Journal of Molecular Structure 994 (2011) 387–391
Contents lists available at ScienceDirect
Journal of Molecular Structure journal homepage: www.elsevier.com/locate/molstruc
Characterization of a hydrogen bonded liquid crystal homologous series: Detailed FTIR studies in various mesophases N. Pongali Sathya Prabu, V.N. Vijayakumar, M.L.N. Madhu Mohan ⇑ Liquid Crystal Research Laboratory (LCRL), Bannari Amman Institute of Technology, Sathyamangalam 638 401, India
a r t i c l e
i n f o
Article history: Received 8 February 2011 Received in revised form 23 March 2011 Accepted 23 March 2011 Available online 30 March 2011 Keywords: Hydroxy benzaldehyde p-n- alkyloxy benzoic acid DSC Temperature dependence FTIR spectra
a b s t r a c t Novel linear hydrogen bonded liquid crystal homologous series has been synthesized and characterized. Hydrogen bond is formed between hydroxy benzaldehyde and various benzoic acids which are varying from pentyl to dodecyl alkyloxy carbon numbers. Synthesized complexes are characterized by FTIR for inferring the formation of hydrogen bonds. Polarizing microscopy and DSC studies reveal various mesophases and their corresponding transition temperatures along with respective enthalpy values. Tilt angle and helix in Smectic C phase have been experimentally elucidated for most of the mesogens. A new technique for the temperature dependent FTIR studies has been proposed. The spectral results obtained by this technique are in good agreement with the conventional method. These FTIR studies bestow information about shifting and occurrence of new peaks in various mesophases thus enriching the knowledge on the chemical molecular environment of the mesogen. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Design of functional organic liquid crystalline materials took a new turn with the use of non-covalent interactions. Synthesis of self assembly systems through hydrogen bonding has turned the scientist’s attention towards the intermolecular hydrogen bonding. Complementary inter molecular hydrogen bonds are formed between a proton donor and an electron acceptor atoms of carboxylic acids. Thus, hydrogen bonding is a powerful tool in assembling molecules for non-covalent interactions. It is noticed [1–4] that in hydrogen bonded liquid crystals (HBLC) lower bonding and activation energies showed a profound influence on their thermal properties, viz, clearing points, enthalpies and mesomorphic phase behavior. Mesogenic properties of HBLC can be tuned easily by changing H- bond donor/acceptor or percentage of respective molar composition. Stable and dynamic molecular complexes can be prepared by simple molecular self assembly processes using such hydrogen bonding. A number of such HBLC have been investigated following the reports of Kato et al. [1–6] which indicates that the mesomorphism results from proper combination of molecular interactions and shape of the molecules. It is inferred [7] that hydrogen bonding has pronounced influence on crystallization and phase behaviors of multi component supra molecular complexes formed by benzoic acids. Fourier transform infrared spectroscopy (FTIR) studies play a pivotal role in understanding the chemical molecular environment ⇑ Corresponding author. Tel.: +91 4295 223 480; fax: +91 4295 223 775. E-mail address:
[email protected] (M.L.N. Madhu Mohan). 0022-2860/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2011.03.056
of the HBLC. FTIR is one of the most powerful tools to understand the relationship between hydrogen bonding and molecular interactions of various mesophases. It is reported [8–11] that hydrogen bonding significantly changes the direction and magnitude of the vibrational dipole transition moments, causing marked changes in the IR dichroic absorbance profiles of hydrogen bonded molecular sub fragments. The conventional recording of the solid state FTIR spectra has its own limitations particularly when the spectra are recorded with respect to temperature. Thus, in the present work, a new technique has been proposed for recording FTIR spectra and studying the molecular environment of the HBLC with respect to the temperature. The self assembly systems formed by alkyloxy carboxylic acids exhibits rich phase polymorphism. It has been earlier reported by us that these self organized systems induce variety of new phenomena like, reentrant phase occurrence [12,13] light modulation [14] optical shuttering action [15–18] and field induced transitions [14,19,20]. The carboxylic acids with other acids are reported [12– 20] to form complementary single and multiple hydrogen bonds. 2. Experimental Optical textural observations were made with a Nikon polarizing microscope (POM) equipped with Nikon digital CCD camera system with 5 mega pixels and 2560 1920 pixel resolutions. The liquid crystalline textures were analyzed and stored with the aid of ACT-2U imaging software system. The temperature control of the liquid crystal cell was equipped by Instec HCS402-STC 200 temperature controller (Instec, USA) to a temperature resolution
