Substitute for Gentleness
I'm not your doll
From the onset, Ayanami Rei is presented as a very stoic girl. The first time she is heard, when Commander Ikari summons her after Shinji initially refused to pilot EVA-01, she only utters one word: "yes." Her manner of speaking is very flat, as if almost lifeless or restraint. She rarely speaks, but when she does, Rei only says what is necessary. Her expressions are also very subdued, more so because she does not know how to express herself than she is trying to hold herself back. The only time she is seem being excited and chattery is in the presence of Commander Ikari.
Her quiet nature results in Shinji's fascination, but also draws out Asuka's dislike. Rei's absolute obedience causes Asuka to regard her as a doll, a puppet. In many ways though, Rei can be looked upon as a doll. She is the clone of Ikari Yui, and she is used by Commander Ikari almost as a surrogate. I'm sure she knows this and values herself less because how Ikari perceives her. In one episode, when Asuka accused Rei of being Commander Ikari's favorite, Rei says, "I'm not favorited either, I know that quite well." I think Rei is always aware that her existence was based on Yui's death, and that Commander Ikari sees her as a vessal for Yui.
Beyond the TV series, in the movie The End of Evangelion, when Ikari Gendou tries to act out his plan of uniting with Yui again, Rei rejects him, telling him she is not his doll. She repeats she is not a doll, and chooses Gendou's son, Shinji, instead. That was the only time she directly disobeyed Commander Ikari Gendou's orders.
There is emotional growth in Rei though. The most happening in Rei II and as a result of her relationship with Ikari Shinji. He tells her when she should smile, and makes her blush when he tells her she seems like a mother. And through their interactions, Rei shows a different side of her, eventually willing to sacrifice her own life to protect Shinji's.