Modeling and Simulation of Capsules and Biological Cells

MODELING AND SIMULATION OF CAPSULES AND BIOLOGICAL CELLS

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MODELING AND SIMULATION OF CAPSULES AND BIOLOGICAL CELLS EDITED BY

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Library of Congress Cataloging-in-Publication Data Modeling and simulation of capsules and biological cells p. cm. — (Chapman & Hall/CRC mathematical biology & medicine series ; v. 2) ISBN 1-58488-359-6 (alk. paper) 1. Cells—Mechanical properties—Mathematical models. 2. Cells—Mechanical properties—Computer simulation. 3. Erythrocytes—Deformability—Mathematical models. 4. Blood—Rheology—Mathematical models. 5. Cell membranes—Mathematical models. I. Pozrikidis, C. II. Series. III. Chapman & Hall/CRC mathematical biology and medicine series ; v. 2. QH645.5.M634 2003 571.6′34—dc21

2003044000

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Preface This collection of contributed chapters addresses the mathematical modeling and numerical simulation of liquid capsules and biological cells. Capsules and cells are distinguished from common bubbles and liquid droplets in that their interfaces exhibit mechanical properties that are more involved than those described by a constant and uniform surface tension. For example, non-uniformities in temperature or interfacial concentration of an insoluble surfactant are responsible for thermocapillary and mass transfer effects that activate the interfaces, in the sense of empowering them with a driving force that contributes to the overall fluid motion, with important consequences and ramifications. The most complex types of particles considered in this book are liquid capsules and biological cells enclosed by structured interfaces with a molecular or shell-like constitution, exhibiting viscoelastic properties under direct mechanical or hydrodynamic loads. Capsules and cells deform and evolve in two ways: passively in response to a flow and because of inter-particle interactions and mechanical stimulus; and actively by means of self-induced motion. For example, under the action of a localized interfacial tension generated by the release or injection of a surfactant, a liquid capsule may self-divide or deform to engulf ambient fluid or another cell or particle residing in its neighborhood. The active motion distinguishes capsules and cells from uncharged solid particles whose surfaces are normally impermeable and inert in the context of hydrodynamics. Natural, artificial, and biological capsules and cells abound in nature, biology, and technology. Examples include the highly flexible, non-nucleated red blood cells, the nearly spherical white blood cells, other types of tissue cells, and various liquid globules encountered in food, cosmetics, and other industrial products. Desirable properties of capsules and cells include the ability to deform and accommodate the shapes of capillaries and microchannels, the ability to withstand the shearing action of an imposed flow, and the capacity to transport material in a protected way, and then release it in a timely fashion for the purpose of achieving a specific goal. In the past three decades, considerable progress has been made in the mathematical analysis, mathematical modeling, and numerical simulation of the fluid dynamics of capsules and cells. Topics of active research include the modeling of interfacial mechanics and transport combined with internal and external hydrodynamics in the context of flow-structure interaction, the unified description of internal and external fluid motion, the coupling of continuum mechanics with molecular processes, and the numerical simulation of large systems accounting for strong hydrodynamic interactions. The chapters in this volume provide an overview of the state of the art on selected topics, also including the results of ongoing research by the individual authors.

v

vi

Capsules and Cells

This book is intended to be a stand-alone reference and a convenient starting point for students and professionals with a general interest in the mathematical and computational sciences, and a specific interest to capsule and cell dynamics and biomechanics. One deliberate restriction is that the discussion remains mostly on the level of a continuum. Molecular processes are discussed in terms of kinetics and only insofar as to provide motivation and justification for the macroscopic equations. The first four chapters are devoted to reviewing the fundamentals of cell and membrane mechanics, and to discussing the behavior in hydrostatics and hydrodynamics. These chapters are suitable for a course in biomechanics. The last two chapters are dedicated to discussing drop and bubble dynamics associated with temperature variations and surfactant transport. These chapters are suitable for an advanced course in interfacial fluid mechanics, interfacial phenomena, and dispersed-phase dynamics. I would like to thank all of the authors who took the time to render their expertise to this unique volume. I am confident that their clear and comprehensive exposition will provide students and researchers with a valuable resource, and will motivate further research in the developing field of mathematical and computational capsule dynamics and biomechanics. C. Pozrikidis San Diego, April 2003

Contributing authors

D. Barth`es-Biesel UTC - Genie Biologique BP 20529 60 205 Compiegne Cedex France Email: [email protected] A. Borhan Department of Chemical Engineering The Pennsylvania State University University Park, PA 16802 USA Email: [email protected] N. R. Gupta Department of Chemical Engineering University of New Hampshire Durham, NH 03824 USA Email: [email protected] H. Liu Division of Computer and Information The Institute of Physical and Chemical Research (RIKEN) Hirosawa 2-1, Wako Saitama, 351-0198 Japan Email: [email protected]

vii

viii

Capsules and Cells

A. Nir and O. M. Lavrenteva Department of Chemical Engineering Technion-Israel Institute of Technology Haifa 32000 Israel Email: [email protected] C. Pozrikidis Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA 92093-0411 USA Email: [email protected] T. W. Secomb Department of Physiology University of Arizona Tucson, AZ 85724-5051 USA Email: [email protected] W. Shyy, N. N’Dri, and R. Tran-Son-Tay Department of Mechanical and Aerospace Engineering 231 Aerospace Building University of Florida Gainesville, FL 32611-6250 USA Email: [email protected] [email protected] [email protected]

Contents

1

2

3

Flow-induced capsule deformation 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1.2 Membrane mechanics . . . . . . . . . . . . . . . . 1.3 Membrane constitutive laws . . . . . . . . . . . . . 1.4 Experimental determination of membrane properties 1.5 Capsule deformation in an ambient flow . . . . . . . 1.6 Numerical simulation of large deformations . . . . . 1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . Shell theory for capsules and cells 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 2.2 Stress resultants and bending moments . . . . . . 2.3 Interface force and torque balances . . . . . . . . 2.4 Surface deformation and elastic tensions . . . . . 2.5 Surface deformation and bending moments . . . . 2.6 Axisymmetric shapes . . . . . . . . . . . . . . . . 2.7 Planar axisymmetric membranes . . . . . . . . . . 2.8 Two-dimensional membranes . . . . . . . . . . . 2.9 Incompressible interfaces . . . . . . . . . . . . . 2.10 Membrane viscoelasticity . . . . . . . . . . . . . 2.11 Discrete models and variational formulations . . . 2.12 Numerical simulations of flow-induced deformation References . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 2 4 11 17 26 30 31

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35 35 38 46 51 58 62 77 79 86 88 89 94 95

Multi-scale modeling spanning from cell surface receptors to blood flow in arteries 103 3.1 Cell adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 3.2 Arterial blood flow . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.3 Immersed boundary method . . . . . . . . . . . . . . . . . . . . . 127 3.4 Leukocyte deformation and recovery . . . . . . . . . . . . . . . . 130 3.5 Rolling of adhering cells . . . . . . . . . . . . . . . . . . . . . . . 134 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

ix

x 4

5

6

Capsules and Cells Mechanics of red blood cells and blood flow in narrow tubes 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Mechanical properties of red blood cells . . . . . . . . . . 4.3 Single-file motion of red blood cells in capillaries . . . . . 4.4 Multi-file motion of red blood cells in microvessels . . . . 4.5 Motion of red blood cells in shear flow . . . . . . . . . . 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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163 163 165 170 179 184 190 191

Capsule dynamics and interfacial transport 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Model of fluid-particle motion with interfacial mass transport 5.3 Boundary-integral formulation . . . . . . . . . . . . . . . . 5.4 Particle motion induced by interfacial mass transport . . . . 5.5 Locomotion induced by the internal secretion of a surfactant 5.6 Drop migration in an ambient concentration gradient . . . . 5.7 Combined effect of gravity and thermo-capillarity . . . . . 5.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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197 197 199 205 208 228 230 249 255 256

Capsule motion and deformation in tube and channel flow 6.1 Capsule motion in tube flow . . . . . . . . . . . . . . . . . 6.2 Numerical methods . . . . . . . . . . . . . . . . . . . . . . 6.3 Capsules with surfactant-induced elasticity in tube flow . . . 6.4 Capsules with temperature-induced elasticity in tube flow . 6.5 Capsules enclosed by elastic membranes in tube flow . . . . 6.6 Capsule motion in channels . . . . . . . . . . . . . . . . . 6.7 Capsules with surfactant-induced elasticity in channel flow . 6.8 Capsules with temperature-induced elasticity in channel flow 6.9 Capsules enclosed by elastic membranes in channel flow . . 6.10 Summary and outlook . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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263 264 276 281 298 303 307 312 318 321 322 323

Index

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329

Chapter 1 Flow-induced capsule deformation

D. Barth`es-Biesel Natural capsules such as cells and eggs, and artificial capsules with various constitutions are encountered in a broad range of biological and engineering applications. In this chapter, the various types of the capsule membrane response under deformation are reviewed, constitutive equations relating the membrane strain to the interfacial tensions for hyperelastic and viscoelastic materials are outlined, and methods for the experimental determination of the membrane properties of large artificial capsules based on squeezing techniques, spinning rheometry, and shearing of a flat sheet, are discussed. The equations describing the motion of a spherical capsule freely suspended in a linear shear flow are then presented, and asymptotic solutions in the limit of small deformation are compared with experimental data. Numerical methods are applied to investigate the large deformation of capsules in simple shear flow, and thereby illustrate the significance of the membrane constitutive equation on the predicted behavior.

1.1 Introduction A capsule consists of an internal deformable substance that is enclosed by a semipermeable membrane. The primary role of the membrane is to confine and protect the encapsulated material, and also control the exchange between the capsule content and the ambient environment. Examples of natural capsules include biological cells and eggs. Artificial capsules are routinely used in the pharmaceutical, cosmetics, and food industries for controlling the release of active substances, aromas, and flavors. Capsule technology finds important applications in the engineering of artificial organs, and in cell therapy where living cells are encapsulated for the treatment of disease such as diabetes and liver failure [26, 32]. Encapsulation of other types of organ cells is still in the early stages of development. Design specifications for artificial capsules intended for biomedical applications arise from several requirements. First and foremost, to be tolerated by the receptor’s body, the membrane must be bio-compatible. In particular, the membrane must pro-

1

2

Capsules and Cells

tect the internal cell culture from rejection reactions triggered by the large macromolecules gamma globulins Second, the membrane must be permeable to small molecules found in nutrients, oxygen, and proteins secreted by cells. Third, the membrane must be resilient enough to withstand manipulation and implantation. In practice, membranes consist of natural and synthetic polymers such as poly-L-lysine and polyacrylates. As a result of the fabrication process, artificial capsules are typically nearly spherical. Measuring and controlling the mechanical properties of the membrane are hindered by its small size and fragility. An additional difficulty is that, in the process of manipulation, a capsule may undergo large deformation. It is then important to be able to determine the constitutive behavior of the membrane in its natural configuration. For this purpose, sophisticated measurement techniques must be used in conjunction with a complete mechanical analysis of the experimental process. In this chapter, the fundamental theoretical concepts pertaining to membrane mechanics will be outlined, classical constitutive laws will be discussed with reference to the tension-strain relations under simple deformation, and experimental techniques used to identify the membrane constitutive equation will be reviewed. Another important aspect of capsule mechanics concerns the motion and deformation under the influence of forces exerted by an ambient flow. To simplify matters, we shall focus our attention on the deformation of artificial capsules with a spherical initial shape. In particular, the motion of a capsule that is freely suspended in a linear shear flow will be considered in detail, and solutions for small deformations will be derived and compared to experimental data on flow-induced deformation. Finally, recent numerical simulations of large deformation will be presented and discussed with reference to experiments.

1.2 Membrane mechanics In the theoretical model, the capsule membrane is regarded as a two-dimensional elastic shell with negligible thickness, allowed to undergo arbitrary and unrestricted deformation under the influence of an imposed surface load. Bending moments are neglected, and the membrane material is assumed to be isotropic. A more general discussion of membrane mechanics including the effect of bending moments is given in Chapter 2.

1.2.1 Membrane deformation Barth`es-Biesel & Rallison [5] showed that large membrane deformation can be conveniently described in terms of three-dimensional Cartesian tensors, which is an alternative to using surface curvilinear coordinates (see also Section 2.4). In the Cartesian formulation, kinematics and dynamics are described in terms of the posi-

Flow-induced capsule deformation

3

tion of membrane material point in the stress-free reference configuration, denoted , and the instantaneous position at time , denoted by

. The unit outby ward normal vector before and after deformation is denoted, respectively, by
and
. The surface relative deformation gradient is defined as








   








(1.2.1)

the left Cauchy-Green surface deformation tensor is defined as


 




Ì

(1.2.2)




and the Green-Lagrange strain tensor is defined as




 




Ì

 

 









(1.2.3)

where  is the identity matrix, and the superscript T denotes the matrix transpose (see also Section 2.4.1). The left Cauchy-Green deformation tensor has two positive definite eigenvalues, 
and  , corresponding to two orthogonal tangential eigenvectors pointing along the local principal axes of deformation. The principal stretches or extension ratios are given by 



 







   


(1.2.4)

where the infinitesimal material vectors 
and  point in the local and instantaneous principal directions. The surface dilation is defined as the ratio between the deformed and undeformed surface area of an infinitesimal material patch, and is given by


 




(1.2.5)



Surface strain invariants are defined in the usual way, 


 







 
 



  





(1.2.6)

(see also Section 2.4).

1.2.2 Elastic tensions and equilibrium Three-dimensional stresses may be integrated over the membrane cross-section

(see also Secto yield the Cauchy elastic tension or stress-resultant tensor tion 2.2). The individual components of

represent the tangential force per

4

Capsules and Cells

unit length of the deformed membrane. When the membrane inertia is negligible, equilibrium requires







Trace








(1.2.7)

where   


is the tangential projection operator, and  is the load exerted on the membrane, defined as the external force per unit area of the deformed membrane surface (see also Section 2.3).

1.2.3 Axisymmetric shapes When the capsule shape and load are axisymmetric, as illustrated in Figure 1.2.1, and in the absence of torsion, the principal directions of stress and strain point in the meridional and azimuthal directions denoted, respectively, by the subscripts 1 and 2. Material point particles over the membrane are identified by the triplet 


before deformation, and


after deformation;  and  are the distances from the axis of revolution, and the arc lengths  and are measured along a meridian curve. The principal extension ratios 
and  are then given by 













 

(1.2.8)




and the membrane load is given by



    



(1.2.9)

where  and
are the meridional tangential and normal unit vectors (see also Section 2.6.4). The normal and tangential components of the vectorial equilibrium equation (1.2.7) take the form






 

 









  

 






(1.2.10)

where the principal tensions (


and principal curvatures (


are measured in the meridional and its conjugate directions (see also Section 2.6.3).

1.3 Membrane constitutive laws When the membrane behaves like a hyperelastic medium, it is possible to introduce a surface strain energy function 


defined with respect to the undeformed surface (see also Section 2.4). The tension tensor derives from the strain energy function from the relation 

 







  
Ì 




(1.3.1)

Flow-induced capsule deformation Before deformation

5

After deformation

j

j

s S R

r

Figure 1.2.1 Schematic illustration of axisymmetric membrane deformation.

Several classical constitutive laws are available for the phenomenological description of common mechanical behavior. To simplify the subsequent discussion, the expression for the principal component
will be given explicitly, with the understanding that the corresponding expression for can be deduced by switching the indices 1 and 2.

1.3.1 Linear elasticity For small deformations, the counterpart of Hooke’s law for a two-dimensional continuum, designated by the superscript  , is given by









 

   






(1.3.2)

where  is the surface shear elastic modulus, and  is the surface Poisson ratio. For an incompressible membrane whose area is locally and globally conserved upon deformation,   . In contrast, the Poisson ratio for an incompressible threedimensional material is    (see Section 2.5). The range of variation of the surface Poisson ratio will be discussed in Section 1.5.

1.3.2 Rubber elasticity In this model, the membrane is regarded as a thin layer of a homogeneous, isotropic, three-dimensional incompressible elastomer obeying a Mooney–Rivlin (MR) law



















  







  








(1.3.3)

where is a material parameter ranging between zero and unity, with   corresponding to a neo-Hookean (NH) solid (e.g., [23]). A local increase in the mem-

6

Capsules and Cells

brane surface area leads to a corresponding decrease in the membrane thickness. By allowing to be a function of the strain invariants, we can describe complex elastic behavior such as strain hardening.

