Interfacial Balance Research Articles

Method for inferring contact angle and for correlating static liquid hold-up in packed beds

Static liquid hold-up (SLH), which is an upper bound measure of the passive liquid volume fraction in packed beds, comes into play in several reaction–separation–transport design models and in correlations of fundamental hydrodynamic variables such as total liquid hold-up, wetting efficiency and pressure gradient. Early reported experimental SLHs in packed beds seemed to level off at a plateau of ca. 5% when capillary force dominates over gravity. Hence, most of the published correlations were constructed taking this asymptotic feature into account, despite the fact that reliable SLH values well beyond 5% were also reported particularly for Bond numbers tending towards zero. Thermodynamics and dimensional analysis of the interfacial momentum balance (Young–Laplace) equation suggests that SLH depends, besides the Bond number ( Bo) and some packing and bed characteristic lengths, on an effective contact angle at the junction where the gas–liquid interface and the packing–gas boundary meet. The contact angle, being difficult to measure, was generally disregarded in most analyses, and in practice, seldom incorporated as a computable quantity in the majority of published SLH correlations. Therefore in this work, a method was proposed for inferring theoretically the contact angle from an energy analysis of the meniscus interfaces’ area obtained from solving the 2-D Young–Laplace equation for two vertically aligned touching equivalent spheres. The approach showed that the intuited contact angles matched the measured ones, when available, with an average error of 8.6%. Following this determination, a new SLH correlation based on the contact angle ( θ c ), the Bond number ( Bo), the solid-to-void volume ratio (1− ε)/ ε, the packing sphericity ( φ), and the bed-to-packing volume ratio, BPR, was proposed and analysed in the light of most published SLH data, i.e., 239 measurements over the past five decades. The SLH correlation yielded an average error of 23%. Sensitivity analysis of its weights showed an important impact of the contact angle as an input variable. The correlation was also found to capture SLH values exceeding 5% in the limit of Bo→0.

Nematostatics of triple lines.

The Landau-de Gennes model for nematic liquid crystal bulk and interfaces has been extended to nematic triple lines involving the intersection of two isotropic fluids and one nematic liquid crystalline phase. A complete set of bulk, interface, and triple line force and torque balance equations has been formulated. The triple line force and torque balance equations have linear, interfacial, and bulk contributions. The bulk contributions appear as junction integrals, the surface contributions as junctions sums, and the line contributions as gradients of stresses. Reduction of dimensionality from three to one dimensional creates the following effects: (a) bulk terms enter interfacial balances as surface jumps and line balances as junction integrals, and (b) surface terms enter linear balances as junction sums. Line stress and torque equations are derived using classical liquid crystal models. The correspondence between line stress and line torque and their surface and bulk analogs is established. The triple line force and torque balance equations are use to analyze the contact angle in a nematic lens lying at the interface between two isotropic fluids, when the preferred surface orientation is tangential. The effect of anisotropy and long range elasticity on triple line phases is established. Under weak anchoring the contact angle is shown to be a function of the anchoring energy at the nematic-isotropic interface, while under strong anchoring conditions the contact angle is a function of the Peach-Koehler force that originates from bulk long range elasticity and acts on the triple line. The use of the complete set of balance equations removes the classical inconsistency in force balances at a contact line by properly taking into account long range (bulk gradient elasticity) and anisotropic (interfacial anchoring elasticity) effects.

Mechanical and Thermodynamical Modeling of Fluid Interfaces. Advances in Mathematics for Applied Sciences Series, Vol 58

