8/18/2019 Fei et al. (1986).pdf

1/9

Co ntrib Mineral P etrol (1986) 94:221-229

ontributions to

M i n era lo g y an d

Pet ro logy

9 Springer-Verlag 1986

So m e binary and ternary s i li cate so lut ion m odels

Y . F e i 1 S . K . S a x e n a 1 a n d G . E r i k s s o n z

1 Dep ar tmen t of Geology, B rooklyn Col lege, Brooklyn, NY , 11210, USA

2 Depar tm ent o f Inorganic C hemist ry, Univers i ty of Umef i , S-901 87 Ume~, Sweden

A b s t r a c t .

F o u r d i f f e r e n t s o l u t i o n m o d e l s , t h e t w o - p a r a m e -

t e r M a r g u l e s , t h e q u a s i - c h e m i c a l ( Q C ) , t h e W i l s o n a n d t h e

n o n - r a n d o m t w o -l iq u i d ( N R T L ) m o d e l , h a v e b e e n u s ed f o r

f i tt i n g t h e c a l o r i m e t r i c e x c e s s e n t h a l p y o f s o l u t i o n f o r t h e

f o l l o w i n g f o u r b i n a r y s i l i c a t e s y s t e m s : a n o r t h i t e - a l b i t e , p y -

r o p e - g r o s s u l a r , d i o p s i d e - e n s t a t i t e a n d d i o p s i d e - C a - T s c h e r -

m a k . A l l m o d e l s e x c e p t th e W i l s o n m o d e l y i el d a s a ti s f a c t o -

r y f it to t h e d a t a b u t t h e N R T L m o d e l g e n e r a ll y re s u lt s

i n t h e l o w e s t r e s i d u a ls . T h e u s e o f N R T L a n d Q C f a c i l it a t e s

t h e s t u d y o f t h e c o n f i g u r a t io n a l a n d n o n - c o n f i g u r a t i o n a l

p a r t s o f t h e e x c e ss e n t r o p y o f m i x i n g .

T h r e e d i f f e re n t m e t h o d s , n a m e l y t h o s e o f K o h l e r , W o h l ,

a n d H i l l e r t , h a v e b e e n u s e d t o c o m b i n e b i n a r y s o l u t i o n

p r o p e r t i e s to p r e d i c t t e r n a r y s o l u t i o n p r o p e r t ie s . C o m p a r i -

s o n o f c o m p u t e d e x c es s fr e e e n e r g y o f m i x in g i n a h y p o t h e t i -

c a l s o l u t i o n s h o w s t h a t a l l t h e t h r e e m e t h o d s a r e v i a b l e

b u t t h e K o h l e r a n d W o h l m e t h o d s a r e si m i l ar to e a c h o t h e r

a n d a r e s i g n i f i c a n t l y d i f f e r e n t f r o m t h e H i l l e r t m e t h o d . T h e

K o h l e r m e t h o d w i t h o n e o r a c o m b i n a t i o n o f d i ff e r e n t b i n a -

r y m o d e l s i s r e c o m m e n d e d f o r p r e d i c t i n g m u l t i c o m p o n e n t

s o l u t i o n p r o p e r t i e s .

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

G e o c h e m i s t s h a v e m a i n l y u s e d t h e M a r g u l e s m o d e l f o r b i -

n a r y s o l i d s o l u t i o n s , p a r t l y d u e t o i t s d e t a i l e d d i s c u s s i o n

b y T h o m p s o n ( 1 96 7 , 1 9 69 ) a n d p a r t l y d u e t o i ts s im p l i c i t y

o f f o r m u l a t i o n . O t h e r s o l u t i o n m o d e l s s u c h a s t h e q u a s i -

c h e m i c a l m o d e l a n d t h o s e d e s c r ib e d h e r e h a v e l a r g e ly b e e n

i g n o r e d ( G r e e n 1 9 7 0 ; S a x e n a 1 9 7 3 ; P o w e l l 1 9 83 ). T h e p u r -

p o s e o f t h i s p a p e r i s f ir s t ly t o c o m p a r e t h e c a p a b i l i t ie s o f

s o m e b i n a r y s o l u t i o n m o d e l s , p o p u l a r i n c h e m i c a l a n d m e -

t a l l u r g i c a l l i t e r a t u r e , i n p r e d i c t i n g t h e e x c e s s f u n c t i o n s o f

m i x i n g i n s e v e r a l b i n a r y s i l i ca t e s a n d s e c o n d l y t o s t u d y t h e

v a r i o u s m e t h o d s a v a i l a b le f o r c o m p u t i n g m u l t i c o m p o n e n t

s o l u t i o n p r o p e r t i e s f r o m t h e d a t a o n b i n a r y s o l u t i o n s . F o r

t e r n a r y o r m u l t i c o m p o n e n t s o l u t io n s t h e p a p e r w i ll b e

m a i n l y c o n c e r n e d w i t h th e e m p i r i c a l m e t h o d s a n d t h e r e

w i ll b e n o d i s c u s s i o n o n t h e p h y s i c a l s i g n if i ca n c e . T h e w o r k

w i ll e x c l u d e m i x i n g o n m o r e t h a n o n e s i te i n m u l t i s it e s o l id s .

Abbreviations. G~ exc ess free energy of mixing; H ex, excess

enthalpy of mixing; Sex, total excess entrop y o f mixing; S~x, confi-

gurational exc ess entro py o f mixing; W ij , interaction energy

para me ter between species i and j ; Xi; mole fraction o f species

Offprint requests to. Y. Fei

i ; QC, quasi -chemical ; NR TL, non-ran dom two- l iquid; M , Ma r-

gules formulat ion; W, Wohl 's formulat ion; RK, Redl ich-Kis ter ;

K, Bertrand-Kohler; H, Hil lert ; Di, diopside (CaMgSi206); En,

enstatite Mg2Si206); Py, p yro pe (M gA12/3SiO4); Gr, grossular

(CaAI2/3SiO~); CaTs, C a-Tsc herm ak CaAlzSiO6); Ab, albite

(NaA1Si3Os); An , ano rthite (CaAlzSi2Os). Othe r abbreviations

and symbols in the text .

B i n a r y s o l u t i o n m o d e l s

W e s h a l l d i v i d e th e m o d e l s t o b e d i s c u s se d u n d e r t w o d i f f e r -

e n t g r o u p s . T h e f i r s t g r o u p o f m o d e l s h a v e t h e i r o r i g i n

i n th e F l o r y - H u g g i n s m o d e l ( F l o r y 1 9 5 3) , i n w h i c h s o l u -

t i o n s a r e c o n s i d e r e d a s a t h e r m a l w i t h z e r o e x c e s s e n t h a l p y

o f m i x i n g . T h e l a t e r r e f in e d v e r s i o n s , w h i c h a r e t h e W i l s o n

m o d e l ( W i l s o n 1 9 6 4 ) , t h e q u a s i - c h e m i c a l ( G u g g e n h e i m

1 95 2) a n d t h e n o n - r a n d o m t w o - l i q u id s ( N R T L ) m o d e l

( R e n o n a n d P r a u s n i t z 1 9 6 8) , d o in v o l v e e n t h a l p y o f m i x i n g

a n d h a v e b e e n r e c e n t l y r e v ie w e d b y A c r e e ( 19 84 ). T h e s e c -

o n d g r o u p o f m o d e l s s i m p l y ex p r e ss f u n c t io n s b y a p o w e r

s e ri e s i n m o l e f r a c t io n . T h e R e d l i c h - K i s t e r a n d t h e t w o -

c o n s t a n t M a r g u l e s m o d e l a m p l y d i s c u s s e d i n g e o c h e m i c a l

l i t e r a tu r e ( T h o m p s o n 1 9 6 7 ; S a x e n a 1 9 73 ) f a ll i n t h i s c a t e g o -

ry.

