## 4 Computational Results for the Dispersion Force on Nanoparticles from Graphene
As opposed to the case of cooled sheet of a pristine graphene [67], for a gapped graphene with sufficiently large \(\Delta\) the nonequilibrium dispersion force remains attractive. Now let us admit that the graphene sheet is heated up to \(T_{g}=500\) K, whereas nanoparticles preserve the environmental temperature \(T_{E}=300\) K. In this case, the computational results for the ratio \(F_{\text{neq}}/F_{c}\) are shown in Figure 2 as the functions of separation by the two red lines plotted for graphene sheets with the mass-gap parameter \(\Delta\) equal to 0.2 and 0.1 eV. The two black lines, which show the ratio \(F_{\text{eq}}/F_{c}\), are reproduced from Figure 1. As explained above, they are plotted for graphene sheets with \(\Delta=0.2\) and 0.1 eV in thermal equilibrium with the environment at \(T_{g}=T_{E}=300\) K.  
From Figure 2, it can be seen that, for a heated graphene sheet, the change in the value of \(\Delta\) has a lesser impact on \(F_{\text{neq}}\) than on \(F_{\text{eq}}\) over the entire separation region being considered. By comparing Figures 1 and 2, one can conclude that the magnitude of a nonequilibrium dispersion force acting on a nanoparticle on the source side of gapped graphene increases with increasing temperature. Figure 1: The ratio of nonequilibrium force acting on a nanoparticle on the source side of cooled to \(T_{g}=77\) 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 blue bottom lines as the function of separation. The two black top lines 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. Now, we investigate the relative role of the first and second contributions \(F_{M}\) and \(F_{r}\) in (2), which represents the total value of \(F_{\rm{neq}}\). We begin with \(F_{M}\), computed by Equations (3) and (4), and the respective expressions (8), (23)-(27) for the polarization tensor calculated at the pure imaginary Matsubara frequencies \(\mathrm{i}^{6}_{EJ}\). Similar to the case of an equilibrium force, \(F_{M}\) is always negative, i.e., contributes to the attraction. The computational results for the ratio \(F_{M}/F_{c}\) are shown as the functions of separation in Figure 3 by the two pairs of blue and red lines computed at the graphene temperature \(T_{g}=77\) K and \(500\) K, respectively. In each pair, the lower line represents a graphene sheet with \(\Delta=0.2\) eV and the upper line represents a graphene sheet with \(\Delta=0.1\) eV. As is seen in Figure 3, at both temperatures, the contribution \(F_{M}\) decreases in magnitude with increasing separation. In the considered separation region, this decrease occurs to the relatively small values at \(T_{g}=77\) K and to the classical limit (29) at \(T_{g}=500\) K. In doing so, the sign of \(F_{M}\) remains negative, which corresponds to the attractive force. The role of the contribution \(F_{r}\) is somewhat different. The sign of \(F_{r}\) in (6) is determined by the sign of the quantity \(\Theta\) in (7) and of the imaginary parts of \(R_{\rm{TM}}\) and \(R_{\rm{TE}}\) defined in (4). Thus, \(F_{r}\) can be both negative and positive. The quantity \(\Theta\) is negative for \(T_{g}>T_{E}\) and positive for \(T_{g}<T_{E}\). As to the sign of \(\mathrm{Im}\,R_{\rm{TM}}\) and \(\mathrm{Im}\,R_{\rm{TE}}\), it depends on the relative contributions of the frequency regions (9) and (10). In the region (9), \(\mathrm{Im}\,R_{\rm{TM,TE}}\) is positive and in the region (10), it is negative. The computational results for \(F_{r}\) are obtained by Equations (6), (4) and respective expressions for the polarization tensor at the real frequencies presented in Section 3.