Thermodynamic model for tricritical mixtures with

Thermodynamic model for tricritical mixtures with application to ammonium sulfate + water + ethanol + benzene Miron Kaufman and Robert B. Griffiths Citation: J. Chem. Phys. 76, 1508 (1982); doi: 10.1063/1.443111 View online: http://dx.doi.org/10.1063/1.443111 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v76/i3 Published by the American Institute of Physics.

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Thermodynamic model for tricritical mixtures with application to ammonium sulfate + water + ethanol benzene

+

Miron Kaufman and Robert B. Griffiths Physics Department. Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

(Received 31 July 1981; accepted 9 October 1981) Procedures are developed for using cOlJlposition data in the three-phase region and variations of meniscus heights to determine the parameters needed to relate a classical thermodynamic model for fluid mixtures near tricritical points to actual experiments. In the case of the mixture ammonium sulfate + water + ethanol + benzene the result is a reasonably good fit to a variety of data, including light scattering measurements of intensity and correlation length. There is some evidence for deviations from the classical theory, but it is not unambiguous.

I. INTRODUCTION

mixtures 1,2

Tricritical pOints in multicomponent fluid arise when three coexisting phases become simultaneously identical to a single critical phase through a variation of temperature, pressure, and composition. In many respects they are analogous to the tricritical points observed in systems with symmetry-breaking phase transitions, such as the metamagnetic transition in FeCI 2, the order -disorder transition in NH4 CI, or the superfluid transition in 4He_ 3He mixtures. 3 The analysis of experimental data is much simpler in the case of symmetry-breaking phase transitions than it is for multicomponent mixtures. The absence of appropriate symmetries in the latter necessitates a rather extensive theoretical framework. It is the purpose of this paper to present the details of such a framework based on a "classical" phenomenological model 4 (the expansion of a free energy in powers of an order parameter) and to apply it to the particular case of the mixture ammonium sulfate + water + ethanol + benzene.

The predictions of the classical model are actually fairly simple when stated in terms of an appropriate set of "scaling fields" (the a J of Sec. II below). The major complication comes about in trying to relate these scaling fields to physical thermodynamic variables such as temperature, pressure, and chemical potentials. One assumes that there is a smooth transformation from one set of variables to the other. However, the actual details of this relationship must be determined by using experimental data to determine explicit values for a large number of phenomenological parameters. In the case of the quaternary mixture mentioned above, a large number of parameters can be determined with the help of the extensive data of Lang and Widom 5 on the compositions of three coexisting phases at several values of overall composition and two different temperatures very near the tricritical point. (A similar analysis has been carried out by Bockos for the mixture water + acetonitrile + benzene + n-hexane.) The phenomenological theory can then be used to predict properties for other points of the three-phase region near the tricritical point; i. e., it serves to extrapolate as well as interpolate the experimental results. 1508

J. Chern. Phys. 76(3),1 Feb. 1982

However, there are also two-phase and one-phase regions in the vicinity of a tricritical point, For an appropriate description of these it is essential to add a "smooth" contribution to the appropriate thermodynamic potential. For the mixture in question we have used data 7,8 on meniscus heights in closed tubes (thus fixed overall composition) as a function of temperature in order to fit the additional parameters present in the smooth part of the potential. The choice of parameters has been checked by comparing the predictions of the classical model with the results of various optical experiments. 7-9 The index of refraction can be calculated using the ClausiusMossotti formula, provided the density of the phase is known along with its composition. Since densities have not been determined directly near the tricritical pOint, we have used a crude but not unreasonable estimate for the volume of mixing. The intensity of light scattering extrapolated to zero angle is determined by the thermodynamiC fluctuation in the (optical) dielectric constant and we have worked out the appropriate formulas and compared them with experiment. The classical model can also be used to predict the correlation length, provided one is willing to supply values for additional phenomenological parameters. We have chosen these in order to provide reasonable agreement with the intensity of scattered light as a function of angle. In an earlier paper, 10 we reported on the results of our computations for the three-phase region and compared them with optical experiments. 9 As a result of the procedures discussed above, we have, in effect, produced a model for the thermodynamic properties of this particular mixture near its tricritical point, a model which can be compared with future experiments or used to generate gedanken experiments. The results of one of the latter, the behavior of a mixture with precisely the tricritical composition, is discussed in Sec. IV G. A classical model of the sort we employ has definite limitations, and it is well to recognize what they are. In the first place, it is well known that classical theories provide results which are quantitatively incorrect at "ordinary" critical points; in particular, the classi-

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© 1982 American I nstitute of Physics


M. Kaufman and R. B. Griffiths: Thermodynamics model for tricritical mixtures

cal critical exponents are not correct. Second, the model is expected to give the correct tricritical exponents (in the sense of scaling) at the tricritical point, but it lacks the logarithmic corrections predicted by modern theory. 11 Third, the classical model does not contain certain divergent susceptibilities which are allowed by scaling and which are observed experimentally in systems with symmetry breaking. 12 And of course the classical model may be deficient in other respects. 13 Despite these limitations our model seems to provide a remarkably good description of the available experimental data near the tricritical point of this mixture. An outline of the paper is as follows. Section II is devoted to an exposition of the classical model, and Sec. III contains the formulas needed to apply it to a quarternalJ mixture at fixed pressure. The application to ammonium sulfate + water + ethanol + benzene will be found in Sec. IV, which describes both the methods we used to fit the parameters in the model, and comparisons between predictions of the model and various experimental data. The prediction for the behavior of a mixture with composition equal to the tricritical composition is at the end of this section. The conclusions of our study are summarized in Sec. V.

