Toward a Simple Predictive Molecular Thermodynamic Model for Bulk

Article pubs.acs.org/IECR

Toward a Simple Predictive Molecular Thermodynamic Model for Bulk Phases and Interfaces Spyros Mastrogeorgopoulos,† Vassily Hatzimanikatis,‡,§ and Costas Panayiotou*,† †

Department of Chemical Engineering, University of Thessaloniki, 54124 Thessaloniki, Greece Laboratory of Computational Systems Biotechnology (LCSB), Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland § Swiss Institute of Bioinformatics (SIB), Lausanne, CH-1015 Switzerland ‡

S Supporting Information *

ABSTRACT: A simple molecular thermodynamic model, based on the QSPR/LSER approach, is presented. The development involves a critical examination of the suitability of available LSER molecular descriptors for calculations of properties of pure-component bulk phases, concentrated solutions, and interfaces. The appropriate descriptors for these calculations are obtained by applying simple and straightforward criteria on freely available data for pure compounds. These criteria arise from a re-examination of basic elements of the solvation picture on which the LSER approach resides. Extensive tables with molecular descriptors are reported. The predictive capacity of the model is tested against a variety of experimental data for mixtures including vapor−liquid equilibria, drug solubilities, and wetting behavior of solid polymer surfaces. Emphasis is given on hydrogen-bonded mixtures. The results are rather satisfactory, in view of the breadth of applications and the nonuse of adjustable parameters. All working equations are expressed in terms of LSER descriptors. The strength and weakness of the model are also critically discussed along with the perspectives of this unified approach to mixture thermodynamics.

1. INTRODUCTION

One of the most widely used QSPR models is Abraham’s QSPR-type model,5−8 known alternatively as the LSER (linear solvation energy relationships) model. It attracts particular interest today, especially since the free online access to the LSER database.9 In the LSER approach, five molecular descriptors, Vx, E, S, A, and B, are used for each compound and stand for the McGowan volume, the excess molar refraction or electron polarizability, the dipolarity/polarizability, the overall hydrogen bond acidity, and the overall hydrogenbond basicity, respectively, of the solute.5,7−9 Successful predictions for generalized solute properties, SP, in a given solvent are made through the classical general equation:

Nowadays a ceaseless strive for novel products with high quality specifications implies an ever increasing need for versatile, reliable, and robust tools for safe predictions of thermodynamic/thermophysical properties in a broad spectrum of fields and applications. This is particularly important when a relatively fast screening tool is required involving thousands of potential compounds/candidates for the examined specific product or production process. In this spirit, modern Quantitative Structure−Property Relationships (QSPR) strive for the judicious selection of a minimum of molecular descriptors, which will lead to the development of reliable predictive methods for a variety of important properties in a multitude of areas of practical and academic interest.1−5 These descriptors may be experimentally determined properties or theoretically derived quantities. The second alternative for the theoretical estimation of molecular descriptors gains strong impetus with today’s unprecedented and ever increasing speed of performing quantum mechanics calculations and molecular simulations. With such calculations, experimental difficulties are avoided and even the system compounds are not required to have ever been synthesized. This, in turn, changes drastically the landscape in today’s design of new products and processes, such as pharmaceuticals, cosmetics, coatings, and foodstuffs. © XXXX American Chemical Society

SP = c + eE + sS + aA + bB + vVx

(1)

The set of coefficients, c, e, s, a, b, and v, are the corresponding solvent properties. There are already available from various quantum-mechanics-based calculation schemes for the LSER molecular descriptors.10−12 Abraham’s model has found successful applications in a remarkably large number of cases involving primarily one or more infinitely dilute phases.5−9 To the best of our knowledge, it has found little or no application Received: Revised: Accepted: Published: A

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publications.13,14,21−25 Those previous publications were focused on the PSP approach which requires information from quantum-chemical calculations and densities or molar volumes. In contrast, the present work does not require this information and uses exclusively the four LSER-type descriptors.

