Mechanical properties of normal and binormal double nanohelices

Abstract

Double helix structures, ubiquitous in nature from nano- to macro- scales, have attracted particular interest due to their unique morphology. Compared to the double nanohelices with the commonest circular cross sections, those with asymmetric cross sections have better mechanical properties for applications in micro-/nanoelectromechanical systems (MEMS/NEMS). In this paper, a novel theoretical basis is proposed based on the extensible Cosserat curve for quantitatively exploring statics and dynamics of the double nanohelices with elliptic cross sections. The normal double nanohelices made up of straight wires rather than single helices are quantitatively confirmed to excel the binormal and rope-like double nanohelices in both load capacity and elasticity, and retain the mechanical stability at the same time. We obtain the interlocking helix angle as well as the minimum boundary value of semiaxes ratio to form a tightly packed double helix. A set of expressions are derived that can be used to measure the mechanical properties of the double helix system under uniaxial stretching, such as the interaction between the strands, tensile modulus and torque. The present work provides useful information for future experimental investigation on normal and binormal double nanohelices as well as their applications in micro-/nanoscale devices.

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I. Introduction

Since James Watson and Francis Crick suggested what is now accepted as the first correct model of the deoxyribonucleic acid (DNA) in 1953,¹ the double helix structure has become a key to open the mysteries of life. Scientists are fascinated by its complex organization and unique characteristics.^2–6 In fact, besides DNA there are many other substance which possess double and even more complex multi-strand helix structures in nature, e.g. from insulin amyloid fibrils,⁷ sickle hemoglobin fibers,⁸ α-amylase⁹ and collagen¹⁰ at the nanoscale, to the twisting mutants of Arabidopsis thaliana at the microscale,¹¹ and then to the intertwined double helix nebula at the macroscopic scale.¹² Two helical chains each coil round the same axis to form the unique double helix structure, which has made itself an attractive architecture for material scientists and engineers interested in micro- and nano- fabrication. It was not until 1990 that the first inorganic double helix was realized in the carbon nanofibers.¹³ Afterwards, carbon nanofibers, SiO₂, Au–Ag alloy and silicon double nano- and micro- helices are obtained in succession.^14–21 The experimental results indicate that these double nano- and micro- helices have excellent elasticity, mechanical strength, stability, conductivity^13–17,21 and provide a wide range of enhanced functionalities for various areas,^22,23 such as engineering, electronics, electromechanics, and optics.

To date, nanohelices with varied morphological diversity, such as twisted, normal and binormal helices, could be obtained using both top down and bottom up strategies.²⁴ According to our previous research, in comparison between helices with circular, square and rectangular cross section, the binormal helical structure has a large linear elasticity regime and is mechanically more stable due to a lower stretch of materials under large axial deformation,²⁵ which is also confirmed by Goriely.²⁶ This situation of single helices makes us wonder whether the asymmetric cross sections can also improve the mechanical properties of double helices. In our former study on the double nanohelices, the rope-like double nanohelices with circular cross sections are tough, relatively elastic, and mechanically stable.²⁷ To form this kind of twisted double helices with two strands in contact along a line, the elliptic cross section is the only available choice among the regular asymmetric shapes. The most obvious advantage of the double helices with elliptic cross section is that the interlocking helix angle²⁸ of a normal double helix in its tightly packed state is larger than that of a double helix with the same area of circular cross section, which implies that a normal double helix can bear more loads.²⁷ From this point of view, the study on double nanohelix with elliptic cross section is of great worth.

In this paper, we present a novel model, within the framework of extensible Cosserat curve theory, for exploring the statics and dynamics of the normal and binormal double nanohelices with elliptic cross sections. We quantitatively demonstrate that it should choose the straight nanowires rather than the single helices to form the double nanohelices with stable tensile modulus and long-range linear elasticity, and moreover the normal double nanohelices have better load capacity and deformability along the helix axis than the binormal ones as well as those with circular cross sections do. By calculating the interlocking helix angle, the critical value of semiaxes ratio to form a twisted double helix is determined. Another important result of our model is that the interaction between the strands and the torque can be predicted by the derived expressions. The presented model can be generalized for the case of double nanohelices with other asymmetric cross sections, such as the rectangle.^29–31