1.3.3 Two-dimensional elasticity Two-dimensional constitutive laws can be derived by invoking general principles of continuum mechanics and thermodynamics. A number of constitutive laws have been proposed under the assumption of plane isotropy. In this section, we focus our attention on two simple such laws originally designed to describe the behavior of the membrane of red blood cells. Skalak et al. [43] proposed a constitutive law (SK) that takes into consideration the elastic response in shearing deformation, with a modulus of elasticity  , and the intrinsic resistance to area dilatation, with a corresponding modulus  . The first principal tension is given by











 








 








(1.3.4)

where   is the ratio of the two moduli (see also Section 2.4.2). The red blood cell membrane consists of a phospholipid bilayer that is lined on the interior side by the cytoskeleton, which is a protein network. Because of the properties of the lipid bilayer, the membrane strongly resists area dilatation. On the other hand, because of the protein network, the interface exhibits elastic response. Consequently, the moduli ratio  is on the order of  . One drawback of the Skalak law is that the effect of area dilatation appears in two terms on the left-hand side of (1.3.4). To isolate the individual effects of shear deformation and area dilatation, Evans & Skalak [21] introduced the alternative law (ES)







  


















  







(1.3.5)

(see also Section 2.4.2). The first term of the right-hand side of (1.3.5) accounts for shear deformation at constant surface area with associated modulus  , whereas the second term accounts for area dilatation in the absence of shearing deformation with associated modulus    , where  is the moduli ratio. Because the membrane is nearly incompressible, the magnitude of  is high. Although the constitutive equations (1.3.4) and (1.3.5) have been derived with reference to the nearly incompressible membrane of red blood cells, they can be used in a more general context to describe the response of membranes with moderate or small area dilatation modulus, whereupon the parameters  and  take values on the order unity [4]. Even more general behavior can be described by allowing the shear modulus  and ratio  or  to be functions of the strain invariants involving additional coefficients which, however, are hard to determine experimentally.

Flow-induced capsule deformation

7

1.3.4 Viscoelasticity Viscoelastic effects in the absence of expansion viscosity can be modeled by a linear law involving the rate of surface deformation. Thus, to any of the preceding elastic laws, the following viscous contribution may be added to yield a viscoelastic material,



 





 




(1.3.6)




where  is the surface viscosity, and  is the material derivative. Further discussion of viscoelasticity for three-dimensional deformation can be found in Section 2.10.

1.3.5 Correspondence between constitutive laws In the asymptotic limit of small deformation where both principal stretches are close to unity, all laws reduce to Hooke’s law discussed in Section 1.3.1. In particular, assuming the same value for  , the asymptotic form of (1.3.3), (1.3.4), and (1.3.5) leads to (1.3.2) with following values for the surface Poisson ratio,


 Mooney–Rivlin 
      

   
 
    




(1.3.7)




It is important to emphasize that, because the preceding hyperelastic laws have different mathematical forms, they lead to different stress-strain relations at large deformation even though they may describe the same asymptotic behavior at small deformation. To demonstrate this clearly, we discuss certain simple modes of deformations that can be realized experimentally, and compare the predicted behavior. Uniaxial extension In one simple experiment, a material sample is stretched in direction 1 while the other direction is free of stress, as illustrated in Figure 1.3.1 (a). The condition   allows the determination of  as a function of the stretch ratio 
in direction 1. The value of the stretching force
is then easily determined as a function of the stretch ratio 
or strain 





. In this case, the surface Young modulus  is defined as



 




½Ë
 





(1.3.8)

8

Capsules and Cells (a)

(b)

t2 =0

t1

t1

Uniaxial Extension

t

t

t

t Isotropic Extension

Figure 1.3.1 Two elementary experiments for assessing the membrane mechanical properties.

The constitutive laws discussed in this section provide us with the following predictions,

     
               Mooney–Rivlin 
 
  
  
   
       !      




(1.3.9)




The corresponding stress-strain response of the membrane material is illustrated in Figure 1.3.2(a) for membranes that behave in the same way at small deformation. For deformation strains larger than 20%, the predictions of the different laws are significantly different. More important, the SK law is strain hardening, whereas the MR and ES laws are strain softening. As the values of the coefficients  and  are raised, the resistance to area dilatation becomes stronger. Accordingly, to reach a certain level of deformation, the required stretching force becomes higher, as shown in Figure 1.3.2(b). However, the SK law still remains strain hardening, and the ES law remains strain softening. Isotropic extension In another simple experiment, the membrane deforms due to an applied isotropic tension, as illustrated in Figure 1.3.1(b). In practice, this mode of deformation can be achieved by raising the osmotic pressure to inflate the capsule. In this case, the

principal tensions and strains are equal in the two principal directions,

and 

 
  " 
, where " is the relative change in the local

Flow-induced capsule deformation

9

(a)

(b)

Figure 1.3.2 (a) Response of different membrane materials in the uniaxial stretching experiment; the Mooney–Rivlin case corresponds to  . (b) Effect of area dilatation resistance.

surface area. The area dilatation modulus  is defined by  

 Ë
 


 

 Ë
 "



(1.3.10)

10

Capsules and Cells

Figure 1.3.3 Isotropic extension of a membrane: Graph of the reduced tension  against the relative area dilatation " .

The following values for  are obtained for the different membrane laws

 
×      
×        Mooney–Rivlin        
  
   

 






(1.3.11)

    


Note that an area incompressible membrane arises for unity Poisson ratio,   , corresponding to infinitely large values of  and . The membrane response under isotropic extension is illustrated in Figure 1.3.3, where the reduced tension  is plotted against the relative area dilatation " . The membranes identified as MR, H(1/2) which is an abbreviation for H(
=1/2), SK(C=1), and ES(A=3) have the same dilatation modulus. The results show that, because the membrane thickness is reduced during deformation, the MR membrane is easy to dilate; strain softening occurs irrespective of the value of the nonlinearity parameter . The SK membrane is strain hardening except when  =0, in which case the ratio  is a linear function of the relative area dilatation. For the same value of area dilatation, the tension of a SK membrane increases with  . For membranes obeying the ES law, the ratio  is always a linear function of " with a slope that is equal to . These comparisons demonstrate that, in situations where local membrane area changes are large, the choice of a constitutive law plays an important role in the predicted behavior.

Flow-induced capsule deformation

11

1.4 Experimental determination of membrane properties It was mentioned earlier in this chapter that the experimental determination of the membrane mechanical properties is hindered by the small capsule size. Because, in general, the membrane constitutive law involves two moduli, two independent measurements must be made. Several techniques have been developed for measuring the response of synthetic capsules or biological cells to external forces. Among these is the aspiration of a portion of the membrane into a micro-pipette, a device used mainly to study the mechanical properties of living cells [24] (see also Section 2.7 and Figure 2.7.1). The technique involves measuring the aspiration length in a micro-pipette under an applied pressure, using a costly and delicate micro-manipulation system. For artificial capsules whose sizes are much larger than those of cells, on the order of a millimeter, the alternative techniques discussed in this section offer alternatives.

1.4.1 Direct measurement of the membrane shear modulus In some cases, it is possible to create a flat sample of the membrane and directly measure the shear elastic modulus. This technique has been used extensively by Rehage and co-workers for measuring the properties of ultra thin cross-linked membranes at an oil–water interface. [1, 10, 33, 41]. Membrane samples can be produced by different methods including self-association due to attractive interactions or crosslinking due to chemical reactions. The generated film is then subjected to harmonic in-plane shear deformation, and the measured complex modulus is decomposed into a storage modulus  and a loss modulus  to directly determine the membrane viscoelastic properties. Using this technique, it is also possible to obtain information on further film properties. In particular, when a small amplitude deformation is applied at constant frequency and for a long period of time (time sweep experiment), it is possible to monitor the kinetics of film formation. After the film has reached a steady state, the mechanical properties are obtained by applying oscillations of increasing amplitude at a fixed frequency (strain sweep experiment). As an example, the data in Figure 1.4.1, reveal that a cross-linked polyamide membrane exhibits a linear visco-elastic response up to deformations on the order of 7 to 8%. For larger values, the shear elastic modulus decreases with deformation. This behavior is most likely due to mechanical damage, although since in this range of deformation the linear visco-elastic model is no longer appropriate, and the viscometer response itself is nonlinear, the exact cause is difficult to identify with precision.

1.4.2 Compression between parallel plates This method involves squeezing a capsule between two rigid parallel plates, while simultaneously measuring the distance  between the plates and the compression

12

Capsules and Cells

G'

G'' (N /m)

10 1

10 2

10 0

10 1 strain γ (%)

10 2

Figure 1.4.1 Measured complex modulus of a flat sample of a cross-linked polyamide membrane. (From Walter, A., Rehage, H., & Leonhard, H., 2000, Colloid Polym. Sci., 278, 169-175. With permission from Springer-Verlag.)

force  . This technique is typically used to measure the force necessary for bursting [7, 19, 42, 50, 51]. However, the compression experiment may also be used to extract further information on the membrane mechanical behavior [25, 44, 45, 46]. Theoretical models have been developed to model the compression of a capsule between two parallel plates. Feng & Yang [22] analyzed the mechanics of an inflated spherical elastic membrane. Subsequently, Lardner & Pujara [29] extended the analysis to the case of a spherical capsule filled with a liquid, and considered the effect of the membrane constitutive law. Their predictions for a Mooney–Rivlin material are in good agreement with measurements of the compression of a fluid-filled rubber ball. Lardner & Pujara [29] also considered area incompressible membranes described by the Skalak law with   , and analyzed the compression of seaurchin eggs. Numerical solutions based on the finite-element method have been presented by other authors on the compression of a thin-walled, liquid-filled sphere squeezed between two surfaces. In particular, Cheng’s computations [14, 15] for a membrane with neo-Hookean properties are in good agreement with Yoneda’s [49] experimental data on sea-urchin eggs. Recently, the approach has been refined to identify possible constitutive laws for the capsule membrane material [11]. Figure 1.4.2(a) shows compression stages of a capsule filled with a saline solution, suspended in another saline solution with the same concentration and enclosed by an HSA-alginate membrane. In the undeformed state, the capsule is a sphere of radius  =1.5 mm, and the thickness of the membrane is   68 m. The photographs demonstrate that the capsule can exhibit substantial

Flow-induced capsule deformation

13

(a)

(b)

Figure 1.4.2 (a) Photographs of the squeezing of a capsule enclosed by an HSAalginate membrane between two parallel plates, and (b) compression force plotted against the relative flattening. (From Carin, M., Barth`es-Biesel, D., Edwards-Levy, F., Postel, C., & Andrei, D., 2003, Biotechnology & Bioengineering, In press. With permission from Wiley.)

deformation before the membrane bursts. The process is fully reversible, that is, the capsule recovers the spherical shape when the compression load is removed. Figure 1.4.2(b) shows a graph of the compression force plotted against the relative flattening of the capsule Æ 









(1.4.1)

for two capsules with different initial membrane thickness,  . To extract the membrane properties from the compression curve, a mechanical analysis of the process can be carried out following the original work of Feng & Yang

14

Capsules and Cells z

z

j

j

1111111111111 0000000000000 0000000000000 1111111111111 1111111111111 0000000000000 n S a

r d

R

s

1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 Before deformation

After deformation

Figure 1.4.3 Schematic illustration of the membrane deformation in the platesqueezing process.

[22]. Because the compression fully axisymmetric, the simplified equations outlined in Section 1.2.3 can be used. The position of a material point in the undeformed state is identified by the initial arc length,  , measured along a meridian line, where    on the axis of revolution, as depicted in Figure 1.4.3. In the deformed configuration, the material point is displaced to the axial position  
at a distance  
from the axis of symmetry. Symmetry about the equator allows us to restrict our attention to the upper half of the meridian curve. The contact region between the membrane and the upper plate is confined between      . The stretch ratios 
and  defined in (1.2.8) are given by 


 



 









  





(1.4.2)

where a prime denotes a derivative with respect to  . Using these expressions in conjunction with one of the constitutive laws (1.3.2) - (1.3.5), we may express the principal tensions
and in terms of the functions  
,  
, and their derivatives. The equilibrium equations (1.2.10) have to be solved with different loads depending on which part of the membrane is under consideration. Assuming no friction between the plates and the membrane, we require    over the contact region confined in the range      . Because this section of the membrane is perfectly flat, the net normal load must also vanish,    for      . In the free region, the load on the membrane is due only to the pressure difference " between the interior and exterior of the capsule: for    , the load is given by   "
. Boundary conditions include continuity of 
and  at the junction  , and the geometrical and symmetry conditions  
 




!




 




 




(1.4.3)

Flow-induced capsule deformation

15

Figure 1.4.4 Theoretical predictions of the squeezing force for different membrane constitutive laws. (From Carin, M., Barth`es-Biesel, D., Edwards-Levy, F., Postel, C., & Andrei, D., 2003, Biotechnology & Bioengineering, In press. With permission from Wiley.)

Finally, mass conservation requires !!





 !

 

 

(1.4.4)

Given the plate separation , the preceding set of equations can be solved numerically to obtain the values of " and contact area, and consequently estimate the magnitude of the squeezing force  . Figure 1.4.4 shows the theoretical predictions for the force normalized by the area dilatation modulus, plotted against the compression ratio Æ . Differences between the constitutive laws become noticeable when Æ  #. Note that, as the magnitude of the constants  and  becomes higher than unity, the curves tend to a limiting asymptotic form. The experimental results can be analyzed in a way that circumvents the use of involved inverse methods to determine the membrane rheological behavior. An important assumption is that the moduli  and  remain constant at various compression ratios. A membrane constitutive law is first assumed; in the case of a SK or a ES law, values to  or  are assigned. For each value of Æ , the comparison between the theoretical and measured force yields an apparent shear elastic modulus  and an apparent dilatation modulus  . The results indicate that a Mooney–Rivlin (MR) law is not able to accurately describe the behavior of the HSA-alginate membrane and is thus unacceptable. The Skalak (SK) law with    and the Evans and Skalak (ES) law with    are both acceptable. The graphs in Figure 1.4.2(b) demonstrate that the agreement between the experimental data and the theoretical prediction is very good even at large deformation.

16

Capsules and Cells

Figure 1.4.5 Deformation of a spherical capsule in a spinning rheometer at several rotation speeds. (From Pieper, G., Rehage, H., & Barth`es-Biesel, D., 1998, J. Coll. Interf. Sci., 202, 293-300. With permission from Academic Press.)

The technique can also be used to measure the viscoelastic properties of the membrane. However, caution should exercised since long-time creep or relaxation experiments may allow for membrane permeability effects, which may be misinterpreted as viscoelastic effects.

1.4.3 Deformation in a spinning rheometer A new approach for the measurement of the mechanical properties of a capsule membrane was recently proposed based on a spinning-drop apparatus. The experimental device was originally designed for measuring the interfacial tension between two immiscible liquids [33]. In the adapted setup, an initially spherical capsule is introduced in the device, and its deformation is observed under increasing rotation rates, as shown in Figure 1.4.5. When gravity forces are small compared to centrifugal forces, that is, for large enough rotation rates, the capsule is axisymmetric. The membrane load is then due

Flow-induced capsule deformation

17

to centrifugal forces alone acting normal to the membrane, given by





"


 "# $ 










(1.4.5)

where " is the unknown internal capsule pressure, $ is the rotation rate, and "# is the difference between the density of the internal and external liquid. The axisymmetric membrane equilibrium equations discussed in Section 1.2.3 are then solved for small deviation from sphericity, whereupon the membrane constitutive law reduces to Hooke’s linear law [33]. The analysis shows that, to leading order, the capsule deforms into an ellipsoid with length along the tube axis %, and breadth & . The Taylor deformation parameter  is given by 

%


& 
"# $

%&

 #  





  

(1.4.6)

Measurements have been made for capsules consisting of an oil droplet enclosed by an ultra-thin cross-linked membrane in an aqueous suspension. The membrane is generated by polymerization of the radicals of surface-active aminomethacrylates. Because the deformation of the capsule in the spinning rheometer is small, the linear asymptotic theory applies. The measured deformation is in excellent agreement with the theoretical predictions, as shown in Figure 1.4.6. Equation (1.4.6) allows us then to infer the value of the property group    
 #  
. Combining this result with independent measurements of the shear elastic modulus of a flat sheet of polymerized aminomethacrylates, we find that  is nearly zero, while  varies between 0.05 and 0.1 N/m depending on the degree of membrane polymerization. Because no contact occurs between a solid surface and the deformed capsule, this experimental technique is both attractive and promising. However, the operating conditions are limited by the intensity of the centrifugal forces that can be achieved, which are controlled by the rotation speed $ . At rotational speeds, mechanical vibrations may significantly perturb the measurement.

1.5 Capsule deformation in an ambient flow The experiments described in Section 1.4 deal with the deformation of the whole or part of the membrane at equilibrium, the objective being to identify a constitutive equation for the membrane material. Obtaining information on the dynamics of a capsule suspended in a flowing liquid requires further considerations from the realm of hydrodynamics and in the specific context of flow-structure interaction. From a practical viewpoint, information on the overall capsule behavior is necessary for manipulating and processing suspensions without inflicting mechanical damage. In the past two decades, a number of experimental and theoretical investigations have been devoted to studying the motion and deformation of capsules subjected

18

Capsules and Cells

Figure 1.4.6 Deformation of a capsule enclosed by a cross-linked membrane, plotted against the centrifugal force; different symbols correspond to different capsules. (From Pieper, G., Rehage, H., & Barth`es-Biesel, D., 1998, J. Coll. Interf. Sci., 202, 293-300. With permission from Academic Press.)

to hydrodynamics forces of various sorts. The system of governing equations and accompanying interfacial conditions are well established [5, 37]. In this section, we outline the mathematical model, discuss asymptotic solutions for small deformations, and compare the theoretical predictions with laboratory observations.

