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Langmuir 1996, 12, 3326-3333
A Regular Solution Theory Treatment of the Surface Tension of the Noncritical Liquid/Vapor Interface in Mixtures of a Dimethylsiloxane or an Alkane + a Perfluoroalkane near a Critical End Point James Bowers and Ian A. McLure* Department of Chemistry, The University, Sheffield S3 7HF, U.K. Received July 24, 1995. In Final Form: March 18, 1996X The solubility parameter and the strictly regular or quasi-crystalline versions of the regular solution theory have been applied to the calculation of the surface tension of the noncritical interface of eight mixtures of the class dimethylsiloxane or an alkane + a perfluoroalkane near their upper critical end points using the experimental upper critical solution temperature TUCS as the only mixture-property input. The solubility parameter version of the regular solution (SPRS) theory predicts the horizontal inflection in the critical isotherm which is both observed experimentally and was first predicted by Widom from a much more detailed treatment than that used here. In contrast to the results from the SPRS theory, the strictly regular solution theory predicts this inflection only at a temperature well below TUCS. For the limited number of cases demonstrated here the agreement with experiment is good for oligomeric mixtures but poor for non-oligomeric mixtures, surprisingly so given their lesser complexity.
I. Introduction In a binary liquid mixture at its critical end point (CEP) the surface tension isotherm for the noncritical interface, i.e., that between the liquid and the vapor which is uninvolved in the phase separation process, often exhibits a horizontal inflection at the critical composition. This inflection was predicted first by Widom1 as one of a number of predictions regarding the tension of the noncritical interface. Some of these predictions, although not this one, were later revised by Ramos-Go´mez and Widom2 using a less restrictive version of the original theory. Telo da Gama and Evans3 subsequently developed a microscopic theory with Lennard-Jones potentials which predicted not only the same horizontal inflection but also the criticalpoint wetting in the two-liquid phase region which by then had been observed experimentally4 but was unaccounted for in either of the Widom theories. One of our present objectives is to predict the occurrence of this horizontal inflection in near-critical mixtures from a simpler theoretical platform than those just discussed afforded by the regular solution theory. A further objective is to predict the occurrence of aneotropy or surface azeotropy from a molecular theory. At an aneotrope the relative adsorption goes to zero, i.e., the surface and bulk compositions are identical. The resemblances between aneotropy and azeotropy are strong and are a particular case of the wider similarities between vapor pressure-composition and surface tension-composition isotherms. For example, the horizontal inflection in the surface tension has its counterpart in the vapor pressure provided that the vapor pressures of the components are sufficiently different. The conditions for both aneotropy and azeotropy have been discussed elsewhere.5,6 The essence of the occurrence of each is that for a given deviation from ideality, no matter how measured, an * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, June 1, 1996. (1) Widom, B. J. Chem. Phys. 1977, 67, 872. (2) Ramos-Go´mez, F. F.; Widom, B. Physica 1980, 104A, 595. (3) Telo da Gama, M. M.; Evans, R. Mol. Phys. 1983, 48, 229. (4) Heady, R. B.; Cahn, J. W. J. Chem. Phys. 1973, 58, 896. (5) Bowers, J.; McLure, I. A. Submitted for publication in J. Chem. Soc., Faraday Trans. (6) Prigogine, I.; Defay, R.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans: London, 1966.
