J. Phys. Chem. 1995,99,13070-13077
13070
FEATURE ARTICLE Ionic Fluids: Near-Critical and Related Properties Kenneth S. Pitzer Lawrence Berkeley Laboratory and Department of Chemistry, University of Califomia, Berkeley, Califomia
94720-I460 Received: April 28, 1995;In Final Form: June 20, 1995”
Precision measurements of the near-critical properties of ionic fluids have been reported recently; they show marked differences from the corresponding properties of nonionic fluids. These very interesting results are summarized and interpreted in terms of theoretical critical exponents and possible crossovers between exponents. Recent calculations for NaCl and the restricted primitive model for near-critical properties and for cluster formation in the vapor are also considered.
Introduction Although ionic fluids such as NaCl are qualitatively similar to nonionic fluids with vapor and liquid phases and critical points, quantitatively they are very different as was first clearly shown in 1962 by Kirshenbaum et al.’ Their result, reproduced here as Figure 1, shows the remarkably larger change in volume with temperature for the ionic liquids. Recent Monte Carlo and molecular dynamics calculations add interesting information conceming the phase diagram which will be discussed briefly. Also, there are interesting predictions conceming dimers and higher clusters in ionic vapors and dilute supercritical fluids. But it is for the near-critical properties, expressed as critical exponents, where especially interesting experimental advances have been made recently. Again, there are major differences between ionic and nonionic fluids. These new results and their interpretation constitute the primary subject of this review. With a critical point above 3000 K for NaC1, precision measurements are not possible. But there are two-component ionic systems which show phase separation and critical points at convenient temperatures. In these fluids Coulombic forces between ions are the primary force driving the phase separation. The general phase diagrams of these ionic systems show a similar difference from nonionic systems to that noted for pure fluids. And the critical exponents for these ionic fluids show very interesting differences from the well established values for nonionic systems. For nonionic fluids it is well established both experimentally and theoretically that, in the limit as the critical point is approached, the differences between individual systems disappear, and the various critical exponents have universal values, specifically those of an Ising lattice. But ionic fluids show differences from nonionic fluids even very close to the critical point-so close that there is uncertainty as to the limiting values of these exponents. Also, some ionic fluids show a sharp “crossover” from an Ising critical exponent near T, to a different (classical or mean-field) exponent over a wide temperature range. While variations in effective exponents are normal and are often called “crossovers”, a sharp crossover between two different theoretically based exponents is rare. Thus, the new examples are of considerable interest. For comparison with the ‘Abstract
published in Aduunce ACS Abstracts, August 1, 1995.
0022-365419512099-13070$09.00/0
PI Figure 1. Reduced density, er = @/ec, for various fluids as a function of reduced temperature, T, = T/Tc (based on Kirshenbaum et a1.I).
crossovers in the ionic systems, the Appendix includes a brief discussion of effective exponent behavior for simple nonionic fluids. The presence of long-range repulsive and attractive ionic forces represents a major difference from the force pattern in neutral-particle fluids. Thus, the theory for ionic near-critical behavior is different and is not yet without uncertainty or disagreement. Several have reviewed one or another aspect of ionic-fluid properties in recent years. FisheF and Stel16 discuss the present status of theory for ionic fluids but do not agree on fm conclusions as to limiting exponents or conceming behavior further from the critical point.
Critical Exponents: General The definitions for the critical exponents in terms of the various near-critical properties are given in Table 1. The coefficients D , B, etc., may vary from substance to substance
0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 35, 1995 13071
Feature Article TABLE 1: Power Laws and Critical Exponents; Definition# property
dependent variable
application
pressure coex curve
one comp one comp twocomp susceptibility one comp twocomp
independent variable AQr A Tr A Tr ATr
P - P,
Aer Ax
K; = pc(aQ/ap)T/e x*
power law
A Tr
=
Alternatively, for the finite differences between pairs of measurements.
(ax/wrP heat capacity one comp CV twocomp Cp, correlation either E length
classical or mean-field king
A Tc A Tr A Tr
(3)
critical exponent values
university class
6
P
a
Y
V
3
0.5
1
0
0.5
4.6
0.325
1.24
0.11
0.63
T, temperature; P,pressure; e, density; x, mole fraction of solute; (e - ev)/ec; ATr = (T- T,)IT,; Ax = XI - x2; 1, liquid; v, vapor; subscript c, critical; 1, concentrated phase; 2, dilute phase. a
A, chemical potential difference @I - pz)/RT,; Aer =
according to the interparticle potentials. It is the exponents 6,
p, etc., that have the same values for all fluids of a given class.
