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SURFACES.THE SURFACE PROPERTIES OF
LIQUID METALS
R. Evans
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque C8, suppZ4ment au n08, Tome 41, aoCt 1980, page
C8-775
THE SURFACE PROPERTIES
OF
L I Q U I D METALS R. EvansH.H. ViZZs Physics Laboratory, U n i v e r s i t y o f BKstoZ, ByYistoZ BSB ITL,
U.K.
A b s t r a c t . - We b r i e f l y review r e c e n t t h e o r e t i c a l and experimental work on t h e s u r f a c e s of v a r i o u s l i q u i d s and l i q u i d m i x t u r e s . While t h e r e has been much p r o g r e s s i n understanding t h e s u r f a c e s t r u c t u r e and thermodynamic p r o p e r t i e s of argon-like l i q u i d s and molten s a l t s , t h e s u r f a c e p r o p e r t i e s of l i q u i d metals and t h e i r . a l l o y s a r e much l e s s understood. We mention t h e relevance of t h e s t u d i e s of o t h e r l i q u i d s t o l i q u i d metal s u r f a c e s and d e s c r i b e some o u t s t a n d i n g problems i n t h i s a r e a .
1. I n t r o d u c t i o n
I n r e c e n t y e a r s t h e r e h a s been c o n s i d e r a b l e p r o g r e s s i n understanding t h e
bulk
thermodynamics and s t r u c t u r e of many l i q u i d m e t a l s and t h e i r a l l o y s . Accurate neutron and X-ray d i f f r a c t i o n experiments have been a s t i m u l u s f o r d e t a i l e d microscopic modelling of t h e s t r u c t u r e f a c t o r s andr a d i a l d i s t r i b u t i o n f u n c t i o n s . For simple (non- t r a n s i t i o n o r r a r e - e a r t h ) m e t a l s i t i s p o s s i b l e t o c a l c u l a t e i n t e r i o n i c p o t e n t i a l s u s i n g
p s e u d o p o t e n t i a l t h e o r y t o d e s c r i b e t h e e l e c t r o n - i o n i n t e r a c t i o n and t h e s e e f f e c t i v e p a i r w i s e p o t e n t i a l s g i v e a r a t h e r good d e s c r i p t i o n of t h e observed s t r u c t u r e and s e v e r a l thermodynamic p r o p e r t i e s e.g. (1, 2, 3 ) . For t r a n s i t i o n and r a r e - e a r t h m e t a l s and t h e i r a l l o y s t h e s i t u a t i o n i s l e s s s a t i s f a c t o r y , s i n c e f o r t h e s e systems t h e e l e c t r o n - i o n i n t e r a c t i o n i s s t r o n g and cannot be modelled by a weak p s e u d o p o t e n t i a l . Consequently t h e r e i s no t r a c t a b l e t h e o r y f o r i n t e r i o n i c f o r c e s i n t h e s e m e t a l s . Nevertheless it i s known t h a t t h e s t r u c t u r e f a c t o r s of many p u r e t r a n s i t i o n m e t a l s can h e c r u d e l y modelled by hard-spheres of t h e a p p r o p r i a t e diameter ( 4 ) and some under- s t a n d i n g of t h e entropy ( 5 ) h a s emerged. By c o n t r a s t t h e s u r f a c e s of l i q u i d m e t a l s a r e much l e s s s t u d i e d and a r e c e r t a i n l y n o t well-understood. There i s a w e a l t h o f experimental d a t a on t h e liquid-vapour s u r f a c e t e n s i o n y f o r b o t h p u r e m e t a l s and a l l o y s (6, 7 ) s i n c e a knowledge of t h i s i n t e r f a c i a l q u a n t i t y i s i m p o r t a n t i n many p r a c t i c a l a n d - t e c h n o l o g i c a l a p p l i c a t i o n s of l i q u i d metals. However, even t h e magnitude of y and i t s
v a r i a t i o n from m e t a l l i c element t o element through t h e p e r i o d i c t a b l e i s n o t s a t i s f a c t o r i l y
e x p l a i n e d by e x i s t i n g t h e o r i e s . When we examine a n o t h e r i m p o r t a n t i n t e r f a c i a l q u a n t i t y , t h e l i q u i d - s o l i d s u r f a c e t e n s i o n , t h e s i t u a t i o n i s much worse;
t h e r e a r e few r e l i a b l e experimental d e t e r m i n a t i o n s and t h e r e i s e s s e n t i a l l y no theory. T h i s means t h a t t h e r e 1s no b a s i s f o r a microscopic d e s c r i p t i o n of a n g l e of c o n t a c t , w e t t i n g p r o p e r t i e s e t c . , which a r e c r u c i a l f o r many m e t a l l u r g i c a l p r o c e s s e s . I n b i n a r y a l l o y s t h e r e i s an important e x t r a f e a t u r e a s s o c ~ a t e d w i t h t h e presence of an i n t e r f a c e and t h i s concerns t h e s u r f a c e c o n c e n t r a t i o n s of t h e two c o n s t i t u e n t s . Many l i q u i d a l l o y s a r e known t o e x h i b i t t h e phenomenon of s u r f a c e s e g r e g a t i o n , i . e . t h e c o n c e n t r a t i o n s o f t h e components i n t h e i n t e r - f a c i a l r e g i o n 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 from t h o s e i n t h e b u l k l i q u i d . T h i s phenomenon g i v e s r i s e t o pronounced e f f e c t s i n t h e c o n c e n t r a t i o n dependence of t h e s u r f a c e t e n s i o n of t h e a l l o y ( 7 ) and i s important f o r s u r f a c e chemistry s i n c e t h e behaviour of adsorbed s p e c i e s depends s t r o n g l y on t h e n a t u r e of t h e s u r f a c e ' l a y e r s ' . The b a s i c f a c t o r s which determine s e g r e g a t i o n i n a l l o y s a r e n o t well-understood from a microscopic viewpoint.
