Embedding Effect on the Mechanical Stability of Pressurised Carbon Nanotubes

Abstract

We elaborate on the cross-sectional deformation of carbon nanotubes embedded into a self-contracting host medium. The continuum elastic approach is used to formulate the mechanical energy of both the embedded nanotubes and the self-contracting outer medium with finite thickness. Our formula allows us to evaluate the critical radial pressure applied on the interface between the embedded nanotube and the outer contracting medium as well as the deformation mode that arises immediately above the critical pressure. An interesting mechanical implication of the embedding effect, that is, the power-law dependence of the critical pressure on the elastic modulus of the medium, is deduced by the theoretical approach established.

1. Introduction

The salient structural feature of a carbon nanotube is its self-repairing behaviour that arises under high-energy beam irradiation [1, 2]. When the kinetic energy transferred from the incident beam to the constituent carbon atoms is sufficiently large, the atoms are pushed away from the original equilibrium positions, leaving vacancies in the host hexagonal lattice [3, 4]. In typical solids, such irradiation-induced vacancies survive without curing as time passes. However, this is not the case in carbon nanotubes; the removal of carbon atoms from the purely hexagonal lattice leads to a local reconstruction that acts to maintain its coherent network structure with cylindrical geometry. For this reason, carbon nanotubes are often referred to as self-repairing (or self-healing) materials [5–7].

This self-repairing nature provides beneficial effects for manipulating the carbon nanotube morphology, especially when combined with heat treatment. For instance, fine-tuning of electron beam irradiation makes it possible to synthesize multiwall carbon nanotubes (MWNTs) with reduced interwall spacings [8]; the spontaneous shrinkage in the radial direction is a result of the knock-on collision of carbon atoms followed by annealing reconstruction of the vacancies. This experimental finding implies that when the outermost carbon walls of an MWNT are eroded selectively by irradiation, the self-contraction of the outermost walls exerts high pressure on the encapsulated, undamaged innermost walls [9–11]. Application of high pressure may then trigger a novel class of cross-sectional transformations of the inner walls [12], similar to the case of pristine (irradiation-free) MWNTs under hydrostatic pressure [13, 14]. Another class of radial contraction has been observed in MWNTs synthesized in the presence of nitrogen [15]. The yielded nanotubes showed polygon-shaped cross-sections rather than ordinary circular ones, a phenomenon that is partly attributed to the interwall thermal contraction upon cooling, as verified numerically by molecular dynamics simulations [16].

From a nanoengineering perspective, the tunability of cross-sectional geometry may be useful for developing nanofluidic [17–19] or nanoelectrochemical devices [20] based on carbon nanotubes, because both utilize the hollow cavity within the innermost tube. A very interesting issue from an academic viewpoint is the effect of the core tube deformation on the physicochemical properties of intercalated molecules confined in the hollow cavity. It is indeed known that various types of intercalated molecules can fill the innermost hollow cavities of nanotubes [18] and exhibit intriguing behaviours that are distinct from their macroscopic counterparts [21–23]. These distinct behaviours originate from the similarity between the intermolecular spacings and the linear dimension of the nanoscale compartment. Therefore, cross-sectional deformation that breaks cylindrical symmetry will provide a clue to improving the performance of nanotube-based devices.

In this paper, we establish the continuum elastic approach that describes the cross-sectional deformation of carbon nanotubes surrounded by a self-contracting host medium. The mechanical energy of both the pressurized nanotubes and the contracting medium with finite thickness are formulated using thin-shell theory. The obtained formula allows us to evaluate the critical radial pressure applied on the interface between the inner nanotube and the outer medium. Our numerical calculations have unveiled a power-law dependence of the critical pressure on the elastic modulus of the medium that is independent of the medium thickness.

2. Methodology

2.1. Relevant Energy Components

Figure 1 illustrates the self-contraction process of MWNTs subjected to high-energetic beam irradiation. Under irradiation, the induced beam kicks off a portion of carbon atoms at outer walls, causing vacancies followed by spontaneous shrinkage as marked by color in Figure 1(b). Even during the outer-walls self-contraction, the inner walls remain undamaged by irradiation, and thus they tend to keep their initial tube radii. As a result, the contracting outerwalls exert a high pressure, designated by 𝑝, on the inner undamaged walls (see Figure 1). A possible consequence of the radiation-induced high-pressure application is a circumferentially wrinkling structure, called radial corrugation [13], in the inner undamaged walls embedded in the eroded region, as examined later in an approximation based on the thin-shell theory.

