Supplemental Material for Synthesis and mechanical behavior of nanoporous nanotwinned copper

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Marylou Bradford
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1 Supplemental Material for Synthesis and mechanical behavior of nanoporous nanotwinned copper 1. Dimensional analysis in porous materials Dimensional analysis has proven useful and is the only practical analytical way of obtaining scalings in porous materials with seminal results in the work of Gibson and Ashby 1. Using dimensional analysis, the mechanical properties of the foam network are obtained by analysis of the deformation mechanisms of a unit cell. While the analytical derivation is found from periodic structures, it has been shown to describe very well the behavior of disordered porous materials by properly changing the proportionality constant. In this supplemental material section we employ an anisotropic unit cell and derive relationship for the foam strength based on strut failure in compression. This derivation is used to infer strut strength from the NP Copper hardness measurements Analysis of NP metal foam strength It has been shown both analytically 2 and experimentally 3 that as the relative density, e.g. solid volume fraction, of a foam increases to 50-60%, the dominant deformation mechanism will change from bending by plastic hinge formation to strut failure by compression 1, 3. Other modes, including non linear elastic behavior e.g. buckling, are not possible for high relative densities (e.g. high t/l ratios) in metals. Figure SM3 shows the anisotropic rectangular unit cell used in this derivation. The overall strut length is l in the planar direction and l h in the through thickness direction, the strut thickness is t/2. We note that the cell length, l differs slightly from the Gibson and Ashby definition so that l G-A =l-t. Figure SM4 shows a stacking arrangements of unit cells. The force, P that each unit cell will carry relates to the remote stress, σ f as follows: / (A.1) This force will be essentially carried by the four through thickness struts one of which is shown on Fig. SM 4(c). The force in each strut will be equal to the critical strut strength as: /4 /2 (A.2) By combining equations A.1-A.2 the overall foam strength is: / (A.3) Where C G1 is a geometric constant and is equal to one for this rectangular unit cell. In practice, the geometric constant is much lower for random foam structures 2. Gibson and Ashby found much similarity in the constant for random ordered foams and that obtained for a space filling 1

2 pentagonal dodecahedron 1. Using the same analysis as described in the work of Gibson 2 the proportionality constant using a pentagonal dodecahedron with 5 vertices, and a crossectional area of A=(2.8l) 2 can be shown to equal C G1 =5/(2.8) 2 ~0.64. This constant is very close to the experimentally obtained value of 0.4 for a compressive mode observed in powder compacted copper foam 3. Therefore the geometric constant can range from 0.4<C G1 < Foam strength relationship with relative density For low relative densities ( / 0.3 the scaling relationship between density and t/l, e.g. 3 (A.4) can be used in equation (A.3) for a rectangular unit cell with l=l h, so as to obtain the relationship between foam strength under compression and the relative density. However, as the relative density increases, equation A.4 is no longer valid and needs to be modified. A modification of the type: 3 1 (A.5) for a rectangular unit cell with l=l h is needed. In the case of the pentagonal dodecahedron, the relationship becomes: (A.6) By substituting equations (A.5) and (A.6) into (A.3) it is possible to obtain a relationship between the foam strength and relative density under the assumption of strut failure by compression. The relationship will be of the type: (A.7) Where the proportionality constant =1/3 for low density rectangular unit cell. The constant will increase as the relative density increases (and equations A.5, A.6 become more appropriate). Its upper bound for the case of a pentagonal dodecahedron is such that ~1. The experimentally obtained constant for equation A.7 for powder compacted copper foam 3 is Therefore the geometric constant for the relative density scaling can range from 0.3 < Foam hardness with relative density Figure SM3 shows the foam hardness as a function of relative density of NT/NC/NP Cu when compared to other NP Cu systems 4-6 reported in the literature. The relative density was obtained through stereographic projections 7 rather than by simply assuming that it equals to the initial alloy composition. The latter assumption is prone to significant errors related to sample 2

3 shrinkage 8. Since NT/NC/NP Cu occupies the same range of relative densities as the literature, the enhanced foam hardness is not due to variations in the relative density. 3. Distribution of twins in struts The high resolution TEM images provided in the main text demonstrate existence of twin boundaries within each imaged grain. In order to provide a more global picture of the distribution of twins we present in Fig. SM4 a low resolution TEM image of NT/NC/NP Cu. It is apparent that the twin lamella are indeed present in nearly every grain. Absence of twins in some grains is due to invisible diffraction conditions. References 1. L. J. Gibson and M. F. Ashby, Cellular solids : structure and properties, 2nd ed. (Cambridge University Press, Cambridge ; New York, 1997). 2. L. J. Gibson, The elastic and plastic behaviour of cellular materials. (University Of Cambridge, Churchill College, England, Cambridge, 1981). 3. M. Hakamada, Y. Asao, T. Kuromura, Y. Chen, H. Kusuda and M. Mabuchi, Acta Materialia 55 (7), (2007). 4. J. R. Hayes, A. M. Hodge, J. Biener, A. V. Hamza and K. Sieradzki, Journal of Materials Research 21 (10), (2006). 5. I. C. Cheng and A. M. Hodge, Advanced Engineering Materials 14 (4), (2012). 6. Z. Qi, C. Zhao, X. Wang, J. Lin, W. Shao, Z. Zhang and X. Bian, The Journal of Physical Chemistry C 113 (16), (2009). 7. R. Liu and A. Antoniou, Acta Materialia 61 (7), (2013). 8. R. Liu and A. Antoniou, Scripta Materialia 67 (12), (2012). 3

4 t/2 l h l Figure SM1. (a) Schematic of the anisotropic unit cell employed in g () p p y this work. The cell thickness (t cell ) and the length in plane (l), and the through thickness length (l h ) are shown. Note that both l and l h are defined from the mid-junction of the unit cell.

5 (a) (b) σ f (c) P/4=σ s (t/2) 2 P=σ f l 2 Figure SM2. Ordered rectangular cell arrangement in space by stacking adjacent rectangular unit cells. Note that l h size has no effect on the compressive strength estimate.

6 Foam hardness(g GPa) NT/NC/NP Cu NT/NC/NP Cu (coarsened) Qi et al. [26] Hayes et al. [25] Cheng et al. [21] Relative density Fi SM3 F h d f NP C f i f l i d i NT/NC/NP C f C Si hibi Figure SM3. Foam hardness of NP Cu as a function of relative density. NT/NC/NP Cu from Cu 0.61 Si 0.39 exhibits at least one order of magnitude greater foam hardness when compared to the data in existing literature.

7 Figure SM4. Low magnification TEM micrograph showing the presence of twins in nearly all grains.

Lecture 7, Foams, 3.054

Lecture 7, Foams, 3.054 Open-cell foams Stress-Strain curve: deformation and failure mechanisms Compression - 3 regimes - linear elastic - bending - stress plateau - cell collapse by buckling yielding