Pressure drop threshold in the foaming of low-density polyethylene, polystyrene, and polypropylene using CO2 and N2 as foaming agents

J. of Supercritical Fluids 103 (2015) 38–47

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Pressure drop threshold in the foaming of low-density polyethylene, polystyrene, and polypropylene using CO2 and N2 as foaming agents Ying Sun, Yumi Ueda, Hiroyuki Suganaga, Masashi Haruki, Shin-ichi Kihara, Shigeki Takishima ∗ Department of Chemical Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashi-Hiroshima, 739-8527, Japan

a r t i c l e

i n f o

Article history: Received 22 January 2015 Received in revised form 28 April 2015 Accepted 28 April 2015 Available online 8 May 2015 Keywords: Foam polymer Pressure drop threshold Interfacial tension Critical bubble radius High-pressure gas

a b s t r a c t Bubble nucleation is a key step in the polymer foaming process using physical agents. Understanding bubble nucleation is vital in order to predict the foam structure based on process conditions. In this work, the influence of sample properties and operating conditions on the pressure drop threshold, Pthreshold , which is the difference between the saturation pressure and the ambient pressure at the onset of bubble nucleation, was studied for polypropylene (PP), low-density polyethylene (LDPE), and polystyrene (PS) using CO2 and N2 as foaming agents. As a result, the Pthreshold was a linear function of the interfacial tension regardless of the foaming gas used for each polymer. Moreover, by correlating the experimental results of bubble number density with the Blander–Katz bubble nucleation rate equation, the pressure inside a critical bubble, PG,cr , was studied. The pressure difference between the inside and outside a critical bubble strongly depended on both the interfacial tension and the temperature. According to these relationships, the bubble nucleation rate could be estimated using interfacial tension for all the polymer/gas systems examined. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In recent years, microcellular plastics (MCP) [1] have been produced by using either CO2 or N2 as a physical blowing agent in order to reduce the amount of polymers used in mass produced items and to improve product performance. MCP are unique polymer foams that are characterized by bubble radii that range from 0.1 to 10 ␮m, number densities of bubbles that range from 109 to 1015 cm−3 , and specific density reductions that range from 5 to 98% [1–3]. In many cases, MCP exhibit high impact resistance, a high stiffness/weight ratio, a low thermal conductivity, and a low dielectric constant compared with conventional polymer foams [4]. Martini et al. [1] has succeeded in generating a large amount of fine bubbles by rapidly reducing the pressure and by heating CO2 -containing polymers in order to promote an unstable thermodynamic state. Many theoretical and experimental studies have been done and summarized by some reviews [2,5–9]. The foam structure and resultant functionality of polymer foams depends mostly on the physical properties of the polymers and on the foaming operation conditions. A fundamental understanding

∗ Corresponding author. Tel.: +81 82 424 7713; fax: +81 82 424 7713. E-mail address: [email protected] (S. Takishima). http://dx.doi.org/10.1016/j.supflu.2015.04.027 0896-8446/© 2015 Elsevier B.V. All rights reserved.

of this dependence is quite important in the manufacture of highquality MCP. Using only experimentation to demonstrate these principles is expensive and time-consuming, which makes simulation of the foaming process a necessity. Studies on bubble nucleation and growth in the foaming process have developed many numerical models. Most models of bubble nucleation have been based on the classical nucleation equation established by Blander and Katz [10,11] wherein the bubble nucleation rate per unit volume is related to the minimum work required to form stable bubbles. On the other hand, most models of bubble growth have been based on three fundamental equations: the momentum balance across the bubble interface, the diffusion of dissolved gas molecules in a polymer, and the overall mass balance of the gas in bubbles and the polymer [12,13]. Bubble growth can be quantitatively predicted using accurate properties of the polymer/gas mixtures such as solubility, diffusion coefficient, viscosity, and interfacial tension. However, in bubble nucleation models, one or more parameters have been used to adjust the pressure drop threshold (Pthreshold , the difference between the saturation pressure and the ambient pressure at the onset of bubble nucleation) and/or the bubble nucleation rate. The Pthreshold greatly affects the subsequent bubble nucleation rate as well as the growth rate of premature bubbles. However, only minimal effort has been made to study the Pthreshold . Leung et al. [14] studied the number

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47 Table 1 Characteristics of the polymers used.

39

Table 2 Surface tension parameters of pure polymer.

