PHASE CDMPOSTTION OF A
PARTIALLY
FROZEN
SOIL^
-By-
Yih-Wu Jame and D. I . Norum
For presentation a t
1972 Fall Annual Meeting
American Geophysical Union
San Francisco, December 4-7, 1972
Department o f Agricultural Engineering
University of Saskatchewan
Saskatoon, Saskatchewan S7N OW0
l ~ e s e a r c hPaper No. 11; Division of Hydrology, College of
Engineering, Univ. o f Saskatche~~an,
Saskatoon, Saskatchewan
INTRODUCTION
I n a f r o z e n s o i l , unfrozen l i q u i d w a t e r can e x i s t i n e q u i l i b r i u m
w i t h i c e over a l a r g e temperature range below O°C.
Equilibrium
thermodynamics t h e r e f o r e a p p e a r s t o be w e l l - s u i t e d t o r e l a t i n g t h e
p r e s s u r e o r energy s t a t e of t h e s o i l w a t e r t o i t s f r e e z i n g - p o i n t
depression.
Many i n v e s t i g a t o r s have s t u d i e d t h i s phenomenon (Kolaian
and Low, 1963; Takagi, 1963; Williams, 1964; Low, Anderson and Hoekstra,
1968).
The t r e a t m e n t s used by many a u t h o r s t o d i s c u s s and deduce t h e
f r e e z i n g - p o i n t d e p r e s s i o n a r e based on t h e assumption t h a t i c e formed i n
f r o z e n s o i l is t h e same a s o r d i n a r y i c e n o t i n f l u e n c e d by t h e p r e s s u r e
s t a t e of t h e s o i l w a t e r and s o h a s t h e p r o p e r t i e s of pure b u l k i c e a t
atmospheric p r e s s u r e .
T h i s assumption h a s been a m a t t e r of c o n s i d e r a b l e
argument (Bolt and M i l l e r , 1958; Takagi, 1963).
It may n o t seem
j u s t i f i e d t o d i s c u s s a phase e q u i l i b r i u m i n which one phase is a f f e c t e d
by t h e f o r c e f i e l d and t h e o t h e r i s not.
N e v e r t h e l e s s , t h e r e a r e two
r e a s o n s which s u p p o r t h i s assumption; f i r s t , t h e r o l e of s u r f a c e energy
and t h e c u r v a t u r e of ice-water i n t e r f a c e which can s u s t a i n w a t e r and i c e
i n e q u i l i b r i u m a t unequal p r e s s u r e , and secondly, t h e i c e i n s o i l s t e n d s
t o £ o m r e l a t i v e l y l a r g e c r y s t a l s o r l e n s e s when t h e s o i l i s n o t r i g i d l y
confined.
It h a s been shown t h a t t h e i c e i n f r o z e n b e n t o n i t e i s p r e s e n t
i n t h e form of normal hexagonal c r y s t a l s (Anderson and Hoekstra, 1965).
Under most c o n d i t i o n s , t h e r e f o r e , t h e assumption i s b e l i e v e d t o be
applicable.
E a r l y concepts of s o i l - w a t e r r e l a t i o n s h i p s were p r i m a r i l y based
on t h e s o - c a l l e d " C a p i l l a r y tube concept" which considered t h a t w a t e r
i s taken up by t h e s o i l a s a r e s u l t of a p r e s s u r e d i f f e r e n c e a c r o s s
t h e curved a i r - w a t e r i n t e r f a c e .
The s u r f a c e t e n s i o n f o r c e i s r e s p o n s i b l e
f o r w a t e r r e t e n t i o n i n s o i l s and t h e w a t e r . i s t h e n i n a s t a t e of n e g a t i v e
pressure o r suction.
To e x p l a i n t h e f r e e z i n g - p o i n t d e p r e s s i o n o f p o r e
water i n t h i s c a s e , i t is n e c e s s a r y t o d i s c u s s t h e r e l a t i o n s h i p between
t h e change of temperature and p r e s s u r e t h a t p e r m i t s t h e phases t o
coexist
.
