Hyperbolic graphene framework with optimum eciency for conductive composites

Constructing conductive ller networks with high eciency is essential to fabricating functional polymer composites. Although two-dimensional (2D) sheets have prevailed in nanocomposites, their eciency in enhancing conductive functions seems to reach the limit, as if merely addressing the dispersion homogeneity. Here, we exploit the unrecognized geometrical curvature of 2D sheets to break the eciency limit of ller systems. The hyperbolic curvature meditates the incompatibility between 2D topology and 3D ller space and holds the ecient conductive path through face-to-face contact. The hyperbolic graphene framework exhibits the record eciency in enhancing electrically and thermally conductive functions of nanocomposites. At volume loading of only 1.6%, the thermal and electrical conductivities reach 31.6 W/(mK) and 13,911 S/m, respectively. Nanocomposites with hyperbolic graphene framework exhibit great potentials in thermal management, sensing and electromagnetic shielding. Our work presents a geometrically optimal ller system to break the eciency limit of multifunctional nanocomposites and broadens the structural design space of 2D sheets by curvature modulation to meet more applications. ller to achieve the record eciency in enhancing conductive properties of nanocomposites. The introduction of curvature design breaks the eciency limit of 2D lamellar ller and extends the neglected parameter space of topologic curvature. The experimental and theoretical analysis reveal that the high eciency roots from the intrinsic face-to-face connections of hyperbolically curved graphene networks, which provide more conductive pathways for phonons and electrons. The enhancement factors of thermal (β 𝛋 =93) and electrical (β σ =10 15 ) conductivity outperform previous 0D, 1D, 2D and 3D functional nanollers for polymer composites. The polymer composites with hyperbolic graphene networks exhibit high thermal (31.6 W/(mK)) and electrical (13,911 S/m) conductivities at an exceptionally low volume fraction of only 1.6%. We have revealed the great values of polymer composites containing hyperbolic llers in electromagnetic shielding and thermal management. The hyperbolically curved graphene network opens the curvature design of 2D sheets and other llers, guides the structural design of functional composites and promotes more functional applications in sensors and energy storage.


Introduction
The composites re ect the ancient wisdom of combining diverse properties into one component and play an increasingly important role in our modern technology 1-3 . In the very example of polymer nanocomposites, nano llers with diverse topologies are introduced into polymeric matrices to enable rich functionalities of their nanocomposites 4-6 , for example, turning insulating into electrically and thermally conductive. The essential for preparing functional polymer composites is constructing e cient conductive llers networks, that is, achieving the optimally high conductivities at the least containing of llers. A natural deduction is that the geometric attribute of networks determines their e ciency. One traditional geometric concept of e cient networks is minimizing the percolation threshold 4,7,8 , at which nano llers become continuous across the microscopic length ( Fig. 1a, b). Most past efforts have followed the percolation concept, by choosing the topology of llers from pseudo-zero dimensional particles to 1D nanowires and 2D sheets 4,[7][8][9][10][11][12] . Beyond the sophisticated percolation notion, the e ciency of formed continuous networks should be considered as another important aspect to develop future functional nanocomposites. A scaling relationship σ (or )∝βϕ α between functional properties and ller containing (ϕ) can be predicted for the fractal feature of networks, in which σ or are electrical or thermal conductivity and the β and α represent enhancement factor and ller e ciency. Following this relationship, higher β and α denote the higher enhancing e ciency of networks (Fig. 1b). Analyzing the ever-achieved advances 4,6-12,13-21 demonstrates that β value has an increasing trend as the topology of nano llers evolves from 0D to 2D, but exhibits a limited range below 3 (Fig. 1c), which becomes a formidable limit for mixing ller system.
Graphene with 2D topology has emerged as a good nano ller to improve the conductive functionalities of polymer composites for its outstanding electrical and thermal transport properties 3 . Following the traditional mixing protocol, graphene has exhibited high e ciency 4 in enhancing conductive functions over other nano llers, in the case of homogenous dispersion. To address the weak tunneling transport of polymer gaped connection, the continuous graphene network with direct connection 22 and no insulating polymer separation has been utilized to promote the e ciency, following an alternative in ltration method to prepare composites. Although the direct contact path in graphene frameworks has extended the e ciency limit in the mixing protocol, the nearly line contact of graphene in polyhedral frameworks is not an ideally conductive path, which sets the new e ciency limit for 2D ller system with β below 10 ( Fig. 1c) [22][23][24][25][26][27][28][29] . The limit of polyhedral card-house framework ( Supplementary Fig. 1a) re ects the incompatibility between the Euclidean 2D topology and the occupying 3D non-Euclidean space, possibly originating from the narrow notion of at consumption of 2D llers. To further break this e ciency limit, it is reasonable to re-consider the geometric curvature of 2D sheets to construct more effective ller networks.
Here, we adopt a hyperbolic graphene continuous network as geometrically optimal ller to fabricate highly conductive polymer composites (PGC). The introduction of hyperbolic curvature of 2D sheets generates a close face-to-face connection con guration and increases the contact area between graphene layers, providing e cient conductive pathway for phonons and electrons. Hyperbolic graphene framework exhibits record e ciency in enhancing electrical and thermal conductivity, and breaks the previous e ciency limit of nano llers ever achieved. The thermal and electrical conductivities of PGC reach 31.6 W/(mK) and 13,911 S/m at a low containing of only 1.6 vol% of hyperbolic graphene network.
The multifunctional PGC with hyperbolic graphene network exhibits highly mechanical exibility, outstanding electromagnetic shielding property, and good thermal conductive capability.

