When Avraham begins the journey to Egypt in Bereshit 12:5, Sarai is identified as his wife and Lot as his brother’s son. By contrast, when the family leaves Egypt in 13:1, Sarai is still identified as his wife but Lot is not given a relational epithet. This implies that Lot’s relationship with Avraham frayed during the time in Egypt, leading to their separation in Ch. 13.
Articles See R. Joseph Soloveitchik, Abraham’s Journey (New York, 2008):117-119, who discusses how Lot's being attracted to Egyptian materialistic culture contributed to the rift between him and Avraham.1