1509

values are associated with two or more coexisting phases. The a's in Eq. (2.3) act as thermodynamiC field variables, and it is convenient to first work out the properties of the model in terms of these scaling fields before considering its relationship to the .Is, the "laboratory fields." Equation (2.2) may be written in the form (2.4) where 1/1 on the right-hand side is the equilibrium value of the order parameter, the value which minimizes for this choice of the a's, and is thus a function of at. ••• , a~_2' If Eq. (2.4) is differentiated with respect to one of the a's, the result is

(:~;). =(:~t~ + (::). (::J.

'

(2.5)

where the subscript a means that the (other) a's are held fixed. Since at the minimum

84)) •=~na"I/I"-1 = 0, (8iF

(2.6)

we see that (2.7)

II. THE CLASSICAL MODEL

The matrix X'J for the singular part of the susceptibility is defined as

A. Basic formulas In this section we present some of the basic formulas for a Landau theory employing a single order parameter. 4 Although we are interested in liquid mixtures, we shall develop the formulas in a form which uses a set of thermodynamic field 14 variables whose nature need not be specified in advance. (In Sec. IlIA we shall apply these formulas to a mixture where the fields are the temperature, pressure, and the chemical potentials.) A fundamental relation15 is provided by expressing one of these fields ~ as a function of the remainder, which we denote by flo 12, .... We assume that

~

--~-'I/IJ-l ~ Ba Ba - } Ba'

Xu-

J

I

where i and j run from 1 to k - 2. Upon differentiating Eq. (2.6) with respect to al (and noting that 1/1 depends on the a's), we obtain 2

B2

8 a2' as, and a(, simplified with the help of Eq. (2.23), are given in Appendix A. We shall adopt Widom'sl convention that a is the phase of lowest density, followed by (3 and then y, and we shall assume that 1/1 has its minimum (most negative) value in phase a, so that (2. 24)

When three phases coexist, /f;,,,, /f;{lJ and /Py all vanish, and 1/1,,,, I/Ia, I/Iy can be expressed in the form

1/10 =A cos 1>0 1200 + (J

(::j L n

1>0= 1200 -

2

a an

af.af,

,)

(2. 17)

(2.25) ,

(2.26)

(J ,

{ (J ,

where

and the divergent part is

(J

lies in the interval (2.27)

and (2.28)

(2.18)

The last equality is obtained with the help of Eqs. (2.8) and (2.10). The correlation length ~ can be obtained within the context of a classical theory by assuming that I/! depends upon position. If one adds a term (2.19)

to the right side of Eq. (... J), with "VI/! the gradient of I/! and g> 0 a smooth function of I/!, becomes a (position-dependent) free energy density. For I/! close to a minimum I/!. of Eq. (2.3), this free energy density is of the form [see Eq. (2.11)] (2.20)

where X'" and ga are X and g evaluated at I/! = I/!a. From this it follows that the corresponding correlation length iS

16

(2. 21)

(It is necessary to have a4 < 0 for three coexisting phases.) Using the expressions in Appendix A one finds that al

= t A 5 cos 3 (J , (2.29)

a2=ftA4, a3=

_~A3cos3(J.

In the case of two coexisting phases, two of the /f;o are zero and one is nonzero. If we let (2.30)

for the nonzero /Po, and let Sand P be the sum and prod. uct, respectively, of the 1/15 values in the two coexisting phases (where ~o = 0), then the a's may be written as follows: al:=: 2PS(P - S2) + APS/2 , a 2 :=: (S2 _p)2 _ A{S + 2P)/4 , a 3 :=:2PS+AS/2,

a 4 ==2(P-S 2)-A/4.

J. Chem. Phys., Vol. 76, No.3, 1 February 1982


(2.31)

M. Kaufman and R. B. Griffiths: Thermodynamic model for tricritical mixtures

1511

Note that by definition .6. $ 0 and 4P $ S2.

(3.1)

Finally, let us note that if lPa = 0, the susceptibility Xa at 1J!= 1J!a, defined by Eq. (2.11), has the form

(3.2)

Xa = {2[(1J!a -1J!s)2 + i)[(1J!a -1J!,i +

IP,.n-1 ,

(2.32)

and analogous expressions hold for Xs or Xy when /[;s= 0 or lPy=O.

for i = 1, 2, and 3. We shall let ~ be the thermodynamic potential, and MI together with T (temperature) and p (pressure) the independent thermodynamic fields. The Gibbs -Duhem equation yields the relation

d~ =

C. Sum rules in the three-phase region

3

-L: xldp.1 + vdp - sdT ,

(3.3)

1·1

In the three-phase region near a tricritical point, the Landau theory predicts that the susceptibilities in the different phases are related by sum rules whose utility will be demonstrated in Sec. IV below. When three phases coexist, lPa = /[;s = lPy= 0, and Eq. (2.32) and the analogous expressions for Xs and Xy are relatively simple. It is then easily shown using Eqs. (2.23), (2.24), and (2.25) that

where XI is the weight fraction of component i, and v and s are the specific volume and entropy. In the notation of Sec. II, P.h M2, P.3, - p, and T are the f' s, while Xl> x2, x 3, v, and s are the corresponding a's. In quaternary mixtures, we expect, in general, to have a one -parameter family of tricritical points, one for every p. Since the tricritical pOint occurs when the scaling fields vanish, i. e. ,

(2.33 )

and

Qx =X~1/2 _Xsl/2 + X;1/2 = -3a,/v'2 . (2.34) We shall refer to Eq. (2.33) as the first and Eq. (2. 34) as the second sum rule. Various other sum rules can be derived from these two. Let I denote the intensity of light, extrapolated to zero scattering angle, scattered by a mixture. Then I is proportional to18 as given by Eq. (2. 18). That is, I should be proportional to X times a term which tends to a constant (independent of phase) at the tricritical pOint. Hence, in the immediate vicinity of the tricritical point, we expect that the scattering intensities in the three phases will satisfy the sum rules

