Interface Evolution in Three Dimensions¶with Curvature-Dependent Energy¶and Surface Diffusion:¶Interface-Controlled Evolution,Phase Transitions,Epitaxial Growth of Elastic Films

When the interfacial energy is a nonconvex function of orientation, the anisotropic-curvature-flow equation becomes backward parabolic. To overcome the instability thus generated, a regularization of the equation that governs the evolution of the interface is needed. In this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk-surface interactions. The theory is based on a superficial mass balance; configurational forces and couples consistent with superficial force and moment balances; a mechanical version of the second law that includes, via the configurational moments, work that accompanies changes in the curvature of the interface; a constitutive theory whose main ingredient is a positive-definite, isotropic, quadratic dependence of the interfacial energy on the curvature tensor. Two special cases are investigated: (i) the interface is a boundary between bulk phases or grains, and (ii) the interfaceseparates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface.     

Sharp-Interface Nematic–Isotropic Phase Transitions without Flow

We derive a supplemental evolution equation for an interface between the nematic and isotropic phases of a liquid crystal when flow is neglected. Our approach is based on the notion of configurational force. As an application, we study the behavior of a spherical isotropic drop surrounded by a radially oriented nematic phase: our supplemental evolution equation then reduces to a simple ordinary differential equation admitting a closed-form solution. In addition to describing many features of isotropic-to-nematic phase transitions, this simplified model yields insight concerning the occurrence and stability of isotropic cores for hedgehog defects in liquid crystals.     

Theory of interfacial dynamics of nematic polymers

 The interfacial momentum and torque balance equations for deforming interfaces between nematic polymers and isotropic viscous fluids are derived and analyzed with respect to shape selection and interfacial nematic ordering. It is found that the interfacial momentum balance equation for nematic interfaces involves bending forces that act normal to the interface, and that interfacial pressure jumps may exist even for planar surfaces. In addition tangential forces on nematic interfaces arise in the presence of surface gradients of the tensor order parameter. The torque balance equation shows that couple stress jumps are balanced by the surface molecular field. The interfacial balance equations are shown to be coupled such that nematic ordering depends on shape and vice versa. The governing dimensionless numbers for deforming nematic polymer interfaces are identified and the limiting regimes are discussed in reference to related experimental data. It is found that the ratio of Frank elasticity to surface anchoring controls whether the surface tensor order parameter deviates from its preferred equilibrium value. Whether the shape is affected, depends on the relative magnitudes of the isotropic surface tension, Frank bulk elasticity, and anchoring energy, and capillary number. Received: 16 April 1999/Accepted: 19 August 1999     

Growth and structure of nematic spherulites under shallow thermal quenches

The growth kinetics, shape, interfacial and internal orientation texture of a submicron nematic spherulite arising during the isotropic-to-nematic liquid crystal phase transformation under shallow thermal quenches is analyzed using theory, scaling, and numerical simulations based on the Landau – de Gennes model (The Physics of Liquid Crystals, 2nd edn. Clarendon, Oxford). The numerical computations from this model yield interfacial cusp formation that relaxes through the nucleation of two disclination lines of topological charge +1/2 and subsequently leads to intra-droplet texturing and a net topological charge within the spherulite of +1. The timing of these events suggests that cusp formation at the interface is intimately associated with the interfacial defect shedding mechanism (J. Chem. Phys. 124:244902, 2006) for shallow quenches. These results are different than predictions for deep quenches (J. Chem. Phys. 124:244902, 2006) where interfacial defect shedding leads to four defects and a net topological charge of +2. A liquid crystal dynamic shape equation is derived from the Landau – de Gennes model to account for the interface shape changes in terms of surface viscosity, the driving forces due to the uniaxial nematic-isotropic free energy difference, capillary forces, and friction forces, and used to semi-quantitatively show that during cusp formation and defect shedding, gradient elasticity, capillary forces and friction play significant roles in decelerating and accelerating the surface. An interfacial eigenvalue analysis shows that during the shallow quench, disclination lines nucleate within the interface itself and then texturize the nematic droplet as they migrate from within the interface to the bulk of the growing nematic droplet. After defect shedding, the spherulite is nearly circular and grows with constant velocity, in agreement with experiments. The results shed new light on intra-spherulite texturing mechanisms in phase ordering under weak driving forces.      

Irreversible thermodynamics of liquid crystal interfaces

A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.     

