Investigation of material characteristics and processing conditions effects on bubble growth behavior in a physical foaming process

1 Introduction

Solid foam has a porous structure in which dispersed gaseous voids are surrounded by a dense phase such as metal, ceramic or polymer phases and can be known as a gas-solid composite. Polymeric foams are a group of foams with unique physical, thermal and mechanical properties built by a combination of two enormously naturally different phases, i.e. polymer and gas (1). Nowadays, this kind of foam is well developed for many industrial applications such as heat and sound insulation, packaging, membranes, etc.

The blowing agent is a substance producing a cellular structure via a foaming process and can be either a physical or chemical type. Foaming of a polymer by a physical foaming agent includes three stages such as implementation of a blowing agent at high pressure, expansion by a pressure release and stabilization (2). In the beginning, implementation of the blowing agent at high pressure forms a thermodynamically stable single phase pressurized polymer/gas solution. The solubility of the blowing agent in the polymer at this time directly affects the expansion ratio in the second stage, and it strongly depends on the pressure, temperature, and interaction between the blowing agent and the polymer (2). The interaction parameter can be determined experimentally or predicted by theoretical models such as Flory-Huggins equation or other advanced models like the Sanchez-Lacombe (3) or modified Simha-Somcynsky models (4).

After implementation of the blowing agent, by a sudden pressure release, the system turns into an unstable state and the gas phase tends to separate from the polymer phase through the nucleation and bubble growth process to reduce the degree of super-saturation and re-establish the equilibrium. In this stage, thermodynamics and fluid dynamics simultaneously become two fundamental principles governing the expansion process. Bubble nucleation and growth at this stage have been studied in some research theoretically (5), (6), (7), (8), (9), (10) and experimentally (11), (12), (13), (14). Although some efforts have been made to present a model for nucleation, a generally applicable and satisfactory nucleation model is not yet available because nucleation mostly takes place heterogeneously on defects and impurities (9). Consequently, most of the theoretical work has focused on bubble growth phenomena from prescribed initial conditions. Street et al. first modeled a single gas bubble grown spherically in a polymer melt media with an unlimited amount of dissolved gas (15). Some researchers developed models for bubble growth in a Newtonian liquid (16). Amon and Denson proposed a cell model in which the bubbles were surrounded by a spherical shell of fluid and expand due to the diffusion of gas into the bubble from the surrounding media (5). The effects of various processing conditions on the final cell size distribution by a model that describes the cell nucleation and growth processes were also studied (17). Other researchers developed some models that predicted the bubble growth in power-law and viscoelastic fluid melts (18), (19), (20). On the other hand, some researchers experimentally evaluated how the material characteristics such as viscosity, surface tension and diffusion coefficient change within the bubble growth as a result of variation of the pressure and blowing agent concentration (21), (22), (23), (24). By solving a blowing agent in a polymer melt, the viscosity and surface tension of the polymers decrease, whereas the diffusion coefficient increases with regard to the plasticizing effect of the solute. The order of polymer characteristics changes is a complex function of solute concentration, and its accurate measurement is difficult due to the high-pressure state of the system (21), (24). These material characteristics also change within gas phase separation during bubble expansion and return to the pure polymer properties.

Although some researchers concentrated on the consequence of the variation of processing conditions and material characteristics in the physical foaming process, no research work has been focused on highlighting the impact of changing material characteristics on the bubble growth phenomenon. In this work, a single bubble growth was modeled by coupled continuity equation, momentum balance and the constitutive equations of diffusion. The integral method was applied to simplify the mass balance equations. The validity of the model was proved using experimental data reported in the literature. Therefore, the impact of material characteristics and processing conditions on the bubble growth rate was evaluated by this model.

