The equilibrium melting temperature

The equilibrium melting temperature (T°) is central to most crystallization theories. The degree of supercooling (AT) is defined with reference to according to the following equation  

The melting point (TJ of samples with a well-defined crystal thickness (L ) can be measured and the data extrapolated to L = 0 using the Thompson—Gibbs equation  

Data for the enthalpy Ah) and entropy (As) of fusion can be obtained for relatively small molecules (oligomers) and the equilibrium melting point is obtained by extrapolation of Ah and As data as a function of degree of polymerization (x) to infinite x  

The Broadhurst equation (applicable to linear polyethylene) is probably the best-known equation of this kind  

The diagram shown in Fig. 8.4 is often referred to as a Hoffman-Weeks plot. It demonstrates that the curve of versus for virgin crystals of a certain 5Lc intersects the graph of = Tc at the equilibrium melting temperature. Curve b (Fig. 8.4) shows the melting point of the initially formed crystals with = 3 nm L = 2 rlAg -h 61, where L is the initial crystal thickness, and 1(71Ag is the minimum crystal thickness possible at the 

As already mentioned, the full characterization of the melting process of polymeric materials requires the estimation of the crystalline fraction, which requires knowing not only the heat fusion for 100% crystalhne polymer but also the equilibrium melting temperature and all the characteristics of the equilibrium melting. 

With the help of measurements of the lamellar thickness, /, by dUatometry, AFM or X-ray, and the melting temperature of the nonequilibrium crystal by DSC, it is possible to obtain the equilibrium melting temperature, T, by extrapolation to infinite lamellar thickness. Next, with Equation (9.7), it is also easy to calculate the fold surface free energy. Usually, the large surface area of the polymer crystal explains irreversibility of the melting process. 

The Hoffman-Weeks method is one of the most common approaches in the literature [1, 3, 39, 40] for estimating the equilibrium melting temperature from the experimental data. This frequently used method was often critically assessed as by Hoffman and Miller in Reference [41] and modified as proposed by Phillips and coworkers [39] for iPP to avoid thickening of lamellae. 

The equilibrium melting temperatures T° for some other polymeric materials are presented in Table 9.1. 

---

We must remember that T in equation (6.161) is the equilibrium melting temperature. Integration of this equation will give an equation that relates melting temperature to activity. Separating variables and integrating... 

In Chapter 6, we derived equation (6.161) shown below, which relates the activity, a, of a component in solution to the equilibrium melting temperature, T, of that substance. 

Typical growth configurations from the simulations are shown in Fig. 4.4, for kT°/s = 0.7 and kT Je = 0.55, respectively (e is the interaction energy between adjacent units, and T° is the equilibrium melting temperature). Notice the increased roughness of the former which has the lower binding energy compared with the temperature. 

Temperature has a complex effect on crystallization rate. Initially, as the temperature falls below the equilibrium melting temperature, the crystallization rate increases because nucleation is favored. However, as the temperature continues to fall, the polymer s viscosity increases, which hampers crystallization. As a rule of thumb, a polymer crystallizes fastest at a temperature approximately mid-way between its glass transition temperature and its equilibrium melting temperature. 

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

Isothermal crystallization was carried out at some range of degree of supercooling (AT = 3.3-14 K). AT was defined by AT = T - Tc, where Tj is the equilibrium melting temperature and Tc is the crystallization temperature. T s was estimated by applying the Gibbs-Thomson equation. It was confirmed that the crystals were isolated from each other by means of a polarizing optical microscope (POM). 

The equilibrium melting temperature T was determined on ECSCs using Wunderlich s method [26]. The Tm of ECSCs was estimated from a tem-... 

Unlike supercooling of liquids, superheating of crystalline solids is difficult due to nucleation of the liquid at surfaces. However, by suppressing surface melting, superheating to temperatures well above the equilibrium melting temperature has... 

As discussed earlier, the amorphous state is a nonequilibrium state at temperatures below the equilibrium melting temperature of a material. Because of the nonequilibrium nature of the amorphous state, various properties of amorphous materials, such as the glass transition, are dependent on time and temperature (Slade and Levine, 1988,1991 Roos, 1995,2003). Therefore,... 

Even though the nonisothermal crystallization leads to just small changes in the subsequent melting behavior of different types of triblock copolymers, isothermal experiments employed to calculate the equilibrium melting temperature, T, have shown that this parameter can exhibit significant changes depending on composition. It has been reported that in PS-fc-PB-fc-PCL tri-... 

Floudas et al. [135] also studied the isothermal crystallization of PEO and PCL blocks within PS-b-PEO-h-PCL star triblock copolymers. In these systems the crystallization occurs from a homogeneous melt Avrami indexes higher than 1 are always observed since the crystallization drives structure formation and does not occur under confined conditions. A reduction in the equilibrium melting temperature in the star block copolymers was also observed. 

Table 20.3 Results obtained for the equilibrium melting temperatures from the Hoffman-Weeks plots of the P(HB60-ET40)/ PET and P(HB80-ET20)/PET blends...

In writing these equations, effects of chain ends are not explicitly taken into account. At the equilibrium melting temperature T, we obtain... 

This simple result for the equilibrium melting temperature offers a guidance in the chemical design of elastomers (low 7 ) and engineering plastics (high T ). For example, if the chain backbone is more flexible, then the change in... 

A plot (called the Hojfman-Weeks plot [36]) of T versus Tc is linear for a constant thickening factor, and the extrapolated intersection of T with Tc is taken to be the equilibrium melting temperature T . While this procedure has been improved [37], the whole concept is also contested [38]. 

The expression for Gi nicely shows its nonmonotonic dependence on the crystallization temperature. Close to the equilibrium melting temperature, the growth is nucleation-dominated and is given essentially by Gi, . For temperatures far below T , but closer to Tq, the Gi d term dominates and the growth rate precipitously decreases with supercooling. 

Fig. 8. Superheating of polyethylene extended chain crystals. Curves 1) at 421.7 K, 2) at 419.2, 3) at 417.7 K, 4) at416.7K, 5) at 414.7 K. The equilibrium melting temperature is 414.6 K. Drawn after Ref.40). "w is the weight fraction molten, obtained by isothermal calorimetry...

Fig. 9. Melting kinetics and crystallization kinetics of polymeric selenium (right) and polyethylene (left). The equilibrium melting temperatures are 494.2 and 414. 6K. The dotted curve indicates that on crystallization of the macromolecule from small molecules Sc2 there is no molecular nucle-ation necessary as in the melt crystallization (see also ref. 43 for a more detailed discussion of Se crystallization and melting). Drawn after Ref. 4,)...

The equilibrium melting temperature, T°m, can be obtained from data for crystals of finite thickness using the Thompson-Gibbs equation. The melting point of crystalline polymers with a well-defined crystal thickness (/c) can be measured and the data extrapolated to 41 = 0 using the Thompson Gibbs equation (Gedde 1995) ... 

Thus the initial crystal thickness can be determined if the equilibrium melting temperature (obtained from extrapolation of Tm versus 4 via eqn 5.5) is known. 

The values of A Hv and A Sv are almost independent of temperature. For freezing at the equilibrium melting temperature, Tm, AGv = 0 so A// = TmASv. Below the melting point,... 

---