An analysis of decoupling procedures in generating thermal Green's functions of O([lambda]â ´) by the Zubarev equation of motion method

The Zubarev equation of motion method has been applied to an anharmonic crystal of O( ,,4). All possible decoupling schemes have been interpreted in order to determine finite temperature expressions for the one phonon Green's function (and self energy) to 0()\4) for a crystal in which every atom is on a site of inversion symmetry. In order to provide a check of these results, the Helmholtz free energy expressions derived from the self energy expressions, have been shown to agree in the high temperature limit with the results obtained from the diagrammatic method. Expressions for the correlation functions that are related to the mean square displacement have been derived to 0(1\4) in the high temperature limit.

Application of [Lambda] to the fourth perturbation theory in calculating the equation of state of rare gas solids and fcc metals

We have calculated the thermodynamic properties of monatomic fcc crystals from the high temperature limit of the Helmholtz free energy. This equation of state included the static and vibrational energy components. The latter contribution was calculated to order A4 of perturbation theory, for a range of crystal volumes, in which a nearest neighbour central force model was used. We have calculated the lattice constant, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the adiabatic and the isothermal bulk modulus, and the Gruneisen parameter, for two of the rare gas solids, Xe and Kr, and for the fcc metals Cu, Ag, Au, Al, and Pb. The LennardJones and the Morse potential were each used to represent the atomic interactions for the rare gas solids, and only the Morse potential was used for the fcc metals. The thermodynamic properties obtained from the A4 equation of state with the Lennard-Jones potential, seem to be in reasonable agreement with experiment for temperatures up to about threequarters of the melting temperature. However, for the higher temperatures, the results are less than satisfactory. For Xe and Kr, the thermodynamic properties calculated from the A2 equation of state with the Morse potential, are qualitatively similar to the A 2 results obtained with the Lennard-Jones potential, however, the properties obtained from the A4 equation of state are in good agreement with experiment, since the contribution from the A4 terms seem to be small. The lattice contribution to the thermal properties of the fcc metals was calculated from the A4 equation of state, and these results produced a slight improvement over the properties calculated from the A2 equation of state. In order to compare the calculated specific heats and bulk moduli results with experiment~ the electronic contribution to thermal properties was taken into account~ by using the free electron model. We found that the results varied significantly with the value chosen for the number of free electrons per atom.

Measurement of forces between and within bilayers of neutral phospholipids

Phospholipids in water form lamellar phases made up of alternating layers of water and bimolecular lipid leaflets. Three complementary methods, osmotic, mechanical, and vapour pressures, were used to measure the work of removing water from lamellar phases composed of frozen dipalmitoylphosphatidylcholine ( DPPC ), melted DPPC, egg phosphatidylethanolamine or equimolar mixtures of DPPC and cholesterol ( DPPC/CHOL ), Concurrently the structural changes that resulted from this water removal were measured using X-ray diffraction. The work was divided into that which forces the bilayers together ( F ) and that which compresses the molecules together within the bilayers ( F )# A large repulsive force exists between bilayers composed of each of the lipids studied and this force increases exponentially as bilayer separation is decreased. F is affected by the nature of the head groups, conformation of the acyl chains and heterogeneity of these chains. In general all of the melted phosphatidylcholines ( melted DPPC, egg lecithin and DPPC/CHOL ) have large equilibrium separations in excess water resulting from large repulsive hydration forces between these bilayers. By comparison, egg PE has an increased attractive force, and frozen DPPC has a decreased hydration force; each results in smaller separations in water for these two lipids. The chemical potentials of the water between the bilayers for all these lipids lie on a continuum, indicating that interbilayer water cannot be characterized by two discrete states, usually referred to as "bound" or "non**bound". For all lipids studied a maximum of 25 % of the total work done on the system goes into deforming the bilayers. The method used here viii to separate repulsion from deformation, developed for us by v. A. Parsegian, provides a unique method for the measurement of lateral pressure of a bilayer and its modulus of deformability ( Y ). Lateral pressure is affected by the nature of the head group, conformation and heterogeneity of the acyl chains. For small changes in molecular surface area ( A ) near equilibrium, both melted and frozen DPPC have similar values for the deformability modulus. Thus in this regime it requires about the same force to change the angle of tilt of frozen chains as it does to compress the fluid bilayer. The introduction of cholesterol into bilayers of DPPC reduces dramatically the lateral pressure of the bilayers over a large range of molecular surface areas ( A ). The variation in the magnitude of bilayer repulsion with different phospholipids provides a basis for the mechanism of lipid segregation in mixed lipid systems and suggests that interacting heterogeneous membranes may influence or modulate the composition of the opposing membrane. The measurements of deformabilities of bilayers provides a direct comparison of them with the properties of monolayers.

