Asymmetric criticality in weakly compressible liquid mixtures · In the present paper, we shall...

Asymmetric criticality in weakly compressible liquid mixturesG. Pérez-Sánchez,1 P. Losada-Pérez,1,a� C. A. Cerdeiriña,1,b� J. V. Sengers,2,c� andM. A. Anisimov21Departamento de Física Aplicada, Universidad de Vigo, As Lagoas s/n, Ourense 32004, Spain2Department of Chemical and Biomolecular Engineering and Institute for Physical Science and Technology,University of Maryland, College Park, Maryland 20742, USA

�Received 27 January 2010; accepted 11 March 2010; published online 16 April 2010�

The thermodynamics of asymmetric liquid-liquid criticality is updated by incorporating pressureeffects into the complete-scaling formulation earlier developed for incompressible liquid mixtures�C. A. Cerdeiriña et al., Chem. Phys. Lett. 424, 414 �2006�; J. T. Wang et al., Phys. Rev. E 77,031127 �2008��. Specifically, we show that pressure mixing enters into weakly compressible liquidmixtures as a consequence of the pressure dependence of the critical parameters. The theory is usedto analyze experimental coexistence-curve data in the mole fraction–temperature,density-temperature, and partial density–temperature planes for a large number of binary liquidmixtures. It is shown how the asymmetry coefficients in the scaling fields are related to thedifference in molecular volumes of the two liquid components. The work resolves the question ofthe so-called “best order parameter” discussed in the literature during the past decades. © 2010American Institute of Physics. �doi:10.1063/1.3378626�

I. INTRODUCTION

Fluid mixtures can exhibit a variety of critical phase-separation phenomena.1–3 Fluids and fluid mixtures belong tothe universality class of Ising-like systems, whose criticalbehavior is characterized by two independent scaling fieldsand one dependent scaling field. Asymptotically close to thecritical point of fluids, the dependent scaling field becomes ageneralized homogeneous function of the independent scal-ing fields which has the same form as that for the Isingmodel.4,5 Hence, the principle of critical isomorphism6–9 as-serts that the thermodynamic behavior near the critical pointsof fluids and fluid mixtures can be mapped onto the criticalbehavior of the Ising model that has only two independentfields. In the Ising model the critical behavior of the orderparameter �spontaneous magnetization� is symmetric with re-spect to the ordering field �magnetic field�. The differenttypes of the critical behavior encountered experimentally influid mixtures, including the observed asymmetric criticalbehavior of physical densities, are determined by the specificanalytic relationships between the Ising fields and the actualphysical fields.

For many years, it was thought that for fluids the twoindependent Ising scaling fields should be analytic functionsof the independent physical fields, while the dependent Isingscaling field was identified with the dependent physical field.This assumption, also referred to as revised scaling,10 wasbased on some simple asymmetric models such as the deco-rated lattice-gas model11 or the penetrable sphere model12

and on some empirical considerations of asymmetricasymptotic critical behavior in one-component fluids near thecritical point.13–15 The implications of this isomorphism as-

sumption for the critical behavior in binary fluid mixtureshave been elucidated by Anisimov et al.16,17 However, a fewyears ago, it was pointed out by Fisher and co-workers18,19

that in order to account for all asymmetric features of criticalphase transitions in fluids, the three Ising scaling fieldsshould be taken as analytic functions of all physical fieldsincluding the dependent physical field, a principle referred toas “complete” scaling.

Fisher and co-workers18–20 formulated complete scalingexplicitly to account for the asymmetric vapor-liquid criticalbehavior in one-component fluids. Complete scaling turnsout to cause a so-called Yang–Yang anomaly for the tempera-ture dependence of the heat capacity in the two-phaseregion.18,19,21,22 Even more importantly, complete scaling in-duces a strong singularity in the temperature dependence ofthe diameter of the coexisting densities,18,20,23–25 in additionto a weak singularity known to follow from revised scaling.10

In two previous publications,26,27 we have extended completescaling to liquid-liquid phase separation and applied the for-malism to incompressible liquid mixtures. In incompressibleliquid mixtures the pressure does not induce critical concen-tration fluctuations. Hence, we were able to evaluate the tem-perature dependence of the solute concentration in the twocoexisting liquid phases for the case of an incompressibleliquid mixture, neglecting the effect of pressure. Neverthe-less, for a complete description of the thermodynamic behav-ior of critical liquid-liquid phase separation, we do need toinclude the pressure. First, the partial molar volumes are re-lated to the pressure derivative of the chemical potentialseven if the compressibility is zero. Second, in nearly incom-pressible liquid mixtures the critical parameters do dependon pressure in practice. In the present paper, we shall showhow complete scaling can be applied to liquid-liquid phaseseparation in binary mixtures, to be referred to as weaklycompressible liquid mixtures, in which the pressure does not

a�Present address: K. U. Leuven, Belgium.b�Electronic mail: [email protected]�Electronic mail: [email protected].

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affect the critical concentration fluctuations but does play arole as a nonordering field,28 causing the presence of a criti-cal locus for phase separation as a function of pressure, asshown schematically in Fig. 1. The procedure to be devel-oped can also serve as a prototype of how complete scalingcan be applied more generally to critical phase transitions inthe presence of a nonordering field.

We shall proceed as follows. In Sec. II we introduce theconcepts of scaling fields, scaling densities, and completescaling for the critical behavior of fluids and illustrate theseconcepts explicitly for the case of one-component fluids nearthe liquid-vapor-critical point. In Sec. III we then formulatein some more detail the concept of complete scaling for thecritical behavior of binary fluid mixtures. In Sec. IV we ap-ply the theory to derive the temperature expansions for vari-ous physical densities of the two coexisting liquid phasesnear the critical point of liquid-liquid phase separation inweakly compressible mixtures. In Sec. V we make somecomments on the role of molecular volumes in liquid-liquidphase-separation asymmetry and possible simplificationswhen the volume of mixing is neglected. In Sec. VI wepresent a comprehensive analysis of the experimentalcoexistence-curve data, yielding evidence for the validity ofcomplete scaling. Our results are summarized and discussedin Sec. VII.

II. SCALING FIELDS AND SCALING DENSITIES

A. General formulation

The critical behavior of Ising-like systems can be de-scribed in terms of two independent scaling fields, a “strong”scaling field h1 �ordering field�, a “weak” scaling field h2,and a dependent scaling field h3, which, asymptotically closeto the critical point, becomes a generalized homogeneousfunction of h1 and h2 of the form

4

h3�h1,h2� � �h2�2−�f�� h1�h2�2−�−�� , �1�where, except for two system-dependent amplitudes, f� is auniversal scaling function with the superscripts � referring

to h2�0 and h2�0, respectively, and where � and � areuniversal critical exponents,5,29

� 0.110, � 0.326. �2�

Associated with these scaling fields are two conjugate scal-ing densities: a strongly fluctuating scaling density �1 �orderparameter� and a weakly fluctuating scaling density �2 suchthat

�1 = � �h3�h1�h2, �2 = ��h3�h2

�h1

. �3�

For positive values of h2, the system is uniform for all valuesof �1. For negative values of h2, there is a region of two-phase equilibrium bounded by a coexistence curve, where�1= ��cxc. Equation �1� implies that along this phaseboundary the order parameter satisfies an asymptotic powerlaw

�1 � � B0�h2��. �4�

In addition, in the two-phase region at h2�0, the secondscaling density �2 will vary as

