Shaun Sellers - Academia.edu (original) (raw)

Papers by Shaun Sellers

Springer eBooks, 2000

We develop a theoretical framework for the diffusion of a single unconstrained species of atoms o... more We develop a theoretical framework for the diffusion of a single unconstrained species of atoms on a crystal lattice that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to account for constitutive nonlinearities, external forces, and the deformation of the lattice. In this framework, we regard atomic diffusion as a microscopic process described by two independent kinematic variables: (i) the atomic flux, which reckons the local motion of atoms relative to the motion of the underlying lattice, and (ii) the time-rate of the atomic density, which encompasses nonlocal interactions between migrating atoms and characterizes the kinematics of phase separation. We introduce generalized forces power-conjugate to each of these rates and require that these forces satisfy ancillary microbalances distinct from the conventional balance involving the forces that expend power over the rate at which the lattice deforms. A mechanical version of the second law, which takes the form of an energy imbalance accounting for all power expenditures (including those due to the atomic diffusion and phase separation), is used to derive restrictions on the constitutive equations. With these restrictions, the microbalance involving the forces conjugate to the atomic flux provides a generalization of the usual constitutive relation between the atomic flux and the gradient of the diffusion potential, a relation that in conjunction with the atomic balance yields a generalized Cahn-Hilliard equation. Dedicated to Roger Fosdick, whose fundamental and broad-reaching work in continuum thermomechanics has provided us with abundant inspiration.

Mathematical Models and Methods in Applied Sciences, 2000

In order to clarify common assumptions on the form of energy and momentum in elasticity, a genera... more In order to clarify common assumptions on the form of energy and momentum in elasticity, a generalized conservation format is proposed for finite elasticity, in which total energy and momentum are not specified a priori. Velocity, stress, and total energy are assumed to depend constitutively on deformation gradient and momentum in a manner restricted by a dissipation principle and certain mild invariance requirements. Under these assumptions, representations are obtained for energy and momentum, demonstrating that (i) the total energy splits into separate internal and kinetic contributions, and (ii) the momentum is linear in the velocity. It is further shown that, if the stress response is strongly elliptic, the classical specifications for kinetic energy and momentum are sufficient to give elasticity the standard format of a quasilinear hyperbolic system.

Molecular Crystals and Liquid Crystals Science and Technology. Section A. Molecular Crystals and Liquid Crystals, 1997

The signs of the Leslie viscosities α2 and α3 for rod-like and disc-like molecules in the uniaxia... more The signs of the Leslie viscosities α2 and α3 for rod-like and disc-like molecules in the uniaxial nematic phase are discussed using a dynamic molecular mean field theory based upon a Fokker-Planck equation. The viscosities are shown to depend on the particle geometry and alignment order parameters. In particular, for positive degree of alignment S 2, the calculations tend to confirm Carlsson's conjecture relating the signs of the two viscosities to the particle geometry. Furthermore, a transition in the signs of α2 for discotics and α3 for nematics is predicted with increasing order parameter S 2 in accord with recent experiments. For small degree of alignment S 2, the results reproduce previous calculations obtained by Hess in a truncation approximation. The qualitative flow behaviour depends upon the signs of these viscosities.

We develop a continuum theory of fluids that consist of orientable permanent dipoles of identical... more We develop a continuum theory of fluids that consist of orientable permanent dipoles of identical geometry subject to the influences of rotary diffusion and externally applied fields. The average alignment of the dipoles may vary in magnitude and in direction. Fundamental to our approach is an extended kinematic description wherein each material element has associated with it a position in space and a distribution of orientations on the unit sphere-the former describing the center of mass of an assembly of dipoles and the latter describing the manner in which the dipoles are oriented about their center of mass. By incorporating transport due to the translatory motion of the mass centers and also to the rotary motion of the dipoles, our proposed balance laws reflect this extended kinematic description. In addition to a rotary mass balance, we formulate separate momentum balances associated with the translatory and the rotary degrees of freedom and a moment of momentum balance that incorporates both translatory and rotary ingredients. For simplicity, we suppress thermal effects, so that the first and second laws of thermodynamics are expressed by an energy imbalance. Our final governing equations result on combining the fundamental balances with constitutive relations restricted to satisfy the energy imbalance in all thermodynamic processes. These partial differential equations provide a framework that encompasses, as specializations reflecting certain idealized constitutive assumptions, equations arising in common statistical models for dipolar fluids, such as the Debye theory of rotary diffusion and various straightforward generalizations thereof

