Thermodynamics of the Clusterization Process of ... - ACS Publications

J. Phys. Chem. B 2009, 113, 4347–4359

4347

Thermodynamics of the Clusterization Process of Cis Isomers of Unsaturated Fatty Acids at the Air/Water Interface Yu. B. Vysotsky,† E. A. Belyaeva,† V. B. Fainerman,‡ D. Vollhardt,*,§ E. V. Aksenenko,| and R. Miller§ Donetsk National Technical UniVersity, 58 Artema Str., 83000 Donetsk, Ukraine; Donetsk Medical UniVersity, 16 Ilych AVenue, Donetsk 83003, Ukraine; Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/ Golm, Germany; and Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky AVenue, 03680, KieV, Ukraine ReceiVed: October 6, 2008; ReVised Manuscript ReceiVed: January 16, 2009

In the framework of the semiempirical PM3 method, the thermodynamic parameters of cis isomers of unsaturated carboxylic acids at the air/water interface are studied. The model systems used are unsaturated cis fatty acid of the composition ∆ ) 12-15 and ω ) 6-11, where ∆ and ω refer to the number of carbon atoms between the functional group and double bond, and that between the double bond and methyl group, respectively. For dimers, trimers, and tetramers of the four acid series, the thermodynamic parameters of clusterization are calculated. It is shown that the position of the double bond does not significantly affect the values of thermodynamic parameters of formation and clusterization of carboxylic acids for equal chain lengths (n ) ∆ + ω). The calculated results show that for cis unsaturated fatty acid with odd ∆ values the spontaneous clusterization threshold corresponds to n ) 17-18 carbon atoms in the alkyl chain, while for monounsaturated acids with even ∆ values this threshold corresponds to n ) 18-19 carbon atoms in the alkyl chain. These differences in the clusterization threshold between the acids with even and odd ∆ values are attributed to the formation of additional intermolecular hydrogen bonds between the ketonic oxygen atom of one monomer and the hydrogen atom linked to the R-carbon atom of the second monomer for the acids with odd ∆ values or between the hydroxyl oxygen atom of one monomer and hydrogen atom linked to the R-carbon atom of the second monomer for the acids with even ∆ values. The results obtained in the study agree satisfactorily with our experimental data for cis unsaturated nervonic (∆15, ω9) and erucic acids (∆13, ω9), and published data for some fatty acids, namely cis-16-heptadecenoic (∆16, ω1), cis-9-hexadecenoic (∆7, ω9), cis-11eicosenoic (∆11, ω9) and cis-9-octadecenoic acid (∆9, ω9). Introduction Fatty carboxylic acids with alkyl chain lengths of 12-24 carbon atoms (the most widespread are those with 16-18 carbon atoms in the alkyl chain) are the constituents of phospholipids and glycolipids.1 Having high biological activity, they maintain the cell membranes structure, they participate in the cholesterol transfer and exchange as well as in the synthesis of prostaglandins and other substances, and they are important for vital processes and are necessary for proper operation of the optic canal and affect the immunity status. In general, fatty carboxylic acids are of enormous importance for the living organism. Unsaturated fatty acids are abundant in biological systems.2,3 Linked in 2-position of phospholipids, the unsaturated chains, varying in type, number, position, and geometric configuration of unsaturated carbon-carbon bonds, are responsible for the broad variation in the biological properties of lipids. They affect essentially the membrane characteristics. This concerns not only the improvement of fluidity but also the control of the phase transition behavior. Model studies of the behavior of unsaturated fatty acids at the air/water interface have provided useful information for * Corresponding author. † Donetsk National Technical University. ‡ Donetsk Medical University. § Max Planck Institute of Colloids and Interfaces. | Institute of Colloid Chemistry and Chemistry of Water.

understanding the related phenomena. In numerous studies it has been found that the main characteristics of unsaturated carboxylic acid monolayers deviate considerably from those of the corresponding saturated fatty acids.4-15 Recently, phase behavior, domain morphology, and two-dimensional lattice structure of some selected unsaturated fatty acid monolayers have been compared, using surface pressure-molecular area (π-A) isotherms, Brewster angle microscopy (BAM), and grazing incidence X-ray diffraction (GIXD) techniques.16 The results suggest an obstructive effect of the double bond in the alkyl chain on the molecular ordering. Simultaneously, the thermodynamic analysis, performed on the basis of the π-A isotherms, have shown that the introduction of unsaturation into the hydrocarbon chain affects the thermodynamic characteristics of the monolayer in a similar way as shortening the alkyl chain length. The objective of the present work is to obtain the thermodynamic function characteristics for the clusterization of the cis isomer of unsaturated carboxylic acids within the framework of PM3 semiemperical method, because only this method predicts the decrease of the Gibbs’ clusterization energy with the increase of the alkyl chain length, which (in perfect agreement with the experimental data) indicates that the formation of clusters becomes possible only if the hydrocarbon chain is long enough. The cis unsaturated carboxylic acids are most widespread in nature, whereas the trans acids are quite

10.1021/jp808834a CCC: $40.75  2009 American Chemical Society Published on Web 03/06/2009

4348 J. Phys. Chem. B, Vol. 113, No. 13, 2009

Vysotsky et al.

scarce.16-18 Therefore, in the present study we focus on the calculations of the thermodynamic parameters of clusterization for the cis unsaturated carboxylic acids at the air/water interface. Unsaturated carboxylic acids with ∆ ) 13 and ∆ ) 15 were chosen as the model systems because of their practical importance. They are constituents of many lipids; also, the application of nervonic acid (∆ ) 15, ω ) 9) for the treatment of disseminated sclerosis has recently been studied.19 In this publication, ∆ and ω refer to the number of carbon atoms between the functional group and double bond, and that between the double bond and methyl group, respectively. The thermodynamic parameters of the clusterization for the cis unsaturated carboxylic acids (∆ ) 14, ω ) 6-11) and (∆ ) 12, ω ) 6-11) are calculated in the present work to compare the clusterizationrelated features of cis unsaturated carboxylic acids with even and odd ∆. Methods The optimization of the geometry for the clusters of cis unsaturated carboxylic acids studied here was performed by the MOPAC2000 software suite using the BFGS (Broyden-FletcherGoldfarb-Shanno) algorithm.20 All calculations were made applying the semiempirical PM3 method that takes proper account for the intermolecular hydrogen-hydrogen interactions. The MOPAC2000 suite provides for the calculation of vibrational frequencies which are subsequently used for the evaluation of entropy values. In this respect, it should be noted that in the present version of the software the contributions from frequencies below 100 cm-1 are neglected. However, these frequencies are essential for the correct description of the systems studied here. Therefore, the relevant contributions from these frequencies were manually calculated using the formulas listed in the MOPAC2000 software manual.20 The quantum chemical methods currently available (also the semiempirical PM3 method) are incapable of a correct description of amphiphilic molecules which are located at the interface and, therefore, partially immersed into each (aqueous and gas) phase. That is why in the present study the air/water interface was accounted for implicitly. The relative orientation of the molecules of the cis unsaturated carboxylic acids was forced to correspond to their relative orientation at the interface. Also, intermolecular hydrogen-hydrogen interactions exist between these molecules, which are characteristic for 2D monolayers. It should be noted that in our previous studies21-26 in which the clusterization of other classes of amphiphilic compounds at the air/water interface was considered, it was shown that this approximate approach is capable of a quite adequate description of the clusterization process that takes place at the interface, and the calculated results agree well with the experimental data. Monomers. The position of the functional acid group relative to the hydrocarbon radical was analyzed earlier,21 where the clusterization of fatty acids was studied. The potential energy minima were located, and an additional optimization of the systems was performed in the vicinity of these minima. In this way, the conformations of carboxylic acids which correspond to the energy minima were determined. It was shown21 that the most energetically preferable conformation of dimers (and also more complicated clusters) is the dimer characterized by the torsion angle ∠HOCC ) 180° and torsion angles ∠OCCC ) 284° (-76°) and 268° (-92°), see dimer 2 in ref 21. The main difference of the unsaturated carboxylic acids studied here from the saturated acids studied earlier, is the presence of the double bond. This bond is located far from the functional group and does not affect it significantly. The analysis of possible relative