11R2. Mechanical and Thermodynamical Modeling of Fluid Interfaces. Advances in Mathematics for Applied Sciences Series, Vol 58. - R Gatignol and R Prud’homme (Lab de Modelisation en Mecanique, Univ Pierre et Marie Curie & CNRS, France). World Sci Publ, Singapore. 2001. 248 pp. ISBN 981-02-4305-7. $58.00.Reviewed by SA Sherif (Dept of Mech Eng, Univ of Florida, 228 MEB, PO Box 116300, Gainesville FL 32611-6300).This book falls into the category of reference books. It deals with an advanced subject area that is beyond the scope of most traditional undergraduate courses in Mechanical or Chemical Engineering. The subject of fluid interfaces is extremely important, but equally complex to understand, and hence the publication of such a book should add good value to the reference shelf of researchers in this field. The book comprises seven chapters. Chapter 1 provides a fundamental analysis of interface and interfacial layers in a thermo-mechanical context. The chapter serves as a nice introduction to readers who would like to get a slow start on the subject and not jump too quickly into the heart of the subject matter. Chapter 2 deals with surface quantities which belong to either a transition layer in which large gradients of densities are present, or a zero-thickness surface. Chapter 3 deals with interfacial balance laws for species, mass, momentum, and energy, as well as interfacial entropy production laws.In Chapter 4, the constitutive relations are derived for two-dimensional interfaces employing linear irreversible thermodynamical relations. Of special interest in this chapter is the coverage of problems related to evaporation and condensation. Chapter 5 introduces the reader to classical three-dimensional constitutive relations. Topics covered include premixed flames with high activation energy as well as shock waves and relaxation zones behind shock waves. Here the gaseous shock wave is treated as an interface. Chapter 6 discusses the application of second gradient theory to interfacial media. Main topics covered in this chapter include the internal capillarity problem in a variety of application areas. Finally, Chapter 7 discusses typical problems involving surface tension and other surface properties. Of particular interest in the chapter is the discussion pertaining to the growth of a bubble, vapor recoil, and diffusive interfaces. The book concludes with three appendices, one dealing with tensor notation, the second dealing with geometry of interfaces and interfacial layers in orthogonal curvilinear coordinates, and the third covering the topic of kinematics of the interface. In general, Mechanical and Thermodynamical Modeling of Fluid Interfaces is well written and should be relatively reader friendly to the scientifically literate individual. The addition of numerical examples on how to apply the different methods could provide additional value to the book and perhaps make the book more useable in the graduate textbook market. Nevertheless, the book serves its stated purpose in its current form in being a solid reference in the area of fluid interfaces.

Interfacial energies for quartz and albite in pelitic schist

Interfacial energies of quartz/quartz (qz/qz), albite/albite (ab/ab), and quartz/albite (qz/ab) boundaries in low-grade pelitic schist were determined based on measured values of dihedral angles. Three kinds of microstructures were investigated, and the interfacial energies were obtained in two independent ways. (1) Relative values of interfacial energy were calculated from dihedral angles formed at quartz and albite triple junctions. (2) Subgrain boundary energy was calculated using the Read–Shockley theory for a boundary connected to an intergranular pore. Dihedral angles formed at the corners of intergranular pores were measured. From the interfacial tension balance equation, the value of the qz/qz grain boundary energy was then obtained. (3) Dihedral angles formed at intersections of either pericline or albite twin boundaries with either ab/ab or qz/ab boundaries were measured. The twin boundary energy was calculated based on a previously derived equation using Landau potential, twin wall thickness, and critical temperature for a phase transition in albite. With a modified interfacial tension balance equation for a twin boundary fixed to a facet orientation, the interfacial energies of ab/ab and qz/ab boundaries were obtained. Energies obtained by methods of (2) and (3) are in good agreement. The interfacial energies for qz/qz, ab/ab, and qz/ab boundaries obtained in this study are 270±110, 300±150, and 250±120 mJ/m2, respectively.

Energy balances at the air/liquid and liquid/solid interfaces with incoming droplets at a moving ice boundary

Interfacial heat balances are investigated for three-phase conditions with impinging droplets on an ice surface. The incoming droplets impart kinetic energy and extract heat from the ice surface due to supercooling. The location where these energy modes are most effectively applied is examined in the modelling of the phase change. Predictions of the ice growth on a heated surface are validated with experimental data.

Expansion of a compressible gas bubble in Stokes flow

The flow-induced deformation of an inviscid bubble occupied by a compressible gas and suspended in an ambient viscous liquid is considered at low Reynolds numbers with particular reference to the pressure developing inside the bubble. Ambient fluid motion alters the bubble pressure with respect to that established in the quiescent state, and requires the bubble to expand or contract according to an assumed equation of state. When changes in the bubble volume are prohibited by a global constraint on the total volume of the flow, the ambient pressure is modified while the bubble pressure remains constant during the deformation. A numerical method is developed for evaluating the pressure inside a two-dimensional bubble in an ambient Stokes flow on the basis of the normal component of the interfacial force balance involving the capillary pressure, the normal viscous stress, and the pressure at the free surface on the side of the liquid; the last is computed by evaluating a strongly singular integral. Dynamical simulations of bubble deformation are performed using the boundary integral method properly implemented to remove the multiplicity of solutions due to the a priori unknown rate of expansion, and three particular problems are discussed in detail: the shrinkage of a bubble at a specified rate, the deformation of a bubble subject to simple shear flow, and the deformation of a bubble subject to a purely elongational flow. In the case of shrinkage, it is found that the surface tension plays a critical role in determining the behaviour of the bubble pressure near the critical time when the bubble disappears. In the case of shear or elongational flow, it is found that the bubble contracts during an initial period of deformation from the circular shape, and then it expands to obtain a stationary shape whose area is higher than that assumed in the quiescent state. Expansion may destabilize the bubble by raising the capillary number above the critical threshold under which stationary shapes can be found.

