**2. Molecular dynamics simulations**

MD simulation is a powerful technique that can be used to solve many physical problems in atomic material research. MD simulation is handled normally all microscopic information and molecular methods have proven to be the product of applied research. Plasmas and complex liquids have various uses, ranging from semiconductor chips, colloids, thin films, and electrochemistry to biochemical films and other important areas where structures play an important role. MD simulation plays an important role in all the advanced sectors, such as textile science, engineering, physics, plasma physics, astronomy, life sciences or organic sciences, and the chemical industry. Computer simulation has become increasingly important in

detecting complex motion systems. Using faster and more sophisticated computer systems, it can be studied the habitat, composition, and behavior of large complex systems. In the 1950s, Alder [19], Wainwright and Rahman [19] used the first MD simulations for liquid argon [1–10], their references herein]. MD simulation has two basic kinds which rely on the properties so far which we can be going to calculate: one is EMD simulations (EMDS) and the other one is NEMD simulations (NEMDS). In the present work, NEMDS is applied to investigate the thermal conductivity of complex plasma at different dusty plasma parameters.

### **2.1 Numerical model and algorithm**

NEMD simulation is used to detect the dust trajectory of an interacting system using Yukawa forces between dust particles. In present case, the HNEMD simulation (HNEMDS) method is used to calculate the thermal flow of complex (dust) plasma formed using Yukawa interaction taking in to account the charged particles with polarization effects and it is given in the form [22]:

$$\phi\_{ij}(|\mathbf{r}|) = \frac{Q\_{\mathbf{d}}^2}{4\pi\varepsilon\_0} \frac{e^{-|\mathbf{r}|/\lambda\_{\mathrm{D}}}}{|\mathbf{r}|} + \frac{d^2}{r^3} \left(\mathbf{1} + \frac{r}{\lambda\_{\mathrm{D}}}\right) e^{-|\mathbf{r}|/\lambda\_{\mathrm{D}}},\tag{1}$$

The first term in Eq. (1) provides the screened charge–charge interaction (form of Yukawa interaction) and the second term gives the screened dipole–dipole interaction. Yukawa potential model of dust particles interaction can be established to take into consideration the polarization effect, the temperature and the screening effects. Here *'r'* is the magnitude of interparticle distance, *Q* is the charge of dust particles, and *λ<sup>d</sup>* is the Debye screening length. We have three normalized (dimensionless) parameters to characterize the Yukawa interaction model *ϕY*ð Þ j j **r** , the Coulomb coupling parameter <sup>Γ</sup> <sup>¼</sup> *<sup>Q</sup>*<sup>2</sup> *=*4*πε*<sup>0</sup> � �*:*ð Þ <sup>1</sup>*=awskBT* , where *<sup>a</sup>*ws is Wigner Seitz radius and it is equal to (nπ) �1/2, here *n* is the number of particles per unit area (*N*/*A*). The *kB* and *T* are Boltzmann constant and absolute temperature of the system, and *A* is the system area. The second is the screening strength (dimensionless inverse) *κ* ¼ *aws=λD*, and additional normalized external force field strength, *<sup>F</sup>* <sup>∗</sup> <sup>¼</sup> ð Þ *FZ : aws=***J***<sup>Q</sup>* � �. GK relations (GKRS) for the hydrodynamic transport coefficients of uncharged particles of pure liquids have applied to calculate the thermal conductivity of 2D complex plasma. Here, **JQ** is the current heat vector at time *t* of 2D case [1–4].

$$\mathbf{J}\_Q(t)A = \sum\_{i=1}^{N} E\_i \frac{\mathbf{p}i}{m} - \frac{1}{2} \sum\_{i \neq j} (\mathbf{r}i - \mathbf{r}j) . \left(\frac{\mathbf{p}i}{m}. \mathbf{F}i\mathbf{j}\right) \tag{2}$$

where **F***ij* is the total interparticle force at time *t,* on particle *i* due to *j*, **r***ij* = **r***<sup>i</sup>* – **r***<sup>j</sup>* are the position vectors (interparticle separation), and **P***<sup>i</sup>* is the momentum vector of the *i*th particle. Where, *Ei* is the total energy of particle *i.*

$$E\_i = \frac{\mathbf{p}^2}{2m} + \frac{1}{2} \sum\_{i \neq j} \phi\_{ij} \tag{3}$$

$$\dot{\mathbf{r}}\_{i} = \frac{\mathbf{p}\_{i}}{m}, \dot{\mathbf{p}}\_{i} = \sum\_{j=1}^{N} \mathbf{F}\_{i} + \mathbf{D}\_{i}(\mathbf{r}\_{i}, \mathbf{p}\_{i}) \mathbf{F}\_{e}(t) - a\mathbf{p}\_{i}. \tag{4}$$

