Equilibrium Melting, Temperature

It is illuminating to rewrite this equation in terms of the degree of undercooling by substituting for Ag. To do that, however, we have to digress a little and define the equilibrium melting temperature. This will also be useful in our discussions of nucleation rates and the melting point, so bear with us  

17 Things either melt or crystallize at a certain temperature, they don t actually hang about in equilibrium. 

We are assuming As/ and Ahf don t vary much with temperature. Remember that this is just the bulk free energy (per unit volume) of the crystal, it does not include any additional free energy from those segments at the surfaces. We can now return to our expression for the critical nucleus size (thickness), which was given previously in Equation 10-25, and substituting for Ag, we obtain Equation 10-30  

Note the inverse dependence of fold period on AT. The larger the undercooling the smaller the fold period. 

The form of the crystallization rate curve is common to many systems and similar expressions have been found for metals, inorganic compounds, sulfur, selenium, antimony, proteins and carbohydrates, graphite silicates and also polymers. 

Central to most crystallization is the idea of an equilibrium melting temperature, J, above which crystallization does not occur. The rate of crystal growth is therefore related to the extent to which supercooling (AT) occurs and is defined by 

6 Variation of melting temperature with crystallization temperature for poly(trimethyl terephthalate) (unpublished data, A. Mckintosh and J. J. Liggat). 

The equilibrium melting temperature can be determined in a number of dilferent ways. The melting point of samples with a well-defined crystal thickness can be measured and the data extrapolated to =0 using the Thompson-Gibbs equation  

An alternative approach is to derive the value of from a study of low molar mass analogues and then extrapolate to the infinitely thick crystal value. Using data for the enthalpy (A//b) and entropy (AAb) of fusion for oligomers, is obtained by extrapolation to infinite molar mass and for linear polyethylene has the form 

---

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. 

Each crystallizable polymer exhibits a characteristic equilibrium melting temperature, at which the crystalline and amorphous states are in equilibrium. Above this temperature crystallites melt. Below this temperature a molten polymer begins to crystallize. 

Primary crystallization occurs when chain segments from a molten polymer that is below its equilibrium melting temperature deposit themselves on the growing face of a crystallite or a nucleus. Primary crystal growth takes place in the "a and b directions, relative to the unit cell, as shown schematically in Fig. 7.8. Inevitably, either the a or b direction of growth is thermodynamically favored and lamellae tend to grow faster in one direction than the other. The crystallite thickness, i.e., the c dimension of the crystallite, remains constant for a given crystallization temperature. Crystallite thickness is proportional to the crystallization 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. 4 Rescaled data from Fig. 3b to show the linear relationship predicted by Eq. 16. The bulk equilibrium melting temperature Ec/k T is chosen to be approximately 0.2. The lines are the results of linear regression, and the symbols are for the variable values of B/Ec [14]...

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-... 

Fig. 5 Equilibrium melting temperature plotted against log Mn. (o) This work, ( ) Wunderlich [26], solid line Hoffman et al. [28]... 

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. 

Measurements by many researchers have shown that PTFE s equilibrium melting temperature is 327°C. ° Once it is heated above its melting temperature, its initial properties are irrecoverable. Compaction of PTFE powder and heating above 327°C results in a partially crystalline solid polymer composed of large crystals with a coexisting noncrystalline phase. Crystal size and perfection depend on die crystallization conditions slow cooling results in larger, more perfect crystals. On this point, we present detailed information from electron microscopy, corroborated by measurements of X-ray line breadth. 

---