110th Anniversary: From Solubility Parameters to Predictive Equation

Article pubs.acs.org/IECR

Cite This: Ind. Eng. Chem. Res. 2019, 58, 12787−12800

110th Anniversary: From Solubility Parameters to Predictive Equation-of-State Modeling Costas Panayiotou,*,† Ioannis Zuburtikudis,‡ and Vassily Hatzimanikatis§,∥ †

Department of Chemical Engineering, University of Thessaloniki, 54124 Thessaloniki, Greece Department of Chemical Engineering, Abu Dhabi University, Abu Dhabi, United Arab Emirates § Laboratory of Computational Systems Biotechnology (LCSB), Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland ∥ Swiss Institute of Bioinformatics (SIB), CH-1015 Lausanne, Switzerland Downloaded via MOUNT ALLISON UNIV on August 3, 2019 at 19:28:21 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: A simple predictive molecular thermodynamic model for bulk phases and interfaces is presented. The model combines features of the quantitative structure−property relationships (QSPR) approach, the equation-of-state approach, and quantum-chemical calculations. Simple formalisms on a sound thermodynamic basis have been developed for the main thermodynamic quantities of pure components and mixtures while the required parameters for the calculations are rather easily obtained from common databases in the public domain. The widely available solubility parameters or the linear solvation energy relationships (LSER) molecular descriptors are exploited for the extraction of the required information. The predictive capacity of the model with a variety of properties and types of systems including highly nonideal hydrogen-bonded mixtures is reviewed. Emphasis is given in this work on predictions of solvation free energies and the results are rather satisfactory. The strength and weakness of the model and its perspectives are also critically discussed.

1. INTRODUCTION

One of the most popular QSPR technique is the Solubility Parameter method,6−9 especially, Hansen’s implementation,8,9 with three molecular descriptors, known as the partial or Hansen Solubility Parameters (HSP). These three HSPs, δd, δp, and δhb, reflect the dispersion, the dipolar, and the hydrogen-bonding intermolecular cohesive interactions, respectively, of each compound. The affinity or the extent to which compound i likes or dislikes compound j is estimated through the radius of solubility, Rij, defined as9

In our era of quantum computers, of the Internet-of-things, of mobile computing, and of the ever increasing computing capacity of pocket devices such as cellular phones, molecular thermodynamics appears to capture the opportunity to exhibit its strength and capacity. Complex calculations and even predictions of a variety of thermodynamic properties are now possible even with the omnipresent mobile phones and similar devices. Such devices may certainly be used in coherent approaches with simple or analytical formalisms but with sound thermodynamic basis. On the other hand, there is an ever increasing need today for fast screening tools and versatile and reliable predictive tools for thermodynamic/ thermophysical properties in various fields and applications. In this respect, particularly useful are modern Quantitative Structure−Property Relationships (QSPR) with a minimum of molecular descriptors and a predictive capacity for a variety of important properties in a multitude of areas of practical and academic interest.1−5 Their descriptors may be experimentally determined properties or theoretically/quantum-chemically derived quantities. This latter alternative is very important in today’s design of new processes and products, such as drugs, cosmetics, coatings, and foodstuffs. © 2019 American Chemical Society

{

R ij 2 = 4 (δdi − δdj)2 + +

1 (δ hbi − δ hbj)2 4

1 (δpi − δpj)2 4

}

(1)

The smaller the radius of solubility the more likely is compound i to be miscible with compound j. There are already available reliable predictive schemes for the HSP descriptors.10,11 Received: Revised: Accepted: Published: 12787

May 29, 2019 June 20, 2019 June 21, 2019 June 21, 2019 DOI: 10.1021/acs.iecr.9b02908 Ind. Eng. Chem. Res. 2019, 58, 12787−12800

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Industrial & Engineering Chemistry Research

significantly broadens the range of applications as well as the range of external conditions. In this direction, a first PSPbased equation-of-state framework was developed and tested against phase equilibrium data.28 These ideas were further developed recently31−35 and the resulting approach was tested against experimental activity coefficients for a variety of solutes including high polymers, copolymers, and polymer mixtures,32 drugs,33 ionic liquids,34 and materials for the conservation of works of art.35 This equation-of-state dimension drastically differentiates the PSP approach from all other widely used QSPR-type approaches. The purpose of the present work is a coherent presentation of the PSP approach, the recent developments, and the prospects. First, the PSP definitions will be recalled along with the way they may act as reference concepts for bridging diverse QSPR approaches. Subsequently, the QSPR ideas will be grafted on a most simple equation-of-state model, namely, the Lattice Fluid with Hydrogen Bonding (LFHB) model.41−44 The care to be exercised in determining the LFHB scaling constants will be emphasized. For convenience, the essentials of LFHB are presented in the Supporting Information along with a simple molecular association approach to correct its calculations near the critical point. Examples of applications will be reviewed for materials characterization and for bulk-phase and interface calculations. The predictive capacity of the model against solvation free energies will be tested. Simplicity with a sound thermodynamic basis will be seen throughout as a key feature of the PSP approach. The advantages, disadvantages, and perspectives will be critically discussed.

With HSPs, the enthalpy of mixing in a binary mixture is given by eq 2: ΔH = x1Vm1ϕ2H12 = x1Vm1ϕ2{(δd1 − δd2)2 + (δp1 − δp2)2 + (δ hb1 − δ hb2)2 }

(2)

In eq 2, xi, ϕi, and Vmi are the mole fraction, the volume fraction, and the molar volume, respectively, of component i. It is clear that eq 2 cannot account for negative heats of mixing, which are often encountered in mixtures of practical interest. Both of the above equations imply that favorable mixing occurs when the solute and solvent have similar solubility parameter components, even the hydrogen-bonding ones (“similarity” principle of solubility). The HSP method does not account for the highly important “complementarity” principle of solubility since it does not distinguish acidic from basic components or proton donor from proton acceptor (hydrogen bonding) molecular sites. Another most widely used QSPR method is Abraham’s QSPR-type model,5,12−15 known alternatively as the Linear Solvation Energy Relationships (LSER) model. Abraham’s model has found successful applications in a remarkably large number of cases involving primarily one or more infinitely dilute phases.12−15 It is particularly attractive due to the free online access to the extensive LSER database.15 In this QSPR/ LSER approach, five molecular descriptors, Vx, E, S, A, and B, are used for each compound standing for the McGowan volume, the excess molar refraction or electron polarizability, the dipolarity/polarizability, the overall hydrogen bond acidity, and the hydrogen-bond basicity, respectively, of the solute.5,12−15 Successful predictions for generalized solute properties, SP, in a given solvent are made through the classical empirical equation: SP = c + eE + sS + aA + bB + vVx