>,

Rl=(I~/2+I~/2)/IJ/2=1,

(2.35) (2.36)

In addition, if we make the plausible assumption that g in Eq. (2.19) tends to a constant near the tricritical point, then Eq. (2.21) and its analogs for phases f3 and y lead to the correlation length sum rules R(=(~a+~y)ns=l ,

Q(

= ~~1 _ ~sl + ~;1 ex: (_ a,) ,

(3.4)

we expect that to a first approximation the aj will be proportional to linear combinations of the deviations of T and the ill from their tricritical values T t and jj.1t, which are, of course, functions of pressure. Since very near the tricritical poi nt there should be considerable flexibility' in choosing we shall assume that

a"

(3.5) where (3.6)

is the deviation of the temperature from its tricritical value, divided by a reference temperature T r • When working at a fixed pressure, we shall let Tr be equal to T t at that pressure. For i = 1, 2, 3, we let 3

where the bJi and FI are assumed to be smooth functions of T (or t) and p, and FI = 0 on the tricriticalline. We shall assume that the regular part of 0, see Eq. (2.1), can be written in the form

Or=~0+tlilal--21 t 1=1

(2.37) (2.38)

in the three-phase region sufficiently near the tricritical point. "I. QUARTERNARY MIXTURES A. Laboratory thermodynamic variables

We shall now show how the approach discussed in Sec.

n can be implemented in the case of a quaternary mixture. Our formulation is chosen in part because of convenience in analyzing the experimental data discussed in Sec. IV below, but it is worth emphasizing that other choices for variables are possible.

(3.7)

al=-L:bJl(p.J-P.Jt)+FI , J.l

1.1

t

J=l

(3.8)

ElJala J

in the nieghborhood of the line of tricritical pOints, where ~o, lil> and EIJ= EJ! are assumed to be smooth functions of T and p. In addition, the matrix EIJ is chosen to be positive definite so that Or is a strictly concave function of the a's. [While the sums in Eq. (3.8) could be chosen to run from 1 to 4, it is just as convenient, given Eq. (3.5), to place the temperature dependence in the coefficients and 0 0, 1 The weight fractions for i = 1, 2, 3 can now be written as [see Eqs. (2.13), (3.7), and (3.8)1

XI=-~=bIO+tblJ(I/JJ-+-EJna) III ~., J.l

,

(3.9)

where

Let III be the chemical potential per unit mass for the ith component, and define J. Chern. PhY5., Vol. 76, No.3, 1 February 1982


(3.10)


1512

Since, of course,

When

G,

x,,

=

4

Eq. (3.15) applied to Eq. (3.9) yields 3

LXI=I,

(3. 11)

X, = b

I·j

,o + L

(3.17)

blJuJ ,

J= 1

it is also possible to write x 4 in the form (3.9), provided the b4.J are chosen so that

where (3. 18)

(3.12) vanishes for j

*" 0 and

is 1 for j

and

= O.

Similar expressions can be written down for v and s, but as we shall not use them in Sec. IV, we shall not provide the explicit formulas here. Note that in computing v it is necessary to take into account the pressure dependence of T, and ji.JI as well as F I , no, etc. [Note that in Ref. 10, Eq. (3.9) occurs with the EJf! terms omitted. This is an adequate approximation in the three -phase region near the tricritical point, but not in the two- and one-phase regions. ]

B. Approximation for v In the experiments discussed in Sec. IV, the actual specific volume v of the mixtures was not directly determined. Since this quantity enters expressions for the index of refractions, we were forced to employ an approximation based on an estimate of the volume of mixing, Eq. (3.14) below. If VI is the specific volume of the ith pure component at the same temperature and pressure as the mixture whose weight fractions are x" the volume of mixing Vm is defined by the equation

v=(I-Vm )Lx,V I ,

(3.13)

I

Since V m vanishes in any pure component, it is a plausible first approximation to suppose that 1 Vm =-2 LLCIJXIXJ

,

(3.14)

I"'J

where C 'J =CJI is a set of temperature and pressure dependent coefficients. They can be determined provided volumes of mixing are known for the various binary mixtures involving the components which make up the quaternary mixture.

(3.19) Once the biJ and the EJn are known, Eq. (3.17) can be used to calculate the overall composition for a given T and p, provided ah a2, as and the A6 are known. The inverse problem of calculating at. a2, as, and the AO' given an overall composition at a particular T and p (and assuming, as before, that blJ and Ejn are known) is much less trivial. Thus while one can solve Eq. (3.17) for the u J , given the x,, it is, in general, difficult or impossible to solve Eqs. (3.18) and (3.19) for the a. and AO in closed form. In order to simplify the inverse problem we shall choose (3.20)

E/I=E!/=O

for all i.

In addition, we suppose that (3. 21)

so that Hr is a strictly concave function of a2 and aa, though not of aj. However [see Eq. (2.12)], n. + nr will be strictly concave in at. a2, and a3• 2. Inverse problem for the three-phase region

The sum Over n in Eq. (3. 18) may be restricted to n := 2, 3 in view of Eq. (3. 20). From Eqs. (2. 29), (2,28), (2.26), and (2.25) we obtain the results a 2 =ft t

2

(3.22)

,

a 3:= _-H-t)3/2cos38:= :=

_t(_t) 3/2

cos3¢0

-2~ -~tlfo ,

(3.23)

where, note, Ifo is the value of If in any of the three phases, Inserting these results in Eq. (3.18) and noting Eq. (3.16) allows us to rewrite Eq. (3.17) as (3.24)

C. Mixtures with several phases 1. Introduction If the fluid in a container consists of several coexisting phases, the value of a density G I for the mixture as a whole is given by the equation

(3. 15)

where G~ is the value of this density in phase 6 and AO is the mass of phase 6 divided by the total mass of the mixture, so that (3. 16)

