**2. Numerical model and simulation techniques**

Secondly dust particles are formed in devices when atoms and molecules are spurted from walls and electrodes into the plasma by electron and ion bombardment. Moreover, the growth of dust particles in the plasma is coagulation, nucleation and surface growth. Thermal fluctuations and Coulomb interaction play the significant role for determination structure of SCCDPs. When the values of Г (>1) increase then system organized from nonideal gases phase to ordered condensed phase. Dust particles are suspended in the gaseous plasma phase with few electrons temperature

134 Plasma Science and Technology - Basic Fundamentals and Modern Applications

structural form even at room temperature. Dust particle has large mass as compared to ion and an electron which gives the results slowly downtime scale and it can easily observed the macroscopic structure and its dynamical behavior directly study in space and real time [10]. Dusty plasma used for nanocrystalline silicon particles grow in the silane plasmas used to increase efficiency and lifetime of the silicon solar cells. It is used for thin film coating applied in plasmaenhanced chemical vapor deposition (PECVD) for the improvement of material surface properties. Carbon-based nanostructure growth in the hydrocarbon plasmas or fluorocarbons used to produce thin carbon films. It is used to improve material properties such as chemical inertness, high hardness and wear resistance. Self-lubrication coating and wear resistance using differ-

). Ar/CH4

plasma) and dusty plasmas depositions techniques devices [11, 13].

diamond films fabricated which as exceptional properties such as chemical inertness, high hardness, and extreme smoothness which used to improve the performance of cutting tools. Diamond whiskers fabricated by the etching in RF plasmas for the enhancement of electron field emission. The reactive ion etching (RIE) process are used to a precise efficiently sharpen micro-tips of diamond [11]. Complex (dusty) plasmas (CDPs) have various advantages in a different industries, technologies, and energy sector due to the existence of dust particles. CDPs are stable under the laboratory condition. The CDPs can also be used for the diagnostic purpose because dust particles are trapped at the room temperature and keep their desire dynamical state for hours. Dusty plasma frequency in the range several hertz and easily observed through CCD cameras. It is produced in the gas discharge tube with natural gas pressure range that varies from 1 to 100 pa, which is subject to moderate damping [8]. Moreover, magnetized dustyplasma device used to produce a number of the verity of magnetic fields configuration with the help of four independent superconducting coils [12]. Magnetized glow discharge dusty plasma device, RF plasma device, ISS experiment and DC glow discharge devices used for different applications in industries and diagnostic purpose of dusty plasma. Dust particles are found in a tokamak (fusion

The dynamical structure factor *S*(*k*,ω)] gives the information about static and dynamic properties of the fluid in simple and complex systems. In hydrodynamic condition, the *S*(*k,*ω) provides experimental calculable quantities such as the thermal diffusivity, adiabatic sound velocity and the ratio of specific heats [8]. These properties of the fluids are measured through light scattering, x-ray and inelastic neutron experiments on a substance such as dense plasmas, liquids, and glasses. Sound waves are generated through *S*(*k,*ω) in strongly coupled CDPs (SCCDPs) and density is more strongly damped at liquid phase [14, 15]. The SCCDP is many body dynamical systems that show different collective excitations and their properties investigated through numerical simulation and theoretical approaches [16]. In condensed