Polymer

Mw [kg/mol]

Tm [◦ C]

Tc [◦ C]

Tg [◦ C]

Tref [◦ C]

Polymer

A [N/(m·K)]

B [N/m]

Data source

LDPE PS PP

220 329 190

110 – 165.5

95 – 130

−78 100 −20

95 100 130

LDPE PS PP

−6.0000 × 10−5 −7.7317 × 10−5 −5.6120 × 10−5

5.1289 × 10−2 6.1955 × 10−2 4.5851 × 10−2

[17]

density of bubbles and the Pthreshold through experimental results of polystyrene/CO2 system. They found that increasing either the gas content or the processing temperature led to a decrease in the Pthreshold , while the pressure release rate did not have a significant effect on the Pthreshold . Moreover, they also used the Blander–Katz equation to estimate the Pthreshold . However, some parameters in the Blander–Katz equation were determined by fitting the experimental results. In the present study, batch foaming experiments of polypropylene (PP), low-density polyethylene (LDPE), and polystyrene (PS) were performed using CO2 and N2 as foaming agents. From the experimental results, the effects of physical properties and operating conditions (saturation temperature, saturation pressure, and pressure release rate) on the Pthreshold were studied. Moreover, estimations of the bubble nucleation rate were examined using the Blander–Katz equation. 2. Experimental 2.1. Materials PP, LDPE, and PS were supplied by Chisso Co., Mitsui Chemicals Co., and PS Japan Co., respectively. The characteristics of the polymers are listed in Table 1. The polymers were molded into sheets and dried under vacuum at 80 ◦ C for approximately five days. Nitrogen (>99.8% purity) and carbon dioxide (>99.5% purity) were obtained from Iwatani Industrial Gas Co. These gases were used as received.

then sealed and evacuated for one hour at 80 ◦ C. The polymer sample was then saturated with a high-pressure gas at a desired temperature and pressure for 3 h. After saturation dissolution was established, the pressure inside the high-pressure cell was released at a prescribed rate to induce bubble nucleation. The images of bubble nucleation were captured through the CCD camera and were recorded along with time, pressure and temperature. The images were analyzed using image analysis software (Mitani Ltd. WinROOF ver. 7.4), and the number density of bubbles per unit polymer volume, NB , was calculated in 0.1 s intervals. In this work, the crystallization temperature for PP and LDPE, and the glass transition temperature for PS were chosen as the reference temperature, Tref , as shown in Table 1. The experiments were performed at saturation temperatures that were 50, 70, and 90 ◦ C higher than Tref , at saturation pressures of 10, 15, and 20 MPa, and at pressure release rates of 0.75 and 1.6 MPa/s. The experiments were repeated more than three times in most of experimental conditions. The scattering in Pthreshold for each condition was less than 0.5 MPa. 3. Theory Based on the classical nucleation theory of Blander–Katz [11], the homogeneous nucleation rate can be expressed as follows.

 Js = CNA



2.2. Foaming apparatus and experimental procedure Fig. 1 shows a schematic diagram of the visual observation apparatus for batch physical foaming used in this work. The apparatus was mainly composed of a section for high-pressure gas supply, a high-pressure cell for batch physical foaming, a visual observation system, and a controlling system for the pressure release rate. The apparatus could be used at pressures as high as 30 MPa and at temperatures as high as 250 ◦ C. A gas injector (Pressure Equipment Industry Co. Ltd., max 30 MPa) was used to introduce gas into the high-pressure cell. The pressure was measured with a pressure gauge (General Electric Co., DPI282). The high-pressure cell for batch physical foaming, in which a polymer sample was located, was surrounded by an aluminum block with four cartridge heaters and a thermal insulator that was comprised of glass wool, as shown in the expanded view. To observe the foaming process, two sapphire windows were installed on the top and bottom of the high-pressure cell. A polymer sample and a C-shaped spacer were compressed between the two sapphire glasses to restrict the sample expansion only in the radial direction. A valve controller (JASCO Co., Ltd., 880-81) and a function generator (Hokuto Denko, HB-111) were used to release the gas inside the high-pressure cell at a constant pressure release rate. The images of the polymer sample were captured using a CCD camera (ELMO, CE421) with a microscope (ELMO, EM-141S). For the foaming experiment, a polymer sample that was 0.6 mm thick and 4.5 mm in diameter and the C-shape spacer that was 0.5 mm thick were first placed between the two sapphire windows inside the high-pressure cell. The high-pressure cell was

NB =

PL



16 3 2 exp −  2 mG 3kB T PG,cr − PL

 dt JS

Psat

dP

dP

 (1)

(2)

where, Js is the bubble nucleation rate per unit polymer volume, NB is the calculated number density of bubbles, C is the molar concentration of dissolved gas, NA is Avogadro’s number,  is the interfacial tension, mG is the molecular mass of dissolved gas, kB is Boltzmann’s constant, T is absolute temperature, PG,cr is the pressure inside a critical bubble, and PL is the ambient pressure. The pressure difference, PG,cr − PL , is the driving force of nucleation. The interfacial tension of gas-polymer phases at a saturation state was calculated by Goel’s equation [15,16], as follows:

