HIGH TEMPERATURE INTERNAL FRICTION IN ICE Ih : A NEW DISLOCATION MODEL

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HIGH TEMPERATURE INTERNAL FRICTION IN ICE Ih : A NEW DISLOCATION MODEL

J. Tatibouët, J. Perez, R. Vassoille

To cite this version:

J. Tatibouët, J. Perez, R. Vassoille. HIGH TEMPERATURE INTERNAL FRICTION IN ICE Ih : A NEW DISLOCATION MODEL. Journal de Physique Colloques, 1985, 46 (C10), pp.C10-339-C10-342.

�10.1051/jphyscol:19851075�. �jpa-00225460�

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JOURNAL DE PHYSIQUE

Colloque C10, supplkment au n012, Tome 46, dkcembre 1985 page C10-339

HIGH TEMPERATURE INTERNAL FRICTION IN ICE Ih : A NEW DISLOCATION MODEL

Groupes d'Etudes de MBtallurgie Physique et de Physique des MatBriaux, st. 502, INSA de Lyon, 69621 Villeurbanne, France

R6sum6 - Un modele d'interaction entre dislocations glissant, dans la glace Ih dans le plan de base et dislocations se d6plasant dans des plans differents du plan de base est developpe. L'application de ce modele au frottement interne haute temperature permet une description satisfaisante des phenomknes observes experimentalement aux tres basses frequences (1-loe3 Hz).

Abstract - An interaction model between dislocations of Ice Ih gliding in basal plane and dislocations moving in non-basal planes is developped. The application of this model to high-temperature internal friction leads to a satisfactory description of phenomena observed by very low frequencies (1-10-3 Hz) experiments.

I - INTRODUCTION

At high temperature (T 2. 200 K) the rise in internal friction were studied by some of the present authors /1/ for a very restricted range of frequency around 1 Hz.

More recently, very low frequency (1 HZ-10-3 Hz) measurements were performed in this temperature range on single crystals. The rise in internal friction was attri- b u t e ~ to the movement of dislocations induced by the cyclic stress /2,3/, but the phenomenon could not be described in a single manner by using the non-crystalline core model previously applied for experiments done at 1 Hz /4/. Microcreep experiments performed on same specimens as studied by internal friction measurements indicates that two types of behaviour occured for dislocation motion (fiq. _{- .} 1). One

6 which is put- in evidence

in short time experiments

r 10' (or high f r e q u e n c i e s )

shows a reversible motion for dislocations inducing anelasticity only

.

^The

o t h e r w h i c h b e c o m e s important with long time e x p e r i m e n t s ( o r l o w

o ²⁰ 40 f r e q u e n c i e s ) is m o r e

.mi

c o m p l e x s h o w i n g t h a t

viscoplastici ty is added t o t h e a n e l a s t i c i t y previously observed. SO, the internal friction in ice has to be interpreted n o t o n l y i n t e r m s o f i s o l a t e d d i s l o c a t i o n s

0 500 1000 motion but interaction

*"'

between dislocations have

Fig.1. Microcreep experiment. pure ice. to be taken into account 6 ^G lo%-2 T=26OK.Elastic parts removed. in the whole network.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851075

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C10-340 JOURNAL DE PHYSIQUE

I1 - ^{THE MODEL}

It is assumed that the observed phenomenon results from the interaction between dislocation glide in basal planes and dislocation climb in non-basal planes. At higher frequencies (1 - lo2 Hz) dislocations are able to move in a reversible way in basal plane. The restoring force is induced by the line tension of dislocations pinned between quasi-imnobile segments of the dislocation network (i.e. jogs, triple nodes, dislocation segments in non-basal planes). Such a movement induces only anelasticity. At lower frequencies (below 10-2 Hz) this reversible motion of dislocation is accompanied by irreversible motion of "pinning points" on dislocations in the basal plane.

Let us consider a single crystal with a schematic configuration of dislocations as shown in figure 2 (n1, T 2 are basal planes). When a stress C i s applied : i) dislocation segment AlBl glides o n n l according to the non-crystalline core model and takes a certain curvature. i i ) dislocation segments A 1 A 2 and B 1 B 2

climb in non-basal planes under the effect of forces existing on A1, B1,

"29 B2..

