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Parachors of liquid/vapor systems: A set of critical amplitudes Article in Fluid Phase Equilibria · June 2005 DOI: 10.1016/j.fluid.2005.03.025
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Fluid Phase Equilibria 233 (2005) 86–95
Parachors of liquid/vapor systems: A set of critical amplitudes D. Broseta a,b,∗ , Y. Mele´an b,c , C. Miqueu a a
Laboratoire des Fluides Complexes, UMR 5150, Universit´e de Pau et des Pays de l’Adour, B.P. 1155, 64013 Pau Cedex, France b Institut Fran¸ cais du P´etrole, 1 and 4 Avenue de Bois Pr´eau, 92582 Rueil-Malmaison Cedex, France c Laboratoire de Tectonophysique, ISTEEM, CNRS-Universit´ e de Montpellier 2, 34095 Montpellier Cedex, France Received 8 September 2004; received in revised form 25 March 2005; accepted 29 March 2005 Available online 31 May 2005
Abstract In light of the available experimental data and of our current understanding of liquid–vapor critical phenomena, we examine the values of the parachors and of the parachor exponent, which are commonly used to estimate surface tension from the density difference between coexisting liquid and vapor phases. This is a controversial issue, as values for the parachor exponent ranging from 3.5 to 4 have been proposed in the literature. The parachor exponent and parachors can be viewed as a critical exponent and critical amplitudes, respectively. The Ising value, equal to 3.88, should be observed for the exponent “close enough” to the liquid/vapor critical point, i.e., for “low enough” tensions and densities. However, a review of experimental data for several fluids suggests an effective value in the range of 3.6, in line with the effective values observed for the exponents that describe the vanishing of the density difference and capillary length with the distance to the critical temperature. In fact, the asymptotic Ising regime is not reached experimentally, as confirmed by an estimation of the parachors very near the critical point. Those (Ising) parachors can be inferred from other critical amplitudes corresponding to bulk properties by using two-scale factor universality. Their values exceed those deduced from off-critical tension and density data by more than 10%, corresponding to surface tension differences larger than 50%. We argue that effective parachors (i.e., corresponding to an exponent in the range of 3.6) can be utilized in combination with two-scale-factor universality for determining the critical behavior of fluid systems in an extended range around their liquid/vapor critical point. © 2005 Elsevier B.V. All rights reserved. Keywords: Critical point; Parachor; Surface tension; Two-scale factor universality
1. Introduction Surface tension plays an important role in processes in which the liquid and vapor phases are finely divided. For instance, heat transfer through the heat exchanging surfaces used in the refrigeration industry depends strongly on the surface tension of the refrigerant. Surface tension also controls a number of porous media transport parameters (capillary pressure, relative permeabilities, and residual liquid saturation) used by reservoir engineers for the simulation of oil recovery processes such as gas injection [1]. One of the most popular method for estimating surface tensions is provided by parachor correlations. Those correla∗
Corresponding author. Tel.: +33 5 59 40 7685; fax: +33 5 59 40 7725. E-mail address:
[email protected] (D. Broseta).
0378-3812/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.03.025
tions consist in a scaling relation between surface tension, σ, and the density difference between the coexisting liquid and vapor phases, ρ =ρl − ρv . In the case of pure compounds, they are usually written as follows: p P σ= ρ , (1) M where p is the parachor exponent, M and P the compound’s molar mass and parachor. This method was introduced in the early 1920s when Macleod [2] and Sugden [3], on the basis of experimental observations, first proposed the value p = 4 for the parachor exponent. These authors examined offcritical systems, i.e., systems far from their liquid/vapor critical point, in which ρv is negligible compared to ρ1 , leading to σ 1/p = Pρ1 /M = P/V1 , where Vl = M/ρl is the liquid molar volume: comparing parachors of liquids with similar surface
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
tensions is therefore equivalent to comparing molar volumes – hence the name of parachors. In the case of mixtures, a scaling relation of the type of Eq. (1), i.e. σ = Cρp ,
(2)
holds experimentally (with p in the range of 4) as well, at least for low enough σ and ρ (C is a constant that depends on the system composition) [1,4–6]. Recent model calculations [7] have shown the robustness of this approach: in a given simple (binary) mixture, Eq. (2) holds with the same values of C and p, independently of the path of approach to the critical point. Another expression, which is a phenomenological extension of Eq. (1) to mixtures, is widely used in the oil and gas industry [8]: p xi yi σ= Pi , (3) − Vl Vv i