(3.25)

b,2 := b'2

,

b,3 := b '3 + 2

t

bIJEJ3 •

J.2

The procedure for obtaining the a's and A'S, given the x's, is then the following. The coefficients bIJ are known (for a given T and p), so Eq. (3. 24) may be solved

J. Chern. Phys., Vol. 76, No.3, 1 February 1982


M. Kaufman and R. B. Griffiths: Thermodynamic model for tricritical mixtures for the u~.

Next, the formula

cos 39"" 4 ( -

t)-3/2

ug -

3 ( - t)-1/2U~

(3026)

,

[which is obtained by multiplying Eq. (3.23) by Xo, summing over Ii, and using Eq. (3.23)] can be solved for 9, and a10 a2, a3, and the lJio calculated from Eqs. (2.29), (2.25), and (2.26). The values of Xo are then obtained from Eqs. (3.19) and (3.16). Of course, if the righthand side of Eq. (3.26) does not lie between -1 and 1, or if any of the Xo are negative, the overall composition does not lie in the three-phase region but rather in the one-phase or two-phase regions. 3. Inverse problem for the two-phase region

so that lJi and the u~ are immediately determined by the Thus for j = 2 and 3, Eq. (3.18) is a pair of simultaneous linear equations in a2 and as, and the solution is straightforward. Inserting these values along with lJi in Eq. (2.6), which has the form

uJ'

(3.31) determines a l • That the result actually represents a minimum of cP can be checked, to begin with, by seeing whether X [Eq. (2. l1)j is positive. To establish that it is a global minimum requires solving Eq. (3.31) numerically and evaluating cp at each of the roots.

the result is

IV. THE MIXTURE AMMONIUM SElFATE+WATER + ETHANOL + BENZENE

~ =: 8(P - 8 2 ) - 6t ,

A. Preliminary remarks

If we insert Eq. (3.5) into Eq. (2.31) and eliminate ~,

1513

al

=:

6PS(P -

s2 - ~t)

,

(3. 27)

a2 = 3[sI - p2 + ~t(82 + 2P)] , a3 =: 8(6P - 482 - 3t) ,

where 8 and P are equal to lJil + lJi2 and lJillJi2, respective1y. One can, in fact, write (3.28) where the + refers to phase 1 and the - to phase 2. If Eq. (3.28) is inserted into Eq. (3.19), and then Eqs. (3.19) and (3.27) are inserted into Eq. (3.18), the result is a set of three equations in the three unknowns X10 P, and 8 (note that X2 "" 1 - XI)' Eliminating XI yields two equations in 8 and P (Appendix B) and eliminating P yields a polynomial equation 6

LB,8

1

(3. 29)

=:0

'=0

in S alone. The coefficients Bit which are given in Appendix B, are functions of the UJo EJn, and t. To solve the inverse problem, we first find the real roots of Eq. (3.29) numerically, and for each S compute the corresponding P. If 4P exceeds s2 or if 4(P _S2) exceeds 3t, the solution is discarded, since lJil and lJi2 must be real, Eq. (3.28), and~, Eqs. (2.30) and (2.31), cannot be positive. Once Sand P are known, the a's can be computed from Eq. (3.27) and inserted into Eq. (3. 18) to provide the u~, and lJil and lJi2 can be calculated from Eq. (3.28). The values of AI and X2 =: 1 - XI are then determined by Eq. (3.19), and of course an acceptable solution requires that both quantities be positive. If there is no solution satisfying all of these criteria, the overall composition does not correspond to two phases at the temperature and pressure of interest. 4. Inverse problem for the one-phase region

In contrast to the two- and three-phase regions, the inverse problem for the one -phase region is straightforward, given Eq. (3.20). Note that u~ in this case is simply lJI', and setting j = 1 in Eq. (3. 18) gives (3.30)

The tricritical point in the mixture ammonium sulfate + water + ethanol + benzene was discovered by Mertslin and Mochalov. 19 Their work, that of Radishevskaya et aZ • • 20 and that mentioned below was carried out at atmospheric or near atmospheric pressure, so we have assumed a constant pressure in our analysis. Lang and Widom 5 carried out a careful study of the compositions of three coexisting phases near the tricritical point. We have used their data in Sec. IV B as a principal source of the b" coefficients defined in Sec. nI, and in what follows we shall always use their convention on labeling the components: i == 1 for ammonium sulfate, i = 2 for water, i = 3 for ethanol, and i = 4 for benzene (the same order as in the title above). We have used published 7 and unpublished 8 data on the heights of meniscuses separating the liquid phases in two sealed samples, here designated A and B, of fixed overall composition in order to estimate the EIJ coeffiCients in Eq. (3. 8), as discussed in Sec. IV D below (following some remarks in Sec. IVe on our procedure for estimating the densities of the phases). Sample A 7 was prepared with a composition (weight fractions) of Xl == 0.0175, x2 == 0.3510, X3 = 0.4500, and x4 = 0.1815, and sample B8,9 with XI"" 0.0179, x 2 = 0.3534, Xs = 0.4471, and x 4 = O. 1817. In order to reconcile the correlation lengths measured in sample B with those found for sample A. we increased the temperatures reported for sample B by 0.2 °e. (This is not inconsistent with a possible uncertainty in the absolute temperature measurements for this sample; the relative temperatures were determined much more precisely.) We also assumed a slightly different set of compositions for sampIe B: 0.0180, 0.3523, 0.4467, and 0.1830. This change is not unreasonable in view of the difficulty of preparing samples of a precise composition, and possible errors in the Lang and Widom results on which we have based our model parameters. No similar adjustments were made for sample A. The correlation lengths for sample B do not include corrections for multiple scattering, and the same is true of the correlation lengths for for sample A as reported in Ref. 9; corrected values for sample A are in Ref. 7. We have used the uncorrected values for sample A in this paper, as these would seem to be more comparable with those for sample




