Jim Throne on September 1st, 1998

Thermoplastic structural foams have been around for several decades now. We all know that, at the same part weight, the stiffness of the structure increases in proportion to the reduction in plastic density. And that at the same part thickness, the weight decreases in proportion to that same reduction in plastic density. Injection molders and extruders have developed new equipment and methodologies to produce structures with uniform density foam and structures with skins. Even blow molders and thermoformers have produced foam parts.

So, why do rotomolders have such a difficult time foaming? After all, we start with powder that is sinter-melted. And as everyone will attest, our biggest problem is getting rid of the bubbles! So, why do we struggle?

In an ANTEC paper (ANTEC 2000) and a RETEC paper (Cleveland 2002), I’ve detailed the dichotomy between the rotomolding process conditions and the conditions necessary to produce controlled bubble nucleation and growth. Furthermore, I’ve described the entire field of foam technology in my 1996 book, Thermoplastic Foams, unfortunately now out of print. So I will not go into these details in this note.

Instead I want to focus on how I believe the rotomolding process needs to be changed in order to produce quality foam parts.

To begin with, consider the various ingredients in the recipe. In general everyday rotational molding, we like a polymer that has relatively little melt elasticity. We know from results of many research papers that elasticity slows the sinter-melting process. Unfortunately, for foams, we need a fair amount of elasticity. The polymer must resist the internal gas pressure of the growing bubble. If it doesn’t, the foam will collapse. I may return to this point later.

Then the blowing agent needs to be properly selected. The blowing agent used in rotational molding is a pure chemical that decomposes to produce the blowing gas at a very specific temperature range. It is imperative that the blowing agent decompose in a temperature range greater than the melt temperature range of the polymer. If the blowing agent decomposes while the sintering step is in progress, the gas will simply escape to join the mold cavity air. I know that many people are using 4,4′-oxybis (benzene-sulfonyl) hydrazine, OBSH or OT, for polyethylene. I prefer azodicarbonamide or AZ. The decomposition temperature for OBSH is 160oC and the gas yield is only 125 cm3/g, whereas the decomposition temperature for AZ is 195oC and the gas yield is 220 cm3/g. The higher decomposition temperature gives me a temperature “cushion” for LLDPE and HDPE. And the higher gas yield means that I don’t need as much blowing agent, which helps reduce the cost.

And the third element is the rest of the ‘stuff’ that we add to the mix. One must be very careful of certain colorants, especially carbon black or iron oxide pigments. These additives can act to adsorb the blowing gases or complex with the chemical blowing agents, thus reducing the blowing agent efficiency. Sometimes dramatically.

Now that brings us to another aspect, the nature of the powder. In other disciplines, such as injection molding, the finely divided blowing agent powder is sometimes added directly to the polymer pellets at the hopper of the injection-molding machine. Although this is not recommended, it is done to determine the optimum level of blowing agent needed to achieve a given density reduction. In rotational molding, the blowing agent must be compounded into the polymer prior to grinding. It is the only sure way of getting uniform dispersion of the sometimes very sticky blowing agent powder.

The nature of the foamed structure depends on its end-use. Foams, in general, have crappy surfaces. [For cosmetically desired surfaces, a two-step process is recommended. Here I’ll only discuss the single-foam process, I’ll deal with the ‘two-step’ in a later technical note.] Does this mean you cannot make an acceptable structure by simply pouring compounded powder into the mold’ It means that you’re going to work a lot harder to achieve a quality surface. And you’ll probably never achieve the surface of an unfoamed rotationally molded part.

Okay, so the only things we need to do is compound the blowing agent into the polymer, grind it to the right size, and drop it in, right? Wrong! For some reason, the “instant gratification” or “instant results” mentality pervades the rotational molding industry. Remember the injection molder? And the extruder? These guys worked dozens of years to perfect their process. In those years, they invented dozens of processes, most of which were not very successful.

I’ve consulted with several rotomolders who insist that “you’ve gotta make it work! Now!” When I recommend process changes they are aghast at my recommendations. I’ve even had a couple refuse to pay me when I told them how to make good quality one-step foam. Their response to me? “That’s ridiculous!”, “That’s stupid!”, “We can’t afford another second on cycle time!” And so on.

So, since many of you won’t hire me or even pay me to help you make quality one-step foam, I’ll give you my approach for free! So listen up!

First, invest in quality tooling. Sheet metal just won’t hack it. One plugged vent tube and the internal gas pressure will turn your part into a sphere. Then increase the wall thickness of your mold. A lot. And replace that inane Teflon tube with the fiberglass that you call a vent. Use a pressure control valve arrangement so that you can control the inner mold pressure.

Wait! You’re just starting! Then make certain that you are using AZ rather than OBSH with LLDPE. And that you have no gas-absorbing pigments compounded in. And use a blowing agent that liberates nitrogen, not carbon dioxide! After all, 79% of the air already in the mold cavity is nitrogen! And then there’s this thing about concentration gradients! Never mind! Just use AZ! It works!

Now we get to the process changes. Run an empty mold into your oven. Measure the time-dependent mold temperature with a hand-held infrared pyrometer. Record the time when the mold temperature gets to 180oC. Now program your oven to hold that temperature for several (or many) minutes. And program your mold inner air pressure to 10 to 15 psi (or whatever your mold is designed for).

Now program your oven to increase the measured mold temperature to 210oC at the rate of 5oC/minute. Now have your oven hold that mold temperature for several (or many) minutes. [Note that I am saying the mold temperature, NOT the air temperature!] Finally, program your system to remove the arm and mold assembly from the oven to quiescent (still) air for several (or many) minutes.

With this profile stored, you are now ready to run your first part. Obviously you will need to play with the hold times, “the several (or many) minutes” things I mentioned earlier. And you’ll also need to play with the cooling sequence. If you hit the hot foam too quickly with cold ambient temperatures, it will collapse. Just like momma’s cake or your first attempt at a soufflé.

Now if you’re like most rotomolders, you’re already screaming at my suggestion of lengthening the already-absurdly-long oven cycle time. “That’s ridiculous!. that’s stupid!” And you’re screaming that your customer won’t stand for the increased cycle time cost or the added expense in the polymer. And you’ll probably rail at me for even suggesting that you have the blowing agent compounded into the polymer. Or that you spend money for more robust molds. Or pressure control valves. “We just can’t afford it!”

And, like the other rotomolders who have ignored my suggestions, you’ll struggle to produce parts without uneven quality surfaces, coarse cell structure, nonuniform density reduction, breakthroughs, blow-outs, and huge voids.

But, if you really want to make quality foam structures, I’m available to guide you through these steps. For a consulting fee, of course.

[In my next epistle, I’ll discuss doing the ‘two-step.’]

Jim Throne on August 16th, 1998

Isotactic polypropylene (iPP or usually just PP) is considered a crystalline commodity polymer of the polyolefin family. It has the following basic molecular structure: (CH2 – CHCH2).

Even though the molecule appears to have a bulky pendant group, there is substantial backbone mobility, allowing the molecule to twist in such a manner as to present a smooth and branchless albeit somewhat bulky molecule. Crystallinity is typically about 65% and crystallization rate is quite slow, probably due to the bulkiness of the molecule. PP crystallizes at about 0.4% the rate of HDPE, as measured at 30oC below their respective melt temperatures. Like polyethylene, PP is translucent in its unpigmented, unfilled state. The translucency is the result of crystallites or spherulites that are larger in dimension than the wavelength of light [0.4 to 0.7 micrometers], thereby interfering with light transmission. Since PP crystallization level is less than that of HDPE, light transmission is greater than that of HDPE. As a result, PP is said to have “contact transparency”. In addition to non-transparency, homo-PP has a relatively high glass transition temperature of about 5oC, and thus has marginal low-temperature [-40oC] impact strength.

PP is considered a competitor to impact polystyrene, PVC and APET in many thin-gage thermoforming applications. It is priced competitively in $/lb with the competition, but its 900 kg/m3 density is 83% of that of PS and two-thirds of that of PVC and APET. Its modulus, about 200,000 psi, is typically about half that of its competitors, at 400,000 psi. PP finds applications where the product must sustain high temperature [100oC] and aggressive environmental conditions.

Traditional PP, usually called homopolymer PP or just homo-PP, can be made quite viscous in the melt state, with fractional melt flow indices [MFIs]. But unlike HDPE, homo-PP has relatively little melt elasticity. It is well-known that thermoforming is basically a solid phase deformation process that depends on the elastic character of the polymer sheet [see J.L. Throne, Technology of Thermoforming, Hanser, 1996 for more information]. Typically, when homo-PP is heated to its melting temperature range of about 165oC, it moves from a stiff-rubbery solid to a floppy, syrupy liquid in only a few degrees, Figure 1.

PP VS PS

It has been said that homo-PP thermoforming window ranges from a degree or two to nothing, depending on the former and the source of the polymer. “Solid phase pressure forming” or SPPF was introduced in the 1970s as a way of pressure-forming thin-gage homo-PP sheet into relatively simple shapes such as drink cups and unit dose cups. SPPF has been probably most successful in the production of hot-fill individual serving juice containers. Technically the process, described elsewhere [Throne, ibid., p. 710+], requires extremely careful monitoring and control of the sheet temperature during heating, and requires 50 psi or more air pressure in order to achieve adequate draw ratios. As a result, homo-PP has never been a widely accepted thermoformable material 1.

Modifications to Homo-PP

The earliest modifications to homo-PP to improve its melt strength or melt elasticity focused on copolymerization with ethylenic molecules to produce ethylene-propylene copolymers. Typically, EP copolymers have lower melting temperatures [about 158oC v. 165oC for pure PP], less ESCR, and greater cost, but superior sag resistance and good to excellent low-temperature brittleness.

Fillers were other early modifications to homo-PP to improve its hot strength. Talc, calcium carbonate and titanium dioxide [TiO2] at dosages of 10% (wt) to 20% (wt) yield improved stiffness at PP melt temperatures, as well as improved room temperature stiffness. It has been shown that, with appropriate coupling agents and compatibilizers, PP will accept filler loadings of 60% (wt) and in certain cases, even more. It must be remembered that fillers do not usually alter the morphological characteristics of the polymer, meaning that the melt temperature and glass transition temperature of homo-PP remain essentially unchanged by filling. If, for example, unfilled homo-PP low temperature impact strength is unacceptable for a given application, filled homo-PP will also be unacceptable. Typically, fillers do not lower the cost of the polymer. This is especially true with homo-PP, since there is a great disparity between the specific gravity of the homo-PP and the filler, and since homo-PP requires coupling agents in order to get the very smooth organic PP molecule to adhere to the inorganic filler.