 = polymer

mix polymer

4



1 − wgas

4

(3)

where,  polymer is the surface tension of a pure polymer, mix and polymer are the densities of the gas-containing polymer and the pure polymer, respectively, wgas is the weight fraction of the gas in the polymer phase. With the method of least squares and Brandrup’s experimental data [17],  polymer were expressed as a linear function of temperature, T [K], as polymer = AT + B

(4)

The values of A and B are given in Table 2. The solubility of gas, wgas , and the mixture density, mix , were calculated using the Sanchez–Lacombe equation of state (S–L EOS)

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Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47

Fig. 1. Schematic diagram of the foaming apparatus: (1) gas injector; (2) buffer tank; (3) vacuum pump; (4) CCD camera; (5) high-pressure cell; (6) monitor; (7) video recorder; (8) cold light; (9) mirror; (10) sapphire window; (11) C-shaped spacer; (12) polymer sample; (13) cell body; (14) thermal insulator; (15) aluminum block; (16) cartridge heater; (17) function generator; (18) valve controller; (19) vent; (20) back pressure regulator.

[18,19]. The S–L EOS was based on the lattice-fluid theory and is expressed below: P˜ = − ˜ 2 − T˜





ln (1 − ) ˜ + 1−

1 r

 ˜

i

T ∗ = P∗

i =



Pi∗ Pj∗

(7)

i

(8)

where, Ti * , Pi * , i * , and ri 0 refer to the characteristic parameters of a pure component i, Mi is the molar mass of component i, kij is a binary interaction parameter between i and j molecules, i 0 and i are segment fractions of component i, and wi is the mass fraction of component i. The pure component parameters used are given in Table 3. The binary parameter k12 had been determined as a function of temperature by minimizing the absolute average deviation between the experimental and the calculated saturated solubility, and the values are listed in Table 4. The molar concentration of dissolved gas was calculated as follows: C=

 0 i

(9)

ri0

i

(12)

wj /j∗

j

 i T ∗ Pi∗

wi /i∗



(6)

j

i

1 = r



i j 1 − kij

(11)

j Pj∗ /Tj∗

j

(5)

where, T* , P* , and * are the characteristic temperature, the characteristic pressure, and the characteristic density, respectively, r is the number of segments in a molecule, and M is the molar mass. For a mixture, these parameters were calculated using the following mixing rules:



i Pi∗ /Ti∗





P  MP ∗ T ˜ = ∗, r = T˜ = ∗ , P˜ = ∗ ,  T P  RT ∗ ∗

P∗ =

i0 =

w 1 i = M Mi

(10)

i

wgas mix Mgas

(13)

The density of pure polymers cannot be calculated accurately using S–L EOS. Therefore, the density of the pure polymer was calculated using the Tait equation, as follows: V (P, T ) = V (0, T )





1 − 0.089 ln 1 +

P B (T )



(14)

Table 3 Characteristic parameters of S–L EOS. Substance

P* [MPa]

* [kg/m3 ]

T* [K]

Parameter source

N2

103.6

803.4

159

[20]

CO2

720.3

1580

208 .9 + 0.459T–7.56 × 10−4 T 2 T in K

[21]

LDPE

349.4

886.1

679

[22]

PS

387

1108

739 .9

[20]

PP

300.7

885.6

690 .6

[23]

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47 Table 4 Binary interaction parameters of S–L EOS. System

k12 (T in K)

Data source

LDPE/CO2 PS/CO2 PP/CO2 LDPE/N2 PS/N2 PP/N2

0.365–1.250 × 10−3 T 0.289–9.800 × 10−4 T 0.433–1.545 × 10−3 T 0.379–4.500 × 10−4 T 0.410–5.000 × 10−4 T 1.136–2.000 × 10−3 T

[24] [25] [26] [27] [28] [27]

pure,Tait =

1 V (P, T )

4.1. Pressure drop threshold

where, V(P, T) is the specific volume of the pure polymer at pressure P and temperature T. V(0, P) and B(T) are functions of the temperature, as shown in Table 5. The values for mixture density that appear in Eqs. (3) and (13) were then calculated using both the Tait and the S–L equations, as follows.