Some issues related to the modeling of interfacial areas in gas–liquid flows I. The conceptual issues

Two-phase flow modeling has been under constant development for the past forty years. Actually there exists a hierarchy of models which extends from the homogeneous model valid for two-phase flows where the phases are strongly coupled to the two-fluid model valid for two-phase flows where the phases are a priori weakly coupled. However the latter model has been used extensively in computer codes because of its potential in handling many different physical situations. The two-fluid model is based on the balance equations for mass, momentum and energy, averaged in a certain sense and expressed for each phase and for the interface between the phases. The difficulty in using the two-fluid model stems from the closure relations needed to arrive at a complete set of partial differential equations describing the flow. These closure relations should supply the information lost during the averaging of the balance equations and should specify in particular the interactions of mass, momentum and energy between the phases. Another requirement for the interaction terms is that they should satisfy the interfacial balance equations. Some of these terms such as the added mass term or the lift force term do not depend on the interfacial area but some others do, such as the mass transfer term, the drag term or the heat flux term. It is then necessary to model the interfacial area in order to evaluate the corresponding fluxes. Another benefit resulting from the modeling of the interfacial area would be to replace the usual static flow pattern maps which specify the flow configuration by a dynamic follow-up of the flow pattern. All these reasons explain why so much effort has been put during the past twenty years on the modeling and measurement of the interfacial area in two-phase flows. This article contains two parts. The first one deals with the conceptual issues and has the following objectives: 1. to give precise definitions of the interfacial area concentrations; 2. to explain the origin of the interfacial area concentration transport equation suggested by M. Ishii in 1975; 3. to explain some paradoxical behaviors encountered when calculating the interfacial area concentration transport velocity.

Saddle-splay elasticity and interfacial nematostatics.

This Brief Report drives the generalized force balance equations of interfacial statics between nematic liquid crystals (NLC) and isotropic fluids (I), using the classical equations of liquid crystal physics, taking into account an important class of gradient surface elasticity, known as saddle-splay elasticity. The objective is to identify the exact nature of the saddle-splay contributions to the fundamental interfacial force balance equations, known as the Laplace-Young equation and the Marangoni force equation. General expressions for the dynamic generalization of these two equations were given by Shih, Mann, and Brown [Mol. Cryst. Liq. Cryst. 98, 47 (1983)], but the specific form of the static terms appearing in these two equations were missing in the literature, and are now given in this paper. It is found that the tensorial order and functional form of the contributions of saddle-splay elasticity to the two force balance equations are congruent with those arising from the interfacial tension. Therefore, to generalize the interfacial equations of nematostatics by including saddle-splay energy, the interfacial tension must be renormalized with the saddle-splay energy contribution. In addition, saddle splay gives rise to distortion stresses, the two-dimensional analog to the bulk Ericksen stresses, which contribute to the tangential Marangoni force. Exact expressions for pressure jumps across NLC/I interfaces and for the tangential Marangoni force are derived and analyzed. These generalized results are expected to be useful in the characterization of nematocapillarity phenomena, such as wetting, spreading, and the mechanics of thin nematic films.

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.

Heat Transfer Modeling during a Vapor Explosion

A steam explosion is the result of the intense heat transfer that can occur when a cold and volatile fluid is brought into contact with a hot fluid. This heat transfer is linked to the fine fragmentation of the hot fluid, so on the explosion timescale, only part of the cold fluid is involved in this heat transfer. In this paper, two different ways of describing this heat transfer are presented. In the first one, i.e., the microinteraction concept, the amount of coolant involved is controlled by the fragmentation kinetics, while in the second one, it is controlled by phase change resulting from interfacial heat balance.

