Molecular statistical theory

A recent publication of Vlasov and Zaitsev reports observations of the non-linear optical activity predicted by Kididi from molecular-statistical theory and by Atkins from quantum scattering theory. 

The expansions (205)—(207) in coiyunction with formulae (191) and (192) define the complete molecular-statisticed theory of the Kerr effect in very dense media. They can be particularized to various cases. ... 

In spite of widespread applications, the exact mechanism of retention in reversed-phase chromatography is still controversial. Various theoretical models of retention for RPC were suggested, such as the model using the Hildebrand solubility parameter theory [32,51-53], or the model supported by the concept of molecular connectivity [54], models based on the solvophobic theory [55,56) or on the molecular statistical theory [57j. Unfortunately, sophisticated models introduce a number of physicochemical constants, which are often not known or are difficult and time-consuming to determine, so that such models are not very suitable for rapid prediction of retention data. 

The retention times of analytes are controlled by the concentration(s) of the organic solvent(s) in the mobile phase. If a relatively small entropic contribution to the retention is neglected, theoretical considerations based either on the model of interaction indices [58], on the solubility parameter theory [51,52] or on the molecular statistical theory [57], lead to the derivation of a quadratic equation for the dependence of the logarithm of the retention factor of a solute. A, on the concentration of organic solvent. aqueous-organic mobile phase ... 

It is very important to obtain accurately the first 2 constants (Ci and C2 or Ki and K ) from the experimental data in order to compare them with those calculated from the molecular-statistical theory. Here, Cl and C2 constants were determined from the experimental isotherm by the least squares method with the aid of a computer. However, for practical purposes, it is necessary to describe the isotherms to a higher adsorption level. From Figures 3 and 6, it can be seen that in order to reproduce the isotherm up to — 75% saturation of the adsorbent and to calculate a at different values of p and T, it is sufficient to determine only 2 more constants, C3 and C4. 

T. Bancewiez. Molecular-statistical theory of the influence of molecular fields in liquids on the spectral distribution and intensity of Rayleigh scattered light in the approach of Racah algebra. Acta Phys. Polonica, A, 56 431-438 (1979). 

Numerous investigators have proposed theories for the temperature dependence of pitch. " However, none completely explain the experimental results obtained for the various cellulosic systems. Of those proposed, Osipov s approach, which is based on a molecular statistical theory,takes into account steric and chiral interactions in solution to predict the influence of temperature and solvent on the pitch and twist sense of cellulose deiivatives. Assuming the cellulosic chain adopts a twisted belt as opposed to a helix, and the persistence length, /, of the cellulose chains is much smaller than that of a rigid chain, the twisting power could be expressed by... 

It is of interest to examine the relationship between the phenomenological model that we have just discussed and the molecular statistical theory of Maier and Saupe. The free energy of the weakly ordered isotropic phase in the presence of an external magnetic field is, according to the mean field theory,... 

In this book the flexoelectric effect is mainly considered from the phenomenological point of view. At the same time it is very interesting and important to reveal the molecular origin of flexoelectricity and, in particular, to consider different types of intermolecular interactions that may be responsible for the dipolar ordering in a deformed liquid crystal, and to study the effects of intermolecular correlations and the molecular structure. This problem can only be solved using a molecular-statistical theory, which eventually allows us to express the flexoelectric coefficients in terms of molecular model parameters using various approximations. 

Here the polar correction to the orientational distribution function h a, b, n) should be determined using a corresponding molecular-statistical theory. 

A molecular-statistical theory of the flexoelectric effect in the nematic phase can be derived in a general way using the density-functional approach to the theory of liquid crystals. In this approach, the free energy of a liquid crystal, F, is a functional of the density po(a ) = Po/(w) where /(w) is the orientational distribution fimction. The general structure of the functional F p) is not known, but the functional derivatives are known and are related to the direct correlation functions of the nematic phase. 

As mentioned above, the first expressions for the flexoelectric coefficients were obtained by Helfrich and Petrov and Derzhanski while a systematic molecular-statistical theory was developed later by Straley. The results of these two approaches were compared by Marcerou and Prost who concluded that the theories of Helfrich and Petrov and Derzhanski and of Straley describe different mechanisms for the dipolar flexoelectric effect because Straley s theory 5nelds values for the flexocoefficients that are two orders of magnitude smaller than the experimental ones, and which therefore can be neglected. 