in systems of concentrated phases, such as in bulk phase equilibria. This is, in fact, a crucial point and a key incentive for this work. The present work is a continuation of two recent publications13,14 in which two widely used physicochemical concepts, namely, the Lewis acidity/basicity15,16 and the solubility parameter17−20 of chemical compounds, were reexamined. Partial Solvation Parameters21−25 (PSPs) were used as the reference concept, in an effort to unify the diverse acidity/basicity scales and widely used QSPR descriptors in the open literature. PSPs are expressed in terms of LSER descriptors through the following equations:14 σd = sd = 100

σp = 100

σGa = 100

S Vm

3.1Vx + E Vm

2. THE THERMODYNAMIC FRAMEWORK WITH LSER DESCRIPTORS In the first part of this section we will recall the pertinent formalism for the thermodynamics of concentrated solutions and in the second part the corresponding formalism for interfaces. The essentials only will be presented that lead to the working equations. Details may be found in the recent relevant literature.13,14,22,25 2.1. Concentrated Solutions. Let us consider a binary mixture of Ni moles of component i (i = 1, 2) at an external temperature T and at a relatively low pressure p. Each molecule of type i consists of ri identical segments. Since our molecules are r-meric (polymeric) in nature, we will adopt the simple lattice picture29−31 for their arrangement in the volume of our system. The numbers of external contacts per molecule are zqi for each components i, z being the lattice coordination number. Besides mole fraction, xi = Ni/N, the composition of the mixture may be expressed by the following alternative fractions: Volume fraction: rN rN i i φi = = i i , i = 1, 2 r1N1 + r2N2 rN (6)

(dispersion PSP) (2)

(polarity PSP) (3)

A Vm

(acidity PSP)

B Vm

(basicity PSP)

(4)

and σGb = 100

(5)

Vm in the above equations is the molar volume of the compound. In view of above one-to-one mapping of LSER descriptors into PSPs, one might envision the direct use of the former into typical applications of the latter including phase equilibria and related properties. This, however, cannot be done properly without recognizing the “free energy” character of LSER descriptors7,13,14 in contrast to the “energy” character of the also widely used Hansen solubility parameters.18−20 Once this distinction is made, one may utilize efficiently a rather large body of freely available information in the open literature9 for the prediction of properties in a variety of cases and system complexities. In this respect, apart from bulk phase properties, many challenges remain in the prediction of surface properties and related interfacial phenomena.26−28 Thus, the present work is attempting to set forth an efficient and robust framework for the estimation of bulk phase equilibria and interfacial properties in terms of LSER-type descriptors. We should make it clear right at the outset that this attempt will be based, heavily, on the already available descriptors,9 which were not intended for this kind of calculations. In other words, it is not the authors’ intention to compete with existing calculation schemes specifically designed from their inception for precise phase equilibria and process design. Having said this, in the next section we will summarize the working equations for the above calculations, which will be used in the section on Applications for the prediction of properties of bulk phases and interfaces. The accompanying Supporting Information, sections SI1 and SI2, should be considered as an integral part of the manuscript. They summarize basic concepts, such as the homosolvation and heterosolvation, the distinction of energy and free energy acidity/basicity parameters, the hydrogen-bonding formalism, provide tables of descriptors and explain with examples how the molecular descriptors are used for the calculations. Details on these concepts may be found in our previous associated

Surface area fraction: θi =

qiNi q1N1 + q2N2

=

qiNi qN

,

i = 1, 2 (7)

The key quantity of our interest is the excess (Gibbs) free energy, GEm, which may be considered to be subdivided into a combinatorial (superscript c) and a residual part (superscript r) as follows:

GmE ΔGmc + ΔGmr = RT RT

(8)

In terms of activity coefficients, eq 8 implies the following relation: ln γi = ln γic + ln γi r

(9)

For the combinatorial part, the widely used Guggenheim− Staverman29−31 expression will be adopted, or ΔGmc = RT

2

⎛ ⎛φ ⎞ ⎛ θ ⎞⎞ z i ⎟ + qi ln⎜⎜ i ⎟⎟⎟⎟ 2 ⎝ φi ⎠⎠ ⎝ ⎝ xi ⎠

∑ xi⎜⎜ln⎜ i=1

(10)

which, for the activity coefficient, implies: ⎛θ ⎞ φj ⎛φ ⎞ z ln γiC = ln⎜ i ⎟ + qi ln⎜⎜ i ⎟⎟ + (lirj − l jri) 2 rj ⎝ xi ⎠ ⎝ φi ⎠

