Dispersant Concentration [mg/ml] Water F /2.0/ Tip sonication (10) Acidic solution ( ph)

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1 Supporting Information. 1. Boron nitride nanoparticle (BN) surface treatment BNs were mixed with a pre-prepared surfactant solution as indicated in Table S1 and then sonicated or stirred. Table S1: Initial concentrations of BNs and surfactants in mixtures. Solvent Dispersant type Dispersant Concentration [mg/ml] Initial BN concentration [mg/ml] Mixing procedure & (duration) [min] Water F /2.0/ Tip sonication (10) Acidic solution ( ph) Silane 1.5/3.0/ Bath sonication (30) Acidic solution ( ph) Silane Magnetic stirrer (30) We used two sonication methods: Bath sonication (BS) was performed in an Elma sonic bath (model S10; 30 W, 37 khz, Singen). The water level in the bath was kept constant and the vial (20 ml) was placed in the center of the bath. Tip sonication (TS) was performed in a VCX 400 instrument (400 W, 20 khz, µtip, Sonics & Materials Inc.) at 20% amplitude. The solution temperature was kept at 0 C. Following sonication the solution was centrifuged (Megafuge 1.0, Heraues, 20 min at 4000g) to accelerate BN aggregate precipitation. We found that longer centrifugation times do not change the concentration of BN. Following sonication and centrifugation, exfoliated (supernatant) and aggregated (precipitate) BN were phase separated by decantation. Following decantation, the aqueous dispersions of BNs (in plastic flasks 40 ml, diameter = 3 cm) were frozen using liquid nitrogen and placed in a lyophilizer (Labconco Freezone 4.5) for 48 h. A thermogravimetric analyzer was used to qualitatively investigate the amounts of surfactants (amino silane and pluronic F127) bonded to the BN surface (after the centrifugation, decantation and lyophilization steps). BNs (with or without surface modifications) and pure surfactants were heated to 900 C (above the non-oxidative degradation temperature of surfactants) under 20 ml/min nitrogen gas flow. The change in weight after heating was recorded and analyzed. TGA results revealed that the amounts of decomposable materials on the as-received BN particles were negligible. Since the surfactants are volatile compared to BNs, the fractional change in weight of the surfacetreated BNs after heating above the non-oxidative degradation temperature of the surfactants relates to the amount of surfactants present.

2 2. Planetary centrifugal mixer Figure S1. (a) The planetary centrifugal mixer (Thinky, ARE-310) (b) Schematic representation of mixer operation. The rotational speed of 2000 rpm applies a powerful acceleration force on the sample that promotes deaeration and generates strong convection of the materials in the container. The zirconia balls significantly increase the compression forces and promote filler dispersion.

3 3. Thermal conductivity determination Figure S2 shows the typical sample arrangement on the DSC sensor. Liquid gallium (150 mg) was poured into a 70-µL alumina crucible and placed on top of a disk shaped sample. To avoid heat loss due to thermal resistances and to achieve the best possible reproducibility, the spaces between the crucible-sample and sample-sensor interfaces were covered by a thin layer (0.5 µl) of silicon heat transfer oil. Figure S2: (a) Sample arrangement on the DSC sensor. The reference crucible was empty. An identical crucible containing gallium was placed on the sample. The yellow dashed lines indicate the silicone heat transfer oil; (b) DSC specimens measuring 5.9 mm in diameter and mm thick: 1 GnP/epoxy composites, 2 stainless steel 316, 3 Pyrex glass 7740 and 4 epoxy disk shapes samples. During heating (28-37 C, 0.5 C/min), when gallium reached its melting point (29.76 C), the temperature remained constant until the melting latent heat was absorbed. The temperature of the upper surface of the sample was thus known during this period. The temperature of the lower surface of the sample and the heat flowing into it were measured by the DSC. The TC of the sample was calculated from the temperature difference between the upper and lower end surfaces of the disk and the heat flow: 1 S = R T + h A TC where S is the slope of the linear side of the melting peak, R T is the combined thermal resistances at the sensor sample and sample crucible interfaces, h is the thickness of the sample disk and A is the cross-sectional area of the sample. (E1)

