Controlling directionality in dipolar colloids via relaxation in alternating rotating fields

Time-varying external fields can program dipolar interactions in soft matter systems. Continuously rotating fields are widely used to promote isotropic self-assembly, but finite relaxation of an induced dipole generically introduces a phase lag that can drive the spinning of both individual particles and clusters. Alternating rotating fields, which reverse the rotation direction every half cycle, suppress net spinning, but need not yield a truly isotropic time-averaged interaction when the dipole relaxes with a finite timescale τ . Here, we propose an extended alternating rotating field in which the field sweeps beyond 2 π within each half cycle to compensate for relaxation-induced undersampling of dipole orientations near the reversal direction. Using paramagnetic colloids under time-varying magnetic fields as a representative realization of a generic first-order relaxation model, we determine the half-cycle sweep angle ϕ iso required to obtain an isotropic cycle-averaged pair potential. The isotropy condition collapses onto a master curve controlled by the dimensionless parameter τ ω , and we derive accurate analytical approximations in the experimentally important small- τ ω regime. We also analyze the residual anisotropy under conventional alternating rotating-field protocols and show that its nonmonotonic dependence on τ ω underlies the single-peaked ϕ iso ( τ ω ) . Because the theory assumes only first-order relaxation, the framework applies broadly to field-responsive soft matter beyond magnetic colloids. It enables switching between isotropic and anisotropic cycle-averaged pair interactions while suppressing net rotation and provides a route to infer relaxation times.