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Principles of Soft-Matter Dynamics pp 499–548Cite as

Molecular and Collective Dynamics in Liquid Crystals and Other Mesophases

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Abstract

In a sense, liquid crystals are strongly related to polymers as concerns molecular dynamics. Analytical treatments on the basis of suitable equations of motions result again in solutions based on collective relaxation modes. The features how order-director fluctuations manifest themselves in NMR relaxation experiments and dynamic light scattering will be figured out in detail. Molecular dynamics in lipid bilayers, often considered as biological model membranes, and other layered structures are further examples belonging to this category. This chapter also addresses less common subjects such as shape fluctuations of vesicles and the consequences of the diffusion of structural defects on dielectric relaxation. Finally, the dynamics in the ordered mesophase of poly(dialkylsiloxane) materials will be examined as a relatively exotic but nevertheless most interesting research area.

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Notes

  1. 1.

    At this point, it should be noted that this sort of polymers must not be confused with liquid-crystalline side or main-chain polymers which actually contain mesogenic groups leading to ordered phases in analogy to conventional liquid crystals.

  2. 2.

    This is in contrast to partly immobilized systems such as isotropic polymer networks, for example (see Sect. 5.3): The temporal average of the restricted orientations of a mesh-chain suggests a finite value of the mean second-order Legendre polynomial in Eq. (6.3) while the (powdery) ensemble average over all mesh chains vanishes. Ergodicity is violated, and the term “order” is ill-defined in this case. On the other hand, if the network is strongly stretched so that ensemble and time averages of mesh orientations approach each other, the term “order” may be appropriate in turn.

  3. 3.

    Note that there is a subtle difference between, on the one hand, the energy of a (variable) magnetization, Eq. (6.7) multiplied by the volume, and the Zeeman energy of a (fixed) nuclear dipole in a magnetic field, Eq. (3.2), on the other!

  4. 4.

    A treatment differentiating all three Frank elastic constants will follow in Sect. 6.7.2.

  5. 5.

    From a molecular point of view, shear stress can also be interpreted as torque per unit volume with a vector direction perpendicular to shear stress.

  6. 6.

    Contrary to the usual representation [1], it was argued in Ref. [17] that the right-hand side of Eq. (6.32) should in principle be supplemented by a term \( - {\mathop\frown\limits_{K} }\left( {{n}\cdot {{\nabla}^2}{n}} \right){n} \), that is, by a vector aligned along the director \( {n} = {{{n}}_0} + \delta {n} \). Since we are referring here to shear deformations in lateral directions, that is, orientational fluctuations, this term will not be pursued any further.

  7. 7.

    Note that we distinguish between electric polarization \( \hat{{P}} \) and wave polarization. The latter is defined by the orientation of the electric field vector \( \hat{{E}} \).

  8. 8.

    This is in contrast to spin–lattice relaxation. In this case, flip-flop spin diffusion (see the discussion following Eqs. 5.169 and 5.247) as an additional exchange mechanism mediates averaging over all heterogeneities. Spin–lattice relaxation curves therefore tend to decay mono-exponentially in a wide range, whereas transverse relaxation is subject to a superposition of virtually two exponential functions.

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  1. Sektion Kernresonanzspektroskopie, Universität Ulm, Ulm, Germany

    Rainer Kimmich

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Kimmich, R. (2012). Molecular and Collective Dynamics in Liquid Crystals and Other Mesophases. In: Principles of Soft-Matter Dynamics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5536-9_6

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