Mixture Equation of State for Water with an Associating Reference Fluid

Article pubs.acs.org/IECR

Cite This: Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Mixture Equation of State for Water with an Associating Reference Fluid Bennett D. Marshall* ExxonMobil Research and Engineering, 22777 Springwoods Village Parkway, Spring, Texas 77389, United States

ABSTRACT: Development of equations of state for water is challenging due to the structural transition to tetrahedral symmetry of liquid water at ambient conditions. Recently, the concept of associated reference perturbation theory (APT) was introduced. APT includes association in the reference fluid to the isotropic attractions. This allows for the structural transition of water to tetrahedral symmetry to be incorporated in the perturbation theory. APT was shown to give substantial improvements over standard SAFT approaches in the description of pure water. In this paper, we further refine this approach by including an associating reference fluid in the polar PC-SAFT equation of state. This allows for a more accurate representation of pure water thermodynamics. It is demonstrated that the new approach gives a substantial improvement in the correlation of liquid densities, hydrogen-bonded structure, and the second virial coefficient as compared to a standard polar PC-SAFT equation of state. The approach qualitatively reproduces the minima in isothermal compressibility and isobaric heat capacity; however it does not reproduce the density maximum. The new approach is also shown to give a good representation of water/hydrocarbon mixturephase equilibria. The perturbed chain statistical associating fluid theory (PCSAFT)2,5 EoS has gained wide popularity and is employed in several commercial thermodynamic packages. Unlike other SAFT versions which consider a reference fluid of spheres, PCSAFT is formulated around a hard chain reference fluid. With this formulation, the effect of chain length is built into the PCSAFT dispersion term. Barker-Hendersen second-order perturbation theory (BH2)8 provides the theoretical basis for the PC-SAFT dispersion term. The reference system integrals of the pair correlation function Iref (and corresponding density derivative) are evaluated empirically by fitting polynomials in the packing fraction to n-alkane phase equilibria data. While PC-SAFT has been successfully applied to a wide range of systems, it provides a poor description of pure water. Water is challenging because it exhibits both energetic and structural cooperativity. Energetically, water is known to exhibit hydrogen-bond cooperativity9 as well as cooperativity between hydrogen bonding and the dipole moment in the liquid phase.10 PC-SAFT models that incorporate hydrogen-bond

I. INTRODUCTION The accurate prediction of multicomponent fluid-phase equilibria is of immense scientific and industrial importance. As the general relations of phase equilibria are known exactly, equality of pressure, temperature, and chemical potentials, the challenge is in approximating the equation of state (EoS) such that accurate multicomponent phase equilibria predictions can be made. An accurate and predictive EoS must account for the fact that molecules have asymmetry in size as well molecular interactions. For instance, water hydrogen bonds and hexane does not. A polymer has two length scales (segment diameter and length), but argon is a spherical atom with a single length scale. The statistical associating fluid theory (SAFT)1−4 class of EoS account for both asymmetry in size and energy scales by treating molecules as chains of spheres with standard dispersion as well as polar energetic degrees of freedom. In all variants of SAFT,1,3,5,6 Wertheim’s first-order thermodynamic perturbation theory (TPT1)7 is used to account for the free energy change due to chain formation as well as association (hydrogen bonding). The different SAFT variants differ primarily in the treatment of the nonpolar dispersive contribution to the free energy. © XXXX American Chemical Society

Received: November 14, 2017 Revised: February 17, 2018 Accepted: March 1, 2018

A

DOI: 10.1021/acs.iecr.7b04712 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research cooperativity11 as well as the coupling between the square well energy ε and the degree of association12 have been recently been developed. The incorporation of structural cooperativity is a more daunting task, as it challenges the very foundation of perturbation theory. Perturbation theories assume that the structure of the fluid is that of the reference fluid. For water, this is a poor approximation because of the tetrahedral symmetry of fully bonded water. As the temperature is decreased and the number of fully hydrogen bonded water molecules increases, the structure of the fluid is changed substantially by hydrogen bonding. This results in the anomalous density maxima as well as minima in the isothermal compressibility as well as isobaric heat capacity.13,14 This changing of the fluid structure with hydrogen bonding violates a fundamental assumption in perturbation theory: the assumption that the structure of the fluid is unchanged from that of the reference fluid. Remsing et al.15 demonstrated that perturbation theory could be successfully applied to water if one includes the short-range attractive interactions in the reference fluid. It is short-range forces (attractive and repulsive) which control fluid structure. Remsing et al.15 decomposed the Columbic contribution of the SPC/E16 potential of water into short-range and long-range contributions using local molecular field theory.17 The shortranged contribution was then evaluated using molecular simulation, and the long-range added as a perturbation. It was demonstrated that the truncated SPC/E potential with long ranged attractions added as a perturbation gave an excellent representation of the full SPC/E potential. This work demonstrated that perturbation theory could be successfully applied to water, if the short-range attractions (hydrogen bonding) where included in the reference fluid. Building on the work of Remsing et al.,15 Marshall18 introduced the idea of associated reference perturbation theory (APT) to describe the thermodynamics of water. Unlike standard perturbation theories which consider a spherically symmetric reference fluid for BH2, APT includes the association contribution in the reference fluid integral Iref to the BH2 perturbation theory. For water, this distinction is particularly important as the coordination number (defined as the number of molecules within a shell of diameter a) of liquid water decreases with decreasing temperature at ambient conditions. See Figure 4 of ref 18 for an illustration of this phenomena. APT exploits the fact that when all water molecules are fully hydrogen bonded the coordination number will reflect tetrahedral symmetry. This allows for the development of a new perturbation theory with minimal additional complexity as compared to standard cases but which gives a much better representation of pure water. While APT provides a good representation of pure water, the theory has not been developed to the point that it can be used for multicomponent phase equilibria calculations. Another variant of SAFT could be developed around this approach. However, it may be more useful to take the ideas of APT and incorporate them into an existing SAFT framework. This will be the subject of this paper. We incorporate APT into the general formalism of the polar perturbed chain SAFT (PPCSAFT) EoS. We call this new approach associated reference polar PC-SAFT or ARPC-SAFT. We demonstrate that ARPCSAFT gives a substantially improved EoS as compared to PPCSAFT.

