On the Theory of the Surface Energy of Heteropolar Crystals

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N A T I O N A L RESEARCH C O U N C I L OF CANADA T s c h n l c ~ l T ~ a n s X a t f o n T T - 1 1 1 T i t l e : On t h e t h e o r y o f the s u r f a c e enepgy o f h e t e r o p o l u r c r y s t a l s , By : A . E, Giauberman, R e f e r e n c e : Zhurn. F f z , Khfm,:

23,

2, 1949, T r ~ a n a l a t e d by: E s t h e ~ Rabkfn,

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The Journal of Physical Chemistry, Vol, 23, No, 2 , Feb, 1949

ON THE THEORY OF THE SURFACE ENERGY OF HETEROPOLAR CRYSTALS by

A, E, Glauberman

Translated by Esther Rabkin.

The surface energy of a h e t e r o p o l a r c r y s t a l of t h e type NaCl was f i r s t c a l c u l a t e d by Born and S t e r n ( I ), L a t e r Lennard-Jones and Taylor c a r r i e d out these c a l c u l a t i o n s by a somewhat d i f f e r e n t method (2)" Both of t h e s e works a r e based on t h e Madelung

( 3 )

method of c a l c u l a t i n g an e l e c t r i c p o t e n t i a 1 , o f a c r y s t a l l i n e h e t e r o p o l a r l a t t i c e . Dent, on the b a s i s of t h a same method, has c a l c u l a t e d t h e c o n t r a c t i o n of

intep-plari'a&% d i s t a n c e s near the s u r f a c e of a c r y s t a l , I n the p r e s e n t work. a s o l u t i o n o f t h i s problem i s obtained by a

method s i m i l a r t o t h e one adopted by the author f o r t h e

a n a l y s i s of metallf c c r y s t a l s ( 5 ) , Prom such an a n a l y s i s a n expression can be obtained f o r the p o t e n t i a l i n the e x t e r n a l space and, i n the region occupied by the c r y s t a l $i,

s o l v i n g t h e problem f i r s t f o r the space l a t t i c e , i t i s t h e n p o s s i b l e t o o b t a i n q u f t e s i m p l y an e x p l i c i t form f o r t h e

general law of t h e c o n t r a c t i o n of i n t e r - p l a n a r d i s t a n c e s . The c o r r e c t i o n f o r the s u r f a c e energy of a c r y s t a l , which appears a s a r e s u l t of t h e deformation of t h e l a t t i c e near t h e s u r f a c e , can a l s o be c a l c u l a t e d q u i t e simply. The f a i l u r e of the Madelung ( 6 ) experiments, which attempted t o r e v e a l a

(4)

double e l e c t r i c a l l a y e r on t h e boundary s u r f a c e of a hetero- p o l a r c r y s t a l , may be explained by the c a l c u l a t i o n of' the deformation of a l a t t i c e by t h e method developed i n t h e p r e s e n t work,

1, The E l e c t r i c P o t e n t i a l of a C r y s t a l l i n e L a t t i ce of a

Heteropolar C r y s t a l

-

We w i l l analyze a c r y s t a l l i n e l a t t i c e of type NaCl occupying a h a l f space. The boundary plane, whi ch subdivides t h e i n f i n i t e c r y s t a l i n t o two e l e c t r i c a l l y n e u t r a l s e c t i o n s , we w i l l s e l e c t a s t h e coordinate plane z = 0,

I n t h e region, e x t e r n a l i n r e l a t i o n t o the c r y s t a l ( z > 0 )

,

t h e p o t e n t i a l i a s a t i s f i e d by t h e Laplace equation

i n t h e i n n e r region of the c r y s t a l ( z < 0) t h e p o t e n t i a l @

can be s a t i s f i e d by the Poisson equation

where p = ek6(z

-

) i s the coordinate of the ion, We

w i 11 express t h e density of the charge b y a t r i p l e Fourier s e r i e s

pagy C O S UX C O S By C 0 6 y ( 2

+

d ) , _{( l o 3 1}

a, 0 9 Y

where a =

-

_d p = - - m _d y=,=, m l , m , n ( 0 , 4 , 2, ...), and d i s the h a l f of the l a t t i c e corntan$, The expansion coef- f i c i e n t s p agy are determined i n the general way

