiment, only its top and bottom surfaces experienced a similar situation, while its side should have extended a much longer distance from the surroundings (i.e., it would have had a longer diffusion path). Therefore, the computer-simulated results would have overestimated the gas depletion rate and, thereby underestimated the bubble lifespan. Nevertheless, the good qualitative agreement between the two sets of results give a good indication that the theory provides a realistic justification of the mechanisms governing bubble growth and collapse behaviors. 5.3. Effect of diffusivity on the sustainability of a bubble Fig. 11 shows the effect of the gas diffusivity in polymer on bubble growth and collapse behaviors. It was observed that the maximum Rbub increased while the bubble lifespan decreased with an increase in the gas diffusivity (i.e., from D to 1.5∗D). With a higher diffusivity, more gas would be accumulated in the gas-rich region around the bubble, and would thereby increase the bubble growth rate in its initial phase. Because faster bubble growth caused a higher gas depletion rate and the promoted gas loss to the surroundings, Rcr rapidly increased. The result was a more rapid bubble collapse rate and an earlier collapse of the generated bubble.
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Fig. 12. Effect of surface tension (lg ) on bubble's sustainability.
Fig. 14. Effect of viscosity on a bubble's sustainability.
Fig. 15. Effect of elasticity on a bubble's sustainability.
Fig. 13. Effect of solubility on a bubble's sustainability.
5.6. Effect of viscosity and elasticity on the sustainability of a bubble
5.4. Effect of surface tension on the sustainability of a bubble Fig. 12 illustrates the effects of surface tension on the sustainability of the CBA-generated bubble. Both the maximum Rbub and the bubble lifespan increased with lower surface tension (i.e., lg to 0.5∗lg ). With a reduction in surface tension, the retarding force on bubble growth would be lower. According to Eq. (2), this would lead to a decrease in Pbub , which means that the gas content at the bubble–polymer interface would be lower. Consequently, during the initial phase of bubble expansion, the concentration gradient between the gas-rich region and the bubble surface was higher. This resulted in a faster diffusion of gas from the gas-rich region into the bubble and a higher bubble expansion rate.
5.5. Effect of solubility on the sustainability of a bubble Fig. 13 shows how the effect of gas solubility in the polymer melt affects the fate of the CBA-blown bubble. The simulation results indicated that higher gas solubility (i.e., KH to 1.5∗KH ) would lead to a larger maximum Rbub and a longer bubble lifespan. With an increase in gas solubility, a larger amount of gas would accumulate in the gas-rich region around the bubble. Therefore, this richer supply of gas would fuel the expansion of the bubble and sustain it longer.
The effects of melt viscosity (i.e., 0 to 2∗0 ) and elasticity (i.e., to 0.5∗) on the sustainability of the generated bubble are illustrated in Figs. 14 and 15. According to Eq. (2), both melt viscosity and elasticity (i.e., measured by ) would influence bubble growth and collapse dynamics. However, the simulation results suggested that the effects of these rheological parameters on the bubble's sustainability were negligible within the ranges of values considered in this study. 6. Conclusions A series of computer simulations for bubble growth and collapse dynamics have demonstrated the continuous change of the critical radius during plastic foaming processes and their relationship to the fate of the generated bubble. The overall patterns of bubble growth and collapse phenomena during various stages have been shown by both the theoretical and experimental results. It is believed that when CBA decomposes, a gas-rich region around the newly formed bubble will develop. This gas-rich region contributes to the bubble's expansion during the initial phase of its life cycle. Meanwhile, the continuous gas loss to the surroundings and the reduction of Pbub will lead to the increase of Rcr . Finally, when Rcr becomes larger than Rbub , the bubble starts to collapse by dissolving its gas into the polymer. Furthermore, the computer simulation results suggested that diffusivity, solubility, and surface tension are important parameters governing the fate of the generated bubble. It is believed that a lower diffusivity, a higher solubility, and a lower surface tension will
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enhance the sustainability of bubbles formed in CBA-based, pressurefree foaming processes. Notation C CR D KH ngen Patm Pbub Psys r Rbub Rcr Rg Rshell t t Tsys x
dissolved gas concentration, mol/m3 dissolved gas concentration at the bubble surface, mol/m3 diffusivity, m2 /s Henry's law constant, mol Pa/m3 number of moles of gas being generated, mol atmospheric pressure, Pa bubble pressure, Pa system pressure, Pa radial position, m bubble radius, m critical radius, m universal gas constant, J/K mol shell radius, m current time, s time at which the bubble forms, s system temperature, ◦ C degree of gas saturation, dimensionless
Greek letters
lg rr
surface tension at the polymer/gas interface, N/m stress component in the r direction, Pa stress component in the direction, Pa
Acknowledgments The authors of this study gratefully acknowledge the financial support of AUTO21, Consortium of Cellular and Micro-Cellular Plastics (CCMCP), and NSERC. References Amon, M., Denson, C.D., 1986. A study of the dynamics of foam growth: simplified analysis and experimental results for bulk density in structural foam molding. Polymer Engineering and Science 26, 255–267. Arefmanesh, A., Advani, S.G., 1991. Diffusion-induced growth of a gas bubble in a viscoelastic fluid. Rheologica Acta 30, 274–283. Arvanitoyannis, I., Biliaderis, C.G., Ogawa, H., Kawasaki, N., 1998. Biodegradable films made from low-density polyethylene (LDPE) rice starch and potato starch for food packaging applications: Part 1. Carbohydrate Polymer 36, 89–104. Chemtura Corporation, 2004. Celogen䉸 OT for plastic data sheet rec., March 25;
Celogen䉸 OT MSDS, Rev. 1.0, July 22, 8pp. www.chemtura.com. Fogler, H.S., Goddard, J.D., 1970. Collapse of spherical cavities in viscoelastic fluids. Physics of Fluids 13, 1135–1141. Fujimoto, Y., Ray, S.S., Okamoto, M., 2003. Well-controlled biodegradable nanocomposite foams: from microcellular to nanocellular. Macromolecular Rapid Communications 24, 457–461.
