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Engineering materials properties in codimension > 0

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Abstract

When thin nanomaterials spontaneously deform into nonflat geometries (e.g, nanorods into nanohelices, thin sheets into ruffled forms), their properties may change by orders of magnitude. We discuss this phenomenon in terms of a formal mathematical concept: codimension c = D - d, the difference between the dimensionality of space D, and that of the object d. We use several independent examples such as the edge stress of graphene nanoribbons, the elastic moduli of nanowires, and the thermal expansion of a modified bead-chain model to demonstrate how this framework can be used to generically understand some nanomaterial properties and how these properties can be engineered by using mechanical constraints to manipulate the codimension of the corresponding structure.

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Acknowledgments

The authors gratefully acknowledge Prof. M. Haataja, Prof. Y.W. Zhang, Prof. S. Safran, Prof. C. Arnold, Dr. C.K. Gan, and Dr. C. Majidi for enlightening discussions and comments.

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Authors and Affiliations

  1. Institute of High Performance Computing, Singapore, 138632, Singapore

    Paulo S. Branicio, Mark H. Jhon & David J. Srolovitz

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  1. Paulo S. Branicio
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Correspondence to Paulo S. Branicio.

Appendix A: Calculation of Graphene Edge Energy and Stress

Appendix A: Calculation of Graphene Edge Energy and Stress

GNR edge energy and stress are calculated in a similar manner as Jun30 and Gan and Srolovitz.32 The edge energy (per unit length) Ee as a function of edge orientation a is defined as

$$E_\text{e} \left( \alpha \right) = \frac {1} {2L} \left[ E_\text{GNR}^N \left(\alpha \right) - NE_\text{B} \right],$$
((A1))

where \(E_\text{GNR}^N\left( \alpha \right)\) is the total energy of a GNR with N atoms, length L, and edge orientation a, and EB is the energy per atom in a bulk (infinite) flat graphene sheet. The factor of 2 in the denominator of Eq. (A1) accounts for the fact that a GNR has two (parallel) edges. For narrow GNRs, Ee(α) depends on the GNR width because of interactions between the edges. This effect decays rapidly with GNR width. Therefore, we perform simulations on GNRs of widths in excess of 100 Å for which Ee(α) reaches the asymptotic value for large widths.

For all edge types, Ee(α) was calculated by minimizing the total energy with respect to all atomic coordinates at T = 0. We employed the limited-memory Broyden–Fletcher–Goldfarb–Shanno49 method to relax the atomic structures (both in c = 0 and 1) until it converged to within 10–10 eV/atom. In the c = 1 case, noise was added to the initial atomic positions (i.e., perturbations in the atom positions from the flat structure) in order to break the symmetry of the problem.

We calculate the edge stress τe by straining the equilibrium (zero stress) GNR and interpolating the resultant system energy versus strain curve at ∈ = 0. For c = 0, both compressive and tensile strains were employed, for c = 1, where the GNR buckles out of plane for compressive strains, only tensile strains were applied in order to determine τe.

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Branicio, P.S., Jhon, M.H. & Srolovitz, D.J. Engineering materials properties in codimension > 0. Journal of Materials Research 27, 619–626 (2012). https://doi.org/10.1557/jmr.2011.306

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