�2 � − A0−1 − � �h2�1−� − Bcr�h2�� , �5�so that the two-phase “isomorphic” heat capacity diverges as

� ��2�h2

��1

� A0−�h2�−� − Bcr. �6�

In Eqs. �4�–�6� B0, A0−, and Bcr are system-dependent ampli-

tudes. The term Bcr in Eq. �6� for the heat capacity arisesfrom an analytic fluctuation-induced contribution to be addedto Eq. �1�30,31

B. Complete scaling in one-component fluids

As a first illustration of the general scaling formulation,we consider its application to the critical behavior of a one-component fluid with the chemical potential � and the tem-perature T as the two independent physical fields and withthe pressure P as the dependent physical field. To formulatethe scaling fields explicitly, it is convenient to introduce thefollowing dimensionless quantities:

�T̂ �T − Tc

Tc, �P̂ �

P − PcckBTc

, ��̂ �� − �ckBTc

, �7�

where kB is Boltzmann’s constant, is the molecular density,and where we adopt the usual convention of designating thevalue of any thermodynamic property at the critical point bya subscript c.

The prototype for the critical behavior in fluids is thelattice gas, which is a simple reformulation of the Isingmodel for one-component fluids.15,32 For the lattice gas, thestrong scaling field h1 is in linear approximation proportional

to ��̂, the weak scaling field h2 proportional to �T̂, and the

dependent scaling field h3 proportional to �P̂. Note that thescaling fields are defined such that at the critical point,h1=h2=h3=0. It then follows from Eq. �3� that for the latticegas the strongly fluctuating scaling density �1 is proportional

FIG. 1. Typical three-dimensional liquid-liquid equilibrium surface of abinary system; T, P, x denote temperature, pressure, and mole fraction,respectively. Solid curves represent the �upper consolute� critical locus andthe T-x coexistence curve for a given pressure terminating at the criticalpoint c.

154502-2 Pérez-Sánchez et al. J. Chem. Phys. 132, 154502 �2010�


to �̂= ̂−1 and the weakly fluctuating scaling density �2 is

proportional to ��̂Ŝ�= ̂Ŝ− Ŝc, where ̂= /c is the dimen-sionless molecular density and Ŝ=S /kB is the dimensionlessmolecular entropy.

Substitution of these expressions for the scaling field h2and for the scaling densities �1 and �2 into Eqs. �4� and �6�yields the well-known asymptotic critical power laws for thecoexisting vapor-liquid densities and for the isochoric heatcapacity in the one-phase region along the critical isochore

=c as a function of �T̂.Like the Ising model, the lattice gas is completely sym-

metric in terms of the order parameter. That is, the average ofthe coexisting vapor and liquid densities, referred to ascoexistence-curve diameter, is equal to the critical densityindependent of the temperature. However, real fluids are notsymmetric and do display a coexistence-curve diameter thatdepends on temperature. Originally,10–15 some vapor-liquidasymmetry was incorporated by introducing revised scalingin which it was assumed that the independent scaling fields

h1 and h2 should both depend on ��̂ and �T̂, while thedependent scaling field h3 was continued to be treated as

proportional to �P̂. However, as was first realized by Fisherand co-workers,18,19 one needs to adopt a principle of com-plete scaling in which all scaling fields should be �analytic�functions of all physical fields. Specifically, complete scalingimplies that in linear approximation the scaling fields for realone-component fluids become18–20

h1 = a1��̂ + a2�T̂ + a3�P̂ , �8�

h2 = b1�T̂ + b2��̂ + b3�P̂ , �9�

h3 = c1�P̂ + c2��̂ + c3�T̂ , �10�

where ai, bi, and ci are system-dependent coefficients. Theresulting asymmetric critical behavior of the one-componentfluids, obtained by substituting these expressions for the scal-ing fields into Eqs. �1� and �3�, has been elucidated in greatdetail by Kim et al.20 and by Anisimov and co-workers24,25,27

and is not repeated here. We only note that revised scalingcauses a singular term in the coexistence-curve diameter pro-

portional to ��T̂�1−� and complete scaling causes an addi-tional more strongly singular term proportional to ��T̂�2�.

III. COMPLETE SCALING IN BINARY FLUIDMIXTURES

Appropriate physical fields to specify the thermody-namic behavior of fluid mixtures are the temperature T, thepressure P, the chemical potential �1 of the solvent, and thedifference �21��2−�1 between the chemical potential �2of the solute and the chemical potential �1 of the solvent. Toformulate the scaling fields explicitly, we introduce againdimensionless quantities similar to those earlier defined inEq. �7�,

�T̂ �T − Tc

Tc, �P̂ �

P − PcckBTc

,

�11�

��̂1 ��1 − �1,c

kBTc, ��̂21 �

�21 − �21,ckBTc

.

Complete scaling now implies that in linear approximationthe scaling fields for binary fluids become27

h1 = a1��̂1 + a2�T̂ + a3�P̂ + a4��̂21, �12�

h2 = b1�T̂ + b2��̂1 + b3�P̂ + b4��̂21, �13�

h3 = c1�P̂ + c2��̂1 + c3�T̂ + c4��̂21. �14�

Equations �12�–�14� are to be supplemented with the Gibbs–Duhem equation, which reads in terms of dimensionlessquantities as

d��̂1 = − Ŝ�T̂ + V̂�P̂ − x2d��̂21, �15�

where V̂�V /Vc� ̂−1 is the dimensionless molecular volumeand x2 is the mole fraction of the solute. As shown by Wanget al.,27 the scaling fields can be simplified by taking

a4 = b1 = c1 = 1, c2 = − 1, c3 = − Ŝc, c4 = − x2,c,

�16�

so that

h1 = ��̂21 + a1��̂1 + a2�T̂ + a3�P̂ , �17�

h2 = �T̂ + b2��̂1 + b3�P̂ + b4��̂21, �18�

h3 = �P̂ − ��̂1 − Ŝc�T̂ − x2,c��̂21. �19�

The zeros of entropies of the two liquid components arearbitrary. We can use this freedom to select a specific valuefor the entropy S of the mixture at the critical point and forits concentration derivative ��S /�x2�P,T at the critical point.Following Wang et al.,27 we choose for the value of the

entropy at the critical point Ŝc=−��̂1 /�T̂�h1=0,c. This choicereduces the number of coefficients in Eq. �17� such that a2 isrelated to a1 by

27

a2 + a1� ��̂1�T̂

�h1=0,c

= a2 − a1Ŝc = 0. �20�

In addition, we shall take the concentration derivative of theentropy at the critical point ��S /�x2�P,T=0, as further dis-cussed in Sec. V.

