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(2)

NATIONAL RESEARCH COUNCIL OF CANADA

T r a n s l a t i o n T T - 1 1 6

THEORY OF THE SURFACE TENSION OF METALS (TEORIYA POVERICHNOSTNOGO NATYAZHENIYA METALLOV)

b y A. E, G l a u b e r m a n T r a n s l a t e d b y E , R a b k f n OTTAWA 1950

(3)

Zhur. Fiz, Khim,, Vol, 23, No, 2,

1949,

po 115

THEORY OF THE SURFACE TENSION OF METALS by

A.

Eo Glauberman

Translated by Esther Rabkin

The basic assumption in the theories of the surface tension of metals, postulated by Gogate and Kothari

(I),

Breger and Zhukhovitskii(2) and Samoflovich

(31,

is the asser- tion that the kinetic energy of the metal electron fulfils the basic and determining function in the value of the surface tension, It is difficult to agree with this assertion, In fact, as has been proposed by Pa,

I,

Frenkel

(4)

as far back as 1917, the determining function belongs to the excess poten- tial energy of the surface particles, which arises because the latter do not have enough neighbors, while the function of the non-Coulombie forces may be reduced to the introduction of a negative correction to the surface tension, analogous to the correction introduced by the so-called repulsion forces in heteropolar c~ystals,

The purpose of the present work is to construct a theory for the surface tension of metals as a development of the elementary Frenkel theory on the basis of a statistical model of a metal, which will take into account the cpystallfne

structure and which will present a possibility of evaluating the contribution of the non-Coulombic forces to the value of the surface tension of a metal.

(4)

E l e c t r i c P o t e n t i a l of a Metal

We w i l l analyze a metal m d e l i n which. the posi- t i v e l y charged ions occupy t h e corners of the c r y s t a l l i n e l a t t i c e , and the r e s t of t h e s u r f a c e i s f i l l e d with an elec- t r o n llliquidig with a charge d e n s i t y

-

=

p"

(x,

y, z ) ,

Let us assume t h a t the metal occupies half of t h e space and we w i l l a l s o assume t h a t above the boundary plane, which i s

a plane subdividing t h e i n f i n i t e metal ( f o e , a metal occupy- ing t h e whole s ~ a c e ) i n t o two n e u t r a l s e c t i o n s , t h e r e i s an

e l e c t r o n "cloudfi c o n s i s t i n g of e l e c t r o n s which have overcome t h e p o t e n t i a l b a r r i e r a t t h e boundary,

I n order t o describe t h e electpon l i q u i d of a metal, we w i l l use the equation of Thomas

-

FermS

-

Birac i n t h e

form given by Jensen

(5)

where n i s t h e d e n s i t y of t h e e l e c t r o a l i q u i d , and

93

i s the p o t e n t i a l of t h e p o s i t i v e ions and e l e c t r o n s , The v a l u e p with an accuracy of the constant f a c t o r c o n s i s t s of t h e chemi- c a l p o t e n t i a l of t h e e l e c t r o n l i q u i d , and Z p and X A a r e constants, the magnitudes of which a r e given by the following

(5)

where a~ = h

,

e and m

-

are the charge and mass of

4

jr' 2me2

an electron,

In equation (1) the first left-hand term corres- ponds to the kinetic energy of electrons, and the second tepm appears as a result of introducing a correction to the poten- tial energy of electron interactions, which takes into account the effects of the electron inte~actions and their redistribu- tion, L e o it is the term corresponding to the so-called

exchange energy,

In the region external in relation to the half infi- nite lattice of the metal, the potential cp can be expressed by the solution of the Poisson equation

where p' is the density of the negative electric charge above the boundary surface, ioe.

Solving equation (1) with respect to n1/3 we obtain

where the selection of the plus sign before the radical en- sures positive values for the density of the electron liquid at any values of

9,

(6)

Introducing the notations

and b = - 3e

5%

9

we wi$l obtain from

(3)

an equation

or, introducing the function

r9'

E a2

+

b y

-

P,

we will

obtain

2

where cl = b ~ b e a 3 , c2 = l2Wbea

,

c3 = 12tlbea and cq = 4Vbe.

