International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 4.4, pp. 549551
Section 4.4.2. The nematic phase ^{a}Division of Engineering and Applied Science and The Physics Department, Harvard University, Cambridge, MA 02138, USA 
The nematic phase is a fluid for which the molecules have longrange orientational order. The phase as well as its molecular origin can be most simply illustrated by treating the molecules as long thin rods. The orientation of each molecule can be described by a symmetric secondrank tensor , where n is a unit vector along the axis of the rod (De Gennes, 1974). For disclike molecules, such as that shown in Fig. 4.4.1.3(f), or for micellar nematic phases, n is along the principal symmetry axis of either the molecule or the micelle (Lawson & Flautt, 1967). Since physical quantities such as the molecular polarizability, or the moment of inertia, transform as symmetric secondrank tensors, either one of these could be used as specific representations of the molecular orientational order. The macroscopic order, however, is given by the statistical average , where is a unit vector along the macroscopic symmetry axis and S is the order parameter of the nematic phase.
The microscopic origin of the phase can be understood in terms of steric constraints that occur on filling space with highly asymmetric objects such as long rods or flat discs. Maximizing the density requires some degree of shortrange orientational order, and theoretical arguments can be invoked to demonstrate longrange order. Onsager presented quantitative arguments of this type to explain the nematic order observed in concentrated solutions of the long thin rods of tobacco mosaic viruses (Onsager, 1949; Lee & Meyer, 1986), and qualitatively similar ideas explain the nematic order for the shorter thermotropic molecules (Maier & Saupe, 1958, 1959).
The existence of nematic order can also be understood in terms of a phenomenological meanfield theory (De Gennes, 1969b, 1971; Fan & Stephen, 1970). If the freeenergy difference between the isotropic and nematic phases can be expressed as an analytic function of the nematic order parameter , one can expand as a power series in which the successive terms all transform as the identity representation of the point group of the isotropic phase, i.e. as scalars. The most general form is given byThe usual meanfield treatment assumes that the coefficient of the leading term is of the form , where T is the absolute temperature and is the temperature at which . Taking a, D and , one can show that for either positive or negative values of B, but for sufficiently large T, the minimum value of occurs for , corresponding to the isotropic phase. For , can be minimized, at some negative value, for a nonzero corresponding to nematic order. The details of how this is derived for a tensorial order parameter can be found in the literature (De Gennes, 1974); however, the basic idea can be understood by treating as a scalar. If we write and if is defined by the condition , then for both and . This value for marks the transition temperature from the isotropic phase, when and the only minimum is at with , to the nematic case when and the absolute minimum with is slightly shifted from . The symmetry properties of secondrank tensors imply that there will usually be a nonvanishing value for B, and this implies that the transition from the isotropic to nematic transition will be first order with a discontinuous jump in the nematic order parameter . Although most nematic systems are uniaxial, biaxial nematic order is theoretically possible (Freiser, 1971; Alben, 1973; Lubensky, 1987) and it has been observed in certain lyotropic nematic liquid crystals (Neto et al., 1985; Hendrikx et al., 1986; Yu & Saupe, 1980) and in one thermotropic system (Malthête et al., 1986).
The Xray scattering cross section of an oriented monodomain sample of the nematic phase with rodlike molecules usually exhibits a diffuse spot like that illustrated in Fig. 4.4.1.2(a), where the maximum of the cross section is along the average molecular axis at a value of , where to 40.0 Å is of the order of the molecular length L. This is a precursor to the smecticA order that develops at lower temperatures for many materials. In addition, there is a diffuse ring along the directions normal to at , where is comparable to the average radius of the molecule. In some nematic systems, the nearneighbour correlations favour antiparallel alignment and molecular centres tend to form pairs such that the peak of the scattering cross section can actually have values anywhere in the range from to . There are also other cases where there are two diffuse peaks, corresponding to both and which are precursors of a richer smecticA morphology (Prost & Barois, 1983; Prost, 1984; Sigaud et al., 1979; Wang & Lubensky, 1984; Hardouin et al., 1983; Chan, Pershan et al., 1985). In some cases, and competition between the order parameters at incommensurate wavevectors gives rise to modulated phases. For the moment, we will restrict the discussion to those systems for which the order parameter is characterized by a single wavevector.
