Cold Regions Science and Technology 34 (2002) 103 – 110
www.elsevier.com/locate/coldregions

Amount of unfrozen water in frozen porous media
saturated with solution
Kunio Watanabe a,*, Masaru Mizoguchi baDepartment of Sustainable Resources, Faculty of Bioresources, Mie University, 1515 Kamihama, Tsu 514-8507, Japanb
Department of Biological and Environmental Engineering, The University of Tokyo, Tokyo 113-8657, Japan
Received 6 February 2001; accepted 17 October 2001

Abstract
In porous media, such as soil, some water do not freeze when cooled below 0 C. The amount of unfrozen water isdependent on temperature and the characteristics of the porous media and solute in water. In this study, three types each ofmonosized glass powder, silty soil, and clay were saturated with various concentrations of NaCl, KCl, and MgCl2 solutions andsubjected to freezing temperatures. The relationship between the reduction in temperature and the amount of unfrozen water
was measured using nuclear magnetic resonance (NMR) technique. The amount of unfrozen water decreased with the loweringof temperature. When a porous medium was saturated with a solution, the amount of unfrozen water increased with increasing
solute concentration, in the order NaCl  KCl < MgCl2. The resulting amount of unfrozen water was considered with respect tovan der Waals and Coulombic interactions and the Gibbs – Thomson effect. The amount of unfrozen water in frozen glass
powder and silty soil saturated with solution at concentrations lower than 0.1 mol l  1 can be estimated when its Hamaker
constant, specific surface area, distribution function, and initial solute concentration are given. D 2002 Elsevier Science B.V.
All rights reserved.
Keywords: Unfrozen water; Solute concentration; Nuclear magnetic resonance methods; Porous media; Supercooling

1. Introduction
Even when cooled below 0 C, some water in porous media, such as soil, remains unfrozen. The amountof unfrozen water in frozen porous media strongly
depends on the temperature, and affects the strengthand the hydraulic and thermal conductivity of the porous media. The hydraulic conductivity of soil drastically decreases as the temperature decreases (Burt and*Corresponding author. Tel.: +81-59-231-9583; fax: +81-59231-9571.
E-mail address: kunio@bio.mie-u.ac.jp (K. Watanabe).

Williams, 1976). The unfrozen water plays an important role in the freezing and thawing processes ofporous media. As soil freezes, water migrates throughsubzero areas and sometimes forms ice lenses thatcause severe frost heave (Horiguchi and Miller, 1980;
Ishizaki, 1995; Watanabe and Mizoguchi, 2001).
The reason why water in porous media does notfreeze at below bulk freezing temperature is attributed,
in the main, to the melting point depression of the water,which is induced by adsorption forces and curvature atthe particle surfaces. Since Faraday (1859) suggested
that premelted (unfrozen) water might exist on thesurface of ice, many studies concerned with surface
melting of ice have been undertaken (Dash et al., 1995).

0165-232X/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 2 3 2 X ( 0 1 ) 0 0 0 6 3 - 5

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K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

It is now understood that a film of unfrozen water exists
between the ice and the particle surface in frozen porous
media, lowering the free energy of the system. The
melting point of water is further depressed if it includes
solutes, and this depends on the concentration of thesolutes. These effects must be considered when considering the behavior and amount of unfrozen water infrozen porous media.
Many methods have been used to measure theamount of unfrozen water in frozen porous media,
including isothermal calorimetry (Anderson and Tice,
1972), differential scanning calorimetry (Handa et al.,1992), nuclear magnetic resonance (NMR) methods(Tice et al., 1978), and time domain reflectometry(Spaans and Baker, 1995). Ishizaki et al. (1996) measured the amount of unfrozen water in frozen porous
glass powder using an NMR technique and thenevaluated the thickness of the layer of unfrozen wateron the particle surface. Tice et al. (1984) introduced atechnique for correcting the amount of unfrozen water
resulting from high salt concentrations in partiallyfrozen soil. The amount of unfrozen water is highly
dependent on the physicochemical properties of the soil
such as specific surface area, surface charge density,and the suite of exchangeable ions (e.g., Anderson andTice, 1972). There are very few soils for which sufficient, high-quality data on unfrozen water exist.
Changes in the amount of unfrozen water due to soluteconcentration are not well understood. Therefore, it isimportant to know how much unfrozen water exists invarious frozen porous media. In this study, using a
NMR technique, we measured the amount of unfrozen
water in frozen glass powders, silty soil, and clay,
saturated with different concentrations of solutions.
The results are considered with respect to van der
Waals and Coulombic interactions and the Gibbs –Thomson effect.

returns to its initial orientation when the pulse iseliminated. NMR measures the voltage induced bychanges in the magnetic field. The voltage decay curveis called free-induction decay (FID). The peak FID
value is proportional to the amounts of water and ice inthe sample; the FID signal resulting from ice decreases
more rapidly than the signal from liquid water. Therefore, we can determine the amount of liquid water(unfrozen water) in a sample from the FID value(Ishizaki et al., 1996).

