The role of interactions between close particles in the shear thickening of concentrated colloids is examined by using a Stokesian dynamics simulation of model systems. The interactions are repulsive thermodynamic forces and lubrication forces. Three different models are contrasted in their thickening behavior: Brownian spheres, polymer coated spheres, and Hookian spheres. Respectively, they show: a "mild" continuous thickening, a "strong" continuous thickening, and a strain thickening with loss of steady state. The relationship of order-disorder transitions and thickening is examined. Depending on the volume fraction and range of repulsive forces, thickening can be observed with or without an order-disorder transition at its onset. The different thickening responses arise from the dependence of the relaxation time of close particle contacts on interparticle gap. A time-scale based criterion for strong thickening is proposed and supported by the simulations. A simple theoretical model based the motion of a pair of particles leads to this criterion, but also predicts the mild thickening of Brownian spheres. It gives a simple fitting of flow curves which includes the details of the interparticle interactions. (C) 2004 The Society of Rheology.