Journal of Crystal Growth 237–239 (2002) 2194–2198

Relationship between growth rate and supercooling in theformation of ice lenses in a glass powder
Kunio Watanabe*
Department of Sustainable Resources Science, Faculty of Bioresources, Mie University, Tsu 514-8507, Japan

Abstract
When a saturated powder is cooled below the freezing temperature of the interstitial water, the water sometimes
forms ice lenses, which are layers of ice from which all particles have been expelled. To clarify the relationship between
the rate of ice lens growth and the temperature at the growth surface, ice lens formation in a water-saturated, ﬁnepowder consisting of uniform-sized, spherical, glass micro-particles was observed in situ, using a microscope and adirectional freezing apparatus. When the powder was frozen rapidly, the ice lenses grew faster, at lower temperatures,and formed thinner layers. The ice lens grew at a rate that was linearly correlated with the degree of supercooling at thegrowth surface, and was signiﬁcantly slower than that of ice in water. The results suggest that the particles near thegrowth surface inhibit ice lens growth. r 2002 Elsevier Science B.V. All rights reserved.
PACS: 47.55.Mh; 44.30.+v; 92.40.Sn; 93.60.S
Keywords: A1. Directional solidiﬁcation; A1. Mass transfer; A1. Segregation

1. Introduction
When water containing ﬁne particles, such as
soil, is frozen directionally, the water sometimes
forms regions of ice that are almost devoid ofparticles; these regions are called ice lens [1,2].
When ice grows in water, its growth rate isexpected to be proportional to the degree ofsupercooling at the growth surface. However, if
there are particles in the water, the particles affect
the growth of the ice. Vignes and Dijkema [3]*Corresponding author. Department of Sustainable Resources Science, Faculty of Biosciences, Mie University, 1515
Kamihama, Tsu 514 8507, Japan. Tel.: +81-59-232-1211-2377;
fax: +81-59-231-9571.
E-mail address: kunio@bio.mie–u.ac.jp (K. Watanabe).

conﬁrmed that the growth rate of ice in a capillarydepends only on the degree of supercooling at thegrowth surface. Ozawa and Kinosita [4] observed
ice growth on a microporous ﬁlter, and showedthat the growth rate is proportional to thedegree of supercooling at the freezing front(3.4 mm s11C1), and is independent of pore size.
Watanabe et al. [5] experimentally determined theproportionality constant of growth rate of an ice
lens in silt (1.0 mm s11C1). However, we still needto understand the growth process of ice lenses inﬁne powder. This paper presents the results of insitu observations of ice lenses in a powder. The
relationship between the growth rate of the ice
lenses and supercooling at the growth surface, andthe inﬂuence of surrounding particles on ice lens
growth, are then determined.

0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.
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K. Watanabe / Journal of Crystal Growth 237–239 (2002) 2194–2198

2195

2. Experiments

3. Results

The sample powder that we used consisted of
uniform spherical glass particles with a diameter of
2.2 mm [6]. The powder was saturated with
distilled, deionized, and degassed water (SampleP). The initial water content was 7872% by mass.
The sample was poured into sample cells thatconsisted of two glass slides and acrylic spacers.
The internal volume of each cell was
20  70  3 mm3. Some sample cells included twothermocouples, accurate to 0.031C, to monitor
internal temperatures. An identical cell was prepared containing only distilled water for comparison (Sample W). Each cell was cooled to 21C
before being frozen.
The samples were frozen using the directional
freezing apparatus (Fig. 1) that was designed byNagashima and Furukawa [7]. The sample cell wasforced to move transversally at an arbitrary
constant rate, R; in a ﬁxed temperature gradient.
In this experiment, the temperature gradient
G ¼ 0:191C mm1 was observed when the sample
cell was pulled at a rate of 0:04pRp2 mm s1. A
microscope was combined with the directional
freezing apparatus. Images were captured at 1-min
intervals; each image was divided into a 12.5-mm
grid by computer, and the location of ice lenses
was recorded. The temperature at the growthsurface of the ice lens was determined by tracing
the temperature gradient onto the image.

