ROLE OF WATER LAYER AT AN ICE SURFACE IN THE KINETIC PROCESSES OF GROWTH OF ICE CRYSTALS - GROWTH OF SNOW CRYSTALS AND FROST HEAVING

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ROLE OF WATER LAYER AT AN ICE SURFACE IN THE KINETIC PROCESSES OF GROWTH OF ICE CRYSTALS - GROWTH OF SNOW CRYSTALS AND

FROST HEAVING

T. Kuroda

To cite this version:

T. Kuroda. ROLE OF WATER LAYER AT AN ICE SURFACE IN THE KINETIC PROCESSES OF GROWTH OF ICE CRYSTALS - GROWTH OF SNOW CRYSTALS AND FROST HEAVING.

Journal de Physique Colloques, 1987, 48 (C1), pp.C1-487-C1-493. �10.1051/jphyscol:1987167�. �jpa-

00226313�

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JOURNAL DE P H Y S I Q U E

C o l l o q u e C1, s u p p l 6 m e n t au n o 3, Tome _{4 8 ,} mars 1 9 8 7

ROLE OF WATER LAYER AT AN ICE SURFACE IN THE KINETIC PROCESSES

OF GROWTH OF ICE CRYSTALS

-

GROWTH OF SNOW CRYSTALS AND FROST HEAVING T . KURODA

Institute of Low Temperature Sciences, Hokkafdo University, Sapporo 060, Japan

&sun6

-

Dans cette 6tude thkrique, les propri&t&s thermodynamiques de la couche d'eau existant de maniere stable h l'interface glace/vapeur ou glace/particule de sol h des t e r a t u r e s infbrieures h O°C sont discutges. I1 est mntr& que la couche d'eau h la surface de la glace joue un r6le important dans la cinbtique de croissance de la glace : 1) croissance des cristaux de neige et 2) pous&e du givre.

Pour ce dernier cas, il existe une surfusion critique

AT^

en-dessous de laquelle une lentille de glace est continuellement form&e sans avanc6e du front de conghlation du &t& du sol non gel&.

A b s t r a c t

-

I n t h i s t h e o r e t i c a l s t u d y , we d i s c u s s e d thermodynamical p r o p e r t i e s of water l a y e r which e x i s t s i n a s t a b l e way a t ice/vapour i n t e r f a c e o r a t i c e / s o i l p a r t i c l e i n t e r f a c e a t t e m p e r a t u r e s below O°C. Then, we showed t h a t t h e water l a y e r a t t h e i c e s u r f a c e p l a y s a n important r o l e i n t h e k i n e t i c p r o c e s s e s of growth of i c e : 1 ) growth of snow c r y s t a l s and 2) f r o s t heaving. A s t o f r o s t heaving, i t was shown t h a t t h e r e i s a c r i t i c a l supercooling ATc below which i c e l e n s i s c o n t i n u o u s l y formed without an advance of f r e e z i n g f r o n t towards t h e unfrozen s o i l .

1. I n t r o d u c t i o n

There a r e s e v e r a l phenomena concerning t h e phase t r a n s f o r m a t i o n i n H20 systems i n which t h e water l a y e r a t an i c e s u r f a c e below O°C p l a y s an important r o l e . For an example, Kuroda and Lacmann[l] t h e o r e t i c a l l y i n v e s t i g a t e d t h e c h a r a c t e r i s t i c s of t h e vapour growth k i n e t i c s of i c e s u r -

f a c e covered w i t h t h e q u a s i - l i q u i d

l a y e r (V-QL-S-mechanism) on t h e b a s i s R frost heaving

over burden h

of a phenomenological model proposed .1

L & I

3.

by Lacmann and S t r a n s k i [ 2 ] . They

- - -

argued t h a t t h e complicated h a b i t

- - -

I c e L e n s

- - -

change of snow c r y s t a l s , i . e . i c e

- - - -

Water Layer

c r y s t a l s grown from t h e vapour, i s

- - - -

/(thickness

6 )

e s s e n t i a l l y caused by t h e change i n t h e e q u i l i b r i u m t h i c k n e s s of t h e q u a s i - l i q u i d l a y e r on b a s a l and prism f a c e s w i t h d e c r e a s i n g temperature.