S0743-7463(95)00610-X CCC: $12.00
extremum is observed only if the vapor pressures or the surface tensions are sufficiently alike. In the remainder of this paper we concentrate solely on mixtures exhibiting positive deviations from ideality which in extreme cases, as discussed below, exhibit positive azeotropy or negative aneotropy. Far from a critical end point in such mixtures the deviations from ideality are small and positive azeotropy therefore generally occurs only if the vapor pressures of the components are sufficiently similar and, similarly, aneotropy occurs only if the surface tensions are sufficiently similar. Near a critical end point matters are less straightforward. Taking the vapor pressure first, there is a necessary horizontal inflection in the isotherm, but if accompanied by azeotropy at a mole fraction xaz close to the critical composition xc, it may, practically, be extremely difficult to observe. For example, the inflection in the mixture propane + octafluoropropane is all but undetectable.7 If the xc and xaz are significantly different, then both the azeotrope and the inflection are apparent, for example, butane + octafluoropropane.7 Turning to the surface tensions, similar observations can be made. Far from a critical end point, aneotropy is not uncommon in positively deviating mixtures of nearly equally tense components. Near a critical end point both theory and experiment suggest that an inflection is strongly likely. If, however, the surface tensions are similar, aneotropy can also occur and is observable if the composition xan is sufficiently far from xc. The mixtures we have chosen to examine in this article show a variety of surface tension behavior, drawing upon examples with and without aneotropy, near the critical end point. Besides seeking a description of the inflection, a further test of the regular solution theories is the ability to predict the occurrence of aneotropes. We have recently measured the orthobaric surface tensions5,8,9 of mixtures from the class a dimethylsiloxane or an alkane + a perfluoroalkane near the upper CEP. These are nonpolar mixtures which characteristically exhibit phase separation over a wide range of upper critical (7) Gilmour, J. B.; Zwicker, J. O.; Katz, J.; Scott, R. L. J. Phys. Chem. 1967, 71, 3259. (8) Whitfield, R.; Bowers, J.; McLure, I. A.; Rodriguez, E. Unpublished results. (9) Ferna´ndez, J.; Williamson, A.-M.; McLure, I. A. J. Chem. Thermodyn. 1994, 26, 897.
© 1996 American Chemical Society
Surface Tensions near a CEP from Regular Solution Theory
solution temperatures, TUCS. Strikingly, TUCS for the mixtures hexamethyldisiloxane + perfluorohexane and octamethyltrisiloxane + perfluorohexane are virtually identical to TUCS for hexane + perfluorohexane and for heptane + perfluorohexane, respectively, and the similarity carries over into other aspects of the bulk thermodynamics.10 This similarity is maintained in the shape of the experimental surface tension isotherms.5,8 From the naı¨ve viewpoint of interpreting the surface tension of binary mixtures solely in terms of cohesive forces, the more volatile componentsdue to its lower cohesionswould normally be expected to be the less tense, i.e., to have the lower surface tension σ*. This is certainly true in unihomologous mixtures like mixtures of nalkanes, but not necessarily so in the bihomologous mixtures studied here. In terms of our nomenclature, at a given temperature T the perfluoroalkane, component 2 throughout, is always the less tense, i.e., σ2* < σ1*, and is often the more volatile, i.e., has the higher vapor pressure p*, so that p2* > p1*smixtures displaying this kind of relationship between the vapor pressures and surface tensions of the components have become known as usual mixtures. However, mixtures are not always usual in this sense and a second, and rarersthus known as unusualsclass exists in which the less volatile component is the less tense, i.e., p2* < p1* and σ2* < σ1*. Both classes of mixture are examined heresalthough data are compared for one unusual mixture only. According to Ramos-Go´mez and Widom the direction of increasing surface tension near the critical composition should be independent of the relative volatilities of the components of the mixture. Thus, again in terms of the mixtures studied here, if σ2* p1* or, less commonly for such mixtures, p2* < p1*. We show that for the mixtures studied here that this is now supported by experiment. There are many theories for predicting the surface tension of mixtures, some are model calculations using a specific intermolecular potentialssuch as the LennardJones potentialsbut few seek to predict the surface tensions of real systems. The easiest approaches are phenomenological using only information from the mixture components or at most a single simple mixture-derived parameter. For example, the corresponding states treatment of Prigogine11 has been applied with excellent qualitative success by Dickinson and McLure to the surface tensions of unihomologous n-alkane mixtures12 and by Dickinson to bihomologous n-alkane + linear dimethylsiloxane mixtures.13 However, the application of the primitive form of this theory near a CEP does not yield a horizontal inflection, unsurprisingly since the Prigogine theory relies on the exchange energy of the mixture being small, which is incompatible with proximity to a critical end point. In the strictly regular or quasi-crystalline solution model of Guggenheim14 there is no limitation on the magnitude of the exchange energy and so their theory can safely be applied to the calculation of bulk thermodynamic properties near a CEP. A detailed investigation of a rather different form of the strictly regular theory, retaining only the quadratic dependence on mole fraction composition of the excess Gibbs function, has been (10) McLure, I. A.; Mohktari, A.; Edmonds, B. J. Chem. Soc., Faraday Trans. In preparation. (11) Prigogine, I. The Molecular Theory of Solutions; North Holland: Amsterdam, 1957. (12) McLure, I. A.; Dickinson, E. J. Phys. Chem. 1974, 80, 1880. (13) Dickinson, E. J. Colloid Interface Sci. 1975, 53, 467. (14) Guggenheim, E. A. Trans. Faraday Soc. 1945, 41, 150.