A continuum model of crystalline solid equilibrium is presented in which the underlying periodic ... more A continuum model of crystalline solid equilibrium is presented in which the underlying periodic lattice structure is taken explicitly into account. This model also allows for both point and line defects in the bulk of the lattice and at interfaces, and the kinematics of such defects is discussed in some detail. A Gibbsian variational argument is used to derive the necessary bulk and interfacial conditions for multi-phase equilibrium (crystal-crystal and crystal-melt) where the allowed lattice variations involve the creation and transport of defects in the bulk and at the phase interface. An interfacial energy, assumed to depend on the interfacial dislocation density and the orientation of the interface with respect to the lattices of both phases, is also included in the analysis. Previous equilibrium results based on nonlinear elastic models for incoherent and coherent interfaces are recovered as special cases for when the lattice distortion is constrained to coincide with the macroscopic deformation gradient, thereby excluding bulk dislocations. The formulation is purely spatial and needs no recourse to a fixed reference configuration or an elastic-plastic decomposition of the strain. Such a decomposition can be introduced however through an incremental elastic deformation superposed onto an already dislocated state, but leads to additional equilibrium conditions. The presentation emphasizes the role of configurational forces as they provide a natural framework for the description and interpretation of singularities and phase transitions.

AIP Conference Proceedings, 1999

We develop a theoretical framework, for the diffusion of a single unconstrained species of atoms ... more We develop a theoretical framework, for the diffusion of a single unconstrained species of atoms on a crystal lattice, that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to account for constitutive nonlinearities, external forces, and the deformation of the lattice. In this framework, we regard atomic diffusion as a microscopic process described by two independent kinematic variables : (i) the atomic flux, which reckons the local motion of atoms relative to the motion of the underlying lattice, and (ii) the time-rate of the atomic density, which encompasses nonlocal interactions between migrating atoms and characterizes the kinematics of phase separation. We introduce generalized forces power-conjugate to each of these rates and require that these forces satisfy ancillary microbalances distinct from the conventional balance involving the forces that expend power over the rate at which the lattice deforms. A me chanica...

Cold Regions Science and Technology, 2000

A model is presented for the transport of water in melting snow where the snow surface and percol... more A model is presented for the transport of water in melting snow where the snow surface and percolation front are treated as propagating singular surfaces. It is based on Colbeck's theory of water transport in bulk snow supplemented with boundary conditions that explicitly include the production of water by snow melting at the surface due to a surface heat supply. The consequent motion of the snow surface leads to a free boundary problem, where the snow surface must be determined as part of the solution, which itself depends on the motion of the snow surface. Explicit relations are obtained for the propagation of the melt surface and percolation front. Numerical examples are given of the propagation of one dimensional meltwater waves in deep snowpacks due to periodic heating of the snow surface. It is shown that, for commonly reported parameter values of deep, homogeneous snow packs, small motions of the snow surface generally lead to small corrections in the water saturation and flux.

The recently proposed neo-classical theory for nematic elastomers is a molecular-statistical gene... more The recently proposed neo-classical theory for nematic elastomers is a molecular-statistical generalization of classical Gaussian network theory. The resulting free-energy density pre-dicts the phenomenon of soft elasticity—the ability of elastomers to undergo large deforma-tions with zero force and energy cost. The theory, however, suffers from several drawbacks: (i) extreme non-uniqueness as zero applied force corresponds to infinitely many possible deformations, (ii) insufficient moduli to model observed experimental behavior, and (iii) physically, a small, but non-zero, force must be applied. Here we propose an alternative continuum model for nematic elastomers that removes these drawbacks. Motivated by the molecular-statistical theory, we identify microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macro-scopic de...

