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Journal o f Low TemperaturePhysics VoL 21 Nos 3 /4 1975

A V ir ial Coeff ic ient A na lys i s o f Hel ium A dsorpt ion

Isotherms

Anthony D Nov aco

La f aye t t e C o llege E as t on P ennsy l van i d f

a n d

S t e v e n s I n s t it u t e o f T e c h n ol o gy H o b o k e n N e w J e r s e y

( R e c e i v e d D e c e m b e r 3 1 , 19 7 4)

A mo del in which the atom s in the second layer o f a mon olayer-plus hel ium

f i lm are assumed to for m a quas i - two-d imensional gas i s used to analyze the

isotherm d ata o f Goellner, D aunt , and Lern er fo r helium on Grafoil . T he

param eters in the model are de termined by a l eas t -squares f i t o f t he model

isotherm to the data. These parameters include the second and third vir ial

coefficients o f the quasi - two-d imens ional gas and the binding energy o f the

secon d-layer a toms to the f i rs t layer. Th e empirical values are com pared to

recent the oret ical predict ions fo r the two-d imens ional hel ium gas. Th e agree-

m en t an d discrepancies be twe en theo retica l an d empirical values ar e discussed.

Th e conclusion i s that the seco nd layer at low densi ties) can be described as an

imperfect tw o-dim ensiona l gas w i th perh aps som e modif ication o f the inter-

atom ic pote nt ial due to sub strate ef fects .

1 . I N T R O D U C T I O N

The v i r i a l expans i on i s a u se fu l t e chn i que fo r s t udy i ng bo t h bu l k and

ad so r be d gases , t-3 ,:~ In t h i s app roa ch , t he r e l evan t t he r m od yn am i c fun c t i ons

a re e xpa nde d i n a po w er se r ie s in p , t he dens i t y o f t he gas . The fund am en t a l

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

pa r t i t i on func t i on , t he expans i on be i ng in pow ers o f 23p t h ree d i m ens i ons )

o r 22p t w o d i m ens i ons ) , w he re 2 = [2n h2/m kT] 1/2 is t he t he rm a l w av e l eng t h

and p i s e i ther the bu lk or a rea l dens i ty .

The v i r i a l expans i on w as u sed i n a r ecen t l y pub l i shed t heo re t i ca l

* W o r k s u p p o r t e d , i n p a r t , b y a g r a n t f ro m t h e N a t i o n a l S c i e nc e F o u n d a t i o n .

t P e r m a n e n t a d d re s s.

:[:S ee R e f . 3 fo r a g e n e r a l d i s c u s s i o n o f t h e t h e r m o d y n a m i c s o f a n d v i r i a l e x p a n s i o n f o r a d s o r b e d

g a s e s .

3 5 9

9 1975 Plenum P ubl i sh ing Corpora t ion , 227 Wes t 17 th St ree t, New Y ork , N .Y. I0011 . No par t o f th is pub l ica t ion

ma y be reproduced, s tored in a retr ieval system, or t ransm itted, in any form or by an y means, electronic, mechanical ,

photo copy ing, microf ilming, recording, or otherwise, with out wri t ten permission of the publisher.



3 6 0 A n t h o n y D .

o v a c o

investigation of the two-dimensional imperfect gas. 4 This study considered

the first-order correction to the thermodynamics of the ideal Boltzmann gas,

that is, it included the second virial coefficient. Both the effects of realistic

two-body interactions and Bose/Fermi statistics were included. The results

of that calculation agree quite well with the specific heat data for the low

density helium monolayer (first layer of helium adsorbed upon Grafoil)fl*

The conclusion of that study was that the low density helium monolayer

could be treated as an imperfect two-dimensional gas. Comparison between

theory and e -~eriment was made over density and temperature ranges of

0.01 < p < 0.03 atom/A 2 and 0.5 < T < 10K.

The virial expansion, however, has not been as useful in helium ad-

sorption studies as it might be. This is, in part, due to the technical difficulty

of making pressure measurements (particularly precise measurements) at

the most interesting temperatures and pressures for studying the two-

dimensional imperfect gas. For instance, the use of the virial expansion for

(helium) isotherm analysis has been mostly restricted to relatively high

temperatures, with as much emphasis on the effects of adatom-substrate

interactions as on the effects of adatom-adatom interactions.6,v Use of the

virial expansion to investigate primarily the effects of adatom-adatom

interactions should involve low temperatures and low densities, so that

adatom-substrate effects do not dominate the physics and obscure adatom-

adatom effects. At these temperatures and densities, the virial coefficients

should be quite sensitive to the adatom-adatom interaction. Furthermore,

if the temperature were low enough, it might be possible to observe the two-

dimensional liquid-gas transition. 4 Such a study must involve both an

experimental technique capable of precise pressure measurements (free of the

usual thermomolecular corrections) and must involve a smooth homogen-

eous substrate for which

qs

the isosteric heat, is not very large (if % is large,

then the pressures will be too low).

