Issue
Eur. Phys. J. Appl. Phys.
Volume 84, Number 3, December 2018
Materials for Energy harvesting, conversion and storage (Icome 2017)
Article Number 30902
Number of page(s) 11
Section Physics of Energy Transfer, Conversion and Storage
DOI https://doi.org/10.1051/epjap/2018180200
Published online 14 January 2019

© EDP Sciences, 2018

1 Introduction

Carbon dioxide is being studied extensively because of many varied reasons: as the focus of sequestration efforts because of its global warming potential, as solvent for enhanced oil recovery (EOR) from oil wells, and as a well-fracturing fluid for the oil and gas industry [1,2]. The prime drawback of using CO2 for EOR is its low viscosity at reservoir conditions, leading to low sweep efficiency [2], which causes CO2 “fingers” toward production wells rather than displacing the oil ahead of it. As CO2 preferentially flows into the high permeability layers, this may also lead to reduced vertical sweep efficiency in stratified reservoirs. By increasing the viscosity of CO2, there can be a remarkable improvement in the oil recovery rate. The increase in viscosity of CO2 would also result in increased well-fracturing efficiency. In the past, various studies have reported increase in viscosity of CO2 using styrene/fluoroacrylate copolymers, introduction of toluene or ethanol to CO2 [3]. One way to enhance its viscosity is through the addition of nanoparticles, which makes it imperative to estimate properties of CO2 nanofluids. Various studies that aim to predict the rheological and thermal properties of CO2 using rigid and flexible models have been reported earlier [4,5]. However, we found that at the temperatures and pressures expected in a typical EOR or well-fracturing operation, the TraPPE model with flexible bond and angle using MORSE potential best predicts the properties of the CO2 base fluid. This particular combination has not been used to predict properties, to the authors' best knowledge. Further, it is complicated and expensive to determine self-diffusion coefficients [6,7] and viscosities [8,9] experimentally.

Nanofluids [10,11] are suspensions composed of a conventional base fluid and nanosized particles scattered in the fluid, primarily aiming at the enhancement of thermal conductivity k and heat capacity c p. But recently, nanofluid applications have expanded to take advantage of their rheological properties (viscosity μ), in addition to their thermal properties. In this study, the nanofluid consists of Al2O3 nanoparticle in CO2 base fluid. Al2O3 is the most commonly used nanoparticle experimentally due to its chemical stability and mechanical strength. Lee et al. [12] investigated the viscosity of nanofluids, in which water is the base fluid. They suggested a relationship between particle volume fraction and temperature with viscosity. They also showed an increase in viscosity with Al2O3 volume fraction and considerable decrease with temperature. Wang et al. [13] have showed up to 86% enhancement in viscosity using 28 nm Al2O3 particles. Most of the studies have shown the effect of particle volume fraction and temperature on viscosity. However, the effect of particle size on viscosity has not been studied as extensively even though it is one of the most crucial properties in nanosystems. Bennacer et al. [14] and Oueslati et al. [15] modified the heat transfer governing equation by considering the variation viscosity when nanofluids were used experimentally. Nguyen et al. [16] reported that μ increases with Al2O3 particle size. However, this finding is contradicted by findings of Pastoriza-Gallego et al. [17], who found that μ decreased with the particle size for Al2O3 nanofluids. Since it is not easy to control the particle size distribution in experimental studies, simulations can be used to estimate thermal and rheological properties under a set of assumptions [18,19]. Furthermore, a more elaborate and useful picture of molecular structural features can be obtained by these simulations, thereby providing additional insights.

To the best of authors' knowledge, the viscosity of nanofluid in gas or supercritical phase has not been reported in open literature so far. Several studies have been done broadly on the diffusion and transport of nanoparticles in gases and supercritical state, which corroborates the presence of nanoparticles in both these phases [20,21]. In this work, molecular dynamics (MD) simulation is performed using large-scale atomic/molecular massively parallel simulator (LAMMPS) [22] to investigate the effect of particle size, temperature and Brownian motion on the enhancement of viscosity of novel Al2O3–CO2 nanofluids. To study the particle size effect on the viscosity of the nanofluid, nanoparticles having 1, 2 and 3 nm diameter are considered. The computer simulation results of the base fluid are compared with the experimental values to evaluate the performance of our flexible model and simulation techniques. In this study, we predict the presence of a nanolayer (denser region), which can be con sidered as one of the reasons for the enhanced viscosity, while the other being the reduced self-diffusion coefficient of CO2 molecules in the nanofluid.