Metallocene PPs seem to offer potential benefits, since their morphological structure apparently can be readily tailored to create higher melt strength, much like MDPEs. At this point, these polymers are relatively new and their costs are not yet in line with other modified PPs.

High-melt strength PPs have been under development for a few years now. Currently, it appears that Montell and Amoco have competitive resins available for thermoformers. The general characteristics of these PPs seem to be short-chain branching, much like HDPE. The short-chain branching yields much greater melt strength and therefore, less sag and much more controllable sag, than homo-PPs. There is an indication that the crystallites that are formed during cooling are smaller and more perfect than those obtained with homo-PPs. Smaller crystallites yield containers that are less opaque than homo-PP containers, and the containers are tougher.

As noted above, the crystallization rate of homo-PP is quite slow. Extensive work on nucleating PP to accelerate the crystallization rate shows that finer crystallites yield better stability of the PP after the sheet has been formed and is cooling on the mold surface. This has recently been verified by Millikan Chemicals Corporation’s development of Millad 3988, which yields a near-haze-free transparent container. The effect of sorbitol on crystallization rate of PP is usually shown as an increase in peak recrystallization temperature, as measured with Differential Scanning Calorimetry as the test specimen is cooling from the melt 2. This is seen in Table 1, taken from Millikan Chemicals datasheets. In addition to Millikan Chemicals, polymer suppliers such as Amoco, are working on or have developed similar nucleating adducts.

Table 1
Adduct Effect on Recrystallization Temperature of
Polypropylenes
Homopolymer
Adduct Xstal Temp Loading Level
No clarifier 102oC
Dibenzylidene sorbitol 115oC @ 1800 ppm
Methyl dibenzylidene sorbitol 120oC @ 1800 ppm
Millad 3988 121oC @ 1200 ppm
Copolymer
No clarifier 92oC
Dibenzylidene sorbitol 105oC @ 1800 ppm
Methyl dibenzylidene sorbitol 107oC @ 1200 ppm
Millad 3988 108oC @ 600 ppm

The Heating of PP

As already noted, PP requires careful heating, no matter whether the polymer is the basic homopolymer or a highly modified one. If we consider the nominal polymer forming temperature to be 165oC, the amount of energy required to raise PP from room temperature of 25oC to a nominal forming temperature of 165oC [140oC temperature increase] is about 100 cal/g or about 180 Btu/lb. However the energy uptake is definitely nonlinear. It requires only 50% of that energy [about 50 cal/g or about 90 Btu/lb] to heat to 130oC [a 105oC temperature increase, or about 75% of the total temperature increase], and the additional 50% to heat the remaining 25% of the way to the forming temperature. Furthermore, it only requires 65 cal/g or about 120 Btu/lb to heat PS or APET to the same forming temperature [65% as much energy] and only 55 cal/g or about 100 Btu/lb to heat PVC to the same forming temperature [55% as much energy]. This means that if the heating cycle for more traditional amorphous polymers [PS, APET, PVC] is optimized, the processing cycle for PP will need to be increased by as much as 50% 3. Of course, increased heating time can be accomplished by using an auxiliary preheater. This unit is placed between the take-off station and the initial engagement of the pin-chain rails.

Keep in mind that the energy added to the sheet during heating must be removed during cooling. Thus if the heating cycle is longer for PP than for amorphous polymers, the cooling cycle, too, must be longer 4. This point will be discussed shortly.

The final thermal conditioning of the sheet is extremely delicate for all but the most highly filled PPs. The primary concern is uncontrolled and uncontrollable sag. Sag bands are continuous polyfluorocarbon-coated wires that are slaved to the rail and index when the rails index. They are used where appropriate. PP is quite sticky when heated above its melt temperature and care must be taken to ensure that the sheet releases cleanly from the sag band as it is indexed into the mold region. Even momentary sticking can cause excessive webbing, non-uniform wall thickness distribution, and rejected parts. Typical forming cycle times are on the order of 3 seconds for 20 mil sheet to 7 seconds for 34 mil sheet.

Mold Design Considerations

As noted, PP is sticky and typically very weak when molten. The sheet can easily tear when contacted with a high-speed plug assist. Heated aluminum plugs seem to work best so long as the plug temperature is very carefully controlled. However, polyfluorocarbon-coated epoxy foam plugs, sometimes called syntactic plugs, are more popular, even though plug mark-off is noticeable.

Because solid PP is slippery-smooth, the mold surface should not be. Parts that are formed on molds with very smooth surfaces may exhibit a series of horizontal banded lines owing to the sheet alternately sticking, then slipping on the mold surface. Large, flat expanses on molds with very smooth surfaces will trap air between the PP sheet and the mold surface. Parts with air trap will have surfaces with characteristic dimples and highly shiny regions.

Grids or cavity isolators are requisite with PP to prevent excessive sheet drawing from one cavity to another. Although many small containers are made with flat lands or lip regions, peripheral cavity dams or moats and dams help minimize the slippery polymeric sheet from being drawn non-uniformly into the mold cavity. Coining or pressing the sheet between two metal surfaces is recommended if a perfectly flat, close-tolerance lip is needed, for heat-sealing for example.

Female molds are recommended with PP. PP shrinkage is about 3 to 8 times that of amorphous polymers such as PVC, PS and APET. Substantial draft [5o for smooth wall parts, more if the mold wall is textured] is required for PP forming on a male mold.

It is recommended that PP be formed against a heated mold, with the mold temperature being on the order of 100oC or so. As noted, the recrystallization temperature range is on the order of 110oC to 120oC. Cooling times become large at mold temperatures above this. Below this temperature, recrystallization may be so slow that the part is released from its fixture, the mold surface, before it has fully crystallized. The results of this early release are part distortion and warpage.

As noted above, cooling times for PP are about 50% longer than those for competitive amorphous polymers. Cooling efficiency therefore becomes paramount. No longer is it sufficient to cool the mold strictly through the mold bottom with a cooling plate. Cooling channels along the sides and into the rim area of deep parts are recommended. The placement of additional cooling channels implies additional tooling cost. Owing to the extreme temperature sensitivity of the recrystallization of PP, care must be taken to provide sufficient cooling channels. It is recommended that the temperature differential from inlet to outlet of a given cooling channel not exceed 3oC or 5oF. If the mold is substantially hotter in one region than elsewhere, differential recrystallization will take place, resulting in warpage and distortion not only of the part in that region, but in the web surrounding it, and by implication, the other parts attached to that web.

Trimming

PP is a fiber former. That is, when PP is elongated locally, it draws rather than fracturing. As a result, every effort must be made to minimize pinching PP between shearing surfaces. For punch-and-die trimming, gaps between the mating tools need to be as close to zero as possible. For compression cutting, such as steel rule die cutting, the die must be kept as sharp as possible. Some molders recommend heated dies for sheet thicknesses of 25 mil or more. Although double hollow-ground beveled dies produce the cleanest trim, single bevel dies are used because they are less expensive and can be leather-stropped to work out feathering that can cause “angel hair” and fuzz. Trimming parts on an in-line canopy trimming press can be problematic, since the web is usually thicker than the molded part and therefore cools more slowly and without support. This means that although the parts are relatively warpage- and distortion-free, the cooling web may have distorted the entire sheet before it is fed into the registering lugs. This distortion will result in out-of-register parts and a substantial generation of unusable parts before appropriate adjustments are made.

In-situ trimming or trimming on the mold becomes difficult with PP, primarily because the trimming area is usually thicker than the rest of the part and therefore, at the time of trimming, it may not have achieved the same degree of crystallization as the rest of the part. Once the part is cut away from the web, support is lost and the trim area is free to distort. And it usually does.

Summary

PP thermoformers have learned that PP requires better process controls and more attention to mold design and trimming. And this points to higher quality operators and maintenance people. According to Biller [see footnote 3], Those thermoforming companies operating their [PP] equipment efficiently and progressively are usually the market leaders in quality and performance. The higher the recrystallization temperature, the more rapidly crystallization is taking place and typically, the smaller the final crystallites will be.

Footnotes

1. * This doesn’t mean that homo-PP cannot be thermoformed successfully. Amoco and Brown Thermoforming Machinery demonstrated thermoforming of thin-gage unit dose cups from 48-inch wide sheet at the 1976 NPE show in Chicago.

2. * The higher the recrystallization temperature, the more rapidly crystallization is taking place and typically, the smaller the final crystallites will be.
3. * Frank Biller, Marbach Werkzeugbach states [“Poly Problem”, World Plastics & Rubber Technology, Issue 10, 1997, p. 54] that While a PVC cup can be produced at 30 cycles per minute, the same cup made in PP often only reaches 20 cycles per minute.
4. * Biller [see footnote 3] attributes the longer cycle to ‘increased cooling requirements’, which, of course, is the other side of the ‘increased heating requirements’ coin.
Jim Throne on July 1st, 1998

Background

Part I detailed the arithmetic used to predict energy transfer to plastic sheet during the heating portion of the thermoforming cycle. In this part, we’ll take a look at heating of thin-gage sheet.

In Part I, we considered the importance of the three modes of heat transfer – conduction, convection and radiation – on the heating of a plastic sheet of nominal thickness. As noted, conduction describes energy transmission from the sheet surface toward its centerline. Convection deals with air motion around the sheet, and radiation is the dominant way in which energy reaches the sheet surface from the heaters. It is well-known that plastics have relatively poor thermal conductivities. As a result, energy transmission into heavy-gage sheet is strongly dependent on conduction. In fact, it is recommended that very heavy-gage sheet be heated in a forced air convection oven, to minimize overheating of the sheet surface before the sheet centerline temperature is within the forming range.

Conduction is relatively less important as the sheet thickness decreases. Typically, the temperature difference across a 60 mil or 1.5 mm sheet may be less than 10OC at the end of the heating cycle, and for 40 mil or 1 mm sheet, it may be less than 5OC. This allows us to simplify the transient heat transfer mathematics, as noted below.

However, another, perhaps more serious concern needs to be addressed for thin-gage sheet, that of sheet semi-transparency to incident radiation.

We explore both of these factors in the Technical Note.

The Thin-Gage Arithmetic

In Part I, we presented the traditional one-dimensional transient heat conduction equation, together with conduction and radiation boundary conditions. For thin-gage sheet, this arithmetic can be substantially modified and simplified. We only need to assume that the sheet has uniform temperature throughout its cross-section. In other words, the center of the sheet has always the same temperature as the surface. Correctly, the arithmetic is referred to as the “lumped parameter method” or LPM, and is found in detail elsewhere [J.L. Throne, Technology of Thermoforming, Hanser, 1966, p. 164+.]. Technically, LPM is applicable when Bi [Biot Number] = happarentL/k, < 0.1, where happarent is the combined radiation and convection heat transfer coefficient, L is the total thickness of the sheet, and k is the thermal conductivity of the plastic. The Biot Number represents the relative importance of energy input to the surface [convection and radiation] to energy transfer into the bulk of the plastic [conduction]. Obviously the thinner the sheet, the smaller the Biot Number.