II. THEORY In this section, we apply the recently developed concept of associated reference perturbation theory18 to develop a new equation of state for water which reduces to the polar PCSAFT equation of state in the fully unhydrogen bonded limit. A. Polar PC-SAFT Equation of State. The statistical associating fluid theory (SAFT) class of equations of state is based on a coarse-grained molecular representation that treats molecules as chains of length m of tangentially bonded spheres of diameter σ. In addition to standard dispersion attractions, there may be hydrogen bonding and long-range polar contributions to the pair potential and the resulting free energy. Figure 1 gives an example molecular representation of a hydrogen bonding chain molecule.

Figure 1. Model representation of chain molecule with two association sites.

The excess Helmholtz free energy is given as a perturbation to a hard chain reference Aex aex = = ahc + aatt + aas + adp Nk bT (1) where N is the number of molecules in the fluid, T is the temperature, kb is Boltzmann’s constant, and ahc is the excess free energy of the hard chain reference fluid: ahc = ma (2) ̅ hs + ach The term ahs is the excess free energy due to hard sphere repulsions, and ach is the excess free energy due to chain formation19 evaluated with Wertheim’s first order perturbation theory7,20 (TPT1). In this work, we follow the simplified approach21 of evaluating these contributions using the pure component Carnahan and Starling22 forms of ahs and the contact value of the pair correlation function ghs ahs =

4η − 3η2 ; (1 − η)2

ach = (1 − m̅ )ln ghs ;

ghs =

1−

η 2

(1 − η)3 (3)

Note that there is no fundamental reason that we chose to use the simplified forms of eq 3 instead of the standard mixture form.2 This was a choice of convenience due to the fact our inhouse model makes this assumption. It has been demonstrated21 that the use of eq 3 results in an equally capable equation of state as compared to the standard approach. The result of this study will be a new PC-SAFT dispersion contribution which can be equally applied to the simplified or normal mixture form. The average chain length is given by m̅ = ∑ xkmk, where xk is the mole fraction of species k. The mixture packing fraction is obtained from the density as B


Article

Industrial & Engineering Chemistry Research η=

π ρ∑ xkmk dk3 6

The term aas gives the change in free energy due to shortrange association (hydrogen bonding) attractions. The association contribution is evaluated24 with Wertheim’s TPT17 for an arbitrary set of sites on each species Γ(k)

(4)

where dk is the temperature-dependent diameter ⎛ ⎛ ε ⎞⎞ dk = σk ⎜⎜1 − 0.12 exp⎜ −3 k ⎟⎟⎟ ⎝ k bT ⎠⎠ ⎝

aas =

(5)

k

2

Gross and Sadowski developed the contribution for dispersion attractions aatt using second order Barker − Hendersen perturbation theory (BH2)8 with a hard-chain reference fluid aatt = a1 + a 2

with the first- and second-order free energies given by

k

a2 = −

j

εkj k bT εkj

⎧ (i , j) ⎪− ε r12 ≤ rc(i , j) and θA1 ≤ θc(,iA) and θB2 ≤ θc(,jB) (i , j) AB φAB (12) = ⎨ ⎪ ⎩ 0 otherwise

2

( )

πρmI̅ 2∑k ∑j σkj3mk mjxkxj 1+

k bT

∂Phc / k bT ∂ρ

(12)

where the first sum is over the number of species and the second sum is over the set of association sites on k. The fractions XA are the fraction of molecules not bonded at association site A.25 The form of eq 12 is independent of the specific form of the association site−site potential of interaction φAB; however, the exact relationship for the fractions XA will depend on φAB. For the site−site potential we assume conical square well (CSW) association sites19

(6)

a1 = −2πρI1∑ ∑ σkj3mk mjxkxj

⎛ X (k) 1⎞ ⎜⎜ln X A(k) − A + ⎟⎟ 2 2⎠ A ∈Γ(k) ⎝

∑ xk ∑

(13)

(7)

where rc is the maximum distance between molecules for which association can occur, θA1 is the angle between the center of site A on molecule 1 and the vector connecting the two centers, and θc,A is the maximum angle for which association can occur. With this, if two molecules are both positioned and oriented correctly, a bond is formed and the energy of the system is decreased by a factor εAB. A diagram of two molecules interacting with conical square well association sites can be found in Figure 2. The benefit of using CSW association sites is

The mixture square well depth εij and hard sphere diameter σij are evaluated with standard Lorentz−Berthelot combining rules εij = εiεj (1 − kij) and σij = (σi + σj)/2. The denominator in a2 gives the compressibility of the hard chain reference fluid evaluated with eqs 2 and 3. Of importance to this work are the reference system integrals I1 and I2. For a pure component fluid with a hard chain reference fluid these are given by λ/σ

I1 =

I2 =

∫

ghc (x)x 2dx

;

x = r /σ

1

(8)

∂ ρI1 ∂ρ

(9)

In eq 8 ghc is the pair correlation function of the hard chain reference fluid and λ is the square well range. The relationship in eq 9 is a result of the local-compressibility approximation of BH.8 In PC-SAFT, the reference integrals are evaluated with the following empirical expansion in packing fraction 6

I1 =

∑ ai(m̅ )ηi i=0

Figure 2. Diagram of interacting conical square well association sites.

