Order parameter mean field functional

This is called the saddle point approximation, and the resulting theory is equivalent to the mean-field theory. In practice, the coarse grained states are defined in terms of an order parameter that by definition has a value 0 in the disordered phase and a nonzero value in the ordered state. In the case of AB diblock copolymers, this order parameter is defined as the deviation of the A monomer fraction from the average value. Since this order parameter is a function of position, it actually represents infinitely many variables. By definition, in the weak segregation regime, the order parameter is small and it is meaningful to expand Fi in a Taylor series. 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

It must be kept in mind, that S represents just a first-order approximation of the distribution function, and this under the additional premise of complete cylindrical symmetry only. It might be an acceptable measure when comparing cases for which a mean-field model applies. However, comparing the order parameters of liquid crystals with those of other partially ordered phases, such as stretched polymers or tribological samples can be misleading due to possibly different types of distribution functions. 

Fig. 10.47. Temporal evolution of mean grain size in phase field model of grain growth (adapted from Chen and Yang (1994)). Plots are of logarithm of average grain area as a function of time, with the two curves corresponding to four (crosses) and thirty-six (circles) different order parameter fields.

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

Fig. 10. Mean hyperfine field as a function of order parameter squared for the FeCo alloy. The solid line is the theoretical expectation. Solid squares, Fr ddowiak (1985) open circles, Eymeiy et al. (1978).

Figure A3.3.5 Thermodynamic force as a function of the order parameter. Three equilibrium isotherms (full curves) are shown according to a mean field description. For T < T, the isotherm has a van der Waals loop, from which the use of the Maxwell equal area construction leads to the horizontal dashed line for the equilibrium isotherm. 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

By virtue of their simple structure, some properties of continuum models can be solved analytically in a mean field approximation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is investigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-paiameter model are described in the review by Gompper and Schick [76]. A very interesting feature of these models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a functional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

Figure 32. Herringbone order parameter for the anisotropic-planar-rotor model (2.5) as a function of the reduced temperature T = TIK. Circles Monte Carlo results [244]. Dotted line mean-field approximation [62, 141]. Solid line triangular cluster-variational method [62]. Arrow first-order transition temperature obtained from a real-space renormalization group treatment of a planar quadrupolar six-state model [345]. (Adapted from Fig. 2 of Ref. 345.)...

Therefore, a diffuse ring of scattering centered at the origin of reciprocal space is attached to each particle. Thus, the nematic order parameter which characterizes the distribution function of a single particle is derived from the interferences among a cluster of particles. This assumption is somewhat similar to a mean-field treatment and tends to overestimate S. 

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