Structural Analysis of Strongly Coupled Dusty Plasma Using Molecular Dynamics Simulation

Abstract

Equilibrium molecular dynamics (EMD) simulation has been used to investigate structural behaviors (order-disorder structures) of three-dimensional (3D) strongly coupled dusty plasmas (SCDPs). The Yukawa (screened coulomb) potential and periodic boundary conditions (PBCs) have been used in the SCDPs algorithm. Two factors have been used to analyze the structural behavior of SCDP which are radial distribution function (RDF), and lattice correlation (LC). The results for these factors have been calculated in a canonical (NVT) ensemble at external electric field strength (E^* = 0.03) for different plasma conditions of Coulomb coupling (Γ) and Debye screening parameters (κ) at the number of particles (N = 500). Their results have shown that the 3D SCDP structure moves from a disordered to an ordered state with increasing Γ, and the long-range order moves to high Γ with an increase of κ. In comparison to earlier numerical, experimental, and theoretical data, the obtained results have been found to be more acceptable.

1. Introduction

Now, complex liquids have become a matter of interest for many researchers. Different techniques such as experimental, theoretical, and simulation are used to study the behavior of complex liquids. Unambiguous models have been used to describe physical properties for a specific range of temperature and pressure. Explicit equations can also be used to calculate the thermophysical properties of liquids when there is not any literature on complex liquids present. The thermophysical properties of complex liquids are altered with the change in temperature, pressure, and material composition, but the chemical properties stay unaffected. The complex liquids’ phase change is described by thermophysical properties. These are divided into two categories transport and thermodynamic properties. Thermodynamic properties explain the system equilibrium conditions which comprise of entropy, heat capacity, internal energy, pressure, density, and enthalpy while the transport properties contain thermal conductivity, viscosity, RDF, waves with its instabilities, and diffusion. These transport properties state the momentum and energy transfer to the system under concern. These properties help in system designing and have knowledge of the physical mechanism. These properties have many applications in industry and laboratory [1].

1.1 Plasma

About 99% of matter that exists in space is plasma and it is called the fourth state of matter. Mainly, plasma occurs in electrified gas form, where atoms are dissociated into positive ions and electrons. It is a form of matter in different areas of Physics such as technical plasma, terrestrial plasma, solid-state plasma, space plasma, and astrophysics. Plasma is produced artificially in laboratories used for numerous applications likely in a display, fluorescent lights, fusion energy research, and others. The term “plasma” was first time used by Irving Langmuir, who is an American physicist and he defined plasma as “plasma is a quasineutral gas of charged and neutral particles which shows collective behavior”. The neutral gas ionization contains equal numbers of positive and negative charge carriers (n_i ≈ n_e ≈ n). Where n_i is ion density, n_e is electron density, and n is number density. Such plasma is called “quasi-neutral”. Collective behavior means that plasma shows a simultaneous response of many particles to an external stimulus. The Debye spheres that surround small particles in plasma are distorted, which allow an externally applied electric field to regulate the interparticle interaction [2]. The main examples of plasma are planetary objects, nuclear fireballs, neon signs, plasma balls, and welding arcs. Plasma is widely used in the field of science and technology. It has a very important part in our daily life. It is used in gas lasers, sterilizing of medical instruments, industries, gas discharge, intense power beams, lightning, water purification plant, and many more [3].

1.2 Plasma history

Irving Langmuir American scientist defined plasma for the first time in 1922. Some academics also worked on plasma physics in 1930. Hans Alfven developed hydromagnetic waves in 1940; these waves are known as Alfven waves, and they were used to explore astrophysical plasma. Beginning around the same time in the Soviet Union, Britain, and the United States in 1950, magnetic fusion energy research was initiated. The study of magnetic fusion energy was regarded as a subfield of thermonuclear power in 1958. By using a Russian Tokomak design, plasma with various plasma properties was produced at the end of 1960. Numerous sophisticated tokamaks that were created between 1970 and 1980 validated the effectiveness of tokamaks. Additionally, the tokamak nearly achieved a fusion break in 1990, and the research on DP physics had begun. DP is defined as “when charged particles absorbed in plasma, becomes four components plasma containing ions, electrons, neutral and dust particles” and dust particles change the plasma properties which is called “dusty plasma” [4].