If this condition is satisfied, the LPM equation becomes a very simple first-order ordinary differential equation:

LPM equation where q convection = h air (T air – T sheet).

The terms given here were, for the most part, defined in Part I. As above, L is the total thickness of the sheet.

Recall in Part I that we defined Fg as the geometric factor or view factor. In Part III, we will address the relevance of geometry on the heating of sheet. We also defined “F” as a non-black body correction factor. We also noted that this correction factor was defined as:

Non-black body correction factor

where εh is the emissivity of the heater, assumed to be 0 < εh < 1 and independent of wavelength. And where εs is the emissivity of the plastic sheet, also assumed to be 0 < εs < 1 and independent of wavelength. As noted, for most heaters εh = 0.9 to 0.95 and εs = 0.9 to 0.95.

In a paper (almost) presented at the 1997 SPE ANTEC, the issue of wavelength-independent plastic sheet emissivities was discussed and examples were (almost) given to illustrate the inappropriateness of this assumption. These results will be given here.

Model Building

The finite difference version of the LPM is much simpler to write, unlike that for the distributed-parameter model given in Part I:

The finite difference version of the LPM

Only one boundary condition, the initial sheet temperature, T [θ = 0] = T0 , is needed. This equation is very easy to solve on even the smallest of computers.

Wavelength-Dependent Emissivity

We rarely are able to measure emissivity. Instead, by means of an infrared scanner, sometimes known as an FTIR device, we measure wavelength-dependent transmission through a thin film of polymer. Now the sum of the transmissivity, τ, absorptivity, α, and reflectivity, ρ, of a material must add to unity: ρ + α + τ = 1.

According to Kirchhoff’s law, absorptivity and emissivity are equal for systems in thermal equilibrium. And usually we can ignore reflectivity, which can be related to the difference in indices of refraction between the plastic and air, and which represents no more than 5% of the total incident energy. Therefore, if we measure transmission, we can assume that absorption is given as 100% – percent transmission. In other words, the chemists’ IR measuring device yields important information about the heating characteristics of thin-gage polymers, as we shall see.

Figure 1: Far-Infrared Spectra for Two Thicknesses of Polystyrene

Far-Infrared Spectra for Two Thicknesses of Polystyrene

Figure 1 shows a typical generic IR curve for polyethylene. Along the x-axis is the logarithmic wavelength, in microns or μm. Percent transmission is given along the y-axis. Before we use this graph for heat transfer, consider certain characteristics on it. First, typical infrared spectra range from 2.5 microns to 10 microns or more. A heater at 1600OF or 870OC produces its peak energy at 2.5 microns. A heater at 1000OF or 540OC produces its peak energy at 3.5 microns. A heater at 700OF or 370OC produces its peak energy at 4.5 microns. And you and I at 98.6OF or 37OC produce our peak energy at about 9.7 microns. Note that polyethylene is completely radiantly opaque in the range of about 3.5 microns. Technically, all carbon-hydrogen organics, including most polymers, absorb all incident radiation at this wavelength, owing to C-H stretching. Other bumps and wiggles along the wavelength path, particularly around 8 microns, are also indicators of molecular absorption characteristics of given polymer types.

Then note that even at 10 mils, 0.010 inches or 0.25 mm, polyethylene is relatively transparent to incident radiation, particularly in the 4 to 7 micron range, where its wavelength-dependent transmission is 60 percent or more.

And finally note that, as expected, thicker films have less infrared transmission and hence greater absorption than thinner films. Most IR scans display spectra for at least two film thicknesses. This allows us to obtain thickness-dependent infrared information.The appropriate model is the Bouguer-Lambert-Beer law which states that wavelength-dependent energy decreases exponentially with increasing thickness, with absorptivity, α (λ), as the proportionality:

Bouguer-Lambert-Beer law

Obtaining α(λ) and Hence α average

The average absorptivity is given as:

Average absorptivity

If you are in control of obtaining transmission curves, set the machine to subtract the transmission value from unity, then integrate the result over the wavelength of the machine. To obtain average absorptivity, correct the data for the appropriate film thickness. The result will be (almost) the appropriate average absorptivity [and through Kirchhoff’s law, average emissivity], which is then applicable for any sheet thickness. This emissivity then becomes the proper one to be used in the F-equation, given earlier.

Now, what happens if the only IR transmission curve you have is one that is published in a book or article? You will need to descritize the transmission curve into dozens of narrow wavelength bands, then subtract each transmission value from unity, then obtain the descritized absorption values. The average absorptivity is obtained from the above equation, where the integrals are replaced with summations:

Average absorptivity with summations

While this is a messy job, it works. Then, the average absorptivity is set equal to the average emissivity and the arithmetic proceeds apace, as before

Results

The descritization method was used to obtain heating efficiencies for both PET and PVC. Tables 1 and 2 show the error generated when the absorptivity [=emissivity] is assumed to be 0.95, compared with the thickness-dependent values obtained from generic IR curves, using the second method described above. This technique has been written into a QBasic computer program.

Computed Data For PVC Heated To 280OF Using Heater at 640OF

Table 1

Sheet
Thickness [mils]
Calc’d Time
When
avg[sec]
Calc’d absorptivity Calc’d Time When absorptivity[sec] Percent Error
1 1.05 0.334 >30 *
3 3.13 0.521 7.77 148
5 5.21 0.601 9.81 88
10 10.52 0.716 15.02 43

Computed Data For PET Heated To 280OF Using Heater at 640OF

Table 2

Sheet
Thickness [mils]
Calc’d Time
When
avg[sec]
Calc’d absorptivity Calc’d Time When absorptivity[sec] Percent Error
1 1.32 0.513 3.28 148
3 3.83 0.753 5.14 34
5 6.38 0.834 7.49 17
10 12.75 0.906 13.51 6
Jim Throne on June 2nd, 1998

Background

In this technical note, we examine the arithmetic used to predict energy transfer to plastic sheet during the heating portion of the thermoforming cycle. Our discussion is restricted to radiantly opaque sheet. There are three modes of heat transfer:

Conduction, where energy is moved through a solid or non-flowing substance. The key to conduction energy transfer is the thermal conductivity and its sister, thermal diffusivity, of the substance. In effect, conduction energy transfer decreases as the thermal conductivity of the substance decreases.

Convection, where energy is transferred by a flowing substance. The key to convection energy transfer is the flow rate of the fluid, with fluid viscosity and thermal properties important secondary influences. A measure of convection energy transfer is the heat transfer coefficient, being the proportional factor between the thermal driving force and the amount of energy transferred.

Radiation, where energy is interchanged between two substances that do not touch. The key to radiation energy transfer is that the two substances have different temperatures. The radiation wavelength-dependent energy absorption characteristics are important secondary influences.

All three modes of heat transfer are relevant in thermoforming. Energy reaches the radiantly opaque surface from the heater source primarily by radiation. Heat may be added to or removed from the sheet surface by convection to the air surrounding the sheet. And energy is transferred from the sheet surface to the sheet interior by conduction.

The Arithmetic

The general approach to modeling the heating process begins with the basic assumption that the thermoforming sheet is arithmetically and thermodynamically thin when compared with its planar dimensions. This allows us to consider the sheet as one-dimensional in the thickness direction only. Since we are considering the sheet to be opaque to incident radiant energy, energy is moved throughout the sheet only by conduction. Convection and radiation energy input is consigned to the sheet surface only, or as will be noted, to the boundary condition of the mathematical model. As a result, the standard textbook transient or time-dependent heat conduction equation is valid:

Time-dependent heat conduction equation

Here, T is the local temperature, T = T(x,), where x is the distance into the sheet and f is the elapsed time. It is common to assume that the polymer thermal conductivity, k, is neither temperature- nor position-dependent. Thus thermal diffusivity, a = k/rcp, becomes the common polymer thermal property, where r is polymer density and c is thermal heat capacity. The equation then becomes:

Thermal diffusivity equation

Boundary Conditions

If the energy input is equal to both sides of the sheet, the thermal gradient in the sheet is symmetrical about the mid-plane. The plane of symmetry is given as:

Plane of symmetry

This means that the heat fluxes from each surface are equal.

The energy input at the sheet surface is given as:

Energy input at the sheet surface

The first term on the right is the convective energy transferred between the sheet, at temperature T, and the environmental air, at temperature Ta. Note that if the air temperature is less than the sheet temperature, energy is transferred from the sheet surface to the environmental air. The second term on the right is the radiative energy transferred between the sheet and the heater, at temperature Th*. The asterisk indicates that the temperatures used here are absolute. That is, for Celsius or Centigrade, T* = T + 273, and for Fahrenheit, T* = T + 460. There are several terms ahead of the bracketed temperatures. σ is the Stefan-Boltzmann constant. F is a non-black body correction factor, discussed below, 0 < F < 1. Fg is a geometric factor, or view factor, also discussed below, 0 < Fg < 1.

Non-Black Body Correction Factor

All radiation theory is based on the maximum amount of energy interchanged between two bodies. The higher temperature body is commonly called the source. The lower temperature body is called the sink. One radiation physical characteristic of a body is its emissivity, ε . The body is called a black body when ε = 1. When ε << 1, the body is called a non-black body. the emissivity can be wavelength-dependent, ε = ε (λ). If the emissivity is not wavelength-dependent, the body is called a gray body. Typically, as a first approximation, radiant heaters used in thermoforming can be assumed to be gray bodies, where εh = 0.90 to 0.95. And most polymer sheet can also be assumed to be gray bodies, where εs = 0.90 to 0.95. As a result, for planar surfaces, F is given as:

Non-Black Body Correction Factor

The Geometric or View Factor

When the source and sink are infinite planes, radiation interchange occurs without any loss due to geometry. When the source and sink are finite however, a portion of the radiation emitted by the source does not impinge on the sink and vice versa. Instead, this radiation is lost from the interchange. In thermoforming, energy interchange is maximum in the center of the sheet and is lowest at the corners of the sheet. This is sometimes referred to as the energy dome effect, Figure 1. The relative effect of sheet dimensions and heater-to-sheet spacing is given in terms of the geometric factor, F.

Energy dome effect

For most thermoforming applications, F is usually greater than about 0.7. In other words, at least 70% of the energy transfer is between the sink and the source. On the other hand, up to 30% of the energy transfer goes into heating non-sheet, viz, rails, clamp frames, oven sidewalls, the outside world, etc.

It should be noted that the value of F is an average

Arithmetically, 0.818 < f < 0.905. in other words, for most thermoforming operations, the energy transmission rate is 80 to 90% of the theoretical maximum rate. d value. For local heating on the sheet surface, it is necessary to devise a more complex arithmetic. This is treated elsewhere {J.L. Throne, Technology of Thermoforming, Hanser, 1996, pp. 155-159}].

