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Study of molecular ordering in liquid crystals: EBBA
Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124 www.elsevier.com/locate/pcrysgrow
Study of molecular ordering in liquid crystals: EBBA M. Mishra, M.K. Dwivedi, R. Shukla, S.N. Tiwari* Department of Physics, D.D.U. Gorakhpur University, Gorakhpur 273 009, India
Abstract Mesomorphism is primarily governed by the nature and strength of molecular interactions acting between sides, planes and ends of a pair of liquid crystal molecules. In view of this fact, intermolecular interactions between a pair of molecules of a nematic liquid crystal namely, p-ethoxybenzylidene-p-nbutylaniline (EBBA) have been evaluated using RayleigheSchrodinger second order perturbation theory along with multicentred-multipole expansion technique. An all valence electron method, CNDO/2 has been employed to compute net atomic charge and corresponding dipole components located at each atomic centre of the molecule. Using the results of stacking and in-plane interaction energy studies, probability calculations at varied angular and positional configurations in a molecular pair of EBBA have been carried out using MaxwelleBoltzmann formula. Results have been used to elucidate the liquid crystalline behaviour of the EBBA molecule. Ó 2006 Elsevier Ltd. All rights reserved. PACS: 61.30.-v; 61.30Gd; 64.43.Bn; 64.70.Md; 82.20.Wt; 31.15.Md; 31.15.Ct Keywords: A1. Computer simulation; B1. Polymers; B2. Dielectric materials (Liquid crystals/Nematogens)
1. Introduction Liquid crystalline phases are stable condensed phases in which molecules pack together with an order that is intermediate between the three-dimensional order of a crystalline solid
* Corresponding author. Tel.: þ91 551 2203182; fax: þ91 551 2340459. E-mail address: [email protected] (S.N. Tiwari). 0960-8974/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pcrysgrow.2006.03.016
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and the disorder of an isotropic liquid. The partial molecular ordering that is a characteristic of liquid crystallinity occurs frequently in natural and synthetic materials. Thus, liquid crystals are of considerable basic and applied interest [1,2]. The peculiar changes, characteristics of mesomorphic behaviour which occur at phase transitions, are primarily governed by the nature and strength of various types of intermolecular interactions acting between sides, planes and ends of a pair of molecules [3]. Using computer simulation and theoretical methods efforts have been made by several workers to understand the role of molecular interactions in accounting for liquid crystallinity [4e8]. It has been observed that the pair potential between such molecules is anisotropic in nature which is in general regarded as the prime requirement for the mesophase formation in thermotropic liquid crystals [5]. Since mesogenic properties are related to molecular aggregation in a specific manner, probability distribution calculations based on interaction energy results are expected to provide information about most probable molecular aggregation and tendency to retain translational and orientational order at different transition temperatures [7,8]. The present paper embodies the results of computational analysis based on actual molecular shape consideration in the case of p-ethoxybenzylidene-p-n-butylaniline (EBBA). The thermodynamic parameters reveal that EBBA shows crystal to nematic transition at 308.5 K and nematic to isotropic melt at 352.5 K [9].
2. Method of calculation The molecular geometry of EBBA has been constructed using crystallographic data from literature and standard values of bond lengths and bond angles [10]. Net charge and corresponding dipole moment components at each of the atomic centres of the molecule have been computed by CNDO/2 method [11]. Modified RayleigheSchrodinger second order perturbation theory along with multicentred-multipole expansion technique has been used to evaluate intermolecular interactions between a couple of EBBA molecules. According to this method, the total interaction energy between two molecules is obtained as the sum of electrostatic, polarization, dispersion and repulsion energy terms. Electrostatic energy is expressed as the sum of monopoleemonopole, monopolee dipole, dipoleedipole and energy terms due to higher order multipoles. In the present calculation electrostatic energy is restricted only up to first three terms while dispersion and repulsion energy terms have been calculated together using Kitaigorodskii formula [12]. The energy minimization has been carried out for both stacking and in-plane (side to side and end to end) interactions separately. One of the interacting molecules is kept fixed throughout the process while both lateral and angular variations are introduced in the other in all respects relative to the fixed one. The first molecule has been assumed to be in the XeY plane with X-axis lying along the long molecular axis while the origin is chosen approximately at the centre of the mass of the molecule. The second molecule has been translated initially along the Z-axis (perpendicular to the molecular plane) and subsequently along Xand Y-axes. Variation of interaction energy with respect to rotation about Z-axis has been ˚ in sliding (translation) and 1 in roexamined in the range of 60 . Accuracies up to 0.1 A tation have been achieved.
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The calculation of probabilities have been carried out using MaxwelleBoltzmann formula [13] as given below: Pi ¼ expðb3i Þ=
X
expðb3i Þ
i
where Pi stands for the probability corresponding to the configuration i; b ¼ 1/kT, where k and T are Boltzmann constant and absolute temperature, respectively and 3i represents the energy of configuration i relative to the minimum energy value in a particular set for which probability distribution is being computed. The details of the mathematical formalism and optimization process may be found in literature [7,8,12].
