Molecular dynamic simulation of Copper and Platinum nanoparticles Poiseuille flow in a nanochannels

Physica E 84 (2016) 152–161

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Molecular dynamic simulation of Copper and Platinum nanoparticles Poiseuille flow in a nanochannels Davood Toghraie a,n, Majid Mokhtari b, Masoud Afrand c a

Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Iran Department of Energy Engineering, Graduate school of Environment and Energy, Science and Research Branch, Islamic Azad University, Tehran, Iran c Department of Mechanical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran b

H I G H L I G H T S


LAMMPS code is used to simulate three-dimensional Poiseuille flow.
Velocity, temperature and density profiles have been obtained, and agglutination of nanoparticles has been discussed.
Using of nanoparticles raises thermal conduction in the channel.

art ic l e i nf o

a b s t r a c t

Article history: Received 30 April 2016 Received in revised form 28 May 2016 Accepted 7 June 2016 Available online 8 June 2016

In this paper, simulation of Poiseuille flow within nanochannel containing Copper and Platinum particles has been performed using molecular dynamic (MD). In this simulation LAMMPS code is used to simulate three-dimensional Poiseuille flow. The atomic interaction is governed by the modified Lennard–Jones potential. To study the wall effects on the surface tension and density profile, we placed two solid walls, one at the bottom boundary and the other at the top boundary. For solid–liquid interactions, the modified Lennard–Jones potential function was used. Velocity profiles and distribution of temperature and density have been obtained, and agglutination of nanoparticles has been discussed. It has also shown that with more particles, less time is required for the particles to fuse or agglutinate. Also, we can conclude that the agglutination time in nanochannel with Copper particles is faster that in Platinum nanoparticles. Finally, it is demonstrated that using nanoparticles raises thermal conduction in the channel. & 2016 Elsevier B.V. All rights reserved.

Keywords: Poiseuille flow Molecular dynamics Agglutination Nanofluid

1. Introduction Common fluids including Water, oil and ethylene glycol are widely used as coolant fluids, which play a significant role in industry with respect to cooling and industrial machinery functions [1–5]. As we know, cooling the machinery and manufacturing processes is considered as a limitation in design and implementations. Fluids play important parts in thermal conduction; hence, using fluids with higher heating capacity and larger better capability of thermal conduction reduces the number of cooling equipment and installations. The literature of thermal conduction has recently focused on the mechanisms that increase the capability of thermal conduction and thermal capacity of fluids used as coolants. We know that metals in solid states have a larger heat n

Corresponding author. E-mail address: [email protected] (D. Toghraie).

http://dx.doi.org/10.1016/j.physe.2016.06.006 1386-9477/& 2016 Elsevier B.V. All rights reserved.

transfer coefficient than fluids. Therefore, it is clear that the heat transfer coefficient of fluids with suspended metal particles is greater than that of the main fluid. This fact is the basis of an interesting invention called “nanofluid”, a kind of fluid in which there is solid nanometric particles. It should be mentioned that in case the size of suspended particles is in millimeters, micrometers or greater scales, no agent might exist to prevent these dimension of suspended particles. Therefore, industrial application of these fluids with metal particles in micro- or millimeter dimensions is not possible. Then, nanofluids attracted a lot of attention provoking nanofluid development of other particles like nanoparticle oxides, nanofibers and nanocarbon pipes. The results obtained by Soltanimehr and Afrand [6] and Toghraie et al. [7] showed that for low nanoparticles concentrations, the thermal conductivity could be increased up to over 20%. This increase depends mainly on such factors as particle form, particle size, particle volume fraction in the solution and the thermal features of nanoparticles and basic fluid.