(11)

where li =

z (ri − qi) − (ri − 1) 2

(12)

There are two distinct types of contributions to the residual part of excess free energy. The first type encompasses B

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Industrial & Engineering Chemistry Research contributions from the London dispersion, dipolar/induced dipolar, and dipolar/dipolar interactions (subscript VES, due to contribution from LSER descriptors Vx, E, S, as we will see below). The second type consists of the specific hydrogen bonding or acid−base (subscript hb) interactions. On adopting this division, the residual free energy may be written as follows: ΔG VES + ΔG hb ΔGmr = RT RT

* − ε1* − ε2*) ΔEm ,VES = − qNθ1θ2(2ε12 = qNθ1θ2( ε1* −

(13)

and, similarly, for the activity coefficients or the chemical potentials. At low pressures we may further assume that ΔG VES ≅ ΔHVES ≅ ΔE VES

⎛ q ε* 1 ≅ x1q1θ2⎜ 1 − ⎜ q 1 ⎝

q2ε2* ⎞⎟ q2 ⎟⎠

⎛ q ε* 1 ≅ x1r1φ2⎜ 1 − ⎜ r1 ⎝

q2ε2* ⎞⎟ r2 ⎟⎠

⎛ q ε* 1 ≅ x1Vx ,1φ2⎜ 1 − ⎜ Vx ,1 ⎝

(14)

ε1*ε2*

=

i = 1, 2

−Em , VES = N11ε11 + N12ε12 + N22ε22

(16)

(17)

(18)

ΔEm ,VES = Em ,VES − E1,VES − E2,VES * − θ1ε1* − θ2ε2*) = −qN (θ12ε1* + θ22ε2* + 2θ1θ2ε12

m

νH =

Vm, i

,

n

∑ ∑ ναβ = α

3.1Vx + E + S σi2,VES = σi2,d + σi2,S = 10000 Vm, i Vm, i

E + S2 3.1 + 2 Vx ,2

⎞2 ⎟ ⎟ ⎠

(19)

On the other hand, from the defining eqs 2 and 3, we may write

qiεi*

RT

This is certainly a simplified picture, especially, as regards to contributions to mixing energy from dipolar interactions (cf. ref 22, for alternative forms and the related discussion), but for the purposes of this work we will confine ourselves to this simple picture. An extensive Table with the LSER descriptors for a number of common compounds is reported in the Supporting Information, section SI1, along with the rationale for their selection. For the hydrogen-bonding contribution to the excess free energy we will use our previous approach,32,33 which has been applied to numerous systems so far in the literature. The essentials of this approach are summarized in Supporting Information 2, section SI2. In this approach 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 of hydrogen bonds in the system:

For the mixing energy, then, we have

* − ε1* − ε2*) = −qNθ1θ2Δε12 * = −qNθ1θ2(2ε12

RT 100Vx ,1φ22

(22)

In the random mixing approximation,29−33 the following equations apply: z Nii = Nq i = 1, 2 i i θi , 2 Nij = Nq i≠j i izθj = Nq j jzθi ,

μ1,VES

⎛ E + S1 ⎜ 3.1 + 1 − ⎜ Vx ,1 ⎝

If Nij is the total number of interactions between sites of types i and j, the corresponding potential energy of the mixture is

=

2 3.1Vx ,2 + E2 + S2 ⎞ ⎟ ⎟ Vx ,2 ⎠

From eq 21 we obtain for the corresponding contribution to the activity coefficient the following equation:

The potential energy of pure component i, due to these interactions, is simply

−Ei ,VES/Ni

2

(21)

ln γ1,VES =

=

2

q2ε2* ⎞⎟ Vx ,2 ⎟⎠

⎛ 3.1V + E + S x ,1 1 1 ⎜ − ⎜ V x ,1 ⎝

(15)

z * −Ei ,VES = Nq i i εii = Nq i iεi , 2

2

= 100x1Vx ,1φ2

The above mixing energies may be calculated by adopting the simple picture of mean-field pairwise interactions of first neighbors in the lattice.30−33 Starting with the nonspecific interactions, we will assume that each segment site of type i interacts with a neighbor segment site of type j with an interaction energy εij and the following equations apply: z z ε11 = ε1*, ε22 = ε2*, 2 2 z z *= ε12 = ε11ε22 = ε12 2 2