One o f t h e main r e a s o n s f o r l a c k o f p r o g r e s s i n t h i s g e n e r a l a r e a i s t h e absence of any d i r e c t e x p e r i m e n t a l probe o f t h e s u r f a c e s t r u c t u r e ; t h e r e i s n o t h i n g comparable t o t h e d i f f r a c t i o n d a t a which h a s p l a y e d a key r o l e i n t h e understanding and development of t h e o r y f o r b u l k l i q u i d s . Indeed t h e r e appears t o be o n l y one measurement, an e s t i m a t e o f t h e s u r f a c e ' t h i c k n e s s ' of t h e l i q u i d - vapo& i n t e r f a c e o f Hg (8) o b t a i n e d by an X-ray t e c h n i q u e , which g i v e s some d i r e c t i n f o r m a t i o n on t h e i n t e r f a c i a l s t r u c t u r e . There a r e s e v e r a l experiments which y i e l d i n t e r e s t i n g and u s e f u l i n d i r e c t information concerning i n t e r f a c i a l s t r u c t u r e . Tl?e s u r f a c e t e n s i o n y i s a s u r f a c e e x c e s s thermodynamic p r o p e r t y and t h e r e f o r e depends on t h e i o n i c and e l e c t r o n d e n s i t y p r o f i l e s i n t h e i n t e r f a c e ; more s p e c i f i c a l l y it i n v o l v e s
JOURNAL DE PHYSIQUE i n t e g r a t i o n s over t h e s e i n t e r f a c i a l p r o f i l e s . S i m i l a r l y t h e temperature c o e f f i c i e n t o f t h e s u r f a c e t e n s i o n
a y / a ~ ,
which i s a measure of t h e s u r f a c e excess e n t r o p y i . e . t h e e x t r a d i s o r d e r i n t r o d u c e d by having an i n t e r f a c e , i s an i n t e g r a l q u a n t i t y . The work f u n c t i o n depends on t h e e l e c t r o n i c s t r u c t u r e o f t h e i n t e r f a c e v i a t h e t o t a l e l e c t r o s t a t i c p o t e n t i a l d i f f e r e n c e , t h e d i p o l e b a r r i e r ( 9 ) . It i s i n t e r e s t i n g t h a t t h e work f u n c t i o n f o r many simple m e t a l s shows almost no change a t t h e m e l t i n g t r a n s i t i o n (10)i n d i c a t i n g t h a t t h e d i p o l e b a r r i e r a t t h e l i q u i d metal s u r f a c e s h o u l d be very s i m i l a r t o t h a t o f t h e p o l y c r y s t a l l i n e s o l i d . T h i s behaviour a p p e a r s somewhat s u r p r i s i n g when one n o t e s t h a t t h e work f u n c t i o n of c r y s t a l l i n e m e t a l s o f t e n v a r i e s by about 10% between d i f f e r e n t s u r f a c e s of t h e same s i n g l e c r y s t a l ( 9 , 11). Another i n d i r e c t experiment, which i s n o t u s u a l l y r e g a r d e d a s a s u r f a c e probe b u t can b e v e r y u s e f u l f o r s t u d y i n g s u r f a c e s e g r e g a t i o n i n l i q u i d (and s o l i d ) a l l o y s , i s p h o t o e l e c t r o n spectroscopy. I n f a v o u r a b l e systems it is p o s s i b l e t o e s t i m a t e t h e e f f e c t i v e s u r f a c e c o n c e n t r a t i o n o f each s p e c i e s by measuring t h e r e l a t i v e i n t e n s i t i e s of prominent e l e c t r o n i c s t r u c t u r e s , e.g. d band f e a t u r e s , a s s o c i a t e d with t h e i n d i v i d u a l s p e c i e s . S e g r e g a t i o n has been i n v e s t i g a t e d i n s e v e r a l b i n a r y a l l o y s (10) u s i n g photoemission. While p r o g r e s s i n understanding t h e s u r f a c e s of o t h e r l i q u i d s h a s a l s o s u f f e r e d from a l a c k of microscopic i n f o r m a t i o n concerning t h e i n t e r f a c i a l s t r u c t u r e , t h e o r y h a s been making some headway over t h e l a s t decade. Consequently i t i s r e a s o n a b l e t o a s k whether t h e t h e o r e t i c a l e x p e r t i s e which has been s u c c e s s f u l l y developed f o r o t h e r l i q u i d systems might be u s e f u l f o r e l u c i d a t i n g t h e s u r f a c e s t r u c t u r e and p r o p e r t i e s o f l i q u i d m e t a l s . I n t h i s a r t i c l e we b r i e f l y d e s c r i b e t h e r e s u l t s o f some r e c e n t t h e o r e t i c a l work and computer s i m u l a t i o n s on t h e s u r f a c e s of pure, monatomic, i n s u l a t i n g ( a r g o n - l i k e ) l i q u i d s , m i x t u r e s of a r g o n - l i k e l i q u i d s and molten s a l t s . We w i l l mention t h e r e l e v a n c e o f t h e s e s t u d i e s t o l i q u i d metal s u r f a c e s and d i s c u s s t h e c u r r e n t s t a t e o f t h e theory f o r m e t a l s . Throughout we c o n c e n t r a t e on t h e l i q u i d - v a p u r i n t e r f a c e a t low t e m p e r a t u r e s ( n e a r t h e t r i p l e p o i n t ) s i n c e t h i s is t h e problem which h a s r e c e i v e d most a t t e n t i o n .
