Monte Carlo simulations reveal the straightening of an end-grafted flexible chain with a rigid side chain

We have studied the conformational properties of a flexible end-grafted chain (length $N$) with a rigid side chain (length $S$) by means of Monte Carlo simulations. Depending on the lengths $N$ and $S$ and the branching site $b$, we observe a considerable straightening of the flexible backbone as quantified via the gyration tensor. For $b=N$, i.e., when attaching the side chain to the free end of the flexible backbone, the effect was strongest.

Figure 1

Two-dimensional sketch of the self-avoiding polymer on a lattice as described in the main text. A rigid side chain (gray) is attached to the fully flexible end-grafted backbone at monomer $b$ (“branching site”). The substrate is impenetrable, i.e., the chain conformations are restricted to the half-space $z>0$. Note that a chain of length $N$ consists of $N+1$ monomers.

Figure 2

(a) The radius of gyration and the end-to-end distance increase like $R_0^2\sim N^{2\nu }$ $(\nu \approx 0.59)$ for a flexible end-grafted chain (without side chain). (b) The probability density function $p\left(\varphi \right)$ of the orientation angle of the polymer’s gyration ellipsoid shows a plateaulike behavior for $\varphi \to 90°$ in the limit of large $N$. Inset: The average angle $⟨\varphi ⟩$, i.e., the first moment of $p\left(\varphi \right)$, depends only weakly on $N$ with an asymptotic value $⟨\varphi ⟩=55°$.

Figure 3

(a) The squared radius of gyration $⟨R_g^2⟩$ increases sigmoidally with the length of the side chain $S∕N$ and levels off at increasing values with increasing $b$ (here $N=100$). (b) All curves collapse to a single master curve when applying the scaling Eq. (2). Open black symbols: $b∕N=0.25$, $N=20$, 60, 80, 100, and 120; filled gray symbols: $b∕N=0.5$, $N=20$, 50, 80, 100, and 120; and crosses: $b∕N=0.75$, $N=20$ and 100.

Figure 4

The ratio $\theta =⟨R_{\text{end}}^2⟩∕⟨R_g^2⟩$ of the squared end-to-end distance and radius of gyration of the backbone increases with increasing length of the rigid side chain $S∕N$ (here $N=100$).

Figure 5

(a) The average angle $⟨\varphi ⟩$ between the longest axis of gyration and the substrate normal decreases for increasing lengths of the side chain $S∕N$, indicating a straightening of the backbone. Moving the branching site $b$ towards the free end of the backbone strongly enhances this straightening. (b) The entire distribution $p\left(\varphi \right)$ shifts and becomes more narrow as $S∕N$ is decreased (here $N=100$).

Figure 6

Normalizing the average angle, i.e., setting $\phi =\varphi ∕⟨\varphi ⟩$, for $b\approx N$ leads to a collapse of all distributions $p\left(\phi \right)$ for various $N,S$ with $S∕N=0.3$ (symbols). The curve is best described by Eq. (3) (full line).