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MELTING UNDER PRESSURE AND THERMAL DEFECTS APPLICATION TO GEOPHYSICAL
MINERALS
A. Migault
To cite this version:
A. Migault. MELTING UNDER PRESSURE AND THERMAL DEFECTS APPLICATION TO GEOPHYSICAL MINERALS. Journal de Physique Colloques, 1984, 45 (C8), pp.C8-113-C8-116.
�10.1051/jphyscol:1984822�. �jpa-00224322�
MELTING UNDER PRESSURE AND THERMAL DEFECTS APPLICATION TO GEOPHYSICAL MINERALS
A. M i g a u l t
Laboratoire d'Energetique at de Detonique*, E.N.S.M.A., rue Guillaume VII, 86034 Poitiers Cedex, France
Abstract -The method of the critical concentration of thermal defects gives a linear correlation between the logarithmic derivative with res- pect to the volume of enthalpy of formation of one defect on the cold- compression isotherm and the pressure derivative of Poisson's ratio.
This relation is applied to three minerals (KC1, NaCl and Mg„ Si 0 J . I - INTRODUCTION
The Lindemann law is one of the oldest and simplest criterion to describe the melting of solids, but her application to polyatomic materials is not even appropriate.
In the following, we utilize another melting criterion which gives a melting curve in the (T,V) plane : the method of the critical concentration of thermal defects ; in this, the dependence of the melting temperature, T , on the pressure P is deter- mined assuming that melting begins when the concentration of thermal defects
(Frenkel or Schottky defects) reaches a critical value which remains constant along the melting curve / I / .
II - METHODE OF THERMAL DEFECTS AND THE KRAUT-KENNEDY MELTING LAW
The criterion of the critical concentration of thermal defects gives a melting law T (V) in the form / I / :
(1)
n is a constant which depends on the interaction law between atoms;
T is the melting temperature at normal pressure (P = 0 ) .
L is the logarithmic derivative with respect to the volume of enthalpy (h ) of formation of one defect on the cold-compression isotherm :
(2)
x is the ratio (V/V„ ) where V„ is the volume at the melting temperature T .
om om 3 r om
*LA 193 CNRS
Résumé -La méthode de la densité critique des défauts thermiques conduit à une corrélation linéaire entre la dérivée logarithmique par rapport au volume de l'enthalpie de formation d'un défaut thermique le long de la courbe de compression au zéro absolu et la dérivée du coefficient de Poisson par rapport à la pression. Cette loi, valable pour des corps monoatomiques est vérifiée dans le cas de NaCl, KC1 et Mg„ Si 0..
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984822
C8-114 JOURNAL DE PHYSIQUE
I f n % 0, e q u a t i o n ( 1 ) g i v e s :
and, f o r small x we can deduce f r o m ( 3 ) :
T h i s i s t h e Kraut-Kennedy m e l t i n g l a w /2/ i f we assume t h a t Loo i s t h e c o n s t a n t C i n t h e Kraut-Kennedy law. E q u a t i o n ( 4 ) i s c o r r e c t even i f n = 0.
From e q u a t i o n (41, we have : dLn T
om
E q u a t i o n ( 3 ) i s a g e n e r a l m e l t i n g law deduced f r o m t h e method o f t h e c r i t i c a l c o n c e n t r a t i o n o f t h e r m a l d e f e c t s : t h i s l a w g i v e s a c u r v e i n t h e (T,V) plane; But, f o r geophysical a p p l i c a t i o n s , i t i s more i n t e r e s t i n g t o have a c u r v e i n (T,P) plane;
so, we must u t i l i z e an e q u a t i o n o f s t a t e .
111 - EQUATION OF STATE (E.O.S.) OF MIE-GRUNEISEN ( / 3 / , / 4 / , / 5 / ) -.
T h i s model d e s c r i b e an E.O.S. where t h e Gruneisen c o e f f i c i e n t Y
,
f o n c t i o n o f t h e volume o n l y , depends on a parameter p which i s a c h a r a c t e r i s t i c o f metal :K i s t h e b u l k modulus a t P = 0 ;
, ,
t h e P o i s s o n ' s r a t i o and oo t h e v a l u e o f 0 a t PO= 0:We see t h a t p i s p r o p o r t i o n a l t o t h e p r e s s u r e d e r i v a t i v e o f P o i s s o n ' s r a t i o a t normal p r e s s u r e .
For p = 0, 2/3 o r 4/3, e q u a t i o n ( 6 ) g i v e s t h e S l a t e r , Dugdale-Mc Donald o r " f r e e volume t h e o r y " f o r m u l a . Only f o r m e t a l s (monoatomic c r y s t a l s ) we have shown t h a t t h e parameter p can be deduced f r o m t h r e e independent methods :
i ) f r o m Hugoniot curves /3/
i i ) f r o m e l a s t i c c o n s t a n t s /4/
i i i ) f r o m m e l t i n g c u r v e s a c c o r d i n g t o t h e Lindemann c r i t e r i o n / 5 / . I V - CORRELATION BETWEEN Lno AND p AND APPLICATION TO GEOPHYSICAL MATERIALS
" -
The method o f c r i t i c a l c o n c e n t r a t i o n o f t h e r m a l d e f e c t s has been a p p l i e d t o copper and g o l d . Up t o 10 GPa, r e s u l t s a r e v e r y c l o s e t o t h o s e o b t a i n w i t h t h e Lindemann c r i t e r i o n /5/. F i g . 1 shows r e s u l t s i n t h e case o f g o l d .
