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ix xiii xv
Chapter 1 Introduction
1
1.1 Importance and distinction of sediments
1
1.2 Outline of the development of cohesive sediment behavior 1.3 Objectives of the book and outline of its contents
Chapter 2 The Mineralogy and the Physicochemical Properties of Cohesive Sediments
3 6
11
2.1 General properties of cohesive sediment suspensions and of cohesive sediment deposits 2.2 The bonding mechanisms
11 14
2.2.1 Interatomic or Primary Bonds
14
2.2.2 Secondary Bonds
22
2.3 The nature and mineralogy of clay particles
24
2.3.1 Introductory Remarks
24
2.3.2 The Basic Clay Minerals
25
2.4 Origin and occurrence of clay minerals and formation of clay deposits
Chapter 3 Forces between Clay Particles and the Process of Flocculation
43
47
3.1 Introductory remarks
47
3.2 The electric charge and the double layer
47
3.2.1 Isomorphous Substitution
48
3.2.2 Preferential Adsorption
48
3.3 The theoretical formulation of the double layer
49
3.3.1 The General Case
49
3.3.2 Surfaces of Constant Potential
55
3.3.3 Surfaces of Constant Charge Density
55
3.3.4 Illustrative Applications
56
Cohesive Sediments in Open Channels Copyright© 2009
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GJ
Contents
3.4 Interaction of two flat double layers
59
3.4.1 Force and Energy Interaction
59
3.4.2 Illustrative Examples
62
3.4.3 Potential Energy of Interaction between Two Flat Double Layers 3.4.4 Illustrative Examples
64 65
3.4.5 Potential Energy of Interaction of Two Flat Double Layers Due to van der Waals Forces
68
3.4.6 Total Potential Energy for Two Particles and the Process of Flocculation
70
3.5 Some important properties of fine particles and aggregates
75
3.5.1 The Counterion Exchange
75
3.5.2 Limitations of the Gouy-Chapman Theory and the Stern Layer
75
3.5.3 The Water Phase
78
3.5.4 Sensitivity and Thixotropy
80
3.6 Internal structure and fabric of floes, aggregates, and cohesive sediment deposits
81
3.6.1 Particle Arrangements within Floes
81
3.6.2 The Microstructure of Deposited Cohesive Sediment Beds
87
Chapter 4 The Hydrodynamic Transport Processes of Cohesive Sediments and Governing Equations
89
4.1 The fundamental transport equations for cohesive sediments
89
4.1.1 The Development of the General Transport Equations 4.1.2 Discussion of the Developed Equations
89 94
4.2 The process and dynamics of flocculation
95
4.2.1 Collisions Due to Brownian Motion
95
4.2.2 Collisions Due to Velocity Gradients
100
4.2.3 Collisions Due to Differential Settling
103
4.2.4 Concluding Remarks 4.3 Review of fundamental properties of turbulent flows
106 106
4.3.1 Significant Stresses and Parameters
106
4.3.2 Collision Rates in Turbulent Flows
116
Contents
4.4 The properties of the aggregates and the aggregate growth equation
118
4.4.1 The Properties of Aggregates and Their Relation to the Controlling Flow Variables
118
4.4.2 Quasi Steady-State Aggregate Distribution and Maximum Aggregate Size
128
4.4.3 Some Additional Research Work on Flocculation and Aggregate Properties 4.4.4 Discussion and Concluding Remarks
Chapter 5 Rheological Properties of Cohesive Sediment Suspensions 5.1 Importance of the subject
140 152
155 155
5.2 Basic properties of sediment suspensions and methods of evaluations 5.3 Concluding remarks
Chapter 6 Erosion of Cohesive Soils
156 169
173
6.1 Introductory remarks
173
6.2 Erosion of consolidated cohesive soils
174
6.2.1 Early Empirical Information
174
6.2.2 More Recent Field and Laboratory Studies
178
6.3 Erosion of soft cohesive sediment deposits
183
6.4 Summary and concluding remarks
200
Chapter 7 Deposition and Resuspension of Cohesive Soils 7.1 Deposition of cohesive sediments 7.1.1 Early Experiments and Preliminary Conclusions
203 203 203
7.1.2 Detailed Studies on Deposition. Part A: The Degree of Deposition
208
7.1.3 Detailed Studies on Deposition. Part B: The Rates of Deposition
224
7.1.4 Variation of Depositional Parameters as the Sediment Sorts during Deposition in Open Conduits
234
7.2 Hydrodynamic interaction of suspended aggregates with the deposited bed
244
Contents
7.3 Resuspension of deposited cohesive sediments
252
7.3.1 Introductory Remarks
252
7.3.2 Fundamental Considerations
253
7.3.3 Experimental Results 7.4 Summary and closing comments
Chapter 8 Engineering Applications of Cohesive Sediment Dynamics
254 271
275
8.1 Areas of application
275
8.2 Design of stable channels
276
8.2.1 Design for Safety Against Scouring 8.2.2 Design for Safety Against Deposition 8.3 Shoaling in estuaries 8.3.1 Fine Sediment Transport Processes in Estuaries 8.4 Illustrative case histories
276 278 278 279 293
8.4.1 The Savannah Estuary
293
8.4.2 The Delaware River Estuary
301
8.4.3 The River Thames Estuary
303
8.4.4 The Maracaibo Estuary
306
8.4.5 The San Francisco Bay Estuary
315
8.4.6 Closing Remarks
317
8.4.7 Applications to Estuarine Modeling
318
8.5 Control of environmental pollution
319
References
323
List of Symbois
333
Greek Symbols
341
About the Author
345
Author Index
347
Subject Index
351
Preface
Cohesive sediments, which consist predominantly of silt and clay with size ranging from a few micrometers to a fraction of a micrometer, enter frequently in several areas within the civil engineering domain as, for example, in soil mechanics and foundations. In the field of water resources, the beds and banks of natural and artificial channels often consist of cohesive sediments subject to erosion. Such channels may also carry such sediments in suspension, and that sediment may eventually deposit in some areas of the canal or in reservoirs. In sanitary and environmental engineering, water purification and sewage treatment involve handling of cohesive sediments. In all the preceding cases and in some others, the physicochemical and colloidal properties of cohesive sediments are of primary importance in their response to external loads, as in the case of soil mechanics and foundations. The same is true in the control of erosion, deposition and transport in open channels, in the maintenance of reservoirs, in water and sewage treatment, and in the control of the environmental quality of natural water systems, such as lakes, rivers, and estuaries. The primary difference between coarse and cohesive sediments lies in the capacity of the latter to form, under the influence of interparticle attractive forces, agglomerations with size, density, and strength much different than those of the original particles. Moreover, these properties are not even constant, but they are functions of the acting forces during their formation and can even vary with time and quality of pore or ambient water. Therefore, any simplified description and modeling of such sediments based only on some gross quantities without taking into consideration the effect of their intricate physicochemical, colloidal, and mineralogical properties may lead to erroneous results. The need to incorporate these properties into engineering design first became obvious in foundations and various soil mechanics problems. This need motivated extensive research, both fundamental and applied, since the early part of the 20th century, which led to significant interdisciplinary advances for a better and more reliable design of foundations and earth structures and to an estimate of their bearing capacity and settlement under external loading. There is indeed a large volume of publications on this subject. Application of the same properties in hydraulic design and/or solution of sedimentation problems started much later, predominantly motivated by the design of open channels safe against scouring and deposition and for the control of
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Preface
shoaling in estuaries. Therefore, it is a relatively new field. Related fundamental research originated about 50 to 60 years ago. A large volume of knowledge has already been obtained while active research is still going on in a number of institutions in the world. The current existing knowledge, however, is sufficient for the formulation of a rigorous hydrodynamic framework for the overall hydraulic behavior of these kinds of sediments. This framework can be and should be used as the basic guideline for a rational approach to hydraulic problems involving cohesive sediments, for the planning of the necessary laboratory experiments and field measurements, and also for further research on these subjects. The objective of this book is to present in a coherent form the entire spectrum of the behavior of cohesive sediments in a flow field, and more specifically in open channels, starting from the process of flocculation and proceeding to the processes of erosion, deposition, resuspension, and transport, always in relation to their physicochemical properties. The main subject is preceded by a brief treatment of the fundamentals of clay mineralogy and clay colloid chemistry for the sake of readers with inadequate background in these fields and as a starting point for those wishing to expend their knowledge with further studies. The subject matter was selected and arranged in a way to contribute to three objectives: first, as an introduction to undergraduate and graduate students of hydraulic, coastal, and environmental engineering; second, as a guideline to practicing engineers; and third, as a starting point and/or an aid for further research. It is hoped that the book will meet all these three goals. Several people from various engineering and scientific areas have so far contributed to the present state of knowledge of cohesive sediment behavior. The subject matter of the book is based on a selection from work related in some way to its primary objective, which is the presentation of a rigorous framework with direct applications. The first major contribution started by Professor R. B. Krone at the University of California in Berkeley in the 1950s with his work on the effect of flow-induced shear stresses on the density and strength of flocs and floc aggregates and his laboratory and field studies on estuarial sediment transport processes. The early work of the author followed in 1960, also at the University of California in Berkeley, focused primarily on the processes of erosion of dense and deposited cohesive estuarine sediments. Work on the deposition phase continued at MIT by Professor J. F. Kennedy, the author and their graduate research assistant from 1963 to 1966. Their fundamental work was significantly enhanced by simultaneous field research on estuarine shoaling in the Bay of Maracaibo in Venezuela by the same people. A special research apparatus was developed and used in that research phase. The latter was furthermore developed and improved at the University of Florida in Gainesville, in 1968 and 1969, and was used from 1968 to 1983 for studies of deposition and resuspension by Mehta, the author, and a number of graduate assistants. Many researchers, mostly sanitary and coastal specialists, made important contributions to the hydrodynamics of floc formation and on the aggregate properties. The work of
Preface
xi
Watanabe, Hozumi, Tambo, and Kusuda and his colleagues in Japan is particularly noteworthy from the practical aspect, and their results are incorporated in this book as most directly related to its main theme and objectives. Additional important work of several other researchers is also mentioned and commented even with only indirect relation to the subject matter of the book.
Acknowledgments
The work by the author and his colleagues on the hydraulic behavior of cohesive sediments has been supported by the following agencies: Ford Foundation supported the author in his doctoral studies and his research at the University of California through a special predoctoral scholarship. The latter part of his research was also partly supported by the Corps of Engineers of the U.S. Army in 1961–1962. Ford Foundation also supported part of his research at the Massachusetts Institute of Technology (MIT) from 1963 to 1965 under a postdoctoral fellowship. The same fundamental research and field investigations in the Gulf of Maracaibo in Venezuela were supported by the U.S. Agency for International Development (AID) from 1963 to 1966. The work at the University of Florida in Gainesville was first supported by the Environmental Protection Agency (EPA) from 1968 to 1970 and from then on by the National Science Foundation until about 1980. Substantial simultaneous support was also provided during that period by the Waterways Experiment Station of the Corps of Engineers in Vicksburg, Mississippi. Finally, the College of Engineering of the University of Florida provided funds for the building of a special room for the housing of the experimental apparatus used for all the fundamental research experiments on the deposition and resuspension of cohesive sediments. All this support, thanks to which the field of cohesive sediment hydrodynamics was brought to its present stage of development, is gratefully acknowledged.
Cohesive Sediments in Open Channels Copyright © 2009
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Dedications
The author wishes to respectfully dedicate this book to the memory of the following outstanding professors with whom he had the privilege to be associated and who had a profound influence in his overall academic work. First, to the memory of Professor H. A. Einstein, founder and one of the most important contributors to Sediment Transport Mechanics, and the PhD thesis supervisor of the author at the University of California at Berkeley. Second, to the memory of Professor Arthur T. Ippen, director of the Hydraulics Laboratory at MIT during the author’s work there for his encouragement and support of his research and for introducing him to the Estuarine Hydrodynamics and shoaling in estuaries. Third but not least, to the memory of Knox Millsaps, Chairman of the Department of Engineering Science at the University of Florida from 1974 to 1986 and in which the author has been a faculty member from 1974 to now, for inspirational leadership and his commitment to academic principles and values.
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Chapter 1
Introduction
1.1 Importance and distinction of sediments The importance of sediments in hydraulic engineering and, in general, in the technical development of water resources is well known. In rivers the total amount of sediment discharge is the most obvious and direct concern. Sediments also affect the roughness and the frictional resistance of natural waterways, thus raising the question of stage-discharge-sediment transport relationships. The stability of beds and banks against scouring and deposition is another important subject, particularly for manmade canals and waterways. The useful life of reservoirs depends on the sediment load of the contributing natural streams. The extent and frequency of maintenance of navigable waterways in estuaries are determined by the rates of deposition of sediments, particularly fine, by the discharging river or rivers into that specific estuary. Another serious problem in estuaries, bays, and lakes is sediment-induced pollution either by increasing the water turbidity or by depositing highly contaminated sediment on ecologically sensitive zones. There are indeed cases where the environmental damage is so severe that the restoration of the original quality becomes either impossible or extremely difficult and expensive. For all these reasons, sedimentation has been the subject of intensive fundamental, applied, and field research since the 19th century and many theories, empirical formulas, and semitheoretical equations have been developed for the prediction of sediment transport rates and the control of channel stability. Sediments have been distinguished into two broad classes: coarse or cohesionless and fine or cohesive. The division has been arbitrarily placed on the grain size distribution. The first class refers to sediments ranging from fine sand to coarse gravel, whereas the second contains silt and clay. It has been observed that in river flows coarse sediments in transport are represented in appreciable quantities in the bed and that their rates of transport are functions of the flow conditions. In contrast, fine sediments are encountered in the bed in only very small quantities in proportion to their total load, and their transport rates appear to be unrelated to the flow parameters and to depend only on their supply rates [25, 26]. The boundary between the Cohesive Sediments in Open Channels Copyright © 2009
Cohesive Sediments in Open Channels
two sediment classes has been established at 50 m. It should be noticed at this point that even the finest sediments eventually deposit under sufficiently low bed shear stresses and, therefore, at some stage their transport and deposition rates have to depend on the flow parameters. The major difference between coarse and fine sediments lies not so much in their size but primarily in the mutual interaction of grains in a water environment. Coarse grains in suspension behave independently from each other, with the exception of mechanical interaction in highly dense suspensions, while as a bed material only forces of interlocking and friction enter into the picture. The settling unit in the sediment transported in suspension and the eroded unit from the coarse bed is the individual sediment unit; therefore, the sediment can be introduced in the relationships describing either the bed stability or sediment transport rates by a representative grain size distribution or by an equivalent distribution of settling velocities. Cohesive sediment grains, which range in size from 50 m to a small fraction of 1 m, are subjected to a set of attractive and repulsive forces of an electrochemical and atomic nature acting on their surfaces and within their mass. These forces are the result of the mineralogical properties of the sediment and of the adsorption of ions on the particle surfaces. The fine sediment grains have in general a flat plate or a needle shape and a high specific area, which is a high surface to volume ratio, so that the total magnitude of the surface forces becomes dominant in comparison to the submerged weight of the particle. Dispersed particles have such a low settling velocity that the finer portion of them can stay in suspension almost indefinitely, whereas even a slight degree of agitation is sufficient to keep the coarser part of them in suspension. When, under certain conditions, the attractive forces exceed the repulsive ones, colliding particles stick together, forming agglomerations known as flocs with size and settling velocities much higher than those of the individual particles. Rapid deposition may then take place. This phenomenon is known as flocculation. In a flocculated cohesive sediment suspension, the settling unit is the floc rather than the individual particle. The same physicochemical forces are responsible for certain properties of consolidated cohesive sediment deposits, such as cohesive strength and plasticity. Flocs join together to form floc aggregates of various orders of magnitude. In a quiescent water environment, the Brownian motion of the water particles provides the only mechanism for interparticle and interaggregate collision. In flowing waters the shear rates and the turbulent velocity fluctuations affect the collision frequency to a much greater extent than the Brownian motion, so that a much higher rate of aggregate formation is expected. At the same time, however, the same forces induce disrupting stresses within the aggregates, thus limiting their maximum size and controlling their basic properties. A quasi steady-state aggregate size distribution is reached, which is a function of the flow parameters themselves. Settling units develop similar bonds with the cohesive bed that have to be broken for the units to be resuspended. These surface forces constitute the enormous difference between coarse and fine cohesive sediments. The flow conditions, which control the degree and rates
Chapter | 1 Introduction
of deposition, also determine the size distribution and the important properties of the aggregates. The deposited cohesive bed is composed of flocs and/or higher order aggregates whose properties have been molded by the flow-induced stresses. Therefore, their erosional resistance and the rates of resuspension and erosion as well as their gross mechanical properties are expected to also be functions of the flow conditions. The same physicochemical forces may attract other suspended matter in the water, such as organic and inorganic toxic substances, so that a highly contaminated bed is formed after deposition. Upon resuspension the polluted sediment may contaminate the entire water environ with detrimental consequences to the aquatic life. To make the situation more complicated, the surface interparticle forces are not even constant, but they may change drastically with small changes in the water quality, temperature, and time. At first glance this interdependence of flow, water quality, aggregate properties, deposition, and erosion gives the impression that a quantitative description of the hydraulic behavior of cohesive sediments constitutes an insurmountable problem. Fortunately, extensive fundamental and applied research, particularly since 1950, has led to a much better understanding of the dynamics of cohesive sediment behavior in a turbulent flow field. Quantitative equations have been developed for the initiation, degree and rates of deposition, erosion, and resuspension in terms of readily determinable flow variables and parameters representing the overall effect of the interparticle physicochemical forces. These relationships and an understanding of the dynamics of floc formation and of the processes of erosion, deposition, resuspension, and transport of cohesive sediments supplemented with some simple laboratory tests and field measurements may lead through mathematical and physical models to reasonable answers to problems involving cohesive sediments. The terms “fines” and “cohesive sediments” have been used in literature meaning essentially the same thing. Both terms will also be used alternatively in this book.
