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Ind. Eng. Chem. Res. 2001, 40, 1748-1754
An Equation of State for Polymers and Normal Fluids Using the Square-Well Potential of Variable Well Width Ma´ rcio L. L. Paredes,† Ronaldo Nobrega,† and Frederico W. Tavares*,‡ Programa de Engenharia Quı´mica/COPPE, Universidade Federal do Rio de Janeiro, C.P. 68502, 21945-970 Rio de Janeiro, Brazil, and Escola de Quı´mica, Universidade Federal do Rio de Janeiro, C.P. 68542, 21949-900 Rio de Janeiro, Brazil
A perturbation theory equation of state for mixtures of freely jointed square-well fluids of variable well width that takes into consideration the influence of the attractive part of the intersegment potential on the chain properties is applied to describe thermodynamic properties of pure normal fluids, polymers, and their mixtures. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference sphere fluid and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of spheres to form chains. Pure polymer parameters are determined from low molecular weight pure substance parameters, from pure polymer volumetric data, and from mixture data. Good results are obtained when this equation of state is used in the calculation of vapor-liquid equilibrium data of highly asymmetric mixtures, with polymer properties determined from low molecular weight substance properties, molecular weight and temperatureindependent parameters, and no binary adjustable parameters. 1. Introduction Thermodynamic properties of chainlike molecules, e.g., polymers and proteins, are of industrial and scientific interest, and many papers1-17 have been published modeling molecules as a chain of connected segments. The equations of state related to this work are based on the Wertheim1 thermodynamic polymerization theory. In Wertheim theory, the thermodynamic properties of sphere chains are obtained from the properties of unbonded segments, and the thermodynamic contribution of the connectivity of the segments to chain properties is accounted for as a perturbation term. This framework has been applied to describe the thermodynamic properties of homopolymers2-8 and copolymers.9-14 The interactions between the chain segments have been described by the Lennard-Jonnes potential7 and by the square-well potential.5,6,9,11-17 The van der Waals approximation2-4,8,10 and perturbation theories5-7,9,11-17 are used in the literature to calculate the thermodynamic properties of the nonbonded segments. Tavares et al.15 have described the properties of pure chain segments that interact by the square-well intersegment potential with variable well width. Paredes et al.16 have extended the work of Tavares et al.15 to mixtures by using the one-fluid mixing rule. Mixture properties obtained with this equation of state are in good agreement with Monte Carlo simulation data.17 The scope of the present work is to apply the equation of state of Paredes et al.16 to describe the properties of polymers, normal fluids, and their mixtures. The equation of state parameters were obtained for a set of pure normal fluids and polymers. Once mean-field theory is
used to calculate thermodynamic properties, the equation of state is not able to reproduce the singular behavior of fluids in the critical region. For this purpose, crossover equations should be used to calculate critical properties of pure18,19 fluids or mixtures.20,21 Following Radosz and co-workers,7,11-14 pure longchain alkane parameters are obtained from low molecular weight alkane parameters. When this procedure is extended, pure addition polymer parameters are obtained from the hydrogenated monomer parameters. Polymer parameters are also obtained from polymer volumetric data3,5,6,8-10 and from mixture data.4 In general, good agreement with experimental data is obtained when the equation of state, with no binary adjustable parameters, was used to represent the properties of mixtures with high asymmetry in length. 2. Equation of State In this section, the analytical equation of state for mixtures of square-well chains proposed by Paredes et al.16 is presented. The interaction potential Φij between two nonadjacent spheres i and j at the distance rij is defined as
{
∞, rij < σij Φij ) -ij, σij e rij < λijσij 0, rij g λijσij where σij is defined in eq 1 and ij and λij are defined in eq 2, following the usual combining rules with no binary adjustable parameters.