Results And Discussion
High enhancement e ciency of PGCs Hyperbolic graphene aerogel (HGA) was prepared by our hydroplastic foaming method 30  exceeds the e ciency ever-achieved in functional nanocomposites of diversely topologic nano llers, as shown in Fig. 1c, Supplementary Fig. 3 and 4. The β value of PGC (93) is nearly two orders of magnitude higher than that of the mixing composites, in which phonon transport is blocked by the gaped polymer between nano llers. The polyhedral network of 2D sheets has exhibited β to approach ~10 in the GA composites, in which adjacent graphene sheets directly contact with each other to form continuous framework. By comparison, the PGC has a breakthrough e ciency, achieving nearly one order of magnitude higher than that of nanocomposites with polyhedral network. In the geometric principle, the hyperbolic network forms face-to-face contact of 2D sheets and provides much more conductive paths than that of polyhedral network with the line-line contact con guration. The PGC introduces a new concept of curvature to reconcile the geometric incompatibility of 2D sheets with the occupied 3D space.

Morphology and properties of HGAs
Structural analysis reveals the hyperbolic geometry of hydroplatically foamed GAs, featuring face-to-face contact of curved graphene sheets (Fig. 2a-e and Supplementary Fig. 5). The hyperbolically curved conformation of graphene sheets is generated by the bubbling mechanism of hydroplastic foaming, in which GO laminates adhere to the bubble surface and form facial tight connection by capillary force. Xray computed microtomography analysis reveals the hyperbolically curved structure of hydroplastic foaming GA, appearing a homogenous porous structure in the long millimeter scale (Fig. 2a). Scanning electron microscope (SEM) inspection demonstrates hyperbolically curved conformation of graphene sheets, including sphere surface and undulating edge. The curved graphene sheets warp together to form porous structure with a spherical diameter (D) ranging from 15 to 200 μm, corresponding to a density (ρ) of 5 to 40 mg/cm 3 , following the relationship of ρ=263.14D -0.710 ( Supplementary Fig. 6), close to the theoretical relation of ρ ~ D -1 (ref. 30). The structural analysis allows the depiction of the 3D structural model of HGA (Fig. 2d). The hyperbolic curvature of graphene sheets permits the characteristic facial contact mode (Fig. 2c, e), and the contact area is estimated to occupy nearly 10-25% of the surface area of curved graphene sheets ( Supplementary Fig. 7). This face-to-face contact mode is different from the line-line contact in the polyhedral GAs and enables much broader conductive paths from the theoretical analysis (Fig. 2h, Supplementary Fig. 1, 8 and 9). After thermal annealing up to 2,800℃, GO completely transforms to graphene through decomposition of functional groups and recrystallization. The annealed graphene sheets restore the high crystalline quality of graphene lattice and have neglectable structural defects, as con rmed by the long-range crystalline order in high resolution transmission electron microscopy (HRTEM) inspection ( Fig. 2f, g) and the trace D peak in Raman spectra ( Supplementary Fig.  10), as well as X-ray diffraction (XRD) and X-ray photoelectron spectroscopy (XPS) results ( Supplementary Fig. 11).
The continuous hyperbolic network of highly crystalline graphene sheets enables the extraordinary electrical and thermal conductivity of GAs, outperforming the pervious neat GAs fabricated by freezedrying and even templated CVD growth ( Fig. 2i and j, Supplementary Fig. 12 and 13). The σ value of HGA along the facial direction gradually increases from 1,304 S/m at ρ of 4.3 mg/cm 3 to 10,703 S/m at ρ of 39.1 mg/cm 3 , following a scaling relation of σ~ρ 1.0 ( Fig. 2i and Supplementary Fig. 12a). At a given density, HGAs have a higher σ as compared with previously reported GAs 22,31-36 . For example, the σ of 8,576 S/m of HGA is 166% higher than that of the CVD-grown GA (3,229 S/m) 22 at an approximately same ρ of 30 mg/cm 3 . More striking is the high thermal conductivity of HGA [37][38][39][40][41] . The thermal diffusion coe cient in the facial direction gradually increases to ~662 mm 2 /s at 39.9 mg/cm 3 , following a scaling law of ~ρ 1.6 ( Fig. 2j and Supplementary Fig. 12b). The highest (18.76 W/(mK)) of HGA achieved is 494% higher than the freezing dried GA (3.16 W/(mK)) after the same heat annealing (Supplementary Fig.  14). By a systematic comparison, the electrically and thermally conductive properties of HGA reach the record e ciency to outperform other carbon networks. This superior e ciency is rooted from the hyperbolically curved conformations of graphene sheets with tight facial contact and broader conductive paths.

