17 IONIC F L U I D S N E A R C R I T I C A L P O I N T S A N D A T HIGH T E M P E R A T U R E S
J.M.H. Levelt Sengers a, A.H.Harvey b, and S.Wiegand a*.
Physical and Chemical Properties Division National Institute of Standards and Technology (NIST) aGaithersburg, MD 20899, U.S.A.," bNISTBoulder, CO 80303, USA.
17.1 17.2
17.3
17.4
17.5
17.6 17.7
Introduction Criticality and Ionic Fluids 17.2.1 Criticality and Critical Exponents 17.2.2 Criticality and Range of Forces 17.2.3 Phase Separation due to Coulombic Interactions 17.2.4 The Restricted Primitive Model, RPM 17.2.5 Criticality of the RPM 17.2.6 Tricriticality and Charge-Density Waves in Ionic Fluids Critical Behavior in One-Component Charged Systems 17.3.1 Water 17.3.2 Liquid Metals near the Vapor-Liquid Critical Point 17.3.3 Molten Salts near the Vapor-Liquid Critical Point Experiments in Partially Miscible Ionic Liquids 17.4.1 The RPM as a Guide 17.4.2 Character of Criticality in Nonaqueous Ionic Solutions 17.4.3 Character of Criticality in Aqueous Ionic Solutions 17.4.4 Viscosity 17.4.5 Summary 17.4.6 Crossover: a Perspective Solution Thermodynamics near Critical Points 17.5.1 Principal Issues 17.5.2 Thermodynamic Properties of Interest 17.5.3 Criticality of Dilute Solutions 17.5.4 Ionic Effects on Critical Behavior Gibbs Energy Models for High-Temperature Aqueous Electrolyte Systems Helmholtz Energy Models; Equations of State 17.7.1 Models Assuming Complete Ionic Dissociation 17.7.2 Models Assuming no Ionic Dissociation 17.7.3 Models with Partial Ionic Dissociation 17.7.4 Solid Solubility Calculations
This chapter deals with aspects of topics treated in other chapters of this volume, namely the critical region (1), and ionic fluids (2). Studies on critical phenomena, however, have until recently mostly avoided the complications introduced by charged species, while traditional work on ionic fluids has avoided the complications that arise in the vicinity of the solvent's or mixture's critical point. Recently, there has been a concerted effort, both experimental and theoretical, to arrive at a better understanding of these issues. In this chapter, we review some of this work and attempt to put it into perspective.
From the standpoint of critical-phenomena theory, the behavior of thermodynamic and transport properties of pure fluids and their mixtures is known and understood in considerable detail. Serious questions exist, however, as to the applicability of the methodology when long- range Coulombic forces are present. These questions have recently generated substantial interest, controversy, and experimental and theoretical research, and some answers are beginning to emerge.
From an engineering perspective, modeling the thermodynamics of aqueous solutions near the vapor-liquid critical line offers special challenges because electrolytes, air constituents, and organic materials may all be present as chemically active solutes in one or more aqueous phases near or above the water critical point. The needs of geochemistry and of the oil and gas industry, and the recent interest in hydrothermal oxidation and hydrothermal chemistry drive the modeling efforts. The behavior near the water critical point is a challenge, because the Gibbs-energy concepts developed for electrolyte solutions during this century provide an awkward framework for property formulations, due to strong divergences of the standard states.
In order to keep the chapter manageable, we have decided to give cursory treatment, but ample bibliography, to many topics that have been covered elsewhere.
After an introduction of theoretical concepts in Section 17.2, Section 17.3 summarizes what is known about critical behavior of one-component charged systems. Section 17.4 describes the experimental critical behavior in nonaqueous, ionic, liquid mixtures where Coulombic forces are believed to be the cause of phase separation, as well as in aqueous, ionic, liquid mixtures. This chapter finishes with a view of an emerging understanding in terms of crossover theory. Section 17.5 treats the critical effects on thermodynamic behavior near the solvent's critical point. Section 17.6 discusses the thermodynamic models for aqueous systems at elevated temperatures, as based on the Gibbs-energy approach, while Section 17.7 discusses those based on the Helmholtz-energy approach. The concluding remarks, Section 17.8, summarize the principal conclusions of this review, and formulate some directions for further research.
17.2 CRITICALITY AND IONIC FLUIDS
17.2.1 Criticality and Critical Exponents
As a one-component fluid approaches the vapor-liquid critical point from above, the compressibility of the system increases to infinity strongly, preparing for the system to separate into two phases of different density but equal pressure. In a partially miscible fluid mixture, an osmotic susceptibility diverges strongly, in preparation for the mixture to separate into two phases of different composition at the same pressure and temperature.
Long-range density or composition fluctuations can develop easily near critical points because they cost little free energy. The spatial extent of such fluctuations, as measured by the correlation length, also diverges at the critical point (1).
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The anomalous behavior of thermophysical properties of fluids and fluid mixtures is a function of the distance from the critical point. The distance is measured in terms of temperature, composition, or other suitable variable, along a path of approach that needs to be specified. The anomalies in the various properties are customarily expressed by means of a power of this distance, called a critical exponent. The factor that multiplies the power law is called a critical amplitude. A brief summary of the properties, variables, paths, power laws and critical exponents used in this chapter is given in Table 17.1 for systems of dimension d = 3. The critical exponents are interrelated by equalities, which are also given in this table. Only two of the critical exponents are independent, and the equalities provide an important check on the accuracy and consistency of measured critical exponents. Likewise, only two critical amplitudes are independent. The term "two-scale-factor tmiversality" expresses this fact, and implies relationships between the many critical amplitudes one can define. The origin of the two different sets of critical exponents in Table 17.1 is described in the next section. For further details, the reader is referred to Chapter 11.
In Table 17.1, the separation into one-and two-component systems is oversimplied, since binary mixtures near vapor-liquid critical points can smoothly go over into one-component criticality by reducing the fraction of one of the components. The crossover from mixture behavior to one-component behavior is a subtle process, some aspects of which are discussed in Section 17.5.
17.2.2 Criticality and Range of Forces
Engineering equations of state, such as the cubic equations modeled on the Van der Waals equation, are analytic in density and temperature at the critical point and are denoted as having classical or mean-field critical behavior. Their critical exponents, defined in Table 17.1, are simple integers or rational fractions. In the standard mean-field theory, the coexistence curve is parabolic, and the critical isotherm is cubic in shape.
Kac et al. (3) showed for the one-dimensional case that the Van der Waals equation would be exact, and therefore mean-field critical behavior would prevail, if each (hard) particle would interact with many others through weak interactions, V(r)=-7a exp(-),r), of a range so long (~, very small) that it always exceeds the range of the fluctuations. The Van der Waals forces between molecules, however, are not long-ranged. Therefore, although classical equations of state may be appropriate for systems far from critical points when fluctuations are small, they break down when the correlation length becomes substantially longer than the molecular interaction range. It is said that the system 'crosses over' from classical to nonclassical critical behavior as the correlation length begins to exceed the range of the forces. One measure of failure of the mean-field theory is the Ginzburg number, or Ginzburg temperature. It represents the reduced temperature distance from the critical point at which the system is in full crossover. In simple one-component molecular fluids, the Ginzburg temperature is typically about 0.01, the equivalent of a few K for systems with critical points near room temperature. For fluctuations to be negligible, the reduced temperature distance from the critical point must be at least ten times the Ginzburg number (1).
Ample experimental evidence indicates that criticality of most fluids and fluid mixtures belong to the Ising-model tmiversality class, as theoretically expected in systems with short-range forces. Experimental critical-exponent values for these fluids are quite different from the mean- field values displayed by engineering equations of state, but are close to those displayed by the Ising model or its lattice-gas variant (Table 17.1).
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Table 17.1 Power laws, paths and critical exponents 1.
Property Application Power laws Path
compressibility one comp.
susceptibility two comp.
* = r l t l " b¢T
* [-r x = F i t
P = P c
X "-- X c
heat capacity one comp.
two comp.
Cv=AI t
* [-Or C p x = A l t
P = P c
X = X c
coexisting densities one comp.
coexisting compositions two comp.
pressure one comp.
p,~q- p,,~, = 2 B 'ltl ~
I II i~ X -X =2B It
* * [8 lAP I = D I A p T=T~
correlation length either
correlation function either
viscosity either
Exponent values Classical Real fluids
7 1 1.24
a 0 0.11
13 1/2 0.326
6 3 4.80
r/ 0 0 .033
v 1/2 0.63
y 0 0.04
- ~01tl -v
G(r) oc 1 / r a-e+"
p = pc, X = Xc
at critical point
Y/crit / T/back -- HI t l y p = pe, x = xc
Exponent equalities
Thermodynamic
7=/3(6-1)
2 - a = 13(6+1)
Fluctuations
7 = v(2-r / )
d v = 2 - a
1T = temperature, p = density, x = composition, d dimensionality, t = (T-Tc)/Tc, Ap* = (p -pc)/pe.
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It has been theoretically proven (4-7) that if the attractive interaction between molecules Vattr as a function of distance r, is written as Vattr "~ - ( l / / ' ) p, then for p > 4.97 the system will have Ising- like non-mean-field critical exponents. For smaller values ofp, there will be departures from Ising exponent values (4-7). For Coulombic interactions, which fall off as 1/r, critical exponent values are not known.
It is tempting to hypothesize (8) that the long-range Coulombic interactions might lead to mean-field critical behavior. It should be appreciated, however, that Coulombic long-range behavior is of a different mathematical form than the Kac potential, so Kac's argument (3) does not apply. Moreover, the range of the bare Coulombic potential is of little practical interest, since, according to the theory of Debye and Hiickel (9), Coulombic interactions are screened. An ion collects around it a "cloud" of counterions. The size of such a cloud is measured by a screening length characterizing the average range of the inter-ion interaction. The inverse of the screening length is indicated by re. For ions of charge _+q, rc is given by
2
2 = q P ~: eoekT (17.1)
with p the ionic number density, e0 the vacuum permittivity, e the relative permittivity or static dielectric constant, k Boltzmann's constant, and T the absolute temperature. Screening modifies the bare Coulomb potential into an effective short-range interaction potential,
2 q 1
4rCeoe r (Coulombic)
2 -~cr q e
4zrc0E r (screened).
(17.2)
The lower the dielectric constant, the lower the temperature and the higher the ion concentration, the more effective the screening and the shorter the distance over which the potential decays (Equation (17.1)). The effective ionic interactions are therefore not long-ranged. This casts doubt on the argument that Coulombic interactions must lead to mean-field behavior because they are long-ranged.
17.2.3 Phase Separation due to Coulombic Interaction
The interest in criticality in ionic systems has initially been driven by the question: is ionic criticality different from that in simple fluids because of the presence of the long-range Coulombic interactions (9-11)? Weing~trtner (12,13) divided systems into two classes, one in which the presence of charges is not essential to the occurrence of the phase transition, and one in which this presence is the origin of the phase transition. See also Friedman (14), Xu et al. (15), and Stell (16). One class consists of cases in which the underlying phase transition occurs even in the absence of ions; this is a non-Coulombic phase transition. An example of this case would be the addition of salt to water near the water vapor-liquid critical point (Section 17.5.4); while the salt may shift the phase transition and critical point, the driving force would continue to be non- Coulombic. An important subclass of non-Coulombic transitions is that of solvophobic separations (12,13,16,17), which may occur in liquid mixtures in which the attraction between
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unlike species is weak: hydrophobic (17), or, more generally, solvophobic (12). An example is the partial miscibility of water and weakly ionized organics. The model for non-Coulombic criticality is the Ising model or its lattice-gas equivalent.
In the case of Coulombic phase separation, however, the charges themselves are the driving force for the phase transition. A critical-point phase separation into a low- and a high- concentration phase may result. A model for Coulombic criticality is the so-called restricted primitive model (RPM), described in Subsection 17.2.4.
Realizing a system with strong ionic interactions is more complicated than one may think. A solvent of low dielectric constant will lead to strong screening and effectively short-ranged forces. Also ions, especially small or divalent ones, will tend to associate, so that few free ions remain in solution. Only monovalent organic salts with large ions are soluble in low-dielectric solvents, and will partially dissociate. But then, the electrostatic forces between the ions are smaller because of their bigger size.
