Question 7 of 7: Short-answer set — mechanism, crystallinity, pressure effects, stiffening PE

Question 7: Short-answer set — mechanism, crystallinity, pressure effects, stiffening PE (Part D — 20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Polymerization mechanism from the polydispersity

The polydispersity index is \(\text{PDI}=\overline{M}_w/\overline{M}_n\). The stated numbers give \(25{,}000/50{,}000=0.5\), which is physically impossible — by definition \(\overline{M}_w\ge\overline{M}_n\) always. The two values are evidently transposed; reading them as \(\overline{M}_n=25{,}000\), \(\overline{M}_w=50{,}000\) gives

\(\displaystyle \boxed{\text{PDI}=\overline{M}_w/\overline{M}_n=2.0.}\)

A polydispersity of 2.0 is the signature of the “most-probable” (Flory) distribution obtained in step-growth polymerization at high conversion, where \(\text{PDI}=1+p\to2\) as \(p\to1\). Chain-growth (free-radical) polymer gives PDI ≈ 1.5 for termination by combination or 2.0 for pure disproportionation, and living/ionic chains give PDI ≈ 1. A clean PDI of 2.0 is therefore most consistent with step-growth polymerization.

(b) Crystallinity of Sample C from density

Given. Two calibration samples of known crystallinity and a two-phase (crystalline + amorphous) density model, in which density is linear in the degree of crystallinity.

1. Calibrate the density–crystallinity line. Using the two-phase relation \(\rho=\rho_a+X(\rho_c-\rho_a)\) with samples A and B: $$\rho_c-\rho_a=\frac{\rho_A-\rho_B}{X_A-X_B}=\frac{0.960-0.930}{0.80-0.70}=0.30\ \text{g/cm}^3,$$ giving \(\rho_a=0.960-0.80(0.30)=0.720\) and \(\rho_c=1.020\ \text{g/cm}^3\).
2. Interpolate Sample C. $$X_C=\frac{\rho_C-\rho_a}{\rho_c-\rho_a}=\frac{0.950-0.720}{0.30} =0.767=\boxed{76.7\%}.$$ Equivalently, straight interpolation between A and B (since \(\rho\) is linear in \(X\)): \(X_C=70\%+\dfrac{0.950-0.930}{0.960-0.930}(10\%)=76.7\%\). The mass-fraction two-phase model (\(1/\rho\) linear in weight-crystallinity) gives \(76.7\%\) as well.

(c) Effect of pressure

(i) Glass-transition temperature — \(T_g\) increases with pressure. Applying hydrostatic pressure compresses the polymer and reduces its free volume. Since the glass transition occurs at an iso-free-volume state, a melt/rubber squeezed to smaller free volume must be heated more to regain the segmental mobility needed for the transition; thus \(T_g\) rises, typically \(dT_g/dP\approx+0.15\)–\(0.3\ \text{K/MPa}\) (consistent with the Ehrenfest relation \(dT_g/dP=TV\Delta\beta/\Delta C_p>0\)).

(ii) Melt viscosity — increases with pressure. By the same free-volume argument (WLF/Doolittle, \(\eta\propto\exp(B/f)\)), raising pressure lowers the fractional free volume \(f\) and raises the effective \(T_g\), pushing the melt closer to its glassy state; segmental friction and hence melt viscosity increase with pressure. This is a real processing concern — injection-moulding pressures can raise melt viscosity appreciably.

(d) Three methods to stiffen room-temperature polyethylene

1. Increase crystallinity/density. Use a more linear, less-branched grade (HDPE rather than LDPE), or anneal/slow-cool to raise the crystalline fraction; the stiff crystallites raise the modulus (the whole basis of part b).
2. Reinforce with a rigid filler or fibre. Compound with glass fibre, talc, carbon black or CaCO&sub3;; the high-modulus phase carries load and stiffens the composite (rule-of-mixtures modulus).
3. Cross-link the chains. Peroxide or radiation cross-linking (as in PEX) ties the network together, restricts chain slippage and raises the modulus (and creep resistance) above \(T_g\).

(Equivalent alternatives: uniaxial drawing/orientation, which greatly stiffens PE along the draw direction; or copolymer/nucleation strategies that raise crystallinity.)