Introduction

Foamed plastics with a high cell density offer superior mechanical properties such as higher impact strength (Matuana et al. 1997; Shimbo et al. 2007) and fatigue life (Seeler and Kumar 1993), as well as better thermal and acoustic insulation properties (Suh et al. 2000; Shimbo et al. 1992), when compared to their solid counterparts. To achieve this, many researchers have investigated the foamability of plastics with a wide range of cell-nucleating agents [e.g., talc (Xu et al. 2003; Wong and Park 2012), calcium carbonate (Yang and Han 1984), and zinc stearate (Colton and Suh 1987a, b; Yang and Han 1984)] to identify effective cell-nucleating agents and their characteristics. Currently, micron-sized cell-nucleating agents are widely used in plastic foaming industry to promote cell nucleation for the production of fine-cell and/or microcellular foams. In recent years, nanoclay has drawn significant research interests as nucleating agents in plastic foaming owing to their small sizes and high aspect ratios; hence, high particle density and large interfacial surface area for heterogeneous nucleation. Okamoto et al. (2001) and Taki et al. (2004) investigated foaming of polypropylene (PP)–nanoclay composites with CO2, and they demonstrated that cell density increased as the nanoclay loading increased (up to 7.5 wt%). Similar results were obtained by Zheng et al. (2010) in a PP-nanoclay extrusion foaming study. Meanwhile, Lee et al. (2007b) demonstrated that the optimal loading of nanoclay to achieve maximum cell density in low density polyethylene (LDPE)–nanoclay foams was lower than 1 wt%. They attributed the decrease in cell density to the insufficient dispersion of nanoclay at high loadings that caused decrease in interfacial area for heterogeneous nucleation. To be specific, nanoclay particles naturally appear as layered stacks of platelets consisting of two silica tetrahedral fused to an edge-shared octahedral sheet of aluminum or magnesium hydroxide (Giannelis 1996). In order to generate a uniform dispersion of nanoclay in plastic (i.e., an exfoliated nanoclay plastic composite) by extrusion compounding, nanoclay platelets needs to be delaminated by sufficient shear forces so that individual platelets could be separated and uniformly distributed in the polymer matrix. To achieve this, strong interfacial interaction between the polymer matrix and nanoclay particles is needed. This could be achieved easily with highly polar plastics such as polyamides (Usuki et al. 1993). However, many commodity plastics used in industries are nonpolar or weakly polar polymers (e.g., polyethylene, polypropylene, and polystyrene), so they have weak interactions with nanoclay. Therefore, it is difficult to achieve complete exfoliation of nanoclay platelets in many plastic/nanoclay nanocomposites (Tanoue et al. 2004; Ton-That et al. 2004). In many cases, nanoclay composites with these plastics are intercalated nanocomposites in which the nanoclay particles exist largely in layered stacks. Consequently, the particle density and interfacial area for heterogeneous cell nucleation for the intercalated case would be lower than the exfoliated case. This reason was given by Lee et al. (2007b) to explain the higher cell density of LDPE–nanoclay composites foams with exfoliated nanoclay as compared to those with intercalated nanoclay.

While these studies have led to valuable insights on the effect of nanoclay on plastic foaming behavior, the final conclusions on the optimal nanoclay content varied from one study to another. This could be due to the different degrees of exfoliation of nanoclay within the plastic nanocomposites in these studies; hence, the effect of nanoclay content could not be examined in an isolated manner. In this context, a fundamental understanding on the effects of exfoliation of nanoclay on cell nucleation and growth behavior is imperative to the effective usage of nanoclay as nucleation agents and to the development of plastic foaming technologies for plastic–nanoclay composites. However, this has yet to be achieved because cell nucleation and growth processes are often not observable in typical foaming equipment (e.g., extrusion foaming or injection foam molding). This study fills this gap through in situ observation of HDPE–nanoclay composites foaming blown with CO2 using a foaming visualization system with a high-temperature/high-pressure view-cell (Wong et al. 2012).

Theoretical framework

Classical nucleation theory

According to the classical nucleation theory (CNT) (Gibbs 1961), a bubble that has a radius (R bub) greater than the critical radius (R cr) grows spontaneously, while one that has a radius smaller than R cr collapses. The R cr is state variable that depends on material, temperature, pressure, and concentration of blowing agent. A critical bubble (i.e., a bubble with radius equals to R cr) is at an unstable equilibrium, where the free energy of the system (ΔF) is at a maximum (see Fig. 1). This maximum ΔF is called the free energy barrier for cell nucleation (W). Cell nucleation occurs when a bubble’s radius (R bub) grows beyond R cr.

