The Significant Structure Theory Applied to a Mesophase System

Proc. Nat. Acad. Sci. USA Vol. 72, No. 1, pp. 78-82, January 1975

The Significant Structure Theory Applied to a Mesophase System (compressibility/coefficient of expansion/specific heat/cooperative phenomenon/liquid crystal) SHAO-MU MA AND H. EYRING Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Contributed by Henry Eyring, August 19, 1974

ABSTRACT The significant structure theory of liquids melting point to the boiling point of the material under in- is extended to the mesophase system with p-azoxyanisole vestigation. as an example. This compound has two different structures, a nematic phase and an isotropic phase, in its liquid state. (2) It gives the volume dependency as well as the tempera- In this study the nematic phase is treated as subject to a ture dependency of the Helmholtz free energy and therefore second volume and temperature-dependent degeneracy of the thermodynamic properties of liquids that depend on formally like that due to melting. The isotropic phase is both volume and temperature. treated as a normal liquid. The specific heat, thermal expansion coefficient, compressibility, volume, entropy of transitions, and heat of transitions are calculated and compared to the observed values. This analysis differs from THEORY previous ones in including the volume dependence as well as the temperature dependence in one explicit expression According to the significant structure theory, there are three for the Helmholtz free energy. important contributions to liquid structure. (1) Molecules with solid-like degrees of freedom. The significant structure theory, which was first proposed by (2) Positional degeneracy in the solid-like structure. Eyring et al. (1), has been successfully applied to many (3) Fluidized vacancies which confer gas-like degrees of different types of liquids, such as normal liquids, molten met- freedom on molecules. The partition function of a normal als, fused salts, liquid hydrogen, water and heavy water, liquid liquid is formed as the product of the partition functions for mixtures, two-dimensional liquids and solid adsorbants, each of these three significant structures. plastic crystals, hydrocarbons and halobenzenes (2-4), and In the formulation of partition functions to characterize a high polymers (5). In this study this theory is extended to liquid crystal like p-azoxyanisole, additional factors must be systems in which the same molecules are arranged in more considered. than one structure. The liquid crystal p-azoxyanisole was (1) Since the structure in the nematic phase differs from chosen for theoretical investigation in the present paper (Fig. that in the isotropic phase, a partition function is used for the 1). nematic phase and a second one for the isotropic phase. X-ray data indicate that the stable crystalline form of (2) A function that takes account of the temperature effect p-azoxyanisole belongs to space group P12/a (c2h5) with four on the molecular rotation is introduced into the partition molecules per unit cell (6). This compound is a thermotropic function for the nematic phase. liquid crystal. It changes from a solid structure to a turbid (3) Since this compound has two transition points, the solid liquid (nematic phase) at its melting point and is converted to nematic transition point (the melting point) and the into a true liquid (isotropic phase) at a higher temperature- nematic to isotropic transition point (the clearing point), the clearing point. For liquid crystal compounds there is a rapid two volume-temperature-dependent degeneracy terms are change in the density, the specific heat, the thermal expan- required in the partition function for the nematic phase. On