## 1 Introduction  
Due to their excellent performance in the charge/discharge rate, power delivery and cycle life, supercapacitors (or electrochemical capacitors) have already been used in parts of hybrid vehicles and emergency systems, and are also reported to be a promising energy source for future energy storage systems [1, 2]. Compared with traditional batteries [3, 4, 5, 6], lower energy density is a critical problem hindering the future successful application of supercapacitors. To overcome this drawback, superior electrode and electrolyte materials, which are two main components of supercapacitors, are urgently needed. Graphene materials [7, 8] have a higher conductivity and larger electrochemical available surface, and thus the ion transport in their 2D plane structure is more efficient. Therefore, compared with other carbon-based materials (such as active carbon [9], carbon nanotubes [10], onion-like carbon [11] and porous carbon materials [12, 13]), graphene materials present an immense potential to improve the energy density of supercapacitors. In addition, ionic liquids [14] could remain stable under a wider range of voltages when used as the electrolyte of supercapacitors. Therefore, based on a combination of these two components, graphene/ionic liquid supercapacitors have become an attractive object in the research field of energy storage [15; 16; 17]. As is well known, for graphene/ionic liquid supercapacitors, the processes of energy storage and release are closely related to the structure of an electrical double layer (EDL), that is, ion distributions on the surface of the graphene electrode. For example, the counterions aggregate in the inner layer of the EDL in the energy storage process and desorb from the electrode surface in the energy release process [18]. For this reason, a thorough understanding of the structure of the EDL and its effects on the capacitance of the ionic liquid EDL is crucial for further improving the efficiency of graphene/ionic liquid supercapacitor applications. An EDL commonly exists in electrolytes at the solid/liquid interface, which usually plays an important role in fluid flow in microchannels or porous materials [19; 20]. An EDL can be represented by a Helmholtz model and by the Gouy-Chapman theory [21; 22]. Note that these models are usually suitable for inorganic electrolytes, where simple inorganic ions are usually regarded as particles. However, the ionic liquid only contains organic cations and organic (or inorganic) anions with a complex structure, which is free of solvent and cannot be simply regarded as a system of 'particles'. This make the ionic liquid EDL more complicated than that of traditional inorganic electrolytes. Consequently, the above theories do not seem to be tenable any more. In order to solve this problem, many researchers have turned to further study of the ionic liquid EDL on the solid surface [23; 24], and they suggest that the EDL of the ionic liquid contains two components: the inner compact layer and the outer diffusive layer. In the former layer, the counterions are packed closely. Outside the compact layer, anions and cations are distributed alternately in several consecutive layers, which form the diffusive layer. From this viewpoint, the complex structures of ions are taken into consideration. Based on this viewpoint, several groups have carried out studies on the ionic liquid EDL on the graphene surface and its structures, as well as the capacitance, in an effort to support the design and optimization of graphene/ionic liquid supercapacitors. [PERSON] et al. [25; 26] examined the EDL structures of different ionic liquids near planar electrodes and found that the size and charge of ions could significantly affect the ions' distribution on the surface of the graphene electrode and the EDL capacitance. [PERSON] et al. [27] showed that an enhancement in the capacitance can be obtained when the length scale of the surface roughness of the graphene electrode is comparable to the ion size.