Dynamic scaling in entangled mean-field gelation polymers

We present a simple reaction kinetics model to describe the polymer synthesis used by Lusignan et al. [Phys. Rev. E 60, 5657 (1999)] to produce randomly branched polymers in the vulcanization class. Numerical solution of the rate equations gives probabilities for different connections in the final product, which we use to generate a numerical ensemble of representative molecules. All structural quantities probed in the experiments are in quantitative agreement with our results for the entire range of molecular weights considered. However, with detailed topological information available in our calculations, our estimate of the “rheologically relevant” linear segment length is smaller than that estimated from the experimental results. We use a numerical method based on a tube model of polymer melts to calculate the rheological properties of such molecules. Results are in good agreement with experiment, except that in the case of the largest molecular weight samples our estimate of the zero-shear viscosity is significantly lower than the experimental findings. Using acid concentration as an indicator for closeness to the gelation transition, we show that the high-molecular-weight polymers considered are at the limit of mean-field behavior—which possibly is the reason for this disagreement. For a truly mean-field gelation class of model polymers, we numerically calculate the rheological properties for a range of segment lengths. Our calculations show that the tube theory with dynamical dilation predicts that, very close to the gelation limit, the contribution to viscosity for this class of polymers is dominated by the contribution from constraint-release Rouse motion and the final viscosity exponent approaches a Rouse-like value.

Figure 1

(a)–(c) Molecular structure of the reactants in polycondensation reaction considered in Ref. 4 (d). Schematic description of the esterification involved in the synthesis.

Figure 2

Fraction of TMP molecules having 1, 2, or 3 of the $\mathrm{OH}$ groups reacted as a function of average molar mass. $k_2∕k_1$ is assumed to be 0.8.

Figure 3

Determining $M_{char}$: the line represents the best fit $\varphi \left(M\right)\sim M^{-3∕2}\exp (-M∕2M_{char})$ indicating the exponent $\tau =5∕2$ and $M_{char}=2\times {10}^5\mathrm{g}∕\mathrm{mol}$. $k_2∕k_1$ is fixed to be $0.8$ in this plot.

Figure 4

(Color online) $M_{char}$ as a function of $M_W$ for several choices of $k_2∕k_1$ ratios superposed with the experimental data.

Figure 5

Ideal radius of gyration for $k_2=0.8k_1$ and $M_W=15200$ (filled circles) and $M_W=226000\mathrm{g}∕\mathrm{mol}$ (open triangles). The line shows the combined behavior of all the data sets from samples with different $M_W$ together. $b$ is the Kuhn length and Kuhn mass is assumed to be $74\mathrm{g}∕\mathrm{mol}$.

Figure 6

Crossover from linear to randomly branched behavior from radius of gyration. The symbols correspond to the radius of gyration determined by making a histogram in mass from all the different $M_W$ samples considered together. The solid line and the broken line correspond to slopes $1$ (linear) and $1∕2$ (randomly branched), respectively. The arrow indicates crossover at $65800\mathrm{g}∕\mathrm{mol}$. $k_2∕k_1$ is fixed at $0.8$.

Figure 7

Probability distribution of linear segments at a fixed acid concentration. $p\left(M_{N,S}\right)$ decays exponentially at large $M_{N,S}$. The line denotes an exponential fit in the shown range.

Figure 8

Estimate of number averaged molar mass of linear segments between branch points as a function of $M_W$. The error bars are estimated from standard error in the exponential fit of $p\left(M_a\right)$ (Fig. 7).

Figure 9

Dependence of characteristic mass $M_{char}$ on the acid fraction $f_a$. The line is a linear fit for high $M_W$ samples. The intersection point with the zero $x$ axis gives the critical acid concentration $f_a^c$, where the characteristic mass diverges.

Figure 10

Zero-shear viscosity $\eta$ as a function of weight-average molar mass. Filled circles are the experimental data from Ref. 4. The triangles and the squares are the results from our calculations using $M_e=1420$ and $1080\mathrm{g}∕\mathrm{mol}$, respectively. The error bars for the $1080\text{−}\mathrm{g}∕\mathrm{mol}$ calculations represent the effect of uncertainty in $\eta$ corresponding to the experimental uncertainty of $10\%$ in determining $M_W$. The line is a power-law fit to the $M_e=1080\text{−}\mathrm{g}∕\mathrm{mol}$ results in the intermediate mass region with the exponent 3.95.

Figure 11

(Color online) Complex viscosity ${\eta }^{*}\left(\omega \right)$ for selected $M_W$. Lines are from calculations in this study, symbols are experimental data from Ref. 4.

Figure 12

Variation of the relaxation exponent $u$ with $M_W$. The line shows the extrapolation used for large $M_W$ to find the value of $u$ at the gel transition.

Figure 13

Radius of gyration for gelation ensemble with $M_{N,S}=22400\mathrm{g}∕\mathrm{mol}$ with $p_b=0.49$. The crossover mass $M_X=314400$ is roughly 14 times larger than $M_{N,S}$.

Figure 14

(Color online) Estimating the exponent $u$: the left panel shows ${\eta }^{*}\left(\omega \right)$ for $N∕N_e=20$ and $ϵ=0.4$, $0.2$, and $0.02$. Data for different $ϵ$ are fitted separately to a power law with exponent $1-u$ in the frequency range ${10}^2–{10}^4{\mathrm{s}}^{-1}$. The exponent $u$ so obtained, as a function of $ϵ$, is plotted in the right panel. A linear fit for $ϵ\le 0.1$ was used to find the limiting value of $u$ at this $N∕N_e$.

Figure 15

Variation of (apparent) relaxation exponent $u$ with $N∕N_e$. The circles are the data obtained from our calculations. The dot—dashed line is the prediction of the hierarchical relaxation model of Ref. 16 and the dashed line is the phenomenological form used in Ref. 4. The triangle corresponds to the exponent $u$ for the branched PTMG polymers, when the segment length is estimated by fitting a Flory distribution to the probability distribution of segment lengths.

Figure 16

(Color online) Zero-shear viscosity $\eta$ and recoverable compliance $J_e^0$ as a function of $ϵ$ for different $N∕N_e$. The left subplot shows $\eta$ for $N∕N_e=3$, $5$, $7$, $10$, and $20$. The right subplot shows $J_e^0$ for only $N∕N_e=15$ and $20$. The lines are the power-law fits in the indicated range with the exponents shown in Figs. 17, 18.

Figure 17

Viscosity exponent $s$ as a function of $N∕N_e$. The circles are results from our calculations and the dashed line is the phenomenological form of Lusignan et al. [4].

Figure 18

Recoverable compliance exponent t as a function of $N∕N_e$. The circles represent results from a direct power-law fit of the form $J_e^0\sim {ϵ}^{-t}$. The error bars are estimated from the variance in $J_e^0$ from three independent sets of molecular ensemble. The squares are calculated by using the hyperscaling relationship [Eq. (16)]. For $N∕N_e=3$ and $5$, the scatter in $J_e^0$ is too large for a direct estimate of $t$.

Figure 19

(Color online) $G\left(t\right)$ (symbols) for $N∕N_e=3$ for $ϵ=0.1$, $0.08$, $0.06$, $0.04$, and $0.02$. As $ϵ$ is made smaller, $G\left(t\right)$ decays slowly. The vertical dotted lines show the range in which the complex viscosity is fitted (in frequency) to find the value of the exponent $u$. The solid line represents this limiting power-law decay $G\left(t\right)\sim t^{-u}$ with the indicated slope $u=0.4253$.