Molecular theory of nematic liquid crystals

Jul 14, 1982 - 'Camille and Henry Dreyfus Teacher-Scholar. ...... bond.) But the Clebsch-Gordan algebra which ensues via .... Department of Chemistry,...
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J. Phys. Chem. 1982, 86, 4298-4307

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FEATURE ARTICLE Molecular Theory of Nematic Liquid Crystals William M. Gelbart’ Department of Chemistry, University of California, Los Angeles, Los Angeles, California 90024 (Received: April 22, 1982; I n Final Form: July 14, 1982)

From a phenomenological point of view, the basic physics underlying the phase transition from an isotropic (“normal”)liquid to a nematic liquid crystal is well understood. At low temperature and high density the molecules align (orientationally order) so that they can lower their potential energy and keep out of each other’s way. On a molecular level, however, rather little is known about the relative stabilities of the liquid crystal phases of different compounds. In particular it is natural to ask which features of the intermolecular forces are most important in determining the various properties of nematic liquid crystals? What are the competing roles of the interparticle attractions and repulsions? How low a nonspherical symmetry (e.g., cylindrical,biaxial) must we ascribe to the constituent molecules in order to account qualitatively for the observed behaviors of nematics? In this article we distinguish between phenomenological and molecular theories of liquid crystals in an effort to answer these sorts of questions. After reviewing some relevant sets of experimental data, we survey briefly the various “lattice” and “continuum”approaches to nematic liquids. In particular we use the generalized van der Waals picture to discuss several molecular “mechanisms”for long-range orientational ordering. The separate contributions from interparticle attractions and repulsions are treated in detail, as is the unique role of molecular biaxiality. We close by outlining some connections between the familiar isotropic-nematic phase transitions in “neat” liquids and those in micellar solutions and polymer systems.

I. Introduction Nematic compounds are perhaps the simplest examples of systems which have long-range order but which retain virtually all of the usual liquidlike properties. Consider, for example, the liquid-crystal-forming molecule p-azoxyanisole (PAA)shown in Figure 1. Just above 408 K, at atmospheric pressure, it exists as a “normal” liquid. AU molecular positions and orientations are equally likely and its compressibility and viscosity, say, are just what one would expect for any other liquid composed of pairs of conjugated benzene rings. When cooled through 408 K, however, it undergoes a first-order phase transition to a liquid crystal in which there is a preferred direction for the molecular axes. This nematic liquid (see Figure 2) is characterized by long-range orientational order even as the distribution of molecular centers-of-mass remains essentially as before. The nematic is only a few tenths of a percent more dense than the isotropic (>408 K) liquid, and-similarly-its compressibility and viscosity are only slightly greater. But now, because of the partial long-range alignment, the liquid crystal is macroscopically uniaxial. And the large regions (domains) of oriented molecules give rise to special curvature elasticity, defect structures, and electro/magneto dynamics. As already mentioned, the transition from isotropic to nematic phases is always observed to be first order. But the corresponding density (p), enthalpy (H),and entropy ( S )changes are found to be very small: A p / p z 0.0035, AH x 0.14 kcal/mol, and AS = 0.34 cal/(mol K).l (Here p is the average of the densities of the coexisting phases.) These discontinuities are ”very small” compared to those characterizing the first-order change which occurs when this nematic is cooled 20 K further to its crystallization ‘Camille and Henry Dreyfus Teacher-Scholar.

temperature (390 K): A p / p = 0.11, AH x 7.1 kcal/mol, and AS i= 18 cal/(mol K).2 The dramatically weak first orderness of the isotropic-nematic phase transition is a point to which we shall return several times in this discussion. The spontaneous alignment of molecular orientations in a liquid crystal is in many ways analogous to the spontaneous magnetization of a spin system. In the latter case the degree of long-range alignment is characterized by u, the average value of Pl(cos 8) = cos 8 where P1is the first Legendre polynomial and u is the angle between an arbitrary spin and the space-fixed z axis. In the case of liquid crystals, on the other hand, the angle-dependent interaction between molecules (“spins”) does not distinguish between “up” and “down”. Accordingly, the longrange order parameter (7)is most naturally defined by the average value of-not P I ,but-the second Legendre polynomial: 7 = (P2(cos 8)) = ( 3 / 2 cos2 8 Here of course 8 is the angle between the long molecular axis and the z direction. In the isotropic (I) phase, where all orientations (e’s) are equally likely, (cos28) = 1 / 3 and 71 = 0. For a completely ordered nematic, where 8 = 0, (cos28) = 1 and 7 = 1. For partial long-range alignment, 0 < 7 < 1: in the thermal range of most nematics (N) 7 ranges from ~ 0 . 3 - 0 . 4a t TNIto ~0.5-0.6 at Tcrystallization. Because of a preferred direction for the long molecular axes, nematic liquids possess special curvature elasticity properties which are totally absent in their isotropic counterparts. Consider the aligned sample depicted schematically at the top (a) of Figure 3. Note that the lines correspond not to the molecular axes themselves but (1) W. Maier and A. Saupe, Z. Naturforsch. A, 14,882 (1959); 15,287 (19601, for A p / p . H. Arnold, Z. Phys. Chem. (Leipzig),226, 146 (1964), for AH and AS. (2) B. Deloche, B. Cabane, and D. Jerome, Mol. Cryst., 15, 1975 (1971).

0022-3654/82/2086-4298$01.25/00 1982 American Chemical Society

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Flgure 1. Structural formula (a) and “space-filling” model (b) for pazoxyanisole (PAA), a prototype liquid-crystal-forming molecule.

Flgure 3. (a) A schematic of an aligned nematic sample; the lines depict the locally preferred direction about which the molecular axes are distributed according to f(8). (b) “Splay”, (c) “twist”, and (d) “bend” modes of curvature deformation; see text for details.

stability in one- and two-component systems. Finally, in section IV we compare these “simple” nematics with long-range orientationally ordered states of soap solutions and polymer melts. m e 2. OversimplM representationof molecular orientational order in the nematic and isotropic phases of a neat liquid. rather to the locally preferred direction-the “director”, ii(r). The sample is macroscopically aligned in the sense that (neglecting fluctuations) the director lies everywhere along the space-fixed z axis, i.e., ii(r) 2. At the same time there is a distribution of molecular axes about this locally preferred direction: f(8) is the fraction of molecules whose long axes lie along 8. As the sample in (a) is “splayed” (see Figure lb)-grad.ii # 0-there is an increase Aa in free energy density. The difficulty with which this curvature deformation is imposed on the system is given by the force constant Ksplay(I K J , the constant of proportionality between Aa and Similarly, K ~ Kt , and Kbnd K3 characterize the remaining “normal modes” (&grad X ii # 0 and ii X grad X ii # 0, respectively) of curvature elasticity (see parts c and d of Figure 3). Normal (isotropic) liquids, of course, have Ki 0 since their lack of alignment allows them to be splayed, twisted, and bent at no thermodynamic cost. As we shall see, the curvature elasticity constants of nematics provide detailed probes into the intermolecular forces and correlations in these liquids. In the present paper we discuss various molecular approaches to the bulk properties of nematic liquid crystals. Section I1 begins with a short discussion of the relative merits of phenomenological vs. microscopic theories for addressing different aspects of this problem. We then provide a systematic but brief survey of the many “lattice” and “continuum” models which have recently been formulated to treat the isotropic to nematic phase transition. Section III examines what can be learned from some of the qualitative failings of the molecular theory. In particular we discuss new probes of competing roles of the interparticle attractions and repulsions. We treat also in considerable detail the biaxial ingredients of liquid crystal