-

paPy

- 1

//1

p cos ax cos gy cos y ( r

+

d ) dx dy dz, dS

(5)

s i n c e b ( r - 5 ) equals 0 a t r

#

rk and = a t rk = r , moreover,

Thus, f o r the expansion c o e f f i c i e n t s , we w i l l obtain

where t h e summation i s c a r r i e d out by the coordinates of a l l t h e ions of the elementary nucleus

( 5 ,

yk9 zk) and ek repre-

s e n t s the carrespondf ng share of the charge on the i on r e l a t i ng

*

t o one nucleus, Since i n a c r y s t a l o f type NaC1, we deal with a eimpIe cube l a t t i c e , then f o r t h e expansion c o e f f i c i e n t s f o r

t h e density of the charge i n t h e Fourier s e r i e s we w i l l o b t a i n

= 0 a t a l l o t h e r values of l ,

rn,

no

Pl,m,n

Expressing the solutions of the equations ( 2 , l ) and ( 2 , 2 ) a s a s e r i e s and conibining t h e obtained s o l u t i o n s on the boundary plane 2: = 0, assuming a contfnuoua p o t e n t i a l and i t s ' f i r s t

d e r i v a t i v e with respect t o z , we w f l l obtain f o r the p o t e n t i a l

*

The t o t a l l a t t i c e we assume as being sub- divided i nto elementary cubes, and according t o Thomson, we w i l l a s s i g n t o a given cube

1/8 of t h e charges located i n t h e apices of t h e cube, 1/4 of t h e charges located a t the c e n t r e of the edges of t h e cube, and 1/2 of the charges located a t the centre of the s u ~ f aces,

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of a c r y s t a l l i n e i n f i n i t e h a l f - l a t t i c e i n t h e e x t e r n a l space

$3. and I n the fnner region $, ( w i t h A a n accuracy up t o the

s u r f a c e l a t t i c e p o t e n t i a l c o n s i s t i n g of t h e charges i ) t h e following expressions: odd odd 0

-

2 cos

$

x cos y o (1.6) I n order t o o b t a i n t h e t o t a l p o t e n t i a l of t h e h a l f - l a t t i c e i n t h e e x t e r n a l region, 1 % i s necessary t o add t o $,

the p o t e n t i a l produced i n t h e e x t e r n a l space of the s u r f a c e e

l a t t i c e , c o n e f s t i n g of charges i <zL'" FOP the p o t e n t i a l of t h e s u r f a c e l a t t i c e f a g expressing t h e s u r f a c e d e n s i t y of t h e charge w by a double Poug~lel~ s e r i e s and t a k i n g i n t o account t h e r e l a t i o n s h f ~

i t i s easy t o o b t a i n

odd

and, hence, f o r the t o t a l p o t e n t i a l of t h e half-sum i n the e x t e r n a l epace we have:

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odd

(108)

By means of the expression obtained f o r t h e p o t e n t i a l

ma

t h e s u r f a c e t e n s i o n f o r the c r y s t a l NaCl can be c a l c u l a t e d , For a comparf son with t h e r e s u l t obtained by Born and S t e r n w e w i 11

c a r r y out a c a l c u l a t f o n f o r the plane ( I , 0 , 0 )

,

f , e o

,

f o r a s u r f a c e of a cube, The s u r f a c e t e n s i o n o i s determined by the