σij ) * To whom correspondence should be addressed. Phone: 562-7650. Fax: 562-7567. E-mail:
[email protected]. † Programa de Engenharia Quı´mica/COPPE. Phone: 5902241. Fax: 590-7135. E-mail:
[email protected]. ‡ Escola de Quı´mica. Phone: 562-7650. Fax: 562-7567. E-mail:
[email protected].
ij ) xiijj
σii + σjj 2
and
λij ) xλiiλjj
(1) (2)
In eqs 1 and 2, σii, ii, and λii are the segment diameter, square-well depth, and square-well width of
10.1021/ie0007116 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/02/2001
Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1749
component i, respectively. The TPT1M16,17 equation of state for square-well chain mixtures is
Zchain ) 1 + (mZR,HS)mix + (mZatt)mix nc
∂
xi(mi - 1)ξ3 ∑ ∂ξ i)1
ln gSWS (3) ii 3
where
π nc ξk ) F xjmjσkjj 6 j)1
∑
where k ) 0, 1, 2, 3
(4)
and the hard-sphere mixture compressibility factor, (mZR,HS), was calculated according to Boublı´k22 and Mansoori et al.23 In eq 4, F is the molecule density and xj and mj are the mole fraction and number of segments of component j, respectively. When the one-fluid-type mixing rule is applied to calculate the attractive square-well contribution to the compressibility factor of nonbonded spheres, the following equation is obtained:
{ (
ij π nc nc xixjmimjσij3 12 Iij + (mZatt)mix ) - F 6 i)1j)1 kT
∑∑
∂Iij ξ3
∂ξ3
) ( )[ +6
ij
kT
2
ξ3(1 - ξ3)4 (1 + 2ξ3)2
(
2
∂Iij ∂ξ3
+ ξ3
(
(1 - ξ3)3(1 - 5ξ3 - 20ξ32 - 12ξ35) (1 + 2ξ3)4
Iij + ξ3
) )]}
∂2Iij
+
∂ξ32 ∂Iij
(5)
∂ξ3
where k is the Boltzmann constant and T is the temperature. In eq 6, the expression for the function Iij is obtained by evaluating the Chang and Sandler24,25 integral function I at the packing fraction ξ3, for a square-well width equal to λij.
Iij ) I(ξ3,λij)
(6)
In the TPT1M equation of state, the radial distribution function at the contact point of square-well spheres gSWS is calculated by ii
) ln gHS ln gSWS ii ii +
ii g1,ii kT gHS
(7)
ii
where the perturbation term g1,ii is expressed in eq 8 and the hard-sphere mixture radial distribution func22 and tion, gHS ii , was calculated according to Boublı´k Mansoori et al.23 In eq 8, the Chang and Sandler24,25 radial distribution function of hard spheres at the distance λiiσii+, + gHS ii (λiiσii ), is used.
(
g1,ii ) -3 Iii + ξ3
)
∂Iii + + λiig3HS ii (λiiσii ) ∂ξ3
(8)
3. Pure-Component Parameters 3.1. Normal Fluid Parameters. In this section the TPT1M equation of state parameters are obtained for
Figure 1. TPT1M parameters for linear alkanes as a function of the number of carbons: (a) /k; (b) σ; (c) λ; (d) m.
pure normal substances. The models present four parameters: and λ are related to the intermolecular attractions and σ and m are related to molecular size and length. The segment parameters , λ, and σ are expected to be molecular weight independent, while the number of segments m is expected to vary linearly with molecular weight. These parameters are determined so that the equation of state gives the minimum root-meansquare relative deviations to the smoothed vapor pressures and saturated liquid densities (Reid et al.26) over a temperature range of 0.6 e T/Tc e 0.98. The obtained parameters , λ, σ, and m for pure alkanes are plotted in Figure 1 against the number of carbons. In Figure 1a-c the TPT1M parameters are almost constant, as is expected. In Figure 1d, the TPT1M parameters vary linearly with the number of carbons, as is expected. The parameters for pure normal fluids are presented in Table 1. In this table, the rootmean-square relative deviations in vapor pressure (RMSDP) and in saturated liquid density (RMSDF) are also presented. 3.2. Polymer Parameters. The strategy used to obtain the normal substance parameters could not be applied to polymers, once vapor pressure data for polymers is not available. In this section, two strategies are used to obtain polymer parameters: from volumetric data and from low molecular weight substance parameters. 