Why is the reported elastic modulus of carbon nanotube so scattered? “Yakobsons Paradox” and Perspective from Huang et. al.

Different authors defined the elastic modulus differently for nanotubes.

This is the reason for the reported data scattering.

I think this is interesting and not at all unrelated to the discussion we are having for journal club in the context of biological molecules. So here's the question:

Is it ever appropriate to quote an elastic modulus for a carbon nanotube?

A nanotube is a structure, not a material. I can draw the analogy to a copper pipe, same geometry. I can define the stiffness of the pipe, but I can't sensibly refer to that as modulus; I can however define and measure the modulus of elasticity of the copper of which the pipe is constructed.

Besides which, another devil's advocate question. Is it really likely that a carbon nanotube "modulus" would be higher than that for diamond?

Strictly speaking, the fabric of a nanotube is not continuum and strain is not uniquely defined as with ordinary interfaces where stress is continuous. A strain component is a function of derivatives of a displacement field. In case of nanotubes (as with graphite sheets, molecules, etc), these derivatives are not single-valued – they depend on how one approaches to the point at which the strain is evaluated. But we have something called engineering stress and it is useful/relevant since, while the point-wise strain definition doesn’t make much sense in nanotubes, the averaged effect resembles strain (deformations) and stress. I think when people are measuring “Young’s modulus,” they are basically characterizing the effect of the defective structure of a nanotube (deviation from a perfect structure) – and since no two defects are identical, there is a spread in “Young’s moduli” measurements .

I agree completely, especially at the point at which you have put "Young's Modulus" in quotation marks.  This is the crux of the point I was trying to make; something is being measured here, it is a stiffness certainly, but it is NOT by definition a "Young's modulus" or "elastic modulus" and as such should not under any circumstances be called as such.  "Effective modulus" is about the best I would give it, although I would be happier with "specific stiffness" or some other term that makes it abundantly clear that we are not in a regime that adheres to the continuum definition of elastic modulus. 

I agree with Michelle Oyen - we need some new terminology for mechanics in nano- and sub-nano-scales. "Specific stiffness" is one good term, another option would be "non-local specific stiffness" since strain is not single-valued in this length-scale (point-wise), but its averaged effect in one length-scale level up is.

As for Preedep Sharma's question, from the SEM/TEM images of carbon nanotubes that I have seen, it appears that there are a lot of "dislocations/irregularities" in lattices, and defects always (?) appear to be present. I'd be interested in knowing if there are some recent manufacturing techniques to make them perfect now.

I focus/limit my discussion on/to strain since stress, as we know it, is derived from strain. I haven't had an opportunity to discuss this with anyone so far, but it seems mechanical force is not even "a universal force" as defined in Physics (the strong nuclear force, the electromagnetic force, the weak nuclear force, and the gravitational force).

Cetin Cetinkaya

Photo-Acoustic Research Lab

Clarkson University

Zhigang and others,

Here are my thoughts on "how the modulus of a carbon sheet can be several times the modulus of diamond".  Basically, it is NOT true.

Let's focus on graphene, a planar layer of carbon atoms.  Such a layer has no thickness.  Therefore one can only define the tension rigidity by the ratio of force to displacement (where the force is for a unit length).  To draw an analogy to a layer of thickness h and Young's modulus E, this tension rigidity is  Eh.  One cannot get E unless the thickness h is known.

For graphene (or carbon nanotubes), the tension rigidity is well defined.  All experimental data and atomistic simulations give (approximately) the same tension rigidity.  However, some researchers want to define the Young's modulus E.  Then the question is "how do we define the thickness h?"

There are two groups of people who are arguing about this (even today).  One group simply takes h as the interlayer spacing of graphite, 0.334nm.  There is no clear physical reason behind this except to say that such a concept works well for multi-wall carbon nanotubes.  The resulting "Young's modulus" is around 1TPa, which is on the same order as diamond or graphite.

The other group argues that E and h can be defined from the linear elastic plate theory.  For a sheet of thickness h, the tension rigidity is Eh, and the bending rigidity is EI=Eh^3/12.  If somehow the tension and bending rigidities are known, then the thickness can be defined from the ratio of bending to tension rigidities by h=sqrt(12*EI/Eh).  Many have used the atomistic simulations to calculate the tension and bending rigidities of a graphene and then use this approach to get the thickness and Young's modulus.  The Young's moduli obtained are very scattered, ranging from 3 to 6 TPa (and are much higher than diamond).  This big discrepancy (1TPa from the first group versus 3~6TPa from the second group) is called the Yakobson's paradox.