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described by Beaglehole.15 The principal new feature of this treatment is the association of the regular solution basis for the mixture with Cahn’s phenomenological theory16 and using the Cahn-Hilliard model17 rather than a restricted layer model to calculate the surface tension. Beaglehole’s chief concerns were with matters related to the concentration profile through the liquid/vapor interface with a view to accounting for his ellipsometric results on the mixture aniline + cyclohexane near its UCEP; however, he also extended the treatment to the Antonow rule and the surface tension along paths of fixed composition close to the critical composition. We have explored for binary mixtures both the strictly regular solution version and the solubility parameter version of the regular solution theory in the form developed by Sprow and Prausnitz18 to evaluate the surface tension in the vicinity of an upper CEP. Both approaches incorporate a model consisting of a distinct surface layer capping a bulk subphase of different properties. A characteristic weakness of such models is that the inclusion of a single surface layer leads to inconsistency with the Gibbs adsorption equation. However, Stecki and co-workers19,20 have shown from calculations in the strictly regular version using multilayer models that the surface tension calculated from single-layer models is deficient only in second and higher order terms; for simplicity therefore we accept here the single-layer model. The extension of the solubility parameter version of the regular solution theory discussed below reproduces the horizontal inflection in the surface tension isotherm at the CEP. II. Solubility Parameter Version of Surface Regular Solution Theory The simple model developed by Sprow and Prausnitz consists of a surface layer of composition xiS but indeterminate thickness superimposed on a bulk phase of composition xiB. This leads to the following expression for the surface tension σ of a binary mixture
σ ) (A1/R1)σ1 + (RT/R1) ln(x1Sγ1S/x1Bγ1B) ) (A2/R2)σ2 + (RT/R2) ln(x2Sγ2S/x2Bγ2B)
(1)
where the Ai is the molar area, Ri the partial molar area, σi the surface tension, xiB the bulk mole fraction, xiS the surface mole fraction, γiB the bulk activity coefficient, and γiS the surface activity coefficient of component i. It is clear that critical effects are generated from the second term of eq 1. The γiB are calculated from the normal equations of the solubility parameter version of the regular solution theory for bulk mixtures
RT ln γiB ) Viφj2(δi - δj)2
(2)
where Vi, is the molar volume and δi is the solubility parameter of component i; φj in a binary mixture is the volume fraction of the other component given by
φj ) xjBVj /
∑ j xjBVj
(3)
The solubility parameter δi is essentially defined as -Ui/Vi, where Ui is the molar configurational energy of the liquid i. The evaluation of δ is discussed below. The γiS are calculated by applying the same arguments to the surface using (15) Beaglehole, D. J. Chem. Phys. 1981, 75, 1544. (16) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667. (17) Cahn, J. W.; Hilliard, J. E. J. Chem. Phys. 1958, 28, 258. (18) Sprow, F. B.; Prausnitz, J. M. Trans. Faraday Soc. 1966, 62, 1105. (19) Bellemans, A.; Stecki, J. Mol. Phys. 1960, 3, 203. (20) Altenberger, A. R.; Stecki, J. Chem. Phys. Lett. 1970, 5, 29.
