Manufacture and model calculation of porcelain-like microcellular low-density polystyrene foam

Abstract

It is a well known that nucleation by gases in the supercritical state lead to high-density nanocellular foams, but the aim of this article was to find a route to microcellular low-density closed-cell foam with expected low heat conductivity and high mechanical properties. Polystyrene was impregnated with carbon dioxide or propane at room temperature and 70 bar pressure. Pentane served as a reference substance. The samples were investigated by electron microscopy. The specimens comprising carbon dioxide and propane showed bubbles surrounded by a ring of lamellas. The samples were foamed in hot bathes of silicone, of saturated NaNO₃ water solution, and in water vapor under stretching conditions at 100℃ or 110℃. The foaming was registered continuously on a balance by buoyancy experimentally and calculated in diffusion as well as viscosity-controlled foaming models. A microcellular foam with 35 kg/m³ density, 1.5 µm or 1500 nm cell diameter, 5.5 MPa tear strength, and 0.027 W/m.K heat conductivity was obtained, which looked like porcelain. The continuous manufacture of low-density carbon dioxide or propane foam by extrusion did not lead to microcellular foam, because the necessary high-pressure, low-temperature nucleation conditions could not be performed.

Keywords

Cell growth cell nucleation expansion ratio extrusion foaming foam density

Introduction

Large-scale cellular polystyrene (PS) is manufactured by two procedures.¹

Expandable PS beads comprising pentane are produced by dispersion polymerization and are expanded in water vapor. After storage, they are further expanded to boards or other items in water vapor. In Europe, 730,000 tonnes per year are produced.

Cellular PS boards X-PS are manufactured by extrusion through a broad slit nozzle. Blowing agents are carbon dioxide and pentane. In Europe, 210,000 tonnes per year are consumed.

Common demands on closed-cell rigid foams are the use of environmentally friendly blowing agents, low heat conductivity, low density, low manufacturing costs, and high mechanical properties.

According to ASTM D 883-80C and DIN 7726, a cellular plastic is defined as a plastic, the apparent density of which was decreased substantially by the presence of numerous cells displaced throughout the mass. Microcellular foams with 10 µm cell diameter and densities between 200 and 800 kg/m³ are published in various literatures.^2–9 Low-density open-cell PS foams with 35–50 kg/m³ are accomplished in extrusion, when mixtures of highly nucleating chlorofluorohydrocarbons, fluorohydrocarbons, and carbon dioxide are used. Though the cell sizes are as small, about 5–80 µm, the K-factors at ambient pressures are 0.5–0.46 W/m.K high.^10–12 Foams with cell sizes of 90–200 µm are obtained, when PS was extruded in the presence of hydrofluoropropenes as a substitute for HFC 134a. The numerical and experimental bubble growth during the microcellular foaming process is described for PS expanded by the blowing agent nitrogen and carbon dioxide.^13,14 The established calculations are based on a diffusion-controlled Newtonian model, a power law model, and a viscoelastic model. In the present paper, an earlier published calculation is used.¹⁵ In the literature,^16–18 the mechanism of extensional stress-induced cell formation in PS foaming is and investigated and visualized.

Nanofoams with pore diameters of less than 100 nm are well known for aero gels, metal, and carbon foams. When polyetherimide is stored at 21℃ and 10 MPa pressure, 12% CO₂ is taken up. After heating at 200℃, a 500 kg/m³ foam with 0.1–0.1 µm pore sizes is obtained.¹⁹

The task was to investigate nanosized low-density closed- cell PS foam with densities of 35 kg/m³ and to find out a route for its production.

Experimental

Materials

PS, with different molecular weights and melt viscosities, was acquired from BASF. The applied blowing agent carbon dioxide, propane, and n-pentane were bought from Haltermann. In the extrusion experiments, talcum from Quarzwerke GmbH was used as a nucleation agent. Foaming was performed in hot silicone oil, in hot saturated solutions of sodium nitrate in water, and in water vapor. All ingredients and their providers were collected in Table 1.

Table 1.

Chemicals and ingredients used.

Chemical	Producer	Molecular weight (g/mole)	Density (g/cm³)110℃; water pressure (bar) 110℃
Sodium nitrate, NaNO₃	Fluka	85	1,53 saturated water solution 0,74
Silicone oil	AK100 Wacker		1,00 0
n‐Pentane C₅H₁₂	Haltermann	72
Propane, C₃H₈	Haltermann	44
Carbon dioxide, CO₂	Haltermann	44
Polystyrene	BASF viscosity number^a	Molecular weight*10⁻³	Melt viscosity* 10⁻⁹ (110℃)
	(ml/g)	(g/mole)	(Poise)
LG 72‐303‐0	67	180	8
143E	83	240	22
168N	96	300	40
Living 190	118	400	130
EF	207	550	400

^aThe viscosity number was measured in the Ubbelohde capillary viscosimeter determining the time of effluence through the capillary for the solvent and for the solution of polystyrene in the solvent: Viscosity number = [(t solution/t solvent) − 1]/c Toluene, 25℃, c = 0.5%.

Equipment

Impregnation of PS with carbon dioxide, propane, and pentane was performed at temperatures from 20℃ to 120℃ at 1 to 100 bar pressure in an autoclave. The uptake of blowing agent was determined by weight and volume measurements.

The expansion b was dimensionless and was defined as (d_o/d_f) −1, where d_o = 1 g/cm³ was the density of the unexpanded PS and d_f was that of the foamed sample. The expansion b of the specimen, with the weight E₁, was measured continuously by buoyancy E₂ using a wire cage containing the sample. The cage was fixed on the electronic microbalance 4130 from Sartorius (Sartorius microbalances: Sartorius, Göttingen, Germany) and dipped into hot silicone oil or saturated aqueous sodium nitrate solution. E₂ was the measured negative weight of the sample in the bath caused by buoyancy E₁*[(b + 1)/d_o]*d₁. After rearrangement the following equation was established

b = (d 0 / d 0 d 1 d 1) * (E 2 / E 2 E 1 E 1) - 1

(1)

where d₁ was the density of the bath at a defined temperature. As the 110℃ hot silicone oil had the density of d_l = 1 g/cm³, equation (1) was simplified to

b = (E 2 / E 1) - 1

(2)

Measurements in silicone oil AK 100 with 1 cm³/g CO₂ solubility and in mercury without gas solubility led to the same results.