388
N. Pongali Sathya Prabu et al. / Journal of Molecular Structure 994 (2011) 387–391
of ±0.1 °C. This unit was interfaced to computer by IEEE –STC 200 to control and monitor the temperature. The liquid crystal sample was filled by capillary action in its isotropic state into a commercially available (Instec, USA) polyamide buffed cell with 5 lm spacer. Optical extinction technique [20] was used for determination of tilt angle. Transition temperatures and corresponding enthalpy values were obtained by DSC (Shimadzu DSC-60, Japan). FTIR spectra was recorded (ABB FTIR MB3000) and analyzed with the MB3000 software. The helicoidal pitch was measured by an optical setup reported [14] earlier by us. The p-n- alkyloxy benzoic acids (nBAO) and hydroxy benzaldehyde were supplied by Sigma Aldrich, (Germany) and all the solvents were of HPLC grade. 2.1. Synthesis of HBLC Intermolecular hydrogen bonded mesogens are synthesized by the addition of one mole of p-n- alkyloxy benzoic acids (nBAO) with one mole of hydroxy benzaldehyde in N,N-Dimethyl formamide (DMF) respectively. They are subject to constant stirring for 14 h at ambient temperature of 30 °C till a white precipitate in a dense solution is formed. These white crystalline crude complexes so obtained, by removing excess DMF, are then recrystallized with dimethyl sulfoxide (DMSO) and the yield varied from 85% to 95%. Yield of higher homologues complexes are observed to be more compared to its lower counterparts. Molecular structure of the present homologous series of p-n- alkyloxy benzoic acid with hydroxy benzaldehyde is depicted in the Fig. 1, where n represents the alkyloxy carbon number. 3. Results and discussion All the hydrogen bonded complexes isolated under the present investigation are white crystalline solids and are stable at room temperature (30 °C). They are insoluble in water and sparingly soluble in common organic solvents such as methanol, ethanol, and benzene and dichloro methane. However, they show a high degree of solubility in coordinating solvents like dimethyl sulfoxide (DMSO), dimethyl formamide (DMF) and pyridine. All these mesogens melt at specific temperatures below 122 °C (Table 1). They show high thermal and chemical stability when subjected to repeated thermal scans performed during POM, DSC and FTIR studies.
benzaldehyde and alkyloxy benzoic acids in the cooling run can be shown as:
Isotropic ! Nematic ! Sm G ! Crystal ðHBA þ 5BAOÞ; ðHBA þ 6BAOÞ Isotropic ! Nematic ! Sm C ! SmG ! Crystal ðHBA þ nBAOÞ; where n ¼ 7 to 12 3.3. DSC studies DSC thermograms are recorded in heating and cooling cycle. The sample is heated with a scan rate of 10 °C/min and held at its isotropic temperature for two minutes so as to attain thermal stability. The cooling run is performed with a scan rate of 10 °C/ min. The respective equilibrium transition temperatures and corresponding enthalpy values of the mesogens of the homologous series are listed separately in Table 1. POM studies also confirm these DSC transition temperatures. 3.3.1. DSC studies of HBA+12BAO The phase transition temperatures and enthalpy values of dodecyloxy benzoic acid and hydroxy benzaldehyde mesogen (HBA + 12BAO) are discussed as a representative case. The DSC thermogram of HBA + 12BAO is illustrated in Fig. 2. From this Fig. 2 and Table 1, it can be inferred that in the DSC heating run exhibits two exothermic peaks at 94.7 °C and 107.7 °C with enthalpy values of 63.59 J/g, and 44.30 J/g respectively. These two peaks correspond to the crystal to crystal and crystal to melt phase transitions respectively. In the cooling run, this sample exhibits four peaks at 109.4 °C, 87.3 °C, 84.2 °C and 81.1 °C with enthalpy values of 1.77 J/g, merged with smectic C, 19.76 J/g and 39.60 J/g respectively. These endothermic peaks corresponds to isotropic to nematic, nematic to smectic C, smectic C to smectic G and smectic G to crystal phases respectively. 3.4. Phase diagram of pure p-n-alkyloxy benzoic acids
3.1. Phase identification
The phase diagram of HBA + nBAO homologous series is constructed through optical polarizing microscopic studies and by the phase transition temperatures observed in the cooling run of the DSC thermogram. The phase diagram of pure p-n- alkyloxy benzoic acid is reported [22,23] to compose of two phases namely, Nematic and smectic C.
