Royal Institute of Technology Department of Chemical Engineering Numerical simulation of multi-dimensional fractal soot aggregates

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Royal Institute of Technology

Department of Chemical Engineering

Numerical simulation of multi-dimensional

fractal soot aggregates

Author:

Andres Suarez

Supervisors:

Matth¨ aus B¨ abler

Lilian de Mart´ın

Stockholm, Sweden

June 30, 2018

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Abstract

Superaggregates are clusters formed by diverse aggregation mechanisms at different scales.

They can be found in fluidized nanoparticles and soot formation. An aggregate, with a single aggregation mechanism, can be described by the fractal dimension, df, which is the measure of the distribution and configuration of primary particles into the aggregates. Similarly, a superaggregate can be analyzed by the different fractal dimensions that are found at each scale.

In a fractal structure aggregate, a self-similarity can be identified at different scales and it has a power law relation between the mass and aggregate size, which can be related to properties like density or light scattering. The fractal dimension, df, can be influenced by aggregation mechanism, particles concentration, temperature, residence time, among other variables. More- over, this parameter can help on the estimation of aggregates’ properties which can help on the design of new processes, analyze health issues and characterize new materials.

A multi-dimensional soot aggregate was simulated with the following approach. The first aggregation stage was modeled with a Diffusion Limited cluster-cluster aggregation (DLCA) mechanism, where primary clusters with a fractal dimension, d_{f 1}, close to 1.44 were obtained.

Then, the second aggregation stage was specified by Ballistic Aggregation (BA) mechanism, where the primary clusters generated in the first stage were used to form a superaggregate. All the models were validated with reported data on different experiments and computer models.

Using the Ballistic Aggregation (BA) model with primary particles as the building blocks, the fractal dimension, d_{f 2}, was close to 2.0, which is the expected value reported by literature.

However, a decrease on this parameter is appreciated using primary clusters, from a DLCA model, as the building blocks because there is a less compact distribution of primary particles in the superaggregate’s structure.

On the second aggregation stage, the fractal dimension, d_{f 2}, increases when the superaggregate size increases, showing an asymptotic behavior to 2.0, which will be developed at higher scales. Partial reorganization was implemented in the Ballistic Aggregation (BA) mechanism where two contact points between primary clusters were achieved for stabilization purposes.

This implementation showed a faster increase on the fractal dimension, d_{f 2}, than without partial reorganization. This behavior is the result of a more packed distribution of primary clusters in a short range scales, but it does not affect the scaling behavior of multi-dimensional fractal structures. Moreover, the same results were obtained with different scenarios where the building block sizes were in the range from 200 to 300 and 700 to 800 primary particles.

The obtained results demonstrate the importance of fractal dimension, d_f, for aggregate characterization. This parameter is powerful, universal and accurate since the identification of the different aggregation stages in the superaggregate can increase the accuracy of the estimation of properties, which is crucial in physics and process modeling.

Keywords: Soot formation, Multi-dimensional fractal aggregates, Diffusion limited cluster- cluster aggregation (DLCA), Ballistic aggregation (BA).

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Acknowledgment

I wish to thank all the people who contributed, supported and gave valuable ideas for this project. I would like to express my very great appreciation to Prof. Matth¨aus B¨abler for his valuable, constructive and never ending ideas and support for this project. His willingness to teach, analyze all the findings, his discipline and endeavor to discover new paths on science from the basics is something I will take with me for future projects.

I would like to thank Prof. Lilian de Mart´ın, from Chalmers University of Technology, for her advices and enthusiasm to share her knowledge and experience. Genuine and honest advices are always hard to find in life but thanks to her, it was possible to work through valuable opinions and discussions which enriched this project.

I wish to acknowledge the help provided by Ramiar Sadegh Vaziri and all the fruitfully discussions spent on the development of the computer model. I would like to thank all the staff from the Chemical Engineering Department for their support when the project was having difficulties. Moreover, advices given by Jairo Mosquera were much appreciated besides the friendship and time spent talking about politics and simulation.

I would like to thank The Royal Institute of Technology KTH for the opportunity to study in one of the pioneer institutions in Europe where I was able to learn so many aspects and developed multiple skills. I wish to thank the Swedish Institute for being part of the scholar- ship programme. This opportunity lets me get the knowledge and experience to help in the development of a sustainable and better society.

Finally but not last, I would like to thank my family for the effort, energy and happiness they have given to my life. In addition, I want to thank Margarita for her infinite and uncon- ditional support, and tricks for my thesis report.

”The whole of science is nothing more than a refinement of everyday thinking.”

Albert Einstein

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List of Figures

1 Micrographs of fractal aggregates . . . 6

2 Agglomerates and aggregates of primary particles . . . 10

3 Evolution of aggregate mobility radius r_m and particle radius due to sintering . . 13

4 Aggregate’s transition based on Geldart’s diagram . . . 14

5 Multi-dimensional fractal aggregates mechanism . . . 20

6 Agglomeration mechanisms in a 3-D space and fractal dimensions associated . . . 22

7 Evolution on soot structure and formation in a combustion system . . . 27

8 Methodology for Superaggregates generation and analysis . . . 29

9 Wraparound effect in boundary conditions . . . 30

10 Diffusion limited cluster-cluster aggregation (DLCA) algorithm . . . 32

11 Distance calculation for moved particle/cluster using cell-list . . . 34

12 Overlap correction algorithm . . . 36

13 Overall methodology for the Diffusion limited cluster-cluster aggregation (DLCA) stage . . . 36

14 Strategies for short-range interactions computation . . . 37

15 Cell-list strategy algorithm . . . 38

16 Pair correlation method . . . 40

17 Initial rotation for the building block in the Ballistic aggregation (BA) method . 43 18 Partial Reorganization for the building block in the Ballistic aggregation (BA) method . . . 44

19 Initial States for the Diffusion Limited cluster-cluster aggregation for different volume fractions and primary particles . . . 47

20 Evolution for the Diffusion Limited Cluster-cluster aggregation (DLCA) model . 47 21 Number of particles in the cluster as a function of Radius of Gyration Rg for the DLCA model . . . 48

