Trigonometric models for scaling behavior near criticality

Parametric scaling representations of the asymptotic equation of state, con-elation lengths, and singular part of the Helmholtz free energy in the critical region of a system with a scalar order parameter are considered with the aim of fitting all 15 available independent universal amplitude ratios and describing naturally van der Waals loops. Defects in previous linear and cubic models and their extensions are described: they cannot represent analytically connected van der Waals loops or approximate satisfactorily all the known amplitude ratios. However, novel trigonometric parametric models for the free energy (and correlations) meet all the desiderata as demonstrated by the explicit numerical data presented.