11. Molecular vs. Phenomenological Theories

A. What Questions Are We Asking and What Answers Are We Looking for? In any disorder-order phase transition it is crucial to ask at the outset: what is the physical mechanism which allows the free energy (A) to be lowered even as the system loses entropy (S)? A = E - TS implies that A is raised as S (>O) decreases. The answer of course is that ordering-alignment in the liquid crystal casedecreases not only the entropy but also the energy ( E ) . This is because the intermolecular potential depends on relative orientations and is lowest for aligned configurations. Thus, at low enough temperature (T),E dominates and the ordered (low S) phase becomes stable. In fact, there are translational-as opposed to rotational-entropy terms which actually increase upon orientational ordering. That is, as the molecules align, there are more ways to pack them without their bumping into one another. This effect becomes important at high density (low temperature) and helps to stabilize the nematic. In phenomenological theories of the isotropic-nematic transition the competition between energy and entropy effects is built into the thermodynamic free energy, taking into account only dimensional analysis and symmetry. Molecular theories, on the other hand, confront explicitly the statistical mechanical details of the interparticle forces and correlations. Each of these two approaches addresses its own level of question, and each pays its separate price. Consider again, for example, the curvature elasticity properties of a nematic. Figure 4a depicts schematically a thin sample aligned between a pair of glass plates (which have been treated so that the director is everywhere perpendicular to their faces: fi(r)= 2). When a static magnetic field is turned on along the x direction, it is found that the z alignment survives until a threshold value (H*) of the field is reached. For H > H* a bend deformation grows in, as shown in Figure 4b, and is characterized by

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of P l a n vs. temperature should give a straightline (- T T*) which crosses the abcissa at T*. This is indeed observed not only for the magnetic l/An (Cotton-Mouton effect) but also for the reciprocal of the electric birefringence (Kerr effect) and of the integrated (over frequency) light-scattering intensity.6 All three experiments have been performed on MBBA,6v7for example, and give the same T*. Again, all of the above analysis is purely phenomenological. In fact the temperature T* is never quite reached. Instead, the isotropic liquid undergoes a first-order transition to a nematic state a t a temperature TNI= T* + A3’/(4A4)just above T*. Only if A , were identically zero would we have a second-order transition (at P ) . That is, the “first 0rderness”-the magnitude of the discontinuities in volume, enthalpy, and entropy, etc.-is controlled by the size of A,. And only a molecular theory can address the questions of how big this coefficient is and how it depends explicitly on the forces between particles. B. Molecular Theories. Molecular theories of nematic N phase transition, can be conliquids, and of the I veniently divided into three categories. ( i ) Attractions Featured. First there are those which ascribe the long-range orientational ordering to angle-dependent intermolecular attractions. Preeminent among such theories is the original work of Maier and Saupes in which van der Waals dispersional interactions between pairs of molecules are assumed to dominate the energy. This potential is mean-field-averaged to give an effective one-body attraction with the form -V2qPz (cos 61, where again 6 is the angle between the molecular long axis and the space-fixed z direction, Pz(x) = ,/,x2 - ’/,, and q = Sdx f ( x ) P2(x);f ( x = cos 6) is the probability of orientation 9 and V, is a strength parameter. In the absence of intermolecular interactions, the entropy would be given by its “ideal gas” form: S = -Nk$dx f (In f ) where N is the number of particles and k is the Boltzmann constant. Thus A = E - TS = Erot- TS,,, = -(N/2)V2q2 + NkTJdx f(ln f ) (3) -

Flgure 4. In (a) the magnetic field H = t%C is insufficient to disturb the uniform 2 alignment Imposed on a nematic by closely spaced (d microns) glass plates. Above threshold-see (b)-a bendlike deformation is induced, allowing determination of the associated curvature elasticity constant.

a sinusoidal variation in the x and z components of the “director”: ii(r) = 2 cos ( a z / d )+ 2 sin ( a z l d ) . The local free energy density thus increases as ‘/2K3(ii X grad X = ‘ / z K 3 ( ~ / d )The ~ . work necessary to impose this deformation comes from the magnetic field H “grabbing on“ to the anisotropic local magnetic susceptibility x . Thus the threshold for appearance of the bend is given by ‘/2K3(a/d)’= I/,xH*’, or H* = ( ~ / d ) ( K , / x ) l /Accord,. ingly, measurement of H*, and knowledge of x and d , provides a direct and accurate determination of K3. By choosing different geometries for the alignment and field directions (relative to the plate normal), K , and K 2 can be determined ~ i m i l a r l y . ~Note ? ~ that this analysis is purely phenomenological in the sense that no mention whatsoever has been made of the microscopic mechanisms which contribute to curvature elasticity. Suppose, however, that we wish to understand why the bend constant K , is sometimes larger than the splay constant K,, whereas for other nematics the splay deformation is “hardest” (i.e., K 1 > K,). Furthermore, we may wish to determine the explicit ways in which each of the curvature elasticity constants depends on the forces between particles. Obviously, a molecular theory is required in these cases. Consider now the nature of the phase transition between isotropic (I) and nematic (N) liquids. With Landau4t5we can assume that the free energy is expressible as a power series in the order parameter: A = A , + Azq2 + A3q3 + A4q4 + ... - XH%J (1) Here A , = 0, in order to guarantee an q = 0 solution to d A / d q = 0, i.e., in order that the isotropic phase ( q E 0) be stable (minimize A ) at zero field and high temperature. As u ~ u a l ,A~, ,has ~ the form A , = a ( T - T*) where T* is the temperature below which the isotropic phase becomes unstable, i.e., the curvature of the free energy at q = 0 changes sign-becomes negative-for T < P.This expansion is valid in the isotropic phase since the magnetically induced long-range order is very small there: 1 >> q k 0. d A / d q = 0 then implies that (neglecting terms of order q2 and higher) 9 xZ@/A, W / ( T - T*) An (2)

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where An = nll- n, is the difference between the refractive indices measured parallel and normal to H. Thus a plot (3) V. Frederiks and V. Zolina, Trans. Faraday Soc., 29,919 (1933); H. &her, ibid., 29,945 (1933); V . Frederihs and V. Zwetkoff, Sou. Phys., 6 , 490 (1930); A. Saupe, 2.Naturforsch. A, 15, 815 (1960); H. Gruler, T. Scheffer, and G. Meier, ibid., 27, 966 (1972). (4) For a comprehensive discussion of these effects see the following monographs: P. G. de Gennes, “The Physics of Liquid Crystals”, Oxford University Press”, Oxford, 1974; S. Chandrasekhar, ’Liquid Crystals”, Cambridge University Press, London, 1977. (5) See D. ter Ham, Ed., “The Collected Papers of L. D. Landau”, Gordon and Breach-Pergamon, New York, 1965; L. D. Landau and E. M. Lifshitz, ’Statistical Physics”, Addison-Wesley, Reading, MA, 1970.

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and minimization with respect to f ( d A / d f = 0) leads to [using S v / S f = P2(x)and (S/Sf)Sdx f(ln f ) = 1 + In f(x)] the self-consistency equation

where = kT/ V, is the dimensionless temp5rature. Equation 4 is satisfied only by q = 0 for all T > 0.22284. For T < 0.22284, an q > 0 solution appears which-like q = 0-corresponds to a local minimum in the free energy. But this nematic ( q > 0) minim- becomes lower thanJhe isotropic ( q = 0) one only for T < 0.22019 E TNI( T = 0.22284 is the upper limi: of metastability of the nematic phase.) Furthermore, at T = TN the value of qnemtiCis 0.44, (6) J. D. Litster in “Critical Phenomena”, R. E. Mills, Ed., McGrawHill, New York, 1971, p 343. (7) MBBA is the acronym for p-methoxybenzylidene-p-butylaniline. Like PAA, shown in Figure 1, it has the structural formula

but with A = -OCH,, X = -CH=N- and B = -C4H9 instead of A = B = -OCHB and X = -NO=N-. (8) W. Maier and A. Saupe, 2.Naturforsch. A , 13, 564 (1958); 14,882 (1959); Ea, 287 (1960).