2 Mathematical modeling

In this model, a polymer melt with a dissolved physical foaming agent in equilibrium at high pressure is assumed. The situation of the bubble is fixed and situated at the origin of a spherical coordinate system. By the pressure release at t=0, the solution becomes supersaturated, and nucleation and bubble growth start. As the bubble growth progresses, the dissolved gas concentration at the bubble surface and the pressure inside the bubble decrease. The reduction of the solute concentration at the gas-liquid interface creates a concentration gradient that is the driving force for bubble growth. A schematic of the bubble growth is demonstrated in Figure 1. The radius of the bubble and the concentration of dissolved gas are denoted by R(t) and c(r,t), respectively. The following assumptions are made in modeling the bubble growth process:

1. The system is isothermal.

2. The bubble is spherically symmetric during the growth process.

3. Mass diffusion is governed by Fick’s law.

4. The fluid is Newtonian.

5. The gas-liquid interface is at equilibrium during expansion and is governed by Henry’s law.

6. Viscosity, Henry’s constant, diffusion coefficient and surface tension are constant.

7. The gas within the bubble obeys the ideal gas law.

8. The initial bubble radius is R₀.

Figure 1:

A schematic of a single bubble growth.

The isothermal growth of a spherical bubble is modeled by the coupled mass equation, the momentum transport and the constitutive diffusion equations with appropriate initial and boundary conditions. The continuity equation for the flow around the growing gas bubble in a spherical coordinate system can be expressed as:

where r is the radial position and u(r) is the fluid velocity at position r. The radial velocity of the liquid, due to the growth of the bubble, can be obtained by integrating the continuity equation:

where R(t) is the radius of the bubble and t is the time. Using equation (2), the momentum equation for the polymer/gas solution which surrounds the gas bubble is expressed as (8):

where P_b(t) is the pressure in the bubble, P_s(t) is the ambient pressure, γ is the surface tension, and μ is the viscosity of the liquid. The initial condition for the last equation is R=(t=0)=R₀. R₀ cannot be assumed zero because P_b tends to infinity.

Mass transfer of gas at the bubble interface is expressed as follows:

here n is the number of moles of the gas inside the bubble and D is the diffusion coefficient. By assuming the ideal gas equation of state for the gas inside the bubble, P_bV=nR_gT, where V is the bubble volume and equals to 43πR3,R_g is the universal gas constant and T is the system temperature, equation [4] can be simplified to:

Besides, the differential mass balance in a binary system assuming a constant diffusion coefficient is:

The boundary and initial conditions for Eq 6 are: c(t=0)=c_i, c(r=∞)=c_i, and c(r=R)=c_s in which c_i is initial gas concentration and c_s is the gas concentration at the bubble surface and is related to the bubble pressure by Henry’s law:

where K_H is Henry’s constant. Equations 3, 5, 6 and 7 should be solved simultaneously by a numerical method to obtain R(t). Nevertheless, simpler equations can be obtained by assuming a thin liquid shell around the bubble with a gas concentration profile. This method is known as the integral method. A second-order polynomial is usually used for the gas concentration profile in this thin layer. The mass balance equation is formulated by integrating the assumed polynomial profile. In this work, the equation proposed by Han and Yoo (11) using the integral method was applied. They assumed a concentration profile as:

where δ is the thin layer thickness. By this assumption, the following equation was derived by combination of equations 5, 6 and 8 as (11):

By using equation 9 instead of equations 5 and 6, the set of partial differential equations (PDE) is simplified to the ordinary differential equations (ODE). These equations were numerically integrated along the time after releasing the pressure by using the Wolfram Mathematica software program (Wolfram Research, Champaign, USA).