Fig. 1
figure 1

ΔF versus R bub plot

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According to the CNT, cell nucleation occurs either within a continuous liquid phase (i.e., homogeneous nucleation), or on a liquid/liquid or liquid/solid interface (i.e., heterogeneous nucleation). By considering the change of ΔF from a state with no bubble to one where a bubble is generated homogeneous or heterogeneously, the equation of R cr for both homogeneous and heterogeneous nucleation have been determined previously as (Gibbs 1961) (see "Appendix" section for details):

$$ R_{\text{cr}} = \frac{{2\gamma_{ \lg } }}{{P_{\text{bub,cr}} - P_{\text{local}} }} $$
(1)

where γ lg is the surface tension along the liquid–gas interface that constitutes the bubble, P bub,cr is the pressure inside a critical bubble, and P local is the local pressure around the critical bubble. The free energy barrier for homogeneous nucleation (W hom) and heterogeneous nucleation (W het) has been determined previously as (Gibbs 1961):

$$ W_{\hom } = \frac{{16\pi \gamma_{\lg }^{3} }}{{3\left( {P_{\text{bub,cr}} - P_{\text{local}} } \right)^{2} }} $$
(2)
$$ W_{\text{het}} = \frac{{16\pi \gamma_{\lg }^{3} F}}{{3\left( {P_{\text{bub,cr}} - P_{\text{local}} } \right)^{2} }} = W_{\hom } F $$
(3)

where F is the ratio of the volume of a heterogeneously nucleated bubble to that of a spherical bubble having the same radius of curvature (see “Cell nucleation from microvoids on nucleating agents” section for details). Since F ≤ 1 in most scenarios, W het is lower than W hom in most cases. Therefore, nucleation is more likely to occur heterogeneously on nucleating agents or impurities as supposed to homogeneously within the bulk phase of a polymer–gas solution.

The concept of R cr, W hom, and W het provides conditions to generate a metastable state necessary for cell nucleation. However, the CNT could not predict when a system would transfer, through molecular perturbation or external work, from a metastable liquid–gas solution to one where a bubble with size R cr is generated within the liquid–gas solution; consideration of kinetics is necessary to determine the rate of cell nucleation, and hence cell density. In this context, Blander and Katz (1975) defined bubble nucleation rate, J, as the rate at which critical bubbles gain gas molecules that trigger their spontaneous growth to become nucleated bubbles. The expressions for J for homogeneous and heterogeneous nucleation, J hom and J het, have been derived previously as (Blander and Katz 1975):

$$ J_{\hom } = N\sqrt {\frac{{2\gamma_{ \lg } }}{\pi m}} \, \exp \left( { - \frac{{W_{ \hom } }}{{k_{\text{B}} T_{\text{sys}} }}} \right) $$
(4)
$$ J_{\text{het}} = N^{\frac{2}{3}} Q\sqrt {\frac{{2\gamma_{ \lg } }}{\pi mF}} \, \exp \left( { - \frac{{W_{\text{het}} }}{{k_{\text{B}} T_{\text{sys}} }}} \right) $$
(5)

where N is the number of gas molecules per unit volume of polymer; Q is the ratio of the surface area of the heterogeneously nucleated bubble to that of a spherical bubble with the same radius of curvature; m is the mass of a gas molecule; and k B is the Boltzmann’s Constant. The expression for cell density, N(t) at time, t is the integral of the total cell nucleation rate:

$$ N\left( t \right) = \int\limits_{0}^{t} {\left\{ {J_{ \hom } \left( {t'} \right) + A_{\text{het}} J_{\text{het}} \left( {t'} \right)} \right\}{\text{d}}t'} $$
(6)

where A het is the total surface area of heterogeneous nucleating sites per unit volume of a polymer–gas solution. CNT describes boiling or cavitation phenomena in many single component systems accurately when extreme care has been used to remove any existing gas bubbles in the liquid phase (e.g., boiling of water). However, in plastic foaming processes, it has been demonstrated by various researchers that CNT often overestimates the degree of supersaturation (P bub,cr – P local) that is needed to initiate cell nucleation; the observed cell nucleation rates were significantly higher than were predicted by CNT (Lubetkin 2003). One of the criticisms of CNT was its assumption of continuum mechanics that ultimately led to overestimation of surface energies of nano-sized bubbles, and hence the energy barrier for cell nucleation (Kim et al. 2011). Other researchers proposed that microvoids exist in polymer–gas solutions as free volumes between polymer chains, or gas pockets on solid particles (e.g., nucleating agents, impurities, etc.) due to the high viscosity of plastic melt and contact angle restraint (Harvey et al. 1944; Levy 1981; Ward and Levart 1984). The free energy required for cell nucleation from the growth of existing microvoids would be lower than that predicted by CNT. Based on this concept, CNT and the corresponding kinetic theory for nucleation discussed above have been extended in this study to include cell nucleation from existing microvoids.

