No. Though one court has said otherwise.
Lawyers clearly have a duty of confidentiality to former clients, that is, a duty not to disclose privileged information learned during the attorney client relationship. And, for that reason, Texas Disciplinary Rule 1.09 prohibits lawyers from being adverse to former clients in matters that are substantially related to the prior representation--recognizing that that situation carries with it too much danger of misuse of confidential information. No actual misuse of confidential information need be shown to disqualify a lawyer or his firm in that situation; rather, the substantial relationship itself is enough.
But what about a duty of loyalty? Certainly, there is a duty of loyalty to current clients. But aside from the duty of preserving confidences, courts generally have not recognized any other sort of loyalty that must be maintained for former clients.
An exception is the Fifth Circuit in In re American Airlines, 972 F.2d 605 (Fifth Cir. 1992). In that case, Vinson Elkins was acting adversely to American Airlines in an antitrust case, and American was a former client. (There was a contention by American that it was even a current client, but that did not appear to be correct based on the evidence and was not a basis of decision of the Court). The Court also did not rely on there having been a substantial relationship between VE's prior representation of American and the case at bar. Rather, the Court stated that VE had violated a "duty of loyalty" to its former client and also invoked the then and now discredited doctrine that a firm should be disqualified when there is even an "appearance of impropriety."
The decision has been strongly criticized by commentators, one of whom was Dean Sutton, who was the principal author of the current Texas Disciplinary Rules. He called the decision's reasoning "unfathomable froth." (Sutton, Introduction to Conflicts of Interest Symposium: Ethics, Law, and Remedies, 16 REV. LITIG. 490, 513 (1997). Another commentator called its reasoning "incoherent." Schneyer, Nostalgia in the Fifth Circuit: Holding the Line on Litigation Conflicts Through Federal Common Law, 16 REV. LITIG. 537, 558 (1997). Shneyer at the time was a professor at the University of Arizona Law School.
I also criticized it in my own article, Boyd Current Trends in Conflict of Interest Law, 53 BAYLOR L. REV. 1 (2001).
Whether the Fifth Circuit would use the same reasoning now is an open question; my guess would be "no."