Gelation threshold of cross-linked polymer brushes

The cross-linking of polymer brushes is studied using the bond-fluctuation model. By mapping the cross-linking process into a two-dimensional (2D) percolation problem within the lattice of grafting points, we investigate the gelation transition in detail. We show that the particular properties of cross-linked polymer brushes can be reduced to the distribution of bonds which are formed between the grafted chains, and we propose scaling arguments to relate the gelation threshold to the chain length and the grafting density. The gelation threshold is lower than the percolation threshold for 2D bond percolation because of the longer range and broad distribution of bonds formed by the cross-linking process. We term this type of percolation problem star percolation. We observe a broad crossover from mean-field to critical percolation behavior by analyzing the cluster size distribution near the gelation threshold.

Figure 1

Visualization of the percolation problem for cross-linked polymer brushes. The dashed green lines are examples of how cross-links can connect two grafting points (a). A brush is called percolating if a continuous path of bonds connects two opposite boundaries of the sample (solid red lines) (b).

Figure 2

Frequency of occurrence of different types of added bonds at the onset of percolation for $N=64$. The fraction of primary cross-links is considerably smaller than one bond per chain (indicated by the horizontal solid orange line at $q=1$), the value expected from classical 2D bond percolation on a square lattice. The theoretical lines are derived and discussed in Sec. 3b.

Figure 3

Symmetrized bond distribution for $N=64$ and $\sigma =0.25$. The height of the column indicates the frequency with which primary cross-links connecting grafting points with the distance ($\Delta x$, $\Delta y$) occur.

Figure 4

Bond length distribution for $N=16$ (circles), 32 (squares), and 64 (diamonds) with $\sigma =0.25$. All distributions show an exponential decay.

Figure 5

Bond length distribution for $\sigma =0.04$ (circles), 0.11 (squares), and 0.25 (diamonds) with $N=64$. All distributions show an exponential decay.

Figure 6

Rescaled bond length distribution of the data in Fig. 4 and 5. The $y$ axis has been rescaled such that all data sets start at $(1,1)$. Almost all data lie on one master curve; the little overlap between chains accounts for the deviations at small grafting densities.

Figure 7

Percolation threshold vs fraction of longer bonds of length $b_2=2$ (squares) and 3 (crosses). The percolation threshold shows an asymmetric dependence on the fraction of longer bonds. The dashed curves represent data obtained from finite size scaling on different data sets.

Figure 8

Percolation threshold $q_c$ for chain length $N=64$ and grafting density $\sigma =0.25$. Finite size scaling yields $q_c=0.662$, which is considerably lower than $q_c=1$ (classical 2D bond percolation). The percolation probability for different lattice sizes is shown as an inset.

Figure 9

Percolation threshold as a function of $N{\sigma }^{1/2\nu }$. The data are shifted by $-0.5$ to reflect the lowest possible percolation threshold $q_c=0.5$, which is given by mean-field gelation. The slope of the fit is $\approx -0.42$.

Figure 10

Cluster size distribution for $N=16$ and $\sigma =0.25$. The transition from mean-field behavior (slope $-2.48$ in good agreement with the theoretical value $-2.5$) to critical behavior (slope $-2.02$ in good agreement with the theoretical value $-2.05$) is clearly visible.