sion coefficient, the compressibility, the viscosity, the dielec- the other hand, the isotropic phase can be treated as a normal tric constant, and a number of other physical properties near liquid with one such degeneracy term. their mesophase transitions. Using the model described above, the partition function for Theoretical interpretations of the thermodynamic proper- one mole of p-azoxyanisole in the nematic phase can be written ties of p-azoxyanisole have been made by several investiga- as: tors. Most of them (7-10) applied the Frenkel heterophase fluctuation theory (11) within 100 of the nematic-isotropic transition. A general theory of phase transition described by a fnematic = e(1-e/T)6 [1 + n(x - single order parameter is proposed by Alben (12). Chandrasek- har et al. discussed the thermodynamic conditions of nematic AN(VslV) stability on the basis of the molecular statistical theory of [1 + n2(x2 - 1)e-a2Es/RT(X21-)]f ib} orientational order (13). The approach of significant structure theory from these approaches in at least two important { (27rmkT)'/2 eV 8r2(87r3ABC) l/2(kT)/112 N((V- Vs)/V) differs N ¢h3 aspects: h3 Jv~~ib (1) Its application can be extended over a range from the + b2(T )b3) [1 Abbreviations: calc., calculated; obs., observed. 78 Downloaded by guest on September 28, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Liquid Crystalline State 79

TABLE 1. The vibrational frequencies (between 100 and 1000 cm-') and moments of inertia used for p-azoxyanisole Vibrational frequencies (cm-') (18) 210 417 611 797 234 474 629 832 317 494 670 848 360 536 725 911 FIG. 1. Chemical structure for p-azoxyanisole. Moments of inertia (g-cm2) (19) A = 7.50 X 10-37 A. Rotational Degrees of Freedom. The three restricted B = 7.06 X 10-31 rotational degrees of freedom, which are three oscillations, are C = 4.71 X 10-38 corrected by a factor (1 + b2((T - Tm)/Tm)b3). This is a correction due to a decrease in the three 0's brought on by a and that for the isotropic phase is: decrease in the barriers to rotation as neighbors oscillate more eE81R T violently with rising temperatures. As discussed by Deloche fisotropic = et al. (14), the interactions between molecules of different e-/T)6 orientations contain two contributions: the van der Waals lN(VJ/1V) attractions, which are expected to be roughly proportional to [1 + n'(x' - 1)e aE,/RT(x'-l)If ib} V/ 1/V2 (10, 15), and the steric repulsions, which have been J(2irmkT) 3/2 eV 8jr2(8i8ABQ '/2(T) 3/2 N(( V- Vs ') V) considered by Onsager (16) to be a linear function of T for the bJ very special case of a hard rod. In our theory, the first effect is hs N Ach3 f volume-dependent and is included in the degeneracy terms, and [2] the second contribution is approximated by this temperature- V V V where X = V X' = -,X2 = V dependent correction of the vibrational barrier. The loosening Vs Vs Vs of 0 with temperature in the nematic phase enables the molecules to gain a greater freedom to rotate and consequently The different symbols are defined as follows: Ea, 0, V,, gives a large temperature increase in the specific heat. V,2, are the energy of sublimation, the Einstein characteristic temperature for the nematic phase, the solid molar volume at B. Vibrational Partition Function. The vibrational partition the melting point and the "effective solid molar volume" for function of a compound usually takes the form the V is the molar volume of the liquid crystal; system; n, a, 1 n2, a2, b2, and bN are dimensionless parameters used to fit the fvib = T 1 - properties in the nematic phase; 0' and Vs' are the Einstein characteristic temperature for the isotropic phase and the where vi are the internal vibrational frequencies. solid molar volume at