2. Factors and theories of ice lensing

2.1 Introduction
Soil is a very complex system, made up of a heterogeneous mixture of solid,
liquid, and gaseous materials.

Some water in soil remains liquid well below themelting point Tm = 0 °C (unfrozen water).

In order to consider ice lensing, we firstneed to understand the mechanisms of water and heat migration and the liquid-solid
phase transition of water during soil freezing.
The classic studies on frost heave associated with ice lensing were carried out
by Conte (1850) and Taber (1929).
early Showa era (1926-1988).

In Japan, frost heave has been studied since the

For instance, the Railroad Bureau studied measures totake against frost heave (Nakaya, 1942).

Later, studying frost heave became a major

topic in civil engineering, with research into measures against frost heave, theutilization of frozen soil, and soil stabilization.
Soil freezing was regarded as a unidirectional heat transfer problem until the
1930’s.

It was then understood to be a problem of coupling the water and heat fluxes

with a boundary at freezing front, although the mechanism of ice lensing was notunderstood.

After 1950, numerous theoretical and semi-empirical models weredeveloped, with the intention of clarifying the mechanism of ice lensing or predicting
and overcoming damage due to frost heave.

In the 1960’s and 70’s, frost heave wasdescribed by capillary theory, which can explain the generation of an ice lens, and bysemi-empirical hydrodynamic models.

However, these models do not sufficiently

explain the formation of intermittent layers of ice lenses and the soil freezing process.
In the mid-1970’s, a secondary frost heave theory that can explain the formation ofintermittent layers was proposed.

Takashi’s theory and concept of segregation- 10 -

potential have also been used to predict and overcome frost heave.

Presently, themodels most often used for frost heave and ice lensing are the capillary theory, thesecondary frost heave theory, and the segregation potential concept.

Nowadays, it isbelieved that these models and theories almost completely explain ice lensing undervarious conditions.

These models and theories, however, still leave much to bephysically explained, and some have not been developed completely.

Moreover, there

are other problems, as too many parameters are required for calculation and thedefinitions are too complex.

Recently, approaches to understanding ice lensing basedon thermodynamics, crystal growth, and intermolecular forces have appeared.
Nevertheless, no theories or models fully explain the mechanism of ice lensing.
This chapter first summarizes the main factors involved in soil freezing and ice
lensing, and the models and theories that have been proposed.

Then, it presents

problems for considering ice lensing and describes the purpose of the study.

2.2 Main factors considered in soil freezing
2.2.1 Unfrozen water
Faraday (1859) believed that liquid water exists on the surface of ice attemperatures below its freezing temperature.

This water, which does not freeze below

the bulk freezing temperature, is called unfrozen water.

Although most soil water

freezes when it is cooled below 0 °C, releasing latent heat, water adjacent to particles
does not freeze, even when it is cooled substantially.
unfrozen water on the surface of particles.

Therefore, there is a film of

It is believed that water flowing throughthe film plays an important role in the growth of ice lenses (e.g. Loch and Kay, 1978;
Horiguchi and Miller, 1980; Ishizaki, 1995; Watanabe and Mizoguchi, 1997).

Theamount of unfrozen water in soil decreases drastically near 0 °C and decreases further
with lowering temperature.

A change in the amount of unfrozen water strongly- 11 -

influences soil properties, such as strength, permeability, and thermal conductivity.
Furthermore, the amount of unfrozen water will change with pressure, the surface areaof soil particles, soil components, and so on.

The factors affecting the amount of

unfrozen water include specific surface area, temperature, overburden pressure, osmotic
potential of the soil solution, fine pore geometry of mineral grains, particle packing
geometry, surface charge density, and exchangeable ions (Anderson and Tice, 1972).
The presence of unfrozen water is accounted for using the free energy concept.
When water is cooled below 0 °C, ice, which has an ordered molecular arrangementwith lower energy, is more stable than liquid water, which has a disordered moleculararrangement.