D. Toghraie et al. / Physica E 84 (2016) 152–161

Here a summary of research works regarding simulation of nanofluids using molecular dynamic is presented. All the relevant fulfilled studies could be divided into two categories: The first category deals with examining thermal conduction mechanism and the second one studies flow properties. Molecular layering in the interface of fluid and solid in a nanofluid by molecular dynamic simulation is being studied. Xue et al. [8,9] studied the effect of fluid layering on the interface of the fluid and solid as well as two states of thermal resistance in the interface of fluid and solid. They were the first to deal with the mechanism of nanofluid thermal conduction. Li et al. [10] simulated Copper molecular particles in the interface of Copper nanoparticles and Argon fluid, and therefore, some physical features of the fluid including density distribution were obtained. Lu and Fan [11] discussed static nanofluids with a volume fraction below 8% by using dynamic molecular simulation. Consequently they found the influence of volume fraction and size of nanoparticles on nanofluid features, and then they compared the results with experimental findings. Using molecular dynamic simulation, Lv et al. [12] studied behavior of nanofluid flow between two parallel screens (sections) under shear stress in the Couette flow and showed nanoparticle and fluid layering. By using molecular dynamic simulation, Sun et al. [13] simulated nanofluid conduction coefficient in the Couette flow under high shear stress. Changes in nanofluid conduction coefficient were determined with regard to the changes in shear stress in this article, which showed constancy of nanofluid conduction coefficient. Using molecular dynamic simulation, Sun et al. [14] showed an unusual increase in nanofluid conduction coefficient as compared to base fluid between two parallel solid walls in the nanoscale. Base fluid, Argon and nanoconstitute the particles used in Copper. This research showed that the wider the channel, the more linear decrease would be seen in nanofluid conduction coefficient. In studies carried out by Sun [13,14], Green–Kobo formula was used to calculate conduction coefficient. Lennard–Jones potential function was also used for interaction between the particles. Xue et al. [9] showed pressure and friction processes in nanofluid. They verified the results by using experimental findings of Copper nano-oil. Kang et al. [15] examined Argon conduction coefficient and Argon and Copper nanofluid by using Green–Kobo method. In this research, Poiseuille flow was simulated at 86° K and Lennard–Jones potential was used for the interaction of Copper particles. Marroo et al. [16,17] calculated temperature distribution with a temperature slope between Argon and Platinum screens. They obtained Argon heat transfer coefficient by calculating the thermal conduction and temperature slope. The objective of this paper is to examine the effect of modified Lennard–Jones potential function on the velocity, temperature and density profiles in order to understand the Poiseuille flow within nanochannel containing Copper and Platinum particles in nanochannel geometry better. It is demonstrated that using nanoparticles raises thermal conduction in the channel. The obtained results show that this study could help us to understand the mechanism of nanoscale flow on micro- and nanoscale better.

2. Molecular dynamic simulation Molecular dynamic simulation method calculates the timedependent behavior of the system and uses equations of motion (2nd law of Newton) to find particle tracks. Therefore, by calculating the time-dependent behavior of the considered system, some system information is obtained that includes velocity and acceleration positions of nanoparticles; moreover, by using statistical methods, these microscopic results determine macroscopic features of the system, such as pressure, energy, thermal capacity,

153

surface tension, density, etc. The molecular dynamic method is a reliable and accurate method in micro- and nanoscales. Regarding the restrictions and limitations of laboratory measurements in nanoscale, molecular dynamic simulation can elaborate the aforementioned phenomena. The most important factor in the accuracy of molecular dynamic method is the choice of potential function. The value of force acted upon the particles is calculated by differentiating the potential function. The behavior of simulation is largely based on the choice of potential function. The simplest two–body potential can be written as the sum of a pairwise interaction of particles i and j at positions ri and r j , respectively,

Φ=

∑ ∑ φ (rij ) i

(1)

j>i

where rij = ri − r j is the inter-particle separation, and i and j denote the particles i and j, respectively. The atomic interaction is governed by the modified Lennard–Jones potential [18]:

⎧ ⎡ ⎛ ⎞12 ⎛ ⎞6⎤ ⎫ ⎧ ⎡ ⎫ ⎛ r ⎞2 ⎛ σ ⎞6⎤ ⎪ ⎪ ⎪ ⎛ σ ⎞12 ⎪ σ σ ⎬ ⎜ ij ⎟ φ (rij ) = ⎨ 4ϵ ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ⎬ + ⎨ 4ϵ ⎢ 6 ⎜ ⎟ − 3 ⎜ ⎟ ⎥ ⎪ ⎥ ⎢ ⎠ ⎝ ⎠ ⎪ ⎝ ⎥⎦ ⎭ ⎝ rc ⎠ ⎢⎣ rc rij ⎠ rij ⎠ ⎦ ⎭ ⎪ r ⎪ ⎝ ⎝ c ⎣ ⎩ ⎩ ⎧ ⎡ ⎛ σ ⎞12 ⎛ σ ⎞6⎤ ⎫ ⎪ ⎪ − ⎨ 4ϵ ⎢ 7 ⎜ ⎟ − 47 ⎜ ⎟ ⎥ ⎬ ⎪ ⎪ ⎠ ⎝ ⎠ ⎝ ⎥ ⎢ r r c c ⎦ ⎣ ⎭ ⎩