1.5.1 Mathematical formulation Consider a spherical capsule of radius  that is filled with a viscous incompressible liquid with viscosity  and is suspended in another viscous liquid with viscosity 
, as depicted in Figure 1.5.1 (a). Here as elsewhere, superscripts and subscripts 1 and 2 will refer, respectively, to the ambient and internal liquid. The membrane will be modeled as an infinitely thin sheet of a viscoelastic material. It is convenient to describe the motion in a frame of reference centered at, and moving with, the center of mass of the capsule. Far from the capsule, the suspending liquid undergoes linear shear flow with velocity field

  '$ 










(1.5.1)

where '$ is the magnitude of the shear rate, 
is the generally time-dependent dimensionless symmetric rate-of-strain or rate-of-deformation tensor, and is the

Flow-induced capsule deformation (a)

19

(b) x2

m1

x2

n L

m2

B

x1

x3

q

x1

x3 x2

(c)

Membrane compression

x1

Membrane compression

x3

Figure 1.5.1 Schematic illustration of capsule freely suspended in (a) a linear flow, and (b, c) a simple shear flow. Frame (c) shows the areas where the membrane undergoes compression.

complementary dimensionless vorticity tensor. Specific flows of interest include the following:



Simple shear flow in the except 


(
(

plane: All components of

 
 %



%
 


 and are zero, (1.5.2)

as illustrated in Figure 1.5.1 (b, c). This flow is easy to generate experimentally in a Couette-flow device [13, 47], and is particularly relevant to the study of suspension rheology.



Plane hyperbolic flow in the (
( plane: All components of  and are zero, except 






 

(1.5.3)

20

Capsules and Cells This irrotational flow is established at the center of a four-roller flow apparatus [8]. The deformation of an artificial capsule suspended in this flow has been studied on two occasions [3, 12].



Axisymmetric straining flow: All components of  and are zero, except


 






   



(1.5.4)

This flow is encountered in experiments pertaining to extrusion processes. Under the influence of the viscous stresses, a capsule deforms to obtain a shape described by the equation   (

(
(




(1.5.5)

where  is the distance from the origin, and the shape function is to be determined as part of the solution. The Reynolds number based on the capsule size  is assumed to be much smaller than unity, ) #
'$  
** , where #
and 
are the suspending fluid density and viscosity. Consequently, the flow of the internal and external liquid is governed by the Stokes equations

    


 
  


(1.5.6)

for +  
, where
is the Newtonian stress tensor. Far from the capsule, the exterior flow tends to the far-field flow given in (1.5.1),


  






(1.5.7)

At the capsule interface,  , we require continuity of velocity,















(1.5.8)

and the kinematic condition











 




(1.5.9)

where the point lies in  , and  is the material derivative following the motion . of interfacial marker points with initial position An interfacial force balance requires that the membrane load  is the balance of the hydrodynamic traction exerted by the flow of the internal and external side of the interface,




















To complete the mathematical formulation, it remains to relate formation following the analysis of Section 1.2.

(1.5.10)

 to the surface de-

Flow-induced capsule deformation

21

Dimensional analysis indicates that the capsule motion and deformation depend  
, and (b) the ratio of the typical magnitude of on (a) the viscosity ratio,  viscous stresses relative to the membrane elastic tension, 
' $ 

,





(1.5.11)




which is the counterpart of the capillary number pertinent to interfaces between immiscible fluids. Further parameters are introduced to specify the type of incident flow described, for example, by the vorticity to strain ratio, the initial particle geometry described, for example, by the aspect ratio and surface to volume ratio, and the membrane properties expressed by the area dilatation modulus, visco-elastic properties, and bending modulus. The problem involves a strong coupling between fluid and solid mechanics in situations where the capsule deformation is large and the hydrodynamic forces due to viscous stresses are high. For large deformations, the solution must be found using numerical methods, as discussed in Section 1.5.3. When the deformation is small, asymptotic solutions based on the method of domain perturbation can be developed.

1.5.2 Small-deformation theory Small capsule deformation occurs when the elongational component of the flow is weak compared to restoring elastic forces developing in the membrane, , ** . Perturbation solutions in this limit have been developed for hyperelastic [2, 5] and linearly visco-elastic capsule membranes [6]. The analysis involves expanding all quantities in the small parameter ,, and computing the deformed capsule shape by the method of successive approximations. To leading order, the displacement of the membrane material points depends on two dimensionless, symmetric, and traceless second-order tensors 
and
,







,





 



 

 - ,


(1.5.12)

The tensor
determines the in-plane deformation, whereas the tensor 
determines the aspect ratio of the ellipsoidal capsule shape according to the equation 

 ,



(1.5.13)



The full set of equations describing the fluid motion, (1.5.6) through (1.5.9), are expanded in , to first order. The membrane equations (1.2.1) through (1.2.7) are expanded in a similar fashion, assuming - ,
deviation from the undeformed spherical shape. The procedure leads to two ordinary differential equations governing the time evolution of and  [6],

 Æ   

# 

.  



+  -

,


, .





(1.5.14)

 Æ  #  

.  






+




-

+  +


,


, .





22

Capsules and Cells

where  is the surface or membrane viscosity, subject to the following definitions:



The left-hand sides of (1.5.14) involve the corrotational Jaumann derivative defined, for example, by

Æ 











 '$





(1.5.15)

The dimensionless parameter .  ' $  expresses the ratio between the membrane characteristic response time and the time scale of the shear flow, and is analogous to the Deborah number pertinent to viscoelastic fluids Depending on the flow conditions, . may vary between zero and infinity. When . is - 
or lower, , must be small for the asymptotic analysis to apply. The dimensionless coefficients + and + depend on the membrane constitutive equation. In early studies, a Mooney–Rivlin law with surface Poisson modulus   , and an area incompressible law corresponding to   were considered [6]. The analysis was recently extended to arbitrary value of  , and thus arbitrary values of the membrane area dilatation modulus, yielding [4] +



 





+  

(1.5.16)

Although a singularity appears as  , corresponding to an infinite area dilatational modulus, a solution for the capsule shape may nevertheless be obtained [6]. In the case of a purely elastic membrane with vanishing surface viscosity,  the deformed capsule profile depends on the rate-of-strain tensor  alone,



# 

  

  






 ,

(1.5.17)

It is clear that  can have a large effect on the capsule deformation if it takes negative values. Zero or slightly negative values of  have been measured by Pieper et al. for thin membranes [33]. Negative values of  correspond to a two-dimensional membrane that is wrinkled in directions perpendicular to its plane. Under uniaxial extension, smoothing of the wrinkles leads to expansion in the lateral direction [9]. The membrane viscosity has a significant effect on the steady capsule deformation only for incident flows with vorticity inducing membrane rotation, such as the simple shear flow illustrated in Figure 1.5.1(b). The small-deformation solution reveals a number of interesting general features regarding the influence of the viscous deforming stresses mediated by the terms involving  in (1.5.14), the competing restoring effect of the elastic forces mediated by the terms involving  and , and the rotational “tank-treading” motion of the membrane around the steady deformed shape. To leading order, the rotation rate of the membrane is equal to the vorticity of the incident flow.

Flow-induced capsule deformation

23

The predictions of the asymptotic theory are useful for analyzing experimental observations on the deformation of artificial capsules, as will be discussed in Section 1.5.4, and for validating numerical solutions in the regime of small deformations.

1.5.3 Simple shear flow In the particular case of simple shear flow whose velocity profile is given in (1.5.2) and is depicted in Figure 1.5.1(b), small-deformation theory predicts that a capsule enclosed by a purely elastic membrane,   , reaches the steady shape described by (1.5.11), where all components of  are zero except 




#    !   

(1.5.18)




according to (1.5.15). The long axis of the capsule is tilted at an angle /  !#Æ with respect to streamlines of the incident flow. The Taylor deformation parameter in the (
( plane that is perpendicular to the vorticity of the incident shear flow is defined %
&
 %  &
, where % and & are the major and minor axis of the by 
deformed profile defined in Figure 1.5.1(b). The asymptotic solution predicts 




# 
' $    !





  



# 
' $







  



(1.5.19)

where  is the Young modulus of elasticity defined in (1.3.8). In the case of a viscoelastic membrane, material points undergo cyclic deformation due to the membrane rotation [6]. When the parameter . is of order unity or higher, corresponding to high membrane viscosity, the Jaumann derivatives in (1.5.14) contribute nonzero values to the diagonal components of  and . When . is large, the deformation parameter tends to an asymptotic limit that depends only on the exterior and membrane viscosity, and is given by 




# 
 



(1.5.20)

1.5.4 Comparison with experiments The objective in most experiments is to document the effect of viscous stresses due to an imposed flow on the overall capsule deformation. A large body of experimental data is available for liquid droplets in the context of emulsion technology and for blood cells in the context of physiology and biomedicine. Few experiments have been conducted with artificial capsules. The first systematic laboratory study of capsule deformation is due to Chang & Olbricht who observed the behavior of a nylon capsule in simple shear flow generated in a Couette-flow apparatus [13]. In particular, measurements were made of the Taylor deformation parameter and orientation angle in the (
( plane as well as in

24

Capsules and Cells

the orthogonal ( ( plane. For a given constant shear rate, the capsule deformation and orientation were observed to oscillate about mean values. These fluctuations are attributed to a slight deviation from sphericity of the undeformed shape. A small amount of permanent deformation was reported after flow cessation, indicating plastic response or else long relaxation times. When the capsule deformation is sufficiently high, pointed shapes appear at the tips, and the membrane fails. A probable cause is local thinning due to excessive stretching. More recently, an extensive study has been conducted for spherical capsules enclosed by polyamide membranes [41, 48]. These investigations complement and extend the earlier work of Chang & Olbricht [13], in that the value of the membrane elastic modulus is controlled by means of the polymerization time. The shear modulus was measured directly by torsion experiments on a flat sample, as discussed in Section 1.4, and was found to vary from 0.08N/m for low polymerization times to the limiting value of 0.2N/m for high polymerization times. In these experiments, the capsule is subjected to a simple shear flow in a Couetteflow device, and the capsule profile is observed perpendicular to the transverse ( axis. The tank-treading of the membrane about the steady deformed shape is studied by observing the motion of interfacial marker points. The deformed capsule takes an ellipsoidal shape with the longest axis inclined with respect to the streamlines of the shear flow, as shown in Figure 1.5.2(a). As in the experiments of Chang & Olbricht [13], small-amplitude oscillations of the deformed shape were observed about the steady state. The deformation versus shear rate curve displayed in Figure 1.5.2(b) shows that the linear prediction (1.5.17) is accurate up to deformations on the order of 20%, which is surprisingly good considering that the theory assumes small deviation from sphericity. For larger shear rates, the theoretical predictions overestimate the deformation. The initial slope of the deformation curve can be used to infer the value of the property group    
   
. Inserting the value of  obtained from torsion experiments, we obtain the surface Poisson ratio   &# irrespective of the polymerization time. This analysis demonstrates that, in order to properly characterize the elastic properties of the membrane, it is imperative to subject the capsule to different types of mechanical forces associated with distinct modes of deformation.

1.5.5 Membrane wrinkling The preceding model overlooks the effect of flexural stiffness mediated by developing bending moments. A consequence of this simplification is that the membrane will tend to buckle by wrinkling in regions where the developing elastic tensions are compressive. To leading-order, the load on a spherical membrane of radius  at steady state is given by



#


' $ 





(1.5.21)

Flow-induced capsule deformation

25

(a)

(b) 0,35 0,30

Asymptotic theory

Deformation D

0,25 0,20 0,15 0,10 0,05 0,00 0

10

20

40

30

50

60

Shear rate g [S ] -1

Figure 1.5.2 (a) Photograph of a capsule enclosed by a polyamide membrane subject to simple shear flow, and (b) capsule Taylor deformation parameter plotted against the shear rate. (From Rehage, H., Husmann, M., & Walter, A., 2002, Rheol. Acta, 41, 292-306. With permission from Springer.)

In the case of simple shear flow described by (1.5.2), solving the equilibrium equations (1.2.7) yields the following expressions for the membrane tensions,

 

 

#


' $  


 # 


' $    /


$  
 
'

#

   /


(1.5.22)

26

Capsules and Cells

Figure 1.5.3 Shear-induced deformation and wrinkling of a spherical capsule enclosed by an organosiloxane membrane at shear rate '$  ! s
. (From Walter, A., Rehage, H., & Leonhard, H., 2001, Coll. Surf. A, Physicochem. Eng. Asp. 183, 123-132. With permission from Elsevier.)

where 
/

define a system of spherical polar coordinates such that the positive semi-axis corresponds to /  , and the (
( plane corresponds to  . In the plane of flow corresponding to /  !, the perpendicular tension 0 is the only nonzero component, taking negative values in the ranges

(

! * * !


! * * !

(1.5.23)

This means that a portion of the membrane undergoes compression, as indicated in Figure 1.5.1(c). The flexural stiffness of a physical membrane prevents wrinkling when the compressing load is less than a critical threshold. This may explain why wrinkling is not observed in the case of capsules enclosed by polyamide membranes, as shown in Figure 1.5.2. On the other hand, capsules enclosed by organosiloxane membranes do exhibit wrinkling even at low shear rates, as shown in Figure 1.5.3, and the direction of folding is consistent with predictions of the asymptotic theory [48]. The reason for the immediate wrinkling of these membranes is not clear, though part of the reason may be that the flexural stiffness is extremely low. An analysis of the developing length scales will be helpful in identifying the bending modulus, which may then be related to the molecular structure of the membrane.

1.6 Numerical simulation of large deformations When the capsule deformation is large, the governing equations must be solved by numerical methods. Available formulations include the immersed boundary method [20] (see also Section 3.3) and the boundary-integral method [16, 27, 31, 35, 36, 38,

Flow-induced capsule deformation

27

39, 52] (see also Sections 5.3 and 6.2). The boundary-integral method is particularly well suited for problems involving interfaces of various sorts in bounded or virtually unbounded domains of flow. The numerical methodology has been applied to study large capsule deformation under various conditions, including the following:

   

Computation of equilibrium capsule shapes and break-up in elongational flow for capsules enclosed by Mooney–Rivlin [17, 27, 31], area incompressible, [34] and viscoelastic membranes [18]. Computation of three-dimensional capsule deformation in a simple shear flow for hyperelastic [36, 40] and area incompressible membranes [52]. Study of the effect of the membrane flexural stiffness in elongational and shear flow [27, 38]. Passage of capsules through small pores with hyperbolic [30] and cylindrical shapes [16, 39].

Capsule deformation in an effectively unbounded simple shear flow, illustrated in Figure 1.5.1(b), has received particular attention. Pozrikidis [36] implemented a boundary-integral method on a structured interfacial grid and performed simulations for capsule to ambient liquid viscosity ratio equal to unity, considering spherical and ellipsoidal unstressed shapes. Ramanujan & Pozrikidis [40] developed a more efficient implementation on an unstructured interfacial grid consisting of curved triangular elements, and simulated the deformation of capsules with spherical, ellipsoidal, and biconcave resting shapes. Their studies extended the earlier results to larger deformations and non-unit viscosity ratios. More recently, Lac et al. [28] implemented a method based on a biparametric surface representation coupled with bicubic & spline interpolation, and investigated the effect of the membrane constitutive law for capsules with spherical resting shapes and unit viscosity ratio. In the remainder of this section, we focus our attention on the deformation of capsules with spherical resting shapes, which is the most common shape of artificial capsules encountered in applications. Because the volume of the capsule remains constant during deformation, the surface area of the membrane must increase from the initial minimal value corresponding to the sphere. Consequently, the membrane constitutive law must allow for area dilatation. The Mooney–Rivlin law, the Skalak (SK) law, and the Evans and Skalak (ES) law fulfill this requirement, the second and third with values of  and  on the order of unity. The simulations show that, following an initial transient period, the capsules may reach a steady shape that is possibly wrinkled in the absence of flexural stiffness, as discussed in Section 1.5.5. Figure 1.6.1 displays the Taylor deformation parameter at steady state, plotted against the capillary number, , 
'$  , for neo-Hookean (NH) membranes described by (1.3.3) with  , as well as for membranes obeying the Skalak law with  = 1 and 10. The results of Ramanujan & Pozrikidis [40] obtained using a slightly different form of the neo-Hookean law are also shown in this figure.

28

Capsules and Cells

Figure 1.6.1 Steady deformation of a spherical capsule in a simple shear flow for    [28].

