Transition discontinuous shape

Fig. 32 Effect of initial position on the discontinuous shape transition, ho = 163 mm. Values of z0 are indicated in the figure...

Fig. 17 shows the adsorption isotherms of all (undimerized and dimerized) particles. Except for a very fast increase of adsorption connected with filling of the first adlayer, the adsorption isotherm for the system A3 is quite smooth. The step at p/k T 0.28 corresponds to building up of the multilayer structure. The most significant change in the shape of the adsorption isotherm for the system 10, in comparison with the system A3, is the presence of a jump discontinuity at p/k T = 0.0099. Inspection of the density profiles attributes this jump to the prewetting transition in the... 

Using the self-consistently obtained solutions of Eq. 19, the calculated chemical shift (Jiso = (To + a(T) is calculated and compared to the experimental data in Fig. 4. Even though the experimentally observed transition is broader than the calculated one, the agreement between theory and experiment is good. As the discontinuity in the lattice-related mode is small at Tc, where Tc corresponds to a = 0, the chemical shift does not show a discontinuity at Tc within numerical accuracy. It is important to note here that the S-shape in the cf T) data is a direct consequence of using the renormalized frequencies as defined in Eq. 19. 

A second-order phase transition is one in which the enthalpy and first derivatives are continuous, but the second derivatives are discontinuous. The Cp versus T curve is often shaped like the Greek letter X. Hence, these transitions are also called -transitions (Figure 2-15b Thompson and Perkins, 1981). The structure change is minor in second-order phase transitions, such as the rotation of bonds and order-disorder of some ions. Examples include melt to glass transition, X-transition in fayalite, and magnetic transitions. Second-order phase transitions often do not require nucleation and are rapid. On some characteristics, these transitions may be viewed as a homogeneous reaction or many simultaneous homogeneous reactions. 

Because these transitions are associated with a mechanism in which one phase gradually evolves into the other, they are also often referred to as continuous or cooperative transitions. The terms second order , lambda , and continuous transitions have often been used interchangeably to refer to the same transition even though a true Ehrenfest second-order heat capacity does not have a lambda shape. We shall use the designation continuous transition (in preference to second order or lambda) for all transitions in which the discontinuity occurs in the second derivative of G. 

Most of the physical properties of the polymer (heat capacity, expansion coefficient, storage modulus, gas permeability, refractive index, etc.) undergo a discontinuous variation at the glass transition. The most frequently used methods to determine Tg are differential scanning calorimetry (DSC), thermomechanical analysis (TMA), and dynamic mechanical thermal analysis (DMTA). But several other techniques may be also employed, such as the measurement of the complex dielectric permittivity as a function of temperature. The shape of variation of corresponding properties is shown in Fig. 4.1. 

As v varies in the opposite way as G, it increases with T (almost discontinuously at secondary transitions), and reaches a value of the order of 0.45 0.01 at Tg — 20 K. Then, it undergoes a rapid increase and attains a value very close to 0.50 in the rubbery region. The shape of temperature variations of v is represented in Fig. 11.6. 

A pronounced hysteresis of ATC = 9.5 K has been observed (Tc = 181.86 K for rising and Tc = 172.33 K for falling temperature). The authors attempted to describe the results with the thermodynamic model for spin transitions of Slichter and Drickamer 91 Different diffraction patterns were recorded above and below Tc, where only one spin isomer was present (see Fig. 39). This provides evidence that a crystallographic phase change accompanies the abrupt spin phase transition in this compound. The temperature dependence of the peak profiles was found to follow that of the HS fraction as derived from the Mossbauer spectra, which shows that the crystallographic phase change is directly associated with the interconversion of the two spin phases. The Debye-Waller factors were evaluated for the two spin phases they showed at Tc a discontinuity of Af 35% on going from the HS to the LS phase. The shape of the transition curve near Tc and Tc itself, were somewhat different in three independently prepared samples. 

Ross and Boyd (II) prepared crystals of sodium chloride in which both the 100 and the HI surfaces were developed the adsorption isotherm of ethane at 90.1° K gave evidence of two type 2—i.e., near-homotattic—surfaces, as well as showing the initial knee, which is evidence of a type 1 surface. This isotherm is shown in Figure 4, which also includes for comparison an ethane isotherm measured at the same temperature for a sodium chloride adsorbent that had only 100 faces developed the pressure characteristic of the phase transition of ethane on the homotattic 190 surface of sodium chloride at 90.1° K shows on both isotherms as the location of a discontinuity at p = 4.5 X 10-3 mm. the homotattic ill portion of the surface is responsible for the convex shape (with respect to the pressure axis) of the isotherm beyond the knee, though the rise is not suffi-... 

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