On the other hand, breaking hydrogen bonds on the ice surface increases

the interfacial free energy, γi.

When liquid water exists between ice and the soil

surface, the interfacial free energy is lower than when ice contacts the soil surface
directly:
γis > γiw + γws,(1)where, γis, γiw, and γws are the interfacial free energy per unit area between ice and soil,
ice and liquid water, and liquid water and soil, respectively.

The film of unfrozen

water exists between ice and soil particles in order to minimize the free energy of thesystem considered.

The presence of unfrozen water has also been explained from theviewpoint of intermolecular forces (Dash et al., 1995).
Solutes and pores in soil induce freezing point depression, so that they have agreat effect on the presence of unfrozen water.

The depression of the freezing point,∆Tf, associated with a solute is known as Raoult’s Law:

- 12 -

∆Tf = -Kf C ,(2)

where, C is the molality, i is Van’t Hoff’s factor, which indicates the electrolytic
dissociation of the solute, and Kf is the molar cryoscopic constant.
1.86 K kg/mol.

Kf in soil is about

If water in salty soil, for instance, contains a solute with aconcentration of 0.025 mol/kg and i = 2, we obtain ∆Tf = -0.1 °C.

The depression of

the freezing point associated with the capillary of a cylindrical pore with radius rp is
explained by the balance between the free energy of the system considered and thesolid-liquid interfacial energy and is given by∆Tf =3vw Tm γiw
,
L rp

(3)

where, vw is the specific volume of water, T m is the bulk melting point of water, and L is
the latent heat. With values vw = 20 × 10-6 m3/mol, Tm = 273.15 K, L = 6000 J/mol, and
γiw = 35 × 10-3 J/m2, the depression of the freezing point is 1.0 °C when rp = 0.1 µm.
Many methods have been used to measure the amount of unfrozen water infrozen soil, e.g. isothermal calorimetry (Anderson and Tice, 1972), differential scanningcalorimetry (Handa et al., 1992; Uchida et al., 1998), nuclear-magnetic resonancemethods (Tice et al., 1982; Tice and Oliphant, 1984; Ishizaki, 1995; Ishizaki et al.,
1996; Watanabe and Mizoguchi, 1997), and time domain reflectometry (Topp and Davis,
1985; Spaan and Baker, 1995).

NMR and DSC are best used for accurate

measurements in the laboratory, while TDR is best used for field observations, due to its
portability. In addition, there are some empirical formulae for predicting the amountof unfrozen water that are based on these experimental results (e.g. Anderson and Tice,
1972; Ishizaki et al., 1996; Watanabe and Mizoguchi, 1997).

- 13 -

2.2.2 Ice formation
There are two types of ice in freezing soil. One is pore ice, which forms
among soil particles in situ and barely causes the particles to move.

The other is an ice

lens, which forms by excluding particles and segregates from the soil matrix.

Bothforms of ice have a crystal structure of Ih and form perpendicular to the direction ofheat flow.

Thin section observation has also shown that the crystals in an ice lens

grow along its a-axis.
Ice formation in soil is regarded as a phase transition phenomenon from liquid
to crystal.

When soil water is cooled below the melting point, ice has a lower energy

than liquid water. Therefore, the change in the free energy explains ice formation:

G = H - TS,

(4)

where, G is the Gibbs free energy, H is enthalpy, T is temperature, and S is entropy.
Ice forms to minimize the free energy G in a temperature and pressure envelope.
Assuming that ice forms at Tm, where the solid and liquid phases have the same freeenergy (∆Gi = ∆Gw), this can be generalized in the Clausius-Clapeyron equation asvw dpw - vi dpi = L dT ,Tm

(5)where, v w and v i are the specific volumes of liquid water and ice, respectively, and pw
and p i are the respective pressures of liquid water and ice.

Ice formation has been

expressed using equation (5) in numerical studies of the mechanism of ice lensing andfrost heave (e.g. Radd and Oertle, 1973; Biermans, 1978; Gilpin, 1980; Nixon, 1991;
Black, 1995; Miyata and Akagawa, 1996, 1997).

