Table of contents

This volume is the outcome of a three-day meeting held 7-10 August, 2004 in St Adèle, Québec honouring Michael F Thorpe, Foundation Professor of Physics, Chemistry and Biophysics at Arizona State University. Michael Thorpe has made many important contributions to condensed matter physics, broadly defined. From the famous Weaire and Thorpe Hamiltonian in 1971 [1] to rigidity percolation theory [2] and flexibility in proteins [3], he has always provided highly original solutions to difficult problems. He has also demonstrated an uncommon gift for selecting and solving problems of remarkably broad significance (for example, rigidity theory is now a powerful tool in glasses, microelectronics and proteins).

Throughout his career, Mike has also made a point of establishing contact with scientists from all disciplines and origins, organizing tens of conferences and maintaining a very active visitor's program both at Michigan State University, where he spent 25 years, and, now, at Arizona State University, where he moved a year ago. It is therefore not surprising that the participants, all with scientific or personal links with Mike, usually both, came from Europe, Asia and North America to celebrate the 60th birthday of an eminent physicist and friend.

Reflecting the impact of Mike's work across the traditional scientific boundaries, the meeting included contributions ranging from studies of concrete by Ed Garboczi (NIST), a hybrid VCA/CPA treatment of the Hubbard model presented by Sir Roger Elliott (Oxford) to the assembly process of viral capsids by Brandon Hespenheide (ASU). Especially memorable talks were given by Rafael Barrio (with his photogenic striped imperial fish), Alex Kolobov (presenting impressive scientific results using equally impressive computer graphics), and Dick Zallen, for a remarkably good `roast' of the honoree and work with his daughter (a molecular biologist) on biophysics.

The meeting had an unusual warmth befitting the birthday celebration, and was also by any measure a scientific success, with innumerable questions and discussions among this diverse group, and the scholarly contributions in this volume.

We thank the participants, and Dr Richard Palmer, the Publisher of Journal of Physics: Condensed Matter and his staff for working to make this volume appear very quickly, thereby increasing the value of the papers to the community. We would also thank the Office of the vice-president for research at Université de Montréal, the Department of Physics and Astronomy at Arizona State University, the Department of Physics and Astronomy at Ohio University and the Canada Research Chair Program for financial assistance that made the meeting more enjoyable.

We were both pleased to be part of this delightful occasion, and wish to take this last opportunity to wish Mike a happy birthday, and urge him to even greater achievements in coming years.

Bibliography

[1] Weaire D and Thorpe M F 1971 Electronic Properties of an Amorphous Solid: I. A Simple Tight Binding Theory Phys. Rev. B 4 2508--20

[2] Thorpe M F 1983 Continuous Deformations in Random Networks J. Non-Cryst. Solids57 355-70

[3] Jacobs D J, Rader A J, Kuhn L A and Thorpe M F 2001 Protein Flexibilty Predictions using Graph Theory Proteins44 150-65

The alpha-helix to coil transition in homogeneous polypeptides has recently been described using a distance constraint model (DCM) that employs network rigidity as an underlying mechanical interaction. The DCM accounts for intramolecular hydrogen bonding, hydrogen bonding to solvent and hydration. These interactions are parametrized to reflect the dependence on the conformational state of the polypeptide backbone. The DCM is capable of describing both heat and cold denaturation. As a function of temperature from low to high, a generic re-entrant response of a homogeneous polypeptide chain in aqueous solution is predicted to make a transition from coil (hydrated state) to helix and then to coil (disordered state). Here we study the thermodynamic stability of heterogeneous polypeptides that include hydrophobic (H) and polar (P) residue types. We explore the nature of the transition by adjusting the overall HP composition using transfer matrix methods that take into account long-range effects due to network rigidity.

It has been shown recently that even a tetrapeptide can form amyloid fibrils sharing all the characteristics of amyloid fibrils built from large proteins. Recent experimental studies also suggest that the toxicity observed in several neurodegenerative diseases, such as Alzheimer's disease and Creutzfeldt–Jakob disease, is not only related to the mature fibrils themselves, but also to the soluble oligomers formed early in the process of fibrillogenesis. This raises the interest in studying the early steps of the aggregation process. Although fibril formation follows the nucleation–condensation process, characterized by the presence of lag phase, the exact pathways remain to be determined. In this study, we used the activation–relaxation technique and a generic energy model to explore the process of self-assembly and the structures of the resulting aggregates of eight KFFE peptides. Our simulations show, starting from different states with a preformed antiparallel dimer, that eight chains can self-assemble to adopt, with various orientations, four possible distant oligomeric well-aligned structures of similar energy. Two of these structures show a double-layer β-sheet organization, in agreement with the structure of amyloid fibrils as observed by x-ray diffraction; another two are mixtures of dimers and trimers. Our results also suggest that octamers are likely to be below the critical size for nucleation of amyloid fibrils for small peptides.

