Kinetic model of two-monomer polymerization

We propose a kinetic gelation model of polymer growth with two monomeric types that have distinct functionalities (reaction sites), and can polymerize using different reaction types. The heterotypic aggregation of two monomer types is modeled using a moment generating function approach by tracking the temporal evolution of a closed system of moment equations up until gelation. We investigate several scenarios of polymerization with two distinct monomers that differ in the types of reactions that can occur. We determine numerical and analytical conditions for finite time blow-up (the emergence of an oligomer of infinite size) that depend on initial conditions, reaction rates, and number of reaction sites per monomer.

Figure 1

Schematic of parameter space explored. Scenario 1 corresponds to the vertical $\varphi$ axis (black) up to 1. Scenario 2 involves exploring the $({\kappa }_{AA},\varphi )$ plane (yellow) for $0\le \varphi \le 1$. The orange plane (scenario 3) explores $({\kappa }_{AA},{\kappa }_{BB})$ parameter space for $0\le \varphi \le 1$. A special case of scenario 3 is shown as the grey arrow, when ${\kappa }_{AA}={\kappa }_{BB}=1$.

Figure 2

Scenario 1 (${\kappa }_{AA}={\kappa }_{BB}=0$): (a) Analytical gel time ${\tau }_{\text{gel}}^{\left(1\right)}$ and (b) numerical gel time ${\tau }_{\text{gel}}^{\text{num}}$ gel time as functions of $\varphi$ and for various functionality combinations: $f_A=3$ (orange), $f_A=4$ (yellow), $f_B=2$ (dashed), $f_B=3$ (dot-dashed), $f_B=4$ (dotted). Vertical lines correspond to lower and upper bounds found in Eq. (46).

Figure 3

Typical oligomers for various parameter combinations for scenario 2, with $f_A=3,f_B=2$. (a) ${\kappa }_{AA}\gg 1, \varphi \approx 1$. (b) ${\kappa }_{AA}\ll 1, \varphi \ll 1$. (c) ${\kappa }_{AA}\approx 1, \varphi$ intermediate.

Figure 4

Scenario 2 (${\kappa }_{BB}=0$): Numerical gel time as a function of $\varphi$ for ${\kappa }_{AA}=1$ and various functionality combinations: $f_A=2$ (black), $f_A=3$ (orange), $f_A=4$ (yellow), $f_B=1$ (solid), $f_B=2$ (dashed), $f_B=3$ (dot-dashed), $f_B=4$ (dotted).

Figure 5

Scenario 2 $({\kappa }_{BB}=0)$: Numerical gel time as a function of $\varphi$ and ${\kappa }_{AA}$. The functionalities are fixed ($f_A=3,f_B=2$). The color bar indicates when finite time blow-up occurred.

Figure 6

Scenario 2 $({\kappa }_{BB}=0)$: ${\tau }_{\text{gel}}^{\text{num}}$ (solid curves) as a function of $\varphi$ with varying ${\kappa }_{AA}(f_A=3,f_B=2)$ with ${\tau }_{\text{gel}}^{\text{ziff}}$, for $f=3$ functionality [5]. Yellow curves indicate ${\kappa }_{AA}<1$ and orange curves indicate ${\kappa }_{AA}\ge 1$. From bottom to top, solid curves are for ${\kappa }_{AA}=10$, 8, 4, 2, 1, 0.5, 0.1, 0.01, 0.001, and 0.

Figure 7

Scenario 2 $({\kappa }_{BB}=0)$: Gel-no gel boundaries similar to Fig. 5 for fixed $f_A$ (a) and $f_B$ (b) functionality. To the left of each curve, gelation does not occur and to the right of the curve, gelation occurs for given $(\varphi ,{\kappa }_{AA})$.

Figure 8

Analytical gel time ${\tau }_{\text{gel}}^{\left(3\right)}$ as a function of $\varphi$ with various functionality combinations: $f_A=2$ (black), $f_A=3$ (orange), $f_A=4$ (yellow), $f_B=1$ (solid), $f_B=2$ (dashed), $f_B=3$ (dot-dashed), $f_B=4$ (dotted). Flat curves ($f_A=f_B$) are equivalent to ${\tau }_{\text{gel}}^{\text{ziff}}$.

Figure 9

Scenario 3 $({\kappa }_{AA}\ne {\kappa }_{BB})$: Numerical gel time for $f_A=4,f_B=3$. Heat map of gel times as a function of ${\kappa }_{AA},{\kappa }_{BB}$ for $\varphi =0.3, {\tau }_{\text{end}}={10}^6$. Black data point corresponds to parameter values found in Fig. 10.

Figure 10

Scenario 3 $({\kappa }_{AA}\ne {\kappa }_{BB})$: Numerical gel time as a function of $\varphi$ for ${\kappa }_{AA}={10}^{-1},{\kappa }_{BB}={10}^{-4}$ for $f_A=4,f_B=3$. Vertical lines indicate gelation bounds from scenario 1. Black curves indicate theoretical Ziff times for $f=4$ and $f=3$.

Figure 11

Gel times for polymerization system with $f_A=3,f_B=2$, varying reaction rates, and $\varphi =0.2$. (a)–(c) on plot marks scenario 1, scenario 2, and theoretical Ziff times for $\varphi =0.2$, respectively. Transitions from scenarios are marked 1–6.

Figure 12

Curves of numerical gel time as a function of $\varphi$ for $f_A=3, f_B=2$ with dashed curves showing transitions as ${\kappa }_{AA}$ and ${\kappa }_{BB}$ vary along paths 1–6 in Fig. 11.