1514

B, for which we did not have corrected values. (Some remarks on these corrections will be found in Sec. IV F.) Light scattering measurements were also reported by Gollub et al. 21 for two samples, and they published the meniscus heights as a function of temperature for one of these. We have not analyzed their data on linewidths or the corresponding data of Wu. 22 The light scattering data 1• 9 for samples A and B is considered in Sec. IV F, following a discussion of indices of refraction in Sec. IVE. Finally, Sec. IVG presents the predictions of our model as to what would happen as the temperature varies in a mixture of composition precisely equal to the tricritical composition. B. The three-phase data of lang and Widom

Lang and Widom 5 measured the composition of each of the three coexisting phases for four samples at a temperature of 20. 9°C, five at 21°C, five at 44. 91°C, four at 48. 04°e, and one at 48. 59°C. The loci of the compositions of these coexisting phases at a given temperature form a (moderately) smooth curve in the composition tetrahedron. At each temperature we analyzed the Lang and Widom data in the following manner. Each sample of three coexisting phases was assigned an angle e, corresponding to the parametric representation Eq. (2. 25), and the mass fraction x~ of component i in phase 1) was assumed to be of the form 3

X~=L J=O

(4.1)

dIJ(cosCPo)J ,

with CPo given by Eq. (2.26), and the dlJ a set of coefficients. The motivation for Eq. (4. 1) comes from Eq. (2.25) and from noting that according to Eq. (3. 24) x~ should be a linear combination of l/!o, ~, and l/!~ [since for phase 0, Ao= 1 in Eq. (3.19) and the other A's are zero].

TAB LE 1. The coefficients dlJ [see Eq. (4. 1) 1 at different temperatures.

)\

1 Salt

2 Water

3 Ethanol

0 1 2 3

0.0244 0.1011 0.1341 0.0568

0.3609 0.5442 - O. 0491 - 0.2495

0.4173 - 0.1205 - 0.3166 0.0902

0.1974 - 0.5249 0.2316 0.1026

21

0 1 2 3

0.0165 0.0939 0.1484 0.0698

0.3467 0.5678 - O. 0219 -0.2936

0.4909 -0.0970 -0.4486 0.0920

0.1459 - O. 5645 0.3221 0.1316

44.91

0 1 2 3

0.0143 0.0331 0.0342 0.0148

0.3514 0.2578 0.0116 - 0.0225

0.4629 -0.0516 -0.1307 0.0045

0.1714 - O. 2393 0.0849 0.0033

48.04

0 1 2 3

0.0160 0.0182 0.0079 0.0021

0.3535 0.1309 0.0023 0.0059

0.4579 - 0.0253 -0.0384 -0.0012

0.1727 - 0.1238 0.0282 -0.0066

T(OC)

20.9

4 Benzene

In practice, what we did was to assign a preliminary value of e to each of the three-phase samples and then adjust the dlJ for a minimum deviation between the experimental compositions and those predicted by Eq. (4. 1). The e's were then varied to improve the fit. Our choices for the d li at the different temperatures are listed in Table I. A comparison of the computed and experimental x~ values is given in Table II in the following form. At each temperature we give the value of e (in degrees) which we assigned to a particular threephase triangle studied by Lang and Widom, followed by the values of x~ computed using Eq. (4. 1). The number beneath each of these values is the difference between it and the corresponding experimental value; the latter can be obtained by adding these two numbers. At 20. 9 and 21°C some of the calculated weight fractions are slightly negative for some cases in which the experimental values are small. The samples are listed in the same order as in Table I in Ref. 5, except that the values at 20. 90°C are all placed together. Figure 1 shows the predicted weight fraction of benzene as a function of cos cp at a temperature of 44. 91°C plotted as a curve with the experimental values shown as pOints. This is typical of the quality of fit possible at the temperatures of 44. 91°C and above. Typical deviations of the measured from the computed weight fractions are O. 005 for the liquid components and O. 001 for the salt. At 21°C the deviations are larger, typically 0.015 for the liquid components and 0.002 for the salt. An improved fit at 21°C can, of course, be obtained by including a term j = 4 in Eq. (4. 1). However, we only used the data above 40°C in determining the b li coefficients. Given the choice in Eq. (3. 5) for a4. it follows from Eqs. (2.28) and (2.25) that l/Jo = ( - t )1 12 cos cP 0

,

(4.2)

where [see Eq. (3.6)] t=(T-Tt)/T t ·

(4.3)

Hence, a comparison of Eqs. (3.24), (3.19), and (4. 1) implies that d

li

= (_ t)J 12bIJ

(4.4)

for Is iS4 and OSj S3. Now the b lJ are smooth functions of the temperature (Sec. IlIA), and thus the same is true of the blJ [Eq. (3.25»). We, therefore, assumed that (4.5) [Equations (4.4) and (4. 5) differ from their counterparts in Ref. 10 because of the omission of the EJn terms in Ref. 10, as noted above in Sec. IlIA.) The quantities hlJO and b(J1 were determined as follows. We assumed a value for the tricritical temperature Ttf thereby fixing t for the Lang and Widom data. The values of iJ lJ were determined at 44. 91 and 48. 04°e using Eq. (4.4), and from these the bw , which are listed in Table III. The procedure is sensitive to the choice of T t because of the way t appears in Eq. (4.4); bad choices for T t result in unreasonably large values



TABLE II. Compositions of the Lang and Widom samples as predicted by our model. The difference between the experimental and computed value is listed in each case directly beneath the latter. Phase T(OC)

20.9

{I

XI

X2

Phase y

Phase (3

Oi

X3

X(

XI

X2

X3

X4

XI

x2

Xa

X4

2.9

0.0000 +0.0009

0.0908 +0.0062

0.3749 -0.0084

0.5344 +0.0013

0.0008 + O. 0001

0.1263 - O. 0084

0.3980 + O. 0041

0.4749 + O. 0044

0.3156 -0.0009

0.6069 +0.0031

0.0711 +0.0043

0.0064 -0.0064

3.7

0.0000 + O. 0002

0.0864 + O. 0062

0.3713 - O. 0113

0.5425 + O. 0047

0.0010 + O. 0001

0.1318 - O. 0112

0.4007 + O. 0080

0.4666 + O. 0030

0.3152 - O. 0010

0.6072 + O. 0034

0.0714 + O. 0038

0.0062 - O. 0062

46.