6

;

I2 =

∑ bi(m̅ )ηi i=0

that the angular and radial degrees of freedom become uncoupled. This will be particularly important in the current development. Also, similar definitions are often used to define hydrogen bonding in neutron diffraction26 and molecular simulation27 studies, allowing for validation of hydrogen bonding predictions of the new approach. For the case of conical square well sites19

(10)

where m̅ − 1 m − 1 m̅ − 2 a1i + ̅ a 2i m̅ m̅ m̅ m−1 m − 1 m̅ − 2 bi(m̅ ) = b0i + ̅ b1i + ̅ b 2i (11) m̅ m̅ m̅ and the constants aji and bji are treated as universal and correlated to n-alkane phase equilibria data. Since the parameters in I1 and I2 are adjusted separately, the relation eq 9 is no longer satisfied. The term adp represents the free energy contribution due to long-range polar interactions. In this work, we employ the longrange polar contribution of Jog and Chapman (JC).23 The JC polar contribution requires the dipole moment μ and the fraction of segments which are polar xp as input. ai(m̅ ) = a0i +

1 X A(i)

= 1 + 4πρ∑ j

∑ B ∈Γ

(j)

(i , j) 3 (i , j) (i , j) (j) σij xjf AB Ihs XB PAB

(14)

where P(i,j) AB is the probability that site A on i and site B on j are oriented correctly for association (i , j) = PAB

i) (1 − cos θc(,A )(1 − cos θc(,Bj))

4

(15)

I(i,j) hs is the integral of the hard sphere reference pair correlation function in the bond volume C


Article

Industrial & Engineering Chemistry Research (i , j) Ihs =

1 σij3

∫σ

rc(i , j)

(i , j) 2 ghs r dr

ambient conditions where water is highly structured with near tetrahedral symmetry. At ambient conditions eq 22 will predict a coordination number which is too large, resulting in the overprediction of the isotropic square well attractions. See Figure 4 of ref 18 for an illustration of this phenomena. For this reason, eq 22 is appropriate as a limiting high temperature case for water. Another important limiting case for water is that of pure tetrahedral symmetry Nref(rc) = 4. For the tetrahedral limit, eq 21 is employed to solve for Iref up to the critical radius rc analytically as

(16)

ij

and f(i,j) AB is the association Mayer function ⎛ ε(i , j) ⎞ (i , j) f AB = exp⎜⎜ AB ⎟⎟ − 1 ⎝ k bT ⎠

(17)

In the typical application of PC-SAFT to associating systems,5 the specific details of the potential φAB are not considered, and eq 14 is evaluated subject to the condition (i , j) (i , j) 4πσij3PAB Ihs =

(i) (j) (i , j) σi3σj3κAB κAB ghs

Itet(rc) =

(18)

where g(i,j) hs is the hard sphere reference correlation function at contact, and κ(i) AB is the pure component association bond volume of component i which is fit to experimental data. In the simplified PC-SAFT21 approach g(i,j) hs is evaluated using the pure component Carnahan and Starling value with the mixture packing fraction (i , j) ghs

≈ ghs =

1−

(19)

Itet(λ) =

B. Associated Reference Perturbation Theory for Water. The PC-SAFT EoS has been successfully applied to a wide range of systems. One area where PC-SAFT performs poorly, is the representation of the properties of pure water at ambient conditions. The poor performance is a result of the fact that water transitions to tetrahedral symmetry at ambient conditions. This transition represents a structural change to the fluid which violates a fundamental assumption of the BH2 perturbation theory; that the structure of the fluid is controlled by reference system repulsions. Recently, Marshall18 introduced the idea of associated reference perturbation theory (APT). In APT the association interactions are included in the reference fluid integral Iref of the BH2 perturbation theory. APT allows for the self-consistent calculation of a more appropriate reference fluid for water. In this section we incorporate the concept of APT into the polar PC-SAFT EoS. Water is taken to be a single sphere (m = 1) of diameter σw with four association sites (two donors and two acceptors). The site−site potentials are given by eq 13 and the reference system attractions in BH2 are that of a square well of depth εw and range λ. APT is formulated around the fact that when water is fully hydrogen bonded it attains tetrahedral symmetry. In perturbation theories it is the reference system integral Iref(a) which controls the coordination number Nref(a) within a shell of radius a Iref (a) =

∫1

a / σw

x 2gref (x) dx

x = r /σw

=

∫1

rc / d

x 2g tet (x) dx +

1 πρσw3

λ/d

∫r /d c

1 + Ihs(λ) − Ihs(rc) πρσw3

(23)

x 2ghs(x) dx (24)

4

Nref (a) = 4πρσw3 ∑ χk Ik(a) k=0

(25)

where χk is the fraction of molecules bonded k times. Comparing eqs 21 and 25 4

Iref (a) =

∑ χk Ik(a) k=0

(26)

The integrals Ik will depend on the hydrogen bonded state of the system; however, Overduin and Patey28 demonstrated that water molecules tend to cluster based on the tetrahedral order parameter. That is, it is more likely that if molecule 0 is fully hydrogen bonded, it is more likely it will be surrounded by other fully hydrogen bonded water molecules than would be the case if molecule 0 were less than fully hydrogen bonded. On this basis, we assume I4 = Itet. This assumes that there is local tetrahedral order surrounding molecules which are fully hydrogen bonded. On the other hand, if molecule 0 is less than fully hydrogen bonded, it is more likely it will be surrounded by other molecules which are not fully hydrogen bonded. As the anomalous properties of water are the result of tetrahedral symmetry,13 for these integrals (k < 4) we revert to the standard perturbation treatment Ik = Ihs. Combining eqs 24−26 with I4 = Itet and Ik = Ihs for k < 4

