Superposition-Additive Approach: Clusterization Thermodynamic

Article pubs.acs.org/JPCC

Superposition-Additive Approach: Clusterization Thermodynamic Parameters of Bifunctional Nonionic Amphiphiles at the Air/Water Interface Yu. B. Vysotsky,† E. S. Fomina,† E. A. Belyaeva,† D. Vollhardt,*,‡ V. B. Fainerman,§ and R. Miller‡ †

Donetsk National Technical University, 58 Artema Str., 83000 Donetsk, Ukraine Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany § Donetsk Medical University, 16 Ilych Avenue, Donetsk 83003, Ukraine ‡

ABSTRACT: The superposition-additive approach developed and tested previously on monofunctional nonionic amphiphiles is updated and used for the calculation of the thermodynamic clusterization parameters for amphiphiles with two functional groups. In the framework of this approach the thermodynamic parameters of formation and clusterization for three classes of bifunctional amphiphiles were calculated using corresponding data for monofunctional amphiphiles. In the example of 2hydroxycarboxylic acids, α-amino acids, and alkyl amides it was shown that the regarded approach adequately reproduces the parameter values of bifunctional amphiphiles without the direct use of quantum chemical calculations. It exploits rather only the calculated or experimental data existing for corresponding monofunctional amphiphiles. The systematic errors for the description of the thermodynamic parameters of formation and clusterization for bifunctional amphiphiles can be attributed to the incomplete reproduction of the values for the increments of intra- and intermolecular interactions between the functional groups in the molecules of bifunctional alkanes and the corresponding values of the increments for monosubstituted alkanes. These errors can be taken into account using an additive correction. Consideration of this correction allows the assessment of the thermodynamic parameter values of formation for 2-hydroxycarboxylic acids, α-amino acids, and alkyl amides with the following values of standard deviations: 0.01, 0.03, and 0.02 kJ/mol for enthalpy; 0.88, 0.50, and 1.16 J/(mol·K) for absolute entropy; and 0.31, 0.29, and 0.37 kJ/mol for Gibbs’ energy, respectively. The values of the thermodynamic parameters of clusterization per one monomer molecule of 2D films for the considered bifunctional amphiphiles are estimated with deviations in the range of 0.10−0.61 kJ/mol, 1.96−2.99 J/(mol·K), and 0.03−0.87 kJ/mol for clusterization enthalpy, entropy, and Gibbs’ energy, respectively.

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INTRODUCTION The postulate about the existence of atoms in molecules1 is the rigorous basis not only for the development of a conception about functional groups but also a tool for the assessment of the transferability of their properties.2 The degree of transferability for atomic groups can be directly calculated using corresponding values of the atomic properties or the bond properties between them. In addition the transferability can be estimated to a certain extent using similarity indices3 based on the comparability of the atomic charge densities in the molecules or the critical points for corresponding bonds.4−6 Such an approach was implemented to the calculation of the atomic transferability in molecules for different classes of organic compounds, particularly hydrocarbons,7,8 alcohols,9 aldehydes,10 ketones,10,11 ethers,12−14 and nitriles.15 These works show that the energetic properties of oxygen, nitrogen, and carbon atoms depend significantly on the nearest environment of the fragment in which regarded atoms are included. In this connection, according to the different additive schemes performed previously16−18 the calculations should be © 2013 American Chemical Society

carried out accounting for the properties of the nearest molecular environment for the considered molecular fragment, i.e., justifying the properties of molecular graphs. One more display of atomic transferability is the superposition-additive approach (SAA) based on the postulate about the existence of atoms in molecules. Now, it is widespread and generally recognized.19−25 For the first time, this approach was developed for the calculations of the electronic structure and physicochemical properties of conjugated systems.19 Later,20 it was applied to the calculation of the thermodynamic parameters of formation and atomization of conjugated systems, their dipole electric polarizabilities, molecular diamagnetic receptivity, π-electronic ring current, etc. In refs 21−25, the superposition-additive approach was used for the description of the thermodynamic parameters of formation and clusterization of substituted Received: May 21, 2013 Revised: July 13, 2013 Published: July 16, 2013 16065

dx.doi.org/10.1021/jp404999u | J. Phys. Chem. C 2013, 117, 16065−16075

The Journal of Physical Chemistry C

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Figure 1. Generalized superposition-additive scheme for the calculation of thermodynamic parameters of monomers of substituted alkanes on the example of amphiphiles with n = 8: X, Y, and Z are schematic denotations of the functional groups.

groups can be the same or different). Structure (3) is the result of superposition of the structures (1) and (2) and also of the structures (4) and (5). As these two superpositions lead to the same result, the properties of any of the four molecules could be expressed as the algebraic sum of the corresponding properties of the other three molecules. Thus, for example, to calculate any thermodynamic parameter of molecule (4) one has to add the corresponding values of the molecules (1) and (2) and to subtract the value of the molecule (5). In refs 21−25, the superposition-additive approach used both calculated and experimental physical-chemical data for the description of the regarded molecular properties. However, the superposition-additive schemes implemented in the present work exploit only values of the thermodynamic parameters calculated previously26−28 in the framework of the quantum chemical semiempiric PM3 method. This is attributed to the fact that the experimental data concerning the considered thermodynamic parameters for unbranched bifunctional amphiphiles are very scarce. The calculated results obtained in the frameworks of superposition-additive schemes (SASs) and illustrating implementation of the superposition-additive approach to the thermodynamics of formation and clusterization of bifunctional alkanes are shown in the example of 2-hydroxycarboxylic acids and listed in the corresponding tables. The structure of the tables is the following. The column “SAS_Scheme No.” lists the values estimated by SAS using the corresponding parameters which were determined earlier by semiempiric PM3 calculations23−31 (they are listed in the “Direct Calculation” column). In our previous studies21−25 devoted to the application of the superposition-additive approach to the description of the thermodynamic parameters of formation and clusterization of monosubstituted alkanes we regarded different SASs, which exploited the values of the corresponding parameters referring both to one class of compounds and to several classes, as well. In this study we regard only the most interesting SASs which use the data for three classes of amphiphiles, i.e., when X ≠ Y ≠ Z (see explanations below in the next section). It should be mentioned that in SASs exploited here the energetically most favored monomer and cluster structures of bifunctional amphiphiles are used (except 2-hydroxycarboxylic acids).26−28 The second favored structures of 2-hydroxycarboxylic acid monomers and clusters have the most similar structure of the hydrophilic part to that of α-amino acids. This enables maximal overlapping of the molecular graphs for the regarded compounds in one of the schemes exploited in the chapters below. This is the reason for using these particular 2-

