The term aerosol refers to an ensemble of liquid or solid particles suspended in a gaseous medium. The study of aerosols is relevant to a variety of fields, including air pollution, combustion, and chemical engineering. Since the 1950s, advances in aerosol science have been driven by investigations into the health effects of radioactive aerosols and industrial aerosols in the workplace and the environment. More recently, much effort has gone into understanding the effect of various natural and anthropomorphic aerosols on global warming. These efforts have largely been aimed at reducing the adverse effects of aerosol particles.
On the other hand, a great deal of knowledge has been recently gained by researchers working with
1. The General Dynamic Equation (GDE) for single-component aerosols
To understand the detailed behavior of aerosol systems, it is necessary to model the dynamic behavior of the aerosol population. Several mathematical models have been developed to simulate the physical processes that affect aerosols. Such processes include nucleation, coagulation, growth due to condensation of gases, shrinkage due to evaporation, sedimentation, and deposition on surfaces. Assuming that a particle in the aerosol can be described by a variable υ to represent its size (e.g., mass, vo1ume, or diameter), the general dynamic equation (GDE) that describes the behavior of a single component aerosol is: 
where n(υ,χ,t)dυ is the continuous distribution function that specifies the concentration of particles in the size range [υ, υ+dυ] at position
These four terms represent processes that may be called external since they involve movement across the walls of an elemental volume. However, particles in a given size range may also be modified as a result of internal processes such as growth and coagulation. The growth term in Equation (1) describes the rate of change of the distribution function
2. Multicomponent aerosols
Even though common aerosols, e.g., atmospheric aerosols, are made of several species rather thana single one, little work has been reported on modeling the evolution of multicomponent to model the particle encapsulation process, and the GDE needs to be generalized to account for processes involving several species. To do so, a new variable
such that O < Yi < 1, and Yi = 1. Now, Equation (1) is the appropriate equation for modeling the particle dynamics with n(υ,
In general, particles of the same size may have different chemical composition and the composition of a particle affects its growth rate. Thus, the computation of size distribution should generally not be performed independent of particle composition. Despite this, it is reasonable to assume that the rate of collision depends only on the sizes of the particles involved and not on their composition. Consequently, the usual expressions for coagulation, diffusion coefficient D, and collision frequency function β can be used. This also implies that interactions such as attractive or repulsive forces developed between particles of different materials are not considered.
Similarly to the case of single component aerosols, the dynamics of particle collisions is reduced to the assumption that the particles stick together with a probability α, which is known as the accommodation or sticking coefficient. For simplicity, α is assumed to be equal to 1. In addition, it is assumed that no chemical reaction occurs among the components in the particles. Consequently, when a collision between a particle of mass υi and composition Yi with another particle of mass υj and composition Yj occurs, a new particle of mass (υi +υj) and composition Yk = (Yi*υi + υj*Yj)/(υi + υj) is formed.
3. Solution methods
From a mathematical point of view, the modeling of multicomponent aerosols and the particle encapsulation process reduces to solving the GDE under the assumptions described in previous sections. The GDE, being a nonlinear integro-differential equation, will be solved numerically. To reach the objectives of this dissertation, the numerical method chosen must:Model coagulation and condensationTreat multicomponent aerosolsInclude the Kelvin effect, particle heating due to condensation, and heat loss from the aerosol
3.1. Solution method for single-component aerosol
Several works have been conducted to solve the GDE. Much of the effort to solve this equation has involved conditions where only coagulation is important. This information is of particular importance because the nonlinearity of the GDE comes from the coagulation term. Reference 58, which is a comparative review of mathematical models for aerosol dynamics, considers four approaches to the solution of the GDE. They are classified according to their representation of the size distribution and differ only in their degree of approximation of the size distribution.
Continuous representation (e.g., J-Space method )
Parameterized representation (e.g., Similarity solution )
Discrete representation (e.g., Sectional method )
Probabilistic methods (e.g., Monte Carlo method )
Those techniques that offer any desired degree of approximation are referred to as
3.2. Solution methods for multicomponent aerosols
Due to the diversity of techniques for computing size distribution dynamics of single-component systems, one approach to the multicomponent problem is to use a single variable, such as the mass of one of the components in the particle, to characterize the particle size and composition. Assuming that the characteristic component is conserved during coagulation, the masses of the nonconserved components in the particle may be determined by an auxiliary constraint, typically thermodynamic equilibrium. Such an approach has been reported for H2SO4-H2O aerosols.  In this case it is assumed that H2SO4 is the characteristic component that is conserved by coagulation, and the water content of the particle is determined by thermodynamic equilibrium. Therefore, only the distribution of H2SO4 with time needs to be determined. Although an attractive feature of the approach is the reduction of the size of the problem, it is greatly limited to systems for which auxiliary constraints are valid and can be obtained.
