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liquid crystal systems: Applicability of continuous thermodynamics
Fluid Phase Equilibria 245 (2006) 102–108
Nematic-isotropic phase behaviours of polydisperse polymer/liquid crystal systems: Applicability of continuous thermodynamics Jung Jin Choi, Young Chan Bae ∗ Division of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Korea Received 1 January 2006; received in revised form 29 March 2006; accepted 10 April 2006
Abstract Phase behaviours of polydisperse polystyrene (PS)/nematic liquid crystals (LCs) P-ethoxy-benzylidene-p-n-butylaniline (EBBA) are investigated by thermo-optical analysis (TOA) technique. In this study, to describe nematic-isotropic transition of the polydisperse PS/EBBA systems, a continuous distribution function to obtain the composition of polydisperse polymers is considered precisely. We apply these molar mass distributions to the extended Flory–Huggins model. The proposed semi-empirical model gives a remarkable agreement with experimental data for the model systems. © 2006 Elsevier B.V. All rights reserved. Keywords: Continuous thermodynamics; Nematic-isotropic transition; Polydisperse polymer; Molar mass distribution function
1. Introduction Mixtures of polymers and nematic liquid crystals (LCs) are subjects of intensive investigation in many laboratories around the world. The interest is motivated by their potential use in many fields of high technology involving electronic equipment, display systems and commutable windows, etc. [1,2]. Many statistical theories of the nematic phase have been presented. The best known theories among them are those suggested by Onsager [3], Maier and Saupe [4,5] and Flory [6] Chandrasekhar [7]. The Onsager treatment has proven successful in the description of rigid molecules of high axial ratio which display lyotropic behavior while the Maier–Saupe treatment has found an application to lower axial ratio molecules displaying thermotropic behaviour [8]. Maier and Saupe [4,5] postulated that the orientation-dependent interaction between nematogenic molecules arises from the anisotropy of their dispersion interaction. They sought to account for the stability and properties of nematic phase solely on the basis of the anisotropy of the forces of attraction, disregarding the steric or space filling, and characteristics of the asymmetric molecules comprising typical nematogenic substances.
∗
Corresponding author. Tel.: +82 2 2298 0529; fax: +82 2 2296 0568. E-mail address: [email protected] (Y.C. Bae).
0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.04.019
Flory and Ronca [9,10] extended the original Flory lattice model to include anisotropic molecular forces. They have created a theoretical frame work which was particularly well suited for the description of mixture containing nematogen [9,11,12]. They combined an orientation-dependent energy with a partition function derived for a system of “hard” rodlike molecules in which the intermolecular energy is the same for all configurations that are devoid of overlaps. Their treatment of a mixture consisting of rods, coil and a solvent meets with gratifying success in a comparison with experimental data. The Flory–Ronca treatment was extended by Ballauff [13] to binary mixtures of a thermotropic nematogen and a coiled species consisting of isodiametric segments linked together flexibly. The free volume which is known to be of utmost importance in describing thermotropic system is introduced by the methods devised previously [11,12]. In their work, the comparison of the model with experimental results was made only for polymers of relatively low molecular weight. Dorgan and Soane [14] reported nematic liquid crystal/wider range of the polymer molecular weight systems. Furthermore, Riccardi et al. [15] extended the Flory–Huggins model to liquid crystal and polydisperse polymer systems. However, although the importance of the compositon-dependence of the Flory–Huggins parameter, χ, has been noted, no further details were presented for phase-equilibrium calculations.
J.J. Choi, Y.C. Bae / Fluid Phase Equilibria 245 (2006) 102–108
In this study, we propose a semi-empirical molecularthermodynamic framework for correlating phase equilibria in polydisperse polymer/nematic liquid crystal systems. Furthermore, we modify χ to take into account both polymer concentration and temperature and employ a continuous thermodynamics [16–18] characterized by a continuous molar mass distribution. We also quantitatively investigate the nematic-isotropic phase transition behaviors of polydisperse polystyrene (PS)/nematic liquid crystal systems and compare our proposed model with experimental results. 2. Experimental 2.1. Materials 2.1.1. Liquid crystals Liquid crystals used in this study were P-ethoxy-benzylidenep-n-butylaniline (EBBA) obtained from Sigma–Aldrich Co. Nematic-isotropic transition temperatures measured by thermooptical analysis (TOA) [19] of EBBA was 76.3 ◦ C.