1.2 Outline of the development of cohesive sediment behavior Because of the outlined importance of sedimentation, the hydraulic behavior of sediments has been a subject of concern and investigation since the inception of the field of hydraulic engineering. In fact studies on sedimentation followed closely the developments in that field and in fluid mechanics in general. A good summary of the historic development of sediment transport mechanics was given Graf [37]. Like many engineering disciplines, hydraulics in general and sedimentation in particular started from pure empiricism and gradually proceeded together with advances in fluid mechanics to more fundamental and universal relationships. This is particularly true for cohesive sediments, whose behavior is much more complicated than that of cohesionless soils. The development of cohesive sediment hydraulics is summarized and discussed in Chapter 6, which deals with the mechanics of erosion of these sediments.
Cohesive Sediments in Open Channels
An earlier analysis and discussion was presented by Paaswell and the author [92, 93, 116, 117]. Only a brief outline of representative examples of the evolution of the hydraulics of cohesive sediments will be given here. The first and earliest phase of the subject consisted in establishing guidelines for the design of stable canals through empirical formulas and/or tables for limiting velocities as the only criterion. The soil properties were described by a mere classification or, at most, by some measure of their density. For instance, in the table of critical velocities recommended by Schoklitsch in 1914, the soil was described only by its type and “the degree of compaction” [125, Vol. I, p. 232]. Similar critical velocities were given by Etcheverry in 1916, also based on a very general soil classification [30]. The average velocity continued to be used as the stability criterion into the early part of the 20th century in spite of the fact that, as early as in 1816, Du Buat introduced for the first time as a criterion of sediment transport the concept of “shear resistance,” that is, essentially the force per unit area of the bed of the stream [19]. This was, in fact, the greatest contribution of Du Buat to the field of sediment transport. A similar concept of tractive force or bed shear stress was introduced in 1879 by Du Boys [18]. Still, however, in 1926 the Special Committee on Irrigation Hydraulics presented estimates of “experienced irrigation engineers” for critical design velocities reported by Fortier and Scobey again on the basis of soil classification [33]. It was only in 1955 that Lane reported data by Russian engineers giving both critical velocities and critical shear stresses for channels with cohesive boundaries and of various densities [70]. Another school of approach within that first phase is known as the “regime theories.” These theories aimed at the development of empirical formulas for velocities for the design of channels in specific areas on the basis of extensive and numerous field data on canals which exhibited various degrees of stability. No laws of mechanics were introduced in the derivation nor were any specific soil data reported. Kennedy’s work on irrigation canals in India is representative of this school of thought. It was presented in 1895, and the channel depth was the only parameter representative of the channel geometry [55]). In 1959 Leliavsky reported later investigations introducing, in addition to the depth of flow, some other variables describing the channel geometry and the boundary resistance [73]. Kennedy’s formulas for stable canal design do not specify whether they imply safety against scouring or shoaling. An analysis in Chapter 6, though, indicated that these formulas and rules really apply for safety against deposition or siltation. Like almost all empirical laws and formulas, these early representative results and criteria may be valid for soil types and soil properties and for general environmental conditions very similar to those they were based on. Otherwise, the results of their application may be erroneous. This is particularly true in the case of cohesive sediments in which even apparently minor changes in one or more aspects may drastically affect their erosional and depositional characteristics. The realization that the flow-induced shear stresses on the bed rather than the average flow velocity is the controlling flow variable for channel stability and the recognition of the related importance of soil properties in addition to its
Chapter | 1 Introduction
composition and density led to the second phase of cohesive sediment research in both the field and in the laboratory. In all studies the objective was to relate the critical bed shear stresses, also referred to as critical tractive forces, to some soil mechanics parameters representative of the soil structure and the gross shear strength. Field investigations conducted by the U.S. Bureau of Reclamation presented in 1953 revealed little correlation between critical tractive force for erosion and mean grain size. The same studies indicated a strong effect of some external factors, such as desiccation, on the critical tractive force [148]. At about the same time, the field studies by Sundborg on the Klarälven river suggested that the critical velocity decreases with decreasing particle size down to the silt range of 50 m, but it increases as the sediment becomes finer [130]. This is the limit below which the interparticle physicochemical forces start causing flocculation. These field investigations may supply valuable data on erosive and depositional trends, but seldom lead to a basic understanding of the particular process, much less to formulations of equations of general validity. The laboratory research of this second phase aimed at the derivation of experimental relationships linking a critical tractive force or boundary shear stress to some readily determinable soil parameters representative of the macroscopic shear strength and other mechanical properties of the soils. The mechanism of erosion and the details of bed structure were not considered. The work by Dunn in 1959 [20], of Smerdon and Beasley in the same year [128], of Moore and Mash in 1962 [90], of Espey in 1963 [29], of Flaxman in 1965 [32], of Berghager and Ladd in 1964 [5], and of Grissinger in 1966 [40] are representative examples of this kind of research effort. In 1966 the Task Committee on Cohesive Sediments of the ASCE published an annotated bibliography on the subject containing the results and conclusions of several other laboratory studies [136]. Most of these studies utilized small, improvised experimental setups, such as cylinders within which an interior cylindrical sample of the soil was subjected to shear by rotation, or jets impinging on a soil sample or relatively small samples of cohesive sediments placed over a section of an open flume. Some other researchers, like Smerdon and Beasley [128] and Abdel-Rahman in 1962 [1] did use a cohesive bed over the entire length of an open flume. The latter work, though, overlaps the third phase. The experimental research of the second phase constitutes an important and indispensable part of the overall research effort in cohesive sediment dynamics. Its direct contribution, however, has been limited by two facts. First, because of their shape and small size, some of the experimental devices do not generate a flow field similar to that in open channels. As a result, the boundary stresses on the sample and the flow structure near the wall may deviate substantially from that in real conduits. In addition, the difference in the shape of the equipment with the associated disparity of boundary conditions makes any comparison between the results of various investigators even more difficult. Nevertheless, these early laboratory experiments on cohesive sediments led to some important conclusions. It was
Cohesive Sediments in Open Channels
made clear that neither the shear strength nor the Atterberg limits could be used as a unique parameter for cohesive soil erodibility. Indeed, samples with comparable strength determined by any standard test used in soil mechanics, and/or comparable Atterberg limits, were found to display erosive resistance differing by orders of magnitude. The third and more detailed phase started from about 1950 and was motivated by the need for a rational control of shoaling in navigable waterways. The erosion and deposition processes were studied in conjunction with flow structure, the dynamics of flocculation, and the interaction between settling suspended flocs and the bed. For the first time the erosion and deposition rates were introduced in addition to the critical limits. Equations have been derived for the initiation, degree, and rates of erosion and deposition of generalized validity, which, used with the appropriate evaluation of certain parameters through field measurements and/or laboratory tests, may lead to reliable estimates for erosion, deposition, and resuspension. This phase started with the pioneering laboratory and field work of Krone on the relationship of floc properties to the flow-induced stresses as well as on the shoaling processes in the San Francisco Bay [61–64]. Studies, primarily on the erosion of dense and deposited cohesive sediment beds, were initiated by the author in 1960 [102, 105]. This work was followed by fundamental studies on erosion, deposition, and resuspension by the author, Kennedy, Mehta, their associates and graduate students, as well as by others along similar lines of approach for several years and are still being continued [16, 31, 81–86, 94, 95, 99–101, 103, 104, 107–109, 112–115, 118, 154, 155]. Parallel research work was developed about the effect of flow-induced stresses on the properties of flocs and higher order aggregates. Many of these studies were conducted by sanitary engineers in their effort to improve the efficiency of water purification and sewage treatment. Their results, though, are equally applicable to any cohesive sediment suspension. A fundamental framework for cohesive sediment dynamics has been thus developed and formulated [101]. This framework can be used as a guideline for rational approaches to problems involving cohesive sediments. It can also serve as a basis for future research in the field of cohesive sediment dynamics. The three outlined phases of cohesive sediment research do overlap, and in fact some of them interact. However, they define three distinct philosophies of approach and the various steps such a complicated subject has gone through to its present state of the art.
1.3 Objectives of the book and outline of its contents The objective of this book is to present in a unified framework current fundamental and applied knowledge on the hydraulic behavior of cohesive sediments in a turbulent flow field and specifically in open channels. It is based on extensive theoretical and laboratory research and on field investigations over the second half of the 20th century. A number of scientists and engineers from various
Chapter | 1 Introduction
specialties have contributed to the present state of knowledge on this subject. The main emphasis is focused on the processes of erosion, transport, deposition, and resuspension of cohesive sediments, and it addresses the following important and frequently encountered problems in water resources projects: 1. Erosion of natural and manmade canals with cohesive beds and/or banks and ways to stabilize them. 2. Control of deposition of fine sediments in suspension in canals and prevention of shoaling. 3. Fine sediment transport processes in tidal estuaries and bays and the relation of these processes to the salinity and the overall regimen of the estuary with the ultimate objective of control of shoaling in navigable waterways. 4. Control of sediment deposition in reservoirs. 5. Design of sedimentation basins in water purification and sewage treatment plants for optimum performance. 6. Prediction and control of turbidity in river and estuarine waters that may have undesirable effects on the marine life. 7. Proper planning of dredging and filling operations in estuaries and bays to avoid sediment pollution by spreading contaminated fine sediments in ecologically sensitive areas. For a rational approach and a successful design of related operations and structures, the behavior of cohesive sediments in a flow field and the basic physicochemical properties of the sediments, which control and determine that behavior, have to be understood. Following are some typical, major questions involved in the problems and the kind of engineering operations listed previously: 1. Under which conditions and water quality do suspended fines flocculate? 2. What are the relevant properties of the flocs and of higher order aggregates, such as density, strength, and settling velocities, and how do they relate to the erosional and depositional processes and criteria? 3. Which hydraulic parameters determine the critical flow conditions for scouring and siltation in open channels? 4. Which sediment properties determine the erodibility and/or the zones of potential shoaling and what kind(s) of test(s) would be representative of these properties? 5. Is the erosive resistance of a cohesive bed related to the gross properties of the soil, such as the macroscopic shear strength and Atterberg limits, and how? One may recall from the brief outline of the second phase of development in the previous section that any correlation to such soil parameters may lead to erroneous results. 6. Are the sediment properties implied in question 4 constant, or do they change with time and/or environmental conditions and how? The fundamental and applied research in the past 40 to 50 years was planned and conducted to provide guidelines, analytical principles, and suggestions for
Cohesive Sediments in Open Channels
laboratory tests and field measurements as a basis for rational answers to these problems and questions. The model of the hydrodynamic behavior of cohesive sediments in a flow field to be presented in this book is based on the results of extensive fundamental and applied research as well as field investigations from 1950 to about the present time by Krone, the author, Mehta, their numerous collaborators and associates, and several others. This book is not meant to be a compilation of all the work related to cohesive sediment behavior. Work even remotely related to the main theme of the book has been analyzed, commented, and integrated in the overall picture. However, work either inconclusive or unrelated or far removed from the primary objectives outlined above is not included. This does not mean that the omitted work is unimportant by any means. It was simply felt in the present phase of the art and in addressing the listed objectives and questions that the inclusion of a large volume of data without an obvious connection and bearing to these objectives and questions would distract rather than benefit the reader. The book is written by a hydraulic and coastal engineer and is addressed to hydraulic and coastal engineers but also to some extent to environmental engineers. A basic knowledge of fluid mechanics and open channel flow at an undergraduate level as well as an introductory course in soil mechanics are considered sufficient for the study of the subject matter. No knowledge in clay mineralogy and colloidal chemistry is assumed from the part of the readers. Since, however, an understanding of the basic properties of the clay minerals and of the clay suspensions is necessary to fully appreciate the hydraulic behavior of cohesive sediments, Chapter 2 and Chapter 3 have been devoted to these subjects. As the title suggests, the subject matter is focused on the hydrodynamic interaction of cohesive sediments with the flow in open channels, and it includes the processes of flocculation.: This book is designed to serve three main objectives: first, to provide hydraulic engineers and scientists with a rigorous basis for rational decisions regarding the appropriate tests and their application to hydraulic problems involving fine cohesive sediments; second, to be used as a textbook for upper division undergraduate and graduate students in civil and/or environmental engineering either for a special course or as a supplement to a course on the general subject of sediment transport; and third, to assist present and future researchers in this field by providing to them a rigorous basis as a guidance to identify other related areas of research and decide about the best methods of approach. There are several special subjects and problems related to cohesive sediments not addressed in this book. Sediment transport by waves, for instance, albeit an important subject to coastal engineering and currently under investigation, is not included in the present edition. Likewise, biological factors affecting flocculation and certain mechanical properties of sediments are excluded. This is a very specialized field requiring extensive additional fundamental and applied research because its present state of knowledge is very inadequate for practical purposes. There have been many field observations regarding in situ behavior of flocs and aggregates of various orders in lakes, estuaries, bays, and rivers, which,
Chapter | 1 Introduction
if properly analyzed, could add substantially to our knowledge on cohesive sediment dynamics. For the same reason these investigations have been left out of the present edition of the book. The subject matter of the book is divided into eight chapters. Chapter 1, the introduction, outlines the importance of the subject, the philosophy of the book, and a brief summary of the historic development of the field. Chapter 2 covers the basic physicochemical and mineralogical properties of clays. Hydraulic engineers need to have a good understanding of these properties and of the nature and the origin of the interparticle physicochemical forces. The chapter starts with an outline and discussion of the bonding mechanism and of the various types of bonds. Some frequently used elementary concepts and definitions regarding the structure of matter are briefly summarized. The mineralogy of clays follows with a section on the most important clay minerals. The chapter closes with the origin and occurrence of clays and clay deposits. Chapter 3 covers the forces between clay particles and the process of flocculation; therefore, it provides the very important background for the understanding of the hydraulic behavior of clays. Particular emphasis is given to the concept of the double layer, which is of primary importance in the process of flocculation, and to the effect of water chemistry on it. All these subjects are treated through mechanistic pictures, when appropriate, and simplified albeit rigorous models representing the bonding mechanisms. The chapter continues with some of the most important properties of fine particles and aggregates and concludes with the internal structure and fabric of flocs and the microstructure of deposited cohesive beds as they are related to erosion and deposition. The material in both Chapters 2 and 3 has been selected from the cited references so as to provide the nonexpert readers with the minimum necessary background to (a) understand the processes and dynamics of flocculation and the hydrodynamic behavior of cohesive sediments; (b) communicate with geologists and clay mineralogists, as the case may be; and (c) expend their knowledge in these fields through additional studies starting from the suggested references. Readers with sufficient background in the areas of clay mineralogy and clay colloid properties may decide to skip these two chapters. Chapter 4 presents and explains the transport processes of sediments by flowing water and related equations. The process and the dynamics of flocculation are treated next with particular emphasis on the relationship of the aggregate properties to the pertinent parameters describing the flow-induced stresses responsible for the molding of fine particles into aggregates of various orders. This chapter also includes an extensive section on the fundamental aspects of turbulent flows that are of direct importance to flocculation. Chapter 5 deals with the special subject of the rheological properties of cohesive sediment suspensions and compares these properties with the erosion process of cohesive sediment beds. Chapter 6 is the first of the three most important chapters of the book covering the essence of the subject matter. For the reasons explained there, the presentation
10
Cohesive Sediments in Open Channels
of the erosion process was treated separately from the deposition and resuspension in spite of the overlap and the fact that both processes constitute two phases of one and the same process. The chapter starts with a critical review of the early empirical information, briefly outlined in Section 1.2, and it proceeds with presentation of the results of the first fundamental research effort on erosion. The material is predominantly based on the early work by Krone [62] and of the author [102, 105]. For the first time it revealed the detailed process of cohesive sediment erosion in relation to the microstructure of the bed and led to the first analytical model and equations describing the hydrodynamic interaction between the surface layer of the bed and the near bed flow structure. Chapter 7 is concentrated on the subject of deposition and resuspension of beds deposited from suspension from flowing waters. The subject matter is primarily based on the work of Mehta, the author, Kennedy, and several of their graduate students and collaborators. The depositional behavior and resuspension processes have been related to the microstructure of the bed with a clear distinction in the erosional behavior of deposited and artificially placed beds of uniform consistency. Analytical equations have been developed for the degree and rates of deposition and resuspension in terms of readily determinable flow variables and sediment parameters representing their cohesive properties. The special experimental equipment for the study of deposition and resuspension of cohesive sediments, developed first by Kennedy, the author, and collaborators at MIT [31, 109, 114, 118] and later on improved at the University of Florida, is outlined together with its operation [31, 81–86, 109, 114, 118]. Similar setups have been developed since then in some other countries for the same purpose. This chapter also contains a section on the hydrodynamic interaction of suspended and bed sediment in general and explains certain differences in the depositional and erosional behavior between coarse and cohesive sediments. Chapter 8 is devoted to engineering applications of the hydraulics of cohesive sediments. Suggestions are presented first, and guidelines are given for the design of stable channels safe against both scouring and siltation. The emphasis, however, is concentrated on the shoaling control of estuarial waterways, which motivated the initiation of more detailed research on erosion and deposition of cohesive soils, as mentioned earlier. Although, as has already been pointed out, mathematical modeling is not part of the objectives of this book, some hints as to the use of the developed analytical expressions to mathematical and particularly physical models are presented and discussed. The chapter closes with a description of five illustrative case histories of partially mixed estuaries with severe shoaling problems and with their associated field investigations and remedial works. There are still many aspects of cohesive sediment behavior of both academic and practical interest that remain inadequately understood and in need of future research. It is hoped that this book will be of help to any such future research effort.
