Cavitation in Non-Newtonian Fluids: With Biomedical and Bioengineering Applications

Cavitation in Non-Newtonian Fluids Emil-Alexandru Brujan Cavitation in Non-Newtonian Fluids With Biomedical and Bioe...

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Cavitation in Non-Newtonian Fluids

Emil-Alexandru Brujan

Cavitation in Non-Newtonian Fluids With Biomedical and Bioengineering Applications

123

Emil-Alexandru Brujan University Politechnica of Bucharest Department of Hydraulics Spl. Independentei 313, sector 6 060042 Bucharest Romania [email protected]

ISBN 978-3-642-15342-6 e-ISBN 978-3-642-15343-3 DOI 10.1007/978-3-642-15343-3 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2010935497 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Cavitation is the formation of voids or bubbles containing vapour and gas in an otherwise homogeneous fluid in regions where the pressure falls locally to that of the vapour pressure corresponding to the ambient temperature. The regions of low pressure may be associated with either a high fluid velocity or vibrations. Cavitation is an important factor in many areas of science and engineering, including acoustics, biomedicine, botany, chemistry and hydraulics. It occurs in many industrial processes such as cleaning, lubrication, printing and coating. While much of the research effort into cavitation has been stimulated by its occurrence in pumps and other fluid mechanical devices involving high speed flows, cavitation is also an important factor in the life of plants and animals, including humans. Several books and review articles have addressed general aspects of bubble dynamics and cavitation in Newtonian fluids but there is, at present, no book devoted to the elucidation of these phenomena in non-Newtonian fluids. The proposed book is intended to provide such a resource, its significance being that non-Newtonian fluids are far more prevalent in the rapidly emerging fields of biomedicine and bioengineering, in addition to being widely encountered in the process industries. The objective of this book is to present a comprehensive perspective of cavitation and bubble dynamics from the stand point of non-Newtonian fluid mechanics, physics, chemical engineering and biomedical engineering. In the last three decades this field has expanded tremendously and new advances have been made in all fronts. Those that affect the basic understanding of cavitation and bubble dynamics in non-Newtonian fluids are described in this book. It is essential to understand that the effects of non-Newtonian properties on bubble dynamics and cavitation are fundamentally different from those of Newtonian fluids. Arguably the most significant effect arises from the dramatic increase in viscosity of polymer solutions in an extensional flow, such as that generated about a spherical bubble during its growth or collapse phase. Specifically, polymers, which are randomly-oriented coils in the absence of an imposed flow-field, are pulled apart and may increase their length by three orders of magnitude in the direction of extension. As a result, the solution can sustain much greater stresses, and pinching is stopped in regions where polymers are stretched. Furthermore, many biological fluids, such as blood, synovial fluid, and saliva, have non-Newtonian properties and

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Preface

can display significant viscoelastic behaviour. Therefore, this is an important topic because cavitation is playing an increasingly important role in the development of modern ultrasound and laser-assisted surgical procedures. Despite their increasing bioengineering applications, a comprehensive presentation of the fundamental processes involved in bubble dynamics and cavitation in non-Newtonian fluids has not appeared in the scientific literature. This is not surprising, as the elements required for an understanding of the relevant processes are wide-ranging. Consequently, researchers who investigate cavitation phenomenon in non-Newtonian fluids originate from several disciplines. Moreover, the resulting scientific reports are often narrow in scope and scattered in journals whose foci range from the physical sciences and engineering to medical sciences. The purpose of this book is to provide, for the first time, an improved mechanistic understanding of bubble dynamics and cavitation in non-Newtonian fluids. The book starts with a concise but readable introduction into non-Newtonian fluids with a special emphasis on biological fluids (blood, synovial liquid, saliva, and cell constituents). A distinct chapter is devoted to nucleation and its role on cavitation inception. The dynamics of spherical and non-spherical bubbles oscillating in non-Newtonian fluids are examined using various mathematical models. One main message here is that the introduction of ideas from theoretical studies of nonlinear acoustics and modern optical techniques has led to some major revisions in our understanding of this topic. Two chapters are devoted to hydrodynamic cavitation and cavitation erosion, with special emphasis on the mechanisms of cavitation erosion in non-Newtonian fluids. The second part of the book describes the role of cavitation and bubbles in the therapeutic applications of ultrasound and laser surgery. Whenever laser pulses are used to ablate or disrupt tissue in a liquid environment, cavitation bubbles are produced which interact with the tissue. The interaction between cavitation bubbles and tissue may cause collateral damage to sensitive tissue structures in the vicinity of the laser focus, and it may also contribute in several ways to ablation and cutting. These situations are encountered in laser angioplasty and transmyocardial laser revascularization. Cavitation is also one of the most exploited bioeffects of ultrasound for therapeutic advantage. In both cases, the violent implosion of cavitation bubbles can lead to the generation of shock waves, high-velocity liquid jets, free radical species, and strong shear forces that can damage the nearby tissue. Knowledge of these physical mechanisms is therefore of vital importance and would provide a framework wherein novel and improved surgical techniques can be developed. This field is as interdisciplinary as any, and the numerous disciplines involved will continue to overlook and reinvent each others’ work. My hope in this book is to attempt to bridge the various communities involved, and to convey the interest, elegance, and variety of physical phenomena that manifest themselves on the micrometer and microsecond scales. This book is offered to mechanical engineers, chemical engineers and biomedical engineers; it can be used for self study, as well as in conjunction with a lecture course. I would like to gratefully acknowledge the advice and help I received from Professor Alfred Vogel (Institute of Biomedical Optics, University of Lübeck),

Preface

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Professor Yoichiro Matsumoto (University of Tokyo), Professor Gary A. Williams (University California Los Angeles), and Professor J.R. Blake (University of Birmingham). I also appreciate fruitful conversations with and kind help I received from Professor Werner Lauterborn (Göttingen University), Dr. Teiichiro Ikeda (Hitachi Ltd), Dr. Kester Nahen (Heidelberg Engineering GmbH), and Peter Schmidt. Bucharest, Romania June 2010

Emil-Alexandru Brujan

Contents

1 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . 1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Newtonian Fluids . . . . . . . . . . . . . . . 1.1.2 Non-Newtonian Fluids . . . . . . . . . . . . 1.2 Non-Newtonian Fluid Behaviour . . . . . . . . . . . 1.2.1 Simple Flows . . . . . . . . . . . . . . . . . 1.2.2 Intrinsic Viscosity and Solution Classification 1.2.3 Dimensionless Numbers . . . . . . . . . . . 1.2.4 Constitutive Equations . . . . . . . . . . . . 1.3 Rheometry . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Shear Rheometry . . . . . . . . . . . . . . . 1.3.2 Extensional Rheometry . . . . . . . . . . . . 1.3.3 Microrheology Measurement Techniques . . . 1.4 Particular Non-Newtonian Fluids . . . . . . . . . . . 1.4.1 Blood . . . . . . . . . . . . . . . . . . . . . 1.4.2 Synovial Fluid . . . . . . . . . . . . . . . . . 1.4.3 Saliva . . . . . . . . . . . . . . . . . . . . . 1.4.4 Cell Constituents . . . . . . . . . . . . . . . 1.4.5 Other Viscoelastic Biological Fluids . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 4 7 7 12 13 15 25 25 29 32 34 34 37 40 41 43 43

2 Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Nucleation Models . . . . . . . . . . . . . . . . . . . . . . . 2.2 Nuclei Distribution . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Distribution of Cavitation Nuclei in Water . . . . . . . 2.2.2 Distribution of Cavitation Nuclei in Polymer Solutions 2.2.3 Cavitation Nuclei in Blood . . . . . . . . . . . . . . . 2.3 Tensile Strength . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49 49 53 53 54 55 57 59

3 Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Spherical Bubble Dynamics . . . . . . . . . . . . . . . . . . . . . 3.1.1 General Equations of Bubble Dynamics . . . . . . . . . .

63 63 63

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65 81 82 86 91 92 98 98 99 101 107 110 112

4 Hydrodynamic Cavitation . . . . . . . . . . . . . . . . . . . . . . 4.1 Non-cavitating Flows . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Drag Reduction . . . . . . . . . . . . . . . . . . . . . 4.1.2 Reduction of Pressure Drop in Flows Through Orifices 4.1.3 Vortex Inhibition . . . . . . . . . . . . . . . . . . . . 4.2 Cavitating Flows . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Cavitation Number . . . . . . . . . . . . . . . . . . . 4.2.2 Jet Cavitation . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Cavitation Around Blunt Bodies . . . . . . . . . . . . 4.2.4 Vortex Cavitation . . . . . . . . . . . . . . . . . . . . 4.2.5 Cavitation in Confined Spaces . . . . . . . . . . . . . 4.2.6 Mechanisms of Cavitation Suppression by Polymer Additives . . . . . . . . . . . . . . . . . . 4.3 Estimation of Extensional Viscosity . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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117 118 118 121 123 123 124 126 129 134 143

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148 149 150

5 Cavitation Erosion . . . . . . . . . . . . . . . . . . . . . . . 5.1 Cavitation Erosion in Non-Newtonian Fluids . . . . . . . 5.2 Mechanisms of Cavitation Damage in Newtonian Fluids 5.3 Reduction of Cavitation Erosion in Polymer Solutions . . References . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

3.3 3.4 3.5

3.1.2 The Equations of Motion for the Bubble Radius . 3.1.3 Heat and Mass Transfer Through the Bubble Wall 3.1.4 Experimental Results . . . . . . . . . . . . . . . 3.1.5 Bubbles in a Sound-Irradiated Liquid . . . . . . . Aspherical Bubble Dynamics . . . . . . . . . . . . . . . 3.2.1 Bubbles Near a Rigid Wall . . . . . . . . . . . . 3.2.2 Bubbles Between Two Rigid Walls . . . . . . . . 3.2.3 Bubbles in a Shear Flow . . . . . . . . . . . . . 3.2.4 Shock-Wave Bubble Interaction . . . . . . . . . Bubbles Near an Elastic Boundary . . . . . . . . . . . . Bubbles in Tissue Phantoms . . . . . . . . . . . . . . . Estimation of Extensional Viscosity . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

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155 156 163 171 172

6 Cardiovascular Cavitation . . . . . . . . . . . . . . . . . . . 6.1 Cavitation for Ultrasonic Surgery . . . . . . . . . . . . . . 6.1.1 Sonothrombolysis . . . . . . . . . . . . . . . . . . 6.1.2 Ultrasound Contrast Agents . . . . . . . . . . . . . 6.2 Cavitation in Laser Surgery . . . . . . . . . . . . . . . . . 6.2.1 Transmyocardial Laser Revascularization . . . . . 6.2.2 Laser Angioplasty . . . . . . . . . . . . . . . . . . 6.3 Cavitation in Mechanical Heart Valves . . . . . . . . . . . 6.3.1 Detection of Cavitation in Mechanical Heart Valves

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175 175 175 177 199 199 202 206 206

Contents

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6.3.2 Mechanisms of Cavitation Inception in Mechanical Heart Valves . . . . . . . . . . . . . . . . . . . . . 6.3.3 Collateral Effects Induced by Cavitation . . . . . . Gas Embolism . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Treatment Strategies for Gas Embolism . . . . . . 6.4.2 Gas Embolotherapy . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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208 209 210 211 211 214

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7 Nanocavitation for Cell Surgery . . . . . . . . . . . . . . 7.1 Cavitation Induced by Femtosecond Laser Pulses . . . 7.1.1 Numerical Simulations . . . . . . . . . . . . . 7.1.2 Experimental Results . . . . . . . . . . . . . . 7.2 Cavitation During Plasmonic Photothermal Therapy . . 7.2.1 Nanoparticles and Surface Plasmon Resonance 7.2.2 Bubble Dynamics . . . . . . . . . . . . . . . . 7.2.3 Biological Effects of Cavitation . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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225 226 226 228 229 231 233 241 245

8 Cavitation in Other Non-Newtonian Biological Fluids . . 8.1 Cavitation in Saliva . . . . . . . . . . . . . . . . . . . 8.1.1 Cavitation During Ultrasonic Plaque Removal . 8.1.2 Cavitation During Passive Ultrasonic Irrigation of the Root Canal . . . . . . . . . . . . . . . . 8.1.3 Cavitation During Laser Activated Irrigation of the Root Canal . . . . . . . . . . . . . . . . 8.1.4 Cavitation During Orthognathic Surgery of the Mandible . . . . . . . . . . . . . . . . . 8.2 Cavitation in Synovial Liquid . . . . . . . . . . . . . . 8.3 Cavitation in Aqueous Humor . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Non-Newtonian Fluids

A fluid can be defined as a material that deforms continually under the application of an external force. In other words, a fluid can flow and has no rigid three-dimensional structure. An ideal fluid may be defined as one in which there is no friction. Thus the forces acting on any internal section of the fluid are purely pressure forces, even during motion. In a real fluid, shearing (tangential) and extensional forces always come into play whenever motion takes place, thus given rise to fluid friction, because these forces oppose the movement of one particle relative to another. These friction forces are due to a property of the fluid called viscosity. The friction forces in fluid flow result from the cohesion and momentum interchange between the molecules in the fluid. The viscosity of most of the fluids we encounter in every day life is independent of the applied external force. There is, however, a large class of fluids with a fundamental different behaviour. This happens, for example, whenever the fluid contains polymer macromolecules, even if they are present in minute concentrations. Two properties are responsible for this behaviour. On one hand, polymers change the viscosity of the suspension by changing their shape depending on the type of flow. On the other hand, polymer have long relaxation times associated with them, which are on same order as the time scale of the flow, and allow the polymers to respond to the flow with a corresponding time delay. Other complex systems consisting of several phases, such as suspensions or emulsions and most of the biological fluids, behave in a similar manner. In the following, we will focus on some of the most important aspects of the flow of this class of fluids.

1.1 Definitions 1.1.1 Newtonian Fluids An important parameter that characterize the behaviour of fluids is viscosity because it relates the local stresses in a moving fluid to the rate of deformation of the fluid element. When a fluid is sheared, it begins to move at a rate of deformation inversely proportional to viscosity. To better understand the concept of shear viscosity we assume the model illustrated in Fig. 1.1. Two solid parallel plates are set on the top E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_1,

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1 Non-Newtonian Fluids

Fig. 1.1 Illustrative example of shear viscosity

of each other with a liquid film of thickness Y between them. The lower plate is at rest, and the upper plate can be set in motion by a force F resulting in velocity U. The movement of the upper plane first sets the immediately adjacent layer of liquid molecules into motion; this layer transmits the action to the subsequent layers underneath it because of the intermolecular forces between the liquid molecules. In a steady state, the velocities of these layers range from U (the layer closest to the moving plate) to 0 (the layer closest to the stationary plate). The applied force acts on an area, A, of the liquid surface, inducing a shear stress (F/A). The displacement of liquid at the top plate, x, relative to the thickness of the film is called shear strain (x/L), and the shear strain per unit time is called the shear rate (U/Y). If the distance Y is not too large or the velocity U too high, the velocity gradient will be a straight line. It was shown that for a large class of fluids F ∼

AU . Y

(1.1)

It may be seen from similar triangles in Fig. 1.1 that U/Y can be replaced by the velocity gradient du/dy. If a constant of proportionality η is now introduced, the shearing stress between any two thin sheets of fluid may be expressed by τ=

U du F =η =η . A Y dy

(1.2)

In transposed form it serves to define the proportionality constant η=

τ , du/dy

(1.3)

which is called the dynamic coefficient of viscosity. The term du/dy = γ˙ is called the shear rate. The dimensions of dynamic viscosity are force per unit area divided by velocity gradient or shear rate. In the metric system the dimensions of dynamic viscosity is Pa·s. A widely used unit for viscosity in the metric system is the poise (P). The poise = 0.1 Ns/m2 . The centipoise (cP) (= 0.01 P = mNs/m2 ) is frequently a more convenient unit. It has a further advantage that the dynamic viscosity of water at 20◦ C is 1 cP. Thus the value of the viscosity in centipoises is an indication of the

1.1

Definitions

3

viscosity of the fluid relative to that of water at 20◦ C. In many problems involving viscosity there frequently appears the value of viscosity divided by density. This is defined as kinematic viscosity, ν, so called because force is not involved, the only dimensions being length and time, as in kinematics. Thus v=

η . ρ

(1.4)

In SI units, kinematic viscosity is measured in m2 /s while in the metric system the common units are cm2 /s, also called the stoke (St). The centistoke (cSt) (0.01 St) is often a more convenient unit because the viscosity of water at 20◦ C is 1 cSt. A fluid for which the constant of proportionality (i.e., the viscosity) does not change with rate of deformation is said to be a Newtonian fluid and can be represented by a straight line in Fig. 1.2. The slope of this line is determined by the viscosity. The ideal fluid, with no viscosity, is represented by the horizontal axis, while the true elastic solid is represented by the vertical axis. A plastic body which sustains a certain amount of stress before suffering a plastic flow can be shown by a straight line intersecting the vertical axis at the yield stress. The relationship between stress and deformation rate given in Eq. (1.3) represents a constitutive equation of the fluid in a simple shear flow. We can generalize this result by saying that, in simple fluids, the stress on a material is determined by the history of the deformation involving only gradients of the first order or more exactly by the relative deformation tensor as every fluid is isotropic. A general constitutive equation which describes the mechanics of materials in classical fluid mechanics can be written as: tij = −pδij + τij + λv ekk δij ,

Fig. 1.2 Rheological behaviour of materials

(1.5)

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or, using the unit tensor I, T = −pI + τ + λv (trE)I,

(1.6)

where T(x, t) denotes the symmetric Cauchy–Green stress tensor at position x and time t, p(x, t) is the pressure in the fluid, τ is the extra stress tensor, λv is the volume viscosity, and E is the rate of deformation tensor of the velocity field u(x, t): 1 eij = 2

∂uj ∂ui + ∂xj ∂xi

(1.7)

or E(x, t) =

1 (∇u) + (∇u)T , 2

(1.8)

where trE = eii = ∂ui /∂xi . The extra stress tensor can be written as τ = η(I1 , I2 , I3 )E.

(1.9)

The apparent viscosity η in the above equation is a function of the first, second and third invariants of the rate of deformation tensor: I1 = eii , I2 =

1 (eii ejj − eij eij ), I3 = det(eij ). 2

(1.10)

For incompressible fluids, the first invariant I1 becomes identically equal to zero. The third invariant I3 vanishes for simple shear flows. The apparent viscosity then is a function of the second invariant I2 alone, and Eq. (1.9) can be written in a simplified form as τ = η(I2 )E.

(1.11)

If the fluid does not undergo a volume change, i.e. it is incompressible, then the last term on the right-hand side of Eq. (1.6) drops out and the volume viscosity has no role to play.

1.1.2 Non-Newtonian Fluids There is a certain class of fluids, called non-Newtonian fluids, in which the viscosity η varies with the shear rate. A particular feature of many non-Newtonian fluids is the retention of a fading “memory” of their flow history which is termed elasticity. Typical representatives of non-Newtonian fluids are liquids which are formed either partly or wholly of macromolecules (polymers), or two phase

1.1

Definitions

5

materials, like, for example, high concentration suspensions of solid particles in a liquid carrier solution. There are various types of non-Newtonian fluids. Pseudoplastic fluids are those fluids for which viscosity decreases with increasing shear rate and hence are often referred to as shear-thinning fluids. These fluids are found in many real fluids, such as polymer melts and solutions or glass melt. When the viscosity increases with shear rate the fluids are referred to as dilatant or shear-thickening fluids. These fluids are less common than with pseudoplastic fluids. Dilatant fluids have been found to closely approximate the behaviour of some real fluids, such as starch in water and an appropriate mixture of sand and water. For pseudoplastic and dilatant fluids, the shear rate at any given point is solely dependent upon the instantaneous shear stress, and the duration of shear does not play any role so far as the viscosity is concerned. Many of these fluids exhibits a constant viscosity at very small shear rates (referred to as zero-shear viscosity, η0 ) and at very large shear rates (referred to as infinite-shear viscosity, η∞ ). Some fluids do not flow unless the stress applied exceeds a certain value referred to as the yield stress. These fluids are termed fluids with yield stress or viscoplastic fluids. The variation of the shear stress with shear rate for pseudoplastic and dilatant fluids with and without yield stress is shown in Fig. 1.3. Viscoelastic fluids are those fluids that possess the added feature of elasticity apart from viscosity. These fluids have a certain amount of energy stored inside them as strain energy thereby showing a partial elastic recovery upon the removal of a deforming stress. In the case of thixotropic fluids, the shear stress decreases with time at a constant shear rate. An example of a thixotropic material is non-drip paint, which becomes thin after being stirred for a time, but does not run on the wall when it is brushed on. By contrast, when the shear stress increases with time at a constant shear rate the fluids are referred to as rheopectic fluids. Some clay suspensions exhibit rheopectic behaviour. Figure 1.4 shows a schematic of the thixotropic

Fig. 1.3 Rheological behaviour of non-Newtonian fluids

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Fig. 1.4 Rheopectic and thixotropic fluids

and rheopectic fluid behaviour. In the case of thixotropic and rheopectic fluids, the shear rate is a function of the magnitude and duration of shear, and the time lapse between consecutive applications of shear stress. Viscoelastic fluids have some additional features. When a viscoelastic fluid is suddenly strained and then the strain is maintained constant afterward, the corresponding stresses induced in the fluid decrease with time. This phenomenon is called stress relaxation. If the fluid is suddenly stressed and then the stress is maintained constant afterward, the fluid continues to deform, and the phenomenon is called creep. If the fluid is subjected to a cycling loading, the stress–strain relationship in the loading process is usually somewhat different from that in the unloading process, and the phenomenon is called hysteresis. There is a distinctive difference in flow behaviour between Newtonian and nonNewtonian fluids to an extent that, at time, certain aspects of non-Newtonian flow behaviour may seem abnormal or even paradoxical. For example, when a rod is rotated in an elastic non-Newtonian fluid, the fluid climbs up the rod against the force of gravity. This is because the rotational force acting in a horizontal plane produces a normal force at right angles to that plane. The tendency of a fluid to flow in a direction normal to the direction of shear stress is known as the Weissenberg effect. Another effect caused by viscoelasticity is the die swell effect of the fluid as it leaves a die exit. This expansion is an elastic response of the fluid to energy stored when its shape changes while entering the die. This energy is released as the fluid leaves the die and causes a swelling effect normal to the direction of flow in the die. It has been also observed that, when a viscoelastic fluid flows in a tube with a sudden contraction, bubbles with a certain diameter come to a sudden stop right at the entrance of the contraction along the centerline before finally passing through after a hold time. This behaviour has been termed the Uebler effect.

1.2

Non-Newtonian Fluid Behaviour

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1.2 Non-Newtonian Fluid Behaviour A Newtonian fluid requires only a single material parameter to relate the internal stress to the applied strain. For non-Newtonian fluids, more elaborate constitutive equations, containing several material parameters, are needed to describe the response of these fluids to complex, time-dependent flows. There exists no general model, i.e., no universal constitutive equation that describes all non-Newtonian fluid behaviour. Currently successful theories are either restricted to very specific, simple flows, especially generalizations of simple shear flow and extensional flow, for which rheological data can be used to develop empirical models, or to very dilute solutions for which the microscale dynamics is dominated by the motion of simple, isolated macromolecules. This section deals with the description of the nature and diversity of material response to simple shearing and extensional flows. The analysis of experimental methods for measuring these quantities is presented in the next section.

1.2.1 Simple Flows We shall now examine some simple flow fields of fluids. Simple flow fields are required to determine the material properties of the fluids and these are separated in three groups: steady simple shear, small-amplitude oscillatory, and extensional flow. 1.2.1.1 Steady Simple Shear Flow The most common flow is steady simple shear flow, represented in rectangular Cartesian coordinates by: ux = γ˙ y,

uy = uz = 0,

(1.12)

where (ux , uy , uz ) are the velocity components in the x, y, and z directions, and γ˙ = dux /dy. For steady shear flow (sometimes called a viscometric flow) the shear rate is independent of time; it is presumed that the shear rate has been constant for such a long time that all the stresses in the fluid are time-independent. The extra stress tensor in such a flow is thus defined by ⎞ τxx τxy 0 τ = ⎝ τyx τyy 0 ⎠ , 0 0 τzz ⎛

(1.13)

where τxy = τyx are called the shear stress components, and τxx , τyy , and τ zz are called the normal stress components. The corresponding stress distribution for a non-Newtonian fluid can be written in the form τxy = τ (γ˙ ) = η(γ˙ )γ˙ ,

(1.14)

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τxx − τyy = N1 (γ˙ ),

(1.15)

τyy − τzz = N2 (γ˙ ),

(1.16)

where N1 and N2 are the first and second normal stress differences. For a Newtonian fluid, η is a constant and N1 and N2 are zero. The variation of η with shear rate and non-zero values of N1 and N2 are manifestations of non-Newtonian viscoelastic behaviour. The second normal stress difference N2 , however, receives less attention due to difficulties in its measurement and for the smallness of its value. For many non-Newtonian fluids, the value of N2 would be usually an order of magnitude smaller than that of N2 . The viscosity function η, the primary and secondary normal stress coefficients ψ1 , and ψ2 , respectively, are the three parameters which completely determine the state of stress in any steady simple shear flow. They are often referred to as the viscometric functions. The normal stress coefficients are defined as follows: τxx − τyy = ψ1 (γ˙ )γ˙ 2 ,

(1.17)

τyy − τzz = ψ2 (γ˙ )γ˙ 2 ,

(1.18)

and

and are also functions of the magnitude of the strain rate. The first and second normal stress coefficients do not change in sign when the direction of the strain rate changes. The primary normal stress coefficient is used to characterize the elasticity of a non-Newtonian fluid. A constant primary normal stress coefficient is obtained when the primary normal stress varies quadratically with shear rate. 1.2.1.2 Small-Amplitude Oscillatory Shear Flow Small-amplitude oscillatory shear flow provides another mean to characterize a viscoelastic fluid. The oscillatory tests belong to the general framework of dynamic characterization of viscoelastic fluids in which both stress and strain vary harmonically with time. The dynamic properties of viscoleastic fluids are of considerable importance because they can be directly related to the viscous and elastic parameters derived from such measurements. Oscillatory tests involve the measurement of the response of the fluid to a small amplitude sinusoidal oscillation. The applied strain and strain rates are given by γ (t) = γ0 sin(ωt),

(1.19)

γ˙ (t) = γ0 ω cos(ωt) = γ˙0 cos(ωt),

(1.20)

and

1.2


9

where γ 0 is the amplitude of the applied strain, γ˙0 is the shear rate amplitude, and ω is the frequency. The resulting shear stress may be given in terms of amplitude, τ0 , and phase shift, δ = π2 − φ, as follows: τxy (t) = τ0 sin(ωt + δ),

(1.21)

τxy (t) = τ0 ω cos (ωt − φ).

(1.22)

and

These equations may be expanded and rewritten in terms of the in-phase and outof-phase parts of the shear stress and placed in terms of four viscoelastic material functions as

τxy (t) = γ0 G sin(ωt) + G cos(ωt) ,

(1.23)

τxy (t) = γ0 ω η cos(ωt) + η sin(ωt) .

(1.24)

The storage modulus, G , is defined as the stress in-phase with the strain in a sinusoidal shear deformation divided by the strain and is a measure of the elastic energy stored in the system at a particular frequency. G represents the solid like response of a material and, for a perfectly elastic solid, is equal to the constant shear modulus, G, for a perfectly elastic solid with the loss modulus equal to zero. Similarly, the loss modulus, G , is defined as the stress 90◦ out-of-phase with the strain divided by the strain, and is a measure of the energy dissipated as a function of frequency. G represents the viscous component or liquid-like response of a material to a deformation. The dynamic viscosity, η , and dynamic rigidity, η , are related to G and G by G , ω G η = . ω

η =

(1.25) (1.26)

The material functions G , G , η , and η are referred to as the linear viscoelastic properties because they are determined from the shear stress which is linear in strain for small deformations. It should be noted that as the frequency approaches zero, η approaches η0 and 2G /ω2 approaches ψ1,0 (the zero-shear-rate value of ψ1 ). Correspondingly, the loss modulus is asymptotic to η0 ω as ω → 0. A method of comparing the storage and loss modulus is made by the calculation of the loss tangent defined as tan δ =

G , G

and represents the phase angle between stress and strain.

