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

Pradeep Sharma's picture

For many mechanicians and materials scientists one of the most confounding things (in the ever increasing literature on carbon nanotubes) is the reported theoretical value of the nanotube elastic modulus. Depending upon the specific paper at hand, the reported numerical values range from 1 -6 TPa! This scatter is known as “Yakobson’s Paradox” after Boris Yakobson (Rice University) who first discussed this. In a recent paper, Young Huang and co-workers address this issue and provide a resolution.

Atomistic calculations (whether empirical molecular dynamics or ab initio approach) provide only an estimate of the “tension rigidity” i.e. “Eh”, where E is the Young’s modulus and “h” is the thickness of the single walled nanotube or the graphene sheet. Most properly done atomistic calculations (from various sources) all seem to agree on the numerical value of Eh. There are many cases where there may be no need to know E. However, if a specific value is desired then an estimate of h is required to compute it. According to Huang et. al., therein lies the source for the wide scatter in reported values of E. If a thickness equivalent to that of graphite interlayer spacing is assumed, E turns out to be roughly 1 TPa. Afew works employ a shell model. In the latter case, one computes both the tension rigidity T=Eh/(1-v2), as well as the bending rigidity B=Eh3/12(1-v2). Their ratio leads to an expression for the thickness h=√12B/T. This exercise results in an estimate of 5-6 TPa for the elastic modulus. Interested reader can get the facts directly from the paper, however briefly the central physical insight is that due to the monolayer nature of graphene, torsion rigidity is zero and the ratio of the bending to tension rigidity is not a constant (depending in fact on the type of loading). This conclusion is in fact quite general for any material (and not just graphene) and will apply to all monolayers with hexagonal symmetry (which have thus ill-defined thickness). Some of these conclusions become quite transparent since Huang’s paper relates the relevant rigidities analytically to the interatomic potential.


Henry Tan's picture

Different authors defined the elastic modulus differently for nanotubes.

This is the reason for the reported data scattering.

MichelleLOyen's picture

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?

Zhigang Suo's picture

Michelle, questions well put. I can imagine that people define stress by force divided by the cross-sectional "area" of the tube, even though the exact area itself may have some fussiness. Then one may define modulus as the ratio of the stress and strain.

Of course, this modulus is useful if it predicts something. For example, if you find that for the same strain the force scales with the cross-sectional area. Then you can correlate the stiffness of single-walled tubes and multi-walled tubes, and tubes of different radii.

Now, your last question raises a point that also puzzled me when I first read Pradeep's post. I'd like to learn how the modulus of a carbon sheet can be several times the modulus of diamond.

Pradeep Sharma's picture

Michelle and Zhigang,

The questions that you both raise are essentially the focus of Young's paper. If there is not need to define a "modulus" like quantity that mechanicians are used to then there is no confusion.......only when we force the notion of elasticity on the graphene monolayer or carbon nanotube does the confusion come into play. As you will read in the paper (which well summarizes the data across different researchers) the different assumptions (shell model, thickness equivalent to graphite interlayer spacing etc.) lead to different answers.


Ying Li's picture

Dibiasio [1] pointed out that the bending modulus of the SWCNT could be larger than that of stretching as high as 19% because the different energy storage mechanisms in each mode and the instability of the tube model to capture these effects. It is not proper to use the torsional deforemation to estimate the thickness. And I think the Young's modulus could be defined as the ratio of nominal stress to nominal strain.

[1] DiBiasio C M. Cullinan M A and Culpepper M L. 2007 Appl. Phys. Lett 90 203116-8

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 .

Pradeep Sharma's picture

I am not sure I understand what you mean by "defective structure". Topologically speaking, a sheet can be deformed into a cylinder without admitting any defects (unlike a sphere which requires presence of a set of disclinations---e.g. buckyball).

MichelleLOyen's picture

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. 