Chapter 2
The Mineralogy and the Physicochemical Properties of Cohesive Sediments
2.1 G eneral properties of cohesive sediment suspensions and of cohesive sediment deposits As pointed out in Chapter 1, in hydraulic engineering the term cohesive sediment implies a mixture of silt and clay with settling diameter less than 50 m and as small as a fraction of 1 m with various degrees of organic matter. The same term applies to such mixtures containing a substantial percentage of sand provided that they still display cohesive properties. Two of the most common properties of clay masses subjected to various degrees of consolidation are plasticity and cohesion. The first is the property of a clay mass to undergo substantial plastic deformation under stress and within a certain range of water content without breaking. The Atterberg limits are used as a measure of this property. Cohesion is the ability of a clay sample to withstand a finite shear stress within its mass without confinement. The concept of cohesive strength of soils can be demonstrated by the following example and in Figure 2.1. If a sample of cohesive soil is submitted to shear stresses under various confining normal stresses to the point of yield or failure, the plot of the ultimate shear stress at failure or at yield, Sh, versus the normal stresses, ph, will fall approximately on a straight line as shown in Figure 2.1. In this diagram the abscissa indicates the normal stresses, ph; and the ordinate, the shear stresses, . This line, if extrapolated, intersects the axis at a point, ch, in general different from the origin. The shear strength, Sh, of the sample can be described by Coulomb’s equation: Cohesive Sediments in Open Channels Copyright © 2009
Sh ch ph tan φh
(2.1) 11
12
Cohesive Sediments in Open Channels
τ �h
ch 0
ph
Figure 2.1 Shear strength envelop for cohesive soils.
The intercept ch is commonly known as the cohesion and φh is defined as the angle of internal friction. It should be noted that the parameters ch and φh depend on the type of the shear test, the drainage conditions, the rate of application of the shear forces, and the degree of saturation. For instance, for undrained conditions, φh becomes approximately zero and ch attains its highest value. For a completely drained triaxial shear test with a slow application of the shearing load, the angle of internal friction attains its highest value, being more representative of the mechanical resistance due to friction and interlocking among the grains, while the value of ch becomes minimum. The latter has been used as a measure of that part of the soil strength, which is due to the interparticle physicochemical bonds. However, the actual strength due to these forces is expected to be somewhat lower due to the expansion of the soil under very low confining pressures. It is doubtful whether any of the classical soil mechanics tests for shear strength give the true measures of these forces. Nevertheless, ch has been used as a convenient parameter and a reasonable measure. The plastic and cohesive properties of fine sediments are due to that part of the soil mass that is fine enough and of specific area sufficiently large for the surface physicochemical forces to become dominant. The size of these particles, also known as colloids, varies from a few micrometers to a small fraction of 1 m, and they normally have the shape of little flat plates or needles or laths, depending on their mineralogical composition. The behavior of colloidal suspensions is well described by van Olphen in the first chapter of Ref. [150]. The most important aspects of this behavior have been briefly summarized and explained by the author elsewhere [93, 99, 117] and are quoted here. In dry form, clays look like a fine powder of various colors, depending on their mineralogy and impurities. When mixed with water, this powder seems to dissolve like a common salt. However, this is not a real solution but actually a dispersion of very small clay particles. Only some of the larger suspended particles
Chapter | 2 The Mineralogy and the Physicochemical Properties
13
can be observed through an ordinary microscope, whereas an ultramicroscopic arrangement is needed for the observation of the finer particles. If light is transmitted to the sample under the microscope in a way that the light beam hits the particles without entering into the objective lens, the particles scatter the light in all directions. Part of this light enters into the objective lens so that the particles appear to the observer as light specs on a dark background. The light specs display a vivid random motion in all directions, known as Brownian motion. This motion is caused by the thermal activity of the water molecules. This phenomenon takes place in the following way. Consider an infinitesimal water particle of mass m close to the surface of a flat solid clay particle. The latter, however small it may be, is by orders of magnitude larger than the group of water molecules composing the particle. Let unB be the average particle velocity normal to the clay surface due to the thermal energy. The collision will impart on the clay particle a force, F, of a magnitude given by the impulse-momentum equation:
F (m)unB
(2.2)
A suspended particle will receive a large number of such random kicks from all directions at any time. Some of the kicks will cancel each other, allowing only a number of unbalanced impulsive forces on the clay particle, which will determine the imparted motion. It is reasonable to assume that, because of the randomness of the impacts, the net unbalanced force on the particle, F, will have little dependence on the size of the particle. The resulting acceleration as a result of that net impact will be
aB
F M
(2.3)
where M is the mass of the solid clay particle. It follows from Equation 2.3 that the Brownian acceleration, aB, decreases rapidly and in inverse proportion to the third power of the particle diameter. In a quiescent water environment, particles with settling velocity smaller than the average velocity imparted by the Brownian motion appear to stay indefinitely in suspension. This is normally the case of clay particles with Stokes diameter equal to or smaller than 1 or 2 m. But even a very slight degree of agitation would be sufficient to keep the coarsest range of fine sediments in suspension. A homogeneous dispersion of clay fine particles is commonly referred to as a clay solution. van Olphen defines this dispersion as colloidal solution or sol when no measurable deposition takes place within a long period of time [150]. Otherwise, he defines it as clay suspension. Regardless of the definition, the Brownian motion will cause particles to collide. In any microscopic arrangement, it will be observed that, under certain conditions, colliding particles will tend to move away from each other, while under some other conditions they will
14
Cohesive Sediments in Open Channels
stick together, forming larger agglomerates. The first is mostly but not always the case of clays dispersed in distilled water, and the sol is defined as stable, peptized, or deflocculated. If, in such a sol, a small amount of an electrolyte, such as ordinary salt, is introduced to the water, the picture changes drastically with colliding particles sticking to each other forming continuously growing agglomerations. These agglomerates eventually grow large enough for rapid deposition to take place. This phenomenon has been defined as flocculation, and the sol is termed as unstable. The described behavior of suspended fines suggests the existence of repulsive interparticle forces in a stable sol and of attractive forces in an unstable one. In reality, however, both sets of interparticle forces coexist in both types of sols. The electrolyte simply changes the relative magnitude of these forces in a way that, depending on the sol, the net effect can be either attraction or repulsion. In an unstable sol in still water, the process of agglomeration will continue at rates increasing with increasing suspended sediment concentration, thus generating larger and larger agglomerates. Eventually a continuous aggregate network will be formed near the bottom of the container, which, if left undisturbed, will continue slowly consolidating with the water escaping through its pores. That network possesses some shear strength. In moving waters, the flow will induce disruptive shear stresses within the flocs and aggregates, thus preventing their volume increase beyond a certain limit. These stresses, as well as the interparticle physicochemical forces, control not only the size distribution of the flocs and aggregates, but also their density, strength, and settling velocity. The interrelationship between the acting and resisting forces and the preceding properties is examined in Chapter 3. Since a cohesive bed is composed of deposited flocs and aggregates, it follows that the same acting and resisting forces responsible for their formation and properties will also determine not only the depositional behavior but also the density and the resistance to erosion of the bed. For this reason the nature and generation of both forces has to be first understood in order to develop rational criteria for the prediction of the degree and rates of erosion and deposition of cohesive sediments.
2.2 The bonding mechanisms The bonding forces between atoms and/or material particles can be distinguished into two general categories: (a) interatomic or primary bonds and (b) secondary bonds. The first are by far the strongest, and they act between all atoms and molecules of any matter; the second are the ones acting between material particles and are much weaker than the former. The attractive and repulsive forces among colloidal particles belong to the second category.
2.2.1 Interatomic or Primary Bonds Interatomic or primary bonds are the bonds that hold the atoms and molecules of any matter together. Their nature lies in the atomic structure of the matter itself.
Chapter | 2 The Mineralogy and the Physicochemical Properties
15
This section summarizes some of the fundamental concepts regarding the origin and the nature of these bonds. According to the Bohr model, developed in 1913, an atom consists of three basic components: electrons, protons, and neutrons. The electron is a particle possessing a negative electric charge equal to 16 1020 coulombs or 4.81010 esu (electrostatic units). The proton is a particle with a positive electric charge equal in magnitude and opposite in sign to the total charge of its electrons. The neutron is a particle without any electric charge. The proton and the neutron have essentially the same mass, which is taken as the unit mass. The mass of an electron is about 1/1648 the mass of one neutron or a proton and, for practical purposes, can be neglected. Neutrons and protons are packed tightly together at the center of the atom to form the nucleus of the latter. The mass of the nucleus is about 99.95 percent of the total mass of the atom. It is surrounded by electrons moving around it in spherical trajectories known as shells with the nucleus as their center, as outlined in Figure 2.2 ([88], Fig. 2.2). The diameter of the nucleus is about 104 times the diameter of the atom. That is, 1012 of the total volume of an atom is occupied by the nucleus. An introduction to the fundamental concepts of the atomic and molecular structure can be found in any elementary textbook of chemistry and physics. The brief summary of these concepts and definitions herewith presented are based on the book titled Chemistry Made Simple by Hess [50]. Two examples of atomic structure are shown in Figure 2.3: the first represents a hydrogen atom, which has the simplest structure; and the second, the much more complex carbon atom. The situation is similar to that of the planets revolving around the sun or of the satellites revolving around the earth. In both cases a steady circular or elliptical trajectory is being maintained, defined by the balanced
Nucleus: contains protons, neutrons, and other particles 99.95% of mass centered in nucleus Electron shells Diameter of atom about 1 Å
Protons and neutrons have same mass Diameter of nucleus about 10�4 Å
Electronic charge � �16.0 � 10�20 coulomb �4.8 � 10�20 esu No. protons � No. electrons � Atomic number No. protons � No. neutrons � Atomic weight Figure 2.2 Simplified representation of the structure of an atom [88, Fig. 2.2].
16
Cohesive Sediments in Open Channels
e– e– e– e–
e–
e–
e–
(a)
Hydrogen At. no. 1 At. wt. 1
(b)
Carbon At. no. 6 At. wt. 12
Figure 2.3 Examples of atomic structure [50].
action of gravitational and centripetal forces. The sum of the potential and the kinetic energies remains constant; therefore, any change of the distance of a satellite from the earth involving an increase or a decrease of its potential energy is accompanied by a corresponding decrease or increase of its kinetic energy. The electronic energy is not continuous but, according to the quantum theory, an electron can have only certain levels of energy. Transition to a different energy level is discontinuous and can be achieved by either absorption or emission of radiant energy. Only two electrons can be at the same energy level spinning in opposite directions. The combined effects of energy quantization and the limitation of the number of electrons at each energy level are responsible for the different bonding mechanisms that develop when the energy level of the electrons of the interacting atoms composing the various molecules is lowered. Atoms are electrically neutral. That means that the number of protons in the nucleus must be equal to the number of the revolving electrons. This number has been defined as the atomic number, and it is one of the main characteristics of each atom. The mass of the atom is determined by the total number of protons and neutrons in the nucleus and is defined as the atomic weight of the atom. The distribution of the electrons around the nucleus generates the necessary bonds between the atoms to form molecules. The electrons revolve in definite distances about the nucleus and in specific patterns. The latter is surrounded by surfaces of electrons, defined as shells, each one of which is capable of containing a definite number of electrons. The shells are designated by order numbers 1, 2, 3, etc. The maximum number N of electrons in a shell of order n is given by the relationship:
N 2 n2
(2.4)
According to this relationship, the first shell can have a maximum of 2 electrons; the second, a maximum of 8; the third, a maximum of 18; and so on.
Chapter | 2 The Mineralogy and the Physicochemical Properties
17
The table of distribution of elements according to the order of their shells can be found in any book or handbook of chemistry and will not be reproduced here. Only a few observations will be made instead. Hydrogen has only one shell. In the first 18 elements, the higher order shell is formed as soon as the one of lower order obtains the maximum possible number of electrons. From then on a higher order shell is formed as soon as the shell of the immediate lower order obtains 8 electrons. In this way in the higher numbered elements, there can be two or even three unfilled shells of electrons, but there can never be more than 8 electrons in their outmost shell. Some elements, known as inert elements, have their shells filled so that, as the name suggests, they cannot form compounds with other substances. The elements with only one shell unfilled are classified as simple elements, whereas elements with two or three unfilled shells are referred to as transition elements or rare earth elements. Only simple elements are of interest for our purposes. In the formation of compounds, normally electrons of the outermost shell are involved although occasionally electrons from the second shell may be affected in some of the higher number elements. In this formation, a rearrangement of the electronic structure takes place so that the structure can obtain an electronic configuration similar to that of a nearby inert element. The property of the elements to form compounds is called valance, and the manner in which they combine is known as the valance mechanism. The number of electrons involved in the process is referred to as the valance number. There are three types of interatomic bonds referred to as primary bonds: (a) electrovalent or ionic bonds, (b) covalent bonds, and (c) metallic bonds. The last is of little importance to the formation of flocs and aggregates and will not be discussed here.
2.1.1.1 Electrovalence or Ionic Bonds Ionic bonds are developed by the electrostatic attraction of elements with opposite electric charges generated in the following way. Consider as an example a sodium and a chlorine atom. In their interaction to form a compound, these two elements will undergo a rearrangement of their electronic configuration similar to that of a nearby inert element. Sodium has to give up the single electron of its outermost shell, thus forming a positively charged sodium cation or simply a cation with electronic structure similar to that of its nearest inert element neon. Chlorine, on the other hand, being closest to the inert element argon, has to add an electron to its outer shell to obtain the electronic structure of the latter, thus forming the negatively charged chlorine ion or anion. In this way two ions of opposite charge can be formed by the transfer of one electron from the sodium to the chlorine atom, resulting in the formation of the electrostatically neutral sodium chloride, NaCl, known as common salt. The valance of each element is indicated as a superscript on the symbol of the element with a sign for anions and a sign for cations. Thus, the symbol of the sodium ion is Na and that for the chlorine ion is Cl. In the same
18
Cohesive Sediments in Open Channels
way symbols for the ions of the bivalent cations calcium, Ca; magnesium, Mg; and aluminum, Al, are indicated by Ca, Mg, and Al or by Ca2, Mg2, and Al3, respectively. For the negatively charged anions, such as sulfur, nitrogen, and oxygen, the symbols are S, N, and O, or S2, N2, and O2, respectively. Compounds formed by ionic bonds can be viewed as ionic agglomerates. Such agglomerates consist essentially of oppositely charged ions packed and held together by forces of electrical attraction with each cation attracting all neighboring anions. It follows that ionic bonds are nondirectional so that compounds formed by such bonds do not display any preferred direction nor any characteristic geometric pattern. For this reason they are defined as amorphous. For example, a cation of sodium chloride may attract as many chlorine anions as will fit around it. Bonding involves energy changes, because it takes energy to remove an electron from one atom and force it into another. Associated with this energy is the concept of the activity of the element. The term indicates the degree of ease with which this transfer takes place. Since this energy change increases with the number of transferred electrons, atoms with one electron in their outer shell are expected to be the most active in forming compounds, followed in activity by atoms with two electrons and so on. By the same reasoning, elements lacking one electron to fill their outer shell, such as chlorine and fluorine, are more active than others lacking two or more electrons in that shell.
2.2.1.2 Covalence and Covalent Bonds Covalence and covalent bonds develop when the shell of one atom penetrates into the shell of another atom in such a way that the electrons of the interpenetrated shells would be affected by the nuclei of both atoms. This is equivalent to a sharing of one or more bonding electrons among a number of atoms in order to complete the outer shell of each of the combining atoms. Depending on the number of shared atoms, two of the same kinds of ions may combine into more than one way to give different substances. Figures 2.4a and 2.4b give such an example of combination for carbon and hydrogen ions to form the two different compounds methane and acetylene. The methane (CH4) consists of one carbon atom with four electrons, indicated by dots, in its outer shell and four hydrogen atoms, each one with one electron in its outer shell, indicated by the symbol x. If the hydrogen shells interpenetrate the outer carbon shell, each electron of the hydrogen atom will be shared also with the carbon atom, and vice versa, each electron of the carbon atom will be shared with the hydrogen atom. In this way each hydrogen atom completes its outer shell with two electrons, thus reaching a stable configuration, and the carbon atom does the same with eight electrons. In acetylene (C2H2) three pairs of electrons are shared between two atoms of carbon forming a triple bond, while a single pair of electrons is shared between a carbon atom and a hydrogen atom. In this way each hydrogen shell has two electrons, and the second shell of each carbon atom has eight electrons.
19
Chapter | 2 The Mineralogy and the Physicochemical Properties
H
H
C
H H
C
C
H
H (a)
Methane
(b)
Acetylene
Figure 2.4 Examples of covalent bonds [50].
In contrast to compounds formed by ionic bonds, molecules formed by covalence contain a definite number of atoms and possess specific properties. In summary, electrovalence leads to ionic agglomerates through a complete transfer of electrons and formation of ions, whereas covalence produces molecules by sharing pairs of electrons. Ionic conglomerates do not display any preferred direction in their internal structure, but covalent bonds are directional with atoms and molecules combining according to specific geometric patterns. This difference is of particular importance to the clay minerals. Ionic bonding causes separation between the centers of positive and negative charges in the compound, thus forming a dipole. The latter is a system of a positive charge, -ne, where n is an integer and e is the unit charge of one electron, separated by a distance d. In an electric field the dipole will orient itself accordingly. The product
Mo dne,
(2.5)
known as the dipole moment, is a measure of its strength. In electrovalent compounds the valence number of an ion is numerically equal to the charge of the ion. In covalent compounds or molecules, the valence number of an atom is numerically equal to the number of electrons shared by the interacting atoms. However and contrary to the ionic compounds, the valence number may vary from molecule to molecule, depending on the number of shared electrons. This is illustrated in the case of the carbon atom in the two examples of Figure 2.4. In methane, the carbon shares all its four electrons in its outer shell with hydrogen; therefore, it has a valence of four. In acetylene, however, it shares only two of its four electrons with two hydrogen atoms and, therefore, it has a valance of two. This property of covalent bonds is responsible for the complexity of organic chemistry. Primary electrovalent and covalent bonds are much stronger in comparison to the secondary bonds to be discussed in Section 2.2.2. The energies of primary bonds per mole of bonded atoms range from 60 103 J to more than 400 103 J, an equivalent range between 15 and 100 kcal ([88], Section 2.3). Considering that every mole of a substance contains 6.03 1023 molecules
20
Cohesive Sediments in Open Channels
(Avogadro’s number), the energy per molecule ranges from 1019 J to a maximum of 66 1019 J. These energies may appear small in absolute terms, but they are very large if the mass of the molecule is taken into account. It follows that high energy is needed to break the primary bonds in comparison to the secondary bonds. The first are significant in understanding the structure and properties of clay minerals, but the latter are the ones of importance to flocculation and to cohesive sediment behavior.