where xi and yi are the molar fractions of component i (with parachor Pi ) in the liquid and vapor phases, respectively. A considerable research effort has been devoted to improve the performance of those correlations by finding out the “best” exponent p and parachors Pi (whose values depend heavily on the choice of the exponent p). In fact, calculated surface tensions are very sensitive to the values of P( = MC1/4 in the case of mixtures, cf. Eq. (2)), as those parachors are raised to a power p ≈ 4. Our purpose in this paper is to gain insight into those exponent and parachors by exploiting the current knowledge of the critical behavior of fluids. It was recognized as early as in 1945, by Guggenheim [9], that Eqs. (1) and (2) are connected to the scaling laws that describe the behavior of thermodynamic properties in the region near the liquid–vapor critical point. Using the modern language of critical phenomena, the parachor exponent is a universal critical exponent, i.e., it does not depend on the chemical nature of the fluid, nor on whether the fluid is a pure compound or a mixture; while parachors P can be viewed as critical amplitudes and are system-dependent. The parachor exponent p is related to the customary critical exponents, µ and β, that describe the vanishing of σ and ρ with the (reduced) distance to the critical temperature t = (Tc − T)/Tc : σ = σ0 t µ ,
(4)
ρ = ρ0 t β ,
(5)
and the parachor P is related to the corresponding critical amplitudes, σ 0 and ρ0 (usually expressed as ρ0 = 2B0 ρc , where ρc is the critical density and B0 a dimensionless number): β/µ
p = µ/β
and P =
Mσ0 . 2B0 ρc
(6)
On the basis of the data available in 1945, Guggenheim [9] proposed the values µ = 11/9 and β = 1/3, leading to the value
87
p = µ/β = 11/3, a value later claimed [10–12] to provide a better description of the surface tension dependence on density than Mac Leod and Sugden’s p = 4, especially in the case of low tensions (less than a few dyn/cm, typically). Other values have been proposed to account for experimental data, including p = 3.6 [13,14] and density-dependent values varying from p = 3.53 very near the critical point (ρ ≈ 0) to values slightly above p = 4 for large ρ [15]. The past three decades have seen considerable progress in the determination of critical exponents, gained through both precise experimental work and powerful calculational techniques [16,17]. The “best” values for β and µ = 2ν (ν is the exponent for the divergence of the correlation length of density fluctuations near Tc , see Section 4) are now considered to be β = 0.326 ± 0.001 and µ = 2ν = 1.260 ± 0.002, leading to p = 2ν/β = 3.87 ± 0.01, very slightly below Mac Leod and Sugden’s value p = 4. In the following, we will stick to the value p = 3.88 proposed previously [4,5,18] and refer to it as the Ising value of the parachor exponent. This exponent characterizes the dependence of σ upon ρ asymptotically close to the critical point, i.e., for “low enough” σ and ρ, of any system belonging to the Ising universality class, which includes not only the liquid–vapor systems of interest here, but also, for instance, demixing liquid–liquid or polymer–solvent binary mixtures close to a consolute point [19,20], and microemulsions near a critical endpoint [21]. In the modern theory of the critical phenomena, this “close enough” distance to the critical point is measured by the Ginzburg number, Gi [22]. The Ising values of the critical exponents are observed only in a reduced temperature range |t| Gi, and at |t| > Gi a so-called cross-over from the asymptotic critical scaling behavior to the regular “mean-field” behavior is observed, characterized by (mean-field) exponents equal to, e.g., µmf = 3/2, βmf = 1/2, pmf = 3. A global crossover model has recently been developed and successfully applied to the prediction of the temperature dependence of surface tension in many fluids [23]. Simple correlations have been proposed that account for such cross-over, such as [24]: σ = σ0 t 1.26 [1 + b0 t 0.24 ],
(7)
where b0 is a numerical constant. In practice, the range |t| in which critical scaling holds varies from one property to another. In the case of ρ this range is very tiny (below around |t| < 10−4 ) [25,26], whereas it is much wider in the case of the squared capillary length (or Sugden parameter) a2 = 2σ/gρ, which is the quantity measured by conventional capillary rise or drop techniques (then σ is deduced from a and ρ). In fact, the behavior of a2 with t is well characterized by the (Ising) exponent φ = µ − β = 0.934 (see Eqs. (4) and (5)) over a wide temperature domain below Tc [27–29]. A rigorous way to extend the range of applicability of critical scaling is to add correction-to-scaling (or Wegner) terms t , t2 , . . . to the leading (asymptotic) scaling term, with ∼ 0.5 for the Ising universality class (see, e.g. Refs.
