Thermomechanical Stresses in Fullerenes at Nanotube

Abstract

The thermomechanical stresses acting between a nanotube and fullerenes encapsulated on it are computed. After a general formulation, based on elasticity, we have applied the analysis to C820𝑥00040(10,10) or C600𝑥00040(10,10) peapods finding stresses in the gigapascal range or vanishing, respectively. The analysis suggests that a thermal control could be used to produce smart fullerenes at nanotube systems, for example, as two-stage nanovectors for drug delivery.

1. Introduction

The Royal Swedish Academy of Sciences awarded the 1996 Nobel Prize in Chemistry jointly to Curl, Kroto, and Smalley for their discovery in 1985, together with Heath and O'Brien [1], of fullerenes. It is common belief that the discovery of carbon nanotubes (CNTs) took place in 1991 thanks to Iijima [2], who reported in Nature the observation of multiwalled CNTs. In 1993, in the same issue of Nature, two independent groups, again Iijima with Ichihashi [3] and Bethune et al. [4], reported the observation of single-walled CNTs. The impact of these papers on the scientific community has been unquestionably tremendous. In spite of this, the first direct observation of a multiwalled CNT (by force subsequent to the production of the transmission electron microscope) was previously reported in 1952 in the Journal of Physical Chemistry of Russia by Radushkevich and Lukyanovich [5], whereas an image, even if controversial, of a single- (or possibly double-) walled CNT was reported in 1976 by Oberlin el al. in the Journal of Crystal Growth [6]. Two editorials, appearing in Carbon in 1997 [7] and 2006 [8], support these pioneering observations.

Aside from the controversy surrounding the discovery of nanotubes, fullerenes and nanotubes have introduced humanity to the new nanoera. In particular, the giant strength and Young's modulus of carbon fullerenes and nanotubes, combined with a low density, promise to revolutionize materials science, as required in the design of super-strong space elevator megacables [9] or super-adhesive Spiderman suits [10]. Combining the two nanostructures even more intriguing systems could be realized.

In this paper, we analyze the thermomechanical stresses acting between a nanotube and encapsulated fullerenes (e.g., see [11, 12]). We have solved the problem treating the fullerenes as a fluid inside an elastic channel (nanotube). We have applied the analysis to a C82@(10,10) system finding huge thermomechanical stresses, in the gigapascal range. A C60@(10,10) system is also considered for comparison and is found free of stresses. Thus the analysis suggests that a thermal control could be used to produce smart fullerenes at nanotube systems, for example, as two-stage nanovectors for drug delivery.

2. The Thermomechanical Elastic Model

Consider a nanotube filled by fullerenes (Figure 1). After a temperature variation, the nanotube diameter will change, interacting with the fullerenes. We treat the nanotube as an elastic cylindrical shell and the fullerenes as elastic spheres. We assume a constant pressure distribution between nanotube and fullerenes, due to the small spacing between fullerenes (0.34 nm), so that their action on the nanotube can be considered as a distributed pressure, for example, as a fluid inside an elastic channel.

Figure 1 

Example of fullerenes at nanotubes (Courtesy of Chiu Po-Wen).

Let us consider a linear elastic isotropic sphere of inner and outer radii a and b subjected to inner and outer pressures 𝑝 and 𝑞, respectively. The radial 𝜎_𝑟 and circumferential 𝜎_𝜗 stresses at a generic radius 𝑎<𝑟<𝑏 are [13]𝜎_𝑟=1𝑏³−𝑎³(−𝑏³𝑞+𝑎³𝑎𝑝+³𝑏³𝑟³𝜎(𝑞−𝑝)),_𝜗=1𝑏³−𝑎³(−𝑏³𝑞+𝑎³𝑎𝑝−³𝑏³2𝑟³(𝑞−𝑝)). (1) Linear elastic isotropic laws for spherical symmetry become 𝜀_𝑟≡d𝑢=𝜎d𝑟𝐸^𝑟−2𝜈𝐸𝜎_𝜗,𝜀_𝜗≡𝑢𝑟=𝜎_𝜗𝐸−𝜈𝐸(𝜎_𝜗+𝜎_𝑟), (2) where E and 𝜈 are the Young modulus and Poisson ratio of the material and 𝑢 is the radial elastic displacement (along 𝑟). We will apply (1) and (2) to fullerenes, thus considering them as elastic spheres.

