Cutoff wave number for shear waves and Maxwell relaxation time in Yukawa liquids

J. Goree, Z. Donkó, and P. Hartmann

Phys. Rev. E 85, 066401 – Published 11 June 2012

Abstract

Because liquids cannot resist shear except over very short distances comparable to the atomic spacing, shear sound waves (i.e., transverse phonons) propagate only for very short wavelengths. A measure of this limit is the cutoff wave number $k_{c}$ , which is sometimes called the critical wave number. Previously $k_{c}$ was determined in molecular dynamics (MD) simulations by obtaining the dispersion relation. Another approach is developed in this paper by identifying the wave number at the onset of a negative peak in the transverse current correlation function. This method is demonstrated using a three-dimensional MD simulation of a Yukawa fluid, which mimics dusty plasmas. In general, $k_{c}$ is an indicator of conditions where elastic and dissipative effects are approximately balanced. Additionally, the crossover frequency for the real and imaginary terms of the complex viscosity of a dusty plasma is obtained; this crossover frequency corresponds to the Maxwell relaxation time.

Received 12 December 2011

DOI:https://doi.org/10.1103/PhysRevE.85.066401

Authors & Affiliations

J. Goree

Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242, USA

Z. Donkó and P. Hartmann

Research Institute for Solid State Physics and Optics of the Hungarian Academy of Sciences, P. O. Box 49, H-1525 Budapest, Hungary

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Vol. 85, Iss. 6 — June 2012

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Figure 1
Example of dispersion relations of shear waves in a 3D Yukawa liquid. The curves for the dispersion relations were determined as the peaks of the phonon spectrum computed using our MD simulation, which is a method used by previous authors, e.g., [4]. Note the absence of a dispersion relation for very small wave numbers: this indicates the cutoff $k_{c}$ . Also shown for comparison is the dispersion relation for propagation along a primitive vector of a bcc lattice for the same $κ$ . The quantities shown are dimensionless; the wave number is normalized $\bar{k} = k a$ where $a$ is the Wigner-Seitz radius, and the frequency is normalized $\bar{ω} = ω / ω_{p}$ where $ω_{p}$ is the plasma frequency.Reuse & Permissions
Figure 2
(a) Example of transverse current autocorrelation function $C_{T}$ for a liquid at a temperature nearly twice as high as the melting point, for various wave numbers $k$ . The hydrodynamic limit is at small wave numbers while at larger wave numbers there are oscillations. (b) Method of determining the cutoff wave number $k_{c}$ . We measure the amplitude $A$ of negative peaks of $C_{T}$ in (a) for various values of $k$ , and plotting these we extrapolate to find the value of $k$ where the amplitude just touches a zero value, i.e., where the curve for $C_{T}$ would barely have a negative oscillation in the absence of noise; we report this value of $k$ as our measure of $k_{c}$ . The vertical axis of (a) is normalized by the value of the correlation function at zero time, $\bar{C} (t) = C (t) / C (0)$ .Reuse & Permissions
Figure 3
Results for cutoff wave number $k_{c}$ , determined from our MD simulation, measured as in Fig. 2 by identifying the smallest value of $k$ that would allow a negative oscillation of the transverse current correlation. Data are shown as a function of the Coulomb coupling parameter $Γ$ , which serves as a measure of inverse temperature, and they are normalized by the melting point $Γ_{melt}$ data from [17]. Generally $k_{c}$ decreases with $Γ$ , i.e., increases with temperature.Reuse & Permissions
Figure 4
Terms of the complex viscosity, where the real term $η^{'}$ indicates dissipation and the imaginary term $η^{''}$ indicates storage of energy. Data points are from the MD simulation. The real and imaginary terms cross over in this case at ${\bar{ω}}_{cross} = 0.240$ . We interpret this crossover of the real and imaginary terms as an indication of a balance of dissipation and elasticity. Also shown are curves for a fit to Eqs. (11) and (12) for the viscoelastic approximation, with fit parameters $\bar{η} = 0.157$ and $τ_{M} = 4.152 ω_{p}^{- 1}$ . Viscosity is normalized here as $\bar{η} = η / m n ω_{p} a^{2}$ .Reuse & Permissions
Figure 5
Crossover frequency $ω_{cross}$ , determined as in Fig. 4. Results are shown in (a) in our usual dimensionless variables. In (b) the frequency is normalized by the Einstein frequency $ω_{E}$ and the inverse temperature $Γ$ is normalized by the melting point, showing a similarity. Data points to the right of the peak are meaningful for viscoelastic effects.Reuse & Permissions
Figure 6
Results from Figs. 3 and 5 combined. Data for liquids are shown in (a) with our usual dimensionless units and in (b) with frequency normalized by the Einstein frequency. A crystalline solid would appear at the origin in this graph.Reuse & Permissions

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