alkanes which have amphiphilic structure and are capable of monolayer formation at the interface. The use of this approach allows quite adequately the reproduction of the thermodynamic parameters of the formation of alkanes, fatty alcohols, carboxylic acids, thioalcohols, amines, cis-monounsaturated carboxylic acids in condensed phases, the parameters of their phase transitions, and the values of their clusterization enthalpy, entropy, and Gibbs’ energy, as well. In this connection, it is interesting to apply the discussed approach to the description of the thermodynamic parameters of formation and clusterization of bifunctional amphiphiles. Contrary to the amphiphiles listed above, their hydrophilic parts possess two functional groups and are capable of monolayer formation structurally different from the monolayers of the amphiphile classes studied earlier. The objective of this work is to modify and to apply the superposition-additive approach developed previously to amphiphiles possessing a bifunctional hydrophilic part and to show that the properties (including the surface) of these compounds can be estimated using the corresponding parameters of the monofunctional amphiphiles. The application of the mentioned approach is illustrated in the example of three classes of homologous amphiphiles: 2-hydroxycarboxylic acids, α-amino acids, and alkyl amides.

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THEORETICAL BASIS OF THE SUPERPOSITION-ADDITIVE APPROACH The postulate about the way in which atoms exist in molecules is the theoretical basis for different superposition-additive schemes.1 Its general idea is that each atom in a molecule retains its individuality in various chemical combinations (i.e., in various molecules). This refers to the transferability of atomic properties. In addition, the atomic values, being summed over all atoms in the molecule, yield the molecular average so that the corresponding molecular characteristics are additive. The main idea of the superposition-additive approach is based on transferability of atomic properties and additivity of molecular properties. The essence of the procedure is the assumption that, when two molecular graphs are superimposed, the properties of the constituent atoms remain unchanged. Then, if the same superposition can be constructed in two different ways, each one involving two entities, it becomes possible to calculate structure and properties of one of these entities, provided the structure and properties of the remaining three entities are known. This principle is graphically illustrated in Figure 1. The molecules (1), (2), (4), and (5) are structures which involve the alkyl chain and the functional groups X, Y, and Z (these 16066



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hydroxycarboxylic acid structures. The geometry of the small aggregates (dimers and tetramers) of α-substituted carboxylic acids is quite similar, as well. The tilt angle of amphiphilic molecules with respect to the normal to the interface was calculated earlier26,28 to be 20−30° for the structures built on the basis of the chosen acid monomers. In addition, there are the equal number of intermolecular CH···HC interactions realized in the regarded aggregates of α-substituted carboxylic acids up to the 2D films. It should be noted that mismatch of the number of the mentioned interactions in the superposition of the molecular graphs for the regarded structures contradicts the main principles of the superposition-additive approach. According to the mentioned fact, such mismatch does not give any satisfactory result. Note, that, in SASs exploited here for the calculation of the thermodynamic parameters of formation and clusterization of bifunctional amphiphiles, we use the monomers and clusters of mono- and unsubstituted alkanes of different structures for αsubstituted acids and amides. Also, the exploited structures of monomers, aggregates, and 2D films for mono- and unsubstituted alkanes could not be most preferable according to the Gibbs’ energy, and they cannot be realized in the experiment. However, the use of such structures is necessary for ensuring a maximal molecular graph overlapping for the structures participating in the superposition. This is the key point for implementation of different superposition-additive schemes. Superposition-Additive Approach for Calculation of the Thermodynamic Parameters of Monomer Formation. It is known20−23 that it is possible to reasonably use a quite wide range of superposition-additive schemes. However, the best results are given by schemes possessing the maximal superimposition of molecular graphs. Therefore, we consider the schemes (see Figure 1) corresponding to the maximal mutual overlapping of hydrocarbon chains (CnH2n−4). The first scheme can be presented as follows

Figure 2. SAS 1 (X = COOH, Y = OH, Z = NH2) for the calculation of the thermodynamic parameters of formation for 2-hydroxycarboxylic acid monomers on the example of 2-hydroxynonanoic acid.

whereas the amide functional group CONH2 can be represented as a combination of the ketonic CO group and the amine NH2 group. Note the simplest superposition scheme for the calculation of the thermodynamic parameters of the alkyl amide formation can be as follows: “amide = aldehyde + amine − alkane”. However, this scheme does not give satisfactory results because of the presence of p-π conjugation of the lone-electron pair of the nitrogen atom and π-electrons of the carbonyl oxygen atom in the hydrophilic headgroup of the amide molecule and its absence in aldehyde. At the same time, carboxylic acids have such a conjugation between the lone-electron pair of a hydroxylic oxygen atom and π-electrons of the carbonyl oxygen atom. As a result, the π-electronic system realized in the compounds described above provides planarity of the regarded atomic groups. This enables maximal superimposition of the molecular graphs in the case of the use of the next scheme “amide = carboxylic acid + amine − alcohol”27 due to the planar structure of the functional groups for the two classes of amphiphiles used in this scheme and the pyramidal structure of two others. Thereby, the superpositionadditive scheme for calculation of the thermodynamic parameters of the formation of alkyl amides is presented as follows