II. The extensible curve model

For a twisted double nanohelix with a straight contact line between two strands, its elongation under axial tensile loading is entirely due to the stretch of the strands. To describe the elasticity of twisted double nanohelices precisely, we regard the strands as Cosserat curves, which are appropriately used to study the statics and the dynamics of continuous rods with extensibility and shear deformation.³² The basic equilibrium equations of each strand can be written as:³³

τ^_α − ε_αβγτ_βW_γ + f_α = 0, (1a)

m̂_α − ε_αβγ(m_βW_γ + τ_βy_γ) = 0, (1b)

where τ and m are the total force and torque across the cross section of the curve, respectively, ε is the permutation tensor, and W and y are the director and position deformation measure, respectively. f is the distributed forces per unit length, which also can be interpreted as the resultant external force per unit length of rod (pressure) acting on the rod,²⁸ i.e., the interaction force between two strands. The Greek subscripts α, β, γ take on the values 1, 2, 3. and λ = ∂s/∂S is the stretch of the curve with S the arc length along a fixed reference configuration and s the one along a deformed configuration.

As presented in Fig. 1(a) and (b), two nanowires with elliptic cross sectional semiminor axis of r₁ and semimajor axis of r₂ (r₂ > r₁) wind around each other, while in contact along a straight line, to form the two types of twisted binormal and normal double helices, respectively.^26,34 Assuming that a uniform twisted double helix H_DI with the radius a₀ = r₁ (binormal) or a₀ = r₂ (normal), pitch b₀, helix angle ζ = arctan(2πa₀/b₀) and number of turns N is held in equilibrium by a moment M₀ at its ends. For a twisted double helix consisting of two freestanding single helices with the same geometry parameters, M₀ = 0; while for that consisting of two straight nanowires with elliptic cross section twisting around each other, M₀ ≠ 0. When H_DI is loaded under a force F along its helix axis, i.e., the straight contact line between the two strands as shown in Fig. 1(c), it transforms to the elongated double helix H_DF with the coil wire semiminor axis (binormal) or semimajor axis (normal) of r, radius a = r, pitch b.

Fig. 1 Schematic illustration of (a) a binormal double helix and (b) a normal double helix H_DI with an elliptic cross section under a constant torque M₀ along its helical axis. (c) Configuration of the elongated normal double helix H_DF after loading by a tensile force F along its helical axis. A section of (d) H_DI and (e) the corresponding elongated section of H_DF.

Fig. 1(d) and (e) present the partial enlargement of the normal double helix H_DI and H_DF with the stretching length ΔL of nanowire. Therefore, in the model H_DI is the fixed reference configuration and H_DF is the deformed one, whose director basis D_i (i = 1, 2, 3) and d_i are defined by a set of Euler angles ϕ₀, θ₀, ψ₀ and ϕ, θ, ψ, respectively. For the normal double helices we chose D₁, d₁ and D₂, d₂ along the direction of the smallest and largest bending stiffness of the cross section in H_DI and H_DF, respectively; while for the binormal double helices the situation is just the reverse. The helix axis is assumed to be along the e₃ axis of the fixed Cartesian basis.

The director deformation measures W⁽⁰⁾ of H_DI and W of H_DF for a normal and a binormal helix are given by

W⁽⁰⁾_1N = −[small psi, Greek, dot above]₀ sin θ₀, W⁽⁰⁾_2N = 0, W⁽⁰⁾_3N = [small psi, Greek, dot above]₀ cos θ₀, (2a)

W_1N = −[small psi, Greek, circumflex] sin θ, W_2N = 0, W_3N = [small psi, Greek, circumflex] cos θ, (2b)

and

W⁽⁰⁾_1B = 0, W⁽⁰⁾_2B = [small psi, Greek, dot above]₀ sin θ₀, W⁽⁰⁾_3B = [small psi, Greek, dot above]₀ cos θ₀, (3a)