- 14 -

2.2.3 Water flow during soil freezing
Soil water flows downward due to gravitation and flows upward due toevaporation and transpiration.
is gravity.

The major driving force of water flow in saturated soil

In saturated soil, the gravitational flow may be written using Darcy’s Law.

In addition, capillary force plays an important role for flow in unsaturated soil.
Frozen soil is permeable, since it contains liquid (unfrozen) water.

If different

parts of frozen soil are at different pressures or temperatures, the soil water will flow
between them.

It has been experimentally confirmed that Darcy’s Law also governs

the water flow in frozen soil (Japanese Society of Geotechnical Engineering, 1994).
The water flow in freezing soil is considered to arise from gravity, ice lensing, and heattransfer.

The full mechanism of such flow, however, is not yet clear, posing a great

problem for frost heave studies.
The hydraulic conductivity of frozen soil depends on the nature of the soil andthe amount of unfrozen water.

Since the amount of unfrozen water declines sharply

with temperature, the hydraulic conductivity strongly depends on the temperature.

It isreported that the hydraulic conductivity of soil decreases by ten times when thetemperature changes from 0 °C to –1 °C (Japanese Society of Geotechnical Engineering,1994).

The hydraulic conductivity and its change with temperature are important

properties when considering the mechanism of frost heave, and a number of researchers
have therefore tried to determine these values (e. g. Loch and Kay, 1978; Konrad, 1980;
Ishizaki et al., 1985; Black and Miller, 1990; Andersland et al., 1996).

2.2.4 Heat transfer during soil freezing
When considering heat transfer in soil, the thermal properties of the soil, suchas specific heat, heat capacity, thermal conductivity, and thermal expansion, areimportant.

The thermal properties depend on the nature and proportions of the soil

- 15 -

elements, i.e. soil minerals, water, ice, and gases.
differ from those of unfrozen soil.

The thermal properties of frozen soil

In addition, latent heat arising from ice formationmust be considered in soil freezing.
In soil, heat is transferred mainly by conduction.

When mass transfer andlatent heat are neglected, the heat transfer in a non-steady state is described by the
Fourier equation:

Q = -k ( T2 - T1 ) A ,d(6)

where, Q is the heat flux, T1 and T 2 are the respective temperatures at points 1 and 2, dis the distance between 1 and 2, A is the area, and k is the thermal conductivity of thesoil. If the change in the thermal conductivity due to the change in temperature is
negligible, equation (6) becomes

k

∂2 T
∂T ,
=ρC
2
∂t
∂x

(7)

where, ρ is the density of the soil, C is the specific heat of the soil, t is time, and x is
location.

Considering the latent heat arising from ice formation in an infinitesimalregion, the heat transfer in a non-steady state is given by

k

∂2 T
∂T
= ρ C - - L ρ wφ .
2
∂t
∂x

(8)

Consequently, heat transfer in freezing soil can be expressed by applying equations (7)and (8) to the frozen and unfrozen regions.

- 16 -

2.3 Ice lensing theories and models
Soil freezing is divided into two types by the condition of the water supply.
One is soil freezing in a closed system, in which soil water is prohibited from external
exchange, and the other is soil freezing in an open system, in which soil water is
exchangeable.

When saturated soil is frozen in a closed system, it expands by about 9

per cent of soil.

When soil freezes in an open system, huge ice lenses form in somecircumstances.

Soil freezing is also separated into water-saturated and unsaturated

conditions. Generally, the soil near the ground surface is unsaturated and soil deeper
underground is saturated.

In order to simplify the analysis, many theories and models

of ice lensing deal with saturated soil in an open system.
Many types of apparatus and experimental conditions are used to observe ice
lenses.