SAS_1:

SAS_2: A(CnH 2nXY) = A(Cn − 1H 2n − 2XZ) + A(Cn − 1H 2n − 1Y) − A(Cn − 2H 2n − 3Z)

A(CnH 2n + 1[XZ]) = A(Cn − 1H 2n − 1[XY]) (1)

+ A(Cn − 1H 2n − 1Z) − A(Cn − 2H 2n − 3Y)

where A is the thermodynamic parameter (absolute entropy, enthalpy, or Gibbs energy of the formation of the compound from elementary substances) at normal conditions (T = 298.15 K); n is the number of atoms in the hydrocarbon chain; and X, Y, and Z define schematically the structural fragments of the functional groups (e.g., X = COOH, Y = OH, Z = H). This scheme can be applied to the description of the thermodynamic parameters of formation of α-substituted acids and presented as: “2-hydroxy(amino)carboxylic acid = carboxylic acid + alcohol (amine) − alkane”29−31 (see Figure 2). It should be noted that one can use other surfactant classes for the calculation of the thermodynamic parameters of the formation of α-substituted acids. For example, it is possible to use a scheme which exploits the value of a given parameter of one substituted acid for the calculation of the corresponding parameter for another one, e.g., “2-hydroxycarboxylic acid = αamino acid + alcohol − amine”.26,28,29,31 This scheme is realized according to eq 1, except Z = NH2. In the case of amides, the functional COOH unit of carboxylic acids can be represented as a combination of the ketonic CO group and the hydroxylic OH group of alcohols,

(2)

where A and n have the same meaning as in the previous expression (1), and X, Y, and Z define schematically the structural fragments of the functional groups, e.g., [XY] = COOH for carboxylic acids, Y = OH for alcohols, Z = NH2 for amines, and [XZ] = CONH2 for amides. This scheme is graphically illustrated in Figure 3. Note that despite the fact that SAS 1 and 2 are optimal due to the maximal molecular graph overlapping, one can use other schemes with a smaller range of overlapping. This was shown in refs 21−23. However, SAS 1 and 2 described above are optimal due to the reproduction of the thermodynamic parameter values of the amphiphile formation with the least standard deviations with respect to the data of the direct calculation using the PM3 method. Consider now the methodology of the calculation according to the superposition-additive scheme 1, in detail. To calculate the thermodynamic parameters of a monomer molecule, containing n carbon atoms in the hydrocarbon chain, one should use the parameters of monomer molecules of two other regarded classes of amphiphiles with (n − 1) number of carbon atoms in their alkyl chains and one with (n − 2) number. For 16067



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(−460.36) (−451.87) (−443.52) (−434.93) (−426.37) (−417.66) (−409.51) (−400.25) (−392.21) (−382.80) (−460.20) (−451.79) (−443.46) (−435.93) (−426.15) (−417.77) (−409.19) (−400.45) (−392.17) (−383.01)

−459.29 −450.80 −442.45 −433.86 −425.30 −416.59 −408.44 −399.18 −391.14 −381.73

kJ/mol

−468.64 −460.30 −451.76 −443.33 −434.90 −426.34 −417.66 −409.22 −400.61 −391.79 −383.27

direct calcd

−462.95 −454.54 −446.21 −437.69 −428.90 −420.52 −411.94 −403.20 −394.92 −385.76 512.26 544.28 576.53 608.12 638.81 670.83 702.24 733.03 765.39 794.94

(500.84) (532.87) (565.12) (596.71) (627.40) (659.42) (690.82) (721.61) (753.98) (783.53)

503.52 535.27 567.41 598.73 630.19 661.11 693.93 722.98 756.13 784.68

(501.20) (532.95) (565.09) (596.41) (627.87) (658.79) (691.61) (720.66) (753.81) (782.36)

469.03 501.29 532.84 564.71 596.58 628.02 659.05 690.88 722.15 752.70 784.24 J/(mol K)

(−725.43) (−748.08) (−770.75) (−793.41) (−816.09) (−838.77) (−861.44) (−884.13) (−906.81) (−929.45)

−724.48 −747.14 −769.82 −792.50 −815.17 −837.85 −860.53 −883.21 −905.90 −928.60

(−725.40) (−748.05) (−770.74) (−793.41) (−816.09) (−838.77) (−861.44) (−884.13) (−906.82) (−929.51)

−702.74 −725.40 −748.06 −770.73 −793.41 −816.09 −838.77 −861.45 −884.13 −906.81 −929.49 kJ/mol

−725.09 −747.74 −770.40 −793.07 −815.75 −838.43 −861.10 −883.79 −906.46 −929.10 system

C7H14O3 C8H16O3 C9H18O3 C10H20O3 C11H22O3 C12H24O3 C13H26O3 C14H28O3 C15H30O3 C16H32O3 C17H34O3

SAS_1 (Z = NH2)

ΔG0298,mon,

SAS_1 (Z = H) direct calcd SAS_1 (Z = NH2)