Retaining a physical picture of the metal in which the ions appear as discrete charges located in the corners of the erys-

talline lattice, we must analyze the functions

9'

as

a

function

of three variables x, y,

z. An

exact solution of equation

(5)

at such an analysis presents insurmountable difficulties. How-

ever, as an approximation which is sufficient for the purpose

of this paper, a solution can be obtained by linearizing the

equation

(5),

The right hand part of

(5)

may be transformed

in the following manner:

moreover the linearization constant A'' can be determined from

the condition

(7)

In this case equation

( 5 )

assumes th4 following form:

where A' = c3

+

2.

Equation (8) corresponds to a space which is

external in pelation to the metal, The solution of equation

(8)

does not present any dfffieulties9 and for tkE external region ( e ) O ) we obtain

- P a

a

where

p

-

and ?C.=

9

,

but

a:

=

7fR

7rm

b b 9

p =

-a

and = d where d is a half of the lattice constant of the metal,

For the region occupied by the metal (z L O ) , the equation for

7 '

rill assume the following Porn:

where

p+

designates the density of distribution of the posi- tively charged ions. If we will expresr

P+

by a triple

Fourier series, then for the potential in the region occupied by the metal we will obtain:

(8)

where

f a

p r

i s the expansion c o e f f i c i e n t of t h e d e n s i t y D+

i n the Fourier s e r i e s , Combining t h e obtained solutions fsr Spa and

qi

on t h e boundary, and assuming t h a t the 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 according t o a a r e continuous ( t h e coordinate system i s s e l e c t e d so t h a t t h e z axis i s perpendi- c u l a r t o t h e surface of t h e metal), we w i l l obtain fop the c o e f f i c i e n t s I a p y and

Dapr

t h e following expression:

n

T

0 -n i + o ~ ~ + / 1 ~ + ~ ~ odd even

The l i m i t i n g conditions f o r cp a r e such t h a t a t z p. + m

ya

;- 0

and a t z = - 0 0

Yi

=

C

krr

GPT

-

cosocx eos p7

hence, i n order t o f u l f i l these conditions, it is necessary t o

3d

assume t h a t

-

7

= O 9 i . e o y = a2

+

-,

o r f i n a l l y f a r ths

3=

chemical p o t e n t i a l V ( t h e chemical pokential d i f f e r s from

p

5 x ~

by a f a c t o r -)s

3 @

The n e u t r a l i t y conditions of t h e nucleus i n s i d e the metal are automatically f u l f i l l e d (

~ ' c f

-

4Tfp+)d

Z

:.0.

(9)

The surface tension of a metal with an

In order t o c a l c u l a t e the surface tensfon. 6 f o r

the assumed model of a metal we

w i l l

make use of the general

expression, formulated i n paragraph 1, f o e , we w i l l determine

t h e surface tension ( a t absolute zero) as an excess energy of t h e metal, whose o r i g i n i s due t o t h e presence of a boundary aurfaee, Therefore,

where U i s the space d e n s i t y of energy i n t h e presence of a surface, depending on t h e d i s t a n c e from t h e l a t t e r , and UO i s the d e n s i t y of t h e energy a t an i n f i n i t e d i s t a n c e f r ~ m the boundary, V' i s the volume of t h e region. occupied by t h e crys-

t a l l i n e l a t t i c e of the metal, V'\S the eolwne of t h e remaining

space ( h a l f space above t h e boundary s u r f a c e ) , and S i s the

area of t h e boundary surface, We w i l l s u b s t i t u t e t h e fntegr-a- t f o n l i m i t s of t h e second i n t e g r a l wf th r e s p e c t t o z (0,

Z

1,

where Z expresses t h e extension boundary of t h e e l e c t r o n cloud of t h e metal On a d i r e c t i o n along t h e

z

a x i s , with t h e himits

(0,

-1,

The e r r o r thus introduced i s insignificant and if

taken i n t o account would imppove the basis f a r q u a l i t a t i v e

eonclusfons from the c a l c u l a z i s n , I f t h e d e n s i t y of the energy i s expressed a s a sum of two components

