Quantum Chemical Analysis of the Thermodynamics of 2-Dimensional

15342

J. Phys. Chem. C 2007, 111, 15342-15349

Quantum Chemical Analysis of the Thermodynamics of 2-Dimensional Cluster Formation of Alkylamines at the Air/Water Interface Yu. B. Vysotsky,† E. A. Belyaeva,† V. B. Fainerman,‡ E. V. Aksenenko,§ D. Vollhardt,*,| and R. Miller| Donetsk National Technical UniVersity, 58 Artema Str., 83000 Donetsk, Ukraine, Medical Physicochemical Centre, Donetsk Medical UniVersity 16 Ilych AVenue, Donetsk 83003, Ukraine, Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky AVenue, 03680 KyiV (KieV), Ukraine, Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany ReceiVed: June 14, 2007; In Final Form: August 2, 2007

The thermodynamic parameters of the formation of monomers, dimers, trimers, tetramers, and one hexamer of alkylamines with the alkyl chain length in the range of 6 to 16 carbon atoms are calculated using the semiempirical PM3 method. The dependencies of potential energy surfaces of monomers and dimers on corresponding torsion angles are analyzed to determine the most stable conformations (local and global minima) of these entities. The thermodynamic parameters of cluster formation (enthalpies, entropies, and Gibbs’ energies) are calculated for the dimers, trimers, tetramers, and the hexamer. The additive approach was further developed to extend the results of the calculations of the thermodynamic properties of small associates (2-6 amine molecules) to infinite clusters. It is shown that these parameter values are stepwise-dependent on the alkyl chain length, and spontaneous cluster formation takes place for this class of compounds when the alkyl chain length becomes 18-19 carbon atoms and higher.

Introduction An essential prerequisite for the study of surface and biologic activity of amines at the air/water interface is the understanding of the mechanism at the formation of a new phase. For example, the bactericidal properties of surface-active amines are determined by their aggregation state and orientation at the cell membranes as well as by their interaction with the proteins that are present in the bacterial cells.1 Therefore, the information obtained from experimental and theoretical studies of the cluster formation thermodynamics is of general interest. Some conclusions about the behavior of these compounds at the air/water interface, particularly, the effect of the alkyl chain length on the cluster formation, can be drawn. In the present work, the cluster formation of a series of saturated n-alkylamines with the general formula CnH2n+1NH2 (n ) 6-16) at the air/water interface is studied. Methods The theoretical calculations were performed using the semiempirical PM3 method. It is known that the PM3 method was formulated as the reparametrization of the AM1 method, which enabled one, in particular, to increase the accuracy of the calculations of the values of formation heats by ∼40%. However, it should be noted that, at present, it cannot be asserted that the PM3 method is more reliable than the AM1 method because the scope of the results obtained by the PM3 method and their comparison with experimental data are, as of yet, too scarce.2 At the same time, our earlier calculations of the thermodynamic parameters of cluster formation, performed for alcohols, fluorinated alcohols, carboxylic acids, and thioalcohols, * To whom correspondence should be addressed. E-mail: vollh@ mpikg-golm.mpg.de. † Donetsk National Technical University ‡ Donetsk Medical University § Institute of Colloid Chemistry and Chemistry of Water | Max Planck Institute of Colloids and Interfaces

have shown that, of all the semiempirical methods, only the PM3 method is capable of a correct description of the behavior of surfactant molecules at the interface.3-8 The optimization of geometric structures of monomers and dimers of alkylamines was performed with the Mopac2000 software package using the BFGS algorithm, generally employed for calculations of systems that involve intermolecular interactions. In this case, the calculation of the correct entropy values should involve the vibrations with characteristic frequencies below 100 cm-1, which were experimentally shown to exist in complexes with intermolecular interactions. Similar to our previous studies,3-8 the relevant contributions were manually calculated. Results and Discussion Monomers. To determine stable conformations of alkylamine monomers, the conformational analysis was performed corresponding to the rotation of the torsion angle between the HNC plane of the amino group and the CCN plane in the range of 0-360°; the angle increment was 5°. The dependencies of the standard enthalpy variation on the CCNH torsion angle value are shown in Figure 1. It is seen that three minima exist, which correspond to a stable conformation of the monomers. In particular, the energetically most-preferable conformer is monomer 1, with a CCNH torsion angle of 60°, whereas monomer 2, with a torsion angle of 180°, is less advantageous. Figure 1 also shows the monomer that is the mirror reflection of monomer 1 with the torsion rotation angle 300° ) -60°. It was proved by additional optimization that these two enantiomers are isoenergetic, so that only monomer 1 was considered. The optimized geometric structures of monomers 1 and 2 are shown in Figure 2. It is seen that monomer 1 is stabilized because of the hydrogen-hydrogen interactions between the R-hydrogen atoms of the alkyl chain and one of the hydrogen atoms involved in the amino group, whereas monomer 2 is stabilized by the interaction of two β-hydrogen atoms of the