ordered. Interestingly, the dust clouds in a dusty plasma formed into the

plasma used for making the nanocrystalline

and charge up to 104

ent compound as a dust particle (MoS<sup>2</sup>

**1.5. Dynamical structure factor**

In this section, we have implemented molecular dynamic (MD) simulation code with Ewald summation for forces and energies which makes it possible to account the long-range Coulomb interparticle interactions. We trace the motion of single charge species and integrated through leapfrog method and assume that the presence of neutralizing homogenous background. In this plasma environment, random fluctuating forces and friction forces are acting on a charged particle in addition to which forces initiating from the interaction of charged particle. Length of simulation cubic box is defined as ( \_\_\_\_ 4*N* <sup>3</sup> ) 1/3 and particles have a random spatial configuration for the beginning of simulation [15, 20]. Fluctuation of microscopic density is observed for different plasma parameters approaching near the equilibrium state [21]. The presented study includes the solution of the equation of motion of a system and particle interacts with each other through Yukawa potential. Provided that an accurate potential can be established for the system of attention under study and equilibrium MD (EMD) can be used irrespective of the phase condition and thermodynamic of the system involved. Yukawa potential is most commonly used potential (screened Coulomb) for SCCDPs including many physical systems such as physics of chemical and polymer, medicine and biology systems, astrophysics, environmental, etc. Major advantage for using this potential is that it reduces the calculation time compared other potentials [22]. The interaction potential energy of a charged particle in Yukawa liquid is given

$$\phi(\mathbf{r}) = \frac{Q^2}{4} \frac{e^{\frac{\mathbf{r}^\circ}{\hbar\_s}}}{\pi \varepsilon\_s} \frac{e^{\frac{\mathbf{r}^\circ}{\hbar\_s}}}{|\mathbf{r}|} \tag{1}$$

Here *Q* is the charge on dust particle, *r* is the distance between interacting particles, λD is Debye screening length that accounts for the screening of interaction of other plasma species and ε<sup>o</sup> is permittivity of free space. The scaling (dimensionless) parameters, which fully characterized the system, one is known as Coulomb coupling parameter [23],

$$
\Gamma = \frac{Q^2}{4\pi\varepsilon\_s} \frac{1}{a\_m k\_g T} \tag{2}
$$

The relation between static structure factors with the dynamic structure factor is

−∞ ∞ <sup>2</sup>*<sup>π</sup>* ∫ ‐∞ ∞

The dynamical structure function is related to longitudinal current correlation function,

*<sup>ω</sup>*<sup>2</sup> *CL*

The EMD simulation has been used for the calculation of *S*(*k,ω*) to understand the dynamic phenomenon of particles for 3D SCCDPs. We have analyzed our simulation results of *S*(*k,*ω) in term of frequency, amplitude and fluctuation rate with respect to these parameters (*κ*, Γ, and *k*). This section shows an overview of our results obtained through EMD simulations for dynamical structure factor *S*(*k*,ω) function of SCCDPs at *κ* = 1, 2, 3 and 5 with *N* = 500. **Figures 1**–**4** shown our results for *S*(*k*,ω) over a wide suitable range of plasma states (Г, *κ*) at four values of wave vectors *k* = (0, 1, 2 and 3). A sequence of dynamical structure factor in increasing of wave vector (*k*) is computed to determine the suitable equilibrium values of *S*(*k*,ω). We have performed 16 EMD simulation with *N* = 500 for each screening parameters at different four values of *k*. There are 64 simulations are carried out for different combination of plasma parameters in order to observe complete behaviors of DSF *S*(*k*,ω) at higher varying frequency (ωp) as compared to earlier simulation results [16]. It is observed that the presented results obtained through EMD simulations for varying parameters have suitable signal-to-noise ratio of the DSF. Our results are satisfactory good agreement with earlier EMD estimations and show that the presented EMD results at higher ωp and earlier EMD simulations have comparable performance with small system size, both yielding the close values of the DSF *S*(*k*,ω). Moreover, the system temperature (1/ Г), strength of Debye screening (*κ*), system run time (total time), system size (*N*), and wave vector (*k*) are changed to

observe how efficiently the presented EMD method computes the DSF *S*(*k*,ω) of SCCDPs.