2. Eure Monatomic L i q u i d s
It i s probably s a f e t o argue t h a t t h e l i q u i d - v a w u r i n t e r f a c e of p u r e a r g o n - l i k e f l u i d s , over t h e f u l l - r a n g e o f liquid-vapour c o e x i s t e n c e , i s now well-understood; f o r r e c e n t reviews s e e (12, 13, 1 4 ) . The s u r f a c e t e n s i o n , t h e form o f t h e atomic d e n s i t y p r o f i l e and even t h e p a i r w i s e d i s t r i b u t i o n f u n c t i o n have been i n v e s t i g a t e d u s i n g a v a r i e t y o f t h e o r e t i c a l t e c h n i q u e s and i n Monte C a r l o and molecular dynamics s i m u l a t i o n s . The b a s i c t h e o r e t i c a l problem i s how t o s o l v e t h e c l a s s i c a l e q u i l i b r i u m s t a t i s t i c a l mechanics f o r a n extremely inhomogeneous f l u i d ; n e a r t h e t r i p l e p i n t t h e d e n s i t y v a r i e s by s e v e r a l o r d e r s of magnitude, from t h e bulk l i q u i d t o t h e b u l k vapour, o v e r a
d i s t a n c e o f a few
2.
Theory and s i m u l a t i o n p r e d i c t monotonic p r o f i l e s whose widths i n c r e a s e smoothly w i t h temperature, d i v e r g i n g a t t h e c r i t i c a l temperature. Near t h e t r i p l e p o i n t t h e'10-90' t h i c k n e s s o f t h e i n t e r f a c e , d e f i n e d a s t h e d i s t a n c e o v e r which t h e d e n s i t y changes from
0.9 ( P Q - ) t o 0 . 1 ( ) where and
e,,
v - P v
a r e t h e bulk l i q u i d and vapour d e n s i t i e s , i s p r e d i c t e d t o b e about 2 molecular diameters. T h i s i s i n good agreement w i t h t h e r e s u l t of a r e c e n t
o p t i c a l r e f l e c t i v i t y ( e l l i p s o m e t r y ) measurement on l i q u i d A r ( 1 5 ) . The v a l u e s of t h e s u r f a c e t e n s i o n o b t a i n e d from v a r i o u s t h e o r i e s a r e i n r e a s o n a b l e agreement w i t h computer r e s u l t s and g i v e a r a t h e r good d e s c r i p t i o n o f t h e measured s u r f a c e t e n s i o n of
A r .
Most of t h e r e c e n t t h e o r e t i c a l work i s based on a d e n s i t y f u n c t i o n a l approach ( 1 2 ) . The s i m p l e s t t h e o r y expands t h e f r e e energy of t h e inhomogeneous f l u i d a s a s e r i e s of d e n s i t y g r a d i e n t s and makes a p r o p e r microscopic
i n t e r f a c e when t h e z c o o r d i n a t e s o f t h e p a r t i c l e s , z1 and z2, l i e i n t h e i n t e r f a c e . The c o r r e l a t i o n l e n g t h i s g i v e n by Ir 5 = Ey/ (pl - Dv)mg
]
which d i v e r g e s a s g , t h e a c c e l e r a t i o n due t o g r a v i t y , goes t o zero. y i s t h e s u r f a c e t e n s i o n . Computer s t u d i e s (18) of t h e p a i r w i s e d i s t r i b u t i o n f u n c t i o n seem t o confirm t h e e x i s t e n c e o f such c o r r e l a t i o n s b u t a d e t a i l e d microscopic model i s s t i l l l a c k i n g . 3 . Mixtures o f Simple L i q u i d s There h a s been r e l a t i v e l y l i t t l e a t t e n t i o n devoted t o t h e s t a t i s t i c a l mechanical d e s c r i p t i o n o f t h e liquid-vapour i n t e r f a c e of c l a s s i c a l m u l t i - component systems; consequently t h e l a r g e amount of experimental d a t a on t h e c o n c e n t r a t i o n dependence of t h e s u r f a c e t e n s i o n and i t s i m p l i c a t i o n f o r s u r f a c e s e g r e g a t i o n l a c k s a d e t a i l e d e x p l a n a t i o n . A s a f i r s t s t e p towards a microscopic t r e a t m e n t of such i n t e r f a c e s Telo d a Gama and Evans (19) have r e c e n t l y c a r r i e d o u t d e t a i l e d c a l c u l a t i o n s of t h e d e n s i t y p r o f i l e s and s u r f a c e t e n s i o n f o r b i n a r y m i x t u r e s o f Lennard- J o n e s f l u i d s u s i n g a g e n e r a l i s a t i o n o f t h e square- g r a d i e n t approximation mentioned i nmodel system should b e a p p r o p r i a t e t o m i x t u r e s of r a r e - g a s l i q u i d s and, perhaps, t o m i x t u r e s o f o t h e r simple l i q u i d s . For parameters a p p r o p r i a t e to. A r
and Kr t h e y f i n d A r s e g r e g a t e s a t t h e s u r f a c e f o r bulk l i q u i d c o n c e n t r a t i o n s of A r l e s s t h a n 0.6. Since t h e s p e c i e s w i t h t h e lower s u r f a c e t e n s i o n i s p r e f e r e n t i a l l y adsorbed a t t h e s u r f a c e t h e e x c e s s s u r f a c e t e n s i o n of t h e m i x t u r e , yex = - Y - X I Y l - X2Y2 (2) where X1, X a r e t h e bulk l i q u i d c o n c e n t r a t i o n s 2 and yl, y a r e t h e s u r f a c e t e n s i o n s o f t h e p u r e 2 components a t t h e same t e m p e r a t u r e , i s n e g a t i v e over much of t h e c o n c e n t r a t i o n range. The
ex
.