THEORY
I f two phases of a s u b s t a n c e a r e i n e q u i l i b r i u m w i t h each o t h e r ,
t h e thermal e q u i l i b r i u m c o n d i t i o n r e q u i r e s t h a t t h e p a r t i a l molar
f r e e energy o f each phase must be t h e same.
Since i n f r o z e n s o i l , t h e
unfrozen l i q u i d w a t e r e x i s t s i n t h e p r e s e n c e of i c e and t h e two w a t e r
phases a r e i n a s t a b l e e q u i l i b r i u m ,
where
F
i s t h e p a r t i a l molar f r e e energy and t h e s u b s c r i p t s i and B
r e f e r t o t h e i c e and l i q u i d w a t e r phase r e s p e c t i v e l y .
I f any change
occurs and e q u i l i b r i u m i s m a i n t a i n e d , i t i s n e c e s s a r y t h a t
S i n c e t h e two e q u i l i b r i u m s t a t e s a r e determined by p r e s s u r e P and
temperature T,
Combining t h e above e q u a t i o n s and u s i n g t h e r e l a t i o n s
where
S
is t h e p a r t i a l molar e n t r o p y and
V
i s t h e p a r t i a l molar volume,
i t follows t h a t
where dPi and dP, a r e t h e t o t a l change i n p r e s s u r e on t h e i c e and t h e
l i q u i d water respectively.
where AH
f
Because
i s t h e molar h e a t of f u s i o n , e q u a t i o n (1) becomes
F u r t h e r , i f i t is t r u e t h a t t h e i c e i n t h e frozen s o i l has t h e p r o p e r t i e s
o f p u r e b u l k i c e , t h e p r e s s u r e on t h e i c e a p p a r e n t l y remains c o n s t a n t ,
t h a t is,
then e q u a t i o n (2) r e d u c e s t o
By i n t e g r a t i n g e q u a t i o n ( 3 ) , assuming AH and
f
R
a r e constant, the s o i l
w a t e r s u c t i o n , APu, can t h e n b e e x p r e s s e d as
where
Tf = f r e e z i n g t e m p e r a t u r e of s o i l w a t e r , and
To = f r e e z i n g t e m p e r a t u r e of p u r e water.
I f t h e v a l u e of t h e r a t i o , T f / ~ i s v e r y c l o s e t o u n i t y , Rn Tf
0
b e approximated by
T
f
/
- ~1 and
~
/ can
~
~
e q u a t i o n (4) becomes
When s o i l w a t e r s u c t i o n i s e x p r e s s e d as an e q u i v a l e n t h e i g h t o f a column
of water, e q u a t i o n (5) i s t h e same a s t h e e q u a t i o n g i v e n by S c h o f i e l d
(1935).
Schofield presented t h i s equation, without derivation, f o r
c a l c u l a t i o n of t h e f r e e z i n g p o i n t d e p r e s s i o n of s o i l m o i s t u r e .
S c h o f i e l d and d a C o s t a (1938) found t h a t t h e i n i t i a l f r e e z i n g - p o i n t s
for
samples w i t h v a r i o u s m o i s t u r e c o n t e n t s (and hence v a r i o u s s u c t i o n v a l u e s )
f i t t e d t h i s relationship w e l l .
More r e c e n t l y , t h e concept of p o t e n t i a l h a s been s u c c e s s f u l l y
a p p l i e d i n t h e s t u d y of s o i l w a t e r systems.
The p o t e n t i a l concept does
n o t c o n s i d e r f o r c e s b u t s p e c i f i e s o n l y t h e energy s t a t e w i t h which w a t e r
i s h e l d by s o i l s .
The r e t e n t i o n of w a t e r i n s o i l s can b e e x p r e s s e d
q u a n t i t a t i v e l y i n terms of t h e f r e e energy of t h e s o i l w a t e r .
