Order parameter discontinuity

The S vs. T relationship (Wang Warner, 1986) is similar to the well-known Maier-Sauipe shape, the temperature scale being T = kBT/ /ue instead of T = kBT/p. The order parameter decreases as T increases until the critical temperature T c = 0.388. The order parameter discontinu-ally jumps from 0.356 to zero. Because T is the reduced temperature the transition temperature T c is a function of the geometric mean of v and e. The greater v, the higher T. The more g, the more stable the nematic phase. These conclusions are consistent with experiments. It was found that the critical order parameter for the semi-flexible liquid crytalline polymers ranges from 0.3 to 0.45. 

Fig. 6.22 Mean field model of the nematic phase temperature dependence of the order parameter. The two branches correspond to stable nematic phase with positive order parameter (solid line), and unstable phase with negative order parameter (dash line). Order parameter discontinuity at 5 = 0.429 indicates the first order N-I transition...

This means that at T( the order parameter discontinuously jumps from... 

A-nematic phase transition (SNT) is of second order, while the NIT is of first order. In Figure 2.16b, the smectic order parameter discontinuously drops to zero with increasing <)> and the SNT is of first order. In Figure 2.16c, the nematic phase disappears and we only have the first-order smectic A-isotropic phase transition (SIT). Owing to McMillan theory for a pure nematogen (<)> = 0), the SNT should be second order for Tsf /T j < 0.87 and first order for larger values of 7 /T j, where shows the SNT temperature of a pure nematogen. 

X-ray, uv, optical, in, and magnetic resonance techniques are used to measure the order parameter in Hquid crystals. Values of S for a typical Hquid crystal are shown in Figure 3. The compound, -methoxyben2yHdene-/) - -butylaniHne (MBBA) is mesomorphic around room temperature. The order parameter ranges from 0.7 to 0.3 and discontinuously falls to 2ero at T, which is sometimes called the clearing temperature (1). 

There are transition temperatures in some Hquid crystals where the positional order disappears but the orientational order remains (with increasing temperature). The positional order parameter becomes zero at this temperature, but unlike i, this can either be a discontinuous drop to zero at this temperature or a continuous decrease of the order parameter which reaches zero at this temperature. 

Although transition across a critical point may proceed without any first-order discontinuity, the fact that there is a change of symmetry implies that the two phases must be described by different functions of the thermodynamic variables, which cannot be continued analytically across the critical point. The order parameter serves to compensate for the reduction in symmetry. Although it is a regular function of temperature it can develop a discontinuous derivative at the critical temperature. Likewise, several measurable... 

It should be noted that the theory described above is strictly vahd only close to Tc for an ideal crystal of infinite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Flere the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc. The reason for this might also be technical It is extremely difficult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat flow that can only occur when the temperature gradient is different from zero. 

The thermodynamics of phase equilibria is reviewed in Chapter 17 and the fundamental thermodynamic differences between conserved and nonconserved order parameters are reinforced with a geometrical construction. These order parameters are used in the kinetic analyses of continuous and discontinuous phase transformations. 

The order of a transition can be illustrated for a fixed-stoichiometry system with the familiar P-T diagram for solid, liquid, and vapor phases in Fig. 17.2. The curves in Fig. 17.2 are sets of P and T at which the molar volume, V, has two distinct equilibrium values—the discontinuous change in molar volume as the system s equilibrium environment crosses a curve indicates that the phase transition is first order. Critical points where the change in the order parameter goes to zero (e.g., at the end of the vapor-liquid coexistence curve) are second-order transitions. 

We all know that when a liquid transforms to a crystal, there is a change in order the crystal has greater order than the liquid. The symmetry also changes in such a transition the liquid has more symmetry than a crystal since the liquid remains invariant under all rotations and translations. Landau introduced the concept of an order parameter, , which is a measure of the order resulting from a phase transition. In a first-order transition (e.g., liquid-crystal), the change in is discontinuous, but in a second-order transition where the change of state is continuous, the change in is also continuous. Landau proposed that G in a second-order (or structured) phase transition is not only a function of P and T but also of and expanded G as... 

NMR measurements in these systems have allowed for the following. A determination of the local-polarization distribution function W(p) and the Edwards-Anderson order parameter qEA in the weak substitutional disorder limit (x=0.5) ongoing through Tc. The results showed that W(p) is asymmetric below Tc and symmetric above Tc and that qEA makes a discontinuous jump on going through Tc in view of first-order nature of this transition. 

Fig. 7.5.3. Variation of the Landau free energy density with the order parameter at temperatures above, at, and below the critical temperature and for changes in direction of the magnetic fields. The heavy dots indicate the value of i) for which the Landau free enei density is minimized. The central row represents a discontinuous first order transition as the temperature is dropped past the its critical value. After Goldenfeld loc. cit.

Figure 9. Strain variations in quartz close to the p o a phase transition. Sohd lines are the Landau solution for a first order phase transition with a small discontinuity in the order parameter at the transition temperature. Solid bars represent the discontinuities expected on the basis of linear expansion data of Bachheimer (1980,...

We now consider the interface between a vacuum and a system that undergoes a first-order (i.e., discontinuous) order-disorder transition in the bulk at a temperature Tc. Due to missing neighbors at a surface, we expect that the order parameter at temperatures T < Tc is slightly reduced in comparison with its bulk value (fig. 67). If this situation persists up to T > T, such that both the bulk order parameter < >(z oo) and the surface order parameter 4> = z = 0) vanish discontinuous the surface stays ordered up to Tc, a situation that is not of very general interest. However, it may happen (Lipowsky, 1982, 1983, 1984, 1987 Lipowsky and Speth, 1983) that the surface region disorders somewhat already at T < Tc, and this disordered layer grows as T T and leads to a continuous... 

Apart from temperature, hydrostatic pressure is the other intensive thermodynamical parameter that can be modified with high-pressure cells to build (T, P) phase diagrams. By changing the relative distances between the atoms and molecules, the strength of the interactions are modified, thereby modifying the transition temperature or even inducing new phases. The change of the transition temperature as a function of pressure depends whether the transition is continuous (second order) or discontinuous (first order). The Clausius-Clapeyron (dTc/dP) and Ehrenfest (dTc/dP)2 relationships apply to first- and second-order phase transitions, respectively,... 

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