Kinetic Gelation

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,

$$\displaystyle {} {K}_{i,j} = i j, $$

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.

Appendix: Derivations

9.1.1 Derivations: Power-Law Kernels

Here we derive the parameters β and q for the power-law kernel

$$\displaystyle \begin{aligned} {K}_{i,j} = (i\, j)^{\nu/2} . \end{aligned} $$

(9.70)

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

$$\displaystyle \begin{aligned} {\bar K}(\mathbf{n}) = \left(\frac{M}{N}\right)^\nu . \end{aligned} $$

([9.58])

The partition function is

$$\displaystyle \begin{aligned} \Omega_{M,N} = \binom{M-1}{N-1} \prod_{g=0}^{M-N-1} \left(\frac{M}{M-g}\right)^\nu = \binom{M-1}{N-1} \left(\frac{M}{M}\frac{M}{M-1} \cdots\frac{M}{N+1}\right)^\nu, \end{aligned}$$

which condenses to

$$\displaystyle \begin{aligned} \Omega_{M,N} = \binom{M-1}{N-1} \left(\frac{M!}{N!}M^{M-N}\right)^\nu . \end{aligned} $$

(9.71)

We obtain β and q in finite-difference form. Starting with β we have

$$\displaystyle \begin{aligned} \beta = \log \frac{\Omega_{M+1,N}}{\Omega_{M,N}} = \nu (M-N) \log \left(\frac{M+1}{M}\right) - \log \left(\frac{M-N+1}{M}\right) . \end{aligned} $$

(9.72)

Using

$$\displaystyle \begin{aligned} \log \left(\frac{M+1}{M}\right)^M\to 1, \end{aligned}$$

we obtain the limiting form of β in the thermodynamic limit:

$$\displaystyle \begin{aligned} \beta = \nu\frac{M-N}{N} - \log \frac{M-N}{N} , \end{aligned}$$

or

(9.73)

For q we find

$$\displaystyle \begin{aligned} q = \frac{\Omega(M,N+1)}{\Omega(M,N)} = \left(1+\frac{1}{N}\right) \left(1-\frac{N}{M}\right) \left(\frac{N+1}{M}\right)^{\nu-1} . \end{aligned} $$

(9.74)

In the thermodynamic limit this becomes

$$\displaystyle \begin{aligned} q = \left(1-\frac{N}{M}\right)\left(\frac{N}{M}\right)^{\nu-1} , \end{aligned}$$

or

(9.75)

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,

$$\displaystyle \begin{aligned} \frac{d q(\theta)}{d\theta} = 0 = (1-\theta )^{\nu -2} (1-\theta\nu) , \end{aligned} $$

(9.76)

whose solution is

$$\displaystyle \begin{aligned} \theta^* = 1/\nu . \end{aligned}$$

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

(9.77)

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.

$$\displaystyle \begin{aligned} & S_0 = \sum_{i=1}^\infty \frac{ i ^{ i } } {i!} e^{-i (a-\log a)} = \frac{a}{1-a} \end{aligned} $$

(9.78)

$$\displaystyle \begin{aligned} & S_1 = \sum_{i=1}^\infty \frac{ i ^{ i-1 } } {i!} e^{-i (a-\log a)} = a \end{aligned} $$

(9.79)

$$\displaystyle \begin{aligned} & S_2 = \sum_{i=1}^\infty \frac{ i ^{ i-2 } } {i!} e^{-i (a-\log a)} = a (1 - a/2) \end{aligned} $$

(9.80)

These can be also expressed in equivalent form as

$$\displaystyle \begin{aligned} & \sum_{i=1}^\infty \frac{(i a )^{ i } } {i!} e^{-i a} = \frac{a}{1-a} \end{aligned} $$

(9.81)

$$\displaystyle \begin{aligned} & \sum_{i=1}^\infty \frac{(i a )^{ i - 1} } {i!} e^{-i a} = 1 \end{aligned} $$

(9.82)

$$\displaystyle \begin{aligned} & \sum_{i=1}^\infty \frac{(i a )^{ i - 2} } {i!} e^{-i a} = \frac{1-a/2} {a} \end{aligned} $$

(9.83)

We outline the derivation of these results. We start with the power series¹³

$$\displaystyle \begin{aligned} \sum_{i=1}^\infty \frac{i^{i-1}}{i!} e^{-b i} = a, \end{aligned} $$

(9.84)

where a is the solution of

$$\displaystyle \begin{aligned} e^{-b} = a e^{-a} . \end{aligned} $$

(9.85)

Taking the log of this expression we have

$$\displaystyle \begin{aligned} b = a - \log a, \end{aligned} $$

(9.86)

which proves Eq. (9.79). We define

$$\displaystyle \begin{aligned} C_k(i) = \frac{i^{i-k}}{i!}, \end{aligned} $$

(9.87)

and express the summations in Eqs. (9.78)–(9.80) as

$$\displaystyle \begin{aligned} S_k(b) = \sum_{i=1^\infty} C_k(i) e^{-b i} . \end{aligned} $$

(9.88)

We now have

$$\displaystyle \begin{gathered} S_0(b) = -\frac{\partial S_1}{\partial b} \end{gathered} $$

(9.89)

$$\displaystyle \begin{gathered} S_1(b) = a(b), \end{gathered} $$

(9.90)

$$\displaystyle \begin{gathered} S_2(b) = -\int S_1 d b \end{gathered} $$

(9.91)

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.