We have modeled dimyristoylphosphatidylcholine bilayer membranes deposited on argon-sputtered glass or polyacrylamide substrates in order to understand the dependence of the lipid self-diffusion coefficient upon temperature. We used a lattice model that has been successful in understanding some of the thermodynamics of lipid bilayers. We considered two modifications of this model : model I in which the bilayer was described by a distribution of noninteracting bilayer domains within and between which lipids could move and model II in which lipid-lipid interactions were weakened in a random way thereby permitting large diffusion coefficients at low temperatures. We modeled the lateral exchange of lipids as being permitted only when either an adjacent pair of lipids or a mutually adjacent trio of lipids were in their excited states. We carried out a direct computer simulation of lipid movement or related the self-diffusion coefficient to pair or triplet correlation functions. We found that the results predicted by the unperturbed model are in general agreement with measurements of the self-diffusion coefficient on oxidized silica wafers as a prototype of a polished surface. This permitted us to apply the perturbed models to study the other cases for which we found that only model I could account for measurements on argon-sputtered glass and on polyacrylamide films. Argon-sputtered glass yielded domain distributions with a most probable size of similar to 250 to 600 lipids per half-bilayer, while polyacrylamide films yielded a most probable domain size of similar to 600 Lipids per half-bilayer. On the basis of these results, we have made predictions about the average number of gauche bonds per molecule as a function of temperature.