References
- Pincus , P. 1969 . Solid State Commun. , 7 : 415 See also Vold, R. R., 1985, Nuclear Magnetic Resonance of Liquid Crystals (NATO ASI Series, Vol. 141, Chap. 11). Edited by J. W. Emsley (Reidel).
- Redfield , A. G. 1957 . I.B.M. Jl , 1 : 19 Slighter, C. P., 1980, Principles of Magnetic Resonance (Springer-Verlag), Chap. 5.
- Polnaszek , C. F. and Freed , J. H. 1975 . J. phys. Chem. , 79 : 2283
- De Gennes , P. G. 1974 . The Physics of Liquid Crystals , Clarendon Press . Chap. 5.
- Landau , L. D. and Lifshitz , E. M. 1986 . Course of Theoretical Physics , 3rd edition Vol. 7 , Pergamon Press . Chap. VI.
- Chandrasekhar , S. Liquid Crystals , Edited by: Woolfson , M. M. and Ziman , J. M. Cambridge Monographs on Physics, Cambridge University Press . Chap. 3
- Freed , J. H. 1977 . J. chem. Phys. , 66 : 4183
- Selwyn , L. S. , Vold , R. R. and Vold , R. L. 1984 . J. chem. Phys. , 80 : 5418 The lowest order theory predicts that only J01 becomes frequency dependent, whereas experiments indicate that J02 also becomes frequency dependent, cf. [stilde]6., Luyten, P. R., Vold, R. R., and Vold, R. L., 1985, J. phys. Chem., 1984, 89, 545.
- Plomp , L. , Schreurs , M. and Bulthuis , J. 1988 . J. chem. Phys. , 88 : 5202 See also Vold, R. L., Vold, R., and Warner, M., 1988, J. chem. Soc. Faraday Trans. II, 84, 997.
- Landau , L. D. and Lifshitz , E. M. 1986 . Course of Theoretical Physics , 3rd edition Vol. 5 , Pergamon Press . Chaps 12 and 13, esp. [stilde]140.
- Bedeaux , D. and Mazur , P. 1974 . Physica , 73 : 431 Even if it is not allowed to separate the time scales progress is still possible, although not along the lines presented in this paper. A more general approach based for the Langevin equations for molecular reorientation and director field motion is currently under investigation. For the method see:
- Abragam , A. 1986, 1974 . Principles of Nuclear Magnetism , Clarendon Press . Chap. 7.
- Rose , M. E. 1954 . Elementary Theory of Angular Momentum , Wiley . Chaps IV and V.
- Weber , G. 1971 . J. chem. Phys. , 55 : 2399 A jump model for rotational reorientation was suggested by:, For the collision model see, e.g., Polnaszek, C. F., Bruno, G. V., and Freed, J. H., 1975, J. chem. Phys., 58, 3185. Rotational diffusion was treated by Favro, L. D., 1960, Phys. Rev., 119, 53. Recently a mixed model was proposed by Vold, R. R., and Vold, R. L., 1988, J. chem. Phys., 88, 1443.
- Onsager , L. 1949 . Ann. N.Y. Acad. Sci. , 51 : 627 We should be cautious here. The vector field n is the ensemble average of a microscopic quantity related to the direction of the z axis of a (liquid crystal) molecular coordinate frame. See e.g., or de Gennes, P. G., 1974, The Physics of Liquid Crystals (Clarendon Press), Chap. 2. A different approach is to construct a tensor field as the average of a tensorial molecular property, e.g., the inertia tensor. This tensor field and the tensor field nn are not necessarily equal since correlations are neglected in the second case. It can be shown, however, that for the nematic phase of a liquid crystal the results are the same. van der Zwan, G., 1988, Physica A, 50, 299.
- Reference [9], especially note 19.
- Reichl , L. E. 1984 . A Modern Course in Statistical Physics , University of Texas press . Chap. 10. This is of course a consequence of expansion to second order in the variables of the free energy, cf., e.g., [10] or
- For a gaussian distribution in x the relation <x2n = (2n)!<x2>n/n!22n holds. This approaches infinity for large values of n.
- For the director field modes the behaviour of the variance for small k is 1/k2, cf. [stilde]4. This has to be summed over all k. The next highest non-zero moment thus behaves as 1/k4, and in that case summation diverges as 1/k for small values of k. If we first sum the complete series the 1/k2 ends up in the exponential and exp (—1/k2) does not diverge at k = 0. See also appendix B.