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Gibbs, J.W., 1961. The Scientific Papers of J. Willard Gibbs, vol. 1. Dover, New York. Glicksman, L., 1992. Notes from MIT Summer Program 4.10S Foams and Cellular Materials: Thermal and Mechanical Properties, Cambridge, MA, June 29–July 1, 1992. Guo, Q., Xu, D., Chang, S.S.Y., Wang, J., Park, C.B., 2005. Study of CBA blown bubble nucleation and life span under high pressure gas. SPE, ANTEC, Technical Papers, Paper #102029, Boston, MA, May 1–4. Krause, B., Diekmann, K., van der Vegt, N.F.A., Wessling, M., 2002. Open nanoporous morphologies from polymeric blends by carbon dioxide foaming. Macromolecules 35, 1738–1745. Lee, J.G., Flumerfelt, R.W., 1995. Nitrogen solubilities in low-density polyethylene at high temperatures and high pressures. Journal of Applied Polymer Science 58, 2213–2219. Lee, J.G., Flumerfelt, R.W., 1996. A refined approach to bubble nucleation and polymer foaming process: dissolved gas and cluster size effects. Journal of Colloid and Interface Science 184, 335–348. Leung, S.N., Park, C.B., Xu, D., Li, H., Fenton, R.G., 2006. Computer simulation of bubble-growth phenomena in foaming. Industrial and Engineering Chemistry Research 45, 7823–7831. Li., L., Yokoyama, H., Nemoto, T., Sugiyama, K., 2004. Facile fabrication of nanocellular block copolymer thin films using supercritical carbon dioxide. Advanced Materials 16, 1226–1229. Ljunggren, S., Eriksson, J.C., 1997. The lifetime of a colloid-sized gas bubble in water and the cause of the hydrophobic attraction. Colloids and Surfaces A: Physicochemical and Engineering Aspects 129, 151–155. Matuana, L.M., Park, C.B., Balatinecz, J.J., 1998. Cell morphology and property relationships of microcellular foamed PVC/wood-fiber composites. Polymer Engineering and Science 38, 1862–1872. ¨ Mitsoulis, E., Schwetz, M., Munstedt, H., 2003. Entry flow of LDPE melts in a planar contraction. Journal of Non-Newtonian Fluid Mechanics 111, 41–61. Nova Chemicals., 2004/2005. NOVAPOL䉸 LS-0522—A LDPE extrusion coating resin
data sheet, rev. April 4, 2005. MSDS NOVAPOL䉸 polyethylene—not colored, Rev. 2.00, July 27, 2004, 9pp. www.novachem.com. Park, C.B., Bladwin, D.F., Suh, N.P., 1995. Effect of the pressure drop rate on cell nucleation in continuous processing of microcellular polymers. Polymer Engineering and Science 35, 432–440. Pop-Iliev, R., Dong, N., Xu, D., Park, C.B., 2007. Visualization of the foaming mechanism of polyethylene blown by chemical blowing agents under ambient pressure. Advances in Polymer Technology 26, 213–222. Street, J.R., 1968. The rheology of phase growth in elastic liquid. Journal of Rheology 12, 103–131. Suh, K.W., Park, C.P., Maurer, M.J., Tusim, M.H., Genova, R.D., Broos, R., Sophiea, D.P., 2000. Lightweight cellular plastics. Advanced Materials 12, 1779–1789. Taleyarkhan, R.P., West, C.D., Cho, J.S., 2002. Evidence for nuclear emissions during acoustic cavitation. Science 295, 1868–1873. Tucker, A.S., Ward, C., 1975. Critical state of bubbles in liquid–gas solutions. Journal of Applied Physics 46, 4801–4808. Tyrrell, J.W.G., Attard, P., 2001. Images of nanobubbles on hydrophobic surfaces and their interactions. Physical Review Letters 87, 176104-1–176104-4. Venerus, D.C., Yala, N., Bernstein, B., 1998. Analysis of diffusion-induced bubble growth in viscoelastic liquids. Journal of Non-Newtonian Fluid Mechanics 75, 55–75. Ward, C.A., Tucker, A.S., 1975. Thermodynamic theory of diffusion-controlled bubble growth or dissolution and experimental examination of the predictions. Journal of Applied Physics 46, 233–238. Xu., D., Pop-Iliev, R., Park, C.B., Fenton, R.G., 2005. Fundamental study of CBA-blown bubble growth and collapse under atmospheric pressure. Journal of Cellular Plastics 41, 519–538. Yokoyama, H., Li, L., Nemoto, T., Sugiyama, K., 2004. Tunable nanocellular polymeric monoliths using fluorinated block copolymer templates and supercritical carbon dioxide. Advanced Materials 16, 1542–1546. Zhu, Z., Park, C.B., 2006. Challenge to formation of nano cells in foaming. In: Society of Polymer Engineering Annual Technical Conference, Technical Papers Proceedings, Paper #103195, Charlotte, NC.