The
r e l a t i v e p a r t i a l m o l a r f r e e energy of s o i l w a t e r i s g i v e n by t h e e q u a t i o n
AF =
where
-F
F0
r e l a t i v e p a r t i a l molar f r e e energy o f s o i l w a t e r ,
= p a r t i a l molar f r e e energy of s o i l w a t e r ,
= f r e e energy of pure b u l k w a t e r a t t h e same t e m p e r a t u r e
a s t h e s o i l sample,
R = universal gas constant,
T = temperature of t h e s o i l sample,
P = vapour p r e s s u r e of w a t e r i n t h e s o i l , and
Po = vapour p r e s s u r e of pure b u l k w a t e r a t t h e same
temperature as t h e s o i l sample.
The concept of s o i l w a t e r p o t e n t i a l can a l s o be adopted t o e x p l a i n
t h e f r e e z i n g - p o i n t d e p r e s s i o n of pore water i n t h e f r o z e n s o i l .
Since
i c e , unfrozen l i q u i d w a t e r and vapour c o e x i s t a t a c e r t a i n t e m p e r a t u r e
i n f r o z e n s o i l s , t h e c o n d i t i o n of phase e q u i l i b r i u m r e q u i r e s t h a t t h e
p a r t i a l molar f r e e energy of t h e t h r e e phases must be equal.
Therefore,
t h e f o l l o w i n g i d e n t i t y must apply
Because i c e and l i q u i d w a t e r phases a r e i n a s t a b l e e q u i l i b r i u m , t h e
vapour p r e s s u r e of unfrozen water must be e q u a l t o t h e vapour p r e s s u r e
of i c e .
A s mentioned p r e v i o u s l y , t h e i c e formed i n f r o z e n s o i l s has t h e
same p r o p e r t i e s a s pure b u l k i c e , t h e r e l a t i v e p a r t i a l molar f r e e energy
of u n f r o z e n w a t e r a t a c e r t a i n t e m p e r a t u r e , t h e r e f o r e , w i l l have a f i x e d
v a l u e which i s e q u a l t o t h e r e l a t i v e f r e e energy of p u r e b u l k i c e a t
t h a t temperature.
Hence, t h e r e l a t i o n s h i p between t h e r e l a t i v e p a r t i a l
molar f r e e energy of unfrozen water and i t s f r e e z i n g - p o i n t temperature
can be e x p r e s s e d a s
-
Fa
-
FO = RT an-
'ice
P O ~
where P
ice
and Po a r e t h e vapour p r e s s u r e of pure b u l k i c e and t h e
T
vapour p r e s s u r e of super cooled water a t a c e r t a i n temperature
respectively.
Both equation (5) and (8) show t h a t a t a f i x e d v a l u e of s o i l w a t e r
s u c t i o n , o r s o i l water p o t e n t i a l , t h e r e i s one corresponding f r e e z i n g p o i n t depression.
This a l s o implies t h a t f o r a c e r t a i n moisture c o n t e n t ,
t h e r e i s a corresponding freezing-point depression.
As freezing occurs
i n t h e s o i l system, water i s transformed i n t o i c e l e n s e s , t h e amount of
w a t e r remaining i n t h e pores thus comes under an i n c r e a s i n g s u c t i o n o r
potential.
This s u c t i o n o r p o t e n t i a l i s probably r e s p o n s i b l e f o r t h e
presence of unfrozen water because of i t s e f f e c t on t h e f r e e z i n g - p o i n t
of t h e l a t t e r .
I f t h e water p o t e n t i a l i s a single-valued f u n c t i o n of
l i q u i d water content and a s i g n i f i c a n t rearrangement of s o i l p a r t i c l e s
does n o t occur when i c e i s formed i n t h e s o i l , t h e unfrozen water c o n t e n t
t h e r e f o r e , w i l l have a f i x e d v a l u e f o r each temperature a t which t h e
i c e and water phases a r e i n e q u i l i b r i u m , r e g a r d l e s s of t h e amount of i c e
present.