<magnitude of y I S , however, small
-
1 dyn. cm.-
1 s i n c e A r and K r a r e n o t t o o d i f f e r e n t a s r e g a r d s t h e i r i n t e r a t o m i c p o t e n t i a l s . For o t h e r m i x t u r e s s e g r e g a t i o n w i l l be more pronounded and yeX l a r g e r i n magnitude. The c a l c u l a t i o n s p r e d i c t t h a t f o r an equimolar b i n a r y m i x t u r e w i t h e q u a l p o t e n t i a l Well-depths, a d i f f e r e n c e i n diameter e x o f n% y i e l d s a y which i s a b o u t n% o f t h e d i f f e r e n c e between t h e s u r f a c e t e n s i o n s of t h e p u r e components, t h e l a r g e r atoms b e i n g adsorbed a t t h e s u r f a c e . For a n equi-diameter m i x t u r e a d i f f e r e n c e i n well-depths of m% y i e l d s a yeXwhich i s roughly -m/2% of t h e d i f f e r e n c e between t h e s u r f a c e t e n s i o n s of t h e p u r e components, t h e atoms w i t h t h e s m a l l e r well-depth b e i n g adsorbed a t t h e s u r f a c e .
U n f o r t u n a t e l y e x c e s s s u r f a c e t e n s i o n s d o n o t appear t o have been measured f o r m i x t u r e s o f =re- g a s f l u i d s s o i n o r d e r t o compare t h e o r y w i t h ,
experiment Telo da Gama and Evans (19) modelled
N and CH using Lennard-Jones p o t e n t i a l s and
2 4
c a l c u l a t e d yeX f o r A r
-
N and A r-
CH m i x t u r e s .2 4
Given t h e crudeness of t h e p o t e n t i a l s , t h e l e v e l o f agreement between t h e o r y and experiment i s encouraging and i t seems t h e r e i s now some p o s s i b i l i t y of t a c k l i n g t h e s u r f a c e p r o p e r t i e s of m i x t u r e s from a microscopic s t a r t i n g p o i n t .
4. Molten S a l t s
The thermodynamic, s t r u c f u r a l and dynamical p r o p e r t i e s of b u l k molten s a l t s have been s t u d i e d i n t e n s i v e l y o v e r t h e l a s t few y e a r s (e.g. ( 2 0 ) ) . Such systems a r e of i n t e r e s t i n s t a t i s t i c a l p h y s i c s because t h e i r c o n s t i t u e n t p a r t i c l e s ( i o n s ) i n t e r a c t v i a t h e long-ranged Coulomb f o r c e . T h i s i n t r o d u c e s f e a t u r e s which a r e a b s e n t i n t h o s e l i q u i d s where t h e i n t e r a c t i o n s a r e of t h e s h o r t - ranged van d e r Waals form. The d i f f i c u l t i e s a s s o c i a t e d w i t h t h e Coulomb i n t e r a c t i o n a r e e s p e c i a l l y r e l e v a n t when we c o n s i d e r t h e s u r f a c e p r o p e r t i e s o f molten s a l t s . I n g e n e r a l t h e r e w i l l b e an e l e c t r i c a l double l a y e r a t t h e liquid-vapour i n t e r f a c e of a s a l t ; t h e c a t i o n s and a n i o n s w i l l a d o p t s l i g h t l y d i f f e r e n t d e n s i t y p r o f i l e s i n ' . o r d e r t o minimize t h e grand p o t e n t i a l of t h e inhomogeneous f l u i d . Thus t h e m o l t e n - s a l t s u r f a c e h a s many o f t h e complications a s s o c i a t e d w i t h e l e c t r o c h e m i c a l systems and any t h e o r y must make some s y s t e m a t i c improvement on the c l a s s i c a l Debye- Htickel orGouy-Chapman t h e o r i e s of inhomogeneous charged f i u i d s . Evans and S l u c k i n (21) have developed a formal d e n s i t y - f u n c t i o n a l t h e o r y f o r charged f l u i d s and d e r i v e d some e x p l i c i t r e s u l t s f o r t h e s u r f a c e p r o p e r t k e s o f a model.qf a q o l t e n s a l t i n which t h e charged p a r t i c l e s a r e t r e a t e d a$
r i g i d i o n s embedded i n a s t r u c t u r e l e s s d i e l e c t r i c medium.
C a l c u l a t i o n s have n o t y e t been performea for
c8-778
JOURNAL DE PHYSIQUEequal radii e.g. NaF, KC1, W C 1 , RhBr, CsCl and CsBr and for these salts we expect on symmetry grounds that the two density profiles are almost the same so there is only a very small double layer. Indeed for a completely 'symmetric' charged fluid where the interionic pairwise potentials are of the form
PA,
(ri = $ir)+
1-1)" 'e2 (3) rand $(r) is the same for all species,the two profiles must be identical and the interface locally electroneutral. Under these circumstances the surface tension reduces to that of a one- component, neutral fluid and can be readily calculated. Calculations of the density profile and surface tension of the restricted primitive model, in which $(r) is taken to be the hard- sphere potential, have been carried out over a wide range of temperature (23). Detailed results will be reported elsewhere but it appears that a simple square-gradient approximation gives a reasonable account of the measured surface tensions for the salts listed above provided the hard-sphere diameters are chosen appropriately. The theory also predicts that the interfacial
thickness w for the 'symmetric' alkali-halide
...
should be very sharp ' 1 hard-sphere diameter near the triple point (23). A retent computer
simulation (24) of the surface of liquid KC1 also predicts a sharp irlterface i.e. somewhat sharper than those discussed in
f
2 for argon-like fluids. we are encouraged by the success of ourcalculations for the 'symmetric' charged fluid to extend the computations to the 'non-symmetric' alkali halides. This necessitates further approximations since theories of bulk fluids do not allow for departures from electroneutrality. Nevertheless we are confident that suitable
schemes can be developed for treating the double layer etc. (22).
