## 4 Computational Results for the Dispersion Force on Nanoparticles from Graphene
Figure 2: The ratio of nonequilibrium force acting on a nanoparticle on the source side of heated to \(T_{g}=500\) K gapped graphene sheets with \(\Delta=0.2\) and \(0.1\) eV to the classical limit of an equilibrium at \(T_{E}=300\) K force acting between the same nanoparticle and an ideal metal plane is shown by the two red lines as the functions of separation. The two black lines reproduced from Figure 1 show the ratio of the equilibrium force between a nanoparticle and the gapped graphene sheets kept at \(T_{E}=300\) K to the same classical limit. In Figure 4a, we plot the ratio \(F_{r}/F_{c}\) as the function of separation at the graphene temperature \(T_{g}=77\) K and in Figure 4b--for \(T_{g}=500\) K (the blue and red pairs of lines, respectively). In both cases, the lower and upper lines are for the graphene sheets with \(\Delta=0.2\) and \(0.1\) eV, respectively. Figure 4: The ratio of the second contribution to the nonequilibrium force acting on a nanoparticle on the source side of (**a**) cooled to \(T_{g}=77\) K and (**b**) heated to \(T_{g}=500\) K gapped graphene sheets to the classical limit of an equilibrium at \(T_{E}=300\) K force acting between the same nanoparticle and an ideal metal plane is shown by the pairs of blue and red lines, respectively, as the function of separation. In each pair, the lower line represents a graphene sheet with the mass-gap parameter \(\Delta=0.2\) eV and the upper line represents a graphene sheet with \(\Delta=0.1\) eV. Figure 3: The ratio of the first contribution to the nonequilibrium force acting on a nanoparticle on the source side of cooled to \(T_{g}=77\) K and heated to \(T_{g}=500\) K gapped graphene sheets to the classical limit of an equilibrium at a \(T_{E}=300\) K force acting between the same nanoparticle and an ideal metal plane is shown by the pairs of blue and red lines, respectively, as the function of separation. In each pair, the lower line represents a graphene sheet with the mass-gap parameter \(\Delta=0.2\) eV and the upper line represents a graphene sheet with \(\Delta=0.1\) eV. From Figure 4a, one can see that, for a graphene sheet with \(\Delta=0.2\) eV at \(T_{g}=77\) K, the contribution \(F_{r}\) remains negligibly small at all considered separations, whereas it increases monotonously with increasing separation for a graphene sheet with \(\Delta=0.1\) eV. The sign of \(F_{r}\) remains negative. Here, the main contribution to \(F_{r}\) given by the frequency region (10) is negative, leading to \(F_{r}<0\). In Figure 4b, the sign of \(F_{r}\) is positive for a graphene sheet with \(\Delta=0.2\) eV and changes from positive to negative for graphene with \(\Delta=0.1\) eV. This means that, at the considered separations, the main contribution to \(F_{r}\) for a graphene sheet with \(\Delta=0.2\) eV given by the frequency region (10) is positive, leading to \(F_{r}>0\). If \(\Delta=0.1\) eV, the relative role of the frequency regions (9) and (10) is different depending on separation. At short distances, the dominant region is (10) and \(F_{r}>0\), whereas at separations exceeding approximately 0.5 \(\upmu\)m, the dominant contribution is given by the region (9) and \(F_{r}<0\). By comparing Figure 1 with Figures 3 and 4a, it is seen that at short separations the major contribution to \(F_{\rm{neq}}\) for a graphene sheet at \(T_{g}=77\) K is given by \(F_{M}\) for both values of the mass-gap parameter. At large separations, the major contribution to \(F_{\rm{neq}}\) is given by \(F_{r}\) for graphene with \(\Delta=0.1\) eV, whereas the relatively small values of \(F_{\rm{neq}}\) for graphene with \(\Delta=0.2\) eV are determined by \(F_{M}\). In a similar way, by comparing Figure 2 with Figures 3 and 4b, we conclude that at short separations the major contributions to \(F_{\rm{neq}}\) for graphene sheets at \(T_{g}=500\) K with both values of \(\Delta\) are also given by \(F_{M}\). These contributions, however, are slightly decreased by the impact of \(F_{r}\), which is of the opposite sign.