2.2.1.3 Some Basic Concepts and Definitions of Chemistry This section will close with the definition of certain basic terms, laws, and concepts of chemistry, which are frequently encountered in soil technology and, therefore, in cohesive sediment behavior. The following are some of the most important of these terms and concepts, which can be found in any introductory book on chemistry: 1. The law of definite proportions. Whether combined by ionic or covalent bonds, the number of electrons involved in the process determines the relative mass of each element in the compound product. That means that a given compound contains the same elements combined in the same proportions by mass. This is the well-known law of definite proportions. As an example, consider the combination of oxygen and hydrogen to produce water. Oxygen has an atomic weight of 16 and 6 electrons in its outermost shell, and hydrogen has an atomic weight of about 1 and 1 electron in its only shell. Since an atom of oxygen needs 2 electrons to fill its outermost shell, it has to combine with two atoms of hydrogen to produce one molecule of water. With the oxygen atom having 16 times the mass of one hydrogen atom, it follows that 8 mass units of oxygen have to combine with one mass unit of hydrogen to produce water. The proportion of atoms in the compound is indicated by subscripts in its constituent element with the subscript 1 omitted. Thus, the chemical formula of water is OH2. From the chemical formula and the atomic weight of the constituent elements, the molecular weight of the compound can be defined and evaluated. The latter is the sum of all protons and neutrons in the elements of the compound. For example, the molecular weight of water is equal to 18; that is 2(atomic weight of hydrogen) one atomic weight of oxygen 2(1) 16 18. 2. The mole. A quantity of a compound equal in weight to its molecular weight is called a mole indicated by M. A mole is furthermore characterized by the units of weight or mass used. For example, it may be indicated as a gram mole, a kilogram mole, and a pound mole. Thus, the gram mole of water is 18.016 grams, and its pound mole is 18.016 lbs. The number of moles that a substance may contain can be found by dividing the actual weight of the substance by its molecular weight. 3. The equivalent weight. This property is defined as that weight or mass of an element which combines with or displaces from any substance a weight or mass of hydrogen equal to the atomic weight of the latter. The equivalent
Chapter | 2 The Mineralogy and the Physicochemical Properties
21
weight is very important in determining the proportions of the various elements that form a specific compound. Since a mole of an element combines with a number of moles of hydrogen equal to their valence number, it follows that the equivalent weight of an element is equal to its atomic weight divided by its valence number. Therefore, one equivalent weight of any element reacts with one equivalent weight of any other element to produce one equivalent weight of the resulting compound. The equivalent weight of a compound is defined as the molecular weight of the substance divided by the net positive valence, that is, by the product of the valence of the positive element of the compound or molecule times its subscript in the chemical formula. This last product represents the number of replaceable hydrogens. In summary:
Equivalent Weight of Element
Equivalent Weight of Mole
Atomic Weight Valence Number
Compound Molecular Weight Net Posiitive Valence
Net Positive Valence (Valence Number ) (Chem. Subscript )
The equivalent weight of oxygen is, for example, 16/2 8, since one mole of oxygen combines with two moles of hydrogen to produce one mole of water. Therefore, the equivalent weight of an element can also be defined as the mass or weight of the element which combines with or displaces eight parts by weight or mass (in this case 8 unit weights) of oxygen. Many elements can manifest more than one valence number, depending on the compound under consideration. Such elements can have more than one equivalent weight, which is shown by the element in a particular reaction. Iron, for example, depending on the conditions of the reaction, may form two oxides: FeO and Fe2O3. In both oxides the oxygen atom has a valence of 2; therefore, the valence of Fe in FeO is 2 but in Fe2O3 is 3. 4. Solutions and concentrations. Substances very often appear in a dissolved state in liquid media. The dissolved substance is defined as the solute and the liquid medium as the solvent. An enormous number of chemical reactions take place in solution. In the domain of cohesive sediment dynamics, the only solvent to be considered is water. The amount of solute per unit volume or per unit mass of solvent is defined as the concentration of the substance. There are various ways to express the latter quantitatively, the following being the most commonly used:
(a) Weight per unit volume. The most frequent units for such a concentration are one gram of substance per liter of solution and one gram of substance per cubic meter of solution. The latter is normally referred to as parts per million symbolized by ppm. Obviously, one gram per liter is equal to 1,000 ppm.
22
Cohesive Sediments in Open Channels
(b) Molarity. This term indicates the number of moles of a solute per liter of solution, and its symbol is M. (c) Normality. This is the number of equivalents of solute per liter of solution, and its symbol is N. Molarity and normality are related by the following equation:
N M(Net Positive Valence) (d) Molality. The number of moles per 1000 grams of solvent is defined as molality, and its symbol is ML. For very low concentrations in water, molality is numerically very close to normality; in general, however, molality and molarity are two distinct terms and they may differ substantially—first, because 1000 grams of a liquid is not necessarily a liter and, second, molality indicates concentration with respect to the weight of solvent, whereas molarity defines concentration with respect to the total volume of solution. (e) Percent of composition. This term may indicate either percentage by weight or percentage by volume according to the relations: Percent by weight Wt. of Dissolved Solute/Wt. of Solvent
Percent by volume Volume of Solute/Volume of Solvent
In solids dissolved in liquids, the first definition is almost exclusively used, while the second is common to solutions of gases in gases or liquids in liquids. Solutes may be distinguished into electrolytes and nonelectrolytes. When dissolved in water, electrolytes produce a solution that conducts electric current, whereas nonelectrolytes do not. Acids, bases and salts are electrolytes. Solutes may affect the properties of the solvent. For example, they may lower the vapor pressure and the freezing point and raise the boiling point. Discussion of these effects is beyond the scope of this limited review; interested readers are referred to any introductory book of chemistry. As already mentioned, ionic and covalent bonds, also known as primary bonds, are the strongest of all bonds and the ones primarily responsible for the formation of compounds and the internal structure of clay minerals. There is a third primary bond, the metallic bond. This is a nondirectional bond generated in metals by loosely held valence electrons that hold the positive metal ions together while at the same time are free to travel through the solid material ([98], Chapter 2). Metallic bonds are irrelevant in most soils and they are mentioned here only for the sake of completion.
2.2.2 Secondary Bonds In addition to the primary bonds, there are other weaker bonds which may affect the final arrangement of atoms in solids and cause attraction between very small particles as well as between solid particles and liquids. These bonds are termed
Chapter | 2 The Mineralogy and the Physicochemical Properties
23
secondary bonds. The following are the most important secondary bonds for cohesive sediments and will be briefly discussed.
2.2.2.1 Hydrogen Bond A hydrogen bond may develop when a hydrogen ion forms the positive end of a dipole being attracted to the negative charge of a molecule at the surface of a particle. These bonds require strongly electronegative atoms, such as oxygen, to produce strong dipoles. The electron transferred from the hydrogen to the oxygen spends most of its time between the two atoms, thus leaving the hydrogen proton to act as the positive end of the dipole and the negative oxygen atom as its negative end. A hydrogen bond is, therefore, a permanent directional dipole and can readily be formed between clay minerals with oxygens on their surfaces; they constitute the mechanism for water adsorption on the surfaces of clay minerals and play an important role in determining the fundamental properties of cohesive sediments. The small size of the hydrogen atom makes it particularly suitable to fit between the spaces of surface atoms of clay particles. For this reason, hydrogen bonds are stronger than any other secondary bond. 2.2.2.2 van der Waals Bonds In contrast to the permanent dipole of hydrogen bonds, the van der Waals bonds amount to a fluctuating dipole generated by the mutual influence of the motion of electrons of atoms. At any time there may be more electrons on one side of the atomic nucleus than on the other, thus creating instantaneous dipoles with oppositely charged ends, which attract each other. The van der Walls forces are always attractive and exist between all units of matter. They are the primary mechanism of flocculation but are one order of magnitude smaller than the hydrogen bonds. There are three reasons for their importance. First, as already mentioned, they exist between any two material units, whereas the development of the hydrogen and cation bonds, to be discussed later, requires special conditions. Second, they are for all practical purposes independent of the water quality and of other physicochemical factors. And third, although individual van der Waals forces are relatively weak, they are nondirectional and additive between atoms, and the overall attractive force and its potential decrease more rapidly with distance than ionic and primary valence bonds. Specifically, for two atoms the van der Waals attractive force is inversely proportional to the seventh power of the interparticle distance, and its potential is inversely proportional to the sixth power of that distance. The net force and potential between two spherical particles are inversely proportional to the third and second power of the distance, respectively. The equation describing the variation of the potential energy of the van der Waals forces is presented in Chapter 3. 2.2.2.3 Cation Bonds Such bonds amount to cations attracting negatively charged particles, as in Figure 2.5. Cation bonds, for example, occur between montmorillonite units
24
Cohesive Sediments in Open Channels
�
�
�
�
�
�
�
�
Figure 2.5 Schematic picture of a cation bond [99].
(Section 2.3) with sodium or potassium as the bonding cations. Cation bonds are weaker than hydrogen bonds and can easily be broken by water adsorption and swelling.
2.2.2.4 Chemical Cementation The type of bonding known as chemical cementation actually amounts to links between clay particles generated by chemical compounds. Such compounds, for example, are used for the treatment of soils as subgrade materials ([88], Chapter 4, Chapter 9, and Chapter 11). Iron oxides, which carry a net positive charge, are strong cementing agents with an iron oxide being attracted by two negatively charged clay particles.
2.3 The nature and mineralogy of clay particles 2.3.1 Introductory Remarks So far we have been using loosely the term fines and clays without any qualification other than their size. For an understanding of their hydrodynamic behavior, these sediments have to be distinguished into certain groups, each one of which is characterized by special mineralogical and physicochemical properties. These properties control the interparticle forces and, therefore, the engineering behavior of aggregates and of cohesive sediment deposits. One may recall that the various atoms within a solid particle are held by strong primary valence bonds, whereas the much weaker secondary valence bonds, specifically the van der Waals, hydrogen, and cation bonds, are responsible for the interparticle attractive forces whether in suspension or in a deposited sediment bed. That means that any particle agglomerate, when subjected to stresses, will deform and eventually break through failure of the interparticle joints rather than by breaking of the particles. Fine sediments have been defined as mixtures of silt and clay possibly with some fine sand and organic matter exhibiting colloidal properties. These colloidal properties are due almost exclusively to the clay fraction of the total sediment. On the basis of their size and according to the most widely accepted classification, silts range in size from 2 to 74 m, whereas the size of clays is below 2 m. However, a mechanical classification by itself does not reveal the basic properties of flocs and of cohesive sediment deposits. The colloidal properties of the clay fraction of the sediment do not depend only on the nominal grain size range, but also on a variety of other factors, such as mineralogy and surface activity.
Chapter | 2 The Mineralogy and the Physicochemical Properties
25
In fact, particles larger than 2 m may display colloidal properties, while particles smaller than 2 m may be virtually inert as far as physicochemical forces are concerned. It is more realistic to distinguish fine sediments into clay and nonclay with silts belonging to the second group. Clay particles have the shape of small flat plates, needles, and tubes with high specific area and are subjected to physicochemical forces that are very large in comparison to their weight. These forces, which are the subject of the following chapter, are determined by the mineralogical characteristics of the clay particles, their crystalline structure, and the dissolved ions in the water.
2.3.2 The Basic Clay Minerals Modern electron microscopy, X-ray diffraction, and differential thermal analysis have revealed that clays are composed essentially of one or more members of a small group of clay minerals [88, 99, 150]. These minerals have a predominantly crystalline arrangement as a result of the directional covalent bonds. That means that the composing atoms are arranged in such an orderly way as to form a definite three-dimensional geometric network defined as the lattice of the crystal. The points of a lattice in which atoms or atomic groups are located are termed lattice points. Only 14 different arrangements of lattice points have been identified and are known as Bravais space lattices. If a clay crystal is continuously subdivided, a minimum size will eventually be reached that has all the characteristics and atomic arrangement of the crystal. That minimum size element is defined as a unit cell. A clay particle, therefore, is composed of unit cells. The 14 unit cells, according to Bravais classification, are shown in Figure 2.6 with dots indicating location of atoms ([88], Chapter 2, Figure 2.4). Each of these arrangements can be described by its three sides a, b, and c and the three angles , , and between them. Crystals have been classified in 32 distinct classes based on the arrangement of atoms and the orientation of their faces. Some of these crystal classes bear close similarities and relationships to each other so that they can be grouped into six crystal systems illustrated in Figure 2.7. The indicated crystallographic axes aa, bb, and cc are parallel to the edges of intersection of the prominent crystal faces, and the relationship between the sides a, b, and c and the angles , , and are indicated for each crystal system. It should be noted that five out of the six crystal systems have three crystallographic axes, a, b, and c, whereas the hexagonal system has four. The following is a brief description of each crystal system with some examples ([88], Chapter 2). 1. Isometric or cubic system. The three axes of this system are perpendicular to each other with equal lengths. Galena, halite, magnetite, and pyrite belong to this system. 2. Hexagonal system. This system has a hexagonal base characterized by three axes of equal length intersecting at 60° and a fourth c axis normal to the hexagonal base. Quartz, brucite, calcite, and beryl are examples of this group.
26
Cohesive Sediments in Open Channels
� � 90�� a�b�c
c
c a a a a p
a
a p
b
c
P a
� � � 90� a�b�c
b P
c
c
a
a I
b C
Tetragonal
a
c
a
� � � 90� a�b�c
a
a b C
F a
a
c a R
Monoclinic a
b F
Rhombohedral
a
� � a�b�c
c a a
c
a
c
I Isometric: a � b � c: � � � 90�
a
a P
a
� � 90� a�c
a
b b I
Hexagonal Cubic
� 120�
Orthorhombic
P Triclinic
a� � � 90� a�b�c
Figure 2.6 Lattice classification according to Bravais [88, Fig. 2.4].
3. Tetragonal system. This system has three mutually perpendicular axes: two horizontal of equal length and one vertical of different length. Zircon belongs to this group. 4. Orthorhombic system. In this system the three mutually perpendicular axes have different lengths. Examples of minerals belonging to this system include sulfur, anhydrite, barite, diaspore, and topaz. 5. Monoclinic system. Two of the three axes of this system are unequal and inclined to each other at an oblique angle, while the third axis is perpendicular
27
Chapter | 2 The Mineralogy and the Physicochemical Properties
c
c
α
β
a
a3
60°
60°
b
b
60°
a2
γ a
a2 a1
abc αβγ90°
a3 a1a2a3c
c
c
Isometric (cubic)
Hexagonal
c
c
α
β
a
b
α
β b
b γ
a1
abc αβγ90°
abc αβγ90°
c
c
Tetragonal
Orthorhombic
c
c
α β
a
b
α
β b
a
b
b
γ a
a1
b
γ a
a1
abc αγ90° β90°
γ cab αβγ90°
a
c
c
Monoclinic
Triclinic
Figure 2.7 The crystalline systems [88, Fig. 2.5].
to the other two. Orthoclase, feldspar, gypsum, muscovite, biotite, gibbsite, and chlorite are among the most common minerals belonging to this group. 6. Triclinic system. All three axes of this system are unequal intersecting each other at oblique angles. Typical examples of this group are plagioclase, feldspar, kaolinite, albite, microcline, and turquoise.
28
Cohesive Sediments in Open Channels
An interesting question is how and why the various crystal structures develop. The answer lies in the principle that atoms tend to take the most stable arrangement in a structure and this arrangement is the one that minimizes the energy per unit volume. The geometry of the lattice determines also the cleavage planes, that is, the planes along which a particular crystal breaks, as well as the external form of a piece of mineral. The cleavage plane lies between planes having the most densely packed atoms because in this case the center-to-center distance between atoms on opposite sides of the plane is greater than across any other plane through the crystal. Therefore, the shear strength along the cleavage plane is smaller than in any other direction. Finally, the orderly atomic arrangement within the crystals allows diffraction of light, X-rays, and electron beams. These properties have been used to identify the minerals. For a more detailed description of crystals and discussion of their properties and their identification, the reader is referred to ([88], Chapter 2). Clay minerals appear as small particles ranging in dimensions from a fraction of 1 m to a few micrometers. Each particle is built by a number of crystal unit cells held together by secondary valence bonds. Depending on the nature and strength of the bonds, clay particles will disintegrate in water to various degrees. Those with the more resistant bonds will form sediment with grains composed of several unit cells in the form of a book or packet, while those with relatively weak bonds and, in particular, with swelling cations between them may disintegrate to almost single unit cells. The clay mineral kaolinite belongs to the first case, while the sodium montmorillonite mineral belongs to the second. Chemically, clay minerals are silicates of aluminum and/or magnesium and iron. From the structural standpoint there are two fundamental building blocks. Each one of these blocks forms two dimensional arrays: the silica tetrahedral unit and the aluminum, magnesium, or iron octahedral unit [88, 99, 150]. The sequence of superposition of these blocks, the cations in the lattice of the mineral, and the adsorbed cations between the unit cells determine the nature of the various clay minerals. The following are the two fundamental building blocks: 1. The silica tetrahedral unit. This unit consists of four oxygens having the configuration of a tetrahedron enclosing a silicon atom (Figure 2.8a). The tetrahedral units are combined in a sheet structure, known as a tetrahedral sheet or a silica sheet, so that the oxygens of their bases are in a common plane with each oxygen shared by two tetrahedral units while all the tips point in the same direction (Figure 2.8b). In this way three of the oxygen atoms of each tetrahedron are shared with three neighboring tetrahedra. Figure 2.9 shows a plan view of such an array of tetrahedra in which the hexagonal symmetry is clearly discerned. The unit cell area is indicated by the dashed line rectangle. This unit has 4 silicon atoms and 10 oxygen atoms with an electric charge of 4, and a composition of (Si4O10)4. Electrical neutrality can be obtained by replacement of four oxygens by hydroxyls or by a union with a positively charged sheet of different composition. In the array,
Chapter | 2 The Mineralogy and the Physicochemical Properties
(a)
Single unit and
(b)
� Oxygens
� Silicons
Figure 2.8 Silica tetrahedral units [88, 99].