This PRB paper shows that the above approach to define the graphene (and carbon nanotube) thickness and Young's modulus is not correct, or at least that it does NOT give to a unique value for the thickness.  The tension rigidity and bending rigidity are obtained analytically from the interatomic potential in the PRB paper.  Their ratio, however, is NOT a constant, and depends on the type of loading.  For example, the ratio of uniaxial bending rigidity to uniaxial tension rigidity is DIFFERENT FROM the ratio of equi-biaxial bending to tension rigidities.  On the contrary, the classical linear elastic theory always give the SAME bending/tension rigidity ratio for all types of loadings.  This is the main reason for the large scattering of Young's modulus reported from the second approach.

Why does the bending/tension rigidity ratio depend on the type of loading?  This is because, as shown analytically in the PRB paper, the graphene has a vanishing torsion rigidity.  More comments on this and its physical reason will be posted in the next comment.

In summary, the tension and bending rigidities of graphene (and also carbon nanotubes) are well defined, but not the thickness.  Some people try very hard to come up with ways to define the thickness (and Young's modulus), which then lead to this high modulus up to 6TPa.  So how should one define the Young's modulus?  Who cares?  We don't need it (for graphene and carbon nanotubes).  The tension and bending rigidies are good enough.

As a follow-up to Dr.Huang et. al.’s recent paper, as well as Dr. Sharma’s comments on the discussion as later referred as "Yakobson Paradox", it is interesting to refresh us with what we had described in a paper by Yao and Lordi (Journal of Applied Physics 84, 1939 (1998)).  It reported that “… In general, we find Young’s modulus of a carbon nanotube to be roughly the same as that for in-plane graphite,~1T Pa (E. J. Seldin and C. W. Nezbeda, J. Appl. Phys. 41, 3389 (1970)), but trends indicative of strain effects and interlayer interactions are evident. Theoretical calculations using a continuum shell model have predicted Young’s modulus as high as 5 T Pa (B. I. Yakobson, C. J. Brabec, and J. Bernholc, Phys. Rev. Lett. 76, 2511 (1996), Overney, W. Zhong, and D. Z. Toma´nek, J. Phys. D 27, 93 (1993)). In these calculations a wall thickness of 0.66 Å, the p orbital extension, was used. Since Young’s modulus is proportional to wall thickness, these high values are actually consistent with our calculations, which use a layer spacing of 3.4 Å as the wall thickness. However, since the shell model does not take into account differences in atomic structure among tubes, variations in Young’s modulus were not observed. MD simulations allow calculations at the atomic level and can account for structure as well. Figure 3 shows how Young’s modulus changes with diameter and helix angle, and the relation to the torsional strain energy described above.   …”.

I would like to add to this interesting discussion another view on the elastic moduli of carbon nanotubes. The notion of elastic modulus is tied to a model of the system as a continuum. The most prevalent view has been to model graphene and nanotubes as thin elastic shells, hence the need to define a thickness, the inconsistencies, the paradoxes and the remedies. However, one can model the two-dimensional arrangement of atoms simply as a surface without thickness. Given that the potential energy only depends on the positions of this 2D (curved) lattice (Born-Oppenheimer assumption), there is no mechanically natural thickness. Such a model has been described in 

Arroyo M, Belytschko T

An atomistic-based finite deformation membrane for single layer crystalline films 

JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 50 (9): 1941-1977 SEP 2002

In another paper, 

Arroyo M, Belytschko T

Finite crystal elasticity of carbon nanotubes based on the exponential Cauchy-Born rule 

PHYSICAL REVIEW B 69 (11): Art. No. 115415 MAR 2004

the issue of the elastic moduli is described with more detail. A full treatment for the elastic moduli of a 2D continuum for graphene is provided. The surface Young's modulus (units of force per unit length), the surface Poisson's ratio and the bending modulus are derived in terms of the atomistic potential. It is shown that the linearized elastic response of graphene to bending is isotropic (as is well known for the in-plane response). In summary, I think this paper shows that a zero-thickness surface model is a natural one for curved 2D lattices, and no mechanics is lost in loosing the thickness.

I really do not see where there is any paradox whatsoever and am curious as to why such a big deal is being made of it. It's really a sophomore level strength of materials issue of material versus structural stiffness. The material stiffness of a nanotube really only has any meaning if the tube is multi-walled. You can only pack the walls so close together (~graphite inter-layer spacing). Doing this calculation will give you a modulus of about 1 TPa, about the same as a graphite crystal in the 'stiff' direction. Thus, if you were to make a bulk material from nanotubes, you would expect a modulus of about 1 TPa since you can only pack them so close together.