2 Computational methods

Many efforts have been devoted to the development of accurate potential model for CO2 molecule, and therefore, many models exist in the literature. A specific CO2 model is usually accurate for predicting certain properties, thus requiring the researchers to select an appropriate model for a specific purpose. The first task of the paper, hereby, is to evaluate the existing models such as MSM_flexible, TraPPe_flexible and EPM2_flexible [4,5] for predicting viscosity under supercritical conditions. Previous studies have shown that flexible models are best for determining CO2 properties and hence in our work, we have used the relatively recent TraPPE_flexible model with MORSE potential (instead of harmonic potential used to model CO2 by previous researchers). The intermolecular potential consists of long-range Coulombic interactions, and a shifted and truncated 12-6 Lennard-Jones (LJ) potential (Eqs. (1)(3)) [23].(1) (2) (3)

where rij is the distance between atoms i and j, εij and σij are LJ potential parameters, and rC is the cutoff radius. The LJ interaction parameters between different types of atoms were calculated from the Lorentz–Berthlot (LB) mixing rule (Eqs. (4) and (5)) [23]:(4) (5)

The Coulombic interactions are given by equation (6):(6)

where qi and qj are the partial charges on atoms i and j; and ε 0 is the dielectric constant of vacuum. In ourwork, we have used the particle–particle particle–mesh solver implemented in LAMMPS for electrostatic interactions [22]. LJ-potential parameters are summarised in Table 1.

For the fully flexible model, additional function is used to describe bond stretching (harmonic potential Eq. (7) or Morse potential Eq. (8) and angle bending of CO2 Eq. (9))(7) (8) (9)where ϕ S , ϕ M and ϕ M describe the bond stretching by harmonic potential, morse potential and angular stretching by harmonic potential; rij is the distance between atoms i and j; θijk is the angle between atoms i, j, k; kS , kM and k B are the force constants. The non-bonding parameters for all models including the TraPPE model are listed in Table 1. The harmonic equation (7) is not able to describe the separation of atoms at longer distance [4]. Hence, a Morse potential for TraPPE_flexible is used and termed as TraPPE_flex2. Besides TraPPE flexible model using MORSE potential, previous researchers have worked on EPM2 rigid and EPM2 flexible model using harmonic and MORSE potential [4], MSM rigid and flexible model using harmonic potential [4] and TraPPE rigid and TraPPE flexible model using harmonic potential [4,5]. Atomic interactions within Al2O3 are modelled using the potential function developed by Vashishta et al. [24]. The interaction between Al2O3–CO2 plays an important role in determining the viscosity of the nanofluid. We used quantum chemistry (Gaussian) calculations to compare the Al2O3–CO2 interaction strength with the potential used in molecular mechanics (MM). To simulate the situation of Al2O3 particle in CO2, a slab model of Al2O3 is constructed (Fig. 1). We used Gaussian 09 [25] to perform a quantum mechanics calculation with B3LYP functional [26,27] and a full-electron 6-311+G(d,p) basis set [28] (Fig. 2). Geometry optimisation and potential energy scan is made before calculating the interatomic force.

Using quantum mechanics, it has been shown that the intermolecular attraction between Al2O3 and CO2 molecules is well characterised by using parameters obtained by LB mixing rule. However, one must verify the different nanoparticles and base fluid system interaction with quantum chemistry analysis before using LB mixing rule.

MD simulations are performed in the canonical ensemble (NVT) and visualised by visual molecular dynamics (VMD) [29]. The Nose–Hoover thermostat is used for maintaining the constant temperature conditions of the system. Spherical region is carved out by inserting Al2O3 nanoparticle in three different configurations. Figure 3 shows the simulation box of 2 nm nanoparticles with 1490 molecules.