2. DEFINITIONS AND THE PSP FRAMEWORK In this section we will, first, recall the definitions of Partial Solvation Parameters (PSP). For the clarity of presentation, we will focus on the recent definitions based on the LSER molecular descriptors.5,12−15 The PSPs are given the dimensions of solubility parameters and are expressed in the same units ( MPa ). In doing so, the PSP approach retains the simplicity of the solubility parameter approach but drastically augments its capacity for thermodynamic calculations. With this capacity, as will be seen, PSPs may bridge and thermodynamically turn rather underutilized QSPR databases into efficient predictive tools in a variety of applications. Though approximate but quite convenient, we will adopt here the common division of intermolecular interactions into weak dispersion interactions, intermediate (di)polar interactions, and strong specific interactions such as Lewis acid/ base or hydrogen-bonding interactions. In the LSER approach,5,12−15 the dispersion interactions are, primarily, expressed by McGowan volume, Vx, and excess refractivity, E. Thus, the dispersion PSP, σd, maps Vx and E LSER descriptors of the compound through the following defining equation:31−34

(3)

The set of coefficients, c, e, s, a, b, and v, are the corresponding solvent properties. There are already available various quantum-mechanics-based calculation schemes for the LSER molecular descriptors.16−18 From their very structure, there is no obvious interconnection between HSP and LSER methods. Various other QSPR type approaches are widely used in the literature, especially databases with acidity/basicity scales,4,19−23 providing useful and rather underexploited information on intermolecular interactions. In a series of recent papers, Panayiotou et al.24−35 have redefined solubility parameters and introduced Partial Solvation Parameters (PSP) in an effort to free solubility parameters from the above-mentioned handicaps and unify the diverse acidity/basicity scales and widely used QSPR descriptors in the open literature. Initially,24−28 PSPs were heavily based on quantum-chemical calculations and the related COSMO-RS model of solutions.36−40 More recently, however, PSPs were primarily based on LSER molecular descriptors.5,12−15 Due to their sound thermodynamic basis, PSPs facilitate the safe exchange of information between the various QSPR approaches. This forms, in turn, a sound basis for the development of a predictive thermodynamic model for bulk phases and interfaces by exploiting information readily available in the open literature. Of particular interest, however, is the prospect of turning PSP into a predictive QSPR/equation-of-state model. This

σd = 100

3.1Vx + E 3Vx + E ∼ 100 Vm Vm

(dispersion PSP)

(4)

Vm in this equation is the molar volume of the compound. The McGowan volume may be considered equivalent to the van der Waals volume.45,46 In fact, there is a good linear 12788

DOI: 10.1021/acs.iecr.9b02908 Ind. Eng. Chem. Res. 2019, 58, 12787−12800

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hydrogen-bonding contribution to the cohesive energy density, cedHB, of pure compounds, given by28

relation between the two, as shown in Figure 1. The excess refractivity, E, represents all the rest of nonpolar interactions

cedHB = −

nHBE HB Vm

(10)

nHB in eq 10 is the number of hydrogen bonds per mole, and EHB is the energy change upon formation of 1 mole of hydrogen bonds. This, in turn, is related to the corresponding free energy change, GHB, at any temperature, T, by the classical equation: G HB = E HB − TSHB

(11)

where SHB is the corresponding entropy change. The following equations are good approximations for these hydrogen bonding quantities:33−35

Figure 1. McGowan volume as a function of the van der Waals volume.46 The straight line through the data is given by the equation: Vx = 0.0187 + 13.831VvdW with R2 = 0.9986.

E HB = −30450 AB

(12)

SHB = −35.1 AB

(13)

and for the free energy change at any temperature T: G HB = −(30450 − 35.1T ) AB

The PSP approach was designed to be expanded in an LFHB-type equation-of-state model41−44 and, thus, the same hydrogen bonding statistics (Veytsman statistics41,47) will be used. For convenience, the essentials of the LFHB model are summarized in the Supporting Information. In this frame, the number of hydrogen bonds per mole, nHB, in eq 10 is obtained from the following equation:33−35,41,47

not accounted for by Vx. It is intimately related to the dispersion HSP, δd, through the following equation:30−32 Vmδd 2 − 3.1Vx (5) 10000 In fact, the factor 3.1 in the defining eq 4 is preferred over the factor 3 in order to strengthen the bridging character of eq 5 between LSER and HSP descriptors. Similarly, the equivalent to δp polarity PSP, σp, maps the polarity, S, descriptor5,12−15 of the compound through the following defining equation:30−35 E=

nHB =

Vmδp S or S = Vm 10000

AH =

(polarity PSP) (6)

A Vm

B Vm

(16)

(17)

In fact, the QSPR/LSER approach does not account for self-association in homosolvated compounds (that is, compounds possessing both acidity and basicity character), and thus, the original A and B descriptors5,12−15 must be corrected for self-association prior to their use for thermodynamic calculations. The above set of eqs 10−17 not only bridges HSPs and LSER descriptors but it also leads to thermodynamically consistent A and B descriptors. Consistent LSER descriptors for a number of common compounds are reported in Table SI2-2.

(acidity PSP) (7)

(basicity PSP) (8)

Subscript G in eqs 7 and 8 indicates that these are Gibbs freeenergy descriptors. Thus, the free energy change upon hydrogen bond formation is30−35 G HB,298 = −2σGaσGbVm = −20000 AB

Vm exp(G HB /RT ) 9.75

cedHB = δt 2 − σd 2 − σp2 = δ hb 2

Similarly, the basicity or hydrogen bonding acceptor PSP, σGb, maps the basicity LSER descriptor, B, through the following defining equation:30−35 σGb = 100

(15)

The above set of eqs 10−16 may bridge the LSER A and B descriptors with Hansen’s δHB through the total solubility parameter, δt, and the following equation:31

It should be made clear that the above equivalence of PSPs and HSPs is a proposed one, not yet an identity. The rest of PSPs have no equivalents in HSPs. The acidity or hydrogen bonding donor PSP, σGa, maps the acidity LSER descriptor, A, and is defined as follows:30−35 σGa = 100

AH + 2 − AH(AH + 4) NH = N 2

where

2

σp = 100

(14)

3. PSPs AND MIXTURE THERMODYNAMICS In this section, we will summarize the working equations for calculations of mixture properties in bulk phases and interfaces. Details may be found in the original references.30−35 The focus will be on binary mixtures. Extension to multicomponent ones is quite straightforward. 3.1. Concentrated Solutions. Let us consider a binary mixture of N1 moles of component 1 with N2 moles of component 2 at temperature T and at a relatively low pressure

(9)

As mentioned already, HSP and LSER are drastically different approaches to hydrogen bonding. Yet, the PSP framework may bring these diverse approaches into a common denominator. As an example, it can do this by requiring from both to estimate in a consistent manner the 12789



ij 3.1V + E x,1 1 ΔEd = 100x1Vx,1ϕ2jjjj − j Vx,1 k

p. Each molecule of type i consists of ri identical segments and has zqi external contacts, z being a coordination number set here equal to 10. Besides mole fraction, xi = Ni/N, the composition of the mixture may be expressed by the following alternative fractions: rN rN i i ϕi = = i i, r1N1 + r2N2 rN