Cellular boards were manufactured using a Polypan LMP R.C.B.3/E machine from Colombo System (Extruder: Polypan, LMP/Polytal, Torino, Italy): corotating twin-screw extruder, 180 mm screw diameter, L/D = 17/1, 3030 kgm torque, rotation 8–26 r/min (LMP/POLTAL Ltd). For the extrusion of boards, a broad slit nozzle with the dimensions 5 mm × 23 mm was used.

The cell structures were determined by light, scanning electron microscopy, and stereo scan sonde. The tear strength of foamed samples was measured on a Shimadzu Autograph tear machine (Tear strength measurements Shimadzu Europe Autograph AG-X Series: Shimadzu Europe Münster, Germany). The heat conductivity was determined with the help of a Dynatech Wärmeleitfähigkeits–Meßgerät Modell K-Matic K75 (Dynatech heat conductivity: Kipp & Zonen, Delft, The Netherlands). The content of the closed cells was measured in the Beckman air pyknometer (Beckman Air Pyknometer: Beckman Instruments GmbH. Munich, Germany) according to ASTM 2856-70.

The polymer permeation analyzer, Dohrmann Div. of Envirotech Corp., provided the constants of permeation and diffusion. The diffusion constant was determined from the instationary part of permeation just after gas contact.

Experiments

Impregnation and characterization of impregnated samples

Solubility data of carbon dioxide, propane, and n-pentane in PS were available in various literatures.^20–22 The measured uptake values of carbon dioxide (CO₂) and propane (C₃H₈) were determined and documented in Figures 1 and 2. PS-143E bars were stored for 3 days under gas pressure in an autoclave. After quick withdrawal, the uptake was measured on a balance. The agreement with solubility data²² was good. The values for the depression of the glass temperature of PS was taken from various studies^23–27 and depicted in Figure 3.

Figure 1.

Carbon dioxide pressure upon PS-143E/CO₂.

Figure 2.

Propane pressure upon PS-143E/propane.

Figure 3.

Glass temperature of PS/CO₂ and critical temperature of CO₂.^23–27

In Table 2, characteristic properties of the pure blowing agents, as temperatures of vaporization T_vap, critical temperatures T_crit, and critical pressures P_crit, the heat conductivities and the heats of condensation H_K were listed, as well as properties obtained, when the blowing agents were dissolved in PS: H_L. The heat of the solution was obtained from the dependence of the vapor pressures on the temperature. The Henry constant k corresponded with the linear rise of the vapor pressures in Figures 1 and 2.

Table 2.

Data of blowing agents alone or in the presence of PS.

Property carbon dioxide propane n-pentane

T_vap (℃) −79 −42 36

T_crit (℃) 31 97 197

P_crit (bar) 74 42 34

Heat conductivity (W/m.K) 0.016 0.017 0.015

H_K (kJ/mol) 15 19 26 (heat of condensation)^21,22

Properties of blowing agents in polystyrene 143E

H_L (kJ/mol) 17 20 29 (heat of solution)

dTglass/wt% (℃) 10 12 9 (decrease in glass temperature)^23–27

k = 1/cL (bar/wt%) (110℃) 8.33 6.67 0.74 (Henry constant)²¹

H (110℃) 1.174 0.939 0.168 (H = k*M*100/(R*T*d_o)

k/H (bar/wt%) (110℃) 7.1 7.1 4.4

D_o*10⁸ (cm²/s) (23℃)¹ 8.0 – 0.008

D_o*10⁸(cm²/s) (20℃) 0.8 0.16 0.008

D_o*10⁸(cm²/s) (110℃) 6.0 4.0 1 (const. of diffusion at c = 0%)

EA (kJ/mol)¹ −34 – −52

EA (kJ/mol) −21 −48 −52

D_c*10⁸ (cm²/s) (110℃) 16.0/6% 12/4.5% 10/7% (const. of diffusion at c%)

a 0.16 0.30 0.33

P *10⁷(N cm³/(cm*s*bar) 7.0 1.0 0.6 (permeation at 110℃)

D_o (110℃) = D_o (20℃)e^−EA/RT D_c = D_o*e^acL

The dimensionless $H = k * M * 100 / (R * T * d o)$ was practicable for further calculations and comprised the Henry constant k, the molecular weight M, the ideal gas constant R = 82 cm³*bar/(grad*mole), the temperature T = 383 K, and the PS density d_o = 1 g/cm³.

D_o was the constant of diffusion without dissolved blowing agent. EA, the energy of activation for diffusion, was determined by the Arrhenius plot: $- EA = 19.15 * dlg D / d (103 / T)$ . For example, D_o (110℃) was calculated from $D o (20 \circ C) = 0.8 : lg D o (110 \circ C) = lg 0.8 + (- EA / 19.1) * d (103 / T) = 0.9031 - 1 + 0.9031 - 1 + 21 * 0.802 / 19.1 = 0.7826, D o = 6$ .

D_c, the diffusion constant at the dissolved gas concentration C, was calculated for the CO₂ concentration 6%

lg D c (100 \circ C) = lg 6 + a * cL / 2.3 = 0.7826 + 0.16 * 6 / 2.3 = 20

Characteristic data of blowing agents in PS were those such as the heat of solution H_L, which was defined as the sum of the heat of condensation and mixing, and the specific glass temperature depression dTglass/wt%. The Henry constant k was equal to the reverse solubility cL of the blowing agent in PS at atmospheric pressure.

