1 Introduction

Polymer chain represents an important class of two-dimensional structures often arising in biology^[1-2] and engineering^[3-6]. They exhibit special shapes that cannot be sufficiently described by one-dimensional helicity theories. Hence, the study for the equilibrium configuration of polymer chain is one of the most challenging issues in polymer physics^[7-13]. However, when modelling this problem, there are some difficulties due to the complicated structures of the polymer and the interactions between the polymer and its environment. Earlier theoretical models used elastic theory to describe the configuration of vital biomolecules. Cui et al.^[14] and Murayama et al.^[15] used the freely jointed chain model and the freely rotating chain model to investigate the elastic stretching of single-strand deoxyribonucleic acid (DNA). Coleman et al.^[16]} and Tobias et al.^[17] discussed the elastic stability of supercoiled DNA configuration with an elastic thin filament model. On the basis of elastic string model, Fain et al.^[18] studied the effects of the thermal fluctuation on the configurations of underwound and overwound DNA. The results reflected the interaction between the DNA and the environment. Using the continuum model, Moroz and Nelson^[19] investigated the entropic elasticity and twisting stiffness of DNA. In general, these models involve the use of elastic coefficients, which depend on the geometric shape of polymer's cross-section and the elastic properties of materials. Nevertheless, up to present, most of these models are merely used to analyze the polymer with a circular cross section. There are numerous experimental observations of polymers with noncircular cross sections, such as cholesterol crystallization^{[1, 20-22]} and the self-assembled gemini surfactants formed into a twisted ribbon^[23]. Therefore, it is unsuitable to apply a model with a circular cross section for these polymers, and improved models are needed with the consideration of the geometric shape of cross sections.

Due to the potential applications ranging from drug delivery systems^[24] to biological force probes^[25-27], during the past two decades, significant progresses have been made in the investigation of the equilibrium configuration and stability for helical and twisted ribbons. With the continuum model based on elastic free energy, the transition of chiral lipid membrane from straight cylindrical tubules to helical ribbons has been revealed^{[20, 28-30]}. Yu et al.^[31] analyzed the twisting stability of multiwalled carbon-nanotube ribbons with a thin walled pipe model. Feoli et al.^[32] and Nesterenko et al.^[33] studied the dependence of protein configuration on curvature and torsion. Zhang et al.^[34] and Thamwattana et al.^[35] extended their results into the cases including the derivation of curvature and torsion. Based on the Kirchhoff rod model, Tu and Ouyang^[36] gave the Euler-Lagrange equations of thin film. In these works, a common characteristic is that, the polymer is regarded as a smooth inextensible curve in the three-dimensional Euclidean space, while the cross-sectional shape effects of the polymer chain are not considered. To generalize the idea used by Feoli et al.^[32] into more complicated polymer structures, we propose an elastic rod model of polymer chain based on a free energy function, which depends not only on the curvature and torsion but also on the twisting angle characterizing the cross-sectional rotation and its derivative. Moreover, the geometric effects of the cross section are involved through the bending rigidity and twisting rigidity of the polymer chain.

The paper is divided into five parts. In Section 2, the three Euler-Lagrange equations of polymer chain are derived by use of the functional minimization theory. In Section 3, with the Euler-Lagrange equations, we analyze the changes of the configuration of helical ribbons with their elastic properties. Using the simplified equation from the Euler-Lagrange equations, in Section 4, we discuss the effect of the intrinsic twisting on the twisted ribbons. Finally, some conclusions are given.

2 Equilibrium equations of polymer chain

In our earlier works^[37-40], the polymer chain was modelled as a thin elastic rod. Assume that the central axis curve R of the rod is a smooth inextensible curve in the three-dimensional Euclidean space^{[32-33, 35]}. Then, the geometrical configuration of the rod is determined by the curvature κ(s), the torsion τ(s) of R, and the twisting angle χ between the Frenet frame and the reference frame on the cross section (see Fig. 1). The orthonormal Frenet frame on R is denoted by (T (s), N (s), B (s)), where T, N, and B are, respectively, the unit tangent, normal, and binormal vectors, respectively. Let the position vector of a point on R be represented by r (s)=(x (s), y (s), z (s)), where s∈(a, b) is the arc-length parameter. With the differential geometry theory^[41], we can write the curvature and torsion of a curve as follows:

(1)

where r′(s), r″(s), and r″′(s) are the first-, second-, and third-order derivatives of r (s) with respect to the parameter s, respectively, and the symbol (, ) denotes the triple product.

