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The importance of near-equilibrium thermodynamics in the growth of TlBaCaCuO materials is well-documented; i.e. the independent control of sample temperature, Tl-oxide partial pressure, and oxygen partial pressure. Here we demonstrate how this is accomplished using a simple version of the crucible process. In addition, we report the relevance of growth kinetics. Superconducting thin films are grown in a two-step process. First, /spl sim/2000 /spl Aring/ thick, amorphous Ba/sub 2/CaCu/sub 2/O/sub x/ and Ba/sub 2/Ca/sub 2/Cu/sub 3/O/sub x/ precursors are sputter-deposited onto LaAlO/sub 3/(100) substrates. Films are then thallinated and annealed using the crucible-pellet process. With this method, it is possible to grow essentially single-phase Tl-1212, Tl-1223, Tl-2212, and Tl-2223 thin films. Under similar thermodynamic conditions, we find that different phases dominate the film growth as a function of reaction time. For example, the Tl-x212 phases generally nucleate first, and then, depending on growth conditions, convert to the Tl-1223 or Tl-2223 phase with extended annealing times. We demonstrate that knowledge of both the thermodynamics and the kinetics of growth can yield high-quality films of each of the major Tl-superconducting phases.