(a) MWNT before irradiation 

(b) MWNT after irradiation

Figure 1 

Schematics of the pressurized carbon nanotubes. (a) A pristine MWNT prior to high-energy beam irradiation. (b) An irradiated MWNT whose outermost carbon walls contract radially because of the self-repairing property of the eroded region (indicated in orange).

The stable cross-sectional shape of the embedded tube is obtained by minimizing its mechanical energy 𝑈 per unit axial length [13] as 𝑈=𝑈_𝐷+𝑈_𝐼+𝑈_𝑚+Ω. (1) The first term 𝑈_𝐷=∑^𝑁_𝑖=1𝑈_𝐷^(𝑖) with the definition 𝑈_𝐷^(𝑖)=𝑟_𝑖2𝐶1−𝜈²₀^2𝜋ℐ²_𝑖𝑑𝜃+𝐷₀^2𝜋ℬ²_𝑖𝑑𝜃 (2) represents the deformation energy of the embedded nanotubes. ℐ_𝑖 and ℬ_𝑖 are, respectively, in-plane and bending-induced strains of the 𝑖th wall, and 𝜃 is the circumferential coordinate. For (2), we supposed that each 𝑖th wall had a radius 𝑟_𝑖 prior to cross-sectional deformation and that the deformation caused a displacement 𝐱_𝑖=(𝑢_𝑖,𝑣_𝑖) of a volume element of the 𝑖th wall at (𝑟_𝑖,𝜃) in the polar coordinate representation. The two strain terms ℐ_𝑖 and ℬ_𝑖 are described in terms of the displacement components by [14] ℐ_𝑖=𝑢_𝑖+𝜕_𝜃𝑣_𝑖𝑟_𝑖+12𝜕_𝜃𝑢_𝑖−𝑣_𝑖𝑟_𝑖²,ℬ_𝑖𝜕=−^2𝜃𝑢_𝑖−𝜕_𝜃𝑣_𝑖𝑟²_𝑖, (3) where 𝜕_𝜃=𝜕/𝜕𝜃.

For quantitative discussions, the elastic coefficients 𝐶 and 𝐷 need to be carefully determined. In conventional thin-shell theory for macroscopic objects, 𝐶 and 𝐷 are related to the Young’s modulus 𝐸 of the wall and its thickness ℎ as 𝐶=𝐸ℎ,𝐷=𝐸ℎ³121−𝜈²formacroscopicshells. (4) However, for carbon nanotubes, the macroscopic relations for 𝐶 and 𝐷 noted earlier fail because there is no unique way of defining the thickness of the graphene wall [24]. Thus, the values of 𝐶 and 𝐷 should be evaluated ab initio from direct measurements or computations of carbon sheets, without reference to the macroscopic relations. In actual calculations, we substitute 𝐶=345 nN/nm,  𝐷=0.238 nN·nm for carbon nanotubes, and 𝜈=0.149 from prior work [25] based on the density functional theory.

The second term, 𝑈_𝐼=∑_{𝑖,𝑗=𝑖±1}𝑈_𝐼^(𝑖,𝑗), in (1) accounts for the van der Waals (vdW) interaction energy, which determines the equilibrium distance between adjacent concentric walls. Thus far, several continuum models for the vdW interactions have been proposed. Expressions for the wall-wall interaction proposed in [26] were based on the surface integration of the vdW force and its derivative over the cylindrical walls, while disregarding the vectorial nature of the force. The significance of the vectorial nature of the force was addressed in [27], where analytical expressions for the wall-wall interaction were obtained by considering only the component of the vdW force normal to the wall. In accordance with the result of [27], we define the interaction energy by 𝑈_𝐼^(𝑖,𝑗)=𝑐_𝑖𝑗𝑟_𝑖+𝑟_𝑗4₀^2𝜋𝑢_𝑖−𝑢_𝑗²𝑑𝜃, (5) where the coefficients 𝑐_𝑖𝑗 are derived through a harmonic approximation of the interwall force [28] associated with the vdW intermolecular potential 𝑉(𝑟)𝛽=4𝛼𝑟¹²−𝛽𝑟⁶ (6) with the definitions of 𝛼=2.39meV and 𝛽=0.3415nm the same as those in [29]. Equation (5) takes into account correctly the normal-to-wall component of vdW forces, and it is valid for infinitesimal deformation, which we address in the present work.

