^{1}, Thomas C. Hammant

^{2}, Ronald R. Horgan

^{2}, Ali Naji

^{2,3,a)}and Rudolf Podgornik

^{4}

### Abstract

The wrapping equilibria of one and two adsorbing cylinders are studied along a semi-flexible filament (polymer) due to the interplay between elastic rigidity and short-range adhesive energy between the cylinder and the filament. We show that statistical mechanics of the system can be solved exactly using a path integral formalism which gives access to the full effect of thermal fluctuations, going thus beyond the usual Gaussian approximations which take into account only the contributions from the minimal energy configuration and small fluctuations about this minimal energy solution. We obtain the free energy of the wrapping-unwrapping transition of the filament around the cylinders as well as the effective interaction between two wrapped cylinders due to thermal fluctuations of the elastic filament. A change of entropy due to wrapping of the filament around the adsorbing cylinders as they move closer together is identified as an additional source of interactions between them. Such entropic wrapping effects should be distinguished from the usual entropic configuration effects in semi-flexible polymers. Our results may be relevant to the problem of adsorption of oriented nano-rods on semi-flexible polymers.

R.P. acknowledges support from Slovenian Research and Development Agency (ARRS) through Research Program P1-0055 and Research Project J1-4297. A.N. acknowledges support from the Royal Society, the Royal Academy of Engineering, and the British Academy through a Newton International Fellowship. We gratefully acknowledge support from Aspen Center for Physics, where this work was initiated during the workshop on *New Perspectives in Strongly Correlated Electrostatics in Soft Matter* (2010). We would also like to thank Martin M. Müller for introducing us to his work on the interaction between colloidal cylinders on membranes, which provided the initial inspiration for this work.

I. INTRODUCTION

II. FILAMENT WRAPPING AROUND CYLINDERS: MODEL AND FORMALISM

A. Filament elastic energy

B. Wrapping adhesive energy

C. Partition function of a filament-cylinder complex

D. Constrained and free wrapping of cylinders

E. Constrained arc-length separation between wrapped cylinders

F. Wrapping-unwrapping transition of cylinders on an elastic filament

III. UNWRAPPING TRANSITION: ONE CYLINDER

IV. UNWRAPPING TRANSITION: TWO CYLINDERS

A. Two cylinders: Constrained wrapping

B. Two cylinders: Unconstrained wrapping

C. Free energy of the unwrapping transition

V. EFFECTIVE INTERACTION BETWEEN TWO CYLINDERS

A. Constrained wrapping angles

B. Unconstrained wrapping angles

VI. SUMMARY AND CONCLUSIONS

## Figures

A schematic drawing of the geometry as well as various quantities defined in the text for one and two cylinders wrapped on the Eulerian filament (shown in a symmetric mode). Specifically, for one cylinder (top) the angle at which the filament first touches the cylinder (advancing along the arc-length to the cylinder) is ψ1 and the angle at which it leaves is ψ2. For two cylinders (bottom), separated by a distance l, the contact angles α1 and α2 are as shown in the figure.

A schematic drawing of the geometry as well as various quantities defined in the text for one and two cylinders wrapped on the Eulerian filament (shown in a symmetric mode). Specifically, for one cylinder (top) the angle at which the filament first touches the cylinder (advancing along the arc-length to the cylinder) is ψ1 and the angle at which it leaves is ψ2. For two cylinders (bottom), separated by a distance l, the contact angles α1 and α2 are as shown in the figure.

Top: Schematic view of the wrapping of two cylinders in an antisymmetric mode where the wrapping angles for the two cylinders differ in sign. Bottom: The symmetric looped configuration (full line) of two wrapped cylinders corresponds to a negative value of the horizontal projected separation d ⊥ (i.e., cylinder 2 lies to the left of cylinder 1) and the symmetric extended configuration (dotted line) with positive d ⊥ (i.e., cylinder 2 lies to the right of cylinder 1).

Top: Schematic view of the wrapping of two cylinders in an antisymmetric mode where the wrapping angles for the two cylinders differ in sign. Bottom: The symmetric looped configuration (full line) of two wrapped cylinders corresponds to a negative value of the horizontal projected separation d ⊥ (i.e., cylinder 2 lies to the left of cylinder 1) and the symmetric extended configuration (dotted line) with positive d ⊥ (i.e., cylinder 2 lies to the right of cylinder 1).

The wrapping transition of a single cylinder on an elastic filament. Exact solution for the average wrapping angle ratio is shown as a function of the dimensionless external tension f = βRF for fixed μ = 1 and σ = 0.75.