Figure 1. Optimized geometrical structures of initial monomers.

orientations of OH group and ketonic oxygen atom in the carboxylic group of the monomer in saturated fatty acids was reported in our publication.21 Therefore, the initial structures of monomers 1 for unsaturated acids were constructed on the basis of the found values of ∠HOCC and ∠OCCC angles. Figure 1 illustrates the structures of two possible monomers of the studied cis unsaturated carboxylic acids: monomer 1 described above, and monomer 2, for which the torsion angle ∠OCCC ) 92°. These monomers 2, similar to monomers 1, participate in the formation of more complicated structures, so that we performed the calculations of the thermodynamic parameters of formation for the homologous series of monomers 2. It was shown that the monomers 2 are isoenergetical to the monomers 1; see Table 1. Therefore, in what follows, only the thermodynamic parameters of monomers 1 were used. To study the effect of the position of double bond on the values of thermodynamic parameters, we have calculated the thermodynamic characteristics (enthalpy and Gibbs’ energy) of the formation of monomers from elementary substances, and their absolute entropies for various ∆ and ω values. The total length of the alkyl chain n ) ∆ + ω was varied from 18 to 26 carbon atoms. The calculated values are listed in Table 1. It is seen from these data that the thermodynamic characteristics of monomers with the same radical length but different position of double bond (i.e., different ∆ and ω values) are the same within the calculation errors; one can conclude therefore that the position of the double bond does not affect significantly the thermodynamic parameters of the monomers. Note that the differences between the values of absolute entropy are most significant. We ascribe this fact to the effects related to the free rotation of the methylene groups.

Cis Isomers of Unsaturated Fatty Acids

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4349

TABLE 1: Thermodynamic Parameters of Formation for Monomers of Cis Unsaturated Carboxylic Acids monomer 1 n/∆

12

18 19 20 21 22 23 24 25 26

-672.80 -695.47 -718.15 -740.84 -763.52 -786.19 -808.88 -831.57 -854.25

18 19 20 21 22 23 24 25 26 18 19 20 21 22 23 24 25 26

13

monomer 2 14

15

15

° , kJ/mol ∆H298,mon -672.79 -672.81 -695.47 -695.43 -718.15 -718.11 -740.83 -740.76 -763.52 -763.45 -786.19 -786.10 -808.88 -808.79 -831.57 -831.40 -854.25 -854.10

-672.72 -695.47 -718.13 -740.79 -763.48 -786.15 -808.83 -831.51 -854.19

-672.79 -695.47 -718.15 -740.82 -763.50 -786.15 -808.83 -831.52 -854.18

792.68 825.66 853.06 886.95 914.92 948.63 980.75 1008.88 1043.09

S298,mon ° , J/(mol · K) 795.00 792.24 824.02 826.19 855.39 851.98 883.30 888.06 914.84 914.34 946.91 948.00 977.02 975.76 1007.61 1010.83 1037.94 1044.22

800.36 830.98 860.46 885.93 916.79 947.69 978.76 1007.06 1035.15

791.86 825.26 852.88 891.83 918.52 946.74 976.88 1006.44 1034.72

-55.91 -47.81 -38.05 -30.23 -20.64 -12.76 -4.41 5.13 12.86

∆G298,mon ° , kJ/mol -56.60 -55.79 -47.32 -47.93 -38.74 -37.69 -29.14 -30.48 -20.62 -20.40 -12.24 -12.47 -3.30 -2.83 5.51 4.71 14.39 12.67

-58.13 -49.39 -40.23 -29.88 -21.15 -12.43 -3.76 5.73 15.28

-55.66 -47.69 -37.99 -31.67 -21.70 -12.15 -3.20 5.91 15.42

For the unsaturated carboxylic acids considered here, neither experimental values of thermodynamic parameters nor values calculated by higher level methods are presently available. Therefore, we cannot immediately compare our results with any independent data. However, in our recent paper,21 where saturated carboxylic acids were studied, the comparison of the thermodynamic parameters calculated using the semiempiric PM3 method with the corresponding experimental data has clearly indicated the applicability of the PM3 method for the characterization of these systems. It could be reasonably supposed that the presence of one localized double bond cannot introduce major biases into the calculated values of the thermodynamic characteristics. As described in detail below, it could be expected that the errors in the thermodynamic characteristics caused by the presence of the localized double bond should compensate each other because the thermodynamic parameters for both the monomers and the clusters are correctly described by the additive scheme. The calculated thermodynamic parameters were used to obtain the correlation dependencies of the enthalpy of formation of ° and their monomers from elementary substances ∆H298,mon ° on the alkyl chain length n: absolute entropy S298,mon ∆H◦298,mon ) [-(22.68 ( 0.003)n - (264.60 ( 0.06)] kJ/mol (R ) 0.99999; S ) 0.04 kJ/mol; N ) 36) (1) ◦ S298,mon ) [(30.55 ( 0.17)n + (244.80 ( 3.74)] J/(mol · K) (R ) 0.9995; S ) 2.61 J/(mol · K); N ) 36) (2)

where R is the regression coefficient, S is the standard deviation, and N is the sampling amount. The calculated values of slopes

agree well with those reported earlier for long-chain saturated carboxylic acids,21 and also for long-chain alcohols,22,23 thioalcohols,24 amines,25 and nitriles.26 Dimers, Trimers, and Tetramers. The initial structures of dimers, trimers, and tetramers of acids studied were constructed on the basis of the optimized structures of the monomers 1 and 2. In particular, the dimers 1, trimers 1, tetramers 1, and tetramers 3 were formed only by the monomers 1, whereas the dimers 2 and 3, the trimers 2, and the tetramers 2 and 4 were formed by the alternating monomers 1 and 2. For example, the structures of the dimers, trimers, and tetramers for carboxylic acids with ∆ ) 15 are shown in Figure 2. The arrows indicate the orientation of dipole moments in the functional groups. For corresponding clusters with ∆ ) 14, 13, and 12, the structures are rather similar to those shown in Figure 2. However, the clusters with odd ∆ ) 13 and 15 are somewhat different from those with even ∆ ) 12 and 14, because in even ∆ clusters two intermolecular hydrogen bonds are present which are absent in the clusters with odd ∆. One of these bonds is located between the ketonic oxygen atom of one molecule and the hydrogen R-atom of the second molecule whereas the other bond links the hydroxyl oxygen of one molecule and the hydrogen R-atom of the second molecule; see Figure 3. To ensure that these bonds are not formed because of edge effects, i.e., that these bonds still persist in the infinite clusters, the optimizations of larger clusters (composed of 16 monomers) were performed, indicating that these bonds exist also in such clusters where they are surrounded by another layer of molecules. Therefore, to obtain the correct description of clusters with ∆ ) 12 and 14, these links between the carboxyl groups should be taken into account. Another difference of the cis unsaturated carboxylic acids from the saturated compounds studied earlier is the presence of the intermolecular interactions between the hydrogen atoms attached to the carbon atoms linked by the double bonds. It was shown earlier21-26 that the “a” type intermolecular hydrogen-hydrogen interactions introduce negative contribution to the Gibbs’ clusterization energy, thus being the main cause of the clusterization process in amphiphilic compounds. In this regard, it seems interesting to elucidate the effect caused by the interactions between the hydrogen atoms attached to the sp3- and sp2-hybridized carbon atoms. For all dimers, trimers, and tetramers with ∆ ) 12-15, the ° - m∆H298,mon ° and entropies standard enthalpies ∆HCl m ) ∆H298,cl ∆SCl ° - mS298,mon ° of clusterization were calculated, where m ) S298,cl m is the number of monomers which comprise the cluster, and the subscript “Cl” refers to clusters. The calculated values are listed in Table 2. The corresponding Gibbs’ energies are easily Cl Cl Cl calculated as ∆Gm ) ∆Hm - T∆Sm . In Table 2 the thermodynamic parameters of clusterization are grouped according to the total radical length and the ω value. Therefore, the structures which possess equal radical length but different ∆ and ω (i.e., different position of the double bond) are listed on the same level. It is seen that, similar to the monomers discussed above, the thermodynamic parameters of clusterization for dimers, trimers, and tetramers with equal chain length are quite similar. Using the calculated thermodynamic parameters (enthalpy and entropy) of clusterization, the corresponding correlation dependencies on the number of the intermolecular hydrogenhydrogen interactions and on the number of intermolecular interactions in the functional groups of various types in the cis unsaturated carboxylic acids were obtained:


∆HCl 298 ) [-(9.2 ( 0.08)Ka - (2.10 ( 0.22)K (21.61 ( 0.37)(nff(u) + nfr(u) + nrf(u) + nff(g)) - (23.73 ( 0.82)nfr(g) (19.90 ( 0.89)nfr(g)] kJ/mol (R ) 0.9993; S ) 1.38 kJ/mol; N ) 222) (3) ∆SCl 298 ) [-(18.4 ( 0.60)Ka - (37.63 ( 3.12)K (132.92 ( 5.03)nff(u) - (134.21 ( 11.23)nfr(u) (95.51 ( 12.02)nrf(u) - (138.02 ( 6.17)nff(g) (161.21 ( 11.23)nfr(g) (111.39 ( 2.02)nrf(g)]J/(mol · K) (R ) 0.99;S ) 9.17 J/(mol · K); N ) 222) (4) Here, similar to our previous studies,25 the increments related to the alcohol skeleton were used. The arrow subscripts which label the numbers of the corresponding intermolecular interactions between the functional groups (nfr, nff, and nrf) indicate schematically the direction of dipole moments in the interacting groups; see Figure 2. The (u) and (g) refer to the clusters with odd

Vysotsky et al. and even ∆ values, respectively. Ka is the number of intermolecular hydrogen-hydrogen interactions, and K is the number of intermolecular H-H interactions between the hydrogen atoms attached to the sp3- and sp2-hybridized carbon atoms; see Figure 4. The values of slopes at Ka agree well with those calculated earlier for saturated carboxylic acids,21 and also for long-chain alcohols,22,23 thioalcohols,24 amines,25 and nitriles.26 This fact indicates that the intermolecular hydrogen-hydrogen bonds responsible for the formation of clusters from corresponding monomers are of the same nature. This is essential for the construction of a general additive scheme (see below) that describes the thermodynamic parameters of clusterization for amphiphilic compounds considered here. It was noted above that cis unsaturated fatty acids with ∆ ) 12 and 14 contain two additional intermolecular hydrogen bonds; see Figure 3. These bonds were accounted for in the correlation parameters, giving rise to the coefficients at nff(g), nfr(g), nrf(g); see eqs 3 and 4. Consider now the energetic contributions from these interactions. It is seen from Figure 5(1) which shows the infinite cluster that the “heads” are oriented in a similar way, and therefore, their orientation leads to equal contributions of additional hydrogen bonds. On the contrary, for the clusters shown

Figure 2. Optimized geometrical structures of dimers, trimers, and tetramers of cis unsaturated carboxylic acids (∆ ) 15, ω ) 6 - 11).



∆SCl 298 ) [-(18.4 ( 0.60)Ka - (37.63 ( 3.12)K (132.92 ( 5.03)nff - (134.21 ( 11.23)nfr (95.51 ( 12.02)nrf] J/(mol · K) (R ) 0.99; S ) 9.17 J/(mol · K); N ) 222) (6)

Figure 3. Orientation of additional intermolecular hydrogen bonds for cis unsaturated carboxylic acids with ∆ ) 12 and 14: (a) q direction (cf. Figure 5); (b) p direction (cf. Figure 5); and (c) view from above.

Cl Cl Cl ) ∆H298 - T∆S298 , where T is the Using the relation ∆G298 absolute temperature in K, one can express the correlation dependence of the Gibbs’ clusterization energy on these parameters:

∆GCl 298 ) [-(3.72 ( 0.26)Ka + (9.11 ( 1.15)K + (18.00 ( 1.86)nff + (18.38 ( 3.72)nfr + (6.85 ( 3.95)nrf] kJ/mol (7)

Figure 4. Intermolecular interactions between hydrogen atoms linked to sp3- and sp2-hybridized carbon atom (schematically).

in Figure 5(2), the orientation of the heads alternates from -92° to 92°, so that the energetic of bonds which are formed between the atoms of the molecules which constitute dimer 2 should be different from that for dimer 3. It is seen from Figure 3 that, in this case, the hydrogen bonds alternate according to the alternation of the monomers, and form the clusters shown in Figure 5(2): the OH · · · H bond always corresponds to dimer 2, whereas the OH · · · H bond always corresponds to dimer 3. The contributions brought up by these bonds can be calculated on the basis of the considerations presented above. Noting that the clusters with even ∆ differ from the clusters with odd ∆ only by additional hydrogen bonds, one can calculate the contributions of these additional bonds because the difference between the energetic contribution from the intermolecular interaction between the functional groups of molecules with even and odd ∆’s (see Table 3) to ensure that these contributions are statistically insignificant. Therefore, it can be assumed that the contributions of “heads” proper for acid molecules with even and odd ∆ values are equal to each other, and thus

nff(u) ) nff(g) ) nff, nfr(u) ) nfr(g) ) nfr, nrf(u) ) nrf(g) ) nrf

It is seen that, in contrast to the contributions from the H-H interactions, the bonds formed between the hydrogen atoms linked to the sp3- and sp2-hybridized carbon atoms give rise to the positive (destabilizing) contribution to the Gibbs’ clusterization energy. Infinite Clusters. We proceed next with the application of the additive scheme defined by eqs 5 and 6 above to the infinite two-dimensional (2D) clusters following the lines of our earlier publications.21-26 In what follows, the application of this scheme to unsaturated fatty acids is described in more detail. Figure 5 illustrates two possible structures of these infinite clusters. The structure presented in Figure 5(1) is based on dimers 1 with all the functional groups oriented in a similar way. On the contrary, the cluster shown in Figure 5(2) is based on dimers 2 and 3, with alternating functional groups. The intermolecular H-H interactions between hydrogen atoms linked to the sp3- and sp2hybridized carbon atoms (the number of these interactions is denoted as K) exist only within the p direction and give rise to the destabilizing contribution to the Gibbs’ energy of the clusterization process (note that the p and q directions are chosen according to the direction of the intermolecular H-H interactions which exist between the hydrocarbon radicals of the unsaturated acid molecules; see Figure 5). To calculate (and then to compare) the clusterization parameters for these two types of clusters, the number of intermolecular interactions in the functional groups (nf) and the number of intermolecular hydrogen-hydrogen interactions (Ka) in the clusters of any dimension, with both even and odd ∆ values, should be calculated. Another relevant quantity is the number of the intermolecular H-H interactions between hydrogen atoms linked to the sp3- and sp2-hybridized carbon atoms (K). These values are

nf ) (p - 1)q + (q - 1)p

(8)

Ka(u) ) [(p - 1)q + (q - 1)p]{(n - 5)/2}

(9)

Ka(g) ) [(p - 1)q + (q - 1)p]{(n - 6)/2}

(10)

K ) 2q(p - 1)

(11)

and eqs 3 and 4 become

∆HCl 298

) [-(9.2 ( 0.08)Ka - (2.10 ( 0.22)K (21.61 ( 0.37)(nff + nfr + nrf)] kJ/mol (R ) 0.9993; S ) 1.38 kJ/mol; N ) 222) (5)

where (u) and (g) refer to the values characteristic for the clusters with even and odd ∆ values, respectively, the braces {...} denote the integer part of a number. To calculate the thermodynamic parameters of clusterization for infinite structures, one should divide the expressions 8-11


Vysotsky et al.