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Chemical Engineering Science journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / c e s
Change in the critical nucleation radius and its impact on cell stability during polymeric foaming processes Siu N. Leung a, ∗ , Anson Wong a , Qingping Guo a , Chul B. Park a , Jin H. Zong b a b
Microcellular Plastics Manufacturing Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario, Canada M5S 3G8 Department of Materials, Mechanical and Automation Engineering, Yanbian University of Science and Technology, Yanji, Jilin, People's Republic of China
A R T I C L E
I N F O
Article history: Received 22 August 2008 Received in revised form 9 June 2009 Accepted 28 July 2009 Available online 4 August 2009 Keywords: Bubble growth and collapse Diffusion Polymer foam Simulation Stability Visualization
A B S T R A C T
The critical radius of cell nucleation is a function of the thermodynamic state that is uniquely determined by the system temperature, system pressure, and the dissolved gas concentration in the polymer/gas solution. Because these state variables change continuously during the foaming process, the critical radius varies simultaneously despite the traditional concept that it is a fixed thermodynamic property for a given initial state. According to classical nucleation theory, the critical radius determines the fate of the bubbles. Therefore, the change in the critical radius during foaming has a strong impact on the stability of foamed cells, especially in the production of microcellular or nanocellular foams. In this study, the continuous change in the critical radius is theoretically demonstrated under atmospheric pressure while bubbles are generated and expanded by the decomposition of a chemical blowing agent. The experimental results observed from the visualization cell are used to support the theoretically derived concept. Sustainability of the nucleated bubbles is also discussed by comparing the bubble size to the critical radius. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Microcellular plastics are characterized by foams with cell densities ranging from 109 to 1015 cells/cm3 , cell sizes ranging from 0.1 to 10 m, and a specific density reduction of 5–95% (Park et al., 1995). Due to their high cell density, tiny cell size, and uniform cell size distributions, these foams have superior mechanical (Matuana et al., 1998), thermal (Glicksman, 1992), and acoustical properties (Suh et al., 2000) when compared with their solid counterparts. In this context, intense research efforts have been made to develop foams with higher cell density and smaller cell size. Successful implementations of nanocellular plastics (i.e., foam with cell density higher than 1015 cells/cm3 and cell sizes less than 0.1 m) have been reported using various batch foaming techniques (Krause et al., 2002; Fujimoto et al., 2003; Li et al., 2004; Yokoyama et al., 2004). But the high cost and slow production rates associated with these foaming techniques have limited commercial applications. Therefore, large-scale production of nanocellular plastics is still technologically challenging and economically unviable. In these circumstances, various researchers have attempted to understand the fate of nano-bubbles. Ljunggren and Eriksson (1997) analyzed the lifetime of a colloid-sized gas bubble in water. Tyrrell and Attard (2001) used atomic force microscopy
∗ Corresponding author. E-mail address: [email protected] (S.N. Leung). 0009-2509/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.07.025
to produce images of nano-bubbles at hydrophobic surfaces in water. Taleyarkhan et al. (2002) investigated the intensive implosive collapse of gas and vapor bubbles experimentally. Despite the valuable insights on bubble growth and collapse processes offered by past researchers (Street, 1968; Fogler and Goddard, 1970; Venerus et al., 1998), only a limited number of publications have focused on identifying the fate and stability of bubbles in polymeric foaming processes. Guo et al. (2005) studied the effects of high-pressure gas on the lifespan of a CBA-blown bubble. Xu et al. (2005) investigated bubble growth and collapse phenomenon in lowdensity polyethylene (LDPE) foaming with chemical blowing agents (CBA) using computer simulation and the empirically observed data in a batch foaming process. Zhu and Park (2006) studied the stability of nano-bubbles and the notion of cell ripening in the polymeric foaming process using a computer simulation system. Their studies applied diffusion phenomena to explain the cell growth and collapse processes. At the same time, these processes can also be explained by classical nucleation theory (CNT) (Gibbs, 1961). According to CNT, a state parameter, called the critical radius (Rcr ), governs the bubble growth and collapse. Theoretically, (Rcr ) is a function of the thermodynamic state that is uniquely determined by the system pressure (Psys ), system temperature (Tsys ), and the dissolved gas concentration (C). In particular, nucleated bubbles larger than Rcr grow spontaneously, whereas those smaller than Rcr collapse. Tucker and Ward (1975) verified this concept experimentally and examined the stability of bubbles in an under-saturated water–oxygen solution.