F l o r y - H u g g i n s a n d W i l s o n m o d e l s

S o l u t i o n s w i t h z e r o e n t h a l p y o f m i x i n g a r e r e f e r r e d t o a s

a t h e r m a l s o l ut io n s . T h e e n t r o p y o f m i x i n g f o r s u ch s o l u -

t i o n s w a s d i s c u s s e d b y F l o r y ( 1 9 5 3 ) . I n t h e F l o r y - H u g g i n s

m o d e l , i t i s a s s u m e d t h a t m s i t es a r e o c c u p i e d c o n s e c u t i v e l y

b y t h e s a m e s p e c i e s a n d i f t h e r e a r e n 2 m o l e s o f t h i s s p e c ie s

a n d n l m o l e s o f t h e s o l v e n t , t h e v o l u m e f r a c t i o n o f th e

s o l v e n t a n d t h e s o l u t e a r e g i v e n b y

~ 1 = n i / n i + n 2 m )

a n d

~ 2 = n 2 m / n l + n 2 m ) . (1)

F o r a t h e r m a l s o l u t i o n t h e n

G imix_- _ T S~i~ =

R T n l l n ~ l +

n21n~2). (2)

E q u a t i o n ( 2 ) i s e x p e c t e d t o b e u s e f u l w h e n t h e m i x i n g

s p e c ie s h a v e i d e n t ic a l m o l a r v o l u m e s . W i l s o n ( 1 9 64 ) c o n s i d -

e r e d m i x i n g o f sp e c i e s w h i c h d i f f e r i n v o l u m e a n d i n t h e i r

i n t e r m o l e c u l a r f o r c e s .

T h e f r e e e n e r g y o f m i x i n g i n t h e W i l s o n m o d e l i s g i v e n

b y

8/18/2019 Fei et al. (1986).pdf

2/9

222

G ~x = RT (Xlln~b 1 + X21n~b2) (3)

w h e r e ~b~ i s th e v o l u m e f r a c t i o n o f a c o m p o n e n t i a b o u t

a c e n t r a l c a ti o n o f t h e s a m e t y p e . T h e v o l u m e f r a c t io n o f

a c o m p o n e n t i s d e te r m i n e d b y c o n s i d e r in g t h e p r o b a b i l i t y

o f c a t i o n d i s t r i b u t i o n s . T h e p r o b a b i l i t y o f fi n d i n g a c a t i o n

o f t y p e 2 r e l a t iv e t o t y p e 1 a r o u n d a c e n t r a l c a t i o n o f t y p e 1

i s g iv e n b y

X 2 1 / X l l = X2 exp ( - w 12/R 7 ) / X l e x p - - w l l / R 7 ) (4)

w h e r e X ~j i s th e l o c a l m o l e f r a c t i o n c o m p o n e n t o f i i n t h e

i m m e d i a t e v i c in i ty o f c o m p o n e n t j, a n d w~j i s p r o p o r t i o n a l

t o t h e in t e r a c t io n e n e r g y b e t w e e n c o m p o n e n t s i a n d j . S i m i-

l a r l y f o r t h e r e v e r s e p r o b a b i l i t y , w e h a v e

X 12/ X 22 = X 1 e x p ( - w 1 2 / R T ) / X 2 e x p - w 2 2 /R 7 ). (5)

T h e v o l u m e f r a c t i o n ~b i s d e f in e d a s

~l=Xll V1/ X11V1~- X21V2)

= X 1 V I e x p ( - wl 1 / R T ) / [X 1 V1 e x p ( - w l I / R 7 )

+ ); -2 V z e x p ( - W l z / R T ) ] . (6)

S i mi l a r l y

~b2 = X 2 V 2 e x p ( - - w22/ R T ) / [X 1 V 1 e x p ( - - w 1 2 / R T )

+ X 2 V 2 e x p ( - w22/ R T ) ] (7)

w h e r e V1 a n d V 2 a r e m o l a r v o l u m e s o f c o m p o n e n t s 1 a n d

2 r e s p e c t iv e l y . T h e e x c e ss o f f r e e e n e r g y o f m i x i n g i s t h e n

g i v e n b y

G ~ = - R T [ X l l n ( XI + X 2 W 2 1 ) + X21n(X2 + X1 W l 2)] (8)

w h e r e

W 2 1 = V 2 / V 0 e x p [ - w 12 - w l 1 )/ R T] (9)

W~ 2 = } 11 / V : ) exp[ - - (w12 - - w~ : ) / R T ] . (10)

O n e o f t h e m a j o r p r o b l e m s w i t h t h e W i l s o n m o d e l s t h a t

i t c a n n o t b e u s e d t o p r e d i c t u n m ix i n g . A c c o r d i n g t o t h e

m o d e l , a s y s t e m w i ll b e c l o se t o s e p a r a t i o n i n t o t w o p h a s e s

w h e n t h e p a r a m e t e r s W ~j a r e c l o s e t o z e r o . I n t h i s li m i t

t h e e x c e s s f re e e n e r g y o f m i x i n g c a n c e l s w i t h t h e i d e a l f r e e

e n e r g y o f m i x i n g , r e q u i r i n g t h a t t h e f r e e e n e r g y o f m i x i n g

t o e q u a l z e r o a t a l l b i n a r y c o m p o s i t i o n s . T h i s i n t u r n m e a n s

a h i g h e x c e s s e n t h a l p y w h i c h w o u l d i n c r e a s e w i t h t e m p e r a -

t u r e bec aus e H ~x equa l s T S ~x.

T h e n o n - r a n d o m t w o -l iq u i d s N R T L ) m o d e l

T o r e m e d y t h e d e f ec t i n th e W i l s o n m o d e l , R e n o n a n d

P r a u s n i t z ( 1 9 6 8 ) i n t r o d u c e d t h e N R T L m o d e l w h e r e t h e y

d e v e l o p e d fu r t h e r t h e c o n c e p t o f n o n - r a n d o m m i x i n g o f

t w o l iq u i d s u b - m i x t u re s . E a c h s u b - m i x t u r e m a y b e c o m p a -

r a b l e t o a d o m a i n ( o r a c e l l ) i n w h i c h t h e c e n t r a l m o l e c u l e

(1 o r 2 ) is s u r r o u n d e d b y a d e f i n i t e o r d e r o f th e t w o t y p e s

o f m o l e c u l e s . T h i s o r d e r d e t e r m i n e s t h e m o l e f r a c t i o n s X 1

a n d 5 (2 . T h e t o t a l G i b b s f re e e n e r g y o f m i x i n g ( G mi~) of

t he so l u t i on i s

G m lx = X I G ( 1) + X 2 G ( 2 ) ( 1 1 )

w h e r e G ~ ) a n d G z) a r e r e l a t e d t o l o c a l m o l e f r a c t i o n c o m -

p o s i t io n s t h r o u g h

G ~1) = X1 l w l 1 + X21w 12 (12)

G ~2) = X1 2w l 2 + X22w22 (13)

w h e r e w~j r e p r e s e n t s a b i n a r y i n t e r a c t i o n p a r a m e t e r a n d

X ~j r e f e r s t o t h e m o l e f r a c t i o n o f c o m p o n e n t i i n t h e i m m e d i -

a t e vi c in i ty o f c o m p o n e n t j . A c c o r d i n g t o R e n o n a n d P r a u s -

n i t z ( 1 9 68 ) t h e e x c e ss f r e e e n e r g y o f m i x i n g i s g i v e n b y

Gex y t ' t - 7 (1 ) ~ (1 ) -~q_~t~ t .G(2 )

G (2) ~ (14 )

- - ~ x 1 u - - ' a p u r e ] 2 k - - p u r e ]

C o m b i n i n g E q s . (1 1 ) - ( 1 4 ), t h e y o b t a i n e d

G e x = x I X 2 1 W I z - w l O + X z X 12 w 1 2 -w 2 2 ). 15 )

I n a n a l o g y w i t h t h e W i l s o n m o d e l , t h e l o c a l m o l e fr a c t i o n s

a r e r e l a t e d t o t h e o v e r a l l m o l e f r a c t i o n s a s f o l l o w s :

X 2 1 / X l l = X 2 e x p - ~ w 1 2 / R 7 ) / X l e x p - ~ w l l / R 7 ) (16)

a n d

X 1 2 / X 2 2 = X l e x p - ~ w 1 2 / R 7 ) / X 2 e xp - O ~ W z 2 / R 7 ) (17)

w h e r e ~ i s a c o r r e c t i o n f a c t o r . T h e e x c e s s f r e e e n e r g y o f

m i x i n g i s g i v e n b y

GeX=RTXlX2{[~21 W 2d X l + X2 W21)]

+ [r~2 w12 / x2 + x l w12)]} 18)

w h e r e

2 12 = ( w 1 2 - w 2 2 ) / R T ;

2 21 = ( W 1 2 - - W 1 1 ) / R T ;

W 12 = e x p ( - ~ 2 12);

W z 1 = e x p ( - ~ 2 21).