The membrane constitutive law is seen to have a considerable influence on the capsule deformation. A capsule enclosed by the Skalak (SK) membrane deforms less than a capsule enclosed by a new-Hookean (NH) membrane for the same value of ,, that is, at the same shear rate and for fixed values of the capsule radius , viscosity 
, and elastic modulus  . The physical reason is the strain hardening behavior of the SK material under extension, as discussed in Section 1.3.5. The overall rotational motion of the membrane around the stationary shape is also found to be sensitive to the membrane constitutive law [28]. At the value of , corresponding to the last data point shown in Figure 1.6.1, the capsule develops high-curvature tips, and the shape of the membrane is difficult to resolve with adequate precision. In these highly curved regions, bending effects are expected to become important and a membrane model that overlooks the flexural stiffness is inadequate. Chang & Olbricht [13] observed capsule breakup preceded by the appearance of similar highly curved tips. However, because membrane rupture was observed to occur over the main body of the capsule where the interface undergoes the highest degree of stretching, the most probable cause of bursting is material failure under large strain. Although the numerical models do not allow for bursting, they do provide us with the distribution of the elastic tensions and strains over the deformed membrane, and thus allow us to determine whether a failure criterion for a specific membrane material is exceeded during the evolution. Ramanujan & Pozrikidis [40] investigated the effect of the viscosity ratio . In the case of a spherical unstressed capsule, the Taylor deformation parameter at steady state for  = 0.2 was found to be roughly 10% larger than that for  =1, as shown in Figure 1.6.2. On the other hand, when   #, the deformation is significantly smaller than that for   , and the deformation parameter tends to an asymptotic value at large shear rates, ,   .

Flow-induced capsule deformation

29

Figure 1.6.2 Effect of the viscosity ratio on the deformation of a spherical capsule enclosed by a neo-Hookean membrane in simple shear flow. The dashed lines represent the predictions of a second-order asymptotic theory. (From Ramanujan, S. & Pozrikidis, C., 1998, J. Fluid Mech., 361, 117-143. With permission from Cambridge University Press.)

Nominally spherical capsules invariably show deviations from the perfect shape. To address the effect of the unstressed shape, Ramanujan & Pozrikidis [40] studied the deformation of capsules with ellipsoidal resting shapes. For resting aspect ratio on the order of 0.9, the simulations showed that the deformation parameter 
exhibits small non-decaying oscillations about the mean steady value of the equivolume spherical resting shape, in agreement with laboratory observations [13, 47]. Raising the shear rate dampens the amplitude of the oscillations, whereas increasing the viscosity ratio promotes the oscillations. Capsules with more eccentric resting shapes exhibit stronger oscillations, although numerical instabilities at large deformations prevent us from drawing definitive conclusions. There is sufficient evidence to indicate that the deformation of capsules enclosed by thin membranes is sensitive to membrane bending effects at small and large deformation. Pozrikidis [38] recently conducted a numerical study of capsule deformation taking into consideration the membrane resistance to bending. For initially spherical capsules enclosed by a membrane that obeys a variant of the neo-Hookean (NH) law, bending stiffness reduces the capsule deformation and prevents the appearance of highly curved shapes. For example, for ,  ',   , and a substantial bending modulus equal to 0.45   , the steady deformation of the capsule is roughly half that in the absence of bending stiffness. Because high internal viscosity limits the deformation altogether for a more viscous capsule,   #, the effect of membrane bending is smaller.

30

Capsules and Cells

On a pragmatic note, accounting for bending effects places strong constraints on the accuracy and stability of the numerical method. In particular, the adequate computation of the curvature of the deformed interface requires a fine mesh, and the explicit time integration of the position of the membrane points requires small time steps, much smaller than those needed in the absence of bending moments. These numerical considerations have limited the investigation of bending effects (see also Chapter 2).

1.7 Summary The study of capsule deformation in flow is a fascinating topic with a wide range of applications. In this chapter, we have considered a special class of liquid-filled capsules enclosed by thin membranes. Following three decades of active research, the general behavior of these capsules is presently well understood in qualitative and quantitative terms. Further studies are needed to describe in more detail the effect of membrane constitutive law, to investigate the capsule behavior in tube and channel flow, to assess the effect of membrane permeability, and to account for the combined effect of viscous shear forces and osmotic pressure. In some cases, the internal content of the capsule is in a gel state [26], whereas in other cases, the capsule contains a nucleus and other inclusions, and example provided by white blood cells. A simple model that assumes an internal Newtonian liquid is not adequate in these situations. Because biological membranes consist of area incompressible lipid bilayers, cell deformability relies on excess surface area defined with respect to that of the equivolume spherical shape. Excess membrane area requires either a non-spherical resting shape, such as the biconcave shape of red blood cells, or spontaneous membrane wrinkling, as in the case of white blood cells. The question then arises as to the proper definition of natural and stress-free membrane state. In conclusion, a considerable amount of further work is needed, and will be forthcoming, to properly understand the mechanics of capsules and cells.

Flow-induced capsule deformation

31

References [1] ACHENBACH , B., H USMAN , M., K APLAN , A., & R EHAGE , H., 2000, Ultrathin networks at fluid interfaces. In: Transport mechanisms across fluid interfaces, Dechema (Ed.), 136, 45-68, Wiley, New York. [2] BARTH E` S -B IESEL , D., 1980, Motion of a spherical microcapsule freely suspended in a linear shear flow, J. Fluid Mech., 100, 831-853. [3] BARTH E` S -B IESEL , D., 1991, Role of interfacial properties on the motion and deformation of capsules in shear flow, Physica A, 172, 103-124. [4] BARTH E` S -B IESEL , D., D IAZ , A., & D HENIN , E., 2002, Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation, J. Fluid Mech., 460, 211-222. [5] BARTH E` S -B IESEL , D. & R ALLISON , J. M., 1981, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech., 113, 251-267. [6] BARTH E` S -B IESEL , D. & S GAIER , H., 1985, Role of membrane viscosity in the orientation and deformation of a capsule suspended in shear flow, J. Fluid Mech., 160, 119-135. [7] BARTKOWIAK , A. & H UNKELER , D., 1999, Alginate-Oligochitosan microcapsules: a mechanistic study relating membrane and capsule properties to reaction conditions, Chem. Mater., 11, 2486-2492. [8] B ENTLEY, B. J. & L EAL , L. G., 1986, An experimental investigation of drop deformation and breakup in steady, two-dimensional linear flows, J. Fluid Mech., 167, 241-283. [9] B OAL , H., S EIFERT, U., & S HILLOCK , J. C., 1993, Negative Poisson ratio in two-dimensional networks under tension, Phys. Rev. E, 48, 4274-4283. [10] B URGER , A., L EONHARD , H., R EHAGE , H., WAGNER , R., & S CHWOERER , M., 1995, Ultrathin cross-linked membranes at the interface between oil and water, Macromol. Chem. Phys., 196, 1-46. [11] C ARIN , M., BARTH E` S -B IESEL , D., E DWARDS -L EVY, F., P OSTEL , C., & A NDREI , D., 2003, Compression of biocompatible liquid filled HSA-alginate capsules: determination of the membrane mechanical properties, Biotechnology & Bioengineering, In press. [12] C HANG , K. S. & O LBRICHT, W. L., 1993, Experimental studies of the deformation of a synthetic capsule in extensional flow, J. Fluid Mech., 250, 587-608. [13] C HANG , K. S. & O LBRICHT, W. L., 1993, Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow, J. Fluid Mech., 250, 609-633.

32

Capsules and Cells

[14] C HENG , L. Y., 1987, Deformation analyses in cell and developmental biology. Part I- Format methodology, J. Biomech. Eng., 109, 10-17. [15] C HENG , L. Y., 1987, Deformation analyses in cell and developmental biology. Part II- Mechanical experiments on cell, J. Biomech. Eng., 109, 18-24. [16] D IAZ , A. & BARTH E` S -B IESEL , D., 2002, Entrance of a bioartificial capsule in a pore, Comput. Model. Eng. Sc., 3, 321-338. [17] D IAZ , A., P ELEKASIS , N., & BARTH E` S -B IESEL , D., 2000, Transient response of a capsule subjected to varying flow conditions: Effect of internal fluid viscosity and membrane elasticity, Phys. Fluids, 12, 948-957. [18] D IAZ , A., P ELEKASIS , N., & BARTH E` S -B IESEL , D., 2001, Effect of membrane viscosity on the dynamic response of an axisymmetric capsule, Phys. Fluids, 13, 3835-3838. [19] E DWARDS -L EVY, F. & L EVY, M. C., 1999, Serum albumin-alginate coated beads: mechanical properties and stability, Biomaterials, 20, 2069-2084. [20] E GGLETON , C. D. & P OPEL , A. S., 1998, Large deformation of red blood cell ghosts in simple shear flow, Phys. Fluids, 10, 1834-1845. [21] E VANS , E. A. & S KALAK , R., 1980, Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton. [22] F ENG , W. W. & YANG , W. H., 1973, On the contact problem of an inflated spherical nonlinear membrane, J. Appl. Mech., 40, 209-214. [23] G REEN , A. E. & A DKINS , J. E., 1970, Large Elastic Deformations, Second Edition, Clarendon Press, Oxford. [24] H OCHMUTH , R. M., 2000, Micropipette aspiration of living cells (review), J. Biomech., 3, 15-22. [25] JAY, A. W. L. & E DWARDS , M. A., 1968, Mechanical properties of semipermeable microcapsules, Canad. J. Physiol. Pharmacol., 46, 731-737. ¨ W. M., L ANZA , R. P., & C HICK , W. L., 1999, Cell encapsu[26] K UHTREIBER lation Technology and Therapeutics, Birkhˆauser, Boston. [27] K WAK , S. & P OZRIKIDIS , C., 2001, Effect of bending stiffness on the deformation of liquid capsules in uniaxial extensional flow, Phys. Fluids, 13(5), 1234-1242. [28] L AC , E., BARTH E` S -B IESEL , D., P ELEKASIS N., & T SAMOPOULOS , J., 2003, Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling, Submitted for publication. [29] L ARDNER , T. J. & P UJARA P., 1980, Compression of spherical cells, Mechanics Today, 5,161-176.

Flow-induced capsule deformation

33

[30] L EYRAT-M AURIN , A. & BARTH E` S -B IESEL , D., 1994, Motion of a deformable capsule through a hyperbolic constriction, J. Fluid Mech., 279, 135-163. [31] L I , X. Z., BARTH E` S -B IESEL , D., & H ELMY, A., 1988, Large deformations and burst of a capsule freely suspended in elongational flow, J. Fluid Mech., 187, 179-196. [32] L IM , F., 1984, Biomedical applications of microencapsulation, CRC Press, Boca Raton. [33] P IEPER , G., R EHAGE , H., & BARTH E` S -B IESEL , D., 1998, Deformation of a capsule in a spinning drop apparatus, J. Coll. Interf. Sci., 202, 293-300. [34] P OZRIKIDIS , C., 1990, The axisymmetric deformation of a red blood cell in uniaxial straining flow, J. Fluid Mech., 216, 231-254. [35] P OZRIKIDIS , C., 1992, Boundary integral and singularity method for linearized viscous flow, Cambridge University Press, New York. [36] P OZRIKIDIS , C., 1995, Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow, J. Fluid Mech., 297, 123-152. [37] P OZRIKIDIS , C., 2001, Interfacial dynamics for Stokes flow, J. Comp. Phys., 169, 250-301. [38] P OZRIKIDIS , C., 2001, Effect of bending stiffness on the deformation of liquid capsules in simple shear flow, J. Fluid Mech., 440, 269-291. [39] Q U E´ GUINER , C. & BARTH E` S -B IESEL , D., 1997, Axisymmetric motion of capsules through cylindrical channels, J. Fluid Mech., 348, 349-376. [40] R AMANUJAN , S. & P OZRIKIDIS , C., 1998, Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: Large deformations and the effect of fluid viscosities, J. Fluid Mech., 361, 117-143. [41] R EHAGE , H., H USMANN , M., & WALTER , A., 2002, From two-dimensional model networks to microcapsules, Rheol. Acta, 41, 292-306. [42] R EHOR , A., C ANAPLE , L., Z HANG , Z., & H UNKELER , D., 2001, The compressive deformation of multicomponent microcapsules: Influence of size, membrane thickness, and compression speed, J. Biomater. Sci. Polym. Ed., 12, 157-170. ¨ [43] S KALAK , R., T OZEREN , A., Z ARDA , P. R., & C HIEN , S., 1973, Strain energy function of red blood cell membranes, Biophys. J., 13, 245-264. [44] S MITH , A. E., M OXHAM , K. E., & M IDDELBERG , A. P. J., 1998, On uniquely determining cell-wall material properties with the compression experiment, Chem. Eng. Sci., 53, 3913-3922.

34

Capsules and Cells

[45] S MITH , A. E., M OXHAM , K. E., & M IDDELBERG , A. P. J., 2000, Wall material properties of yeast cells. Part II. Analysis, Chem. Eng. Sci., 55, 20432053. [46] S MITH , A. E., Z HANG Z., & T HOMAS , C. R., 2000, Wall material properties of yeast cells. Part I. Cell measurements and compression experiments, Chem. Eng. Sci., 55, 2031-2041. [47] WALTER , A., R EHAGE , H., & L EONHARD , H., 2000, Shear-induced deformations of polyamide microcapsules, Colloid Polym. Sci., 278, 169-175. [48] WALTER , A., R EHAGE , H., & L EONHARD , H., 2001, Shear-induced deformations of microcapsules: shape oscillations and membrane folding, Coll. Surf. A, Physicochem. Eng. Asp. 183, 123-132. [49] YONEDA , M., 1964, Tension at the surface of sea urchin egg: a critical examination of Cole’s experiment, J. Exp. Biol., 41, 893-906. [50] Z HANG , Z., B LEWETT, J. M., & T HOMAS , C. R., 1999, Modeling the effect of osmolality on the bursting strength of yeast cells, J. Biotechnology, 71, 17-24. [51] Z HANG , Z., S AUNDERS , R., & T HOMAS , C. R., 1999, Mechanical strength of single microcapsules determined by a novel micromanipulation technique, J. Microencapsulation, 16, 117-124. [52] Z HOU , H. & P OZRIKIDIS , C., 1995, Deformation of liquid capsules with incompressible interfaces in simple shear flow, J. Fluid Mech., 283, 175-200.

Chapter 2 Shell theory for capsules and cells

C. Pozrikidis Classical thin-shell theory provides us with a natural framework for describing the stresses developing over the membranes enclosing capsules and cells. Under the auspices of this theory, the interfaces are regarded as distinct two-dimensional curved media embedded in three-dimensional space over which tangential and transverse shear tensions and in-plane bending moments develop as the result of deformation from a reference configuration. A variety of results may then be obtained regarding equilibrium shapes, stability, deformation, and dynamics subject to an external to internal pressure difference and under the action of an ambient viscous flow. In this chapter, the theory of thin elastic shells is reviewed with emphasis on its application to the hydrostatics and hydrodynamics of capsules and cells. Extensions of the classical approach to account for the mechanical behavior exhibited by biological membranes including area incompressibility and viscous dissipation are emphasized, and numerical methods for solving the governing equations are reviewed and further developed.

2.1 Introduction Clean interfaces between two immiscible fluids and interfaces hosting surfactants exhibit a macroscopic isotropic surface tension that may be regarded as kind of a “surface pressure,” and is a function of temperature and local concentration of surface active agents known as surfactants. Molecular investigations have shown that, in reality, a clean interface between two immiscible fluids that is devoid of surfactants is a thin transition zone spanning several molecular layers, across which large differences in the normal stresses are spontaneously established (e.g., [1, 65]). A schematic illustration of the interfacial stratum and associated distribution of the , is shown in Figure 2.1.1. In hydrostatics, tangential normal stress, denoted by
, where
is the pressure. The horizontal dashed line in this figure marks the nominal position of the interface located at  
, whereas the discontinuous vertical broken line illustrates the idealized distribution of the normal stress as seen by a macroscopic observer.

35

36

Capsules and Cells (1)

s xx y

Fluid 1

y

s

I

xx

x

Fluid 2

(2) xx

s

Figure 2.1.1 Illustration of the surface stratum of a clean interface between two immiscible fluids.