the clearing point; n' is the dimension- For p-azoxyanisole there are 33 atoms per molecule and less parameter used to fit the properties in the isotropic phase, hence 3 X 33 - 6 = 93 normal modes of vibration. In Raman respectively. N, Tm, T. m, R k, h, a are Avogadro's number, spectra some 50 lines were observed by various investigators the absolute temperature, the melting point, the mass of the (17, 18). However, not all of these lines are normal modes and molecule, the gas constant, the Boltzmann constant, Planck's to assign these lines is not possible with current techniques. constant, and the symmetry number (= 1), respectively. Because of this limitation, we chose an approximate method as A, B, and C are the principal moments of inertia of the iso- follows: lated p-azoxyanisole molecule. (1) For frequencies between 0 and 100 cm-': A density-of- The vibrational frequencies, between 100 and 1000 cm-', state function, g(v), was obtained from the intensity-frequency and moments of inertia used for p-azoxyanisole are given in curve of Raman spectra (19). I = 2c(1/v) [1 + n(v)]g(v) or Table 1. The numerical constants for this compound, which g(v) = c'Iv/ [1 + n(v) ] where n(v) = [ehv/kT 1 ]-1 and c and were given in literature, are listed in Table 2. c' are constants. The vibrational partition functions for TABLE 2. The numerical constants used for p-azoxyanisole frequencies in this range can then be calculated using graph- ical integration. Molecular weight 258.3 '/c = 100 Solid-nematic transition point (Tm) 118.20 (27), 117.6° (28) (fvib)0-100 cm' = J g(v)eh/kT dv Nematic-isotropic transition point (T,) 135.30 (27), 133.90 (28) Volume parameters (ml/mol) (21) Since the Raman spectrum for the nematic phase correspond- Tm Te ing to frequencies between 0 and 100 cm-' is not the same as V, solid 199.35 V, nematic 255.04 that for the isotropic phase, the vibrational partition functions V, nematic 221.34 V, isotropic 225.84 for these two phases are also different and treated accord- AV = 11.03% AVc = 0.36' % ingly. Heat of transition (cal/mol) This is essentially the method used by Shuker and Gammon AHm = 7067 (22), 8610 (14) (20) for calculating the dielectric correlation function in tH, = 137.2 (22), 413 (29) glasses. Specific heat at constant volume (cal/g') (2) For frequencies between 100 and 1000 cm-': The 16 (a) C, isotropic phase 0.47 (27, 30), 0.488 (7) (b) C, nematic phase 0. 46 (27, 30), 0.525 (7) observed lines in the Raman spectra (18) in this frequency range are assumed to be normal modes. Downloaded by guest on September 28, 2021 80 Chemistry: Ma and Eyring Proc. Nat. Acad. Sci. USA 72 (1976)

TABLE 3. Parameters for p-azoxyanisole using melting point properties. We have three conditions to satisfy. They are: Nematic phase Isotropic phase (1) The double tangent for solid and nematic phase must n = 85.8 n2 = 5 n' = 86.3 correspond to 1 atmosphere. a = 5.34 X 10-3 a2 = 1.92 X 10-3 a =5.34 X 10-3 (2) The correct volume for the nematic phase must be V, = 197 V.2 = 221.0 Vs' = 200.7 obtained. E. = 27500 b2 = 7.5 E= 27500 (3) The calculated AS at the melting point must be close to 6 = 67 b3 = 3 0' = 54 the observed value. Ovib = 0.9 Tm O Vib =0.9 T, Parameters a2, V82 and n2 were adjusted to give the best b = 25 b= 25 value of the thermodynamic properties V, a and j# from the melting point (Tm) to the nematic-isotropic transition point 1000 1 or clearing point (T,). Since the correction term for the rota- (fvib)100-100cm0 = /ctloo 1 - ehP/kT tional degrees of freedom is volume-independent, it does not affect the calculated values of a and j#. (3) For frequencies larger than 1000 cm-': The contribu- In determining parameters for fisotropic we take Es and a to tion to the vibrational partition function for normal modes be the same in both the nematic phase and the isotropic phase. with frequency higher than 1000 cm-' is very small. The par- VS' is taken as (V, + volume change between the two transi- tition function for such degrees of freedom is, therefore, taken tions) by considering the isotropic phase as a normal liquid as unity. with a single melting point T,. 