Phase transitions free energy density

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)... 

One sfrategy is phenomenological and rooted in the Landau theory of phase transitions. The basic idea is to assert that the homogeneous free energy density... 

The quantity m is a scaled volume fraction of the NPs. For rod-like NPs of diameter D and length L it is roughly given by u L D. This term enforces a first-order structural transition from the isotropic Snp=0) to an orientationally liquid-crystal-like ordered phase (5 Afp>0) at the critical scaled concentration u=u =2.1. For the LC free-energy density f b f e simplified... 

Figure 7.9 Schematic plot of local free energy density and crystal order parameter fr for various temperatures showing the metastable energy barrier for phase transition from the melt (t/r = 0) to the crystalline state < 1) at a given crystallization temperature. The crystal order parameter less than unity implies the imperfect crystals containing some noncrystaUine components.

Free energy density of a system undergoing phase transition... 

Fig. 6.5 (a) Temperature dependence of the order parametru in the Landau-de Gennes model (B) and (C) are coefficients of the expansion. Tjvj Tc is experimental value of the isotropic—nematic phase transition temperature corresponding to equality of free energy densities for the two phases, (b) Experimental dependence of the order parameter for 5CB and the characteristic temperature points Tc, Tc and Tc defined in accordance with the model of panel (a)... 

A qualitative picture, Fig. 10.4, shows the distance dependencies of the orientational order parameter for homeotropically aligned nematic liquid crystal at the solid substrate. The problem is to explain such dependencies [6]. The influence of the surface on the orientational order parameter may be discussed in terms of the modified Landau-de Gennes phase transition theory. Consider a semi-infinite nematic of area A being in contact with a substrate at z = 0 and uniform in the x and y directions. When writing the free energy density a surface term -W8(z)S must be added to the standard expansion of the bulk free energy density ... 

In the proper ferroelectrics, the spontaneous polarisation appears as a result of the polarisation catastrophe or, in other words, due to electric dipole-dipole interactions. There are also improper ferroelectrics, in particular, liquid crystalline ones, in which a structural transition into a polar phase occurs due to other interactions and, consequently, appears as a secondary phenomenon. We shall discuss this case later. For simplicity, the square of spontaneous polarisation vector can be taken as a scalar order parameter for the transition from the higher symmetry paraelectric phase to the lower symmetry ferroelectric phase. Therefore, in the absence of an external field, we can expand the free energy density in a series over P (T) and this expansion for ferroelectrics is called Landau-Ginzburg expansion ... 

Fig. 11. Effect of network force on the free energy density (16) 2 K below the phase transition, Sc phase (o) calculated potential of the Sc phase, (O) force due to the network, and ( ) superposition of both.

The electroclinic or soft mode effect could be explained within the framework of the Landau theory of phase transitions [4]. We can write the expansion of the free energy density g of smectic A up to the 0 term as follows ... 

In this equation, y is the interaction strength, c(r) the crosslink concentration, the smectic order parameter, and Vz (r) the relative displacement of the rubber matrix. Witkowski and Terentjev [132] evaluated (15) for (r) = 1, which is valid deep in the smectic phase, i.e., far below the smectic-nematic transition. Using the so-called replica trick, they integrated out the rubbery matrix fluctuations and obtained an effective free-energy density that depends only on the layer displacements M(r). Under the restriction that wave vector components along the layer normal dominate over in-layer components, q q, and considering only long-... 

In the temperature range close to the transition temperature JsmA-N, the phase transition from the smectic A to the nematic phase can phenomenologically be described by the Landau de Geimes theory [12], [42], [43]. The free energy density is expanded in a power series of an order parameter i/r ... 

Figure 10.3. Free-energy density as a function of the order parameter for different temperatures (top) and the corresponding relation between the temperature and the order parameter (bottom) in the vicinity of (a) a second-order and (b) a first-order phase transition.

We now consider a second-order phase transition. Simple examples include the smectic C-smectic A transition, which is characterized by a continuous decrease in an order parameter describing molecular tilt. The smectic A (lamellar)-isotropic transition can also be second order under certain conditions. In this case symmetry means that terms with odd powers of are zero. To see this consider the smectic C-smectic A transition. The appropriate order parameter is given by Eq. (5.23) and is a complex quantity. However, the free energy density must be real thus only even terms ijnjr and etc., remain. Then the free energy can be written as... 

Using this order parameter, the free energy density in the nematic phase close to a transition to the smectic phase can be shown to be given by... 

Depiction of a first order transition in terms of the variation of the Landau free energy density with the order parameter at several reduced temperatures. For details see text. Reproduced from N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Copyright (c) July 21, 1992 reprinted with permission of Westview Press of the Perseus books Group. 

Mean field approximation neglects all spatial inhomogeneities. The order parameter is thus considered as constant all over the sample. In the vicinity of the N-SmA phase transition, a Landau free energy density may be expanded in invariant combinations of the order parameter F (the subscript 1 referring to the first term of the Fourier expansion will be omitted in the following). Translational invariance requires that the phase O does not enter the free energy. The Landau expansion thus reduces to ... 

In the vicinity of fluctuation dominated phase transitions, the temperature dependence of thermodynamic parameters such as the specific heat at constant pressure, Cp=e , and the order parameter, y/ e, are all related to through a free energy density giving rise to scaling relations. For example a=2-vdmdp=(d-2) v/2 [2]. Despite the variety of their continuous broken symmetries, most liquid crystal phase transitions are expected to fall in the 3D-XY (helium) universality class with, a=-0.01, V-0.67 and y8-0.33. 

Electroclinic Effect Near the Smectic A—Smectic C Phase Transition. The electric field may induce a tilt of the director in the orthogonal smectic A phase near the smectic A-smectic C phase transition (electroclinic effect [189, 191]). The electroclinic effect may be understood within framework of the Landau phase transition theory [7]. If the dielectric anisotropy is negligible, the free energy density of the smectic A phase may be expanded over the field-induced tilt angle 6 ... 

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