Experimental study of molten bubble layer growth and bubble size distribution in post-flame-spread PMMA samples

Although thermoplastics possess many desirable traits as solidified materials, they are combustible and burn readily. Post burning samples of polymethylmethacrylate (PMMA) are the focus of this study. During flame spread, PMMA forms a bubble containing molten surface layer that influences the spread rate and the surface mass efflux. This study examines the formation and distribution of bubbles inside a PMMA sample that has previously been subjected to flame spread and then re-hardened into its solid state. Experiments discussed herein were conducted in a Narrow Channel Apparatus (NCA), which reduces the influences of gravitation (buoyancy) on flame spread. Bubble sizes and counts were determined using digital image analysis (DIA). The burnt samples were analyzed by dividing sample images into eight equal area sections. Frequency distributions of bubble size (area) were compared. Distribution functions were fitted against the empirical probability density function (PDF) for the bubble size distribution. The log-normal distribution predicts the bubble size distribution for all segments. The bubble distribution function can be used to describe physical processes inside the polymeric material as it undergoes thermal transformation by the spreading flame.


| INTRODUCTION
Thermoplastics have uses in the construction industry, in transport vehicles, in the electro-technical industry, in aircraft, and in common household materials. However, they are usually highly flammable and therefore constitute a potential fire hazard. Thermoplastics often support uniform, regular, measurable rates of flame spread and thus are commonly used as test materials.
The combustion of thermoplastics is a complicated process that involves both solid and gas phase reactions. When heated, thermoplastic solids can form internal bubbles, undergo bubble bursting, exhibit sputtering, or swell. temperature line, where T = T g , moves into the solid phase, creating a finite-thickness melt layer in which T > T g , and inside which the polymeric thermoplastic degrades by a process called unzipping into monomeric units. At the high temperatures T > T g in the molten layer, the superheated liquid monomer generates vapor nucleates, some of which become bubbles. As additional liquid monomer vaporizes at the bubble/liquid interface, the bubbles grow larger until they begin rising in the liquid layer toward the heated surface. The dominant mechanism of bubble rise is the surface tension gradient, which is itself a consequence of the temperature gradient in the liquid monomer layer.
Ultimately, the bubbles transport gaseous volatile monomer molecules, that is, fuel, to the surface where they burst, meet the surrounding oxidizer, react, and support further flame spread. The molten bubble layer can reduce the effective thermal conductivity of the condensed phase due to the generation of low-conductivity gases near the thermoplastic surface, which may serve to slow the flame spread process. Nevertheless, whether the spread process is slowed or not, the transport of volatile fuel species to the surface of the sample supports continued flame spread over the material surface.
Concerning the class of materials of interest, attention here is restricted to bubble formation in non-charring thermoplastic polymethylmethacrylate (PMMA). The constituents of PMMA thermal degradation (or decomposition) are approximately 80% methylmethacrylate monomer (MMA) when the temperature is greater than 500 C. 1 Cast PMMA does not char or drip during combustion or flame spread, even in downward spread. By contrast, extruded PMMA drips because its aligned polymer chains produce a relatively low glass transition temperature (375 K) compared with a pyrolysis temperature of approximately 670 K. 2 Therefore, cast PMMA yields a harder, sturdier, more homogeneous material with a higher molecular weight and longer polymer chains.
Concerning testing configuration, the PMMA samples in this work were tested in the horizontal opposed-flow flame-spread configuration.
Concerning sample thickness, a fuel specimen can be categorized using its actual thickness τ along with the preheat length L s defined according to where α g is the thermal diffusivity in the gas, α s is the thermal diffusiv-

| EXPERIMENT DESCRIPTION AND IMAGE ACQUISITION
Flame spread experiments were conducted on thermally thick samples of PMMA. Section 3.1 describes the original flame spread facilities and testing procedure. In Section 3.2 the bubble examination process, including visualization, is described.

| Flame spread facility and tests
The flame spread experiments were conducted in the horizontal MSU Narrow Channel Apparatus (NCA). The setup of the test facility and diagnostics is shown in Figure 1. During each flame spread test the sample was surrounded (two sides and bottom) by 1 in thick Marinite insulation boards, see Figure 1: the thermal conductivity of calcium silicate ranges between 0.05-0.12 W/mK.
At the downstream side of the sample (with respect to the opposed oxidizer inflow), a 28-gauge Kanthal wire connected to a DC power supply was used to ignite the sample. For improved ignition the trailing edge was machined into a pointed shape, as discussed previously 12,13 and as seen in Figure 2B After the samples had cooled to room temperature it was possible to analyze the hardened (frozen) sample bubble distribution, the sample shape and other quantities related to the material response. Procedures related to the bubble distribution are described below.     Figure 4B shows the normalized bubble count for the three samples. In addition to largely eliminating the offset in Figure 4A Each identified bubble area is summed in order to compute the total bubble area for a given segment. A fraction of the surface area is occupied by bubbles. Figure 5B shows the percentage of bubble area for the three samples. The percent of bubble area calculation shows that segment 1 has the maximum bubble count as well as the highest bubble area percentage, approximately $22% for all three samples. Figure 5B shows a similar trend as Figure 5A: the overall percentage of bubble area lies between 12 and 22%. This is sufficiently low that it avoids substantial bubble overlap in the images, which could skew the observed size distributions.

