Model of Enhanced Strength and Ductility of Metal/Graphene Composites with Bimodal Grain Size Distribution

Abstract

A model is proposed that describes plastic deformation in metal/graphene composites with a bimodal grain size distribution of the metallic matrix. Within the model, dislocation pile-ups are generated in large grains at Frank–Read sources, and their stresses promote dislocation motion within the nanocrystalline/ultrafine-grained phase. Also, the presence of graphene gives rise to the mechanisms of strengthening, such as the load transfer to graphene platelets, thermal-mismatch-induced strengthening and Orowan strengthening, as well as to back stress hardening. We demonstrated that the strengthening and strain hardening in bimodal metal/graphene composites are dominated by the Orowan strengthening and back stress hardening. The results also indicate that regardless of the lateral size of graphene platelets, bimodal metal/graphene composites can simultaneously have high yield strength and large uniform deformation but the values of the yield strength and critical uniform deformation are higher in the case of small graphene platelets.

Appendix A

In this Appendix, we calculate the average effective distances, $l_{0}$ and $l$, between the edges of neighboring graphene platelets for the cases where they lie inside a grain or in a GB, respectively. First, we cast the average distance $l_{0}$ between the projections of the points B and D of the intersection of the edges of neighboring graphene platelets and the dislocation slip plane on the initially straight line of moving dislocation (Figure A1(a)).

Fig. A1

Geometry of graphene platelets and moving dislocations in bimodal metal/graphene composites. (a) Graphene platelets are located in a grain interior. The left and right parts of the figure depict the dislocation slip plane and graphene platelet plane, respectively. (b) Graphene platelets lie in the GB

To do so, we denote the number of graphene platelets per unit volume as $N_{V}$. Then, in the examined case of platelets in the form of discs of diameter $L$ and thickness $H$, we have the following relation between $N_{V}$ and the volume fraction $f_{V}$ of graphene: $f_{V} = \pi L^{2} HN_{V} /4$. In turn, the number of graphene platelets that intersect a dislocation slip plane per its unit area is $N_{S} = N_{V} L < \cos \theta >$, where $\theta$ is the angle between the platelet plane and the normal to the dislocation slip plane. In the approximation where $\theta$ represents a random variable in the range $0 \le \theta \le \pi /2$, we have $< \cos \theta > = 2/\pi$, and so $N_{S} = (2/\pi )N_{V} L = 8f_{V} /(\pi^{2} LH)$.

The average distance $\lambda$ between the projections of the centers, K and O, of the segments, AB and DE, produced by the intersections of neighboring platelets with the dislocation slip plane, on the dislocation line is $\lambda = 1/\sqrt {N_{S} } = \pi \sqrt {LH/(8f_{V} )}$. In turn, the average total length of the two segments, BK and DO (see Figure A1) is $p = L < \cos \varphi > = (2/\pi )L$, where $\varphi$ is the angle displayed in Figure A1(a). Then the average total length of the projections of segments BK and DO onto the dislocation line is $p < \cos \delta > = (2/\pi )p = (4/\pi^{2} )L$, where $\delta$ is the angle between the segment BK and the dislocation line (see Figure A1(a)). As a result, we obtain

$$ l_{0} = \lambda - \frac{4L}{{\pi^{2} }} = \pi \sqrt {\frac{{L{\kern 1pt} H}}{{8f_{V} }}} - \frac{4L}{{\pi^{2} }}, $$

(A1)

which coincides with formula [20] of the main text.

Now calculate the average distance $l$ between the points of the intersection of the edges of two neighboring graphene platelets, lying in the same GB, with the straight line of a GB dislocation (Figure A1(b)). Let $f_{\text{gr}}$ be the fraction of the GB occupied by graphene platelets. Then the number of graphene platelets per unit GB area $N_{\text{S}}^{\text{GB}}$ is $N_{\text{S}}^{\text{GB}} = 4f_{\text{gr}} /(\pi L^{2} )$. To calculate the distance $l$, consider the simplified model case where the GB has the shape of a square with the size $d_{\text{GB}}$ and a straight line with the coordinate $y = y_{l}$ is parallel to a side of the square GB (Figure A1(b)). Then the number of graphene platelets per unit length of this line is $N_{\text{S}}^{\text{GB}} d_{\text{GB}}$. The average number of platelets that intersect this line (per its unit length) is $N_{\text{S}}^{\text{GB}} d_{\text{GB}} P_{\text{av}}$, where $P_{\text{av}}$ is the probability that a random graphene platelet intersects the line with a random value of the coordinate $y_{l}$.