Although there has been speculation that asymptotic Coulombic criticality might not be Ising-like, neither experiment nor theory has confirmed this, as will be shown in what follows. Nevertheless, when comparing Coulombic with non-Coulombic systems, there are important differences in the location of the phase boundary, and in the character of the crossover from Ising to mean-field behavior. Thus, we will briefly summarize what is known about the RPM, and make use of this model to select those experimental systems in which Coulombic forces are believed to drive the phase separation.
17.2.4 The Restricted Primitive Model, RPM
In this model the ionic fluid is described as a system of equal-sized, charged, hard spheres imbedded in a homogeneous dielectric medium of relative permittivity e. Phase separation was expected in RPM-like systems as early as the 1960s, and Monte-Carlo simulations in the early 1970s confirmed this. Stell and coworkers developed the first systematic approach in the mid- 1970s (18), and localized the region of phase separation, including the critical point. The most striking features are the low values of the critical temperature and concentration. Stell admirably reviewed the history (16). Especially in the past decade, great strides have been made in the refinement of the estimates of the phase boundary and critical point location of the RPM, and the identification of the most important physical effects. Amongst these a dominant effect is ion association, the formation of mostly neutral aggregates of ions. Near the critical point, few free ions are present. Thus, Shelley and Patey showed that a system of hard, dipolar dumbbells behaves like the RPM (19). Stell and coworkers have refined their integral-equation-based theories (16,20). Fisher and coworkers have developed an alternative approach, successively adding excluded-volume effects, ion association, and the solvation energy of dipolar pairs to the Debye-Hiickel flee energy (21-23). Computer simulation of the phase boundary continues to lead to lower critical temperatures as methods improve (24-28).
Results of RPM theory are conveniently displayed in a corresponding-states plot, as introduced by Friedman and Larsen (29). The basic parameters of the theory are the hard-core diameter a, and the Bjerrum length b, the latter being the distance at which the ionic-interaction energy equals kT. They define a reduced density, or ion concentration c*, and a reduced temperature T*
c = p / a 3 • T =a/b=akT(4rCe, oe,)/q 2, (17.3)
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Figure 17.1 shows such a 'Friedman plot', on which several estimates are indicated of the location of the coexistence curve of the RPM.
17.2.5 Criticality of the RPM
At present, the evidence is growing that the asymptotic critical behavior of the RPM is Ising- like. The accuracy of computer simulations (24-28), though steadily improving, is not high enough to yield reliable estimates of critical exponents. The evidence for Ising behavior comes from theoretical considerations, and from calculations of the Ginzburg number for the mean-field models that have been formulated.
"1"--.
0.09
0.08
0.07
0.06
0.05
0.04 1E-3
c ' ~ 0.070
~ T -- 0.049
•
. . . < > - "
° - ,
~,.." -",.~.
• . . . . . . . . . . . " . . . . . . . . . . . .
/
/ | | | | | | | i ] | ! ! | | i | ! I
0.01 0.1
C
Figure 17.1 Predictions of the coexistence curve of the RPM, in reduced coordinates (29): full curve, Stell (16); long-dashed curve, Fisher and Levin (23); short-dashed curve and diamonds, Valleau (24); dotted curve, simulations: solid triangle, Caillol (27), open triangle, Panagiotopoulos (25); dot-dash curve, simulations: open circle, Orkoulas (26), solid circle, Orkoulas (28).
Hafskjold and Stell (30), on the basis of integral-equation theory, have argued that charge fluctuations are decoupled from the mass-density fluctuation and remain finite; thus the RPM must have Ising-like critical behavior. Stell warns, however, that the special synlmetry of the RPM (ions of the same size and charge) causes the decoupling, so that Ising behavior cannot be taken for granted in real fluids which seldom if ever display this symmetry (16).
Estimates for the Ginzburg temperature have come from Leote de Carvalho and Evans (31), Fisher and Lee (32,33), and Schrrer and Weiss (34). The results are converging to values that, if anything, are larger than those of simple fluids. We refer to Section 17.4.6 for more detail.
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17.2.6 Tricriticality and Charge-Density Waves in Ionic Fluids
Repeatedly, in the past 15 years, there have been reports in the literature that two order parameters may play a role in charged systems. Nabutovskii et al. (35), on the basis of a phenomenological Landau-Ginzburg treatment, distinguish an order parameter for the vapor- liquid phase transition, and one for a transition to a periodic charge-density wave. Ho3,e and Stell (36) found such a transition in a system of ions in a dipolar solvent. Ciach and Stell (37) recently performed analytic calculations for the lattice RPM. To their surprise, the phase diagram was different from that of the continuum RPM. A tricritical point appeared, with a lambda line emerging from it, as happens in polymeric systems of increasing molecular weight, or in colloidal systems (c f , Section 17.4.6). Panagiotopoulos and Kumar (38) simulated the lattice RPM, confirmed the Stell results, and also showed that decreasing the lattice size well below the ion size returns the phase envelope characteristic of the continuous RPM. Next, Ciach and Stell (39), adding Van der Waals forces to the continuum RPM, found similar more complex phase behavior as in the lattice RPM.
Theoretical evidence is accumulating that certain charged systems are capable of a phase transition to an organized state which in principle can compete with the ordinary vapor-liquid or liquid-liquid phase separation and give rise to some sort of tricritical point. It is obvious that the presence of a nearby real or virtual tricritical point will affect the character of crossover to a critical point. See also Section 17.4.6.
17.3 CRITICAL BEHAVIOR IN ONE-COMPONENT CHARGED SYSTEMS
In this section we summarize what is known experimentally about the criticality of three categories of conducting one-component fluids: water, liquid metals and molten salts.
17.3.1 Water
Water is a slightly conducting fluid. Near its critical point, the ion product, on a per unit mass basis, is at least three orders of magnitude larger than at room temperature, and the dielectric constant is low, about 5, so that Coulombic forces are strong. The critical behavior of water has nevertheless been shown to be fully Ising-like, no different from that of nonpolar fluids (40,41).
17.3.2 Liquid Metals near the Vapor-Liquid Critical Point
Even the most accessible metallic systems have vapor-liquid critical points well above 1500 K, with pressures exceeding 100 MPa, which calls for fearless and pioneering experimentation. Dillon et al. (42) measured the vapor-liquid coexistence curve of sodium, and Hensel and Franck (10) that of mercury. In these cases the high critical temperatures were only approximately known, the temperature uncertainty was 10 K or more, and the approach to the critical point no closer than 50 K, so that the asymptotic region was probably not probed.
Nevertheless, a composite logarithmic plot in Buback's Ph.D. thesis (43) is quite provocative. It is shown in Figure 17.2. For a variety of substances, the coexisting density difference versus the temperature difference from critical is plotted. The two metallic systems show a distinctly different slope compared to the Ising-like slopes for simple gases. Subsequent more refined and detailed studies of Hensel and coworkers on the vapor-liquid coexistence curves of cesium and rubidium (44), and on that of mercury (45), however, revealed Ising-like
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critical exponent values. Measurements of the conductivity show complex features for the near- and supercritical
regimes for these metallic systems. Mercury, cesium and rubidium all undergo a nonmetal-to-metal phase transition at densities substantially higher than the critical density. Thus the top of the coexistence curve does not correspond to an ionic state at all. Moreover, acoustic studies by Kozhevnikov et aL (46) in mercury have revealed evidence of a wetting-nonwetting phase transitions close to the coexistence curve on the vapor side, and in the supercritical regime (47).
10
¢?, E 0 03
._u-
0.1
I I I
_ f H g
J ~ [3 : 0.40
..olo--O BiCl3 Xe ~
- ~:Z~...(9.~ ~HF~~.50 NH4cl 0 . 3 4
o. o
I , I
1 10 100 Tc- T,K
I
1000
Figure 17.2 Log-log plot for coexisting densities of a variety of fluids (43), including polar, ionic and metallic ones.
17.3.3 Molten Salts near the Vapor-Liquid Critical Point
Some low-melting salts have critical temperatures below 1000 K, and their critical points are therefore more accessible than those of metals, even though the critical pressures are still well over 100 MPa. Johnson and Cubiciotti (48) studied the vapor-liquid coexistence curve of bismuth chloride. Buback and Franck (49) did this for ammonium chloride. Both cases are displayed in Buback's thesis, Section 17.3.2 and Figure 17.2, with bismuth chloride showing Ising-like, and ammonium chloride mean-field behavior. In neither case, however, is the critical point approached to within what is presently considered to be the asymptotic regime. Buback's conductivity measurements indicate that liquid ammonium chloride is mostly ionized even as close as 32 K fi'om its critical point of 1155 K, and he estimates that this is also true at the critical point. This is, however, not the case for bismuth chloride which is estimated to be less than 1% ionized at its critical point. Buback and Franck stress that although ammonium chloride may display a Van-der-Waals-like critical-exponent value, this does not imply it obeys a Van der Waals equation. They draw attention to the critical compressibility factor Pc Vc/RTc, which, at
814
1.86, is about a factor of five higher than that of the Van der Waals equation. This very high critical ratio is apparently not caused by an unusually low critical density, as in the RPM: the ratio of the critical to the triple-point liquid density is approximately 0.37, just about what it is for ordinary liquids.
17.4 EXPERIMENTS IN PARTIALLY MISCIBLE IONIC LIQUIDS
We summarize the experiments, and their outcomes, in partially miscible, ionic binary mixtures, as performed during the past two decades. The experimental situation is complicated, but a picture begins to emerge. Reviews from the mid-1990s (11,50) emphasize experimental aspects of these systems.
Table 17.2 lists all systems, nonaqueous and aqueous, discussed in this section. Several different properties, experimental methods, and paths of approach to criticality were used in the experiments listed in Table 17.2. Properties studied include turbidity, intensity and dynamics of scattered light (usually at a 90-degree angle), and the coexistence curve (composition or refractive index of coexisting phases).
Table 17.2 also includes information on the nature of criticality and on the approach to it. For the vast majority of the data, Ising-like limiting critical behavior (I) is found. Only exceptionally, mean-field (MF) behavior is reported. In many cases, the Ising behavior is limited to a small region near the critical point, and corrections to scaling, as proposed by Wegner (51), are needed. If an effective exponent moves uniformly towards the Ising limiting value, the Wegner correction is positive (W+). In simple fluids, for instance, the effective exponent, beginning with the Ising value at the critical point, moves monotonically (W+) away from this point over many decades in reduced temperature, never to reach the mean-field limiting value. If the approach is nonuniform, "overshooting" the Ising value and then turning towards it, the Wegner correction is negative (W-). In a few cases, the full crossover (Cr) from the Ising value nearby to the mean-field value further out has been noted, within a typical reduced temperature range of t < 5.10 -2.
Although individual experimental uncertainties vary, some general remarks pertain to the amount of information available in each of the properties investigated. Turbidity is the property which is the least informative. The reason is that turbidity diminishes rapidly away from the critical point, while window reflections then become a dominant source of error; close to criticality, turbidity requires large Omstein-Zemike corrections. The main source of uncertainty in light scattering studies is multiple scattering, which is otien large in systems with unmatched refractive indices and large correlation length amplitude. It needs to be carefully corrected for, to the extent possible. The most informative property is the coexistence curve composition or refractive index, which can usually be accurately determined over many decades of reduced temperature.
17.4.1 The R P M as a Guide
A starting point for bringing some order into the wealth of systems studied is the Friedman diagram of corresponding states in the RPM, introduced in Section 17.2.3. Figure 17.3 shows, in reduced variables, the critical points of the ionic binary mixtures of Table 17.2, both nonaqueous and aqueous.
Table 17.2 and Figure 17.3 have drawn heavily upon the systematic studies by Weingartner (17) on the critical points of many ionic solutions that show liquid-liquid demixing; he compared
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Table 17.2 Experiments in near-critical ionic fluids.
