3. Numerical results and discussion

This section provides the outcomes of thermal conductivity of 3D complex dusty plasmas by using HNEMD simulations over suitable plasma couplings Γ (�1, 200) and screening strengths κ (�1.4, 4) at constant external force strength of Fext (�0.005). It is noted that we have already reported our similar results with higher system sizes [13] and with different varying force fields [13]. In this present work, we have reported our HNEMD outcomes for different low to intermediate system sizes at fixed force field.

Figures 1–4 show our main outcomes of plasma thermal conductivity (λ) by employing HNEMD approach. Here, the thermal conductivity is normalized by plasma frequency (ωp) as λ<sup>0</sup> = λ/nkBωpaws, or by the Einstein frequency (ωE) as λ\* = λ/√3nkBωEaws of SCCNPs, at the normalized external field strength F\* = (Fz) (aws/JQ), where aws is radius of Wigner-Seitz (WS) radius with n being the equilibrium particle number density, k<sup>B</sup> is Boltzmann constant and JQ. It should be noted that these normalizations have been employed for classical Coulomb onecomponent plasmas (COCPs) [24] and SCCNPs [7].

Diverse series of the plasma λ<sup>0</sup> subsequent to a decreasing series of external force field F\* are computed to establish the linear system of the SCCNPs under the influence of the normalized force field strength. The current HNEMD outcomes allow investigation for the complete range of plasma parameters (Γ, κ) with variation of external force field F\* . In this work, a feasible suitable value of the external

Non-Newtonian Dynamics with Heat Transport in Complex Systems DOI: http://dx.doi.org/10.5772/intechopen.82291

### Figure 1.

r • <sup>i</sup> <sup>¼</sup> <sup>p</sup><sup>i</sup> m

F<sup>i</sup> þ D<sup>i</sup> ri; p<sup>i</sup>

� � is the tensor phase variable that describes the coupling of system to

� �:Feð Þ<sup>t</sup> � �:p<sup>i</sup>

<sup>J</sup>Q zð Þ<sup>t</sup> <sup>J</sup>Q zð Þ <sup>0</sup> � �dt

the fictitious external force field Feð Þt . A mechanical work is performed through the external applied force field Feð Þt , and thus, the equilibrium cannot be maintained. In the above Eq. (9), α is the Gaussian thermostat multiplier that maintains the

<sup>i</sup>¼<sup>1</sup> <sup>F</sup><sup>i</sup> <sup>þ</sup> <sup>D</sup><sup>i</sup> <sup>r</sup>i; <sup>p</sup><sup>i</sup>

∑<sup>N</sup> <sup>i</sup>¼<sup>1</sup>p<sup>2</sup> <sup>i</sup> =mi

∞ð

0

The external force field parallel to the z-axis is of the form FeðÞ¼ t ð Þ 0; 0; FZ ;

� JQz ð Þt D E

where JQZ ð Þt is the z-component of the heat flux vector and the external force

This section provides the outcomes of thermal conductivity of 3D complex dusty plasmas by using HNEMD simulations over suitable plasma couplings Γ (�1, 200) and screening strengths κ (�1.4, 4) at constant external force strength of Fext (�0.005). It is noted that we have already reported our similar results with higher system sizes [13] and with different varying force fields [13]. In this present work, we have reported our HNEMD outcomes for different low to intermediate system

Figures 1–4 show our main outcomes of plasma thermal conductivity (λ) by employing HNEMD approach. Here, the thermal conductivity is normalized by plasma frequency (ωp) as λ<sup>0</sup> = λ/nkBωpaws, or by the Einstein frequency (ωE) as λ\* = λ/√3nkBωEaws of SCCNPs, at the normalized external field strength F\* = (Fz) (aws/JQ), where aws is radius of Wigner-Seitz (WS) radius with n being the equilibrium particle number density, k<sup>B</sup> is Boltzmann constant and JQ. It should be noted

Diverse series of the plasma λ<sup>0</sup> subsequent to a decreasing series of external force field F\* are computed to establish the linear system of the SCCNPs under the influence of the normalized force field strength. The current HNEMD outcomes allow investigation for the complete range of plasma parameters (Γ, κ) with varia-

. In this work, a feasible suitable value of the external

that these normalizations have been employed for classical Coulomb one-

component plasmas (COCPs) [24] and SCCNPs [7].