Therefore, when we deal with ice lensing theories and models, we have to pay

attention to many other factors, such as the boundary conditions of temperature, water,
and pressure, the freezing direction, sample cell size, friction between sample and cell,and total elapsed time.
To study ice lensing, we first need to find the conditions that initiate ice lensing.
These conditions should correspond to the criteria under which frost heave occurs.
Many empirical studies have sought to determine these conditions, especially from theviewpoint of finding factors affecting frost susceptibility.

The criteria for frost heave

include various properties, such as pore size, specific surface area, pF, height ofcapillary rise, and consistency. For example, Chamberlain (1981) found more than
100 criteria.

Recently, criteria based on particle size, and criteria coupled to particle

size and other factors, are often used.

Now, it is believed that the amount of ice lens

growth (i.e. the amount of frost heave) can be roughly estimated using a formula
incorporating these criteria and temperature.

- 17 -

2.3.1 Capillary theory
The capillary theory was proposed and developed by Sill and Skapski (1956),
Gold (1957), and Everett (1961).

Experimental studies that applied this theory havebeen presented by Penner (1966, 1967), Jackson and Chalmers (1958), Jackson andUhlmann (1966), Jackson et al. (1966), Vignes and Dijkema (1974), Vignes (1977),
Biermans et al. (1978), and Ozawa and Kinosita (1989).
The capillary theory regards an ice lens as ice growth at the freezing front. At
the solid-liquid interface at equilibrium, the pressures of solid and liquid differ due to
The pressure difference, ∆psl, is given by the Young-Laplace

the surface tension.
equation:∆psl = γsl ( r1 + r1 )
s

l

,

(9)

where, γsl is the surface tension and rs and rl are the radii of the interfacial curvature of
the solid and liquid, respectively.

Assuming a spherical interface between the ice andliquid water in a pore with radius rp, the pressure difference is given by2γ∆pil = pi - pl = r il .

(10)

p

In other words, equation (10) indicates the pressure under which ice can penetrate apore.

If the pressure difference between ice and water satisfies the following

inequality, the ice will penetrate the pore.
2γ
pi - p w > r iw
p

.

(11)

Therefore, the ice will not move any particles and no ice lens will occur.

On the other

hand, if the pressure difference satisfies the following inequality, the ice will not be able
to penetrate the pore.
2γ
pi - p w < r iw
p

.

(12)

- 18 -

Now the ice will exclude particles and an ice lens will form.

From equations (5) and(10), the temperature that will generate an ice lens is given by∆T =2γiwTmvw
,
rp L

(13)

where, vw is the specific volume of water and L is the latent heat of melting.
Summarizing capillary theory, it is obvious that pore size is the most importantfactor contributing to ice lensing.

In this theory, an ice lens occurs when the ice-water

meniscus cannot penetrate a pore with radius of rp, as shown in Figure 3.

The ice lens

grows at the freezing front.
The frost heave model based on capillary theory regards the heaving pressure
as the pressure at the ice lens’ growth surface.

From equation (12), the heavingpressure is given by2γ
pi = p w + r iw
p

.

(14)

According to equation (14), however, the heaving pressure depends only on soil
characteristics and does not depend on temperature, temperature gradient, or freezingrate.

It was found that this model underestimates the maximum freezing pressure in

non-colloid soils (Penner, 1967; Radd and Oertle, 1973).

Furthermore, although thismodel can explain the generation of an ice lens, it cannot explain why an ice lens forms
intermittent layers.

Miller (1972) assumed that an ice lens grows at a site colder than

the freezing front, and called the ground expansion due to the ice formation associatedwith capillary theory preliminary frost heave.

- 19 -

Fig. 3.

(a)Spheres, (b) schematic drawing of ice-water interface prior to propagating

through pore restriction between three touching spheres (after Penner, 1966).

- 20 -

2.3.2 Hydrodynamic model
In the hydrodynamic model, soil freezing is described by water migration
coupled with heat transport (Harlan, 1973; Guymon and Luthin, 1974; Guymon et al.,
1980; Hromadka et al., 1981).