The observed phase variants, transition temperatures and corresponding enthalpy values obtained by DSC in the cooling and heating cycles for the HBA + nBAO complexes are presented in Table 1. These studies are in concurrence with POM data.
3.4.1. Phase Diagram of HBA+nBAO Phase diagram of hydroxy benzaldehyde and alkyloxy benzoic acids are depicted in Fig. 3. The following points can be elucidated from this figure. (i) The HBA + nBAO hydrogen bonded homologous series exhibits nematic as orthogonal phase and smectic C, and smectic G as tilted phases. (ii) The total thermal range of the mesogenic phases increased with increase in the alkyloxy carbon number up to nonyloxy carbon and then starts to decrease till dodecyloxy benzoic acid.
3.2. HBA+nBAO homologous series The mesogens of the hydroxy benzaldehyde and alkyloxy benzoic acid homologous series are found to exhibit characteristic textures [21], viz., nematic (threaded texture), Smectic C (broken focal conic texture) and Smectic G (smooth multi colored mosaic texture) respectively. The general phase sequence of the hydroxy
OH
CHO
HOOC
Fig. 1. Molecular structure of HBA + nBAO.
OCnH2n+1
389
N. Pongali Sathya Prabu et al. / Journal of Molecular Structure 994 (2011) 387–391 Table 1 Phase transition temperatures of HBA + nBAO homologues series obtained by various techniques along with enthalpy values in J/g. Complex
Phase variance
Study
Crystal to Melt
N
HBA + 12BAO
NCG
DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c) DSC (h) DSC (c) POM (c)
107.7 (44.30)
# 109.4 110.8 # 118.6 119.5 114.5 111.1 112.9 # 106.5 107.8 # 116.3 117.6 # 106.1 107.5 # 107.7 108.8 # 122.1 123.2
HBA + 11BAO
NCG
HBA + 10BAO
NCG
HBA + 9BAO
NCG
HBA + 8BAO
NCG
HBA + 7BAO
NCG
HBA + 6BAO
NG
HBA + 5BAO
NG
107.0 (23.09)
85.6 (41.17)
93.6 (62.85)
100.0 (36.23)
92.4 (79.88)
101.2 (36.56)
104.3 (73.07)
(1.77)
(0.06) (1.26) (0.13)
(3.83)
(Merged with C)
(0.03)
(1.04)
(Merged with G)
C
G
# 87.3 (Merged with G) 88.1 # 114.2 (0.23) 115.1 # 108.2 (6.14) 109.3 # 90.3 (17.35) 91.7 # 101.4 (1.44) 102.9 # 99.7 (0.97) 100.9
# 84.2 (19.76) 84.9 # 108.0 (4.85) 108.7 95.8 (20.78) 92.6 (Merged with crystal) 93.5 # 74.3 (33.14) 74.9 # 94.8 (25.64) 95.3 # 88.2 (24.27) 89.1 # 93.2 (30.47) 94.3 114.7 (21.31) 110.2 (50.61) 111.4
Crystal 81.1 (39.60) 81.8 91.4 (112.69) 91.8 82.5 (22.41) 82.9 62.0 (43.27) 62.4 75.1 (27.96) 75.5 83.3 (93.05) 83.8 83.5 (38.60) 84.0 88.0 (56.77) 88.7
(c) Cooling run, (h) heating run, # not resolvedtransition.
15 Crystal
*^
Iso.
N
HBA+12BAO ^ Smectic C * Smectic F
Heat flow / Jg
-1
10
5
0
(iii) Smectic G phase is observed in all the complexes of the present homologous series. (iv) The Smectic C phase is induced in the higher homologous members from heptyloxy benzoic acid and continued till dodecyloxy carbon. (v) The liquid crystalline thermal range is largest for nonyloxy carbon and narrowest for undecyloxy carbon. (vi) A systematic decrease in the crystallization temperatures is observed up to nonyloxy benzoic carbon number from then the crystallization temperatures starts to increase proportionally along with its corresponding carbon number, with an exception of dodecyloxy carbon number.