The size of the simulation domain (90 × 90 × 90 Å) having periodic boundary conditions in all the three directions is varied to have constant bulk density of dense CO2 (150 kg/m3). The two phases (i.e. CO2 and solid nanoparticle) present in the domain are grouped separately. Minimisation is done to remove close contacts and thus avoid high potential energy collisions. Sufficient time steps are performed to achieve equilibrium state for CO2 molecules surrounding the nanoparticle while keeping the nanoparticle immobile, which is under the microcanonical ensemble (NVE) and Langevin thermostat. To achieve equilibrium state for the nanoparticle separately, vice versa is done. Then, canonical ensemble (NVT) is used for the whole system before switching to NPT. The pressure and temperature ranges are 54–200 bar and 300–700 K, respectively, in the simulation setup with different nanoparticle size, such that a constant bulk fluid density is maintained. Then, fluctuation of autocorrelations is performed under the microcanonical ensemble (NVE) for data computation to calculate viscosity for each nanofluid system. The same procedure is followed for all the three systems. Newton's equations of motion are integrated using the velocity Verlet algorithm [22] with a sufficient time step of 4 fs.

The mean square displacement (MSD) was calculated for gas molecules and solid nanoparticles using the atom's position at different intervals of time. The distance moved by an atom/molecule was measured using MSD, which is defined as follows:(10)where r i (t) is the position of ith atom at time t and (ri(t) − ri(0)) is the displacement of ith atom over a time interval t. An estimate of the number of colliding atoms in the simulation was determined by MSD.

MD method calculates the viscosity μ of fluid based on Green–Kubo formalism by integral of the auto correlation function of the pressure tensor [23] via equation (11).(11)

where V is the system volume, k B is the Boltzmann constant and T is the temperature. Pαβ is an off-diagonal (αβ) element of the pressure tensor, which for an N-particle system is calculated using equation (12) [30].(12)

where mi , vi , ri and fi are mass, velocity, position and force of the atom i, respectively. The first term of equation (12) represents kinetic energy, and the second is the virial term. The atom position, force and velocity information are recorded in each timestep.

Table 1

LJ parameters used for carbon–carbon and oxygen–oxygen interaction for several CO2 models. Parameters for flexible models with force constants [4].

thumbnail Fig. 1

(a) Al of Al2O3 facing O of CO2 and distance is varied. (b) O of Al2O3 facing O of CO2 and distance is varied.

thumbnail Fig. 2

Interatomic force between atoms of alumina and CO2 are compared using quantum mechanics (QM) and molecular mechanics (MM) via LAMMPS. (a) Al–O being aluminium atom facing oxygen atom of CO2. (b) O–O being oxygen atom of alumina facing oxygen atom of CO2.

thumbnail Fig. 3

Cross-sectional view of the Al2O3–CO2 nanofluid with 2 nm diameter under investigation.

3 Results and discussion

3.1 Transport property (viscosity) of models

First, we checked the fidelity of our simulations by comparing our results for the base fluid with corresponding experimental values and with other simulations that use four different CO2 models (Fig. 4), i.e. TraPPE_flex2 (using MORSE potential), EPM2_flex (using MORSE potential), TraPPE_flex1 (using harmonic potential) and MSM_flex (using harmonic potential), for 300 < T < 700 K. The viscosity predicted from the TraPPE_flex2 model is closest to the experimental values [31] in the temperature and pressure ranges used in this work, with a maximum relative error of 7.66%. In addition, the viscosity calculation is more accurate at higher temperatures for this model. Therefore, TraPPE_flex2 appears to be the best model for predicting viscosity of CO2. As seen, the introduction of molecular flexibility by MORSE potential to the existing TraPPE flexible model appears to slightly improve the accuracy of viscosity prediction.

thumbnail Fig. 4

(a) Comparison of simulated viscosity of the base fluid (gaseous phase) of different flexible models with the experimental data [31] at different temperatures and pressures. (b) Comparison of simulated viscosity of the base fluid (supercritical phase) of different flexible models with the experimental data [31] at different temperatures and pressures.