Cell nucleation from microvoids within bulk phase of polymer–gas solution

Given a scenario where a gas bubble exists within the bulk phase of a polymer/gas solution (see Fig. 2ii). The free energy change (ΔF hom) from a bubble with volume of V g,1 and surface area of A g,1 (State 1) to a larger gas bubble with volume of V g,2 and surface area of A g,2 (State 2) within the polymer/gas solution could be given as:

$$ \begin{gathered} \hfill \\ \Updelta F_{\hom } = - \left( {P_{\text{bub,2}} - P_{\text{local}} } \right)V_{\text{g,2}} + \gamma_{ \lg } A_{\text{lg,2}} - \left[ { - \left( {P_{\text{bub,1}} - P_{\text{local}} } \right)V_{\text{g,1}} + \gamma_{ \lg } A_{{{\text{lg,1}}}} } \right] \hfill \\ \hfill \\ \end{gathered} $$
(7)

where and P bub is the pressure inside the bubble. Assuming that the bubble is spherical, Eq. 7 could be simplified to:

$$ \Updelta F_{ \hom } = - \frac{{4\pi R_{\text{bub,2}}^{3} }}{3}\left( {P_{\text{bub,2}} - P_{\text{local}} } \right) + \gamma_{ \lg } \left( {4\pi R_{\text{bub,2}}^{2} } \right) - \left[ { - \frac{{4\pi R_{\text{bub,1}}^{3} }}{3}\left( {P_{\text{bub,1}} - P_{\text{local}} } \right) + \gamma_{ \lg } \left( {4\pi R_{\text{bub,1}}^{2} } \right)} \right] $$
(8)

where R bub is the bubble radius. Both P bub,1 and R bub, 1 are independent of R bub,2. By taking the derivative of ΔF hom with respect to R bub,2 and equating it to zero, the expression of R cr could be shown to be the same as the CNT case (see Eq. 1). By substituting the expression of R cr to Eq. 8, the free energy barrier of cell nucleation from an existing microvoid (W hom) within the bulk phase of a polymer–gas solution could be simplified to:

$$ W_{ \hom } = \frac{{16\pi \gamma_{ \lg }^{3} }}{{3\left( {P_{\text{bub,cr}} - P_{\text{local}} } \right)^{2} }} - \frac{{4\pi R_{\text{bub,1}}^{2} }}{3}\left[ {\left( {P_{\text{local}} - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] $$
(9)
Fig. 2
figure 2

Homogeneous nucleation of CNT versus nucleation through growth of a microvoid in polymer–gas solution

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The first term on the right hand side is the free energy barrier to nucleate a bubble homogeneously within a polymer–gas solution without the presence of an existing microvoid (same as Eq. 2). The second term on the right hand side is the free energy change from a metastable polymer–gas solution to the homogeneous formation of a gas bubble with a size of R bub,1 within the polymer–gas solution. As demonstrated by this equation, the overall W hom is decreased when cell nucleation occurs through the growth of an existing microvoid. Therefore, if there are microvoids within a polymer–gas solution, cell nucleation is likely to occur via the growth of these microvoids as supposed to homogeneous nucleation within the bulk phase of the polymer–gas solution. Due to the presence of existing microvoids, the cell nucleation rate and hence the cell density are expected to increase. Based on the expression for the classical homogeneous cell nucleation rate (J het) (Eq. 4) and Eq. 9, the cell nucleation rate from the growth of microvoids within a bulk phase of polymer–gas solution could be derived to be:

$$ J_{ \hom } = \int\limits_{{R_{\text{bub,1}} }} {\rho_{R} N\sqrt {\frac{{2\gamma_{ \lg } }}{\pi m}} \, \exp \left\{ { - \frac{1}{{k_{\text{B}} T_{\text{sys}} }}\left( \begin{array}{l} \frac{{16\pi \gamma_{ \lg }^{3} }}{{3\left( {P_{\text{bub,cr}} - P_{\text{local}} } \right)^{2} }} \\ - \frac{{4\pi R_{\text{bub,1}}^{2} }}{3}\left[ {\left( {P_{\text{local}} - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] \\ \end{array} \right)} \right\}} {\text{d}}R_{\text{bub,1}} $$
(10)

where ρ R (R bub,1) is the probability density function of the size of existing microvoids (R bub,1). If no microvoid exists (i.e., homogeneous nucleation of CNT), then ρ R (R bub,1 = 0) = 1. In that case, Eq. 10 would be reduced to the original form (Eq. 4).