(1.27)

10


For more detail on these and other linear viscoelastic properties, standard references should be consulted (for example, Bird et al. 1987). 1.2.1.3 Extensional Flow Shear measurements are not sufficient to characterize the behaviour of nonNewtonian liquids and must be supplemented by measurements obtained in extension or extension-like deformations. An extensional flow is one in which fluid elements are stretched or extended without being rotated or sheared. Extensional flow can be visualized as that occurring when a material is longitudinally stretched as in fiber spinning. In this case, the extension occurs in a single direction and the related flow is termed uniaxial extension. Extension of material takes place in processing operation as well, such as film blowing and flat-film extrusion. Here, the extension occurs in two directions and the flow is referred to as biaxial extension in one case and planar extension in the other. In biaxial extension, the material is stretched in two directions and compressed in the other. In planar extension, the material is stretched in one direction, held to the same dimension in a second, and compressed in the third. A schematic representation of the three types of extensional flow fields is shown in Fig. 1.5. Uniaxial Extensional Flow In a uniaxial extensional flow, the dimension of the fluid elements changes in only one direction. The velocity components are: ux = ε˙ x,

ε˙ uy = − y, 2

where ε˙ = dux /dx is a constant strain tensor is ⎛ τxx τ=⎝ 0 0

ε˙ uz = − z, 2

(1.28)

rate, and the corresponding extra stress ⎞ 0 0 τyy 0 ⎠ . 0 τzz

Fig. 1.5 Extensional flow fields: (a) uniaxial, (b) biaxial, (c) planar

(1.29)

1.2


11

The corresponding stress distribution can be written in the form τxx − τyy = τxx − τzz = ηE (˙ε)˙ε ,

(1.30)

τij = 0, i = j,

(1.31)

where ηE is the uniaxial extensional viscosity. Fluids are considered extensionalthinning if ηE decreases with increasing ε˙ . They are considered extensionalthickening if ηE increases with ε˙ . These terms are analogous to shear-thinning and shear-thickening used to describe changes in viscosity with shear rate. The uniaxial extensional viscosity is frequently qualitatively different from shear viscosity. For example, highly elastic polymer solutions that posses a shear viscosity that decreases in shear often exhibit uniaxial extensional viscosity that increases with strain rate. In most applications, the extensional viscosity is presented in terms of a Trouton ratio which is defined conveniently to be the ratio of extensional viscosity to the shear viscosity, Tr = ηE /η. For calculating Trouton ratio in uniaxial extensional flow, √ the shear viscosity should be evaluated at a shear rate numerically equal to 3˙ε . This result is obtained by comparing extensional and shear viscosities at equal values of the second invariant of the rate of deformation tensor. The Trouton ratio, which takes the constant value 3 for Newtonian liquids and shear-thinning inelastic liquids, is found to be a strong function of strain rate ε˙ in many viscoelastic liquids, with very high values, of about 104 , possible in extreme cases.

Biaxial Extensional Flow In biaxial extensional flow, the dimensions of the fluid elements change drastically but they change only in two directions. The velocity field in simple biaxial extensional flow is given by ux = ε˙ B x,

uy = ε˙ B y,

uz = −2˙εB z.

(1.32)

The corresponding stress distribution is τxx − τzz = τyy − τzz = ηEB (˙εB )˙εB ,

(1.33)

where ηEB is the biaxial extensional viscosity. The Trouton number for the case of biaxial extensional flow can be calculated as TrEB = ηEB /η. For calculating Trouton ratio in a biaxial extensional √ flow, the shear viscosity should be evaluated at a shear rate numerically equal to 12˙ε. For a Newtonian fluid, TrEB = 6.

12


Planar Extensional Flow Planar extensional flow is the type of flow where there is no deformation in one direction. The velocity field is represented by ux = ε˙ P x,

uy = −˙εP y,

uz = 0.

(1.34)

In this case, the stress distribution is given as τxx − τyy = ηEP (˙εP ) ε˙ P ,

(1.35)

where ηEP is the planar extensional viscosity. The Trouton number for the case of planar extensional flow can be calculated as TrP = ηP /η. For calculating Trouton ratio in a panar extensional flow, the shear viscosity should be evaluated at a shear rate numerically equal to 2˙ε . For a Newtonian fluid, TrP = 4. It is difficult to generate planar extensional flow and experimental tests of this type are less common than those involving uniaxial or biaxial extensional flows.

1.2.2 Intrinsic Viscosity and Solution Classification The intrinsic viscosity is another parameter that characterize the behaviour of nonNewtonian fluids. The intrinsic viscosity, [η], of a polymer solution is defined as the zero concentration limit of the reduced viscosity, ηred = ηsp /c, where c is the polymer concentration and ηsp is the specific viscosity. The specific viscosity is defined as the relative polymer contribution to viscosity ηsp = (η0 − ηs )/μs , where η0 is the zero-shear viscosity and ηs is the solvent viscosity. The intrinsic viscosity can thus be expressed as: [η] = lim ηred = lim c→0

c→0

η0 − ηs . cηs

(1.36)

Note that the intrinsic viscosity has dimensions of reciprocal concentration. The intrinsic viscosity is determined graphically by plotting ηred versus c and extrapolating to zero concentration. It is also found that extrapolation to zero concentration of the inherent viscosity, ηinh = 1c ln(ηsp + 1), can also be used to determine the intrinsic viscosity and the same result for [η] must be achieved. The most common relation between specific viscosity and polymer concentration is that of Huggins (1942), ηsp = [η] + k [η]2 c, c

(1.37)

where k is the Huggins slope constant. The alternative expression of Kraemer (1938)

1.2


13

1 ηsp ln = [η] − k [η]2 c, c c

(1.38)

where k is the Kraemer constant, may also be used. Huggins slope constant and Kraemer constant are related by k + k = 0.5. The intrinsic viscosity can be used to determine the viscosity molecular weight, Mη , using the Mark-Houwink equation as follows (Bird et al. 1987) [η] = kMηα ,

(1.39)

where k and α are determined from a double logarithmic plot of intrinsic viscosity and molecular weight. These parameters have been published for many systems by Bandrup and Immergut (1975). The polymer solutions are regarded as dilute when there is no interaction between molecules. A standard method to evaluate whether a polymer solution is dilute is to determine a dimensionless concentration of polymer solution which can be given by either [η]c (Flory 1953) or cNA V/Mw (Doi and Edwards 1986), where c is the polymer concentration, NA is the Avogadro’s number, V is the volume occupied by a polymer molecule, and Mw is the average molecular weight. Flexible polymers tend to occupy a spherical region in solution such that V = 4π R3h /3. In the case of rigid molecules, the spherical region required such that the large aspect ratio molecule can freely rotate without interaction with its neighbours is calculated from the molecule length such that V = π L3 /6, where L is the length of the molecule. The length L can be determined using relations and

given by Broersma (1960) Young et al. (1978) for rigid molecules, L = Rh 2δ − 0.19 − (8.24/δ) + 12/δ 2 , where δ = ln (L/r) is the aspect ratio of a rod and r is the radius of the rigid rod. The polymer solution is regarded as dilute when both dimensionless concentrations are less than unity. When one of the dimensionless concentrations is larger than 1, the polymer solution is considered semi-dilute.

1.2.3 Dimensionless Numbers Fluid dynamics is parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. These include, for example, the Reynolds number, addressing inertial effects, the Froude number, describing gravity-driven flows, the Weber number, addressing the importance of surface tension forces, the Grashof number, addressing buoyancy effects, or the Mach number, describing the importance of liquid compressibility. In the specific case of non-Newtonian fluids, three additional sets of non-dimensional parameters are generated, namely the Weissenberg number, the Deborah number, and the elasticity number, describing elastic effects. The dimensionless numbers are particularly useful for scaling arguments, for consolidating experimental, analytical, and numerical results into a compact form, and also for cataloging various flow regimes.

14


1.2.3.1 Reynolds Number Of all dimensionless numbers encountered in fluid dynamics, the Reynolds number is the one most often mentioned in connection with non-Newtonian fluids. The Reynolds number represents the ratio of inertia forces to viscous forces and has the expression: Re =

LU ρLU = , ν η

(1.40)

where L is a linear dimension that may be any length that is significant in the flow pattern and U is the flow velocity. For example, for a pipe completely filled, L might be either the diameter or the radius, and the numerical value of Re will vary accordingly. 1.2.3.2 Weissenberg Number The Weissenberg number is defined as Wi = τfluid e˙ or

τfluid γ˙ ,

(1.41)

which relates the relaxation time of the viscoelastic liquid to the flow deformation time, either inverse extension rate 1/˙ε or shear rate 1/γ˙ . When Wi is small, the liquid relaxes before the flow deforms it significantly, and perturbations to equilibrium are small. As Wi approaches 1, the liquid does not have time to relax and is deformed significantly. 1.2.3.3 Deborah Number Another relevant time scale, τflow , characteristic of the flow geometry may also exist. For example, a channel that contracts over a length L0 introduces a geometric time scale τflow = L0 /U0 required for a liquid to transverse it with velocity U0 . The flow time scale τflow can be long or short compared with the liquid relaxation time, τfluid , resulting in a dimensionless ratio known as the Deborah number De =

τfluid . τ flow

(1.42)

For small De values, the material responses like a fluid, while for large De values, we have a solid-like response. In the limit, when De = 0 one has a Newtonian fluid, and when De = ∞, an elastic solid. The usage of De and Wi can vary. Some references use Wi exclusively to describe shear flows and use De for the general case, whereas others use Wi for local flow time scales due to a local shear and De for global flow time scales due to residence time in flow.

1.2


15

1.2.3.4 Elasticity Number As the flow velocity increases, elastic effects become stronger and De and We increase. However, the Reynolds number Re increases in the same way, so that inertial effects become more important as well. The elasticity number El = De/Re =

τfluid η , ρL2

(1.43)

where L is a dimension setting the shear rate, expresses the relative importance of elastic to inertial effects. Significantly, El depends only on the geometry and material properties of the fluid, and is independent of flow rate. For example, extrusion of polymer melts corresponds to El >> 1, whereas processing flows for dilute polymer solutions (such as spin-casting) typically correspond to El 1 the fluid is shear-thickening. The power-law model is the most well-known and widely-used empiricism in engineering work, because a wide variety of flow problems have been solved analytically for it. One can often get a rough estimate of the effect of the non-Newtonian viscosity by making a calculation based on the power-law model. One shortcoming of the power law model is that it does not describe the low shear and high shear rate constant viscosity data of

16


shear-thinning or shear-thickening fluids. For n < 1, this model presents a problem when the shear rate tends to zero because the fluid viscosity becomes infinite. The Carreau Model A more sophisticated model is the Carreau model given as η0 − η∞ η = η∞ +

N , 1 + (λc γ˙ )

(1.45)

where λc is a time constant and N is a dimensionless exponent. At low shear rates, the model predicts Newtonian properties with a constant zero-shear viscosity, η0 , while at high shear rates, it predicts a limiting and constant infinite-shear viscosity, η∞ . The Carreau model can be modified to include a term due to yield stress. For example, the Carreau model with a yield term given by η=

ηp τ0 +

N , γ˙ 1 + (λc γ˙ )

(1.46)

where τ0 is the yield stress and ηp is the plateau viscosity, was employed in the study of the rheological behaviour of glass-filled polymers (Poslinski et al. 1988). The Casson Model The Casson model given by √ γ˙ =

0

√ τ − τ0 2 √ η

, for τ ≥ τ0 , , for τ < τ0

(1.47)

where τ0 is the yield stress, captures both the yield stress and shear dependent viscosity of a fluid. This model reduces to a Newtonian fluid when τ0 = 0. Equation (1.47) indicates that a finite yield stress is required before flow can start. This yield stress results in a plug flow and the velocity distribution shaped like a blunted parabola that is so typical of blood flow in small diameter vessels. The Casson model was originally developed to describe the flow of printing ink through capillaries and was later applied to other fluids containing chain like particles. The Casson equation has also proven useful for the description of the flow of blood on both glass and fibrin surfaces. 1.2.4.2 Viscoelastic Fluids A large number of constitutive equations have been proposed to describe the viscoelastic behaviour of non-Newtonian fluids. The Maxwell and Oldroyd-B models have had a popularity far beyond expectation and anticipation. Their relative simplicity has obviously been an attraction, especially in the case of numerical

1.2


17

simulation of viscoelastic flows, where simple models have been essential in the development of numerical strategies. Other important viscoelastic models that have been used extensively are the dumbbell models and the KBKZ model. The Maxwell Model The simplest constitutive model to account for fluid elasticity is the Maxwell model which considers the fluid as being both viscous and elastic. The Maxwell equation is given by: τ+λ

∂τ = 2ηE, ∂t

(1.48)

where λ is the relaxation time and η is the constant shear viscosity. For steady-state motions this equation simplifies to the Newtonian fluid with viscosity η. By replacing the time derivative with the convected time derivative, the upper convected Maxwell model is obtained which is given as ∇

τ + λ τ = 2ηE,

(1.49)

∇

where the upper convected derivative τ is defined by ∇

τ=

∂τ + (u · ∇)τ − (∇u)T τ − τ (∇u) . ∂t

(1.50)

For steady simple shear flow, the Maxwell relaxation time is λ=

ψ1 N1 , = 2 2η 2ηγ˙

(1.51)

while in small-amplitude oscillatory flow, the viscoelastic properties for this model are given by G =

ληω2 , 1 + λ2 ω 2

(1.52)

η =

η . 1 + λ2 ω 2

(1.53)

and

At low frequency, G is predicted to vary quadratically with frequency while it approaches a constant value at high frequencies. The uniaxial extensional viscosity for the upper convected Maxwell model is ηE = 3η

1 . (1 + λ˙ε) (1 − 2λ˙ε )

(1.54)

18


This model predicts strain rate thickening behaviour, but the predicted extensional viscosity asymptotes to infinity when ε˙ = 1 (2λ) . The upper convective Maxwell model exhibits many of the qualitative behaviours of viscoelastic fluids, including normal stresses in shear, extension thickening, and elastic recovery. However, it does not exhibit shear thinning. To get a reasonable match to viscoelastic behaviour, one must introduce some additional nonliniarities by altering the model in the form ∇

Y · τ + λ τ = 2ηE.

(1.55)

Two models that are widely used are the Giesekus model, which has Y=I+

αλ τ, η

and the Phan-Thien-Tanner model, for which ελ Y = exp tr(τ) I. η

(1.56)

(1.57)

Each of these models adds another dimensionless parameter, α or ε, that control the nonlinearity. A multi-mode Maxwell model may also be used to allow the material functions to be predicted more accurately by adjusting the parameters in each mode. The extra stress tensor is expressed, in this case, as a combination of several relaxation times as τ=

n

τi ,

(1.58)

i=1

where each τi is described by ∇

τι + λi τi = 2ηi E.

(1.59)

The Oldroyd-B Model The Maxwell model may be extended to obtain a more useful constitutive equation by including the convected time derivative of the rate of deformation tensor. This way the Oldroyd-B constitutive model is obtained which is described by ∇ ∇ τ + λ1 τ = 2η E + λ2 E ,

(1.60)

where λ1 and λ2 are the time constants (relaxation and retardation) and the viscosity has also a constant value. We observe that, by setting λ2 = 0, the above equation

1.2


19

reduces to the upper convected Maxwell model. The Oldroyd-B model qualitatively describes many features of the so-called Boger fluids (elastic fluids with almost constant viscosity). The material functions of this model are defined as ψ1 = 2η (λ1 − λ2 ) ,

ψ2 = 0,

(1.61)

while the linear viscoelastic properties are given by G =

(λ1 − λ2 )ηω2 , 1 + λ21 ω2

(1.62)

and

η =

1 + λ1 λ2 ω 2 η 1 + λ21 ω2

.

(1.63)

As in the case of Maxwell model, the Oldryod-B model predicts that at low frequencies the storage modulus varies quadratically with frequency while at high frequencies a constant value is obtained. The equation for the uniaxial extensional viscosity is given by ηE = 3η

1 − λ2 ε˙ − 2λ1 λ2 ε˙ 2 , 1 − λ1 ε˙ − 2λ21 ε˙ 2

(1.64)

and, therefore, the extensional viscosity asymptotes to infinity when ε˙ = 1 (2λ1 ) . In non-convected form, the Oldroyd-B model is referred to as the Jeffreys model which is given by ∂τ ∂E = 2η E + λ2 . τ + λ1 ∂t ∂t

(1.65)

It is interesting to note that this equation was originally proposed for the study of wave propagation in the earth’s mantle (Jeffreys 1929). The Dumbbell Model In elastic dumbbell models a polymer is described as two beads connected by a Hookean spring. The beads represent the ends of the molecule and their separation is a measure of the extension. The beads experience a hydrodynamic drag force, a Brownian force due to thermal fluctuations of the fluid, and an elastic force due to the spring connecting one bead to the other. It can be further assumed that the polymer solution is sufficiently dilute that the polymer molecules do not interact with one another. The polymer contribution to the stress tensor is ∇

τp + λH τp = nkB TλH E,

(1.66)

20


where n is the number density of molecules, kB is the Boltzman constant, T is the temperature, and λH is the relaxation time for a Hookean dumbbell. The Hookean relaxation time is defined in terms of a friction coefficient of the beads, ς , and a Hookean spring constant, H, as λH = ς/(4H). The Oldroyd-B constitutive equation may be derived from the elastic dumbbell model with the following relations used to determine the material functions for steady shear flow: η = ηs + nkTλH

,

ψ1 = 2nkTλ2H

,

ψ2 = 0,

(1.67)

while the relaxation time is given as a function of the intrinsic viscosity as λ1 =

[η] ηs Mw , Rg T

(1.68)

The retardation time in the where Rg is the universal gas constant (8.314 J/(kg·mol)). Oldroyd-B model is given by λ2 = λ1 ηs η . The relaxation time in the Maxwell constitutive model, λ, is related to the Oldroyd characteristic times by λ = λ1 − λ2 . The elastic dumbbell model is only suitable to use for flexible polymers, such as polyacrylamide. The rigid dumbbell model may be used to describe rigid or semi-rigid molecules (such as, DNA in a helix configuration, xanthan gum, or carboxymethylcellulose) in solution. It accounts for the orientability of the rod-like macromolecules in the flow field while ignoring molecular stretching and bending motions which are not considered significant for this class of macromolecules. The macromolecule is represented by two beads joined by a massless rod, with the solvent presumed to only interact at the beads. The rigid dumbbell relaxation time is given by λD =

1 m [η] ηs Mw , m= , Rg T m1 + m2

(1.69)

where the values of m1 and m2 depend on the details of the model, as listed by Ferry (1980). The extensional viscosity approaches a constant at relatively low extension rates such that as ε˙ → ∞ the limiting extensional viscosity is ηe = 3ηs + 6ckB TλD ,

(1.70)

where c is the polymer concentration. The BKBZ Model The KBKZ model is an integral type constitutive equation proposed by Kaye (1962) and Bernstein et al. (1963). The time-integral constitutive equation of the KBKZ model is

1.2


t σp =

21

μ t − t H (I1 , I2 ) B t, t dt ,

(1.71)

−∞

where σp is the polymer contribution to the extra stress tensor, μ t, t is the linear memory function, H(I1 , I2 ) is a non-linear damping function, and viscoelastic B t, t is the Finger strain tensor given by ⎛ 2 ⎞ ς t, t 0 0 B t, t = ⎝ 0 ς 2m t, t 0 ⎠, −2(m+1) 0 0 ς t, t

ς t, t = exp ε˙ 0 t − t ,

(1.72)

(1.73)

where m = –0.5 for uniaxial extension, m = 0 for planar extension, m = 1 for biaxial extension, and ς is the extension ratio. The memory function is expressed as an exponentially fading term while the strain function can be written as (Papanastasiou et al. 1983) a t − t μ t, t = exp − , λ λ H(I1 , I2 ) =

α , (α − 3) + βI1 + (1 − β) I2

(1.74) (1.75)

where α and β are adjustable parameters determined from the shear and extensional results, respectively. The set of parameters {λ, a} are the conventional relaxation time and weight, which can be evaluated from simple rheological tests such as stress relaxation or sinusoidal oscillations. Several other forms of the damping term are known in literature such as those provided by Wagner (1976), Wagner and Demarmels (1990): 1 , √ 1 + α (I1 − 3)(I2 − 3)

H(I1 , I2 ) = exp −β αI1 + (1 − α)I2 − 3 . H(I1 , I2 ) =

(1.76) (1.77)

This constitutive equation has been found to accurately predict transient and shear modes of simple shear, uniaxial extension and biaxial extension at low, moderate, and high strains and rates of strain (Papanastasiou et al. 1983). Example: Material Functions for the Oldroyd-B Model In a steady simple shear flow ux = γ˙ y,

uy = 0,

uz = 0,

with γ˙ = dux dy .

(1)

22


The rate of deformation and extra stress tensors have the following expressions: ⎛ 01 1 1 ∇u + ∇uT = ⎝ 0 0 E= 2 2 00

⎞ ⎛ ⎞ ⎛ ⎞ 0 000 010 1 1 0 ⎠ γ˙ + ⎝ 1 0 0 ⎠ γ˙ = ⎝ 1 0 0 ⎠ γ˙ , 2 000 2 000 0

⎛ ⎞⎛ ⎞ 010 010 1 du E= − ∇u · E + E · ∇uT = − ⎝ 0 0 0 ⎠ ⎝ 1 0 0 ⎠ γ˙ 2 dt 2 000 000 ⎛ ⎞ ⎛ ⎞⎛ ⎞ 100 010 000 1 − ⎝ 1 0 0 ⎠ ⎝ 1 0 0 ⎠ γ˙ 2 = − ⎝ 0 0 0 ⎠ γ˙ 2 , 2 000 000 000

(2)

∇

(3)

⎞ τxx τxy 0 τ = ⎝ τyx τyy 0 ⎠ , 0 0 τzz

(4)

⎛ ⎞⎛ ⎞ 010 τxx τxy 0 dτ − ∇u · τ + τ · ∇uT = − ⎝ 0 0 0 ⎠ ⎝ τyx τyy 0 ⎠ γ˙ τ= dt 000 0 0 τzz ⎛ ⎞⎛ ⎞ ⎛ ⎞ τxx τxy 0 000 2τxy τyy 0 − ⎝ τyx τyy 0 ⎠ ⎝ 1 0 0 ⎠ γ˙ = ⎝ τyy 0 0 ⎠ γ˙ . 000 0 0 τzz 0 0 0

(5)

⎛

and ∇

Replacing these results into the equation of the rheological model, we can write ⎛ ⎛ ⎞⎤ ⎞ ⎞ ⎞ ⎡ ⎛ τxy τyy 0 100 τxx τxy 0 010 1 ⎝ τyx τyy 0 ⎠ − λ1 γ˙ ⎝ τyy 0 0 ⎠ = 2ηγ˙ ⎣ ⎝ 1 0 0 ⎠ − λ2 γ˙ ⎝ 0 0 0 ⎠⎦ , (6) 2 000 000 0 0 τzz 0 0 0 ⎛

from which it follows that τxx − 2λ1 γ˙ τxy = −2ηλ2 γ˙ 2 τxy − λ1 γ˙ τyy = ηλ2 γ˙

,

(7)

so that τxy = ηγ˙ , τxx − τyy = N1 = 2η(λ1 − λ2 ) γ˙ 2 , τyy − τzz = N2 = 0,

(8)

1.2


23

and ψ1 = 2η (λ1 − λ2 ) , ψ2 = 0.

(9)

For an upper convective Maxwell fluid, λ2 = 0, and λ = ψ1 2η . For a small amplitude oscillatory shearing flow

γ˙ (t) = γ0 ω cos (ωt) , τxy = γ0 G sin (ωt) + G cos (ωt) .

(10)

The Oldroyd-B equation becomes

G sin (ωt) + G cos (ωt) + λ1 ω G cos (ωt) − G sin (ωt) = ηω [cos (ωt) − λ2 ω cos (ωt)] .

(11)

We obtain

G − λ1 ωG = −λ2 ω2 η G + λ1 ωG = ωη

,

(12)

and 1 + λ1 λ2 ω2 ηω (λ1 − λ2 ) ηω2 = , G = , 1 + λ21 ω2 1 + λ21 ω2 1 + λ1 λ2 ω 2 η (λ1 − λ2 ) ηω η = , η = . 2 2 1 + λ1 ω 1 + λ21 ω2

G

(13)

For the Upper Convective Maxwell fluid we have G =

ληω2 ηω η ληω , G = , η = , η = . 1 + λω2 1 + λω2 1 + λω2 1 + λω2

(14)

In a steady uniaxial extensional flow 1 1 ux = ε˙ x, uy = − ε˙ y, uz = − ε˙ y. 2 2

(15)

The rate of deformation tensor and the extra stress tensor become ⎞ ⎞ ⎛ ⎛ ⎛ ⎞ 1 0 0 1 0 0 2 0 0 1⎝ 1 1 0 −1 2 0 ⎠ ε˙ + ⎝ 0 −1 2 0 ⎠ ε˙ = ⎝ 0 −1 0 ⎠ ε˙ , E= 2 0 0 2 0 0 2 0 0 −1 −1 2 −1 2 (16)

24


⎞⎛ ⎞ 1 0 0 2 0 0 1 0 ⎠ ⎝ 0 −1 0 ⎠ ε˙ 2 E = − ⎝ 0 −1 2 2 0 0 0 0 −1 −1 2 ⎞ ⎛ ⎞⎛ ⎛ 1 0 0 2 0 0 40 1⎝ 1 0 −1 0 ⎠ ⎝ 0 −1 2 0 ⎠ ε˙ 2 = − ⎝ 0 1 − 2 0 0 −1 2 00 0 0 −1 2 ⎛

∇

⎞ 0 0 ⎠ ε˙ 2 , 1

(17)

⎛

⎞ τxx 0 0 τ = ⎝ 0 τyy 0 ⎠ , 0 0 τzz

(18)

and ⎞⎛ ⎞ 1 0 0 τxx 0 0 τ = − ⎝ 0 −1 2 0 ⎠ ⎝ 0 τyy 0 ⎠ ε˙ 0 0 τzz 0 0 −1 2 ⎞ ⎛ ⎛ ⎞⎛ ⎞ τxx 0 0 1 0 0 τxx 0 0 1 − ⎝ 0 τyy 0 ⎠ ⎝ 0 −1 2 0 ⎠ ε˙ = −2 ⎝ 0 − 2 τyy 0 ⎠ ε˙ . 0 0 τzz 0 0 −1 2 0 0 − 12 τzz ⎛

∇

(19)

Replacing these results into the equation of the rheological model, we can write in the case of a steady uniaxial extensional flow ⎞ ⎛ ⎞ 0 τxx 0 τxx 0 0 1 ⎟ ⎝ 0 τyy 0 ⎠ − 2λ1 ε˙ ⎜ ⎝ 0 − τyy 0 ⎠ 2 0 0 τzz 0 0 − 12 τzz ⎡ ⎛ ⎛ ⎞⎤ ⎞ 400 2 0 0 1 1 = 2η˙ε ⎣ ⎝ 0 −1 0 ⎠ − λ2 ε˙ ⎝ 0 1 0 ⎠⎦ , 2 0 0 −1 2 001 ⎛

(20)

and ⎧ 2 ⎪ ⎨τxx − 2λ1 ε˙ τxx = 2η˙ε − 4ηλ2 ε˙ τyy + λ1 ε˙ τyy = −η˙ε − ηλ2 ε˙ 2 ⎪ ⎩ τzz + λ1 ε˙ τzz = −η˙ε − ηλ2 ε˙ 2

.

(21)

Thus, the uniaxial extensional viscosity is given by ηE =

τxx − τyy 1 − λ2 ε˙ (1 + 2λ1 ε˙ ) = 3η . ε˙ (1 − 2λ1 ε˙ ) (1 + λ1 ε˙ )

(22)

1.3

Rheometry

25

For an upper convective Maxwell model, λ2 = 0, and ηE = 3η

1 . (1 − 2λ˙ε ) (1 + λ˙ε)

(23)

1.3 Rheometry While the rheological behaviour of Newtonian fluids is completely determined by the constant viscosity, η, the situation for non-Newtonian fluids is much more complicated. The rheological characterization of non-Newtonian fluids is widely acknowledged to be far from straightforward. Even the apparently simple determination of a shear rate versus shear stress relationship is difficult as the shear rate can only be determined directly if it is constant throughout the measuring device employed. Rheological measurements may be further complicated by nonlinear and thixotropic properties. The most commonly techniques used for measuring the viscoelastic properties of non-Newtonian fluids are considered below.