Ohtomo and Wakahama[3] a s c r i b e d t h e measured l a r g e v a l u e of d i f f u s i o n c o n s t a n t i n t h e g r a i n boundary a t 272.7K t o g r a i n boundary melting.

Recently, Kuroda[4] a p p l i e d t h e theo- r e t i c a l s t u d y of t h e V-QL-S-mechanism

111 t o t h e k i n e t i c p r o c e s s e s of f r o s t heaving occuring a t water l a y e r between i c e l e n s and s o i l p a r t i c l e s

(Fig. 1). F i g . 1 Schematic r e p r e s e n t a t i o n of k i n e t i c

p r o c e s s e s a t a water l a y e r between i c e l e n s and s o i l p a r t i c l e s , i . e . f r e e z i n g of t h e l a y e r and s u c t i o n of p o r e w a t e r ( a f t e r [ 4 ] ) .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987167

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C1-488 JOURNAL DE PHYSIQUE

I n o r d e r t o pursue t h e k i n e t i c p r o c e s s e s w i t h which t h e water l a y e r a t i c e sur- f a c e i s concerned, we need thermodynamical f u n c t i o n s of t h e water l a y e r such a s chemi- c a l p o t e n t i a l Ap(6) depending on i t s t h i c k n e s s 6 . Forms of t h e f u n c t i o n s depend on t h e model which e x p l a i n s t h e o r i g i n of t h e s t a b l e e x i s t e n c e of t h e l a y e r below O°C.

I n t h i s t h e o r e t i c a l s t u d y , however, we d i s c u s s t h e g e n e r a l thermodynamical p r o p e r t i e s of t h e l a y e r without a model, a s f a r a s p o s i b l e ( s e c t i o n 2) and pursue t h e r o l e of t h e w a t e r l a y e r a t t h e i c e s u r f a c e i n t h e k i n e t i c p r o c e s s e s of growth of i c e c r y s t a l s , e.g. growth of snow c r y s t a l s ( s e c t i o n 3) and f r o s t heaving ( s e c t i o n 4 ) .

2. Thermodynamical p r o p e r t i e s of a w a t e r l a y e r a t an i c e s u r f a c e

On t h e b a s i s of s e v e r a l p i e c e s of experimental ( s e e R e f . [ l ] ) and t h e o r e t i c a l s t u d i e s [ l , 2 , 5 ] , i t i s w e l l known t h a t an i c e s u r f a c e , i . e . ice/vapour i n t e r f a c e i s covered w i t h a q u a s i - l i q u i d l a y e r i n e q u i l i b r i u m a t t e m p e r a t u r e s a p p r e c i a b l y below m e l t i n g p o i n t Tm (=O°C) of i c e .

I n a d d i t i o n , a s t o f r o s t heaving, measurements u s i n g c a l o r i m e t r y [ 6 ] and NMR[7]

show a l s o t h a t a t h i n water l a y e r e x i s t s between i c e l e n s and s o i l p a r t i c l e s i n a s t a b l e way a t t e m p e r a t u r e s below O°C.

L e t Ap(6) denote t h e chemical p o t e n t i a l of a water l a y e r (so c a l l e d q u a s i - l i q u i d l a y e r a t i c e / v a p o u r i n t e r f a c e o r water l a y e r a t i c e - l e n s / s o i l - p a r t i c l e i n t e r f a c e ) whose t h i c k n e s s is 6 and Avw denote t h e chemical p o t e n t i a l of bulk w a t e r . Chemical p o t e n t i a l is s o scaled that it is z e r o f o r bulk i c e . Because of molecular i n t e r a c - t i o n s among i c e , water and vapour o r s o i l p a r t i c l e s , Av(6) i s considered t o depend on t h e t h i c k n e s s 6 of t h e water l a y e r .

