Cross-over of exponent values near the critical double point

Physwa 103A (1980) 316-324 © North-Holland Pubhshmg Co

C R O S S - O V E R OF E X P O N E N T VALUES N E A R THE CRITICAL DOUBLE POINT H M J BOOTS Phthps Research Laboratorws, Emdhoven, The Netherlands

and A C MICHELS Van der Waals Laboratormm, Unwersttett van Amsterdam, Amsterdam, The Netherlands

Received 24 March 1980 Mixtures displaying gas-gas equdtbrm of the second kind are discussed on the basis of the geometry of the coexistence surface If one approaches the crlhcal double point isothermally the values of the critical exponents a,/3, y and v are twice their "usual" values, whereas ~ and rl are "usual" In a regmn above the cnhcal-double-pomt temperature a cross-over effect occurs for Isothermal approach of the cnhcal hne close to crltlcahty the exponents are "usual" while further away from the crlhcal state a,/3, y and v are doubled A simple expression for the cross-over pressure ~s dertved Consequences for the scahng relatmns are inveshgated

I. Introduction I n m i x t u r e s d i s p l a y i n g g a s - g a s e q u i l i b r i a o f t h e s e c o n d k i n d ~-4) t h e c o e x i s t e n c e c u r v e h a s a s h a p e w h i c h is p a r t l y s h o w n in fig 1. I n e a c h p l a n e o f c o n s t a n t p r e s s u r e m this figure t h e r e is a c o e x i s t e n c e c u r v e f o r s e p a r a t i o n o f t w o p h a s e s o f d i f f e r e n t c o n c e n t r a t i o n a n d t h e r e is a c r i t i c a l p o i n t at w h i c h t h e s e p h a s e s b e c o m e i n d i s t i n g u i s h a b l e . T h e c r i t i c a l h n e is g i v e n b y t h e p r e s s u r e d e p e n d e n c e o f t h e c r i t i c a l t e m p e r a t u r e To(p) a n d t h e c r i t i c a l c o n c e n t r a t i o n xc(p). O n e o f t h e i n t e r e s t i n g f e a t u r e s o f g a s - g a s e q u i l i b r i a o f t h e s e c o n d k i n d is t h e t e m p e r a t u r e m i n i m u m o f t h e c r i t i c a l h n e , w h i c h b y d e f i n i h o n is t h e c r i t i c a l d o u b l e p o i n t ( C D P ) w i t h c o o r d i n a t e s (PD, Tc(pD) =-- TD, xc(PD) = XD). T h e e x i s t e n c e o f C D P s w a s p r e d i c t e d b y V a n d e r W a a l s in 1894~), but, d u e to t h e high v a l u e s o f t h e C D P p r e s s u r e , Pd, a v e r i f i c a t i o n o f this p r e d i c t i o n w a s n o t o b t a i n e d until 19402). S i n c e t h e n m a n y s y s t e m s w i t h C D P s h a v e b e e n found. T h e w i d e s p r e a d i n t e r e s t in c r i t i c a l p h e n o m e n a in g e n e r a l h a s s t i m u l a t e d w o r k o n t h e d e t e r m i n a t i o n o f t h e c r i h c a l e x p o n e n t s in t h e n e i g h b o u r h o o d o f t h e C D P . T h e e x p e r i m e n t a l s i t u a t i o n to d a t e is a s f o l l o w s . T r a p p e n i e r s , D e e r e n b e r g a n d S c h o u t e n 3) h a v e d e t e r m i n e d t h e b e h a v i o u r o f t h e e x p o n e n t / 3 316

EXPONENTVALUESNEARTHECRITICALDOUBLEPOINT 1267

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317

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Ne-Xe 1039

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867-

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Fig 1 A part of the coexistence s u r f a c e in the nelghbourhood of the critical double point of the s y s t e m N e - X e T a k e n f r o m D e e r e n b e r g ' s 3) thesis with the a u t h o r ' s permission T h e variables are the p r e s s u r e p, the t e m p e r a t u r e t and the molar concentration x T h e thick d a s h e d h n e is the critical curve, its t e m p e r a t u r e m i n i m u m is the C D P

very carefully This exponent is related to the shape of the coexistence surface according to I x ' - x " I ~ IATI~p

at constant p,

Ix'--X"I--IAplaT

at constant T,

(1.1)

where x' and x" are the concentrations of the coexisting phases and A T =-- T - T o ( p ) , A p =- p - pc(T),

(I 2)

are the temperature and pressure distances to the critical points at constant p and T, respectively (see fig. 2). Trappeniers et al. 3) draw the following conclusions from the experimental data" (i) /3p =/3o for all p where/3o is the "usual" value of/3 pertaining to simple liquids, binary hquid mixtures, etc.