0' and n' were then adjusted (4) The missing normal modes: A term of the form (T/ to give 1 atmosphere pressure at a given volume and tempera- Ovib)b is used to take account of the missing normal modes. ture. Here Ovib = hv/k and b is the equivalent number of harmonic 0vib and 0'vib were chosen as 0.9 Tm and 0.9 T,, respec- oscillators. tively. The available data do not suffice to determine them The total vibrational partition function is, therefore more accurately. b and b' were adjusted to give the best c, values. fvib = (fvib)0-100 cm-' X (fvib)100-1000 cm' X (T/Ovib)' The calculated V,, E,, 0, 0' and b (or b') are further dis- C. Parameter Determination (Table 3). It was found that the cussed in the following: ratio n/V, was nearly a constant for the inert gases, methane, (1) Vs: and nitrogen (average = 0.473). The parameter n for p- X-ray data (6) indicate that the unit cell of p-azoxyanisole azoxyanisole can be calculated (4) from the experimental V8 at at room temperature has the dimensions a = 15.776, b = the melting point (21) and the n/V8 ratio of argon. n = n- 8.112, c = 11.018, and, = 114.570. The V, calculated (argon) X V8(compound)/V8(argon) = 0.432(V, compound) from these data is 192.49 ml. On the other hand, the V, = 86.2. The calculated V8 is slightly different from the V8 at at the melting point measured by dilatometric method was the melting point; fine adjustment gives a value of 85.8 for n. reported (21) as 199.35 ml. A comparison of V, with these two Parameters V8, 0, Es, and a for fnematic were determined by values indicates our calculated V, (= 197 ml) is about right. TABLE 4. Molar volumes and heat capacity of p-azoxyanisole V (ml/mol) C. (cal/mole 0C) C,, (cal/mole 0C) T ('C) CalC. Obs. (10) Error % CalC. Obs. (7) Error % CalC. Obs. (27) Error % Nematic phase 117.6 (Tm) 221.34 221.34 0 106.69 119.65 120 221.67 221.74 -0.03 108.50 120.90 120.9 +0.00 122 221.94 222.1 -0.072 110.00 122.04 122.0 +0.03 124 222.2 222.46 -0.117 111.60 123.30 123.0 +0.24 126 222.47 222.83 -0.166 113.20 124.95 124.3 +0.52 128 222.73 223.21 -0.215 114.80 127.10 126.4 +0.55 130 223.03 223.62 -0.264 116.58 130.00 128.81 +0.92 132 223.36 224.06 -0.312 118.32 134.4 133.4 +0.75 133 223.56 224.30 -0.330 119.21 137.80 136.9 +0.66 134 223.81 224.53 -0.321 120.01 142.67 144.9 - 1.53 135 (T,) 224.62 225.04* -0.187 121.00 135.60 -10.76 321.90 - Isotropic phase 135 (T,) 225.66 225.86* -0.089 107.71 126.05 -14.55 118.31 - 136 225.78 225.83 -0.022 107.75 118.20 - - 137 225.90 226.03 -0.058 107.79 118.11 122.85 -3.86 138 226.01 226.21 -0.088 107.84 118.02 122.28 -3.48 140 226.25 226.57 -0.141 107.92 117.87 121.71 -3.16 142 226.48 226.91 -0.189 108.00 117.74 121.43 -3.04 144 226.70 227.26 -0.246 108.10 117.64

* From ref. 21. Downloaded by guest on September 28, 2021 Proc. Nat. Acad. Sci. USA 72 (1975) Liquid Crystalline State 81 TABLE 5. Thermal expansion coefficient and compressibility of p-azoxyanisole

a X 104 (degree-') # X 105 (atmosphere-) T (0C) Calc. Obs. (10) Error % Calc. Obs. (31) Error % Nematic phase 117.6 (Tm) 6.33 7.6 -16.7 6.48 - - 120 6.14 7.8 -21.3 6.40 6.0 +6.67 122 6.00 6.35 6.15 +3.25 124 5.91 8.35 -29.2 6.34 6.35 -0.16 126 5.95 - 6.47 6.6 -1.97 128 6.18 8.7 -28.9 6.81 6.95 -2.01 130 6.85 9.2 -25.5 7.62 7.4 +2.97 132 8.22 9.8 -16.1 9.21 7.8 +18.08 133 9.49 11 +13.7 10.65 8.1 +31.48 134 11.49 12.5 +8.08 12.91 8.8 +46.70 135 (T,) 100.3 - - 111.0 - - Isotropic phase 135 (T,) 5.36 9.3 -42.4 6.03 7.5 -19.6 136 5.31 8.5 -37.5 6.01 7.15 -15.9 137 5.25 8.0 -34.3 5.99 7.0 -14.4 138 5.21 7.8 -33.2 5.98 6.9 -13.3 140 5.12 7.6 -32.6 5.96 6.9 -13.6 142 5.04 7.5 -32.8 5.96 7.0 -14.9 144 4.98 - 5.94 - -