S i n c e t h e b u l k water i s u n s t a b l e a t temperatures below Tm, i t s chemical poten-

t i a l Avw = Om

AT IT^ ---

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i s p o s i t i v e a t T < Tm (Qm: l a t e n t h e a t of f u s i o n p e r molecule, AT=Tm-T:supercooling;l N e v e r t h e l e s s , s e v e r a l experimental r e s u l t s confirmed t h a t a w a t e r l a y e r can e x i s t a t an i c e s u r f a c e i n a s t a b l e way even T < ^Tma s mentioned above. It means t h a t t h e chemical p o t e n t i a l Av(6) of t h e t h i n water l a y e r h a s t o be s m a l l e r t h a n Avw. However, Av (6) must approach Apw a s 6 + m:

AP (6-) = Avw

,

s i n c e t h e water l a y e r w i t h i n f i n i t e t h i c k n e s s can b e regarded a s b u l k water. Thus, Ap(6) i s considered t o be a n i n c r e a s i n g f u n c t i o n of t h e l a y e r t h i c k n e s s 6 (Fig.2).

Namely, Ap(6) d e c r e a s e s w i t h d e c r e a s i n g 6 and i t is expected t h a t t h e r e i s a equi- l i b r i u m t h i c k n e s s 6eq f o r which t h e chemical p o t e n t i a l of t h e water l a y e r i s equal t o t h a t of b u l k i c e , i . e .

Av(6eq) = 0

---

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This c o n d i t i o n i s n o t h i n g but minimizing t h e t o t a l f r e e energy of t h e system w i t h r e s p e c t t o 6. We can o b t a i n t h e e q u i l i b r i u m t h i c k n e s s 6, as a f u n c t i o n of super- c o o l i n g AT from eq. ( 2 ) , i f Ap(6) i s known a s a f u n c t i o n

01

^6. ^6eqi s expected t o d e c r e a s e w i t h i n c r e a s i n g AT, because t h e i n s t a b i l i t y of bulk l i q u i d i n c r e a s e s w i t h i n c r e a s i n g AT(e.g. s e e eq. ( 6 ) ) .

We can q u a l i t a t i v e l y d i s c u s s a l s o t h e dependence on t h e t h i c k n e s s 6 of t h e equi- l i b r i u m vapour p r e s s u r e p ( 6 ) of a t h i n water l a y e r u s i n g a g e n e r a l r e l a t i o n between p ( 6 ) and Ap(6):

- - -

-

i c e

- -

- - - - - - - - -

- - - - i c e - -

,+ bi/w

water layer

$+a

Fig.2 Chemical p o t e n t i a l Ap(6) Fig.3 Schematic r e p r e s e n t a t i o n of i n t e r - of a w a t e r l a y e r r e l a t i v e t o f a c i a l f r e e energy of t h e i c e and a-phase b u l k i c e a s a f u n c t i o n of t h e system ( a ) without w a t e r l a y e r and (b) l a y e r t h i c k n e s s 6. w i t h water l a y e r .

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k~ l n [ p ( 6 ) l p i 1 = A I J ( ~ ) ,

---

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where p i is t h e e q u i l i b r i u m vapour p r e s s u r e of i c e . p ( 6 ) i s considered t o be equal t o pi f o r 6 = 6eq, t h e n t o i n c r e a s e w i t h i n c r e a s i n g 6 and t o approach t h e e q u i l i b r i u m vapour p r e s s u r e pw of b u l k water a s 6 ^-+rn.

It should be noted t h a t c o n c r e t e e x p r e s s i o n s f o r t h e f u n c t i o n s Ap(6), 6eq(AT) and p ( 6 ) depend on t h e model which e x p l a i n s t h e s t a b l e e x i s t e n c e of a water l a y e r a t T<Tm

,

even though t h e y show t h e same behaviour a s d i s c u s s e d above i n any models.