You can also view a single single-walled nanotube as a structure with some structural properties. In the simplest case, there are two structural constants--the extensional stiffness, C, and the bending stiffness, D. As Marino and Yonggang said, there is really no reason to go any further to try to back out E and h based on the C and D you get from deriving shell theory from 3D elasticity. There is no thickness with which to integrate through. In fact, nanotubes mechanics seems to be the best argument for using, so-called, direct shell theories (ala Green and Naghdi). If you do try to back out E and h from C and D, you get a much larger E. Why would anyone do this? Probably because some finite element package wanted them to enter E and h for it's shell elements. (Although Poisson's ratio should be in there too and I have seen values from ~0.1 to ~0.4 reported for it.)

Why this requires an entire journal publication to rectify is beyond me.

Continuum theory is frequently used to model CNTs because it may lead to concise solutions of problems we considered. This seems the origin why one needs to have a thickness for a SWCNT. However, this parameter is unnecessary in a shell model (e.g. by Ru) or a membrane model (e.g. by Arroro), except that a classical FE procedure is needed (as said in ericmock's post).

In my opinion, a specific value of the effective thickness of a SWCNT has very limited meaning because different physical requirements would result in different values for a same tube. We have briefly discussed this in some of our papers, e.g. JMPS2003 and PRB2006.   

One of important contributions of Professor Huang's paper, I think, is explicitly show how the ill-defined tube thickness is extracted from classical relationships of continuum mechanics (CM).  

In fact, some of CM relationships may be no longer retained at nanoscale. As an example, we show in a RSPA paper by a molecular mechanics model that the elastic modulus and shear modulus of a SWCNT can not be related by the classical CM relationship.

As a followup, I would like to point out that the vast majority of the molecular potentials people are using to find material/structural properties are not at all accurate for bending. These include both Brenner potentials (Tersoff and REBO) and MM3. In all cases, the bending stiffness predicted by the molecular potential is off by about a factor of two from what quantum mechanics (DFT) says it should be.

The reason for this is fairly obvious when one investigates how these potentials 'create' bending stiffness and the applications for which they were developed. First, bending stiffness results in these potential from next nearest neighbor interactions. However, the next nearest neighbor interaction model was not fit to bending stiffness measurements/calculations. They were included to essentially model how equilibrium bond lengths change when bond angles change. Depending on how delocalized the wave function is, one could include longer and longer range interactions and this is often necessary for other molecules.

Now, if one thinks about how bending stiffness in a graphene sheet arises, it is a completely different mechanism than is being modeled in standard C-C potentials. In graphene, the bending stiffness is basically a result of distorting the out-of-plane p-orbitals (and pi bonds). On one side of a bent sheet they are compressed and on the other they are stretched. Thus, in my opinion, better potentials that attempt to model this should be developed. Hopefully, they will predict the bending modulus to within better than a factor of two.

Anyone interested in writing a paper presenting a modified version of, say, REBO to better predict bending resistance?

The purpose of the PRB paper is not to evaluate any interatomic potential (Brenner potential br REBO).  It is to link the tension and bending rigidities analytically to any interatomic potential.  If the potential is not accurate, then of course the predicted tension and bending rigidities are not either, and neither is atomistic simulations.  The point of this paper is to present analytical method that bypasses the atomistic simulations to obtain the tension and bending rigidities directly from any interatomic potential.

The paper has a direct implication on the continuum modeling of carbon nanotubes as thin shells.  It is shown in the paper that the bending rigidity is NOT a constant but depends on the type of loading.  For example, the torsion rigidity is zero, and so is the bending rigidity when the graphene is subjected to curvatures k11=-k22.  This implies that one cannot assume a universal nanotube thickness for all loadings.  If one really wants to use a thin shell model for carbon nanotubes, the thin shell thickness may depend on the loading, i.e., may take a different value when the shell is subjected to bending or internal pressure or tension.

Just like Eric pointed out, "There is no thickness with which to integrate through", which is the reason that the graphene has a vanishing torsion rigidity and also vanishing bending rigidity for curvatures k11=-k22.  One may be curious that how come the bending rigidity does not vanish for equi-biaxial curvatures k11=k22.  The reason is that the interatomic intential is a multi-body potential.  For a pair potential the bending rigidity would also be zero.

Has anyone considered the effect of quantum confinement for nanotubes? Because the thickness of the graphene sheet is less than 1 Å, which approaches the Bohr excitation radius.

I support Arroyo's viewpoint that the concepts of E, v, h are the concepts for a solid shell with 3D isotropic materials. Graphene or single-walled carbon nanotubes are 2D objects. If they can be regarded as 2D isotropic, then there should be four elastic parameters to describe their mechanical properties as a result of the symmetric argument [Please see Eq.(123) in http://arxiv.org/abs/0706.0001]. Using these four parameters, we can avoid defining E, v, h. However, we can still recover the previous theoretical or numerical results on the instability behavior of carbon nanotubes.

Therefore,  Yakobsons Paradox can be solved by abondoning the concepts E,v,h and replacing them with four elastic parameters.