1.3 Types of plasma

The plasma can be categorized on the bases of the Coulomb coupling factor (Γ) into two categories.

1.3.1 Ideal plasma (weakly coupled plasma)

A plasma is called ideal plasma if Γ ∼ ${10}^{-4}\ll 1.$ It is a kind of plasma whose temperature is very high and density is low, and in this plasma average kinetic energy (K.E) is considerably more than the interaction potential energy (P.E) of particles, even if we can ignore this interaction P.E due to increase of average distance between the particles. It is also called as weakly coupled (ideal) plasma (WCP) because the Debye sphere is densely occupied. Because of small interaction, this plasma does not have any organization of particles or self-organization similar to plasma crystal, and plasma behaves as a nonideal gas. Its example is ordinary plasma and hot plasma [5].

1.3.2 Nonideal plasma (strongly coupled plasma)

A plasma is called nonideal plasma if Γ = ${10}^4\gg 1$. It is a kind of plasma whose temperature is very low and density is high and in this plasma average K.E is considerably small than the interaction P.E of particles. It is also called as SCP and cold plasma. They can have the structure of a liquid for weak to intermediate Γ values and the structure of a crystalline lattice or solid for higher Γ values. It appears in various physical systems such as electrons drifting of helium liquid on its surface, and astrophysical systems such as giant planetary interiors, neutron stars, ions in the interior of white dwarfs, dusty plasma, and cryogenic traps. An example is laboratory plasma [5].

1.4 Complex (dusty) plasmas (CDPs)

DP is a complex many-component plasmas containing ions, electrons, neutral particles, and dust particles. Some plasmas naturally contain dust particles or they may be introduced artificially through different mechanisms e.g., sputtering, etching, etc. CDPs contain dust particles of sizes ranging from tens of nanometers to hundreds of microns. The size of the particle is 3e⁻⁸g. DP properties become more complex when charged particles are absorbed in plasma and also when used on the laboratory scale. DP is called as CDP in similarity to the condensed matter field of “complex liquids” in soft matter (polymers, colloidal suspensions, surfactants, etc.). Studies of solitons, shock waves, crystallization and melting fronts, and crystal dislocations are just a few examples of the experiments that have been inspired by this concept. CDP, which can be utilized for similar reasons but differ from colloidal suspensions in having weak damping, make it possible to investigate various processes on their inherent dynamic time scale [6]. Dust particles can be in solid (metallic), liquid, gaseous, crystal, or dielectric form. On the basis of the ordering of many radii and characteristic lengths between particles interacting (r_d, λ_D), plasma with dust particles can be called either “dusty plasma” or “dust in plasma”. If the λ_D > r_d then it is called “dusty plasma” and if λ_D < r_d then it is called “dust in plasma”. Here λ_D is dust particles Debye length and r_d is the interparticle distance [4, 7].

1.5 History of dusty plasma

In 1924, the term plasma was first time defined by Irving Langmuir. In 1980, a very exciting incident happened in the field of DP for the Saturn ring and in laboratory measure of DP dynamical structure parameter was performed. Hannes Alfven and Lyman Spitzer proved dust particles as an important component and images from the Saturn ring have shown that dust particles are rotating around the Saturn ring, having the shape of spokes. The Ulysses European spacecraft passed by Jupiter in 1992, detecting dust particles and measuring their weights and collision speeds. In 1995, NASA’s Galileo spacecraft detected the source of streams of dust around Jupiter. 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 dynamic and physical behaviors. Other displays of DP were noctilucent clouds, origin nebula, zodiac light, etc. The research from the last analysis has shown that these dust particles are fine particles. The present condition (2000–2017) of DP is stable and is playing a very dominant role in science, industries, technology, energy sectors, and medical stores [8].