22 Fractal dimension distribution for clusters between 200 to 300 size . . . 49

23 Building block pair correlation curve with a fractal dimension of 1.35 . . . 49

24 Building block pair correlation curve with a fractal dimension of 1.59 . . . 49

25 System pair correlation evolution as collision percentage . . . 50

26 Cluster size distribution evolution as collision percentage . . . 50

27 Standard deviation as a function of mean cluster size when system evolves . . . . 51

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28 Cluster size distribution evolution as collision percentage . . . 51 29 Box counting method for fractal dimension . . . 52 30 Pair correlation for Ballistic Particle cluster aggregation (BA) . . . 52 31 Aggregate’s mass as a function of the Radius of Gyration R_g for the BA model . 53 32 Pair correlation for Ballistic Particle cluster aggregation (BA) using primary

clusters . . . 53 33 Radius of Gyration Rg as a function of the aggregate’s size for the BA model

using primary clusters . . . 54 34 Fractal dimension, d_f, as a function of cluster’s mass. . . 54 35 Pair correlation for Ballistic Particle cluster aggregation (BA) using primary

clusters . . . 55 36 Aggregate’s mass as a function of Radius of Gyration Rg for the BA model using

primary clusters . . . 55 37 Pair correlation for Ballistic Particle cluster aggregation (BA) using primary

clusters . . . 56 38 Pair correlation for Ballistic Particle cluster aggregation (BA) using primary

clusters . . . 56 39 Aggregate’s mass and Aggregate’s density as a function of Radius of Gyration

R_g for the BA model using primary clusters . . . 57 40 Fractal dimension, df, as a function of cluster’s mass. . . 57 41 Aggregate’s mass as a function of Radius of Gyration R_g for the 2-stages super-

aggregate . . . 58 42 Aggregate’s density as a function of Radius of Gyration R_g for the 2-stages su-

peraggregate . . . 58 43 Fractal structures in a tree . . . 67 44 Self-similarity behavior for the void ratio of a soot superaggregate . . . 68

List of Tables

1 Fractal dimensions of clusters generated by different mechanisms. . . 24

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1 Introduction

Patterns are ubiquitous in nature. The characterization of those patterns can help us to understand physical and chemical phenomena which can be useful in industrial applications. More- over, the recognition of patterns at different scales shows the self-similarity in structures where a special ordering is presented. This self-similarity is related to the fractal dimension, introduced by the mathematician Benoit B. Mandelbrot in 1977 [45]. When a structure is self-similar, if a portion of the aggregate is extracted and magnified, it would look virtually the same as the whole aggregate. For instance, for a mathematical fractal, such as the Koch snowflake, the magnified portion of the structure looks exactly the same as the original. This behavior has a huge implication on properties like density which for fractal aggregates decreases as it sizes increases.

Fractal geometries are presented in particle aggregation and coagulations systems, important in chemical processes. Fractal aggregates are recognized in soot, cosmic dust, microalgae, nanoparticles, and bacteria as shown in Figure 1. The identification of fractal aggregates and their analysis in industrial applications like combustion systems and fluidization is important for the design and operation of processes because their properties can be calculated accurately.

Figure 1: Micrographs of fractal aggregates. a) soot; b) snow; c) cosmic dust; d) gold nanoparticles; e) the bacteria M. luteus; f) colonie of the microalgae B. braunii. Taken from [30]

Particle dynamics is the study of the motion of particles and it involves multiple phenomena.

Particles can move by two mechanisms: diffusion and convection. First, diffusion is related to the molecular interaction between the solid particles and the molecules of the suspended medium, e.g. the gas in the case of soot or aerosols. It is expressed by the Brownian motion which is a random movement of particles in the medium. Second, convection is related to the coherent motion of particles influenced by a motion force exerted to the whole system which can be

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colloidal interactions such as Van der-Waals and electrostatic forces.

In addition, the particles are influenced by other processes like sintering, surface growth, chemical reaction, coagulation and fragmentation. As a consequence, particle dynamics’ models can be complex and expensive to perform because to many factors are involved in a realistic model. However, there are simplified models which bring an adequate approach on the understanding of particle dynamics. Those models have shown to be in agreement with experimental data and they indicate that some mechanisms have a lower influence than others and they can be neglected for modeling purposes.

One case in which particle behavior is relevant is on soot aggregates in industry, since they are involved in combustion systems. Soot has an impact on climate change, health and process performance [32, 35]. The presence of soot in the atmosphere absorbs solar radiation which produce an increase on earth surface temperature. On the other hand, soot emissions are linked to health issues, specifically respiratory and cardiovascular diseases in cities and areas with a high particulate air pollution. Finally, the presence of soot can reduce the efficiency of industrial processes due to equipment fouling, pressure drop increase and lack of particle removal.

Therefore, the understanding of how soot aggregates are formed from primary soot particles is important to improve health, environmental and process issues related.

Soot aggregates are known to form superaggregates [7, 6, 35]. They are defined as multi- dimensional fractal aggregates with a differentiated aggregation mechanism at different scales.

Soot aggregates are formed by the aggregation of primary particles at the beginning. Then, the clusters that are formed by the primary clusters interact in a different mechanism, with a change on the structure at large scales. In order to characterize those structures, the fractal dimension, found at different scales, is a powerful measurement on how the soot structure changes as the aggregation mechanism evolves at different scales.

In order to characterize the properties of an aggregate, the fractal dimension needs to be determined as a measure of self-similarity. This applies in particular to properties like density or scattering of light, which crucially influence the velocity, drag and deposition or adsorption of the soot particles.

1.1 Problem Statement

Fractal aggregates are presented in a wide range of industrial applications. In addition, the development of advanced characterization techniques has revealed complex structures that can be described by multiple fractal dimensions. Those structures, called superaggregates, are found in soot formation and nanoparticles fluidization.

For combustion systems, soot aggregates are formed by complex mechanisms related to nucleation and growing kinetics. However, most of the computer models that are used for soot

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formation, are focused at a molecular level where primary particles are formed by chemical reactions, surface growth and sintering mechanisms. Then, the soot particles aggregate to form primary clusters that undergo additional aggregation between developed clusters to create a multi-dimensional fractal aggregate or superaggregate with two or more defined mechanisms at different scales.

Soot is demonstrated to form hierarchical superaggregates [7,6,35]. They are produced due to a change of mechanism or interaction between particles at different scales in the aggregation process to create larger aggregates. As a consequence, a method to estimate the properties accurately of those structures is required. The fractal dimension, at different scales, can be used for the estimation of superaggregates properties at different sizes, with the capability to identify the stages involved in the aggregation processes.

The aggregation mechanism and structural properties for superaggregates can be analyzed by modeling. For this project, cluster-cluster aggregation models, based on the Monte Carlo algorithm, are an adequate approach for modeling soot aggregation at the meso and macro scale.

Moreover, using this model, it is possible to estimate the fractal dimensions at different scales and system properties, like aggregate size distribution, which are crucial in the characterization of aggregates and the system evolution for health, environment and process purposes.

1.2 Purpose

Main aim

The main aim of this project is to understand how aggregation of primary particles are carried out in the formation of multi-dimensional fractal soot superaggregates and how the properties of these structures are affected, like the pre-factor, k0, in the fractal scaling law.