H was dimensionless and resulted from $H = k * M * 100 / (R * T * d o)$ , where R was the gas constant, T the temperature in K, d_o the PS density, M the molecular weight of blowing agent, and p_at the atmospheric pressure. D_o was the diffusion constant in PS at 23℃ and 110℃ without dissolved blowing agent. EA was the activation energy according to $D = D o * e - EA / RT$ . The diffusion constant in PS with dissolved blowing agent D_c was calculated by $D c = D o * e acL$ , where a was a constant and cL the amount of dissolved blowing agent. The permeation constant P allowed the losses of the blowing agents to be estimated. Carbon dioxide with the highest losses had, by far, the highest P.

PS was stored in liquid n-pentane at the room temperature (RT). In the case of gaseous blowing agents carbon dioxide and propane, PS was impregnated in an autoclave at a pressure of 70 and 120 bars at temperatures of between RT and 100℃ for 3 days. The procedure was exemplified with carbon dioxide: immediately after the removal from the pressure vessel, the amount of dissolved carbon dioxide cL and the specific volume V₁ of the sample was measured. V₁ followed equation (3) and was the sum of the volumes of PS and liquid carbon dioxide

V 1 = (100 - cL / 100) * VPS + (cL / 100) * V C O 2

(3)

After storage, the samples lost their blowing agent quickly and reached the specific volume V₂ according to the following equation

V 2 = VPS + [cL / (100 - cL)] * V C O 2 = V 1 * 100 / (100 - cL)

(4)

The sample stored at RT and 7 MPa pressure was investigated with different enlargement by scattering electronic microscopy, as shown in Figure 4.

Figure 4.

Electron microscopy of PS-143E/CO₂ sample delivered from the pressure vessel after storage at RT and 7 MPa CO₂ pressure for 3 days.

The structure of the cavities was astonishing: the whole assembly consisted of a bubble surrounded by a circular ring of fanned cavities. In Figure 4, 1.5 assemblies were registered per 10 µm or 150/ mm. When the fanned cells, with a geometry of small discs with an area of r_o²*π and a height of r_o = 0.1 µm, were counted, a 10 time larger number of 15/10 µm or 1500/1 mm was determined. These figures were important for the calculation of the expected cellular structure of foams with low densities of about 30 kg/m³ according to equation (5). For instance, in the case of 150 cells per 1 mm (Z_o = 150) with a density of 100 kg/m³ according to equation (5), a cell number of 46.5 (Z_f = 36.5) resulted in a foam with a density of 30 kg/m³

Z f = (d f / d o) 1 / 3 * Z o = (30 / 1000) 1 / 3 * 150 = 46.5

(5)

In equation (5), Z_f is the expected number of cells per 1 mm of the foam. Z_o is the number of cells per 1 mm in the unexpanded specimen. For the case of Z_o = 1500/mm, a 10 times higher number of cells, Z_f = 465/mm, was determined. The specific volumes of the samples loaded by CO₂, as well as the number of cells Z_o per distance, were determined and collected in Table 3.

Table 3.

PS-143E/CO₂ storing conditions in the pressure vessel, cL amount of dissolved CO₂, specific volumes VCO₂, VPS, V₁, V₂, and cell number Z_o.

p (MPa)	T (℃)	cL (%CO₂)	VCO₂ (mL/g)	VPS (mL/g)	V₁ (cm³/g)	V₂ (cm³/g)	Z_o (1 mm)
7	20	11	1.08	0.923	0.943	1.06	1500
7	40	9	5	0.934	1.3	1.429	1000
7	50	8	5.9	0.938	1.335	1.451	500
7	60	7.5	6.7	0.947	1.379	1.491	300
7	80	6.5	8.3	0.958	1.435	1.535	200
7	100	4	9.1	0.97	1.295	1.349	150
10	120	6	6.7	0.982	1.325	1.41	10

In the foaming process, expansion had to start from an initial expansion of b_o. This starting expansion was given by the number of spherical holes multiplied by the volume of holes. For the circular ring of fanned cavities with the geometry of discs, b_o was calculated by equation (6a) and amounted to 0.011. The averaged radius of the holes was r_o and had the dimension of about 0.1 µm.

b o = Z_{o}^{3} * r o 3 π = 1.53 * 0.13 * 3.14 = 0.011

(6a)

In the reference sample of PS/pentane, only the large spherical holes with a diameter of 2r = 20 µm in wide distances of about 65 µm were detectable by electronic microscopy. Therefore, in the case of Z_o = 0.015/µm, the initial expansion started at b_o according to

b o = Z_{o}^{3} * (20 r) 3 * π / 6 = 0.0153 * 23 * 3.14 / 6 = 0.014

(6b)

Though the cell numbers were very different, the initial expansions of b_o were quite similar and about 0.01–0.015. The knowledge of b_o was important for the calculation of foaming according to the viscosity model.

Foaming in silicone oil with vapor pressure 0 bar at 110℃

The wire cage of the Sartorius balance was charged with samples of PS comprising blowing agent, which were immersed in the 110℃ hot bathes of silicone oil.

In Figure 5, the foaming in silicone oil at 110℃ was documented for PS-143E impregnated with c_o = 5% carbon dioxide, propane, and pentane. The cell numbers Z_f were determined by light microscopy and by electronic microscopy on cuts of the maximum foamed samples and are recorded in Table 4.

Figure 5.

Expansion b of PS-143E 0.1 g samples with 5% blowing agent in an oil bath at 110℃.

Table 4.

Foaming of 0.1 g PS-143E with 5% blowing agent in silicone oil with p_w=0 at 110℃.