θi =

q1N1 + q2N2

=

qiNi qN

and

ji S1 ΔEp = 100x1Vx,1ϕ2jjjj − j Vx,1 k

i = 1, 2 (volume fraction)

(19)

ln γ1,d =

The excess (Gibbs) free energy, GEm, which is the key quantity of our interest, may be considered subdivided in a combinatorial (superscript c) and a residual part (superscript r) as follows: GmE ΔGmc + ΔGmr = RT RT

ij θ yzzy ji i ϕ y z ∑ xijjjjjlnjjjjj i zzzzz + qi lnjjjj i zzzzzzzzz x 2 i=1 k k i { k ϕi {{

(21)

ln γ1d =

yz zz zz z {

2

S2 Vx,2

yz zz zz z {

2

(30)

Vm,1ϕ2 2 RT

(σd1 − σd2)2

(31)

(σp1 − σp2)2

(32)

and

(22)

ln γ1p =

Vm,1ϕ2 2 RT

Regarding hydrogen-bonding contribution to the excess free energy and in the general case of a system with m types of donors and n types of acceptors, we may write for the average per segment number, νH, of hydrogen bonds in the system:25,41−44,47

(23)

m

(24)

νH =

n

∑ ∑ ναβ = α

β

m

n

∑∑ α

β

H Nαβ

rN

(33)

In this equation, the summations extend over all α−β donor− acceptor combinations in the system. ναβ values are obtained from equations analogous to eq 15 (cf. Part SI1, Supporting Information). If dki is the number of hydrogen bond donors of type k (k = 1, m) in each molecule of type i (i = 1, 2) and αkj is the number of hydrogen-bond acceptors of type k (k = 1, n) in each molecule of type j, να0, and ν0β is the per-segment number of free (non-hydrogen-bonded) donors and acceptors of type α and β, respectively, the hydrogen bonding contribution to the chemical potential of component i in the mixture is given by41,44,47,51 (cf. Part SI1, Supporting Information):

(25)

and, similarly, for the activity coefficients or the chemical potentials. At low pressures we may further assume that ΔGd ≅ ΔHd ≅ ΔEd

E 3.1 + 2 Vx,2

In terms of PSPs, the above equations become25,30−35

The above-mentioned three types of intermolecular interactions have their distinct types of contributions to the residual part of excess free energy, namely, dispersion, (di)polar, and hydrogen-bonding or (Lewis) acid/base, indicated by subscripts d, p, and hb, respectively. On adopting this division, the residual free energy may be written as follows: ΔGd + ΔGp + ΔG hb ΔGmr = RT RT

RT

100Vx,1ϕ2 2 ijj S1 jj ln γ1,p = = jj V − RT RT k x,1

where z (ri − qi) − (ri − 1) 2

(28)

μ1,d

μ1,p

which, for the activity coefficient, implies the following equation:

li =

2

and

2

ij θ yz ϕj z ji ϕ zy ln γic = lnjjj i zzz + qi lnjjjj i zzzz + (lirj − l jri) j xi z 2 rj k { k ϕi {

zyz zz zz {

(29) (20)

The widely used Guggenheim-Staverman48−50 expression will be adopted for the combinatorial part or ΔGmc = RT

S2 Vx,2

100Vx,1ϕ2 2 ijj jj 3.1 + E1 − = jj RT Vx,1 k

In terms of activity coefficients, eq 20 implies the following relation: ln γi = ln γic + ln γi r

2

From eqs 27and 28, we obtain for the corresponding contributions to the activity coefficient the following equations:

,

i = 1, 2 (surface area fraction)

3.1Vx,2 + E2 yzz zz zz Vx,2 {

(27)

(18)

qiNi

Article

(26)

and, similarly for the dipolar contribution. The above mixing energies may be calculated by using the very definitions of the corresponding PSPs. The key equations in terms of LSER descriptors are as follows:25,30−35

μi ,H RT

m

= ri(νH − νH0) −

∑ dαi ln α=1

να00 − να 0

n

∑ aβi ln β=1

ν00β ν0β (34)

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jijij 1 + AH,12 − nHB,2 yzz r1 0 zz (nHB,2 − nHB,2 ) − d1 lnjjjjjjj 0 0 zz jj 1 + AH,12 r2 − n HB,2 { kk 0 0 y i y i y AH,12 zz j 1 + AH,21 − nHB,2 zz AH,21 zz zz − a1 lnjjjjjjj zz zz z jjjj 0 0 z zz AH,21 zz AH,12 + − 1 A n H,21 HB,2 { { kk { | 0 l o o ji 1 − nHB,1 zyzo o 0 zzo −o − 2 lnjjjj nHB,1 − nHB,1 m } o z o o j 1 − nHB,1 zo o o (37) k { n ~

where the superscript 0 indicates quantities in a fictitious

ln γ1H, ∞ =

system of identical composition but with zero interaction freeenergies. Thus, the overall equation for the residual part of the chemical potential (or the activity coefficient) is μ1,p μ1,d μ1,H 10000Vx,1ϕ2 2 μr ln(γ1r) = 1 = + + = RT RT RT RT RT ÄÅ 2 ÅÅi y ÅÅjj ÅÅjj 3.1 + E1 − 3.1 + E2 zzzz ÅÅjj ÅÅ Vx,1 Vx,2 zz ÅÅÇk { É Ñ 2Ñ ij S S2 yzzz ÑÑÑÑ j 0 1 j + jj − z ÑÑ + r1(νH − νH) j Vx,1 Vx,2 zz ÑÑÑ k { ÑÑÖ m

−

∑ dα1 ln α=1

να00 − να 0

n

∑ a1β ln β=1

(cf. eq 15 for the numbers, nHB, of hydrogen bonds). Among other applications, the formalism of this subsection may be used for the characterization of materials through the correlation of thermodynamic properties at infinite dilution, such as retention volumes by inverse gas chromatography (IGC). Of particular interest is the calculation of solvation free energies, which may be obtained from the activity coefficient by recalling their definition in the mole−mole convention44,53−55 or

ν00β ν0β

(35)

S S ΔG1/2 − ΔG1/1

If desired, PSPs may, of course, be used as HSPs.8,9,25,35 In this case, eq 35 becomes ln(γ1r) =

Vm,1ϕ2 2 4RT

=

[4(σd1 − σd2)2 + (σp1 − σp2)2 ] m

+ r1(νH −

νH0)

−

∑ α=1

dα1

ν0 ln α 0 − να 0

−

n

∑ β=1

aβ1

ln

RT μ1(T , P , x1 → 0) − μ1IG (T , P , x1 → 0) RT μ10 (T , P) − μ10,IG (T , P)

3.2. Infinitely Dilute Systems. Of very much interest in numerous practical applications are expressions for the activity

S ΔG1/1

coefficient at infinite dilution. The total activity coefficient

RT

(21a)

S ΔG1/2

RT

In the case of binary systems, the Guggenheim−Staverman equation

(eq 23) becomes in the limit:

ln γ1c, ∞ = ln

(G−S) (23a)