10.1021/jp074615v CCC: $37.00 © 2007 American Chemical Society Published on Web 10/03/2007

Thermodynamics of 2D Cluster Formation of Alkylamines

Figure 1. Dependence of the enthalpy variation of the monomer formation on the CCNH torsion angle.

alkyl chain and two atoms involved in the amino group. For these stable conformations of the monomers, the thermodynamic parameters (standard enthalpy of formation, Gibbs’ energy of formation, and absolute entropy) were calculated; the theoretical and experimental9-12 values are summarized in Table 1. It is to be noted that the correction for the free rotation of alkyl groups, amounting to 7.08 J/(mol K) per one group, was additionally introduced into the standard entropy values. This essentially improves the agreement between the calculated and experimental values. The correction is quite similar to those calculated for alkylalcohols and alkylthioalcohols with 6.6 J/(mol K)4 and 7.03 J/(mol K),8 respectively, which shows that the nature of the functional group only slightly affects the free rotation of the methylene groups. The root-mean-square errors corresponding to the deviations between the calculated and experimental enthalpy values for monomers 1 and 2 are 15.6 and 10.8 kJ/mol, respectively. These are than the values calculated earlier for alkylalcohols (23 kJ/mol),4 whereas they are somewhat higher than the inaccuracy for alkylcarboxylic acids (5.9 kJ/mol)7 and thioalcohols (5.7 and 3.9 kJ/mol).8 The values of the thermodynamic characteristics summarized in Table 1 were used to determine the correlation dependencies of these values on the alkyl chain length. These dependencies are linear (similar to those obtained for alcohols,4 carbon acids,7 and thioalcohols8). The corresponding correlation parameters and standard deviations (S) are listed in Table 2. It is seen that the parameter that characterizes the slope of the dependence is almost equal to those calculated earlier for carboxylic acids and thioalcohols, indicating that the main additive contribution to the thermodynamic parameter values comes from the methylene groups of the alkyl chain. Dimers, Trimers, and Tetramers. The initial structures of dimers were built on the basis of monomers 1 and 2. Therefore, the optimization resulted in two types of dimers, namely those with the ‘a’-type of H-H interactions between the hydrocarbon skeletons (Dimers 1-6), and those with the ‘e’-type of these interactions (Dimer 7). These interactions are schematically represented in Figure 3. For these types of dimers, the angles (φ) between the alkyl chains and functional groups were varied in each monomer. The potential surfaces obtained in this way are shown in Figures 4 and 5. At each of these surfaces, nine minima were found. The dimer geometry was additionally optimized in these minima to show that all the minima correspond to stable conformations of dimers. However, the dimers with the coordinates (58°, 304°) and (304°, 58°), (58°, 175°) and (175°, 58°), and (180°, 306°) and (306°, 180°) are the respective mirror isomers.

