EXPERIMENTS IN 3He-4He LIQUID MIXTURES NEAR THE TRICRITICAL POINT

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EXPERIMENTS IN 3He-4He LIQUID MIXTURES NEAR THE TRICRITICAL POINT

J. del Cueto, R. Johnson, T. Rohde, F. Wirth, E. Graf

To cite this version:

J. del Cueto, R. Johnson, T. Rohde, F. Wirth, E. Graf. EXPERIMENTS IN 3He-4He LIQUID

MIXTURES NEAR THE TRICRITICAL POINT. Journal de Physique Colloques, 1980, 41 (C7),

pp.C7-133-C7-136. �10.1051/jphyscol:1980721�. �jpa-00220158�

JOURNAL DE PHYSIQUE CoZloque C7, suppl6ment au n o 7, Tome 41, juiZZet 1980, page C 7 - 1 3 3

EXPERIMENTS I N 3 ~ e - 4 ~ e L I Q U I D MIXTURES NEAR THE T R I C R I T I C A L P O I N T J. d e l Cueto, R.L. Johnson, T. ~ o h d e * , F.H. ~ i r t h * * and E.H. Graf

Department of Physics, SUNY at Stony Brook, Stony Brook, New York 11794, USA.

R6sumG.- Nous p r 6 s e n t o n s des mesures r e c e n t e s du diagramme de phases des mglanges l i q u i d e s 3 ~ e - 4 He prGs du p o i n t t r i c r i t i q u e 3 p r e s s i o n s 6lev6es. Nous e x p l i q u o n s nos r e s u l t a t s s u i v a n t un mod2le c l a s s i q u e avec c o r r e c t i o n s quantiques. La p o s i t i o n du p o i n t t r i c r i t i q u e e s t t r 6 s s e n s i b l e

a

ces c o r r e c t i o n s e t d e v r a i t donc s e r v i r de bon i n d i c a t e u r des e f f e t s de p o l a r i s a t i o n dans l e s melanges.

A b s t r a c t . - We p r e s e n t some r e c e n t measurements o f t h e phase diagram o f l i q u i d 3 ~ e - 4 He m i x t u r e s n e a r t h e t r i c r i t i c a l p o i n t a t e l e v a t e d p r e s s u r e s . We e x p l a i n o u r r e s u l t s by means o f a c l a s s i c a l model w i t h quantum c o r r e c t i o n s . The l o c a t i o n o f t h e t r i c r i t i c a l p o i n t i s v e r y s e n s i t i v e t o t h e s i z e o f t h e s e c o r - r e c t i o n s and s h o u l d t h e r e f o r e s e r v e as a good i n d i c a t o r o f p o l a r i z a t i o n e f f e c t s i n t h e m i x t u r e s . 1 . I n t r o d u c t i o n . - The t r i c r i t i c a l p o i n t i s an unusu-

a l thermodynamic phenomenon, found i n h e l i u m m i x - t u r e s and i n c e r t a i n m a g n e t i c systems / I / . The pos- s i b i l i t y o f such a p o i n t , which i s a c r i t i c a l p o i n t o f a l i n e o f h i g h e r - o r d e r phase t r a n s i t i o n s , seems t o have o c c u r r e d first t o Landau /2/, and was s t u d - i e d i n terms o f h e l i u m m i x t u r e s by Cohen and van Leeuwen /3/, b e f o r e b e i n g e x p e r i m e n t a l l y observed /4/ i n 1967. I n t h i s communication we p r e s e n t mea- surements o f t h e phase diagram o f h e l i u m m i x t u r e s a t e l e v a t e d p r e s s u r e s and d i s c u s s t h e i r s i g n i f i - cance t o h e l i u m m i x t u r e s i n which t h e 3 ~ e compo- n e n t i s s p i n p o l a r i z e d .

2. Apparatus.- Measurements were p e r f o r m e d i n sam- p l e c e l l s o f v a r y i n g geometry, under c o n d i t i o n s o f c o n s t a n t p r e s s u r e o r c o n s t a n t volume. The concen- t r a t i o n s o f t h e m i x t u r e s were d e t e r m i n e d b y measur- i n g t h e i r d i e l e c t r i c c o n s t a n t w i t h a c a p a c i t o r t h a t was p a r t o f a 1 2 MHz l o w - t e m p e r a t u r e t u n n e l - d i o d e o s c i l l a t o r /4/. Frequency s t a b i l i t y on t h e o r d e r o f one p a r t i n 107 r e s u l t e d i n c o n c e n t r a t i o n s e n s i t i - v i t y o f a b o u t m o l e - f r a c t i o n 3 ~ e .

