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Article: An introduction to mathematical models of coagulation–fragmentation processes: A discrete deterministic mean-field approach

Authors: Jonathan A.D. Wattis

Reviewed by: Dustin Keck

All work shown below can be assumed to originate from Wattis unless otherwise cited.

Executive Summary

This paper focuses on fundamental discrete deterministic mean-field models for both coagulation and fragmentation processes. On the other hand, the paper does not describe in any detail stochastic modeling for these processes nor does it cover a host of other ways of modeling coagulation and/or fragmentation. Throughout the paper, the authors make several major assumptions. First, the systems described are composed of a large number of fundamental and identical units ('atoms') that cannot be broke down any farther. Second, these fundamental units can join together to form clusters. Third, the authors can then use law of mass action to describe the interactions of these clusters. Fourth, in each of the models discussed, the authors assume a discretization of the size of the 'atoms', while they assume time is a continuous variable. The paper then takes on three distinct sections. It first reviews Smoluchowski's coagulation equation (done in 1917) and describes some updated results. Next, this paper explains the Becker-Doring equations that account for both coagulation and fragmentation. The paper then concludes by describing a few, more complex models than either Smoluchowski's equation or the Becker-Doring model.


Context/Biological Phenomenon under Consideration

Coagulation-fragmentation processes occur across of breadth of areas. This paper mentions the formation of aerosols, colloidal aggregates (such as shaving cream or mayonnaise)[1], polymers, which are large molecules composed of repeating structural units[2], and celestial bodies on astronomical scales. Of particular interest to us, coagulation-fragmentation processes also occur in bacterial pathogens such as Klebsiella pneumoniae, see APPM4390:Bacterial Aggregates in the Bloodstream. The coagulation process consists of the dynamics occurring when 'atoms' of a particular substance join together to form larger clusters of those 'atoms'. Conversely, fragmentation entails the dynamics involved when pieces of these clusters break away from their original cluster to form smaller clusters of their own. Again, as discussed in APPM4390:Bacterial Aggregates in the Bloodstream, understanding these dynamics can lead to better treatments for conditions such as blood-stream infections.

History

Marian Smoluchowski was an ethnic Polish scientist in the Austro-Hungarian Empire. He was a pioneer of statistical physics and an avid mountaineer [3]. He actually described Brownian motion independently of Albert Einstein. Smoluchowski's work, inculding that on coagulation, had dramatic consequences in the early 1900's. Fulinski writes, "Let us remember that at the beginning of our century the 'molecular hypothesis' was just a hypothesis, which needed proofs, and that the first experimental proofs — honoured in 1926 by Nobel prizes in physics (for Perrin) and in chemistry (for Svedberg) — were based on Smoluchowski’s theory of Brownian motion, criticalopalescence, sedimentation and coagulation." [4]

Becker and Doring were both physists. Their equations used in this paper added complexity to the model used by Smoluchowski by making coagulation a reversible process which introduces fragmentation. On the other hand, their model greatly simplified Smoluchowski by assuming that reaction of clusters only included interactions with monomers. In other words, a cluster of size 7 only includes a coagulations between clusters of size 6 and monomers (clusters of size 1) to form clusters of size 7, and clusters of size 7 joining with monomers to form clusters of size 8. The reverse process, fragmentation, of each of these are also included.

Of course, there have been modifications made to these models as time has evolved. Fournier and Laurencot have considered models with homogeneous kernels of mass-conserving type. Also, Menon and Pego have shown a number of interesting results for the continuous Smoluchowski coagulation equations with a constant kernel. Regarding the Becker-Doring equations, Dreyer and Duderstadt, as well as Hermann et al. take a different approach with respect to the available free energy of the system. The last section of the paper discuss a few more even more general approaches taken to these types of problems.

Full model with results from the paper

The first set of models discussed in this paper are derived from Smoluchowski's coagulation equation, which models the process of binary aggregation. Aggregation and coagulation are used interchangeably throughout the paper. If we want to know how cluster size distributions change over time, we can describe the process as:

C_{r}+C_{s}\rightarrow C_{{r+s}}

where, C_{r} represents a cluster of size r fundamental units. Denoting the reaction rate constant as a_{{r,s}} and the concentration at time t of C_{r}, the law of mass action gives

{\frac  {dc_{r}}{dt}}={\frac  {1}{2}}\sum _{{s=1}}^{{r-1}}a_{{s,r-s}}c_{s}c_{{r-s}}-\sum _{{s=1}}^{{\infty }}a_{{r,s}}c_{r}c_{s}