$Research paper thumbnail of Anomalous diffusion in simulations of pumping tests on fractal lattices$

Geological Society, London, Special Publications, 2005

The recent interest in fractals in the geosciences literature has led to several proposed theoret... more The recent interest in fractals in the geosciences literature has led to several proposed theoretical models for the hydraulic testing of fractured-rock systems that exhibit a fractal-like geometric structure. There is, however, no agreement on the correct form of the resulting model equations. In order to gain some insight into the range of possible behaviours to be expected from pumping tests on such systems, as well as the type of theoretical models needed, extensive simulations of pressure diffusion for transient groundwater flow, modelled by random walks on both deterministic and random fractal lattices were performed. For simplicity, the focus was on measurements of the random-walk dimension for generalized Sierpinski carpets, a proposed model for porous and fractured media. In addition to the expected anomalous slow down in diffusion in fractals as measured by the random-walk dimension, the simulations show further novel and unexpected anomalous behaviour due to the presence of internal boundaries at all scales. None of the proposed theoretical models for pumping tests on fractals appears consistent with all of the observed anomalous behaviours. The simulations suggest that interpretation of experimental pumping tests in terms of well-defined non-integer dimensions can be difficult, even when finite-size effects are negligible.

Mol Cryst Liquid Cryst, 1997

Physical Review E Statistical Nonlinear and Soft Matter Physics, 2006

Motivated by studies of comblike structures, we present a generalization of the classical diffusi... more Motivated by studies of comblike structures, we present a generalization of the classical diffusion equation to model anisotropic, anomalous diffusion. We assume that the diffusive flux is given by a diffusion tensor acting on the gradient of the probability density, where each component of the diffusion tensor can have its own scaling law. We also assume scaling laws that have an explicit power-law dependence on space and time. Solutions of the proposed generalized diffusion equation are consistent with previously derived asymptotic results for the probability density on comblike structures.

The Journal of Chemical Physics, 1995

Advances in Continuum Mechanics and Thermodynamics of Material Behavior, 2000

Physical Review E Statistical Physics Plasmas Fluids and Related Interdisciplinary Topics, Nov 1, 1999

Based on the definition of the mesoscopic concept by Blenk et al. [Physica A 174, 119 (1991); J. ... more Based on the definition of the mesoscopic concept by Blenk et al. [Physica A 174, 119 (1991); J. Noneq. Therm. 16, 67 (1991); Mol. Cryst. Liq. Cryst. 204, 133 (1991)] an approach to calculate the Leslie viscosity coefficients for nematic liquid crystals is presented. The approach rests upon the mesoscopic stress tensor, whose structure is assumed similar to the macroscopic Leslie viscous stress. The proposed form is also the main dissipation part of the mesoscopic Navier-Stokes equation. On the basis of the correspondence between microscopic and mesoscopic scales a mean-field mesoscopic potential is introduced. It allows us to obtain the stress tensor angular velocity of the free rotating molecules with the help of the orientational Fokker-Planck equation. The macroscopic stress tensor is calculated as an average of the mesoscopic counterpart. Appropriate relations among mesoscopic viscosities have been found. The mesoscopic analysis results are shown to be consistent with the diffusional model of Kuzuu-Doi and Osipov-Terentjev with the exception of the shear viscosity α4. In the nematic phase α4 is shown to have two contributions: isotropic and nematic. There exists an indication that the influence of the isotropic part is dominant over the nematic part. The so-called microscopic stress tensor used in the microscopic theories is shown to be the mean-field potential-dependent representation of the mesoscopic stress tensor. In the limiting case of total alignment the Leslie coefficients are estimated for the diffusional and mesoscopic models. They are compared to the results of the affine transformation model of the perfectly ordered systems. This comparison shows disagreement concerning the rotational viscosity, whereas the coefficients characteristic for the symmetric part of the viscous stress tensor remain the same. The difference is caused by the hindered diffusion in the affine model case.

Journal of the Mechanics and Physics of Solids, Jul 1, 2004

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-... more The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.

Molecular Crystals and Liquid Crystals Science and Technology Section a Molecular Crystals and Liquid Crystals, 1997

The remarkable ability of nematic elastomers to exist in multiple stable equilibrium configuratio... more The remarkable ability of nematic elastomers to exist in multiple stable equilibrium configurations in the absence of force and energy cost is known as soft elasticity.