Normally, people ask who would you like to face in the next round. I'm asking the opposite question: who would each of the teams least like to face in the quarter finals of the Champions League? Here's my penny's worth.
The 4 non-EPL teams (Barcelona, Roma, Fenerbache and Schalke) would each probably take the same view: anyone but one of the EPL's Big Four.
Arsenal would least like to face Manchester United. United play in a way which causes Arsenal a lot of problems, especially in the really big games. With the memory of the FA Cup humiliation still buring bright, Arsene Wenger would say "Non" to United if he had the choice.
United would hate to meet Chelsea. Chelsea are a big strong side, with a powerful midfield and have been the side which has given United most trouble over the past 3 to 4 years, bar none. United have often struggled to score goals against Chelsea; and have been well beaten on occasion. Fergie will be choking on his glass of Merlot if he ends up with Chelsea.
Chelsea would not want to meet Liverpool again. Twice in the past 3 seasons, Liverpool have done them over in the later stages of the Champions League. Liverpool are arguably a better team now (with Torres) than at any time under Benitez. Chelsea with Mourinho couldn't overcome a fired up Liverpool on big European nights at Anfield. Would Avram Grant succeed where Mourinho at his best couldn't? Probably not. Were Chelsea to have the ill-fortune to face Liverpool, then not for the first time this season, Grant would have to try (in that unconvincing way of his) to persuade us that he was really quite happy with the draw. Don't think so, Avram.
Liverpool would not want to face Arsenal. Of all the teams Liverpool face in the EPL or Europe, Arsenal on song is the one which discombobulates Benitez's careful game-plans more than any other. The pace and movement of Arsenal's sweet passing is likely to sweep Liverpool out of the competition should they meet. Don't expect any "Ole"s from Benitez should Arsenal be pulled out of the hat to face his team.
It will be an intriguing draw at the end of the week. I believe that one of the EPL's Big Four will win the Champions League this season. But please, do me a favour, don't ask me which one.

