Frenkel line

The Frenkel line is a line of change of thermodynamics, dynamics and structure of fluids. Below the Frenkel line the fluids are "rigid" and "solid-like", whereas above it fluids are "soft" and "gas-like".

Two types of approaches to the behavior of liquids are present in the literature. The most common one is due to van der Waals. It treats the liquids as dense structureless gases. Although this approach allows one to explain many principal features of fluids, in particular, the liquid-gas phase transition, it fails to explain other important issues such as, for example, the existence in liquids of transverse collective excitations such as phonons.

Another approach to fluid properties was proposed by Yakov Frenkel. It is based on the assumption that at moderate temperatures the particles of liquid behave in a similar manner as a crystal, i.e. the particles demonstrate oscillatory motions. However, while in crystal they oscillate around theirs nodes, in liquids after several periods the particles change the nodes. This approach is based on postulation of some similarity between crystals and liquids, providing insight into many important properties of the latter: transverse collective excitations, large heat capacity and so on.

From the discussion above one can see that the microscopic behavior of particles of moderate and high temperature fluids is qualitatively different. If one heats up a fluid from a temperature close to the melting point up to some high temperature, a crossover from the solid-like to gas-like regime appears. The line of this crossover was named Frenkel line after Yakov Frenkel.

Several methods to locate the Frenkel line were proposed in the literature. The exact criterion of Frenkel line is the one based on comparison of characteristic times in fluids. One can define a 'jump time' via

where ${\displaystyle a}$ is the particles size and ${\displaystyle D}$ is the diffusion coefficient. This is the time necessary for a particle to move a distance comparable to its own size. The second characteristic time corresponds to the shortest period of transverse oscillations of particles within the fluid, ${\displaystyle \tau ^{*}}$. When these two time scales are roughly equal one cannot distinguish between the oscillations of the particles and theirs jumps to another position. Thus the criterion for Frenkel line is given by ${\displaystyle \tau _{0}\approx \tau ^{*}}$.

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