Requiring that the tangential force exerted on a cross-section of the interface extending over the interval      due to the actual distribution of normal stresses is equal to: (a) the tangential force due to the idealized distribution, and (b) a phenomenological contribution due to the “surface tension”  , we write (e.g., [24])  














(2.1.1)

Rearranging, we obtain a definition for the surface tension,




 






 


 





 

(2.1.2)

where the superscripts (1) and (2) denote the corresponding fluid, as illustrated in Figure 2.1.1, and the limits of integration  and  are set at positions where the two integrands on the right-hand side of (2.1.2) virtually vanish. In hydrostatics, equation (2.1.2) takes the form




 





 













  

(2.1.3)

The preceding two equations make an important distinction between the integral of the normal stress across the interfacial stratum and the surface tension. For example, if the lower and upper pressures

 and
 are identical and equal to
, expression (2.1.3) yields

Membrane theory for capsules and cells





37







(2.1.4)

where   is the interface thickness. To compute the nominal location of the interface, 
, we write a  -moment equivalence that is analogous to the tangential force equivalence displayed in (2.1.1), 

 











 
 


(2.1.5)

which can be rearranged to give

 





 






  


 





  

(2.1.6)

Equations (2.1.2) and (2.1.6) provide us with a basis for computing  and 
from knowledge of the actual distribution of the normal stress . In practice, we encounter contaminated, polymerized, and biological interfaces consisting of lipids, proteins, and other macromolecules. These interfaces exhibit a macroscopic mechanical behavior that is more involved than that described by isotropic surface tension. For example, red blood cells are enclosed by a biological membrane consisting of a lipid bilayer and a supportive network of proteins (e.g., [31, 58, 86]). An assortment of other proteins transverses the triple structure anchoring the bilayer to the cytoskeleton. The bilayer is responsible for incompressible behavior in which the surface area of any infinitesimal or finite portion of the membrane is conserved during deformation, whereas the cytoskeleton is responsible for elastic behavior that causes the cell to return to the resting shape of a biconcave disk in the absence of a persistent mechanical load. When a red blood cell is subjected to hydrodynamic stresses, the membrane develops anisotropic elastic tensions and a position-dependent isotropic tension that ensures area preserving deformation. The excess surface area of a healthy membrane, combined with its low shear modulus of elasticity and bending, allows the red blood cells to readily deform and squeeze through the microcapillaries. Membranes of capsules and vesicles consisting of lipid bilayers exhibit bending elasticity, that is, resistance to bending from an equilibrium configuration mediated by bending moments. If the lipid bilayer is chemically symmetric, the equilibrium shape of the membrane possesses zero mean curvature [54, 83]. The development of bending moments endows a shell-like membrane with flexural stiffness whose magnitude depends on a bending modulus that is generally distinct from, and unrelated to, the modulus of elasticity corresponding to in-plane deformation. In the absence of bending stiffness, an elastic membrane may wrinkle to develop corrugations of arbitrarily small wave length under compression. More important, a membrane may fold without resistance to form cornered shapes of vanishing curvature. The presence of bending stiffness imposes limits on the minimum attainable length scale by

38

Capsules and Cells

restricting the minimum radius of curvature above a certain threshold. Steigmann and Ogden [90, 91] studied the deformation of membranes coated on the surface of a two- or three-dimensional elastic medium, and concluded that a surface model that does not account for bending stiffness cannot be used to simulate local surface features engendered by the response of solids to compressive surface stress of any magnitude. Membranes separating viscous fluids are expected to behave in a similar fashion. The mathematical modeling of tensions and bending moments developing over interfaces with a membrane-like constitution relies on the classical theory of thin shells developed and widely used in structural engineering [10, 34, 35, 37, 38, 53, 56, 59, 61, 64]. In this formulation, the membrane is regarded as a curved two-dimensional medium with small or infinitesimal thickness, and the mid-surface of the membrane is described in parametric form, typically in surface curvilinear coordinates. Three approaches are available for describing the membrane deformation, for deriving equilibrium conditions, and for computing the stresses and moments developing due to deformation. In the first approach, the membrane is regarded as a thin sheet of a three-dimensional material, and asymptotic forms of the governing equations and boundary conditions are derived in the limit of zero thickness [50]. In the second approach, specific assumptions are made regarding the deformation of material fibers oriented normal the mid-surface of the membrane. In the third approach, the third dimension is abandoned at the outset, and the membrane is regarded as a curved two-dimensional medium embedded in three-dimensional space. The third approach, to be discussed in the remainder of this chapter, has significant advantages. Most important, it circumvents certain inconsistencies encountered in the first two approaches [18], and it is appropriate for molecular membranes where the assumption of continuum in the normal direction is not meaningful. Recently, Steigmann and Ogden [89, 90, 91] established a rigorous theoretical foundation that allows the consistent computation of the membrane tensions and bending moments from a unified membrane strain energy function.

2.2 Stress resultants and bending moments Consider a membrane in a specified reference state, and label the constituent point particles using two “convected” surface curvilinear coordinates  , so that a line of constant , a line of constant  , and a line along the unit normal vector  define a system of right-handed, but not necessarily orthogonal, curvilinear coordinates, as depicted on the left of Figure 2.2.1. The position of point particles at the reference state is denoted by
  . Assume now that the membrane deforms under the action of a localized or distributed load to obtain a new shape, and denote the position of the point particles in the new state by
 , as illustrated on the right of Figure 2.2.1. The developing

Membrane theory for capsules and cells n

39

R

n x

R

x

h

x

x

Reference state

h

Deformed state

Figure 2.2.1 Illustration of a three-dimensional membrane at the reference and at the deformed state.

in-plane stress resultants, also called tensions, , transverse shear tensions  , and bending moments  , are shown in Figure 2.2.2. If the membrane is a thin sheet of an elastic material, tensions arise by integrating the stresses over the cross-section to obtain stress resultants, whereas bending moments arise because of the non-uniform distribution of the stresses over the crosssection, as will be demonstrated by example in this section. In the “membrane approximation” of thin-shell theory, the transverse tensions and bending moments are neglected, and only the in-plane stress resultants are retained in the analysis (e.g., [23]). However, this approximation is not appropriate for polymerized capsules and biological membranes where bending moments make an important, and in some cases essential, contribution.

2.2.1 Cylindrical shapes In the case of a two-dimensional (cylindrical) membrane that is unstressed in the direction of the generators, we work with the in-plane tension , the transverse shear tension  , and the bending moment , as shown in Figure 2.2.3(a). To simplify the analysis, we shall assume that the membrane is a thin elastic sheet of uniform thickness, as depicted in Figure 2.2.3(b), where the dashed line represents the midsurface. In this case, the scalar in-plane stress resultant is the integral of the in-plane stress over the cross-section, 





(2.2.1)

The associated bending moment is given by



 



  

(2.2.2)

40

Capsules and Cells (a) t

q

xx

t

(b)

x

x

x

xh

t t

n

n q

m h

xx

m

xh

hx

m hh

m

hx

h

hh

h

Figure 2.2.2 Illustration of (a) in-plane and transverse shear tensions (stress resultants), and (b) bending moments developing around the edges of a patch of a three-dimensional membrane. where    describes the location of the mid-surface with respect to arc length  measured along the mid-surface, defined such that  



  



(2.2.3)

yielding  
. To demonstrate the relation between the bending moments and the curvature of the mid-surface, we consider the bending of a flat piece of rubber into a circular arc of aperture angle , as shown in Figure 2.2.3(c). A horizontal material line of length  becomes a circular arc of length ¼  , where  
  is the radius of the material line in the deformed state, and  is the radius of the centerline. When   , we obtain the length of the centerline, ¼
 . The stretch or extension ratio of the material line is given by




¼ 

¼ ¼  ¼

 










(2.2.4)

where  ¼   is the stretch of the mid-surface. Assuming now that the tangential stresses are given by the linear Hooke’s law   , where  is a modulus of elasticity, and using the definition (2.2.3), we find that the bending moment defined in (2.2.2) is given by



 



  

Performing the integration, we find



    

   





 


(2.2.5)

(2.2.6)

Membrane theory for capsules and cells

41

(a) q y

t

t

n

m

l

m x

(b)

y s

t

b

yc

h

m

l

a

(c) L

y b

a

R s

m q

s

Figure 2.2.3 (a) Illustration of a two-dimensional (cylindrical) membrane showing the in-plane tension , the transverse tension  , and the bending moment  developing due to deformation. (b) In-plane tension, , and bending moment, , developing in a thin cylindrical elastic sheet. The dashed line represents the mid-surface. (c) Bending of a flat piece of rubber into a circular arc, illustrating the relation between bending moments and mid-plane curvature.

42

Capsules and Cells

  is the sheet thickness in the deformed state, and   is where

the curvature of the centerline. If the sheet is made of an incompressible material,  
 .   and the bending moment is given by  
As a further example, we consider the stresses developing in a thick-walled infinite cylinder with inner radius  and outer radius , subject to internal pressure
and outer pressure
. The classical Lam´e solution yields the radial and circumferential principal stress distributions











where  

















































 (2.2.7)






is the transmural pressure (e.g., [93], p. 247). Note that
and   
, as required. Using the definitions (2.2.1)

and (2.2.2), we find that the tangential tension normal to the generators is given by 

where









 




(2.2.8)

 is the wall thickness. The associated bending moment is given by 

 





 

    
     









(2.2.9)


 is the radius of the mid-surface. In the limit as  where  (2.2.8) yields Laplace’s law

 


 ,

(2.2.10)

and (2.2.8) yields












 

(2.2.11)

The right-hand side is proportional to the cross-sectional curvature, .

2.2.2 Axisymmetric membranes Figure 2.2.4(a) illustrates an axisymmetric membrane whose mid-surface is generated by rotating of curve around the  axis. In this case, it is natural to introduce polar cylindrical coordinates comprised of the axial position , the distance from the  axis denoted by , and the meridional angle measured around the  axis with origin in the  plane, denoted by . The membrane tensions and bending moments are all assumed to be axisymmetric.

Membrane theory for capsules and cells (a)

43

y q

ts

ts

n

s

ms

tj

mj

z

j

(b)

x

r

Rs sj

c

Rj

ss

x

Figure 2.2.4 (a) Illustration of an axisymmetric membrane showing the principal elastic tensions and bending moments. (b) A section of a membrane consisting of a thin sheet, confined by planes that define the principal curvatures.

As a preliminary, we introduce the arc length measured along the contour of the membrane in a meridional plane, denoted by , the unit vector that is tangential to the membrane and lies in a meridional plane defined by a certain value of the meridional angle , denoted by  , and the conjugate meridional unit vector, denoted by  . The unit vector normal to the membrane, , points outward, as illustrated in Figure 2.2.4(a). Working under the auspices of thin-shell theory, we introduce the azimuthal and meridional tensions  and  , which are the principal tensions of the in-plane stress resultants, the transverse shear tension  exerted on a cross-section of the membrane that is normal to the  axis, and the azimuthal and meridional bending moments  and  , as illustrated in Figure 2.2.4(a). Note that the axisymmetric system of tensions and bending moments is a simplification of that depicted in Figure 2.2.2 for a genuinely three-dimensional membrane.

44

Capsules and Cells

Figure 2.2.4(b) depicts a patch of an axisymmetric membrane consisting of a thin sheet, confined by: (a) two meridional planes that pass through the  axis, and (b) two conjugate planes that are normal to the mid-surface described by the dashed lines. To leading order approximation, the trace of the mid-surface on the front conjugate plane is a circular arc of radius  centered at the  axis, where  is a principal curvature. Using the notation of Figure 2.2.4(b), we find that the tangential force exerted at the corresponding cross-section is given by





 



  



 







 







 

Æ



Æ Æ






 








Æ Æ

  

(2.2.12)

where    define the outer and inner surface of the sheet,

  is the sheet thickness, and Æ
  . Realizing that   is the differential arc length along the mid-surface, we identify the expression inside the large parentheses on the left-hand side of (2.2.12) with the azimuthal principal tension  , and obtain

 where 


 



(2.2.13)

 is the conjugate principal curvature, 








 Æ

(2.2.14)

 

and










Æ Æ

(2.2.15)

 

is the azimuthal principal bending moment. Working in a similar fashion, we find

 where 


 



(2.2.16)

 is the azimuthal principal curvature, 








 Æ

(2.2.17)

 

and










Æ Æ

(2.2.18)

 

is the meridional principal bending moment. Expressions (2.2.13) and (2.2.16) appear to have been first derived by Evans & Yeung [32].

Membrane theory for capsules and cells Fluid 1 (external)

45

n

n Membrane patch

b

C

s Fluid 2 (internal)

Figure 2.2.5 Force and torque balances are performed over the cross-section of the contour of a membrane patch.

2.2.3 Cartesian formulation To facilitate the coupling of the membrane mechanics to the hydrostatics or hydrodynamics on either side of a membrane, it is convenient to describe the membrane tensions and bending moments in global Cartesian coordinates. To do this, we extend the domain of definition of the membrane tensions and bending moments into the whole three-dimensional space subject to appropriate constraints, as follows [39, 53, 71, 94]:



 

The in-plane tensions are described in terms of the Cartesian tensor defined such that the in-plane tension exerted on a cross-section of the membrane that is normal to the tangential unit vector  is given by   , as illustrated in Figure 2.2.5. Furthermore, to ensure that the tension lies in the tangential  and  . For example, if the membrane plane, we require  exhibits isotropic tension  , then  , where   is the tangential projection operator, and  is the identity matrix. The transverse shear tension is described in terms of the Cartesian vector  defined such that the transverse shear tension exerted on a cross-section of the membrane that is normal to the tangential unit vector  is given by   . Furthermore, we require the obvious condition   . The bending moments are expressed in terms of the Cartesian tensor  defined such that the bending moment vector exerted on a cross-section of the membrane that is normal to the tangential unit vector  is given by    . Furthermore, to ensure that the moment vector lies in the tangential plane, we . require    and  

To make the preceding definitions more concrete, we consider the two-dimensional (cylindrical) membrane depicted in Figure 2.2.3, and identify the unit vector  with

46

Capsules and Cells

 and  , which the unit tangential vector . By definition then,   suggests that

 , meaning    . Similar considerations suggest that    and    , so that       , where  is the unit vector perpendicular to the  plane. In the case of the axisymmetric membrane depicted in Figure 2.2.4, the Cartesian tension and bending moment tensors are given by

  
  



  
  

(2.2.19)

and the transverse shear tension vector is given by



  

(2.2.20)

2.3 Interface force and torque balances Consider a liquid capsule that is suspended in an ambient fluid labeled 1 and encloses another fluid labeled 2. A membrane patch confined by the contour  is illustrated in Figure 2.2.5. Assuming that the mass and thus the inertia of the membrane is negligible, we perform force and torque balances over the patch, and thereby derive expressions between (a) the hydrodynamic traction playing the role of a distributed load, and (b) the membrane tensions and bending moments.

2.3.1 Cartesian curvature tensor To prepare the ground for performing interfacial force and torque balances in global Cartesian coordinates, we introduce the Cartesian curvature tensor defined as the gradient of the unit vector that is normal to the interface and points outward from the capsule, properly extended off the interface into the three-dimensional space,




(2.3.1)

where 
 is the tangential projection operator, and  is the identity matrix. By definition, and because of the constancy of the length of the unit normal vector,

 



(2.3.2)

which shows that the normal vector is an eigenvector of and its transpose corresponding to the null eigenvalue. The two remaining eigenvectors lie in a plane that is tangential to the interface and are parallel to the mutually orthogonal directions of the principal curvatures. If  , 
are the principal curvatures and  ,
 are the corresponding tangential unit eigenvectors, then can be expressed in the form



  



 

(2.3.3)

Membrane theory for capsules and cells

47

If the surface is locally spherical at a point, the two principal curvatures are equal to the local mean curvature,  
 , and   at that point. The mean curvature of the interface, denoted by  , derives from the trace of

from the relation

 

Trace 

(2.3.4)

which is clearly satisfied when

 . However, the Gaussian curvature is not related to the determinant of the Cartesian curvature tensor in a simple fashion, as it is related to the determinant of the two-dimensional intrinsic curvature tensor defined in terms of derivatives of the normal vector with respect to surface curvilinear coordinates. To evaluate the curvature tensor at a point, we consider the variation of the Cartesian components of the position vector
and unit normal vector along two generally non-orthogonal surface curvilinear coordinates  and , and require

 


 

 


  

(2.3.5)

Appending to these vector equations the constraint  , we obtain three systems of three linear algebraic equations for the three columns of .

2.3.2 Balances in Cartesian coordinates In global Cartesian coordinates, the force balance over the patch depicted in Figure 2.2.5 reads 
























   

(2.3.6)

where
is the hydrostatics or hydrodynamics stress tensor,  is the unit vector tangential to  ,    is the unit vector that is tangential to the membrane and lies in a plane that is normal to the contour  , and  is the arc length along  . Using the divergence theorem to convert the contour integral to a surface integral on the right-hand side of (2.3.6), and taking the limit as the size of the patch becomes infinitesimal, we find that the jump in the hydrodynamic traction across the membrane is given by [71]







     
 





Trace  


  

(2.3.7)

The right-hand side of (2.3.7) expresses the surface divergence of the generalized elastic tension tensor in Cartesian coordinates. The tangential derivatives are taken with respect to two isometric orthogonal rectilinear coordinates that are tangential to the membrane at the point where the divergence is evaluated.

48

Capsules and Cells

An analogous torque balance with respect to the arbitrary point
 requires 



    










 




      
   






     

(2.3.8)

Next, we use the divergence theorem to convert the contour integral to a surface integral on the right-hand side of (2.3.8), let the size of the patch become infinitesimal, and use the force balance (2.3.7) to derive an expression for the transverse shear tension,

      



Trace    

(2.3.9)

and another expression for the antisymmetric part of the in-plane tension tensor 

   

(2.3.10)

where the superscript ! denotes the matrix transpose, and is the Cartesian curvature tensor (e.g., [17, 96, 71]). In the case of a two-dimensional (cylindrical) membrane, we refer to Figure 2.2.3 and find that the two-dimensional curvature tension is given by   , where  is the curvature of the membrane in the  plane. Recalling that     shows that the right-hand of (2.3.10) vanishes and confirms that the tension tensor is symmetric. Similar results are obtained for an axisymmetric membrane.