5. Liquid Metals
A pure liquid metal is a two-component fluid; the ions form a dense classical liquid and the valence (conduction) electrons form a dense Fermi liquid. It is, of course, the
quantum mechanical nature of the electron liquid which distinguishes the liquid metal problem from the molten salt problem and makes it more
difficult! A proper theory of the liquid metal surface must be capable of describing, in
a
self-consistent way, a highly inhomogeneous distribution of strongly interacting ions andelectrons since we expect both the ion and electron density profiles to vary extremely rapidly at the interface. For a liquid-vapour interface there is the additional complication that somewhere in the vapour side there must be a transition from metallic-like bonding (ions plus electrons) to an insulating phase. Before reviewing various theoretical approaches it is probably worthwhile to list some questions for which theory should provide answers:
i) What form do the density profiles adopt and what is their extent?
ii) Which interactions determine the surface tension and its variation through the periodic table?
iii) What is the surface contribution to the work function of a liquid metal?
While i) and ii) are the same as we asked for other liquids there are important new aspects for metals. Do the electrons 'follow' the ions and vice-versa or is there a pronounced electrical double layer with the electrons extending further than the ions? In crystalline metals, where the ions are rigidly fixed in lattice planes, the electron density exhibits oscillatory behaviour in the surface layers (Friedel
oscillations). Do the mobile ions in the liquid state mimic this oscillatory behaviour or are both profiles monotonic as a result of the increased thermal fluctuations? Unlike the surface tension of the liquid rare-gases which varies from 13 dyn.
-1 -1
cm. for Ar to 19 dyn. cm. for Xe and that of the alkali-halides which lies in the range
-1
80
-
180 dyn. cm.,
the surface tension of metals exhibits an enormous variation from element to element. For example, y for Cs near its melting-1
temperature is about 71 dyn. cm
,
a value close -1 to that of water, y for A1 is 914 dyn. cm. while y for the high melting-point transition-1
metal Re is about 2700 dyn. cm. (6). The temperature dependence of y in liquid metals is also of considerable interest because of reports of positive values of dy/d~ in some experiments
l o c a l o r d e r i n g of atoms i n t h e s u r f a c e . Question
iii) i s s p e c i f i c t o m e t a l s and, a s mentioned i n t h e i n t r o d u c t i o n , concerns t h e magnitude of t h e s u r f a c e d i p o l e b a r r i e r and how t h i s v a r i e s from t h e c r y s t a l l i n e t o t h e l i q u i d m e t a l .
T h e o r e t i c a l work p r i o r t o 1977 h a s been reviewed i n papers by t h e p r e s e n t a u t h o r (26, 27,
3) s o h e r e we w i l l o n l y summarize some of t h e main c o n c l u s i o n s . The well-known screened-pseudoatom model which r e p r e s e n t s t h e p o t e n t i a l energy of a m e t a l a s a sum of e f f e c t i v e p a i r w i s e i n t e r i o n i c p o t e n t i a l s p l u s a volume dependent ' s e l f - e n e r g y ' c o n t r i b u t i o n , and which i s known t o g i v e a r a t h e r good d e s c r i p t i o n of t h e
3
p r o p e r t i e s of many simple l i q u i d m e t a l s , appears t o be Inadequate f o r d e s c r i b i n g t h e i n t e r f a c e . The s u r f a c e t e n s i o n can be e s t i m a t e d by making approximations t oP '2' ( ~ ~i n t h e i n t e r f a c e . ~ 4 ) For t h e a l k a l i s t h e c a l c u l a t e d s u r f a c e t e n s i o n s a r e t y p i c a l l y h a l f of t h o s e which a r e measured while i n t h e p o l y v a l e n t m e t a l s t h e c a l c u l a t e d v a l u e s a r e t y p i c a l l y 10-30% o f t h e measured v a l u e s (26, 27, 3 ) . The e f f e c t i v e p a i r w i s e p o t e n t i a l s , a s o b t a i n e d from p s e u d o p o t e n t i a l t h e o r y , do n o t have s u f f i c i e n t l y deep minima t o e x p l a i n t h e l a r g e s u r f a c e t e n s i o n s of m e t a l s . T h i s i s n o t t o o s u r p r i s i n g i f we r e c a l l t h a t most of t h e cohesive energy of a bulk simple metal r e s i d e s i n t h e s t r u c t u r e independent ' s e l f - e n e r g y ' term ( 3 ) . D e t a i l e d c a l c u l a t i o n s of t h e i o n d e n s i t y p r o f i l e could, i n p r i n c i p l e , b e c a r r i e d o u t f o r t h i s model u s i n g t h e t e c h n i q u e s developed f o r monatomic
l i q u i d s s i n c e t h e pseudoatoms form an e f f e c t i v e one-component f l u i d ? The average e l e c t r o n d e n s i t y could t h e n b e o b t a i n e d u s i n g l i n e a r response t h e o r y
( 3 ) . Such computations have n o t been performed and, i n t h e l i g h t o f e x p e r i e n c e w i t h t h e s u r f a c e t e n s i o n , i t i s d o u b t f u l whether t h e y would y i e l d r e a l i s t i c r e s u l t s e x c e p t , perhaps, f o r a l k a l i m e t a l s .