*

formula

where S = d2, and u q 2 i s the energy of the fn$eapactfon of an i n f i n i t e h a l f - l a t t i c e w i t h t h e column of the second l a t t i c e of an f n f i n i t e c r y s t a l , %he base 0 % whi eh f s a square havfng a

s i d e d, and the cen%s.e i s i n one of the i o n s of the boundary s u r f a c e P a t t f c e , The e P e c t s o s t a t i c p o r t i o n of t h e energy of i n t e r a c t i s n ~ $ w i l l be: 2

( I " 10)

and, t h e r e f o r e , the e l e c t r o s t a t i c p o r t i o n of the s u r f a c e t e n s i o n oe can be expressed by fhe formula:

if i n t h e e r p r e s s f o n f o r t h e p o t e n t i a l cPa w e w i l l assume t h a t

16111,

x = C O B

g

y = 1

eos

d

*

See Born and Gseppert-Mayer, Theory of a S o l i d Body,

(8)

and, i f ' we w i 11 r e t a i n only the main terms of the r a p i l l y converging s e r i e s determining Rumerf c a l l y we o b t a ~ n

I f t h e t o t a l energy of a c r y s t a l l i n e l a t t i c e f o r one i o n i s i s expressed i n t h e form

where k and p a r e const&qts, and the exponent n = .g, then i f we consi der only t h e n e a r e s t nefghbours t o the l a t t i c e MaC1, the r~umberc p c h a r a c t e r i z i n g such neighbours w i l l be 6, and

k = ? 750 Thus f o r t h e a d d i t i o n a l s e c t f on of the s u r f a c e t e n s i o n a, (corresponding t o t h e r e p u l s i o n f o r c e s ) we w i l l

o b t a i n

and, hence, the t o t a l value f o r t h e s u r f a c e t e n s i o n w i l l be equal t o :

A t d = 2,81 o 4 ~ ' em. and e = 4,8 CGSE we w i l l o b t a i n

a = -I 50 dynes/cm,

This r e s u l t agrees very well with t h e r e s u l t obtained f o r NaCl by Born and S t e r n , using the Madelung method.

(9)

2, The Deformation of the L a t t i ce near the Surface

We w i l l analyze one-half of the c r y s t a l under the a c t i o n of an e x t e r n a l e l e c t r i c f i e l d , opposite t o t h e f i e l d of the absent h a l f of the c r y s t a l (we a r e i n t e r e s t e d i n the component of the f i e l d perpendicular t o the boundary plane

The a-component of t h e f i e l d fop the absent h a l f - l a t t i c e can be expressed by the formula

where a = 2d,

This f i e l d d e s i p a t e s the r e s u l t i n g e l e c t r i c a l f o r c e experienced b y the ions I n undisplaced posi%ions, Analyzing %he displacements a r i s i n g i n t h e network of the ions located along an a x i s p a r a l l e l t o t h e z-axis, we w i l l neglect the changes i n the forces of the plane i o n i c s u r f a c e s of the

c r y s t a l , due t o the l a t e r a l i n t e r a c t i o n s of the ions displaced a s a whole. Prom an analysf s of such a network of f o m i n the f i e l d E z , i t i s possible t o c a l c u l a t e , approximately, the

displacements of the ions which we w i l l designate 2&, ( t h e

subscript n designates the ions of the network), I n the d i s - placements of the ions, an equilibrium is e s t a b l i s h e d between the changes i n the repulsion f o r c e s f r o m the nearest neighbours

(10)

of a g i v e n i o n and t h e " e x t e r n a l " f o r c e of t h e f i e l d E, a c t i n g on a g i v e n ion, Considering t h e df splacements

<

a s b e i n g

s m a l l , we w i l l o b t a i n f o r t h e changes i n p o t e n t i a l energy of t h e r e p u l s i o n f o r c e s A$ t h e f o l l o w i n g e x p r e s s i o n :

where

Q

i s t h e energy of the r e p u l s i o n f o r c e s ,

rk, ri

a r e t h e displacements of t h e k and i i o n s correspondingly, and zk and

Zf a r e t h e e q u i l i b r i u m c o o r d i n a t e s of t h e i o n s ( i n t h e non-

d i s p l a c e d p o s i t i o n s ) , The c o n d i t i o n of equilibrpium can be expressed in t h e f o l l o w i n g form:

Taking i n t o account t h a t the enepgy r e p r e s e n t s a sum of f u n c t i o n s , depending on the r e l a t i v e d i s t a n c e s of p a i r s of i o n s of t h e f o m and s u b s t i t u t i n g t h e e x p r e s s i o n f o r t h e f i e l d E k , a f t e r some s f m p l i f i c a t i o n s we w i l l o b t a i n from ( 2 , 3 ) an i n f f n f t e system of a l g e b p a i c e q u a t i o n s e x p r e s s i n g c k ( k = 1 , 2,

3 , .

.

) , which a f t e r expansion of ~ ~

+

r k ) f o r t h e o r d e r of s m a l l q u a n t i t i e s ( z ~ of

gk

and r e t a i n i n g t h e terms of t h e f i r s t o r d e r w i l l assume t h e form (for" k > I )

a

m

--

(11)

2xm

interatomic distance, and it i s aesumed t h a t cos%x= c o s ~ y = 3 .

The Pfmi%ing ease of t h i s problem i s given by an equatf on of t h i s system when k = 1, Neglecting i n %he r i g h t p a r t of the equation the term proportional t o

gk

(ck

is small

and e n t e r s i n t o the r i g h t p a r t of the equation with a c o e f f i c i e n t

-

proportional t o e P k ) 9 we w i l l r e t a i n i t only i n the l i m i t i n g case, i , e , , w h e n k = d o E,xpressing% i n t h e form

we w i l l analyze the system of equatf ons

O O O O o O O o D o O 0 . O a . o

O D O O O O O D O O O O O D O O D O 0

A s o l u t i o n f o r t h i s wylstem has the following form:

where the constant

t;im

characterizes the displacement of the c r y s t a l as a whole (t;_ i s determined from the l i m i t i n g case) and can be neglected, When analyzing an i n f i n i t e c r y s t a l i n an

(12)

equilibrium s t a t e

( r

= r o = a ) , each h a l f of the c r y s t a l can

*

be assumed i n an e l e c t r i c a l sense a s an equivalent charge e c r e a t i n g a f i e l d a t a distance P O equal t o the f i e l d o f the

t o t a l h a l f - l a t t i c e a t a distance P O f r o m the boundary plane ( i n a d i m e t i o n along t h e z-axis), then on the b a s i s o f ( A ,

43)

f o r u" we w i l l obtain (when n = 9 and ro = d) the formula

Retaining i n a l l the sums the main term ( l a

+

m2 = 2) we w i l l obtain f OP the df splacements t h e numeri c a l values:

T s

=

+

0.008 d, i o e o ,

i & l

= 0.8% of d,

=

-

0,000097 d, i e , l < s I = O o 0 O 9 ~ . . o i d o (20 8 )

As the distance from the surface decreases, the displacements

decrease according t o an exponential law, From ( 2 , 8 ) i t i s

evident t h a t the displacements b r i n g about B e o n t r a c t i o n i n

t h e surface layer, Our c a l c u l a t i o n s give somewhat higher

values f o r t h e contractions i n the inter-planar d i s t a n c e s than the values c a ~ c u l a t e d by Bept, who used anofbher metho&?

