Quantum-Chemical Description of the ... - ACS Publications

J. Phys. Chem. B 2009, 113, 13235–13248

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Quantum-Chemical Description of the Thermodynamic Characteristics of Clusterization of Melamine-type Amphiphiles at the Air/Water Interface Yu. B. Vysotsky,† A. A. Shved,† E. A. Belyaeva,† E. V. Aksenenko,‡ V. B. Fainerman,§ D. Vollhardt,*,| and R. Miller| Donetsk National Technical UniVersity, 58 Artema Strasse, 83000 Donetsk, Ukraine, Institute of Colloid Chemistry and Chemistry of Water, 42 Vernadsky AVenue, 03680 KyiV (KieV), Ukraine, Donetsk Medical UniVersity, 16 Ilych AVenue, Donetsk 83003, Ukraine, and Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany ReceiVed: May 17, 2009; ReVised Manuscript ReceiVed: August 18, 2009

The semiempiric PM3 method is used to calculate the thermodynamic parameters of the formation of monomers, dimers, trimers, and tetramers of the amphiphilic melamine-type series of 2,4-di(n-alkylamino)-6-amino1,3,5-triazine (2CnH2n+1-melamine) with n ) 9-16. The most stable conformations are determined, which then are used to construct the clusters. The peculiar feature of these structures is the existence of a bend at one of the alkyl chains. Thus, the formation of infinite films becomes possible because of their spatial arrangement. From the calculation of the relative amount of various conformers in the mixture, it follows that, if the alkyl chain length is lower than 11-12 carbon atoms, the mixture is composed mainly of the monomers that do not contain any intramolecular interactions, whereas for higher alkyl chain lengths the monomers that involve such interactions prevail in the mixture. For all clusters thus considered, the thermodynamic parameters (enthalpies, entropies, and Gibbs’ energies) of clusterization are calculated. It is shown that the dependencies of these parameters on the alkyl chain length either exhibit stepwise shape or are represented by the combination of a linear and stepwise function. This depends on the different number of hydrogen-hydrogen interactions in the structures considered. Five types of clusters that are capable of the formation of infinite 2D films are considered in detail. For each of these types, the dependencies of the clusterization enthalpy, entropy, and Gibbs’ energy on the alkyl chain length in the constituting monomers are derived. Using these dependencies, it becomes possible to calculate these thermodynamic characteristics for clusters of any size, and also for infinite 2D films. It is shown that the spontaneous clusterization of 2CnH2n+1-melamine becomes possible if the alkyl chain length exceeds 9 carbon atoms. Introduction The successful design of supramolecular architectures by combination of melamine with pyrimidine derivatives1-10 in organic media and in the solid state has focused the attention on artificial recognition models and crystalline architectures11 with high effectiveness of complementary hydrogen bonds in melamine-type systems. A large number of host molecules have been synthesized to comprehend artificial molecular recognition by complementary hydrogen bonding. The use of twodimensional arrangements at the air-solution interface turned out to be an interesting approach to regulate the recognition process by incorporating strong directional interactions of hydrogen bonds with a dissolved component for the manufacture of thin films.12,13 Correspondingly, amphiphilic melamine derivatives have been synthesized and used as host-component in monolayers and bilayers for interfacial molecular recognition of barbiturates.14-17 It has been shown that systems consisting of an amphiphilic melamine-type monolayer and a pyrimidine derivative dissolved in the aqueous subphase are good candidates for the formation of interfacial supramolecular entities by hydrogen-bond-based molecular recognition of nonsurface-active * Corresponding author. E-mail: [email protected]. † Donetsk National Technical University. ‡ Institute of Colloid Chemistry and Chemistry of Water. § Donetsk Medical University. | Max Planck Institute of Colloids and Interfaces.

substrates. The molecular recognition of nonsurface-active pyrimidine derivatives dissolved in the aqueous subphase changes drastically and in specific way the characteristic features (π-A isotherms, morphology of the condensed phase domains) of these special melamine-type monolayers.18-24 It is interesting to note that the striking differences in the main characteristics between the supramolecular assemblies can be related to their different chemical structures depending on the chemical structure of the pyrimidine derivatives.18-20,24 The headgroup of 2,4-di(n-alkylamino)-6-amino-1,3,5-triazine (2CnH2n+1melamine) has two binding faces on both sides. The binding faces of the studied pyrimidine derivatives are different. Therefore, two pyrimidine derivatives can be bonded by 2CnH2n+1-melamine if they have only one binding face (thymine, uracil), whereas a linear supramolecular network between 2CnH2n+1-melamine and the pyrimidine derivative is formed by a linear extension of the hydrogen-bond interaction if the pyrimidine derivative has two binding faces (barbituric acid). The present work focuses on the quantum chemical approach to the calculation of the clusterization parameters of this amphiphilic melamine-type host-component of the monolayer. 2CnH2n+1-melamine with n ) 9-16 was selected for this study. The amphiphilic structure of the molecule consists of two alkyl chains fixed at one polar headgroup. The present study should provide additional theoretical insight into the processes related to the formation of a new condensed phase governed by relevant

10.1021/jp904598k CCC: $40.75  2009 American Chemical Society Published on Web 09/17/2009

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Figure 1. Orientation of melamine derivative molecules relative to the air/water interface.

intra/intermolecular interactions ranging from C-H · · · O hydrogen bonds to H-H bonds intermolecular interactions.25,26 In the present study, quantum chemical methods are applied for the thermodynamic description of the aggregation of pure amphiphilic melamine-type monolayers and the interaction of the amphiphilic melamine-type monolayer with a pyrimidine derivative having two binding faces (barbituric acid). Methods All calculations were made applying the semiempirical PM3 method. The optimization of all geometric structures studied was performed by the MOPAC2000 software suite using the BFGS algorithm.29 The thermodynamic parameters of molecular systems that involve intermolecular interactions (intermolecular hydrogen bonds and intermolecular hydrogen-hydrogen bonds) were calculated. For the complexes studied here, essential are the entropy contributions from vibration frequencies below 100 cm-1, which are neglected in the present version of the software. Therefore, the contributions from these frequencies were calculated manually similar to our previous studies.30-36,38,39 It should be noted that the air/water interface was accounted for implicitly. The molecules of 2CnH2n+1-melamine were located relative to each other according to their orientation at the interface; see Figure 1. Also, as the surface exhibits the expanding properties (the hydrophilic parts of the molecules are drawn into the aqueous phase, whereas the hydrophobic parts are repulsed from this phase), only linear conformations of the molecules were considered. The solvation effects are neglected, because the molecule is mostly located in the air phase, while the clusterization processes are governed primarily by the interactions between the alkyl chains, which becomes evident from the experimental fact that clusterization takes place only for the alkyl chain lengths larger than a certain value. The monolayers at the interface were modeled by an infinite twodimensional regular array. Note that, although this approach does not involve the aqueous phase explicitly, it was shown30-36,38,39 that this approximation is capable of a quite reliable description of the clusterization at the interface for a vide variety of surfactants, and the results of the calculations agree well with the experimental data. Results and Discussion Monomers. As mentioned above, the clusterization of 2CnH2n+1-melamine is considered in the present study. To calculate the thermodynamic parameters of clusterization for these amphiphiles, the thermodynamic parameters of formation of the corresponding monomers should be determined. Therefore, we begin with the calculation of enthalpy and Gibbs’

energy of the formation of monomers from elementary substances, and their absolute enthalpies. It was noted above that the melamine molecules are oriented at the interface in a way that the hydrophilic polar triazine group (“head” of the molecule) is immersed into the aqueous phase, whereas the hydrophobic alkyl chains are expelled from water and can form the 2D clusters if the concentration exceeds a certain value. In our previous papers,30-36 the position of the functional group relative to the alkyl chain was analyzed to determine which conformation of the molecule is the most stable one. In the present study, such analysis is irrelevant, because the plane conjugate system of triazine determines the only existing conformation of the polar headgroup of the molecule; see Figure 2. It was shown26-32 that infinite 2D films are formed on the basis of the monomer 2 structure shown in Figure 2. However, some other isomers can exist. Relevant possible structures are shown in Figure 2 (the structures were built using the ChemOffice 9.0 software). The geometry of the monomer 1-monomer 4 structures, as optimized using the Mopac2000 software, is shown in Figure 3. It is seen that these monomers cannot be arranged to 2D infinite clusters. In fact, as the polar headgroup of the molecules is immersed into water, while the alkyl chains are located in the air phase (even this is impossible for monomer 4), only one or two intermolecular hydrogen-hydrogen bonds can be formed between the alkyl chains of two adjacent monomers. In our previous publications,30-36 it was shown that the intermolecular H-H interactions mainly contribute to the Gibbs’ clusterization energy, and clusterization becomes possible if 5-9 interactions of the “a” type are present (possible types of interactions are illustrated by Figure 4). Therefore, the existence of one or two such interactions is insufficient for a stable infinite extension. Note that, in our earlier studies,30-36 this question could not arise, because the amphiphiles considered therein have only one alkyl chain. It is evident that an infinite cluster on the basis of 2CnH2n+1melamine can only be formed by the monomer in which the two alkyl chains are parallel to each other. In this case, these two parallel alkyl chains behave as a single entity, making the formation of 2D structures possible. Two chains can only be parallel to each other if one of these chains form a bend; see monomers 5, 6, and 7, as shown in Figure 2. The optimized geometry of these structures is illustrated by Figure 3 (note that monomers 3 and 7 are the enantiomers of monomers 1 and 5, respectively, and are therefore omitted in Figure 3). It should be stressed that the optimized structure of monomer 5 is