Property	CO₂	C₃H₈	C₅H₁₂	Melt viscosity of PS at 110℃
Z_f (mm⁻¹)	700	150	4	v_o*10⁻⁹ (Poise)
Z_f ^1/6	3.0	2.3	1.3	22
b_max*Z_f^1/6/(c_o‐cL) cal/ex	7.1/3.1	7.1/7.6	4.4/4.6	22
t_maxc_o = 1%* c_o²*Z_f	22,750	3000	240	22
t_maxc_o = 1%* c_o²*Z_f			1000 168N t_max(min) = 10	40
t_maxc_o = 1%* c_o²*Z_f			200 LG t_max(min) = 2	8
t_maxc_o = 1%* c_o²*Z_f			2000 Living t_max(min) = 20	130
t_maxc_o = 1%* c_o²*Z_f			4000 EF t_max(min) = 40	400
EA (kJ/mole) t_max	–	–	9.4 (8%) ‐7.5 (4%)	22
EA (kJ/mole) t_max	150–200	117–200	90–200	22
t_max (min)	4	0.8	2.4	22
b _max	1.3	16	13	22
d_f (kg/m³)	170	53	73	22
cL (%)	0.12	0.15	1.35	22

The reduced data of Table 4 were derived from foaming experiments, such as that shown in Figure 5. Vice versa, the data of Figure 5 could be calculated from the reduced data in Table 4, under the use of equations (7) and (8): for instance b_max and t_max were calculated for the sample 143E/CO₂. In order to obtain b_max for 5% CO₂, the reduced value $b \max * Z f 1 / 6 / (c o - cL) = 3.1$ was multiplied with $(c o - cL) / Z f 1 / 6 = (5 - 0.12) / 3 = 5$ . In order to obtain for the same sample t_max, the reduced value $t \max * c o 2 * Z f = 22.750$ was divided by c_o²*Z_f = 25*700 = 1.3 min. For inter- and extrapolations, the reduced values $b \max * Z f 1 / 6 / (c o - cL)$ and $t \max * c o 2 * Z f$ were tabulated, so that the maximum expansion b_max as well as the time needed to reach the maximum expansion t_max, could be calculated for other cellular structures Z_f and amounts of blowing agents c_o

b \max = [(b \max * Z f 1 / 6) / (c o - cL)] * (c o - cL) / Z f 1 / 6

(7)

t \max = [t \max * c o 2 * Z f] / [c o 2 * Z f *]

(8)

Pentane was the best investigated blowing agent and served as reference: the influence of PS molecular weights on b_max was negligible, but on t_max the maximum foaming time as well as the reduced value increased with v_o, the melt viscosity of PS. t_max was obtained, when t_max.c_o²*Z_f = 420 was divided by c_o²*Z_f = 5²*7 = 175 then t_max = 2.4 min.

The experimental data indicated a linear relationship of t_max with the viscosity v_o

t \max * (c o 2 * Z f (\min / mm) = 23 * 10 - 9 * v o (\min) or t \max * c o 2 * Z f (s / mm) = 1380 \times 10 - 9 v o (s)

(9)

Samples with melt viscosities higher than 50 × 10⁹ expanded very slowly and did not reach b_max. The activation energy EA −7.5 kJ/mole (4% pentane) and −9.4 kJ/mole (8% pentane) of Table 4 were determined in detail in Table 5.

Table 5.

Clausius–Clapeyron treatment of b_max in dependence of temperature for PS-143E/pentane with 4% and 8% pentane.

T (℃)	1000/T (K⁻¹)	b_max4%	log b_max	b_max8%	log b_max
75	2.874	7	0.8451	10.5	1.0212
85	2.793	7.8	0.85	12.2	1.0864
95	2.717	8.7	0.9069	13.6	1.1335
105	2.646	9.4	0.9763	14.8	1.1703
115	2.577	10	1	15.8	1.1987
125	2.513	10.4	1.017	16.6	1.2201
135	2.451	10.6	1.0253	17.2	1.2355
145	2.392	10.8	1.0334	17.7	1.248
Difference	0.482		0.1883		0.2268

-EA (4% pentane) = 0.1883*19.15/0.482 = 7.5 kJ/mol.

-EA (8% pentane) = 0.2268*19.15/0.482 = 9.4 kJ/mol.

The pentane saturation in PS, with 12% −EA = 7.5*12/4 = 22.5 kJ/mol corresponded with the heat of vaporization H_v = −29 kJ/mol minus the heat or work per mole of the viscoelastic counter forces EA = 6.5 kJ/mol. The activation energies EA 90–200 kJ/mol, derived from the t_max dependence on temperature, corresponded with the activation energy of the viscosity of PS loaded with definite amounts of blowing agent of 206 kJ/mol (0% n-pentane), 132 kJ/mol (4% n-pentane), and 90 kJ/mol (8% n-pentane) in Table 4.

For the case of an ideal gas, b_max could be calculated by

b \max = [R * T * d o / (M * 100 * p at)] * (c o - cL)

(10)

When the ideal gas constant R = 82 cm³*bar/(K*mol), the temperature T = 383 K, d_o the density of the unexpanded sample with 1 g/cm³, the molecular weight M, and the atmospheric pressure p_at = 1 bar, were introduced in equation (10).

The calculated data were listed in Table 4. Experimental and calculated values of PS–C₅H₁₂ and PS–C₃H₈ were in good agreement. The bad agreement of PS–CO₂ was reasonable, when great losses of CO₂ were taken into account.

Foaming in aqueous saturated sodium nitrate with vapor pressure p_w = 0.75 bar at 110℃

In Figure 6, samples of PS-143E with 5% pentane and different cell structures were foamed in aqueous saturated sodium nitrate with a density of 1.53 g/cm³ at 110℃ and vapor pressure of 0.74 bar.

Figure 6.

Expansion b of PS-143E/pentane with 5% pentane and different cellular structure in a saturated solution of sodium nitrate in water with a density of 1.53 g/cm³ at 110℃, p_w = 0.74 bar.

In Table 6, the figures were given for the reduced values

b \max * Z f 1 / 6 (p at - p w) / [c o - cL (p at - p w)]

and

t \max * c o 2 * Z f * (p at - p w)

obtained from samples with a different cellular structure in saturated sodium nitrate solution.

Table 6.