The p r e s e n t a u t h o r (26) attempted t o improve upon t h e l i n e a r - s c r e e n i n g approach, which i s i n h e r e n t i n t h e above model, by a l l o w i n g t h e e f f e c t i v e p a i r w i s e p o t e n t i a l s and t h e ' s e l f - energy' o f t h e i o n immersed i n i t s cloud o f conduction e l e c t r o n s t o depend on t h e l o c a l e l e c t r o n d e n s i t y . BY assuming some e l e c t r o n p r o f i l e n ( z ) i t was p o s s i b l e t o c a l c u l a t e t h e i o n i c p r o f i l e u s i n g a thermodynamic p e r t u r b a t i o n t h e o r y (27) and a l s o t h e s u r f a c e t e n s i o n . The v a l u e s o f t h e l a t t e r were i n q u i t e good
agreement w i t h experiment f o r Na, K and A l . The i o n i c p r o f i l e was p r e d i c t e d t o b e extremely s h a r p ; t h e c a l c u l a t e d i n t e r f a c i a l t h i c k n e s s w i s about
12
f o r Na and K and o n l y0.38
f o r A l .These c a l c u l a t i o n s were n o t s e l f - c o n s i s t e n t and no a t t e m p t was made t o r e c a l c u l a t e t h e e l e c t r o n d e n s i t y . Consequently it i s n o t p o s s i b l e t o d i s c u s s t h e magnitude o f t h e d i p o l e b a r r i e r , e t c .
A somewhat more s y s t e m a t i c way of proceeding i s t o f o l l o w t h e r o u t e i n t r o d u c e d by Lang and Kohn (28) ( s e e a l s o ( 9 ) ) f o r t h e s u r f a c e s of c r y s t a l l i n e m e t a l s . T h i s i s based on t h e ' j e l l i u m ' model of a s u r f a c e i n which t h e i o n s a r e r e p l a c e d by a uniform p o s i t i v e background o f charge w i t h a s t e p - f u n c t i o n p r o f i l e t o r e p r e s e n t t h e s u r f a c e . The e l e c t r o n d e n s i t y p r o f i l e and t h e s u r f a c e energy ( t h i s i s a z e r o temperature t h e o r y ) can be c a l c u l a t e d u s i n g t h e n o n - l i n e a r , s e l f - c o n s i s t e n t Hohenberg-Kohn-sham d e n s i t y f u n c t i o n a l a2proach. The i o n s a r e r e - i n t r o d u c e d u s i n g p e r t u r b a t i o n t h e o r y , i . e . t h e p o s i t i v e charge d i s t r i b u t i o n i s r e p l a c e d by a semi- i n f i n i t e l a t t i c e of pseudo-ions g i v i n g r i s e t o a change 6v&) = 1 v ( r
-
R_R)
- V + ( r )II
PS-
( 4 ) i n t h e t o t a l e l e c t r o n - i o n p o t e n t i a l . Here%
d e n o t e s t h e i o n i c c o o r d i n a t e s . v i s t h e pseudo- P s p o t e n t i a l of a s i n g l e i o n and V + ( r ) i s t h e p o t e n t i a l of t h e o r i g i n a l p o s i t i v e background. The change i n s u r f a c e energy i s c a l c u l a t e d t o f i r s t o r d e r i n 6V s o t h a t t h e e f f e c t s of e l e c t r o n s c r e e n i n g of t h e i o n s a r e n o t i n c l u d e d . The d i r e c t ion-ion i n t e r a c t i o n g i v e s r i s e t o a ' c l a s s i c a l cleavage e n e r g y ' ; t h i s i s j u s t t h e change i n Madelung energy a s s o c i a t e d w i t h t h e presence of a s u r f a c e . For many simple m e t a l s Lang and Kohn's c a l c u l a t i o n s gave r e s u l t s f o r t h e s u r f a c e energy which were i n r a t h e r good agreement w i t he s t i m a t e s based on e x t r a p o l a t i n g measured s u r f a c e t e n s i o n d a t a on l i q u i d m e t a l s t o a b s o l u t e zero.
( ~ t i s d i f f i c u l t t o measure t h e s u r f a c e t e n s i o n o f a c r y s t a 1 , s o l i t t l e d a t a i s a v a i l a b l e . ) 1t i s n o t completely obvious why t h i s approach should work s i n c e 6V can have q u i t e l a r g e F o u r i e r components l e a d i n g t o l a r g e c o r r e c t i o n terms
2
0 (6V ) e t c . ( 2 9 , 3 )
.