(2)

where ε is the maximum potential depth and a parameter corresponding to the energy, σ is the length scale parameter and rc is the cut-off radius beyond which the intermolecular interaction could be ignored. Hence, for the purpose of physical understanding, in the present study, Argon is used as the LJ fluid with the following potential parameters: m = 6.63 × 10−26 kg , s¼ 3.045 Å and ϵ = 1.67 × 10−21 J . The solid walls are represented by three layers of face centered cubic surface of Platinum molecules with parameters as: ms = 3.24 × 10−26 kg , s ¼ 2.475 Å and ϵs = 8.35 × 10−20 J. To study the wall effects on the surface tension and density profile, we placed two solid walls, one at the bottom boundary and the other at the top boundary. For solid–liquid interactions, the following modified Lennard–Jones potential function was used:

⎧ ⎡ ⎛ σsf ⎞12 ⎛ σsf ⎞6⎤ ⎫ ⎪ 4ϵsf ⎢ ⎜ ⎟ − ⎜ ⎟ ⎥ ⎬ φw (rij ) = ⎨ ⎪ ⎝ z ⎠ ⎦⎭ ⎣⎝ z ⎠ ⎩

⎪

⎪

⎧ 2 ⎡ ⎛ σ ⎞12 ⎛ σsf ⎞6⎤ ⎫ ⎪ ⎪⎛ z ⎞ sf + ⎨ 4ϵsf ⎢ 6 ⎜ ⎟ − 3 ⎜ ⎟ ⎥ ⎬ ⎟ ⎜ ⎪ ⎝ rc ⎠ ⎝ rc ⎠ ⎥⎦ ⎭ ⎢⎣ ⎝ rc ⎠ ⎪ ⎩ ⎧ ⎡ ⎛ σ ⎞12 ⎛ σsf ⎞6⎤ ⎫ ⎪ ⎪ sf − ⎨ 4ϵsf ⎢ 7 ⎜ ⎟ − 47 ⎜ ⎟ ⎥ ⎬ ⎪ ⎪ ⎠ ⎝ ⎠ ⎝ ⎥ ⎢ r r c c ⎦⎭ ⎣ ⎩

σ Sf

⎡ ⎢ 1 εσ12 = ⎢ 13 2 εσ 6εsσs6 ⎢⎣

εsf =

(3)

1

1 ⎞13⎤ 6 ⎛ ⎜ 1 + ⎛ εsσs12 ⎞ 13 ⎟ ⎥ ⎜ 12 ⎟ ⎟ ⎥ ⎜ ⎝ εσ ⎠ ⎝ ⎠ ⎥⎦

εσ 6εs σs6 σsf 6

(4)

(5)

where, the Kong mixing rule is applied to the system for calculating σsf and εsf . For calculating σsf and εsf , there are other mixing rules such as Lorenthz–Berthelot and Waldman–Kugler rules. These mixing rules are represented as follows, Lorenthz–Berthelot:

εSf =

εS εf ,

σ Sf =

σ S + σf 2

(6)

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Waldman–Kugler:

ϵf σ f6 ϵs σs6

ϵsf =

σsf6

⎛ σ6 + S , σ Sf = ⎜⎜ 2 ⎝

1 σ f6 ⎞ 6

⎟⎟ ⎠

(7)

When a two-body potential model is chosen, the interaction force between a pair of molecules can be derived from a potential model such as the following relation:

→ Fi =

phase with the positions. The initial condition for each molecule is usually assigned by using the Gaussian distribution. However, in order to make sure that the steady state can be easily reached after some reasonable time, the randomly distributed velocities should satisfy the following formula,