S0298,mon,

SAS_1 (Z = H) direct calcd SAS_1 (Z = NH2)

example, to calculate the thermodynamic parameter for 2hydroxyoctanoic acid one should add the values of the corresponding parameter (enthalpy, entropy, or Gibbs’ energy) for heptanoic acid and heptanol and subtract from the calculated sum corresponding to the thermodynamic parameter for hexane. To calculate the thermodynamic parameter for 2hydroxynonanoic acid one should add the values of the thermodynamic parameter for octanoic acid and octanol and subtract from the calculated sum the corresponding parameter for heptanol, etc. The calculation according the SAS 2 applies analogously. The results of the calculation of the thermodynamic parameters of formation for regarded bifunctional amphiphiles using schemes 1 and 2 described above are listed in Table 1. The standard deviations for the description of enthalpy, absolute entropy, and Gibbs’ energy are shown in Table 2. The data analysis suggests that the results calculated according to the superposition-additive scheme 1 agree well with the corresponding results of direct calculations for 2hydroxycarboxylic acids. Note, that SAS 1 with Z = NH2 provides calculated data with significantly lower values of standard deviations in comparison to SAS 1 with Z = H. This can be attributed to the fact that SAS 1 with Z = NH2 exploits the values of two monosubstituted and one bisubstituted alkanes for the calculation of the required parameters of another bisubstituted alkane. As mentioned above, the structures of the monomers for α-amino and 2-hydroxycarboxylic acids participating in the regarded scheme coincide virtually with each other. The values of the torsion angles for the functional groups of these monomers coincide in the range of 10−15°.26,28 This enables maximal possible overlapping of the molecular graphs for the compounds involved in the superposition and obtaining values of the required parameters with higher accuracy. Undoubtedly, the contributions of the intermolecular interactions realized between the functional groups in the molecules of 2-hydroxycarboxylic and α-amino acids in the thermodynamics of their formation are different. Nevertheless, scheme 1, which uses compounds with the most similar structures, gives the most accurate results. As one can see from Table 2, the value of the systematic error for absolute entropy of formation for 2-hydroxycarboxylic acids according to SAS 1 with Z = NH2 is five times smaller than that for SAS 1 with Z = H. Analogously to 2-hydroxycarboxylic acids described above, there is a systematic error for α-amino acid parameters obtained according to SAS 1 with Z = H. The values of the systematic errors of the thermodynamic parameters of formation for these

ΔH0298,mon,

Figure 3. SAS 2 ([XY] = COOH, Y = OH, Z = NH2) for the calculation of the thermodynamic parameters of formation for aliphatic amide monomers on the example of nonyl amide.

SAS_1 (Z = H)

Table 1. Comparison of the Thermodynamic Parameters of 2-Hydroxycarboxylic Acid Formation Calculated Using the Superposition-Additive Approach with Data Calculated Using the Semiempiric PM3 Method

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Table 2. Values of Standard Deviations of Thermodynamic Parameters of Formation of Bifunctional Amphiphiles Calculated According to the SAS 1and 2 from the Values Calculated Using the PM3 Methoda system SAS_1 SAS_1 SAS_1 SAS_2 a

(X = COOH, Y = OH, Z = H) (X = COOH, Y = NH2, Z = H) (X = COOH, Y = OH, Z = NH2) ([XY] = COOH, Y = OH, Z = NH2)

ΔH0298,mon, kJ/mol 0.34 0.49 0.91 12.97

S0298,mon, J/(mol K)

(0.02) (0.03) (0.01) (0.02)

11.41 9.52 2.32 8.79

(0.68) (0.50) (0.88) (1.16)

ΔG0298,mon, kJ/mol 2.75 3.82 1.07 16.13

(0.21) (0.29) (0.31) (0.37)

N 10 10 10 10

N is sample size.

Figure 4. Geometric structure of aggregates of bifunctional amphiphiles.

α-substituted acids are quite similar. It might be attributed to the fact that the most energetically favored conformers of these acids have almost the same values of the torsion angles for their functional groups with respect to the hydrocarbon chain. In both cases, the incompleteness of the molecular graph overlapping in the regarded superpositions affects mostly the entropy factor. In the case of alkyl amides, one can see that the regarded systematic errors for the description of enthalpy, entropy, and Gibbs’ energy of formation have the highest values. It might be attributed to the nonequivalence of the contributions for the increments of the functional groups of amides and carboxylic acids into the values of the thermodynamic parameters of formation. In addition, it is impossible to obtain the complete molecular graph overlapping for compounds involved in scheme 2 because of conformational differences between carboxylic acids and amides induced by the presence of the π-electronic system described above. Nevertheless, taking into account the systematic errors found in all the described cases decreases significantly the values of the standard deviations of the thermodynamic parameters of formation for bifunctional amphiphiles. The corrected values of these parameters are listed in the braces in Table 2.

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SUPERPOSITION-ADDITIVE APPROACH FOR THE CALCULATION OF THE THERMODYNAMIC PARAMETERS OF THE FORMATION FOR AGGREGATES OF BIFUNCTIONAL ALKANES Calculations of the thermodynamic parameters of formation and clusterization for bifunctional amphiphiles are shown on the example of dimers and tetramers of alkyl amides, α-amino, and 2-hydroxycarboxylic acids. The structure of these aggregates is illustrated in Figure 4. As seen from Figure 4, the structure of the dimers for the considered amphiphiles is similar, and they have an equal number of intermolecular CH···HC interactions. However, the tetramers of the regarded compounds are different because of the fact that α-substituted acid molecules orientate with a higher value of the tilt angle with respect to the normal to the interface than alkyl amide molecules. In addition, the number of the intermolecular CH···HC interactions realized in the tetramers of alkyl amides and α-substituted acids is different. This is caused by the fact that both of the functional groups in the molecule of the αsubstituted acids act as a single hydrophilic part. The size of this single hydrophilic part of the acids is significantly larger than that for amides. As noted recently21 regarding the application of the proposed approach to the calculation of the thermodynamic parameters 16069



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of formation and clusterization of amphiphile aggregates, it should be kept in mind that, in contrast to the individual monomers, these systems involve also intermolecular CH···HC interactions (see Figure 5). Thus, the CH···HC interactions