I f t h i s i s t h e c a s e , t h e r e l a t i o n s h i p between f r e e z i n g - p o i n t
d e p r e s s i o n and moisture content can a l s o r e p r e s e n t t h e r e l a t i o n s h i p
between t h e temperature and t h e unfrozen water c o n t e n t .
EXPERIMENTAL PROCEDURE
Two types of a r t i f i c i a l s o i l were s e l e c t e d f o r f r e e z i n g s t u d i e s .
The Type 1 s o i l was a ill40 mesh S i l i c a Flour, w i t h a s p e c i f i c a t i o n a s
follows :
0.2% r e t a i n e d on a #lo0 mesh s i e v e
1.2% r e t a i n e d on a a140 mesh s i e v e
6.2% r e t a i n e d on a /I200 mesh s i e v e
10.8% r e t a i n e d on a #270 mesh s i e v e
10.4% r e t a i n e d on a /I325 mesh s i e v e
71.4% r e t a i n e d i n t h e pan
The Type 2 s o i l was a m i x t u r e of S i l i c a Flour w i t h 50% (by weight) k a o l i n i t e .
These s o i l s were s e l e c t e d because of t h e d i f f e r e n c e i n f r e e z i n g b e h a v i o r
t h a t could be expected.
The c a l o r i m e t r i c method was used t o determine t h e
amount of unfrozen w a t e r i n a molded specimen of known m o i s t u r e c o n t e n t ,
d e n s i t y , and subzero temperature.
The f r e e z i n g - p o i n t of water i n t h e
s o i l w i t h v a r i o u s m o i s t u r e c o n t e n t s were a l s o measured, from which a c u r v e
showing t h e dependence of t h e f r e e z i n g - p o i n t d e p r e s s i o n of a s o i l on i t s
s o i l m o i s t u r e c o n t e n t was c o n s t r u c t e d .
I n p r e p a r i n g t h e t e s t specimens, t h e oven-dried
(105aC) s o i l s were
mixed w i t h s u f f i c i e n t d i s t i l l e d w a t e r t o make samples c o n t a i n i n g t h e
desired moisture content.
The m o i s t s o i l was then packed i n t o a p l e x i g l a s s
tube of i n s i d e diameter 3 . 8 1 cm and l e n g t h 10.16 cm.
was 1 . 3 3 gm/cm3 f o r each specimen.
The d r y d e n s i t y
To achieve uniform d e n s i t y throughout
each sample i n t h e t u b e , t h e s o i l was packed i n f i v e l a y e r s , each
approximately 2 cm t h i c k .
The t u b e was threaded a t each end and f i t t e d
w i t h screw-on c a p s t o p r e v e n t e v a p o r a t i o n of water.
Two thermocouples
were i n s e r t e d i n t o t h e s o i l f o r each t u b e , one i n t h e c e n t e r of t h e s o i l
sample and t h e o t h e r j u s t i n s i d e t h e w a l l of t h e tube.
The tube was
suspended i n a c y l i n d r i c a l copper c o n t a i n e r which was s e a l e d w i t h a
rubber s t o p p e r .
The whole assembly was then immersed i n a temperature
c o n t r o l l e d b a t h which was connected t o a r e f r i g e r a t i o n system.
The
b a t h t e m p e r a t u r e was g r a d u a l l y lowered and t h e sample t e m p e r a t u r e was
monitored by t h e thermocouples.
g i v e n by Jame (1972).
The p r o c e d u r e , o u t l i n e d i n d e t a i l , i s
F i g u r e 1 and F i g u r e 2 are some t y p i c a l f r e e z i n g
c u r v e s o b t a i n e d from t h e two s o i l s .
A f t e r n u c l e a t i o n had o c c u r r e d i n t h e s o i l sample, t h e b a t h
t e m p e r a t u r e was set t o a c e r t a i n t e m p e r a t u r e a t which t h e u n f r o z e n
w a t e r c o n t e n t was t o b e determined and t h e s o i l sample was allowed t o
come t o t h e r m a l e q u i l i b r i u m a t t h i s temperature.