Foaming of 0.1 g PS-143E/pentane with 5% n-pentane and different cellular structure in saturated sodium nitrate water solution at 110℃ and p_w = 0.74 bar.

Z_f (mm⁻¹)	2	5	10	15
Z_f ^1/6	1.12	1.31	1.47	1.57
$b \max * Z f 1 / 6 * (p at - p w) / [c o - cL (p at - p w)] cal / ex$	4.4/4.7
$t \max * c o 2 * Z f * (p at - p w) (min / mm)$	234
b_max cal/ex	75/78	60/65	54/58	50/54
t_max cal/ex	18/16	7.0/9.0	3.6/4.5	2.4/2.5
cL*(p_at – p_w)	0.35

The reduced values allowed the determination of the experimental data of Figure 6, for instance for PS-143E/C₅H₁₂, Z_f = 2

b \max = b \max * Z f 1 / 6 (p at - p w) / [c o - cL (p at - p w)] * [c o - cL (p at - p w)] / [Z f 1 / 6 * (p at - p w)] = 4.7 * 4.65 / (1.12 * 0.26) = 75

(11)

t \max = t \max * c o 2 * Z f * (p at - p w) / [c o 2 * Z f * (p at - p w)] = 234 / (25 * 2 * 0.26) = 18

(12)

Foaming in water vapor p_w=1 at 100℃

As experiments with the balance allowed only foaming of small specimens, larger samples were expanded in hot water vapor. In Figure 7, the same samples were foamed in water vapor at 100℃ and vapor pressure of 1 bar. In the case of PS–CO₂, the expansion curve showed a dependence on the sample size, which could be understood, especially when small samples large losses of carbon dioxide were taken into account.

Figure 7.

Expansion b of PS-143E with 5% blowing agent in water at 100℃.

In Table 7, the cellular structure was determined by light and electronic microscopies.

Table 7.

Foaming of PS-143E with 5% blowing agent in water vapor at 100℃.

Property	CO₂	CO₂	C₃H₈	C₅H₁₂
Weight (g)	20	0.1	0.1	0.1
Z_f (mm⁻¹)	500	650	100	5
Z_f ^1/6	2.8	2.9	2.1	1.3
b_max*Z_f^1/6/c_o	15.7	4.6	16.2	12.4
t_max*c_o²*Z_f	187,500	487,500	25,000	2500
b _max	28	8	38.5	47.5
t_max (min)	15	30	10	20
EA (kJ/mol) b_max				−56
EA (kJ/mol) t_max				230

Figure 8(a) shows the surface of the 35 kg/m³ PS-143E/ CO₂ foam, which was coated by vaporized coal: about five cells were counted over a distnce of 10 µm corresponding with 500 cells per mm and a cell size of 2000 nm. Figure 8(b) was taken from the same sample using a stereo scan microsonde: about seven cells were counted over a distance of 10 µm, which corresponded with a cell number of 700 and an average cell size of 1400 nm.

According to equation (11), the maximum expansion should reach infinity, when the water vapor pressure rose to 1 bar and was equal to the atmospheric pressure. Only the viscoelastic counter forces limited the expansion. In Table 7, the activation energy −56 kJ/mol derived from the temperature dependence of b_max corresponded with the sum of −16 kJ/mol of the heat of vaporization of pentane and −41 kJ/mol of the heat of vaporization of water.

Higher b_max was reached, when foaming took place in aqueous salt solutions or in water vapor. This observation was used in extrusion, where higher b_max was achieved, when the extruded foam passed a bath of aqueous salt solution or of water glycol mixtures installed just after the die.²⁸

Shrinkage and orientation

A PS-143E bar was elongated to twice its length at 130℃. Shrinkage of 100% was observed at 140℃ after 1 h. By stretching, the tear strength increased from 38 MPa to 70 MPa.

Samples with cellular structures between Z_f = 1 and 16 mm⁻¹ but the same specific volume of 54 cm³/g, which had been manufactured by foaming PS-143E with 5% pentane and different nucleation in water vapor at 100℃, were stored at 110℃ for 40 h. The amount of shrinkage was proportional with the number of cells Z_f

% shrinkage = 6 * Z f (m m - 1)

(13)

Equation (12) meant that a sample with Z_f = 16.7 shrunk to its virgin volume.

The time for 100% shrinkage obeyed

t (h) (100 % shrinkage) = 10.000 / Z f 2 (mm - 1)

(14)

A tear strength of 0.5 MPa was determined for the PS/pentane foam with 40 kg/m³ and Z_f = 5. For the PS/CO₂ foam with 35 kg/m³ and Z_f = 500, a tear strength of 5.5 MPa was measured.

A 10 times higher tear strength should be comprehensive, when stretching under orientation had occurred. A measure of orientation was given by the surface increase of the cellular sample.

The surface O was a function of b_max and the cell number Z_f. For cubic foam, the surface of one cell was 6a². When 6a² = 6 × 10²/Z_f² was multiplied by the numbers of cells in 1 g foam n = Z_f³*10³*(b_max+1)/d_o, the surface O was obtained.

As most of the cell walls belong to two cells, O was divided by 2

O (cm 2 / g) = 30 * Z f (mm - 1) * (b \max + 1) / d o

(15)

For the case that equation (14) was valid, the PS/CO₂ foam with Z_f = 700/mm had a 100 times higher surface or a triaxial orientation than the PS/pentane foam. In one direction the orientation was 33, which correlated with an 11 times higher tear strength.

The relevant data were collected in Table 10, displayed later in this article.

Extrusion

As in extrusion as well as in foaming in silicone oil, the same pressure conditions of p_at = 1 bar and p_w = 0 bar were valid, for both cases the same b_max was expected. The extrusion equipment of Colombo, together with a broad slit nozzle of the dimension Qo = 23 mm*So = 5 mm, was used. PS comprising 4.5% propane, 7% n-pentane, and 6% CO₂ was extruded under the parameters of Table 8. The conditions of extrusion in Table 7 represent those which allowed the production of foams of good appearance and low densities.

