The rate of volatile gas generation due to polymer degradation at a given temperature can be described by the Arrhenius expression in equation (1). Given this chemical model, the quantity of gas produced in a given time increment and within a given radial shell can be treated as a known quantity. One question that arises is whether it is reasonable to distribute this gas immediately among nearby bubbles (i.e. assuming mass diffusion time rapid compared to the thermal time scale), or whether some amount must be partitioned off into a reservoir of individual molecules dispersed throughout the polymer melt.
For a bubble in a single component fluid, the growth process is controlled by a succession of forces. Initial growth from the critical size is dominated by surface tension forces and very slow. As inertial forces become dominant, growth is driven primarily by the difference between the vapor pressure within the bubble and the external pressure, and the growth rate is linear with time. Finally, evaporation caused by heat transfer from the surrounding liquid to the lower temperature of the bubble wall predominates, and the growth rate becomes proportional to t½.
In a liquid-gas solution, the gas concentration is a critical variable, and diffusion is often the controlling factor. A simple diffusion-controlled bubble growth problem, neglecting surface tension and convective mass transport, was solved analytically by Epstein and Plesset [40], who found the growth rate of a spherical gas bubble in an oversaturated solution to be
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(18) |
where D is the coefficient of diffusivity and S the gas solubility. An approximate solution to this equation is [37]
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(19) |
where γ2 = S[(P0 / PV) − 1]/2π. The rate of change of volume (4π / 3)R2 dR / dt for a bubble under fixed conditions of D, S, and P0 / PV is therefore approximately proportional to R.
A rigorous model of diffusion-induced growth in viscoelastic fluids including convection, diffusion, surface tension, and inertial effects for a variety of viscoelastic constitutive relations was performed by Venerus et al. [41]. This analysis showed that the bubble growth rate in a viscoelastic fluid is bounded below by growth in a Newtonian fluid and above by diffusion-controlled growth, and that the effects of nonlinear fluid rheology are minor relative to elasticity effects.
This and other literature on diffusion-induced growth could be used to develop a reasonable partitioning into bubbles and solution for gases generated by polymer degradation at high temperatures. Alternatively, a simpler assumption of instantaneous distribution to bubbles has a theoretical and experimental basis for melts with elastic properties.
Devolatilization experiments performed by Yarin et al. [38] on a variety of polymers with shear-thinning behavior showed a large number of "microblisters" a few tenths of micrometers in size located on the walls between large primary bubbles one or two magnitudes larger in size. This is a secondary nucleation phenomenon, in which fluid motions due to the growth of a primary bubble in a polymer melt cause deformation of nearby macromolecules and introduce elastic stresses. The accumulation of elastic energy associated with these stresses leads to rapid mechanical degradation of nearby macromolecules and lowers the free energy barrier for the formation of new bubble nuclei. These secondary nuclei grow due to diffusion or internal gas pressure and may coalesce with the primary bubble, considerably accelerating the effective growth of this bubble. Theoretical analysis confirms a high secondary bubble nucleation rate in the neighborhood of primary bubbles larger than a few micrometers. When stress relaxation in the polymer melt is included, the nucleation rate remains high as long as the growth rate of the primary bubble is sufficiently large. Significant secondary nucleation is therefore expected to occur over a limited period in the development of a primary bubble, when both its size and growth rate are sufficiently high. During this time period, theory [38] shows that it is possible for secondary bubble nucleation and growth to occupy a volume as much as two orders of magnitude larger than that occupied by the primary bubble, resulting in flashlike devolatilization in agreement with experiments.
Making the simple assumption for this model that gases are instantaneously transported to bubbles upon generation, then, is a reasonable assumption, especially for polymers such as PMMA that retain long macromolecules during heating and therefore maintain strong elastic properties.