318

H M J BOOTS AND A C MICHELS

curve

cr•hcol D)

1 ) Pl )

~T

Fig 2 Projechon of the cnhcal curve on the plane x = XD The pressure distance ,:lp~ at constant temperature T~ and the temperature distance AT~ at constant p~ are indicated for an arbitrary point (Pl, TI) with T~ > TD 0i) At t e m p e r a t u r e s m u c h higher than T o / 3 r =/3o, w h e r e a s at t e m p e r a t u r e s shghtly higher than To /3T = /30 f o r

IApl ~ Iap¢o(T)l

and /3T = 2/30 for

IApl ~ IApco(T)l

with a s m o o t h c r o s s - o v e r at the p r e s s u r e difference Apco(T). (lii) T h e value o f this c r o s s - o v e r p r e s s u r e difference is not c o m m e n t e d u p o n in detail in ref. 3 H o w e v e r , it is pointed out that Ap¢o(T)-*O as T--* To, so that o n e m a y state that /3T = 2/3O at T = TD T u f e u , K e y e s and DanIels 4) h a v e d e t e r m m e d the b e h a v l o u r o f the e x p o n e n t y in three planes near the C D P . Their results are m o r e vague, but at least not m c o n s l s t e n t with the relation y/y0 =/3/]3o f o r a n y isobaric or isothermal c r o s s - s e c t i o n o f p, T, x - s p a c e . This relation will be derived in the p r e s e n t article A theoretical basis for u n d e r s t a n d i n g the b e h a v i o u r of critical e x p o n e n t s m m u l t i - c o m p o n e n t s y s t e m s ts the w e l l - k n o w n t h e o r y o f Grifliths and WheelerS). F o r the p r e s e n t case it predicts that all e x p o n e n t s o b t a i n e d f r o m isothermal and isobaric e x p e r i m e n t s will h a v e the usual values, e x c e p t , possibly, t h o s e o b t a i n e d in an isothermal e x p e r i m e n t at T = To. Bartis and Hall 6) h a v e p r o p o s e d a lattice model, w h i c h r e p r o d u c e s the qualitative b e h a v i o u r o f

EXPONENT VALUES NEAR THE CRITICAL DOUBLE POINT

319

g a s - g a s equihbrla of the first as well as the second kind r e m a r k a b l y well o v e r a range which extends far b e y o n d that part of p, T, x-space, which is shown m fig. 1. T h e y conclude that/3 =/30 for any isbbaric or isothermal experiment, except for an e x p e r i m e n t at T = To, w h e r e / 3 r = 2/3o. This article extends an Idea which Griffiths c o m m u n i c a t e d to Barbs and Hall4'6). The b e h a v l o u r of the critical e x p o n e n t s In Isothermal situations Is determined by consideration of the isobaric crihcal e x p o n e n t s and the g e o m e t r y of the critical hne. Because of the central role of the g e o m e t r y of the critical hne a geometric a p p r o a c h seems to be preferable to an ad hoc extension of the scaling hypotheses. We shall investigate the isothermal critical e x p o n e n t s , the c r o s s - o v e r point and the scaling relations for these exponents As an illustration of the method let us consider the situation sketched in fig. 2. For simplicity the concentration variable has been left out. The point (Pl, TO m a y be regarded either as a point at t e m p e r a t u r e distance AT~ f r o m the critical point on the hne p = pj or as a point at pressure distance Ap~ f r o m the crihcal point on the line T - - TI. F r o m this figure one sees that for small values of IATII

ATI ~ Apt

(1.3)

if T~ is not the C D P t e m p e r a t u r e , and

AT~ ~ Ap~

(1.4)

if Ti = TD (and if d2Tc(p)/dp 2 ~ 0 at p = PD). In e v e r y isobaric plane I x ' - x " [ is proportional to IAT~[~° with a proportionality constant which is insensitive to small changes in the pressure of the isobaric plane. F r o m eqs. (1.3) and (1.4) we see t h e r e f o r e that

[ x ' - x " [ - ] A p , [ ~° if T, ~ To, IX'-- X"[ ~ [Ap,[ 2~° if T1 = To.