(b) (a)
Legend Oxygen Slicon (c)
Combination
(d)
Figure 2.9 Top view of a tetrahedral sheet [150, Ch. 6].
29
30
Cohesive Sediments in Open Channels
OH OH
OH
OH
OH
OH
OH
OH OH
OH OH
OH OH (a)
OH Single unit
OH OH (b)
OH and OH � Hydroxyls
OH
OH OH
Combination � Aluminum, Iron and Magnesium
Figure 2.10 Octahedral units [88, Ch. 3; 99].
the oxygen-to-oxygen distance is 2.55 Å, the space available for the silicon atom is 0.55 Å, and the thickness of the sheet is 4.63 Å ([150], Chapter 6). The hexagonal oxygen arrangement in the silica sheet leaves holes within each hexagonal ring of six oxygens. These holes play a dominant role in the adsorption of ions and in the stability of clay minerals in water. 2. Aluminum, iron or magnesium octahedral. This second building block consists of six hydroxyls and/or oxygens having the configuration of an octahedron enclosing an aluminum, iron, or magnesium atom, as shown in Figure 2.10a The octahedral units combine together into a sheet structure, known as an octahedral sheet or an alumina or a magnesia sheet, with each hydroxyl shared by two units as in Figure 2.10b. The sheet may be viewed as two layers of densely packed hydroxyls with the cation between them in octahedral coordination. The neutral three-layer structure is the basic model known as pyrophyllite. If the cation is trivalent aluminum, only two of the three cation spaces are filled. Such a structure is called dioctahedral with composition Al2(OH)6 and the corresponding mineral is known as gibbside. If the cation is divalent, such as magnesium, normally all three cation spaces are filled with the cation, and the structure is defined as trioctahedral with composition Mg3(OH)6. The mineral corresponding to this trioctahedral structure is defined as brucite ([88], Chapter 3). Figure 2.11 shows a plan view of an octahedral sheet together with a horizontal projection of an octahedral unit. It is observed that the oxygen atoms and hydroxyl groups form a hexagonal close packing with “holes” considerably smaller than those in the tetrahedral sheet. Cations other than aluminum, Al3, and magnesium, Mg2, can also be present in the octahedral like Fe2, Fe3, Mn2, Ti4, Ni2, and Cr3. The oxygen-tooxygen distance in the octahedral sheet is 2.60 Å, the space available for the cation is 0.61 Å, and the thickness of the octahedral sheet is 5.05 Å. The various clay minerals are formed by superposition of these two fundamental building blocks. The structure of each clay mineral is determined by the cation in the octahedral layer, the sequence and number of building blocks in the unit cell
Chapter | 2 The Mineralogy and the Physicochemical Properties
31
(b)
(a)
Legend Oxygen
Hydroxyl
(c)
Al or Mg
(d)
Figure 2.11 Top view of an octahedral sheet [150, Ch. 6].
of the mineral, and the cations between the unit cells. A unit cell of a clay mineral is the smallest part of the structure, which repeats itself in both directions of the sheet and normal to it. Tetrahedral and octahedral sheets combine by sharing oxygen atoms in a common plane. Depending on whether two layers (one tetrahedral and one octahedral sheet) or three layers (two tetrahedral and one octahedral sheet) are involved in the unit cell, the minerals are distinguished into two-layer minerals or 1:1 layer minerals and three-layer minerals or 2:1 layer minerals, respectively. Unit layers are stacked parallel to each other (Figure 2.12). There are many factors involved in the development of the clay minerals. The most important of these factors are the cations in the lattice of the building blocks, the unit layer of adsorbed cations and water between the layers, and the development of the crystal structure. The latter may range from very poor to almost perfect. There is, therefore, an enormous variety of clay minerals, so that clay mineralogy amounts to a distinct field in itself. This discussion is limited to the description of a simplified structural model to be used for an explanation of the fundamental characteristics and engineering behavior of the most commonly encountered groups of clay minerals. For an in-depth study, Refs. [88, 150] are recommended.
32
Cohesive Sediments in Open Channels
Charge 6 O 12 4 Si 16
Tetraheral sheet octahedral sheet
4O 2 OH
10
4 Al
12
6 OH 6
(a)
Surface of the unit cell: 5.15 8.9 Å2 Formula of unit cell: [Al2 (OH)4 (Si2 O5)]2 Unit cell weight: 516
O OH Si Al
28 28 C–spacing 7.2 Å
Distance between atom centers Tetraheral sheet
Charge
0.60 Å
6O 4 Si
1.60 Å
16
4 O 10 2 OH
Octahedral sheet
2.20 Å
Tetraheral sheet
1.60 Å
4 Al
O OH Si Al
12
2 OH 10 4O
0.60 Å
(b)
12
Surface of the unit cell: 5.15 8.9 Å2 Formula of unit cell: [Al2 (OH)4 (Si2 O5)2]2 Unit cell weight: 720 Hydroxyl water: 5%
4 Si 6O
12
16
44 44 C–spacing 9.2 Å
Figure 2.12 Outline of building blocks: (a) two-layer mineral; (b) three-layer mineral [150, Ch. 6].
Clay minerals can be grouped according to their crystal structure and stacking sequence of the various layers. Those belonging to the same group have similar engineering properties and are expected to display similar sedimentological behavior. In relation to the lattice cations, there is an important process known as isomorphous substitution. This term implies replacement within the lattice of the unit layer of one cation by another of the same or different valence without altering the crystal structure of the mineral. In this way, some of the tetravalent silica, Si4, in the tetrahedral sheet may be replaced by trivalent aluminum, Al3. It is more common, however, to have a substitution in the octahedral sheet of trivalent aluminum by divalent magnesium, Mg2, and of magnesium by ferrous iron. The cation distribution in the octahedral and tetrahedral layers may develop either
33
Chapter | 2 The Mineralogy and the Physicochemical Properties
OH
OH
OH
OH
OH
Oxygens
7Å
OH Hydroxyls
Aluminiums OH
OH OH
OH
Silicons
OH
Figure 2.13 Kaolinite structural unit [99, 100].
during the initial stage of formation of the clay mineral or later on. An isomorphous substitution of a higher valence cation by one of lower valence leads to the development of a negative electric charge on the clay particle. This charge plays a dominant role in the process of flocculation. As already discussed, the unit layer of a clay mineral consists of a sequence of tetrahedral and octahedral sheets with two such sheets sharing oxygens and hydroxyls on the same plane with both covalent and ionic bonding of the primary valence type. Unit layers are held together by much weaker secondary valence bonds, some of which may vary widely from relatively strong to very weak; in addition, they can be drastically affected by physicochemical changes in the water environment. van der Waals forces and hydrogen bonds are relatively stable; cation bonds, however, can be significantly affected by water adsorption and swelling, which may push the unit layers apart. Potassium ions provide the most stable cation bonds because they fit well into the holes of the bases of the silicon tetrahedron sheet. In contrast, sodium between octahedral units does not fit in any base holes, hydrates, and swells in the presence of water, thus resulting in extreme cases in an almost complete disintegration of the clay mineral to its unit cells. There are two basic structural groups of tetrahedral and octahedral sheets: the kaolinite and the montmorillonite. 1. The kaolinite structural or two-layer group. The unit of this group consists of an aluminum (or iron or magnesium) octahedral layer with a parallel superimposed silica tetrahedral layer so that the tips of the silica sheet and one of the layers of the octahedral unit form a common plane as in Figure 2.13. The unit is about 7 Å thick and may extend indefinitely in the other two directions. Kaolinite, dictite, nacrite, and halloysite are members of this group. That mineral
34
Cohesive Sediments in Open Channels
may thus be viewed as a succession of oxygens, silicons, hydroxyls plus oxygens, aluminums (or irons or magnesiums), and hydroxyls. Particles are formed by a regular stacking of such 7 Å units in a book-like form. 2. The montmorillonite structural unit. This structural unit is made up of sheetlike layers with each of them composed of two silica tetrahedral layers and one octahedral layer between them like a sandwich. The oxygen atoms of the tetrahedral tips combine with the hydroxyls of the octahedral layer bonded by primary valence bonds into a continuous plane. The order of the elements in the unit layer of this group is as follows: oxygens, silicons, oxygens plus hydroxyls, aluminums (or irons or magnesiums), oxygens plus hydroxyls, silicons, and oxygens (Figure 2.14). The thickness of the sheet is approximately
OH
9.5 A
OH OH
Exchangeable cations and water
Oxygens OH Hydroxyls
Aluminum, Iron, and Magnesium
Silicon, Occasionally Aluminum
Figure 2.14 Montmorillonite structural unit [88, Section 3.7].
35
Chapter | 2 The Mineralogy and the Physicochemical Properties
9.5 Å with unlimited dimensions in the other two directions. The sheets are stacked one above the other like the leaves of a book in an arrangement similar to that of the kaolinite group. The various principal mineral groups to be described and discussed here consist of different combinations of these two fundamental structural groups stacked one on the other and held together by secondary valence bonds. The kinds of bonds, the nature of the bonding cations, and the cations within the structural group determine the type of clay mineral, its physicochemical characteristics, and its engineering properties, specifically its hydrodynamic behavior in a flow field. The following are the most important clay minerals. 1. The kaolinite-serpentine group or the 1-1 minerals ([88], Chapter 3, Section 3.6). The minerals of this group are composed of the structural kaolinite unit, which consists of alternating layers of silica tetrahedral and octahedral sheets with the tips of the tetrahedra coinciding with one of the planes of atoms of the octahedral sheet and pointing toward the center of the latter. Two thirds of the atoms in that plane are oxygens shared by both the tetrahedral and the octahedral layers. The remaining atoms in that plane are hydroxyls (OH) with each one of them located below the hole of the hexagonal network formed by the bases of the silica tetrahedra, as shown in Figure 2.9. The mineral is called kaolinite if the octahedral layer is dioctahedral gibbsite, and it is called serpentine if the octahedral layer is trioctahedral brucite. The structural formula of the kaolinite is (OH)8Si4Al4O10 and is outlined schematically in Figure 2.15. As shown, the unit cell has zero charge and its exact dimension is 7.2 Å. Dioctahedral minerals of the kaolinite group are by far the most common; in contrast, trioctahedral 1:1 minerals are rather rare, and they are normally encountered mixed with kaolinite and illite.
B
G
B
G
7.2 Å
7.2 Å B
G
(a)
(b)
Figure 2.15 Atomic arrangement in a unit cell of (a) kaolinite, (b) (serpentine) [88, Fig. 3.9].
36
Cohesive Sediments in Open Channels
Successive kaolinite layers are held together by strong hydrogen and van der Waals bonds. Therefore, very little water can enter between the individual unit layers, and very few cations can be adsorbed between them. Because of this bonding, kaolinite clays do not swell, have a stable structure and, although they cleave fairly easily along the plane surface of the 7.2 Å unit cell, do not disintegrate and split in water like other clay minerals. They form the largest subgroup of clay minerals in nature in the form of six-sided flat plates with lateral dimensions ranging from 0.1 to 4 m and with a thickness ranging from 0.05 to 2 m. The specific area of a kaolinite particle ranges from 10 m2 to 20 m2 per gram of dry clay. This is the lowest specific area in clay minerals; therefore, kaolinites are characterized by the lowest surface activity, which is the lowest intensity of interfacial physicochemical forces in comparison to their weight. A good measure of surface activity is the cation exchange capacity. As it will be shown in Chapter 3, the surfaces of the clay particles carry a negative charge either from adsorption of anions or from isomorphous substitution. To preserve electrical neutrality, cations are attracted on the surfaces and the edges of clay particles. Some of these cations can be replaced by others of a different type in the solution. The quantity of such exchangeable cations per unit mass of clay is defined as the cation exchange capacity, it is symbolized by CEC, and it is commonly expressed in milliequivalents (meq) per 100 g of clay. Kaolinite has the lowest CEC ranging from 3 to 15 meq/100 g. Although isomorphous substitution in kaolinites is believed to be rather rare and uncertain, it takes the replacement of one trivalent silicon by a bivalent aluminum in every 400 units to account for an exchange capacity of the above order ([150], Chapter 6). Moreover, broken bonds around the edges of the clay particles may give rise to unsatisfied charges that are also balanced by adsorbed cations. It is noteworthy that, because separation does not take place between unit layers of kaolinite, the balancing cations can be adsorbed only on the faces and edges of particles but never in the inbetween layers. A variation of the kaolinite subgroup is the mineral halloysite [88, 150]. It is structurally similar to kaolinite, except that its unit layers are separated from each other by a single layer of water molecules, when hydrated. Its composition is (OH)8Si4Al4O10.4(H2O), and the spacing of the unit layer is 10.1 Å, which is considerably higher than the 7.2 Å spacing of the kaolinite. The difference of 2.9 Å is the approximate thickness of one water molecule. Because of this layer of water molecules, the interlayer bonds are reduced, resulting in a curvature of the mineral and the formation of tubes with the hydroxyls of the octahedra on the inside and the basis of the tetrahedra on the outside. The outside diameter of the tubes ranges from 0.05 to 0.20 m, and the average wall thickness is 0.02 m. Dehydration of halloysites at low temperature removes the interlayer water and reduces the mineral to its kaolinitic form; however, this may cause splitting and unrolling of the tubes. The separation of the unit layers by water molecules makes possible the adsorption of cations there, thus increasing the cation exchange capacity of the mineral to a range between 40 and 50 meq/100 g. This is quite higher than that
37
Chapter | 2 The Mineralogy and the Physicochemical Properties
of kaolinite. The specific area of halloysites ranges from 35 to 70 m2g1. This is about 3.5 times the specific area of kaolinites. Therefore, from the standpoint of surface activity, halloysites are more active than kaolinites. 2. The smectite or montmorillonoid structural subgroup. This subgroup includes expending three-layer clays of the montmorillonite structural group shown in Figure 2.14. The theoretical composition in the absence of isomorphous substitution is (OH)4Si6Al4O20.n(interlayer)H2O [88, p. 30]. The hydroxyls of the octahedral sheet fall directly above and below the hexagonal holes formed by the bases of the silicon tetrahedral units. This neutral three-layer structure represents the mineral pyrophyllite, and it has a net spacing of 9.2 Å. When the octahedral layer is dioctahedral gibbsite with aluminum as its cation, the subgroup is defined as montmorillonite. When the octahedral layer is trioctahedral brucite with magnesium as its cation, the mineral is defined as saponite. The structures of these two subgroups are shown schematically in Figure 2.16. In either case the unit layer has a thickness of about 9.6 Å with indefinite dimensions in the other two directions. The unit layers are stacked one above the other like the leaves of a book in a way similar to the kaolinite. This higher spacing is due to the adsorption of cations and water molecules between the unit layers. There is, however, an important bonding difference between the kaolinite-serpentine group and the smectite group. Unlike kaolinites with strong interlayer hydrogen bonds, both sides of a smectite unit layer end in oxygens, so that the only possible interparticle bonds are the weaker van der Waals and cation bonds. As a result, cations and water can enter between the unit cells. The thickness of the water layer can vary depending on the nature of the adsorbed cations and on the hydration energy
G
(a)
B
G
9.6 Å → ∞
B
G
n-H2O � cations in interlayer regions
B
9.6 Å → ∞
(b)
Figure 2.16 Schematic diagram of smectite structure: (a) montmorillonite, (b) saponite [88, Fig. 3.15].
38
Cohesive Sediments in Open Channels
involved. If that energy is able to overcome the energy of the van der Waals attraction, the layers will split apart, leading in extreme cases to complete disintegration of the clay to its individual unit layers of a thickness of 9.6 Å. In smectites, there is an isomorphous substitution of silicon and aluminum by other cations. Thus, aluminum in the octahedral layer can be replaced by magnesium, iron, zinc, nickel, lithium, or other cations ([88], Chapter 6). The resulting structure is almost either exactly dioctahedral (montmorillonite subgroup) or trioctahedral (saponite subgroup). Figure 2.17 shows the charge distribution in a unit cell for the structural type of montmorillonite, where the indicated electric charges are in terms of the charge of an electron ([88], Chapter 2). These substitutions cause a positive charge deficiency and an excess of negative charge in the range of 0.5 to 1.2 with an average of 0.66 electronic unit charges per unit cell. Montmorillonite clays usually occur in thin flakes with a minimum thickness ranging from that of the unit layer of 10 Å to about 1/100 of their width and a maximum length of 1 to 2 m. The specific area of the smectites is even more illustrative of their activity. Their primary surface, excluding the interlayer zones, ranges from 50 to 100 m2g1, while the specific area including the interlayer zones, which are exposed to water penetration and to cation adsorption, can be as high as 840 m2g1 ([88], p. 31). Therefore, smectite minerals are electrochemically the most active. A special type of montmorillonite with commercial applications is the bentonite. It is the main clay mineral in the bentonite rock, which originates from the volcanic ash. It is the most active clay, it can form highly colloidal glue type suspensions, and it has the highest degree of swelling with a liquid limit of 500 and higher. Because of these properties, it has several practical applications, specifically as a stabilizing material in drilling in loose soils. That is why it is commonly 6O
�12
4 Si
�16
6O �10 2 (OH) 4 AI
�12
4O �10 2 (OH) 4 Si
�16
6O
�12
Net charge �44�44�0 Figure 2.17 Charges in a montmorillonite unit [88, Fig. 3.17].