1.3.1 Shear Rheometry In the case of shear rheometry, the shear flow is generated between a moving and a fixed rigid surface or by a pressure difference over a tube (Fig. 1.6). Classic examples of shear flow geometries belonging to the first group include cone and plate and concentric cylinder. An example of shear flow geometry belonging to the second group is capillary or Poiseuille flow.

Fig. 1.6 Shear rheometers. (a) Cone and plate, (b) Concentric cylinder, (c) Capillary rheometer

26


1.3.1.1 Cone and Plate Rheometer The principal features of the cone and plate rheometer are shown schematically in Fig. 1.6a. The fluid sample, whose rheological properties are to be measured, is trapped between the circular conical disc at the top and the circular horizontal plate at the bottom. Two types of cone and plate rheometers are used for steady shear measurements, either a constant rate rheometer or a constant stress rheometer. In the constant rate instrument, the plate is rotated at a constant rate and the resulting shear stress is determined from the measurement of torque, M, on the cone. The shear rate, shear stress, τxy , and viscosity, η are given by γ˙ ∼ = , θ τxy =

μ (γ˙ ) =

3M , 2π R3

τxy 3M θ = , γ˙ 2π R3

(1.78)

(1.79)

(1.80)

where is the angular rotation rate of the plate, θ is the cone angle and R is the radius of the cone and plate. The θ angle between the cone and the plate is assumed to be small. Typically, θ is less than 4◦ . The small cone angle ensures that the shear rate is constant throughout the shearing gap, this being of particular interest when investigating time-dependent systems because all elements of the fluid sample experience the same shear history, but the small angle can lead to serious errors arising from eccentricities and misalignment. In the characterization of viscoelastic fluids, a force may result from the rotation of the plate which acts to separate the plates. The total thrust, F, on the bottom plate may then be used to determine the primary normal stress difference and typically placed in terms of a primary normal stress coefficient as N1 =

2F = ψ1 (γ˙ ) γ˙ 2 , π R2

(1.81)

and ψ1 (γ˙ ) =

2F θ 2 . π R2 2

(1.82)

All the above relationships are obtained under the assumption of negligible fluid inertial and edge effects, including surface tension. For accurate measurements, corrections for these possible errors are recommended in Carreau et al. (1997). The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the fluid sample is deformed by an oscillatory driver and the amplitude of the sinusoidal deformation is measured by a strain transducer. The force

1.3

Rheometry

27

deforming the fluid sample is measured by the small deformation of a relatively rigid spring to which is attached a stress transducer. On account of the energy dissipated by the fluid, a phase difference develops between the stress and the strain. The material functions are determined from the amplitudes of stress and strain and the phase angle between them. The major advantage of this type of viscometer is the constant shear rate throughout all the liquid. This is because, at a fixed radial position, the circumferential viscosity varies linearly across the gap between the cone and the plate. A consequence of this fact is that the cone and plate viscometer is well suited for time-dependent measurements, such as dynamic measurements or transient measurements that imply a step change in the rate of shear strain. On the other hand, it should be noted that measurements can usually be made at relatively low shear rates. With increasing shear rate, there is a tendency for the development of secondary flows in the fluid and the fluid will crawl out of the instrument under the influence of centrifugal forces and elastic instabilities. 1.3.1.2 Concentric Cylinders Rheometer Another rheometer commonly used to determine the apparent viscosity of nonNewtonian fluids is the concentric cylinder or Couette flow rheometer, schematically depicted in Fig. 1.6b. Typically the outer cylinder rotates with an angular speed and the torgue M on the inner cylinder, which is usually suspended from a torsion wire or bar, is measured. The apparent viscosity of the fluid is given by η=

M R2o − R2i 4π LR2o R2i

,

(1.83)

where Ro and Ri are the radii of the outer and inner cylinder, respectively. A narrow gap approximation is typically involved to avoid a priori selection of a rheologicalmodel. It is recommended that the narrow gap approximation only be used for Ri Ro ≥ 0.99 . The main sources of error in the concentric cylinder rheometer arise from end effects, wall slip, inertia and secondary flows, viscous heating effects and eccentricities due to misalignment of the geometry. To minimize end effects the lower end of the inner cylinder is a truncated cone. Secondary flows are of particular interest in the controlled stress instruments which usually employ a rotating inner cylinder. In this case, inertial forces cause an axisymmetric cellular secondary motion (Taylor vortices). The dissipation of energy by these vortices leads to overestimation of the torque. For a Newtonian fluid in a narrow gap, the stability criterion is (Macosko 1994) ρ 2 2 (Ro − Ri )3 Ri < 3400, whereas, for non-Newtonian fluids, the stability limit increases.

(1.84)

28


The maximum value of the shear rate achievable with concentric cylinders viscometers is, in most of the cases, not an instrument limitation but depends on the viscosity of the fluid sample. For high viscosity fluids, viscous heating may become a problem. For low viscosity fluids, the upper limit may be set by the occurrence of secondary flows. Usually, the maximum value of the shear rate is about 102 s–1 . At the other end of scale, it is possible to go down to shear rates as low as 10–2 s–1 , especially with high viscosity fluids. 1.3.1.3 Capillary Rheometer This method involves the laminar flow of a fluid through a small tube (Fig. 1.6c). In this case, the shear rate γ˙ has a maximum at the wall and zero in the centre of the flow. The flow is therefore non-homogeneous and capillary rheometers are restricted to measuring steady shear functions, i.e. steady shear stress – shear rate behaviour for time independent fluids. For an ideal viscometer, the flow rate is given by π R3 Q= 3 τw

τw τ 2 γ˙ (τ ) dτ ,

(1.85)

0

where τw = (R/2) (−p/L) is the shear stress at the wall of the tube, R is the tube radius, L is the tube length, and p is the pressure drop over the length L. For a Newtonian fluid, γ˙ (τ ) = τ/η, this equation yields π R4 Q= 8η

p − , L

(1.86)

from which η can be calculated using a value of Q obtained for a single value of (−p/L). For flow of unknown form, Eq. (1.85) yields (see, for example, White 1995) γ˙ (τw ) =

3n + 1 4Q , 4n π R3

(1.87)

with n =

d log τw

, d log π4Q 3 R

(1.88)

which is known as the Weissenberg–Rabinowitsch equation. For shear-thinning fluids, the apparent shear rate at the wall is less than the true shear rate. Thus at some radius, ς R, the true shear rate of a non-Newtonian fluid of apparent viscosity equals that of a Newtonian fluid of the same viscosity. The stress at this radius, ς τw , is independent of fluid properties and thus the true viscosity at this radius equals the apparent viscosity at the wall and the viscosity calculated from Eq. (1.86) is the true viscosity at a stress ς τw . Laun (1983) indicated that this method for correcting viscosity is as accurate as the Weissenberg–Rabinowitsch

1.3

Rheometry

29

method. Errors may occur due to wall slip, e.g. in the case of concentrated dispersions where the layer of particles may be more dilute near the wall than in the bulk flow. The layer near the wall has a much lower viscosity, resulting in an apparent slippage of the bulk fluid along the wall. In addition to these effects, viscous dissipation heating, fluid compressibility, change of viscosity with pressure, and flow instabilities can introduce errors into capillary viscometer measurements. The temperature rise associated with viscous dissipation can be reduced by using a smaller diameter capillary, since for shear-thinning fluids the rate of heat conduction to the wall increases more rapidly than viscous heat generation with decreasing tube radius. The intrinsic viscosity of polymer solutions is typically determined by measuring the viscosity using capillary viscometers due to their ability to precisely detect small differences in viscosity at low polymer concentration. Incorrect determination of the viscosity and, therefore, intrinsic viscosity can arise if the polymer solution is shear thinning and measurements must be made at low shear rates such that the viscosity equates to the zero-shear viscosity.

1.3.2 Extensional Rheometry As discussed in Sect. 1.2.1, extensional flow is fundamentally different from shear flow and extensional viscosity is a different material function from shear viscosity. The major difficulty in this type of rheometry is to generate a purely extensional flow, especially for low-viscosity fluids. In most of the cases, different measuring techniques give different results. There are, however, several types of flow geometries that can approximate a purely extensional flow, such as squeezing flow, stagnation point, entrance flow, or sheet stretching (Macosko 1994). Here we limit our attention to the opposed-jet and filament stretching techniques which are the most popular and promising methods for studying extensional properties of non-Newtonian fluids. 1.3.2.1 Opposed-Jet Rheometer The opposed jet rheometry was first introduced by Fuller et al. (1987) and a schematic diagram of the device is shown in Fig. 1.7. Fluid is drawn into opposed jets with the right nozzle arm fixed while the left nozzle arm is free to rotate about a pivot. The fluid exerts a hydrodynamic force onto the nozzles during flow, which is balanced by applying a torque, TM , to the pivot arm to prevent movement of the left arm. The force, FR used to balance the hydrodynamic force is related to the fluid extensional viscosity and can be used to define an apparent extensional stress difference (τc = τzz − τxx ) as: τc =

TM FR = , A AL

(1.89)

where L is the length of the lever arm between the nozzle and transducer, R is the nozzle radius, and A = π R2 is the area of the nozzle opening. Assuming a uniform

30


Fig. 1.7 Opposed-jet rheometer

jet entrance velocity, the apparent extensional rate, ε˙ , in the flow field defined in terms of the volumetric flow rate through a nozzle, Q, and the gap between the nozzles, dn , is ε˙ =

Q . Adn

(1.90)

The apparent extensional viscosity can then be calculated according to ηEa =

τc . ε˙

(1.91)

One problem with the opposed-jet apparatus is that an upturn in the extensional viscosity is measured at high rates of strain for Newtonian fluids of low viscosity which Hermansky and Boger (1995) associated with fluid inertia. By introducing a correction coefficient, they indicated the following relationship between the measured and corrected Trouton ratio ηa a(R) ρdn2 ηc = E − ε˙ , η η 4π LR2 η

(1.92)

where ηc is the corrected extensional viscosity and η is the shear viscosity. The parameter a(R) was found to be constant for a particular jet. It should be noted here that before the correction is applied to a viscoelastic fluid, it is recommended that it be applied to a Newtonian fluid with a similar shear viscosity to the viscoelastic fluid, in order to ensure accuracy. Corrections are not required for fluids with a shear viscosity larger than about 50 mPa·s.

1.3

Rheometry

31

1.3.2.2 Filament Stretching Rheometer The filament stretching device for determining the steady uniaxial extensional viscosity was first developed by Tirtaatmadja and Sridhar (1995). A depiction of the instrument is shown in Fig. 1.8. The fluid sample is held between two disks which move apart at an increasing rate so that the extension rate along the filament midpoint is held constant. The instrument has the advantage over the opposed-et apparatus by producing a flow field which is a very close approximation to pure uniaxial extensional flow. The extensional stress is derived from the surface tension of the fluid, σ , reducing the calculation of extensional viscosity to ηEa =

σ σ/Rm =− , ε˙ 2dRm /dt

(1.93)

where Rm is the midpoint radius of the filament. The instantaneous deformation rate experienced by the fluid element at the axial midplane can be determined in a filament rheometer in real-time using the high resolution laser micrometer which measures Rm (t). Numerous experimental variants have been developed, and by precisely controlling the endplate displacement profile it is now possible to reliably attain the desired kinematics. The results can be best represented on a “master curve” showing the evolution of the imposed axial strain on the endplate versus the resulting radial strain at the midplane. The excellent review by McKinley and Sridhar (2002) surveys some of the recent developments in filament stretching extensional rheometry.

Fig. 1.8 Filament stretching rheometer

32


1.3.3 Microrheology Measurement Techniques The traditional rheometers described so far measure the rheological properties of fluids using milliliter-scale material samples. In contrast, microrheology is tipically concerned with flows around microscale and nanoscale particles that are embedded within a very small (microliter or even nanoliter) volume of the test fluid. Moreover, conventional rheometers provide an average measurement of the bulk response, and do not allow for local measurements in inhomogeneous systems. To address these issues, a new type of microrheology measurement techniques has emerged. Two classes of microrheology tests can be distinguished. The first class of microrheology tests exploits the Brownian motion of the tracer particles and is termed passive microrheology. Because no external forces are applied, passive experiments always operate in the linear viscoelastic regime and are suitable for soft media. The second class uses active manipulation of the particles by applying an external driving force. These techniques include, for example, the use of magnetic or optical tweezers. Active methods are useful if large stresses have to be applied to stiff media as well as for investigations of nonlinear response and non-equilibrium phenomena. 1.3.3.1 Passive Measurement Methods Passive microrheology is based on an extension of the concepts of Brownian motion of particles in simple liquids. The motion of particles within a liquid can be quantified with the diffusion coefficient, D, which is a measure of how rapidly particles execute a thermally driven random walk. Given the particle size, temperature, and viscosity, η, the diffusion coefficient in a viscous liquid can be determined by the Stokes–Einstein relation D=

kB T , 6π aη

(1.94)

where kB is Boltzmann’s constant, a is the particle radius, and T is the absolute temperature. In the above equation, it is assumed that particles are spherical and rigid and no heterogeneities exist. The dynamics of particle motion are usually described by the time-dependent mean-square displacement, r2 (t). When particles diffuse through a test fluid or are transported in a non-diffusive manner the mean-square displacement becomes nonlinear with time and can be described with a time-dependent power law, r2 (t) ∝ tα . The slope of the log–log plot of the r2 (t), denoted by α (also referred to as the diffusive exponent), describes the mode of motion a particle is undergoing and is defined for physical processes between 0 ≤ α ≤ 2. The time-dependent mean-square displacement can be used to obtain rheological properties of a complex fluid microenvironment. The Stokes–Einstein relation correlates the particle radius, a, and the term r2 (t) provide the creep compliance D=

π a r2 (t) . kB T

(1.95)

1.3

Rheometry

33

The time-dependent creep compliance, or material deformation under a stepincrease in stress, can be directly obtained from the mean-square displacement. This provides a measure of the viscosity or the elastic modulus in viscous or elastic samples, respectively. In addition, a method was developed to estimate the elastic, storage modulus and the loss modulus based on the logarithmic slope of the mean-square displacement (Mason 2000). Another passive microrheology test is the fluorescence correlation spectroscopy which is based on the principles of dynamic light scattering. Fluorescence correlation spectroscopy uses a laser beam focused on a small volume within the test fluid and photon detectors to record fluctuations in fluorescence resulting from the movement of fluorescent molecules into and out of the volume (Hess et al. 2002). The method is well suited for the study of viscoelasticity within a cell. 1.3.3.2 Active Measurement Methods In addition to the passive techniques described above, fluids may be externally manipulated using active microrheology techniques. Externally applied forces, acting on particles in a test fluid, result in local stress and movement of the particles through more elastic regions. Active forces can be applied to particles in a test fluid through magnetic and laser tweezers. For example, paramagnetic and ferromagnetic microbeads can be manipulated by magnetic-field gradients and used to apply large forces in a viscoelastic fluid. Magnetic-field gradients applied to paramagnetic beads can generate pulling forces (Bausch et al. 1998; Karcher et al. 2003), whereas their application to ferromagnetic particles can generate torsional forces (Fabry et al. 2001, 2003). Forces up to 10 nN (Karcher et al. 2003) can be generated using paramagnetic beads, and forces of several pN (Trepat et al. 2007) can be generated using ferromagnetic beads. Similarly, laser tweezers have been used to manipulate particles, cells, and bacteria (Ashkin et al. 1987; Ashkin and Dziedzic 1987) by applying small forces to them and then measuring their displacements with high accuracy. Trapped particles can be restricted to a specific region and passively monitored (Tolic-Norrelykke et al. 2004), or an active force can be locally applied and its effects on internal structure measured. Laser tweezers have been used to trap spherical, polymeric particles or naturally occurring granules within cells (Tolic-Norrelykke et al. 2004). To measure rheological properties, optical tweezers are used to apply a stress locally by moving the laser beam and dragging the trapped particle through the surrounding material; the resultant bead displacement is interpreted in terms of viscoelastic response. Elasticity measurements are possible by applying a constant force with the optical tweezers and measuring the resultant displacement of the particle. Alternatively, local frequencydependent rheological properties can be measured by oscillating the laser position with an external steerable mirror and measuring the amplitude of the bead motion and the phase shift with respect to the driving force (Ou-Yang 1999). This method produces forces lower than 100 pN and can be used to measure the cell’s linear response (Peterman et al. 2003). Magnetic tweezers have the advantage over their optical counterparts that they generate no heat in the sample examined, can have a

34


uniform force field over the entire field of view, and can orient objects regardless of their geometry. They do have the disadvantage that it is difficult to make multiple independent traps.

1.4 Particular Non-Newtonian Fluids We turn now to a particular class of non-Newtonian fluids, namely the biological fluids. They are rheologically complex due to their multi-component nature. The complexity of the biological fluids relies on the fact that such fluids are active, and can rearrange their microstructure to produce different properties, in order to achieve a precise function. As a result they are both elastic and viscous. The biological fluids whose rheology has been most studied are human blood, synovial fluid, saliva, and the cell constituents, cytoplasm and cytoskeleton. Blood is undoubtedly the most important biological fluid and its rheology is interesting from both theoretical and applied points of view.

1.4.1 Blood Blood is a suspension of cells in plasma. Plasma represents about 55% of the total blood volume. It is composed of mostly water (92% by volume), and contains dissolved proteins (6–8%), glucose, lipids, mineral ions, hormones, and carbon dioxide. Blood plasma has a density of approximately 1,025 kg/m3 (Lentner 1979). The cells are red blood cells (erythrocytes), white blood cells (leukocytes) of several types, and plateles. The red blood cells are biconcave disks, some 8.5 μm in diameter and of maximum thickness 2.5 μm. The cells consist of a highly flexible membrane, filled with a concentrated haemoglobin solution. The membrane, consisting of a lipid bilayer and a cytoskeleton (a network of protein molecules), exhibits viscoelastic properties. The elastic shear modulus (about 6×10–6 N/m) is several orders of magnitude lower that the modulus of isotropic dilatation (about 0.5 N/m) and so the membrane shears rapidly but resists area changes. Also, bending resistance is small unless very small radii of curvature are involved; the bending modulus is about 1.8×10–19 Nm (Evans 1983). Normal human blood has a hematocrit (volume fraction of red cells) of about 45%, and so red cells strongly influence the flow properties of blood. White blood cells are comparable in size to red blood cells but much less numerous. They are much stiffer than red blood cells and may contribute significantly to microvascular flow resistance (Schmid-Schönbein et al. 1981). There are several classes of white blood cells, e.g., granulocytes, which include neutrophils, basophils and eosinophils, monocytes, lymphocytes, macrophages, and phagocytes. They vary in size and properties, e.g., a typical inactivated neutrophil is approximately spherical in shape with a diameter of about 8 μm. The mechanical properties of white blood cells have been discussed by Schmid-Schönbein (1990). Platelets are discoid particles with a diameter of about 2 μm. Platelets, which are essential to the blood clotting process, are much

1.4

Particular Non-Newtonian Fluids

35

smaller than the red cells and do not contribute significantly to flow resistance. They are preferentially distributed near microvessel walls, probably as a result of hydrodynamic interaction with red cells (Tangelder et al. 1985). Under physiological conditions white blood cells occupy 1/600 of total cell volume while platelets occupy approximately 1/800 of total cell volume. The viscosity of plasma has been shown to be invariant with γ˙ (Newtonian fluid) and is dependent mainly on protein content and temperature. In normal conditions, the viscosity of plasma is about 1.1 cP. Whole blood is a non-Newtonian fluid. At small values of shear rate (γ˙ < 0.5 s–1 ), the apparent viscosity of whole blood shows a zero-shear plateau followed, at higher shear rates, by a decrease of viscosity with the shear rate. Blood is therefore a shear-thinning fluid. At very high values of shear rate (γ˙ > 102 s–1 ), the apparent viscosity of blood is almost constant indicating an infinite-shear plateau. At normal hematocrit content and at a temperature of 37◦ C, the zero-shear viscosity of blood is as high as 120 cP (Chmiel and Walitza 1980) while the infinite-shear viscosity is about 4 cP (MacKintosh and Walker 1973; Lowe and Barbenel 1988). The rheology of blood is primarily determined by the behaviour of the red blood cells at different shear rates. At sufficiently low shear rates the red blood cells agglomerate into column-like structures called rouleaux and the concentration of fibrinogen and immunoglobulins in the plasma is known to have an important role in this process (Baskurt and Meiselman 2003). At yet lower shear rates these rouleaux may develop branches and at even lower shear rates complex networks of red blood cells may be observed (Samsel and Perelson 1982; Baskurt and Meiselman 2003). At rest, human blood cells form a gel all over the sample and some researchers claim experimental evidence for the existence of a yield stress (Thurston 1993; Picart et al. 1998). Red cell aggregation or rouleaux formation is induced by many macromolecules, and particularly, fibrinogen and imunoglobulins contained in plasma (Brooks et al. 1970). On the other hand, increase in shear breaks down the bridging lattice and reduces the rouleaux length, with minimal aggregation for shear rate above 102 s–1 . As aggregates are broken down with increasing shear, the number of individual red cells increases. These align with the flow direction, causing further reduction in viscosity for larger values of shear rates. Chien (1970) has shown that for shear rate values up to 1 s–1 , aggregation dominates the viscous behaviour, whereas in the range 1–100 s–1 , deformation of red blood cells is the dominant factor. A comparison of this shear thinning characteristic of blood viscosity in the presence and absence of aggregating agents suggests that about 75% of the viscosity decrease is a result of the disruption of red cell aggregates, and 25% is due to red cell deformation in response to increased shear stresses (Lipowski 2005). At a given shear rate, blood viscosity rises exponentially with increasing red blood cell concentration (hematocrit) to a degree dependent on prevailing γ˙ . Blood viscosity is higher in men than in women because of the men’s higher hematocrit level. Furthermore, blood viscosity decreases with increasing temperature (Eckmann et al. 2000). The blood viscosity decreases with decreasing the diameter of the vessel. Fahreus and Lindqvist (1931) were the first to indicate that the flow resistance of blood in a cylindrical tube cannot be predicted on the basis of the viscosity of the blood as

36


measured in large scale rheometers. A large number of publications has addressed the dependence of apparent blood viscosity on tube diameter and hematocrit. The results of 18 studies were combined to a parametric description of apparent blood viscosity relative to the viscosity of plasma (relative apparent blood viscosity) as a function of tube diameter, D, and hematocrit, H, according to the equation (Pries et al. 1992), η = 1 + (0.45 − 1)

(1 − H)c − 1 . (1 − 0.45)c − 1

(1.96)

Here, η0.45 the relative apparent blood viscosity for a fixed hematocrit of 45%, is given by η0.45 = 220e−1.3D + 3.2 − 2.44e−0.06D

0.645

,

(1.97)

and c describes the shape of the viscosity dependence on hematocrit

−0.075D

c = 0.8 + e

1 −1 + 1 + 10−11 D12

+

1 . 1 + 10−11 D12

(1.98)

The apparent blood viscosity exhibits a strong decrease with decreasing tube diameter reaching a minimum at about 7 μm. At this value, the apparent viscosity of blood with a hematocrit of 45% is only 25% higher than that of plasma. Only at diameters below about 3.5 μm does the apparent viscosity increase above the level seen in large vessels. In the absence of shear, red blood cells only collide rarely so that aggregation tends to be a slow process (Samsel and Perelson 1982). Aggregation and disaggregation take place over differing non-zero time scales. Blood is therefore thixotropic (Huang et al. 1975), in the sense that when a step increase in shear rate is applied to blood the viscosity is a decreasing function of time. Blood thixotropy is exhibited mostly at low shear rates (up to 10 s–1 ) (Huang et al. 1987) and has a fairly long time scale. For example, initial resistance to flow start-up returns only slowly to normal blood, with well over 1 min of standing required (Mewis 1979). This suggests that thixotropy is of secondary importance in pulsatile blood flow, which has a time scale of approximately 1 s. Whole blood viscosity is an important physiological parameter (see, for a detailed discussion, Baskurt 2003 and the references therein). For example, the viscosity of whole blood was associated with coronary arterial diseases. Whole blood viscosity is significantly higher in patients with peripheral arterial disease than that in healthy controls. Other researchers investigated correlation between the hemorheological parameters and stroke. They reported that stroke patients showed two or more elevated rheological parameters, which included whole blood viscosity, plasma viscosity, red blood cell and plate aggregation, red blood cell rigidity, and hematocrit. It was also reported that both whole blood viscosity and plasma viscosity are significantly higher in patients with essential hypertension than in

1.4

Particular Non-Newtonian Fluids

37

healthy people. In diabetics, whole blood viscosity, plasma viscosity, and hematocrit are elevated, whereas red blood cell deformability is decreased. There is also a direct connection between whole blood viscosity and smoking, age, and gender. It was found that smoking and aging might cause the elevated blood viscosity. In addition, it was reported that male blood possessed higher blood viscosity, red blood cell aggregability, and red blood cell rigidity than premenopausal female blood, which may be attributed to monthly blood-loss. Blood is also a viscoelastic fluid. The deformations of the erythrocytes in flow and the storage and release of elastic energy that this implies, as well as the dissipation in blood due primarily to evolution of the erythrocyte networks (at low shear rate) and internal friction (at higher shear rates) (Anand and Rajagopal 2004), give rise to the viscoelastic character of blood (Chien et al. 1975). At normal hematocrit values, the viscous component, η , of the complex viscosity predominates over the elastic component, η (Chmiel and Walitza 1980). This suggests that blood viscoelasticity also has a secondary impact on blood flow at physiological hematocrit values. Thurston (1979) indicated that both components of the complex viscosity have relatively constant values for shear rates below 1.5 s–1 . In this range, the elastic and viscous components are approximately 3.9 mPa·s and 11.5 mPa·s, respectively. As the shear rate is increased beyond this level, blood displays a nonlinear viscoelastic behaviour, i.e. η and η are dependent on shear rate. The value of η starts dropping rapidly as the shear rate is increased beyond this level, diminishing to 0.1 mPa·s by 16 s–1 . This sharp decrease is connected to the breakdown of the blood microstructure formed by red blood cell aggregates. The speed of sound in blood was investigated by Bakke et al. (1975), where the following equation is given for the dependence of whole blood on hematocrit at a temperature of 37◦ C c = 1541.82 + 0.98 × H,

(1.99)

where c is sound speed in m/s and H is the hematocrit. At normal hematocrit content, this equation gives a value of c of 1,586 m/s. This is in close agreement with values reported by other authors; e.g., 1,590 m/s (Hughes et al. 1979) and 1,584.2 m/s (Collings and Bajenov1987). The sound speed in whole blood increases with temperature at a rate of c T = 1.3 m/s/◦ C. The surface tension of whole blood at normal hematocrit content is 5.6×10–2 N/m (Lentner 1979). This value was, however, measured at 24◦ C and no data at normal body temperature are available in literature. The density of whole blood is approximately 1,060 kg/m3 (Lentner 1979).

