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We give a mean-field, continuum treatment of ballistic aggregation on a seed and a line. The treatment is deterministic, except for one statistical assumption, the so-called tangent rule which determines the mean direction of growth. Our treatment represents progress toward the explanation of the columnar microstructure.PACS nUmbers: 68.55. + b, 05.70.Ln, 81.15.Jj In recent years much interest has focused on nonequilibrium aggregation processes, that is, the formation of structures by the irreversible addition of subunits from outside. An example of such a process is diffusion-limited aggregation (DLA) where fractals are formed. A simpler problem than the diffusionlimited case (where the aggregating particles perform random walks) is ballistic aggregation. In this process particles moving in straight lines are added to a structure whenever they touch a previously added particle. Early work on this problem seemed to show that fractals were produced, but it is now believed both on the basis of more detailed numerical studies and from analytical results that ballistic aggregates are amorphous solids of fixed density. Nevertheless, the patterns formed in this simple problem are both intriguing theoretically and technologically interesting. In Fig. 1 we show two types of ballistic aggregates: one (a "fan") formed by attachment to a seed4 and another by attachment to a plane of a beam of nonnormal incidence. 5 In both cases the particles all move in parallel straight lines, as shown, from random launching points. The similarities of the patterns are striking.The peculiar long open streaks are the unexpected feature.For the case of attachment to a plane these patterns are known as the columnar micro structure. The columns form both in computer simulations (as shown in Fig. 1) and in the real world in vapor-deposited thin films of both metals and insulators. For example, aluminum films deposited on cold substrates often show this morphology. An understanding of this structure is of particular technological interest as the surface properties, notably electrical and optical, are substantially modified from the bulk properties of the material by the surface microstructure. See Ref. 5 for more detail and actual experimental results. Even in the presence of short-range attractive forces among the atoms, which curve the straight-line trajectories, columns still form in numerical simulations. The streaks and columns have not heretofore been explained.At first glance, it seems that any such explanation would be very complicated because the voids clearly arise from shadowing of one part of the structure by another. In fact, this is an interesting feature of the system; precisely the same sort of nonlocal shadowing produces the fractals of the DLA problem. However, we will show here that many features of the structures can be explained in a remarkably simple way.In the next section we propose a kind of mean-field treatment for aggregation both on a point and on a line. We will always consider a situation in which the particles ...

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