3.1. Directional freezing of water-saturated powder

Temperature

Temperature (˚C)

IIIII

R = 0.06 µm s-1
R = 0.3 µm s-1
R = 0.5 µm s-1
R = 0.7 µm s-1

0.4
0.2
00

400
800
Elapsed time (min)1200

Fig. 2. Change in temperature in powder during directional
freezing experiment. The pulling rate R is measured in mm s1.
In regions I, II and III, the sample was frozen without sample
cell movement, with R before ﬁxed temperature gradient wasestablished, and with R in the ﬁxed temperature gradient,
respectively.

0˚C32mm

THCopperBlocksSaturatedPowder

CopperCold Blocks

0.6IZIce Lens

RI

0.8

TH

TL
TL

Fig. 2 shows the temperatures measured by thethermocouples in Sample P during the experiment.
Fig. 3 shows the ice lenses observed in Sample P,
when R ¼ 0:4 mm s1 was applied for 250 min. At
time zero, the sample cell was placed in thedirectional freezing apparatus and the freezing

WarmFig. 1. Schematic illustration of the unidirectional freezing
experiment. A temperature gradient was established along theZ-direction. The sample cell was pulled at constant rate, R;
through the temperature gradient.

Fig. 3. Ice lenses in water-saturated powder. Right-hand iswarmer and left-hand is colder. The temperature gradientestablished along the Z-direction. IL1-4 are ice lenses observed
during R ¼ 0:4 mm s1 was applied. The ice lenses appear blacksince background can be seen through ice lenses, and thepowder appears white.

K. Watanabe / Journal of Crystal Growth 237–239 (2002) 2194–2198

front began to advance in the sample. It took
about 100 min to establish a ﬁxed temperature
gradient (Fig. 2-I). During this stage of freezing, aseries of ice lenses formed from the cold side as anintermittent layer (Fig. 3-I). After 150 min, R wasapplied to the sample. After a short initialtransition stage (Fig. 2-II), the temperature decreased linearly with time (Fig. 2-III), since thethermocouples moved at a constant rate. The
length of the initial transition stage decreased withincreasing R: Under R; the freezing front wasvirtually ﬁxed in the view of the microscope, andrepeated generation and growth of ice lenses wasobserved at the freezing front (Fig. 3-III, IL1-4).
At this stage of freezing, ice lenses grew at aconstant rate and formed with virtually the same
thickness and spacing.

thickness
Mean length (mm)2196

2spacing

100

0.2

0.4
0.6
R (µm/s)

0.8

1Fig. 5. Mean thickness of the ice lens and mean spacing
between them as a function of R:

3.2. Ice lens formation under a constant pullingrate, RFig. 4 shows the growth rate of an ice lens andthat of ice in water. The ice lens grew more slowly
than R; while the ice in water grew at almost thesame rate as R: When R was slower than0.1 mm s1, a single ice lens continued to growuntil it extended more than 4 mm, and the growthrate was close to R: On the other hand, when R
was faster than 0.2 mm s1, new ice lenses appeared
repeatedly at the freezing front, and formed
several layers while the sample was moved above
4 mm. With increasing R; the growth rate of

Growth rate (µm/s)

Sample P
Sample W

0.3
0.2
0.1
000.2

0.4
0.6
R (µm/s)

0.81

Fig. 4. Relationship between growth rate of ice and R: The
Sample P and Sample W are the ice lens in water-saturated
powder and ice in water, respectively.

Growth rate (µm/s)

Sample P
Sample W

0.6
0.4
0.2
00

0.1

0.4
0.2
0.3
Supercooling, T0 - T(˚C)

0.5

Fig. 6. Relationship between growth rate of ice and degree ofsupercooling at the growth surface.

individual ice lenses lagged behind R; and approached 0.24 mm s1 asymptotically. When R was
faster than 2.0 mm s1, no ice lenses were observedin the sample.
Fig. 5 shows the mean thickness of ice lenses
and the mean spacing between them as a functionof R: Ice lenses were thinner when R was high,while no good relationship between spacing and R
was found. Fig. 6 shows the relationship between
the growth rate and the degree of supercoolingT0  T at the growth surface. When R was high,ice lenses grew faster, at lower temperatures. The
growth rate of the ice was linearly correlated withthe degree of supercooling (0.43 mm s11C1), andwas about one-thirteenth the growth rate of ice inwater (12.3 mm s11C1).