1.4.2 Synovial Fluid Synovial fluid is found in the diarthrodial joints where it forms a thin viscous film over the surface of the synovium and articular cartilage in the joint space. The composition of the synovial fluid is almost identical to that of plasma with

38


the exception of the large polymers like fibrinogen or larger globulins, which are reduced or not found at all in the synovial fluid due to the sieving action of the synovial capillary walls. Synovial fluid can be distinguished from plasma by the presence of hyaluronic acid and lubricin. These two molecules are the major determinants of synovial fluid viscosity and are of key importance for one of its main functions, which is to act as lubricant of the joint surfaces. Hyaluronic acid, or hyaluronan, is the most abundant glycosaminoglycan in mammalian tissue. It is present in high concentrations in connective tissue, such as skin, vitreous humor, cartilage, and umbilical cord, but the largest single reservoir is the synovial fluid of the diarthrodial joints, where concentrations of 0.5–4 mg/ml are achieved (Laurent and Fraser 1992; Fraser et al. 1997). The high concentration of hyaluronic acid in synovial fluid is essential for normal joint function, because hyaluronan confers exceptional viscoelasticity and lubricating properties to synovial fluid, particularly during high shear conditions. Under dynamic loading of diarthrodial joints, shear thinning and a reduction in viscosity occur because of decreased physical entanglements of hyaluronan molecules and their realignment to directions more parallel with the axis of articulation. These unique non-Newtonian rheological properties of hyaluronan not only reduce wear and attrition of articular cartilage during joint motion (Balazs et al. 1967; Balazs and Denlinger 1985) but also stabilize joints at low shear rates (Cullis-Hill and Ghosh 1987). At high loads, however, hyaluronan is not an effective lubricant. Here lubricin, which interacts with surfaceactive phospholipids, seems to have an important role (Simkin 1985). Furthermore, hyaluronan in the synovial fluid also bonds the opposing surfaces of the joints to each other. This creates tensile strength with little or no shear strength and enables opposing surfaces to slide freely across each other but limits their distraction (Wooley et al. 2005). The volume of the synovial fluid in normal human joints is quite small with approximately 0.5–2.0 ml (Dewire and Einhorn 2001; Mason et al. 1999). The synovial fluid undergoes continuous turnover by trans-synovial flow into synovial lymph vessels. As a result, water and protein in the synovial fluid are replaced within a period of 2 h or less. The turnover of hyaluronan is considerably slower with complete replacement of hyaluronan within about 38 h (Mason et al. 1999). The density of the synovial fluid is ρ = 1008–1015 kg/m3 (Duck 1990). Rheological studies have documented three types of non-Newtonian properties for the synovial fluid: shear-thinning, elasticity, and rheopexy. King (1966) seems to be the first who tested synovial fluids of bullocks using a cone-and-plate rheometer. He found that, at very small shear rates ( 103 Hz, the loss modulus becomes dominant. Experiments were also performed by introducing various drugs into the cell in order to create contraction or relaxation in the cytoskeleton. Similar qualitative properties were observed. The storage modulus increased with increasing frequency as a power law and the loss modulus also increased with increasing frequency with the same power law and same exponent up to frequencies of 10 Hz.

1.4.5 Other Viscoelastic Biological Fluids Some other biological fluids have non-Newtonian properties as well. For example, mucus from the respiratory tract is a viscoelastic fluid. Its viscoelasticity is influenced by bacteria and bacterial DNA. Cervical mucus and semen are other examples of viscoelastic biological fluids. A rheological description of these fluids is, however, beyond the scope of this book because there is no biomedical or bioengineering application where cavitation might occur in these fluids. The reader interested in the viscoelastic properties of these fluids are referred to the book of Fung (1993).

References Anand, M., Rajagopal, K.R. 2004 A shear-thinning viscoelastic fluid model for describing the flow of blood. Int. J. Cardiovasc. Med. Sci. 4, 59–68. Ashkin, A., Dziedzic, J.M., Yamane, T. 1987 Optical trapping and manipulation of single cells using infrared-laser beams. Nature 330, 769–771. Ashkin, A., Dziedzic, J.M. 1987 Optical trapping and manipulation of viruses and bacteria. Science 235, 1517–1520. Bakke, T., Gytre, T., Haagensen, A., Giezendanner, L. 1975 Ultrasonic measurement of sound velocity in whole blood. Scand. J. Clin. Lab. Invest. 35, 473–478. Balazs, E.A. 1968 Viscoelastic properties of hyaluronic acid and biological lubrication. Univ. Mich. Med. Center J. 9, 255–259. Balazs, E.A., Denlinger, J.L. 1985 Sodium hyaluronate and joint function. Equine Vet. Sci. 5, 217–228.

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Balazs, E.A., Watson, D., Duff, I.F., Roseman, S. 1967 Hyaluronic acid in synovial fluid. I. Molecular parameters of hyaluronic acid in normal and arthritic human fluids. Arthritis Rheum. 10, 357–376. Bandrup, J., Immergut, E.H. 1975 Polymer Handbook. Wiley, New York. Baskurt, O.K. 2003 Pathophysiological significance of blood rheology. Turk. J. Med. Sci. 33, 347–355. Baskurt, O.K., Meiselman, H.J. 2003 Blood rheology and hemodynamics. Semin. Thromb. Hemost. 29, 435–450. Bausch, R.A. Moller, W., Sackmann, E. 1999 Measurement of local viscoelasticity and forces in living cells by magnetic tweezers. Biophys. J. 76, 573–579. Bausch, A.R., Ziemann, F., Boulbitch, A.A., Jacobson, K., Sackmann, E. 1998 Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry. Biophys. J. 75, 2038–2049. Bernstein, B., Kearsley, E.A., Zapas, L. 1963 A study of stress relaxation with finite strain. Trans. Soc. Rheol. 7, 391–410. Bird, R.B., Curtiss, C.F., Armstrong, R.C., Hassanger, O. 1987 Dynamics of Polymeric Liquids: Fluid Mechanics. Wiley, New York. Broersma, S. 1960 Rotational diffusion constant of a cylindrical particle. J. Chem. Phys. 32, 1626–1631. Brooks, D., Goodwin, J.W., Seaman, G.V. 1970 Interactions among erythrocyes under shear. J. Appl. Physiol. 28, 172–177. Carreau, P.J., De Kee, D.C.R., Chhabra, R.P. 1997 Rheology of Polymeric Systems. Hanser, Cincinnati. Chien, S. 1970 Shear dependence of effective cell volume as a determinant of blood viscosity. Science 168, 977–979. Chien, S., King, R.G., Skalak, R., Usami, S., Copley, A.L. 1975 Viscoelastic properties of human blood and red cell suspensions. Biorheology 12, 341–346. Chmiel, H., Walitza, E. 1980 On the Rheology of Human Blood and Synovial Liquids. Research Studies Press, Chichister. Collings, A.F., Bajenov, N. 1987 Temperature dependence of the velocity of sound in human blood and blood components. Australas. Phys. Eng. Sci. Med. 10, 123–127. Cullis-Hill, D., Ghosh, P. 1987 The role of hyaluronic acid in joint stability – a hypothesis for hip displasia and allied disorders. Med. Hypotheses 23, 171–185. Davies, D.V., Palfrey. A. J. 1968 Some of the physical properties of normal and pathological synovial fluids. J. Biomechanics 1, 79–88. Dembo, M, Harlow, F. 1986 Cell motion, contractile networks, and the physics of interpenetrating reactive flow. Biophys. J. 50, 109–122. Dewire, P., Einhorn, T.A. 2001 The joint as an organ. In Osteoarthritis. Diagnosis and Medical/Surgical Management (Eds. R.W. Moskowitz, D.S. Howell, R.D. Altman, J.A. Buckwalter, and V.M. Goldberg). Saunders, Philadelphia, pp. 49–68. Doi, M., Edwards, S.F. 1986 The Theory of Polymer Dynamics. Clarendon, Oxford. Duck, F.A. 1990 Physical Properties of Tissue. Academic Press, London. Eckmann, D.M., Bowers, S., Stecker, M., Cheung, A.T. 2000 Hematocrit, volume expander, temperature, and shear rate effects on blood viscosity. Anesth. Analg 91, 539–545. Evans, E.A. 1983 Bending elastic modulus of red bllod cell membrane derived from buckling instability in micropipet aspiration test. Biophys. J. 43, 398–405. Evans, E., Yeung, A. 1989 Apparent viscosity and cortical tension of blood granulocytes determined by micropipet aspiration. Biophys. J. 56, 151–160. Fabry, B., Maksym, G., Butler, J., Glogauer, M., Navajas, D., Fredberg, J. 2001 Scaling the microrheology of living cells. Phys. Rev. Lett. 87, 148102. Fabry, B., Maksym, G.N., Butler, J.P., Glogauer, M., Navajas, D., et al. 2003 Time scale and other invariants of integrative mechanical behavior in living cells. Phys. Rev. E 68, 041914. Fahreus, R., Lindqvist, T. 1931 The viscosity of the blood in narrow capillary tubes. Am. J. Physiol. 96, 562–568.

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Macosko,C.W. 1994 Rheology: Principles, Measurements and Applications. VCH Publishers, New York. Mason, T. G. 2000 Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation. Rheol. Acta. 39, 371–378. Mason, R.M., Levick, J.R., Coleman, P.J., Scott, D. 1999 Biochemistry of synovium and synovial fluid. In Biology of the Synovial Joint (Eds. C.W. Archer, M. Benjamin, B. Caterson, and J.R. Ralphs). Harwood Academic, Amsterdam, pp. 253–264. Mastro, A.M., Keith, A.D. 1984 Diffusion in the aqueous compartment. J. Cell Biol. 99, 180–187. McKinley, G.H., Sridhar, T. 2002 Filament-stretching rheometry of complex fluids. Annu. Rev. Flui Mech. 34, 375–415. Mewis, J. 1979 Thixotropy – a general review. J. Non Newt. Fluid Mech. 6, 1–20. Mitchison, T., Kirschner, M. 1984 Dynamic instability of microtubule growth. Nature 312, 237–242. Nordbo, H., Darwish, S., Bhatnagar, R.S. 1984 Salivary viscosity and lubrication: influence of pH and calcium. Scand. J. Dent. Res. 92, 306–314. Oates, K.M.N., Krause, W.E., Jones, R.L., Colby, R.H. 2006 Rheopexy of synovial fluid and protein aggregation. J. R. Soc. Interface 22, 167–174. Ou-Yang, H.D. 1999 Design and applications of oscillating optical tweezers for direct measurements of colloidal forces. In Colloid-Polymer Interactions: From Fundamentals to Practice (Eds. R.S. Farinato and P.L. Dubin). Wiley, New York, pp. 385–405. Papanastasiou, A.C., Scriven L.E., Macosko, C.W. 1983 An integral constitutive equation for mixed flows: viscoelastic characterization. J. Rheol. 27, 387–410. Peterman, E.J.G., van Dijk, M.A., Kapitein, L.C., Schmidt, C.F. 2003 Extending the bandwidth of optical-tweezers interferometry. Rev. Sci. Instrum. 74, 3246–3249. Picart, C., Piau, J.-M., Galliard, H., Carpentier, P. 1998 Human blood shear yield stress and its hematocrit dependence. J. Rheol. 42, 1–12. Poslinski, A.J., Ryan, M.E., Gupta, R.K., seshadri, S.G., Frechette, F.J. 1988 Rheological behaviour of filled polymeric systems. 1. Yield stress and shear-thinning effects. J. Rheol. 32, 703–735. Pries, A.R., Fritzsche, A., Ley, K., Gaehtgens, P. 1992 Redistribution of red blood cell flow in microcirculatory networks by hemodilution. Circ. Res. 70, 1113–1121. Rantonen, P.J.F., Meurman, J.H. 1998 Viscosity of whole saliva. Acta Odontol. Scand. 56, 210–214. Rwei, S.P., Chen, S.W., Mao, C.F., Fang, H.W. 2008 Viscoelasticity and wearability of hyaluronate solutions. Biochem. Eng. J. 40, 211–217. Samsel, R.W., Perelson, A.S.1982 Kinetics of rouleau formation. Biophys. J. 37, 493–514. Sato, M., Wong, T.Z., Brown, D.T., Allen, R.D. 1984 Rheological properties of living cytoplasm: a preliminary investigation of squid axoplasm (Loligo pealei). Cell Motil. 4, 7–23. Sato, S., Koga, T., Inoue, M. 1983 Degradation of the microbial and salivary components participating in human dental plaque formation by proteases elaborated by plaque bacteria. Arch. Oral Biol. 28, 211–216. Schipper, R.G., Silletti, E., Vingerhoeds, M.H. 2007 Saliva as research material: biochemical, physicochemical and practical aspects. Arch. Oral Biol. 52, 1114–1135. Schurz, J. 1996 Rheology of synovial fluids and substitute polymers. J. Mat. Sci. Pure Appl. Chem. A 33, 1249–1262. ScottBlair, G.W., Williams, P.O., Fletcher, E.T.D., Markham, R.L. 1954 On the flow of certain pathological human synovial effusions through narrow tubes. Biochem. J. 56, 504–508. Schmid-Schönbein, G.W. 1990 Leukocyte biophysics. Cell Biophys. 12, 107–135. Schmid-Schönbein, G.W., Sung, K.P., Tözeren, H., Skalak, R., Chien, S. 1981 Passive mechanical properties of human leukocytes. Biophys. J. 36, 243–256. Schwartz, W.H. 1987 The rheology of saliva. J. Dent. Res. 66, 660–664. Silvers, A.R., Som, P.M. 1998 Salivary glands. Radiol. Clin. North Am. 36, 941.

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Chapter 2

Nucleation

Cavitation is critically dependent on the existence of nucleation sites. Cavitation starts when these nuclei enter a low-pressure region where the equilibrium between the various forces acting on the nuclei surface cannot be established. As a result, bubbles appear at discrete spots in low-pressure regions, grow quickly to relatively large size, and suddenly implode as they are swept into regions of higher pressure. In most conventional engineering contexts, the prediction and control of nucleation sites is very uncertain even when dealing with a simple liquid like water. Here we present data on the nuclei distribution in more complex fluids, such as polymer aqueous solutions and blood.

2.1 Nucleation Models Nucleation is the onset of a phase transition in a small region of a medium. The phase transition can be the formation of a tiny bubble in a liquid or of a droplet in saturated vapour. There are two main types of nucleation models: homogeneous nucleation and heterogeneous nucleation. Homogeneous nucleation takes place in a liquid phase without the prior presence of additional phases (Fuerth 1941; Church 2002). It is a consequence of the distribution of thermal energy among the molecules comprising a volume of liquid. Because some molecules will be more energetic than others, random processes will occasionally produce groupings of higher energy molecules. If the average energy is high enough, such a grouping of molecules represents an inclusion consisting of gas and vapour in the bulk of the liquid. Because statistical fluctuations in the distribution of thermal energy occur continuously, the small gas or vapour inclusions are constantly forming and disappearing. Such cavitation nuclei are, however, unstable. A gas bubble will dissolve in an undersaturated solution and the effect of surface tension will cause it to dissolve in a saturated solution. In supersaturated solutions, a bubble can be in equilibrium because the tendency for the bubble to dissolve due to surface tension is opposed by the tendency for the bubble to grow by diffusion of gas into it. This equilibrium is unstable; the bubble will grow or dissolve depending on whether the perturbation increases or decreases the bubble’s radius relative E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_2,

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50

2

Nucleation

to its equilibrium radius (Epstein and Plesset 1950). Therefore, a liquid would be free of bubbles after a short period of time. This does not imply that gas bubbles could not serve as cavitation nuclei. It does imply, however, that in order for gas bubbles to serve as cavitation nuclei, they must be stabilized at a size small enough to prevent their rising to the surface of the liquid, yet large enough so that they will grow when exposed to negative pressure as low as a few bars. In other words, a stabilization mechanism must exist for a gas bubble before it can act as a cavitation nucleus. Various types of stabilizing skins have been proposed. These skins usually consist of contaminants which somehow deposit themselves on the bubble’s surface and counteract the surface tension. Fox and Herzfeld (1954) proposed that surface active organic molecules could form a rigid skin around a gas bubble. This skin would be impermeable to gas diffusion and would be mechanically strong enough to withstand moderate hydrostatic pressures. Rather than the rigid skin of organic molecules proposed by Fox and Herzfeld, Yount (1979, 1982) has developed a stabilization theory in which the dissolution of gas bubbles is halted by a non-rigid organic skin. This so-called varying-permeability model, which employs a skin of surface-active molecules to stabilize the nucleus, has been mainly applied to bubble formation in supersaturated liquids. Although this model was originally used to explain bubble formation in gelatin upon rapid decompression, Yount noted that it can be applied to bubbles in water as well. It can also be applied to polymer solutions because polymers have surfactant properties. Surfactants present a barrier to mass transport and reduce the surface tension at a liquid-gas interface (Borwankar and Wassan 1983). Both mechanisms increase the stability of nuclei against dissolution (Porter et al. 2004). Furthermore, since surfactants are prelevant in biological systems, this model may be particularly important for applications involving decompression sickness and medical ultrasonics. In heterogeneous nucleation small pockets of gas are stabilized at the bottom of the cracks or crevices found on hydrophobic solid impurities in the liquid (Strasberg 1959; Apfel 1970; Atchley and Prosperetti 1989). Liquids normally contain a large number of solid impurities with a very irregular surface consisting of grooves or pits (Crum 1979). As is schematically shown in Fig. 2.1, a crevice stabilized gas nucleus can have an interface that is concave towards the liquid. Due to surface tension, the pressure of the gas in the nucleus can therefore be less than the pressure in

Fig. 2.1 The crevice model of nucleation: (a) Stabilization mechanism of nuclei. (b) Nucleus starts to grow into a bubble when the pressure in the surrounding liquid is reduced

2.1

Nucleation Models

51

the liquid, and if gas diffuses from the nucleus, so long as the contact line is pinned, the concavity will increase, reducing the pressure of gas. Hence such a nucleus can persist without dissolving completely into the liquid. The origin of such nuclei has been explained by considering the flow of a liquid onto a hydrophobic surface with crevices (Atchley and Prosperetti 1989). The crevice model is useful for explaining the hysteresis effect of pressurization on cavitation threshold (Crum 1980). The cavitation threshold increases because pressurization causes the crevice to shrink and gas diffuses into the surrounding liquid. After the pressure is released, a smaller pocket of gas exists in the crevice requiring a larger negative pressure to produce nucleation. Another explanation of the origin and persistence of nuclei is that ordering of liquid molecules adjacent to solid surfaces leads to local hydrophobicity in regions of concavity of an otherwise non-hydrophobic surface (Mørch 2000). This explanation suggests that the resulting voids have interfaces which are convex toward the liquids, and that their persistence is due to a resonant behaviour forced by ambient vibrations. Cavitation nuclei are not always permanently stabilized. Short-lived nuclei can also formed by radiation. Although many theories have been proposed to explain this phenomenon, the one that seems to have the most experimental support is the thermal spike model (Seitz 1958). In this model, a positive ion is created by the radiation-matter interaction. This ion quickly liberates its energy, generating neighbouring atoms that are thermally excited. If tension exists within the liquid, this region can produce a vapour bubble that expands and eventually results in a cavitation event. Example: Classical Theory of Homogeneous Nucleation According to classical theory of homogeneous nucleation (see, for example, Frenkel 1955), a nucleus is spontaneously generated as a result of density fluctuations in the metastable liquid phase in the form of a small vapour bubble of radius r. A minimum reversible work required to form a nucleus of new phase depends on the radius of the bubble and arrives a maximum at the critical radius rc . The nucleation rate J, which determines the average number of nuclei formed in a unit volume of the metastable phase per unit time, is proportional to the probability of having a critical nucleus J = J0 exp (−Wmin /kB T),

(1)

where Wmin =

4 2 πr σ 3 c

(2)

determines the nucleation barrier, which is equal to the minimum reversible work required to form a critical size nucleus, σ is the surface tension, kB is the Boltzmann constant, and J0 is a factor which does not depend on the critical radius rc and changes only slightly with the depth of penetration into the metastable state. All

52

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Nucleation

modifications introduced in the theory later do not change the general result of the classical theory given by Eqs. (1) and (2) and concern only details of calculations of the kinetic prefactor J0 and the nucleation barrier Wmin in different cases of metastable states (see, for example, Debenedetti 1996). In the classical theory of homogeneous nucleation the critical radius and the nucleation barrier can be calculated with the Gibbs equations (Landau and Lifshitz 1980) rc = Wmin =

2σ ν , μ(P) − μ (P)

σ 3ν2 16π , 3 [μ(P) − μ (P)]2

(3) (4)

where P is the bulk phase pressure, v is a specific volume of the nucleus, and μ (P) and μ(P) are the chemical potentials of the nucleus and of the metastable bulk fluid phase, respectively. A nucleus with radius less than a critical size rc requires energy for further growth and usually disappears without reaching the critical size. A nucleus with radius larger than rc grows freely with decrease of free energy, and a phase transition into a thermodynamically stable vapour phase takes place. Equations (3) and (4) can be applied for the formation of liquid droplets in supercooled vapour as well as for the formation of vapour bubbles in superheated liquid at moderate positive pressures and in the critical region. In metastable liquids at low temperatures considerable negative pressures are observed. In this case, Eqs. (3) and (4) are not more applicable. Particularly for large negative pressures the equations for rc and Wmin developed by Fisher (1948) can be used rc = − Wmin =

2σ , P

16π σ 3 . 3 P2

(5) (6)

However, these equations in turn fail at zero and small positive pressures where they give an unphysical divergence rc → ∞ and Wmin → ∞. A better result in this region can be obtained with the theory developed by Blander and Katz (1975). They obtained that the nucleation barrier Wmin is defined by Eq. (2), and found for the critical radius rc =

2σ v ∼ 2σ , = ∗ PV − P (P − P) δ

(7)

where P∗ is the saturation pressure at given temperature T. The correction factor δ takes into account the effect of the pressure P in the metastable liquid on the vapour pressure pv in the nucleus and is given by the equation

2.2

Nuclei Distribution

53

ρV 1 δ∼ + =1− ρL 2

ρV ρL

2 ,

(8)

where ρ L is the density of the liquid and ρ V the density of the vapour. Equations (7) and (8) are accurate for values of P at least up to 0.1 MPa, but are not valid in the critical region where the ratio ρV /ρL ∼ = 1 is not small and analytical expansion (8) is not more applicable. In the theory of homogeneous nucleation the mean time of formation of a critical nucleus in a volume V (9) tM = (JV)−1 determines the lifetime of the metastable state. The homogeneous nucleation limit of the metastable state is determined as a locus of the constant lifetime tM = const.

2.2 Nuclei Distribution The basic questions we want to answer in this section are how big are the nuclei and how many are these of each size. Data are presented for water, various polymer solutions, and blood. No information is available in the literature for the case of synovial liquid and saliva.

2.2.1 Distribution of Cavitation Nuclei in Water Several methods have been used to investigate the distribution of cavitation nuclei in water. Yilmaz et al. (1976) and Ben-Yosef et al. (1975) used the light scattering method, Gates and Bacon (1978) used a holographic technique, while Gavrilov (1969) used acoustic methods. Measurements of nuclei distribution using a Coulter counter were performed by Ahmed and Hammitt (1972), Pynn et al. (1976) and Oba et al. (1980). The Coulter counter detects change in electrical conductance of a small aperture as fluid containing cavitation nuclei is drawn through. A typical apparatus has one or more microchannels that separate two chambers containing electrolyte solutions. When a nucleus flows through one of the microchannels, it results in the electrical resistance change of the liquid filled microchannel. This resistance change can be recorded as voltage pulses, which can be correlated to the size of cavitation nuclei. Another direct measurement of the presence of cavitation nuclei is achieved when a liquid sample is passed through a region of known low pressure. Nuclei with radii that exceed a certain value radius will cavitate. The event rate of these cavitating bubbles can then be counted by visual observations. Moreover, when a cavitating bubble is convected to a region of higher pressure downstream, it will collapse producing an acoustic emission. The noise pulses can be detected and counted, giving another independent measurement of the nuclei. Devices that measure nuclei through inducing cavitation events are called cavitation susceptibility meters (Chambers et al. 1999).

54

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Nucleation

Some attempts were made to obtain a relationship that describes the distribution of cavitation nuclei in water. Gavrilov (1969) reported that the number of bubble nuclei is inversely proportional to the nuclei radius. Ahmed and Hammitt (1972) indicated that the distribution of cavitation nuclei can be described by pV , 4.838 σ v2/3 + pv

N(v) =

(2.1)

where N(v) is the number of nuclei of volume v, V the total gas content, σ the surface tension, and p is the ambient pressure. If 4.838 σ v2/3 > pv then N(v) ∝ d−2 , where d is the gas nuclei diameter. Shima et al. (1985) indicated that, in the range 2 μm < d < 20 μm, the gas nuclei distribution in water can be described by N(d) =

M , dn

(2.2)

where M is a constant. They found that the values of the exponent n lies between 2 and 4, in agreement with the results of Gavrilov (1969) who indicated n = 3.5. In a later study, Shima and Sakai (1987) obtained a more general equation for the size distribution of bubble nuclei in the form: N (r) =

M − nK (ln r−ln α)2 e 2 , rn

(2.3)

where r is the nuclei radius and M, n, K, and α are constants. They found good agreement with the experimental results reported by Ahmed and Hammitt (1972), Ben-Yosef et al. (1975), and Klaestrup-Kristensen et al. (1978).

2.2.2 Distribution of Cavitation Nuclei in Polymer Solutions Oba et al. (1980) and Shima et al. (1985) have measured the distribution of cavitation nuclei in water and various polymer solutions using a Coulter counter. They indicated that the size range of the nuclei is from 2 to 50 μm in radius, and the number of small nuclei below 7 μm represents more than 50% from the total number of nuclei. Oba et al. (1980) investigated the influence of polyethylene oxide concentration on the nuclei size distribution (Fig. 2.2). They found that the number of nuclei increases with the polymer concentration for nuclei diameters smaller 14 μm. For a diameter of about 12 μm, the number of bubble nuclei is one order of magnitude larger than in the case of water. However, for nuclei diameters larger than 14 μm, a significant reduction of the number of cavitation nuclei was observed. For a diameter of 35 μm, the number of nuclei in the 100 ppm polyethylene solution is one order of magnitude smaller than in the case of water.

2.2

Nuclei Distribution

55

Fig. 2.2 Nuclei distribution in a polyethylene oxide (PEO) aqueous solution. Adapted from Oba et al. (1980)

Shima et al. (1985) measured the cavitation nuclei distribution, in the range 2–20 μm, in three polymer aqueous solutions, namely a 100 ppm polyethylene oxide (Polyox) aqueous solution, a 2,000 ppm hydroxyethylcelullose aqueous solution, and a 50 ppm polyacrylamide aqueous solution (Fig. 2.3). For nuclei diameters larger than 3 μm they also found a decrease of the number of bubble nuclei in comparison to the case of water. The largest reduction was observed in the polyacrylamide and polyethylene solutions, while the results obtained in the hydroxyethylcelullose solution are almost similar to the case of water. They also indicated that the scaling law between the number of bubble nuclei and the nuclei diameter is not affected by the polymer additives.

2.2.3 Cavitation Nuclei in Blood The first attempts to detect cavitation in blood within the abdominal aorta of dogs exposed in vivo to lithotripsy have not proved successful although cavitation was observed in blood under in vitro conditions (Williams et al. 1988). Similar observations have been made by Deng et al. (1996). Lee et al. (1993) investigated bubble formation in the inferior vena cavae of dead rats after 6–15 h exposures to air at 12.3 MPa and decompression to 0.1 MPa at 1.36 MPa/min. Bubbles were detected by light microscopy, buoyancy, and underwater dissection. No bubbles were formed in 42 blood-filled vena cavae that were

56

2

Nucleation

Fig. 2.3 Nuclei distribution in various polymer aqueous solutions. Adapted from Shima et al. (1985)

isolated from the minor circulation by ligatures, but bubbles were always observed in unisolated vena cavae. Their results indicate that nuclei are not present in blood, even at supersaturations that are significantly higher than those experienced in vivo. One explanation for this result is that the continuous filtration of impurities by the body allows the presence of cavitation nuclei in only minute amounts, and only in particular sites. This observation concurs with the finding that the cavitation threshold for water doubles upon filtration to 2 μm (Greenspan and Tschiegg 1967). More recently, Chambers et al. (1999) investigated the nuclei characteristics of blood using a cavitation susceptibility meter in an ex vivo sheep model. This hydrodynamic method measures the nuclei threshold pressure by subjecting the fluid to a certain characterized flow. All nuclei with a critical pressure higher than the minimum pressure within the device will cavitate, and the number of activated nuclei was determined by counting the cavitation events. The nuclei concentration of blood was measured to be at most 2.7 nuclei per litre and the authors estimated that the radius of the nuclei is on the order of 0.3 μm. However, they noted that these values may be even lower in an in vivo situation. Chappel and Payne (2006) suggested that cavitation nuclei could originate from tissues or microcapillaries and migrate into blood circulation. The contact between adjoining endothelial cells on the capillary walls could be a site for crevice nuclei. The effect of muscular contraction on crevices might be expected to squeeze the gas pocket and potentially cause the release of bubbles. While the concept of in

2.3

Tensile Strength

57

vivo hydrophobic crevices remains a theoretical possibility, none have yet been identified. No bubble formation was observed when isolated endothelium in contact with blood was decompressed (Lee et al. 1993). The extravascular space could be an alternative location: as extravascular gas nuclei expand, they might rupture capillaries, thereby seeding the blood with gas (Vann 2004). It has been also suggested that musculoskeletal activity could generate surfaceactive molecules that stabilize the nuclei and increase their lifetime (Hills 1992). On the other hand, there have been studies that demonstrate the beneficial effect of surfactants on bubble elimination. The addition of surfactants to blood makes it feasible to manipulate interfacial stresses and prevent or reduce formation of the adhesion responsible for trapping intravascular gas bubbles. In vivo studies have shown that the addition of surfactants favorably alters the patterns of deposition and accelerate the rates of clearance of bubbles (Suzuki et al. 2004). While surfactants could play a role first in nuclei stabilization, they could also be involved at last in vascular bubble elimination.

