MOF-derived multifractal porous carbon with ultrahigh lithium-ion storage performance

Porous carbon is one of the most promising alternatives to traditional graphite materials in lithium-ion batteries. This is not only attributed to its advantages of good safety, stability and electrical conductivity, which are held by all the carbon-based electrodes, but also especially ascribed to its relatively high capacity and excellent cycle stability. Here we report the design and synthesis of a highly porous pure carbon material with multifractal structures. This material is prepared by the vacuum carbonization of a zinc-based metal-organic framework, which demonstrates an ultrahigh lithium storage capacity of 2458 mAh g−1 and a favorable high-rate performance. The associations between the structural features and the lithium storage mechanism are also revealed by small-angle X-ray scattering (SAXS), especially the closed pore effects on lithium-ion storage.


SI.1 Characterization of Zn-MOF
In Figure S1a, the XRD pattern of Zn-MOF exhibits a high crystallinity, which is similar to the reference report s1 . The nitrogen adsorption isotherms of Zn-MOF display characteristics of type Ⅰ ( Fig. S1b) with a rapid uptake of adsorption at low relative pressure (P/P 0 < 0.1), indicating that the pores in Zn-MOF are micropores s2 . The type of the pores is further confirmed by the pore size distribution calculated using the non-local density functional theory (NLDFT) model (inset of Fig.   S1b). The specific surface area from BET method is also obtained to be 2137 m 2 g -1 . The total pore volume of Zn-MOF calculated from the amount adsorbed at P/P 0 = 0.994 is 1.15 cm 3 g -1 .
The optical microscope images were taken by the polarized-light microscope (Olympus BX51M), as illustrated in Fig. S1c&d.

SI.3 Porous property characterization of FPC and VFPC by SAXS
Porod's law is one of the basic theories in SAXS, which can be used to describe the porous properties of the samples s3 . In Fig. S4a&b, the Porod plots of both FPC and VFPC show a positive deviation, indicating a quasi two-phase system with micro-fluctuations of electron density within any phase of a two-phase system. For the positive deviation from the Porod's law, one obtains s3 : where I(q) is the scattering intensity, q is the scattering vector, q = 4πsinθ/λ , 2θ is the scattering angle, K is the Porod constant, and b is a constant related to the size of the regions with micro-fluctuations of electron density. The fitting results show that K for FPC and VFPC are 103.5 and 280.0, respectively.
The specific surface area of the scatterers can be calculated by Porod method s3 : (1 ) where S V is the total surface per unit of volume, P is the porosity of the sample, and Q is the invariant constant, which is given by s3 : The SAXS method can also be used to simulate the scatterer size distribution. The scattered intensity I(q) for a polydisperse system, which can be expressed by s3,s4 :

SI.4 Calculation of micropore volume of FPC and VFPC by SAXS s5
Usually, the micropore volume V mic can be calculated from the density of carbon phase with micropores dispersed in it, eg. the microporous backbone phase ρ mc and the density of amorphous carbon phase ρ c . The relationship of the parameters can be expressed by: In order to obtain the ρ mc , the bulk can be treated as a two-phase system consisting of microporous backbone phase and the macro/meso-pore scatterers phase; thus, the relationship between the macroscopic density ρ and the microporous backbone phase ρ mc can be expressed as: where the constant C = 8.504×10 11 m kg -1 that connects the mass density of the scattering entities to the scattering cross section. s3 SAXS method has been widely used to investigate the fractal characteristics of the structure of irregular objects. Briefly in the theory of small-angle X-ray scattering, the SAXS intensity from fractal objects has a power-law form: For porous fractals, the porous fractal dimension D p is given by:

SI.5 Fractal characterization of FPC and VFPC by SAXS
(9) whereas for surface fractals, the surface fractal dimension D s is given by: (b) lnI-lnq plots of VFPC, and the curves display three linear domains which can be ascribed to two surface fractal areas and a porous fractal.

SI.6 Structure characterization of Li-FPC and Li-VFPC by SAXS
For the negative deviation from the Porod's law, the correction form can be written as s3 : or -2 2 3 2 2 3

( ) exp( ) ln[ ( )] ln
where σ is the standard deviation of the Gaussian smoothing function, which is a parameter related to the thickness of the transition zone. The thickness of the transition zone E can be expressed by s3 :

0.5
(2 ) E   (12) The electrical density of the scatterers can also be obtained from SAXS, and the electrical density is related to K by the equation s6 :   Figure S7. EIS results of FPC and VFPC.

SI.8 TEM images of FPC and VFPC in full lithium insertion state
As illustrated in HRTEM mode, large amounts of nanoparticles dispersing in carbon matrix densely can be seen in Fig