-5 4. Technique to record temperature dependent FTIR spectra
-10 40
60
80
100
120
140
o
Temperature / C Fig. 2. DSC thermogram of HBA + 12BAO complex.
HBA+nBAO Isotropic
o
Temperature / C
120
100
N
C
80
G
60
Crystal
4
6
8
10
12
Alkyloxy carbon number Fig. 3. Phase diagram of HBA + nBAO homologous series.
To elicit the molecular environment of the hydrogen bonded liquid crystal, temperature influenced FTIR study has been carried out. It has been reported by Kato [4–6] that a sample placed in a cell comprising of two KBr pellets without any surface treatment are heated to yield the FTIR spectra. The problem of temperature holder and its accuracy compounded by the sandwiching of the pellets makes it a difficult task to record FTIR spectra. This conventional method of recording FTIR spectra with KBr pellet has been replaced by a novel technique which is described in the following section. The technique proposed by us consists of a thin layer of liquid crystalline sample evenly spread in its isotropic state over a 5 lm spacer glass cell which is transparent to IR radiation. Thus the cell with sample can be placed in an Instec temperature controller equipped with hot and cold stage. This setup allows for a greater thermal accuracy with repeatable results. The reference spectra in this case would be the empty glass cell with out the liquid crystalline sample. With the above technique, the FTIR spectra is recorded for all the samples (HBA + nBAO) of the present homologous series and compared with the conventional solid state KBr spectra. As a respective case, spectrum of HBA + 12BAO complex is shown in Fig. 4. Spectra obtained by the above techniques when compared it is noted that there is no variation either in the relative peak
390
N. Pongali Sathya Prabu et al. / Journal of Molecular Structure 994 (2011) 387–391
120
HBA+12BAO 2360
HBA+8BAO
Wavenumber /cm-1
Transmittance
100 Conventional method
80 60 40 20 Proposed method
2350
2340
2330
Iso.
0
Cryst.
G
C
N
2320 4000
3500
3000
2500 -1
120
2000
100
Wavenumber / cm
Fig. 4. FTIR spectra of HBA + 12BAO complex recorded by conventional and proposed method.
intensities or in the peak assignments. Hence it may be argued that this new technique is much more convenient for the liquid crystalline research. 4.1. Analysis of the temperature dependent FTIR data As a representative case, the intermolecular hydrogen bond of HBA + 12BAO complex recorded at room temperature is discussed. A peak of the carbonyl band at 1690 cm1 of the HBA + 12BAO complex along with a broad peak at 3600 cm1 shows the formation of inter molecular hydrogen bonding [24,25,23] and non-appearance of the same in HBA and 12BAO clearly suggests the formation of hydrogen bond upon complexation. Further more, the carbonyl band in the pure 12BAO is at 1685 cm1 and the corresponding band for HBA + 12BAO complex is observed at 1690 cm1 the shifting of this band manifests the formation of inter hydrogen bonding between 12BAO and HBA. The sharp high intense band at 2924 cm1 in the HBA + 12BAO complex clearly suggests the stretching vibration of CH2 groups. As a representative case, Fig. 5 depicts the FTIR spectra in various mesogenic phases of HBA + 8BAO complex in the range 3200– 2100 cm1. Two new bands centered at 2400 cm1 and 2425 cm1 are strong evidence of the unionized-type inter molecular hydrogen bonding between hydroxy benzoic acid and carboxylic group [3,8]. The origin of these two new peaks are attributed to the carbonyl [C6H6C@HACOOA] bond. The corresponding shift of wave
40 Sm C
30
Sm G Crystal
20 10 0 4000
5. Optical tilt angle The optical tilt angle has been experimentally measured by optical extinction method [20] in smectic C phase of all the
2715 HBA+9BAO
2700 2685 2670 2655 2640
3500
3000
2500
Wavenumber / cm-1 Fig. 5. FTIR spectra of HBA + 8BAO complex recorded at various mesophases.
