# Semiflexible Polymers Interacting with Planar Surfaces: Weak versus Strong Adsorption

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## Abstract

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## 1. Introduction

## 2. Simulated Model

## 3. Numerical Results for the Properties of Adsorbed Chains

## 4. Distributions of Trains and Loops

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Hard Rod Limit

## References

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**Figure 1.**Snapshot pictures of $\mathcal{N}=50$ single semiflexible polymers, described by a bead-spring model, with $N=250$ monomers each for the case of a chain stiffness $\kappa =25$ and wall potential depth ${\u03f5}_{wall}=0.60$ (

**left**) and $0.80$ (

**right**)—see Section 2 for a precise description of the chosen model. The adsorbing surface is shown in green, and monomeric units within the range of the adsorption potential are shown in blue while those outside of this range are shown in yellow. Note that the root mean square gyration radius in the z-direction $\sqrt{\langle {R}_{gz}^{2}\rangle}$ is only about $2.14$ for ${\u03f5}_{wall}=0.60$, and about $0.48$ for ${\u03f5}_{wall}=0.80$, implying that in both cases the polymer conformations are almost two-dimensional while $\sqrt{{\langle {R}_{g}^{2}\rangle}_{\left|\right|}}\approx 48$ in both cases. All chains are grafted at the point $x=0,\phantom{\rule{0.166667em}{0ex}}y=0,\phantom{\rule{0.166667em}{0ex}}z=0.97$ so the superposition of the snapshots should not be confused with pictures of star polymers.

**Figure 2.**(

**a**) adsorption Mie-potential, Equation (6)—broken curves, plotted for z and five choices of ${\u03f5}_{wall}$, as indicated. The corresponding monomer density distributions $\rho \left(z\right)$ for $N=500,\phantom{\rule{0.277778em}{0ex}}\kappa =25$, normalized as ${\int}_{0}^{\infty}\rho \left(z\right)dz=1$, are indicated by full curves, (

**b**) probability distribution $P\left({z}_{CM}\right)$ of the center-of-mass position ${z}_{CM}$ of a chain with $N=500,\kappa =25$, for six choices of ${\u03f5}_{wall}$. Weakly bound chains (${\u03f5}_{wall}=0.55$ and $0.60$) have very asymmetric distributions while strongly bound chains (${\u03f5}_{wall}=0.75$ and $0.80$) have almost Gaussian distributions with only small asymmetry in the tails, whereas the peak position exceeds ${z}_{min}$ distinctly. Inset shows the variation of $\langle {z}_{CM}\rangle $ with ${\u03f5}_{wall}$ for five choices of N.

**Figure 3.**(

**a**) Ratio of perpendicular and parallel parts ${\langle {R}_{g}^{2}\rangle}_{\perp}/{\langle {R}_{g}^{2}\rangle}_{\left|\right|}$ of the mean square gyration radius of semiflexible macromolecules with $\kappa =25$, plotted vs. strength ${\u03f5}_{wall}$ of the adsorption potential for five choices of N. A second order adsorption transition should show up as universal crossing point of the curves for large N. The resulting estimate ${\u03f5}_{wall}^{cr}=0.47\pm 0.01$ is indicated by an arrow, (

**b**) log-log plot of the adsorbed fraction of monomers f vs. chain length for $\kappa =25$ and five choices of ${\u03f5}_{wall}$, as indicated. Broken straight lines denote power laws, the slopes show the resulting effective exponents.

**Figure 4.**Conformations of strongly adsorbed chains ($N=250,\phantom{\rule{0.277778em}{0ex}}\kappa =25$) showing the z coordinate ${z}_{i}$ of the monomers plotted vs. their index i labeling them along the chain contour. Three choices of ${\u03f5}_{wall}$ are shown: ${\u03f5}_{wall}=0.60$ (

**a**); $0.70$ (

**b**); and $0.80$ (

**c**). Monomers in trains are shown in blue, those in loops in yellow. (

**d**) probability distribution $W\left(\lambda \right)$ plotted vs. deflection length $\lambda $ for $N=250,{\u03f5}_{wall}=0.60$, and five choices of $\kappa $, as indicated. The dashed line denotes the function $W\left(\lambda \right)=0.18ln\left(\right)open="("\; close=")">\frac{{\lambda}_{0}}{\lambda}$ with ${\lambda}_{0}\approx 7.39$.

**Figure 5.**Variation of the deflection length $\lambda $ with chain stiffness $\kappa $ at ${\u03f5}_{wall}=0.60$ for chains of length $N=250$ and 500 in log-log coordinates, testing different definitions of the deflection length, as indicated. Slopes are displayed above the respective straight lines. The inset shows $\lambda /{\ell}_{b}$ vs. stiffness $\kappa $ for three different strengths of ${\u03f5}_{wall}$ in normal coordinates.

