2018-(I)-Critical Contribution, Dynamic Scaling and Crossover Theory
Article Index
The crossover theory in binary mixtures (General conceptions of crossover theory):
The classical theories by van der Waals, Bertholt, and Dieterici describe pretty well the hydrodynamic and thermodynamic behavior of classical fluids in the mean-field region. Moreover, all their classical equations show the existence of a critical point. Unfortunately, they do not predict the non-analytic behavior in real systems. This fluid domain has been treated by Wilson, Fisher and Wagner within the framework of the renormalization-group theory. The conceptions and formalisms of this theory lead to a description in terms of scaling laws near the consolute point. Furthermore, renormalization-group theory has been quite successful in calculations and predictions of critical exponents. However, due to crossover effects this theory has a limited range of validity.
Unfortunately, the theoretical descriptions are valid only in a range extremely close to the critical point when ε → 0. There is no conception of extrapolation from the mean-field to the critical region. This so-called crossover-range has been reated 1986 in a paper by Albright at al, consistent with the renormalization-group theory. Their crossover descriptions for properties of fluids take into account, that besides the contributions from critical fluctuations to the critical behavior, there are further degrees of freedom, such as changes of molecular conformations and of the extent of hydrogen bonding. These kinds of effects do not couple to the critical fluctuations. Therefore, it is adequate to divide the modes of the considered liquid system into those that show only short range order and high frequency fluctuations and are only weakly coupled and into such which show long range fluctuations and are coupled strongly. These last ones lead to non-analytic behavior near the consolute point. Consequently, the existence of a cut-off Λ in the wave numbers of fluctuations, has to be taken into account when fluctuation dominated behavior of the system is studied. This view has been successfully applied to the van der Waals gas.
The Crossover corrections:
In the case of the dynamic light scattering, the shear viscosity as well as ultrasonic attenuation spectroscopy crossover corrections have to be taken into account when T is not sufficiently close to T. It has been shown, that close to the critical
point, in the asymptotic limit, the shear viscosity can be described by the expression:
here Q0 denotes the system-dependent critical amplitude and ξ the fluctuation correlation length. The background viscosity ηbg is given by the relation:
with the system specific parameters A_η , B_η, and T_η and with the absolute temperature T. The inverse critical amplitude of the viscosity Q−1, can be written as:
where qc and qD are the noncritical cut-off wave numbers. Eq.(4.28) is correct only in a region close to the critical point. Therefore, when treating data over a large temperature range, it is essential to consider the crossover corrections as has been presented by Burstyn at al. In that paper Burstyn et al. introduce a crossover function H (ξ,q_c ,q_D,), which is also dependent on the noncritical cut-off wave numbers q_c ,q_D as well as the correlation length ξ:
For large ε, the crossover function behaves as H (ξ(ε),q_D,q_c) → 0, so that η → ηbg. In the asymptotic limit the equation simplifies to the power law in. The influence of the crossover function is not only restricted to the shear viscosity. The cut-off wave numbers qc and qD play also an important role in the mutual diffusion coefficient, which is given by:
The value ∆D represents the singular contribution, which is shaped by the Kawasaki function ΩK:
with x = qξ. The mutual diffusion is then represented by:
Another important consequence resulting from crossover corrections is the significant influence of the values ξ0 and qc on the reduced temperature ε. Bhattacharjee and Ferrell have presented a correction expression that is based on the use of an effective reduced e temperature ε, which is given by:
with the parameter β = 1.18.