# The Smoluchowski Ensemble—Statistical Mechanics of Aggregation

## Abstract

**:**

## 1. Introduction

## 2. The Smoluchowski Ensemble

#### 2.1. Kinetics

#### 2.2. Probabilities

#### 2.3. Smoluchowski Equation

## 3. Thermodynamic Formalism

#### 3.1. Partition Function and Selection Functional

#### 3.2. Shannon Entropy

#### 3.3. The Selection Functional

#### 3.4. Propagation Equations

## 4. Scaling Limit

#### 4.1. Most Probable Distribution

#### 4.2. Thermodynamics

## 5. Gibbs Distributions

#### 5.1. Constant Kernel

#### 5.2. Sum Kernel

#### 5.3. Quasi-Gibbs Kernels—The Product Kernel

## 6. Phase Behavior

#### 6.1. Stability

#### 6.2. Phase Splitting—The Sol-Gel Transition

**Pre-Gel Region**$0\le \theta <{\theta}^{*}$

**Post-Gel Region**${\theta}^{*}\le \theta <1$

- Obtain ${\theta}_{\mathrm{sol}}$ by solving$$q\left({\theta}_{\mathrm{sol}}\right)=q\left(\theta \right),\phantom{\rule{1.em}{0ex}}{\theta}_{\mathrm{sol}}\le {\theta}^{*}.$$
- Obtain ${\varphi}_{\mathrm{sol}}$ and ${\overline{x}}_{\mathrm{sol}}$ from$${\varphi}_{\mathrm{sol}}=\frac{1-\theta}{1-{\theta}_{\mathrm{sol}}},\phantom{\rule{1.em}{0ex}}{\overline{x}}_{\mathrm{sol}}=\frac{1}{1-{\theta}_{\mathrm{sol}}}.$$
- Obtain the gel fraction from mass balance:$${\varphi}_{\mathrm{gel}}=1-{\varphi}_{\mathrm{sol}}=\frac{\theta -{\theta}_{\mathrm{sol}}}{1-{\theta}_{\mathrm{sol}}}.$$The mean size of the gel cluster is ${\overline{k}}_{\mathrm{gel}}={\varphi}_{\mathrm{gel}}M$, where M is the total mass in the system. In the scaling limit the gel fraction is 1 and the size of the gel cluster is ∞.

