It's time to have a heart-to-heart with your first child and get to the bottom of why she's so hard on her younger sister. Since you want her to feel like she can really open up to you, don't have the talk right after a fight. Instead, take her out for a little one-on-one time with Mom. Go to the mall and have the chat over ice cream.
It's possible that she's behaving well all day in school and just doesn't yet have the emotional maturity to keep it up once she gets home. Or maybe there's a problem at school that's causing her to come home with a lot of pent-up hostility. Regardless of the issue, she probably targets your middle child when she needs to blow off steam because she may feel that the youngest is off-limits. Explain to her gently, but firmly, that it's not okay to take her unhappiness out on other people -- and that if she's had a difficult day, the grown-up thing to do is to talk to you or Dad about it. If you work, tell her she can always call you at the office after school.
There could also be a bit of sibling rivalry or jealousy going on. Perhaps your older daughter feels that her younger sister is prettier, smarter, more popular, or something else that would cause her to feel envious or inferior. Although this can be a hard thing for a parent to hear (since you obviously want your girls to have a close and loving relationship), explain to your daughter that everyone has different strengths and weaknesses and that you love each of them as individuals. Help her brainstorm and focus on some of her own strengths and skills instead.