Henry Tan's picture

"“Young's modulus" or "elastic modulus" are originally defined to express the constitutive relation between stress and strain. To define a modulus for nanotube, we first must be very clear about what are the stress and strain we are talking about.

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

Yonggang Huang's picture

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.   …”.

Marino Arroyo's picture

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 


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.

ericmock's picture

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.

Yonggang Huang's picture

This is a reply of Eric Mock's question or comment "Why this requires an entire journal publication to rectify is beyond me".

The main purpose of this paper is not on the "Yakobson's paradox".  Instead, it is just a byproduct.  It is absolutely correct that we just need the tension and bending rigidities of graphene and carbon nanotubes, not their Young's modulus and thickness.  However, one still has to do atomistic simulations (e.g., molecular dynamics) to get the tension and bending rigidities for a chosen interatomic potential.  The purpose of this PRB paper is to bypass all atomistic simulations and get the tension and bending rigidities directly from the interatomic potential -- no calculations needed at all (except to use a calculator to get values of some sin, cos, exp functions).  For any interatomic potential (not limited to Brenner potential), the analytical expressions of the tention and bending rigidities are given in terms of the interatomic potential.  In other words, one does not need to do atomistic simulations to get the tension and bending rigidities.  Just give me the interatomic potential and one can get the tension and bending rigidities analytically.



Marino Arroyo's picture

Dear Yonggang,
I have been going through your PRB paper. Sometime ago (PHYSICAL REVIEW B 69, 115415 2004), we also derived analytical expressions for the elastic moduli of graphene, in particular the bending rigidity, in terms of the potential. Unlike the well known in-plane moduli, I think that such expressions for the bending rigidity cannot be found in the literature, and are very useful (as you say, no atomistic calculation is needed to get a modulus, avoiding the error-prone numerical differetiation). We checked our formula against atomistic calculations, as described in the paper.
A first look at your formula in p 5 seems to indicate that it does not coincide with that in our paper (Eq. 23 for a bond order potential). I was wondering if you had computed the numerical value, say for Brenner's potential, so that we can compare.

ericmock's picture


Sorry for seemingly dismissing the importance of your paper. I did not realize "Young Huang" in Pradeep's original post was you when I replied. If I had made the connection I would have downloaded the paper and carefully read this entire thread before replying.

The reply also stems from a great deal of frustration with the shear volume of papers that are being published. I thought yet another person was jumping into this issue. I see so much garbage being published (and submitted). It seems like without experimentalists being able to do detailed experiments to validate models, it's easy for many theorists to jump on the nano-bandwagon without having a sense of what has been done. As I know you know the history (and were indeed part of it), I would not have assumed this was just another paper with no new insights.


Tienchong Chang's picture

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.


Marino Arroyo's picture

I did not know the RSPA paper you refer to, but the issue you report about the shear moduli is most likely related to the discrepancy between Eqs (18) and (19) in the PRB paper I referred to in my previous post. 

Tienchong Chang's picture

Hi Marino,

I did not notice your PRB2004 work, although we referenced your AIAA2002 and PRL2003 works. It seems that the results given by you and us are not the same thing.

Eqs (18) and (19) in your paper seems to show that the shear-tension relation you obtained does not consist with that from CM for a GRAPHENE SHEET. (Am I right?) However, our results showed that the relations from molecular mechanics model and from CM are not the same for CNTs, but identical for graphene sheets. The curvature effect is the main reason lead to the discrepancy for CNTs.



Marino Arroyo's picture

I misunderstood your paper. Now I see what you mean, that the moduli are not constant with deformation (curving the sheet is imposing a deformation). And yes, I was talking about the moduli around the planar graphene configuration (in which the elasticity is isotropic, this is not the case for finitely curved sheets). What I was mentioning is that the usual relatioships between the Lamé coefficients and Y and nu valid for bulk materials need to be carefully modified for a 2D continuum.

ericmock's picture

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?

Yonggang Huang's picture

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.

Henry Tan's picture

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.

zhanchun Tu's picture

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]. 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.

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