4.2.4 Bubble Bursting

A critical factor determining the effects of bubbles on the thermal and mass transport behavior of burning thermoplastic materials is the behavior of the bubbles upon reaching the sample surface. If the bubble bursts immediately, the effects are limited to internal transport by the individual bubble. A significant delay in bursting not only slows the delivery of volatile gases to the flame, but could also lead to the development of a foamy insulating surface layer and/or the formation of large bubbles with violent bursting events. Although not addressed by this model, the debris expelled by bursting is of particular interest in microgravity, where gases forceably released by an unattached burning object propel the object in the opposite direction, and where burning droplets may be expelled at high speed.

Upon reaching the surface of a liquid, a bubble may not burst immediately. The deformation of the bubble and the surface interface forms a thin film that must drain and rupture in order for the bubble to release its gases into the surroundings. The physics on the microscopic scale of the liquid membrane is quite different from that controlling the macroscopic bubble motion. At these thicknesses, Londonvan der Waals and surface tension forces become very important in determining fluid behavior. As was mentioned in the previous subsection, the shape of the thin film is critical to the drainage process. Films whose thinnest point is in the center drain rapidly, while those that are thinnest in a ring near the rim, or "dimpled", are much slower. A dimpled film can develop with time during drainage or be formed at the initial stage. The presence of a surfactant immobilizes the fluid in the film, causing slow drainage regardless of shape.

A theoretical analysis of coalescence time for a draining film between two bubbles in a system without surfactant has been developed by Li and Liu [44]. For a bubble approaching a large planar surface, their expression in CGS measurement units¹ is

(25)

where σ is the surface tension between the polymer and gas, κ = µ_g / µ_p is the ratio of viscosities, B is the London-van der Waals constant, and F is the force on the bubble. The parameter K is the curvature of the thin film at the point along the rim at which the pressure within the film equals its local hydrostatic value, calculated as a function of the mobility coefficient M:

where M = R_B / (κR₀) with R₀ the initial rim radius of the film (R_B / R₀ ≈ 4).

For immobile films (viscosity ratio κ → ∞, resulting in M → 0), the second term in equation (25) is negligible. However, for a bubble in a pure liquid without surfactants, κ   1, and the second term dominates. The mobility coefficient M is large in this case, and the value of K is about 16. Estimating other parameters as B ≈ 10⁻¹⁹ erg-cm, σ ≈ 50 dynes/cm, and µ_g ≈ 2 x 10⁻⁴ poise, neglecting the first term, and substituting equation (20) for F, the bursting time is approximately

(27)

Large bubbles therefore take longer to burst, as do bubbles travelling with higher velocity. As the melt viscosity decreases with the rising temperature, the bursting time decreases.

Using these expressions, Li and Liu found that coalescence times were on the order of milliseconds for bubbles in a pure liquid in agreement with experiment, as compared to hundreds of seconds for immobile films. Since the effects of thermal degradation of the film are not considered in this coalescence model, however, these timescales are likely to be considerably longer than those actually encountered during bubbling pyrolysis.

The bursting process is accompanied by release of a spray of tiny film droplets around the rim and sometimes by the forceful expulsion of one or more large jet droplets from the rebounding center [22], [23]. In a normal gravity environment, these liquid droplets would fall back into the sample; however, in microgravity the loss of material is permanent. Since these phenomena are not accounted for in this model, the rate of mass loss is expected to be underpredicted.

¹The policy of the National Institute of Standards and Technology is to use SI units of measurement in all its publications. In this document, however, the CGS system of units is sometimes used because of the scales and ranges of the quantities and the wide use of such units in the materials modeling field. For clarification, erg = 10⁻⁷ J, dyne = 10⁻⁵ N, and poise = 0.1 N-s/m².