Scalar order parameter

If the scalar order parameter of the Ising model is replaced by a two-component vector n = 2), the XY model results. All important example that satisfies this model is the 3-transition in helium, from superfiuid helium-II... 

Here difference between the oil and water concentrations go is the strength of surfactant and /o is the parameter describing the stability of the microemulsion and is proportional to the chemical potential of the surfactant. The constant go is solely responsible for the creation of internal surfaces in the model. The microemulsion or the lamellar phase forms only when go is negative. The function/(<))) is the bulk free energy and describes the coexistence of the pure water phase (4> = —1), pure oil phase (4> = 1), and microemulsion (< ) = 0), provided that/o = 0 (in the mean-held approximation). One can easily calculate the correlation function (4>(r)(0)) — (4>(r) (4>(0)) in various bulk homogeneous phases. In the microemulsion this function oscillates, indicating local correlations between water-rich and oil-rich domains. In the pure water or oil phases it should decay monotonically to zero. This does occur, provided that g2 > 4 /TT/o — go- Because of the < ), —<(> (oil-water) symmetry of the model, the interface between the oil-rich and water-rich domains is given by... 

In this simple model characterized by a single scalar order parameter, the structures with periodic surfaces are metastable. It simply means that we need a more complex model including the surfactant degrees of freedom (its polar nature) in order to stabilize structures with P, D, and G surfaces. In the Ciach model [120-122] indeed the introduction of additional degrees of freedom stabilizes such structures. 

Let us consider a dynamically symmetric binary mixture described by the scalar order parameter field < )(r) that gives the local volume fraction of component A at point r. The order parameter < )(r) should satisfy the local conservation law, which can be written as a continuity equation [143] ... 

According to the well-known Landau theory, the eigenvector of the order parameter in any second order solid-solid phase transition transforms according to an irreducible representation of the space group of the parent phase state. Furthermore, the free energy F=U -TS can be expanded around the transition temperature Tc in terms of the scalar order parameter p, which... 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

Tumbling regime At very low shear rates, the birefringence axis (or the director) of the nematic solution tumbles continuously up to a reduced shear rate T < 9.5. While the time for complete rotation stays approximately equal to that calculated from Eq. (85), the scalar order parameter S,dy) oscillates around its equilibrium value S. Maximum positive departures of S(dy) from S occur at 0 n/4 and — 3n/4, and maximum negative departures at 0 x — k/4 and — 5it/4, while the amplitude of oscillation increases with increasing T. 

Figure 12-5. Gibbs energy as a function of the (scalar) order parameter.

The phase behaviour of blends of homopolymers containing block copolymers is governed by a competition between macrophase separation of the homopolymer and microphase separation of the block copolymers. The former occurs at a wavenumber q = 0, whereas the latter is characterized by q + 0. The locus of critical transitions at q, the so-called X line, is divided into q = 0 and q + 0 branches by the (isotropic) Lifshitz point. The Lifshitz point can be described using a simple Landau-Ginzburg free-energy functional for a scalar order parameter rp(r), which for ternary blends containing block copolymers is the total volume fraction of, say, A monomers. The free energy density can be written (Selke 1992)... 

As a particular case, Eq. (4.351) at 1 = 0 renders the equation for the scalar order parameter of the particles along the field direction ... 

In their original theory, Maier and Saupe supposed that the molecular interactions responsible for the nematic state are anisotropic van der Waals interactions (discussed in Section 2.3), in which case mms should be temperature-independent. However, it is now recognized that shape anisotropy is also important, even for small-molecule thermotropic nematics. By making mms temperature-dependent, the Maier-Saupe potential can, in principle, accommodate both energetic and entropic effects. In fact, if the function sin(u, u) in the purely entropic Onsager potential Eq. (2-5) is approximated by the expansion 1 — V2 cos (u, u)+. . ., then to lowest order the Maier-Saupe potential (2-7) is obtained with C/ms — Uo bT/S, where we have defined the dimensionless Maier-Saupe energy constant by Uus = ums/ksT, Thus, the Maier-Saupe potential can be used as an approximation to describe orientational order in either lyotropic (solvent-based) or thermotropic nematics. For a thermotropic melt, the Maier-Saupe theory predicts a first-order transition from the isotropic to the nematic phase when mms/ bT = U s — t i.MS = 4.55, and at this transition the scalar order parameter S jumps from zero to 0.43. S increases toward unity with further increases in Uus- The spinodal point at which the isotropic phase is unstable to even small orientational perturbations occurs atU — = 5 for the Maier-... 

Here 8 is the unit tensor, and ( ) denotes the average over the distribution function f [see Eq. (2-4)]. The scalar order parameter S is related to S by... 

As the shear rate increases, the numerical solutions of the Smoluchowski equation (11-3) begin to show deviations from the predictions of the simple Ericksen theory. Tn particular, the scalar order parameter S begins to oscillate during the tumbling motion of the director (for a discussion of tumbling, see Sections 11.4.4 and 10.2.6). The maxima in the order parameter occur when the director is in the first and third quadrants of the deformation plane i.e., 0 < 9 — nn < it j2, where n is an integer. Minima of S occur in the second and fourth quadrants. The amplitude of the oscillations in S increases as y increases, until S is reduced to only 0.25 or so over part of the tumbling cycle. 

The degree of ordering in a uniaxial liquid crystal is given by the scalar order parameter S. It is the largest eigenvalue of the order tensor ... 

We illustrate the behavior for a first order transition between a vapor and a dense liquid in the framework of a simple Lennard-Jones model. The condensation of a vapor into a dense liquid upon cooling is a prototype of a phase transition that is characterized by a single scalar order parameter - the density, p. The thermodynamically conjugated field is the chemical potential, p. The qualitative features, however, are general and carry over to other types of phase coexistence, e.g., Sect. 3.4. 