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.     

A Continuum Theory of Multispecies Thin Solid Film Growth by Chemical Vapor Deposition

A continuum theory for the chemical vapor deposition of thin solid films is proposed, in which a flowing, chemically reacting, gaseous mixture is coupled to the bulk of a growing thin film via the equations that govern the morphological evolution of the interface separating them. The vapor-film interface is viewed as a surface of zero thickness capable of sustaining mass and endowed with thermodynamic variables that account for its distinct structure. We consider situations in which species diffusion and heat conduction occur in all three phases (vapor, bulk and surface), with the former mechanism augmented by the convective transport of particles in the gas. Special attention is given to the chemical reactions that occur both in the vapor and on the film surface. Ours is a conceptual framework based on conservation laws for chemical species, momentum and energy, together with a separate balance of configurational forces. These balances are supplemented by an appropriate version of the second law which is used to develop suitable constitutive relations for each of the phases. In particular, we investigate the case of an elastic film, deposited on a rigid substrate and in contact with a reacting, multispecies, ideal vapor, whose surface behaves like an anisotropic, chemically reactive, multicomponent, ideal lattice gas. In addition to recovering the standard equations that describe the behavior of the gas and film phases, we derive the coupled PDE's that govern the interfacial morphological, chemical, and thermal evolution. In particular, the constitutively augmented interfacial configurational force balance provides a “kinetic relation” linking the thermodynamic “driving force” at the film surface to the growth rate. The special cases of (i) negligible interfacial species densities, and (ii) local (mechanical) equilibrium of both multi- and single-species films are investigated.     

Equilibrium of Phases with Interfacial Energy: A Variational Approach

The paper proves the existence of equilibrium two phase states with elastic solid bulk phases and deformation dependent interfacial energy. The states are pairs (y,E) consisting of the deformation y on the body and the region E occupied by one of the phases in the reference configuration. The bulk energies of the two phases are polyconvex functions representing two wells of the substance. The interfacial energy is interface polyconvex. The last notion is introduced and discussed below, together with the interface quasiconvexity and interface null Lagrangians. The constitutive theory and equilibrium theory of the interface are discussed in detail under appropriate smoothness hypotheses. Various forms of the interfacial stress relations for the standard and configurational (Eshelby) interfacial stresses are established. The equilibrium equations are derived by a variational argument emphasizing the roles of outer and inner variations.     

Kinetics of thermally induced swelling of hydrogels

We present a continuum model for thermally induced volume transitions in stimulus–responsive hydrogels (SRHs). The framework views the transition as proceeding via the motion of a sharp interface separating swollen and collapsed phases of the underlying polymer network. In addition to bulk and interfacial force and energy balances, our model imposes an interfacial normal configurational force balance. To account for the large volume changes exhibited by SRHs during actuation, the governing equations are developed in the setting of finite-strain kinematics. The numerical approximations to the coupled thermomechanical equations are obtained with an extended finite element/level-set method. The solution strategy involves a non-standard operator split and a simplified version of the level-set update. A number of representative problems are considered to investigate the model and compare its predictions to experimental observations. In particular, we consider the thermally induced swelling of spherical and cylindrical specimens. The stability of the interface evolution is also examined.     

Sharp-crack limit of a phase-field model for brittle fracture

A matched asymptotic analysis is used to establish the correspondence between an appropriately scaled version of the governing equations of a phase-field model for fracture and the equations of the two-dimensional sharp-crack theory of Gurtin and Podio-Guidugli (1996) that arise on assuming that the bulk constitutive behavior is nonlinearly elastic, requiring that surface energy provides the only factor limiting crack propagation, and assuming that the fracture kinetics are isotropic. Consistent with the prominence of the configurational momentum balance at the crack tip in the latter theory, the approach capitalizes on the configurational momentum balance that arises naturally in the context of the phase-field model. The model developed and utilized here incorporates irreversibility of the phase-field evolution. This is achieved by introducing a suitable constraint and by carefully heeding the influence of that constraint on the kinetics underlying microstructural changes associated with fracture. The analysis is predicated on the assumption that the phase-field variable takes values in the closed interval between zero and unity.     

