Is it permissible to separate from one's wife without divorce, in order to preserve the children, while continuing to provide for her, given that he is unable to have marital relations with her or treat her with the same feelings as his second wife? And is there any injustice or sin in that?

Justice among wives in love, emotional inclination, and sexual intercourse is not obligatory. However, justice is obligatory in providing for them and in ensuring their chastity. If the husband is capable of being just in terms of dwelling, spending, lodging, and sexual intercourse, then there is no blame upon him for preferring one of them in sexual intercourse and emotional inclination. However, if he does not wish to spend the night with one of them and cannot fulfill her right to sexual intercourse despite her need, then this is impermissible due to the injustice it entails. Allah Almighty says: "So do not incline completely [toward one] and leave her [the other wife] hanging." Nevertheless, if the wife agrees to waive her right to spend the night, there is no blame. It is permissible for a woman to give her share of the division to her husband or her co-wives with his consent.