Numerical simulations are used as the main tool to study this statement. A soot superaggregate built by two aggregation stages is chosen. In the first stage, the Diffusion Limited Cluster-cluster (DLCA) aggregation mechanism is used to generate primary clusters. Then, a Ballistic particle-cluster (BA) aggregation mechanism is used in the second stage where the primary clusters generated by the DLCA mechanism are used to create the superaggregate.

Both models are based on the Monte Carlo algorithm which ensures statistical validation on phenomena like Brownian motion.

Specific objectives

• Develop and investigate a numerical model to generate and analyze muti-dimensional fractal aggregates and their influence on property estimation.

• Find out an adequate and optimal algorithm for the modeling of an off-lattice, 2-dimensional Diffusion Limited cluster-cluster (DLCA) aggregation using a large number of primary particles.

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• Create an algorithm for the modelling of Ballistic particle-cluster (BA) aggregation for generating multi-dimensional fractal superaggrgeates from primary clusters.

• Calculate the main properties for an aggregate such as the fractal dimension and their relation on properties and system behavior.

1.3 Relevance

The understanding and model development of nanoparticle aggregation mechanisms allows the measurement of aggregate properties such as density as a function of aggregate size, given by the radius of gyration. The behavior of such properties can be related with the fractal dimension.

For an aggregate, the density decreases as the size increases which is a different behavior from homogeneous materials.

In addition, when a superaggregate is considered, a change of density occurs as the aggregate grows at different decreasing rates, according to the aggregation stage involved. Therefore the identification of the hierarchical stages on the aggregation process enables an accurate estimation of properties which can improve processes like particle separation, coalescence, fluidization and filtration. In addition, it can help to understand the influence of superaggregates on health, environment and process performance.

Moreover, the generalization of the proposed method can be extended to other systems like particle dynamics, where computer models are used to generate multi-dimensional fractal aggregates and their characterization by fractal dimension. Therefore, this method could improve the process design of fluidization systems, relevant in the pharmaceutical and specialties chemical industry.

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2 Background

2.1 Agglomerates and Aggregates

Agglomerates and aggregates are structures formed by the coupling of nanoparticles in the gas phase. They are used in the production of nanostructured commodities, mainly in paint, paper, pharmaceutic and electronic industry [18,11]. An aggregate or agglomerate is a cluster of primary particles that can be formed by sintering, surface growth, chemical reaction, aerosol coagulation and fragmentation. The understanding of the formation mechanism can be complex because all the latter mechanisms can be presented simultaneously [18,75].

The main differences between an aggregate and an agglomerate are that meanwhile aggregates are bounded by strong chemical forces (metallic, ionic or covalent), in agglomerates, the primary particles are sticked together by weak physical forces (Van derWaals, capillary or electrostatic). Moreover, aggregates are presented in catalysts and electronic devices production and agglomerates are used in nanocomposites and suspensions. [65, 28]. In real scenarios, a cluster can be formed by weak or strong bonds as presented in Fig. 2.

Figure 2: Formation of aggregates and agglomerates by primary particles.[11]

These structures can be characterized by microscopy, Transmission Electron Microscopy (TEM), electromagnetic scattering and mass mobility measurements, in order to obtain some properties like density or conductivity which are important for process design and operation [11].

Agglomerates and aggregates are formed in aerosols, which are suspensions of small particles in gases, mainly due to the formation of particles from gases and powdered material suspensions. Soot is a type of aerosol, formed by small carbon particles generated in combustion and mixed with inorganic oxides [18].

A primary particle diameter can go from molecular clusters, about 10 ˚A, to dust particles as large as 100 µm. Furthermore, the complex structure of an agglomerate cannot be assumed as a circumference because their behavior and properties are related to the particle size and density [18,83]. As a consequence, a parameter to describe the size of an aggregate or agglomerate is required. There are two main parameters: the radius of gyration, R_g, and the mobility radius, Rm [11].

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The mobility radius, R_m, is the equivalent radius of a sphere experiencing the same drag force as an agglomerate [18, 58]. Sorensen [68] summarized simulations and experiments for cluster-cluster agglomerates (With d_f = 1.8 − 1.9 and k_n= 1.3 in a 3-D system) and suggested the following equations to calculate the mobility radius, Rm, [39,76].

• Continuum regime (k_n→ 0, f ∼ R_m) [11].

Rm,c = rpN_p^0.46 f or Np < 100 Rm,c = 0.65rpN_p^0.56 f or Np > 100

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• Free molecular regime (k_n→ ∞, f ∼ R²_m) [11].

R_{m,f m} = r_pN_p^0.46 f or all N_p (2)

Where k_n is the Knudsen number, r_p is the particle’s radius and N_p is the total number of particles in the aggregate.

The radius of gyration, R_g, is equivalent to the radius of a sphere experiencing the same resistance to a change in angular velocity [18, 40] when in vacuum (no drag). This depends only on the geometric configuration and regime. For a single sphere, the radius of gyration can be defined as Eq. 3. [63,69]

R_g,p = r3

5R_p (3)

For an agglomerate composed by spherical primary particles, the radius of gyration is defined as the root-mean square particles’ radius from the cluster center of mass normalized by the cluster mass and it can be calculated as Eq. 4. [39].

R_g = v u u

tR²_g,p+ 1 2N_p²

Np

X

m,j=1

r_mj² (4)

Where N_p is the total number of primary particles in the cluster and r_mj are the distances of the primary particle ”m”, to the primary particle ”j”. The center of mass, CMi, is the point where all the mass distribution is balanced and it can be calculated by summing all the coordinates of every primary particle and dividing by the total cluster mass or number of particles N_p, as presented in Eq. 5.

CM_x= 1 N_p

N p

X

i=1

x_i CM_y = 1 N_p

N p

X

i=1

y_i (5)

Before the fractal dimension was introduced, the agglomerates were treated as compact spheres or formless amorphous structures. Then, Meakin et al. [47,82], using computer simulation, demonstrated, over a finite scale range for agglomerates and aggregates, a self-similar or

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scale invariant structure. This self-similarity in the structure is identified by the fractal dimension, d_f, and can be related with the mass of the agglomerate as presented in Eq.6. Moreover, the fractal dimension can be related with properties like density being scaled as shown in Eq.7.

N = k₀ Rg

R_p

df

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ρ = k₀ Rg

R_p

3−df

(7) Wang and Sorensen [77] concluded that the characteristic lengths of aggregates are proportional to its mass, when the Knudsen number, kn 1 (Stokes regime), and the number of primary particles in the aggregate is above 100. As a result, the calculation of the fractal dimension is based on statistical methods and higher accuracy is expected at large samples. [77,69,68]

Another characteristic length that is used for describing fractal aggregates is the perimeter radius, Rpc, which is the radius of a sphere that can enclose an aggregate and it is related to the radius of gyration, R_g, and fractal dimension as presented in Eq.8 [59,34,80].