3. Results and discussion The molecular geometry of EBBA has been shown in Fig. 1 while net atomic charge and dipole components corresponding to each atomic centre of the molecule are given in Table 1. As expected, electronegative atoms like nitrogen and oxygen possess negative charges while carbon atoms exhibit both positive and negative charges according to their position in the molecule. The variation of probability with respect to translation of one of the stacked molecules along the Z-axis corresponding to four sets of axial rotation viz. X(0 )Y(0 ), X(0 )Y(180 ), X(180 )Y(0 ) and X(180 )Y(180 ) has been depicted in Fig. 2. It is observed that the configurations X(0 )Y(0 ) and X(0 )Y(180 ) show a sharp peak at the different interplanar separation, ˚ and 4.0 A ˚ with probability values nearly 98% and 97%, respectively. Below 4.5 A ˚ and 4.5 A ˚ 4.0 A the probability of occurrence of these configurations reduces to zero. The configurations ˚ and 4.5 A ˚ with probability X(180 )Y(0 ) and X(180 )Y(180 ) show their maxima at 5.0 A values nearly 68% and 96%, respectively. An analysis of the relative probability of being at maximum point corresponding to four temperatures viz. 300 K (room temperature), 308.5 K (solid to nematic), 352.5 K (nematic to isotropic) and 500 K (above room temperature), is shown in Table 2. As evident from this table, the configuration X(0 )Y(0 ) has almost 73% probability of occurrence at nematic to isotropic transition temperature while configurations X(0 )Y(180 ), X(180 )Y(0 ) and X(180 )Y(180 ) have probabilities nearly 26%, 0.01% and 0.77%, respectively.
Fig. 1. Molecular geometry of p-ethoxybenzylidene-p-n-butylaniline with its index numbers.
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Table 1 Net atomic charge and dipole moment components of EBBA molecule Atom no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
Atom symbol
C C C C C C N C C C C C C C O C C C C C C H H H H H H H H H H H H H H H H H H H H H H H
Charge (e.u.)
0.030 0.010 0.003 0.008 0.024 0.113 0.163 0.120 0.007 0.027 0.054 0.191 0.053 0.038 0.227 0.180 0.022 0.028 0.049 0.045 0.013 0.044 0.006 0.016 0.014 0.008 0.014 0.005 0.015 0.017 0.019 0.019 0.006 0.021 0.006 0.009 0.001 0.007 0.002 0.027 0.027 0.012 0.009 0.012
Atomic dipole components (Debye) X
Y
Z
0.065 0.066 0.127 0.065 0.055 0.052 0.695 0.080 0.050 0.085 0.062 0.178 0.070 0.065 0.633 0.026 0.208 0.031 0.106 0.092 0.090 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.119 0.090 0.022 0.103 0.114 0.021 1.199 0.155 0.007 0.080 0.098 0.002 0.088 0.096 0.842 0.167 0.046 0.141 0.132 0.135 0.181 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.012 0.055 0.245 0.027 0.033 0.022 1.068 0.105 0.003 0.079 0.086 0.028 0.096 0.104 0.837 0.167 0.044 0.046 0.058 0.057 0.007 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Total dipole moment ¼ 2.87 Debye.
Fig. 3 represents the variation of probability with respect to translation of one of the stacked molecules along the long molecular axis (X-axis) for configuration X(0 )Y(0 ) in ˚ corresponding to four selective rotations about Z-axis by Z(0 ), the range of 10 A ˚. Z(90 ), Z(180 ) and Z(270 ).The interplanar separation between the molecules is 4.5 A
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TRANSLATION (Å) 120
3.5
4
4.5
5
5.5
6
X(0) Y(0) X(0) Y(180)
100
PROBABILITY (%)
X(180) Y(0) X(180) Y(180)
80
60
40
20
0
Fig. 2. Variation of probability with respect to translation along Z-axis.
It may be observed that configuration Z(0 ) shows a relatively sharp peak with a maximum probability of 92% at room temperature while Z(90 ), Z(180 ) and Z(270 ) curves exhibit values nearly 47%, 87% and 82%, respectively. Comparative probability of minimum energy stacked complexes with respect to translation along X-axis is shown in Table 3 which shows that configuration Z(0 ) has maximum probability of occurrence 99% at nematic to isotropic transition temperature while configurations Z(90 ), Z(180 ) and Z(270 ) have probabilities nearly zero. A graphical representation of the probability distribution with respect to translation of one of the stacked molecules along an axis perpendicular to the long ˚ has been molecular axis and lying in the molecular plane (Y-axis) in the range of 2.0 A shown in Fig. 4 which indicates an overlapped structure as the most probable one. Since the ˚ ) is probpeak is not very sharp, translation along this axis in a very small range (<0.5 A able at increased thermal agitations. Fig. 5 exhibits the variation of probability with respect to rotation about Z-axis in the range of 60 at 300 K. It is evident from this figure that maximum probability (approximately 51%) lies at perfectly aligned structure. The variation of probability at 300 K with respect to in-plane (side to side) translation of one of the
Table 2 Comparative probability of the minimum energy stacked complexes of EBBA molecules with respect to translation along Z-axis corresponding to various rotational sets Configurations
X(0)Y(0) X(180)Y(0) X(0)Y(180) X(180)Y(180)
˚) Separation (A
4.5 4.0 5.0 4.5
Energy (Kcal/mol)
10.62 9.91 4.10 7.43
Probability (%) T ¼ 300 K
T ¼ 308.5 K
T ¼ 352.5 K
T ¼ 500 K
76.61 23.02 0.00 0.36
75.98 23.60 0.00 0.42
72.75 26.15 0.01 0.77
65.45 31.81 0.09 2.65
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TRANSLATION (Å) 100
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(0) Z(90)
PROBABILITY (%)
80
Z(180) Z(270)
60
40
20
0
Fig. 3. Variation of probability with respect to translation along X-axis.