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i.e., the order parameter at the (first-order) transition is predicted to be universal-independent of intermolecular forces, applied pressure, etc. This prediction is borne out quite well by experiment. The true beauty of the Maier-Saupe theory lies, however, in its simplicity: it is the direct analogue of the Weiss theory of ferromagneti~m.~ This analogy with single-spin mean field theory has been exploited by Sheng and Wojtowiczlowho have extended the constant coupling approximation” from a ferromagnetic to liquid crystalline context, and by Madhusudana and Chandrasekharl2and Ypma and Vertogen13who have similarly generalized the Bethe-Peierls-Weiss a p p r ~ a c h . ~ Short-range correlation effects are thereby included in these calculations. The Maier-Saupe theory has also been “improved upon” in an additional direction by Luckhurst et al.14 and Chandrasekhar and co-workers15 who have included P,(cos e) terms in the representation of the effective one-body attraction; this work has been discussed critically elsewhere.16 (ii)Repulsions Dominant. A significant departure from the Maier-Saupe approach arises, however, only when short-range repulsions are explicitly taken into account, to the exclusion of attractive forces. On the one hand, there are “continuum” theories inspired by Onsager’s original hard-rod-gas modeP7for the I-N transition in dilute suspensions of anisotropic colloidal particles [e.g., tobacco mosaic virus (TMV) molecules]. Here the onset of longrange orientational ordering occurs a t very low concentration-the particles are so long and thin that they begin to “bump into” each other at very low volume fractions. Accordingly the free energy can be well-represented by a truncated virial expansion. Onsager evaluated the leading term-the pair excluded volume, or second virial coefficient-whereas Zwanzig calculatedla the first seven after restricting the principle axes of each molecule to lie along the space-fixed directions. Using these results, Runnels and Colvinlgconstructed the corresponding Pad6 approximant. For neat liquids composed of hard anisotropic particles, as opposed to dilute colloidal suspensions of infinitely long and thin ones, the scaled particle theorym has been applied by Cotterz1to treat the I N transition. Similarly, Barboy and Gelbart have developedzz a new series representation for thermodynamic functions of hard particle fluids which is quickly convergent up through the full liquid range.

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(9)See the monograph by J. S. Smart, “Effective Field Theories of Magnetism”, Saunders, New York, 1966. It is of interest to note that the first molecular theory of the ordeldisorder transition in liquid crystals was due to M. Born, Sitzungsber. Phys. Math., 25,614 (1916),using an electric dipolar-rather than dispersional-interaction. (10)P. Sheng and P. J. Wojtwoicz, Phys. Reu. A14, 1883 (1976). (11)P. W. Kasteleijn and J. van Kranendonk, Physica, 22,317 (1956). (12)N. V. Madhusudana and S.Chandrasekhar, Sol. State Commun., 13,377 (1973)and Pramana 1, 12 (1973). (13)J. G. T.Ypma and G. Vertogen, Sol. State Commun., 18, 475 (1976)and J. G. J. Ypma, G. Vertogen and H. T. Koster, Mol. Cryst. Liq. Cryst., 37,57 (1976). (14)R. L. Humuhries, P. G. James, and F. G. Luckhurst, J. Chem. Soc., Faraday Trim. 2,68,1031 (1972). (15)S.Chandrasekhar and N. V. Madhusudana, Acta Crystallogr.,27, 303 (1971). (16)W. M. Gelbart and B. Barboy, Mol. Cryst. Liq. Cryst., 55, 209 (1979). (17)L. Onsager, Ann. N.Y. Acad. Sci., 51, 627 (1949). (18)R. W. Zwanzig, J. Chem. Phys., 39, 1714 (1963). (19)L. K. Runnels and C. Colvin, J. Chem. Phys., 53, 4219 (1970). (20)See review by H. Reiss in “Statistical Mechanics and Statistical Methods in Theory and Application”, U.Landman, Ed., Plenum Press, New York, 1977. (21)M. Cotter in “The Molecular Physics of Liquid Crystals”, G. R. Luckhurst and G. W. Gray, Eds., Academic Press, London, 1979, and references contained therein [in particular, G. Lasher, J. Chem. Phys., 53, 4141 (1970);R. M. Gibbons, Mol. Phys., 18,809 (1970)l. (22)B. Barboy and W. M. Gelbart, J. Chem. Phys.,71,3053 (1979); J. Stat. Phys., 22, 709 (1980).

An alternate approach to the hard-anisotropic-particle liquid has been inspired by the early “lattice” theories of Floryz3and D i M a r z i ~ . Here ~ ~ the molecular centers-ofmass are confined to the sites of a simple cubic lattice, thereby allowing analytical evaluations of the number of ways of packing hard rods with arbitrary orientational d i s t r i b u t i ~ n .These ~ ~ calculations have been extended to include semiflexible “tails” on the hard cores; it can be shownz6that their effect is to “soften” the first orderness of the I N transition, i.e., to decrease theoretical estimates of the discontinuities in volume and entropy, etc. Note that, in all of the (“continuum” and “lattice”) models of the hard-particle liquid crystals, there is no energy contribution to the Helmholtz free energy: all allowed configurations have zero potential energy (all others have infinite energy, and are thus excluded entirely). The entropy then consists of two parts, associated alternately with orientational and positional degrees of freedom. The rotational entropy is the “ideal gas”, “entropy of mixing” mentioned earlier: S,,, = -RSdQ f(ln f). The “packing” entropy, Strans, on the other hand, corresponds to the number of ways of translationally realizing the system for a given f . This latter term is evaluated within the lattice and continuum approaches by calculating inthe appropriate excluded volumes. Note that Strans creases as the rods align: the hard rods can move more ways, without “bumping into” each other, if they first “line up”. Strans = B(p2), whereas S,,,-which decreases upon alignment-goes as p . Thus the liquid orientationally orders at sufficiently high density. (iii) T h e van der Waals Picture. Earlier, in our discussion of the Maier-Saupe-type theories, we had seen how the decrease in S, upon alignment is offset instead by the lowering of the attraction energy E. In reality, of course, both of these mechanisms are operative. Thus it is important to develop theories in which both anisotropic hard-core repulsions and angle-dependent attractions are explicitly included. This has been done within the lattice-model framework by Alber1,2~Flory and R ~ n c aand ,~~ Warner.z8 Dowe1lZ9has carried this work still further by including the effects of “soft” repulsions and multiple-site attractions; again, as with semiflexible “tails”, the first orderness of the I N transition is found to be significantly moderated. Within the continuum framework, anisotropic repulsions and attractions have been treated by means of the generalized van der Waals (GVDW) theory. Since this approach has already been discussed at length in earlier reviews,30we shall only outline the basic idea here. In the following section we interpret the results of GVDW theory in an attempt to sort out the molecular mechanisms underlying the isotropic to nematic phase transition. In the original spirit of van der Waals31we break up the intermolecular pair potential into two parts: U = U,, +

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(23)P. J. Flory, Proc. R. SOC.London, Ser. A , 234,73 (1956). (24)E.A. DiMarzio, J. Chem. Phys., 35, 658 (1961). (25)R. Alben, Mol. Cryst. Liq. Cryst., 13,193 (1971);M. A. Cotter and D. E. Martire, ibid., 7, 295 (1969);M. A. Cotter, ibid., 35, 33 (1976). (26)G. I. Agren and D. E. Martire, J. Chem. Phys., 61,3959 (1974); F. Dowell and D. E. Martire, ibid., 68, 1088, 1094 (1978). (27)P.J. Flory and G . Ronca, Mol. Cryst. Liq. Cryst., 54, 289, 311 (1979). (28)M. Warner, J. Chem. Phys., 73,6327 (1980). (29)F. Dowell and D. E. Martire, J. Chem. Phys., 68, 1088, 1094 (1978),and references contained therein. (30)(a) W. M. Gelbart and B. Barboy, Acc. Chem. Res., 13,290(1980); (b) M. A. Cotter In “The Molecular Physics of Liquid Crystals”, G. W. Gray and G. R. Luckhurst, Eds., Plenum Press, New York, 1978,and references contained therein. (31)J. D. van der Waals, Dissertation, Leiden, 1873;English translation in Phys. Memoirs, 1, 333 (1890).

