Abstract
Understanding of thermophysical properties of complex liquids under various conditions is of practical interest in the field of science and technology. Thermal conductivity of nonideal complex (dusty) plasmas (NICDPs) is investigated by using homogeneous nonequilibrium molecular dynamics (HNEMD) simulation method. New investigations have shown, for the first time, that Yukawa dusty plasma liquids (YDPLs) exhibit a non-Newtonian behavior expressed with the increase of plasma conductivity with increasing external force field strength Fext. The observations for lattice correlation functions Ψ (t) show, that our YDPL system remains in strongly coupled regime for a complete range of plasma states of (Γ, κ), where (Γ) Coulomb coupling and (κ) Debye screening length. It is demonstrated, that the present NICDP system follows a simple scaling law of thermal conductivity. It has been shown, that our new simulations extend the range of Fext used in the earlier studies in order to find out the size of the linear ranges. It has been shown that obtained results at near equilibrium (Fext = 0.005) are in satisfactory agreement with the earlier simulation results and with the presented reference set of data showed deviations within less than ±15% for most of the present data points and generally overpredicted thermal conductivity by 3–22%, depending on (Γ, κ).
Keywords
- thermophysical properties
- thermal conductivity
- nonlinear effects
- lattice correlation functions
- nonideal complex (dusty) plasmas
- nonequilibrium molecular dynamics
1. Introduction
The exact numerical investigation of transport properties of complex liquids is a fundamental research task in the field of thermophysics, as various transport data are closely related with setup and the confirmation of equations of state. A reliable knowledge of transport data is also important for optimization of technological processes and apparatus design in various engineering and science fields (incl. thermoelectric devices) and, in particular, when provision of precise data for parameters of heat, mass, and momentum transport is required [1–3]. In thermophysical properties of fluids, chemical properties remain unaffected, but physical properties of material are changed by variable temperature, composition, and pressure. These properties of simple and complex liquids explain the phase transition [4]. These fluids can be examined experimentally, theoretically, and by simulation techniques. Thermophysical properties (thermodynamics and transport coefficients) include thermal conductivity, thermal expansion, thermal radiative properties, thermal diffusivity, enthalpy internal energy, Joule-Thomson coefficients, and heat capacity, as well as, thermal diffusion coefficients, mass coefficients, viscosity, speed of sound, and interfacial and surface tension. Thermophysical properties of gases and liquids, such as hydrogen, H2; oxygen, O2; nitrogen, N2; and water, H2O are different from ideal gas at high pressure and low temperature. Specific models are required for the calculation of these properties in the widest range of pressure and temperature. Different fluids, such as gases and liquids are used as a power generation source in different power plants. For example, heavy water, steam, air, and different gases are used for power generation in nuclear power plants, gas turbine plants, and internal combustion plants. Also, for cooling in refrigerators and fast nuclear reactors, ammonia and sodium in liquid phase are used as a cooling agent.
1.2. Dusty plasma
Nowadays dusty plasma refers to as complex plasma in analogy to the condensed matter field of “complex liquids” in soft matter (colloidal suspensions, polymers, surfactants, etc.). The dust particles combine physics of nonideal plasmas and condensed matter, and this field has played an important role in both newly system designs and advance development micro- and nanotechnology. This complex plasma system has four components, i.e., ions, electrons, neutral atoms, and dust particles with high charges, as compared to other species, which are responsible for the extraordinary plasma properties. The study of complex (dusty) plasmas reveals rich variety of interesting phenomena and extends knowledge on fundamental aspects of plasma physics at the microscopic level. Among these, the freezing (gaseous-liquid-solid) phase transition is of particular interest. Complex plasma is called strongly coupled plasma, in which thermal energy (kinetic energy) of nearest neighbors is much smaller than their Coulomb interaction potential energy, whereas plasma is called weakly coupled when Coulomb interparticle potential energy of nearest particles is much smaller than their kinetic energy [5–7].
Plasma is the fourth state of matter, and usually, it is said, that there are three states of matter, but another state was also found to exist, named as plasma. Irving Langmuir (American physicist) defined plasma as “it is a quasi-neutral gas of charged and neutral particles, which exhibits collective behavior,” and he got the Nobel Prize in 1927 because firstly he was using the term plasma [8]. In this definition, quasi-neutral means that plasma is electrically neutral and has approximately equal ion and electron density (
1.2.1. Types of plasma
There are different types of plasma that are described by many characteristics, such as temperature, degree of ionization, and density.
