## 2 Material Preparation and Characterization  
The thermal composites tested in this research were prepared from several common commercial TIMs (e.g., \"Ice Fusion\" and \"Arctic Alumina\"). These composite materials use silver, aluminum oxide, and aluminum nitrite particles--or combinations of thereof--as fillers to enhance the thermal conductivity of TIM. As an additional filler, we have utilized commercial liquid phase exfoliated (LPE) graphene and FLG mixture (Graphene Supermarket, Graphene Laboratories Inc., Calverton, NY, USA). The mixture had a large concentration of FGL flakes with the thickness between three and eight monolayers (total thickness below 3 nm) and lateral dimension in the range of 2-8 um. The quality of material has been verified with Raman spectroscopy, optical and scanning electron microscopy (SEM) following the procedures reported by some of us earlier [41; 43; 45]. One should note that LPE is a scalable method for producing inexpensive graphene--FLG solution with the quality sufficient for thermal applications. We used a microscale to weigh the TIM and LPE graphene with \(\pm 0.01\%\) accuracy. The mixture of commercial TIMs with added graphene fillers was placed in a vial to the two-axis rotating mixer. The composite was mixed at 2500 rpm for three minutes. A vacuum pump was used to eliminate the bubbles in the composite for two minutes. This mixing and vacuuming process was repeated three to four times to ensure uniform filler dispersion with minimum concentration and size of air pockets. The percentage of additional graphene fillers in TIMs is calculated as wt \(\%=W_{\mathrm{g}}/(W_{\mathrm{B}}+W_{\mathrm{g}})\), where \(W_{\mathrm{g}}\) and \(W_{\mathrm{B}}\) are the weights of graphene--FLG fillers and base material, respectively. In this case, the base material is the initial commercial TIM. The volume loading fraction of graphene fillers was determined from the equation: vol \(\%=(W_{\mathrm{g}}/\rho_{\mathrm{g}})/((W_{\mathrm{g}}/\rho_{\mathrm{g}})+(W_ {\mathrm{B}}/\rho_{\mathrm{B}}))\), where \(\rho_{\mathrm{g}}\) and \(\rho_{\mathrm{B}}\) are the mass density of graphene and base material, respectively. In order to make sure that addition of graphene--FLG fillers was done successfully and the thermal conductivity of TIMs increased, we conducted measurements of the apparent thermal conductivity of the resulting composite between two metal plates. The measurements were conducted using the industry standard TIM tester method (Analysis Tech TIM Tester, Boston, MA, USA) with the software-controlled contact pressure and temperature. The electronic 'in situ thickness measurement' function provided accurate values of the thickness of the composite layer for further analysis. It is important to note here that the TIM tester values of the apparent thermal conductivity are lower than the bulk values owing to the effect of the thermal resistance with the contacting surfaces. However, the TIM tester thermal conductivity is more relevant for practical applications of TIMs. Figure 2 shows representative results of the thermal measurements for two commercial TIMs with added graphene fillers. The thermal conductivity data are presented for graphene filler loading of up to 6 wt %. The sample temperature was fixed at 50 \({}^{\circ}\)C while the pressure maintained at 100 kPa. One can see that, in one composite, the thermal conductivity increases by 130% as compared to commercial TIM without extra graphene fillers. In another composite, the thermal conductivity increases by 77%and starts to saturate gradually for the loading fractions above 4 wt %. It is interesting to note that a stronger enhancement due to graphene was observed in the least expensive commercial TIM. The fact that higher loading fraction of additional graphene fillers does not improve thermal conductivity of some TIMs is not surprising. Commercial TIMs already have high loading of other fillers and the addition of graphene fillers above certain fraction (\(\sim\)4 wt % in this case) can degrade viscosity and other parameters.