  We present a new, physically-intuitive model of ice-lens formation and growth
during the freezing of soils and other dense, particulate suspensions.
Motivated by experimental evidence, we consider the growth of an ice-filled
crack in a freezing soil. At low temperatures, ice in the crack exerts large
pressures on the crack walls that will eventually cause the crack to split
open. We show that the crack will then propagate across the soil to form a new
lens. The process is controlled by two factors: the cohesion of the soil, and
the geometrical supercooling of the water in the soil; a new concept introduced
to measure the energy available to form a new ice lens. When the supercooling
exceeds a critical amount (proportional to the cohesive strength of the soil) a
new ice lens forms. This condition for ice-lens formation and growth does not
appeal to any ad hoc, empirical assumptions, and explains how periodic ice
lenses can form with or without the presence of a frozen fringe. The proposed
mechanism is in good agreement with experiments, in particular explaining
ice-lens pattern formation, and surges in heave rate associated with the growth
of new lenses. Importantly for systems with no frozen fringe, ice-lens
formation and frost heave can be predicted given only the unfrozen properties
of the soil. We use our theory to estimate ice-lens growth temperatures
obtaining quantitative agreement with the limited experimental data that is
currently available. Finally we suggest experiments that might be performed in
order to verify this theory in more detail. The theory is generalizable to
complex natural-soil scenarios, and should therefore be useful in the
prediction of macroscopic frost heave rates.