Alternatively, the classical Flory−Huggins equation r1 r +1− 1 r2 r2

(Flory−Huggins)

P10Vm1 RT

(39)

= ln

P10Vm2 + ln γ1∞ RT

(40)

γhb = 2 γaγb

26,52

(41)

where the acidic and basic surface tension components, γa and γb, respectively, satisfy the equation: γa γ γVES = b = A B 3.1Vx + E + S (42)

becomes in the limit: ln γ1c, ∞ = ln

= ln

This is a most useful equation for the prediction of solvation free energies. 3.3. Surface Energy Components and Wetting Phenomena. The defining equation for the hydrogenbonding contribution to the surface tension of pure components, γhb, is31−35

q r2 r1 z z (q r1 − q1r2) + q1 ln 1 + 2 q2r1 2r2 2 r2

r zy ji + jjj1 − 1 zzz j r2 z{ k

(38)

where P01 is the vapor pressure of the solute at temperature T. Replacing eq 39 in eq 38, we obtain

consists again of combinatorial and residual contributions or

48−50

Vm2 Vm1

Superscript IG in eq 38 indicates the chemical potential of the solute in the ideal-gas limit. However, the self-solvation free energy, ΔG*1/1S , at ambient to moderately high temperatures and pressures is given by the following equation:55

ν0β (35a)

ln γi∞ = ln γic, ∞ + ln γi r, ∞

RT

ν00β

= ln γ1∞ + ln

(36)

The non-hydrogen-bonding component, γVES, of the surface tension, when added to the above hydrogen-bonding component gives the total surface tension, γ, of the compound or

The dispersion and polar contributions are obtained from eqs 29 and 30 or 31 and 32 by setting ϕ2 = 1. The hydrogenbonding contribution may be obtained from eq 34 either

γ = γVES + γhb = γVES + 2 γaγb

numerically (by setting ϕ1 very close to zero) or by the following approximate analytical form:35

(43)

and is obtained from the following equation:31−35 12791


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Industrial & Engineering Chemistry Research γVES = γ

3.1Vx + E + S 3.1Vx + E + S + 2 AB

system of two equations with two unknowns, ε* and vsp* or ρ*, namely, the scaling constants for the interaction energy and hard-core volume (or density), respectively. Thus, when the LSER descriptors are known, one may calculate in a straightforward manner the equation-of-state scaling constants. This has been done already31−34 and will be done again in section 4.2 below. One of the key features of eq 47 is that it permits the use of density or volumetric data for the characterization of compounds and materials. This was valuable for the integral thermodynamic characterization of polymers,32 ionic liquids,34 drugs,33 and dyes.35 It also permits a reasonably safe division of cohesive energy density into hydrogen-bonding and nonhydrogen-bonding components.34 Very accurate volumetric (PVT) data over a broad range of temperature and pressure are now available in the open literature for a large number of compounds and materials. It is important to point out, however, that mean-field type approximate equations of state, such as the LFHB, have limitations and, thus, the interconnection of LSER descriptors and equation-of-state scaling constants should be accounted for with due care. The LSER descriptors are often obtained from properties measured at or near ambient conditions. In contrast, the scaling constants may be obtained from measurements at near ambient but also at far remote conditions. All known and widely used equation-of-state models are approximate and the obtained scaling constants depend, to some extent, on the data used for their determination. When the compound is included in comprehensive databases, such as the DIPPR database,46 which reports various thermodynamic properties over a broad range of external conditions, this should be the first choice for data source. When only PVT data are available, more caution should be exercised with the obtained scaling constants. In order to make this point clear, let us take the simple LFHB equation-of-state model (cf. Part SI1, Supporting Information) and determine the scaling constants for the hydrogen-bonded compound, ethylene glycol. This molecule has two equivalent proton-donor and two proton-acceptor sites (homosolvated compound).31 The scaling constants for ethylene glycol may be obtained in three different ways. First, from eqs 46 and 47 above along with eq 23 of Part SI1 and with Vm data46 at 25 °C. These scaling constants are reported in Table 1 as LSER/Vm/ ambient. Second, from a simultaneous least-squares fit of densities, vapor pressures and heats of vaporization46 nearly up to the critical point (see below and Part SI1). The obtained scaling constants are reported in Table 1 as LSER/ all/DIPPR. However, for ethylene glycol we also have goodquality PVT data56 for a broad range of pressures up to 95 MPa. Thus, in Table 1 are also reported the scaling constants obtained from these PVT data by using either all of them or data up to a certain pressure indicated in Table 1 as max pressure. It is assumed here that the acid/base LSER descriptors (A = 0.56 and B = 0.78) remain constant over the full range of temperatures and pressures. In Table 1 are also reported the estimated Vm, total solubility parameter, δtot, and its non-hydrogen bonding component, δdp, at 25 °C. Several comments are in order regarding these calculations. As shown in Table 1, the LFHB scaling constants depend on the data set used for their determination. However, this variation does not lead to substantial variation of the calculated solubility parameters. It was this invariance that

(44)

Replacing in eq 42, we also obtain γa, γb, and thus γhb. The experimental surface tensions, γ, of many common liquids are available in widely used compilations, such as the DIPPR database.46 A most useful equation for the characterization of solid surfaces is the following one31−35 for the contact angle, θ, of a liquid, L, with a solid surface, S:

{

γL(1 + cos θ ) = 2

γVES, LγVES,S +

γa,Lγb,S +

γa,Sγb,L

}

(45)

From contact angles with known liquids, one may determine with eq 45 the surface energy components of the studied solid surface.

4. APPLICATIONS The formalism developed in the previous section has been used already in a variety of applications including high polymers and polymer blends,32 pharmaceuticals,33 ionic liquids,34 dyes, and materials for the conservation of works of art.35 In this section, we will highlight some of these applications and report some new. In particular, it will be shown how this formalism may be combined with the LFHB equation-of-state approach41−44 (cf. LFHB essentials in Part SI1, Supporting Information) and how useful this combination can be for calculations in various areas of much practical interest. The purpose is to show how widely available QSPR databases may become rich sources of valuable information for extensive thermodynamic calculations and for the development of predictive equation-of-state models that drastically augment their range of applications. An extensive list with LSER molecular descriptors is reported in Table SI2-2. LSER descriptors for additional compounds may be found in the open literature.5,12−15,30−35 It is recalled that the hydrogen bonded compounds in the PSP approach are divided into homosolvated (compounds that may self-associate as well as cross-associate) and heterosolvated (compounds that cannot self-associate; they crossassociate only).25−35 At the outset of applications, it is essential to establish the interconnection between the PSPs or the LSER molecular descriptors and the scaling constants of the LFHB equation-of-state model. This is emphasized in the next subsection, especially, the care to be exercised in obtaining the equation-of-state scaling constants. 4.1. From LSER Descriptors to Equation-of-State Scaling Constants. The cohesive energy density of a pure fluid is, by definition, equal to the ratio of the potential energy over the molar volume. In the LFHB framework (cf. Part SI1 for nomenclature), it is obtained from the following equation: ced =

rρε̃ * − E HBnHB −E HBnHB ε* = ρ 2̃ + Vm 9.75 Vm

= σdp2 + σhb 2

(46)