3 Results and discussion

3.1 The validity of the model

For the numerical solution of the equations, physical properties of a polymer including the diffusion coefficient, surface tension, and viscosity, are needed. These material characteristics can be typically found in data handbooks for any polymer. However, to realize the accuracy of the model, the predicted bubble growth by the model was compared with two experimental works performed on the foaming of polypropylene and polystyrene by using supercritical CO₂ (8), (21). Supercritical CO₂ and N₂ are widely used as physical foaming agents. The former can be applied for producing high-expansion foams due to higher solubility of CO₂ in polymers. The material properties and processing conditions data in these experimental works are listed in Table 1. A comparison between the experimental observations and our theoretical model is shown in Figure 2. Although the model slightly underpredicts the actual bubble growth, there is a reasonable agreement between experimental data and the theoretical prediction. The only inadequacy of the model is that it is not able to predict the end of bubble growth while the bubble growth stops at a given moment. Anyhow, this model seems appropriate to consider the effect of material characteristics and processing conditions on the bubble growth process.

Table 1:

Polymer properties and processing conditions in research (8), (21).

Property	Unit	PP [Ref. (8)]	PS [Ref. (21)]
Viscosity, μ	Pa.s	16,500	4000
Diffusion coefficient, D	m2/s	8.07×10-9	5.5×10-10
Surface tension, γ	mJ/m2	12.3	28
Henry’s constant, KH	mol/(Pa.m3)	1.15×10-4	7.4×10-5
Temperature, T	°K	473.15	473
Pressure release rate	MPa/s	0.33	–

Figure 2:

Comparison of the model prediction and experimental data on (A) PP/CO₂ [Ref. (8)] and (B) PS/CO₂ [Ref. (21)].

3.2 Effect of initial bubble radius (R₀)

Bubble growth and bubble pressure profiles obtained by varying the initial bubble radii, R₀, over the range of 0.01–1 μm, are illustrated in Figure 3. The predicted bubble growth profiles were almost indistinguishable from each other. Only at very early stage of initial profiles, the bubble with smaller R₀ values had a little higher bubble pressure which led to a higher initial growth rate. This difference was quite negligible, and any initial bubble radius in the mentioned range was confirmed to be acceptable for the modeling.

Figure 3:

Effect of initial bubble radius on bubble growth and bubble pressure profiles.

3.3 Influence of diffusivity (D)

The diffusion coefficient of carbon dioxide in PP depends on temperature and pressure, and it is approximately in the order of 10^-9 m²/s. In the polymer foaming processing, the diffusion coefficient typically first increases as a result of the plasticizing effect of the dissolved gas in the polymer. This value recovers during the expansion stage due to a drop in the gas concentration. The quantitative relationship between diffusion coefficient and solute concentration has been assessed earlier in some research works (21), (25). The effect of diffusivity on the bubble pressure and bubble growth profiles was studied by varying its value over a range of 4×10^-9–16×10^-9 m²/s and the predicted results are presented in Figure 4. The bubble growth rates were just proportional in a very early stage of the curves below 0.2 s in which the bubble radii reached up to about 5 μm. After that time, the bubble radii were raised with substantially different rates. The higher diffusivity led to a faster diffusion rate of gas molecules into the bubble and therefore, the higher growth rate. This fact indicated that the variation of diffusion coefficient by solving the gas and also during phase separation cannot be ignored. As we assume the diffusion coefficient is constant; this may be the main reason that why the model is underestimating the bubble radius (Figure 2).

Figure 4:

The predicted bubble growth and bubble pressure profiles at different diffusion coefficients.

Bubble pressure is mainly a function of ambient pressure, and because the pressure release rate was the same (0.33 MPa/s) for all three different samples, the bubble pressure was obtained as equivalent.

3.4 Effect of surface tension (γ)

In contrast to the diffusivity, surface tension decreases by dissolution of gas into the polymer. The surface tension of any polymer is a function of temperature and pressure and is usually decreased linearly with increasing temperature or pressure (21). The effect of surface tension on the predicted bubble growth profiles was investigated by varying its value over a range of 4–50 mJ/m². Figure 5 shows that the overall bubble growth profile was not so sensitive to the variety of the surface tension. In other words, although the surface tension of the polymer/gas system has a complex relationship with a solute concentration (21), (26), its variation in a wide range did not make a noticeable effect on the bubble growth profile.

Figure 5:

Effect of surface tension variation on the predicted bubble growth profile.