Analyzing the same problem, Madelung a r r i v e d a t t h e conelusion t h a t on the bouadary of a c r y s t a l t h e ions of

d i f f e r e n t sf gn a r e d i aplaced i n opgasi t e d i r e c t i one, perpendi

-

e u l a r t o the boundary plane, Because of t h i s , according t o Madelung, on the boundary plane t h e r e must a r i s e a double electrical l a y e r , Attempts t o reveal e x p e ~ f m e n t a l l y the

presence of such a layer have not been successful, This f a c t

i s of t e n accepted as proof of exeeedi ngly small deformation

*

brought about by the presence of t h e surface,

_-

*

See Born and Goeppert-Mayer, Theory of a S o l i d Body, page 32b0

(13)

We must note, t h a t from t h e c a l c u l a t i o n s of the ion displacements by %he method presented i n t h i s work, i t i s

evident t h a t the f i e l d a c t i n g on the Sons changes p e r i o d i c a l l y i n the x and y d i r e c t i o n s with a perf od equal t o the constant of the l a t t i c e , and thus e l e c t r i c a l f o r @ e s opposite i n d i r e c t i o n ac% on the ions of opposite sign, Therefore the t o t a l i o n i c l a t t i c e i s displaced as a whole, and i t does 30% seem possjlble

t h a t a double e l e c t r i c a l l a y e r can a r i s e on t h e boundary,

Hence, the f a i l u r e of the Madelung experiments can be explained a s belng due not t o the f a c t t h a t the deformation o f t h e

l a t t i c e near $he surface i s very small, But Ro the f a c t t h a t the double l a y e r s , which he attempled to d e t e c t , c a m o t e x i s t a t a l l ,

3

The Correction t o %he Value sf the Surface Tensi on

I n @onclusfon we w i l l analyze the energy eorkection f o r the value of the surface tension of a ckys$al NaCl t y p e ,

arksnng a s a r e s u l t o f the defoma%ion of t h e s u r f a c e l a y e r , We w i l l write down the force a c t i n g on each i o n of the analyzed Ionic network as a d i f f e r e n c e

Fk =

Fk

-

f k r ( 3 . 1 )

where Fk i s t h e e l e c t r o s t a t i c f o r c e , and f k i s the difference of the ~ e p u l s i on f orces from the nearest nef ghbours. A t

gk

= 0,

Fk

= 0 and

Fk

= _Fk We w i l l express

Fk(<)

as an expansion of small values of

C,

f o e , ,

(14)

If we l i m i t ourselves t o the terms o f the f i r s t order, then Zmax

where

Ek

is the maximum displacement of t h e k - i on i n the max

network, The work associated with the displacements of a l l

i 0,- i n the network can be expressed i n the f o m

=

( F ~ ( O ) Z ~

max

l9m n

Retaining i n the e a l e u l a t f o n s the main terms of t h i s s e r f e e corresponding t o e =

rn

= 1 , we w i l l o b t a i n

and, t h e r e f o ~ e ,

This c o ~ r e c t i o n mst be sntroduced t o t h e value of the surface tension with a minus s n g n since the deformation of the c r y s t a l ns spontaneous, and, t h e r e f o r e , must b r i n g about a

decrease i n energy, Taking i n t o accoust t h e c a l c u l a t e d correc- t i o n , t h e value obtafned fop the surface tensf o n of the NaCl c r y s t a l is

On the Theory of the Surface Energy of Heteropolar Crystals

NRC Publications Archive

Archives des publications du CNRC

23,

( 3 )

-

,

-

+

-

-

- 1

//1

+

#

( 5 ,

*

rn,

*

-

$

ma

,

,

*

g

d

*

<

Q

rk, ri

3 , .

.

.

+

gk

--

gk

(ck

-

t;im

( r

*

43)

+

T s

+

i & l

-

-

*

-

*

3

Fk

-

gk

Fk

Fk

Fk(<)

C,

Ek

( F ~ ( O ) Z ~

rn

In conclusion I wish to express

gratitude to

E, F~enkel

for his assistance with this work,

Sent to the editor, 25 May

1948.

The Leningrad Polytechnical Institute, Livov State University,

Bibliography

I

Fn, Born, Atomtheorfe

festen Zustandes,

Lennard-Jones and Taylor, Proc, ROY, Soc, No,

CIX,

1925,

Page

476.

_-