  In the present work, the calculations have been performed for dispersion
interaction between heavy elements (Zn$^+$, Cd$^+$, Hg$^+$, Pb$^+$, Zn, Cd, Hg,
and Pb) with graphene and carbon nanotubes by evaluating van der Waals $C_3$
coefficients using well-known Lifshitz theory. The dispersion coefficients are
expressed in terms of reflection coefficients of graphene and carbon nanotubes
which are calculated within the framework of Dirac model. In addition, accurate
values of dynamic dipole polarizabilities at imaginary frequencies of
considered ions and atoms which are vital, have been calculated using
relativistic methods. The comparisons of our calculated static dipole
polarizabilities for the considered elements with the values reported in the
literature are also presented. The dispersion interactions of the considered
heavy elements with the graphene and carbon nanotubes of different radii in a
wide range of separation distances have been studied. The results have also
been analysed with another subtle variable, i.e., gap parameter of graphene
wall. The interaction coefficients obtained for both the materials, i.e., for
graphene and carbon nanotubes are mutually compared and it is found that
graphene can be said as preferable material for adsorption of these toxic heavy
elements.