3.2 Density distribution

The nanolayer surrounding the nanoparticle (Fig. 5) can be considered as a region having higher density than the bulk fluid density. The thickness of the nanolayer can be estimated by analysing the density distribution of CO2 molecules within the domain. For this, the computational domain is divided into several (hollow) spherical bins, and the average density of CO2 molecules in each bin is estimated.

Figure 6 shows the effects of nanoparticle diameter and temperature on density distribution. In the vicinity of the nanoparticle, CO2 density is the highest (ρ max) (relative to the bulk density ρ bulk). Based on the density distribution of CO2 molecules, highest thickness of nanolayer is observed for 3 nm diameter nanoparticle and lowest for 1 nm diameter nanoparticle. Therefore, thickness of the nanolayer can be considered to be a function of nanoparticle diameter. Table 2 shows the approximate nanolayer thicknesses for different nanoparticle diameters. It is seen that with increase in temperature, the density ratio (ρ max/ ρ bulk) decreases, which is due to the weak adsorption. Also, it can be observed from the figure that the thickness of the nanolayer is unaffected by changes in temperature. This means that temperature plays a significant role in changing the viscosity of the system but not the nanolayer thickness. Additional peaks are observed in density ratio curve for higher nanoparticle diameters, indicating an increasing nanolayer thickness.

thumbnail Fig. 5

Schematic of the Al2O3 nanoparticle and nanolayer formation around the nanoparticle.

thumbnail Fig. 6

Effect of nanoparticle diameter and temperature on the density of CO2. (a) at 300 K, (b) at 330 K, (c) at 550 K and (d) at 700 K. Similar observation is seen with other temperatures (not shown).

Table 2

Nanolayer thickness values for different nanoparticle diameter.

3.3 Mean square displacement

The mean square displacement (MSD) of base fluid particles in the nanofluid are calculated and then compared with that of the base fluid (in the absence of nanoparticle, Fig. 7). It is observed that the MSD of CO2 molecules in nanofluid is lower compared to the base fluid. The non-bonded interactions between CO2 molecules present in the nanolayer and the bulk fluid increase with nanoparticle diameter due to more CO2 molecules in the nanolayer. As expected, the investigation revealed a decrease in MSD slope for base fluid having larger particle diameter, which indicates that Brownian motion does not contribute to the increase in μ. The analysis was carried out to confirm the effect of MSD slope on increased viscosity.

The Einstein's relation [22] is used to calculate the self-diffusion coefficient (D), which is given as(13)

Using equation (13), D of the base fluid is calculated in the absence and presence of dnp and compared as shown in Figure 8. It is observed that the diffusion coefficient of CO2 increases almost linearly with increase in particle diameter and temperature, but the values are less than those in the absence of nanoparticle. This shows that nanofluid with larger nanoparticle size at higher temperatures is more viscous. These findings are in good agreement with the Ar–Cu nanofluid results shown by Lee et al. [32].

thumbnail Fig. 7

(a) Compares the MSD of base fluid at 300 < T < 700 K. (b), (c) and (d) compare the MSD of base fluid in presence of 1, 2 and 3 nm diameter at 300 < T < 700 K. (e) Compares the MSD of base fluid with and without nanoparticle diameter at 330 K.

thumbnail Fig. 8

Self-diffusion coefficient variation of CO2 with and without nanoparticle diameter at different temperatures.

3.4 Radial distribution function

The radial distribution functions (RDFs) of mass centres for different CO2 molecules are calculated to explore the structure of CO2 fluid in gaseous and supercritical phase. The RDF peak in Figure 9 decreases with increase in temperature, being the highest at 300 K. Then, the RDF for CO2 containing various nanoparticle diameters are calculated as shown in Figure 10. The addition of nano particles changes the structure of the gas near the interface to a denser form (as compared to the RDF in absence of nanoparticles). RDF peaks increase with increase in nanoparticle size, demonstrating that CO2 is more structured in the presence of a larger particle. It is observed again that with increase in temperature, RDF peak decreases, but the position of the peak does not change. Among the three configurations studied, the number of CO2 molecules interacting with the nanoparticle is higher for larger particle diameter. This is likely to be the reason for the variation in the peak density observed for different diameter nanoparticles. At higher temperatures and pressures (supercritical), the first peak disappears, which indicates that sc-CO2 starts to behave more like a rarefied gas at these conditions. Two regimes can be noted from Figure 11, which shows the first peak height as a function of temperature: one with a fast change (gas regime, 300 < T < 330 K) and the other with a slow change (supercritical regime, 450 < T < 700 K) of RDF peaks as a function of temperature. A similar trend has been observed for the second peak height of the RDF.