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However, no study has used the critical radius concept to explain bubble growth and collapse processes in polymeric foaming. It was previously believed that Rcr is constant during a polymeric foaming process. However, Psys or C or even both change continuously and effect a continuous change in Rcr . As Rcr governs a nucleated cell's stability, it is interesting to study its evolution during the plastic foaming process and to examine the sustainability of its bubbles under different conditions. In our previous work, a numerical simulation system to model the bubble growth process in polymeric foaming was developed and verified using in situ plastic-foaming visualization data (Leung et al., 2006). As an extension of the investigations conducted by Guo et al. (2005) and Xu et al. (2005), in the current study, we have modified our numerical simulation system to examine bubble growth and collapse processes in LDPE foaming with a CBA. Our goal is to elucidate the mechanisms that govern the following processes: bubble growth and collapse behaviors during plastic foaming; the relationship between the dynamic change of Rcr and the fate of the generated bubble; and the dependences of bubble lifespan on various thermo-physical parameters. These processes were captured in situ in batch foaming experiments to qualitatively compare them with those obtained in the numerically simulated results. 2. Theoretical framework In this study, bubble growth and collapse phenomena were theoretically investigated in a CBA-based, pressure-free foaming process. The continuous change of Rcr was simulated to demonstrate its role in governing the sustainability of nucleated bubbles. In general, a large number of bubbles grew in close proximity to each other in the polymer/gas solution during plastic foaming. Our previous work (Leung et al., 2006) verified that the well-known cell model (Amon and Denson, 1986) is appropriate for the simulation of bubble expansion in a physical blowing-agent-based foaming process. To simulate bubble expansion and collapse behaviors of CBA-blown bubbles, we modified the cell model by adjusting its mathematical models and simulation algorithm. Instead of simulating an isothermal process, the modified simulation program examines temperature increases during the heating process and their effect on the various thermophysical parameters. 2.1. Cell model Based on the cell model, Amon and Denson (1986) studied the diffusion-induced growth of a gas bubble surrounded by a thin film of Newtonian liquid to simulate cases where a large number of bubbles grow in close proximity during foaming. Polymer melts, however, are viscoelastic in nature. Therefore, instead of considering that bubbles grow in a Newtonian fluid, we adopted the strategy used in our previous work (Leung et al., 2006) and assumed that each bubble was surrounded by a shell of a viscoelastic plastic fluid with a finite volume. A schematic diagram of the cell model is illustrated in Fig. 1. In Fig. 1, Patm is the atmospheric pressure; Pbub (t, t ) is the bubble pressure at time t for the bubble nucleated at time t; Rbub (t, t ) is the bubble radius at time t for the bubble nucleated at time t; Rshell (t, t ) is the radius of the corresponding polymer/gas solution shell; C(r, t ,t) is the dissolved gas concentration at the radial position r; and CR (t, t) is the dissolved gas concentration at the bubble surface. To implement the cell model in the simulation algorithm of the CBA-based bubble growth and collapse processes, the following assumptions were made: (1) The bubble is spherically symmetric throughout the bubble growth and collapse processes. (2) The polymer/gas solution is incompressible.
Fig. 1. A schematic diagram of the cell model.
(3) The inertial forces are negligible. (4) The accumulation of adsorbed gas molecules on the bubble surface is negligible. (5) The gas inside the bubble obeys the ideal gas law throughout the bubble growth and collapse processes. (6) The polymer/gas solution is a weak solution. (7) The dissolved gas concentration at the polymer/gas interface can be related to the Pbub using Henry's law: CR (t, t ) = KH Pbub (t, t )
(1)
where KH is Henry's law constant. (8) After the system has been heated up to the set point temperature, the bubble growth and collapse processes are isothermal. (9) The effects of gravity on the bubble growth and collapse processes are negligible. (10) The initial accumulated stress around the growing bubble is zero. (11) The initial bubble volume is the same as the volume of the decomposed CBA particle. 2.2. Mathematical formulations The bubble expansion and shrinking mechanisms can be described by a standard group of governing equations that include the following: (i) the momentum equation; (ii) the mass balance equation over the bubble; and (iii) the gas diffusion equation in the surrounding polymer melt. Using Assumption 3 and considering the surrounding pressure to be Patm , the dynamics of the aforementioned system are governed by the conservation of momentum in the radial direction. The corresponding momentum equation can be written as (Arefmanesh and Advani, 1991) Pbub − Patm −
2lg Rbub
+2
Rshell
Rbub
rr − r
dr = 0
(2)
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Laplace equation: Rcr =
where lg is the surface tension at the polymer/gas interface; and rr and are the stress components in the r and directions, respectively, of the spherical coordinates. The integration term in Eq. (2) can be evaluated by considering the quasi-linear, upperconvected Maxwell model with the Lagrangian coordinate transformation (Arefmanesh and Advani, 1991). Using Assumption 4, the change rate of gas content inside the expanding/collapsing bubble must balance with both the rate at which gas diffuses in and out of the bubble and the rate of gas generated through CBA decomposition. In this way, with the conditions of Assumption 5, the mass conservation equation can be obtained: 4 d 3 dt
Pbub R3bub Rg Tsys
2 *C = 4R D *r
+ r=Rbub
dngen dt
(3)
where Rg is the universal gas constant; ngen is the number of moles of gas being generated; and D is the gas diffusivity in the polymer. With a knowledge of the concentration gradient at the bubble surface, Eq. (3) can be solved with Eq. (2) to obtain Pbub and Rbub at a particular instant. Therefore, it is necessary to determine the concentration profile around the gas bubble. The diffusion equation can be written as (Arefmanesh and Advani, 1991) follows: •
*C dngen R2bub Rbub *C D *c 2 *C + = + r dt r2 *t *r r 2 *r *r
for r ⱖ R
(4)
Eq. (4) can be solved by imposing the following boundary and initial conditions: For r > Rbub ,
C(r, 0) = xKH Patm
(8)
where Pbub,cr is the pressure inside the critical bubble. Because a critical bubble is in an unstable equilibrium state with its surrounding polymer/gas solution, Pbub,cr can be determined by considering the equilibrium between the chemical potential of the gas inside the bubble (i.e., g ) and that of the gas in the polymer/gas solution outside the bubble (i.e., g,sol ). Using Assumptions 5 and 6, g and g,sol can be expressed as Eqs. (9) and (10), respectively (Ward and Tucker, 1975). P g (T, Pbub,cr ) = g (T, Psys ) + kT ln bub,cr (9) Psys
Fig. 2. Change in free energy vs. bubble radius.