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or cooling (3 ◦ C/min) was used. In this way, the approximate transition point of the sample was found. The sample was then repeatedly heated and cooled over a temperature range near the approximate transition temperature of the sample while the light intensity was monitored. If the measured transition points upon heating and cooling agreed with each other, the point was taken as the equilibrium transition temperature. If these temperatures were different, slower heating and cooling rates (0.3 ◦ C/min) were employed. In the case of a slight discrepancy (less than 1 ◦ C), the transition temperature upon cooling, corresponding to the traditional cloud point, has been taken as the equilibrium temperature. Table 1 shows the transition temperatures upon heating and cooling for various EBBA concentrations at the rate of 0.3 ◦ C/min. As shown in table, the measured transition points upon heating and cooling agreed with each other at less than 1 ◦ C, and the reproducibility is very reasonable. That is, the repeatedly nematic-isotropic transition temperatures of EBBA were correspondent. 2.4. Theoretical considerations
2.1.2. Polymers Polystryene samples were obtained from Aldrich Chemi¯ w ) and cal company. The weight-average molecular weight (M ¯ number-average molecular weight (Mn ) were determined by gel permeation chromatography (GPC), which was operated under condition of PS as a standard sample and tetrahydrofuran as the mobil phase. Their molecular weights were listed in Table 3. 2.2. Sample preparation Samples were prepared by the solvent casting method with various compositions: the materials were weighed-out into clean sample vials. Polymer/EBBA samples were dissolved in chloroform and stirred at 50 ◦ C for 15 h until samples became homogeneous. The samples were then dried in vacuum oven at 60 ◦ C for 30 h. 2.3. Experimental procedure A sample was placed in the microscope heating stage. To find the range of transition points of the sample, a scan rate of heating
We consider a system consisting of a liquid crystal and a polymer. Diameters of the components are assumed to be equal to the size of the lattice cell. The axial ratio of the liquid crystal is xl and the contour length of the ith component of the polymer is xi . The combinatorial analysis along the lines given by Flory [8] may be readily adapted for the system under consideration. As usually [10–12], the mixing partition function ZM is subdivided into three contributions: a liquid crystal, a polymer, and an orientation. ZM = ZLC Zorient Zpolymer
(1)
ZLC and Zpolymer are partition functions representing the method of packing the lattice with liquid crystal and polymer, respectively. First, lattice sites are occupied by polymer and then the liquid crystal is packed. The quantity y below specifies the disorientation of the liquid crystals. For a molecule with the long axis at an angle ψ with respect to the domain axis [10]: y=
4 xl sin ψ π
(2)
Table 1 Transition temperature upon heating and cooling for various EBBA concentrations ¯ w = 137, 800 of PS in g/mol M
¯ w = 198, 900 of PS in g/mol M
Weight fraction of EBBA
Transition temperature upon heating (K)
Transition temperature upon cooling (K)
Weight fraction of EBBA
Transition temperature upon heating (K)
Transition temperature upon cooling (K)
1 0.95 0.89 0.84 0.79 0.75 0.70 0.64 0.60
349.25 350.05 350.15 348.45 346.45 341.15 343.25 337.25 331.95
349.45 350.05 349.75 348.85 346.15 340.45 342.95 336.45 331.05
1 0.95 0.9 0.85 0.80 0.75 0.70 0.65 0.60
349.25 349.83 349.65 348.96 348.06 347.15 346.57 340.01 337.95
349.45 350.22 350.16 349.13 348.90 347.98 347.20 341.67 338.75
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with the orientational distribution function for liquid crystal being given by the ratio nly /nl , i.e. the number of molecules with disorientation y, nly , to the total number nl . The mean value y¯ follows as nly y¯ = y (3) nl y ZLC and Zpolymer are given by Zpolymer = ZLC =
n 0 − n l xl −
(n0 − nl xl ) ! [n (x −1)] ni xi ! ni !n0 i i
(n0 − nl (xl − y¯ ))! (n0 − nl xl )!nl !nn0 l (¯y−1)
Eq. (13) does not consider an isotropic interaction between the liquid crystal and the polymer. The reduced free energy is extended with isotropic energetics [14] as follows:
(5)
HM = nl xl φp χ = nl xl φp D(T )B(φp )
(7)
where ωy is a priori probability of the disorientation y for a liquid crystal [9], εy is the orientation-dependent interaction energy and kB is a Boltzmann constant. The inclusion of anisotropic molecular dispersion forces is accomplished by introducing the mean energy of a rodlike segment expressed as [8,9] 3 εy = −kB T ∗ φl s 1 − sin2 ψ (8) 2 where the characteristic temperature T* is the strength of the orientation-dependent forces for a given nematic liquid crystal [9]. The order parameter s is expressed as [13] 2 2
sin ψ (9) s=1− 3 The total number of lattice sites n0 is n0 = nl xl + n i xi φl = φi =
(13)
(4)
where n0 and ni are the total number of lattice sites and the number of the ith component of the polymer, respectively. Eq. (4) expresses the expected number of configurations for the polymer in the empty lattice. Eq. (5) takes into account the configurations accessible to the nl liquid crystals which have been added to the lattice subsequently. The orientational partition function Zorient is expressed as [10,12]: ω y nl −xl εy nly Zorient =Π (6) exp y nly 2kB T ωy = sin ψ
is obtained [20] y¯ n l xl A =− ni xi + nl y¯ ln 1 − 1− kT n0 xl nl ni + nl ln + (ni (xi − 1)) + ni ln n0 n0 nly nly n l xl 2 + nl (¯y − 1) + nl ln − s n n ω 2θ l l y y
(10)
n l xl nl xl + ni xi
(11)
ni xi nl xl + n i xi
(12)
where φl and φi are the volume fraction of the liquid crystal and the ith component of the polydisperse polymer, respectively. The total volume fraction φp of the polydisperse polymer is φp = φ i . Combining Eqs. (3), (4), (11), and (12) and employing Stirling’s approximation for the factorials, the reduced free energy
(14)
where the interaction parameter χ is given by the product of a temperature-dependent term, D(T), and a concentrationdependent term, B(φp ) [21–23]. χ(T, φp ) = D(T )B(φp ) D(T ) = d0 + B(φp ) =
d1 T
1 1 − bφp
(15) (16) (17)
where, d0, d1 and b are adjustable model parameters. Combining Eqs. (13) and (14), the reduced free energy is obtained A y¯ φl φl = −φp ln 1 − φl 1 − + ln n0 kT xl xl xl φi φi φi φl + + φp − + (¯y − 1) ln xi xi xi xl φ2 s φl s + ln f1−1 − l 1− + φ l φp χ (18) xl θ 2 where the reduced temperature θ, the order parameter s and disorientation index y¯ and defined as follow: θ=
