Large Scale Simulations for Dust Acoustic Waves in Weakly Coupled Dusty Plasmas

Abstract

Dust acoustic wave of three-dimensional (3D) dusty plasmas (DPs) has been computed using equilibrium molecular dynamics (EMD) simulations for plasma parameters of Coulomb coupling strength (Γ) and Debye screening (κ). New simulations of wave properties such as longitudinal current correlation (LCC) CL(k, t) function have been investigated for 3D weakly DPs (WCDPs), for the first time. EMD results, CL (k, t) have been simulated for four normalized wave numbers (k = 0, 1, 2, and 3). Our simulations illustrate that the frequency and amplitude of oscillation vary with increasing of Γ and κ. Moreover, present simulations of CL (k, t) illustrate that the varying behavior has been observed for changing (Γ, κ) and system sizes (N). Current investigation illustrates that amplitude of wave oscillation increases with a decrease in Γ and N. However, there are slightly change in the value of CL (k, t) and its fluctuation increases with an increasing k. The obtained outcomes have found to be more acceptable than those that of previous numerical, theoretical, and experimental data. EMD simulation has been performed with an increasing sequence for WCDPs and it serves to benchmark improved approach for future energy generation applications.

1. Introduction

Nowadays, it has been seen complex liquids have attracted considerable interest for many researchers. Complex liquids have introduced themselves by emerging the technology of all processes. Different techniques such as experimental, theoretical, and simulation are used to study the behavior of the complex liquid. Unambiguous models have been used to describe physical properties for specific ranges of temperature and pressure. Explicit equations can also be used to calculate the thermophysical properties of fluids when there is not any literature on complex fluids present. To study the behavior of complex fluids, information on thermophysical properties should be required. Transport properties of complex fluids play an essential role in laboratory and industrial applications. Moreover, thermophysical properties have played a very dominant role in optimization and system design. For the past few years, research on transport properties has occurred through several techniques such as experimental, theoretical, and simulation. Experimental research on transport properties has resulted in non-absolutely convergent expression because there is no accurate and precise experimental calculation of transport properties [1].

1.1 Plasma

Plasma makes up 99% of the matter in space and is referred to as the fourth state of matter. In essence, plasma is a sort of electrified gas in which atoms have split into positive ions and electrons. It is a type of matter in a variety of branches of physics, including astrophysics, technical plasma, and terrestrial plasma. Plasma is created artificially in a lab and used for a variety of technical applications, including fusion energy research, display, fluorescent lighting, and more. The term “plasma” was initially used by American scientist Irving Langmuir, who defined it as “a quasi-neutral gas of charged particles that exhibits collective behavior.” When the number of ions equals the number of electrons (n_i ≈n_e ≈n), a gas is said to be “quasi-neutral,” meaning it becomes electrically neutral. Where n is the number density, n_i is the ion density, and n_e is the electron density. The electric field and coulomb potential cause charged particles to collide when they exhibit collective behavior. The usage of plasma in research and technology is widespread. It is extremely important in daily life. Plasma is employed in many aspects of daily life, including lasers, sterilizing medical equipment, lightning, high-intensity beams, purifying water, and many more [2].

1.2 History of plasma

Plasma was initially defined in 1922 by American scientist Irving Langmuir, who was the only one to do so. Various academics began studying plasma physics in 1930 after being motivated by some particle puzzles. Hanes Alfven developed hydromagnetic waves in 1940; these waves are known as Alfven waves. He also mentioned that astrophysical plasma would be studied using these waves. Beginning around the same time in the Soviet Union, Britain, and the United States in 1950, research on magnetic fusion energy was initiated. The study of magnetic fusion energy was regarded as a subfield of thermonuclear power in 1958. This study was initially conducted under strict confidentiality, but after it became clear that the military did not appreciate controlled fusion research, the above-mentioned three countries made the research publicly available. Because of this, additional nations may take part in plasma-based fusion research. By the end of 1960, a Russian Tokomak configuration produces plasma with various plasma characteristics. Numerous sophisticated tokamaks that were created between 1970 and 1980 validated the effectiveness of the tokamak. Additionally, the Tokamak experiment nearly succeeded in achieving fusion break, and in 1990, work on DP physics started. When charged particles are absorbed by plasma, they split into four different components: electrons, ions, neutral particles, and dust particles and dust particles change plasma properties which are called as “Dusty plasma” [3].