Table 8.

Conditions of extrusion.

No.	Blowing agent	Cone	Output	Barrel		Nozzle	va/ve
				temperature/pressure		temperature/pressure
		(%)	(kg/h)	(℃/bar)	(℃/bar)	(℃/bar)
1	CO₂	6	150	150/160	130/135	110/110	0.5
2	C₃H₈	4.5	150	160/110	130/45	110/30	0.6
3	C₅H₁₂	7	100	190/15	130/6.5	110/5	0.8

Propane and carbon dioxide exhibited high pressures and, therefore, the extruder had to resist pressures of up to 150 bar in order to avoid foaming in the barrel. The maximum expansion was determined from the density of the foam as well as from the profiles using the following equation

Q \max / Qo * S \max / So * L \max / Lo = b \max

(16)

where L_max/Lo correlated with va/ve, the speed of conveyor carrying the foam divided by the speed of extrusion. va/ve was kept at 0.8, 0.6, and 0.5. Pictures were taken from the profiles in Figure 8. L_max was the distance from the nozzle to the point of fully expanded foam b_max in Figure 8. The obtained cellular boards are characterized in Table 9.

Figure 8.

Electron microscopy of the cut surface of PS-143E/CO₂ with a foam density of 35 kg/m³ derived from a sample previously stored at RT under 7 MPa CO₂ pressure and foamed in 100℃ hot water vapor.

Table 9.

Cellular structure, density of extruded boards, and b_max in comparison with data obtained in 110℃ silicon.

No.	Z_f	d_f	b_max (d_f)	b_max^a (bath)	b_max cal^b	b_max cal^c	L _max Figure 9	t _max	Z_f ^1/6
	(mm⁻¹)	(kg/m³)					(m)	(s)
1	13	40	24	11	6.57.50.5 = 24.4	27	0.05	0.27	1.6
2	10	38.5	25	22	670.6 = 25.2	22	0.12	0.55	1.5
3	6	39	24.5	20	560.8 = 24.0	18	0.18	0.98	1.3

$b \max * Z f 1 / 6 / (c o - cL) * (c o - cL) / Z f 1 / 6$ .

$b \max cal = Q \max / Qo * S \max / So + L \max / Lo$ .

$b \max cal = R * T * d o (c o - cL) / (M * 100 * p at * Z f 1 / 6)$ .

Table 10.

Properties of PS foams: density d_f, maximum expansion b_max, cell structure Z, tear strength TS, surface O, heat conductivity HC, and cell wall thickness W.

No.	Blowing agent	d_f	b _max	Z_f	Z ₂₀	TS ^a	O	Heat conductivity^b	W
		(kg/m³)		(mm⁻¹)	(mm⁻¹)	(MPa)	(cm²/g)	(W/m·K)	(μm)
1	PS/CO₂ extrusion	40	24	13	10	0.5	7.650	0.034	0.95
2	PS/C₃H₈ extrusion	39	25	10	8	0.55	6.240	0.033	1.30
3	PS/C₅H₁₂ extrusion	39	25	6	5	0.4	3.750	0.034	2.10
4	PS³⁰	1000	0	0	0	38	0	0.175	‐
5	PS (100% shrinkage)	1000	0	0	0	70	6	0.175	‐
6	PS/CO₂	35	27.5	500	412	5.5	427.500	0.027	0.02
7	PS/C₃H₈	30	32	100	87	1.5	99.000	0.031	0.10
8	Styropor (BASF)	20	49	4	4	0.3	6.000	0.035	1.67
9	Roofmate (DOW)	40	24	5	4	0.5	3.750	0.030	2.65

DIN 53530.

DIN 52612.

In Table 9, the time until the foam achieved its highest expansion t_max was calculated using the following equation

t \max (s) = L \max (m) * 60 / va (m/min)

(17)

The maximum expansion b_max in the extrusion were compared with those obtained in the silicone bath in Table 9. On average, they were comparable with the exception PS-143E/CO₂ caused by loss of blowing agent.

Figure 9.

Profiles of the extruded foams comprising PS-142E + 1% talcum with 6% CO₂, 4.5% C₃H₈, and 7% C₅H₁₂ material temperature at the nozzle 110℃ (1:1 natural size, 1:0.5 half natural size).

t_max (extrusion) was not compared with t_max (silicone bath), because no agreement was expected. t_max in extrusion was determined by diffusion and t_max in silicon bath was determined by heating and diffusion.

No microcellular foam was obtained by extrusion process. Also, dies with a higher pressure drop would not have improved the results very much as suggested in Rohsenow and Choi.²⁹ There, a pressure drop of 10 MPa resulted in a cell density of 10¹⁰ cells/cm³, which corresponded with a cell number of the unexpanded foam Z_o = 100 mm⁻¹, or Z_f = 31 at a foam density of 30 kg/m³. The characteristic data of the extruded boards are collected in Table 10.

Characterization of the obtained foams

PS impregnated with carbon dioxide or propane at 7 MPa pressure at RT was foamed to densities of 35 kg/m³ in water vapor under stretching conditions. The cell structure of the cut surface of a 35 kg/m³ foam was in nanoscale and had to be investigated by electron microscopy. Measurements with the air pyknometer confirmed the closed-cell structure.

The low-density microcellular foam sample No. 6 exerted tear strength TS up to 5.5 MPa owing to the high degree of orientation, which correlated with the increase in surface O. The propane-expanded samples with lower cell numbers reached only 1.5 MPa. Nanocellular foams with density d_f of 35 kg/m³, a cell size of 2 µm, a cell wall thickness W of 0.02 µm or 20 nm, and tear strength of 5.5 MPa showed a reduced heat conductivity of 0.027 W/m.K, which could be explained by the small cell dimension of about 0.2 µm.^31–34 The restricted gas movement produced by the Knudsen effect was expected for a mean free path of 0.07 µm at ambient pressure. After foaming, the highly nucleated samples looked like porcelain with a tear strength of 25 MPa.

