Order parameter curves

The orientational transition [211, 212] occurs in the classical model near 38 K as judged from the inflection point of the order parameter curve. Quantum fluctuations effectively soften the potential and decrease the transition temperature by 10%. In addition, the transition is broadened as can be judged from the heat capacity anomaly, which is found to decay to the required ground-state value zero at low temperatures. The ground-state saturation value of the order parameter is not unity as in the classical case, but... 

Lowering the temperature increases Szz for all systems to the same limiting value of Szz 0.65 in the nematic phase [35, 99, 100]. Note, that the order parameter curve of polymer 1 exhibits a horizontal slope at the glass transition. A further jump of Szz. observed for the polymers 2, indicates an additional smectic A phase with a limiting value of Szz 0.9. No such discontinuity is detected for the monomeric liquid crystal 5, exhibiting a smectic A phase, likewise. Comparison of the order parameters of Fig. 17 with those of NMR [108-110] and birefringence studies... 

Figure 4. Field-induced changes in order parameters for a smectic A liquid crystal. Q, orientational order parameter curves ag, bg, and cq show the temperature dependence of Q at various fields. R, order parameter describing the coupling of translational and orientational order, curves Ug, b, and show the temperature dependence of R. The field increases from curves a to curves c and results in a change in the N-C transition from first to second order [24],...

Figure A2.5.16. The coexistence curve, = KI(2R) versus mole fraction v for a simple mixture. Also shown as an abscissa is the order parameter s, which makes the diagram equally applicable to order-disorder phenomena in solids and to ferromagnetism. The dotted curve is the spinodal.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

The central quantity is the order parameter as a function of temperature (see Fig. 13). The phase transition temperature Tq of the classical system can be located around 38 K. At high temperatures, the quantum curve of the order parameter merges with the classical curve, whereas it starts to deviate below Tq. Qualitatively, quantum fluctuations lower the ordering and thus the quantum order parameter is always smaller than its classical counterpart. The inclusion of quantum effects results in a nearly 10% lowering of Tq (see Fig. 13). 

Furthermore, one can infer quantitatively from the data in Fig. 13 that the quantum system cannot reach the maximum herringbone ordering even at extremely low temperatures the quantum hbrations depress the saturation value by 10%. In Fig. 13, the order parameter and total energy as obtained from the full quantum simulation are compared with standard approximate theories valid for low and high temperatures. One can clearly see how the quasi classical Feynman-Hibbs curve matches the exact quantum data above 30 K. However, just below the phase transition, this second-order approximation in the quantum fluctuations fails and yields uncontrolled estimates just below the point of failure it gives classical values for the order parameter and the herringbone ordering even vanishes below... 

Additional isothermal treatments at neighbouring temperatures small step annealing) yield plateau values of resistivity corresponding to equilibrium values at certain temperatures which reflect the order parameter in thermal equilibrium as a function of temperature ( equilibrium curve , curve 4 in Figure 1). This study can be used for an analysis of the kinetics of order-order relaxations (see Figure 3 below). 

A is a constant and p is the critical exponent which adopts values from 0.3 to 0.5. Values around p = 0.5 are observed for long-range interactions between the particles for short-range interactions (e.g. magnetic interactions) the critical exponent is closer to p 0.33. As shown in the typical curve diagram in Fig. 4.2, the order parameter experiences its most relevant changes close to the critical temperature the curve runs vertical at Tc. 

Alternatively, proton double quantum (DQ) NMR, based on a combined DQ excitation and a reconversion block of the pulse sequence, has been utilized to gain direct access to residual DCCs for cross-linked systems.69,83-89 For this purpose, double-quantum buildup curves are obtained with use of a well-defined double-quantum Hamiltonian along with a specific normalization approach. Residual interactions are directly proportional to a dynamic order parameter Sb of the polymer backbone,87... 

Attempts to interpret the corresponding build-up curves according to the Lipari-Szabo approach lead to inconsistent results (for instance, order parameters greater than unity). This indicates that these remote correlations are probably not of infra-molecular origin but would rather arise from znfer-molecular dipolar interactions which could become significant when some contacts exist between neighbouring aliphatic chains. This... 

The mechanism, albeit somewhat more complicated than in the preceding case, also involves competition between a first- and second-order follow-up reaction. For this reason, a similar analysis applies and the yields vs. competition parameter curves can be derived from those pertaining to the preceding case. 

Figure 5.55 Mutual dependence of Q i and Q d order parameters. In the upper part of the figure is outlined the T dependence of substitutional disorder Qod for different values of Qdi and, in the lower part, the T dependence of the displacive disorder parameter Qdt for different values of The heavy lines on the surface of local curves represent the solution for thermal equilibrium. From E. Salje and B. Kuscholke, Thermodynamics of sodium feldspar II experimental results and numerical calculations. Physics and Chemistry of Minerals, 12, 99-107, figures 5-8, copyright 1985 by Springer Verlag. Reprinted with the permission of Springer-Verlag GmbH Co. KG.

Figure 4.25 Comparison of the different order parameters for the 151 K second-order transition of PrAlOj. (After Sturge et al., 1975.) Unbroken line is the smooth curve through the internal displacement order parameter, cos 2(, from ESR measurements. Black circles represent the electronic order parameter from optical absorption studies. Squares represent the reduced macroscopic strain from elastic neutron scattering.

Fig. 12a-c. Polymer concentration dependence of the orientational order parameters S for three liquid-crystalline polymer systems a PBLG-DMF [92,93] b PHIC-toluene [94] c PYPt-TCE [33], Marks experimental data solid curves, theoretical values calculated from the scaled particle theory. The left end of each curve gives the phase boundary concentration cA... 

When each peak in the reference spectrum has been matched with a corresponding peak in the spectrum acquired, the mass difference is calculated for each pair of peaks (see Section 3.1.2). These mass differences are plotted as points on a graph each data point has the mass of the acquired peak as its x coordinate, and the mass difference above as its coordinate, and a smooth curve is drawn through the points (Figure 13.10) [5]. The polynomial order parameter controls the type of curve that is drawn and can be set to any value between 1 and 5 ... 

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

Calcd. quantities Distribution curves P(9 of the angle 9 defined by the two terminal bond vectors Order parameter S... 

Figure 12-6. Potential valleys for (C-C°) as a function of the order parameter tj. Curve parameter is the...

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. 

The composition or the number fraction of component B, Xb, is an example of an order parameter that is conserved in a closed system. Figure 17.6 shows a molar free energy versus composition curve for a binary solution. The molar free energy for a solution at any composition Xb can be written in terms of its partial molar quantities, Fa(Xb) and Fb(Xb) 4... 

At low temperatures the bend in the V(T) curves indicates a glass transition. It is of interest, whether this transition can be described in analogy to amorphous polymers on the basis of the concept of using one internal order parameter and whether the liquid crystalline structure is maintained in the glassy state. This will be discussed in Chap. 2.3.4. 

In Fig. 14 the calculated order parameters according to Eq. 6 are plotted vs. temperature for the three systems M2, M3, C4 (refer to Table 4). For the 1-l.c. the well known curve is observed. At the reduced temperature of T = 0.95 we find S = 0.62. Comparing this order parameter of the dye with the order parameter of the... 

---