2.3 Tensile Strength It is important to realise that cavitation is not necessarily a consequence of the reduction of pressure to the liquid’s vapour pressure, the latter being the equilibrium pressure, at a specified temperature, of the liquid’s vapour in contact with an existing free surface. Cavity formation in a homogeneous liquid requires a stress sufficiently large to rupture the liquid. This stress represents the tensile strength of the liquid at that temperature (Brennen 1995; Trevena 1987; Young 1989). Several methods have been employed to obtain the tensile strength of water. The first to be used was the Berthelot tube technique: a vessel is filled with liquid water at high temperature and positive pressure, then sealed and cooled down at constant volume. The liquid sample follows an isochore and is brought to negative pressure. Berthelot claimed that he had reached –5 MPa in a glass ampoule completely filled with pure water (Berthelot 1850). Another method was designed by Briggs (1950): by spinning a glass capillary filled with water, he obtained a minimum value of the tensile stress of –27.7 MPa at 10◦ C; the tension falls to a much smaller value at lower temperature (down to –2 MPa at 0◦ C). Shock tube and bullet piston experiments generate negative pressure by reflection of a compression wave travelling in water at an appropriate boundary. This type of experiments has been reconsidered several times and the presently accepted results are around –10 MPa (Williams and Williams 2000). It is worth noting here that even larger values of the tensile strength of water were obtained. Zheng et al. (1991) used an improved version of the static Berthelot method by using synthetic water inclusions in quartz. A quartz crystal with cracks is autoclaved in the presence of liquid water. Water fills the cracks which then heal at high temperature, thus providing low density water in a small Berthelot tube. They reported a maximum tension of –140 MPa at 43◦ C. This result is similar to that obtained by Roedder (1967) who reached –100 MPa with water inclusions in natural rocks.

58

2

Nucleation

Despite numerous studies, the precise role of non-Newtonian properties in determining cavitation threshold remains unclear. Most previous work in this area has considered polymer solutions – fluids made non-Newtonian by polymeric additives (Trevena 1987). Under conditions of dynamic stressing by pulses of tension there is evidence that polymer additives can lower cavitation threshold. An example has been reported by Sedgewick and Trevena (1978) who studied the cavitation properties of water containing polyacrylamide additives by the bullet-piston reflection method. Williams and Williams (2000) have shown that the latter method, which involves the conversion of a compressional pulse to a rarefaction at the free surface of a column of liquid, provides realistic estimates of tensile strength for water and other Newtonian fluids (Williams and Williams 2002). The experimental arrangement used by Williams and Williams (2002) consists of a cylindrical, stainless steel tube closed at its lower end by a piston (Fig. 2.4a). The piston’s lower surface is coupled to a stun-gun which generates a pressure pulse in a column of liquid within the tube. The upper flange connects the tube to a regulated oxygen-free nitrogen supply and a pressure gauge. Pressure changes within the liquid are monitored using three dynamic pressure transducers mounted

Fig. 2.4 The bullet-piston method for estimating the tensile strength of liquids. (a) Schematic of the cavitation threshold apparatus. (b) Pressure record obtained from a pressure transducer in a experiment on a sample of distilled water. (c) Cavitation threshold of distilled water. Reproduced with permission from Williams and Williams (2002). © IOP Publishing Ltd

References

59

in mechanically isolated ports in the wall of the tube. The main features of a typical pressure record obtained from a pressure transducer in an experiment on a sample of distilled water are shown in Fig. 2.4b, in which the data are presented in terms of transducer output in unscaled ADC units (positive values correspond to positive pressure and vice versa). A pressure pulse (feature “1” in Fig. 2.4b) is followed immediately by a tension pulse (“2”) and thereafter the record comprises “secondary” pressure-tension cycles (“3–4”, “5–6”, etc.) associated with cavitational activity. The method involves regulating a static pressure, Ps , in the space above the liquid, Ps being increased gradually in a series of dynamic stressing experiments. From the dynamic pressure records a measurement is made of the time delay, τ i , between the peak incident pressure (“1” in Fig. 2.4b) and the first pressure pulse arising from cavity collapse (“3” in Fig. 2.4b). Under tension, cavities grow from pre-existing nuclei within the liquid and eventually collapse and rebound, emitting a pressure wave into the liquid as they do so. Hence the interval τ i , which encompasses the attainment of maximum cavity radius and its subsequent decrease to a minimum value, is reduced by increasing Ps (τ i therefore provides a convenient measure of cavitational activity). The experiment involves the transmission of tension by the liquid to the face of the piston and it follows that in the case of experiments in which cavitation is detected, the magnitude of the tension transmitted by the liquid is sufficient to develop a transient, net negative pressure in the presence of a background static pressure Ps . Thus an estimate of the magnitude of tension capable of being transmitted by the liquid can be obtained from a knowledge of Ps . The time delay, τ o , between pulses corresponding to “1” and “2” in Fig. 2.4b represents the time required for the upward travelling pressure wave to return, as tension, to the lower transducer location. It also represents the smallest time interval for which a cavity growth-collapse cycle could occur (given that a bubble would have to grow and collapse infinitely quickly in order that τ i = τ o ). Thus the tensile strength can be estimated by extrapolation of the data in Fig. 2.4c to that value of the pressure Ps at which τ i = τ o , this condition representing the complete suppression of cavitation. Bullet-piston work has demonstrated a reduction of liquid effective tensile strength in non-Newtonian polymer solutions, the reduction increasing with increasing polymer concentration (Williams and Williams 2002). However, when this system was investigated using an ab initio technique, the cavitation threshold was found to be increased by the same polymer additive (Overton et al. 1984; Brown and Williams 2000). When subjected to quasi-static stressing (in a modified Berthelot tube) the presence of polymer made no discernible difference to the effective tensile strength of the liquid (Trevena 1987).

References Ahmed, O., Hammitt, F.G. 1972 Cavitation nuclei size distribution in high speed water tunnel under cavitating and non-cavitating conditions. Univ. Michigan ORA Rep. UMICH 013570-23-T. Apfel, R.E. 1970 The Role of impurities in cavitation-threshold determination. J. Acoust. Soc. Am. 48, 1179–1186.

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Atchley, A.A., Prosperetti, A. 1989 The crevice model of bubble nucleation. J. Acoust. Soc. Am. 86, 1065–1084. Berthelot, M. 1850 Sur quelques phenomenes de dilatation force des liquids. Ann. Chim. Phys. 30, 232–237. Ben-Yosef, N., Ginis, O., Mahlab, P., Weity, A. 1975 Bubble size distribution measurement by Doppler viscometer. J. Appl. Phys. 46, 738–740. Blander, M., Katz, J.L. 1975 Bubble nucleation in liquids. AIChE J. 21, 833–848. Borwankar, R.P., Wassan, D.T. 1983 The kinetics of adsorption of surface active agents at gasliquid interface. Chem. Engng Sci. 25, 1637–1649. Brennen, C.E. 1995 Cavitation and Bubble Dynamics. Oxford University Press, Oxford. Briggs, L.J. 1950 Limiting negative pressure of water. J. Appl. Phys. 21, 721–722. Brown, S.W.J., Williams, P.R. 2000 The tensile behaviour of elastic liquids under dynamic stressing. J. Non Newt. Fluid Mech. 90, 1–11. Chambers, S.D., Bartlett, R.H., Ceccio, S.L. 1999 Determination of the in vivo cavitation nuclei characteristics of blood. ASAIO J. 45, 541–549. Chappel, M.A., Payne, S.J. 2006 A physiological model of gas pockets in crevices and their behaviour under compression. Respir. Physiol. Neurobiol. 152, 100–114. Church, C.C. 2002 Spontaneous homogeneous nucleation, inertial cavitation and the safety of diagnostic ultrasound. Ultrasound Med. Biol. 10, 1349–1364. Crum, L.A. 1979 Tensile strength of water. Nature 278, 148–149. Crum, L.A. 1980 Acoustic cavitation threshold in water. In Cavitation Inhomogeneities in Underwater Acoustics (Ed. W. Lauterborn), Springer, New York, pp. 84–89. Debenedetti, P.G. 1996 Metastable Liquids. Concepts and Principles. Princeton University Press, Princeton. Deng, C.X., Xu, Q., Apfel, R.E., Holland, C.K. 1996 In vitro measurements of inertial cavitation thresholds in human blood. Ultrasound Med. Biol. 22, 939–948. Epstein, P., Plesset, M. 1950 On the stability of gas bubbles in liquid-gas solutions. J. Chem. Phys. 18, 1505–1509. Fisher, J.C. 1948 The fracture of liquids. J. Appl. Phys. 19, 1062–1067. Frenkel, J. 1955 Kinetic Theory of Liquids. Dover, New York. Fox, F., Herzfeld, K. 1954 Gas bubbles with organic skin as cavitation nuclei. J. Acoust. Soc. Am. 26, 984–989. Fuerth, R. 1941 On the theory of the liquid state. Proc. Camb. Philosph. Soc. 37, 252–290. Gates, E.M., Bacon, J. 1978 A note on the determination of cavitation nuclei distributions by holography. J. Ship Res. 22, 29–31. Gavrilov, L.R. 1969 On the size distribution of gas bubbles in water. Sov. Phys. Acoust. 15, 22–24. Greenspan M., Tschiegg, C.E. 1967 Radiation-induced acoustic cavitation; apparatus and some results. J. Res. NBS 71C, 299–312. Hills, B.A. 1992 A hydrophobic oligolamellar lining to the vascular lumen in some organs. Undersea Biomed. Res. 19, 107–120. Klaestrup-Kristensen, J., Hansson, I., Mørch, K.A. 1978 A simple-model for cavitation erosion of metals. J. Phys. D Appl. Phys. 11, 899–912. Landau, L.D., Lifshitz, E.M. 1980 Statistical Physics. Pergamon, New York. Lee, Y.C., Wu, Y.C., Gerth, W.A., et al. 1993 Absence of intravascular bubble nucleation in dead rats. Undersea Hyperb. Med. 20, 289–296. Mørch, K.A. 2000 Cavitation nuclei and bubble formation – a dynamic liquid-solid interface problem. J. Fluids Eng. 122, 494–498. Oba, R., Kim, K.T., Niitsuma, H., Ikohagi, T., Sato, R. 1980 Cavitation-nuclei measurements by a newly made Coulter-counter without adding salt in water. Rep. Inst. High Speed Mech. Tohoku Univ. 43, 163–176. Overton, G.D.N., Williams, P.R., Trevena, D.H. 1984 The influence of cavitation history and entrained gas on liquid tensile strength. J. Phys. D Appl. Phys. 17, 979–987. Porter, T.M., Crum, L.A., Stayton, P.S., Hoffman, A.S. 2004 Effect of polymer surface activity on cavitation nuclei stability against dissolution. J. Acoust. Soc. Am. 116, 721–728.

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Pynn, J.J., Hammitt, F.G., Keller, A. 1976 Microbubble spectra and superheat in water and sodium, including effect of fast neutron irradiation. J. Fluids Eng. 98, 87–97. Roedder, E. 1967 Metastable superheated ice in liquid-water inclusions under high negative pressure. Science 155, 1413–1417. Sedgewick, S.A., Trevena, D.H. 1978 Breaking tensions of dilute polyacrylamide solutions. J. Phys. D Appl. Phys. 11, 2517–2526. Seitz, F. 1958 On the theory of bubble chambers. Phys. Fluids 1, 2–13. Shima, A., Tsujino, T., Tanaka, J. 1985 On the equation for the size distribution of bubble nuclei in liquids. Rep. Inst. High Speed Mech. Tohoku Univ. 50, 59–66. Shima, A., Sakai, I. 1987 On the equation for the size distribution of bubble nuclei in liquids (Report 2). Rep. Inst. High Speed Mech. Tohoku Univ. 54, 51–59. Strasberg, M. 1959 Onset of ultrasonic cavitation in tap water. J. Acoust. Soc. Am. 31, 163–169. Suzuki, A„ Armstead, S.C., Eckmann, D.M. 2004 Surfactant reduction in embolism bubble adhesion and endothelial damage. Anesthesiology 101, 97–103. Trevena, D.H. 1987 Cavitation and Tension in Liquids. Adam Hilger, Bristol. Vann, R.D. 2004 Mechanisms and risks of decompression. Diving Medicine, Saunders, Philadelphia, pp. 127–164. Williams, A.R., Delius, M., Miller, D.L., Schwarze, W. 1988 Investigation of cavitation in flowing media by lithotripter shock-waves both in vitro and in vivo. Ultrasound Med. Biol. 15, 53–60. Williams, P.R., Williams, R.L. 2000 On the anomalously low values of the tensile strength of water. Proc. R. Soc. A 456, 1321–1332. Williams, P.R., Williams, R.L. 2002 Cavitation of liquids under dynamic stressing by pulses of tension. J. Phys. D Appl. Phys. 35, 2222–2230. Yilmaz, E., Hammitt, F.G., Keller, A. 1976 Cavitation inception thresholds in water and nuclei spectra by light-scattering technique. J. Acoust. Soc. Am. 59, 329–338. Young, F.R. 1989 Cavitation. McGraw-Hill, New York. Yount, D. 1979 Skins of varying permeability: a stabilization mechanism for gas cavitation nuclei. J. Acoust. Soc. Am. 65, 1429–1439. Yount, D. 1982 On the evolution, generation, and regeneration of gas cavitation nuclei. J. Acoust. Soc. Am. 71, 1473–1481. Zheng, Q., Durben D.J., Wolf G.H., Angell, C.A. 1991 Liquids at large negative pressures: water at the homogeneous limit. Science 254, 829–832.

Chapter 3

Bubble Dynamics

The main goal of the investigations on bubble dynamics is to describe the velocity field and the pressure distribution in the liquid surrounding the bubble. In this section we describe the effect of the viscoelastic properties of the liquid on the behaviour of cavitation bubbles situated in a liquid of infinite extent or near a rigid boundary. As a special case, we will consider the interaction of individual cavitation bubbles situated in water with boundary materials with elastic/plastic properties. Due to the difficulty of the problem most of the theoretical work on bubble dynamics in non-Newtonian fluids was restricted to the case of spherical bubbles. The experimental studies, on the other hand, made use of high speed cameras to observe the growth and collapse of both spherical and non-spherical bubbles in nonNewtonian fluids. Two types of experiments have been conducted: in the first type, a cavitation bubble is collapsed after its expansion by the ambient pressure in the surrounding fluid. In this case, the bubble is generated by laser or electrical discharge. Alternatively, a stable gas bubble, usually of a size large enough to be visible, is compressed by a positive pressure pulse.

3.1 Spherical Bubble Dynamics The investigation of the dynamics of spherical cavitation bubbles is of no direct interest for the explanation of cavitation erosion, because bubbles close enough to a boundary to cause damage will always collapse aspherically. Nevertheless, it provides the basis for the interpretation of data obtained for the asymmetrical collapse of bubbles in non-Newtonian fluids and is to date the only means of comparing experimental results with theory.

3.1.1 General Equations of Bubble Dynamics Consider a spherical bubble of initial radius R0 situated in a compressible viscoelastic liquid. Until the reference time, t = 0, the pressure is uniform at p∞ and the liquid is at rest. At t = 0, the pressure inside the bubble is decreased instantaneously E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_3,

63

64

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to p0 and the bubble begins to collapse due to the pressure difference between the inside and outside of the bubble. The bubble keeps its spherical shape throughout the motion and the centre of the bubble remains fixed and is the centre of a spherically symmetric coordinate system. In principle, the quantities associated with the bubble collapse, such as velocity and pressure, can be determined from the solution of the conservation equations of continuum mechanics inside and outside of the bubble joined together by suitable boundary conditions at the bubble interface. Neglecting the effects of gravity, gas diffusion and heat conduction through the bubble wall, the governing equations may be expressed as follows: (i) Continuity: ∂p ∂(ρvr ) ρvr + +2 = 0, ∂t ∂r r

(3.1)

1 ∂p 1 ∂vr ∂vr + vr =− − (∇ · τ)r , ∂t ∂r ρ ∂r ρ

(3.2)

(ii) Momentum:

where ν r is the radial component of the velocity field, ρ, the liquid density, p(r, t) is the pressure in the liquid, and τ is the extra stress tensor. (iii) Equation of state for the liquid: A widely used equation of state for liquids is the Tait form:

p+B = p∞ + B

ρ ρ∞

n ,

(3.3)

where the subscript ∞ refers to the values at infinity, and B and n are constants having, for water, the values n = 7.15 and B = 3,049.13 atm. (iv) Equation of state for the gas inside the bubble: pi = p0

R0 R

3κ ,

(3.4)

where κ is the polytropic index. (v) Boundary conditions at the bubble wall (r = R(t)): Kinematic boundary condition: vr (t) =

dR ˙ = R, dt

(3.5)

Dynamic boundary condition: pB (t) = pi (t) −

2σ − (τrr )r=R , R

(3.6)

where pB is the pressure on the liquid at the bubble wall and σ is the surface tension.

3.1

Spherical Bubble Dynamics

65

Several comments relevant to bubble dynamics in non-Newtonian liquids are appropriate here. In a compressible liquid the extra stress tensor consists of two parts. The first part is the shear stress tensor τs that depends on the rate-of-strain tensor. For a purely viscous liquid, this tensor has the form tr(γ˙ )I , τs = 2η γ˙ − 3

(3.7)

where η is the shear viscosity of the liquid, I the unit tensor, and γ˙ is the shear rate. The second part is the isotropic tensor τi = f0 I with f0 being a function of invariants of the rate-of-strain tensor, i.e., f0 = f0 (I1 , I2 , I3 ), where I1 = tr(γ˙ ),

2 I2 = tr(γ˙ ) − tr γ˙ 2 , and I3 = Det(γ˙ ). For Newtonian and linear viscoelastic liquids τi has the form τi = λv tr(γ˙ )I,

(3.8)

where λv is the second coefficient of viscosity. For non-linear viscoelastic liquids, where the shear stress tensor has a finite trace, tr(τ) = 0, there is an additional contribution to the mean pressure p¯ = −tr[−pI + τ] that results in its variation from the pressure p in the liquid surrounding the bubble. We further note that Eq. (3.3) applies only to isentropic changes, but can be applied with reasonable accuracy in general since n is independent of entropy and B and ρ∞ are only slowly varying functions of entropy. Finally, Eq. (3.6) assumes that the gas-liquid interface is “clean” i.e., the only molecules present are those of the gas and the surrounding liquid. However where surfactants are adsorbed onto the bubble surface, a surface stress term needs to be added to Eq. (3.6) which includes the effects of surface viscosity and surface tension gradients. The latter occurs when the concentration of surfactant molecules on bubble surface is not constant resulting in an additional radial force that arise from the variation in the concentration of surface active molecules. A further approximation that was introduced in (3.6) is the neglect of the surface viscous term which, in the case of a spherical symmetric motion, is defined as ˙ 2 , where αs is the surface dilatational viscosity (Aris 1989). While τrr,s = 4αs R/R this procedure is justified for dilute surfactant solutions, it may be noted here that the predictions of a pure interface model are of interest in themselves in view of the frequent use of such a model in the study of bubble dynamics in non-Newtonian liquids.

3.1.2 The Equations of Motion for the Bubble Radius Here we shall restrict ourselves only to the case of linear viscoelastic liquids for which the extra stress tensor is traceless i.e., the sum of the normal stress components is zero. It should be emphasized here that these models are not entirely satisfactory for the description of viscoelastic flow behaviour. However, studies of idealized models may provide a qualitative insight for more realistic systems, and also quantitative results about their intermediate asymptotic behaviour. Moreover,

66

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these models have the main advantage of being tractable and, thus, they allow us to obtain an elegant solution by reducing the problem to a non-linear differential equation. The “near field” is a region surrounding the bubble with typical dimension R, the bubble radius, the “far field” scales with a typical length c∞ T, where c∞ is the speed of sound in the liquid and T a characteristic time, such as the collapse time. If ˙ with R˙ a typical radial velocity of the bubble one assumes that R is of the order RT, wall, the ratio of length scales is just the Mach number of the bubble wall motion. Once cast in these terms it is clear that, to lowest order, the near-field dynamics are essentially incompressible while the far field is governed by linear acoustics. The picture becomes considerably more intricate for a non-linear viscoelastic liquid, however (Khismatulin and Nadim 2002). The analysis leads unambiguously to the following equation for the radius of a spherical bubble situated in a linear viscoelastic liquid (Brujan 1998, 1999, 2001, 2009a): ∞

∂τrr 3τrr 1 1 2 ... 3 2 3 ˙ ¨ ˙ ˙ ¨ + dr, (3.9) R R + 6RRR + 2R = H − RR + R − 2 c∞ ρ∞ ∂r r R

where H is the liquid enthalpy at the bubble wall n(p∞ + B) H= (n − 1)ρ∞

!

pB + B p∞ + B

(n−1)/n

" −1 ,

(3.10)

˙ 2. with τrr evaluated in the near-field where vr = R2 R/r The striking feature of Eq. (3.9) is the appearance of the third-order derivative of the bubble radius with respect to time. This is just a consequence of using ¨ − R/c∞ ) ≈ Taylor series expansions to express retarded-time quantities, e.g. R(t ... ¨R(t) − (R/c∞ ) R . A similar term arises in Lorentz’s theory of electrons. Lorentz was ... considering periodic displacements x at frequency ω and thus set x ≈ −ω2 x˙ and identified this term with radiation damping. Later researchers, however, were deeply puzzled by this third derivative although there is nothing mysterious about it (Brujan 2001). For c∞ → ∞ the incompressible formulation is recovered, namely: 3 1 RR¨ + R˙ 2 = H − 2 ρ∞

∞ R

∂τrr 3τrr + dr, ∂r r

(3.11)

which, in the case of a Newtonian fluid, is known as the Rayleigh–Plesset formula˙ = α(R2 R) ˙ + (1 − α)(R2 R) ˙ and uses the tion. Furthermore, if one writes (R2 R) incompressible formulation in the form ˙ ˙ 2 1 (R2 R) 1 (R2 R) − =H− R 2 R4 ρ∞

∞ R

∂τrr 3τrr + dr ∂r r

(3.12)

3.1


67

to evaluate the first term and (3.11) to express the third derivative of the radius which appears on expanding the second term, one finds 3α + 1 α+1 3 2 1−α R ¨ ˙ ˙ ˙ ˙ ˙ RR 1 − R + R 1− R =H 1+ R + H c∞ 2 3c∞ c∞ c∞ ∞ ∞ ∂τrr ∂τrr 3τrr 3τrr 1−α 1 1 R d + + R˙ 1+ − dr − dr, ρ∞ c∞ ∂r r ρ∞ c∞ dt ∂r r R

R

(3.13) which represents an extension of the general Keller–Herring equation to the case of a bubble in a linear viscoelastic liquid. For a Newtonian liquid, by taking α = 0, Eq. (3.13) becomes identical to the equation proposed by Keller and Kolodner (1956), while the value α = 1 brings it into the form suggested by Herring (see, for example, Trilling 1952). It will be noted that, by dropping terms in c−1 ∞ , Eq. (3.13) reduces to Eq. (3.11), which is therefore seen to have an error of the order c−1 ∞ . The arbitrary parameter α (which does not seem to have any physical meaning) must, of course, be of order 1 so as not to destroy the order of accuracy of the approximate Eq. (3.13). Because of the presence of the third time derivative of the radius, the form (3.9) of the radial equation is hardly more attractive than (3.13), if for nothing else than for ¨ Actually, this is a minor difficulty the need to prescribe an initial condition for R. since, to the same order of accuracy in the bubble wall Mach number, an initial condition for R¨ can be obtained by substituting the given initial conditions for R and R˙ in the incompressible formulation (6). However, in view of its uniqueness (Brujan 1999), it is proper to consider Eq. (3.9) the fundamental form of the motion equation of a spherical bubble in a compressible linear viscoelastic liquid. With reference to Eq. (3.13) it should be noted that a related equation is that due to Gilmore (see, for example, Prosperetti and Lezzi 1986): R˙ 3 R˙ R R˙ R˙ ˙ + R˙ 2 1 − =H 1+ + 1− H RR¨ 1 − C 2 C C C C ∞ (3.14) ∞ R˙ ∂τrr ∂τrr 3τrr 1 R d 3τrr 1 1+ + dr − + dr, − ρ∞ C ∂r r ρ∞ C dt ∂r r R

R

whereby C is the speed of sound at the bubble wall C = [c2∞ + (n − 1)H]1/2 ,

(3.15)

and whose derivation relies on the Kirkwood–Bethe approximation (Kirkwood and Bethe 1942; Knapp et al. 1970). In this approach, the speed of sound C is not constant, but depends on H. This allows one to model the increase of the speed of sound with increasing pressure around the bubble, which leads to significantly reduced Mach numbers at bubble collapse.

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To close the mathematical formulation an equation for the shear stress in terms of the rate-of-strain is necessary. Several examples on obtaining the equation of motion for the bubble radius for some constitutive models are given below. Example 3.1: Equation of Motion for Bubble Radius in Terms of Pressure #p Using the Taylor series expansion and the definition of enthalpy, h = p∞ dp/ρ, we may write H=

pB − p∞ ρ∞

1 p − p∞ 1− . 2 ρ∞ c2∞

(1)

With this result and using the dynamic boundary condition (3.6), the equation of motion for the bubble radius in terms of pressure is found to be ∞

3 1 3 2 τrr 1 2 ... 2σ 3 ˙ ¨ ˙ ¨ ˙ pi (t) − − p∞ − dr, R R + 6RRR + 2R = RR+ R − 2 c∞ ρ∞ R ρ∞ r R

(2) where, in the case of an adiabatic evolution of the gas inside the bubble, pi (t) is given by Eq. (3.4). Example 3.2: Equation of Motion for Bubble Radius for a Newtonian Fluid In the case of a Newtonian fluid τrr = −2η

R2 R˙ ∂vr = 4η 3 , ∂r r

(1)

and ∞ 3

R˙ τrr dr = 4η . r R

(2)

R

Thus, the equation of motion for the bubble radius in a Newtonian fluid written in terms of pressure becomes

3 2 1 1 2 ... 2σ R˙ 3 ˙ ¨ ˙ ¨ ˙ RR + R − pi (t) − − p∞ − 4η . (3) R R + 6RRR + 2R = 2 c∞ ρ∞ R R After some time of oscillation, due to acoustic and viscous dissipation, the trajectory R = R(t) move towards an equilibrium position characterized by the equi... librium radius of the bubble Re . This value may be obtained imposing R˙ = R¨ = R = 0 in Eq. (3) to find

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R3γ e +

69

2σ 3γ −1 p0 3γ R + R = 0. p∞ e p∞ 0

(4)

Suppose now that the bubble oscillates with small amplitude. Then a solution R of this equation may be given as R = Re (1 + δ), |δ| 1 N/m for SonovueTM microbubbles, while for the BR14 microbubbles the corresponding values are χ = 1 N/m, κ s = 7.2×10–9 Ns/m, and σ break-up = 0.13 N/m (Marmottant et al. 2005). The translational motion of a microbubble in a fluid during insonation can be studied by solving a particle trajectory equation (Dayton et al. 2002): du 2 ˙ r − πρL R3 r − 2πρL |ur |ur R2 cd , ρb V X¨ = −V∇pa − 2πρL R2 Ru 3 dr

(6.7)

where ur = X˙ + ∇pa /ρL .

(6.8)

In the above equations, X˙ is the bubble translation velocity, ρb , ur , and V are the density of gas inside the bubble, the relative velocity between the bubble and liquid and the volume of the bubble, respectively. Cd is the drag coefficient determined by Reynolds number of liquid around the oscillating bubble, as defined in Dayton et al. (2002) and Watanabe and Kukita (1993): cd =

24 2R|uL −ub | ν

!