1.2.1.1. Cold plasma
In laboratory, in the positive column of a glow discharge tube, there exists plasma in which the same number of ions and electrons is present. When gas pressure is low, collision between electrons and gas molecules is not frequent. So, nonthermal equilibrium between energy of electrons and gas molecules does not exist. So, energy of electrons is very high as compared to gas molecules and the motion of gas molecules can be ignored. We have
1.2.1.2. Hot plasma
When gas pressure is high in the discharge tube, then electrons collide with gas molecules very frequently and thermal equilibrium exists between electrons and gas molecules. We have
1.2.1.3. Ultracold plasma
If plasma occurs at temperature as low as 1 K, then such type of plasma is known as ultracold plasma, and it can be formed by photoionizing laser-cooled atoms and pulsed lasers. In ultracold plasmas, the particles are strongly interacting because their thermal energy is less than Coulomb energy between neighboring particles [9].
1.2.1.4. Ideal plasma
There are mainly two types of plasma according to plasmas’ ideality and properties study, nonideal plasmas (weakly coupled and strongly coupled plasmas) and ideal plasmas (very weakly coupled plasmas). Whenever the kinetic energy of plasma is much larger than the potential energy and plasma has a low temperature and high density, then such type of plasma is known as ideal plasma. Ideal plasma is one in which Coulomb collisions are negligible. If the average distance among the interacting particles is large, then the interaction potential can be ignored due to this large-distance ideal plasma that does not have any arrangement of particles [2].
1.2.1.5. Nonideal (complex) plasma
Nonideal plasmas are often found in nature, as well as, in technological services. They can be shown as electron plasma in solid and liquid metals and electrolytes, the superdense plasma of the matter of white dwarfs, the sun and the interiors (deep layers) of the giant planets of the solar system, and astrophysical objects, whose structure and evolution are defined by plasma characteristics [2]. Further examples of nonideal plasmas are brown dwarfs, laser-generated plasmas, capillary discharges, plasma-opening switches, high-power electrical fuses, exploding wires, etc. On the bases of Coulomb coupling, nonideal plasma can be divided into two families: weakly coupled plasma (WCP, Г < 1) and strongly coupled plasma (SCP, Г ≥ 1).
Nonideal complex plasmas are found in daily life and can be found in processing industries to manufacture many products that we deal in our everyday life directly or indirectly at moderate temperature, such as plastic bags, automobile bumpers, airplane turbine blades, artificial joints, and, most importantly, in semiconductor circuits. Moreover, nonideal plasmas (terrestrial plasmas) are not hard to find. They occur in gas-discharge lighting, such as neon lighting used for commercial purposes and fluorescent lamps, for instance, compact fluorescence light sources, which have a higher performance than the traditional incandescent light sources, a variety of laboratory experiments, and a growing array of industrial processes. Modern display methods contain plasma screens, in which small plasma discharges are used to stimulate a phosphor layer, which then emits light [2, 7].
1.2.2. Complex (dusty) plasma
Dust is present everywhere in the universe and mostly it is present in solid form. It is also present in gaseous form, which is often ionized, and thus the dust coexists with plasma and forms “dusty plasma.” In dusty plasma, dust particles are immersed in plasma, in which ions, electrons, and neutrals are present. These dust particles are charged and then affected by electric or magnetic fields and can cause different changes in the properties of plasma. The presence of dust component gives rise to new plasma phenomena and allows study of fundamental aspects of plasma physics at the microscopic level. Dust particles are charged due to the interaction between dust particles and the surrounding plasmas. Due to this interaction, grains are charged very rapidly. The charge on grains depends on the flow of ions and electrons. These charged grains enhance plasma environments, for example, setting up space charges. Also, to determine the charge on dust grain, it is assumed, that a spherically symmetric isolated dust grain is injected in plasma and only the effect of ion and electron is considered. Moreover, there are many other charging processes, such as secondary emission, electron emission, thermionic emission, field emission, radioactivity, and impact ionization. Complex plasma is condensed plasma characterized by strong interaction between existing molecules and atoms; it is also called strongly coupled complex plasma. Dusty plasma is complex plasma which includes many components: ions, electrons, neutral particle, and dust particles. Last 20–25 years, strongly coupled plasmas were mainly studied theoretically, due to lack of suitable laboratory tools and equipment. However, experimental strongly coupled plasma studies became more common with the discovery of ways to find dusty plasma [10], laser-cooled ion plasmas in a penning trap [11], and ultracold neutral plasmas [12]. Plasma systems can be treated theoretically in a straightforward way in the extreme limits of both weakly coupled and strongly coupled plasmas [2].