As noted in Table 1, there are two sets of theoretical values for these exponents. We now review briefly the bases for these theories. If the system is described by a mathematically analytic equation of state, however complex, that expression can be expanded around the critical point in a power series in temperature and density (or volume). By the definition of the critical point, the first and second derivatives of pressure with respect to density at constant T are zero. Hence, the first nonzero term has the third power of density and the exponent 6 is 3. More complex, but essentially similar, arguments give the other values on the line denoted “classical” or “mean-field’. The latter term is used because it was shown by Kac et aL7 that this behavior is expected if there are long-range attractive forces between all particles. While Coulombic forces are long-range in that sense, ionic systems have different, positive and negative, particles; hence, the Kac theory does not apply. For many decades it was known that for simple, nonionic fluids /3 was about l/3. The difference from the classical value of remained a puzzle. Then more complex statistical theory showed, initially for lattice models with short-range forces, that the fluctuations from most probable distributions were greater than that expected from simple theory. The resulting expressions are very complex and nonanalytic. Those for a particular lattice, the king lattice, are believed to apply also to simple fluids, and they are given in Table 1. Most recent experiments very close to the critical point for simple fluids, both onecomponent and two-component, agree with the king values. There is a question, however, concerning the effect of a gravitational field. This was ignored in the theory, and it would tend to reduce fluctuations. There is an indication8 of classical exponents for SFs which has a very large mass in relation to its low critical temperature. While this possibility of a gravitational field effect is interesting, it is not an important question for the present consideration of ionic fluids. In the interpretation of experimental measurements, it is useful to convert the definitions of Table 1 to logarithmic form. Then, for AQ, and ATr one has ln(A@,) = 4, In IATrI
+ In B
gives p unambiguously. For most real systems, however, such a graph shows some change of slope, and it is the limiting slope for very small ATr or Aer that is related to the basic theory. It is useful to define an effective exponent as the differential slope; thus, for the liquid-vapor density difference one has
(1)
with corresponding expressions for the other exponents. If the values of ln(Aer) vs lnlATrI fall on a straight line, the slope
Similar expressions pertain for the other exponents. In a few systems, one finds a clear shift of effective exponent from one value near the critical point to a different value over a wide range away from the critical point; this is called a crossover. Examples of systems showing a crossover are given below. Another treatment is frequently used for finite ranges of variables around a critical point. This is a Wegner series expansion; again for the liquid-vapor density difference, this is Aer = B,IAT,IP(l
+ BllATrlA+ B,IATr12A+ ...)
(4)
where the exponent A is 0.5 in classical theory and approximately 0.5 in Ising theory. The Wegner series is especially useful for the near-critical region, but it tends to diverge for large IATJ. Thus, the effective exponent method, eqs 2 and 3, is preferable for a wide range of IATrI. In addition to the properties listed in Table 1, the “diameter” is of interest, and in this case it is useful to consider initially a three-term expression
Since the classical value of a is zero, the second and third terms become identical, and a linear diameter is predicted. Even on an king basis the exponent 1 - a = 0.89, and the nonlinearity is often not significant. Historical Background As noted in the Introduction, Kirshenbaum et al.’ reported in 1962 liquid densities of NaCl and KCl over a wide range of temperature. They noted that the rate of expansion was much greater for these ionic liquids than that for nonionic liquids. They presented the graph, reproduced here as Figure 1, showing the marked difference between ionic and nonionic fluids. Thus, the particles, which are comparably packed in the lowtemperature liquid, are much less closely packed at the critical point for an ionic fluid as compared to a nonionic fluid. A decade later, Bubach and Franckg reported density measurements for NH4Cl extending to its critical point at 1155 K. They found not only a large expansion coefficient for the liquid, which is generally very similar to KC1, but also a near-critical phase diagram with a distinctly different shape. This is shown in Figure 2 , where the slope, which is the effective p, is 0.50 in contrast to about ‘/3 for CH4 or other nonionic fluids. While the precision of measurement above 1000 K was limited, there was no indication of a decrease in p as the critical point was approached . The near-critical behavior of the two-component system NaNH3 was investigated by Chieux and Sienko’O in 1970. Here the concentrated phase is metallic, but the dilute phase is similar to the vapor of a pure ionic fluid with the ion-pair species dominant. For Na-NH3, a striking crossover was observed
Pitzer
13072 J. Phys. Chem., Vol. 99, No. 35, 1995 325
Po =320.4 bar 416.2.C 315
310
-1
5 -3
-4
-2
0
-1
305
-
300
-
!Og,J-AT, )
Figure 2. Logarithmetric plot of Aer or Axr vs lATrl, whose slope yields the exponent p for three fluids as indicated. ,
405.2T “V
-
285
Po 280.7 bar 400.2%
P
50 1
0
“qu8ds
P
0
I
0.2
I
0.4
2
4
(
1
8
1
0
1
2
t 8 1 - x , ~ *1~0 4
1
I
I
0.6
0.0
I
I .o
Figure 3. Phase diagram for the system tetra-n-butylammoniumpicrate and I-chloroheptane (from ref 11).