2.3.3 Balances in surface curvilinear coordinates The Cartesian formulation described in Section 2.3.2 requires that the membrane tensions and bending moments be extended into the whole space in an appropriate fashion. This extension can be circumvented by working in surface curvilinear coordinates (e.g., [4]). To develop up this formulation, we introduce the generally non-unit tangential vectors













(2.3.11)

 

(2.3.12)

and the corresponding arc length metric coefficients



 



The first fundamental form of the surface is the square of the length of an infinitesimal fiber whose end-points are separated by a vector corresponding to the coordinate differentials  and  ,






  

   
  


(2.3.13)

Membrane theory for capsules and cells

49

where










  




 

(2.3.14)

The surface area of a patch that is confined between two  and segments with  infinitesimal increments  and  is equal to  , where 

 
 . The second fundamental form of the surface is the quadratic form

  


"

   

  


(2.3.15)

where

 

 

  





  





(2.3.16)

  

are the coefficients of the second fundamental form,

      

(2.3.17)

is the unit normal vector, and  is the symmetric surface curvature tensor. The normal curvature of the surface in the direction of an infinitesimal vector whose endpoints are separated by a distance correspond to the infinitesimal increments  and  is equal to the ratio of the second and first fundamental form of the surface. Next, we introduce the surface contravariant components of the tension tensor denoted by , the surface contravariant components of the transverse shear tension vector denoted by   and   , and the surface contravariant components of the bending moment tensor  denoted by  ; Greek superscripts and subscripts stand for  or . The corresponding covariant components are illustrated in Figure 2.2.2. Subject to the preceding definitions, the force equilibrium equation (2.3.7) may be resolved into normal and tangential components as















# 
#  
#  

(2.3.18)

where

#  

#








 



#

 






(2.3.19)

 

(e.g., [59], p. 165; [96], eq. (3.5)). The vertical bar denotes the covariant derivative taken with respect to the subscripted variable and defined in terms of the Christoffel symbols (e.g., [3]).

50

Capsules and Cells

Correspondingly, the torque equilibrium equations (2.3.9) and (2.3.10) take the form



  





  



(2.3.20)

and

  

   

(2.3.21)

The mixed derivatives  derive from the pure covariant derivatives  by the relation     (e.g., [59], p. 177; [96], eq. (3.9)).

2.3.4 Balances in the lines of principal curvatures Considerable simplifications can be achieved by referring to surface curvilinear coordinates whose tangential vector at every point points in the direction of the principal curvatures, defined as the lines of principal curvatures. A line of constant , a line of constant  , and a line directed along the unit normal vector define a right-handed system of orthogonal curvilinear coordinates. In the case of an axisymmetric membrane developing axisymmetric tensions, to be discussed in Section 2.6, the lines of principal curvatures are the contours of the membrane in meridional and azimuthal planes. In surface curvilinear coordinates that are lines of principal curvatures, the decomposition (2.3.18) can be recast into the preferred form




  # 
#  
#   (2.3.22) where 
   and 
   are unit tangential vectors along the surface 




¼

¼

curvilinear coordinates. The normal and tangential components of the jump in traction are given by

# 

     
   

       

# 

    
  
 

       

# 

    
  

     

¼



 
 

¼

 

 

 

 

  


   

(2.3.23)

  (2.3.24)

  (2.3.25)

where

 are the principal curvatures.













(2.3.26)

Membrane theory for capsules and cells

51

Moreover, expressions (2.3.20) and (2.3.21) simplify to

 

     
   
  

               
   

       

    

(2.3.27)

  


   

(2.3.28)

and



 
 



(2.3.29)

(e.g., [59], p. 33). In Section 2.6, we shall present the specific forms of these expressions in cylindrical polar coordinates for axisymmetric shapes.

2.4 Surface deformation and elastic tensions As a prelude to evaluating the elastic tensions, we refer to Figure 2.2.1 and introduce the three-dimensional Cartesian relative deformation gradient tensor , which is a Cartesian tensor with components



$ 

  





(2.4.1)



Let the infinitesimal vector   describe a small fiber that is either tangential or normal to the membrane at the reference state. After deformation, the fiber has rotated and stretched or compressed to its image described by



   

(2.4.2)

The nine components of the relative deformation gradient tensor may be evaluated from knowledge of the images of two material fibers that are tangential to the membrane at a point, and the image of a fiber that is normal to the membrane at that point. In the present formulation, the image of a fiber that is normal to the membrane is assumed to vanish, so that the deformation of this fiber enters the computation of the elastic tensions only by means of the deformation of the tangential fibers according to constitutive laws expressing the membrane material properties. For the purpose of computing the elastic tensions, equation (2.4.2) is written as





  

(2.4.3)

where




 

 



(2.4.4)

52

Capsules and Cells

is the surface relative deformation gradient, and the superscript “S” stands for “surface.” Because  is an eigenvector of corresponding to a vanishing eigenvalue, the matrix is singular. If   is a tangential fiber at the reference state, then  is also a tangential fiber in the deformed state, and this requires       . However,  , which suggests that since the orientation of  is arbitrary, it must be     or






   

 



(2.4.5)

Thus, is an eigenvector of the transpose of corresponding to the vanishing eigenvalue. Using the polar decomposition theorem, we write      , where  is an orthogonal matrix expressing plane rotation, and ,  are the positivedefinite and symmetric right or left stretch tensors expressing pure deformation. Following standard procedure in the theory of elasticity [38, 37, 10], we introduce the positive-definite, symmetric, left Cauchy-Green surface deformation tensor












(2.4.6)

where the superscript T denotes the matrix transpose. The eigenvalues of 
are equal to 
 , 

, and 0, corresponding to the orthogonal tangential eigenvectors  , 
and to the normal vector ;   and 
 are the principal stretches or extension ratios. The eigenvectors of 
are also eigenvectors of the tension tensor . In terms of the principal elastic tensions ! and
! and the unit tangential eigenvectors 
   and



, the tension tensor is given by the spectral decomposition

!  

!



(2.4.7)

When bending moments are significant, the tension tensor has an additional antisymmetric component, as will be discussed in Section 2.5.

2.4.1 Elastic membranes Next, we proceed to relate the tensions to the surface strains by means of a constitutive equation. As a prelude, we consider a three-dimensional elastic medium and express the force exerted on a small material patch of surface area  that is perpendicular to the unit normal vector in terms of the Eulerian Cauchy stress tensor
, in the familiar form








(2.4.8)

Furthermore, we introduce the first Piola-Kirchhoff tensor , also known as the Lagrange or nominal stress tensor, and the Piola-Kirchhoff tensor , defined by the relations














    



(2.4.9)

Membrane theory for capsules and cells

53

where  is the unit vector normal to the patch in a reference state, and   is the corresponding surface area (e.g., [35] p. 438; [64] p. 106). The Eulerian stress tensor
is related to the first Piola-Kirchhoff tensor  and to the Piola-Kirchhoff tensor  by the equation




        %

(2.4.10)

%

where %   is the dilatation of an infinitesimal material parcel after deformation; if the material is incompressible, % . For a Green-elastic three-dimensional medium, the first Piola-Kirchhoff tensor and the Piola-Kirchhoff tensor derive from a volume strain-energy function &"   by the relations

! 

&" $ 



&"  

(2.4.11)

where



     

(2.4.12)

is the Green-Lagrange strain tensor, also known as the material or Lagrangian strain tensor (e.g., [10]; [35], p. 449; [37], p. 7; [64], pp. 204–209).

2.4.2 Surface tension tensors To develop the two-dimensional analog of the preceding equations over the curved surface of a membrane in the absence of bending moments, we replace relations (2.4.10) by

  %

   %






(2.4.13)

where %  
is the dilatation of an infinitesimal membrane patch, and  ,  are the surface Piola-Kirchhoff tensors. The counterparts of relations (2.4.11) are

! 

& $ 



&  

(2.4.14)



   




 



(2.4.15)

where


is the surface Green-Lagrange strain tensor, also known as the material or Lagrangian strain tensor, and & is the surface strain-energy function or Helmholtz free-energy    and  density of the membrane. In the absence of deformation, vanishes.

54

Capsules and Cells

Referring to local Cartesian coordinates with two axes parallel to the local principal directions of the tension tensor, we use equations (2.4.13) and (2.4.14) and find that the principal tensions are given by

 & 


!

 &   



!

(2.4.16)

Combining expression (2.4.7) with equations (2.4.16) we obtain a complete description of the elastic tensions.

2.4.3 Surface strain invariants Kinematic constraints require that the surface strain-energy function & must depend on the surface deformation gradient by means of surface strain invariants. Skalak et al. [87] introduced the invariants

'
  



'

%









 







 (2.4.17)







and recast expressions (2.4.16) into the form

!


!

 




 






& ' & '


  




& '



  




(2.4.18)

&  '


Substituting (2.4.18) into (2.4.7), we find

 
  &
   
 & ' '

 







































(2.4.19)

or

 & 

   &  
 
' '




(2.4.20)

Note that, if & ' , the tensions are isotropic. Moreover, Skalak et al. [87] proposed the following strain energy function for the membrane of a red blood cell,

&

 '
'   

(






'







'



(2.4.21)

where ( and  are physical constants with estimated values on the order of ( 0.005 dyn/cm and  100 dyn/cm. The large magnitude of the constant  compared to that of ( ensures that a small deviation of '
from unity generates large

Membrane theory for capsules and cells

55

elastic tensions. Consequently, the membrane is nearly incompressible and the developing tensions are nearly isotropic. Barth`es-Biesel & Rallison [8] introduced the alternative strain invariants

  '
  
 
    '  
   



















(2.4.22)



Substituting these expressions into (2.4.16), we find

!


!

  &  
 




  &  
 











&   


(2.4.23)

&   


Substituting further expressions (2.4.23) into (2.4.7), we find

 
 &

  &     



















or

  &   
 



















&   




(2.4.24)

(2.4.25)




When &  
, the tensions are isotropic. In the limit of small deformation, the strain energy function obtains the standard Mooney–Rivlin form

&

) 


 )
)   ¾
)  








 






(2.4.26)

where ) , )
, and ) are material constants. Evans & Skalak [31] noted that the invariants

)


 '



*


'
  '





  








  




(2.4.27)



describe, respectively, the change in the area of a surface patch and the change in the element aspect ratio: if  
, then * . Differentiating aided by the chain rule, we find

& '

 &  )
 *

& '


 )


& )

*
 &  )

* 

(2.4.28)

56

Capsules and Cells

Substituting these expressions into (2.4.18) and rearranging, we derive the alternative expressions for the principal tensions

!

& )








!

& )

 






  &










*

  &










*

(2.4.29)



When & * , the tensions are isotropic. Materials that exhibit a constant value of & * at large deformations, such as natural rubber, are called “hyperelastic.”

2.4.4 Thin elastic shells It is instructive to compare the preceding results derived in the context of membrane theory with corresponding results of classical elasticity for the stress resultants developing in a thin shell consisting of a three-dimensional incompressible elastic material with uniform thickness (e.g., [37] pp. 156–159, [56] p. 399). For this purpose, we introduce the volume strain invariants

'"




 













'
"






 













(2.4.30)

and express the principal elastic tensions in terms of the volume strain energy function &" as

!


!

   &"
 ' "

 
 



 



 
 





   &"
 ' "


 












&"  '
" &"  '
"

(2.4.31)

The Mooney–Rivlin strain-energy function is given by

&"

" 



+ '"


+ ' " 


(2.4.32)

where " is the volume modulus of elasticity, and + is a material parameter vary corresponds to a linear neo-Hookean material ing between zero and unity; + (e.g., [64], p. 221). Substituting (2.4.32) into expressions (2.4.31), we compute the principal elastic tensions

!




!





 

 


     











   


 



    




+ 
+ 

(2.4.33)

+ 
+ 


Membrane theory for capsules and cells

57

where 
"  is the surface modulus of elasticity. For small deformations,

 " 

'"


 

¾





(2.4.34)




where  and 
are the surface invariants defined in (2.4.22). Expression (2.4.32) for a neo-Hookean material (+ ) then reduces to (2.4.26) with



)

 

)




 

)



(2.4.35)

yielding the surface strain-energy function [8]



¾ 
 

&



 




(2.4.36)



2.4.5 Decomposition into isotropic and deviatoric tensions In certain cases, it is useful to express the principal membrane tensions in terms of an isotropic tension,  , and a deviatoric tension,  ¼ , defined as




  !
!   




¼


  ! 


!




! 



(2.4.37)

Thus,

!


¼

¼ 

(2.4.38)

Physically, the deviatoric tension is the maximum shear tension exerted on a crosssection of the membrane that is inclined at an angle of , with respect to the directions of the principal tensions. Expressions (2.4.29) show that, in terms of the surface strain-energy function,



& )

 &  *

¼










 




(2.4.39)

On the other hand, if the membrane consists of a neo-Hookean elastic material whose principal tensions are given by (2.4.33) with + ,

 ¼

 
  












   






(2.4.40)

   














These expressions will find application later in this chapter in our discussion of axisymmetric equilibrium membrane shapes.

58

Capsules and Cells

2.5 Surface deformation and bending moments When bending moments are significant, the in-plane tension tensor is modified with the addition of an antisymmetric component according to equations (2.3.10), (2.3.21), and (2.3.29), and with the alteration of the symmetric part as required by the functional dependence of the strain energy function on properly defined measures of bending for an elastic membrane. With regard to the second modification, it is commonly assumed that the bending moments have a negligible effect on the symmetric part of the elastic tension tensor. It should be noted that the reference state concerning the bending moments, defined as the state where the bending moments vanish, is not necessarily the same as that of the elastic tensions, reflecting differences in the physical mechanics that are responsible for their respective development. For example, in the case of a membrane with a dual molecular structure or a laminated interface consisting of multiple molecular layers or thin shells, the relaxed state of the individual constituents may correspond to different configurations.

2.5.1 Small deformation First, we consider the most tractable case of small deformation (e.g., [59], p. 17). Referring to orthogonal curvilinear coordinates   which are the lines of principal curvatures in the reference configuration, as discussed in Section 2.3.3, we introduce strain invariants defined in terms of the displacement of a material point particle over the membrane, ,

- -

 
 

  

-

-

 
 

  

  
  

          

(2.5.1)

The invariant - expresses the elongation of a fiber in the direction of the  axis, the invariant - expresses the elongation of a fiber in the direction of the axis, and the invariant - is a measure of the deformation of an infinitesimal patch. Corresponding measures of bending,  ,  , and  can be defined in terms of the rotation of a surface patch due to the deformation (e.g., [59], pp. 21, 25). The strain and bending measures may now be used to define the global strain measure expressed by the vector


- - -    

(2.5.2)

and the surface strain energy function

&

      

(2.5.3)

Membrane theory for capsules and cells

59

where  is a positive-definite matrix expressing membrane material properties (e.g., [59], p. 45). For example, if the membrane is a thin shell of a three-dimensional isotropic elastic material, the strain energy function is given by Love’s first approximation describing the infinitesimal displacement of a thin plate of thickness ,

&

   .  -
 -
- 
. -
-        .
 #   .  
 
 
. 
 


























(2.5.4)

where

 

#

 

.


(2.5.5)

is the plate modulus of bending,  is the volume modulus of elasticity, and . is the Poisson ratio (e.g., [35], p. 461). For a homogeneous material, the Poisson ratio takes values in the range    ; if the material is incompressible, . . In terms of the strain energy function, the stress resultants and bending moments are given by

 

& -



& 







 &  -



 &  

& - 

&  

(2.5.6)

Using Love’s strain energy function given in (2.5.4), we obtain



 and





-
.-  .
 

  
. - 

 .
-
.- 

(2.5.7)

 

  . 
.   

   
.    
.     . 

(2.5.8)

















Note that, when . , in which case deformation in one principal direction does not induce stresses in the perpendicular direction, the first and last of the equations in (2.5.8) are consistent with (2.2.6).

60

Capsules and Cells

2.5.2 Large deformation Consider now a small material membrane patch at the resting state and then at a deformed state. The bending moments developing along the edges of the patch in the deformed state depend on the instantaneous edge curvature, as well as on the edge curvature at the resting state. Evaluation of the latter requires knowledge of the rotation that the edge has undergone due to the deformation, and necessitates an involved formulation in terms of the surface deformation gradient [91]. If, however, the undeformed surface patch has uniform curvature, knowledge of the patch rotation is not required; no matter how much the edge of a material patch has rotated, the curvature of the edge at the undeformed state is constant and independent of orientation. Motivated by this simplification, we focus our attention on membranes that are isotropic at the reference bending state. In the case of materially homogeneous and isotropic membranes, the assumption of isotropy requires that the directional curvature at the reference state is independent of orientation in the tangential plane, which is true for the flat and spherical shape. For sufficiently small bending deformations, but not necessary small in-plane deformations, the bending moments may be approximated with the linear constitutive equation



# 

   

(2.5.9)

where # is the scalar bending modulus, allowed to be a function of the invariants of the strain and curvature tensors  , 
,  , and $ [91],  is the mean curvature, and $ is the Gaussian curvature. The reference mean curvature,   , is zero for the flat resting shape and nonzero for the spherical resting shape. Since the bending moment tensor  is symmetric, the antisymmetric part of vanishes, and the bending moments affect only the transverse shear tensions by means of equation (2.3.9). In practice, # is assumed to be a constant, independent of its arguments listed on the right-hand side of equations (2.5.9). Substituting (2.5.9) with a constant bending modulus # into (2.3.9), and observing that

      

  

(2.5.10)

we find that a constant reference curvature has no effect on the transverse shear tension and is thus inconsequential to the hydrostatics and hydrodynamics. Thus, when evolving under the action of the bending moments alone, a capsule will tend to the spherical shape irrespective of the initial shape and reference curvature. To account for the qualitative effect of bending moments in the more general case of arbitrary resting shapes, the following generalized version of (2.5.9) may be employed,



# 

   

(2.5.11)

where   is the resting mean curvature. Physically, (2.5.11) is expected to be apply for small deformations from the resting configuration.

