Structural order parameters

Considering Fig. 17.4, the development of the B2 structure creates two sublattices from the original A2 structure. One of the B2 sublattices consists of the b.c.c. unit-cell centers (indicated by /3 in Fig. 17.4) are displaced from the b.c.c. corners (a in Fig. 17.4) by a/2(lll). An ordering transformation produces sublattices, a and /3, with differing site fractions, xB and Xg. Their difference becomes a structural order parameter ... 

These probabilities must all lie between zero and one this sets bounds on physical values for the structural order parameter... 

For the disordered (A2) phases, rjeq = 0, Eq. 17.19 is satisfied automatically, and equilibrium tie-lines are present if W < 0 and T < Tcrit = W/k, as illustrated in Fig. 17.5. In the nearest-neighbor model, ordered B2 solutions can appear at any uniform composition Xb Nonzero equilibrium structural order parameters appear only if W > 0 at temperatures T satisfying... 

The development of the miscibility gap for W < 0 and the antiphases ( Tjeq) for W > 0 have entirely different kinetic implications. For decomposition, mass flux is necessary for the evolution of two phases with differing compositions. Furthermore, interfaces between these two phases necessarily develop. The evolution of ordered phases from disordered phases (i.e., the onset of nonzero structural order parameters) can occur with no mass flux macroscopic diffusion is not necessary. Because the 77+q-phase is thermodynamically equivalent to the 7/iq-phase, the development of 77+q-phase in one material location is simultaneous with the evolution of r lq-phase at another location. The impingement of these two phases creates an antiphase domain boundary. These interfaces are regions of local heterogeneity and increase the free energy above the homogeneous value given by Eq. 17.14. The kinetic implications of macroscopic diffusion and of the development of interfaces are treated in Chapter 18. 

To describe quantitatively the disorder present in a material, it is often convenient to introduce a structural order parameter. This term refers to a metric that can detect the development of order in a many-body system, perhaps by employing the tools of pattern recognition (Brostow et al., 1998). In many cases, such a measure is constructed to serve as a reaction coordinate for a thermodynamic phase transition (van Duijneveldt and Frenkel, 1992). However, since the form of the order parameter clearly depends on the phenomenon of interest, the development of such measures can be a difficult and subtle matter. 

Given that the supercooling of a liquid can lead to structurally distinct possibilities (the stable crystal or a glass), structural order parameters are especially valuable in understanding low-temperature metastabiUty. In particular, it has been demonstrated (van Duijneveldt and Frenkel, 1992) that the bond-orientational order parameters introduced by Steinhardt et al (1983) are well suited for detecting crystalline order in computer simulations of simple supercooled liquids. The bond-orientational order parameters are so named because they focus on the spatial orientation of imaginary bonds" that connect molecules to their nearest neighbors defined as above with... 

In contrast to the bond-orientational order parameters mentioned above, scalar measures for translational order [that is, of the tendency of particles (atoms, molecules) to adopt preferential pair distances in space] have not been well studied. However, a number of simple metrics have been introduced recently (Truskett et al., 2000 Torquato et al., 2000, Errington and Debenedetti, 2001) to capture the degree of spatial ordering in a many-body system. In particular, the structural order parameter t. 

The structural order parameter r should not be confused with the relaxation time r defined in Eq. (2). 

Fig. 9. Two-parameter ordering phase diagram for a system of 500 identical hard spheres (Truskett et ai, 2000 Torquato et ai, 2000). Shown are the coordinates in structural order parameter space (r, ) for the equilibrium fluid (dot-dashed), the equilibrium FCC crystal (dashed), and a set of glasses (circles) produced with varying compression rates. Here, r is the translational order parameter from (26) and is the bond-orientational order parameter Q( from (25) normalized by its value in the perfect FCC crystal ( = Each circle...

More recently, Lacks and Weinhoff [265] have examined the mechanical stability of soft spheres as a function of size polydispersity. In their study they minimize the potential energy of an fee array of polydisperse spheres, and examine how the structure of the minimum-energy configuration changes with increasing polydispersity. An appropriate structural order parameter for the fee solid is found to drop precipitously for polydispersities in the range of 10% to 15%. The absence of energy minima in the ordered phase indicates that such a system is not mechanically stable, and therefore cannot be thermodynamically stable. 

The supercooled liquid catastrophe, if it exists, would necessarily be associated with diverging fluctuations in the structural order parameter F. This stems from the fact that the Y surface develops a vanishing curvature in the F direction as this endpoint is approached. Because the bicyclic octamer elements are bulky, fluctuations in their coiKentration amount to density fluctuations. Diverging density fluctuations then imply diverging isothermal compressibility. Furthermore the infinite slope of the metastable liquid locus at its endpoint implies the divergence of thermal expansion. Potential energy fluctuations remain essentially normal, so constant-volume heat capacity remains small. But the volumetric divergence creates an unbounded constant-pressure heat capacity. 

Andrich, M.P., and Vanderkooi, J.M., Temperature dependence of 1,6-diphenyl-1,3,5-hexa-triene fluorescence in phospholipid artificial membranes, Biochemistry, 15, 1257, 1976. Blitterswijk, W.J.V., Hoeven, R.P.V., and Dermeer, B.W.V., Lipid structural order parameters (reciprocal of fluidity) in biomembranes derived from steady-state fluorescence polarization measurements, Biochem. Biophys. Acta, 644, 323, 1981. 

In simulation, we can also measure the structural order parameters that provides local structural arrangements. The orientational order among the bonds present in the system is characterized by local orientational order Qi which is given by... 

In this work we have simulated a system consisting of rigid dimers in which each monomer interacts with monomers from the other dimers through a core-softened shoulder potential. In order to check how the anisotropy induced by the dimeric structure affects the presence of density, diffusion and structural anomalies, we have obtained the pressure temperature phase diagram, the diffusion constant and the structural order parameter of the system for different A, distance between the bonded particles in each dimer, values. 

The main stmcture parameter of cluster model-nanoclusters relative fraction (p, which is polymers structure order parameter in strict physical sense of this tern, can be calculated according to the Eq. (1.11). In its turn, the polymer structure fractal dimension value is determined according to the Eqs. (1.9) and (2.20). 

A model independent form for the structural order parameter S has been derived [17] that has the form... 

Figure 15 Dependence of the structural order parameter of DPH in microsomal membranes derived from chiUing-resistant tomato fruit (a) and chilling-sensitive tomato fruit (b). The phase transition temperature of these biological membranes is an indication of the susceptibility of the fruit to cold temperature damage. (Data reanalyzed from Ref. 22.)...

---