First, examine the case where $d_{\text{GB}} > 2L$. In this case, for $L < y_{l} < d_{\text{GB}} - L$, the probability $P_{1}$ of the intersection of this line with a random platelet is $P_{1} = L/(d_{\text{GB}} - L)$. Here $d_{\text{GB}} - L$ is length of the interval of all possible coordinates of platelet centers (lying in the range $L/2 < y < d_{\text{GB}} - L/2$), and $L$ is the length of the interval of the possible coordinates of the centers of the platelets that intersect the straight line. For $y_{l} < L$ and $y_{l} > d_{\text{GB}} - L$, the probability of the intersection is $P_{2} = y_{l} /(d_{\text{GB}} - L)$ and $(d_{\text{GB}} - y_{l} )/(d_{\text{GB}} - L)$, respectively. The probability of the intersection between the line with a random coordinate $y_{l}$ and a random graphene platelet is $P_{\text{av}} (d_{\text{GB}} > 2L) = \frac{{(d_{\text{GB}} - 2L)P_{1} \, + 2L < P_{2} > }}{{d_{\text{GB}} }}$, where $< P_{2} >$ is the average value of $P_{2}$ in the interval $0 < y_{l} < L$. Since $< P_{2} > = L/[2(d_{\text{GB}} - L)]$, from the latter relation for $P_{\text{av}}$ we obtain $P_{\text{av}} (d_{\text{GB}} > 2L) = L/d_{\text{GB}}$.

Now consider the case where $d_{\text{GB}} \le 2L$ and, for definiteness, assume that the straight line lies in the lower half of the GB, that is, $y_{l} \le d_{\text{GB}} /2$. In this case, the probability of the intersection of the straight line and a random platelet is $P_{3} = 1$ for $d_{\text{GB}} - L \le y_{l} \le d_{\text{GB}} /2$ and follows as $P_{4} = y_{l} /(d_{GB} - L)$ for $0 \le y_{l} < d_{GB} - L$. The probability of the intersection between the line with a random coordinate $y_{l}$ and a random graphene platelet is $P_{\text{av}} (d_{\text{GB}} \le 2L) = \frac{{P_{3} [d_{\text{GB}} /2 - (d_{\text{GB}} - L)] + < P_{4} > (d_{\text{GB}} - L)}}{{d_{\text{GB}} /2}}$, where $< P_{4} >$ is the average value of $P_{4}$ in the interval $0 \le y_{l} < d_{\text{GB}} - L$ . Since $< P_{4} > = 1/2$, we obtain $P_{av} (d_{GB} \le 2L) = L/d_{GB}$.

Thus, regardless of the value of the GB length $d_{\text{GB}}$, we have $P_{\text{av}} = L/d_{\text{GB}}$. Consequently, the number $N_{\text{S}}^{\text{GB}} d_{\text{GB}} P_{\text{av}}$ of the graphene platelets that intersect a random straight line spanning the GB (per unit length of this line) equals $N_{\text{S}}^{\text{GB}} L$. As a result, the average distance $\lambda_{\text{GB}}$ between the centers, K′ and O′, of the segments produced by the intersection of neighboring graphene platelets with a specified straight line in the GB follows as $\lambda_{\text{GB}} = 1/(N_{\text{S}}^{\text{GB}} L) = \pi L/(4f_{\text{gr}} )$. The average total length of the segments A’K’ and B’O’ (see Figure A1(b)) is $(2/\pi )L$. As a consequence, we obtain

$$ l = \lambda_{\text{GB}} - (2/\pi )L = \frac{{(\pi^{2} - 8f_{\text{GB}} )L}}{{4\pi f_{\text{GB}} }} $$

(A2)

which coincides with formula [23] of the main text.