Combining eqs 4, 6, and 46, we obtain 10000

3.1Vx + E + S ε* = ρ 2̃ = σdp2 Vm 9.75

(47)

This equation, when combined with the LFHB equation of state (eq 4 or 23 in Part SI1, Supporting Information) form a 12792


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Industrial & Engineering Chemistry Research Table 1. Scaling Constants and the Solubility Parameters at 25 °C for Ethylene Glycol Obtained from PVT Data Sets with Varying Maximum Pressure56 max pressure/ MPa

εa/J mol−1

vspa/ cm3mol−1

Vmol /cm3mol−1

δdp/ MPa0.5

δtot/ MPa0.5

PVT Calculations 5713 0.865 56.01a 23.05 34.19 5624 0.8636 55.96 22.97 34.14 5557 0.8625 55.92 22.91 34.10 5475 0.8611 55.88 22.80 34.03 5355 0.8587 55.73 22.60 33.91 5212 0.8556 55.69 22.36 33.77 Vapor Pressure, Heat of Vaporization, Densities (DIPPR) LFHB Calculations LSER/Vm/ 5397 0.8585 55.91 22.44 33.77 ambient LSER/all/ 5484 0.8610 55.91 22.66 33.93 DIPPR 95 50 20 10 2 1

Figure 3. Variation of the non-hydrogen-bonding solubility parameter, δdp, of ethylene glycol as a function of the corresponding * ε* (cf. Table 1). The line through the data is an aid for product, vsp

Estimated molar volumes at 25 °C and 1 atm. Corresponding experimental values:46,9 55.908 and 55.80 cm3 mol−1.

a

the eye. Off of the line is the corresponding δdp from the ε* and vsp* pair from Vm46 and LSER descriptors12−15,31 that were used for the LFHB calculations.

was exploited for the integral thermodynamic characterization of polymers,32 ionic liquids,34 drugs,33 and dyes.35 In contrast, the variation of ε* may be as large as 10%. Thus, the PVT data at far remote conditions may not be useful for the characterization of compounds with simple equations of state such as LFHB. More scaling constants or temperature/ pressure dependent ones may be needed. The above are shown in graphical form in Figures 2−4. As shown in Figures 3 and 4, the calculated solubility parameters

Figure 4. Variation of the estimated total solubility parameter, δtot, ε*, of ethylene glycol as a function of the corresponding product, * ε* (cf. Table 1). The line through the data is an aid for the eye. vsp Off of the line is the corresponding δtot from DIPPR.46

correlated when scaling constants (LSER/all/DIPPR) are determined. Similar is the correlation of densities (not shown). Although the correlations appear satisfactory, we should keep in mind that these equations of state fail near the critical point, and thus, experimental data at around 30 degrees or less below Tc should not be used for obtaining scaling constants. Possible ways of overcoming this problem are discussed in Part SI1 by modeling the well-known longrange volume fluctuations or molecular clustering near the critical point as a cooperative association process with Veytsman statistics.47 Similar approximate calculations can also be done when reliable volumetric or density data are not available but one has access to COSMO-RS type databases.36,40 In that case, the LFHB scaling constants may be predicted via artificial neural networks, as discussed previously.44 It should be pointed out that the selected compound (ethylene glycol) is not a simple one, and the prediction of its thermodynamic behavior is not a

Figure 2. Variation of the interaction energy scaling constant, ε*, as a function of the corresponding specific hard-core volume, vsp*, from the various experimental data sets (cf. Table 1). The numbers near the data indicate the maximum pressure in the data set. The line through the data is an aid for the eye. Off of the line is the corresponding ε* and vsp* pair from Vm46 and LSER descriptors12−15,31 that were used for the LFHB calculations.

from PVT data depart from the corresponding calculations from LSER/Vm data or from the experimental46 values but these differences are rather within the usual experimental uncertainties.9 Apart from the above limitations, the LFHB has some additional limitations, common to all widely used mean-field type equations of state, to be kept in mind. Figures 5 and 6 show how vapor pressures and heats of vaporization are 12793


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Industrial & Engineering Chemistry Research F12 =

F21 =

Figure 6. Experimental46 (symbols) and calculated (line) heats of vaporization of ethylene glycol. The LFHB calculations were done with the LSER/all/DIPPR set of scaling constants (cf. Table 1).

trivial one but the due care in the determination of scaling constants applies to simpler molecules as well. With these precautions, the above extension to a predictive equation-ofstate framework is quite useful as will be shown in the next section. 4.2. Prediction of Solvation Free Energies. With the molar volumes and the LSER descriptors or the LSER/PSPs known, we may follow two alternative routes for the prediction of solvation free energies. The first is through the direct application of eq 40. However, as discussed in the previous section 4.1, with the very same information, we may obtain the scaling constants by combining eq 46 or 47 above with eq 4 or 23 of Part SI1. Once the scaling constants are available, the equation for the solvation free energy of solute 1 by solvent 2, ΔG*1/2S , may be obtained from the LFHB equation-of-state model as shown previously44 or RT

= ln

ω1IG ε* − r1 ln(1 − ρ2̃ ) − 2r1ρ2̃ 12 − d1 ln F12 RT ω1

− a1 ln F21

0 0 AH,12 1 + AH,12 − nHB,2 0 1 + AH,21 − nHB,2 AH,21 0 0 AH,21 1 + AH,21 − nHB,2

(49)

The first term in the rhs of eq 48 arises from the conformational changes of the solute molecule upon transfer from the pure gas state to the infinitely dilute state in solvent 2. The second term reflects the cavitation work needed to accommodate the solute molecule in the solvent. The remaining terms are the charging terms and reflect the intermolecular interactions of the solute with its neighboring * , is obtained solvent molecules. The cross-interaction term, ε12 * = ε1*ε2* . The quantum by the simple Berthelot’s rule: ε12 chemical considerations needed to account for the conformational changes are omitted here for simplicity, and the corresponding term in eq 48 is neglected or it is lumped in the cavitation term by replacing the number of segments, r1, with an effective number: re1 = kr1, k being a cavitation correction parameter near one. In Table 2 are reported the experimental57 and predicted solvation free energies with eq 40 and eq 48 (k = 1) for various solutes in representative solvents. This is part of the complete Table SI2-1 of Supporting Information Part SI2, where more solute/solvent pairs are reported. All parameters needed for these calculations are reported in Part SI2. The equation-of-state approach (eq 48) provides also with estimations of the cavitation and charging components of the solvation free energy. This capacity to account for cavitation contribution leads to somewhat superior predictive capacity compared to the plain LSER/PSP approach (eq 40). As shown in Table 2 and Table SI2-1, the predictive capacity, especially of the equation-of-state approach (eq 48), is rather very good and nearly always the predictions are within or close to the range of the experimental data. Of much interest are the calculations of hydration free energies reported in Table 3. It is known44,58 that the conformational contributions to the hydration free energies are not always negligible, and thus, the latter were calculated here with the equation-of-state approach (eq 48) where corrected cavitation contributions may be accounted for. The solutes in Table 3 are divided in five classes. The first class consists of all types of alkanes whose hydration free energy may be predicted by setting k = 1.335. This is the most hydrophobic class studied here, and the high value of k indicates an augmented cavitation contribution due to severe restructuring of water in order to accommodate these solute molecules. The second class, alkenes, is somewhat less hydrophobic, and the required correction parameter is k = 1.305. Similarly, the third class of alkynes and aromatic hydrocarbons, being even less hydrophobic, requires a correction parameter k = 1.260. The fourth major class consists of relatively low molecular-weight solutes possessing O or N atoms in their molecule and being, thus, able to interact with water via hydrogen-bonding. In their case, the cavitation contribution does not require significant correction and, thus, their hydration free energies may be predicted satisfactorily with k near unity (k = 1.05). The fifth class of amides and dimethyl sulfoxide is a most interesting class of highly hydrophilic solutes requiring a cavitation correction

