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Study about the nanoparticle agglomeration in a magnetic nanofluid by the Langevin dynamics simulation model using an effective Verlet-type algorithm

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Abstract

This paper presents a modelling study about the nanoparticle agglomeration in magnetic nanofluids. The colloidal magnetic nanoparticles size distribution is subjected to simultaneous translation and rotation movements under the action of conservative and dissipative forces, with their respective moments. In order to obtain the numerical solution of the coupled equations of motion, we use a Langevin dynamics stochastic method based on an effective Verlet-type algorithm. The presented model is based on an easy-to-implement integrator. We apply a number of analytical techniques to assess the performance of the method. The model has been tested on a magnetite nanoparticle-based nanofluid. Finally, the paper presents a number of structures obtained in various physical conditions, discussing the retrieved results of modelling and simulation. In weak external magnetic field, the nanoparticles form arrangements like linear chains or dense globes and rings, with magnetic moments rotating in both directions (both clockwise and counterclockwise). These arrangements, in vortex and toroidal states, are reported in actual scientific literature and open new perspectives for understanding the behaviour of nanofluids, with applications in engineering and medicine. Chains are predominant in high external magnetic, with local magnetic moments mainly orientated mainly along the direction of the applied field.

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Abbreviations

m i (kg):

The ith nanoparticle mass

\(\vec{v}_{i}\) (m/s):

The ith nanoparticle linear speed

\(\vec{f}_{i}\) (N):

The resultant of the conservative forces acting on the ith nanoparticle

\(\alpha_{{i,{\text{tr}}}}\) (N s)/m):

The translational friction coefficients of the ith nanoparticle

\(\alpha_{{i,{\text{rot}}}}\)(N s m):

The rotational friction coefficients of the ith nanoparticle

η (Pa·s):

The dynamic viscosity coefficient

r i (m):

The ith nanoparticle radius

\(\beta_{{i,{\text{tr}}}} (t)\) (N):

The random Brownian force of the ith nanoparticle

\(\beta_{{i,{\text{rot}}}} (t)\) (N m):

The random Brownian torque of the ith nanoparticle

I i (kg m2):

The moment of inertia of the ith nanoparticle

\(\vec{\omega }_{i}\) (rad/s):

The angular speed of the ith nanoparticle

\(\vec{T}_{i,c}\) (N m):

The resultant of the conservative torques acting on the ith nanoparticle

δ(t):

The Dirac delta function

\(\hat{n}_{ij}\) :

The versor of the direction connecting the ith and jth particles

D ij (m):

The distance between the centres of the ith and jth nanoparticles

\(\vec{D}_{ij}\) (m):

The vector of the direction connecting the centres of the ith and jth nanoparticles

\(s_{ij}\) (m):

The surface-to-surface separation between the ith and jth nanoparticles

A eff (J):

The Hamaker effective constant for iron-oxide nanoparticles in water

\(\vec{F}_{{{\text{el}},{\text{DL}},ij}}\) (N):

The electrostatic force acting between ith and jth nanoparticle, in a double layer system

\(V_{{{\text{el}},{\text{DL}},ij}}\) (J):

The electrostatic potential energy acting between ith and jth nanoparticle, in a double layer system

\(\varPhi_{0i}\) (C/m):

The surface potential of the ith nanoparticle at infinite separation

κ (m−1):

The thickness of the screening ionic layer

e (C):

Electron charge

n 0 (ions/m3):

The concentration of ions in bulk solution

z i :

The valence of the ith ions from electrolyte

ε (F/m):

The electrical permittivity of the solvent

k B (J/K):

The Boltzmann constant

T (K):

The absolute temperature

c i (ions/m3):

The concentration of the ith ion species in bulk solution

q (C):

The effective charge of the particles

\(V_{\text{steric}}^{\text{ES}} (s_{ij} ,r_{i} )\) (J):

The steric potential between two equal spheres

\(V_{\text{steric}}^{\text{ES}} (s_{ij} ,r_{i} ,r_{j} )\) (J):

The steric potential between two unequal spheres

\(\vec{F}_{{{\text{steric}},ij}}\) (N):

The steric force

\(d_{i}\) (m):

The diameter of the ith nanoparticle

\(\xi\) (m−2):

The surface density of the polymers

M s (A/m):

The spontaneous magnetization

V i (m3):

The particle volume

\(\hat{\mu }_{i}\) :

The unit vector of the magnetic moments

\(\vec{\mu }_{i}\)(A m2):

The magnetic moment of the i nanoparticle

\(E_{\text{vdw}}\) (J):

The van der Waals’ interaction energy between spherical particles i and j

\(\vec{F}_{\text{vdw}}\) (N):

The van der Waals’ force

\(\vec{F}_{{{\text{md}},ij}}\) (N):

The dipolar magnetic force exerted between the magnetic moments of the nanoparticles i and j

\(\vec{\tau }_{i}\) (N m):

The magnetic torque acting on an ith nanoparticle

μ0 :

The vacuum magnetic permeability

\(\vec{H}_{i}\) :

The local magnetic field on each nanoparticle

\(\vec{H}_{ext}\) (A/m):

The external magnetic field applied

\(\vec{H}_{\text{id}}\) (A/m):

The internal dipolar magnetic field

t (s):

Time

dt (s):

Time integration step

\(\bar{v}^{2}\) (m2/s2):

The mean-square velocity

\(\phi_{i}\) (rad):

Rotation angle

\(\bar{\omega }^{2}\) (rad2/s2):

The mean square of the angular velocities

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Authors and Affiliations

  1. Department of Electrical Engineering and Industrial Informatics, “Politehnica” University of Timisoara, Revolutiei Nr.5, 331128, Hunedoara, Jud. Hunedoara, Romania

    Mihaela Osaci

  2. DICEAM, University “Mediterranea” of Reggio Calabria, Via Graziella Feo di Vito, 89100, Reggio Calabria, Italy

    Matteo Cacciola

Authors
  1. Mihaela Osaci
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  2. Matteo Cacciola
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Correspondence to Mihaela Osaci.

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Osaci, M., Cacciola, M. Study about the nanoparticle agglomeration in a magnetic nanofluid by the Langevin dynamics simulation model using an effective Verlet-type algorithm. Microfluid Nanofluid 21, 19 (2017). https://doi.org/10.1007/s10404-017-1856-0

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