Analogously, let us consider a linear elastic isotropic cylinder (of length 𝐿) of inner and outer radii 𝑎 and 𝑏 subjected to inner and outer pressures p and q, respectively. The radial and circumferential stresses at a generic radius 𝑎<𝑟<𝑏 are [13]𝜎_𝑟=1𝑏²−𝑎²(−𝑏²𝑞+𝑎²𝑎𝑝+²𝑏²𝑟²𝜎(𝑞−𝑝)),_𝜗=1𝑏²−𝑎²(−𝑏²𝑞+𝑎²𝑎𝑝−²𝑏²𝑟²(𝑞−𝑝)). (3) Linear elastic isotropic laws for cylindrical symmetry become𝜀_𝑟≡d𝑢=𝜎d𝑟𝐸^𝑟/−𝜈^/𝐸^/𝜎_𝜗,𝜀_𝜗≡𝑢𝑟=𝜎_𝜗𝐸^/−𝜈^/𝐸^/𝜎_𝑟,𝐸^/=𝐸,𝑣^/𝐸=𝜈forplanestress;_/=𝐸1−𝑣²,𝑣^/=𝑣1−𝑣forplanestrain. (4) We can assume plain stress condition (i.e., 𝜎_𝑧=0) or plane strain condition (i.e., 𝜀_𝑧=0, for which in addition 𝜎_𝑧=𝜈(𝜎_𝑟+𝜎_𝜗)). Assuming a nanotube with free ends, the first case is the most realistic, whereas the second case better describes fixed ends; thus we will treat nanotubes as elastic cylinders, by (3) and (4).

Consider a single fullerene at temperature T as a linear elastic isotropic sphere (elastic properties with subscript “𝑓”) having inner and outer radii 𝑟_𝑖 and 𝑟_𝑜, respectively. The confinement from the nanotube is described by the external pressure 𝑝. According to (1) and (2)𝜎_𝑟(𝑟)=−1−𝑟³_𝑖/𝑟³1−𝑟³_𝑖/𝑟³_𝑜𝜎𝑝,_𝜗(𝑟)=−1+𝑟³_𝑖/2𝑟³1−𝑟³_𝑖/𝑟³_𝑜𝑢𝑝,_tot(𝑟)=𝑢(𝑟)+𝛼_𝑓𝜎(𝑇)Δ𝑇𝑟=[^𝜗(𝑟)𝐸_𝑓−𝜈_𝑓𝐸_𝑓(𝜎_𝜗(𝑟)+𝜎_𝑟(𝑟))+𝛼_𝑓(𝑇)Δ𝑇]𝑟, (5) where 𝑢_tot is the total (elastic + thermal) radial displacement, 𝛼_𝑓 is the thermal expansion coefficient for fullerene (in general a function of the temperature, see Section 3), and Δ𝑇 is the temperature variation.

Thus the fullerene elastic outer radius 𝑟^∗_𝑜 (𝑟_𝑜 if unstressed) is𝑟^∗_𝑜=𝑟_𝑜+𝑢_tot(𝑟=𝑟_𝑜). (6)

Now consider a nanotube at temperature 𝑇 as a linear elastic isotropic cylinder (elastic properties with subscript “𝑛”) having inner and outer radii 𝑅_𝑖 and 𝑅_𝑜, respectively. The confinement from the fullerenes is described by the internal pressure 𝑝. According to (3) and (4):𝜎_𝑟𝑅(𝑟)=^2𝑖/𝑅²_𝑜−𝑅²_𝑖/𝑟²1−𝑅²_𝑖/𝑅²_𝑜𝑝,𝜎_𝜗𝑅(𝑟)=^2𝑖/𝑅²_𝑜+𝑅²_𝑖/𝑟²1−𝑅²_𝑖/𝑅²_𝑜𝑢𝑝,_tot(𝑟)=𝑢(𝑟)+𝛼_𝑛𝜎(𝑇)Δ𝑇𝑟=[^𝜗(𝑟)𝐸^/_𝑛−𝜈^/_𝑛𝐸^/_𝑛𝜎_𝑟(𝑟)+𝛼_𝑛(𝑇)Δ𝑇]𝑟. (7)(8) Thus the nanotube elastic inner radius 𝑅^∗_𝑖 (𝑅_𝑖 if unstressed) is𝑅^∗_𝑖=𝑅_𝑖+𝑢_tot(𝑟=𝑅_𝑖). (9) The compatibility of the displacements (elastic contact between nanotube and fullerenes) implies𝑟^∗_𝑜(𝑝,Δ𝑇)=𝑅^∗_𝑖(𝑝,Δ𝑇). (10) The solution of (10) gives the effective internal pressure 𝑝^∗; the fullerenes/nanotube internal/external radius as a function of temperature are simply given by 𝑟^∗_𝑖(𝑝^∗,Δ𝑇)=𝑟^∗_𝑜(𝑝^∗,Δ𝑇)−𝑡𝑅^∗_𝑜(𝑝^∗,Δ𝑇)=𝑅^∗_𝑖(𝑝^∗,Δ𝑇)+𝑡, where t is the shell thickness (0.34 nm); moreover, 𝑟^∗₀(𝑝^∗,Δ𝑇)=𝑅^∗_𝑖(𝑝^∗,Δ𝑇) can be derived from (9). From the computed value of the pressure 𝑝^∗ the thermomechanical stresses and strains in the fullerenes and nanotubes can be calculated via (1)–(4).