(16)

The solubility of the gases in the polymers is shown in Fig. 2. The solubility of CO2 in the polymers was much higher than that of N2 . When saturation temperature increased, the solubility decreased for CO2 systems and increased for N2 systems, as shown in Fig. 2(a). The solubility of both the gases in polymers was in the order of PP > LDPE > PS, as shown in Fig. 2(b). The interfacial tension of the polymer/gas systems at a saturation state is plotted in Fig. 3. The interfacial tension decreased gradually as temperature and pressure were increased in N2 systems. However, for the CO2 systems, the interfacial tension decreased greatly as saturation pressure was increased due to the high solubility of CO2 in polymers, while the effect of saturation temperature on the interfacial tension was small. The order of interfacial tension was PS > LDPE > PP for both gases, as shown in Fig. 3(b). In the Blander–Katz equation, because the pressure inside a critical bubble, PG,cr , cannot be measured via experiments, the saturation pressure, Psat , has alternatively been used in some of previous studies. However, PG,cr should be greater than Psat in a very small critical bubble due to the interfacial tension, as described by the Laplace equation. PG − PL =

2 R

the bubbles via the Blander–Katz equation for each experiment. The influence of physical properties and operating conditions on PG,cr was then investigated. 4. Results and discussion

(15)

mix = pure,Tait + mix,S−L − pure,S−L

(17)

In the present study, PG,cr was first determined so as to accurately correlate the experimental results of the number density of

Fig. 4 shows a series of foaming process images for the LDPE/CO2 system at a saturation temperature of Tref + 50 ◦ C, a saturation pressure of 10 MPa, and a pressure release rate of 0.75 MPa/s. The ambient pressure is shown under each image. In this figure, the ambient pressure was reduced from the saturation pressure, Psat = 10.19 MPa (Fig. 4(a)). A small number of bubbles could be observed at an ambient pressure of 6.27 MPa, as shown in Fig. 4(c), and the number of bubbles was gradually increased with a reduction in ambient pressure. After reaching a maximum nucleation rate at 6.15 MPa (Fig. 4(d)), bubble nucleation had almost stopped at 6.07 MPa (Fig. 4(e)). After that, generated bubbles continued to grow and bubble coalescence was observed, as shown in Fig. 4(f)–(i). By analyzing these images, values for the number density of bubbles and the radius of each bubble were obtained. Corresponding to the experiment shown in Fig. 4, the change in the number density of bubbles for the LDPE/CO2 system with ambient pressure is shown in Fig. 5. To determine the ambient pressure at the onset of bubble nucleation, PL * , we used the intersection of the horizontal axis and a tangent line of the number density of bubbles at the maximum nucleation rate, as shown in Fig. 5. The experimental results of the pressure drop thresholds, Pthreshold = Psat − PL * , are shown in Fig. 6. The Pthreshold was decreased with increases in the saturation pressure, Psat , from 10 to 20 MPa. As shown in Fig. 2, a higher saturation pressure led to a higher solubility of the gases in the polymers. Moreover, as shown in Fig. 3, with the solubility of gases in polymers increasing, the interfacial tension was decreased, which led to a reduction in the energy barrier of bubble nucleation. Hence, these results explain why the Pthreshold was decreased with increase in the saturation pressure, as shown in Fig. 6. However, because the solubility of N2 in the polymers was increased and interfacial tension in the polymer/N2 systems was decreased with increases in temperature, as shown in Figs. 2 and 3, the Pthreshold decreased as saturation temperature increased in the polymer/N2 systems. However, in the polymer/CO2 systems, the Pthreshold was almost independent of the temperature, because the effect of temperature on the interfacial tension was small. Moreover, the Pthreshold in the 200

(a)

Solubility [g-gas/kg-polymer]

Solubility [g-gas/kg-polymer]

200

150

CO2 100

N2

50

0

41

10

20

Saturation pressure, Psat [MPa]

30

(b) 150

100

CO2

50

0

N2 10

20

30

Saturation pressure, Psat [MPa]

Fig. 2. Solubility of CO2 and N2 in (a) LDPE at saturation temperature of Tref + 50 ◦ C (solid line), Tref + 70 ◦ C (dashed line) and Tref + 90 ◦ C (dotted line); and in (b) PP (solid line), LDPE (dashed line) and PS (dotted line) at saturation temperature of Tref + 70 ◦ C.

42

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47

Table 5 Tait equation parameters. V(0, T) [cm3 /g] (t in ◦ C)

Polymer

−4

1.1484 exp(6.950 × 10 t) 0.9287 exp(5.131 × 10−4 t) 1.1606 exp (6.7 × 10−4 t)

LDPE PS PP

(a) 30

Interfacial tension, [mN/m]

Interfacial tension, [mN/m]

Parameter source

192.9 exp(−4.701 × 10−3 t) 216.9 exp(−3.319 × 10−3 t) 149.1 exp (−4.177 × 10−3 t)

[29] [30] [29]

40

40

N2

20

10

0

B(T) [MPa] (t in ◦ C)

CO2

10

20

30

Saturation pressure, Psat [MPa]

(b) 30

20

N2

10

CO2

0

10

20

30

Saturation pressure,Psat [MPa]

Fig. 3. Interfacial tension for (a) LDPE/CO2 systems at saturation temperature of Tref +50 ◦ C (solid line), Tref + 70 ◦ C (dashed line) and Tref + 90 ◦ C (dotted line); and for (b) PS/N2 and PS/CO2 (solid line), LDPE/N2 and LDPE/CO2 (dashed line) and PP/N2 and PP/CO2 (dotted line) at saturation temperature of Tref + 70 ◦ C.