(20)

(21)

Iref controls the number of molecules for which a given molecule will share isotropic square well attractions. Most perturbation theories assume a hard sphere reference Iref = Ihs

x 2g tet (x) dx =

Equations 20 and 21 are developed by considering a molecule (labeled 0) at the origin (r = 0). For water, molecule 0 could be hydrogen bonded k = 0−4 times. For the case that molecule 0 is bonded k times, the number of molecules within a shell a will be Nk(a) = 4πσ3wρIk(a), where Ik is the reference fluid integral for the case that molecule 0 is bonded k times. For the hydrogen bonding reference fluid, Nref is taken as an average over all possible bonding states

The integral Iref is related to Nref through the following relation Nref (a) = 4πρσw3Iref (a)

rc / d

It has been enforced in eq 23 that if all water molecules are fully bonded there will be tetrahedral coordination within the shell rc. The correlation function gtet is the pair correlation function between two water molecules in a fully hydrogen bonded fluid. To extend Itet over the full square well range λ, the standard hard sphere reference is used for r > rc

η 2

(1 − η)3

∫1

(22)

While this is often a reasonable approximation for nonassociating simple fluids, it will lose accuracy for water at D


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Industrial & Engineering Chemistry Research Iref (λ) = (1 − χ4 )Ihs(λ) + χ4 Itet(λ)

I1̅ =

⎞ ⎛ 1 = Ihs(λ) + χ4 ⎜ − Ihs(rc)⎟ 3 ⎠ ⎝ πρσw

∑ xkI1(k) k

I2̅ =

(27)

∑ xkI2(k) (31)

k

18

In the original formulation of APT, Marshall evaluated the integrals Ihs(λ) and Ihs(rc) using Chang and Sandler’s real function solution of the Ornstein−Zernike relation within the Percus−Yevick approximation.29,30 In this work it is desired to incorporate the reference integral eq 27 into the general PCSAFT formalism. To make the connection between the two approaches, we simply identify Ihs(λ) = I1, where I1 is the PCSAFT integral in eq 10. We then write a new PC-SAFT integral from eq 27 for water as ⎛ 1 ⎞ I1(w) = Iref (λ) = I1 + χ4 ⎜ − Ihs(rc)⎟ 3 ⎝ πρσw ⎠

where I (j k)

⎞⎫ ∂ ⎧ ⎛ 1 ⎨ρχ4 ⎜ − Ihs(rc)⎟⎬ 3 ∂ρ ⎩ ⎝ πρσw ⎠⎭ ⎪

⎪

⎪

⎪

a1 = −2πρI1̅∑ ∑ σkj3mk mjxkxj (28)

k

a2 = −

j

εkj k bT εkj

2

( )

πρmI̅ 2̅ ∑k ∑j σkj3mk mjxkxj 1+

k bT

∂Phc / k bT ∂ρ

(33)

All contributions which explicitly depend on the packing fraction (ahs, ghs, I(rc), Ij) also require input for the temperature dependent hard sphere diameter eq 5. However, as previously noted,12 the hard sphere diameter in PC-SAFT performs very poorly for water. This is due to the fact that the structure of water at ambient conditions is controlled by hydrogen bonding, not solely repulsive forces. It was shown12 that if one simply uses the hard sphere diameter of water, instead of the temperature dependent diameter, one obtains a much more accurate equation of state. Here we continue this treatment and assume the following temperature-dependent diameter

(29)

Equations 28 and 29 reduce exactly to the standard PCSAFT reference integrals I1 and I2 in the absence of fully hydrogen-bonded water molecules. The reference integral Ihs(rc) in eqs 28 and 29 as well as eq 16 is still evaluated with Chang and Sandler’s real function solution.29 The Chang and Sandler solution of Ihs and the corresponding density derivatives are algebraically complex. However, for any specific range rc in eq 20, Ihs can be easily correlated with a polynomial in η. In the appendix we correlate Ihs(rc) to the critical radius regressed in section III. The fraction of water molecules which are fully bonded is evaluated using the standard TPT1 approach31

χ4 = (1 − X A(w))4

(32)

In eq 32, j = 1, 2 with Ij given by eq 10 and I(w) given by eqs 28 j and 29. Combining eqs 7 and 31 we obtain the first- and second-order free energies

Calculating I(w) from I(w) using eq 9 and evaluating I2 with 2 1 the PC-SAFT correlation in (10) I2(w) = I2 +

⎧ I (w) for k = water ⎪ j =⎨ ⎪ Ij otherwise ⎩

⎧ σk k = water ⎪ ⎪ ⎛ dk = ⎨ ⎛ ε ⎞⎞ ⎪ σk ⎜⎜1 − 0.12 exp⎜ −3 k ⎟⎟⎟ otherwise ⎪ ⎝ ⎝ k bT ⎠⎠ ⎩

(30)

X(w) A

where is the fraction of water association sites which are unbonded, calculated self-consistently through eq 14. In the original formulation of APT, the compressibility in the denominator of the second order BH2 free energy (denominator of a2 in eq 7) included a contribution due to association. However, it is known8 that the second order term corrects the theory in the low density limit, with the first-order term dominating in dense fluids where association is the strongest. Hence, to avoid undue complexity, we neglect the association contribution to the compressibility leaving the denominator in a2 unchanged. As a quick summary, we have incorporated the effect of the transition to tetrahedral symmetry on the reference integrals I1 and I2 of the BH2 perturbation theory in the PC-SAFT dispersion contribution. However, the associated reference fluid defined by the free energy aref = ahc + aas does itself not reflect a transition to tetrahedral symmetry. This results from the fact that aas is evaluated in Wertheim’s thermodynamic perturbation theory which assumes an unchanging fluid structure with hydrogen bonding. C. Associated Reference Fluid for Multicomponent Mixtures. To extend this approach to multicomponent mixtures we employ the simple mixing rules on the reference integrals

