^{1}, Stefano Giordano

^{2,3,a)}, Pier Luca Palla

^{2,4}, Fabrizio Cleri

^{2,4}and Luciano Colombo

^{1}

### Abstract

Recent developments of microscopic mechanical experiments allow the manipulation of individual polymer molecules in two main ways: *uniform stretching* by external forces and *non-uniform stretching* by external fields. Many results can be thereby obtained for specific kinds of polymers and specific geometries. In this work, we describe the non-uniform stretching of a single, non-branched polymer molecule by an external field (e.g., fluid in uniform motion, or uniform electric field) by a universal physical framework, which leads to general conclusions on different types of polymers. We derive analytical results both for the freely-jointed chain and the worm-like chain models based on classical statistical mechanics. Moreover, we provide a Monte Carlo numerical analysis of the mechanical properties of flexible and semiflexible polymers anchored at one end. The simulations confirm the analytical achievements, and moreover allow to study the situations where the theory cannot provide explicit and useful results. In all cases, we evaluate the average conformation of the polymer and its fluctuation statistics as a function of the chain length, bending rigidity, and field strength.

We acknowledge computational support by CASPUR (Rome, Italy) under project “Standard HPC Grant 2011/2012.” F.M. acknowledges the Department of Physics of the University of Cagliari for the extended visiting grant, and the IEMN for the kind hospitality offered during the part of this work.

I. INTRODUCTION

II. GENERAL THEORETICAL FRAMEWORK

III. FREELY-JOINTED CHAIN MODEL UNDER EXTERNAL FIELD

A. Average values of positions

B. Covariances and variances of positions

IV. WORM-LIKE CHAIN MODEL UNDER EXTERNAL FIELD

V. ACTION OF A PULLING FORCE NOT ALIGNED WITH THE EXTERNAL FIELD

VI. CONCLUSIONS

### Key Topics

- Polymers
- 85.0
- External field
- 32.0
- Statistical mechanics models
- 15.0
- Monte Carlo methods
- 13.0
- Elasticity
- 9.0

## Figures

A polymer chain in an external field. The first monomer is clamped at position while the others are free to fluctuate. Each monomer is subjected to an external force (different in strength and direction for any *K*): all these forces mimic an external field. Another external force, playing the role of a main pulling load, , is applied to the last monomer at the position .

A polymer chain in an external field. The first monomer is clamped at position while the others are free to fluctuate. Each monomer is subjected to an external force (different in strength and direction for any *K*): all these forces mimic an external field. Another external force, playing the role of a main pulling load, , is applied to the last monomer at the position .

Average values of the longitudinal component of the positions induced by the external field for the 2D FJC case. The red solid lines correspond to the analytical results Eqs. (28) and (32), MC results are superimposed in black circles. Top panel: each curve corresponds to different chain lengths *N* = 10, 20, 30, 40, 50 for a fixed value *gl*/(*k* _{ B } *T*) = 1 (e.g., corresponding to *l* = 1 nm, *g* = 4 pN at *T* = 293 K). Bottom panel: each curve corresponds to the different values *gl*/(*k* _{ B } *T*) = 0.1, 0.25, 0.5, 1, 2, 10 for a fixed chain length *N* = 20.

Average values of the longitudinal component of the positions induced by the external field for the 2D FJC case. The red solid lines correspond to the analytical results Eqs. (28) and (32), MC results are superimposed in black circles. Top panel: each curve corresponds to different chain lengths *N* = 10, 20, 30, 40, 50 for a fixed value *gl*/(*k* _{ B } *T*) = 1 (e.g., corresponding to *l* = 1 nm, *g* = 4 pN at *T* = 293 K). Bottom panel: each curve corresponds to the different values *gl*/(*k* _{ B } *T*) = 0.1, 0.25, 0.5, 1, 2, 10 for a fixed chain length *N* = 20.

