McMillan model

In the McMillan model, the smectic A-nematic transition can be continuous or discontinuous. If a is less than 0.7, then o decreases to zero continuously and S is continuous at the smectic A- nematic transition. If a is between 0.7 and 0.98, then a jumps to zero discontinuously and S has a small discontinuity at the smectic A-nematic transition. When a is greater than 0.98, the smectic phase transforms directly into the isotropic phase with discontinuities in both order parameters. So just as in the extended Landau-de Geimes theory for the smectic A phase, a tricritical point is predicted at a=0.7, which corresponds to a ratio in the smectic A—nematic transition temperature to the nematic-isotropic transition temperature of 0.87. A great deal of experimental work has been done on the smectic A-nematic transition, and the results seem to indicate that the tricritical point occurs when the ratio of the two transition temperatures is significantly larger than 0.87. 

One of the most important theories for the nematic-smectic A phase transition is the Landau-de Gennes model. Another is the McMillan model, which was discussed in Section 5.5.2. The Landau-de Gennes theory is applied in the case of a second-order phase transition by combining a Landau expansion (Section 1.5) for the free energy in terms of an order parameter for smectic layering with the elastic energy of the nematic phase (Eq. 5.19). A suitable order parameter for the smectic structure allows both for the layer periodicity and the fluctuations of layer position (r) ... 

In the McMillan model the pair interaction potential is specified as... 

In the McMillan model the parameter a characterizes the strength of the interaction that induces the smectic ordering. The parameter a decreases with the increasing smectic period d=27t/k which is of the order of molecular length. Thus a is supposed to increase with increasing chain length. 

In some compounds (usually the lower members of a homologous series), the transition to the Nd phase takes place directly from the crystal, while in others (the higher members of the series) it takes place via a columnar phase (Table 2) [31, 32]. The trend is somewhat analogous to the behaviour of the smectic A-nematic transition for a homologous series, and as will be discussed in Sec. 3 this trend can be explained by an extension of the McMillan model of the A-N transition to systems of discotic molecules. There is also X-ray evidence of skew cybotactic groups in the Np phase prior to the transition to the D, phase, very much like the situation seen near a nematic-smectic C transition [34]. 

Substitution of Eq. [17] into Eq.[15 ] gives the McMillan model of the potential in the form... 

Equation (2.5) can also lead to the McMillan model [32] for a smectic A phase of polymer-rod blends [33] (Section 2.3.5). When the system is spatially non-uniform... 

From these results, the thennodynamic properties of the solutions may be obtamed within the McMillan-Mayer approximation i.e. treating the dilute solution as a quasi-ideal gas, and looking at deviations from this model solely in temis of ion-ion interactions, we have... 

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

McMillan s model [71] for transitions to and from tlie SmA phase (section C2.2.3.2) has been extended to columnar liquid crystal phases fonned by discotic molecules [36, 103]. An order parameter tliat couples translational order to orientational order is again added into a modified Maier-Saupe tlieory, tliat provides tlie orientational order parameter. The coupling order parameter allows for tlie two-dimensional symmetry of tlie columnar phase. This tlieory is able to account for stable isotropic, discotic nematic and hexagonal columnar phases. 

McMillan W L 1971 Simple molecular model for the smectic A phase of liquid crystals Phys.Rev A 4 1238-46... 

The commonly used method for the determination of association constants is by conductivity measurements on symmetrical electrolytes at low salt concentrations. The evaluation may advantageously be based on the low-concentration chemical model (lcCM), which is a Hamiltonian model at the McMillan-Mayer level including short-range nonelectrostatic interactions of cations and anions [89]. It is a feature of the lcCM that the association constants do not depend on the physical... 

Smectic A and C phases are characterized by a translational order in one dimension and a liquid-like positional order in two others. In the smectic A phase the molecules are oriented on average in the direction perpendicular to the layers, whereas in the smectic C phase the director is tilted with respect to the layer normal. A simple model of the smectic A phase has been proposed by McMillan [8] and Kobayashi [9] by extending the Maier-Saupe approach for the case of one-dimensional density modulation. The corresponding mean field, single particle potential can be expanded in a Fourier series retaining only the leading term ... 

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

Marquez, AC, Blanchar C (2004) The procurement of strategic parts. Analysis of contracts with suppliers using a system dynamics simulation model. International Journal of Production Economics 88 (1) 29-49 Mason S (2002) Simulation software buyer s guide. HE Solutions May 45-51 McAfee R, McMillan J (1987) Auctions and Bidding. Journal of Economic Literature 25 699-738... 

The study of McMillan-Mayer level models, in which the solvent coordinates have been averaged over so that only solvent-mediated ion-ion forces need be treated, is relatively well developed. However the real forces at this level are even more poorly known than the forces at the Born-Oppenheim level referred to above. It is found that McMillan-Mayer level models can be brought into good agreement with solution thermodynamic data. 

In a McMillan-Mayer level model (MM-level) for a solution, the particles are the solute particles (i.e. the ions with positive, negative, or zero charge). The ion-ion potentials can, in principle, be generated by calculations in which one averages over the solvent coordinates in a BO-level model which sees the solvent particles. (k,5,12) Pairwise additivity (we use overbars for solvent-averaged potentials)... 