T T∗
(19)
f2 f1 3 f3 s=1− 2 f1 y¯ =
4 xl π
where
fp = 0
π/2
(20) (21)
4 3 xl φl s sinp ψ exp − xl a sin ψ− sin2 ψ dψ π 2 θ (22)
2.5. Distribution function Compositions of polydisperse polymers are usually expressed by a continuous distribution function W(I), where I is a distribution variable such as a length r or a relative molar
J.J. Choi, Y.C. Bae / Fluid Phase Equilibria 245 (2006) 102–108
mass M. In this study, we employ the log-normal molar mass distribution function. The log-normal distribution function [24] is A I 2 1 W(I) = √ exp − 2 ln (23) γ I0 γ πI where γ, I0 , and A are adjustable model parameters. The lognormal distribution has a maximum value: 2 −γ IMAX = I0 exp (24) 4 From Eq. (24), we determine A, which represents the experimental peak height, γ and I0 are defined (25) γ = 2 ln(σ + 1) ¯ + 1)1/2 I0 = I(σ
(26)
where σ = µ2 /I¯ 2 . µ2 and I¯ are the variance and the average molecular weight of a given sample, respectively. Distribution function W(I) satisfies the following normalization constraint:
∞
∞ W(I) dI = 1, Φp W(I) dI = Φp (27) 0
0
The discrete multicomponent framework is transferred into the continuous thermodynamic framework as follows: A y¯ φl φl + ln = −φp ln 1 − φl 1 − n0 kT xl xl xl
∞ φp W(I) φp W(I) + ln dI + φp x(I) x(I) 0
∞ φp W(I) φl φl − dI + (¯y − 1) + ln f1−1 x(I) x xl l 0 φ2 s s − l (28) 1− + φ l φp χ θ 2 2.6. Equilibrium conditions For calculating the isotropic-nematic transition curve, we need chemical potentials for the liquid crystal and the polymer. Expressions for chemical potentials are derived directly from Eq. (28). µl ∂(A/kT ) = kT nl φ p − x l φp y¯ − xl φl + ln + + ln f1−1 xl φp + y¯ φl xl xn + y¯ − 1 xl s s − 1− φl (1 + φp ) + (1 − b)xl φp2 D(T )B(φp )2 θ 2 (29)
= −xl φp2
∂(A/kT ) µ(I) = kT ni y¯ − xl y¯ = −x(I) ln φp + φl + x(I)φp φl xl xl φp + y¯ φl
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x(I)φp φp W(I) x(I) φl − + ln + x(I) xl xn x(I) x(I)sφl2 s + 1− + x(I)φl2 D(t)B(φp )2 θ 2 ∞ where xn = Φp / 0 Φp W(I)x(I)−1 dI The equilibrium conditions are given by
(30)
µil = µnl
(31)
µ(I)i = µ(I)n
(32)
−
where superscripts i and n denote two phases at equilibrium. 2.7. Numerical calculations Biphasic equilibrium between the nematic and the isotropic phase also requires µ(I)i = µ(I)n for all components present in the system. The number of Eq. (32) becomes excessively large and the following conservation of mass is introduced: VΦi = V i Φii + V n Φni
(33)
where Vi and Vn are the volumes of the phase poor and rich in liquid crystal, respectively, and V is the volume of the original mixture. Thus, the isotropic-nematic transition curve is obtained from solving Eqs. (31)–(33), simultaneously. In this procedure, if the average molar mass of the polydisperse polymer is high, the PS is practically excluded from the liquid crystal rich phase [14,15]. Calculations are, therefore, accordingly performed by setting the composition of the ordered phase to unity and by solving one remaining equation for the isotropic phase composition. 3. Results and discussion Continuous thermodynamics gives us a new impetus not only because of its theoretical integrity and its consistent framework for multicomponent systems but also because it requires a comparatively simple mathematical procedure in comparison with the tedious determinant derivations in the usual discrete approach when calculating the spinodal and the critical-point criteria. When the continuous thermodynamic framework is applied to phase-equilibrium calculations for polydisperse polymer and liquid crystals, a molar mass distribution for describing the composition of polydisperse polymers is considered precisely. In this study, we use three adjustable model parameters (d0 , ¯ w /M ¯ n = 1) d1 and b) for monodisperse PS/EBBA system (M and then calculate the isotropic-nematic transition curve. These values are listed in Table 2. Fig. 1 shows a calculated nematic-isotropic transi¯ w = 22, 000, M ¯ w /M ¯n< tion curve for monodisperse PS(M 1.05)/EBBA system. The solid line represents a nematicisotropic transition curve. Open circles are experimental data [14]. The calculated transition curve provides a good fit to the experimental data. Fig. 2 shows a calculated nematic-isotropic transi¯ w = 113, 000, M ¯ w /M ¯n< tion curve for monodisperse PS(M
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Table 2 Adjustable model parameters for monodisperse PS/EBBA systems Parameters
b d0 d1
¯ w of PS in g/mol M
Table 3 Model parameters for molar mass distribution function Parameters
22,000
113,000
−0.607 −0.049 49.147
0.221 0.488 −117.068
γ I0 A
¯ w of PS in g/mol M ¯ w /M ¯ n = 2.3) 137,800 (M
¯ w /M ¯ n = 2.3) 198,900 (M
1.612 140099.0 73184.9
1.620 202563.7 304351.6
¯w= Fig. 1. Nematic-Isotropic transition curve for monodisperse PS(M 22, 000)/EBBA system. Open circles are experimental data. The solid line is a nematic-isotropic transition curve calculated by the proposed model.
¯ n = 59, 900, M ¯ w = 137, 800). Open Fig. 3. Molar mass distribution of PS(M circles are experimental data and the solid line is a calculated curve.
1.05)/EBBA system. The solid line represents a nematicisotropic transition curve. Open circles are experimental data [13]. The calculated transition curve also shows a good agreement with experimental data. For polydisperse PS/EBBA systems, the log-normal distribution function is applied to represent the molar mass distribution of each sample determined by GPC. Parameters used in these continuous distribution functions, γ and I0 , are obtained from Eqs. (25) and (26), respectively. Parameter, A, which expresses the experimental peak height, is obtained from Eq. (24). Values
of model parameters for polydisperse PS used in this study are listed in Table 3. Figs. 3 and 4 show the comparison of GPC data for polydisperse PS with the calculated molar mass distributions. The calculated log-normal distributions give fairly good agreement with GPC results. Fig. 5 shows experimental nematic-isotropic transition data with different molar masses and polydispersities of PS. On figure, one of the green crosses represents a local minimum. This minimum may be caused by the unexpected experimental
¯w= Fig. 2. Nematic-isotropic transition curve for monodisperse PS(M 113, 000)/EBBA system. Open circles are experimental data. The solid line is a nematic-isotropic transition curve calculated by the proposed model.