J. Phys. Chem. C, Vol. 111, No. 42, 2007 15343 Therefore, in the subsequent calculations only dimers 1-6 were considered. The thermodynamic parameters of dimerization were also calculated for dimer 7. It can be seen that, in contrast to the dimers that involve the a-type H-H interaction, for the dimers with the e-type H-H interaction the Gibbs’ energies of dimerization become higher with the increase of the alkyl chain length. The structures of the calculated entities are shown in Figure 2. Table 3 summarizes the thermodynamic parameters of the cluster formation for the alkylamines studied. Enthalpy, entropy and Gibbs’ energy of the cluster formation were calculated as 0 0 Cl 0 ∆HCl m ) ∆H298,Cl - m × ∆H,298,mon ∆Sm ) S298,C - m × 0 Cl Cl Cl S,298,mon and ∆Gm ) ∆Hm - T × ∆Sm respectively, where ∆H0298,Cl and S0298,Cl are the enthalpy of formation and absolute entropy of corresponding clusters, respectively; ∆H0298,mon and S0298,mon are the enthalpy of formation and absolute entropy of the monomers, respectively, with the corresponding type of hydrogen-hydrogen interactions between the hydrogen atoms of the functional group and the hydrocarbon radical hydrogen atoms, and m is the number of monomers in the cluster. Using these calculated values, the dependencies of ∆HCl m, Cl ∆SCl m and ∆Gm for the studied compounds on the alkyl chain length were plotted, which show a step-like shape similar to those found for alkylalcohols, alkylcarboxylic acids, and alkylthioalcohols.4,7,8 The parameters of the corresponding regression dependencies on the H-H interaction number (ka) are summarized in Table 4. It is seen from Table 4 that the correlation coefficients for enthalpy, entropy, and free energy of the cluster formation exceed the values of 0.997, 0.97, and 0.93, respectively. The regression slopes for alkylamines are quite similar to those calculated earlier for alkylalcohols, alkylcarboxylic acids, and alkylthioalcohols.4,7,8 In the case of enthalpy, these coefficients are -10.06 to -10.30 kJ/mol for amines, -10.24 to -10.36 kJ/mol for alcohols, -10.22 to -10.34 kJ/mol for carboxylic acids, and -10.08 to -10.24 kJ/mol for thioalcohols. The fact that, for all the systems studied, these values are almost equal indicates that all these systems exhibit the same character of H-H interactions. This in turn should enable us to integrate all of the thermodynamic parameters into a common additive scheme. In the present publication, similar to that assumed in ref 8, we distinguish between four types of interactions between the functional groups: nNH2-NH2 is the number of intermolecular hydrogen bonds in a cluster with sequentially arranged NH2 bonds (dimer 4, trimer 1, tetramer 1); nN-N is the number of interactions between NH2 groups in a cluster with their parallel arrangement (dimer 1, trimer 2, tetramer 2); ntri is the number of NH2 hydrogen bonds with their mutual arrangement (as in Trimer 3), and nlac is the number of lacunas formed because of the interaction between two trimers 1 under formation of hexamer 1. The correlation dependencies for dimer 4 (as an example) are as follows:

∆HCl m ) -(10.11 ( 0.08)ka - (1.31 ( 0.47)nNH2-NH2 (3.65 ( 0.47)nN-N + (1.40 ( 0.53)ntri - (6.53 ( 2.37)nlac (R ) 0.9998, S ) 3.01) ∆SCl m ) -(24.49 ( 1.04)ka - (80.36 ( 5.94)nNH2-NH2 (85.61 ( 5.94)nN-N - (28.19 ( 6.65)ntri (149.87 ( 29.68)nlac (R ) 0.997, S ) 37.72)

15344 J. Phys. Chem. C, Vol. 111, No. 42, 2007

Vysotsky et al.

Figure 2. Optimized geometric structures of monomers and dimers of alkyl amine.

For alkylamines, the standard deviation related to the description of entropy was 37.72 J/(mol K), which is almost equal to that for alkylalcohols of 36.1 J/(mol K)4 and is somewhat higher

than the values for alkylcarboxylic acids and alkylthioalcohols of 29.58 J/(mol K).8 The dependencies summarized above were used to construct the additive scheme for the thermodynamic


J. Phys. Chem. C, Vol. 111, No. 42, 2007 15345

TABLE 1: Thermodynamic Parameters for Amine Monomers system

monomer 1

monomer 2

C3H7NH2 C4H9NH2 C5H11NH2 C6H13NH2 C7H15NH2 C8H17NH2 C9H19NH2 C10H21NH2 C11H23NH2 C12H25NH2 C13H27NH2 C14H29NH2 C15H31NH2 C16H33NH2

∆H°298, kJ/mol -74.78 -69.06 -97.47 -91.65 -120.12 -114.33 -142.79 -137.00 -165.46 -159.68 -188.14 -182.36 -210.82 -205.04 -233.50 -227.72 -256.18 -250.40 -278.86 -273.09 -301.54 -295.77 -324.23 -318.45 -346.91 -341.13 -369.59 -363.82


S°298, J/(mol K) 328.58 329.79 368.11 369.30 407.90 409.05 447.52 448.57 486.92 488.13 526.14 527.52 565.91 566.98 605.27 606.02 643.93 644.87 683.63 683.44 722.71 723.49 761.17 761.48 799.97 801.89 839.22 838.57


∆G°298, kJ/mol 37.25 42.61 43.38 48.85 49.48 54.93 55.61 61.09 61.80 67.23 68.04 73.42 74.12 79.58 80.31 85.87 86.71 92.21 92.81 98.64 99.09 104.63 105.55 111.23 111.91 117.11 118.14 124.11

experiment9-12 -70.5 -92.0 -113.0 -133.3 -153.0 -174.6 -195.2 -215.8 -236.0 -256.0 -298.2 Figure 3. Types of hydrogen-hydrogen interactions.

