Entropic force of cone-tethered polymers interacting with a planar surface

Computer simulations are used to characterize the entropic force of one or more polymers tethered to the tip of a hard conical object that interact with a nearby hard flat surface. Pruned-enriched Rosenbluth method Monte Carlo simulations are used to calculate the variation of the conformational free energy $F$ of a hard-sphere polymer with respect to the cone-tip-to-surface distance $h$ from which the variation of the entropic force $f\equiv |dF/dh|$ with $h$ is determined. We consider the following cases: a single freely jointed tethered chain, a single semiflexible tethered chain, and several freely jointed chains of equal length each tethered to the cone tip. The simulation results are used to test the validity of a prediction by Maghrebi et al. [Maghrebi et al., Europhys. Lett. 96, 66002 (2011); Phys. Rev. E 86, 061801 (2012)] that $f\propto ({\gamma }_{\infty }-{\gamma }_0)h^{-1}$, where ${\gamma }_0$ and ${\gamma }_{\infty }$ are universal scaling exponents for the partition function of the tethered polymer for $h=0$ and $h=\infty$, respectively. The measured functions $f\left(h\right)$ are generally consistent with the predictions, with small quantitative discrepancies arising from the approximations employed in the theory. In the case of multiple tethered polymers, the entropic force per polymer is roughly constant, which is qualitatively inconsistent with the predictions.

Figure 1

Illustration of the system examined in this study. A polymer is tethered to the tip of a hard conical object in the vicinity of a hard planar surface. The cone and the surface both extend to infinity. The cone half angle $\alpha$ and the cone-tip-to-surface distance $h$ are both illustrated. We examine cases of single flexible and semiflexible chains as well as multiple flexible chains tethered to the cone tip.

Figure 2

(a) Free energy difference $\mathrm{\Delta }F\equiv F(h=0)-F(h=\infty )$ vs polymer length $N$ for a single freely jointed chain tethered to a cone of half angle $\alpha$. Results for various values of $\alpha$ are shown. The dashed lines show fits of the data in the region $N\in [150,5000]$ to the function $\mathrm{\Delta }F=a_0+a_{1}lnN$. The fitted curves extend to lower values of $N$ to highlight the discrepancy in this region. (b) Variation of $\mathrm{\Delta }\gamma$ with $\alpha$, where $\mathrm{\Delta }\gamma \equiv {\gamma }_{\infty }-{\gamma }_0$ was obtained from the fits of the data in (a). The inset shows a close-up of the data for low $\alpha$ plotted using a linear scale for $\mathrm{\Delta }\gamma$. The solid line is a guide for the eye. Overlaid on these results are data for semiflexible chains with $\kappa =4$ for calculations described in Sec. 5b. Note that the uncertainties in the data points are smaller than the size of the symbols.

Figure 3

Free energy $F$ vs tip-to-surface distance $h$ for a cone angle of $\alpha ={50}^{\circ }$. Results for various polymer lengths are shown. The inset shows the variation of the entropic force $f\equiv |dF/dh|$ with $h$. The green dashed line shows the theoretical prediction using the value of $\mathrm{\Delta }\gamma$ for $\alpha ={50}^{\circ }$ in Fig. 2. The black dotted line shows the fit of the $N=5000$ curve to the function $f=c{N}^{\mu }$ in the region $h\in [4,20]$.

Figure 4

Free energy $F$ vs tip-to-surface distance $h$ for a polymer of length $N=5000$. Results for different cone angles are shown. The inset shows the variation of the entropic force $f\equiv |dF/dh|$ with $h$. The dashed lines show the functions proportional to $h^{-0.88}$ and $h^{-1}$.

Figure 5

Entropic force $f$ vs $\mathrm{\Delta }\gamma$ for $N=5000$ and $h=10$ obtained from the data of the inset of Fig. 4. The red line is a linear fit with a slope of 0.141. The blue line is the prediction using Eq. (6) using $h=10$ and $\nu =0.588$. The uncertainties in the data points are smaller than the symbol size.