Specifically, this paper uses this model to answer the following questions. Is the total mass in the system conserved? Do self-similar solutions exist and what is their form? The authors emphasize that for a specific application, the form of a_{{r,s}} is not necessarily easy to determine. While many different forms of the aggregation kernel a_{{r,s}} have been used, this paper describes three specific cases in detail. In all three cases, this paper uses a couple of important concepts and their resultant equations. First they use a generating function approach. With this approach, they define

C(z,t)=\sum _{{r=1}}^{{\infty }}c_{r}(t)e^{{-rz}}

where M_{0}(t)=C(0,t)=\sum _{{r=1}}^{{\infty }}c_{r}(t) is the total number of cluster in the system, M_{1}(t)=-C_{z}(0,t)=\sum _{{r=1}}^{{\infty }}rc_{r}(t) is the total mass of the system, and the kth moment of the cluster size distribution function is

M_{k}(t)=\left.(-1)^{k}{\frac  {\partial ^{k}}{\partial z^{k}}}C(z,t)\right|_{{z=0.}}

All three cases also use the following initial conditions:

c_{r}(0)=0\;{\textrm  {forall}}\;r>1,\;\;\;c_{1}(0)=1.

The first specific form of a_{{r,s}} models Brownian coagulation in which case a_{{r,s}}=a. Here the rate constant does not depend on the size of the cluster. Specifically for this example, the authors use a_{{r,s}}=1. In combination with the generating function this results in the single partial differential equation (PDE)

{\frac  {\partial C}{\partial t}}={\frac  {1}{2}}C(z,t)^{2}-C(0,t)C(z,t)

which after substituting the initial conditions leads to a self-similar solution of the form

c_{r}(t)={\frac  {4}{t^{2}}}e^{{-2r/t}}\;{\textrm  {as}}\;t\rightarrow \infty \;{\textrm  {with}}\;r=O(t).

A self-similar solution means that the system eventually "forgets" its initial conditions, so that given the initial mass and compactly supported initial conditions, the long-term solution is not too complex.

The second specific form of a_{{r,s}} models a limited case of gravitational coagulation in which case a_{{r,s}}=a(r+s). Specifically, the authors use a_{{r,s}}={\frac  {r+s}{2}}. The PDE in this case is

{\frac  {\partial C}{\partial t}}=-{\frac  {1}{2}}CC_{z}+{\frac  {1}{2}}C_{z}M_{0}(t)-{\frac  {1}{2}}M_{1}(t)C.

which after substituting the initial conditions leads to a self-similar solution of the form

c_{r}(t)\sim e^{{-2t}}g\left({\frac  {r}{e^{t}}}\right),\;{\textrm  {with}}\;g(\eta )={\frac  {e^{{\eta /2}}}{(2\pi \eta ^{3})^{{1/2}}}}.

The third specific form of a_{{r,s}} models branched chain polymerisation in which case a_{{r,s}}=ars. Specifically, the authors use a_{{r,s}}=rs. The resulting PDE is

{\frac  {\partial C}{\partial t}}={\frac  {1}{2}}C_{z}^{2}+M_{1}(t)C_{z}.

which after substituting the initial conditions leads to a self-similar solution of the form

c_{r}(t)\sim {\begin{cases}{\frac  {e^{{-r(t-1-log(t))}}}{r^{{5/2}}t{\sqrt  {2\pi }}}}\;t<1\\{\frac  {1}{r^{{5/2}}t{\sqrt  {2\pi }}}}\;t\geq 1\end{cases}}.

In this situation, the solution introduces the concept of gelation, which will be discussed further in the Analysis section of this review.

The paper then goes on to show that systems with more general assumptions behave somewhat like those discussed above. Considering only what happens at large times and large cluster sizes, kernels of the following form are considered:

a_{{r,s}}=a(r^{{\mu }}s^{{\nu }}+r^{{\nu }}s^{{\mu }}),

where r,s>>1 with a\in {\mathbb  {R}}_{+}. Four types of behavior are observed under these assumption, all of which are discussed in the Analysis section of this review.

The next set of models described in this paper are derived from the Becker-Doring equations. These equations allow for both aggregation and fragmentation of a single particle at a time. In this case the reaction looks like

C_{r}+C_{1}\leftrightharpoons C_{{r+1}}

where the forward rate is denoted a_{r} and the backward rate is denoted b_{{r+1}}. The paper provides a helpful example of what this means. For instance, the only reaction that involve C_{4} are C_{3}+C_{1}\leftrightharpoons C_{4} and C_{4}+C_{1}\leftrightharpoons C_{5} where the rate of change is

{\frac  {dc_{4}}{dt}}=a_{3}c_{3}(t)c_{1}(t)-b_{4}c_{4}(t)-a_{4}c_{4}(t)c_{1}(t)+b_{5}c_{5}(t). (1)

They also define the general rate of change of concentrations as

{\frac  {dc_{r}}{dt}}=J_{{r-1}}(t)-J_{r}(t),\;\;(r\geq 2)

where J_{r}(t) is defined as

J_{r}(t)=a_{r}c_{r}(t)c_{1}(t)-b_{{r-1}}c_{{r+1}}(t).