interacting molecules along X-axis has been plotted in Fig. 6. The intermolecular separation ˚ and one of the molecules possesses X(0 )Y(0 )Z(0 ) configuration. As along Y-axis is 7.0 A evident from this figure, maximum probability (nearly 28%) occurs when one of the inter˚ along X-axis with a simultaneous rotation of 180 acting molecules is displaced by 6.0 A about Y-axis. Fig. 7 shows a plot of probability distribution for translation of one of the ˚. interacting molecules along Y-axis during in-plane interactions at an interval of 0.2 A ˚ This figure shows a sharp peak at an intermolecular separation of 7.0 A. Further, it may be observed here that like the case of stacking interactions, distinct maximum exists during in-plane interactions in both the cases. The variation of probability with respect to change of inter-group separation during terminal interactions for the energetically most stable interacting terminal groups viz (eC4H9/H5C2Oe) configuration has been shown in Fig. 8. It is observed from this figure that probability of any stable end to end configuration is very small (<9%) even at room temperature. The optimum end to end separation in ˚ whereas the centre of mass positions of this case (eC4H9/H5C2Oe) is 4.16 A ˚ . Further other possible interacting interacting EBBA molecules are separated by 16.8 A groups i.e. (eC4H9/H9C4e) and (eOC2H5/H5C2Oe) show negligible probability of occurrence.
Table 3 Comparative probability of the minimum energy stacked complexes of EBBA molecules with respect to translation along the long molecular axis corresponding to four selective rotations about Z-axis Rotation about Z-axis
0 90 180 270
Energy (Kcal/mol)
10.62 4.44 3.59 6.40
Probability (%) T ¼ 300 K
T ¼ 308.5 K
T ¼ 352.5 K
T ¼ 500 K
99.91 0.00 0.00 0.08
99.89 0.00 0.00 0.10
99.74 0.02 0.00 0.24
98.32 0.20 0.08 1.41
M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124
TRANSLATION (Å)
PROBABILITY (%)
50
-2
-1.2
0.4
-0.4
1.2
2
40
30
20
10
0
Fig. 4. Variation of probability with respect to translation along Y-axis.
ROTATION (Degree) -60
-40
-20
0
20
40
60
PROBABILITY (%)
60 50 40 30 20 10 0
Fig. 5. Variation of probability with respect to rotation about Z-axis.
TRANSLATION (Å) 30
PROBABILITY (%)
120
-26 -22 -18 -14 -10 -6 -2 2
6 10 14 18 22 26
20
10
0
Fig. 6. Variation of probability with respect to translation along X-axis during in-plane interactions.
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121
TRANSLATION (Å) -1 50
-0.6
-0.2
0.2
0.6
1
PROBABILITY (%)
40
30
20
10
0
Fig. 7. Variation of probability with respect to translation along Y-axis during in-plane interactions.
Translational and rotational rigidities corresponding to most probable configurations during stacking, in-plane and terminal interactions at different temperatures have been shown in Table 4. It should be mentioned here that rotational rigidity has been defined as the probability ratio of being at maximum to that corresponding to 10 of rotations. Translational rigidity has ˚, been defined as the probability ratio of being at maximum to that corresponding to 2.0 A ˚ ˚ 0.4 A and 0.5 A of translations of one of the stacked molecules along X-, Y- and Z-axes, respectively, during stacking interactions. During in-plane (side to side) interactions, transla˚ for X-sliding and 0.2 A ˚ for Y-sliding tional rigidity has been calculated in a range of 2.0 A ˚ whereas the same has been considered in the range of 0.2 A for translation along X-axis during in-plane (end to end) interactions. As evident from Table 4, both rotational and translational rigidities decrease with increase in temperature in each case. The probability of any stable configuration during terminal interactions is poor. Variation of translational rigidity with respect to translation along X-axis during stacking, in-plane and terminal interactions at various
TRANSLATION (Å)
PROBABILITY (%)
10
-1
-0.6
-0.2
0.2
0.6
1
8 6 4 2 0
Fig. 8. Variation of probability with respect to translation along X-axis during in-plane (end to end) interactions corresponding to (eC4H9/H5C2Oe) interacting groups.