R_pc≈ (1 + 2/d_f)^0.5R_g (8)

The main characteristic length is the radius of gyration, R_g, because it depends only on the spatial distribution of the primary particles in the agglomerate without any external influence.

Therefore it is an intrinsic property. On the other hand, the mobility radius, R_m, depends on the agglomerate structure and on features from the space, like the medium properties. Therefore, it is not an intrinsic property and both radius are not directly proportional. As a consequence, the radius of gyration, R_g is the best characteristic length for calculating the fractal dimension, df [59,28].

2.1.1 Application of agglomerates and aggregates

The main applications for agglomerates and aggregates are in the production of composites, sensors and processing fuels. Agglomerates as composites are used as fillers for tires, furniture, synthetic implants and paper coating. For tires, carbon black is an agglomerate that is used to reinforce and increase the life cycle due to its ramified structure that serves as an entanglement with rubber [68,61,60].

Agglomerates as sensors are presented in electronic devices containing nanostrutured materials like gas sensors, battery electrodes and solid oxide fuel cells. The structure of the agglomerate affects properties like conductivity or resistance. Particle laden fluids are used as chemical-mechanical polishing. They are formed by fractal-like aggregates that burnish the surface of microchips for multilayer deposition processes. Moreover, the use of fractal-like agglomerates slurries are used to enhance heat transfer in thermal fluids [68,60,8,11].

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2.2 Formation of nanoparticles agglomerates

When primary particles are polydisperse (size diversity), they have a size distribution which can affect the final cluster structure and the fractal dimension, d_f. Therefore, when the cluster packing fraction increases, more particles are required to reach an asymptotic limit where some properties become constant on time. [56,76]

Moreover, aggregates are formed by sintering of neighboring primary particles or surface growth by deposition or chemical reaction. In cluster formation, the main driving force is the minimization of Gibbs free energy, so the cluster tries to eliminate any defect or interface as presented in Fig. 3. During the cluster formation, there is a release of energy that can increase the temperature and the coagulation rate as a consequence. [52,21,13]

Figure 3: Evolution of aggregate mobility radius rm and particle radius due to sintering.[11]

If sintering is faster than collision, spherical particles are formed. Sintering is an important mechanism for particle growth and dominates the formation of primary particles. Later, as particle grows, the sintering rate decreases while the particle coalescence rate increases, producing fractal shaped aggregates. Therefore, the cluster structure depends on the relative rates between particle collision and coalescence, which is affected by process conditions and material [82,18].

In most computer models, agglomerates are formed from primary particles that are assumed to be spherical and uniform in size (monodisperse) [50]. Therefore, an agglomerate behavior differs from a sphere with the same mass because its structure affects properties such as the rate of collisions. As a result of this structure, it is found that the agglomerate can be characterized by the fractal dimension, df. [44,52]

The aggregation process has two limiting cases. At the beginning, particles are dispersed in space and they follow Brownian motion. No interactions are presented and the Gibb’s free energy per particle is at its maximum. Then, when the system evolves, it starts to minimize the Gibb’s free energy forming close structures[79, 68, 21]. The bonds between the primary particles are an energy barrier that stabilizes the structure and prevents reorganization [79,18].

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2.3 Aerosol Dynamics

In an aerosol system composed of several particles, the size distribution and chemical com- position, change with time and position because of different phenomena such as coagulation, agglomeration, fragmentation, surface growth, etc. Therefore, an understanding of particle transport or movement is basic for process design, operation and for finding new measurements techniques. Moreover, the awareness on how interactions between particles happen is also important because it will define the agglomerates’ nature and behavior. [18,75,60].

Aerosol transport processes can take place at two different scales. In first place, in an individual particle scale, the particles exchange mass, heat, momentum and charge between the nearby particles and surroundings. In second place, in a large scale the group of particles move due to a gradient of concentration, temperature, pressure or electrical field [18,60].

In fluidization, particles are suspended in a gas stream, exhibiting a fluid-like behavior, resulting in an adequate mixing. The fluidization is influenced by the particle size and gas-solid density difference. A classification of the fluidization behavior for particles is summarized by the Geldart’s diagram shown in Fig.4. In Fig. 4 the x-axis refers to the particle size, d_p and the y-axis is the gas-solid difference, ρp− ρ. According to the diagram, nanoparticles belong to group C (Cohesive), and they cannot be fluidized. However, they are fluidized because they form aggregates with large sizes and low densities moving to group A (Aeratable) [27,18,84].

Figure 4: Aggregate’s transition based on Geldart’s diagram.[84]

The main mechanism for particles’ transport is diffusion. In diffusion, small particles move in random steps due to fluctuating forces exerted on the particles and surrounding gas molecules.

As a result, there is a net migration of particles from regions of high to low concentrations. The equation of conservation of species in terms of the flux vectors in a three dimensional space is

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presented in Eq. 9 [18,77].

∂n

∂t = − ∂Jx

∂x +∂Jy

∂y +∂Jz

∂z

= −∇J (9)

Where n is the concentration of particles in the system and the flux, J , is carried by diffusion and can be expressed as Eq. 10 for each dimension.

Jx = −D∂n

∂x; Jy = −D∂n

∂y; Jz = −D∂n

∂z (10)

Where D is the diffusion coefficient and depends on the particle size, temperature and concentration. An assumption is to consider the diffusion coefficient as isotropic [60,61]. Therefore, the value does not depend on the location, and Eq. 9 can be used to express the Fick’s second law shown in Eq.11.

∂n

∂t = D ∂²n

∂x² + ∂²n

∂y² + ∂²n

∂z²

= D∇²n (11)

The diffusion coefficient, D, is only a function of particle size at isotropic conditions. There- fore, small clusters diffuse faster than larger ones. In addition, diffusion is driven by two main forces: frictional resistance of the fluid, that is proportional to cluster size and velocity, and the fluctuating forces due to thermal motion, which is size independent [18,83,39].

In diffusion, there is an important assumption made by A. Einstein [12], which relates the motion of small particles to the motion of gas molecules. It is called ”principle of equipartition”

and states that the energy in a system is distributed to translational energy of the particles only [12, 18]. As a result, the diffusion coefficient can be calculated with the Einstein-Stokes equation, presented in Eq. 12.