0.0007 + O. 0001

0.0145 + O. 0011

0.1537 - O. 0088

0.8311 + O. 0075

0.0637 - O. 0020

0.5019 +0.0192

0.3619 + O. 0210

0.0725 - O. 0381

0.1784 + O. 0062

0.6316 - 0.0143

0.2111 - O. 0163

- O. 0210 + O. 0243

57.6

0.0006 - O. 0006

0.0169 - O. 0002

0.1317 + O. 0122

0.8507 - 0.0113

0.1057 - O. 0018

0.5777 + 0.0031

0.3025 - 0.0042

0.0142 + O. 0028

0.1258 - O. 0014

0.6000 - O. 0081

0.2757 - O. 0044

- 0.0015 + O. 0139

11.9

- 0.0007 +0.0007

0.0451 +0.0217

0.3283 -0.0226

0.6274 + O. 0001

- O. 0004 + O. 0015

0.1769 - 0.0129

0.4750 - O. 0093

0.3484 + O. 0208

0.3159 - O. 0050

0.6062 + O. 0061

0.0527 + O. 0241

0.0252 - 0.0252

:0

19.2

0.0002 +0.0003

0.0316 + O. 0116

0.2674 - O. 0216

0.7007 + O. 0097

0.0037 + O. 0001

0.2414 - O. 0225

0.4928 + O. 0143

0.2621 + O. 0081

0.2963 + O. 0018

0.6160 + O. 0014

0.0767 + 0.0078

0.0108 - O. 0108

G) ..,

24.1

0.0007 + O. 0007

0.0284 + O. 0040

0.2263 - O. 0156

0.7446 + O. 0110

0.0083 - O. 0017

0.2884 - O. 0122

0.4961 + O. 0237

0.2072 - O. 0098

0.2790 + O. 0057

0.6234 - O. 0025

0.0986 - O. 0042

-0.0009 + O. 0009

~

7'

II>

C .....

3

II>

:::J

II>

:::J

0-

21 '--

9 T f is impossible in our model, since a4 is a positive constant times T - T I , Eq. (3.5).]

u.

If the u. are asymptotically proportional to t, the same is true of the coefficients Bo, Bit and B2 (Appendix B) in Eq. (3. 29}, whereas B a, B" B 5, and Bs tend to constant values at the tricritical point. Hence, we conclude that in the two-phase region (if it exists), S will vary

46

48

50

T (OC) FIG. 11. Calculated fractional meniscus heights as a function of temperature for a sample with overall composition precisely that of a tricritical mixture. Note the break in scale; at all temperatures most of the sample is in phase (3.

asymptotically as S

rx ltl 1l3 ,

whereas P (Appendix B) and as prxt, ~rxltI2/3.

~

(4.22) lEq. (3. 27)J will behave (4.23)

These results can be used to show that the order parameter susceptibility X [see Eq. (2.32) and also Eqs. (2.23) and (2.30)] dlVerges as

xrx ltl-413

(4.24)

upon approaching the tricritical pOint through the twophase region and that, moreover, the ratio of X in the {3 phase (defined within the model by Eq. (2.24)] to that in the coexisting phase tends to 2 at the tricritical pOint. The asymptotic behavior of X in the one-phase region (if it exists) can be determined usingthe methods of Sec. III e 4. In particular, Eq. (3.30) implies that ljJ is proportional to t, and Eq. (3.18) that a2 and a 3 are proportional to t. Hence, by use of Eq. (2.11) we infer that (4.25 ) It should be noted that these results for the one- and two-phase regions depend on the assumption (3.20). Whereas we do not think they would be altered in the presence of nonzero E,l> we have ilot checked this.

Unfortunately, we have not been able to confirm that the situation shown in Fig. 11, two phases coexisting just below and only one phase just above the tricritical temperature, is a general feature of the classical model, nor to show the opposite, that other situations are possible by an alternative selection of the parameters. V. CONCLUSIONS

The basic conclusion of our calculation is that all the available experimental data on the static properties of the mixture ammonium sulfate + water + ethanol + benzene