O t h e r e x c e s s f u n c t i o n s a r e

H e x = R T X 1 X z { [ X ~ + X 2 W21)Vza W21

- x I ~ L w 2 1 ] / x , + x 2 w 2 1 2

+ [ x , + x l w ,2)~12 w12

- - X 2 0{ 17 22 W 1 2 ] / ( X ' 2 ~ ~ ('1 m 1 2 ) 2 } ( 1 9 )

s ~ = - RXlX2[X1 ~r L W21/ X l + X2 W~ 1 )2

+ X2~r~2 W I2 / X2 + ) 1/4112)1]. (20)

F o r a c t i v i t y c o e f f ic i e n ts , w e h a v e

RZqnyl = RrX~[~21 W L / X l + X 2 W 2 0 2

+ 2712 Wl2/ X2 ~-X1W 1 2 ) 2 ] . ( 2 1 )

T h e e x p r e s s i o n f o r ~ '2 i s o b t a i n e d b y e x c h a n g i n g t h e s u b -

sc r i p t s 1 and 2 .

Q u a s i - c h em i c a l m o d e l

I n t h e z e r o t h a p p r o x i m a t i o n , G u g g e n h e i m ( 1 9 5 2 ) c o n s i d -

e r e d a r a n d o m m i x i n g o f t w o c o m p o n e n t s , T h u s , t h e a v e r -

a g e n u m b e r o f

A B

p a i r s i n a p a r t i c u l a r c o n f i g u r a t i o n i s

g i v e n b y

X = N A N R / N A + N B ) (22)

w h e r e N A a n d N B r e f e r t o t h e n u m b e r o f m o l e s o f t h e s p e -

c i es A a n d B r e s p e c t i v e ly . T h e q u a s i - c h e m i c a l m o d e l r e -

p l a c e s t h e p h y s i c a l l y u n r e a l i s ti c r a n d o m v a l u e o f X w i t h

~/-2= ( NA - - X) (NB - X) ex p ( - - 2 w / Z k T ) (23)

w h e r e k i s B o l t z m a n n c o n s t a n t a n d 2 w / Z i s t he ene r gy r e -

q u i r e d t o c h a n g e a n A A p a i r a n d a B B p a i r i n t o t w o A B

p a i r s a n d X , 1 / 2 ( N A - - X ) a n d 1 / 2 ( N s - X ) a r e p r o p o r t i o n a l

t o t h e n u m b e r o f A B , A A a n d B B pa i r s r e spec t i ve l y . I t

i s p o s s i b l e t o d i r e c t l y c o m p a r e t h e q u a s i - c h e m i c a l m o d e l

w i t h t h e z e r o t h a p p r o x i m a t i o n b y d e f in i n g a p a r a m e t e r fl,

s u c h t h a t

X = [ N A N s / N A + Ns)][ 2/(fl + 1)]. (24)

I f f l = 1 , w e o b t a i n t h e z e r o t h a p p r o x i m a t i o n . I f f l i s g r e a t e r

t h a n u n i t y , t h e r e w o u l d b e a t e n d e n c y f o r c l u s t e r i n g . A

8/18/2019 Fei et al. (1986).pdf

3/9

l e ss t h a n u n i t y v a l u e o f f l i n d i c a t e s a t r e n d t o w a r d s c o m -

p o u n d f o r m a t i o n . F r o m (2 3) , o n e m a y o b t a i n

f l = {1 +

4 X t X 2 [ e x p ( 2 W / Z R T ) -

1]} 1/2. (25)

I n o r d e r t o c o n s i d e r m i x i n g o f m o l e c u l e s o f d if f e r e n t

s iz e s, G u g g e n h e i m ( 1 95 2 ) i n t ro d u c e d t h e c o n c e p t o f p o -

l y m e r s o c c u p y i n g t h e s it e s. A m o n o m e r o c c u p i e s o n e s it e ,

a d i m e r t w o a n d a r - m e r o c c u p i e s r s it es . A v o l u m e f r a c t i o n

~b i s d e f i n e d a s t h e f r a c t i o n o f s i te s o c c u p i e d b y a p o l y m e r .

F o r n o n - a t h e r m a l s o l u t i o n s , G u g g e n h e i m ( 1 9 5 2 ) a r g u e d

t h a t t h e c o n f i g u r a t i o n a l p o t e n t i a l e n e r g y m a y b e e x p r e s se d

a s a s u m o f c o n t r i b u ti o n s f r o m p a i r s o f n e i g h b o r in g e l e -

m e n t s , t h e c o n t r i b u t io n f r o m e a c h p a i r d e p e n d i n g o n t h e

n a t u r e o f b o t h e l e m e n t s f o r m i n g t h e p a ir . H e u s e d t h e w o r d

c o n t a c t t o d e n o t e t h e g e o m e t r ic a l r e la t i o n o f a n e l e m e n t

t o t h e e l e m e n t o f s o m e o t h e r m o l e c u l e i n a n e i g h b o r i n g

s it e. T h u s a n o p e n - c h a i n r - m e r c o n t a i n i n g r e l e m e n t s h a s

Z q c o n t a c t s w h e r e q is r e l a te d t o r b y

( 1 / 2 ) Z ( r - - q ) = r - 1 . ( 26)

T h e f o l lo w i n g e q u a t i o n s o f t h e Q C m o d e l a r e a d o p t e d f r o m

G u g g e n h e i m ( 19 5 2) . T h e y h a v e a l s o b e e n d i s c u s s e d b y

G r e e n ( 1 9 7 0 ) a n d S a x e n a ( 19 7 3) . T h e e x c e ss f u n c t i o n s o f

m i x i n g a r e g i v e n b y ( S a x e n a 1 9 7 3 ):

G ~x = ( 1 / 2 ) Z R T { X t q t l n [ ( f l + ~ - ~ 2 ) / q ~ ( f l +

1)]

+ X2qa l n [ ( f l + ~ 2 - - a s j / ~ q ~ + 1 )]} ( 27 )

H ~x = [4Xa x z

W/ f l ( f l

+ 1)]exp(2

W / Z R T )

9 x tq tc ~dr + ~ t - ~ ) + x~q2~ t / r + ~ -

ah) ] 28)

S ~ = ( H ~ - G ~ X ) /T (29)

w h e r e Z i s c o o r d i n a t i o n n u m b e r , ~b~ a n d ~ 2 a r e c o n t a c t

f r a c t io n s , q l a n d q z c o n t a c t f a c t o r s , a n d W i s t h e i n t e r a c t i o n

p a r a m e t e r . T h e q u a n t i t i e s q a n d ~b a r e r e l a t e d a s

~ t = X t q t / ( X t q a + X 2 q 2 )

a n d

q ~ = X ~ q 2 / ( X ~ q t + X z q ~ ) (30)

and f l i s g i ven by ( 25) . D i f f e r en t i a t i on o f ( 27), g i ves :

R T l nT ~ = R T In [ 1 + ~ ( f l - - 1)/q~t(fl + 1 ]Zqd2. (31)

T o o b t a i n R T l n ~ z , e x c h a n g e t h e s u b s c r i p t s 1 a n d 2 .