88
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
[26,28]). For instance, the following extension of the scaling Eq. (4) for σ: σ = σ0 t 1.26 (1 + a1 t 0.5 + a2 t)
(8)
provides over a wide temperature interval an accurate representation of surface tensions of a large number of compounds, even when using unique values of a1 and a2 (those constants are, in principle, system-dependent) [30]. Another convenient way to extend the range of applicability of critical scaling consists in the use of effective or apparent critical exponents. For instance, the use of a value βeff in the range of 0.35–0.36 considerably extends the range of applicability of Eq. (5) [26,31,32]. According to many authors [6,10–12,33], Guggenheim’s value p = 11/3 can be used as an effective parachor exponent, at least for sufficiently low tensions or densities. However, the use of the Ising value, p = 3.88, was recently advocated by Schechter and Guo [5], who examined a large set of data and suggested that the range of applicability of Ising scaling is large. This paper proposes a clarification of this controversy based on a careful analysis of the existing surface tension and density data and on our current understanding of liquid/vapor critical phenomena. In particular, we use the twoscale-factor universality (TSFU) principle. TSFU goes well beyond simple scaling and assumes that there exists only two independent critical amplitudes, just as there are two independent critical exponents. Once these two amplitudes (or scale factors) are known, all other critical amplitudes can be determined from a few universal critical amplitude ratios (specific to a given universality class, here the Ising class). For instance, the critical amplitudes of thermodynamic “bulk” properties (such as specific heat, isothermal compressibility, etc.) are connected to the critical amplitudes of surface tension [27–29], capillary length [24], as well as to parachors, as we discuss below. This connection allows us to estimate the Ising parachors, i.e., the parachors that should be observed very near the critical point. The outline is as follows. In Section 2, we examine closely the dependence of σ with ρ in a dozen of pure compounds, chosen for the availability of experimental σ, ρ and other
bulk properties (specific heat, compressibility, etc.) in a range of temperature approaching Tc closely. We confirm that, as a rule, an effective exponent lower than the Ising value – in the range of p ≈ 3.6 – is observed for low enough surface tensions or densities; while higher apparent exponents, close to the Ising value p ≈ 3.88 or to Mac Leod and Sugden’s p = 4, hold for high tensions and densities. In Section 3, we show that the effective value for p is consistent with the effective values of the critical exponents describing the vanishing with t of the density difference ρ and capillary length a. In Section 4, we use TSFU to infer the Ising parachors of those compounds from the (Ising) critical amplitudes of isochoric specific heat, isothermal compressibility, etc. In the last section, we compare the values of those Ising parachors to the experimental values (=Mσ 1/3.88 /ρ) and propose to use TSFU in combination with parachors to infer the whole thermodynamic behavior of fluids in a extended range around the liquid/vapor critical point.
2. Experimental data In this section we examine how closely the asymptotic (Ising) scaling behavior, σ ∼ ρ3.88 , is verified by the experimental σ and ρ values of the following pure compounds: Ar, Kr, CO2 , N2 , O2 , SF6 , Xe, and the low-molecular-weight alkanes: CH4 , C2 H6, C3 H8 , i-C4 H10 and n-C4 H10 . For those compounds reliable σ and ρ data are available over a wide temperature range (including in most cases the near-critical region, i.e., small values of t), in addition to near-critical data of other thermodynamic properties, such as the correlation length of density fluctuations, the isochoric specific heat, and the isothermal compressibility. The latter information will be used in Section 4 in combination with TSFU to provide an estimation of the Ising parachor PIsing . Table 1 summarizes the available information, extracted from a larger data base [30], and lists, for each compound, the range of σ, ρ and temperatures, together with the literature source. For each compound, we have chosen among the data gathered and analyzed in Ref. [30] the set of data that covers the largest interval of t or ρ. In the case of C3 H8
Table 1 σ, ρ and t data considered in this study Ar Kr Xe CH4 O2 N2 C2 H6 C3 H8 i-C4 H10 n-C4 H10 SF6 CO2
σ range (dyn/cm)
ρ range (g/cm3 )
t range
Data sources
0.55–5.0 0.89–15.4 0.19–15.6 0.009–3.81 1.19–12.1 0.014–5.21 0.013–33.5 0.28–13.0 0.15–13.0 0.11–10.35 0.0005–11.1 0.0013–4.44
0.59–1.10 1.11–2.40 0.86–2.82 0.058–0.31 0.61–1.14 0.14–0.71 0.082–0.65 0.19–0.55 0.18–0.60 0.16–0.55 0.13–1.82 0.087–0.83
0.036–0.20 0.04–0.40 0.012–0.35 0.0015–0.523 0.064–0.415 0.0026–0.265 0.0013–0.71 0.015–0.32 0.0095–0.63 0.007–0.27 0.00012–0.295 0.00015–0.10
[36] [36] a2 : [37] ρ: [38] a2 : [29] ρ: [39] a2 : [29] ρ: [40] a2 : [29] ρ: [41] a2 : [42] ρ: [43] a2 : [34,44] ρ: [34,45] a2 : [34,44] ρ: [34,45] [46] [35,47] a2 : [48] ρ: [49]
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
Fig. 1. Experimental σ and ρ data for N2 , CO2 , and O2 . The arrows correspond to σ = 5 dyn/cm. The interval between the horizontal lines corresponds to the range of Ising values (PIsing /M)3.88 estimated in Section 4.
