Original English dissertation · scientific equations reproduced directly from the source PDF.

4.5 The crossover theory in ternary mixtures

ηbg : background viscosity; qc : cut off wave number; ˜qc : 1 qc + 1 2·qD; ΩK(x) : Kawasaki function Eq.(4.34)); ξ : correlation length.

Another important consequence resulting from crossover corrections is the significant in- fluence 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 temperature eε, which is given by:

with the parameter β = 1.18. The significance of the cut-off wave number qc has been demonstrated in [70]; due to a large value of qc it was found |eε−ε|/ε < 5·10−3.

4.5 The crossover theory in ternary mixtures

The crossover theory has been developed for binary mixtures. It is not self-evident that this theory is also valid for ternary mixtures. Moreover, the renormalization group ε- expansion predicts a dependence of the critical exponent of viscosity Zη upon the dimen- sion of the considered system, [71]:

with ε (note, in this case ε is not the reduced temperature), the critical dimension dc = 4, and ε = dc −d, where d is the dimension of the system. Consequently, Eq.(4.37) predicts the value ε/19 for a 3d-Ising system, in first order for Zη. Taking into account the second order corrections, Zη = 0.065 results. This value has been predicted by the mode-coupling theory. In the case of tricritical point the critical dimension is reduced to dc = 3, as has been given by Pfeuty [72]. As a consequence, ε = 0 and therefore Zη = 0. It seems that there is no critical divergence of the viscosity. However, the ternary system, that has been investigated it this work is of type 2a (see Section (2.4.2)) and belongs to the same universality class for dynamical properties as the binary fluids. Thus, there is no reason to treat the experimental data different from the binary mixtures. Furthermore, it can be expected, that the crossover corrections are likewise valid.

View original page 67 (authoritative equations, figures and layout)