HGA lled polymer nanocomposites
The neat HGA provides an optimally e cient ller network to improve the conductive properties of composites. We prepared the model PGCs by vacuum in ltrating precursors followed by thermal curing (Fig. 3a). The large-area PGCs can be facilely prepared in batches, as shown as the composite mat with an area of ~1,500 cm 2 ( Fig. 3b and Supplementary Fig. 15). The hyperbolically curved structure was maintained during the ltration and curing processes. SEM inspections reveal that the graphene network in the composite maintains the spherical shape on the surface and keeps indulging continuous connections on the section (Fig. 3c and Supplementary Fig. 16), that is similar to the structure of neat HGA (Fig. 2a-c).
The model PGC exhibits an overwhelmingly high ller e ciency in enhancing thermal and electrical conductivity, outperforming the e ciency of composites with nano llers extending from 0D particles, 1D nanowires, 2D sheets to conventional GAs ( Fig. 3d and Supplementary Fig. 17). For all nano llers, the monotonously increases as the ller content increases. But the PGCs follow the steepest trend and have the highest thermal conductivity at the same ller content, exhibiting the highest e ciency than any other ller composites 4,6-9,11,16,22,23,25 . For example, the of PGC achieves 31.6 W/(mK) only at a containing of 1.6%, and the peak value is 5, 40, 60 and 150 folds higher than those of 3D GAs (~6 W/(mK)), 2D graphene (0.8 W/(mK)), 1D carbon nanotube (CNT, 0.5 W/(mK)) and 0D carbon black (0.2 W/(mK)), respectively. More impressively, the PGC with optimized geometrical structure shows an ultrahigh thermal conductivity enhancement (TCE) of 15,700%, which is calculated by ×100%, where and are the thermal conductivity of composite and matrix, respectively ( Supplementary Fig. 18), corresponding to the specific TCE (TCE per 1 vol% filler addition) of 9,813%. Beside the predominant , the PGCs also exhibit overwhelmingly high σ values, and the data line is located at the top to break through the limit of previous

Electromagnetic properties of exible PGCs
The PGC bulks remain high resilience as the pristine PDMS elastomer, under stretching, bending and compression. The tensile strength of PGC barely decays and loading-unloading curves are almost overlapped for 1,000 cycles at a strain of 10% (Fig. 4a). The cycle bending test at small bending radius reveals the stable bending elasticity (Fig. 4b).
As to the compression, the PGC recovers to its original height at high strain of 20% for 1,000 cycles ( Fig. 4c and Supplementary Fig. 19). Besides, PGC behaves great sensitivity to these mechanical behaviors (Supplementary Fig. 20).
The high electrical conductivity of PGC enables itself as a predominant electromagnetic interference (EMI) shielding material. With the increase of the graphene content from 0.17% to 0.98%, the average shielding effectiveness (SE) of PGCs increases from ~47 dB to ~100 dB in the frequency range of 2-18 GHz, far exceeding the commercial standard of 20 dB (Fig. 4d and Supplementary Fig. 21a). In addition, the EMI shielding performance of PGC almost keeps stable after 1,000 bending cycles ( Supplementary   Fig. 21b), only with a reduction by ~5 dB. The superior e ciency in electrically conductive PGC allows the superiority in EMI shielding. As shown in Fig. 4e Thermal management of exible PGC The overwhelming thermal conductivity of PGC with the least lling content provides new opportunities for thermal management, especially for high power electronics. As a proof, we use the PGC mat with 0.2 vol% graphene loading as heat diffusion and heat transfer materials for a high-power LED lamp (Fig. 5a).
For heat diffusion, a high-power LED was placed on the pure PDMS (as a contrast) and PGC (Supplementary Movie 1). As displayed in Fig. 5b and c, the LED lamp equipped with the PGC mat as a thermal spreader exhibit lower heat-spot temperature than pure PDMS after 180 s work. For heat transfer, PGC and PDMS were vertically put between a high-power LED and an aluminum heat sink, respectively. The exible thermal interfacial material of PGC can effectively transfer the heat to the bottom heat sink (Supplementary Movie 2), which brings dramatically temperature drop in the LED lamp (Fig. 5d, e). These results indicate the as-prepared exible PGC not only holds higher thermal conductivity, but also exhibits great potential to relieve the urgently thermal management demands in electronic devices.

