INTERFACIAL TRANSPORT PHENOMENA 2""* Edition

Transcription

1 INTERFACIAL TRANSPORT PHENOMENA 2""* Edition

2 INTERFACIAL TRANSPORT PHENOMENA 2"^ Edition John C. Slattery Department of Aerospace Engineering Texas A&M University Leonard Sagis Department of Agrotechnology & Food Science Wageningen University Eun-Suok Oh LG Chem, Research Park South Korea Springe]

3 John C. Slatteiy Department of Aerospace Engineering Texas A &M University College Station, TX Leonard Sagis Department ofagrotechnology and Food Science Wageningen University The Netherlands Eun-Suok Oh LG Chem, Research South Korea Park Interfacial Transport Phenomena, Second Edition Library of Congress Control Number: ISBN e-isbn ISBN Printed on acid-free paper Springer Science+Business Medisi, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights springer.com

4 Contents Kinematics and Conservation of Mass Motion Body Stretch and Rotation [19, p. 17] Motion of Multiphase Bodies What are Phase Interfaces? Three-Dimensional Interfacial Region Dividing surface Dividing Surface as a Model for a Three-Dimensional Interfacial Region Motion of Dividing Surface Stretch and Rotation within Dividing Surfaces More about Surface Velocity Rate of Deformation Moving Common Lines: Qualitative Description Moving Common Lines: Emission of Material Surfaces [16] Moving Common Lines: Velocity is Multivalued on a Rigid SoUd Moving Common Lines: Quantitative Description Mass Conservation of Mass Surface Mass Density Surface Transport Theorem Transport Theorem for Body Containing Dividing Surface Jump Mass Balance Location of Dividing Surface Transport Theorem for Body Containing Intersecting Dividing Surfaces Mass Balance at a Common Line 79

5 Contents Comment on Velocity Distribution in Neighborhood of Moving Common Line on Rigid Sohd More Comments on Velocity Distribution in Neighborhood of Moving Common Line on Rigid Sohd Frame Changes of Frame Frame Indifferent Scalars, Vectors, and Tensors Equivalent Motions Principle of Frame Indifference 105 Foundations for Momentum Transfer Force What are Forces? Momentum and Moment of Momentum Balances Ill Body Forces and Contact Forces Momentum Balance at Dividing Surfaces Surface Stress Tensor Jump Momentum Balance T^*"^ is Symmetric Tangential Tensor Surface Velocity, Surface Stress, and Surface Body Forcel Momentum Balance at Common Line Momentum Balance at Common Line on Relatively Rigid Sohd Factors Influencing Measured Contact Angles Relationships for Measured Contact Angles More Comments Concerning Moving Common Lines and Contact Angles on Rigid Solids and Their Relation to the Disjoining Pressure Correcting Material Behavior for Intermolecular Forces from Adjacent Phases [20] The Correction One Unbounded Dividing Surface: View (iv) One Thin Lens or Fracture: View (iv) One Thin Film: View (v) A Discontinuous Thin Film: View (v) One Unbounded Common Line: View (iv) 157 Applications of the Differential Balances to Momentum Transfer Philosophy Structure of Problem Approximations Only Interfacial Tension Classes of Problems Spinning Drop Interfacial Tensiometer [21] 164

6 Contents vii Meniscal BreakofF Interfacial Tensiometer Pendant Drop Sessile Drop Applications of Our Extension of Continuum Mechanics to the Nanoscale Supercritical Adsorption [22] Static Contact Angle [20] A Review of Coalescence (with J. D. Chen) Coalescence [23-25] Moving Common Line and Receding Contact Angle Nanoscale Fracture [26] 248 Foundations for Simultaneous Momentum, Energy, and Mass Transfer Viewpoint Viewpoint in Considering Multicomponent Materials Body, Motion, and Material Coordinates of Species A Motion of Multicomponent Dividing Surface More about Surface Velocity of Species A Mass Balance Species Mass Balance Concentrations, Velocities, and Mass Fluxes Location of Multicomponent Dividing Surface Further Comments on Viewpoint Further Comments on Viewpoint of Multicomponent Materials Mass Conservation of Mass Force Momentum and Moment of Momentum Balances Jump Momentum Balance T^'^) is Symmetric, Tangential Tensor Energy Rate of Energy Transmission Energy Balance Radiant and Contact Energy Transmission Jump Energy Balance Entropy Entropy Inequality Radiant and Contact Entropy Transmission Jump Entropy Inequality Behavior as Restricted by Entropy Inequality Behavior of Multicomponent Materials Bulk Behavior: Implications of Entropy Inequality 304