For a n example, according t o Lacmann-Stranskimodel[2], o r i g i n of a q u a s i - l i q u i d l a y e r below Tm i s p o s i t i v e v a l u e of t h e w e t t a b i l i t y parameter AobD defined by t h e following equation:

ham = a i l a

-

( a i l w

+

a,/,).

---

^{( 4 )}

Here O i P

i s t h e i n t e r f a c i a l t e n s i o n ( s p e c i f i c i n t e r f a c i a l f r e e energy) between i c e and a-p a s e without a water l a y e r a s shown i n Fig.3a(a-phase corresponds t o t h e vapour i n t h e c a s e of growth of snow c r y s t a l s and t o s o i l p a r t i c l e i n t h e c a s e of f r o s t heaving), ailw i s t h a t between i c e and water and awla i s t h a t between water and a-phase a s shown i n Fig.3b.

The e x p r e s s i o n s f o r Ap(6) and 6," derived from t h e model a r e a s follows[2,4]:

where vm i s t h e volume of a water molecule, and A and n a r e t h e parameters concern- i n g molecular i n t e r a c t i o n s among i c e , water and vapour of s o i l p a r t i c l e . Ap(6) i s smaller t h a n t h e chemical p o t e n t i a l of b u l k w a t e r ( f i r s t term) by a n amount of t h e second term i n c l u d i n g t h e w e t t a b i l i t y parameter Aam.

3. Growth of snow c r y s t a l s a t temperatures not. f a r below O°C

Snow c r y s t a l s a r e i c e c r y s t a l s grown from water vapour i n t h e a i r . Therefore, q u a s i - l i q u i d l a y e r on an i c e s u r f a c e i s expected t o i n f l u e n c e t h e growth mechanism of t h e snow c r y s t a l s a t temperatures n o t f a r below O'C. The growth mechanism of an i c e s u r f a c e covered w i t h t h e q u a s i - l i q u i d l a y e r was f i r s t proposed by Lacmann and StranskiL21 and i t was a p p l i e d t o t h e i n t e r p r e t a t i o n of complicated h a b i t change of snow c r y s t a l s by Kuroda and Lacmann[l].

I f t h e a c t u a l p r e s s u r e p of water vapour is l a r g e r t h a n t h e e q u i l i b r i u m vapour p r e s s u r e p(6) of a q u a s i - l i q u i d l a y e r w i t h a t h i c k n e s s 6 , water vapour i s condensed i n t o t h e q u a s i - l i q u i d l a y e r and consequently t h e t h i c k n e s s 6. i n c r e a s e s . The con- d e n s a t i o n r a t e Rc i s expressed by Hertz-Knudsen equation, i . e .

where m i s t h e mass of a water molecule. Since p ( 6 ) i s a i n c r e a s i n g f u n c t i o n of 6, Rc d e c r e a s e s w i t h i n c r e a s i n g 6.

On t h e o t h e r hand, t h e chemical p o t e n t i a l Ap(6) of t h e q u a s i - l i q u i d l a y e r becomes p o s i t i v e , when i t s t h i c k n e s s 6 exceeds t h e e q u i l i b r i u m t h i c k n e s s 6eq because of t h e condensation of water vapour i n t o t h e q u a s i - l i q u i d l a y e r . Therefore, f r e e z - i n g occurs a t t h e i n t e r f a c e between i c e and q u a s i l i q u i d l a y e r , s o t h a t 6 decreases.

The f r e e z i n g r a t e Rf(Ap(6)) i n c r e a s e s a s Ap(6) i n c r e a s e s . Namely, i t i n c r e a s e s w i t h i n c r e a s i n g 6.

Under s t e a d y s t a t e c o n d i t i o n s , both r a t e s Rc and Rf must be equal, i . e . From t h i s equation, we can e v a l u a t e t h e r a t e R,(T, Ap) of vapour growth of a n i c e s u r f a c e covered w i t h a q u a s i - l i q u i d l a y e r and i t s t h i c k n e s s 6 s t ( ~ , Ap) under steady s t a t e c o n d i t i o n s a s f u n c t i o n s of temperature T and e x c e s s vapour p r e s s u r e Ap.