Conclusions
We have developed a hyperbolic graphene continuous network as a geometrically optimal ller to achieve the record e ciency in enhancing conductive properties of nanocomposites. The introduction of curvature design breaks the e ciency limit of 2D lamellar ller and extends the neglected parameter space of topologic curvature. The experimental and theoretical analysis reveal that the high e ciency roots from the intrinsic face-to-face connections of hyperbolically curved graphene networks, which provide more conductive pathways for phonons and electrons. The enhancement factors of thermal (β =93) and electrical (β σ =10 15 ) conductivity outperform previous 0D, 1D, 2D and 3D functional nano llers for polymer composites. The polymer composites with hyperbolic graphene networks exhibit high thermal (31.6 W/(mK)) and electrical (13,911 S/m) conductivities at an exceptionally low volume fraction of only 1.6%. We have revealed the great values of polymer composites containing hyperbolic llers in electromagnetic shielding and thermal management. The hyperbolically curved graphene network opens the curvature design of 2D sheets and other llers, guides the structural design of functional composites and promotes more functional applications in sensors and energy storage.

Materials And Methods
Preparation of HGAs. Aqueous GO dispersions with the average size of ~50 µm were produced from Hangzhou Gaoxi Technology Co. Ltd. (www.gaoxitech.com). Self-supporting GO lm was acquired by casting GO dispersions onto the PET substrate. Then the dried GO lms were soaked into 10% N 2 H 4 aqueous solution to realize hydroplastic foaming GAs 30 . The GA density can be tuned by the foaming time. The HI/HAc (1:1) reduction and high-temperature treatment (2,800℃ for 2 h) were conducted to obtain conductive HGAs.
Preparation of PGCs. A typical vacuum impregnant method in industrial engineering was adopted for the fabrication of polymer composites. As-prepared HGAs with different densities were placed into the PDMS prepolymer, followed by degassing in a vacuum environment for 30 min and thermally curing at 60℃ for 10 min. Then PGCs with various graphene loadings were achieved.
Characterizations. The morphology and microstructure of HGAs and PGCs were characterized by eldemission SEM (Hitachi S4800), TEM (FEI Titan G2 60-300) and CT with a resolution of 200 nm (skysCan2211). Raman measurements were performed on a Renishaw inVia-Re ex using a 532 nm laser beam. XPS characterization was conducted using an Escalab 250Xi instrument. XRD patterns was measured with a X-pert Powder diffractometer. The compressive test was performed on a microcomputer control electronic universal testing machine (RGWT-4000-20, REGER). The tensile and bending measurements were used by the Instron Legend 2344 machine. The electrical properties of HGAs and PGCs were evaluated by the Keithley 2400 Source Meter. The EMI shielding performances were measured by Vector Network Analyzer (ZNB40). The thermal diffusivities were directly measured using the LFA 467 HyperFlash. The thermal conductivity was calculated using the equation = α × C p × ρ, where α is the thermal diffusivity, C p is the speci c heat capacity, and ρ is the density of the sample. The heat capacity (C p ) was measured by DSC (TA Q200) at a heating rate of 10 ℃/min under nitrogen atmosphere. The IR thermal photos were captured by a high-resolution infrared imager ((FLIR T630sc).

Declarations
Schematic illustrations of polymer nanocomposites with 0D, 1D, 2D and 3D nano llers. (a) Representative diagram of continuous conductive paths in polymer nanocomposite and (b) its typical evolution of conductive performances. (c) Thermal enhancement factor of polymer nanocomposites with different nano llers, divided into mixing and lled systems.