So g l ~ d e onlT 1 is reversible when stress vanishes under the effect of

/

line tension lanelasticitv) and climb motion on A ~ A and B ~ B ~ - ~ S non

BZ

"/

reversible motion fviscoplasticity) if we neqlect the variation of free

/ ma enthalpy-of the crystal resulting

f r o m t h e m o d i f i c a t i o n o f t h e Fig.2. Schematic configuration dislocation network.

1) Glide on basal plane. At point A1 when stress C is applied, the line tension T is in equilibrium with the resulting force FA^ on dislocation segment AlBl and T = FA^ = 1/2 G B ~ ; G = shear modulus, b = Burgercvector.

T. ( d ^2y/ 3 ^x2j⁼⁶b, then y = G b(1-x)x/2 T if AlBl = 1 and tano(= Qb!/2T. In regard with time, the variation of the curvature of AlBl can be given by : BU1 + KU1

= Gb.

U1 is the mean displacement of the segment and B is the damping constant i.e. the inverse of mobility F/vd where F is the applied force and vd the dislocation velocity (B =bb/vd). In the present model vd is taken as the dislocation velocity deduced from the non-crystalline core model /3/. K is the "spring constant" due to the line tension /5/. K = 6 ~b2/12.

2) Climb motion in -non-basal plane (fig. 3). The force acting on A1 and A2 can be written /6/ Fc = GbL.2 c o s g / 2 h with h the distance between two planesT1 andtrz. For small angles, c o s (3 ⁼ s i n d 3 t a n N = s l / G b and F c = e b l / h

per unit length.

Under the action of the force Fc on A1A2, this segment will climb thanks to jog diffusion. One jog over each segment is r e a s o n a b l e v a l u e a c c o r d i n g t o c o n c e n t r a t i o n e s t i m a t i o n /6/. Jog

, diffusion will be envisaged by thermal

,

A1 activation favorised by the force on

A1A2. If 3 is the frequency of these Fig.3. Climb in non-basal plane events leading to a displacement of A1A2

equal to a= b.b/h we get :

=yo

[e;p

-

^(EFV⁺EM" -rb31/h)/k~) - ^exp^(-^(EFV+ EHV +bb3l/h)/k~)]

and U2 = V.a = 2 DSD sinh (6b31/h k T)/h

is the climb velocity, DSD the self-diffusion coefficient.

In fac; this relation corresponds only to the case where A B1 is in equilibrium i . . U~=O). Mhen it is not the case ( U l c U1 equilibrium\ it is necessary to replacegb by K. U1 : indeed, the instant position U1 corresponds to an equilibrium resulting from the application of a fictive stress Gfictive as K U1 =Gfictive.b.

Thus the global motion is given by solving the system and with U 1 replacedby (U1

-

^u21.

= ( 6 b

-

^{K (U1}- ^U2))/B

U2 = 2 DSD sin h (K (U1

-

U2) b2 l/hkT)/h

1

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The deformation due to the dislocations motion will be taken as b disl.

According to our experiments a distribution in restoring force as to be =

rb

^aken^81.^into

account /7/, that is to say that a distribution over the length of dislocation segments in the basal plane has been introduced.To simplify our calculation an uniform distribution has been choosen. L

N(l) dl = C dl with

dl

^{C dl.}

and 10-5 4 lo < 2.10-4 m.

If we consider the displacement of dislocation segment not in the basal plane (AlA7) - - the velocity will give a non-linear effect when

K (UI

-

U?) b2 11hkT 3 1

or i b 3 llhii r 10-4 l/h 1

in our conditions (T r 240 K) it corresponds to 10-8 4 h L 5. m and the dependence of the internal friction whenCmaX increases will be taken into account.

Limits of the model are obtained when Franck & Read sources are activated that is to say that dislocation density increases. It is obtained when glide in basal planes leads to a curvature as x = 112, y = 112.

or G l i m i t = 2 Gb/l 2 lo5 ~ . m ~ .

and when the time where stress is applied,is tlimit = Ul/vd*500 s at 250 K.