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U,, where U, is the hard (anisotropic) core repulsion and Uattris the long-ranged angle-dependent attraction. The (Helmholtz) free energy consists similarly of two parts: AGVDW[~(Q)I = Ah,tf(Q)l

+ ( N / 2 ) 1 d Q f(Q)$(Q)

(5)

where $ ( Q ) = pJdQ’

f(Q’)j’dr

ua&,QQ’)

(6)

is the mean attraction (“field”) felt by a molecule when it has orientation Q. f ( Q ) denotes the fraction of molecules with orientation Q and the prime on the r integral restricta the averaging to those values for which a pair of hard cores, with orientations Q and a‘, do not penetrate each other: J’dr = Jdr exp[-U,(r,QQ’)/kZ‘‘J]. Finally, Ahc[f(Q)]is the free energy associated with a “reference” fluid whose molecules interact only via hard core repulsions and whose orientational distribution is f ( Q ) . The remaining thermodynamic functions follow from ACMW[f(Q);T,V,IVIin the usual way: S = -(dA/d?3vf14,,,, P = -(~A/C~V),~,,,,, and I.L = ( ~ 3 A / d n ? , ~ , ~ (Thus , , . a complete specification of the phase transition requires a proper calculation of Ah, and $. Note that in the limit of zero anisotropy for the hard cores, Le., Uhc(r,QQ’) Ubd sphere(r), the GVDW theory outlined above reduces directly to the Maiel-Saupe theory described earlier. In the case of vanishing attractions (U,,, and hence $ 0), on the other hand, our theory reduces trivially to the hard-particle theory. The calculation of A&(Q)] requires a statistical mechanical evaluation of the “packing”entropy associated with a liquid-density system of hard anisotropic particles. As mentioned earlier this can be done, within a continuum context, via scaled particle theoryz1or the “y expansion”.22 Calculation of the effective attraction $(a) appears to be more straightforward, but it is not. If, for example, we take the anisotropic hard cores to be “spherocylinders” (righbcircular cylinders capped by half-spheres), then J’dr = Jdr (r outside Se,,) where SeI2is an explicitly specified surface32which surrounds one particle and contains the volume excluded to a second whose long axis makes an angle d12 with respect to the first. For reasonable choices of hard core dimensions and estimates of Ua*(r,QQ’), the effective attraction is found to acquire its angle dependence primarily from the anisotropy of the Uhc rather than that of Uam That is, $(a) depends on Q through the anisotropic short-range correlations associated with the excluded volume factor exp[-Uhc(r,QQ’)/kr]. When we evaluate J d r Uatt,(rQQ’)exp[-Uhc(r,QQ’)/kT]we assert implicitly that it comes close to approximating Jdr Uattr(rQQ’). ghc(rQQ’); ghc is the actual orientation-dependent radial distribution function for the reference system and is, of course, quite nontrival to compute. Nevertheless there do exist computational methods for determining this quanand it will be important to compare results of this kind for $(Q) with those calculated from putting ghc exp[-&,/kT]. We suspect that eq 6 overestimates considerably the amount of angle dependence in the mean attraction 9. But we also believe that this error is made tolerable by the considerable simplicity-and qualitative accuracy-of the basic theory involved.

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111. Molecular “Mechanisms”

A. What’s Wrong? In all the GVDW calculations performed to date, the hard core is taken to be a sphero(32) W. M. Gelbart and A. Gelbart, Mol. Phys., 33, 1387 (1977). (33)See,for example, the review by D. W. Chandler, Annu. Reu. Phys. Chem., 29, 441 (1978), and references t o related work cited therein.

Gelbart

cylinder and the pair attraction is approximated by the angle-dependent dispersional potential: U,, is uniquely characterized by the choice of a length (L)and width ( W), and Ua&by the anisotropy ( a l l / a Land ) trace (Tr a)of the molecular polarizability tensor. In all these cases the densities (pu and pr) of the coexisting (uniaxial and isotropic) phases are predicted to be too small, whereas the discontinuities in volume, entropy, and order parameter (AV/V, AS, and 77) are uniformly too large (by factors of ten to fifty!). These results are related: the molecular theory exaggerates the differences between the isotropic (I) and uniaxial nematic (U)phases. What is the significance of these qualitative failings? Consider again the simplifying assumptions behind our calculations. First we treat only crudely-via truncated y expansionsn or scaled particle theory21-the short-range order due to hard-core repulsions. Similarly, as just noted at the end of section 11, our mean field average involves not the actual radial distribution function for the reference system but rather its zero-density limit. Alternatively, Woo and c o - ~ o r k e r have s ~ ~ calculated those pair correlations which follow from an orientationally averaged potential. Instead of explicitly breaking up an a priori pair interaction U(rQQ’)into repulsions and attractions, they consider an orientationally averaged potential o(r) u&) + uz(r)q2 whose defining parameters are determined empirically. More precisely they use the equilibrium hierarchy plus Kirkwood superposition35to solve for the effective isotropic radial distribution g(r) corresponding to o(r). g ( r ) takes the place of our exp[-Uhc(r,QQ’)/kr]and leads to the Helmholtz free energy as an explicit function of q,N,V and T. So, spatial correlations are included, but only after assuming a decoupling of positional and orientational degrees of freedom. The anisotropy of short-range repulsions is either neglected altogether or taken to be weak (thereby allowing a perturbative analysis). For this reason we prefer the GVDW theory, which explicitly includes the coupling between radial and angular variables and allows for arbitrarily large anisotropy in the hard cores. A clear weakness of the GVDW theory as presently formulated is the neglect of multiple centers of attraction (which effect should become significant when molecular lengths become large compared to the average distance between centers-of-”), semiflexibility of end chains, and softness of repulsions. These complications can be handled in the lattice model mentioned earlier; within this context they have been shownmto decrease considerably the discontinuities at the I U transition. One could try to include these effects within a continuum context as well, but obviously the real pair potential is too complicatedand too poorly known-to expect quantitatiue accuracy from the molecular theory. Even if U(r,QQ’)were known exactly, we could not do good enough statistical mechanics with it. The molecular theory should be looked to instead to provide qualitative insight into the microscopic mechanisms of long-range orientational ordering. For example, we shall show a little later how the very small magnitude of the I U discontinuities in volume, entropy, and order parameter can be accounted for by considering small deviations of particle symmetry from uniaxial. First, though, it is instructive to inquire how the molecular theory can be used to determine the relative importance of attractions and repulsions in stabilizing the nematic and establishing

-

-

(34)V. T.Rajan and C. W. Woo, Phys. Reu. A , 17, 382 (1978); F. Feijoo, V. T. Rajan, and C. W. Woo, ibid., 19,1263 (1979); J. Shen, L. Lin, L. Yu, and C. W. Woo,Mol. Cryst. Liq. Cryst., 70, 301 (1981). (35) T.L. Hill “Statistical Mechanics”, McGraw-Hill,New York, 1956.