2lg Pbub,cr − Psys
(5)
For t > 0,
C(Rshell , t) = KH Patm
(6)
For t ⱖ 0,
C(Rbub , t) = KH Pbub
(7)
where x is the degree of gas saturation in the polymer melt. Eqs. (2)–(7) constitute a complete set of equations that describes bubble growth and collapse behaviors in the polymer/gas solution. 2.3. Determination of critical radius According to classical nucleation theory (Gibbs, 1961), and as shown in Fig. 2, a critical bubble is in an unstable equilibrium state with its surroundings. Fig. 2 illustrates how nucleated cells larger than the critical radius (Rcr ) grow spontaneously, whereas those smaller than Rcr collapse. Theoretically, Rcr is a function of the thermodynamic state that is uniquely determined by the system pressure (Psys ), the system temperature (Tsys ), and the dissolved gas concentration (C). Past studies showed that Rcr can be estimated by the
g,sol (T, Psys , CR ) = g (T, Psys ) + kT ln
CR Csat
(10)
where Psys is the pressure inside the polymer/gas solution and Csat is the saturated gas concentration corresponding to Psys . As a result, the chemical equilibrium condition (i.e., g = g,sol ) can be solved to determine Pbub,cr : Pbub,cr = Psys
CR CR = Csat KH
(11)
Using Eq. (11), Eq. (8) can be rewritten as follows: Rcr =
2lg CR − Psys KH
(12)
Eq. (12) can be employed to determine the continuously changing Rcr during bubble expansion and collapse processes. The sizes of Rcr at different times can be compared with Rbub to investigate the relationship between Rcr and the sustainability of nucleated bubbles under different conditions. 3. Implementation of a computer simulation 3.1. Numerical simulation algorithm Due to the nonlinearity and coupling of the governing equations that describe bubble growth and collapse behaviors, an analytical solution was not possible. Hence, a numerical simulation algorithm, which integrates the explicit finite difference scheme and the fourthorder Runge–Kutta method, was employed to solve Eqs. (2)–(4), and thereby simulated the expanding and shrinking phenomena of the formed bubble. The simulation involved two major numerical difficulties: (i) a moving boundary and (ii) a large concentration gradient at the bubble–polymer interface. The free-moving boundaries for the governing equations were immobilized by the Lagrangian coordinate transformation (Arefmanesh and Advani, 1991). A variable mesh, which had grid points clustered near the interface, was used to overcome the numerical challenges caused by the steep concentration gradient at the interface. The convergence of the finite difference scheme was verified by simulating bubble growth and collapse profiles using a different number of mesh points. It was found that the simulated bubble-growth profiles converged when 100 or more mesh points were used. As a result, 100 mesh points were used throughout this study to simulate the bubble growth and collapse phenomena. In CBA-based plastic foaming, bubble generation is initiated by heating the system to a temperature that is higher than the CBA's decomposition temperature (Tdecomposition ). To reflect this heating process, the simulation program was modified to take into account the
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Fig. 4. TGA curve of Celogen䉸 OT at heating rates of 10 and 20 ◦ C/min.