(10)

where Ue is t h e d e n s i t y of t h e electrostatic: energy, UFg l a

t h e density of t h e k i n e t i c and exchange energies of t h e

e l e c t r o n s (including the so-called Wigner c o r r e l a t i o n energy

(611,

then i t i s possible t o c a l c u l a t e each component i n d i - v i d u a l l y o For

r e ,

i o s o f o r t h e s e c t i o n of t h e s u r f a c e t e n - s i o n corresponding t o t h e Coulombic f o r c e s , we obtain:

If t h e p o t e n t i a l i n s i d e t h e metal

c p i

i s expressed i n t h e fopm

where

cp?

i s t h e p o t e n t i a l a t an i n f i n i t e d i s t a n c e from t h e

boundary, and i s t h e a u x i l i a r y term brought about by

t h e presence of a surface, which decreases exponentially a s t h e d i s t a n c e from t h e l a t t e r . I n view of t h e r e l a t i o n s h i p

an analogous expression i n the form of a sum may a l s o be obtained f o r t h e d e n s i t y charge of e l e c t r o n s

P r

=

,QG+

APG

After c a l c u l a t i n g t h e - i n t e g r a l s and sunamOng up

0

t h e s e r i e s , obtained by s u b s t i t u t i n g f o r t h e functions

D

(11)

In order to calculate the portlion of the surface tension determined by the non-electrostatic forces, which

we have designated by A c , we will rewrite the term eorres-

ponding to the Wigner correlation energy in notations used in

the ppesent

HOT^.

In

the Wigner notations this correction

term has the following form:

v

where

rs

is determined from the relationship y(r,aHl3 =

w

,

and Ry is the ionization energy for %,

N

is the total mmber

of electrons, and V is the total volume, Expressing Ry by the

Ry =

,

and -9

00916

a~ a~

where m is the electron density, we will obtain in our

notations

50

la^

where

a

= this so-called correlation eo~rection

takes into account the mutual position of the electrons with antiparallel spins. Thus the complete expression for the

(12)

energy of the non-electrostatic forces is obtained in the form:

Expressing the electron density inside the metal in the form

I

n

=

n o - n

,

we obtain an expression for the excess non-electrostatic energy

Aulur9

(211

Considering that no

>

n'

,

we can simplify this by expanding into series according to the small order of

n o o

Aftep simpli- Ifcations and integration we obtain (if we retain in the series only the w i n terms corresp~nding to the indices of sumation

R = r n = n m 0 )

Thus, the total value for the surface tension 61- 6,

+

A 6

is

(13)

The equations (22) and (23) show t h a t t h e n o n - e l e c t r o s t a t i c f o r c e s introduce a negative term t o t h e value of the surface tension of a metal. This r e s u l t f u l l y agrees with t h e con-

cept regarding t h e pressure of an electpon gas i n a metal as

*

repulsion f o r c e s , ensuring t h e equilibrium of t h e l a t t i c e , I n a d d i t i o n , equation (23) i l l u s t r a t e s an e x a c t f u l f i l m e n t o f t h e v i r i a l theorem. I n f a c t , t h e v i r i a l theorem i n our case should b w r i t t e n i n t h e following form;

where AEF

-

i s t h e change i n the k i n e t i c energy of" e l e c t r o n s ,

Age

-

i s t h e change i n t h e e l e c t r o s t a t i c energy of t h e i n t e r - a c t i n g charges, and

AEA

-

i s t h e change i n t h e exchange energy

(including t h e Wigner c o r r e c t i o n ) . The last equation can be r e w r i t t e n thus:

1

A t g

+ x A )

=

- ?

