Models and Calculations for Polymer Modulus
An example of a relatively simple equation for a two-phase composite model that has been employed to model polyurethane properties is the Kerner equation . Like many such equations, the derivation of the final equation assumes a macroscopically homogeneous and isotropic solid with properties defined by the properties and concentrations of the components in the bulk. One component is assumed to be on average spherical and randomly distributed within a uniform matrix. Key considerations in this model are that the composite inclusions are bonded to the matrix and— they are isolated from each other—thus, they do not form a co-continuous structure within the composite. The approach used by Kerner is termed a generalized self-consistent field approach. Kerner assumed that the spherical particle is embedded in a spherical shell of matrix surrounded by an "infinite" body with the average composite properties. A basis for the equation is also that the adhesion between the filler and the matrix is perfect, so that displacements across boundaries are continuous and invariant. The spatially averaged properties of the embedded particle must match the properties of the matrix. Analysis of the cell and its properties and distributions are iterated until there is a match between the modeled volume element and the bulk reference . The resulting Kerner equation is given by Equation 4.8 where Gc is the shear modulus of the composite, Gp the shear modulus of the matrix phase, Gf the shear modulus of the filler phase,vp and vf are the volume fraction of polymer and filler respectively, and v the Poisson's ratio.
The form of Equation 4.8 is the same as the Clausius-Mossotti equation for defining the properties of two-component systems, each component having definable and unique properties . The terms G.(7 - 5v)/(8 - 10v) are derived for the purpose of setting upper and lower bounds on the bulk moduli . The value of the Poisson's ratio, v, will have a quantitative effect on the result, but not a qualitative effect. In principle, v should be a function of the filler volume fraction for the usual case that the matrix and the filler have different Poisson's ratio values, but in practice this is usually not a significant effect.
In contrast to the isolated particle assumption of Kerner's equation 4.8, the Davies equation 4.9 assumes that the filler phase and the matrix phases are co-continuous and adhesion between the phases is perfect . Other than the co-continuity assumption, the physical foundation of Davies' model is similar to that of the Kerner equation :
In Equation 4.9 0 . represents the fraction of component I while G represents the respective shear modulus. The Davies equation has been used successfully to model the composition dependence of numerous two-phase systems including semi-crystalline polymers , block copolymers , and block copolymers with an interstitially polymerized third polymer phase [32, 39].
The simplicity and success of the Davies equation has inspired modifications to take account of physical phenomena associated with the phase separation process such as phase percolation associated with the transition from the spherical to the cylindrical phase transition (Fig. 4.6). An example might be typified by Equation 4.10  that has demonstrated additional utility in evaluating polyurethane elastomer data.
In Equation 4.10, 0 p is the hard segment weight fraction that is reinforcing, and 0 sC is the hard segment fraction at which the phase morphology transitions from a spherical to a cylindrical phase morphology which can be interpreted as a percolation threshold.
Figure 4.9 Graphical representations of the Kerner and Davies equations using the assumption of a 103 difference between filler and polymer moduli (relevant to polyurethane systems).
A visual comparison of the Davies and Kerner equations is provided in Figure 4.9. The rapid and continuous increase of modulus in the Davies equation is reflective of the percolation assumption. The slow increase in modulus observed for systems modeled by Kerner's equation reflects the noninteraction of particles and their non-continuous phase status.
Both the Davies and Kerner equations lack recognition that at some hard segment fraction there can occur a phase inversion in which the hard segment is the matrix phase and the soft segment is in effect the filler. Budiansky  proposed an alternative two-phase composite model that would provide a means for such inversion to occur. The Budiansky model differs fundamentally in that his model embeds the filler particle (hard segment in a polyurethane) directly into the effective medium rather than in the matrix polymer. In this manner, the physics and the equation are symmetric with replacement of matrix and filler fractions (Eq. 4.11).
In Equation 4.11, 0 2 is the variable volume fraction, G is the composite modulus, and G1 and G2 are the filler and matrix moduli. A graphical representation of the Budiansky equation is shown in Figure 4.10. This graph shows the realistic case for systems intermediate to the Kerner and Davies case in that the filler may at some range of volume fraction exists in isolated domains. In the Budiansky representation, the modulus function begins to show deviation from an initial linear increase at
Figure 4.10 Graphical representation of the Budiansky equation using a soft phase modulus of 1 MPa and a hard phase modulus of 1 GPa as relevant to polyurethane systems.
approximately 30% volume percentage filler. At a volume fraction of about 60%, the function again assumes a linear increase suggesting a phase inversion has occurred. While a phase inversion is not a requirement of a specific composite system in this range, and the specific details of the derivative in any specific range may be inexact, the basic form of the Budiansky equation is quite similar to that observed in actual polyurethanes as a function of hard segment volume. The actual values obtained in the moderate to high hard segment volumes are often better approximated by the Davies equation [32, 41].
Another commonly invoked equation to predict composite modulus is the Halpin-Tsai equation (including the numerous variations and refinements that have been developed). This equation is unique in that it allows for distinction of filler shape. It has been well applied to many composite materials and materials that can be modeled as composites, extending from semi-crystalline polymers to nanocomposite filled thermosets [39, 42—15]. For polyurethanes, it is possible to use a simple form of the equation such as Equation 4.12.
Where A is the filler aspect ratio, 0 is the filler volume fraction, and the other symbols are the respective moduli. The aspect ratio is the ratio of the filler's length to width. Thus a spherical particle would have an aspect ratio of 1 while a fiber might have an aspect ratio of 100 or higher. An assumption of Equation 4.12 is that anisotropic fillers are perfectly aligned. The modulus for the direction perpendicular
Figure 4.11 Graphical representation of the Halpin-Tsai equation (4.12) using a soft phase modulus of 1 MPa, a hard phase modulus of 1 GPa, and with aspect ratio (A) of either 100 or 1.
to fiber orientation can be calculated with the alternative assumption of A = 1 and the modulus of the overall random composite calculated by taking the average. A graphical representation of the Halpin-Tsai expression is given in Figure 4.11 showing the significant orientation effect of high aspect ratio fillers. The similarity between the Kerner result (Fig. 4.9) and the Halpin-Tsai result for A = 1 reflects the parallel initial assumptions.
An approximate expression (Eq. 4.13) that can also be applied is based on the expressions of Kolarik  and Bicerano  and is based on a simple percolation equation.
Where 0 perc is the volume fraction at which the filler percolates as a co-continuous phase. For polyurethanes, the percolation threshold can be construed at the spherical to cylindrical phase transition or even the cylindrical-to-lamellar phase transition (see Fig. 4.6). The exponent p is sometimes called the percolation exponent and is an adjustable variable on the order of 2. An obvious simplification in this calculation is the invitation to neglect the soft segment since its modulus is much smaller than that of the hard segment. The equation does not have a defined result below the percolation threshold but can be refined to compensate if desired. Results are qualitatively similar to the Davies equation (Fig. 4.12).
Figure 4.12 Graphical representation of the modulus predictions based on a percolation equation (4.13) with a filler modulus of 1 GPa and a percolation threshold of 0.25 filler volume fraction.