from an effective p of close to the critical point to l/2 further from T,. Figure 2 includes this system and shows that each region extends over at least a 10-fold change in ATr. These results presented a challenge to find an ionic fluid showing phase separation in a range where more precise measurements were feasible. The critical temperature should be in the range 300-400 K or about one-tenth that of NaC1. The critical temperature of an ionic fluid is inversely proportional to the closest interionic distance and to the dielectric constant (or relative permittivity) if a solvent is present. By use of large ions, one can increase the closest distance by about a factor of 3. Then the presence of a solvent of dielectric constant 3-4 might yield the desired properties. The first system of this type was prepared and studied by Pitzer, de Lima, and Schreiber” in 1985 and in more detail by Schreiber et a1.’* in 1987. The ionic component was tetra-n-butylammonium picrate, and the solvent was 1-chloroheptane. The phase diagram is shown in Figure 3. The critical point is at 414.4 K and 0.085 mole fraction picrate. This very low mole fraction indicates the same loose packing of the ions that was noted above for pure ionic fluids. The coexistence curve is represented by the equation x I - x2 = 0.028(414.4 -
n”2
(6)
Thus, the effective exponent @ is l/2 for this ionic system, but the precision was limited. Also, the diameter is linear. Soon thereafter several similar systems were identified, and highprecision, near-critical measurements have been made; these very interesting results are presented below. Before proceeding to that presentation, however, a few remarks are appropriate about clusters in dilute ionic fluids and about other two-component systems that contain ions and show phase separation. We fist consider two types of two-component fluids, both aqueous but otherwise very different.
Figure 4. Test of eq 7 relating pressure to difference in mole fraction. Data and figure are from ref 13.
The high dielectric constant E = 80 of water at room temperature reduces 1- 1 or even 2-2 ionic interactions to levels too small to cause phase separation. But at high temperatures E is greatly reduced. Thus, in supercritical steam containing NaC1, the ionic forces play a major role. Important measurements of phase composition as a function of pressure at constant T were reported by Bischoff and R0senba~er.I~ For measurements well separated from the critical point of pure H20, it is expected that the applicable equation is x,
- x2 = IP - P,lp, T constant
(7)
Figure 4 shows the results presented as (XI - ~ 2 vs) P; ~ hence, the linear behavior indicates /? = I/*, the classical value. For a temperature close to Tc of pure H20, the apparent p increases above I/*. The complexities of interpretation of this particular system near to T, of H20 are discussed by Harvey and Levelt SengersI4 and by Pitzer and TangerI5 and more generally for this type of system by Levelt Sengers and Given.3 Another type of system comprises a salt with large, tetraalkylammonium ions and small negative ions with water (or a glycol) as solvent. Tetra-n-pentylammonium bromide in water is an example; Japas and Levelt SengersI6 found a critical point at 409 K. Here the primary force driving phase separation is the hydrophobic effect of the large alkyl groups in water. In both dilute and concentrated phases, the salt is ionized, and the bromide ion is strongly hydrated. With E large, the ionic forces are weak and would not cause phase separation. The nearcritical phase compositions of these “hydrophobic” ionic systems show Ising exponents. In 1989 Weingartneri7 and in 1991 Weingartner et a1.I8 presented interesting discussions of these aqueous and similar solvophobic systems. It has been known for many years that, while NaCl molecules are most abundant, a large fraction of the saturated vapor of NaCl is dimerized as Na2C12 molecules and that there are very few individual Na+ and C1- ions. References to this literature and appropriate thermodynamic functions for each species are given in the JANAF Tables (Chase et al.I9). For the model
Feature Article
0’4
J. Phys. Chem., Vol. 99, No. 35, 1995 13073
t
clusters neutro‘
yJ loglo lAT,I
Figure 6. Alternate nonlinear least-squares fits of eq 8 to the experimental measurements. The curves are calculated, whereas the experimental points are shown as solid circles with the T, for P = 0.5 or as open circles with the T, for P = 0.326 (from ref 25). C*
Figure 5. Distribution of particles among clusters for the RPM at r* = 0.059 @* = l / P = 17) as a function of reduced concentration c* = g* (from ref 22). The vertical distance between lines gives the fraction of particles, (s t)C?;/C*,in the indicated cluster. The charged clusters are grouped at the bottom, and the solid line gives the division between the totals for charged and for neutral clusters. See the text for further details.
