Introduction

A nanofluid can be defined as a fluid having nanosized particles, which is named nanoparticles. The colloidal suspensions of nanoparticles are designed by these fluids [1, 2]. Oxides, metals, silicon carbides, and carbon nanotubes are the components of nanoparticles, which are used in nanofluids. Further, oil, ethylene glycol, and water are some of the common fluids [3]. Innovative features of nanofluids cause them potentially effective for lots of uses in heat transmitting, such as fuel cells, microelectronics, pharmaceutic procedures, and hybrid-powered engines, electronic cooling systems, household fridges, chillers, heat exchangers, in machining, grinding equipment and in boiler flue gas temperature decreasing [4, 5]. One of the most importance of these structures is good thermal conductivity of them [6,7,8]. In comparison with the common fluids, they indicate better thermic power of conducting and convicted heat transmitting efficiency [9]. So various types of these nanostructures are used in many reality applications for achieving high rate of thermal manner of optimized structures with nanofluids [10,11,12,13]. Reason for being critical about determining the appropriateness to convective heat transferring uses is a rise in knowledge of nanofluids rheological behavior [14, 15]. Unlike many benefits of nanofluids, some adverse phenomenon can disrupt their properties. One of the most important inappropriate phenomena is the aggregation of nanoparticles in nanofluid structures. Previously, some researchers investigated the effect of both nanoparticle size and aggregation upon viscosity [16,17,18]. Nkurikiyimfura et al. [19] proposed that nanoparticle accumulation may cause increasing thermic conductivity for nanofluids. The transfer electron micrograph picture of copper–water nanofluids was observed by Xuan et al. [20] in experiments. Not only did they fulfill a simulation of nanoparticle aggregation, but they also regarded that Brownian motion resulted in abnormal movement of nanoparticles. Murshed et al. [21] made nanofluids ready by various volume fractions of TiO2 nanoparticles. By the study of previous reports in nanofluid structures [22, 23] at various conditions [11, 48,49,50,51,52,53,54,55,56,57,58], we can say atomic manner of these structure is important for prediction of nanoparticle aggregation. Today, molecular dynamics (MD) simulation is one of the exact methods to study atomic systems manner. Historically, MD simulations were initially expanded in the late 1950s, following the earlier accomplishments of Monte Carlo simulations [24,25,26]. Its growth was based on Laplace’s foundational work [27]. In 1957, Wainwright and Alder applied a computer for simulating completely elastic collisions between hard structures [28]. In 1960, Gibson and others simulated radiation damage of solid copper by using a Born–Mayer type of repulsive interaction in addition to the united surface force [29]. The landmark simulation of liquid argon, which applied a Lennard-Jones (LJ) potential, was released by Rahman in 1964. Technically, molecular systems contain plenty of particles and determining the features of such complicated systems is not possible. By using numerical methods, MD simulation avoids this problem. Nevertheless, long MD simulations are mathematically inappropriate and they generate a lot of mistakes in numerical integration which could be decreased by appropriate choice of parameters and algorithms, but they could not be omitted completely. This computational approach is applicable to the study of various types of systems. In this work, we used this computational method to study nanofluid manner [59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78], composed of Al2O3 nanoparticles in argon fluid in the presence of external magnetic field. Atomic barrier with cone shape to nanofluid medium is a novelty in our computational study which improves or the simulation results and more adapted by reality cases.

Method

MD simulations based on atomic representations are among the most commonly used methods, to investigate nanofluids manner [30,31,32,33]. We used MD simulations to calculate the aggregation of hybrid nanoparticles in liquid argon with the first conditions. MD is a computer simulation procedure to trace evolution of molecules and atoms for a period of time. In this way, for each time step, particles can be free for interactions, giving the perception of the mechanical development of the atomic system. In general version of MD simulation, the particle trajectories are settled by solving Newton’s equations for the particle systems, at which forces between particles and their possible energies are frequently computed by interatomic force field https://en.wikipedia.org/wiki/Interatomic_potential. The results of the MD simulations are based on selection of interatomic force field. Interatomic interaction between nanofluids atoms in platinum microchannel is accounted by universal force field (UFF) and embedded atom model (EAM) force field [34, 35]. UFF is a whole atom potential having parameters for each atom. These force field parameters are calculated by using usual rules base on the element, its connectivity and hybridization. The philosophy of UFF force field is using general force constant and geometric parameters that are based on simple hybridization considerations instead of individual force constant and geometric parameters which are depended on special atom combinations using in the angle terms. In this force field, LJ potential applied for non-bond interaction between various atoms [36]:

$$\phi (r_{\text{ij}} ) = 4\varepsilon \left[ {\left( {\frac{\sigma }{{r_{\text{ij}} }}} \right)^{12} - \left( {\frac{\sigma }{{r_{\text{ij}} }}} \right)^{6} } \right]\quad r_{\text{ij}} \le r_{\text{c}}$$
(1)

where ε is the depth of the potential, σ is the limited distance at which the interatomic potential is zero, and r is the distance between atoms. In this equation, cutoff radius is indicated by rc (see Table 1 for other constant amounts).

Table 1 Length and energy of LJ potential [35, 36]
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Furthermore, the EAM potential energy of an atom, i, is given by [34, 35]:

$$E_{\text{i}} = F_{\upalpha} \left( {\mathop \sum \limits_{{{\text{i}} \ne {\text{j}}}} \rho_{\upbeta} \left( {r_{\text{ij}} } \right)} \right) + \frac{1}{2} \mathop \sum \limits_{{{\text{i}} \ne {\text{j}}}} \phi_{{\upalpha \upbeta }} \left( {r_{\text{ij}} } \right)$$
(2)

The distance between atoms i and j is shown by rij, ϕαβ is a pair-wise possible function, ραβ is the contribution to electron charge density from atom j of type β at the place of atom i, and Fα is an embedding function which represents the energy that is necessary for placing atom i of type α into the electron cloud. To calculate the particle’s movement via simulation time, Newton’s second law at the atomic level is used as the gradient of the interatomic potential in Eq. (3),

$$F_{\text{i}} = \sum\limits_{{{\text{i}} \ne {\text{j}}}} {F_{\text{ij}} = m_{\text{i}} \frac{{{\text{d}}^{2} r_{\text{i}} }}{{{\text{d}}t^{2} }} = m_{\text{i}} \frac{{{\text{d}}v_{\text{i}} }}{{{\text{d}}t}}}$$
(3)

After that, Gaussian distribution is fulfilled for calculating the temperature of particles that is shown in this formula:

$$\frac{3}{2}k_{\text{B}} T = \frac{1}{{N_{\text{atom}} }}\sum\limits_{{{\text{i}} = 1}}^{N} {\frac{1}{2}mv_{\text{i}}^{2} }$$
(4)

Newton’s law:

$$v(t + \delta t) = v(t) + a(t)\delta t$$
(5)
$$r(t + \delta t) = r(t) + v(t)\delta t$$
(6)

r(t) and v(t) are the initial rates of these mechanical parameters. In our study, every MD simulation was doing by applying Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) simulation package released by Sandia National Laboratories [37,38,39,40]. Finally, we can say that our MD simulations were done in two steps:

Step A Argon-Al2O3 nanofluid was simulated with atomic details. For this procedure, atomic structure temperature was fixed at 300 K with 1 femtosecond time step and periodic boundary condition implemented in x and y directions and fixed one used for z direction [41]. By definition of initial settings of MD simulations, atomic structure equilibrated for 1 ns. After that, the system reached to equilibrium state and computational running continued to 2 ns. In this step of MD simulation, the potential energy of atomic structures was reported to verify our MD simulations.

Step B In the second step, external magnetic field was inserted to simulation box. The simulated structures equilibrated for 1 ns at 300 K with nose–Hoover thermostat [42, 43]. In our calculations, the temperature damping rate was set to 0.01 rate. Then, the atomic interaction between atoms fulfilled for 2 ns. Finally, to analyze nanoparticle aggregation phenomena, physical parameters such as potential energy, distance of structures, and radial distribution function (RDF) were reported.