3.2.1. Polymer Parameters Obtained from Volumetric Data. Pure polymer parameters were obtained using density data (Tait equation) at a large range of temperature and pressure (Danner and High27). These parameters are determined so that the equation of state gives the minimum root-mean-square relative deviations to polymer densities. This procedure is commonly used in the literature to obtain polymer parameters.3,5,6,8-10 The parameters for polyethylene (PE), polystyrene (PS), poly(vinyl acetate) (PVAc), and poly(dimethylsiloxane) (PDMS) are presented in Table 2. In this table, RMSDF values are also presented. Very low deviations in density are obtained with the TPT1M equation of state. The comparison between the parameters presented in Tables 1 and 2 shows the discrepancy between the parameters , λ, and σ for the polymers and for the hydrogenated monomers, e.g., the discrepancy between parameters for PE and ethane, PVAc and ethyl acetate, or PS and ethylbenzene. The behavior of the curves in Figure 1 suggests that the PE parameters should be close to parameters obtained for the alkane series. The
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Table 1. TPT1M Parameters for Pure Substances Obtained from Fitting the Calculated Vapor Pressure and Saturated Liquid Density to the Experimental Data
a
substance
/k (K)
σ (Å)
λ
m
RMSDPa
RMSDFb
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane ethyl acetate methyl ethyl ether methyl isobutyrate ethylbenzene benzene toluene p-xylene cyclohexane acetone
70.9044 88.0442 94.5548 100.7422 100.7895 104.1677 109.3651 107.8091 107.1975 108.4273 109.0903 111.2230 94.2033 86.9172 100.9109 127.7645 121.5111 121.4538 121.9878 124.0394 91.2194
2.8486 2.8360 2.9410 3.0238 3.0609 3.1125 3.1533 3.1606 3.1672 3.1903 3.2144 3.2379 2.6841 2.7096 2.8420 3.1183 2.9588 3.0336 3.0841 3.1530 2.5994
1.7934 1.8441 1.8677 1.8792 1.9038 1.9083 1.8989 1.9146 1.9247 1.9271 1.9326 1.9293 1.9505 1.9383 1.9307 1.9052 1.9171 1.9222 1.9220 1.8961 1.9919
1.7811 2.5404 3.0361 3.4729 3.9863 4.4135 4.8453 5.3706 5.8952 6.3250 6.7270 7.1355 5.1210 4.0706 5.1130 4.4095 3.6641 4.1131 4.5797 3.6664 4.3301
0.93 1.11 1.17 1.37 1.53 1.47 1.42 1.63 1.90 1.73 1.61 1.60 1.44 0.89 1.31 1.44 1.31 1.61 1.86 1.16 2.24
1.98 2.01 1.98 2.09 2.25 2.19 2.06 2.20 2.27 2.28 2.33 2.26 2.35 2.26 2.29 2.30 2.31 2.33 2.47 2.17 2.84
RMSDP: root-mean-square deviations in pressure )
100 b
x∑( ) nexp
Pi - Pexp i
i)1
Pexp i
RMSDF: root-mean-square deviations in saturated liquid density )
100
Table 2. TPT1M Parameters for Pure Polymers Obtained from Fitting the Calculated Densities to the Experimental Data polymer
Mn
/k (K)
σ (Å)
PE PS PVAc PDMS PDMS PDMS
52 000 90 700 84 000 594 7 856 47 200
417.4495 731.8082 365.2562 348.6600 363.2519 374.8885
4.3779 6.1643 3.7675 4.8567 4.8887 5.0571
a
m
λ
1.3938 1104.0657 1.2605 665.7995 1.4286 1930.9956 1.3593 8.3239 1.3605 103.6497 1.3555 569.6996
RMSDFa 0.0055 0.031 0.012 0.037 0.022 0.027
RMSDF: root-mean-square deviations in density )
100
x∑( ) nexp
Fi - Fexp i
i)1
Fexp i
2
2
/nexp
x∑( ) nexp
Pi - Pexp i
i)1
Pexp i
2
/nexp
is to use coherent sets of pure-component parameters in the mixing rule, because chemically similar segments (e.g., segments in alkane chains) should be represented by similar parameters. As shown in Figure 1, the , λ, and σ parameters for alkanes are almost constant and the parameter m varies linearly with the number of carbons. For this reason, the long-chain alkane parameters , λ, and σ are set equal to those for dodecane. The parameter m is obtained by extrapolation of the curve in Figure 1d. Excluding the methane parameter, the m parameter and the ration moles per meter for alkanes can be obtained from the relation
/nexp
polymer parameters presented in Table 2 are used in this work only to represent the properties of PE in Figure 4a. 3.2.2. Polymer Parameters Obtained from Low Molecular Weight Substance Parameters. In recent works based on the TPT1 equation,7,11-14 equation of state parameters for long alkanes have been obtained by extrapolations of parameter curves obtained for low molecular weight alkanes. Following this idea, it is proposed to obtain polymer parameters from the parameters of the substances in the series in the degree of polymerization of the corresponding polymer, e.g., PE parameters obtained from low molecular weight alkanes. When these data are not available, it is proposed to use pure hydrogenated monomer data in order to obtain the correspondent homopolymer parameters. Moreover, when the monomer data is not available, it is proposed to use mixture data in order to obtain the polymer parameters. The main reason for this procedure