thumbnail Fig. 9

RDF for bulk CO2 (ρ base = 150 kg/m3) in two regimes: (a) gaseous phase, 300 < T < 330 and (b) supercritical phase 450 K < T < 700 K, other temperatures are not shown.

thumbnail Fig. 10

RDF for bulk CO2 in the presence of different nanoparticle diameters 1, 2 and 3 nm at (a) 300 K, (b) 330 K, (c) 550 K and (d) 700 K.

thumbnail Fig. 11

(a) and (b) First and second peak height of RDF as a function of temperature.

3.5 Nanofluid viscosity

Finally, viscosity of nanofluid is calculated via Green–Kubo formalism with different nanoparticle loadings at various temperatures and pressures while keeping the base fluid density constant at ρ base = 150 kg/m3. Accuracy of the estimated μ values is determined by observing the convergence of viscosities with time. Figure 12 illustrates that the μ NF > μ base, and further that μ NF has an increasing trend with increased d np. The trend of nanofluid viscosity variation with temperature is similar to that of the base fluid, i.e. viscosity increases with increasing temperature. This is because the transfer of momentum between atoms increases at higher temperatures, which in turn strengthens the resistance of motion in gas and supercritical phase. The results shown in this paper indicate that the increase in viscosity is due to the presence of nanolayer, and the contribution of Brownian motion (of CO2 particles) to viscosity enhancement appears to be negligibly small.

thumbnail Fig 12

Viscosity of nanofluid at different temperatures compared with its base fluid (diameter = 0 nm).

4 Conclusion

MD simulations are performed to determine the viscosity of Al2O3–CO2 nanofluid using Green–Kubo formalism in the temperature range of 300 < T < 700 K and pressures up to 200 bar for the first time. Generally, viscosity predictions using MD simulations with different interaction potential models show good agreement with the experiments. However, the TraPPE_flex2 model is found to be most appropriate for the temperature and pressure range used in the current work. Furthermore, the model can be used with other nanoparticles to determine the viscosity of CO2-based nanofluids. Influences of parameters such as temperature and nanoparticle size are investigated on the nanofluid viscosity. Using quantum mechanics, it is shown that the intermolecular attraction between Al2O3 and CO2 molecules is well characterised by using parameters obtained by LB mixing rule. An increase in density of CO2 molecules is found around the nanoparticle, and the thickness of this nanolayer is estimated with the aid of density distribution curves. The layer thickness for 1, 2 and 3 nm particle diameter systems are 0.6, 0.9 and 1.2 nm, respectively. With the increase of temperature, density ratio peaks decrease due to weak adsorption at higher temperatures. It is also seen that the nanoparticle effect is very low at the distance far away from it and density of the system is same as bulk density of CO2 (150 kg/m3). The RDF for CO2 molecules in the presence of different nanoparticle sizes is compared with the RDF of bulk base fluid, and it is shown that the bulk structure of the CO2 is altered with the addition of nanoparticle; its density increases with nanoparticle diameter. However, this trend is weakened at relatively higher temperatures. The most structured CO2 is found to be at the lowest temperature investigated (300 K), with d np = 3 nm. MSD analysis shows that the presence of Al2O3 particle leads to a decreased motion of the CO2 molecules (and hence to a decreased D). Brownian motion of CO2 molecules does not appear to be the cause of the enhanced viscosity of the nanofluid. This enhancement can rather be attributed to nanolayer formation. This “thickened” CO2 has the potential to increase the sweep efficiency of CO2 flooding, concomitantly increasing the oil recovery rate.