Model Building

There are two general approaches to the arithmetic needed to model the heating process. The older technique is called finite difference equation or FDE. The newer technique is called finite element analysis or FEA. For conduction heat transfer, FDE is simpler to understand and to implement than FEA. While there are sophisticated FDE models for minimizing error generation, the speed of modern computers allows the use of the simplest model, that of explicit FDE. In this model, the partial differential equation is written as:

T Θ+1 = T Θ + αƒ [ T Θ X i , T Θ ( X i – 1 ), T Θ ( X i + 1 ) ]

for i = 1 to N

The boundary conditions become quite simple, as well. In order to maintain computational stability, the time step, ΔΘ, must be less than:

Δθ ≤ [ Δx² / 2α ]

where Δx  is the differential slice of sheet parallel to the sheet surface. As is apparent, if the temperature profile must be very accurate or if the sheet is very thin, Δx must be small and ΔΘ is reduced in proportion to the square of Δx. In other words, the differential time step must be very small and so the number of iterations needed to complete the computation must be very large. Again, however, with the speed of modern microcomputers, the actual computation time is quite short. For example, TF202, a computer program written in QBasic, takes approximately 4 seconds to run the example below, on a Pentium 166MHz computer.

Example

A 20 inch x 30 inch x 0.060 inch thick polystyrene sheet is to be heated to its forming temperature using the following environmental settings:

Top and bottom heater temperature, Th= 600F
Top and bottom air temperature, Ta = 175F
Top and bottom heat transfer coefficient, h = 2
Top and bottom sheet-to-heater spacing = 6 in
Sheet and heater emissivity, εs = εh= 0.9

The sheet thickness is divided into 10 elements [ Δx = 0.060 in/10.] The calculated time when the average sheet temperature equals the lower forming temperature of 260F is 43.4 s. The calculated time when the average sheet temperature equals the normal forming temperature of 300F is 56.6 s. The calculated time when the average sheet temperature equals the upper forming temperature of 360F is 82.4 s.

Jim Throne on March 13th, 1998

In my Rotomolding Tech Minute I, I commented in general on model building for the heating and cooling process in rotational molding. And further, I remarked on the six technical papers presented at the 1997 SPE ANTEC that focused on this aspect of rotational molding. According to the 1998 SPE ANTEC advance program, there will be another spate of model building programs in Atlanta. While this activity brings much-needed technical interest to a process that has suffered a dearth of attention, I believe that there are some simple schemes that can aid the engineer to understand the heating process. The following is a thought problem to help bring some order to the subject.

Forget the Powder!

Really! At least insofar as energy uptake is concerned. Andy Rao and I determined a long time ago that, insofar as heat sinks in the process, the mold absorbs ten to 50 times more energy than the plastic. So, as the mold is heating through to the final plastic temperature, its temperature is essentially unaffected by the small amount of thermal heat sink offered by the sticking, densifying plastic. Thus, modelers should feel reasonably safe in assuming, for all intents, that when the mold is inserted in an isothermal oven, the outer mold surface will heat in a first-order fashion according to

where Ta is the isothermal oven temperature, To is the initial mold temperature, h is the heat transfer coefficient for forced air convection, σ is the thermal diffusivity of the metal, =k/ρcp, L is the thickness of the mold, k is the thermal conductivity of the metal, ρ is its density, cp is its heat capacity, and Θ is time. For more information on the equations above and below, check out the chapter on Conduction in the Handbook of Heat Transfer, McGraw-Hill.

Note that the inside mold cavity temperature always lags the outside mold cavity temperature. In fact, the time at which the inside mold cavity temperature first begins to increase from To is given by:

Really! This little equation incorporates the type of mold material, the rate of heating, the wall thickness, everything! It should be apparent that the mold cavity temperature lag time is weakly dependent on the energy input from the oven [h, the heat transfer coefficient], but extremely dependent on the mold wall thickness, L [to the almost square] and the mold thermal diffusivity, α.

Although not theoretically correct, the inside mold wall temperature, TL, can be assumed to lag the outside mold wall temperature, Tw, according to:

TL ≈ TW – h(Tα – TW)(L/2k)

The reader should note that this equation is correct for constant heat flux. The heat flux in rotational molding slowly decreases as the mold temperature increases. And further, this equation assumes that the inside mold cavity surface is insulated, which it obviously is not! However, since this is just a thought problem and not an academic exercise, this assumption is okay, at least for the moment.

Again, note that the temperature offset is proportional to the convection air heat transfer coefficient and the thickness and thermal properties of the mold material. The asymptotic temperature difference increases with more rapid air motion, thicker molds and lower mold thermal conductivity.

Furthermore, it can be shown that the inner and outer mold surface temperatures become parallel when θ asymptate ≈ 0.45L2/α.

So what, you say? Well, this approach adequately defines the inner mold cavity temperature throughout the heating [and cooling] cycle! The inner and outer mold temperatures look something like curves A and B of Figure 1.

What About the Mold Cavity Air Temperature?

Devices such as Rotolog and Rotolog II contain thermocouples that monitor internal temperatures. Mold cavity air temperature is one of the temperatures that these devices measure. Consider energy transfer between the mold and the mold cavity air temperature for a moment 4. And further, assume that [horrors!], we forgot to put plastic powder in the mold cavity. The energy uptake by the air depends on convection through a relatively stagnant air layer at the interface between the mold cavity air and the inner mold cavity surface. Thus, the air temperature will lag that of the inner mold cavity surface. Since the volume of air in a given mold cavity is known, the exact air temperature can be calculated by solving the traditional transient heat conduction equation with an appropriate adiabatic inner mold cavity surface boundary condition. For now, we’ll leave this to the academics. Suffice it to say that the time-dependent mold cavity air temperature will quickly parallel that of the inner mold cavity surface, as Crawford et al have experimentally verified.This is shown in our thought Figure 1 as curve C.

So, What Happens When Powder Starts to Stick to the Mold?

The first question is ” When does powder start to stick to the mold? ” or more correctly, “At what temperature does powder start to stick to the mold? ” In earlier writings, I’ve defined that temperature as the tack temperature. It’s most certainly a temperature around the melting temperature of a crystalline polymer, although its exact value depends on subtle things such as particle shape and particle surface characteristics. For the purposes of this thought problem, consider polymer sticking when its melting temperature is reached. This is given as the horizontal line D in Figure 1.

Okay, but now a most difficult problem is posed. How do we model the sticking rate, the thermal properties of the initially fluffy, now stuck-together powder, the heat of melting, and so on? Aha! We don’t!

Instead, we move to the other end of the time-dependent temperature scheme and consider what happens after all the powder is stuck to the mold and melted and densified. At this condition, the polymer is molten and has uniformly coated the inner mold cavity wall surface. The energy transfer now is through the mold wall, through the liquid polymer layer and into the mold cavity air. And once again, the air temperature should be paralleling the outer mold surface temperature. The offset temperature between the inner liquid polymer surface and the outer mold surface temperature is given as 5:

TP ≈ TW -[h(Tα-Tw)-(L/2k+Δ/2kp)]

where now Δ is the thickness of the polymer layer and kp is the thermal conductivity of the liquid polymer. In other words, the air temperature now lags the outer mold temperature by both the thermal resistance through the metal, L/k, and that through the polymer, Δ/kp. It is immediately apparent that the thicker the polymer layer becomes, the greater the thermal lag will become. Thus, in Figure 1, the air temperature is shifted away from its original curve C to a new curve E by the amount of the thermal resistance through the polymer.

Okay, So What?

Well, in Figure 1, we have now defined our thought problem in a series of curves and lines [A through E]. We know that as soon as the inner mold cavity temperature reaches curve D, the air temperature will begin to deviate away from curve C toward curve E. The rate at which it asymptotically reaches E will depend on those fussy little things like bed configuration and rates of coalescence and densification and air migration from the polymer and latent heat of melting. But, given sufficient time in the oven, the inner air temperature will asymptotically reach curve E.

And it seems that we’ve developed is a molding diagram of sorts for rotational molding, all based on a thought problem!

An Aside

Consider the following variant to this problem. Instead of charging the mold with polymer powder, leave a part in the mold and let it be reheated and remelted [horrors II!]. Should the time-dependent air temperature curve look like Curve E right from the beginning of the heating time?

Jim Throne, 13 March 1998

Jim Throne on October 1st, 1997

Introduction

In the first part of this dissertation, we considered how the kinetic energy of an incoming mass is imparted and distributed to the shock mitigating foam. We assumed that the footprint of the incoming mass, that is, its impacting area, was the same size as the foam slab. But, what happens when the incoming mass footprint is substantially smaller? It would seem that this situation occurs more frequently than the latter. First, we need to mentally separate the behavior of the foam under the inbound mass from that everywhere else. What we would like to do, but cannot, is to assume the foam under the inbound mass to be a series of isolated springs and dashpots, totally unconnected to each other and to the rest of the foam structure. If we could assume this, then the analysis would be easy. The foam under the inbound mass would behave exactly as described in Part 1. And the foam away from the inbound mass would be totally unaffected by the presence of the inbound mass. At best, however, we can assume that the foam at some far distance, albeit undefined here, is unaffected by the presence of the inbound mass. And that the foam immediately under the geometric center of the inbound mass area is described by the discourse in Part 1. And, unfortunately, everywhere else, the presence of the inbound mass affects and is affected by the foam cell structure. So, let’s see how to go about analyzing this problem.

The Net, Envisioned

Consider an elastic fishing net with its edges rigidly fastened. Call it an elastic mesh with, say, square elements. An inbound mass, impacting the mesh, will act to stretch all elements of the net, with the possible exception of the area directly under the mass. The mesh free of the mass contact area is stretched in what is known as “plane strain” fashion. That is, if you look perpendicularly at the surface of the mesh, all the elements will still look rectangular. But from the side, the elements will stretch between the edges and the mass contact area. (The arithmetic for plane strain deformation is given in J.L. Throne, Technology of Thermoforming, Hanser/Gardner, 1996.) In other words, the mesh is deformed everywhere by the incoming mass impact. The elastic mesh material response is given not in terms of compressive stress-strain but in terms of tensile stress-strain. Again, the higher the modulus of the mesh material becomes, the greater the material resistance to incoming kinetic energy and the lower the deformation of the elastic mesh will be when the incoming mass kinetic energy has been completely converted to elastic potential energy in the mesh.