Clusterization of Melamine-type Amphiphiles

J. Phys. Chem. B, Vol. 113, No. 40, 2009 13237

Figure 2. Schematic representation of possible structures of melamine derivative monomers, as built by using the ChemOffice 9.0 software.

Figure 3. Geometric structures of monomers of some amphiphilic melamine-type components (2CnH2n+1-melamine) optimized by the semiempiric PM3 method.

essentially different from the schematic structure shown in Figure 2. It is seen that the bend (shown by the red arrow in Figure 3) is formed in one of the alkyl chains. The characteristic feature of this structure is the formation of hydrocarbon bonds between the two methylene groups of the bended alkyl chain and the free electron pairs of the nitrogen atom in the triazine ring. This in turn leads to the displacement of two carbon atoms of methylene groups out of the molecular plane. Thus, the intramolecular hydrogen-hydrogen interactions of the (ther-

modynamically preferable) “a” type are formed. This results in the formation of a compact monomer 5 structure, for which, in contrast to the branched monomer 1-4 structures (cf., Figure 3), no steric hindrances exist, preventing the formation of infinite films. The same is true for monomer 6, which has also parallel alkyl chains. Therefore, in the case considered here, monomers 5, 6, and 7 can form infinite films at the air/water interface. Note that the fatty radicals exist in the trans-configuration for

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Figure 4. Types of hydrogen-hydrogen interactions.

TABLE 1: Thermodynamic Parameters of 2CnH2n+1-Melamine Molecules n

monomer 1

monomer 2

monomer 5 ∆Hmon 298 ,

9 10 11 12 13 14 15 16 9 10 11 12 13 14 15 16 9 10 11 12 13 14 15 16

Standard Enthalpy of Formation -263.59 -263.60 -276.74 -308.99 -308.96 -330.56 -354.32 -354.33 -377.80 -399.63 -399.69 -431.71 -445.05 -445.02 -478.96 -490.37 -490.43 -532.84 -535.82 -535.79 -580.12 -581.19 -581.16 -633.96

monomer 6 kJ/mol -262.11 -304.54 -375.37 -397.38 -447.98 -494.29 -547.07 -593.62

Standard Entropy of Formation ∆Smon 298 , J/(mol · K) -2503.57 -2501.62 -2553.80 -2617.71 -2713.44 -2712.61 -2773.18 -2833.99 -2920.80 -2919.34 -2998.34 -3085.93 -3133.42 -3131.38 -3202.59 -3265.79 -3340.94 -3338.06 -3433.09 -3486.47 -3548.68 -3545.48 -3646.41 -3708.02 -3752.53 -3760.14 -3865.61 -3936.17 -3960.96 -3968.41 -4074.97 -4179.36 Standard Gibbs’ Energy ∆Gmon 298 , kJ/mol 482.47 481.88 484.29 499.61 499.39 495.85 516.08 515.64 515.71 534.13 533.46 522.66 550.55 549.72 544.10 567.13 566.13 553.79 582.43 584.73 571.83 599.18 601.43 580.38

517.97 539.98 544.24 575.83 590.99 610.70 625.90 651.83

monomers 5 and 7, and in the cis-configuration for monomer 6. This orientation of the radicals determines the type of the intramolecular hydrogen-hydrogen interactions shown in Figure 4, which arise between them. More specifically, the “a” type interactions are formed for monomers 5 and 7, whereas the “e” type interactions are formed for monomer 6. It was shown30 that the formation of clusters with “e” type of intramolecular H-H interaction is disadvantageous (the clusterization Gibbs’ energy becomes higher with the increase of the alkyl chain length). It should be expected therefore that, due to the intramolecular H-H interactions, monomer 6 is less stable than monomer 5 (cf., Table 1), and the formation of clusters formed

on the basis of monomer 6 because of the intermolecular H-H interactions is impossible. Therefore, only monomer 5 is considered here as the building element of 2D films. The schematic notation for the two alkyl chains, used herein for monomer 5 and the clusters thus built, is explained in Figure 5. Here, the radicals A and B correspond to the bended and nonbended chains, respectively. For all of these conformers, the formation enthalpies, absolute entropies, and Gibbs’ energies of formation were calculated; see Table 1. The thermodynamic parameters for monomer 3 and monomer 7 are equal to those calculated for monomer 1 and monomer 5, respectively (because of the fact that these structures are corresponding mirror isomers). The alkyl chain length in the structures studied here was 9-16 carbon atoms. Note that, for lower alkyl chain lengths in monomers that comprise intramolecular H-H interactions (i.e., monomers 5, 6, and 7), edge effects arise due to the formation of additional interactions, which lead to changes in the relative orientation of alkyl chains in the molecule. These additional interactions are irregular, which prevents a consistent description of 2D films composed of the monomers with low alkyl chain lengths. It is seen from Table 1 that monomer 5 is most favorable. For monomer 6, the calculation of the thermodynamic parameters once again has shown that the formation of structures (also films) on the basis of molecules that involve the “e” type of H-H interactions is thermodynamically unfavorable. For monomers 1, 2, and 4, the distance between their alkyl chains is too high, and these alkyl chains do not significantly interact with the triazine ring. This results in equal thermodynamic characteristics of these monomers; cf., Table 1. It was earlier shown for fatty alcohols,30-33 carboxylic acids,34 thioalcohols,35 and amines36 that the contributions to thermodynamic quantities from CH2 groups are additive. It becomes possible, therefore, to determine the corresponding regression dependencies of the thermodynamic characteristics on the number of CH2 groups and number of intramolecular H-H interactions on the basis of the calculated values of formation mon mon enthalpies ∆H298 and absolute entropies S298 of monomers. These dependencies were built for the homologous series of

Figure 5. Schematic representation of monomers for seven melamine derivatives.



Figure 6. Dependence of standard Gibbs’ energy of the monomer formation on the alkyl chain length for the monomers indicated in the figure.

Figure 7. Optimized structures of dimers, trimers, and tetramers of 2CnH2n+1-melamine.

monomers 1, 2, 5, and 6. The correlation coefficients were found mon mon and 0.999 for S298 to exceed 0.9999 for ∆H298 ; for monomers 1, 2, and 5, standard deviations were 0.02-0.04 kJ/mol for mon mon and 1.94-4.34 J/(mol · K) for S298 . These standard ∆H298 mon deviations were higher for monomer 6: 7.37 kJ/mol for ∆H298 mon and 15.88 J/(mol · K) for S298 . This difference results from the fact that the “e” interaction type is less favorable than the “a”

type, and, therefore, the relative location of alkyl chains for “e” interactions is less firmly fixed. Of all of the monomers studied, monomer 5 is most energetically favorable and capable of the formation of infinite films. The correlation dependencies of its thermodynamic parameters on the number of CH2 groups nCH2 and the number of intramolecular H-H interactions nH-H are:

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mon ∆H298 ) [-(23.63 ( 0.01) · nCH2 - (6.60 ( 0.04) · nH-H + (161.89 ( 0.11)] kJ/mol (R ) 0.999; S ) 0.03 kJ/mol; N ) 10)