System Ref. Csolvent To, K c2 T~* ~ ~0, Nature of Symbol (To) nm criticality
Na/NH3 (56,57) 22.9 231.6 0.065 0.127 0.04 Cr []
N4444Pic/chloroheptane (60,61) 4.00 414.400 0.061 0.061 0.07 MF o
Nn444Pic/1-tetradecanol (66) 3.60 351.090 0.068 0.046 0.25 I, W+ zx
Figure 17.3 Critical points of ionic binaries in their relation to simulation results for the RPM, dotted curve (26,28). For symbols, see Table 17.2.
the reduced critical parameters of these solutions with the predictions of the restricted primitive model (RPM). Those systems whose critical parameter values agree with the predictions of RPM are considered to be 'Coulombic' (12,16). This excludes the aqueous binaries with monovalent ions, which have much higher reduced critical temperatures because of the large dielectric constant of water; for these solvophobic systems the RPM is not an appropriate model.
For comparison, recent computer-simulation results for the coexistence curve in RPM (26,28) are also shown. It is obvious that the critical points of the ionic salts in solvents with low dielectric constants lie mostly in the general vicinity of the predicted range for the RPM critical point (Section 17.2.3). Only the critical concentration of ethyl-ammonium nitrate in octanol, c* = 0.4, is at a value much higher than that of the RPM.
For all these systems the value of the reduced critical concentration is quite uncertain because of difficulty in representing real molecules as hard spheres, and of uncertainty in estimating the effective sphere radius. Some possible choices for the ionic diameter, a, are the distance between cation and anion, or the radius of a sphere with the Van der Waals volume of the molecule. In some cases, these choices may differ by as much as 50%, leading to quite a large uncertainty in c* = p.a 3, see Equation (17.3). An alternative way of checking conformity with RPM predictions, which does not rely on the value of a, was proposed in Figure 11 of reference 52, in which (co) 1/3 / Tc is plotted versus e, leading to a straight line for the RPM.
For both nonaqueous and aqueous, nonionic binaries, the experimental critical composition, if indicated as a volume fraction, is usually near 0.5. For nonaqueous ionic binaries, the critical volume fraction of the salt tends to be considerably below 0.5, see Table 17.2. This type of asymmetry is also observed in polymeric solutions.
817
We have given no uncertainties for the reduced critical temperature T* = (4neo)e a kT/(q2), because there is no way of evaluating the uncertainty of e ; in Figure 17.3, we used a literature value of the dielectric constant of the pure solvent and took into account the temperature dependence. The dielectric constant of a solution containing a polar solute may increase steeply with concentration (53), but there is no good way of taking this effect into account within the framework of the RPM.
In principle the RPM might be used to predict critical points and give guidelines for finding appropriate systems, but reality appears to be much more complicated. There are several ionic systems for which the behavior contradicts the expectations of the RPM (54). Compare, for instance, the cases of the ionic salts, tetra-butyl-ammonium bromide (N4444Br) and tetra-butyl-ammonium picrate (Naa44Pic). In the RPM, the product of dielectric constant, ionic diameter and critical temperature is a constant according to Equation (17.3). The RPM predicts, in the same solvent, a higher consolute point (poorer miscibility) for N4444Br, because the ions are smaller than those of Nan44Pic. While N4444Pic has a consolute point in 1-tridecanol and 1-dodecanol, however, N4444Br is completely miscible with these solvents. Likewise, the solubility of tri-butyl-heptyl-ammonium iodide (N4447I) in o-xylene and toluene (55) contradicts the predictions of the RPM: N4447I has an upper critical solution point at 390 K in o-xylene, while it is completely miscible with toluene. The dielectric constants of toluene and o-xylene are very similar and the chemical structure of the solvents is also very similar. These examples show the limitations of the RPM: it contains no structural information about the solvent or the salt, although it is known that the solubility depends strongly on chemical structure. Even adding a CH3-group can make a big difference, so this structural information is certainly relevant for an understanding of the phase stability of these systems.
We will now proceed to discuss the different ionic binaries and temaries that have been studied near consolute points. We begin with nonaqueous solutions, and finish with aqueous ones. The systems discussed are characterized in Table 17.2. For some of the organic ions discussed, schematic chemical structures are displayed in Figure 17.4.
17.4.2 Character of Criticality in Nonaqueous Ionic Solutions
Sodium-ammonia (Na/NH3). Chieux and Sienko (56) reported a precise study of the coexistence curve of a partially miscible solution of sodium in liquid ammonia near its consolute point. The sodium-ammonia solution splits into a low-conducting sodium-poor phase and a high-conducting sodium-rich phase. At a reduced temperature of about 10 ' , this coexistence curve, Figure 17.5, middle, shows a sharp crossover (Cr) from mean-field behavior far from the critical point to Ising behavior nearby. In the past decade, Chieux and coworkers have done further detailed neutron-scattering studies of this system (57). Crossover from Ising-like to mean-field-like behavior has been noted to occur for the susceptibility and correlation length as well, in a range fully consistent with that reported for the coexistence curve more than 20 years ago. For a review of metal-ammonia and metal-molten salt systems, and for further references, see (58). The authors mention the possibility of the metal-nonmetal phase transition interferering with the consolute phase transition. They speculate that the sharp crossover observed at a correlation length of about 5 nm might be connected with the existence of ionic clusters of roughly that size. Only recently has such a sharp crossover been found in other nonaqueous ionic solutions (see below).
Figure 17.3 and Table 17.2 indicate that the reduced critical temperature and composition of
818
o. P '~C(16) 0,4% o o . / . " 0(41)
0-.,.7- -'~Cll~l C(21) ~ (~,,C(18) 0(20)
~ ( 2 ) 0(31) 'IroN(3) ~) -0(21)
0(30)
t: )-.o
fo)
(c)
Br--
Figure 17.4 Structure of some of the organic ions discussed in Section 17.4 and listed in Table 17.2: (a) tetra-n-butylammonium picrate; (b) tri-ethyl-n-hexyl-ammonium tri-ethyl-n-hexyl-boride; (c) tri- methyl-ethyl-ammonium bromide. Those under (a) and (b) are schematic only; in reality, the alkyl chains are not straight.
Na/NH3 are high (17) compared to those of the RPM, even when we take into account the uncertainty of the estimates. The high value of the reduced critical temperature is due to the high dielectric constant of pure ammonia, e = 22.9 at the consolute temperature, and to the relatively large ionic diameter a. For the latter, we took a value of a = 0.4 nm, the sum of the radii of the sodium ion, 0.10 nm, and that of the electron cavity, 0.30 nm (59).
Tetra-n-butylammonium picrate (Na444Pic) in various solvents and solvent mixtures. The ionic salt N4444Pic was first studied by Pitzer et al. (60,61) in chloroheptane. They found a mean-
field (MF) critical exponent ~ = 0.5 for the coexistence curve, but the precision of the experiment was limited because of the high critical temperature, 414.4 K. The first light-scattering measurements for the same salt in tridecanol were published by Weingartner et al. (62). The critical temperature of this system is close to 342 K and the critical mole fraction is xc = 0.156. The analysis of the scattered-light intensity in the range of 3.10 -4 < t < 3.10 -2 ruled out a spherical-model exponent (sph) of ? = 2, which had been predicted by Kholodenko and
819
Beyerlein, see (21) for references and critique. It was not possible to decide between mean-field (MF) and Ising (I) behavior, but the analysis favored mean-field behavior. The reduced critical concentration, c~* = 0.024, and temperature, T~* = 0.035, are close to the values predicted by the RPM after judicious choice of the ionic diameter a. From X-ray analysis of the solid salt (54) we calculate a = 4.1/~; this value is consistent with the dipole moment as obtained from dielectric measurements in benzene (48), and has been used to calculate the reduced critical concentration and temperature.
10 -1
x'-
10 -2
0.5 O4
×~ o.2
0.1
0.05
0.0100
0.0050
0.0020
0.0010
0.0005
T/K 404.8 404.5 400.9 382.5
N5555 Br
=
I 1 I I I I I I I
10 -4 10 -3 10 -2 10 -1
I f - Tc l / T c 1 1 I I I I I I I /
Na - NH 3
_
/
_
1 I J, I I I I I 1 I
10 -4 10 -3 10 -2 10 -1
I f - Tc l / T c i i i i 1 1 i l
N2226 B2226 in diphenyl e t h e r
4> f c = 44,541°C o T c -- 44.543°C
, ~ Slope = 0.475
J 0.0002
0.0001 ........ l c~.i02 " J I. 01.2 r 0.005 0.01 u 0.05 01 0.5
I T ToI, K I i
10 .4 10 .3
i i
1 2
I
10 -2
I T - To I I T c
Figure 17.5 Log-log plots of coexistence curves showing three different types of observed critical behavior Ising, top (82); crossover, center (56); and mean-field, bottom (69).
820
Narayanan and Pitzer (63-65) performed precise turbidity measurements of N4444Pic in long-chain alkanols CnH2n+IOH (n = 11,12,13) and in different mixtures of dodecanol and 1,4-butanediol. The data show a crossover (Cr) from mean-field to Ising behavior as the critical point is approached. The crossover, as evident from the behavior of an effective exponent y~ff, deduced from the turbidity data, was reported to be sharp, see Figure 10 in (64). The authors reported that the mean-field region becomes larger when the dielectric constant of the solvent is decreased.
The turbidity, contrary to the coexistence curve (Figure 17.5), lacks a region of visible pure-power-law behavior because it saturates near the critical point due to Ornstein-Zemike anisotropy in the scattering. Instead, we have backed out an effective exponent yeff from the turbidity and/or light scatering data. See Figure 17.6b for yeff crossover in one of the picrate systems of (64), and compare with the simple crossover in xenon, Figure 17.6a.
1.2
1.1
1.3
o o
TPDB
1.1
1.0
1.3
1.2
1.0 (c)
Xe
IBA/H20
(d)
"\ '.(.. ',
\ ,.".. ,, '\ " . . . ,
3MP/H20/NaBr ~\ '\iiiii]~'i,12 k,q,...
!
log 1: log 1:
Figure 17.6 Various types of crossover behavior as reflected in the behavior of the effective exponent 7eft. (a) Xenon: positive Wegner correction and monotonic decrease, never reaching the mean-field value. (b) Isobutyric acid and water: negative Wegner correction. (c) Tetrabutylammonium picrate in 1,4 butanediol plus 1-dodecanol: negative Wegner correction and crossover. (d) 3-Methyl pyridine, water and sodium bromide: from negative Wegner corrections to sharp crossover as salt concentration increases, in the order of 1-5.
821
Kleemeier et al. (66) measured the coexistence curves of N444aPic in five different alkanols, with carbon chain lengths of 3, 10, 12, 13, and 14, respectively (Table 17.2). With the exception of the shortest-chain solvent, 2-propanol (which has a dielectric constant value of almost 17 at the critical temperature), the critical points of the solutions all were close to the RPM prediction. Ising limiting behavior is found in all cases. The range of measurements is typically 104< I tl < 10-1. Departures from Ising scaling are seen in this range. The deviations from asymptotic Ising behavior are negative for the propanol system (W-), and positive (W+) for the others. The authors warn, however, that conclusions about corrections to scaling are very sensitive to the choice of order parameter: volume fraction or mol fraction.
Oleinikova and Bonetti (67) measured the coexistence curve of N4444Pic in 1-dodecanol. They report Ising behavior in the temperature range of 4.10 -5 < I tl < 4.10 -3, with no evidence of crossover in this close-in region.
Tri-ethyl-n-hexyl-ammonium tri-ethyl-n-hexyl-boride (N2226B2226) in diphenyl ether. Singh and Pitzer (68,69) synthesized this symmetric organic salt. N2226B2226 has a consolute point in diphenyl ether at 317 K, at a critical mole fraction of x~ = 0.053 _+ 0.001. To determine the reduced variables c~ and T~* the ionic diameter, a, must be estimated. The ions in N2226B2226 should be no larger than those in Na444Pic. If the X-ray analysis for N4444Pic (54) and the molecular structure of the N2226B2226 are taken into account, it is reasonable to choose a no larger than 0.6 nm, which is the distance between the nitrogen atom N(1) in Figure 17.4a and the center of the ring for the picrate. For N2226B2226 this leads to c~* = 0.035 and T f = 0.042, both values being consistent with the RPM prediction.