¯ n = 85, 600, M ¯ w = 198, 900). Open Fig. 4. Molar mass distribution of PS(M circles are experimental data and the solid line is a calculated curve.
J.J. Choi, Y.C. Bae / Fluid Phase Equilibria 245 (2006) 102–108
107
Table 5 Adjustable model parameters for polydisperse PS/EBBA systems Parameters
b d0 d1
¯ w of PS in g/mol M 137,800
198,900
0.196 0.628 −167.981
0.087 0.355 −69.526
Fig. 5. The experimental nematic-isotropic transition data for PS/EBBA systems. The triangle (), circle (), cross (+) and rectangle () are ¯ w /M ¯ n < 1.05), ¯ w = 22, 000 (M the nematic-isotropic transition data for M ¯ w /M ¯ n < 1.05), Mw = 137, 800 (M ¯ w /M ¯ n = 2.3) and M ¯w= ¯ w = 113, 000 (M M ¯ w /M ¯ n = 2.3), respectively. 198, 900 (M
error—spatial or temporal inhomogeneities, impurities on thin film, and so on. ¯w= As the molar mass of the monodisperse PS(M ¯ 22, 000, Mw = 113, 000) is greater, a miscibility gap becomes ¯ w = 137, 800, M ¯ w = 198, 900), wider. For polydisperse PS(M the results are similar. However, despite the lower molar mass ¯ w = 113, 000), the transition temperature shows higher of PS(M ¯ w = 137, 800 (M ¯ w /M ¯ n = 2.3) of PS. This value than that of M is probably due to the polydispersity of PS. That is, the average molar mass is different from that of the highest molar mass distribution, which has the major effect on the nematic-isotropic transition. Values of molar masses with maximum value in molar mass distribution are listed in Table 4. As shown in Fig. 5, despite the similar molar masses between ¯ w = 113, 000) and the polydisperse the monodisperse PS(M PS(Mw = 114,000) of the highest molar mass distribution), the miscibility gap of the polydisperse PS is broader than that of the monodisperse PS. This is the expected result as the phase behavior for liquid–liquid equilibria [25]. Three adjustable model parameters (d0 , d1 and b) used for the polydisperse PS/EBBA system are listed in Table 5. Fig. 6 shows a calculated nematic-isotropic transi¯ w = 137, 800, M ¯n= tion curve for polydisperse PS(M 59, 900)/EBBA system. The solid line represents a nematicisotropic transition curve. Open circles are experimental data. The calculated transition curve provides fairly good fit to the experimental data.
¯n= Fig. 6. Nematic-Isotropic transition curve for polydisperse PS(M ¯ w = 137, 800)/EBBA system. Open circles are experimental data. 59, 900, M The solid line is a nematic-isotropic transition curve calculated by the proposed model.
Fig. 7 shows a calculated nematic-isotropic transition ¯ w = 198, 900, M ¯ n = 85, 600)/ curve for polydisperse PS(M EBBA system. The solid line represents a nematic-isotropic transition curve. Open circles are experimental data. The calculated transition curve also shows a good agreement with experimental data.
Table 4 Molar masses of the highest molar mass distribution Average molar mass of PS in g/mol
Molar mass of the highest molar mass distribution of PS in g/mol
137,800 198,900
77,400 114,000
¯n= Fig. 7. Nematic-Isotropic transition curve for polydisperse PS(M ¯ w = 198, 900)/EBBA system. Open circles are experimental data. 85, 600, M The solid line is a nematic-isotropic transition curve calculated by the proposed model.
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4. Conclusion The log-normal distribution offers more accurate description of the distribution of polymer species. To introduce the proper distribution function, we employ the extended Flory–Huggins model that takes into account both temperature and composition dependence in interaction parameter. It is simply able to represent the nematic-isotropic transition. The proposed model presented here has little theoretical basis; it is essentially semiempirical. Its advantage follows from its simplicity; a simple algebraic form with a few adjustable parameters appears to be suitable for representing nematic-isotropic transition of polydisperse polymer and liquid crystal systems. References [1] J. West, in: R.B. McKay (Ed.), Technological Applications of Dispersions, Marcel Dekker, New York, 1994, p. 345. [2] J.W. Doane, in: B. Bahadur (Ed.), Liquid Crystals: Their Applications and Uses, World Scientific, Teaneck, N.J., 1990, p. 361. [3] L. Onsanger, Ann. N.Y. Acad. Sci. 51 (1949) 627. [4] W. Maier, A. Saupe, Z. Naturforsch. 14a (1959) 882. [5] W. Maier, A. Saupe, Z. Naturforsch. 15a (1960) 287. [6] P.J. Flory, Proc. R. Soc. London Ser. A 234 (1956) 73.