Figure 6. SAS 4 ([XY] = COOH, Y = OH, Z = NH2) for the calculation of the parameters for aliphatic amide dimers on the example of undecyl amide. Figure 5. SAS 3 (X = COOH, Y = OH, Z = NH2) for the calculation of the parameters for 2-hydroxycarboxylic acid dimers on the example of 2-hydroxyundecanoic acid.

correspondingly for dimers with even n structures with even n. This ensures maximal overlapping of the molecular graphs. Enthalpy and absolute entropy for dimers and tetramers of bifunctional alkanes are calculated using SAS 3 and 4. The obtained values are illustrated in the example of 2hydroxycarboxylic acids and listed in Table 3. The standard deviations of the regarded parameters of the formation obtained from the results of the direct calculation are shown in Table 4 for all considered amphiphilic compounds. It can be clearly seen that the standard deviation of the enthalpy description for 2-hydroxycarboxylic acid aggregates according to SAS 3 with Z = H is lower than that for α-amino acids. In the case of the entropy the error of the description of dimers and tetramers for 2-hydroxycarboxylic acids is 2.5 and 5 times as much as for α-amino acids. However, the standard deviation of the dimers for 2-hydroxycarboxylic acids using SAS 3 with Z = NH2 is lower than that using SAS 3 with Z = H. It will be recalled that SAS 3 with Z = NH2 uses the parameters of monoand bifunctional alkanes, whereas SAS 3 with Z = H exploits the parameters of mono- and unsubstituted alkanes for the calculation of the corresponding parameters of bifunctional alkanes. As noted by the authors of ref 32, for amphiphiles with two closely located functional groups (no larger distance than four methylene units), these functional groups act as a single hydrophilic part of the molecule. In this connection the presence of mentioned systematic errors for the thermodynamic parameters of the aggregate formation, as in the case of monomers, is caused by the following fact. The sum of the increments for individual functional groups of the amphiphiles involved in the superposition is only partially reproduced by the increments of the interactions between the single bifunctional hydrophilic parts of the molecule. The systematic errors are the highest for alkyl amides among all the classes of bifunctional alkanes considered here. As shown in ref 27, it is caused by incomplete overlapping of the molecular graphs of the structures involved in this scheme. This can be attributed to the impossibility to represent intermolecular interactions of the hydrophilic head groups in the amide associates only as a combination of the corresponding interactions realized in the clusters of carboxylic acids, amines, and alcohols. The p-π-conjugated system realized in the functional group of the amide molecule differs from that present in carboxylic acids, and it is absent in the molecules of amines and alcohols. The account of the described errors improves significantly the agreement between the results of direct calculations using the PM3 method and SAS 3 and 4. The corrected values of the thermodynamic parameters of the aggregate formation for 2hydroxycarboxylic acids are listed in braces in Table 3.

realized in clusters of amphiphiles with an odd number of carbon atoms in the monomer chains do not overlap with those realized in the clusters of amphiphiles with an even number of carbon atoms in the monomer chains. This provokes inapplicability of the superposition-additive schemes 1 and 2 to the calculation of the homologue with n carbon atoms out of two homologues with (n − 1) and one with (n − 2) carbon atoms in the alkyl chains. Therefore, in this case the schemes provided with the principle of mutual molecular graphs overlapping would be correct. The molecular graph overlapping is possible if one uses thermodynamic properties of clusters with an odd (even) number of methylene units for the calculation of the corresponding parameters only for clusters having an odd (even) number of CH2 groups. To calculate the parameters of the α-substituted acids we apply SAS 3 with maximal overlapping of the molecular graphs of the structures involved in the mentioned scheme (see Figure 5) SAS_3: A(CnH 2nXY)m /m = A(Cn − 2H 2n − 4XZ)m /m + A(Cn − 2H 2n − 3Y)m /m − A(Cn − 4 H 2n − 7Z)m /m

(3)

where A is the thermodynamic parameter of formation or clusterization of associates per one monomer molecule; X, Y, and Z are the schematic definitions of the structural fragments of the functional groups as in expression 1; m is the number of monomers in a cluster (m = 2 for dimers, m = 4 for tetramers); and n is the number of carbon atoms in the hydrocarbon chain of the monomer. The superposition-additive scheme for the calculation of the thermodynamic parameters of formation and clusterization of alkyl amide aggregates is as follows (see Figure 6) SAS_4: A(CnH 2n + 1[XZ])m /m = A(Cn − 2H 2n − 3[XY])m /m + A(Cn − 2H 2n − 3Z)m /m − A(Cn − 4 H 2n − 7Y)m /m

(4)

where the parameters A, m, and n have the same meaning as in eq 2; X, Y, and Z are the schematic denotations of the structural fragments of the functional groups similar to those used in eq 2. The main difference of SAS 3 and 4 from the corresponding schemes for monomers (SAS 1 and 2) is that for the calculation of the parameters for dimers (and larger aggregates up to 2D films) with odd n SAS 3 and 4 use structures with odd n but 16070



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Table 3. Comparison of Formation Enthalpy and Absolute Entropy for Clusters Per One Monomer Molecule of 2Hydroxycarboxylic Acid with Data Obtained by Direct Calculation Using the PM3 Method dimers system

SAS_3 (Z = H)

C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH

−818.80 −842.14 −869.36 −892.67 −919.89 −943.24 −970.45

509.77 544.59 559.51 598.74 618.41 650.35 669.01

(−818.38) (−841.72) (−868.94) (−892.25) (−919.48) (−942.83) (−970.03)

(484.67) (519.48) (534.40) (573.63) (593.31) (625.25) (643.91)

tetramers

SAS_3 (Z = NH2)