4 After measuring at least three randomly orientated samples of different thicknesses ( mm) the TC and R T were determined by linear regression of 1/S and h/a (Figure S3). Figure S3. TC measurements by DSC. (a) The curves (colors represent different sample thicknesses; Black: h = 1.48 mm, Green: h = 0.93 mm, Blue: h = 0.29 mm 1.48) show the DSC thermograms of φ H15 = GnP composite samples as a function of sample temperature. (b) TC measurements of H 15 -based composites, TC = slope -1 (calculated based on equation 1). Figure S3a shows gallium melting curves for composites filled with φ H15 = (Figure 1). As expected, the slope of the linear side of the melting peak decreased with sample thickness (h). Linear regressions of 1/S and h/a is presented in Figure S3b. Based on equation (1), the TC of the samples is slope -1, and the combined thermal resistance (R T ) at the sensor sample and sample crucible interfaces are the intersection with the y axis. As a control experiment, we measured the TC values of reference materials for which the TC values are known (Fisure S2b), namely, epoxy (0.18 ± 0.01 W/mK), Pyrex 7740 glass (1.22 ± 0.02 W/mK) and stainless steel 316 (15.21 ± 0.55 W/mK). The results were in excellent agreement with the vendor s specifications with an experimental error of < 5%. Moreover, the TCs of highly conductive samples, namely TC>4W/mK, were measured and compared with the results using the Hot Disc method (ISO ).

5 4. Characterization of as-received materials Figure S4: SEM micrographs of as-received materials :(a) M 5, (b) M 15, (c) M 25 and (d) BNs.

6 Table S2. GnP atomic composition as derived from XPS measurements. GnP type C[%] O[%] Si[%] N[%] M M H M C Thermal and electrical conductivity models 5.1 Nans model [1] 3 + φ ( 2(K11 c TC m ) c + K c 33 1) K TC = TC m ( 11 + TC m TC m 3 φ 2(K 11 c TC m ) c + TC m ) (E2) K 11 c K 11 = TC GnP c = 1 + 2a ktc GnP hk m ; K 33 TC GnP 1 + 2a ktc GnP dtc m ; a k = ITR TC m For H 15 GnPs dispersed in epoxy the following parameters were assumed: GnP thickness of h = 15 nm, GnP lateral dimension of d = 15 µm, GnP TC of TC GnP = 3000 W/mK and epoxy TC of TC m = 0.18 W/mK 5.2 Adjusted percolation critical power law [2, 3] TC GnP network = TC 0 ( φ φ t 1 φ t ) t TC = TC matrix + TC GnP network = TC m (1 φ) + TC 0 ( φ φ t 1 φ t ) t (E3.1) (E3.2) with a thermal percolation threshold of φ t = 0.166, a pre-exponential factor TC 0 = 91 W/mK, and a critical exponent t = 0.84; the epoxy TC is TC m = 0.19 W/mK.

7 5.3 Critical power law [2]: EC~( φ φ c 1 φ c ) t (E4) with a percolation threshold at φ c = 0.05 and a critical exponent t = Figure S5: SEM micrographs of gaps closure in composites with GnPs loading above the thermal percolation threshold (φ M15 = 0.19). Figure S6: SEM micrographs of fractured Hybrid 1 surface. (a-b) composite loaded with with φ M15 = 0.15 and φ BN = 0.02; (c) Magnification of the marked area in (b) shows BN nanospheres (marked with white arrows) homogenously dispersed on the surface of M 15 GnPs.

8 References: 1. Nan, C.-W.; Liu, G.; Lin, Y.; Li, M.Appl. Phys. Lett.2004, 85, Stauffer, D.; Aharony, A.Introduction to Percolation Theory, 2 nd ed.; Taylor & Francis: London, Bonnet, P.; Sireude, D.; Garnier, B.; Chauvet, O.Appl. Phys.Lett.2007, 91,

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