**4. Results and discussion**

Fig. 1 shows the aggregation patterns of magnetic nanoparticles formed in the absence of an external magnetic field obtained from computer simulations with the modified DEM methodology. It may be seen that small isolated aggregates of nanoparticles are observed at low solid volume fractions while at high solid volume fractions, an extended network of nanoparticles usually referred to as a percolated network is observed to form spontaneously. The former is typically associated with gelation experiments carried out at insufficient concentrations of nanoparticles resulting in simple destabilisation of the suspension and formation of a collapsed structure.

numbers of nanoparticles simulated were 485, 975, 1460 and 1950 which correspond to solid volume fractions of 0.05, 0.10, 0.15 and 0.20 respectively. Here, solid volume fraction is defined to be the ratio of the total volume of all nanoparticles present to the volume of the pseudo-three-dimensional domain. To ensure numerical stability and accuracy, a relatively small time step of 10 ps was applied for all simulations carried out in this study. Table 1 summarizes the values of pertinent material properties and system parameters applied in the simulations. At the start of each simulation, the positions of all nanoparticles were assigned randomly within the computational domain such that no overlap between any two nanoparticles occurred. Periodic boundary conditions were applied on all four sides of the computational domain so as to eliminate any possible effects that may arise due to the presence of boundaries. The application of such boundary conditions also allowed the possibility of simulating a large system using a significantly smaller computational domain

which leads to more efficient utilization of computing resources.

Shape of particles Spherical

Particle diameter, d 70 nm Particle density, <sup>p</sup> 1000 kg m-3 Spring constant in force model, 1.0 10-3 N m-1 Viscous contact damping coefficient, 1.0 10-12 Coefficient of restitution 0.99 Coefficient of friction 0.5

Saturation magnetization, Md 1.0 105 A m-1 Hamaker constant, Ha 1.0 10-19 J Surface charge, q 1.6 10-15 C Ion concentration, nb 1.0 M Temperature, T 298 K

Simulation time step, t 10 ps

Table 1. Material properties and system parameters for DEM simulations

**4. Results and discussion** 

suspension and formation of a collapsed structure.

Domain size 5 m 5 m 70 nm

Fig. 1 shows the aggregation patterns of magnetic nanoparticles formed in the absence of an external magnetic field obtained from computer simulations with the modified DEM methodology. It may be seen that small isolated aggregates of nanoparticles are observed at low solid volume fractions while at high solid volume fractions, an extended network of nanoparticles usually referred to as a percolated network is observed to form spontaneously. The former is typically associated with gelation experiments carried out at insufficient concentrations of nanoparticles resulting in simple destabilisation of the

Number of particles, N 485, 975, 1460, 1950 Solid volume fraction 0.05, 0.10, 0.15, 0.20

Fig. 1. Aggregation patterns of magnetic nanoparticles in the absence of an external magnetic field at 10-3 s physical time obtained from the modified DEM simulations. The solid volume fractions applied were (a) 0.05, (b) 0.10, (c) 0.15 and (d) 0.20.

Gelation of Magnetic Nanoparticles 223

To observe gelation of nanoparticle suspensions, the present simulations have shown that solid concentrations must be sufficiently high and beyond the percolation threshold in order for a stable network structure to form. Although the simulations presented here are computationally expensive and so have been carried out at smaller length and time scales than those associated with experiments, it may be seen that the main qualitative features of the type of gel networks formed in the absence of an external magnetic field have been reproduced computationally. In particular, Fig. 2 shows that the gelation process takes place with the initial formation of small random aggregates throughout the domain which then join to form a fairly open network with no specific orientation of the various