In addition, in each panels of **Figures 1**–**4**, we have shown the behaviors of *S*(*k*, ω) at four values of *k*. All these data are excellent due to the good statistics and allow the analysis of the structure–function within a broad dynamical range. It is examined that the peaks of *S*(*k*,ω) are decay at lower Г values for different four values of *κ* and *k*. It can be noted that the peaks survive up to higher *k* values at intermediate to higher Г values and shift toward the higher frequency (ωp). It is further observed that the wave spectra of *S*(*k*,ω) shifts toward sinusoidal waves form, at the intermediate values of Г = 50 and wave spectra shifts toward square type waves forms at higher values of (Г = 100,200). Moreover, the panels (a) to (d) of each figures represent the results of *S*(*k*,ω) in nonideal gases state to liquid and crystalline order state

*F*(*k*, *t*) *ei<sup>t</sup> dt* (7)

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137

(*k*, *ω*) (9)

*S*(*k*, *ω*)*d* = *S*(*k*) (8)

Numerical Approach to Dynamical Structure Factor of Dusty Plasmas

S(*k*, *ω*) = \_\_\_<sup>1</sup>

*S*(*k*, *ω*) = \_\_\_<sup>1</sup>

This satisfies the sum rule as

which is expressed as [22].

**3. Results and discussion**

∫

where, *a*ws is the "Wigner-Seitz" radius and it is defined as *<sup>n</sup>* \_\_−1 2 with *n* is the equilibrium dust number density, *kB* is the Boltzmann constant and *T* is absolute system temperature. It is noted that the Γ is measured as the ratio of average potential energy to average kinetic energy per particle. Second scaling parameter is Debye screening parameter and it is given as *κ* = *a*ws/λD.

The EMD simulations are performed for a particle number that is chosen between 500 and 1000 particles in a microcanonical ensemble using periodic boundary conditions and minimum image convention of the dust particles. It is to be mentioned that the number density (*n*) is defined as *n* = *N*/*V,* here *N* is the number of particles and *V* is the volume of simulation box and it is calculated as *V =* 4π*N*/3. On our case, most of simulations are performed for *N* = 500 and it is observed that the mentioned number of particles is suitable for EMD computations with statistical uncertainty limits. In presented case, the simulation time step is d*t* = 0.0001 and total simulation time limit was 425,000 step units. The EMD simulations are run between 4.25 10<sup>6</sup> (1/ωp) to 3.25 10<sup>6</sup> (1/ ωp) time units for each combination of (Γ, *κ*) in the series of recording dynamical structure factor (DSF), *S*(*k*,ω), of SCCDPs. It can be seen that the first patch of *S*(*k*,ω) results is obtained after the time limit of (38000) step unit. For our case, 13 patches of *S*(*k,*ω) results are obtained and results show that each patch has nearly behavior of *S*(*k,*ω) showing the accuracy of numerical algorithm. In this work, the dynamical structure factor computations of SCCDPs are reported for a wide range of Coulomb coupling parameters Γ ≡ (1, 200) and the Debye screening strength *κ* ≡ (1.4, 4).

#### **2.1. Model of DSF [***S***(***k,***ω)]**

The number density of single species also known as mass density and dimensionless quantity in molecular dynamic units can be written as

$$\rho(r,t) = \sum\_{\uparrow} \delta(r - r\_{\uparrow}(t)) \tag{3}$$

here *r* is the point at time *t*. from the practical point of view, it is known as local density. The average occupancy in the small volume of *r* space and it calculates over the short interval of time. The definition of density is satisfied matter conserved requirement.

$$f\rho(r,t)dr = N\_w\tag{4}$$

The space and time-dependent density correlation are explain from the van Hove correlation function which is define as *G*(*r, t*) = *Gs* (*r, t*) + *Gs* (*r, t*) and further detail is given in Ref. [23]. Fourier transform of density fluctuation and new expression is given as

$$
\rho(\mathbf{k}, t) = f \rho(r, t) \, e^{-\mathbf{k} \cdot \mathbf{r}} \, dr = \sum\_{\uparrow} e^{-\mathbf{k} \cdot \mathbf{r} \prime 0} \tag{5}
$$

where *k* is the wave number and becomes equal to *k* = 2π/*L* and *L* is the length of simulation box. The intermediate function can be defined as

$$F(k,t) = \left\langle \rho(k,t)\rho(-k,t) \right\rangle \tag{6}$$

The relation between static structure factors with the dynamic structure factor is

$$\mathcal{S}(k,\omega) = \frac{1}{2\pi} \iint\_{\ast} F(k,t) \, e^{i\omega t} \, dt \tag{7}$$