+
fluid of charged, hard spheres, the “restricted primitive model” (RPM), BjermmZotreated the formation of ion pairs in 1926. More recently, GillanZ1 made extensive calculations by MC methods for various clusters up to a neutral sextet. He used the designations ( l , l ) , (1,2), (2,1), (2,2) for the numbers of positive and negative ions in a cluster. He defined C* = Nu3/ V, with N the total number of individual ions and u the hardsphere diameter, and also a reduced reciprocal temperature /3* = q2/kTu,with q the charge of an ion. We later use r* = 1/p* and the symbol e* for C*. Gillan then considered equilibrium mixtures on an ideal gas basis. Pitzer and Schreiber22improved the mixed-gas calculations for higher densities by including appropriate cluster interaction effects; their result for p* = 17 (P= 0.059) is shown as Figure 5. As will be shown below, this is a slightly supercritical temperature. At C*less than the ion pair (1,l) species dominates but with a significant portion of the neutral (2,2) species. At higher concentrationsthe neutral (3,3) species and the charged (2,1), (1,2), (3,2), and (2,3) species also become significant. But at still higher concentrationsmany clusters merge, and the cluster representation becomes inappropriate. Then MC calculations for an entire sample are required; these are discussed in a later section. Also, Bresme et al.23have very recently reported highly detailed calculations including even larger clusters for a few vaporlike densities and values of P.
Near-Critical Exponents: First Precise Measurements In 1988, Singh and P i t ~ e rreported ~~ a system with almost purely ionic character and with a critical point near 317 K. The salt was triethyl-n-hexylammonium triethyl-n-hexylboride (N2226B2226). Here, the charges on the nitrogen and boron atoms are buried within identical arrays of alkyl groups. A similar tetraalkyl salt might seem even better, but then the melting point is too high. As a solvent, diphenyl ether is very satisfactory and yields the phase separation below 317 K. Although the salt must be prepared and handled with complete exclusion of air and water, the solutions were completely stable over the course of measurements. The general phase diagram is similar to that
TABLE 2: Alternate Sets of Parameters for the Wegner Expansion (Ea 8) for the System N2226B2226-Diphenyl Ether
P
TJK
Bo
BI
B2
0.5 0.326
317.6932 317.6910
0.351 f 0.009 0.058 f 0.002
-3.4 f 1.2 24.5 f 2.7
27 f 14 -123.3 i 28.4
shown in Figure 3, but the critical temperature is much lower for the new system. Near critical measurements of p were made for this system by Singh and PitzerSZ5Subsequently, Zhang et a1.26determined the exponent y by turbidity measurements. For the phase composition measurements the refractive index was determined. The sample was in a prism-shaped cell thermostated to f1 mK for the short term and f 3 mK over 24 h periods. The refractive indices of the top and bottom phases were measured to 3~0.0001. Equilibrium was sluggish near T, but was verified by approach from both higher and lower temperatures. The difference in refractive index, In1 - n2(,is a very good approximation to the difference in volume fraction in this type of system and is, therefore, a good order ~arameter.~’On this basis, eq 4 for the Wegner series becomes
+
+
In, - n21 = BolAT,lB(l B,~ATJ* B2~ATr~2A...) (8) This equation was fitted to the measurements within about 0.3 K of T, with limiting ,8 values alternatively of 0.50 and 0.326 and with the exact Tc adjusted. The results are shown in Table 2 and Figure 6. The fitted values of T, fall within the measured range 317.693 f 0.003 K. Note, however, that the expansion parameter Bl is small for p = 0.50 (mean field) but large for /3 = 0.326. For the Ising case with /? = 0.326, this involves a crossover. The midpoint (where& = (0.50 0.326)/2 = 0.413) is very close to T, at IAT,,,I = 0.9 x lop4. Thus, the effective p either remains close to 0.50 or shows a crossover from 0.326 to near 0.5 at very small IAT,l. In the range further from the critical point, the refractive index measurements and mole fraction measurements give Be in the range 0.47-0.50. Figure 7 shows the near-critical befrom the two Wegner series expressions with extensions to larger AT,; also shown is the essentially constant De for a two-component nonionic fluid in this range. There is also some interest in the diameter, which in this case is the mean refractive index. An expression linear in AT, fits well within experimental uncertainty; it is
+
(nl
+ nJ2
= 1.5504
+ 0.16(ATr)
(9) This result is consistent with classical exponents. Even if the
Pitzer
13074 J. Phys. Chem., Vol. 99, No. 35, 1995 2000
-
0.5 IONIC r
1600
-A1200 8
-.
h
NONIONIC
c I
1
I
t
103 T I ( K ' )
t
!
