1 Introduction

The standard Smoluchowski coagulation model^[1] provides a mean field description of several processes of mass aggregation in nature. It has found applications in such a variety of processes as the formation of clouds and smog^[2],the clustering of planets,stars,and galaxies^[3],the kinetics of polymerization^[4],the schooling of fishes^[5] and the formation of ‘marine snow’^[6]. So the analysis of aggregation through Smoluchowski’s equation has attracted considerable attention over the past several decades. While the majority of studies are focused on the temporal evolution of size (or mass) distribution of particles with only one composition,in most cases with practical interest,the aggregation particles consist of multiple composition (or component),e.g.,in atmospheric aerosols^[7],in the nanoparticle synthesis from the gas phase^[8] and in the granulation of pharmaceutical powders^[9]. There are a limited number of studies addressing multicomponent aggregation compared to single-component aggregation. The discrete population balance equation for multicomponent system is given by extending Smoluchowski’s equation for single-component aggregation as follows:

where m = (m₁,m₂,...,m_k),and mi (1 ≤ i ≤ k) represents the mass of the ith species. The sum on the above equation is carried over all k variables. The kernel K(m,m') is,in general, a matrix of rate constants for different species with different mass (or size). Here,we limit the case where the kernel is independent of particle composition and only depends on the total mass of the reacting particles. Such simplification of the kernel is still of practical interest,and particularly,it makes theoretical analysis of multicomponent aggregation possible.

The development of asymptotic scaling solutions which describe self-preserving universal particle size distributions in the long time limit,has been an important advance in the understanding of the process of aggregation^{[10, 11]}. In analogy with single-component aggregation, these scaling solutions have recently been extended to the case of multicomponent aggregation for constant kernel^{[12, 13]} and non-constant kernel^[14]. In ^[14],the kernel is proportional to the sum of mass of each component. Recently,Fernandez-Diaz and Gomez-Garcia derived exact solution of continuous multicomponent equation with the sum kernel^[15] and the product kernel^[16]. Furthermore,the system with continuously injection also stimulates the interests of researchers where steady solution can be reached^{[17, 18]}.

The numerical methods to (1) mainly fall into two classes,deterministic methods^{[19, 20]} based on differential equations and stochastic methods^{[8, 21, 22]}. They are another important way to investigate the evolution of the multicomponent coagulation.

In this paper,we will derive the self-similar solution of multicomponent system for the constant,sum and product kernels by use of a different generating function,and then we seek the steady solution with continuous injection of monomer particles in the lower end.

This paper is organized as follows. In Section 2,we present exact solutions for the constant kernel,sum kernel and product kernel in multicomponent coagulation system. The steady-state solution for large masses in the multicomponent coagulation system with continuously injection is derived in Section 3. Finally,we give out summary and suggestions for further research in Section 4. 2 Exact solutions without injection

As mentioned in the previous section,we concentrate on the component-independent kernel. Following Loshnikov’s method^[12],we introduce a new variable: m₊ = m₁ + m₂ + · · · + m_k. It is obvious that c(m₊,t) obeys one-component Smoluchowski coagulation equation,i.e.,

For monodisperse initial conditions,the solution of (1) can be written as

where the combinatorial function g(m) reads

Now,in this simplified case,the solution of (1) is related to the solution of (2) via (3). To solve (2),we can introduce the following generating function^[23]:

Now,the moments of sum variable can be easily calculated by virtue of exponential function in the generating function as follows:

Substitution of (5) into (2) yields a governing equation for F,whose solution depends on the detail of the kernel. We solve F for the case of constant kernel,sum kernel of total mass and product kernel of total mass in Sub-sections 2.1-2.3,respectively. 2.1 Solution for constant kernel

For the constant kernel,substituting the generating function into (2) yields

Obviously,with solution M₀ = 2/(t + 2),we have

With Taylor’s expansion,it yields

In the long time limit,the corresponding distribution is given by

To demonstrate the asymptotic properties of c(m,t),we concentrate on the case k = 2. For large masses m₁,m₂,the combinatorial function reads

