#### Modeling Airflow

We compared our experimental airflow measurements through the foam (described in the Testing page) to an Kozeny-Carman (KC) model for ﬂuid ﬂow through a packed particulate bed. The KC model relates ﬂuid ﬂow to key parameters as displayed in the equation below.

\(Q = {\Delta P \Psi^2 D_p^2 \phi^3A \over 180 L \eta (1-\phi)^2}\)

where Q is ﬂow rate, ΔP is pressure drop across the sample, Ψ is PDMS spherical, D_{p} is the spacing between pores, Φ is sample porosity, A is sample cross-sectional area, L is sample length, and η is viscosity of the ﬂuid.

Of these parameters, we directly measured Q, ΔP, A, and L from the airﬂow experiment measurements and sample geometry.

We used X-ray μCT to extract Ψ, D_{p }, and Φ for each of the tested foams. Using the freely available ImageJ image processing software with the BoneJ plugin (version 1.4.0), we determined the porosity, pore surface area, and average spacing between pores from the scans using the Volume Fraction, Isosurface, and Thickness functions, respectively. Since the foam microstructure is relatively similar to that of cancellous bone, the BoneJ plugin is a particularly useful software package.

From these measurements, we calculated the sphericity (Ψ) of PDMS from the following equation with measured values of PDMS volume (V) and surface area (S.A.) from BoneJ.

\(\psi = {\pi^{1/3} (6V)^{2/3}\over S.A.}\)

When input into the KC model, using no free parameters, the predicted ﬂow rates are in close agreement with the measured rates through rigid foams. The soft foams, however, exhibited much higher ﬂow rates than both the rigid foams and the ﬂow rate predicted using the KC model. To determine the cause of this discrepancy, we compared µCT scans of the sample with and without air ﬂowing through it. Visually, when air ﬂowed through the foam samples, we noted that the foam cylinder deformed to create hemispherical surfaces in the direction of airﬂow. The µCT scans were consistent with this behavior in that they showed an apparent 12% increase in porosity when permeating at a 7 L min^{−1} ﬂow rate relative to the porosity at 0 L min^{−1}. We attribute the apparent increase in porosity to the expansion of pores in accommodating the strain of the hemispherical macroscopic deformation. Further verifying this phenomenon, our calculations showed that the porosity was highest in the center of the sample (the region furthest from ﬁxed edges) and became incrementally less porous in regions closer to the perimeter. As the porosity is a heavily weighted parameter in the KC equation, we attribute the increase in ﬂow through soft foams to this strain-induced porosity increase.