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Table 4 Translational and rotational rigidities corresponding to most probable configuration during stacking, in-plane (side to side) and terminal interactions in case of EBBA at various temperatures Temperature (K)
Translational rigidity
Rotational rigidity
Stacking
300.0 308.5 352.5 500.0
Side to side
Terminal
X-axis ˚) (2.0 A
Y-axis ˚) (0.4 A
Z-axis ˚) (0.5 A
X-axis ˚) (2.0 A
Y-axis ˚) (0.2 A
X-axis ˚) (0.2 A
Z-axis (10 )
1.67 1.53 0.94 0.48
0.96 0.95 0.88 0.75
54.25 45.72 25.95 14.45
1.27 1.24 1.11 0.88
1.94 0.95 0.93 0.86
0.57 0.56 0.55 0.54
1.07 1.06 1.05 1.03
temperatures is shown in Fig. 9. As evident here, rigidity decreases in each case with temperature. Similarly there is a lesser probability of existence of stable configurations corresponding to the translation along an axis perpendicular to the long molecular axis (Y-axis). The minimum ˚ is shown energy stacked complex with energy e12.76 kcal/mol at interplanar separation 4.1 A in Fig. 10(a) while the in-plane minimum energy configuration with energy e5.62 kcal/mol at ˚ is shown in Fig. 10(b), respectively. It seems pertinent to mention here that induced di6.2 A poleeinduced dipole (dispersion) interactions are solely responsible for the stability of the packing of EBBA molecules in crystals which is in agreement to MaiereSaupe theory and other observation [14,15]. 4. Conclusion It seems that EBBA exhibits a strong preference for aligned structure at transition temperature. In a stacked molecular pair both orientational flexibility and transitional freedom are
2 Stacking
Translational rigidity
In-plane 1.5
Terminal
1
0.5
0
300
308.5
352.5
500
Temperature (K) Fig. 9. Translational rigidity parameter along X-axis as a function of temperature during stacking, in-plane (side to side) and terminal interactions.
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Fig. 10. Minimum energy configurations for a pair of EBBA molecules resulting from (a) stacking interactions with ˚ and (b) in-plane interactions with energy e5.62 kcal/mol energy e12.76 kcal/mol at an interplanar distance of 4.1 A ˚. at intermolecular distance of 6.2 A
found to be small corresponding to minimum energy configuration. Other configurations show greater translational flexibility along with their intrinsic preference for aligned structure, which accounts for the nematic behaviour of the molecule.
Acknowledgements SNT is grateful to late Professor Nitish K. Sanyal for his encouragement and fruitful suggestions. MKD is thankful to UGC, New Delhi for awarding Junior Research Fellowship.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Oxford University Press, London, 1993. S. Chandrasekhar, Liquid Crystals, Cambridge University Press, London, 1992. G.W. Gray, Molecular Structure and Properties of Liquid Crystals, Academic Press, New York, 1962. K. Tokita, K. Fujimura, S. Kondo, M. Takeda, Mol. Cryst. Liq. Cryst. Lett. 64 (1981) 171. G.R. Luckhurst, S. Romono, Liq. Cryst. 26 (1999) 871. S. Sarman, Mol. Phys. 98 (2000) 27. S.N. Tiwari, M. Mishra, N.K. Sanyal, Proc. Natl. Acad. Sci. 73A (2003) 159. D.P. Ojha, Ann. Phys. 13 (2004) 357; J. Mol. Struct. 716 (2005) 39. E.M. Barrell, J.F. Johnson, in: G.W. Gray, P.A. Winsor (Eds.), Liquid Crystals and Plastic Crystals, John Wiley, New York, 1974, p. 254. I. Bar, J. Bernstein, Acta Crystallogr. B33 (1977) 738. J.A. Pople, D.L. Beveridge, Approximate Molecular Orbital Theory, Mc-Graw Hill, New York, 1970. P. Claverie, in: B. Pullman (Ed.), Intermolecular Interactions e From Diatomics to Biopolymers, John Wiley, New York, 1978, p. 69. R.C. Tolman, The Principles of Statistical Mechanics, Oxford University Press, London, 1938. W. Maier, A. Saupe, Z. Naturforsch 13A (1958) 568; 14A (1959) 882; 15A (1960) 287. R.K. Mishra, R.S. Tyagi, in: J.F. Johnson, R.S. Porter (Eds.), Liquid Crystals and Ordered Fluids, vol. 2, Plenum Press, New York, 1973, p. 759.
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M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124 Dr. M. Mishra (Madhurenda Mishra) obtained his M.Sc. and Ph.D. degrees in Physics from D.D.U. Gorakhpur University, Gorakhpur. Presently he is a Post-Doctoral Fellow in the Department of Physics and has published five research papers.
Mr. M.K. Dwivedi is a Ph.D. student.
Mr. Rajeshwer Shukla (R. Shukla) is a Ph.D. student.
Dr. S.N. Tiwari (Sugriva Nath Tiwari) obtained his M.Sc. and Ph.D. degrees in Physics from D.D.U. Gorakhpur University, Gorakhpur and is presently working as Reader in the Department of Physics, D.D.U. Gorakhpur University, Gorakhpur, India. He has been conferred ISCA Young Scientist Award in 1988. His research areas of interest are Material Science (Liquid Crystals), Molecular Biophysics, Biosensors and Radiation Biophysics. Dr. Tiwari has successfully completed two research projects and has published more than forty research papers in various national and international journals.