From ( 2 ) and ( 7 1 , i t i s c l e a r t h a t i t must have a c o r r e l a t l i o n between p and Loo because t h e s e two q u a n t i t i e s depend on t h e d e r i v a t i v e w i t h r e s p e c t t o t h e volume of t h e c o h e s i v e energy o f c r y s t a l .
We have s t u d i e d t h i s c o r r e l a t i o n f o r 10 m e t a l s (Mg, A l , I n , Pb, Ag, Cu, Au, Cd, N i , and P t ) : L i s deduced f r o m ( 5 ) and r e f e r e n c e /6/, p i s t a k e n f r o m /5/. With t h e method o f lea!? square f i t , we o b t a i n a l i n e a r l a w :
FIG.1 - G o l d : comparison between Lindemann law and c r i t e r i o n o f t h e r m a l d e f e c t s . 0.99
0.98 0.97 0.96 0.95 0.94
I I
MELTING
:Thermal Defects
I- -
-
-
- - -
I I I I
&
I 1FORST.
-
-
--
-THERMAL DEFECTS (11
-
LINOEMANN LAW : p=-1 ; p=- 1.2
-
--P=10.5 GPa
I I I I I I I 1
FIG.2
-
Experimental p o i n t s (Loo, p ) and t h e l i n e a r l a w ( 8 ) .1.0 1.04 1.08 1.12 1.16 1.20 1.24 1.28 1.32 1.36 I.W)Tm/Tom
A t normal pressure, Mukherlee / 7 / has shown t h a t i t e x i s t s a l i n e a r r e l a t i o n between t h e energy o f f o r m a t i o n o f a thermal d e f e c t ( E f ) and t h e q u a n t i t y
eD2 I 3 ! ' !
where O D i s t h e Debye t e m p e r a t u r e and V, t h e volume.So, a t normal pressure, t h i s r e l a t i o n g i v e s :
Loo = 2 y o
-
2/3 ( 9 )where uo i s t h e Gruneisen parameter a t normal p r e s s u r e / 3 / .
F o r Morse o r Born-Mayer i n t e r a t o m i c p o t e n t i a l , r e s u l t s o f /3/ g i v e :
where A i s a q u a n t i t y which i s r e l a t e d t o c o n s t a n t s o f metal under c o n s i d e r a t i o n . So, r e l a t i o n ( 1 0 ) shows t h a t t h e l i n e a r r e l a t i o n ( 8 ) i s n o t f o r t u i t u s .
C8-116 JOURNAL DE PHYSIQUE
We have i n v e s t i g a t e d s e v e r a l m a t e r i a l s o f g e o p h y s i c a l i n t e r e s t w i t h two o r more atoms i n u n i t c e l l : S y l v i t e (KC1
1,
H a l i t e (NaCl), F o r s t e r i t e (Mg S i 01.
These 3 m i n e r a l s have a w e l l - d e f i n e d m e l t i n g round about P = 0. Data, d?ssoci&ed w i t h a m a t e r i a l , necessary f o r comparison w i t h ( 8 ) a r e r e p o r t e d i n t a b l e I .TABLE I
F o r NaCl and KCl, L i s deduced f r o m e q u a t i o n ( 5 ) and r e s u l t s o f /11/ ; p i s g i v e n i n /8/ f r o m e q u a t i o n P 0 ( 7 ) . F o r Mg2 S I 04, p i s deduced f r o m e q u a t i o n s ( 7 ) and d a t a g i v e n i n /9/.
NaCl KC 1 Mg2 S i O4
Figufe ( 2 ) shows t h e c o o r d i n a t e s ( L
,
p ) f o r t h e s e 3 m i n e r a l s . We can see t h a t t h e y a r e i n good accordance w i t h e q u a t i o f i 0 ( 8 ) .We have a l s o i n v e s t i g a t e d f a y a l i t e (Fez S i 04) which i s another m a t e r i a l o f geophy- s i c a l i n t e r e s t ( a b o u t 20 volume p e r c e n t i n o l i v i n e w i t h 80 volume p e r c e n t o f f o r s - t e r i t e ) . R e s u l t s of Sumino /12/ and e q u a t i o n ( 6 ) g i v e : p = 2.4 + 1.2 and t h o s e o f /12/ and /13/ g i v e : L = 3.64. T h i s r e s u l t i s c o n s i s t e n t w i t h P ( 8 ) i f i t i s t a k e i n t o account l a r g e e r r o P O i n p . But we must n o t i c e t h a t m e l t i n g o f f a y a l i t e i s n o t w e l l - d e f i n e d i n t h e r a n g e o f 0-7 GPa
/ l o / .
Tom (K) 1074 /11/
1045 /11/
2163
/ l o /
CONCLUSION
The c r i t e r i o n o f t h e r m a l d e f e c t s , which i s n o t a m e l t i n g t h e o r y , g i v e s a m e l t i n g l a w i n t h e (T,V) p l a n e . T h i s method i s p h y s i c a l l y v a l i d f o r monoatomic as w e l l as m u l t i a t o m i c u n i t c e l l s . From t h i s method, we a r e a b l e t o o b t a i n t h e Kraut-Kennedy m e l t i n g law, and t h e knowledge o f t h e p r e s s u r e d e r i v a t i v e o f P o i s s o n ' s r a t i o g i v e s some i n f o r m a t i o n s on t h e m e l t i n g o f m i n e r a l s .
P
-0.22 /8/
0.16 /8/
1.65 /9/
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