## 3 Results and Discussion
### Lithiation  
Evaluation of the anode efficiency is related to its capacity and the formation of the SEI layer. This work does not discuss the formation of the SEI layer, however, in the experimental works [46; 49], where similar topologies of carbon and carbon-silicon anodes were studied, the formation of an SEI layer was observed in the first charge/discharge cycles. During subsequent cycling the anode capacity is reduced by 8% (10 cycles for the carbon-silicon anode) and by 0.74% (5 cycles for the carbon anode). We assume that the formation of the SEI layer at the interface of the studied anode will have a similar character because of the similarity of the anode material. To evaluate the efficiency of the anode material (excluding the SEI) we have successively performed numerical experiments on the lithiation of the composite with various amounts of silicon clusters in the pores of the pillared graphene. To find the optimal concentration of the silicon mass fraction in the LIB anode material, which would ensure the maximum capacity, we considered the filling of the carbon frame with Si16 clusters in amount of 5 to 59. For the model I (Figure 1a), this filling corresponded to the range 6.34-44.5 wt.% silicon of the total mass, for model II (Figure 1c)--to the range of 5.9-42.6 wt.%. Each numerical experiment of lithiation with a different mass fraction of silicon in the composite was performed according to the following algorithm. At the first stage, the composite with a certain number of silicon clusters was completely filled with positively charged lithium ions according to the rule--the distance between the framework and the neighboring lithium ion should be at least 3.2A.By the LAMMPS software package we removed one electron from lithium atoms to form lithium ions. The required number of electrons was transferred to the carbon-silicon frame so that the total charge was zero. In the second step the supercell was optimized by the molecular dynamics method. At the third stage, the capacity of the composite was calculated. The capacity was calculated based on the amount of lithium ions that interact with silicon and carbon. Further, according to the Faraday principle [50], the capacities  
Figure 1: Pillared graphene: (**a**) supercell of model I—with short tube; (**b**) view from above; (**c**) supercell of model II—with long tube; (**d**) supercell of model I with silicon clusters Si16 (purple color). of Li-Si and Li-C were calculated separately, and then their capacities were summed on the basis of their mass fractions. This approach was applied in [12], where it was studied the intercalation of lithium into a layer of silicon and silicon oxide \"wrapped\" in a carbon framework that could retain the silicon volume. The following electrochemical reactions in the anode were studied:  
\[3C+xLi^{+}+xe^{-}\leftrightarrow LixC_{3} \tag{1}\]  
\[Si+xLi^{+}+xe^{-}\leftrightarrow LixSi\ (0<x>4.4) \tag{2}\]  
Figure 2a demonstrates the first step of the algorithm at the example of model I, where there are only five silicon clusters of Si16 in the composite pores (purple color) and the entire free space of the pores is filled with lithium ions (green color). The number of lithium ions incorporated into the cell was corresponded to maximal Li-Si and Li-C bond to assess the capacity of the anode. A numerical experiment in this case showed that lithium ions \"settled\" on the carbon frame and on the silicon atoms. Here, the carbon frame demonstrated mechanical stability, while silicon clusters did not. Si16 clusters lost their initial shape and, in some cases, lost their chemical bond with the carbon framework. When the chemical bond is lost, there is no electronic contact either, this is due to the low concentration of silicon clusters. Figure 2b clearly shows that a layer or several layers of lithium have been appeared between silicon and carbon. In this case, the capacity was 1378.6 mAh-g\({}^{-1}\) that exceeds such value for the graphite anode (372 mAh-g\({}^{-1}\)) [51].