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.

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