and i-C4 H10 , we have added the values recently published by Lin and Duan [34] at high t or ρ. All those data correspond to conventional measurements by capillary rise or drop techniques. In the case of SF6 , we have added the values obtained very close to the critical point by Wu and Webb using quasi-elastic light scattering from capillary waves [35]. To magnify possible deviations from the asymptotic (Ising) scaling behavior (σ ∼ ρ3.88 ), we have divided out surface tensions by the leading singularity ρ3.88 . Fully logarithmic plots of the quantities σ/ρ3.88 versus ρ are displayed in Figs. 1–3. Clearly, the near-critical (Ising) regime where σ ∼ ρ3.88 is not observed in the range of ρ and σ investigated. Instead, a negative slope [equal to peff − 3.88, where peff = d log()/d log(ρ) refers to the effective parachor exponent] is observed over the lowest tensions investi-
89
Fig. 3. Experimental σ and ρ data for CH4 , C2 H6 , C3 H8 , i-C4 H10 , and n-C4 H10 (for the meaning of arrows and lines, see caption of Fig. 1).
gated, which persists until σ ∼ 1–5 dyn/cm, corresponding to reduced temperatures in the range of, typically, t ∼ 0.1–0.2. This slope is fairly constant in a low-to-intermediate range of surface tensions (from ∼10−3 to ∼1–5 dyn/cm), and lies between −0.13 and −0.32, or peff ∼ 3.56–3.75. For high values of ρ or σ (σ above around 1–5 dyn/cm), σ/ρ3.88 levels off or even increases very slightly with ρ or σ in all compounds but xenon, indicating in that region an effective parachor exponent close to, or slightly higher than, 3.88. It is worth mentioning here that large uncertainties (not displayed in Figs. 1–3) are associated with the measurements by conventional drop or capillary rise techniques at low σ or ρ: they are clearly apparent (see Fig. 1), for instance, in the CO2 data points with ρ ∼ 0.1 g/cm3 , σ ∼ 0.001 dyn/cm. In this region, the quasi-elastic light scattering technique is more appropriate for measuring surface tensions. In the low-to-intermediate surface tension region (from σ ∼ 10−3 to ∼1–5 dyn/cm) such observations of an effective parachor exponent peff ∼ 3.56–3.75 are in line with Guggenheim’s value p = 11/3–3.67 adopted by several authors [6,10–12,33]. There is no fundamental reason, however, why p should be fractional: the choice of p = 11/3 corresponds to an assumption in the early literature on critical phenomena that critical exponents should be fractional.
3. Effective parachor exponent
Fig. 2. Experimental σ and ρ data for Ar, Kr, SF6 , and Xe (for the meaning of arrows and lines, see the caption in Fig. 1). The line corresponds to SF6 surface tensions obtained by the quasi-elastic light scattering technique (given in Ref. [35] in the form of a power law in t).
In this section, we show that those values for the effective parachor exponent, peff ∼ 3.56–3.75, are consistent with the values observed in the corresponding range of t (i.e., from t ∼ 10−3 to ∼10−1 ) for the critical exponents characterizing the behavior of the measured quantities, i.e., the difference in densities ρ and the squared capillary length a2 (from which surface tension is deduced: σ = 1/2gρa2 ).
90
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
By elimination of t between Eq. (5) and the scaling equation for the squared capillary length a2 = a02 t φ ,
(9)
where φ = µ − β we obtain σ β/(β+φ) =
β/(β+φ)
(a02 g/2) ρ. (2ρc B0 )φ/(β+φ)
(10)
This expression is the most appropriate for discussing the values of the effective parachor exponent peff , since it is related to measured quantities: ρ and a2 . The exponent (φ + β)/β = 1 + φ/β can be identified with the parachor exponent and the quantity: β/(β+φ)
M
(a02 g/2) 2ρc B0 φ/(β+φ)
(11)
with the parachor P. Let us now examine the effective values observed for the exponents β and φ. On the one hand, the scaling Eq. (5) holds for the density difference ρ of liquid/vapor systems over a wide temperature range (of order 10−3 < t < 0.03) if one uses an effective or apparent exponent βeff in the range of 0.35–0.36. βeff = 0.355 ± 0.010 is the value recommended in the NIST publications (see, e.g., Refs. [31,32])—this value was in fact believed to characterize the scaling behavior of ρ close to Tc [31] until measurements in the very close neighborhood of the critical point (t below ∼10−4 ) could be carried out in the mid-1970s [25]. On the other hand, the squared capillary length of liquid/vapor systems, a2 , conforms better to Ising scaling: an effective exponent φeff equal to (or very slightly below) the Ising value µ − β = 0.934 provides a description of a2 over a very wide temperature range (extending to close to the triple point) [27,29]. If we use the value βeff = 0.355 and the Ising value for φeff , we find an effective parachor exponent peff = 1 + φeff /βeff ∼ 3.63 in a range consistent with the observations reported in Section 2. If we adopt the set of effective critical exponents recommended by the NIST, i.e., βeff = 0.355 and αeff = 0.1 (α is the exponent for the specific heat divergence, see next section), the exponents of interest here are then φeff = 2νeff − βeff = 2(2 − αeff )/3 − βeff ∼ 0.912 and peff = 1 + φeff /βeff ∼ 3.57,1 again consistent with experimental data (albeit in the lower end of the range of observed values). Since a2 conforms to Ising scaling in an extended region around the critical point, the range where the asymptotic behavior, σ ∼ ρ3.88 , should be observed coincides with the range over which ρ obeys Ising scaling, i.e., t below ∼10−4 or, using the ρc and B0 values listed in Table 2, ρ( = 2B0 ρc tβ ) below about 0.16 g/cm3 (SF6 ), 0.15 g/cm3 (xenon), 0.14 g/cm3 (Kr), 0.13 g/cm3 (CO2 ), 0.12 g/cm3 (Ar), 0.11 g/cm3 (O2 and N2 ), and 0.09–0.10 g/cm3 (all alkanes). 1 The following equality holds between the critical exponents α and ν: 2 − α=3ν (see Section 4).
