We consider what should happen when surface adsorbed atoms are created at a rate faster than they can be accommodated at kink sites on a surface, such as graphite, for which the nucleation of new growth steps and kinks is difficult. We suppose that this is the condition of common ion beam sputtering systems when the beam current exceeds a critical value. We conclude that, following the nucleation of a second phase as a basal structure, whiskers should grow in the direction of the ion beam, whatever its orientation to the surface normal. Provided that possible attachment sites on the sides of the whiskers are bound, as by termination of dangling bonds by the hydrogen generally present in sputtering systems, the whiskers should grow in length at a constant rate with little or no increase in their diameter. This is because the tips of the whiskers would provide the only nonsaturable source of attachment sites for the surface adsorbed atoms produced by subsequent sputtering. We suggest that this mechanism is responsible for the growth of carbon whiskers with diameters of order 15 nm and remarkably high aspect ratios on graphite surfaces in 1 keV sputtering systems. Such whiskers have been grown to a length in excess of 20 μm and have been demonstrated to be clearly different from the scrolled-graphite structure whiskers obtained by catalytic pyrolysis of hydrocarbons. © 1987.