A microfluidic technique is associated to X-ray fluorescence in order to investigate the origin of the so-called synergy effect observed in liquid-liquid extraction of rare earth elements (REE) when special combinations of two extractants-one solvating and one ionic-are used.The setup enables kinetic studies by varying the two phases' contact time. Results obtained are compared to those obtained using standard batch extraction method with equal contact time. We then determine variation of free energies of transfer for five rare earth elements present in solution together with a non-target ion (Fe$^{3+}$) at different pH. Analysis of the effect of temperature, as well as of surface charge density of the coexisting cations, allow separating electrostatic from complexation effects. We finally show that all non-linear (synergic) effects are quadratic in mole fraction. This demonstrates that in-plane mixing entropy of the bent extractant film, in the first nanometer around rare earth ions, is the determining term in the synergy effect. Surprisingly, even when the third phase is present, free energies of transfer could still be measured in the diluted phase, which is reported for the first time, to our knowledge. We hence show that the extractive power of the dense third phase is stronger than conventional reverse aggregates in equilibrium with excess water.