Does a divorce take effect if the wife travels after the husband, in a moment of anger, said, "By divorce, you will not travel to your family," when he intended to annoy her and did not actually object to her travel, and is expiation required for this?

The majority of scholars hold that swearing an oath by divorce results in divorce if the oath is broken. So, if you swore to your wife that she would not travel, and she did travel, divorce occurs. However, if you did not intend divorce but rather to vex her, then you owe an expiation for breaking an oath, according to Shaykh al-Islam Ibn Taymiyyah and those who agree with him. You cannot revoke a suspended divorce according to the majority of scholars, and if it occurs, you may take her back before her waiting period (iddah) ends, as long as it is not the third divorce. The divorce of an angry person does not occur if their anger is so severe that they do not comprehend what they are saying. Deliberately vexing one's wife is contrary to "retaining her with kindness."