The final term Ω in (1) is the negative of the work done by 𝑝 during cross-sectional deformation; it can be written as [13] Ω=𝑝₀^2𝜋𝑟_𝑁𝑢_𝑁+𝑢²_𝑁+𝑣²_𝑁−𝑣_𝑁𝜕_𝜃𝑢_𝑁+𝑢_𝑁𝜕_𝜃𝑣_𝑁2𝑑𝜃. (7) Note that all three terms are functions of 𝑢_𝑖(𝑝,𝜃) and 𝑣_𝑖(𝑝,𝜃) of the 𝑖th wall under 𝑝.

The remaining term, 𝑈_𝑚, in (1) is the elastic energy of the eroded medium surrounding the inner part of the nanotubes. To derive it, we assume that the medium has finite thickness of 𝑟_𝑚−𝑟_𝑁, where 𝑟_𝑚 is the outmost radius of the cylinder-shaped surrounding medium and 𝑟_𝑁 is the outmost tube radius of the embedded nanotube. In addition, the medium is assumed to be homogeneous and isotropic with Young’s modulus 𝐸_𝑚 and Poisson’s ratio 𝜈_𝑚. The validity of the latter assumption depends on the following two effects of irradiation on the mechanical stiffness of MWNTs. Irradiation reduces the axial stiffness because it creates vacancies [30, 31], and it simultaneously enhances the radial stiffness, owing to the production of covalent bonding between adjacent walls [32]. The possible value of 𝐸_𝑚 ranges from 100 GPa (for amorphous carbon) [33, 34] to much less. An explicit form of 𝑈_𝑚 is presented in the next section.

2.2. Corrugation Mode Analysis

Our objectives are to determine (i) the optimal displacements 𝑢_𝑖 and 𝑣_𝑖 that minimize 𝑈 under a given value of 𝑝 and (ii) the critical pressure 𝑝_𝑐 above which the circular cross section of the embedded nanotube is elastically deformed into a noncircular one. These are accomplished by decomposing the displacement 𝑢_𝑖(𝑝,𝜃) just after buckling as follows: 𝑢_𝑖𝑝,𝜃=𝑢_𝑖⁽⁰⁾𝑝+𝛿𝑢_𝑖(𝜃). (8) In (8), the term 𝑢_𝑖⁽⁰⁾(𝑝)∝𝑝 describes a uniform radial contraction at 𝑝<𝑝_𝑐, at which the cross section remains circular and thus the displacement is independent of 𝜃. The other term 𝛿𝑢_𝑖(𝜃) describes a deformed (noncircular) cross-section observed immediately above 𝑝_𝑐. Note that the superscript (0) attached to 𝑢_𝑖⁽⁰⁾(𝑝) differentiates it from the 𝜃-dependent displacement 𝑢_𝑖(𝑝,𝜃) observed at 𝑝>𝑝_𝑐. As to 𝑣_𝑖(𝑝,𝜃), we can write 𝑣_𝑖𝑝,𝜃=𝛿𝑣_𝑖(𝜃), (9) because no circumferential displacement arises at 𝑝<𝑝_𝑐 (i.e., 𝑣_𝑖⁽⁰⁾(𝑝)≡0).