Cell nucleation from microvoids on nucleating agents

As mentioned earlier, nucleating agents (e.g., nanoclay, talc, etc.) are often added to polymer matrix to enhance bubble nucleation. These nucleating agents and/or their agglomerates (e.g., intercalated nanoclay particles) might have non-uniform surfaces that could be modelled as conical cavities (Cole 1974; Leung et al. 2006). This scenario is illustrated in Fig. 3. Based on this scenario, the free energy change (ΔF het) from a microvoid with a radius of curvature of R bub,1 on a nucleating site (State 1) to a larger microvoid with a radius of curvature of R bub,2 on the same nucleating site (State 2) within a polymer–gas solution could be determined using a similar approach described above:

$$ \begin{aligned} \Updelta F_{\text{het}} = & - \left( {P_{\text{bub,2}} - P_{\text{local}} } \right)V_{\text{g,2}} + \left( {\gamma_{\text{sg}} - \gamma_{\text{sl}} } \right)A_{\text{sg,2}} + \gamma_{ \lg } A_{\text{lg,2}} \\ & - \left[ { - \left( {P_{\text{bub,1}} - P_{\text{local}} } \right)V_{\text{g,1}} + \left( {\gamma_{\text{sg}} - \gamma_{\text{sl}} } \right)A_{\text{sg,1}} + \gamma_{ \lg } A_{\text{lg,1}} } \right] \\ \end{aligned} $$
(11)

where V g is the volume of the bubble; A sg and A lg are the area of the solid–gas and liquid–gas interfaces, respectively; and γ sg, γ sl, and γ lg are the surface tension along the solid–gas, solid–liquid, and liquid–gas interfaces, respectively. Using the Young’s equation, which relates the interfacial energies and the contact angle (θ c) between the liquid and gas phase (measured in the liquid phase) (Ward and Tucker 1975):

$$ \gamma_{\text{sg}} - \gamma_{\text{sl}} = \gamma_{ \lg } \cos \theta_{\text{c}} $$
(12)

and the expressions for V g, A sg, and A lg for a microvoid on a conical cavity (Cole 1974), Eq. 11 could be simplified to:

$$ \Updelta F_{\text{het}} = - \frac{{4\pi R_{\text{bub,2}}^{3} }}{3}\left( {P_{\text{bub,2}} - P_{\text{local}} } \right)F + 4\pi R_{\text{bub,2}}^{2} \gamma_{ \lg } F - \left[ { - \frac{{4\pi R_{\text{bub,1}}^{3} }}{3}\left( {P_{\text{bub,1}} - P_{\text{local}} } \right)F + 4\pi R_{\text{bub,1}}^{2} \gamma_{ \lg } F} \right] $$
(13)
Fig. 3
figure 3

Heterogeneous nucleation of CNT versus nucleation through growth of a microvoid on a conical cavity

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The expression for F describing a conical cavity has been derived to be (Cole 1974):

$$ F\left( {\theta_{\text{c}} ,\beta } \right) = \frac{1}{4}\left[ {2 - 2\sin \left( {\theta_{\text{c}} - \beta } \right) + \frac{{\cos \theta_{\text{c}} \cos^{2} \left( {\theta_{\text{c}} - \beta } \right)}}{\sin \beta }} \right] $$
(14)

where β is the semi-conical angle of a conical cavity. Both P bub,1 and R bub,1 are independent of R bub,2. By taking the derivative of ΔF het with respect to R bub,2 and equating it to zero, the expression of R cr could be shown to be the same as the CNT case (see Eq. 1). By substituting R cr to Eq. 13, the free energy barrier of cell nucleation from an existing microvoid on a conical cavity (W het) could be determined to be:

$$ W_{\text{het}} = \frac{{16\pi \gamma_{ \lg }^{3} F}}{{3\left( {P_{\text{bub,cr}} - P_{\text{local}} } \right)^{2} }} - \frac{{4\pi R_{\text{bub,1}}^{2} F}}{3}\left[ {\left( {P_{\text{local}} - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] $$
(15)

The first term on the right hand side is the free energy barrier to nucleate a bubble heterogeneously on a conical cavity within a polymer–gas solution and without the presence of an existing microvoid (same as Eq. 3). The second term on the right hand side is the free energy change from a state with a conical cavity within a metastable polymer–gas solution to the formation of a gas bubble with a size of R bub,1 on the conical cavity within the polymer–gas solution. As shown in Eq. 15, the overall W het is decreased if cell nucleation occurs through the growth of an existing microvoid on a conical cavity. Therefore, if there are microvoids on cavities (e.g., on nucleating agents, impurities, or equipment surfaces, etc.), cell nucleation are likely to occur through the growth of these microvoids as supposed to heterogeneous nucleation without the presence of microvoids. These microvoids are more likely to exist if there are poor interface between the polymer–gas mixture and the surfaces of nucleating agents. Moreover, in the presence of flow, tensile, and/or shear stresses might cause the polymer–gas mixture to be dewetted from the nucleating agent, thus generating more microvoids for cell nucleation. Therefore, due to the presence of these microvoids, the cell nucleation rate and hence the cell density are expected to increase. Based on the expression for classical heterogeneous cell nucleation rate (J het) (Eq. 5), the heterogeneous cell nucleation rate from microvoids could be derived to be:

