Approaching diamond’s theoretical elasticity and strength limits

Diamond is the hardest natural material, but its practical strength is low and its elastic deformability extremely limited. While recent experiments have demonstrated that diamond nanoneedles can sustain exceptionally large elastic tensile strains with high tensile strengths, the size- and orientation-dependence of these properties remains unknown. Here we report maximum achievable tensile strain and strength of diamond nanoneedles with various diameters, oriented in <100>, <110> and <111> -directions, using in situ transmission electron microscopy. We show that reversible elastic deformation depends both on nanoneedle diameter and orientation. <100> -oriented nanoneedles with a diameter of 60 nm exhibit highest elastic tensile strain (13.4%) and tensile strength (125 GPa). These values are comparable with the theoretical elasticity and Griffith strength limits of diamond, respectively. Our experimental data, together with first principles simulations, indicate that maximum achievable elastic strain and strength are primarily determined by surface conditions of the nanoneedles.

(1) In describing the bent diamond nanoneedles, the imaging and illustrations throughout the manuscript depict the tensile strains on the convex (outer) side of the nanoneedles while the concave (inner) side is labeled as having zero strains. This cannot be the case since there must be compressive, i.e., negative, strains on the concave side, which need to be properly considered and described.
(2) Given the existence of compressive strains on the concave side of the nanoneedles, a natural follow-up question is then where does the fracture initiate? Does it always occur on the convex side, or the concave side, or is it size and orientation dependent?
(3) There is obviously a large strain gradient in the bent nanoneedle. How would such a gradient affect the fracture mechanism and breaking strain and peak stress?
(4) There are some descriptions and discussions in the manuscript that need correction, further consideration and/or clarification. 4.5 -The Griffith theoretical strength given by E/10 or E/9 is non-directional, but this quantity is being compared to the measured maximal strength in the [100] direction. 4.6 -Line 214-216, discussion on possible further increasing fracture strain by reducing needle size and surface defects, but this would lead to maximal strength exceeding the Griffith limit since the currently measured tensile strength along the [100] direction has already reached the theoretical limit. 4.7 -The reported maximal strain on [100] needle is 13.4%, which is close to the theoretical strain of 13.5% calculated for the needle with 2-atom steps, but the stress was derived from bulk diamond stress-strain relation at 13.4% to reach the conclusion that the extracted, again from the bulk diamond stress-strain relation, stress value of 125 GPa at 13.4% matches the Griffith limit of 122 GPa. These comparisons are inconsistent. The authors performed in-situ TEM nanomechanical experiments on diamond nanoneedles and systematically explored the size-and orientation-dependence of elastic strain and strength of nanoscale diamond. The <100>-oriented nanoneedles with a diameter of 60 nm was found to have the highest elastic tensile strain of 13.4% and the corresponding tensile strength was estimated to be 125 GPa. The reported values are comparable with the theoretical elasticity and Griffith strength limits of diamond. Careful first principles modeling results suggest that maximum achievable elastic strain and the corresponding strength are closely correlated to surface defects of the diamond nanoneedles. The authors' main findings are indeed convincing and robust, however there are a few additional suggestions and clarifications that the authors should consider.
1) The authors adopted the elastic parameters used in ref.
(3) for FEM simulations to estimate the maximum strength (stress) corresponding to the in-situ TEM observations. However, the elastic parameters used in ref.
(3) were along [111] direction. As we know single crystalline diamond has a finite elasticity anisotropy along different directions, the FEM simulations along [100] and [110] should consider the influence due to elasticity anisotropy, so the estimated maximum strength values would be more accurate.
2) Although the assumed friction probably won't significantly change the results, the description on the friction in estimating the maximum strength was not clear. The choice of friction may have an impact on the compression force experienced by the nanoneedle and consequently affect the stress distribution within the nanoneedle. Some discussions should be provided.

