Tricritical state

S functions is taken into account. As a result of the conservation of the total momcnliiiii, we have fci — + Pi + t 2 = 0. 

Section 1.6 h2is described the tricritic.al state of a thermodynamic system whose potential expansion coefficients in terms of even powers of the order parameter are functions of temperature and pressure. 

In a more general case, for determination of the properties of the tricritical state, a Hamiltonian with even powers of the order parameter and coefficients depending, except temperature and pressure, on the nonorienting field / (sc e section 1.6) and the tricritical interaction parameter (see further) is considered (Riedel tuid Wegner, 1972). 

The behaviour of such systems has been studied using the renormalization group method. In particular, the tricriticeil indices have been estimated. Kg. for 31) space (d = 3) 

Asymmetric tricritical points arc observed in mixtures of several components (Griffiths, 1974), and will be considered in detail later using polymer systems as examples (section 3.10). 

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It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

Second, there is a line of charge-ordering in the T -p plane, where the charge-charge correlation function begins to oscillate. This line, as established from GDH theory, passes close to the critical point and may generate a virtual tricritical state. A charge-density wave scenario also arises from r-dependent cavity interactions. 

Let us consider a solution of partially repulsive chains. The polystyrene-cyclohexane solutions in the vicinity of 34 °C (temperature TF) are of this type, and have been studied a great deal. We shall interpret the observations with the help of the basic parameters of the standard continuous model, which are the two-body interaction b and the three-body interaction c. In the sequel (Section 2.2), we also use the osmotic parameters g and h which are more adapted to a study of the tricritical state. 

There are also special temperatures which are associated with a solvent-solute system of area S and which have the same limit TF when S - oo. This fact reveals that the system is in a tricritical state. 

The temperatures Tr of the most current solvent-solute system were measured during the years 1960-70, and at that time the theories concerning polymer solutions in the vicinity of TF were still inaccurate, if not erroneous. This explains why the experimentalists who determined /, by using various methods did not obtain really concordant results. The renormalization theory applied to the tricritical state of polymer solutions explains the systematic discrepancies... 

Equations 154, 156, and 157 are followed by the conditions of the tricritical state, expressed through the vertex functions... 

It was (le Gennes who first pointed to the correspondence between the 6 point of tin polymer problem and the tricritical state of general-type systems. 

Most likely, there exist fine differences between the 0 state of a molecular coil and the tricritical state in the Lagrangian theoretical field model (see the end of section 4.1) (Oono and Freed, 1981a). 

The concentration range, corresponding to the tricritical state of dilute solutions, follows from E(]uatioii 151... 

The theory of the tricritical state is valid where cso is small to meet condition 129. In view of Equation 186,... 

The tricritical state theory is not applicable in the neighbourhood of the critical concentration c, but only when c > Cf. 

Duplaiitier (1987) correlated the results of different approaches in renormalization theory to clear up a number of contradictions dtiscribed by different authors in the cours< of development of the tricritical state theory. 

As Duplantier et al. (1989) conclude on the basis of Duplantier s theory discussed above, the validity range of the tricritical state theory (tricritical theory) is near 6... 

For long chains in the tricritical state of dilute solutions, the renormalization procedure allowed researchers to calculate ir (Equation 5.-5-183) and >it(S) (Equation 5.5 176), and for (fi ), Equation 5.5 122 with a distinct concentration-dependent correction (Equation 5.5 181) was obtaimxl. 

The theory of tricritical phenomena was developed by different methods the direct renormalization, the dimensional regularization in momentum p2tce, etc. There are some contradictions between different authors which can be explained by certain specific features of different approsiches. However, Duplantier performed comparative calculations and an analysis of the tricritical state of polymer systems by different methods (in particular, by the method of dimensional regularization and using a cut-off) and achieved equivalent results. 

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