# Thermomechanical Stresses in Fullerenes at  Nanotube

Authors: Nicola M. Pugno
Journal: Journal of Nanomaterials
 (2008)

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## 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 C82@(10,10) or C60@(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.


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## Body

## 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 (Figure1). 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 radiia and b subjected to inner and
outer pressures p and q, respectively. The radial σr and circumferential σϑ  stresses at a generic radius a<r<b  are [13](1)σr=1b3−a3(−b3q+a3p+a3b3r3(q−p)),σϑ=1b3−a3(−b3q+a3p−a3b32r3(q−p)). Linear elastic isotropic laws for
spherical symmetry become (2)εr≡dudr=σrE−2νEσϑ,εϑ≡ur=σϑE−νE(σϑ+σr), 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 lengthL) 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](3)σr=1b2−a2(−b2q+a2p+a2b2r2(q−p)),σϑ=1b2−a2(−b2q+a2p−a2b2r2(q−p)).  Linear elastic isotropic laws for cylindrical
symmetry become(4)εr≡dudr=σrE/−ν/E/σϑ,εϑ≡ur=σϑE/−ν/E/σr,E/=E,v/=νfor   plane   stress;E/=E1−v2,v/=v1−vfor   plane   strain.  We can assume plain stress condition
(i.e., σz=0) or plane strain condition (i.e., εz=0,
for which in addition σz=ν(σr+σϑ)). 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 temperatureT as a linear elastic isotropic sphere
(elastic properties with subscript “f”) having inner and outer radii ri and ro, respectively. The confinement from
the nanotube is described by the external pressure p. According to
(1)
and (2)(5)σr(r)=−1−ri3/r31−ri3/ro3p,σϑ(r)=−1+ri3/2r31−ri3/ro3p,utot(r)=u(r)+αf(T)ΔTr=[σϑ(r)Ef−νfEf(σϑ(r)+σr(r))+αf(T)ΔT]r,
where utot is the total (elastic + thermal) radial
displacement, αf 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
radiusro* (ro if unstressed) is(6)ro*=ro+utot(r=ro).Now consider a nanotube at temperatureT as a linear elastic isotropic cylinder
(elastic properties with subscript “n”) having inner and outer radii Ri and Ro, respectively. The confinement from
the fullerenes is described by the internal pressure p. According to (3)
and (4):(7)σr(r)=Ri2/Ro2−Ri2/r21−Ri2/Ro2p,σϑ(r)=Ri2/Ro2+Ri2/r21−Ri2/Ro2p,(8)utot(r)=u(r)+αn(T)ΔTr=[σϑ(r)En/−νn/En/σr(r)+αn(T)ΔT]r. Thus the nanotube elastic inner
radius Ri* (Ri if unstressed) is(9)Ri*=Ri+utot(r=Ri). The compatibility of the displacements (elastic
contact between nanotube and fullerenes) implies(10)ro*(p,ΔT)=Ri*(p,ΔT). 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 ri*(p*,ΔT)=ro*(p*,ΔT)−tRo*(p*,ΔT)=Ri*(p*,ΔT)+t,
where t is the shell thickness (0.34 nm); moreover, r0*(p*,ΔT)=Ri*(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 thicknesst and plane stress condition and both nanotube
and fullerenes composed by the same material, for which we assume Ef=En=E≈1⁢TPa, νf=νn=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 by(11)Rn=Ro+Ri2≈0.0392n2+m2+nm⁢nm,Rf=r=ro+ri2≈0.0458lnm, so that Ri=Rn−t/2 and ro=Rf+t/2 with t  = 0.3 nm
(close to the van der Waals spacing [14]). For ro>Ri mechanical stresses will be present.In addition, the thermal volumetric expansion coefficientβf of carbon fullerenes is [15]
(12)βf≈3αf(T)≈{−1×10−5T70T<70⁢K,1×10−5T−1508070⁢K≤T<400⁢K,
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:(13)αn≈−1.2×10−5T400T<400⁢K. Thus, according to these numerical solutions,
following the analysis reported in Section 2, we derive
(14)p=p*={E(Rf+t/2)(1+αfΔT)−(Rn−t/2)(1+αnΔT)Rn/t(Rn−t/2)+Rf/(2t)(Rf+t/2)=χ⁢ifχ>0(contact),0⁢ifχ≤0(nocontact),σrf,n(max⁡)=−p,σϑf=−Rf2tp,σϑn=Rntp,Rf,n*=Rf,n(1+σϑf,nE+αf,nΔT),Rf,n/=Rf,n(1+αf,nΔT),where σr(max⁡) is the maximum radial stress, thus evaluated
at ro and Ri (at ri and Ro we have σr=0) and Rf,n/ are the radii of the nanotube and fullerenes