Singh and Pitzer measured the refractive-index difference of coexisting phases in the last 1 K from the critical point, with temperature control on the rnK level (69). The slope of this curve • • < - 4 is mean-field-like in the temperature range ]tl>10 4, see Figure 17.5. For Itl 10 , mean-field behavior is consistent with the data to within their uncertainty, but crossover to Ising behavior cannot be excluded, as stressed by the authors (69).
Zhang et al. (70) measured the turbidity in the same system, but they found the consolute point in the range of 288-295 K and at a critical mole fraction of x~ = 0.049 _+ 0.001. In the reduced temperature range of 10 -4 < t < 10 -1, they fotmd 7 = 1.01 _+ 0.01 . This system thus shows apparent mean-field behavior for the temperature range studied. A crossover to Ising behavior at temperatures closer to Tc is not excluded, but did not appear in the range experimentally accessed. The original cell did not survive the measurements. Later (71), turbidity measurements on a remaining fraction of the original sample which had been sealed inside a small cell indicated a substantial increase of turbidity, leading to suspicion that the original sample had not been chemically stable.
Wiegand and coworkers (71-73) undertook a comprehensive study of this system, for a sample of the salt synthesized by a different company. They measured turbidity (71,72), refractive index in coexisting phases (72), dynamic light scattering (72) and viscosity (73) in a range of 10 4 < t < 10 -~. Contrary to the turbidity results in the earlier sample, all experimental data were consistent with Ising-like behavior throughout the experimental range, with no indication of crossover. The correlation length amplitude ~0 is very large, approximately 1.4 nm. The turbidity results differed substantially from those of Zhang et al. (70), and the coexistence curve had no resemblance to that of Singh and Pitzer (69).
The authors confirmed that the critical amplitudes for the correlation length, coexistence curve, and susceptibility of the new sample were in agreement with two-scale-factor universality to within their uncertainty.
There has as yet not been an explanation of such essentially different behavior in two
822
different samples of what should have been the same system. Ethylammonium nitrate (NZHHHNO3) in n-octanol. Static and dynamic light-scattering
studies were reported by Weingartner, Schrrer and coworkers (74,75). The (T-x) coexistence curve of this system is highly skewed (74), but, in contrast to all other cases, the critical point lies in the salt-rich regime at a mole fraction of the salt xc = 0.773 + 0.0005. It is quite difficult to estimate e, since it makes little sense to use the value of e obtained for the pure solvent. If we
nonetheless assume this, we find T* = 0.07, which is near the RPM prediction in Figure 17.3; the critical concentration c* =0.4, however, is displaced towards much higher densities. Light-scattering data, taken in a range of, 10 -4 < t < 10 2 yield a critical exponent v = 0.61, with a critical correlation-length amplitude {0 = 0.47 nm. The conductance of the system can only be understood by assuming an extraordinary stability of the C2 H5 NH3 - NO3 pair, which could be due to hydrogen bonding (75). These ion pairs will have high dipole moments and their immersion in a solvent of low polarity could drive the phase separation (14). The interaction would then be short-ranged, favoring Ising behavior.
Bonetti and Oleinikova (76) reported the coexistence curve for this binary mixture from detailed refractive-index measurements in coexistent phases in the temperature range of 2.6.10 -5 < t < 4.2.10 2. They report the need for corrections to scaling. The effective exponent 13 approaches the Ising limiting value from below (negative first Wegner correction).
Tri-methyl-ethyl-ammonium bromide (N2iJimBr) in chloroform. Wiegand et al. (77) published the results of measurements of the intensity and line width of Rayleigh light scattered from this system with 1 mass % ethanol added for stabilization and to prevent suppression of the phase separation by crystallization. The system has a lower critical temperature near 298 K, and the critical mole fraction is xe = 0.0503 _+ 0.0002, which corresponds to the reduced variables T* = 0.036 and c~ = 0.029, in good agreement with critical values predicted by the RPM. A range of i0 5 < t < 4.10 -2 was spanned. The analysis of the scattering intensity at the critical composition gives the Ising-like value v = 0.621 + 0.003 for the critical exponent of the correlation length ~, with an amplitude of~0 = 0.80 + 0.01 nm. The system is a solution of a salt of essentially spherical ions of almost equal size in a simple, low-dielectric polar liquid. The critical parameters are very close to predictions of the RPM. Nevertheless it shows only pure Ising critical behavior, while most other systems show at least a beginning of departures. The reason, in this case as well as in the previous one, could be a strong specific interaction between the bromide ion and the solvent. Evidence is that if single crystals are formed from a chloroform solution, they contain the salt and the solvent at a stoichometric ratio (1:1).
Tetra-n-butylammonium naphthyl sulfonate (N4444NS) in toluene. Schrfer et aL (78) measured the light scattering intensity at a 90-degree angle, the turbidity and the time correlation function at the critical composition above the critical point, and also the refractive indices of coexisting phases, in the temperature range 2.10 .5 <It I < 6"10 "2. This particular system is Coulombic by the criterion of having phase separation in the range of that of the RPM. The solvent has a very low dielectric constant. The coexistence curve is roughly symmetric, independent of the choice of order parameter. The limiting behavior is Ising-like for all properties investigated. The amplitude of the correlation length, ~0 = 2.2 nm, is about the largest found for ionic binaries so far. There is no indication of crossover to mean-field behavior.
Tributylheptylammonium dodecyl sulfate (N4447DS) in cyclohexane. For this system, light scattering results are mentioned in passing by Schrrer et al. (78). It has Ising-like critical exponent values and a correlation length amplitude ~0 of about 1 nm.
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17.4.3 Character of Criticality in Aqueous Ionic Solutions
Many binary aqueous solutions of organics show separation into two liquid phases. Typically, the organic compound carries an OH, amine or carboxylic group, which is polar and/or weakly ionic, acidic or basic, and capable of hydrogen-bonding with water, and with each other. If the interaction between unlike pairs is energetically unfavorable compared to that between like pairs, phase separation may result below a consolute point, the system segregating into a mostly aqueous and a mostly organic phase. This non-Coulombic phase transition is called hydrophobic or solvophobic demixing (12,16,17). In a large subclass of these systems, the phenomenon of closed-loop coexistence curves exists, in which a region of two-phase coexistence is terminated by both an upper and a lower consolute point. Many weakly dissociated aqueous mixtures have been studied at upper, and several at lower consolute points. For reviews and references, see (11) and (79). Here we limit ourselves to a few systems that will allow us to make some important points.
Isobutyric acid (IBA) in water. Highly accurate light scattering data were obtained by Shanks (80). An analysis of the effective exponent 7 by Anisimov et al. (81) shows that this system approaches the Ising value nonuniformIy (W-), see Figure 17.6c. Figure 17.3 shows that the RPM is not an appropriate model for this ionic binary.
Tetra n-pentyl ammonium bromide (N5555Br) in water. This system, an organic salt in water, is substantially ionized at the critical point. Japas and Levelt Sengers (82) reported Ising-like behavior, see Figure 17.5, with a negative Wegner correction (W-).
Tri-butyl-propyl ammonium iodide (N4443I) in water. Schrrer et al. (83) found that this system shows a tiny solubility loop in the range of 55 °C to 75 °C. On adding an alcohol such as glycerol, the loop shrinks to a double critical point with exponent doubling of v and 7. The critical exponents for the coexistence curve are 13 = 0.331 for the upper (U) and 13 = 0.322 for the lower (L) part. The Ising behavior of the system was also confirmed in light-scattering measurements.
3-methylpyridine (3-MP), water and sodium bromide. (3-MP) and water are fully miscible. Adding a small amount of NaBr leads to formation of a closed-loop region of phase separation. As more salt is added, the lower critical temperature decreases. Jacob et al. (84) performed light scattering measurements as a function of temperature, along five isopleths at the respective critical compositions and in the reduced temperature range 2.10 -4 < t < 3.10 2. In the data analysis, the apparent critical exponents must be corrected for Fisher renormalization due to the presence of the third component. The behavior of the corrected effective exponent 7 is shown in Figure 17.6d. At the lowest salt concentration, (1), 7 remains essentially at its Ising value until the reduced temperature reaches 10 -2. Then it begins to fall off below 7 < 1.2. As the salt concentration increases, on the next three isopleths the apparent exponent first rises (W-) and then falls off sooner and sharper than on the first isopleth, and to lower values. At the highest salt concentration, there is complete crossover (Cr) towards the mean-field value. The data are well fitted by a crossover function with two length parameters, the cutoff parameter A, associated with
a characteristic length {D, and a coupling constant u which is basically constant. The five adjusted parameters per isopleth turn out to be smooth functions of the concentration. The most striking features are: the increase of the correlation length amplitude as the salt concentration increases; the negative value of the first Wegner correction, and the supermolecular size of {D, increasing from 1 to 7.5 nm as the salt concentration rises. It was predicted that this length will
824
diverge at a salt concentration slightly higher than the experimental range, at which point the exponent ~/would be entirely mean-field. Measurements have since located this singular point, and have been extended beyond it.
Dodecylammonium chloride (DAC), water and potassium bromide. Rubio and coworkers (85) studied the phase separation of this micellar system for three salt concentrations, 0.26, 0.28 and 0.30 M. This system has a critical endpoint (CEP) near 292 K and 0.24 M of salt, at which endpoint the surfactant solidifies. The principal objective of the authors was to put to rest a lingering controversy about the nature of criticality in micellar systems. The authors state that the salt concentration is the same in coexisting phases, so the system behaves as a binary. For each of the three salt concentrations, the temperature of phase separation is measured for varying DAC concentrations. By interpolating for chosen temperatures, three 'coexistence curves' are obtained. By power-law, and by crossover analysis, the authors find that, asymptotically, the coexistence curve for each of the three salt concentrations is consistent with Ising exponents, be it in a range no larger than I tl - 10 "~, and smallest for the lowest salt concentration. The first Wegner correction is large and positive. The correlation length amplitude, indicative of the size of the interacting particles, is in excess of 2 nm.
17.4.4 Viscosity
For a variety of ionic binaries, the viscosity has been measured near the critical point. Simple fluids and binaries show, along with Ising-like static critical behavior, a weak divergence in the viscosity, characterized by a critical exponent y = xnv, with xn = 0.066, y = 0.0416 theoretically, and y between 0.04 and 0.044 experimentally (86). Absence of such an anomaly might indicate leading mean-field behavior (87). In several ionic binaries, the viscosity has been measured in recent years. Kleemeier et al. (88) measured a viscosity anomaly in the system N4444Pic in 1-tridecanol. Wiegand et al. (73) measured such an anomaly in the system N2226B2226 in diphenyl ether. Oleinikova et al. (89,90) did likewise in the system 3-methylpyridine in heavy water, with and without added NaC1. In all cases, the critical exponent of the anomaly was in agreement with the theoretical prediction to within experimental uncertainty.
17.4.5 Summary
The RPM has been used as a heuristic tool to select experimental systems with Coulombic phase separation, by criteria such as: low dielectric constant of the solvent, leading to strong charge interactions; a critical point near that of RPM in the Friedman plot; and a low critical volume fraction of the salt. Considerations of this type have produced the collection of nonaqueous ionic systems discussed here, and displayed in Table 17.2. Only one system, N2226B2226 in diphenyl ether (68-70), has shown apparent mean-field behavior throughout the entire range 10-'> Itl > 10 4, but the results of (70) have since been retracted (71), and a new sample showed Ising behavior (72). Two more systems, studied with relatively low accuracy, were consistent with mean-field behavior, namely N4444Pic in chloroheptane (60,61) and in 1-tridecanol (62). Several systems, Na/NH3 (56-58), and Naa44Pic in various long-chain alkanols, have shown crossover from asymptotic Ising-like to mean-field behavior, and others have shown only Ising behavior in a similar-size range: N2226B2226 in diphenyl ether (71,72), Nan44NS in toluene (78), N2HHHNO3 in n-octanol (74), and N2HHHNO3 in chloroform (77). A nonuniform approach to Ising behavior, that is, a negative first Wegner correction, was reported in several cases: N2HHHNO3 in n-octanol (76); Na444Pic in 2-propanol (66), and in a mixed alkanol solvent
825
(64). Positive Wegner corrections were found for Naa44Pic in the four long-chain alkanol solvents in (66), and for an ionic ternary micellar system (85). Aqueous ionic fluids, which all have Ising limiting behavior, tend to have negative first Wegner corrections: isobutyric acid and water (80), N5555Br and water (82), and NaBr in 3-MP/water.
