Universal scaling of transverse sound speed and its isomorphic property in Yukawa fluids

Molecular dynamical simulations are performed to investigate the scaling of the transverse sound speed in two-dimensional (2D) and 3D Yukawa fluids. From the calculated diagnostics of the radial distribution function, the mean-squared displacement, and the Pearson correlation coefficient, the approximate isomorphic curves for 2D and 3D liquidlike Yukawa systems are obtained. It is found that the structure and dynamics of 2D and 3D liquidlike Yukawa systems exhibit the isomorphic property under the conditions of the same relative coupling parameter $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=\mathrm{const}$. It is demonstrated that the reduced transverse sound speed, i.e., the ratio of the transverse sound speed to the thermal speed, is an isomorph invariant, which is a quasiuniversal function of $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$. The obtained isomorph invariant of the reduced transverse sound speed can be useful to estimate the transverse sound speed, or determine the coupling strength, with applications to dusty (complex) plasma or colloidal systems.

Figure 1

Calculated RDF $g\left(r\right)$ for (a) 2D and (b) 3D Yukawa fluids, for the same relative coupling parameter of $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.50$. Clearly, for either 2D or 3D Yukawa fluids, the obtained RDF $g\left(r\right)$ with different $\kappa$ values nearly collapse to one universal curve in reduced units, implying the same translational structure. Note that the obtained RDF results for 2D Yukawa fluids exhibit more substantial oscillations at larger $r$ values, as compared to 3D Yukawa fluids.

Figure 2

Calculated $\mathrm{MSD}\left(\tau \right)$ for (a) 2D and (b) 3D Yukawa fluids, for the same relative coupling parameter of $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.50$. For either 2D or 3D Yukawa fluids, all obtained MSD results with different $\kappa$ values nearly collapse to one universal curve in reduced units, implying the invariant dynamical property. The dashed and dot-dashed lines correspond to the scaling exponents of the ballistic and diffusive motions, respectively.

Figure 3

Calculated Pearson correlation coefficient $R$ between the virial and the potential energy for 2D liquidlike Yukawa systems under various conditions. All of the $R$ values are larger than 0.9, clearly indicating that 2D liquid-like Yukawa systems are the Rosklide-simple systems.

Figure 4

Calculated reduced instantaneous transverse sound speed $C_T/v_{\text{th}}$ for (a) 2D and (b) 3D Yukawa fluids under various conditions. For fixed $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$ values, the obtained $C_T/v_{\text{th}}$ values are nearly the same, clearly indicating that $C_T/v_{\text{th}}$ is an isomorph invariant. For each $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$ value, the average of the obtained $C_T/v_{\text{th}}$ values under various $\kappa$ values is shown as the dashed line. Note, when $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.05$, the obtained $C_T/v_{\text{th}}$ values for both 2D and 3D Yukawa fluids are close to 1.4.

Figure 5

Calculated dispersion relations of the transverse mode in the 2D Yukawa fluid at $\kappa =0.5$ for (a) $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.98$ and (b) 0.50. For each wave number $k$, the corresponding frequency spectrum Eq. (7) is fit to the damped harmonic oscillator (DHO) model first, resulting in the peak frequency ${\omega }_T$ denoted by the cross symbol. For comparison, the frequency corresponding to the highest intensity for each wave number in the frequency spectrum of Eq. (7) is denoted by the plus symbol. The obtained dependence ${\omega }_T\left(k\right)$ is fitted by a linear function as the solid line shown. The slope of this line is identified as the effective transverse sound speed $C_T^{★}$.

Figure 6

Calculated reduced effective transverse sound speed $C_T^{★}/v_{\text{th}}$ from the transverse mode dispersion relation for 2D Yukawa fluids under various conditions. For the same relative coupling parameter $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$, all obtained $C_T^{★}/v_{\text{th}}$ values are almost the same, suggesting that $C_T^{★}/v_{\text{th}}$ is an isomorph invariant. For each relative coupling parameter, the dashed line corresponds to the averaged value of $C_T^{★}/v_{\text{th}}$ for different $\kappa$ values, while the dotted line just corresponds to the averaged value of $C_T/v_{\text{th}}$ obtained from the dashed line in Fig. 4, respectively. The average values of $C_T^{★}/v_{\text{th}}$ here are slightly different from those in Fig. 4. The error bars come from the linear fitting of the dispersion relation.

Figure 7

Reduced instantaneous transverse sound speed $C_T/v_{\text{th}}$ for (a) 2D and (b) 3D Yukawa fluids, as the relative coupling parameter $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$ varies. The plus and cross symbols correspond to the averaged values of $C_T/v_{\text{th}}$ for 2D and 3D Yukawa fluids with varying $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$, obtained from the locations of the dashed lines in Figs. 4 and 4, respectively. The dashed curves are fittings of Eq. (9), with the fitting coefficients of ${\delta }_{2\mathrm{D}}\simeq 19.8$ and ${\delta }_{3\mathrm{D}}\simeq 22.3$.