1.3 Dusty plasma

When charged particles are submerged in plasma, the plasma properties become more complicated. This causes the plasma to be referred to as dusty plasma, which is also known as complex plasma. DP typically contains an extra charged particle and is electron-ion plasma. DPs contain dust particles of sizes ranging from tens of nanometers to hundreds of microns. The size of the particle is 3e-8g. This charged component can also be thought of as a micron-sized dust particle. Ice or metallic particles could make up dust particles. Different sizes and types of dust particles can be found throughout the universe and in the atmosphere. Although it typically takes the form of a solid, it can also be liquid or gaseous. The movement of ions and electrons can charge up dust particles. The electric potential of charged dust particles, which ranges from 1 to 10 V, is altered by the electric and magnetic fields. In a lab, dust particles can be grown. The existence of dust particles in plasma has exposed advanced fields for researchers and scientists. The majority of plasma in the universe is DP. It occurs frequently in the atmospheres of stars, the sun, planetary rings, galaxies, cosmic radiation, and the earth’s magnetosphere and ionosphere. Based on the ordering of many radii and characteristic lengths between particles interacting (rd, λD), plasma with dust particles can be called either “dusty plasma” or “dust in plasma”. If the λD > rd then it is called “dusty plasma” and if λD < rd then it is called “dust in plasma”. Here, λD is dust particles’ Debye length and rd is the interparticle distance. These are the conditions for DP [3, 4].

1.3.1 History of dusty plasma

Plasma is a very advanced field of science and technology. Many researchers have been motivated by their achievements in the field of science, medical field, and technology. In 1924, the term “plasma” first time defined by Irving Langmuir. Contemporary research on DP also has motivated many researchers and has become an interesting field of science and technology. In 1980, very exciting incident happened in the field of DP for the Saturn ring. In 1997, Mendis discovered a bright comet by a distant ancestor. The “distant ancestor” is an extraordinary comic laboratory for the investigation and study of interactions between dust particles and their physical and dynamic behaviors. Further manifestations of DPs were noctilucent clouds, origin nebula (which can be seen from the naked eye), zodiac light, etc. Images that are taken from the Saturn ring have shown that dust particles rotating around the Saturn ring have shape of spokes. The research from the last analysis has shown that these dust particles are fine particles. In 2000–2017, the present condition of DP is stable and is playing a main dominant role in science, industries, technology, energy sectors, and medical stores [5].

1.3.2 Charge on dust particle

Charging of dust grains can be drained through different processes involving the background plasma (electron and ion) being bombarded on surface of dust particles, secondary electron production, ion sputtering, photoelectronic emission by UV radiations, etc. Mostly, the charge on dust particles is negative in low-temperature laboratory plasma. When an electron from the background plasma strikes the surface of dust particle and is attached and lost from the plasma. Because electrons are more moveable than ions, the surface dust grain collects electrons, attracts ions, and repels electrons until the state of immobility is attained. This charge is accountable for a long lifetime of particles and confinement in plasma [4]. Other collective phenomena and wave instabilities are created due to interactions between these particles.

1.4 Types of dusty plasma

Dusty plasma is classified by the Coulomb coupling parameter.

• Weakly coupled plasma

• Strongly coupled plasma

1.4.1 Weakly coupled plasma

Weakly coupled plasma (WCP) is described by Coulomb coupling parameter Г. Plasma is referred to as WCP when the coupling parameter’s value becomes negligible. Hot plasma is another name for WCP. Plasma is referred to be hot plasma or ideal plasma when the temperature of the electron reaches the same level as the temperature of the ion (Te ≅Ti). In a laboratory discharge tube with high gas pressure, hot plasma is produced. The flame, atmospheric arcs, and sparks are examples of hot plasma. WCDP lacks a defined shape due to the extremely weak interactions between interacting particles at high temperatures and low densities. The EMD simulation has been used to examine the occurrence of diffusion motion in WCDPs [6].

1.4.2 Strongly coupled plasma

Strongly coupled plasma (SCP) is defined by the Yukawa potential, also known as the screened coulomb coupling potential (Γ). Coulomb coupling potential (Γ) is the ratio of potential energy to kinetic energy. Whenever potential energy exceeds unity, then kinetic energy becomes as Γ > 1. Its mean SCP is also defined as cold plasma since potential energy increases from inter-particle kinetic energy and plasma particles change into crystalline form. Cold plasma is used for teeth sterilization and food processing. Charged particles in SCP are influenced by the electric field, but the influence of the magnetic field is ignored for this kind of cold plasma.