McMillan DE, Snodgrass SH. (1991). Effects of acute and chronic administration of delta 9-tetrahydrocannabinol or cocaine on ethanol intake in a rat model. Drug Alcohol Dependence. 27(3) 263-74. 

G 8. Gordon, M., B. M. Grieveson, and I. D. McMillan Monomer reactivity ratios in a model unsaturated polyester system. J. Polymer Sci. 18, 497... 

Scholtz MT, McMillan AC, Slama C, Li YF, Ting N, Davidson K (1997) Pesticides emission modelling. Development of a North American pesticide inventory. Report CGEIC-1997-1, 1-242, Canadian Global Emissions Interpretation Centre, Toronto, ON, Canada... 

Clear evidence of L-L transitions has been found only in /-Si modeled by the SW potential [269]. Sastry and Angell [288] performed MD simulations of supercooled /-Si using the SW potential. After cooling at ambient pressure, the liquid (HDL) was transformed to LDL at 1060 K. The Nc in LDL is almost 4, and the diffusivity is low compared with that in HDL. The structural properties of LDL, such as g(r) and Nc, are very close to those of LDA, which indicates that this HDL-LDL transition is a manifestation of the multiple amorphous forms (LDA and HDA) of Si. McMillan et al. [264] and Morishita [289] have also found structural fluctuations between LDL-like and HDL-like forms in their MD calculations for /-Si at 1100 K. Morishita has demonstrated that such a structural fluctuation induces spatial and temporal dynamical heterogeneity, and this heterogeneity accounts for the non-Debye relaxation process that becomes noticeable in the supercooled state [289]. 

CONTENTS Introduction, Thom H. Dunning, Jr. Electronic Structure Theory and Atomistic Computer Simulations of Materials, Richard P. Messmer, General Electric Corporate Research and Development and the University of Pennsylvania. Calculation of the Electronic Structure of Transition Metals in Ionic Crystals, Nicholas W. Winter, Livermore National Laboratory, David K. Temple, University of California, Victor Luana, Universidad de Oviedo and Russell M. Pitzer, The Ohio State University. Ab Initio Studies of Molecular Models of Zeolitic Catalysts, Joachim Sauer, Central Institute of Physical Chemistry, Germany. Ab Inito Methods in Geochemistry and Mineralogy, Anthony C. Hess, Battelle, Pacific Northwest Laboratories and Paul F. McMillan, Arizona State University. 

It has been seen that reliable conductivity values are known only at low electrolyte concentrations. Under these conditions, even conductance equations for models such as the McMillan-Mayer theory (Sections 3.12 and 3.16) are known. However, the empirical extension of these equations to high concentration ranges has not been successful. One of the reasons is that conductivity measurements in nonaqueous solutions are still quite crude and literature values for a given system may vary by as much as 50% (doubtless due to purification problems). 

Akaogi, M., N. L. Ross, P. McMillan, and A. Navrotsky (1984). The Mg2Si04 polymorphs (olivine, modified spinel, spinel)—thermodynamic properties from oxide melt solution calorimetry, phase relations, and models of lattice vibrations. Amer. Mineral. 69, 499-512. 

Summarizing, one can say that the lattice theories need improvement and compact macromolecules need more refined treatment. We shall develop in this paper a refined and unified theory of macromolecular solutions with special emphasis on dilute solutions. We shall put our standpoint on the general theory of solutions developed by McMillan and Mayer in 1945 and Kirkwood and Buff in 1951 (9). TTiese theories do not use the lattice model and are more natural for application especially to dilute solutions. The theories extend statistical theories on gases and this is the reason why we used the name gas theories (70) in the beginning of this Introduction. 

Many models are available for describing the thermodynamic behavior of solutions. " However, so far no one could satisfactorily simulate the solution behavior over the whole concentration range and provide the correct pressure and temperature dependencies. This generated interest in the thermodynamically rigorous theories of Kirkwood—Buff and McMillan—Mayer. In the present paper, the emphasis is on the application of the Kirkwood—Buff theory to the aqueous solutions of alcohols, because it is the only one which can describe the thermodynamic properties of a solution over the entire concentration range. The key quantities in the Kirkwood-Buff theory of solution are the so-called Kirkwood-Buff integrals (KBIs) defined as... 

In this section we consider the application of the concept of ion association to describe the properties of electrolyte solutions within the ion or McMillan-Mayer level approach. In this approach the effects of solvent molecules are taken into account by introducing the dielectric constant into Coulomb interaction law and by appropriately choosing the short-range part of ion-ion interactions. To simplify, we consider here the restrictive primitive model (RPM)... 

Figure 10. Osmotic coefficient as a function of the reduced density of monomer units pp = pm.03, where prn is the number density of monomer units. Solvent primitive model (continuous lines) McMillan-Mayer model results (broken lines). From top to bottom a = 0.125,0.5, and 1.0 (each bead is charged).

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