7 Contents Surface Behavior: Implications of Jump Entropy Inequality Surface Behavior: Adsorption Isotherms and Equations of State Alternative Forms for the Energy Balances and the Entropy Inequalities Behavior as Restricted by Frame Indifference Other Principles to be Considered Alternative Independent Variables in Constitutive Equations Bulk Behavior: Constitutive Equations for Stress Tensor, Energy Flux Vector and Mass Flux Vector Surface Behavior: Constitutive Equations for Surface Stress Tensor Boussinesq Surface Fluid Simple Surface Material Surface Isotropy Group Isotropic Simple Surface Materials Simple Surface Solid Simple Surface Fluid Fading Memory and Special Cases of Simple Surface Fluid Simple Surface Fluid Crystals Surface Behavior: Constitutive Equations for Surface Energy Flux Vector Surface Behavior: Constitutive Equations for Surface Mass Flux Vector Intrinsically Stable Equihbrium [27] Stable Equilibrium Constraints on Isolated Systems Implications of ( ) for Intrinsically Stable Equilibrium Implications of ( ) for Intrinsically Stable Equilibrium Thermodynamics of Single-Component, Elastic, Crystalline Surface Solids [28] Thermodynamics of Surface Crystals Constraints on Isolated Systems Implications of Equilibrium Stress-Deformation Behavior of Single-Walled Carbon Nanotubes 423 Applications of the Differential Balances to Momentum, Energy and Mass Transfer Philosophy 429

8 Contents ix Structure of Problems Involving Momentum Transfer Structure of Problems Involving Energy Transfer Structure of Problems Involving Mass Transfer Problems Involving Momentum Transfer Boussinesq Surface Fluid in a Knife-edge Surface Viscometer Generalized Boussinesq Surface Fluid in a Deep Channel Surface Viscometer Simple Surface Fluid in Curvilineal Surface Flows [29] Simple Surface Fluid in a Deep Channel Surface Viscometer [29] Simple Surface Fluid in an Oscillating Deep Channel Surface Viscometer [29] Limiting Cases when Effects of Interfacial Viscosities Dominate Displacement in a Capillary [30] Several Interfacial Viscometers Suitable for Measuring Generalized Boussinesq Surface Fluid Behavior [31] Stochastic Interfacial Disturbances Created by Thermal Noise and the Importance of the Interfacial Viscosities [32] Capillary Rise [30, 33] Common Line Motion in Systems with Simple Surface Fluid Material Behavior: Implications of the Entropy Inequality [34, 35] More on Common Line Motion in Systems with Simple Surface Fluid Material Behavior: Implications in Polymer Extrusion [36] Limiting Cases of Energy Transfer Motion of a Drop or Bubble [37; with D. Li] Limiting Cases of Mass Transfer Motion of a Drop or Bubble [38; with D. Li] Longitudinal and Transverse Waves [32] 587 A Differential Geometry 611 A.l Physical Space 611 A.1.1 Euclidean Space 611 A.1.2 Notation in (E^, V^) 613 A.1.3 Surface in {E^,V^) 617 A.2 Vector Fields 617 A.2.1 Natural Basis 617 A.2.2 Surface Gradient of Scalar Field 624 A.2.3 Dual Basis 625 A.2.4 Covariant and Contravariant Components 625 A.2.5 Physical Components 626

