Eliot Fried

Continuum theory of thermally induced phase transitions based on an order parameter

Abstract Using balance laws for accretive force and energy in conjunction with constitutive equations restricted so as to be compatible with the second law, we develop a theory for the study of solid-liquid and solid-solid phase transitions where accretion and heat conduction dominate mass diffusion and deformation. Our theory furnishes generalizations of the Ginzburg-Landau equation and the phase-field equations, generalizations that allow for anisotropically induced preferred growth and nonlinear transition kinetics.

Dynamic solid-solid transitions with phase characterized by an order parameter

Abstract We develop a thermodynamically consistent continuum theory for the study of solid-solid phase transitions where deformation dominates heat and mass transfer. We use an order parameter to characterize the notion of phase and identify phase interfaces with thin transition zones within which the strain and order parameter exhibit large gradients. To model the growth of one phase at the expense of another we introduce configurational forces, subject to their own balance, that work against changes in the order parameter. By studying traveling waves and the asymptotics of a transition layer, we establish connections between our theory and more standard approaches that identify phase interfaces with surfaces of strain discontinuity.

Chemically induced swelling of hydrogels

We consider a continuum model for chemically induced volume transitions in hydrogels. Consistent with experimental observations, the model allows for a sharp interface separating swelled and collapsed phases of the underlying polymer network. The polymer chains are treated as a solute with an associated diffusion potential and their concentration is assumed to be discontinuous across the interface. In addition to the standard bulk and interfacial equations imposing force balance and solute balance, the model involves a supplemental interfacial equation imposing configurational force balance. We present a hybrid eXtended-Finite-Element/Level-Set Method for obtaining approximate solutions to the governing equations of the model. As an application, we consider the swelling of a spherical specimen whose boundary is traction-free and is in contact with a reservoir of uniform chemical potential. Our numerical results exhibit good qualitative comparison with experimental observations and predict characteristic swelling times that are proportional to the square of the specimen radius. Our results also suggest several possible synthetic pathways that might be pursued to engineer hydrogels with optimal response times.

Phase Separation in Biological Membranes: Integration of Theory and Experiment

Lipid bilayer model membranes that contain a single lipid species can undergo transitions between ordered and disordered phases, and membranes that contain a mixture of lipid species can undergo phase separations. Studies of these transformations are of interest for what they can tell us about the interaction energies of lipid molecules of different species and conformations. Nanoscopic phases (<200 nm) can provide a model for membrane rafts, specialized membrane domains enriched in cholesterol and sphingomyelin, which are believed to have essential biological functions in cell membranes. Crucial questions are whether lipid nanodomains can exist in stable equilibrium in membranes and what is the distribution of their sizes and lifetimes in membranes of different composition. Theoretical methods have supplied much information on these questions, but better experimental methods are needed to detect and characterize nanodomains under normal membrane conditions. This review summarizes linkages between theoretical and experimental studies of phase separation in lipid bilayer model membranes.

An elementary molecular-statistical basis for the Mooney and Rivlin-Saunders theories of rubber elasticity

By relaxing the assumption that the end-to-end vectors of molecules transform as macroscopic material line elements, we arrive at a generalization of the molecular-statistical theory of rubber elasticity. This generalization includes as special cases continuum-mechanical theories proposed by Mooney and by Rivlin and Saunders as improvements upon the classical neo-Hookean theory.

A unified treatment of evolving interfaces accounting for small deformations and atomic transport with emphasis on grain-boundaries and epitaxy

Publisher Summary This chapter presents a unified treatment of several topics at the intersection of continuum mechanics and materials science; the thrust concerns processes involving evolving interfaces, focusing on grain-boundaries, solid–vapor interfaces (with emphasis on epitaxy), and coherent phase transitions. Central to the discussion is the interaction of deformation, atomic transport, and accretion within a dissipative, dynamical framework, but as the interest is crystalline materials, the focus is restricted to small deformations. To avoid geometrical complications associated with surfaces in three-dimensional space, two space-dimensions are used when discussing interfaces, but three space dimensions while discussing the theory in bulk.

