4.2.1 Bubble Nucleation

Current understanding of the process of bubble nucleation in polymer foams [31], [32] suggests the following steps for a thermoplastic material undergoing pyrolysis:

• The energy added to the material during heating translates to molecular motion, so that the polymer is considered to be in a melted state.
• A thermal fluctuation causes fragmentation of polymer molecules in a local region. Volatile products of degradation form a cluster of gas molecules which constitutes the bubble "seed" or nucleus.
• If the cluster exceeds a critical size for which free energy of cluster formation is a maximum, it grows by kinetics as gas molecules collide with the cluster. The free energy may be thought of as an energy barrier that must be overcome in order for nucleation to occur.
• The growing cluster reaches the critical size for bubble formation, for which the bubble is in thermal, chemical, and mechanical equilibrium with the polymer. At this point the gas molecules are separated from the liquid molecules by an interface.
• Growth of the supercritical bubble proceeds by diffusion in addition to surface tension, inertial, and heat-transfer forces.

This process is known as homogeneous nucleation. The alternate process of heterogeneous nucleation requires the presence of physical impurities to provide the seeds of bubble formation. Although commercial polymers may contain such sites, heterogeneous nucleation is not necessary for significant bubbling to take place, as suggested by recent advances in nucleation theory and by experiments in which adding fine particles to a polymer melt has only minor effects on bubbling behavior [38].

The classical molecular theory of nucleation assumes that critical bubbles can be described in terms of bulk thermodynamic properties. The nucleation rate per unit volume J is related linearly to the number of molecules per unit volume of the metastable (liquid) phase M and a frequency factor B, and exponentially to the free energy change ΔF_cr required to form the critical bubble [33], [34]:

(14)

where k_B is Boltzmann's constant and T is temperature. The frequency factor describes the frequency with which the bubble seed comes into contact with gas molecules. For boiling of a single component liquid, the frequency factor is

where σ is the surface tension at the liquid-gas interface, m the mass of the gas molecule, and C a coefficient related to the ratio of liquid phase pressure P_L to vapor pressure P_V. For a mixture in which one component is volatile, assuming Henry’s law relating pressure to concentration holds, B is estimated as [33]

where D is the diffusion coefficient for the volatile in the liquid and C_V and C_L are equilibrium concentrations at P_V and P_L respectively.

Direct application of the classical theory to bubble formation in polymer melts resulted in predictions of nucleation rate many orders of magnitude smaller than those observed in experiments [35]. Han and Han [34] modified the classical theory to include the presence of macromolecules and the degree of supersaturation in the free energy term, introduce empirical relations for the frequency factor and bubble growth, and account for the consumption of a finite quantity of gas:

(17)

where B₁ and B₂ are empirically determined constants, n is the number of gas molecules in a critical bubble, ΔF* is the modified change in free energy, and D = D(T) is the diffusivity of the volatile molecule in the polymer solution [36]. This model predicts nucleation rates on the order of J ≈ 10¹³ to 10¹⁹ bubbles/cm³-s at temperatures from 150 ºC to 180 ºC. Good agreement with experiment is not yet achieved by this model. Bubble population density from experiment was found to be roughly four orders of magnitude smaller than that determined by the modified theory, possibly attributable in part to bubble coalescence, which is not included in the theory.

As a final note, bubble nucleation is also affected by the presence of elastic stresses, which lower the free energy required for critical cluster formation. Since elastic stresses result from the deformation of macromolecules in the vicinity of an existing bubble, the formation of small satellite bubbles near a primary bubble, also referred to as secondary nucleation, may occur. This phenomenon is noted for elastomers by Gent and Tompkins [37] and studied in detail for viscoelastic fluids by Yarin et al. [38], who include molecular relaxation in their analysis. Secondary nucleation may provide an explanation for the "frosted" appearance of some bubbles formed within burning PMMA in experiments by Olson [39]. The model for bubbles used in this work does not include secondary nucleation, but its implications for bubble growth are discussed in the next subsection.