W_1B = 0, W_2B = [small psi, Greek, circumflex] sin θ, W_3B = [small psi, Greek, circumflex] cos θ, (3b)

where the subscript ‘N’ and ‘B’ identify the normal and binormal physical quantities. For the configuration of helix θ₀, [small psi, Greek, dot above]₀, θ, [small psi, Greek, circumflex] are all constants,³⁵ and [small psi, Greek, circumflex] = [small psi, Greek, dot above]₀.³⁶ We choose the third director D₃ of H_DI along the tangent to the centerline of the coil wire axis. For such a case the force and torque in eqn (1) yield:

τ₁ = E₁y₁, τ₂ = E₂y₂, τ₃ = E₃(y₃ − 1), (4a)

m₁ = A(W₁ − W⁽⁰⁾₁), m₂ = B(W₂ − W⁽⁰⁾₂), m₃ = C(W₃ − W⁽⁰⁾₃), (4b)

where E₁ = E₂ = K₂Gπr₁r₂, E₃ = πr₁r₂, A = EI₁, B = EI₂ and C = 4GI₁I₂/(I₁ + I₂). E and G are the Young's and shear moduli, respectively. I₁ = (πr₂³r₁)/4 and I₂ = (πr₂r₁³)/4 are the moment of inertia of the cross section. K₂ is the Timoshenko shear coefficients and related to the Poisson's ratio ν throughfor a normal double helix, andfor a binormal double helix.³⁷

According to eqn (2)–(4), W and τ is the function of [small psi, Greek, circumflex] and θ, which leads to:

Ŵ_i = 0, τ^_i = 0 (i = 1, 2, 3). (6)

By virtue of eqn (1b), (4a) and (6) and W⁽⁰⁾_1B = W_1B = 0, W⁽⁰⁾_2N = W_2N = 0, we obtain:

y_1B = 0, y_2N = 0, (7)

τ_1B = 0, τ_2N = 0. (8)

Considering eqn (6) and (7), we can rewrite the equilibrium eqn (1b) as:

(A − C)W_1NW_3N − AW⁽⁰⁾_1NW_3N + CW⁽⁰⁾_3NW_1N + (E₁ − E₃)y_1Ny_3N + E₃y_1N = 0, (9a)

(B − C)W_2BW_3B − BW⁽⁰⁾_2BW_3B + CW⁽⁰⁾_3BW_2B + (E₂ − E₃)y_2By_3B + E₃y_2B = 0 (9b)

From eqn (1a), (6) and (8) and W_1B = 0, W_2N = 0, it is found that the interaction force f is along the direction of d₂ and d₁ for normal and binormal double helices, respectively:

f_1N = f_3N = 0, f_N = f_2Nd₂, (10a)

f_2B = f_3B = 0, f_B = f_1Bd₁. (10b)

Since d₂⊥e₃ for the normal double helix and d₁⊥e₃ for the binormal double helix, eqn (10a) and (10b) show that the interaction f, marked by the red arrows in Fig. 1(d) and (e), is always perpendicular to the e₃ axis, i.e. the helix axis of the double helix, which is the same as the situation of the rope-like double helices with circular cross sections.^27,28 The action line (or pressure line) of the interaction force f between two stands is the contact straight line, i.e., the helix axis.

We use the director deformation measures W of H_DF and eqn (6) in the equilibrium equation eqn (1a) and obtain:

τ_1N[small psi, Greek, circumflex] cos θ + τ_3N[small psi, Greek, circumflex] sin θ = −f_2N, (11a)

−τ_2B[small psi, Greek, circumflex] cos θ + τ_3B[small psi, Greek, circumflex] sin θ = −f_1B. (11b)

The axial force balance for each nanowire is F′ = F/2, where the loading force F is along the helix axis of the double helix. We combine the relationship of τ·e₃ = F′ with eqn (2) and (3) and get

−τ_1N sin θ + τ_3N cos θ = F′, (12a)

τ_2B sin θ + τ_3B cos θ = F′. (12b)

Then the eqn (4a), (11a), (11b), (12a) and (12b) and the derived conditions of y_1B = y_2N = 0, τ_1B = τ_2N = 0 give the position vectors of H_D:

andFrom eqn (13a) and (13b), we can not only have the stretch λ:but also the radius and pitch of H_D in terms of the Euler angles:where for normal double helix f = f_2N and for binormal double helix f = f_1B. Using eqn (2), (3), (13a) and (13b) and [small psi, Greek, circumflex] = [small psi, Greek, dot above]₀, eqn (9a) and (9b) can be rewritten as:where Δ ≡ I₂/I₁, i = 1 (or i = 2) for a normal (or binormal) helix and δ_i2 is the Kronecker delta.