(1.5)

This intuitive argument will be made more precise in sechons 2 and 3. It will be shown that the c r o s s - o v e r pressure difference [Apco(T1)[ mentioned a b o v e is equal to 2[p~(T0--pD[. Section 4 contains a discussion of the results.

2. The isobaric and isothermal distances to the critical curve

The discussion in the introduction shows that the isothermal critical e x p o n e n t s depend strongly on the local f o r m of the critical c u r v e [Tc(p),xc(p)]. In c o n s i s t e n c y with the experimental observation 3) we shall

320

H M J BOOTS ANDAC MICHELS

assume T~(p) to be analytic with a non-vanishing second-order derivative at the critical double point We choose a state (p~, T], x]) with Tt/> To and define p c ( T ) as the Inverse function of T~(p). (For the inversion we choose the branch for which sgn [pc(T1)--PD] = sgn(pl--PD); see fig. 2 ) F r o m a Taylor series expansion we find ATI = - T ' ( p O A p j + ~?(Ap]) 2 = - A p l [ p ' ( T 0 + G(/tp]) 2

(2 1)

Here the prime indicates differentiation with respect to the argument. Our mterest focuses on the region around the C D P , where T ' ( p ) vanishes Expansion of Tc(p) about this point yields T~(pO - To = ½T"(pD)(Pl -- pD)2 + t~(p] -- pD) 3, Ti -- To = Tc[p~(TO]

-

TD

=

~T"(pD)[p~(TO - PD] + ~?[pc(T1) - pD] 3

(2 2a) (2 2b)

Neglecting the higher-order terms in these equations we find after subtraction AT] = --~T"(pD)Ap1{Ap~ + 2[pc(T0 - PD]}

(2 3)

This leads to AT1

ap,

A T ] - Ap~

for lap,[ ~lapco[,

(2 4a)

for [,ap,I >> [zapcol,

(2.4b)

where the c r o s s - o v e r pressure is given by the simple expression

IZlPco( T,)[ = 2lpc(T,) - PDI

(2 5)

Thus we see that Apco(TD) = 0; the region m which eq. (2.4a) is apphcable goes to zero as T1 a p p r o a c h e s TD Experimentally this region can only be investigated for those t e m p e r a t u r e s T~ for which pressure differences smaller than Apco(T~) can still be m e a s u r e d ; otherwise only the quadratic behavlour e x p r e s s e d in eq. (2 4b) is o b s e r v e d The results obtained in ref. 3 for the exponent fir confirm this. In w r m n g eq (2 3) we have neglected contributions which m a y be shown to be of order IAp~l+lpc(TO--pD[. It is convenient to define a pressure difference 7rHo(T0 such that these contributions are neghgible if and only if JAper < rrHo(T0. Far f r o m the C D P t e m p e r a t u r e [Pc(TO--PDI is large and eq (2 3) is invalid for all values of Zapl; in such case ~'Ho(T0 = 0 and eq. (2 1) must be used instead of eqs (2.3) and (2.4). We conclude this section by summarizing the situation as follows: a) If T~ = TD, then AT] ~ ApE for [Apl[ < "trio(TO b) If Ti > TD and [ZlPco(Tt)[ ~ ~'Ho(T0, then the c r o s s - o v e r behaviour expressed in eqs (2 4a) and (2.4b) applies for [Ap][ < 7trio(T0. c) If T1 > To and [zap¢o(T0[/> ~'Ho(T0, then LIT~ -- Ap~ for pressures for which

EXPONENT VALUES NEAR THE CRITICAL DOUBLE POINT

321

higher-order terms in eq. (2.1) may be neglected, this is the region far from the CDP temperature.

