Mechanical Equilibrium Conditions Research Articles

Heterogeneous nucleation in multi-component vapor on a partially wettable charged conducting particle. II. The generalized Laplace, Gibbs-Kelvin, and Young equations and application to nucleation

Based on the results of a previous paper [M. Noppel, H. Vehkamäki, P. M. Winkler, M. Kulmala, and P. E. Wagner, J. Chem. Phys. 139, 134107 (2013)], we derive a thermodynamically consistent expression for reversible or minimal work needed to form a dielectric liquid nucleus of a new phase on a charged insoluble conducting sphere within a uniform macroscopic one- or multicomponent mother phase. The currently available model for ion-induced nucleation assumes complete spherical symmetry of the system, implying that the seed ion is immediately surrounded by the condensing liquid from all sides. We take a step further and treat more realistic geometries, where a cap-shaped liquid cluster forms on the surface of the seed particle. We derive the equilibrium conditions for such a cluster. The equalities of chemical potentials of each species between the nucleus and the vapor represent the conditions of chemical equilibrium. The generalized Young equation that relates contact angle with surface tensions, surface excess polarizations, and line tension, also containing the electrical contribution from triple line excess polarization, expresses the condition of thermodynamic equilibrium at three-phase contact line. The generalized Laplace equation gives the condition of mechanical equilibrium at vapor-liquid dividing surface: it relates generalized pressures in neighboring bulk phases at an interface with surface tension, excess surface polarization, and dielectric displacements in neighboring phases with two principal radii of surface curvature and curvatures of equipotential surfaces in neighboring phases at that point. We also re-express the generalized Laplace equation as a partial differential equation, which, along with electrostatic Laplace equations for bulk phases, determines the shape of a nucleus. We derive expressions that are suitable for calculations of the size and composition of a critical nucleus (generalized version of the classical Kelvin-Thomson equation).

Spontaneous penetration of a non-wetting drop into an exposed pore

We consider the penetration process of a liquid drop approaching an exposed pore along the axis of symmetry, which is intended to model the penetration of non-wetting drops into a porous medium. Inertia and gravity are neglected at the current stage. In addition to the penetration into a capillary tube in the literature, the drop may spread on the outer surface of the porous medium. Based on the mechanical equilibrium states, we find the critical drop radius, below which the drop penetration is spontaneous. We further identify five penetration regimes based on the drop radius and the static contact angle, all of which are exemplified by phase-field simulations. The free energy as a function of penetration depth reveals only two stable equilibrium states: the drop either enters the pore completely (maximum penetration) or stays at the pore inlet (zero penetration). For a non-penetrating drop radius, the free energy has a local maximum which constitutes an energy barrier that prevents spontaneous penetration. Finally, we modify the Lucas-Washburn equation to describe the dynamic process of penetration. Due to the neglect of dissipation from moving contact lines and entry flow, the modified Lucas-Washburn equation greatly overestimates the penetration rate, especially at the initial stage.

Contact network topology in tapped granular media

We analyze the contact network of simulated two-dimensional granular packings in different states of mechanical equilibrium obtained by tapping. We show that topological descriptors of the contact network allow one to distinguish steady states of the same mean density obtained with different tap intensities. These equal-density states were recently proven to be distinguishable through the mean force moment tensor. In contrast, geometrical descriptors, such as radial distribution functions, bond order parameters, and Voronoi cell distributions, can hardly discriminate among these states. We find that small-order loops of contacts-the polygons of the network-are especially sensitive probes for the contact structure.

Mechanical Analysis and Stability Design of Extrusion Forming Machine of Thorn Pile

In the paper basic knowledge of thorn pile and the working principle characteristics are introduced in different directions. Mechanical equilibrium conditions of thorn pile are that Adjacent to the piston must be each other angle of 60 °and Hydraulic cylinders are assembled In the longitudinal. The paper laid the foundation for the design and use of thorn pile extrusion forming machine.

Effect of macroscopic relaxation on residual stress analysis by diffraction methods

To know the residual stress state within a component is essential for predicting its service life time. Neutron diffraction is a reliable tool for non-destructive determination of macroscopic residual stresses, even deep inside a component. One standard procedure in neutron residual stress analysis is to cut out small macrostress free coupons from the strained component, which serve as reference samples. Redistribution of intergranular stresses due to relaxation when the coupon is cut out, however, may lead to spurious macroscopic stresses. A quantitative study of the change of intergranular stresses shows that, contrary to conventional thought, even the {311} Bragg reflection is affected strongly by this phenomenon. In this case, the use of reference values calculated using boundary conditions of mechanical equilibrium enables to extract reliable macroscopic residual stress values.