$$ J_{\text{het}} = \int\limits_{\beta } {\int\limits_{{R_{\text{bub,1}} }} {\rho_{\beta } \rho_{R} N^{\frac{2}{3}} Q\sqrt {\frac{{2\gamma_{ \lg } }}{\pi mF}} \, \exp \left\{ { - \frac{1}{{k_{\text{B}} T_{\text{sys}} }}\left( \begin{array} {l} \frac{{16\pi \gamma_{ \lg }^{3} F}}{{3\left( {P_{\text{bub,cr}} - P_{\text{local}} } \right)^{2} }} \\ - \frac{{4\pi R_{\text{bub,1}}^{2} F}}{3}\left[ {\left( {P_{\text{local}} - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] \\ \end{array} \right)} \right\}} {\text{d}}R_{\text{bub,1}} {\text{d}}\beta } $$
(16)

where ρ β (β) is the probability density function of the semi-conical angle (β) of conical cavities and ρ R (R bub,1) is the probability density function of the size of existing microvoids (R bub,1). Note that the value of β could also influence the distribution of R bub,1. If no microvoid exists (i.e., the case for the classical heterogeneous nucleation), then ρ R (R bub,1 = 0) = 1. In that case, Eq. 16 would be reduced to the original form (Eq. 5).

Experimental materials and procedure

The HDPE–nanoclay composites were prepared using melt compounding with an extrusion system (Lee et al. 2007a). The materials forming the composites were: HDPE (Sclair 2607, Nova Chemicals, MFI = 5.0 g/10 min), HDPE-g-maleic anhydride (PE-g-MAn, Fusabond MB-100D, Dupont, MFI = 0.96 g/10 min), and organoclay (Cloisite 20A, Southern Clay Products). The PE-g-MAn served as a compatibilizer to achieve exfoliation of nanoclay. The intercalated HDPE-nanocomposite was prepared by mixing HDPE with 0.5 wt% nanoclay (labeled as HNC0.5). The exfoliated sample was prepared by compounding HDPE (49.5 wt%), PE-g-MAn (50 wt%), and nanoclay (0.5 wt%) and was labeled as HWC0.5. Neat HDPE and a mixture of HDPE (50 wt%) and PE-g-MAn mixture (50 wt%) were also studied (labeled as HNC0 and HWC0, respectively). By wide-angle X-ray diffraction (XRD) measurements, Lee et al. (2007a) demonstrated that the HWC0.5 has a significantly improved degree of exfoliation of nanoclay over HNC0.5. To be specific, the diffraction peak that characterized the interlayer spacing of stacks of nanoclay platelets was not found in the HWC0.5 sample, and the corresponding TEM image showed that delaminated nanoclay platelets were uniformly distributed in the sample (see Fig. 4). Therefore, it was considered as the exfoliated sample in this study. On the other hand, the diffraction peak that characterized the interlayer spacing of stacks of nanoclay platelets was found in the HNC0.5 sample, and the nanoclay particles were significantly larger and thicker, as shown in the corresponding TEM image. This demonstrated that the HNC0.5 sample was not exfoliated. The angle of diffraction peak was smaller than that of nanoclay, which indicated that the interlayer spacing increased as a result of the extrusion compounding process. Therefore, it was considered as the intercalated sample in this study.

Fig. 4
figure 4

XRD diffraction patterns and TEM of a HWC composites; b HNC composites (Lee et al. 2007a)

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By comparing the foaming behavior of the HNC and HWC composites, the exfoliation effect on foaming were elucidated. Experiments were conducted with a foaming visualization with a high-temperature/high-pressure view-cell (see Fig. 5) to capture the foaming processes under high-speed and micro-scale (Wong et al. 2012). To conduct an experiment, a thin HDPE film (6.3 mm in diameter and 0.4 mm in thickness) was placed inside the view-cell to allow dissolution of CO2 at high-temperature and pressure for 30 min. The plastic sample was placed on top of a thin PET film with a 1-mm hole at the center; observation and characterization was made at the suspended region to eliminate the heterogeneous nucleating effect of the PET films and the sapphire lens underneath. Foaming was induced by a rapid depressurization and was captured with a high-speed camera. Three saturation pressures (P sat) were used: 6 MPa (870 psi), 7.6 MPa (1100 psi), and 10.3 MPa (1500 psi). Meanwhile, the foaming temperature (T sys) was kept at 180 °C for all cases. The T sys was chosen at a high temperature to ensure that polymer is in the melt state, thus eliminating any effect of crystals on cell nucleation and growth. Each experimental case was conducted three times. For analysis, cell density data was extracted from the foaming visualization data. To achieve this, N(t), the number of cells within a superimposed circular boundary with an area of A c at time t, was counted at each time frame. The radii of 10 randomly selected bubbles at each time frame (i.e., R bub,i (t), where i = 1…10) were also measured. The cell density with respect to the foamed volume, N foam(t), and the cell density with respect to the unfoamed volume, N unfoam(t), were calculated using the following equations:

$$ N_{\text{foam}} \left( t \right) = \left( {\frac{N\left( t \right)}{{A_{\text{c}} }}} \right)^{\frac{3}{2}} $$
(17)
$$ N_{\text{unfoam}} \left( t \right) = N_{\text{foam}} \left( t \right) \times {\text{VER}}\left( t \right) $$
(18)
$$ {\text{VER}}\left( t \right) = 1 + \left( {\frac{4}{3}\pi \left( {\sum\limits_{i}^{10} {\frac{{R_{{{\text{bub,}}i}} \left( t \right)^{3} }}{10}} } \right) \times N_{\text{foam}} \left( t \right)} \right) $$
(19)

where VER(t) is the volume expansion ratio of the foam at time t. The data was collected from t = 0 to after the completion of cell nucleation process to extract the cell density versus time data. Note that the smallest cells that could be observed were approximately 2–5 μm in diameter (Wong et al. 2007). Therefore, there could be a small time delay between cell nucleation and the instant at which cells were observed. Therefore, the cell density profiles were based on the observable cells only. Also, a line-of-best-fit was generated for the cell radius versus time data of each measured cell (30 cells for each experimental case). Subsequently, the average bubble growth rate of each case was obtained by taking the average of the slopes of the lines-of-best-fit.

Fig. 5
figure 5

Foaming visualization system with a high-temperature/high-pressure view-cell (Wong et al. 2012)

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Results and discussion

Snapshots of the foaming visualization data for P sat = 6.0 MPa (870 psi), 7.6 MPa (1100 psi), and 10.3 MPa (1500 psi) are shown in Fig. 6a–c, respectively. For the P sat = 6.0 and 7.6 MPa cases, no cell was nucleated in the suspended region of the samples with no nanoclay added (HNC0 and HWC0). Meanwhile, cells were nucleated in the surrounding regions, where the samples were in contact with the PET surface because the PET surfaces act as heterogeneous nucleation sites, which decreased the energy barrier for nucleation. This demonstrated that the heterogeneous nucleation effect of the PET surface was significant, and observation had to be made in the suspended region in order to study the effect of nanoclay in an isolated manner. For the samples with nanoclay (HNC0.5 and HWC0.5) at these pressures, cell nucleation occurred in the suspended region for the HNC0.5 sample (with intercalated nanoclay), but not for the HWC0.5 sample (with exfoliated nanoclay). For the P sat = 10.3 MPa cases (see Fig. 6c), all four samples were foamed in the suspended region due to the increased gas concentration that led to higher supersaturation level and lower γ lg that resulted in lower W hom and W het. As shown in Fig. 7a, b, the maximum cell density of the HNC0.5 and HWC0.5 cases (N unfoam = 3.2 × 106 and 2.6 × 106 cells/cm3, respectively) were higher than the HNC0 and HWC0 cases (N unfoam = 5.2 × 105 and 9.1 × 105 cells/cm3, respectively), which demonstrated the nucleating power of nanoclay. The HWC0 sample resulted in slightly higher maximum cell density than the HNC0 sample. On the other hand, the maximum cell density of the HNC0.5 case was higher than the HWC0.5 case. Figure 7c summarized the cell density data for all experimental cases; it demonstrated that the HNC0.5 samples resulted in the highest cell densities at all three P sat. This result might seem counter-intuitive because it is commonly believed that the particle density and interfacial area for heterogeneous nucleation should be significantly higher in the sample with exfoliated nanoclay (HWC0.5). This should have led to a higher cell density, according to Eq. 6.

Fig. 6
figure 6

Snapshots of in situ foaming visualization data: a P sat = 6.0 MPa; b P sat = 7.6 MPa; c P sat = 10.3 MPa

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Fig. 7
figure 7

Cell density data: a N unfoam versus time (P sat = 10.3 MPa); b maximum cell density (P sat = 10.3 MPa); c maximum cell density (all cases). (Note: 100 signify zero cell density)

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There are two main reasons for the superior cell nucleating performance of the intercalated nanoclay over the exfoliated one. The first reason is the stress-induced nucleation effect. As cells nucleated and grew near a nanoclay particle, the local pressure (P local) around the particle might not equal to the system pressure (P sys) as the flow field of the polymer–gas solution would be changed by the presence of the nanoclay particle. Consequently, the polymer–gas solution could be subjected to local pressure variations (∆P local) in these regions. Wang et al. (2010) demonstrated this behavior by conducting a numerical simulation of the pressure field of a polymer–gas solution around a solid particle near the presence of a growing bubble; it was found that tensile stresses could exist in some regions around the particle. Due to the tensile stresses, the local system pressure would be lowered, which increased the supersaturation level in those regions. Consequently, cell nucleation would likely occur in these regions. This stress-induced nucleation theory was first proposed by Albalak et al. (1990), and was later confirmed by Leung et al. (2012) and Wong et al. (2011) via foaming visualization studies of PS–talc composites. To account for these local pressure variations (∆P local), the expression for R cr (Eq. 1) has been previously modified as follows (Leung et al. 2012; Wong et al. 2011):

$$ R_{\text{cr}} = \frac{{2\gamma_{ \lg } }}{{P_{\text{bub,cr}} - \left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right)}} $$
(20)