In a more general approach, the techniques for single-component aerosols have been extended to include condensation/evaporation of the same or of a second material. Only the sectional method has been considered because the fundamental assumptions of the continuous and parameterized representation approaches are not valid when condensation occurs. Gelbard and Seinfelds8 extended the sectional method to account for composition in multicomponent aerosols. They assumed that all the particles within a section have a constant size and composition and that coagulation and condensation are independent of particle composition. Then the governing equations are reduced to a set of m x z ODEs, where m is the number of sections for the size range and
At high rates of condensation, three main problems appear : the first is numerical diffusion. This problem arises because growth by condensation is a hyperbolic differential equation. Numerical diffusion lowers the peak value and broadens the size distribution, as shown in Figure 1. The second problem is conservation of mass of both gas-phase and particles during growth. Often, growth models subtract off the amount of gas removed by aerosol growth. However, subtracting can result in negative gas concentrations, requiring subsequent adjustments. The third problem is associated with the wide range of timescales for aerosol processes. Since the particle size domain may extend over several orders of magnitude in particle diameter, the characteristic timescales for particle growth may also have large variations. This results in stiff systems of differential equations. Such systems require special solution techniques that are not robust and are computationally expensive. 
The sectional method creates numerical diffusion because when mass moves to larger (condensation) or smaller (evaporation) sections, it distributes itself uniformly throughout the section. As a result, the distributed mass can quickly grow or evaporate to the next larger or smaller section the very next time step. This error can be reduced by drastically refining discretization. However, the computational effort for such approach is prohibitive. Quadrupling the number of sections that span the size range will halve the numerical diffusion.  By studying the resulting linear differential equations for a constant condensable vapor concentration, Gelbard  showed that the fixed-grid approaches cannot overcome numerical diffusion. Applying the
Jacobson and Turco  advanced this idea and developed a model that uses a hybrid size grid. They used a moving size grid for condensation/evaporation and a stationary grid for other processes, such as coagulation, and nucleation. With this model, particles are assumed to be composed of an involatile core and a coating of a second condensable phase. The assumption of same size and composition for all particles within a grid still remains. Initially, the size distribution is organized within the sections in order of increasing core mass as opposed to increasing total mass. When coagulation among uncoated cores occurs, the mass of the new formed core is distributed evenly among the sections having cores with masses immediately larger and smaller than the new formed core. For example, if sections A, B, C, and D contain particles with cores of masses 2, 4, 8, and 16 units, respectively, coagulation among particles coming from sections B and C will result in a core of mass 12 units. Then half of the mass of the particle is added to section C and the other half to section D. Consequently, the number of particles in section B is reduced by one, section C is reduced by half, and section D increased by half. On the other hand, the moving grid contains the total mass of the particles (core plus coating). Then, during condensation or evaporation, particles are not transferred between sections; instead, their total masses increase or decrease to their exact sizes. To simulate coagulation among coated cores, it is assumed that when two coated cores combine, they form a third particle of a composition that falls between the characteristic compositions of two adjacent sections, i.e., the methodology for uncoated cores is applied regardless of the composition of the new formed particle. Clearly, this leads to problems in the conservation of mass of the condensable phase.
In summary, it has been shown that the coagulation equation must be solved through numerical methods. Among the many techniques available, the sectional method, although is not the simplest, is well accepted due to its generality. Contrary to the case of coagulation, there is a general analytical solution for the condensation equation,  but the numerical techniques available for its solution possess several inherent problems like numerical dispersion, numerical diffusion, and an inability to conserve mass of the components. Only with a moving-grid-type technique can a proper numerical solution be obtained. However, this technique is not compatible or applicable when there is simultaneous coagulation and condensation.
Then there exists a need for a robust numerical technique to solve for coagulation and condensation simultaneously. In addition, the technique should be able to handle multicomponent aerosols with no restrictions in particle size and composition.
4. The solution method proposed
In the process of selecting the most convenient numerical technique to simultaneously solve the coagulation and condensation equation to study the encapsulation process, it was found that the Monte Carlo method possesses several advantages that make it an attractive route:
The Monte Carlo method is intrinsically compatible with probabilistic phenomena like the coagulation process.
The Monte Carlo method can be coupled to other probabilistic and deterministic approaches.
The rate of particle growth is a simple ODE that can be solved very easily in a deterministic manner.
The proposed methodology is to simulate coagulation with the Monte Carlo method and couple it through time to a deterministic solution of the condensational growth equation. The coupling is obtained by using the time step for coagulation as the time step for condensation. The following chapter describes the Monte Carlo method adopted. Chapter 5 describes the solution of the equations for particle growth by condensation and the methodology used to couple the two solutions.