Report of Reviewer #1 (Remarks to the Author):
This paper is a follow-up of reference [3] in the manuscript. The same experimental techniques are used to manufacture the diamond nano needles, to test the mechanical behaviour and to determine the maximum strain (FEM and DFT). The only novelty of the paper is that needles with different orientations (<100>, <110> and <111>) have been analysed an in all cases the ideal strength has been achieved because of the lack of defects in the nanoneedles. Moreover, it was found that fracture always took place along the {111} facets of the diamond crystals.
All these results are interesting and the paper is well written, I think that advance in knowledge provided by the paper is very limited (in comparison with reference [3]) and does not justify its publication in Nature Communications. I recommend that the authors submit the paper to other journal related to nanotechnology or extreme mechanics Response: We thank the referee for the critical comments. However, we respectfully disagree with the comments "that advance in knowledge provided by the paper is very limited (in comparison with reference [3])".
Please allow us to summarize the major differences between our study and Reference We would also like to draw an attention to the significance of technical development reported in our work. As a matter of fact, our project was initiated several years ago.
However, it was challenging to perform the desired mechanical tests on diamond using commercial in-situ TEM holders inside the TEM, because the mN-level forces needed to break a nanoscale diamond require high-precision displacement control. We spent more than two years to develop our own in situ mechanical testing system inside the TEM (X-Nano in-situ TEM instrument, Fig. S9). Thanks to this X-Nano in-situ TEM instrument, significant progress has been made, enabling us to gain more comprehensive insight into the strength and deformation limits of diamond.

Report of Reviewer #2 (Remarks to the Author):
In this work the authors report on an experimental study, supported by computational simulations and theoretical analysis, of large tensile strain and stress achieved in diamond nanoneedles that approach theoretical limits. This phenomenon was recently reported by another group, but the present findings expand considerably the range and depth of the observed behaviors, especially the crystal size and orientation dependence of the measured mechanical properties. The reported results are interesting and insightful, providing a useful foundation for further exploration and application of this remarkable phenomenon. The experimental measurements seem to have been done carefully and presented well. There are, however, some issues with the analysis and discussion, which are listed below, that need to be addressed before a decision regarding a final recommendation for this manuscript can be reached.
(1) In describing the bent diamond nanoneedles, the imaging and illustrations throughout the manuscript depict the tensile strains on the convex (outer) side of the nanoneedles while the concave (inner) side is labeled as having zero strains. This cannot be the case since there must be compressive, i.e., negative, strains on the concave side, which need to be properly considered and described.

Response:
We thank the reviewer for the constructive comments. Following the reviewer's suggestion, we have included the maximum compressive strain on the concave side of the diamond nanoneedles in the revised manuscript ( Figure 2c, Figure  3b, Figure 4c, Figure 4d, Figures S3b, and Figure S4b). Corresponding discussion has also been included in the revised manuscript. "Besides the tensile strain on the convex side of the nanoneedles, there must be a compressive strain on the concave side for mechanical equilibrium. FEM calculations show that the absolute values of the compressive strain on the concave side are slightly larger than those of the corresponding tensile strain at the convex side (Fig. 2c, Fig.3b, Fig.4c, Fig.4d, Fig. S3b and Fig. S4b). The tensile strength of a material is usually much lower than the compressive strength because compressive loading tends to suppress microcracks. It has been experimentally demonstrated that compressive strains of -19% and -16% can be achieved in the <100> and <111>-oriented diamond pillars with a diameter of 200 nm in uniaxial compression, respectively 19 . Here, the compressive strains at the concave side of diamond nanoneedles are still far below the limits of the nanoneedles. Therefore, fracture tends to initiate from the convex side and the tensile limits of the nanoneedles can be well expressed by the bending experiments 3 ." (Line 134-146, Page 7) (2) Given the existence of compressive strains on the concave side of the nanoneedles, a natural follow-up question is then where does the fracture initiate? Does it always occur on the convex side, or the concave side, or is it size and orientation dependent?
Response: Thanks for this insight. Although the absolute values of the compressive strain on the concave side are slightly larger than those of the corresponding tensile strain at the convex side of diamond nanoneedles, we believe the fracture should initiate on the surface of the convex side with maximum tensile strain. The main reasons are as follows: (1) In practice, the compressive deformation limit of diamond is higher than the tensile one because compressive loading tends to suppress microcracks. It has been experimentally demonstrated that -19% and -16% compressive strains can be respectively achieved in the <100> and <111>-oriented diamond pillars with a diameter of 200 nm in uniaxial compression (Nano Letters, 16, 812-816, 2016). In this work, compressive strains of <100> and <111>-oriented diamond nanoneedles only approached -14% and -9.6%, respectively, which are believed still far below their limits of the nanoneedles. In contrast, the tensile strain at the convex side, which has a parallel value to the compressive one at the concave side, almost approached the tensile deformation limit of diamond.
(2) The first-principle calculations also verify the diamond can withstand a larger compressive strain than tensile one before failure. We performed the first-principle calculations on the compressive deformation of diamond with consideration of a 2-atom surface flaw. The initial model is the same one used for tensile deformation in Figure S7c. As shown in Fig. R1, the diamond supercell was compressed along the <100> direction. The bond around the surface flaw highlighted by the red arrow was stretched with the increase of compressive strain. Figure R1a-d show the charge density distribution of the diamond with different compressive strains. We can see that the charge is depleted at a compressive strain of -20%, indicating the breakage of the bond near the surface. As shown in Figure R1e, when the compressive strain approached 20%, the bond length reached 1.75Å, which was believed as a critical value indicating the bond breaking and fracture initiation. For the diamond crystal with 2-atom surface flaws, its compressive strain limit could approach 20%, which is significantly higher than the tensile one of 13.5%.
Therefore, fracture tends to initiate on the surface of the convex side with maximum tensile strain and the tensile limits of the nanoneedles can be well expressed by the bending experiments. Corresponding simulations have been included in the supplementary as Fig S8. Corresponding discussion has also been included in the revised manuscript. "In contrast, the diamond with a two-atom step surface can withstand a critical compressive strain of -20% when loading along the [100] direction ( Supplementary  Fig. S8)." (Line 224-226, Page 11) (3) There is obviously a large strain gradient in the bent nanoneedle. How would such a gradient affect the fracture mechanism and breaking strain and peak stress?