In a recent experimental study of the adsorption of the helium isotopes

on Grafoil, precise pressure measurements were made using an i n s i t u capaci-

tance gauge to measure the pressure. 8 Although it was not possible to measure

pressures in the imperfect gas region for the first layer (empirical estimates

place this pressure at 10-1~ Torr),9 it was possible to measure the

pressure for adsorbed volumes V slightly above Vm, the adsorbed volume

corresponding to a completed monolayer. Furthermore, the precision of

these measurements is about 0.005 Torr. This raises the possibility of de-

termining the virial coefficients appropriate to a quasi-two-dimensional

imperfect gas of helium atoms adsorbed on a substrate of helium-plated

Grafoil. It is the purpose of this communication to show that such an analysis

is practical and to report the numerical values for the second and third virial

S e e R ef. 5 f o r a s u m m a r y o f re c e n t w o r k o n h e l i u m m o n o l a y e r s ,



A Vir ia l Coe f f ic ien t na ly s i s o f H el iu m dsorpt ion I so therms 361

c o ef fi ci en ts w h i c h r e s u lt f r o m t h e a n a ly s is o f t h e G o e l l n e r - D a u n t - L e r n e r

d a t a . F r o m t h e se v i ri al c o e ff ic ie n ts , t h e a p p r o p r i a t e v a n d e r W a a l s c o n s t a n t s

a r e c a l c u l a t e d .

T h e b a si c a g r e e m e n t b e t w e e n t h e o r y a n d e x p e r i m e n t is g o o d , a l t h o u g h

s o m e d i s c r e p a n c i e s d o e x i s t . I t i s f o u n d t h a t t h e d a t a a n a l y s i s w i l l p r o d u c e

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

( h e l iu m ) t o th e s u b s t r a t e ( h e l i u m - p l a t e d G r a f o i l) , a n d t h e s e c o n d a n d t h i r d

v i r i a l c oe f f i c i e n t s f o r t he i m pe r f e c t ga s.

2 . I S O T H E R M A N A L Y S I S A N D T H E V I R IA L E X P A N S I O N

T h e a n a ly s is o f t h e a d s o r p t i o n i s o t h e r m v ia a v i ri al e x p a n s i o n ( f o r t h e

2 D g a s) c a n b e f o u n d in R o s s a n d O l i v i e r s t r e a ti se o n a d s o r p t i o n . ~~ W e

o u t l i n e t h e t r e a t m e n t h e r e f o r t h e s a k e o f c o m p l e t e n e s s a n d t o e m p h a s i z e t h e

s a l i e n t po i n t s .

T h e a d s o r b e d g a s is t r e a t e d a s a t w o - d i m e n s i o n a l i m p e r f e c t g a s w h i c h is

i n e q u i l i b r i u m w i th a n i d e a l b u lk v a p o r . T h e i n t e r a c t i o n o f t h e a d a t o m w i t h

t h e s u b s t r a t e is d e s c r ib e d b y a s i ng le n u m b e r , t h e b i n d i n g e n e r g y t o t h e

s u b s tr a te . T h i s m e a n s t h a t t h e t e m p e r a t u r e m u s t b e l ow e n o u g h s o t h a t t h e

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

s u r f ac e . 11 T h e i n t e r a c t i o n b e t w e e n t h e a d a t o m s a n d t h e i r s t a ti s ti c s a r e

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

o b t a i n e d b y r e l a ti n g t h e p r e s s u r e o f th e b u l k g a s to t h e d e n s i ty o f th e a d s o r b e d

a t o m s t h r o u g h t h e u n i q u e n e s s o f t h e ch e m i c a l p o t e n t i a l/ ~ . T h e p r e s s u r e o f

t h e b u l k g a s is

P = k T 2 - 3 e ~ /k r

(1)

I f t h e a d s o r b e d g a s w e r e a n i d e a l B o l t z m a n n g a s, th e n t h e c h e m i c a l p o t e n t i a l

w o u l d b e

t~ = e k T

In

p ) 2 )