TFz

p • <sup>i</sup> ¼ ∑ N J¼1

� �

system temperature and it is given as [16, 23]:

<sup>α</sup> <sup>¼</sup> <sup>∑</sup><sup>N</sup>

<sup>λ</sup> <sup>¼</sup> <sup>V</sup> 3kBT<sup>2</sup>

> ¼ lim Fz!0 limt!∞

therefore, the thermal conductivity is calculated as:

3. Numerical results and discussion

where <sup>F</sup><sup>i</sup> ¼ �∂ϕij=∂ri

Non-Equilibrium Particle Dynamics

D<sup>i</sup> ¼ D<sup>i</sup> ri; p<sup>i</sup>

field FeðÞ¼ t ð Þ FZ .

sizes at fixed force field.

tion of external force field F\*

176

(8)

(10)

(11)

� �:FeðÞ�<sup>t</sup> <sup>α</sup>p<sup>i</sup> (9)

is the total interparticle force acting on particle i and

Comparison of normalized plasma thermal conductivity (λ0), computed by various numerical approaches for plasma coupling states Γ (1, 200). Results investigated by Shahzad and He for homogenous nonequilibrium MD (HNEMD) [13], Salin and Caillol for equilibrium MD (EMD) [21], Donko and Hartmann for inhomogenous NEMD [17]. Shahzad and He for homogenous perturbed MD (HPMD) [18] and Faussurier and Murillo for variance procedure (VP) [25], (a) for N = 256, (b) N = 1372 and at κ = 1.4.

force field strength F\* (=0.005) for the computation of the steady state values of the plasma normalized thermal conductivity is to be selected, for small varying practical system size. This feasible suitable external force field provides the steady state plasma thermal conductivity estimations, which are satisfactory over the whole range of the plasma state points (Γ, κ).

Figures 1–3 display that the computed plasma thermal conductivity is in acceptable conformity with former HNEMD investigations by Shahzad and He [13], EMD calculations of Salin and Caillol [21], inhomogenous NEMD estimations of Donkó and Hartmann [17], homogenous perturbed molecular dynamics simulations (HPMD) measurements of Shahzad and He, and theoretical predictions of Faussurier and Murillo for variance procedure (VP) [18, 25]. It can be seen from Figure 1 that our results are slightly lower as compared to earlier known numerical results based on different numerical techniques, at lower Γ. However, the present

#### Figure 2.

Comparison of normalized plasma thermal conductivity (λ0), computed by various numerical approaches for plasma coupling states Γ (1, 300). Results investigated by Shahzad and He for homogenous nonequilibrium MD (HNEMD) [13], Salin and Caillol for equilibrium MD (EMD) [21], Donko and Hartmann for inhomogenous NEMD [17]. Shahzad and He for homogenous perturbed MD (HPMD) [18] and Faussurier and Murillo for variance procedure (VP) [25], (a) for N = 500, (b) N = 864 and at κ = 3.

of data points are far away from present data that are not mentioned here but most of data points are within limited statistical range, as expected. At higher screening κ = 4, it is examined from Figure 3 that the present results are definitely lower as compared to earlier EMD computations of Salin and Caillol and HNEMD estimations at higher N of Shahzad and He. Moreover, it can be noted that the present outcomes are slightly lower at intermediate Γ and well matched at higher Γ,

Comparison of normalized plasma thermal conductivity (λ0), computed by various numerical approaches for plasma coupling states Γ (1, 300). Results investigated by Shahzad and He for homogenous nonequilibrium MD (HNEMD) [13], Salin and Caillol for equilibrium MD (EMD) [21], Shahzad and He for homogenous perturbed MD (HPMD) [18] and Faussurier and Murillo for variance procedure (VP) [25], (a) for N = 500,

Non-Newtonian Dynamics with Heat Transport in Complex Systems

DOI: http://dx.doi.org/10.5772/intechopen.82291

It is suggested from these figures that measured outcomes are satisfactory well matched with previous outcomes at intermediate to high Γ; however, some results diverge at the lower Γ points but all within statistical unlimited uncertainty range. Figures 1–3 show that the presented HNEMD method may precisely calculate the

confirming earlier results.

(b) N = 864 and at κ = 4.

Figure 3.

179

results are well matched with earlier results for intermediate to higher Γ at two different system sizes N = (256 and 1372) and it is clearly shown that our results are very close EMD and HNEMD results. It is observed from Figure 2 that HNEMD results are between EMD (at lower N) results and HNEMD (at higher N) computations at low value of Γ but our outcomes are satisfactorily matched with earlier results at intermediate and higher Γ. It can be seen from Figure 2 that our HNEMD data for low intermediate system size it is increasing behavior at higher Γ, confirming earlier HPMD results [18]. It is observed that the deviation of data from earlier known measured data based on different methods of EMD, HPMD, and InNEMD is also computed and the outcomes of plasma λ<sup>0</sup> are within range of 3– 22% for EMD, 7–20% for NEMD, and 10–35% for HPMD. It is noted that some

Non-Newtonian Dynamics with Heat Transport in Complex Systems DOI: http://dx.doi.org/10.5772/intechopen.82291

### Figure 3.

results are well matched with earlier results for intermediate to higher Γ at two different system sizes N = (256 and 1372) and it is clearly shown that our results are very close EMD and HNEMD results. It is observed from Figure 2 that HNEMD results are between EMD (at lower N) results and HNEMD (at higher N) computations at low value of Γ but our outcomes are satisfactorily matched with earlier results at intermediate and higher Γ. It can be seen from Figure 2 that our HNEMD

and Murillo for variance procedure (VP) [25], (a) for N = 500, (b) N = 864 and at κ = 3.