A Continuum-Mechanical Theory for Nematic Elastomers

We develop a continuum theory for the mechanical behavior of rubber-like solids that are formed by the cross-linking of polymeric fluids that include nematic molecules as elements of their main-chains and/or as pendant side-groups. The basic kinematic ingredients of this theory are identical to those arising in continuum-level theories for nematic fluids: in addition to the deformation, which describes the trajectories of material particles, an orientation, which delineates the evolution of the nematic microstructure, is introduced. The kinetic structure of our theory relies on the precept that a complete reckoning of the power expended during the evolution of a continuum requires the introduction of forces that act conjugate to each operative kinematic variable and that to each such force system there should correspond a distinct momentum balance. In addition to conventional deformational forces, which expend power over the time-rate of the deformation and enter the deformational (or linear) momentum balance, we, therefore, introduce a system of orientational forces, which expend power over the time-rate of the orientation and enter an additional orientational momentum balance. We restrict our attention to a purely mechanical setting, so that the thermodynamic structure of our theory rests on an energy imbalance that serves in lieu of the first and second laws of thermodynamics. We consider only nematic elastomers that are incompressible and microstructurally inextensible, and a novel aspect of our approach concerns our treatment of these material constraints. We refrain both from an a priori decomposition of fields into active and reactive components and an introduction of Lagrange multipliers; rather, we start with a mathematical decomposition of the dependent fields such as the deformational stress based on the geometry of the constraint manifold. This naturally gives rise to active and reactive components, where only the former enter into the energy imbalance because the latter automatically expend zero power in processes consistent with the constraints. The reactive components are scaled by multipliers which we take to be constitutively indeterminate. We assume constitutive equations for the active components, and the requirement that these equations be consistent with the energy imbalance in all processes leads to the active components being determined by an energy density response function of the deformation gradient, the orientation, and the orientation gradient. We formulate the requirements of observer independence and material symmetry for such a function and provide, as a specialization, an expression that encompasses the energy densities used in the Mooney-Rivlin description of rubber and the Oseen-Zöcher-Frank description of nematic fluids.

Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interfaces

We establish transport relations for integrals over evolving fluid interfaces. These relations make it possible to localize integral balance laws over non-material interfaces separating fluid phases and, therefore, obtain associated interface conditions in differential form.

Coherent Solid-State Phase Transitions with Atomic Diffusion: A Thermomechanical Treatment

Using the framework of modern continuum thermomechanics, we develop sharp- and diffuse-interface theories for coherent solid-state phase transitions. These theories account for atomic diffusion and for deformation. Of essential importance in our formulation of the sharp-interface theory are a system of “configurational forces” and an associated “configurational force balance.” These forces, which are distinct from standard Newtonian forces, describe the intrinsic material structure of a body. The configurational balance, when restricted to the interface, leads to a generalization of the classical Gibbs–Thomson relation, a generalization that accounts for the orientation dependence of the interfacial energy density and also for a broad spectrum of dissipative transition kinetics. Our diffuse-interface theory involves nonstandard “microforces” and an associated “microforce balance.” These forces arise naturally from an interpretation of the atomic densities as macroscopic parameters that describe atomistic kinematics distinct from the motion of material particles. When supplemented by thermodynamically consistent constitutive relations, the microforce balance yields a generalization of the Cahn–Hilliard relation giving the chemical potentials as variational derivatives of the total free energy with respect to the atomic densities. A formal asymptotic analysis (thickness of the transition layer approaching zero) demonstrates the correspondence between versions of our theories specialized to the case of a single mobile species for situations in which the time scale for interface propagation is small compared to that for bulk diffusion. While the configurational force balance is redundant in the diffuse-interface theory, when integrated over the transition layer, the limit of this balance is the interfacial configurational force balance (i.e., generalized Gibbs–Thomson relation) of the sharp-interface theory.

Configurational stress, yield and flow in rate-independent plasticity

The role of configurational stress in yield and plastic flow is discussed for a macroscopic model of rate–independent, finite–strain plasticity. The model is based on the traditional elastic–plastic decomposition of the deformation gradient, on integral balance laws and on thermodynamically restricted, rate–independent constitutive relations. Its formulation emphasizes the intermediate configuration in both the development of constitutive relations and the expression of balance laws. In addition to the usual balance laws, a couple balance is included to represent the action of plastic couples in the intermediate configuration. In particular, it is shown that the internal couple decomposes into a non–dissipative configurational stress and a dissipative couple that resists plastic flow. The couple balance thus determines a relation between the configurational stress and the plastic–flow resistance, a relation that can be interpreted as a generalized yield condition. A dissipation function is introduced and a maximum–dissipation criterion is used to obtain additional constitutive restrictions, which lead to a counterpart in the intermediate configuration of the classical normality conditions. The versatility of the framework is illustrated by applying it to rigid–plastic flow, in which case a nonlinear generalization of the classical Levy–von Mises theory is obtained.