Ethene adsorption, dehydrogenation and reaction with Pd(110): Pd as a carbon 'sponge'

The interaction of ethene with the Pd(110) surface has been investigated, mainly with a view to understanding the dehydrogenation reactions of the molecule and mainly using a molecular beam reactor. Ethene adsorbs with a high probability over the temperature range 130 to 800 K with the low-coverage sticking probability dropping from 0.8 at 130 K to 0.35 at 800 K. The adsorption is of the precursor type, with a weakly held form of ethene being the intermediate between the gas phase and strong chemisorption. Dehydrogenation begins at approximately 300 K and is fast above 350 K. If adsorption is carried out at temperatures up to approximately 380 K, adsorption saturates after about 0.25 monolayer have adsorbed, but above approximately 450 K, adsorption continues at a high rate with continuous hydrogen evolution and C deposition onto the surface. It appears that, in the intermediate temperature range, the carbonaceous species formed is located in the top layer and thus interferes with adsorption, whereas the C goes subsurface above 450 K, the adsorption is almost unaffected, and the C signal is significantly attenuated in XPS. However, the deposited carbon can easily be removed again by reaction with oxygen, thus implying that the carbon remains in the selvedge, that is, in the immediate subsurface region probably consisting of a few atomic layers. No well-ordered structures are identified in either LEED or STM, though some evidence of a c(2 x 2) structure can be seen. The Pd surface, at least above 450 K, appears to act as a "sponge" for carbon atoms, and this effect is also seen for the adsorption of other hydrocarbons such as acetaldehyde and acetic acid.     

Acetaldehyde adsorption and catalytic decomposition on Pd(1 1 0) and the dissolution of carbon

The reaction of acetaldehyde with the Pd(1 1 0) surface has been studied using a molecular beam reactor, TPD and LEED. Below 270 K acetaldehyde sticks to the surface with a high initial probability (∼0.8), but no gas phase products evolve. When the reaction is run at >270 K, hydrogen evolves into the gas phase early in the reaction together with methane in a non-steady-state fashion, but above 300 K there is a very efficient steady-state catalytic reaction at the surface; this reaction is the decarbonylation of acetaldehyde to produce methane and carbon monoxide in the gas phase. This behaviour continues up to about 400 K. However, when acetaldehyde is dosed at 423 K, the reaction rate slowly evolves through a maximum to a very low catalytic rate. Upon carrying out reactor experiments at 473 K and above, the reaction mechanism changes to total dehydrogenation, and CO and H₂ are produced at high steady-state rate, not withstanding the fact that carbon is continually being deposited onto the surface. This carbon does not appear to affect the reaction, which takes place on a surface with a c(2 × 2)-C layer present, since the extra carbon is lost from the reaction zone by diffusion into the bulk of the crystal.     

Partition identities and the coin exchange problem

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a and b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.     

Some Circumstances Where Extra Updates Can Delay Mixing

Peres and Winkler proved a ‘censoring' inequality for Glauber dynamics on monotone spins systems such as the Ising model. Specifically, if, starting from a constant-spin configuration, the spins are updated at some sequence of sites, then inserting another site into this sequence brings the resulting configuration closer in total variation to the stationary distribution. We show by means of simple counterexamples that the analogous statements fail for Glauber dynamics on proper colorings of a graph, and for lazy transpositions on permutations, answering two questions of Peres. It is not known whether the censoring property holds in other natural settings such as the Potts model.     

Enumeration of acyclic and unicyclic nets with four types of self-duality

A net is a graph in which each point and line is given a sign. The point, line, and simple duals of a net are obtained by reversing the signs of the points, lines, or both. If a net possesses two of the three types of self-duality, it possesses all three and is said to be doubly self-dual. Enumeration formulas are derived for nets and point, line, simply, and doubly self-dual nets, whose underlying graphs are acyclic and unicyclic. The numbers are tabulated up to 12 points (24 for doubly self-dual nets) in each case.     

A sharper threshold for bootstrap percolation in two dimensions

Two-dimensional bootstrap percolation is a cellular automaton in which sites become ‘infected’ by contact with two or more already infected nearest neighbours. We consider these dynamics, which can be interpreted as a monotone version of the Ising model, on an n × n square, with sites initially infected independently with probability p. The critical probability p _c is the smallest p for which the probability that the entire square is eventually infected exceeds 1/2. Holroyd determined the sharp first-order approximation: p _c ~ π ²/(18?log?n) as n → ∞. Here we sharpen this result, proving that the second term in the expansion is ?(log?n)^?3/2+o(1), and moreover determining it up to a poly(log?log?n)-factor. The exponent ?3/2 corrects numerical predictions from the physics literature.     

Strengthening Hadwiger's Conjecture

We consider the following strengthening of Hadwiger's Conjecture. Let G be any graph of chromatic number k, S any subset of V(G)which takes all k colours in each proper k-colouring of G. Thenthere are k pairwise adjacent connected subgraphs of G, eachof whose vertex sets has non-trivial intersection with S. We show that the truth of this conjecture for all graphs ofchromatic number k implies the truth of Hadwiger's Conjecturefor all graphs of chromatic number k + 1. We also show thatits truth implies the following statement (which is at firstsight even stronger). For any graph G of chromatic number k and any subset S of V(G),define (S; G) to be the least number of colours that can appearon S in any proper k-colouring of G, and h(S; G) to be the largestnumber of pairwise adjacent connected subgraphs of G each havingnon-trivial intersection with S. Then (S; G) h(S; G). We define the number w(S; G) to be the largest cardinality ofa subset T of S such that, however T is partitioned into pairs(possibly with one spare element), there are vertex-disjointpaths linking the elements in each pair, none passing throughthe spare element if it exists. We show that (S; G) (|S| +w(S; G))/2 for any graph G and subset S of V(G). Finally, we show that for any graph G, (S; G) h(S; G) whenever(S; G) 3. 1991 Mathematics Subject Classification 05C15.