Bilan d'aire interfaciale dans un mélange diphasique : approche locale vs approche particulaire

The interfacial area balance equation in a two-phase mixture looks quite different depending whether the area per unit volume is defined as: i) the sum of all the areas contained in the unit volume or ii) the sum of the areas of all the particles with their centers in this unit volume. We prove here the complete compatibility between the two balance equations and show their respective merits.

Water‐air flux of dimethylsulfide

The water‐air exchange of dimethylsulfide (DMS) has been measured in a laboratory wind‐wave tank in fresh and seawater. To understand the transport behavior of DMS, its exchange was measured simultaneously with that of O2, SF6, Ne, CH4, and He under varying wind speeds and hydrodynamic conditions. No unpredictable differences between fresh and seawater were found, indicating that DMS surface affinities do not exist in seawater. Results also indicate that Schmidt number corrections can be applied to DMS gas exchange. While the rate of transfer of relatively insoluble or sparingly soluble permanent gases between the ocean and the atmosphere is primarily controlled by the rate of flux through the aqueous boundary layer, interfacial mass balances indicate that increased DMS solubility increases the significance of the airside control of flux. The hypothesis that DMS transport across the water‐air interface is subject to waterside control for moderate environmental conditions is supported. However, for ocean‐atmospheric DMS exchange with low sea surface temperature or moderate wind speeds, there may be a significant influence by the atmospheric mass boundary layer. An atmospheric gradient fraction γa applied to the waterside air‐sea gas transfer velocity will correct for these effects. Estimates of ocean‐atmospheric DMS transfer velocities for these conditions are provided.

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.

Viscoelastic Properties of Lipopolymers at the Air−Water Interface: A Combined Interfacial Stress Rheometer and Film Balance Study

Poly(ethylene glycol) (PEG) is a molecule that exhibits unique behavior when compared with polymers in its homologous family. Depending on its environment, it may show hydrophilic, hydrophobic, or amphiphilic properties. We have studied several PEG lipopolymers, where a PEG chain with a molecular weight (MW) of 2000 g/mol or 5000 g/mol is covalently attached to 1,2-dipalmitoyl- or 1,2-distearoyl-sn-glycero-3-phosphoethanolamine, with a Langmuir film balance and a recently developed interfacial stress rheometer. In particular, we have determined how the rheological properties of PEG molecules anchored at the air−water interface change when the polymer chains are forced into highly stretched brush conformations. Pressure−area isotherms of monolayers of PEG lipopolymers exhibit two phase transitions: a desorption transition of the PEG chains from the air−water interface at 10 mN/m and a high film pressure transition at 20−40 mN/m, but the nature of the latter transition is still poorly understood. We have o...

Spreading Dynamics of Surfactant Solutions

The spreading behavior of microdrops of surfactant solutions at solid surfaces has been studied. The influence on the spreading of different factors, such as the drop lifetime prior to surface contact, surface tension dynamics, surface energy, and surfactant properties, was systematically investigated. The results obtained suggest the existence of two spreading regimes exhibiting different spreading characteristics: In the first, nondiffusive regime, the spreading is very rapid and controlled to different extents by inertia, gravity, and capillarity, depending on the drop size, impact energy, and interfacial tension balance. It is shown in this study that the initial drop surface tension, which is set by the surface tension decay rate and the drop lifetime prior to the surface impact, strongly influences the maximum spreading distance in the nondiffusive spreading regime. The second, diffusion-controlled regime, is characterized by slower concentration-dependent spreading rates. The spreading rate is, here, mainly controlled by the diffusive transport of surfactant to the expanding liquid−vapor interface. In this regime, the drop base radius exhibits an approximate rb2 ∝ t dependence on time. The spreading kinetics at hydrophobic surfaces has been discussed in the framework of a simplified theory. Depending on the assumptions regarding the drop shape, rb5/2 ∝ t to rb2 ∝ t spreading laws are obtained. This agrees reasonably with the experimentally observed relationship.

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.