TABLE 2: Thermodynamic Parameters of Clusterization for Cis Unsaturated Carboxylic Acids ∆13 n

Cl ∆H298 ,

kJ/mol

∆15

Cl ∆S298 ,

J/(mol · K)

Cl ∆H298 ,

kJ/mol

18 19 20 21 22 23 24 25 26

-88.13 -89.43 -98.54 -99.95 -108.87 -110.40 -

-307.32 -310.51 -329.70 -332.02 -358.85 -356.38 -

-98.51 -99.85 -108.91 -110.25 -119.29 -120.68

18 19 20 21 22 23 24 25 26

-89.13 -90.41 -99.54 -101.01 -109.91 -111.40 -

-304.16 -306.38 -326.38 -330.66 -355.43 -355.25 -

-89.13 -90.41 -99.54 -101.01 -109.91 -111.40

18 19 20 21 22 23 24 25 26

-89.13 -90.41 -99.54 -101.01 -109.91 -111.40 -

-304.16 -306.38 -326.38 -330.66 -355.43 -355.25 -

-98.51 -99.85 -108.91 -110.25 -119.29 -120.68

18 19 20 21 22 23 24 25 26

-176.13 -178.28 -196.93 -199.44 -217.36 -220.58 -

-605.87 -654.59 -662.41 -703.40 -791.44 -763.71 -

-196.96 -199.05 -217.76 -220.33 -238.50 -241.30

18 19 20 21 22 23 24 25 26

-177.01 -179.18 -197.65 -200.31 -218.38 -221.45 -

-602.22 -651.02 -675.35 -696.06 -759.16 -758.49 -

-197.87 -200.03 -218.66 -221.34 -239.45 -242.30

18 19 20 21 22 23 24 25 26

-264.00 -267.03 -295.08 -298.75 -326.18 -330.34 -

-1000.56 -977.12 -1057.27 -1060.59 -1183.71 -1131.06 -

-295.32 -298.29 -326.44 -330.06 -357.66 -361.55

18 19 20 21 22 23 24 25 26

-264.65 -267.73 -295.78 -299.44 -326.88 -331.01 -

-995.26 -973.72 -1056.37 -1062.66 -1141.66 -1125.17 -

-296.02 -299.15 -326.78 -330.34 -357.82 -361.92

18 19

-341.51

-1139.34

-

∆12

Cl ∆S298 ,

J/(mol · K)

Cl ∆H298 ,

kJ/mol

dimer 1 -78.50 -80.43 -88.91 -323.71 -91.01 -329.39 -99.26 -352.76 -100.67 -350.84 -375.26 -365.93 dimer 2 -82.13 -83.30 -92.54 -304.16 -93.81 -306.38 -102.88 -326.38 -102.03 -330.66 -355.43 -355.25 dimer 3 -80.40 -79.07 -90.81 -323.71 -89.60 -329.39 -101.16 -352.76 -102.52 -350.84 -375.26 -365.93 trimer 1 -158.84 -160.88 -179.64 -709.30 -181.87 -700.27 -200.40 -759.83 -200.28 -768.18 -800.84 -802.09 trimer 2 -157.51 -159.60 -178.33 -711.57 -180.62 -705.80 -199.06 -764.53 -198.76 -774.25 -806.74 -809.36 tetramer 1 -237.98 -240.77 -269.11 -1054.71 -272.43 -1048.17 -300.21 -1133.79 -302.22 -1142.19 -1205.20 -1203.69 tetramer 2 -241.20 -244.11 -272.40 -1068.66 -275.83 -1065.03 -303.60 -1135.73 -305.31 -1143.12 -1203.04 -1201.17 tetramer 3 -305.74 -312.23

∆14

Cl ∆S298 ,

J/(mol · K)

Cl ∆H298 ,

Cl kJ/mol ∆S298 , J/(mol · K)

-292.06 -297.62 -311.42 -332.97 -339.36 -340.87 -

-89.72 -90.73 -100.12 -101.76 -110.40 -112.03 -

-340.75 -325.93 -358.83 -365.65 -384.50 -391.86 -

-307.34 -305.66 -327.18 -334.30 -353.87 -356.95 -

-92.63 -94.14 -103.06 -104.73 -113.24 -115.09 -

-319.82 -334.32 -334.14 -357.26 -368.03 -388.51 -

-303.74 -307.29 -323.74 -335.92 -352.89 -348.29 -

-88.65 -89.55 -98.96 -100.55 -109.15 -110.92 -

-332.83 -312.69 -355.88 -360.64 -377.93 -386.72 -

-634.52 -631.37 -739.04 -734.09 -782.72 -817.66 -

-179.82 -181.85 -200.66 -203.08 -221.41 -224.07 -

-734.87 -758.56 -807.17 -785.28 -831.87 -874.38 -

-612.39 -609.82 -717.83 -718.31 -765.16 -794.47 -

-180.31 -182.56 -199.15 -203.92 -219.76 -224.86 -

-715.99 -727.35 -760.16 -809.61 -802.04 -835.36 -

-1015.57 -1055.00 -1118.76 -1085.13 -1151.04 -1192.10 -

-269.61 -272.58 -300.77 -304.14 -331.95 -335.49 -

-1081.78 -1134.88 -1197.80 -1154.32 -1227.71 -1239.30 -

-1000.18 -1001.04 -1064.04 -1077.86 -1137.99 -1217.18 -

-241.20 -244.11 -272.40 -275.83 -303.60 -305.31 -

-1000.18 -1001.04 -1064.04 -1077.86 -1137.99 -1217.18 -

-1040.78 -1126.49

-

-



TABLE 2: Continued ∆13 n

Cl ∆H298 ,

∆15

Cl ∆S298 ,

kJ/mol

J/(mol · K)

Cl ∆H298 ,

kJ/mol

20 21 22 23 24 25 26

-347.97 -383.33 -390.25 -424.91 -432.14 -

-1151.50 -1236.42 -1227.30 -1304.25 -1316.81 -

-383.10 -389.08 -424.86 -431.42 -466.58 -473.30

18 19 20 21 22 23 24 25 26

-342.20 -354.19 -384.10 -392.13 -425.37 -433.81 -

-1152.74 -1185.48 -1230.47 -1239.86 -1317.93 -1332.15 -

-383.89 -390.69 -425.70 -433.08 -467.31 -474.88

∆12

Cl ∆S298 ,

J/(mol · K)

Ka∞(g)/m ) 2{(n - 6)/2}; nf∞ /m ) 2; K∞ /m ) 2 (12)

Next, introducing eq 12 into eqs 5-7, one obtains the dependencies of the thermodynamic characteristics of clusterization for infinite clusters on the basis of the structures shown in Figure 5(1):