(AE*

-

A E ~ ) ) ( 2 U

from which i t i s evident t h a t t h e change in t h e energy of the a o n - e l e c t r o s t a t i c f o r c e s i s somewhat l e s s than h a l f of t h e

change i n t h e e l e c t r o s t a t i c energy of t h e i n t e r a c t i n g particles

having

an

opposite sign, which corresponds t o t h e numerical

r e l a t i o n s h i p (23), A t small values of t h e l a t t i c e constant, t h e c o r r e c t i o n AEA i s small, which i s t r u e f o r t h e example

- - - -- - - - - - - - -- - - -- -- - - - -

*

This treatment of repulsion f o r c e s i n metals was formulated

(14)

calculated i n t h i s paper, s i n c e t h e numerical r e s u l t s a r e f o r copper, Therefore, a s u f f i c i e n t l y c l o s e approximation f o r the value of t h e surface t e n s i o n can be obtained by

dividing t h e value of 6, by 2, which would take i n t o account t h e fopces of n o n - e l e c t r o s t a t i c o r i g i n , f o e , t h e value of t h e surface tension would correspond only t o t h e Coulombic f o r c e s ,

Correction i n t h e volume o c c u ~ i e d by ions

and a comparisog with experiment

The c a l c u l a t i o n of t h e s l e c t r o s t a t i c portion of t h e surface tension can a l s o be mads by a method somewhat d i f f e - r e n t from t h e one used above, For t h i s purpose we w i l l analyze an i n f i n i t e metal according t o t h e Frqnkel model, f o e o , we w i l l describe i t a s a l a t t i c e c o s s i s t i n g of p o s i t i v e ions immersed

i n a uniformly d i s t r i b u t e d , negatively charged e l e c t ~ o n l f q u i d , Then by imagining a plane which subdivides t h e f a f i n i t e metal 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 g and s e l e c t i n g t h i s plane a s t h e coordinate plane z = 0 , it would be easy t o f i n d a value f o r tpe e1ektf);fcal p o t e n t i a l created by one half' l a t t l c e i n t h e region occupied by t h e o t h e r h a l f l a t t i c e , For t h i s i t

i s necessary t o f i n d a s o l u t i o n f o r t h e Laplace equation

a2rp=

0 , (251

f o r t h e region, external i n r e l a t i o n t o t h e s e l e c t e d half Pat- t i c e , and a s o l u t i o n f o r t h e Poisson equation

(15)

-

13

-

for the region occupied by the selected half lattice, More- over,

p

=

p+

-

P - ,

whqre

p-

= a constant

,

and is the

density of the electron charges, and

P+

is the density of the positive charges ,o+ = e&(r

-

rk)? if % is the coordf- nate of the corresponding ion. The continuity conditions of

the potential and its first derivative along

z

assuming

p*

as a triple Fourier series permit us to find a solution for the potential in the form of serfes which have good converg- ence, For the determination of the electrostatic portion of the surface tension, the interaction energy of the two half lattices of the metal U12 is calculated, and then the surface tension, or more accurately its Coulombfe portion C,, is

equal to

where S is the surface of the subdivision boundary, The cal- culation for a face-centered lattice gives a formula which coincides with formula

(18).

Taking into account the forces of non-elect~ostatic origin and using the vfrial relationship in agreement with the conclusions of the previous paragraph,

we obtain for the total value of the surface tension

analogous results are easily obtained for a cubic-centered lattice,

*see the determination of the surface tension for solid bodies of heteropolar erystals~ Born and Goeppert-Mayer, Theory of a Solid Body, page 326.

(16)

Before carrying out numerical comparisons w i t h

experimental data, it is necessary to introduce a co~reetion, aecomting for the fact that valency electpons are gengrally distributed in the space outside the actual volume of the metal ions, The additional terms in the expression far the

total space energy, which, as was shown by Gombas (7) in

1935,

leads to an energetic unsuitabtlity of this process, is con- nected with the penetration of the electron liquid of the metal inside of an ion,

If, for simplification, we assume that the volume occupied by ions, impermeable to the valency electrons (such a simplification does not distort much the actual picture), then we can describe the metal by means of a model in which the ion is expressed as a sphere having a definite padius Ri, in the center of which is found the exact charge ze, equal to the charge of the ion, The space outside of this sphe~e, we will assume as being filled with an electron liquid having a constant charge density p P 0

In

an electr~static sense, this picture for the points, occupied by ions, located outside %he sphere, is equivalent to the picture in which a somewhat large^

positive charge z e t is located in the center of the sphere, and the

psst

of the space including the sections formerly

occupied by spheres of radii Ri eo~responding to the dimensions of ions, is filled with an electron liquid having a densi ty of a negative charge ,uQo If