Shape of an evaporating completely wetting extended meniscus

The microscopic details of fluid flow and heat transfer in the contact line region of an evaporating curved liquid film were experimentally and theoretically evaluated. The evaporating film thickness profiles were measured optically using null ellipsometry and image analyzing interferometry. These thickness profiles were analyzed using the augmented Young-Laplace equation to obtain the pressure field. Using the liquid pressure field, the evaporative mass flux profile was obtained from a Kelvin -Clapeyron model for the local vapor pressure. A correlation for the local slope (apparent contact angle) at a film thickness of 6 = 20 nm as a function of a dimensionless contact line heat sink was thereby obtained for a group of completely wetting fluids. This change in local slope leads to a decrease in the maximum value of the possible capillary suction at the base of the meniscus. A complementary macroscopic interfacial force balance was also used to describe the effects of viscous losses and interfacial forces on the local values of the apparent contact angle and curvature that are functions of the film thickness and heat flux. These two perspectives give a complete description of an evaporating, nonpolar, completely wetting curved film in the contact line region.

Atomic force microscopy: a key to direct wafer bonding technology

Wafer bonding is the dedicated joining of two semiconductor wafer surfaces without the use of adhesives. Specific applications include silicon-oninsulator (SOI), sensors and actuators, and threedimensional integrated microstructures [1-5]. Although a complete theory has not been presented to explain the bonding process, it is generally accepted that the attractive interfacial force between the two surfaces that are separated only by atomic distances is responsible for the initial bonding. These attractive interracial forces are believed to be either electrostatic or chemical in nature. When the two wafers are contacted a relatively weak adhesive force is developed between the surfaces. This initial bond can be strengthened by additional processing such as thermal annealing. The sum of the attractive chemical or ionic interfacial forces balance against the mechanical forces exerted by surface contaminants, roughness, waviness, and wafer bow to determine if an initial bond will form [6]. It is obvious, therefore, that surface-related defects such as particulate contamination, surface roughness, and surface waviness must be minimized and controlled in order to optimize the initial contact for successful bonding to take place. Due to this fact the development of wafer bonding has been highly focused on smooth films such as thermal oxides and bare silicon surfaces. Chemically vapour deposited (CVD) oxides are widely used for the development of integrated circuit (IC) technologies. These CVD oxides may be good candidates for low-temperature wafer bonded surfaces. Structures that do not have surfaces that naturally bond may still be bonded by using deposited films such as oxides as an intermediate layer. P r e v i o u s l y , Si3N 4 and TiN deposited films have been found to form low-temperature bonds [7, 8]. Two other possibilities for low-temperature bondable deposited films are borophosphosilicate glass (BPSG), and low pressure chemically vapour deposited (LPCVD) undoped SiO2. BPSG is used extensively in microelectronics device fabrication as an interlayer dielectric film due to its excellent planarization, gettering and flow properties [9]; LPCVD SiO2 is used in applications where there are stringent step coverage and gap-filling requirements [10]. The successful and reliable bonding of these films would be a very important milestone in the realization of novel structures such as three-dimensional ICs

Erratum to Beam focusing characteristics and alloying element effects on high-intensity electron beam welding

Effects of focusing characteristics of the beam as well as concentrations of a volatile alloying element in the workpiece on the shape of the cavity produced by a high-energy beam are systematically and quantitatively investigated. The energy flux of the focused energy beam is independently specified by the convergence angle, the energy distribution parameter at the focal spot, and the focal spot location relative to the workpiece surface. Energy flux at any cross section of the beam is a Gaussian distribution. The geometry of the cavity is determined by satisfying interfacial energy and momentum balances. By accounting for beam focusing characteristics, the cavity surface temperatures, depths of penetration, and cavity shapes are found to agree with experimental data. The opening diameter and depth of the cavity depend primarily upon the energy distribution parameter at the workpiece surface for a surface-focused weld t increase in the content of the volatile alloying element zinc in aluminum exhibits a pronounced influence on the shape of the cavity.

Beam focusing characteristics and alloying element effects on high-intensity electron beam welding

Effects of focusing characteristics of the beam as well as concentrations of a volatile alloying element in the workpiece on the shape of the cavity produced by a high-energy beam are systematically and quantitatively investigated. The energy flux of the focused energy beam is independently specified by the convergence angle, the energy distribution parameter at the focal spot, and the focal spot location relative to the workpiece surface. Energy flux at any cross section of the beam is a Gaussian distribution. The geometry of the cavity is determined by satisfying interfacial energy and momentum balances. By accounting for beam focusing characteristics, the cavity surface temperatures, depths of penetration, and cavity shapes are found to agree with experimental data. The opening diameter and depth of the cavity depend primarily upon the energy distribution parameter at the workpiece surface for a surface-focused weld t increase in the content of the volatile alloying element zinc in aluminum exhibits a pronounced influence on the shape of the cavity.