Model calculation

Model calculations were performed under the assumption of a cubic or spherical cellular structure and viscosity or diffusion determined expansion. The viscosity model took the melt viscosity v_o and its change by the solved blowing agent vs= 90*v_o/(cl +2)^6,5 into account but no influence of the cellular structure Z.

The importance of orientation caused by the cellular structure and expansion was considered by calculating the surface produced by the numbers of cells and by expansion. As known from experiments, the maximum expansion b_max, as well as the tear strength TS, were influenced by orientation caused by stretching: b_max was reduced by Z_f^1/6 and the tear strength T increased with increasing Z_f. In the diffusion-controlled model, the influence of melt viscosity on foaming was neglected.

Diffusion-controlled foaming

Ramesh et al.³⁵ published a numerical and experimental study of bubble growth during the microcellular foaming process. The wall thickness W of cells was calculated dependent on the expansion b and the number of cells at the foam density of 20 kg/m³ Z₂₀ for cubic cell structure in

W (μ m) = (103 / 3) * (d f / d o) * (20 / d f) 1 / 3 * (1 / Z 20) = (0.9048 / Z 20) * d f 2 / 3 = 90.48 / [(Z 20 * (b + 1) 2 / 3]

(18)

A diffusion-controlled model was set up in equation (18) using a cubic cellular structure and Fick's law for diffusion in plates, of which the graphical solution was given in Crank.³⁶

b / b \max = function D c * t / (W / 2) 2 = function D c * t / [(40.24 / [Z 20 * (b + 1) 2 / 3] 2

(19)

In detail, for chosen b/b_max values, the corresponding D_c*t/(W/2)² values were taken from the graphical solution of Fick's law. b_max was calculated using equation (10). When b_max was known, the b values were also known. W was calculated using equation (18). D_c values are taken from Table 11. The time for foaming t was determined by dividing D_c*t/(W/2)² with D_c and multiplying with (W/2)².

Table 11.

Calculation of PS/6% CO₂, PS/4.5% C₃H₈, and PS/7% C₅H₁₂ according to the diffusion model.

		6% CO₂	Z₂₀=10
b/b_max	D_c*t/(W/2)²	B	W (µm)	t (s)	D_c*10⁸ (cm²/s)
0.125	0.015	3	4.2	0.004	16
0.25	0.05	6	2.7	0.008	11
0.33	0.09	7.9	2.4	0.013	10
0.5	0.2	12	1.7	0.016	9
0.75	0.475	18	1.25	0.022	8.5
0.9	0.85	21.6	1.1	0.036	7
1	1	24	1	0.042	6
		4.5% C₃H₈	Z₂₀ = 5
0.125	0.015	3	8.6	0.023	12
0.25	0.05	6	5.5	0.040	9.5
0.33	0.09	7.9	4.4	0.052	8.4
0.5	0.2	12	3.0	0.056	8.0
0.75	0.475	18	2.4	0.092	7.4
0.9	0.85	21.6	2.3	0.206	5
1	1	24	2	0.250	4.0
		7% C₅H₁₂	Z₂₀ = 5
0.125	0.015	3.1	8.4	0.026	10
0.25	0.05	6.25	5.1	0.074	3.5
0.33	0.09	8.33	4.2	0.132	3
0.5	0.2	12.5	2.9	0.183	2.3
0.75	0.475	18.75	2.4	0.489	1.4
0.9	0.85	22.5	2.2	0.857	1.2
1	1	25	2	1.000	1

In Table 12, the time required for maximum expansion t_max was compared with those calculated using the diffusion model. The best agreement was achieved in the case of pentane.

Table 12.

Comparison of experimental data of extrusion with calculated data according to the diffusion model.

Blowing agent	Conc. (%)	Z_f (mm¹)	Z₂₀ (mm¹)	t_max ex(s)	t_max cal (s)	b_max ex (s)	b_max cal eqn (10)
CO₂	6	13	10	0.27	0.04	24	42
C₃H₈	4.5	10	8	0.55	0.25	25	31
C₅H₁₂	5	6	5	‐	2.0	14.5	12
C₅H₁₂	7	6	5	0.98	1.0	24.5	25

Viscosity-controlled foaming

For spherical foam, a formula for foaming¹⁵ was set up in equation (19). Curves of expansion were calculated for samples of PS/pentane under the influence of melt viscosity v_o and amount of blowing agent c_o. The dimensionless parameters H = k*M*100/(R*T*d_o) and the Henry constant k bar were taken from Table 2

dt = 12 * 10 - 6 * x y * v o * db / b * [(1 + H * (b - b o)) / (c o + x * (1 + H * (b - b o))] y * (1 + H * (b - b o)) / [k * co - p at * (1 + H * (b - b o)]

(20)

x and y were parameters, y = 6.5 and x = 2, and resulted from 18 divided by the specific depression of glass temperature dTglass/wt%, which was 9 K/wt% for pentane. After introducing these parameters, the following equation was obtained

dt = 1.086 * 10 - 3 * vo * db / b * [(1 + H * (b - b o) / (c o + 2 * (1 + H * (b - b o))] 6.5 + (1 + H * (b - b o)) / [k * c o - p at * (1 + H * (b - b o))]

(21)

When equation (19) was discussed by differentiation db/dt = 0, the solution for the maximum was

t \max = infinite and [(k * c o - p at * (1 + H * b \max)] / (1 + H * b \max) = 0 or b \max = 1 / H * (kco / p at - 1) = k / (H * p at) * (c o - 1 / k * p at) = (R * T * d o / M * 100) * (c o - cL)

(22)

Equation (22) was identical to equation (10).