After equilibrium,
t h e whole assembly was removed fram t h e b a t h and t h e s o i l sample was
f o r c e d o u t of t h e t u b e and q u i c k l y i n s e r t e d i n t o a c a l o r i m e t e r t h a t
c o n t a i n e d approximately 800 gm of water.
The change i n t e m p e r a t u r e o f
t h e c o n t e n t s of t h e c a l o r i m e t e r was t h e n used t o c a l c u l a t e t h e amount
of i c e p r e s e n t i n t h e sample.
Once a g a i n , complete d e t a i l s are g i v e n
by Jame (1972).
RESULTS AND DISCUSSION
S e v e r a l specimens o f Type 1 and Type 2 s o i l w i t h v a r i o u s w a t e r
c o n t e n t s were p r e p a r e d and t h e i r corresponding f r e e z i n g - p o i n t d e p r e s s i o n s
were measured.
F i g u r e 3 shows t h e r e l a t i o n s h i p between t h e m o i s t u r e
c o n t e n t and i t s corresponding f r e e z i n g - p o i n t d e p r e s s i o n f o r t h e two
t y p e s of s o i l .
T e s t r e s u l t s of t h e unfrozen water content a t d i f f e r e n t subzero
t e m p e r a t u r e s f o r b o t h Type 1 s o i l and Type 2 s o i l w i t h v a r i o u s i n i t i a l
m o i s t u r e c o n t e n t s a r e shown i n F i g u r e 4.
When u s i n g t h e c a l o r i m e t r i c method d e s c r i b e d i n t h e preceding
s e c t i o n , i t was n o t p o s s i b l e t o d i s t i n g u i s h any e f f e c t of t h e i n i t i a l
w a t e r c o n t e n t upon t h e unfrozen w a t e r c o n t e n t .
However, t h e r e s u l t s
shown i n F i g u r e 4 do i l l u s t r a t e t h a t f o r a given s o i l t h e l i q u i d w a t e r
c o n t e n t h a s a f i x e d v a l u e f o r each subzero temperature, r e g a r d l e s s of
t h e i n i t i a l water c o n t e n t o f t h e s o i l sample.
a s o b t a i n e d by Nersesova and T s y t o v i c h (1963).
T h i s i s t h e same r e s u l t
Comparing t h e f r e e z i n g -
p o i n t d e p r e s s i o n of s o i l w a t e r i n Figure 3 w i t h t h e unfrozen w a t e r
c o n t e n t of t h e s e two t y p e s of s o i l i n Figure 4 , i t i s c l e a r t h a t t h e
r e l a t i o n s h i p between f r e e z i n g - p o i n t d e p r e s s i o n and w a t e r c o n t e n t a l s o
r e p r e s e n t s t h e r e l a t i o n s h i p between subzero temperature and u n f r o z e n
water content.
F i g u r e 5 shows t h e s e two r e l a t i o n s h i p s .
This is i n
agreement w i t h t h e o p i n i o n of Low, Anderson and Hoekstra (1968).
The important consequence of t h e s e r e s u l t s i s t h a t i t p e r m i t s one
t o p r e d i c t t h e amount o f unfrozen w a t e r as a f u n c t i o n of temperature
from t h e m o i s t u r e c h a r a c t e r i s t i c curve of t h e s o i l .
A moisture
c h a r a c t e r i s t i c c u r v e , showing t h e r e l a t i o n between water c o n t e n t and
w a t e r p o t e n t i a l , i s u s u a l l y determined a t room temperature.
For a given
subzero t e m p e r a t u r e , t h e p r e s s u r e o r energy s t a t e of l i q u i d w a t e r can
b e o b t a i n e d from e q u a t i o n (6) o r ( 9 ) .