1.6 Charge on dust particle

Dust particles have almost e⁴ order of charge. Their charge is mostly negative in low-temperature laboratory plasma but depending on the charging process charge may be positive. Charging of dust particles can be drained through different processes involving the background plasma (electron and ion) being bombarded on the surface of dust particles, secondary electron production, ion sputtering, photoelectronic emission by UV radiations, etc. When an electron from the background plasma strikes the surface of a dust particle, it gets an electric charge. As electrons are more mobile than ions, the surface dust particle collects electrons, attracts ions, and repels electrons until the state of immobility is attained. Other collective phenomena and wave instabilities are created due to interactions between these particles. This charge is accountable for a long lifetime of particles and confinement in plasma. Dust particle potential is reliant on some parameters such as grain composition, grain size, plasma condition, temperature, and velocity [7].

1.7 Formation of dusty plasma in laboratory

In-lab manufacturing of DP has been expanded using a variety of techniques. Various techniques, including modified Q machines, DC discharges, and RF discharges, have been applied recently. The modified Q machine, a single-ended device used to produce DP, permits the dust particles to disperse throughout a portion of the cylindrical plasma column. Dust particles smaller than a micron form DP in the stratum of dc neon glow discharge. Cold electrodes in a glass tube with a cylinder shape are used to create the discharge. The electrodes are 40 cm apart. Neon pressure varies from 0.2 to 1 Torr, while discharge current varies from 0.4 to 2.5 mA. In a symmetric cylindrical rf plasma system, DP is restrained. The grounded electrodes, the hollow outer electrode, capacitively connected to a 14 MHz rf power amplifier, and the glass window make up the system of RF discharges. Thermonuclear fireballs, dust precipitators, rocket exhaust, dust in space stations, and laboratory-generated DP are examples of man-made DP. Microelectronics fabrication, Plasma Enhanced Chemical Vapor Deposition, flat-panel displays, solar cells, and semiconductor chips are among further uses for DP. DP devices (DPDs) are used in laboratories to create DP. Ordinary flames, candle flames, intense desire, fire, and blaze created by burning gas are essentially weakly ionized plasma with dust particles. The degree of ionization in typical hydrocarbon burns is increased by the thermionic electron emission of 10 nm dust particles. DP also contains in-charged snow and volcanos [9, 10].

2. Methodology

To evaluate the interactions and movements of atoms and molecules at a given time period and to investigate the time relating role of the molecular system a computational method MD simulation is used. It is reliant on Newton’s second law of motion, stated as F = ma. 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. In this study, each simulation has considered 500 particles for a face-centered cubic (fcc) lattice in an NVT ensemble. PBCs are imposed on this lattice. Particles interact with each other through Yukawa (screened Coulomb) potential. When there is no E^* Yukawa potential which is an ideal potential for CDPs study is given as

$$\varphi \left(\left|\mathbf{r}\right|\right)=\frac{Q^2}{4\mathrm{\pi }{\epsilon }_0}\frac{e^{-\left|\mathbf{r}\right|/{\mathrm{\lambda }}_{\mathrm{D}}}}{\left|\mathbf{r}\right|}$$E1

In the above equation, r represents interparticle distance, λ_D is Debye screening length, Q is a charge on dust grains and ε_o is the permittivity of free space [5, 11].

However, particle interaction occurs under the effect of E^* whose value is 0.03, which causes electric force which is given as $F_e=Q_{d}E$. This force causes the drifting of negative dust particles. Basically, there are four forces which are the force of gravity, the electric force, the neutral drag force, and the thermophoretic force acting on dust particles in the direction of E^*. In this work, we have assumed that E^* makes chains or strings of plasma causing structural order resulting in plasma crystal is obtained. The drift velocity due to dust particles flow is given as v_d = $E/\mathit{mv}$, where m indicates mass and $v$ is the frequency of collision with neutrals. As E^* acts along the z-direction whose z-component is given as

$$E=\left(0,0,\mathit{Ez}\right)=\mathit{eEr}cos\theta$$E2

where z indicates unit vector direction along the z-axis and $\theta$ gives an idea of the direction of E^*. Now, there are two terms included in interaction potential which are Yukawa potential which is repulsive interaction, and electric potential whose interaction is attractive [12].