Response:
The strain gradient is usually generated in the bending or buckling pillars. In our work, there do exist obvious strain gradient in the bent diamond nanoneedles before their fractures. However, differing from the materials with good ductility (plasticity) like metals, the diamond nanoneedles exhibited a sudden brittle fracture, giving no time for the observation of the strain gradient's role in the fracture or cracking of the diamond at present temporal resolution inside TEM. In spite of various strain gradient existing among different diamond nanoneedles, the diamond nanoneedles exhibit common feature of brittle fracture of {111} cleavage. Since the strain gradient is complicatedly interwoven with the size, curvature, and stress state of the sample, it is a challenge to get a direct answer to the correlations between the strain gradient and breaking stain as well as the peak stress. However, the strain gradient is no doubt a subject worthy of our future study, requiring a rational design of series of experiments.
(4) There are some descriptions and discussions in the manuscript that need correction, further consideration and/or clarification.  Fig. 5b?

-line 396-397, the statement "For tension along [100], all bonds belong to the type-I category" is incorrect.
Response: Such a statement was made according to the results of our first-principles calculations, which indicate that all bonds are elongated when stretching the diamond supercell along the [100] direction. This can also be understood from the perspective of geometric relationships. As shown in Fig. R1 below, A Fig. S5

-The Griffith theoretical strength given by E/10 or E/9 is non-directional, but this quantity is being compared to the measured maximal strength in the [100] direction.
Response: Griffith theoretical strength is derived from a continuum model for the crack extension, assuming the isotropy and homogeneity of the substance, as well as the linearity (Phil. Trans. Roy. Soc. Lond. A 221, 163-198, 1921). It is a well-known theoretical strength for brittle materials, and regarded as a rough "benchmark" to judge whether the material approaches in a theoretical strength scope. Although anisotropy and specific crystal structures were not considered in his model, Griffith agreed that the best arrangement for withstanding tension is where the strongest directions of all the "molecules" are parallel to the axis of tension (Phil. Trans. Roy. Soc. Lond. A 221, 163-198, 1921 Response: Thanks again for the comments. As we mentioned in the above response, the Griffith limit of E/9 is a roughly estimated value by approximation. In addition, numerous attempts were made to determine the theoretical strength more accurately. However, there are rather large differences in its estimates depending on the approximations and calculation procedures. For example, Slutsker obtained values of theoretical strength for 15 metals fall in the range E/6-E/10 (Fiz Tverd Tela 46, [1606][1607][1608][1609][1610][1611][1612][1613]2004). Mecholscky used fractal geometry to estimate the theoretical strength of brittle materials approximately E/8 (Materials Letters 60, [2485][2486][2487][2488]2006). We can see that the Griffith limit is not strictly an absolute criterion for the upper limits on the theoretical tensile strength. The comparison between our experimental strength and Griffith theoretical strength is just one of the ways to show how far the strength of the diamond can go. Correspondingly, first principle calculations, which is widely recognized as a more accurate way to predict the ideal strength of diamond, is also present in our work. The ideal strength of <100>-orientated diamond calculated by the first principle calculations is as high as 225 GPa (Figure 5a), which is much higher than our experimental value. When considering the effect of surface (flat surface, 1-atom step, and 2-atom step), the calculated strength of diamond is still as high as 205 GPa (28% strain), 157 GPa (18% strain), and 125.4 GPa (13.5% strain) ( Figure S7), respectively, which are still greater than the experimentally achieved value. Our first principle calculations indicate that higher strength than the Griffith theoretical limit could be achieved if the defect-free diamond nanoneedles are with a perfectly smooth surface, offering guidance for the ultra-strong materials design.