Membrane theory for capsules and cells

61

Bending strain-energy functions Finding the strain energy function that corresponds to the stipulated constitutive equation (2.5.9) is a nontrivial exercise. The linear dependence of the bending moment tensor on the Cartesian curvature tensor suggests that the underlying strain energy function is somehow related to the bending energy functional introduced by Canham [20] and generalized by Helfrich [44] for biological membranes, given by

#

/





0 



$

$ 

(2.5.12)

where the integration is performed over the instantaneous membrane shape, 0 is the “spontaneous curvature,” which is the counterpart of twice the reference curvature presently employed, # is the bending modulus associated with the mean curvature, and $ is the bending modulus associated with the Gaussian curvature. The concept of spontaneous curvature was introduced by Helfrich to account for possible asymmetries in the bilayer molecular structure of a biological membrane; in the case of a symmetric membrane, 0 . According to the Gauss–Bonnet theorem of differential geometry (e.g., [57]), the last term on the right-hand side of (2.5.12) depends only on the topological genus of the membrane and is thus identical for topologically equivalent shapes and may thus be ignored in further analysis. It is illuminating to expand the quadratic in (2.5.12) and rearrange to obtain

/

#









0

 






0






# 0

 


 (2.5.13)

 

 is the mean curvature, where  and 
are the principal curvatures, 
and  is a topological constant also incorporating the spontaneous curvature. The first term on the right-hand side of (2.5.13) is a generalization of Canham’s energy , the second term on functional involving the spontaneous curvature. When 0 the right-hand side of (2.5.13) vanishes, and the generalized Canham and Helfrich energy functionals are equivalent. Now, the Helfrich formulation relies on a somewhat arbitrary and seemingly unphysical choice for the bending energy functional. Indeed, a noticeable inconsistency of the corresponding energy density function is apparent at a point where the mean  0 , but the principal curvature is equal to half the spontaneous curvature, 
 curvatures are not equal to the mean curvature; when 0 , this occurs at a saddle point. In such cases, the integrand of the first term in (2.5.12) predicts zero contribution to the bending energy, which seems physically implausible. When 0 , this conceptual difficulty can be resolved by using the Gauss-Bonnet theorem to restate the energy density function in the physically acceptable Canham form expressed by the first integrand on the right-hand side of (2.5.13).  , where In the case of a two-dimensional (cylindrical) membrane, we set 
 is the curvature of the membrane in the  plane, and obtain

  
#  

0 


(2.5.14)

62

Capsules and Cells

In Section 2.8, we will show that (2.5.14) reproduces the constitutive relation (2.5.9) with a constant bending modulus for an inextensible membrane.

2.6 Axisymmetric shapes In the case of an axisymmetric membrane whose mid-plane is generated by rotating of curve around the  axis, it is convenient to work in polar cylindrical coordinates consisting of the axial position , the distance from the  axis denoted by , and the meridional angle  measured around the  axis with origin in the  plane, as illustrated in Figure 2.2.4. The fluid stresses inside and outside the membrane, the membrane tensions and bending moments developing due to the deformation, are all assumed to be axisymmetric.

2.6.1 Geometrical preliminaries Using fundamental relations of differential geometry, we find that, if the radial position of the membrane is described by the equations

 

(2.6.1)

then the principal curvatures are given by





 and





 

 
  

¼¼

(2.6.2)





¼


  



¼




 







(2.6.3)

where ¼
  and 
 (e.g., [69], p. 162). The plus sign of selected when   , and the minus sign otherwise. Expressions (2.6.2) and (2.6.3) are consistent with Codazzi’s equation



   

is

(2.6.4)

which allows us to compute one of the principal curvatures in terms of the other (e.g., [59], p. 9). Rearranging (2.6.4), we obtain

 







(2.6.5)

, and using the rule de l’Hˆospital to Applying (2.6.5) at the axis of symmetry, evaluate the right-hand side, we find  %  % . Differentiating (2.6.5) with respect to  and working in a similar fashion, we find  
 
% 
 
% .

Membrane theory for capsules and cells

63

2.6.2 Capsule shape in terms of the curvature To compute the contour of a capsule in terms of the meridional curvature  , we regard the  and coordinates of point particles along the trace of the membrane in a meridional plane as functions of the meridional arc length , writing    and 

. By definition then, ¼

¼

, which can be differentiated ¼ ¼
¼¼
, where a prime denotes a derivative with respect to . Using to yield  ¼¼ elementary differential geometry, we derive the relations

¼¼ ¼

¼ ¼¼




¼¼ ¼


¼¼
 ¼

(2.6.6)

Next, we introduce the functions 
¼ and 
¼
satisfying 


 obtain the following system of four nonlinear differential equations,

 






 



 

 

, and

   (2.6.7)

The second pair of equations is decoupled from the first pair and can be integrated independently. Once the solution has been found, the first pair can be integrated to generate the membrane shape. For example, to compute a biconcave shape that is symmetric with respect to the , as illustrated in Figure 2.6.2, we may express the meridional mid-plane  curvature in the form



,  

Æ

 , 

(2.6.8)

where     ,  is the total arc length of the cell contour in a meridional plane and Æ is a specified dimensionless amplitude, and then integrate system (2.6.7) using, for example, a Runge–Kutta method (e.g., [70]) with initial conditions

 








 



 



(2.6.9)

where  is an arbitrary position. Cell contours computed in this manner are displayed in Figure 2.6.2 for Æ = 0 (sphere), 0.5, 1.0, 1.5, 2.0, and 2.3, on a scale that has been adjusted so that all cells have the same surface area. The shape for Æ = 2.0 is similar to the average normal blood cell shape reported by Evans & Fung [30].

2.6.3 Force and torque balances Equilibrium equations can be derived by considering a small section of the membrane that is confined between (a) two adjacent meridional planes passing through the  axis, and (b) two parallel planes that are perpendicular to the  axis and enclose a small section of the membrane in a meridional plane, as depicted in Figure 2.6.1. Performing a force balance over this section, we find that the jump in the traction across the membrane, that is, the membrane load, is given by

 





  

# 
#  

(2.6.10)

64

Capsules and Cells

1

0

-1

-1

0

1

Figure 2.6.2 Contours of oblate and biconcave cells whose curvature is given by equation (2.6.8) with Æ = 0 (sphere), 0.5, 1.0, 1.5, 2.0, and 2.3. The shapes have been scaled so that all cells have the same surface area.

where
 is the stress tensor in the surrounding fluid, and
 is the stress tensor inside the cell. The normal jump is given by

# 



 
 

   

(2.6.11)

and the tangential jump is given by

# 

  



   

 

 

  

 




 

 

(2.6.12)

An analogous torque balance provides us with an expression for the transverse shear tension in terms of the bending moments,








   

  





 




   





(2.6.13)

(e.g., [59], p. 33). Substituting the right-hand side of (2.6.13) in place of the shear tension in (2.6.11) and (2.6.12), we obtain expressions for the jump in traction in terms of the in-plane tensions and bending moments alone. It is reassuring to confirm that expressions (2.6.11) through (2.6.13) are consistent with the more general equilibrium equation for a three-dimensional membrane

Membrane theory for capsules and cells

65

discussed in Section 2.3. To see this, we identify the surface curvilinear coordinate  with the meridional arc length , and the curvilinear coordinate with the meridional angle ; the corresponding metric coefficients are given by   and

 . Since all tensions and moments are axisymmetric, the principal directions point along the chosen curvilinear axes, and equations (2.3.23) through (2.3.28) reproduce equations (2.6.12) through (2.6.13).

2.6.4 Constitutive equations for the elastic tensions To derive relations for the elastic tensions, we introduce the principal stretches or extension ratios



 







(2.6.14)

where the subscript , subsequently also used as a superscript, denotes the reference state. If the area of the membrane is locally and thus globally conserved,

 



(2.6.15)

as will be discussed in Section 2.8. To this end, we have two main choices reflecting the assumed nature of the membrane. First, we may regard the membrane as a distinct two-dimensional elastic medium and express the principal stress resultants in terms of the surface strain energy function & using equations (2.4.18), subject to the substitutions

!


!












 

(2.6.16)

Alternatively, we may regard the membrane as a thin sheet of a three-dimensional incompressible material and work with the strain invariants shown in (2.4.30). In this case, the principal elastic tensions derive from the volume strain energy function &" by equations (2.4.31). In the case of a neo-Hookean material, the principal tensions are given by (2.4.33).

2.6.5 Constitutive equations for the bending moments To compute the bending moments developing in an elastic membrane, we introduce the bending measures of strain

1

 

 

1

 

 

(2.6.17)

where the superscript  denotes a reference configuration corresponding to the unstressed shape where the bending moments vanish [76, 77, 78, 79, 98]. The bending moments may then be expressed in terms of the surface bending strain energy function &# in a form that is analogous to that shown in equations (2.4.16),



 &#  1



 &#   1

(2.6.18)

66

Capsules and Cells

Love’s first approximation expressed by the second term on the right-hand side of (2.5.4) yields the bending energy function

&#

   1
 . 1 1
1      # 





(2.6.19)

Substituting (2.6.19) into (2.6.18), we find



#  1
. 1  



#  1
. 1  

(2.6.20)

The bending measures (2.6.17) have been designed so that self-similar deformations do not induce bending moments; an example is provided by the expansion of a sphere. This choice is appropriate for molecular membranes whose bending moments depend exclusively on the solid angles of the molecular bonds. For membranes comprised of thin elastic sheets whose thickness changes as a result of the deformation, we may replace the constitutive equations (2.6.20) with the alternative linear relations



# 

  



# 

  

(2.6.21)

In the case of a spherical membrane with reference radius  and deformed radius &  , where 
 is the  '  extension ratio. Note that the bending moments are negative in the case of expansion and positive in the case of shrinkage. An in-depth discussion of constitutive equations for the bending moments can be found in recent articles by Steigmann and Ogden [89, 90, 91].

, equations (2.6.21) yield 

2.6.6 Capsule shapes in hydrostatics Consider now the equilibrium shape of a deformed capsule that is enclosed by an elastic membrane in hydrostatics. When the effect of gravity is insignificant, the pressure inside and outside the capsule is constant denoted, respectively, by
 and
 . Setting

  and

 , where  is the identity matrix, we find that the jump in hydrodynamic traction across the membrane is given by  
 , where 



 is the transmural pressure. The equilibrium equations (2.6.10) and (2.6.11) require

  
  




   

(2.6.22)

 

(2.6.23)

and

   



 

where the transverse shear tension  is given in terms of the bending moments by (2.6.13). In the case of spherical capsule of radius , equal principal curvatures

Membrane theory for capsules and cells

67

 , and mean curvature  , equations (2.6.22) and (2.6.23) are satisfied for   
   and  . Solving (2.6.22) for  , substituting the result in (2.6.23), rearranging and multiplying the derived expression by , we find





  
     










 




   (2.6.24)

Using now Codazzi’s relation (2.6.4) to eliminate the curvature  from the lefthand side and rearranging, we obtain

  or

 




 











 





 




 

   







  

  

  

(2.6.25)

(2.6.26)

When the bending moments and thus the transverse shear tensions are negligible, the right-hand side of (2.6.26) is zero. Integrating the left-hand side with respect to from a fixed point up to an arbitrary point, we find



 




 


 
   %¼  









(2.6.27)

which shows that an unphysical singularity appears when   and  . This result reveals that stress resultants alone are not capable of supporting shapes where the  position reaches a local maximum along the trace of the membrane in the meridional plane, including the biconcave shape assumed by relaxed red blood cells. In the absence of bending moments, the membrane must be unstressed,   and 
 .

2.6.7 Equilibrium shapes with isotropic tension When the membrane tensions are isotropic,  (2.6.23) obtain the simplified forms

   

 

  

 , equations (2.6.22) and  

(2.6.28)

where  

 is the mean curvature. Solving the first equation in (2.6.28) for  and substituting the result in the second equation, we obtain

 





 




  


     

(2.6.29)

68

Capsules and Cells

It is convenient for computational purposes to introduce the reduced tension 2

 # satisfying the differential equation



2 

#



 

#






   



  

(2.6.30)

where # is a constant bending modulus. The right-hand side of (2.6.30) arises by expressing the transverse shear tension in terms of the bending moments using the equilibrium equation (2.6.13). The first of equations (2.6.28) then becomes

   



  # 

2  



(2.6.31)

To this end, we adopt the constitutive equations (2.6.21), and recast (2.6.30) and (2.6.31) into the more specific forms

2 












  



 



 

 (2.6.32)

and











  

   

 





     #

2  

 (2.6.33)

involving the curvatures and reduced tension. Isolating the terms containing the reference curvature on the right-hand side and rearranging, we obtain

2 





   





    2
   
  
       
    

     



 



   #






  

   (2.6.34)

2

   





   (2.6.35) 


The second expression in (2.6.34) was derived using (2.6.5). When the reference   the rightshape of the membrane is a sphere of radius  ,    hand sides of (2.6.34) and (2.6.35) vanish, and the resulting simplified equations are distinguished by the absence of the reference curvature. Eliminating the reduced tension 2 from (2.6.32) and (2.6.33), we obtain a thirdorder differential equation for the curvatures with respect to meridional arc length

Membrane theory for capsules and cells

69

describing the capsule shape,

 








 









  



 

   





  

 




 


  

 

# 



(2.6.36)

Far from the axis of symmetry,  and   tend to vanish, and (2.6.36) takes the simplified asymptotic form


   
     

 

   # 



(2.6.37)

which involves only the meridional curvature in the deformed and reference state. Equation (2.6.37) is the point of departure for computing buckled shapes of twodimensional (cylindrical) shells under a negative transmural pressure (e.g., [33, 73]), as will be discussed in Section 2.8. Given the distribution of the reference curvatures around the deformed contour,  expressed by the functions    and  , equations (2.6.2), (2.6.3), (2.6.7), (2.6.34), and (2.6.35) provide us with a complete system of coupled ordinary differential equations for the functions  , 
,  ,  ,  ,  , and 2. An equivalent system of first-order equations arises by recalling the definition   , 
, denoting 
 ,  
 , and 2  , and collecting the governing equations into the form

 

#

(2.6.38)

for 3     . Using equation (2.6.3) to write   
and Codazzi’s equation (2.6.5) to write    

 

, we find that the phase-space velocities are given by

#



#




#

 

#

   
  

#





 #

  


 

#



   

      
   

# 


       
    

  

  





   
  



  

   


   

(2.6.39)

70

Capsules and Cells

The accompanying boundary conditions are

 

 
         

 (2.6.40)

at the axis of symmetry where   and ;  is an arbitrary position along the  axis. For shapes with left-to-right symmetry, such as those displayed in Figure 2.6.2, we also require

 



 



(2.6.41)

where  is the total arc length of the cell contour in a meridional plane. The expressions for the phase-space velocities # and # become indeterminate at the axis of symmetry where

. Careful consideration of the limit of the corresponding differential equations assisted by Codazzi’s equations (2.6.4) and (2.6.5) shows that



    
      

#

 

# 







# 









(2.6.42)



and these values are used to initialize the computation. A numerical method was implemented for solving system (2.6.38) using the fourthorder Runge–Kutta method [74]. The shooting variables are adjusted using Newton’s method, with the    Jacobian matrix computed by numerical differentiation. The solution of the boundary-value problem for each set of parameters requires only a few seconds of CPU time on a 1.7 GHz Intel processor running Linux. Because multiple solution branches exist for a specified set of conditions, as will be discussed later in this section, the converged capsule shape can be notably sensitive to the initial guesses for the transmural pressure and for the value of the reduced tension  . At high transmural pressures, parameter continuation with a very small step is necessary to successfully trace a branch. Non-dimensionalization In the space of dimensionless functions, the solution of system (2.6.38) can be parametrized by one of the following three dimensionless negative transmural pressures,


(

 

(

#






 

#




"

 

"

#



(2.6.43)

( is the mean curvature of the equivalent spherical shape whose where ( , ( ;  perimeter 4 in a meridional plane is equal to that of the capsule, 4  is the mean curvature of the equivalent spherical shape whose surface area