1 Natm

Natm

∑ i=1

1 3 m vi2 = kB T 2 2

(14)

→ ∑ Fij + Fext i≠j

→ → Fij = − ∇ (ϕ + ϕw )

(8)

→ → where Fext = Fext i is the inlet external driving force per molecule in the x direction to characterize the forced flow in nanochannels. The cut-off radius for the LJ potential in all simulations is rc = 4σ . By integrating Eq. (8), the position vector ri and velocity vector vi can be calculated, and then the thermodynamical properties of simulation domain can be calculated using ri and vi . Under such a situation, a great amount of heat was applied to the system and thermostats were applied to all wall molecules or fluid molecules, but the obtained results were not realistic. In order to extract heat from the system, velocity rescaling (Berendsen thermostat) is used for the bins near the walls. In this method, the near wall region is divided into three bins. (Along the flow direction) ( 1σ in width). Also the velocity rescaling method acts in each of them independently. In this method, the wall atoms are assumed to have an infinite mass and therefore remain static in their original positions. This does not allow for heat exchange via the walls, and the artificial thermostats (near the walls) extract heat from the system. For velocity rescaling, the velocities are first updated from the forces acting on the particles and then rescaled at each time step according to:

1

⎛ Tt arget ⎞ 2 → → ⎟ Vnew = λ Vold where λ = ⎜ ⎝ Tglobal ⎠

(9)

where Tt arget is the target temperature and Tglobal is the global temperature of the fluid particles in the system. Because the total kinetic energy of a system fluctuates, the instantaneous temperature is defined as a fluctuation kinetic energy per particle at each slab per degree of freedom;

T (z ) =

1 2

(10)

where N (z ) is the number of atoms at the z height. The engine of a molecular dynamics program is its integration algorithm, required to integrate the equations of motion of the interacting particles and follow their trajectories. Integration algorithms are based on the finite difference methods with discretized time and the time step equal to δt . Knowing the positions and some of their time derivatives at time t, the integration scheme gives the quantities at a later time (t + δt ). With such procedure, the evolution of the system can be followed for a long time. Velocity Verlet method is used for the integration of the Newtonian equation,

r (t + δt ) = r (t ) + v (t )δt +

F (t ) δt 2 m 2

⎛ δt ⎞ 1 F (t ) δt v ⎜ t + ⎟ = v (t ) + ⎝ 2⎠ 2 m

⎛ δt ⎞ 1 F (t + δt ) δt v (t + δt ) = v ⎜ t + ⎟ + ⎝ 2⎠ 2 m

Boundaries along the x-and y-directions are periodic, which means that as a particle moves out of the simulation box, an image particle moves in to replace it. But in the z direction, the system is bounded by two solid walls. The thermal wall boundary condition is applied for particles near the walls. In this method, when the liquid particle is near the wall surface, the repulsive force will push the liquid particle away from the wall surface, but some liquid particles may collide with the wall surface. When a particle collides with a wall at the temperature Twall , all three components of velocities are reset to a biased Maxwellian distribution,

vxi =

kB Twall ψ1 mi

(15)

vyi =

kB Twall ψ2 mi

(16)

−2kB Twall ln ψ3 mi

vzi = ±

(17)

where kB is Boltzmann constant, mi is the mass of each molecule, ψ3 is a uniformly distributed number in range of ( 0, 1) and ψ1 and ψ2 are Gaussian-distributed random numbers with zero mean and unit variance. In order to compute density, streaming velocity and temperature profiles, the domain is divided into Nbin bins along the z-direction, each bin with a volume of Lx  Ly  Δz , where Δz is equal to ( L z /Nbin ). The number of bins depends on the particle number of particles. For a large number of particles, more bins are divided. For adding molecules in each layer, Hn (zi ) is defined as,

Hn (zi, j ) = 1 (n − 1)Δz ≤ zi ≤ nΔz

2 ∑ mi (Vxi + V2yi + Vzi2 )

3N (Z ) KB

3. Boundary conditions

(11) (12) (13)

Hn (zi, j ) = 0 otherwize

where the subscript j represents the ( Δp/L )2 time step. Hence, the average density in the nth slab from time step Istart to Iend is, I