(1) mon S298 ) [(23.87 ( 1.77) · nCH2 + (13.50 ( 6.63)nH-H + (445.69 ( 19.13)] J/(mol · K) (R ) 0.999; S ) 4.58 J/(mol · K); N ) 10) (2)

where R is the regression coefficient, S is the standard deviation, and N is the sampling amount. Figure 6 illustrates the dependencies of Gibbs’ energies of the monomer formation on the alkyl chain length. For monomers 1 and 2, these dependencies are linear and almost coincide with each other. For monomers 5 and 6, the dependencies are the superpositions of linear and stepwise functions. It is seen from Figure 6 that for n > 11, the monomers that involve the “a” type intramolecular interactions (monomer 5) are more preferable as compared to other monomers. Now the relative concentration of the monomers that contain the intramolecular H-H interactions and those that do not contain these interactions in the mixture should be determined. This percentage could be calculated from the Gibbs’ standard energies of formation for coexisting monomers.37 Here, it should be kept in mind that (as mentioned above; cf., also Table 1), as monomers 1, 2, 3, and 4 do not contain intramolecular H-H interactions, their thermodynamic parameters are the same, and, therefore, their equilibrium concentrations are equal to each other, whereas monomers 5 and 7 are corresponding enantiomers with equal thermodynamic parameters. Thus, the calculated percentages of equilibrium concentrations of monomers 1, 2, 3, and 4 are 20.16, 7.62, 15.82, 0.48, 3.22, 0.23, and 0.67 for the alkyl chain length 9-15, while the values for monomers 5 and 7 for these alkyl chain lengths are 9.67, 34.76, 18.37, 49.04, 43.55, 49.55, and 48.65. It is seen that the concentrations of monomers 5 and 7, which contain intermolecular H-H interactions for the alkyl chain length above 11-12, are higher than the concentrations of monomers in which these interactions are absent. It becomes evident from the structural and thermodynamic features of the monomers shown in Figure 3 that only the optimized structures of monomer 5 can be used to construct the clusters capable of the formation of infinite 2D films. On the other hand, there are two reasons that prevent the formation of infinite structures on the basis of monomer 1, its enantiomer monomer 3, and monomer 2. First, the spatial geometry of these monomers enables them to arrange only into spherical micelles and micelles that have the shape of an infinite cylinder. Second, their formation is less advantageous from the thermodynamic considerations; see Table 1. Also, for alkyl chain lengths above 11-12, the composition of the mixture is essentially different from that characteristic for lower alkyl chain lengths. Dimers, Trimers, and Tetramers. The initial structures of dimers, trimers, and tetramers were built on the basis of the optimized structure of monomer 5, as shown in Figure 7. All of these structures can be divided into three groups according to the type of the relative orientations of monomers in the cluster: (i) dimers, trimers, and tetramers with “stacked” structure; these are dimers 1 and 2, trimers 1 and 2, and tetramers 1 and 2; (ii) dimers and trimers with “linear” structure; these are dimers 3, 4, and 5, trimer 3; and (iii) all other dimers, trimers, and tetramers; these are dimers 6, 7, and 8, trimers 4 and 5, and tetramers 3 and 4. In these entities, “linear” interactions are present, whereas “stacked” interactions between the heads

groups of the molecules are absent. Also, it is seen that, in this case, H-H interactions other than the “stacked” interactions are formed; that can be further subdivided into the “internal” (shown in Figure 7 by arrows for dimers 6, 7, and 8) and “external” interactions (located on the periphery of the molecules). For all compounds, shown in Figure 7, the thermodynamic parameters of their formation and clusterization were calculated. To calculate the clusterization parameters, similar to our previous publications,30-36,38,39 standard expressions were used: 0 mon Cl ∆HCl 298 ) ∆H298 - m · ∆H298 for the clusterization enthalpy, ∆S298 0 mon Cl ) S298 - m · ∆S298 for the clusterization entropy, and ∆G298 ) Cl Cl - T · ∆S298 for the clusterization Gibbs’ energy. Here, m ∆H298 0 0 and S298 are is the number of monomers in the cluster, ∆H298 the enthalpy of formation and absolute entropy of the corremon mon and S298 are enthalpy of the sponding cluster, and ∆H298 formation and absolute entropy of the constituting monomer(s). The calculated values are listed in Table 2. Using the data thus obtained, the partial correlation dependencies of the clusterization thermodynamic parameters on the number of intermolecular H-H interactions and on the number of intermolecular interactions between the functional groups were calculated. The correlation parameters for each thermodynamic quantity considered are listed in Table 3. It is seen that the slopes of these correlations are quite similar to the values calculated earlier5-10,27 for other classes of surfactants. This indicates the similar character of the intermolecular H-H interactions involved in these compounds. Also, the slopes of these partial correlations are close to each other (see Table 3). Therefore, all of the correlations can be generalized into one. As the most probable route for the formation of the clusters includes the formation of “stacked” structures, the “linear” clusters are disregarded in the correlation. The generalized correlation dependencies of clusterization enthalpy and entropy on the number of intermolecular H-H interactions and the number of interactions between functional groups for those clusters are shown below that are capable of the formation of clusters with large dimensions: ∆HCl 298 ) [-(7.03 ( 0.13) · Ka′ - (10.50 ( 0.13) · Ka + (1.11 ( 1.31) · n1 - (4.47 ( 1.34) · n2 - (8.08 ( 1.09) · n3 (10.45 ( 0.99) · n6 - (10.14 ( 1.52) · n7 - (3.47 ( 1.22) · n8] kJ/mol (3) (R ) 0.999; S ) 3.55 kJ/mol; N ) 86)

∆SCl 298 ) [-(18.89 ( 1.16) · Ka′ - (18.68 ( 1.16) · Ka (175.43 ( 11.81) · n1 - (173.96 ( 12.03) · n2 (78.37 ( 9.78) · n3 - (147.63 ( 8.96) · n6 (122.83 ( 13.64) · n7 - (130.46 ( 10.96) · n8] J/(mol · K) (R ) 0.999; S ) 31.80 J/(mol · K); N ) 86)

(4)

where Ka is the number of intermolecular “stacked”, “linear”, and “internal” interactions; K′a is the number of intermolecular “external” interactions in clusters similar to dimers 6, 7, and 8; and ni is the number of interactions between the triazine rings in the cluster as in dimers 1, 2, 3, 6, 7, and 8; see Figure 7. The values of slopes relative to Ka agree well with the parameters reported earlier for alcohols,30-33 carboxylic acids,34 thioalcohols,35 amines,36 nitriles,38 and cis-monounsaturated carboxylic acids.39 This fact will enable the generalization of the superposition-additive scheme onto all of these classes of surfactants. Large and Infinite Clusters. Now one can calculate the thermodynamic parameters of clusterization for any regular (also



0 TABLE 2: Standard Thermodynamic Characteristics of Melamine Derivative Clusters: ∆H298 , Enthalpy of Cluster Formation; 0 Cl Cl Cl S298 , Absolute Entropy of Cluster; ∆H298 , ∆S298 , and ∆G298 , Clusterization Enthalpy, Entropy, and Gibbs’ Energy, Respectively

n

∆H0298, kJ/mol

∆HCl 298, kJ/mol

9 10 11 12 13 14 15 16

-635.92 -754.55 -858.55 -977.70 -1081.79 -1200.87 -1305.02 -1424.04

-82.43 -93.44 -102.95 -114.28 -123.88 -135.19 -144.79 -156.11

9 10 11 12 13 14 15 16

-623.87 -734.25 -845.26 -957.02 -1068.10 -1179.84 -1291.21 -1402.66

9 10 11 12 13 14 15 16

S0298, J/(mol · K)

∆SCl 298, J/(mol · K)