if assumed to be not interacting.The cases of a (10,10) nanotube (Rn≈0.68nm) coupled with C60 (Rf≈0.35⁢nm) or C82 (Rf≈0.41⁢nm) 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.
T [K]αf [K-1]αn [K-1]ro/ [nm]Ri/ [nm]ro* [nm]Ri* [nm]2900.000006−0.0000090.5000000.5300000.5000000.5300002800.000005−0.0000080.4999730.5300450.4999730.5300452700.000005−0.0000080.4999500.5300860.4999500.5300862600.000005−0.0000080.4999310.5301240.4999310.5301242500.000004−0.0000080.4999170.5301590.4999170.5301592400.000004−0.0000070.4999060.5301910.4999060.5301912300.000003−0.0000070.4999000.5302190.4999000.5302192200.000003−0.0000070.4998980.5302450.4998980.5302452100.000003−0.0000060.4999000.5302670.4999000.5302672000.000002−0.0000060.4999060.5302860.4999060.5302861900.000002−0.0000060.4999170.5303020.4999170.5303021800.000001−0.0000050.4999310.5303150.4999310.5303151700.000001−0.0000050.4999500.5303240.4999500.5303241600.000000−0.0000050.4999730.5303310.4999730.5303311500.000000−0.0000050.5000000.5303340.5000000.5303341400.000000−0.0000040.5000310.5303340.5000310.530334130−0.000001−0.0000040.5000670.5303310.5000670.530331120−0.000001−0.0000040.5001060.5303240.5001060.530324110−0.000002−0.0000030.5001500.5303150.5001500.530315100−0.000002−0.0000030.5001980.5303020.5001980.53030290−0.000003−0.0000030.5002500.5302860.5002500.53028680−0.000003−0.0000020.5003060.5302670.5003060.53026770−0.000003−0.0000020.5003670.5302450.5003670.53024560−0.000003−0.0000020.5003290.5302190.5003290.53021950−0.000002−0.0000020.5002860.5301910.5002860.53019140−0.000002−0.0000010.5002380.5301590.5002380.53015930−0.000001−0.0000010.5001860.5301240.5001860.53012420−0.000001−0.0000010.5001290.5300860.5001290.530086100.0000000.0000000.5000670.5300450.5000670.530045Table 2
Computed values for
C82@(10,10), thermomechanical stresses are in the gigapascal range.
T
[K]ro/
[nm]Ri/
[nm]p [MPa]σrf,σrn [MPa]σϑf [MPa]σϑn [MPa]ro*  [nm]Ri*   [nm]2900.560000.5300018939.39−18939.39−12941.9242929.290.554690.559192800.559970.5300418892.14−18892.14−12909.6342822.180.554680.559162700.559940.5300918849.84−18849.84−12880.7242726.290.554660.559142600.559920.5301218812.49−18812.49−12855.2042641.640.554650.559122500.559910.5301618780.09−18780.09−12833.0642568.210.554650.559112400.559900.5301918752.65−18752.65−12814.3142506.010.554640.559092300.559890.5302218730.16−18730.16−12798.9542455.040.554640.559092200.559890.5302418712.63−18712.63−12786.9642415.300.554640.559092100.559890.5302718700.05−18700.05−12778.3742386.780.554650.559092000.559900.5302918692.42−18692.42−12773.1642369.490.554660.559101900.559910.5303018689.75−18689.75−12771.3342363.440.554670.559111800.559920.5303118692.03−18692.03−12772.8942368.610.554690.559131700.559940.5303218699.27−18699.27−12777.8342385.010.554710.559151600.559970.5303318711.46−18711.46−12786.1642412.630.554730.559171500.560000.5303318728.60−18728.60−12797.8842451.490.554750.559201400.560040.5303318750.69−18750.69−12812.9742501.570.554780.559231300.560070.5303318777.74−18777.74−12831.4642562.890.554810.559271200.560120.5303218809.75−18809.75−12853.3342635.430.554850.559321100.560170.5303118846.70−18846.70−12878.5842719.200.554890.559361000.560220.5303018888.62−18888.62−12907.2242814.190.554930.55942900.560280.5302918935.48−18935.48−12939.2442920.420.554970.55947800.560340.5302718987.30−18987.30−12974.6543037.880.555020.55953700.560410.5302419044.07−19044.07−13013.4543166.560.555080.55960600.560370.5302219033.19−19033.19−13006.0243141.910.555040.55956500.560320.5301919020.96−19020.96−12997.6643114.180.554990.55951400.560270.5301619007.37−19007.37−12988.3743083.360.554940.55946300.560210.5301218992.41−18992.41−12978.1543049.470.554890.55940200.560140.5300918976.10−18976.10−12967.0043012.490.554830.55933100.560070.5300418958.43−18958.43−12954.9242972.430.554760.55927Considering 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.



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*Source: 156724-2008-04-14.xml*