Is it permissible for a female physical therapist to treat men in addition to women, knowing that direct physical contact is necessary for the nature of the work, and is her touching of patients considered impermissible? If so, what is its expiation?

A woman's work is permissible, and it may even be recommended or obligatory, provided she adheres to the Sharia-compliant manners, such as wearing the hijab, avoiding seclusion (khalwa) and mixing (ikhtilat), and treating only women or mahrams, except in cases of compelling necessity or urgent need. So, if you work in a general hospital and treat non-mahram men, you must avoid that except in cases of necessity. The expiation for what is not permitted by the Sharia is through repentance, abandoning it, resolving not to return to it, and increasing good deeds, for Allah the Almighty says: "Indeed, good deeds do away with evil deeds. That is a reminder for those who remember."