The main goals of this chapter are to study thermal conductivity (
2. HNEMD model and simulation approach
In this section, we will introduce theoretical background needed in this work. We start by introducing the model system, which is used in our HNEMD simulations. We consider a cubic box of edge length
The particles interact through screened Coulomb potential, which depends on the physical parameters and the background plasmas. Average interparticle interaction is frequently considered to be isotropic and basically repulsive and approximated by Yukawa interaction potential [1–3]. Yukawa model has been employed in many physical and chemical systems (for instance, biological and pharmaceutical sciences, colloidal and ionic systems, space and environment sciences, physics and chemistry of polymers and materials, etc.) [1–8]. In the present case, the interaction of potential energy of particles is in Yukawa form:
where charge on dust particle is
In HNEMD technique, in order to measure thermal conductivity and nonlinearity of NICDPs, the system will be perturbed by applying the external field along
Here,
In this equation, r
where ∅
where J
3. Computer simulation outcomes
3.1. Particle lattice correlation
The structural information of Yukawa system is given by lattice correlation. For the calculation of lattice correlation, density of given material at point
where
System arrangement (ordered or disordered) is calculated by simulation based on Eq. (7). When value of lattice correlation approaches to |Ψ| ≈ 1, then the system will be in ordered state, and if value becomes |Ψ| ≈ 0, then the system will be in liquid or gas (nonideal gas) state. In Eq. (7),
Lattice correlation was examined in 3D NICDPs in the limit of appropriate constant near equilibrium external force field strength
3.2. Normalized thermal conductivity
We now turn attention to the key results obtained through HNEMD simulations. Obtained computer-simulated data confirm, that thermal conductivity of Yukawa system can be calculated with satisfactory statistics by an extended HNEMD approach. Figures 2–5 display the main results calculated from HNEMD method for various plasma states for Yukawa liquids at
Different sequences of
Figures 2–5 show, that measured thermal conductivity is in satisfactory agreement with earlier HPMD simulations by Shahzad and He [1], inhomogeneous NEMD computations by Donko and Hartmann [20], and EMD measurements by Salin and Caillol [19]. The present results are also higher than Salin and Caillol [19] results at lower Γ = 1 and 2, for
This comparison shows, that our data remain within the limited statistical uncertainty range. Panels (b) of Figures 2–5 compare the present simulation results of thermal conductivity, normalized by reference data, calculated here from HNEMD approach for different sets of external force field strengths with reference set of data and earlier known simulation data of HPMD, EMD, inhomogeneous NEMD, and VP techniques [1, 19–21]. A series of different sequences of HNEMD simulations are performed with
For the cases of
It is concluded from figures, that obtained results agree well with earlier results at intermediate and high Γ values; however, some data points deviate at the lower Γ values. Figures 2–5 depict, that extended HNEMD approach can accurately predict thermal conductivity of Yukawa system (dusty plasma). We have used the present developed homogeneous NEMD method, which has an excellent performance; its accuracy is comparable to that of EMD and inhomogeneous NEMD techniques. The first conclusion from above Figure 5 is that thermal conductivity depends on plasma parameters Γ and
4. Summary
Thermal conductivity of NICDP system was investigated for wide range of Coulomb coupling parameter (1 ≤ Γ ≤ 300) and screening parameter (1 ≤
Acknowledgments
This work was sponsored by the National Natural Science Fund for Distinguished Young Scholars of China (NSFC no. 51525604) and partially sponsored by the Higher Education Commission (HEC) of Pakistan (no. IPFP/HRD/HEC/2014/916). 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 High Performance Computing Center of Xi'an Jiaotong University and National Advanced Computing Center of the National Centre for Physics (NCP), Pakistan, for allocating computer time to test and run our MD code.
References
- 1.
Shahzad A, He M-G. Thermal conductivity calculation of complex (dusty) plasmas. Phys. Plasmas. 2012; 19 (8):083707. DOI: 10.1063/1.4748526 - 2.