2R|uL − ub | 1 + 0.197 ν

0.63

−4

+ 2.6 × 10

2R|uL − ub | ν

1.38 " , (6.9)

where ν is the kinematic viscosity of the liquid surrounding the microbubble. The term on the left in Eq. (6.7) is the product of the mass of the bubble and its acceleration. The four terms on the right side of the equation describe the radiation force on a highly compressible bubble as a result of the acoustic pressure wave, the added mass as a result of the oscillating bubble wall, the added mass required to accelerate a rigid sphere in the surrounding fluid, and the quasistatic drag force, respectively. Example 6.1: Equation of Motion for a Microbubble Encapsulated with a Thin Membrane For a contaminated gas/liquid interface with a surface active substance, such as a surfactant, the interfacial stress is a function of two intrinsic properties of the

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interface, the surface shear viscosity, μs , and the surface dilatational viscosity, κ s . Consider, for simplicity, the case of a spherical bubble and a Newtonian interface, i.e. an interface for which the relationship between the viscous part of the surface stress and the surface rate of deformation is linear. The surface shear viscosity does not come into play in the present situation, because of the radial motion of the bubble. If the bubble surface is expanded at a constant dilatational rate λ˙ =

1 dA , A dt

(1)

where A is the area of the bubble, the constitutive law for the isotropic part of the surface stress is given by ˙ τrrs = σ + κ s λ.

(2)

We also note that in compression or expansion deformation of an insoluble monolayer, an elastic modulus is defined as the increase in surface tension for a small increase in area of a surface element at constant shape and curvature χ=

dσ . d ln A

(3)

The variation of surface tension with the bubble radius R is thus expressed as ' σ (R) = σ (R0 ) + χ

) R2 −1 , R20

(4)

which, for |R − R0 | R2 , v is the velocity of the surrounding liquid. The assumption of incompressible shell gives the following equations: R32 − R31 = R320 − R310 , R21 R˙ 1 = R22 R˙ 2 ,

(3)

where R10 and R20 are, respectively, the inner and the outer radii of the bubble shell at rest. Conservation of radial momentum yields ρ

∂v ∂v +v ∂t ∂r

=−

∂p ∂τrr 3τrr + + , ∂r ∂r r

(4)

where ρ is equal to ρS or ρL , ρS and ρL , are respectively, the equilibrium densities of the shell and the liquid, p is the pressure, and τrr is the stress deviator in the shell or the liquid. The boundary conditions at the two interfaces are given by pg (R1 , t) = pS (R1 , t) − τrrS (R1 , t) +

2σ1 , R1

(5)

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pS (R2 , t) − τrrS (R2 , t) = ρL (R2 , t) − τrrL (R2 , t) +


2σ2 + p0 + pa (t), R2

(6)

where pg (R1 , t) is the pressure of the gas inside the bubble, σ1 and σ2 are the surface tension coefficients for the corresponding interfaces, and pa (t) is the driving acoustic pressure at the location of the bubble. Integrating Eq. (4) over r from R1 to R2 using the parameters appropriate for the encapsulating layer and from R2 to ∞ using those appropriate or the surrounding liquid, assuming that the liquid pressure at infinity is equal to the hydrostatic pressure, p0 , and combining the resulting equation with Eq. (2), one obtains ! ) " ' 3 4R2 − R31 R1 3 ρL − ρS R1 ρL − ρS 2 + R˙ 1 + R1 R¨ 1 1 + ρS R2 2 ρS R2 2R32 ⎤ ⎡ R2 S ∞ L 1 ⎢ τrr (r, t) τrr (r, t) ⎥ 2σ1 2σ2 dr + 3 dr⎦ = − +3 ⎣pg (R1 , t) − (p0 + pa ) − ρS R1 R2 r r R1

R2

(7) Assuming that the surrounding liquid is a viscous Newtonian fluid, τrrL (r, t) is written as τrrL = 2ηL

∂v , ∂r

(8)

where ηL is the shear viscosity of the liquid. By using Eqs. (8) and (2), the second integral term in Eq. (7) is found to be ∞ 3 R2

R2 R˙ 1 τrrL (r, t) dr = −4ηL 1 3 . r R2

(9)

Consider now a viscoelastic shell whose rheology is described by the linear Maxwell constitutive equation τrrS + λ

∂v ∂τrrS = 2ηS , ∂t ∂t

(10)

where λ is the relaxation time and ηS is the shear viscosity of the shell. Substituting Eq. (2) into Eq. (10), one has τrrS + λ

R2 R˙ 1 ∂τrrS = −4ηS 1 3 . ∂t r

(11)

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Cavitation for Ultrasonic Surgery

187

This equation suggests that τrrS (r, t) can be written as τrrS = −4ηS

D(t) , r3

(12)

and, therefore, the function D(t) obeys the equation ˙ = R21 R˙ 1 . D(t) + λD(t)

(13)

Using Eqs. (12) and (2), the first integral term in Eq. (7) is calculated as R2 3 R1

D(t)(R320 − R310 ) τrrS (r, t) dr = −4ηS . r R31 R32

(14)

Substitution of Eqs. (9) and (14) into Eq. (7) yields ! ) " ' 3 4R2 − R31 R1 3 ρL − ρS ρL − ρ S R 1 2 ˙ ¨ + + R1 R1 R1 1 + ρS R2 2 ρS R2 2R32 ! " (15) D(t)(R320 − R310 ) R21 R˙ 1 2σ1 2σ2 1 − − 4ηL 3 − 4ηL pg (R1 , t) − (p0 + pa ) − , = ρS R1 R2 R R3 R3 2

1 2

where the function D(t) is calculated from Eq. (13). Resonance Frequency The linear resonance frequency of microbubble oscillation f0 is the frequency at which the bubble first harmonic response (linear amplitude-frequency response) has a local maximum. The linear resonance frequency of encapsulated microbubbles in the Church model is (Church 1995): % 2σ2 R301 ρL − ρS R01 −1/2 2σ1 1+ − 3κp0 − ρS R02 R01 R402 ! ) "&1/2 ' 3R301 R302 2σ1 VS GS 2σ2 1 +4 3 + . 1+ 1+ 3 4GS R01 R02 VS R02 R02

1 f0 = 2π

$

ρS R201

(6.10)

Equation (6.10) refers to a breathing mode of oscillation where the bubble simply pulsates. This is the frequency of oscillation of a zero-order spherical harmonic perturbation upon a spherical bubble. Putting ρL = ρS = ρ, R01 = R02 = R0 , σ2 = 0, and GS = 0 in this equation yields the natural frequency of the bubble in a Newtonian liquid (Lauterborn 1976). The resonance frequency of the encapsulated microbubbles increases approximately as the square root of the modulus of

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rigidity GS . Encapsulated microbubbles will resonate at higher frequencies than free bubbles of the same size, and therefore will tend to appear acoustically smaller than they actually are. QuantisonTM , which has the thickest and most rigid shell shows increased resonance frequency, while SonoVueTM , which has an encapsulation more flexible has a lower resonance frequency (Boukaz and de Jong 2007). Effects due to the density and surface tension of the shell are relatively minor by comparison to that produced by its elasticity (Church 1995). Khismatullin and Nadim (2002) found a decrease in the maximal resonance frequency with decreasing the speed of sound in the liquid (Fig. 6.3a). They also noted that the maximal resonance frequency is larger in a viscoelastic liquid than in a Newtonian liquid (Fig. 6.3b). Both effects are, however, small compared to the shell effect (Fig. 6.4). The resonance frequency of the encapsulated microbubbles in the Morgan model may be approximately expressed as (Wu et al. 2003): 1 f0 = 2π

&1/2 2(σ + χ ) 4μ + 12εμS /R0 3κ 2σ + 6χ − . (6.11) p0 + − R0 ρR20 ρR20 ρR30

It can be seen that the resonance frequency of the microbubble increases with increasing the shell elasticity and decreasing the thickness and viscosity of the shell. The most influential parameter is, however, the shell elasticity (Wu et al. 2003). Scattering Cross Section As scatter and reflection are exploited by ultrasound imaging, a contrast agent material has to possess a high scattering cross section in order to provide a significant scatter enhancement compared to the surrounding tissue. The scattering cross section may be defined as the ratio of the total acoustic power scattered by a microbubble at a particular frequency to the incoming acoustic intensity WS , Iinc

(6.12)

4πr2 |PS |2 , 2ρc

(6.13)

|Pa |2 , 2ρc

(6.14)

σS = with WS = and WS =

where PS is the amplitude of the scattered wave at a distance r from the emission center.

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189

Fig. 6.3 Effects of liquid compressibility and viscoelasticity on the resonance frequency of microbubble oscillation. Plots (a) and (b) show the resonance frequency as a function of bubble radius for different values of sound velocity c for a Newtonian liquid and of relaxation time λ1 at c = 1,500 m/s when the retardation time λ2 = 0, respectively. Other parameters are Gs = 88.8 MPa, μs = 1.77 kg/(ms), and shell thickness 15 nm. Reproduced with permission from Khismatullin and Nadim (2002). © American Institute of Physics

190 Fig. 6.4 Resonance frequency versus bubble radius for an encapsulated microbubble in a compressible Newtonian liquid (c = 1,500 m/s) for different values of (a) shell elasticity, Gs , and (b) shell viscosity, μs . Shell thickness 15 nm. Reproduced with permission from Khismatullin and Nadim (2002). © American Institute of Physics

6


6.1


191

Fig. 6.5 Effect of liquid viscoelasticity on the second-harmonic scattering cross section. The amplitude of the acoustic pressure is pa = 0.3. Other parameters are Gs = 88.8 MPa, μs = 1.77 kg/(ms), and shell thickness 15 nm. Reproduced with permission from Khismatullin and Nadim (2002). © American Institute of Physics

The final form of the expressions for the scattering cross section depends on the model used to describe the radial oscillations of the microbubble. The resulting expressions for a thick-shelled microbubble can be found in Khismatullin and Nadim (2002) and for a thin-shelled microbubble in Wu et al. (2003). The numerical results obtained by Church (1995), Khismatullin and Nadim (2002), and Wu et al. (2003) indicate that the scattering cross section increases with increasing shell elasticity and decreasing shell viscosity and thickness. Scattering is higher in a viscoelastic liquid than in a Newtonian liquid but this effect is minor as compared to the shell effect (Fig. 6.5). 6.1.2.3 Potential Therapeutic Applications of Microbubble Ultrasound Contrast Agents As microbubble contrast agents developed, interest grew in understanding their interaction with propagating ultrasound waves and nearby biological tissue. Hypotheses of potential benefits from these interactions suggested that microbubble contrast agents loaded with therapeutic substances could be targeted for destruction with ultrasound and thus enhance diffusion-mediated delivery by increasing localized concentration of the substances. The ability to increase tissue permeability and concomitantly augment localized drug concentrations through targeted microbubble destruction has fuelled interest in developing efficient methods for delivering drugs and genetic material. Damage of cell membrane is a well-known biological effect of cavitation (Miller et al. 2002). The mechanical action of the cavitation bubbles typically causes cell

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lysis and disintegration. However, sub-lethal membrane damage also occurs, in which large molecules in the surrounding medium are able to pass in or out of the cell, followed by membrane sealing and cell survival. This allows foreign macromolecules to be trapped inside the cell. This ultrasound-mediated increase in cell membrane permeability has been termed sonoporation. It should be noted that sonoporation represents transient permeabilization, which can be indicated by trapping large fluorescent molecules inside the viable cells, and is different from the commonly noted permeabilization indicated by trypan blue or propidium iodide stains, which stain lysed, nonviable cells. Most investigators who have used ultrasound contrast agents for therapeutic applications worked with perfluorocarbon bubbles stabilized by an albumin or lipid shell. The main advantage of this type of contrast agents is their fragility when exposed to ultrasound. Microbubbles can be produced together with the bioactive substance, thus potentially incorporating it into the microbubble shell or lumen (Shohet et al. 2000; Frenkel et al. 2002; Erikson et al. 2003; Unger et al. 2002) (Fig. 6.6a, b), or microbubbles can be incubated with the bioactive substance, thus attaching the substance to the microbubble shell, presumably by electrostatic or weak non-covalent interactions (Lawrie et al. 2000; Pislaru et al. 2003; Mukherjee et al. 2000) (Fig. 6.6c). In several other studies microbubbles and the bioactive

Fig. 6.6 Illustrating the transfer modalities of active substances (drugs or genes) to tissue using microbubble ultrasound contrast agents. (a) Active substances are included in the gas-core region of the microbubble, (b) Active substances are incorporated in the shell of the microbubble, (c) Active substances are attached to the microbubble shell, (d) Microbubbles and the active substances are co-administrated in the targeted region

6.1


193

substance were co-administered (Price et al. 1998; Song et al. 2002; Kondo et al. 2004) (Fig. 6.6d). The most widely investigated application is for gene transfer/gene therapy. A second application is for drug and protein delivery. Finally, ultrasound targeted microbubble destruction alone has been studied for therapeutic effects without any transported substance. Blood vessels are obvious targets for microbubbles and ultrasound, because they are the first tissue exposed to the microbubbles. Several in vitro and in vivo studies have been performed to evaluate transfection and the physiologic response to ultrasound-target microbubble destruction in vessels. Cultured vascular smooth muscle cells and endothelial cells were transfected with plasmids and microbubbles, showing 3,000-fold higher expression than obtained with naked DNA alone (Lawrie et al. 2000). Rat carotid arteries were transfected with anti-oncogene plasmids and microbubbles, resulting in a significant reduction of intimal proliferation (Taniyama et al. 2002a). Similarly, oligodeoxynucleotides were used with microbubbles to reduce intimal proliferation in balloon-injured rat carotids (Hashiya et al. 2004). Hynynen et al. (2001) has shown that transcranial application of ultrasound combined with intravenous administration of microbubbles in rabbits reversibly open the blood-brain barrier. They indicate that the mechanism responsible for opening the blood-brain barrier is most likely due to cavitation of microbubbles with ultrasound. Many potent drugs with severe adverse effects may be used more beneficially if local concentrations could be increased while keeping systemic concentrations low. Several studies demonstrated the potential for using ultrasound–microbubble interactions to deliver therapeutically functional substances to treat various cardiac pathologies through myocardial microcirculation. Figure 6.7 shows a schematic diagram of drug delivery or gene therapy to the heart. A diagnostic ultrasound transducer is placed on the patient’s chest. An ultrasound contrast agent bearing drug or genetic material has been administered intravenously. As the microbubbles enter the region of insonation, they distribute within the myocardial tissue via the vascular bed. The microbubbles cavitate within the capillaries of the myocardial tissue releasing the drug or genetic material (Unger et al. 2001). Vascular endothelial growth factor bound to albumin microbubbles was delivered to the heart using ultrasound. A 13-fold augmentation of cardiac vascular endothelial growth factor uptake was seen compared with systemic vascular endothelial growth factor administration (Mukherjee et al. 2000). A study using lipid microbubbles with luciferase protein demonstrated up to seven-fold augmented cardiac uptake of luciferase compared with systemic administration (Bekeredjian et al. 2005a). In a rat model of acute myocardial infarction, Kondo et al. (2004) utilized ultrasonic microbubble destruction to transfer systemically injected hepatocyte growth factor plasmid into myocardial cells to enhance capillary density and limit or negate left ventricular remodeling. Erikson et al. (2003) used low-frequency ultrasound (1 MHz) to release antisense oligonucleotides from albumin-shelled microbubbles, thereby facilitating oligonucleotide delivery to the myocardium. Recently, the vascular endothelial growth factor protein and its encoding gene have been administered in a canine (Zhou et al. 2002) and rat (Zhigang et al. 2004) model of myocardial infarction, respectively. In vitro studies have shown that microbubbles contrast agents can also

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Fig. 6.7 Schematic diagram of drug delivery or gene therapy to the heart. Reproduced with permission from Unger et al. (2001). © John Wiley and Sons

be used to deliver an antibiotic (Tiukinhoy et al. 2004) or a radionuclide (van Wamel et al. 2004). Studies of the interaction of microbubbles contrast agents with skeletal muscle are also pertinent to cardiovascular treatment because of the similarity of cardiac and skeletal muscle as target tissues. Two different strategies have been described to transfect skeletal muscle. Direct injection of microbubbles and green fluorescent protein encoding plasmids into the skeletal muscle with ultrasound application increased green fluorescent protein expression compared with intra-muscular naked plasmid injection alone and, at the same time, reduced muscle damage (Lu et al. 2003). This study also demonstrated an enhanced transfection of DNA by microbubbles without ultrasound, although the mechanism for this finding was not elucidated. In a second approach, intravascular infusion of cytomegalovirus-luciferase encoding plasmids bound to microbubbles with ultrasound was able to achieve luciferase expression in rat skeletal muscle, with intra-arterial application more efficient than intravenous infusion (Christiansen et al. 2003). Taniyama et al. (2002b) demonstrated increased capillary density in rabbit skeletal muscle using hepatocyte growth factor plasmid combined with microbubble contrast agents. Gene delivery to the myocardium of rats was obtained with harmonic mode diagnostic ultrasound, a microbubble contrast agent and a viral β-galactosidase

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195

vector (Shohet et al. 2000). Three frames from a 1.3 MHz transducer destroyed the microbubbles evident in the second-harmonic image, and three frame bursts were triggered to allow refill of the tissue between scans. Expression of the reporter gene was assayed in histological sections and by measurement of enzyme activity. Staining and enzyme activity was detected in the myocardium after echocardiographic destruction of the microbubbles mixed with the viral vector at about ten times the levels found in controls (bubbles plus ultrasound, no vector; bubbles plus vector, no ultrasound; vector alone, no bubbles, no ultrasound). Cavitation activity was clearly responsible for the effect because the procedure involved destruction of contrast agent microbubbles. However, it is uncertain whether the viral vector was delivered by sonoporation or by some other process. Echocardiographic microbubble destruction followed by vector infusion generated about twice the gene expression of controls, indicating that disruption of the endothelial barrier during microbubble destruction might be a factor in the enhanced viral transduction. The potential application of microbubble contrast agents as an adjuvant to thrombolytic therapy is also promising. Ultrasound at frequencies ranging from 20 to 3 MHz has been shown to enhance the thrombolytic efficacy of urokinase and tissue plasminogen activator (Lauer et al. 1992; Francis et al. 1992; Tachibana and Tachibana 1995; Porter et al. 1996). Acceleration of thrombolysis with ultrasound is probably due to local cavitation that may weaken the clot surface and/or improve clot penetration by the fibrinolytic agents. This process can be enhanced greatly by the presence of microbubbles. In vitro studies have shown that ultrasound energy at high acoustic pressures combined with microbubble administration enhances the thrombolytic efficacy of urokinase from 1.5- to over 3-fold (Tachibana and Tachibana 1995; Porter et al. 1996), and can even result in efficient clot lysis in the absence of thrombolytic therapy (Porter et al. 1996). The mechanisms by which ultrasound-contrast agent interactions induce an increase in cell and microvessel permeability are poorly understood, although several hypotheses exist. Postema et al. (2004) describe the different effects of ultrasound on microbubbles and demonstrate these effects by experiments using high-speed photography. Depending on the applied ultrasound amplitude and frequency, effects such as stable oscillation of microbubbles, inertial cavitation, coalescence, fragmentation, ultrasound induced damage of the shell causing gas to escape from microbubbles (sonic cracking), and jetting are ascribed. Sustained oscillatory motion of bubbles (stable cavitation) induces fluid velocities and exert shear forces on the surrounding tissues and cells (Suslick 1988). In the presence of a high-power, low-frequency ultrasound beam, microbubble contrast agents expand and contract nonlinearly, a phenomenon known as inertial cavitation, which often leads to bubble fragmentation (Chomas et al. 2000; de Jong et al. 2000; Boukaz et al. 2005; Boukaz and de Jong 2007). An example of microbubble fragmentation is shown in Fig. 6.8 for the case of the experimental contrast agent MP1950 containing C4 F10 encapsulated by a phospholipid shell (Chomas et al. 2000). The effects of various factors including the ultrasound driving frequency, pulse length, peak negative pressure, bubble size and shell properties on the fragmentation of microbubbles were investigated by Chomas et al. (2001) and Bloch et al. (2004).

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6


Fig. 6.8 Optical frame images corresponding to the oscillation and fragmentation of a contrast agent microbubble. The initial diameter of the microbubble is 3 μm (frame (a)). The streak image in frame (h) shows the diameter of the bubble as a function of time, and dashed lines indicate the times at which the two-dimensional frame images in frames (a)–(g) were acquired relative to the streak image. Reproduced with permission from Chomas et al. (2000). © American Institute of Physics

The constrained boundary also has a significant effect on microbubble fragmentation. Zheng et al. (2007) demonstrated that microbubbles within smaller tubes have a higher fragmentation which may result from the decreased radial oscillation, and decreased wall velocity and acceleration within the small tube. The collapse of inertial cavitation bubbles generates shock waves with amplitude exceeding 5 GPa (Pecha and Gompf 2000; Brujan et al. 2008). Although cavitation-induced shock waves persist for a very short period of time, the large spatio-temporal pressure gradients associated with shock waves can disrupt tissue. Rapid collapse of microbubbles near a boundary will lead to asymmetric movements that can form high velocity fluid microjets (Brujan 2004; Brujan et al. 2005). Microjet formation during collapse of OptisonTM microbubbles in the vicinity of a boundary was experimentally observed by Prentice et al. (2005) (Fig. 6.9). They also noted that the jetting and the microbubble translation towards the boundary are dependent on the relative distance between microbubble and boundary. This jetting is associated with high pressures at the tip of the jet that are sufficient to penetrate any cell membrane. It is widely proposed that jetting is responsible for the transient nanopores which were observed in cell membranes by electron microscopy immediately after destruction of microbubbles (Tachibana et al. 2002; Miller et al. 2002). Translation of microbubbles was also observed by Zheng et al. (2007) (Fig. 6.10). Cavitating microbubble contrast agents may also induce significant but transient thermal fluctuations (Wu 1998) as well as toxic chemical production (Kondo et al. 1998). During the collapse, the temperature of the bubble core can increase by more than 1,000 K and induce chemical changes in the surrounding medium, an effect termed sonochemistry (Suslick 1988). Of particular

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197

Fig. 6.9 Illustrating the formation of a microjet during the collapse of a microbubble contrast agent near a solid boundary. The microjet is indicated by the white arrow. The initial position of the microbubble is indicated by the black arrow. It is 26.5 μm in the top sequence and 19 μm in the bottom sequence. Frame size is 163 μm × 110 μm. Reproduced with permission from Prentice et al. (2005). © Macmillan Publishers Ltd

Fig. 6.10 Microbubble translation under high-pulse repetition frequency ultrasound within microtubes observed by a microscope video camera system at 240 frames/s. The microbubble with an initial radius of 1.2 μm is moving fom the center of the microtube to the wall. Reproduced with permission from Zheng et al. (2007). © Elsevier B.V.

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importance is the generation of highly reactive species, such as free radicals, that can induce chemical transformations in the medium which may be involved in the enhancement of permeability of endothelial cell layers. A significant increase in free radical production in endothelial cells after exposure to ultrasound was demonstrated by Basta et al. (2003). Finally, another mechanism by which the use of encapsulated microbubbles could facilitate deposition of drugs and genes in a cell is fusion of the phospholipid microbubble coating with the bilayer of the cell membrane. This could result in delivery of the microbubble substances directly into the cytoplasm of the cell (Dijkmans et al. 2004). All or a combination of these events may alter, displace, or destroy cells, possibly resulting in cell microporation (Deng et al. 2004) and gaps between neighboring cells. For example, Ohl et al. (2006) demonstrated that the collapse of microbubbles cause membrane poration to cells plated on a substrate through a complex sequence of events. When the jet developed during bubble collapse impacts onto the boundary, it spreads out radially along the substrate causing a strong gradient in the velocity component parallel with the substrate. The resulting shear stress leads to the detachment of cells. Cells at the edge of the area of detachment were found to be permanently porated, whereas cells at some distance from the detachment area undergo viable cell membrane poration. The high shear stress caused by violent microstreaming or microjets developed during microbubble collapse may explain the maximum transfection efficiency and lowest cell viability obtained at high ultrasound pressures (Wu 2002; Wu et al. 2002). Several hypotheses on the mechanism of blood-brain barrier disruption with microbubbles and ultrasound have been proposed (Sheikov et al. 2004). Since an ultrasound wave causes microbubbles to expand and contract in the capillaries, the expansion of larger microbubbles could fill the entire capillary lumen, resulting in a mechanical stretching of the vessel wall which, in turn, could result in the opening of the tight junctions. This interaction could create a change in the pressure in the capillary to evoke biochemical reactions that trigger the opening of the bloodbrain barrier. Moreover, bubble oscillation may also reduce the local blood flow and induce transient ischemia, which could also trigger blood-brain barrier opening. Extensive reviews of the therapeutic applications ultrasound-targeted microbubble destruction, including ultrasound–microbubble interactions, are currently available in literature (Lindner 2004; Unger et al. 2004; Liu et al. 2006; Chappell and Price 2006; Bekeredjian et al. 2005b, 2006; Ferrara et al. 2007; Shengping et al. 2009). 6.1.2.4 Collateral Effects Induced by Cavitation The collateral effects induced by microbubble contrast agents in the cardiovascular applications of ultrasound have been recently summarized by Dalecki (2007). Diagnostic ultrasound can produce premature cardiac contractions in laboratory animals and humans when microbubble contrast agents are present in the blood with end-systolic triggering. Myocardial damage in humans has not been reported to result from the interaction of ultrasound and contrast agents. Premature atrial contractions, ventricular contractions and ventricular tachycardia were observed in

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animals exposed to ultrasound in the presence of microbubble contrast agents. The interaction of diagnostic ultrasound and microbubble contrast agents can produce damage to the vasculature in kidneys. Lower ultrasound frequencies produce more damage than higher frequencies. Diagnostic ultrasound imaging devices were also reported to produce capillary damage in muscle in laboratory animals when contrast agents are present in the blood.

6.2 Cavitation in Laser Surgery Whenever laser pulses are used to ablate, cut, or disrupt tissue inside the human body, cavitation bubbles are produced that interact with the tissue. In cardiovascular laser applications, this situation is encountered in myocardial laser revascularization and laser angioplasty.

