ON THE PHENOMENOLOGY OF PHASE TRANSITIONS

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ON THE PHENOMENOLOGY OF PHASE TRANSITIONS

L. Wilcox, W. Estler

To cite this version:

L. Wilcox, W. Estler. ON THE PHENOMENOLOGY OF PHASE TRANSITIONS. Journal de Physique Colloques, 1971, 32 (C5), pp.C5a-175-C5a-179. �10.1051/jphyscol:1971521�. �jpa-00214741�

JOURNAL DE PHYSIQUE Colloque C5a, supplkment au no 10, Tome 32, Octobre 1971, page C5a-175

ON THE PHENOMENOLOGY OF PHASE TRANSITIONS

L. WILCOX and W. T. ESTLER

Department of Physics, State University of New York at Stony Brook Stony Brook, New York 11790

~ksumb. - Nous introduisons une nouvelle formulation paramktrique de la description phkno- mknologique d'un fluide statique (OU d'un corps ferromagnktique) prks d'un point critique. On se rappelle et on manipule facilement les nouvelles kquations qui font intervenir six parametres.

pour un choix particulier de l'un des paramktres (A), nos kquations se ramknent 9. la formulation g cinq parametres suggerke et appliquke par Ho et Litster. Si on laisse libre le sixikme paramktre, il apparaft une amklioration sensible dans l'accord entre thkorie et donnks expkrimentales, pour un faible en complication. On formule I'analyse prksente d'une faqon telle qu'elle facilite l'interprktation des mesures interfkromktriques telles que celles qu'ont dkveloppees Wilcox et Balzarini.

Abstract. - We introduce a new parametric form for the phenomenological description of a static fluid (or ferromagnet) near a critical point. The new equations involving six parameters are easily remembered and manipulated. For a particular choice of one parameter ^(A), our equations reduce to the five parameter form suggested and applied by Ho and Litster. There appears to be a significant improvement in the fit to experimental data at small cost in complication if the sixth parameter is free. The present analysis is formulated in such a way as to facilitate the interpretation of interferometric measurements such as developed by Wilcox and Balzarini.

Review of Phenomenology. - Van der Waals' theory : simple, elegant, productive

...

where M(T) is the value of p(p, T) on the critical isochore (p = p,) if (T > T,) or the value at the boundary of coexistent phases if (T < T,). For fixed pl # p,, the first two terms are assumed to be analytic functions of temperature T. The interesting singularity is associated with the integrand Ap ⁼ ,u(p, T) - M ( T ) upon which attention is focused. Following usual practice we introduce reduced, dimensionless variables :

P" = (PCIPC) AP (2) and then drop (*) henceforth.

For small values, an expansion of Van der Waals' equation takes the form :

The boundary of coexisting phases is defined by f = 0 so that ^{P 2} cc E on the boundary and this is clearly contradicted by observations in fluids and magnets. In 1965 B. Widom [I] prescribed a partial remedy. One might replace eq. (3), he suggested, by an analogous expression :

He did not specify the function f. The locus f = 0 is

now 1 p I1/fi cc e and if p

- ^4,

nature is much improved. For small e and p, Widom argued, eq. 4 accords with known facts, notably : the inequalities of Griffiths and Rushbrooke are satisfied as equalities. Ensuing discussions by Domb and Hunter [2], Griffiths [3], and by Kadanoff [4] based upon different considerations, deepened insight and increased the plausibility of Widom's hypothesis.

Whatever the ultimate fate of the scaling hypothesis as a theoretical construct, it well serves the experimenter for the organization and presentation of his data.

Green, Missoni and Sengers [5] were quick to exploit the new mode of presentation, examining older data in the new light. But that data of dubious precision derived from various fluids did not provide the sharp test which the scaling hypothesis needed. Nor did their

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971521

C5a-176 L. WILCOX A1 rTD W. T. ESTLER analysis reveal the form of the scaled state function

f (811 p IllP) with satisfying precision. It was clear by then that the situation would be improved by an adequate paramatrization for f (&/I p This could then be tested against uniform high precision data.