(2)

w h e r e - e is t h e b i n d i n g e n e r g y ( p e r a t o m ) o f t h e a d a t o m t o t h e s u b s t r a t e

a n d p is t h e a d a t o m n u m b e r d e n s i ty . F o r a B o s e o r F e r m i s y s te m , th is is t h e

l e a d i n g te r m a t l o w d e n si ty . F o r t h e i m p e r f e c t g a s a n d / o r f o r B o se o r F e r m i

s t a t i s t i c s , a c o r r e c t i o n t o t h e a b o v e e x p r e s s i o n c a n b e w r i t t e n a s a p o w e r

s e r ie s i n t h e d e n s i t y p :

/~ = e + k T [ l n

p ) 2 ) + 2 B p ~ C p 2 ~ D p 3

+ . . . ] (3)

i

T o r e d u c e t h is t o t h e u s u a l v i ri a l e x p a n s i o n f o r t h e p r e s s u r e , it is o n l y n e c e s s a r y

t o u s e

~ r = p ~u ep

4 )



362 Anthony D. Novaco

w h e r e 4~ is t h e s p r e a d i n g ( o r t w o - d i m e n s i o n a l ) p r e s s u r e . T h u s

~ ) / k T = p + B p 2 + C O 3 + D p '* + . . .

(5)

w i t h B a s t he s e c o nd v i r i a l c oe f f i c ie n t , C a s t he t h i r d , e tc . T h e s e c oe f f i c ie n t s

a r e , i n g e n e r a l, d e p e n d e n t u p o n t e m p e r a t u r e . T h e r e s u l t in g i s o t h e r m is

P = [ p k T 2 - 1 e x p ( ~ / k T ) ] e x p ( 2 B p + 3 C p 2 + - }D p 3 + . . . ) (6)

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

w h i c h t h e a b o v e t r e a t m e n t is re l a te d t o t h e v a n d e r W a a l s e q u a t i o n o f s ta te ,

a n d h o w t h es e c o n s i d e r a t i o n s a f f ec t t h e i s o st er ic h e at . T h e v a n d e r W a a l s

e q u a t i o n o f st a te i s

[tip + a p 2]

[1 -

b p ] = p k T

(7)

w h e r e t h e c o n s t a n t s a a n d b a r e in d e p e n d e n t o f t e m p e r a t u r e . R e a r r a n g i n g

t h e a b o v e a s a p o w e r s e r ie s in p , w e o b t a i n

d p / k T = p + [ b - ( a / k T ) ] p 2 + b a p 3 + b 3 p 4 + 9 (8)

f r o m w h i c h t h e v i r i a l c o e f f ic i e n ts c a n b e r e a d .

T h e i s o s t er i c is d e f i n e d b y

_ O I n P w h e re fl 1

= t?fl o = ~ (9)

T h e r e f o r e , c o m b i n i n g (1 0) a n d (6 ), w e o b t a i n

q st = - e + - } k T - 2 p ( ~ B / O f l ) - ~ p 2 ( O C / ~ f l) + . . (10)

I f t h e g a s is d e s c r i b e d b y t h e v a n d e r W a a l s e q u a t i o n o f s ta t e, t h e n OC/Of l = 0

a n d

O B / ~ f l = - - a ,

s o t h a t

% = - ~ + ~ k T + 2 a p

(11)

3 . S E C O N D - L A Y E R M O D E L A N D T H E D A T A A N A L Y S I S

T h e a p p l i c a t i o n o f t h e i s o t h e r m e q u a t i o n in (1) t o t h e e x p e r i m e n t a l

d a t a f o r V > V ,, is b a s e d u p o n t h e a s s u m p t i o n o f a n i n e r t f ir s t l a y e r a n d t h e

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

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

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

s e c o n d l a y e r i s

p = [ (V -

V , , ) / A V o ] N o

(12)



A V i r ia l C o e f f i c i e n t A n a l y s i s

o f H e l i u m d s o r p t io n I s ot h e r m s

3 6 3

w h e r e V i s t h e v o l u m e o f t h e a d s o r b e d g a s a t S T P , Vm is t h e s a m e f o r a " c o m -

p l e t e d " m o n o l a y e r , V is t h e v o l u m e o f o n e m o l e a t S T P , N o is A v o g a d r o ' s

c o n s t a n t , a n d A is t h e a r e a o f t h e s u r f ac e .

