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Nonlinear mechanics of single-atomic-layer graphene sheets
Qiang Lu and Rui Huang
Department of Aerospace Engineering and Engineering mechanics, University of Texas, Austin,
TX 78712, USA
Abstract: The unique lattice structure and properties of graphene has drawn tremendous interests recently.
By combining continuum and atomistic approaches, this paper investigates the mechanical
properties of single-atomic-layer graphene sheets. A theoretical framework of nonlinear
continuum mechanics is developed for graphene under both in-plane and bending deformation.
Atomistic simulations are carried out to deduce the effective mechanical properties. It is found
that graphene becomes highly nonlinear and anisotropic under finite-strain uniaxial stretch, and
coupling between stretch and shear occurs except for stretching in the zigzag and armchair
directions. The theoretical strength (fracture strain and fracture stress) of perfect graphene lattice
also varies with the chiral direction of uniaxial stretch. By rolling graphene sheets into
cylindrical tubes of various radii, the bending modulus of graphene is obtained. Buckling of
graphene ribbons under uniaxial compression is simulated and the critical strain for the onset of
buckling is compared to a linear buckling analysis.
This work has been submitted for review, and the manuscript is attached.
Update on September 7, 2009: This manuscript has been published as: Q. Lu and R. Huang, Nonlinear mechanics of single-atomic-layer graphene sheets. Int. J. Applied Mechanics 1, 443-467 (2009).
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summary of findings
RH
A comment on your paper
Dear Qiang, Rui,
I have read with interest your paper. Since the conclusions partially refer to some of my previous work, I wanted to share my view with you. The discussion you include in the paper regarding whether the bending modulus should be probed with or without relaxing the radius of the tube is an interesting one. You suggest that this might contribute to the factor of 2 difference you get on the bending modulus as compared to our work and to the work of others. From my experience and understanding, it is not the case. This is only a minor effect. Figure 3 in reference 26 compares indeed fully relaxed tubes with the analytical formula that we obtained for the bending modulus in 26, later published by others in 27. Similar calculations were performed with "rigidly bent" atomistic models, only finding slight differences. On the other hand, simple nonlinear elastic models with three elastic parameters, one being the bending stiffness reported in 26, have been tested against atomistic calculations, with excellent agreement for moderate deformations, before nonlinear material effects kick in (Arroyo and Belytschko, Meccanica, 40:455-469 (2005), available at http://www-lacan.upc.es/arroyo).
For these reasons, I am puzzled by Table I in your paper. While the in-plane moduli all agree, we obtain (before conversion to the awkward units in 26) 0.13 nN nm for the bending modulus, far apart from your calculation. Have you tried to either fully relax the atomistic models of CNTs of different radii, or prevent any relaxation at all while rolling?
Best regards,
Marino Arroyo
bending modulus of graphene
Dear Marino,
Thanks for your comment. This is indeed a big question I have as well. I would leave to Qiang to address the details of his MM simulations. Just as you suggested, Qiang is working on simulations that relax in both the circumferential and axial directions of CNTs. We would like to see how much difference it could make in terms of the strain energy. The paper you mentioned will be helpful.
There is one question I would like to ask you as well. When studying your paper (Ref. 26), I am a little puzzled by the term b' in the equation for the bending modulus. As we know, the bond order parameter (b) depends on more than just one bond angle. When taking derivatives, does it only consider one of the bond angles? In the appendix of your paper, there was a summation over all bond angles before the final equation was reached. Unfortunately I was not able to follow through the last step of the derivation, and thus your help would be greatly appreciated.
One more thing to point out. I am new in molecular simulations or atomistic modeling in general. I felt suspisious about the MM results at first, but later gained some confidence in it by looking at the buckling results (Fig. 12). The MM simulations of buckling may serve as an independent check of the bending modulus, if you trust the linear analysis at least for the critical strain. I hope through this discussion we can all understand this problem better.
Regards,
RH
RE: bending modulus of graphene
Dear Rui,
I can help with the derivation in the appendix. Indeed, In eq B7 there is a summation on the three inequivalent bonds and the three inequivalent angles. The sum over the bonds vanishes at equilibrium precisely because of the equilibrium equations that determine the equilibrium bond length. In the sum over the angles, at equilibrium, one can factor out from the sum \partial W / \partial \theta_i since all angles are equal in this configuration. We are left with this factor times the second object in Eq B3, from which the result follows.
Best regards,
Marino
counting bond angles
Dear Marino,
Thanks for your help with the derivation. Now I seem to understand that the summation over the three bond angles results in a factor of 2 in the second equation of (B8) in your PRB paper, because each of the bond angle must be counted twice for the three bonds in the unit cell. However, if you use the hexagonal unit cell shown in Fig. 2, you would have six half-bonds, equivalent to three full bonds. In calculating the bond energy, each C-C bond is associated with four bond angles since B_bar = (B_ij + B_ji)/2. Therefore, the three full bonds in the unit cell would involve 12 counts of the bond angles, four for each inequivalent bond angle. If I am not mistaken here, the factor of 2 in the second equation of (B8) should be a factor of 4 instead? Please correct me if I am wrong. Thanks.