Figure 6

(a) Free energy vs cone-tip-to-surface distance $h$ for a polymer of length $N=5000$ tethered to a cone. Values for various values of $\alpha$ are shown. The dashed line is the predicted power-law scaling for slit confinement in the de Gennes regime. (b) Entropic force calculated using the data of (a). The dashed lines show the prediction for slit confinement in the de Gennes regime ($f\sim h^{-2.7}$) and the observed scaling for $\alpha \lesssim {60}^{\circ }$ noted in Fig. 4 ($f\sim h^{-0.88}$).

Figure 7

(a) Free energy difference $\mathrm{\Delta }F\equiv F(h=0)-F(h=\infty )$ vs polymer length $N$ for a single semiflexible chain with bending rigidity $\kappa =4$ tethered to a cone of half angle $\alpha$. Results for various values of $\alpha$ are shown. The dashed lines show fits of the data in the region $N\in [300,2000]$ to the function $\mathrm{\Delta }F=a_0+a_{1}lnN$. (b) Same as in (a) but for fixed cone angle of $\alpha ={40}^{\circ }$ and various values of $\kappa$.

Figure 8

Free energy $F$ of a semiflexible tethered chain vs tip-to-surface distance $h$ for a cone angle of $\alpha ={40}^{\circ }$. Results for various polymer lengths are shown. The inset shows the variation of the entropic force $f\equiv |dF/dh|$ with $h$. The green dashed line shows the theoretical prediction using the value of $\mathrm{\Delta }\gamma$ extracted from the fit in Fig. 7. The black dotted line shows a fit of the $N=5000$ curve to the function $f=c{N}^{\mu }$ in the region $h\in [4,30]$.

Figure 9

Entropic force $f$ vs tip-to-surface distance $h$ for a polymer of length $N=5000$ and bending rigidity $\kappa =4$. Results for different cone angles are shown. The dashed lines show the prediction of Eq. (6) ($f\sim h^{-1}$) as well as the prediction for slit confinement in the de Gennes regime ($f\sim h^{-2.7}\approx h^{-1/\nu -1}$).

Figure 10

Entropic force $f$ vs tip-to-surface distance $h$ for a polymer of length $N=5000$ and cone angle of $\alpha ={40}^{\circ }$. Results for various values of the bending rigidity are shown. The inset shows the difference $\mathrm{\Delta }f=f-f^{*}$ vs $h$, where $f^{*}\left(h\right)$ is the fit of each curve to the function $f^{*}=c_0{h}^{\mu }$ in the range $h\in [8,30]$.

Figure 11

Free energy difference $\mathrm{\Delta }F\equiv F(h=0)-F(h=\infty )$ for a star polymer with $n_{\mathrm{arm}}$ arms and with the branch point tethered to the tip of the cone. Results are shown for a cone angle of $\alpha ={45}^{\circ }$ and a star polymer with the length of each arm up to $N=1000$. Results are shown for $n_{\mathrm{arm}}=1\text{−}5$. The inset shows the variation of $\mathcal{A}/n_{\mathrm{arm}}$ vs $n_{\mathrm{arm}}$, where $\mathcal{A}\equiv \mathrm{\Delta }\gamma /\nu$ and where $\mathrm{\Delta }\gamma$ is obtained from the fit $\mathrm{\Delta }F=\mathrm{const}+\mathrm{\Delta }\gamma N$ in the region $N\in [600,1000]$. The blue line is a linear fit to the data.

Figure 12

Entropic force $f$ per arm vs tip-to-surface distance $h$ for a star polymer with $n_{\mathrm{arm}}$ arms and with the branch point tethered to the tip of the cone. Results are shown for an arm length $N=1000$ and cone angle of $\alpha ={45}^{\circ }$. Results for various values of $n_{\mathrm{arm}}$ are shown. The inset shows $f$ vs $h$ for the same data sets.