324.2 363.0 402.0 443.2 480.0 525.1 561.1 600.4 638.0 678.0 757.6

41.7 49.3 57.3 65.4 75.3 81.9 90.2 98.5 108.0 116.5

Figure 4. Potential energy surface for the dimer on the basis of monomer 1.

131.8

TABLE 2: Coefficients of Correlation Equations y ) (a ( ∆a)n + (b ( ∆b) for the Thermodynamic Characteristics of Amine Monomersa conformer

a

characteristics

a ( ∆a

b ( ∆b

S

1

∆G°298, kJ/mol -22.68 ( 0.01 -6.73 ( 0.01 0.014 S°298, J/(mol K) 39.28 ( 0.04 211.66 ( 0.44 0.65 ∆G°298, kJ/mol 6.22 ( 0.01 18.32 ( 0.12 0.18

2

∆G°298, kJ/mol -22.68 ( 0.01 -0.95 ( 0.02 0.024 S°298, J/(mol K) 39.20 ( 0.06 213.24 ( 0.59 0.86 ∆G°298, kJ/mol 6.25 ( 0.02 23.63 ( 0.17 0.24

The amount of sampling is equal to 14.

parameters of alkylamines, from which the thermodynamic characteristics for the formation of large and infinite clusters were calculated. Large and Infinite Clusters. Two types of infinite clusters are analyzed: rectangular clusters constructed on the basis of linear tetramers (Figure 6) and triangular clusters formed on the basis of trimer 3 (Figure 7). First, rectangular clusters are considered. It is seen from Figure 6 that these entities involve two types of intermolecular

Figure 5. Potential energy surface for the dimer on the basis of monomer 2.


Vysotsky et al.

TABLE 3: Standard Thermodynamic Characteristics of the Formation of Dimers, Trimers, and Tetramers of Thioalcohols as Calculated by PM3 Parametrization molecule

∆HCl m kJ/mol

∆SCl m J/(mol K)

∆GCl m kJ/mol

C6H13NH2 C7H15NH2 C8H17NH2 C9H19NH2 C10H21NH2 C11H23NH2

-32.05 -34.51 -42.42 -44.90 -52.80 -55.31

-148.82 -163.20 -178.65 -190.06 -206.79 -216.60

12.30 14.13 10.82 11.73 8.82 9.24


-31.73 -34.26 -42.14 -44.66 -51.93 -55.07

-155.77 -165.42 -182.48 -193.09 -199.27 -218.17

14.69 15.03 12.24 12.88 7.46 9.95


-29.80 -32.37 -39.71 -42.72 -50.06 -53.12

-152.73 -163.38 -169.88 -190.70 -202.99 -215.54

15.71 16.31 10.92 14.11 10.43 11.12


-29.97 -32.54 -40.09 -42.91 -50.47 -53.31

-150.27 -162.57 -169.96 -190.63 -202.20 -216.82

14.81 15.91 10.56 13.90 9.79 11.31


-30.91 -33.50 -40.76 -43.85 -51.11 -54.21

-154.95 -163.37 -165.97 -191.46 -198.37 -216.73

15.27 15.18 8.70 13.20 8.01 10.37


-32.05 -34.51 -42.41 -44.90 -52.51 -55.31

-150.76 -163.26 -178.99 -190.28 -201.13 -216.42

12.88 14.14 10.93 11.80 7.42 9.18


-16.63 -18.92 -22.05 -24.34 -27.47 -29.77

-205.57 -203.05 -239.04 -237.06 -274.41 -270.19

44.63 41.59 49.18 46.30 54.31 50.74


-59.89 -64.66 -80.72 -85.42 -101.52 -106.29

-304.65 -330.37 -361.34 -384.10 -419.53 -421.61

30.89 33.79 26.96 29.05 23.50 19.35


-64.48 -69.12 -85.25 -89.94 -106.06 -110.75

-304.38 -331.21 -364.88 -384.80 -420.23 -437.20

26.23 29.58 23.49 24.73 19.16 19.54


-89.90 -92.14 -118.48 -120.65 -147.00 -149.19

-368.56 -368.54 -416.90 -415.01 -462.23 -457.39

19.93 17.69 5.75 3.03 -9.26 -12.89

molecule

∆HCl m kJ/mol

∆SCl m J/(mol K)