The intermediate states of the gel during its formation process that are unobservable experimentally with present day technology are readily available from DEM simulations. With the advent of computing power, this computational technique is expected to become more important in this research field as such information will be necessary for more fundamental and mechanistic understanding of nanoparticle gelation processes. Fig. 3 shows that in the presence of an external magnetic field, the aggregates of nanoparticles are aligned along the direction of the magnetic field due to the anisotropic nature of the magnetic forces exerted on each nanoparticle and aggregate. At low solid volume fractions, individual elongated strands of aggregates are formed while at high solid volume fractions, such aggregates are capable of joining together due to smaller distances between aggregates. In comparison with the previous case where an external magnetic field was absent, the branches of the network that is beginning to form here are composed of more particles and are thus longer. This can be understood from inspection of the intermediate states of

Fig. 4 shows that the aggregation process in the presence of an external magnetic field starts, as in the previous case, with the formation of random aggregates throughout the domain. However, due to the anisotropic magnetic forces, aggregates formed are rotated to align along the direction of the magnetic field. Elongation of aggregates occurs as the growth of these aggregates also occurs along the direction of the magnetic field imposed. The final network structure consisting of long, parallel chains of nanoparticles is also in good agreement with structures of gels obtained experimentally with an applied

Fig. 5 shows quantitatively the time evolution of the average sizes of clusters formed by the magnetic nanoparticles both in the absence and presence of an external magnetic field. Here, average cluster size is defined as the average number of nanoparticles forming a cluster or aggregate. It may be observed that average cluster sizes increase with increasing total number of nanoparticles present within the domain or, equivalently, the overall solid volume fraction. Interestingly, the average size of clusters formed at each solid volume fraction evolves in a similar fashion with respect to time regardless of the presence or absence of an external magnetic field. This is despite the fact that the morphologies of the clusters or aggregates formed are significantly different as seen earlier. At the end of 1 ms, the average cluster sizes for N = 485 and N = 975 both in the absence and presence of an external magnetic field have reached more or less steady values. In contrast, the clusters formed for N = 1460 and N = 1950 are still growing in size, indicating that the percolation

branches.

magnetic field.

aggregation obtained from the simulations.

process is not completed yet at the end of 1 ms.

Fig. 2. Aggregation process of magnetic nanoparticles of solid volume fraction 0.20 in the absence of an external magnetic field. The states of aggregation correspond to (a) 0.0 s, (b) 2.0 × 10-4 s, (c) 4.0 × 10-4 s, (d) 6.0 × 10-4 s, (e) 8.0 × 10-4 s and (f) 10-3 s.

Fig. 2. Aggregation process of magnetic nanoparticles of solid volume fraction 0.20 in the absence of an external magnetic field. The states of aggregation correspond to (a) 0.0 s, (b)

2.0 × 10-4 s, (c) 4.0 × 10-4 s, (d) 6.0 × 10-4 s, (e) 8.0 × 10-4 s and (f) 10-3 s.

(e) (f)

(a) (b)

(c) (d)

To observe gelation of nanoparticle suspensions, the present simulations have shown that solid concentrations must be sufficiently high and beyond the percolation threshold in order for a stable network structure to form. Although the simulations presented here are computationally expensive and so have been carried out at smaller length and time scales than those associated with experiments, it may be seen that the main qualitative features of the type of gel networks formed in the absence of an external magnetic field have been reproduced computationally. In particular, Fig. 2 shows that the gelation process takes place with the initial formation of small random aggregates throughout the domain which then join to form a fairly open network with no specific orientation of the various branches.

The intermediate states of the gel during its formation process that are unobservable experimentally with present day technology are readily available from DEM simulations. With the advent of computing power, this computational technique is expected to become more important in this research field as such information will be necessary for more fundamental and mechanistic understanding of nanoparticle gelation processes. Fig. 3 shows that in the presence of an external magnetic field, the aggregates of nanoparticles are aligned along the direction of the magnetic field due to the anisotropic nature of the magnetic forces exerted on each nanoparticle and aggregate. At low solid volume fractions, individual elongated strands of aggregates are formed while at high solid volume fractions, such aggregates are capable of joining together due to smaller distances between aggregates. In comparison with the previous case where an external magnetic field was absent, the branches of the network that is beginning to form here are composed of more particles and are thus longer. This can be understood from inspection of the intermediate states of aggregation obtained from the simulations.