The wrapping transition of a single cylinder on an elastic filament. Exact solution for the average wrapping angle ratio is shown as a function of the dimensionless external tension f = βRF for fixed μ = 1 and σ = 0.75.

The wrapping transition as a function of the actual force F and for temperatures T = 210 K up to 390 K in steps of 30 K. The physical values for the parameters are given in the text. In both cases the curves move right as T increases. Left panel: we have σ/μ = 0.15, σ^{′} < 0. The wrapping transition is relatively smooth and as T increases it becomes stronger and moves to larger values of F. Right panel: σ/μ = 0.35, σ^{′} > 0. The wrapping transition is much stronger and sharper as compared with the former case and, as predicted, the transition moves to larger values of F as T increases. It is this behavior in both cases which might be considered counter-intuitive.

The wrapping transition as a function of the actual force F and for temperatures T = 210 K up to 390 K in steps of 30 K. The physical values for the parameters are given in the text. In both cases the curves move right as T increases. Left panel: we have σ/μ = 0.15, σ^{′} < 0. The wrapping transition is relatively smooth and as T increases it becomes stronger and moves to larger values of F. Right panel: σ/μ = 0.35, σ^{′} > 0. The wrapping transition is much stronger and sharper as compared with the former case and, as predicted, the transition moves to larger values of F as T increases. It is this behavior in both cases which might be considered counter-intuitive.

Dependence of the average horizontal separation ⟨d ⊥⟩ (normalized to d max = l + 2R) on the external dimensionless tension f = βRF for different sets of parameters in the case of constrained symmetric wrapping with α1 = α2 = π. Left panel: μ = 10 (high rigidity). Right panel: μ = 5 (low rigidity). Negative values of the average horizontal separation indicate the presence of a looped phase, i.e., the cylinder at larger distance along the elastic filament lies to the left of the other cylinder. Curves from left to right correspond to l = nR for integer n = 3, …, 8.

Dependence of the average horizontal separation ⟨d ⊥⟩ (normalized to d max = l + 2R) on the external dimensionless tension f = βRF for different sets of parameters in the case of constrained symmetric wrapping with α1 = α2 = π. Left panel: μ = 10 (high rigidity). Right panel: μ = 5 (low rigidity). Negative values of the average horizontal separation indicate the presence of a looped phase, i.e., the cylinder at larger distance along the elastic filament lies to the left of the other cylinder. Curves from left to right correspond to l = nR for integer n = 3, …, 8.

Dependence of the average horizontal separation ⟨d ⊥⟩ (normalized to d max = l + 2R) on the external dimensionless force f = βRF for different sets of parameters in the case of constrained symmetric wrapping α1 = α2 = 3π/8. Left panel: μ = 10 (high rigidity). Right panel: μ = 5 (low rigidity). Clearly, for this set of parameters only the extended phase is allowed for large values of force regardless of rigidity. Curves from left to right correspond to l = nR for integer n = 3, …, 8.

Dependence of the average horizontal separation ⟨d ⊥⟩ (normalized to d max = l + 2R) on the external dimensionless force f = βRF for different sets of parameters in the case of constrained symmetric wrapping α1 = α2 = 3π/8. Left panel: μ = 10 (high rigidity). Right panel: μ = 5 (low rigidity). Clearly, for this set of parameters only the extended phase is allowed for large values of force regardless of rigidity. Curves from left to right correspond to l = nR for integer n = 3, …, 8.

Mean projected separation, ⟨d ⊥⟩ (blue curves), and mean wrapping angle, ⟨α⟩ (red curves), for unconstrained wrapping of elastic filament around two cylinders. Left panel: σ = 1. Right panel: σ = 10. Solid curves correspond to μ = 10 and dashed curves correspond to μ = 1. Note that the results have been normalized to maximum displacement, d max = l ^{′} + 2R, or to maximum angle.

Mean projected separation, ⟨d ⊥⟩ (blue curves), and mean wrapping angle, ⟨α⟩ (red curves), for unconstrained wrapping of elastic filament around two cylinders. Left panel: σ = 1. Right panel: σ = 10. Solid curves correspond to μ = 10 and dashed curves correspond to μ = 1. Note that the results have been normalized to maximum displacement, d max = l ^{′} + 2R, or to maximum angle.