[7] S. Chandrasekhar, Liquid Crystals, Cambridge University Press, Cambridge, 1977. [8] P.J. Flory, Macromolecules 11 (1978) 1138. [9] P.J. Flory, G. Ronca, Mol. Cryst. Liq. Cryst. 54 (1979) 311. [10] P.J. Flory, G. Ronca, Mol. Cryst. Liq. Cryst. 54 (1979) 289. [11] P.J. Flory, P. Irvine, J. Chem. Soc. Faraday Trans. I 80 (1984) 1807. [12] M. Ballauff, P.J. Flory, Ber. Bunseges Phys. Chem. 88 (1984) 530. [13] M. Ballauff, Mol. Cryst. Liq. Cryst. 136 (1986) 175. [14] J.R. Dorgan, D.S. Soane, Mol. Cryst. Liq. Cryst. 188 (1990) 129. [15] C.C. Riccardi, J. Borrajo, R.J.J. Williams, H. Masood Siddiqi, M. Dumon, J.P. Pascault, Macromolecules 31 (1998) 1124. [16] H. Kehlen, M.T. R¨atzsch, J. Bergmann, AIChE J. 31 (1985) 1136. [17] R. Cotterman, R. Bender, J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 24 (1985) 194. [18] R. Cotterman, J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 24 (1991) 434. [19] Y.C. Bae, S.M. Lambert, D.S. Soane, J.M. Prausnitz, Macromolecules 24 (1991) 4403. [20] M. Ballauff, Mol. Cryst. Liq. Cryst. Lett. 4 (1) (1986) 15. [21] C. Qian, S.J. Munby, B.E. Eichinger, Macromolecules 24 (1991) 1655. [22] C. Qian, S.J. Munby, B.E. Eichinger, J. Polym. Sci. Part B 29 (1991) 635. [23] Y.C. Bae, J.J. Shim, D.S. Soane, J.M. Prausnitz, J. Appl. Polym. Sci. 47 (1993) 1193. [24] B. Chu, Laser Light Scatterig, Academic Press, 1974. [25] Y. Hu, X. Ying, D.T. Wu, J.M. Prausnitz, Fluid Phase Equilib. 104 (1995) 229.
Nematic-isotropic phase behaviours of polydisperse polymer/liquid crystal systems: Applicability of continuous thermodynamics Jung Jin Choi, Young Chan Bae ∗ Division of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Korea Received 1 January 2006; received in revised form 29 March 2006; accepted 10 April 2006
Abstract Phase behaviours of polydisperse polystyrene (PS)/nematic liquid crystals (LCs) P-ethoxy-benzylidene-p-n-butylaniline (EBBA) are investigated by thermo-optical analysis (TOA) technique. In this study, to describe nematic-isotropic transition of the polydisperse PS/EBBA systems, a continuous distribution function to obtain the composition of polydisperse polymers is considered precisely. We apply these molar mass distributions to the extended Flory–Huggins model. The proposed semi-empirical model gives a remarkable agreement with experimental data for the model systems. © 2006 Elsevier B.V. All rights reserved. Keywords: Continuous thermodynamics; Nematic-isotropic transition; Polydisperse polymer; Molar mass distribution function
1. Introduction Mixtures of polymers and nematic liquid crystals (LCs) are subjects of intensive investigation in many laboratories around the world. The interest is motivated by their potential use in many fields of high technology involving electronic equipment, display systems and commutable windows, etc. [1,2]. Many statistical theories of the nematic phase have been presented. The best known theories among them are those suggested by Onsager [3], Maier and Saupe [4,5] and Flory [6] Chandrasekhar [7]. The Onsager treatment has proven successful in the description of rigid molecules of high axial ratio which display lyotropic behavior while the Maier–Saupe treatment has found an application to lower axial ratio molecules displaying thermotropic behaviour [8]. Maier and Saupe [4,5] postulated that the orientation-dependent interaction between nematogenic molecules arises from the anisotropy of their dispersion interaction. They sought to account for the stability and properties of nematic phase solely on the basis of the anisotropy of the forces of attraction, disregarding the steric or space filling, and characteristics of the asymmetric molecules comprising typical nematogenic substances.
∗
Corresponding author. Tel.: +82 2 2298 0529; fax: +82 2 2296 0568. E-mail address: [email protected] (Y.C. Bae).
0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2006.04.019
Flory and Ronca [9,10] extended the original Flory lattice model to include anisotropic molecular forces. They have created a theoretical frame work which was particularly well suited for the description of mixture containing nematogen [9,11,12]. They combined an orientation-dependent energy with a partition function derived for a system of “hard” rodlike molecules in which the intermolecular energy is the same for all configurations that are devoid of overlaps. Their treatment of a mixture consisting of rods, coil and a solvent meets with gratifying success in a comparison with experimental data. The Flory–Ronca treatment was extended by Ballauff [13] to binary mixtures of a thermotropic nematogen and a coiled species consisting of isodiametric segments linked together flexibly. The free volume which is known to be of utmost importance in describing thermotropic system is introduced by the methods devised previously [11,12]. In their work, the comparison of the model with experimental results was made only for polymers of relatively low molecular weight. Dorgan and Soane [14] reported nematic liquid crystal/wider range of the polymer molecular weight systems. Furthermore, Riccardi et al. [15] extended the Flory–Huggins model to liquid crystal and polydisperse polymer systems. However, although the importance of the compositon-dependence of the Flory–Huggins parameter, χ, has been noted, no further details were presented for phase-equilibrium calculations.
J.J. Choi, Y.C. Bae / Fluid Phase Equilibria 245 (2006) 102–108
In this study, we propose a semi-empirical molecularthermodynamic framework for correlating phase equilibria in polydisperse polymer/nematic liquid crystal systems. Furthermore, we modify χ to take into account both polymer concentration and temperature and employ a continuous thermodynamics [16–18] characterized by a continuous molar mass distribution. We also quantitatively investigate the nematic-isotropic phase transition behaviors of polydisperse polystyrene (PS)/nematic liquid crystal systems and compare our proposed model with experimental results. 2. Experimental 2.1. Materials 2.1.1. Liquid crystals Liquid crystals used in this study were P-ethoxy-benzylidenep-n-butylaniline (EBBA) obtained from Sigma–Aldrich Co. Nematic-isotropic transition temperatures measured by thermooptical analysis (TOA) [19] of EBBA was 76.3 ◦ C.