∆H∞Cl /m ∆S∞Cl /m

) [-9.20Ka∞ /m - 47.42] kJ/mol

) [-3.72Ka∞ /m + 54.22] kJ/mol

∆14

Cl ∆S298 ,

J/(mol · K)

Cl ∆H298 ,

Cl kJ/mol ∆S298 , J/(mol · K)

-1201.15 -1205.53 -1289.90 -1290.19 -

-347.47 -353.63 -389.31 -395.90 -431.14 -437.99 -

-1204.99 -1214.39 -1291.70 -1287.72 -1364.83 -1368.49 -

-1089.12 -1168.76 -1250.84 -1252.11 -1308.59 -1333.42 -

-349.78 -356.32 -391.67 -398.71 -433.45 -440.85 -

-1216.02 -1214.68 -1309.83 -1301.67 -1394.91 -1390.74 -

and Figure 5(1b), or Figure 5(2a) and Figure 5(2b), these parameters are different. For infinite linear cluster in the p direction, where the bonds between the hydrogen atoms of the methyl and methylene groups exist, K∞/m ) 2, whereas in the q direction such bonds do not exist, and therefore K∞/m ) 0. Then, for linear infinite clusters aligned in the p direction the dependencies of thermodynamic parameters of clusterization on the number of H-H interactions are

∆H∞Cl /m ) [-9.20Ka∞ /m - 25.81] kJ/mol

(20)

∆S∞Cl /m ) [-18.40Ka∞ /m - 190.12] J/(mol · K) (21)

(13)

) [-18.40Ka∞ /m - 341.10] J/(mol · K) (14)

∆G∞Cl /m

kJ/mol

-347.54 -1200.86 -354.40 -1244.80 -389.07 -1293.50 -396.67 -1301.19 -1364.20 -1422.62 tetramer 4 -308.44 -314.77 -350.25 -1196.72 -356.93 -1220.73 -391.84 -1287.19 -399.22 -1314.00 -1365.86 -1378.46 -

by the number of monomers in a cluster p · q and calculate the limits of values at m f ∞, thus obtained. This gives the parameters of an infinite cluster per monomer:

Ka∞(u)/m ) 2{(n - 5)/2};

Cl ∆H298 ,

(15)

while for the clusters shown in Figure 5(2) these dependencies are

∆G∞Cl /m ) [-3.72Ka∞ /m + 30.84] kJ/mol

(22)

whereas for linear infinite clusters aligned in the q direction these dependencies are

∆H∞Cl /m ) [-9.20Ka∞ /m - 21.61] kJ/mol

(23)

∆S∞Cl /m ) [-18.40Ka∞ /m - 114.86] J/(mol · K) (24) ∆H∞Cl /m ) [-9.20Ka∞ /m - 47.42] kJ/mol

(16) ∆G∞Cl /m ) [-3.72Ka∞ /m + 12.62] kJ/mol

∆S∞Cl /m

(25)

) [-18.40Ka∞ /m - 304.98] J/(mol · K) (17)

∆G∞Cl /m

) [3.72Ka∞ /m + 43.46] kJ/mol

(18)

The parameters for infinite linear clusters aligned in the p and q directions can be calculated in a similar way considering the structures shown in Figure 5(1a) and 5(1b):

Ka∞(u)/m ) {(n - 5)/2};

Ka∞(g)/m ) {(n - 6)/2}; nf∞ /m ) 1 (19)

It should be noted again that the parameter K that characterizes the number of intermolecular H-H interactions between hydrogen atoms linked to the sp3- and sp2-hybridized carbon atoms (see Figure 4) depends on the type of the cluster considered. In particular, for clusters presented in Figure 5(1a`)

It should be noted that the spontaneous clusterization threshold for the structures with p ) 1, q ) ∞ exists at lower alkyl chain length than for the structures with p ) ∞, q ) 1, because in the latter case there exist intermolecular interactions between the hydrogen atoms linked to the carbon atoms that form the double bond. Consider next the 2D structures. Figure 6 shows the dependencies of clusterization enthalpy on the length of the alkyl chain segment located between the double bond and the methyl group (ω), as calculated from eqs 5, 16, 20, and 23. Note that, in contrast to our previous publications,21-26 the dependencies are plotted vs ω rather than vs n, to prevent overlapping the curves with equal Ka values, which would make it difficult to distinguish between each other. For the same reason, only the dependencies for acids with ∆ ) 15 are presented in Figure 6. The lines show the correlation dependencies on ω, whereas the points correspond to the results obtained by direct calculations.


Vysotsky et al.

Figure 5. Geometrical structures of two possible infinite clusters: 1, 1a, 1b: view from above, p and q directions for clusters formed by monomers 1; 2, 2a, 2b: view from above, p and q directions for clusters formed by alternating monomers 1 and 2.

It is seen that the predicted and calculated values agree with each other quite well. Also, for dimers, trimers, and tetramers with different relative orientation of the acid groups and equal ∆ and ω, the clusterization enthalpies are equal. It follows from Table 2 that this is true for all acids considered here. Note that, for acids with even ∆, some scattering of the enthalpy values occurs (up to 2 kJ/mol), that should be attributed to small imperfections in the structures caused by the presence of additional intermolecular hydrogen bonds; see Figure 3. For these clusters, the values of Gibbs’ energy of clusterization are

essentially more different, because their clusterization entropies also differ significantly from each other. The dependencies of Gibbs’ energy of clusterization per monomer on ω for different ∆ are shown in Figure 7a. Note that only the dependencies for infinite clusters are shown in Figure 7 because the intermolecular hydrogen-hydrogen interactions between the hydrogen atoms linked to the sp3- and sp2-hybridized carbon atoms exist only in the p direction, while their presence in small clusters (e.g., tetramers) is the edge effect that leads to small deviations in the Gibbs’ energy dependencies on ω. It would be difficult to



TABLE 3: Contributions of Additional Intermolecular Hydrogen Bonds to the Clusterization Energetics coefficients at

∆HCl 298, kJ/mol

∆SCl 298, J/(mol · K)

∆GCl 298, kJ/mol

nff(u) nfr(u) nrf(u) nff(g) nfr(g) nrf(g) nOH(u) ) nO(u) ) nff(g) - nff(u) nOH(g) ) nfr(g) - nfr(u) nO(g) ) nrf(g) - nrf(u)

-21.61 ( 0.37 -21.61 ( 0.37 -21.61 ( 0.37 -21.61 ( 0.37 -23.73 ( 0.82 -19.90 ( 0.89 0 -2.12 ( 1.19 1.71 ( 1.26

-132.92 ( 5.03 -134.21 ( 11.23 -95.51 ( 12.02 -138.02 ( 6.17 -161.21 ( 1.23 -111.39 ( 2.02 -5.10 ( 11.20 -27.00 ( 12.46 -15.88 ( 14.04