All experimental data points displayed in Figs. 1–3 fall out of this range, except for a few points in the case of CO2 (Fig. 1), SF6 (Fig. 2) and CH4 (Fig. 3). The latter CO2 and CH4 points are considerably scattered, as they correspond to very low surface tensions (below 10−3 dyn/cm) not easily measurable by conventional drop techniques. The Ising parachors, corresponding to the amplitudes of the asymptotic Ising behavior, can however be estimated by using TSFU, as we show in the next section.
4. Asymptotic Ising behavior inferred from other near-critical data and TSFU According to TSFU, the thermodynamical potential (or, more precisely, its singular part) is a universal function of suitably defined variables and is characterized by two nonuniversal scale factors. As a consequence combinations between critical amplitudes can be formed that take universal values (within a given universality class). Some of those combinations involving the surface quantity σ are briefly presented below. They are then utilized for deriving expressions for the parachors in terms of other critical amplitudes. Near the critical point, the correlation length of density fluctuations, ξ, is the relevant length scale (this quantity is readily measured by radiation scattering techniques). ξ diverges on the critical isochore (ρ = ρc ) like ξ ± = ξ0± |t|−ν ,
(ν = 0.63),
(12)
where ξ + and ξ − (and the corresponding amplitudes ξ0+ and ξ0− ) refer to values measured, respectively, above Tc (onephase region, t < 0) and below Tc (two-phase region, t > 0). Surface tension, an energy per surface unit, behaves near Tc like σ=
U − kTc (ξ − )2
,
(13)
where kTc (k is Boltzmann’s constant) is the energy scale near Tc . Widom’s equality between critical exponents follows, µ = 2ν, together with the following combination of critical amplitudes: σ0 (ξ0− ) = U− kTc 2
(14)
in which, according to TSFU, U− is a universal dimensionless number. In practice, the correlation length is more readily measured in the one-phase region (T above Tc ) and, since the ratio ξ0+ /ξ0− is also a universal number (equal to 1.96 [17]), a test of TSFU is also provided by forming the ratio: σ0 (ξ0+ ) σ(ξ + ) = = U +, kTc kTc 2
2
(15)
where ξ + is measured in the one-phase region at the reduced temperature −t (t is where σ is measured). The recommended
Table 2 Estimation of the Ising parachors using various literature sources for the Ising critical amplitudes, as well as Eq. (26) from Leneindre and Garrabos. P3.88 is the parachor estimated from the σ and ρ data farthest from the critical point where σ/ρ3.88 levels off (see Figs. 1–3) Kr
Xe
CH4
O2
N2
C 2 H6
C3 H8
i-C4 H10
n-C4 H10
SF6
CO2
150.69 4.863 74.59 0.536 0.289 0.00 1.42 [16], 1.47
209.34 5.505 91.99 0.911 0.291 0.00 1.42 [16]
190.55 4.60 98.63 0.163 0.286 0.011 1.48 [58], 1.60 [59]
154.58 5.04 73.44 0.436 0.288 0.022 1.51 [29]
126.2 3.398 73.44 0.314 0.289 0.037 1.48 [60], 1.58 [29]
407.85 3.65 258.28 0.225 0.278 0.186 1.64 [16,27] 1.70 [59]
425.38 3.809 254.06 0.229 0.274 0.200 1.71 [59]
318.72 3.755 196.79 0.742 0.279 0.208 1.62 [29], 1.74 [27]
304.18 7.386 94.16 0.467 0.275 0.225 1.59–1.68 [16,27,29,64]
1.88 [27], 2.0 [16] 3.52 [16], 4.03 [29] 0.058 [16] 0.0119 [29] 0.14 [16]