In Eq. (4), **<sup>F</sup>***<sup>i</sup>* ¼ � *<sup>d</sup>ϕ*Yð Þ j j **<sup>r</sup>** *dri* is the Yukawa interaction force acting on particle *i*, where *ϕ*Y|**r**| is given from Eq. (1), and **D***<sup>i</sup>* **= D***i*{(**r***i*, **p***i*), *i* = 1,2, … ,*N*} is the phase *Polarized Thermal Conductivity of Two-Dimensional Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.100545*

space distribution function with **r***<sup>i</sup>* and **p***<sup>i</sup>* being the coordinate and momentum vectors of the *i*th particle in an *N*-particle system. A Gaussian thermostat multiplier (*α*) has been used to maintain the system temperature at equilibrium position and it is given as

$$a = \frac{\sum\_{i=1}^{N} \left[ \mathbf{F}\_i + \mathbf{D}\_i(\mathbf{r}\_i, \mathbf{p}\_i) . \mathbf{F}\_i(t) \right] . \mathbf{p}\_i}{\sum\_{i=1}^{N} p\_i^2 / m\_i} \tag{5}$$

When an external force field is selected parallel to the z-axis **F**e(*t*) = (0, *Fz*), in the limit *t* ! ∞. Then the thermal conductivity is given as [8].

$$\lambda = \frac{1}{2k\_{\text{B}}AT^{2}} \Bigg\vert \left\langle \mathbf{J}\_{Q\_{x}}(t)\mathbf{J}\_{Q\_{x}}(0) \right\rangle dt = \lim\_{F\_{x} \to 0} \lim\_{t \to \infty} \frac{-\left\langle \mathbf{J}\_{Q\_{x}}(t) \right\rangle}{TF\_{x}} \tag{6}$$

where **J***Qz* is the z-component of the heat flux vector (energy current). All time series data are recorded during HNEMD and used in Eq. (6) to calculate λ. The detail of present scheme with all parameters (Gaussian thermostat multiplier, external force, **D***i, Fi*, etc) for Yukawa interaction has been reported in our earlier works [8]. The most time consuming part of the algorithm is the calculation of particle interactions (energy and interaction forces). It has been shown in our previous work that the proposed method has the advantage of calculating Yukawa interaction and its associated energy with the appropriate computational power at the right time of the computer simulation. The actual HNEMD simulations are performed between 1.5 x 105 /*ω*<sup>p</sup> and 3.0 x 10<sup>5</sup> /*ω*<sup>p</sup> time units in the series of data recording of thermal conductivity (λ). Here, *ω*<sup>p</sup> is the plasma frequency and it is defined as *ω*pd = (*Q*<sup>d</sup> 2 /2π*ε*0*mda*<sup>3</sup> ) 1/2, where *m*<sup>d</sup> is the mass of a dust particle.

## **3. Simulation results and discussion**

In this section, we have discussed the preliminary results obtained through HNMED simulation for Coulomb coupling parameters of Г (= 10, 100), polarization values Г<sup>d</sup> = (0, 1, 10, 20, 50 and 100) and Debye screening (*κ* = 1.4, 2 and 3) at constant external force field (*F*\* = 0.02) of 2D strongly coupled dusty plasmas. We have chosen suitable number of particles (*N* = 400) in the simulation box with edge length (*Lx*, *Ly*). Periodic boundary conditions (PBCs) are applied along with minimum image convection in a simulation box of length *L*.

There are different conditions to improve the efficient results of thermal conductivity under polarization effects. These conditions include the system size (*N*), Coulomb coupling (Г), Debye screening length (*κ*), system total length (*t*), simulation time step (d*t*), and external force field (*F*\* ) strength and polarization values (Гd). We have chosen suitable parameters for precise results of thermal conductivity with increasing Г<sup>d</sup> and *κ*. The simulation data are for a suitable system size (*N* = 400) with different Г, which covers the values of strongly coupled plasma states (nonideal gases-liquid to crystalline) at constant force field strength (*F*\* = 0.02) with varying polarization values Г<sup>d</sup> (0 to 100).