Results and discussions

Equilibrium MD consequences

At the first section, not only the atomic structure of non-ideal Pt microchannel and fluid/nanofluid but also the accuracy of atomic structure and used force fields are studied. The length of Pt microchannel in our simulations is 300 × 300 × 1000 Å3 in x, y and z directions, respectively. Further, Ar/Al2O3 nanofluid is simulated in interior space of simulated microchannel. The atomic structure can be seen in Fig. 1 [44]. In Fig. 1, it should be noted that this figure is related to Ar base fluid in non-ideal microchannel which is depicted in top, front, and perspective views. OVITO is a scientific visualization and analysis package for atomistic and particle-based simulations. Our MD simulations showed that the initial position of atoms in microchannel and fluid/nanofluid structures is adopted with UFF and EAM force fields. Physically, the stability of atomic structures is described by reporting potential energy of these structures at 300 K. Figure 2. shows the potential energy of atomic structures as a function of simulation time. Base on this figure, one can see the atomic structures energy converged after 1 ns simulation and this parameter of base fluid reached to − 272 eV.

Fig. 1
figure 1

Schematic of simulated non-ideal platinum microchannel and argon fluid by using the LAMMPS at a top, b front, and c perspective directions

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Fig. 2
figure 2

Potential energy variation of Ar fluid as a function of simulation time

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In this section, atomic structure of microchannel and Ar/Al2O3 nanofluid is studied. Figure 3 shows the simulated nanofluid with various numbers of Al2O3 nanoparticles (N = 2, 3, and 4). Our MD simulations show that the initial position of atoms in nanofluids is adopted with UFF and EAM force fields. The stability of atomic structures is described by reporting potential energy of these structures. Figure 4. shows the possible energy of atomic structures as a function of simulation time. Base on this figure, one can see the atomic structures energy converged after 1 ns simulation to − 411 eV, − 442 eV, and − 470 eV for nanofluid with 2, 3, and 4 nanoparticles, respectively. Numerically, the potential energy value increases by the number of nanoparticles increasing. This manner shows that, by Al2O3 nanoparticle adding to Ar fluid, the stability of structure rises. Physically, in atomic structures with negative potential energy, the atoms bonded to each other with attractive atomic force and so we conclude attractive force between nanofluid atoms is bigger than fluid one.

Fig. 3
figure 3

Atomic representation of the Ar/Al2O3 nanofluid structure with a 2, b 3, and c 4 nanoparticles by using the LAMMPS package

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Fig. 4
figure 4

Potential energy variation of Ar/Al2O3 nanofluid according to the simulation time and number of nanoparticles

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Dynamical evolution of atomic structures

After initial MD simulation and temperature equilibration of Pt microchannel and Ar/Al2O3 nanofluid, external force with 0.002 eV Å−1 magnitude is inserted to nanofluid and micro-canonical ensemble implemented to the next 2 ns. In statistical mechanics, a micro-canonical ensemble is the statistical condition which is used to show the probable states of a mechanical system that contains a completely specified energy. It is supposed that the system is isolated, i.e., it cannot exchange energy or particles with its environment; thus, as time passes, by saving of energy, the energy of the system remains exactly the same. In the first step of this section, we will report the center of mass (COM) distance between nanoparticles. The center of mass of a distribution of mass in space is the special point that the weighted relative position of the distributed mass reaches to zero [45]. From Fig. 5, we can see that the interatomic force is attractive one and so nanoparticles get closer to each other. Numerically, the distance of nanoparticles varies from 50 to 2.1 Å at 300 K. From Fig. 6 we can see the slope of particles distance is changing non-uniformly. This atomic manner arises from atomic interaction between particles which is described by LJ force field. In this force field, the interatomic force gets positive and negative in various atomic distances. Further, this physical parameter decreases by N increasing as reported in Table 2. This manner of atomic structures arises from increasing potential energy and attraction force between Al2O3 nanoparticles. Physically, increasing the interatomic force between nanoparticles causes more Al2O3 atoms penetration together. In next step, the time-dependent external magnetic field is inserted to Ar/Al2O3 nanofluid by the following equation:

$$B = q{\mathbf{vB}}\sin \left( {\omega t} \right)$$
(7)

q is electrical charge, v is atoms velocity, B is magnetic field magnitude, ω is field frequency, and t is the MD simulation time. By adding this magnetic field, the aggregation process occurs in shorter time. In Tables 3 and 4, the time of aggregation phenomena for different simulated structures with time-dependent magnetic field is reported. By more magnitude of external field, the time of aggregation phenomena increases. Furthermore, by increasing frequency, the time of Al2O3 nanoparticles aggregation increases and it happens because of atoms fluctuation.