#### 6.3. Monte Carlo Simulations

## 7. Continuous Limit

## 8. Summary

## Supplementary Materials

## Funding

## Conflicts of Interest

## References

- Smoluchowski, M. Versuch einer mathematischen Theorie der Koagulationkinetic kolloider Loesungen. Z. Phys. Chem.
**1917**, 92, 129–168. [Google Scholar] - Leyvraz, F. Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Rep.
**2003**, 383, 95–212. [Google Scholar] [CrossRef] [Green Version] - Ziff, R.M.; Hendriks, E.M.; Ernst, M.H. Critical Properties for Gelation: A Kinetic Approach. Phys. Rev. Lett.
**1982**, 49, 593–595. [Google Scholar] [CrossRef] - Hendriks, E.M.; Ernst, M.H.; Ziff, R.M. Coagulation Equations with Gelation. J. Stat. Phys.
**1983**, 31, 519–563. [Google Scholar] [CrossRef] [Green Version] - Ziff, R.M.; Stell, G. Kinetics of polymer gelation. J. Chem. Phys.
**1980**, 73, 3492–3499. [Google Scholar] [CrossRef] - Marcus, A. Stochastic Coalescence. Technometrics
**1968**, 10, 133–143. [Google Scholar] [CrossRef] - Lushnikov, A.A. Coagulation in finite systems. J. Colloid Interface Sci.
**1978**, 65, 276–285. [Google Scholar] [CrossRef] - Lushnikov, A.A. Exact kinetics of a coagulating system with the kernel K = 1. J. Phys. A Math. Theor.
**2011**, 44, 335001. [Google Scholar] [CrossRef] - Lushnikov, A.A. Exact kinetics of the sol-gel transition. Phys. Rev. E
**2005**, 71, 046129. [Google Scholar] [CrossRef] - Lushnikov, A.A. Exact particle mass spectrum in a gelling system. J. Phys. A Math. Gen.
**2005**, 38, L35. [Google Scholar] [CrossRef] - Lushnikov, A.A. From Sol to Gel Exactly. Phys. Rev. Lett.
**2004**, 93, 198302. [Google Scholar] [CrossRef] [PubMed] - Aldous, D.J. Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli
**1999**, 5, 3–48. [Google Scholar] [CrossRef] - Stockmayer, W.H. Theory of Molecular Size Distribution and Gel Formation in Branched-Chain Polymers. J. Chem. Phys.
**1943**, 11, 45–55. [Google Scholar] [CrossRef] - Spouge, J.L. Analytic solutions to Smoluchowski’s coagulation equation: A combinatorial interpretation. J. Phys. Math. Gen.
**1985**, 18, 3063. [Google Scholar] [CrossRef] - Spouge, J.L. Equilibrium polymer size distributions. Macromolecules
**1983**, 16, 121–127. [Google Scholar] [CrossRef] - Hendriks, E.M.; Spouge, J.L.; Eibl, M.; Schreckenberg, M. Exact solutions for random coagulation processes. Z. Phys. B Condens. Matter
**1985**, 58, 219–227. [Google Scholar] [CrossRef] - Spouge, J.L. The size distribution for the A
_{g}RB_{f-g}Model of polymerization. J. Stat. Phys.**1983**, 31, 363–378. [Google Scholar] [CrossRef] - Matsoukas, T. Statistical Thermodynamics of Irreversible Aggregation: The Sol-Gel Transition. Sci. Rep.
**2015**, 5, 8855. [Google Scholar] [CrossRef] - Matsoukas, T. Abrupt percolation in small equilibrated networks. Phys. Rev. E
**2015**, 91, 052105. [Google Scholar] [CrossRef] - Matsoukas, T. Statistical thermodynamics of clustered populations. Phys. Rev. E
**2014**, 90, 022113. [Google Scholar] [CrossRef] [Green Version] - Flory, P.J. Molecular Size Distribution in Three Dimensional Polymers. I. Gelation. J. Am. Chem. Soc.
**1941**, 63, 3083–3090. [Google Scholar] [CrossRef] - Matsoukas, T. Generalized Statistical Thermodynamics: Thermodynamics of Probability Distributions and Stochastic Processes; Springer International Publishing: Berlin/Heidelberg, Germany, 2019. [Google Scholar] [CrossRef]
- Berestycki, N.; Pitman, J. Gibbs Distributions for Random Partitions Generated by a Fragmentation Process. J. Stat. Phys.
**2007**, 127, 381–418. [Google Scholar] [CrossRef] [Green Version] - Kelly, F.P. Reversibility and Stochastic Networks; Cambridge University Press: Cambridge, UK, 2011; (Reprint of the 1979 edition by Wiley). [Google Scholar]
- Smith, M.; Matsoukas, T. Constant-number Monte Carlo simulation of population balances. Chem. Eng. Sci.
**1998**, 53, 1777–1786. [Google Scholar] [CrossRef] - Hendriks, E.M. Cluster size distributions in equilibrium. Z. Phys. B Condens. Matter
**1984**, 57, 307–314. [Google Scholar] [CrossRef] - Lushnikov, A.A. Evolution of coagulating systems. J. Colloid Interface Sci.
**1973**, 45, 549–556. [Google Scholar] [CrossRef] - Matsoukas, T. Thermodynamics Beyond Molecules: Statistical Thermodynamics of Probability Distributions. Entropy
**2019**, 21, 890. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The aggregation graph for $M=7$. Each layer contains all feasible distributions in that generation.