9.6 to 21 Å
Chapter | 2 The Mineralogy and the Physicochemical Properties
39
referred to as driller’s mud. It should be noted that the name bentonite refers to the rock and not to the mineral itself. Montmorillonites constitute the most common mineral of the smectite group. The electric charge deficiency results from replacement in the octahedral sheet of every sixth trivalent aluminum atom (Al3+) by a divalent magnesium atom (Mg2+) with no substitution in the tetrahedral layer. The excess charge is balanced by exchangeable cations adsorbed not only on the surfaces of the clay particles, as in kaolinites, but also between unit cell layers. The chemical formula for the montmorillonite is [150, Appendix II]
Si8 (Al333 Mg067 )O20 (OH)4 ↓ Na 066
with the downward arrow indicating the deficiency and the balancing cation. In the case of sodium in the saponite subgroup, there is isomorphous substitution of silicon (Si4) by aluminum (Al3) in the tetrahedral sheet and iron (Fe3) for magnesium (Mg2) in the octahedral sheet. Its formula per unit cell is ([88], Chapter 3, Table 3.2):
(OH)4 (Si734 Al066 )Mg6 O20 ↓ Na 066
These formulas are indicative of the fundamental crystal structure of the clay mineral. A number of variations may exist within the same crystal structure ([88], Chapter 3, Table 3.2). 3. The mica group or nonexpending three-layer clays. Minerals of this group have the basic pyrophylite structure and consist of stacks of the three-layer units of Figure 2.16 held together by van der Waals forces. Since these forces are weak relatively to the primary valence bonds, cleavage parallel to the faces of the unit layers is easier. As a result, these minerals occur in the form of flakes. However, in the absence of interlayer swelling, these flakes are rather thick and large in comparison to clay particles of the kaolinite and smectite groups, and they do not disintegrate to a size typical for the clay minerals. For this reason they are classified as mica-type minerals. There are two main subgroups of the mica group: (a) the moscovite and illite and (b) the vermiculite. Their structures are schematically outlined in Figure 2.18. In moscovite the octahedral layer is gibbsite ([88], Section 3.8). About onefourth of the silicon positions in the pyrophylite unit are filled with aluminum, thus resulting in a negative charge balanced by potassium between the unit layers. Potassium atoms have a radius of 1.33 Å, which is about equal to the 1.32 Å radius of the holes in the bases of the tetrahedral sheets; therefore, the potassium
40
Cohesive Sediments in Open Channels
G K
K
Ca
K
K
K
K
G K
K
B
Mg
Exchangeable Ca
10 Å K Fixed
Mg
B
10 to 14 Å (14 Å as shown)
Ca
Mg
G B (a)
(b)
Figure 2.18 Structure of mica type minerals [88, Fig. 3.19]: (a) Moscovite and illite, (b) vermiculite
atoms can be buried into the structure of the pyrophylite. The unit cell formula then becomes (OH)4K2(Si6Al2)Al4O20. Moscovite is the dioctahedral end member of the micas with only Al3 in the octahedral layer. There are other types of micas, such as phlogopite or brown mica, which is the trioctahedral end of the mica subgroup with the octahedral layer filled entirely by magnesium. In between these two extreme clay minerals, there is the black mica or biotite, which is trioctahedral with the positions in the octahedral layer filled mostly with magnesium or iron. Since micas are relatively large flakes, they do not display colloidal behavior; however, they may contribute to the compressibility of the soil. On the other hand, although of a structure basically similar to that of micas, illites differ from the latter in certain details, which cause a drastically different formation of particles. Specifically, fewer of the silicon (Si4) positions are filled with aluminum (Al3) in the tetrahedral layer of illites and, consequently, there are fewer potassium atoms between the unit layers. There is also some randomness in the stacking of the unit layers. Some illites may contain magnesium and iron in the octahedral sheet. The charge deficiency in illites ranges from 1.3 to 1.5 electronic units per unit cell located primarily in the tetrahedral sheet ([88], Section 3.8). This deficiency is balanced by nonexchangeable potassium atoms between the unit layers, as in micas. However, because of the lesser charge, the number of the potassiums between the unit layers is lower in illites than in micas so that the bond between the former is weaker than that in the latter. Illite clays may expand and break into small flaky particles with a long axis ranging from 0.1 m to a few micrometers and a plate thickness as small as 30 Å. The only potassium ions that can be exchanged by other cations are the ones adsorbed on the external surfaces of the illite particles. In contrast, in smectites, such an exchange can take place also between the unit layers. The specific area of illites ranges from about 65 to 100 m2g1, and their cation exchange capacity (CEC) varies from 10 to 40 meq/100 g.
41
Chapter | 2 The Mineralogy and the Physicochemical Properties
Table 2.1 Clay Mineral
Lateral Dimensions (microns)
Thickness (microns)
Specific Area (m2/gm)
CEC (meq/ 100 gm)
Kaolinite
0.1–4
0.05–2
10–20
3–15
Illite
0.1–few
0.03
65–100
10–40
Montmorillonite
1–2
0.01– 1/100width
840
80–150
In vermiculites the octahedral layer is biotite with magnesium and calcium between the layers as charge compensating cations. In addition and very important, there is a double layer of water between the unit layers giving a basal spacing of 14 Å. The charge deficiency in vermiculites varies between 1 and 1.4 electronic units per unit cell, which is slightly less than that of the illites. In vermiculites the cations are exchangeable even between the unit layers due to spacing and the water layer; therefore, their cation exchange capacity (CEC) is higher than that in the illites, ranging from 100 to 150 meq/100 g. The specific area of the particles ranges from 40 to 80 m2g1, but the secondary or interlayer specific surface may be as high as 870 m2g1, that is, about equal to that of montmorillonites. Illite is the most common mineral found in soils of interest to sedimentation and to soil engineering. It is sometimes referred to as hydrous mica. In summary, the three most important clay minerals in the order of diminishing size and increasing physicochemical activity, as measured by their CEC, are kaolinites, illites and montmorillonites with the properties shown in Table 2.1. 4. Other clay minerals. In addition to the described primary clay minerals, there are some others less frequently encountered. Only the following most important of these minerals will be outlined: (a) Chlorites. The structure of the chlorites is similar to that of the vermiculites, and it is outlined in Figure 2.19. The mineral consists of alternate layers of mica and brucite with a basal spacing of 14 Å. An octahedral sheet replaces the water layer that exists between the mica layers. There is substitution of Mg2 by Al3, Fe2, and Fe3 as well as Al3 for Mg2 in the brucite layer. Chlorites are normally encountered mixed with other clay minerals in the soils in the form of plate-like microscopic grains of undetermined size and with a cation exchange capacity (CEC) between 40 and 100 meq/100 g ([88], Section 3.9). (b) Chain structure of clay minerals. These minerals include attapulgite and imogolite. They are formed of double chains of silica tetrahedra, and they have a thread-like or lath-like morphology with a diameter between 50 and 100 Å and a maximum length up to 5 m.
42
Cohesive Sediments in Open Channels
B–G B B–G
14 Å
B B–G
Figure 2.19 Structure of chlorite mineral [88, Fig. 3.23].
5. Amorphous (noncrystalline) materials. This group contains all materials encountered in soils without any definite crystalline structure. They are distinguished into (a) allophanes and (b) oxides. The first have no definite shape or composition, and their physical properties may vary widely. The most common oxides, contained to some degree in almost all soils, are hydroxides of aluminum, silicon, and iron. Oxides, as will be seen in Chapter 6, may affect the properties of the flocs and of the soils; however, at present, little is known about these effects. 6. Summary and general remarks. In this section the fundamental mineralogical properties of the basic clay minerals have been presented in a concise way and discussed. These properties are related to the interparticle physicochemical forces and to the mechanical properties of flocs and of cohesive sediment deposits; therefore, an understanding of them is essential in the study of the flocculation process and of the engineering behavior of cohesive soils. Specifically, clay mineralogy plays a dominant role in determining size, shape, and surface physicochemical properties of clay particles as well as their interaction with water. In engineering practice, mineralogical analysis is seldom performed except as a general description of the nature of the sediments. Instead, the mineralogical effects on the sediment and the soil properties can be represented by some parameters that can be readily evaluated by a few simple tests. The background material presented here is needed in order to appreciate the significance and reliability of these parameters, to properly control and interpret the related tests and, equally important, to communicate with the specialists in clay mineralogy and soil science whenever they may be involved in special cases.
Chapter | 2 The Mineralogy and the Physicochemical Properties
43
2.4 Origin and occurrence of clay minerals and formation of clay deposits Clays originate from the chemical weathering of rocks of various mineralogical compositions. The chief agent for weathering is the atmosphere, which uses energy derived from the heat of the sun. The gases of the atmosphere penetrate into the openings of the rocks and come into contact with a large internal surface of the latter. Oxygen and moisture are the most important ingredients of the atmosphere responsible for weathering. The process is furthermore enhanced by temperature variations ([88], Chapter 4, Section 4.2). Rock weathering can be distinguished into mechanical and chemical. The former amounts to a breaking of the rocks through mechanical forces generated mainly by temperature variations, frost action, stresses induced by drying and shrinking, abrasion, and by growth of plant roots in existing fractures. The product of mechanical weathering is sand and gravel. Chemical weathering, on the other hand, can eventually produce clays by chemically dissolving and removing various parent substances or through chemical reactions, bond breaking, and freeing cations. Practically all chemical weathering processes require the presence of water. The products of chemical weathering can readily be eroded by water and wind. In particular, clays, after having been subjected to flocculation, are transported by the flowing water and eventually are deposited in areas of favorable hydraulic conditions, such as flats, lakes, and estuaries. Such clay deposits can attain great depths. Throughout their geological history, clay deposits have been subjected to overburden pressures and high temperatures due to sediment accumulation, action of glaciers, and geotectonic processes. As a result, various degrees of metamorphism and changes of mechanical properties of the original clay deposits have taken place. Other clays, such as bentonites, have been formed essentially by alteration of in situ volcanic ash. Whatever the origin of the clay deposits, the geotectonic changes and the lifting of lands have brought above the water level older clay strata previously deposited under water and subsequently subjected to high overburden pressures. Erosion by rain, overland flow, waves, and rivers have exposed enormous masses of cohesive soils to the surface. The stability of these masses against further erosion may be very important in many cases. Such, for example, is the case of soil erosion by overland flow and of the stability of natural and artificial channels against scouring and/or siltation. If erosion continues, the eroded fine material will be transported and eventually deposited in other places, and the cycle will continue. In summary, there is a continuous process of decomposition, erosion, transport, deposition, and resuspension, known as the geologic cycle, outlined in Figure 2.20. A simplified version of the rock cycle is outlined in Figure 2.21 ([88], Chapter 4).
44
Cohesive Sediments in Open Channels
Denudation
Sediment formation
Deposition
Crustal movements Figure 2.20 Geologic cycle [88, Ch. 4].
Weathering environment Erosion, transportation, and deposition
Uplift and erosion
Metamorphicigneous environment
Sedimentary environment
Deep burial or igneous intrusion
Burial and heating Diagenetichydrothermal environment
Figure 2.21 Rock cycle [88, Ch. 4].
The geologic cycle consists of the following phases: (a) Denudation, which indicates the process of wearing down the land masses. (b) Weathering, which includes all the mechanical and chemical processes that break down in situ rock masses. (c) Transport, of eroded soil from its original position to the areas of deposition.
Chapter | 2 The Mineralogy and the Physicochemical Properties
45
(d) Deposition, which is the accumulation of sediment from its site to places where favorable conditions exist for deposition. (e) Sediment formation, which includes all the processes by which accumulated sediments are consolidated, densified, altered in composition, and eventually transformed into metamorphic rock. (f) Crystal movement, involving two processes: (i) the epirogenic movement that amounts to a gradual rising of unloaded areas and slow subsidence of basins where deposition takes place; (ii) the abrupt movements associated with faulting and earthquakes defined as tectonic movement. It should be noted, however, that more than one process may act simultaneously. For example, erosion can take place during periods of crystal movement; likewise, regional subsidence may occur during periods of filling of the basin with sediment. Hydraulic and soil engineers are almost exclusively interested in recent sediment deposits subjected to various degrees of consolidation and to a depth affected by externally imposed stresses. The engineering behavior of these surficial cohesive soils, and specifically their resistance to erosion, depends not only on the physicochemical and mineralogical characteristics of the clay itself, but also on the previous stress history of the deposits. Related to the latter is the fabric of the clay mass, which is the relative orientation and spacing of the clay particles. The soil fabric is determined first by the fabric of the depositing aggregates and, second, by the stresses and the degree of disturbance of the soil after deposition. It is reminded that the properties of the flocs and their aggregates depend also strongly on the flow-induced shear stresses during deposition. This aspect of sedimentation together with the types of flocculation and the soil fabric will be presented after the discussion in the following chapter of the forces between clay particles and of the process of floc formation.
Chapter 3
Forces between Clay Particles and the Process of Flocculation
3.1 Introductory remarks The basic properties of colloidal suspensions in water, the process of flocculation, the factors affecting flocculation, and some important and frequently encountered definitions were outlined and discussed in Section 2.1. The section showed that suspended particles may either repel or attract each other depending on the chemical properties of the solvent. This change from repulsion to attraction indicates the existence of both attractive and repulsive interparticle forces acting simultaneously on each colloidal particle. The interparticle forces are essentially secondary bond forces of an electrochemical nature and were outlined in Section 2.2. The origin of the repulsive forces lies in the net electric charge of the sediment particles. Although a sol is electrically neutral, suspended particles may have a negative or positive net electric charge. This charge can readily be demonstrated by applying an electric field to the sol, in which case negatively charged particles move toward the positive electrode and vice versa. This transport is known as electrophoresis. Almost without exception, clay mineral particles carry a negative electric charge on their faces and sometimes a positive charge on their edges. Positive charges occur in ferric hydroxide sols and other metal hydroxide sols. The latter, however, are not of direct importance to cohesive sediment transport mechanics, unless they interfere with the clay minerals themselves.
3.2 The electric charge and the double layer The electric charges on the surfaces and edges of clay mineral particles can be generated in two ways: (1) by isomorphous substitution, outlined in Section 2.3, and (2) by preferential adsorption of certain specific ions on the surfaces and the edges of clay particles. Cohesive Sediments in Open Channels Copyright © 2009
47
48
Cohesive Sediments in Open Channels
3.2.1 Isomorphous Substitution This origin of electric charges lies in the imperfections in the crystal lattice of the mineral. A single plane of atoms exists in both the tetrahedral and the octahedral sheets, which compose the clay mineral. The sheets are bonded together by strong primary bonds and the various structural units by secondary valance bonds. The latter are of several types and may be sufficiently weak to allow some chemical changes in these bonds. Specifically, a layer of one cation can be replaced by another of lower valence without altering the structure of the mineral, resulting in a deficiency of positive charge and in excess of negative charge. This substitution was discussed in Section 2.3. It was defined as isomorphous substitution, and it is a permanent feature of the clay mineral. Thus, in the tetrahedral sheet tetravalent, silicon, Si4, can be replaced by trivalent aluminum, Al3, and in the octahedral sheet, aluminum can be replaced by divalent magnesium, Mg2. The negative electric charge created by the isomorphous substitution is compensated by the adsorption on the surface layer of cations that are too large to be accommodated within the lattice. Isomorphous substitution is very common in montmorillonite but seldom in kaolinite. In the same section, an important property of the clay particles associated with the isomorphous substitution was introduced as the cation exchange capacity, symbolized as CEC. When clays are submerged in cation containing water, the compensating cations on the particle surfaces may be readily exchanged by other cations dissolved in the suspension known as exchangeable cations. This capacity can be determined chemically, and it is measured in milliequivalents per 100 g of clay mineral (meq/100 g). CEC is characteristic of the activity of the clays and their potential for flocculation. The following are some typical values of CEC: kaolinite, 3–8 meq/100 g; illite, 40 meq/100 g; montmorillonite, 80 meq/100 g. Montmorillonite is the most active and kaolinite the least active of these clays. The replacement depends on the valance, the relative abundance of ions of different types, and the ion size. Normally, small cations tend to replace larger ones. The following order is representative of the replacement ability of the cations [88, Chapter 7]:
Na , Li , K , Rb , Cs , Mg2 , Ca 2 , Ba 2 , Cu2 , Al3 , Fe3 , Th 4
The rate of exchange depends on the type of clay and on some other factors. It increases with decreasing physicochemical activity and vice versa. In kaolinite minerals the exchange is almost instantaneous; in illite clays a few hours are needed for complete exchange, whereas for montmorillonites it takes the longest time because the major part of the exchange capacity is located in the interlayer regions.
3.2.2 Preferential Adsorption This charge is created by the adsorption of specific ions on the particle surfaces. The valences of the lattice atoms that are exposed to the surface are not as completely compensated as those in the interior atoms of the lattice. These uncompensated
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
49
valances, referred to as broken bonds, are the ones responsible for the adsorption of the specific ions defined as peptizing ions, because they contribute to the creation of a deflocculated sol. The peptizing ions constitute an outer coating of the particle, and adsorption can take place by three processes: (1) by chemical bonds (chemisorption), (2) hydrogen bonds, and (3) by van der Waals attraction. The last two mechanisms are known as physical adsorption and are particularly responsible for the adsorption of organic ions. The adsorbed ions can be either negative or positive. Normally, negative stabilizing ions are adsorbed on the surfaces of the clay particles and positive ones on their edges. This last type of electric charge generation is important for kaolinites, where little isomorphous substitution takes place. There are several theories for ionic adsorption well discussed by Mitchell [88, Chapter 7]. In systems containing both monovalent and bivalent cations, the ratio of divalent to monovalent cations is much higher in the adsorbed layer than in the solution. A suggested practical guide is the Gapon equation:
1 Na Na k k (SAR)(meq / liter ) 2 2 (3.1) 2 1 [(Ca 2 Mg 2 ) / 2] 2 Ca Mg s e
The quantity on the right side in parentheses is defined as the sodium adsorption ratio (SAR) and can be determined by chemical analysis. The subscript s refers to the exchange complex of the clay and e to the equilibrium solution. k is the selectivity constant and has a value of 0.017 for most soils. Thus, if the composition of the pore fluid is known, the relative amounts of monovalent to divalent ions in the adsorbed cation complex can be estimated [88, p. 129]. Regardless of the origin of the electric charges, any clay particle in an ion containing water will attract ions of opposite charges. These ions are defined as counterions or gegenions. At the same time the ions tend to diffuse away from the surface because of their thermal activity. The diffusion takes place from a zone of high concentration to a zone of lower concentration in a way analogous to the diffusion of air molecules in the atmosphere. A clay particle can then be visualized as a thin rectangular plate surrounded on either side by a diffused layer of counterions with a distribution determined by the balance between the electrostatic attraction and their thermal activity. This layer is defined as the double layer and plays a dominant role in flocculation and in aggregate properties. The system of the particle and the double layer, defined as clay micelle, is electrically neutral and is outlined in Figure 3.1.