18.00 ( 2.20 18.38 ( 3.71 6.85 ( 3.95 19.52 ( 2.21 24.31 ( 1.19 13.29 ( 4.47 1.52 ( 3.34 5.93 ( 4.90 6.44 ( 5.44

distinguish between the corresponding curves for finite clusters, “piled” over each other, if plotted in the figure. Also, we are mainly interested in monolayers, for which the experimental data are known. It is seen that the curves for ∆ ) 13 and 15 coincide with each other. This is because, in these cases, the number of intermolecular hydrogen-hydrogen interactions for equal ω is the same. Also, the additional contribution from intermolecular hydrogen bonds which exist for ∆ ) 14 was disregarded because of its statistical insignificance whereas the contributions from the intermolecular interactions between the functional groups are also equal to each other. In the framework of this model, the spontaneous clusterization threshold for acids with ∆ ) 12 corresponds to ω ) 6-7 (n ) 18-19), for ∆ ) 13 the value ω ) 4-5 (n ) 17-18), for ∆ ) 14 the value ω ) 4-5 (n ) 18-19), and for ∆15 the value ω ) 2 - 3 (n ) 17-18). These results agree well with the available experimental data.11-16 It is important to note once more that, according to the approach developed here, the thermodynamic parameters of clusterization of unsaturated fatty acids depend only on the number of carbon atoms in the hydrocarbon chain (n ) ∆ + ω), and do not depend on the position of the double bond. This type of dependence arises from the fact that, according to the additive scheme (see e.g. eqs 6 and 7), the interactions between the carboxyl groups and the number of the hydrogen-hydrogen interactions do not depend on the position of the double bond. The interactions of hydrogen atoms which belong to the ethylene groups with alkyl chains do also not depend on the position of the double bond in the radical. Note that, if the double bond is located at the end of the chain (ω ) 1), then the number of interactions formed between the hydrogen atoms which belong to the CdC bond and the hydrogen atoms of the hydrocarbon radical becomes lower by one. This fact should be accounted for in the calculation of the thermodynamic parameters of

clusterization of such compounds. From eq 7 it is seen that, in this case, the ∆GCl ∞ /m value will change by -9.11 kJ/mol. It also follows from Figure 7 that the stepwise dependence of the Gibbs’ clusterization energy on ω for the clusters which involve H-H interactions between the hydrogen atoms linked to the sp3- and sp2-hybridized carbon atoms (2D films and linear clusters in p direction) is less obvious for compounds with ω ) 1. As explained above, this inconsistency is due to the fact that, in these cases, one intermolecular H-H interaction at the CdC bond is absent, and the clusterization free energy value becomes lower as compared with that which would follow from the additive scheme. Note that this energy decrease is more pronounced for linear clusters, whereas for 2D films this decrease is compensated by the negative contributions from intermolecular H-H interactions, the number of which in the films is twice as large as in the linear clusters. For the case that ∆ ) 2, the number of interactions between the hydrogen atoms which belong to the CdC bond and the hydrogen atoms of the alkyl chain becomes also lower by one. In this case, however, a new hydrogen bond is formed between the ketonic oxygen atom and the hydrogen atom of the ethylene group. Also, for small ∆ values (1-3) the ethylene fragment is immersed into water, and its solvation should additionally contribute to Gibbs’ clusterization energy. In this study we do not consider cis unsaturated carboxylic acids with small ∆ values (1-3). Consider next the possible influence of additional hydrogen bonds which arise in the even ∆ systems in comparison to systems with odd ∆. Note that, in the framework of the additive scheme developed here, this influence was found to be insignificant. To elucidate the effect caused by additional intermolecular hydrogen bonds (see Figure 3) we calculate their contributions to the thermodynamic parameters of clusterization (although it was noted above that these contributions are statistically insignificant, cf. Table 3). The number of additional hydrogen interactions for the clusters with similar direction of

Figure 6. Dependencies of standard enthalpy of clusterization per monomer molecule on ω for cis unsaturated carboxylic acids with ∆ ) 15.


Vysotsky et al.

Figure 7. Dependencies of standard Gibbs’ energy of clusterization per monomer molecule on ω for infinite clusters of various structures: (a) with account for contributions from additional intermolecular interactions between functional groups and alkyl chains; (b) without this account.

dipole moments in the functional groups is nOHff ) nOff ) pq/4, i.e., 1/4 per monomer for the infinite cluster. Then, by introduction the corresponding values listed in Table 3 into eqs 13-15 one obtains the expressions for the thermodynamic parameters of clusterization for infinite clusters on the basis of structures shown in Figure 5(1):

nOHfr ) nOrf ) pq/4, i.e., 1/4 per monomer for the infinite cluster. Then, eqs 16-18become

∆H∞Cl /m ) [-18.4Ka∞ /m - 45.66] kJ/mol

(29)

∆S∞Cl /m ) [-36.80Ka∞ /m - 307.65] J/(mol · K) (30) ∆H∞Cl /m ) [-18.40Ka∞ /m - 47.42] kJ/mol

(26) ∆G∞Cl /m ) [-7.44Ka∞ /m + 43.54] kJ/mol

∆S∞Cl /m ) [-36.80Ka∞ /m - 341.74] J/(mol · K) (27) ∆G∞Cl /m ) [-7.44Ka∞ /m + 54.41] kJ/mol

(28)

To obtain the expressions for clusters with alternating directions of dipole moments in the functional groups one has to note that the OH · · · H bond always corresponds to dimer 2, whereas the O · · · H bond is characteristic for dimer 3. Therefore, two types of interactions can exist: nOHfr and nOrf. The number of interactions between the acid groups in these clusters is equal to the corresponding number of interactions in the infinite cluster considered above. The only difference between the clusters is related to the relative orientation of the functional groups. Therefore, the number of the corresponding interactions is the same as for the case considered above:

(31)

The dependencies of the Gibbs’ clusterization energy per monomer on ω calculated from eq 31 are shown in Figure 7b. It is seen that the spontaneous clusterization threshold for the acids with ∆ ) 13 and 15 corresponds to the radical carbon atoms length of 17, in agreement with the experimental data.11-15 The spontaneous clusterization threshold for the acids with ∆ ) 12 and 14 corresponds to the radical with 19 carbon atoms, i.e., by 2 atoms higher than for clusters with odd ∆ values. This also agrees with the experimental data.11-15 We attribute this fact to the entropy factor which introduces some destabilization to the clusterization energetics. It is seen from the comparison of the spontaneous clusterization threshold parameters calculated here with those reported previously,19 that for saturated acids this threshold corresponds to 5-6 intermolecular hydrogen-hydrogen interactions,21 whereas for unsaturated acids with odd ∆ values this threshold corre-


Figure 8. Surface pressure isotherms of nervonic acid monolayers (curves labeled by the temperature values); solid lines, experiment; dotted lines, theory.

sponds to the same Ka values but longer chain length, and for even ∆ values it corresponds to a Ka value by one unit higher because of the destabilizing influence of additional intermolecular hydrogen bonds. In our case the radical length which corresponds to the spontaneous clusterization threshold is larger than that characteristic for saturated acids. This should be attributed to the fact that the hydrogen atoms linked to four sp3- or sp2-hybridized carbon atoms enter the intermolecular hydrogen-hydrogen interactions (see Figure 4), and one carbon atom belongs to the acid group, whereas in saturated acids one extra carbon atom that is involved in the acid group is present in the chain length. Also, for the acids with even ∆ values one carbon atom forms the intermolecular hydrogen bond with the oxygen atom of the neighboring molecule. This affects the correlations (7), (15), (18) due to the destabilizing contribution to the clusterization Gibbs’ energy caused by the intermolecular hydrogen-hydrogen interactions between the hydrogen atoms linked to four sp3- or sp2-hybridized carbon atoms. Note that, if one assumes K ) 0 in eq 7 (which would correspond to the absence of double bonds in the acid molecule), then the clusterization threshold would occur for the molecules with the chain length shorter by 5 carbon atoms, in accordance with the experimental data for fatty acids; see e.g. ref 21. The molecules of fatty acids are linear and do not have any bends similar to that shown in Figure 4; generally they do not form any intermolecular hydrogen bonds between the functional groups and hydrogen atoms of the alkyl chain. Therefore, for equal radical length, the molecules of saturated acids form more intermolecular hydrogen-hydrogen interactions than the molecules of unsaturated acids. In this study, two possible models of clusters are considered (see Figure 5). Figures 6 and 7 illustrate the dependencies only for the structures with alternating directions of dipole moments in the functional groups (see Figure 5(2)) because it appears from the calculated values of Gibbs’ clusterization energy that these clusters are energetically more preferable. Comparison with Experiment. The measurements of the surface pressure (π-A) isotherms were carried out in a selfmade computer-interfaced film balance. The surface pressure was measured, using the Wilhelmy method with a roughened glass plate, with a reproducibility (0.1 mN/m. The equilibrium π-A isotherms were recorded at a compression rate of 0.01 nm2 molecule-1 min-1.16 Figures 8 and 9 illustrate the experimental surface pressure-area per molecule (π-A) isotherms for monolayers of nervonic (∆ ) 15, ω ) 9, n ) 24) and erucic


Figure 9. Surface pressure isotherms of erucic acid monolayers (curves labeled by the temperature values); solid lines, experiment; dotted lines, theory.

