In this paper we consider explicit substitutions calculi that allow open terms. In particular, we propose a variant of the $\lambda_\sigma$-calculus, that we call $\lambda_\phi$. For this calculus and its simply-typed version, we study its meta-theoretical properties. The $\lambda_\phi$-calculus enjoys the same general characteristics as $\lambda_\sigma$, i.e. a simple and finitary first-order presentation, confluent on terms with meta-variables, with a composition operator and with simultaneous substitutions. However, $\lambda_\phi$ does not have the non-left-linear surjective pairing rule of $\lambda_\sigma$ which raises technical problems in some frameworks.