- Values of λ can be obtained from quadrupolar line splittings and are generally between 0·5 and 2kT, cf. table 1.
- Blinc , R. N.M.R. Basic Principles and Progress , Edited by: Pintar , M. M. Vol. 13 , 97 – 111 . Springer Verlag . See also It is not always appreciated that there are only two fluctuating modes for a nematic, cf., e.g., Berreman, D. W., 1975, J. chem. Phys., 62, 776.
- Haller , I. 1975 . J. chem. Phys. , 57 : 1400 The values for K obtained from equation (4.14) does not differ greatly from the average value of the the Ks used in the one-constant approximation. For instance for 4-methoxybenzylidene-4′-n-butylaniline we have K1 = 5·7 pN, K2 = 3·5 pN and K3 = 7·2 pN at T - TNI = 16·6 K, see; this gives for the average K: 5·47 pN and for the K from eqation (4.14): 5·20 pN. See also Faber, T. E., 1977, Proc. R. Soc. A, 353, 247.
- van Kampen , N. G. 1977 . Stochastic Processes in Physics and Chemistry , North-Holland Physics Publishing . Chap. 1.
- From the context it should be clear that we use the symbol <…> for two different types of averaging. In this section up to equation (4.18) it was used to represent the average over the equilibrium fluctuations of the director field; in this and the following paragraph it symbolizes an average over molecular orientations including director fluctuations.
- In fact this point deserves a more detailed investigation, which is beyond the scope of this paper. The condition is only marginally satisfied for liquid crystal molecules and cut-off wavelengths of the order of magnitude indicated.
- The appropriate viscosities are complicated k-dependent combinations of the Leslie coefficients and Miesowicz viscosities, see e.g., [21] and [4], Chap. 5, p. 180; or Groupe d'Etude des Cristaux Liquides (Orsay), 1969, J. chem. Phys., 51, 816.
- Reference [23], Chap. 4.
- Plomp , L. , Loman , A. C. and Bulthuis , J. J. chem. Phys. , 84 6591 Explicit expressions for this integral can be found in
- Faber , T. E. 1977 . Proc. R. Soc. A , 353 : 277
- To be submitted.
- For instance for xylene it was found that δ = 1·72 and ϱ - 0·35; Schreurs, M. (unpublished results).
- Warner , M. 1984 . Molec. Phys. , 52 : 677 In this paper the influence of director fluctuations is severely over estimated since the fundamental restriction on director fluctuations was overlooked. On the basis of these results, which can be restated in our notation as <D2 00(x)/<D2 00)(α = 0) = 1 - 3α2/2 the cut-off wavelength is also overestimated in Zanoni, C., 1986, J. chem. Phys., 84, 426. For ℷ = 5 (where the molecular order parameter is ≈ 0·65) and α = 0·05 we find <D2 00>(α)/<D2 00>(α = 0) = 0·97, whereas Zanoni would find 0·925 for this ratio which shows that at these high λ values renormalization is still relatively small without having to resort to extemely high cut-off values for kc. This reduces the cut-off wavelength found in this last paper by a factor of 3, which makes it, however, still too large to explain the experimentally found frequency dependence in the spectral density J01. It is interesting to note that in the (improper) limit where we let the fluctuating term dominate (αλ → ∞) our molecular order parameter approaches 0·25, whereas Zanoni finds 0·24; see also Faber, T. E., 1977, Proc. R. Soc. A, 353, 247.
- Maier , W. and Saupe , A. 1958 . Z. Naturforschg , a13 : 564 For <θ2<1/2 ≈ 10°-20°.; 1958, a13, 566; 1959, a14, 882; 1960, a15, 287.; 1958, a13, 566; 1959, a14, 882; 1960, a15, 287.
- Gierer , A. and Wirtz , K. 1953 . Z. Naturforschg. , A8 : 532 The translational diffusion constant used in the earlier references is given by Dtr = kT/6≈a, sometimes taken with corrections for size of the particles; for a different approach see Miyajima, S., Nakamura, N., and Chihara, H., 1983, Bull. chem. Soc. Japan, 56, 400.
- Plomp , L. and Bulthuis , J. 1988 . Liq. Crystals , 3 : 927