RH
RE: counting bond angles
Dear Rui,
As you say, the unit cell contains 3 inequivalent bonds and three inequivalent angles. The factor of 2 in Eq B8 comes from looking at Eq 7. Note that i,j,k is an even permutation of 1,2,3 (this is the way we wrote the Brenner potential once one assumes the bond network is fixed). Thus, the dependence of this expression on say \theta_1 shows up when i=2 and i=3. Once differenciation has been done, we evaluate at the graphene equilibrium cofig where all angles and lengths are equal, and hence we have twice the same term. Hope this helps.
Marino
why even permutation only?
Dear Marino,
It becomes clearer now that the difference may lie in our understanding of Brenner's potential. In my understanding, i, j, k do not have to be an even permutation. Therefore, \theta_1 appears twice in B_bar (j = 1 and k = 1) for each i (=2 or 3). Unless you modified the original definition of B_bar in the Brenner's potential, for each C-C bond ij, there are four nearest neighbours (say k = 1-4) that give four bond angles: ij1, ij2, ji3, ji4. Among the four angles, two are inequivalent, i.e., ij1 ~ ji3 ~ theta_1 and ij2 ~ ji4 ~ theta_2. Therefore, if B_bar' is the the derivative of B_bar w.r.t. theta_1, it is twice of the derivative w.r.t. each individual angle ijk. In this regard, I agree that Eq. (23) in your PRB paper is correct, but the value in your Table I seems to account for only one of the four bond angles and thus should be doubled. Make sense?
Thanks.
RH
the brenner potential is rewritten
Dear Rui,
In the PRB paper, the Brenner potential is written in a way adapted to the continuum treatment based on the exponential Cauchy-Born rule. In this setting, you can easily check that B_ij = B_ji = \bar{B}_ij because of the symmetries introduced by the kinematic rule relating continuum and atomistic deformations. Therefore, for each bond ij, B_ij depends on two angles, the angles \theta_ijk in the notation of Brenner, wich in the setting of our theory coincide with the angles looking at the bond as ji. The \bar{B} function of our paper is this function with two arguments (for moderate deformations, it does not depend on bond lengths). I agree that this is a specialization of the potential, but the relevant one for the elastic moduli around the graphene configuration. In taking derivatives of \bar{B} in our paper, these matters should be carefully accounted for. Of course, in the fully atomistic code that we used to check the results, the Brenner potential without any extra assumption was implemented.
In any case, the function \bar{B} and its derivatives also appear in the expressions for Young's modulus and the Poisson ratio, and there we have no differences, so I do not think that this is the source of the disagreement.
Best regards,
marino
modified Cauchy-Born rule
Dear Qiang, Rui
Interesting work. After reading your paper, I have some questions. First I noted that the Green-Lagrange strain tensor E and a curvature tensor K are defined, however, I did not find how these tensors are related to the variations of the bond lengths and angles. I try to understand the paper as follows: the stress tensors can be obtained via atomistic simulations under the prescribed strain through a incremental progress, and the incremental relationships are given by Eq (17) (18) form which we can get the tangent modulus. If this is the case, I would understand your ideas.
And then I have another question, only using deformation gradient tensor F to link the lattices before and after deformation for atoms structure is just the standard Cauchy-Born rule, which would fail when decribing the inhomogeneous deformation of 2D membrane atom structure in 3D space, and this has been pointed out by many authors. To overcome this, Arroyo and Belytschko proposed the exponential Cauchy-Born rule and later Guo etal used higer Cauchy-Born rule to investigate the mechanics of CNTs. I want to know that have you ever considered this effect.
To be honest, I did not follow your paper completely, if I have anything wrong, please let me know.
Best regards
Teng zhang
Re: modified Cauchy-Born rule
Teng: Let me try to answer your questions one by one.
(1) The Green-Lagrange strain E and curvature K are defined for a continuum sheet, without referring to the atomistic structures. Their relation to the bond lengths and bond angles of graphene has been established in previous studies, e.g., Ref. 27 for the case of infinitesimal deformation. In the present work, we do not need such an explicit relationship.
(2) The stress/moment tensors are calculated either from an energy method or from viral stresses based on molecular mechanics simulations under prescribed strain, but not necessarily through an incremental progress. For each strain the MM simulation is independent of any previous steps. Once a stress-strain curve is obtained, the tangent modulus is determined from the incremental relationship (17) and (18).