Furthermore, the torque M along the helix axis of the double helix is expressed by:

M = 2{EI₁[1 − (Δ − 1)δ_i2][small psi, Greek, dot above]₀(sin θ − sin θ₀)sin θ + C[small psi, Greek, dot above]₀(cos θ − cos θ₀)cos θ}. (17)

The semi axis of strand can be regarded as constant during the loading.¹⁴ Under this condition the spring constant of the double helix is deduced from eqn (15b), (15c) and (16) according to the Hooke's law h = dF/d(Nb):

where

If the semiminor axis of r₁ equals to the semimajor axis of r₂, eqn (18) represents the spring constant of the rope-like double nanohelices with circular cross section.²⁷ (The mathematical expression of double helix H_DI is available in the ESI.†)

To quantitatively understand the mechanical properties of normal and binormal double nanohelices, eqn (14)–(18) are used to obtain the semiaxis r, the pitch b, the stretch λ, the interaction f between the two strands, the spring constant h of the elongated normal and binormal double helix H_DF as well as the torque M along the helix axis with the conservation of the nanowire length and that of the nanowire volume, or just with the approximation of unchanged semiaxes. It requires the knowledge of the geometry parameters of r₁, r₂, b₀ of double helix H_DI, the loading force F, and the material parameters of Young's or shear modulus and the Poisson's ratio.

III. Interlocking helix angle ζ_max

When a double nanohelix is tightly packed, it can hold the maximum load because of the largest area of transection. Therefore it is crucial to obtain the characteristic geometry parameter of the tightly packed double nanohelices, i.e., the interlocking helix angle ζ_max.²⁸ For the normal and binormal twisted double nanohelices with elliptic cross sections, ζ_max is determined by the semiminor axis of r₁and semimajor axis of r₂. Fig. 2(a) shows a section of a tightly packed normal double helix. The black line is the centre line of the rod and T is a contacting point on the helix axis. We have

ρ = ρ_T + r₂, (19)

where ρ is the radius of curvature of the rod centre line, i.e. the curvature radius of the double helix, and ρ_T is the radius of curvature along the tangent line direction of the blue rod centre line at the contacting point T. The schematics on the left panel in Fig. 2(b) displays a tightly packed normal double helix with the number of turns 1/4. The black and red dash-dotted lines are along the tangent line direction of the yellow and blue rod centre line at the points of the corresponding contacting point T, respectively. The schematics on the right panel in Fig. 2(b) presents the elliptic cross section of the yellow rod with semi-axes r₂, r₃ along the tangent line direction of the blue rod centre line, i.e. the red dash-dotted line, therefore the semiaxis r₃ satisfies

Fig. 2 (a) A section of a tightly packed normal double helix. (b) The elliptic cross section of the yellow rod along the tangent line direction of the blue rod centre line. (c) The theoretical calculations of the interlocking helix angle ζ_max versus semiaxes ratio η smaller than 2.

For a tightly packed normal double nanohelix, ρ_T is equal to the radius of curvature of the yellow elliptic cross section of the rod at the contacting point T, shown in Fig. 2(b), which leads to

Combining eqn (19) and (21) with the radius of curvature of the double helix ρ = (4π²r₂² + b₀²)/(4π²r₂) and the relationship of tan ζ_max = (2πr₂)/(b₀), we can obtain for a normal double helix. Similarly, for a binormal double helix can be derived (details on modeling are available in the ESI†). By defining the semiaxes ratio η = r₁/r₂ for normal double helix and η = r₂/r₁ for binormal double helix, we give the following final expression of the helix angle for a tightly packed double nanohelix with elliptic cross section

Fig. 2(c) shows the theoretical calculations of the interlocking helix angle ζ_max versus semiaxes ratio η smaller than 2 based on eqn (4). Since ζ_max is just equal to zero when η = 0.5, η must be larger than 0.5 for a double nanohelix with elliptic cross section. In the region of binormal double nanohelix, 0.5 < η < 1 and 0 < ζ_max < 45°; while in the region of normal double nanohelix, η > 1 and ζ_max > 45°. When η = 1, the cross section of a double nanohelix is circular and ζ_max = 45°, which is in agreement with the conclusion given by Ji and Olsen.^38,39 According to the red upward sloping curve, ζ_max = 60° for η = 2 and increases with further increasing η.