| Bubble volume
In the previous two subsections, the bubble count and area were measured by viewing the projection of the bubbles from the sample rear, as seen in Figure 2B. The volume of an individual bubble can be calculated when its shape is identified, which is not possible from the current 1-sided, 2D non-intrusive measurements. The theoretical methodology used to estimate the bubble volume from the bubble area is now described.
Side observation as shown in Figure 2B revealed first, that the bubbles reside within a surface layer, and second, that they can be broadly categorized into two classes of shapes: (1) round bubbles that can be assumed to be spherical and (2) elongated bubbles that can be considered to be cylindrical and extending vertically through a cylinder height that is approximately equal to the molten layer depth. The average bubble diameter for these two postulated geometries is determined from the projected (surface) area.
The projected area is that of a circle whose diameter represents an overall average characteristic length. The bubble count and area are used to calculate this characteristic diameter. The equation for the average bubble diameter d avg is where A is the total bubble area in mm 2 and N is the number of bubbles. Figure 6 shows the average bubble diameter for all three samples and all eight segments. Sample 1 shows an increase to segment 3 followed by a plateau for the remaining segments 4-8. Sample 2 shows an increase to segment 5 followed by a slightly higher plateau for the remaining segments 6-8. Sample 3 shows similar behavior to sample 1 (peak at segment 3), though a higher average diameter, followed by a plateau and then a fall-off over the final two segments 7-8. The overall behavior is similar but the plateau diameters range from 0.29 cm for sample 1 to 0.32 cm for sample 2 to 0.36 cm for sample 3.
The average bubble diameter is used to calculate the total volume of the bubbles in each segment. If all bubbles are assumed to be spherical, the equation that determines the total bubble volume is If all the bubbles are assumed to be cylindrical the total bubble volume is where h b is the average molten layer depth. This average molten layer depth in all different segments varies between 3 and 4 mm.
The common factor appearing in Equation (3) and Equation (4) is 1 2 πd 2 avg N. Therefore, the bubble volume is dictated by the factors 1 3 d avg and 1 2 h b when the postulated bubble shapes are either spherical and cylindrical respectively. In this work, these quantities are called the "shape factors". As the h b $ O(10 0 mm) > > d avg $ O(10 À1 mm), the shape factor for assumed cylindrical shape bubbles is higher than spherical shaped bubbles.
The higher shape factor results in a significantly higher bubble volume for the assumed cylindrical shape. In actuality, the bubbles are neither always spherical nor cylindrical, but some fraction of these shapes assuming these are the dominant two. However, it is not possible to know their respective counts from the current work, which measures only their projections. In order to estimate the bubble volume, a fraction of the bubbles, 0 ≤ η ≤ 1, is assumed to be cylindrical. Based on this assumption, the total bubble volume can be estimated from the following equation, Finally, the percentage of total bubble volume in each segment, ψ, is given by the ratio of total bubble volume, V b and the volume of a segment, V slice F I G U R E 6 Average bubble diameter versus the segment number. The blue circles, red triangles, and green rectangles denote samples 1, 2, 3 respectively. The blue, red, and green dashed lines are the average between segments 2 to 8 for samples 1, 2, and 3 respectively. The error percentages (variations in average diameter) from segments 2 to 8, based on the least-squares method, are 1.7%, 3.8%, and 4.3% for samples 1, 2, and 3 respectively. Note the relative plateau for all samples after segment 1, as the errors are relatively low where w b is the sample segment width, l b is the sample segment length, and t b is the maximum molten liquid layer depth. Figure 7 shows the percentage of bubble volume for all three samples. Here, η = 0.2 is used as a visual compromise between a 10% fraction, which seemed too low, and a 33% fraction, which seemed too high. This figure exhibits a slow decay along its length from 6% to 3% for all samples.