Average values of the longitudinal component of the positions induced by the external field for the 3D FJC case. The red solid lines correspond to the analytical results Eqs. (29) and (33), MC results are superimposed in black circles. Top panel: each curve corresponds to different chain lengths *N* = 10, 20, 30, 40, 50 for a fixed value *gl*/(*k* _{ B } *T*) = 1. Bottom panel: each curve corresponds to the different values *gl*/(*k* _{ B } *T*) = 0.1, 0.25, 0.5, 1, 2, 10 for a fixed chain length *N* = 20.

Average values of the longitudinal component of the positions induced by the external field for the 3D FJC case. The red solid lines correspond to the analytical results Eqs. (29) and (33), MC results are superimposed in black circles. Top panel: each curve corresponds to different chain lengths *N* = 10, 20, 30, 40, 50 for a fixed value *gl*/(*k* _{ B } *T*) = 1. Bottom panel: each curve corresponds to the different values *gl*/(*k* _{ B } *T*) = 0.1, 0.25, 0.5, 1, 2, 10 for a fixed chain length *N* = 20.

Longitudinal (top panel) and transversal (bottom panel) component of the variance of positions for the 3D FJC case. The red solid lines correspond to the analytical result Eq. (42), MC results are superimposed in black circles. Each curve corresponds to different chain lengths *N* = 10, 20, 30, 40, 50 for a fixed value of the external field defined by *gl*/(*k* _{ B } *T*) = 1.

Longitudinal (top panel) and transversal (bottom panel) component of the variance of positions for the 3D FJC case. The red solid lines correspond to the analytical result Eq. (42), MC results are superimposed in black circles. Each curve corresponds to different chain lengths *N* = 10, 20, 30, 40, 50 for a fixed value of the external field defined by *gl*/(*k* _{ B } *T*) = 1.

Longitudinal (top panel) and transversal (bottom panel) component of the variance of positions for the 3D FJC case. The red solid lines correspond to the analytical result Eq. (42), MC results are superimposed in black circles. Each curve corresponds to different values of the external field amplitude defined by *gl*/(*k* _{ B } *T*) = 0.1, 0.25, 0.5, 1, 2, 10 for a fixed chain length *N* = 20.

Longitudinal (top panel) and transversal (bottom panel) component of the variance of positions for the 3D FJC case. The red solid lines correspond to the analytical result Eq. (42), MC results are superimposed in black circles. Each curve corresponds to different values of the external field amplitude defined by *gl*/(*k* _{ B } *T*) = 0.1, 0.25, 0.5, 1, 2, 10 for a fixed chain length *N* = 20.

Force-extension curves of a FJC polymer in an external field (or external force) with N = 20. The red line corresponds to the approximated expressions given in Eqs. (32) and (33) while the black circles have been obtained through MC simulations. The 2D (Eq. (30)) and 3D (Eq. (31)) FJC expressions (without an external field) are plotted for comparison with *f* = *g* and *f* = *Ng*.

Force-extension curves of a FJC polymer in an external field (or external force) with N = 20. The red line corresponds to the approximated expressions given in Eqs. (32) and (33) while the black circles have been obtained through MC simulations. The 2D (Eq. (30)) and 3D (Eq. (31)) FJC expressions (without an external field) are plotted for comparison with *f* = *g* and *f* = *Ng*.

Force-extension curves of a WLC polymer in an external field (or external force) with N = 20. The red line corresponds to the approximated expressions given in Eqs. (46) and (48) while the black circles have been obtained through MC simulations. The 2D (Eq. (44)) and 3D (Eq. (47)) WLC expressions (without an external field) are plotted for comparison with *f* = *g* and *f* = *Ng*. The value of the bending spring constant is κ = 0.4 × 10^{−19} Nm ≃ 10*k* _{ B } *T* at *T* = 293 K.