ε2* )2

β

m

n

∑∑ α

β

H Nαβ

rN

(23)

In this equation, the summations extend over all α−β donor− acceptor combinations in the system. 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 the number of hydrogen-bond acceptors of type k (k = 1, n) in each molecule of type j, the hydrogen bonding contribution to the chemical potential of component i in the mixture is given by (cf. section SI2 for nomenclature):

i = 1, 2 (20)

By further assuming that the number of segments is proportional to the McGowan volume, Vx, of the molecule, eq 19 may be expressed as follows: C

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m

n

νdα − να 0

∑ dαi ln

= riνH −

α=1

∑ aβi ln β=1

νaβ ν0β

μ1 − μ10 RT

(24)

where the average numbers of hydrogen bonds per segment, ναβ, satisfy the minimization conditions:

+

να 0ν0β

for all pairs of donors−acceptors(α , β)

H H H Gαβ = Eαβ − TSαβ

r1νH,0 = r1ν11,0 =

(27)

It is precisely at this point that one should make the distinction between “free-energy” and “energy” acid−base LSER descriptors. The LSER A and B descriptors, used in defining eqs 4 and 5 and reported in section SI1, are “free energy” descriptors.14,25 The rationale for obtaining these descriptors as well as the corresponding “energy” descriptors is presented in SI1 along with detailed examples of calculations. Replacing from the defining eqs 4 and (5) in eq 27 we obtain H Gαβ = −2σGaασGbβ VmiVmj = −20000 Aα Bβ

RT

=

RT

+

⎛ E + S1 ⎜ 3.1 + 1 − ⎜ Vx ,1 ⎝ m

−

⎛ νi ⎞ d ⎟− ⎝ νi0 ⎠

∑ di1 ln⎜ i

n

RT

γa γb

(at 25 °C)

r1(r1 + 4K11)

2K11

(31)

(32)

=

A B

(33)

γ = γVES + γhb = γV + γE + γS + 2 γaγb

(34)

and is obtained from the following equation:14 γVES =

10000Vx ,1φ22

⎛ νj ⎞ a ⎟ ⎟ ⎝ ν0j ⎠

r1 + 2K11 −

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

RT 2 E 2 + S2 ⎞ ⎟ 3.1 + + r1νH Vx ,2 ⎟⎠

∑ a1j ln⎜⎜ j

=

(30)

The acidic and basic surface tension components, γa and γb, respectively satisfy the following equation:

The acid−base (hydrogen bonding) contribution to the chemical potential (cf. eq 24) may be obtained once the freeenergy acid/base LSER descriptors are available (e.g., Table S1 in section SI1). Thus, the overall equation for the residual part of the chemical potential (or the activity coefficient) is ln(γ1r ) =

⎞2 ⎟ + r1νH ⎟ ⎠

γhb = 2 γaγb

(28)

μ1,H

E + S2 3.1 + 2 Vx ,2

The general case of systems of self-associated and crossassociated molecules is handled in a similar manner as discussed in section SI2. 2.2. Surface Energy Components and Wetting Phenomena. The way the surface energy (tension) components may be evaluated from PSPs and, thus, from LSER descriptors has been presented in ref 14. Here we will confine ourselves to the basic definitions and the working equations. For pure compounds, the defining equation for the hydrogen-bonding surface tension, γhb, is14

Alternatively, it is given in terms of the acid/base free-energy PSPs by eq 27:13,14,25 H Gαβ = −2σGaασGbβ VmiVmj = −σGhb, αβ VmαVmβ

RT

Subscript H, 0 means hydrogen bonding in pure-fluid state, and

(26)

μ1,VES

10000Vx ,1φ22

⎛ ⎞⎞ ⎛ x1 ⎞ ⎛ 1 ⎟⎟⎟⎟ − 2 ln⎜ ⎟ − ⎜⎜r1νH,0 − 2 ln⎜⎜ ⎝ x1 − rνH ⎠ ⎝ ⎝ 1 − r1νH,0 ⎠⎠

(25)