∆GCl 298, kJ/mol

Dimer 1 1514.22 1588.24 1708.47 1774.64 1890.54 1960.35 2080.33 2105.19

-300.50 -332.75 -307.25 -377.61 -345.77 -394.35 -381.02 -482.48

7.12 5.72 -11.39 -1.75 -20.84 -17.67 -31.24 -12.33

-70.38 -73.13 -89.66 -93.60 -110.19 -114.15 -130.98 -134.73

Dimer 2 1519.49 1629.79 1735.18 1818.71 1912.77 2012.73 2111.75 2144.43

-295.24 -291.20 -280.54 -333.55 -323.54 -341.97 -349.60 -443.23

17.60 13.65 -6.06 5.79 -13.77 -12.24 -26.80 -2.65

-592.01 -701.99 -803.89 -914.17 -1016.53 -1124.94 -1228.13 -1337.42

-38.52 -40.87 -48.30 -50.75 -58.60 -59.24 -67.90 -69.48

Dimer 3 1613.11 1717.50 1829.39 1922.77 1968.32 2081.82 2167.79 2230.84

-201.60 -203.49 -186.33 -229.48 -267.98 -272.87 -293.55 -356.82

21.55 19.76 7.23 17.62 21.25 22.07 19.57 36.84

9 10 11 12 13 14 15 16

-591.16 -708.70 -802.91 -921.17 -1015.31 -1133.78 -1227.86 -1346.28

-37.67 -47.59 -47.31 -57.76 -57.40 -68.09 -67.63 -78.35

Dimer 4 1609.87 1686.74 1814.93 1881.29 2012.56 2034.76 2154.55 2217.08

-204.85 -234.25 -200.79 -270.96 -223.74 -319.94 -306.79 -370.58

23.37 22.22 12.52 22.99 9.28 27.25 23.79 32.08

9 10 11 12 13 14 15 16

-594.77 -702.76 -806.55 -914.92 -1018.95 -1127.55 -1231.53 -1339.91

-41.28 -41.64 -50.95 -51.50 -61.03 -61.86 -71.30 -71.98

Dimer 5 1642.39 1699.80 1792.09 1910.41 1935.18 2041.90 2121.77 2220.56

-172.34 -221.20 -223.64 -241.84 -301.13 -312.80 -339.58 -367.11

10.07 24.27 15.69 20.57 28.71 31.35 29.90 37.42

-342.67 -389.29 -378.20 -451.36 -428.42 -487.04 -493.46

5.19 4.26 -8.04 -1.94 -17.73 -16.21 -23.25

9 10 11 12 13 14 15

-650.41 -772.86 -876.34 -999.87 -1103.32 -1227.04 -1330.53

-96.93 -111.75 -120.74 -136.45 -145.40 -161.35 -170.30

Dimer 6 1472.05 1531.71 1637.53 1700.90 1807.89 1867.66 1967.89

9 10 11 12 13 14 15

-662.65 -778.15 -887.13 -1003.23 -1112.65 -1229.29 -1338.91

-109.16 -117.04 -131.53 -139.81 -154.73 -163.60 -178.67

Dimer 7 1459.43 1534.30 1616.33 1698.62 1784.78 1857.94 1932.57

-355.29 -386.69 -399.39 -453.63 -451.53 -496.76 -528.78

-3.29 -1.80 -12.51 -4.63 -20.18 -15.57 -21.10

9 10 11 12 13 14 15 16

-652.78 -773.89 -877.88 -999.65 -1103.69 -1225.46 -1329.58 -1451.31

-99.29 -112.78 -122.29 -136.24 -145.77 -159.77 -169.35 -183.38

Dimer 8 1465.90 1524.99 1651.95 1690.32 1816.03 1855.79 1974.72 2018.46

-348.82 -396.00 -363.77 -461.93 -420.28 -498.91 -486.63 -569.21

4.66 5.23 -13.88 1.42 -20.53 -11.09 -24.34 -13.76

infinite) clusters with relative orientations of molecular head groups summarized in the regression dependencies eqs 3 and 4. Figure 8 illustrates five types of clusters formed by combining eight stable conformations of dimers shown in Figure 7. To determine the values of thermodynamic parameters of cluster-

n

∆H0298, kJ/mol

∆HCl 298, kJ/mol

9 10 11 12

-1000.36 -1178.37 -1339.28 -1523.78

-170.13 -186.70 -205.89 -228.65

9 10 11

-969.41 -1148.59 -1310.53

-139.18 -156.92 -177.13

12 13 14 15 16

-1492.60 -1654.24 -1836.94 -1997.64 -2182.00

-197.48 -217.36 -238.41 -257.30 -280.11

9 10 11

-913.22 -1074.51 -1234.23

-82.99 -82.83 -100.83

12

-1397.99

-102.86

9 10 11 12 13 14 15

-1026.47 -1208.05 -1376.06 -1558.82 -1727.03 -1910.00 -2078.91

-196.25 -216.38 -242.67 -263.69 -290.16 -311.47 -338.56

16

-2261.52

-359.63

9 10 11 12 13 14 15

-1015.43 -1193.02 -1364.70 -1541.65 -1714.80 -1890.58 -2063.18

-185.20 -201.35 -231.30 -246.52 -277.93 -292.05 -322.84

9 10 11 12 13 14

-1349.91 -1590.83 -1815.71 -2057.79 -2282.87 -2525.00

S0298, J/(mol · K) Trimer 1 2073.93 2232.70 2361.69 2454.53 Trimer 2 2155.18 2306.98 2362.76 2501.79 2593.30 2773.90 2943.41 3017.22 Trimer 3 2244.92 2378.22 2530.70 2667.79 Trimer 4 2094.23 2204.44 2275.25 2386.16 2511.30 2625.94 2695.54 2857.41 Trimer 5 2032.12 2167.34 2239.28 2364.78 2474.64 2592.69 2707.12

9

-1361.94

Tetramer 1 2690.37 2861.87 3052.70 3209.13 3377.03 3529.75 Tetramer 2 -254.97 2597.07

10 11 12

-1600.02 -1823.47 -2068.87

-277.79 -312.27 -342.03

9 10 11 12

-1387.93 -1633.77 -1857.65 -2105.50

-280.96 -311.54 -346.45 -378.66

13 14

-2329.63 -2577.05

-413.80 -445.67

9 10 11 12 13

-1358.39 -1611.43 -1833.91 -2083.90 -2306.40

-251.42 -289.20 -322.72 -357.06 -390.56

-242.94 -268.60 -304.52 -330.95 -367.03 -393.62

2804.11 2921.78 3105.34 Tetramer 3 2607.39 2732.20 2904.93 3040.89 3212.46 3323.04 Tetramer 4 2610.78 2726.43 2900.01 3050.16 3188.82

∆SCl 298, J/(mol · K)

∆GCl 298, kJ/mol

-648.15 -648.79 -661.89 -773.86

23.02 6.64 -8.64 1.96

-566.90 -574.51 -660.83

29.75 14.29 19.79

-726.59 -761.16 -758.15 -748.61 -864.28

19.05 9.46 -12.48 -34.21 -22.55

-477.16 -503.27 -492.88

59.21 67.14 46.05

-560.59

64.20

-627.85 -677.05 -748.34 -842.22 -843.17 -906.10 -996.48

-9.15 -14.62 -19.66 -12.71 -38.89 -41.45 -41.61

-1024.08

-54.45

-689.96 -714.16 -784.31 -863.60 -879.82 -939.36 -984.91

20.41 11.47 2.42 10.83 -15.74 -12.12 -29.34

-939.07 -980.12 -978.74 -1095.38 -1095.59 -1179.64

36.90 23.47 -12.85 -4.53 -40.54 -42.08

-1032.37

52.68

-1037.88 -1109.66 -1199.17

31.50 18.41 15.32

-1022.05 -1109.79 -1126.52 -1263.62

27.83 0.16 -42.10 -55.15

-1260.16 -1386.36

-103.43 -119.71

-1018.66 -1115.56 -1131.43 -1254.35 -1283.80

52.15 43.24 14.45 16.73 -7.99

ization for any cluster, one should first calculate the number of intermolecular H-H interactions and the number of interactions between the functional groups in this cluster. If the number of monomers of any cluster in one direction is p, and in the other direction is q (see Figure 1), it is straightforward to determine

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TABLE 3: Parameters of Partial Correlation Dependencies of Thermodynamic Parameters of Clusterization y ) (a ( ∆a) + (b ( ∆b) · (Ka + K′a) on the Number of Intermolecular H-H Interactions (Shown Are Clusters Listed in Table 2)a cluster dimer 1

dimer 2

dimer 3

dimer 4

dimer 5

dimer 6

dimer 7

dimer 8

trimer 1

trimer 2

trimer 3

trimer 4

trimer 5

tetramer 1

tetramer 2

tetramer 3

tetramer 4

characteristic

a ( ∆a

b ( ∆b

S

∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl ∆S298, J/(mol · K) ∆GCl 298, kJ/mol Cl , kJ/mol ∆H298 ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl , J/(mol · K) ∆S298 ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl ∆S298, J/(mol · K) Cl , kJ/mol ∆G298 ∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl ∆S298, J/(mol · K) ∆GCl 298, kJ/mol Cl ∆H298 , kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl ∆S298 , J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl ∆S298, J/(mol · K) Cl ∆G298 , kJ/mol ∆HCl 298, kJ/mol ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol ∆HCl 298, kJ/mol Cl ∆S298, J/(mol · K) ∆GCl 298, kJ/mol Cl , kJ/mol ∆H298 ∆SCl 298, J/(mol · K) ∆GCl 298, kJ/mol