2. Materials and methods

2.3. Test procedure

2.1. Pulsed NMR methods

For NMR measurements, the prepared sample wascompacted in a Teflon tube that was 25 mm in diameter and then sealed with a rubber stopper to prevent
water evaporation. The tubes with samples were
immersed in a temperature-controlled bath containing
an ethylene glycol –water mixture as a coolant andallowed to reach a specified temperature. To measure

A proton of water bears a slight magnetic moment
originating from its spin and acts as a magnet thattends to align along a fixed external magnetic field.
The magnetic moment reorients when a pulse is applied to a sample set in a fixed magnetic field and2.2. Samples
The porous media that we used included three glass
powders (GP9.7, GP5.3, and GP2.2), fujinomori soil
(FJ), and bentonite (BN). Table 1 lists some characteristics of the porous media and Fig. 1 shows the poresize distribution of each medium. The mean diameter
was estimated from electron micrographs. The poresize distribution, specific surface area, and specificnanopore volume were determined by nitrogen adsorption. Each glass powder consists of monodispersed
particles and has frost susceptibility (Mutou et al.,
1998; Watanabe and Mizoguchi, 2000). The particles
are spherical with roughly the same diameter and theirsurfaces have nanopores with a uniform diameter(Fig. 1). Fujinomori soil is a silty soil with high frost
susceptibility (Watanabe et al., 1997); bentonite is aswelling clay.
Distilled, deionized, and degassed water and NaCl,KCl, and MgCl2 of 99.9% purity were used. The saturated porous medium was prepared by placing the
porous medium in a sealed chamber with the solution,
evacuating the chamber, allowing a few days for solid –vapor equilibrium to be established, adding solution,
and allowing the system to equilibrate for 1 day. The
initial water contents and the solute concentrations ofpore water are shown in Tables 1 and 2.

K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

105

Table 1
Characteristics of the porous media
Name

Glass powder

Fujinomori soil

Bentonite

GP9.7

GP5.3

GP2.2

FJ

BN

Gravel (%)
Sand (%)
Silt (%)
Clay (%)

0
0
97.3
2.78

0
0
45.9
54.1

0
0
12.6
87.4

0
15
61
24

0
0
0
100

Mineral components

SiO2, Na2O
(little), etc.
(very little)

SiO2, Na2O
(little), etc.
(very little)

SiO2, Na2O
(little), etc.
(very little)

montmorillonite,etc.(very little)

Mean diameter ofparticle (mm)Mean diameter ofnanopore (nm)Specific nanopore
volume (cm3/g)
Specific surface area (m2/g)
Specific gravity ofparticle (g/cm3)
Dry powder density (g/cm3)
Initial water content (gwater/g)

9.7

5.3

2.2

quartz, feldspar(little), kaolin, mica,
montmorillonite,
etc. (very little)
–

3.3

3.3

3.1

–

–

0.256

0.253

0.253

0.049

0.173

150.5
2.12

146.4
2.12

128.6
2.12

29.9
2.61

80.9
–

0.791
0.8

0.82
0.8

0.853
0.8

1.18
0.5

0.21
5

–

GP9.7, GP5.3, and GP2.2 are uniform-sized glass powders with mean diameters of 9.7, 5.3, and 2.2 mm, respectively. FJ and BN are fujinomori
soil and bentonite, respectively.

the sample temperature, not the coolant temperature, aplatinum resistance thermometer was inserted in a dry
sample packed in an identical Teflon tube placed next
to the sample tubes. After an equilibrium temperature
was reached, the sample tube was removed from thebath, the peak FID value was measured, and then the
tube was returned to the bath (Ishizaki et al., 1996).

A series of FID peak values was measured as the
sample thawed from  50 to + 10 C, using a
PRAXIS-2 pulsed NMR instrument. The applied magnetic field was 2.5 kG, the radio frequency was 10.7
MHz, and the pulse width was 12 mm. The amount of
unfrozen water was then calculated from the FID value(Ishizaki et al., 1996). We assumed negligible melting
and water migration in the sample during NMR measurement since the procedure was performed within 1
min, at an ambient temperature of 2 C.

Table 2
Initial solute concentration of pore water in FJ and BN

Fig. 1. Distribution curves of pores in the samples obtained by thenitrogen adsorption method.