In the Frenet frame, the variation of R can be represented as^[35]

(2)

where ε_i(s) (i=1, 2, 3) are the arbitrarily constants, and ψ_i(s) (i=1, 2, 3) are the arbitrarily functions that are compactly supported on (a, b), i.e., both they and their derivatives vanish at the end of the curve.

Take account of the contributions of the twisting angle χ and its derivative with respect to the arc-length. The free energy E takes the form as follows:

(3)

where χ'(s)=dχ /ds, which is the rotation rate of the cross section along the central axis and named as the internal twist. Γ is the free energy density, and L is the contour length of R.

With the help of the minimum potential energy principle, we have the following Euler-Lagrange equations:

(4)

Substituting Eq. (3) into Eq. (4) leads to

(5)

Combining Eqs. (1) and (2), we calculate the partial derivatives of κ, τ, and with respect to the parameters ε_i(s) (i=1, 2, 3). The results read

(6)

(7)

(8)

The details to derive Eqs. (6) -(8) can be found in Ref. ^[35]. Similarly, the variation of the twist angle is given by

(9)

where the variation of the twist angle is assumed to be a small quantity, which is denoted by ψ₄. Substituting Eqs. (6) -(9) into Eq. (5) leads to

(10)

which is valid for the arbitrary functions ψ₁, ψ₂, ψ₃, and ψ₄ if and only if

(11)

(12)

(13)

(14)

From Eq. (14), it is easy to find that Eq. (11) is an identity. Therefore, Eqs. (12) -(14) are the governing equations in the elastic rod model of the polymer chain. They provide an explicit description for the equilibrium configurations of the polymer chains.

3 Helical ribbons

Using the above equations, we investigate some helical ribbons with different cross-sectional shapes. The elastic ribbons are modelled as the inextensible ribbons whose elastic properties are characterized by the bending and twisting rigidities. Following Panyukov and Rabin^[42], Kessler and Rabin^[43], and Marko and Siggia^[44], under the action of small disturbances, the corresponding free energy density of these elastic helical ribbons with the intrinsic twisting is

(15)

where A is the bending rigidity of the cross section relative to the minor inertia axis, B is the bending rigidity relative to the major inertia axis, and C is the twisting rigidity. Clearly, A>B. ω₃⁰ is the intrinsic twisting. When ω₃⁰=0, the initial configuration of the elastic ribbon comes to a straight elastic ribbon.

As a simple example, the curvature κ and the torsion τ are assumed to be independent of the arc length s. Substituting Eq. (15) into Eqs. (12) -(14), we find that Eq. (13) is an identity, whereas Eqs. (12) and (14) can be simplified to

(16)

(17)

For simplicity, χ' is regarded as a constant. Under this case, Eq. (17) can be reduced to sinχ cosχ=0. Therefore, we have

(18)

With Eq. (18), Eq. (16) leads to

(19)

(20)

The two equations are the same in form. They characterize the bending around the two principal inertia axes, respectively. However, a ribbon is easy to bend around the major one of the two principal inertia axes. Therefore, in the following, we only concern Eq. (20). Liu et al.^[45] have proven that the twist rate ω₃(ω₃=τ +χ') is a constant when χ=jπ/2. It is marked by ω₃₀ (ω₃=τ=ω₃₀) so as to distinguish it from the intrinsic twisting ω₃⁰.

Since a helix can be characterized by the radius r and the pitch p=2πh, where h is a constant for the helix, from Eq. (1), we have

(21)

Inserting Eq. (21) into (20) gives

(22)

From Eq. (22), we can obtain that the pitch angle φ, which is defined by tan φ=h/r, can be represented by

(23)

where μ=C/B, and η=ω ₃⁰/ω₃₀. The pitch angle φ is the key role to determine the configuration of the ribbon. It relies on the elastic properties of the material and the geometry of the cross section. Therefore, we examine the effects of the elastic properties and the geometry of the cross section on the pitch angle. For the analysis convenience, we choose two symmetrical cross sections, i.e., rectangle and ellipse, in the following text.

3.1 Elastic ribbon with rectangular cross section

The rigidity of structure can be expressed in terms of Young's modulus E or shear modulus G and the cross-sectional geometry. As shown in Fig. 2, let c₁ and c₂ be the side lengths of the rectangular cross section. Then, we have^[45]

Since G=E/2(1+υ), we can rewrite Eq. (23) as follows:

(24)

where υ is Poisson's ratio. Equation (24) shows that the pitch angle of the helical ribbon depends on the ratio of the intrinsic twisting ω ₃⁰ to the twist rate ω₃₀ and Poisson's ratio.