The polarized thermal conductivity of SCCDPs stated here may be scaled as λ<sup>0</sup> = λ/*nmω*p*a*<sup>2</sup> (by the plasma frequency). It is demonstrated that the *ω*<sup>E</sup> decreases with increasing *κ*, *ω*<sup>E</sup> ! *ω*p/√3 as *κ* ! 0, for the 3D case [1–10]. Furthermore, for

the assessment of appropriate equilibrium range (nearly-equilibrium of external field strength) of 2D HNEMD scheme, various sequences of the polarized thermal conductivity corresponding to an increasing order of an external force field [**F**e(*t*) = (0, *Fz*) *<sup>F</sup>*\* = (*Fz*) (*a*/J*Q*)] are planned to measure the nearly equilibrium values of the λ0. This possible appropriate *F*\* value gives the steady state λ<sup>0</sup> investigations, which are appropriate for the whole range of plasma states of Γ (10, 100) and *κ* (1.4, 3).

**Figures 1**–**3** illustrate the normalized polarized thermal conductivity (plasma frequency, *ω*p), as a function of Coulomb coupling (system temperature 1 / Γ) for the cases of *κ* = 1.4, 2 and 3, respectively, at constant force *F*\* with varying six values of polarizations Г<sup>d</sup> = (0, 1, 10, 20, 50 and 100). For three cases, the simulations are performed with setting *N* = 400 for *κ* = 1.4, 2 and 3, respectively, at constant

**Figure 1.**

*Variations of thermal conductivity as a function of Coulomb coupling of strongly coupled complex plasma at* κ *= 1.4 with* N *= 400 and (a) Г<sup>d</sup> = 0, (b) Г<sup>d</sup> = 10, (c) Г<sup>d</sup> = 20, (d) Г<sup>d</sup> = 50, (e) Г<sup>d</sup> = 50 and (f) Г<sup>d</sup> = 100.*

**Figure 2.**

*Variations of thermal conductivity as a function of Coulomb coupling of strongly coupled complex plasma at* κ *= 2.0 with* N *= 400 and (a) Г<sup>d</sup> = 0, (b) Г<sup>d</sup> = 10, (c) Г<sup>d</sup> = 20, (d) Г<sup>d</sup> = 50, (e) Г<sup>d</sup> = 50 and (f) Г<sup>d</sup> = 100.*

*F*\* = 0.02. Performing HENMD simulations with varying polarizations at constant *F*\* we examined the efficiency and reliability of the polarized λ<sup>0</sup> measurements. For three cases, we evaluate the six various simulation data sets covering from nonideal state (Γ = 10) to a strongly coupled liquid regime (Γ = 100). Figures show that the effects of polarization on the thermal conductivity have no significant changes and it is seen that the thermal conductivity remains constant under varying polarizations. However, the present results of thermal conductivity under varying polarizations are satisfactory agreement with earlier know available numerical data for a complete range of plasma parameters.

**Figure 4** shows comparisons with earlier available 2D and 3D numerical data of thermal conductivity with setting *N* = 1024 and Г<sup>d</sup> = 1. The current results are

**Figure 3.**

*Variations of thermal conductivity as a function of Coulomb coupling of strongly coupled complex plasma at* κ *= 3.0 with* N *= 400 and (a) Г<sup>d</sup> = 0, (b) Г<sup>d</sup> = 10, (c) Г<sup>d</sup> = 20, (d) Г<sup>d</sup> = 50, (e) Г<sup>d</sup> = 50 and (f) Г<sup>d</sup> = 100.*

generally excellent agreement for the whole Coulomb coupling range and plot show overall the same behaviors as in the earlier simulation method s of 2D plasma systems. Figure involve the earlier work of 3D HNEMD and HPMD by Shahzad and He [8, 16, 23], EMD investigations of Salin and Caillol [12] and 3D theoretical prediction of Faussurier and Murillo [13] as well as 2D GKR-EMD of Khrustalyov and Vaulina [24].

*Polarized Thermal Conductivity of Two-Dimensional Dusty Plasmas DOI: http://dx.doi.org/10.5772/intechopen.100545*

**Figure 4.**

*Comparison of thermal conductivity as a function of Coulomb coupling of strongly coupled complex plasma at* κ *= 2.0 with* N *= 1024 and Г<sup>d</sup> = 1.*