On the equation of state and atomic mean square displacement of crystals

We have presented a Green's function method for the calculation of the atomic mean square displacement (MSD) for an anharmonic Hamil toni an . This method effectively sums a whole class of anharmonic contributions to MSD in the perturbation expansion in the high temperature limit. Using this formalism we have calculated the MSD for a nearest neighbour fcc Lennard Jones solid. The results show an improvement over the lowest order perturbation theory results, the difference with Monte Carlo calculations at temperatures close to melting is reduced from 11% to 3%. We also calculated the MSD for the Alkali metals Nat K/ Cs where a sixth neighbour interaction potential derived from the pseudopotential theory was employed in the calculations. The MSD by this method increases by 2.5% to 3.5% over the respective perturbation theory results. The MSD was calculated for Aluminum where different pseudopotential functions and a phenomenological Morse potential were used. The results show that the pseudopotentials provide better agreement with experimental data than the Morse potential. An excellent agreement with experiment over the whole temperature range is achieved with the Harrison modified point-ion pseudopotential with Hubbard-Sham screening function. We have calculated the thermodynamic properties of solid Kr by minimizing the total energy consisting of static and vibrational components, employing different schemes: The quasiharmonic theory (QH), ).2 and).4 perturbation theory, all terms up to 0 ().4) of the improved self consistent phonon theory (ISC), the ring diagrams up to o ().4) (RING), the iteration scheme (ITER) derived from the Greens's function method and a scheme consisting of ITER plus the remaining contributions of 0 ().4) which are not included in ITER which we call E(FULL). We have calculated the lattice constant, the volume expansion, the isothermal and adiabatic bulk modulus, the specific heat at constant volume and at constant pressure, and the Gruneisen parameter from two different potential functions: Lennard-Jones and Aziz. The Aziz potential gives generally a better agreement with experimental data than the LJ potential for the QH, ).2, ).4 and E(FULL) schemes. When only a partial sum of the).4 diagrams is used in the calculations (e.g. RING and ISC) the LJ results are in better agreement with experiment. The iteration scheme brings a definitive improvement over the).2 PT for both potentials.

Structural effects of neutral lipids on the divalent cation-induced interactions of phosphatidylserine - containing bilayers

As Ca2+ and phosphatidylserine (PS) are known to induce the adhesion of bilayer vesicles and form collapsed multibilayer structures in vitro, it was the aim of this study to examine how that interaction and the resultant structures might be modified by neutral lipid species. X-ray diffraction data from multilamellar systems suggest that phosphatidylcholine (PC) and diacylglycerol (DG) might be in the collapsed phase up to a concentration of -30 mole % and that above this concentration these neutral lipids may modify Ca2+-induced bilayer interactions. Using large unilamellar vesicles and long incubations in excess Ca2+ to ensure equilibration, similar preliminary results were again obtained with PC, and also with phosphatidylethanolamine (PE). A combination of X-ray diffraction, thin-layer chromatography, density gradient centrifugation and freeze-fracture electron microscopy, used in conjunction with an osmotic stress technique, showed that (i) -30 mole % PC can be accomodated in the Ca(DOPS)2 phase; and (ii) higher PC levels modify Ca2+-induced bilayer interactions resulting in single lamellar phases of larger dimension and reduced tendency for REV collapse. Importantly, the data suggest that PC is dehydrated during the rapid collapse process leading. to Ca(DOPS)2 formation and exists with this dehydrated phase. Similar results were obtained using PS isolated from bovine brain. Preliminary studies using two different phosphatidylethanolamine (PE) species indicated accomodation by Ca(DOPS)2 of -25-30 mole 0/0 PE and bulk phase separation, of species favouring a non-bilayer phase, at higher levels. Significantly, all PS/PE vesicles appear to undergo a complete Ca2+-induced collapse, even with contents of up to 90 mole % PE. These data suggest that PE may have an important role in fusion mechanisms in vivo. In sum the data lend both structural and stoichiometric evidence for th~ existence of laterally segregated neutral lipid molecules within the same bilayers as PS domains exposed to Ca2+.

Some Travelling Wave Solutions of KdV-Burgers Equation

In this paper we study the extended Tanh method to obtain some exact solutions of KdV-Burgers equation. The principle of the Tanh method has been explained and then apply to the nonlinear KdV- Burgers evolution equation. A finnite power series in tanh is considered as an ansatz and the symbolic computational system is used to obtain solution of that nonlinear evolution equation. The obtained solutions are all travelling wave solutions.