E q u a t i o n ( 12 ) c o m b i n e d w i t h E q . (6 ) r e q u i r e s t h a t f iv e c o n s t a n t s b e

d e t e r m i n e d . O n e o f t h e s e c o n s t a n t s , n a m e l y t h e a r e a , se r v e s t o s c a le t h e

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

r e m a i n i n g f o u r c o n s t a n t s , n a m e l y Vm, e, B , a n d C , a r e d e t e r m i n e d b y a l e a s t

s q u a r e s f it o f E q . (6 ) t o t h e d a t a o f G o e l l n e r

e t a t 8

T h e v a l u e s f o r t h e s e c o n -

s t a n t s a r e l i s te d i n T a b l e I , a n d a r e b a s e d o n t h e 4 .2 4 K i s o t h e r m f o r e a c h

i s o t o p e .

T h e a r e a w a s c a l c u l a t e d b y u s i n g t h e f it te d v a l u e o f V,, a n d t h e e x p e r i -

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

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

p , , f o r t he f i r s t l a ye r i s 0 . 115 / ~ - 2 f o r 4H e . S T h i s pa r t i c u l a r va l ue i s a l s o t ha t

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

F o r 4 H e , V,~ is f o u n d t o b e 6 .4 3 c m 3 p e r g r a m o f G r a f o i l , i m p l y i n g a n a r e a

o f 1 5 .0 m 2 p e r g r a m o f G r a f o i l . T h i s is t h e a r e a f o u n d b y G o e l l n e r

e t a l

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

c h e c k e d b y u s i n g t h e a b o v e v a l u e o f t h e a r e a a n d t h e f i tt e d v a l u e o f Vm

f o r 3 H e ( 6. 02 2 c m 3 p e r g r a m o f G r a f o i l ) to c a l c u l a t e t h e d e n s i t y o f t h e c o m -

p l e t ed 3 H e m o n o l a y e r . T h e r e s u lt o f t h is c a l c u l a t i o n is P m = 0 . 10 8 ~ - z 0

a r e s u lt w h i c h a g r e e s b o t h w i t h t h e o r y a n d w i t h o t h e r e x p e r i m e n t s , s 't s

E q u a t i o n (6 ) w a s f i tt e d to t h e d a t a o f G o e l l n e r

e t a L

u s i n g a s t a n d a r d

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

a n d t h e i r s t a n d a r d d e v i a t i o n s w e r e c a l c u l a t e d a l o n g w i t h t h e p r e d i c t e d

d e v i a t i o n o f e a c h d a t u m f r o m t h e f i tt ed c u rv e . T h e s t a n d a r d d e v i a t io n s a r e

l is t e d in T a b l e I . T a b I e I I li st s t h e t a b u l a t e d d a t a 8 a n d t h e a p p r o p r i a t e v a l u e s

o f t h e f it te d c u r v e. T h e p r e d i c t e d d e v i a t i o n s w e r e u s ed t o d e t e r m i n e t h e d a t a

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

S i nc e t h e e x p e r i m e n t a l r e s u lt s e x t e n d b e y o n d t h e r e g i o n o f v a li d it y

o f the i s o t h e r m [ E q . (6 )], i t is ne c e s s a r y t o s pe c i f y r e a s o na b l e c r i t e r i a f o r t he

i n c l u s i o n o r e x c l u s i o n o f d a t a i n t h e l e as t s q u a r e s f it. T h e s e c r i t e ri a a r e b a s e d

T A B L E I

B e s t F i t V a l u e s f o r V ,~ p e r G r a m G r a f o i l ) , e,

B

a n d C

B a s e d u p o n t h e D a t a i n R e f . 8 )

4 H e 3 H e

Vm, cm 3 ST P) 6 .4 3 3 _+ 0 .0 0 4 6 .0 2 3 • 0 .0 0 3

a/k

K - 3 0 . 0 • 0 .1 - 2 5 . 4 • 0 .1

B ~2

- 0 . 3 3 • 0 . 2 2 7 . 0 • 0 . 6

C , ~4 5 3 • 1 2 3 7 • 2 8



364

A n t h o n y D . N o v a c o

T A B L E I I

F i t t e d I s o t h e r m v s . D a t a o f R e f . 8

~ H e ~ H e

A d s o r b e d P r e s s u r e P r e s s u r e A d s o r b e d P r e s s u r e P r e s s u re

v o l u m e f l (fitted), (actual), volume, (f it ted~, (actual),

c m 3 S T P T o r r T o r r c m 3 S T P T o r r T o r r

6.681 0.3887 0.388

6.784 0.4947 0.488

6.816 0.6028 0.597

6.883 0.7098 0.703

6.950 0.8176 0.816

7.017 0.9262 0.928

7.157 1.155 1.159

7.287 1.373 1.380

7.417 1.595 1.600

7.547 1.821 1.821

7.675 2.050 2.054

7.804 2.286 2.278

6.265 0.889 0.902

6.321 1.111 1.109

6.376 1.334 1.323

6.429 1.556 1.545

6.533 2.009 2.000

6.633 2.469 2.477

6.732 2.948 2.964

6.833 3.463 3.460

V a l ue s f o r a d s o r b e d v o l u m e a r e p e r g r a m o f G r a f o i l.