Applying the variation method to 𝑈 with respect to 𝑢_𝑖 and 𝑣_𝑖, we obtain a system of 2𝑁 linear differential equations with regard to 𝛿𝑢_𝑖(𝜃) and 𝛿𝑣_𝑖(𝜃). To derive the 2𝑁 linear differential equations, quadratic or cubic terms in 𝛿𝑢_𝑖 and 𝛿𝑣_𝑖 are omitted since we consider elastic deformation with sufficiently small displacements. In addition, the terms consisting only of 𝑢_𝑖⁽⁰⁾ and 𝑝 are also omitted; the sum of such terms should be equal to zero since 𝑢_𝑖⁽⁰⁾ represents an equilibrium circular cross-section under 𝑝. In fact, the function form of 𝑢_𝑖⁽⁰⁾(𝑝) is determined by the fact that the sum of those terms equals zero. The differential equations can be solved using the Fourier series expansions 𝛿𝑢_𝑖(𝜃)=^∞_𝑛=1𝛿𝜇_𝑖^(𝑛)cos𝑛𝜃,𝛿𝑣_𝑖(𝜃)=^∞_𝑛=1𝛿𝜈_𝑖^(𝑛)sin𝑛𝜃, (10) wherein we took into account that 𝛿𝑢_𝑖, 𝛿𝑣_𝑖, and their derivatives are periodic in 𝜃. Substituting the expansions into the differential equations results in the matrix equation 𝐌𝐮=𝟎; the vector 𝐮 consists of 𝛿𝜇_𝑖^(𝑛) and 𝛿𝜈_𝑖^(𝑛) with all possible 𝑖 and 𝑛, and the matrix 𝐌 involves one variable 𝑝 and other material parameters. It should be noted that, due to the orthogonality of cos𝑛𝜃 and sin𝑛𝜃, the matrix 𝐌 can be expressed by a block diagonal matrix of the form 𝐌=𝐌_𝑛=1⊕𝐌_𝑛=2⊕⋯. (11) Here, 𝐌_𝑛=𝑘 for arbitrary integer 𝑘 is a 2𝑁×2𝑁 submatrix that satisfies 𝐌_𝑛=𝑘𝐮_𝑛=𝑘=𝟎, where 𝐮_𝑛=𝑘 is a 2𝑁-column vector composed of 𝛿𝜇_𝑖^(𝑘) and 𝛿𝜈_𝑖^(𝑘). As a result, the secular equation det(𝐌)=0 is rewritten by det𝐌_𝑛=1det𝐌_𝑛=2⋯=0. (12) Solving (12) with respect to 𝑝, we obtain a sequence of discrete values of 𝑝. Among these values, the minimum one serves as the critical pressure 𝑝_𝑐 that is associated with a specific integer 𝑘. From the definition, the 𝑝_𝑐 associated with a specific 𝑘 allows only 𝛿𝜇_𝑖(𝑛=𝑘) and 𝛿𝜈_𝑖(𝑛=𝑘) be finite, but it requires 𝛿𝜇_𝑖(𝑛≠𝑘)≡0 and 𝛿𝜈_𝑖(𝑛≠𝑘)≡0. Immediately above 𝑝_𝑐, therefore, the cross section of embedded nanotubes becomes radially deformed as described by 𝑢_𝑖(𝜃)=𝑢_𝑖⁽⁰⁾𝑝_𝑐+𝛿𝜇_𝑖^(𝑘)cos𝑘𝜃,𝑣_𝑖(𝜃)=𝛿𝜈_𝑖^(𝑘)sin𝑘𝜃, (13) where the value of 𝑘 is uniquely determined by the one-to-one relation between 𝑘 and 𝑝_𝑐.

3. Mechanical Energy of the Embedding Medium

3.1. Total Energy Cost

In polar coordinates, the radial and circumferential components of normal stress in the medium are denoted by 𝜎_𝑟 and 𝜎_𝜃, respectively, and the shear stress is denoted by 𝜏_𝑟𝜃; all three quantities are functions of 𝑟 and 𝜃. Then, 𝑈_𝑚 is determined by 𝜎_𝑟 and 𝜏_𝑟𝜃 at 𝑟=𝑟_𝑁 as 𝑈_𝑚=𝑈_𝑚⁽⁰⁾+Δ𝑈_𝑚^(𝑛),𝑈_𝑚⁽⁰⁾=𝑟_𝑁2₀^2𝜋𝜎_𝑟⁽⁰⁾_{𝑟=𝑟𝑁}𝑢_𝑁⁽⁰⁾𝑑𝜃,Δ𝑈_𝑚^(𝑛)=𝑟_𝑁2₀^2𝜋𝜎_𝑟^(𝑛)_{𝑟=𝑟𝑁}𝛿𝑢_𝑁+𝜏^(𝑛)_{𝑟𝜃𝑟=𝑟𝑁}𝛿𝑣_𝑁𝑑𝜃, (14)(15)(16) where 𝛿𝑢_𝑁 and 𝛿𝑣_𝑁 describe the corrugation amplitudes of the outermost wall of the embedded nanotube; see (10). The superscripts (0) and (𝑛) indicate the quantities corresponding to uniform contraction and radial corrugation, respectively. In other words, 𝑈_𝑚⁽⁰⁾ represents the energy required for uniform radial contraction of the surrounding medium keeping in contact with the embedded nanotube, and Δ𝑈_𝑚 represents the energy required for radial corrugation with mode index 𝑛.

