A Polyester That Misses Its Molecular Weight Target and Shows a High-MW Shoulder
A melt transesterification of dimethyl terephthalate and 1,4-butanediol is run with 250 ppm titanium tetrabutoxide under a nitrogen sparge, programmed to 245 C over 4 h with vacuum applied in the final hour. The charge is calculated for exactly 1.00:1.00 mol ratio and a target Mn of 20,000 g/mol (repeat unit 220 g/mol). GPC of the final resin against polystyrene standards gives Mn = 8,200 g/mol and PDI = 2.3, with a distinct high-molecular-weight shoulder at about four times the peak maximum. End-group analysis by 1H NMR (CH2OH at 3.65 ppm versus aromatic at 8.1 ppm) gives Mn = 9,000 g/mol, consistent with GPC. The diol is drum grade containing 0.4 wt% water and 0.3 mol% glycerol; the nitrogen sparge is estimated to remove about 2 mol% of the butanediol over the run. The customer requires Mn of at least 18,000 with PDI not more than 2.2 for fibre spinning, and the plant has no chain extender available.
1. From the measured Mn, calculate the number-average degree of polymerisation and the stoichiometric imbalance it implies. Does the butanediol loss account for it?
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Degree of polymerisation. Xn = Mn / repeat unit = 8,200 / 220 = 37 (end-group correction is small at this length; the NMR value of 9,000 gives 41, so use 37-41).
Imbalance from Carothers. For step-growth with stoichiometric imbalance ratio r (moles of minority to majority functional groups) and extent of reaction p: Xn = (1 + r) / (1 + r - 2rp). Taking p = 0.99, which is realistic for a Ti-catalysed melt process at 245 C with vacuum:
37 = (1 + r) / (1 + r - 1.98r) 1 + r = 37(1 - 0.98r) = 37 - 36.26r 37.26r = 36, so r = 0.966.
That is an acid excess of about 3.4 mol%, equivalent to a 3.4 mol% diol deficit.
Consistency check. The nitrogen sparge removes about 2 mol% of the butanediol, and the vacuum stage at 245 C, above the diol boiling point of about 230 C, strips more. Together these plausibly reach 3-3.5 mol%. The 0.4 wt% water hydrolyses methyl ester to acid but does not remove acid equivalents, and the 0.3 mol% glycerol contributes only about 0.9 mol% of extra hydroxyl, which is minor and partly offsets the deficit. The conclusion is that diol loss, dominated by stripping, is the principal cause.
How to confirm. Weigh and analyse the distillate for butanediol, and measure the acid value and hydroxyl value of the resin directly; the ratio should be about 1.035.
2. Explain the high-molecular-weight shoulder and the PDI of 2.3. What experiments would confirm branching, and why does branching not rescue the molecular weight?
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Origin of the shoulder. Glycerol at 0.3 mol% is trifunctional and creates branch points. In a step-growth system, a small trifunctional fraction produces a subpopulation of branched molecules whose hydrodynamic volume and molecular weight are far above the linear chains, eluting as a shoulder at high MW, while the bulk of the resin remains linear and short. The PDI of 2.3, above the theoretical value of about 2.0 for a linear step-growth polymer at high conversion, is consistent with the same broadening.
Why there is no gel. With only 0.3 mol% trifunctional units, the extent of reaction needed for network formation is far above 1, so the batch cannot gel. Branching here is a distribution problem, not a gelation risk.
Confirming experiments. GPC with viscometry or multi-angle light scattering: at equal molecular weight, branched chains have a lower intrinsic viscosity and a smaller radius of gyration than linear chains, and the Mark-Houwink exponent drops. 13C NMR for glyceride and glycerol ether carbons, and hydrolysis of the resin followed by GC for glycerol, quantify the trifunctional content directly.
Why branching does not help Mn. Branching increases the weight of a few chains but does not change the number of chain ends. The number-average molecular weight is set by the number of molecules, which is fixed by the stoichiometric imbalance. A branched resin with r = 0.966 still has Mn around 8,000-9,000; the branch points simply add a high-MW tail on top of a short-chain matrix.
3. Recommend corrective actions and in-process controls to reach Mn of at least 18,000 with PDI not more than 2.2, given that no chain extender is available.
Read the worked answer
What the target requires. For Mn = 18,000, Xn = 82. With p = 0.995, solve 82 = (1 + r)/(1 + r - 1.99r): 1 + r = 82(1 - 0.99r), giving r = 0.987. So the stoichiometry must be held within about 1.3 mol%, and conversion above 0.995 must be achieved. The current r of 0.966 cannot be compensated by any practical increase in conversion, which is why stoichiometry is the primary lever.
Corrective actions. (1) Charge the diol at a 2-4 mol% excess, calculated from the measured stripping loss, rather than at 1.00:1.00. (2) Reduce stripping: lower the sparge rate, use a partial condenser or a closed system, and add the diol in a staged charge. (3) Dry the diol or use a Dean-Stark type removal, because 0.4 wt% water makes the effective stoichiometry irreproducible. (4) Set a diol specification of not more than 0.05 mol% glycerol, or use a polymer-grade diol. (5) Drive the final stage harder: deeper vacuum, higher torque endpoint, longer hold at 245 C, since p above 0.995 is mandatory.
In-process controls. Track hydroxyl value and acid value (or 1H NMR end-group ratio) before the vacuum stage and correct the stoichiometry then. Use melt viscosity or torque as the end-point control instead of a fixed time, and release on Mn by GPC plus end-group NMR, with glycerol content by 13C NMR or hydrolysis-GC.