Recently, Monnier and Perdew (30) have improved upon t h e o r i g i n a l Lang-Kohn scheme by modifying t h e i n i t i a l j e l l i u m c a l c u l a t i o n . The p e r t u r b a t i o n i n t r o d u c e d by t h e pseudoions 6V i s t h e n much s m a l l e r and t h i s shouldc8-780
JOURNAL DE PHYSIQUEl e a d t o b e t t e r convergence. The Lang and Kohn approach a l s o g i v e s a good account o f t h e work f u n c t i o n 0 of t h e simple m e t a l s . I n g e n e r a l a can be w r i t t e n a s t h e sum of a b u l k and s u r f a c e c o n t r i b u t i o n :
@ = - ; + A +
( 5 )
where
ii
i s t h e ,bulk chemical p o t e n t i a l and A$, t h e d i p o l e b a r r i e r , i s t h e d i f f e r e n c e between t h e e l e c t r o s t a t i c p o t e n t i a l i n t h e vacuum and t h a t deep i n s i d e t h e metal. From t h e i r c a l c u l a t i o n s Lang and Kohn deduce t h a t t h e magnitude of Q i s p r i m a r i l y determined by t h e j e l l i u m c o n t r i b u t i o n i . e . i n c l u d i n g d i s c r e t e l a t t i c e c o n t r i b u t i o n s v i a t h e p e r t u r b a t i o n 6V changes 0 by 10% o r l e s s from i t s v a l u e f o r j e l l i u m ; t h e change can be p o s i t i v e o r n e g a t i v e depending on t h e p a r t i c u l a r c r y s t a l f a c e . I n t h e j e l l i u m c a l c u l a t i o n a p p r o p r i a t e t o a p o l y v a l e n t m e t a l such a s A l , t h e d i p o l e b a r r i e rA +
i s l a r g e s i n c e it has t o c a n c e l t h e l a r g e p o s i t i v eiT:
For t h e a l k a l i sA 4
i s small andi
i
i s n e g a t i v e .I n p r i n c i p l e i t i s p o s s i b l e t o extend t h e Lang and Kohn approach t o t h e l i q u i d m e t a l s u r f a c e problem ( 3 ) . By invoking t h e Born- Oppenheimer approximation one can c o n s t r u c t an e f f e c t i v e Hamiltonian f o r t h e i o n s i n t e r a c t i n g i n t h e presence o f t h e inhomogeneous e l e c t r o n d i s t r i b u t i o n and a t t e m p t t o s o l v e t h e c l a s s i c a l s t a t i s t i c a l mechanics, i . e . c a l c u l a t e t h e e q u i l i b r i u m i o n d e n s i t y p r o f i l e and s u r f a c e t e n s i o n . The d i f f i c u l t y a r i s e s i n modelling ( 2 ) r ) f o r t h i s system of i o n s which Q ( L ~ , -2 i n t e r a c t v i a t h e bare Coulomb i n t e r a c t i o n ; it i s d i f f i c u l t t o c a l c u l a t e t h e analogue o f t h e ' c l a s s i c a l cleavage e n e r g y ' . I t may, n e v e r t h e l e s s , be p o s s i b l e t o develop u s e f u l approximation schemes i n t h e s p i r i t o f t h o s e mentioned i n
$
2. Once t h e i o n d e n s i t y were e v a l u a t e d t h e e l e c t r o n d e n s i t y could be r e c a l c u l a t e d and t h e p r o c e s s continued towards s e l f - c o n s i s t e n c y . To o u r knowledge such a1 '
programme h a s n o t been c a r r i e d t h r o u g h s o we do n o t have r e s u l t s f o r t h e p r o f i l e s , d i p o l e b a r r i e r e t c . A l l e n and Rice (31) have performed s t r i c t l y j e l l i u m c a l c u l a t i o n s i n which t h e p o s i t i v e background, 6 s w e l l a s t h e e l e c t r o n d i s t r i b u t i o n , i s allowed t o r e l a x . They f i n d t h a t t h e i o n p r o f i l e s c l o s e l y match t h e e l e c t r o n p r o f i l e s ; both p r o f i l e s e x h i b i t o s c i l l a t o r y ' h e h a v i o u r f o r bulk e l e c t r o n d e n s i t i e s corresponding t o a l k a l i m e t a l s w h i l e f o r h i g h e r d e n s i t i e s t h e p r o f i l e s a r e monotonic. The i n t e r f a c i a l t h i c k n e s s e s a r e i n t h e range 3.5
-
6.48. We d o n o t f i n d t h i s r e s u l t s u r p r i s i n g ; t h e minimum energy s i t u a t i o n f o r a j e l l i u m o f t h i s t y p e should correspond t o l o c a l e l e c t r o n e u t r a l i t y . Such a model g i v e s , however, a completely u n r e a l i s t i c d e s c r i p t i o n o f a s e a l l i q u i d m e t a l s u r f a c e . The d i s c r e t e c h a r a c t e r o f t h e i o n s i s o f c r u c i a l importance and must be t a k e n i n t o account. For example, A l l e n and R i c e ' s c a l c u l a t i o n s would presumably p r e d i c t a n e g a t i v e work f u n c t i o n f o r h i g h d e n s i t y m e t a l s s i n c e A+ would be almost zero. Furthermore, t h e s e c a l c u l a t i o n s r e f e r t o a b s o l u t e z e r o s o t h e r e a r e no e n t r o p yc o n t r i b u t i o n s ; t h e s u r f a c e t e n s i o n of a l i q u i d m e t a l , and c e r t a i n l y i t s temperature dependence, must b e s t r o n g l y i n f l u e n c e d by t h e e n t r o p y a s s o c i a t e d w i t h t h e c o n f i g u r a t i o n a l d i s o r d e r o f t h e i o n s . I n a subsequent paper ( 3 2 ) , A l l e n and Rice used t h e f i r s t o r d e r p e r t u r b a t i o n t h e o r y o f Lang and Kohn t o i n v e s t i g a t e d i s c r e t e i o n e f f e c t s . They modelled t h e s u r f a c e by c l o s e packed p l a n e s o f c u b i c l a t t i c e s and allowed t h e p o s i t i o n s o f t h e two outermost p l a n e s t o r e l a x . They a r g u e t h a t t h e F r i e d e l o s c i l l a t i o n s i n t h e e l e c t r o n d e n s i t y p r o f i l e induce s i m i l a r o s c i l l a t i o n s i n t h e
p o s i t i o n s o f t h e i o n p l a n e s . The magnitude of t h e r e l a x a t i o n i s v e r y s m a l l , t y p i c a l l y
-
1% of t h e bulk l a t t i c e spacing. These c a l c u l a t i o n s r e f e r t o a b s o l u t e z e r o and it is n o t c l e a r whether thermal motion of t h e i o n s i n a l i q u i d would enhance o r reduce such e f f e c t s .A s mentioned i n t h e ' i n t r o d u c t i o n , t h e r e i s one experiment which i s d i r e c t l y r e l e v a n t t o t h e i s s u e s r a i s e d h e r e . Lu and Rice (8) have
determined t h e i o n d e n s i t y p r o f i l e of t h e l i q u i d - vapour i n t e r f a c e o f Hg a t room t e m p e r a t u r e u s i n g a t e c h n i q u e based on t h e t o t a l r e f l e c t i o n o f X-rays. They conclude t h a t t h e p r o f i l e i s t a n h - l i k e and t h a t t h e i n t e r f a c i a l t h i c k n e s s i s about 98 o r 3 atomic d i a m e t e r s . This i s of t h e same o r d e r a s t h a t , m e a s u r e d and p r e d i c t e d f o r a r g o n - l i k e f l u i d s
( s e e
8
2) and i s v e r y much l a r g e r t h a n t h e t h i c k n e s s e s p r e d i c t e d by t h e pseudoatom approach( 2 7 ) d i s c u s s e d above. I f t h e i o n d e n s i t y p r o f i l e does have a r e l a t i v e l y - l a r g e e x t e n t i n Hg t h e n
See, however, S . Amokrane, J.P. B a d i a l i , 1I.L.
