Mean-field transition temperature

Fig. 2.5 Schematic showing the variation of inverse scattering intensity and domain spacing (as determined from SAXS or SANS) across the order-disorder transition of a block copolymer melt. The mean field transition temperature has been identified operationally as the point where, on heating, the inverse intensity crosses over to a linear dependence on T (after Sakamoto and Hashimoto 1995).

From this expression one finds that f(T) approaches 1 for T —> 0 (by integration) and /(T) 2(1 - T/TCMF) - Z 2 for T -> TCMF. TCMF denotes the mean-field transition temperature. For quasi one-dimensional systems p1 has an additional factor C-2 (the inverse area perpendicular to the chain). 

In the following we use a minimal model for the low energy, long wave length excitations of the condensed charge density wave. Since fluctuations in the amplitude Z are suppressed, because they are massive, we take only fluctuations of the phase cp (cf. eq. 2) into account. Clearly, such an approach breaks down sufficiently close to the mean-field transition temperature TCMF. Neglecting fluctuations in Z, the Hamiltonian for our model is given by... 

A divergence in (40) identifies a mean-field transition temperature to a state with LRO. Below the transition temperature, there will be an order parameter describing this LRO in which all chains are synchronized. If it is incommensurate, it will be able to slide (the Frolich mode). Close to the transition temperature, there is a critical region in which fluctuations will play an important role [4,5,37], The RG is a key ingredient in the... 

FIG. 4. Excess conductivity vs reduced temperature of KjCw). The slope of this curve has been proposed for a zero-dimensional system. A three-dimensional system exhibits fluctuation conductivity indicated by curve 3D, based on the measured coherence length gl(0)-26 A. Inset The divergence of the conductivity in unrenormalized quantities. The two lines indicate the theoretical prediction for a OD and a 3D system, respectively. The arrow indicates the mean-field transition temperature T,o-... 

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... 

In the systems with diamagnetic M(mnt) units, only a tetramerisation of the perylene chains, corresponding to the 2kp Peierls transition, is observed. In the other cases, except for M = Fe, where the dimerisation is of stronger chemical nature, a dimerisation of the M(mnt) units takes place simultaneously with the distortion in the perylene chains. This distortion in the M(mnt)J chains with 5=1/2 can be associated with a spin-Peierls-like transition, and this association is especially tempting in view of the typical spin-Peierls exponential decrease of the susceptibility below f for directions parallel and perpendicular to the chain axis. However for this transition to be spin-phonon driven, it lacks the usual criteria for a spin-Peierls transition requiring an antiferromagnetic interaction J k Tsf, where T p° is the mean field transition temperature [80] for instance in the case of M = Pt with a Tg = 8.2 K, J/ks was estimated as 15 K while Tgp° should be 25 K, the temperature at which one-dimensional... 

X = A + BIT, where A and B are constants). Thus S(q )should change linearly with 1/7. t his was indeed observed by Hashimoto etal. (1983b) at high temperatures however, at a temperature associated with the transition from the homogeneous disordered phase to the ordered phase, a deviation from linear behaviour was found. Such deviations are now ascribed to the effects of composition fluctuations (Bates et al. 1988 Lodge et al. 1996), and the crossover from linear to non-linear dependence of S(q ) on 1/7 does not correspond to the order disorder transition, rather the mean-field to non-mean-field transition (see Section 2.2.1 for block copolymer melts). 

The temperature dependence of the spin susceptibility is one of the most intriguing problems of the conducting charge transfer salts. Basing their arguments on T measurements, some authors maintain that the large decrease of observed between room temperature and the metal-insulator transition temperature is related to the development of a pseudo-gap in the density of states at the Fermi level, as temperature becomes smaller than a mean-field Peierls temperature /3o,... 

In the inset of Fig. 9 we show the mean field frequency 0 = 0// as a function of density for T = 1. At this temperature the system undergoes a phase transition from a paramagnetic to a ferromagnetic fluid at a density whose mean field value is p mf = 0-4- For densities below this value we obtain 0 = cjq, which agrees with the frequency value of the low-order virial expansion (see Eq. (34)). For p > Pc,mF) increases with the density due to increase of the magnetization. 

FIG. 14 Phase diagram of the quantum APR model in the Q -T plane. The solid curve shows the line of continuous phase transitions from an ordered phase at low temperatures and small rotational constants to a disordered phase according to the mean-field approximation. The symbols show the transitions found by the finite-size scaling analysis of the path integral Monte Carlo data. The dashed line connecting these data is for visual help only. (Reprinted with permission from Ref. 328, Fig. 2. 1997, American Physical Society.)... 

In order to finally address the question whether our system has a reentrant phase transition, as predicted by the mean-field study the low temperature region was analyzed by the cumulant intersection finite-size scahng method described in Sec. IV A. For the rotational constant 0 = 0.6109 an... 

The rapid rise in computer speed over recent years has led to atom-based simulations of liquid crystals becoming an important new area of research. Molecular mechanics and Monte Carlo studies of isolated liquid crystal molecules are now routine. However, care must be taken to model properly the influence of a nematic mean field if information about molecular structure in a mesophase is required. The current state-of-the-art consists of studies of (in the order of) 100 molecules in the bulk, in contact with a surface, or in a bilayer in contact with a solvent. Current simulation times can extend to around 10 ns and are sufficient to observe the growth of mesophases from an isotropic liquid. The results from a number of studies look very promising, and a wealth of structural and dynamic data now exists for bulk phases, monolayers and bilayers. Continued development of force fields for liquid crystals will be particularly important in the next few years, and particular emphasis must be placed on the development of all-atom force fields that are able to reproduce liquid phase densities for small molecules. Without these it will be difficult to obtain accurate phase transition temperatures. It will also be necessary to extend atomistic models to several thousand molecules to remove major system size effects which are present in all current work. This will be greatly facilitated by modern parallel simulation methods that allow molecular dynamics simulations to be carried out in parallel on multi-processor systems [115]. 

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

The mean-field phase diagram in the WSL calculated by Matsen et al. [138] predicts a transition from C to the disordered state via the bcc and the fee array with decreasing /N. This was not followed here. Transitions from the C to S (at 115.7 °C), to the lattice-disordered sphere - where the bcc lattice was distorted by thermal fluctuations - and finally to the disordered state (estimation > 180 °C but not attained in the study) were observed. It was reasoned to consider the lattice-disordered spheres as a fluctuation-induced lattice disordered phase. This enlarges the window for the disordered one and causes the fee phase to disappear. Even if the latter should exist, its observation will be aggravated by its narrow temperature width of about 8 K and its slow formation due to the symmetry changes between fee and bcc spheres. 

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