Extending the Net Concept

Certainly if we replace the elastic mesh with a monolithic very thin rubber sheet that has been inscribed with square elements, we should not expect to see any difference between the way in which the sheet and the elastic mesh deforms. Now, let us increase the sheet thickness. The tensile load bearing characteristic of a polymer, or any material, for that matter, is proportional to the cross-sectional area of the specimen. In other words, tensile stress is the applied force per unit area. As we increase the thickness, we increase the “unit area,” as it were. Therefore, the thicker the rubber sheet becomes, the lower the deformation of the sheet will be under the same inbound kinetic energy. Hold this thought as we continue to explore the nature of the rubbery sheet.

Now, consider the sheet to be foamed, very lightly, at first. As seen earlier, foaming reduces the effective modulus. Since the tensile modulus is one of the design criteria for the material resistance to tensile load, one would expect the tensile modulus of a foam to decrease in proportion to the square of the density, as with the compressive modulus. And it does. Therefore, foaming increases the extent of deflection under constant inbound kinetic energy

What about Thick, Highly Foamed Sheet?

So far, we have considered only the tensile aspect of the sheet. And that’s because we assumed the sheet to be thin and only lightly foamed. In actuality, the mechanical behavior of thin, low-density foam, such as polyethylene furniture wrap, is very well characterized by the plane strain model. It’s when the thickness becomes large, say, greater than about 3 mm, that a second mechanism begins to influence the mechanical response to the applied load. Why is this?

First, the foam directly under the applied load is compressed in the fashion described in Part 1. Recall that the polymer properties become very important if the applied stress collapses the foam to an extent greater than about 65%. If the foam is thick enough, compressive levels do not exceed about 65% and the rules described in Part 1 apply, but again, only directly under the applied load. Then, at the edge of the applied load, the foam is undergoing a transition between compression and tension. The edge of the applied load represents one of the “fastening points” for tension in the foam. The other is the far edge of the foam. We’ll discuss the nature of fastening at the far edge of the foam momentarily. The tension is applied here to the top of the foam (assuming that the incoming load is impacting the foam from above). The tension at the other end is usually applied to the underside of the foam (assuming again that the other end is, in fact, fastened to something). As the foam is stretched under applied load, the tensile load is therefore applied through the foam, from the top edge on one “end” to the bottom edge on the other “end.” This causes a shearing effect through the foam. In other words, in order to accommodate the applied load, the foam cell walls are not compressed or stretched, but are bent by differential forces parallel to the foam surface. Consider the simple example of pushing on the top of a cleaning sponge while the underside is against a solid surface. This is the shearing action.

So, there we have it. Compression of the foam under the incoming load. Tensile stretching of the foam surfaces at or near the edges. And shearing forces on the foam between. Oh, and there is one more factor that we need to include. We are assuming at this point that the foam underside is unencumbered and can freely. Later we consider what happens when the foam underside is rigidly in contact with a solid substrate.

So, how do we characterize the foam response to incoming load? Again, we look at potential elastic and dissipative modes in the foam. These are the same as described in Part 1. Energy is converted to heat by cell compression and by internal heating of the polymer. And energy is elastically stored in the spring action of unfolded, unbuckled cell walls and in the P-V-T characteristics of the cell gas. However, we need to keep certain aspects of the stretching behavior of the foam in mind. For example, uniaxially stretching of foam does not necessarily decrease cell volume, and may, in fact, increase it. Therefore the P-V-T cell gas effect may be absent. Now tensile distortion, buckling and folding of the cell walls do take place and as a result, we should expect some internal heating and energy dissipation. Shearing of the foam cells again may not effect individual cell volume and therefore P-V-T effects may not be significant. But, again, shear distortion, buckling and folding of cell walls do take place and again, some internal heating and energy dissipation does take place. However, from all but the highest stretching cases, the tensile and shear dissipative characteristics should be small. This means that for all intents, the net/mesh/membrane characteristic of foam sheet should be more elastic than dissipative. In short, one should expect the incoming mass to rebound from a foam sheet as fully as if it were solid.

The Role of the Substrate

Aha, you say, but what happens if the foam sheet is rigidly supported? Or if it’s really, really thick? First, it is apparent that the underside of the foam is no longer under tension and it therefore represents the neutral tensile axis plane. The top surface is still divided into the region of compression directly under the incoming load and the tensile region stretching from the edge of the load to the edge of the foam, which we are still assuming to be rigidly affixed (to something). Unlike the earlier case, it is uncertain whether plane strain can be assumed. In fact, from the simple observation described earlier for the net/mesh/membrane deformation, it appears that rectangular elements scribed on the undeformed foam surface show substantially more distortion in the region or the load than in regions far from the load. One apparent reason for this must focus on the compressible nature of the foam itself, particularly when rigidified from the underside, either by a rigid support or simply because the foam is very thick. In other words, the tensile stretching of the foam at the upper surface is compromised by compression of the foam beneath.

This complex interaction makes detailed technical analysis difficult. But a heuristic analysis goes something like this. The maximum amount of energy dissipation in the foam is determined by assuming all the foam is compressed to the same level as that directly under the incoming load. In other words, consider the mass footprint to be the same size as the foam sheet. The minimum amount of energy dissipation assumes that the underside of the foam is unsupported, as detailed above. In fact, in the limit, one can assume that the mass footprint is negligible compared with the surface area of the now-very thin sheet. As a result, this yields the maximum stored kinetic energy in the foam. The real case is somewhere in between. Certainly as the mass footprint increases in size as compared to the foam surface area, the stored kinetic energy in the foam will decrease and the foam will behave more like a uniformly compressed foam.

What about Edge Supports?

In some cases, the foam edges are not adhered or fastened to other rigid structures. In other cases, the foam edges are free to distort under incoming loads. How does edge support effect the performance of the foam under incoming load?

Well, in the heuristic analyses considered so far, we’ve assumed that the edges of the foam have been rigidly fixed. As a result, for the net/mesh/membrane stretching under load, we noted that the membrane was stretched between two fixed points, the edge of the foam and the edge of the incoming load. Certainly if the edge of the foam is not fixed, neither the top of the foam or the underside of the foam can be under tension. As a result, the foam is simply folded inward as the load impacts it. Obviously this is a useless situation. But what about rigidly supported or very thick foam?

The top surface of the foam is usually connected to the underside by cell walls. As the top begins to follow the impacting load downward, the tension is applied first to the top surface or skin, then to the cell walls that make up the cut edges that are at right angles to the top surface. The tensile strength of the cut edge is related to the modulus of the polymer and the amount of polymer in the cut edge. The tensile strength of foam polymers is usually assumed to be proportional to the density ratio to the 3/2-power (not the 2-power for elastic modulus). Now since the top surface of a foam usually has a skin, the tensile strength of the surface is usually higher than that of the cut edges. This means that the cut edges will stretch at least as much if not more than the top surface. The result of the stretching is a combination of shear and compression on the foam at or near the top layer and the corner of the foam slab. Again the general effect of compression is one of dissipation rather than elasticity. As a result, energy absorption of the inbound mass should always be greater with foams with free surfaces than with foams with fixed surfaces.

There is one more aspect of shock mitigation to consider. Consider the fact that at high impact energy, the inbound load generates an energy wave that propagates through the foam at a sonic velocity. Although the wave is dissipated to some degree by the non-homogeneous nature of the structure, viz, cells and cell walls, it propagates to the reverse side of the foam slab, where it is reflected back into the foam. The returning wave acts as a reinforcing agent to the foam cell structure itself, thereby momentarily increasing the foam resistance to the inbound load. It also results in an enhanced localized heating and consequently localized failure of the cell wall. In higher density foams, the effect is dramatic, with the foam exhibiting a remarkable ductile-to-brittle failure layer that clearly demarks the specific plane where the sonic wave is locally reinforced. Furthermore, this sonic reinforcement is used to an advantage in military structures such as tanks, where laminate structures of differing sonic velocities are used to attenuate incoming small-arms fire.

Shock Mitigation for Open-Celled Foams

It is well-documented that the stress-strain curve of an open-celled foam is very similar to that for a closed-cell foam, particularly at high impacting speeds. Why is this? After all, there should be no closed cell gas compressibility factor, right? This assumption depends entirely on the degree of “open-celledness.” Consider the foam to generally be closed except for a hole in each of the cell walls. When the cell is compressed by the incoming mass, the cell gas must flow through the holes to the surrounding cells. Of course if all the cells in a given area are being compressed to the same degree, the cell gas pressure in each cell equals that in the neighboring cells and so there is no flow. This would generally be the case if the mass footprint is the same as the area of the foam. If the mass footprint is much smaller than the area of the foam, then cell gas can flow from the more compressed region to the less compressed region. It is surprising that cell gas flow under this type of impact is perpendicular to the impacting foam surface and in a direction opposite that of the incoming mass. To confirm this, you can sprinkle a fine film of talc on the surface of the foam prior to impact. Upon impact, the ejected cell gas will blow talc from the surface at a short distance from the edges of the impacting mass.

The rate of cell gas flow depends entirely on the size of the hole in the cell wall. If the hole is very small, the cell gas flow is very low. As a result, the cell gas is compressed essentially as if there are no holes! Even if the holes are relatively large, there should still be some measure of cell gas compression. As discussed earlier, any cell gas compression provides kinetic energy dissipation. And of course, the internal energy dissipation in the polymer cell walls is the same, whether or not the cells are open. In other words, as the holes in the cell walls become larger and larger, the adiabatic energy dissipation effect of the cell gas slowly diminishes. The true effect of open cell foams is noted when the mass becomes static, since the auxiliary cell gas “pillow” support is absent.

Summary

So, what have we learned about shock mitigating foams? The compressive stress-strain curve teaches us the maximum amount of energy that can be transferred to the foam from the inbound mass. The extent of energy dissipation depends on the internal energy of the polymer as it is bend, folded or buckled by the force of the incoming mass, and on the adiabatically compressible nature of the cell gas. For masses with footprints smaller than the foam, the surface of the foam that is impacted by the mass is put under tension. Energy absorption depends on the thickness of the foam, whether the foam is supported on the reverse side, and whether the foam edge is fixed or free. And open cell foams show energy absorption curves that are similar to those of closed cell foams.

Jim Throne on September 10th, 1997

Introduction

Low-density foams are used in shock mitigation, cushioning, thermal insulation and vibration isolation. Typically, these foams are made by mixing gas-generating small molecules with molten polymer at elevated pressure, then rapidly dropping the pressure to allow the gas to come from solution to produce discrete bubbles or cells. In most cases, the blowing gas is not normally found in large concentrations in the environment. And in most cases, air is not normally used as a blowing gas for thermoplastic foams. As a result, for some time, perhaps years, air is diffusing into the foam and the blowing gas is diffusing out. Although there may be some justification for considering effects such as “sweep gas diffusion”, being the condition where one gas aids another during diffusion, the simplest approach is to assume that the gases are migrating independent of one another.