Our m o s t r e c e n t sample chamber i s shown i n F i g . ( 1 ) . I t was equipped w i t h a m a g n e t i c a l l y d r i - ven metal p l u n g e r which made i t p o s s i b l e t o a d j u s t t h e p o s i t i o n o f t h e i n t e r f a c e i n phase-separated samples by a b o u t 1 cm. W i t h t h i s t e c h n i q u e , t h e c o n c e n t r a t i o n o f b o t h phases c o u l d be observed, as c o u l d t h e v e r t i c a l c o n c e n t r a t i o n p r o f i l e due t o g r a v i t y . The c a p i l l a r y f i l l - l i n e was equipped w i t h a

*

Deceased

**

P r e s e n t address: Department o f P h y s i c s and A s t r o - nomy, U n i v e r s i t y o f Massachusetts, Amherst, Massa- c h u s e t t s 01003 USA.

low-temperature v a l v e t o i n s u r e t h e c o n c e n t r a t i o n i n t e g r . i t y o f t h e sample. Temperature was measured b y germanium r e s i s t a n c e thermometry c a l i b r a t e d ag- a i n s t He vapor p r e s s u r e . 3

STAINLESS STEEL- DIAPHRAGM

THERMOMETER

F i g u r e 1

.-

Sample Chamber

3. R e s u l t s

.-

Our e x p e r i m e n t a l r e s u l t s a r e shown i n F i g s . ( 2 ) and ( 3 ) . T r i c r i t i c a l b e h a v i o r o f t h e m i x - t u r e s o c c u r s a t a l l observed p r e s s u r e s between t h e s a t u r a t e d vapor p r e s s u r e and 22 atm, i n d i c a t i n g t h a t t h e p h a s e - s e p a r a t i o n i s d r i v e n p r i m a r i l y by t h e sup- e r f l u i d o r d e r i n g t r a n s i t i o n r a t h e r t h a n b y d i f f e r - ences i n t h e e f f e c t i v e m o l e c u l a r i n t e r a c t i o n s o f t h e components, w h i c h i s t h e usual cause o f phase-sepa- r a t i o n i n l i q u i d m i x t u r e s . As t h e p r e s s u r e P i s i n - creased f r o m i t s s a t u r a t e d - v a p o r v a l u e , t h e t r i c r i -

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980721

JOURNAL DE PHYSIQUE

t i c a l p o i n t moves t o higher He c o n c e n t r a t i o n and 3 l o w e r temperature, reaching a maximum c o n c e n t r a t i o n a t P=9.9 atm. A t h i g h e r pressures, t h e t r i c r i t i c a l c o n c e n t r a t i o n X t diminishes. I t s behavior para1 1 e l s the pressure dependence o f t h e low-temperature 1 im- i t o f s o l u b i l i t y /5/. A connection between these two regions o f t h e phase diagram appears t h e o r e t i - c a l l y i n t h e hard-sphere model o f Cohen and van Leeuwen /3,6/ and i s discussed i n the c o n t e x t o f h i g h a p p l i e d magnetic f i e l d s by M i l l e r / 7 / .

4. T h e o r e t i c a l Model

.-

We f i n d t h a t we can r o u g h l y describe o u r r e s u l t s i n terms o f an elementary c l a s s i c a l model w i t h quantum c o r r e c t i o n s . We assume we have N4 4 ~ e atoms mixed w i t h N3 3 ~ e atoms and t h a t the 4 ~ e component i s s u b j e c t t o an o r d e r i n g t r a n s i t i o n . Following Blume, Emery and G r i f f i t h s (BEG) /8/, we approximate ( o r l'mimic') t h e s u p e r f l - u i d o r d e r i n g by a f i c t i t i o u s magnetic o r d e r i n g i n which the He atoms a r e assumed t o have "up" and 4

"down" s p i n s t a t e s and an o r d e r parameter o r "mag- n e t i z a t i o n " p r o p o r t i o n a l t o (N1-N2)/V, where N1 and N2 a r e t h e number o f He atoms w i t h up and down 4 s p i n s r e s p e c t i v e l y and V i s t h e sample volume. Obvi- ousl y N1 +N2=N4.

I f we were d e a l i n g w i t h an i d e a l c l a s s i c a l mix- t u r e o f t h r e e kinds o f p a r t i c l e s o f numbers N1, N2 and N3, the Gibbs f r e e energy G would be

where

fiO

and p4,, a r e t h e chemical p o t e n t i a l s o f the pure components, T i s the temperature and k i s Bol tzmann's constant. Xi=Ni/(N3+N4). The l a s t t h r e e terms i n Eq. (1 ) a r e due t o t h e c l a s s i c a l e n t r o p y o f mixing.