Substituting Eqs. �17�–�19� into the definition �3� of theIsing densities and using the Gibbs–Duhem equation �15�,one obtains the following expressions for the physical den-sities:

Ŝ =Ŝc + a2�1 + �2

1 + a1�1 + b2�2, �21�

154502-3 Asymmetric criticality in liquid mixtures J. Chem. Phys. 132, 154502 �2010�


V̂ =1 − a3�1 − b3�31 + a1�1 + b2�2

, �22�

x2 =x2,c + �1 + b4�21 + a1�1 + b2�2

, �23�

̂ =1 + a1�1 + b2�21 − a3�1 − b3�2

, �24�

̂x2,c =x2,c + �1 + b4�21 − a3�1 − b3�2

. �25�

We note that at phase coexistence, h1=h1,=0, so thatfor liquid-liquid phase-coexistence data at constant pressure,Eqs. �17�–�19� reduce to

0 = ��̂21 + a1��̂1 + a2�T̂ , �26�

h2, = �T̂ + b2��̂1 + b4��̂21, �27�

h3, = − ��̂1 − Ŝc�T̂ − x2,c��̂21, �28�

where h2, and h3, are the scaling fields h2 and h3 at coex-istence. Solving Eq. �26� for ��̂21 and substituting the resultinto Eqs. �27� and �28�, we obtain

h2, = �1 − b4a2��T̂ + �b2 − b4a1��̂1, �29�

h3, = − �1 − a1x2,c��̂1 − �Ŝc − a2x2,c��T̂ . �30�

Since at coexistence h3,��h2�2−�f−�0�, in accordance withEq. �1�, we obtain from Eq. �30�,

��̂1 = −Ŝc − a2x2,c1 − a1x2,c

�T̂ −1

1 − a1x2,c�h2�2−�f−�0� . �31�

If we insert Eq. �31� into Eq. �29� and solve iteratively forh2,, we find that in leading order h2, varies with tempera-ture as

h2, � �0�T̂ �32�

with

�0 = 1 − b4a2 −�b2 − b4a1��Ŝc − a2x2,c�

1 − a1x2,c, �33�

which, upon substitution of the relationship �16� between thecoefficients a1 and a2, reduces to

27

�0 = 1 − b2Ŝc. �34�

We note that the choice Ŝc=−��̂1 /�T̂�h1=0,c for the criticalentropy entering into Eq. �34� for �0 has only been made forconvenience. The resulting expressions for all physically ob-servable quantities do not depend on this particular choice.

IV. APPLICATION TO WEAKLY COMPRESSIBLELIQUID MIXTURES

In two earlier publications,26,27 we have considered someconsequences of complete scaling for the case of incom-

pressible liquid mixtures by neglecting the contribution of

the pressure �P̂ from the pressure in Eqs. �13� and �14� forthe two independent scaling fields, h1 and h2 �i.e., by takinga3=b3=0�. Thus, instead of the more complete expressions�17� and �18�, we started from the simpler expressions26,27

h1 = ��̂21 + a1��̂1 + a2�T̂ , �35�

h2 = �T̂ + b2��̂1 + b4��̂21, �36�

while retaining expression �19� for the dependent scalingfield h3 so that the molecular volume V̂ can be defined ther-modynamically consistent with Eq. �15�.

However, most liquid mixtures are weakly compressiblefor which the critical point of liquid-liquid phase separationis not a single point, but is located on a critical locus thatdoes depend on pressure, as was indicated in Fig. 1. Hence,the critical parameters, as well as the coefficients in expres-sions �35� and �36� for the scaling fields, depend on the pres-sure as a hidden field,

h1 = ��̂21�P̂� + a1�P̂��̂1�P̂� + a2�P̂��T̂�P̂� , �37�

h2 = �T̂�P̂� + b2�P̂��̂1�P̂� + b4�P̂��̂21�P̂� . �38�

If we now expand Eqs. �37� and �38� in terms of �P̂= P̂− P̂c

0, where P̂c0 is the experimental reference pressure and

only retain the leading-order terms, we recover Eqs.�17�–�19� for the scaling fields, where the coefficients a1, a2,b2, and b4 in Eqs. �17� and �18� are to be identified with thevalues ai�P̂c

0� and bi�P̂c0� at the reference pressure P̂= P̂c

0,while the asymmetry coefficients a3 and b3 are given by

a3 = − �d�̂21,cdP̂

+ a1d�̂1,c

dP̂+ a2

dT̂c

dP̂� , �39�

b3 = − �dT̂cdP̂

+ b2d�̂1,c

dP̂+ b4

d�̂21,c

dP̂� . �40�

In these expressions the pressure derivatives are to be taken

along the critical locus at P̂= P̂c0. For the dependent scaling

field h3, we continue to use Eq. �19� which did not containthe incompressibility approximation. From Eqs. �39� and�40�, we see that in weakly compressible liquid mixtures thepressure does contribute to the scaling fields through thepressure dependence of the critical parameters.

We now consider the expansions of some physical den-sities as a function of temperature along the phase boundary.They are derived by substituting the asymptotic power laws�4� and �5� into Eqs. �21�–�25� and expanding the results into��T̂�. For any physical density Z, one then obtains an expan-sion of the form



Z� � Zc � B̂0Z��T̂��1 + B̂1

Z��T̂��1 + B̂2Z��T̂�2��

+ D̂2Z��T̂�2� + D̂1

Z� Â0−1 − �

��T̂�1−� − B̂cr��T̂�� . �41�The amplitudes Â0

− and B̂cr in Eq. �41� are related to the Isingamplitudes in Eqs. �4� and �5� by

Â0− = A0

−��0�2−�, B̂cr = Bcr��0�2, �42�

with �0 given in Eq. �34�. The amplitudes Â0− and B̂cr now

specify the asymptotic divergence of the dimensionless two-

phase molecular heat capacity ĈP=CP /kB,

ĈP � Â0−��T̂�−� − B̂cr. �43�

In Eq. �41� we have included a symmetric correction-to-scaling term with exponent �10.5, as was done by Kim etal.20 for a one-component fluid.

It is convenient to distinguish between the difference andthe sum of the values of the physical densities in the twocoexisting phases: the “width” of the phase boundary�Z+−Z−� /2��Zcxc and the “diameter” �Z++Z−� /2�Zd,which will be expressed in terms of the deviation function��Z++Z−� /2�−Zc��Zd so that

�Zcxc � � B̂0Z��T̂��1 + B̂1

Z��T̂��1 + B̂2Z��T̂�2�� 44�

and

�Zd � D̂2Z��T̂�2� + D̂1

Z� Â0−1 − �

��T̂�1−� − B̂cr��T̂�� . �45�In this paper, we shall consider specifically the mole

fraction x2, the dimensionless density ̂, and the dimension-less partial molar density ̂x2. The expressions for the coef-

ficients B̂0Z, B̂2

Z, D̂2Z, and D̂1

Z for these properties in the expan-sions �44� and �45� are given by

B̂0x2 = �1 − a1x2,c�B0��0��,

�46�B̂0

̂ = �a1 + a3�B0��0��, B0̂x2 = �1 + a3x2,c�B0��0��,

B̂2x2 =

a12�B̂0

x2�2

�1 − a1x2,c�2,

�47�

B̂2̂ =

a12�B̂0

̂�2

�a1 + a3�2, B̂2

̂x2 =a3

2�B̂0̂x2�2

�1 + a3x2,c�2,

D̂2x2 = −

a1�B̂0x2�2

1 − a1x2,c, D̂2

̂ =a3�B̂0

̂�2

a1 + a3, D̂2

̂x2 =a3�B̂0

̂x2�2

1 + a3x2,c,

�48�

D̂1x2 = �b2x2,c − b4��0�−1,

�49�D̂1

̂ = − �b2 + b3��0�−1, D̂1̂x2 = − �b4 + b3x2,c��0�−1.