M a r g u l e s m o d e l

T h i s m o d e l h a s b e e n d i s c u s s e d e x t e n s i v e l y b y T h o m p s o n

( 1 9 6 9 ) a n d t h e r e f o r e , n o n e w r e v i e w s i s n e c e s s a r y . T h e e x -

c e ss f re e e n e r g y o f m i x i n g ( S a x e n a 1 9 7 3 ) i s g i v e n b y

G ~ = X~X 2( X 2 W ~z + XI W~z~). (32)

O t h e r e x c e s s f u n c t i o n s m a y b e e x p r e s s e d s i m i l a r l y :

H ~ = X~X 2(X2 W~2 + X~ Wzn~) (33)

S ~ = X t X 2( X z W~2 + XI W~2~). (34)

A c t i v i t y c o e f f i c ie n t s a r e g i v e n b y

R ~ n ) , , = x ~ [ w ? ~ + 2 x ~ w ~ -

w ? ~ ) ] 3 5 )

RTIn~ ,2 = X~[W2~t

+ 2 X 2 ( W ~ a 2 -

W~2 )]. (36 )

R e d l i c h - K i st e r m o d e l

G u g g e n h e i m ( 1 9 37 ) s u g g e s t e d t h a t G ~ c a n b e e x p r e s s e d

a s a p o l y n o m i a l i n X a s

223

G ~ = X ~ X 2 [ A o + A t ( X t - X 2 ) + A 2 ( X 1 - - X 2 ) 2 + . . . ] (37)

w h e r e A o , A t a n d A s a r e c o n s t a n t s . T h e a c t i v i ty c o e ff i c ie n t s

u s i n g th e R e d l i c h - K i st e r m e t h o d a r e g i v en b y

R T l n y~ = X ~ [A o + a , ( 3 x t - x z )

+ A~(X~ -

X ~ ) S X l - x ~ ) + . . . 1 3 8 )

R T l n T z = X ~ t A o +

A 1(3X2 - X1)

+ A 2 ( X 2 - X~ )( 5X2 - X~ ) + . . . ] . ( 39)

N o t e t h a t t h e t w o - p a r a m e t e r M a r g u l e s f o r m u l a t i o n i s a

s p e ci a l c a s e o f th e p o l y n o m i a l m o d e l w i t h t w o c o n s t a n t s

A o a n d A t w h i c h a r e r e l a t e d t o W ~ b y

Ao = (W~zt + W~12)/2 (40 )

A t = (W~z, - W~2)/2 (41)

I f t e m p e r a t u r e d e p e n d e n c e o f t h e c o n s t a n t s A o , A , , A 2 ,

e t c ., i s kn ow n, H ~x and S ~x can be c a l cu l a t ed a s f o l l ow s :

H ~ = X1 X2 {Ao - -

T ( O A o / O T ) + [ A~ - - T ( O A t / a T ) ] ( X 1

- Xz)

q - [ A 2 - -

T OA2/OT)I Xl -

X 2 ) 2 k . . . } ( 4 2 )

SeX= - X t X z [ ( OAo / OT) + ( aA , / OT ) (X t - X 2 )

q- OA 2/O T) X 1 -

) ( 2 ) 2 q - . . . 1 . ( 4 3 )

ernary models

E x c e l l e n t re v i e w s o f e m p i r i c a l m e t h o d s o f p r e d i c t in g t e r n a r y

s o l u t i o n p r o p e r t i e s f r o m b i n a r y s o l u t i o n d a t a h a v e a p -

p e a r e d r e c e n t l y ( H i l l e rt 1 9 8 0 ; B e r t r a n d e t a l . 1 9 8 3; A c r e e

1 98 4 ). O n t h e b a s i s o f t h e s e r ev i e w s a n d o u r o w n e x p e r i e n c e

w i t h t e r n a r y m o d e l s ( S a x e n a 1 9 7 3 ; G a n g u l y a n d S a x e n a

1 9 84 ), w e h a v e s e l e c te d t h r e e m o d e l s o f c o m b i n i n g t h e b i n a -

r y d a t a t o p r e d i c t t e r n a r y p r o p e r t i e s . T h e s e a r e t h e W o h l

m o d e l , t h e B e r t r a n d - K o h l e r m o d e l , a n d t h e H i l l e r t m o d e l .

T h e r e a r e f e w te r n a r y d a t a a v a i l a b l e o n s o l id s o l u t i o n s a n d

t h e r e i s n o w a y t h e d i f f e r e n t p r e d i c t iv e b e h a v i o u r o f t h e

m o d e l s . H o w e v e r , s e v e r a l st u d i e s o f p h a s e e q u i l i b r iu m i n

e x p e r i m e n t a l a n d n a t u r a l s y s t e m s r e qu i re t h e t e r n a r y m o d -

e l s f o r i n t e r - a n d e x t r a - p o l a t i o n . I t i s, th e r e f o r e , i m p o r t a n t

t o d o c u m e n t t h e p r e d i c t i o n s o f t h es e m o d e l s .

W o h l s m o d e l

A c c o r d i n g t o W o h l ( 1 9 53 ) t h e e x c es s f r e e e n e r g y o f m i x i n g

i s g i v e n b y

Gex= xIX2 [W 1 2 X2 +

W 2 1 X 1 ] + X - 2 X 3 [ W 2 3 X 3 + W 3 2 . u

+ X I X 3 [ W 1 3 X 3 + W 3 1 X 1 ]

+ x ~ . v ~ J G [ o . 5 w ~ + w ~ + w ~ 3

+ W 3 1 + W a 3 + W 3 2 ) - C ] (4 4)

w h e r e C i s a t e r n a r y c o n s t a n t t o b e d e t e r m i n e d f r o m t h e

t e r n a r y s o l u t io n d a t a ( G a n g u l y a n d S a x e n a 1 9 84 f o r e s ti m a -

t i o n ) . F o r a c t i v i t y c o e f f i c ie n t o f c o m p o n e n t 1 , w e h a v e

R ~qnyl = r~[ w12 + 2 x l vG 1 - ~vl 2)]

+ ;r w t 3 + 2 x ~ w 3 ~ - w l ~ ) ]

+ x ~ x d o . 5 w 2 1 + w ~2 + w 3 1 + w 1 3 - w ~ - w ~ 2)

+ x t w 2 ~ - w ~2 + W ~ t - w t ~)

q - ( X z - - X 3 ) ( W 2 3 - - W 3 z ) - (1 - 2 X 1 ) C ] . ( 4 5 )

F o r ) 2 r ep l ace 1 by 2 , 2 by 3 and 3 by 1 . S i mi l a r l y , f o r

Y3 r ep l ace 1 by 3 , 2 by 1 and 3 by 2 .

8/18/2019 Fei et al. (1986).pdf

4/9

224

Table 1 . Excess enthalpy of mixing a t 970 ~ K for fou r b ina ry s i l ica te so l id solu t ions

D i - C a T s D i - E n P y - G r A n - A b

XD~ H~/kJ XDi /P~ /kJ Xpy H~ XAn H X/kj

0 .900 4 .058_+1.136 0 .900 3 .55 6_+ 1.74 1 0 .910 5 .98 3+_ 1.69 8

0.800 5 .85 8+_ 1.08 9 0 .800 6 .443+_0.969 0 .820 6 .527+ 1.335

0.650 5 .941+ 1.193 0 .700 9 .916-+1.386 0 .725 8 .49 4_+ 1.45 8

0.500 4 .017+_1.442 0 .600 12 ,05 0- + .953 0.530 7.113-+1.424

0.300 3.096-+1.249 0.500 12.678+_1.146 0.200 3.515+_1.598

0.400 12.678+-0.809 0.100 4.01 7+ 1.66 9

0.300 12.468 __ 1.349

0.220 10.000 +- 1.628

0.900 3 .891 +0.88 0

0.830 2.427 _+ 1.426

0.770 3.807___1.545

0.700 5.397 -+ 0.728

0.650 5 .230--+0.841

0 . 600 4 . 602_+ 0 . 781

0.500 4.728 -+ 1.762

0.420 3.724 + 1.764

0.400 3.891 + 2.198

0.330 2.720 + 0.698

0.300 2,634_+ 1.588

0.250 3,180___0.804

0.200 3 .138+0.762

0.150 2.176_+1.124

0.110 0.293 +- 1.399

0 . 100 0 . 293+ _1 . 123

0.050 0.084 +_0.923

Abbreviations. Ab, a lb i te (NaA1SiaOs) ; Di , d iops ide (CaMgSizO6) ; Gr , grossular (CaAlz/3SiO4) ; CaTs, Ca-Tschermak (CaAlzSiOt) ;