Figure 5. Experimental46 (symbols) and calculated (line) vapor pressures of ethylene glycol. The LFHB calculations were done with the LSER/all/DIPPR set of scaling constants (cf. Table 1).

*S ΔG1/2

0 1 + AH,12 − nHB,2 AH,12

(48)

where d1 and a1 are the numbers of donors and acceptors in the solute molecule and in our case (cf. eq 37 above), 12794


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Table 2. Experimental57 and Predicted Solvation Free Energies, ΔG1/2 *S , and Their Cavitation, ΔGcav *S , and Charging, ΔGchr *S , Components at 298 K (in kcal/mol) predicted by eq 48 solute

ΔG*cavS

ΔG*chrS

n-hexane acetone chloroform carbon tetrachloride benzene ethanol 1-butanol methyl ethyl ketone n-pentane n-octane cyclohexane pyridine water

10.50 7.35 7.44 8.75 8.20 5.61 8.39 8.71 9.18 13.21 9.20 8.26 2.12

−14.58 −10.22 −11.14 −12.87 −12.26 −8.12 −12.09 −12.16 −12.65 −18.50 −13.49 −12.54 −2.93

benzene acetone chloroform carbon tetrachloride n-propyl acetate ethanol methyl ethyl ketone methyl acetate n-pentane n-octane pyridine water

10.16 8.41 9.16 10.89 12.79 6.56 10.20 8.98 12.08 16.94 9.87 2.04

−14.76 −12.30 −13.52 −15.50 −17.76 −10.01 −14.64 −12.96 −15.23 −22.27 −15.09 −4.22

acetone chloroform carbon tetrachloride benzene ethanol methyl ethyl ketone methyl acetate n-pentane n-octane pyridine water

8.08 8.84 11.30 9.85 6.40 9.89 8.66 12.40 17.71 9.29 1.76

−12.22 −13.69 −15.40 −14.66 −10.92 −14.54 −12.88 −15.13 −22.12 −14.99 −6.05

ethanol acetone chloroform n-propyl acetate benzene 1-propanol 1-butanol methyl ethyl ketone methyl acetate n-pentane n-octane pyridine water

7.37 9.39 10.18 13.99 11.23 9.54 11.49 11.27 9.99 13.25 18.83 10.52 2.81

−12.45 −13.19 −14.07 −18.79 −15.34 −15.12 −17.85 −15.64 −13.81 −15.81 −23.12 −16.12 −7.82

predicted by eq 40 ΔG*1/2S

Solvent: n-Hexane −4.08 −2.87 −3.70 −4.12 −4.06 −2.52 −3.70 −3.45 −3.47 −5.29 −4.29 −4.28 −0.80 Solvent: Benzene −4.60 −3.89 −4.36 −4.61 −4.97 −3.44 −4.43 −3.98 −3.15 −5.33 −5.23 −2.18 Solvent: Acetone −4.14 −4.85 −4.10 −4.81 −4.52 −4.65 −4.22 −2.73 −4.41 −5.70 −4.29 Solvent: Ethanol −5.08 −3.80 −3.89 −4.80 −4.11 −5.58 −6.36 −4.37 −3.82 −2.56 −4.28 −5.59 −5.01

ΔG*1/2S −4.05 −3.20 −3.27 −3.82 −3.97 −2.96 −4.20 −3.82 −3.34 −5.48 −4.23 −4.20 −0.63 −4.55 −3.74 −4.10 −4.33 −4.93 −4.02 −4.36 −3.81 −3.22 −5.40 −5.32 −1.30 −4.15 −4.49 −4.43 −4.40 −4.28 −4.71 −4.19 −3.03 −4.96 −5.25 −3.56 −5.08 −4.04 −3.42 −4.81 −3.60 −6.97 −7.83 −4.57 −4.07 −2.79 −4.48 −5.07 −5.24

experimental ΔG1/2 *S from

−4.12 −2.17 −3.39 −3.28 −5.41 −4.23 −0.66

−3.74 −4.19 −4.41 −3.16

−5.19

−4.59 −4.11 −4.36

−2.7 −4.37

−4.06 −3.69

−4.25 −2.48 −4.23 −5.05

to −4.05 −2.68 −3.67 −4.15 −4.05 −2.61 −3.75 −3.48 −3.36 −5.45 −4.26 −4.19 −0.82 −4.56 −3.85 −4.2 −5.15 −3.42 −4.46 −3.88 −3.14 −5.35 −5.25 −2.19 −4.14 −4.79 −4.39 −4.42 −4.65 −4.16 −2.78 −4.44 −5.4 −4.34 −5.08 −3.72 −4.08 −4.75 −4.03 −5.61 −6.35 −4.33 −3.85 −2.59 −4.32 −5.65 −5.14

Doubtless, eq 48 provides a rather too simple or crude account for the cavitation contribution to hydration free energy. Yet, the above values of the correction parameter, k, are rather meaningful and may form the basis for the

parameter below unity (k = 0.75). Apparently, these solutes not only interact with water via hydrogen bonding but they also allow for an energetically favorable restructuring of neighboring water molecules. 12795


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Table 3. Experimental57 and Predicted Hydration Free Energies, ΔG1/2 *S , and Their Cavitation, ΔGcav *S , and Charging, ΔGchr *S , Components at 298 K (in kcal/mol) experimental ΔG1/2 *S

predicted by eq 48 solute propane n-butane n-pentane n-hexane n-heptane n-octane 2,2,4-trimethylpentane isopentane neopentane cyclohexane cyclooctane 1-butene isobutene 1-pentene 1-hexene 1-heptene 1-octene 1-pentyne 1-hexyne 1-heptyne benzene toluene p-xylene acetone methyl ethyl ketone 2-pentanone 1-propanal 1-butanal methanol ethanol 1-propanol 1-butanol 1-pentanol isopropanol 2-butanol ethyl acetate n-propyl acetate methylamine ethylamine n-propylamine n-butylamine pyridine formamide acetamide dimethyl sulfoxide

