Shear modulus of two-dimensional Yukawa or dusty-plasma solids obtained from the viscoelasticity in the liquid state

Langevin dynamical simulations of two-dimensional (2D) Yukawa liquids are performed to investigate the shear modulus of 2D solid dusty plasmas. Using the known transverse sound speeds, we obtain a theoretical expression of the shear modulus of 2D Yukawa crystals as a function of the screening parameter $\kappa$, which can be used as the candidate of their shear modulus. The shear relaxation modulus $G\left(t\right)$ of 2D Yukawa liquids is calculated from the shear stress autocorrelation function, consisting of the kinetic, potential, and cross portions. Due to their viscoelasticity, 2D Yukawa liquids exhibit the typical elastic property when the time duration is much less than the Maxwell relaxation time. As a result, the infinite frequency shear modulus $G_{\infty }$, i.e., the shear relaxation modulus $G\left(t\right)$ when $t=0$, of a 2D Yukawa liquid should be related to the shear modulus of the corresponding quenched 2D Yukawa solid (with the same $\kappa$ value), with all particles suddenly frozen at their locations of the liquid state. It is found that the potential portion of the infinite frequency shear modulus for 2D Yukawa liquids at any temperature well agrees with the shear modulus of 2D Yukawa crystals with the same $\kappa$ obtained from the transverse sound speeds. Thus, we find that the shear modulus of 2D Yukawa solids can be obtained from the motion of individual particles of the corresponding Yukawa liquids using their viscoelastic property.

Figure 1

The shear modulus of Yukawa crystals as a function of the screening parameter $\kappa$, Eq. (3), derived from the transverse sound speeds in [36]. The star indicates the shear modulus reported in [3], obtained from the definition of the shear modulus when $\kappa =0.73$. The reported shear modulus in [3] agrees well with our obtained shear modulus results of Eq. (3). Later, this shear modulus expression of Eq. (3) can be used as the candidate of the shear modulus of 2D Yukawa crystals.

Figure 2

Shear relaxation modulus $G\left(t\right)$ of 2D dusty plasmas with a constant liquid temperature of $\mathrm{\Gamma }=68$, while $\kappa$ changes. Clearly, for 2D liquid dusty plasmas, the shear relaxation modulus $G\left(t\right)$ decays gradually to zero finally.

Figure 3

A comparison of the infinite frequency shear modulus $G_{\infty }$ from the simulation of 2D Yukawa liquids of $\mathrm{\Gamma }=68$ and the shear modulus of 2D Yukawa crystals of Eq. (3), for different $\kappa$ values. They follow the same trend, but with nearly a constant difference, no matter how $\kappa$ varies. This difference may come from the constant temperature of $\mathrm{\Gamma }=68$ in our simulations.

Figure 4

Three components in the infinite frequency shear modulus, obtained from the simulation of 2D Yukawa liquids of $\mathrm{\Gamma }=68$ as presented in Fig. 3. These three components are the kinetic portion $G_{\infty }^{kk}$, the potential portion $G_{\infty }^{pp}$, and the cross portion $G_{\infty }^{kp}$, respectively. Here, $G_{\infty }^{kp}$ is close to zero, which means that the correlation between the kinetic and the potential parts is negligible. Since in our simulation runs we specify a constant liquid temperature of $\mathrm{\Gamma }=68$, $G_{\infty }^{kk}$ is near unchanged for different $\kappa$ values. The $G_{\infty }^{pp}$ almost fall on the obtained candidate of the shear modulus of 2D Yukawa crystals of Eq. (3). This result clearly suggests that the potential portion in the infinite frequency shear modulus $G_{\infty }^{pp}$ of a Yukawa liquid should be equivalent to the shear modulus of the corresponding quenched solid.

Figure 5

The different components of the infinite frequency shear modulus for different higher temperatures of $\mathrm{\Gamma }=20$ (a) and $\mathrm{\Gamma }=8$ (b). At higher temperatures, the kinetic portion $G_{\infty }^{kk}$ becomes larger, even larger than the potential portion $G_{\infty }^{pp}$ for all data in (b). However, the results of the potential portion $G_{\infty }^{pp}$ still always fall around the solid curve, which is the obtained candidate of the shear modulus of the 2D Yukawa crystals of Eq. (3), well consistent with our finding of the shear modulus above.

Figure 6

The potential portion of the infinite frequency shear modulus $G_{\infty }^{pp}$, obtained from the simulation of 2D Yukawa liquids at various conditions, i.e., different values for the coupling parameter of $\mathrm{\Gamma }=8$ (square), 20 (circle), and 68 (triangle) and the melting points ${\mathrm{\Gamma }}_m$ (diamond), while the screening parameter $\kappa$ also changes. All these data well agree with the candidate of the shear modulus of 2D Yukawa crystals of Eq. (3). Thus, we can draw a conclusion that the shear modulus of 2D solid dusty plasmas can be obtained from the potential portion of the infinite frequency shear modulus $G_{\infty }^{pp}$ of 2D liquid dusty plasmas at any temperatures.