| Bubble statistics
As previously mentioned, three PMMA samples designated herein as  Figure 8 shows the variation of the bubble cross-sectional area distribution in the ROI of sample 1 that was shown in Figure 2D. To construct the frequency distributions, the discrete bubble sizes were tabulated in ascending order of area. Then they were grouped into size bins of 5 Â 10 À3 mm 2 over the range 2 Â 10 À3 À 52 Â 10 À3 mm 2 . The smallest area here corresponds to a total of four camera pixels; measuring below this area was not possible. For the sake of clarity, bubble areas greater than 52 Â 10 À3 mm 2 were eliminated due to their statistical

| Probability density functions (PDF) for bubble area distributions
The empirical Probability Density Function (PDF) for sample 1 was estimated for each frequency distribution. The histogram of Figure 8 shows that the bubble area distributions are typically non-normal, positively skewed, and positive-definite (areas are always positive). Figure 9 shows the estimated PDF as a function of bubble area. Here, the PDF is determined from the histogram of the bubble sizes.
Equation (7) below is used to compute the empirical PDF: The bin width is 5 Â 10 À3 mm 2 for all segments. For clarity, the total number of bubbles in every segment is indicated in the legend.
In Figure 9 the PDFs in each bin are plotted against the median of the corresponding group.
It is seen that the sample 1 PDFs are non-symmetrical and strongly skewed toward smaller rather than larger bubble areas. In addition, the trends are nearly identical for all segments. Although the frequency distribution shows that segment 1 has the maximum number of smallest size bubbles, the PDF that contains the smallest range of bubble sizes for segment 1 is the minimum for this bin. This result demonstrates that the frequency of smaller size bubble formation is higher in segment 1 than in any of the other segments, because this minimum is still larger than all of the other frequencies shown.
where n is the number of data points, i is the variable, x i is the empirical PDF from the experiment, and b x i is the estimated PDF from the distribution function.  The calculations of the preceding section are extended to calculate the bubble average area from the PDF of the bubble distribution. By the above described fitting procedure (Section 4.2.3), it was determined that the bubble area distribution was best described with the log-normal function. The PDF equation for this distribution function is where A is the bubble area, μ and σ are the mean and the standard deviation of the bubble area's natural logarithm. By multiplying this PDF by the bubble area and integrating over the assumed total area, the average bubble area is calculated from Equation (11)  for each segment. Here, the average bubble area was analyzed over the range of (2-52) Â 10 À3 mm 2 due to the statistical insignificance of the outlying ranges below and above these limits.
Similarly, the bubble average area can be determined from the experimental measurement. The average area will be the total bubble area over the total number of bubbles.
Average bubble area ¼ Total bubble area of a segment, A Total bubble count of that segment, N : ð12Þ A semi-empirical scaling analysis was conducted in order to estimate the bubble volume. First, an average diameter is determined from the bubble count and area. Then, two different postulated bubble shapes were considered, the sphere and the cylinder, in order to calculate the bubble volume and the bubble volume fraction in each segment.
Since the segment thickness τ is of order 10 times the molten layer thickness (i.e., sample thickness τ ≈ 24.5 mm and molten layer thickness ≈3À4 mm) this analysis focuses on the molten layer that contributes to the volatile mass efflux from the degrading thermoplastic.
The bubble area frequency distribution of the post-burn PMMA samples, characterized herein as "frozen," is determined from experimental measurements for the eight segments. Each histogram is positively skewed and the distributions (empirical PDFs) are all qualitatively similar. Segment 1 has a higher number of bubbles compared with the other segments. The PDF from the empirical histogram is compared with fittings provided by three distribution functions. A two-parameter log-normal function best described the density functions for bubble areas less than 0.052 mm 2 . Bubble sizes above and below this value were considered statistically insignificant due to low count numbers.
The authors believe that it will be increasingly necessary in future research to address the changes in material structure and morphology during flame spread, and to somehow categorize these changes in F I G U R E 1 2 Performance of the three distribution functions fitted to the PDF of the bubble area in terms of the Root Mean Square Deviation (RMSD) (see Equation (8)). Eight different markers are used to indicate the eight (8) separate segments F I G U R E 1 3 Average bubble area for sample 1 from experimental measurement and PDF integration. Note the plateau from segment 3 onward terms of the material chemical composition. Work on these questions has begun for charring, crack-forming materials. [14][15][16] ACKNOWLEDGMENTS This research was funded by National Aeronautics and Space Administration Cooperative agreement NNX16AC24A for which the authors are grateful. The comments by the reviewers were very helpful, and much appreciated.

CONFLICT OF INTEREST
The authors have norelevant conflicts of interest to disclose.

DATA AVAILABILITY STATEMENT
Peer review of empirical data will be conducted to confirm the quality of the shared data, for example, that sample sizes match, that the variables described in the article are present as fields in the data repository, that data is complete; that data is properly labelled and described; and that it has the appropriate metadata for the kind of data being shared.