Table 3 Numerical values of physical properties of LDPE and N2 system at 160–190 ◦ C (Arvanitoyannis et al., 1998; Lee and Flumerfelt, 1995, 1996; Mitsoulis et al., 2003; Pop-Iliev et al., 2007). Physical properties 2
D (m /s) KH (mol/m3 Pa) lg (N/m) 0 (N s/m2 ) (s)
Fig. 3. The overall simulation algorithm (Leung et al., 2006). Table 1 Properties of LDPE (Nova Chemicals, 2004/2005). Properties of LDPE
LDPE (LC0522A)
Melt index (g/10 min) Density (g/cm3 ) Melting temperature (◦ C)
4.50 0.922 110
Table 2 Properties of Celogen䉸 OT (Chemtura Corporation, 2004). Properties of CBA
Celogen䉸 OT
Gas, yield (cm3 /g) Specific gravity Decomposition temperature (◦ C)
N2 , 125 1.55 158–160
ramping up of temperature in the experiment's initial phase and the changes in the thermo-physical properties of the polymer/gas system caused by the temperature increase. The overall numerical simulation algorithm is illustrated in Fig. 3. 3.2. Materials and physical parameters The polymer and CBA considered in this study were low-density polyethylene (LDPE) supplied by Nova Chemicals (i.e., Novapol䉸 LC0522A) and Celogen䉸 OT supplied by Chemtura Corporation, respectively. Tables 1 and 2 show a summary of the material properties of the LDPE and CBA. A thermogravimetric analysis was done to study the decomposition behavior of Celogen䉸 OT using a thermogravimetric analyzer
Value 3.10×10−9 –6.04×10−9 4.18×10−5 –5.13×10−5 0.026–0.028 1431.99–2450.0 0.00634–0.00909
(TGA) (TA Instruments Q50). About 10 mg of pure CBA sample was loaded into the TGA and heated from 30 to 100 ◦ C at heating rates of 10 and 20 ◦ C/min. A stream of N2 was used for purging. The sample weight was recorded as a function of temperature. Fig. 4 illustrates the results. It can be observed that the onset temperature of CBA decomposition was measured as 155–160 ◦ C. Moreover, the observed decomposition rate of Celogen䉸 OT was extremely high regardless of the heating rate. Therefore, in the computer simulation, the CBA particle was assumed to decompose and generate the respective amount of gas instantaneously. Because the majority of gas released by decomposing Celogen䉸 OT was N2 , all the physical constants were estimated based on the system of LDPE and N2 . The diffusivity (D) and Henry's law constant (KH ) for N2 in LDPE were estimated based on the data and charts presented by Arvanitoyannis et al. (1998) and Lee and Flumerfelt (1995). The surface tension (lg ) values for LDPE melts were obtained from Lee and Flumerfelt (1996). The zero-shear viscosity (0 ) and the relaxation time () of LDPE melts were measured by our group and Mitsoulis et al. (2003). Table 3 provides a summary of the physical constants employed in this research. 3.3. Initial conditions For the decomposition of CBA particles, it was assumed that the amount of gas being released was in proportion to the CBA mass. For a single (i.e., 3 m×3 m×3 m) Celogen䉸 OT particle, the number of moles of N2 being generated, which were calculated from the specific gravity (i.e., 1.55) and the gas yield (i.e., 125 cm3 /g: 91% N2 and 9% H2 O) (Nova Chemicals, 2004/2005), was found to be 2.335×10−13 mol. Moreover, the average particle size of Celogen䉸 OT was about 3 m (Xu et al., 2005). By assuming that the volume of the initial bubble was equal to the free volume generated by the decomposing CBA particle, that the CBA particle had a cubic shape, and that the generated bubble was spherical and occupied the same volume of the CBA particle, the initial Rbub generated by a 3 m×3 m×3 m
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Fig. 6. Simulated lifespan of a CBA-generated bubble at various degrees of saturation (x).
Fig. 5. A schematic diagram of the experimental setup.
CBA particle was estimated to be 1.86 m. Consequently, based on Assumption 5, the initial Pbub was determined. At the beginning of each experiment, the LDPE sample was equilibrated at 150 ◦ C and would be saturated with N2 in the atmosphere under Patm . During the rapid temperature increase, the solubility of N2 in LDPE became higher, resulting in an under-saturation of N2 . When the system temperature was increased to 190 ◦ C, the degree of saturation was about 80%. Finally, when the CBA decomposed, the generation of N2 might have led to saturation or over-saturation of N2 in the LDPE melt. It was impossible, however, to determine precisely the degrees of saturation of N2 in the LDPE melt. Consequently, the various degrees of saturation—(x), 80%, 100%, 105%, and 110%—were considered in the computer simulation to cover different possible scenarios.
4. Experimental verification In order to verify the theory describing bubble growth and collapse phenomena, the computer simulation results were compared with the experimental data of bubble growth and collapse as observed using a hot-stage optical microscope-based image processing system. Fig. 5 illustrates the schematic diagram of the experimental setup. The hot-stage (Linkam HFS 91) with a precision temperature controller (Linkam TP 93) was used to heat the system temperature to the desired level at a controlled rate. An optical system, which consists of a high speed CMOS camera coupled with a high magnification zoom lens and an optic fiber transmissive light source, was installed to allow for bright-field observation and video recording of the plastic sample during the foaming process.
4.1. Sample preparation Film samples of LDPE with 0.25 and 0.50 wt% Celogen䉸 OT were prepared using a compression molding machine equipped with a digital temperature controller (Fred S. Carver Inc.). LDPE powders were dry-blended with the specific amount of Celogen䉸 OT powders. The mixture was then molded into a 500 m thick film by using a hot press, which was pre-heated to a temperature above the LDPE's melting point and below the CBA's decomposition temperature.