The x-ray data for the solid give a volume for the solid which D. Thermodynamic Properties. The data presented for should only agree with the solid-like structure in the liquid if V, CP,CO a, and # in Table 4 and Table 5, are all calculated there is no substantial volume change in V8 on melting. For at p = 1 atmosphere. The melting point and clearing point example, the ice -- water transition exemplifies a shrinkage of properties of this compound are given in Table 6. the solid-like structure. Our calculated Ct and Cp agree quite well with the observed (2) E,: values except at the nematic-isotropic transition point. The Although we have no direct experimental value for E, the calculated a and jB are somewhat less satisfactory. There is a calculated value can be justified by estimating E, as follows: 46.7% error in # at 1340 (10 below To) and about 29% error in (i) Es -- (AH) fusion + (AH) vaporization (AH) fusion for a between 124 and 1280. this compound = 7067 to 8610 cal/mol (14, 22). (AH) This discrepancy at T, is partly due to the method used in vaporization can be roughly calculated by using Trouton's fixing the parameters. The melting point technique used in rule. Since a compound this study is usually less accurate than a triple-point technique (24) or the melting point-boiling point technique (25). CHO 0 N=NS{) However, it is used because the triple point properties and the boiling point properties are not available for p-azoxyanisole. has a boiling point of 3400 (23), the boiling point of The calculated a and # are less accurate for the isotropic 0 phase than for the nematic phase. Our explanation is as CHO- 0N~N--- OH3 follows: should be (1) The changes in volume and entropy at T, are small. considerably higher. Therefore, for the latter (AlH) Thus the parameters based on the properties at this transition vaporization >22 X (340 + 273) - 13500 cal/mol and E, > 20,000 cal/mol. are not very accurately fixed. (ii) for benzene (2) Since the substance changes its structure at Tc, Es and E, is about 11,000 cal/mol and E, for p- a may not be the same in the nematic and the azoxyanisole must be considerably more than twice 11,000 isotropic phases cal/mol. as we have assumed. The results could be improved by ad- b justing these parameters. (3) (or b'): Other The value of b (or b') should be interpreted as the equivalent improvements could be made by finding a better and not the actual number of harmonic oscillators, since inter- expression for the vibrational partition functions. However, nal barriers to rotation and rotation of the molecule with this treatment goes far beyond anything previously attempted respect to neighbors may involve partition functions depend- TABLE 6. M3elting point and clearing point properties of ing on the temperature to higher or lower powers of T than the p-azoxyanisole first. (4) 0, 0': AS,,, = 16.9 (obs. 18.1) (32) 0' for the isotropic phase is lower than 0 for the nematic AHm = 6603 cal/mol [obs. 7067 (22)-8610 (14) cal/mol] phase because the molecules in the isotropic phase are expected ASC = 0.97 (obs. 0.336) (32) to have more rotational freedom than those in the nematic AHc = 396 cal/mol [obs. 137.2 (22)-413 (29) cal/mol] phase. AVc = 1.04 ml (obs. 0.82 ml) (21) Downloaded by guest on September 28, 2021 82 Chemistry: Ma and Eyring Proc. Nat. Acad. Sci. USA 72 (1975)