103
or cooling (3 ◦ C/min) was used. In this way, the approximate transition point of the sample was found. The sample was then repeatedly heated and cooled over a temperature range near the approximate transition temperature of the sample while the light intensity was monitored. If the measured transition points upon heating and cooling agreed with each other, the point was taken as the equilibrium transition temperature. If these temperatures were different, slower heating and cooling rates (0.3 ◦ C/min) were employed. In the case of a slight discrepancy (less than 1 ◦ C), the transition temperature upon cooling, corresponding to the traditional cloud point, has been taken as the equilibrium temperature. Table 1 shows the transition temperatures upon heating and cooling for various EBBA concentrations at the rate of 0.3 ◦ C/min. As shown in table, the measured transition points upon heating and cooling agreed with each other at less than 1 ◦ C, and the reproducibility is very reasonable. That is, the repeatedly nematic-isotropic transition temperatures of EBBA were correspondent. 2.4. Theoretical considerations
2.1.2. Polymers Polystryene samples were obtained from Aldrich Chemi¯ w ) and cal company. The weight-average molecular weight (M ¯ number-average molecular weight (Mn ) were determined by gel permeation chromatography (GPC), which was operated under condition of PS as a standard sample and tetrahydrofuran as the mobil phase. Their molecular weights were listed in Table 3. 2.2. Sample preparation Samples were prepared by the solvent casting method with various compositions: the materials were weighed-out into clean sample vials. Polymer/EBBA samples were dissolved in chloroform and stirred at 50 ◦ C for 15 h until samples became homogeneous. The samples were then dried in vacuum oven at 60 ◦ C for 30 h. 2.3. Experimental procedure A sample was placed in the microscope heating stage. To find the range of transition points of the sample, a scan rate of heating
We consider a system consisting of a liquid crystal and a polymer. Diameters of the components are assumed to be equal to the size of the lattice cell. The axial ratio of the liquid crystal is xl and the contour length of the ith component of the polymer is xi . The combinatorial analysis along the lines given by Flory [8] may be readily adapted for the system under consideration. As usually [10–12], the mixing partition function ZM is subdivided into three contributions: a liquid crystal, a polymer, and an orientation. ZM = ZLC Zorient Zpolymer
(1)
ZLC and Zpolymer are partition functions representing the method of packing the lattice with liquid crystal and polymer, respectively. First, lattice sites are occupied by polymer and then the liquid crystal is packed. The quantity y below specifies the disorientation of the liquid crystals. For a molecule with the long axis at an angle ψ with respect to the domain axis [10]: y=
4 xl sin ψ π
(2)
Table 1 Transition temperature upon heating and cooling for various EBBA concentrations ¯ w = 137, 800 of PS in g/mol M
¯ w = 198, 900 of PS in g/mol M
Weight fraction of EBBA
Transition temperature upon heating (K)
Transition temperature upon cooling (K)
Weight fraction of EBBA
Transition temperature upon heating (K)
Transition temperature upon cooling (K)
1 0.95 0.89 0.84 0.79 0.75 0.70 0.64 0.60
349.25 350.05 350.15 348.45 346.45 341.15 343.25 337.25 331.95
349.45 350.05 349.75 348.85 346.15 340.45 342.95 336.45 331.05
1 0.95 0.9 0.85 0.80 0.75 0.70 0.65 0.60
349.25 349.83 349.65 348.96 348.06 347.15 346.57 340.01 337.95
349.45 350.22 350.16 349.13 348.90 347.98 347.20 341.67 338.75
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with the orientational distribution function for liquid crystal being given by the ratio nly /nl , i.e. the number of molecules with disorientation y, nly , to the total number nl . The mean value y¯ follows as nly y¯ = y (3) nl y ZLC and Zpolymer are given by Zpolymer = ZLC =
n 0 − n l xl −
(n0 − nl xl ) ! [n (x −1)] ni xi ! ni !n0 i i
(n0 − nl (xl − y¯ ))! (n0 − nl xl )!nl !nn0 l (¯y−1)
Eq. (13) does not consider an isotropic interaction between the liquid crystal and the polymer. The reduced free energy is extended with isotropic energetics [14] as follows:
(5)
HM = nl xl φp χ = nl xl φp D(T )B(φp )
(7)
where ωy is a priori probability of the disorientation y for a liquid crystal [9], εy is the orientation-dependent interaction energy and kB is a Boltzmann constant. The inclusion of anisotropic molecular dispersion forces is accomplished by introducing the mean energy of a rodlike segment expressed as [8,9] 3 εy = −kB T ∗ φl s 1 − sin2 ψ (8) 2 where the characteristic temperature T* is the strength of the orientation-dependent forces for a given nematic liquid crystal [9]. The order parameter s is expressed as [13] 2 2
sin ψ (9) s=1− 3 The total number of lattice sites n0 is n0 = nl xl + n i xi φl = φi =
(13)
(4)
where n0 and ni are the total number of lattice sites and the number of the ith component of the polymer, respectively. Eq. (4) expresses the expected number of configurations for the polymer in the empty lattice. Eq. (5) takes into account the configurations accessible to the nl liquid crystals which have been added to the lattice subsequently. The orientational partition function Zorient is expressed as [10,12]: ω y nl −xl εy nly Zorient =Π (6) exp y nly 2kB T ωy = sin ψ
is obtained [20] y¯ n l xl A =− ni xi + nl y¯ ln 1 − 1− kT n0 xl nl ni + nl ln + (ni (xi − 1)) + ni ln n0 n0 nly nly n l xl 2 + nl (¯y − 1) + nl ln − s n n ω 2θ l l y y
(10)
n l xl nl xl + ni xi
(11)
ni xi nl xl + n i xi
(12)