Missoni, Sengers and Green [6] devised a six parameter formula which we later discuss.

Josephson [7] and Schofield [8] stressed the advan- tages which accrue when Widom's hypothesis is stated in parametric form. We follow this path.

The R-T Plane. - We choose to write parametric equations in the form :

Here H and M are homogeneous functions in R and E

of degree p. In the second form of eq. (5) and (6) we replace the pair of functions H and M by another pair, Y(R, E ) and W(0), in which Yis homogeneous of degree 1, and W of degree 0, i. e., a function of 0 = E/R only.

Evidently :

and

The desirability of representation in terms of Wand Y will emerge presently.

For the parameter R, one is free to choose any positive, homogeneous function R(E, I p l1IP) ofdegree 1, which, in some sense, measures distance from the critical point. Static susceptibility, magnetic or fluidic, is inherently positive and generally easy to measure. In fluid studies, for example (8p/dp)-'

--

If the exponent y ^1!1.25 has its usual significance in this field, then the stated requirements on R are satisfied.

We are thus led to consider thermodynamic functions on the R-T plane, the geometry of which is depicted in figure 1. The physically accessible region of the R-T plane lies above straight lines which are the loci p = 0.

The lines are : E = ROO, the critical isochore, and

E = Re,, the boundary of coexisting phases. The cri- tical point is mapped into the vertex. Straight lines radiating from the vertex are loci of constant 8.

Coexisting

FIG. 1. - The R-Tplane. The accessible portion of the plane lies above the straight lines 0 = 00 and 0 = Ox, which are loci of

p = 0.

The W-Function. - The function W(0), defined by eq. (9), possesses some simple properties. At the coexistence boundary, (8 = ex), we have H = 0, but M > 0. Therefore, from eq. (9), W(0,) = 0. On the critical isochore in the one-phase region :

Lim = Lim - H = 1 , for T > T c , (11)

P + O (+I~P)T ^{P + O} M

so W(0,) = co. On the critical isotherm (0 = O), p cc and HIM = 116. Using the scaling relation 6 - 1 = yip, we learn : W(0) = 1. The three obvious conditions upon W(0) are, then :

The Phase Function. - The homogeneous function Y(R, E) of degree 1 has desirable properties. It will turn out that YP is determined directly by optical inter- ferometric measurements, it being proportional to an cr optical phase ⁾⁾ at stationary points [9]. Furthermore, the functions Wand Yare simply related. Differentiat- ing YP in eq. (8) with respect to the first variable (R), we obtain :

a Legendre transformation. The bracketed quantity vanishes by virtue of the definition, eq. (10). Thus, with eq. ( 5 ) , we find

This is easily integrated to yield Y when W is known

where Yo = Y(l, 8,) is an independent constant.

ON THE PHENOMENOLOGY OF PHASE TRANSITIONS C5a-177

The Definition of Tc. - In laboratory studies of critical phenomena, the experimenter is inevitably obliged to arrive at some operational definition for Tc on his laboratory scale. The history of this subject includes many definitions, for example : T, is the temperature at which : << the meniscus disappears ⁾⁾ ;

(< susceptibility is infinite ⁾⁾ ; << heat capacity is infi-

nite n ; <( spontaneous magnetization vanishes )). These definitions require an extrapolation of experimental data, needing an exponent which is to be simultaneously determined. Kouvel and Fisher [lo] recognized this problem, and accordingly introduced a bootstrap technique in which they defined a quantity T*, deter- mined from data alone. From a plot of T* vs. T, the parameter Tc and the exponent of interest is extracted.

The value of T, is obtained by extrapolation ; errors are introduced unless data close to the critical point are available.

In the interferometric method [9], with which we are primarily concerned, experimental data consists of measured triplets (YP, Ry, T) for values of T which include T,. From eq. (14) we derive :

This permits a new operational definition of T, : for fixed T, plot In Y%s In RY. The slope on this plot is (Ply) W(&/R). If the slope is constant, independent of R, then necessarily T = T,. Furthermore, this constant slope is ply. This definition assumes scaling, but is otherwise free of bootstrap difficulties which extrapolat- ing definitions entail. This definition does not << point ⁾⁾ to T, ; rather, it signals arrival at T,, and it determines the ratio y/P ⁼ 6 - 1 from data at T = T,. In this light, the interferometric technique appears capable of yielding an especially precise determination of the critical temperature in a fluid.