∆GCl m kJ/mol

C12H25NH2 C13H27NH2 C14H29NH2 C15H31NH2 C16H33NH2

-63.19 -65.72 -73.56 -76.12 -83.99

-233.03 -243.47 -259.12 -271.31 -284.89

6.25 6.83 3.65 4.73 0.91


-62.28 -65.46 -72.68 -74.10 -83.07

-228.79 -245.68 -255.09 -247.37 -282.07

5.90 7.75 3.34 -0.39 0.98


-60.45 -63.48 -70.84 -72.28 -81.24

-229.43 -244.52 -255.61 -247.68 -282.21

7.92 9.38 5.33 1.53 2.86


-60.85 -62.29 -71.25 -72.67 -81.64

-227.34 -223.35 -253.37 -252.93 -278.58

6.90 4.27 4.26 2.71 1.37


-61.48 -64.61 -71.85 -73.31 -82.24

-225.21 -245.04 -252.06 -247.97 -279.13

5.64 8.41 3.27 0.59 0.94


-62.87 -65.72 -73.27 -74.71 -83.66

-228.34 -242.96 -255.37 -247.53 -278.16

5.17 6.68 2.83 -0.95 -0.77


-32.90 -35.23 -38.34 -40.66 -43.79

-307.15 -304.15 -340.18 -335.44 -374.02

58.63 55.41 63.03 59.30 67.67


-122.38 -127.07 -143.20 -147.94 -164.07

-454.75 -477.77 -498.74 -527.58 -558.17

13.14 15.30 5.42 9.28 2.26


-126.87 -131.57 -147.68 -152.42 -168.50

-468.38 -489.11 -523.23 -543.64 -574.08

12.71 14.18 8.24 9.58 2.58


-175.51 -177.66 -203.97 -206.20 -232.53

-505.08 -503.83 -552.68 -549.30 -592.30

-24.99 -27.52 -39.28 -42.51 -56.03

Dimer 1

Dimer 2

Dimer 3

Dimer 4

Dimer 5

Dimer 6

Dimer 7

Trimer 1

Trimer 2

Trimer 3


J. Phys. Chem. C, Vol. 111, No. 42, 2007 15347

TABLE 3 (Continued) molecule

∆HCl m kJ/mol

∆SCl m J/(mol K)

∆GCl m kJ/mol

∆HCl m kJ/mol

∆SCl m J/(mol K)

∆GCl m kJ/mol


-89.84 -96.87 -121.07 -127.99 -152.33 -159.24

-457.74 -500.15 -551.49 -583.74 -637.06 -653.74

46.57 52.17 43.27 45.96 37.52 35.57

Tetramer 1 C12H25NH2 C13H27NH2 C14H29NH2 C15H31NH2 C16H33NH2

-183.49 -190.48 -214.89 -221.68 -246.20

-678.18 -722.94 -757.94 -801.11 -897.71

18.61 24.96 10.97 17.05 21.32


-97.04 -103.86 -128.19 -135.07 -159.46 -166.32

-464.54 -506.63 -554.71 -586.91 -640.63 -665.53

41.40 47.11 37.11 39.83 31.45 32.01

Tetramer 2 C12H25NH2 C13H27NH2 C14H29NH2 C15H31NH2 C16H33NH2

-190.66 -197.59 -221.93 -228.86 -253.22

-713.70 -743.97 -794.26 -821.97 -927.55

22.02 24.11 14.76 16.09 23.19

C6H13NH2 C7H15NH2 C8H17NH2 C9H19NH2

-124.78 -134.54 -165.82 -176.28

-546.70 -582.27 -626.61 -671.81

38.14 38.97 20.91 23.92

Tetramer 3 C10H21NH2 C11H23NH2 C12H25NH2 C13H27NH2

-207.58 -218.09 -249.37 -256.85

-722.76 -762.45 -812.56 -823.40

7.81 9.12 -7.23 -11.48

C6H13NH2 C7H15NH2

-240.64 -264.35

-941.98 -975.13

40.07 26.24

Hexamer 1 C9H19NH2 C10H21NH2

-318.34 -342.15

-1072.75 -1127.20

1.34 -6.25

molecule

TABLE 4: Coefficients of Correlation Equations y ) (a ( ∆a)ka + (b ( ∆b) for the Thermodynamic Characteristics of Amine Cluster Formationa system