When there is a tensile stress, ∆P local would be negative, hence the P local would decrease and the supersaturation level would increase. Similarly, the expression of W hom (Eq. 9), J hom (Eq. 10), W het (Eq. 15), and J het (Eq. 16) could be modified, respectively, to be:

$$ W_{ \hom } = \frac{{16\pi \gamma_{ \lg }^{3} }}{{3\left( {P_{\text{bub,cr}} - \left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right)} \right)^{2} }} - \frac{{4\pi R_{\text{bub,1}}^{2} }}{3}\left[ {\left( {\left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right) - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] $$
(21)
$$ J_{ \hom } = \int\limits_{{R_{\text{bub,1}} }} {\rho_{R} N\sqrt {\frac{{2\gamma_{ \lg } }}{\pi m}} \, \exp \left\{ { - \frac{1}{{k_{\text{B}} T_{\text{sys}} }}\left( \begin{array} {l} \frac{{16\pi \gamma_{ \lg }^{3} }}{{3\left( {P_{\text{bub,cr}} - \left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right)} \right)^{2} }} \\ - \frac{{4\pi R_{\text{bub,1}}^{2} }}{3}\left[ {\left( {\left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right) - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] \\ \end{array} \right)} \right\}} {\text{d}}R_{\text{bub,1}} $$
(22)
$$ W_{\text{het}} = \frac{{16\pi \gamma_{ \lg }^{3} F}}{{3\left( {P_{\text{bub,cr}} - \left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right)} \right)^{2} }} - \frac{{4\pi R_{\text{bub,1}}^{2} F}}{3}\left[ {\left( {\left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right) - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] $$
(23)
$$ J_{\text{het}} = \int\limits_{\beta } {\int\limits_{{R_{\text{bub,1}} }} {\rho_{\beta } \rho_{R} N^{\frac{2}{3}} Q\sqrt {\frac{{2\gamma_{ \lg } }}{\pi mF}} \, \exp \left\{ { - \frac{1}{{k_{\text{B}} T_{\text{sys}} }}\left( \begin{array} {l} \frac{{16\pi \gamma_{ \lg }^{3} F}}{{3\left( {P_{\text{bub,cr}} - \left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right)} \right)^{2} }} \\ - \frac{{4\pi R_{\text{bub,1}}^{2} F}}{3}\left[ {\left( {\left( {P_{\text{sys}} + \Updelta P_{\text{local}} } \right) - P_{\text{bub,1}} } \right)R_{\text{bub,1}} + 3\gamma_{ \lg } } \right] \\ \end{array} \right)} \right\}} {\text{d}}R_{\text{bub,1}} {\text{d}}\beta } $$
(24)

The intercalated nanoclay particles had larger particle sizes than the exfoliated ones (see Fig. 4), and were also expected to be stiffer due to their multilayer structures. Because of this, the intercalated nanoclay particles would be more restricted to move with the flow of the polymer–gas solution as cells grew around its vicinity. This would result in higher tensile stresses (e.g., negative ∆P local) in localized regions around the particles, which decreased the energy barrier for nucleation, and hence the cell nucleation rate would be increased (see Eq. 24). This result is similar to the PS foaming visualization study by Wong and Park (2012), where talc with different sizes were used as nucleating agents to investigate the effect of talc sizes on cell nucleating performance. In that study, it was observed that the type of talc with the largest sizes resulted in the highest cell density despite its lowest particle density when the same weight content of talc was used. Similar to this study, it was believed that the primary reason for this behavior was the stress-induced cell nucleation phenomena. Because of this foaming mechanism, intercalated nanoclay was more effective in generating cells despite its lower particle density and interfacial area for heterogeneous nucleation. On the other hand, Lee et al. (2007a) showed that HWC0.5 possessed higher viscosity and elasticity than HNC0.5 due to the well-dispersed exfoliated nanoclay and the PE-g-MAn. In theory, this could lead to higher level of tensile stresses in presence of nearby growing bubbles, and hence stress-induced nucleation could be enhanced. However, the particle size and stiffness effects on stress-induced nucleation seemed to be more dominant at the experimental conditions used in this study.