Many foam properties are influenced by the composition of gases in the foam. These include thermal conductivity, fire retardancy, moisture transmission, dimensional stability, compressive properties, and shock mitigating properties.

The concentration and make-up of gases in any given cell depends on several parameters, including the age of the foam, its temperature, its thickness, the nominal cell dimension, the permeability of the gases in the plastic, the role of adducts in inhibiting or promoting gas transfer, the extent of closed cells or complete membranes, and most importantly, the polymer itself.

It is not possible in this technical minute to elaborate on many of these parameters. Instead I want to focus on the general effects of gas migration.

Permeability of Gases in Plastics

Consider the case of a relatively thick foam sheet of substantial area. Consider further that the gas migration is only occurring perpendicular to the sheet surfaces. The blowing gas at the centerline of the sheet migrates to the surface via concentration gradient. The air in the environment at the sheet surface migrates into the sheet via concentration gradient. Although concentration gradient is the driving force, diffusivity is the material property that dictates the rate at which gas moves through the plastic. In polymer film technology, permeability is the parameter. Permeability, P, by definition, is the product of solubility, S, and diffusivity, D:

P = S x D

Solubility is the equilibrium up-take of gas in plastic. Solubility is both temperature- and pressure-dependent and is frequently written as:

S = H(T) x P

where H(T) is temperature-dependent Henry’s law constant and P is absolute pressure. Typically, gases in polymers decrease in solubility with increasing temperature. And the crystalline portion of a polymer dissolves relatively little gas compared with the amorphous portion of the polymer.

Diffusivity is also highly temperature-dependent and is strongly dependent on the projected size of the gas molecule. Hydrogen and helium, for example, are very small molecules and diffuse very rapidly through all polymers. Aliphatic hydrocarbons such as hexane and heptane, are relatively large molecules and diffuse much more slowly. Owing to the size of the chlorine atom, chlorocarbons and chlorofluorocarbons are very large molecules and therefore diffuse extremely slowly. It is projected that trichlorofluoromethane, CCl3F or R11, has a half-life of 70 years in 2-inch thick 2 lb/ft3 PS foam.

As noted, permeability is the product of diffusivity and solubility. Permeability is expected to be high for small molecules in amorphous polymers and low for large molecules in highly crystalline polymers. Note that the level of crystallinity of the polymer affects gases permeating into and out of the polymer to the same degree! Thus the expected rate of change of the time-dependent properties discussed earlier can be influenced by the degree of crystallinity, but not the final effect!

Pressure Profile During Cooling

Extruded foams stop expanding when the cell gas pressure drops to about the environmental pressure. As the foam continues to cool, the internal cell gas pressure drops below one atmosphere absolute. Simple PvT models show that if the cell remains fixed in volume, the internal cell gas pressure can be in the range of 0.3 to 0.6 atmospheres, absolute. If the foam is rigid, as with PS, mPPO and most amorphous polymers, the cell walls are stiff enough to prevent wholesale cell collapse. Some shrinkage may be seen, but not gross distortion. If the foam is very flexible or the polymer has a low modulus, as with LDPE, PP, EVA, and certain TPOs, the cell walls will not be stiff enough to prevent excessive shrinkage or cell collapse. For soft foams, the internal cell gas pressure will drop to a relatively low level, then remain constant as the foam collapses. Nevertheless, in both cases, for very fresh closed cell foam, the internal cell gas pressure will be less than one atmosphere.

Fresh foam is also hot foam, meaning that the gas inside the foam is at elevated temperature. Even though the internal cell gas pressure is less than one atmosphere, the gas inside the cell has a higher concentration than that in the environment and so the blowing gas can permeate quite rapidly from the foam. Similarly, air in contact with the hot plastic quickly permeates into the foam. This very rapid initial gas interchange slows dramatically as the foam cools.

Temperature Profile During Cooling

The temperature along the centerline of freshly formed thick foam is always hotter than the temperature at the surface, due simply to conduction of heat to the foam surface. Similarly, the concentration of blowing gas in the foam is always lower at the surface than at the centerline and the concentration of air is always greater at the surface than the centerline, due simply to molecular migration.

A snapshot of the concentrations of blowing gas and air sometime after the foam has been made would show an exponentially decreasing concentration of air from the surface to some distance into the foam and an exponentially decreasing concentration of blowing gas from a short distance from the surface to the surface of the foam. Since the pressure of environmental air is one atmosphere, the internal cell gas pressure of cells at the foam surface would be essentially one atmosphere. If air migrates more rapidly than the blowing gas, the pressure would increase above one atmosphere at some distance into the foam, then fall again to the partial pressure of the blowing gas at the centerline, where the air had yet to diffuse. This is shown in Figure 1 for CFCl3-blown 35 kg/m3 PS foam [J.L. Throne, Thermoplastic Foams, Sherwood Publishers, Hinckley OH, 1996, Figure 9.112, p. 520.].

CFCl3-blown 35 kg/m3 PS foam

Figure 1

On the other hand, if the blowing gas diffuses faster than air, the internal cell gas pressure would fall from essentially one atmosphere to a very low subatmospheric pressure at some distance into them foam, then rise again to the partial pressure of the blowing gas at the centerline, where the air had yet to diffuse. This roller-coaster variation in pressure through the foam is a known cause of bowing, distortion, shrinkage, and surface irregularities.

Time-Dependent Thermal Conductivity

Here we consider the effects of time-dependent gas migration on thermal conductivity. The typical time-dependent thermal conductivity of foams is seen in Figure 2 [J.L. Throne, Thermoplastic Foams, Sherwood Publishers, Hinckley OH, 1996, Figure 9.110, p. 518.]

Typical time-dependent thermal conductivity of foams

Figure 2

Keep in mind that the inverse of thermal conductivity is thermal resistance. The smaller the value of thermal conductivity, the slower energy is conducted. Thermal conductivity of foams is considered to be a simple sum of the thermal conductivities of polymer and gas, and microconvection within the cell and radiation through the cell walls:

kfoam = kplastic + kgas + k’convection + kr,radiation

For low-density foams at room temperature, the relative contributions of these elements are:

Polymer 6%
Gas conduction 74%
Cell convection 0%
Radiation 20%

When the cell contains a mixture of gases, say, blowing gas and air, the gas conductivity is assumed to be a simple weighted sum:

kgas = x kblowing gas + (1-x) kair

where x is the weight fraction of blowing gas in the cell. If we use a simple transient one-dimensional equation to determine individual gas concentrations throughout the thickness of the foam, we can then obtain a very simple summation model for heat conduction. If Q/A is the heat flux [Btu/ft2.hr for example] and ΔT is the thermal driving force across the foam slab from centerline to surface, then the summation model is:

Q / A = ΔT / [L1 / k1 + L2 / k2 + … + LN / kN]

where L1 is the dimension of the ith cell containing a gas mixture having a thermal conductivity of k1 and N is the number of cells across the sheet half-thickness. If the blowing gas has a lower thermal conductivity than air, the foam will retain a relatively low thermal conductivity, or in insulation terms, a higher R-value, substantially longer than a foam blown with a high thermal conductivity gas. Note, however, that given long enough, the internal cell gas pressure will reach a uniform one atmosphere and the only gas in the cell will be air.

In the second part of this Technical Minute, we’ll explore the more complex ramifications of time-dependent gas migration on some of the other factors such as dimensional and property changes.

Background

Low-density thermoplastic foams are frequently used as energy-absorbing foams. That is, kinetic energy from an incoming mass is mostly dissipated in the foam, resulting in little, if any, throwback or reverse propulsion of the mass from the foam. Typically, energy absorption is described in terms of the area under the foam stress-strain curve. The typical stress-strain curve for a low-density foam is depicted as having three general components:

At low compression, say, less than about 5%, the foam acts as a Hookean elastic spring. That is, the extent of compression, ε, is directly proportional to the applied stress, σ , with the proportionality, Ef, being the modulus of the foam: σ = Ef ε.

For essentially all foams, the modulus of the foam is directly related to the modulus of the polymer, Ep, and the ratio of squares of the foam density, Ep, and polymer density, ρP: Ef = Epf / ρo)².

At high compression, say, greater than about 65%, the foam cells have collapsed to such an extent that the strut supports are interfering with one another. Now the resistance to further compression is mostly related to the compressive character of the polymer itself and not so much to the compressive nature of the foam.

In between, from about 5% to about 65% compression, the compressive nature of the foam is given as a combination of the bending and/or buckling characteristics of the strut supports and the compressive nature of the gas in the foam cells.

First, consider that the impact loading does not compress the foam more than 65% or so, or into the region where the stress-strain curve begins to turn up. Then, consider that compression is not the idealized uniform birdcage strut bending pictured in most textbooks on the subject. Instead, compression usually occurs only in the impact region of the incoming mass and then usually in a plane essentially parallel to the foam surface in that region. The latter is sometimes referred to as “conjugated cell collapse.” Planar compression usually begins with bending and folding of the thinner and therefore weaker cell walls. If there is a preponderance of weak cells in a given plane, the initial compression will be centered there. Elsewhere, the foam cells will not seem compressed at all. Basically, the struts will bend and fold until further compression requires compression of the polymer itself. Then the incoming mass will force weaker cell walls in other portions of the foam to begin to collapse.

Of course, as the mass continues into the foam, conjugated cell collapse will occur in more and more planes parallel to the foam surface. The general shape of the stress-strain curve in this region tends to be nearly linear with the load-bearing characteristics of the foam gradually increasing as the foam is compressed. The region from about 25% compression to about 65% compression is sometimes referred to as the “comfort zone,” and the shape of the stress-strain curve in this region is most important in seating and bedding.

It is somewhat surprising to learn that stress-strain curves for most polymeric foams have very similar shape, with the primary difference being the effect of foam density on the foam resistance to incoming mass. Thermoplastic foams have stress-strain curves that are very similar to those of thermosetting foams, such as urethanes and silicones. And open-celled foams have stress-strain curves that are similar but not identical to closed-cell foams. The energy absorption of open-celled foams is the subject of a subsequent paper in this area.

Energy Absorption in Foams

So the question is: How is the energy dissipated from the incoming mass? Consider two scenarios. In the first, the area or footprint of the mass is essentially the same as the area of the foam. This allows the foam to be compressed essentially uniformly across its surface. In the second, the footprint of the mass is substantially smaller than the area of the foam. Therefore the foam is compressed locally under the mass while the foam at some distance from the mass is essentially uncompressed. Then consider the nature of the impact. Again there are two scenarios. In the first, the impact is a singular event. That is, the mass impacts the foam once, and comes to rest atop the foam. In the second, the impact is part of a multiple event. In other words, the mass impacts the foam over and over. For purpose of this discussion, consider that the repetition is periodic and that the amount of energy to be dissipated is the same each time. In other words, consider vibrational impact as an example of this case. Of course, there are other types of multiple impacts, such as a mass that rebounds, then re-impacts, only to rebound again. And again and again.