6.2.1 Transmyocardial Laser Revascularization 6.2.1.1 The Basic Principles of Transmyocardial Laser Revascularization Transmyocardial laser revascularisation is used to treat patients with severe coronary disease. Although surgical procedures such as coronary angioplasty and coronary artery bypass grafting are proven methods of treating heart disease, many patients have conditions that are not amenable to these therapies. Transmyocardial laser revascularisation was proposed as a means of bypassing the coronary circulation altogether, instead perfusing the myocardium with oxygenated blood directly from the left ventricular chamber, in a similar manner to the embryonic and reptilian cardiac circulation. Up to 50 narrow channels are drilled in the left ventricular myocardium, which are closed at the epicardial surface and open to the left ventricular cavity at the endocardial surface. These channels are typically about 1 mm in diameter and are created approximately 1 cm apart (Horvath et al. 1995). How long these channels remain open and to what extent the blood flows through them to contribute to angina relief remains a matter of controversy. The types of lasers currently used for transmyocardial revascularisation are mainly the carbon dioxide (CO2 ) which delivers light pulses at 10.6 μm wavelength with 20–90 ms duration and energies of up to 40 J, and the Holmium-Yag (Ho:Yag) which emits light pulses at 2.1 μm wavelength with 100–500 μs pulse duration and energies up to 30 J. Another type of laser is the Excimer laser (XeCl) emitting shorter light pulses (150 ns) at a wavelength of 308 nm with energies between 20 and 40 mJ. The CO2 and Ho:Yag lasers are infrared lasers exerting their effect by vaporising water molecules. These lasers have frequencies similar to the vibrational frequency of water and absorption of laser energy by water molecules results in heating, evaporation, and tissue ablation. The XeCl laser, on the other hand, operates in the ultraviolet spectrum and exerts its effect by dissociating the dipeptide bonds of

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proteins. Because the myocardium is composed predominantly of water and proteins, these types of lasers are the most used for creating transmyocardial channels (Cooley et al. 1996; Klein et al. 1998; Lange and Hillis 1999). With CO2 laser, a single light pulse is used to create the laser channel. The laser light is delivered to the beating heart through an intercostal incision 15–25 cm long using an articulated mirror arm and long focusing optics. With the Ho:Yag and XeCl lasers, multiple pulses guided by a silica fibre are required. The fiber is directed to the myocardium through a very small incision or even percutaneously through the femoral artery, as in balloon angioplasty. The laser energy causes tissue ablation and vaporisation that can be detected as a puff of smoke on transesophageal echocardiogram when the laser transverses the free wall of the left ventricle (Horvath et al. 1996). 6.2.1.2 Collateral Effects Induced by Cavitation The expansion of gaseous products produced during tissue ablation creates a cavitation bubble in the medium surrounding the ablation site (Duco Jansen et al. 1996; Brinkmann et al. 1999; Vogel and Venugopalan 2003). When the optical fibre is not in contact with tissue, a cavitation bubble is formed by absorption of infrared laser radiation in the liquid separating the fibre tip and the tissue surface. This bubble is essential for the transmission of the optical energy to the tissue. A similar event occurs during ultraviolet ablation when the surrounding fluid is blood, because hemoglobin and tissue proteins absorb strongly in the ultraviolet range (van Leeuwen et al. 1992). When the fibre tip is placed in contact with the tissue surface, the ablation products are even more strongly confined than when they are surrounded by liquid alone, resulting in considerably higher temperatures and pressures within the tissue. The cavitation bubble dynamics influence the ablation efficiency in two ways. First, the bubble creates a transmission channel for the laser radiation. Second, the forces exerted on the tissue as a consequence of the bubble dynamics may also contribute to the material removal. The dynamics of channel formation within tissue has been studied in the context of transmyocardial laser revascularisation by Brinkmann et al. (1999). They noted that the shape and lifetime of the transmitted channel depend on the laser pulse duration and the optical penetration depth. For example, a 2.2 ms pulse of a Ho:Yag laser, transmitted through an optical fibre at the surface of a tissue phantom, creates an elongated bubble that partially collapses during the laser pulse, such that the light path from the fibre to the target is partially blocked. Tissue was ablated at the bubble wall opposite to the fibre tip but even at the largest value of the pulse energy used in their experiment the channel to the surface of the tissue phantom is almost closed at the end of the laser pulse (Fig. 6.11). By contrast, a 15 ms CO2 laser pulse generates an oscillating vapour channel that remains opened at the end of the laser pulse (Fig. 6.12). The confinement of the ablation products by the ablation channel leads to an increase of the collateral damage because of the high pressure and heat contained in the ablation products. This effect is clearly visible in Fig. 6.12 and is manifested by the formation of a large cavity at a depth of about 10 mm from the surface of the tissue phantom. Cavitation can thus

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Fig. 6.11 A series of pictures illustrating the interaction of a holmium laser pulse with a polyacrylamide sample in water environment. The energy of the laser pulse is 12 J and the pulse duration is 2.2 ms. Times indicated are delay times of the photograph relative to the onset of the laser pulse. Reproduced with permission from Brinkmann et al. (1999). © IEEE

Fig. 6.12 A series of pictures illustrating the interaction of a CO2 laser pulse with a polyacrylamide sample in water environment. The power of the laser pulse is 800 W and the pulse duration is 15 ms. Times indicated are delay times of the photograph relative to the onset of the laser pulse. Reproduced with permission from Brinkmann et al. (1999). © IEEE

lead to a structural deformation of the tissue adjacent to the ablation site that is much more pronounced than the ablative tissue effect itself and compromises the high precision of the original ablation. In addition, the authors concluded, from experiments on porcine heart tissue, that the orientation of the myocardial fibrils significantly influences the dynamics of cavitation bubbles, the shape of the ablated cavities, and

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the thermo-mechanical collateral damage areas. Deep and straight channels were found for fibrils running perpendicular to the endocardium, while much smaller but widely dissected tissue was found for horizontal channels. The fragmentation of cavitation bubbles during implosion may result in many gaseous microemboli that may persist briefly in the circulation. The formation of such microemboli, during transmyocardial laser revascularisation on human subjects, was observed in the middle cerebral artery by von Knobelsdorff et al. (1997). However, none of the patients exhibited major neurological deficits on the first day after surgery, indicating that transmyocardial laser revascularisation does not cause significant cerebral ischemia. The authors explained this result by the very small size of the induced microemboli. Indeed, Feinstein et al. (1984) found that only arterial emboli larger than 15 μm lead to temporary oclusion of more than 1 min, whereas emboli of less than 10 μm in diameter pass the capillary vasculature unrestricted. Multiple microembolic signals were also detected in the ophthalmic artery during transmyocardial laser revascularisation in pigs by Gerriets et al. (2004). They demonstrated that the microembolic load can be reduced by ventilation with 100% oxygen and by decreasing the laser pulse energy.

6.2.2 Laser Angioplasty 6.2.2.1 The Basic Principles of Coronary Angioplasty The main goal of coronary angioplasty is to recanalize the blood vessels that are obstructed by fatty or artheroscopic plaque. Angioplasty is designed to relieve the chest pain a person usually feels when the heart is not getting enough blood and oxygen. Percutaneous transluminal coronary angioplasty or ballon angioplasty is the most frequently applied interventional technique for treatment of coronary artery disease (Bittl 1996). Plastic deformation of the obstructive plaque with the creation of splits, intimal tears and dissections is the main mechanism of percutaneous transluminal coronary angioplasty for lumen widening. Limiting dissections and acute vessel closure can unpredictably occur resulting in myocardial infarction and urgent bypass surgery. Moreover, long-term success of percutaneous transluminal coronary angioplasty is limited by restenosis. In order to overcome these limitations, alternative interventional techniques were developed. These techniques include directional angioplasty (Bittl 1996), ultrasound angioplasty (Rosenschein et al. 1991), laser angioplasty (Lee and Mason 1992), and high-speed rotational angioplasty (Safian et al. 1993). During directional coronary atherectomy, artherosclerotic tissue is extracted from the coronary artery with a cutting blade spinning at 5,000 rpm in the tip of the atherectomy device. In ultrasound angioplasty, direct mechanical contact between an oscillating tip and vascular plaque results in fragmentation and ablation of material into microscopic particles. Flexible biological materials such as healthy arterial wall or skin easily distend with the oscillation of the distal-tip. In contrast, the rigid calcium plaque matrix lacks flexibility and is disrupted (Demer et al. 1991). During excimer-laser angioplasty, short light pulses ( R2np ), the expression for the critical laser fluence becomes (Egerev et al. 2009)

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Fc = 4π Rnp cl ρl χl τL Tboil /σabs .

(7.10)

Here, Vnp , cnp , and ρnp are the volume, specific heat capacity, and density of the nanoparticle. The term σabs represents the absorption cross section given by ( (2 ( (2

2π (2n + 1) (an ( Cn + (bn ( (n + 1)Cn−1 + nCn+1 , 2 k0 |ε| n=1 ∞

σabs =

(7.11)

with Cn = Im

-√ √ . √ ε∗ j n k0 Rnp ε jn−1 k0 Rnp ε∗ ,

(7.12)

where an and bn are the Mie coefficients for the transmitted field, k0 = 2π/λ is the wave number, jn is the Riccati-Bessel function, and the superscript ∗ indicates the operation of complex conjugation. The mathematical formulation proposed by Egerev et al. (2009) requires the knowledge of the optical absorption cross section for the determination of the initial bubble radius. It is well known that the optical absorption and scattering properties of gold nanoparticles can be tuned by changing their size and shape (Jain et al. 2006). For example, gold nanospheres with a diameter of 20 nm show essentially only surface plasmon enhanced absorption with negligible scattering (Jain et al. 2006). However, when the nanoparticle diameter is increased from 20 to 80 nm, the relative contribution of surface plasmon scattering to the total extinction of the nanoparticle increases. Thus, larger nanoparticles are more suitable for light-scattering-based applications. Calculations show that, for gold nanoparticles irradiated at λ = 532 nm, the maximum absorption corresponds to the nanoparticles with radius of 10–40 nm (Jain et al. 2006). This is the optimal size of a gold nanosphere to achieve maximum energy absorption per unit volume for the specific incident laser wavelength. For a gold nanosphere with a radius of 10 nm the calculated optical cross-section equals the geometrical cross-section. The calculated optical cross-section increases with the diameter of the nanospheres and is about three times larger than the geometrical cross-section for a nanoparticle radius of 40 nm (Jain et al. 2006). Another interesting property of gold nanoparticle surface plasmon resonance is its sensitivity to the local refractive index or dielectric constant of the environment surrounding the nanoparticle surface. The nanosphere plasmon resonance shifts to higher wavelengths with increasing refractive index of the surrounding medium (Lee and El-Sayed 2006). The nanoparticle surface plasmon resonance can also be red-shifted by the self-assembly or aggregation of nanoparticles (Sönnichsen et al. 2005). Oraevsky (2008) has indicated that for a nanoparticle inside a bubble irradiated at the wavelength close to its plasmon resonance optical absorption, the absorption cross section can be approximately estimated as equal to its geometric cross section.

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It is also worth noting here that, for very short laser pulses, the critical laser fluence is independent of the pulse shape. Furthermore, for a specific value of the pulse duration there is an optimal particle size, which has the minimal value of Fc . For large particles, heat exchange with the surroundings is negligible, and Fc ∝ Rnp . In contrast, for small particles (χl τL >> R2np ), Fc ∝ R−1 np (Egerev et al. 2009).

7.2.3 Biological Effects of Cavitation The first thorough study using pulsed laser radiation and gold nanospheres was performed in 2003 by Lin and co-workers for selective and highly localized photothermolysis of targeted lumphocytes cells (Pitsillides et al. 2003). Lumphocytes incubated with gold nanoparticles conjugated to antibodies were exposed to nanosecond laser pulses (Q-switched Nd:YAG laser, 565 nm wavelength, 20 ns duration) showed cell death with 100 laser pulses at an energy of 0.5 J/cm2 . Adjacent cells just a few micrometers away without nanoparticles remained viable. Their numerical calculations showed that the peak temperature lasting for nanoseconds under a single pulse exceeds 2,000 K at a fluence of 0.5 J/cm2 with a heat fluid layer of 15 nm. The cell death was attributed mainly to the cavitation damage induced by the generated cavitation bubbles around the nanoparticles. In the same year, Zharov et al. (2003) performed similar studies on the photothermal destruction of K562 cancer cells. They further detected the laserinduced bubbles and studied their dynamics during the treatment using a pump– probe photothermal imaging technique. Later they demonstrated the technique in vitro on the treatment of some other type of cancer cells such as cervical and breast cancer using the laser induced-bubbles under nanosecond laser pulses (Zharov et al. 2004, 2005b, c). Recent work has demonstrated the treatment modality for in vivo tumor ablation in a rat (Hleb et al. 2008). Intracelullar bubble formation resulted in individual tumor cell damage. The formation of cavitation bubbles around nanoparticles also caused physical damage to the Staphylococcus aureus bacterium as confirmed by the images presented by Zharov et al. (2006) (Figs. 7.10 and 7.11). At relatively low laser energies, they observed a very slight penetration of nanoparticles in the cell wall (Fig. 7.11b) compared to the control without laser exposure (Fig. 7.11a). Higher laser energies, or the formation of nanoparticle clusters, led to a deeper penetration of nanoparticles inside the bacterial wall (Fig. 7.11c). High laser energy and/or formation of nanoclusters coupled with multi-pulse exposure produced local cell-wall damage (Fig. 7.11d) and finally complete bacterial disintegration (Fig. 7.11e, shows fragmented bacteria). These data demonstrate that, despite the relatively high thickness and density of the bacterial cell wall, bubble formation around nanoparticles may potentially cause irreparable damage to bacteria. The photothermolysis of living EMT-6 breast tumor cells triggered by gold nanorods was investigated by Chen et al. (2010). In the absence of gold nanoparticles, the cells survived under the excited energy fluence of 93 mJ/cm2 . However, cell mortality was observed at 113 mJ/cm2 energy fluence. Results of

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Fig. 7.10 Images of Staphylococcus aureus with attached gold nanoparticles: (a) phase contrast image; (b) photothermal image of bacteria alone; (c) photothermal images of bacteria with 40-nm gold particles irradiated at a laser fluence of 0.4 J/cm2 ; and (d) photothermal images of bacteria with 40-nm gold particles irradiated at a laser fluence of 2 J/cm2 . Dashed lines represent the bacterial boundary in (c) and (d). Arrows in (d) indicate photothermal images of single nanoparticles, whereas the arrowhead shows a bubble around one nanocluster. Reproduced with permission from Zharov et al. (2006). © Elsevier B.V.

Fig. 7.11 Images of Staphylococcus aureus conjugated with gold nanoparticles before (a) and after (b–e) multilaser exposure of 100 pulses, wavelength of 532 nm, and pulse duration of 12 ns at a different conditions: (b) laser fluence of 0.5 J/cm2 and no clusters; (c) laser fluence of 0.5 J/cm2 with clustered nanoparticles; and (d) laser fluence of 3 J/cm2 at one and several (e) nanocluster numbers. A dashed line represents the bacterial boundary in (e). Arrows in (b) and (c) indicate penetration of nanoparticles into the wall, and in (d), arrows indicate local cell-wall damage. Reproduced with permission from Zharov et al. (2006). © Elsevier B.V.

the cells with gold nanoparticles, under excitation at energy fluences of 113 and 93 mJ/cm2 , are shown in the series of images in Fig. 7.12; the images were taken within a period of 60 s. Upon reaching an energy fluence of 113 mJ/cm2 , the whole cell was seriously destroyed (Fig. 7.12a–d). At an energy fluence of 93 mJ/cm2 , a discernible internal explosion phenomenon occurred upon excitation (Fig. 7.12e–h). Meanwhile, the formation of characteristic cavities (shadows indicated by arrows) was especially pronounced at nanoparticle cluster locations (cluster size between 2 and 3 μm). The diameter of the cavities can reach as large as 10 μm. The results showed that localized photothermal effect of gold nanoparticles was large enough to trigger a considerable explosion, resulting in the formation of cavitation bubbles inside cells. These bubbles are responsible for the perforation or sudden rupture of plasma membrane. Their study also indicates that the energy threshold for cell therapy depends significantly on the number of nanoparticles taken up per cell. For an ingested gold nanoparticle cluster quantity N ∼ 10–30 per cell, it was found that energy fluences larger than 93 mJ/cm2 led to effective cell destruction within

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Fig. 7.12 Photothermolysis of the EMT-6 tumor cell triggered by gold nanoparticles under different energy fluences. (a–d) 113 mJ/cm2 ; (e–h) 93 mJ/cm2 . The shadows indicated by arrows are attributed to the formation of transient cavitation bubbles. The gold nanoparticles inside the cell can be seen in (a) and (e). Reproduced with permission from Chen et al. (2010). © Elsevier B.V.

a very short period. As for a lower energy level (18 mJ/cm2 ) with N ∼ 60–100, a non-instant, but progressive cell deterioration, was observed. The photothermolysis of lung carcinoma cells (A549) triggered by gold nanospheres with a diameter of 50 nm was investigated by Lukianova-Hleb et al. (2010). They found that at laser fluences below the bubble generation threshold, the nanoparticles in cells still were significantly heated by the laser pulse but did not cause detectable damage to the cells. Also, the exposure of the cell to 16 pump laser pulses (at 15 Hz frequency), instead of a single pulse, did not influence the cell viability and the level of the damage threshold fluence, which suggests that the cell damage results from a single event rather than from an accumulative effect of the sequence of the bubbles. They concluded that the bubble damage mechanism is mechanical and non-thermal: a single laser pulse induces an expanding bubble that disrupts the cellular cytoskeleton and plasma membrane causing visible membrane blebs. Blebbing of the plasma membrane for various cell types was also observed by Tong et al. (2007) (Fig. 7.13). The authors noted that bleb formation could not be the direct product of cavitation, as the rates of growth were several orders of magnitude slower than the timescale for microbubble expansion. They hypothesized that the blebbing response was due to the disruption of actin filaments, which form a dense three dimensional network beneath the cell membrane to provide mechanical support and sustain cell shape. However, an important conclusion of their study is that the cell death is attributed to the disruption of the plasma membrane as a

244


Fig. 7.13 Photothermolysis mediated by gold nanorods with longitudinal plasmon resonances centered at 765 nm. Cells were irradiated at 765 nm using a Ti:sapphire laser which could be switched between fs-pulsed and cw mode. (a, b) Cells with membrane-bound gold nanorods exposed to cw near infrared laser irradiation experienced membrane perforation and blebbing at 6 mW power. The loss of membrane integrity was indicated by EB staining (light grey, yellow online). (c, d) Cells with internalized gold nanorods required 60 mW to produce a similar level of response. (e, f) Gold nanorods internalized in KB cells labeled by folate-Bodipy (lighter grey, green online) were exposed to laser irradiation at 60 mW, resulting in both membrane blebbing and disappearance of the gold nanorods. (g, h) NIH-3T3 cells were unresponsive to gold nanorods, and did not suffer photoinduced damage upon 60 mW laser irradiation. (i, j) Cells with membranebound gold nanorods exposed to fs-pulsed laser irradiation produced membrane blebbing at 0.75 mW. (k, l) Cells with internalized gold nanorods remained viable after fs-pulsed irradiation at 4.50 mW, as indicated by a strong calcein signal (grey, green online). Reproduced with permission from Tong et al. (2007). © Wiley-VCH Verlag GmbH & Co. KGaA

consequence of gold nanoparticles mediated cavitation. Membrane perforation led to an influx of extracellular Ca2+ followed by degradation of the actin network, producing a dramatic blebbing response. Lin et al. investigated the thresholds for cell death produced by cavitation induced around absorbing microparticles irradiated by nanosecond laser pulses (Lin et al. 1990; Leszczynski et al. 2001). They observed that an energy of 3 nJ absorbed by a single particle of 1-μm diameter produced sufficiently strong cavitation to kill a trabecular meshwork cell after irradiation with a single laser pulse. Pulses with 1-nJ absorbed energy produced lethality after several exposures (Lin et al. 1990). Viability was lost even when no morphological damage was apparent immediately after the collapse of a transient bubble with a maximum radius of about 6 μm.

References

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It is clear that the laser-induced cavitation bubbles represent an important damaging factor in plasmonic photothermal therapy. The generation of cavitation bubbles may occur simultaneously in several locations inside the cell volume. The most effective cell killing occurs when the nanoparticles are located on or inside the cell membrane to provide membrane rupture. When the bubble reaches the size comparable to that of the cell it would definitely damage cellular membrane causing necrosis and lysis. Smaller bubbles may also induce apoptosis without rupturing the membrane. However, nanometer-sized cavitation bubbles that emerge around nanoparticles located at a distance from the cell membrane do not damage the cells due to their limited diameter of less than a micrometer. The threshold of pulsed laser interaction with clusters of nanoparticles is significantly lower than that for a single nanoparticle. Superheating of the nanoparticle clusters generates a much larger cavitation bubble capable of damaging even large cells. Thus, the creation of nanoclusters, consisting of many small nanoparticles, on the cell membrane or inside the cell is one potential way to overcome the limitations of using single nanoparticles which are due to the lower efficiency of bubble formation in the case of small nanoparticles or to the difficulties with their selective delivery to the target in the case of large nanoparticles. More effective bubble formation in a cluster of gold nanoparticles is associated with optical and thermal amplification effects and, especially, with overlapping nanobubbles from a single nanoparticle as separate nucleation centers or the generation of one large bubble around a gold nanoparticles cluster as a single nucleation center due to rapid heat redistribution between very closely located gold nanoparticles within a cluster (Zharov et al. 2005a). An alternative damage mechanism that should be considered is the mechanical destruction of cell structures by high tensile stresses. The numerical results presented by Volkov et al. (2007) indicate that the pressure waves emitted from the nanoparticles do not have any significant tensile stress component. However, particle reflection of the compressive pressure wave from internal cellular structures may result in the generation of the tensile stresses and associated cell damage. They estimated that, for a laser pulse duration of 200 fs, the maximum amplitude of the tensile stress exceeds 1 MPa for particles larger than 25 nm and laser fluences larger than 20 J/m2 . Additionally, the effective therapeutic effect for cancer cell killing may be achieved owing to nonlinear phenomena that accompany the thermal explosion of the gold nanoparticles, such as the generation of nanoparticle explosion products with high kinetic energy as well as strong shock waves with supersonic expansion in the cell volume.

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Chapter 8

Cavitation in Other Non-Newtonian Biological Fluids

In the previous chapters we have described the effects of cavitation in the cardiovascular system and cell surgery. There are an increasing number of biomedical contexts where cavitation takes place in other non-Newtonian biological fluids, such as saliva or synovial fluid. In saliva, cavitation occurs during some medical applications of lasers and ultrasound. In synovial liquid, cavitation is responsible for the cracking noise emitted from joints and may also damage the articular cartilage. In this chapter, we provide a qualitative description of cavitation and some of its associated bioeffects encountered in clinical applications. The archival literature in these cases is not as impressive as in the case of blood. Threshold conditions for the onset of cavitation in various biological fluids require more precise definition, preferably mathematical models underpinned by an extensive body of experimental evidence. The conditions associated with the onset of morphological damage also merit a more precise description. Nevertheless, we invite the reader to appreciate how cavitational activity can help address some of the present therapeutic challenges in several non-Newtonian biological fluids.

8.1 Cavitation in Saliva In some dentistry applications, an ultrasonically vibrating probe is placed in close proximity to the biological tissue or rigid material. The cavitation induced at the tip of this probe (or around the probe) creates the desired effect when it is placed close to the tissue or rigid material. Cleaning the teeth by dislodging plaque is one of the earliest applications of such an ultrasonic probe in dentistry. Other current applications include passive irrigation of the root canal and orthognathic surgery of the mandible.

8.1.1 Cavitation During Ultrasonic Plaque Removal The old-fashioned technique of plaque removal is the hand instrumentation. In this case, a curette must be placed below the deposit to be effective. When deep calculus approaches the bottom of the pocket, positioning the instrument may damage E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3_8,

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the periodontal attachment. In the attempt to create smooth roots, free of any deep accretions, the hand instruments tear into the fragile periodontal ligament and scrape off tooth structure. Unlike the curette, an ultrasonic scaler tip works from the top of the deposit downward, so there is no need to violate the attachment. Thus, ultrasonic instrumentation is now the first choice over hand instrumentation for most patients. The ultrasonic scaler has been used for about 30 years. In most practices it is used primarily for gross removal of supra-gingival calculus. New super-thin tips are now available that fit into deep pockets and small furcal areas where a standard curette is ineffective. Some manufacturers have designed machines with a far wider power range, so they can create effective cavitation at the low power settings needed for sub-gingival use (O’Leary et al. 1997). Ultrasonic scalers are now the preferred method for sub-gingival debridement. Recent research has shown that sub-gingival ultrasonic scaling not only removes calculus as well as traditional hand instrumentation, but that it also kills bacteria and reduces the level of endotoxins. Back in the early 70’s researchers noticed that the ultrasonic scaler cleaning ability dropped significantly when the water flow was interrupted. They speculated that this was due to the irrigating effect of the water (Clark 1969). Later research indicated that no matter which tip was used, and no matter at what angle it touched the tooth, the amount of plaque-free surface increased by 500–800% when the water was turned on (Walmsley et al. 1988). They noted that the dry tip removed plaque only where it contacted the tooth. However, when a water cooled tip was used they observed that surfaces as much as a half millimeter away from the tip were completely plaque-free. The tip’s high-frequency vibrations create cavitation bubbles. When the energized spray from the hand-piece contacts the tooth surface, these bubbles collapse and release short bursts of energy which literally blast the plaque from the surface and tear apart bacterial cell membranes in the process (Walmsley et al. 1988; Lea et al. 2005; Felver et al. 2009). The ultrasound field generated by the scaler is comprised of a series of compressions and rarefactions (regions of high and low pressure) which cause small cavitation nuclei to expand and contract (Fig. 8.1a). Inertial cavitation bubbles oscillate violently and may expand to many times their original size before imploding. The collapse of such a bubble can result in shock

Fig. 8.1 Cavitation and acoustic streaming around an ultrasonic scaler. (a) The oscillating pressure field causes a cavitation nucleus to expand and contract. (b) When the cavitation bubble is located close to the scaler, it may collapse accompanied by the formation of a liquid jet

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waves associated with massive temperatures and pressures. If the bubble collapses or implodes near to the surface of a tooth or a scaler tip then the collapse is asymmetrical resulting in an inrushing jet of liquid targeted at the surface (Fig. 8.1b). This jet of liquid is powerful enough to potentially remove calculus and other materials from the tooth surface (Walmsley et al. 1984, 1988). Furthermore, the force of these jets is enough to visibly roughen the metallic surface of the ultrasonic scaler tip. A clear visualization of the spatial distribution of cavitation bubbles around three scaler tips, observed using sonochemiluminescence from a luminol solution, is given in Fig. 8.2 (Felver et al. 2009). The highest levels of cavitation activity were observed around vibration antinodes close to the bend in each tip. Surprisingly,

Fig. 8.2 Luminol photography of three scaler tips (Piezon miniMaster, Electro Medical Systems). Light regions indicate areas of high cavitation activity, with dark regions indicating little or no activity. Reproduced with permission from Felver et al. (2009). © Elsevier B.V.

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while the displacement amplitude was greatest at the free end of the tip, little to no cavitation was observed at the free end using luminol photography. It is also interesting to note here that cavitation does not occur around powered tooth brushes. Lea et al. (2004) tested five commercial brushes and monitored the formation of the hydroxyl radical that occurs during cavitation bubble collapse. Operating the toothbrushes for periods up to 20 min resulted in no cavitational activity being detected.

8.1.2 Cavitation During Passive Ultrasonic Irrigation of the Root Canal The goal of ultrasonic irrigation of the root canal is to remove pulp tissue and microorganisms from the root canal system as well as smear layer and dentim debris that occur following instrumentation of the root canal (van der Sluis et al. 2007). Passive (non-cutting) ultrasonic irrigation is based on the transmission of energy from an ultrasonically oscillating instrument to an irrigant in the root canal (van der Sluis et al. 2005). After the root canal has been shaped to the master apical file, a small file (or wire) is inserted in the centre of the root canal, as far as the apical region. The root canal is then filled with an irrigant, usually a sodium hypochlorate solution, and the ultrasonically vibrating file activates the irrigant in order to penetrate more easily into the apical part of the root canal system (Krell et al. 1988). The file is driven to operate in transverse mode at frequencies of 25–30 kHz. During passive ultrasonic irrigation, acoustic microstreaming and cavitation can occur which cause a streaming pattern within the root canal from the apical to the coronal (Ahmad et al. 1987; Roy et al. 1994). Because of this microstreaming, more dentine debris can be removed from the root canal compared with syringe delivery of the irrigant (Lea et al. 2004), even from remote places in the root canal (Goodman et al. 1985). A detailed investigation on the behaviour of cavitation bubbles generated during passive ultrasonic irrigation of the root canal was conducted by Roy et al. (1994). They indicated that transient cavitation bubbles only occur when the file can vibrate freely in the canal or when the file touches un-intentionally (or for a short duration) the canal wall (Fig. 8.3) (see also Lumley et al. 1993). Intentional (long duration) contact with the canal wall suppresses the formation of transient cavitation bubbles. The authors also noted that a smooth file with sharp edges and a square cross-section produced significantly more transient cavitation than a normal file. The transient cavitation was visible at the apical end and along the length of the file. When the file came in contact with the canal wall, stable cavitation was affected less than transient cavitation and was mainly seen at the midpoint of the file. A pre-shaped file brought into a curved canal is more likely to produce transient cavitation rather than a straight file. Other researchers claim that cavitation provides only minor benefit in ultrasonic irrigation, or that it does not occur at all (Walmsley 1987; Ahmad et al. 1988; Lumley et al. 1988).

