4. Nanotubes Under Hydrostatic Pressure 65
4.2. Elastic properties of carbon nanotubes
The relation between a macroscopic stressσi j applied to a material and the resulting micro-scopic deformationεkl is usually described with the help of the elastic constant tensor Ci jkl (repeated indices are to be summed over)103
σi j=Ci jklεkl. (4.1)
The symmetry of a crystal restricts the number of non-zero and linearly independent con-stants Ci jkl. For example in nanotubes obviously Cxxxx=Cyyyy, since the tubes are isotropic in the plane perpendicular to the z axis. The elastic constants can be measured by a variety of techniques like ultrasound experiments or direct measurements of the lattice constants under stress. Despite the large interest in the mechanical properties of carbon nanotubes, in particular as reinforcement materials,104 successful measurements of the elasticity constants have not been reported up to now. Young’s modulus E was investigated several times both theoretically and experimentally; it was found to be on the order of the graphite value.105–107 For some selected tubes elastic constants were calculated with an empirical force constants
Reference Normalized pressure dependence (TPa−1)
Semiconducting tubes 1592 cm−1 1570 cm−1 1555 cm−1
Venkateswaran et al.29 3.5 3.4 –
Thomsen et al.11 3.6 3.7 3.7
Peters et al.98 4.6 4.5 2.0
Teredesai et al.97 3.7 3.6 3.7
Venkateswaran et al.101 3.7 3.9 3.7
Metallic tubes 1592 cm−1 1565 cm−1 1544 cm−1
Reich et al.99 3.8 4.1 3.5
Table 4.1: Logarithmic pressure derivatives of the high energy modes reported in the literature. The ambient pressure frequencies in the headings of the columns differ somewhat in the references; the number represents a label rather than an absolute frequency. The data of Venkateswaran et al.29 were fitted by a straight line instead of the quadratic fit performed by the authors. Likewise, the two slopes by Peters et al.98 were replaced by a single straight line.
model by Lu.108, 109 Here I chose another way for finding the elastic response of nanotubes under pressure. I approximate a nanotube by a hollow cylinder with closed ends and a finite wall thickness made out of graphene. In addition to yielding the deformation of a nanotube under pressure this model provides nice insight into the problem whether the different elastic properties of graphite and carbon nanotubes follow plainly from the geometry of the tube or some additional enforcement, e.g., by rehybridization.
4.2.1. Continuum model
The starting point of the continuum mechanical description is the equilibrium condition110, 111
∂σi j
∂xk
=0, (4.2)
where xk are the normal coordinates. The generalized Hooke’s law in an isotropic medium is given by
σik= E 1−ν
ν
1−2νεllδik+εik
(4.3a) or – the inverse relationship –
εik= 1 E
(1+ν)σik−ν σj jδik
, (4.3b)
where E is again Young’s modulus andν Poisson’s ratio. The strain εik is defined by the displacement vector uuu describing the shift of a point in the deformed material
εik=εki= 1 2
∂ui
∂xk
+∂uk
∂xi
. (4.4)
Inserting Hooke’s law (4.3) into Eq. (4.2) the fundamental equation of continuums mechanics can be derived110
(1−2ν)∆uuu+grad div uuu=2(1−ν)grad div uuu−(1−2ν)rot rot uuu=0. (4.5) Since the rotation of uuu vanishes in our problem, Eq. (4.5) is further simplified, to yield, in cylindrical coordinates,
div uuu= 1 r
∂(r ur)
∂r + 1 r
∂uθ
∂θ +
∂uz
∂z =const. (4.6)
Fig. 4.2 shows the continuum approximation of a single or multiwalled nanotube. The tube is modeled as a finite hollow graphene cylinder of length l with closed caps and inner and outer radius Riand Ro, respectively. At z=0 the displacement uz=0 and increases continuously
Figure 4.2: Continuum mechanical model of a nanotube - a closed cylinder of length l with inner radius Riand outer radius Ro. The boundary conditions un-der hydrostatic pressure are indicated in the figure; A is the ratio between the cap area and the area supported by the cylin-der walls A=R2o/(R2i −R2o).