From (3),(10),and (11),it arrives at the asymptotic form of the multicomponent mass distribution

with m_± = m₁ ± m₂. By the central limit theorem,since each of its components is conserved, the conserved mass difference m_- = m₁ − m₂ is normal,and thus,m_- ~ . Furthermore, using the scaling variable

where s(t) is the mean particle size,and by the scaling hypothesis of ^[10],s(t) ~ t for the constant kernel. Consequently,the solution can be written in a convenient scaling form, with scaling profile The result is consistent with the solution in ^[13]. 2.2 Solution for sum kernel

Here,we assume K(m,m') = m₊ + m₊'. We also investigate c(m₊,t) first. With the generating function (5),we obtain the equation

where M₁(t) is the first moment of c(m₊,t). Being a non-gelling system,without loss of generality,we can set M₁ = 1 for all time. The corresponding zeroth moment equation can be obtained by setting z = 0

It gives M₀ = exp(−t/2 ) for the monodisperse initial conditions. Substituting it into (16),a simple differential equation is obtained as

For this PDE,the following expression is obtained by using the method of characteristics

With Lagrange’s expansion,the distribution function can be given as

In the long time limit,the corresponding distribution is given by

For (m₊)!,we use Stirling’s formula,and get

For two species case,from (3) and (11),it arrives at the asymptotic form of the multicomponent mass distribution

From scaling hypothesis in ^[10],we know that the mean particle size s(t) ~ et,and m_- ~ . Hence,using the scaling variable

we can write the solution in a convenient scaling form, with scaling profile This result agrees with the Vigil-Ziff conjecture in ^[14]. 2.3 Self-similar solution for product kernel

Now,we investigate the product kernel K(m,m') = m₊m₊'. We can obtain the equation for generating function F as follows:

Explicit derivation for the solution can refer to ^[23]. We can write them as

as m₊→ +∞.

Similarly,we can write out the multivariable distribution function

As the definition of ^[10],it is a gelling system. When t approaches the critical time t_c (= 1), the mass distribution would arrive to a steady state. According to the scaling hypothesis,before gelation,s(t) ~ (1 − t)⁻²,and m_- ~ ,similarly,we can arrive

It is the product of a normal distribution and the scaling function for the homogenous coagulation. This result suggests that for some gelling kernel the Vigil-Ziff conjecture is valid.

Figure 1 shows profiles of self-similar solutions for above mentioned three kernels. These functions are the product of a normal distribution and the scaling solution for the single variable coagulation. For a fixed y value,φ is monotonous for constant and sum kernels,while it is not for the product kernel.

3 Steady-state solution with injection

We now investigate the steady-state properties of the aggregation process by introducing the input of particles into the system. Here we focus on the case that the number of free monomers at a specific concentration for all time by continually introducing monomers into the system, that is ċ(I,t) = 0,I = (1,1,· · · ,1). Obviously,every mass distribution expect m = I obeys (1). For simplicity,we only consider the case with constant kernel. Similarly,one can find the steady-state solution for the sum kernel.

We denote steady-state distribution by c(m) and the generating function by F(z) = By eliminating the time derivative,we arrive at the generating equation

The solution to the generating equation reads F(0) = 2 and F(z) = 2(1−). With the similar process,we find that the steady-state distribution is

with

For large masses,the asymptotic form of the steady-state distribution is

It is clearly seen again that the similar solution is the product of a normal distribution and the scaling solution for the single variable coagulation. 4 Conclusions

In this paper,we study solutions of the multicomponent coagulation system under the assumption that the coagulation kernels neglect the interactions between different species. With the generating function,we obtain the self-similar solution of the constant kernel,sum kernel and product kernel as m → ∞. For the constant monomer system,we find out the steadystate solution for the constant kernel. The asymptotic process is governed by two distinct size scales. A typical coagulation scale m₊ characterizes the total mass,while a diffusive scale m− characterizes individual masses.