I t should be n o t i c e d t h a t t h e v a l u e of t h e edge energy of a two-dimensional n u c l e u s a t t h e q u a s i - l i q u i d l i c e i n t e r f a c e i s much s m a l l e r t h a n a t a v a p o u r / i c e i n t e r f a c e without q u a s i - l i q u i d l a y e r , and consequently t h e q u a s i - l i q u i d l a y e r promotes t h e growth of i c e c r y s t a l s from t h e vapour phase a t lower s u p e r s a t u r a t i o n s

F. 7

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JOURNAL DE PHYSIQUE

4. F r o s t heaving

4 . 1 R a t e of f r o s t heave d u r i n g continuous formation of i c e l e n s 141

K i n e t i c p r o c e s s e s of f r o s t heaving c a n b e d i s c u s s e d a l s o on t h e b a s i s of t h e thermodynamical p r o p e r t i e s of a t h i n water l a y e r between i c e l e n s and s o i l p a r t i c l e s ( F i g . l ) , i n t h e same way a s t h e k i n e t i c s of vapour growth of i c e c r y s t a l s covered w i t h t h e q u a s i - l i q u i d l a y e r .

A t f i r s t , l e t t h e t h i c k n e s s of t h e water l a y e r b e e q u a l t o t h e e q u i l i b r i u m t h i c k n e s s rSeq f o r which t h e chemical p o t e n t i a l

AM(^,^)

of t h e w a t e r l a y e r r e l a t i v e t o bulk i c e i s zero. On t h e o t h e r hand, t h e chemical p o t e n t i a l Av(rp) of p o r e water which f i l l s p o r e s among s o i l p a r t i c l e s i s l a r g e r t h a n t h a t of t h i n water l a y e r f o r T<Tm, s i n c e t h e d i s t a n c e rp from t h e s u r f a c e of s o i l p a r t i c l e t o t h e c e n t r a l p a r t of of t h e p o r e water i s l a r g e r t h a n t h e t h i c k n e s s 6eq of t h e water l a y e r ( F i g . 1 and 2 ) . I n g e n e r a l , m a t e r i a l t e n d s t o f l o w from t h e s t a t e of h i g h e r chemical p o t e n t i a l t o t h e s t a t e of lower chemical p o t e n t i a l . T h e r e f o r e , t h e p o r e water i s drawn towards t h e water l a y e r s o t h a t t h e t h i c k n e s s of t h e water l a y e r becomes l a r g e r t h a n t h e e q u i l i b r i u m t h i c k n e s s 6eq and consequently i c e l e n s i s pushed up. The g r a d i e n t of t h e chemical p o t e n t i a l a t t h e water l a y e r which i s a d r i v i n g f o r c e f o r t h e s u c t i o n of p o r e water i s approximately expressed a s

s i n c e t h e d i s t a n c e between water l a y e r and p o r e water i s of t h e o r d e r of mean r a d i u s rs of t h e s o i l ~ a r t i c l e s ( F i g . 1 ) ~ and t h e chemical p o t e n t i a l Ap(r ) o f p o r e w a t e r i s roughly assumed t o be e q u a l t o AM, of b u l k w a t e r ( F i g . 2 ) . Thus, t h e r a t e Rs of P t h i c k e n i n g of t h e water l a y e r by s u c t i o n of p o r e water may b e given by

where K i s t h e h y d r a u l i c c o n d u c t i v i t y n e a r t h e water l a y e r . The e s t i m a t e d v a l u e s of K from t h e experiment by Horiguchi(unpub1ished) a r e of t h e o r d e r of 1 0 ~ & 1 0 ~ cm2 s-l erg-1.

