  We propose a new statistical physics model to study equilibrium solute
segregation at grain boundaries and the resulting embrittlement effect. This
low-temperature expansion model is general and efficient, and its parameters
can be obtained from atomistic calculations. It is possible to take into
account multiple species, multiple segregation sites with different segregation
free energies, account for configurational entropy, grain radius and site
competition between solutes. As an example, the model is then applied to the
study of phosphorus and hydrogen co-segregation at $\Sigma3$
$109.5\deg[0\bar{1}1]\{111\}$ twin boundaries in $\alpha$-Fe, using energetic
parameters from density-functional theory calculations. We show that P--H
interactions may lead to increased P segregation at grain boundaries and cause
additional embrittlement compared to the case where P and H are considered
separately.