D = kT

f = kT

6πµR_h (12)

Where, k is the Boltzman constant, T is the absolute temperature and f is the friction coefficient. For spherical particles, the friction coefficient can be expressed based on the particle size and medium viscosity [12,81, 4]. Therefore, the particles’ transport mechanism is mainly caused by to thermal diffusion caused by random collisions of primary particles with surrounded gas molecules. In the overall system, this phenomena changes the particle number density,

˜

n(vi, t), that can be described by the Smoluchowski equation, in Eq. 13, where vj and vi are the volumes of the colliding particles and β is the collision frequency [11].

∂ ˜n(v_i, t)

∂t = 1 2

Z vi

0

β (v_j, v_i− v_j) ñ (v_j, t) ñ (v_i− v_j, t) dv_j− ñ(v_i, t) Z ∞

0

β (v_j, v_i) ˜n (v_j, t) dv_j (13) The particle friction coefficient, f , depends on collisions between particles and gas molecules.

In addition, by increasing the cluster radius, occurs a transition from free molecular regime (f ∼ r²) to the Stokes or continuum regime (f ∼ r). The collision frequencies, β, is the probability of two clusters colliding which is proportional to the product of their densities and can

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be determined by simulations using Monte Carlo methods [68,50,52].

The Smoluchowski equation describes the rate of change of the concentration, ˜n, of particles containing i number of monomers. The collision frequencies, β, can also be referred as the aggregation kernel, which defines the rate of aggregation of particles with size i, and particles of size j [18,16].

The collision frequency, β, is an independent homogeneous function that is used for solving the Smoluchowski equation giving a self-preserving solution, where the shape of the function is the same through time and the time dependent factor is the mean cluster size [20,76]. The collision frequencies, β are important for describing an aerosol system since it has embedded information like the aggregation rate, size distribution and particle morphology.

All particles in a medium are subject to drag forces. In order to describe the particle motion in a medium (regime), the most used parameter is the Knudsen number, kn, which is defined as the ratio between the mean free path of the medium molecules, λ_m, and the particle radius, Rp as presented in Eq. 14 [59,77].

k_n= λ_m Rp

(14) If kn 1, the particles are in the continuum regime. It is also known as ”Stokes” or hy- drodynamic regime. In this regime, particle motion disturbs the isotropism of the distribution of molecular velocities of the medium [59, 67,50]. In addition, in this regime the agglomerate diffuses a small fraction of the primary particle’s diameter before changing direction. The diffusion coefficient can be calculated according to the Eq. 12 with the Einstein-Stokes equation.

At the continuum regime, the medium is dense and the path of the medium’s molecules which will hit the particle’s surface will be affected by the medium’s molecules leaving the surface [17,59,56].

On the other hand, when kn 1, the particles are at the free molecular regime. In this regime the particle has a little effect on the medium since most of the medium molecules impacting the particle surface come from far away from the medium and the agglomerate moves several primary particle’s diameters before changing direction [59,17, 65]. Moreover, particles can move in a diffusion (random motion) or ballistic mode. At the free molecular regime, the drag force coefficient for a spherical particle can be calculated using the Epstein equation presented in Eq.15 [18].

f = 8

3R²ρ 2πkT M

0.5

1 +βπ 8

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Where ρ is the medium density, m is the gas molecular mass and β is the momentum ac- commodation factor. This factor represents the fraction of molecules that leave the surface of the particle after collision. At the free molecular regime, the medium is light and the medium’s

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molecules path are not affected by the molecules that are leaving the particle’s surface [59,52].

The slip regime is the transition between continuum and free molecular regime. The diffusion coefficient can be calculated with the use of the Cunningham correction, to the Stokes-Einstein diffusion equation in Eq. 12. The Cunningham correction depends on the Knudsen number as showed in Eq. 16 [50,59].

C(kn) = 1 + 1.257kn+ 0.4 exp(−1.1/kn) (16) The regime behavior can be calculated for fractal aggregates if the characteristic length is used instead of the particle’s radius, like the radius of gyration, Rg, or the mobility radius, Rm

[68].

2.3.1 Agglomerate diffusion coefficient

The diffusion coefficient of an agglomerate differs for each particle because of the different structures formed by the primary particles creating a cluster. Experimentally, it was found a power law relationship between the cluster’s size and the fractal dimension expressed in Eq. 6 [52,46,68,50].

The fractal dimension has a range between 1 (one-dimensional chainlike structure) and 3 (homogeneous 3-dimensional structures, typically more compact than fractal structures). For compact agglomerates, the diffusion coefficient can be estimated from Stokes-Einstein Eq. (12) [12].In contrast for open structures, the gas can flow through the cluster, increasing the drag and reducing the diffusion coefficient[68].

On the other hand, without any external force, the primary particles are attracted by Van der-Waals forces, that are short-range and stabilize when the particles collide with each other.

Van der-Waals forces are caused by fluctuations in electron clouds at the atomic scale of a particle. Although the particle is electrically neutral, instantaneous dipoles are formed due to these fluctuations, which are propagated to neighboring particles [18,52].

The diffusional Knudsen number, k_nd, is the ratio of the agglomerate persistence length to a characteristic length. It is an indicator of the cluster’s diffusion magnitude and how an agglomerate moves effectively in a straight line [25, 47]. If k_nd → 0, the motion is diffusive and if knd → ∞ the motion is ballistic. If the aggregation system is diluted (V_f < 0.001), the characteristic length of the aggregate can be defined by the radius of gyration [25].

In a cluster-cluster aggregation system with high particle’s volume fractions, for example highly sooting flames, there is a transition between diffusion and ballistic motion which occurs due to cluster growth, affecting the kinetics of aggregation, cluster size distribution and agglomerate structures. Therefore, an impact on the fractal dimension is expected when the cluster size is larger [50,4,34].

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2.4 Stability of nanoclusters

Agglomerates do not possess a rigid structure. Their form can change due to condensation and evaporation, heating or mechanical stress. Those changes have an important influence on transport and light scattering properties [51,28,1].

Agglomerates can change and fragment their structure under shear and tension. Moreover, they can behave as elastomers, with a reversible behavior under stretching. This behavior is useful for the development of elastic materials [18]. For soot clusters, they are hard to break as their covalent forces produce sintering in their necks. Kutz and Schmidt [37], explored the restructuring of soot agglomerates using different solvents. Their results showed a more compact structure at saturation condition and no effect using water due to the hydrophobic nature on soot [10,50].

On the other hand, restructuring due to thermal forces can produces two limiting behaviors.

In first place, for strongly bonded particles, the structure remains (same d_f) but the size of the cluster is reduced. In addition, in polydisperse systems, the mean size of the particle size distribution is increased [59, 68, 48]. In second place, for weakly bonded particles, primary particles tend to restructure and the cluster becomes more compact (higher df) [82,65].