Equation 25 is a set of (m × n) quadratic equations that must be solved simultaneously in order to obtain the number of each type of hydrogen bonds. GHαβ is the free enthalpy of formation of the hydrogen bond of type α−β and is given in terms of the energy (E), and entropy (S) of hydrogen bond formation (assuming negligible volume change) by the equation:

μ1r

r2

(l1r2 − l 2r1) +

⎛ E + S1 ⎜ 3.1 + 1 − ⎜ Vx ,1 ⎝

⎛ GH ⎞ αβ ⎟⎟ = Kαβ = exp⎜⎜ − ⎝ RT ⎠

ναβ

φ2

⎛φ ⎞ z ⎛θ ⎞ = ln(γ1x1) = ln(x1) + ln⎜ 1 ⎟ + ln⎜⎜ 1 ⎟⎟ 2 ⎝ φ1 ⎠ ⎝ x1 ⎠

γ 1+

2 AB 3.1Vx + E + S

(35)

The above eqs 32−35, when combined, give all required components, γVES, γa, γb and γhb. The experimental surface tensions of many common liquids are available in the open literature or in critical compilations such as the DIPPR database.34 From these total surface tensions one may, then, obtain the partial surface tensions, reported in Table S2 of section SI1, by using the above set of simple equations. The above equations hold for both homosolvated (self- and crossassociated) and heterosolvated (cross-associated only) compounds.14,21,22 Table S2 may be used for calculations at liquid interfaces but also for the characterization of solid surfaces for which the direct experimental measurement of surface tension remains a challenge, as will be shown in the next section on Applications.

(29)

In section SI2 the case of alkanol−alkane mixtures is used as an example in order to show how the hydrogen bonding equations may be used in practice. In this case, the full expression for the chemical potential or the activity coefficient of component 1 (and similarly for component 2) becomes (cf. eq (T-21) of section SI2): D

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the figure, the predicted vapor−liquid equilibrium lines are in rather good agreement with the experimental data.35 In contrast to the previous mixture, the mixture of basic acetone with the acidic chloroform does interact with hydrogen bonding. In this case, all three types of interactions, dispersion, polar, and hydrogen-bonding, are present in the system and all original LSER descriptors (not the new ones, cf. SI1) should be used in the calculations. To understand this, we should recall that some of the acetone sites involved in hydrogen bonding in this system were involved in polar and dispersion interactions only in the previous mixture with benzene. This means that the polarity descriptors, S, for acetone in the mixture with chloroform should be less than the value obtained from its cohesive energy density. The same holds true for chloroform. Thus, when the heterosolvated compound is involved in hydrogen bonding interactions, the appropriate set of LSER descriptors is the original set7−9 including acidity/basicity descriptors. When the heterosolvated compound is not involved in hydrogen-bonding in a mixture, one should use its new S value obtained from total solubility parameter and should not use acidity/basicity descriptors. Figure 2 compares the experimental35 with the predicted vapor pressures of the acetone−chloroform mixture at three

3. APPLICATIONS The formalism developed in the previous section will be used for calculations in three areas of applications, namely, vapor− liquid equilibria over a temperature range, solubility of solids in different solvents, and contact angles of various solvents with polymeric solid surfaces. The required LSER molecular descriptors in all these applications are reported in Table S1 of section SI1. The hydrogen bonded compounds in Table S1 are divided into homosolvated (compounds that may selfassociate as well as cross-associate) and heterosolvated (compounds that cannot self-associate−they cross-associate only).14,21−25 3.1. Prediction of Vapor−Liquid Equilibria. The purpose of the applications in this section is to verify the capacity of the model to handle any type of VLE calculations including VLEs of highly complex hydrogen-bonded systems. Thus, representative systems will be studied with varying degrees of (hydrogen-bonding) complexities. Since the external pressures, P, in the systems to be studied are not very high, the VLE calculations will be made via the classical equilibrium equation: γixiPi0 = yP i

(36)

Equation 29 will be used for the activity coefficient γi, and the vapor pressures at various temperatures will be obtained from DIPPR.34 Before going to complex hydrogen-bonded systems, it is worth testing the model with a representative nonhydrogen-bonded system. Vapor−liquid equilibria in mixtures of aliphatic or aromatic hydrocarbons are predicted rather satisfactorily. However, hydrogen-bonding interactions do not occur also in systems of heterosolvated compounds unless one component is acidic and the other basic. In Figure 1 are compared the experimental