Repulsive Casimir interaction between conducting and permeable corrugated plates

We consider the Casimir interaction between perfectly conducting and infinitely permeable corrugated parallel plates. We first study smooth plates which face each other at the distance $H$ with mixed boundary conditions (BCs) and show that the Casimir force is repulsive between them. Then we consider the effect of corrugation on the Casimir interaction. We investigate two geometries with mixed BCs, one with a corrugated plate and a flat plate (cf) and another with two corrugated surfaces. For the geometry of cf, at small distances the correction to the Casimir energy is ${E}_{\mathrm{cf}}^{\mathrm{N},\mathrm{D}}\ensuremath{\sim}\frac{+\ensuremath{\hbar}c{a}^{2}}{{H}^{5}}$, while at large distances ${E}_{\mathrm{cf}}^{\mathrm{N},\mathrm{D}}\ensuremath{\sim}\frac{+\ensuremath{\hbar}c{a}^{2}}{{H}^{4}\ensuremath{\lambda}}$, where $a$ and $\ensuremath{\lambda}$ are amplitude and wavelength of the sinusoidal corrugated surface respectively. On the other hand, when both surfaces are sinusoidally corrugated with a wavelength of $\ensuremath{\lambda}$, they experience a periodic sinusoidal lateral casimir force ${F}_{l}\mathrm{sin}(2\ensuremath{\pi}l/\ensuremath{\lambda})$, where $l$ is the lateral displacement of the plates with respect to each other. In the limit of $\frac{H}{\ensuremath{\lambda}}\ensuremath{\ll}1$, ${F}_{l}\ensuremath{\sim}\ensuremath{-}\frac{\ensuremath{\hbar}c{a}^{2}}{{H}^{5}\ensuremath{\lambda}}$ while at the region of $\frac{H}{\ensuremath{\lambda}}\ensuremath{\gg}1$, ${F}_{l}$ decays exponentially as ${F}_{l}\ensuremath{\sim}\ensuremath{-}\frac{\ensuremath{\hbar}c{a}^{2}}{H{\ensuremath{\lambda}}^{5}}{e}^{\ensuremath{-}2\ensuremath{\pi}H/\ensuremath{\lambda}}$. For fixed $H$ and $\ensuremath{\lambda}$, stable and unstable mechanical equilibrium states of the system occur when $l=n\ensuremath{\lambda}$ and $l=(n+\frac{1}{2})\ensuremath{\lambda}$ respectively, where $n$ is an integer number.

Fabric evolution and accessible geometrical states in granular materials

We analyze the geometrical states of granular materials by means of a fabric tensor involving the coordination number and fabric anisotropy as the lowest-order descriptors of the contact network. In particular, we show that the fabric states in this representation are constrained by steric exclusions and the condition of mechanical equilibrium required in the quasi-static limit. A simple model, supported by numerical data, allows us to characterize the range of accessible fabric states and the joint evolution of fabric parameters. The critical state in this framework appears as a jammed state in the sense of a saturation of contact gain and loss along the principal strain-rate directions.

Trench curvature and deformation of the subducting lithosphere

SUMMARY The subduction of oceanic lithosphere is generally accompanied by downdip and lateral deformation. The downdip component of strain is associated with external forces that are applied to the slab during its sinking, namely the gravitational force and the mantle resistance to penetration. Here, we present theoretical arguments showing that a tectonic plate is also subject to a predictable amount of lateral deformation as a consequence of its bending along an arcuate trench zone, independently from the long-term physical processes that have determined the actual curvature of the subduction zone. In particular, we show that the state of lateral strain and the lateral strain rate of a subducting slab depend from geometric and kinematic parameters, such as trench curvature, dip function and subduction velocity. We also demonstrate that the relationship between the state of lateral strain in a subducting slab and the geometry of bending at the corresponding active margin implies a small component of lateral shortening at shallow depths, and may include large extensional lateral deformation at intermediate depths, whereas a state of lateral mechanical equilibrium can only represent a localized exception. Our formulation overcomes the flaws of the classic ‘ping-pong ball’ model for the bending of the lithosphere at subduction zones, which lead to severe discrepancies with the observed geometry and style of deformation of the modern subducting slabs. A study of the geometry and seismicity of eight modern subduction zones is performed, to assess the validity of the theoretical relationship between trench curvature, slab dip function, and lateral strain rate. The strain pattern within the eight present-day slabs, which is reconstructed through an analysis of Harvard CMT solutions, shows that tectonic plates cannot be considered as flexible-inextensible spherical caps, whereas the lateral intraslab deformation which is accommodated through seismic slip can be explained in terms of deviations from the mechanical equilibrium.

Directed transport in equilibrium: A model study

It is generally believed that sustained directed transport in mechanical systems is not achievable under equilibrium conditions. In this article, by analyzing a simple model, we will show that directed motion under equilibrium conditions can take place and is not inconsistent with conservation of energy and the second law of thermodynamics. Our model demonstrates a novel symmetry breaking mechanism sustainable under equilibrium conditions and average uniform motion of the center of mass is a consequence of the sustained broken symmetry. The most important consequence of our results is that, no current condition for mechanical equilibrium is possibly non-universal.