1.5 Formation of dusty plasma in the laboratory

Different methods have been expanded for the production of DP in laboratories. Modified Q machines, rf discharges, and dc discharges are different methods that have been used in the last few years. Modified Q machine is single-ended machine used for the production of DP which allows the dust grains to dispersion over a cylindrical plasma column portion. In dc neon glow discharge stratum, DP is produced by a micron-sized dust particle. The discharge is created with cold electrodes in cylindrical glass tube. The distance between electrodes is 40 cm. The neon pressure is varied from 0.2 to 1 Torr and the fluctuation of the discharge current is 0.4–2.5 mA. DP is restrained in a cylindrical symmetric rf plasma system. The system of rf discharges consists of grounded electrodes, hollow outer electrode capacitive coupled to a 14 MHz rf power amplifier, and glass window. Man-made DP or DP generated in laboratory is dust in fusion devices, rocket exhaust, dust in space stations, dust precipitators, and thermonuclear fireballs. Other applications of DP are microelectronics fabrications; plasma enhanced chemical vapor deposition (PECVD), flat panel displays, solar cells, semiconductor chips, dusty plasma devices (DPDs) are used to produce DP in the laboratory. Ordinary flames, flame of candlelight, strong passion, fire, and blaze produced by burning gas are basically weakly ionized plasma that contains dust particles. Thermionic electron emission of 10 nm dust particles increases the degree of ionization in ordinary hydrocarbon flames. DP also contains volcanos and charged snow [7].

1.6 Acoustic modes

Acoustic modes are categorized into two different types such as collision-less, uniform, and unmagnetized dusty plasma. These are dust acoustic (DA) and ion-acoustic (IA) waves.

1.6.1 Dust acoustic (DA) wave

Dust Acoustic Wave (DAW) is a low-frequency type of longitudinal wave. Basically, DAW is a sound wave when it travels through complex fluid which causes the oscillation between charged particles. The mass of dust particles is considered very important because it provides inertia which is used to sustain DAW, pressure of ions and electrons affords the restoring force. The dust acoustic wave is examined in DP and presented the dispersion relation using the Boltzmann distribution of ion and electron density [8].

1.6.2 Ion acoustic (IA) wave

The propagation of sound waves in ordinary gas is absurd due to the absence of collision between particles. Thus, in ionized gas, there is no possibility of the occurrence of sound waves due to the absence of collision. However, in the case of plasma, ions can transmit vibrations to each other due to their charges. IA wave is a constant velocity wave. The occurrence of IA wave depends upon the thermal motion of ions. It is a longitudinal wave in magnetized plasma. It demonstrates reflection, diffraction, and interference phenomena. The phase velocity of electron thermal speed is much greater than the IA wave. Particle displacement, particle velocity, sound intensity, and sound pressure are essential quantities of IA waves.

2. Models and numerical simulations

Interactions of the dust particle are studied by many mathematical models such as “in the center of potential well and repulsion potential, in rigid sphere”. There are a number of interaction potentials for the calculations of complex systems for instance DP. However, in the best of accurate experimental and theoretical balance, the Yukawa potential is more favorable interaction potential for WCDPs and is also used in many other physical systems (for instance, medicine and biology systems, chemicals, astrophysics environmental and physics of polymers, etc.) [9]. For charged particles, Yukawa potential has the following form [5]:

$$\varphi \left(\left|r\right|\right)=\frac{Q^2}{4\pi {\epsilon }_o}\frac{e^{\frac{-\left|r\right|}{{\lambda }_D}}}{\left|r\right|}$$E1

In the above equation, r represents inter-particle distance, Debye screening length is represented by ${\lambda }_D$, the permittivity of free space is represented by $\epsilon$₀, and charge on dust particle is represented by Q. System particle is expressed by plasma parameters. Coulomb coupling strength (Γ) is defined as the potential energy of interaction between particles and the average kinetic energy of the particles. The Γ defines the distribution of plasma. Γ is given in this form of:

$$\Gamma =\left(\frac{Q^2}{4\pi {\epsilon }_o}\frac{1}{a_{\mathit{WS}}{k}_{B}T}\right)$$E2