Finally, while there are a number of choices for c_{1}(t), this paper considers two formulations of the Becker-Doring equations. In the first case, the authors assume c_{1}(t) is a constant, so c_{1}(t)\equiv c_{1}. For the second case, total mass of the system is fixed, so the monomer concentration, c_{1}(t) varies. As with the Smoluchowski equations, ideally we would like explicit solutions to the Becker-Doring equations, but here again, the authors emphasize in actual application this can be extremely difficult, so studying long term behavior of systems takes precedence. They also make an important distinction between equilibrium and steady-state. Equilibrium implies the the forward and backward rates of reaction are equal vs. steady-state where the time derivatives are zero. The following highlights some of the more important models/parts of the models for the two different assumptions on the monomer concentration with the results explained in the next section.

First, we must address some notation. At equilibrium, all of the fluxes, J_{r} are zero, so the equilibrium concentrations are

c_{{eq,r}}={\frac  {a_{{r-1}}}{b_{r}}}C_{{eq,1}}c_{{eq,r-1}}=\cdots ={\frac  {a_{{r-1}}\dots a_{q}}{b_{r}\dots b_{2}}}c_{{eq,1}}^{r}=Q_{r}c_{{eq,1}}^{r}. (2)

On the other hand, at steady-state, J_{r}=J for some constant J. We then have

c_{{sss,r}}=Q_{r}c_{1}^{r}\left(1-J\sum _{{k=1}}^{{r-1}}{\frac  {1}{a_{k}Q_{k}c_{1}^{{k+1}}}}\right).

Now in the case of constant monomer concentration we also define \theta =ac_{1}/b which shows which process, aggregation or fragmentation, is dominating the reaction. When \theta <1, fragmentation dominates aggregation. After some derivation, the authors show that for \theta >1, the steady-state solution c_{{sss,r}}=c_{1} is approached at large times, whereas for \theta <1, the equilibrium solution c_{{eq,r}}=\theta ^{{r-1}}c_{1} is approached at large times. The paper then examines these solutions for three cases. When \theta <1

c_{r}(t)\sim {\frac  {b}{2a}}\left({\frac  {ac-1}{b}}\right)^{r}\;{\textrm  {erfc}}\;\left({\frac  {r-(b-ac_{1})t}{{\sqrt  {2(b+ac_{1})t}}}}\right)\;{\textrm  {as}}\;t\rightarrow \infty . (3)

When \theta =1 we have

c_{r}(t)=c-1\;{\textrm  {erfc}}\;\left({\frac  {r}{2{\sqrt  {bt}}}}\right)\;{\textrm  {as}}\;t\rightarrow \infty . (4)

Finally, when \theta >1 we have

c_{r}(t)\sim {\frac  {1}{2}}c_{1}\;{\textrm  {erfc}}\;\left({\frac  {r-(ac_{1}-b)t}{{\sqrt  {2(b+ac_{1})t}}}}\right)\;{\textrm  {as}}\;t\rightarrow \infty . (5)

With the other approach, the authors assume constant total mass in the system, so

{\frac  {dc_{1}}{dt}}=-J_{1}(t)-\sum _{{r=1}}^{{\infty }}J_{r}(t) (6)

At this point, the paper summarizes the results of Ball, Carr, and Penrose as they sought solutions to (6). There are four cases, based on specific combinations of aggregation and fragmentation rate coefficients, in which metastability occurs in the last two. In this context, metastability is when the system is resistant to many perturbations away from that state, but still evolves on some trajectory. For case 1, we have R=\infty , where R is the radius of convergence of

\varrho _{{eq}}(z)=\sum _{{r=1}}^{{\infty }}rc_{{eq,r}}(z)=\sum _{{r=1}}^{{\infty }}rQ_{r}z^{r}.

where \varrho _{{eq}}(z) is the total mass of the equilibrium solution and z=c_{{eq,1}}. Then choosing rates such that fragmentation dominates,

\varrho _{{eq}}(z)=\sum _{{r=1}}^{{\infty }}{\frac  {z^{r}}{r!}}=e^{z}-1.

