Reading E. Randolph Richards's Paul and First-Century Letter Writing, I was introduced to the concept of philophronesis, which Richards attributes to Heikki Koskenniemi's studies of letter writing. The idea is that private correspondence tends to have at least one, sometimes more, of three purposes:
(1) philophronesis: the maintaining of friendly relations
(2) parousia: being present even when not physically so
(3) homilia: carrying on an ongoing conversation, each letter being one-half of a dialogue.
In effect, philophronesis in this context means the use of correspondence to establish, strengthen, or restore good personal relations.
'Philophronesis' is itself an interesting word; the Greek is often translated as 'showing kindness'. It's also the name of a rhetorical approach, known in Latin as 'benevolentia', in which you use gentle speech in order to pacify an angry interlocutor. We might perhaps gloss it as taking thought for friendship (or friendliness).