^{1}, Li-Zhen Sun

^{1}, Chao Wang

^{1}and Meng-Bo Luo

^{1,2,a)}

### Abstract

The effect of crowded environment with static obstacles on the translocation of a three-dimensional self-avoiding polymer through a small pore is studied using dynamic Monte Carlo simulation. The translocation time τ is dependent on polymer-obstacle interaction and obstacle concentration. The influence of obstacles on the polymer translocation is explained qualitatively by the free energy landscape. There exists a special polymer-obstacle interaction at which the translocation time is roughly independent of the obstacle concentration at low obstacle concentration, and the strength of the special interaction is roughly independent of chain length *N*. Scaling relation τ ∼ *N* ^{1.25} is observed for strong driving translocations. The diffusion property of polymer chain is also influenced by obstacles. Normal diffusion is only observed in dilute solution without obstacles or in a crowded environment with weak polymer-obstacle attraction. Otherwise, subdiffusion behavior of polymer is observed.

This work was supported by the National Natural Science Foundation of China under Grant Nos. 20874088 and 21174132.

I. INTRODUCTION

II. MODEL AND SIMULATION METHOD

III. SIMULATION RESULTS AND DISCUSSIONS

IV. CONCLUSION

### Key Topics

- Polymers
- 91.0
- Diffusion
- 27.0
- Free energy
- 21.0
- Chemical potential
- 7.0
- Strong interactions
- 7.0

## Figures

A 2D sketch of 3D SAW bond-fluctuation model on SC lattice. (a) Original configuration. There are three NN empty sites marked “A”, “B”, and “C” for the selected monomer 4. But only the site “B” is a “good” site as it satisfies: (1) empty site, (2) without bond crossing, and (3) bond length being allowed. On 2D SC lattice, bond length can be 1 or . (b) New configuration as monomer 4 moving to a new site. Here, the bond length is changeable.

A 2D sketch of 3D SAW bond-fluctuation model on SC lattice. (a) Original configuration. There are three NN empty sites marked “A”, “B”, and “C” for the selected monomer 4. But only the site “B” is a “good” site as it satisfies: (1) empty site, (2) without bond crossing, and (3) bond length being allowed. On 2D SC lattice, bond length can be 1 or . (b) New configuration as monomer 4 moving to a new site. Here, the bond length is changeable.

Semi-logarithmic plot of the translocation time τ vs the concentration of obstacle ϕ_{ c } at the *cis* side for polymer with length *N* = 100 at the chemical potential difference Δμ = 0 and Δμ = 0.5. Pure excluded volume effect of obstacles is considered. The inset presents τ vs the concentration of obstacle ϕ_{ t } at the *trans* side for polymer with length *N* = 100 at Δμ = 0.5.

Semi-logarithmic plot of the translocation time τ vs the concentration of obstacle ϕ_{ c } at the *cis* side for polymer with length *N* = 100 at the chemical potential difference Δμ = 0 and Δμ = 0.5. Pure excluded volume effect of obstacles is considered. The inset presents τ vs the concentration of obstacle ϕ_{ t } at the *trans* side for polymer with length *N* = 100 at Δμ = 0.5.

(a) Semi-logarithmic plot of the translocation time τ vs the polymer-obstacle interaction ɛ_{ c } at different concentrations of obstacle ϕ_{ c } at the *cis* side; (b) Plot of the translocation time τ vs the obstacle concentration ϕ_{ c } for the polymer-obstacle interaction ɛ_{ c } = 0.5, −0.3, −0.6, −0.8, and −1. Horizontal dashed line represents τ_{0} at ϕ_{ c } = 0, while the vertical one indicates the point . Other parameters are: chain length *N* = 100, chemical potential difference Δμ = 0.5, and obstacle concentration at the *trans* side ϕ_{ t }=0.

(a) Semi-logarithmic plot of the translocation time τ vs the polymer-obstacle interaction ɛ_{ c } at different concentrations of obstacle ϕ_{ c } at the *cis* side; (b) Plot of the translocation time τ vs the obstacle concentration ϕ_{ c } for the polymer-obstacle interaction ɛ_{ c } = 0.5, −0.3, −0.6, −0.8, and −1. Horizontal dashed line represents τ_{0} at ϕ_{ c } = 0, while the vertical one indicates the point . Other parameters are: chain length *N* = 100, chemical potential difference Δμ = 0.5, and obstacle concentration at the *trans* side ϕ_{ t }=0.

Semi-logarithm plot of the translocation time τ vs the polymer-obstacle interaction ɛ_{ t } at different obstacle concentrations at the *trans* side. Other parameters are: *N* = 100, Δμ = 0.5, and ϕ_{ c } = 0.