Shahzad A, He M-G. Computer Simulation of Complex Plasmas: Molecular Modeling and Elementary Processes in Complex Plasmas. 1st ed. Saarbrücken, Germany: Scholar’s Press; 2014. 170 p. - 3.
Shahzad A, He M-G. Thermal conductivity of three-dimensional Yukawa liquids (dusty plasmas). Contrib. Plasma Phys. 2012; 52 (8):667. DOI: 10.1002/ctpp.201200002 - 4.
Fortov VE, Vaulina OS, Lisin EA, Gavrikov AV, Petrov OF. Analysis of pair interparticle interaction in nonideal dissipative systems. J. Exp. Theor. Phys. 2010; 110 :662–674.DOI: 10.1134/S1063776110040138 - 5.
Shahzad A, He M-G. Thermodynamic characteristics of dusty plasma studied by using molecular dynamics simulation. Plasma Sci.Technol. 2012; 14 (9):771–777. DOI: 10.1088/1009-0630/14/9/01 - 6.
Shahzad A, He M-G. Interaction contributions in thermal conductivity of three-dimensional complex liquids. In: Liejin GUO, editor. AIP Conference Proceedings; 26–30 October; Xi’an, China. USA: AIP; 2013. p. 173–180. DOI: 10.1063/1.4816866 - 7.
Wigner E. Quasiparticle treatment of quantum-mechanical perturbation theory in the free electron gas. Phys. Rev. 1934; 46 : 1002. DOI: 10.1007/BF01029223 - 8.
Chen FF. Introduction to Plasma Physics and Controlled Fusion. 2nd ed. New York: Spring verlag; 2006. 200 p. - 9.
Killian T, Pattard T, Pohl T, Rost J. Ultracold neutral plasmas. Phys. Rep. 2007; 449 :77. DOI:10.1016/j.physrep.2007.04.007 - 10.
Kalman GJ, Rommel JM, Blagoev K. Strongly Coupled Coulomb Systems. New York: Plenum; 1998. 198 p. - 11.
Jensen MJ, Hasegawa T, Bollinger JJ, Dubin DHE, et al. Rapid heating of a strongly coupled plasma near the solid-liquid phase transition. Phys. Rev. Lett. 2005; 94 :025001. DOI: 10.1103/PhysRevLett.94.025001 - 12.
Robicheaux F, Hanson James D. Ultra cold neutral plasmas. Phys. Plasmas 2003; 10 (2217). DOI: 10.1063/1.1573213 - 13.
Toukmaji AY, Board Jr, John A. Comput. Phys. Commun. 1996; 95 :73–92. - 14.
Rapaport DC. The Art of Molecular Dynamics Simulation. New York: Cambridge University Press; 2004. 250 p. - 15.
Evans DJ, Morriss GP. Statistical Mechanics of Non-Equilibrium Liquids. London: Academic; 1990. 20–300 p. - 16.
Shahzad A, He M-G. Homogeneous nonequilibrium molecular dynamics evaluations of thermal conductivity 2D Yukawa liquids. Int. J. Thermophys. 2015; 36 (10–11):2565. DOI: 10.1007/s10765-014-1671-8 - 17.
Shahzad A, He M-G. Calculations of thermal conductivity of complex (dusty) plasmas using homogeneous nonequilibrium molecular simulations. Radiat. Eff. Defect. S. 2015; 170 (9):758–770. DOI: 10.1080/10420150.2015.1108316 - 18.
Pierleoni C, Ciccotti G, Bernu B. Thermal conductivity of the classical one-component plasma by nonequilibrium molecular dynamics. Europhys. Lett. 1987; 4 :1115. - 19.
Salin G, Caillol JM. Equilibrium molecular dynamics simulations of the transport coefficients of the Yukawa one component plasma. Phys. Plasmas. 2003; 10 (5):1220. DOI: DOI: 10.1063/1.1566749 - 20.
Donko Z, Hartmann P. Thermal conductivity of strongly coupled Yukawa liquids. Phys. Rev. E. 2004; 69 (1):016405. DOI: 10.1103/PhysRevE.69.016405 - 21.
Faussurier G, Murillo MS. Gibbs-Bogolyubov inequality and transport properties for strongly coupled Yukawa fluids. Phys. Rev. E. 2003; 67 (4):046404. DOI: 10.1103/PhysRevE.67.046404