The molecular-statistical theory of flexoelectricity, presented in the previous sections, does not allow us to establish a direct relation between the flexocoefficients and the details of a particular molecular structure (except for permanent electric and steric dipoles) because the theory is based on simple model interaction potentials. A different version of the mean-field theory, which takes into consideration the real molecular shape, has recently been proposed by Ferrarini et This approach is based on the... 

Maier and Saupe, in their well-known molecular-statistical theory, described the intermolecular orientational forces by a mean field method. The Maier-Saupe theory successfully predicts the relationship between the molecular orientation parameter S and the nematic potential D as a function of temperature [10,14]. 

W. Maier and A. Saupe, A simple molecular statistical theory for nematic liquid crystal phase. Part H,... 

The molecular-statistical theory of polymer solutions in confined media was also applied to the conventional chromatographic theory of gradient elution [ 163]. This approach leads to the prediction of the special mode of interactive polymer chromatography gradient elution at critical point of adsorption. It was demonstrated theoretically and experimentally that under appropriate conditions elution of each compositionally homogeneous fraction of copolymer occurs at the critical mobile phase composition. This critical mobile phase composition depends only on the local structure of the copolymer chain and is independent of its molecular weight. As a consequence, gradient elution produces the chemical com-... 

The current state of the molecular-statistical theory of liquid crystals does not yet permit the reliable prediction of transition temperatures between... 

From Table 1.3 it can be seen that there is a definite correlation between the increase in polarizability of the substituent (Aa) and the increase in Tni- Such a dependence would, generally speaking, be expected from the Maier-Saupe molecular-statistical theory if the role of steric factors were not allowed for. In the case of reasonably bulky substituents these factors prove to be significant, and, in particular, they can explain the fall-out points for CH3O and C2H5O groups. This correlation is also confirmed for compounds of the form (l.xx), where X, Y = —COO—, —OCO, —N=N—, or a simple bond [63]. 

Ma Y, Zha L, Hu W, Reiter G, Han CC (2008) Crystal nucleation enhanced at the diffuse interface of immiscible polymer blends. Phys Rev E 77(6) 061801 Maier W, Saupe A (1959) A simple molecular statistical theory of the nematic crystalline-liquid phase. Zeitschrift fur Naturforschung 14 882-889 Mandelkem L (1964) Crystallization of polymers. McGraw-Hill, New York Matsuoka S (1962) The effect of pressure and temperature on the speciflc volume of polyethylene. J Polym Sci 57(165) 569-588... 

Recently the molecular statistical theory has been used to examine the limitation of the Kelvin equation and predicts that the critical width between the micropore and the mesopore is 1.3-1.7 nm (corresponding to 4-5 layers of adsorbed N2), which is smaller than 2 nm [96]. 

Calamitic and discotic compounds can be described theoretically using the same basic model, i.e. the cylinder, by elongation or compression, respectively, of the central axis of the cylinder [23]. Of course, the cylinder model is a oversimplification of real mesogenic molecules. As most of the compounds contain ring systems, which are somewhat flat, it could be proposed that parallelepipeds provide better models. In fact, such models have been used for molecular-statistical theories, and these have allowed the description of, in addition to calamitic and discotic molecules, flat elongated (board-like, sanidic) molecules [24]. 

In order to synthesize liquid crystals with predicted properties, one of the basic questions for the chemist is Which intermolec-ular interactions are responsible for the stabilization of liquid crystalline phases The known dependence of mesogenity on the molecular geometry prompted the derivation of molecular-statistical theories, based on the shape-induced anisotropy of the repulsion of the molecules [25], Despite the remarkable success of such theories, it is clear that real condensed phases always eon-tain strong attraetive interactions, which are completely neglected by this procedure. Therefore, attempts have been made to... 

Molecular-statistical theories are available for several different smectic phase types [25]. In addition to the ingredients of the nematic phases, the theories incorporate parameters responsible for the formation of layers. The clearing temperatures, however. 

In view of the desirable molecular design of liquid crystalline phases, the final goal of microscopic theories is not only to predict the temperature dependence of elastic coefficients, but also to reveal relations between the molecular and elastic properties of the mesophases. Since the original work of Saupe in 1960 [241], a large number of microscopic theories have been developed [181-183, 242-270]. The first molecular statistical theories of the Frank elastic constants were formulated by Priest [242] and Straley [243], and a significant. step was made by Poniewierski and Stecki [244,245] who applied density-functional theory in order to relate in a direct quantitative way the elastic constants to characteristic properties of a statistical ensemble of molecules with... 

S. Chandrasekhar and N. V. Madhusudana, Molecular Statistical Theory of Nematic Liquid Crystals, Acta Crysl., Vol. A27, p. 303 (1971). 

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