Dynamic scaling study of vapor deposition polymerization: A Monte Carlo approach

The morphological scaling properties of linear polymer films grown by vapor deposition polymerization are studied by $1+1\text{D}$ Monte Carlo simulations. The model implements the basic processes of random angle ballistic deposition $\left(F\right)$, free-monomer diffusion $\left(D\right)$ and monomer adsorption along with the dynamical processes of polymer chain initiation, extension, and merger. The ratio $G=D/F$ is found to have a strong influence on the polymer film morphology. Spatial and temporal behavior of kinetic roughening has been extensively studied using finite-length scaling and height-height correlations $H\left(r,t\right)$. The scaling analysis has been performed within the no-overhang approximation and the scaling behaviors at local and global length scales were found to be very different. The global and local scaling exponents for morphological evolution have been evaluated for varying free-monomer diffusion by growing the films at $G=10$, ${10}^2$, ${10}^3$, and ${10}^4$ and fixing the deposition flux $F$. With an increase in $G$ from 10 to ${10}^4$, the average growth exponent $\beta \approx 0.50$ was found to be invariant, whereas the global roughness exponent ${\alpha }_g$ decreased from 0.87 (1) to 0.73 (1) along with a corresponding decrease in the global dynamic exponent $z_g$ from 1.71(1) to 1.38(2). The global scaling exponents were observed to follow the dynamic scaling hypothesis, $z_g={\alpha }_g/\beta$. With a similar increase in $G$ however, the average local roughness exponent ${\alpha }_l$ remained close to 0.46 and the anomalous growth exponent ${\beta }_{\ast }$ decreased from 0.23(4) to 0.18(8). The interfaces display anomalous scaling and multiscaling in the relevant height-height correlations. The variation in $H\left(r,t\right)$ with deposition time $t$ indicates nonstationary growth. A comparison has been made between the simulational findings and the experiments wherever applicable.

Figure 1

(Color online) Schematic of the $1+1\text{D}$ growth model. 1(a) Free-monomer deposition at random angles; 2(a) adsorbed free monomer diffuses along the substrate; 2(b) adsorbed free monomer diffuses on a polymer chain; 3(a) and 3(b) polymer chain initiation resulting from random angle deposition; 3(c) and 3(d) polymer chain initiation resulting from free-monomer diffusion on substrate and polymer chain respectively; 4(a,b) free monomer deposits onto the active end of a polymer chain resulting in chain propagation; and 4(c) chain propagation due to free-monomer diffusion.

Figure 2

(Color online) Snapshots of the polymer films grown on a $L=512$ substrate. (a)–(c) were obtained for $G=10$ at deposition times $t=15$, 30, and 60 units respectively; whereas (d)–(f) were obtained for $G={10}^6$ and deposition times $t=15$, 30, and 60 units, respectively. One deposition time unit corresponds to a deposition of $L$ free monomers. For a clear view, the growth front (shown in red) is displaced vertically by 100 pixels and the longest polymer chain is highlighted in black.

Figure 3

(Color online) Generalized correlation function $H_q\left(r,t\right)$ calculated for $L=2000$ and $t=1000$. The data are averaged over 200 independent runs and the error bars are smaller than the symbol size.

Figure 4

(Color online) Variation in $h_{avg}$ as a function of deposition time $t$ for different $G$ and $L=512$. The data are averaged over 500 independent runs. The statistical errors are smaller than the symbol size.

Figure 5

(Color online) Variation in $r\left(G\right)$ as a function of $G$. The data were obtained for $L=512$ and averaged over 500 independent runs. The statistical errors are smaller than the symbol size.

Figure 6

(Color online) Lateral film density profiles $\rho \left(y\right)$ of polymer films grown on a $L=512$ substrate with $G=10$ at deposition times of $t=10$, 20, 40, and 60 respectively. The statistical errors are smaller than the symbol size.

Figure 7

(Color online) Time variation of the interface width $w\left(L,t\right)$ for (a) $G=10$ and (b) $G={10}^3$ with $L=100$, 200, and 300. The data are averaged over $1.8\times {10}^3$ independent runs.

Figure 8

(Color online) Variation in $w_{sat}$ with $L$ for $G$ varying from 10 to ${10}^4$. The data are averaged over $1.8\times {10}^3$ independent runs and the statistical errors are smaller than the symbol size. The error in the exponents were obtained from the curve fit.

Figure 9

(Color online) Rescaled plots of $w\left(L,t\right)/L^{{\alpha }_g}$ versus $t/L^{z_g}$. The “collapsed” curves shown in the plots are the scaling functions for (a) $G=10$, (b) $G={10}^3$ and indicate the presence on dynamic scaling. The data are averaged over $1.8\times {10}^3$ independent runs.

Figure 10

(Color online) Autocorrelation function $C\left(r,t\right)$ calculated for $L=2000$ and $t=1000$ (averaged over 200 independent runs).

Figure 11

(Color online) Estimates for column separation $\lambda$ and lateral correlation length $\xi$ obtained for $G=10\left(circles\right)$ and $G={10}^3\left(triangles\right)$. Open symbols correspond to $\xi$ and filled symbols correspond to $\lambda$. The data are averaged over 200 independent simulations with $L=2000$ and $t=1000$. The error in the exponents were obtained from the curve fit.

Figure 12

(Color online) Height-height correlation function $H\left(r,t\right)$ for (a) $G=10$, (b) $G={10}^3$ using $L=2000$ at $t=500$, 750, and 1000. The data are averaged over 200 independent runs and error in the exponents were obtained from the curve fit.

Figure 13

(Color online) Plots of rescaled $H\left(r,t\right)$ showing the data collapse for (a): $G=10$, and (b): $G={10}^3$ using $L=2000$ at $t=500$, 750, and 1000. The data are averaged over 200 independent runs and error in the exponents were obtained from the curve fit. The straight lines are plotted as a guide to the eye. The scaling agrees with Eq. (21).

Figure 14

(Color online) Variation in exponents ${\alpha }_g$, $\beta$, $z_g$, ${\alpha }_l$, and ${\beta }_{\ast }$ with $G$. The global exponents were calculated after $t=5000$ and the data were averaged over $1.8\times {10}^3$ independent runs. The local exponents were calculated using $L=2000$ and $t=1000$. The data were averaged over 200 runs and the error in the exponents were obtained from the curve fit.