IV. Results and discussions

Based on the proposed Cosserat curve model, we now quantitatively analyze the mechanical properties of the normal and binormal twisted double nanohelices with the aid of the carbon double nanohelices, which were obtained by overtwisting a singles yarn and then allowing it to relax around itself until it reached a torque-balanced state.²¹ The carbon double nanohelix has a circular cross section with radius of r₀ = 1 μm.²¹ In the following calculation we set the area of cross sections to πr₀², and change the semiaxes ratio η to get the double nanohelices with elliptic cross section. There are two methods to acquire a twisted double nanohelix: one is joining two single helices of the same geometry parameters together, the other is twisting two straight wires. Obviously the carbon double nanohelix was prepared by the later. It is necessary to figure out which method is more benefit for application in MEMS/NEMS. Fig. 3(a) and (a’) show how the tensile stress F/A depends on the tensile strain of (b − b₀)/b₀ across the region of 0.5 ≤ η ≤ 2 for the tightly packed double nanohelices made up of single helices with M₀ = 0, as well as those made up of straight wires with M₀ ≠ 0. The modeling results are deduced from eqn (15a)–(17) with the material parameters of G = 2.5 GPa, υ = 0.27 ⁴⁰ and the approximation of unchanged semiaxes. The theoretical value of the ultimate tensile strength (UTS) of carbon double nanohelices is E/10, i.e., 0.635 GPa.⁴¹ It is found that the maximum load F_max = A·UTS of a normal double nanohelix with η = 2 is 2.22 and 2.98 times as that of a double nanohelix of circular cross section with η = 1 and the straight wires with η = 0.5, respectively. According to our calculation, when the cross section deforms from an ellipse of η = 0.5 to a circle, the maximum tensile strain will be raised by a factor of 3.2 for M₀ = 0 and 2.7 for M₀ ≠ 0. As for the binormal double nanohelices with η = 2, a factor of 2.8 and 5.1 will be added, respectively. It suggests that a larger semiaxes ratio η leads to a stronger load capacity and a better elasticity. We further compare the two figures and find that for the double nanohelices made of single helices, the stress–strain relationship remain linear only in the binormal domain. However for the double nanohelices made of straight wires, the linearity covers all elasticity regions including that of the normal double nanohelices and the double nanohelices of circular cross sections, which is consistent with the experimental results of the carbon double nanohelix in the low strain region, as the material deformation can be considered as elastic.²¹

Fig. 3 (a) and (a’) Tensile stress and (b) and (b’) tensile modulus versus tensile strain and semiaxes ratio η for the carbon double nanohelices made up of single helices with M₀ = 0 and straight wires with M₀ ≠ 0, respectively.

Fig. 3(b) and (b’) illustrate the tensile modulus defined as (hNb)/A versus tensile strain and semiaxes ratio η for the double nanohelices made up of single helices and straight wires by the use of eqn (18). According to the changing trend of the color map, straight wires should be the better choice to prepare a double nanohelix with stable tensile modulus during the loading. With increasing η from 0.5 to 2, the no-load tensile modulus decreasing from 6.35 GPa to 0.24 GPa for M₀ = 0 and to 0.49 GPa for M₀ ≠ 0. This implies that under the same loading force, the normal double nanohelices are more deformable with larger elongation. Therefore, compared to the rope-like double nanohelices in our previous research,²⁷ the normal double nanohelices made up of straight wires are tougher, more elastic and remain the mechanical stability.