40
number 2300 cm1 with respect to various phases is depicted in Fig. 6. At the phase transition from one mesogenic phase to the other, the shift appears to be more prominent while in a particular phase the shift is almost nominal. The magnitude of the shift of the 2330 cm1 wave number from isotropic to nematic is around 5 cm1 and it is unaltered in the entire thermal span of the nematic phase. As the temperature is decreased the phase transition from nematic to smectic C is evinced through a shift of 10 cm1 and it is again unaltered in the entire thermal span of the smectic C phase. As the temperature is further reduced, the phase transition from smectic C to smectic G is observed which is associated with a large wave number shifted of 15 cm1 and in the final phase transition from smectic G to crystal is considerably low. This implies that the OAH bond reflect the molecular environment changes in all the phase transitions except for nematic to isotropic transition. The shift is also attributed to the changes of inter and intra molecular interactions between aromatic hydrogen and the carbonyl group [8–10]. Similar trends of results are obtained for the entire homologous series of HBA + nBAO complexes. As a representative case the shift of the wave numbers with various temperatures in the FTIR spectra of HBA + 9BAO is shown in Fig. 7.
HBA+8BAO
Isotropic N
60
Fig. 6. Frequency shift of HBA + 8BAO complex in various mesophases.
Wavenumber / cm-1
Transmittance
50
80
Temperature / oC
Iso.
120
N 100
C
G
80
Cryst. 60
40
Temperature / o C Fig. 7. Frequency shift of HBA + 9BAO complex in various mesophases.
N. Pongali Sathya Prabu et al. / Journal of Molecular Structure 994 (2011) 387–391
391
saturated magnitude of tilt angle corresponding to various hydrogen bonded complexes increases with the increment in alkyloxy carbon number.
16
Smectic C
Tilt angle / θ
12 6. Helical pitch
8 HBA+7BAO HBA+8BAO HBA+9BAO HBA+10BAO HBA+11BAO
4
0 0
1
2
3
4
5
(T-Tc) / OC Fig. 8. Temperature variation of tilt angle in Smectic C phase for various hydrogen bonded complexes.
The pitch of liquid crystalline materials in smectic C phase is typically in the order of 1–100 lm [30], whereas the thickness of one layer is of the order 20–30 Å. The helical pitch is measured by diffraction of He–Ne red laser light on sample, filled in a commercial cell. This method can be used for measurement of the helical pitch of limited length. For pitch shorter than 0.8 lm the diffraction ring is diffused or completely disappears. The magnitude of the helical pitch in smectic C phase is found to increase with decreasing temperature and attains a saturated value. This unwinding of the helix with temperature is studied for the HBA + 9BAO, HBA + 10BAO and HBA + 11BAO complexes and the results are shown in Fig. 9. These results are in good agreement with the reported [17–20] data on similar HBLC systems. Acknowledgements
Normalized Helix
1.0
The authors acknowledge the financial support rendered by Department of Science and Technology, New Delhi. Infrastructural support provided by Bannari Amman Institute of Technology is gratefully acknowledged.
0.8 0.6
References
Smectic C 0.4
HBA+ 9BAO HBA+10BAO HBA+11BAO
0.2 0.0 0
2
4
6
8
o
(T-Tc) / C Fig. 9. Temperature variation of helical pitch in Smectic C phase for various hydrogen bonded complexes.
members of the present HBA + nBAO homologous series. Fig. 8 depicts such variation of optical tilt angle with temperature for HBA + nBAO (where n = 7 to 11) respectively. In the above figures the theoretical fit obtained from the Mean Field theory is denoted by the solid line. It is observed from the Fig. 8, that the tilt angle increases with decreasing temperature and attains a saturation value. These large magnitudes of the tilt angle are attributed to the direction of the soft covalent hydrogen bond interaction which spreads along molecular long axis with finite inclination [26]. Tilt angle is a primary order parameter [27] and the temperature variation is estimated by fitting the observed data of h (T) to the relation.