Fig. 4. A series of foaming process images for the LDPE/CO2 system at a saturation temperature of Tref + 50 ◦ C, a saturation pressure of 10 MPa, and a pressure release rate of 0.75 MPa/s: (a) PL = Psat = 10.19 MPa, saturation pressure; (b) PL = 6.56 MPa; (c) PL = 6.27 MPa, nucleation start; (d) PL = 6.15 MPa, maximum nucleation rate; (e) PL = 6.07 MPa, nucleation stop; (f) PL = 6.05 MPa; (g) PL = 6.00 MPa; (h) PL = 5.95 MPa; (i) PL = 5.89 MPa, bubble coalescence.

43

12

500

LDPE PS 400

∆Pthreshold [MPa]

Number density of bubbles, NB[1/mm3]

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47

(e)

300

(d)

200

100

(a) 0 11

(b)

10

8

6

4

(c) 2 410

10

9

8

7

6

5

4

3

2

Fig. 5. Number density of bubbles as a function of the ambient pressure for the LDPE/CO2 system at a saturation temperature of Tref + 50 ◦ C, a saturation pressure of 10 MPa, and a pressure release rate of 0.75 MPa/s: (a)–(e) are corresponding to those shown in Fig. 4.

polymer/N2 systems was higher than that in the polymer/CO2 systems, as shown in Fig. 6. This was due to the lower solubility of the gas, and, hence, the higher interfacial tension in the polymer/N2 systems led to a higher energy barrier for bubble nucleation. Fig. 7

430

440

450

460

470

Fig. 7. Effect of the pressure release rate on the pressure drop thresholds, Pthreshold , for PS/N2 and LDPE/N2 systems at a saturation pressure of 10 MPa.

shows the Pthreshold for the different pressure release rates in the LDPE/N2 and PS/N2 systems. In this figure, the Pthreshold remained approximately the same by raising the pressure release rate from 0.75 MPa/s to 1.6 MPa/s at the same saturation temperature and pressure in both systems. The effects of process conditions on the Pthreshold were similar for all systems examined in the present

10

10

(a)

LDPE N2 6

4

2

CO2 10

15

20

Saturation pressure, Psat [MPa]

(b)

PS

∆Pthreshold [MPa]

8

∆Pthreshold [MPa]

420

Saturation temperature[°C]

Ambient pressure, PL[MPa]

0

0.75 MPa/s 1.60 MPa/s

8

N2 6

4

CO2

2

0

10

15

20

Saturation pressure, Psat [MPa]

∆Pthreshold [MPa]

10

(c)

PP 8

6

N2

4

2

CO2 0

10

15

20

Saturation pressure, Psat [MPa] Fig. 6. Effect of the saturation pressure and temperature on the pressure drop thresholds (Pthreshold ) for (a) LDPE, (b) PS, and (c) PP using CO2 (open symbols) and N2 (closed symbols) as foaming gases at saturation temperature Tref + 50 ◦ C (circle), Tref + 70 ◦ C (triangle) and Tref + 90 ◦ C (rectangle).

44

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47

Fig. 8. Relationship between the pressure drop thresholds, Pthreshold , and the interfacial tension at a saturation state.

work and also similar to those for the PS/CO2 system examined by Leung et al. [14]. From these experimental results, the interfacial tension seemed to have a great effect on the Pthreshold . The relationship of the Pthreshold and interfacial tension in a saturation state was plotted, as shown in Fig. 8. In this figure, the Pthreshold increased almost linearly with the interfacial tension for each polymer regardless of the type of gas. The straight lines could be expressed as follows: for PP/gas systems Pthreshold = Psat − PL∗ = 0.379 − 1.877

(18)

for LDPE/gas systems Pthreshold = Psat − PL∗ = 0.475 − 5.202

(19)

for PS/gas systems Pthreshold = Psat − PL∗ = 0.549 − 8.759

(20)

where, the Pthreshold and  are in the units of MPa and mN/m, respectively.