Study of molecular ordering in liquid crystals: EBBA M. Mishra, M.K. Dwivedi, R. Shukla, S.N. Tiwari* Department of Physics, D.D.U. Gorakhpur University, Gorakhpur 273 009, India
Abstract Mesomorphism is primarily governed by the nature and strength of molecular interactions acting between sides, planes and ends of a pair of liquid crystal molecules. In view of this fact, intermolecular interactions between a pair of molecules of a nematic liquid crystal namely, p-ethoxybenzylidene-p-nbutylaniline (EBBA) have been evaluated using RayleigheSchrodinger second order perturbation theory along with multicentred-multipole expansion technique. An all valence electron method, CNDO/2 has been employed to compute net atomic charge and corresponding dipole components located at each atomic centre of the molecule. Using the results of stacking and in-plane interaction energy studies, probability calculations at varied angular and positional configurations in a molecular pair of EBBA have been carried out using MaxwelleBoltzmann formula. Results have been used to elucidate the liquid crystalline behaviour of the EBBA molecule. Ó 2006 Elsevier Ltd. All rights reserved. PACS: 61.30.-v; 61.30Gd; 64.43.Bn; 64.70.Md; 82.20.Wt; 31.15.Md; 31.15.Ct Keywords: A1. Computer simulation; B1. Polymers; B2. Dielectric materials (Liquid crystals/Nematogens)
1. Introduction Liquid crystalline phases are stable condensed phases in which molecules pack together with an order that is intermediate between the three-dimensional order of a crystalline solid
* Corresponding author. Tel.: þ91 551 2203182; fax: þ91 551 2340459. E-mail address: [email protected] (S.N. Tiwari). 0960-8974/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.pcrysgrow.2006.03.016
M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124
115
and the disorder of an isotropic liquid. The partial molecular ordering that is a characteristic of liquid crystallinity occurs frequently in natural and synthetic materials. Thus, liquid crystals are of considerable basic and applied interest [1,2]. The peculiar changes, characteristics of mesomorphic behaviour which occur at phase transitions, are primarily governed by the nature and strength of various types of intermolecular interactions acting between sides, planes and ends of a pair of molecules [3]. Using computer simulation and theoretical methods efforts have been made by several workers to understand the role of molecular interactions in accounting for liquid crystallinity [4e8]. It has been observed that the pair potential between such molecules is anisotropic in nature which is in general regarded as the prime requirement for the mesophase formation in thermotropic liquid crystals [5]. Since mesogenic properties are related to molecular aggregation in a specific manner, probability distribution calculations based on interaction energy results are expected to provide information about most probable molecular aggregation and tendency to retain translational and orientational order at different transition temperatures [7,8]. The present paper embodies the results of computational analysis based on actual molecular shape consideration in the case of p-ethoxybenzylidene-p-n-butylaniline (EBBA). The thermodynamic parameters reveal that EBBA shows crystal to nematic transition at 308.5 K and nematic to isotropic melt at 352.5 K [9].
2. Method of calculation The molecular geometry of EBBA has been constructed using crystallographic data from literature and standard values of bond lengths and bond angles [10]. Net charge and corresponding dipole moment components at each of the atomic centres of the molecule have been computed by CNDO/2 method [11]. Modified RayleigheSchrodinger second order perturbation theory along with multicentred-multipole expansion technique has been used to evaluate intermolecular interactions between a couple of EBBA molecules. According to this method, the total interaction energy between two molecules is obtained as the sum of electrostatic, polarization, dispersion and repulsion energy terms. Electrostatic energy is expressed as the sum of monopoleemonopole, monopolee dipole, dipoleedipole and energy terms due to higher order multipoles. In the present calculation electrostatic energy is restricted only up to first three terms while dispersion and repulsion energy terms have been calculated together using Kitaigorodskii formula [12]. The energy minimization has been carried out for both stacking and in-plane (side to side and end to end) interactions separately. One of the interacting molecules is kept fixed throughout the process while both lateral and angular variations are introduced in the other in all respects relative to the fixed one. The first molecule has been assumed to be in the XeY plane with X-axis lying along the long molecular axis while the origin is chosen approximately at the centre of the mass of the molecule. The second molecule has been translated initially along the Z-axis (perpendicular to the molecular plane) and subsequently along Xand Y-axes. Variation of interaction energy with respect to rotation about Z-axis has been ˚ in sliding (translation) and 1 in roexamined in the range of 60 . Accuracies up to 0.1 A tation have been achieved.
116
M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124
The calculation of probabilities have been carried out using MaxwelleBoltzmann formula [13] as given below: Pi ¼ expðb3i Þ=
X
expðb3i Þ
i
where Pi stands for the probability corresponding to the configuration i; b ¼ 1/kT, where k and T are Boltzmann constant and absolute temperature, respectively and 3i represents the energy of configuration i relative to the minimum energy value in a particular set for which probability distribution is being computed. The details of the mathematical formalism and optimization process may be found in literature [7,8,12].