The coefficient B̂1Z of the correction-to-scaling term in

the expansion �44� is the same for all properties: B̂1x2 = B̂1

̂

= B̂1̂x2. We note that for all properties the diameter �Zd has a

singular term proportional to ��T̂�2� and another singularterm related to the two-phase heat capacity. However, thephysical origin of the leading singularity is different, depend-ing on the property considered. For the mole fraction x2, the

amplitudes B̂0x2, B̂1

x2, B̂2x2, D̂2

x2, and D̂1x2 are independent of the

coefficients a3 and b3 representing the pressure contributionsin expressions �17� and �18� for the scaling fields. Hence, theexpansion for the mole fraction x2 of a weakly compressibleliquid mixture is identical to that for an incompressible liquid

mixture.26,27 However, the coefficients D̂2̂ and D̂2

̂x2 for the

singular term proportional to ��T̂�2� in the diameters for ̂and ̂x2 are proportional to a3 given by Eq. �39�. The pres-ence of a singular term proportional to ��T̂�1−� in the expan-sion of the diameters has been recognized for many yearssince it is already a consequence of revised scaling.10–15

However, a singular term proportional to ��T̂�2�, sometimesobserved phenomenologically, was traditionally thought tobe an artifact due to an “incorrect” choice of the orderparameter.33 We now see that the contribution proportional to

��T̂�2� in the expansion of the diameter of the mole fractionx2 is due to complete scaling �a1�0� as pointed outpreviously,26,27 while a similar singular term in the expansionfor the densities ̂ and ̂x2 is a consequence of the depen-dence of the critical parameters on pressure through the co-efficient a3�0 given by Eq. �39�.

We conclude this section by considering as an exampleexplicitly the expansions for the mole fraction x2. For thedifference �x2,cxc= �x2

+−x2−� /2 and the diameter �x2,d

= �x2++x2

−� /2−x2,c, we obtain

�x2,cxc � � B̂0x2��T̂��1 + B̂1x2��T̂��1

+ � a11 − a1x2,c

�2�B̂0x2�2��T̂�2�� , �50�

�x2,d � −a1

1 − a1x2,c�B̂0

x2�2��T̂�2�

+b2x2,c − b4

1 − b2Ŝc� Â0−

1 − ��T̂�1−� − B̂cr��T̂�� . �51�

In our previous papers,26,27 we have retained in Eq. �50� onlythe leading term proportional to ��T̂��, which already yieldsa good description of the asymptotic behavior of �x2,cxc.However, complete scaling with a1�0 yields not only a sin-

gular contribution in the diameter proportional to ��T̂�2� butalso a confluent singularity proportional to ��T̂�2� in the dif-ference between the mole fractions of the coexistingphases.20 Equation �51� can be written in the form



�x2,d � aeff�B̂0x2�2��T̂�2� − beff� Â0−1 − � ��T̂�1−� − B̂cr��T̂��

�52�

with

aeff = −a1

1 − a1x2,c, �53�

beff = −b2x2,c − b4

1 − b2Ŝc. �54�

Thus, the singular behavior of the coexistence-curve diam-eter is determined by two asymmetry coefficients aeff andbeff, as previously pointed out by Wang et al.

27

V. ROLE OF MOLECULAR VOLUMES

In a previous analysis of the coexistence-curve data forliquid solutions of nitrobenzene in a series of hydrocarbons,we found a close relationship between the asymmetry coef-ficients aeff and beff in Eq. �52� and the solute-solventmolecular-volume ratios.26,27 Hence, it is of interest to con-sider the relationship between the asymmetry coefficientsand the molecular volumes of the two liquid components.For this purpose, we note that

d�i = V̄idP − S̄idT , �55�

where V̄i and S̄i are the partial molar volume and partialmolar entropy of component i. From Eq. �55�, it follows thatfor the pressure derivatives of the chemical potentials at thecritical point

d�idP

= V̄i,c − S̄i,cdT

dP, �56�

where V̄i,c and S̄i,c are the values of the partial molar volumeand partial molar entropy at the critical point of the mixture.Substitution of Eq. �56� into expression �39� for the coeffi-cient a3 yields

a3 = �1 − a1�V̂1,c − V̂2,c − �a2 + �1 − a1�Ŝ1,c − Ŝ2,c�dT̂c

dP̂,

�57�

where V̂i= V̄i /Vc, Ŝi= S̄i /kB, and dT̂c /dP̂=ckBdTc /dP. FromEq. �20�, we note that

a2 = a1Ŝc = a1��1 − x2,c�Ŝ1,c + x2,cŜ2,c� , �58�

so that upon substituting Eq. �58� into Eq. �57�,

a3 = �1 − a1�V̂1,c − V̂2,c + �1 − a1x2,c��Ŝ2,c − Ŝ1,c�dT̂c

dP̂.

�59�

The experimental values of the dimensionless slope dT̂c /dP̂are usually small for liquid-liquid systems34 �see Table I�.However, as earlier mentioned in Sec. III, the reference en-tropies of the two components are arbitrary so that we can

select a value not only for the entropy at the critical point inaccordance with Eq. �20�, but we are also free to select avalue zero for its concentration derivative ��S /�x2�P,T= S̄2− S̄1 at the critical point. With this choice, we obtain fromEq. �59�,

a3 = �1 − a1�V̂1,c − V̂2,c. �60�

If we assume that the volume of mixing VE is negligiblysmall, then

a3 �1 − a1�V̂10 − V̂2

0, �61�

where V̂i0=Vi

0 /Vc with Vi0 being the molecular volumes of the

two individual components i=1,2. Actually, in the ideal-volumetric approximation, Eq. �61� follows directly from Eq.�57� due to an exact relation35,42

dTcdP

= limT→Tc,x→x2,c

T��2V/�x2

2�T,P��2H/�x2

2�T,P, �62�

from which one infers that dTc /dP=0 when V= �1−x2�V10

+x2V20. �H denotes the enthalpy in Eq. �62�.� Alternatively,

one can start from

̂ =1

�1 − x2�V̂10 + x2V̂2

0, �63�

so that asymptotically

̂ � 1 � �V̂20 − V̂1

0�B̂0x2��T̂��. �64�

By equating the amplitude of the ��T̂�� in Eq. �64� with theamplitude B̂0

̂ in Eq. �46�, one obtains Eq. �61�.It is interesting to consider three special cases. The first

is the incompressible limit �a3=0� so that

TABLE I. Dimensionless slope of the critical line dT̂c /dP̂ for a number ofliquid-liquid systems.

Systema dT̃c /dP̃

NP-C10 0.0063b

NP-C12 0.0073b

NP-C14 0.0081b

AN-CH 0.0057f

NB-C6 −0.0162h

NB-C12 0.0071c

NB-IO −0.0079d

NE-3MP 0.0030e

DMC-C10 0.0124g

EtOH-C12 0.0174g

aAbbreviated nomenclature: 1-nitropropane–decane �NP-C10�,1-nitropropane–dodecane �NPC12�, 1-nitropropane–tetradecane �NP-C14�,aniline–cyclohexane �AN-CH�, nitrobenzene-hexane �NB-C6�,nitrobenzene–dodecane �NB-C12�, nitrobenzene–isooctane �NB-IO�,nitroethane–3-methylpentane �NE-3MP�, dimethyl carbonate–decane�DMC-C10�, and ethanol-dodecane �EtOH-C12�.bReference 35.cReference 38.dReference 39.eReference 40.fReference 36.gReference 41.hReference 37.



a1 � 1 − V20/V1

0. �65�

Thus, the coefficient a1 is related to the solute-solventmolecular-volume ratio V2