9 X Contents A.2.6 Tangential and Normal Components 627 A.3 Second-Order Tensor Fields 629 A.3.1 Tangential Transformations and Surface Tensors 629 A.3.2 Projection Tensor 631 A.3.3 Tangential Cross Tensor 633 A.3.4 Transpose 636 A.3.5 Inverse 637 A.3.6 Orthogonal Tangential Transformation 639 A.3.7 Surface Determinant of Tangential Transformation 641 A.3.8 Polar Decomposition 643 A.4 Third-Order Tensor Fields 646 A.4.1 Surface Tensors 646 A.5 Surface Gradient 647 A.5.1 Spatial Vector Field 647 A.5.2 Vector Field is Explicit Function of Position in Space A.5.3 Vector Field is Explicit Function of Position on Surface 649 A.5.4 Second-Order Tensor Field 660 A.5.5 Tensor Field is Explicit Function of Position in Space A.5.6 Tensor Field is Explicit Function of Position on Surface 662 A.6 Integration 666 A.6.1 Line Integration 666 A.6.2 Surface Integration 668 A.6.3 Surface Divergence Theorem 669 B Sumrriciry of Useful Equations 673 B.l Useful Equations for Single Component Systems 673 B.1.1 Bulk Phases 673 B.l.2 Dividing Surfaces 675 B.l.3 Common Lines 693 B.2 Useful Equations for Multicomponent Systems with Simultaneous Momentum, Energy, and Mass Transfer 694 B.2.1 Concentrations, Velocities, and Fluxes 694 B.2.2 Jump Mass, Jump Energy, and Jump Entropy Balance. 700 B.2.3 Specific Forms 704 C Applications of integral averaging to momentum, energy, and mass transfer 735 C.l Integral balances 735 C.1.1 Integral overall mass balance 736 C.1.2 The Integral Mass Balance for Species A 738 C.1.3 Integral momentum balance 739 C.1.4 Integral mechanical energy balance 742 C.1.5 The Integral Energy Balance 749 C.1.6 The Integral Entropy Inequality 753

10 Contents xi Notation 757 References 773 Author Index 809 Index 821

11 Preface to Second Edition Our objective in preparing this second edition was not only to update the material but also to focus it more closely on the fundamentals. In addition to carefully re-editing everything and placing more emphasis on the common line or three-phase line of contact, the primary changes are these. In chapter 2, we have added section 2.2, which describes how material behavior within the immediate neighborhood of interfaces can be corrected for the effects of long-range intermolecular forces from adjacent phases. We have also strengthened the discussion of interfacial behavior by moving it to chapter 4, following the discussion of entropy. Since the interfacial viscosities are zero for single-component systems or in the absence of surfactants, we have moved the discussion of these problems from chapter 3 to chapter 5. We have added a discussion of problems that support the theory for the extension of continuum mechanics to the nanoscale developed in section 2.2. In view of our concerns about focus, the discussion of the integral balances in the original chapters 4 and 7 have been scaled back in Appendix C. The original chapter 5 has now become chapter 4. With our emphasis on focus, we have both trimmed the discussion of stable equilibrium and expanded it to include a two-dimensional surface solid, such as a carbon nanotube. The new chapter 5 now consolidates our discussion of problems involving surface stress-deformation behavior. We hope that you find the result as useful as we found it interesting to write. John C. Slattery, Leonard M. C. Sagis, and Eun-Suok Oh March 7, 2006

12 Preface to the First Edition Transport phenomena is used here to describe momentuin, energy, mass, and entropy transfer [1, 2]. It includes thermodynamics, a special case of which is thermosiaiics. Interfacial transport phenomena refers to momentum, energy, mass, and entropy transfer within the immediate neighborhood of a phase interface, including the thermodynamics of the interface. In terms of qualitative physical observations, this is a very old field. Pliny the Elder (Gains Plinius Secundus, A.D.; Pliny [3]) described divers who released small quantities of oil from their mouths, in order to damp capillary ripples on the ocean surface and in this way provide more uniform lighting for their work. Similar stories were retold by Benjamin Franklin, who conducted experiments of his own in England [4]. In terms of analysis, this is a generally young field. Surface thermostatics developed relatively early, starting with Gibbs [5] and continuing with important contributions by many others (see Chap. 4). Derjaguin and Landau [6] and Verwey and Overbeek [7] indicated how London-van der Waals and electrostatic double-layer forces were to be incorporated in continuum mechanics, now often referred to as DLVO theory. But prior to 1960, there were relatively few notable papers concerned with the analysis of dynamic systems. Two stand out in my mind. Boussinesq [8] recognized the surface stress tensor and proposed the constitutive equation that we now refer to as the Boussinesq surface fluid model (Sect ). Unfortunately, he did not carry out an experiment in which the effects of the interfacial viscosities could be clearly recognized. While many studies of the surface viscosities followed, the corresponding data analyses were not convincing. Brown et al. [9] appear to have been the first to demonstrate how the interfacial shear viscosity could be measured in a limit where the viscous effects in the adjacent phases could be neglected with respect to those in the interface. More recently, interest in analysis has begun to flourish within this area. Since many people have made important contributions, the best that I can do briefly is to indicate a few papers that have had particular meaning for me. Scriven [10] restated the Boussinesq surface fluid model in a form more