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its various thermal and elastic properties. B. Relative Roles of Attractions and Repulsions. For example, if the molecules “felt” each other only through hard-core interactions, the transition temperature TNI would be directly proportional to the external pressure. Experimentally it is found-for PAA2, say-that T N I increases by ”only” about 25 K (from 408 K) as the pressure is increased 1000-fold from atmospheric. From reasonable estimates of Uhc(i.e., L and W) and of U,, (i.e., aI1/a1and Tr a) this value of dTNI/dPhas been accounted for36,37 within the GVDW theory. Similarly it is interesting to consider the quantity 7’-

p(dS/ap)T

T(aq/ar),

(7)

which measures the relative dependence of the orientational order parameter on density and temperature. Again, in the case of hard-core interactions only, y = 00 since the system is “athermal”; for Uh, sphere, on the other hand (i.e., only angle-dependent attractions), y = 1. In “the real world”, e.g., from measurements on PAA,38y is found to be =4, in accord with the GVDW based on the same choices of Uh, and Vat, which gave reasonable values for dTNI/dP. Furthermore, the theoretical calculations show that y is very sensitive to up, the dimensionless density (“packing fraction”-here u is the hard-core volume associated with a single particle); y increases dramatically with up attesting to the enhanced role of hard-core interactions at high density. A property which is still more sensitive to the competing roles of intermolecular attractions and repulsions is the curvature elasticity of the nematic liquid crystal. More explicitly, within the GVDW theory it is possible to write each elastic constant as a sum of two terms: Ki= K? + Kiattr,i = 1,2, 3. The relative magnitudes of K1,K2,and K3 provide sensitive probes of the corresponding (“splay”, “twist”, and “bend”) anisotropies in the pair potential U(r,QQ’).In particular it can be shown39that K P L4 (W fixed) W ( L fixed)

-

--

whereas

Kiattr

- LO N

w-1

Tr a (8) Also, the sign of (KShC/KlhC) - 1 is generally4O positive whereas the sign of (K3attr/Klattr) - 1 is negative. These “facts” allow us to understand how the relative magnitudes of K3 and Kl can change by moving through a homologous series or upon increasing the temperature. For example, if L / W is large enough and Tr a is small enough, the hard-core interactions will dominate and Kbnd (EK3)will exceed Ksplay(Kl), and so In this connection it is interesting to consider again the dilute colloidal suspension of very long molecules. If the molecules are sufficiently long and rigid, then the Onsager theory mentioned earlier becomes essentially exact.41 I t can then be shown42that K3

%

7Ki

%

20 K2

(36)M. A. Cotter, J. Chem. Phys., 67,1098 (1977). (37)B. A. Baron and W. M. Gelbart, J. Chem. Phys., 67,5745(1977). (38)J. R. McColl and C. S. Shih, Phys. Rev. Lett., 29, 85 (1972). (39)W.M. Gelbart and A. Ben-Shaul, J.Chem. Phys., 77,916 (1982). (40)For p4> 0,see discussion in ref 39. (41)J. P.Straley, Mol. Cryst. Li9. Cryst., 24,7 (1973). (42)J. P.Straley, Phys. Rev. A, 8,2181 (1973);W.M. Gelbart and A. Ben-Shaul, unpublished results.

and, as before, Ki L4 for all i. If, on the other hand, the molecules are sufficiently long and flexible, then de Gennes43has argued that Kl >> K z , K 3 Lo N

-

As far as we are aware there are no data reported in the literature on the curvature elastic constants of lyotropic liquid crystals, i.e., of colloidal suspensions of long molecules. Preliminary measurements by DuPrgU and Meyera on aqueous PBLG solutions solutions indicate that these have considerable flexibility, but more comprehensive length and temperature variations are clearly desirable. C. Biaxiality. We have commented earlier on the role of multiple-site attractions, “soft” repulsions, and endchain flexibility in decreasing the first orderness of the isotropic-to-nematic phase transition. Surely another contributing factor is the lower-than-cylindrical symmetry of the molecules which make up real liquid crystals. In the overwhelming majority of cases they involve two (or more) conjugated benzene rings, e.g.

where the bridging group X maintains the planarity and rigidity of the ?r-electron system and A and/or B enhances the overall polarizability. The most often-encountered prototypes are PAA (A = B = OCH3,X = NN-O), MBBA (A = C4H9,B = OCH3, X = CN-H), and the alkyl cyanobiphenyls (A = CnHzn+l,B = CN, X = nothing). Because the benzene rings are essentially coplanar, and their width (57 A) exceeds significantly their thickness ( 2 4 A), the molecules are inescapably biaxial. Accordingly, they should be treated as such; the fact that they are observed to rotate fast about their long axis is irrelevant to any a priori treatment of their equilibrium thermodynamics. Consider now the macroscopically uniaxial phases of a nematic liquid, Le., those where there exists a direction about which the system is rotationally invariant. If the constituent molecules are themselves uniaxial and prolate, say, then a uniaxial state of the liquid derives from alignment of the long axes. If, on the other hand, the particles are uniaxial and oblate, then a uniaxial state of the orientationally ordered liquid corresponds to alignment of the short axes. For a biaxially symmetry molecule, either the long or the short axis will order, depending on whether the particle is more prolate or more oblate. Thus as we go from a “cigar” to a “pancake”,passing continuously through all intermediate biaxial shapes, we move from long- to short-axis alignment. More explicitly we suppose that the particle is an ellipsoid whose three principal axes are all different: c > b > a. We can go from cigar to pancake by fixing c and a and then varying b from b = a (cigar) to b = c (pancake). Only in these two limits is the particle uniaxial; for all intermediate values it is biaxial. Figure 5i shows the corresponding phase diagram, plotted as transition temperature ( T J vs. shape b. For sufficiently prolate molecules ( b < b*), the liquid undergoes a first-order phase transition from isotropic to positive (i.e., long-axis ordered) uniaxial; for b > b* the I U change is again first-order but now the nematic involves alignment of the short axes. As b approaches b* from either side the first orderness becomes vanishingly small. For the special, intermediate biaxial shape b = b*, the orientational ordering tendencies

-

(43)P.G. de Gennes, Mol. Cryst. Li9. Cryst., 34, 177 (1977). (44)D. DuprB, private communication;Mol. Cryst. Li9. Cryst. Lett., in press. (45)R. B. Meyer, private communication.

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The Journal of Physical Chemistry, Vol. 86, No. 22, 1982

-

b=o

\1

I

b=b*

b=c

“rodness”

I

L

/

x,.l

/‘

-

1

xr=xy*=1/2

xr=o

“rodness“

Figure 5. (i) The dependence of liquid crystal transition temperature on b , the middle axis of a hard (a < b < c ) biaxial particle. b = a and b = c correspond to the “rod” (prolate) and ”plate” (oblate) limits; b is the special value for which the I (isotropic) U (uniaxial) B (biaxlal) sequence sidesteps the uniaxial phase. The sola and dashed lines refer to the first- and second-order coexistence curves, respectively. The +(-) subscript on U describes the sign of the longaxis order parameter (9).(ii) A similar “phase diagram” for a binary mixture of hard rods and plates whose anisotropies are equal; X , is the mole fraction of rods. -+

-

of the long- and short-axes are equal and the isotropic liquid undergoes directly a second-order transition to a biaxial (B) nematic. Whenever b is not identically equal to b* the path followed is I U B, the successive steps being first- and second-order, respectively.& Here the biaxial state corresponds to both the long and short molecular axes being aligned-for b < b*, for example, not only do the long axes tend to point ,up” but the plane normals (short axes) prefer to head “right” say. To date of course no biaxial state of a one-component nematic liquid has ever been observed. Apparently the liquid freezes before a biaxial state can appear. So the “B” regions of the phase diagram in Figure 5i are somewhat academic. Instead the important point is that the deviations of molecular symmetry from cylindrical ( b # a and b # c ) effect significantly the behavior of the isotropic to uniaxial coexistence. In particular the first orderness of this transition is found to be drastically decreased by particle biaxiality. Recall that all a priori theories of rodlike molecules lead to volume and entropy discontinuities that are one to two orders of magnitude too large. Allowing for c:b:a = 4:2:1 instead of c:b:a = 4:1:1, for example, decreases these values by a factor of ten.16v30aThus molecular biaxiality is seen to provide a key mechanism for the weak first orderness which characterizes the isotropic to nematic phase transition. It is instructive in this context to consider a binary mixture of uniaxial molecules, one prolate and the other oblate. This system is largely isomorphous to the “neat” liquid of biaxial particles discussed above. In the onecomponent liquid, biaxiality arises directly because the

--

(46)M. J. Freiser, Phys. Reu. Lett., 24, 1041 (1970);Mol. Cryst. L ~ Q . Cryst., 13, 281 (1972);C. S.Shih and R. Alben, J. Chem. Phys., 57,3055 (1972);R.Alben, Phys. Rev. Lett., 30,778 (1973);J. P.Straley, Phys. Reu., 10, 1881 (1974).