3. The critical exponents It will now be considered how the cross-over behavlour described m the preceding sections influences the values of the critical exponents and the exponent relations that are derived from scaling theory For a binary fluid system the scahng hypothesis leads to the following equation of state 7) Aix(p, T, x) = AxlAxlS-'h(y),

(3.1)

where y is defined as y =- (aTITc)/(Axlx~) l/~

(3 2)

and where Ax and A/z are the concentration and chemical-potentml differences between the states (p, T , x ) and (p, T, xc) The scahng function h(y) may be one of several heurishc expressions such as the (restricted) linear-model equation, the cubic-model equation or the NBS-equation 7) We shall assume that the pressure dependences of the non-universal parameters in the function h(y) as well as the pressure dependences of Tc and x~ may be neglected over the pressure range Ap for which the discussion in section 2 apphes We will also neglect the pressure dependence of a x , a justification for this will be given at the end of this section. The pressure d e p e n d e n c e of A T = - T - T ~ ( p ) is Important for small AT and cannot be neglected After these approximations A/z may be written as Atx = Atz[AT(p, T), Ax],

(3 3)

or, in view of eq (2.3), Al.t = Atx[AT(Ap, T), Ax]

(3.4)

The power laws Involving A T are derived from an explicit formulation of eq. (3.3) and similar equations for the free energy and the correlahon function. These laws contain /3, 6 and 77 as critical exponents. A simple example for the behaviour of these exponents near a CDP has been given in eq (1.1) For Isobaric cross-sections all available evidence leads to /3p =/30For isothermal cross-sections it is found from eqs. (2.3) and (2.4) that ,St = z/30 where

(3.5)

322

H M J BOOTS AND A C MICHELS Z= 1

If [ A p l ~ l A p c o ( T ) [ ,

z = 2

if [APl >> [APco(T)I.

(3.6)

Another example is presented by the power-law for the specific heat, which is derived to be 8) C -IAT1-2+°0(~°+1', where the e x p o n e n t is conventionally defined as - a 0 sections this leads to an e x p o n e n t relation

(3 7) For Isobaric cross-

% = 2 -/3p(6 + 1).

(3.8)

It can be shown qmte s i m p l y - s e e the a p p e n d i x - t h a t the e x p o n e n t s 6 and rl must have the same values for all cross-sections; hence subscripts on these e x p o n e n t s are omitted. In isothermal cross-sections this relation then becomes O~r = 2Z --/3T(6 + 1),

(3.9)

where the value of z is again given by eq (3.6). The e x p o n e n t relations y = / 3 ( 8 - l) = v ( 2 - rt)

(3.10)

will be valid throughout the critical region since f r o m IATI ~ [Apl z

(3 11)

and laTJ',

= ]ATI a"'~-'~ --" laTI ~"'2-'~

(3.12)

it follows that l a r t "~ =

laT?~"-" -= IATI ~'2-~,

(3 13)

independent of z. It IS interesting to note that If eq. (3.10) IS vahd, we have aT = Zao, /3T = Z/30, YT = Z'/0 and VT = ZVo. T h e r e f o r e the ratio of any two of these four e x p o n e n t s must be independent of the direction of a p p r o a c h to the critical line F u r t h e r m o r e , determination of the c r o s s - o v e r b e h a v l o u r for different e x p o n e n t s , e g. /3[/30 and Y/T0 must yield exactly the same results. L e t us now reconsider the three cases mentioned m section 2 a) T = TD; then for lap[ < ~'Ho(TD) we have aT = 2a0 and similar relations for /3T, TT and b'T, b) T > TD and 21p~(T)-PDI ~ 1r.o(T); then aT, /3T, TT and VT c r o s s - o v e r f r o m values a0, for I A p l ~ 21p~(T)- PD[ to values 2a0 . . . . for I~p[ >>21p~(T)- PD[ c) T > TD and 2 [ p ~ ( T ) - p D [ ~ r . o ( T ) ; then all e x p o n e n t s have their usual values throughout the range of power-law b e h a v i o u r

EXPONENT VALUES NEAR THE CRITICAL DOUBLE POINT

323

In this section we h a v e neglected the pressure d e p e n d e n c e of the concentration difference dx. This is allowed ff

lax/ ~p l >>JXc'[pc(r)]l. Thus the results of this section are invalid if one a p p r o a c h e s the critical point isothermally at t e m p e r a t u r e T via, or very near to, the line x = xc[pc(T)] Practically this a p p r o a c h is only relevant if one determines T and a in the one-phase region. H o w e v e r , e x p o n e n t s deviating f r o m their usual values do not o c c u r in this region since [Ap[ <½[Apco(T)[.

