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.].
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.]
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:
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.