Fig. 4 shows that the aggregation process in the presence of an external magnetic field starts, as in the previous case, with the formation of random aggregates throughout the domain. However, due to the anisotropic magnetic forces, aggregates formed are rotated to align along the direction of the magnetic field. Elongation of aggregates occurs as the growth of these aggregates also occurs along the direction of the magnetic field imposed. The final network structure consisting of long, parallel chains of nanoparticles is also in good agreement with structures of gels obtained experimentally with an applied magnetic field.

Fig. 5 shows quantitatively the time evolution of the average sizes of clusters formed by the magnetic nanoparticles both in the absence and presence of an external magnetic field. Here, average cluster size is defined as the average number of nanoparticles forming a cluster or aggregate. It may be observed that average cluster sizes increase with increasing total number of nanoparticles present within the domain or, equivalently, the overall solid volume fraction. Interestingly, the average size of clusters formed at each solid volume fraction evolves in a similar fashion with respect to time regardless of the presence or absence of an external magnetic field. This is despite the fact that the morphologies of the clusters or aggregates formed are significantly different as seen earlier. At the end of 1 ms, the average cluster sizes for N = 485 and N = 975 both in the absence and presence of an external magnetic field have reached more or less steady values. In contrast, the clusters formed for N = 1460 and N = 1950 are still growing in size, indicating that the percolation process is not completed yet at the end of 1 ms.

(c) (d)

Fig. 3. Aggregation patterns of magnetic nanoparticles in the presence of an external magnetic field at 10-3 s physical time obtained from the modified DEM simulations. The orientation of the simulated magnetic field was in the vertical direction. The solid volume fractions applied were (a) 0.05, (b) 0.10, (c) 0.15 and (d) 0.20.

Gelation of Magnetic Nanoparticles 225

Fig. 4. Aggregation process of magnetic nanoparticles of solid volume fraction 0.20 in the presence of an external magnetic field. The states of aggregation correspond to (a) 0.0 s, (b)

2.0 × 10-4 s, (c) 4.0 × 10-4 s, (d) 6.0 × 10-4 s, (e) 8.0 × 10-4 s and (f) 10-3 s.

(e) (f)

(a) (b)

(c) (d)

Fig. 3. Aggregation patterns of magnetic nanoparticles in the presence of an external magnetic field at 10-3 s physical time obtained from the modified DEM simulations. The orientation of the simulated magnetic field was in the vertical direction. The solid volume

fractions applied were (a) 0.05, (b) 0.10, (c) 0.15 and (d) 0.20.

(c) (d)

(a) (b)

Fig. 4. Aggregation process of magnetic nanoparticles of solid volume fraction 0.20 in the presence of an external magnetic field. The states of aggregation correspond to (a) 0.0 s, (b) 2.0 × 10-4 s, (c) 4.0 × 10-4 s, (d) 6.0 × 10-4 s, (e) 8.0 × 10-4 s and (f) 10-3 s.

Gelation of Magnetic Nanoparticles 227

the simulations performed using a modified Discrete Element Method. Gelation occurred by the formation of random aggregates of nanoparticles within the domain which then joined with one another to form a network. However, in the presence of anisotropic magnetic forces, these aggregates were rotated to align along the direction of the magnetic field. Elongation of aggregates occurred and the final network formed consisted largely of such elongated

This study has been supported by the National University of Singapore under Grant

Brunet, E.; Degre, G.; Okkels, F.; Tabeling, P. (2005). Aggregation of Paramagnetic Particles

Climent, E.; Maxey, M. R.; Karniadakis, G. E. (2004). Dynamics of Self-Assembled Chaining

