Hoffman-Weeks equation

The lamellar habit adopted by crystalline polymers adds surface terms to the specific Gibbs function (chemical potential), most importantly the fold surface free energy, ae, which contributes 2ae/Xg for a lamella of thickness k and crystalline density q. In consequence melting points are lowered from T, for infinite thickness, to Tm according to the Hoffman-Weeks equation... 

With = const we get immediately the famous Hoffman-Weeks equation (Hoffman and Weeks, 1962a Hoffman and Weeks, 1962b) ... 

A method was described for measuring the heat and rate of crystallisation following partial melting. The method uses a specific sequence of temperatures with a differential scanning calorimeter, and the melting and recrystallisation processes were confirmed by optical observations. The method was applied to poly(butylene terephthalate). The melting temperatures obtained from recrystallisation were used in the Hoffman-Weeks equation to deduce 236 C as the equilibrium melting temperature for PBTP. 22 refs. 

FIGURE 8 Hoffman-Weeks plots for PCL of= 80 000 g/mol(H), data after Chong et al. (2004), and PEO = 13 000 g/mol (o), data after Alfonso and Russell (1986), the arrow marks the distance to equilibrium after Equation (20). 

It demonstrates that Gibbs-Thomson and Hoffman-Weeks approach are completely equivalent. The extrapolation of / - °o equals in thermodynamic terms extrapolation of T ° after Equation (28). Use of Equation (7) in Equation (28) results in ... 

Gibbs-Thomson equation Hoffman-Weeks approach Isothermal crystallization Non-isothermal crystallization Rates of crystallization... 

Estimates of equilibrium melting temperatures of PCL crystallites were made, for various binary and ternary systems, with the aid of Hoffman-Weeks plots. The observed melting points of PCL were between 55 ° C and 65 ° C and varied by about 3 °C with changes in from 38 °C to 48 °C. From the melting-point depressions, as a function of composition and pressure-volume temperature data, Kim and Paul estimated equation-of-state parameters. During the course of this study Kim and Paul determined specific volume data for PCL at a series of hydrostatic pressures (Fig.51) [87]. 

Melting point depressions in isothermally crystallised blends were analysed in terms of the Nishi-Wang equation (Eq. 10) [40] using equilibrium values of the melting point determined from Hoffman-Weeks plots melting points were determined by DSC and optical microscopy [137]. Values of the melting-point... 

Li et al. [81] and Nojima et al. [145] noted very small equilibrium melting point depressions for PCL. The former, using equilibrium values of T from Hoffman-Weeks extrapolations, found a melting point depression of about -0.14 K (wt %) PS. Tanaka and Nishi [146] showed that the depressions were consistent with the Nishi-Wang equation (Eq. 9). 

The crystallization of PEO was also unveiled at the level of individual lamellae in ultrathin films (278-280). It was observed that although lamellar growth rates were retarded in films thinner than 200 nm because of interfacial interactions, SFM provided the same qualitative and quantitative information as typically applied bulk characterization techniques on lamellar growth rates, lamellar thicknesses, and melting points. The Hoffman-Weeks extrapolation, the Gihhs-Thompson equation, and the Hoffman-Lauritzen theory were apphed and the results compared favorably to the corresponding hulk data. 

This equation has been widely used although the validity of applying the WLF equation to spherulitic growth rate is merely a repetitive assertion (Hoffman et al., 1959 Hoffman and Weeks, 1962), not involving any direct proof of substantiation, as Mandelkern has stated. 

Equation (76) is the expression originally given by Hoffman and Weeks who made the explicit assumption that the lamellar crystallites were made up of regularly folded chains. As the general derivation indicates, these are restrictive conditions. In fact lamellar crystallites are not even required for equation (76) to be valid. Based on either equations (75) or (76) a plot of against should be linear and the intersection with the 45"" line gives T, However, only when equation (76) is applicable will the slope of the straight line be 0.5. 

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