  It is known that in a water solution with multivalent counterions (Z-ions),
two likely charged macroions can attract each other due to correlations of
Z-ions adsorbed on their surfaces. This "correlation" attraction is
short-ranged and decays exponentially with increasing distance between
macroions at characteristic distance A/2\pi, where A is the average distance
between Z-ions on the surfaces of macroions. In this work, we show that an
additional long range "polarization" attraction exists when the bare surface
charge densities of the two macroions have the same sign, but are different in
absolute values. The key idea is that with adsorbed Z-ions, two insulating
macroions can be considered as conductors with fixed but different electric
potentials. Each potential is determined by the difference between the entropic
bulk chemical potential of a Z-ion and its correlation chemical potential at
the surface of the macroion determined by its bare surface charge density. When
the two macroions are close enough, they get polarized in such a way that their
adjacent spots form a charged capacitor, which leads to attraction. In a salt
free solution this polarization attractive force is long ranged: it decays as a
power of the distance between the surfaces of two macroions, d. The
polarization force decays slower than the van der Waals attraction and
therefore is much larger than it in a large range of distances. In the presence
of large amount of monovalent salt, when A/2\pi<< d<< r_s (r_s is the
Debye-H\"{u}ckel screening radius), this force is still much stronger than the
van der Waals attraction and the correlation attraction mentioned above.