Davis, S. W.; McCausland, W.; McGahagan, H. C.; Tanaka, C. T.; Widom, M. (1999). Clusterbased Monte Carlo Simulation of Ferrofluids. *Phys. Rev. E*, Vol. 59, pp. 2424–2428 Dominguez-Garcia, P.; Melle, S.; Pastor, J. M.; Rubio, M. A. (2007). Scaling in the Aggregation Dynamics of a Magnetorheological Fluid. *Phys. Rev. E*, Vol. 76, pp. 051403 Duncan, P. D.; Camp, P. J. (2006). Aggregation Kinetics and the Nature of Phase Separation in Two-Dimensional Dipolar Fluids. *Phys. Rev. Lett.*, Vol. 97, pp. 107202 Furlani, E. P. (2006). Analysis of Particle Transport in a Magnetophoretic Microsystem. *J.* 

Furlani, E. P.; Ng, K. C. (2008). Nanoscale Magnetic Biotransport with Application to

Huang, J. P.; Wang, Z. W.; Holm, C. (2005). Computer Simulations of the Structure of

Kantorovich, S.; Cerda, J. J.; Holm, C. (2008). Microstructure Analysis of Monodisperse Ferrofluid Monolayers: Theory and Simulation. *Phys. Chem. Chem. Phys.*, Vol. 10, pp. 1883–1895 Lattuada, M.; Alan Hatton, T. (2007). Preparation and Controlled Self-Assembly of Janus Magnetic Nanoparticles. *J. Am. Chem. Soc.*, Vol. 129, pp. 12878–12889 Lattuada, M.; Wu, H.; Sandkuhler, P.; Sefcik, J.; Morbidelli, M. (2004a). Modelling of

Lattuada, M.; Wu, H.; Morbidelli, M. (2004b). Experimental Investigation of Colloidal Gel

Lim, E. W. C.; Wang, C. H., Yu, A. B. (2006a). Discrete Element Simulation for Pneumatic

Lim, E. W. C.; Zhang, Y.; Wang, C. H. (2006b). Effects of an Electrostatic Field in Pneumatic

Lim, E. W. C.; Wang, C. H. (2006). Diffusion Modeling of Bulk Granular Attrition. *Ind. Eng.* 

Lim, E. W. C.; Wong, Y. S.; Wang, C. H. (2007). Particle Image Velocimetry Experiment and

Conveying of Granular Material. *AIChE J.*, Vol. 52, pp. 496–509

Fluidized Bed. *Ind. Eng. Chem. Res.*, Vol. 46, pp. 1375–1389

Aggregation Kinetics of Colloidal Systems and its Validation by Light Scattering

Conveying of Granular Materials through Inclined and Vertical Pipes. *Chem. Eng.* 

Discrete-Element Simulation of Voidage Wave Instability in a Vibrated Liquid-

in the Presence of a Hydrodynamic Shear. *J. Colloid Interface Sci.*, Vol. 282, pp. 58–68

branches of magnetic nanoparticles arranged more or less parallel to one another.

in Magnetorheological Fluids. *Langmuir*, Vol. 20, pp. 507–513

**6. Acknowledgment** 

**7. References** 

Number R-279-000-275-112.

*Appl. Phys.*, Vol. 99, pp. 024912

Magnetofection. *Phys. Rev. E*, Vol. 77, pp. 061914

Colloidal Ferrofluids. *Phys. Rev. E*, Vol. 71, pp. 061203

Measurements. *Chem. Eng. Sci.*, Vol. 59, pp. 1783–1798

Structures. *Langmuir*, Vol. 20, pp. 4355–4362

*Sci.*, Vol. 61, pp. 7889–7908

*Chem. Res.*, Vol. 45, pp. 2077–2083

Fig. 5. Time evolution of average size of clusters formed by magnetic nanoparticles (a) in the absence of an external magnetic field and (b) in the presence of an external magnetic field.

#### **5. Conclusions**

The process of gelation with and without the application of an external magnetic field giving rise to the different internal pore structures could be understood mechanistically by results of

(a)

(b)

**0.0 0.2 0.4 0.6 0.8 1.0**

**time (×10-3 s)**

Fig. 5. Time evolution of average size of clusters formed by magnetic nanoparticles (a) in the absence of an external magnetic field and (b) in the presence of an external magnetic field.