(3) For the Cauchy-Born rule, we realize that the deformation gradient F is inhomogeneous for bending of a graphene sheet, thus violating the basic assumption of homogeneous deformation in the standard Cauchy-Born rule. I am not certian how the modified Cauchy-Born rules handle inhomogeneous deformation. In the bending simulations of present study, we bypass this difficulty by prescribing bending curvature from a continuum point of view and determing the bending energy from MM calculations. Since the bond lengths and bond angles are calculated based on the atomic positions in the MM simulations, we do not need direct mapping from bond to bond.
RH
Thanks prof. Huang
Thanks prof. Huang. I have a clearer understanding of your paper now.
Teng zhang
The difference in bending stiffness is due to the dihedral term
Thanks to Marino, I have looked at the reason for the difference in our bending stiffness calculation. The reason why we have a higher bending stiffness is due to the dihedral term. This dihedral term does have a significant contribution to the bending stiffness. However, in Ref. [26], apparently, this dihedral term was ignored.
I have written a short document showing the contribution of the dihedral term to the bending stiffness. I don't know how to attach a file with a comment. So please take a look at here
http://imechanica.org/node/4249
I received great help from Dr. Huang and Marino. Thank you very much.
effect of dihedral angle on bending modulus of graphene
Qiang seems to have convinced me that the difference between the bending modulus of graphene from the present molecular mechanics calculations and that from the analytical solution in the previous studies (Arroyo and Belytschko, PRB 69, 115415, 2004) comes from the term associated with the dihebral angle in the second-generation Brenner potential for the carbon-carbon bond. Such a term does not exist in the first-generation Tersoff-Brenner potential, and thus no difference in the bending modulus there. However, reading Brenner's paper (J. Phys. 14, 783-802, 2002) again, I am not sure whether the dihedral term should be included for graphene. Quoting from that paper, "The value of the second term b_{ij}^{DH} depends on the dihedral angle for carbon-carbon double bonds." and "This function was parameterized such that for carbon-carbon bonds that are not double bonds, this contribution to the bond order is zero." With very limited knowledge in chemistry, my question is then: Are the carbon-carbon bonds in graphene all double bonds? In Arroyo and Belytschko (PRB 2004), they discarded the dihedral term in their derivation of the bending modulus, although they did point out that the derivative of the dihedral angle with respect to the bending curvature is not zero. Qiang's derivation shows exactly this derivative led to the difference in the bending modulus.
RH
Suggestions from one of the coauthors of REBO
I have discussed the issue of dihedral term with Prof. Sinnott at University of Florida. Her opinion is that the function of Tij in the dihedral term was parameterized in the way that it automatically characterizes the effects of single bond/double bond. So in the calculation it should always be there.
This is true for single bond. If you try to put diamond in the calculation, Tij is zero. And for graphene, according to Prof. Sinnott, the delocalized bonding effect is considered in the potential. So we should let the dihedral term work as it was designed.
RE: effect of dihedral angle on bending modulus of graphene
Thanks Qiang and Rui for the discussion. It seems that indeed, the difference is in the dihedral contribution, but in my opinion the potential being considered does not have such contribution. In our original paper in 2004, we considered the 1st generation Brenner potential, which does not depend on dihedral contributions, and also on the 2nd generation Brenner potential, which in our view does not depend on dihedral angles either. I must admit that I am not 100% sure about this, because Brenner's paper is not very clear. But it seems to me that from the values in Table 5 and the fact that T_ij is described using cubic splines, the dihedral contributions actually vanish for graphene and moderately deformed graphene. This is why in our paper, we did not go through the nice derivation of Qiang for the relation between the curvature and the dihedral angles. Of course, this derivation is very relevant for general potentials that do have dihedral contributions.
marino
dihedral contribution for graphene and CNTs
Dear Marino,
Thank you for following up on this discussion, from which I learned a lot more about the Brenner potential. I agree with you that the original paper by Brenner et al (2002) was not very clear about the dihedral term, and thus my question above about the C-C bonds in graphene (same question for CNTs too). Apparently the potential was not intended for graphene (or graphite, CNTs) as in the paper diamond and carbon-hydrogen bonds were considered, but the authors seem to suggest that the potential is applicable for other C-C bonds including those in graphite. The question is then whether the dihedral term contribute to the bond energy and mechanical properties of graphene . My understanding is that the function T_{ij} is not zero for graphene but the dihedral angle is zero for a perfectly flat graphene sheet. Consequently, the dihedral term does not contribute to the bond energy of flat graphene, neither does it contribute to the in-plane modulus of graphene. However, bending of graphene introduces non-zero dihedral angles, and as the derivative of the dihedral angle with respect to the bending curvature does not vanish even for an infinitesimal bending from the ground state, the dihedral term does contribute to the bending modulus of graphene. By this I assume all the C-C bonds in graphene can be considered as double bonds due to delocalization of the p-electrons. This may also lead to interesting questions about single-wall CNTs: how would the dihedral contribution affect the bond energy and mechanical properties of CNTs?
RH