u p o n t h e s ta t is t ic a l a n a ly s i s o f t h e d a ta . T h e m a x i m u m n u m b e r o f p o i n t s

w a s i n c l u d e d i n t h e a n a l y s i s s u b j e c t to t w o c o n s t r a i n t s . F i r s t , t h e q u a l i t y

o f th e f it w a s n o t s i g n i fi c a n t ly a l te r e d b y t h e i n c l u s i o n o f a d a t u m . S e c o n d , a

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

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

t ri al fi ts in o r d e r t o d e t e r m i n e t h e s e n s i t i v i ty o f t h e i s o t h e r m p a r a m e t e r s t o

t h e d a t a r a n g e u s e d . I n c r e a s i n g t h e d a t a r a n g e b y a b o u t 2 5 ~ h a d , a t m o s t , a

2 0 ~ e ff e ct u p o n a n y p a r a m e t er . H o w e v e r , s u c h a n i n c re a s e h a d a b o u t a n

o r d e r - o f - m a g n i t u d e e ff ec t u p o n s t a n d a rd d e v i a t i o n s a n d t h e g e n e r al q u a l it y

o f th e f it . T h e f i n a l v a l u e s f o r t h e p a r a m e t e r s w e r e d e t e r m i n e d f r o m 1 2 p o i n t s

4 H e ) a n d 8 p o i n t s 3 H e ) w i th th e ty p i c a l d e v i a t i o n s b e t w e e n d a t a a n d t h e

f i tt e d c u r v e b e i n g 0 . 0 0 5 T o r r 4 H e ) a n d 0 .0 0 8 T o r r 3 H e ) . A l l p o i n t s w i t h i n a

g i v e n r a n g e w e r e i n c l u d e d . H o w e v e r , d a t a c o r r e s p o n d i n g t o s e c o n d - l a y e r

d e n s i t i e s l e s s t h a n 0 .0 3 pr o o r g r e a t e r t h a n 0 .3 p ro w e r e e x c l u d e d .

T h e e x c l u s i o n o f h i g h d e n s i t y d a t a p o i n t s c a n b e j u s ti fi e d b y e s t im a t e s o f

t h e e f fe c t s o f D , t h e f o u r t h v i r ia l c o e f fi c ie n t . T h e e x c l u s i o n o f p o i n t s n e a r t h e

m o n o l a y e r c o m p l e t i o n p o i n t i m p l ie s t h a t th e a s s u m p t i o n o f t h e in e rt f ir st

l a ye r m a y n o t b e v a l id f o r ve r y l o w s e c o n d - l a y e r d e n s it ie s . B o t h o f t h e se p o i n t s

a r e e x a m i n e d i n t h e l a s t s e c t i o n .

T h e a s s u m p t i o n t h a t t h e ef f e ct iv e a r ea f o r t h e s e c o n d l a y e r is t h e s a m e

a s t h e a r e a o f t h e fi rs t l a y e r c o u l d b e a s o u r c e o f a s y s t e m a t i c e r r o r i n th e

v a l u e s o f t h e c o e f f ic i e n t s . T h i s u n c e r t a i n t y d o e s p l a c e a li m i t u p o n t h e r e li a -

b i l it y o f t h e v a l u e s f o r t h e v ir i a l c o e f f i c ie n t s w h i c h w e e s t im a t e t o b e a b o u t



A V i r i a l C o e f f ic i e n t A n a l y s i s o f H e l i u m A d s o r p t io n I s o t h er m s

6 5

1 0 - 1 5 ~o. T h e e f fe c t u p o n e is m u c h s m a l l er . I f s e v e r a l i s o t h e r m s ( a t d i f f e r e n t

t e m p e r a t u r e s ) a r e a n a l y z e d , t h e n t h e a r e a f o r th e s e c o n d l a y e r c a n b e d e t e r-

m i n e d d i r e c t l y .

