PS
r
Ice
B
Ice
Water
Water
A
E q u i l i b r i u m i n t e r fa c e
b e t w e e n i c e a n d w a t e r.
w h e n PL = PS
w h e n PL PS < 2 / r
PL
w h e n PL PS = 2 / r
Figure 5. Pistoncylinder model of ice growth. (After Everett 1961.)
Using the equilibrium condition of eq 47 (and not-
Once the top cylinder is ice-filled, further heat loss will
ing that a hemispherical cap has the maximum ( Ar/
result in either 1) upward movement of the upper pis-
ton, with flow of water from the bottom cylinder, or 2)
this maximum pressure (with Pl = 0) is the maximum
propagation of ice along the capillary. If the pressures
heaving pressure that can be reached in porous media.
on the cylinders are equal, Ps = Pl = P, then the inter-
Thus, he concluded that the maximum heaving pres-
face between phases is planar and ice cannot penetrate
sure is a function of pore size and interfacial energy
into the capillary. Ice will form in the top cylinder until
between the ice and water. If this heaving pressure
all water is removed from the bottom cylinder; i.e., frost
heave will occur. An example of this is needle ice*
exceeds the overburden pressure in a freezing soil, then
frost heave will occur. Here is a basis for understand-
growing at the soil surface. When the needles first begin
ing why ice can grow against an overburden pressure.
to grow, there is no overburden and no significant self-
Because of the pressure difference across the curved
weight, thus no chance for Ps to develop.
interface, the water can exist at a lower pressure than
If the pressure on the ice phase, Ps, can be main-
the ice on the other side of it. For a hemispherical ice
tained at a higher level than the liquid, then the chemi-
front in a pore, Ar/ V = 2/r; therefore, Everett (1961)
cal potential of the ice in the cylinder (bulk ice) will
concluded that heaving pressure is inversely propor-
increase. (For example, as the needle ice at the ground
tional to the size of the pore radius, r. Note that the
surface grows in length, the weight of the ice exerts a
pressure difference maintained across an ice/water inter-
positive ice pressure in the pores at the soil surface.)
face can arise from a reduction of the liquid water pres-
The freezing temperature becomes depressed and ei-
sure as well as from an increase in the bulk ice pressure.
ther the ice will melt or heat will be withdrawn until, at
Everett's model considered the mechanical equilib-
the new equilibrium temperature, there is a curved inter-
rium between ice and water in porous materials, but
face between the two phases. If the pressure difference
ignored the soil particle surface effects on the adsorbed
is constant between the ice and the water while further
water. As mentioned earlier, a complete thermodynam-
heat is removed, the bulk ice will again grow in the top
ic equilibrium formulation of the problem would con-
cylinder as described above. If Ps increases to the point
sider thermal, mechanical, and chemical equilibrium.
at which the chemical potential of the ice in the piston
This was the approach taken by R.D. Miller and his
exceeds that of a hemispherical cap of ice between the
students (e.g., Miller et al. 1960, Miller et al. 1975),
ice and water in the pore, then ice growth proceeds down
discussed in the next section.
the capillary.
Miller and Loch
* Everett (1961) referred to the needle-ice as "hoarfrost"; however,
Miller et al. (1960) and Miller et al. (1975) accounted
hoarfrost refers to the deposition of ice crystals on objects by direct
for the osmotic effects related to films adsorbed on soil
sublimation from water vapor.
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