This satisfies the sum rule as

<sup>Γ</sup> <sup>=</sup> *<sup>Q</sup>*<sup>2</sup>

**2.1. Model of DSF [***S***(***k,***ω)]**

in molecular dynamic units can be written as

function which is define as *G*(*r, t*) = *Gs*

where, *a*ws is the "Wigner-Seitz" radius and it is defined as *<sup>n</sup>*

136 Plasma Science and Technology - Basic Fundamentals and Modern Applications

limit was 425,000 step units. The EMD simulations are run between 4.25 10<sup>6</sup>

\_\_\_\_ 4 *<sup>o</sup>*

number density, *kB* is the Boltzmann constant and *T* is absolute system temperature. It is noted that the Γ is measured as the ratio of average potential energy to average kinetic energy per particle. Second scaling parameter is Debye screening parameter and it is given as *κ* = *a*ws/λD. The EMD simulations are performed for a particle number that is chosen between 500 and 1000 particles in a microcanonical ensemble using periodic boundary conditions and minimum image convention of the dust particles. It is to be mentioned that the number density (*n*) is defined as *n* = *N*/*V,* here *N* is the number of particles and *V* is the volume of simulation box and it is calculated as *V =* 4π*N*/3. On our case, most of simulations are performed for *N* = 500 and it is observed that the mentioned number of particles is suitable for EMD computations with statistical uncertainty limits. In presented case, the simulation time step is d*t* = 0.0001 and total simulation time

ωp) time units for each combination of (Γ, *κ*) in the series of recording dynamical structure factor (DSF), *S*(*k*,ω), of SCCDPs. It can be seen that the first patch of *S*(*k*,ω) results is obtained after the time limit of (38000) step unit. For our case, 13 patches of *S*(*k,*ω) results are obtained and results show that each patch has nearly behavior of *S*(*k,*ω) showing the accuracy of numerical algorithm. In this work, the dynamical structure factor computations of SCCDPs are reported for a wide range of Coulomb coupling parameters Γ ≡ (1, 200) and the Debye screening strength *κ* ≡ (1.4, 4).

The number density of single species also known as mass density and dimensionless quantity

here *r* is the point at time *t*. from the practical point of view, it is known as local density. The average occupancy in the small volume of *r* space and it calculates over the short interval of

*f*(*r*, *t*)*dr* = *Nm* (4)

The space and time-dependent density correlation are explain from the van Hove correlation

where *k* is the wave number and becomes equal to *k* = 2π/*L* and *L* is the length of simulation

*F*(*k*, *t*) = 〈*ρ*(*k*, *t*)*ρ*(‐*k*, *t*)〉 (6)

*j*

(*r, t*) + *Gs*

*δ*(*r* − *rj*

*ρ*(*r*, *t* ) = ∑ *j*

time. The definition of density is satisfied matter conserved requirement.

Fourier transform of density fluctuation and new expression is given as

*ρ*(*k*, *t*) = *f*(*r*, *t*) *e* <sup>−</sup>*ik*.*<sup>r</sup> dr* = ∑

box. The intermediate function can be defined as

\_\_\_\_\_\_ 1

*aws kB <sup>T</sup>* (2)

2 with *n* is the equilibrium dust

(1/ωp) to 3.25 10<sup>6</sup>

(*t* ) (3)

(*r, t*) and further detail is given in Ref. [23].