Rosinberg and J. Goodisman ( t h e s e conference proceedings)
.
t h e e l e c t r o n p r o f i l e should have an even g r e a t e r e x t e n t i n o r d e r t o produce a s i g n i f i c a n t d i p o l e b a r r i e r . The work f u n c t i o n of Hg does n o t change on m e l t i n g (10) s o A$ should be a t l e a s t 2 o r 3 eV i n t h e l i q u i d .
I t should be c l e a r from t h e s e d i s c u s s i o n s t h a t t h e t h e o r y o f t h e l i q u i d m e t a l i n t e r f a c e , even f o r p u r e simple m e t a l s , i s n o t very s a t i s f a c t o r y . A
f r e s h approach i s probably r e q u i r e d . One s e n s i b l e l i n e of a t t a c k is t o develop t h e o r i e s which t r e a t t h e e l e c t r o n s and t h e i o n s on an e q u a l f o o t i n g . Indeed i t i s e a s y t o w r i t e down a formally e x a c t r e s u l t f o r t h e s u r f a c e t e n s i o n and e q u a t i o n s f o r t h e d e n s i t y p r o f i l e s u s i n g t h e d e n s i t y f u n c t i o n a l formalism (21) mentioned i n
§
4 f o r molten s a l t s . The d e r i v a t i o n which l e a d s t h e r e s u l t (21) f o r t h e s u r f a c e t e n s i o n o f a multi-component charged f l u i d does n o t depend on t h e s t a t i s t i c s ; it i s v a l i d f o r quantum a s w e l l a s c l a s s i c a l s t a t i s t i c s provided t h e r e l e v a n t c o e f f i c i e n t s a r e i n t e r p r e t e d a p p r o p r i a t e l y ( 3 3 ) . These c o e f f i c i e n t s a r e d i r e c t l y r e l a t e d t o i n v e r s e s o f t h e interacting d e n s i t y - d e n s i t y response f u n c t i o n s of t h e inhomogeneous f l u i d , s o f o r a l i q u i d m e t a l t h e y would i n v o l v e Xion-ion, and'el-ion ' e l - e l - A t p r e s e n t t h e r e i s no f e a s i b l e scheme f o r c a l c u l a t i n g t h e s e r e s p o n s e f u n c t i o n s - even t h e r e s p o n s e f u n c t i o n o f t h e inhomogeneous e l e c t r o n g a s a t a j e l l i u m s u r f a c e i s n o t well-understood. I f , however, one i s w i l l i n g t o make a g r a d i e n t expansion, a s f o r t h e molten s a l t s , it might be p o s s i b l e t o make some headway. What i s t h e n r e q u i r e d i s a scheme f o r c a l c u l a t i n g t h e f r e e energy d e n s i t y and t h e d e n s i t y - d e n s i t y r e s p o n s e f u n c t i o n s o f a uniform mixture o f i o n s and e l e c t r o n s f o r a wide range of d e n s i t i e s . For simple m e t a l s t h i s might b e achieved u s i n g low o r d e r p e r t u r b a t i o n t h e o r y ( p s e u d o p o t e n t i a l s ) (34)
.
Such a t h e o r y would g i v e a self-consistentd e s c r i p t i o n o f both t h e i o n s and e l e c t r o n s a n d
would be c a p a b l e o f answering, a t l e a s t q u a l i t a t i v e l y , many of t h e q u e s t i o n s we have posed. It would n o t be c a p a b l e of d e s c r i b i n g F r i e d e l o s c i l l a t i o n s i f t h e s e were p r e s e n t . I t
C8-782
JOURNAL DE PHYSIQUEconcentrations of the two species would also be extremely useful.
Our current understanding of the liquid-solid interface is very primitive; even for relatively simple insulating substances statistical theories are still crude (40). For metals we have the additional complications associated with the electron distribution. It will be some time before theoretical physicists can explain why solid Fe is extremely well wetted by molten Cu but is badly wetted by molten Ag!
The author is grateful to his collaborators in Bristol, T.J. Sluckin and M.M. Telo da Gama, for many helpful discussions and permission to refer to unpublished work. He also acknowledges useful conversations with F. Flores and M. Parrinello.
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1131.Addendum
Since the completion of this article we have learnt that K.K. Mon and D. Stroud (to be
published) have calculated the '.pn density profile and the surface tension using the pseudoatom model. These authors use a square-gradient approximation of the type described inj2; the direct correlation function which enters such an approach is