A fundamental equation of this model is the Clausius-Clapeyron equation (5), which is used to calculate the pressure of unfrozen water in apartially frozen region.

Then, the water migration is described using a modified

Darcy’s Law, with the assumption that ice in the partially frozen region does not
influence the pressure gradient for water migration.
temperatures below freezing point.

The amount of unfrozen water is given by aformula that depends on the temperature.
transport.

In this model, an ice lens grows at

The water migration is coupled with heat

In this model, frost heave occurs when the ice content exceeds somefraction of the porosity (85~90%).

Although the hydrodynamic model can calculate

water conditions and the temperature profile of the system, it cannot explain thepressure condition and the mechanism by which intermittent layers of ice lenses form.

2.3.3 Secondary frost heave theory
In 1972, Miller proposed the secondary frost heave theory.

Miller assumed

that an ice lens grows at a site colder than the freezing front, and named the region
between the ice lens growth surface and the freezing front ‘the partially frozen region’(frozen fringe).

In this theory, an ice lens occurs where the effective stress of soil

particles in the frozen region becomes zero.

This is virtually the only theory that can

describe the formation of intermittent layers of ice lenses, so it has been widely reported
and applied (e.g. Miller and Koslow, 1980; Gilpin, 1980; Hopke, 1980; O’Neill andMiller, 1982, 1985; Holden, 1983; Black and Miller, 1990; Nixon, 1991; Padilla andVilleneuve, 1992).

Moreover, Hoekstra (1966) and Loch and Kay (1978) performedexperimental studies to confirm the presence of the frozen fringe.

- 21 -

In the secondary frost heave theory, the structure of the frozen fringe isbelieved to be that shown in Figure 4. The pores in the frozen fringe are saturated withpore ice and unfrozen water.

The pore ice, which is connected to the growing ice lens,exists as a single rigid body throughout the frozen zone and moves along the heat flux
with a uniform velocity (the heaving rate).
pore by regelation.

In this case, pore ice moves from pore toRegelation occurs continuously, while water and heat circulation

happen locally.
Summarizing the secondary frost heave theory, it is obvious that the stress ineach phase is the most important factor.

Miller (1978) considered that frost heave will

occur if the neutral stress exceeds the overburden pressure.

Therefore, he determined

the neutral stress from the pressures acting on pore ice and pore water.
pi = σe + σn = σe + χ pw + (1 - χ) pi

,

(15)

where, p is the overburden pressure, σe is the effective stress, σn is the neutral stress, andχ is the stress partition factor.

The secondary frost heave theory explains themechanism of ice lensing as shown in Figure 5 (Miller, 1980).

First, the system canalmost heave; the neutral stress σe is almost high enough to lift the load σ.

To initiate

heaving, we have only to lower the temperature, thereby increasing the ice pressure pi
until σn reaches σ and σe reaches zero.

At this time, an ice lens will appear in thelocation where the ice pressure pi exceeds the overburden pressure p.

The ice pressure

declines to equal σ at the base of the ice lens, since the ice lens supports the pressure,thereby decreasing pi and σn, and increasing σe.

Once the ice lens starts growing,

however, we find that the surface temperature must be lowered further to sustain
heaving.

- 22 -

Fig. 4.

Schematic diagram of the frozen fringe, with ice lens above (after O'Neill andMiller, 1985).

- 23 -

Fig. 5. Principles of the secondary frost heave theory.
(a) Schematic representation of conditions near the freezing front when secondary
heaving is in progress.

(b) Profiles of stress near the freezing front during secondary

frost heaving. At left, profiles represent the moment when a new lens is about toappear. At center, these profiles have shifted moments after a new lens has been
initiated. At right, another new lens is about to appear at a lower level at some latertime (from Miller, 1980).

- 24 -

In the secondary frost heave theory, the heaving pressure is described by thetemperature at the base of the ice lens and the pore water pressure.