13 xvi Contents convenient for analysis. Burton and Mannheimer [11-15] analyzed and demonstrated the deep channel surface viscometer, which is still the recommended technique for measuring relatively small surface shear viscosities. Dussan V and Davis [16], through both analysis and experiment, pointed out with unusual clarity the contradictions to be reconciled in describing a moving common line (Sects through and 1.3.9). By analyzing a thin film, Israelachvili [17] derived an expression for interfacial tension that is in excellent agreement with experimental measurements, demonstrating that continuum mechanics can be usefully extended to regions having molecular dimensions. With the appearance of these papers, there were also questions. Were the surface viscosities real physical parameters or were they artifacts of the manner in which the surface viscometer was analyzed? Was the measured value of the surface shear viscosity consequently dependent upon the viscometer used to measure it? Was the introduction of the surface stress tensor consistent with some general view of continuum mechanics? Could the effects of the surface viscosities be observed in any situations judged to be of practical importance? Was there really slip in the neighborhood of a moving common hue? Was it possible to successfully apply continuum mechanics to the very thin films within the neighborhood of a common line? In trying to answer questions like these for my students, I decided to prepare this book. This book is written both as a guide for those preparing for active research in transport phenomena and as a reference for those currently working in the area. The emphasis is upon achieving understanding starting from the fundamental postulates. The dominant theme is the translation of physical problems into mathematical terms. I normally introduce my students to this book after they have completed the first semester of lectures from my first book [18]. The text is self-contained, but I would prefer to see the reader already conversant with analogous discussions for single phases. Although I have lectured from this text here at Texas A & M, it is written with the intention of being sufficiently complete to be used for self-study. This is the manner in which most of my students have employed the text as it was being written. All of the exercises have answers. Where appropriate, the reader is led through an exercise, since the objective is not to test his comprehension of the preceding text. The exercises are used as a literary device to transmit information relevant to the text without overwhelming the reader with additional details. In many respects this book was a group effort. Many colleagues have influenced and directed my thinking through conversations, by listening to their talks at meetings, and by reading their papers. While I have not been able to provide complete answers to all of their questions, I have been able to finish this book only through the continued probing, encouragement, and active help of my students. Jing-Den Chen and M. Sami Selim offered comments on portions of the final manuscript. My wife Bea and Brenda Wilson cheerfully typed and retyped through many revisions over many years, never questioning whether the book would finally be completed. The final manuscript

14 Contents xvii was prepared by Cheri Sandlin, with assistance from Ruth Heeremans and Izora Brown. Alfred Li provided invaluable help and support through the long months of proof reading, correcting the final manuscript, and preparing indices. The Peregrine Falcon Company made available a test copy of THE EGG BOOKMAKER INTERFACE (The Peregrine Falcon Co., P. O. Box 8155, Newport Beach, CA ), in which the camerarready copy was typed. David Adelson further modified this test copy, permitting me to use boldface greek, boldface script, boldface brackets (for jumps at interfaces), and boldface parentheses (for jumps at common lines). Joel Meyer and Peter Weiss prepared the final forms of the figures. Stephen H. Davis shared with me the original photographs from his work with Elizabeth B. Dussan in Sect Richard Williams and the David Sarnoff Research Center provided both the previously published and the previously unpublished photographs from his work that also appear in Sect My friends and colleagues at Northwestern University, where most of this book was written between 1972 and 1989, gave me their patience and encouragement. Thanks to you all. John C. Slattery College Station, Texas July 10, 1990