Interviews
With respect to admissions interviews, doctoral graduate programs fall somewhat between medical schools that require formal interviews of their candidates and law schools that have no interview component at all.
Rather than requiring interviews as part of admissions decisions by doctoral programs, research intensive universities are much more likely to hold recruiting weekends for candidates who have been or are likely to be admitted. It the recruiting event is held prior to final decisions, an applicant's behavior can influence the outcome, but the true intention of the department is to present and emphasize the intellectual strengths of its faculty and quality of resources, as well as to give applicants an impression of its academic and social environment. The weekend will include social events, presentations of research and the opportunity to speak with multiple faculty. Your goal at such a weekend is to confirm the perceptions that led you to apply to the department, to establish contacts with current graduate students for later assistance with questions and to make yourself known to and to assess your prospective doctoral thesis advisors. You should be prepared to have natural conversations about your undergraduate academic experience as well as the successes and/or frustrations of your individual research and writing projects and to convey a sense of the breadth of your training and experience in the field.
Applicants and prospective applicants for admission to Ph.D. programs are generally welcome to arrange their own visits to the department and to schedule an appointment with one or more prospective faculty mentors. You might make such visits early enough to inform your decision making about where to apply. Note that undergraduates who travel to professional society meetings to present papers or posters have an opportunity to meet faculty from other institutions, a perfect time to discuss their interest in those programs.
Whether you plan your own trip to visit a graduate program or are invited to a weekend arranged for applicants, you should know that such events are much less formal than interviews for professional schools or for employment. The dress code for campus visits is upper end of everyday college attire - neat pants/skirt and shirt/sweater, no suits. The graduate teaching assistants in related Duke departments provide good models for expected dress. However, if you are invited for an interview for a distinguished scholarship or a combined degree program, consider it to be more formal, probably requiring business dress.