On cooling, many nematic systems undergo a secondorder phase transition to a smecticA phase and as the temperature approaches the nematic to smecticA transition the widths of these diffuse peaks become infinitesimally small. De Gennes (1972) demonstrated that this phenomenon could be understood by analogy with the transitions from either normal fluidity to superfluidity in liquid helium or normal conductivity to superconductivity in metals. Since the electron density of the smecticA phase is (quasi)periodic in one dimension, he represented it by the form where d is the thickness of the smectic layers lying in the plane. The complex quantity is similar to the superfluid wavefunction except that in this analogy the amplitude describes the electrondensity variations normal to the smectic layers, and the phase describes the position of the layers along the z axis. De Gennes proposed a meanfield theory for the transition in which the freeenergy difference between the nematic and smecticA phase was represented by This meanfield theory differs from the one for the isotropic to nematic transition in that the symmetry for the latter allowed a term that was cubic in the order parameter, while no such term is allowed for the nematic to smecticA transition. In both cases, however, the coefficient of the leading term is taken to have the form . If , without the cubic term the free energy has only one minimum when at , and two equivalent minima at for . On the basis of this free energy, the nematic to smecticA transition can be second order with a transition temperature and an order parameter that varies as the square root of . There are conditions that we will not discuss in detail when D can be negative. In that case, the nematic to smecticA transition will be first order (McMillan, 1972, 1973a,b,c). McMillan pointed out that, by allowing coupling between the smectic and nematic order parameters, a more general free energy can be developed in which D is negative. McMillan's prediction that for systems in which the difference is small the nematic to smecticA transition will be first order is supported by experiment (Ocko, Birgeneau & Litster, 1986; Ocko et al., 1984; Thoen et al., 1984). Although the meanfield theory is not quantitatively accurate, it does explain the principal qualitative features of the nematic to smecticA transition.
The differential scattering cross section for Xrays can be expressed in terms of the Fourier transform of the density–density correlation function . The expectation value is calculated from the thermal average of the order parameter that is obtained from the freeenergy density . If one takes the transform the freeenergy density in reciprocal space has the form and one can show that for the cross section obtained from the above form for the free energy is where the term in has been neglected. The meanfield theory predicts that the peak intensity should vary as and that the half width of the peak in any direction should vary as . The physical interpretation of the half width is that the smectic fluctuations in the nematic phase are correlated over lengths .
One of the major shortcomings of all meanfield theories is that they do not take into account the difference between the average value of the order parameter and the instantaneous value , where represents the thermal fluctuations (Ma, 1976). The usual effect expected from theories for this type of critical phenomenon is a `renormalization' of the various terms in the free energy such that the temperature dependence of correlation length has the form , where , is the secondorder transition temperature, and is expected to have some universal value that is generally not equal to 0.5. One of the major unsolved problems of the nematic to smecticA phase transition is that the width along the scattering vector q varies as with a temperature dependence different from that of the width perpendicular to q, ; also, neither nor have the expected universal values (Lubensky, 1983; Nelson & Toner, 1981).
The correlation lengths are measured by fitting the differential scattering cross sections to the empirical form: The amplitude , where the measured values of are empirically found to be very close to the measured values for the sum . Most of the systems that have been measured to date have values for and to 0.2. Table 4.4.2.1 lists sources of the observed values for , and . The theoretical and experimental studies of this pretransition effect account for a sizeable fraction of all of the liquidcrystal research in the last 15 or 20 years, and as of this writing the explanation for these two different temperature dependences remains one of the major unresolved theoretical questions in equilibrium statistical physics.

It is very likely that the origin of the problem is the QLRO in the position of the smectic layers. Lubensky attempted to deal with this by introducing a gauge transformation in such a way that the thermal fluctuations of the transformed order parameter did not have the logarithmic divergence. While this approach has been informative, it has not yet yielded an agreedupon understanding. Experimentally, the effect of the phase can be studied in systems where there are two competing order parameters with wavevectors that are at and (Sigaud et al., 1979; Hardouin et al., 1983; Prost & Barois, 1983; Wang & Lubensky, 1984; Chan, Pershan et al., 1985). On cooling, mixtures of 4hexylphenyl 4(4cyanobenzoyloxy)benzoate (DB_{6}) and N,N′(1,4phenylenedimethylene)bis(4butylaniline) (also known as terephthalbisbutylaniline, TBBA) first undergo a secondorder transition from the nematic to a phase that is designated as smecticA_{1}. The various smecticA and smecticC morphologies will be described in more detail in the following section; however, the smecticA_{1} phase is characterized by a single peak at owing to a onedimensional density wave with wavelength d of the order of the molecular length L. In addition, however, there are thermal fluctuations of a secondorder parameter with a period of 2L that give rise to a diffuse peak at . On further cooling, this system undergoes a second secondorder transition to a smecticA_{2} phase with QLRO at , with a second harmonic that is exactly at . The critical scattering on approaching this transition is similar to that of the nematic to smecticA_{1}, except that the preexisting density wave at quenches the phase fluctuations of the order parameter at the subharmonic . The measured values of (Chan, Pershan et al., 1985) agree with those expected from the appropriate theory (Huse, 1985). A meanfield theory that describes this effect is discussed in Section 4.4.3.2 below.
It is interesting to note that even those systems for which the nematic to smecticA transition is first order show some pretransitional lengthening of the correlation lengths and . In these cases, the apparent at which the correlation lengths would diverge is lower than and the divergence is truncated by the firstorder transition (Ocko et al., 1984).
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