ambient pressure using the pressure inside a critical bubble, PG,cr , as an adjustable parameter for each experiment. As an example, the correlated results of the number density of bubbles for the LDPE/CO2 and PS/N2 systems are shown in Fig. 9. In this figure, the correlated results agreed well with the experimental results as far as the intermediate stage of the foaming process. In the final stage, however, the correlated lines increased to infinity with decreases in ambient pressure, while the experimental results reached finite values. This was because neither the growth nor the coalescence of the bubbles was considered in this correlation. In fact, a portion of the dissolved gas was consumed by the growth of the bubbles, which resulted in the termination of bubble nucleation. In order to improve this disagreement, a simultaneous simulation of bubble nucleation and growth should be considered in future studies. The differences between the pressure inside a critical bubble and the ambient pressure at the onset of bubble nucleation, PG,cr − PL * , are shown in Fig. 10. PG,cr − PL * decreased with increases in saturation pressure, Psat , from 10 to 20 MPa. As shown in Fig. 3, the interfacial tension was decreased as the solubility of gases in the polymers increased, which should have led to the reduction of PG,cr − PL * , as described by the Laplace equation. On the other hand, because interfacial tension of the polymer/N2 systems was decreased with increasing temperature, as shown in Fig. 3, PG,cr − PL * decreased with increasing saturation temperature in the polymer/N2 systems. However, in the polymer/CO2 systems, PG,cr − PL * was almost independent of temperature, because the effect of temperature on interfacial tension was small. Moreover, in the polymer/N2 systems, PG,cr − PL * was higher than that in the polymer/CO2 systems, as shown in Fig. 10, because of the higher interfacial tension of the polymer/N2 systems. As described above, PG,cr − PL * strongly depended on interfacial tension, . Therefore, the relationship between PG,cr − PL * and  was also investigated. Fig. 11 shows the relationship between (PG,cr − PL * )2 × T and  3 based on the exponential term of the Blander–Katz equation. In this figure, (PG,cr − PL * )2 × T increased linearly with  3 regardless of the polymer/gas systems. The straight line could be expressed as follows: 3 (PG,cr − PL ∗ ) T

The Blander–Katz equation was fitted to the experimental results of the number density of bubbles as a function of

where, PG,cr − PL * , T and  are in the units of MPa, K, and mN/m, respectively. Eq. (21) denotes the inside of the exponential term of

2

2000

(a)

LDPE/CO2

1500

1000

500

0 25

20

15

10

5

Ambient pressure, PL [MPa]

0

Number density of bubbles, NB [1/mm3]

Number density of bubbles, NB [1/mm3]

4.2. Correlation of the bubble nucleation by the Blander–Katz equation

= 0.0503

(21)

5000

(b)

PS/N2

4000

3000

2000

1000

0 25

20

15

10

5

0

Ambient pressure, PL [MPa]

Fig. 9. Comparison of experimental and correlation results for the number density of bubbles for (a) LDPE/CO2 and (b) PS/N2 systems. Plots are experiments and lines are correlations by the Blander–Katz equation at saturation temperature of Tref + 50 ◦ C (circle, solid line), Tref + 70 ◦ C (triangle, dashed line) and Tref + 90 ◦ C (rectangle, dotted line).

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47

40

45

30

N2

25 20 15

CO2

10 10

12

35

N2

30 25 20

CO2

15 14

16

18

20

Saturation pressure, Psat [MPa]

(b)

PS

40

PG,cr-PL* [MPa]

PG,cr-PL* [MPa]

(a)

LDPE

35

5

45

10

10

15

20

Saturation pressure, Psat [MPa]

PG,cr-PL* [MPa]

30

(c)

PP

25 20

N2

15 10

CO2

5 0

10

12

14

16

18

20

Saturation pressure, Psat [MPa] Fig. 10. Effect of the saturation pressure and temperature on the differences between the pressure inside a critical bubble and the ambient pressure at the onset of bubble nucleation, PG,cr − PL * , for (a) LDPE, (b) PS, and (c) PP using CO2 (open symbols) and N2 (closed symbols) as foaming gases at saturation temperature of Tref + 50 ◦ C (circle), Tref + 70 ◦ C (triangle) and Tref + 90 ◦ C (rectangle).

the Blander–Katz equations was almost a constant value of −62 at the onset of bubble nucleation. According to Eq. (21) as well as Eqs. (18)–(20), the pressure inside a critical bubble, PG,cr , can be estimated using interfacial tension, , in a saturation state. Moreover, by substituting Eqs.

(18)–(21) in the Blander–Katz equation, the change in the bubble nucleation rate with ambient pressure reduction can be estimated. Fig. 12 compares the estimated and experimental number density of bubbles for the six polymer/gas systems. In this figure, the estimated results agreed well with the experimental results with the exception of the final stage of the foaming process.

4.3. Critical bubble radius The critical bubble radius, Rcr , is the minimum bubble radius that will sustain stable growth. In the present study, the critical bubble radius at the onset of bubble nucleation, Rcr * , was calculated by the Laplace equation. PG,cr − PL∗ =

Fig. 11. Relationship between (PG,cr − PL * )2 × T and  3 for PP (circle), LDPE (triangle), and PS (rectangle) using CO2 (open symbols), and N2 (closed symbols) as foaming gases.