400
crossover to Ising exponents is the true picture, the deviation from eq 9 would not be expected to exceed experimental uncertainties. The turbidity measurements of Zhang et a1.26 on the N2226B2226-diphenyl ether system yield the exponent y for temperatures above Tc with mole fraction fixed at the critical value. The interpretation is more complex because the conversion of turbidity to osmotic susceptibility XT involves the Omstein-Zemike correction, which in tum involves the correlation length with its own critical exponent Y (0.63 king, 0.50 mean field). This correction becomes very significant for turbidity very close to T,. Zhang et a1.26 were unable to determine T, for their sample more closely than f 6 mK. Nevertheless, they demonstrate clearly that their results over the range 10-4-10-' in ATr are consistent with y = 1.01 f 0.01. Zhang et al. do not give an altemate Wegner series expression with the king limiting exponent and a crossover. But with the uncertainty in T,, and the complexity of the Omstein-Zemike correction, I believe that a crossover from y = 1.24 with IAT,,,I at 0.9 x low4is an allowed alternative. Thus, my conclusion for the N2226-B2226 system from both investigations is that either the limiting exponents have the classical values or there is a to Ising exponents. Further crossover with lATr,xl= 1 x from T, the effective exponents are close to the classical values, as had been noted earlier for similar ionic systems. In addition to the results for the N2226B2226 salt, near-critical measurements have been made for less ideally ionic systems. Weingartner et al? measured light scattering for the system tetra-n-butylammonium picrate in 1-tridecanol, which shows phase separation below 342 K. With the OH group of the alcohol solvent there is a fairly strong attractive interaction with peripheral oxygens on the picrate. Thus, this is a less ideally ionic system than those described above, but it is still primarily ionic. The light scattering results were consistent either with y GX 1.00, Y = 0.50 (classical) throughout or with a crossover from the Ising y = 1.24, Y = 0.63 to effective y 2 1.0, Y = 0.50. These results were important in suggesting that a crossover might be clearly identified by further studies of this type of system. At the time of the Weingartner et a1.28measurements, there was a special interest in the exponent y . Kholodenko and B e ~ e r l e i nhad ~ ~claimed that ionic fluids would show sphericalmodel exponents. This model gives 0.5 for /3, which agreed with the Singh-Pitzer results. But the spherical model predicts y = 2.0. Thus, results of Weingartner et a1.28 and the slightly later results of Zhang et a1.26 excluded the spherical model. Fisher30 also challenged the arguments of Kholodenko and Beyer1ei1-1.~~
0 2.80
2.84
2.88 2.92 2.96 i03 T" (K.')
3.00
Figure 8. Reciprocal of Omstein-Zemike Aa) corrected turbidity as a function of inverse temperature for TPDD system. Linear behavior of the graph is a direct consequence of x;' = T - Tc,,,,r.Departure from linearity demonstrates the crossover behavior with T, 330.0 K, corresponding to Tc.mr= 332.28 K. The inset depicts the king behavior near T, (from ref 32).
-
Crossover Confirmed for Some Systems For the most fully ionic system, "jB2226-diphenyl ether, the limiting pattem at the critical point remains uncertain, even with very precise measurement^.^^.^^ Either the limiting exponents are classical or there is a crossover to Ising exponents extremely close to T,. Since the resolution of this question experimentally appears not to be feasible with currently available methods, it seemed useful to explore less ideally ionic systems where a crossover, if present, might be clearly measurable. Very recently, Narayanan and P i t ~ e r ~ Iadopted - ~ ~ this path using turbidity as the precision measurement. The salt for all of their systems is tetra-n-butylammonium picrate (TP), which is easily prepared and purified. As a primary set of solvents, 1-mdecanol (TD), 1-dodecanol (DD), and I-undecanol (UD) were selected; combined symbols TPTD, TPDD, and TPUD will be used. The first of these is the system studied by Weingartner et a1.28 As the hydrocarbon chain is shortened, the dielectric constant increases and the system is less strongly ionic. The first system measured by Narayanan and P i t ~ e r , ~ ' . ~ ~ TPDD, showed a very clear crossover with a near-critical region with y G 1.24, the Ising value, and a distant region with ye close to the classical value 1.00. Figures 8 and 9 show this clearly. That the mean-field (classical) pattern is correct away from the near-critical region is clear from Figure 8, while Figure 9 and the inset i n Figure 8 show that king behavior is followed close to T,. Figure 8 shows agreement on the mean-field basis for T'-= 2.98 x or AT, 0.01. Figure 10 shows values of the effective y calculated from pairs of measurements; they were calculated using for y the equivalent to eq 3 for p. The uncertainties are large for the individual points, but the crossover is very clear. And for AT, > 0.03, all points agree, within their uncertainties, with ye = 1.00. The TPTD and TPUD systems33also show clear crossovers. Figure 11 presents the results for the three systems (including TPDD) in a manner to show the region of Ising behavior and the beginning of the crossover. As expected, the crossover is closest to T, for TPTD where the dielectric constant is smallest and the ionic forces are strongest. The trend is regular with the crossover in TPUD farthest from T,.
