Subject order parameter

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Figure 2 Time sequence of th< spin configuration on a (100) plane at 50% when the system at T=2.5 (snapshot a) is quenched down to T—1.7 and is subject to an isothermal aging. Snapshots demonstrated in figs, b, c and d correspond to time t=20,000, 43,000 and 50,000. The long range and short range order parameters input from the PPM calculations and resultant ones in the simulated lattice are also demonstrated [22, 24, 28]. ...

Fluorescence polarization is the subject of Chapter 5. Factors affecting the polarization of fluorescence are described and it is shown how the measurement of emission anisotropy can provide information on fluidity and order parameters. 

Naturally, the fixed composition phase transformations treated in this section can be accompanied by local fluctuations in the composition field. Because of the similarity of Fig. 17.3 to a binary eutectic phase diagram, it is apparent that composition plays a similar role to other order parameters, such as molar volume. Before treating the composition order parameter explicitly for a binary alloy, a preliminary distinction between types of order parameters can be obtained. Order parameters such as composition and molar volume are derived from extensive variables any kinetic equations that apply for them must account for any conservation principles that apply to the extensive variable. Order parameters such as the atomic displacement 77 in a piezoelectric transition, or spin in a magnetic transition, are not subject to any conservation principles. Fundamental differences between conserved and nonconserved order parameters are treated in Sections 17.2 and 18.3. 

The equilibrium order parameters X g and rjeq minimize AF subject to any system constraints. Supposing that the system s composition is fixed, the method of Lagrange multipliers leads to a common-tangent construction for AF with respect to XB—or equivalently, equality of chemical potentials of both A and B. Two compositions, Xjj and X +, will coexist at equilibrium for average compositions XB in the composition range Xe < XB < -X Bq+ if they satisfy... 

We derive here the governing equations necessary to describe the structure factor, S (q, t), for a complex liquid mixture subject to flow. Since this observable is the Fourier transformation of the spatial correlation of concentration functions, it is first required to develop an equation of motion for 5c (r, t). The approach described here employs a modified Cahn-Hilliard equation and is described in greater detail in the book by Goldenfeld [91]. To describe the physical system, an order parameter, q/ (r, ), is introduced. In a complex mixture, this parameter would simply be /(r, 0 = c(r, r)-(c), where c(r, t) is the local concentration of one of the constituents and mean concentration. The order parameter has the property of being zero in a disordered, or on phase region, and non-zero in the ordered or two-phase region. The observed structure factor, which is the object of this calculation, is simply... 

Such order can be described in terms of the preferential alignment of the director, a unit vector that describes the orientation of molecules in a nematic phase. Because the molecules are still subject to random fluctuations, only an average orientation can be described, usually by an ordering matrix S, which can be expressed in terms of any Cartesian coordinate system fixed in the molecule. S is symmetric and traceless and hence has five independent elements, but a suitable choice of the molecular axes may reduce the number. In principle, it is always possible to diagonalize S, and in such a principal axis coordinate system there are only two nonzero elements (as there would be, for example, in a quadrupole coupling tensor). In the absence of symmetry in the molecule, there is no way of specifying the orientation of the principal axes of S, but considerable simplification is obtained for symmetric molecules. If a molecule has a threefold or higher axis of symmetry, its selection as one of the axes of the Cartesian coordinate system leaves only one independent order parameter, with the now familiar form ... 

In the literature on the subject it is conventional to introduce a quantity termed the degree of order, or order parameter. sp, defined by... 

The problem of eliminating fast variables plays a decisive role in the subject of synergetics. The concept of order parameters and slowing in nonequilibrium systems has been proved by Haken to be a powerful tool for analyzing instabilities far from thermal equilibrium. Furthermore, Haken used this concept to elucidate important analogies among many-component systems from completely difierent disciplines. For these reasons the Haken school has also been motivated to devdop a systematic procedure of adiabatic elimination. ... 

Micelles of low molecular weight surfactants are known to be very dynamic structures, although the various fragments of the molecules within a micelle are subject to some restrictions of mobility in comparison to molecular solution [23, 379-381]. As the hydrophilic head groups are anchored at the micellar surface , NMR-studies show that the mobility and the order parameter decreases along the alkyl tail from its end towards the head group [379-385]. 

Assume now that the interfaces are positioned at x = h/2 and x = — h/2 and that x](h/2) = —r ( — /i/2) = x 0. The minimization of free energy density given by equation 20 subject to these boundary conditions results in the following form for the order parameter ... 

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