Clearly, Eq. (24) requires . Figure 3 exhibits the change of the pitch angle φ with the ratio η. One can see that the pitch angle φ decreases with an increase in the ratio η and approaches the intrinsic twisting ω₃⁰. Figure 3 also shows that all curves intersect at a point (1, 35°). This is because that, the pitch angle φ is independent of Poisson's ratio when η=1, as indicated by Eq. (24).

For a given elastic ribbon, the intrinsic twisting ω₃⁰ is fixed. Therefore, we discuss the pitch angle changing with the twist rate of the ribbon (see Fig. 4). The result indicates that the pitch angle increases with an increase in the twist rate, which is consistent with the helical spring^[45].

3.2 Elastic ribbons with elliptical and circular cross sections

If the cross section is an ellipse with the major axis b and the minor axis a (see Fig. 5), the bending rigidity and twisting rigidity of the ribbon can be written as^[46]

Let b=ka, where k is the shape factor characterizing the axial ratio of a symmetrical cross section, and Eq. (23) be rewritten as follows:

(25)

Clearly, the cross section is circular if k=1. Take υ=0.23. Then, the pitch angle φ is completely determined by the values of η and k. When k is given, the change of φ with η is shown in Fig. 6. We find that, with the increase in η, the pitch angle φ decreases and tends asymptotically to zero. From Eq. (25), it is easy to see that φ approaches a constant (35°) as k tends to infinity, that is, τ/κ(τ /κ=h/r) comes close to a constant. This is an interesting result. It means that bending and twisting always appear or vary simultaneously if the bending rigidity is much greater than the twisting rigidity.

This conclusion is consistent with the results in Ref. ^[43], which showed that the deformation of the helical springs under increasing tensile force was continuous for a sufficiently small ratio of twisting to bending rigidity. Meanwhile, in the opposite limit, increasing the force resulted in a sequence of discontinuous upward jumps of the deformation, i.e., the change of bending or twisting was transilient and first-order phase transitions took place.

3.3 Effects of elastic properties

Equation (23) is valid for either isotropic or anisotropic materials. We analyze the pitch angle changing with the ratio μ (=C/B). As shown in Fig. 7, when η is smaller than 1 (namely, the intrinsic twisting ω₃⁰ is less than the twist rate ω₃₀), the pitch angle φ increases with the increase in μ. On the contrary, when η exceeds 1, φ decreases with the increase in μ. Therefore, besides the cross-sectional geometry, the elastic property of the material is also a key factor to influence the configuration of the polymer chain, which coincides with the arguments in Ref. ^[1] that two kinds of helices with two distinctive pitch angles have different elastic properties.

4 Twisted ribbons

Neglecting the change of the curvature and torsion of the ribbons caused by deformation (i.e., κ=0 and τ=0), we merely take account of the twisting contributions in the ribbons. Equation (16) becomes

(26)

In the case of the twisting rigidity C not equal to zero, Eq. (26) becomes

(27)

which shows that the derivative of the twisting angle only depends on a constant intrinsic twisting. Noticing κ=0, τ=0, κ'=0, τ '=0, and Eq. (27), we can find that the Euler-Lagrange equations (12) -(14) always hold when the energy density takes the form of Eq. (15). Therefore, the configuration of the twisted ribbons is only determined by the intrinsic twisting of the ribbon, just like a straight rod subjected to twisting. Clearly, the total twisting angle of the ribbon is equal to ω₃⁰L.

5 Conclusions

Based on the variational principle, the equilibrium equations of the polymer chain are derived from a free energy functional associated with the curvature, torsion, twisting angle, and their derivatives with respect to the arc-length. The configurations of the helical ribbons with different cross-sectional shapes, elastic properties, and intrinsic twisting angles are discussed. Several conclusions are summarized as follows:

(i) To more accurately characterize the configuration of the polymer chain, it is necessary to consider the twisting angle and its derivative with respect to the arc-length in the elastic rod model of the polymer chain.

(ii) For helical ribbons, the pitch angle decreases with the increase in the ratio of twisting rigidity to bending rigidity and approaches the intrinsic twisting ω₃⁰. When η=1, the pitch angle φ is independent of Poisson's ratio.

(iii) If the bending rigidity is much greater than the twisting rigidity, the bending and twisting of a helical ribbon always appear simultaneously.

(iv) The equilibrium configuration of a twisted ribbon merely depends on the intrinsic twisting of the ribbon, and the twisting angle per unit length is a constant.