z= 0
- /2l l/2
Ro Ri
s( ) =Ro -p
s(± /2) =l -Ap s( )= 0Ri
in both directionsεzz=∂uz/∂z =const. The circumferential displacement uθ is independent ofθ, since our problem is rotationally symmetric, i.e., ∂uθ/∂θ =0. Finally, according to Eq. (4.6) 1/r·∂(r ur)/∂r is again constant. The strains εii in cylindrical coordinates are therefore given by11
εrr= ∂ur
∂r =a− b
r2, εθθ = 1 r
∂uθ
∂θ + ur
r =a+ b
r2, and εzz= ∂uz
∂z =const. (4.7) The constants a, b, and εzz are determined by the boundary conditions for σ under hydro-static pressure. I assume that the pressure media cannot enter the nanotubeσrr(Ri) =0 and σrr(Ro) =−p, where p is the applied pressure. Along the z direction the pressure transmits a force−p·πR2oto the caps of the tube. The area supported by the wall isπ(R2o−R2i)and henceσzz=−R2o/(R2o−R2i)·p=−Ap. Inserting the boundary conditions into Eq. (4.3a) I obtain the constants in Eq. (4.7)
a=εzz =−(1−2ν)pA
E b=−(1+ν)pA Ri
E . (4.8)
Reinserting the intergration constants into Eq. (4.7) finally yields the strain tensor in a nan-otube under external hydrostatic pressure p
εrr=−Ap E
h(1−2ν)−(1+ν)R2i r2
i εθθ =−Ap E
h(1−2ν) + (1+ν)R2i r2 i
εzz=−Ap
E (1−2ν). (4.9)
The mixed strain components εi j,(i6= j), vanish. The strain tensor in a nanotube under pressure is thus given by two elastic constants E andν and the cylinder geometry.
Before substituting typical values into Eq. (4.9) I briefly discuss the general implications of the strain tensor found by the continuum approximation. The change in the tube length or the translational periodicity along the z axis is described by εzz, while εθθ is the radial or circumferential deformation. Both components are always negative for positive pressure.
Moreover, the circumferential deformation is always larger than the axial deformation εθθ
εzz
=1+ 1+ν 1−2ν
R2i
r2 >1, (4.10)
as expected for an anisotropic system. Note that the differences are plainly a consequence of the cylindrical geometry; I assumed the nanotubes wall to be isotropic. The change in the wall thickness, εrr, can be positive or negative. In particular, it depends on r when going across the wall and at the inner radius εrr(Ri)>0. For isolated single walled nan-otubes a varying wall thickness is unlikely, since they consists only of one graphene sheet.
This unphysical result is omitted by choosing r0=p
(1+ν)/(1−2ν)Ri, i.e., εrr(r0) =0.
However, as long as reasonable values for r are considered, e.g., another natural choice is the mean value of Ri and Ro, εrr and εθθ do not depend very much on r in single walled nanotubes.
The strain components which are responsible for the experimentally observed frequency shift of the high-energy Raman modes are the circumferential and tangential strain compo-nents. As I discussed in Chapter 2. the high-energy vibrations are parallel to the nanotube’s wall. Hence εrr is negligible both for single and multiwall nanotubes. Consider now un-wrapping the tube to a rectangle; the strain along the narrower, circumferential direction is thenεθθ and the longer side is deformed according toεzz. With typical values for the radii and the elastic constants of single walled nanotubes (Ri=5.2 ˚A, Ro=8.6 ˚A, E =1 TPa−1, andν =0.14)11, 106I findεθθ(p) =−2.04 TPa−1p andεzz(p) =−1.07 TPa−1p. Within the elastic continuum model the ratio between the circumferential and the axial strain is 1.9.
From this two-dimensional strain pattern the change in phonon frequency follows from a linear expansion of the dynamical equation in the presence of strain. Before deriving the phonon slopes under pressure I compare the results of the elastic continuum model to other calculations of the elastic properties of carbon nanotubes.