For case 2, we have 0<R<\infty ,\;\varrho _{{eq}}(R)=\infty . Here for any z\in [0,R) we have \varrho _{{eq}}(z)=\varrho _{0} for any intial mass \varrho _{0}. For case 3, we have 0<R<\infty ,\;\varrho _{{eq}}(R)<\infty . Here \varrho _{{eq}}(z) has a maximum value defined as \varrho _{c}=\varrho _{{eq}}(R)<\infty . Now, if \varrho _{0}<\varrho _{c} we get similar results as case 2, but if \varrho _{0}>\varrho _{c} we get weak convergence. Then with very specific aggregation and fragmentation rates, we have

\varrho _{{eq}}(z)={\frac  {2z}{1+{\sqrt  {1-az/b}}}}.

Then for case 4, we have R=0. In this case, the only equilibrium solution is c_{{eq,r}}=0 which means for all r,c_{r}(t)\rightarrow 0 as t\rightarrow \infty , so that for every t we have \sum _{{r=1}}^{{\infty }}rc_{r}(t)=\varrho _{0}.

The models in the final section of the paper reflect more generalized processes of coagulation and fragmentation. They mention three processes in particular. First, the authors consider binary fragmentation which generalizes the step-wise fragmentation of the Becker-Doring equations. In this case, we have

C_{r}\rightarrow C_{s}+C_{{r-s}} (7)

where under the assumption that b_{{s,r-s}}=b_{{r-s,s}}, we get

{\frac  {dc_{r}}{dt}}=-{\frac  {1}{2}}c_{r}\sum _{{s=1}}^{{r-1}}b_{{s,r-s}}+\sum _{{s=1}}^{{\infty }}b_{{r,s}}c_{{r+s}}. (8)

Second, the authors briefly describe Smoluchowski coagulation-fragmentation, in which case

C_{r}+C_{s}\leftrightharpoons C_{{r+s}},\;{\textrm  {forall}}\;r,s\geq 1.

Here, they are still only considering binary fragmentation where both the aggregation rate, a_{{r,s}} and the fragmentation rate, b_{{r,s}} are symmetric. Then applying the law of mass action,

{\frac  {dc_{r}}{dt}}=-{\frac  {1}{2}}\sum _{{s=1}}^{{r-1}}W_{{s,r-s}}+\sum _{{s=1}}^{{\infty }}W_{{r,s}} (9)

W_{{r,s}}=a_{{r,s}}c_{r}c_{s}-b_{{r,s}}c_{{r+s}},\;{\textrm  {with}}\;{\dot  {c_{1}}}=-\sum _{{s=1}}^{{\infty }}W_{{1,s}}. (10)

Third, the authors examine multi-component coagulation-fragmentation, where multiple species form heterogeneous clusters. A couple of examples are offered. In one such example, if each cluster, c_{{r,s}}(t) has r units of type A and s units of type B, we have

{\frac  {dc_{{r,s}}}{dt}}=J_{{r-1,s}}-J_{{r,s}}+I_{{r,s-1}}-I_{{r,s}},\;\;(r>1,\;s>1), (11)

where

J_{{r,s}}=a_{{r,s}}c_{{r,s}}c_{{1,0}}-b_{{r+1,s}}c_{{r+1,s}}\;{\textrm  {and}}\;I_{{r,s}}=\alpha _{{r,s}}c_{{r,s}}c_{{0,1}}-\beta _{{r,s+1}}c_{{r,s+1}} (12)

defined on the region \Omega =\{(r,s)\in {\mathbb  {Z}}^{2}:\;r\geq 0,\;s\geq 0\}\backslash \{0,0\}

Analysis/interpretation

The Smoluchowski equations only allow for coagulation, and from the three specific cases of aggregation kernels discussed, we learn a little about the form of the distribution of large clusters at large times. In the first two cases where a_{{r,x}}=1 and a_{{r,s}}=(r+s)/2 we had a conservation of mass and the shape of the distribution decays exponentially in \eta . In the third case where a_{{r,s}}=rs we had gelation. When gelation occurs, there is a finite time at which points all of the clusters form a single mass (or gel) and then continues to grow.

For the more general assumptions, where a_{{r,s}}=a(r^{{\mu }}s^{{\nu }}+r^{{\nu }}s^{{\mu }}), we had four types of behaviors. First, there was no gelation. This occurs when 0<\mu <1, 0<\nu <1 and \lambda <1, which was like the first two specific cases above. The second behavior, which occurs when 0<\mu <1, 0<\nu <1 and \lambda >1,was delayed gelation just as we saw in the third specific case above. The third behavior was instantaneous gelation, which occurs when -1<\mu -\nu <1 and max\{\mu ,\nu \}>1. The fourth behavior was non-existence of solutions which occurs when |\mu -\nu |>1. These four regions of behavior are illustrated in Fig. 1.