0 /V10. If we label the liquid with the

smaller molecular volume as solvent 1, the coefficient a1 willbe negative. The asymmetry coefficient a1 vanishes asV2

0 /V10→1, whereas it becomes large for mixtures with very

dissimilar molecular volumes �i.e., when V20 /V1

0 is large�. An-other interesting case is a1=0 so that from Eq. �61�,

a3 = V̂10 − V̂2

0, �66�

and the coefficient a3 takes on negative values whose mag-nitude depends on the difference between the molecular vol-umes of the two individual components. Finally, whenV1

0=V20, Eq. �61� turns into

a3 = − V̂10a1. �67�

VI. ANALYSIS OF THE EXPERIMENTAL DATA

A. Overview

Visual methods �I� or refractive-index measurements �II�are the methods most frequently used for determining liquid-liquid coexistence curves.43 In method I, the phase-separation temperature of a mixture with a known composi-tion x is visually determined by cooling or heating thesample until mixing or demixing is observed. To obtain thex-T coexistence curves, this process needs to be repeated fora number of samples with different x. A limitation of themethod is that the set of phase-transition temperatures of thevarious samples can exhibit inconsistencies because of im-purity effects. In method II, one needs only one mixture witha near-critical composition for which the refractive index nof the two coexisting phases is measured as a function oftemperature. These measurements are converted to values forthe compositions of the coexisting phases by two main ap-proaches. In the first approach �IIA�, one uses the Lorentz–Lorenz �LL� equation for the refractive index. This proce-dure assumes that the LL equation does not exhibit anysingular behavior at the critical point.44 The validity of thisassumption is not obvious,45,46 but some experiments seem toindicate that deviations may be negligible in practice.47,48

Another approach �IIb� is to convert n�T� data into x�T� byusing the previously experimentally determined n�T ,x� func-tion that describes the behavior of n as the one-phase regionclose to the phase boundary, as adopted by An et al.49 Theaccuracy of this procedure depends on the accuracy of thecalibration function n�T ,x�. Method II has the advantage thatit directly provides data for the coexisting phase as a func-tion of temperature for a single sample.

In selecting coexistence-curve data, we have taken thelimitations associated with the various experimental proce-dures into consideration. As for method I, we have carefullychecked the internal consistency of the available x-T data.Moreover, for each original data point, the corresponding xvalue on the opposite branch of the coexistence boundarywas obtained by interpolation. As for method II, we haverestricted our analysis to the data obtained with method IIb�for simplicity, referred to as method II� to avoid any prob-

lems that may be associated with the use of the LL equation.In addition, data for perfluorohexane-alkane mixtures, ob-tained with an optical technique called “narrow-beam

-raying,”50,51 have been included for analysis. This experi-mental method, in which a regular volume of mixing wasassumed in the treatment of the original results, has all theadvantages of method II.

Experimental mole-fraction data have been converted todata for the volume fraction � via

�i =xiVi

0

V

xiVi0

x1V10 + x2V2

0 , �68�

if one neglects any volume of mixing �i.e., VE=0�. Then thepartial density is obtained by using the exact relation

xi = �i/Vi0. �69�

In previous publications,26,27 we have verified the pres-ence of complete scaling with a1�0 by analyzing the expan-sions for the mole fractions of a number of liquid-liquidsystems. To verify the presence of pressure mixing in com-plete scaling, i.e., whether a3�0, we prefer first to analyzeexperimental density data not possibly contaminated by anyapproximation. As far as we are aware, such data exist fornitroethane-3-methylpentane �NE-3MP� �Ref. 52� and forisobutyric acid–water.53,54 However, the system isobutyricacid–water exhibits little asymmetry. Hence, the first candi-date system for checking pressure mixing unambiguously isNE-3MP to be analyzed in Sec. VI B.

In Sec. VI C we report the results obtained from ananalysis of the x-T and x-T data for 37 systems listed inTables II and III. For the analysis we considered the originalcoexistence-curve data obtained by methods I and II, but

restricted to �T−Tc��10 K �or, equivalently, ��T̂��0.03�. Asan example, we show in Figs. 2 and 3 the coexistence-curvedata for methanol-heptane,60 a system in which asymmetryeffects are clearly present. In Fig. 2 we show thecoexistence-curve of this system for both the mole fractionx2 and the partial density ̂x2 as a function of temperature.Figure 3 shows the diameters �x2,d and ��̂x2�d as a functionof ��T̂� for this system. In Sec. VI C we shall describe theprocedures adopted for determining the asymmetry coeffi-cients a1 and a3 and the relationship between the observedasymmetries and the molecular volumes of the components.

B. Evidence for pressure mixing

Figure 4�a� shows the temperature dependence of thedensity diameter �̂d of NE-3MP.

It is evident that this diameter exhibits singular criticalbehavior. To check whether a contribution proportional to

��T̂�2� is present due to pressure mixing, we consider theexpansion �45� for the density diameter

�̂d � D̂2̂��T̂�2� + D̂1

̂ Â0−1 − �

��T̂�1−� − B̂cr��T̂�� . �70�To obtain �̂d, we first determined the critical temperature Tcas the temperature at which the coexistence-width vanishesand then the critical density c. From the literature data,

74



for the heat capacity of this system we find Â0−=5.23 and

B̂cr=2.66.75 A fit of Eq. �70� to the density diameter then

yielded D̂2̂=0.896 and D̂1

̂=−0.067. Although not shown ex-plicitly here, we have verified in preliminary fits that for anappropriate description of the density-diameter data a leading

term proportional to ��T̂�2� is required. This observation iscorroborated by the value 0.75 of the effective exponent �eff

,

defined as76 log �̂d=D0+�eff log��T̂�, which lies between

2�0.652 and 1−�0.89. We are thus led to conclude thatin accordance with theoretical predictions, density data forNE-3MP call for the presence of pressure mixing in the or-dering field.

C. Asymmetry coefficients a1 and a3

To design an adequate, physically reliable strategy fordetermining a1 and a3, it is useful to focus on x2 and x2.First, it is instructive to analyze the values for the effectivecritical exponents of the coexistence-curve diameters. Againwe first determined Tc and x2,c as before, namely, the formerwas identified with the temperature at which the coexistence-curve width vanishes, while x2,c is the value of the diameterat the critical temperature. From log-log plots �see Fig. 5 forillustration� we found that values for the effective exponent�eff

x2 of many mixtures �shown in Tables II and III� are closeto 2�0.65. Hence, it appears that for many weakly

TABLE II. Critical parameters Tc and x2,c, effective critical exponents �effx2 , reduced volumes V̂i,c of the pure liquids at the critical point, and results of fits to

Eqs. �71� and �72� �the relevant information here are the mixing coefficients a1 and a3� for systems whose coexistence curves were determined by visualmethods �see text�; superscript “calc” means “calculated values from Eq. �61�.”