3. Application to the C82(10,10) or C60(10,10) Peapods

According to the previous analysis, we consider nanotube or fullerenes as elastic cylindrical or spherical thin shells; we assume a constant thickness t and plane stress condition and both nanotube and fullerenes composed by the same material, for which we assume 𝐸_𝑓=𝐸_𝑛=𝐸≈1TPa, 𝜈_𝑓=𝜈_𝑛=𝜈≈0 (carbon). For an (𝑛,𝑚) nanotube and for C𝑙 fullerenes, the mean radii (at room temperature, i.e., at 290 K) are respectively given by𝑅_𝑛=𝑅_𝑜+𝑅_𝑖2≈0.0392√𝑛²+𝑚²𝑅+𝑛𝑚nm,_𝑓𝑟=𝑟=^𝑜+𝑟_𝑖2≈0.0458√𝑙nm, (11) so that 𝑅_𝑖=𝑅_𝑛−𝑡/2 and 𝑟_𝑜=𝑅_𝑓+𝑡/2 with t = 0.3 nm (close to the van der Waals spacing [14]). For 𝑟_𝑜>𝑅_𝑖 mechanical stresses will be present.

In addition, the thermal volumetric expansion coefficient 𝛽_𝑓 of carbon fullerenes is [15] 𝛽_𝑓≈3𝛼_𝑓(𝑇)≈−1×10⁻⁵𝑇70𝑇<70K,1×10⁻⁵𝑇−1508070K≤𝑇<400K, (12) which thus changes sign around 70 K, whereas the thermal expansion coefficient of a carbon nanotube, according to [15], is always negative in the considered temperature range:𝛼_𝑛≈−1.2×10⁻⁵𝑇400𝑇<400K. (13) Thus, according to these numerical solutions, following the analysis reported in Section 2, we derive 𝑝=𝑝^∗=𝐸(𝑅_𝑓+𝑡/2)(1+𝛼_𝑓Δ𝑇)−(𝑅_𝑛−𝑡/2)(1+𝛼_𝑛Δ𝑇)𝑅_𝑛/𝑡(𝑅_𝑛−𝑡/2)+𝑅_𝑓/(2𝑡)(𝑅_𝑓𝜎+𝑡/2)=𝜒if𝜒>0(contact),0if𝜒≤0(nocontact),_{(max)𝑟𝑓,𝑛}=−𝑝,𝜎_𝜗𝑓𝑅=−^𝑓2𝑡𝑝,𝜎_𝜗𝑛=𝑅_𝑛𝑡𝑅𝑝,_∗𝑓,𝑛=𝑅_𝑓,𝑛𝜎(1+^{𝜗𝑓,𝑛}𝐸+𝛼_𝑓,𝑛𝑅Δ𝑇),_/𝑓,𝑛=𝑅_𝑓,𝑛(1+𝛼_𝑓,𝑛Δ𝑇), (14)where 𝜎_𝑟^(max) is the maximum radial stress, thus evaluated at 𝑟_𝑜 and 𝑅_𝑖 (at 𝑟_𝑖 and 𝑅_𝑜 we have 𝜎_𝑟=0) and 𝑅^/_𝑓,𝑛 are the radii of the nanotube and fullerenes if assumed to be not interacting.

The cases of a (10,10) nanotube (𝑅_𝑛≈0.68nm) coupled with C60 (𝑅_𝑓≈0.35nm) or C82 (𝑅_𝑓≈0.41nm) fullerenes cooled from 290 K to 10 K are reported in Tables 1 and 2, respectively. Note that the diameter variation is very small, but stresses are huge for C82@(10,10), in the gigapascale range, but are vanishing for the C60@(10,10) peapod. Note the maximum circumferential tensile stress in the nanotube around 70 K of 43 GPa, whereas the circumferential compression in the fullerenes is of 13 GPa; the calculated contact pressure is of 19 GPa.

Table 1 

Computed values for C60@(10,10), thermomechanical stresses are vanishing.

Table 2 

Computed values for C82@(10,10), thermomechanical stresses are in the gigapascal range.

Considering different elastic constants (𝐸,𝜈) would correspond to slightly different values, whereas the thermal expansion coefficients and in general the radii, unfortunately not fully defined for an atomistic object (see (8) and (11)), play a dramatic role on the computed thermomechanical stresses. Thus the analysis is accurate in the procedure but the deduced thermal stresses must be viewed just as an example of calculation. Slightly changing the definition of the radii (e.g., considering a different value for 𝑡) we have found an intermediate behaviour in which the interaction fullerenes-nanotube vanishes only in a given temperature range. This suggests that fullerenes (other types of nanoparticles can be envisioned too) could be released from the nanotube by thermal activation, a perhaps useful concept for producing innovative two-stage nanovectors capable of smartly delivering the fullerenes/drugs by a remote thermal control.

4. Conclusions

According to our analysis, the thermomechanical stresses of a fullerenes at nanotube peapod can be tuned by varying the temperature in a controllable way. Tunable stiffness and band structure (metallic, semiconductor) of a nanotube could thus be achieved by embedding fullerenes on it and by controlling the temperature. The analysis suggests that smart fullerenes at nanotube peapods, such as two-stage nanovectors for drug delivery, could in principle be realized by remote thermal activation.

Acknowledgment

The author is supported by the ‘‘Bando Ricerca Scientifica Piemonte BIADS 2006’: novel biomaterials for intraoperative adjustable devices for fine tuning of prostheses shape and performance in surgery.