Another reason for the better cell nucleating performance of the intercalated nanoclay was its lower polymer–nanoclay surface affinity when compared to the exfoliated one. As mentioned earlier, PE-g-MAn has been used for the HWC0.5 sample (exfoliated case) to improve the distribution and exfoliation of nanoclay in the HDPE matrix. The PE-g-MAn would increase the affinity between the nanoclay surface and the HDPE matrix. This would decrease the contact angle (θ c) as the polymer tended to wet on the surface of the nanoclay. Consequently, the geometric factor F (i.e., the ratio of the volume of a nucleated bubble to the volume of a spherical bubble with the same radius of curvature) would increase. To illustrate this, a 3D plot of F as a function of θ c and β has been generated using Eq. 14 (see Fig. 8). This figure demonstrates that F increased significantly as θ c decreased. It is believed this is an important reason behind the decreased nucleating power of the exfoliated nanoclay in the HDPE nanocomposites because cell nucleation rate and cell density is highly sensitive to changes in θ c. For example, a numerical simulation of PS foaming with CO2 demonstrated that the cell density increased by four orders of magnitude (i.e., from 105 to 109 cells/cm3) due to a small change of θ c from 85.5° to 87.5° (Leung et al. 2006). Also, with an increase of the polymer–nanoclay surface affinity, the polymer–gas solution would be more likely to wet on any nano- or micro-sized crack or cavities on the nanoclay particles; thus, it would less likely for microvoids to exist under high-temperatures and pressures. Therefore, nucleation from microvoids, which was thermodynamically favorable, would be less dominant in the HWC0.5 case. Meanwhile, intercalated nanoclay particles would have irregular surfaces due to their multi-layer structure of clay platelets. Gas clusters could be trapped on surfaces or between layers of clay platelets to act as seeds for nucleation. This further explained the inferior nucleation performance of the exfoliated nanoclay when compared to the intercalated ones.

Fig. 8
figure 8

F versus θ c and β plot

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The average cell growth rates for all four samples (HNC0, HWC0, HNC0.5, and HWC0.5) were characterized based on the method described in “Experimental materials and procedure” section. Comparison between all samples was made at P sat = 10.3 MPa (1500 psi) only because that was the only experimental condition at which all samples were foamed (see Fig. 9). Compared to HNC0, the HWC0 seemed to have slightly lower cell growth rate, which could be attributed to the higher viscosity of the HWC0 sample due to the addition of PE-g-MAn. However, the differences were not statistically significant due to the large variations of the cell growth rate data. Also, the addition of intercalated nanoclay did not seem to hinder cell growth (see HNC0.5 in Fig. 9) as the average cell-growth rate remained at a similar level. On the other hand, the exfoliated nanoclay seemed to have effectively hindered the average cell-growth rates (see HWC0.5 in Fig. 9).

Fig. 9
figure 9

Average bubble growth rate (P sat = 10.3 MPa). (Note: error bars signify one standard deviation among 30 sets of cell radius data)

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Moreover, during bubble growth, nanoclay particles tended to align to cell boundary due to biaxial extensional flow of polymer–gas solution (see Fig. 10). This has previously been demonstrated by Okamoto et al. (2001) via TEM imaging of foamed PP–nanoclay composites. The alignment of nanoclay particles increased the melt strength of the polymer nanocomposites along cell walls, which prevented cell wall rupture, and hence cell coalescence. Exfoliated nanoclay had a higher particle number and better polymer interaction than the intercalated counterpart; hence, the melt strength improvement of the exfoliated nanoclay case was expected to be higher than the intercalated case. Also, the alignment of nanoclay particles generated tortuous structures that would act as a barrier to hinder gas diffusion. This would effectively prevent cell coarsening (i.e., the collapse of small cells adjacent to larger ones due to gas diffusion from the small cells to the large cells) (see Fig. 11). This behavior was believed to be more significant for the exfoliated nanoclay case due to its higher particle density.

Fig. 10
figure 10

Alignment of nanoclay particles during bubble growth

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Fig. 11
figure 11

Aligned nanoclay particles act as gas diffusion barrier

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Due to the superior cell stabilization characteristics of the exfoliated nanoclay, the cellular morphology of HDPE foams with exfoliated nanoclay might be better (e.g., cells with uniform cell sizes, closed-cell morphology, etc.) than those with intercalated nanoclay, despite of their inferior cell nucleating characteristics. However, due to the high processing temperature used and the difficulty to instantly cool the foam sample to stabilize the cellular structure, cell deterioration was unavoidable in this study. Therefore, the stabilized foam samples could not be used to further investigate the effect of exfoliation of nanoclay on the cell stabilization characteristics and the final cell structure. Nevertheless, the cell growth rate data provided evidences of the superior cell stabilization characteristics of the exfoliated nanoclay.

Conclusion

Through in situ foaming visualization via a high-temperature/high-pressure view-cell system, the cell nucleation and growth behaviors of HDPE–nanoclay composites have been examined to elucidate the effects of intercalation and exfoliation of nanoclay. Intercalated nanoclay was found to be more effective in inducing cell nucleation, despite its lower particle density and smaller interfacial area for heterogeneous nucleation. This was caused by stress-induced cell nucleation, lower polymer–nanoclay affinity and higher number of existing gas cavities for cell nucleation. On the other hand, the cell stabilization capability of exfoliated nanoclay was superior to the intercalated nanoclay. This could be attributed to its higher melt strength and gas barrier properties, which would suppress cell coalescence and coarsening during cell stabilization.