Boundary layer viscosity of CNT-doped liquid crystals: effects of phase behavior

Carbon nanotube (CNT)-doped liquid crystals (LCs) have attracted intensive research studies as prospective materials in optic display devices, microfluidic sensors, and lubricants due to their unique molecular structures and properties. In this paper, the interaction between the doped CNTs and the hosting 4-cyano-4 $^{\prime }$ -pentylbiphenyl (5CB) molecules (in both nematic and isotropic phases) was investigated and we focused on the boundary layer rheological properties of the CNT-doped 5CB under external electric field. The experiments were performed by using a quartz crystal microbalance (for boundary layer viscosity investigation) and a rheometer (for bulk viscosity measurement). The results indicate that the bulk viscosity of the CNT-doped 5CB presents an obvious electroviscous effect in its nematic phase while no electroviscous effect in its isotropic phase. Additionally, we found that the boundary layer viscosity of the CNT-doped 5CB demonstrated significant electroviscous effects both in its nematic phase and isotropic phase. The enhanced electroviscous effects on the boundary layer viscosity of CNT-doped 5CB can be attributed to the highly ordered structures of LC molecules and CNTs on the substrate under external electric field. The unique properties of the boundary layer rheology of CNT-doped 5CB LC were further discussed in view of the ordering of LC molecules induced by the electric field, the polarity of CNTs, and the aligning interaction between CNTs and LC molecules.     

Evolution of the balance equations in saturated thermoelastic porous media following abrupt simultaneous changes in pressure and temperature

A mathematical model is developed for saturated flow of a Newtonian fluid in a thermoelastic, homogeneous, isotropic porous medium domain under nonisothermal conditions. The model contains mass, momentum and energy balance equations. Both the momentum and energy balance equations have been developed to include a Forchheimer term which represents the interaction at the solid-fluid interface at high Reynolds numbers. The evolution of these equations, following an abrupt change in both fluid pressure and temperature, is presented. Using a dimensional analysis, four evolution periods are distinguished. At the very first instant, pressure, effective stress, and matrix temperature are found to be disturbed with no attenuation. During this stage, the temporal rate of pressure change is linearly proportional to that of the fluid temperature. In the second time period, nonlinear waves are formed in terms of solid deformation, fluid density, and velocities of phases. The equation describing heat transfer becomes parabolic. During the third evolution stage, the inertial and the dissipative terms are of equal order of magnitude. However, during the fourth time period, the fluid's inertial terms subside, reducing the fluid's momentum balance equation to the form of Darcy's law. During this period, we note that the body and surface forces on the solid phase are balanced, while mechanical work and heat conduction of the phases are reduced.     

流场中聚合物共混体系液滴形变的理论模型

讨论了两相聚合物共混体系中，悬浮于另一种牛顿（或粘弹）液体中的牛顿（或粘弹）液滴的形变理论模型．影响液滴形变的主要因素有两相的组成、粘度比和弹性比、动态界面张力、临界界面张力系数，外流场形式及其强度．对于两相均为牛顿流体的体系，理论预测能够与实验相符；对于两相（或其中一相）为粘弹流体的体系，由于弹性的影响而使液滴形变的研究变得复杂，理论模型尚需完善．建立完整的液滴形变理论模型还需深入研究界面层、微观分子形变、液滴之间及液滴和连续相介质之间的相互作用对液滴形变的影响     

Interface balance laws,phase growth and nucleation conditions for multiphase solids with inhomogeneous surface stress

In this paper, the thermodynamic configurational force associated with a moving interface is used to derive the conditions for phase growth and nucleation in bodies with multiple diffusing species and arbitrary surface stress at the phase interface. First, the mass, momentum and energy balances are derived on the evolving phase interface. The thermodynamic conditions that result from free energy inequality at the interface are derived leading to the analytical form of the configurational force for bodies subject to mechanical loads, heat and multiple diffusing species. The derived second law condition naturally extends the Eshelby energy–momentum tensor to include species diffusion terms. The above second law restriction is then used to derive the condition for the growth of new phases in a body undergoing finite deformation subject to inhomogeneous as well as anisotropic interface stress, and multiple diffusing species. The growth conditions are derived in both current and reference configurations. The statistical temperature-dependent growth velocity is next derived using the Boltzmann distribution. The derived finite deformation form of growth requirement is simplified to obtain the small deformation diffusive void growth condition. Next, a general, finite deformation, arbitrary surface stress form of phase nucleation condition is derived by considering uncertainty in growth of a small nucleus. The probability of nucleation is shown to naturally depend on a theoretical estimate of critical volumetric energy density, which is directly related to the surface stress. The classical nucleation theory is shown to result from a simplified special case of the general criterion. As an application of the developed theory, the classical Blech electromigration experiment is simulated to estimate the critical energy density corresponding to the onset of electromigration voids at Al–TiN interface.     