J. Chern. Phys., Vol. 76. No.3, 1 February 1982


1522


near its tricritical point can be fit remarkably well by a classical model based on the Landau expansion (2. 3). Since this is a phenomenological model, it is necessary, in the nature of the case, to choose values for a large number of phenomenological constants which are free parameters within the framework of the model. Our choices are given in Tables lIT and IV, and with these we find that the classical moo el performs reasonably well for temperatures within three or four degrees of the tricritical temperature (I t I :;; 0.01) and compositions sufficiently close to the tricritical value to exhibit three-phase coexistence within this temperature range. The earlier analysis by Lang and Widom 5 showed that the three-phase region near the tricritical point was in good qualitative and fairly good quantitative agreement with the classical model. Our own calculations have confirmed this result. The compositions given in Table I are based purely on the classical model, and we find no evidence of any substantial departure from classical behavior. Our estimate of the tricritical temperature (49.1 °C) is slightly higher than that of Lang and Widom, and we suspect that their choice of a lower value may have been the source of their conclusion that their exponent {31 was 0.4 instead of the classical O. 5. In addition, however, we have shown that the classical model is in good agreement with the results of lightscattering measurements in the three-phase region. The prediction of the intensity I of light scattered at zero angle (Fig. 4) is particularly impressive when one notes that the calculation requires a rather careful analysis of the dependence of the optical dielectric constant on the order parameter, and this analysis was carried out within the framework of the classical model. It is true that we found it necessary, in the absence of experimental data on the densities of the coexisting phases, to make a somewhat crude estimate for the volume of mixing: nonetheless, this estimate involved no additional free parameters since the CIJ in Table IV were taken from other experiments. To be sure, agreement between theory and experiment is not perfect. This is particularly apparent in the dimensionless quantity RIJ defined in Eq. (2.35) and plotted in Fig. 6. The theoretical curves decrease with increasing temperature, and there is no evidence for such a decrease in the experimental values. Nonetheless, one needs to be cautious, both because the theoretical curves could be shifted somewhat (though they would probably still be monotone decreasing) if we had better estimates of the densities of the coexisting phases, and because of the substantial multiple -scattering effects in the experiments. The agreement between our calculated values for the correlation length (Fig. 8) and the experimental values determined by light scattering is, once again, fairly good. We had to choose three adjustable parameters, the gj in Eq. (4. 18), in order to carry out the comparison, but this choice could be made at a single temperature. (The comparison actually involves correlation lengths which have not been corrected for the effects of multiple scattering in the original data, for rea-

sons explained in Sec. IV A. However, as noted in Sec. IV F, a different choice of the gt's leads to an equally good fit to the corrected correlation lengths 7 of sample A. It should also be noted that we made an adjustment to the temperature scale and the composition for one of the samples, as discussed in Sec. IV.) The agreement between our model predictions and available data in the two-phase region near the tricritical point is much less impressive. The reason is that that we had very little data to work with and a large number of adjustable parameters (the Ejj and bl12 of Table IV). Nonetheless, we think it is Significant that the model was able to provide reasonable predictions for the meniscus height in the two-phase region for two samples of constant composition. Additional experiments in the two-phase region would help to refine the model parameters and might well uncover some deficienCies in the classical approach. It would also be interesting if exper iments could be carried out atthe tricritical composition or extremely close to it, in order to test the predictions of the model given in Sec. IV G. Current theoretical ideas, and experiments near tricritical points in systems with symmetry breaking, strongly suggest that departures from classical tricritical behavior should occur in liquid mixtures of the sort considered in this paper. Thus the success of our analySis, whereas it is in one sense very gratifying, is in another sense a bit disappointing in that we find no solid evidence for nonclassical tricritical effects. It would be very useful to have an appropriate nonclassical equation of state near the tricritical point which could be fitted to available experimental data and which could provide indications ofthe magnitude and location of the most prominent nonclassical effects. We ourselves, however, ha.ve not been successful in developing such an equation of state. ACKNOWLEDGMENTS

Dr. K. K. Bardhan was a collaborator at an early stage in the computations reported here and we have benefited from his work. Dr. Goldburg, Dr. Kim, Dr. Levelt Sengers, Dr. Esfandiari, and Dr. Wu have given us major assistance by providing us with their numerous comments and answers to our questions. Dr. Levelt Sengers assisted us in our calculations of the optical dielectric constant. Our research has been supported by the National Science Foundation through Grant Nos. DMR 76-23071 and DMR 78-20394. APPENDIX A: EXPRESSIONS FOR THE aj

By multiplying out Eq. (2.22), comparing it with Eq. (2.3), and using Eq. (2.23), one obtains the expressions at =

if!",if!sif!JI/},. + 1fa + if!~+ 2(~ + ~ + ~;)l :J.2-a -a :;2 :;1 :J.2 -2(if!",'f'sif!y+ I/Isif!ot'f'y+ if!y'f',;.I/Js) ,

aa = t(1/},. + 1fa + if!~)a - (if!~ + ife + I/?,.)(~ + ~ + ~)

+ 3(t~ + lfaij + I/?,.~) + ~~~ + ij~+ ~~, as = 2 (l/i",f;. + l/isij + l/i'l'i[l; -l/i",l/isl/i,,) • a4 = ~

+ ~ + ¢; - (if!; + fa + I/?,.) •




1523

APPENDIX B: COEFFICIENTS B; IN EO. (3.29) In order to write the coefficients in a compact form, let us introduce the following symbols, employed (in the manner defined here) only in this appendix: b=E22,

C=E23,

/ = 10 33,

g=E~3-€22E33'

where me subscripts x and il mean that the remaining x's or jj.'s are held fixed. Similarly, from Eq. (C2), we obtain

a=ul

(C4)

The elimination procedure described in Sec. III C 3 leads to the following two equations in Sand P: P[e - ba (3

+ (6g - b)S] = by -

c(3

+ (2g8 + ea)S - baS 2 + 4gS3 ,

+ P - as + eS(6P - 4S2 - 28) + b[3S 4 - 3P2 + 8(5 2 + 2P)1 = 0 .

Combining Eq. (C3) with (C4) yields

( ail) alJ!

,,= z=(av) ax J

J x

(~) [( alJ!) alJ! ,,+ ~ jj.