2.0 [16]
305.33 4.87 155.45 0.193 0.298 0.099 1.57 [27], 1.66 [59] 2.22 [27]
370.4 4.25 199.42 0.221 0.275 0.152 1.68 [59]
A+ 0
289.73 5.84 118.28 1.11 0.287 0.00 1.42 [16], 1.475 [57] 1.84 [27], 2.0 [16] 3.52 [27,29]
30.3 [61]
2.8 [16,27]
2.86 [59]
3.20–3.37 [16,27,29,62,63]
2.43–3.06 [16,27,62]
4.09 [27]
5.78 [61]
5.215 [27]
5.93–6.20 [27,29,63]
5.32–5.56 [27,29]
0.061 [58]
0.055 [16]
0.046 [16,27]
0.046–0.052 [16,27,64]
0.0094 [29]
0.0094 [29]
0.171 [56]
0.188 [29]
0.153 [27]
61.7–68.6
83.0–84.0
83.5
60.7–62.1
63.7–72.4
132.2–136.2 165.7
214.0
204.5–209.3
153.8–166.2
86.2–90.0
59.5–64.6 64.05
77.3–83.9 80.65
0.0594 [57] 0.0119 [29] 0.184 [27] 109-2112.2 102.6–111.3 107.4
79.1–85.2 82.5
58.9–63.9 60.5
65.8–68.0 68.0
133.8–145.2 163.8–177.7 138.7 165
209.4–227.2 206.7–224.2 218 214
157.2–170.6 162
83.6–90.7 87.2
55.3
70.5
95.0
74.0
53.7
60.4
114
194
149
78.7
A− 0
Γ0+ Γ0− ξ0+
(nm)
PIsing Eqs. (16), (22) and (24) PIsing (Eq. (26) PIsing (median value) P3.88 a
0.06 [56]
2.07 [58]
4.45 [29]
0.0125 [29]
3.53 [29]
0.047 [60] 0.0131 [29]
0.010 [60] 0.18 [27]
155
193
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
Ar Tc (K) Pc (MPa) Vc (cm3 /mol) ρc (g/cm3 ) Zc ω B0 a
Values corresponding to β = 0.325, 0.326 or 0.327; all other amplitudes correspond to the Ising exponent values given in the text.
91
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D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
value of U+ is 0.39 [28] (in agreement with theoretical estimates [50,51], see also [52]: U+ = 0.37 ± 0.03). Combining Eqs. (15) and (6), the following expression holds for the Ising parachor in terms of the critical parameters Tc and Vc = M/ρc and amplitudes B0 and ξ0+ : β/µ Vc U + kTc PIsing = . (16) 2B0 ξ0+2 From a similar argument, the singular part of the free energy density, an energy per unit volume, behaves on the critical isochore (ρ = ρc ) near Tc like f
±
=
C± kTc (ξ ± )3
,
(17)
where again, superscripts − and + refer to values below and above Tc and C+ and C− are universal numbers. The quantity measurable experimentally is not f + or f − but rather the specific heat per unit volume Cv± =
∂2 f ±
−T ∂T 2
,
(18)
which on the critical isochore (ρ = ρc ) diverges near Tc like Cv± =
A± 0 Pc −α |t| , α Tc
(19)
where from Eqs. (12) and (17)–(19), α = 2–3ν = 0.11. In Eq. (17) Pc /Tc (Pc is the critical pressure) has been introduced to − render the amplitudes A+ 0 and A0 dimensionless. From Eqs. (17) and (19), it turns out that the following combinations of critical amplitudes: + A+ 0 (ξ0 )
3
Pc kTc
(20a)
Pc kTc
(20b)
and + A− 0 (ξ0 )
3
are universal numbers. The accepted values for those numbers − are, respectively, 0.0188 and 0.0360 (A+ 0 /A0 ∼ 0.523) [17]. + The elimination of ξ0 between Eqs. (15) and (20) leads to Y
+
σ0 = kTc
P c A+ 0 kTc
−2/3
P c A− 0 kTc
−2/3
(21a)
and Y−
σ0 = kTc
number): PIsing =
Vc ± 2/3 β/µ Y (kTc )1/3 (Pc A± 0) 2B0 ± 2/3 β/µ NAv Zc (A0 )
= (Y ± )
2B0
(kTc )1+β/3µ /Pc2β/3µ . (22)
Other expressions for the Ising parachor can be obtained in terms of the dimensionless critical amplitudes for the isothermal compressibility Γ0+ and Γ0− , defined on the critical isochore as follows: χ± = (∂ρ/∂µ)T = Γ0±
ρc2 −γ |t| , Pc
γ = 1.24.