1.18 ( 0.86 -126.4 ( 56.6 38.83 ( 16.11 -10.38 ( 3.10 -181.5 ( 51.1 43.70 ( 13.06 -10.84 ( 1.62 -57.8 ( 40.5 6.39 ( 11.28 3.36 ( 0.48 -4.9 ( 42.8 4.82 ( 12.31 -1.03 ( 0.69 26.7 ( 34.7 -8.99 ( 9.86 -6.20 ( 1.25 -158.2 ( 32.2 40.93 ( 10.40 -16.22 ( 1.03 -136.2 ( 31.8 24.36 ( 8,79 -13.32 ( 1.69 -132.3 ( 46.2 26.12 ( 15.10 -13.78 ( 7.92 -185.0 ( 87.9 160.56 ( 19.6 -16.62 ( 1.77 -336.8 ( 54.1 83.75 ( 15.79 -26.10 ( 3.59 -380.7 ( 128.7 87.33 ( 35.02 -14.78 ( 2.02 -193.8 ( 38.5 42.49 ( 11.09 -28.29 ( 3.51 -341.9 ( 33.5 73.59 ( 12.39 -16.45 ( 6.01 -595.9 ( 64.3 161.11 ( 14.30 -15.87 ( 15.26 -551.2 ( 131.2 148.38 ( 27.39 -31.72 ( 4.39 -514.7 ( 94.0 121.67 ( 23.70 -28.41 ( 11.56 -598.4 ( 89.8 149.89 ( 17.76

-10.48 ( 0.07 -20.8 ( 4.8 -4.27 ( 1.37 -10.19 ( 0.33 -16.8 ( 5.5 -5.20 ( 1.41 -9.64 ( 0.34 -43.0 ( 8.7 3.19 ( 2.43 -10.18 ( 0.07 -43.6 ( 7.0 2.81 ( 2.01 -10.07 ( 0.12 -54.4 ( 6.2 6.14 ( 1.76 -8.18 ( 0.08 -16.9 ( 2.0 -3.13 ( 0.65 -7.72 ( 0.06 -18.6 ( 1.9 -2.19 ( 0.53 -7.74 ( 0.10 -18.8 ( 2.7 -2.13 ( 0.89 -10.49 ( 0.27 -15.9 ( 3.0 -5.76 ( 0.87 -10.07 ( 0.09 -19.5 ( 2.8 -4.26 ( 0.81 -9.47 ( 0.51 -18.6 ( 18.1 -4.03 ( 4.95 -9.55 ( 0.07 -23.2 ( 1.4 -2.62 ( 0.40 -9.17 ( 0.14 -20.4 ( 1.4 -3.10 ( 0.49 -10.39 ( 0.19 -15.5 ( 2.1 -5.78 ( 0.46 -9.86 ( 0.53 -19.1 ( 4.6 -4.17 ( 0.95 -9.60 ( 0.13 -19.71 ( 2.7 -3.72 ( 0.68 -9.86 ( 0.38 -18.9 ( 3.0 -4.23 ( 0.59

0.47 31.3 8.90 2.11 34.8 8.91 1.10 27.6 7.69 0.27 24.2 6.97 0.39 19.6 5.55 0.62 16.0 5.18 0.50 15.3 4.22 0.99 27.1 8.87 3.30 36.6 8.17 1.17 35.9 10.47 1.02 36.3 9.91 1.16 22.0 6.35 1.87 17.9 6.61 3.07 32.9 7.31 3.57 30.7 6.40 1.82 38.9 9.80 4.24 32.9 6.52

a a, b, and S are the regression absolute term, slope, and standard deviation, respectively.

that, for cluster 1 (taken here as an example; see Figure 8, cluster 1), the number of head groups in the p and q directions is n1 ) q · (p - 1) and n3 ) p · (q - 1), respectively. (Note that it is seen from Figure 8 that the head groups as in dimer 1 correspond to cluster 1 in the p direction, and the head groups as in dimer 3 correspond to cluster 1 in the q direction.) Also, the number of H-H interactions in the p dimension of cluster 1 is equal to that in dimer 1, and the number of these interactions in the q dimension is equal to that in dimer 3. These numbers are n 1 and {(n - 3)/2}, where n is the number of carbon atoms per one monomer chain, and the braces {...} denote the integer part of a number. Next, the number of intermolecular interactions in cluster 1 is Ka ) (n - 1) · q · (p - 1) + {(n - 3)/2} · p · (q -1). The numbers of intermolecular interactions for four other clusters can be calculated in a similar way. The general formulas for the dependencies of the number of intermolecular interac-

tions between the functional groups in a cluster on its dimensions (p and q) and the dependence of the number of intermolecular H-H interactions on the alkyl chain length and the dimensions of the cluster are presented in the upper part of Table 4. Using these expressions, one can formulate the dependencies of clusterization enthalpy, entropy, and Gibbs’ energy for any cluster. In particular, for cluster 1:

∆HCl 298 ) [-(10.50 ( 0.13) · ((n - 1) · q · (p - 1) + {(n - 3)/2} · p · (q - 1)) + (1.11 ( 1.31) · q · (p - 1) (8.08 ( 1.09) · p · (q-1)] kJ/mol (5) Cl ∆S298 ) [-(18.68 ( 1.16) · ((n - 1) · q · (p - 1) + {(n - 3)/2} · p · (q - 1)) - (175.42 ( 11.82) · q · (p - 1) (78.37 ( 9.78) · p · (q - 1)] J/(mol · K) (6)

∆GCl 298 ) [-(4.93 ( 0.48) · ((n - 1) · q · (p - 1) + {(n - 3)/2} · p · (q - 1)) + (53.38 ( 4.83) · q · (p - 1) (15.27 ( 4.00) · p · (q - 1)] kJ/mol (7) Cluster 2 (see Figure 8, cluster 2) involves intermolecular interactions between the alkyl chains and between the head groups, identical to those in dimer 2 (p direction) and in dimer 3 (q direction). Therefore, using the expressions for the H-H interactions and interactions between the head groups listed in Table 4, one obtains the correlation dependencies for the clusterization enthalpy, entropy, and Gibbs’ energy for a cluster with any p and q:

∆HCl 298 ) [-(10.50 ( 0.13) · (2 · {(n - 3)/2} · q · (p - 1) + {(n - 3)/2} · p · (q - 1)) - (4.87 ( 1.34) · q · (p - 1) (8.08 ( 1.09) · p · (q - 1)] kJ/mol (8) Cl ) [-(18.68 ( 1.16) · (2 · {(n - 3)/2} · q · (p - 1) + ∆S298 {(n - 3)/2} · p · (q - 1)) - (173.95 ( 12.03) · q · (p - 1) (78.37 ( 9.78) · p · (q - 1)] J/(mol · K) (9)

∆GCl 298 ) [-(4.93 ( 0.48) · (2 · {(n - 3)/2} · q · (p - 1) + {(n - 3)/2} · p · (q - 1)) + (46.97 ( 4.92) · q · (p - 1) (15.27 ( 4.00) · p · (q - 1)] kJ/mol (10) The third type of 2D films (see Figure 8, cluster 3) is built on the basis of dimers 1 and 2 (p direction), and dimer 3 (q direction). Similar to what was explained above, using the parameters listed in Table 4, one obtains the correlation dependencies of the thermodynamic parameters of clusterization for this cluster 3 with any p and q: Cl ∆H298 ) [-(10.50 ( 0.13) · ((n - 1) · {p/2} · q + 2 · {(n - 3)/2} · {(p - 1)/2} · q) + {(n - 3)/2} · p · (q - 1)) + (1.11 ( 1.31) · {p/2} · q - (4.87 ( 1.34) · {(p - 1)/2} · q (8.08 ( 1.09) · p · (q - 1)] kJ/mol (11) Cl ∆S298 ) [-(18.68 ( 1.16 · ((n - 1) · {p/2} · q + 2 · {(n - 3)/2} · {(p - 1)/2} · q) + {(n - 3)/2} · p · (q - 1)) (175.42 ( 11.82) · {p/2} · q - (173.95 ( 12.03) · {(p - 1)/2} · q(78.37 ( 9.78) · p · (q - 1)] J/(mol · K) (12) Cl ∆G298 ) [-(4.93 ( 0.48) · ((n - 1) · {p/2} · q + 2 · {(n - 3)/2} · {(p - 1)/2} · q + {(n - 3)/2} · p · (q - 1)) + (53.38 ( 4.83) · {p/2} · q + (46.97 ( 4.92) · {(p - 1)/2} · q + (15.27 ( 4.00) · p · (q - 1)] kJ/mol (13)



