The cause of that smile? Yuuta temporarily readopting his Dark Flame Master persona to bestow Rikka with a cool e-mail address— "black raison d'etre." He's basically saying he acknowledges Rikka's existence, which is a big deal for someone who's always felt (mostly) alone.
Episode 7. Yuuta's concern for Rikka when she behaves strangely during the summer trip. After Yuuta learns what happened, he consoles her when she's depressed and cheers her up. During the scene on the roof, Yuuta catches Rikka in his arms when she jumps down (although they both hit a bush). He goes along with Rikka's delusions and bikes both of them to Rikka's old house because he thought it would make her happy.
Even though Shinka and Sanae attack each other often, Shinka was still seen taking care of Sanae twice in episode 7.
EPISODE 8. JUST EPISODE 8.
Episode 9 as well.
In context, a positively adorable embrace between Rikka and Yuuta, made even more romantic by the fact that its set to the backdrop of a patented Kyo Ani golden-sunset.
The confession scene in Episode 10.
Nibutani hugging Dekomori to help her weeping enemy cope with essentially losing her best friend.
In the Season Finale, when Yuuta invoked Rikka's Chuunibyo imagination the way it was meant to have been invoked to help her say goodbye to her father and move on rather than escape the reality of his loss.