The mechanics of solid-state nanofoaming

Abstract

Solid-state nanofoaming experiments are conducted on two polymethyl methacrylate (PMMA) grades of markedly different molecular weight using CO₂ as the blowing agent. The sensitivity of porosity to foaming time and foaming temperature is measured. Also, the microstructure of the PMMA nanofoams is characterized in terms of cell size and cell nucleation density. A one-dimensional numerical model is developed to predict the growth of spherical, gas-filled voids during the solid-state foaming process. Diffusion of CO₂ within the PMMA matrix is sufficiently rapid for the concentration of CO₂ to remain almost uniform spatially. The foaming model makes use of experimentally calibrated constitutive laws for the uniaxial stress versus strain response of the PMMA grades as a function of strain rate and temperature, and the effect of dissolved CO₂ is accounted for by a shift in the glass transition temperature of the PMMA. The maximum achievable porosity is interpreted in terms of cell wall tearing and comparisons are made between the predictions of the model and nanofoaming measurements; it is deduced that the failure strain of the cell walls is sensitive to cell wall thickness.

Keywords: PMMA nanofoams; deformation mechanism maps; molecular weight; porosity limit; solid-state foaming; void growth model.

Figure 1.

Reported porosity f versus void size l of high porosity (PMMA-based) nanofoams produced via solid-state foaming. The ‘open circles’ refer to results obtained in the present study. The ‘filled circles’ refer to data retrieved from [,–17]; see the electronic supplementary material information for the reference corresponding to a data point.

Figure 2.

SEM micrographs of the low M_w nanofoams at (a) $T_{\text{f}}={60}^{\circ }\text{C}$, (b) $T_{\text{f}}={100}^{\circ }\text{C}$ and of the high M_w nanofoams at (c) $T_{\text{f}}={60}^{\circ }\text{C}$ and (d) $T_{\text{f}}={100}^{\circ }\text{C}$.

Figure 3.

Nanofoaming experiments on the low M_w and high M_w PMMA grades: (a) measured average cell size l versus foaming time t_f for T_f = 60°C, (b) measured porosity f versus foaming time t_f for T_f = 60°C and T_f = 100°C, (c) measured porosity f versus foaming temperature T_f for the range of explored foaming times (t_f = 60–600 s) for the low M_w nanofoams and (d) measured f versus T_f for the range of explored foaming times (t_f = 60–600 s) for the high M_w nanofoams.

Figure 4.

Measured open cell content O_v as a function of porosity f for (a) the low M_w PMMA nanofoam and (b) high M_w PMMA nanofoam.

Figure 5.

Spherical void in (a) undeformed configuration with initial radius a₀ and initial outer radius b₀ and (b) deformed configuration at time t of the void with radius a, outer radius b and gas pressure p. (Online version in colour.)

Figure 6.

The normalized glass transition temperature $T_{\text{g}}/T_{\text{g}}^0$ of PMMA as a function of CO₂ mass concentration C, as reported by Chiou et al. [44], Wissinger & Paulaitis [45] and Guo & Kumar [46]. The $T_{\text{g}}/T_{\text{g}}^0$ versus C curve is given by the calibrated version of equation (4.17).

Figure 7.

Deformation mechanism maps for (a) low M_w PMMA and (b) high M_w PMMA (for a reference strain ε_ref = 0.05), for contours of effective strain rate ${\dot{\epsilon }}_{\text{e}}$. The predicted effective stress at the surface of the cavity ${\sigma }_{\text{e}}^{}$ is plotted as a function of T/T_g for foaming temperatures T_f = 25°C and T_f = 80°C and for a foaming time up to 600 s.

Figure 8.

Predicted and measured porosity f versus foaming time t_f, for T_f = 25°C to T_f = 80°C for (a) the low M_w PMMA nanofoams and (b) the high M_w PMMA nanofoams. The ductility-governed porosity limit f_f is plotted via equation (5.3) for an initial porosity f₀ = 10⁻³. The minimum cell wall thickness-governed porosity limit f_c is plotted via equation (5.5) for f₀ = 10⁻³ and h_c/a₀ = 3 (low M_w PMMA) and h_c/a₀ = 4.2 (high M_w PMMA).

Figure 9.

Measured true tensile stress σ versus true tensile strain ε curves for the low M_w and high M_w PMMA grades in uniaxial tension for a nominal strain rate $\dot{\epsilon }=5.9\times {10}^{-2}\hspace{0.17em}{\text{s}}^{-1}$ and for temperatures ranging from (a) T = 90°C to T = 120°C and (b) T = 130°C to T = 170°C. A cross at the end of the curve denotes specimen failure.

Figure 10.

Loading–unloading true stress versus true strain curves for the low M_w PMMA and high M_w PMMA grades in uniaxial tension, at selected values of T/T_g, for a nominal strain rate $\dot{\epsilon }=5.9\times {10}^{-4}{\text{s}}^{-1}$.

Figure 11.

Deformation mechanism maps of the low M_w and high M_w PMMA grades. Flow strength σ_y (=σ_e) versus T/T_g is plotted, with the curve fits of the constitutive models included for a reference strain ε_ref = 0.05.

Figure 12.

The measured true tensile failure strain ε_f as a function of normalized testing temperature T/T_g for the low M_w and high M_w PMMA grades, at a nominal strain rate $\dot{\epsilon }=5.9\times {10}^{-2}\hspace{0.17em}{\text{s}}^{-1}$.