where φl and φi are the volume fraction of the liquid crystal and the ith component of the polydisperse polymer, respectively. The total volume fraction φp of the polydisperse polymer is φp = φ i . Combining Eqs. (3), (4), (11), and (12) and employing Stirling’s approximation for the factorials, the reduced free energy
(14)
where the interaction parameter χ is given by the product of a temperature-dependent term, D(T), and a concentrationdependent term, B(φp ) [21–23]. χ(T, φp ) = D(T )B(φp ) D(T ) = d0 + B(φp ) =
d1 T
1 1 − bφp
(15) (16) (17)
where, d0, d1 and b are adjustable model parameters. Combining Eqs. (13) and (14), the reduced free energy is obtained A y¯ φl φl = −φp ln 1 − φl 1 − + ln n0 kT xl xl xl φi φi φi φl + + φp − + (¯y − 1) ln xi xi xi xl φ2 s φl s + ln f1−1 − l 1− + φ l φp χ (18) xl θ 2 where the reduced temperature θ, the order parameter s and disorientation index y¯ and defined as follow: θ=
T T∗
(19)
f2 f1 3 f3 s=1− 2 f1 y¯ =
4 xl π
where
fp = 0
π/2
(20) (21)
4 3 xl φl s sinp ψ exp − xl a sin ψ− sin2 ψ dψ π 2 θ (22)
2.5. Distribution function Compositions of polydisperse polymers are usually expressed by a continuous distribution function W(I), where I is a distribution variable such as a length r or a relative molar
J.J. Choi, Y.C. Bae / Fluid Phase Equilibria 245 (2006) 102–108
mass M. In this study, we employ the log-normal molar mass distribution function. The log-normal distribution function [24] is A I 2 1 W(I) = √ exp − 2 ln (23) γ I0 γ πI where γ, I0 , and A are adjustable model parameters. The lognormal distribution has a maximum value: 2 −γ IMAX = I0 exp (24) 4 From Eq. (24), we determine A, which represents the experimental peak height, γ and I0 are defined (25) γ = 2 ln(σ + 1) ¯ + 1)1/2 I0 = I(σ
(26)
where σ = µ2 /I¯ 2 . µ2 and I¯ are the variance and the average molecular weight of a given sample, respectively. Distribution function W(I) satisfies the following normalization constraint:
∞
∞ W(I) dI = 1, Φp W(I) dI = Φp (27) 0
0
The discrete multicomponent framework is transferred into the continuous thermodynamic framework as follows: A y¯ φl φl + ln = −φp ln 1 − φl 1 − n0 kT xl xl xl
∞ φp W(I) φp W(I) + ln dI + φp x(I) x(I) 0
∞ φp W(I) φl φl − dI + (¯y − 1) + ln f1−1 x(I) x xl l 0 φ2 s s − l (28) 1− + φ l φp χ θ 2 2.6. Equilibrium conditions For calculating the isotropic-nematic transition curve, we need chemical potentials for the liquid crystal and the polymer. Expressions for chemical potentials are derived directly from Eq. (28). µl ∂(A/kT ) = kT nl φ p − x l φp y¯ − xl φl + ln + + ln f1−1 xl φp + y¯ φl xl xn + y¯ − 1 xl s s − 1− φl (1 + φp ) + (1 − b)xl φp2 D(T )B(φp )2 θ 2 (29)
= −xl φp2
∂(A/kT ) µ(I) = kT ni y¯ − xl y¯ = −x(I) ln φp + φl + x(I)φp φl xl xl φp + y¯ φl
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x(I)φp φp W(I) x(I) φl − + ln + x(I) xl xn x(I) x(I)sφl2 s + 1− + x(I)φl2 D(t)B(φp )2 θ 2 ∞ where xn = Φp / 0 Φp W(I)x(I)−1 dI The equilibrium conditions are given by
(30)
µil = µnl
(31)
µ(I)i = µ(I)n
(32)
−
where superscripts i and n denote two phases at equilibrium. 2.7. Numerical calculations Biphasic equilibrium between the nematic and the isotropic phase also requires µ(I)i = µ(I)n for all components present in the system. The number of Eq. (32) becomes excessively large and the following conservation of mass is introduced: VΦi = V i Φii + V n Φni
(33)
where Vi and Vn are the volumes of the phase poor and rich in liquid crystal, respectively, and V is the volume of the original mixture. Thus, the isotropic-nematic transition curve is obtained from solving Eqs. (31)–(33), simultaneously. In this procedure, if the average molar mass of the polydisperse polymer is high, the PS is practically excluded from the liquid crystal rich phase [14,15]. Calculations are, therefore, accordingly performed by setting the composition of the ordered phase to unity and by solving one remaining equation for the isotropic phase composition. 3. Results and discussion Continuous thermodynamics gives us a new impetus not only because of its theoretical integrity and its consistent framework for multicomponent systems but also because it requires a comparatively simple mathematical procedure in comparison with the tedious determinant derivations in the usual discrete approach when calculating the spinodal and the critical-point criteria. When the continuous thermodynamic framework is applied to phase-equilibrium calculations for polydisperse polymer and liquid crystals, a molar mass distribution for describing the composition of polydisperse polymers is considered precisely. In this study, we use three adjustable model parameters (d0 , ¯ w /M ¯ n = 1) d1 and b) for monodisperse PS/EBBA system (M and then calculate the isotropic-nematic transition curve. These values are listed in Table 2. Fig. 1 shows a calculated nematic-isotropic transi¯ w = 22, 000, M ¯ w /M ¯n< tion curve for monodisperse PS(M 1.05)/EBBA system. The solid line represents a nematicisotropic transition curve. Open circles are experimental data [14]. The calculated transition curve provides a good fit to the experimental data. Fig. 2 shows a calculated nematic-isotropic transi¯ w = 113, 000, M ¯ w /M ¯n< tion curve for monodisperse PS(M
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Table 2 Adjustable model parameters for monodisperse PS/EBBA systems Parameters
b d0 d1
¯ w of PS in g/mol M
Table 3 Model parameters for molar mass distribution function Parameters
22,000
113,000
−0.607 −0.049 49.147
0.221 0.488 −117.068
γ I0 A
¯ w of PS in g/mol M ¯ w /M ¯ n = 2.3) 137,800 (M
¯ w /M ¯ n = 2.3) 198,900 (M
1.612 140099.0 73184.9
1.620 202563.7 304351.6
¯w= Fig. 1. Nematic-Isotropic transition curve for monodisperse PS(M 22, 000)/EBBA system. Open circles are experimental data. The solid line is a nematic-isotropic transition curve calculated by the proposed model.