Comparison of normalized plasma thermal conductivity (λ0), computed by various numerical approaches for plasma coupling states Γ (1, 300). Results investigated by Shahzad and He for homogenous nonequilibrium MD (HNEMD) [13], Salin and Caillol for equilibrium MD (EMD) [21], Donko and Hartmann for inhomogenous NEMD [17]. Shahzad and He for homogenous perturbed MD (HPMD) [18] and Faussurier

Figure 2.

Non-Equilibrium Particle Dynamics

178

data for low intermediate system size it is increasing behavior at higher Γ,

confirming earlier HPMD results [18]. It is observed that the deviation of data from earlier known measured data based on different methods of EMD, HPMD, and InNEMD is also computed and the outcomes of plasma λ<sup>0</sup> are within range of 3– 22% for EMD, 7–20% for NEMD, and 10–35% for HPMD. It is noted that some

Comparison of normalized plasma thermal conductivity (λ0), computed by various numerical approaches for plasma coupling states Γ (1, 300). Results investigated by Shahzad and He for homogenous nonequilibrium MD (HNEMD) [13], Salin and Caillol for equilibrium MD (EMD) [21], Shahzad and He for homogenous perturbed MD (HPMD) [18] and Faussurier and Murillo for variance procedure (VP) [25], (a) for N = 500, (b) N = 864 and at κ = 4.

of data points are far away from present data that are not mentioned here but most of data points are within limited statistical range, as expected. At higher screening κ = 4, it is examined from Figure 3 that the present results are definitely lower as compared to earlier EMD computations of Salin and Caillol and HNEMD estimations at higher N of Shahzad and He. Moreover, it can be noted that the present outcomes are slightly lower at intermediate Γ and well matched at higher Γ, confirming earlier results.

It is suggested from these figures that measured outcomes are satisfactory well matched with previous outcomes at intermediate to high Γ; however, some results diverge at the lower Γ points but all within statistical unlimited uncertainty range. Figures 1–3 show that the presented HNEMD method may precisely calculate the

Acknowledgements

Abbreviations

time to test and run our MD code.

DOI: http://dx.doi.org/10.5772/intechopen.82291

Γ Coulomb coupling κ Debye screening length F\* external force field strength

MD molecular dynamics

λ thermal conductivity

VP variance procedure

N number of particles

Aamir Shahzad1,2\* and Fang Yang2

provided the original work is properly cited.

Author details

181

SCP strongly coupled plasma

SCCNPs strongly coupled complex nonideal plasmas

Non-Newtonian Dynamics with Heat Transport in Complex Systems

NEMD nonequilibrium molecular dynamics

EMD equilibrium molecular dynamics

λ<sup>0</sup> normalized thermal conductivity PBCs periodic boundary conditions

HPMD homogenous perturbed MD

HNEMD homogenous nonequilibrium molecular dynamics

InHNEMD inhomogenous nonequilibrium molecular dynamics

1 Molecular Modeling and Simulation Laboratory, Department of Physics, Government College University Faisalabad (GCUF), Faisalabad, Pakistan

2 College of Physics, Civil Aviation University of China, Tianjin, P. R. China

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

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

The authors thank Z. Donkó (Hungarian Academy of Sciences) for providing his thermal conductivity data of Yukawa Liquids for the comparisons of our simulation results, and useful discussions. We are grateful to the National Advanced Computing Center of National Center of Physics (NCP), Pakistan, for allocating computer

### Figure 4.

Trend of normalized potential energy (P.E.) with four values of screening parameters (κ = 1, 2, 3, and 4) for different coupling states Γ (1, 5, 10, 20, 50, 100, 200, and 300) and at N = 256.

plasma thermal conductivity of strongly coupled complex plasmas. We have shown that the present method has good performance and its accuracy is very close to earlier EMD and InHNEMD methods. It is concluded that our outcomes depend on the plasma parameters of Coulomb coupling and Debye screening strength, confirming earlier simulations. Moreover, it is shown that the position of minimum value of thermal conductivity shifts toward higher Γ with an increase of screening κ, as expected. Presently, we have demonstrated our results for a wide range of plasma parameters, ranging from nonideal gaseous state to strongly coupled range. It is noted that the extended HNMED method is excellent for lower system sizes with constant external force field strength, where signal-to-noise ratio is acceptable for equilibrium plasma thermal conductivity.