Finally consider the classic case of an incoming mass impacting the foam surface, compressing the foam, and coming to rest atop the foam without rebounding free of the surface.

And then consider the general energy absorption picture insofar as the foam is concerned. As noted above, the key to foam energy absorption is the stress-strain curve. As the incoming mass impacts the foam surface, compression begins and the foam resistance is dictated by following up the stress-strain curve. The foam continues to compress until the mass velocity reaches zero. In technical terms, the foam has compressed to ε dynamic. The extent of stress at this condition is σ dynamic. In other words, the kinetic energy of the mass has been converted into heat and potential energy in the foam.

This is obviously a metastable state, since the foam has substantial elastic character and the “foam spring has been compressed,” so to speak. Now the elastic character of the foam pushes against the now-stationary mass, reversing its direction. Consider the following two scenarios.

If the foam were completely elastic, the mass would be propelled or rebounded from the foam with the same kinetic energy it had when it first contacted the foam. The foam, in essence, would resemble a bed of perfect springs. On the other hand, if the foam were completely dissipative, the foam would remain compressed by the mass. The foam, in essence, would resemble a bed of Play-DohTM.

Quite obviously, no foam is either a bed of perfect springs or perfect Play-Doh. Some, hopefully, most of the incoming kinetic energy is dissipated but typically, some of the kinetic energy is returned to the mass. Certainly the greater the elastic modulus of the foam, the greater the “spring effect” of the foam will be. Again, since the elastic modulus of a foam is the product of the modulus of the polymer and the square of the foam density, the “spring effect” is diminished by using a low-modulus polymer such as EVA or LDPE, or by foaming the polymer to a lowered density.

In addition to the elastic effect of the foam, the compressibility of the gas in the cell structure must also be considered. Assume for the moment that the foam is closed cell. And assume that the incoming mass has already buckled or bent the supporting struts. The buckling or bending of the supporting struts acts to reduce the individual cell volume. Since the air in the individual cells is compressible, the result is an increase in cell gas pressure. If the compression is isothermal, that is, the gas and polymer temperatures remain constant, the cell gas pressure is inversely proportional to the cell volume: Pl / Po – (Vl / Vo)-1.

Now, as the cell is compressed, the internal pressure increases, thus offering resistance to the incoming mass. This resistance is in addition to the polymer elastic resistance. Extensive technical studies have shown that the easiest way to consider the effects of both cell gas pressure and polymer elasticity is to simply add the effects together:

Foam resistance = Polymer elasticity + cell gas pressure

Again, polymer elasticity dominates the early portion of compression whereas cell gas pressure influences the later portion of compression. Now technically, polymer elasticity is reversible and cell gas compression can be reversible. The above discussion assumes that the polymer and gas temperatures remain constant. In certain cases, such as very high-speed impact, the impacting energy is dissipated as heat, raising both polymer and gas temperatures. The gas temperature is raised adiabatically, and the cell gas pressure is no longer inversely proportional to the cell volume to the first power. More importantly, since energy is dissipated as heat, the amount of energy remaining in the foam to rebound the mass is diminished. And the more energy that is dissipated as heat, the less energy remains to rebound the mass. In other words, the key feature to achieving Play-Doh-like dissipation is heat absorption by the gas and the polymer.

So, how does the polymer absorb heat? Two general ways. First, through energy interchange with the adiabatically compressed cell gas. It is admitted, however, that heat transfer between a quiescent gas and a polymer is notoriously poor. The second way is through localized bending or buckling of the polymer struts. Recall the old experiment of rapidly stretching a rubber band and quickly placing it against your lip to feel the heat? This is a similar situation. Molecular stretching due to folding, bending, or buckling results in internal heat generation in the polymer. Heat dissipation in polymers is extremely poor, since the primary mode of energy transfer is conduction from the folding or bending source. This means that the bent or folded area remains “hot” for awhile after impact. This also means that the elastic characteristic of the polymer can be mitigated by this.

Consider what the local heating does to the polymer morphology. Low-density foam cell walls are films that are biaxially oriented as much as four to five times (4X to 5X). When the film is bent elastically, it recovers elastically. However, when the film is heated locally, the film is “de-oriented” locally. If the energy absorbed is sufficiently high, the film may actually melt locally, then cool into a new, unoriented state. And for crystalline polymers, the new state may be a different crystalline morphology. This localized softening or melting effect happens primarily at compression levels far beyond the 5% elastic condition discussed earlier. And this effect is also one of the primary causes of “compression set,” where the foam simply does not recover its full thickness after the mass has been removed.

Furthermore, if the foam is subjected to repeated impacts, as in vibrational applications, the local heat is never fully dissipated and thermal damage and de-orientation can continue throughout the foam cell structure.

[It should be noted parenthetically, however, that insofar as I can tell, the type of permanent damage described above has not been observed or photographed. For those seeking confirmation of this type of thermal damage, I suggest careful 30X examination of singularly impacted “crystal” or unmodified polystyrene foam cell walls, using crossed polarizers. Localized softening and recooling should yield much lower strain fields than what can be observed in the surrounding biaxially oriented film.]

Energy Balance in Shock Mitigation

Okay, now the foam is compressed to its maximum. The total kinetic energy of the incoming mass is now zero, with the energy having been transferred to the foam. Part of the energy has been converted into heat, through adiabatic compression of the cell gas and internal heating of the polymer at the to-be-permanent folds, bends or buckles in the cell walls. This energy is dissipative, or unrecoverable. Whatever energy is left must be elastic or completely recoverable. This energy, such as the springiness of the elastically bent cell walls and the compressibility of the cell gas, pushes back against the now-still mass. Certainly, the mass itself provides a static stress against the foam, given by the weight of the mass divided by its footprint area. Thus, the energy balance for the inbound mass is given as:

Incoming kinetic energy of mass = Heat dissipated to gas and polymer + Static stress of the mass + Kinetic energy of rebounded mass.

It is apparent that if the mass is not rebounded from the foam, viz, the Play-Doh example, all the inbound energy must be converted to dissipative heat or must act to simply support the static mass. Therefore, the rebound or kinetic energy imparted to the mass must be related to the elastic potential energy stored in the foam, given as: Rebound energy = Elasticity of cell walls + Differential gas pressure.

Nature of Cell Gas Compression

Before continuing, consider the importance of cell gas pressure on energy absorption. It is well known that the cell gas pressure of fresh foam at room temperature is sub-atmospheric. This is because foam stops expanding when the polymer and cell gas is quite warm, near the polymer melting or glass transition temperature and the cell gas pressure is about one atmosphere. And as the foam cools, the cell gas pressure drops below atmospheric. There follows a gas diffusional interchange, with air diffusing into the cell structure while the foaming gas diffuses out. Typically, cell gas pressures of mature foams are greater than one atmosphere. Certainly cell gas pressures greater than one atmosphere will support the static load better than cell gas pressures less than one atmosphere.

Next

So far, we’ve considered energy distribution of an inbound mass that occupies the same footprint as the foam. In the next section, we consider what might happen if the foam is substantially larger than the footprint of the inbound mass.

Jim Throne on August 29th, 1997

Summary of Part I

In part I, I discussed the general types of thermoplastic polyesters, noting that polyethylene terephthalate or PET is a slowly crystallizing polymer that has a melting temperature of about 260 oC. It is considered a likely candidate for many high-temperature applications, particularly since it is relatively inexpensive when compared with other high-temperature polymers such as polyamides. Foamed crystalline PET is considered a candidate for higher temperature thermal insulation applications. Part II of this Technical Minute focuses on the desired properties of foamed PET.

Packaging Applications

Crystalline polyethylene terephthalate is called CPET in the written literature. When the polymer crystallinity is in the 20% range, the polymer is tough at room temperature. At temperatures greater than the glass transition temperature of about 70 oC, it is pliable but retains its shape until temperatures in excess of about 200 oC. CPET at 40% crystallization level is quite brittle. Unfoamed CPET has been thermoformed into “TV dinner trays” for more than a decade, where the tray is exposed to “recommended” convection oven temperatures of 175 oC to 180 oC for up to 60 minutes without deterioration. However, the traditional unfoamed tray suffers from several limitations:

  • It is difficult to keep PET from crystallizing during extrusion when the sheet thickness is in excess of 0.060 inches or 1.5 mm. As a result, thermoforming is restricted to relatively shallow shapes. This obviates certain applications, such as bakery containers, which require relatively deep containers.
  • In addition to the restriction on the depth of draw, container stiffness becomes a problem when the side walls of the container becomes too thin. Keep in mind that flexural stiffness, S, is given as:S = EIwhere E is the modulus of the polymer and I is the moment of inertia. For a thin beam, b in width and t in thickness, the moment of inertia is given as:I = bt3/12It is apparent that stiffness is proportional to the container wall thickness to the cube power. For economy, it is desired to keep the wall thickness as thin as possible. Therefore the container wall can become quite flexible, particularly above Tg, where the PET modulus drops dramatically.
  • When unfoamed PET is thermoformed, two types of crystallization occur. The first, thermal crystallization, dominates, effectively locking the shape into its final configuration until the product temperature is raised to the PET melting temperature of about 260 oC. However the PET sheet is also biaxially stretched, and a second type of crystallization, orientation crystallization also occurs. Certainly, the level of orientation increases as the draw depth increases. This added crystallization level may make the CPET product more brittle than tough.
  • And, in a typical food container application, the product placed in the CPET container is flash frozen at temperatures of about -30 oC. PET at this temperature is 100 oC below its glass transition temperature. It is brittle, much like PS is at room temperature. As a result, special care must be taken to minimize “rough handling” such as impact during filling, flash-freezing, and shipping of frozen product. Otherwise, the product may be compromised by broken corners and split sealing regions.

Impact and Foaming

Even though it is well-documented that biaxial orientation of a brittle polymer increases its toughness, it is also well-known that the impact strength of a brittle polymer does not necessarily increase by foaming. However, foaming does alter the nature of flexural impact failure. Foamed CPET fails by crushing under impact, instead of failing catastrophically, as unfoamed CPET does. That is, individual CPET membranes bend, then break under impact, with impact energy being dissipated first to the intersections of the membranes, then to neighboring cell membranes. As a result, even though many membranes rupture under impact, the foam structure remains essentially intact, thereby protecting the product. This effect is well-known for polystyrene foam, where the container of low-density polystyrene foam is used to protect refrigerated eggs, for example.