V

is the volume of the

(17)

elementary nucleus, and

v Q

i s t h e volume of a l l the ions belonging t o the elementary nucleus, then

and f o r t h e Itreduced1# charge e n we o b t a i n

f o r a face-centered l a t t i c e . For the p o t e n t i a l of an i n f i n i t e h a l f - l a t t i c e i n an e x t e r n a l space t h e following expression i s o b t a i ned :

Thus, by introducing a c o r r e c t i o n f o r the volume of ions, w e o b t a i n f o r the surface t e n s i o n t h e following formulae:

(18)

f o r a face-centered l a t t i c e and

f o r a cubic-centered l a t t i c e ,

A s i s evident from t h e formulae (32) and ( 3 3 ) , t h e e s s e n t i a l function i s f u l f i l l e d by t h e number of free e l e c t r o n s per atom i n a given metal, The number of f r e e e l e c t r o n s per atom, which must be taken i n t o account does. not coinci de w i t h t h e number of e l e c t r o n s outside t h e closed s h e l l s of t h e atom, For metals with a face-centered s t r u c t u r e , such a s Cu, Ag, Au, P t and so on, t h e number of f r e e e l e c t r o n s exceeds t h e number of e l e c t r o n s o u t s i d e t h e closed s h e l l s of t h e atom, The same can be s a i d not only of t h e above mentioned metals, but a l s o of Mg, a l l t h e a l k a l i n e - e a r t h metals and many others, As t h e number of f r e e e l e c t r o n s p e r atom i n a metal, i t 1s q u i t e s a f e t o t a k e t h e average value of t h e chemical valency of t h e atom," Taking i n t o account t h i s f a c t , i t i s p o s s i b l e t o c a l c u l a t e the surface t e n s i o n f o r a number of metals by considering t h e

f o r c e s of non-electrostatic o r i g i n , The r e s u l t s a r e compiled i n a t a b l e , i n which t h e c a l c u l a t e d values f o r the. surface %

See Bethe and Somrnerfeld, HElectron Theory of Metalste, United S c i e n t i f i c and Technical p u b l i s h e r s , 1938,

(19)

tension 6 r e l a t e t g t h e surface of a cube, f , e, c =

c

(1,60),

Table 1**

**

I n t h e calnulatfons of 6 we have neglected t h e small cor- r e c t i o n , connected with t h e deformat,ion of a c r y s t a l l i n e l a t t i c e near t h e surface. This c o r r e c t i o n i s smaller by an order of magqitude than t h e value of t h e surface ten- s i o n d-,

he

experimental value of t h e surface t e n s i o n f o r IC i s taken from t h e work by A, Samoilovich according t o t h e d a t a by Prof. Semenchenko.

The discrepancy i n t h e t h e o r e t i c a l value of

c

with

t h e experimental f o r P t may be explained by t h e f a c t t h a t it i s probable t h a t i n t h i s case 2 - 2 , i n s t e a d of a = 2 a s

assumed f o r t h e c a l c u l a t i o n s . It should a l s o be assumed t h a t

2 ) 2 f o r t h e metals Ag and Au, since Ag and Au a r e seldom

monovalent and a r e more o f t e n d i - and t r i - v a l e n t , Good agree- ment between t h e t h e o r e t i c a l and experimental values of t h e surface tension f o r metals with a face-centered s t r u c t w e supports t h e f a c t t h a t t h e Frenkel metal model, taken a s t h e basis of t h e present work, i s f a i r l y c l o s e t o r e a l i t y f o r t h e

(20)

ease of compact s t r u c t u r e s . These s t r u c t u r e s appear t o be a d i r e c t conclusion of the Frenkel metal model, From t h i s -view- point metals w i t h a compact s t r u c t u r e may be considered a s being more " t y p i c a l l y metallictd than the a l k a l i n e metals. Worse agreement f o r a l k a l i n e metals may f i n d an explanation

i n t h e same causes a s t h e i r non-compact cubic-centered s t r u c - t w e , one of which may be t h e nonuniform d i s t r i b u t i o n of the e l e c t r o n l i q u i d i n the l a t t i c e , We must a l s o point out the f a c t t h a t i n t h e case of a l k a l i n e metals, characterized by a small number of f r e e e l e c t r o n s (one e l e c t r o n per atom), t h e re-evaluation of t h e energy of t h e mutual repulsion of elee- t r o n s , i n which t h e e l e c t r o n r e t a i n s f o r i t s e l f a portion of the repulsion energy, i s of much g r e a t e r importance than i n t h e o t h e r eases,