2.4.1 Fractal dimension d_f

The fractal dimension, d_f can be defined as the measure of the ”stringiness” of an agglomerate. It is related to the configuration and location of primary particles within the agglomerate.

However, the fractal dimension, df, does not define completely the structure of an aggregate, giving a partial description of the geometry and spatial distribution [56,50].

Furthermore, the fractal dimension, d_f, can be an indicator of the cluster structure self- similarity, which brings statistical scaling properties because all the characteristic lengths in the cluster, for example the particle radius and radius of gyration, are proportional. The mean radius and standard deviation from those characteristic lengths have a similar behavior [2].

However, the agglomerates are fractal-like in a statistical sense since the self-similarity cannot be identified at infinite scales. Therefore, the agglomerate can be described at an specific reso- lution and it can have a lower limit which is the primary particle size [47,39,11].

The fractal dimension, d_f, can be affected by the agglomeration mechanism formation, concentration, temperature, residence time, among other variables. The agglomeration mechanism has the highest effect on the fractal dimension because it defines how the particles collide and interact between each other [62,40]. The system concentration can also affect the collision frequency between clusters. Gonzalez et al. [23], found a dependency between concentration and the fractal dimension for diffusion limited cluster-cluster aggregation (DLCA). A linear increase in the fractal dimension applies for 2-D systems, while the increase has a square root behavior

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for 3-D systems. For 2-D off-lattice DLCA system the correlation between concentration and fractal dimension, d_f, is presented in Eq.17 [23].

d_f = d_{f o}+ aφ^c (17)

Where d_{f o} is 1.445, according to the fractal dimension obtained for 2-D DLCA systems experimentally, a is 1.005 and c is 0.999, values obtained from the data fitting of several simulations [23]. At high concentrations, the space is overcrowded with clusters which collide in the middle of other cluster instead of the branches as expected in highly diluted systems. Therefore, the produced agglomerate will be more compacted [34,44].

In addition, the fractal dimension, d_f, increases with increasing polydispersity of primary particles, decreasing the fractal dimension of the agglomeration mechanism (From DLA to RLA) and increasing the space dimension [10]. A more detailed explanation about the fractal dimension, d_f, is presented in Appendix A.

2.4.2 Anisotropy and cluster shape

The fractal dimension, d_f, and the radius of gyration, R_g, provide an average description of the cluster structure with no details on shape. For shape characterization, the symmetric moment of inertia tensor, T_I is calculated for the cluster with components expressed in Eq.18 [20,19].

T_ij = Z

ρ(r)q_iq_jdr f or i, j = 1, ..., d, i 6= j Tii=

Z

ρ(r)(r²− q²_i)dr + Im f or i = 1, ..., d

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Where r is the particle’s radial distance from the cluster’s center of mass, q_i is referred to x or y coordinates and i goes from 1 until the euclidean dimension of the system. For larger cluster sizes, the primary particle moment of inertia, I_m can be neglected. Those components lead to the square of principle radii of gyration, R²_i, from 1 until the euclidean dimension [20,52].

Therefore, the cluster shape and overall size are related by Eq. 19.

R²_g = 1

2[R²₁+ R₂²+ R²₃] (19)

The cluster shape can be analyzed with the mean anisotropy which is defined as the ratios of the squares of the principle radii of gyration as shown in Eq. 20. The mean anisotropy is the degree of the cluster’s symmetry at different dimensions. It is a measurement of the cluster’s shape and primary particles distribution on space [20,69,5].

hA_iji = hR²_i

R²_ji (20)

In addition, Fry at al. [20] proposed an alternate way to calculate anisotropy based on an equivalent ellipsoid which encloses the cluster.

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2.4.3 Multi-dimensional fractal aggregates

De Mart´ın et al. [8, 9, 46] found that big agglomerates (200 µm-400 µm) are not formed by a single particle-cluster mechanism but they are formed by the interaction of several clusters from different sizes. Therefore, a multihierarchical structure in fluidized agglomerates are formed, which can be described by different fractal dimensions, d_f [9, 77]. Several studies [8,9,46,13,65,83,42] have identified three scale levels. De Mart´ın et al. [9,8], demonstrated two fractal dimensions in the scale from 20 nm to 20 µm for fluidized nanoparticles. As a consequence from this multi-dimensional structures, a new approach on the relation between fractal dimension and size is required because the characteristic lengths are different for each level. The multi-dimensional aggregates are usually referred as superaggregates [69, 35].

In soot formation, the first mechanism is cluster dilute aggregation, where an experimental fractal dimension of 1.8 has been found for aggregate’s sizes ≤ 1µm. Soot superaggregates are formed by subsequent cluster-dense aggregation of low density aggregates (generated by the first mechanism) resulting in a fractal dimension of 2.6 for aggregate’s sizes ≥ 1µm [35, 6].

Moreover, a soot aggregate usually consists of 3000 monomers approximately and overlapping may occur with an increase of the first scale fractal dimension to 1.9 [35,55]. Superaggregates are created due to the transition where cluster-dilute aggregates enter into a cluster-dense aggregation regime, which is characterized by a small ratio between the mean nearest neighbor separation distance and the aggregate size [35,7].

The formation of soot superaggregates involves two stages. In the second stage, the building blocks are the primary clusters generated in the first stage mechanism [57,69]. In addition, the building blocks for the primary clusters are the primary particles. Therefore, superaggregates of size R_g2 are built by a number N_2,1 building blocks (primary clusters) with an aggregation mechanism which can be described by the fractal dimension df 2. The primary clusters (building blocks) are composed of N_p,1 primary particles with an aggregation mechanism described by the fractal dimension df 1. This argument can be extended to higher multi-fractal aggregates [9,69]. The mechanism for superaggregates with two fractal scales is summarized in Fig.5.

Figure 5: Multi-dimensional fractal aggregates mechanism

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As a result, the relation of the fractal dimension with the size of the superaggregate at different scales can be expressed as shown in Eq.21.

N1,p = k1

Rg1

R_p

df 1

N_2,1 = k₂Rg2

R_g1

df 2 (21)

Assuming monodisperse building blocks, the total number of primary particles in the superaggregate can be calculated as presented in Eq.22.

N_p= k_nR_g2 Rp

d_{f 2}

k_n= k₁k₂Rp^d^{f 2}^−d^{f 1}R^d_g1^{f 1}^−d^{f 2}

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Where k_n is the overall aggregate pre-factor and it is a function of the size of the building blocks and the mechanism at different stages [70]. De Mart´ın et al. [9] found that assuming a pre-factor of unity can lead to errors as it is assumed that agglomeration is performed only by individual primary particles, instead of a cluster-cluster aggregation mechanism [42]. The overall pre-factor, k_n, has information about the building blocks at different levels, like shape, size distribution, anisotropy, etc [70,69].