Figure 2. Experimental35 (symbols) and predicted (lines) vapor− liquid equilibria for the system acetone (1)−CHCl3 (2) at three temperatures.

temperatures. The predictions were made by using the full set of the original LSER descriptors for both compounds, as explained above. As observed, the agreement is rather satisfactory for this azeotropic system. This is a typical system showing the strength of this approach. The negative deviation here amounts to the tendency of the molecules to favor the liquid state (lower escaping tendency) in which they may multiply their strong specific interactions. Since both heterosolvated compounds found their complementary counterpart, the appropriate descriptors are the original ones which account for both their polarity as well as their acidity/basicity. The next two studied systems are mixtures of a homosolvated compound with an inert one. As explained in SI1, the original acidity/basicity LSER descriptors7−9 for homosolvated compounds are, in general, different from the corresponding descriptors obtained from their cohesive energy density. Thus, in part C of Table S1 are also reported these new A and B descriptors for homosolvated compounds. In all cases involving homosolvated compounds the new A and B

Figure 1. Experimental35 (symbols) and predicted (lines) vapor− liquid equilibria for the system acetone (1)−benzene (2) at two temperatures. For VLE calculations, eq 29 was used for the liquid phase while the vapor phase was assumed ideal.

vapor pressures35 with the predicted ones of the acetone− benzene mixture at two temperatures. The interactions in the mixture are of the same kind as in their pure state. The polarity descriptor, S, for both compounds are obtained from their cohesive energy density or total solubility parameter as explained in section SI1. Thus, eq 29 without the hydrogenbonding terms was used for the VLE calculations. As shown in E

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The next type of systems to be studied is mixtures involving homosolvated compounds. A representative example is the mixture of water with ethanol. In Figure 5 the predictions with

descriptors were used along with the original non-hydrogenbonding descriptors Vx, E, and S. Figure 3 compares the experimental35 with the predicted vapor pressures of the ethanol−n-hexane mixture at three

Figure 5. Experimental35 (symbols) and predicted (lines) vapor− liquid equilibria for the system ethanol (1)−water (2) at three temperatures. The lines were calculated by attributing two equivalent proton donors and two proton acceptor sites to each water molecule.

35

Figure 3. Experimental (symbols) and predicted (lines) vapor− liquid equilibria for the system ethanol (1)−n-hexane (2) at three temperatures.

temperatures. This system forms an azeotrope and it is rather satisfactorily described by the predicted lines in the figure. Similarly, Figure 4 compares the experimental35 with the

experimental data35 at three temperatures are compared. This is a highly nonideal system of two compounds which selfassociate and cross-associate. Ethanol has one proton donor site and one proton acceptor site. Water may be handled in two nearly equivalent ways: as possessing either one donor site and one acceptor site or two donors and two acceptors but with half the hydrogen-bonding strength each.25 The agreement in Figure 5 is rather satisfactory in view of the complexity of the studied system. Another representative system is shown in Figure 6 and involves one homosolvated compound, propanol, and one heterosolvated compound, chloroform. The basic site of propanol forms a hydrogen bond with the acidic site of another propanol molecule (self-association) or with the acidic site of a chloroform molecule (cross-association). As shown in

Figure 4. Experimental35 (symbols) and predicted (lines) activity coefficients (VLE data) for the system methanol (1)−toluene (2) at 318.15 K.

predicted activity coefficients of the methanol−toluene mixture at 45 °C. As explained in section SI1, the original9 S descriptor for aromatic hydrocarbons is rather high and does not match the cohesive energy density of the compounds in their pure state. In fact, in the original set9 of LSER descriptors of aromatic hydrocarbons, besides the relatively high S value, a small value is also attributed to their basicity B descriptor. This would lead to a still higher S value matching the cohesive energy density, and this is difficult to explain. We will come back to this issue in a later section. For consistency, the new S values for aromatic hydrocarbons reported in Table S1 and obtained from total solubility parameters 34 were used throughout this work.