Py, pyrope (MgA12/3SiO4); En, ens ta t i te (Mg2Si2Ot) ; An, an or th i te (CaA L2SizOs)

Data sources. Ne wto n e t a l . (1977) , Ne wto n e t a l . (1979) , and Ne wto n e t a l . (1980)

The errors in excess enthalpy o f mixing we r e c ompu t e d f r om t he f o r mu l a

flip ~= ]~ XIOH1)2 + Xe6HE) 2 + (0Hmi~)2

whe r e 8Ht , 8Hz and ~/ - /~ are the er rors in the enthalpies of the two end-m embers an d o f the mixing, respective ly , an d ) (1 and

X2 a r e mo l e f r a c t ions

Bertrand-Kohler model

B e r t r a n d e t a l. ( 1 9 83 ) p r o p o s e d a g e n e r a l m o d e l o f w h i c h

t h e K o h l e r f o r m u l a t i o n b e c o m e s a s p e c ia l ca s e. A c c o r d i n g

t o t h e m o d e l o f B e r t r a n d e t a l. , a n e x ce s s p r o p e r t y o f a

m u l t i c o m p o n e n t s o l u ti o n i s g i v en b y

zex =K'N X'N Xi+Xj) fi_.}_fj) Aze.~), ( 46 )

1 2 . . . N . d . a i = 1 / j > i

i n w h i c h

AZ~])*

i s t h e m o l a r e x c e s s p r o p e r t y ( e n t h a l p y ,

e n t r o p y , v o l u m e , f r e e e n er g y , e tc . ) o f t h e b i n a r y s y s t e m

w i t h c o m p o n e n t s a t t h e s a m e m o l a r r a t i o a s in m u l t i c o m -

p o n e n t s y s t e m a n d f a n d J ] a re w e i g h t e d m o l e f r a c t i o n s

u s i n g w e i g h i n g f a c t o r s b a s e d o n t h e e x c e s s p r o p e r t i e s o f

t h e b i n a r y s y s t em s . X i is u s e d a s t h e m o l e f r a c t i o n i n t h e

m u l t i c o m p o n e n t s y s t e m .

F o r t h e a c t i v i ty c o e f f ic i e n t o f c o m p o u n d i, w e h a v e

(RT lnTi) 12 .. .N = ~j (X i + Xj) (1 - - f~- -J~ )(A G~])*

- Y ~ j ,k x j + x ~ ) ~ + A ) ~ 6 ; ~ ) *

+ Z j ( f + j ~ ) ( R T ln y 0 ~ j i r 1 6 2 (47)

w h e r e t h e b i n a r y f u n c t i o n s d e n o t e d b y a n a s t er i sk , r e p r e s e n t

t h e e x c es s p r o p e r t y o f t h e b i n a r y s y s t e m .

The Hillert model

H i l l e rt ( 1 9 8 0 ) s u g g e s te d a m o d i f i c a t i o n o f T o o p ' s ( 1 9 6 5)

m e t h o d . T h e e x c e s s fr e e e n e r g y o f m i x i n g f o r a t e r n a r y

m i x t u r e is g i v e n b y

- - X1) I { X1 X~ [ A 12 + AIa X~ -- X2)]}

~ .

o ~ . ,

+ A~3( X~ - - X3) ] }[ X 3 / ( 1 - X ~ ) ] { X * X * [ A ~ 1 , ,

+ X2Xa)[A~ +

A2~3(V2a - V32)] (48 )

w h e r e

V 23 = ( 1 + X 2 - X a ) /2

// '32 = (1 + Xa - X= )/2.

X~i i s u s e d a s t h e m o l e f r a c t i o n i n t h e b i n a r y s y s t e m . T h e

c o n s t a n t s A~ AIz e tc . a r e s i m i la r t o t h e c o n s t a n t s A o a n d

A , i n t h e R e d l i c h - K i s t e r f o r m u l a t i o n .

A note on entropy of mixing

F o r a b i n a r y s i m p l e m i x t u r e ( G u g g e n h e i m 1 9 6 7) , w e h a v e

G ex = X 1X2 W ~ ( 49 )

w h e r e W is th e i n t e r a c t i o n e n e r g y p a r a m e t e r . P a r t i a l d i f fe r -

e n t i a t i o n o f E q . ( 4 9 ) y i e l d s

H ex = XIX2 [ V~ - T OW~/OT)] (50)

S e X = -X1X2[OW~/07]. ( 51 )

T h e s q u a r e b r a c k e t e d t e r m s a r e d e fi n e d b y T h o m p s o n

( 1 9 67 ) a s W n a n d W s , l e a d i n g t o t h e p o p u l a r y u s e d r e l a t io n

W ~ = W n - T W s. (52)

S i n c e OWa/OT i s u s u a l l y n e g a t i v e , W s b e c o m e s a p o s i t i v e

q u a n t i t y w h i c h w h e n s u b s t i t u te d i n E q . ( 5 1) y ie l d s a p o s it i v e

e x c es s e n t r o p y o f m i x i n g . S u c h S e~ i m p l ie s a t o t a l e n t r o p y

o f m i x in g g r e a t e r t h a n t h e r a n d o m m i x i n g c o n f i g u r a ti o n a l

e n t r o p y . N o t e t h a t t h e e x c e ss e n t r o p y o f m i x i n g c o n s i s t s

o f S ex a r i s in g d u e t o u n e q u a l t e m p e r a t u r e d e p e n d e n c e o f

t h e p a i r p o t e n t i a l e n e r g y o f in t e r a c t i o n s i n t h e s o l u t i o n a n d

t h e e x c e s s c o n f i g u r a t i o n a l e n t r o p y , S ~ x, w h i c h s h o u l d a l -

w a y s b e l e ss t h a n z e r o .

F r o m t h e a b o v e d i s c u s s i o n , i t i s a p p a r e n t t h a t t h e o n l y

p a r t o f th e e x c es s e n t r o p y o f m i x i n g t h e m o d e l s c a n p r e d i c t

( E q s . 1 5 a n d 1 9 ) f r o m i s o t h e r m a l h e a t o f s o l u t i o n d a t a i s

8/18/2019 Fei et al. (1986).pdf

5/9

8/18/2019 Fei et al. (1986).pdf

6/9

226

12 i , i i ,

. . . . . . b l

. . . . . Q c

N R T L

0

8

-12

. 1 . 2 i ; . 6 . 7 . ; . ;

P y X f r ) 5 r

Fig. 3. Calculated excess functions of mixing for the pyrope-grossu-

lar join at 970~ K. Solid curves 1 and 2 are calculated from the

NRTL model. The solution parameters are given in Table 2 for

curve 1. The parameters for curve 2 are (w12-w22)=18.277 kJ,

(w12- w~ ~)= 2.823 k J, c~= 0.25. Experimental data are from New-

ton et al. (1977)

8

-

%

-4

1/ 1

I---

16

12

x •

4

--, 0

F -

i i i i # i i i

. . . . . . N

. . . . 0 s

NRTL

/ / \ \

i

/ / / / /

\ /

\ /

\ x I

/

N N f .

/

t

/

2 5 6 7

8 0 . I . . 3 l~ 8 1 9 110

D i X En ) En

Fig. 4. Calculated excess functions of mixing for the join CaMg-

Si206-Mg2Si206 at 970~ K. Experimental data are from Newton

et al. (1979)

Since the Redlich-Kister model with three or more con-

stants is bou nd to be the better fitting than the two-cons tant

Margules, we shall not discuss it separately. With increasing

number of constants, the Redlich-Kister model, although

superior for calculating phase diagrams, is not readily inter-

pretable in terms of crystal-chemical parameters.

A n o r t h i t e - a l b i t e .