2 ∗ Rcr

(22)

The relationship between Rcr * and interfacial tension at saturation state, , are plotted in Fig. 13 for all the polymer/gas systems. In this figure, the critical bubble radii that were obtained from experimental PG,cr and PL * are plotted, while the solid line denotes the estimates using Eqs. (21) and (22). Rcr * was well represented by the interfacial tension and ranged from1.7 to 3.5 nm under our experimental conditions.

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Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47

Fig. 12. Comparison of experimental and estimated results for the number density of bubbles for (a) LDPE/CO2 , (b) PS/CO2 , (c) PP/CO2 , (d) LDPE/N2 , (e) PS/N2 , and (f) PP/N2 systems. Plots are experiments and lines are estimated results by the Blander–Katz equation as well as Eqs. (18)–(22) at saturation temperature of Tref + 50 ◦ C (circle, solid line), Tref + 70 ◦ C (triangle, dashed line) and Tref + 90 ◦ C (rectangle, dotted line).

Fig. 13. The relationship of critical bubble radius, Rcr * , and (T/)0.5 . Plots are experimental results and the solid line denotes the estimated results by Eqs. (18)–(22).

5. Conclusions The effects of polymer properties and operating conditions on the bubble nucleation process were established based on the results of batch foaming experiments for polypropylene (PP), low-density polyethylene (LDPE), and polystyrene (PS) using CO2 and N2 as foaming agents. The pressure drop threshold, Pthreshold , which is the difference between the saturation pressure and the ambient pressure at the onset of bubble nucleation, was decreased with increases in saturation pressure. The Pthreshold was also decreased with increases in saturation temperature for the polymer/N2

systems, while the Pthreshold was almost independent of temperature for the polymer/CO2 systems. Moreover, because higher gas solubility, and hence a lower interfacial tension of the polymer/CO2 systems led to a lower energy barrier for bubble nucleation, the Pthreshold for the polymer/CO2 systems was smaller than that for the polymer/N2 systems. Based on these experimental results, the correlation equation for the Pthreshold was formulated as a linear function of the interfacial tension for each polymer regardless of the type of gas. Moreover, the Blander–Katz equation was fitted to the experimental results of the number density of bubbles using the pressure inside a critical bubble, PG,cr , as an adjustable parameter for each experiment. The correlated results agreed well with the experimental results as far as the intermediate stage of the foaming process. The relationship between PG,cr and  was also investigated and (PG,cr − PL * )2 × T increased linearly with  3 regardless of the polymer/gas systems. This indicated that the inside of the exponential term of the Blander–Katz equation was almost a constant value of −62 at the onset of bubble nucleation. By using the correlation equations for PG,cr and PL * with the Blander–Katz equation, the change in the bubble nucleation rate with a reduction in ambient pressure could be estimated quantitatively. Furthermore, by using the Laplace equation, the critical bubble radius at the onset of bubble nucleation, Rcr * , was also estimated within a range from 1.7 to 3.5 nm.

References [1] J. E. Martini-Vvedensky, N.P. Suh, F.A. Waldman, Microcellular closed cell foams and their method of manufacture, United States Patent 4 (473), 665 (1984). [2] V. Kumar, Microcellular polymers: novel materials for the 21st century, Cellular Polymers 12 (1993) 207–223.