−824.10 −847.89 −874.61 −898.45 −925.30 −949.04 −975.86

504.64 529.18 553.67 586.61 609.34 640.11 660.98

(−818.57) (−842.37) (−869.09) (−892.93) (−919.77) (−943.51) (−970.34)

(489.04) (513.58) (538.07) (571.01) (593.74) (624.51) (645.38)

direct calcd ΔHm/m, kJ/mol −717.61 −741.40 −768.02 −791.93 −818.55 −842.47 −869.10 −893.02 −919.65 −943.59 −970.21 Sm/m, J/(mol·K) 390.93 417.81 442.31 468.59 492.60 518.52 541.96 569.49 592.55 619.90 642.98

SAS_3 (Z = H)

−839.57 −867.97 −895.33 −923.74 −951.08 −979.53 −1006.88

418.19 442.26 466.64 493.29 517.49 541.79 562.84

SAS_3 (Z = NH2)

−848.35 −875.37 −903.65 −931.11 −959.27 −986.85 −1015.27

(−841.26) (−869.65) (−897.01) (−925.43) (−952.77) (−981.21) (−1008.56)

(390.43) (414.50) (438.88) (465.53) (489.74) (514.03) (535.08)

414.01 437.49 459.38 490.59 506.25 540.74 554.74

(−841.99) (−869.01) (−897.29) (−924.76) (−952.91) (−980.50) (−1008.91)

direct calcd −731.65 −758.39 −786.65 −813.37 −841.86 −869.05 −897.26 −924.76 −953.04 −980.56 −1008.84 311.09 334.68 353.70 380.83 397.31 424.28 443.97 466.72 485.18 508.98 526.87

(390.99) (414.48) (436.36) (467.58) (487.90) (517.73) (531.72)

Table 4. Values of Standard Deviations of Thermodynamic Parameters of Aggregate Formation for Bifunctional Amphiphiles Calculated According to the SAS 3 and 4 from the Values Calculated Using the PM3 Methoda system

a

SAS_3 SAS_3 SAS_3 SAS_4




ΔHm/m,mon, kJ/mol Dimers 0.29 (0.42) 5.47 (0.20) 5.52 (0.09) 11.08 (1.35) Tetramers 1.69 (0.51) 7.91 (0.50) 6.36 (0.08) 16.28 (3.21)

Sm/m,mon, J/(mol·K)

N

25.10 9.89 15.60 34.38

(4.90) (5.33) (3.44) (4.68)

7 7 7 7

27.76 5.13 23.02 38.41

(6.37) (3.26) (6.57) (6.95)

7 7 7 7

N is sample size.

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the corresponding values for α-amino acids. Whereas, as it is simple to see from Table 6, the application of scheme 3 with both Z = H and Z = NH2 gives approximately the same results, although this scheme involves different classes of bi- and monofunctional amphiphiles and unsubstituted alkanes, as well. In the case of alkyl amides, the systematic errors of the thermodynamic clusterization parameters for dimers are comparable with those for α-substituted acids, but for tetramers they are mostly higher than for α-substituted acids. The corrected values of the thermodynamic clusterization parameters of α-hydroxylic acids with account of the described systematic errors are listed in braces in Table 6. The final point of this work is devoted to the description of the thermodynamic clusterization parameters for monolayers of bifunctional alkanes in the framework of the superpositionadditive approach. It will be recalled that, as in the case of small clusters for the calculation of the thermodynamic clusterization

SUPERPOSITION-ADDITIVE APPROACH FOR CALCULATION OF THE THERMODYNAMIC PARAMETERS OF CLUSTERIZATION FOR AGGREGATES OF BIFUNCTIONAL ALKANES The calculation of enthalpy, entropy, and Gibbs’ energy of clusterization per one monomer of bifunctional amphiphiles is also carried out according to SAS 3 and 4. The results are shown in Table 5 on the example of 2-hydroxycarboxylic acids. The values of the thermodynamic parameters obtained using scheme 3 with Z = H and Z = NH2 are in good agreement among themselves and with the data obtained by direct calculation. The superposition-additive approach reproduces the values for the clusterization parameters with a systematic error comparatively to the directly calculated data using the PM3 method. Note that the values of the systematic errors of entropy and Gibbs’ energy of clusterization for the 2hydroxycarboxylic acid dimers and tetramers exceed 3-fold 16071



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Table 5. Comparison of Thermodynamic Clusterization Parameters for Small Aggregates of 2-Hydroxycarboxylic Acid Per One Monomer Molecule Obtained Using SAA with Data Obtained from Direct Calculations Using the PM3 Method dimers system C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH

SAS_3 (Z = H)

−25.72 −26.39 −30.93 −31.56 −36.12 −36.77 −41.30

−99.25 −95.96 −112.12 −103.88 −120.58 −114.69 −128.96

3.86 2.21 2.48 −0.60 −0.18 −2.60 −2.87

(−25.27) (−25.94) (−30.49) (−31.11) (−35.67) (−36.33) (−40.86)

(−111.70) (−108.42) (−124.57) (−116.34) (−133.04) (−127.14) (−141.41)

(7.80) (6.15) (6.41) (3.33) (3.75) (1.34) (1.06)

tetramers

SAS_3 (Z = NH2)