The structure of the fourth type of clusters is different from those considered above (see Figure 8, cluster 4). This cluster is built on the basis of dimer 3 (p direction) and dimer 6 (q direction). Dimer 6 involves the “external” and “internal” H-H interactions between the alkyl chains. The number of these interactions is denoted by K′a (in Table 4, this interaction type, and also the Ka type, are listed in the second column for corresponding cluster). The correlation dependencies for the thermodynamic functions of clusterization for cluster 4 with any p and q are:

Cl ∆H298 ) [-2 · (7.03 ( 0.13) · {(n - 2)/2} · {p/2} (10.50 ( 0.13) · ({(n - 3)/2} · p · (q - 1) + {n/2} · (q · (p - 1) {(p - 1)/2}+ {(n + 1)/2} · (q · (p - 1) - {p/2})) (8.08 ( 1.09) · p · (q - 1)- (10.14 ( 1.51) · q · (p - 1) (17) (3.47 ( 1.21) · (p - 1) · (q - 1)] kJ/mol

∆SCl 298 ) [-2 · (18.89 ( 1.16) · {(n - 2)/2} · {p/2} (18.68 ( 1.16) · ({(n - 3)/2} · p · (q - 1) + {n/2} · (q · (p - 1) {(p - 1)/2} + {(n + 1)/2} · (q · (p - 1) - {p/2})) (78.37 ( 9.78) · p · (q - 1) - (122.83 ( 13.64) · q · (p - 1) (18) (130.45 ( 10.96) · (p - 1) · (q - 1)] J/(mol · K)

Cl ∆H298 ) [-(7.03 ( 0.13) · {p/2} · (n - 1)(10.50 ( 0.13) · ({(n-3)/2} · p · (q - 1) + {(n - 2)/2} · (p - 1) · (2q - 1)) - (8.08 ( 1.09) · p · (q - 1) (10.45 ( 0.99) · (p - 1) · (2q - 1)] kJ/mol (14) Cl ∆S298 ) [-(18.89 ( 1.16) · {p/2} · (n - 1) (18.68 ( 1.16) · ({(n - 3)/2} · p · (q - 1) + {(n - 2)/2} · (p - 1) · (2q - 1)) - (78.37 ( 9.78) · p · (q - 1) (15) (147.63 ( 8.96) · (p - 1) · (2q - 1)] J/(mol · K)

) [-(1.40 ( 0.47) · {p/2} · (n - 1) (4.93 ( 0.48) · ({(n - 3)/2} · p · (q - 1) + {(n - 2)/2} · (p - 1) · (2q - 1)) + (15.27 ( 4.00) · p · (q - 1) + (33.54 ( 3.66) · (p - 1) · (2q - 1)] kJ/mol (16)

Cl ∆G298

The fifth type of clusters involves the same types of intermolecular interactions as in dimer 3 (q direction), and dimers 7 and 8 (p direction). Similar to cluster 4, this type of cluster involves both “external” and “internal” H-H interactions between the alkyl chains that should be accounted for in the calculations of the thermodynamic parameters of smaller clusters, whereas (as shown below) these interactions could be neglected in the calculations of these parameters for infinite 2D films. Using the values listed in Table 4, one obtains:

∆GCl 298 ) [-2 · (1.40 ( 0.47) · {(n - 2)/2} · {p/2} (4.93 ( 0.48) · ({(n - 3)/2} · p · (q - 1) + {n/2} · (q · (p - 1) {(p - 1)/2} + {(n + 1)/2} · (q · (p - 1)-{p/2})) + (15.27 ( 4.00) · p · (q - 1) + (26.46 ( 5.57) · q · (p - 1) + (19) (35.40 ( 4.47) · (p - 1) · (q - 1)] kJ/mol

To calculate the parameters for infinite clusters, one should determine the number of H-H interactions and interactions between the head groups in an infinite cluster per one 2CnH2n+1-melamine molecule. This is done by dividing the expressions for the number of interacting triazine rings and the number of intermolecular H-H interactions in a corresponding cluster by p · q and calculating the limit of the expressions thus obtained, for p and q tending to infinity. These dependencies of the number of intermolecular interactions on the alkyl chain length are listed in the lower part of Table 4. Introducing these expressions into the correlation dependencies for the clusterization enthalpy (1) and entropy (2), one obtains the expressions for the 2D film (p ) ∞, q ) ∞) per one monomer molecule. Next, one can calculate the clusterization Gibbs’ energy for an infinite cluster as ∆GCl(∞) 298 / Cl(∞) Cl(∞) /m - T · ∆S298 /m. For cluster 1, these dependm ) ∆H298 encies are:

TABLE 4: Number and Type of Intermolecular Interactions in Clusters 1-5 cluster cluster 1

Finite Clusters Ka ) (n - 1) · q · (p - 1) + {(n - 3)/2} · p · (q -1)

cluster 2

Ka ) 2 · {(n - 3)/2} · q · (p - 1) + {(n - 3)/2} · p · (q -1)

cluster 3

Ka ) (n - 1) · {p/2} · q + 2 · {(n - 3)/2} · {(p - 1)/2} · q) + {(n - 3)/2} · p · (q - 1)

cluster 5

Ka ) {(n - 3)/2} · p · (q - 1) + {(n - 2)/2} · (p - 1) · (2q - 1) K′a ) {p/2} · (n - 1) Ka ) {(n - 3)/2} · p · (q - 1) + {n/2} · (q · (p - 1) - {(p - 1)/2} + {(n + 1)/2} · (q · (p - 1) - {p/2}) K′a ) 2 · {(n - 2)/2} · {p/2}

cluster 1

Infinite Clusters K(∞) a /m ) (n - 1) + {(n - 3)/2}

cluster 2

K(∞) a /m ) 3 · {(n - 3)/2}

cluster 3

K(∞) a /m ) (n - 1)/2 + 2 · {(n - 3)/2}

cluster 4

K(∞) a /m ) {(n - 3)/2} + 2 · {(n - 2)/2} K′(∞) a /m ) 0 K(∞) a /m ) n +{(n - 3)/2} ) 0 K′(∞) a /m

cluster 4

cluster 5

interactions between triazine rings

H-H interactions n1 n3 n2 n3 n1

) ) ) ) )

q · (p - 1) p · (q - 1) q · (p - 1) p · (q - 1) {p/2} · q

n2 n3 n3 n6 n3 n7

) ) ) ) ) )

{(p - 1)/2} · q) p · (q - 1) p · (q - 1) (p - 1) · (2q - 1) p · (q - 1) q · (p - 1) - {(p - 1)/2}

n8 ) q · (p - 1) - {p/2} n(∞) 1 /m n(∞) 3 /m n(∞) 2 /m n(∞) 3 /m n(∞) 1 /m n(∞) 2 /m n(∞) 3 /m n(∞) 3 /m n(∞) 6 /m n(∞) 6 /m n(∞) 7 /m

) ) ) ) ) ) ) ) ) ) )

1 1 1 1 1/2 1/2 1 1 2 1 1

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∆HCl(∞) 298 /m ) [-10.50 · ((n - 1) + {(n - 3)/2}) 6.97] kJ/mol (20)

For infinite clusters 3, the dependencies of the thermodynamic parameters of clusterization are:

∆SCl(∞) 298 /m ) [-18.68 · ((n - 1) + {(n - 3)/2}) 253.79] J/(mol · K) (21)

∆HCl(∞) 298 /m ) [-10.50 · ((n - 1)/2 + 2 · {(n - 3)/2}) 9.96] kJ/mol (26)

∆GCl(∞) 298 /m ) [-4.93 · ((n - 1) + {(n - 3)/2}) 6.97] kJ/mol (22)

∆SCl(∞) 298 /m ) [-18.68 · ((n - 1)/2 + 2 · {(n - 3)/2}) 253.06] J/(mol · K) (27)

The parameters for the infinite clusters 2-5 can be calculated in a similar way. Using the parameters listed in Table 4, one obtains the dependencies of the clusterization enthalpy, entropy, and Gibbs’ free energy per one monomer molecule for cluster 2 with p f ∞, q f ∞:

∆GCl(∞) 298 /m ) [-4.93 · ((n - 1)/2 + 2 · {(n - 3)/2}) + 65.45] kJ/mol (28)

∆HCl(∞) 298 /m ) [-10.50 · (3 · {(n - 3)/2}) - 12.95] kJ/mol (23) ∆SCl(∞) 298 /m ) [-18.68 · (3 · {(n - 3)/2}) 252.32] J/(mol · K) (24) ∆GCl(∞) 298 /m

) [-4.93 · (3 · {(n - 3)/2}) + 62.24] kJ/mol (25)

Note that, in the clusters of this type, the layers with even and odd p exhibit different inclination with respect to the interface. Therefore, the angle between the alkyl chains A and B that interact along the p direction is ∼15°, not zero. For the large alkyl chain length n, this should result in a lower interaction between the alkyl chains and, hence, in a decrease of the stability of the film. However, by direct calculations, it can be shown that for n < 17 this fact does not affect the calculated thermodynamic characteristics.