3.3 The theoretical formulation of the double layer 3.3.1 The General Case The mathematical description of the double layer is based on the distribution of the electrical potential and electric forces around the surfaces of the clay particles, and it is well presented by van Olphen [150, Appendix III]. The theoretical model was
50
Cohesive Sediments in Open Channels
Diffused Double layer Figure 3.1 The clay micelle [99].
first developed by Gouy in 1910 and Chapman in 1913. It is known as the GouyChapman theory, and the diffused ionic layer is also referred to as the Gouy layer. The theory has been developed for both planar (one-dimensional) and spherical surfaces. Only the one-dimensional case will be presented here based on the assumptions of (a) zero volume for ions (point charges), (b) uniform distribution of electric charge on the surface of the particle, and (c) infinitely large particle surface in comparison to its thickness (one-dimensional assumption). The derived equations have been shown to hold strictly for smectite particles and for monovalent solutions at low electrolyte concentrations, specifically less than 100 M/m3. Nevertheless, the model is adequate for the understanding of the flocculation process in general and of the behavior and properties of cohesive soils. A concise summary of the GouyChapman theory and some examples are presented in this section. A more complete treatment can be found in Ref. [88, Chapter 7] and in Ref. [150, Appendix III]. The counterions in Figure 3.1 are subjected to two opposing tendencies: electrostatic attraction by the negatively charged particle surface, which tends to bring them close to it, and diffusion due to thermal activity, which tends to drive them away from the particle surface. The result of these two opposing tendencies is a counterion distribution that decreases with distance from the surface reaching asymptotically the mean water concentration. The opposite is true for the negatively charged ions. However, at any distance within the double layer, both positive and negative ions are present. The ionic distribution within the double layer is given by the following equations developed by Boltzmann:
e * n n exp kT
(3.2a)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
e * n n exp kT
51
(3.2b)
where n and n are the local concentrations of the positive and negative ions, ∗ ∗ respectively; n and n are the corresponding concentrations within the equilibrium liquid and sufficiently far from the particle; is the electric potential with respect to the water medium at a distance x from the particle surface; T is the absolute temperature in degrees Kelvin (K); and k is the Boltzmann constant. On the SI system this constant has a value equal to 1.38 1023 J°K1. and are the valances of the positive and negative ions, respectively, and e is the elementary ionic charge equal to 16 1020 coulomb (C) or 4.8 1010 esu. It is observed that when is negative, n n and vice versa. The concentrations are expressed as numbers of ions per cm3. The local electric charge, e, is then given by
e en en
(3.3)
The relationship between the electric density and the electric potential was developed by Poisson on the assumption of uniform charge distribution and of a flat plate shape. Its general form for ions of several valances was presented and discussed by Mitchell [88, Chapter 7]. For the simplest case of monovalent ions and for n∗ n∗ n∗ Poisson’s equation obtains the simplified form d 2
dx
2
e e ed eo Dr
(3.4a)
in which d 2 /dx 2 is the rate of change of the strength of the electric field, d/dx , and ed is the dielectric constant of the medium given by
ed eo Dr
(3.5)
In Equation 3.5, Dr is the relative dielectric constant of the medium with respect to that of the vacuum and eo is the dielectric constant for vacuum equal to 8.8542 1012 C2J1m1 with C indicating coulomb units. In the esu system, eo has the value of 1/4. For water, Dr has the value of 80. The dielectric constant is defined in terms of the electric force, Fe, between two particles with electric charges, Q and Q9, and at a distance d by the relationship
Fe
QQ9 ed d 2
(3.6)
If the units in Equation 3.3 are in the esu-CGS system, Equation 3.4a can be written as 4 e Dr dx
d 2
2
(3.4b)
52
Cohesive Sediments in Open Channels
Introduction of Equation 3.2a, Equation 3.2b, and Equation 3.3 into Equation 3.4a leads to the following equation for the ionic distribution within the double layer: 2 n∗ e e e sinh kT 2 dx d
d 2
(3.7a)
for the SI-Coulomb system of units and 8 n∗ e e D sinh kT 2 dx r
d 2
(3.7b)
for the esu-CGS system of units. The preceding two equations can be transformed to a more convenient form by introducing the following dimensionless variables:
y
eo e ,z kT kT
and x
(3.8)
o is the potential energy at the surface of the particle and 2 n∗ e2 2 2 m ed kT
(3.9a)
8 n∗ e2 2 in cm2 Dr kT
(3.9b)
2 for the Coulomb-SI system, and 2
for the esu-CGS system. Introduction of Equation 3.8, Equation 3.9a, and Equation 3.9b to the appropriate Equation 3.7a or Equation 3.7b leads to the following dimensionless equation, which is independent of units: d2y
(d )2
sinh y
(3.10)
A first integration taking into consideration that for , y 0, and dy /d 0 – that is, the potential energy tends asymptotically to zero – yields
y dy 2sinh (2cosh y 2)½ 2 d
(3.11)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
53
The minus sign in Equation 3.11 was selected because y diminishes with increasing . Integration of Equation 3.11 subject to the boundary conditions for 0, o, or y z leads to the final equation: e
2
( 1 (e
) 1) e
e 2 1 e 2 1 e z
y
e
z
2
z
z
2
(3.12)
in which e is the basis for the Neperian logarithms. A more simplified equation can be derived for small surface potentials, that is, for o ,, 25 mV, z ,, 1, and y ,, 1, which for practical purposes is a case of interest. Under this condition, the hyperbolic sine in Equation 3.7a can be approximated by y, in which case the equation reduces to d 2 dx 2
2
(3.13)
whose solution is o exp(x )
(3.14)
Equation 3.14 means that for small surface values the electric potential is approximately an exponentially decaying function of the distance from the surface of the particle, as represented in Figure 3.2. A good measure of the double layer thickness is the center of mass of the electric charge, defined as the characteristic length of the double layer: xc
1
(3.15)
Of importance is the total charge on each side of the double layer, , per unit area of the particle surface. This charge can be evaluated as follows, considering that at infinity d/dx 0 :
∫
∞ 0
e dx
D d Dr ∞ d 2 dx r ∫ 2 0 4 4 dx x0 dx
(3.16a)
for csu-CGS units and
d ed dx x0
(3.16b)
54
Cohesive Sediments in Open Channels
Imaginary plane of infinite charge and potential
Surface of clay plate
y�
υeΦ kT
ξ0 �� x0 ξ�ξ0 �� (x�x0) ξ�� x ξ0 Figure 3.2 Variation of electric potential with distance from the particle surface [88, Fig. 7.4].
for C-SI units. It can be thus concluded that the initial slope of the potential distribution determines the charge on the surface of the particle. To express the derivative d/dx x 0 , we have to go back to Equation 3.11; substitute y and in terms of , , e, k, and T, as given by Equation 3.8; and use the chain rule of differentiation (d / d ) (d/dx )(dx /d ) (1/) d/dx . Next, in the resulting equation, we introduce from Equation 3.9a for the case of Equation 3.16a and from Equation 3.9b for the case of Equation 3.16b. Finally, the expressions thus derived for d/dx x 0 are introduced to Equation 3.16a or Equation 3.16b, depending on the units, to obtain
for the esu-CGS system, and
D n∗ ½ r ( eo ) 2 kT
(3.17a)
2n∗ e ½ d ( eo ) kT
(3.17b)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
55
for the SI-Coulomb system of units. For the general case of large values of electric potential, can be readily derived from Equation 3.11 and the proper substitutions to yield
n∗ D kT ½ ∗ ½ r ( 2 cosh z 2 )½ 2 n Dr kT sinh z 2 2
(3.18a)
for the esu-CGS system and
(
2 n∗ ed kT
½
)
½
z
½ ( 2 cosh z 2) ( 8 n∗ kT ed ) sinh (3.18b) 2
for the Coulomb-SI system. These equations reduce to Equation 3.17a and Equation 3.17b for sinh z z. In both equations, the coefficients of o are equal to the square root of 1/, that is, the characteristic length, which can also be interpreted as the capacity of the double layer. There are the following two cases of electric charges on the particle surfaces [150, Chapter 3]:
3.3.2 Surfaces of Constant Potential Such surfaces have a constant concentration of potential determining ions due to adsorbed cations. According to Equation 3.17 or Equation 3.18, the charge, , increases in proportion to the square root of the electrolyte concentration, n*, while, according to Equations 3.16a and Equation 3.16b, is also proportional to the gradient of the potential density, d/dx at x 0. Therefore, as indicated in Figure 3.3a, the initial slope of the potential curve increases with increaseing surface charge, o. Also the distance at which the becomes zero, that is, when the electrolyte concentration becomes about equal to n*, decreases with decreasing n*.
3.3.3 Surfaces of Constant Charge Density The density of the electric charge in this case is determined by the imperfections in the interior of the crystal lattice of the clay particle and is independent of the electrolyte concentration. According to Equation 3.16a and Equation 3.16b, the gradient d/dx remains constant since is a constant quantity. The variation of the electric potential with distance from the surface of the particle is schematically outlined in Figure 3.3b. In both cases, according to Equation 3.9, the double layer thickness, as measured by 1/, decreases with increasing n*. In the first case, starts from a constant value, o, but the slope d/dx increases with n*. In the second case, the potential energy, , decreases with n*, but the near-surface potential gradient remains constant so that the curves for high and low n* remain qualitatively parallel.
56
Cohesive Sediments in Open Channels
�0 � ��0 Electrical potential, �
�
High concentration (n*�)
0
0
�0 Electrical potential, �
�
Low concentration (n*)
(a)
(b)
� ∝ �1 � � ∝ �2 �0 � ��0
Distance from surface
�
��0
�∝�∝� �0 � ��0
� Low concentration (n*) High concentration (n*�)
0
0
Distance from surface
Figure 3.3 (a) Potential energy distribution in the double layer for constant surface potential; (b) potential energy distribution in the double layer for constant surface charge [150, Ch. 3].
3.3.4 Illustrative Applications The use of the developed equations for the estimate of the effect of electrolyte concentration on the double layer thickness are demonstrated by the following examples for both sets of units and for normal ambient conditions. The following constant quantities enter into the equations: Na Avogadro’s number 6.02 1023 ions/mole k Boltzman’s constant 1.38 1023 J°K1 or 1.38 1016 erg°K1 kT 4 1014 erg or 4 1021 J for room temperature e electron charge 1.602 1019 coulomb (C) or e 4.8 1010 esu dielectric constant ed eoDr 8.8542 1012 C2J1m1Dr or 1 ed Dr esu2erg1cm1 4 Dr is the relative dielectric constant, which for water has the value of 80.
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
57
The salt content in the water is normally given in terms of normality, N, that is, number of equivalents per liter. To convert it to ions per unit volume, n*, we have to multiply by Avogadro’s number 6.02 1023 moles per liter according to the relation: n* N 6.02 1023 (ions per mole) The cation exchange capacity (CEC) given in meqL1 should be divided by the specific area of the clay mineral to obtain the surface charge density, , which can be converted to coulombs through multiplication by Faraday number Fa 96.5. It follows from Equation 3.9 and Equation 3.15 that the center of mass of the double layer thickness is inversely proportional to the square root of normality, N, and inversely proportional to the valence, . Therefore, by increasing the first by an order of magnitude of 100, we reduce the thickness of the double layer by a factor of 10. Such an increase is very easy to achieve in relatively fresh river and lake waters with very low original salinity by adding small amounts of salt. The following numerical examples using both the esu-CGS and Coulomb-SI units will further demonstrate the use of the double layer equations. For a constant charge density, , of 15 coulombs cm2 on a flat surface and a normality N 0.001 of NaCl in the solvent water, determine the representative thickness of the double layer, 1/. Solution: The surface density in esu cm2 is
esu / cm2
(
)
mmC / cm 2 106 4 0.3 10 10 3.33 10
where on the right side is in coulombs cm2 and the denominator is the factor converting Coulomb units to esu units. For the present example, 15 106 0.3 1010 4.5 104 esu/cm2 and n* (concentration) (Avogadro’s number). The solute concentration, Cs, is expressed in number of equivalents per liter of solution. Therefore, it has to be multiplied by 103 to give the mole concentration per cm3. The ion content per cm3 then becomes n* N 103 6.02 1023 6.02N 1020 with N in expressed in terms of normality. For N 0.001, n* 6.02 1017. (8 )6.02 N 1020 ½ 4.8 1010 From Equation 3.9b, we obtain 80 4 1014 33 106 N . For 1 and N 0.001, we have 1.04 106 cm1 and 1/ 100 Å.
58
Cohesive Sediments in Open Channels
Next, from Equation 3.18a, it follows that z sinh ∗ 2 2 n Dr kT
1
2
20 14 2 6.02 N 10 80 4 10
0.0082 107 z sinh 2 N
1
2
0.0286 10−3
N
1
2
or
,
z where 0.3 104(15) 4.5 104, sinh 40.7, z 8.80. From the 2 definition of z in Equation 3.8, we obtain o
zkT z 300 103 25 220 mV , e
where mV is in mV for a valence, , equal to 1. For a solute concentration Cs 0.1 N. n* 6.02 1019 ions cm3 and 33 106 N 33 106 0.1 10.4 106 0.1 1 or xc 107 cm 10 Å That is a tenfold decrease of the double layer length. From Equation 3.18, we observe the sinh(z/2) is inversely proportional to the square root of n*. Therefore, a hundredfold increase of n* will decrease sinh(z/2) by a factor of 10. So for Cs 0.1 N, sinh(z/2) 4.07, z 4.22, and o 105 mV. 1 Likewise, for Cs 105 N, = xc = 1000 Å, sinh(z/2) 406.9, z 13.4, and o 335 mV. In conclusion, for a particle surface of constant electric charge, an increase of salinity by a factor of 100 decreases the thickness of the double layer by a factor of 10 and the surface potential by a factor of about 2. Both effects are very important for flocculation. The same problem will be now solved using the Coulomb-SI units. Here, n* has to be expressed in terms of ions m3; that is, n* N 6.02 1023 with N again expressed in equivalents per liter. Therefore, the right side of the last equation for n* has to be multiplied by 103 to become equivalents per m3; that is, n* 0.001 103 6.02 1023 or n* 6.02 1023. Also kT 4 1021 J and from Equation 2.29a for Coulomb-SI units, we obtain
59
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
1
2 6.02 1023 (16 1020 )2 12 2 1.043 108 m1 , 8.8542 1012 80 4 1021 1 xc 0.96 108 m 100 Å From Equation 3.18b the surface potential can be obtained as follows: ½ z 1 sinh 271 2 8 6.02 1023 4 1021 8.8542 1012 80 15mmC /cm 2 15 106 z 10 4 C /m 2 0.15C /m 2
and
z sinh 40.65, z 8.80 2 and from Equation 3.8 the following value of o is obtained as follows: o
8.80 4 1021 1 16 1020
0.22 V 220 mV
For surfaces of constant potential energy, the procedure is the same except that is evaluated from Equation 3.18a or Equation 3.18b for a known z given from Equation 3.9b.
3.4 Interaction of two flat double layers 3.4.1 Force and Energy Interaction The interaction of force and energy is based on the assumption of the validity of the Gouy-Chapman theory and of a double layer thickness much smaller than the particle size. The electric potential distribution between two such layers together with xc 1/ is indicated schematically in Figure 3.4, where 2d is the distance between the clay particles and d is the midway potential. The theory was first developed by Verwey and Overbeek [151] and is well presented in an abbreviated form by van Olphen in Appendix III of Ref. [150]. Using the dimensional variables defined by Equation 3.8, we indicate by ϕd the dimensionless potential at midpoint as
ϕd
ed kT
(3.19)
60
Cohesive Sediments in Open Channels
0
0
(x)
d
(x)
0
x d
2d
0
x
1/�
Figure 3.4 Schematic distribution of electric potential between two clay particles [151, Fig. 8].
The governing differential equation is still 3.10, but with the boundary conditions set at the midpoint between the plates, that is, at x xd d, dy/d 0, d, and y ϕd, because at that point the potential attains its minimum value. Equation 3.10 can also be written as 2 1 d dy d (cosh y ) 2 d d d
(3.20)
which, upon integration, introduction of the stated boundary conditions, and the fact that the gradients of y and normally have a negative value, takes the following form: 1 dy (2 cosh y 2 cosh ϕd ) 2 d
(3.21)
A second integration yields d
∫ d −d 0
ϕd
∫0
1
(2 cosh y 2 cosh ϕd )
2
dy
(3.22)
Equation 3.22 gives the midway dimensionless potential, ϕd, for any specific value of surface potential, as represented by z, and a given electrolyte concentration, which determines the value of in Equation 3.9a and Equation 3.9b. For most practical cases, knowledge of only ϕd is sufficient. The integral in Equation 3.22 is elliptic of the first order and cannot be evaluated in closed form, but only numerically. Table 3.1 and Figure 3.5, developed by Vervey and Overbeek, can be used for an estimate of d for given values of z and ϕd and, vice versa, an estimate of ϕd from z and xd. In Table 3.1 the first line gives d from the simplified Equation 3.24 that follows, and the second line is based on the numerical integration of
61
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
Table 3.1 κd as a function of z and ϕd according to Equation 3.23 and 3.24 [151, Table X, p.71] z 6, y 0.905 ϕd
0.1
0.25
xd (eq. 3.24) 4.28
0.50
3.365 2.67
z 8, y 0.964 1.0
2.0
0.1
0.25
0.50
1.0
2.0
1.98
1.29
4.35
3.43
2.73
2.04
1.35
xd (eq. 3.22) 4.280 3.354 2.635 1.876 1.061
d �
4.343 3.417 2.698 1.939 1.124
υe d kT
10 9
z � 10
8
z�8
7 6 5 z�6
4
z�4
3
z�2
2
z�1 z � 0.5
1 0
0
0.5
1.0
2.0
3.0
4.0
5.0 6.0 d
Figure 3.5 Graphical solution of Equation 3.22 [151, p. 72].