On t h e o t h e r hand, t h e chemical p o t e n t i a l Au(6) of t h e water l a y e r becomes p o s i t i v e , s i n c e i t s t h i c k n e s s becomes l a r g e r t h a n 6eq because of s u c t i o n of p o r e water. Therefore, f r e e z i n g o c c u r s a t t h e i n t e r f a c e between i c e l e n s and s o i l p a r t i c l e s , s o t h a t t h i c k n e s s 6 d e c r e a s e s . The r a t e Rf of t h i n n i n g of t h e water l a y e r by f r e e z i n g i s expressed a s

where D i s t h e s e l f - d i f f u s i o n c o n s t a n t of water molecules and a i s t h e molecular d i s t a n c e i n t h e water l a y e r .

When t h e t h i c k e n i n g r a t e Rs by s u c t i o n and t h e t h i n n i n g r a t e Rf by f r e e z i n g a r e e q u a l , f r e e z i n g f r o n t does n o t advance towards t h e unfrozen s o i l , but i c e l e n s i s c o n t i n u o u s l y formed. T h e r e f o r e , from t h e c o n d i t i o n s f o r s t e a d y s t a t e :

we c a n o b t a i n t h e l a y e r t h i c k n e s s 6 and t h e r a t e Rh of f r o s t heave d u r i n g c o n t i n - uous f o r m a t i o n of i c e l e n s a s f u n c t % n s of s u p e r c o o l i n g AT a t t h e water l a y e r [ 4 ] :

The o b t a i n e d r a t e R~ c o n s i s t s of a n o v e r a l l d r i v i n g f o r c e f o r f r o s t h e a v i n g ( s e e t h i c k arrow i n Fig.2) p r o p o r t i o n a l t o AT i n t h e numerator and a sum of t h e r e s i s t a n c e akT/D of t h e f r e e z i n g p r o c e s s and t h e r e s i s t a n c e r s / ~ of t h e s u c t i o n p r o c e s s i n t h e denominator.

Using numerical v a l u e s , Kuroda showed t h a t t h e r e s i s t a n c e akT/D i s n e g l i g i b l y small a g a i n s t t h e r e s i s t a n c e r s / K [4]. T h e r e f o r e , Rh can b e expressed a s

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Namely, t h e r a t e of f r o s t heave i s s u b s t a n t i a l l y c o n t r o l l e d by t h e r a t e Rs of s u c t i o n of pore water a t t h e water l a y e r .

We d e r i v e d t h e r a t e of f r o s t heave under t h e assumptions t h a t supercooling AT i s g i v e n a t f r e e z i n g f r o n t and s o i l is s a t u r a t e d w i t h water. To pursue t h e r a t e of f r o s t heave i n a c t u a l system, we must combine t h i s s t u d y on k i n e t i c s a t t h e water l a y e r between i c e l e n s and s o i l p a r t i c l e s w i t h s t u d i e s on h e a t conduction p r o c e s s c o n t r o l l i n g AT and macroscopic mass t r a n s p o r t p r o c e s s from water t a b l e f o r compen- s a t i n g a d e c r e a s e i n w a t e r c o n t e n t due t o s u c t i o n of p o r e w a t e r n e a r f r e e z i n g f r o n t . 4.2 Conditions f o r continuous formation of i c e l e n s

I n s e c t i o n 4.1, we d e r i v e d t h e r a t e Rh of f r o s t heave d u r i n g continuous forma- t i o n of i c e l e n s which i s r e a l i z e d when o n l y t h e water drawn towards t h e f r e e z i n g f r o n t by s u c t i o n f r e e z e s and f r e e z i n g f r o n t does n o t advance downwards.