Gelbart

middle molecular axis ( b ) is different from the short ( a ) and long (c) ones. In the mixture, on the other hand, biaxiality enters via the rod-plate intera~tion.~’Suppose for example, that the rods and plates have equal hard-core volumes and anisotropies: v, = vp and a,/c, = c p / a p(with b, = a, and bp = cp)-for simplicity we neglect here the effects of interparticle attractions. Then whichever species in the binary mixture is more abundant will have a greater tendency to order. In the case where the mole fraction X , of rods is greater than 0.5, say, the rods will order-i.e., point along z , say-as soon as the temperature is sufficiently low. Then, because of the anisotropy of the rodplate (excluded volume) interaction, the plates will be forced to keep out of the rods’ way by “slipping” in between them with their short axes in the x y plane, i.e., perpendicular to the preferred direction for the rods. As the temperature is lowered further this uniaxial state will give way continuously to a biaxial one in which the short axes of the plates are ordered further, pointing preferentially along the x direction, say. Figure 5ii shows the phase diagram obtained47bfor the “equal particle anisotropy” case outlined above. Again, as in Figure 5i, the solid and dotted lines refer to first- and second-order coexistence curves, respectively. For any “lopsided” (x, # 0.50) mixtures, cooling at constant pressure leads to the series of transitions I U B. In the 50-50 ( X I= 0.50) case, the system passes directly from an isotropic into a biaxial state where the rods and plates are ordered to equal extents. Note that XI= 0.50 in Figure 5ii corresponds to the point b = b* in Figure 5i. Again, U transition is found the strength of the first-order I to decrease as X , 0.5 from either side, vanishing identically at X,= X,*= 0.5. For mixtures of rods and plates having unequal anisotropies, the T vs. X , phase diagrams are somewhat more ~omplicated.~’These behaviors, and in particular the dependence of T,, on the mole fraction (both transition temperature “depressions” and “elevations”),account47bfor many of the recently compiled data48 on the “doping” of nematic liquids of “rods” (“plates”) with “plates” (”rods”).

--

-

-

IV. Nematic Behavior in Other Systems A . Soap Solutions. In all of the above we have been dealing with long-range orientational ordering (LROO) in liquids composed of small rigid, anisotropic molecules. We have seen that LROO (e.g., isotropic to nematic phase transition) occurs because-at low temperature and high density-the decrease in rotational (“mixing”)entropy is offset by a lowering of the potential energy and an increase in the translational (“packing”)entropy. The microscopic basis for the anisotropic interaction derives from both angle-dependent attractions (dispersional, multipolar) and repulsions (electron-charge-cloud overlap) between pairs of particles. The details of the isotropic-nematic phase transition depend on the specific features of the attractions and repulsions, and on the extent of molecular biaxiality, etc. But the key qualitative ingredient-the basic driving force for the LROO-is simply the anisotropy of the interacting particles. Thus it is not surprising that isotropic-nematic phase transitions have recently been observed in systems which have no direct physical connection with the Yusualnliquid crystal circumstances. Consider first the case of aqueous solutions of amphiphilic (“soap”) molecules.49 A typical example is sodium (47)R. Alben, J. Chem. Phys., 59, 4299 (1973); Y. Rabin, W. McMullen, and W. M. Gelbart, Mol. Cryst. L ~ QCryst., . in press. (48) R. E. Goozner and M. M. Labes, Mol. Cryst. L ~ QCryst., . 56,75 (1979);F.Hardouin, G. Sigaud, M. F. Achard, and H. Gasparoux, Mol. Cryst. Liq. Cryst., 58, 155 (1979);Chem. Phys. Lett., 48, 122 (1977).

The Journal of Physical Chemistry, Vol. 86, No. 22, 1982 4305

Feature Article

0 II

Na-0-S-0-C12

II

H25

0

(a 1

o-(b) Figure 6, (a) Structural formula for the amphiphile sodium dodecyl sulfate and (b) Its schematic depiction, showing the hydrophilic (sulfate) “head” and hydrophobic (alkyl) “tail”.

dodecyl sulfate (SDS), shown in Figure 6a. The molecule dissociates in water, leaving an ionic “head” (SO4-) and alkyl “tail”, as depicted schematically in Figure 6b. Due to hydrophobic interactions the molecules assemble at high enough concentrations into aggregates of various shapes and sizes. The most commonly considered shapes for these micelles are spherical, “cylindrical”, and disklike. The cylinders are thought of as having half-spherical caps at their ends, and similarly each of the “disks” is ringed by a half-torus-these structures enable the hydrophobic “tails” to avoid the aqueous phase, “protected” from it by the hydrophobic “heads”. Which particular shape will be favorable is determined primarily by packing constraints associated with geometrical properties of the individual molecules,50and secondarily by the concentrations of amphiphile (and of salt and cosurfactants like decanol) in the aqueous solution. Over a limited range of concentrations, isotropic suspensions of rodlike micelles have been observed to undergo a transition to a nematic state51(similarly for suspensions of disklike aggregates). It is tempting to treat this new instance of LROO by applying some or all of the theories discussed above in the context of “usual” liquid crystalline behaviors. This cannot be done, however, because we are now dealing with a system of interacting particles where the fundamental anisotropic units are (not rigid bodies but rather) aggregates of many molecules. Molecules-or groups of them-can detach from one aggregate and join another, the size distributions being determined by a chemical-equilibrium-like condition. Consequently, we need to employ a statistical mechanical formalism for the concentration and temperature dependence of the thermodynamic properties of interacting anisotropic “particles” which do not maintain their integrity. Ben-Shaul and Gelbart53have taken a first step in this direction by treating the case of dilute isotropic suspensions of cylindrical (and disklike) micelles. If the micelles were rigid, then the thermodynamic properties (including LROO) induced by their mutual interactions could be straightforwardly handled by the theory of Onsagerl’ mentioned earlier. If, on the other hand, intermicelle forces are assumed negligible, then the size distribution of aggregates can be described by any of several current approache~.4~@’b~ What Ben-Shad and Gelbart have done (49)A. Ben-Naim, “Hydrophobic Interaction”, Plenum, New York, 1980,C. Tanford, ’The Hydrophobic Effect”, 2nd ed., Wiley, New York, 1980. (50)J. N. Israelachvili, D. J. Mitchell, and B. W. Ninham, J . Chem. SOC.,Faraday Tram. 2,72, 1525 (1976);J. N. Israelachvili,S.Marcelja, and R. G. Horn, Q.Reu. Biophys., 13,121 (1980);D. J. Mitchell and B. W. Ninham, J . Chem. SOC.,Faraday Trans. 2,77,601 (1981). (51) (a) J. Charvolin and Y. Hendrikx in ‘Liquid Crystals of One and Two Dimensional Order”, W. Helfrich and G. Heppke, Ed., SpringerVerlag, Heidelberg, 1980, (b) L. J. Yu and A. Saupe, J.Am. Chem. Soc., 102,4879 (1980);Phys. Reu. Lett., 45,lo00 (1980). (52)P. J. Missel, N. A. Mazer, G. B. Benedek, C. Y. Young, and M. C. Carey, J. Phys. Chem., 84,1044 (1980).