In Eq. (2), T is absolute temperature, k_B is Boltzmann constant, and a_ws represents Wigner Seitz (WS) radius and its value is (3/4πn)^1/3 with n representing number density (n = N/V) of dust particle. There is another parameter, k equal to 2π/L which affects the system of WCDPs. Where L represents the computational box length and k*= 2a_wsπ/L is the normalized value of k. The F_i= ${\sum }_{j}F$_ij is the force experienced on all dust particles. The force on ith particle is exerted by other particles. Debye screening strength is the ratio of the interparticle distance a (Wigner Seitz radius) to the Debye length${\mathrm{\lambda }}_{\mathrm{D}}$ [3, 5, 7, 10]. The value of κ can be represented in this form:

$$\kappa =\frac{a}{{\lambda }_D}$$E3

In this part, the EMD simulation is reported of C_L (k, t) of WCDP for large domain of plasma parameter of (0.9$\le$Г$\le$0.1) and (1$\le$κ$\le$3) along with the varying value of wave number (k = 0, 1, 2 and 3). The accuracy and correctness of our proposed model for DAWs C_L (k, t) is checked with different system sizes (N = 500, 2048, and 4000).

For the system of 3D Yukawa system, time scales in WCDP are categorized by the inverse of plasma frequency as (ω_p⁻¹) [1].

$${\omega }_p^{-1}={\left(\frac{n{Q}_d^2}{{\epsilon }_{o}m}\right)}^{\frac{-1}{2}}$$E4

In Eq. (4), dust particle mass and number density are represented by m and n. In EMD simulation, the number of particles is N=500 to 4000 (the system size will be increased on the validity of computational power) placed in computational box.

2.1 Current correlation function

Local density provides information about the distribution of atoms. It is possible to study the local variations in motion of atoms. The momentum and particle current for single atomic particles are calculated by molecular dynamics simulation [9].

$$\pi \left(r,t\right)=\sum _j{v}_j\delta \left(r-r_j\left(t\right)\right)$$E5

With Fourier transformation

$$\pi \left(r,t\right)=\sum _j{v}_j{e}^{-\mathit{ik}.r_j\left(t\right)}$$E6

Fourier transform equation is used to measure the LCC function.

$$C_{\mathit{\alpha \beta }}\left(k,t\right)=\frac{k^2}{N_m}\left({\pi }_{\alpha }\left(k,t\right){\pi }_{\beta }(-k,0)\right)$$E7

In Eq. (7), k is the wave vector. The equation given above can also be stated in form of the longitudinal and transverse current correlation function.

$$C_{\mathit{\alpha \beta }}\left(k,t\right)=\frac{k_{\alpha }{k}_{\beta }}{k^2}{C}_L\left(k,t\right)+\left({\delta }_{\mathit{\alpha \beta }}-\frac{k_{\alpha }{k}_{\beta }}{k^2}\right)C_L\left(k,t\right)$$E8

Longitudinal and transverse current functions are acquired in X, Y, and Z directions by locating k are equal to kẑ.

2.2 Normalized longitudinal current correlation function C_L (k, t)

Fourier transformation is used to write Eq. (7) into LCC functions which are normalized

$$C_L\left(k,t\right)=\frac{k^2}{N^m}⟨{\pi }_Z\left(k,t\right){\pi }_Z\left(-k,0\right)⟩$$E9

By putting π_z value in Eq. (9), we obtain

$$C_L\left(k,t\right)=\frac{k^2}{N^m}⟨\sum v_j{e}^{-\mathit{ik}.Z_j\left(t\right)}\left(k,t\right)\sum v_j{e}^{-\mathit{ik}.Z}(k,t)⟩$$E10

Eq. (10) explains charge particle transports in longitudinal motion. Longitudinal current correlation function describes that wave is propagating in negative and positive directions of the Z-axis.

2.3 Parameters and simulation techniques

This segment shows that system contains N = 500–4000 particles. Yukawa potential causes the interaction between these numbers of particles which is shown in Eq. (1) and contain in the cubic region. Moreover, this segment illustrates an overview of results attained through EMD simulation for longitudinal current correlation function which is shown in Eq. (10). The simulation takes place in simulation box and the simulation box dimension is selected as L_X, L_Y, and L_Z. In our case, we have used the EMD simulation to calculate the longitudinal current correlation function for N = 500 number of particles at plasma coupling parameter Г = (0.1–0.9) with κ = (1.4, 2.0 and 3.0) at four varying values of wave numbers k = (0, 1, 2 and 3). The EMD simulation has been used to investigate shear viscosity and dynamical structure S(k,ω) of SCDPs [5, 7, 10, 11, 12].