4.2. Experimental procedure In the experiments, the sample was first heated up and equilibrated at 150 ◦ C on the hot stage. Then, the system temperature was rapidly ramped up to 190 ◦ C to initiate bubble generation. Experiments were conducted at two CBA contents (i.e., 0.25 and 0.50 wt%). 5. Results and discussion 5.1. Computer simulation Fig. 6 shows simulated bubble growth and/or collapse behaviors under different degrees of saturation. The results indicate that the bulk gas concentration, which was affected by the CBA content and changed with time, influenced the maximum bubble size and the bubble lifespan. When x was low (e.g., 80% and 100%), the simulated lifespan was extremely short (i.e., < 1 s). The rapid dissolution of these small gas bubbles would quickly lead to the saturation or oversaturation (i.e., increased x) of N2 in the LDPE melt, especially in the regions where the CBA particles were dispersed densely. Once the degree of saturation was high enough (e.g., x = 110%), the CBAblown bubbles would sustain and grow. Figs. 7(a)–(f) illustrate the proposed mechanisms of bubble growth and collapse in CBA-based foaming processes. Initially, during the heating process, the CBA decomposes, and the gas generated increases the dissolved gas content in the polymer matrix. Once the dissolved gas content is sufficiently high, the subsequent CBA decomposition will then form a bubble that can be sustained and grow. Due to its small size, this newly generated bubble has a high internal pressure; therefore, a high concentration gradient will develop around it and cause a rapid diffusion of gas from the bubble to its surroundings. As a result, a thin gas-rich layer forms around the bubble. Meanwhile, the high pressure inside the bubble will lead to rapid hydrodynamic-controlled bubble growth. When the bubble grows larger, its internal pressure reduces. Therefore, the gas concentration at the bubble surface will decrease dramatically. At that moment, the gas-rich layer surrounding the bubble will become the source of gas sustaining its continuous expansion. Finally, due to the gas loss from the sample surface to its surroundings, the bubble will shrink. To elucidate the relationship between the Rcr and bubble growth and collapse behaviors, the actual bubble radii (Rbub ) and Rcr were compared at different times. Fig. 8 shows the evolution of Rcr during bubble growth and collapse processes when the LDPE melt was oversaturated with N2 at a saturation level of 110%. The simulated result shows that Rcr was infinitely large initially because the LDPE melt was fully saturated with N2 . As T increased
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Fig. 7. The proposed mechanism of bubble growth and collapse in CBA-induced foaming is as follows: (a) heating, (b) bubble generation, (c) bubble expansion, (d) maximum bubble growth, (e) bubble collapse, and (f) bubble disappearance.
increase due to the gas depletion around the bubble while it grew and the gas was lost to its surroundings. Finally, when Rcr became larger than Rbub (i.e., 6 s), the bubble started to collapse. As the bubble was collapsing, its internal pressure increased and led to the reduction of Rcr . Nevertheless, as Rcr remained larger than Rbub , the bubble continued to shrink and finally disappeared. This result can also be used to explain the pre-nucleation phase of CBA-based plastic foaming suggested by Pop-Iliev et al. (2007). Within the region where the CBA particles were dispersed densely, many tiny bubbles nucleated as CBA decomposed. Nevertheless, because these cells were extremely small, the time for these bubbles to collapse was very short. Eventually, only a small number of bubbles could survive, which helps to explain why the number of survived bubbles were much less than the number of CBA particles. Fig. 8. Simulated bubble size (Rbub ) and critical radius (Rcr ) (x = 110%).
and the CBA decomposed, a large amount of N2 was generated within a small volume, which dramatically increased the N2 concentration at the polymer/gas interface. Hence, the Rcr reduced rapidly and allowed for the generation of the bubble. After that, Rcr started to
5.2. Computer simulation vs. experimental simulation Figs. 9(a) and (b) show a series of micrographs taken in situ during the CBA-based foaming, with the addition of 0.25 and 0.50 wt% CBA, respectively. A bubble, which is indicated by an arrow, has been chosen from each set of experiments to demonstrate its growth and collapse phenomena. Based on observation, a typical curve describing
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Fig. 9. Bubble growth and collapse phenomena with different CBA contents: (a) 0.25 wt% Celogen䉸 OT and (b) 0.50 wt% Celogen䉸 OT.
Fig. 10. Simulated vs. experimentally observed lifespan of bubbles.
Fig. 11. Effect of diffusivity (D) on bubble's sustainability.