The maximum

heaving pressure is given bypmax = ρi (pw L∆T
+
) ,
ρw
T

(16)

where, ρ is density.
The secondary frost heave theory is the theory currently used most often,
because it can deal with the formation of intermittent layers of ice lenses.

However,

there are some problems with this theory, as its assumptions have not been demonstrated
adequately. For example, the presence of a rigid-body of pore ice and a frozen fringe
has not been confirmed experimentally.

Moreover, the stress partition factor, which isimportant for calculating the neutral stress, has not been theoretically verified. In
addition, we have to solve simultaneous partial differential equations in this model.

2.3.4 Osmotic model
Numerical models applying the secondary frost heave theory have been
proposed.

Horiguchi (1987) developed the osmotic model from the viewpoint of adiffuse double layer around soil particles.

In the secondary frost heave theory, it isbelieved that the effective stress propagates through connections among soil particles.
However, it is doubtful that soil particles are connected in highly frost-susceptible soils,
such as silty clay, since there is a film of unfrozen water around the particles.
Therefore, Horiguchi thought that soil particles have diffuse double layers on theirsurfaces, and are not connected when secondary frost heave occurs.

In addition, hecalculated the pressure conditions based on the osmotic pressure resulting from theconcentration difference between the ice lens, the diffuse double layer, and pore water.

- 25 -

Water flow depends on the calculated pressure gradient in the frozen fringe and thetemperature gradient of the liquid phase.

This model also estimates the influence ofoverburden pressure and the heat and water fluxes on the heaving rate.

2.3.5 Takashi’s theory
Takashi et al. (1974) experimentally measured the relationship between the
frost heaving pressure σu and the temperature of the ice lens growth surface:

σu = - ρi LT .
Tm

(17)

He assumed that the driving force of water transport is proportional to the difference
between the maximum frost heaving pressure σu and the effective stress σe (= p 0 - p w).
Then, he postulated that suction heaving occurs when σu > σe and drainage heaving
occurs when σu < σe (Takashi et al., 1979).

Takashi’s theory was expressedmathematically and the heaving rate is calculated fromVh = Vh0 + σ0 (1 +
σ

Vf0
Vf

,

(18)

where, Vh is the heaving rate, Vf is the rate of advance of the freezing front, andvariables with a subscript zero are experimentally obtained constants (Takashi, 1982).
In Japan, Takashi’s theory is often used for engineering, the design of underground
LNG tanks, and so on.

2.3.6 Adsorption force theory- 26 -

The mechanism of ice lensing has been considered from the viewpoint ofadsorption force (Takagi, 1980).

Taber (1930) also remarked on the adsorption force

as the driving force of frost heave.

In the adsorption force theory, it is believed that aninternal solid-like stress is formed in the adsorption water on the surface of soil particles.
With lowering temperature, the adsorption water freezes and the water film on theparticles becomes thinner.

In this case, water flows towards the freezing front in orderto keep the film thickness constant.

This theory describes how an ice lens grows near

the freezing front when this kind of water flow occurs.

In other words, this kind ofwater freezing means freezing of the heterogeneous film of water adsorbed between theparticle surface and the ice surface.
the absorption force.

The equilibrium thickness of the film depends on

An ice lens is generated when soil freezes with suction derived

from intermolecular forces.

In contrast, the freezing of homogeneous free pore water

is termed in situ freezing, since this mechanism does not create suction, i.e. the in situ
freezing front advances as in situ freezing progresses.

This model is incomplete andhas not been applied in practice.

2.3.7 Segregation potential concept
Konrad and Morgenstern (1980, 1981) defined the segregation potential (SP)as the ratio of the rate of moisture migration to the temperature gradient in a frozen soil
near the 0 ˚C isotherm.

In the models based on this concept, frost heave has beentreated as a problem of water supply to ice lenses.
In these models, water flow depends on suction derived from phase equilibrium
at the growth surface of the ice lens and hydraulic conductivity near the surface (in thefrozen fringe).