ΔG*cavS

ΔG*chrS

ΔG*1/2S

Linear-, Branched-, and Cycloalkanes, k = 1.335 14.53288 −12.5808 1.95 16.76819 −14.6786 2.09 19.71855 −17.3396 2.38 22.78635 −20.3907 2.40 25.74195 −23.1065 2.64 28.77373 −25.8771 2.90 28.93511 −26.0353 2.90 20.14578 −17.7638 2.38 20.75266 −17.9133 2.84 19.76544 −18.871 0.89 24.89146 −24.0352 0.86 Alkenes, k = 1.305 ± 0.005 15.86164 −14.4979 1.36 15.91556 −14.6438 1.27 18.86155 −17.2134 1.65 21.55647 −19.8809 1.68 24.53015 −22.6647 1.87 27.57663 −25.4104 2.17 Alkynes, Aromatic Hydrocarbons, k = 1.26 ± 0.01 16.66526 −16.6344 0.03 19.71185 −19.4093 0.30 22.76217 −22.144 0.62 15.91883 −17.1487 −1.23 19.12889 −19.9619 −0.83 22.14776 −22.8607 −0.71 Low MW Ketones, Aldehydes, Alcohols, Esters, Amines, k = 1.05 ± 0.07 11.0703 −14.9223 −3.85 14.06622 −17.6805 −3.61 17.07002 −20.3815 −3.31 11.36102 −14.6194 −3.26 14.40111 −17.4374 −3.04 5.687065 −10.7827 −5.10 8.971159 −14.0356 −5.06 12.12524 −17.0614 −4.94 14.08913 −18.7844 −4.70 16.63953 −21.2023 −4.56 12.07132 −17.0657 −4.99 15.01772 −19.6633 −4.65 14.83027 −17.721 −2.89 18.22817 −21.1433 −2.92 6.362829 −10.9233 −4.56 9.560575 −14.0928 −4.53 12.27566 −16.7033 −4.43 15.37156 −19.6717 −4.30 13.13674 −17.84 −4.70 Amides, Sulfoxides, k = 0.75 4.728528 −15.31 −10.58 6.752343 −16.4616 −9.71 8.25435 −17.5728 −9.32

development of more refined predictive schemes. As an example, higher molecular weight compounds of the third class in Table 3 with longer hydrocarbon tails are expected to require higher values for k, and in the limit of very long tails, the k values should approach the corresponding hydrocarbon k values. These values may be handled by a plain groupcontribution predictive scheme.

from 1.96 2.07 0.71 0.97 0.94 2.22 2.87 2.36 2.95 0.87 0.84

to 1.98 2.15 2.34 2.57 2.78 3.14 2.38 2.68 1.26

1.366 1.266 1.653 1.674 2.156 0.004 0.304 0.623 −0.741 −0.7 −0.53 −3.73 −3.51 −3.18 −3.15 −2.89 −4.91 −4.8 −4.79 −4.49 −4.49 −4.72 −4.57 −2.75 −2.74 −4.57 −4.51 −4.40 −4.30 −4.63

−1.31 −0.88 −0.82 −4.25 −3.64 −3.39 −3.29 −3.10 −5.14 −5.14 −5.08 −4.83 −4.63 −5.09 −4.66 −3.13 −2.94

−4.75

−10.15 −9.71 −9.32

The parameters needed for the calculations in Tables 2 and 3 are reported in Part SI2. Once again, it should be emphasized that the predictions in Table 2 and 3 were made with scaling constants based on molar volumes and LSER descriptors. Taking this into account, along with the variety of the studied solute/solvent pairs, the above results are rather quite satisfactory. 12796


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Industrial & Engineering Chemistry Research Table 4. Integral Thermodynamic Characterization of Representative Compounds31−34 LSER descriptors

scaling constants

surface energy components

compound

E

S

A

B

Vx

ε*

vsp*

γVES

γa

γb

aspirin acetaminophen cyclosporin A symvastatin polyethylene polystyrene PMMAa PVCb [C4C1im][PF6]c [C6C1im][PF6] [C8C1im][PF6] [C4C1im][BF4]

0.80 1.093 4.23 1.35 0.31 0.19 1.501 0 4.921 5.117 5.474 2.125

1.245 1.663 7.72 4.28 0.02 0.056 0.332 0.427 0.807 0.690 0.637 1.336

0.59 0.58 0.70 0.20 0 0 0 0.09 0.24 0.25 0.26 0.25

0.66 0.77 4.5 0.7 0 0.06 0.65 0 0.14 0.14 0.14 0.18

1.288 1.172 10.02 3.430 0.327 0.938 0.790 0.453 1.968 2.225 2.546 2.024

5245 4670

0.715 0.936

5100 5342 6558 7481 6105 5759 5517 5662

1.105 0.903 0.802 0.698 0.711 0.744 0.774 0.799

40.09 12.89 13.19 57.36 33.2 42.0 40.2 41.5 43.52 35.34 30.78 37.02

3.92 1.17 0.21 0.71 0 0. 0 1.6 3.38 2.78 2.37 4.14

4.38 1.55 1.38 2.47 0 0.8 6.6 0 2.06 1.56 1.2 3.11

Poly(methyl methacrylate). bPoly(vinyl chloride). c[Butyl methyl imidazolium]+[hexafluorophosphate]−.

a

4.3. Integral Thermodynamic Characterization of Materials. As mentioned already, one key feature of the PSP approach is its capacity to provide with an integral characterization of compounds and materials, such as polymers, drugs, ionic liquids, etc. By integral characterization, we mean the determination of the LSER descriptors or PSPs, equation-of-state scaling constants, and surface energy components. For this purpose, the formalism presented in the previous sections is used in order to correlate the appropriate experimental information, such as IGC (inverse gas chromatography) retention volumes and sorption measurements, volumetric/PVT data, and contact angles on solid surfaces. Solubility data may be used instead of (or with) IGC data in order to obtain LSER molecular descriptors of the studied compound.33−35 On the other hand, IGC measurements of sorption of inert or moderately polar probe molecules lead to a reasonably accurate determination of the non-hydrogen-bonding surface energy component, γVES, (cf. section 3.3).33 This, in turn, may be used in eqs 42−44 and obtain all surface energy components of the studied surface. Of course, if the total surface tension, γ, is known, IGC measurements are not needed and eqs 42−44 may be used directly for the determination of surface energy components. In the absence of IGC measurements, contact angle data of solvents may be used with eq 45 in order to obtain the surface energy components of the studied surface. The contact angles of water, formamide, and diiodomethane are, typically, used for this purpose. In Table 4 are reported the surface energy components and LSER descriptors of some representative studied drugs,31,33 polymers,32 and ionic liquids.34 The scaling constants are also reported for most of the materials as obtained by using the full range of available experimental volumetric data. This, then, constitutes a global or integral characterization of these materials in regards to their thermodynamic behavior. Thus, PVT, IGC (or solubility), and wetting/contact-angle measurements, when simultaneously available, permit the full or integral thermodynamic characterization of the studied new compounds/materials. This can be done with various (co)polymers, (nano)composites, pharmaceuticals, dyes, ionic liquids, etc. The publically available LSER database15 does not include, of course, such compounds and materials. The overall rationale, the steps, and their interconnection for the integral characterization of compounds/materials are

shown graphically in Figure 7. We are not aware of anything analogous in the open literature.