Experiments in silicone oil derived equation 23, which was valid for foaming PS/C₅H₁₂ with PS with different melt viscosities v_o at 110℃

t \max * c o 2 * Z f = 1.38 * 10 - 6 * v o ->

(23)

Using equation (20), results of some relevance and agreement with foaming in a silicone bath were expected for small samples with low Z_f and high heating rate, such as with pentane and Z_f = 5. Expansion curves were calculated for PS-143E comprising 5%, 8%, and 10% pentane by introducing H = 0.168, k = 0.74 bar, p_at = 1 bar, v_o = 22 × 10⁹ Poise in an adjustment of equation (20). The integration was performed from b_o = 0.015 to b_max in the time 0 to t_max for 110℃ and visualized in Figure 10. The same calculation was performed for different melt viscosities: 143E v_o = 22 × 10⁹, PS 1v_o = 27 × 10⁹, and 168 N v_o = 40 × 10⁹ Poise, keeping the pentane content constant at 5% in Figure 11.

Figure 10.

Calculated expansion b of PS with the same viscosity as PS-143E comprising 5%, 8%, and 10% pentane at 110℃ according to the viscosity model with b_o = 0.015.

Figure 11.

Calculated expansion b of PS with different melt viscosities with 5% pentane according to the viscosity model with b_o = 0.015.

In Table 12, the maximum time of expansion t_max determined at extrusion was compared with the values calculated by the diffusion model. In Table 13, t_max obtained for PS-pentane samples with Z_f = 5/mm in a silicone bath was compared with calculated values according to the viscosity model.

Table 13.

Comparison experimental data of foaming PS-143E/pentane in a silicone bath with calculated data according the viscosity model.

Blowing agent	Conc.	v_o*10⁻⁹	Z_f	b_max ex		b_max cal	t_max ex	t_max cal (90%)
	(%)	(Poise)	(mm¹)	Z_f	Z_f = 1	eqn (10)	(s)	(s)b_o = 0.015
CO₂	6	22	700	6	18	42	168	‐
C₃H₈	4,5	22	150	14	33	31	60	‐
C₅H₁₂	5	22	5	14	17	16	144	150
C₅H₁₂	7	22	5	21	26	25	73	140
C₅H₁₂	8	22	5	25	31	29	56	100
C₅H₁₂	10	22	5	32	40	38	36	80
C₅H₁₂	5	27	5	17	14	16	228	185
C₅H₁₂	5	40	5	17	14	16	343	270

The increase of t_max with increasing melt viscosities was expected and was already predicted by the energy of activation for the maximum foaming time, which correlated with 220 kJ/mol, which is the activation energy of the melt viscosity as shown in Table 5.

In the experiment, b_max was dependent on Z_f^1/6. Both models included no influence of Z_f on b_max, because b_max was calculated by the ideal gas equation. In the viscosity-controlled model t_max was infinite, the only way to get figures for t_max was to determine the time at 90% b_max.

In the experiment, t_max was proportional to 1/Z_f. In the diffusion model t_max was proportional to 1/Z_f². Also, shrinkage was proportional to 1/Z_f². The increase of t_max with the viscosity of the PS was determined experimentally and was calculated by the viscosity model. In Table 13, experimental data of small samples with high heating rates in silicone oil were compared with those calculated according to the viscosity model.

Conclusion

Of special interest was the production of low-density closed-cell microcellular foam with favored properties of high mechanical properties and low thermal conductivity, which could be realized when PS was impregnated with carbon dioxide or propane at RT, and high pressures near or above the critical pressures. After removal from the pressure vessel nucleation occurred, and assemblies comprising a hole surrounded by a ring of fanned cavities were observed. During foaming, these fanned cavities grew to cells. The foaming in water vapor at 100℃ took place under stretching conditions, therefore very rigid, porcelain-like low density foams of 35 kg/m³ and tear strength of 5.5 MPa were obtained. Owing to the small cell size, the gas movement was restricted and the heat conductivity reduced to 0.027 W/m.K.

In the extrusion process, the condition for high nucleation, at a low temperature of 20℃ and high pressure of 7 MPa, could not be established, and in no case was microcellular low-density foam obtained. The extruded foams with 35 kg/m³ foam density showed a cell diameter of 100 µm, 0.5 MPa tear strength, and 0.035 W/m.K heat conductivity.

Equations were set up for the maximum expansion b_max and the foaming time t_max, until the maximum expansion was reached, for foaming in baths

t \max = t \max * c o 2 * Z f * (p at - p w) / [c o 2 * Z f * (p at - p w)]

(24)

b \max = b \max * Z f 1 / 6 (p at - p w) / [c o - cL (p at - p w)] * [c o - cL (p at - p w)] / [Z f 1 / 6 * (p at - p w)]

(25)

The maximum blowing time t_max was in reverse proportional to Z_f, but in the diffusion-controlled model t_max as well as the maximum time of shrinkage were in reverse proportional with Z_f².

In the diffusion-controlled model, b_max was calculated by the ideal gas equation and t_max was obtained by applying Fick's law for the diffusion in plates. The calculated data were in agreement with the data from the extrusion process.

When the equation for viscosity-controlled foaming was used, t_max for 90% b_max could be calculated for small samples with high heating rates. In this model, the influence of the amount of blowing agent and the influence of the melt viscosity v_o of PS on t_max was correctly reported.

Microcellular foams with desired properties could only be achieved by storing the samples at RT under high pressure followed by foaming in water vapor.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Manufacture and model calculation of porcelain-like microcellular low-density polystyrene foam

Abstract

Keywords

Introduction

Experimental

Materials

Equipment

Experiments

Impregnation and characterization of impregnated samples

Foaming in silicone oil with vapor pressure 0 bar at 110℃

Foaming in aqueous saturated sodium nitrate with vapor pressure p w = 0.75 bar at 110℃

Foaming in water vapor p w =1 at 100℃

Shrinkage and orientation

Extrusion

Characterization of the obtained foams

Model calculation

Diffusion-controlled foaming

Viscosity-controlled foaming

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

References

Foaming in aqueous saturated sodium nitrate with vapor pressure p_w = 0.75 bar at 110℃

Foaming in water vapor p_w=1 at 100℃