The cowpea chlorotic mottle virus (CCMV) has a protein cage, or capsid, which encloses its genetic material. The structure of the capsid consists of 180 copies of a single protein that self-assemble inside a cell to form a complete capsid with icosahedral symmetry. The icosahedral surface can be naturally divided into pentagonal and hexagonal faces, and the formation of either of these faces has been proposed to be the first step in the capsid assembly process. We have used the software FIRST to analyse the rigidity of pentameric and hexameric substructures of the complete capsid to explore the viability of certain capsid assembly pathways. FIRST uses the 3D pebble game to determine structural rigidity, and a brief description of this algorithm, as applied to body–bar networks, is given here. We find that the pentameric substructure, which corresponds to a pentagonal face on the icosahedral surface, provides the best structural properties for nucleating the capsid assembly process, consistent with experimental observations.

The complexity and functionality of proteins requires that they occupy an exponentially small fraction of configuration space (perhaps 10⁻³⁰⁰). How did evolution manage to create such unlikely objects? Thorpe has solved the static half of this problem (known in protein chemistry as Levinthal's paradox) by observing that for stress-free chain segments the complexity of optimally constrained elastic networks scales not with expN (where –1000 is the number of amino acids in a protein), but only with N. Newman's results for diffusion in N-dimensional spaces provide suggestive insights into the dynamical half of the problem. He showed that the distribution of residence (or pausing) time between sign reversals changes qualitatively at . The overall sign of a protein can be defined in terms of a product of curvature and hydrophobic(philic) character over all amino acid residues. This construction agrees with the sizes of the smallest known proteins and prions, and it suggests a universal clock for protein molecular dynamics simulations.

Convergent extension is the cell-rearrangement process by which a developing embryo elongates to establish the head-to-tail body axis. In the early Drosophila embryo, this process occurs within a one-cell-thick epithelial layer. Using confocal microscopy, images were collected of the two-dimensional cell pattern at four stages during convergent extension in wild-type embryos and at one stage in two classes of mutant embryos. The cellular topology was analysed in terms of the statistical distribution p(n), the frequency of occurrence of n-sided cells. For wild-type embryos, the results demonstrate progressive cell-pattern disordering during convergent extension. The second moment (variance) of p(n) triples to 1.1 while the peak at p(6) drops from 0.65 to 0.38. The fraction of fourfold vertices (four edges meeting) increases from 2% to 8%. Quantitative analysis of interface orientations reveals that the initial degree of hexatic edge-orientational order essentially disappears during the course of convergent extension. The degree of cell-pattern disordering in the two mutants resembles distinct stages in the wild type and correlates with the extent of axis elongation.

We investigate the formation of Turing patterns (arising from a diffusion driven instability) when diffusion is anomalous. We analyse a model that contains the important features of Turing systems and that has been extensively used in the past to model interesting biological patterns. We concentrate on the case of asymmetric anomalous diffusion, and we cast a version of the model, suitable for numerical calculations, using a discrete version of the fractional derivatives defining the anomalous diffusion operator. The results are interesting in the sense that patterns are no longer stationary but acquire a velocity that depends on the exponent of the fractional derivatives. Extensive numerical calculations in one and two dimensions exhibit many of the features predicted by the analysis of the equations.

X-ray and neutron powder diffraction data as a function of temperature are analysed for the colossal dielectric constant material CaCu₃Ti₄O₁₂. The local structure is studied using atomic pair distribution function analysis. No evidence is found for enlarged oxygen or Ti displacement parameters suggesting that short range octahedral tilt disorder and off-centre Ti displacements are minimal. However, an unusual temperature dependence for the atomic displacement parameters of calcium and copper is observed. Temperature dependent modelling of the structure, using bond valence concepts, suggests that the calcium atoms become underbonded below approximately 260 K, which provides a rationale for the unusually high Ca displacement parameters at low temperature.