3.2. Stress-Strain-Displacement Relation

The mechanics of an elastic medium is governed by the stress function 𝜙, which satisfies the so-called compatibility equation [35] 𝜕²_𝑟+𝑟⁻¹𝜕_𝑟+𝑟⁻²𝜕_𝜃²𝜙(𝑟,𝜃)=0, (17) where 𝜕_𝑟=𝜕/𝜕𝑟 and 𝜕_𝜃=𝜕/𝜕𝜃. Once 𝜙 is obtained, we can deduce the stress components as follows: 𝜎_𝑟=𝑟⁻¹𝜕_𝑟+𝑟⁻²𝜕²_𝜃𝜙,𝜎_𝜃=𝜕²_𝑟𝜏𝜙,_𝑟𝜃=𝜕_𝑟𝑟⁻¹𝜕_𝜃𝜙. (18) By definition, the strain components 𝜀_𝑟, 𝜀_𝜃, and 𝛾_𝑟𝜃 are given by the matrix form 𝜀_𝑟𝜀_𝜃𝛾_𝑟𝜃𝜎=𝐆𝜎^𝑟_𝜃𝜏_𝑟𝜃=𝜕_𝑟0𝑟⁻¹𝑟⁻¹𝜕_𝜃𝑟⁻¹𝜕_𝜃𝜕_𝑟−𝑟⁻¹𝑢𝑣1,𝐆=𝐸_𝑚1−𝜈²_𝑚−𝜈_𝑚1+𝜈_𝑚0−𝜈_𝑚1+𝜈_𝑚1−𝜈²_𝑚00021+𝜈_𝑚, (19)(20) where 𝑢=𝑢(𝑟,𝜃) and 𝑣=𝑣(𝑟,𝜃) are, respectively, the radial and circumferential displacements of a volume element in the elastic medium.

The general solution of (17) is given by 𝜙(𝑟,𝜃)=^∞_𝑛=0𝜙_𝑛(𝑟,𝜃)=^∞_𝑛=0𝑓_𝑛(𝑟)cos𝑛𝜃+𝑔_𝑛(𝑟)sin𝑛𝜃. (21) The zeroth component 𝜙₀ represents a uniform contraction of the circular cross section, thus corresponding to the energy 𝑈_𝑚⁽⁰⁾ that we have introduced in (15). The first one, 𝜙₁, implies a rigid-body translation, which is irrelevant to our consideration. Other components, 𝜙_𝑛 for 𝑛≥2, describe radial corrugations with mode index 𝑛, thus providing the energy Δ𝑈_𝑚^(𝑛) given by (16). In the following, we set [36] 𝑓₀=𝑎₀log𝑟+𝑐₀𝑟²,𝑓_𝑛=𝑎_𝑛𝑟^−𝑛+𝑏_𝑛𝑟^2−𝑛+𝑐_𝑛𝑟^2+𝑛+𝑑_𝑛𝑟^𝑛,(𝑛≥2) (22) in order to obtain physically relevant solutions of 𝜎_𝑟, 𝜎_𝜃, and 𝜏_𝑟𝜃 that decay with increasing 𝑟. Without loss of generality, we set 𝑔_𝑛≡0 in (21) according to our assumption of cosine-type radial displacement 𝛿𝑢_𝑖(𝜃); see (10). We emphasize that the hypothesized solution forms of 𝑓₀ and 𝑓_𝑛≥2 differ from those in our earlier work [12]; in the present study, we introduce the positive power terms of 𝑟 in (22) to take into account correctly the boundary condition at 𝑟=𝑟_𝑚, whereas these terms were omitted in [12].

3.3. Energy Cost under Uniform Contraction

We now evaluate the explicit form of the energy 𝑈_𝑚⁽⁰⁾. It results from the uniform contraction of the medium described by the 𝑟-dependent medium displacement 𝑢⁽⁰⁾(𝑟). The elastic nature of the embedding medium assures that 𝜎_𝑟⁽⁰⁾(𝑟)=𝜅(𝑟)𝑢⁽⁰⁾(𝑟) (23) with the 𝑟-dependent stiffness coefficient 𝜅(𝑟). In addition, complete contact at the interface of the medium and the embedded nanotube implies 𝑢⁽⁰⁾𝑟_𝑁=𝑢_𝑁⁽⁰⁾. (24) Hence, the previous expression of 𝑈_𝑚⁽⁰⁾, given in (15), can be rewritten as 𝑈_𝑚⁽⁰⁾=𝜅₀2₀^2𝜋𝑢_𝑁⁽⁰⁾²𝑟_𝑁𝑑𝜃with𝜅₀≡𝜅𝑟_𝑁. (25) The remaining task is, therefore, to represent 𝜅₀ in terms of already-known material parameters such as 𝐸_𝑚,𝜈_𝑚,𝑟_𝑚, and 𝑟_𝑁.