Ionic fluids, though almost exclusively Ising-like in their asymptotic critical behavior, differ from simple molecular fluids in the ways they approach criticality. Neither the RPM nor the value of the dielectric constant appears to be a proper guide for organizing or understanding this variety of behavior.
17.4.6 Crossover: a Perspective
Theorists expect that both aqueous and nonaqueous ionic fluids should display asymptotic Ising behavior (1,4,91). The different experiments performed during the past decade in several laboratories on a large number of ionic binaries and some temaries have mostly confirmed this expectation. Ionic systems vary, however, in the ways they cross over from the Ising fluctuation- dominated regime to states further from criticality where fluctuations are not important and mean-field theories should apply. All systems must eventually cross over: departures from the Ising asymptotic scaling laws must set in when the critical fluctuations no longer dominate the behavior.
A first way of characterizing crossover is through the Ginzburg nunaber, Section 17.2.2 and (1,92), the reduced temperature distance from the critical point at which mean-field theories begin to fail due the growth of fluctuations. Calculations for square-well fluids (34) have shown that the Ginzburg number depends on the range of the forces: the longer the range, the closer one can approach criticality with a mean-field description. Simulations of the Ising model with non- nearest-neighbor interactions have shown that the Ginzburg number varies inversely with the 6th power of the interaction range (93). On the other hand, the Ginzburg temperature for the RPM appears to be, if anything, considerably larger than that in simple molecular fluids (32-34). Debye screening thus makes the ionic interactions effectively short-range (4,91).
The Wegner corrections indicate when departure from limiting Ising behavior sets in, beginning at the critical point (Section 17.4 and Table 17.2). In simple molecular fluids, and in several of the nonaqueous ionic mixtures, the first Wegner correction is positive, and the effective critical exponent moves away monotonically from its Ising value. In systems with very strong, short-range forces, such as the Ising model, and hydrogen-bonded or aqueous ionic binaries the Wegner correction is negative, and the effective critical exponent passes through an extremum before turning over and moving towards mean-field behavior.
Modem crossover theory (1,92,93,95,96) incorporates both the Ginzburg number and the Wegner corrections in terms of one coherent global description. In addition, it describes more complex crossovers in systems with additional length scales or competing order parameters. Chapter 11 gives a full description of this theory, and we refer to that chapter for details.
An example of competing order parameters is that of the superfluid phase transition in mixtures of He-3 and He-4. The lambda line for the superfluid phase transition as a function of the concentration of He-3 ends at a tricritical point, beyond which a first-order demixing into a He-4 rich superfluid phase and a He-3 rich normal takes place. At a tricritical point, the susceptibility is mean-field-like. An example of an additional length scales is that of partially miscible polymer solutions for which the chain length N, and therefore the radius of gyration, is varied. For very large N, such systems develop a strongly skewed coexistence curve with a very small Ising region. The limiting case, for infinite N, corresponds to the theta point, which is a tricritical point. Nearness to the theta point will powerfully influence the nature of crossover of
826
the apparent critical exponents (97). As mentioned in Section 17.2, simulations of the RPM on a lattice have also shown
tricriticality (38) if the lattice size is comparable with the ionic size. For smaller lattice size, the phenomenon disappears, and the lattice RPM behaves like the fluid RPM.
Light scattering experiments in the aqueous ionic temary of Jacob et al. (84) have provided evidence of a profound change in crossover, and an approach to some form of tricritical point as the salt concentration increases (Figure 17.6d). There is evidence of the presence of microstructure near that singular point.
The presence of charges may lead to the formation of microstructures characterized by additional length scales. Van Leeuwen and Smit (98) showed in their simulation of dipolar fluids with low dispersion forces that the dipoles align in chains. Chen and Kachaturyan (99) simulated a 2-dimensional binary alloy of particles with dispersive interactions leading to phase separation, and added in Coulombic forces. They showed how the second-order phase transition turned first- order at a tricritical point, and demonstrated formation of dispersed mesoscale structures in one of the phases. Ermi and Amis (100) found domain structure in aqueous polyelectrolyte solutions with low concentrations of added salt.
• Crossover theory (Chapter 11) starts from a Landau-Ginzburg expansion of the free energy in terms of 2 na and 4 th powers of the order parameter, plus a square-gradient term. If one integrates over all wavelengths of the fluctuations, one obtains 'naive' crossover such as typical for xenon.
Strictly speaking, however, one needs to cut off the integration at the molecular scale, which is small in simple fluids. This cut-off is an additional parameter in the theory when the molecular interaction range is not small. For instance, crossover theory has been able to describe the results of molecular simulations of an Ising model in which interactions extend beyond nearest neighbors (93).
The additional parameter is essential in explaining crossover in the presence of a supermolecular structure such as the radius of gyration in a polymeric system, and crossover near tricriticality. The normalized coefficient of the 4 th power term is called u, which is a constant equal to unity at the so-called fixed point of the renormalization-group theory. Positive Wegner corrections correspond to u < 1, and negative Wegner c_orrections to u > 1. The Ginzburg temperature is a combination of the cut-off parameter and u.
Crossover theory incorporates all features we have noted in criticality of ionic fluids. Thus, the naive integration over all wavelengths results in the Ginzburg temperature becoming a function of only one crossover parameter (see Chapter 11). This description is suitable for the Coulombic systems listed in Table 17.2 as having positive Wegner corrections.
In aqueous (ionic and nonionic) binaries with very short-ranged forces, and in some of the Coulombic systems in Table 17.2, the first Wegner correction is negative. If the experimental temperature range does not extend far from the critical point, the exponents are effectively constant, and only Ising behavior, no crossover, is noted (Figure 17.6c). Once the effective exponent passes through a shallow extremum, the system begins its crossover towards mean-field behavior.
In the presence of a supramolecular length, such as the radius of gyration in polymers, or an association of polar molecules (98), the cut-off length is large, and crossover is distinctly different. On approach to criticality, the system will only enter the Ising regime when the fluctuations exceed the supramolecular scale in size. Thus the Ising regime shrinks, the Ginzburg temperature becomes smaller, and a negative first Wegner correction becomes the rule rather than the exception.
Crossover theory has been able to fully describe the susceptibility crossover in a polymeric
827
system as a function of the radius of gyration (97) near to and away from the theta point. The susceptibility crossover of the ionic ternary (84) has been similarly well described, and the existence of a singular point on the critical line predicted. The nature of this singular point is being investigated.
Interactions in ionic fluids, such as those listed in Table 17.2, are complex, as we repeatedly stressed. Variants of the RPM, and charged systems of even greater complexity (99, 100) have shown tricriticality and supramolecular structures. A physical explanation of the additional crossover length scale is an urgent challenge. We suggest that nearness of a competing phase transition, and ordered mesoscale structures, may be part of the explanation in several cases.
17.5 SOLUTION THERMODYNAMICS NEAR THE SOLVENT CRITICAL POINT
In earlier sections, we made a distinction between vapor-liquid criticality of pure fluids, and that of liquid mixtures (see, for instance, Table 17.1, and Sections 17.3 and 17.4). At this point, we will shift attention to vapor-liquid behavior of fluid mixtures. The present Section, 17.5, forms the transition, in the sense that the emphasis is still on near-criticality. At the same time, the fluid considered is in a transition between pure-fluid and mixture behavior.
Supercritical fluids as media for chemical analysis and chemical processing have received much attention in the past decade. In applications such as supercritical chromatography and supercritical extraction, the solutes are typically at low to modest concentrations. It becomes then important to know in how far the properties of the supercritical solvent are modified by the presence of the solute. Near the solvent's critical point, the effect of the solute on the solvent density becomes hugely magnified and difficult to control or estimate if, as is usually the case, the pressure and temperature (but not the volume) are kept constant.
In this section, we summarize the peculiar features of solutions near the solvent critical point, as derivable from the application of thermodynamics. Simple expressions for the anomalous behavior close to the solvent critical point can be obtained if one concentrates on the density rather than the pressure as an independent variable. Apart from the pure-solvent properties, only one experimental constant, the Krichevskii parameter, is needed to describe many dilute-mixture properties. Although many of our results are general, attention will be focussed mostly on high-temperature aqueous systems. In the last part of the section, we will return to the issue of the effect of ions on criticality, with NaC1 in supercritical water as an example. The actual thermodynamic modeling of high-temperature aqueous systems is the topic of Sections 17.6 and 17.7.
17.5.1 Principal issues
In our coverage of ionic, partially miscible, binary liquids, the principal issue was the nature of criticality in the presence of charges. This issue remains of interest in the case of vapor-liquid criticality of aqueous solutions. Other issues, however, especially those connected with the large compressibility, are of pressing practical importance and must come to the foreground. The weak anomalies, not represented by classical models, are small compared to the very noticeable anomalies induced by the strongly diverging compressibility, about which even a classical model can give a qualitative insight. Thus, the main emphasis of this section will be on these strong anomalies.
Characteristics which supercritical aqueous mixtures share with other supercritical mixtures include: the large range of densities encountered (from steam-like to water-like), a compressibility much larger than that of noncritical water and steam, and some quite noticeable
828
peculiarities near the solvent's critical point. A notable feature is the great diversity of solutes that can be present in supercritical water. Gaseous solutes may have critical temperatures less than 1/5 of that of water, but are fully miscible above the water critical temperature. Organics mix freely with supercritical water. Salts may have critical temperatures more than 5 times higher than that of water, and deposit solid phases in supercritical water. How to model the thermodynamics of such complex systems is the topic of Sections 17.6 and 17.7.
17.5.2 Thermodynamic Properties of Interest
The prime interest of the power industry and of geochemistry is the distribution of species of varying volatility between aqueous vapor and liquid phases. The traditional approach is to extend the methods used for water below the boiling point to higher temperatures. It is invariably found that strong variations occur in Henry's constant and the distribution coefficient or K-factor as water is heated above 573 K. Also, it is found that the composition dependences of the activity coefficient and of the distribution coefficient become more complex above 573 K. Therefore, it seems sensible to assume a different point of view and first determine what happens near or at the critical point of water, so that extrapolations from lower temperatures are firmly anchored. In recent literature, there is a great deal of emphasis on infinite-dilution partial molar properties, which have critical divergences driven by the diverging compressibility of the solvent. These properties have moved to center stage because of their linkage to the standard states that have been traditionally used in aqueous physical chemistry. Section 17.5.3 is devoted to the criticality of dilute solutions.
17.5.3 Criticality of Dilute Solutions
Dilute solutions near the vapor-liquid critical point of the solvent share peculiarities in their thermodynamic behavior that will be summarized here. For a more detailed review see reference 101. The behavior of dilute binary mixtures near the solvent's critical point is governed by the
6 Krichevskii parameter (102), defined (101) as: -A~x=_-(aZA/aVOx)C=(oP/ax)v,r, where A is
the molar Helmholtz energy, x is the mole fraction of the solute, the superscript c indicates that the derivative is taken at the critical point of the pure solvent (the x ~ 0 limit is taken first, followed by the limit to the critical point). It has long been recognized (103-105) that this derivative plays a crucial role in the thermodynamics of dilute solutions near critical points.
Critical line. The Krichevskii parameter may be calculated from the initial pressure- composition (P-x) and temperature-composition (T-x) slopes of the critical line as follows:
c c c c (oP/Ox)~,~ = (dP/dx)c~- (dP/drlcxc. (dr /dxlc~. (17.4)
Here the subscript CRL indicates the direction of the critical line, the subscript CXC that of the vapor-pressure curve of the pure solvent, and x denotes the mole fraction of the solute.