-
The reported maximal strain on [100] needle is 13.4%, which is close to the theoretical strain of 13.5% calculated for the needle with 2-atom steps, but the stress was derived from bulk diamond stress-strain relation at 13.4% to reach the conclusion that the extracted, again from the bulk diamond stress-strain relation, stress value of 125 GPa at 13.4% matches the Griffith limit of 122 GPa. These comparisons are inconsistent.
Response: Thanks for the insight. I think it has been explained well for this comment in the response for comments 4.5 and 4.7. The Griffith limit of E/9 was an estimated value by an approximation based on energy consideration. It stands to reason the maximum value (125 GPa) of our experimental data goes little beyond the value of E/9 (122 GPa), as E/9 here is usually roughly considered as "benchmark" to judge whether the material approaches in an ideal strength scope. As for the first principle calculations, when considering a 2-atom-step flaw on the surface, the calculated elastic strain limits and strength are about 13.5% and 125.4 GPa, respectively, which are also closely approached by the highest values obtained in our experiments. Response: Thank the reviewer for the kindly concern. The relationships between fracture and bond length that we talked about are all about the type-I <111> bonds that are elongated. We added one word in the caption of Figure 5 for the clarification: "The bond lengths discussed in the main text belong to type-I bond." (Line 423-424, Page 21)

Report of Reviewer #3 (Remarks to the Author):
The Where σ cr is the Euler's critical stress, E is the elastic modulus of diamond, K is the effective length factor, L is the unsupported length of the needle, and r is the radius of gyration. Given E = 1050 GPa, K = 0.7, L = 4000 nm and r = 50 nm, the resultant σ cr is ~3 GPa, without considering anisotropy of diamond. It means that only a compression stress on the order of 3 GPa can reach Euler buckling instability of the diamond nanoneedles. Compared to a strength of 125 GPa measured in the diamond, the value of σ cr can be included in the tolerance range in our work. Therefore, the friction effect can be negligible, as it only affects a negligible compressive stress. The corresponding discussion has been included in the revised manuscript: "Friction during bending tests may have an impact on the compression force and consequently affect stress distribution within the nanoneedle. Here, we estimate friction effects through Euler instability of a slender pillar (the diamond nanoneedle).
To the first-order approximation by neglecting crystal anisotropy, the longitudinal compression stress on the needle is given by Euler's critical stress σ cr , according to the following formula 35 : (1) where E is the isotropic elastic modulus of diamond, K is effective length factor, L is the unsupported length of column, r is the radius of gyration. Given E = 1050 GPa, K = 0.7, L = 4000 nm and r = 50 nm, the resultant σ cr is ~3 GPa. Compared to the strength on the order of 100 GPa measured in diamond, the friction effect appears negligible and only contributes to uncertainties." (Line 287-299, Page 14 and 15) 3) Lines 165-168: It is not clear how the projection is done from [100] or [110] to [111].

Response:
The projection is meant to resolve the stress onto the <111>-orientated C-C bonds. Figure R2 shows the detailed stress decomposition process for the <100>-loading case, where the lattice of diamond is elongated into a tetragonal configuration at the largest strain of 38%. The right panel of a is a sectional view corresponding to the plane outlined by the red lines in the left panel. The (111) plane is viewed along edge-on in the right panel. The normal stress on the (111)  For the <110>-loading case, the projection process is similar to that of <100>-loading case, as shown in Fig. R2b [111]. The projection illustrates that t fracture is controlled by the breakage of <111>-orientated C-C bonds, explaining the fractures along the {111} planes. We have added supplementary Fig. S5 to illustrate these projection processes.