(34)

Equations 31−34 summarize the modifications made to the BH2 contribution. The treatment of the hydrogen-bonding fractions also merits discussion. Evaluating eq 14 for the case where water is the only hydrogen-bonding species in the fluid gives 1 X A(w)

(w,w) 3 (w,w) (w) = 1 + 4πρPAB σwf AB X A Ihs(rc)

(35)

As with eqs 28 and 29, Ihs(rc) is evaluated with the real function solution of Chang and Sandler. For mixtures we evaluate the pure component form of Ihs(rc) with the mixture packing fraction given by eq 4. For all other hydrogen-bonding molecules, we formulate the theory such that it reduces to eq 18 subject to eq 19 in the absence of water, hence retaining the standard PC-SAFT treatment (or more specifically here, the simplified PC-SAFT treatment). For the simplified treatment of the pair correlation function the combining rule in eq 18 can be recast as (k , i) (k , i) 4πσki3PAB Ihs ρ =

(i) (k) ρghsσi3κAB ρghsσk3κAB =

pp i k

(36)

ρghsσ3i κ(i) AB

where the product simply gives the probability that two molecules are positioned and oriented correctly to form an E


Article

Industrial & Engineering Chemistry Research association bond pi. For water, using the approach described above we have (w,w) pw = 4πσw3PAB ρIhs(rc)

The new ARPC-SAFT approach gives a substantial improvement in the prediction of liquid densities over PPC-SAFT. The APT of Marshall gives the best agreement with the data for both vapor pressure and liquid densities. This can be observed in the phase diagram given in Figure 3. The left panel shows the full phase diagram, while the right panel focuses on the low T liquid branch. Both APT and ARPC-SAFT do a good job of describing the low-temperature liquid branch of the phase diagram, while PPC-SAFT is not accurate for the prediction of liquid densities. APT clearly outperforms ARPC-SAFT. The reason for this is that APT is a cohesive theoretical formalism based on BH2 perturbation theory while in ARPC-SAFT we have restricted the theory such that the reference integrals reduce to those of the PC-SAFT EoS (eq 10) in the absence of hydrogen bonding. In short, in ARPC-SAFT we cannot control the square well range λ in the reference integral eq 27, as this is not defined in the empirical PC-SAFT approach. For this reason, inclusion of the polar contribution to the free energy is a requirement to obtain an accurate equation of state. APT suffers no such limitation. In many ways, implementation of eq 27 with the SAFT-VR36 EoS would be more natural than with PC-SAFT. We will consider this extension in a future work. The improved performance of ARPC-SAFT over PPC-SAFT in the prediction of liquid densities can be traced back to the reference integrals eqs 28−29. The left panel of Figure 4 plots the reference integrals I(w) 1 and I1 for a saturated liquid while the right panel plots the fraction of water molecules bonded 4 times χ4. The standard PC-SAFT reference integral I1 shows a increases with monotonic increase with decreasing T. I(w) 1 decreasing T at high temperatures, exhibits a maximum near T = 550 K, and then begins to decrease with decreasing T. This behavior results from the increase in the fraction χ4 as temperature is decreased; signifying an increase in tetrahedral order. While both APT and ARPC-SAFT show a very substantial improvement in the representation of liquid densities, for the current choice of parameters they do not reproduce the density maximum. As discussed in ref 18 APT can be forced to yield a density maximum, but this comes at the expense of the representation of higher temperature liquid densities as well as vapor pressures. In this work, we have made no attempt to “force” ARPC-SAFT to yield a density maximum. Now we focus on the model predictions of the degree of hydrogen bonding. First, the fraction of free OH groups as measured by Luck is considered.34 It has been noted by several authors27,37,11,38 that hydrogen-bonding parameters that give good a representation of Luck’s data give poor predictions of phase equilibria within PC-SAFT. Figure 5 compares model predictions for the fraction of free OH groups (XA calculated with eq 35) to the experimental data of Luck for a saturated liquid. APT overpredicts and PPC-SAFT under predicts the fraction of free OH groups. On the other hand, ARPC-SAFT is in reasonable agreement with the data far from the critical point. Given uncertainties in measurement and data interpretation, ARPC-SAFT predictions for the fraction of free OH groups are entirely consistent with Luck’s data far from the critical point. Further validation of ARPC-SAFT predictions of hydrogen bonding structure can be found in Figure 6, which compares ARPC-SAFT predictions of the fraction of molecules hydrogen bonded k times χk in a saturated liquid to molecular simulations27 using the i-AMOEBA39 force field. As can be seen, ARPC-SAFT gives a very good representation of the

(37)

Combining eqs 36 and 37, we then obtain the association combining rule for cross-associating mixtures with water (w, i) (w, i) 4πσw3iPAB Ihs =

(w,w) (i) 4πσw3PAB Ihs(rc)σi3κAB ghs

(38)

With eqs 18, 19, and 38 the model reduces to the simplified PC-SAFT approach in the absence of water.