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RT ln γiS ) Aiθj2(δiS - δjS)2
Bowers and McLure
(4)
where θj is the surface fraction of the other component j given by
∑ j xjSAj
θj ) xjSAj /
(5)
and δiS, the surface solubility parameter, is given by
(δiS)2 ) δi2(Vi/Ai) - σi + T(dσi/dT)
(6)
Deviations from the geometric-mean assumptionsused to combine the pure component solubility parameters, δi or δiS, into the interaction parameter δijsare incorporated by replacing the appropriate (δi - δj)2 term with (δi - δj)2 + 2l12δiδj where l12 is a correction parameter related but not equal to the correction term ξ often used to modify the Berthelot rule for the calculation of the unlike interaction well-depth 12 ) ξ(1122)1/2, where ii is the like interaction well-depth. We have chosen for convenience and internal consistency in our application of the theory to evaluate l12 from the upper critical solution temperature TUCS using the regular solution equations
x1c/x2c ) ((V12 + V22 - V1V2)1/2 - V2)/(V1 - (V12 + V22 - V1V2)1/2) (7)
σ ) σ0(1 - T/Tc)1.26
in which xic is the critical composition, and
RTUCS ) 2A12(x1cx2cV12V22/(x1cV1 + x2cV2)3)
ature variations from the tabulated T taking the difference in the heat capacities of the liquid and coexisting vapor ∆Cσ as approximated by that for cyclohexane.23 The dimethylsiloxane oligomers, hexamethyldisiloxane apart, are too involatile to readily yield experimental determinations of ∆Hvap(T ) 298 K) either by calorimetry or vapor pressures using the Clapeyron equation. Furthermore, the large magnitude of ∆Cσ for such large molecules renders unsafe the assumption of a temperature independent ∆Hvap over the large temperature range between say, the normal boiling temperature and 298 K. Accordingly, for the higher dimethylsiloxanes, the solubility parameter was evaluated from the thermal pressure coefficient γV ) (∂p/∂T)V via δ2 ≈ TγV using thermal pressure coefficients for the dimethylsiloxanes previously measured in this laboratory.24 The densities for methane, hexane, heptane, and 2-methylpentane were taken from Francis,25 for perfluorohexane from Dunlap et al.,26 for methylcyclohexane and perfluoromethylcyclohexane from Heady and Cahn,4 for perfluorotributylamine from Edmonds and McLure,27 for perfluoromethane from Knobler and Pings,28 and for the linear dimethylsiloxanes from McLure et al.20 The pure component surface tensions were taken either from Jasper’s compilation29 or from the mixture measurements cited here. The surface tensions were nonlinear least squares fitted to expressions of the form
(8)
where A12 ) (δi - δj)2 + 2l12δiδj. We have assumedsfaute de mieuxsthat the value of l12 at the surface is the same as in the bulk. Since (δS)2 is the superficial cohesive energy density whereas δ2 is the conventional cohesive energy density, this approximation is inexact, but for reasons outlined in the Appendix it serves as justifiable within the scope of this investigation. The surface tension and the surface mole fractions are obtained by numerical simultaneous solution of eqs 1 and 4 using Brent’s method of root finding.21 Although regular solution theory without l12 predicts TUCS badly for alkane + perfluoroalkane mixturessand in fact this failure was originally the signal for a wider exploration of such mixtures22snonetheless it is easy to obtain l12 experimentally from TUCS. For all the mixtures considered here the calculated critical composition, xc, is in excellent agreement with the experimental values. This is fortunate since in this theory the calculated xc cannot be reconciled by artifice with discordant experimental values. As we shall see, this vitiates the extension of the theory to mixtures containing a polar molecule for which xc is particularly poorly predicted. For the calculation of the surface tension the following information for each component is required: the density, the solubility parameter, and the surface tension. The bulk solubility parameter δ can be evaluated in a number of ways but for common substances principally from the molar enthalpy of vaporization via δ2 ≈ (∆Hvap - RT)/V, where ∆Hvap is the molar enthalpy of vaporization. The enthalpies of vaporization for the alkanes and perfluoroalkanes were taken from the CRC Handbook of Solubility Parameters and other Cohesion Parameters, where necessary applying small corrections for temper(21) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, 1986. (22) Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand Reinhold Company: New York, 1970.