m ) 1.636 + 0.464n
and mol 14.027n + 2.016 ) (9) m 1.636 + 0.464n
where n is the number of carbons. In this work, squalane (2,6,10,15,19,23-hexamethyltetracosane) was considered to be a long-chain linear alkane, and its parameters were obtained from eq 9 and from dodecane parameters. The same strategy was used to obtain equation of state parameters for PE. Squalane and PE parameters are presented in Table 3. In general, pure substance vapor-liquid equilibrium (VLE) data for a series in degrees of polymerization of a given polymer are not available. In the absence of data for constructing the curves in Figure 1 for any polymer, in the present work the parameters , λ, and σ are considered to be constant for any degree of polymerization. Within this assumption, the pure hydrogenated vinyl monomer parameters can be used as the polymer parameters for addition polymers. In the absence of data for constructing the m parameter curve of a given series
Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1751 Table 3. TPT1M Equation of State Parameters for Squalane and Polymers polymer
/k (K)
σ (Å)
λ
mol/m
squalane PE PVAc PVME PMMA PS PDMS
111.2230a 111.2230a 94.2033b 86.9172c 100.9109d 127.7645e 167.3660f
3.2379a 3.2379a 2.6841b 2.7096c 2.8420d 3.1183e 3.2419f
1.9293a 1.9293a 1.9505b 1.9383c 1.9307d 1.9052e 1.5393f
27.1809g 30.2306g 43.9411h 37.7053h 51.0164h 61.4922h 24.9008f
a Dodecane parameter. b Ethyl acetate parameter. c Ethyl methyl ether parameter. d Methyl isobutyrate parameter. e Ethylbenzene parameter. f Obtained from mixture data. g Extrapolation from an m parameter alkane curve. h mol/m extrapolation.
in degrees of polymerization, in this work the following empirical relation is used:
(mol/m)addition polymer (mol/m)hydrogenated monomer
)
(mol/m)PE (mol/m)ethane
(10)
This procedure was used to determine the pure polymer parameters of PVAc, poly(vinyl methyl ether) (PVME), poly(methyl methacrylate) (PMMA), and PS. The PDMS parameters were determined from mixture data. The obtained polymer parameters are presented in Table 3. 4. Results for Binary Mixtures The TPT1M model was used to predict VLE data of binary low molecular weight28,29 mixtures and of polymer solutions30 with different solvents, at different temperatures, and for different polymer number-average molecular weights (Mn). In the polymer solution calculations, it was considered that in the vapor phase there was no polymer; the equilibrium pressure was determined so that the solvent chemical potential was the same in the vapor and liquid phases, for a given liquid composition and temperature. For low molecular weight mixtures, the pressure and vapor composition were determined so that the chemical potentials of all components were the same in the vapor and liquid phases, for a given liquid composition and temperature. 4.1. Alkane and PE Mixtures. 4.1.1. Ethane/ Decane Mixture. The TPT1M equation of state was used in the calculation of VLE of the ethane/decane mixture at temperatures below and above the ethane critical temperature. The TPT1M predictions and experimental data28 are shown in Figure 2. The TPT1M predictions for the equilibrium pressure are in very good agreement with the experimental data, below and above the ethane critical temperature. 4.1.2. Hexane/Squalane Mixture. The TPT1M equation of state was used to describe the VLE of the mixture hexane/squalane29 at temperatures above and below the hexane critical point, as shown in Figure 3. The TPT1M predictions in Figure 3 are in very good agreement with the experimental data, below and above the hexane critical temperature. This result indicates that the TPT1M equation of state is appropriate to describe the properties of mixtures of substances highly asymmetric in length, obtaining the pure long-chain alkane parameters from low molecular weight alkane parameters. 4.1.3. Binary PE Mixtures. To compare the TPT1M predictions using the different sets of parameters for PE, the equilibrium pressure for the mixture PE/pxylene is plotted in Figure 4a, where “From Table 3”
Figure 2. VLE experimental data for decane and ethane28 and TPT1M prediction curves.