Figure 7. Interconnection of steps for the integral thermodynamic characterization of materials.

5. DISCUSSION PSPs are developed with one goal in mind: use readily available information, such as densities, total solubility parameters, and LSER descriptors, in order to predict the thermodynamic behavior of fluids and their mixtures over a broad range of temperature, pressure, and composition. This goal is not a trivial one and, in this respect, the current developments are by no means complete. The PSP approach, as presented in this work, may be used at various levels for extensive thermodynamic calculations. In a first level, it may utilize freely available QSPR type molecular descriptors, such as the LSER descriptors, and calculate activity coefficients of concentrated multicomponent mixtures or at infinite dilution. In a second level, it may combine information on molecular descriptors with volumetric data and perform equation-of-state calculations over a broad range of external conditions. In a third level, PVT, IGC/solubility, and wetting/contact-angle measurements may be used to provide with a coherent and integral thermodynamic characterization of compounds including materials of not necessarily known detailed molecular constitution. This permits the prediction of a variety of their thermodynamic properties in bulk phases and interfaces. PVT, IGC/solubility, and wetting/contact-angle measurements are not always 12797


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tal data has, in general, an influence on the equation-of-state scaling constants and care should be exercised, in this regard, for reliable predictions. Among others, the model was used for solvation free-energy predictions for a variety of solute/solvent pairs and the results are rather satisfactory. A key feature of the model is its capacity to provide with an integral characterization of compounds and even of materials with not well-known composition. This feature could make the model a most useful tool for the efficient exploitation of experimental information from widely used techniques, especially, IGC, solid solubilities, or surface wetting measurements. Although bound to the simple LFHB equation-of-state model, the developments in this work could form the basis for alternative more refined expansions and quantitative predictions.

available. Depending, then, on this availability, the approach of this work may be used to obtain a full or partial thermodynamic characterization of new materials. For the extension of the LSER/PSP approach to a predictive LFHB equation-of-state framework, the knowledge of molar volumes suffices, as has been shown in section 4.2. However, one should keep in mind that LFHB is an approximate equation-of-state model and its scaling constants depend, to some extent, on the used experimental data for their determination. This was clearly shown in section 4.1 where the influence of the pressure range of experimental data on scaling constants was examined. Of course, one may introduce a pressure dependent volume parameter, v*sp, and greatly reduce the influence of the pressure range of experimental data, but this is outside the aim and spirit of the present work, at least, at this stage. In any case, the prediction of key quantities, such as the solvation free energies, just from LSER descriptors and the molar volume at ambient conditions is rather quite satisfactory, as shown in section 4.2. Based on what has been presented in the previous section on applications, our goal to use readily available information and perform predictive equation-of-state calculations seems to have been reached to an appreciable degree with a rather simple formalism nearly always analytical. However, although good in general, the accuracy of the predicted properties is not always very high in order to use the approach for rational design calculations of processes and products. Yet, we are not aware of any analogous approach in the open literature in order to make extensive comparisons. COSMO-RS36−40 type models, besides requiring access to quantum chemical suites, they may not be used for systems of high molecular weight compounds or for the characterization of complex materials such as polymer composites and nanocomposite formulations. UNIFAC type group-contribution models59 are well suited for design calculations but achieve it with rather dedicated sets of group-contribution values. In our approach, quantitative predictions may arise by reworking and refining the full LSER database and/or the scaling constants but, again, this was not the goal of this work. Of course, the reported calculations/results are associated with a specific equation-of-state model, the LFHB.41−44 This model is by no means unique, and any other analogous equation-of-state models that would base their parameters on volumetric properties and/or vapor pressures and/or the cohesive energy densities and heats of vaporization may in principle be adapted to the present approach and replace LFHB. Similarly, the QSPR/LSER database is by no means unique and other analogous databases or appropriate combinations of complementary databases may replace the LSER one. There is, thus, able space of improvement of our approach, and it is hoped that the present work will stir analogous interest in the literature.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.9b02908.

■

Part 1 (SI1), summary of basic concepts and the essentials of the LFHB equation of state model along with the Veytsman statistics for hydrogen-bonding; Part 2 (SI2), two tables: the complete table with the solvation free energies and the table with LSER descriptors and scaling constants for the studied solutes (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone/fax: +302310-996223. ORCID

Costas Panayiotou: 0000-0002-0579-3244 Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Karelson, M. Molecular Descriptors in QSAR/QSPR; Wiley: New York, 2000. (2) Manoharan, P.; Vijayan, R. S. K.; Ghoshal, N. Rationalizing fragment based drug discovery for BACE1: Insights from FB-QSAR, FB-QSSR, multi objective (MO-QSPR) and MIF studies. J. Comput.Aided Mol. Des. 2010, 24, 843−864. (3) Castro, E. A.; Haghi, A. K. Advanced Methods and Applications in Chemoinformatics; Engineering Science Reference: Hershey, PA, 2011. (4) Katritzky, A. R.; Dobchev, D. A.; Karelson, M. Physical, chemical, and technological property correlation with chemical structure: The potential of QSPR. Z. Naturforsch., B: Chem.Sci. 2006, 61, 373−384. (5) Zissimos, A. M.; Abraham, M. H.; Klamt, A.; Eckert, F.; Wood, J. A comparison between the two general sets of linear free energy descriptors of Abraham and Klamt. J. Chem. Inf. Comput. Sci. 2002, 42, 1320−1331. (6) Hildebrand, J.; Scott, R. L. Regular Solutions; Prentice Hall: Englewood Cliffs, NJ, 1962. (7) Barton, A. F. M. Handbook of Solubility Parameters and Other Cohesion Parameters; CRC Press: Boca Raton, FL, 1983. (8) Hansen, C. M. Hansen Solubility Parameters, A User’s Handbook; CRC Press: Boca Raton, FL., 2007. (9) Abbott, S.; Yamamoto, H.; Hansen, C. M. Hansen Solubility Parameters in Practice, Complete with Software, Data and Examples, 3rd ed., v. 3.1.20; Hansen-Solubility, 2010.

6. CONCLUSIONS A simple coherent thermodynamic model for bulk phases and interfaces with a nearly always analytical formalism has been presented in this work. With a minimum of rather readily available information, such as densities, LSER molecular descriptors, and solubility parameters, one may turn this model into a predictive equation-of-state model for thermodynamic calculations over a broad range of external conditions. The range of temperature/pressure of experimen12798


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