As one could expect, the number of “external” H-H interactions per one monomer between alkyl chains in the infinite cluster 4 (and also in the infinite cluster 5) tends to zero. Therefore, for cluster 4:

∆HCl(∞) 298 /m ) [-10.50 · ({(n - 3)/2} + 2 · {(n - 2)/2}) 28.98] kJ/mol (29) ∆SCl(∞) 298 /m ) [-18.68 · ({(n - 3)/2} + 2 · {(n - 2)/2}) 373.63] J/(mol · K) (30) ∆GCl(∞) 298 /m ) [-4.93 · ({(n - 3)/2} + 2 · {(n - 2)/2}) + 82.36] kJ/mol (31) and for cluster 5, the dependencies of clusterization enthalpy, entropy, and Gibbs’ energy are:

∆HCl(∞) 298 /m ) [-10.50 · (n + {(n - 3)/2}) - 21.69] kJ/mol (32)

Figure 8. Schematic representations of fragments of possible structures of 2D films formed by 2CnH2n+1-melamine.


∆SCl(∞) 298 /m ) [-18.68 · (n + {(n - 3)/2}) 331.65] J/(mol · K) (33) ∆GCl(∞) 298 /m ) [-4.93 · (n + {(n - 3)/2}) + 77.14] kJ/mol (34) Figure 9 illustrates the dependencies of the clusterization Gibbs’ energy per one monomer for infinite clusters on the alkyl chain length in the constituting monomers. It is seen that the spontaneous clusterization for clusters 1-5 can take place if the alkyl chain length is equal or higher than the values 10-11, 12-13, 11-12, 14-15, and 11-12 carbon atoms, respectively. It follows from the calculated results that the formation of cluster 1 is most energetically advantageous, which is supposed to take place actually. The formation of other types of clusters is less probable. When considering the structures of the clusters shown in Figure 8, it may be supposed that the thermodynamic parameters for clusters 1 and 5 are close to each other (and equal to each other for infinite clusters), because the intermolecular H-H interactions in both of these clusters are formed between the A and A, and B and B alkyl chains, and cluster 5 can be transformed into cluster 1 by the parallel translation of the rightangled triangle shown in Figure 8 (left side of cluster 5). This translation would lead to the loss of (p - 1) interactions between the translated triangle and the initial cluster, if one compares the cluster, thus obtained, with cluster 1. At the same time, the number of intermolecular H-H interactions per one monomer in cluster 1 is by one unit smaller than that in cluster 5. Therefore, it would seem that cluster 5 is more energetically advantageous than cluster 1. However, the differences in the orientation of the head groups in clusters 1 and 5 make the formation of cluster 1 more advantageous than the formation of cluster 5. Comparing the corresponding correlation parameters of eqs 1 and 2, one can see that the contributions brought by the head groups of clusters 1 and 5 into the clusterization

J. Phys. Chem. B, Vol. 113, No. 40, 2009 13245 enthalpy are essentially different. Also, for clusters 2 and 4, a similar relationship exists. Therefore, it could be presumed that the 2D film with relative orientation of monomers as in infinite cluster 1 is the most probable infinite structure. We present in Figures 10 and 11 the dependencies of clusterization enthalpy and Gibbs’ energy per one monomer on the alkyl chain length for infinite cluster 1 and finite clusters with corresponding structure. In these figures, the lines show the correlation dependencies, and the points correspond to the results of direct calculations. The shape of the dependencies is determined by the number of intermolecular H-H interactions; in particular, for 2D films (p > 1) the curves are seen to be the superposition of the linear and stepwise functions; cf., eqs 5-7. Also shown in the figures are the dependencies for linear clusters (p ) 1), which are clearly stepwise, as the number of H-H interactions depends on the alkyl chain length proportional to {(n - 3)/2}; see eqs 5-7. To summarize, spontaneous clusterization of 2CnH2n+1melamine at the interface can take place if the alkyl chain length is 10 and more carbon atoms, in agreement with the experimental data reported in ref 24. The clusterization process involves the formation of stacked dimers (see Figure 6), which then arrange into more complicated stacked clusters. When the number of intermolecular H-H interactions in these clusters becomes high enough, they combine to form clusters that contain linear sections (see Figure 8, cluster 1). This stage follows the stage of the formation of stacked clusters because the linear arrangement is less advantageous. However, when the number of advantageous stacked interactions is high enough, this process becomes probable. Also, the formation of clusters 5, 3, and 2 (see Figure 8) is possible but less probable. Comparison with Experiment. The surface pressuremolecular area (π-A) isotherms for three 2CnH2n+1-melamines at different temperatures (5-20, 12-32, and 25-39 °C for n ) 10, 11, and 12, respectively) were reported.22 In these

Figure 9. Dependence of the variation of clusterization free energy per one monomer on the alkyl chain length for five types of 2D films formed by 2CnH2n+1-melamine: 1, cluster 1; 2, cluster 2; 3, cluster 3; 4, cluster 4; 5, cluster 5.

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Figure 10. Dependence of the variation of clusterization enthalpy per one monomer on the alkyl chain length for clusters formed by 2CnH2n+1melamine.

Figure 11. Dependence of the variation of clusterization free energy per one monomer on alkyl chain length for clusters formed by 2CnH2n+1melamine.

temperature ranges, the π-A isotherms exhibit the existence of a critical point, which corresponds to the two-dimensional fluid (gaseous, liquid expanded, G, LE)-condensed (liquidcondensed, LC) phase transition. This result agrees with the theoretical calculations of Gibbs’ energy for cluster 1 shown Cl(∞) /m in Figures 9 and 11: for n ) 10 and above, the ∆G298 value becomes negative. From the experimental value of area per one molecule in the critical point, one can estimate the clusterization Gibbs’ energy per one mole of the monomers using the relationship derived in ref 40:

∆GCl ) RT ln(ω/Ac)

(35)

where R is the gas constant, T is the temperature, ω is the area per monomer in a cluster, and Ac is the area per one

molecule in the initial point of the phase transition. The values of ω and Ac for 2CnH2n+1-melamines at different temperatures are reported in ref 22. In the paper,39 a more rigorous equation for ∆GCl was derived that assumes enthalpy and entropy nonideality of the two-component surface layer. Cl(∞) Now the ∆G298 /m values obtained above for the infinite cluster 1 can be compared to the ∆GCl values estimated from eq 35 and from the more rigorous equation presented in ref 39. For the 2CnH2n+1-melamines considered here, the results obtained from quantum chemical calculations are -0.51, -5.46, and -15.33 kJ/mol, respectively, whereas eq 35 gives ∆GCl values -0.6 kJ/mol for n ) 10 at 20 °C, -0.8 kJ/mol for n ) 11 at 25 °C, and -1.2 kJ/mol for n ) 12 at 25 °C. The more rigorous model yields ∆GCl ) -5.2, -5.6, and -6.1 kJ/mol, respectively.39