Equation 3.22 [151, Table IX] Table 3.3 [151, Table XI] gives d for a wider range of z and ϕd. In either case, interpolation will be needed, and the approximation is satisfactory for most practical cases [151, pp. 67–72]. For large values of electrolyte concentrations and, therefore, large values of d, Equation 3.11 for y ϕd can be reduced to the much simpler form [151, Appendix III]: z
d
ϕd 4 γ e
where γ
e 2 1 z
e 2 1
(3.23)
62
Cohesive Sediments in Open Channels
For this case, the two interacting double layers can be assumed as practically unperturbed and, therefore, linearly superimposed. Accordingly, the dimensionless midway potential is about double of that for a single particle at the same distance d, that is:
ϕd 2 yd 8γ ed
(3.24)
3.4.2 Illustrative Examples Case 1: Constant surface potential. As an application of Table 3.1 and Figure 3.5, we consider two particles with surface charge of the previous example in Section 3.3.4, except that instead of a constant charge, the potential is taken as constant and equal to 200 mV or 0.200 volts. We want to find the distance for which the potential d reaches the value of 0.05 volts for three electrolyte concentrations: 103, 105, and 101 N. In the first case, it was found that 1.043 108m1 and the center of mass of the double layer, xc, is 1/ 100 Å. For normal ambient conditions, e / kT ≈ 40 ; therefore, z 40 (0.200) 8.00 and for d 0.05 volts ϕd 40 0.05 2.00. From Table 3.1, we obtain d 1.124 and d 1.124 100 Å 112 Å. For the second concentration, 1/ 1000 Å and d 1124 Å, while in the last case of the highest concentration, d is about 12 Å. Very close but more approximate values can be verified from Figure 3.5. So far, we have discussed the case of constant surface potential. The same procedure could be followed when the surface charge rather than the surface potential is constant, as in Section 3.3.4. The latter case is the most common for clays whose surface charge is determined by isomorphous substitution. The approach could be the same as in the previous example but in reverse, like in case 2, which follows. Case 2: Constant surface charge. The data are the same as in the previous case but with a constant surface charge of 15 C cm2, as in the example in Section 3.3.4. For an electrolyte concentration of 103 N, it was found that o was about 0.220 V, z 8.8, 1.043 108m1, 1/ 100 Å, and e / kT 40 . The distance, d, at which the potential energy becomes d 0.05 V can be found, as in the case of constant surface potential. ϕd is equal to 2. From Table 3.1, at z 9 from extrapolation as an approximation and in the column of ϕd 2, we obtain d 1.14 and d 114 Å. For electrolyte concentration of 101 N, was found to be equal to 10.4 106, 1/ equal to about 10 Å, z 4.22, and o equal to 105 mV. ϕd is still the same at 2. Then from Table 3.3 or from Equation 3.22 for z approximated by 4, we obtain d 0.884 and d 8.8 Å. Finally, for electrolyte concentration equal to 105 N, 1/ 1000 Å, z 13.4, and o 335 mV, while ϕd still has the value of 2. From Table 3.3 and Equation 3.22, we then find d 1.15 and a distance d 1150 Å.
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
63
Similar approximate results can be found from Figure 3.5, as in the case of constant potential. One can easily observe from both Table 3.3 of Section 3.4.3 and Figure 3.5 that the effect of z on d becomes negligible for z higher than 9. Moreover, xc for a single particle is the same for both constant surface charge and constant surface potential because it depends only on the electrolyte concentration. A final important observation is the comparison of d with xc, that is, the center of mass and the representative double layer length. For the 103 N case, d somewhat exceeds xc, the center of mass of the double layer; for the 101 N case, d is somewhat shorter than xc; while for the last case of 105 N, d is longer than xc. For the latter case of constant surface charge, van Olphen developed a different approach [150, p. 260]. Using Equation 3.11, he computed dy/d for several values of the charge, , from Equation 3.16a and Equation 3.8, that is d
∫ e dx 0
D n∗ kT r 2
1
2
Dr d D dy kT r 4 dx x0 4 d 0 e
(2 cosh z 2 cosh ϕd )
1
2
(3.25)
Therefore, for a constant surface charge and, consequently, constant electrolyte concentration, n*, (n*)½(2 cosh z 2 cosh ϕd)½ is constant and independent of d. The problem of evaluating ϕd for different distances, d, and for a given electrolyte concentration, amounts to finding a set of values for z and ϕd that satisfy the following equation:
dy d
0
(2 cosh z 2 cosh ϕd )
1
2
2 D r n∗ kT
1
2
constant (3.26)
For the values of Dr and kT, given for normal conditions and used in the illustrative example in Section 3.3.4, we obtain the derivativze (dy/dξ)ξ50 5 4peυs/DrkTk, which, upon substitution of the appropriate values, becomes (dy/dξ)ξ50 5 1885(s/k). In Equation 3.26, is in esu cm2, in cm1, Dr in (esu)2 (erg)1 cm1, and kT in erg. van Olphen developed tables for dy/d of 840, 560, 280, 187, 94, 84, 56, 28, 18.7, 9.4, 8.4, 5.6, 2.8, 1.87, and 0.94 [150, Appendix III]. The table for the first three values of (dy/d) is reproduced here as Table 3.2. The procedure is as follows. From the known charge, , in esu cm2 and the computed value on n* from the electrolyte concentration, we evaluate (dy/d) at the surface of the particle. We then enter the table at the two values of (dy/dξ)ξ50 containing the actual value. For each value and for the closest value of z, going horizontally, we determine both ϕd and d, from which the
64
Cohesive Sediments in Open Channels
Table 3.2 Surface potential, z, and midway potential, ϕd, for three surface charges, , at various electrolyte concentrations, , and plate distance, 2d [150, p. 262] (dy/d)0 9.4 d
ϕd
(dy/d)0 8.4 z
(dy/d)0 5.6
d
ϕd
z
d
ϕd
z
8.7760
0.001 4.50
15.659
1026
4.28
15.545
1026
3.51
6.4734
0.01
4.50
13.356
25
10
4.28
13.242
25
10
3.51
3.2424
0.25
4.51
11.054
1024
4.28
10.940
1024
3.51
2.5240
0.5
4.51
8.7513
10
4.28
8.637
23
10
3.51
2.0868
0.75
4.51
6.4487
0.01
4.28
6.335
0.01
3.51
1.7650
1.0
4.52
4.1436
0.1
4.28
4.030
0.1
3.51
1.5080
1.25
4.52
2.4993
0.5
4.29
2.386
0.5
3.51
1.2939
1.5
4.53
1.7406
1.0
4.30
1.6292
1.0
3.54
0.9540
2.0
4.56
1.2700
1.5
4.32
1.1621
1.5
3.58
0.6979
2.5
4.61
0.9309
2.0
4.36
0.8289
2.0
3.66
0.5029
3.0
4.69
0.6760
2.5
4.41
0.5824
2.5
3.78
0.3551
3.5
4.80
0.4827
3.0
4.51
0.4003
3.0
3.94
0.2449
4.0
4.96
0.2684
3.5
4.17
0.1646
4.5
5.18
0.1756
4.0
4.45
0.1079
5.0
5.47
0.1124
4.5
4.80
23
estimate of d follows. Interpolation will be required for the best approximation of the last two parameters.
3.4.3 P otential Energy of Interaction between Two Flat Double Layers By definition, the potential energy of interaction is the work necessary to bring two charged particles from infinity to a distance 2d. A quantitative expression for this energy is important in determining the net force between the two interacting double layers because the net force of interaction is the negative derivative of the potential energy. Before proceeding with the attractive potential due to the van der Waals forces, we will elaborate a little on the repulsive potential, VR, at the middle of the distance 2d between the two layers. Associated with the latter is the free energy, indicated by VF, which, according to Verwey and Overbeek, “represents the amount of work associated with some isothermal and reversible process of building up the double layer” [150, Chapter 3]:
VR 2(VF d VF ∞ )
(3.27)
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
65
where VF is the free energy per cm2 of the single (noninteracting) double layer and VFd is the free energy per cm2 of the double layer when the distance is 2d. An analysis for the free energy was developed by Verwey and Overbeek and is well presented in Chapter 5 of Ref. [151] with a table and some graphs for an estimate of VR for constant surface electric potential. This table is reproduced here as Table 3.3. For a given electrolyte concentration and known and z, the table gives the function f ( z, ϕd ) ( 2 /)VR in 107 dynes. All the variables in the table are expressed in esu-CGS units. Therefore, the value of VR obtained is in erg cm2. For given concentration, can be evaluated from Equation 3.9a. z is determined by the constant surface charge. Then entering Table 3.3 at the appropriate z and proceeding horizontally, we determine both ϕd and d for any particular value of the dimensionless midway potential ϕd. Repeating the procedure for sufficient number of values of ϕd, we can obtain a graph of potential versus distance 2d or half distance d. Remember that the same table can be used instead of Table 3.1 for a more accurate estimate of the distance between two flat particles at which the electric potential has a prescribed value, as in the examples of Section 3.4.2. One should observe that for any given value of f(z,ϕd), VR is inversely proportional to the square of the ion valence . Therefore, for a divalent ion at the same distance d, VR will have one-quarter of the value corresponding to monovalent ions with the other variables remaining the same.
3.4.4 Illustrative Examples Example 1. Let us consider two flat plates under the data and the conditions of the illustrative example in Section 3.3.4, which is for a concentration of 103 N and a surface charge of 15 C cm2 or 4.5 104 esu cm2. The following values were found for standard conditions and for a value of Dr of 80, which is quite close to 78.55 of Table 3.3: kT 4 1014 erg 1.04 106 and 1/ 106 cm or 100 Å z 8.8 then for a ϕd equal to 2, for example, and a z about equal to 9, we obtain f(z, ϕd) 1.26 107 dynes and d 1.139. The following values of VR and d are then computed: V R 1.26 107 106 0.126 ergs cm2 d 1.139 106 114 Å.
Table 3.3 Repulsive potential f (z, ϕd)VR of double layers around two parallel flat plates at a distance 2d for different values of z and ϕd 25 °C temperature and a value of Dr equal to 78.55 [151, p. 82] ϕd z z 10 f(ϕd·z) 268.3 κd z 9
0.0000
f(ϕd·z) 161.5 κd
0.0000
z 8
f(ϕd·z) 96.52
z 7
f(ϕd·z) 57.13
κd κd z 6
0.0000
0.0000
f(ϕd·z) 33.27 κd
0.0000
z 5
f(ϕd·z) 18.83
z 4
f(ϕd·z) 10.13
z 3
f(ϕd·z) 4.962
z 2
f(ϕd·z) 1.993
κd κd κd κd z 1
0.0000
0.0000
0.0000
0.0000
f(ϕd·z) 0.4575 κd
0.0000
8
7
6
5
4
3
2
1
0.5
0.25
0.1
127.1
75.4
44.1
25.4
14.1
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0204
0.0437
0.0813
0.143
0.244
0.412
0.690
1.148
1.962
2.721
3.440
4.366
95.6
76.3
44.3
25.4
14.1
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0337
z 0.1
z 0.3
z 0.6
228.2
192.6
160.0
0.00434
0.00836
0.0134
135.2
115.2
9
0.0073
0.0138
0.0221
0.0721
0.134
0.236
0.403
0.679
1.139
1.953
2.712
3.431
4.357
80.56
68.56
56.60
44.8
25.4
14.1
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0555
0.0121
0.0227
0.0364
0.119
0.221
0.388
0.665
1.124
1.939
2.698
3.417
4.343
47.46
40.18
32.89
25.8
14.17
7.36
3.42
1.26
0.26
0.06
0.015
0.0023
0.0915
0.0199
0.0375
0.0600
0.196
0.364
0.641
1.101
1.915
2.674
3.393
4.318
27.47
23.04
18.66
14.38
7.39
3.42
1.26
0.26
0.06
0.015
0.0023
0.1509
0.0327
0.0618
0.0990
0.323
0.601
1.061
1.876
2.635
3.354
4.280
15.32
12.69
10.07
7.52
3.43
1.26
0.26
0.06
0.015
0.0023
0.2488 0.533
0.995
1.811
2.570
3.290
4.215
1.26
0.26
0.06
0.015
0.0023
0.0541
0.1018
0.1632
8.07
6.51
4.97
3.50
0.4105 0.884
0.0891
0.1680
0.2692
3.793
2.913
2.061
1.291 0.681
1.702
2.462
3.181
0.4107
0.26
0.06
0.015
0.0023
0.1471
0.2774
0.4455
1.518
2.280
2.998
3.924
1.413
0.966
0.584
0.265
0.06
0.015
0.0023
1.178
0.2435
0.4643
0.751
1.915
2.680
3.608
0.280
0.135
0.0348
0.063
0.015
0.0023
0.4353
0.855
1.537
1.283
2.035
2.971
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
67
For the same data but for a divalent cation, the value of becomes, according to Equation 3.9a, twice that for monovalent ions, that is, about 2 106; and the value of 1/ or xc becomes equal to 50 Å or 0.5 108 cm. z remains the same and equal to about 9. Then for ϕd 2, f(z, ϕd) and d have the same values as for monovalent cations so that VR 1,26 107 2 106 22 0.063 102 erg cm2 at a distance of 50 Å, which is one-half of the previous value. This change is indicative of the rapid rate of decrease of the repulsive potential with increasing ionic valence. As already mentioned, the outlined table and procedures are valid only for constant surface electric potential, whereas in the most common clay minerals, in which the charge is generated by isomorphous substitution, the surface charge is constant rather than the surface potential. For a direct computation of the interaction force between two double layers, van Olphen outlined an analysis by Langmuir [71] based on the concept of the osmotic pressure [150, Appendix III]. The assumption was that the repulsive force is given by the osmotic pressure at the midpoint of the distance between the two plates, which is determined by the excess of the midway ionic concentration. Using Equation 3.2a, Equation 3.2b, Equation 3.3, and the dimensionless electric potential, ϕd, from Equation 3.19 for equal valence and concentration of cations and anions, the total excess concentration of cations at the midpoint is given by
ϕdd
n* (eϕd 1) n∗ (e
1) 2 n∗ (2 cosh ϕd 1)
(3.28)
The repulsive force per unit area is
pre 2 n∗ kT (cosh ϕd 1)
(3.29)
In contrast to the first method based on the free potential energy, Equation 3.29 is valid for both constant charge and constant potential surfaces. Thus, the pressure pre can be evaluated from the solute ion concentration and the midpoint potential for a single surface. For small interactions and high ion concentrations, the analysis can be further simplified by introducing Equation 3.23 and Equation 3.24 for ϕd into Equation 3.29 and integrating to find the following expression for the repulsive potential energy per unit surface:
64 n∗ kT γ 2 exp(2d ) V R
(3.30)
Example 2. As an illustration of the application of Equation 3.30, we will compute the interaction energy of two double layers at a distance of 100 Å for an electrolyte concentration 103 N of NaCl and a surface charge density 0.50 104 esu cm2 assuming week interaction.
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Cohesive Sediments in Open Channels
In the example in Section 3.3.4 for the same electrolyte concentration, was 106, 1/ xc was 100 Å or 106 cm, with all the other variables remaining the same. Then, according to Equation 3.26, we have dy d
0
0.50 10 4 1885 1885 9.42 106
Next, we enter Table 3.2, [150, Appendix III] in the column for 9.4 (a good approximation) to d of 0.954, which is the closest to the actual 1, and moving horizontally, we find the values ϕd 2 and z 4.56. Equation 3.23 then gives
γ
e e
z z
2 2
1 1
e2.28 1 e2.28 1
0.81 and γ 2 0.663
e2z e2 0.1353 VR
64 103 103 6.02 1023 0.4 1013 106
(0.663 0.1353)
0.138 erg/cm 2 and pre 2 103 103 6.02 1023 0.4 1013 (3.762 1) 13.3 10 4 dynes/cm 2 0.127 atm
3.4.5 P otential Energy of Interaction of Two Flat Double Layers Due to van der Waals Forces As explained earlier, the van der Waals forces are the dominant attractive forces with a long range of action. For very small distances, particularly when two particles are about to come into contact, other factors, such as residual chemical valence fields, may also have a measurable influence. However, as far as flocculation is concerned and for the reasons to be analyzed shortly, the long-range effect is the one of primary importance. Moreover, these forces are universal acting between all atoms, molecules, and ions and are generated by the mutual influence of the electronic motion on two interacting atoms. The fluctuating charge in one atom causes, as a first approximation, a fluctuating dipole in the second atom. The latter induces a dipole in the first, and the net result of the interaction of these dipoles is attraction. The attractive potential between two dipoles is proportional to r3, where r is the distance between the molecules. The attractive force, therefore, is proportional to r4. Then, according to London’s
Chapter | 3 Forces between Clay Particles and the Process of Flocculation
69
theory, the van der Waals forces and attractive potential between two atoms will be proportional to r7 and r6, respectively. The attractive potential energy for two atoms is given by the equation VA p
r6
(3.31)
in which the constant depends on the properties of the atoms or molecules under consideration [151, Chapter 6]. The attractive potential between two particles can then be evaluated by integrating that potential between all pairs of atoms [151, Chapter 6 and Chapter 11]. The integration involves some constants that are very difficult to evaluate precisely. For particles of the order of clay particles, and more specifically between two unit layers of three-layer clay minerals, the following approximation was accepted as satisfactory by van Olphen [150, Appendix III]:
Aa 1 1 2 VA 48 d 2 (d )2 2 d 2
(3.32)
is the plate thickness and Aa is a constant whose value depends on the number of atoms contained in 1 cm3 of the substance building up the particles, and 2d is the distance between the two particles. For practical purposes Aa has been approximately accepted as 1012. Equation 3.32 can be reduced to the following simpler approximations for special cases [151, Chapter 6]:
VA VA
2 Aa 32 d 4
for d >>
Aa 1 7 2 2 for d