Dew-bubble curve. We will discuss the shape of this curve in pressure-composition (P-x) coordinates near the critical point of the pure solvent, see Figure 17.7. To fix ideas, we consider the case of a less-volatile solute in a more-volatile solvent (such as NaC1 in water). Above the solvent's critical temperature, the phase envelope formed by the dew and bubble curves is a closed loop that does not intersect the x = 0 (pure solvent) axis. The pressure maximum on such a loop is a point on the mixture critical line. Below the solvent's critical temperature, the dew and bubble curves meet at an angle (related to the relative volatility of the two components) at the
829
x = 0 axis. The transition between these two forms occurs at the solvent's critical temperature; there the dew and bubble curves form a cusp, or 'bird's-beak', in which they both meet the x = 0 axis with the same slope. This slope is equal to the Krichevskii parameter discussed above. This behavior was first derived early in the century (105); see also (101).
24 .0 I I t I
23 .5 -
o
23 .0
A rl . . . . . . - . - - - - - - . - . - - - _
z ; 22 .5
eL-
< 22 .0 ~'"
I " " " - - -
" " - - 647.1 K (Tc)
~ " 646.6 K
21 .5
653.1 K vo------____o_
21.01 1 I i 0 0 .04 0 .08 0 .12 0 .16
x, mo le % NaCI
Figure 17.7 Limiting behavior of the dew-bubble curve of NaCl-water near the water critical point (117); data from reference 116.
Henry's constant and K-factor. These quantities are defined in coexisting phases at infinite dilution as
Henry'sconstant kH= lim f___zz. RT In (kH / f * ) = lim OA----~ ~. (17.5) x~0 X ' x-~0 0y '
Ks - fac to r K~=l im Y" R T l n K ~ = l i m [ OA--~r OA--~r], (17.6) x--,o x ' x--,o Ox Oy
with subscript 1 indicating the solvent, 2 the solute, x the mole fraction of solute in the liquid phase, y that in the vapor phase, f the fugacity, oo infinite-dilution, and the asterisk a pure-solvent property. Both Henry's constant and the infinite-dilution K-factor are simply related to the composition derivative of the residual Helmholtz energy Ar('~,T,x) in the limit of infinite dilution, as indicated in Equations (17.5) and (17.6), see reference 106. Near the critical point, the leading density dependencies of the x and y derivatives in Equation (17.6) become equal in
830
magnitude but of opposite sign. The strong variations of Henry's constant and the K-factor with temperature or pressure along the coexistence curve come about because the dominant variation of the composition derivatives of A r is asymptotically linear in volume (or density) near the critical point of the solvent (106), with the Krichevskii parameter as a prefactor. Thus the asymptotic behavior of Henry's constant and the K-factor is given by
_ 2 RT In(kH / f*) * Cl + h~,x ( P l Pc) / Pc (17.7)
e 2 RT In K ~ Aw (Pv - Pl) / PC (17.8)
where Cl is an undefined constant and the subscripts 1, v, and c denote the solvent's saturated liquid, vapor, and critical densities, respectively. The simple, linear, density dependence, common to classical and nonclassical critical behavior, results in a diverging temperature derivative along the coexistence curve because
(d p v,Z /d T )cxc oc (T¢ - T) ('-~), (17.9)
with 13-1 equal to-0.5 (classically) or -0.674 (real fluids): a strong divergence. By concentrating on the asymptotic critical behavior, and choosing the density as variable instead of temperature, the strong curvatures of Henry's constant and the K-factor in temperature representation are revealed as simple linear relations in the density. Engineering equations of state have the behavior given by Equations (17.5)-(17.9) built in correctly on the classical level.
Partial molar properties. A striking feature of dilute solutions near the solvent's critical point is the divergence of the various partial molar properties of the solute. Figure 17.8 shows data (107) for the apparent molar volume of dilute (0.01 molal) NaC1 in water as a function of temperature at a pressure (28 MPa) not far above water's critical pressure. The isothermal compressibility of water is plotted on the same graph. The divergences of the infinite-dilution partial molar volume V~' and the partial molar enthalpy H7 of the solute are proportional to that of the solvent's isothermal compressibility K r; the sign and amplitude of the divergences are
c related to the Krichevskii parameter-Avx
V7 "~- A~x Kr; (17.1 O)
H7 ~- A~,x K* T(dP/dT)*. (17.11)
The diverging partial molar properties thus do no more than reflect the diverging compressibility of pure water, which is, of course, a very well tmderstood and characterized property. Measuring
¢ and correlating the relatively well-behaved derivative -Avx might be a more promising avenue than trying to capture strongly diverging infinite-dilution properties. Promising progress is being made along these lines (108,109).
Solid solubility. Solids show a finite solubility in near- and supercritical solvents. As a consequence, the incipient critical line of the solution is terminated at an upper critical end point (UCEP) where the solution becomes saturated. At this UCEP, the solubility of the solute increases with infinite slope as a function of pressure along the UCEP isotherm, or as a function
831
of temperature along the UCEP isobar. The infinity is due to the criticality condition at the UCEP, not to the nearness to the solvent critical point. When the solute is poorly soluble, the UCEP may be close to the solvent's critical point, but the two critical points should not be confused (101).
More than 30 years ago, Martynova (110) recognized that the solubility in supercritical water of salts and oxides of interest to the power industry could be represented by a plot linear in the logarithm of the solvent density. Kumar and Johnston (111) derived an asymptotically linear relation between In y2 and In(9/9c), on the critical isotherm as the critical density pc was approached. For small values of the argument, In(p/pc) is linear in (p/p~). Strictly speaking, however, in a saturated solution the critical point of the solvent cannot be reached and the solution cannot be infinitely dilute (112).
0 0 i i
- 1 o o o - E
E o
Lt] - 2 0 0 0 -
d o >
m - 3 0 0 0 -
o
- 4 0 0 0 - Z W
- 5 0 0 0 -
- 6 0 0 0 -
6 0 0
\ \ O \ \
\ \ \
I I
I
N a C I - H 2 0
• M a j e r e t a l . ( 2 8 M P a , 0.01 m)
- - W a t e r C o m p r e s s i b i l i t y
I 6 2 0
|
I I / / /
I /
I I / I #!
~ 1 / | I / I | I
I I I I i I
i .,I
I I • I
6 4 0 6 6 0 6 8 0
T E M P E R A T U R E , K
"7, t'lzi
n
- - 0 . 0 4 u d 133 O3 O0 W
- 0 . 0 8 rr 13._
o O
._J <
- - 0 . 1 2 "/S r r W t I-- o 09
- - 0 . 1 6 - - r r w F-
- 0 . 2 0 7O0
0 1 1 /
i
Figure 17.8 Measured partial molar volumes for dilute NaC1 in water (106) compared to compressibility of pure water.
Harvey (113) showed more generally (for small solubility, and for departm'es from the critical point not necessarily along the critical isotherm) that
c 2
RT In Y2 '~ c2 + Avx (P - P¢) / P~ ( 1 7 . 1 2 )
with C2 a constant. Equation (17.12) again demonstrates the crucial role of the Krichevskii parameter. The UCEP should be close enough to the solvent's critical point that the Krichevskii parameter may be substituted for (OP/Ox)vr, and the temperature should be no lower than the
832
UCEP temperature. The solubility should be low enough that the distinction between the solvent density and the solution density can be ignored.
In summary, we have shown that the Krichevskii parameter contains the key to the peculiar dilute-mixture critical behavior. Often, diverging slopes are eliminated if properties are related to density rather than pressure. Engineering equations of state will have this limiting density dependence built in correctly.
17.5.4 Ionic Effects on Critical Behavior
The issue of whether dissolved salts such as NaC1 in water might produce classical (or effectively classical) critical behavior by suppressing long-range fluctuations was first raised by Pitzer (114). The vast majority of the data relevant to that question are for the NaCl-water binary; we will therefore concentrate on that system.
The hypothesis appeared to receive some support when high-quality supercritical VLE data were published (115,116) for NaC1 in water. When these data were analyzed according to the asymptotic scaling relationship
I Xl- xv I °c (Pc -P)O, (17.13)
here expressed in terms of the pressure, rather than the temperature distance from the critical point (Table 17.1), the resulting plots (115) indicated values of 13 near and in some cases greater than the mean-field value of 0.5. While the values greater than 0.5 were puzzling, the overall results were taken as evidence for the classical hypothesis.
Harvey and Levelt Sengers (116) pointed out that the unexpectedly high apparent values of [3 were an artifact due to a peculiarity of dilute mixtures near the solvent critical point. Fundamentally, the exponent [3 characterizes the shape of the coexistence curve in P-p (pressure- density) coordinates according to
(Pl" Pv ) °c (Pc - e)/3. (17.14)
While for a binary mixture the P-x (pressure,composition) coexistence curve is normally characterized by the same exponent, the P-x curve becomes distorted near the solvent's critical point. This distortion is exactly the "oird's-beak' noted in the preceding section and shown in Figure 17.7. At the solvent's critical temperature, the limiting behavior of the P-x curve is actually characterized by the exponent [3+1, which equals 1.5 for engineering equations.
Pitzer and Tanger (118) did classical model calculations to estimate the extent to which the influence of this higher exponent persists in nearby regions. In the context of the experimental isotherms, the apparent exponent of the 653 K isotherm is badly distorted by the bird's-beak effect, and the distortion will diminish but not be negligible for the three higher isotherms (673 K, 678 K, and 688 K). This is consistent with analysis of these data via Equation (17.13) which shows apparent exponents near 1.1 at 653 K and near 0.45 for the three higher temperatures.
Povodyrev et al. (119) revisited this issue and examined the entire collection of P-x and P-p coexistence curves. Their model has Ising limiting behavior built in, and allows for two- parameter crossover as discussed in Section 17.4.6, the second parameter indicative of the existence of an additional supermolecular length scale other than the correlation length. The authors obtained a close fit to all coexistence data over the entire range from 15 to 60 MPa, up to
833
30 mass fraction NaC1, and temperatures from 648 to 873 K. For the lowest salt concentrations, the change of the effective exponent 13 away from the Ising values occurred smoothly over many decades in reduced temperature, but for higher salt concentrations, the Ising region shrank and the crossover occurred more sharply.
Thus, the critical behavior of aqueous salt solutions near the critical line has similar features as to what was found to occur in binary and ternary, aqueous and nonaqueous, ionic binary systems (Section 17.4.6). The differences between systems are not in asymptotic behavior, which is always Ising-like, but in the nature of crossover. For the NaCl-water system, the second parameter is phenomenological, and its interpretation in terms of a structural length remains to be considered.
17.6 GIBBS ENERGY MODELS FOR HIGH-TEMPERATURE AQUEOUS ELECTRO- LYTE SYSTEMS
At room temperature and atmospheric pressure, there is a multitude of data for the thermodynamics of electrolytes in water and there are many well-established semi-empirical methods for modeling both the activities of electrolyte species and the equilibria of reactions involving ions. These methods, which are not reviewed here, are perfectly adequate for calculations such as describing the effects of strong electrolytes on the vapor pressure of water at 298 K.
There are, however, many important problems and processes for which traditional methods are not adequate. In steam power plants, aqueous electrolytes may be a concern in liquid phases at temperatures up to 623 K, and salts may be carded over into superheated vapors, which can reach temperatures of almost 900 K. Geothermal brines (some of which contain dissolved methane, a potential energy source) may reach temperatures of 573 K and pressures above 100 MPa. In the emerging environmental technology of supercritical water oxidation (SCWO), pressures and temperatures are typically above water's critical point, and it is important to be able to predict the precipitation of various salt species from the complex mixture of water, salts, gases such as air components, and organic solutes. Such high-temperature problems present severe challenges for thermodynamic modeling.
An additional complication, which brings these problems into the purview of this chapter, is that the relevant range of conditions often includes or approaches the critical point of water. While it is usually not crucial to obtain a precise representation of the critical divergences, it is vital that any model be of sufficient fundamental soundness so as not to give nonsense in the vicinity of the critical point.