$$\varphi \left(\left|\mathbf{r}\right|,\theta \right)=\frac{Q^2}{4\mathrm{\pi }{\epsilon }_0}\frac{e^{-\left|\mathbf{r}\right|/{\mathrm{\lambda }}_{\mathrm{D}}}}{\left|\mathbf{r}\right|}-{\mathit{eE}}_{z}rcos\theta$$E3

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:

$$\mathrm{\Gamma }=\left(\frac{Q^2}{4{\mathit{\pi \epsilon }}_o}\frac{1}{a_{\mathit{WS}}{k}_{B}T}\right)$$E4

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 the number density (n = N/V) of dust particle. Debye screening strength is the ratio of the interparticle distance a (Wigner Seitz radius) to the Debye length ${\mathrm{\lambda }}_{\mathrm{D}}$ [1, 5, 8]. The value of κ can be represented in this form:

$$\kappa =\frac{a}{{\lambda }_D}$$E5

In this velocity verlet algorithm is used to allot initial values of acceleration and velocity to the atom. It is a 3D Ewald summation method for calculation of RDF, and LC. This method is used for long-range interactions in periodic systems [13]. In this section, the EMD simulation is described for RDF, and LC of SCDP for an extensive range of plasma parameters of (1 $\le$κ$\le$ 3) and (100 $\le$ Г $\le$ 5) along with E^* = 0.03 for N = 500.

The lack of any permanent structure characterizes the fluid condition. Nonetheless, there are well-structural correlations that may be studied experimentally to reveal essential insights regarding molecular structure. For systems that are spatially homogeneous, simply relative separation is significant, causing a sum over atom pairs,

$$g\left(r\right)=\frac{2V}{N_m^2}<\sum _{i<j}\mathrm{\delta }\left(r-r_{\mathit{ij}}\right)>$$E6

and it is possible to average the function over angles without information loss. The g(r) – RDF is defined as a function that defines the spherically averaged local grouping around some certain atom. The description of g(r) suggests that ρg(r) dr is related to the probability of locating an atom in the dr volume element at a distance r from a certain atom, and in 3D, 4πρg(r)r²Δr is the atoms’ mean number in a shell of thickness Δr and radius r surrounding atom [14]. The RDF of the system of particles (dust grains) in equilibrium shows numerous sharp peaks, which further supports the medium’s crystalline state [15].

LC gives an idea of structural behavior (order-disorder (OD) structure). In this, we used the EMD technique for the simulation of long-term order and Yukawa potential. In this, we will find density at a point r by the following formula

$$\mathrm{\rho }\left(r\right)=\sum _{j=1}^N\mathrm{\delta }\left(r-r_j\right)$$E7

Long-term crystalline order will be calculated using Fourier transform

$$\mathrm{\psi }\left(\mathrm{\kappa }\right)=\frac{1}{N}\sum _{i=1}^{N}exp\left(-\mathit{ik}.r_j\right)$$E8

The 3D Yukawa liquids (DP) motion is simulated by means of the above equation and SCCDPs OD structures can be found, and in this equation k denotes the OD position reciprocal lattice vector (RLV) of the existing Yukawa system. Here k is given as k = 2π/(1,-1,1)λ, when the structure is fcc then k is RLV and has been used in our EMD numerical code, where edge length is denoted by λ and k* = 2a_wsπ/λ is the normalized value of k. The value of LC |Ψ(k)| ∼ 1 if there is a completely ordered state of the Yukawa system, then, there is a strongly coupled system. On the other side, when the LC |Ψ(k)| ∼ 0, there is a disordered state of the Yukawa system, that is, a gas nonideal state. Between upper and lower bounds 0 < |Ψ(k)| < 1 the Yukawa system rests more possibly overhead the intermediary line, then, there is a moderately ordered state of the system. It is significant that the disordered state disappears with |Ψ(k)| = O(N^−1/2) [5, 14].

