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The existence of long-range attractive electrostatic forces between particles of like charge is one of the great current controversies of colloid science. The established theory (Derjaguin±Landau± Vervey±Overbeek; DLVO) of colloidal interactions predicts that an isolated pair of like-charged colloidal spheres in an electrolyte should experience a purely repulsive screened electrostatic (coulombic) interaction 1,2 . Direct measurements of such interactions have shown quantitative agreement with DLVO theory 3±5 . Recent experiments, however, provide evidence that the effective interparticle potential can have a long-range attractive component in more concentrated suspensions 6,7 and for particles con®ned by charged glass walls 3,5,8±10 . It is apparent that the longrange attraction in concentrated systems is due to multi-body interactions and may have a similar explanation to the attraction observed for otherwise con®ned colloids. Theoretical explanations have been proposed 11±13 but remain the subject of controversy 14,15 . Here we present a quantitative theoretical explanation of these attractive forces between con®ned colloidal particles, based on direct solutions of the nonlinear Poisson± Boltzmann equation for two like-charged spheres con®ned in a cylindrical charged pore. The calculations show that the attraction may be explained by the redistribution of the electric double layers of ions and counterions in solution around the spheres, owing to the presence of the wall; there is thus no need to revise the established concepts underlying theories of colloidal interactions.The calculation of the interaction between two spheres close to a planar wall, which has been measured experimentally 6,10 , requires a full three-dimensional solution of the governing equations. However, consideration of the case of like-charge spheres con®ned in a long, charged cylindrical pore simpli®es the calculations by reducing the dimensions of the problem to two. Nevertheless, the calculations remain directly relevant to the experimental situation and have been carried out for conditions of sphere-radius/poreradius (sphere-radius/sphere-wall separation) ratio, sphere surface potential, pore surface potential and Debye length comparable to those occurring experimentally 6,10 . We have solved the nonlinear Poisson±Boltzmann equation using an adaptive ®nite-element method with error minimization 16 , and have then calculated the electrostatic forces acting on the spheres.The normalized Poisson±Boltzmann equation for a two-dimensional electrostatic double layer in cylindrical coordinates may be written aswhere the non-dimensional space-charge density is writtenwith C i c i =I being the relative concentration. Other variables are de®ned in Methods. The summation is taken over all ionic species present in the electrolyte. For a symmetric 1:1 electrolyte system, equation (2) reduces to j sinhª. The dimensionless parameters and coordinates are de®ned in accordance with the usual practice in colloid science 16 , R kr; Z kz; ª ew=kT and k e 2 ...

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