  We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that
allows to distinguish smooth real algebraic surfaces up to birational
diffeomorphism. As an application, we construct infinite families of smooth
rational real algebraic surfaces with trivial homology groups, whose real loci
are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally
diffeomorphic. There are thus infinitely many non-trivial models of the
euclidean plane, contrary to the compact case.