To accomplish this task, we consider a specific solution of (17) that has the form 𝜙=𝜙₀, and then we substitute it back into (18) to obtain 𝜎_𝑟⁽⁰⁾=𝑎₀𝑟⁻²+2𝑐₀,𝜎_𝜃⁽⁰⁾=−𝑎₀𝑟⁻²+2𝑐₀, (26) and 𝜏⁽⁰⁾_𝑟𝜃=0. The coefficients 𝑎₀ and 𝑐₀ are determined by imposing the boundary conditions of 𝜎_𝑟⁽⁰⁾ at 𝑟=𝑟_𝑁 and 𝑟=𝑟_𝑚 as follows: 𝑎₀𝑟_𝑁⁻²+2𝑐₀=𝑝,𝑎₀𝑟_𝑚⁻²+2𝑐₀=0. (27) Eliminating 𝑎₀ and 𝑐₀ from (27), we have 𝜎_𝑟⁽⁰⁾(𝑟)=𝑟²_𝑁𝑝𝑟²_𝑚−𝑟²_𝑁𝑟²_𝑚𝑟²,𝜎−1_𝜃(0)(𝑟)=−𝑟²_𝑁𝑝𝑟²_𝑚−𝑟²_𝑁𝑟²_𝑚𝑟²+1. (28) The obtained 𝜎_𝑟⁽⁰⁾(𝑟) and 𝜎_𝜃⁽⁰⁾(𝑟) lead to the displacements 𝑢⁽⁰⁾(𝑟)=−1+𝜈_𝑚𝐸_𝑚⋅𝑟²_𝑁𝑝𝑟²_𝑚−𝑟²_𝑁1−2𝜈_𝑚𝑟𝑟+^2𝑚𝑟 (29) and 𝑣⁽⁰⁾(𝑟)=0 at the medium region (𝑟_𝑁<𝑟<𝑟_𝑚), as confirmed by (19). By comparing (23), (28), and (29), we conclude that 𝜅₀𝐸=−^𝑚1+𝜈_𝑚⋅𝑟²_𝑚𝑟²_𝑁−1⋅(1−2𝜈_𝑚)𝑟_𝑁+𝑟²_𝑚𝑟_𝑁⁻¹. (30)

3.4. Energy Cost under Radial Corrugation

Next, we consider the energy Δ𝑈_𝑚^(𝑛)(𝑛≥2) that corresponds to the radial corrugation of the 𝑛th order. A similar procedure to the case of 𝑛=0 yields 𝜎_𝑟^(𝑛)()=()𝑎𝑟,𝜃−𝑛𝑛+1_𝑛𝑟⁻²−(𝑛−1)(𝑛+2)𝑏_𝑛𝑟^−𝑛+()𝑑cos𝑛𝜃−𝑛𝑛−1_𝑛𝑟⁻²−(𝑛+1)(𝑛−2)𝑐_𝑛𝑟^𝑛𝜎cos𝑛𝜃,_𝜃(𝑛)()=𝑛()𝑎𝑟,𝜃𝑛+1_𝑛𝑟⁻²+(𝑛−1)(𝑛−2)𝑏_𝑛𝑟^−𝑛+𝑛()𝑑cos𝑛𝜃𝑛−1_𝑛𝑟⁻²+(𝑛+1)(𝑛+2)𝑐_𝑛𝑟^𝑛𝜏cos𝑛𝜃,_{(𝑛)𝑟𝜃}()=𝑛()𝑎𝑟,𝜃𝑛+1_𝑛𝑟⁻²+𝑛(𝑛−1)𝑏_𝑛𝑟^−𝑛−𝑛()𝑑sin𝑛𝜃𝑛−1_𝑛𝑟⁻²+𝑛(𝑛+1)𝑐_𝑛𝑟^𝑛sin𝑛𝜃, (31) leading to the results 𝑢^(𝑛)(𝑟,𝜃)=1+𝜈_𝑚𝐸_𝑚𝑛𝑎_𝑛𝑟⁻²+𝜉₊𝑏_𝑛𝑟^1−𝑛−cos𝑛𝜃1+𝜈_𝑚𝐸_𝑚𝑛𝑑_𝑛𝑟⁻²+𝜉₋𝑐_𝑛𝑟^1+𝑛𝑣cos𝑛𝜃,_(𝑛)(𝑟,𝜃)=1+𝜈_𝑚𝐸_𝑚𝑛𝑎_𝑛𝑟⁻²+𝜂₋𝑏_𝑛𝑟^1−𝑛+sin𝑛𝜃1+𝜈_𝑚𝐸_𝑚𝑛𝑑_𝑛𝑟⁻²+𝜂₊𝑐_𝑛𝑟^1+𝑛sin𝑛𝜃, (32)(33) with 𝜉_±=𝑛±2(1−2𝜈_𝑚) and 𝜂_±=𝑛±4(1−𝜈_𝑚). Of the four coefficients 𝑎_𝑛, 𝑏_𝑛, 𝑐_𝑛, and 𝑑_𝑛 involved in the previous equations, two of them are eliminated by considering the boundary conditions 𝜎_𝑟^(𝑛)(𝑟_𝑁,𝜃)≡0 and 𝜏^(𝑛)_𝑟𝜃(𝑟_𝑁,𝜃)≡0.

