Polymer Morphology and Crystallinity

Crystallinity and crystal morphology

Semi-crystalline polymers consist of crystalline lamellae embedded in amorphous regions. The degree of crystallinity is the fraction of the material that is crystalline, usually expressed as a weight or volume fraction. Crystallization requires chains to pack into a periodic lattice, so crystallinity depends on three factors: chain regularity (tacticity and sequence), the symmetry of the repeat unit, and the strength of intermolecular forces. Random sequences cannot pack and remain essentially amorphous. Stereoregular chains can crystallize, but bulky side groups and slow kinetics limit the amount of crystal formed. Chains that carry strong interchain hydrogen bonds or have symmetric, planar repeat units crystallize readily.

• Atactic — random placement of side groups along the chain, which prevents lattice packing. e.g. An atactic vinyl polymer is almost fully amorphous.
• Syndiotactic — regular alternation of side groups, which allows crystallization. e.g. Syndiotactic polystyrene crystallizes, but more slowly and to a lower extent than polyesters or polyamides.
• Interchain hydrogen bonding — strong secondary bonds between amide groups that stabilize the lattice. e.g. Polyamide chains pack tightly into sheets and show high crystallinity.

The heat of fusion gives a direct route to crystallinity: X_c = \Delta H_f / \Delta H_f^{\circ}, where \Delta H_f^{\circ} is the value for a 100 % crystalline sample. A change in crystallinity is a difference in percentage points, not a relative percentage. Rapid cooling from the melt leaves little time for crystal growth, so a quenched sample has a smaller heat of fusion and a lower crystallinity than a slowly cooled sample of the same polymer.

In the melt, chain entanglements act as physical obstacles that restrict segmental motion. Crystallization is therefore favoured when entanglement density is low, and only polymers with an ordered, regular molecular arrangement can crystallize.

Biaxial orientation produces layered crystalline structures that are uniform through the film thickness. A uniform refractive index means fewer interfaces that scatter light, so the film has low haze and high clarity.

Density-based crystallinity calculation

A semi-crystalline polymer is treated as a two-phase system whose specific volume is additive. For a weight fraction w_c of crystals with density \rho_c and amorphous density \rho_a, the relation is \dfrac{1}{\rho} = \dfrac{w_c}{\rho_c} + \dfrac{1 - w_c}{\rho_a}. Solving for the weight fraction gives w_c = \dfrac{\rho_c}{\rho}\cdot\dfrac{\rho - \rho_a}{\rho_c - \rho_a}. Density is measured by a density gradient column or by displacement, and the result is a mass-based crystallinity. A volume-based crystallinity is obtained with \rho = \phi_c \rho_c + (1-\phi_c)\rho_a, which is linear in volume fraction rather than in its reciprocal.

If the specific volume is given directly, the same additivity applies: v = w_c v_c + (1 - w_c) v_a, so w_c = \dfrac{v_a - v}{v_a - v_c}. Check the basis of the question carefully, because the answer may be requested on a mass or volume basis.

When the crystal and amorphous densities are unknown, two specimens with different crystallinities give two linear equations in 1/\rho_c and 1/\rho_a, which can be solved simultaneously. The amorphous density can then be found from the result.

Worked example: with \rho_c = 1.0 g cm^{-3}, \rho_a = 0.8 g cm^{-3} and \rho = 0.9 g cm^{-3}, w_c = \dfrac{1.0}{0.9}\cdot\dfrac{0.1}{0.2} \approx 0.56.

X-ray diffraction of polymer crystals

Diffraction from crystalline polymers follows Bragg's law, n\lambda = 2d\sin\theta, where d is the interplanar spacing and \theta is the Bragg angle. The shape of the diffraction pattern reveals the orientation of the crystallites.

• Isotropic specimen — crystallites are randomly oriented, so diffraction forms continuous Debye–Scherrer rings.
• Highly oriented fibre or film — diffraction collapses into discrete spots or arcs.
• Partially oriented sample — a mixture of rings and spots or arcs.