Historically, aqueous electrolyte solutions at low and moderate temperatures and pressures have been successfully described with an approach beginning with standard-state (infinite- dilution) Gibbs energies for solute species and then modeling the deviation from this standard state with an activity-coefficient model. For simplicity, we first consider a system containing a single solvent (species 1) and some number (species 2 to N) of nondissociating solute species, using a mole-fraction concentration scale. The molar Gibbs energy of the mixture can be formally divided into that of an ideal dilute solution and an excess, nonideal part:
~ID ~E g=g +g , (17.16)
N N ~ID 0
g = XlY1 + ~"~xi ]'~i "Jr" e z EX i In X i " (17.17) i=2 i=1
834
Here the superscript * indicates that we have taken an infinite-dilution standard state for the 0 solute species, la~ is the chemical potential of the solvent in its standard state, which is the pure
solvent at the temperature and pressure of interest, la/ is the standard-state chemical potential of
solute i; the standard state is a hypothetical one of infinite-dilution behavior extrapolated to a mole fraction of one. The excess Gibbs energy in this convention can be related to the activity coefficients of each species:
*E N , g =RT[x~ lny 1+ ZX / lny/ ] , (17.18)
/=2
where ? is an activity coefficient and the superscript * indicates that those activity coefficients are relative to an infinite-dilution standard state. The derivatives of this excess Gibbs energy with respect to its natural variables are the apparent molar volume, enthalpy, and higher derivatives, which are closely related to partial molar properties discussed in Section 17.5. In standard-state condition, the partial molar volume and enthalpy of the solute diverge strongly, and the next derivatives even more so. For the solvent, the second-order derivatives are the first to diverge strongly. The properties of the solution are thus expressed with respect to standard states that 'run away' at the solvent critical point.
For electrolyte solutions, one must take into account the dissociation of the species. It is also common to use the molality (moles solute per kg solvent) concentration scale, which means that
the extrapolation to la/ is to a hypothetical state of unit molality rather than unit mole fraction.
Both of these factors complicate the bookkeeping, but do not change the basic nature of the excess-Gibbs-energy modeling approach. See references 120 and 121 for fimher exposition of standard states and concentration scales. We note in passing that, in systems where the amount of water can approach zero, the usual molality concentration scale must be abandoned in favor of mole fraction. Mole-fraction approaches for electrolytes have been developed and applied to completely miscible systems such as nitric acid/water (122-125).
For chemical reactions, the thermodynamic equilibrium constant is related to the standard- state Gibbs energy change, AG ° of the reaction at constant temperature and pressure:
A G ° = - R T InK, (17.19)
where the equilibrium constant K is:
N
K = I-I(a/ )v, . (17.20) i=1
Here vi is the stoichiometric coefficient of species i in the reaction, ai is the activity of species i, defined as the concentration (in mole fraction, molality, or whatever units are convenient for the problem and consistent with the choice of standard states) multiplied by the activity coefficient. Often vapor-liquid equilibria for solute species are also written in this 'reaction' form; in that case the activity of the vapor species is usually taken to be the fugacity. Computing equilibrium constants, then, becomes a matter of computing the differences of standard-state Gibbs energies between the reactants and products, or, equivalently, computing the Gibbs energies of hydration
835
for aqueous solutes from an ideal-gas state and the ideal-gas Gibbs energies of formation of each species.
This approach for modeling aqueous solutions can be viewed as consisting of two pieces: 1) Calculation of equilibrium constants from the standard-state Gibbs energy of reaction. 2) Calculation of activity coefficients.
Both of these present substantial challenges for calculations at high temperatures. First, we consider the equilibrium constant. Most equilibrium constants of interest are
known accurately at 298 K and often over a modest range of temperatures. One might therefore think of simply extrapolating to predict high-temperature equilibria. This is not recommended, however, because the behavior of AG ° with temperature for reactions involving ions is often highly nonlinear. The simplifying assumption of linearity of In K in 1 / T ceases to be valid, especially for reactions in which the number of charged species changes. Some relief maybe obtained by writing the reactions in 'isocoulombic' form (126,127), where each side of the equilibrium (perhaps after combining with well-known equilibria such as the ionization of water) contains the same amount of positive and negative charges. Equilibria written in this form behave more manageably with temperature; this has been exploited to estimate many equilibria of interest in the steam-power industry (126,128). Another approach that has met with some success is to model the temperature and pressure dependence of the Gibbs energy of hydration of each solute species; these can then be combined with ideal-gas values to produce an equilibrium constant. Expressions have been developed (128-132) that provide good extrapolations for these values to conditions close to water's critical point.
While either of the above approaches may be successful at temperatures up to 573 K, or perhaps even 623 K, they must ultimately fail as water's critical temperature is approached, because of the inherent critical divergences in the standard states.
Second, we consider activity coefficients. Many models, some wholly empirical and others semi-theoretical, exist for activity coefficients of electrolytes at room temperature. Many of these have been extended to higher temperatures (133-136), but even in the best cases temperature extrapolation of aqueous ionic activity coefficients above 573 K is not reliable. A slightly better approach at very high temperatures is to use a 'model substance' approach, in which the known activity coefficients of a specific electrolyte (usually NaC1) are correlated empirically and it is assumed that all other electrolytes behave like the model substance.
It is not always appreciated that the activity coefficient will behave in an anomalous way because it must try to make up for the runaway standard state. It is relatively simple to show, however, that on the critical isotherm-isobar of the solvent the activity coefficient must vary as x 1/~ (137,139), with 6 close to 5 for real fluids (Table 17.1). This implies that at the solvent's critical temperature and pressure, the activity coefficient of a 10 ppm impurity is as far from unity as that of a 10% impurity in the liquid solvent.
For some high-temperature problems, modeling the activity coefficients is not important. This would be the case in systems where the water is nearly pure and not near its critical point, so that ionic species would be close enough to infinite dilution for their activity coefficients to be unity. In such cases, only the modeling of the reaction equilibria would be important.
The use of the Debye-Htickel limiting law in the Gibbs energy formalism presents a problem in that the limiting law, derived for the Helmholtz energy (9), can only be used in the Gibbs energy for relatively incompressible states. This could not be ft~her from the truth near the critical point of the solvent (139,140). An estimate of the concentration below which criticality overtakes Debye-Htickel effects on the critical isotherm-isobar of NaC1-H20 was presented by Morrison (141) and deserves to be revisited. The conductivity measurements by Wood and coworkers (142,143) in dilute NaC1 solutions near the water critical point may provide an
836
experimental answer to this question. In summary, the Gibbs energy approach is optimally designed for incorporating chemistry
and ionic dissociation in low-compressibility states. If proper care is taken, one can successfully model the thermodynamics of aqueous electrolytes up to 573 K and perhaps a little beyond. However, the problems inherent in such an approach become nearly intractable as the critical point of water is approached, and in a supercritical regime where water is very compressible. A more appropriate way to model thermodynamics near a critical point is to use the Helmholtz energy where the natural independent variables are temperature and density. This is most conveniently done with an equation of state.
17.7 HELMHOLTZ ENERGY MODELS; EQUATIONS OF STATE
Helmholtz energy models are optimally designed for describing fluid mixtures where the continuity of states in the supercritical regime is an important aspect. The strong critical anomalies are properly imbedded, albeit with classical exponents. There is a large choice of Helmholtz energy models available for Complex mixtures, and phase equilibrium calculations have been developed to a fine art. The disadvantages of the approach are that the ideal-gas standard state may not be known for components of low volatility at the experimental temperatta'e, and that the Helmholtz energy has traditionally not been much used as the vehicle for incorporating chemistry. Nevertheless, the application of equation-of-state (EOS) technology to ionic systems has become an active area of research.
The approach is to begin with some EOS valid for normal (non-ionic) fluid mixtures. Additional terms are then superimposed to account for electrostatic effects not seen in normal fluids. This is done by writing the EOS in Helmholtz-energy form in which the electrostatic terms are additional contributions. The general form is:
A = A "°"~ + A charging + A i°n-i°n (17.21)
The first term represents the non-ionic aspects of the system. It may be a simple cubic EOS, or a more complex molecularly based EOS with terms for non-ionic electrostatic interactions such as those involving dipoles. EOS parameters for ions (for example, Lennard-Jones collision diameters and energy parameters) cannot be obtained in a straightforward manner as for pure uncharged species. Typically, the size parameter for an ion is related to the crystallographic radius, and the energy parameter is estimated from the polarizability.
The second term is the energy of charging the ions, and is necessary because the reference state for the EOS is an uncharged ideal gas. In order to compute this term, it is usually assumed that the ion is a charged hard sphere immersed in a dielectric continuum; the problem is then simple electrostatics (144). For most purposes, the only practical effect of this charging term is to keep the ions from partitioning into the vapor phase, so often it can be omitted and the ions simply constrained a priori to the liquid phase. It would be wrong, however, to make this simplification in a situation where ions were entering the vapor phase or partitioning between coexisting liquid phases.
The third term accounts for electrostatic interactions among ions. The model is again that of charged hard spheres in a dielectric continuum. The interaction energies in that model were first
837
approximated by Debye and Htickel (9); more sophisticated approximations are used in the equations of state reviewed below.
Copeman and Stein (145) began with a perturbed-hard-sphere EOS and superimposed ion- ion interactions in the form of an explicit approximation to the Mean Spherical Approximation (MSA). To fit the experimental data for single salts in water, they had to introduce a binary parameter into the nonelectrolyte portion of the EOS for interactions between ions and solvent molecules. Since the effects of cations and anions cannot be separated, these were actually salt- water parameters.
Jin and Donohue (146) began with a sophisticated perturbed-anisotropic-chain EOS. Ion-ion interactions were represented by a perturbation expansion due to Henderson (147). Their model also included a perturbation-theory representation of charge-dipole interactions between ions and solvent molecules. They found that one additional adjustable parameter (applied to the ionic diameters) was needed to fit data for each salt in water. Fair predictions were obtained for the salting-out of argon and methane from aqueous solution. The work was later extended to systems containing multiple salts (148).
Harvey and Prausnitz (149) began with a perturbed-soft-sphere EOS. They incorporated a charging term using effective ion diameters (150), and used a simplification of the MSA for ion- ion interactions. They also found that one salt-solvent binary parameter was required. Their work was particularly directed toward gas solubilities in salt solutions at high temperatures and pressures. Fair quantitative results were obtained with a single salt-solute binary parameter, which could be obtained from the Setchenow constant that describes salting-out at low concentrations and room temperature.
Aasberg-Petersen et al. (151) began with a modem cubic EOS. They represented ion-ion interactions with a modified Debye-Htickel term. They also used a single adjustable parameter in the nonelectrolyte portion of the EOS for salt-solvent interactions and, for salting-out calculations, fit a single salt-solute parameter to low-pressure data. Their results for gas solubilities in high-temperature salt solutions were superior to those of Harvey and Prausnitz; this is in part because they began with a better EOS representation of the non-ionic gas/water systems. A similar approach, with similar results, was taken independently by Zuo and Guo (152), who later extended this work to high-pressure systems containing hydrocarbons and brines (153).
Fiirst and Renon (154) began with a cubic EOS. They used a modification of the MSA for ion-ion interactions, and added a specific interaction term between ions and water. Their model was applied to a number of salts in water at room temperature, and was later extended to gas solubilities at higher temperatures by adding an adjustable cation-gas parameter (155), and to vapor-liquid equilibria in mixed-solvent systems (156).
Zhao and Lu (157) also began with a cubic EOS, but they introduced the effect of electrolytes in the EOS mixing rules. They used a mixing rule based on excess Gibbs energies (158), and described those Gibbs energies with the electrolyte NRTL model (159). Good results were obtained for vapor-liquid equilibria at ambient temperatures. However, they were unable to use electrolyte NRTL parameters fitted directly to activity coefficients, which somewhat defeats the purpose of using an excess-Gibbs-energy mixing rule.