the lifespan of a bubble had two stages: bubble growth and bubble collapse. The bubble growth process was both hydrodynamic- and diffusion-controlled. The hydrodynamic-controlled bubble growth was predominant at the onset of the process, while the diffusioncontrolled process sustained the subsequent growth of the nucleated bubble. As gas was continuously lost to the surroundings through the sample surface, the bubble eventually shrank and completely dissolved into the polymer matrix. Fig. 10 shows how the computer-simulated bubble growth and collapse phenomenon and the experimental simulations both follow a similar trend and have good qualitative agreement. This confirms the validity of the computer simulation model and the theory that supports it. On the other hand, a quantitative discrepancy between the computer simulated results and those observed experimentally is noted. There are many possible reasons for the differences between the two sets of results. First, although the heating rate employed in the experiment was 1.5 ◦ C/s, the actual heating rate of the LDPE sample might have been lower due to the low thermo-conductivity of polymer. To reflect the possible effect of the lower heating rate on bubble lifespan, a simulation was run at a heating rate of 0.5 ◦ C/s for comparison. Fig. 10 illustrates the result. Second, the bubble-to-bubble interactions in the experiments were not considered in the computer simulations. Third, the initial thickness of the shell of LDPE melt around the bubble being studied in the computer simulation was estimated
based on half of the sample thickness (i.e., 500 m/2 = 250 m). However, when a bubble was formed in the LDPE film sample during the experiment, only its top and bottom surfaces experienced a similar situation, while its side should have extended a much longer distance from the surroundings (i.e., it would have had a longer diffusion path). Therefore, the computer-simulated results would have overestimated the gas depletion rate and, thereby underestimated the bubble lifespan. Nevertheless, the good qualitative agreement between the two sets of results give a good indication that the theory provides a realistic justification of the mechanisms governing bubble growth and collapse behaviors. 5.3. Effect of diffusivity on the sustainability of a bubble Fig. 11 shows the effect of the gas diffusivity in polymer on bubble growth and collapse behaviors. It was observed that the maximum Rbub increased while the bubble lifespan decreased with an increase in the gas diffusivity (i.e., from D to 1.5∗D). With a higher diffusivity, more gas would be accumulated in the gas-rich region around the bubble, and would thereby increase the bubble growth rate in its initial phase. Because faster bubble growth caused a higher gas depletion rate and the promoted gas loss to the surroundings, Rcr rapidly increased. The result was a more rapid bubble collapse rate and an earlier collapse of the generated bubble.
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Fig. 12. Effect of surface tension (lg ) on bubble's sustainability.
Fig. 14. Effect of viscosity on a bubble's sustainability.
Fig. 15. Effect of elasticity on a bubble's sustainability.
Fig. 13. Effect of solubility on a bubble's sustainability.
5.6. Effect of viscosity and elasticity on the sustainability of a bubble
5.4. Effect of surface tension on the sustainability of a bubble Fig. 12 illustrates the effects of surface tension on the sustainability of the CBA-generated bubble. Both the maximum Rbub and the bubble lifespan increased with lower surface tension (i.e., lg to 0.5∗lg ). With a reduction in surface tension, the retarding force on bubble growth would be lower. According to Eq. (2), this would lead to a decrease in Pbub , which means that the gas content at the bubble–polymer interface would be lower. Consequently, during the initial phase of bubble expansion, the concentration gradient between the gas-rich region and the bubble surface was higher. This resulted in a faster diffusion of gas from the gas-rich region into the bubble and a higher bubble expansion rate.
5.5. Effect of solubility on the sustainability of a bubble Fig. 13 shows how the effect of gas solubility in the polymer melt affects the fate of the CBA-blown bubble. The simulation results indicated that higher gas solubility (i.e., KH to 1.5∗KH ) would lead to a larger maximum Rbub and a longer bubble lifespan. With an increase in gas solubility, a larger amount of gas would accumulate in the gas-rich region around the bubble. Therefore, this richer supply of gas would fuel the expansion of the bubble and sustain it longer.
The effects of melt viscosity (i.e., 0 to 2∗0 ) and elasticity (i.e., to 0.5∗) on the sustainability of the generated bubble are illustrated in Figs. 14 and 15. According to Eq. (2), both melt viscosity and elasticity (i.e., measured by ) would influence bubble growth and collapse dynamics. However, the simulation results suggested that the effects of these rheological parameters on the bubble's sustainability were negligible within the ranges of values considered in this study. 6. Conclusions A series of computer simulations for bubble growth and collapse dynamics have demonstrated the continuous change of the critical radius during plastic foaming processes and their relationship to the fate of the generated bubble. The overall patterns of bubble growth and collapse phenomena during various stages have been shown by both the theoretical and experimental results. It is believed that when CBA decomposes, a gas-rich region around the newly formed bubble will develop. This gas-rich region contributes to the bubble's expansion during the initial phase of its life cycle. Meanwhile, the continuous gas loss to the surroundings and the reduction of Pbub will lead to the increase of Rcr . Finally, when Rcr becomes larger than Rbub , the bubble starts to collapse by dissolving its gas into the polymer. Furthermore, the computer simulation results suggested that diffusivity, solubility, and surface tension are important parameters governing the fate of the generated bubble. It is believed that a lower diffusivity, a higher solubility, and a lower surface tension will
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enhance the sustainability of bubbles formed in CBA-based, pressurefree foaming processes. Notation C CR D KH ngen Patm Pbub Psys r Rbub Rcr Rg Rshell t t Tsys x
dissolved gas concentration, mol/m3 dissolved gas concentration at the bubble surface, mol/m3 diffusivity, m2 /s Henry's law constant, mol Pa/m3 number of moles of gas being generated, mol atmospheric pressure, Pa bubble pressure, Pa system pressure, Pa radial position, m bubble radius, m critical radius, m universal gas constant, J/K mol shell radius, m current time, s time at which the bubble forms, s system temperature, ◦ C degree of gas saturation, dimensionless
Greek letters
lg rr
surface tension at the polymer/gas interface, N/m stress component in the r direction, Pa stress component in the direction, Pa
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