4 . C O M P A R I S O N O F T H E R E S U L T S W I T H T H E O R Y

S i n c e o n l y o n e i s o t h e r m w a s a n a l y z e d f o r e a c h i s o t o p e , it is no t. p o s s i b l e

t o d o a d e t a i le d c o m p a r i s o n b e t w e e n l :h e o r y a n d t h e le a s t s q u a r e s a n a l y s is .

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

c o e f f ic i e n ts o n t h e b a s is o f p r e d i c t e d v a n d e r W a a l s c o n s t a n t s , a t l e a s t f o r

4 H e . T h i s i s a r e a s o n a b l e c o m p a r i s o n s i n c e t h e t h e o r e t i c a l B T ) b e h a v e s i n

t h e m a n n e r p r e d i c t e d b y t h e v a n d e r W a a l s f o r m :

B T ) = b - a / k T . 4

I t i s

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

t h e o r e t i c a l p r e d i c ti o n s a n d o t h e r e x p e r im e n t s .

A q u i c k c h e c k o f th e t h e o r e t i c a P v a l u e o f t h e 4 H e B a t T = 4 .2 4 K

( - 4 . 3 •2 ) s h o w s t h a t t h e e m p i r i c a l v a l u e - 0 . 3 3 A 2) is a f a c t o r o f t e n s m a l le r .

F u r t h e r m o r e , t r ia l f its o f t h e d a t a u s i n g t h e t h e o r e t i c a l v a l u e o f B w e r e n o t

s a t i s fa c t o r y . T h e t y p i c a l d e v i a t i o n f o r s u c h a f i t is 0 .1 T o r r i n s t e a d o f t h e

0 .0 0 5 T o r r f o u n d f o r th e b e s t fit. T h e t h e o r e t i c a l v a l u e s fo r th e v a n d e r W a a l s

c o n s t a n t s a a n d b a r e a / k = 5 3 / ~ Z K a n d b = 8 A 2. T h e r e s u l t in g B o y l e

t e m p e r a t u r e [ t e m p e r a t u r e w h e r e B ( T ) = 0 ] is 6 .6 K . T h e e m p i r i c a l v a lu e s o f

a a n d b a r e a / k = 3 2 A 2 K a n d b = 7 .3 A 2. T h e c o r r e s p o n d i n g B o y l e te m -

p e r a t u r e is 4 .2 K . T h e o r y a n d a n a l y s i s d i ff e r, t h e n , o n t h e v a l u e o f a b u t n o t

on t he va l ue o f b, s i nc e the d i f f e r e n c e i n t he va l ue o f b is le ss t ha n t he s t a n da r d

d e v i a t i o n f o r t h e e m p i r i c a l v a l u e . T h e d i f f e re n c e i n t h e v a l u e f o r a is sig -

n i fi ca n t, b u t t h i s t e r m c o u l d b e s e n s il iv e to t h e b e h a v i o r o f t h e a d a t o m -

a d a t o m p o t e n t ia l in th e r e g io n o f t h e m i n i m u m a n d t h e p o te n t ia l c o u l d b e

s e n si ti v e t o s u b s t r a t e m e d i a t e d i n t e r a c t io n s 12 a n d / o r t h e a v e r a g i n g o f t h e

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

s u rf a ce .1 3 A m a j o r c o n s e q u e n c e o f t h e s m a l l e r v a l u e o f a is a l o w e r i n g o f t h e

e s t i m a t e d c r it ic a l t e m p e r a t u r e f r o m f i e t h e o r e t i c a l r a n g e 4 o f 1 . 4- 2 .0 K t o

t h e r a n g e o f 0 . 9 - 1 .2 K . T h i s k i n d o f e f fe c t i n th e s e c o n d l a y e r w a s s e e n b y

B r e tz , ~4 w h o m e a s u r e d t h e h e a t c a p a ci l y o f t h e s e c o n d l ay e r . T h e m a x i m u m

i n t h e h e a t c a p a c i t y f o r th i s l a y e r occu ~rred a t a l o w e r t e m p e r a t u r e t h a n t h e

c o r r e s p o n d i n g m a x i m u m in t h e f i rs t -l a y e r h e a t c a p a c it y . S in c e th e s u b s t r a te

p o t e n t i a l t e n d s t o l o c a l iz e t h e fi rs t l a y e r (i n t h e d i r e c t i o n n o r m a l t o t h e s u r f a c e )

m o r e t h a n t h e s e c o n d l a y e r , i t is t h e f ir s t l a y e r w h i c h s h o u l d b e m o r e l ik e

t h e th e o r e t i c a l t w o - d i m e n s i o n a l c a l c u la t io n . * F u r t h e r m o r e , t h e s e c o n d l a y e r

r id e s o n a l a y e r o f h e l iu m , a n d t h is s u b s t r a t e m i g h t c a u s e i m p o r t a n t

m o d i f ic a t io n s i n th e a d a t o m - a d a t o m i n t e ra c t io n i n t he s e c o n d l ay e r.