We now add another term t o t h e f r e e energy t o d e s c r i b e t h e magnetic o r d e r i n g :

G~~~ = -J(N -N ) ²

1 2

2v (2)

where J i s a ferromagnetic "exchange energy".

D e f i n i n g g=G/(N3+N4) and approximating t h e volume o f t h e l i q u i d by

v

⁼ V4(P) [N4+a(P)N3) (3)

where V4(P) i s the volume taken up b y a He atom 4 i n t h e l i q u i d a t pressure P and a(P) i s a f a c t o r

ATM. P TEMP

LAMBDA SURFACE

''"A

-TRICRITICAL LINE

. 3.4 AT".

10 8 .' ' . 'b.9 ATM.

12 .. '., '.,

14 ... .. >

3 4

Figure 2.- O v e r a l l Phase Diagram o f -He- He M i x t u r e s

MOLE % ~e~

F i g u r e 3.- D e t a i l o f t h e T r i c r i t i c a l P o i n t ' s Pres- sure Dependence.

(-4/3 a t vapor pressure) t o describe t h e ( l a r g e r ) volume taken up by a He atom, we have 3

D e f i n i n g = (XI-X2)/2 and expanding the m i x i n g terms i n powers o f

r]

near the t r a n s i t i o n (by sym- metry X1 = (1-X3)/2 + Tl and X2 = (1 -X3)/2

-

^{Ti )}

,

we o b t a i n

where go i n c l u d e s a l l terms independent o f Q . This expansion i s o f Landau form 191, w i t h a t r a n - s i ti o n temperature given by

where T ~ ~ ( P ) i s the t r a n s i t i o n temperature a t pres- sure

P

w i t h X3=0. This i s t h e equation f o r the lambda l i n e , which i s l i n e a r f o r t h e s i m p l e s t case a=l

.

The f r e e energy o f Eq. ( 4 ) becomes u n s t a b l e w i t h r e s p e c t t o phase-separation under c e r t a i n con- d i t i o n s o f c o n c e n t r a t i o n and temperature; i.e., t h e

3 4

chemical p o t e n t i a l s o f the He and He components p3 and p4, take on p a r t i c u l a r values f o r more than one value o f X3. The m i x t u r e s t r a t i f i e s i n t o two c o e x i s t i n g phases, w i t h upper and lower concen- t r a t i o n s X3, and X31 which s a t i s f y the e q u i l i b r i u m c o n d i t i o n s

from which t h e phase diagram can be c a l c u l a t e d . For the case a=l, we o b t a i n a phase diagram t h a t i s

2-

e n t i c a l w i t h t h a t o f the BEG model solved i n t h e mean-field approximation.

We can improve o u r model by m o d i f y i n g the form o f G4ex i n Eq. ( 2 ) . The c o r r e c t i o n t o the c l a s s i c a l expression f o r the f r e e energy o f p a r t i c l e s begin- n i n g t o f e e l t h e e f f e c t s o f quantum s t a t i s t i c s i s /1

o/

where g i s the Lande f a c t o r and m* i s the e f f e c t i v e mass o f the p a r t i c l e s . The (+I s i g n i s f o r fermions and the (-1 s i g n f o r bosons. This suggests t h a t Eq.

( 2 ) should be o f t h e form

t o b e t t e r approximate t h e e f f e c t s o f Bose degene- racy. Using Eq. ( 9 ) i n p l a c e o f Eq. ( 2 ) i n Eq. ( 5 ) r e s u l t s i n an expression f o r the f r e e energy t h a t

i s s t i l l o f Landau form, b u t which now has a lamb- da l i n e given b y ( f o r a = l )

which agrees w i t h experiment b e t t e r than does-Eq.

( 6 ) 1111.

S i m i l a r c o n s i d e r a t i o n s f o r t h e He p a r t i c l e s 3 l e a d t o a c o n t r i b u t i o n t o the f r e e energy o f the form

where TFo i s t h e Fenni temperature o f pure 3 ~ e l i q u - i d a t T=O.

Using Eqs. (9) and (11 ) w i t h t h e vapor-pressure values o f a and TFo 1121, we f i n d the phase diagram shown i n Fig. ( 4 ) . The experimental vapor-pressure phase diagram i s shown f o r comparison.

- - -- ^- EXP .

I /

/ \

/ I

I

I t I 1 I I I I

0

0 0.2 0.4 0.6 0 . 8 1

MOLE

- ^FRACTION

^{3 ~ e}

Figure 4.- T h e o r e t i c a l Phase Diagram

5. Discussion.- Our t h e o r e t i c a l r e s u l t s a r e most s a t i s f a c t o r y f o r T t T t . A t l o w e r temperatures, Eq.

