Mean-field transition temperature equation

The equation determining the mean-field transition temperature Tc is obtained by setting A=0 in Eq. (6) this yields I(T)s1, where I(T) is the integral... 

The above reasoning can easily be generalized to the case of a phase transition to a spatially modulated ground state, characterized by non-zero magnetization Mq. The corresponding mean-field value of the ordering temperature Tc(q) is given by the solution of the equation (Diet et al. 1999)... 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

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. 

It has been the merit of Picken (1989, 1990) having modified the Maier-Saupe mean field theory successfully for application to LCPs. He derived the stability of the nematic mesophase from an anisotropic potential, thereby making use of a coupling constant that determines the strength of the orientation potential. He also incorporated influences of concentration and molecular weight in the Maier-Saupe model. Moreover, he used Ciferri s equation to take into account the temperature dependence of the persistence length. In this way he found a relationship between clearing temperature (i.e. the temperature of transition from the nematic to the isotropic phase) and concentration ... 

In a mean field approximation based on the electron interaction on different lines we obtain the following equation for the dielectric transition temperature (2) ... 

Using the mean field approximation, one can find the critical temperature of the transition between the ferromagnetic (FM) and paramagnetic (PM) phases as it is shown in Fig. 4.19. In the considered case it is determined from the condition that the exchange energy JiiiR) should be equal to the thermal energy, J23(R) = kBT. In accordance with Fig. 4.18 there exist either two (ryi [Pg.215]

Phase transitions are defined thermodynamically. However, to model them, we must turn to theories that describe the ordering in the system. This is often done approximately, using the average order parameter (here we assume one will suffice to describe the transition) within a so-called mean field theory. The choice of appropriate order parameter is discussed in the next section. The order parameter for a system is a function of the thermodynamic state of the system (often temperature alone is varied) and is uniform throughout the system and, at equilibrium, is not time dependent. A mean field theory is the simplest approximate model for the dependence of the order parameter on temperature within a phase, as well as for the change in order parameter and thermodynamic properties at a phase transition. Mean field theories date back to when van der Waals introduced his equation of state for the liquid-gas transition. 

If the director is already parallel to the electric field, the free energy decreases as increases. This means an ordering of the anisotropic state, which in the isotropic phase is equivalent to a field-induced transition to the nematic liquid crystal phase. This is similar to the case when the pressure induces a transition to the lower symmetric phase. Such a situation is described by the Clausius-Clapeiron equation that relates the increment of the phase transition temperature, to the pressure. In our case this is equivalent to the field-induced increase of the isotropic-nematic transition as ... 

Second, for the response of the interacting system described by Xeff, the coupling constant g is extremely important. In the mean-field approximation, a divergent susceptibihty Xeff leads to a phase transition at a finite temperature Tc, MF [57, p. 8, Eq. 1.23]. As can be seen in the equation for Xeff,g>0 may create a divergence in Xeff due to a vanishing denominator even when xo is finite (note that xo < 0). On the other hand, g < 0 reduces the bare susceptibihty xo- In an HEG, for... 

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... 

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