Viscosity of a Newtonian fluid calculated from the deformation of droplets covered with a surfactant under a linear shear flow

The viscosity of small fluid droplets covered with a surfactant is determined using drop deformation techniques. This method, proposed by Hu and Lips, is here extended to the case of the presence of a surface-active adsorpted at the liquid–liquid interface, to consider more general scenarios. In these experiments, a droplet is sheared by another immiscible fluid of known viscosity, both Newtonian liquids. From the steady-state deformation and retraction mechanisms, the droplet viscosity is calculated using an equation derived from the theories of Taylor and Rallison. Although these theories were expressed for surfactant-free interfaces, they can be applied when a surfactant is present in the system if the sheared droplet reaches reliable steady-state deformations and the surfactant attains its equilibrium adsorption concentration. These determinations are compared to bulk viscosities measured in a rheometer for systems with different viscosity ratios and surfactant concentrations. Very good agreement between both determinations is found for drops more viscous than the continuous phase.     

Disclinations in a homogenously deformed nematic elastomer

We consider a nematic elastomer which has been cross-linked in an isotropic state. As an application, we consider the time-independent, isochoric, homogeneous deformation of a right circular cylinder wherein each circular cross section of the specimen is deformed into an ellipse. We explore the possibility of the existence of disclinations when the molecular conformation is uniaxial. Numerical solution of the governing boundary-value problem indicates the presence of an isotropic core (the disclination core) of material surrounding the cylinder axis in which the polymeric chains are shaped as spherical coils. The isotropic core is bounded by a narrow transition layer across which the molecular conformation changes from spherical to uniaxial. The material thereby becomes anisotropic away from the disclination. The anisotropic regions show a markedly different distribution for the free energy. This distribution of energy determines the radius of the core to be on the order of 10⁻² μm. This work is a first step toward gaining an understanding of the existence of defective states in homogenous deformed nematic elastomers.     

Continuum theory for nematic liquid crystals

This paper presents a formulation of continuum theory for nematic liquid crystals based upon the balance laws for linear and angular momentum, that derives directly expressions for stress and couple stress in these transversely isotropic liquids. This approach therefore avoids the introduction of generalised forces or torques associated with the director describing the axis of transverse isotropy.     

Design of three dimensional isotropic microstructures for maximized stiffness and conductivity

The level-set method of topology optimization is used to design isotropic two-phase periodic multifunctional composites in three dimensions. One phase is stiff and insulating whereas the other is conductive and mechanically compliant. The optimization objective is to maximize a linear combination of the effective bulk modulus and conductivity of the composite. Composites with the Schwartz primitive and diamond minimal surfaces as the phase interface have been shown to have maximal bulk modulus and conductivity. Since these composites are not elastically isotropic their stiffness under uniaxial loading varies with the direction of the load. An isotropic composite is presented with similar conductivity which is at least 23% stiffer under uniaxial loading than the Schwartz structures when loaded uniaxially along their weakest direction. Other new near-optimal isotropic composites are presented, proving the capablities of the level-set method for microstructure design.     

Dynamics and rheology of the morphology of immiscible polymer blends – on modeling and simulation

 The material properties of heterogeneous polymer blends are crucially influenced by their morphology, i.e., by the spatial structure of the blend components and by the specific configuration of the interfaces separating the phases. Hence, in order to understand the behavior of experimentally obtained morphologies, one is interested in modeling the relevant dynamics of the morphology subject to external flow. Thus one can study, e.g., through the interfacial stress tensor the rheological properties due to the interfaces. The balance equations used for that purpose are based on a Cahn-Hilliard equation for the local concentration, the continuity equation, and a modified Navier-Stokes equation for the local velocity. The essential material and processing parameters such as surface tension, viscosity and volume fraction of both polymers, and imposed shear rate are taken into consideration as model coefficients. By regarding hydrodynamic interaction, which is proved to be important in case of immiscible blends, the interfacial relaxation is described properly. Simulations in both three and two dimensions agree at least qualitatively with experimental results concerning droplet deformation, droplet coalescence, and interfacial rheological properties of the blend. Received: 25 September 2000 Accepted: 24 April 2001