Except for the Wilson model, all four mod-

els can be fit with similar residuals. Figure 1 shows the

excess e nthalpy of solution at 970 ~ K calculated acco rding

to the three models, Margules quasi-chemical and the non-

random two-liquid (referred to as M, QC and NRTL re-

spectively). As noted before, the QC and NR TL models

also yield the conf iguration al excess ent rop y of mixing (Si x)

s e e Eqs. 20 and 29). The term -T S~ x is shown in Fig. 2

for the two models. For the M model,

S ~ t o t a l )

has been de-

rived from Eq. (34) where

~ 2 = - w ~ 2 - ~ 2 ) / r 5 3)

and similarly for 1~21. The values for ~ are from Saxena

and Ribbe (1972) who used Orville's (1972) phase equilibri-

um data. The Saxena-Ribbe formulation was used because

Ganguly and Saxena (1984) found it to be consistent with

data on garnet-plagioclase phase equilibria. There would

be little difference if Ghiors o's (1984) fo rmulation is used.

In addition, we have also plotted -TS

ex

calculated with

the Al-avoidance model of Kerrick and Dark en (1975) as:

- - TS ex = R T{X A bln [A A b 2 -X A b)]

+ XA=ln[(1 -- XA,)2/4]}. (54)

Note that -TS

ex

derived from the combination of phase

equilibrium data of Orville (1972) and enthalpy o f solution

data should be comparable if the model of Al-avoidance

is appropriate. As demo nstrated by Newton et al. (1980)

and shown in Fig. 2, the model produces a close approxima-

tion of Sex. Comparison of

TS ~x

and

TS~ x

as predicted by

the NR TL and QC models gives an indication of the magni-

tude o f the non-configurational part of the excess entropy.

P y r o p e - g r o s s u l a r .

For this binary solution, Table 2 and

Fig. 3 show that all models reproduce the data on excess

enthalpy of solution rather well. The fit with the NRTL

model is best, followed by the QC and by the Margules

model. For the NRTL model, we calculated two sets of

solution parameters by chan ging ~. Both reproduce the data

on excess enthalpy of solution rather well, but the configu-

rational excess entropies of mixing predicted by the two

sets of parameters are significantly different. This may im-

ply that the factor ~ has some connotation with crystal

chemical parameters. To predict a reliable configurational

entropy, one might determine the factor ~ separately in

terms o f crystal chemical parameters. Figure 3 shows plots

of -TS ~ x for the QC and NRTL models and -T S ex for

the M model, for which a Ws of 6.276 J/cation has been

taken from Haselton and Newton (1980). Again, we may

8/18/2019 Fei et al. (1986).pdf

7/9

227

i i t i i i

. . . . . . N

. . . . ( ~ C

I N R T L

N . \

~ N .

\ \ , Z

\ \

N N

\ \ / / /

I I I I I I I I

- 6 . 1 . 2 . . / . 5 . 6 . 7 . 8 . 9 1 . 0

Bi X EaTs ) CaTs

Fig. 5. Calculated excess functions of mixing for the diopside-CaTs

join at 970 ~ K. Experimental data are from N ewto n et al . (1977)

r

- , - 2

I

c o n s i d e r t h e d i f f e re n c e b e t w e e n ~ a n d S ~ a s r e p r e s e n t i n g

t h e n o n - c o n f i g u r a t i o n a l p a r t o f S ~x.

D i o p s i d e - e n s t a t i t e . T h e N R T L a n d M m o d e l s f it t h e en -

t h a l p y d a t a b e t t e r t h a n t h e Q C m o d e l . F o r t h e Q C m o d e l ,

a h ig h c o o r d i n a t i o n n u m b e r ( Z ) o f 1 1 w a s n e e d e d . T h e

M 2 s i te i s n o m i n a l l y 8 - c o o r d i n a t e d . W h e n a v a l u e o f 8

w a s u s e d f o r Z , t h e r e s u l t s o f t h e f it w e r e q u i t e u n a c c e p t -

a b l e , t h e f i t t e d c u r v e t o t a l l y m i s s e d s e v e r a l d a t a p o i n t s .

F i g u r e 4 sh o w s t h a t t h e Q C m o d e l t e n d s t o p r o d u c e a l e ss

s y m m e t r i c c u r v e t h a n r e q u i r e d b y t h e d a t a . F i g u r e 4 a ls o

s h o w s a p l o t o f - T S ~

x

a s c a l c u la t e d b y t h e N R T L a n d

Q C m o d e l s . A c c o r d i n g t o L i n d s l e y e t a l . (1 9 81 ), W s f o r

t h i s s y s t e m i s z e r o , w h i c h w o u l d r e q u i r e a S e~ ( n o n - c o n f i g u r -

a t i o n a l ) o f s i m i l a r m a g n i t u d e t o S ~~, bu t opp os i t e i n s i gn .

D i o p s i d e - C a - T s c h e r m a k . F i g u r e 5 a n d T a b l e 2 s h o w t h a t

b o t h t h e Q C a n d N R T L m o d e l s a r e e q u a l l y w e l l s u i t e d

t o t h e s o l u t i o n d a t a a n d r e s u l t i n l o w e r r e s i d u a l s t h a n t h e

M a r g u l e s m o d e l . F r o m G a s p a r i k a n d L i n d s l e y ' s ( 1 9 8 0 )

a n a l y s i s o f t h e p h a s e e q u i l i b r i u m d a t a , a n d t h e h e a t o f s o lu -

t i on da t a o f N ew t on e t a l . (1977) , t he so l u t i on ha s a wS 2~=

3 0 .9 0 a n d V ~ 1 2 = 1 2 .2 5 . F i g u r e 5 s h o w s - T S ~ x f o r t h e

N R T L a n d Q C m o d e ls a nd - T S ex f o r t h e M a r g u l e s m o d e l .

F o r t h i s s o l u t i o n t h e r e i s a s u b s t a n t i a l d i f f e r e n c e b e t w e e n

t h e t w o q u a n t i t ie s , s u g g e s t i n g a l a r g e n o n - c o n f i g u r a t i o n a l

e x c e s s e n t r o p y o f m i x i n g .

G e n e r a l d i s c u s s i o n . T h e n a t u r e o f ex c e s s f u n c t i o n s p r e d i c t e d

b y t h e t w o - p a r a m e t e r M a r g u l e s m o d e l h a v e b e e n d i s cu s s e d

am pl y i n l i t e r a t u r e ( Ne wt on e t a l . 1977 , 1980 ; L i nds l ey e t a l.

1 9 8 1 ) . T h e r e f o r e , w e s h a l l c o n c e n t r a t e o n u n d e r s t a n d i n g

t h e a p p li c a b il i ty o f Q C a n d N R T L m o d e l s . T h e Q C p a r a m -

e t er s q l a n d q2 k n o w n as c o n t a c t f a c t o r s a r e s u p p o s e d

t o b e r e l a t e d t o t h e g e o m e t r y a n d s iz e o f t h e m i x i n g u n i t s ,

w i t h t h e p r o p e r t y t h a t

q l / q 2 ~ l

a s e i t h e r b e c o m e s u n i t y

( G u g g e n h e i m 1 9 5 2 ; G r e e n 1 9 7 0 ). I n o u r c a se w e h a v e u s e d

q l + q z = 2 . 0 w h i c h s a ti sf ie s t h e a b o v e r e q u i re m e n t . H o w -

e v e r , t o o b t a i n t h e d e s i r e d a s y m m e t r i c f it q~ a n d q 2 h a d

t o b e m a d e c o n s i d e r a b l y d i f f e r e n t . T h e r e i s n o r e l a t i o n s h i p '

b e t w e e n q ~ / q 2 a n d a s i z e p r o p e r t y , e . g . , m o l a r v o l u m e r a t i o

o f t h e m i x i n g u n i t s . I n v i e w o f G r e e n ' s ( 1 9 7 0) s u c c e s s i n

u s i n g t h e Q C m o d e l , t h e r e s u l t s f r o m t h i s m o d e l a r e d i s a p -

p o i n t i n g , p a r t i c u l a r l y i f w e c o n s i d e r t h a t t h e q l / q 2 r a t i o

s e e m s t o b e a r n o r e l a t i o n t o t h e v o l u m e d i f f e r e n c e s o f t h e

m i x i n g s p e c i e s. I n a d d i t i o n t o t h e p h y s i c a l ly m e a n i n g l e s s

c o n t a c t f a c t o r s, t h e c o o r d i n a t i o n n u m b e r s z u s e d , a l so s e e m

u n s a t i s f a c t o r y . S i m i l a r l y , f o r t h e N R T L m o d e l , w h i c h f i t s