*C om pare the substrate potential in Ref. 11 (Fig. 2) to that in Ref. 15 (Fig. 3).



366 Anthony D o v a e o

The binding energy (-z/k) of a 4He atom in the second layer to the

helium-plated Grafoil substrate is 30 K. This number agrees with the

empirical value of Elgin and Goodstein9 (29 K) and with the theoretical

value (30 K).I 5

The comparison between theory and analysis is not as rewarding for

3He as for 4He. This is due, in large part, to the large standard deviation

found for the third virial coefficient. The empirical values for the second and

third virial coefficients (at 4.24 K) are 7.0/~2 and 37 A 4, respectively. Varia-

tions of 50% in the value of C produced only small (< 10%) variations in

the value of B. This indicates that the value of B is reliable even if the value

of C is not. The empirical value of B is only 10 % below the theoretical value

(8.1 A2). The empirical value of

e / k

is 25.4 K, which is good agreement

with the theoretical value, x5

5. IMPLICATIONS AND CONCLUSIONS

Estimates of the effects of higher virial coefficients can be made by usiiag

the van der Waals description. Examination of Eq. (8) shows that D = b 3.

The value of b is 7.3 A 2 for 4He. If p is equal to 0.025 A 2 (this is at the high

end of the density range), then the effect of D upon the pressure is less than

0.02 Tort. This is to be compared to a typical deviation of 0.005 Torr,

indicating that the effect of the fourth virial coefficient is just noticeable at

high densities. Thus it is reasonable to exclude data at higher densities in a

fit which excludes the fourth and higher virial coefficients.

The contribution of the third virial coefficient to the pressure is im-

portant throughout most of the density range considered. Yet the calculation

of Siddon and Schick (which excludes this term) agrees very well with the

specific heat data in this very range. An explanation of this paradox

is suggested by the van der Waals description, for which C = b 2 is indepen-

dent of temperature. A temperature-independent C does not contribute

to the specific heat, just as it does not contribute to the isosteric heat [see

Eqs. (10) and (11)]. The contributions to % due to the nonideal behavior

can be calculated from Eq. (11). The value of

elk

is -30 K_3T/2 is about

6 K and

pa/k

< 1 K. Thus the corrections to qst due to the virial coefficients

is about 3 %.

The necessity of excluding points near the monolayer completion

point could be an indication that the first layer is not inert near this point.

In fact, near monolayer completion, it is reasonable to assume that both

the density of the first and the density of the second layer are affected by

addi tion of atoms to the film. The analysis then implies that after the density

of atoms in the second layer is greater than about 0.03pro, the density of the

first layer is unchanged by further addit ion of atoms to the film and only the



A V irial

Coefficient nalysis of Hel ium dsorption

isotherms 367

d e n s i t y o f t h e s e c o n d l a y e r c h a n g e s . F u r t h e r m o r e , t he o v e r a ll q u a l i t y o f t h e

f it i n d i c a t e s t h a t s u b s t r a t e i n h o m o g e n e i t y e ff ec ts a r e n o t i m p o r t a n t f o r t h e

s e c o n d l a y e r ( a t l e a s t a f t e r p > 0 . 0 3 p , , ) ,

T h e a n a l y s is s h o w s t h a t t h e l o w d e n s i t y s e c o n d l a y e r c a n t r u l y b e

t h o u g h t o f a s a q u a s i - t w o - d i m e n s i o n a l i m p e r fe c t g as o n a s m o o t h s u b s t ra t e .

T h e e m p i r i c a l v a l u e s f o r th e v i ri a t c o e f f ic i e n t s a n d b i n d i n g e n e r g i e s a r e i n

g o o d a g r e e m e n t w i t h t h e t h e o re t i c a l v a l u es , a l t h o u g h t h e r e is a g o o d i n d i c a -

t i o n t h a t t h e i n t e r a t o m i c p o t e n t i a l f o r a t o m s in th is l a y e r is m o d i f i e d b y t he

s u b s tr a t e . E m p i r i c a l v a n d e r W a a l s c o n s t a n t s w e r e f o u n d f o r * H e , a n d t h e se

v a l u e s i n d i c a t e t h a t a l i q u i d g a s c r i ti c a l p o i n t m a y e x i st a t a b o u t 1 K .