The hydraulic conductivity is determined by the amount of unfrozen

water. Empirically, the rate of moisture migration to the growth surface of the ice lens,V(t), is proportional to the temperature gradient in frozen soil near the 0 ˚C isotherm, i.e.

- 27 -

in the frozen fringe.

V(t) = SP(t) grad T f(t) .

(19)

The SP in field observations is defined in the same way as in laboratory
experiments.

The rate of moisture migration is calculated from the expansion ratio ofwater, porosity, and the amount of unfrozen water at the temperature at which an ice
lens is expected to segregate.

Konrad and Morgenstern (1982) reported that the SP

decreased exponentially with increasing overburden pressure.

The SP has been used toclassify the frost susceptibly of soil (Kujala, 1991).
Although these models cannot describe the details of the ice lensing
phenomenon, they are very good models for engineering, because if we measure the SP
we can easily calculate the moisture migration in soil and the amount of frost heave.
The SP has been applied to various conditions and experiments (e.g. Nixon, 1982;
Konrad, 1987, 1989; Konrad and Duguennoi, 1993).

2.3.8 Kinetic model
Kuroda (1985) considered that water is induced to flow toward an ice lens by
the chemical potential difference between the film of unfrozen water (quasi-liquid waterlayer) that exists between the ice lens and soil particles, ∆µq, and water that exists
among the soil particles, ∆µw. When the unfrozen water reaches equilibrium thickness,the chemical potential ∆µq equals the chemical potential at the base of the ice lens, ∆µi.
In this model, the relationship between the rate of water migration, V, and the chemical
potential difference between water and ice, ∆µqw, is derived in terms of a kinetic process,in which the rate of increase of the thickness of unfrozen water due to suction equals therate of decrease due to freezing.

- 28 -

V=∆µiw,
Rh + Rf(20)

where, R h is the resistance in the suction process and Rf is the resistance in the freezing
process of the water film.

Ozawa and Kinosita (1989) have confirmed this relationshipexperimentally.

2.3.9 Thermomolecular pressure model
The adsorption force theory describes how an unfrozen water film exists onparticles, and water migrates in a film that maintains a constant thickness due tointermolecular forces.

The kinetic model also considers that water flow to an ice lensoccurs so that the film can maintain its equilibrium thickness.
based on intermolecular forces has been proposed.

Recently, a new model

In this model, the intermolecularforces (van der Waals forces) in the water film are divided into thermomolecularpressure and hydrodynamic pressure, then the migration of unfrozen water and thegrowth of an ice lens are explained.
Dash (1989) explained the existence of unfrozen water between solid andliquid on the basis of intermolecular forces, and suggested that flow in the unfrozen
water film results from a temperature gradient. He regarded the thermomolecularpressure in the unfrozen water film as the driving force of frost heave.

Wettlaufer and

Worster (1995) explained the fluid mechanics of the unfrozen water film using thethermomolecular pressure, then described ice growth in a capillary among particles.
Wilen and Dash (1995) showed experimentally that the flow in the unfrozen water film
occurs along a temperature gradient.

Wettlaufer et al. (1996) proposed the lubrication

theory and modeled the flow. The fundamental mechanism of ice lens has been
discussed and explained from the perspective of such a thermomolecular pressure and- 29 -

lubrication theory (Worster and Wettlaufer, 1998).

This model considers that an ice

lens grows at below 0 ˚C, and gives the relationship between the rate of ice lens growth
and the degree of supercooling of the growth surface.

The thermomolecular pressure

model theoretically derives water and heat flows from intermolecular forces and can
explain various kinetic phenomena.

However, it is an unfinished model, and it cannot

explain why ice lenses form intermittent layers.