III. PARAMETERIZATION AND PURE WATER In this section, we develop model parameters for the associated reference polar PC-SAFT theory (ARPC-SAFT) developed in section II. The equation of state is described by the chain length mw = 1, hard sphere diameter σw, dispersion energy εw, critical angle for association θc,w, critical radius for association rc,w, association energy ε(w) AB and finally the long-range polar parameter xp (the dipole moment is set to 1.85 D). Hence, six parameters need to be adjusted to experimental data which consist of vapor pressure and saturated liquid density data in the temperature range 273.15−580 K. In addition to pure component data, we follow Liang et al.32 and include a limited amount of binary LLE data with a hydrocarbon (n-hexane) to help guide the specific ratio of polar to nonpolar energy scales. The regressed parameters are physically reasonable and can be found in Table 1. The water diameter σw is consistent with Table 1. Model Parameters for ARPC-SAFT σ(Å) 2.796

ε/kb (K) 218.90

εAB /kb (K) 1795.27

θc 38.70

◦

rc (Å)

xp

3.71

0.641

the location of the first maximum of the oxygen−oxygen pair correlation function in liquid water which was measured by neutron diffraction26 to be at a diameter 2.75 Å. The square well energy ε/kb is consistent with the value obtained through molecular simulation (ε/kb = 220 K)33 by adjusting the simulation force field to reproduce the solubility of water in hydrocarbon. The hydrogen bond energy is close to the liquid phase hydrogen bond energy measured by Luck34 of 1862 kb. The critical radius for hydrogen bonding rc is close to the first minima in the oxygen−oxygen pair correlation in liquid water, which was measured26 to be at a distance of 3.5 Å. Table 2 shows the average absolute deviations (AAD) between model and experimental results for ARPC-SAFT, the recent associated reference perturbation theory (APT) of Marshall,18 which employs a rigorous BH2 perturbation theory, and finally a standard polar PC-SAFT model (PPC-SAFT) as described by Fouad et al.27 Table 2. Average Absolute Deviations for Three Equations of State with Vapor Pressure (Psat) and Saturated Liquid Density (ρL) in the Temperature Range 273.15 K < T < 580 K method

AAD% ρL

AAD% Psat

ARPC-SAFT APT18 PPC-SAFT27

1.05 0.3 4.7

1.98 0.82 1.1 F


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Industrial & Engineering Chemistry Research

Figure 3. (Left) Full T−ρ diagram for pure water. (Right) Low-temperature liquid branch. Red circles are experimental data,35 and curves are model results: PPC-SAFT27, long dashed green; APT, short dashed black; ARPC-SAFT, solid blue.

Figure 4. (Left) Reference integrals I(w) (eq 28, solid curve) and I1 (eq 10, dashed curve) versus temperature for a saturated liquid. (Right) 1 Corresponding fraction of fully hydrogen-bonded water molecules calculated through eqs 30 and 35.

hydrogen bonding structure predictions which are consistent11 with Luck’s data,27 as well as the more recent neutron diffraction (1999) data of Soper et al.26 Both SPC/E40 and TIP4P/200527,40 have been shown to predict stronger hydrogen bonding than Luck’s data. Both the SAFT-VR Mie EoS40 and PPC-SAFT27 have been shown to predict the degree of hydrogen bonding consistent with TIP4P/2005. Dufal et al.40 invoked the earlier neutron diffraction data of Soper and Phillips,41 who estimated that, on average, each water molecule in the liquid phase at ambient conditions participated in ∼3.8 hydrogen bonds. However, the more recent neutron diffraction results of Soper et al.26 obtained an updated estimate that each water molecule participated in 3.58 hydrogen bonds (on average). It was recently shown by Marshall11 that the updated results of Soper et al.26 are consistent with both Lucks data as well as i-AMOEBA. At the end of the day, hydrogen bonding in both molecular simulation and neutron diffraction data is defined in an ad hoc manner. A proper comparison between various results relies on a consistent definition of what it means to be hydrogen bonded. This point is missed in previous comparisons of SAFT type theory predictions of hydrogen bonding fractions to molecular simulation data.27,40 In the current work, there is a slight discrepancy between the potential parameters θc and rc used to define a hydrogen bond in ARPC-SAFT, and those used to define a hydrogen bond in the simulations. The simulations define a hydrogen bond to form when two water molecules are both oriented such than θ < 30◦ and are positioned such that r < 3.5 Å27, while the regressed parameters in ARPC-SAFT define a hydrogen bond to form when θ < 38.7° and are positioned such that r < 3.71 Å (Table 1). However, given the similarities between these two hydrogen bond definitions, the qualitative comparison between ARPC-SAFT and simulation is justified.

Figure 5. Comparison of model predictions (curves with same meaning as Figure 3) to Luck’s data34 (circles) for the fraction of free OH groups in saturated liquid water

Figure 6. Comparison of ARPC-SAFT predictions (curves) to molecular simulations27 using the i-AMOEBA39 force field (symbols) for the fraction of water molecules hydrogen bonded k times in a saturated liquid.

simulation data, very accurately predicting the temperature dependence of the various bonding fractions. i-AMOEBA is a polarizable water model which has been shown to give G


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We conclude this section with a comparison between model and experiment for the second derivative properties isothermal compressibility κ and the isobaric heat capacity Cp. Figure 9 compares model results for ARPC-SAFT and PPC-SAFT to experimental data for these quantities. While neither approach yields quantitatively accurate results, ARPC-SAFT is qualitatively correct in that it predicts the anomalous minimum in each of these quantities. On the contrary, PPC-SAFT does not exhibit these minima, instead showing a monotonic decrease for both κ and Cp with decreasing temperature. Table 3 compares the experimental temperatures at which the minima occurs to the ARPC-SAFT predictions.

Figure 7 compares model results (APT, ARPC-SAFT, PPCSAFT) to experimental data for the heat of vaporization ΔHvap.