(a ( ∆a)

(b ( ∆b)

R

S ∆HCl m,

system

(a ( ∆a)

(b ( ∆b)

R

S

dimer 1 dimer 2 dimer 3 dimer 4 dimer 5 dimer 6 dimer 7

-10.27 ( 0.26 -10.06 ( 0.27 -10.10 ( 0.27 -10.11 ( 0.23 -10.07 ( 0.28 -10.13 ( 0.25 -5.32 ( 0.24

-2.62 ( 1.42 -3.10 ( 1.48 -0.95 ( 1.50 -1.08 ( 1.28 -2.14 ( 1.54 -3.12 ( 1.38 -1.95 ( 1.30

0.997 0.997 0.997 0.998 0.997 0.997 0.999

1.36 1.42 1.45 1.24 1.48 1.33 1.25

kJ/mol trimer 1 trimer 2 trimer 3 tetramer 1 tetramer 2 tetramer 3

-10.29 ( 0.24 -10.30 ( 0.24 -9.47 ( 0.07 -10.30 ( 0.24 -10.30 ( 0.23 -10.24 ( 0.23

-5.34 ( 2.65 -0.75 ( 2.67 -5.94 ( 1.24 -1.01 ( 3.90 -8.13 ( 3.88 -7.34 ( 5.14

0.998 0.998 0.9997 0.998 0.998 0.998

2.55 2.57 1.19 3.76 3.74 4.95

dimer 1 dimer 2 dimer 3 dimer 4 dimer 5 dimer 6 dimer 7

-26.47 ( 1.24 -22.87 ( 1.77 -24.56 ( 1.56 -24.05 ( 1.37 -23.96 ( 1.83 -23.86 ( 1.38 -33.65 ( 0.73

-78.27 ( 6.81 -94.31 ( 9.75 -84.56 ( 8.59 -84.82 ( 7.55 -86.39 ( 10.08 -88.11 ( 7.60 -103.48 ( 2.06

0.99 0.97 0.98 0.99 0.97 0.99 0.99

∆SCl m , J/(mol K) 6.56 trimer 1 9.39 trimer 2 8.28 trimer 3 7.27 tetramer 1 9.71 tetramer 2 7.33 tetramer 3 1.98

-26.12 ( 1.12 -23.98 ( 1.16 -14.99 ( 0.12 -25.86 ( 1.52 -27.97 ( 1.24 -21.33 ( 0.81

-164.64 ( 12.29 -177.86 ( 12.74 -234.80 ( 2.05 -249.30 ( 25.14 -232.09 ( 20.52 -309.67 ( 18.00

0.992 0.99 0.9997 0.99 0.991 0.993

11.84 12.28 1.97 24.22 19.77 17.30

0.99 0.93 0.95 0.97 0.95 0.97 0.98

∆GCl m , kJ/mol 0.67 trimer 1 2.20 trimer 2 1.55 trimer 3 1.36 tetramer 1 1.76 tetramer 2 1.33 tetramer 3 1.81

-3.15 ( 0.17 -2.51 ( 0.11 -5.00 ( 0.10 -2.60 ( 0.35 -1.97 ( 0.27 -3.88 ( 0.87

0.99 0.991 0.998 0.93 0.93 0.998

1.86 1.17 1.56 5.58 4.28 1.84

dimer 1 dimer 2 dimer 3 dimer 4 dimer 5 dimer 6 dimer 7 a

-2.39 ( 0.12 -3.24 ( 0.41 -2.78 ( 0.29 -2.94 ( 0.26 -2.93 ( 0.33 -3.02 ( 0.25 4.71 ( 0.34

20.71 ( 0.68 25.00 ( 2.27 24.25 ( 1.61 24.20 ( 1.41 23.61 ( 1.83 23.13 ( 1.38 28.89 ( 1.88

52.25 ( 1.93 43.72 ( 1.21 64.03 ( 1.61 73.28 ( 5.79 61.03 ( 4.45 84.95 ( 1.91

The amount of sampling is equal to 11.

hydrogen interactions: NH2-NH2 and N-N. The number of these interactions are given by eqs 1.From eq 1 it is straight-

nNH2-NH2 ) (p - 1)q nN-N ) (q - 1)p

(1)

forward to determine the number of intermolecular hydrogen interactions in a cluster (eq 2),

ka ) [(p - 1)q + (q - 1)p]{n/2}

(2)

where n is the number of carbon atoms in the alkyl chain, the braces denote the integer part of a number, and p and q are the

Figure 6. Geometric structure of the rectangular p-q cluster.