ρ (z ) =

N

∑i =atm m ∑ jend Hn (zi, j ) = Istart 1 L x L y Δz (Istart − Iend + 1)

(19)

where m is the mass of each molecule and zi is the coordinate of the mid-point of the nth slab. When sampling begins, the molecule number in each slab is accounted and the local density can be considered. The simulation produces steady-state velocity profiles. The slab average velocity from time step Istart to Iend is computed as,

vx =

Iend

1 (Istart − Iend + 1)

N ∑i =atm 1

Hn (zi, j )

Natm

∑ ∑ j = Istart i = 1

Hn (zi, j ) vix, j

(20)

vix, j

where is the velocity in x component (flow direction) of particle i at time step j, and the temperature of the system is defined as:

10−15

A time step δt is about 10 fs (1 femtosecond¼ 1 fs¼ s). Smaller time steps will not result in any change in the flow field. The advantage of this algorithm is to calculate the velocities in

(18)

N

T=

∑i =atm m ( vxi2 + vyi2 + vzi2) 1 3kB Natm

(21)

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4. Simulation and results

Xu et al. Present data

1.4

In this simulation LAMMPS code is used to simulate three-dimensional Poiseuille flow. The following features are used for the simulation: Lennard–Jones potential with the parameters in Table 1. FCC network with network constant 0.7. Constant temperature thermostat (T* ¼0.71). Integration algorithm of velocity. Periodic boundary condition aligned with flow (X). Exerting the forces 0.066, 0.033, F* ¼0.1.

Many studies have been conducted about simulation of flow within Poiseuille and Couette nanochannels in recent years [18– 20]. Xue and Zhou [20] studied liquid Argon between Platinum screens (Fig. 1). They used Lennard–Jones potential function to examine the interaction of existing particles in solution domain and applied Maxwell thermal wall model as well because of lack of absorption of liquid Argon particles within walls. 1600 Platinum particles were applied in the wall and 1372 liquid Argon particles were located in the middle of the flow. Dimensionless forces 0.002, 0.1 and 0.5 acted upon the three dimensional Poiseuille flow, and the appropriately adapted results were obtained. These results are shown in Fig. 1.

1

U*









1.2

0.8 0.6 0.4 0.2 0

0

2

4

6

Y/σ

8

10

12

Fig. 1. Velocity profile aligned with the flow by exerting dimensionless force 0.1 and comparing with [20].

4.1. Simulation of Poiseuille flow within nanochannel Assume a channel with dimensions of 30s × 24s × 8s and with smooth walls as in Fig. 2. Particles are put in the solution domain by using FCC network, and accordingly 4000 Argon particles and 1440 Copper or Platinum particles are put within the walls. The flow simulation is studied regarding the conditions mentioned in previous sections, and then velocity and temperature profiles for various dimensionless forces F* ¼0.1, 0.066, 0.033 are examined (Figs. 3 and 4). As in Figs. 3 and 4, with increased force exerted to the flow, the maximum velocity in the channel increases, because by increasing the force, the acceleration of particles rises too, and with increased acceleration, a raise in velocity can also be seen. Velocity of Argon particles adjacent to channel wall is low because of interaction with Copper or Platinum particles within the wall. This happens because the energy depth of wall particles (Copper and Platinum) is more than that of Argon, and influence further particles near the wall; therefore, their velocity falls and the velocity near the wall is also reduced. Since the kinetic energy is in a direct relation with the system or particles temperature, it can be concluded that with the faster particles, or in other words, the stronger force exerted to the particles, the fluid temperature within the nanochannel will be higher. 4.2. Simulation of Poiseuille flow within nanochannel by adding copper and platinum nanofluids Smooth channel is simulated in the simulation in two positions regarding two, three and four Platinum and Copper nanofluids. Based and according to FCC network, the number of particles in each simulation is indicated in Table 2. The number within parentheses in the column related to number of nanoparticle Table 1 The parameters of Lennard–Jones potential.

