^{1,a)}, Ying-Cai Chen

^{1}, Yan-Li Zhou

^{1}and Meng-Bo Luo

^{2}

### Abstract

The escape of polymer chains from an attractive channel under external electrical field is studied using dynamical Monte Carlo method. Though the escaping process is nonequilibrium in nature, results show that the one-dimensional diffusion theoretical model based on the equilibrium assumption can describe the dependence of the average escaping time (*τ* _{0}) on the polymer–channel interaction (*ɛ*), the electrical field (*E*), the chain length (*n*), and the channel length (*L*), qualitatively. Results indicate that both *ɛ* and *E* play very important roles in the escaping dynamics. For small *ɛ*, the polymer chain moves out of the channel continuously and quickly. While for large *ɛ*, the polymer chain is difficult to move out of long channels as it is trapped for a long time (*τ* _{trap}) when the end segment is near the critical point*x* _{ C }. These results are consistent with the theoretical results for the free energy profiles at small *ɛ* and large *ɛ*, respectively. The dependence of *x* _{ C } and *τ* _{trap} on *ɛ* and *E* are discussed, and specific relations are obtained. The configurational properties of polymer chain are also investigated during the escaping process.

This work was supported by the National Natural Science Foundation of China under Grant No. 20874088.

I. INTRODUCTION

II. MODEL AND SIMULATION METHOD

III. ESTIMATE OF THE ESCAPING TIME FROM THE THEORETICAL MODEL

IV. SIMULATION RESULTS AND DISCUSSION

V. CONCLUSION

### Key Topics

- Polymers
- 83.0
- Free energy
- 35.0
- Critical point phenomena
- 9.0
- Diffusion
- 8.0
- Monte Carlo methods
- 5.0

## Figures

Four states of the translocation for polymer chain through a channel. Solid filled circles represent the leading and the end segment of the polymer chain, respectively.

Four states of the translocation for polymer chain through a channel. Solid filled circles represent the leading and the end segment of the polymer chain, respectively.

A schematic of the model geometry and the polymer model used in the simulation. *L* and *h* represent the length and height of the channel, respectively. An external electrical field with strength *E* is applied inside the channel.

A schematic of the model geometry and the polymer model used in the simulation. *L* and *h* represent the length and height of the channel, respectively. An external electrical field with strength *E* is applied inside the channel.

Free energy *F* as a function of the end segment position *x* inside the channel for both ε < ln μ and ε > ln μ, where the electrical field *E* = 0.1 and the chain length *n* = 100. For ε > ln μ, the free energy *F* reaches its minimum *F* _{min } at *x* = *x* _{ m }.

Free energy *F* as a function of the end segment position *x* inside the channel for both ε < ln μ and ε > ln μ, where the electrical field *E* = 0.1 and the chain length *n* = 100. For ε > ln μ, the free energy *F* reaches its minimum *F* _{min } at *x* = *x* _{ m }.

The dependence of the escaping time *τ* _{0} on the polymer-channel interaction *ɛ* with electrical field *E* = 0.08 and 0.12, chain length *n* = 100, 200, and 300, here channel length *L* = 80.

The dependence of the escaping time *τ* _{0} on the polymer-channel interaction *ɛ* with electrical field *E* = 0.08 and 0.12, chain length *n* = 100, 200, and 300, here channel length *L* = 80.

The dependence of the escaping time *τ* _{0} on the channel length *L* for (a) *ɛ* = 2.0 and (b) *ɛ* = 6.0, where electrical field *E* = 0.1 and chain length *n* = 100.

The dependence of the escaping time *τ* _{0} on the channel length *L* for (a) *ɛ* = 2.0 and (b) *ɛ* = 6.0, where electrical field *E* = 0.1 and chain length *n* = 100.