Figure 1: The Four Types of Gelation.

The Becker-Doring equations add complexity to the modelling in that now fragmentation is allowed, but only cluster-monomer interactions are allowed, which actually simplifies the modelling as shown in equation 1.

As mentioned in the modeling section, the Becker-Doring equations are divided under two different assumptions. First, the authors assume that the monomer concentration is constant. They then use \theta as a measure of the relative strength of aggregation to fragmentation. In each of the three cases for values of \theta we get different distributions of the concentrations of cluster sizes (equations 3, 4, and 5). When \theta <1, the evolution of the evolution of the distribution function c_{r}(t) to its equilibrium state can be seen in Fig. 2.

Figure 2: Evolution of Distribution Function.

Three regions can be seen. First, when r is small, c_{r}(t) is already at its equilibrium. Then there is a transition region where c_{r}(t) goes to its equilibrium, equation 2. Finally, in the region where r is large, c_{r}(t) stays close to its initial conditions. The authors highlight that if one wants to study the evolution of c_{R}(t) for some large R, the concentration of clusters size R remains low for a long time, before it goes to and remains at its equilibrium value.

When \theta =1, evolution of the distribution function is a stationary diffusive wave which can be seen in Fig. 3. When \theta >1 the equilibrium solution diverges, but the system converges to a steady-state solution as shown in Fig. 4.

Figure 3: Evolution of Distribution Function.
Figure 4: Evolution of Distribution Function.


The authors then change the assumption on the monomer concentration to one that varies (equation 6), but where the total mass of the system remains constant. As solutions are sought to equation 6, four cases arise under different combinations of aggregation and fragmentation rates. Since the total mass at equilibrium should be the same as the initial mass, solving \varrho _{{eq}}(z)=\varrho _{0} for the equilibrium monomer concentration, z, in each of these four cases provides the results shown in Fig. 5.

Figure 5: Equilibrium Data in Constant Mass Becker-Doring System.

In the first two cases, given any \varrho _{0}\geq 0 there is a unique solution z. The difference is that in case 1, z\rightarrow \infty as \varrho _{0}\rightarrow \infty , but in case 2, z\rightarrow R^{-} as \varrho \rightarrow \infty . On the other hand, analysis gets more complicated in cases 3 and 4. In case 3, z has a maximum and at that maximum the total mass is finite. When the initial mass is less than the mass at maximum z, results are similar to case 2. Conversely, when the initial mass is greater than the mass at maximum z, we have weak convergence to a metastable process. The authors offer this comparison. In the Smoluchowski model, when mass was not conserved we had gelation (a kind of super-particle). With the Becker-Doring model, the kinetics are slower and gelation can't occur at some finite time. This is where they make an important distinction. Metastable states last a long time and evolve quite slowly, but they are not equilibrium states.

In the final section of the paper, the authors show how the previous models can be generalized. In the first generalized model, binary fragmentation is considered described by equations 7 and 8. They point out that solutions to those equations have been shown to exist, but they do not conserve mass. In the next generalized model, the authors study Smoluchowski coagulation-fragmentation. They use the law of mass action, symmetric rate constants, and derive a system of equations (9 and 10) which behave much like the Becker-Doring equations. They point out that under a variety of different aggregation and fragmentation kernels, many authors have proven existence and uniqueness of solutions. Finally, the authors examine multi-component coagulation-fragmentation where multiple species form heterogeneous clusters. In the example defined by equations 11 and 12, they make several new modifications to the systems and expect to conserve quantities of both components A and B.

With all of these models, the authors remind us of several challenges. One is determining appropriate rate coefficients. Another is the understanding of multi-component systems. They conclude with the motivation that this area of study has many unanswered questions with a myriad of applications.

References/External Links

  1. Wikipedia contributors. "Colloid." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 29 Mar. 2012. Web. 31 Mar. 2012.
  2. Wikipedia contributors. "Polymer." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 29 Mar. 2012. Web. 31 Mar. 2012.
  3. Wikipedia contributors. "Marian Smoluchowski." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 12 Feb. 2012. Web. 31 Mar. 2012.
  4. Fuliński, A. "ON MARIAN SMOLUCHOWSKI’S LIFE AND CONTRIBUTION TO PHYSICS." ACTA PHYSICA POLONICA B 29.6 (1998): 1524-537. Web. 31 Mar. 2012.