2022: SklogWiki celebrates 15 years on-line

# Frenkel line

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Frenkel line is a line of change of microscopic dynamics of fluids. Below the Frenkel line the fluid are "rigid" and "solid-like" while above it fluids are "soft" and "gas-like".

Two types of approaches to the behavior of liquids 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 to explain many principle features of fluids, in particular, the liquid-gas phase transition, it fails in explanation of other important issues, such as, for example, existence in liquids of transverse collective excitations such as phonons.

Another approach to fluid properties was proposed by J. Frenkel [1]. It is based on an assumption that at moderate temperatures the particles of liquid behave similar to the case of 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 based on postulation of some similarity between crystals and liquids allows to explain many important properties of the later: transverse collective excitations, large hear 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 one 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 J. Frenkel.

Several methods to locate the Frenkel line were proposed in the literature. The most detailed reviews of the methods are given in Refs. [2], [3]. The exact criterion of Frenkel line is the one based on comparison of characteristic times in fluids. One can define a 'jump time' via :$\tau_0=\frac{a^2}{6D}$, where $a$ is the particles size and :$D$ - diffusion coefficient. This is the time necessary for a particle to move to it's own size. The second characteristic time is the shortest period of transverse oscillations of particles of fluid: $\tau^*$. When these two time scales become comparable one cannot distinguish the oscillations of the particles and theirs jumps to another position. Therefore the criterion for Frenkel line is given by $\tau_0 \approx \tau^*$.

There are several approximate criteria to locate the Frenkel line in $(P,T)$ Refs. [2], [3], [4]. One of these criteria is based on velocity autocorrelation function (vacf): below the Frenkel line vacf demonstrate oscillation behavior while above it vacfs monotonically decay to zero. The second criterion is based on the fact that at moderate temperature liquids can sustain transverse excitations which disappear on heating the liquid up. One more criterion is based on isochoric heat capacities measurements. The isochoric heat capacity per particle of a monatomic liquid close to the melting line is close to $3 k_B$ ( $k_B$ is Boltzmann constant). The contribution to the heat capacity of potential part of transverse excitations is $1 k_B$. Therefore at the Frenkel line where transverse excitations vanish the isochoric heat capacity per particle should be $c_V=2 k_B$.

Crossing the Frenkel line leads also to some structural changes in fluids [5].

Currently Frenkel lines of several model liquids (Lennard-Jones and soft spheres [2], [3], [4] and real ones (liquid iron [6], hydrogen [7], water [8], $CO_2$ [8], $CH_4$ [8] were reported in the literature.

## References

1. [J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, 1947]
2. V.V. Brazhkin, A.G. Lyapin, V.N. Ryzhov, K. Trachenko, Yu.D. Fomin, E.N. Tsiok, Phys. Usp. 55, 1061 (2012) Cite error: Invalid <ref> tag; name "ufn" defined multiple times with different content Cite error: Invalid <ref> tag; name "ufn" defined multiple times with different content
3. V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, and K. Trachenko, Phys. Rev. E 85, 031203 (2012) Cite error: Invalid <ref> tag; name "frpre" defined multiple times with different content Cite error: Invalid <ref> tag; name "frpre" defined multiple times with different content
4. V. V. Brazhkin, Yu. D. Fomin, A. G. Lyapin, V. N. Ryzhov, E. N. Tsiok, and Kostya Trachenko, Phys. Rev. Lett. 111, 145901 (2013) Cite error: Invalid <ref> tag; name "frprl" defined multiple times with different content
5. C. Yang, V. V. Brazhkin, M. T. Dove, and K. Trachenko, Phys. Rev. E, 91, 012112 (2015) Cite error: Invalid <ref> tag; name "kostya3" defined multiple times with different content Cite error: Invalid <ref> tag; name "kostya3" defined multiple times with different content