By means of x-ray absorption fine structure and Raman scattering spectroscopies we demonstrate that the structure of amorphous GeTe is likely to be a mixture of 4(Ge):2(Te) and 3(Ge):3(Te)-coordinated structural units. Upon crystallization, a rhombohedral (distorted rocksalt) structure is established with about 10% of vacancies occurring on Ge sites. The vacancies are believed to play an important role in determining the ratio of 3(Ge):3(Te) and 4(Ge):2(Te) structural units.

Gaussian positional disorder has been introduced into the structure of various 2D and 3D crystal lattices. As the degree of positional disorder is increased, the weights of the highest-k peaks in the orientationally-averaged structure factor are decreased first, eventually leaving only a single peak having significant weight, that with the lowest k-value. The orientationally-averaged real-space pair-correlation function for lattices with such high levels of positional disorder exhibits a corresponding power-law-damped series of oscillations, with a single period equal to the separation between the furthest-separated, lowest-k lattice planes. This last surviving peak in the structure factor is at a k-value nearly identical to that of the experimentally-observed first sharp diffraction peak for the corresponding real amorphous phases of such lattices (e.g. Si, SiO₂). These results also have a bearing on the unsolved Gauss circle problem in mathematics.

Ge_xP_xS_1−2x glasses in the compositional range have been synthesized and examined in temperature modulated differential scanning calorimetry (MDSC) and Raman scattering experiments. Trends in the non-reversing enthalpy ΔH_nr(x) near T_g show the term to almost vanish in the 0.090(5)<x<0.135(5) range, and to increase by an order of magnitude at x<0.09, and at x>0.135. In analogy to previous results on chalcogenide glasses, we identify compositions at x<0.09 to be elastically floppy, those in the 0.090<x<0.135 range to be in the intermediate phase, and those at x>0.135 to be stressed rigid. MDSC results also show that the ΔH_nr term ages in the stressed-rigid and floppy phases but not in the intermediate phase. The intermediate phase is viewed to be a self-organized phase of a disordered network. It consists of at least four isostatically rigid local structures: corner-sharing GeS₄, edge-sharing GeS₂, pyramidal P(S_1/2)₃ and quasi-tetrahedral S = P(S_1/2)₃ units for which evidence comes from Raman scattering. The latter method also shows the existence of P₄S₇ and P₄S₁₀ molecules in the glasses segregated from the backbone. These aspects of structure contribute to an intermediate phase that is significantly narrower in width than in the corresponding selenide glasses.

The performance and reliability of aggressively-scaled field effect transistors are determined in large part by electronically-active defects and defect precursors at the Si–SiO₂ and internal SiO₂-high-k dielectric interfaces. A crucial aspect of reducing interfacial defects and defect precursors is associated with bond-strain-driven bonding interfacial self-organizations that take place during high temperature annealing in inert ambients. The interfacial self-organizations and intrinsic interface defects are addressed through an extension of bond constraint theory from bulk glasses to interfaces between non-crystalline SiO₂ and (i) crystalline Si, and (ii) non-crystalline and crystalline alternative gate dielectric materials.

We make here an effort to find commonality between the athermal constraint approach to glassiness due to Phillips and Thorpe, in which composition is the key variable, and the more conventional temperature-induced softening approaches to the glass transition at constant composition. A starting point for our discussion is the parallel in behaviour of the boson peak, derived from the vibrational density of states, which is enhanced both by increasing glass fictive temperature (potential energy), and by decreasing glass coordination number (through a rigidity threshold), e.g. in chalcogenide glasses. We relate the potential energy of the glass to a topological defect concentration, and see defect formation as a means of lifting constraints, and hence promoting flow in formally overconstrained glasses. This viewpoint is supported by observations on irradiation of glasses, in which the athermal introduction of defects, or lifting of barriers, may induce flow, or relaxation/annealing. These considerations emphasize the importance of taking temperature, and fictive ('structural') temperature, considerations into account in evaluating the properties of laboratory glasses for comparison with constraint theory predictions.