J. Phys. Chem., Vol. 99, No. 35, I995 13075
Feature Article 20000
18000 16000 14000 t
-5 12000
5 10000 v
;8000
v
6000 4000 2000 n ”
325
0
,
,, ,
-
IShg - - - - Mean field
i
1o’2
i0-3
10”
ATr Figure 9. Background-corrected turbidity data of the TPDD system at critical mole fraction showing Ising behavior near T, and the crossover (from ref 31).
0.80 1o.4
t 0.3
335
345
355
T (K)
Experimental . . . . Isingrmrrec1,On
1o.*
10.‘
AT1 Figure 10. Effective ye for the TPDD system based on the measurements of Narayanan and Pit~er.~’ Reference 33 describes these results in detail with figures for TPTD and TPUD corresponding to Figures 8 and 9 for TPDD and tables of parameters for Wegner series representations. Also, details are given for the Omstein-Zernike correction f(a) to turbidity involving the correlation length 6. Wegner expansions of both 6 and the osmotic susceptibility XT are employed in the most complete treatment with the appropnate values of the critical exponent for each. The range over which a crossover occurs is a significant property and one of theoretical interest. For TPDD (see Figure to 3 x in ATr, Le., about lo), it is from about 3 x 1.0 in log(ATr). The ranges for TPTD and TPUD are less clearly defined than for TPDD but are consistent with 1.0 in log(AT,). For the N2226B2226 system, and if the Ising limiting exponent is assumed, there is a crossover. Its range is less clearly defined than that of TPDD but is consistent with 3 x or the same 1.0 in log IAT,(. For Na-NH3 to 3 x the original figure of Chieux and Sienko’O (included in Figure 2 above) shows a very abrupt crossover, but examination of the original data indicates no disagreement with a more gradual crossover such as that of TPDD. Thus, a range of about 1.0 in log IATrl is consistent with the data for all ionic systems and
Figure 11. Evidence of the beginning of the crossover for the systems TPTD, TPDD, and TPUD. Note that the onset is closest to T, in TPTD with the smallest E where Coulombic forces are strongest. The ordinate involving the critical part of the turbidity tc,etc., is chosen to give the linear pattern near T,. Measurements from ref 33.
for Na-NH3. This is a sharper crossover than some theories have suggested, but there is no theory directly relevant to ionic systems or to Na-NH3. Systems with larger dielectric constants than that of l-undecan01 were also measured by Narayanan and P i t ~ e r . ~ ’A- ~ ~ three-component system using a solvent 75 vol % 1-dodecanol and 25 vol % 1,4-butanediol with the same TP salt also showed a crossover, but one located about 3 times as far from the critical point as that of TPDD.32 The range of verified classical behavior was more limited, but it was still clear. Other system^^'^^^ with still higher dielectric constant, such as TP with pure butanediol, were primarily solvophobic in character with clear Ising behavior. Although the effective y departs somewhat from the limiting value, the change from the Ising value 1.24 is small, and there is no clear crossover in these cases. Weingartner, Schroer, and their associates have also made measurements on several systems that involve both ionic and nonionic interactions; these results are discussed in ref 5.