Owing to the complete contact condition, we have 𝑢^(𝑛)𝑟_𝑁,𝜃=𝛿𝜇_𝑁^(𝑛)cos𝑛𝜃,𝑣^(𝑛)𝑟_𝑁,𝜃=𝛿𝜈_𝑁^(𝑛)sin𝑛𝜃. (34) Besides, the elastic nature of the medium at the contact interface implies the relations 𝜎_𝑟^(𝑛)_{𝑟=𝑟𝑁}=𝜅₁𝛿𝜇_𝑁^(𝑛)+𝜅₂𝛿𝜈_𝑁^(𝑛)𝜏cos𝑛𝜃,_{(𝑛)𝑟𝜃𝑟=𝑟𝑁}=𝜅₂𝛿𝜇_𝑁^(𝑛)+𝜅₁𝛿𝜈_𝑁^(𝑛)sin𝑛𝜃. (35) From (35), the stiffness coefficients are given by 𝜅₁=𝜎_𝑟^(𝑛)|_{𝑟=𝑟𝑁,𝛿𝜇𝑁}^(𝑛)_{=1,𝛿𝜈𝑁}^(𝑛)₌₀cos𝑛𝜃,𝜅₂=𝜎_𝑟^(𝑛)|_{𝑟=𝑟𝑁,𝛿𝜇𝑁}^(𝑛)_{=0,𝛿𝜈𝑁}^(𝑛)₌₁.cos𝑛𝜃 (36) Note that a set of two normalization conditions of 𝛿𝜇_𝑁^(𝑛)=1,𝛿𝜈_𝑁^(𝑛)=0,𝛿𝜇_𝑁^(𝑛)=0,𝛿𝜈_𝑁^(𝑛)=1, (37) which is indicated by (36), allows us to describe the two remaining undetermined coefficients, from among 𝑎_𝑛, 𝑏_𝑛, 𝑐_𝑛, and 𝑑_𝑛, in terms of materials parameters such as 𝐸_𝑚,  𝜈_𝑚, and 𝑟_𝑚; see the statement immediately below (33). Hence, 𝜅₁ and 𝜅₂ derived as mentioned are those expressed by already-known parameters, though the explicit forms are not shown here to save space. Eventually, we obtain Δ𝑈_𝑚^(𝑛) by substituting the results into (16).

4. Results and Discussions

Figure 2 shows the critical pressure 𝑝_𝑐 required for the cross-sectional deformation of embedded carbon nanotubes. Different types of curves in the plots (dashed, dashed-dotted, and solid) correspond to different values of Young’s modulus ratio 𝐸_𝑚/𝐸, where 𝐸=1 TPa is assumed to be the nanotube’s modulus. We found that, independent of the 𝐸_𝑚/𝐸 value, the 𝑝_𝑐 curves are upward convex as functions of the medium radius 𝑟_𝑚 in units of 𝑟_𝑁. The growth of 𝑝_𝑐 is rapid for 𝑟_𝑚/𝑟_𝑁<2, and then, it saturates for larger 𝑟_𝑚/𝑟_𝑁. The rapid growth in 𝑝_𝑐 indicates a “hardening effect” caused by the surrounding medium, that is, an enhancement in the radial stiffness of the embedded nanotube by encapsulation. This hardening effect disappears with a further increase in 𝑟_𝑚/𝑟_𝑁; the results imply that the surrounding medium with thickness 𝑟_𝑚>5𝑟_𝑁 no longer enhances the hardening effect and thus can be identified as a medium with infinitely large thickness (𝑟_𝑚→∞), which is the case considered in our earlier work [12].

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Critical pressure 𝑝𝑐 for cross-sectional deformation of embedded carbon nanotubes. Monotonic increases with 𝑟𝑚/𝑟𝑁, the ratio between the radii of the medium (𝑟𝑚) and the outermost tube (𝑟𝑁), are observed regardless of the Young’s modulus ratio 𝐸𝑚/𝐸 between the elastic medium (𝐸𝑚) and the carbon nanotubes (𝐸): 𝐸𝑚/𝐸=10−3 (dashed curves), 𝐸𝑚/𝐸=10−2 (dash-dotted curves), and 𝐸𝑚/𝐸=10−1 (solid curve).