While there have been many studies based on models of amorphous silicon, there have been surprisingly few (perhaps only one) that have seriously addressed the radial distribution function at low temperature. Our work is based in part on the so-called NRL tight binding method using parameters for silicon determined by Bernstein et al. As we have recently shown in the case of 216-atom models, upon including zero-point motion good agreement is obtained with very accurate low temperature x-ray diffraction measurements by Laaziri et al of the radial distribution function, although, as also found by Herrero who used the Stillinger–Weber potential, a slight asymmetry of the first peak in the RDF is predicted and this asymmetry has not been observed experimentally. Upon use of an estimate of zero-point broadening from our previous work we show here that 1000-atom models lead to good agreement with experiment for the RDF. Perhaps fortuitously, we obtain models that agree with the experimentally determined second peak in the RDF for both annealed and unannealed samples: our tight binding relaxed models based on topologies derived from the Wooten–Winer–Weaire method and the Barkema–Mousseau method yield unannealed-sample results, whereas our tight binding relaxed model based on an MD quench of the liquid using the semi-empirical interatomic potential, EDIP, of Kaxiras and coworkers yield the annealed-sample results. Finally, the significant effect of zero-point motion on the first peak in the radial distribution that we obtain in the case of amorphous silicon could also have implications for other amorphous materials, e.g. SiO₂.

We propose a novel approach to model amorphous materials using a first principles density functional method while simultaneously enforcing agreement with selected experimental data. We illustrate our method with applications to amorphous silicon and glassy GeSe₂. The structural, vibrational and electronic properties of the models are found to be in agreement with experimental results. The method is general and can be extended to other complex materials.

Many properties of disordered materials can be understood by looking at idealized structural models, in which the strain is as small as is possible in the absence of long-range order. For covalent amorphous semiconductors and glasses, such an idealized structural model, the continuous random network, was introduced 70 years ago by Zachariasen. In this model, each atom is placed in a crystal-like local environment, with perfect coordination and chemical ordering, yet longer-range order is nonexistent. Defects, such as missing or added bonds, or chemical mismatches, however, are not accounted for. In this paper we explore under what conditions the idealized CRN model without defects captures the properties of the material, and under what conditions defects are an inherent part of the idealized model. We find that the density of defects in tetrahedral networks does not vary smoothly with variations in the interaction strengths, but jumps from close to zero to a finite density. Consequently, in certain materials, defects do not play a role except for being thermodynamical excitations, whereas in others they are a fundamental ingredient of the ideal structure.

The idea of quantifying order in disordered systems has been introduced recently by Torquato and co-workers (2000 Phys. Rev. E 62 993–1001). We are interested in the application of this idea to measure structure in non-equilibrium systems. Here we focus on gels, using as a model system colloidal gels formed from hard spheres with polymer added to the systems to induce a controlled, weak attraction. To describe the structure of the gels we use real space imaging via confocal microscopy to obtain the full three-dimensional structure. We measure experimentally both translational order and bond angle correlations, defining a new (refined) translational order parameter that is sensitive to long range order in these non-random packings. This metric is also sensitive to anisotropy, which should be important in the many physical situations where an external force is present. The bond angle distribution shows coordinated organization. To give a clearer physical picture for gels, we compare the experimental data to computer generated hard sphere systems.

The conformation and drift properties of a telechelic chain moving through a porous host of randomly distributed static obstacles are studied using Brownian dynamics simulation for a Grest–Kremer bead-spring model of polymer. Static chain conformations exhibit non-monotonic behaviour as a function of the impurity density ρ_imp at zero bias. The chain initially shrinks due to an entropic barrier as the density of the obstacles ρ_imp is increased. In the presence of the obstacles it requires only a tiny bias for a long chain to be mostly stretched. We then study the drift property of the chain as a function of the bias and impurity density. The drift velocity of the chain saturates beyond a threshold bias F_x^crit. This observation can be useful in designing DNA sequencing methods based on electrophoretic mobility.

Diffusion of molecules in brain and other tissues is important in a wide range of biological processes, such as the delivery of drugs, and for measurements, such as diffusion-weighted magnetic resonance imaging. The time-dependent diffusion coefficient D(t) of mobile molecules confined in pores or cells carries information about the confining geometry. At early times, D(t) gives, irrespective of details, the pore surface to volume ratio (S/V_p), and the cell-wall permeability κ. At long times, D(t) reaches a limiting value D₀/α, where tortuosity α is a characteristic of the geometry of the medium.

We propose a new treatment of the Hubbard model that is based both on the coherent-potential approximation (CPA) and the virtual-crystal approximation (VCA). It is well known that the equilibrium found using the one-particle CPA Green functions does not predict an ordered magnetic ground state, while Stoner's mean-field treatment, which is equivalent to the VCA on the Hubbard model, does so for a wide range of parameters. A hybrid treatment, the τ-CPA, is developed, in which a particle is assumed to be scattered from an array of static opposite spins for a time τ related to the inverse of the band width. The propagation is treated in the CPA over this period; thereafter the particle sees the time-averaged effect of the scatterers and hence can be treated in the VCA. This model, with suitable approximations, does predict magnetism for a modified Stoner criterion.

