The order parameter

Order leads to all virtues But what leads to order  

It is convenient to begin with a simple one-component system having three phases - gas, liquid and solid. At the primitive level these phases differ by a density n, i.e., the number of particles per unit volume. If we fix now for simplicity the pressure in this system, in the thermodynamical equilibrium. 

The critical point is one of many examples of higher-order phase transitions including the second-order transitions in ferromagnetics and ferroelectrics and A-transition in liquid He. Unlike the first-order transition, the heat of the 

Let us consider now the gas-liquid system near the critical point (Fig. 1.3). At r Tc both phases coexist, their densities nt (liquid) and nc (gas) could be formally written as ni = ric + 5n/2, nc = nc — 6n/2, where ric is density at the critical point. Note that in the physics of critical phenomena the order parameter is often defined subtracting the background value of nc, i.e., as the order parameter the difference of densities, 8n = n — nQ, could be used rather than these individual densities themselves. Such an order parameter is zero at T Tc and becomes nonzero at T Tc. Another distinctive feature of the order parameter is that for all simple systems the algebraic law 5n oc (Tc — T) holds, where (3 is constant. 

It is useful to check whether this kind of relations is valid for other systems like ferromagnetics and ferroelectrics too. Here the order parameters are the magnetization M and the polarization P, respectively. At high temperatures (T Tc), and zero external field these values are M = 0 (paramagnetic phase) and P = 0 (paraelectric phase) respectively. At lower temperatures close to the phase transition point, however, spontaneous magnetization and polarization arise following both the algebraic law M, P oc (Tc - Tf. 

Unlike a solid crystal, in which all molecules in the lattice are consistently oriented to form the crystal structure, in the liquid crystal phase there is a significant amount of freedom per molecule. The phases are fluid-like in that molecules are not confined to lattice positions and may diffuse throughout the bulk. They are, however, subject to certain packing constraints. 

Liquid crystal materials can be grouped into two main classifications, thermotropics and lyotropics. A thermotropic phase is one that can form by heating or cooling a material. Just as we see a phase transition between solid and liquid as we heat and melt ice, in thermotropic liquid crystals, additional melting points can be observed in between the solid and the liquid phases. These are the thermotropic liquid crystalline phases. A lyotropic liquid crystal phase is formed by molecules dissolved in a solvent, and phases form at certain concentrations in that solvent. In this chapter, we focus on descriptions of the thermotropic liquid crystal phases. Lyotropics, although also liquid crystals, are described in detail in Chapter 3 on surfactants. 

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Assume that the free energy can be expanded in powers of the magnetization m which is the order parameter. At zero field, only even powers of m appear in the expansion, due to the up-down symmetry of the system, and... 

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 A2.5.19. Isothemis showing the reduced external magnetic field B. = P Bq/ZcTj versus the order parameter s = for various reduced temperatures J = TIT. ...

For the kind of transition above which the order parameter is zero and below which other values are stable, the coefficient 2 iiiust change sign at the transition point and must remain positive. As we have seen, the dependence of s on temperature is detemiined by requiring the free energy to be a miniimuii (i.e. by setting its derivative with respect to s equal to zero). Thus... 

In both cases the late stages of kinetics show power law domain growth, the nature of which does not depend on the mitial state it depends on the nature of the fluctuating variable(s) which is (are) driving the phase separation process. Such a fluctuating variable is called the order parameter for a binary mixture, tlie order parameter o(r,0 is tlie relative concentration of one of the two species and its fluctuation around the mean value is 5e(/,t) = c(r,t) - c. In the disordered phase, the system s concentration is homogeneous and the order... 

Here we shall consider two simple cases one in which the order parameter is a non-conserved scalar variable and another in which it is a conserved scalar variable. The latter is exemplified by the binary mixture phase separation, and is treated here at much greater length. The fonner occurs in a variety of examples, including some order-disorder transitions and antrferromagnets. The example of the para-ferro transition is one in which the magnetization is a conserved quantity in the absence of an external magnetic field, but becomes non-conserved in its presence. 

For a one-component fluid, the vapour-liquid transition is characterized by density fluctuations here the order parameter, mass density p, is also conserved. The equilibrium structure factor S(k) of a one component fluid is... 

Hamiltonian, but in practice one often begins with a phenomenological set of equations. The set of macrovariables are chosen to include the order parameter and all otlier slow variables to which it couples. Such slow variables are typically obtained from the consideration of the conservation laws and broken synnnetries of the system. The remaining degrees of freedom are assumed to vary on a much faster timescale and enter the phenomenological description as random themial noise. The resulting coupled nonlinear stochastic differential equations for such a chosen relevant set of macrovariables are collectively referred to as the Langevin field theory description. 

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...

Figure A3.3.6 Free energy as a function of the order parameter cji for the homogeneous single phase (a) and for the two-phase regions (b), 0.

Equation (A3.3.57) must be supplied with appropriate initial conditions describing the system prior to the onset of phase separation. The initial post-quench state is characterized by the order parameter fluctuations characteristic of the pre-quench initial temperature T.. The role of these fluctuations has been described in detail m [23]. Flowever, again using the renomialization group arguments, any initial short-range correlations should be irrelevant, and one can take the initial conditions to represent a completely disordered state at J = xj. For example, one can choose the white noise fomi (i /(,t,0)v (,t, 0)) = q8(.t -. ), where ( ) represents an... 

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

This time development of the order parameter is completely detenninistic when the equilibrium p(r) = const is reached the dynamics comes to rest. Noise can be added to capture the effect of themial fluctuations. This leads to a Langevin dynamics for the order parameter. 

With all-atom simulations the locations of the hydrogen atoms are known and so the order parameters can be calculated directly. Another structural property of interest is the ratio of trans conformations to gauche conformations for the CH2—CH2 bonds in the hydrocarbon tail. The trans gauche ratio can be estimated using a variety of experimental techniques such as Raman, infrared and NMR spectroscopy. 

If all the molecules are perfectly parallel, S would equal 1. In an isotropic Hquid, f 6) is constant so that < cos 0 > equals 1/3 and S is therefore 2ero. The order parameter for Hquid crystals falls somewhere between these limits and decreases somewhat with increasing temperature. 

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. 

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