K. Watanabe / Journal of Crystal Growth 237–239 (2002) 2194–2198

4. Discussion
4.1. Ice lens thickness
Some studies have examined particle rejection
from the advancing ice surface, and an inverse
relationship between the critical freezing rate forparticle rejection and particle radius has been
proposed [8–10]. As water containing suspended
ﬁne particles freezes directionally, the particles are
rejected from the ice growth surface and thenaccumulate as a layer near the surface; the ice thensuddenly engulfs all the particles once the layer
reaches a certain thickness [11]. A similar situation
occurs during ice lensing. To generate a new ice
lens, the ice must engulf the layer of particles that
accumulated at the surface of the previous ice lens.
The thickness of the particle layer depends on thethickness of the previous ice lens, Lil : If we assume
that the critical growth rate of an ice lens isdetermined by Lil (as is particle radius for therejection of an isolated particle), the relationship isexpressed as Ril ¼ A=Lil : Substituting the experimental results (Fig. 5) into this relationship, thenA¼ 0:86  104 s1 : Although the physical meaning of A has not been investigated, it may dependon the difference in the surface energy between
particle–water and water–ice, and on the viscosityof the water and the packing of the particles near
the ice lens.
4.2. Ice lens growth rateConsidering the difference between the chemical
potential of the pre-melted water ﬁlm at thegrowth surface of an ice lens and that of the waterin the pores between particles, Kuroda [12]represented the growth rate of an ice lens, Ril ; asfollows:
Ril ¼

Qm ðT0  TÞ=T0  pDv;
akT=D þ d=k2197

pre-melted water ﬁlm, K is the hydraulic permeability of the powder, and d is the length in which
K is effective. According to Kuroda’s theory, thegrowth rate of an ice lens is affected by theresistance to the freezing process and the diffusion
process, akT=D; and the hydraulic resistance,
d=K: In our experiment, the growth rate of anice lens was Ril ¼ 0:43 (T0  T)+const. mm s1(Fig. 6), and p could be ignored. Using
T0 =Qm ¼ 2:71  1022 kJ1,d ¼ 0:02 m[6],a ¼ 3:00  1010 m, k ¼ 1:38  1023 J K1, andK ¼ 6:54  1016 m2s J1 [6], we obtain D ¼ 1:3 1014 m2 s1 : The value of D is ﬁve orders ofmagnitude smaller than that of bulk water at 01C[13], and is about one-twentieth that of the premelted water ﬁlm at the ice–air interface at –1.51C[14]. The results show that surrounding particles,
which surface has an attractive interaction withwater molecules, inhibit the mobility of watermolecules near the ice lens.

5. Conclusions
Directional freezing experiments were carriedout on an unconﬁned, water-saturated powder
consisting of uniform-sized, glass micro-particles.
The formation of ice lenses in the powder wasobserved; the thickness and growth rate of the ice
lenses, and the temperature at the growth surface,were investigated under a constant pulling rate, R:
When R was high, the ice lenses grew faster and atlower temperatures. The growth rate of the ice
lenses was linearly correlated with the degree ofsupercooling at the growth surface, and wassigniﬁcantly slower than that of ice in water. These
results suggest that particles near to the growthsurface inhibit ice lens growth through the inhibition of the mobility of water molecules.

ð1Þ

where Qm is the latent heat of fusion of ice permolecule, T0 is the melting temperature of ice, p is
the overburden pressure, Dv is the molecularvolume difference between ice and water, a is themolecular distance, k is the Boltzmann factor, D isthe self-diffusion coefﬁcient of a water molecule inReferences[1] J.G. Dash, H.-Y. Fu, J.S. Wettlaufer, Rep. Prog. Phys. 58(1995) 115.
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