Solute

NaCl

KCl

MgCl2

Solute concentration(mol/l)00.00170.0086
0.017
0.086
0.17
0.34
0.51
1.7

0
0.00130.0067
0.013
0.067
0.13
0.27
0.4
1.3

0
0.00110.0053
0.011
0.053
0.11
0.21
0.32
1.1

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K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

3. Results
The unfrozen water fraction, x, is calculated as theratio of liquid water to total water in a porous
medium; the ice fraction is 1  x.
Fig. 2 shows the unfrozen water fractions in theglass powders saturated with pure water (GP9.7,
GP5.3, and GP2.2). In Fig. 2, there were two significant temperatures, near  1 and  15 C, at which
the slope of the unfrozen water fraction curve changed
dramatically. As the temperature dropped below 0 C,
the amount of unfrozen water initially decreased
steeply to half of the total amount of water, and subsequently decreased gradually. Even at  10 C, morethan 30% of the water remained unfrozen. Below 15 C, the amount of unfrozen water decreased
more slowly and reached about 5% of the initialamount at  40 C. As described in the following
section, the melting point of water in nanopores withradii between 2 and 4 nm was depressed to  13 and 6.5 C, respectively. Since the glass powders had
many pores about 3 nm in radius (Fig. 1), they would
contain large amounts of unfrozen water at between
 1 and  15 C. The glass powders GP9.7 andGP5.3 had similar unfrozen water fraction curves inspite of their different particle sizes. This may result
from these powders having similar pore distributions(Fig. 1) and similar specific surface areas (  150 m2
g  1). On the other hand, GP2.2, which had a different
pore distribution (Fig. 1) and a smaller specific surface
area (128.6 m2 g  1), had slightly less unfrozen water

Fig. 2. Unfrozen water fraction in the glass powders. The initialwater content of the sample is 80%, which corresponds to an unfrozen water fraction x = 1.

Fig. 3. Unfrozen water fraction in fujinomori soil (FJ). The soil was
saturated with (a) NaCl, (b) KCl, and (c) MgCl2 solutions. The initialwater content was about 50%. In the figure, the Arabic numerals
indicate the concentration of the solutes that the FJs are saturated
with. The solid, dotted, and dashed lines indicate the concentration of
1, 0.1, and 0.01 mol l  1 evaluated by interpolation.

at below  20 C and slightly more unfrozen water
near 0 C. These results suggest that the amount of
unfrozen water in a frozen porous medium is strongly
affected by the pore distribution and the specific surface area. In actual soil, there is a range of particle
sizes with a consequent range in the distributions ofpore space and surface area, which affect the amountof unfrozen water in the soil.
Figs. 3a– c and 4a– c show the respective unfrozen
water fractions in fujinomori soil (FJ) and bentonite(BN) saturated with various concentrations of NaCl,

K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

107

concentration was also seen in FJ and BN saturated
with KCl and MgCl2 solutions. The unfrozen water
fraction curves for FJ and BN saturated with solutions
at concentrations of 1, 0.1, and 0.01 mol l  1 were
evaluated by interpolation (solid, dotted, and dashedlines in Figs. 3 and 4, respectively). Comparing the
curves for fujinomori soils, FJ + MgCl2aq had themost unfrozen water, and FJ + NaClaq had slightly
more unfrozen water than FJ + KClaq (i.e., FJ +MgCl2aq > FJ + NaClaq  FJ + KClaq). This tendency
was stronger at higher solute concentrations. In bentonite, the amount of unfrozen water was in thefollowing order: BN + MgCl2aq > BN + KClaq  BN +NaClaq. The melting point of water is depressed
according to the number of ions it contains. If thesolutes have similar magnitudes of electrolytic dissociation, the MgCl2 solution contains more anions thanother solutions, so that FJ and BN saturated with
MgCl2 solution would contain more unfrozen water.
The magnitude of electrolytic dissociation of the solutes, however, would vary with concentration andtemperature. Some cations in solution would be exchanged on the clay surface. Differences in the electrolytic dissociation of the solute and the interaction
between ions and clay surface may also affect theunfrozen water fraction curves.

4. Discussion

Fig. 4. Unfrozen water fraction in bentonite (BN). Bentonite wassaturated with (a) NaCl, (b) KCl, and (c) MgCl2 solutions. The
initial water content was about 500%. In the figure, the Arabic
numerals indicate the concentration of the solutes that the BNs are
saturated with. The dotted and dashed lines indicate the concentration of 0.1 and 0.01 mol l  1 evaluated by interpolation.

