Rigorous results in the mean-field theory of freezing

The mean-field theory of freezing, of which the Kirkwood-Monroe theory is a special case, is formulated in terms of a variational principle which can be derived from a statistical model. New upper and lower bounds on the free energy are found, and used to prove the existence of and to locate a freezing transition. The Kirkwood-Monroe theory is shown to obey a scaling law which implies that the freezing transition has no critical point and that the pressure-temperature phase diagram is a parabola. For a suitable choice of interaction potential, the free energy and density distribution of a crystalline state are calculated with very small error. The variational principle yields an integral equation found previously by van Kampen. This equation is derived rigorously, and shown to admit solutions of any chosen periodicity. The equation is transformed into a nonlinear integral equation of the Hammerstein type, and some standard uniqueness theorems applied. The equation is also transformed into a set of linear equations closely resembling the Kirkwood-Salsburg equations. Many unsolved problems are raised.     

An alternative formulation of rigorous mean-field theory

It is shown how to derive rigorous mean-field theory from a type of many-body interaction.     

The van der Waals limit for classical systems. I. A variational principle

We consider the thermodynamic pressurep(, ) of a classical system of particles with the two-body interaction potentialq(r)+ ^v K(r), where is the number of space dimensions, is a positive parameter, and is the chemical potential. The temperature is not shown in the notation. We prove rigorously, for hard-core potentialsq(r) and for a very general class of functionsK(s), that the limit 0 of the pressurep(, ) exists and is given by where the limit and the supremum can be interchanged. Here is a certain class of nonnegative, Riemann integrable functions,D is a cube of volume |D|, anda ⁰() is the free energy density of a system withK=0 and density . A similar result is proved for the free energy.     

Analysis of the products of the anionic oligomerisation of a phosphaalkene using MALDI-TOF mass spectrometry

The first use of MALDI-TOF MS in the study of the products of RLi (R = Me, Bu) initiated oligomerisation of P[double bond, length as m-dash]C bonds is reported. These studies may be considered as models for analogous polymerisation reactions. The detected linear products with R-P and C-H end-groups are consistent with a chain growth mechanism similar to that known for olefins. Interestingly, the results suggest that backbiting may compete with chain growth.     

Short Distance Expansion from the Dual Representation of Infinite Dimensional Lie Algebras

We develop a method for computing the short distance expansion of fields or operators that live in the coadjoint representation of an infinite dimensional Lie algebra by using only properties of the adjoint representation and its dual. We explicitly implement this method by computing the short distance expansion for the duals of the Virasoro algebra, affine Lie algebras and the geometrically realized N-extended supersymmetric Virasoro algebra. This method can also be used to compute short distance expansions between fields that transform in the adjoint and those that transform in the coadjoint representations.Supported in part by National Science Foundation Grant PHY-0099544 and PHY-0244377     

Reactivity of site-isolated metal clusters: propylidyne on gamma-Al2O3-supported Ir4.

To contrast the reactivity of supported metal clusters with that of extended metal surfaces, we investigated the reactions of tetrairidium clusters supported on porous gamma-Al2O3 (Ir4/gamma-Al2O3) with propene and with H2. Infrared, 13C NMR, and extended X-ray absorption fine-structure spectroscopy were used to characterize the ligands formed on the clusters. Propene adsorption onto Ir4/gamma-Al2O3 at 298 K gave stable, cluster-bound mu3-propylidyne. Propene adsorbed onto Ir4/gamma-Al2O3 at 138 K reacted at approximately 219 K to form a stable, highly dehydrogenated, cluster-bound hydrocarbon species approximated as CxHy (such as, for example, C3H2 or C2H). H2 reacted with Ir4/gamma-Al2O3 at 298 K, forming ligands (likely hydrides), which prevented subsequent reaction of the clusters with propene to form propylidyne. Propylidyne on Ir4 was stable in helium or H2 as the sample was heated to 523 K, whereupon it reacted with oxygen of the support to give CO. Propylidyne on Ir4 did not undergo isotopic exchange in the presence of D2 at 298 K. In contrast, the literature shows that propylidyne chemisorbed on extended metal surfaces is hydrogenated in the presence of H2 (or D2) and exchanges hydrogen with gaseous D2 at room temperature; in the absence of H2, it decomposes thermally to give hydrocarbon fragments at temperatures much less than 523 K. The striking difference in reactivities of propylidyne on clusters and propylidyne on extended metal surfaces implies the requirement of ensembles of more than the three metal surface atoms bonded to propylidyne in the surface reactions. The results highlight the unique reactivity of small site-isolated metal clusters.     

Experimental Investigation on Upgrading of Lignin‐Derived Bio‐Oils: Kinetic Analysis of Anisole Conversion on Sulfided CoMo/Al2O3 Catalyst

Kinetics of the hydroprocessing of anisole, a compound representative of lignin‐derived bio‐oils, catalyzed by a commercial sulfided CoMo/Al₂O₃, was determined at 8–20 bar pressure and 573–673 K with a once‐through flow reactor. The catalyst was sulfided in an atmosphere of H₂ + H₂S prior to the measurement of its performance. Selectivity‐conversion data were used as a basis for determining an approximate, partially quantified reaction network showing that hydrodeoxygenation (HDO), hydrogenolysis, and alkylation reactions take place simultaneously. The data indicate that these reactions can be stopped at the point where HDO is virtually completed and hydrogenation reactions (and thus H₂ consumption) are minimized. Phenol was the major product of the reactions, with direct deoxygenation of anisole to give benzene being kinetically almost insignificant under our conditions. We infer that the scission of the C_methyl–O bond is more facile than the scission of the C_aromatic–O bond, so that the HDO of anisole likely proceeds substantially through the reactive intermediate phenol to give transalkylation products such as 2‐methylphenol. The data determine rates of formation of the major primary products. The data show that if oxygen removal is the main processing goal, higher temperatures and lower pressures are favored.