2.5 Computer simulation of Agglomerate formation

Aerosols are challenging to handle, because they can exhibit different behaviors under multiple conditions. The technical understanding of aerosols is low compared with other systems [18,48].

Therefore, there is a need to develop new theories on suspensions of solids in gas phase [50].

Simulations have permitted the statistical description of agglomerates and the development of theories for their formation and growth. The need of computer simulations for aerosol dynamics is justified because first of all, there is no comprehensive theory on aerosols systems which predicts the behavior with high precision [63]. Secondly, experiments are expensive, time consuming and difficult to measure. Moreover, with the aid of simulation, it is possible to identify new variables affecting the phenomena. Finally, experimental results can be related to an industrial process application [18,60,50].

In an aerosol, primary particles move in a specific manner until they collide. Therefore, assumptions are required to create a model on this phenomena. They are related to the motion of particles and agglomerates, as well as to the nature of the collision process [55, 48, 1].

There are three computational algorithms for agglomeration processes, that are subdivided by particle-cluster mechanisms and cluster-cluster mechanisms. Aggregates generated by cluster- cluster mechanisms are anisotropic, and their structure is not influenced by its size. In contrast, particle-cluster mechanisms have an anisotropy ratio which tend to 1, and the aggregates are influenced by their size [5]. Fig. 6 shows the main agglomeration mechanisms which are explained in the next sections.

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Figure 6: Agglomeration mechanisms in a 3-D space and fractal dimension associated.[18,9]

2.5.1 Difussion limited Aggregation (DLA)

In the first case, particle-cluster, a primary particle is set at the origin. Then, another particle is released at a random site, related to the location of the primary particle. The new particle moves in a random path until it collides the fixed particle or leaves the space. Then, another particle is released and the process is repeated until a desired size is achieved. Usually, the fractal dimension, d_f, is 2.5 for 3-D systems and 1.44 for 2-D systems with a well-defined center [50,49]. The generated structures have a random branching and an open structure, which look self-similar. The small fluctuations are enhanced at the branches, and this instability together with the randomness of Brownian motion can lead to complex structures [62].

In an aerosol, particle-cluster agglomeration happens at the beginning to form small agglomerates. Then, they collide in a cluster-cluster agglomeration. The agglomerates move in a random path until they collide forming a bigger structure, which is chain like and has a lack of a well-defined center [48,52].

Moreover, in cluster-cluster DLA, the resulting clusters have the largest anisotropy among all agglomeration mechanisms. In the cluster-cluster mechanisms, there is no seed particle like in the particle-cluster case. Instead, each cluster participates in the system with Brownian motion. As the system evolves, the number of clusters decrease and large randomly branching structures appear in the system [82,28].

The DLA mechanism is governed by the spatial distribution of particles which is non-local.

Therefore, it can satisfy the Laplace equation with moving boundary conditions in order to get an analytical solution. The main assumption in this mechanism is that particles stick together

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irreversibly.[62]. This model was introduced by Witen and Sander [82].

2.5.2 Ballistic aggregation (BA)

In Ballistic aggregation, the mean free path of the particle which collides is large compared with its size. The path is more linear compared to the Brownian motion in DLA. As a result, the primary particles can stick deeply into a fixed agglomerate, producing very compact structures with a fractal dimension, df, close to 3 for 3-D systems and close to 2 for 2-D systems [49,70].

On the other hand, cluster-cluster ballistic aggregation produces chain-like structures with a fractal dimension, d_f, of 1.95 approximately for a 3-D system [18].

In the particle-cluster mechanism, a single stationary particle is placed in the system and other particles are allowed to follow ballistic ”linear” trajectories until collision. Upon contact, the particles stick together [25,73]. Ballistic aggregation (BA) mechanisms can be found when the gas pressure is very low or when particles are too large [49].

2.5.3 Reaction limited aggregation (RLA)

For DLA and ballistic aggregation, all collisions are successful, increasing the cluster size, which is the case for most of aerosols without electrical charges that can repel particles [44]. The DLA can be extended to the Reaction limited aggregation (RLA) mechanism, adding a restriction on the number of collisions required before a sticking. This mechanism is a result of the repulsive barrier in the particle-particle pair potential that must be overcome before having an attractive interaction [49]. If the attractive force is not strong enough, the formation of the cluster can follow reorganization or dissociation. This is called reversible aggregation [62,15].

RLA produces more compact structures and the fractal dimension, d_f, is affected by the number of collisions before sticking. RLA is common in colloids and it is not usual in aerosols [68]. For cluster-cluster mechanisms, the overall shape is not spherical and its anisotropy is pronounced as the cluster becomes bigger [62].

Different cluster-cluster strategies exist to recreate physical phenomena like reorganization after aggregation, rotational diffusion or random bond breaking. However, these phenomena have little or no effect on the fractal dimension at a large scale and they only modify the structure on short length scales [51,69,65].

Finally, on cluster-cluster aggregation simulation, there are two methods on how cluster collision can be controlled: the first strategy is the polydisperse way where clusters are chosen randomly for motion. The second strategy is called hierarchical and only clusters of the same size are chosen for collision. In the hierarchical method, 2^N particles are in the system and they form 2^{N −1} binary clusters. Then, the binary clusters form 2^{N −2} clusters of 4 particles and the system continues until a single cluster is generated [49,28,48].

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Table 1: Fractal dimensions of clusters generated by different mechanisms [50,49,68].

Table 1 summarizes the characteristic fractal dimension of clusters generated by different mechanisms and space dimensions.

2.6 Statistical mechanism and Monte Carlo algorithm

Aerosols consist of a large number of particles that interact based on short range forces. Their dynamics are governed by Newton’s equations of motion for the center of mass coordinates and the Euler angles of its particles i, where i = 1, .., N_p, presented in Eq. 23 [61,16].

∂²r_i

∂t² = 1

m_iF_i(r_j, v_j, ϕ_j, w_j)

∂²ϕ_i

∂t² = 1 Ji

M_i(r_j, v_j, ϕ_j, w_j)

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Where Force, F_i, and torque M_i act on particle i of mass m and the tensorial moment of inertia, Ji, is a function of the particle positions, r_j, their angular orientation, ϕ_i, and their corresponding velocities, v_j and w_j. This set of equations is the core for molecular dynamics algorithms and they cannot be solved analytically [29,36].

Moreover, several simplifications are used for the agglomeration system modeling. The main assumption on the modeling of this system is that classical mechanics are used to describe the motion of particles, and for agglomeration dynamics, long-range fields are neglected. In addition, properties are calculated for the whole system using statistical mechanics [61].