is to allow explicitly for interactions between the anisotropic micelles, even as they undergo exchange of amphiphile molecules. The interactions are assumed to be of the excluded volume type and therefore depend directly on the aggregate dimensions, i.e., on the size of the rodlike or platelike micelles. And the aggregate dimensions depend in turn on the strength and anisotropy of the intermicelle forces! This self-consistent coupling between aggregation size and intermicelle forces is contained in the defining relation for the chemical potential per molecule, p, pJs, where s is the number of amphiphiles in an aggregate:

p, = p>

+ ( k T / s ) In p S + x,

(9)

Here p , is the number of “s aggregates” per unit volume of solution and p> is the standard chemical potential ap~~-~~ propriate to a single s aggregate in the s o l ~ e n t . The second term involves the usual entropy of mixing, and the third accounts for intermicelle forces: x, = ( k T / s )In ys, where 7,-the activity coefficient of an s aggregate-can be written as a power series in the aggregate densities: yS = CB(s,r)p, + ob2)

(10)

r

Here B(s,r) is the orientational average of the volume excluded to an s aggregate by an r aggregate. Combining the chemical equilibrium condition p, = p, (for all r and s) with eq 9 we obtain the size distribution ((p,)) of interacting micelles.53 At low concentrations the N / V dependence of the size distribution enters solely through the “mixing entropy” terms ( ( l / s ) k TIn p,) in (9). At higher densities the interaction term ( ( l / s ) k TIn 7,)in (9) introduces an additional “translational entropy” contribution associated with the finite volume occupied by the aggregates. For long cylindrical micelles it is easy to sh0W5~that the excluded volume B(s,r) goes BS the length of an s aggregate and hence as s. Thus ( - ( l / s ) B ) is independent of s and intermicelle forces are expected to have little effect on the size distribution. That is, the free energy corresponding to n, long rodlike aggregates of length 1, is essentially the same as that of a system containing 2n, aggregates of length lJ2! For long disklike micelles, on the other hand, the form of B(s,r) leads53to the dependence x, S - ~ I ~ . Accordingly, excluded volume effects favor the formation of still larger disklike aggregates. Furthermore these effects can be significant, even at concentrations close to the critical value for micellization. Finally, for small aggregates, interaction contributions to the micelle size distribution can be important for both disks and rods. At much higher concentrations, intermicelle interactions become the driving force for the transition from an isotropic to an ordered solution of anisotropic aggregates. Depending on the precise conditions of temperature and of amphiphile, cosurfactant, and salt concentration, lyotropic nematic phases are found which are uniaxial with positive ( “ ~ a l a m i t i c ”or~ negative ~~) (“discotic”) character, or biaxial. (Lamellar, hexagonal, and other kinds of positionally ordered phases are also seen.) To study these systems from the above-described point of view, we must generalize the chemical potential of eq 9 to allow explicitly for an anisotropic distribution of aggregate orientations. Also, the Onsager virial expansion must be abandoned in favor of a representation which involves summing the density to all orders; the y expansion, developed earlierz2 to treat the statistical thermodynamics of neat liquid crystals, should be useful here. Many of the same con-

-

(53)A. Ben-Shaul and W. M. Gelbart, J. Phys. Chem., 86,316 (1982).

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The Journal of Physical Chemistry, Vol. 86, No. 22, 1982

siderations (e.g., E, and S-~,,, vs. Smhtiod) should apply as before, but now there is an all-important coupling between short- and long-range orientational ordering, i.e., between micellization and the isotropic to uniaxial phase transition. The aggregation into anisotropic micelles is enhanced by the I U ordering, and conversely the liquid crystal formation is made easier by bigger (and hence more anisotropic) aggregates. For example, the onset of a nematic phase should imply a “growth” of disklike micelles. Again, the microscopic mechanism for this self-consistent coupling derives from the interactions whose changing strength (orientation dependence) is controlled by the changing anisotropy of the micelles. B. Polymer Systems. An analogous situation arises in the case of nematic polymers. Here the fundamental units are still less well defined than the micelles in soap solutions. More explicitly, consider a “melt” or bulk “liquid” sample of flexible polymer molecules. Along each chain we can proceed only so far before the local direction is significantly different than from where we started. This “orientational persistence length” is proportional to the intrinsic rigidity of the flexible polymer chain and defines the dimensions of the basic interacting “particle”. As in the case of the anisotropic micelles discussed above, the size of these anisotropic units depends on the density and temperature-and on the extent of long-range orientational ordering-via their mutual interactions. That is, there is again a self-consistent coupling between the short-range organization (here the ”local rigidity” or effective angular persistence length) and the long-range collective behavior (the transition from isotropic to nematic phases of the polymer melt). The first problem which must be addressed is then: what happens to the anisotropy of a semiflexible polymer chain in the presence of a nematic field? We can attempt to answer this question on a molecular level by considering a chain whose conformational statistics are described by the rotational isomer state model. That is, we treat a molecule like polyethylene where the angle between adjacent monomers (C-C bonds) is fixed (e.g., tetrahedral bonding along the C-C-C-C backbone) and where conformations of the polymer are then defined by specifying the dihedral angles ((cpi)) that each monomer makes with respect to the plane of the previous two. Allowing only for pi = 0 (“trans”) or kl2O0 (“& gauche”), for example, gives rise to the usual three-state rotational isomer model,” which is isomorphous to the one-dimensional spin-1 Ising problem.55 Estimates of the root-mean-square end-to-end distance (and higher-order radial moments)56and of the monomer-monomer orientational correlation^^^ then follow naturally. Suppose we now “turn on” a nematic field (e.g., we dissolve our chain in a liquid crystalline solvent) and inquire into its effect on the polymer anisotropy. We could do so by just adding a -Vi77 CiP2(cosOJ term to the usual conformational energy and using the same transfer matrix techniques as in the zero field case.57 (Here Oi is the angle between the ith monomer and the special space-fixed direction imposed by solvent, 7 is its nematic P2 order parameter, and V2’ is the monomer-solvent coupling strength-it measures how strongly the field “grabs on”

-

(54) P. J. Flory, ‘Statistical Mechanics of Chain Molecules”, Wiley, New York, 1969. (55) H. E. Stanley, ‘Introduction to Phase Transitions and Critical Phenomena”, Wiley, New York, 1971. (56) J. Freire and M. Fixman, J . Chem. Phys., 69, 634 (1978), and references contained therein. (57) A. Baram and W. M. Gelbart, J . Chem. Phys., 66, 617, 4666 (1977).

