Abstract
Binary aggregation usually leads to stable populations whose mean size increases indefinitely as aggregation advances, but under special conditions it is possible to observe a phase transition that produces a giant cluster of macroscopic size. In the polymer literature this is known as gelation and is observed experimentally in some polymer systems. Polymerization is perhaps a standard example of aggregation: starting with monomers, polymer molecules of linear or branched structure are formed by joining smaller units together, one pair at a time. The aggregation kernel in this case is determined by the functionality and reactivity of the chemical sites where polymers can join by forming bonds. A kernel that is known to lead to gelation is the product kernel,
The classical symptom of gelation is the breakdown of the Smoluchowski equation: within finite time the mean cluster size diverges and the Smoluchowski equation ceases to conserve mass. Gelation has the qualitative features of a phase transition and is often discussed qualitatively in the thermodynamic language of phase equilibrium.
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- 1.
This scaling excludes distributions that contain a gel cluster. A necessary though not sufficient condition for Eq. (9.4) is that the distribution must be a single sol phase.
- 2.
Details are given in the appendix, section “Derivations—Power-Law Kernels.”
- 3.
The recursion for the sum kernel is
$$\displaystyle \begin{aligned} w_i = \frac{i}{2(i-1)} \sum_{j=1}^{i-1} w_{i-j} w_j,\quad \text{(sum)} \end{aligned}$$Using a i = iw i the recursion for the product kernel can be written as
$$\displaystyle \begin{aligned} a_i = \frac{i}{i-1}\sum_{j=1}^{i-1} a_i,\quad \text{(product)} \end{aligned}$$Comparing the two recursions we establish the relationship between the cluster functions of the two kernels,
$$\displaystyle \begin{aligned} w_i^{\text{prod}} = \frac{2^{i-1}}{i} w_i^{\text{sum}}, \end{aligned}$$which allows us to obtain w i of the product kernel from that of the sum kernel.
- 4.
See Eq. (8.22) on p. 252.
- 5.
See Smith and Matsoukas (1998).
- 6.
On the other hand, the average mass of the gel cluster is somewhat lower than the position of the maximum, as indicated by the fact that in the vicinity of the gel point ϕ gel from simulation lies below the theoretical value in Fig. 9.4.
- 7.
- 8.
- 9.
Stockmayer uses w i for this number but here we use a i to avoid confusion with the cluster function w i = a i∕i!
- 10.
To economize in notation we adopt Stockmayer’s Ω for the microcanonical weight, even though we have reserved it for the partition function. We do this knowing that in the thermodynamic limit their logarithms are equal.
- 11.
In the absence of cycles, it is easy to show that if we move a monomer to a different position on the chain the number of free sites (that is, sites that are bonded to other monomers) does not change. This means that the number of free sites is the same for all polymers with the same number of monomers, regardless of structure. We may then establish this number by looking at a linear chain: the i − 2 monomers in the interior of the chain have f − 2 free sites and the two monomers at the ends of the chain have f − 1 each. The total number is (i − 2)(f − 2) + 2(f − 1).
- 12.
Mayer and Mayer (1940).
- 13.
Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL (2018).
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Appendix: Derivations
Appendix: Derivations
9.1.1 Derivations: Power-Law Kernels
Here we derive the parameters β and q for the power-law kernel
This includes the product kernel as a special case for ν = 1. For distributions without a gel phase, the mean kernel in distribution n is
The partition function is
which condenses to
We obtain β and q in finite-difference form. Starting with β we have
Using
we obtain the limiting form of β in the thermodynamic limit:
or
For q we find
In the thermodynamic limit this becomes
or
With ν = 0, 1, or 2, Eqs. (9.71), (9.73), and (9.75) revert to those for the constant kernel, sum kernel, or product kernel, respectively.
9.1.2 I Locus of Gel Points
The system is unstable if q(θ) contains a branch where dq∕dθ < 0. The onset of instability is defined by the condition,
whose solution is
Since θ must be between 0 and 1, instability may occur only if ν > 1. The condition that defines the locus of gel points is
The case ν = 1 represents a singular case: the system forms a gel phase but the gel point is at θ = 1, which corresponds to \({\bar x}=\infty \) and is not observed under finite \({\bar x}\).
9.1.3 I Some Useful Power Series
Here are some useful results involving infinite series that are appear in the solution of the sum and the product kernel.
These can be also expressed in equivalent form as
We outline the derivation of these results. We start with the power seriesFootnote 13
where a is the solution of
Taking the log of this expression we have
which proves Eq. (9.79). We define
and express the summations in Eqs. (9.78)–(9.80) as
We now have
with the relationship a = a(b) from Eq. (9.86). By differentiation and integration of S 1(b) with respect to b we obtain Eqs. (9.78) and (9.80), respectively. Equations (9.81)–(9.83) follow by straightforward manipulation.
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Matsoukas, T. (2018). Kinetic Gelation. In: Generalized Statistical Thermodynamics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-04149-6_9
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