Fig. 5
figure 5

Time evolution of Al2O3 nanoparticles aggregation in MD simulations

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Fig. 6
figure 6

COM distance variation of Al2O3 nanoparticles as a function of simulation time

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Table 2 COM distance of Al2O3 nanoparticles as a function of simulation time and nanoparticle number without external magnetic field
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Table 3 COM distance of Al2O3 nanoparticles as a function of simulation time and nanoparticle number with external magnetic field (B = 1 and ω = 0.1)
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Table 4 COM distance of Al2O3 nanoparticles as a function of B and ω variation at t = 1 ns MD simulation
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After equilibration procedure of atomic structures, the radial distribution function (RDF) of Al2O3 nanoparticles is calculated to investigate the Al2O3 aggregation phenomena in non-ideal Pt microchannel. In this step, g(r) is defined as follows [46, 47]:

$$g\left( r \right) = \frac{{dn_{\text{r}} }}{4\pi dr\rho }$$
(8)

In this equation, r is the distance between the Al2O3 nanoparticles respect to each other. g(r) defines the probability of finding a nanoparticle at a distance r from central nanoparticle. dnr is a function that computes the number of particles within a shell of thickness dr and ρ is the density of atomic structure. Physically, the radial distribution function is useful to analyze the spheroid distribution of the Al2O3 molecules around every nanoparticle since it is normalized by the volume of the spheroid shell. In RDF curve, the distance from the initial atomic structure, at which there is the most possibility to find a neighbor target atom, creates the first peak. It shows the most possible and the closest neighbor distance. Over large radius, the oscillations in g(r) are decreased, and it approaches the value of unity when normalized the mean density. The first and second peak of nanoparticles RDF as a function of various time-dependent magnetic field is reported in Tables 5 and 6. In addition, from Fig. 7a we can see that the attraction force between nanoparticles decreases when B increases. Further, ω is reciprocal relation with interatomic force between nanoparticles (Fig. 7b). Numerically, the most possible distance to find nanoparticles molecules varies from 1.3 to 1.7 Å by increasing B = 1 to B = 3 and this physical parameter varies from 1.3 to 1.8 Å by increasing w = 0.1 to w = 0.4. This manner of atomic structures arises from decreasing the amplitude of the atomic fluctuations by b/w increasing. Finally, increasing the amplitude of atomic fluctuation causes decreasing penetration of atoms to each other and so the aggregation of nanoparticles in simulated nanofluids occurs in bigger simulation time. This manner of nanofluids which reported in this section can be used in heat transfer applications to improve their efficiency.

Table 5 Position of the first and second peak of Al2O3 nanoparticle’s RDF as a function of B magnitude of external magnetic field
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Table 6 Position of the first and second peak of Al2O3 nanoparticle’s RDF as a function of ω rate of external magnetic field
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Fig. 7
figure 7

Radial distribution function of Al atoms as a function of sample a B and b ω parameters of external magnetic field

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Conclusions

We used the molecular dynamics method for simulating the aggregation process of hybrid nanoparticles which is inserted to Ar base fluid in non-ideal microchannel with pyramidal barriers. Our computational results from atomic simulations are as follows:

  • Universal force field and embedded atom model force field are the suitable interatomic potential to simulation of nanofluid atomic structures.

  • By increasing nanoparticles into liquid Argon, the stability of nanofluid increases and potential energy of these structures raises to − 470 eV from − 272 eV.

  • Al2O3 center of mass decreases by time passing in our simulations from 50 Å to 2.1 Å in which, by the optimal magnetic field inserting to nanofluid, this parameter changes to 2.7 Å.

  • Generally, increasing amplitude and frequency of external magnetic field leads to the aggregation delay of nanoparticles and these phenomenon occur after 2.3 ns.

  • Involving the optimal magnetic field, the first peak of this atomic structure radial distribution function varies from 1.3 to 1.8 Å.

These numerical results from our computational study can be used in industry for the design of new heat transfer apparatus.