Meier theory

For comparison, the results obtained using the Maier-Meier theory [4] are also shown this is a generalization of the Onsager model [13] to uniaxial media. The same dipole moment used for the calculations with the molecular shaped cavity was assumed, and the radius a was taken to be 3.9 A, a value derived from the density of the system. Improvement of the predictions, when the sphere is replaced by a molecular shaped... 

Figure 2.33 Dielectric permittivity of 5CB. Experimental data [2] (solid line), and theoretical results obtained with the IEF method (filled diamonds) and with the Maier-Meier theory [4] (open diamonds).

X values for the solvent cast polymer can also be calculated using the theory of Meier (8). We assumed (8s bd) = 0.6 and M = Mn(PS) + 0.5 Mn(PBD) = 35000 and obtained X 0.5. Considering the above referred to uncertainties in the accuracy of the radioluminescence data and the only limited applicability of Meiers theory (the blocks do not have equal length and the concentration gradient is sigmoidal rather than linear as assumed here), one must be surprised about the closeness of the X values obtained from theory and experimental data. 

Hesselink has argued that whilst this procedure ensures that bonds 1 to A do not transgress the interfaces, it places no restrictions on the bonds k 1) to n. The summation in equation (11.25) thus includes some conformations that are forbidden by the presence of the impenetrable barriers. Since the smaller is k, the more likely are the subsequent segments to cross the plates, the counting of these forbidden conformations should lead to a segment distribution that is biased towards conformations close to the interface, i.e. the segment distribution is compressed too close to the interface. The result of this artifact is that the Meier theory predicts values for the interaction between plates that are usually ca 20-30% too low. 

Hesselink, Vrij and Overbeek s extension of the Meier theory... 

In experiment on nematic liquid crystals, both positive and negative anisotropy <, is observed, the sign depending on chemical structure. The magnitude of e is often proportional to orientational order parameter S. In the isotropic phase the anisotropy disappears. Typical temperature dependencies of n and , are shown in Fig. 7.5. These observations can be accounted for by the Maier-Meier theory [5]. The latter is based on the following seven assumptions ... 

Fig. 7.7 Location of a molecular dipole moment with respect to the longitudinal molecular axis of a molecule. Note that in the Maier-Meier theory the dipole moment forms angle P with the axis of maximum polarizability of a spherical molecule...

This effect originates from the anisotropic dipole-dipole correlations not accounted for by the Maier-Meier theory operating with a single particle distribution function. When, with decreasing temperature, the smectic density wave p(z) develops (even at the short-range scale) the longitudinal dipole moments prefer to form antiparallel pairs and the apparent molecular dipole moment becomes smaller. This would reduce positive s. Theoretically, dipole-dipole correlations may be taken into account by introducing the so-called Kirkwood factors. 

Recently, Sharma has proposed some extension of the Maier-Meier approach to the case of nematogens with antiparallel dipole-dipole correlations of the molecules. He treated a polar LC material as a mixture of unpaired molecules with a finite dipole moment /u. and antiparallel pairs with zero dipole moment. The molecules interact with each other through a combination of the generalized Maier-Saupe pseudopotential for nematic mixtures and a reaction field energy term calculated from an extension of the Maier-Meier theory. Additionally, it was assumed that a dipole with dipole moment fi is embedded in a spherical cavity of dielectric permittivity n, which is surrounded by a medium of average dielectric permittivity e. In that case the expressions for the cavity field factor h and the reaction field factor / are given by h + n ), /= (e - rt")/[2rre a (2e-i-n )] and the left sides of... 

The dielectric anisotropy Ae of LC-materi-als is defined by Ae= j -ex, where and ej. are the dielectric constants parallel and perpendicular to the director. From the Maier and Meier theory it can be seen that both the polarizability anisotropy Aa and the permanent dipole movement /i of the LC molecule determine the dielectric anisotropy... 

EQNS (1), (2) and (3) are commonly used as the basis of the molecular interpretation of the static permittivities measured as a function of temperature (e.g. [6,12-15,20,21,28-30,36,37]) and/or pressure [30,38]. Recently, quite successful predictions of anisotropic optical and dielectric constants from molecular modelling calculations were achieved with the aid of the Maier-Meier theory [39,40]. It seems worthwhile, therefore, to analyse these equations in order to point out the weak and strong points of the theory. EQN (3) is the most convenient for this purpose (both components of the permittivity are discussed by Jadzyn et al in a recent paper [41]). The parameters N, F and h in the Maier-Meier equations vary little with temperature. Therefore, the contribution from the polarisability anisotropy Aa to A8 varies with temperature in the same way as the order parameter S, whereas that connected with the orientation polarisation varies like S/T. Especially interesting seems to be the case of constant temperature discussed in [38] where As was measured as a function of pressure, p. The discussion of the measured permittivities, Sj and the anisotropy As as a function of the order parameter S obtained from the independent experiment seems to be the best way of verifying the assumptions on which the theory is based. 