Y. Sun et al. / J. of Supercritical Fluids 103 (2015) 38–47 [3] J.S. Colton, N.P. Suh, Nucleation of microcellular foam: theory and practice, Polymer Engineering and Science 27 (1987) 500–503. [4] K.A. Seeler, V. Kumar, Tension–tension fatigue of microcellular polycarbonate: initial results, J. Reinforced Plastics and Compos 12 (1993) 359–376. [5] A.K. Blendzki, O. Faruk, H. Kirschling, J. Kuhn, A. Jaszkiewicz, Microcellular polymer and composites part II. Properties of different types of microcellular materials, Polymer 52 (2007) 3–12. [6] L.J.M. Jacobs, M.F. Kemmere, J.F. Keurentjes, Sustainable polymer foaming using high pressure carbon dioxide: a review on fundamentals, processes and applications, Green Chemistry 10 (2008) 731–738. [7] M. Antunes, J.I. Velasco, Multifunctional polymer foams with carbon nanoparticles, Progress in Polymer Science 39 (2014) 486–509. [8] D. Raps, N. Hossieny, C.B. Park, V. Altstädt, Past and present developments in polymer bead foams and bead foaming technology, Polymer 56 (2015) 5–19. [9] C. Forest, P. Chaumont, P. Cassagnau, B. Swoboda, P. Sonntag, Polymer nanofoams for insulating applications prepared from CO2 foaming, Progress in Polymer Science 41 (2015) 122–145. [10] M. Blander, J.L. Katz, The thermodynamics of cluster formation in nucleation theory, J. Statistical Physics 4 (1972) 55–59. [11] M. Blander, J.L. Katz, Bubble nucleation in liquids, AICHE J. 21 (1975) 833–848. [12] P. Payvar, Mass transfer-controlled bubble-growth during rapid decompression of a liquid, International J. Heat and Mass Transfer 30 (1987) 699–706. [13] S.N. Leung, C.B. Park, D. Xu, H. Li, R.G. Fenton, Computer simulation of bubblegrowth phenomena in foaming, Industrial & Engineering Chemistry Research 45 (2006) 7823–7831. [14] S.N. Leung, A. Wong, C.B. Park, Q. Guo, Strategies to estimate the pressure drop threshold of nucleation for polystyrene foam with carbon dioxide, Industrial & Engineering Chemistry Research 48 (2009) 1921–1927. [15] S.K. Goel, E.J. Beckman, Generation of microcellular polymeric foams using supercritical carbon dioxide. I: Effect of pressure and temperature on nucleation, Polymer Engineering and Science 34 (1994) 1137–1147. [16] S.K. Goel, E.J. Beckman, Generation of microcellular polymeric foams using supercritical carbon dioxide. II: Cell growth and skin formation, Polymer Engineering and Science 34 (1994) 1148–1156. [17] J. Brandrup, E.H., Immergut, E.A., Grulke, Polymer Handbook, fourth ed., 1999, VI521–VI541. [18] I.C. Sanchez, R.H. Lacombe, Elementary molecular theory of classical fluids–pure fluids, J. Physical Chemistry 80 (1976) 2352–2362.

47

[19] I.C. Sanchez, R.H. Lacombe, Statistical thermodynamics of polymer solutions, Macromolecules 11 (1978) 1145–1156. [20] N.-H. Wang, S. Ishida, S. Takishima, H. Masuoka, Sorption measurement of nitrogen in polystyrene at high pressure and its correlation by a modified dual-sorption model (in Japanese, with English summary), Kagaku Kogaku Ronbunshu 18 (1992) 226–232. [21] N.-H. Wang, K. Hattori, S. Takishima, H. Masuoka, Measurement and prediction of vapor–liquid equilibrium ratio for solutes at infinite dilution in CO2 +poly(vinyl acetate) system at high pressures (in Japanese, with English summary), Kagaku Kogaku Ron-bunshu 17 (1991) 1138–1145. [22] P.A. Rodgers, Pressure–volume–temperature relationship for polymeric liquids: a review of equations of state and their characterictic parameters for 56 polymers, J. Applied Polymer Science 48 (1993) 1061–1080. [23] Y. Sato, M. Yurugi, T. Yamabiki, S. Takishima, H. Masuoka, Solubility of propylene in semicrystalline polypropylene, J. Applied Polymer Science 79 (2001) 1134–1143. [24] N. Sumoto, Solubility and Diffusion Coefficient Of Carbon Dioxide In Polymers Under High Pressures And Temperatures (in Japanese, B. Eng. thesis), Hiroshima University, Higashi-Hiroshima, Japan, 1999. [25] Y. Sato, T. Takikawa, S. Takishima, H. Masuoka, Solubilities and diffusion coefficients of carbon dioxide in poly(vinyl acetate) and polystyrene, J. Supercritical Fluids 19 (2001) 187–198. [26] M. Sorakubo, Solubility and Diffusion Coefficient Of Carbon Dioxide In Polymers Under High Pressures And Temperatures (in Japanese, M. Eng. thesis), Hiroshima University, Higashi-Hiroshima, Japan, 2000. [27] Y. Sato, K. Fujiwara, T. Takikawa, Sumarno, S. Takishima, H. Masuoka, Solubilities and diffusion coefficients of carbon dioxide and nitrogen in polypropylene, high-density polyethylene, and polystyrene under high pressures and temperatures, Fluid Phase Equilibria 162 (1999) 261–276. [28] K. Fujiwara, Solubility and Diffusion Coefficient Of Gas In Polymers Under High Pressures And Temperatures (in Japanese, M. Eng. thesis), Hiroshima University, Higashi-Hiroshima, Japan, 1997. [29] P. Zoller, Pressure–volume–temperature relationships of solid and molten polypropylene and poly(butene-1), J. Applied Polymer Science 23 (1979) 1057–1061. [30] A. Quach, R. Simha, Pressure–volume–temperature properties and transitions of amorphous polymers; polystyrene and poly(orthomethylstyrene), J. Applied Physics 42 (1971) 4592–4606.