d a t a f o r a l l t h e s o l u t i o n s v e r y w e l l , t h e f a c t o r ~ d o e s n o t

s h o w a n y c o r r e l a t i o n w i t h v o l u m e r a t i o o f t h e s p e c ie s . T h e

p a r a m e t e r c~ c h a r a c te r i ze s t h e t e n d e n c y o f t h e c o m p o n e n t s

t o m i x i n a n o n r a n d o m m a n n e r s e e A c r e e 1 9 8 4 ) . W h e n

c~ equa l s z e r o , t he l oca l m ol e f r a c t i on s ( X ij , s ee E q . 2 ) be -

c o m e e q u a l t o t h e o v e r a l l m o l e f r a c t i o n s ( X 0 a n d m i x i n g

i s c o m p l e t e l y r a n d o m . T h e r e f o r e i t i s d i s t u r b i n g t o n o t e

t h a t t h e D i - E n s o l u ti o n h a s t h e s m a l l e s t e a n d P y - G r t h e

l a r g es t in d i c a ti n g t h a t n o p h y s i c a l m e a n i n g c a n b e a t t a c h e d

t o ~ . A c o m p a r i s o n o f t h e v a r i o u s p l o t s o f - T S ~x a g a i n s t

m o l e f ra c t i o n s h o w s t h a t f o r a ll t h e s o l u t i o n s , t h e q u a n t i t y

TS; i s h i g h f o r t h e c o m p o s i t i o n a l s i d e w i t h h i g h H ex.

T h u s f o r t h e p y r o p e - g r o s s u l a r s o l u t i o n t h e m o d e l s N R T L

a n d i n p a r t i c u l a r Q C p r e d i c t - T S ~ ~ w h i c h i s c lo s e r t o z e r o

f o r p y r o p e - r i c h c o m p o s i t io n s t h a n f o r t h o s e r i c h in g r o s s u -

l a r. I n th i s r e g a r d - T S ~ x b e h a v e s s i m i l a r l y t o ( b u t w i t h

o p p o s i t e s i g n) t h e e x c e ss v o l u m e o f m i x i n g ( H a s e l t o n a n d

N e w t o n 1 9 80 ). T h i s i m p l i e s t h a t v o l u m e o f m i x i n g c a n b e

b e t te r m o d e l le d e m p l o y i n g t h e N R T L o r Q C m o d e l t h a n

b y M a r g u l e s f o r m u l a t io n .

2 . T e r n a r y s o l u t i o n m o d e l s

A s e x p l a i n e d e a r li er , t h r e e m e t h o d s o f c o m b i n i n g b i n a r y

m o d e l p a r a m e t e r s t o p r e d i c t t e r n a r y s o l u t i o n p r o p e r t i e s a r e

c h o s e n . A s a n e x a m p l e , l e t u s c o n s i d e r a t e r n a r y s o l u t i o n

w i t h t h e f o ll o w i n g d a t a : W l z = 1 0 , 4 6 0 , W 2 1 = 8 3 7 , W 1 3 =

3 ,617 , W 31 =9 ,1 32 , 17 /23=11 ,529 an d W 32= 10 ,437 . T he so -

l u t i o n p a r a m e t e r s a r e p a r t l y t h o s e u s e d b y G a n g u l y a n d

S a x e n a ( 1 9 8 4 ) f o r g a r n e t . T h e c a l c u l a t i o n s a r e n o t n e c e s s a r i -

l y f o r g a r n e t s i n c e a ll t h e d a t a o n g a r n e t s o l u t i o n a r e n o t

y e t a v a i l a b l e . F i g u r e 6 s h o w s e x c e s s fr e e e n e r g y o f m i x i n g

c a l c u l a t e d a c c o r d i n g t o K o h l e r f o r m u l a t i o n . T h e G ex d a t a

r e s u l ti n g f r o m t h e u s e o f W o h l ' s f o r m u l a t i o n a r e v e r y si na i-

l a r a n d , th e r e f o r e , h a v e n o t b e e n p l o t t e d s e p a r a t e l y . H o w -

e v e r , ti le d a t a f r o m t h e H i l l e r t m o d e l a r e s i g n i f i c a n tl y d i ff e r -

e n t . T h e s e d a t a a r e p l o t t e d i n F i g . 7 w h i c h s h o w s t h a t t h e

G ex t e n d s t o b e h i g h e r f o r s o m e o f t h e t e r n a r y c o m p o s i t i o n s

t h a n t h o s e c o m p u t e d b y e i th e r th e K o h l e r o r W o h l f o r m u l a -

t io n . R e m e m b e r i n g t h a t a s i m p l e s u m m a t i o n o f t h e b i n a r y

G ~x wo ul d y i e l d t he h i ghes t va l ues o f t he t e r n a r y G ex , t he

K o h l e r a n d W o h l f o r m u l a t i o n s d e p a r t m o s t a n d t h e H i l l e r t

m o d e l l e a s t o f t h e t h r e e f r o m p r e d i c t i n g s u c h v a l u e s . T h e

q u e s t i o n a s t o w h i c h m o d e l i s m o s t a p p r o p r i a t e c a n b e

8/18/2019 Fei et al. (1986).pdf

8/9

228

Fig. 6. Calculated excess free energies in

a hypothetical ternary system according

to the Kohler model

Fig. 7. Calculated excess free energies in

a hypothetical ternary system according

to the Hillert model

answered only with additional data on ternary solutions.

It is possible to consider addition of ternary or higher con-

stants to any of the above models if suitable data are avail-

able. Ganguly and Saxena (1984) discussed empirical esti-

mate of such constants to any of the above models if suit-

able data are available. Ganguly and Saxena (1984) dis-

cussed empirical estimate of such con stants for Wohl s

model.

onc lus ions

For the purpose of calculating phase diagrams and repro-

ducing experimental equilibria, it is obvious that the best

model is the polynomial expression as proposed by Guggen -

heim (1937) which has now come to be popularly known

as Redlich-Kister model after the authors who wrote the

expression for the activity coefficients. The polynomial

when truncated after two constants is equivalent to the two-

parameter Margules model which may be used in many

cases. It is easy to handle computationally. The QC model,

even after unrealistic adjustments in the physical parame-

ters, does not work well. The NRTL model also is not

an improvement over the Redlich-Kister model because,

although it is functionally useful in data representation,

it appears to have coefficients that are physically mean-

ingless.

8/18/2019 Fei et al. (1986).pdf

9/9

229

P u r e l y a s a m e t h o d f o r f it t in g s o l u ti o n d a t a , t h e N R T L

m o d e l h a s b e e n f o u n d t o b e a s u s e f u l a s th e M a r g u l e s m o d e l

f o r tw o s o l u t io n s a n d s o m e w h a t b e t t e r f o r t he r e m a i n i n g

t w o . I t s d i r e c t u s e w i t h t h e p h a s e e q u i l i b r i u m d a t a s h o u l d

b e p o s s i b l e w h e r e v e r s e p a r a t e e s t i m a t e s o f H cx a n d S ~x a r e

n o t n e e d e d . T h e m o d e l i s e a s il y e x t e n d e d t o a m u l t i c o m p o -

n e n t s y s t e m a s f o l l o w s ( A c r e e 1 9 8 4 , p 9 9 ) :

~6~ .. . .

N N

= RTEN= 1 X i ( E j = i X j ~ j i m j l ) l ( E m = l W m i X m ) ] . ( 5 5 )

T h e c o m b i n a t i o n o f t h e b i n a r y d a t a t o p r e d i c t t e r n a r y

o r m u l t i c o m p o n e n t d a t a m a y b e d o n e u s i n g an y o f th e

t h r e e m o d e l s d i s c u s s e d i n t he t e xt . W e r e c o m m e n d t h e

K o h l e r f o r m u l a t i o n b e c a u s e o f t h e ea s y f o r m u l a t io n . A n y

b i n a r y m o d e l M a r g u l e s , R e d l i c h - K i s t e r , Q C o r N R T L o r

a n y c o m b i n a t i o n t h e r e o f ca n b e u s e d i n t h is f o r m u l a t i o n .