Induced Thermal Stresses in Core Shell Magnetic Particles

In this paper, the induced thermal stresses that appear during the cooling process from the room temperature to a lower temperature for core shell magnetic particles are analyzed. Using a finite element method for solving the Fourier heat transport equation, the thermal stresses in the system were calculated taking into account of the temperature dependence of thermal conductivity, heat capacity and different thermal expansion coefficients of the two materials (iron core + alumina shell) in contact. The thermal stresses depend on the particle's radius and the thickness of the alumina shell, having negative values due to the thermal contraction. It was estimated the influence of shell thickness on thermal stresses and on the speed of reaching the mechanical and thermal equilibrium state.

Human annulus fibrosus dynamic tensile modulus increases with degeneration

The annulus fibrosus (AF) of the intervertebral disc experiences cyclic tensile loading in vivo at various states of mechanical equilibrium. Disc degeneration is associated with alterations in the biochemical composition of the AF including decreased water content, decreased proteoglycan concentration, and increased collagen deposition that affect mechanical function of the AF in compression and shear. Such changes may also affect the dynamic viscoelastic properties of the AF and thus alter the disc’s ability to dissipate energy under physiologic loading. The objectives of this study were to quantify the dynamic viscoelastic properties of human AF in circumferential tension and to determine the effect of degeneration on these properties. Nondegenerate and degenerate human AF tensile samples were tested in uniaxial tension over a spectrum of loading frequencies spanning 0.01–2 Hz at several states of equilibrium strain to determine the dynamic viscoelastic properties (dynamic modulus, phase angle) using a linear viscoelastic model. The AF dynamic modulus increased at higher equilibrium strain levels. The AF behaved more elastically at higher frequencies with a decreased phase angle. Degeneration resulted in a higher dynamic modulus at all strain levels but had no effect on phase angle. The findings from this study elucidate the effect of degeneration on the dynamic viscoelastic properties of human AF and lend insight into the mechanical role of the AF in cyclic loading conditions.

The equivalence between volume averaging and method of planes definitions of the pressure tensor at a plane

It is shown analytically that the method of planes (MOP) [Todd, Evans, and Daivis, Phys. Rev. E 52, 1627 (1995)] and volume averaging (VA) [Cormier, Rickman, and Delph, J. Appl. Phys. 89, 99 (2001)] formulas for the local pressure tensor, P(α, y)(y), where α ≡ x, y, or z, are mathematically identical. In the case of VA, the sampling volume is taken to be an infinitely thin parallelepiped, with an infinite lateral extent. This limit is shown to yield the MOP expression. The treatment is extended to include the condition of mechanical equilibrium resulting from an imposed force field. This analytical development is followed by numerical simulations. The equivalence of these two methods is demonstrated in the context of non-equilibrium molecular dynamics (NEMD) simulations of boundary-driven shear flow. A wall of tethered atoms is constrained to impose a normal load and a velocity profile on the entrained central layer. The VA formula can be used to compute all components of P(αβ)(y), which offers an advantage in calculating, for example, P(xx)(y) for nano-scale pressure-driven flows in the x-direction, where deviations from the classical Poiseuille flow solution can occur.

Influence of substrate stiffness on cell–substrate interfacial adhesion and spreading: A mechano-chemical coupling model

Cell interactions with extracellular matrix, such as cell adhesion and spreading, are crucial for many biological functions and processes. Recent experimental progresses have demonstrated that substrate rigidities exert a remarkable influence on cell–substrate interfacial adhesion and spreading behaviors. The underlying biophysical mechanism, however, remains elusive. Based on the classical Bell-Dembo’s theory, this paper develops a mechano-chemical coupling model to physically describe cell adhesion and spreading mediated by substrate stiffness. By investigating the competitive nature between cell–substrate specific attraction and non-specific repulsion, the kinetic relation of receptor–ligand interplay is established, in which the influences of receptor–ligand separation, substrate elasticity and non-specific repulsion on cell adhesions are especially addressed. According to mechanical equilibrium conditions between cell membranes and underlying elastic substrates, an analytical expression is then deduced to relate the cell–substrate interfacial adhesion strength to the substrate rigidity. Moreover, by means of the conventional wetting theory, the dependence of steady-state cell spreading on substrate stiffness is also quantitatively studied. Comparisons with the existing experimental data show that the proposed model can be used to explore cell–substrate interactions regulated by substrate rigidities.