Figure 6. Experimental35 (symbols) and predicted (lines) activity coefficients (VLE data) for the system propanol (1)−chloroform (2) at 328.15 K. F

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Industrial & Engineering Chemistry Research the figure, the agreement of model predictions with experimental activity coefficients35 is again rather satisfactory. 3.2. Prediction of Drug Solubilities. The solubility (mole fraction) of solid 1 in solvent 2 over a temperature range below its melting point may be obtained from the classical (though approximate) equation36,37 y1 =

⎛ ΔH m ⎛ T ⎞⎞ 1 exp⎜⎜ 1 ⎜ m − 1⎟⎟⎟ γ1 ⎠⎠ ⎝ RT ⎝ T1

ΔHm 1

(37)

Tm 1

where, is the enthalpy of fusion of solid 1 and its melting point. Melting points and heats of fusion for numerous solids are freely available from the NIST database (webbook.nist.gov). If heats of fusion are not available, one may use Yalkowsky’s38 approximation and replace the term in the bracket in eq 37 by the term A = −0.02278 (Tm 1 − T), or 1 y1 = exp( −0.02278(T1m − T )) γ1 (38)

Figure 8. Experimental39 (symbols) and predicted solubilities of ibuprofen and acetaminophen in ethanol as a function of temperature. The LSER predictions are shown by stars near the corresponding data.

In Figure 7 the experimental 39 solubilities of aspirin (acetylsalicylic acid) and ibuprofen in isopropyl alcohol with

Figure 9. Experimental39 (symbols) and predicted (solid lines) solubilities of ibuprofen and acetylsalicylic acid in acetone as a function of temperature. The LSER predictions are shown by stars near the corresponding data.

Figure 7. Experimental39 (symbols) and predicted (solid lines) solubilities of ibuprofen and acetylsalicylic acid in isopropyl alcohol as a function of temperature. The LSER predictions are shown by stars near the corresponding data.

uncertainty in the reported experimental data39 for ibuprofen, and safe conclusions for the model performance cannot be drawn. In contrast, the model predictions are much better for the solubility of paracetamol in ethanol and of aspirin in acetone. These model predictions are also better than the corresponding typical LSER predictions with eq 39. 3.3. Prediction of Solvent−Polymer Contact Angles. In this subsection we will test the capacity of the model to predict the wetting behavior of (solid) polymer surfaces. Before predicting contact angles, the surface tension components of the polymer surfaces are required. Surface tension and contact angle data for polyethylene (PE), polystyrene (PS), poly(methyl methacrylate) (PMMA), and poly(vinyl chloride) (PVC) have been reported by Owens and Wendt.40 These data are widely used as reference data in the literature and will be used in the present work as well. By applying the methodology developed in section 2.2, we obtain the surface tension components of the polymers, which are reported in Table 1.14 The work of adhesion of a solid, S, with a liquid-probe, L, forming a contact angle θ with the solid surface, is given by the combined Young−Dupre equation:41−43

the predicted ones as a function of temperature are compared. In the same figure are also shown the corresponding typical LSER predictions with eq 39 for the solubility of solute (in solvent, s, over the solubility in water, w): log

Ss = c + eE + sS + aA + bB + vVx Sw

(39)

The required coefficients for use of eq 39 are also available from the data set of Endo et al.,9 while data for Sw (mol/L) may be obtained from Yalkowsky’s compilation.38 As observed in Figure 7, the predictions are in a rather qualitative agreement with experimental data. A more or less similar picture emerges from Figures 8 and 9. The solubilities of ibuprofen and paracetamol (acetaminophen) in ethanol are reported in Figure 8, and the solubilities of ibuprofen and aspirin in acetone are reported in Figure 9. One common characteristic in Figures 7−9 is that the present model overestimates the solubility of ibuprofen in all three solvents. However, as shown in Figures 8 and 9, there is a rather large G

DOI: 10.1021/acs.iecr.7b02286 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

