Melting equilibrium

This brief summary reveals a too complex issue to be discussed in all details in a single book chapter. More information is available through the 2879 screens of the computer course Thermal Analysis of Materials, available by downloading from the Internet [1] and the reference work of 2547 pages on Thermal Characterisation of Polymeric Materials [2], or in the treatises on Thermal Analysis of Polymeric Materials [3] and Macromolecular Physics [4-6]. General information and data can be found in Calorimetry [7], the Encyclopaedia of Polymer Science and Engineering [8], the Polymer Handbook [9] and the ATHAS Data Bank [10]. All these sources should give you access to the information needed for interpretation of the instrumental and polymeric materials problems beyond this chapter. 

Section 2 of this chapter contains the basics needed to understand melting and crystallisation, mainly using equilibrium and irreversible thermodynamics and kinetics. Section 3 comprises a summary of the details on instrumentation and data treatment. Both of these sections can be bypassed initially when the main goal is to get started quickly on experiments. As the need arises, the basic material can then be filled in by reading Sections 2 and 3 and consulting the references. 

Combining MTDSC and polymer science is a challenge, which when met, yields so much additional information on the subjects that the added effort to understand it is well worthwhile. This chapter can, naturally, only point the way and help in avoiding the most common pitfalls. 

The equilibrium description of melting and crystallisation is a subject of the field of thermodynamics. The basic quantity of calorimetry is the heat capacity, Cp (at constant pressure, in J moL ), which represents the amount of heat, Q (in joules, J), needed to be added to raise the temperature by 1 K or to be extracted to lower the temperature by 1 K for 1 mol of material. If the material analysed has a mass of 1 g, one calls this quantity the specific heat capacity, Cp (at constant pressure, in J K g ). In the more precise differential notation, one writes for the heat capacity that 

Cp = (9if/9r)p where H is the enthalpy or heat content, and the subscripts p and n signify that the partial differential is taken at a constant pressure (usually atmospheric pressure) and without changing the amount of material which is expressed in number of moles n, respectively. 

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For a pure substance, the melting point is identical to the freezing point It represents the temperature at which solid and liquid phases are in equilibrium. Melting points are usually measured in an open container, that is, at atmospheric pressure. For most substances, the melting point at 1 atm (the normal melting point) is virtually identical with the triple-point temperature. For water, the difference is only 0.01°C. 

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... 

Line db in Figure 8.1 represents the equilibrium melting line for C02. Note that the equilibrium pressure is very nearly a linear function of T in the (p, T) range shown in this portion of the graph, and that the slope of the line, (d/ /d7 )s ], is positive and very steep, with a magnitude of approximately 5 MPa-K-1. These observations can be explained using the Clapeyron equation. For the process... 

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)...

Fig-1 Schematic illustration of the crystallization and melting processes of polymers. The crystallization process corresponds to processes of disentanglement and chain sliding diffusion. The melting process is the reverse of the crystallization process. Between equilibrium melt and ideal crystal, there exists metastable melt and crystal. Cross marks indicate entanglement... 

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]... 

The copolymers consist of strictly alternating sequences of diene and olefin. C-NMR measurements Showed the microstructure of the butadiene units in BPR to be exclusively of the trans-1,4 configuration (Figure 8). The isoprene units in isoprene-ethylene copolymer (IER) contain 84 % trans-1,4, 15 % cis-1,4, and 1 % 3,4 structures (Figure 9). Spontaneous crystallization in unstretched BPR samples was detected by dilatometry and confirmed by X-ray diffraction and DSC measurements. The extrapolated equilibrium melting point is about -10 °C. 

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,... 

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