(∆ ) 13, ω ) 9, n ) 22) acids, respectively.16 The curves are labeled by the temperature values. It is clearly seen that the monolayers of these acids undergo the fluid (G, LE)/condensed phase transition. The theoretical curves, also shown in Figures 8 and 9, were calculated from the model discussed previously.27,28 This model was also used in our previous publications.21-25 It is seen that the theoretical model matches well the experimental dependencies. It follows from the parameters of the theoretical model that, in the region of fluid monolayer, i.e., for A > Ac, where Ac is the area per molecule in the initial point of phase transition, the monolayers of these cis unsaturated carboxylic acids are composed of dimers. This result agrees with the quantum chemical calculations of the dimerization energy of nervonic and erucic acids, for which the Gibbs’ free energy of dimerization is negative; see Table 2. For these acids the free energy of dimerization depends on the dimer geometry and is equal to -(2.5-5.7) and -(1.1-2.5) kJ/mol, respectively. The thermodynamic characteristics of clusterization of these acids can also be estimated from the π-A isotherms. The value of the standard free (Gibbs’) energy ∆GCl for the monomer-cluster or oligomer-cluster transitions, calculated per mole of the monomers, can be written in the following form:29

∆GCl ) RT ln(ω/Ac)

(32)

where R is the gas constant, T is temperature, and ω is the area per monomer in a cluster. The results obtained for ∆GCl at various temperatures can be used to estimate the values of standard enthalpy (∆HCl) and standard entropy of aggregation (∆SCl)

∆GCl ) ∆HCl - T∆SCl -

( )

∆HCl ∂ ∆GCl ) 2 ∂T T T

(33)

(34)

In particular, the Gibbs’ energy for nervonic acid, as calculated from eq 32 assuming the formation of clusters by dimers22,23 within the temperature range of 20-27 °C varies from -4.0 to -2.2 kJ/mol, whereas the average value of ∆HCl is -63 kJ/mol. For erucic acid the Gibbs’ energy in the temperature range of 3.6-11.6 °C varies from -3.0 to -1.5


Vysotsky et al.

kJ/mol, with the average value of ∆HCl -59 kJ/mol. The clusterization entropy values for both these acids are about -200 J/(mol · K), which is significantly lower (by absolute value) than the values calculated for infinite clusters by quantum chemical methods, where the values of enthalpy, entropy, and Gibbs’ energy of clusterization per monomer molecule -213.02 and -194.62 kJ/mol, -636.18 and -599.38 J/(mol · K), -16.07 and -23.50 kJ/mol for nervonic acid. This discrepancy could possibly be ascribed to the approximate nature of eq 32 which, instead of the monomers’ activity, involves the degree of monolayer coverage by monomers in the initial point of the phase transition: θc ) ω/Ac. Instead of eq 32, more rigorous theory which involves the activity coefficients of the monolayer arising from the nonideality of enthalpy,27 entropy,30 and the relationship between molar fraction and area fraction31 leads to the equation

[(

∆GCl ) RT ln

)

θc (ω/ω0)(1 - θc) + θc Πcohω ω0 θc 1 (35) ω kT

(

)

]

Here ω0 is the area per solvent molecule (the area per water molecule is about twice as small as the ω values for the acids studied here), Πcoh is the cohesion pressure, and k is the Boltzmann constant. The first term on the right-hand side of eq 35 expresses the molar fraction of the surface layer occupied by the amphiphilic molecules at the initial point of the phase transition, the second term corresponds to the nonideal entropy contribution to the activity, whereas the third term describes the contribution from the nonideal enthalpy. For the case when the molecular area of the solvent is equal to that of the amphiphilic molecule (ω0 ) ω), the second term on the righthand side vanishes, and the first term becomes equal to the righthand side of eq 32. The calculations made with eq 35 assuming the formation of clusters by dimers, have resulted in the ∆GCl values of -9.6 to -7.0 kJ/mol and -7.9 to -5.6 kJ/mol for the nervonic and erucic acid, respectively, in the temperature ranges indicated above. In this case, the average ∆HCl values for the nervonic and erucic acid were -120 and -95 kJ/mol, and the clusterization entropy ∆SCl was -380 and -320 J/(mol · K), respectively. These values are much higher (by absolute value) than those calculated from the approximate relation (32), but are somewhat lower than those obtained by the semiempiric PM3 method. This fact could be attributed to the structural peculiarities of the studied compounds. In the case considered here, the existence of infinite 2D monolayers would require the formation of intermolecular bonds between the hydrogen atoms attached to the sp2- and sp3-hybridized carbon atoms (see Figure 4). These interactions exist in the p direction (see Figure 5), while they cannot be formed along the q direction. Obviously, in the case of cluster formation along the p direction (in contrast to the q direction) the curliness of the hydrocarbon radical cis structure makes it more difficult for the monomers to approach each other close enough to form intermolecular interactions. Therefore, one can expect that actual clusters possess dendritic structure comprising both linear and compact sections. It may be expected that the values of thermodynamic parameters of clusterization for such monolayers will be intermediate between the corresponding parameters of linear and compact infinite clusters. It is of interest to perform a comparison with experimental results obtained in the literature. π-A isotherms for the isomers