The phase behavior of binary blends becomes much more complex if one relaxes the assumption of incompressibility. Then both liquid-liquid demixing and liquid-vapor phase separation are possible and the system is described by two scalar order parameters - density and composition (p, 4>) or the two densities of the species (pcos j Phd)- The interplay between the two types of phase... 

The simulation techniques presented above can be applied to all first order phase transitions provided that an appropriate order parameter is identified. For vapor-liquid equilibria, where the two coexisting phases of the fluid have the a similar structure, the density (a thermodynamic property) was an appropriate order parameter. More generally, the order parameter must clearly distinguish any coexisting phases from each other. Examples of suitable order parameters include the scalar order parameter for study of nematic-isotropic transitions in liquid crystals [87], a density-based order parameter for block copolymer systems [88], or a bond order parameter for study of crystallization [89]. Having specified a suitable order parameter, we now show how the EXEDOS technique introduced earlier can be used to obtain in a particularly effective manner for simulations of crystallization [33]. The Landau free energy of the system A( ) can then be related to P,g p( ((/"))... 

Since the identification of universality classes for surface layer transitions needs the I-andau expansion as a basic step, we first formulate Landau s theory (Toledano and Toledano, 1987) for the simplest case, a scalar order parameter density

phase transition and slowly varying in space. It can be obtained by averaging a microscopic variable over a suitable coarsc-graining cell Ld (in d-dimensional space). For example, for the c(2x2) structure in fig. 10 the microscopic variable is the difference in density between the two sublattices I (a and c in fig. 10) or II (b and d in fig. 10), ,- = pj1 — pj. The index i now labels the elementary cells (which contain one site from each sublattice I, II). Then... 

Fig. 13. Schematic variation of the Landau free energy at transitions of (a) second order and (b), (c) first order as a function of the (scalar) order parameter 0. Cases (a) and (b) assume a symmetry around 0 = 0, whereas case (c) allows a cubic term.

The alternative mechanism by which a first-order transition arises in the Landau theory with a scalar order parameter is the lack of symmetry of F against a sign change of (f>. Then we may add a term id3 to eq. (14), with another phenomenological coefficient w. For u > 0, / ((/>) may have two minima (fig. 13c) again the transition occurs when the minima are equally deep. For r = r T — T0) this happens when... 

In the previous section, we have seen that it cannot suffice to consider the order parameter alone. A crucial role is played by order parameter fluctuations that are intimately connected to the various singularities sketched in fig. 11. We first consider critical fluctuations in the framework of Landau s theory itself, and return to the simplest case of a scalar order parameter (j ) with no third-order term, and u > 0 [eq. (14)], but add a weak wavevector dependent field <5 H(x) = SHqexp(iq x) to the homogeneous field H. Then the problem of minimizing the free energy functional is equivalent to the task of solving the Ginzburg-Landau differential equation... 

We return here to the simple mean field description of second-order phase transitions in terms of Landau s theory, assuming a scalar order parameter cj)(x) and consider the situation T < Tc for H = 0. Then domains with = + / r/u can coexist in thermal equilibrium with domains with —domain with exists in the halfspace with z < 0 and a domain with 4>(x) = +

0 (fig. 35a), the plane z = 0 hence being the interface between the coexisting phases. While this interface is sharp on an atomic scale at T = 0 for an (sing model, with = -1 for sites with z < 0, cpi = +1 for sites with z > 0 (assuming the plane z = 0 in between two lattice planes), we expect near Tc a smooth variation of the (coarse-grained) order parameter field (z), as sketched in fig. 35a. Within Landau s theory (remember 10(jc) 1, v 00 01 < 1) the interfacial profile is described by... 

We return here to mean field theory with a scalar order parameter thick film geometry, assuming hard walls or surface against vacuum, respectively) at z = 0 and z = L. Starting again from eq. (14), we may disregard the x and y-coordinates [as in our treatment of the interfacial profile, eqs. (177)—(181)], but now we have to add a perturbation 2Fv(bare) to... 

In the present conribution, we develop a continuum-based model to describe experimentally observable interphases in thin adhesive films. The model is based on an extended contiuum theory, i.e. the mechanical behaviour in these interphases is captured by an additional field equation. The introduced scalar order parameter models the microscopical mechanical properties of the film phenomenologically. 

Consider an expansion of the excess free energy of any ordered system in powers of a scalar order parameter s in the following form ... 

Earlier, Homreich, Luban and Shtrikman had proposed a new type of higher order critical point, which they called the Lifshitz point, by considering a Landau-Ginsburg free energy expansion in terms of a scalar order parameter M for a magnetic system... 

In some instances, the single-order parameter model (16) is not sufficient. This is the case when fluctuations of the amphiphile concentration play an important role, such as in the calculation of the scattering intensity in film contrast. A second scalar order parameter field, p(r), which describes the deviation of the local amphiphile concentration from the average p, has to be included in the model. For a balanced system, the free energy functional... 

Ginzburg-Landau models that in addition to the scalar order parameter for the concentrations of oil, water, and amphiphile contain a vector order parameter for the amphiphile orientation have also been studied [41, 45, 49-53]. 

It is useful to rewrite the quantities into a dimensionless form. Usually, coordinates are measured in terms of the film thickness d (or another t5q)ical dimension in the case of a non-planar geometry) and the correlation length = Qd= y/27CL/B fs 10 nm. The order parameter is rescaled in units of the scalar order parameter of the nematic phase at the phase transition temperature, = 2B/Z /6C 0.2 — 0.6, and the temperature is controlled by d = (T — — T ) the reduced temperatures 0 = 1, 0, and 9/8... 

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