5.1 Geometry

The geometry of the spherical sample is shown in Figure 6. The sample is divided radially into N − 1 elements bounded by N nodes forming concentric shells around the center of the sphere. Element thicknesses and node locations change with time, so that the radius of the sample at any time t is given by r_N = r_S(t).

At time t = 0, the sample radius is r_N = r₀, with a total volume The sphere is assumed to be homogeneous initially, and each of the N − 1 elements is assigned an equal volume,

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This sets the initial node positions. With node 1 at the center, r₁ = 0, node i is initialized at the radial location

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for i = 2, 3,..., N. Element i is bounded by nodes i − 1 and i.

Each element consists of a material volume of polymer plus the gases contained within that region at that timestep. As the polymeric sample gasifies, the nodes defining the element relocate to enclose whatever remains of the polymer originally contained within. For purposes of calculating the bulk material properties of the element, the gas contained by all bubbles within the element is assumed to be evenly distributed throughout the entire spherical shell of the element. For purposes of determining the transport of gases through the sample, however, each bubble k is represented by a sphere of radius R_k located at (r_k, θ_k, φ_k), as shown in Figure 7. At the time of "nucleation", the new bubble is assigned a radial location within the element, and its volume is set to zero. The angular coordinates (θ_k, φ_k) of the bubble are selected randomly from a distribution for which bubbles are uniformly scattered within the spherical sample volume.

Figure 6: Radial geometry for an element within the spherical sample.

Figure 7: Location of bubble k within the spherical sample.