Asymptotics of pressure tensor in the wedge-shaped film of simple fluid: 1. Calculation of Irving-Kirkwood tensor

Based on the distribution functions calculated previously, the Irving-Kirkwood pressure tensor in the wedge-shaped cavity of solid containing fluid in the presence of only dispersion forces is calculated. All four (three diagonal and one off-diagonal) components of pressure tensor are calculated as functions of position inside the cavity. It is shown that these components satisfy condition for mechanical equilibrium.

European Society of Biomechanics S.M. Perren Award 2010: An adaptation mechanism for fibrous tissue to sustained shortening

The mechanism by which fibrous tissues adapt upon alterations in their mechanical environment remains unresolved. Here, we determine that periosteum in chick embryos resides in an identical mechanical state, irrespective of the developmental stage. This state is characterized by a residual tissue strain that corresponds to the strain in between the pliant and stiffer region of the force-strain curve. We demonstrate that periosteum is able to regain that mechanical equilibrium state in vitro, within three days upon perturbation of that equilibrium state. This adaptation process is not dependent on protein synthesis, because the addition of cycloheximide did not affect the response. However, a functional actin filament network is required, as is illustrated by a lack of adaptation in the presence of cytochalasin D. This led us to hypothesize that cells actively reduce collagen fiber crimp after tissue shortening, i.e. that in time the number of recruited fibers is increased via cell contraction. Support for this mechanism is found by visualization of fiber crimp with multiphoton microscopy before the perturbation and at different time points during the adaptive response.

A Variational Approach and Finite Element Implementation for Swelling of Polymeric Hydrogels Under Geometric Constraints

A hydrogel consists of a cross-linked polymer network and solvent molecules. Depending on its chemical and mechanical environment, the polymer network may undergo enormous volume change. The present work develops a general formulation based on a variational approach, which leads to a set of governing equations coupling mechanical and chemical equilibrium conditions along with proper boundary conditions. A specific material model is employed in a finite element implementation, for which the nonlinear constitutive behavior is derived from a free energy function, with explicit formula for the true stress and tangent modulus at the current state of deformation and chemical potential. Such implementation enables numerical simulations of hydrogels swelling under various constraints. Several examples are presented, with both homogeneous and inhomogeneous swelling deformation. In particular, the effect of geometric constraint is emphasized for the inhomogeneous swelling of surface-attached hydrogel lines of rectangular cross sections, which depends on the width-to-height aspect ratio of the line. The present numerical simulations show that, beyond a critical aspect ratio, creaselike surface instability occurs upon swelling.

Jamming I: A volume function for jammed matter

We introduce a “Hamiltonian”-like function, called the volume function, indispensable to describe the ensemble of jammed matter such as granular materials and emulsions from a geometrical point of view. The volume function represents the available volume of each particle in the jammed systems. At the microscopic level, we show that the volume function is the Voronoi volume associated to each particle and in turn we provide an analytical formula for the Voronoi volume in terms of the contact network, valid for any dimension. We then develop a statistical theory for the probability distribution of the volumes in 3d to calculate an average volume function coarse-grained at a mesoscopic level. The salient result is the discovery of a mesoscopic volume function inversely proportional to the coordination number. Our analysis is the first step toward the calculation of macroscopic observables and equations of state using the statistical mechanics of jammed matter, when supplemented by the condition of mechanical equilibrium of jamming that properly defines jammed matter at the ensemble level.

Controllable equilibrium of an inhomogeneously heated liquid in a vibrational field

Equations of thermal vibrational convection of an inhomogeneously heated liquid are analytically solved for arbitrary orientation of vectors determining the heating direction, force of gravity, and vibrations in a closed cavity in the shape of an infinite horizontal cylinder. It is demonstrated that for any orientation of these vectors in the plane perpendicular to the cylinder, the gravitational and vibrational convection mechanisms can completely compensate each other. The ratio is determined between the vibrational and the gravitational Grashof numbers such that mechanical equilibrium conditions are satisfied. This ratio is proved by direct numerical calculation of complete equations of thermal vibrational convection.

Quanta Energy Level Structure and the First-Order Freedericksz Transition of Weak Anchoring NLC Cells

The energy problem of weak anchoring nematic liquid crystal (NLC) cell is investigated. For given system, there are four solutions satisfying mechanical equilibrium conditions simultaneity under certain condition. Liquid crystal system exist quanta energy level. The values and sequence of energy levels related to the change of external field and anchoring parameters. The state with the smallest energy is the stable state and the others are metastable states, which are separated by the energy barriers. The Freedericksz transition can be considered as the director slippage through these barriers to the lowest energy level and the first order transition and bistable state result from the existence of the multiple disturbed solutions.

On the calculation of pressure tensor in wedge-shaped cavity

A new simplified expression for the components of a pressure tensor in a wedge-shaped cavity of a solid with dispersion forces is derived. All four (three diagonal and one off-diagonal) components of the pressure tensor are calculated as functions of their positions inside the cavity. It is verified that these functions satisfy the conditions for mechanical equilibrium.