(23)
Those expressions, which are derived from Eq. (22) and the + 2 universal amplitude combinations: A+ 0 Γ0 /B0 = 0.0581 and + − Γ0 /Γ0 = 4.95 [17], are PIsing =
Vc + 2/3 β/µ [Y (kTc )1/3 (0.0581Pc B02 /Γ0+ ) ] 2B0
(24)
and the same expression, in which Γ0+ is replaced by 4.95 Γ0− (Eq. (24)). TSFU also holds with effective critical exponents, such as the NIST exponents mentioned in the previous section: βeff = 0.355 and αeff = 0.1, corresponding to φeff = 0.912 and an effective parachor exponent peff = 3.57. The values of the universal amplitude ratios are then slightly different than those reported above: for instance, Y+ and Y− are greater than the Ising values 5.73 and 3.74 [27,32]. Relations similar to Eqs. (22) and (24) hold between effective parachors Peff and critical amplitudes B0 eff , A0 eff , Γ 0 eff , that present more practical interest, as discussed in next section. It is worth mentioning here the recent work by Leneindre and Garrabos [24], who used TSFU to propose a simple description of the amplitudes B0 and σ 0 in terms of a few parameters characterizing the thermodynamic surface near the critical point. They define universal constants zB = 0.468 ± 0.004 and zσ = 1.147 ± 0.085 that relate B0 and σ 0 to those parameters. By combining the expressions B0 and σ 0 (Eqs. (16) and (21) in Ref. [24]) and using Eq. (6) above, a very simple expression is obtained for the Ising parachor: β/µ 1/3 2/3 z (kT ) P c σ c M PIsing = Zc1/2 (25) ρc 2zB that may be rewritten in terms of Pc , Tc and Zc :
.
(21b)
We will stick here to the values Y + = 5.73 ± 0.16 and Y − = 3.74 ± 0.09 quoted by Moldover and Rainwater [32]. Then, from Eqs. (21) and (6) we obtain the following expressions for the Ising parachors as a function of the ampli− tudes B0 , A+ 0 or A0 , Tc , Pc and critical volume Vc or critical compressibility factor Zc = Pc Vc /NAv kTc (NAv is Avogadro’s
PIsing = (1.105 ± 0.044)NAv Zc3/2
kTc 1+β/3µ 1−2β/3µ
Pc
.
(26)
This expression, as well as Eqs. (22) and (24), obey a corresponding-states principle, that takes the general form 1+1/3p 1−2/3p (for an arbitrary p = µ/β): P ∼ Tc /Pc . In Eq. (26), the deviation from the corresponding-states principle is ac3/2 counted for by the term Zc , that varies from one compound
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
to the other. (Another way to account for this deviation consists in using Pitzer’s acentric factor ω [33], which in the case of normal fluids is linearly related to Zc [53].) Using Eqs. (16), (22), (24) and (26), and the critical pa− rameters (Pc , Tc and Zc ) and amplitudes (B0 , ξ0+ , A+ 0 , A0 and + Γ0 ) taken from various sources, we have calculated the Ising parachors PIsing of the compounds of Section 2 (Table 2). The scatter in the calculated values of PIsing reflects (slight) differences in the values of the critical amplitudes found by various authors. The values (or ranges of values) of (PIsing /M)3.88 are reported in Figs. 1–3.