Nomenclature

k B : Boltzmann’s constant (J/K)

r : Distance between two atoms (m)

θ : Angle between two atoms (rad)

f : Interaction between particles

σ : Interatomic length scale between atom

ε : Interaction strength (J)

: Potential (J)

m : Mass of the particle

U : Pair potential interaction (J)

T : Thermodynamic temperature (K)

N : Total number of atoms

v : Velocity of particle (m/s)

V : Volume (m3)

μ : Viscosity (Pa.s)

D : Self-diffusion coefficient (m2/s)

q : Partial charge (e)

Subscripts

i,j : denotes atoms

α, β : denotes different types of atom

Author contribution statement

Z. Ahmed carried out the simulations and wrote the manuscript with inputs from all the authors. A. Bhargav provided the idea of investigating the cause of enhanced viscosity and supervised the findings of this work. Sairam S. Mallajosyula conceived the idea of validation of potentials using quantum chemistry and also helped Z. Ahmed to carry out the quantum chemistry simulations.

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Cite this article as: Zeeshan Ahmed, Atul Bhargav, Sairam S. Mallajosyula, Estimating Al2O3–CO2 nanofluid viscosity: a molecular dynamics approach, Eur. Phys. J. Appl. Phys. 84, 30902 (2018)

All Tables

Table 1

LJ parameters used for carbon–carbon and oxygen–oxygen interaction for several CO2 models. Parameters for flexible models with force constants [4].

Table 2

Nanolayer thickness values for different nanoparticle diameter.

All Figures

thumbnail Fig. 1

(a) Al of Al2O3 facing O of CO2 and distance is varied. (b) O of Al2O3 facing O of CO2 and distance is varied.

In the text
thumbnail Fig. 2

Interatomic force between atoms of alumina and CO2 are compared using quantum mechanics (QM) and molecular mechanics (MM) via LAMMPS. (a) Al–O being aluminium atom facing oxygen atom of CO2. (b) O–O being oxygen atom of alumina facing oxygen atom of CO2.

In the text
thumbnail Fig. 3

Cross-sectional view of the Al2O3–CO2 nanofluid with 2 nm diameter under investigation.

In the text
thumbnail Fig. 4

(a) Comparison of simulated viscosity of the base fluid (gaseous phase) of different flexible models with the experimental data [31] at different temperatures and pressures. (b) Comparison of simulated viscosity of the base fluid (supercritical phase) of different flexible models with the experimental data [31] at different temperatures and pressures.

In the text
thumbnail Fig. 5

Schematic of the Al2O3 nanoparticle and nanolayer formation around the nanoparticle.

In the text
thumbnail Fig. 6

Effect of nanoparticle diameter and temperature on the density of CO2. (a) at 300 K, (b) at 330 K, (c) at 550 K and (d) at 700 K. Similar observation is seen with other temperatures (not shown).

In the text
thumbnail Fig. 7

(a) Compares the MSD of base fluid at 300 < T < 700 K. (b), (c) and (d) compare the MSD of base fluid in presence of 1, 2 and 3 nm diameter at 300 < T < 700 K. (e) Compares the MSD of base fluid with and without nanoparticle diameter at 330 K.

In the text
thumbnail Fig. 8

Self-diffusion coefficient variation of CO2 with and without nanoparticle diameter at different temperatures.

In the text
thumbnail Fig. 9

RDF for bulk CO2 (ρ base = 150 kg/m3) in two regimes: (a) gaseous phase, 300 < T < 330 and (b) supercritical phase 450 K < T < 700 K, other temperatures are not shown.

In the text
thumbnail Fig. 10

RDF for bulk CO2 in the presence of different nanoparticle diameters 1, 2 and 3 nm at (a) 300 K, (b) 330 K, (c) 550 K and (d) 700 K.

In the text
thumbnail Fig. 11

(a) and (b) First and second peak height of RDF as a function of temperature.

In the text
thumbnail Fig 12

Viscosity of nanofluid at different temperatures compared with its base fluid (diameter = 0 nm).

In the text

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