Calculating Foam Properties
The properties of polyurethane foams are a convolution of the polymer physical properties with the mechanical properties of foam structure. Almost any solid can be foamed and so the concepts that qualitatively and quantitatively describe generic foam mechanical properties can be applied to polyurethane foams, at least as a starting point. In fact, it has been shown in numerous places that polyurethane foam properties are often well described by a picture of a polyurethane elastomer at lower density [59-61].
Observations of properties translating from an elastomer to a foamed elastomer are accurate, but only in a limited sense. The fact is that morphological factors that distinguish polyurethane from nonmicrophase separating materials (for instance polystyrene/StyrofoamTM) can have significant effects on the foaming properties. The chemical and physical processes that occur during the foam process can be quite
Figure 4.18 Graphical representation of Equation 4.16 calculating the maximum stress as a function of maximum elongation and molecular weight between crosslinks. The maximum elongation at break is arbitrarily, though reasonably, set at 400%.
different from those that occur when making a polyurethane elastomer, and they can modify the foam properties by resulting in greater or lesser phase separation, differing levels of network connectivity and molecular weight between crosslinks. All the physical concepts that create a specific polyurethane elastomer translate to the polymer morphology of the polyurethane foam, but processing conditions can result in unexpected variance. Furthermore, foam properties are highly influenced by differing choices for catalysis, and surfactancy, which can affect physical properties and cannot easily be accounted for in a predictive calculation .
For open cell foams, the equation relating foam's Young's and shear modulus to the modulus of unfoamed material is given by Equation 4.17. These equations are useful for a wide variety of polymers, densities, and polymer thermal properties .
Where Ei is the Young's modulus, G is the shear modulus and p the density. Measured variances from these simple relationships can be significant and can result from imperfections in foam cell struts, and the presence of closed cells or partially closed cells within the foam. Deviations from expected properties might well be expected to be sensitive to structural imperfections since the foams are usually tested in compression, and the flexural modulus Eb, bending stress ob and maximum strain at the bottom surface of a bending beam eb are exceptionally sensitive to geometry (Eq. 4.18)
In Equation 4.18, L is the support span, b is the width of the beam, h is the height of the beam, m is the slope of the linear elastic portion of the load deflection curve, and P is a given point on the load deflection curve.
Closed cell foams are more complicated than open cell foams from a fundamental point of view, if only slightly more difficult from a computational perspective. Polyurethane foams are made from an initial liquid state. As the chemical and physical processes proceed to form a foam, material will be drawn from the expanding cell bubble into the edges. In open cell foam, cells matriculate as closed cells up until the thinning cell face membrane ruptures and open cells percolate a continuous open channel from one end of the foam to the other. In the case of closed cell foams, the internal gas pressure that builds within the individual cell bubbles is insufficient to break the thinning cell face membrane prior to the system vitrifying . In this case, a substantial amount of the foam tensile properties will be due to the foamed material contained within the cell faces.
An additional contribution to cell tensile properties will be made by the gas that is trapped within the closed cells. If the sample is compressed by given strain e, the volume of the foam decreases given by Equation 4.19 where v is Poisson's ratio.
However, the gas only occupies the space not occupied by the foam bubble edges and faces, so that the gas volume decreases by
where p/ps is the ratio of the foam density to the solid density. The change of the cell internal gas pressure upon compression is given by Boyle's law.
The pressure resisting compression and contributing to the foam stiffness is the difference between the calculated p and the internal gas pressure under no strain (p ').
Thus the contribution of the gas trapped within closed cells to the foam modulus is given by Equation 4.22.
Thus, the modulus of the closed cell foam can be calculated from a simple additivity assumption as equal to the sum of moduli from the cell edges, cell faces, and internal gas pressure (Eq. 4.23) .
Where q> is the fraction of solid in the bubble edges and 1- q> is the fraction in the cell faces. There is no shear modulus contribution from the cell internal gas pressure so it does not appear in that equation. Figure 4.19 compares the modulus ratio of the foam
Figure 4.19 Graphical representations of Equations 4.17 and 4.23 showing the modulus ratio of foam to its solid polymer as a function of density and the fraction of polymer in the cell edges (q>).
to the unfoamed polymer as a function of the ratio of the foam to the solid density using Equations 4.17 and 4.23. For reference, a rigid (closed cell) polyurethane foam has been reported to have a q> value of 0.8 .