Moreover, it is important to determine the torque required to prepare a tightly packed double nanohelix by two straight nanowires as well as the effects of loading force on torque. Fig. 4(a) and (a’) present how the torque varies with tensile strain in the range of 0.5 ≤ η ≤ 2 for the double nanohelices made up of single helices and straight wires, respectively, based on eqn (17). In Fig. 4(a), for η < 1 the torque decreases from zero and for η > 1 the torque increases from zero during the loading, which indicates that a binormal double nanohelix contrarotates about its helix axis, while a normal double nanohelix revolves clockwise round its helix axis. In Fig. 4(a’), the no-load torque varies non-monotonically with the semiaxes ratio η. The largest torque of 1.37 × 10⁻⁹ Nm required to form a tightly packed double helix appears at η = 1.04. Increasing loading force can decrease the torque along the helix axis. In addition, to quantitatively study the influence of the stretching length of strands ΔL on the elongation of double nanohelices, Fig. 4(b) and (b’) present the ratio of ΔL/(NΔb) during the loading in the region of 0.5 ≤ η ≤ 2 for the double nanohelices made up of single helices and straight rods, respectively, derived from eqn (15a) and (14). For a uniform double nanoheilx . According to the diagram all the ratio is above 50%. Thus the stretch of strands cannot be omitted and the extensible Cosserat curve is an efficient tool to investigate the mechanical behavior of double nanohelices.

Fig. 4 (a) and (a’) Torque, (b) and (b’) the ratio of the stretching length ΔL of nanowire to the increment of pitch Δb and (c) and (c’) distributed forces per unit length f versus tensile strain and semiaxes ratio η for the carbon double nanohelices made up of single helices with M₀ = 0 and straight wires with M₀ ≠ 0, respectively.

Considering that the interaction force is the characteristic parameter of double nanohelices, as shown in Fig. 4(c) and (c’), finally, we study the dependence of the distributed forces per unit length, f, on the tensile strain in the range of 0.5 ≤ η ≤ 2 from eqn (15a)–(16) for the double nanohelices made up of single helices and straight rods, respectively. We have f_B = fd₁ and f_N = fd₂ with d₁ and d₂ along the direction of the largest bending stiffness of the cross section of H_DF. In both figures f is negative, which indicates that the interaction force f should be along the helix radial axis as shown in Fig. 1(d) and (e).²¹ Obviously, stretching a double nanohelix will enhance the interaction force between the strands. Since there is no interaction between two helices for a freestanding double nanohelices made up of single helices, the interaction force f increases from zero at every fixed semiaxes ratio η during the loading. When two straight wires twine around each other under a torque to form a double helix, there is bound to be interaction force between the rods without loading.

V. Conclusions

In summary, we used the Cosserat curve theory to obtain a comprehensive model for investigating the mechanical properties of normal and binormal double helix structures. A case of the twisted double helices with elliptic cross sections was considered. We derived the interlocking helix angle ζ_max for the tightly packed normal and binormal double helices, and confirmed the expression by ζ_max = 45° for the double helices with circular cross sections. It is found that the semiaxes ratio must be larger than 0.5 to form a twisted double helix with elliptic cross section, and a double nanohelix with larger semiaxes ratio is more deformable with larger elongation. By quantitatively comparing the tightly packed double nanohelices made up of single helices with those made up of straight wires, we revealed that the later ones have more stable tensile modulus during the loading and their linearity covers both normal and binormal elasticity regions. In particular, the normal double nanohelices made up of straight wires are better at load capacity and elasticity than the rope-like double nanohelices are, and remain the mechanical stability. The characteristics of the interaction between two strands and torque are also precisely described and explained in the entire stretching region. We hope that our analysis may stimulate experimentalists to further investigate the normal and binormal double nanohelices and use them as building blocks for micro-/nanoscale devices. In addition, our work supplies a basis for extending the present Cosserat curve model to the framework of more complex double helix structures such as DNA.

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant no. 11347136, no. 11305035, no. 11347180, the Science Foundation of Suzhou University of Science and Technology with the Project no. XKQ201406, the National Natural Science Funds of China for Young Scholar with the Project no. 61305124 and the Early Career Scheme (ECS) grant from the Research Grants Council of Hong Kong SAR, with the Project no. 439113.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4ra11373j

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