The unfrozen w a t e r c o n t e n t f o r
t h i s temperature i s then simply found by f i n d i n g t h e water c o n t e n t f o r
t h e corresponding s u c t i o n o r p o t e n t i a l v a l u e from t h e m o i s t u r e
c h a r a c t e r i s t i c curve of t h e s o i l .
I n t h i s connection, i t i s assumed
t h a t t h e temperature e f f e c t on t h e m o i s t u r e c h a r a c t e r i s t i c curve i s n o t
significant.
This i s a c c e p t a b l e because i t was p o i n t e d o u t t h a t t h e
m o i s t u r e p o t e n t i a l of t h e s o i l i s n o t v e r y s e n s i t i v e t o t e m p e r a t u r e
(Taylor and S t e w a r t , 1960).
For p r a c t i c a l purposes, t h e temperature
e f f e c t on t h e m o i s t u r e c h a r a c t e r i s t i c curve, r e l a t e d t o t h e h e a t of
w e t t i n g , can be n e g l e c t e d .
A m o i s t u r e c h a r a c t e r i s t i c curve can t h u s
b e considered as a r e l a t i o n s h i p between w a t e r c o n t e n t and s o i l w a t e r
p o t e n t i a l over a f a i r l y l a r g e temperature range, (Edlefsen and Anderson,
1943; Hoekstra, 1969).
From h i s e x p e r i m e n t a l o b s e r v a t i o n s , W i l l i a m s
(1964) found a unique r e l a t i o n s h i p e x i s t e n c e between t h e n e g a t i v e
temperature a t which a g i v e n unfrozen m o i s t u r e c o n t e n t o c c u r s and t h e
s u c t i o n corresponding t o a s i m i l a r m o i s t u r e c o n t e n t a t room temperature.
He a l s o p r e s e n t e d t h e same method t o p r e d i c t t h e unfrozen w a t e r c o n t e n t
i n a frozen s o i l .
ACKNOWLEDGEMENT
The a u t h o r s acknowledge r e s e a r c h g r a n t s given by t h e Canada
Department of A g r i c u l t u r e and I m p e r i a l O i l Limited which made t h i s
study possible.
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-
l*lrF:-.T
-.
.-.-.-.-
FPD
0.174OC
0.255OC
-
\
FPD
1.22OC
1
I
\
\
\
\
-
\
\
'.
I
\
\
'
\
\
-
W=5
OO
/
W = WATER CONTENT
(BY
DRY W E I G H T
-
TIME (HOURS)
Figure
1
TEMPERATURE
VS. T l M E CURVE
OF FREEZING TYPE I S O I L .
-
.
.
1
-
FPD
0.874 O
-
C
w
*.
I
-
'
I
\
\
1
-
W = 10.56
OO
/
b
I
FPD
\
2.22OC
\.
I
1
\ I
W = WATER CONTENT
. ...
1'
-
( B Y DRY WEIGHT )
...
*.
I
1
I
0
I
I
2
TIME
Figure
2
.w = 6 . 2 6 %
'*..
I
I
I
3
1
4
(HOURS
TEMPERATURE VS. T l M E CURVE
OF FREEZING T Y P E 2 SOIL
I
I
5
I
2
FREEZING POINT DEPRESSION OC
Figure
3
EXPERIMENTAL RESULTS OF RELATIONSHIP BETWEEN
FREEZING POINT DEPRESSION AND WATER CONTENT
LA.
Z
3
LA.
LEGEND
o
:
FREEZING POINT DEPRESSION OF SOlL WATER
UNFROZEN WATER CONTENT, AT SUBZERO TEMPERATURES
TYPE 2 SOlL
-
I
2
3
4
FREEZING POINT DEPRESSION
Figure
5
5
6
(DEGREES BELOW 0.C 1
RELATIONSHIP BETWEEN FREEZING POINT DEPRESSION AND
WATER CONTENT AND RELATIONSHIP BETWEEN
UNFROZEN WATER CONTENTS AND SUBZERO TEMPERATURES