Vysotsky et al.

Figure 7. Geometric structures of clusters on the basis of trimer 3: (a) trimer 3; (b) hexamer 1; (c) first layer of an infinite hexagonal cluster; (d) two internal layers of an infinite hexagonal cluster; (e) three internal layers of an infinite hexagonal cluster.

Figure 8. Dependence of the enthalpy variation of the linear clusters formation on the alkyl chain length.

Figure 9. Dependence of the entropy variation of the linear clusters formation on the alkyl chain length.

numbers of monomers in a plane rectangular cluster in two mutually perpendicular directions; see Figure 5. To determine the parameters for infinite clusters, the limits of the number of intermolecular hydrogen bonds in a cluster with sequential arrangement of the NH2 bonds, the number of interactions between NH2 bonds with their parallel arrangement, and the number of intermolecular H-H interactions per amine molecule should be calculated when the number of molecules approaches infinity. For a linear cluster that involves NH2-NH2 interactions (p ) ∞, q ) 1), the above expressions become eqs 3.

nNH2-NH2 ) 1

nN-N ) 0

ka ) {n/2}

(3)

To obtain the expressions for the thermodynamic characteristics per amine molecule, one has to introduce eqs 3 into the correlation dependencies for enthalpy and entropy of the cluster formation. For the linear cluster (p ) ∞, q ) 1) the resulting expressions are as follows.

dHCl ∞ /m ) -10.11{n/2} - 1.31 dSCl ∞ /m ) -24.49{n/2} - 80.36 dGCl ∞ /m ) -2.81{n/2} + 22.64

(4)

Figures 8 and 9 illustrate the dependencies of the enthalpy and entropy variations of the cluster formation on the alkyl chain length. Similar to the systems studied earlier,3-8 these dependencies are step-like. It is seen that the dependencies predicted from eqs 4 (shown by lines) agree well with the results obtained

Figure 10. Dependence of the variation of free energy per monomer for the formation of rectangular clusters on the alkyl chain length.

from direct calculations (points). Figure 10 displays the dependence of the cluster formation free-energy variation per amine monomer on the alkyl chain length. It is seen that, for alkylamines, a spontaneous cluster formation becomes possible if the alkyl chain length becomes equal to or higher than 18-19 carbon atoms, whereas for alcohols this threshold is 10-12,4 and for carboxylic acids7 and thioalcohols8 the threshold value is 14-15 carbon atoms. This difference could be attributed to the fact that the intermolecular hydrogen bonds N-H‚‚‚N are less strong than the O-H‚‚‚O bonds, whereas the bonds formed between the amines and water molecules are stronger. Therefore, the formation of aggregates between amines becomes possible when the solubility of amines is small, (i.e., for larger alkyl chain length).


J. Phys. Chem. C, Vol. 111, No. 42, 2007 15349 Π/A isotherms. The free energy for the two-dimensional (2D) cluster formation could be calculated from the characteristic values (critical point coordinates, area per molecule in the condensed state, and aggregation number for small aggregates in the LE monolayer)3,4 of ∼2 kJ/mol for eicosylamine and ∼5 kJ/mol for docosylamine. These values, estimated from the experimental data,13-16 qualitatively agree with the data shown in Figure 10 and support the possibility of the formation of 2D rectangular infinite clusters. Conclusions

Figure 11. Dependence of the variation of free energy per monomer for the formation of hexagonal clusters on the alkyl chain length.

Other types of infinite clusters are illustrated in Figure 7; these clusters involve the ‘tri’ and ‘lac’ interactions. To analyze these clusters, the number of layers ‘r’ is introduced. For example, the entities shown in Figure 7, panels c-e, consist of one, two, and three layers, respectively. Thus, the general expressions become those shown in eqs 5:

ka ) 6r{n/2} + 6r(3r - 1)n,

ntri ) 18r2 nlac ) 3(3r2 - r) (5)

Similar to the case considered above, the limits should be calculated for the number of molecules in a cluster approaching infinity (eqs 6).

ka∞ ) n

ntri ) 1

nlac∞ ) 0.5

(6)

Next, the calculated values were introduced into the regression equations for entropy, enthalpy, and the Gibbs’ energy of the cluster formation (eqs 7).