σAr − Ar = 3.4*10−10

εAr − Ar = 1.65*10−21

σAr − pt = 3.2425*10−10

εAr − pt = 1.10538*10−21

mAr = 6.63*10−26

mPt = 4.95mAr = 32.82*10−26

Fig. 2. A perspective of modeled nanochannel.

indicates the same number of particles constituting spherical nanoparticles. A section of simulated channels including the number of various nanoparticles is shown in Fig. 5. It should be emphasized that the diameter of each added nanoparticle is 3s. The number of particles before accurate speculation of velocity profiles, etc. and the time of simulation equilibrium should be determined in order to calculate the mean of the obtained results, and hence to derive the velocity, temperature and density profiles. Velocity profiles for the number of various nanoparticles for exerting dimensionless force F¼0.046 is shown in Fig. 6. According to this curve, it can be

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F*=0.1 F*=0.066 F*=0.036

0.8

U*

0.6

0.4

0.2

0 0

5

10

Y/σ

15

20

Fig. 3. Velocity profile of nanochannel by exerting different forces.

F*=0.1 F*=0.066 F*=0.036

0.8

ρ∗

0.6

0.4

0.2

0

0

5

10

Y/σ

15

20

Fig. 4. Temperature profile of nanochannel by exerting different forces.

Table 2 Number of particles existing in solution domain in smooth channel simulation.

Smooth channel

Number of nanoparticles (same Number of number of particles constitut- Argon particles ing spherical nanoparticles)

Number of Copper particles

2 (176) 3 (268) 4 (361)

1440 1440 1440

3604 3512 3419

concluded that calculating the mean and extracting the diagrams are possible in 1 million iterations. Figs. 6 and 7 show the effect of various forces on velocity profile across the nanochannel. As can be seen, with increased force, the velocity profile and the rate of maximum velocity increase, too. Based on the equation of Newton's second law, particle acceleration also rises with increased force, and therefore, particle velocity increases too as it can be observed in these diagrams.

Fig. 5. Front view of modeled nanochannels regarding number of various nanoparticles.

D. Toghraie et al. / Physica E 84 (2016) 152–161

157

Iterations

500,000 1,000,000

0.4

2,000,000 1,500,000

0.3

Cu-2NP Cu-3NP Cu-4NP

0.8

ρ∗

U*

0.6

0.2

0.4

0.1

0

0.2

0

5

10

Y/σ

15

0

20

Fig. 6. Velocity profile of smooth channel by exerting dimensionless force 0.046 in different time iterations for two copper nanoparticles.

5

10

Y/σ

15

20

Fig. 8. Temperature profile of smooth channel by exerting a dimensionless force 0.036 with copper nanoparticles.

1.6

F*=0.036 F*=0.066 F*=0.1

1.4

0

F*=0.1 Pt-2NP Pt-3NP Pt-4NP

1.4

1.2 1.2

1 0.8

U*

U*

1 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

5

10

Y/σ

15

20

0

0

5

10

Y/σ

15

20

Fig. 7. Velocity profile of nanochannel by exerting different forces with two copper nanoparticles.

Fig. 9. Velocity profile of smooth channel by exerting a dimensionless force 0.036 considering number of different platinum nanoparticles.

According to Figs. 8 and 9, it is clear that the velocity and temperature in nanochannel increase, with increased number of nanofluid or Platinum or Copper nanoparticles in the Argon flow. This happens because with increased number of nanoparticles in nanochannel, the number of Argon particles decreases. Hence, in a constant force, regarding the reduction in the number of Argon particles and therefore mass of Argon particles, particle acceleration increases, and subsequently, temperature and velocity rise, too (according to Newton's second law). As kinetic energy is directly related to second power of velocity and on the other hand energy is in a positive relation with the temperature, it can be concluded that with increased force, velocity and therefore temperature increase as indicated in Fig. 9. As in Fig. 10, by increasing the force, the velocity and consequently the temperature increase, too. As mentioned before, with increased force, velocity rises too, and as kinetic energy is directly

related to second power of velocity and, on the other hand, energy is in a positive relation with the temperature. It can be concluded that velocity and therefore temperature increase by increasing the force. As can be seen in Fig. 11, the larger the number of nanoparticles in the nanochannel, the lower the number of Argon particles in the nanochannel would be and, as a consequence, the lower the density profile of the fluid. In a nanochannel with four nanoparticles, the number of Argon particles is lower because of nanoparticles in solution domain, and density profile is lower than the position with two or three nanoparticles. Since the energy depth of Copper particle is 73.4 as larger as than Platinum particle and the size of Copper particle is 0.758 as larger as Platinum, and considering that the force is calculated by potential function derivation, Lennard–Jones potential could indicate that force exerted to Copper particles is larger than that actedacting upon Platinum