Systema�Tb

�K�Tc�K� x2c �eff

x2 V̂1,c0 V̂2,c

0 a1 a3 a3calc B̂0 D̂1,eff

̂x2

DMC-C10c 5.3 286.602 0.314 0.59 0.707 1.640 −1.27 −0.02 −0.03 0.57 −0.04

DMC-C12c 7.7 297.842 0.264 0.60 0.691 1.864 −1.71 0.03 0.01 0.46 −0.01

DMC-C14c 9.7 307.663 0.223 0.65 0.683 2.100 −3.05 0.74 0.67 0.38 −0.13

DMC-C16c 9.2 316.212 0.198 0.69 0.672 2.316 −2.90 0.35 0.31 0.38 −0.03

NB-C5d 2.0 297.481 0.619 0.957 1.026 −0.26 0.18 0.18 0.65 0.12

NB-C6d 2.0 292.700 0.576 0.862 1.101 0.26 −0.46 −0.46 0.80 0.14

NB-C7d 3.2 292.221 0.534 0.965 1.031 −0.14 0.07 0.07 0.69 0.03

NB-C8d 4.6 292.625 0.487 0.777 1.234 −0.47 −0.08 −0.09 0.65 0.11

NB-C9d 3.1 294.130 0.455 0.746 1.304 −0.70 −0.02 −0.03 0.58 0.02

NB-C10d 1.0 296.111 0.421 0.724 1.380 −0.73 0.23 −0.12 0.65 0.12

NB-C11d 4.8 298.270 0.396 0.58 0.703 1.453 −0.89 −0.12 −0.12 0.55 0.01

NB-C12d 4.8 300.287 0.358 0.66 0.695 1.546 −2.04 0.58 0.57 0.42 −0.02

NB-C14d 5.1 305.052 0.319 0.62 0.683 1.676 −2.35 0.56 0.54 0.38 −0.12

PN-C5e 1.3 277.525 0.563 0.739 1.202 −0.03 −0.44 −0.44 0.87 0.11

PN-C6e 3.7 284.657 0.511 0.697 1.290 −0.21 −0.44 −0.45 0.83 0.11

PN-C7e 2.4 292.604 0.463 0.64 0.667 1.387 −0.55 −0.35 −0.35 0.73 0.06

PN-C8e 1.1 300.209 0.422 0.645 1.486 −0.51 −0.51 −0.51 0.77 0.11

PN-C10e 7.2 314.163 0.346 0.57 0.623 1.712 −1.43 −0.18 −0.20 0.57 0.00

MO-C5f 12.8 303.842 0.539 0.796 1.172 −0.08 −0.29 −0.31 0.89 0.16

MO-C6f 6.3 311.306 0.499 0.749 1.252 −0.05 −0.45 −0.46 0.96 0.14

MO-C7f 4.6 319.758 0.445 0.720 1.350 −0.56 −0.22 −0.22 0.78 0.11

MO-C8f 6.1 328.110 0.404 0.697 1.447 −0.89 −0.12 −0.13 0.71 0.03

MO-C10f 10.2 343.334 0.341 0.65 0.664 1.648 −1.04 −0.28 −0.29 0.67 0.08

MO-C12f 9.7 356.735 0.284 0.66 0.651 1.878 −1.79 −0.07 −0.09 0.55 −0.01

MO-br-C8g 4.3 319.341 0.423 0.61 0.682 1.437 −0.68 −0.29 −0.29 0.73 −0.01

MeOH-C7h 10.0 324.008 0.389 0.65 0.495 1.795 −1.03 −0.77 −0.79 0.60 0.07

MeOH-C10i 11.8 363.762 0.257 0.63 0.510 2.412 −2.31 −0.87 −0.73 0.47 0.19

EtOH-C14i 11.0 307.801 0.271 0.68 0.518 2.300 −2.54 −0.41 −0.47 0.39 −0.07

aAbbreviated nomenclature: dimethyl carbonate–decane �DCM-C10�, dimethylcarbonate-dodecane �DCM-C12�, dimethyl carbonate–tetradecane �DCM-C14�,dimethyl carbonate–hexadecane �DCM-C16�, nitrobenzene-pentane �NB-C5�, nitrobenzene-hexane �NB-C6�, nitrobenzene-heptane �NB-C7�, nitrobenzene-octane �NB-C8�, nitrobenzene-nonane �NB-C9�, nitrobenzene-decane �NB-C10�, nitrobenzene-undecane �NB-C11�, nitrobenzene-dodecane �NB-C12�,nitrobenzene-tridecane �NB-C13�, propanonitrile-pentane �PN-C5�, propanonitrile-hexane �PN-C6�, propanonitrile-heptane �PN-C7�, propanonitrile-octane�PN-C8�, propanonitrile-decane �PN-C10�, 2-methoxyethanol–pentane �MO-C5�, 2-methoxyethanol–hexane �MO-C6�, 2-methoxyethanol–heptane �MO-C7�,2-methoxyethanol–octane �MO-C8�, 2-methoxyethanol–decane �MO-C10�, 2-methoxyethanol–dodecane �MO-C12�, 2-methoxyethanol–2,2,4-trimehylpentane�MO-br-C8�, methanol-heptane �MeOH-C7�, methanol-decane �MeOH-C10�, and ethanol-tetradecane �EtOH-C12�.bExperimental data within �T−Tc��T have been included in the fit.cReference 55.dReference 56.eReference 57.fReference 58.gReference 59.hReference 60.iReference 61.



compressible liquid mixtures terms beyond ��T̂�2� are of mi-nor importance within the temperature range under consider-ation. It has been noticed previously for the �x2,d ofnitrobenzene-alkane solutions26,27 that, while a term propor-

tional to ��T̂�1−� is significant for NB-C5, its significancedecreases with increasing alkane number. With regard to��̂x2�d, some systems display nearly constant diameterswith �eff

x2 values, not shown here explicitly, that fall between2� and 1−� for a few systems �namely, NE-3MP, alcohol–alkane, and perflurorohexane-alkane systems�. In the moregeneral cases, the value of �eff

x2 varies from 0 to 0.5. We have

observed that the deviations of ��̂x2�d from zero are usuallyvery small �unlike for �x2,d�, thus making it difficult to

determine the nature of the behavior �singular or regular� of��̂x2�d. We expect singular contributions to be present, buthigher experimental accuracy would be required in order toidentify them unambiguously.

While our main interest is focused on coexistence-curvediameters, it will be useful to also study the widths of thecoexistence curves since their behavior is related to that ofdiameters, as was noted in Sec. IV. This way, we can obtaina complete analysis of the shapes of the coexistence curves.We thus have chosen to fit data for the diameter and for thewidth of both x2-T and ̂x2-T coexistence curves to the cor-responding theoretical expansions simultaneously. In accor-dance with the preliminary observations discussed in the pre-ceding paragraph, the following expressions were adopted:

TABLE III. Critical parameters Tc and x2,c, effective critical exponents �effx2 , reduced volumes V̂i,c

0 of the pure liquids at the critical point, and results of fits toEqs. �71� and �72� �the relevant information here is the mixing coefficients a1 and a3� for systems whose coexistence curves were determined fromrefractive-index, buoyancy, and -raying measurements �see text�; superscript “calc” means “calculated values from Eq. �61�.”