hðTÞaðT C TÞb
ð1Þ
The critical exponent b value estimated by fitting the data of h (T) to the above Eq. (1) is found to be 0.50 to agree with the Mean Field prediction [28,29]. The solid lines in the Fig. 8 depict the fitted data for various mesogens. Further, the agreement of magnitude of b (0.5) with Mean Field value (0.5) infers the long-range interaction of transverse dipole moment for the stabilization of tilted smectic C phase. It is noteworthy to point out that the
[1] T. Kato, J.M.J. Frechet, J. Am. Chem. Soc. 111 (1989) 8533. [2] T. Kato, H. Kihara, T. Uryu, S. Ujiie, K. Iimura, J.M.J. Frechet, U. Kumar, Ferroelectronics 148 (1993) 161. [3] T. Kato, T. Uryu, F. Kaneuchi, C. Jin, J.M.J. Frechet, Liq. Cryst. 14 (1993) 1311. [4] T. Kato, J.M.J. Frechet, Macromolecules 22 (1989) 3818. [5] U. Kumar, T. Kato, J.M.J. Frechet, J. Am. Chem. Soc 114 (1992) 6630. [6] H. Kihara, T. Kato, S. Ujiie, T. Uryu, U. Kumar, J.M.J. Frechet, D.W. Bruce, D.J. Price, Liq. Cryst. 21 (1996) 25. [7] X. Lu, C. He, A.C. Griffins, Macromolecules (2003) 5195. [8] Y. Tian, X. Xu, Y. Zhao, X. Tang, T. Li, J. Sun, C. Li, A. Pan, Thin Solid Films 284 (1996) 603. [9] A. Ghanem, C. Noel, Mol. Cryst. Liq. Cryst. 150 (1987) 447. [10] S. Jiang, W. Xu, B. Zhao, Y. Tian, Y. Zhao, Mater. Sci. Eng: C 11 (2000) 85. [11] W.G. Jang, C.S. Park, K.H. Kim, M.A. Glaser, N.A. Clark, Phys. Rev. E 62 (2000) 5027. [12] V.N. Vijayakumar, K. Murugadass, M.L.N. Madhu Mohan, Mol. Cryst. Liq. Cryst. 517 (2010) 43. [13] T. Chitravel, M.L.N. Madhu Mohan, Mol. Cryst. Liq. Cryst. 524 (2010) 131. [14] V.N. Vijayakumar, M.L.N. Madhu Mohan, Mol. Cryst. Liq. Cryst. (2010) 113. [15] V.N. Vijayakumar, M.L.N. Madhu Mohan, J. Optom. Electr. Adv. Mater. 11 (8) (2009) 1139. [16] V.N. Vijayakumar, M.L.N. Madhu Mohan, J. Mol. Struct. 991 (2011) 60. [17] V.N. Vijayakumar, M.L.N. Madhu Mohan, Sol. State. Sci. 4 (2009) 482. [18] V.N. Vijayakumar, M.L.N. Madhu Mohan, Braz J. Phys. 39 (4) (2009) 677. [19] V.N. Vijayakumar, K. Murugadass, M.L.N. Madhu Mohan, Mol. Cryst. Liq. Cryst. 537 (2011) 22. [20] V.N. Vijayakumar, M.L.N. Madhu Mohan, Sol State. Commun. 149 (2009) 2090. [21] G.W. Gray, J.W.G. Goodby, Smectic Liquid Crystals: Textures and Structures, Leonard Hill, London, 1984. [22] M. Srinivasulu, P.V.V. Satyanarayana, P.A. Kumar, V.G.K.M. Pisipati, Z. Naturforsch A: Phys. Sci 56a (2002) 685. [23] P. Swathi, P.A. Kumar, V.G.K.M. Pisipati, A.V. Rajeswari, S. Sreehari Sastry, P. Narayana Murty, Z. Natur Forsch 57a (2002) 797. [24] K. Nakamoto, Infrared and Raman Spectra of Inorganic and Co-ordination Compounds, Interscience, New York, 1978. [25] A.G. Cook, U. Baumeister, C. Tschierske, J. Mater. Chem. 5 (2005) 1708. [26] E.B. Barmatov, A. Bobrovsky, M.V. Barmatova, V.P. Shibaev, Liq. Cryst. 26 (1999) 581. [27] P.G. de Gennes, The Physics of Liquid Crystals, Oxford Press, London, 1974. [28] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, New York, 1977. [29] H.E. Stanley, Introduction to Phase Transition and Critical Phenomena, Clarendon Press, New York, 1971. [30] R. Karapinar, Turk. J. Phys. 24 (2000) 115.