Table 3. Comparison of ARPC-SAFT Predictions to Data for the Location of Temperature Minima in the Isothermal Compressibility and Isobaric Heat Capacity

Figure 7. Comparison of model predictions (curves with same meaning as Figure 3) to experimental data42 (circles) for the heat of vaporization of pure water.

κ CP

As can be seen, each equation of state gives a good representation of the data far from the critical point. Near the critical point, APT and ARPC-SAFT are more accurate than PPC-SAFT because of their improved representation of liquid densities. Now the second virial coefficient B is considered. Figure 8 compares model predictions to NIST correlation data for B.

Tmin(data) (K)

Tmin(ARPC-SAFT) (K)

319 308

290 364

In this section, we have demonstrated that ARPC-SAFT allows for a consistently accurate description of water properties. ARPC-SAFT gives a substantial improvement over PPC-SAFT; however, APT does allow for a more accurate representation of liquid density and vapor pressure. The entire purpose of developing ARPC-SAFT is to extend the concept of associated reference perturbation theory to multicomponent mixtures. We demonstrate this feature of ARPC-SAFT in section IV.

IV. WATER−HYDROCARBON MIXTURES In this section, we validate the model for use in multicomponent noncross-associating mixtures. A stringent test of any EoS is the prediction of the mutual solubilities in waterhydrocarbon liquid−liquid equilibria (LLE). Figure 10 compares ARPC-SAFT predictions (kij = 0) versus experimental data for the mutual solubilities of water with several nonaromatic hydrocarbons. Note, hexane LLE data was included in the development of the water parameters; however, results for all other hydrocarbons are predictions. Overall, the theory is in good agreement for both hydrocarbon solubility in water as well as water solubility in hydrocarbon. The theory does not predict the hydrocarbon solubility minima in the aqueous phase. The solubility minima of hydrocarbons in water results from the structural transition of water to tetrahedral symmetry. It is disappointing that this phenomenon is not reproduced here.

Figure 8. Model predictions (dashed curve, PPC-SAFT; solid curve, ARPC-SAFT) compared to NIST correlation43 (circles) for the second virial coefficient of water.

ARPC-SAFT gives a good representation of B, substantially outperforming PPC-SAFT. The improved performance of ARPC-SAFT is the result of the physically justified parameters (Table 1) used in the model.

Figure 9. Comparison of model predictions (dashed curve, PPC-SAFT; solid curve, ARPC-SAFT) to data (circles) for the isothermal compressibility44 (left) and isobaric heat capacity35 (right) of liquid water. H


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Figure 10. Comparison of ARPC-SAFT model predictions (curves) to experimental (symbols) data for the mutual solubilities of water with hexane45 (top left), decane46 (top right), cyclohexane45 (bottom left), and ethylcyclohexane47 (bottom right).

V. SUMMARY AND FUTURE WORK In this work, we incorporated the recently developed associated reference perturbation theory (APT) of Marshall18 into a generalized associated reference polar PC-SAFT EoS (ARPCSAFT). ARPC-SAFT modifies the standard PC-SAFT attractive BH2 free energy by accounting for the relative decrease in reference system coordination number which accompanies water becoming tetrahedrally coordinated at ambient conditions. Both APT and ARPC-PCSAFT where shown to give substantial improvements in the representation of water as compared to the standard polar PC-SAFT (PPCSAFT) EoS. This improvement results from the explicit treatment of the structural transition to tetrahedral symmetry of water at ambient temperatures in the BH2 reference integrals. Comparing APT and ARPC-SAFT, APT gives an improved description of water vapor pressures and liquid densities, although ARPC-SAFT gives a better representation of hydrogen bonding structure. ARPC-SAFT was shown to predict (qualitatively) the minima in both the isobaric heat capacity as well as volume expansivity, but (at least for this choice of parameters) does not reproduce the density maximum. The accuracy of the approach for the description of mixtures containing water and hydrocarbon molecules was demonstrated by the prediction of water/hydrocarbon mutual solubilities in Figure 10 as well as the binary vapor−liquid equilibria of a methane-water mixture in Figure 11. While overall agreement was very good, ARPC-SAFT did not predict the hydrocarbon solubility minima in the aqueous phase. This points to a shortcoming of this approach. Of all the free energy contributions in eq 1, the only term which “feels” the structural transition to tetrahedral symmetry is the PC-SAFT attractive contribution aat. This lack of feed back to the hard chain contribution ahc means the incorporation of the structural transition is incomplete. We plan to address this deficiency in future work.

The shortcoming is likely the result of the fact that in ARPCSAFT, we only correct for the tetrahedral symmetry of water in the BH2 reference integrals, and not in the ahc contribution which gives rise to repulsions. Future work will focus on correcting this deficiency. As a further test, Figure 11 compares ARPC-SAFT predictions to experimental data for the composition of

Figure 11. ARPC-SAFT predictions (curves) for the vapor-phase composition at the dew point in a binary mixture of water and methane versus experimental data48 (circles) at several temperatures.

dewpoint gas versus pressure for several isotherms in a water/methane binary mixture. Again, the theoretical calculations are performed without tuning to binary data (kij = 0). As can be seen, ARPC-SAFT accurately predicts the composition of dew point gas as a function of temperature and pressure. The results in Figures 10 and 11 demonstrate the accuracy of ARPC-SAFT for the prediction of multicomponent phase equilibria. This validates the mixing rules proposed in eqs 31 and 32. I