3. EMD simulation results

3.1 Time-dependent longitudinal current correlation function C_L (k, t), at κ = 1.4

3.2 Time-dependent longitudinal current correlation function C_L (k, t), at κ = 2.0

Computational results using EMD simulation have shown in Figures 1 and 2 for weakly coupled Yukawa system LCC, CL (k, t) as a function of time at κ (= 1.4, 2.0 and 3.0) for six different values of Г (= 0.1, 0.2, 0.3, 0.5, 0.7 and 0.9) and size of system is N=500 and four values of wavenumbers k (= 0, 1, 2 and 3) which are normalized. Figures 1 and 2 explain main results for every three different values of the Debye screening parameter (κ = 1.4, 2.0, 3.0) of 3D Yukawa complex liquids. It is obtained from six panels of Figure 1 that the amplitude and wavelength of LCC CL (k, t) increase with decrease of Г (0.9–0.1). It is also obtained that by increasing wave numbers k (0, 1, 2, and 3), the amplitude of longitudinal current increases. It is noted that the CL (k, t) value for k = 2 and 3 shows maximum sinusoidal wave behavior at lower value of Г from Figures 1a–d and 2a–c. The sinusoidal wave behavior decreases with an increase of Г and damping of the wave increases with an increase of time and Г.

Figure 1.

The fluctuation of C_L (k, t) as function of time of WCDP at system size N = 500, κ = 1.4, wave numbers (k = 0, 1, 2 and 3) for six different plasma parameters (a) Г = 0.1, (b) Г = 0.2, (c) Г = 0.3, (d) Г = 0.5, (e) Г = 0.7, and (f) Г = 0.9.

Figure 2.

The fluctuation of C_L (k, t) as function of time of WCDP at system size N = 500, κ = 2.0, wave numbers (k = 0, 1, 2 and 3) for six different plasma parameters (a) Г = 0.1, (b) Г = 0.2, (c) Г = 0.3, (d) Г = 0.5, (e) Г = 0.7, and (f) Г = 0.9.

4. Summaries

Time-dependent LCC function, C_L (k, t) has been studied for weakly coupled Yukawa system through EMD simulation over a wider range of plasma parameters (0.9$\le$ Γ $\le$0.1), and (1.4 $\le \kappa \le$ 3.0) along with varying wave numbers values (k = 0, 1, 2, 3). The behavior of the Weakly Coupled Yukawa system using EMD simulation has not been studied yet. It is observed that the amplitude and frequency C_L (k, t) of wave of oscillation increase when the value of Г (0.9–0.1) decreases. C_L (k, t) value also increases when wave number k increases. The amplitude of C_L (k, t) steadily decreases with an increase in the number of particles N. It is implied that the presented simulation is based on Yukawa potential which has been employed to investigate the propagation of waves in WCDP. The presented simulation provides a great understanding of the propagation of the wave in WCDP. The employed simulation affords further respectable data for propagation in WCDP. It has been observed from these simulations that waves are repeatedly propagated at intermediate and higher values of Γ. The frequency and amplitude of waves are examined at different mode of waves.

Acknowledgments

This work was partially sponsored by the National Research Program for Universities (NRPU) with (No. 20-15251/NRPU/R&D/HEC/2021) for utilization of manpower resources. Moreover, this work also was partially supported by the industrial collaborative work between Government College University Faisalabad (GCUF) and Metal Industries Development Complex (MIDC) Sialkot, TEVTA, Govt. of Punjab, Pakistan, as a position of Consultant in a proposed project, for providing facilities to test and run our computer Experiment.

Abbreviations

EMD	equilibrium molecular dynamics
Г	Coulomb coupling
κ	Debye screening strength
MD	molecular dynamics
PBCs	periodic boundary conditions
CL (k, t)	longitudinal current correlation function
DP	dusty plasma
EMD	equilibrium MD
LCC	longitudinal current correlation
k	wave number
WCDPs	weakly coupled dusty plasmas
WCP	weakly coupled plasma
SCP	strongly coupled plasma
DA	dust acoustic
IA	ion acoustic
DAW	DUST ACOUSTIC WAVE