Bending rigidities of tensionless balanced liquid-liquid interfaces as occurring in microemulsions are predicted using self-consistent field theory for molecularly inhomogeneous systems. Considering geometries with scale invariant curvature energies gives unambiguous bending rigidities for systems with fixed chemical potentials: The minimal surface Im3m cubic phase is used to find the Gaussian bending rigidity,κ, and a torus with Willmore energy W = 2π 2 allows for direct evaluation of the mean bending modulus, κ. Consistent with this, the spherical droplet gives access to 2κ +κ. We observe thatκ tends to be negative for strong segregation and positive for weak segregation; a finding which is instrumental for understanding phase transitions from a lamellar to a sponge-like microemulsion. Invariably, κ remains positive and increases with increasing strength of segregation.Interfaces characterized by dense surfactant packings, such as microemulsions [1][2][3] and biological membranes [4] that are found naturally or manipulated artificially to be in a state of near-zero tension, have extensive areas. Often such interfaces feature a spontaneous curvature that manifests in spherical or cylindrical (swollen) micelles [5,6]. When a system is tensionless and precisely balanced-typical for single component bilayers and expected for the middle-phase microemulsions-the interface's spontaneous curvature vanishes [7] and ultra-low interfacial energies can be achieved [8,9]. Here, the elastic moduli, mean (κ) and Gaussian (κ) bending rigidities, control the interface fluctuations and topology, respectively. Such systems show a first-order phase transition from lamellar to sponge-like phases, e.g., upon an increase of the temperature for nonionic systems, and a change of the salinity for ionic systems [3,8,10]. A preeminent challenge is to predict, from a molecular model for such interfaces, a means to induce a sign change in thē κ from negative to positive; this signals the loss of stability of the lamellar, L α , oil-surfactant-water ordering in favor of a phase with saddles, L 3 or sponge-like. Another long-standing problem is understanding the relation between surfactant chain architecture and corresponding bending rigidities [11,12].Earlier theoretical methods [13,14], experiments [15][16][17], and simulations [18,19] that attempted to link bending rigidities to molecular properties did not provide information onκ; moreover, the results for κ were not consistent with each other. Therefore uncertainties prevail and these persist also because internal checks for presented rigidities are rarely provided. As a result, there exists no accepted molecular level theory that convincingly links molecular characteristics to both mechanical parameters of the interfaces (κ andκ). Notably, the missing information forκ is remarkable as its magnitude and, in particular, its sign are fundamental to the understanding of microemulsions.The primary obstacle in establishing a molecular model for determining bending rigidities is the requirem...