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(8) Barker, J. A.; Henderson, D. Perturbation Theory and Equation of State for Fluids: The Square-Well Potential. J. Chem. Phys. 1967, 47, 2856. (9) Ojamaee, L.; Hermansson, K. Ab Initio Study of Cooperativity in Water Chains: Binding Energies and Anharmonic Frequencies. J. Phys. Chem. 1994, 98, 4271. (10) Bakó, I.; Mayer, I. Hierarchy of the Collective Effects in Water Clusters. J. Phys. Chem. A 2016, 120, 631. (11) Marshall, B. D. A second order thermodynamic perturbation theory for hydrogen bond cooperativity in water. J. Chem. Phys. 2017, 146, 174104. (12) Marshall, B. D. On the cooperativity of association and reference energy scales in thermodynamic perturbation theory. J. Chem. Phys. 2016, 145, 204104. (13) Errington, J. R.; Debenedetti, P. G. Relationship between structural order and the anomalies of liquid water. Nature 2001, 409, 318. (14) Debenedetti, P. G. Supercooled and glassy water. J. Phys.: Condens. Matter 2003, 15, R1669. (15) Remsing, R. C.; Rodgers, J. M.; Weeks, J. D. Deconstructing Classical Water Models at Interfaces and in Bulk. J. Stat. Phys. 2011, 145, 313−334. (16) Berendsen, H. J.; Grigera, J. R.; Straatsma, T. P. The missing term in effective pair potentials. J. Phys. Chem. 1987, 91, 6269−6271. (17) Chen, Y.-g.; Kaur, c.; Weeks, J. d. Connecting Systems with Short and Long Ranged Interactions: Local Molecular Field Theory for Ionic Fluids. J. Phys. Chem. B 2004, 108, 19874−19884. (18) Marshall, B. D. Perturbation theory for water with an associating reference fluid. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2017, 96, 52602. (19) Jackson, G.; Chapman, W. G.; Gubbins, K. E. Phase equilibria of associating fluids. Mol. Phys. 1988, 65, 1−31. (20) Wertheim, M. S. Thermodynamic perturbation theory of polymerization. J. Chem. Phys. 1987, 87, 7323. (21) von Solms, N.; Michelsen, M. L.; Kontogeorgis, G. M. Computational and Physical Performance of a Modified PC-SAFT Equation of State for Highly Asymmetric and Associating Mixtures. Ind. Eng. Chem. Res. 2003, 42, 1098−1105. (22) Carnahan, N. F.; Starling, K. E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys. 1969, 51, 635. (23) Jog, P. K.; Sauer, S. G.; Blaesing, J.; Chapman, W. G. Application of Dipolar Chain Theory to the Phase Behavior of Polar Fluids and Mixtures. Ind. Eng. Chem. Res. 2001, 40, 4641−4648. (24) Chapman, W. G. Ph.D. Dissertation, Cornell University, 1988. (25) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Phase equilibria of associating fluids. Mol. Phys. 1988, 65, 1057−1079. (26) Soper, A. K.; Bruni, F.; Ricci, M. A. Site−site pair correlation functions of water from 25 to 400 °C: Revised analysis of new and old diffraction data. J. Chem. Phys. 1998, 106, 247. (27) Fouad, W. A.; Wang, L.; Haghmoradi, A.; Asthagiri, D.; Chapman, W. G. Understanding the Thermodynamics of Hydrogen Bonding in Alcohol-Containing Mixtures: Cross-Association. J. Phys. Chem. B 2016, 120, 3388−3402. (28) Overduin, S. D.; Patey, G. N. Understanding the Structure Factor and Isothermal Compressibility of Ambient Water in Terms of Local Structural Environments. J. Phys. Chem. B 2012, 116, 12014− 12020. (29) Chang, J.; Sandler, S. I. A completely analytic perturbation theory for the square-well fluid of variable well width. Mol. Phys. 1994, 81, 745−765. (30) Chang, J.; Sandler, S. I. A real function representation for the structure of the hard-sphere fluid. Mol. Phys. 1994, 81, 735−744. (31) Ghonasgi, D.; Chapman, W. G. Theory and simulation for associating fluids with four bonding sites. Mol. Phys. 1993, 79, 291− 311. (32) Liang, X.; Tsivintzelis, I.; Kontogeorgis, G. M. Modeling Water Containing Systems with the Simplified PC-SAFT and CPA Equations of State. Ind. Eng. Chem. Res. 2014, 53, 14493−14507.

In this work, attention was focused on the model development and validation for pure water properties as well as mixtures of water with nonassociating mixtures. The combining rule eq 36 was proposed to allow for ARPC-SAFT to reduce to a standard treatment in the absence of water. However, this relationship (and ARPC-SAFT in general) need to be validated for cross associating mixtures. This author will not pursue this further. It is hoped that other researchers will validate ARPC-SAFT for cross-associating mixtures. Given the good simultaneous description of water density, hydrogenbonding structure, and mixture phase equilibria with hydrocarbon molecules, it is expected that ARPC-SAFT will perform well for cross-associating mixtures.

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APPENDIX

Correlation of Ihs(rc)

Chang and Sandler provide a general solution of the integral over the hard-sphere correlation function Ihs(rc) within the Percus−Yevick approximation.29,30 Ihs(a) =

∫1

a / σw

ghs(x)x 2 dx ;

x = r /σw

(A1)

In the general formulation of ARPC-SAFT, the value of this integral is needed only at the critical radius for association rc. For the value of rc and σw listed in Table 1, this integral can be correlated with the following polynomial in packing fraction Ihs(rc) = − 6.008621η 4 + 3.339672η3 − 0.077981η2 + 0.823879η + 0.445403

(A2)

For the parameters given in Table 1, eq A2 allows for a simpler and computationally faster implementation of ARPCSAFT.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Bennett D. Marshall: 0000-0002-3079-5946 Notes

The author declares no competing financial interest.

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REFERENCES

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K


Mixture Equation of State for Water with an Associating Reference Fluid

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