Semi-logarithm plot of the translocation time τ vs the polymer-obstacle interaction ɛ_{ t } at different obstacle concentrations at the *trans* side. Other parameters are: *N* = 100, Δμ = 0.5, and ϕ_{ c } = 0.

Sketches of free energy landscape of translocation at different obstacle concentrations. The free energy increases for ϕ_{ c } > 0 at *cis* side (a) and for ϕ_{ t } > 0 at *trans* side (b). The attractive interaction lowers the free energy. *z* is along the translocation direction of chain.

Sketches of free energy landscape of translocation at different obstacle concentrations. The free energy increases for ϕ_{ c } > 0 at *cis* side (a) and for ϕ_{ t } > 0 at *trans* side (b). The attractive interaction lowers the free energy. *z* is along the translocation direction of chain.

Plot of the special interactions (open circle) and (solid triangle) versus the chemical potential difference Δμ.

Plot of the special interactions (open circle) and (solid triangle) versus the chemical potential difference Δμ.

Log-log plot of the translocation time τ versus the polymer chain length for three cases: (1) ϕ_{ c } = 0.125 at = −0.2 (circle); (2) ϕ_{ t } = 0.125 at (cross); and (3) ϕ_{ c } = ϕ_{ t } = 0 (solid line) at different chemical potential difference Δμ = 0.2, 0.5, and 1.0. Dashed line has a slope 1.25.

Log-log plot of the translocation time τ versus the polymer chain length for three cases: (1) ϕ_{ c } = 0.125 at = −0.2 (circle); (2) ϕ_{ t } = 0.125 at (cross); and (3) ϕ_{ c } = ϕ_{ t } = 0 (solid line) at different chemical potential difference Δμ = 0.2, 0.5, and 1.0. Dashed line has a slope 1.25.

Configurational properties of chain during the translocation at ϕ_{ c } = ϕ_{ t } = 0.125. Mean square end-to-end distance 〈*R* ^{2}〉 (a), mean square radius of gyration 〈*S* ^{2}〉 (b), and segment-obstacle NN pair number *N* _{pair} (c) at different chemical potential differences. Plot (d) shows *N* _{pair} at different interaction ɛ and at Δμ = 0.5. Value *m* _{ c(t)} is the length of partial chain at *cis* (*trans*) side during the translocation. Chain length is =100. ɛ = 0 is used for plots (a)–(c).

Configurational properties of chain during the translocation at ϕ_{ c } = ϕ_{ t } = 0.125. Mean square end-to-end distance 〈*R* ^{2}〉 (a), mean square radius of gyration 〈*S* ^{2}〉 (b), and segment-obstacle NN pair number *N* _{pair} (c) at different chemical potential differences. Plot (d) shows *N* _{pair} at different interaction ɛ and at Δμ = 0.5. Value *m* _{ c(t)} is the length of partial chain at *cis* (*trans*) side during the translocation. Chain length is =100. ɛ = 0 is used for plots (a)–(c).

Log-log plot of the mean square displacement of center of mass 〈Δ*r* ^{2}〉 versus the simulation time *t* at different obstacle concentrations. Solid lines are the linear fit of the simulation data. The inset presents the exponent β at different obstacle concentrations. Other parameters are: chain length *N* = 100, polymer-obstacle interaction ɛ = 0, and system size *L* _{ x } = *L* _{ y } = *L* _{ z } = 80.

Log-log plot of the mean square displacement of center of mass 〈Δ*r* ^{2}〉 versus the simulation time *t* at different obstacle concentrations. Solid lines are the linear fit of the simulation data. The inset presents the exponent β at different obstacle concentrations. Other parameters are: chain length *N* = 100, polymer-obstacle interaction ɛ = 0, and system size *L* _{ x } = *L* _{ y } = *L* _{ z } = 80.

Log-log plot of the mean square displacement of center of mass 〈Δ*r* ^{2}〉 versus the simulation time *t* at different polymer-obstacle interactions. The solid lines are guide for eyes. The inset presents the exponent β at different polymer-obstacle interactions. Concentration of obstacle is ϕ = 0.05 and chain length is *N* = 100.

Log-log plot of the mean square displacement of center of mass 〈Δ*r* ^{2}〉 versus the simulation time *t* at different polymer-obstacle interactions. The solid lines are guide for eyes. The inset presents the exponent β at different polymer-obstacle interactions. Concentration of obstacle is ϕ = 0.05 and chain length is *N* = 100.

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