**Figure 6.**Plot of ${\langle {R}_{g}^{2}\rangle}_{\perp}$ vs. $1/N$ for five choices of $\kappa $, and ${\u03f5}_{wall}=0.80$ (

**a**); and $0.70$ (

**b**). Straight lines through the data are guides for the eye.

**Figure 7.**Plot of f vs. $1/N$ for ${\u03f5}_{wall}=0.80$ (

**a**); and $0.70$ (

**b**). Straight lines through the data are tentative extrapolations. Six values of $\kappa $ are included, as indicated. $\kappa =16$ data without excluded volume are systematically larger, while for larger $\kappa $ presence or absence of excluded volume does not cause appreciable difference.

**Figure 8.**Deviation of the bond orientational order parameter $\eta $ from value $-1/2$, i.e., from perfectly parallel orientation, plotted vs. $1/N$ for 6 stiffnesses $\kappa $, as indicated, for ${\u03f5}_{wall}=0.80$ (

**a**), and ${\u03f5}_{wall}=0.70$ (

**b**). Presence or absence of excluded volume interaction does not cause any difference on the data. Straight lines show tentative extrapolations.

**Figure 9.**(

**a**) log-log plot of $3/2\langle {\alpha}^{2}\rangle =1/2+\eta $ vs. stiffness $\kappa $ at ${\u03f5}_{wall}=0.80$ for several choices of N, as indicated. Only data for $\kappa \ge 16$ are included so as to restrict attention to the strongly adsorbed case; (

**b**) the same as (

**a**), but for ${\langle {R}_{g}^{2}\rangle}_{\perp}$ vs. $\kappa $; (

**c**) log-log plot of $1/2+\eta $ vs. ${\left(\right)}^{{\langle {R}_{g}^{2}\rangle}_{\perp}}$ for ${\u03f5}_{wall}=0.80$; (

**d**) the same as in (

**c**), but for ${\u03f5}_{wall}=1.00$.

**Figure 10.**Probability distribution ${P}_{train}\left(n\right)$ to observe trains having a length of n monomers, for $N=750$ and two choices of $\kappa $, $\kappa =5$ (

**a**); and $\kappa =8$ (

**b**). Five choices of ${\u03f5}_{wall}$ are included, as indicated. In case (

**b**), symbols indicate data with EV shut off while curves show data including EV. Inserts show the average train length, defined from ${n}_{av}=\int dn{P}_{\mathrm{train}}\left(n\right)n.$

**Figure 11.**Probability distribution ${P}_{loop}\left(n\right)$ to observe loops having a length of n monomers, for $N=750$ and two choices of $\kappa $, $\kappa =5$ (

**a**); and $\kappa =8$ (

**b**). Five choices of ${\u03f5}_{wall}$ are included, as indicated. In case (

**b**), symbols indicate data with EV shut off while curves show data including EV. Insets show the average loop length, defined as ${n}_{av}=\int dn{P}_{loop}\left(n\right)n$.

**Figure 12.**Density distribution function $\rho \left(z\right)$ vs. z for several choices of $\kappa $ for $N=500$ and ${\u03f5}_{wall}=0.8$ (

**a**) and ${\u03f5}_{wall}=0.65$ (

**b**); For $\kappa =5$, the predicted decay proportional to $-4/3$ is seen; (

**c**) log-log plot of the mean square width $\langle \Delta {H}^{2}\rangle $ of the density distribution versus stiffness $\kappa $, at ${\u03f5}_{wall}=1.0$ and five choices of the chain length N. Tentative phenomenological power law fits are indicated by a full straight line and a broken straight line as explained in the legend.

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**MDPI and ACS Style**

Milchev, A.; Binder, K.
Semiflexible Polymers Interacting with Planar Surfaces: Weak versus Strong Adsorption. *Polymers* **2020**, *12*, 255.
https://doi.org/10.3390/polym12020255

**AMA Style**

Milchev A, Binder K.
Semiflexible Polymers Interacting with Planar Surfaces: Weak versus Strong Adsorption. *Polymers*. 2020; 12(2):255.
https://doi.org/10.3390/polym12020255

**Chicago/Turabian Style**

Milchev, Andrey, and Kurt Binder.
2020. "Semiflexible Polymers Interacting with Planar Surfaces: Weak versus Strong Adsorption" *Polymers* 12, no. 2: 255.
https://doi.org/10.3390/polym12020255