**0.0 0.2 0.4 0.6 0.8 1.0**

**time (×10-3 s)**

The process of gelation with and without the application of an external magnetic field giving rise to the different internal pore structures could be understood mechanistically by results of

**5. Conclusions** 

**0.0**

**0.0**

**2.0**

**4.0**

**6.0**

**8.0**

**10.0**

**Average cluster size**

**12.0**

**14.0**

**16.0**

**18.0**

**N = 485 N = 975 N = 1460 N = 1950**

**2.0**

**4.0**

**6.0**

**8.0**

**10.0**

**Average cluster size**

**12.0**

**14.0**

**16.0**

**18.0**

**N = 485 N = 975 N = 1460 N = 1950** the simulations performed using a modified Discrete Element Method. Gelation occurred by the formation of random aggregates of nanoparticles within the domain which then joined with one another to form a network. However, in the presence of anisotropic magnetic forces, these aggregates were rotated to align along the direction of the magnetic field. Elongation of aggregates occurred and the final network formed consisted largely of such elongated branches of magnetic nanoparticles arranged more or less parallel to one another.

#### **6. Acknowledgment**

This study has been supported by the National University of Singapore under Grant Number R-279-000-275-112.

#### **7. References**


Lim, E. W. C. (2007). Voidage Waves in Hydraulic Conveying through Narrow Pipes. *Chem. Eng. Sci.*, Vol. 62, pp. 4529–4543

**0**

**12**

*Chile*

**Inelastic Collisions and Hypervelocity Impacts at Nanoscopic Level: A Molecular Dynamics Study**

In this chapter we present an atomic level study of nano-particle impact using molecular dynamics simulation. Two cases have been considered. First, we simulate the bouncing of a ball over a surface due to a constant force (which mimic the gravity force), modeling the inter-atomic interaction by a modified Lennard-Jones potential, where the ball-surface atom interaction is represented by a purely repulsive term. The analysis of the results makes it possible, among other aspects, to determine the restitution coefficient in each bounce as well as to understand the processes of energy loss in inelastic collisions, which are actually not a loss, but a transfer to thermal and vibrational energy. The second simulation describes the impact mechanisms of a solid projectile hitting a target at high velocity. Both the projectile and the target are made of copper, which is modeled by a realistic many-body tight-binding potential. The projectile velocity is kept constant during all the simulation, representing an extreme condition, where the momentum and hardness of the projectile is much higher than the momentum and hardness of the target. In this regime, we identify two different behavior in dependence of the projectile velocity: at low velocities (less than 4 km/s) the target basically recover its structure after the passage of the projectile, but at higher velocities, the projectile

Both problems, inelastic collisions and hypervelocity impacts, are non-equilibrium related phenomena which are important from a basic and applied point of view, in several areas of science: physics, materials science, aeronautics, mechanics, among others. From a theoretical point of view, they have been extensively treated in the macroscopic level, by using continuum hydrodynamic simulation, and only recently researchers are using molecular dynamic simulation, intended to an understanding of these phenomena at the scale of inter-atomic interactions. Besides the calculation of equilibrium properties and their associated fluctuations, molecular dynamics allows for a wider range of problems to be tackled: given that we have access to the atomic trajectories we can study the transit to equilibrium, as well as purely non-equilibrium phenomena (where we are interested not in the final state but in the process itself ), for instance, shock-induced plasticity and fracture of materials. In this regard, Non-Equilibrium Molecular Dynamics (NEMD) has emerged recently as a branch dealing with, and promising to shed light on, the mechanism behind

**1. Introduction**

left a permanent hole in the target.

these (and other similar) irreversible processes.

\*www.gnm.cl

G. Gutiérrez, S. Davis, C. Loyola, J. Peralta, F. González,

*Group of NanoMaterials*\**, Departamento de Física, Facultad de Ciencias,*

Y. Navarrete and F. González-Wasaff

*Universidad de Chile*