Figure 3. VLE experimental data for hexane and squalane29 and TPT1M prediction curves.
means the TPT1M predictions using the PE parameters in Table 3 and “From Table 2” means the TPT1M predictions using the PE parameters in Table 2. As can be seen in Figure 4a, the TPT1M predictions using the parameters in Table 3 are in good agreement with the experimental data and in better agreement with these data than the TPT1M predictions using the parameters in Table 2. Although the mixture pressure predictions for the following polymer mixtures using the parameters in Table 2 are not shown, similar results are obtained. The equilibrium pressures for the mixtures PE/toluene and PE/heptane are plotted in Figure 4b. The TPT1M predictions using the PE parameters in Table 3 are in very good agreement with experimental data. The agreement with experimental data was achieved with extrapolated values for PE, with no binary adjustable parameters and for different solvents, temperatures, and polymer molecular weights.
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Figure 4. VLE experimental data30 and TPT1M prediction curves for PE mixtures using different sets of parameters for the polymer. “From Table 2” refers to the TPT1M predictions using the PE parameters in Table 2. “From Table 3” and “TPT1M” refer to the TPT1M predictions using the PE parameters in Table 3: (a) p-xylene, Mn ) 2650; (b) toluene and heptane.
Figure 6. VLE experimental data30 and TPT1M prediction curves for PVME mixtures.
Figure 7. VLE experimental data30 and TPT1M prediction curves for PMMA mixtures.
Figure 5. VLE experimental data30 and TPT1M prediction curves for PVAc mixtures: (a) acetone and ethyl acetate, Mn ) 8600, T ) 303.15 K; (b) acetone and ethyl acetate, Mn ) 109 100, T ) 303.15 K; (c) benzene, Mn ) 158 000, T ) 313.15 and 333.15 K.
4.2. Binary PVAc Mixtures. The TPT1M parameters for PVAc were determined from pure ethyl acetate parameters using eq 10. The TPT1M equation of state predictions and the experimental equilibrium pressure for the mixtures PVAc/acetone and PVAc/ethyl acetate at 303.15 K, for polymer molecular weights equal to 8600 and 109 100, are plotted in parts a and b of Figure 5, respectively. The TPT1M predictions are in very good agreement with the experimental data. The TPT1M predictions and experimental equilibrium pressure for the mixture PVAc/benzene with polymer molecular weight equal to 158 000, at 313.15 and 333.15 K, are plotted in Figure 5c. The TPT1M predictions are in very good agreement with experimental data.
4.3. Binary PVME Mixtures. The TPT1M parameters for PVME were determined from pure ethyl methyl ether parameters using eq 10. The TPT1M equation of state predictions and the experimental equilibrium pressure for the mixtures PVME/ethylbenzene at 398.15 K, PVME/toluene at 343.15 K, and PVME/benzene at 298.15 K, for polymer molecular weight equal to 14 600, are plotted in Figure 6. The TPT1M predictions are in very good agreement with the experimental data. 4.4. Binary PMMA Mixtures. The TPT1M parameters for PMMA were determined from pure methyl isobutyrate parameters using eq 10. The TPT1M equation of state predictions and the experimental equilibrium pressure for the mixtures PMMA/benzene (polymer molecular weights equal to 574 and 8560) and PMMA/toluene (polymer molecular weights equal to 574 and 36410), at 343 K, are plotted in Figure 7. The TPT1M predictions are in good agreement with the experimental data. 4.5. Binary PS Mixtures. The TPT1M parameters for PS were determined from pure ethylbenzene parameters using eq 10. The TPT1M equation of state predictions and the experimental equilibrium pressure for the mixtures PS/benzene at 323.15 K, PS/cyclohexane at 303.15 K, and PS/ethylbenzene at 343.15 K (polymer molecular weight equal to 3600) and PS/ toluene at 373.15 K, PS/benzene at 323.15 K, and PS/ cyclohexane at 303.15 K (polymer molecular weight equal to 103 800) are plotted in parts a and b of Figure 8, respectively. The TPT1M predictions are in good agreement with experimental data, except for the mixture PS/toluene.