3. Results and discussion The molecular geometry of EBBA has been shown in Fig. 1 while net atomic charge and dipole components corresponding to each atomic centre of the molecule are given in Table 1. As expected, electronegative atoms like nitrogen and oxygen possess negative charges while carbon atoms exhibit both positive and negative charges according to their position in the molecule. The variation of probability with respect to translation of one of the stacked molecules along the Z-axis corresponding to four sets of axial rotation viz. X(0 )Y(0 ), X(0 )Y(180 ), X(180 )Y(0 ) and X(180 )Y(180 ) has been depicted in Fig. 2. It is observed that the configurations X(0 )Y(0 ) and X(0 )Y(180 ) show a sharp peak at the different interplanar separation, ˚ and 4.0 A ˚ with probability values nearly 98% and 97%, respectively. Below 4.5 A ˚ and 4.5 A ˚ 4.0 A the probability of occurrence of these configurations reduces to zero. The configurations ˚ and 4.5 A ˚ with probability X(180 )Y(0 ) and X(180 )Y(180 ) show their maxima at 5.0 A values nearly 68% and 96%, respectively. An analysis of the relative probability of being at maximum point corresponding to four temperatures viz. 300 K (room temperature), 308.5 K (solid to nematic), 352.5 K (nematic to isotropic) and 500 K (above room temperature), is shown in Table 2. As evident from this table, the configuration X(0 )Y(0 ) has almost 73% probability of occurrence at nematic to isotropic transition temperature while configurations X(0 )Y(180 ), X(180 )Y(0 ) and X(180 )Y(180 ) have probabilities nearly 26%, 0.01% and 0.77%, respectively.
Fig. 1. Molecular geometry of p-ethoxybenzylidene-p-n-butylaniline with its index numbers.
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Table 1 Net atomic charge and dipole moment components of EBBA molecule Atom no.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
Atom symbol
C C C C C C N C C C C C C C O C C C C C C H H H H H H H H H H H H H H H H H H H H H H H
Charge (e.u.)
0.030 0.010 0.003 0.008 0.024 0.113 0.163 0.120 0.007 0.027 0.054 0.191 0.053 0.038 0.227 0.180 0.022 0.028 0.049 0.045 0.013 0.044 0.006 0.016 0.014 0.008 0.014 0.005 0.015 0.017 0.019 0.019 0.006 0.021 0.006 0.009 0.001 0.007 0.002 0.027 0.027 0.012 0.009 0.012
Atomic dipole components (Debye) X
Y
Z
0.065 0.066 0.127 0.065 0.055 0.052 0.695 0.080 0.050 0.085 0.062 0.178 0.070 0.065 0.633 0.026 0.208 0.031 0.106 0.092 0.090 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.119 0.090 0.022 0.103 0.114 0.021 1.199 0.155 0.007 0.080 0.098 0.002 0.088 0.096 0.842 0.167 0.046 0.141 0.132 0.135 0.181 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.012 0.055 0.245 0.027 0.033 0.022 1.068 0.105 0.003 0.079 0.086 0.028 0.096 0.104 0.837 0.167 0.044 0.046 0.058 0.057 0.007 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
Total dipole moment ¼ 2.87 Debye.
Fig. 3 represents the variation of probability with respect to translation of one of the stacked molecules along the long molecular axis (X-axis) for configuration X(0 )Y(0 ) in ˚ corresponding to four selective rotations about Z-axis by Z(0 ), the range of 10 A ˚. Z(90 ), Z(180 ) and Z(270 ).The interplanar separation between the molecules is 4.5 A
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TRANSLATION (Å) 120
3.5
4
4.5
5
5.5
6
X(0) Y(0) X(0) Y(180)
100
PROBABILITY (%)
X(180) Y(0) X(180) Y(180)
80
60
40
20
0
Fig. 2. Variation of probability with respect to translation along Z-axis.
It may be observed that configuration Z(0 ) shows a relatively sharp peak with a maximum probability of 92% at room temperature while Z(90 ), Z(180 ) and Z(270 ) curves exhibit values nearly 47%, 87% and 82%, respectively. Comparative probability of minimum energy stacked complexes with respect to translation along X-axis is shown in Table 3 which shows that configuration Z(0 ) has maximum probability of occurrence 99% at nematic to isotropic transition temperature while configurations Z(90 ), Z(180 ) and Z(270 ) have probabilities nearly zero. A graphical representation of the probability distribution with respect to translation of one of the stacked molecules along an axis perpendicular to the long ˚ has been molecular axis and lying in the molecular plane (Y-axis) in the range of 2.0 A shown in Fig. 4 which indicates an overlapped structure as the most probable one. Since the ˚ ) is probpeak is not very sharp, translation along this axis in a very small range (<0.5 A able at increased thermal agitations. Fig. 5 exhibits the variation of probability with respect to rotation about Z-axis in the range of 60 at 300 K. It is evident from this figure that maximum probability (approximately 51%) lies at perfectly aligned structure. The variation of probability at 300 K with respect to in-plane (side to side) translation of one of the
Table 2 Comparative probability of the minimum energy stacked complexes of EBBA molecules with respect to translation along Z-axis corresponding to various rotational sets Configurations
X(0)Y(0) X(180)Y(0) X(0)Y(180) X(180)Y(180)
˚) Separation (A
4.5 4.0 5.0 4.5
Energy (Kcal/mol)
10.62 9.91 4.10 7.43
Probability (%) T ¼ 300 K
T ¼ 308.5 K
T ¼ 352.5 K
T ¼ 500 K
76.61 23.02 0.00 0.36
75.98 23.60 0.00 0.42
72.75 26.15 0.01 0.77
65.45 31.81 0.09 2.65
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119
TRANSLATION (Å) 100
-10
-8
-6
-4
-2
0
2
4
6
8
10
Z(0) Z(90)
PROBABILITY (%)
80
Z(180) Z(270)
60
40
20
0
Fig. 3. Variation of probability with respect to translation along X-axis.