4. Discussion In this study we have investigated how critical e x p o n e n t s for isothermal e x p e r i m e n t s In a binary mixture and the relations b e t w e e n these e x p o n e n t s will be influenced by the existence of a C D P in such a mixture It was shown that the simple p o w e r laws associated with the e x p o n e n t s a, fl, y and v b r e a k down as a result of a c r o s s - o v e r effect in the vicinity of this point whereas the e x p o n e n t s 6 and r/ are unaltered. Any ratio, however, b e t w e e n two of the f o r m e r e x p o n e n t s will also be constant. One of the two e x p o n e n t e q u a h h e s that were studied, the one derived f r o m an expression for the specific heat, was found to be altered consistent with the results obtained for the exponents. It should be noted that these equalities were originally derived as inequalities. T h e s e inequalities are influenced in the s a m e way, as m a y be verified by inspection of the original derivations9). A quantitative verification of these results can only scarcely be obtained Deerenberg3), who made a power-law analysis of his experimental data on the s y s t e m N e - X e , has evaluated the b e h a v l o u r of x ' - x " for a n u m b e r of isothermal cross-sections of p, T, x - s p a c e Only two of these belong to case (b) as mentioned in sections 2 and 3 In this analysis the fir-value at the c r o s s - o v e r pressure, 2 [ p c ( T ) - p d , IS 0.52, whereas /3p=0.348 This IS in perfect a g r e e m e n t with the a b o v e results. The doubhng of critical e x p o n e n t s occurs in m a n y systems. T o name one very different example" there IS one set of e x p o n e n t s for all direchons of a p p r o a c h to the gelatlon curve, except for the tangent approach, for which doubling must be e x p e c t e d The special p r o p e r t y of g a s - g a s equilibria of the second kind, h o w e v e r , is that in C D P this tangent a p p r o a c h is isothermal.

324

H M J BOOTS AND A C MICHELS

Acknowledgement T h e a u t h o r s t h a n k Dr. J . A . S c h o u t e n f o r i n t r o d u c i n g t h e m into this p r o b l e m a n d f o r v a l u a b l e d i s c u s s i o n a n d Dr. A. D e e r e n b e r g f o r his p e r m i s s i o n to u s e fig 1, w h i c h w a s t a k e n f r o m his t h e s i s . H . B . t h a n k s Dr. K W e i s s a n d D r M.F.H. Schuurmans for reading the manuscript.

Appendix The exponents ~ and "0 T h e e x p o n e n t ~ d o e s n o t d e p e n d o n t h e d i r e c t i o n o f a p p r o a c h to t h e c r i t i c a l line, b e c a u s e it is r e l a t e d to t h e e x p o n e n t f o r t h e s p a t i a l - d i s t a n c e v a r i a b l e in t h e c o r r e l a t i o n f u n c t i o n ; this d i s t a n c e is n o t a t h e r m o d y n a m i c v a r i a b l e . In a n y i s o b a r i c p l a n e t h e e x p o n e n t 3 is r e l a t e d to t h e i s o t h e r m a l a p p r o a c h to t h e c r i t i c a l c u r v e , T~ = Tc(pl). T h e i s o b a r i c i s o t h e r m a l a p p r o a c h is j u s t t h e i s o b a r i c a p p r o a c h , p~ = p c ( T 0 , in a n i s o t h e r m a l p l a n e In o t h e r w o r d s , in t h e d e r i v a t i o n o f t h e p o w e r l a w i n v o l v i n g 6 o n e p u t s A T = 0. T h u s this s c a l i n g l a w d o e s n o t c o n t a i n A T a n d is t h e r e f o r e u n a f f e c t e d b y t h e s u b s t i t u t i o n s u s e d in t h e a p p l i c a t i o n o f e q (2.4).

References 1) J D Van der Waals, Zlttmgsverslag Kon Acad Wet Amsterdam, Nov 1894, 133 2) I R Knchevskn, Acta Physlcachlmlca USSR 12 (1949) 480 3) N J Trappemers, A Deerenberg and J A Schouten, Physlca, to be pubhshed A Deerenberg, thesis Umv of Amsterdam, 1977 (In Dutch) 4) R J Tufeu, P H Keyes and W B Darnels, Phys Rev Lett 35 (1975) 1004 5) R B Gnffiths and J C Wheeler, Phys Rev A2 (1970) 1047 6) J T Barbs and C K Hall, Physlca 78 (1974) 1 7) J M H Levelt-Sengers, W L Greer and J V Sengers, J Phys Chem Ref Data 5 (1976) 1 8) H E Stanley, Introduchon to Phase Translhons and Critical Phenomena (Clarendon, Oxford, 1971) 9) R B Gnflits, J Chem Phys 43 (1965) 1958