J'l

Solving the first equation for P and inserting it in the second yields Eq. (3.29) with the coefficients B j given by the following expressions:

We expect the term in square brackets in Eq. (CS) to remain finite at the tricritical point, due to the properties of the Xj and the functions previously discussed and the fact that we do not expect any divergences in av/axl' On the other hand,

B o =3(by-c(3j2+(ba -c)l28(by-c(3)+y+a(31,

v

Bl = (by - c(3)(1 + 6/ + 2b8) + (ba - ell( a + 2c8)2

- 2/8(1 + 2b8) + 12cy - 2(1

+ 6/)(3]

(C5)

,

B2 = - 6ba(by - c(3) + (b - 6g)[(a + 2c8)2

- 2/8(1 + 2b8) + 6cy - (1 + 6/)(3]

(:tl)';;2'';;3'P'T

+ 3(ea + 2g8)2 + 3(ba - c)[2/a - (ba - c )8] , B3 = 2{2c/ + 12g(by - c(3) + a[c(e - 4ba) + /(b - l8g)) + 2 8[c(b - 8g) - bcdb - 5g))} , B4 == 4ca(2b + 9g) - 8(b - 2g)(b -18g) _7c 2 - 4j(b - 6g) ,

(C6)

should be proportional to X [Eq. (2.11)] and diverge strongly at the tricritical pOint. Hence, it should be a good approximation to retain only the first sum on the right side of Eq. (CS). But then differentiating Eqs. (CI) and (3.9) with respect to lJ! and inserting the results in Eq. (C5) yields (in this approximation) Eq. (4.17).

Bo == lOc(b - 6g) - 6ba (b - 2g) , B6== -3(b -2g)(b -lOg).

APPENDIX C: DERIVATION OF EO. (4.17) If we choose i = 4 in Eq. (2. 13) and make the identification (Sec. lIlA) of 04 with v and/4 with -p, the result is

(Cl) where vr stands for a~~r/ap. The dependence of the function on lJ! is that given explicitly in Eq. (Cl), while its dependence on il j , p, and T when lJ! is fixed comes about through the dependence of vr and aa,j ap on these quantities. Thus when lJ! is fixed, is a smooth function of all its other arguments, and the same is true of its partial derivatives at fixed lJ!. In a similar manner we can write Eq. (3.9) in the form

v

v

XI

=

xl(il,p, T, 1/1) ,

(C2)

where the dependence of XI on lJ! is that shown explicitly in Eq. (3.9), and for fixed lJ!, is a smooth function of its other arguments.

x,

Of course, the equilibrium lJ! is itself a specific function of p, T!. and the iL's. Thus v is some function of p, T, and the }.l's or, equivalently, of p, T, and xl> X2, x3' In the calculation which follows, we shall keep p and T fixed and therefore omit them from the arguments and subscripts. By differentiating (Cl) with respect to iLl at fixed iL2 and ;l3 we obtain

lBrief introductions to the subject will be found in B. Widorn, J. Phys. Chern. 77, 2196 (1973); R. B. Griffiths, Physica (Utrecht) 73, 174 (1974). 2A number of reference will be found in R. B. Griffiths and B. Widom, Phys. Rev. A 8, 2173 (1973); B. Wid om , Kinam; J. Phys. Chern. (to be published). 3See J. F. Nagle and J. C. Bonner, Annu. Rev. Phys. Chern. 27, 291 (1976), for a review of theoretical and experimental papers on phase transitions and critical phenomena, including tricriticality. 4R • B. Griffiths, J. Chern. Phys. 60, 195 (1974). 5J . C. Lang, Jr. and B. Widom, PhysicaA 81, 190 (1975). Gp. Bocko, Phys. status Solidi A 103, 140 (1980). 1M. W. Kim, W. 1. Goldburg, P. Esfandiari, and J. M. H. Levelt Sengers, J. Chern. Phys. 71, 4888 (1979). BE. -So Wu (private communication). 9 M• W. Kim, W. I. Goldburg, P. Esfandiari, J. M. H. Levelt Sengers, and E. -So Wu, Phys. Rev. Lett. 44, 80 (1980). 10M• Kaufman, K. K. Bardhan, and R. B. Griffiths, Phys. Rev. Lett. 44, 77 (1980). 11 M • J. Stephen, E. Abrahams, and J. P. Straley, Phys. Rev. B 12, 256 (1975); M. J. Stephen, ibid. 12, 1015 (1975); Y. Yamazaki and M. Suzuki, Progr. Theor. Phys. 58, 1367 (1977); M. J. Stephen, J. Phys. C 13, L83 (1980). 12R. B. Griffiths, Phys. Rev. Lett. 24, 715 (1970). 13M. E. Fisher and S. Sarbach, Phys. Rev. Lett. 41, 1127 (1978); S. Sarbach and M. E. Fisher, Phys. Rev. B 18, 2350 (1978); 20, 2797 (1979). 14In the terminology of R. B. Griffiths and J. C. Wheeler, Phys. Rev. A 2, 1047 (1970). ISH. B. Callen, Thermodynamics (Wiley, New York, 1960), Chaps. 1 and 5.

J. Chern. PhY5., Vol. 76, No.3, 1 February 1982


1524


t6L. D. Landau and E. M. Lifshitz, Statistical Physics, 2nd ed. (Pergamon, London, 1969), Chap. XII. 17The actual form of M depends on the choice of thermodynamic variables. Some examples are given in Ref. 16. t8L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), Chap. XIV. t~. V. Mertslin and K. I. Mochalov, Zh. Obshch. Khim. 29, 3172 (1959). 20G. S. Radyshevskaya, N. I. Nikurashina, and R. V. Mertslin, J. Gen. Chem. USSR 32, 673 (1962). 21J. P. Gollub, A. A. Koenig, and J. S. Huang, J. Chem.

Phys. 65, 639 (1976). 22 E• -So Wu, Phys. Rev. A 18, 1641 (1978). 23CRC Handbook of Chemistry and Physics, 59th ed., edited by R. C. Weast (Chemical Rubber, Cleveland, 1978). 24J • Timmermans, Physico-Chemical Constants of Pure Organic Compounds (Elsevier, Amsterdam, 1965), Vol. 2. 25J. Timmermans, The Physico-Chemical Constants of Binary Systems in Concentrated Solutions (Interscience, New York, 1959, 1960), Vols. 2 and 4. 26J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 155.