KCl, and MgCl2 solutions. In FJ and BN saturated withpure water, the amounts of unfrozen water decreased
with temperature. No significant temperature, such as
those shown in Fig. 2, was observed. In FJ and BN
saturated with NaCl solution, the unfrozen water fractions maintained their initial value at temperatures a
few degrees lower than 0 C because of the melting
point depression induced by the solute. Similar dependence of the unfrozen water fraction on the solute

4.1. Amount of unfrozen water and the degree ofsupercooling
When ice contacts a foreign substrate below its
bulk melting temperature, a thin film of unfrozen
water exists on the surface of the substrate, which
depends on intermolecular interactions (Dash et al.,1995). Consider a semi-infinite amount of ice separated by unfrozen water from a semi-infinite substrate.
The van der Waals intermolecular energy per unit area
between ice and the substrate is
FvdW ðdÞ ¼ 

A,
12pd 2

ð1Þ

where d is the equilibrium thickness of the unfrozen
water film and A is the Hamaker constant, which
depends on the dielectric properties of all threematerials in the layered system (Israelachvili, 1985).

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K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

The relationship between d and the degree of supercooling, Tm  T, is given by:ATm
d3 ¼ 
,ð2Þ
6pqs q Tm  T
where Tm is the bulk melting temperature of water, qs
is the density of ice, and q is the latent heat of solidification per unit mass (Worster and Wettlaufer, 1999).
When any electrolyte is present in the film of
unfrozen water between ice and a charged surface,
the ions screen the Coulombic interaction with anefficiency that depends on the finite charge density qs
of the surface and the ion density of the electrolyte.
For a monovalent electrolyte and a surface potential
less than 25 mV, the excess interfacial free energy
per unit area across an unfrozen water film of thickness d is approximated using the Debye – Hückel limit
FDH 

2q2s jd
e ,
jee0

ð3Þ

where j  1 = (ee0kT/e2nb)1/2, e is the dielectric constant of the film, e0 is the free space permittivity, k is
the Boltzmann factor, and nb is the bulk ion density(Wettlaufer, 1999). When the unfrozen water film isvery thin, the approximation must be modified, but wewill not deal with these complications here. Within thelimits described, the total excess interfacial free energyof relevance to the unfrozen water film in the presenceof electrolytes is the sum of FvdW(d ) and FDH(d ) (Eqs.
(1) and (3)). Considering the situation in which theunfrozen water film includes a single species of monovalent nonvolatile electrolyte, there are a fixed numberof electrolyte ions in the film, which maintain equilibrium with the interfacial charge. Wettlaufer (1999)
presented the relationship between supercooling andthe thickness of the unfrozen water film as"rﬃﬃﬃﬃﬃ
Tm RTm Ni Dcr2 Dcc Ni
Tm  T ¼
 3 
2
ql q
d
d
d
!#
ﬃﬃﬃﬃﬃ
r

r
Ni
 1 þ exp cðd  rÞ ,ð4Þddthe order of molecular diameter, and c is a constant(7.237  107 m  1/2 mol  1/2) when j = c(Ni/d)1/2.
In the case of water in a porous material, somewater will not freeze at below its bulk freezing point
as a result of the melting point depression due to curvature. The shift in freezing temperature, Tm  T, for acylinder of ice of radius r is known as the Gibbs –Thomson effect:
Tm c
,
ð5Þ
Tm  T ¼
qs qr
where c is the ice – liquid water interfacial free energy.
4.2. Calculating the amount of unfrozen water in frozen
porous media
When a water-saturated porous medium is cooled
below the bulk freezing temperature of water, as
shown in Eqs. (2) and (4), unfrozen water films witha thickness of d(T ) will exist on the particle surfaces.
Assuming that water in the vicinity of the surface has
the same density, qw, as bulk water, therefore, a watersaturated porous medium with specific surface area, a,
has an amount of unfrozen water per unit mass of
Us = ad(T )qw. On the other hand, if the particles have
nanopores on the surface, the water in the pores will
not freeze until it has cooled to a temperature determined by the pore size. Here, we assume that all thepores on the surface are cylindrical and that the
distribution of the internal radii of the cylinders
follows the distribution function g(r), as shown in
Fig. 1. If the interfacial and size effects act independently, the water in a cylindrical pore with radius r(T )will remain unfrozen at a temperature, T, as describedin Eq. (5). In this case, the porous medium has an
amount of unfrozen water per unit mass of:Z rðT ÞUp ¼
gðrÞdr:ð6Þ
0Then, the total amount of unfrozen water per unit
mass in a porous medium Uall can be calculated fromthe sum of these amounts of unfrozen water:
Uall ¼ Us þ Up

where ql is the molar density of the solvent, R is the gas
constant, Ni is the number of moles per unit area of theelectrolyte, Dc is the difference in free energy between
the dry and wet surfaces, r is a short-range cutoff of¼ qw adðT Þ þ