Figure 2-38B. Net expansion factor, Y, for compressible flow through nozzles and orifices. By permission, Crane Co., Technical Paper 410, Engineering Div., 1957. Also see 1976 edition and Fluid Meiers, Their Theory and Application, Part 1, 5th Ed., 1959 and R. G. Cunningham, Paper 50-A-45, American Society of Mechanical Engineers.

Meier D.J., Theory of block copolymers Domain formation in A-B block copolymers, J. Polym. Sci., Part C, 26, 81, 1969. 

Memory effects play an important role for the description of dynamical effects in open quantum systems. As mentioned above, Meier and Tannor [32] developed a time-nonlocal scheme employing the numerical decomposition of the spectral density. The TL approach as discussed above as well as the approaches by Yan and coworkers [33-35] use similar techniques. Few systems exist for which exact solutions are available and can serve as test beds for the various theories. Among them is the damped harmonic oscillator for which a path-integral solution exists [1], In the simple model of an initially excited... 

Dale Meier has been one of the firsts if not the first who presented a theory of domain formation in block copolymers39. In its original version39, Meier s theory was restricted to AB block copolymers and spherical domains. In a series of following papers40-45, however, Meier has refined his theory considering different shapes of domains, the effect of the presence of a solvent, the dimensions of the interface, the interfacial properties of block copolymers and the solubilization of homopolymers by copolymers. 

Helfand (25,26,27,28,29) has formulated a statistical thermodynamic model of the microphases similar to that of Meier. This treatment, however, requires no adjustable parameters. Using the so-called mean-field-theory approach, the necessary statistics of the molecules are embodied in the solutions of modified diffusion equations. The constraint at the boundary was achieved by a narrow interface approximation which is accomplished mathematically by applying reflection boundary conditions. 

A number of statistical thermodynamic theories for the domain formation in block and graft copolymers have been formulated on the basis of this idea. The pioneering work in this area was done by Meier (43). In his original work, however, he assumed that the boundary between the two phases is sharp. Leary and Williams (43,44) were the first to recognize that the interphase must be diffuse and has finite thickness. Kawai and co-workers (31) treated the problem from the point of view of micelle formation. As the solvent evaporates from a block copolymer solution, a critical micelle concentration is reached. At this point, the domains are formed and are assumed to undergo no further change with continued solvent evaporation. Minimum free energies for an AB-type block copolymer were computed this way. 

Gaylord and Lohse (10) have calculated the stress-strain relation for cilia and tie molecules in a spherical domain morphology using the same type of three-chain model as Meier. It is assumed that the overall sample deformation is affine while the domains are undeformable. It is predicted that the stress increases rapidly with increasing strain for both types of chains. The rate of stress rise is greatly accelerated as the ratio of the domain thickness to the initial interdomain separation increases. The results indicate that it is not correct to use the stress-strain equation obtained by Gaussian elasticity theory, even if it is multiplied by a filler effect correction term. No connection is made between the initial dimensions and the volume fractions of the domain and interdomain material in this theory. 

M. Meier, R. Preuss, V. Dose Interaction of CH3 and H with amorphous hydrocarbon surfaces Estimation of reaction cross-sections using Bayesian probability theory. New J. Phys. 5, 133 (2003)... 

P.H.E. Meier and E. Bauer, Group Theory , North-Holland, Amsterdam, 1962. 

Rod—coil block copolymers have both rigid rod and block copolymer characteristics. The formation of liquid crystalline nematic phase is characteristic of rigid rod, and the formation of various nanosized structures is a block copolymer characteristic. A theory for the nematic ordering of rigid rods in a solution has been initiated by Onsager and Flory,28-29 and the fundamentals of liquid crystals have been reviewed in books.30 31 The theoretical study of coil-coil block copolymer was initiated by Meier,32 and the various geometries of microdomains and micro phase transitions are now fully understood. A phase diagram for a structurally symmetric coil—coil block copolymer has been theoretically predicted as a... 

More elaborate models, including the rheology, and the drift of sea ice have to be applied to describe the formation of different ice classes, transports of sea ice, and a forecast potential for ship navigation. Dynamic-thermodynamic sea ice is applied in three-dimensional models of the Baltic Sea by Haapala and Lepparanta (1996), Meier et al., (1999,2(X)2a,b), Lehmann and Hinrichsen (2000), and Schrum et al. (2003). A comprehensive overview of the theory and application of sea ice drift is given by Lepparanta (2005). Models based on the Flexible Model System (FMS), including the Modular Ocean Model (version 4), may also apply a dynamical ice module (Griffies et al., 2004 Balaji, 2004). 

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