Structures and diffusion of two-dimensional dusty plasmas on one-dimensional periodic substrates

Using numerical simulations, we examine the structure and diffusion of a two-dimensional dusty plasma (2DDP) in the presence of a one-dimensional periodic substrate (1DPS) as a function of increasing substrate strength. Both the pair correlation function perpendicular to the substrate modulation and the mean-squared displacement (MSD) of dust particles are calculated. It is found that both the structure and dynamics of 2DDP exhibit strong anisotropic effects, due to the applied 1DPS. As the substrate strength increases from zero, the structure order of dusty plasma along each potential well of 1DPS increases first probably due to the competition between the interparticle interactions and the particle-substrate interactions, and then decreases gradually, which may be due to the reduced dimensionality and the enhanced fluctuations. The obtained MSD along potential wells of 1DPS clearly shows three processes of diffusion in our studied 2DDP. Between the initial ballistic and finally diffusive motion, there is the intermediate subdiffusion discovered here, which may result from the substrate-induced distortion of the caging dynamics.

Figure 1

Snapshots of the simulated dust particles positions (black dots) within different 1DPS (solid curves). When the depth of 1DPS is small, $U_0/k_{B}T=0.525$ (a), the distribution of dust particles is random with the typical liquid state. When the depth of 1DPS is larger, $U_0/k_{B}T=10.50$ (b), the dust particles are confined in the different columns provided by the 1DPS, forming several 1D or quasi-1D chains. For each panel, the inset on the bottom left corner is the 2D distribution function [22] $g(x,y)$. Note that only $\approx 36\%$ of the total simulated region is plotted here.

Figure 2

Averaged pair correlation functions $g\left(y\right)$ in the $y$ direction, parallel to the potential wells of 1DPS, for various depths of 1DPS, $U_0/k_{B}T=0$, 0.525, 1.575, 2.1, 2.625, 5.25, and 10.5, from bottom to top. The curves have been offset vertically for clarity. As the depth of the 1DPS $U_0$ increases gradually from zero, the number of peaks in $g\left(y\right)$ first increases and then gradually decreases as the structure transitions from the initial disordered liquid state to an intermediate modulated order state, followed by a reentrant melting transition into a modulated disordered state.

Figure 3

The averaged pair correlation $g\left(y\right)$ (symbols) along with fits (solid lines) to $g\left(y\right)-1\sim y^{-\gamma }$ for the peaks in $g\left(y\right)$. The curves have been offset vertically for clarity. (a) Substrates with $U_0/k_{B}T=0.525$, 2.625, and 10.50, from bottom to top. For the intermediate substrates with $U_0/k_{B}T=2.625$, we find that in this modulated ordered state $g\left(y\right)$ decays the most slowly with the smallest exponent of $\gamma =0.463$. (b) Substrates with $U_0/k_{B}T=52.5$, 105.0, and 157.5, from bottom to top. For strong substrates, the exponents are nearly constant, $\gamma \approx 0.63$, so that further increase in the strength of substrates has little effect on the structure in the $y$ direction.

Figure 4

The averaged structure factor $S\left(q\right)$ in the $y$ direction, along the potential wells of 1DPS, for $U_0/k_{B}T=10.5$ (squares), 5.25 (right triangles), 2.625 (diamonds), 2.1 (up triangles), 1.575 (down triangles), 0.525 (circles), and 0 (stars). We find the sharpest peaks for intermediate values of $U_0/k_{B}T$ ranging from 2.1 to 5.25, where the particles develop the strongest 1D ordering along the substrate troughs.

Figure 5

The heights of the first and second peaks in $g\left(y\right)$ and $S\left(q\right)$ for different depths of a 1DPS. We can clearly observe the values of peaks first increase, corresponding to the more ordered state, and then decrease gradually, corresponding to the reentrant melting into a modulated disorder state.

Figure 6

Mean-squared displacement (MSD) of different directions of 2D dusty plasmas for different depths of 1DPS. (a) MSD calculated from the motion in both the $x$ and $y$ directions. (b) MSD calculated from the motion in only the $x$ direction (XMSD), perpendicular to the potential wells. (c) MSD calculated from the motion in only the $y$ direction (YMSD), along the potential wells. In the $x$ direction, with stronger strength of 1DPS, the diffusion is completely constraint. In the $y$ direction, the diffusion is suppressed gradually by the 1DPS. When the strength of 1DPS is slightly larger enough, $U_0/k_{B}T\ge 2.625$, all YMSD curves overlap together, suggesting YMSD will not be substantially changed as the strength 1DPS increases more.

Figure 7

For the motion in the $y$ direction, the obtained exponent $\alpha \left(t\right)$ in the relation $\text{MSD}=4D{t}^{\alpha \left(t\right)}$ for different depths of 1DPS at different times. Here the exponent $\alpha \left(t\right)$ in the $y$ direction is calculated using the data from Fig. 6. The diffusion of 2DDP motion in the $y$ direction can be divided into three processes, due to the 1DPS. Initially, the exponent decays from 2, due to the ballistic motion of dust particles, since they are still inside the cages formed by neighboring dust particles. At the intermediate timescale, the exponent of diffusion is between 0.5 and 1, corresponding to the subdiffusion. This subdiffusion feature is obvious, even when the depth of the 1DPS is not so large, such as $U_0/k_{B}T\ge 1.575$. For the long timescale, the diffusion all changes to normal diffusion ($\alpha =1.0$) for different depths of 1DPS.