Free energy as a function of the dimensionless external tension f for μ = 10 and σ = 4.5. Left: Red solid curve corresponds to a single cylinder α = π, dashed curves correspond to antisymmetric double cylinder, α1 = −α2 = π. The lower (blue) curve corresponds to l = 2πR and the upper (black) curve is limit of increased separation. Right: Red solid curve corresponds to a single cylinder α = π, dashed curves correspond to symmetric double cylinder, α1 = α2 = π. The upper (blue) curve corresponds to l = 2πR and the lower (black) curve is limit of increased separation. Note the double cylinder (black) curve that coincides with the single cylinder (red) curve at free energy equal to zero gives the limit of infinite separation.

Free energy as a function of the dimensionless external tension f for μ = 10 and σ = 4.5. Left: Red solid curve corresponds to a single cylinder α = π, dashed curves correspond to antisymmetric double cylinder, α1 = −α2 = π. The lower (blue) curve corresponds to l = 2πR and the upper (black) curve is limit of increased separation. Right: Red solid curve corresponds to a single cylinder α = π, dashed curves correspond to symmetric double cylinder, α1 = α2 = π. The upper (blue) curve corresponds to l = 2πR and the lower (black) curve is limit of increased separation. Note the double cylinder (black) curve that coincides with the single cylinder (red) curve at free energy equal to zero gives the limit of infinite separation.

Free energy as a function of the horizontal displacement projection ⟨d ⊥⟩ is shown for the case of constrained wrapping with α1 = ±α2 = 5π/8, f = 0.4, 1, 2, 3 (from lower to upper curve), μ = 50 and σ = 13.0. In the antisymmetric case (left) the effective interaction is attractive and in the symmetric case (right) it is repulsive.

Free energy as a function of the horizontal displacement projection ⟨d ⊥⟩ is shown for the case of constrained wrapping with α1 = ±α2 = 5π/8, f = 0.4, 1, 2, 3 (from lower to upper curve), μ = 50 and σ = 13.0. In the antisymmetric case (left) the effective interaction is attractive and in the symmetric case (right) it is repulsive.

Left: The free energy as a function of the horizontal displacement projection ⟨d ⊥⟩ (normalized to d max = l + 2R). Right: the required externally applied force, λ, as a function of the horizontal displacement projection ⟨d ⊥⟩. Here we consider the case of constant wrapping with α = 5π/8 > π/2 and α = 3π/8 < π/2 for f = 0.4, μ = 50, σ = 13.0, and l = 4πR.

Left: The free energy as a function of the horizontal displacement projection ⟨d ⊥⟩ (normalized to d max = l + 2R). Right: the required externally applied force, λ, as a function of the horizontal displacement projection ⟨d ⊥⟩. Here we consider the case of constant wrapping with α = 5π/8 > π/2 and α = 3π/8 < π/2 for f = 0.4, μ = 50, σ = 13.0, and l = 4πR.

The free energy for the case of unconstrained wrapping angles. Left: The free energy is shown as a function of external force f for a single cylinder and for two cylinders with symmetric wrapping. The upper dashed curve is for l ^{′} = πR and the lower dashed curve corresponds to l ^{′} = 20 πR, being effectively infinite. As one expects, for large separation and low force, the free energy of two cylinders is double that of a single cylinder (red curve). Right: The free energy as a function of projected horizontal distance d ⊥, for f = 0.01, 1, 2, 3 (from bottom to top). The free energy decreases with separation (entropic effect) and increases with force. In both cases μ = 1 and σ = 1.25.

The free energy for the case of unconstrained wrapping angles. Left: The free energy is shown as a function of external force f for a single cylinder and for two cylinders with symmetric wrapping. The upper dashed curve is for l ^{′} = πR and the lower dashed curve corresponds to l ^{′} = 20 πR, being effectively infinite. As one expects, for large separation and low force, the free energy of two cylinders is double that of a single cylinder (red curve). Right: The free energy as a function of projected horizontal distance d ⊥, for f = 0.01, 1, 2, 3 (from bottom to top). The free energy decreases with separation (entropic effect) and increases with force. In both cases μ = 1 and σ = 1.25.

The required externally applied force, λ, as a function of the horizontal displacement projection ⟨d ⊥⟩ (normalized to d max = l + 2R) is shown for the case of unconstrained wrapping with f = 3, μ = 1, σ = 1.25, and l ^{′} = 2πR.

The required externally applied force, λ, as a function of the horizontal displacement projection ⟨d ⊥⟩ (normalized to d max = l + 2R) is shown for the case of unconstrained wrapping with f = 3, μ = 1, σ = 1.25, and l ^{′} = 2πR.

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