interacting molecules along X-axis has been plotted in Fig. 6. The intermolecular separation ˚ and one of the molecules possesses X(0 )Y(0 )Z(0 ) configuration. As along Y-axis is 7.0 A evident from this figure, maximum probability (nearly 28%) occurs when one of the inter˚ along X-axis with a simultaneous rotation of 180 acting molecules is displaced by 6.0 A about Y-axis. Fig. 7 shows a plot of probability distribution for translation of one of the ˚. interacting molecules along Y-axis during in-plane interactions at an interval of 0.2 A ˚ This figure shows a sharp peak at an intermolecular separation of 7.0 A. Further, it may be observed here that like the case of stacking interactions, distinct maximum exists during in-plane interactions in both the cases. The variation of probability with respect to change of inter-group separation during terminal interactions for the energetically most stable interacting terminal groups viz (eC4H9/H5C2Oe) configuration has been shown in Fig. 8. It is observed from this figure that probability of any stable end to end configuration is very small (<9%) even at room temperature. The optimum end to end separation in ˚ whereas the centre of mass positions of this case (eC4H9/H5C2Oe) is 4.16 A ˚ . Further other possible interacting interacting EBBA molecules are separated by 16.8 A groups i.e. (eC4H9/H9C4e) and (eOC2H5/H5C2Oe) show negligible probability of occurrence.
Table 3 Comparative probability of the minimum energy stacked complexes of EBBA molecules with respect to translation along the long molecular axis corresponding to four selective rotations about Z-axis Rotation about Z-axis
0 90 180 270
Energy (Kcal/mol)
10.62 4.44 3.59 6.40
Probability (%) T ¼ 300 K
T ¼ 308.5 K
T ¼ 352.5 K
T ¼ 500 K
99.91 0.00 0.00 0.08
99.89 0.00 0.00 0.10
99.74 0.02 0.00 0.24
98.32 0.20 0.08 1.41
M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124
TRANSLATION (Å)
PROBABILITY (%)
50
-2
-1.2
0.4
-0.4
1.2
2
40
30
20
10
0
Fig. 4. Variation of probability with respect to translation along Y-axis.
ROTATION (Degree) -60
-40
-20
0
20
40
60
PROBABILITY (%)
60 50 40 30 20 10 0
Fig. 5. Variation of probability with respect to rotation about Z-axis.
TRANSLATION (Å) 30
PROBABILITY (%)
120
-26 -22 -18 -14 -10 -6 -2 2
6 10 14 18 22 26
20
10
0
Fig. 6. Variation of probability with respect to translation along X-axis during in-plane interactions.
M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124
121
TRANSLATION (Å) -1 50
-0.6
-0.2
0.2
0.6
1
PROBABILITY (%)
40
30
20
10
0
Fig. 7. Variation of probability with respect to translation along Y-axis during in-plane interactions.
Translational and rotational rigidities corresponding to most probable configurations during stacking, in-plane and terminal interactions at different temperatures have been shown in Table 4. It should be mentioned here that rotational rigidity has been defined as the probability ratio of being at maximum to that corresponding to 10 of rotations. Translational rigidity has ˚, been defined as the probability ratio of being at maximum to that corresponding to 2.0 A ˚ ˚ 0.4 A and 0.5 A of translations of one of the stacked molecules along X-, Y- and Z-axes, respectively, during stacking interactions. During in-plane (side to side) interactions, transla˚ for X-sliding and 0.2 A ˚ for Y-sliding tional rigidity has been calculated in a range of 2.0 A ˚ whereas the same has been considered in the range of 0.2 A for translation along X-axis during in-plane (end to end) interactions. As evident from Table 4, both rotational and translational rigidities decrease with increase in temperature in each case. The probability of any stable configuration during terminal interactions is poor. Variation of translational rigidity with respect to translation along X-axis during stacking, in-plane and terminal interactions at various
TRANSLATION (Å)
PROBABILITY (%)
10
-1
-0.6
-0.2
0.2
0.6
1
8 6 4 2 0
Fig. 8. Variation of probability with respect to translation along X-axis during in-plane (end to end) interactions corresponding to (eC4H9/H5C2Oe) interacting groups.