Conclus i ons

A theory f o r the surface tension of metals has been

constructed on t h e basis of the idea proposed by Frenkel i n

t h e f i r s t wopk on t h i s problem, Within the frame of t h i s theory i t i s possible t o explain the c r y s t a l l i n e s t r u c t u r e of a metal and the function of t h e non-Coulombic f o r e e s , It has been shown t h a t the value of t h e surface tension of metals i s determined by t h e excess p o t e n t i a l energy of t h e surface parc- t i c l e s , which agrees f u l l y with the treatment of t h e problem

(21)

it fsPYows that the funetion of the nan-sl,e-,~ros%atfe forces may be reduced to the introduction of a txega;-,b,rre correctfon, whi ch is in agreement with the virial &itdeorem,

A

eompa~f son sf the calculated values cf the surface tension for various metals with the existing experimental da$a permits us to speak of good agreement between theory and experiment (keeping in mind that the experimental data ~ e ~ d % e to n s e d metals, and that the theory is constructed for absolute zero, we must speak of agreement in the order of values),

In

csnelusion the author wi shes to express his sineeq.e appreciation to Ya,

I,

F r e e e l for his guidance sip.

this work,

Sen% to the editor May 25,

1948,

Kalinin Leningrad Polyteehnical Institute Lgvov State Unive~sity,

Bibliography F

D o

V o Gogate and B, S o Kothari, P h d l , Mag.,

2,

20, 1935,

Breger and Zhukhovitski i 9 ZhpaPo F i z , . U ~ f m , 209

b-5,

1946,

I,

Frenke1, Phi$, Mag,, April

1917,

(22)

DISCUSSION ON TIIE PAPER BY A,

E,

GLAUBEFMAB ENTITLED I0THE THEORY OF THE SURFACE TENSION OF METALStf

by

Ao G o S m o f l o v f ~ h

I n his paper (1) A. E o Glauberman affirms t h a t t h e t h e o r i e s of t h e surface t e n s i o n of metals postulated by

~ o g a t e and Kothari, Breger and Zhulchovitskif, and Samoilovieh i n which without s u b s t a n t i a t f ~ n they a s s i g n t h e determining function

t o

t h e k i n e t i c energy On the value of t h e surface tension of metals, and t h e r e f o r e these a r e f n e o r r e c t , S'upport f o r t h i s he f i n d s i r r h i s erroneous caleulatfoaao

Glauberman in v a i n comec$s t h e s e t h r e e completely d i f f e r e n t works, A s has f i r s t been noted i n t h e work by

Samoilovich (21, the theory of Gogate and Kotbari i s based on a rough mistake i n c a l c u l a t i o n s ( i n c o r r e c t d i f f e r e n t f a t i o n ) and should have been discontinued a s a ~ e f erenee Bong ago,

As regards t h e work of Glauberman, he r e p e a t s an

e r r o r of s e v e r a l o l d e r papers, a c r % t i c i s m of which was given e a r l i e r (21, This e r r o r c o n s i s t s

i n

ignoring t h e equilibrium conditions of a c a p i l l a r y l a y e r , and which, i f not accounted f o r 9 make it impossible t o c a l c u l a t e a c c u r a t e l y t h e surfaee tension, s i n c e i t i s a thermodynamical value determined f o r an equilibrium system, It' 'has .been shown (21, t h a t i n prin- c i p l e i t i s impossible t o o b t a i n

9:

p o s i t i v e value f o r t h e

(23)

surface tension without taking i n t o account t h e anisotropic

i

portion of the pressure of an e l e c t r o n gas, And i f GPauber- man has succeeded i n accomplishing t h i s , then i t must f o l l o w t h a t h i s c a l c u l a t i o n s contain a numerical e r r o r ,