Acknowledgements

T h e r e s e a r c h h a s b e e n s u p p o r t e d b y a N S F

g r a n t ( E A R : I 8 41 5 8 00 ) a n d b y a P S C - B H E g r a n t ( 6 - 6 4 1 9 1 ) .

E r i k s s o n a c k n o w l e d g e s t h e r e s e a r c h s u p p o r t f r o m t h e S w e d i sh N a t -

u r a l S c i e nc e C o u n c i l ( N F R ) .

e f erences

A c r e e W E J r ( 19 8 4) T h e r m o d y n a m i c p r o p e r t i e s o f n o n e l e c tr o l y t e

so lu t ions . Acade mic Press , New Yo rk , p 308

B e r t r a n d G L , A c r e e W E J r , B u r c h fi e ld T ( 19 8 3) T h e r m o d y n a m i c a l

e x c e s s p r o p e r t i e s o f m u l t i c o m p o n e n t s y s t e m s : r e p r e s e n t a t i o n

a n d e s t i m a t i o n f r o m b i n a r y m i x i n g d a t a . J S o l u t i o n C h e m

12: 327-340

Flo ry PJ (1953) M olecu la r conf igu ra t ion o f po lye lec t ro ly tes . J

Ch em Phy s 21 : 162-163

G a n g u l y J , S a x e n a S K ( 19 84 ) M i x i n g p r o p e r t i e s o f a l u m i n o s i l ic a t e

g a r n e t s : c o n s t r a i n ts f r o m n a t u r a l a n d e x p e r i m e n t a l d a t a , a n d

a p p l i c a t i o n s t o g e o t h e r m o - b a r o m e t r y . A m M i n e r a l 6 9 : 8 8 - 9 7

Gaspar ik T , L inds ley DH (1980) Phase equ i l ib r ia a t h igh p ressure

o f p y r o x e n e s c o n t a in i n g m o n o v a l e n t a n d t r i v a le n t io n s . R e v

M inera l 7 : 309-340

G h i o r s o M S ( 19 8 4) A c t i v i t y / c o m p o s i t i o n r e l a ti o n s i n t h e t e r n a r y

f e l d sp a r s . C o n t r i b M i n e r a l P e t r o l 8 5 : 1 8 6 - 1 9 6

G r e e n E J ( 19 7 0) P r e d i c t i v e t h e r m o d y n a m i c m o d e l s f o r m i n e r a l s y s -

tems . 1 . Quas i -chem ica l ana lys i s o f the ha l i t e - sy lv i t e subso l idus .

A m M i n e r a l 5 5 : 1 6 9 2 - 1 7 1 3

G u g g e n h e i m E A ( 19 37 ) T h e o r e t i c a l ba s i s o f R a o u l t ' s l a w . T r a n s

Fa rad ay Soc 33 : 151-159

G u g g e n h e i m E A ( 1 9 5 2) M i x t u re s . C l a r e n d o n P r e s s , O x f o r d , p 3 6 5

G u g g e n h e i m E A ( 19 6 7) T h e r m o d y n a m i c s . N o r t h H o l l a n d P u b l i s h -

i n g C o . , A m s t e r d a m , p 4 3 7

H a s e l t o n H T , N e w t o n R C ( 1 9 80 ) T h e r m o d y n a m i c s o f p y r o p e -

grossu la r ga rne t s and the i r s t ab i l i t i e s a t h igh t emp era tu res an d

pressures . J Geophys Res 85 :6973-6982

H i l l e r t M ( 19 8 0) E m p i r i c a l m e t h o d s o f p r e d i c t in g a n d r e p r e s e n t in g

t h e r m o d y n a m i c p r o p e r t i e s o f t e r n a r y s o l u t i o n p h a s e s . C A L -

P H A D 4 : 1 - 1 2

K e r r i c k D M , D a r k e n L S ( 19 75 ) S t a t i s t i c a l t h e r m o d y n a m i c m o d e l s

fo r idea l ox ide and s i l i ca te so l id so lu t ions , w i th app l ica t ion

t o p l a g i o cl a s e . G e o c h i m C o s m o c h i m A c t a 3 9 : 1 4 3 1 -1 4 4 2

L i n d s l e y D H , G r o v e r J E , D a v i d s o n P M ( 19 81 ) T h e th e r m o d y n a m -

i cs o f t h e M g 2 S i 2 0 6 - C a M g S i 2 0 6 j o i n : a r e v ie w a n d a n i m -

proved mode l . Ad v Phys Geo chem 1 : 149-176

N e w t o n R C , C h a r l u T V , K l e p p a O J ( 19 7 7) T h e r m o c h e m i s t r y o f

h i g h p r e s su r e g a r n e t s a n d c l i n o p y r o x e n e s i n t h e s y s t e m C a O -

M g O - A l z O a - S iO 2 . G e o c h i m C o s m o c h i m A c t a 4 1 : 3 6 9 -3 7 7

N e w t o n R C , C h a r l u T V , K l e p p a O J ( 19 8 0) T h e r m o c h e m i s t r y o f

t h e h i g h s t r u c t u r a l s t a t e p l a g i o c l a s e s . G e o c h i m C o s m o c h i m

A c t a 4 4 : 9 3 3 -9 4 1

N e w t o n R C , C h a r l u T V , A n d e r s o n P A M , K l e p p a O J (1 9 79 ) T h e r -

m o c h e m i s t r y o f s y n th e t i c c l i n o p y r o x e n e s o n t h e j o i n C a M g -

S i z O 6 -M g 2 S i 2 0 6. G e o c h i m C o s m o c h i m A c t a 4 3 : 5 5 - 6 0

Orv i l l e PM (1972) P lag ioc lase ca t ion eq u i l ib r ia wi th a queous ch lo -

r ide so lu t ion : Resu l t s o f 700 ~ C a nd 2 ,000 ba rs in the p resence

of quar tz . A m J Sc i 272 :234 272

P o w e l l R ( 1 9 8 3 ) T h e r m o d y n a m i c s o f c o m p l e x p h a s e s . A d v P h y s

G e o c h e m 3 : 2 4 1 - 26 6

R e n o n H , P r a u s n i t z J M ( 1 9 68 ) L o c a l c o m p o s i t i o n s i n t h e r m o d y -

namic excess func t ions fo r l iqu id mix tu res . Am Ins t Chem Eng

J 1 4 : 1 3 5 - 1 4 4

S a x e n a S K ( 19 73 ) T h e r m o d y n a m i c s o f r o c k - f o r m i n g c r y s t a ll i n e s o -

lu t ions . Spr inger He ide lbe rg Ber l in New Y ork , p 189

S a x e n a S K , R i b b e P H ( 19 72 ) A c t i v i t y - c o m p o s i t i o n r e l a t io n s i n

f e l d s pa r s . C o n t r i b M i n e r a l P e t r o l 3 7 : 1 3 1 - 1 3 8

T h o m p s o n J B J r ( 19 6 7) T h e r m o d y n a m i c p r o p e r t i e s o f s i m p l e s o l u -

t ions . In : Abe lson PH (ed) Resea rches in geochemis t ry , vo l 11 .

John W i ley , Inc , New York , pp 340-~361

T h o m p s o n J B J r , W a l d b a u m D R ( 19 6 9) M i x i n g p r o p e r t ie s o f s a n i -

d ine c rys ta l l ine so lu t ions . I I I . Ca lcu la t ion bas ed on teo -phase

d a t a . A m M i n e r a l 5 4 : 8 1 1 -8 3 8

T o o p G W ( 19 65 ) P r e d i c ti n g t e r n a r y a c t iv i t ie s u s i n g b i n a r y d a t a .

T r a n s A m I n s t M i n E n g 2 2 3 :8 5 0 8 5 5

W i l son GM (1964) A new express ion fo r the excess f ree energy

o f m i x in g . J A m C h e m S o c 8 6 : 1 2 7 - 1 3 0

W o h l K ( 1 9 5 3) T h e r m o d y n a m i c e v a l u a t i o n o f b i n a r y a n d t e r n a r y

l iqu id sys tems . Chem E ng Pro g 49 : 218-219

Rece ived December 17 , 1985 /Accep ted June 13 , 1986