It i s h i g h l y d e s i r e d t o f i n d t h e c o n d i t i o n s f o r determining whether t h e f r e e z - i n g f r o n t advance, p e n e t r a t i n g through narrow water c h a n n e l s among s o i l p a r t i c l e s , o r n o t . There a r e twokey f a c t o r s concerned w i t h t h i s problem. One i s chemical p o t e n t i a l of i c e w i t h a s m a l l r a d i u s of c u r v a t u r e and t h e o t h e r i s s t a b l e e x i s t e n c e of a water l a y e r between i c e and s o i l p a r t i c l e s . I f a p a r t of f r e e z i n g f r o n t begins t o advance towards unfrozen s o i l , r a d i u s rt of c u r v a t u r e of t h e t i p of i c e g r a d u a l l y d e c r e a s e s ( s e e 1, 2 and 3 i n F i g . 4 ) . T h e r e f o r e , t h e chemical p o t e n t i a l Ap(rt)of t h e t i p of i c e becomes l a r g e r t h a n t h e chemical p o t e n t i a l of b u l k i c e because of Gibbs- Thomson e f f e c t based on i n t e r f a c i a l t e n s i o n O i l w between i c e and water:

where v i i s t h e volume of a m o l e c u l e ' i n i c e . On t h e o t h e r hand, t h i n water l a y e r e x i s t s between i c e and s o i l p a r t i c l e s so a s t o lower t h e t o t a l f r e e energy of t h e system. Thus, t h e minimum r a d i u s r t of c u r v a t u r e of t h e t i p of i c e corresponding t o t h e p o s i t i o n 3 i n Fig.4 may be roughly given by

rt = rP - Beq(AT)

where r p i s t h e mean s i z e of pore among s o i l p a r t i c l e s and 6, (AT) i s t h e e q u i l i b r i u m t h i c k n e s s ( e q . ( 6 ) ) . T h e r e f o r e , t h e l a r g e s t chemical p o t e n t i a ? of convex p a r t of i c e i s expressed a s

I

i c e ^lensI no ice lens

Fig.4 Schematic r e p r e s e n t a t i o n of Fig.5 Schematic r e p r e s e n t a t i o n of chemical advance of f r e e z i n g f r o n t through p o t e n t i a l Aui(rt) of t i p of i c e and chemi- narrow water c h a n n e l s and change c a l p o t e n t i a l Au(rp) of pore water a s of i t s r a d i u s rt of c u r v a t u r e . f u n c t i o n s of supercooling AT.

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JOURNAL DE PHYSIQUE

Since 6, (AT) i n c r e a s e s w i t h d e c r e a s i n g AT, i t becomes e q u a l t o rp a t a c e r t a i n supercoofing AT; and consequently Aui(rt) d i v e r g e s a t AT;. For AT > AT:, Ap+ ( r t ) d e c r e a s e s w i t h i n c r e a s i n g AT(F'ig.5). For AT < AT;, f r e e z i n g f r o n t never p e n e t r a t e through c h a n n e l s of p o r e s f i l l e d w i t h w a t e r , because 6, (AT) > rp.

On t h e o t h e r hand, t h e chemical p o t e n t i a l Au(rp)

02

p o r e water which i s l o c a t e d i n f r o n t of t h e t i p of i c e i s roughly assumed t o b e e q u a l t o t h e chemical p o t e n t i a l Apw of b u l k water ( s e e Fig.2). A s shown i n Fig.5, Au(rp) i s p r o p o r t i o n a l t o AT.

T h e r e f o r e , t h e r e i s a c r i t i c a l supercooling AT= f o r which

I f s u p e r c o o l i n g AT a t f r e e z i n g f r o n t exceeds t h e c r i t i c a l v a l u e AT^, t h e convex p a r t of iceshown i n Fig.4 b e g i n s t o advance towards t h e unfrozen s o i l , s i n c e Aui(rt) <