in calculating the volume dependence of the Helmholtz free 12. AlbenLwR. (1970) AMol. Cryst. Liquid Cryst. 10, 21-29. 13. S. & energy A (V,T), for a liquid crystal as well as the coefficients Chan4asekhar, Madhusudana, N. V. (1972) Liquid Crysta 3, eds. Brown, G. H. & Labes, M. M. (Gordon and of expansion and compressibility and cp. The viscosity of p-. J Breach Science Publishers, London, New York, Paris), azoxyanisole (26) and other nematic liquids shows practically part 1, pp. 251-261. normal viscous behavior. This tells us that there is nothing 14. Deloche, B., Cabane, B. & Jerome, D. (1971) Mol. Cryst. unusual about the translational degrees of freedom in the Li ,uih Cryst. 15, 197-209. 15. 4fier, W. & Saupe, A. (1959) Z. Naturforsch. A 14, 882-889. mesomorphic state. The large contribution that increase in 16. Cnsager, L. (1949) Ann. N.Y. Acad. Sci. 51, 627-659. volume makes to the degeneracy and, therefore, to the en- 17. Zhdanova, A. S., Morozova, L. F., Peregudov, G. V. & tropy in both the nematic and the isotropic states is largely Sushchinskii, M. M. (1969) Opt. Spectrosc. 24, 112-114. due to the easing of rotational restrictions. We have for this 18. Amer, N. WI. & Shen, Y. R. (1972) J. Chem. Phys. 56, 2654- reason not included a treatment of viscosity as it would follow 2664. 19. Bulkin, B. J. & Prochaska, F. T. (1971) J. Chem. Phys. 54, our standard procedures (2). Chow and Martire (33) pointed 635-639. out that there exists a metastable second solid phase, which 20. Shuker, R. & Gammon, R. W. (1970) Phys. Rev. Lett. 25, is not treated in this study because of insufficient relevant 222-225. data. 21. McLaughlin, E., Shakespeare, M. A. & Ubbelohde, A. R. (1964) Trans. Faraday Soc. 60, 25-32. The authors wish to thank the National Institutes of Health, 22. Chistyakov, 1. G. (1967) Sov. Phys. Usp. 9, 551-573. Grant GP 12862, National Science Foundation, Grant GP 28631, 23. Weast, R. C., ed. (1970-1971) Handbook of Chemistry and and the Army Research-Durham, Contract DA-ARO-D-31-124- Physics (Chemical Rubber Co., Cleveland), pp. C-129, 72-G15, for support of this work. 51st ed. 24. Chang, S. & Park, H. S. (1963) J. Korean Chem. Soc. 7, 1. Eyring, H., Ree, T. & Hirai, N. (1958) Proc. Nat. Acad. Sci. 174-178. USA 44, 683-688. 25. McLaughlin, D. R. (1965) Phi1). Dissertation, University 2. Eyring, H. & Jhon, M. S. (1969) Significant Liquid Struc- of Utah. tures (John Wiley and Sons, Inc., New York). 26. Porter, R. S. & Johnson, J. F. (1963) J. Appl. Phys. 34, 3. Wu, Shuh-Wei (1972) Ph.D. Dissertation, University of 51-54. Utah. 27. Arnold, H. (1964) Z. Phys. Chem. 226, 146-156. 4. Faerber, G. L. (1971) Ph.D. Dissertation, University of 28. Barrall, E. M., Porter, R. S. & Johnson, J. F. (1964) J. Phys. Utah. Chem. 68, 2810-2814. 5. Ma, Shao-mu, Eyring, H. & Jhon, Mu Shik (1974) Proc. 29. Kreutzer, C. & Kast, W. (1937) Naturwissenschaften 25, Nat. Acad. Sci. USA 71, 3096-3100. 233-234. 6. Krigbaum, W. R., Chatani, Y. & Barber, P. G. (1970) Acta Crystallogr. Sect. B 26, 97-102. 30. Barrall, E. M., Porter, R. S. & Johnson, J. F. (1967) J. Phys. 7. Hoyer, W. A. & Nolle, A. W. (1956) J. Chem. Phys. 24, Chem. 71, 895-900. 803-811. 31. Kapustin, A. P. & Bykova, W. T. (1966) Sov. Phys. Crystal- 8. Arnold, H. (1964) Z. Chem. 4, 211-216. logr. 11, 297-298. 9.. Torgalkar, A. & Porter, R. S. (1968) J. Chem. Phys. 48, 32. Johnson, J. F., Porter, R. S. & Barrall, E. M., II (1969) 3897-3901. Liquid Crystals 2, ed. Brown, G. H. (Gordon and Breach 10. Maier, W. & Saupe, A. (1960) Z. Naturforsch. A, 15, 287- Science Publishers, London, New York, Paris), part 1, pp. 292. 169-175. 11. Frenkel, J. (1956) in Kinetic Theory of Liquids (Dover 33. Chow, L. C. & Martire, D. E. (1969) J. Phys. Chem. 73, Publications, Inc., New York), pp. 382-390. 1127-1132. Downloaded by guest on September 28, 2021