3.5 Effect of viscosity (μ)

The viscosity is a retarding force against the bubble growth, and it depends on the temperature, pressure, and solute concentration. Figure 6 illustrates the effects of viscosity on the predicted bubble growth profile when it is varied from 100 to 30,000 Pa.s. As expected, the bubble growth rate at the earliest stage of the curve was slower at higher μ. The maximum difference between radii was at about 0.2 s when its value was about 13 μm. After that time, the difference between radii was dropped gradually to a final value of 3 μm at 10 s. The overlapping of the predicted bubble growth profiles from the middle stage (after 2 s) showed that the effect of viscosity on the bubble growth rate can be neglected for large bubble size foams. Indeed, the initial stage of bubble growing is controlled by viscosity. Then, the diffusivity predominantly controls the bubble growing. On the other hand, it is known that polymer foaming is strongly influenced by the elongational melt viscosity. A polymer with high elongational viscosity usually shows a strain hardening behavior which leads to high volume expansion ratio, less cell coalescence, and a more homogeneous cell structure (27), (28). However, the predicted results demonstrated that the elongational viscosity should not have a significant effect on the bubble growth rate.

Figure 6:

The bubble radius and bubble pressure curves vs. time at different viscosities.

At the first stage of the bubble pressure profile, the bubble pressure decreased sooner at lower viscosity as a consequence of the higher bubble growing rate.

3.6 Effect of temperature (T)

Besides material characteristics, the foaming processing conditions such as temperature and pressure release rate would affect the bubble growth profile. In the foaming process, the temperature can vary in a limited range above the melting point and below the degradation temperature. By changing the temperature, material properties such as viscosity, surface tension and diffusivity change, as well. However, here, to study the effect of temperature independently, we presumed that the temperature varies only while the polymer properties such as viscosity, diffusion coefficient and surface tension are constant. The effect of temperature on the bubble growth profile is depicted in Figure 7. The variation of temperature by 40 K did not have an evident effect at the start of the bubble growth curve until the bubble radius reached to about 50 μm. After that, the curves of different foaming temperatures rose away gradually so that bubble radii were finally separated by about 17 μm. This fact showed that the temperature itself has a minor influence on the bubble growth profile. However, its effect on the material properties, especially on the viscosity and diffusion coefficient may alter the bubble growth, significantly.

Figure 7:

Effect of variation of temperature on the bubble growth profile.

3.7 Effect of pressure release rate

Another important processing parameter is the pressure release rate. Figure 8 represents its effect on the bubble growth and bubble pressure profile. A wide diversity of bubble growth and pressure curves indicated that the pressure release rate had a substantial influence on both profiles. The higher pressure release rate resulted in the higher bubble growth rate and the lower bubble pressure. This result was confirmed by some experimental work that stated that the pressure release rate is the most important processing parameter which controls the foam morphology (29), (30).

Figure 8:

The bubble growth rate and bubble pressure profiles at different pressure release rates.

4 Conclusions

In this work, a simplified single bubble growth in a physical foaming process was modeled by the coupled mass and momentum transport and constitutive diffusion equations. The precision of the model was considered and confirmed by comparing the predicted results with experimental observation data reported in the literature. Then, the model was applied to analyze the impact of either the physical properties of the polymer or processing conditions on the bubble growth and bubble pressure profile. It was found that the surface tension variation has almost no impression on the bubble growth rate. The viscosity at the start of foaming altered the profiles slightly and its impact was weakened, in the following. On the other hand, the variation of diffusion coefficient significantly changed the bubble growth rate. Thus, for more accurate modeling, its variation during bubble growing should be considered. It seems that the early stage of bubble growth is mostly controlled by the viscosity of the polymer, and then (after about 1 s) the diffusivity predominantly controls the growth rate. As a processing parameter, the temperature independently had a minor effect on the bubble growth rate. However, varying pressure release rate substantially caused to transform the bubble growth rate and bubble pressure profiles.