Ap(rp) f o r AT > AT, (Fig.5). Namely, i c e l e n s i s never formed f o r AT > AT,. On t h e o t h e r hand, any p a r t s of f r e e z i n g f r o n t cannot advance towards t h e unfrozen s o i l f o r AT < ATc, because Api(rt) > Au(rp). Under t h e circumstances, o n l y t h e water drawn towards t h e f r e e z i n g f r o n t by s u c t i o n f r e e z e s so t h a t i c e l e n s can be formed c o n t i n u o u s l y . T h i s t h e o r e t i c a l p r e d i c t i o n i s q u a l i t a t i v e l y i n good agreement w i t h such experimental r e s u l t s t h a t i c e l e n s i s e a s i l y formed when t h e r a t e of h e a t removal i s s m a l l o r s o i l f r e e z e s slowly(8],becausesmaller r a t e of h e a t removal reduces t h e s u p e r c o o l i n g AT a t t h e f r e e z i n g f r o n t . It should be n o t i c e d t h a t t h e c r i t i c a l s u p e r c o o l i n g ATc determined by eqs. (6) and (15) depends on kind of s o i l through w e t t a b i l i t y parameter A% d e f i n e d by eq. (4) and on s i z e rs of s o i l p a r t i c l e s . Acknowledgement

The a u t h o r wishes t o e x p r e s s h i s c o r d i a l t h a n k s t o P r o f . R. Lacmann f o r s t i m u l a t i n g h i s i n t e r e s t i n t h e growth k i n e t i c s w i t h which a water l a y e r a t an i c e s u r f a c e i s concerned. The a u t h o r i s a l s o g r a t e f u l t o Prof. S. Kinosita, D r s . K. Horiguchi and M. Fukuda f o r u s e f u l d i s c u s s i o n on t h e f r o s t heaving.

References

[ I ] Kuroda,T. and Lacmann,R., J. C r y s t a l Growth 56 (1982)189 -205.

[ 2 ] Lacmann,R. and S t r a n s k i , I . , J. C r y s t a l Growth 13/14(1972)236-240.

[ 3 ] Ohtomo,M. and Wakahama,G., J. Phys. Chem. 87(1983)4139-4142.

[4] Kuroda,T., i n Ground Freezing, ed. S.Kinosita and M.Fukuda(Balkema, Rotterdom)

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v o l . 1 , 1985, 39-45.

151 F l e t c h e r .N.H., P h i l . Mag 7 (1962)255-269; 18(1968)1287-1300.

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-

^-

[6] ~ o r i ~ u c h i , ~ .

,

- i n Ground ~reezi-ng , ed. S .K i n o s i t a and M.Fukuda(Balkema, Rotterdom) v o l . 1 , 1985, 33-38.

[ 7 ] Xiaozn, Xu e t a l . , i n Ground Freezing, ed. S.Kinosita and M.Fukuda(Balkema, Rotterdom) vo1.2, 1985, 83-87.

[8] Takeda,K., Nakazawa,J. and Kinosita,S., i n Ground F r e e z i n g , ed. S.Kinosita and M.Fukuda(Balkema, Rotterdom) vo1.2, 1985, 89-94?

COMMENTS E. GAFFNEY

I n your d i s c u s s i o n of t h e d i v i s i o n of t h e energy balance l e a d i n g t o f r o s t heaving ( p a r t i t i o n i n g between s u c t i o n and l a t e n t h e a t of f r e e z i n g ) you seem t o have ignored t h e energy r e q u i r e d t o r a i s e t h e m a t e r i a l above t h e f r e e z i n g l a y e r . T h i s w i l l e q u a l t h e overburden p r e s s u r e . A t Im depth t h a t w i l l b e on t h e o r d e r o f 200 J/mol.

Answer :

Yes, you are r i g h t , overburden p r e s s u r e makes t h e chemical p o t e n t i a l X)J^

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of a

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thin water layer relative to bulk ice smaller. So, it reduces the rate of frost heave (see Ref. ( 4 ) in the paper for Proceedings).

V.F. PETRENKO

When you calculate a thickness of 'liquid-like' layer at the Ice surface do you use the thermodynamic parameters of ordinary water ? I mean that now when we know that this layer is in superionic state you have to take into account a unique property of this layer for chemical potential calculations.

Answer :

Yes, it is necessary for precise calculation based on the microscopic point of view.