Domain-induced control of latent heat in freestanding BaTiO$_3$ membranes
Thin ferroelectric BaTiO$_3$ films often exhibit continuous transitions instead of the first-order behavior of bulk crystals, a discrepancy usually attributed to epitaxial strain or dimensionality. Using quasi-adiabatic nanocalorimetry on freestanding BaTiO$_3$ membranes-free of clamping and substrate heat sinking-we show that domain morphology, not thickness or boundary conditions, controls the transition order. Thick membranes with large, monodomain-like regions display clear latent heat, whereas thinner membranes with dense 180$^{\circ}$ domain patterns show a continuous transition despite undergoing the same tetragonal-cubic structural change confirmed by x-ray diffraction. Piezoresponse force microscopy links this behavior to domain-size evolution, and a Ginzburg-Landau analysis demonstrates how reduced domain size lowers the free-energy barrier, rounding a nominally first-order instability. These results identify domain morphology as the key determinant of ferroelectric transition order in oxide membranes and establish design guidelines for enhancing caloric effects through domain engineering.