Phase transitions Landau theory

Chen J-FI and Lubensky T C 1976 Landau-Ginzburg mean-fieid theory for the nematio to smeotio C and nematio to smeotio A phase transitions Phys.Rev. A 14 1202-7... 

Toledano J-C, Toledano P (1987) The Landau theory of phase transitions. World Scientific, Singapore... 

Kuchanov SI, Panyukov SV (2006) A new look at the Landau theory of phase transitions in polydisperse heteropolymer liquids (paper to be published)... 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

Fig. 29). Using Landau theory, Bak et al. (BMVW) have shown that it is the wall crossing energy A which determines the symmetry of the weakly incommensurate phase and the nature of the phase transition ... 

Salje (1985) interpreted overlapping (displacive plus Al-Si substitutional) phase transitions in albite in the light of Landau theory (see section 2.8.1), assigning two distinct order parameters Q n and to displacive and substitutional disorder and expanding the excess Gibbs free energy of transition in the appropriate Landau form ... 

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... 

This relation is analogous to the equation of state for spin systems in the famous Landau theory of phase transitions. It reads, H = A0(T — Tc)t]/ + u0ij/3, where H is a magnetic field and i// is an order parameter, A0 and u0 being constants. In our problem, the left hand side of Eq. (2.41), which is a function of T and corresponds to H in spin systems. On the right hand side, the coefficient Vic — Vi which is determined by the degree of ionization, corresponds to the temperature parameter A0(T — Tc) in spin systems. Therefore, at yt = yu, we find T — Tc cc — c a with 8 = 3 near the critical point of ionic gels, where T— Tc plays the role of H in spin systems. 

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. 

Within Landau-Ginzburg theory, the free energy functional near a second-order or weakly first-order phase transition is expanded in terms of an order parameter rj>(q) ... 

A primary focus of our work has been to understand the ferroelectric phase transition in thin epitaxial films of PbTiOs. It is expected that epitaxial strain effects are important in such films because of the large, anisotropic strain associated with the phase transition. Figure 8.3 shows the phase diagram for PbTiOs as a function of epitaxial strain and temperature calculated using Landau-Ginzburg-Devonshire (lgd) theory [9], Here epitaxial strain is defined as the in-plane strain imposed by the substrate, experienced by the cubic (paraelectric) phase of PbTiOs. The dashed line shows that a coherent PbTiOs film on a SrTiOs substrate experiences somewhat more than 1 % compressive epitaxial strain. Such compressive strain favors the ferroelectric PbTiOs phase having the c domain orientation, i.e. with the c (polar) axis normal to the film. From Figure 8.3 one can see that the paraelectric-ferroelectric transition temperature Tc for coherently-strained PbTiOs films on SrTiOs is predicted to be elevated by 260°C above that of... 

The ferroelectric phase transition of second-order in tgs at 0C = 49° C can be described in the framework of the Landau-theory (e.g. [4]) by the thermodynamical potential... 

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

In Fig. 13 the result of the field dependent calculation of the heat capacity using Landau theory for fields within the tetragonal plane is shown. In agreement with the experiment (Fig. 5), the magnetic field broadens the anomaly observed at the Neel temperature TN and lowers the transition point T of the commensurate to incommensurate phase transition. The increase of the calculated heat capacity while lowering the temperature below T, is a consequence of the approach to the temperature T of the proper instability of the ferromagnetic subsystem. As such, the same increase is observed in the experimental data too and it is possible to assume that the broad maximum at low temperature is connected with the subsequent phase transition in the magnetic subsystem of copper metaborate. 

Dynamical Self-Organization. When the parameter X passes slowly through X (l),the bifurcation picture of the previous section accurateiy describes the system. However, in Fucus, and probably in many other examples, this time scale separation between the characteristic time on which X varies and the time to obtain the patterned state does not hold. Thus a dynamical theory allowing for the interplay of these two time scales is required to characterize the developmental scenario. A natural formalism to describe this process is that of time dependent Ginzburg-Landau (tdgl) equations used successfully in other contexts of nonequilibrium phase transitions (27). 

A superconductor exhibits perfect conductivity (See Section 7.2) and the Meissner effect (See Section 7.3) below some critical temperature, Tc. The transition from a normal conductor to a superconductor is a second-order, phase-transition which is also well-described by mean-field theory. Note that the mean-field condensation is not a Bose condensation nor does it require and energy gap. The mean-field theory is combined with London-Ginzburg-Landau theory through the concentration of superconducting carriers as follows ... 

The variation of 5(7) near the N-I phase transition will be measured in this experiment and will be compared with the behavior predicted by Landau theory, " " which is a variant of the mean-field theory first introduced for magnetic order-disorder systems. In this theory, local variations in the environment of each molecule are ignored and interactions with neighbors are represented by an average. This type of theory for order-disorder phase transitions is a very useful approximate treatment that retains the essential features of the transition behavior. Its simplicity arises from the suppression of many complex details that make the statistical mechanical solution of 3-D order-disorder problems impossible to solve exactly. 

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