Soft order parameters transitions

A basic concept in the reconstruction theory of solid surfaces is the soft phonon approach of displacive structural transitions. An essential property of these structural phase transitions is the existence of an order parameter which... 

It should be noted that the theory described above is strictly vahd only close to Tc for an ideal crystal of infinite size, with translational invariance over the whole volume. Real crystals can only approach this behaviour to a certain extent. Flere the crystal quality plays an essential role. Furthermore, the coupling of the order parameter to the macroscopic strain often leads to a positive feedback, which makes the transition discontinuous. In fact, from NMR investigations there is not a single example of a second order phase transition known where the soft mode really has reached zero frequency at Tc. The reason for this might also be technical It is extremely difficult to achieve a zero temperature gradient throughout the sample, especially close to a phase transition where the transition enthalpy requires a heat flow that can only occur when the temperature gradient is different from zero. 

The soft mode concept can be extended to all distortive phase transitions (transitions with relatively small atomic displacements), even if they are only close to second order. In the case of a ferro-distortive transition, as for example in BaTiOs or KDP, the order parameter is proportional to the spontaneous electric polarization Fj. d F/ dp is not only proportional to co, but also to the dielectric susceptibility. This does not, however, mean that all components of the order parameter eigenvector must contribute to Ps. 

If, however, the transition is of a pure displacive nature, the fluctuation amplitude of the order parameter is critical and is by no means temperature-independent. Since the soft mode is an under-damped lattice vibration (at least outside the close vicinity of Tc), defined by its frequency a>s and damping constant Tj, the spectral density is a Lorentzian centred at s and the... 

They can serve therefore as a test for Ti dispersion. In Fig. 12 the relaxation results are shown for D-RADP-15. The solid lines are a fit of the theory [19] to the data. Above Tc the lit is excellent, whereas below Tc it probably suffers from the fact that the phase transition is already diffuse and only nearly of second order. This proves that a soft mode component is needed to explain the data. Furthermore, the fact that the ratio ti/t2 remains unchanged above and below Tc proves that the order parameter fluctuations are in the fast motion regime on both sides of the transition. 

The fact that the order parameter vanishes above does not mean that Nature does not have an inkling of things to come well below (or above) T. Such indicators are indeed found in many instances in terms of the behaviour of certain vibrational modes. As early as 1940, Raman and Nedungadi discovered that the a-) transition of quartz was accompanied by a decrease in the frequency of a totally symmetric optic mode as the temperature approached the phase transition temperature from below. Historically, this is the first observation of a soft mode. Operationally, a soft mode is a collective excitation whose frequency decreases anomalously as the transition point is reached. In Fig. 4.4, we show the temperature dependence of the soft-mode frequency. While in a second-order transition the soft-mode frequency goes to zero at T, in a first-order transition the change of phase occurs before the mode frequency is able to go to zero. 

Schmahl WW, Swainson IP, Dove MT, Graeme-Barber A (1992) Landau free energy and order parameter behavior of the a-p phase transition in ciistobalite. Z Kristallogr 201 125-145 Sollich P, Heine V, Dove MT (1994) The Ginzburg interval in soft mode phase transitions Consequences of the Rigid Unit Mode picture. JPhys Condensed Matter6 3171-3196 Strauch D, Domer B (1993) Lattice dynamics of a-quartz. 1. Experiment. J Phys Condensed Matter 5 6149-6154... 

Some insight into the mechanism by which Pc is changed is provided by the frequency of the soft mode responsible for the phase transition. In the stability field of the high symmetry (tetragonal) phase, the inverse order parameter susceptibility, x of the order parameter varies as... 

Phase transitions in which the square of the soft-mode frequency or its related microscopic order parameter goes to zero continuously with temperature can be defined as second order within the framework of the Ginzburg-Landau model [110]. The behavior is obviously classical and consistent with mean field... 

Fig. 8.5. Profile of the lowest order parameter mode (thick line) and of one of the highly excited modes for a LC heterophase system in contact with (a) ordering and (b) disordering substrates in the proximity of the phase transition. Dashed line denotes the spatial variation of the mean-field scsdar order parameter. In all cases T — Tjvj- Inset The corresponding spectrum of the relaxation rates chciracterized by a soft lowest order parameter mode.

Ha)nvard et al. (2005) performed a comprehensive investigation of LaAlOs in the temperature range of 10-750 K and determined the crystal structure, dielectric relaxation, specific heat, birefringence, and the frequencies of the two soft modes via Raman spectroscopy. While all these experiments show that the behaviour at the critical point around Tc = 813 K is consistent with a second-order transition, some evidence for an additional anomalous behaviour below 730 K have been shown. This anomaly was explained by a biquadratic coupling between the primary order parameter of the transition and the hopping of intrinsic oxygen vacancies. 