of cis-octadecanoic acid (n ) 18) with ∆ ) 2-17 are reported in ref 10. The π-A isotherms of the isomers with ∆ ) 2, 15, 16, and 17 have an inflection point which enables one to estimate the Gibbs’ clusterization energy using eqs 32 and 35. It was found that the ∆GCl value for the octadecanoic acid isomers increases as ∆ increases from 15 to 17: -0.9 (-3.0) kJ/mol; -1.5 (-3.9) kJ/mol, and -2.3 (-5.0) kJ/mol, respectively (the values in parentheses refer to the results of the calculations made using eq 35). The π-A isotherm for the isomer with ∆ ) 2 is almost coincident with the isotherm for ∆ ) 17.10 Therefore, the ∆GCl values for these isomers calculated by using eqs 32 and 35 are also quite similar. Note that the π-A isotherms for the isomers with ∆ ) 4-14 shown in ref 10 do not depend on the position of the double bond and are described by a single curve which is close to the π-A isotherm for the isomer with ∆ ) 15. These ∆GCl data obtained on the basis of the experiments of ref 10 agree satisfactorily with the results of the quantum chemical calculations. In particular, it follows from eq 18 that for the isomers with ∆ ) 4-16 the value ∆GCl ) -1.18 kJ/mol, whereas for the isomer with ∆ ) 17 the value ∆GCl ) -5.74 kJ/mol. It was noted above that for this isomer the change of the number of interactions between the hydrogen atoms which belong to the CdC bond and the hydrogen atoms of the hydrocarbon chain was taken into account, which leads to the variation of the ∆GCl value by -4.56 kJ/mol. Conclusions In the present study, the thermodynamic parameters of clusterization for cis unsaturated carboxylic acids at the air/ water interface are calculated that involve calculations of the thermodynamic parameters of formation of monomers from elementary components and the calculations of absolute entropies. It is shown that, for the monomers with equal radical chain lengths, the thermodynamic parameters (enthalpy, entropy, and Gibbs’ free energy) do not depend on the position of the double bond. The regression dependencies of the thermodynamic parameters on the radical chain length are calculated. The regression slopes were found to be -22.68 kJ/mol and 30.55 J/(mol · K) for enthalpy and entropy, respectively. These values are almost the same as those found earlier21-26 for other amphiphilic compounds. The regression coefficients exceed 0.999, whereas the standard deviations do not exceed 0.04 kJ/ mol and 2.61 J/(mol · K) for the enthalpy of formation and absolute entropy, respectively. The thermodynamic parameters of clusterization are calculated for dimers, trimers and tetramers of cis unsaturated carboxylic acids of different structures. These clusterization parameters are used to construct the additive scheme for the cis unsaturated carboxylic acids capable of the prediction of the spontaneous clusterization threshold. There are two possible models of infinite clusters with different relative orientation of functional groups. The calculations show that the spontaneous clusterization threshold for cis unsaturated carboxylic acids with ∆ ) 12, 13, 14, and 15 corresponds to the total radical chain length of 18-19, 17-18, again 18-19, and 17-18 carbon atoms, respectively, i.e., 18-19 carbon atoms for molecules with even ∆, and 17-18 carbon atoms for molecules with odd ∆, in agreement with the experimental data available; see ref 3. These results could be expected to be applicable for any cis unsaturated carboxylic acids. In particular, they agree with the experimental data for cis-16-heptadecenoic acid (∆16, ω1);11,12 cis-9-hexadecenoic acid (∆7, ω9);11,13 cis-11-eicosenoic acid (∆11, ω9);11,14 and cis-9-octadecenoic acid (∆9, ω 9).11,14,15 From the comparison of the calculated results with corresponding parameters of clusterization for saturated carboxylic

Cis Isomers of Unsaturated Fatty Acids acids,8 it is shown that, for cis unsaturated carboxylic acids with even and odd ∆ values, the spontaneous clusterization threshold corresponds to the number of carbon atoms in the radical by 6 and 7 units higher, respectively, than for the saturated acids. This fact is attributed to the differences in the structures of cis unsaturated fatty acids and saturated fatty acids, namely to the presence of sp3- and sp2-hybridized carbon atoms in the cis unsaturated fatty acids that contribute to additional intermolecular H-H interactions; see Figure 4. Also, the number of carbon atoms in the saturated carboxylic acids alkyl chain is by one larger than in the unsaturated acids radical. This difference is due to the presence of an additional carbon atom involved in the acid group (because, according to the current classification of organic compounds, the carbon atom which belongs to the acid group of unsaturated acid is accounted for in the ∆ value, whereas for saturated acids this atom is not considered to be the constituent of the fatty radical). Moreover, for even ∆ values, one hydrogen atom in the radical chain participates in the formation of an additional intermolecular hydrogen bond what does not exist for saturated acids and cis unsaturated acids with odd ∆. Also, the number of “a” type intermolecular H-H interactions21-26 corresponding to the spontaneous clusterization threshold is 5-6 for both saturated and unsaturated acids showing that this is the type of interaction which governs the clusterization. The approach proposed in this study could be applied to the description of the clusterization process in monounsaturated acids with disconnected double bonds, and then, generalized to polyunsaturated acids with disconnected double bonds. References and Notes (1) Surface, Interface, and Colloids; VCH Publishers, Inc.: New York, 1991; p 429. (2) Hann, R. A. In Langmuir-Blodgett Films; Roberts, G., Ed.; Plenum Press: New York, 1990; Chapter2. (3) Birdi, K. S. Self-Assembly Monolayer Structures of Lipids and Macromolecules at Interfaces; Kluwer Academic/Plenum Publishers: New York, 1999; Chapter 4. (4) Hughes, A. H.; Rideal, E. K. Proc. R. Soc. London 1933, A140, 253.

J. Phys. Chem. B, Vol. 113, No. 13, 2009 4359 (5) Hughes, A. H. J. Chem. Soc. 1933, 338. (6) Marsden, J.; Rideal, E. K. J. Chem. Soc. 1938, 1163. (7) Schneider, V. L.; Holman, R. T.; Burr, G. O. J. Phys. Chem. 1949, 53, 1016. (8) Glazer, J.; Goddard, E. D. J. Chem. Soc. 1950, 3406. (9) Goddard, E. D.; Alexander, A. E. Biochem. J. 1951, 47, 331. (10) Welles, H. L.; Zografi, G.; Scrimgeour, C. M.; Gunstone, F. D. In Monolayers; Goddard, E. D., Ed.; Advances in Chemistry Series 144; American Chemical Society: Washington, DC, 1975; Chapter10. (11) Mingotaud, A. F.; Mingotaud, C.; Patterson, L. K. Handbook of Monolayers. V.1; Academic Press, Inc.: San Diego, CA, 1993; p 1385. (12) O’Brien, K. C.; Rogers, C. E.; Lando, J. B. Thin Solid Films 1983, 102, 131. (13) Bishop, D. G.; Kenrick, J. R.; Bayston, J. H.; Macpherson, A. S.; Johns, S. R. Biochim. Biophys. Acta 1980, 602, 248. (14) Peltonen, J. P.; Rosenholm, J. B. Thin Solid Films 1989, 179, 543. (15) Tomoasia-Cotisel, M.; Zsako, J.; Mocanu, A.; Lupea, M.; Chifu, E. J. Colloid Interface Sci. 1987, 117, 464. (16) Vollhardt, D. J. Phys. Chem. C 2007, 111, 6805. (17) ComprehensiVe Organic Chemistry. The Synthesis and Reactions of Organic Compounds; V.5 Biological Compounds; Haslam, E., Ed.; Pergamon Press: Oxford, UK, 1986; p 735. (18) ComprehensiVe Organic Chemistry. The Synthesis and Reactions of Organic Compounds; V.2 Carboxylic Acids, Phosphorus Compounds; Haslam, E., Ed.; Pergamon Press: Oxford, UK, 1983; p 727. (19) Fatty Acid and Lipid Chemistry; Gunstone, F. D., Ed.; Blackie Academic: London, 1996; p 246. (20) Mopac 2000 Manual; Stewart, J. J. P., Ed.; Fujitsu Limited: Tokyo, 1999, p 535. (21) Vysotsky, Yu. B.; Muratov, D. V.; Boldyreva, F. L.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. B 2006, 110, 4717. (22) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B. J. Phys. Chem., B 2002, 106, 11285. (23) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B. J. Phys. Chem., B 2002, 106, 121. (24) Vysotsky, Yu. B.; Belyaeva, E. A.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem., C 2007, 111, 5374. (25) Vysotsky, Yu. B.; Belyaeva, E. A.; Fainerman, V. B.; Aksenenko, E. V.; Vollhardt, D.; Miller, R. J. Phys. Chem. C 2007, 111, 15342. (26) Vysotsky, Yu. B.; Belyaeva, E. A.; Vollhardt, D.; Aksenenko, E. V.; Miller, R. J. Colloid Interface Sci. 2008, 326, 339. (27) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145. (28) Vollhardt, D.; Fainerman, V. B. J. Phys. Chem. B 2008, 112, 10514. (29) Vollhardt, D.; Fainerman, V. B.; Siegel, S. J. Phys. Chem. B 2000, 104, 4115. (30) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2006, 110, 10436. (31) Lucassen-Reynders, E. H. Colloids Surf. A 1994, 91, 79.

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