Theory FisheP and Stel16 discuss the current status of theory for the critical exponents of Coulombically dominated fluids. They make many suggestions but do not agree on firm conclusions. Stel16 argues that the RPM, with ions of exactly the same size and charge, would have no coupling between the fluctuations in mass and those in charge. The mass fluctuations would then yield Ising exponents. But without exact symmetry, charge and mass fluctuations would be coupled, and mean-field exponents are expected in Stell’s view. Even though the N2226+ and B2226ions are very similar, they are not identical in size and they differ considerably in their interaction with the diphenyl ether solvent. Thus, the Stell theory for the RPM would be questionable for this system. All other real systems here considered are even less symmetrical. Then, the Ising exponents observed for some other systems are presumably related to the short-range forces present. Fisher, in his re vie^,^ discusses several theories, including Stell’s, but finds serious weaknesses. He states, “thus Pitzer’s view (in reference 2) that the current situation is ‘inconclusive’ seems fair.” Fisher and his collaborators have published even more recently in this field, including theory for systems in 2 or other d i m e n ~ i o n a l i t y . ~But ~ . ~he ~ has not reached a different general conclusion as to the exponents for ionic fluids in 3 dimensions. Theory for crossover behavior in primarily ionic systems appears to be even less firm than that for the limiting exponents.
Pitzer
13076 J. Phys. Chem., Vol. 99, No. 35, 1995 TABLE 3: Critical Properties of NaCl .
00550
-A
A
A
!I
0 0525
GG-EOP GG adjusted40 RPM, 0 ~ 3 8 RPM, eqs 10, 11 Kirshenbaum et al.'
TJK
e,/g cm-'
3068 3300
0.17, 0.18 0.058 0.115 0.22
3210
3060 3400
T'
I 0 0450
&
0 0425000 005 0 10 0 15 020 025 030 0 3 5
P' Figure 12. Near-critical region for the RPM. Points are GEMC calculated values: circles from ref 38 and triangles from ref 37. The fit shown was selected as a compromise which yields a linear diameter (dashed) and a B = '12 density-difference dependence for the vapor and liquid curves (solid) as expected for an ionic fluid.
Phase Diagram over a Greater Range of Temperature As a supplement to the presentation of the limiting critical exponents of ionic fluids, it is desirable to review briefly the near-critical, vapor-liquid phase diagram over a somewhat extended range and to note recent advances. The remarkably large expansion coefficient of the liquid was noted in 1962 by Kirshenbaum et al. (see Figure 1). This difference from nonionic fluids is equally striking in the properties of the restricted primitive ionic model (RPM). Gibbs ensemble Monte Carlo (GEMC) calculations for the RPM have been reported by Panagiotopo~los~~ and more recently by CallioP and by Orkoulas and Panagiotopoulo~.~~ This model comprises charged hard spheres of the same diameter u and magnitude of charge q for and - ions. This is a purely ionic model in that there are no short-range attractive forces or any softness of repulsive forces. For the RPM a reduced density e* = Nu3/V and a reduced temperature P = kTu/q2are defined. The Gibbs Monte Carlo method involves transfers of particles from one phase to the other. For an ionic system the two earlier calculation^^^^^^ shifted individual ions with a correction to maintain charge neutrality over all in each phase. The vapor at significantly subcritical temperatures, however, comprises primarily ion pairs with few individual ions, and this introduces very serious problems of rate of convergence in the calculations. Orkoulas and Panagiotopoul~s~~ (OP) transferred ion pairs instead of single ions. They first showed that for canonicalensemble (NVT) Monte Carlo the convergence was much more rapid for pairs than for single ions. Thus, I believe, their GEMC calculations for the RPM are the best now available. Figure 12 shows the near-critical GEMC results of Calli01~~ (triangles) and of OP3x(circles). The earlier GEMC results of Panagiotopo~los~~ are similar to those of Calliol; also, an approximate equation of Fisher and L e ~ i gives n ~ ~ a generally similar coexistence curve. All of these investigations place the RPM critical point in the range 0.050-0.057 in P and 0.0250.05 in e*. Further examination of the GEMC results indicates fairly good agreement among all three calculations for temperatures up to P = 0.0475 but serious differences at higher P.Indeed, OP38give no separate vapor and liquid results above 0.050 in P,while Callio13?reports distinct vapor and liquid densities
+
through 0.055. The shape of the Calliol curve above 0.050, however, is peculiar in relation to that at lower P. While further GEMC calculations are clearly needed to resolve the RPM properties in the near-critical region, it is interesting to apply to the present results the general expectations of a linear diameter and a critical exponent /3 = l/2. The vaporliquid curve and the diameter shown in Figure 12 seem reasonable on this basis. The liquid density is between the two GEMC values at P = 0.050 and somewhat closer to the more recent and probably the more accurate one. The vapor density is certainly small at P = 0.050 and lower temperatures, although the uncertainty on a percentage basis is considerable. The,curve in Figure 12 yields for the critical point, T,* = 0.0505, ec* = 0.05, and corresponds to the following equations:
(e; - et)2= 0.616 - 1 2 . 2 P (e: + $)I2
= 0.96 - 1 8 P
(10)
(1 1) These equations are valid only above 0.0475 in P ;additional terms would be needed for a wider temperature range. Orkoulas and Panagiotopoulos3xrecommend Tc* = 0.053 and eC* = 0.025; this is shown as a square in Figure 12. Their rather small value for the liquid density at P = 0.050 presumably influenced their choice of 0.025 for the critical density. Indeed, a vapor-liquid curve of reasonable curvature through their points at P = 0.050 is consistent with this density, but it would have a critical temperature considerably lower than 0.053. The GEMC calculation^^^-^^ extend to temperatures lower than the range of Figure 12. Also, there are other MC calculations for the liquid. Rather than discussing this area in detail, I tum now to the situation for the real fluid NaCl. Guissani and Guillot40 (GG) recently reported molecular dynamics calculations for fluid NaCl using a set of interionic potentials that had been developed by others from the extensive data base for crystalline NaCl. While the GG calculations cover the entire temperature range from the melting point to well above the critical point, we will give primary attention to the range above 1900 K where they fitted a numerical equation of state (EOS). From this EOS they obtained critical properties that are listed in Table 3. In addition, they made a small adjustment in both T and e in order to more accurately fit the experimental densities of the liquid from Kirshenbaum et al.' Their adjusted critical properties are also given in Table 3. It is interesting to compare these values of GG with values from the RPM discussed above. For this purpose an effective u for NaCl must be chosen. OP selected u = 0.276 nm, which is very close to the nearest-neighbor distance 0.282 nm in crystalline NaCl. On this basis, the calculated RPM liquid densities are consistent with the measured NaCl values of Kirshenbaum et al.' This basis then yields the values in Table 3 for both the OP critical values T,* = 0.053, e,* = 0.025 and the altemate values Tc* = 0.0505, ec* = 0.05 of the Figure 12 phase diagram above. The near identity of the T, values of GG-EOS and of "RPM, eqs 10, 11" is interesting, but more significant is the general agreement of all four recent values. One can have considerable
Feature Article
J. Phys. Chem., Vol. 99, No. 35, 1995 13077 References and Notes
io 1
104
I 06
07
0 8
09
d
Figure 13. Curves of effective /Ifor several fluids; see text for further details (from ref 44).
confidence that T, for NaCl is within the range 3200 & 200 K. It is now clear that values4' I proposed in the early 1980s were too high. The range of critical density values in Table 3 is much greater than those of Tc. The value based on eqs 10 and 11 is at about the geometrical mean of the highest and lowest values, which is some recommendation. Although the uncertainty remains great, I recommend ec = 0.11 g cm-3 for NaCl. Much could be said about NaCl properties at lower temperatures, including the extensive experimental data'9.42 and theory43concerning the dimer Na2C12 in the vapor, but this is beyond the scope of the present review.
Concluding Remarks For the practical representation of properties over a substantial range of temperature, ionic fluids may be assumed to have classical (mean-field) effective exponents. Thus, mathematically analytic equations of state suffice. But the exact behavior of ionic fluids very near the critical point constitutes an interesting challenge to theory. A crossover to Ising exponents definitely occurs if there are significant short-range attractive forces in addition to the Coulombic forces, and this crossover moves further away from T, as these short-range forces become more important. Finally, as these short-range forces decrease to zero, the limiting exponents are uncertain. For the best example, very precise experiments do not distinguish between classical limiting exponents or a crossover with midpoint at about 1 x in IATrI.
Acknowledgment. I thank Dr. J. M. H. Levelt Sengers and Dr. T. Narayanan for valuable comments. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division, of the U.S. Department of Energy under Contract DE-AC03-76SF00098. Appendix: Effective Exponents for Nonionic Fluids For comparison with the sharp crossovers exhibited by certain Be are shown i n Figure 13 for Xe and CO2 and for the mixed fluids C7F14-CC14 and C H ~ C N - C ~ H(from I ~ Singh and Pitze?). For the pure fluids, Be increases initially from the king value of 0.325 as T, decreases from 1.O, but it does not approach the mean-field value of 0.5. Instead, Be has a maximum of about 0.35 and then decreases. For the two-component nonionic fluids /ledecreases slowly from the Ising value, and nothing remotely like a crossover occurs. The dashed line is for an Ising lattice; Monte Carlo calculated values for a 6-12 Lennard-Jones fluid are also shown. ionic systems and Na-NH3, curves of
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