The dependence of the mean square radius of gyration of the outside part on the number of segments *m* outside the channel. The top *x* axis represents end segment position inside the channel when there are *m* segments out of the channel eventually, i.e., *x* = −(*n* − *m*)*l* _{0}. The dashed line represents the dependence of on length *m* for tethered chain in equilibrium state.

The dependence of the mean square radius of gyration of the outside part on the number of segments *m* outside the channel. The top *x* axis represents end segment position inside the channel when there are *m* segments out of the channel eventually, i.e., *x* = −(*n* − *m*)*l* _{0}. The dashed line represents the dependence of on length *m* for tethered chain in equilibrium state.

(a) The dependence of the escaping time τ_{0} on the polymer–channel interaction *ɛ* with electrical field *E* = 0.08 and 0.12, chain length *n* = 100, 200, and 300, channel length *L* = 80. (b) The dependence of the escaping time τ_{0} on the channel length *L* for different *ɛ* and different *E*, where the chain length *n* = 100.

(a) The dependence of the escaping time τ_{0} on the polymer–channel interaction *ɛ* with electrical field *E* = 0.08 and 0.12, chain length *n* = 100, 200, and 300, channel length *L* = 80. (b) The dependence of the escaping time τ_{0} on the channel length *L* for different *ɛ* and different *E*, where the chain length *n* = 100.

The evolution of the position of the end segment inside the channel for (a) small *ɛ* and (b) large *ɛ*, where electrical field *E* = 0.1, chain length *n* = 100 and channel length *L* = 80.

The evolution of the position of the end segment inside the channel for (a) small *ɛ* and (b) large *ɛ*, where electrical field *E* = 0.1, chain length *n* = 100 and channel length *L* = 80.

(a) The dependence of the ratio *t* _{ r }(*x*)/τ_{0} and (b) Δ*t*(*x*)/τ_{0} on the end segment position *x* for large *ɛ* at three different *E*s, where the chain length *n* = 100 and the channel length *L* = 80.

(a) The dependence of the ratio *t* _{ r }(*x*)/τ_{0} and (b) Δ*t*(*x*)/τ_{0} on the end segment position *x* for large *ɛ* at three different *E*s, where the chain length *n* = 100 and the channel length *L* = 80.

The dependence of the product of *x* _{ C } and *x* _{ A } with *E* (*Ex* _{ C } and *Ex* _{ A }) on *ɛ*, where the chain length *n* = 100 and the channel length *L* = 80. The solid lines represent the linear fit, which can be written as *Ex* _{ C } = −0.8(ε − 1.5) and *Ex* _{ A } = −1.85(ε − 1.76), respectively.

The dependence of the product of *x* _{ C } and *x* _{ A } with *E* (*Ex* _{ C } and *Ex* _{ A }) on *ɛ*, where the chain length *n* = 100 and the channel length *L* = 80. The solid lines represent the linear fit, which can be written as *Ex* _{ C } = −0.8(ε − 1.5) and *Ex* _{ A } = −1.85(ε − 1.76), respectively.

(a) The dependence of the trapped time τ_{trap} on *ɛ* for four *E*'s, where the chain length *n* = 100 and the channel length *L* = 80. (b) The dependence of *E* ln(τ_{trap}) on exp(*ɛ*), where the data are the same as (a).

(a) The dependence of the trapped time τ_{trap} on *ɛ* for four *E*'s, where the chain length *n* = 100 and the channel length *L* = 80. (b) The dependence of *E* ln(τ_{trap}) on exp(*ɛ*), where the data are the same as (a).

## Tables

The values of *x* _{ C }, *x* _{ A }, and τ_{trap} for polymer chains with length *n* = 100, 400, 800, and 1000 at *ɛ* = 3.0, 3.2, 3.4, and 3.6 under *E* = 0.1.

The values of *x* _{ C }, *x* _{ A }, and τ_{trap} for polymer chains with length *n* = 100, 400, 800, and 1000 at *ɛ* = 3.0, 3.2, 3.4, and 3.6 under *E* = 0.1.

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