jats:titleAbstract</jats:title>jats:pRecent directional solidification experiments with aqueous suspensions of alumina particles (Anderson & Worster, jats:italicLangmuir</jats:italic>, vol. 28 (48), 2012, pp. 16512–16523) motivate a model for freezing colloidal suspensions that builds upon a theoretical framework developed by Rempel jats:italicet al.</jats:italic> (jats:italicJ. Fluid Mech.</jats:italic>, vol. 498, 2004, pp. 227–244) in the context of freezing soils. Ice segregates from the suspension at slow freezing rates into discrete horizontal layers of particle-free ice, known as ice lenses. A portion of the particles is trapped between ice lenses, while the remainder are pushed ahead, forming a layer of fully compacted particles separated from the bulk suspension by a sharp compaction front. By dynamically modelling the compaction front, the growth kinetics of the ice lenses are fully coupled to the viscous flow through the evolving compacted layer. We examine the periodic states that develop at fixed freezing rates in a constant, uniform temperature gradient, and compare the results against experimental observations. Congruent with the experiments, three periodic regimes are identified. At low freezing rates, a regular periodic sequence of ice lenses is obtained; predictions for the compacted layer thickness and ice-lens characteristics as a function of freezing rate are consistent with experiments. At intermediate freezing rates, multiple modes of periodic ice lenses occur with a significantly diminished compacted layer. When the cohesion between the compacted particles is sufficiently strong, a sequence of mode-doubling bifurcations lead to chaos, which may explain the disordered ice lenses observed experimentally. Finally, beyond a critical freezing rate, the regime for sustained ice-lens growth breaks down. This breakdown is consistent with the emergence of a distinct regime of ice segregation found experimentally, which exhibits a periodic, banded structure that is qualitatively distinct from ice lenses.</jats:p>