(9)
where σ0 is a critical amplitude; this expression readily yields the necessary temperature dependence. The gasliquid critical temperatures Tc were taken from Ambrose’s compilation.30 III. Strictly Regular Version of the Regular Solution Theory The basis of this version of the theory is a quasicrystalline model in which first the grand partition function of the outermost layer, taken as the interfacial monolayer, and secondly from it the chemical potentials of the components are calculated. The latter are equated to those of the components in the bulk obtained from the customary quasi-crystalline theory of mixtures. Once the chemical potentials in the surface layer are known, the surface tension σ and the surface mole fractions xiS emerge directly. The calculation assumes inter alia that the crystal structure of the surface layer is that of the interior of the crystal.6 The surface tension of the mixture is given by
σ ) σ1 + (RT/A) ln(x1S/x1B) + (RL/A)((1 - x1S)2 (1 - x1B)2) - (RM/A)(1 - x1B) ) σ2 + (RT/A) ln(x2S/x2B) + (RL/A)((1 - x2S)2 (1 - x2B)2) - (RM/A)(1 - x2B)2 (10) where A is an effective molar area and M and L are lattice (23) Barton, A. F. M. CRC Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1983. (24) McLure, I. A.; Pretty, A. J.; Sadler, P. A. J. Chem. Eng. Data 1977, 22, 372. (25) Francis, A. W. Chem. Eng. Sci. 1959, 10, 37. (26) Dunlap, R. D.; Murphy, C. J., Jr.; Bedford, R. G. J. Am. Chem. Soc. 1958, 80, 83. (27) Edmonds, B.; McLure, I. A. J. Chem. Eng. Data 1977, 22, 127. (28) Knobler, C. M.; Pings, C. J. J. Chem. Eng. Data 1965, 10, 129. (29) Jasper, J. J. J. Phys. Chem. Ref. Data 1972, 1, 841. (30) Ambrose, D. Vapour-Liquid Critical Properties, Report Chem 107; National Physical Laboratory: London, 1980. Also supplement of same name and by same author, University College London, 1983.
Surface Tensions near a CEP from Regular Solution Theory
Figure 1. Surface tension isotherms at, from top to bottom, 25, 30, 40, and 50 °C for (1 - x)hexamethyldisiloxane + xperfluorohexane calculated from the solubility parameter version of the regular solution theory. TUCS ) 23.8 °C.
parameters, taken heresfaute de mieuxsto be M ) 1/6 and L ) 4/6, appropriate to a cubic lattice. The quantity R, the molar exchange energy, is defined at the CEP by
R ) NAω ) 2RTUCS
(11)
where ω is the exchange energy ω ) 12 - 0.5(11 + 22). All nonideality is introduced through the exchange energy. Equation 10 is solved numerically by Brent’s root finding method.21 One slight difference between the two models used here that merits emphasis is the degree of restrictivity of the theories to the nature of the surface layer. In the strictly regular theory the surface layer is strictly defined as a monolayer, although as noted earlier Stecki and others have explored the relaxation of this surface description to accommodate multilayer surfaces. By contrast, in their development of the solubility parameter version of the theory Sprow and Prausnitz invoked no limitation on the thickness of the surface layer. In this as in other aspects of the theory, for example, the restriction of criticality to equimolar mixtures, the solubility parameter version of the theory is the more flexible and thus more likely to describe real mixtures better. IV. Results and Discussion 1. The Solubility Parameter Version. The agreement between the calculated and experimental dependence of the surface tension on composition and temperature is generally good. Specifically, the theory yields both a critical composition in good agreement with experiment and, like the Widom treatment, a horizontal inflection in the critical surface tension isotherm at that point. As the temperature moves from TUCS further into the single-liquid-phase region, the slope at xc increases until finally all traces of the critical inflection are lost. This is demonstrated in Figure 1 for hexamethyldisiloxane + perfluorohexane and is in qualitative agreement with experiment5 and with the theory of Telo da Gama and Evans.3
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Figure 2. Surface tension isotherm at 47.15 °C for (1 - x)methylcyclohexane + xperfluoromethylcyclohexane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 46.11 °C.