That is to say that for internal friction measurement the limits of the model will be attained when frequency decreases below loe3 Hz.

A program of simulation has been developped for internal friction and microcreep experiments. For internal friction we calculate the deformation resulting from a sinusoi'dal stress G =Go s i n w t by an incremental method. We obtained couples of values ( G i , E i ) when Svaries. Internal friction is obtained from the area of cycle (%E).~he values are taken after the second cycle in order to obtain steady state

regime.

From microcreep simulation, the best fit with experimental curves is obtained when 1, = 10-4 m and h = 2.10-8 m ; the dislocation density is taken between 109 and 5.

109 m-2 in respect with X-ray topography done on our specimen.

Thus, the main features (temperature, frequency and stress dependance of internal friction - comparison between anelastic and viscoplastic behaviour ) have been explained elsewhere, but the effect of HF doping was still to be considered.

The dislocation velocity vd is taken as those deduced from the non-crystalline core model 131. V d = ( 2 / 9 ~ ) . ( ~ b ~ / 6 ~ k T) exp (- 11C) where C is the concentration of broken bonds in the core.

Fig.4.Internal friction vs.T

pure ice. Dashed 1 ines=sirnulation. F i g . 5 . Internal friction vs. T HF-doped ice.Dashed lines=sirnulation.

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~1::-342 JOURNAL DE PHYSIQUE

For pure i c e C = C ( ~ T / E ) ~ w i t h k = 1.4 and E d i f f e r e n c e between energy o f bonds i n t h e c o r e and broken bonds (E =! 0.1 eV). F o r HF-doped ice,C has t h e same f o r m depending on HF concentration. With 10 ppm HF, C s ( ~ T / E ) ~ + 2.5 / 3 / .

The s e i f - d i f f u s i o n c o e f f i c i e n t DSD does n o t v a r y when doped and i t i s taken as DSD = 1 5 . 1 0 - ~ exp (- 0.62/kT) m2 s-1 /8/.

S i m u l a t i o n s g i v e s curves as shown i n f i g . 4 f o r pure i c e w i t h p = 1.4 and a d i s l o c a t i o n d e n s i t y equal t o 5. 109 F o r HF doped i c e comparison i s more d i f f i c u l t because HF-doping i s known t o i n c r e a s e a l s o d i s l o c a t i o n density. However, as shown i n f i g . 5, s i m u l a t i o n s w i t h

P

= 5 l o 9 m-2 f o r a 10 ppm HF doped i c e a r e q u a l i t a t i v e l y i n agreement w i t h 'experiment i n p a r t i c u l a r , t h e r i s e i n i n t e r n a l f r i c t i o n begins a t lower temperature than i n pure ice.

REFERENCES

/1/ Perez, J., Mai, C., Tatibouet, J. and Vassoil l e R., Nuovo Cimento 833 (1976) 86-95.

/2/ Vassoile, R., Mai, C. and Perez J., J. G l a c i o l . 85 (1978) 375-384.

/3/ Perez, J., Mai, C., Tatibouet, J. and V a s s o i l E , R., J. G l a c i o l . 9 1 (1980) 133-149.

/4/ T a t i b o u e t , J., Perez, J. and V a s s o i l l e , R., J. de Physique Q, t. 44 (1983), 799-803.

/5/ Benoit, W., Gremaud, G. and S c h a l l e r , R., P l a s t i c Deformation o f amorphous and semi-crystal 1 in e m a t e r i a l s , Les Houches, E d i t i o n de Physique (1982) 65-91.

/6/ F r i e d e l J., D i s l o c a t i o n s , S o l i d S t a t e Physics v o l . 3, Pergamon Press, N-Y (1964) 105-111.

/7/ T a t i b o u e t , J., Perez, J. and V a s s o i l l e , R., J. Phys. Chem. 87 (1983) 4050-4054.

/8/ Ramseier, R.O., J. Appl. Phys. 2 (1967) 2553-2556.