Figure 3 provides the index of radial deformation modes observed immediately above 𝑝_𝑐. Successive transformation of deformation modes with increasing medium thickness 𝑟_𝑚 was confirmed, as a result of the fact that the energy required to deform the surrounding medium needed to be responsible for determining the stable corrugation pattern of the embedded nanotube. Again, we found that the mode variation disappears for 𝑟_𝑚/𝑟_𝑁>3, and a larger value of 𝑛 arises for a higher modulus ratio 𝐸_𝑚/𝐸. The nonmonotonic variance in the corrugation mode 𝑛 observed within 1<𝑟_𝑚/𝑟_𝑁<2 is what can be clarified for the first time by the present work; it is applicable to the condition 𝑟_𝑚∼𝑟_𝑁, which lay beyond the scope of our earlier work [12].

(a) SWNT 

(b) 10-walled NT

Figure 3 

Mode index 𝑛 of the radial corrugation observed in the embedded nanotubes. The modulus ratio is set to 𝐸𝑚/𝐸=10−3, 10−2, and 10−1 from the bottom to the top.

Figure 4 shows the 𝑝_𝑐 dependence on the modulus ratio (𝐸_𝑚/𝐸), where the radius ratio 𝛼≡𝑟_𝑚/𝑟_𝑁 is fixed to be 𝛼=1.1 or 5.0. For every choice of 𝛼 and 𝑁, the 𝑝_𝑐 curves in this plot obey a power law represented by 𝑝_𝑐∝𝐸_𝑚𝐸^𝛽2with𝛽≃3. (38) We have confirmed that the power-law behaviour holds for any values of 𝛼 and 𝑁 within the ranges of 1.1<𝛼<5 and 1<𝑁<10, respectively. Recall here that, for 𝛼>5, 𝑝_𝑐 becomes almost constant even with increasing 𝛼. These facts lead to the conjecture that the power law described by (38) is universal to all embedded nanotubes, at least under the numerical conditions that we have addressed. The power law of 𝑝_𝑐 represented by (38) is an interesting manifestation of the embedding effect of nanotubes into an elastic medium.

Figure 4 

Double logarithm plot of the critical pressure 𝑝𝑐 as a function of the Young’s modulus ratio 𝐸𝑚/𝐸 between the elastic medium (𝐸𝑚) and the nanotubes (𝐸). The parameter 𝛼≡𝑟𝑚/𝑟𝑁 represents the ratio of the medium radius 𝑟𝑚 to the outermost tube radius 𝑟𝑁, where 𝑁=1 for single-walled nanotubes and 𝑁=10 for MWNTs. For all the parameter values chosen, the two-third power law of 𝑝𝑐 given by (38) is observed.

It is worthy to note that the two-third power law of 𝑝_𝑐 has been also proposed for the buckling of macroscopic “tunnels" embedded in an infinite soft ground [36]. The radial collapse of tunnel liners, which is one of the major civil engineering disasters, was analysed to reveal a closed form of 𝑝_𝑐; several approximations led to the conclusion of a power-law of 𝑝_𝑐 similar to (38), though an infinitely large medium and only the case of one cylindrical shell (i.e., 𝑁=1) are hypothesized. In this context, our results suggest that the approximation theory proposed for macroscopic tunnels [36] is valid for nanoscopic cylinders, even though they consist of more than one concentric walls.

The previous discussion may pose a further question: does the power law of 𝑝_𝑐 hold true for macroscopic counterparts of “many”-walled nanotubes? In fact, macroscaled pipe-in-pipe structures (i.e., a pipe inserted inside another pipe) are known to be an efficient design solution for subsea-pipeline developments in deep water [37, 38], wherein buckling resistance to large amounts of external hydrostatic pressure is a key structural design requirement. Addressing these problems is expected to shed light on the development of multiple-cylindrical structures from multidisciplinary viewpoints and will be considered in our future work.

5. Conclusion

We have demonstrated a continuum elastic approach that describes the cross-sectional deformation of carbon nanotubes surrounded by a self-contracting host medium. The approach enables quantitative discussions of the critical pressure for radial corrugation and the stable corrugation mode of the nanotube surrounded by the contracting medium. Numerical calculations based on the established theory revealed the power-law dependence of the critical pressure on the elastic modulus of the medium, which is independent of the medium thickness. Further studies are expected to shed light on other mechanical properties of embedded nanotubes and to suggest applications based on their unique cross-sectional deformations.

Acknowledgments

This work was supported by JSPS KAKENHI (Grant nos. 24686096, 11J03636, 22760058, and 25390147). One of the authors (H. Shima) appreciates the financial support provided by the Asahi Glass Foundation.