The qualitative shapes of the isotherms calculated are in generally good agreement with experiment for mixtures of oligomers but rather surprisingly the agreement is poor for less complex mixtures. The details discussed below are summarized in Table 1. 1.1. Usual Mixtures, i.e., in which the less tense component is the more volatile. Methylcyclohexane + Perfluoromethylcyclohexane. The surface tension of the alkane σA exceeds that of the perfluoroalkane σF, i.e., σA > σF. TUCS ) 46.11 °C which gives l12 ) 0.060. The agreement of the theoretical and experimental xc (xc ) 0.36)4 and surface tension isotherms9 is excellent. Comparable isotherms at 47.15 °C are shown in Figure 2. Aneotropy is found in both the experimental and calculated data. It is worth noting here that this mixture can be regarded as consisting of pseudospherical molecules. Hexane + Perfluorohexane. σA > σF. TUCS ) 22.65 °C which gives l12 ) 0.062. The calculated xc is 0.35, which compares well with the experimental value31 of xc ) 0.36. Again the qualitative agreement between the calculated surface tension and the experimental data8 is good, both displaying an aneotrope around x ) 0.9, as shown in Figure 3 for isotherms at 25 °C. Heptane + Perfluorohexane. σA > σF. TUCS ) 43 °C which gives l12 ) 0.051. The calculated xc is 0.36, the same as the experimental value.32 In this mixture no aneotrope is found either by experiment8 or from calculation. The calculated isotherm shape is in qualitative agreement with experiment. Figure 4 shows comparable isotherms for 45 °C. Hexamethyldisiloxane + Perfluorohexane. The dimethylsiloxane surface tension σDMSI exceeds that of the perfluoroalkane, σDMSI > σF. TUCS ) 23.8 °C which gives l12 ) 0.055. The calculated xc is 0.52, which agrees well (31) Bedford, R. D.; Dunlap, R. G. J. Am. Chem. Soc. 1958, 80, 282. (32) Whitfield, R.; McLure, I. A. Unpublished results.
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Bowers and McLure
Table 1. Summary of TUCS and Associated l12 Values, Calculated (xc(SP)) and Experimental Critical Compositions (xc(expt)), and Comment on the Agreement between Experimental and Theoretical Surface Tension Isotherms near TUCS mixture
TUCS °C
l12
xc(SP)
xc(expt)
σ-x agreement
(1 - x)methylcyclohexane + xperfluoromethylcyclohexane (1 - x)hexane + xperfluorohexane (1 - x)heptane + xperfluorohexane (1 - x)hexamethyldisiloxane + xperfluorohexane (1 - x)octamethyltrisiloxane + xperfluorohexane (1 - x)decamethyltetrasiloxane + xperfluorohexane (1 - x)2-methylpentane + xperfluorotributylamine (1 - x)methane + xperfluoromethane
46.11 22.65 43.0 23.8 42.2 59.8 45.04 -178.35
0.060 0.062 0.051 0.055 0.047 0.042 0.050 0.066
0.36 0.35 0.36 0.52 0.60 0.70 0.20 0.40
0.36 0.36 0.36 0.53 0.63 0.70 0.22 0.43
excellent good good good good excellent poor poor
Figure 3. Surface tension isotherm at 25 °C for (1 - x)hexane + xperfluorohexane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 22.65 °C.
Figure 4. Surface tension isotherm at 45 °C for (1 - x)heptane + xperfluorohexane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 43 °C.
with the experimental value of xc ) 0.53.33 The qualitative agreement of the calculated surface tension isotherm is again in reasonable agreement with experiment.5 Figure 5 shows comparable isotherms for 25 °C. However, the experimentally observed aneotropy is not predicted by the theory. Toward the dimethylsiloxane extreme of the composition range the calculated surface tension has a tendency to approach the dimethylsiloxane surface tension before reaching the pure dimethylsiloxane composition, this is common to all the dimethylsiloxane-containing mixtures we have examined here. This effect is hinted by the experimental datasalthough there are insufficient data in this composition region to tell whether this effect is real. More than likely this foreign inflection is a result of the numerical method used. Nevertheless, such detail in surface tension isotherm shapes has been observed before so there is the possibility of using this theory to add justification to these observations. Octamethyltrisiloxane + Perfluorohexane. σDMSI > σF. TUCS ) 42.2 °C, which gives l12 ) 0.047. The calculated xc ) 0.60 compares well with the experimental value xc ) 0.63.33 Figure 6 shows the qualitative agreement of the theoretical and experimental5 surface tensions for isotherms of 45 °C. Even more obvious for this mixture is the flattening of the isotherm slope as x approaches zero resulting, again, with a foreign inflection. Unfor-
tunately, there are few experimental data in this region, but the polynomial fitted to the experimental results has a tendency to assume the qualitative shape calculated from the theory. However, in this case and for the following mixture, we suspect this is again due to the computational method. Decamethyltetrasiloxane + Perfluorohexane. σDMSI > σF. TUCS ) 59.8 °C, which gives l12 ) 0.042. The calculated xc is 0.70, which is the same as the experimental value.33 There is excellent agreement between the theoretical and experimental5 isotherms both qualitatively and quantitatively as demonstrated in Figure 7 for isotherms of 60 °C. The flattening of the isotherm as x approaches zero is found again, this time the experimental data could possibly do the same; however, there are insufficient results to determine whether this phenomenon occurs in reality. Methane + Perfluoromethane. σF > σA. TUCS ) -178.85 °C, which gives l12 ) 0.066. (xc ) 0.43 from experiment34 and xc ) 0.40 from the theory). The agreement between theoretical and experimental35 surface tension isotherms is poor as shown in Figure 8 for isotherms of -175 °C, despite xc being correctly predicted. Aneotropy is found by experiment, and it is worth noting again how the aneotrope obscures the presence of the
(33) McLure, I. A.; Mohktari, A.; Bowers, J. Submitted for publication in J. Chem. Soc., Faraday Trans.