Chains of quantum spins with open ends and isotropic Heisenberg exchange are studied. By diagonalizing the Hamiltonian for chains of finite length N and obtaining all the energy eigenvalues, the magnetic susceptibility χ, the specific heat C_v, and the partition function Z can be calculated exactly for these chains. The high-temperature series expansions of these are then evaluated. For χ and C_v it is found that the terms in the series consist of three parts. One is the normal high-T series already known in great detail for the ring (chain with periodic boundary conditions). The other two consist of a 'surface' term and a correction term of order (1/T)^N. The surface term is found as a series up to and including (1/T)⁸ for spin S = 1/2 and 1. Simple Padé approximant formulae are given to extend the range of validity below T = 1.

A theoretical treatment is presented of the electromagnetic transmission properties of a photonic crystal waveguide that interacts with off-channel dielectric impurities, impurity clusters, neighbouring waveguides, or neighbouring waveguide networks. The photonic crystal studied is a two-dimensional square lattice array of parallel axis dielectric cylinders (formed of linear dielectric medium) in vacuum, and impurities and waveguides are created by cylinder replacement. The waveguide whose transmission is calculated is formed from linear dielectric media, but the off-channel features that it interacts with may be formed from either linear or Kerr nonlinear dielectric media. The off-channel features may also be composed of dissipative or amplifying media. Waveguide transmission resonances, associated with resonant scattering from electromagnetic modes on the off-channel features, are found. Modes present on off-channel features formed from linear dielectric media include both propagating and localized modes. Modes present on off-channel features formed from Kerr nonlinear dielectric media include simple localized and intrinsic localized (soliton-like) modes.

Steady-state and transient photoconductivity methods are used to investigate the electronic density of states in evaporated layers of amorphous selenium. From the temperature dependence of the steady-state photocurrent and, independently, from an analysis of the post-transit currents of time-of-flight transients, energy levels in the gap at 1.43 ± 0.02 eV and 0.40 ± 0.02 eV above the valence band have been determined for the occupied state of the negative-U centres. An absorption band around 1.50 eV is seen in the spectral distribution of the photocurrent. The distribution of tail states may—to first approximation—be described by a steep exponential with a characteristic width of  meV at the valence band and a more steeply declining functional of similar width at the conduction band.

Published electron and hole drift-mobility measurements in hydrogenated amorphous silicon (a-Si:H), amorphous silicon alloys (a-SiGe:H and a-SiC:H), and microcrystalline silicon (μc-Si:H) are analysed in terms of the exponential bandtail trapping model. A three-parameter model was employed using an exponential bandtail width ΔE, the band mobility μ₀, and the attempt-to-escape frequency ν. Low-temperature measurements indicate a value around μ₀ = 1 cm² V⁻¹ s⁻¹ for both the conduction and valence bands over the entire range of materials. High temperature-measurements for electrons in a-Si:H suggest a larger value of 7 cm² V⁻¹ s⁻¹. These properties and those of the frequency ν are discussed as possible attributes of a mobility edge.

Lead chalcogenide salts PbSe and PbTe are IV–VI narrow gap semiconductors whose study over last several decades has been motivated by their importance in infrared detectors, lasers and thermoelectrics. Recently a class of systems, AgSbPb_2n−2Te_n, has been found to be excellent high temperature thermoelectrics. Since electronic properties of semiconductors in general and thermoelectric behaviour in particular, are dominated by defects, we have carried out ab initio electronic structure calculations of Ag and Sb substitutional defects in PbTe. We find that these defects give rise to deep defect states and the electronic structure near the gap depends sensitively on the micro structural arrangements of these defects. The modified electronic structure may be responsible for the observed high temperature thermoelectric properties of the above compounds.

In this paper we study electron dynamics and transport in models of amorphous silicon and amorphous silicon hydride. By integrating the time-dependent Kohn–Sham equation, we compute the time evolution of electron states near the gap, and study the spatial and spectral diffusion of these states due to lattice motion. We perform these calculations with a view to developing ab initio hopping transport methods. The techniques are implemented with the ab initio local basis code SIESTA, and may be applicable to molecular, biomolecular and other condensed matter systems.