Polymer Blends and Composites

Rule of mixtures for fiber composites

The rule of mixtures expresses a composite property as the volume-weighted average of the properties of its constituents. For a unidirectional fibre composite loaded along the fibre direction, the fibre and matrix undergo the same strain (iso-strain), so the longitudinal modulus is E_c = V_f E_f + V_m E_m, where V_f + V_m = 1. The stress is shared in proportion to each phase's stiffness, so even a moderate fibre volume fraction makes the fibres carry most of the load.

When the stiffness of the fibre is far greater than that of the matrix, the ratio E_c/E_m becomes large and quantifies the reinforcement effect. This ratio is always greater than one for a fibre-reinforced system loaded along the fibres.

The same parallel-path logic applies to thermal conductivity for heat flow along the fibres, giving k_c = V_f k_f + V_m k_m. Heat flows through the fibres and the matrix in parallel, so the more conductive phase dominates in proportion to its volume fraction.

In a tensile test along the fibre direction, the strain of the composite is obtained from the composite modulus, \varepsilon = \sigma / E_c, and not from the matrix modulus alone. Converting units consistently, with 1\ \text{GPa} = 10^{9}\ \text{Pa} and 1\ \text{MPa} = 10^{6}\ \text{Pa}, is the most common source of error in these calculations.

For a given strain, the stress is first found as \sigma = E_c \varepsilon, and the force is then F = \sigma A, where A is the cross-sectional area. The area must be converted to \text{m}^2 before the force in newtons is computed.

Key pitfalls:

• Iso-strain (longitudinal) — equal strain in fibre and matrix; use the volume-weighted sum of moduli.
• Iso-stress (transverse) — equal stress in each phase; the inverse rule 1/E_c = V_f/E_f + V_m/E_m applies and gives a much lower value.
• Volume fraction vs. weight fraction — the rule of mixtures uses volume fractions; convert if weight fractions are given.

Example: a composite with a fibre volume fraction of 0.2, fibre modulus 200 GPa and matrix modulus 2 GPa has E_c = 0.2(200) + 0.8(2) = 41.6 GPa along the fibres.

Polymer blends and interfacial compatibility

Most polymer pairs are immiscible, so a blend consists of dispersed domains in a continuous phase separated by a weak, high-energy interface. Poor adhesion across this interface reduces stress transfer and makes the blend brittle. A compatibilizer is an additive that locates at the interface, lowers the interfacial tension and improves adhesion between the phases.

Physical compatibilizers are usually block or graft copolymers whose segments are each miscible with one component of the blend. Like a surfactant, the copolymer anchors at the interface with each segment in its own phase. For a blend of polyethylene and polypropylene, a block copolymer of ethylene and propylene is the natural choice because its segments match the two homopolymers.

Reactive compatibilizers carry functional groups that react with functional end groups of the other polymer during melt mixing, forming covalent bonds at the interface. A polyolefin grafted with maleic anhydride, or with glycidyl methacrylate, has a polyolefin backbone that is miscible with the polyolefin phase and a reactive group that bonds to the amine end groups of a polyamide.

A compatibilizer must interact favourably with both phases. A segment can be chosen for specific interactions such as dipole or hydrogen bonding rather than for similarity alone. For example, a poly(methyl methacrylate) segment interacts with poly(vinylidene fluoride) through carbonyl and CH_2 contacts and is also compatible with the styrene-acrylonitrile phase of ABS, so it can bridge that blend.

Interfacial wetting is also controlled by additives. A polar additive that hydrogen-bonds to the functional groups of a filler surface can cover the sites where the polymer matrix should wet the filler. Loss of wetting blocks stress transfer across the interface and lowers the mechanical properties even though the filler is nominally functionalised.