# Adsorption of Wormlike Chains onto Partially Permeable Membranes

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## Abstract

**:**

## 1. Introduction

## 2. Perfectly Penetrable Free-Standing Film

#### 2.1. Adsorption Threshold

#### 2.2. Adsorbed Layer Structure and Concentration Profile

#### 2.3. Dependencies of Energy, Free Energy and Concentration Profiles on the Attraction Strength u

## 3. Quantitative Approach

## 4. Dependence of the Adsorbed Layer Structure on the Attraction Strength

## 5. Preliminary Discussion

## 6. Partially Penetrable Membranes

## 7. Discussion

#### 7.1. Discussion on Two Models of Polymer/Membrane Interactions

#### 7.2. Notes Related to the Model of Solid Film with Holes

#### 7.3. Loop and Tail Distributions

#### 7.4. Non-Ideality Effects

#### 7.5. Estimates of the Layer Thickness

## 8. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

c | Polymer concentration |

$c\left(x\right)$ | Monomer concentration profile |

${c}_{l}\left(x\right)$ | Concentration profile of monomers belonging to loops |

${c}_{t}\left(x\right)$ | Concentration profile of monomers belonging to tails |

D | Mean hole diameter |

E | Chain potential energy |

${F}_{conf}$ | Chain confinement free energy |

$f\left(\eta \right)$ | Angle-dependent factor of the partition function, Equation (39) |

h | The terminal thickness of adsorbed layer |

L | Chain contour length |

${L}_{c}$ | The total polymer length in the attractive layer |

${L}_{t}$ | The mean tail length |

${l}_{p}$ | Persistence length |

l | Kuhn segment |

N | Number of monomer units (or $\lambda $-segments) per chain |

${n}_{c}$ | Number of polymer/membrane contacts |

p | Fraction of polymer segments in the attraction layer, Equation (59) |

R | Coil size |

T | Temperature in energy units |

U | Attraction potential (per unit length), Equation (4) |

u | Membrane attraction strength |

${u}^{*}$ | Critical adsorption threshold |

w | Porosity of nano-membrane |

$\left|x\right|$ | Distance to the membrane |

$\tilde{x},{x}^{*},\tilde{L}$ | Crossover length-scales |

${Z}_{l}\left(s\right)$ | Partition function of a loop of length s |

${Z}_{t}\left(s\right)$ | Partition function of a tail of length s |

${Z}_{l}(s,w)$ | Partition function of a loop s for porosity w, Equation (70) |

$\alpha $ | Critical exponent, Equation (39) |

${\alpha}_{0}$ | The value of $\alpha $ corresponding to a repulsive membrane |

${\alpha}_{1}$ | The exponent corresponding to critically adsorbing membrane |

$\beta $ | Critical exponent for proximal concentration profile, Equation (81) |

$\Delta $ | Membrane attraction range |

$\u03f5$ | Reduced adsorption energy per unit length, Equation (34) |

$\eta $ | Reduced tilt angle, $\eta =\theta /{\theta}_{x}$ |

${\theta}_{x}$ | Typical bending angle of a loop at distance x from the film |

$\Lambda $ | Crossover length-scale |

$\lambda $ | Characteristic segment length in the contact layer, Equation (7) |

$\rho \left(\eta \right)$ | Distribution of the reduced tilt angle, Equation (48) |

$\tau $ | Relative deviation from the critical point, $\tau =u/{u}^{*}-1$ |

$\varphi $ | The crossover critical exponent, Equation (1) |

$\psi $ | The chain partition function |

$\Psi $ | The Tricomi function, Equation (43) |

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**Figure 1.**(

**a**) A wormlike chain with two ends located in the membrane (at $-\Delta <x<\Delta $). The chain segments are oriented at small angles $\theta $ to the membrane plane $(x=0$). Typically, $\left|\theta \right|\lesssim {\theta}_{\Delta}$ in the ‘train’ sections, and $\left|\theta \right|\gg {\theta}_{\Delta}$ in the ‘loops’ (see Equation (10)). (

**b**) A wormlike chain of contour length L with free ends; the chain is aligned by attraction to a penetrable membrane. The right-hand end is located at distance x from the membrane mid-plane; its orientation is defined by the unit vector $\underline{n}$. The chain involves ‘train’ sections trapped in the short-range attraction layer representing the membrane, ‘loops’ of different sizes away from it, and two ‘tails’.

**Figure 2.**Two chains with the same orientation at $x=-\Delta $ defined by the angle $\theta >0$. One chain ends at point ${A}_{-}$ just below the membrane. The second chain ends at point ${A}_{+}$ slightly above the membrane intersecting it with the end segment of length ${s}_{int}$.

**Figure 3.**The normalized distribution (see Equation (49)) of the reduced angle $\eta =\theta {\left(l/x\right)}^{1/3}$ in the proximal aligned layer, where $\theta $ is the angle of a chain segment and x is its distance to the membrane. Inset: The same function in the log-log scale. The dashed line indicates the asymptotic law, $\rho \propto {\eta}^{-4}$ for $\left|\eta \right|\gg 1$ (cf. Equation (51)).

**Figure 4.**The structure of loops formed by a wormlike chain near a penetrable membrane at a near-critical adsorption strength, $u\approx {u}^{*}$, $u<{u}^{*}$. There are long semiflexible isotropic loops (shown in cyan) everywhere, while shorter aligned loops (shown in black) are present only close enough to the membrane. Aligned loops show a fractal distribution for $\left|x\right|\lesssim \Lambda $, and they dominate by mass in the proximal layer at $\left|x\right|\lesssim \tilde{x}$, where $\tilde{x}<\Lambda $. The aligned loops virtually merge with long isotropic loops at the crossover distance $\left|x\right|\sim \Lambda $ (see green and pink fragments as examples).

**Figure 5.**A wormlike chain near a porous solid nanofilm of effective thickness $2\Delta $. The chain can freely pass from one side of the film to the other through the holes of diameter $\sim D$.

**Figure 6.**Clarification on Equation (71): The chain ending at point ${A}_{+}$ above the membrane intersects it through a hole. Otherwise, the conformation of this chain is similar to that of the chain ${A}_{-}$ which ends below the membrane thus avoiding the last intersection. In analogy with Figure 2 the partition functions $(\psi $) of the chains ${A}_{-}$ and ${A}_{+}$ are nearly equal, and the same is true, of course, for the chains ${A}_{-}$ and ${B}_{-}$. By contrast, the partition function of the chain ${B}_{+}$ is zero since intersections of a solid part of the membrane are not allowed. On averaging over the film area, these relations lead to Equation (71).

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**MDPI and ACS Style**

Semenov, A.; Nyrkova, I.
Adsorption of Wormlike Chains onto Partially Permeable Membranes. *Polymers* **2023**, *15*, 35.
https://doi.org/10.3390/polym15010035

**AMA Style**

Semenov A, Nyrkova I.
Adsorption of Wormlike Chains onto Partially Permeable Membranes. *Polymers*. 2023; 15(1):35.
https://doi.org/10.3390/polym15010035

**Chicago/Turabian Style**

Semenov, Alexander, and Irina Nyrkova.
2023. "Adsorption of Wormlike Chains onto Partially Permeable Membranes" *Polymers* 15, no. 1: 35.
https://doi.org/10.3390/polym15010035