Systema�Tb

�K�Tc�K� x2,c �eff

x2 V̂1,c0 V̂2,c

0 a1 a3 a3calc B̂0 D̂1,eff

̂x2

DMC-C10c 9.1 287.102 0.311 0.709 1.644 −1.27 −0.02 −0.04 0.58 0.04

DMC-C14c 9.7 307.754 0.225 0.693 2.055 −2.24 0.14 0.12 0.44 −0.02

NB-C5d 10.1 297.104 0.612 0.958 1.026 0.14 −0.20 −0.21 0.79 0.13

NB-C6e 10.1 293.100 0.572 0.863 1.102 −0.05 −0.18 −0.19 0.74 0.11

NB-C7e 8.9 291.900 0.529 0.965 1.031 −0.10 0.03 0.03 0.75 0.02

NB-C8f 10.0 293.052 0.495 0.61 0.914 1.088 −0.19 0.00 0.00 0.72 0.02

NB-C10d 10.0 295.963 0.425 0.62 0.838 1.219 −0.57 0.09 0.09 0.64 0.01

NB-C11g 10.0 298.009 0.395 0.63 0.704 1.454 −0.77 −0.20 −0.21 0.61 0.05

NB-C12h 10.4 300.368 0.369 0.689 1.533 −0.86 −0.23 −0.25 0.58 0.04

NB-C13i 9.0 302.995 0.347 0.70 0.676 1.611 −1.18 −0.12 −0.14 0.53 0.01

NB-C14j 9.9 304.940 0.324 0.62 0.680 1.669 −1.51 0.00 −0.02 0.48 −0.01

NB-C16k 8.7 309.690 0.284 0.64 0.66 1.890 −1.80 −0.03 −0.04 0.45 −0.06

BN-C8l 5.7 283.225 0.539 0.762 1.203 −0.16 −0.31 −0.32 0.72 0.08

BN-C12m 11.0 293.028 0.416 0.67 0.664 1.472 −0.76 −0.29 −0.30 0.59 0.03

BN-C16n 9.4 304.400 0.324 0.64 0.624 1.783 −1.55 −0.17 −0.19 0.48 −0.03

BN-C18o 8.6 309.600 0.289 0.62 0.614 1.949 −2.04 −0.06 −0.08 0.43 −0.02

C6-PFC6p 6.7 295.691 0.359 0.62 0.746 1.143 −0.60 1.10 1.10 0.72 0.15

C7-PFC6q 11.3 316.266 0.389 0.70 0.811 1.120 −0.44 0.83 0.80 0.75 0.14

NE-3MPr 2.5 299.605 0.500 0.73 0.710 1.290 −0.32 −0.18 −0.35 0.81 0.20

aAbbreviated nomenclature: dimethyl carbonate–decane �DMC-C10�, dimethyl carbonate–tetradecane �DMC-C14�, nitrobenzene-pentane �NB-C5�,nitrobenzene-hexane �NB-C6�, nitrobenzene-heptane �NB-C7�, nitrobenzene-octane �NB-C8�, nitrobenzene-decane �NB-C10�, nitrobenzene-undecane�NB-C11�, nitrobenzene-dodecane �NB-C12�, nitrobenzene-tridecane �NB-C13�, benzonitrile-octane �BN-C8�, benzonitrile-dodecane �BN-C12�, benzonitrile-tetradecane �BN-C14�, benzonitrile-octadecane �BN-Cl8�, hexane-perfluorohexane �C6-PFC6�, heptane-perfluorohexane �C7-PFC6�, nitroethane–3-methylpentane �NE-3MP�.bExperimental data within �T−Tc��T have been included in the fit.cReference 62.dReference 63.eReference 64.fReference 65.gReference 66.hReference 67.iReference 68.jReference 49.kReference 69.lReference 70.mReference 71.nReference 72.oReference 73.pReference 49.qReference 50.rReference 51.



�Zd �Z+ + Z−

2− Zc � D̂2

Z��T̂�2� + D̂1,effZ ��T̂�1−�, �71�

�Zcxc �Z+ − Z−

2� B̂0

Z��T̂��1 + B̂1Z��T̂��1 + B̂2

Z��T̂�2�� ,

�72�

where Z can be either x2 or ̂x2. In the absence of heat-capacity data for many of the systems, it is not advisable inthe expansion �45� for the diameter to use the coefficients ofthe terms proportional to ��T̂�1−� and ��T̂� both as adjustableparameters since these two terms are highly correlated23 andthey have opposite signs. Hence, we have replaced these two

terms by one term proportional to ��T̂�1−� with an effectiveamplitude D̂1,eff

Z . Initially, the fitting parameters were a1, a3,

B̂0=B0��0��, B̂1x2 = B̂1

̂x2, D̂1,effx2 , and D̂1,eff

̂x2 . However, for most

systems, we have found that B̂1x2, B̂1

x2, and D̂1,effx2 are not

needed for a reasonably good thermodynamic description�see, e.g., Fig. 6�. Hence, to keep the number of parametersto a minimum so as to reduce correlation effects, the coeffi-

cients B̂1x2, B̂1

x2, and D̂1,effx2 were set equal to zero in the sub-

sequent fits. Actually, symmetric corrections-to-scaling areexpected to be small within the temperature rangeconsidered,17 while the net effect of the contributions propor-

tional to ��T̂�1−� and ��T̂�, which have opposite sign, appearsto be of minor significance for �x2,d, as discussed above.

However, for ��̂x2�d an �effective� ��T̂�1−� term, in additionto the ��T̂�2� contribution, is generally required.

All relevant information deduced from the fits—obtainedby labeling the liquid with lower volume per particle V ascomponent 1—is displayed in Tables II and III. The values of

the coefficients B̂2Z and D̂2

Z in Eqs. �71� and �72� can bereadily deduced from the information in Tables II and IIIwith the aid of Eqs. �47� and �48�. Before discussing theresults, we first note that, in all cases, the x-T diameter bendstoward the phase rich in the liquid with lower V. As Tables IIand III show, the a1 values are always negative, ranging from−3 to 0.

It is informative to analyze the results obtained for a1and a3 from different sources. This is possible fornitrobenzene-alkane and dimethyl carbonate–alkane mix-tures, for which data observed by both methods I and II areavailable. The results, depicted in Fig. 7 for thenitrobenzene-alkane series, exhibit a good internal consis-tency at a quantitative level. Hence, we confirm that mea-surements for the same system performed in different labo-ratories by using distinct experimental techniques yieldessentially the same results. It is then fair to assert that theobserved asymmetry effects are not due to experimental er-rors and/or numerical artifacts in data treatment and pointstoward the reliability of the adopted fitting approaches. In-consistencies between the values of the asymmetry coeffi-cients from different data sets arise when the diameter isalmost linear and the asymmetry effects are small.

The results for a3 must be considered with some cautionbecause they have been calculated under the assumption VE

=0, disregarding any excess volume effects in the estimationof the critical density. Regarding the singular behavior, it isdifficult to judge precisely to which extent any nonzero vol-

FIG. 2. Coexistence curves of methanol-heptane �Ref. 60� in the molefraction–temperature �x2-T� and �dimensionless� partial density–temperature�̂x2-T� planes and the corresponding coexistence-curve diameters.

FIG. 3. Mole-fraction diameter �x2,d and dimensionless partial diameter

��̂x2�d of methanol-heptane �Ref. 60� as a function of ��T̂�= ��T−Tc� /Tc�.

FIG. 4. �a� Reduced coexistence-curve diameter in the density temperatureplane �̂d for NE-3MP as a function of ��T̂�= ��T−Tc� /Tc�: the points indi-cate experimental data �Ref. 52�, while the solid and dashed curves representvalues calculated from Eq. �70�. �b� The corresponding log-log plot: data fitto log �̂d=D0+�eff

log��T̂�; the numerical value of �eff is provided.

