Crystallization of Polymers. Volume 1, Equilibrium Concepts

56 Fusion of homopolymers
Table 2.3. Parameters governing fusion of linear polyethylene a (55)
M n x T m,e ( ◦ C) ζ e ζ e /x σ eq (cal mol −1 ) x − ζ e
1586 113 124.5 95 ± 1 0.84 ± 0.01 1298 ± 200 18
2221 159 126.0 140 ± 2 0.88 ± 0.01 2024 ± 200 19
3769 269 132.0 242 ± 3 0.90 ± 0.01 2551 ± 300 27
5600 400 134.2 368 ± 4 0.92 ± 0.01 3485 ± 500 32
a Uncertainties calculated by assuming T m,e =±1 ◦ C.
Table 2.4. Parameters governing fusion of poly(ethylene oxide) a
T 0 m = 80 ◦ C
T 0 m = 76 ◦ C
σ eq
M n x T m ( ◦ C) ζ e ζ e /x (cal mol −1 ) x − ζ e ζ e ζ e /x (cal mol −1 ) x − ζ e
1110 25 43.3 23 0.90 1413 2 22 0.88 1186 3
1350 31 46.0 28 0.91 1734 3 28 0.90 1447 3
1890 43 52.7 39 0.91 1995 4 38 0.89 1588 5
2780 63 57.6 58 0.93 2567 5 57 0.90 1954 6
3900 89 60.4 84 0.94 3410 5 82 0.92 2523 7
5970 136 63.3 129 0.95 4776 7 127 0.93 3389 9
7760 176 64.3 169 0.96 6080 7 166 0.94 4261 10
a Melting temperature data from Ref. (18).
σ eq
The results of the analyses for linear polyethylene and poly(ethylene oxide) are
summarized in Tables 2.3 and 2.4 respectively. In order to perform the calculation
it is necessary to assume a value for Tm 0. For polyethylene 145 ± 1 ◦ C was taken for
Tm 0, while for poly(ethylene oxide) either 76 ◦ Cor80 ◦ C was assumed. Although the
precise values of the parameters deduced will depend on the value of Tm 0 the trends
with molecular weight are unaffected by the choice. Similar results are obtained
for both polymers. Over the molecular weight range for which appropriate data are
available σ eq varies three- to four-fold. Put another way, the parameter b in Eq. (2.26)
varies by about a factor of two over the molecular weight range appropriate for
analysis. Booth and coworkers performed a similar analysis with the low molecular
weight poly(ethylene oxides).(56) The σ eq values were of the same magnitude as
reported here, and they also increased with chain length. It is evident that Eq. (2.27)
cannot be used to extrapolate the melting temperatures of chains of finite length to
T mṪhe 0 increase in σ eq with chain length is caused by the first term on the right in
Eq. (2.25). It stems from the number of ways the equilibrium sequence of crystalline
units ζ e can be chosen from among the x units, if the chain ends are excluded. It