This conclusion may a l s o be drawn fpom the follow-

ing : Glaube rman found t h e portion i n t mdueed

the k i n e t i c energy t o t h e surface tensfon i s negative and

one half of t h e portion introduced by t h e e l e e t ~ o s t a t i e energy, and i n t h i s he s e e s an exact f u l f i l l m e n t of t h e v i r i a l theorem, It i s e x a c t l y the v i r i a l theorem which makes it impossible t o dpaw such a r e s u l t , The f a c t ' i s t h a t i n c a l c u l a t i n g t h e elec- t r o s t a t i c portion of the s u r f a c e tension, Glauberman e a l e u l a t e s t h e t o t a l e l e c t r o s t a t i c energy of the metal, while only the e l e c t r o n p o t e n t i a l energy e n t e r s i n t o the v i r i a P theorem,

Glauberman has i n c o r r e c t l y w r i t t e n down formula (21)

f o r t h e excess n o n e l e c t r o s t a t i c energy,

Thus we b e l l eve t h a t t h e paper of Glauberman d i d not disprove anything contained i n t h e t h e o r i e s of Breger and

Zhukhovitskii, and Samoilovieh and has not introduced arnythicg

new i n t o t h i s problem.

Sent t o t h e Editor A p r i l 19, 1949,

Lf t e r a t w g

---

--

l o A, E, Glauberman, Zhur, F i z , KhOm.

,

2Jp

11s9

1969,

2, Ao G o Samoilovich, Zhur, Exper, and Theor, Fizo, l6,

135,

(24)

GLAUBERMAN S REPLY TO THE ARTICLE BY A. G , SAMOILQVLGH

It appears t o u s t h a t t h e objections of A. G o

Samoilovich t o some assumptions i n our paper (1) seem t o be without foundatiou.

The t h e o r i e s of ~ o ~ a t e and Kothari, Breger and Zhukovitskii, and Samoilovich, i n s p i t e of e s s e n t i a l d i f f e r - ences i n t h e models of the rne t a l and t h e methods of caleula- t i o n , s u f f e r from a common i n c o r r e c t treatment of t h e problem regarding the surface tension of metals. The i n c o r r e c t con- cept regarding the determining function of t h e k i n e t i c energy of e l e c t r o n s i n the value of s u r f a c e tension of metals, which .

i s c b a ~ a c t e r f s t i c of the above mentioned t h e o r i e s , requires from a bas;ic viewpoint t h a t they be placed i n one category, independent of t h e d e t a i l s and e r r o r s i n c a l e u l a t i o n ,

The objections of Samoilovich a r e based on

an

appeal t o h i s i n c o r r e c t work (21, The d i seussion on t h e v i r i a l

theorem cfted a t t h e end of t h e Samoilovich paper i s not c o r r e c t , Contrary t o t h e opinion of Samoflovieh, t h e poten- t i a l energy which e n t e r s i n t o t h e v i r i a l c o r r e l a t i o n must express t h e t o t a l p o t e n t f a l energy of t h e metal and not t h e

p o t e n t i a l energy of t h e e l e c t r o n s only, As has been shown (l),

t h e surface tension of metals i s determined mainly by t h e excess p o t e n t f a l energy of a n s t a l associated with t h e pres- ence of a boundary surface, The k i n e t i c f o r c e s introduce a

(25)

negative contribution t o the value of t h e surface tension of a metal, which i s completely understandable from a physical viewpoint, since t h e k i n e t i c f o r c e s i n metals c o n s t i t u t e f o r c e s of ~ e p u l s i o n . The fact. t h a t a prinef- p a l l y i n c o r r e c t treatment of t h e problem on t h e nature of

t h e surface tension of metals continues t o be defeqded

makes i t evident t h a t it i s necessary t o publish a d e t a i l e d c r f t i e i s m of t h e SamoflovOch theory.

Sent t o t h e Editor June

7,

1949,

Literature

1, A. E. Glauberman,

mur.

Fizo Khim,,

a,

115,

1949,

2. A, G , Samoflovich, Z h u ~ , Exper, and Them, FfzO9

a,

135,

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