However, one must be careful to recognize that the transition from catastrophic failure to non-catastrophic, crushing damage occurs at relatively high foaming levels. For example, polymeric structural foam, where density reduction usually does not exceed 50%, fails under impact in much the same way as the unfoamed polymer. Density reduction of more than 50% and usually 90% is needed in order to achieve crushing rather than catastrophic failure.

Stiffness and Foaming

As noted above, container stiffness is proportional to the container wall thickness to the cube power and to the modulus of the polymer to the first power. At the same container weight, foaming increases the effective wall thickness of the container. But foaming decreases the effective modulus of the container. It is well-known that the modulus of a foamed polymer decreases in proportion to the square of the foam density:

Ef = Eof / ρo

At the same container weight, the sidewall thickness increases in inverse proportion to the foam density:

tf = tof / ρo) -1

Therefore, we can easily show that at the same container weight, the container stiffness increases with decreasing foam density:

S = Ef If = Eo Iof / ρo) -1

In short, by foaming, we can improve two critical aspects of container performance – impact strength, particularly at freezer temperature, and sidewall stiffness, particularly for deep-draw containers.

The real question that remains is: Can we produce a CPET foam having up to 90% density reduction?

The Practical Aspects of Foam Density Reduction

The foam industry has long believed that certain polymers foam to what has been called “natural densities”. For example, polystyrene and HDPE seem to foam well at a natural density of about 2.2 lb/ft3 = 35 kg/m3, whereas PP seems to foam well at about 10 to 15 kg/m3. PVC, on the other hand, is difficult to foam to densities less than about 100 kg/m3. And, it is thought, PET should also be difficult to foam to densities less than about 100 kg/m3.

The natural density belief has some basis in fact. The production of low-density foams involves, as the terminal phase of the process, the biaxial stretching of membranes. As discussed in detail elsewhere [JLThrone, Thermoplastic Foams, Sherwood Publishers, 1996 – see this Web page for ordering information], the available stretching force is directly related to the differential pressure between the cell gas and the environment of the forming foam. The resistance to the stretching force is viscoelastic character of the polymer film, which is experiencing strain hardening and rapid cooling. If the stretching force is too high or the resistance to the stretching force is too low, the film may rupture, which may result in catastropic foam collapse. If the stretching force is too low or the resistance is too high, the film may not stretch sufficiently, and the foam density may not achieve a desired low value.

Many polymers can be uniaxially stretched to 4X or more near but above their transition temperatures, before rupturing. This implies that foams of 30X density reduction are achievable before catastrophic collapse. For PS, for example, this predicts a foam density of 35 kg/m3. This further imples that for CPET, and PVC for that matter, foam densities of 45 kg/m3 are achievable, given the correct processing conditions. The key phase here, of course, is “correct processing conditions”. [Some processing conditions were discussed in Part I, but additional discussion will be found in an upcoming Technical Minute.]

So, to summarize this part, foaming of CPET to low densities for commercial container applications is justified by improved performance at low temperatures and improved stiffness for deeply formed containers.

What about thermal insulation? Does CPET have adequate insulating properties? Part III addresses this question.

Jim Throne

29 August 1997

Jim Throne on June 25th, 1997

Introduction

The class of thermoplastic polyesters has two major subclasses – polybutylene terephthalate or PBT, and polyethylene terephthalate or PET – and two minor subclasses – PET copolymer such as Eastman’s PETG, and polyethylene naphthanate or PEN. With the exception of the copolymers, all polyesters can be crystallized, albeit very slowly when compared with polyethylenes. Of the majors, PBT resin is more expensive to produce than PET and so has found use as a highly filled or reinforced engineering resin that can be foamed to about 70% of the unfoamed resin density. Virgin PET is currently selling for less than $0.50/pound and finds extensive use as amorphous sheet in packaging and as biaxially oriented crystalline shapes in barrier containers. Crystalline PET has a glass transition temperature of about 70oC and a melting temperature of about 265oC. All thermoplastic polyesters are “condensation polymers”, meaning that the polymer molecular weight is increased by eliminating a small molecule such as water. Like most condensation polymerizations, the PET polymerization is reversible. PET molecular weight is usually determined by wet chemistry solution viscosity, and is usually reported as inherent viscosity, [η], or intrinsic viscosity or IV. One relationship between [η] and number-average molecular weight is:

[η] = 7.50 x 1-4 (Mn)0.68

The standard relationship between shear viscosity and number-average molecular weight is:

ηs = α(Mn

where α is dependent on the type of polyester and α is on the order of 3.4. The fiber grade PET has an IV range of about 0.6 to 0.7. Bottle grade PET has an IV range of 0.7 to perhaps 0.9. Special techniques are needed to achieve IVs in excess of about 0.9. The most common technique is solid state polymerization. Traditional solid stating operations require hours of elevated temperature and relatively high levels of vacuum to achieve IVs in the 1.2 to 1.5 range.

One of the earliest uses of the crystalline form of PET was in the ubiquitous thermoformed ovenable “TV dinner tray”, first commercialized in the mid-1980s. The technique for producing this product is quite unique. A PET of about 1.0 that has been doped with nucleants and impact modifiers is extruded as a sheet and quenched to minimize crystallization. The sheet, with no more than 5% crystalline, is then heated very rapidly to the forming temperature of around 150oC to 160oC. Since PET is a very slowly crystallizing polymer, only a small amount of crystallization, up to about 10% or so, has taken place before the sheet is stretched under pressure against a mold heated to 170oC to 180oC. The PET crystallizes against the hot mold to about 20 to 25%. This level of crystallinity is sufficient to rigidify the shape against 200oC hot air oven use for one hour or more. Currently, approximately 100 million pounds of CPET or “Crystallized PET” are consumed in the US each year, with a projected growth of about 5% APR.

Until earlier this year, the historically high cost per unit volume of PET, when compared with PS, restricted the market development as a PS replacement in other packaging areas, including foams. This was also valid for products such as generic insulation board and pipe insulation, this despite the much higher continuous use temperature capability of PET. The current worldwide excess capacity of PET has driven virgin bottle- and fiber-grade prices to record lows. While higher IV grades have not seen price reductions as dramatic, certain higher performance styrenic foam applications are now economically feasible targets for PET foam. Three general markets have been targeted:

  • Transportation, for head liners, automotive insulation pads and coolant fluid hose insulation covers. The typical density range is 100 to 300 kg/m 3. The polymeric competition is SMA. The annual US market is estimated to be about 10,000 tons.
  • Packaging, for niche applications between compact CPET and olefin and styrenic foams. The typical density range is 350 to 700 kg/m 3. The product requires high-temperature FDA approval. The market size is estimated to be about 25,000 tons.
  • Construction, for high-temperature metal surface insulation applications. The anticipated density range is 30 to 100 kg/m 3. The competition is glass fiber and mineral wool. The market size is estimated to be about 20,000 tons.

The Development of a PET Foam

It is well-known that a melt viscosity is a primary material property in the formation of stable bubbles. Early attempts to foam bottle- and fiber-grade PET were unsuccessful, since no processing window could be found. The polymer could not be cooled sufficiently to prevent bubble collapse before it crystallized. PET copolymers and PETs with IVs in excess of 1.0 could be foamed but were too expensive. These limitations have been relaxed recently, with less expensive copolymers and more rapid molecular weight appreciation techniques. Thin PET foam sheets (less than 0.100 inch) with nominal densities of 100 kg/m3 have been commercially made, and foams with densities of 60 kg/m3 have been made experimentally. Crystallinity levels of 15% or so have been achieved, although very thin sheet tends to have much lower crystallinity. Thick PET foam plank (1 inch or so) has been continuously extruded at densities of 80 kg/m3 and crystallinity levels of 25% or so.

PET foams has been successfully produced on the three major types of foam equipment – twin-screw, two-stage single screw and tandem. PET has been foamed with hydrocarbons, HCFCs, carbon dioxide, and combinations thereof. Classically, a chemical nucleating agent/foaming agent such as AZNP, 5PT and / or an endothermic foaming agent, is added at 0.1% to 0.2% (wt). Care must be taken using talc as a bubble nucleator since it also acts as a crystallite nucleator. An excess of talc will yield a brittle foam. In fact, care must be taken with all adducts to ensure that the adduct does not carry adsorbed water into the extruder or produce water as part of its decomposition reaction. As a result, it is recommended that PET be dried at 160oC for at least 4 hr prior to extrusion. All reclaimed regrind must be dried to the same intensity. A small quantity of water, typically less than 50 parts per million, will cause substantial deterioration in PET IV. [Chemical blowing agents such as NaHCO3 or any of the modified bicarbonates produce water as a decomposition product and so must not be used. AZNP and 5PT can also be problems since they produce by-products such as ammonia, which will degrade PET.]

The standard PET plasticating extruder temperature profile is 200(oC)-280-270- through the melt pumping stage. Melt pressure at the physical foaming agent inlet should be in excess of 100 atm. Although sophisticated plasticating screw sections are commonly used in PS and PE foams, they should be avoided in PET foaming, since excessive shear leads to thermal degradation. The same is true for excessive dissipative mixing downstream of the blowing agent inlet. PET seems to run satisfactory on a standard PS screw, and in fact plasticizes like PS, albeit at melt temperatures that are about 100oC higher.

As with all foamable melts, cooling is critical. The gas-laden PET melt should be delivered to the extrusion die at about 185oC or about 85oC or more below the traditional PET melt temperature and about 10oC below its recrystallization temperature. As with all foamable melts, the melt pressure at the die entrance should be 100 atm or so. For PET, a higher melt pressure is needed to prevent premature foaming inside the die, an effect that leads to poor final product surface quality and fibrillation at the die exit.

Twin-screw extruders, two-stage or long single-screw extruders, tandem single-screw extruders and twin-screw-to-single-screw tandem extruders have all been used to produce production quality foams. The key to effective tandem foam extrusion seems to lie in quality gas seal at the secondary extruder screw bearing. Reverse flighted melt seals are only partially effective in achieving quality gas seal. Since the blowing agent dosage for PET is usually quite low [on the order of 3% (wt) or so], any gas loss through any aperture in the barrel will dramatically affect the foam quality.

Since PET foams at much higher gas-laden melt temperatures than either PE or PS, higher boiling point blowing agents are needed. HCFCs that have been used successfully include 134a, 142a and 152a. Their properties are given in Thermoplastic Foams. Although isopentane or i-C5 produces an adequate foam, hexane produces a much more uniform foam with the classic halo around the foam die. Since there are environmental concerns about using C6, a combination of heptane and isopentane seem to produce a foam quality similar to that foamed with C6. Carbon dioxide, by itself, does not produce a useful foam. Although the cell size of a PET foamed with CO2 is extremely fine, the corrugations normally produced with a very low boiling foaming agent are so severe as to render the product useless. [The causes and mitigation for corrugation is given in another Technical Minute.]

Jim Throne, 25 June 1997