The W-Hypothesis. - Within the constraints sup- plied in eq. (121, an obvious guess for W(8) is

We claim that this is in fact an excellent guess. It will require a sophisticated analysis of very precise data to detect any departure from eq. (171, which we shall call the W-hypothesis. Using eq. (17), it follows from eq. (1 5) that :

where

A = 1 - e0/8, . _{(1 Sb)}

Given R and T, then p and ,u are determined when the following six parameters are specified (e,, A, Yo, y, p, T,). The first parameter, 80, is a scale factor for

temperature. The parameter Yo defines the boundary of coexisting phases :

The constant G suggests E. A. Guggenheim [ll], who showed that G

-

This ratio is subject to some controversy. In their first survey of fluid data, Green et al. set this ratio at

--

Preliminary optical measurements by Wilcox and Balzarini found the value to be

-

PA ⁼ 3, and /3 = 0.36, which implies that the corres- ponding ratio in CrBr, is (A

- -

For comparison with other representations, it is useful to consider the ratio :

With the W-Hypothesis, this becomes :

Note : Z(8,) = 161, Z(0) = 116, Z(8,) = 0. If A = 6, then Z(0) is linear in 8. Since A is in fact nearly equal to 6, the choice we have made for the definition of 8 appears fortunate. To emphasize small departures of Z(0) from linearity, we calculate from eq. (20) :

This difference vanishes at 8 = 8, and again at 8 = 0.

Comparison with Schofield, Ho and Litster. - The W-Hypothesis identically reproduces the parametric model of Ho and Litster [12], if PA = 3. Our para- meters are associated with those in that model, accord- ing to the following translation :

This paper Ho and Litster (eq. 6).

- -

8/80 1 - b2

e2

Roo r (22b)

C5a-178 L. WILCOX AND W. T. ESTLER In figure 2, we display Z, - 2, from eq. (21), plotted

for several values of the parameter A . The curve labelled PA ⁼ 1.5 applies to the model of Ho and Litster, or equally to ours for that particular choice of A .

This limit is zero ! The behavior of the continued isochore is depicted by the computer plot, figure 3. In the present context there is, then, no spinodal in the sense of analytic continuation.

FIG. 2. - Plot of ZA(0) - Z8(8). Comparison of our parametric equations with the model of Ho and Litster, and with Missoni, Sengers, and Green. The best value of PA probably

lies somewhere between 1.45 and 1.55.

Comparison with Missoni, Sengers and Green. -

For the particular choice, PA = 1.467 our equation, coincides well, but not perfectly, with the equation suggested by M. S. G. [6]. In figure 2 we display 2, - Z, computed from their eq. (5.3) using our defini- tion eq. (19). The algebraic expressions which one encounters when dealing with M. S. G.'s equation are unwieldy. We therefore give only results from the computer. For comparison the curve generated from our equation with P A = 1.467 is plotted. The two curves sensibly coincide except for 8 < - .2. Near the coexistence boundary, the curve acquires a large derivative. This probably indicates the defect in their ad hoc formula which M. S. G. discuss.

Isochores. - We conclude by considering general features of an isochore ( M = M, ⁼ constant) on the R-T plane. From eq. (6) with Y given by eq. (18) we derive the general isochore in parametric form :

(See Fig. 3 for definition of symbols).

It is interesting to speculate about the existence of a

<< spinodal D. By ^<( spinodal ⁾⁾ we mean an analytic continuation of the isochore to R = 0, beyond the coexistence boundary. Analytic continuation is simply continuation in the case since the factors in eq. (23) are positive in the range

-

FIG. 3. - Isochores on the R-T plane. The continuation3f our equation of state into the non-physical region yields no spinodal.

The limiting slope at the coexistence boundary 8 = Ox is about a factor of 2 greater than the slope of the critical isochore.