*e* <sup>−</sup>*ik*.*r*(*t*) (5)

(1/

\_\_−1

$$\underset{\omega}{\stackrel{\nu}{\quad}} S(k,\omega)d\alpha = S(k) \tag{8}$$

The dynamical structure function is related to longitudinal current correlation function, which is expressed as [22].

$$\mathcal{S}\langle k,\omega\rangle = \frac{1}{\omega^2} \mathbb{C}\_{\mathbb{L}}\langle k,\omega\rangle \tag{9}$$

## **3. Results and discussion**

The EMD simulation has been used for the calculation of *S*(*k,ω*) to understand the dynamic phenomenon of particles for 3D SCCDPs. We have analyzed our simulation results of *S*(*k,*ω) in term of frequency, amplitude and fluctuation rate with respect to these parameters (*κ*, Γ, and *k*).

This section shows an overview of our results obtained through EMD simulations for dynamical structure factor *S*(*k*,ω) function of SCCDPs at *κ* = 1, 2, 3 and 5 with *N* = 500. **Figures 1**–**4** shown our results for *S*(*k*,ω) over a wide suitable range of plasma states (Г, *κ*) at four values of wave vectors *k* = (0, 1, 2 and 3). A sequence of dynamical structure factor in increasing of wave vector (*k*) is computed to determine the suitable equilibrium values of *S*(*k*,ω). We have performed 16 EMD simulation with *N* = 500 for each screening parameters at different four values of *k*. There are 64 simulations are carried out for different combination of plasma parameters in order to observe complete behaviors of DSF *S*(*k*,ω) at higher varying frequency (ωp) as compared to earlier simulation results [16]. It is observed that the presented results obtained through EMD simulations for varying parameters have suitable signal-to-noise ratio of the DSF. Our results are satisfactory good agreement with earlier EMD estimations and show that the presented EMD results at higher ωp and earlier EMD simulations have comparable performance with small system size, both yielding the close values of the DSF *S*(*k*,ω). Moreover, the system temperature (1/ Г), strength of Debye screening (*κ*), system run time (total time), system size (*N*), and wave vector (*k*) are changed to observe how efficiently the presented EMD method computes the DSF *S*(*k*,ω) of SCCDPs.

In addition, in each panels of **Figures 1**–**4**, we have shown the behaviors of *S*(*k*, ω) at four values of *k*. All these data are excellent due to the good statistics and allow the analysis of the structure–function within a broad dynamical range. It is examined that the peaks of *S*(*k*,ω) are decay at lower Г values for different four values of *κ* and *k*. It can be noted that the peaks survive up to higher *k* values at intermediate to higher Г values and shift toward the higher frequency (ωp). It is further observed that the wave spectra of *S*(*k*,ω) shifts toward sinusoidal waves form, at the intermediate values of Г = 50 and wave spectra shifts toward square type waves forms at higher values of (Г = 100,200). Moreover, the panels (a) to (d) of each figures represent the results of *S*(*k*,ω) in nonideal gases state to liquid and crystalline order state

**Figure 1.** Variation of dynamical structure factor *S*(*k,*ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 1.4, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Г = 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

**Figure 3.** Variation of dynamical structure factor *S*(*k,*ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 3, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Г = 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

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**Figure 4.** Variation of dynamical structure factor *S*(*k*,ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 4, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Γ= 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

**Figure 2.** Variation of dynamical structure factor *S*(*k,*ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 2, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Г = 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

Numerical Approach to Dynamical Structure Factor of Dusty Plasmas http://dx.doi.org/10.5772/intechopen.78334 139

**Figure 3.** Variation of dynamical structure factor *S*(*k,*ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 3, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Г = 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

**Figure 1.** Variation of dynamical structure factor *S*(*k,*ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 1.4, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Г = 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

138 Plasma Science and Technology - Basic Fundamentals and Modern Applications

**Figure 2.** Variation of dynamical structure factor *S*(*k,*ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 2, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Г = 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