Gelbart

to the axially symmetric polarizability tensor of the C-C bond.) But the Clebsch-Gordan algebra which ensues via this approach is obfuscating. By taking overlapping trimers (i.e., triplets of adjacent C-C bonds) rather than monomers as the basic unit, and by defining each monomer state with respect to spacefixed diamond-lattice directions (rather than with respect to the plane of the previous two), it is possible to cast the problem of the rotational isomer state model in a nematic field into a particularly simple form of random walk. More explicitly,even with the inclusion of third-nearest neighbor (Le., “pentane effect””) interactions between bonds, we can sidestep the usual transformations between local coordinate frame^",^^^^ and solve for the conformational statistics completely in terms of intrachain Boltzmann factors. It is then straightforward to calculates P2 order parameters for each bond as a function of the dimensionless confor- E,,)/kT-and field interaction mation energy-(E strength-Viq/krj[ the CH bonds, for example, are found to become increasingly aligned as one heads “inside” from the end of a solute chain. The C-C P,’s also increase from the outside to the inside, showing58an “odd-even’’ effect reminiscent of that reported in other contexts.59 These long-range order parameters, which jump discontinuously from zero to a few hundredths at the nematic transition of the solvent, have recently been measured by SamulskiG0for “short”-chain alkanes (CnD2n+2, with n I 36) in various liquid crystalline hosts. He has deduced values for the CD P2’s by numerically fitting deuterium quadrupolar splittings to a model which constrains the semiflexible chain to a hypothetical cylinder. In principal, one could measure as well the change, upon nematic ordering of the solvent host, of the solute polymer size (root-mean-square end-to-end distance) or persistence length (monomer-monomer angular correlation range). But both of these properties are nonzero and essentially the same58 in the isotropic and ordered environments, thereby making their changes difficult to resolve experimentally. Important contributions to the problem of polymer conformations in nematic fields are also contained in the work of M a r ~ e l j awho , ~ ~provided an essentially complete numerical analysis of the self-consistent coupling between the orderings of molecular hard cores and their alkyl chain substituents in “neat” liquid crystals. More recently, JahnigG1has treated a related physical situation-a lipid bilayer-in which the nematic (long-range) order arises from the fact that the “heads” of the interacting alkyl chains are “anchored” at the membrane surface. Again, the nematic order enhances local anisotropy of the semiflexible chains, and vice versa. In the case of single chains dissolved in liquid crystalline solvents, this self-consistent coupling-between the nematic ordering and the polymer anisotropy induced by it-is overwhelmed by an opposite one in which the long-range orientational order of the solvent is destroyed by the polymer flexibility. That is, because the angular persistence range of the solute is short compared to its actual chain length, and because of the large curvature elasticity force constants of the nematic, the aligned solvent is incompatible with the flexible polymer. Correspondingly, the I U transition temperature is dramatically

-

(58) Y. Rabin, A. Ben-Shaul, and W. M. Gelbart in “Liquid Crystals and Ordered Fluids”, J. F. Johnson and A. C. Griffin, Ed., Vol. 4, Plenum, in press. (59) S. Marcelja, J. Chem. Phys., 60, 3599 (1974). (60) E. T. Samulski in “Ferroelectrics”, Vol. 30, Gordon and Breach. New York, 1980, pp 83-93. (61) F. Jahnig, J . Chem. Phys., 70, 3279 (1979).

J. Phys. Chem. 1982, 86, 4307-4312

decreased as the sparingly soluble polymer is dissolved. This “disordering” of the nematic host has recently been studied both experimentally62and t h e ~ r e t i c a l l y . ~ ~ The self-consistent coupling between short- and longrange orientational ordering is of course not overwhelmed in the case of bulk polymer melts.64 Here each chain is “dissolved” not in a nematic of small molecules but in a sea of other long flexible polymers. Long-range orientational ordering-a nematic state-arises from the mutual (62) B. Kronbere. I. Bassienana. and D. Patterson. J. Chem. Phvs.. 82. 1714 (1978); A. Dibault, CrCas&rande, M. VeyssiC, and B. Deloche; Phys. Reu. Lett., 45, 1645 (1980). (63) F. Brochard-Wyart, C. R. Acad. Sci. (Paris), 289, 229 (1979). (64) See, for example, the following review discussions of the experimental situation: F. Cser, J. Phys. Colloq. (Orsay, Fr.), C3, Supplement No. 4, Tome 40, C3-459(1979); E. T. Samulski and D. B. DuPrB, Adu. Liq. Cryst., 4, 121 (1979). Also, many papers on the subject have been assembled in “Liquid Crystalline Order in Polymers”, A. Blumstein, Ed., Academic Press, New York, 1978.

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interaction of the closely packed anisotropic units (persistence lengths) of each chain. In turn these “particles” have their size enhanced by the nematic field and thereby stabilize further the long-range ordering.& Similar effects have recently been considered@as well in rubber elasticity contexts, where the flexible polymer chains are cross-linked to form a network.

Acknowledgment. Over the course of the past several years I have had the great pleasure of collaborating with Drs. Boris Barboy, Avinoam Ben-Shaul, and Yitzhak Rabin on the researches described in this review. I thank them heartily for making these efforts so scientifically fruitful and personally enjoyable. This work was supported in part by NSF Grant CHE80-24270. (65) P. Pincus and P. G. de Gennes, J . Polym. Sci.: Polym. Symp., 65, 85 (1978); Y. Kim and P. Pincus, ACS Symp. Ser. No. 74 (1978). (66)J. P. Jarry and L. Monnerie, Macromolecules, 12, 317 (1979).

ARTICLES Excited States and Photochemistry of Saturated Molecules. 11. Potential Energy Surfaces in Low-Lying States of Ethane James W. Caldwell and Mark S. Gordon’ Department of Chemistry, North Dakota State University, Fargo, North Dakota 58105 (Received:Februaty 1, 1982; I n Final Form: June 1, 1982)

Ab initio calculations at both the restricted Hartree-Fock (RHF) and singly excited configuration interaction (SECI)levels have been carried out on sections of the potential energy surfaces for a number of low-lying excited singlet electronic states of ethane. All vertical states below the first ionization potential are predominantly Rydberg in character, as are all minima detected on the potential energy surfaces. The lowest IE, state is predicted to have only a very small (- 1kcal/mol) barrier separating its minimum-energy structure from the 1A” ethylidene + IXg+ H2products. Since the vertical E, molecule has more than enough energy to surpass this barrier, the state is effectively dissociative. The fact that it is also electronically forbidden may explain why the UV spectrum of ethane exhibits vibrational structure.

Introduction The vacuum-UV photolyses of the light normal alkanes are dominated by molecular elimination1 of H2at threshold energies. The threshold absorption spectra2are generally broad and featureless, with the striking exception of ethane, which possesses a clearly resolvable vibrational fine str~cture.~ The lack of features in the methane spectrum is evidently due to an allowed absorption to a state which is directly diss~ciative.~ The lowest vertical singlet state is (1) P. Ausloos and S. G. Lias in “Chemical Spectroscopy and Photochemistry in the Vacuum UV”, C. Sandorfy, P. Ausloos, and M. Robin, Eds., Reidel, Dordrecht, Netherlands, 1974, p 465. (2) B. A. Lombos, P. Sauvageau, and C. Sandorfy, J.Mol. Spectrosc., 24, 253 (1967). (3) E. F. Peamon and K. K. Innes, J. Mol. Spectrosc., 30,232 (1969). (4) M. S. Gordon and J. W. Caldwell, J. Chem. Phys., 70,5503 (1979); M.S.Gordon, Chem. Phys. Lett., 52, 161 (1977). 0022-3654/82/2086-4307$01.25/0

11T2which distorts from Tdto C2”symmetry and dissociates with no barrier to methylene (CH,) and HP. For C3 and higher alkanes, the lack of vibrational structure may be due to excitation into dissociative states or to a high density of vibronically coupled allowed states. The case of ethane is intriguing since the threshold photolyses5*were performed with 8.4-eV radiation, whereas the presumed 0-0 peak in the (allowed) absorption spectrum occurs at 8.65 eV. This indicates that the predominant threshold reaction may arise from absorption to a formally dipole-forbidden state. Calculations on the vertical spectrum of ethane6,7 (5) (a) R. F. Hampson, J. R. McNesbey, H. Akimoto, and I. Tanaka, J . Chem. Phys., 40, 1099 (1964); (b) S. G. Lias, G. J. Collins, R. E. Rebbert, and P. J. Ausloos, ibid., 52, 1841 (1970).

0 1982 American Chemical Society