5.3 Bubble Growth

To consider the effects of bubbles on the transport of gases to the surface of the sample, the full amount of gas generated in an element during one timestep is considered to instantly diffuse to any bubble extending within the element. This is a simplification that is also consistent with an assumption that secondary nucleation due to elasticity in the melt is significant, as discussed in Section 4.2.2. Bubble dynamics then determine the movement of the gas.

In reality, of course, gases diffuse to a nearby bubble, rather than one that may be located on the opposite side of the spherical shell element. The treatment used here is equivalent to assuming that the rate of gas added to a bubble is determined solely by the temperature at its center, and it neglects the complexity of diffusion from the bubble's nonuniformly heated surroundings. Since more gas will be generated in the hottest region around the bubble, the assumption of a uniformly expanding bubble introduces a bias that slows bubble migration toward the heated surface.

If an element contains more than one bubble, the gases produced during the timestep must be distributed in a reasonable way. For the simple diffusion-controlled bubble growth from equation (18), the growth in bubble volume is linear with radius. An argument can also be made for distributing gases by bubble volume. According to secondary nucleation theory, the rate of change of the mass _s of gas within the secondary bubbles is proportional to the volume of the primary bubble times the nucleation rate I, which increases strongly with radius both in the pre-exponential factor and in the exponent [38]:

(37)

If secondary nucleation is a dominant factor in this problem, therefore, the gas volume generated in a given element should be distributed among all bubbles within the element in proportion to their volume.

Another approach to distributing gases to bubbles within an element must be taken if the model includes homogeneous nucleation. If bubbles nucleate within an element that already contains bubbles of finite size, distributing gases by bubble size alone eliminates growth for the new bubbles. An alternative distribution method considers any bubble within the element to draw all gas within a zone extending a radial distance ΔR_z from the surface of the bubble. The thickness of this diffusion zone for bubbles within a given element is calculated such that the total volume within all zones contains all of the polymeric melt. Distribution of gases is then made according to the fraction of the polymer volume represented by the diffusion zone around an individual bubble.

Options are also available for distributing gases evenly among bubbles and for distributing by bubble surface area.