Clusterization of Melamine-type Amphiphiles It is seen that for n ) 10 and 11 the estimations of clusterization Gibbs’ energy on the basis of the location of the π-A isotherms are close enough to the values obtained from quantum chemical calculations. The discrepancy between the experimental and theoretical results obtained for n ) 12 could possibly be ascribed to the fact that the semiempiric calculations assumed an ideal model with regular structure of the monolayer, for which the entropy factor leads to significant gains in the stability of the system. At the same time, in real systems the formation of a more loose monolayer could be expected, for which the entropy contribution results in higher values of the clusterization entropy leading to lower (by absolute value) values of Gibbs’ free energy. For the three 2CnH2n+1-melamines (n ) 10, 11, and 12), the clusterization Gibbs’ energies calculated from the experimental data (-5.2 to -6.1 kJ/mol) are rather close to the value -5.46 kJ/mol obtained by semiempiric PM3 calculations for n ) 11. It could be supposed, therefore, that for the formation of cluster 1 type in the loose monolayer of 2CnH2n+1-melamines (at least with n values close to 11) each molecule forms the average of 28 intermolecular H-H interactions with its neighbors; cf., Table 4, cluster 1. However, for n ) 12 the formation of cluster 1 type may possibly be accompanied by the formation of clusters 3 and 5, for which the Gibbs’ free energy is also negative, whereas it is much lower (by absolute value) than that for cluster 1 (see Figure 9). Conclusions To summarize, the semiempiric PM3 method is used for quantum chemical calculations of the thermodynamic parameters (enthalpy, entropy, and Gibbs’ energy) corresponding to the clusterization of 2CnH2n+1-melamines at the air/water interface. The structural features of the monomers of these compounds caused by the presence of two alkyl chains are considered in detail. In particular, one alkyl chain (denoted as A here) is bended because of the formation of hydrogen bonds between the hydrogen atoms of two methylene groups in this bended alkyl chain with free electron pairs of the nitrogen atom contained in the triazine ring, which results in the formation of thermodynamically advantageous intramolecular hydrogenhydrogen interactions of the “a” type. It is shown that only the monomers that contain this bended radical are capable of the formation of infinite 2D films. The correlation dependencies of the formation enthalpy, absolute entropy, and the formation Gibbs’ energy for these monomers on the alkyl chain length are derived. These dependencies are shown to be linear, stepwise, or superpositional (combined linear and stepwise) depending on the number of intramolecular H-H interactions. It is shown that, similar to the case of “e” type intermolecular H-H interactions,36 the intramolecular “e” type H-H interactions lead to the formation of structures that are thermodynamically less advantageous than those formed by “a” type intramolecular H-H interactions. The composition of the monomer mixture is calculated showing that, for alkyl chain lengths of above 11-12 carbon atoms, the monomers comprising these intramolecular H-H interactions prevail in the system, whereas for lower alkyl chain lengths the mixture is composed mainly of monomers in which these monomers are absent. The structures of dimers, trimers, and tetramers are composed on the basis of monomers with a bended alkyl chain. For all of these entities, the clusterization enthalpy, entropy, and Gibbs’ energy are calculated, and partial correlation dependencies of these quantities on the structural parameters of clusters are

J. Phys. Chem. B, Vol. 113, No. 40, 2009 13247 derived. The parameters of these partial correlation dependencies are shown to be in good agreement with each other, which enabled us to generalize these into one correlation dependence that was next used to calculate the thermodynamic parameters of clusterization for various structures with any dimensions, including the 2D films. Five types of the clusters with different relative orientations of monomers are considered. Each of the five studied structures can, in principle, be used as a building block of the 2D film formed by the molecules of 2CnH2n+1-melamine on the air/water interface. The calculations have shown that the formation of films by cluster 1, shown in Figure 8, is thermodynamically most advantageous and, therefore, the most probable. The spontaneous clusterization of this type becomes possible if the alkyl chain length exceeds 10 carbon atoms. This agrees with the experimental data.22,24 In the next stage of these studies, the introduction of the barbituric acid molecules into the structure of the monolayer of the first type (Figure 8, cluster 1) will be considered. This should be a theoretical contribution to characterize the processes of the molecular recognition. References and Notes (1) Zerkowski, J. A.; MacDonald, J. C.; Seto, C. T.; Wierda, D. A.; Whitesides, G. M. J. Am. Chem. Soc. 1994, 116, 2382. (2) Zerkowski, J. A.; Whitesides, G. M. J. Am. Chem. Soc. 1994, 116, 4298. (3) Zerkowski, J. A.; Mathias, J. P.; Whitesides, G. M. J. Am. Chem. Soc. 1994, 116, 4305. (4) Yagai, S.; Higashi, M.; Karatsu, T.; Kitamura, A. Chem. Commun. 2006, 14, 1500. (5) Johal, M. S.; Cao, Y. W.; Chai, X. D.; Smilowitz, L. B.; Robinson, J. M.; Li, T. J.; McBranch, D.; Li, D. Q. Chem. Mater. 1999, 8, 11. (6) Marchiartzner, V.; Artzner, F.; Karthaus, O.; Shimomura, M.; Ariga, K.; Kunitake, T.; Lehn, J. M. Langmuir 1998, 14, 5164. (7) Koyano, H.; Bissel, P.; Yoshihara, K.; Ariga, K.; Kunitake, T. Langmuir 1997, 13, 5426. (8) Koyano, H.; Bissel, P.; Yoshihara, K.; Ariga, K.; Kunitake, T. Chem.-Eur. J. 1997, 3, 1077. (9) Cao, Y. W.; Chai, X. D.; Chen, S. G.; Jiang, Y. S.; Yang, W. S.; Lu, R.; Ren, Y. Z.; Blancharddesce, M.; Li, T. J.; Lehn, J. M. Synth. Met. 1995, 71, 1733. (10) Cao, Y. W.; Chai, X. D.; Li, T. J.; Smith, J.; Li, D. Q. Chem. Commun. 1999, 1605. (11) Desiraju, G. R. Angew. Chem. 1995, 107, 2541. (12) Ikeura, Y.; Kurihara, K.; Kunitake, T. J. Am. Chem. Soc. 1991, 113, 7342. (13) Kusmenko, I.; Buller, R.; Bouwman, W. G.; Kjaer, K.; Als-Nielsen, J.; Lahav, M.; Leiserowitz, L. Science 1996, 274, 2046. (14) Ebara, Y.; Itakura, K.; Okahata, Y. Langmuir 1996, 12, 5165. (15) Matsuura, K.; Ebara, Y.; Okahata, Y. Langmuir 1997, 13, 814. (16) Kimizuka, N.; Kawasaki, T.; Kunitake, T. J. Am. Chem. Soc. 1993, 115, 4387. (17) Kimizuka, N.; Kawasaki, T.; Kunitake, T. Chem. Lett. 1994, 33. (18) Vollhardt, D.; Liu, F.; Rudert, R. ChemPhysChem 2005, 6, 1246. (19) Vollhardt, D.; Liu, F.; Rudert, R.; He, W. J. Phys. Chem. 2005, 109, 10849. (20) Vollhardt, D.; Liu, F.; Rudert, R. J. Phys. Chem. B 2005, 109, 17635. (21) Fainerman, V. B.; Vollhardt, D.; Aksenenko, E. V.; Liu, F. J. Phys. Chem. B 2005, 109, 14137. (22) Vollhardt, D.; Fainerman, V. B.; Liu, F. J. Phys. Chem. B 2005, 109, 11706. (23) Kovalchuk, N. M.; Vollhardt, D.; Fainerman, V. B.; Aksenenko, E. V. J. Phys. Chem. B 2007, 111, 8283. (24) Vollhardt, D. AdV. Colloid Interface Sci. 2005, 116, 63. (25) Wolstenholme, D. J.; Cameron, T. S. J. Phys. Chem. A 2006, 110, 8970. (26) Kaplan, I. G. Intermolecular Interactions. Physical Picture. Computation Methods and Model Potentials; Wiley: New York, 2006; p 380. (27) Kim, J. M.; Lee, B. L.; Kwon, Y.-S. Bull. Korean Chem. Soc. 1997, 18, 1056. (28) Johal, M. S.; Cao, Y. W.; Chai, X. D.; Smilowitz, L. B.; Robinson, J. M.; Li, T. J.; McBranch, D.; Li, D. Q. Chem. Mater. 1999, 11, 1962. (29) Stewart, J. J. P. MOPAC 2000.00 Manual; Fujitsu Limited: Tokyo, 1999; p 555.

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(30) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B. J. Phys. Chem. B 2002, 106, 121. (31) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B. J. Phys. Chem. B 2002, 106, 11285. (32) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D.; Miller, R. Colloids Surf., A 2002, 209, 1. (33) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B. Prog. Colloid Polym. Sci. 2002, 121, 72. (34) Vysotsky, Yu. B.; Muratov, D. V.; Boldyreva, F. L.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. B 2006, 110, 4717. (35) Vysotsky, Yu. B.; Belyaeva, E. A.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. B 2007, 111, 5374.

Vysotsky et al. (36) Vysotsky, Yu. B.; Belyaeva, E. A.; Fainerman, V. B.; Vollhardt, D.; Miller, R. J. Phys. Chem. C 2007, 111, 15342. (37) Isaacs, N. S. Physical Organic Chemistry; Longman Group UK Limited: London, 1995; p 871. (38) Vysotsky, Yu. B.; Belyaeva, E. A.; Vollhardt, D.; Aksenenko, E. V.; Miller, R. J. Colloid Interface Sci. 2008, 326, 339. (39) Vysotsky, Yu. B.; Belyaeva, E. A.; Fainerman, V. B.; Vollhardt, D.; Aksenenko, E. V.; Miller, R. J. Phys. Chem. B 2009, 113, 4347. (40) Vollhardt, D.; Fainerman, V. B.; Siegel, S. J. Phys. Chem. B 2000, 104, 4115.

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