−25.81 −26.93 −30.97 −32.13 −36.31 −37.36 −41.50

−94.04 100.48 −107.84 −105.71 −119.19 −114.55 −127.71

2.21 3.01 1.16 −0.63 −0.78 −3.23 −3.44

direct calcd

ΔHclm/m, kJ/mol −14.87 −16.00 −19.95 −21.20 (−25.15) −25.14 (−26.27) −26.38 (−30.31) −30.33 (−31.47) −31.58 (−35.65) −35.52 (−36.70) −36.78 (−40.84) −40.72 ΔSclm/m, J/(mol·K) −78.09 −83.39 −90.38 −95.98 (−106.71) −103.95 (−113.16) −109.58 (−120.51) −117.32 (−118.38) −121.45 (−131.87) −129.54 (−127.23) −132.60 (−140.38) −140.85 ΔGclm/m, kJ/mol 8.40 8.85 6.98 7.40 (6.62) 5.84 (7.42) 6.28 (5.57) 4.63 (3.77) 4.61 (3.62) 3.08 (1.18) 2.73 (0.96) 1.25

SAS_3 (Z = H)

−46.48 −52.22 −56.91 −62.63 −67.31 −73.06 −77.74

(−47.85) (−53.58) (−58.27) (−63.99) (−68.67) (−74.42) (−79.10)

−190.82 (−205.83) −198.29 (−213.29) 204.99 (−219.99) −209.33 (−224.33) −221.50(−236.51) −223.25 (−238.25) −235.12 (−250.13)

10.38 6.88 4.18 −0.25 −1.30 −6.53 −7.67

(13.77) (10.27) (3.15) (2.10) (−3.14) (−4.28)

SAS_3 (Z = NH2)

−50.06 −54.41 −60.01 −64.80 −70.35 −75.18 −80.91

−184.67 −192.17 −202.12 −201.72 −217.61 −213.92 −233.95

4.97 2.86 0.22 −4.69 −5.50 −11.43 −11.20

direct calcd

(−48.56) (−52.91) (−58.51) (−63.29) (−68.85) (−73.68) (−79.41)

−28.91 −32.99 −38.59 −42.64 −48.45 −52.96 58.50 −63.31 −68.91 73.75 −79.34

(−204.78) (−212.27) (−222.23) (−221.83) (−237.72) (−234.02) (−254.05)

−157.94 −166.61 −166.61 −183.88 −199.27 −203.74 −215.08 224.16 −236.97 243.72 −257.37

(12.41) (10.29) (7.65) (2.75) (1.93) (−4.00) (−3.76)

18.15 16.66 14.79 12.16 10.94 7.76 5.60 3.49 1.71 −1.12 −2.65

Table 6. Values of Standard Deviations of Thermodynamic Clusterization Parameters for Bifunctional Amphiphiles Calculated According to the SAS 3 and 4 from Values Calculated Using the PM3 Methoda system

a

ΔHclm/m, kJ/mol



0.45 0.33 0.66 1.49



1.36 2.72 1.50 0.69



6.52 3.83 10.57 1.45

Dimers (0.30) (0.22) (0.09) (0.10) Tetramers (0.51) (0.49) (0.06) (0.35) 2D films (0.61) (0.10) (0.71) (0.26)

ΔSclm/m, J/(mol·K)

ΔGclm/m, kJ/mol

N

12.45 4.51 12.67 17.59

(5.11) (4.18) (3.29) (4.75)

3.94 1.38 4.41 3.82

(1.26) (0.99) (0.95) (1.41)

7 7 7 7

15.00 8.50 20.11 26.53

(5.87) (3.63) (6.16) (5.00)

3.39 5.05 7.43 7.49

(1.90) (2.39) (1.81) (1.72)

7 7 7 7

9.35 52.31 42.20 24.21

(1.96) (2.30) (4.26) (2.99)

3.77 19.42 23.07 11.01

(0.03) (0.59) (0.56) (0.87)

7 7 7 7

N is sample size.

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Table 7. Comparison of Thermodynamic Clusterization Parameters Per One Monomer of 2-Hydroxycarboxylic Acid 2D Films Obtained Using SAS 3 with Data Calculated Earlier in the Framework of the Quantum Chemical Model system

SAS_3 (Z = H)

SAS_3 (Z = NH2)

quantum chemical model

ΔHclm/m, kJ/mol C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18OHCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH

−92.37 −102.78 −113.19 −123.61 −134.02 −144.43 −154.85

−383.27 −404.27 −425.27 −446.27 −467.27 −488.27 −509.27

(−85.84) (−96.26) (−106.67) (−117.08) (−127.50) (−137.91) (−148.32)

−96.28 −106.74 −117.20 −127.66 −138.12 −148.58 −159.04 ΔSclm/m, J/(mol·K)

−328.70 −350.85 −372.99 −395.14 −417.29 −439.44 −461.59

(−373.92) (−394.92) (−415.92) (−436.92) (−457.92) (−478.92) (−499.92)

(−85.71) (−96.17) (−106.63) (−117.09) (−127.55) (−138.01) (−148.47)

−46.36 −56.47 −66.58 −76.68 −86.79 −96.90 −107.01 −117.11 −127.22 −137.33 −147.43

(−370.89) (−393.04) (−415.19) (−437.34) (−459.49) (−481.64) (−503.79)

−296.96 −316.98 −337.01 −357.03 −377.05 −397.08 −417.10 −437.13 −457.15 −477.17 −497.20

ΔGclm/m, kJ/mol C6H12OHCOOH C7H14OHCOOH C8H16OHCOOH C9H18O = HCOOH C10H20OHCOOH C11H22OHCOOH C12H24OHCOOH C13H26OHCOOH C14H28OHCOOH C15H30OHCOOH C16H32OHCOOH

21.85 17.69 13.54 9.38 5.23 1.07 −3.08

(25.62) (21.46) (17.31) (13.15) (9.00) (4.84) (0.69)

1.67 −2.19 −6.05 −9.91 −13.77 −17.63 −21.49

(24.74) (20.88) (17.02) (13.16) (9.30) (5.44) (1.58)