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1.4

1.8

Pt-4NP Pt-3NP Pt-2NP

1.3

F*=0.1 Pt-2NP Pt-3NP Pt-4NP Cu-2NP Cu-3NP Cu-4NP

1.6 1.4

1.2 1.2 1

U*

T*

1.1

0.8

1

0.6

0.9

0.4

0.8 0.2

0.7

0

5

10

Y/σ

15

0

20

Fig. 10. Temperature profile of nanochannel with the dimensionless force F¼ 0.1 and platinum nanoparticles.

Pt-4NP Pt-3NP Pt-2NP

1

ρ∗

0.8

0

5

10

Y/σ

15

20

Fig. 12. Velocity profiles obtained from two copper and platinum nanochannels for two and four suspended nanoparticles.

Table 3 Time or number of agglutination iteration of nanoparticles in nanochannel with roughness ratio of 2.5.

Two Three Four

Platinum

Copper

Agglutination 890,000 650,000

Agglutination 680,000 500,000

0.6

0.4

0.2 Fig. 13. A view of rectangular roughness and a/b ratio.

0

0

5

10

Y/σ

15

20

Table 4 Different kinds of modeled roughness in simulated nanochannel. Channel

I

II

III

a/b

2

2.5

3

Fig. 11. Density profile of channel considering various platinum nanoparticles.

particles. There upon, velocity profiles of Copper nanochannels shown in Fig. 12 are higher than those of Platinum. In this graph, velocity profiles related to a nanochannel in which there are nanoparticles are indicated by a filled-in sign, and lines and hollow sign show a nanochannel with Platinum nanoparticles. Accordingly, with increased number of nanoparticles, the maximum velocity increases too, as described before. 4.3. Agglutination of nanoparticles Agglutination indicates attachment of particles including the factors existing constantly in simulation of the flow within nanochannels in the case of low distance between particles or large number of nanoparticles. As stated, in some results such as velocity results, velocity was maximized because of agglutination. It is crystal clear that in the experienced simulations, the fastest agglutination relates to nanochannels with four nanoparticles.

Table 3 shows the agglutination time of each nanoparticle. This table shows that Copper nanoparticles are agglutinated faster than Platinum. Regarding the fact that the energy depth and Copper thickness parameters are respectively 73.4 and 0.758 as larger as Platinum, it can be said that the repulsion and absorption forces in Copper particles are stronger those in Platinum, and the force exerted from Copper to Argon and Copper particles is larger than the same state in Argon and Platinum. Finally, we can conclude that the agglutination time in nanochannel with Copper particles is faster that in Platinum nanoparticles (Fig. 13). 4.4. Simulation of Poiseuille flow within the rough channel by adding copper and platinum nanofluids A nanochannel with previous nanochannel dimensions is considered in this section and rough walls are assumed instead of

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1

a/b = 2.0 a/b = 2.5 a/b = 3.0

0.8

ρ∗

0.6

0.4

0.2

0

0

5

10

Y/σ

15

20

Fig. 15. Density profile of nanochannel with four copper nanoparticles for different rectangular kinds of roughness.

F*=0.036 - Cu - 4NP 300,000 800,000 1,000,000 1,200,000 1,500,000

0.8

U*

0.6

0.4

0.2

0 0

5

10

Y/σ

15

20

Fig. 16. Velocity profiles of nanochannels with four copper nanoparticles for rectangular roughness 2 in different time steps.