(34) Simon, M.; Fannin, A. A., Jr.; Knobler, C. M. Ber. Bunsen-Ges. Phys. Chem. 1972, 76, 321. (35) Higgins, R. MSc Thesis, University of Sheffield, 1978.
Surface Tensions near a CEP from Regular Solution Theory
Figure 5. Surface tension isotherm at 25 °C for (1 x)hexamethyldisiloxane + xperfluorohexane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 23.8 °C.
Figure 6. Surface tension isotherm at 45 °C for (1 x)octamethyltrisiloxane + xperfluorohexane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 42.2 °C.
critical inflection, but the theory fails to account for it. The constituent molecules are pseudospherical rather than oligomeric, for which we would expect the theory to work well after the success of the theory for methylcyclohexane + perfluoromethylcyclohexane. However, in light of the shapes of the other calculated σ-x isotherms this discrepancy is not totally surprising.
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Figure 7. Surface tension isotherm at 60 °C for (1 x)decamethyltetrasiloxane + xperfluorohexane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 59.8 °C.
Figure 8. Surface tension isotherm at -175 °C for (1 - x)methane + xperfluoromethane. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) -178.15 °C.
1.2. Unusual Mixture, i.e., in which the less tense component is the less volatile. 2-Methylpentane + Perfluorotributylamine. σA > σF. TUCS ) 45.04 °C, which gives l12 ) 0.050. The calculated xc is 0.20, which compares well with the experimental value36 of xc ) 0.22. In the theoretical isotherm the presence of the aneotrope obscures the horizontal inflection. Figure 9 shows isotherms at 46 °C. (36) Munson, M. S. B. J. Phys. Chem. 1964, 68, 796.
3332 Langmuir, Vol. 12, No. 13, 1996
Figure 9. Surface tension isotherm at 46 °C for (1 - x)2methylpentane + xperfluorotributylamine. The line is calculated from the solubility parameter version of the regular solution theory; the data points are from experiment. TUCS ) 45.04 °C.
The qualitative agreement between the calculation and experiment8 is fair. Aneotropy is found by experiment and predicted by the theory, although the xan differ. 2. The Strictly Regular Version. As a demonstration of this version we have chosen the mixture hexane + perfluorohexane. The exchange energy is calculated from the TUCS ) 22.65 °C.31 The isotherms, for temperatures of 50, 22.65, 0, -25, and -50 °C, shown in Figure 10, demonstrate that it is necessary to reduce the temperature by ca. 50 °C below the TUCS to be able to detect a horizontal inflection in the isotherm. This discrepancy is due to the restrictive model form of the excess Gibbs function, which does not yield a surface layer of the required composition to produce the horizontal inflection in the isotherm. The calculated surface compositions for these two versions are tabulated together with the experimental composition determined by specular neutron reflection37 in Table 2. The theoretical values of xS are not directly comparable with experiment since the surface, in reality, can be modeled as a composite of a perfluorohexane monolayer and a perfluorohexane-enriched surface phase. Although regular solution theory assumes that the excess volume VE and excess entropy SE of mixing are zero, Sprow and Prausnitz used an excess volume term in the calculation of the partial molar volumes at the surface. This provides a small correction, ca.