Coexistence of FM order and superconductivity under pressure The experimental phase diagram of FM collapse under pressure and simultaneous appearance of superconductivity is shown in fig. 43. The critical pressure for disappearance of FM order is pc2 = 16-17 kbar. The SC phase appears between pc = 10 kbar and pc2 = 16 kbar which is also the critical pressure for the FM-PM transition. The critical temperature Tx p) of the jc-phase hits the maximum of Tdp) at the optimum pressure pm = 12.5 kbar. As mentioned before the nature of the order parameter in the jc-phase remains elusive. The coincidence of maximum Tc with vanishing jc-phase order parameter suggests that the collective bosonic excitations of the X-phase which supposedly become soft at pm mediate superconductivity and not quantum critical FM spin fluctuations which are absent due to the persisting large FM... 

In this chapter, intermolecular forces that are the basis of self-assembly are considered in Section 1.2. Section 1.3 outlines common features of structural ordering in soft materials. Section 1.4 deals similarly with general considerations concerning the dynamics of macromolecules and colloids. Section 1.5 focuses on phase transitions along with theories that describe them, and the associated definition of a suitable order parameter is introduced in Section 1.6. Scaling laws are defined in Section 1.7. Polydispersity in particle size is an important characteristic of soft materials and is described in Section 1.8. Section 1.9 details the primary experimental tools for studying soft matter and Section 1.10 summarizes the essential features of appropriate computer simulation methods. 

In this section we consider a general model that has broad applicability to phase transitions in soft materials the Landau theory, which is based on an expansion of the free energy in a power series of an order parameter. The Landau theory describes the ordering at the mesoscopic, not molecular, level. Molecular mean field theories include the Maier-Saupe model, discussed in detail in Section 5.5.2. This describes the orientation of an arbitrary molecule surrounded by all others (Fig. 1.5), which set up an average anisotropic interaction potential, which is the mean field in this case. In polymer physics, the Flory-Huggins theory is a powerful mean field model for a polymer-solvent or polymer-polymer mixture. It is outlined in Section 2.5.6. 

Controlling the spontaneous formation of ordered domains in soft materials such as block copolymers [189] may lead to the development of stimuli-responsive materials for applications such as actuators [190] and photonics [191] due to the reversible nature of order formation. However, the stimuli that are typically used to control the morphology of block copolymers are e.g., temperature, pressure, solvent type and concentration... Pioneering work by Abbott and co-workers used the chemical oxidation approach to control the self-assembly of small-molecule amphiphiles containing ferrocene [192]. Rabin and co-workers have shown that the introduction of dissociated charges on one of the blocks of a diblock copolymer leads to stabilization of the disordered phase [193]. They also quantified the increase in x at the order-disorder transition (ODT), xodt, due to the entropic contribution of the dissociated counterions. The Flory-Huggins parameter,x, that is used to quantify interactions between polymer chains is assumed to be proportional to the difference in the polarizibility of the blocks [194]. The polarizibility of polyferrocenyldimethylsilane, which is larger than that of either polystyrene or polyisoprene [195], must increase upon oxidation due to the presence of the NO ions. 

Phase transitions in a material can be classified as first order and continuous (second order). At a first-order phase transition, we observe a discontinuity of some physical property representative of the degree of order in the system. For example, this could be material density or a calculated measure of order in the material (i.e., an order parameter). If we measure this parameter across the phase boundary, there will be a step, or discontinuity, at the transition point. An example of a typical first-order phase transition is ice melting. At the transition point, the density of the material abruptly changes as we go from ice to liquid water. First-order transitions also have a measureable latent heat. Some phase transitions can be described as weakly first order. In this case, the enthalpy change associated with that transition is very small. This is often true of phase transitions in soft matter systems. As a result, the enthalpy change may be difficult to measure, thus making the phase change difficult to detect by thermal properties alone. 

The situation is much different, if a phase transition from Sm-A to the crystalline B phase (Xtal-j5 ) is considered. A Xtal-B phase does have in-plane positional order and is, thus, a really three-dimensional (albeit rather soft) crystal with still some layered structure. The phase transition from Sm-A to Xtal-B is described by a two-dimensional (in-plane) density wave as order-parameter. The amplitude of this density wave, S describing the degree of ordering is a scalar quantity with properties quite similar to 5 of the Sm-A to Hex-B phase transition. Especially the dynamic equation (1) also applies to 5, with A only slightly different from Eq.(2) in the Xtal-B phase... 

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