Z rðT Þ dðT ÞrðT Þ  dðT Þ
rðT Þ

2

!
gðrÞdr ,ð7Þ

K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

where (r(T )  d(T ))2/r(T )2 is a term that corrects theamount of unfrozen water on the surface in the pores.
The amount of unfrozen water in the glass powder
with a diameter of 9.7 mm was calculated (Fig. 5:
GP9.7) using Eq. (7) and the following parametervalues: a = 150.5 m2 g  1, c = 0.029 J m  2, qw =998,000 g m  3, qs = 917,000 g m  3, and Tm =273.15 K. The distribution function g(r) was obtained
from Fig. 1 and d(T ) and r(T ) were given by Eqs. (2)and (5), respectively. The Hamaker constant, A, forwater on a solid surface generally takes a value from 10  19 to  10  20 J. We tried various values of the
Hamaker constant and found that the amount of unfrozen water in the glass powder was best predictedwith A =  10  19.5 (Fig. 5). In the range Tm  T = 0– 5
K, most water in surface pores remained unfrozen andmade up the majority of the total amount of unfrozen
water. Pore water proceeded to freeze at Tm  T = 5–
18 K, causing the amount of unfrozen water to decrease. The strong peak in the distribution of pore size
shown in Fig. 1 contributed to the sudden decrease in
the amount of unfrozen water around Tm  T = 10 K in
Fig. 5. When Tm  T exceeded 20 K, the amount of
unfrozen water virtually followed Eq. (2) and decreased in proportion to (Tm  T )  1/3. With specificsurface areas a = 146.5 and 128.6 m2 g  1 and therespective g(r) shown in Fig. 1, the amount of unfrozen
water in GP5.3 and GP2.2 was also calculated. The
calculated values predict the amount of unfrozen waterin water-saturated glass powders well (Fig. 5).

109

Fig. 6. Calculated amount of unfrozen water in fujinomori soil,
compared with the value measured by the NMR technique.

With a = 24.5 m2 g  1, R = 8.314 J mol  1 K  1,
r = 1.7  10  10 m (Azouni et al., 1997), Dc = A/
12pr2 J m  2 (Wettlaufer and Worster, 1995), andd(T ) obtained from Eq. (4), the amount of unfrozen
water in fujinomori soil saturated with various concentrations of NaCl solution was calculated in asimilar way (Fig. 6). Soil is a very complex andheterogeneous system, and would not have a single
value of the Hamaker constant. Using the value of the
Hamaker constant for glass powders, the unfrozen
water amount in fujinomori soil saturated with purewater was well predicted, so that we assume that themean value of the Hamaker constant for fujinomori
soil corresponds to the value used for the glass
powders. With the lowering of temperature, the waterin the soil started freezing at a supercooling degree,Tm  T, according to the solute concentration; subsequently, the amount of unfrozen water decreased. The
calculated results were in good agreement with theexperimental data at concentrations lower than 0.1
mol l  1. The calculation tends to underestimate theamount of unfrozen water at higher concentrations,although the mechanism is still obscure. In addition,
the calculation cannot deal with swelling clays, suchas bentonite.

5. Conclusion
Fig. 5. Calculated amount of unfrozen water in glass powder. The
calculated values for GP9.7 with three values of the Hamaker
constant, A, are shown in the figure. (NMR) is the value measured
using a NMR technique.

Using an NMR technique, the amounts of unfrozen
water in frozen porous glass powders, fujinomori soil,
and bentonite were measured. The amount of unfro-

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K. Watanabe, M. Mizoguchi / Cold Regions Science and Technology 34 (2002) 103–110

zen water decreased with temperature. In the glass
powders, a large amount of unfrozen water caused bynanopores was observed. When the porous medium
was saturated with different solutions, the amount of
unfrozen water increased with increasing solute concentration, in the order NaCl  KCl < MgCl2.
The amounts of unfrozen water in glass powder
saturated with pure water and in fujinomori clay
saturated with NaCl solution were calculated whileconsidering van der Waals and Coulombic interactions and the distribution of pores on the surface of theparticles. It was found that the amount of unfrozen
water in a porous medium at a given temperature
could be estimated, if its Hamaker constant, specificsurface area, distribution function, and initial soluteconcentration are known.

Acknowledgements
The authors thank S. Akagawa and Y. Miyata forobtaining data on the sample characteristics.

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