Fig. 14. A view of modeled rough nanochannels with four nanoparticles and difa a a ferent rough rectangular geometries A: =2. 0 , B: =2. 5, C: =3. 0 . b

b

b

smooth walls. Different ratios of rectangular roughness are shown in the following table: (Table 4). A view of modeled nanochannels in different times of simulation is shown in Fig. 14. Regarding the geometry of all kinds of roughness in the desired nanochannels, the density profile for a nanochannel in which there are four Copper nanoparticles (Fig. 15) is obtained. Regarding the roughness geometry and the number of Argon particles existing in the roughness, it can be said that the largest number of particles are incarcerated in the roughness with ratios “2” and “2.5”, and the smallest number of particles are stuck in nanochannel with the roughness ratio of “3”. It should also be mentioned that in the roughness ratio of “3”, a lower density of Argon particles could be observed in solution domain, which

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F*=0.1 - Pt 4NP

3

F*=0.036 - Cu - 4NP

a/b=3.0 a/b=2.5 a/b=2.0

2.5

0.6

U*

2

U*

a/b = 3.0 a/b = 2.5 a/b = 2.0

0.8

1.5

0.4

1

0.2 0.5

0 0 0

5

10

Y/σ

15

0

20

Fig. 17. Velocity profiles with four platinum nanoparticles and various kinds of rectangular roughness.

10

Y/σ

15

20

Fig. 19. Velocity profiles of copper nanochannels for four suspended nanoparticles.

a/b = 3.0 a/b = 2.5 a/b = 2.0

2.5

5

a/b = 3.0 a/b = 2.5 a/b = 2.0

1

0.9

T*

T*

2

1.5

0.8

1 0.7

0.5

0

0

5

10

Y/σ

15

20

Fig. 18. Temperature profiles with four platinum nanoparticles and all kinds of rectangular roughness.

indicates the agglutination of Copper nanoparticles in this nanochannel. Relevant suitable “time step” needs to be calculated in order to achieve the results. Therefore, in the nanochannel with four Copper nanoparticles, the velocity profiles in different iterations are obtained (Fig. 16). To prevent similar repetitive results, only those related to rough nanochannels are considered, with three types of roughness and four nanoparticles in the solution domain. As can be observed in Figs. 17–20, a larger number of Argon particles are placed in the roughness as compared to roughness ratio “3”, in compared with the of roughness ratios of “2” and “2.5”. Regarding the same number of particles in three channels, the number of Argon particles in the center of the channels with roughness ratios of 2 and 2.5 decreases, and with reduced number of particles in the center of nanochannel, the particles in the center are accelerated and

5

10

Y/σ

15

20

Fig. 20. Temperature profiles of copper nanochannels for four suspended nanoparticles.

velocity is increased. As stated earlier, with increased velocity, temperature also increases. The mentioned figures indicate this fact. In roughness ratio of 3, agglutination or conglomeration of nanoparticles leads Argon particles towards nanochannel walls and the number of Argon particles in the middle of channel is reduced because of accumulation of Copper nanoparticles; and for exerting constant forces, Argon particles are accelerated and therefore, velocity rises. According to previous section and the existing analysis, since energy depth of Copper particle is 73.4 as large as that of Platinum particle and at the same time Copper particle size is 0.758 as large as that of Platinum, it can be concluded by using Lennard–Jones potential that the force acting upon Copper particles is larger than the force that acting on the Platinum particles. Hence, the velocity profiles of Copper nanochannel shown in Fig. 21 are higher than those of Platinum nanochannel.

D. Toghraie et al. / Physica E 84 (2016) 152–161

1.4

a/b = 2.0, F*=0.066, 4NP Cu-4NP Pt-4NP

1.2 1

U*

0.8 0.6 0.4 0.2 0 0

5

10

Y/σ

15

20

Fig. 21. Velocity profiles of two nanochannels with copper and platinum nanoparticles for four suspended nanoparticles.

5. Conclusion In this research, Poiseuille flow of liquid Argon with dimensions of 30s × 24s × 8s was modeled and simulated by using molecular dynamic simulation method. Smooth and rough nanochannels were simulated by considering three kinds of rectangular roughness and by putting one, two, three or four Copper and Platinum particles in the solution domain or simulated channel; then velocity, temperature and density profiles were obtained by exerting different forces to the solution domain or the nanochannel. All the results were obtained after equilibrium of the solution domain. The results of all simulation positions were compared. Finally, according to the drawn diagrams, it can be found that, velocity and consequently temperature of nanochannel increase with increased number of nanofluid, which is in turn an effective factor in increasing the thermal conduction and energy optimization. The extension of this paper and our previous works [21–29] affords engineers a good option for nanochannel simulation.

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