4. DISCUSSION Abraham’s LSER approach7−9 is already proven to be a very useful and often remarkably successful predictive tool for numerous applications. The LSER molecular descriptors are now freely available for thousands of compounds.9 In fact, they may be obtained for any compound through its SMILES form.9 It is then reasonable to exploit further this valuable source of information and broaden further its range of applications. This is precisely the motivation and rationale of the present work. What has been attempted in the previous sections was the introduction of LSER descriptors into a coherent and consistent thermodynamic framework for the prediction of thermodynamic properties in bulk phases and interfaces. This exercise has revealed a number of features of the original LSER descriptors as regards to their capacity to describe the (inter)molecular interactions. The complex spectrum of these interactions is divided in the LSER approach6−8 in polarizability/polarity, excess refraction, and acidity/basicity interactions. The obvious question is what current quantum chemical calculations attribute to each of the above portions of the interaction spectrum or how this division compares with the analogous divisions of alternative widely used QSPR-type approaches? An answer to this more general question was attempted in two recent publications13,14 and in the present work. When comparing different approaches one needs a reference framework. Such a framework could act as a bridge for the safe exchange of information between the different approaches. This is precisely the rationale behind the partial solvation parameter (PSP) approach.14,21−25 It is the PSP approach through which the above-mentioned features of the LSER descriptors were studied. We have seen already (cf. eq 2) which combination of the E and Vx LSER descriptors represents the dispersion interactions in a way entirely equivalent to the dispersion solubility parameter,18−20 δd. The focus in the present work was on the S, A, and B descriptors. From their very nature and provenance,6−9 the acidity/ basicity descriptors A and B are “free-energy” descriptors. But, when attempting to quantify hydrogen-bonding energies through A and B descriptors we should recall that, in contrast to the hydrogen-bonding energy and entropy which may be considered constant for a temperature range, the hydrogenbonding free energy is much sensitive to temperature changes. An appropriate, then, expression for hydrogen-bonding free energy in terms of the involved A and B descriptors should also indicate the temperature dependence. This is what eq 42

Table 1. Surface Tension Components (mN/m) of Polymers Based on Experimental Data by Owens and Wendt40 polymer

γtot

γVES

γa

γb

polyethylene, PE polystyrene, PS poly(methyl methacrylate), PMMA poly(vinyl chloride), PVC

33.2 42.0 40.2 41.5

33.2 42.0 40.2 41.5

0 0.002 0 0.69

0 0 5.68 0

adh W SL = γS + γL − γSL = γL(1 + cos θ )

(40)

In terms of the surface tension components of solid and liquidprobe, this equation leads to the following working equation:14,27,28,44 γL(1 + cos θ ) = 2{ γVES,LγVES,S +

γa ,Lγb ,S +

γa ,Sγb ,L } (41)

This equation is most useful for the characterization of solid surfaces such as polymer and drug surfaces. The contact angles of the solvent/solid-surface pair may be predicted by applying eq 41 with the known surface tension components of solvents and solid surfaces. The predicted contact angles are reported in Table 2 along with the experimental ones45 for comparison. As observed, the experimental contact angles are in rather reasonable agreement with the predicted ones for almost all cases. The contact angles of ethylene glycol with PVC and PMMA are rather overpredicted when compared with the experimental data from ref 46. It is worth mentioning here that a single set of LSER descriptors was used for water throughout this work. This is important because it underlines the coherence of the approach and it indicates that one may use LSER descriptors derived from bulk phase equilibria to applications involving interfaces. Of course, the calculations required the prior knowledge of polymer PSPs out of which one may obtain the corresponding LSER-type descriptors, as explained previously.14 To the best of our knowledge, there are no extensive tables in the open literature with such LSER descriptors for polymers. However, one may rework the formalism of the inverse gas chromatography (IGC) technique and obtain directly the polymer LSER descriptors from the corresponding LSER descriptors of the probe molecules with a procedure analogous to the one described previously47 and which will be reported in a forthcoming publication.

Table 2. Experimental (expt) and Predicted (pred) Contact Angles on Four Common Polymers. Experimental Data from the Compilation of Lyklema45 unless Otherwise Specified contact angle, θ polyethylene

a

polystyrene

PVC

PMMA

solvent

pred

expt

pred

expt

pred

expt

pred

expt

water diiodomethane formamide glycerol ethylene glycol DMSO

105 53 82 92 79 41

104 53 77−81 79−94

99 37 74 84 71 9

91 35 69−80 73−80 75

93 33 67 77 63