(11) ceases t o be a good approximation (even a t Tt i t i s probably good o n l y t o w i t h i n 20-30%). I t i s because we do n o t take the low-temperature Fermi s t a t i s t i c s p r o p e r l y i n t o account t h a t we g e t a zero l i m i t o f s o l u b i l i t y o f He i n 4 ~ e . 3

The pressure dependence o f t h e t r i c r i t i c a l p o i n t i s q u i t e w e l l described by t h i s model, how- ever. Experimental l y, as P increases, t h e parameters a and TFo b o t h decrease I 1 31. I n our model

,

a de- crease i n a has t h e e f f e c t o f i n c r e a s i n g t h e t r i c r i -

JOURNAL DE P H Y S I Q U E

t i c a l c o n c e n t r a t i o n Xt. Decreasing TFo has t h e o p p o s i t e e f f e c t . A t low pressures, a changes r a p i d l y and dominates t h e e f f e c t s on Xt. When P210 atm, however, a changes much more s l o w l y and we move i n t o a regime where v a r i a t i o n s i n TFo dominate. The pressure dependence o f TFo, u n l i k e t h a t o f a, does n o t d i m i n i s h a t h i g h e r pressures. The n e t r e s u l t o f t h i s i s t o cause the t r i c r i t i c a l p o i n t t o reverse i t s motion a t higher pressures.

I n t h i s d i s c u s s i o n we have emphasized t h e ef- f e c t s o f pressure v a r i a t i o n s o f TFo on t h e l o c a - t i o n o f Xt. !4e have, f o r present purposes, neglected any pressure e f f e c t s on t h e n a t u r e o f the lambda t r a n s i t i o n which may a l s o s h i f t Xt.

Takagi /14/ has considered a l a t t i c e model f o r helium m i x t u r e s i n which j u s t such e f f e c t s appear.

We s h a l l consider t h i s q u e s t i o n as w e l l as t h e question o f u s i n g a more r e a l i s t i c expression than Eq. ( 3 ) f o r the molar volumes /15/ o f the m i x t u r e s i n a l a t e r communication.

5. Conclusion.- The l o c a t i o n o f t h e t r i c r i t i c a l p o i n t i s v e r y s e n s i t i v e t o t h e magnitude o f TFo;

chanqes of-10% i n TFo r e s u l t i n changes i n Xt o f -19, which a r e v e r y e a s i l y observable. We t h e r e f o r e

expect t h e phase diagram o f l i q u i d helium m i x t u r e s t o be a good i n d i c a t o r o f p o l a r i z a t i o n e f f e c t s i n the 3 ~ e component.

7. Acknowledgements.- We wish t o thank M. Blume, E.

6 . D. Cohen, V. J. Emery, R. B. G r i f f i t h s , J. M.

K i n c a i d and G. S t e l l f o r h e l p f u l discussions a t var-

i o u s stages o f t h i s work. We thank t h e O f f i c e o f Naval Research f o r making gaseous helium a v a i l a b l e t o our l i q u e f a c t i o n f a c i l i t y and t h e National S c i - ence Foundation and t h e New York S t a t e Research Foundation f o r t h e i r support o f t h i s research.

8. References.

-

/1/ For a review, see J. M. K i n c a i d and E. G. D.

Cohen, Physics Reports

22

62, 57 (1975).

/2/ L. Landau, Phys. Z f t . der Sowjet Union

11,

26 (1 937).

/3/ E. G . D. Cohen and J. M. J. van Leeuwen, Physi- ca

26,

1171 (1960).

/4/ E. H. Graf, D. M. Lee and J. D. Reppy, Phys.

Rev. L e t t . - 19, 417 (1967).

/5/ G. E. Watson, J. D. Reppy and R. C. Richardson, Phys. Rev. 188, 384 (1969).

/6/ E. G . D. Cohen and J. .!t J. van Leeuwen, Phys.

Rev. Q, 385 (1968).

/7/ M. D. M i l l e r , these proceedings.

/8/ M. Blume, V. J. Emery and R. B. G r i f f i t h s , Phys.

Rev. - A4, 1071 (1971).

/9/ See, f o r example, Landau and L i f s h i t z , S t a t i s t i - c a l Physics (Pergamon Press 1958), p . 436.

-

--

/ l o / s.,

p . 159.

/11/ See J. Wilks,

The

P r o o e r t i e s o f L i o u i d and So-

l i d Helium (Oxford, 1967), p . 7 2

- -

--

/12/

E.,

pp. 672 and 471.

/13/

E.,

pp. 673 and 471.

/14/ S. Takagi, Prog. Theor. Phys.

9,

22 (1972).

/15/ t!. Kierstead, Jour. Low Temp. Phys.

a,

497 (1 976).