**Figure 4.** Variation of dynamical structure factor *S*(*k*,ω) as a function of plasma frequency (ω) of strongly coupled complex plasma at *κ* = 4, *N* = 500 and waves number *k* = 0, 1, 2, and 3 for (a) Γ= 1, (b) Г = 50, (c) Г = 100, (d) Г = 200.

corresponding to Г (1, 200). In case of crystalline order state, the amplitude of vibrating particle decreases significantly and it converts completely in square waveform. It is interesting to note that the fundamental behavior of dust particles is different and it shows decaying trends in nonideal gases state, sinusoidal form in liquid state and square wave form in crystalline state at fixed screening value. Figures show that the results of *S*(*k*,ω) depend on the plasma parameters (*κ*, Г), as expected. Furthermore, it is investigated that in the ideal form of dusty plasma with high temperatures (low coupling values) the dust particles are exponentially decaying from high to low amplitude. One of justification is the dust particles transferred their energy to the surrounding particles that at high plasma temperature values. Furthermore, in panels (b) represent the EMD results of *S*(*k*,ω) in the liquefied state of dusty plasma. In this regime, dust particles comparatively less transfer their energy to the system. At this regime amplitude is maximum for simulation box size (*L* = max) and frequency of oscillating of dust particles is low and in the sinusoidal waveform. The frequency and amplitude of oscillating dust particles increase with increasing wave numbers *k =* 2, 3 and exhibit in the periodic wave form. Panels (c) show the results of *S*(*k*,ω) of dusty plasma in the nearly crystalline states. We have analyzed that in this regime the particles are tightly bound. In this case, the dust particles have small amplitude as compared to the gaseous and liquefy forms of dusty plasma.

**Acknowledgements**

**Abbreviation and symbols**

MD molecular dynamics

CDPs complex (dusty) plasma

**DSF** dynamical structure factor

S(*k,*ω) dynamical structure factor

*κ* Debye screening strength

Aamir Shahzad1,2\*, Muhammad Asif Shakoori1

\*Address all correspondence to: aamirshahzad\_8@hotmail.com

College University Faisalabad (GCUF), Faisalabad, Pakistan

*k* wave number

*N* number of particles

ω<sup>p</sup> plasma frequency

Г Coulomb coupling

Xi'an Jiaotong University, P. R. China

**Author details**

EMD equilibrium molecular dynamics

SCCDPs strongly coupled complex dusty plasmas *WCCDPs* weakly coupled complex dusty plasmas

PECVD plasma-enhanced chemical vapor deposition

MD code.

The authors are grateful to the National Advanced Computing, National Center of Physics (NCP), Pakistan, and National High Performance Computing Center of X'ian Jiaotong. The authors thank the University, P.R. China, for allocating computer time to test and run our

Numerical Approach to Dynamical Structure Factor of Dusty Plasmas

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141

, Mao-Gang HE2

1 Molecular Modeling and Simulation Laboratory, Department of Physics, Government

3 Center for Soft Condensed Matter Physics and Interdisciplinary Research, College of

Physics, Optoelectronics and Energy, Soochow University, Suzhou, China

2 Key Laboratory of Thermo-Fluid Science and Engineering, Ministry of Education (MOE),

and Yan Feng<sup>3</sup>

It is observed from each panel of **Figures 1**–**4**, the dynamic of dust particles increases with increasing wave number. It is to be noted that the values of amplitude of the *S*(*k,ω*) are 3 = 0.7406, 0.6662, 0.8004 and 0.7902 respectively, for four wave numbers (*k* = 0, 1, 2, and 3) at the same values of *κ*, Г and *N*. It is observed that the dynamical structure factor of SCCDPs depends on plasma parameters (*κ*, Г). The frequency mode of *S*(*k,ω*) is high at low *κ* for SCCDPs. It is observed that *κ* is equally affecting on the dynamic of dust particles either the dusty plasma in any phase (gaseous, liquid and crystalline).