2.3.10 Thermodynamics approach
The growth of ice lenses and its mechanism have also been actively discussed
from the viewpoint of thermodynamics, although there are no complete models.
Fremond and Mikkola (1991) separated saturated soil into soil particles, liquidwater, and ice, and modeled frost heave on the basis of its energy balance and entropy.
This model handles water supply due to the freezing of pore water and transfers of porewater and heat, and it can describe the total amount of ice lens growth.
Ozawa (1997) found that frost heave is consistent with the second law ofthermodynamics.

He explained that water movement to the ice lens results from athermodynamic tendency to increase entropy in the whole system, which is kept in asupercooled state.
To express the state of the ice lens, most models use the generalized ClausiusClapeyron equation (5).

However the equation (5) was originally for a static state, andcan therefore only express limited conditions, at which the ice lens stops growing.
Miyata (1998) derived a dynamic equation of state from a thermodynamic model andexpressed its pressure condition.

- 30 -

2.4 Problems for theories and models of ice lensing
Summarizing the historical studies, there are two problems associated with
developing models and theories for ice lensing.

One is to clarify the microstructurenear the freezing front, which has been dealt with like a black box, i.e., to clarify thewater conditions and particle migration in the frozen fringe.

The other is to explain the

dynamic mechanism of ice lensing, in which the generation and growth of an ice lens isrepeated to form intermittent layers.

This study made experimental observations toclarify the main factors of ice lensing and the microstructure near the growth surface ofan ice lens.

Based on the experimental results, we developed a model to simulate ice

lensing in water-saturated porous media.
There are experimental problems involved in observing ice lens development
in soil.

Soil is heterogeneous in terms of texture, particle shape, electrical charge, andchemical composition.

Ice lensing is a complex crystal growth phenomenon that iscaused by various water and heat-related factors.

Observing the microstructure near

the ice lenses and clarifying the mechanism of formation are difficult because of the
non-uniformity of the sample and the complexity of the phenomenon, whereas to clarify
the ice lensing mechanism requires continuous direct observation without nonuniformity or complexity.

To be specific, for ideal freezing of a sample, the following

conditions are required: (1) Water in the sample does not solidify until the temperature
is below its bulk melting point.

(2) The size, shape, and distribution of pores are

uniform and it is easy to measure the liquid-solid phase transition in the sample.

(3)

The electric properties and surface conditions are known well enough to model.

(4) Itis easy to control the experimental conditions, e.g. temperature gradient, freezing rate,pressure conditions, and water conditions.
Ice lensing is known to occur in several different porous media (Derjaguin and
Churaev, 1986; Hiroi et al., 1989; Hiroi and Mizusaki, 1991; Wilen and Dash, 1995;- 31 -

Muto et al., 1998).

In this study, we prepared two porous media for experiments.

One was actual soil and the other a porous medium consisting of micro glass particles
that satisfies the required conditions (1) to (3).
Many methods using microscopic techniques for direct observation of water
migration and microstructure in soil and other porous media have been reported.
Kumai (1979) observed the ice configuration in frozen clay by scanning electron
microscopy. Colbeck (1982, 1985) directly observed the freezing front in a watersaturated, unconsolidated medium consisting of glass beads a few millimeters indiameter. Lu et al. (1994a, 1994b, 1995) observed the water interface that infiltrated a
porous media consisting of glass beads 0.1 to 0.01 millimeters in diameter, and showed
its interfacial shapes among the beads and hysteresis.

Since the surface moves duringfreezing, the growth surface of an ice lens could not be observed continuously, even if
these observation methods were applied to ice lensing.

In the field of crystal growth,

Uhlmann et al. (1964), Uhlmann and Jackson (1985), and Köber et al. (1992) developed
a unidirectional freezing apparatus that can control temperature gradient and freezing
rate independently, and used it to continuously observe crystal growth from liquid withor without encapsulating particles.

Using a similar unidirectional freezing apparatus,

Nagashima and Furukawa (1997) estimated the nonequilibrium effect of the anisotropicinterface kinetics of ice crystals.

In this study, we observed ice lensing using amodified unidirectional freezing apparatus that satisfies required condition (4).

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