3. EMD simulation results

3.1 Lattice correlation as a function of time Ψ(t)

3.1.1 Radial distribution function as a function of distance g(r)

Computational results using EMD simulation have shown in Figure 1 for SCDP LC, Ψ(t) as a function of time at κ = 1, 2, and 3 for four different values of Γ = 5, 20, 50, and 100 and system size is N = 500 and external electric field E^* = 0.03 which is normalized. Computational results using EMD simulation have shown in Figure 2 for SCDP RDF, g(r) as a function of distance at κ = 1, 2, and 3 for four different values of Γ = 5, 20, 50, and 100 and system size is N = 500 and E^* = 0.03 which is normalized and windows within graph have shown elaborated peaks. It has been obtained from Figure 1 that long-range order moves to high Γ with increasing κ and for moderate and larger values of Γ, a moderate to high ordered structure is obtained. The collective effect is one of the reasons for the larger values of Γ (e.g., Γ> > 100), and the motion of particles is weak sufficiently that particles incline to stay caged or trapped near-equilibrium points which are centered among near neighbors and a crystalline-like lattice is formed by self-organizing of particles [5, 16, 17]. It has been obtained from Figure 2 that κ value decreases no of peaks for moderate values of Γ which means structure shifts to disorderness. The high peak has been obtained at a small distance and the number of peaks decreases with increasing distance. It has been obtained that for moderate to higher coulomb coupling values, a moderate to high degree of order occurs as a greater no of sharp and fine peaks are formed. Because of repulsion among the particles, there is “correlation hole” for function g(r) at short separations, which prohibits it from getting to close due to the absence of an adequate amount of K.E. At a distance equivalent to the most likely first neighbor distance, the first coordinating shell is marked by a peak which is exhibited by g(r). This is a high point. After then, there’s a depleted region and a series of other coordinated shells, which emerge in g(r) in the form of smaller amplitude peaks due to correlations decay above longer distances. Fluids have a finite spatial organization across a finite distance the fluid “keeps” the lattice structure within this distance. Because of this, the g(r) peak positions of g(r) vary only to some extent with an increase of Γ. Because of the system’s significant structure for large Γ, the particles oscillate in localized potential wells of the potential surface formed by other particles, which is known as quasi-localization. The change in the potential surface owing to particle diffusion is substantially slower as compared to the time scale of these oscillations for (Γ> > 1). The quasi-localized charge approximation is based on this behavior, with the dynamical features extracted from the interaction potential and the pair correlation function [18, 19, 20]. Screening effects have an important role in electron-electron interaction potential and atomic systems show more stability when the particular screening effects are considered [21].

Figure 1.

The variation of LC, Ψ(t) as a function of time of SCDP at system size N = 500, E^* = 0.03, and Γ = 5, 20, 50, and 100 for three different plasma factors (a) κ = 1, (b) κ = 2, and (c) κ = 3.

Figure 2.

The variation of RDF, g(r) as a function of distance of SCDP at system size N = 500, E^* = 0.03, and Γ = 5, 20, 50, and 100 for three different plasma factors (a) κ = 1, (b) κ = 2, and (c) κ = 3.

4. Conclusions

Two factors LC and RDF have been used for the structural analysis of 3D SCDPs, for a wide range of plasma parameters (Γ, κ) at E^* = 0.03 and N = 500 using EMD simulation. It has been obtained that the SCDP structure moves from disordered or moderate to completely ordered conditions with increasing Γ, and the long-range order moves to high Γ with an increase of κ. The presented EMD method has an outstanding performance; its consistency and accuracy are comparable to those of previous findings.

Abbreviations

DP	Dusty plasma
CDP	Complex dusty plasma
EMD	Equilibrium molecular dynamics
Г	Coulomb coupling
κ	Debye screening strength
MD	Molecular dynamics
PBCs	Periodic boundary conditions
SCDPs	Strongly coupled dusty plasmas
WCP	Weakly coupled plasma
E^*	External electric field
SCP	Strongly coupled plasma
LC	Lattice correlation
RDF	Radial distribution function
PBC	Periodic boundary conditions
3D	Three-dimensional
NVT	Canonical ensemble
N	Number of particles
K.E.	Kinetic energy
P.E.	Potential energy
fcc	Face-centered cubic
OD	Order-disorder
RLV	Reciprocal lattice vector