**Figure 4.**Phase diagram of power-law kernels: In the shaded region the system is stable and is represented by its MPD. The unshaded region is unstable and the system is split into two phases, a sol phase and a gel phase, each represented by its own MPD. (

**a**,

**b**) provide equivalent criteria of stability.

**Figure 5.**(

**a**) Gel fraction and (

**b**) mean sol cluster size as a function of the progress variable $\theta $. Past the gel point the mean size in the sol retraces its pre-gel history back to its initial size ${\overline{x}}_{\mathrm{sol}}=1$. The dashed lines are Monte Carlo (MC) simulations with $M=200$ particles.

**Figure 6.**Monte Carlo snapshots of the mean distribution of the product kernel with $M=200$ particles (open circles are MC results, solid lines are calculated from theory). The gel phase emerges at ${\theta}^{*}=0.5$ and moves towards ever larger sizes (arrows mark the theoretical predictions). The distribution of the sol grows in the pre-gel region range $0<\theta <0.5$ but contracts once past the post-gel point ($\theta >0.5$).

Brownian coagulation | ${K}_{i,j}=\frac{1}{4}\left(2+{\left(\frac{i}{j}\right)}^{1/3}+{\left(\frac{j}{i}\right)}^{1/3}\right)$ |

Constant kernel | ${K}_{i,j}=1$ |

Flory/Stockmayer kernel | ${K}_{i,j}=\frac{(fi-2i+2)(fj-2j+2)}{{f}^{2}}$ |

Product kernel | ${K}_{i,j}=ij$ |

Sum kernel | ${K}_{i,j}=\frac{i+j}{2}$ |

Most Probable Distribution | $\frac{{n}_{k}^{*}}{N}={w}_{k}^{*}\frac{{e}^{-\beta k}}{q}$ | Equation (39) |

Partition Function | ${\mathsf{\Omega}}_{M,N}=\beta M+(logq)N$ | Equation (40) |

$\beta ={\left(\frac{\partial log\mathsf{\Omega}}{\partial M}\right)}_{N}$ | Equation (42) | |

$logq={\left(\frac{\partial log\mathsf{\Omega}}{\partial M}\right)}_{M}$ | Equation (43) | |

Gibbs-Duhem Equation | $Md\beta +Ndlogq=0$ | Equation (44) |

Variational Condition(Second Law) | $\frac{log{\mathsf{\Omega}}_{M,N}}{N}$$\ge -{\sum}_{i}{p}_{i}log\frac{{p}_{i}}{{w}_{i}^{*}}$ | Equation (48) |

Constant Kernel | Sum Kernel | Product Kernel ${}^{\u2020}$ | |
---|---|---|---|

${K}_{i,j}$ | 1 | $(i+j)/2$ | $ij$ |

$\mathsf{\Omega}$ | $\left(\genfrac{}{}{0pt}{}{M-1}{N-1}\right)$ | $N!\frac{{M}^{M-N}}{M!}\left(\genfrac{}{}{0pt}{}{M-1}{N-1}\right)$ | ${\left(N!\frac{{M}^{M-N}}{M!}\right)}^{2}\left(\genfrac{}{}{0pt}{}{M-1}{N-1}\right)$ |

$\beta $ | $-log\theta $ | $\theta -log\theta $ | $2\theta -log\theta $ |

q | $\frac{\theta}{1-\theta}$ | $\theta $ | $\theta (1-\theta )$ |

${w}_{k}$ | 1 | $\frac{{k}^{k-1}}{k!}$ | $\frac{{2}^{k-1}{k}^{k-2}}{k!}$ |

MPD | $(1-\theta ){\theta}^{k-1}$ | $\frac{{k}^{k-1}}{k!}{\theta}^{k-1}{e}^{k\theta}$ | $\frac{{\left(2\theta k\right)}^{k-2}}{k!}\frac{2\theta}{1-\theta}{e}^{-2\theta k}$ |

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Matsoukas, T.
The Smoluchowski Ensemble—Statistical Mechanics of Aggregation. *Entropy* **2020**, *22*, 1181.
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Matsoukas T.
The Smoluchowski Ensemble—Statistical Mechanics of Aggregation. *Entropy*. 2020; 22(10):1181.
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2020. "The Smoluchowski Ensemble—Statistical Mechanics of Aggregation" *Entropy* 22, no. 10: 1181.
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