References and Notes

dHCl ∞ /m ) -10.11{n/2} - 1.87 dSCl ∞ /m ) -24.49{n/2} - 103.13 dGCl ∞ /m ) -2.81{n/2} + 28.86

To summarize, the thermodynamic parameters for the cluster formation of alkylamines at the air/water interface are calculated using the semiempirical PM3 method. The calculated values agree well with the experimental data. It is shown that the formation of dimers on the basis of the e-type of H-H interactions is disadvantageous from the energetic considerations. Therefore, the consideration of more complicated structures formed on the basis of units of this of interaction type can largely be ignored. The slopes of correlation dependencies of enthalpy and entropy for the cluster formation on the alkyl chain length of alkylamines are almost the same as those for alkylalcohols,4 alkylcarboxylic acids,7 and alkylthioalcohols,8 indicating the similarity of the intermolecular H-H interactions involved in these classes of compounds. It is shown that, similar to the systems studied earlier, the dependencies of enthalpy, entropy, and Gibbs’ energy for the cluster formation on the alkyl chain length are step-like, with the corresponding correlation coefficients of above 0.99, 0.97, and 0.93, respectively. Spontaneous cluster formation of alkylamines takes place when the alkyl chain length exceeds 18-19 carbon atoms, which is higher than the values characteristic for alcohols, carboxylic acids, and thioalcohols.

(7)

The dependencies of free energy for the cluster formation on the alkyl chain length for two-dimensional clusters are shown in Figure 11. Spontaneous cluster formation for these types of entities becomes possible for alkyl chain lengths equal to or higher than 13-14 carbon atoms. However, because the simultaneous collision of three particles is far less probable, the cluster formation should take place preferentially with the formation of rectangular infinite clusters with the process threshold of 18-19 carbon atoms in the alkyl chain. Experimental results concerning cluster formation by saturated amines are quite scarce and refer to alkyl chain lengths of 1822 carbon atoms.13-16 The surface pressure/area per one molecule (Π/A) isotherm for octadecylamine (C18) at 25 °C does not exhibit any critical points relevant to a transition from the liquid-expanded (LE) to liquid-condensed (LC) monolayer state.13 At the same time, the analysis of this isotherm using the thermodynamic model proposed in refs 3 and 4 indicates that the formation of small aggregates (aggregation number 4-8) is probable in the LE monolayer range. The studies of eicosylamine (C20)14,15 and docosylamine (C22)16 monolayers indicate the existence of a LE/LC phase transition point in the

(1) ComprehensiVe Organic Chemistry. The Synthesis and Reactions of Organic Compounds; Sutherland, I. O. Ed.; Pergamon Press Ltd: Oxford, 1979; Vol 2. (2) Soloviov, M. E.; Soloviov, M. M. Computational Chemistry; SOLON-Press: Moscow, 2005 (in Russian). (3) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. B 2002, 106, 121. (4) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2002, 106, 11285. (5) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D.; Miller, R. Colloids Surf. A 2002, 209, 1. (6) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D.; Miller, R. Progr. Colloid. Polym. Sci. 2002, 121, 72. (7) Vysotsky, Yu. B.; Muratov, D. V.; Boldyreva, F. L.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. B 2006, 110, 4717. (8) Vysotsky, Yu. B.; Belyaeva, E. A.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. C 2007, 111, 5374-5381. (9) Daubert, T. E.; Danner, R. P.; Sibul, H. M.; Stebbins, C. C. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation, Parts 1-5; Taylor & Francis: Pennsylvania, 1998. (10) Stull, D. R.; Westrum, E. F., Jr.; Sinke, G. C. The Chemical Thermodynamics of Organic Compounds; John Wiley & Sons: New York, 1969. (11) Zwolinski, B. J; Wilhoit, R. Heats of Formation and Heats of Combustion In American Institute of Physics Handbook, 3rd ed; D. E. Gray; Mc. Graw-Hill: New York, 1972. (12) Rihani, D. N. Hydrocarbon Process. 1968, 47, 111. (13) Puggelli, M.; Gabrielli, G. Colloid Polym. Sci. 1980, 261, 82. (14) Nakahara, H.; Mo¨bius, D., J. Colloid Interfaces Sci. 1986, 114, 363. (15) Vogel, V.; Mo¨bius, D., J. Colloid Interfaces Sci. 1988, 126, 408. (16) Johnston, D. S.; Coppard, E.; Parera, G. V.; Chapman, D. Biochemistry, 1984, 23, 6912.

Quantum Chemical Analysis of the Thermodynamics of 2-Dimensional

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