Ind. Eng. Chem. Res., Vol. 40, No. 7, 2001 1753
Figure 8. VLE experimental data30 and TPT1M prediction curves for PS mixtures: (a) benzene, cyclohexane, and ethylbenzene, Mn ) 3600; (b) toluene, benzene, and ethylbenzene, Mn ) 103 800.
Figure 9. VLE experimental data30 and TPT1M correlation and prediction curves for PDMS mixtures: (a) correlation curves, Mn ) 89 000, T ) 303.04 K; (b) prediction curves, Mn ) 1540, T ) 313.15 K.
4.6. Binary PDMS Mixtures. The PDMS parameters were determined so that the TPT1M equation of state gave the minimum root-mean-square relation deviation to the equilibrium pressures for the mixtures of PDMS (polymer molecular weight equal to 89 000) with pentane, hexane, benzene, cyclohexane, and heptane at 303.04 K and with a polymer weight fraction in the range 0.9-1. The TPT1M correlation and experimental data are presented in Figure 9a. In Figure 9, the TPT1M correlation is denoted by solid lines and the TPT1M predictions are denoted by dashed lines. The obtained parameters were used to calculate the TPT1M pressure predictions for PDMS (Mn ) 1540) mixtures at 313.15 K and with different solvents and with polymer weight fractions in the range 0.6-1. The results are plotted in Figure 9b. Both the TPT1M correlation and predictions are in very good agreement with the experimental data.
proposed to use pure hydrogenated monomer parameters in order to determine the pure addition homopolymer parameters. The TPT1M equation of state was used to describe the properties of mixtures highly asymmetric in length with no binary adjustable parameters. This model was used to describe the properties of ethane/decane and hexane/squalane mixtures at temperatures above and below the critical temperatures of ethane and hexane, respectively. The predictions were in very good agreement with the experimental data. The TPT1M equation of state was used to describe the properties of binary PE solutions. The TPT1M parameters for PE were obtained from low molecular weight alkane parameters, and the equation of state predictions were in very good agreement with experimental data. The TPT1M equation of state was used to describe the properties of binary PVAc, PVME, PMMA, PS, and PDMS mixtures. The pure PVAc, PVME, PMMA, and PS parameters were obtained from the pure hydrogenated vinyl monomer parameters. The PDMS parameters were obtained from mixture data. The TPT1M predictions and correlation were, in general, in good agreement with the experimental data. The TPT1M equation of state showed to be able to predict the properties of polymer mixtures with no binary adjustable parameters, for different polymer molecular weights, with different solvents, at different temperatures, and in the hole range of composition. The predictions involving high molecular weight compounds were possible with only low molecular weight substance data. List of Symbols g ) radial distribution function at the contact point m ) number of segments Mn ) number-average molecular weight, g/mol n ) number of carbons nc ) number of components Z ) compressibility factor Greek Symbols ) square-well depth, J Φij ) interaction potential between two nonadjacent spheres i and j, J λ ) square-well length σ ) hard-core diameter, Å ξ3 ) mixture packing fraction Subscripts chain ) sphere chain
5. Conclusions
Superscripts
The TPT1M analytical perturbation theory equation of state for mixtures of square-well homonuclear chain fluids of variable well width was used to describe the properties of polymers and normal substances. Pure polymer parameters were obtained from volumetric data and from low molecular weight substance parameters. The parameters obtained from volumetric data are discrepant from the segment parameters obtained for their hydrogenated monomers, and the mixture properties calculated with these parameters are not in agreement with experimental data. For this reason, it is
R ) residual att ) attractive SWS ) square-well system HS ) hard sphere Abbreviations RMSDP ) root-mean-square relative deviations in vapor pressure RMSDF ) relative root-mean-square deviations in saturated liquid density
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Received for review July 28, 2000 Revised manuscript received December 11, 2000 Accepted December 12, 2000 IE0007116