F u r t h e r e x p e r im e n t a l w o r k n e e d s t o b e d o n e f o r b o t h 3 H e a n d 4 H e ,

a l t h o u g h e x p e r i m e n t s v e r y c lo s e to t h e (* H e ) c r it ic a l p o i n t m i g h t b e i m -

p o s s i b l e s i n c e t h e p r e s s u r e w i lt d e c r e a s e a s e x p F . ( - 3 0 K ) ( 1 / T - 1 /4 K ) ] .

T h i s m e a n s t h a t a t d e n s i ti e s n e a r t h e c r it ic a l d e n s i t y ( a b o u t 0 .0 3 A - 2 ) a n d

e v e n a t 2 K , t h e p r e s s u r e is a b o u t 1 0 - a T o r r . N e v e r t h e l e s s , i t is s ti ll i m p o r t a n t

t h a t m o r e w o r k b e c a rr i e d o u t s o t h a t t he e q u a t i o n o f s ta te b e d e t e r m i n e d

( o v e r a s l a r g e a r e g i o n a s i s p r a c t i c a l ) . S u c h a n i n v e s t i g a t i o n w o u l d b e v e r y

u s ef u l to t h e s t u d y o f l o w d e n s i t y m o n o l a y e r s .

C K N O W L E D G M E N T S

T h e a u t h o r w o u l d l ik e to th a n k P r o f. J. G . D a u n t a n d D r . G . G o e l l n e r

f o r a c c e ss to t h e ir d a t a p r i o r t o p u b l i c a t i o n a n d f o r m a n y s t i m u l a t i n g d is -

c u s s i o n s w h i l e t h e a u t h o r w a s a g u e s t a t t h e C r y o g e n i c s C e n t e r a t S t e v e n s

I n s t i tu t e o f T e c h n o l o g y d u r i n g t h e s u m m e r o f i 97 4 .

R E F E R E N C E S

~. B. Kahn and G. E. Uhlenbeck, Physica 5, 399 (I938).

2. W . A. Steele and M . Ro ss,

J. Chem. Phys.

35, 850 (1961).

3. W . A. Steele, The Interaction of Gases with Solid Sur/aces (Pergam on, N ew York~ i974).

4. R. L. Siddon and M. Schick, Phys. Rev. A 9, 90 7, t7 53 (1974).

5. J. G. D aun t and E. Lerner, eds., Monolayer and Submonolayer He lium Films (Plenum, New

York, 1973); also see M. Bretz, J. G. Dash, D. C . Hickernel, and E. W . Vitches, Phys. Rev. A

8, 1589 (1973).

6. W . A. Steele and M. Ro ss, J. Chem. Phys. 35, 862 (1961), see also E~ J. De rderian, Thesis,

Pennsylvania State U niversity (1974), unpublished

7. B. K. Oh and S. K. K im,

J. Chem. Phys.

61, 1797, 1808 (1974).

8. G. J. Go ellner, J. G. D aunt, and E. Lerne r, 9 .

Low Temp. Phys

21, 347 (1975), this issue,

preceding paper.

9. R. L. Elgin and D. L. Goo dstein, in Monolayer andSubm ono yer Helium Films, J. G. Dau nt

and E. Lerner, eds. (Plenum, New York, 1973).

10. S. Ross and J. P. Olivier,

On Physica l Adsorption

(Interscience, New Y ork, 1964).

il . A. D. No vaco and F. J. Milford, Phys . Rev .A 5, 783 (1972).

12. M . Schick and C . E. Cam pbell,

Phys. Rev.

A 2, 1591 (1970).



3 6 8 A n t h o n y D N o v a c o

13 . A . D . N ovaco ,

Phys. Rev. A

8 , 3065 1973) ; s ee a l s o A . D . N ova co an d C . E ; C am pbe l l ,

Phys. Rev. B

11, 2525 1975).

14. M. Bretz, in

Monolayer and Submonolayer He lium Films

J . G . D aun t and E . Le r ne r , eds .

P l enum , N ew Y or k , 1973) .

15 . C . E: Campbel l ,

et al. Phy s. Rev.

A 6, 1648 1972) .

journal of low temperature physics volume 21 issue 3-4 1975 [doi 10.1007%2fbf01141332] anthony d....

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