This model is time and resource consuming. The main bottlenecks are: the calculation of short-range distances for all the particles, data extraction and post-processing. As a dynamic system, initial and boundary conditions are required [29,16].

There are several methods for modeling the formation of agglomerates and aerosol dynamics: Event-drive Molecular dynamics, direct simulation Monte Carlo and Rigid body dynamics.

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Moreover, there are advanced techniques like cellular automata, bottom to top reconstruction and Langevin dynamics [60].

In the event driven method, the main assumption is that at any time instant of the system, there is at least one very fast collision. During the time step, particles move with ballistic trajectories. Therefore, the system can be described by discrete events with long duration until collision happens. This method is not suitable for Brownian motion simulations due to the lack of random motion of the particles [60,61].

The rigid body dynamics is based on the opposite idea of evaluating interaction forces. The interaction forces are determined from consistency requirements on the behavior of particles, related to contact, deformation and no attractive normal forces. This method is suitable for rigid body motion on a plane and it is a simplified method for handling a large number of particles [60, 16].

The cellular automata is a discrete dynamical system which is composed of finite states automata (cells) that change their states according to the state of their neighbors based on deterministic rules. This method is useful for the simplification of complex dynamic behaviors and big systems where parallel computing can be used. It is suitable for homogeneous systems that are not applicable in aggregation. It can be used on granular heap growth or avalanche statistics [61,60,18].

In the bottom to top reconstruction, the particle motion is considered sequentially until a condition is reached. This is different from considering all particles simultaneously. In addition, only the particles that can physically interact with a selected particle are considered.

This method is a modification of the event driven method which does not solve the Newton’s equations of motion. This method is not suitable for dynamic systems as it is a very fast but highly limited [61,16].

The Langevin dynamics method solves the Newton’s equations of motion using stochastic differential equations as a strategy to simplify the system. This method is very accurate because it tracks the energy balance of the entire system given by the Hamiltonian. It is suitable for small systems where detailed results are required. However, it is highly computing expensive [61,60].

Finally, the Direct Monte Carlo is the most widely used method for particle dynamics. It is more efficient than the event driven method and it is more accurate since it determines the time-dependent distribution function, f (r, v, t), which can quantify the number of particles, the location at a specific volume and the velocity [16, 60]. The dynamic distribution is governed by the Boltzman equation, which does not describe particle’s motion but the time dependent behavior of probability due to random sampling [29, 36]. This method is suitable for large

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systems governed by the Brownian motion. Therefore, this method is widely used on the simulation of Diffusion Limited Aggregation (DLA) models [1,59, 50].

2.7 Soot formation

Soot is a common byproduct of fuel-rich combustion systems above a certain fuel to oxidizer range. The kinetics affecting combustion can also be related to the sooting process [78]. The most important factors for soot formation are: acetylene concentration which is the most abun- dant intermediate, Hydrogen atom which is the driving force on chain branching and combustion reactions and temperature which enhances the soot formation [7,35]. The thermodynamic and kinetics of the sooting process can be explained by the hydrogen-abstraction-carbon-addition (HACA) mechanism, developed by Frenklach et al. [17].

The soot production starts when the heat of the flame pyrolyzes the fuel, which breaks the fuel into smaller species like radicals, acetylene and oleofins. Then, those species combine to form benzene rings which subsequently form polycyclic aromatic hydrocarbons (PAH) [35]. The soot formation starts with the dimerization of first particles of PAH’s. These primary particles grow and nucleate to form nanoscale organic carbon particles, known as nascent soot [66]. As this particles spend time in the high temperature region of the flame, they lose hydrogen and become more carbonized with a graphite structure. This soot particles are know as mature soot [33].

Mature soot consists of aggregates of primary particles which are spherical in size from 15 to 50 nm in diameter. These primary particles are composed of an amorphous carbon core with a graphite shell [33, 54]. The primary particles grow due to surface reaction, gas phase PAH condensation or coagulation. On the other hand, the soot structure can grow attaching primary particles. Then, soot travels thorough the combustion system and it can be burned completely or escape through the exhausting system [66,72].

Nascent and mature soot particles have different properties and structures. First, nascent soot has a more reactive surface and it does not have any pattern in the internal structure [33,78]. In contrast, mature soot has a core-shell structure that can change due to process conditions like pressure, type of fuel, temperature and residence time [78,6]. In Fig.7 the evolution on soot structure and formation in a combustion system is shown.

The collision and coalescence mechanism for soot formation is based on multiple steps: the first one, converting the aerosol precursors into condensed molecules by chemical or physical processes [54]. Then, self-nucleation of the condensed molecules to form a cloud of stable nuclei.

Next, the collision and coalescense of stable nuclei to form primary particles. The coalescense rate decreases as primary particle size increases. Finally, the formation of fractal-like agglomerates by means of slight coalescence of some particles [17,4].

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Figure 7: Evolution on soot structure and formation in a combustion system [33]

The steps can go simultaneously in a defined system and they can produce agglomerates and aggregates at the same time. In soot formation, the primary particle size is determined by the different mechanisms, characterized by the coalescence time, τf, and the time taken for two particles to collide, τ_c. The interplay between these two mechanisms also affects the nature and strength of the bonds between primary particles [18].

During the time where, τ_f τ_c, coalescence is fast and particles are spherical. As τ_f approach to τc two limiting types of neck formation processes can occur:

When ^dτ_dt^f ^dτ_dt^c, and when τ_f ∼ τ_c, the sudden increase in τ_f freezes the particles between collisions. Therefore, coalescence is not produced on collisions. This leads to agglomerates that are created by weak forces [18,69,59].

On the other hand, when ^dτ_dt^f > ^dτ_dt^c, and when τ_f ∼ τ_c, the particle has time to coalesce partially. The resulting aggregate is sticked together by strong chemical forces [18,17].

Mitchell and Frenklach [56, 55], modeled the growth of soot using dynamic Monte Carlo method, including coagulation and deposition of gaseous species in the particle surface. From the simulation, fractal-like structures were formed when coagulation dominates and spherical structures when surface deposition controls. In addition, Balthasar and Frenklach[3], analyzed the effect of realistic particle size distribution on the morphology of primary soot particles in a laminar premixed flame using the Monte Carlo method [11,54]. Among other studies about soot agglomeration mechanism, Morgan et al.[57], simulated the evolution of mass, surface area and structural details of soot in laminar premixed flames, validating the computer model with experimental results [54].

Royal Institute of Technology Department of Chemical Engineering Numerical simulation of multi-dimensional fractal soot aggregates