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Fig. 8.3 A glass root canal model showing the file at rest (left) and file in operation displaying cavitation bubbles. Reproduced with permission from Roy et al. (1994). © John Wiley and Sons

Cavitation bubbles can also be used for the delivery of antibacterial nanoparticles into dentinal tubules. Persistent root canal infection has been associated with bacterial presence in the dentinal tubules. Studies have shown that bacteria can penetrate into dentinal tubules, and the depth of penetration varies from 300 to 1,500 μm (Love and Jenkinson 2002). However, the bacteria within the dentinal tubules are inaccessible to the conventional root canal irrigants, medicaments, and sealers because they have limited penetrability into the dentinal tubules. Although the application of ultrasound produces better results compared with syringe irrigation in cleaning and delivering irrigants into the anatomic complexities, ultrasonic irrigation does not debride the root canal system completely. In a very recent study, Shrestha et al. (2009) have indicated that the collapsing cavitation bubbles treatment using high-intensity focused ultrasound can result in a significant penetration up to 1,000 μm of antibacterial nanoparticles into the dentinal tubules. The cavitation bubbles produced using high-intensity focused ultrasound can be used as a potential method to deliver antibacterial nanoparticles into the dentinal tubules to enhance root canal disinfection. The mechanism responsible for the delivery of antibacterial nanoparticles is illustrated in Fig. 8.4 in the case of a spark-generated cavitation bubble. The bubble grows to a maximum size (with maximum radius of 3.3 mm) in a time of 0.46 ms (Fig. 8.4b). The collapse of cavitation bubble (Fig. 8.4c–g) generates a high-speed jet, which moved toward the channel at about 68 m/s. This jet delivers the bead of plaster (with a mass of approximately 6 mg), which was originally placed about 2 mm from the top of the channel, into the whole length of the channel (Fig. 8.4e–g).

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Fig. 8.4 The collapse of a cavitation bubble with a maximum bubble radius of 3.3 mm on top of a tubular channel of 3.3-mm diameter. The time is indicated at the bottom right. The bubble collapses at time t = 0.73 ms with a jet speed of about 68 m/s. The rubber plaster balls are centered initially 2 mm from the top of the channel opening. It can be seen from frames corresponding to times t = 1.3 to t = 5.8 ms that the rubber plaster is pushed by the flow down the entire length of the channel. Reproduced with permission from Shrestha et al. (2009). © Elsevier B.V.

8.1.3 Cavitation During Laser Activated Irrigation of the Root Canal The first laser use in endodontics was reported by Weichman and Johnson (1971) who attempted to seal the apical foramen in vitro by means of a high power-infrared (CO2 ) laser. Although their goal was not achieved, sufficient relevant and interesting data were obtained to encourage further study. Subsequently, attempts have been made to seal the apical foramen using the Nd:YAG laser (Weichman et al. 1972). Although more information regarding this laser interaction with dentine was obtained, the use of the laser in endodontics was not feasible at that time. Since then, many papers on laser applications in dentistry have been published (see, for example, Wigdor et al. 1995 and the references therein). Nevertheless, in dentistry and in

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endodontics in particular, acceptance of this technology by clinicians has remained limited, perhaps partly due to the fact that this technology blurs the border between technical, biological, and dental research. Lasers, such as the Er,Cr:YSGG laser, have been also proposed as an alternative for the conventional approach in cleaning, disinfecting and even shaping of the root canal or as an adjuvant to conventional chemo-mechanical preparation in order to enhance debridement and disinfection (Kimura et al. 2000; Stabholz et al. 2004). The high-speed recordings obtained by Blanken et al. (2009) have demonstrated that vaporization of the liquid inside a root canal model will result in the formation of cavitation bubbles, which expand and implode with secondary cavitation effects. At the beginning of the laser pulse, the energy is absorbed in a thin liquid layer that is instantly heated to boiling temperature at high pressure and turned into vapour. This vapour at high pressure starts to expand at high speed leading to the formation of a cavitation bubble. A free expansion of the bubble laterally is not possible in the root canal model, and hence the liquid is pushed forward and backward in the canal. The forward pressure can be easily observed in the Fig. 8.5 showing an air bubble, present in the canal, being compressed to a flat disk. As the energy source stops, the vapour cools and starts condensing, while the momentum of expansion creates a lower pressure inside the bubble. Liquid surrounding the bubble is accelerated to fill in the gap. Secondary cavitation bubbles are also be induced at irregularities along the root canal wall. The implosion of the primary and secondary bubbles creates microjets in the fluid aimed at the wall with very high forces locally. This mechanism might also contribute to the disruption of cells and the smear layer at the wall.

Fig. 8.5 An air inclusion being compressed when a laser-induced cavitation bubble grows and expands in an artificial root canal. The maximum compression of the air inclusion is visible in the third frame. Reproduced with permission from Blanken et al. (2009). © John Wiley and Sons

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8.1.4 Cavitation During Orthognathic Surgery of the Mandible Ultrasonic devices might be also effective in minimizing the hazard of surgical trauma in maxillofacial surgery. Particularly in elective orthognatic surgery of the mandible protection of the inferior alveolar nerve is important to reduce surgical morbidity. Gruber et al. (2005) have recently presented some preliminary results on using an ultrasonic bone cutting device in bilateral sagittal split osteotomies of the mandible. They noted that the cavitation phenomenon is responsible for the good visibility of the surgical site due to a cleaning effect of the microstream towards the rigid boundary of the surface of the bone. The effect of cavitation on cells of the adjacent tissue such as periosteal cells or bone cells is still not fully understood.

8.2 Cavitation in Synovial Liquid Cavitation in human joints has been linked with the sharp cracking noise emitted from some joints, particularly from the metacarpophalangeal joint (Unsworth et al. 1971). When a synovial joint is distracted, the pressure in the synovial liquid can drop below its vapour pressure (approximately 6,500 Pa), and the fluid evaporates spontaneously forming a bubble in the joint space (Fig. 8.6) (Unsworth et al. 1971).

Fig. 8.6 Roentgenogram of a metacarpophalangeal joint after cracking showing the bubble present in the joint space. Reproduced with permission from Unsworth et al. (1971). © BMJ Publishing Group Ltd

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A very nice visualisation of bubble formation in the metacarpophalangeal joint is also given by Watson et al. (1989). This phenomenon is known as viscous adhesion or tribonucleation (Campbell 1968). It causes bubble formation as a result of the large negative pressure generated by viscous adhesion between surfaces separating in liquid. It occurs when two closely opposed surfaces separated by a thin film of viscous liquid are pulled rapidly apart. Viscosity prevents the liquid from filling the widening gap, resulting in negative pressure. Cavitation may also occur when the articular surfaces are separated through the elastic recoil of the synovial fluid above a critical velocity, causing the synovial fluid to fracture like a solid. A proportion of the cracking noise during cavitation of synovial fluid may therefore be considered as synonymous with the inception of the cavitation (Chen and Israelachvili 1991; Chen et al. 1992). Cavitation is not the only mechanism of all cracking noises emitted from joints. Some sounds are produced by patellofemoral crepitus (a fine crunching noise, usually on bending the knee from standing, which is said to be due the kneecap cartilage rubbing against the underlying cartilage of the femur) or when the plica (a thin wall of fibrous tissue that are extensions of the synovial capsule of the knee) snaps over the end of the femur (Beverland et al. 1986; McCoy et al. 1987). Other studies provide clear evidence that the anatomic source of the cracking sound associated with spinal high-velocity low-amplitude thrust manipulations is associated with cavitation of the synovial fluid (Watson and Mollan 1990; Evans 2002). The audible “crack” is often viewed as signifying a successful manipulation. Several authors suggested that cavitation during in vivo conditions can take two forms: (a) that which produces the familiar cracking noise (called macrocavitation), and (b) microbubble activity that may be occurring because of the bubbles remaining in the synovial fluid after the crack (called microcavitation) (Watson et al. 1989; Unsworth et al. 1971). The existence of gas bubbles in synovial joints (after macrocavitation) has been demonstrated by radiography as a dark, intra-articular radiolucent region since early in the twentieth century (Unsworth et al. 1971; Fuiks and Grayson 1950; Kramer 1990). Damage of the articular cartilage is a possible consequence of cavitation in the synovial liquid. Watson et al. (1989) investigated the effects of cavitation on bovine knee joint articular cartilage. Cavitation was generated using a vibrating tip operating at 20 kHz with a maximum amplitude of 0.127 mm. During the first 20 s of exposure to cavitation, no significant damage of the specimen was observed. The specimen exposed to cavitation for 1 min presented shallow depressions, approximately 20 μm in diameter, covering the surface. After 10 min of exposure to cavitation, the specimen displayed considerable surface disruption with distinct craters that have approximately 100 μm in diameter. Obviously, such large collateral effects are unlikely in an in vivo situation but they emphasize a possible role of cavitation in damaging the articular cartilage. Watson et al. (1989) noted that the cumulative effects of cavitation may be, over a period of time, sufficient to damage the articular cartilage. Although these authors proposed this theory as a cause of direct damage to the joint cartilage, there is, so far, little clinical evidence to support this mechanism. In a recent in vivo study, the effect of extracorporeal shock waves on joint cartilage was evaluated in 24 rabbits

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(Väterlein et al. 2000). It is well known that the combined effects of the shock waves and cavitational collapse induce harmful side effects on the adjacent biological structures, such as that observed in lithotripsy (Delius et al. 1988). However, macroscopical radiological and histological analysis at 0, 3, 12 and 24 weeks after treatment showed no pathological changes in the joint cartilage. Arthroscopic cartilage ablation (see, for example, Smith 1993) is another medical application where cavitation takes place in the synovial liquid. The dynamics of cavitation bubbles generated by a pulsed holmium laser radiation (wavelength 2.12 mm, pulse duration between 100 and 1,000 ps), transmitted through an optical fiber, and their impact on medical laser use for cartilage ablation have been investigated by Asshauer (1996). Shock waves were observed at the bubble collapse several hundred microseconds after the start of the laser pulse and peak pressures up to several kilobars were measured. The observed complex bubble dynamics and pressure transient generation was explained by a two stage model: in the first stage of the bubble formation process, a water volume at the fiber tip is superheated by the laser radiation, until an explosive vaporization induces an isotropic vapour bubble expansion. In the second stage, a quasi-continuous ablation through the bubble takes place. The relative importance of the second stage increases for higher fluences and longer pulse durations, perturbing the initial nearly spherical symmetry of the bubbles. The angle of incidence of the laser radiation was identified as an important additional parameter for cartilage ablation. It was shown that shallow angles of incidence reduce pressure transient amplitudes as well as thermal side effects of cartilage ablation. Ultrasonically induced cavitation may also have a clinical benefit to control synovial proliferation and inflammation or some other disorders of joints. Nakaya et al. (2005) investigated the effect of a microbubble-enhanced ultrasound treatment on the delivery of methotrexate (an antimetabolite and antifolate drug used in treatment of cancer and autoimmune diseases) into synovial cells. They found that the methotrexate concentration in synovial tissue was significantly higher in the presence microbubbles while the synovial inflamation was less prominent. Saito et al. (2007) reported that the expression of plasmid DNA and small interfering RNA in the synovium was significantly enhanced by ultrasound in combination with microbubbles. In a more recent study, Nakamura et al. (2008) observed that ultrasound treatment in combination with microbubbles increased cellular uptake of enzymes (histone deacetylase) into human rheumatoid synovial cells.

8.3 Cavitation in Aqueous Humor An interesting application of cavitation in non-Newtonian fluids is encountered in ultrasound phacoemulsification. In cataract surgery, the turbid nucleus and cortex of the lens of the eye are removed and an artificial lens is implanted into the capsular bag to restore vision. After the cornea and the anterior lens capsule are surgically opened, an ultrasound tip similar to a Mason-horn, operating at frequencies between

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20 and 40 kHz, is used to emulsify or fragment the lens nucleus (whether it is emulsification or fragmentation depends on the hardness of the nucleus). The fragments are then removed by means of an irrigation suction system that is integrated into the phaco-tip. After the lens capsule is cleaned, the artificial lens is implanted and the eye is closed again. Manual extraction of the nucleus demands a large wound of approximately 9 mm chord length. Phacoemulsification can be done through a smaller wound of approximately 3 mm length. Wound length is also governed by the size of the intraocular lens that is inserted. Conventional polymethylmethacrylate lenses require the phacoemulsification incision to be enlarged to 6 mm to allow their insertion. Intraocular lenses made from different materials such as hydroxymethyl methacrylate or silicone can be folded to allow their insertion. This further facilitates the use of small incisions. Besides the intended surgical effect, some unwanted collateral effects are observed, such as damage of the corneal endothelium (Walkow et al. 2000), rupture of the posterior capsule (Martin and Burton 2000) and damage of the phaco-tip itself with metal particulate often left in the eye after surgery (Gimbel 1990; Kreiler et al. 1992). A typical lesion on human corneal endothelium is shown in Fig. 8.7. The most serious ocular complication of phacoemulsification lens extraction is dropping the nucleus into the vitreous cavity. This may result in visual loss due to inflammation and retinal detachment. Fortunately, this complication is unusual in experienced surgeons and sight loss can be prevented by vitrectomy and nucleus removal. Phacoemulsification predominates as the procedure of choice for cataract extraction. The reasons are rooted in improved outcome for the patient. The main advantage is reduced corneal astigmatism after cataract surgery. Since the cornea is the major refracting surface of the eye, minor disturbances to its shape may result in marked astigmatism with serious consequences for vision. All corneal surgery has the tendency to produce astigmatism with less intervention producing less distortion than the more disruptive procedures. However, small cataract incisions produce less astigmatism than large incision cataract surgery.

Fig. 8.7 Scanning-electron micrograph of human endothelium lesion resulting from a 5-min exposure to ultrasound. Bar marker: 20 μm. Reproduced with permission from Olson et al. (1978). © BMJ Publishing Group Ltd

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The phaco-tip vibrations are strong enough to generate both transient and stable cavitation bubbles which probably produce most of the fragmentation and emulsification of the lens and which may also cause a removal of material from the phaco-tip. An example of cavitation bubble formation as a result of phaco-tip vibrations is illustrated in Fig. 8.8. Several authors cite the formation of free radicals as evidence of cavitation during phacoemulsification. These species are thought to be generated when the heat from the implosion of cavitation bubbles causes the decomposition of water (Augustin and Dick 2004; Shimmura et al. 1992; Takahashi et al. 2002). Holst et al. (1993) used a single photon counting apparatus and luminol in rabbit eyes to demonstrate chemoluminescence secondary to the production of free radicals during phacoemulsification. They also obtained data correlating the amount of free radicals produced with the amount of ultrasonic power used. Topaz et al. (2002) demonstrated sonoluminescence under simulated phacoemulsification in aqueous medium using electron paramagnetic resonance spectroscopy and photon detection. They also noted reduction of cavitation intensity and elimination of sonoluminescence by saturation of the solution with carbon dioxide. Cavitation around the phaco tip was also observed by Zacharias (2008). However, his study found strong evidence that cavitation plays no role in phacoemulsification, leaving the jackhammer effect as the most important mechanism responsible for the lens-disrupting power of phacomeulsification. Current surgical procedures, particularly ultrasound phacoemulsification for cataract surgery and other operations involving the anterior chamber of the eye, have benefited from the use of ophthalmic viscoelastic substances (Silver et al. 1992; Behndig and Lundberg 2002). The primary goal of these substances is to protect the corneal endothelium during surgical procedures. The viscoelastic substances should offer minimal thixotropy in order to aid retention within the eye and yet, following implantation, should possess high-equilibrium viscosity to ensure that there is an appropriate maintenance of the ocular space (Andrews et al. 2005). The viscoelastic properties of these substances may also reduce the cavitation intensity and, thus, the

Fig. 8.8 Ultrasonic phaco-tip showing wave propagation and presence of presumed cavitation bubbles. Reproduced with permission from Packer et al. (2005). © Elsevier B.V.

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addition of a suitably viscoelastic substance to the eye seems to be a potential way of preventing or mitigating the negative collateral effects induced by cavitation in ultrasound phacoemulsification. Although no direct evidence is available in literature, numerous experimental results indicate the reduction of cavitation damage in viscoelastic liquids for conditions similar to those encountered during ultrasound phacoemusification (see, for a detailed list of references, Chap. 3).

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Index

A Ablation products, 200, 205 Absorption cross section, 230, 238, 240 Acoustic droplet vaporization, 211, 213 Acoustic power, 188 Acoustic pressure, 133–134, 179–180, 183, 186, 191, 195, 212–213 Added mass, 183 Added pressure, 149 Angioplasty laser, 175, 199, 202–206, 210 percutaneous transluminal, 202–203 rotational, 202–203, 206 ultrasound, 202 Annular flow, 98, 102, 105 Artheroscopic plaque, 202 Arthroscopic cartilage ablation, 258 Articular cartilage, 37–38, 249, 257 Attenuation stage, 155–156 B Berthelot tube, 57, 59 Bifurcation, 89–91, 212–214 Bjerknes force, 102, 104 Blood density, 34, 37 elasticity, 34, 37 infinite-shear viscosity, 35 sound speed, 37 structure, 35, 37 surface tension, 37 thixotropy, 36 zero-shear viscosity, 35 Blood-brain barrier, 193, 198 Blunt bodies, 129–134, 148 Boger fluid, 19, 146–147 Boiling temperature, 239, 255 Boltzmann constant, 51, 239 Boundary integral methods, 97

Boundary layer transition, 126, 131 Bubble cloud, 129, 166–167, 177, 206–207, 209 Bubble splitting, 98, 101–105 Bullet-piston method, 58 C Capillary rheometer, 28–29 Cataract surgery, 258–260 Cavitation erosion, 63, 92, 101, 155–172 fixed, 117 hydrodynamic, 117–150 incipient, 117, 127, 142–143, 149 inertial, 195–196, 212, 250 jet, 126–129 noise, 128 number, 124–128, 130–131, 133, 135, 137–139, 142–145, 149 tip vortex, 117, 134–142, 148 traveling, 117 vortex, 117, 134–142, 148, 209 Cavitation damage mechanisms polymer solutions, 54, 84–85 water, 54, 84–85 Cavitation nanobubbles, 225 Cavitation susceptibility meter, 53, 56 Cell constituents, 34, 41–43 cytoplasmic viscosity, 42 cytoskeleton rheology, 41 Chaotic oscillations, 89–91 Circulation, 56, 123, 135, 138–139, 148, 177, 199, 202, 209 Clot lysis, 175, 195 Concentric cylinder rheometer, 25, 27 Cone and plate rheometer, 26–27 Confined space, 144–145, 149, 213 Constitutive equation

E-A. Brujan, Cavitation in Non-Newtonian Fluids, C Springer-Verlag Berlin Heidelberg 2011 DOI 10.1007/978-3-642-15343-3,

265

266 Carreau, 16 Casson, 16 elastic dumbbell, 19–20 Giesekus, 18 Jeffreys, 19 KBKZ, 20–21 linear Oldroyd, 71 Maxwell, 17–18 Oldroyd-B, 18–19 Phan-Thien-Tanner, 18 power law, 15–16 rigid dumbell, 20 upper convective Maxwell, 23, 25 Williamson, 70–71, 76 Contrast particles, 177 Convected time derivative, 17–18, 81 Corneal endothelium, 259–260 Coulter counter, 53–54 Counterjet, 92 Cracking noise, 117, 249, 256–257 Creep, 6, 32–33, 97 Critical break-up tension, 182 Critical laser fluence, 238–239, 241 Critical nucleus, 51, 53 D Damaged area, 159–161 Dentinal tubules, 253 Depolymerization, 42, 157 Desinent cavitation number, 127–128, 138–139 Diagnostic ultrasound, 175, 193–194, 198–199 Dilatational rate, 184 Dimensionless number Deborah, 13–14, 76, 79, 81, 89–90 elasticity, 13, 15 Reynolds, 13–15, 75, 79, 81, 85, 89, 118–119, 122–124, 127–128, 131–134, 143–145, 148–149, 183 Weissenberg, 13–14 Dirac function, 73 Drag coefficient, 183 Drag reduction, 118–121, 123, 144, 148 Dynamic rigidity, 9 E Elastic boundary, 101–107 Elastic compression modulus, 182 Elastic modulus, 33, 42, 102–108, 120, 168, 184 Elastic solid, 3, 9, 14 Encapsulated microbubbles buckling radius, 182 mathematical formulations, 238–241

Index translational motion, 183 Endodontics, 254–255 Enthalpy, 66, 68 Equation of state, 64, 169, 239 Equilibrium radius, 50, 69, 71, 86, 182 Erosion pattern, 158, 162 Extensional rheometry, 29–31 Extensional viscosity, 11–12, 17–20, 24, 29–31, 96, 110–112, 118–119, 127, 132, 148–150, 171 estimation, 110–112, 149–150 Extra stress tensor, 4, 7, 10, 15, 18, 21–23, 64–65, 80 F Fahreus effect, 35 Femtosecond optical breakdown, 226–227 Filament stretching rheometer, 31 Flow biaxial extensional, 11 oscillatory shear, 8–10 planar extensional, 12 simple shear, 3–4, 7–8, 17, 21 uniaxial extensional, 10–11, 23–24, 31, 41, 97 Flow time scale, 14 Fluid ideal, 1, 3 Newtonian, 1–4, 11–12, 15–16, 25, 27–28, 30, 35, 58, 66, 68, 71, 76, 97, 118, 137, 144, 146, 150, 163, 171, 186 non-Newtonian, 1–43, 63, 97, 118, 124, 137, 144, 148–149, 156–163, 258 real, 1, 5 rheopectic, 5–6 shear-thickening, 5, 16 shear-thinning, 5, 28–29, 35, 70–71, 79, 146 thixotropic, 5–6 viscoelastic, 5–6, 8, 16–21, 26, 30, 33, 37, 43, 78, 98, 120, 137, 146–148 viscoplastic, 5 Fluorescence correlation spectroscopy, 33 Free radicals, 198, 229, 260 Free-stream turbulence, 125–126 Frequency response curve, 86–89 G Gas content, 54, 125, 127–128, 135 embolism, 210–214 embolotherapy, 211–214 Geometric focusing effects, 168

Index Gibbs equations, 52 Gilmore equation, 67, 74 Green fluorescent protein, 194 H Harmonic resonance, 87–88, 90 Hookean relaxation time, 20 Huggins slope constant, 12–13 Hyperbaric oxygen therapy, 211 Hysteresis, 6, 51 I Imaging techniques, 179, 241 Inception cavitation number, 127, 131, 135, 137, 145, 149 Incubation stage, 155 Internal energy, 239 Intracorporeal stones, 165 Irrigant, 252–253 J Jet formation, 91, 100–101, 104, 164, 166, 168, 196 Jet velocity, 92–93, 96–97, 104–105, 121, 127, 166 K Keller-Herring equation, 67 Kinetic spinodal, 227 L Lamb vortex, 135 Laminar separation point, 126 Laplace transform, 73 Laser fluence threshold, 231 Loss modulus, 9, 33, 39, 43 Loss tangent, 9 Lumley hypothesis, 120 M Magnetic tweezer, 33 Mark-Houwink equation, 13 Mean-square displacement, 32–33 Mechanical heart valve, 206–210 Membrane blebbing, 244 Metacarpophalangeal joint, 256–257 Microemboli, 202, 209–211 Microrheology active methods, 32–34 passive methods, 32–33 Mie coefficients, 240

267 N Nanoparticle, 225, 230–245, 253 Normal stress coefficients, 8 Nucleation barrier, 51–52 heterogeneous, 49–50 homogeneous, 49, 51–53 rate, 51 Nuclei distribution blood, 55–57 polymer solutions, 54–55 water, 53–54 stabilization mechanisms, 50 Numerical methods, 11–12, 14, 17, 78, 80–82, 85, 87–90, 98, 105–106, 110, 119, 142, 148, 167, 181, 191, 213, 226–229, 234, 241, 245 O Opposed jet rheometer, 29–30 Optical tweezer, 32–33 Orifice flow, 121–123, 126, 148 Orthognatic surgery, 256 P Period-doubling cascade, 90 Perturbation approach, 97 Photodynamic therapy, 229 Photothermal therapy, 225, 229–245 Plasma, 34–38, 226, 242–243 Plasmonic photothermal therapy, 225, 229–245 Plastic flow stress, 168, 170–171 Polymer injection, 142 solutions dilute, 15, 82, 85, 110, 131, 149 semi-dilute, 13, 138, 143 ultrasonic degradation, 156–157 Polytropic index, 64 Pressure attenuation, 78–79 coefficient, 124, 135 drop, 28, 118, 121–123, 139, 143, 228 gradient, 91–92, 102, 118, 137, 144, 178, 196 Protein inactivation, 232 R Rankine vortex, 142 Rate of deformation tensor, 4, 11, 18, 23, 70 invariants, 4, 65 Rayleigh-Plesset equation, 180, 184, 233, 234 Red cell aggregation, 35

268 Relaxation time, 1, 14, 17–18, 20–21, 41, 71–72, 88–89, 146–147, 186, 189 Resonance frequency, 86, 187–190 Retardation time, 20, 71, 79, 81, 88–89, 189 Riccati-Bessel function, 240 Root canal, 249, 252–255 infection, 253 S Saddle-node bifurcation, 90 Saliva elasticity, 40–41 relaxation time, 41 structure, 40 viscosity, 40–41 Saturation pressure, 52, 144 Scattering cross section, 188, 191 Schiebe body, 131 Secondary flow, 27–28 Shear rheometry, 25–29 Shear waves, 120 Shock -induced collapse, 99 -induced jet, 99–101, 171 pressure, 96, 128, 164, 166 wave, 78, 82–83, 93–95, 99–101, 109, 164, 166–168, 171–172, 196, 212, 232, 245, 257–258 emission, 167, 232 Sonoporation, 192, 195 Sonothrombolysis, 175–177 Specific heat capacity, 240 Spherical acoustic wave, 78 Spherical bubble collapse time, 66, 76, 83–85, 99–100 dimensionless variables, 74 energy, 63–68 natural frequency, 71, 187 pressure distribution, 71, 78, 147, 226 scaling laws, 84–85, 108–109 thermal effects, 81, 227 Spherical bubble dynamics compressible formulation, 77, 79–80 general equations, 63–65 incompressible formulation, 66–67, 77, 79–80, 86 Splash effect, 165 Squeeze flow, 209 Stagnation point, 29, 131–132 Stagnation pressure, 170 Stokes-Einstein relation, 32 Storage modulus, 9, 19, 33, 39, 43 Strange attractor, 89–90

Index Streamwise velocity fluctuations, 120 Stress relaxation, 6, 21 tensor, 4, 7, 10, 18–19, 22–23, 64–65, 80 Subharmonic resonance, 87–88, 90 Surface dilatational viscosity, 65, 182, 184 plasmon resonance, 231–233, 240 Surfactant, 50, 57, 65, 97, 178, 183, 211 Synovial fluid density, 38 elasticity, 38–39 rheopexy, 38–39 structure, 39 viscosity, 38–39 Synovial proliferation, 258 T Tensile strength polymer solutions, 58–59 water, 58–59 Tensile stress, 57, 107, 111, 147, 166, 226–227, 245 Therapeutic ultrasound, 175 Thermal diffusivity, 239 Thermal expansion coefficient, 226 Thermo-elastic stress, 226 Threshold fluence, 235–236, 243 Thrombogenesis, 210 Thrombolitic agents, 176 Tissue ablation, 199–200 oxygenation, 229 plasminogen activator, 176, 195 Tooth brush, 252 Transmyocardial laser revascularisation, 199–200, 202 Tribonucleation, 257 Trouton ratio, 11–12, 30, 94–95, 148–150, 172 U Ultraharmonic resonance, 87, 90 Ultrasonic irrigation, 252–254 Ultrasonic scaler, 250–251 Ultrasound contrast agents, 106, 177–199 Ultrasound phacoemulsification, 258, 260–261 V Van der Waals equation, 239 Vascular endothelial growth factor, 193 Vena contracta, 121 Ventricular pressure, 206 Vibratory devices, 156 Viscoelastic effects

Index die swell, 6 Uebler, 6 Weissenberg, 6, 13–14, 28 Viscoelastic shell, 186 Viscometric functions, 8 Viscosity apparent, 4, 27–28, 35–36, 38–40, 42–43, 76 biaxial extensional, 11–12 dynamic, 2, 9 infinite-shear, 5, 16, 35, 70, 76, 87 intrinsic, 12–13, 20, 29 kinematic, 3, 146, 183 molecular weight, 13 planar extensional, 12 shear, 1–2, 5, 11–12, 16–17, 29–30, 35, 38, 65, 70, 76, 87, 90, 100–101, 112, 120, 127, 144–146, 149, 184, 186 specific, 12

269 uniaxial extensional, 11, 17, 19, 24, 31, 111 volume, 4 zero-shear, 5, 12, 16, 29, 35, 70, 146 Viscous adhesion, 257 Viscous sublayer, 119 Vortex chamber, 142–143 inhibition, 118, 123 strength, 135 W Wall turbulence, 120 Wave number, 240 Weight loss, 155–160, 163 Weissenberg-Rabinowitsch equation, 28 Y Yield strength, 107, 165, 168

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