42.13 37.99 33.85 29.71 25.57 21.43 17.29 13.15 9.01 4.87 0.73

obtained using SAS 3 and 4 and the quantum chemical model. Different from the 2-hydroxycarboxylic acid monomers, the values of the systematic error of the thermodynamic parameters for monolayer formation are considerably higher in the case of use of SAS 3 with Z = NH2 than of use of SAS 3 with Z = H. Taking into account the mentioned systematic errors improves essentially the agreement between the calculated data and decreases the values of the standard deviations for the thermodynamic clusterization parameters per one monomer of monolayer (see the values in the braces in Tables 6 and 7). As seen from the data listed above, the schemes involving compounds having structures most similar to the structure of the required compound give the most accurate results for the thermodynamic parameters of monomer formation. That means, if it is required to find the value of the thermodynamic formation parameter for one α-substituted acid (2-hydroxycarboxylic acid) using different superpositions, one should exploit a scheme that involves another α-substituted acid (αamino acid). The structures of these acids should be as similar as possible. However, for the calculation of thermodynamic clusterization parameters of 2D films, it is better to use a scheme that involves the corresponding parameters for

parameters per one monomer of monolayers with an odd (even) number of methylene units (n), we use these parameters for structures having an odd (even) number of carbon atoms in the alkyl chain of the amphiphiles involved. The calculated thermodynamic clusterization parameters per one monomer of 2D films are listed in Table 7 in the example of 2-hydroxycarboxylic acids. The column “Quantum chemical model” shows the clusterization parameters per one monomer molecule calculated recently in the frameworks of the quantum chemical approach.28 This approach is capable of the calculation of the thermodynamic clusterization parameters of amphiphilic aggregates of any dimension up to 2D films. The data listed in Tables 6 and 7 reveal the presence of the systematic error for the description of the thermodynamic clusterization parameters of monolayers, as in the case of small aggregates. It is attributed to the incomplete reproduction of the increments of intermolecular interactions of the hydrophilic parts of bifunctional alkanes by the sum of the corresponding increments for the hydrophilic parts of monofunctional alkanes. As seen from Table 6, this incomparability affects most significantly the clusterization entropy. This gives rise to a lower agreement between the values of the Gibbs’ energy 16073



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for bifunctional amphiphiles can be taken into account using additive corrections. The consideration of these corrections allows the assessment of the thermodynamic parameter values of formation for 2-hydroxycarboxylic acids, α-amino acids, and alkyl amides with the following values of standard deviations: 0.01, 0.03, and 0.02 kJ/mol for enthalpy, 0.88, 0.50, and 1.16 J/ (mol·K) for absolute entropy, and 0.31, 0.29, and 0.37 kJ/mol for Gibbs’ energy, respectively. The values of the thermodynamic clusterization parameters per one monomer molecule of 2D films for the considered bifunctional amphiphiles are estimated with deviations in the range of 0.10−0.61 kJ/mol, 1.96−2.99 J/(mol·K), and 0.03−0.87 kJ/mol for enthalpy, entropy, and Gibbs’ energy of clusterization, respectively.

monosubstituted alkanes (carboxylic acids and alcohols or amines). Thus, different superposition-additive schemes can be reasonably used to calculate the thermodynamic parameters of formation and clusterization of bifunctional amphiphiles. However, it is necessary to observe several requirements. The structural peculiarities for monolayers involved in one or another scheme should be as similar as possible. That means the number of the intermolecular CH···HC interactions in the aggregates has to be equal, and the tilt angles of the amphiphile molecules with respect to the normal to the interface should be close together. Implementation of the mentioned conditions allows realizing a maximal overlapping of the molecular graphs for compounds involved in the superposition.

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CONCLUSIONS This work shows the possibility to apply the superpositionadditive approach to the calculation of the thermodynamic parameters of formation and clusterization for bifunctional amphiphiles using calculated data existing for corresponding monofunctional compounds. The application of this approach is illustrated in the example of 2-hydroxycarboxylic acids, αamino acids, and alkyl amides in the framework of different schemes. These schemes involve molecular graphs overlapping for different classes of bi-, mono-, and unsubstituted alkanes. As already previously shown, the best agreement between the results of direct calculations and exploited schemes is provided by schemes with the maximal molecular graph overlapping of corresponding compounds. It was found that the thermodynamic parameters of formation and clusterization of α-substituted acids can be calculated as a sum of the corresponding parameters of monosubstituted alkanes after subtraction of unsubstituted alkanes in the framework of the superposition-additive approach. Along with this scheme it is possible to use another one. It exploits the parameters of one class of bifunctional amphiphiles and two classes of monofunctional compounds. It was concluded that the last scheme gives the most accurate results for the thermodynamic parameters of 2-hydroxycarboxylic acid formation. This is caused by the fact that this scheme involves conformers of two α-substituted acids with most similar structure providing a maximal molecular graph overlapping for compounds participating in the superposition. However, the first scheme is more appropriate for the calculation of the thermodynamic clusterization parameters. In addition when calculating the thermodynamic parameters of formation and clusterization for aggregates of different dimension up to 2D films the structural peculiarities of the aggregates involved in one or another scheme should be as similar as possible. That means the number of the intermolecular CH···HC interactions in the aggregates should be the same, and the tilt angles of the amphiphile molecules with respect to the normal to the interface should be close to each other. This provides maximal overlapping of the molecular graphs as the key point for the realization of the superpositionadditive approach. It was shown that the superposition-additive approach partially reproduces the values of the increments of the intraand intermolecular interactions between the functional groups in the molecules of bifunctional alkanes by the corresponding values of the increments for monosubstituted alkanes. It was demonstrated that the systematic errors for the description of the thermodynamic parameters of formation and clusterization

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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