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Table 4 Translational and rotational rigidities corresponding to most probable configuration during stacking, in-plane (side to side) and terminal interactions in case of EBBA at various temperatures Temperature (K)
Translational rigidity
Rotational rigidity
Stacking
300.0 308.5 352.5 500.0
Side to side
Terminal
X-axis ˚) (2.0 A
Y-axis ˚) (0.4 A
Z-axis ˚) (0.5 A
X-axis ˚) (2.0 A
Y-axis ˚) (0.2 A
X-axis ˚) (0.2 A
Z-axis (10 )
1.67 1.53 0.94 0.48
0.96 0.95 0.88 0.75
54.25 45.72 25.95 14.45
1.27 1.24 1.11 0.88
1.94 0.95 0.93 0.86
0.57 0.56 0.55 0.54
1.07 1.06 1.05 1.03
temperatures is shown in Fig. 9. As evident here, rigidity decreases in each case with temperature. Similarly there is a lesser probability of existence of stable configurations corresponding to the translation along an axis perpendicular to the long molecular axis (Y-axis). The minimum ˚ is shown energy stacked complex with energy e12.76 kcal/mol at interplanar separation 4.1 A in Fig. 10(a) while the in-plane minimum energy configuration with energy e5.62 kcal/mol at ˚ is shown in Fig. 10(b), respectively. It seems pertinent to mention here that induced di6.2 A poleeinduced dipole (dispersion) interactions are solely responsible for the stability of the packing of EBBA molecules in crystals which is in agreement to MaiereSaupe theory and other observation [14,15]. 4. Conclusion It seems that EBBA exhibits a strong preference for aligned structure at transition temperature. In a stacked molecular pair both orientational flexibility and transitional freedom are
2 Stacking
Translational rigidity
In-plane 1.5
Terminal
1
0.5
0
300
308.5
352.5
500
Temperature (K) Fig. 9. Translational rigidity parameter along X-axis as a function of temperature during stacking, in-plane (side to side) and terminal interactions.
M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124
123
Fig. 10. Minimum energy configurations for a pair of EBBA molecules resulting from (a) stacking interactions with ˚ and (b) in-plane interactions with energy e5.62 kcal/mol energy e12.76 kcal/mol at an interplanar distance of 4.1 A ˚. at intermolecular distance of 6.2 A
found to be small corresponding to minimum energy configuration. Other configurations show greater translational flexibility along with their intrinsic preference for aligned structure, which accounts for the nematic behaviour of the molecule.
Acknowledgements SNT is grateful to late Professor Nitish K. Sanyal for his encouragement and fruitful suggestions. MKD is thankful to UGC, New Delhi for awarding Junior Research Fellowship.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Oxford University Press, London, 1993. S. Chandrasekhar, Liquid Crystals, Cambridge University Press, London, 1992. G.W. Gray, Molecular Structure and Properties of Liquid Crystals, Academic Press, New York, 1962. K. Tokita, K. Fujimura, S. Kondo, M. Takeda, Mol. Cryst. Liq. Cryst. Lett. 64 (1981) 171. G.R. Luckhurst, S. Romono, Liq. Cryst. 26 (1999) 871. S. Sarman, Mol. Phys. 98 (2000) 27. S.N. Tiwari, M. Mishra, N.K. Sanyal, Proc. Natl. Acad. Sci. 73A (2003) 159. D.P. Ojha, Ann. Phys. 13 (2004) 357; J. Mol. Struct. 716 (2005) 39. E.M. Barrell, J.F. Johnson, in: G.W. Gray, P.A. Winsor (Eds.), Liquid Crystals and Plastic Crystals, John Wiley, New York, 1974, p. 254. I. Bar, J. Bernstein, Acta Crystallogr. B33 (1977) 738. J.A. Pople, D.L. Beveridge, Approximate Molecular Orbital Theory, Mc-Graw Hill, New York, 1970. P. Claverie, in: B. Pullman (Ed.), Intermolecular Interactions e From Diatomics to Biopolymers, John Wiley, New York, 1978, p. 69. R.C. Tolman, The Principles of Statistical Mechanics, Oxford University Press, London, 1938. W. Maier, A. Saupe, Z. Naturforsch 13A (1958) 568; 14A (1959) 882; 15A (1960) 287. R.K. Mishra, R.S. Tyagi, in: J.F. Johnson, R.S. Porter (Eds.), Liquid Crystals and Ordered Fluids, vol. 2, Plenum Press, New York, 1973, p. 759.
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M. Mishra et al. / Progress in Crystal Growth and Characterization of Materials 52 (2006) 114e124 Dr. M. Mishra (Madhurenda Mishra) obtained his M.Sc. and Ph.D. degrees in Physics from D.D.U. Gorakhpur University, Gorakhpur. Presently he is a Post-Doctoral Fellow in the Department of Physics and has published five research papers.
Mr. M.K. Dwivedi is a Ph.D. student.
Mr. Rajeshwer Shukla (R. Shukla) is a Ph.D. student.
Dr. S.N. Tiwari (Sugriva Nath Tiwari) obtained his M.Sc. and Ph.D. degrees in Physics from D.D.U. Gorakhpur University, Gorakhpur and is presently working as Reader in the Department of Physics, D.D.U. Gorakhpur University, Gorakhpur, India. He has been conferred ISCA Young Scientist Award in 1988. His research areas of interest are Material Science (Liquid Crystals), Molecular Biophysics, Biosensors and Radiation Biophysics. Dr. Tiwari has successfully completed two research projects and has published more than forty research papers in various national and international journals.