Equilibrium melting temperature equation

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. 

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). 

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

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) ... 

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 ... 

The simple method based on Equation 10.13, however, is not commonly used because 100% pure crystalline samples for most polymers are not available. One alternative approach is to use the fusion enthalpy, the latent heat of melting, of chemical repeated units (A Hu) to replace A/fioo in the calculations. AHu can be calculated using the I dory relationship for the depression of the equilibrium melting temperature of a homopolymer due to the presence of low molecular mass diluents. The AHu values of some polymers are available in literature. 

What is important to the present study is to examine the effect of PEC comonomer, not i-PS copolymer structure, on the thermodynamics of interaction with i-PS. Consequently, we limited crystallization temperatures to values less than 180 °C to Inhibit the Tb enhancement process. As can be seen from Figures 5 and 6, good straight lines, which appear to follow Equation 6, can be drawn through the solid data points corresponding to these lower crystallization temperatures, and reasonable equilibrium melting temperatures result from the intersection of these lines with the Tb = Tc lines for each blend fraction of PEC. 

This equation is a general form of the relation between the experimental and equilibrium melting temperature of the blend. 

The T m of iPP crystallized from iPP/elas tomers mixtures increases linearly with T, according to the relation of Hoffman . At a given Tq the observed melting temperatures of the blends are always lower than that of the pure iPP. In particular, for iPP/PiBLM and iPP/PiB M blends, T m decreases monotgnically with the elastomer content, whereas a minimum is observed in the iPP/PiBjjj blends at the 90/10 composition (Fig. 8). The variation of the equilibrium melting temperature, Tm> obtained from the Hoffman equation, with the elastomer content is shown in Fig. 9. 

The values of the equilibrium melting temperature T, determined from Eq. (3), decrease in a continuous way with increasing PMMA content (Fig, 6). The depression observed for the equilibrium melting temperature of pure PEO, AT = T° - T j (where T° refers to pure PEO) has then been related to the volume fraction of the non-crystallizing polymer according to the equation derived by Nishi and Wang for compatible binary blends ... 

Thermal analysis data on lamellar crystals of polyethylene over a wide range of thicknesses are plotted in Fig. 2.90. The Gibbs-Thomson equation is a good mathematical description of the observed straight line and can be used to calculate the equilibrium melting temperature by setting C = (t ° = 414.2 K). Also, the ratio of the surface free energy to the heat of fusion can be obtained from the equation. 

Show that the equilibrium melting temperature of a pure material varies with interfacial curvature in accordance with the following equation (cf. the last term of Equation 6.66 and the accompanying discussion) ... 

When a solid melts at its equilibrium melting temperature, the system absorbs heat reversibly, because the solid and liquid are in equilibrium under these conditions. Dnring the melting process, both temperature and pressure remain fixed. Because this is a constant-pressure process, the heat absorbed by the system can be equated to the enthalpy of fnsion, A fus (see Section 6.6), that is, Because... 

The melt temperature will rise initially and then level off as the viscous dissipation reduces with increasing melt temperature. When a steady state is achieved, the melt temperature no longer changes along the length of the extruder. This is called the fully developed melt temperature or equilibrium melt temperature. This temperature Te can be determined from a simple energy balance equating the viscous dissipation to the conductive heat loss ... 

When qc > qvo the equilibrium melt temperature rise will be negative when q,. < qvo the equilibrium melt temperature rise will be positive. Equation 7.413 provides a simple and convenient expression from which the effect of the various factors influencing the melt temperature becomes clear. These effects can be easily quantified as discussed next. 

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]. 

An experimental phase diagram of polyethylene dissolved in 1,2,4,5-tetrachlorobenzene (TCB) is shown in Figure 16b (132). Both polymer and low molar mass solvent have similar equilibrium melting temperatures. On the left-hand side of the phase diagram, the liquidus line follows equation 47, the right-hand side does not follow equation 46. Again, this indicates the usual nonequilibrium state of polymer crystals. Besides too low melting temperatures, low... 

There are several definitions of the equilibrium melting temperature currently in use. According to equation (6.42) the highest melting temperature (and presumably the equilibrium melting temperature) is reached at infinite molecular weight. Another definition assumes infinitely thick crystalline lamellae (153) see Section 6.7.4. 

Equation (2.39) represents the relation between the melting temperature and crystallite thickness f for different chain lengths. The crystallite thickness is not constrained to its equilibrium value and cTec is characteristic of the particular interface that is developed in the crystallite under the specific set of crystallization conditions. The melting temperature depression, Eq. (2.39), is calculated from the equilibrium melting temperature of the infinite chain, T. For high molecular weights Eq. (2.39) reduces to... 

Here and AHf are the equilibrium melting temperature, and enthalpy of fusion between the crystalline polymer and its anisotropic melt. Equation (3.43) is analogous to Eq. (3.44) for an isotropic melt. 

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