4.2.2. Ab initio, tight-binding, and force-constants calculation
Two general approaches may be considered for theoretically finding the elastic properties of a material. Either the lattice constants under stress are calculated by directly incorporating the applied stress tensor into the calculation or the elastic constants are found from the second derivatives of the energy in a strained unit cell. The former approach was used in my ab initio calculation as well as by Venkateswaran et al.29 in tight-binding molecular dynamics;
the latter by Lu108, 109 in his force constants calculation.
In the ab initio calculations I considered three small diameter nanotubes, an armchair (6,6), a zig-zag (10,0), and a chiral (8,4) nanotube. Details on the computational scheme and the parameters are given in Appendix I.. The equilibrium lattice constant and the atomic posi-tions were first obtained under ambient pressure by a conjugate gradient minimization. Then
Figure 4.3: Ab initio calculation of the axial (closed symbols) and circumferential (open sym-bols) strain in single walled nanotubes bundles.
Circles refer to an (8,4), up triangles to a (6,6), and down triangles to a (10,0) nanotube. The full lines show a least square fit of the strain components in the three tubes; the broken lines are for the contin-uum approximation [see Eq. (4.9), r=4.05 ˚A].
0 1 2 3 4 5 6 7 8 9
0 2 4 6 8 10 12 14
εθθ = -1.5 TPa-1p εzz = -0.9 TPa-1p ab initio calculation
-Strain (10-3 )
Pressure (GPa)
I relaxed the geometry under the constraint of an hydrostatic stress tensor for several high-pressure points. The circumferentialεθθ= [r(p)−ra]/raand axial strainεzz= [a(p)−aa]/aa
are shown in Fig. 4.3; r(p)and a(p) are the stress dependent radius and translational peri-odicity, raand aa are the ambient pressure values. As in the continuum approximation the circumferential strain is larger than the strain along the nanotube axis. Moreover, the strain components are found to a very good approximation to be independent of the chiralities of the nanotubes, a parameter which is completely neglected within the continuum approxima-tion. The calculated pressure slopes of the radial and axial strain areεθθ=−1.5 TPa−1p and εzz=−0.9 TPa−1p (full lines in Fig. 4.3). This is in excellent agreement with the elasticity model for r=4.05 ˚A as can be seen in Fig. 4.3, where the strains obtained from Eq. (4.9) εθθ=−1.42 TPa−1p andεzz=−0.86 TPa−1p are shown by the broken lines. The similarity between the elastic continuum model and the first principle calculations is quite remark-able. The continuum mechanical approximation works well even in the limit of a single atomic layer. A similar good agreement was found in two-dimensional semiconductors, see Ref. 112.
Two other studies of the elastic properties of armchair tubes were reported in the litera-ture. Lu109calculated the elastic constants Ci jkl from force constants fitted to reproduce the phonon dispersion in nanotubes. From the elastic constants for a two-layer (10,10) tube in Ref. 109 the strain components are obtained with Eq. (4.1); they are given in Table 4.2. The strains in both directions are somewhat smaller than within the continuum model. The split-ting between the axial and circumferential strain is predicted to be 3.5. Venkateswaran et al.29 performed molecular dynamics simulations of (9,9) single tubes and bundles of tubes under pressure. The normalized axial and circumferential strain they reported in the bun-dle is shown in Fig. 4.4. The compressibility along the axis is similar to the continuum
Figure 4.4: Molecular dynamics simulation of the axial (closed circles) and the circumferential (open circles) strain under hydrostatic pressure. The data points are taken from Fig. 4 of Ref. 29. The cir-cumferential deformation shown here corresponds to the average of the two radii under pressure re-ported by Venkateswaran et al.29
0 1 2 3 4 5
0 5 10
15 εθθ = -3.41 TPa-1p εzz = -0.91 TPa-1p
molecular dynamics calculation
-Strain (10-3 )
Pressure (GPa)
value, whereas the circumferential strain was found to be much larger than in the continuum approximation.
Despite some differences in the absolute values of the predicted strains all four calculation agree in the following fundamental points: Under hydrostatic pressure the circumferential strain is by a factor of 2-4 larger than the axial strain. The linear compressibility along the nanotubes axis is similar to graphite (−0.8 TPa−1p).