Tricritical exponents

B. Duplantier, H. Saleur. Exact tricritical exponents for polymers at the 0-point in two dimensions. Phys Rev Lett 59 539-542, 1987. 

Thus Landau s theory predicts the following set of tricritical exponents (Griffiths, 1970 Sarbach and Lawrie, 1984)... 

As a final point of this section, we return to tricritical phenomena in d = 2 dimensions. The tricritical exponents are known exactly from conformal invariance (Cardy, 1987). For the Ising case, the results are (Pearson, 1980 Nienhuis, 1982)... 

Also the tricritical 3-state Potts exponents (for a phase diagram, see fig. 28c) can be obtained from conformal invariance (Cardy, 1987). But in this case the standard Potts critical exponents are related to an exactly solved hard core model, namely the hard hexagon model (Baxter, 1980), and not the tricritical ones. The latter have the values crt = 5/6, = 1/18, Yt = 19/18, <5t = 20, ut = 7/12, rjt = 4/21,

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The air-water interface must be considered as a good solvent for the polyvinyl acetate used in this experiment. Vilanove and Rondelez11 also used another polymer namely methyl-polymethacrylate, but in this case, the air-water interface must be considered as a poor solvent, and a different exponent v is obtained. This exponent is interpreted as a tricritical exponent (see Chapter 14, Section 7.1). The value which they found for the exponent is... 

Finally, in 1987, Duplantier and Saleur found the exact values21 of the tricritical exponents v, and y, in two dimensions. These values are... 

However, as given by group renormalization theory (45), the values of the universal exponents depend on the (thermodynamic) dimensionality of the system. For four dimensions (as required by the phase rule for the existence of tricritical points), the exponents have classical values. This means the values are multiples of 1/2. The dimensions of the volume of tietriangles are (31)... 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

What does your best value for the exponent j8 indicate about the nature of the critical behavior underlying the N-I transition (mean-field second-order or tricritical) Note that /3 and P describe the temperature variation of the nematic order parameter S, which is a basic characteristic of the liquid crystal smdied. Thus, the same j8 and P values could have been obtained from measurements of several other physical properties, such as those mentioned in the methods section. 

The Landau model for phase transitions is typically applied in a phenomenological manner, with experimental or other data providing a means by which to scale the relative terms in the expansion and fix the parameters a, b, c, etc. The expression given in Equation (9) is usually terminated to the lowest feasible number of terms. Hence both a second-order phase transition and a tricritical transition can be described adequately by a two term expansion, the former as a 2-4 potential and the latter as a 2-6 potential, these figures referring to those exponents in Q present. 

A fit of the data for anorthite to Equations 7 and 8 describes the change in chemical shift with temperature in the PI phase (Fig. 18) and yields a value for the critical exponent, P = 0.27( 0.04), that is consistent with measurements using techniques sensitive to muchjonger length scales, such as X-ray diffraction (Redfem et al. 1987), that indicate the PI -71 transition in Si,Al ordered anorthite is tricritical. 

Thus we conclude that the order parameter exponent at the point ht,Tt, which is called a tricritical point (Sarbach and Lawrie, 1984), has a different... 

These effective exponents can be re-expressed in terms parameter h. Let us set of the tricritical... 

Tricritical swelling exponent in two dimensions estimations and exact values... 

For d — 3 the three-body interaction is just marginal. This is not the case for d < 3 (and in particular for d = 2) since, then, the dimensionless parameter y = cS3 t becomes infinite when S - oo. In this case, a system with two-body and three-body interactions is tricritical when the second virial coefficient A2 vanishes (or z = 0). Then, for an isolated chain, the dependence of R2 with respect to S (for large S) is characterized by an exponent v, (v, > 1/2)... 

From (5.5.20) and (5.5.21) it is seen that Beet, where = 2vj — v, and D oc fK If V = 2, B should be finite at the transition temperature. However, experimentally, it appears that B at the transition is almost vanishingly small within experimental limits. Few measurements are available on D to draw any definite conclusions. In any case, as pointed out earlier, the exponents are neither universal nor do they agree with the predictions of any of the theoretical models. Vithana et a/. have suggested that the widely differing values of the exponents for the different compounds may be a consequence of the fact that one is measuring effective values associated with crossover effects between the XY class and a tricritical point. A further complication is that the experiments of Evans-Lutterodt et appear to indicate that the occurrence of different... 

Exceptions to these general rules about behaviour in the critical region occur at special points on a binary mixture critical line - examples are extrema in pressure or temperature, points where azeotrope lines join critical lines, or double points (intersections of two critical lines). Moreover, in the special higher-order critical points ( tricritical points ) found in systems with a greater number of variables (three- and four-component fluid mixtures He + He), different exponents may be found. ... 

Griffiths has given a phenomenological (Landau) treatment of tricritical points which expresses the free energy as a sixth-order polynomial in an order parameter (which is some suitable linear combination of the physical densities , e.g. the mole fractions). The scaling properties of the singular part of the polynomial lead to four numbers = 5/6, 2 = 4/6 = 2/3, 3 = 3/6 = 1/2, = 2/6 = 1/3, in terms of which various critical exponents are expressed. Because this is an analytic (mean field) formulation, these exponents are classical , but it is believed that for experimental tricritical points in three dimensions they should be. ( Nonclassical logarithmic factors may exist, but these do not alter the exponents.)... 

It should be noted that the combination of these two sets of experiments suffices to determine all four of the numbers i, 2, 3, and 4. While more experiments are obviously desirable, there seems little reason to doubt that the exponents governing the change of appropriate properties as the tricritical point is approached are indeed the classical ones deduced from the phenomenological theory. 

In the limit when the number N of monomer nnits goes to infinity, there is sharp transition from the swollen to collapsed phase at a critical value of u. This transition is described as a critical phenomena analogous to a tricritical point for magnetic system [2]. For large polymers, the average gyration radius R at the transition behaves as Rjv N <> where the exponent vg is intermediate between the value v for swollen state and the value Vc = i-/D for compact globule on a lattice of fractal dimension D. 

The collapse transition occurs at u = u. At this fixed point, all the values A, B, ... are non-zero. This is a tricritical fixed point, with two eigenvalues larger than one. The exponent Vt = 0.63250 is very close to i>c. The exponent a is negative, a = —4.0269, showing that the singularity in the specific heat at u = is very weak. 

A, B same as for the DS fixed point. The linearization of recursion relations about this fixed point gives two eigenvalues Ai = 3.1319 and A2 = 2.5858 greater than one. The line 0Jc u, t) is therefore a tricritical line. The crossover exponent (j> = 0.8321. 

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