  In complex oxide heteroepitaxy, strain engineering is a powerful tool to
obtain phases in thin films that may be otherwise unstable in bulk. A
successful example of this approach is mixed phase bismuth ferrite (BiFeO3)
epitaxial thin films. The coexistence of a tetragonal-like (T-like) matrix and
rhombohedral-like (R-like) striations provides an enhanced electromechanical
response, along with other attractive functional behaviors. In this paper, we
compare the energetics associated with two thickness dependent strain
relaxation mechanisms in this system: domain walls arising from monoclinic
distortion in the T-like phase, and the interphase boundary between the host
T-like matrix and tilted R-like phases. Combining x-ray diffraction
measurements with scanning probe microscopy, we extract quantitative values
using an empirical energy balance approach. The domain wall and phase boundary
energies are found to be 113 $\pm$ 21 and 426 $\pm$ 23 mJ.m$^{-2}$,
respectively. These numerical estimates will help us realize designer phase
boundaries in multiferroics, which possess colossal responses to external
stimuli, attractive for a diverse range of functional applications.