Force-extension curves of a WLC polymer in an external field (or external force) with N = 20. The red line corresponds to the approximated expressions given in Eqs. (46) and (48) while the black circles have been obtained through MC simulations. The 2D (Eq. (44)) and 3D (Eq. (47)) WLC expressions (without an external field) are plotted for comparison with *f* = *g* and *f* = *Ng*. The value of the bending spring constant is κ = 0.4 × 10^{−19} Nm ≃ 10*k* _{ B } *T* at *T* = 293 K.

Action of a pulling force *f* (along the *y*-axis) perpendicular to the applied field *g* (along the *z*-axis). We adopted different values of the bending spring constant: κ = 0.08, 0.6, 2, 8 × 10^{−19} Nm. The chain length is fixed (*N* = 20), the external field amplitude is *g* = 4 pN and the force applied to the last monomer of the chain corresponds to *f* = 8 pN. The red solid lines correspond to the analytical results for the FJC case (see Eqs. (29) and (42)). Black circles correspond to the MC simulations with the different bending spring constants. In the top panel, we reported the average positions, while in the others the three variances of the *x*, *y*, and *z* components.

Action of a pulling force *f* (along the *y*-axis) perpendicular to the applied field *g* (along the *z*-axis). We adopted different values of the bending spring constant: κ = 0.08, 0.6, 2, 8 × 10^{−19} Nm. The chain length is fixed (*N* = 20), the external field amplitude is *g* = 4 pN and the force applied to the last monomer of the chain corresponds to *f* = 8 pN. The red solid lines correspond to the analytical results for the FJC case (see Eqs. (29) and (42)). Black circles correspond to the MC simulations with the different bending spring constants. In the top panel, we reported the average positions, while in the others the three variances of the *x*, *y*, and *z* components.

Average positions of the chain for different angles between the external traction force *f* and the direction of the applied field *g*. We adopted *N* = 20, *g* = 4 pN, and *f* = 60 pN. The red solid lines correspond to the FJC analytical result, Eq. (29). The symbols represent the MC results for the WLC model with κ = 0.08, 0.6, 2 × 10^{−19} Nm (circles, triangles, and squares, respectively). For both FJC and WLC models, we used different values of the angle between the applied field and the traction force θ = π/2, 3π/4, 5π/6, 15π/16 from the right left.

Average positions of the chain for different angles between the external traction force *f* and the direction of the applied field *g*. We adopted *N* = 20, *g* = 4 pN, and *f* = 60 pN. The red solid lines correspond to the FJC analytical result, Eq. (29). The symbols represent the MC results for the WLC model with κ = 0.08, 0.6, 2 × 10^{−19} Nm (circles, triangles, and squares, respectively). For both FJC and WLC models, we used different values of the angle between the applied field and the traction force θ = π/2, 3π/4, 5π/6, 15π/16 from the right left.

Monomer variances versus the position along the chain (*i*) and the angle between force and field (0 < θ < π) for the FJC model. As before, we used *N* = 20, *g* = 4 pN, and *f* = 60 pN.

Monomer variances versus the position along the chain (*i*) and the angle between force and field (0 < θ < π) for the FJC model. As before, we used *N* = 20, *g* = 4 pN, and *f* = 60 pN.

Monomer variances versus the position along the chain (*i*) and the angle between force and field (0 < θ < π) for the WLC model. As before we used *N* = 20, *g* = 4 pN, and *f* = 60 pN. We also adopted a bending stiffness κ = 0.6 × 10^{−19} Nm.

Monomer variances versus the position along the chain (*i*) and the angle between force and field (0 < θ < π) for the WLC model. As before we used *N* = 20, *g* = 4 pN, and *f* = 60 pN. We also adopted a bending stiffness κ = 0.6 × 10^{−19} Nm.

## Tables

Asymptotic forms of the force-extension curves for all cases described in the paper: FJC and WLC models in 2D and 3D geometry with force applied *f* or field applied *g*.

Asymptotic forms of the force-extension curves for all cases described in the paper: FJC and WLC models in 2D and 3D geometry with force applied *f* or field applied *g*.

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