FIG. 5. Log-log plots of the mole fraction–temperature �x2,d and partialdensity–temperature reduced diameters ��̂x2�d for methanol-heptane �Ref.60�. Data fit to a straight line: log �Zd=D0

Z+�effZ log��T̂� �with Z=x2 or ̂x2

and ��T̂�= ��T−Tc� /Tc��; numerical values for �effZ are provided.



ume of mixing matters. Nevertheless, a rough estimation ispossible in the case of NE-3MP, for which, as mentioned,density data at coexistence are available. Figure 8 shows theV-T coexistence curve for this system. As can be seen, thediameter varies 0.3 g cm−3 within �T−Tc��2.3 K. Experi-ments for this system77 reveal that the width of the VE-Tcoexistence curve at �T−Tc�2.8 K is approximately0.2 g cm−3. Estimating generously that the anomaly in theexcess volume diameter is not larger than the 25% of thewidth at �T−Tc�2.8 K, we get a maximum error of0.05 g cm−3. Clearly, such an error associated with the VE

=0 assumption does not change the picture significantly. Inany case, within the ideal-volumetric-mixing approximation,the following features of a3 are observed. When nonzero,this mixing coefficient is usually negative and small in mag-nitude �compared to a1�. However, some systems �e.g.,alcohol-alkane mixtures� do display a significant ��T̂�2� sin-gularity in their x2 diameters �see Fig. 3�. We note from Eq.�46� that negative values of a1 and a3 imply that the ampli-tude B̂0

̂ for the difference between the coexisting densities isalso negative. This means that the solute-rich phase has alower density than the solvent-rich phase.

We conclude with some comments regarding the role ofmolecular volumes. For all systems studied we have calcu-

lated a3 by introducing both the fitted a1 and the V̂i,c0 values

into Eq. �61�. The results are included �with the values of

V̂i,c0 � in Tables II and III. As can be seen, the a3 values pre-

dicted from Eq. �61� are in very good agreement with thoseobtained from the T-x data directly. This fact may not besurprising since the experimental data were determined inthe VE=0 approximation, which leads to Eq. �61�, but it doessuggest that the employed procedures for obtaining a1 and a3from experimental data are fairly adequate. As regards theparticular case a3=0, it is observed that values of a1 as afunction of V2,c

0 /V1,c0 for systems with nearly vanishing a3

values �say, �a3� less than 0.3� are in agreement with Eq. �65�:as Fig. 9 shows, they do follow a straight line with consistentvalues for both the slope and the intercept. On the otherhand, the case a1=0 �Eq. �66�� is unlikely to be of highinterest since we are not aware of any system exhibiting suchbehavior. Indeed, we have observed that a30 only whena10 �with V2,c

0 �V1,c0 �; this is the case of a fully symmetric

system �see, e.g., results for nitrobenzene-pentane�.

VII. DISCUSSION

We have shown how pressure is incorporated into thecomplete-scaling formulation of asymmetric liquid-liquidcriticality in weakly compressible binary mixtures. Pressuremixing in the scaling fields is found to originate from thedependence of the critical parameters on pressure. We have

demonstrated that contributions proportional to ��T̂�2� in the

FIG. 6. The coexistence-curve width �Zcxc and diameter �Zd for dimethyl

carbonate–tetradecane �Ref. 62� as a as a function of ��T̂�= ��T−Tc� /Tc�. ��Z=x2; �� Z= ̂x2. The solid curves represent values calculated from Eqs.�71� and �72�.

FIG. 7. Mixing coefficients a1 and a3 for the nitrobenzene-alkane �NB-CN�series obtained from the experimental data of type—see text—I �� and II�using LL equation �� and the methodology explained in Ref. 49 ��.

FIG. 8. The coexistence curve of NE-3MP in the molar volume–temperature�Vm-T� plane and the corresponding coexistence-curve diameter as a func-tion of ��T̂�= ��T−Tc� /Tc�.

FIG. 9. Mixing coefficient a1 as a function of the solute/solvent molecularvolume ratio V2,c

0 /V1,c0 at the critical temperature for a variety of systems.

Data fit to a straight line with slope and intercept as indicated.



coexistence-curve diameters in the x-T and x-T planes arisefrom �1 mixing and P mixing into the ordering field,respectively.

Experimental data for a large number of mixtures sup-port the existence of these singularities in the coexistence-curve diameters. They are especially relevant for the case ofx-T coexistence curves, from which we have found reliablevalues for the mixing coefficient a1. The situation regardinga3 is somewhat less definitive since only data for one system�NE-3MP� could be investigated without any assumption ofideal-volumetric mixing. That is, actual experimental meth-ods provide good x-T data, while, in the majority of cases,-T and x−T coexistence curves are obtained indirectly ei-ther by assuming that VE=0 or by using approximated, regu-lar VE�x� functions. Therefore, it would be desirable to obtainaccurate -T measurements for systems for which good x-Tdata are available.

We have found that the molecular volume plays an im-portant role in liquid-liquid coexistence asymmetry. In par-ticular, the x-T diameter always bends toward the phase richin the component with lower molecular volume. �For pure-fluid criticality the diameter curves toward the less densevapor phase�. More fundamentally, thermodynamics pro-vides useful insights on the role of molecular volumes fromfairly simple arguments: by appropriately choosing the zero-point entropy values of the pure components, and assuming avanishing volume of mixing, we have found that a1 and a3are mutually related via the molecular volumes. As a matterof fact, for the �quite frequent� a30 case, it has been shownthat a1 correlates remarkably well with the solute/solventmolecular volume ratio.

To put the above findings in an appropriate context, wefirst note that the notion that “asymmetry of liquid-liquidcoexistence curves is �in part� driven by molecular volumes”is not new. It is essential, however, to characterize preciselywhat is meant by asymmetry. Before the development of themodern theory of critical phenomena, the deviation of thecritical composition from 0.5 and the value of the slope ofwhat was at that time called “linear” diameter were appro-priate measures of asymmetry. The issue has been discussedby Damay and LeClercq78 from a more contemporary point

of view. Specifically, they suggested that the ��T̂�2� singular-ity is due to what they called the size effect, while the

��T̂�1−� singularity originates from field mixing. Our analysisagrees with theirs in that, under certain restrictive assump-

tions, the ��T̂�2� anomaly in the x-T diameter is controlled bythe solute/solvent molecular volume ratio �or size effect�.However, as we have demonstrated, both sources of asym-metry arise from field mixing. In other words, complete scal-ing incorporates, in a natural way, also the size effect into thethermodynamics of the system in the critical region.

In this work, we have confined ourselves to mixturescomposed of molecular liquids. Further work will entailstudying the coexistence curves of mixtures involving com-plex fluids. Polymer solutions are interesting in that there is alarge difference between the molecular volume of the poly-mer and that of the solvent; however, care must been takensince these systems display also crossover from Ising critical

behavior to �mean-field-like� theta-point tricriticality as thepolymer size is increased.79–81 The critical behavior in ionicsolutions—which is Ising-like29—also displays large asym-metry effects82 with some special subtle features.83 Applyingcomplete scaling to these problems will be another interest-ing problem.

ACKNOWLEDGMENTS

Financial support from the “Xunta de Galicia” underGrant No. PGIDIT-06PXIB3832828PR has been greatly ap-preciated. The research of G.P.-S. and of P.L.-P. was sup-ported by the “Ministerio de Educación y Ciencia” of Spainunder the “Programa Nacional de Formación del ProfesoradoUniversitario” �Grant Nos. AP-2005-1959 and AP-2004-2947�.

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