Crystal phase order parameter

Polymer crystallization has been described in the framework of a phase field free energy pertaining to a crystal order parameter in which = 0 defines the melt and assumes finite values close to unity in the metastable crystal phase, but = 1 at the equilibrium limit (23-25). The crystal phase order parameter (xj/) may be defined as the ratio of the lamellar thickness (f) to the lamellar thickness of a perfect polymer crystal (P), i.e., xlr = l/P, and thus it represents the linear crystallinity, that is, the crystallinity in one dimension. The free energy density of a polymer blend containing one crystalline component may be expressed as... 

Xo[l + ecos(40)], where Xq is a constant and e is the strength of surface energy anisotropy. Further, Kobayashi [12] coupled the time evolution equation pertaining to the crystal phase order parameter to the energy balance (i.e., heat conduction) equation and demonstrated the evolution of side-branched dendritic structures growing into an undercooled melt. 

An interesting aspect of many structural phase transitions is the coupling of the primary order parameter to a secondary order parameter. In transitions of molecular crystals, the order parameter is coupled with reorientational or libration modes. In Jahn-Teller as well as ferroelastic transitions, an optical phonon or an electronic excitation is coupled with strain (acoustic phonon). In antiferrodistortive transitions, a zone-boundary phonon (primary order parameter) can induce spontaneous polarization (secondary order parameter). Magnetic resonance and vibrational spectroscopic methods provide valuable information on static as well as dynamic processes occurring during a transition (Owens et ai, 1979 Iqbal Owens, 1984 Rao, 1993). Complementary information is provided by diffraction methods. 

A molecular dynamics simulation has been performed on 4-n-pentyl-4(-cyanobiphenyl (5CB) in the nematic phase. Order parameters and dipolar couplings have been calculated and used to test theoretical models. Theoretical models have also been developed to explain the shielding of a noble-gas atom in an anisotropic environment and applied to explain the medium-induced shielding of the noble gases Xe and Ne in the nematic liquid crystal 4(-ethoxybenzylidene-4-n-butylaniline (EBBA). ... 

The order parameter S = 0.5(3(cos ) — I) characterizes the long-range order of molecules in a mesophase, where 0 is the momentary angle between the long axis of the molecule and the director. In an ideal crystal the order parameter S equals 1, and it equals 0 in an isotropic liquid. In a nematic phase the order parameter hes in the range 0.5-0.7. 

A physical system in which phase transition(s) can occur is usually characterized by one or more long range order parameters (order parameter for short). For example, in nematic liquid crystals the order parameter is the quantity S = (P2(cos 0)) as defined in previous chapters " in ferromagnets the order parameter is the magnetization in a single domain and in liquid-gas systems the order parameter is the density difference between the liquid and gas phases. In each of the above cases the state of the system, at any fixed temperature, can be described by an equilibrium value of the order parameter and fluctuations about that value. A phase transition can be accompanied by either a continuous or a discontinuous change in the equilibrium value of the order parameter when the system transforms from one phase to the other. (For simplicity we will consider temperature as the only thermodynamic variable in this paper the pressure depedence of the various phenomena will be neglected). 

Thennotropic liquid crystal phases are fonned by rodlike or disclike molecules. However, in the following we consider orientational ordering of rodlike molecules for definiteness, although the same parameters can be used for discotics. In a liquid crystal phase, the anisotropic molecules tend to point along the same direction. This is known as the director, which is a unit vector denoted n. 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

We will use it here in order to derive an analytical form for a crystal profile with a rough interface as an exphcit example. An order parameter

crystalline phase with 0 > 0 and the gaseous (or hquid) one with 0 < 0. 

The anisotropy of the liquid crystal phases also means that the orientational distribution function for the intermolecular vector is of value in characterising the structure of the phase [22]. The distribution is clearly a function of both the angle, made by the intermolecular vector with the director and the separation, r, between the two molecules [23]. However, a simpler way in which to investigate the distribution of the intermolecular vector is via the distance dependent order parameters Pl+(J") defined as the averages of the even Legendre polynomials, PL(cosj r)- As with the molecular orientational order parameters those of low rank namely Pj(r) and P (r), prove to be the most useful for investigating the phase structure [22]. 

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