Some experimenters have sought to employ the spinodal concept in another sense : as the temperature to which the isochore ((points ^)> as it intersects the coexistence boundary. The limiting slope of the general isochore at 8 = 0, is

the last expression follows from the homogeneity of M.

With the W-hypothesis the result is

Lim

[g]

⁺

^.

8-e, P 60

Then

where EL is the intercept by linear extrapolation (see Fig. 3). Thus, the compressibility of a general isochore

<< apparently ⁾⁾ diverges at a negative temperature

ON THE PHENOMENOLOGY OF PHASE TRANSITIONS C5a-179 Summary and Conclusions. - The significant diffe-

rence between equations proposed and used by Ho and Litster and those employed by Missoni et al. correspond to a change in our parameter A of some 2 %. Ho and

THE R -T PLANE

C L A S S I C A L XENON DATA

titical lsochore

Accsrriblrto Interfemmrtric Mea8uramrntt

-200 0 200 400 800 MX)

60th S C O I ~ S U ~ I ~ B of lo-'

( € / m -

FIG. 4. - A catalogue of classical xenon data. The cross-hatched portion corresponds to the region of the R-T plane explored by optical measurements near completion at S. U. N. Y., Stony Brook. The iso-,u curve corresponds to a height of 1 cm in earth's gravity. This represents the fundamental limitation on the range of variables accessible to the optical method. Note that the full range Ox < 0 < 00 is accessible and for 6 = 8, or 80 the full

range of R is accessible.

Litster's equations are easier to manipulate than M.

S. G'. s, and they are free of unwelcome singularities at the boundary of coexisting phases. Their form is derivable from ours if FA is fixed atj1.50. However, we see a positive advantage to leaving A a free experimental parameter unless and until its value can be deduced from fundamental theory.

We advocate representation of thermodynamic quantities on the R-T plane. We give a central impor- tance to the function W(8) since this appears to us to be the most economical way to represent the equation of state. W(8) necessarily has a pole and a zero. We hypothesize that is has a simple zero and a simple pole ; i. e. it possesses its obvious and necessary properties and no others are needed for the phenomenological description of experimental results.

In figure 4, on the R-T plane, we locate data points for xenon which have been tabulated by Missoni et al.

No points for 8 < .2 are included in their tabulation.

The data from Habgood and Schneider are sparse for xenon. The cross hatched portion near the vertex corresponds to the region of the R-T plane explored by the interferometric measurements of Estler, Wilcox, Hocken and Charlton. These measurements determine all six parameters (y,

p,

Acknowledgements. - Our associates in this work were Robert Hocken and Tom Charlton. We have enjoyed discussions with Professors George Stell and Nandor Balazs of S. U. N. Y. at Stony Brook.

References

El] WIDOM (B.), J. Chem. Phys., 1965,43,3898. [7] JOSEPHSON (B. D.), J. Physique, 1969, C 2,1113.

121 DoMB (C.1 and HUNTER (D. La), P ~ o c . Phys. Soc.9 1965, [8] SCHOFIELD (P.), Phys. Rev. Letters, 1969, 22, 606.

86, 1147.

[3] GRIFFITHS (R. B.), Phys. Rev., 1967,158, 176. [9] WILCOX (L.) and BALZARINI (D.), J. Chem. Phys., [4] KADANOFF (L. P.), Physics, 1966,2,263. 1968,48,753.

151 GREEN (M. S.), VINCENTINI-MISSONI (M.) and LEVELT ¹¹⁰¹ KOUVEL (J. S.? and FISHER (M. E.1, Phys. Rev., 1964, SENGERS (J. M. H.), Phys. Rev. Letters, 1967, 18, 136,1626.

1113. [ l l ] GUGGENHEIM (E. A.), J. Chem. Phys., 1945, 13, 253.

261 VINCENTINI-MISSONI (M.), LEVELT SENGERS (J. M. H.),

and GREEN (M. S.), J. Res. Natl. Bur. Std. (U. S.), ¹¹²¹ Ho JOHN (T.) and LITSTER (J. D.), Phys. Rev., 1970,

1969, 73A, 563. B 2, 4523.