The recent efforts of Wu and Prausnitz (160) and of Liu et al. (161) both started with the Statistical Associating Fluid Theory (SAFT) for their nonelectrolyte EOS; otherwise, they are similar to the models described above. Liu et al. (161) only considered ionic activity coefficients at room temperature; Wu and Prausnitz (160) also considered systems with dissolved gases at high temperatures and pressures.
Two additional EOS efforts are those ofRaatschen et al. (162) and Simon et al. (163). Both described ion-ion interactions with a form ofPitzer's ionic virial expansion (164). More notably,
838
both were extended to mixed-solvent systems. Raatschen's EOS gave, after substantial parameter fitting, a fair description of the water/methanol/LiBr system up to 120 °C. Simon did not compare extensively with experiment, but showed fair agreement with some ambient- temperature data for the water/ethanol/NaC1 system.
One notable feature of all of the above equations of state is that they require the fitting of an empirical parameter to describe single salts in water. In most cases, this was expressed as a modification of the attractive interaction in the nonelectrolyte portion of the EOS. The necessity of these parameters testifies to a deficiency of the second and third terms in Equation (17.21). These terms assume that ions in water behave as charged hard spheres in a dielectric continuum; in reality, the specific forces involved in ion hydration are not nearly so simple. Explicitly accounting for these interactions is infeasible, so an empirical correction is used to compensate for the deficiency of the models. This empiricism, while necessary, greatly reduces their predictive and extrapolative capabilities.
While none of these EOS studies performed calculations above about 200 °C, the framework allows for calculations at higher temperatures, especially if there are data to which parameters can be fit. The major problem at high temperatures lies in the assumption of complete dissociation. It is known that, at densities near or below the critical density of water, salts in aqueous solution primarily exist in their undissociated, molecular form. Any framework that assumes full dissociation will not give good results in this region. We therefore conclude that equations of state based on complete dissociation are only directly useful for phase-equilibrium calculations at conditions where the electrolytes are fully dissociated in the liquid phase and not significantly present in the vapor. For most salts in water, this limitation would be at about 300 °C.
17.7.2 Models Assuming no Ionic Dissociation
Since the ionic dissociation becomes small at high temperatures, a promising approach for modeling this region is to completely ignore dissociation in the EOS. This will, of course, fail to describe the low-temperature region properly, but, for some applications, those temperatures are not of interest.
Gallagher and Levelt Sengers (165,166) used an extended corresponding-states model for the water-NaC1 system. While one would not expect corresponding states to work well in a system where the species are so different, they managed to obtain semi-quantitative agreement with data.
Pitzer and coworkers (114,167,168) presented equations of state for water-salt mixtures. They began with an accurate EOS for water, and then added a small number of perturbation terms in powers of (undissociated) salt concentration and in the difference between the solvent density and its critical density. They were successful at describing phase equilibria for NaC1 and KC1 solutions in a substantial region around water's critical point, but their liquid densities were unsatisfactory.
Anderko and Pitzer (169) took a more general approach with the water-NaC1 system, beginning with an EOS based on a truncated virial expansion. The EOS also included an explicit term from thermodynamic perturbation theory (170,171) for dipolar interactions; this term contributed significantly since both water and the tmdissociated salt have large dipole moments. EOS parameters were fitted to reproduce the properties of pure water closely. For NaC1, some parameters were set to zero; the rest (along with some binary parameters) were fitted to limited data for pure NaC1 and a large amotmt of data for the binary mixture. The result was an excellent description of the system over a temperature range from 300 °C to 900 °C. Figure 17.9, taken from (169), shows the agreement with VLE data for four isotherms both above and below water's
839
critical temperature. This approach has been extended to aqueous solutions of KC1 and of KC1 with NaC1 (172) and, with the addition of perturbation terms for interactions involving quadrupoles, to the water-CaC12 system (173). Duan et al. (174) have extended the approach of Anderko and Pitzer to the H20-NaC1-CO2 system. They have obtained good agreement with high-temperature data, though they had to fit some binary and ternary parameters to ternary data. The analysis of the NaC1-H20 equation by Pitzer and Jiang (175) shows good agreement with some geological equilibria, but also warns that the EOS does not give absolute quantitative values for the salt's fugacity, nor is it quantitative for mole fractions of salt above 0.4. This primarily reflects the lack of data in this region when the equation was constructed, rather than any limitation of the EOS itself.
26 []
24
22
20
: ~ ,
~ 1 8 ~,
I
16 n
14
i, 1 2
i t ~ i I l i n u J i l z I i l
10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 100
X NaCI
Figure 17.9 Dew-bubble curves for the NaCl-water system at 633.15 K (,), 643.15 K (A), 653.15 K (11) and 663.15 K (o), as described by the Anderko-Pitzer Helmholtz energy (169).
Economou et al. (176) based their model on an associated-perturbed-anisotropic-chain equation of state (177, 178). The EOS uses perturbation-theory results for interactions involving dipoles and quadrupoles. In addition, it allows for association equilibria. Such equilibria were used to represent the properties of pure water, and also to describe the hydration of salt molecules. VLE were calculated for ten alkali halides in water. The quantitative accuracy was not as good as that of Anderko and Pitzer for the systems covered by both groups, but this is at least in part because Economou et al. incorporated fewer adjustable parameters.
Gallagher (179) modified the Anderko-Pitzer model for a quadrupolar salt, Na2SO4, in near- critical and supercritical water, strongly reducing the number of adjustable parameters by taking them from the NaCl-water system as far as possible, and by reducing the number of terms in the virial expansions. He was able to fit P V T x and partial molar volume data, as well as the three- phase curve solid-liquid-vapor. A limitation of the EOS was the poor fit to the pure-water data in the supercritical regime.
840
The EOS approaches of Pitzer and coworkers (169,172,173) and ofEconomou et al. (176) are promising for high-temperature applications where the assumption of no dissociation is warranted. One advantage is that the EOS form can be extended to include multiple salts and also other solutes such as the air components and combustion products that would be present in a SCWO process. While their predictive capability is limited, their molecular basis is strong enough that only a relatively small amount of data is necessary to obtain a fair representation of the phase behavior at high-temperature conditions. A disadvantage, of course, is that the neglect of dissociation causes a loss of accuracy at lower temperatures.
17.7.3 Models with Partial Ionic Dissociation
Since fairly good EOS models exist assuming complete dissociation and also no dissociation, a logical next step would be to incorporate a dissociation equilibrium. In theory, this would allow a single model to work in both the low- and high-temperature regions. Such a weak electrolyte approach would be difficult in practice because the equilibrium constants for the dissociation of salts in high-temperature water are not known accurately (the definition of the dissociation constant is itself somewhat ambiguous), and the calculation of ionic fugacity coefficients is quite sensitive to the modeling of the electrostatic effects. Thus far, the only studies in which weak electrolyte equilibria have been superimposed on an EOS have been in cases where the reaction equilibria (in an infinite-dilution standard state) were known and the EOS was used to provide activity coefficients. Applications have been limited to single weak electrolytes in water (163,180) and to the water-ammonia-carbon dioxide system (181).
17.7.4 Solid Solubility Calculations
In principle, any EOS that allows calculation of a fugacity for the associated salt species can be used to calculate solid-fluid equilibria, provided the fugacity for the pure solid is known at the conditions of interest. However, for salts in high-temperature water, this information is seldom available. Instead, some experimental data are necessary. The equation describing the equilibrium is
s s s P~2 x2= P2dP2 exp[(P- P2)v2 / RT] (17.22)
where it has been assumed for simplicity that v~, the molar volume of the solid salt, is
independent of pressure. The subscript 2 refers to the solute, superscript s refers to saturation conditions for the pure salt, and ~2 is a solute fugacity coefficient.
If one solubility-data point (for example, the solubility at the vapor-liquid-solid three-phase $
line) is known at a particular temperature, that can be used to obtain the product P~ ~b 2 at that
temperature. The EOS can then be used to compute values of ~2 at other pressures, allowing the
calculation of x2 via Equation (17.22). Figure 17.10, from reference 169, shows solid solubilities calculated by this procedure on one isotherm (723 K). The procedure must be repeated for each
temperature of interest, although it would not be too inaccurate to use values of P~ ~ ~ obtained
from data at only two temperatures to extrapolate on the assumption that, like the vapor pressure itself, the logarithm of this product should be approximately linear in reciprocal temperature. Both the Anderko (169,172) and Economou (176) equations of state have been used for solid solubility calculations.
841
10 -4
10 -5
10 -6
10 -7
10 -8
10 -9 t i • i I i i t i I I
10 P, MPa
Figure 17.10 Solubility of solid NaC1 in supercritical steam at 773.15 K, as described by the Anderko-Pitzer model (169). Full curve, model prediction; symbols, experimental data sets, see (169) for references.
17.8 CONCLUSIONS
As shown in the first half of the chapter, the theoretical and the experimental understanding of the critical behavior of ionic fluids are converging to Ising-like asymptotic critical behavior. For the simplest model of ionic fluids, the restricted primitive model, theoretical expectations tend to favor asymptotic Ising behavior, as in other fluids. Recent experiments carried out close to the critical point for a dozen Coulombic, ionic solutions have, with only a few exceptions, shown Ising-like asymptotic critical behavior. Non-Coulombic aqueous ionic binary liquids have long been known to behave Ising-like. The vapor-liquid transition of aqueous NaC1 above the water critical point, as well, shows Ising critical behavior. There are, however, characteristic differences between the various systems as far as crossover behavior is concerned. A few appear to show no evidence of crossover to mean field in the experimental range, some show monotonic crossover from Ising-like nearby to mean-field behavior far away from criticality, others show non-monotonic crossover behavior. Crossover theory (Chapter 11) can make sense of this variety of behavior. The most challenging cases of ionic criticality appear to be models and experimental systems with competing order parameters or additional length scales, in which the possibility of higher-order critical points and charge-density waves exist. In one aqueous ionic ternary, a singular point of this sort has been recently discovered.
The second half of the chapter has shitled to the modeling of mostly aqueous ionic systems at high temperatures, including the supercritical regime. The thermodynamic behavior of dilute
842
mixtures near the solvent critical point in terms of the Krichevskii parameter forms the bridge to this part of the chapter.
The modeling of electrolyte systems, especially aqueous systems, at low and moderate temperatures is now routine. The Gibbs energy modeling techniques, while often containing a heavy measure of empiricism, have been refined to the point where they have substantial correlative and fair extrapolative capability. Even chemically reacting systems can be handled if sufficient data exist to evaluate the reaction equilibrium constants. For several reasons, most notably the divergence of the standard states used in the models, this approach breaks down in the highly compressible fluid in the region around the solvent critical point.
A more appropriate approach is to use the Helmholtz energy in an equation-of-state model. Although existing applications almost exclusively imply mean-field, rather than Ising critical behavior, they do not give nonsense at the critical point of the solvent. Helmholtz energy models have been developed for aqueous systems containing fully ionized electrolytes and for systems at high temperatures where the electrolyte is assumed not to dissociate at all. As these models are further developed and optimized, they should provide an alternative to Gibbs energy models for near- and supercritical aqueous-electrolyte systems. One challenging area for fiLrther development would be the incorporation of reaction chemistry into this framework. The successful inclusion of reaction equilibria, while far from trivial, would allow a single model to span the range from dense-solvent conditions in which a salt is completely dissociated, to supercritical conditions where the salt is essentially undissociated.
Acknowledgements
Professors M.E. Fisher, K.S. Pitzer, W. Schrrer and G. Stell, as well as Dr. J. Douglas and Dr. M. Kleemeier, generously shared with us ideas and prepublication results. We received useful comments and advice from Professors K.S. Pitzer, J.M. Prausnitz, G. Stell, M.A. Anisimov, M.L. Japas, J.V. Sengers, and Dr. A. Anderko. We profited greatly from a thorough review by Dr. J. Douglas. Dr. R.F. Kayser lent support to this work in many ways and served as a critical reader. Dr. G. Orkoulas, Dr. J. Jacob, and Mr. V.A. Agayan provided input to figures and tables. One of us (SW) received a postdoctoral research grant from the Alexander von Humboldt-Stiftung.
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