¯ n = 59, 900, M ¯ w = 137, 800). Open Fig. 3. Molar mass distribution of PS(M circles are experimental data and the solid line is a calculated curve.
1.05)/EBBA system. The solid line represents a nematicisotropic transition curve. Open circles are experimental data [13]. The calculated transition curve also shows a good agreement with experimental data. For polydisperse PS/EBBA systems, the log-normal distribution function is applied to represent the molar mass distribution of each sample determined by GPC. Parameters used in these continuous distribution functions, γ and I0 , are obtained from Eqs. (25) and (26), respectively. Parameter, A, which expresses the experimental peak height, is obtained from Eq. (24). Values
of model parameters for polydisperse PS used in this study are listed in Table 3. Figs. 3 and 4 show the comparison of GPC data for polydisperse PS with the calculated molar mass distributions. The calculated log-normal distributions give fairly good agreement with GPC results. Fig. 5 shows experimental nematic-isotropic transition data with different molar masses and polydispersities of PS. On figure, one of the green crosses represents a local minimum. This minimum may be caused by the unexpected experimental
¯w= Fig. 2. Nematic-isotropic transition curve for monodisperse PS(M 113, 000)/EBBA system. Open circles are experimental data. The solid line is a nematic-isotropic transition curve calculated by the proposed model.
¯ n = 85, 600, M ¯ w = 198, 900). Open Fig. 4. Molar mass distribution of PS(M circles are experimental data and the solid line is a calculated curve.
J.J. Choi, Y.C. Bae / Fluid Phase Equilibria 245 (2006) 102–108
107
Table 5 Adjustable model parameters for polydisperse PS/EBBA systems Parameters
b d0 d1
¯ w of PS in g/mol M 137,800
198,900
0.196 0.628 −167.981
0.087 0.355 −69.526
Fig. 5. The experimental nematic-isotropic transition data for PS/EBBA systems. The triangle (), circle (), cross (+) and rectangle () are ¯ w /M ¯ n < 1.05), ¯ w = 22, 000 (M the nematic-isotropic transition data for M ¯ w /M ¯ n < 1.05), Mw = 137, 800 (M ¯ w /M ¯ n = 2.3) and M ¯w= ¯ w = 113, 000 (M M ¯ w /M ¯ n = 2.3), respectively. 198, 900 (M
error—spatial or temporal inhomogeneities, impurities on thin film, and so on. ¯w= As the molar mass of the monodisperse PS(M ¯ 22, 000, Mw = 113, 000) is greater, a miscibility gap becomes ¯ w = 137, 800, M ¯ w = 198, 900), wider. For polydisperse PS(M the results are similar. However, despite the lower molar mass ¯ w = 113, 000), the transition temperature shows higher of PS(M ¯ w = 137, 800 (M ¯ w /M ¯ n = 2.3) of PS. This value than that of M is probably due to the polydispersity of PS. That is, the average molar mass is different from that of the highest molar mass distribution, which has the major effect on the nematic-isotropic transition. Values of molar masses with maximum value in molar mass distribution are listed in Table 4. As shown in Fig. 5, despite the similar molar masses between ¯ w = 113, 000) and the polydisperse the monodisperse PS(M PS(Mw = 114,000) of the highest molar mass distribution), the miscibility gap of the polydisperse PS is broader than that of the monodisperse PS. This is the expected result as the phase behavior for liquid–liquid equilibria [25]. Three adjustable model parameters (d0 , d1 and b) used for the polydisperse PS/EBBA system are listed in Table 5. Fig. 6 shows a calculated nematic-isotropic transi¯ w = 137, 800, M ¯n= tion curve for polydisperse PS(M 59, 900)/EBBA system. The solid line represents a nematicisotropic transition curve. Open circles are experimental data. The calculated transition curve provides fairly good fit to the experimental data.
¯n= Fig. 6. Nematic-Isotropic transition curve for polydisperse PS(M ¯ w = 137, 800)/EBBA system. Open circles are experimental data. 59, 900, M The solid line is a nematic-isotropic transition curve calculated by the proposed model.
Fig. 7 shows a calculated nematic-isotropic transition ¯ w = 198, 900, M ¯ n = 85, 600)/ curve for polydisperse PS(M EBBA system. The solid line represents a nematic-isotropic transition curve. Open circles are experimental data. The calculated transition curve also shows a good agreement with experimental data.
Table 4 Molar masses of the highest molar mass distribution Average molar mass of PS in g/mol
Molar mass of the highest molar mass distribution of PS in g/mol
137,800 198,900
77,400 114,000
¯n= Fig. 7. Nematic-Isotropic transition curve for polydisperse PS(M ¯ w = 198, 900)/EBBA system. Open circles are experimental data. 85, 600, M The solid line is a nematic-isotropic transition curve calculated by the proposed model.
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4. Conclusion The log-normal distribution offers more accurate description of the distribution of polymer species. To introduce the proper distribution function, we employ the extended Flory–Huggins model that takes into account both temperature and composition dependence in interaction parameter. It is simply able to represent the nematic-isotropic transition. The proposed model presented here has little theoretical basis; it is essentially semiempirical. Its advantage follows from its simplicity; a simple algebraic form with a few adjustable parameters appears to be suitable for representing nematic-isotropic transition of polydisperse polymer and liquid crystal systems. References [1] J. West, in: R.B. McKay (Ed.), Technological Applications of Dispersions, Marcel Dekker, New York, 1994, p. 345. [2] J.W. Doane, in: B. Bahadur (Ed.), Liquid Crystals: Their Applications and Uses, World Scientific, Teaneck, N.J., 1990, p. 361. [3] L. Onsanger, Ann. N.Y. Acad. Sci. 51 (1949) 627. [4] W. Maier, A. Saupe, Z. Naturforsch. 14a (1959) 882. [5] W. Maier, A. Saupe, Z. Naturforsch. 15a (1960) 287. [6] P.J. Flory, Proc. R. Soc. London Ser. A 234 (1956) 73.
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