First-order Transition Analysis

A simple analysis of an irreversible first-order transition is the cold crystallization, defined in Sect 3.5.5. For polymers, crystallization on heating from the glassy state may be so far from equilibrium that the temperature modulation will have little effect on its rate, as seen in Fig. 4.122. The modeling of the measurement of heat capacity in the presence of large, irreversible heat flows in Fig. 4.102, and irreversible melting in Figs. 3.89 and 4.123, document this capability of TMDSC to separate irreversible and reversible effects. Little needs to be added to this important application. 

A similar analysis of a melt-crystallized poly(ethylene terephthalate), PET, of the typical molecular mass of a polyester showed a surprising reversing melting peak, as seen in Fig. 3.92. On comparison with an amorphous PET, one finds that the reversing peak depends on crystallization history, as is shown in Fig. 4.136. The change of the glass transition with crystallization is typical for polymers. It shows a 

Quasi-isothermal cure of an epoxy-amine resin at 343 K, moduiation 0.5 K, p = 60s 

In summary, the applications of TMDSC look quite different from those of DSC. Many important apphcations can only be solved by TMDSC. A good number of others are, however, still better, and sometimes only, solvable by DSC. Fortunately any TMDSC can also be run without modulation under DSC conditions. The development of TMDSC has also led to long-awaited progress in the hardware and software improvements of DSC as illustrated with Fig. 4.54 and Appendix 11. 

The variables of state for thermomechanical analysis are deformation (strain) and stress. The SI units of deformation are based on length (meter, m), volume, (cubic meter, m ) and angle (radian, rad, or degree) as listed in Fig. 4.143 (see also Fig. 2.3). Stress is defined as force per unit area with the SI unit newton m , also called by its own name, pascal. Pa. Since these units are not quite as frequently used, some conversion factors are listed below. The stress is always defined as force per area. 

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Phase transitions, whether first-order or second-order, are potent sources of instability of solid drugs and can usually be detected and studied by thermal methods of analysis (e.g., DSC, TGA, TMA, ODSC, DMA, DEA). In crystalline solids, typical first-order transitions are polymorphic or desolvation transitions. In amorphous solids, second-order transitions, such as glass transitions, are common. 

If the ranges of homogeneity of the phases taking part in the transformation are wider than those of line compounds, the kinetic coefficients in Eqns. (12.22) and (12.23), that is v jf, yb, and A b, are certainly not composition independent. It may then be questionable if transport across the boundary (Eqn. (12.22)) and the simultaneous structure change (Eqn. (12.23)) are independent processes as was tacitly assumed by formulating the kinetic relations in Eqns. (12.22) and (12.23). Let us emphasize that the foregoing analysis is meant to clarify the physico-chemical conceptual frame in which first-order transitions which include matter transport should be discussed. Pertinent experiments are still rare. 

The Statistical Rate Theory (SRT) is based on considering the quantum-mechanical transition probability in an isolated many particle system. Assuming that the transport of molecules between the phases at the thermal equilibrium results primarily from single molecular events, the expression for the rate of molecular transport between the two phases 1 and 2 , R 2, was developed by using the first-order perturbation analysis of the Schrodinger equation and the Boltzmann definition of entropy. 

The thermal variation of specific heat measured by adiabatic calorimetry is showninfig 1 At about 347 K a jump of occurs which corresponds to a first order transition. This is confirmed by 0. S. C. measurement which gives an enthalpy of transition AH = 1160+ 100 cal/mole. This transition which presents an endothermic signal is however irreversible, at least down to 4 K, To obtain the initial phase it is necsssary to crystallize again the trensformed product in eceto-nitrile. However the irreversibility is not due to solvent inclusion as evidenced by mass spectroscopy analysis,... 

Adsorption of nonionic surfactants on porous solids has been studied by Huinink et al. in a series of p ers [ 149,150]. They elaborated a thermodynamic approach that accounts for the major features of experimental adsorption isotherms. It is a very well known fact that during the adsorption of nonionic surfactants there is a sharp step in the isotherm. This step is interpreted as a change from monomer adsorption to a regime where micelle adsorption takes place. Different surfactants produce the step in a different concentration range. The step is more or less vertical depending on the adsorbate. The thermodynamic analysis made by Huinink et al. is based on the assumption that the step could be treated as a pseudo first order transition. Their final equation is a Kelvin-like one, which shows that the change in chemical potential of the phase transition is proportional to the curvature constant (Helmholtz curvature energy of the surface). 

The introduction of Xe in a layer of Kr stabilizes the V3xV3 commensurate structure [84]. The analysis of the corresponding binary phase diagram shows a first order transition between the commensurate and incommensurate sohds [85,86]. Mixtures of Ar and Xe exhibit a large tendency to form commensurate alloys [87]. A first order commensurate-incommensurate phase transition is also observed in that case. 

Differential scanning calorimetry is not an absolute measuring technique, calibrations are thus of prime importance. Calibrations are necessary for the measurement of temperature, T (in K) amplitude, expressed as temperature difference, AT (in K) or as heat-flow rate, dQ/dt (in J s or W) peak area AH (in J) and time, t (in s or min). Figure 4.62 shows the analysis of a typical first-order transition, a melting transition. 

Polyethylene data are shown in Fig. 2.23. At the equilibrium melting temperature of 416.4 K, the heat of fusion and entropy of fusion are indicated as a step increase. The free enthalpy shows only a change in slopes, characteristic of a first-order transition. Actual measurements are available to 600 K. The further data are extrapolated. This summary allows a close connection between quantitative DSC measurement and the derivation of thermodynamic data for the limiting phases, as well as a connection to the molecular motion. In Chaps. 5 to 7 it will be shown that this information is basic to undertake the final quantitative step, the analysis of nonequilibrium states as are common in polymeric systems. 

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