Tricritical coexistence in three dimensions: The multicomponent limit

Abstract

The asymptotic tricritical equation of state, including the three=phase coexistence monohedron, is analyzed in detail for the exactly soluble multicomponent or spherical limit, n~, of the continuous-spin model with terms of order s, s, and s, in d =3 spatial dimensions. Various nonuniversal scaling functions and amplitude ratios, depending on the range parameter z~1/Ro, are evaluated explicitly and reveal the nature and magnitude of the deviations from the classical, phenomenological theory of tricriticality (which is developed systematically in an Appendix). The relationship to results for finite n is discussed briefly.

Scaling of demixing curves and crossover from critical to tricritical behavior in polymer solutions

“…( 22) and ( 24) we can form the universal ratio φ sd /φ c = 3, which, since λ ∼ T − Θ, measures the ratio of the slopes of the wing critical line and the phase separation boundary in the semi-dilute limit. The result φ sd /φ c = 3 coincides with Flory theory but differs from the result φ sd /φ c = 5/2 [25,26] that one finds from the Landau expansion by keeping h 2 constant instead of N in the calculation of the wing critical line. To our knowledge the only attempt to independently obtain this universal ratio was made in simulations by Frauenkron and Grassberger [27,28].…”

Optical and calorimetric studies near the smectic-A to smectic-C tricritical point

“…Conversely, the parameter z exhibits a relatively smaller value for the fit range-C. The extracted z values for the smaller fit ranges in the present system reveal a well agreement with the reported values at the tricritical N-Sm-A phase transition for polar cyanobiphenyls [73], while they are found to be quite higher than the reported values of z = 0.12 for 3 He- 4 He mixtures and z = 0.21 for the metamagnet Dy 3 Al 5 O 12 [Dysprosium Aluminum Garnet (DAG)] [75,76]. Therefore, it can be concluded that the order character of the Sm-A-Sm-C transition for the mixtures x 7OCB = 0.1, 0.125, 0.15, 0.158, 0.167 and 0.174 are truly tricritical in nature.…”

“…Hence, in the present system the extracted values of the quotient / -+ A A are non-universal over the variation of fit ranges. According to Fisher and Sarbach [75,76], this type of amplitude ratio at the tricritical point can be well described by an exactly solvable spherical (n = ∞) model, where / -+ A A is a function of the single variable…”

Bifurcation analysis and phase diagram of a spin-string model with buckled states

“…At the tricritical point K, the coefficient of A 3 in the bifurcation equation (26) vanishes. We can unfold this bifurcation by expanding the free energy up to O( 6 ) terms [32,33] and rescaling the bifurcation parameters. If we set…”

“…Again, C eq is found by looking for the extrema of ∆F , and C eq = O( ). By introducing C eq = A, we have that A is the solution of the biquadratic equation (33). Let us denote by A 2 ± the two solutions of Eq.…”