Linear Dimensions of Adsorbed Semiflexible Polymers: What Can Be Learned about Their Persistence Length?

Conformations of partially or fully adsorbed semiflexible polymer chains are studied varying both contour length $L$, chain stiffness, $\kappa$, and the strength of the adsorption potential over a wide range. Molecular dynamics simulations show that partially adsorbed chains (with “tails,” surface attached “trains,” and “loops”) are not described by the Kratky-Porod wormlike chain model. The crossover of the persistence length from its three-dimensional value (${\ell }_p$) to the enhanced value in two dimensions ($2{\ell }_p$) is analyzed, and excluded volume effects are identified for $L\gg {\ell }_p$. Consequences for the interpretation of experiments are suggested. We verify the prediction that the adsorption threshold scales as ${\ell }_p^{-1/3}$.

Figure 1

(a) Snapshot of an adsorbed chain with $N=500$ for $\kappa =16$, ${\epsilon }_{\text{wall}}=0.65$. Loops and a tail are shown in green, trains are in dark blue. (b) Decay length ${\ell }_p^{\mathrm{eff}}/{\ell }_b$ vs stiffness $\kappa$ for $N=250$ and several choices of ${\epsilon }_{\text{wall}}$. Here, $n=1$ means an angle between nearest bonds, $n=2$ stands for next-nearest bonds. Data for $n=1$, 2 indicate that ${\ell }_p^{\mathrm{eff}}$ increases rather gradually with $\kappa$ for adsorbed chains. The inset illustrates the geometry of the $x$, $z$ coordinates of two subsequent bonds where the $X$ axis is chosen such that the bond from ${\overrightarrow{r}}_{j-1}$ to ${\overrightarrow{r}}_j$ lies in the $X$, $Z$ plane. The angles ${\alpha }_j=(\pi /2)-{\vartheta }_j$ are the complements to the polar angles ${\vartheta }_j$ of the bonds with the $Z$ axis.

Figure 2

(a) Semilog plot of $C(n)=⟨\cos \theta (n)⟩$, vs $n$ in semilog coordinates for $\kappa =8$ (main panel) and $\kappa =16$ (inset). Several choices of ${\epsilon }_{\text{wall}}$ are shown, as indicated. All data are for $N=250$. (b) The same as in (a) but for strongly adsorbed (${\epsilon }_{\text{wall}}=0.80$, 1.00) chains with stiffness $\kappa =5$, 8, 16, 25 without EV interactions. The inset indicates the gradual crossover in the decay of $C(n)$ with $n$ for $\kappa =8$ and $n=1$, 2 from ${\ell }_p^{\mathrm{eff}}\approx 8$ to ${\ell }_p^{\mathrm{eff}}=12.3$ for large $n$.

Figure 3

Mean-square lateral gyration radius for $N=250$ vs ${\epsilon }_{\text{wall}}$ for seven choices of $\kappa$ with (full symbols) and without (open symbols) excluded volume (EV) interactions. EV is more important for small $\kappa$ ($\kappa =5$, 8). Horizontal straight lines show KP predictions for $d=3$ (for small ${\epsilon }_{\text{wall}}$) and $d=2$ (for larger ${\epsilon }_{\text{wall}}$). The shaded transition region from nonadsorbed to adsorbed chains narrows down with growing $\kappa$.

Figure 4

(a) Fraction $f$ of adsorbed monomers vs ${\epsilon }_{\text{wall}}$ for $\kappa =16$ (left panel) and $\kappa =25$ (right panel), for the four chain lengths $N=50$, 100, 250, and 500, respectively. Tentative linear extrapolations indicate the estimated location of the adsorption transition, ${\epsilon }_{\text{wall}}^{\mathrm{cr}}$, (nonzero $f$ for ${\epsilon }_{\text{wall}}<{\epsilon }_{\text{wall}}^{\mathrm{cr}}$ is a finite-size effect). Also, the estimation of ${\epsilon }_{\text{wall}}^{\text{sat}}$, where $f$ crosses over to saturation value $f=1$, is indicated. (b) Adsorbed fraction $f$ plotted vs ${\epsilon }_{\text{wall}}$ for $N=250$ and four choices of $\kappa$ (left), and orientational order parameter of the bonds $\eta =\frac{3}{2}⟨{\cos }^2\vartheta ⟩-\frac{1}{2}$ vs ${\epsilon }_{\text{wall}}$ (right). Data with no EV interaction (open symbols) are also included. (c) Variation of the critical adsorption potential ${\epsilon }_{\text{wall}}^{\mathrm{cr}}$ with chain stiffness $\kappa$ for $N=250$.