5. Discussion and prospects As is apparent in Figs. 1–3, the experimental values of σ/ρ3.88 are consistently below the values (PIsing /M)3.88 calculated from other critical amplitudes and TSFU in the previous section, except maybe very near the critical point. In fact, when t, σ and ρ are below approximately 10−3 , 10−3 dyn/cm and 10−1 g/cm3 (this range is very difficult to reach experimentally, and truly concerns here only CO2 , SF6 and CH4 , see Figs. 1–3), then σ/ρ3.88 coincides with (PIsing /M)3.88 , notwithstanding the important scatter in both the experimental and calculated values. Beyond those limits, and up to t ∼ 0.1, σ ∼ 1–5 dyn/cm and ρ ∼ 1 g/cm3 , an effective exponent in the range of 3.6 describes the σ versus ρ behavior. Those Ising parachors, either inferred from TSFU or determined experimentally from surface tension and density data extremely close to the critical point, are significantly different from those inferred using the Ising exponent p = 3.88 and off-critical surface tensions and density differences (in the range of the dyn/cm and g/cm3 , respectively). The quantities P3.88 = Mσ 1/3.88 /ρ corresponding to the high-ρ, usually flat portions of the curves in Figs. 1–3 are much lower than the asymptotic Ising values at low ρ. In the last line of Table 2, the values of P3.88 are listed for each one of the 12 compounds examined in this paper: they are lower than the median values of the Ising parachors by more than 10% (which corresponds to differences in surface tensions larger than 50%) for all compounds but propane and SF6 . The above quantities P3.88 (determined from off-critical experimental data) are in fact very close to the conventional Mac Leod and Sugden’s parachors P4 = Mσ 1/4 /ρ inferred from σ and ρ data far from the critical point [2,3]. Most of the σ versus ρ data shown by Schechter and Guo [5] in favor of the use of the Ising exponent concern either liquid/vapor systems quite remote from the critical point (σ > 1 dyn/cm and ρ > 0.3 g/cm3 for all but very few points in Figs. 1 and 5 of Ref. [5]), or liquid/liquid systems close to a consolute point. Those latter systems are known to conform to Ising scaling in a wider range of t than liquid/vapor systems [26]. The near-critical data (σ < 1 dyn/cm) of liquid/vapor systems shown by Schechter and Guo [5], such as those displayed in their Fig. 7, clearly obey a scaling law σ ∼ ρp with an
93
exponent significantly lower than p = 3.88. It is interesting to note that such behavior is very different from the meanfield behavior (expected far enough from the critical point) characterized by an exponent pmf = 3 [7]. In the regime of low-to-intermediate surface tensions (10−3 dyn/cm < σ < 1 dyn/cm, corresponding to, approximately, 10−3 < t < 10−1 , effective parachor exponents peff in the range of 3.56–3.75 are usually observed, consistent with the commonly accepted value p = 11/3 and with the observed scaling behaviors of the density difference ρ and squared capillary length a2 (see Section 3). In this region, a single measurement of surface tension and density difference between the liquid and vapor phases gives not only access to the whole σ versus ρ behavior in a large domain, but also to the whole effective thermodynamic behavior by the use of TSFU. For instance, adopting for convenience (because of the availability of the effective critical amplitudes) the NIST critical exponents, corresponding to peff = 3.57, equations analog to Eqs. (22) and (24) are obtained: NAv Zc (A− 0 eff ) = 1.52 2B0 eff
2/3peff
P3.57
(kTc )1+1/3peff Pc1+2/3peff , (27)
P3.57 = 0.0034
NAv Zc + 2/3peff 1−4/3peff 2(Γ0 eff ) B0 eff
× (kTc )1+1/3peff Pc1+2/3peff
(28)
The prefactors 1.52 [equal to (Yp−eff ) eff , 0eff = 3.57, see Eq. (22)] and 0.0034 have been obtained by adjustment, using available values [31,32,54,55] of the effective critical am+ plitudes B0 eff , A− 0 eff and Γ0 eff and values of P3.57 inferred from the experimental values of σ and ρ (in the low-tointermediate surface tension regime) depicted in Figs. 1–3. Here, we suggest, following Moldover and Rainwater [32], to use this kind of relations to determine the critical-point (effective) singularities of the thermodynamic potential of fluids and their thermodynamic behavior in an extended region around the critical point. For instance, the critical am+ plitudes A− 0 eff and Γ0 eff of a given fluid system are easily obtained using Eqs. (27) and (28) from the knowledge of its critical parameters Tc , Pc and Zc , its parachor, and its amplitude B0 eff (itself determined from ρ measurements or, in the case of pure compounds, inferred from, e.g., the acentric factor ω [26,53]). Parachors, unlike other critical amplitudes, are readily available in the literature for a large variety of compounds and, in conclusion, we suggest to use them for estimating not only surface tension, but also the whole nearcritical behavior of liquid/vapor systems. 1/p
List of symbols a capillary length f free energy density g acceleration due to gravity k Boltzmann constant
94
M NAv p peff P Pc t T Tc V Vc xi yi Zc
D. Broseta et al. / Fluid Phase Equilibria 233 (2005) 86–95
molar mass Avogadro number parachor exponent effective parachor exponent parachor critical pressure reduced temperature [=(Tc − T)/Tc ] temperature critical temperature molar volume critical molar volume molar fraction of component i in the liquid phase molar fraction of component i in the vapor phase critical compressibility factor
Greek letters α critical exponent of heat capacity β critical exponent of the coexistence curve γ critical exponent of the compressibility correction-to-scaling exponent ρ density difference between liquid and vapor phases φ critical exponent of the squared capillary length Γ0 amplitude of the isothermal compressibility ν critical exponent of the correlation length of density fluctuations µ critical exponent of the surface tension ρ density ρc critical density σ surface tension σ0 critical amplitude of the surface tension ξ correlation length of density fluctuations ξ0 amplitude of the correlation length of density fluctuations ω acentric factor Subscripts c critical l liquid v vapor eff effective mf mean field Superscripts + one-phase region (t < 0) − two-phase region (t > 0)
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