Membrane theory for capsules and cells

71

,
; and " " is the mean curvature is identical to that of the capsule, of the equivalent spherical shape whose volume 5 is same as that of the capsule, 5 ) " . In practice, a family of solutions is found by specifying the perimeter , the curvature of the capsule at the axis,  , and the bending modulus, # , and then solving the boundary-value problem by the shooting method, where the trial variables are the transmural pressure and the initial value  . Spherical unstressed shapes Consider first the deformation of a capsule with a spherical resting shape. The solid lines in Figure 2.6.3(a) show a family of oblate and dimpled deformed shapes for centerline curvature (   = 0.99, 0.95, 0.90, 0.80,   , -1.00, plotted on a scale that has been adjusted so that all capsules have the same surface area. The continuation of this family to shapes with lower negative centerline curvature yields unphysical self-intersecting profiles, as shown by the dashed line in Figure 2.6.3(a) . Half of these intersecting shapes, however, can corresponding to (   be identified with a deformed hemispherical cap fitted to the end of a semi-infinite circular tube, buckling inward due to a difference between the low tube pressure and the high ambient pressure. Figure 2.6.3(b) shows a second family of deformed 0.98, shapes with more convoluted geometry for centerline curvature (   0.95, 0.90, 0.80, 0.60,   , -3.40, -3.60. Figure 2.6.4 displays the volume of the first and second family of shapes drawn, respectively, with thin and thick lines, plotted against the reduced centerline curvature (  . The solid lines show the volume normalized by ) ( , and the dashed lines show the volume normalized by )  . The information contained in this figure can be used to identify the shape of a spherical capsule after a certain amount of fluid has been withdrawn from its interior with a syringe, or else diffused through the membrane due to high internal osmotic pressure. The results reveal that the volume of a cell enclosed by an incompressible membrane with constant surface area, such as the membrane of a vesicle enclosed by a lipid bilayer, decreases monotonically at a nearly quadratic rate with respect to the deviation of the centerline curvature from the reference value. Given the ambient pressure, the interior capsule pressure and thus the transmural pressure is different for each one of the shapes displayed in Figure 2.6.3. Figure 2.6.5
( (solid line) shows a graph of the negative of the reduced transmural pressure  and 
(dashed line), both defined in (2.6.43), plotted against the reduced centerline curvature (  . The results reveal that the spherical shape corresponding to (    is possible for any value of the transmural pressure. Bifurcations into the first and second family of deformed shapes displayed in Figure 2.6.3 occur at the
(  
 = 8 and 36. critical points  The structure of the solution space displayed in Figure 2.6.5 is similar to that of an elastic cylindrical tube with a circular resting shape buckling inward due to low tube pressure. In the case of the tube, bifurcating solution branches are known to originate from the critical transmural pressures &*¿
6
, where + is the 

72

Capsules and Cells (a)

1

0

-1

-1

0

1

-1

0

1

(b)

1

0

-1

Figure 2.6.3 Two families of deformed shapes of a capsule with spherical resting shape for reduced centerline curvature (a) (   0.99, 0.95, 0.90, 0.80,   , -1.00, and (b) 0.98, 0.95, 0.90, 0.80, 0.60,   , -3.40, -3.60. The dashed . The scale in line in (a) shows a self-intersecting shape with (   both figures has been adjusted so that all capsules have the same surface area.

curvature of the undeformed shape and 6 is the wave number of the circumferential mode (e.g., [36], p. 177; [97]; see also Section 2.8.3). The two solution branches  and 4. The displayed in Figure 2.6.3 correspond to the meridional modes 6 present numerical results suggest that critical bifurcation points for a spherical cell

Membrane theory for capsules and cells

73

1

Centerline curvature

0

-1

-2

-3

-4

0

1

0.5

1.5

Volume

Figure 2.6.4 Volume of the first (thin lines) and second (thick lines) family of shapes displayed in Figure 2.6.3, plotted against the reduced centerline curvature (  . The solid lines show the volume normalized by ) ( , and the dashed lines show the volume normalized by )  .

are given by the formula 
(   6

6 , and this can be confirmed by carrying out a formal analysis for slightly deformed shapes [14]. More direct information on the transmural pressure of deflated capsules is presented in Figure 2.6.6, showing a graph of the negative of the transmural pressure 
for the first and second family of shapes, drawn, respectively, with the thin and thick line, plotted against the capsule volume and reduced so that all shapes have the same surface area. These results clearly demonstrate that withdrawing an infinitesimal amount of fluid from the capsule causes the internal pressure to assume quantum levels according to the prevailing mode of deformation. Although a rigorous proof is not available, the first mode corresponding to the biconcave shape is most likely to occur in practice. Biconcave unstressed shapes To compute deformed shapes of capsules with nonspherical resting shapes, we must have available the distributions of the reference principal curvatures    and   , where  is the arc length around the deformed contour. In general, to obtain these distributions, it is necessary to introduce constitutive equations for the elastic tensions and simultaneously solve for the principal extension ratios. Doing this considerably complicates the mathematical formulation by introducing further terms involving the principal stretches in the governing equations (2.6.38).

74

Capsules and Cells 1

Centerline curvature

0

-1

-2

-3

-4

0

20 40 60 Negative of the transmural pressure

80

Figure 2.6.5 The vertical axis measures the reduced centerline curvature (  , and the horizontal axis measures the dimensionless negative transmural pressure 
( (solid lines) or 
(dashed lines) defined in (2.6.43) for capsules with spherical undeformed shapes.

, and therefore      As a compromise, we may assume that    , whereupon the meridional arc lengths  and  vary over the and          same range. Conversely, this assumption may be regarded as an artificial constitutive equation that can be used to make a correspondence between the position of point particles in the reference and deformed state. When the incompressibility constraint    is also required,  , and point particles along the membrane are displaced parallel to the  axis. Figure 2.6.7 shows a family of deformed shapes for a capsule whose resting shape , for centerline curvature (   is described by equation (2.6.8) with Æ -1.3, -1.2, -1.1, -1.0 (resting shape), -0.9,   , 0.6. The scale has been adjusted so that the shapes displayed have the same surface area. Figure 2.6.8 shows the
plotted against the capsule volume reduced dimensionless transmural pressure  by ,
, which is the maximum volume of a spherical capsule with a given surface area. The undeformed shape corresponds to a reduced volume 0.614. In agreement with physical intuition, negative and positive values of the transmural pressure occur, respectively, in the case of deflation or inflation. In particular, as a capsule enclosed by an incompressible membrane is inflated, the internal pressure rapidly escalates toward a large, but most certainly finite limit.

Membrane theory for capsules and cells

75

50

Negative transmural pressure

40

30

20

10

0

0

0.2

0.4 0.6 Reduced volume

0.8

1

Figure 2.6.6 Negative of the transmural pressure 
plotted against the capsule volume for the first (thin line) and second (thick line) family of deformed shapes shown in Figure 2.6.3.

Red blood cells Zarda et al. [98] computed the shapes of deflated and inflated capsules with spherical and biconcave resting shapes resembling red blood cells, on the basis of the equilibrium equations (2.6.10) through (2.6.13). In their analysis, these equations are written in terms of the angle  subtended between the  axis and the normal to the membrane, defined such that     and     . Their . transverse shear tension " is the negative of the one presently employed, " Zarda et al. [98] expressed the principal membrane tensions in terms of isotropic and deviatoric components  and  ¼ , as  
 ¼ and    ¼ . The isotropic tension was computed to satisfy the inextensibility condition   , while the deviatoric component arises from (2.4.16) as

¼


   

 

    

& 

 &   

(2.6.44)

where the surface strain energy function & is given in (2.4.21). Because the constant ( is smaller by five orders of magnitude than the constant  , the tensions are nearly isotropic. The principal bending moments are given by the constitutive equations (2.6.18) involving the principal stretch ratios. To compute equilibrium shapes, Zarda et al. [98] traced the membrane contour in a meridional plane with marker points and solved the governing equations indirectly by minimizing a properly constructed energy functional using a finite-element method.

76

Capsules and Cells

1

0

-1

-1

0

1

Figure 2.6.7 Deformed shapes of a capsule with a biconcave resting shape drawn with the heavy line.

40

Transmural pressure

30

20

10

0 0.4

0.5

0.6

0.7 0.8 Reduced volume

0.9

1

Figure 2.6.8 Dimensionless transmural pressure 
plotted against the reduced capsule volume for the shapes depicted in Figure 2.6.7.

Zarda et al. ([98], Figures 8 and 9) presented shapes of deflated spherical capsules that are qualitative similar to those depicted in Figure 2.6.3(a), and produced a graph

Membrane theory for capsules and cells

77

of the transmural pressure against the capsule volume, as shown in Figure 2.6.6. Their results show that the transmural pressure diverges to infinity as the reduced volume approaches unity, which means that, if an infinitesimal amount of fluid is withdrawn from the capsule, the capsule pressure immediately becomes very large or infinite. This is in contrast with our present results and at odds with physical intuition. The results presented in Figures 2.6.7 and 2.6.8 are qualitatively similar to those presented in Figures 4 and 5 of Zarda et al. [98] for sphered red blood cells. In their Table 1, these authors list the cell volume and transmural pressure for surface area 141.6 7
. The reduced volume of the undeformed shape is 0.58, which is close to the value 0.614 corresponding to the cell depicted with the heavy line in Figure
 . Taking 2.6.7. The present results show that at the reduced volume 0.92,   7, corresponding to the surface area of 141.6 7
, and #  
dyn cm, we find the transmural pressure 0.38 dyn/cm
. Considering the important differences in the constitutive equations for the bending moments and in the assumed resting shapes, this prediction is reasonably close to the value 3.6 dyn/cm
reported by Zarda et al. [98].

2.7 Planar axisymmetric membranes In the case of a planar membrane supporting an axisymmetric distribution of tensions, the right-hand of (2.6.23) vanishes, yielding the simplified equilibrium equation

   





(2.7.1)

Writing  
 ¼ and    ¼ , where 

 
  is the isotropic tension, and  ¼

     is the deviatoric tension, we recast (2.7.1) into the form

  


 

¼



(2.7.2)

An immediate corollary of (2.7.2) is that, when the deviatoric tension vanishes, the isotropic tension is uniform over the membrane. Substituting now the second equation in (2.4.39) for the deviatoric tension into (2.7.2), we find

   where  ¼


&

¼









 




* is the shear modulus of elasticity.

(2.7.3)

78

Capsules and Cells

, as will be discussed in Section 2.9,

If the membrane is incompressible,   and (2.7.3) becomes

  




¼







¼






























(2.7.4)

Note that by requiring incompressibility, we are forced to abandon the constitutive equation for the isotropic tension shown, for example, in the first equation in (2.4.39). To be able to integrate the ordinary differential equation (2.7.4), we require the inverse function  8  mapping the distance of a material point particle from the  axis along the membrane from the deformed state to the reference state. This function must be found by integrating the constraint    expressed in the differential form    , along the meridional contour of the deformed shape. As an application, we consider the aspiration of an infinite planar sheet representing the surface of a locally flat vesicle into a micropipette of radius , as illustrated in Figure 2.7.1. Assuming that the aspirated shape consists of: (a) a hemispherical cap of radius , (b) a cylindrical body of length , and (c) a planar sheet attached to the rim of the micropipette, and requiring that the area of the membrane is conserved from the axis up to an arbitrary point on the deformed membrane over the planar sheet, we write

,
,


, 


, 










(2.7.5)

which can be rearranged to yield

 














(2.7.6)

Substituting this expression into (2.7.4), we obtain the differential equation

  

¼




 








   







(2.7.7)

To make further progress, we assume that  ¼ is constant and then integrate (2.7.7) with respect to from  9  to infinity where the tensions vanish, to obtain

 





½ ¼






 








     







 

(2.7.8)

Moreover, we assume that the meridional tension at the rim just outside the micropipette (pointing in the radial direction) is equal to the meridional tension just inside the micropipette (pointing in the axial direction), and write the global force balance

4 ,




,  



(2.7.9)

where 

4 4 is the aspiration pressure, 4 is the suction pressure, and 4 is the cell pressure, as illustrated in Figure 2.7.1. Combining the last two equations, we

Membrane theory for capsules and cells

79

sR s Ps

a

Pc

x

L

Figure 2.7.1 Aspiration of an infinite planar sheet representing a locally flat vesicle into a micropipette.

derive an algebraic relation between the aspiration pressure, the micropipette radius, and the projection length ,

4 

½

¼




 








   







 

(2.7.10)

Evaluating the integral with the aid of standard tables (e.g., [11], p. 241), and rearranging, we obtain




 4 ¼


 
! 

(2.7.11)

where 
 is the reduced aspiration length. The solid line in Figure 2.7.2 is the graph of  plotted against the reduced aspiration pressure , and the circles represent data on the aspiration of a flaccid red cell for  ¼     dyn/cm, adapted from Figure 5.18B of Evans & Skalak [31]. The reasonable qualitative agreement supports this best-fit estimate for the red blood cell membrane shear modulus of elasticity in the particular case of aspiration.

2.8 Two-dimensional membranes Consider a two-dimensional (cylindrical) membrane, and assume that the tensions and bending moments are independent of the : coordinate that is normal to the  plane. When deformed from a reference state, the membrane develops an in-plane tension , a transverse shear tension  , and a bending moment , as illustrated in Figure 2.2.3. The vectorial tension exerted on a cross-section is given by






(2.8.1)

80

Capsules and Cells 4

l

3

2

1

0

0

2

4

8

6

c

10

Figure 2.7.2 Graph of the reduced projection length   plotted against the reduced aspiration pressure predicted by the theoretical model.

where  is the unit tangent vector pointing in the direction of increasing arc length , and is the unit normal vector. Performing a force balance over an infinitesimal section of the membrane, we find that the discontinuity in the surface traction across the membrane, that is, the membrane load, is given by


# 
#  

 

  
  

(2.8.2)

Expanding out the derivatives of the products on the left-hand side of (2.8.2) and using the relations

 



 



(2.8.3)

where  is the curvature of the membrane in the  plane, we obtain the normal and tangential loads

# 



 

# 

 

 

(2.8.4)

which are simplified versions of the more general forms (2.3.23) and (2.3.24) written for   and : .

Membrane theory for capsules and cells

81

An analogous torque balance yields an expression for the transverse shear tension in terms of the bending moment,



 

(2.8.5)

which can be recognized as a simplified version of (2.3.27).

2.8.1 Constitutive equations for the elastic tensions To develop constitutive equations for the elastic tension , we introduce the extension ratio or stretch




 

(2.8.6)

where  is the arc length at the position of point particles in the reference state. For an elastic membrane,

& 

(2.8.7)

where & is the surface strain-energy function or Helmholtz free energy. In the case    
and   , where  is the of a linearly elastic material, &
modulus of elasticity.

2.8.2 Constitutive equations for the bending moments A popular constitutive equation for the bending moments is given by the linear relation

# 



  

(2.8.8)

where # is the bending modulus, and   is the curvature of the membrane in a resting configuration where the bending moments vanish (e.g., [90]). The surface bending-energy density function underlying (2.8.8), denoted by & and defined such     
. that  & , is given by &
# To provide a foundation for the constitutive equation (2.8.8) in the case of an inextensible membrane, we introduce the bending-energy functional

  
#         
 
   
 
 
      #    
































(2.8.9)

where
denotes the position of point particles along the membrane, the integration is performed along the instantaneous membrane contour, and the curvature has been

82

Capsules and Cells

expressed in the form  


. The energy variation due to an infinitesimal virtual displacement Æ
that preserves the arc length between any two point particles along the membrane is given by

Æ

#



)




Æ
  



(2.8.10)

where )
  . Integrating the right-hand side of (2.8.10) by parts twice, we derive the preferred form



 

)
 Æ
 # 
 
 #     Æ
 


Æ

(2.8.11)

which may be recast into the form

Æ

#



   

  


 



 



 Æ


(2.8.12)

The inextensibility condition requires that the virtual displacements are subject to the constraint

 which suggests the identity



Æ




(2.8.13)

 #    Æ
 



(2.8.14)

where #  is an arbitrary function. Now, the principle of virtual displacements provides us with an integral equation of the first kind for the membrane load  ,

Æ



  Æ


(2.8.15)

Comparing (2.8.15) with (2.8.11) and taking into consideration (2.8.14), we find



#

   

 
#  


 



  



(2.8.16)

Comparing further (2.8.16) with (2.8.4) and (2.8.5), we deduce the linear constitutive for the bending moments given in equation (2.8.8). An analogous deduction for the in-plane tension is prohibited by the presence of the eigenfunction # . Carrying out the differentiation on the right-hand side of (2.8.16), we find that the normal component of the load is given by

# 




 
 # 


    


(2.8.17)

Membrane theory for capsules and cells

83

When the resting curvature vanishes, equation (2.8.16) takes the simpler form



#

 
 
  #  
    

(2.8.18)

Previous authors have used expressions (2.8.17) and (2.8.18) with specific choices for the resting curvature and for the indeterminate function # . Harris & Hearst [43] studied the dynamics of a polymeric molecule that develops spring-like forces and bending moments, and identified the function #  with a constant playing the role of a Lagrange multiplier, denoted by . Setting the load  equal to the rate of change of momentum of an effective distributed molecular mass, they transformed the equilibrium equation (2.8.18) into an equation of motion for the molecule centerline, and then developed a relation between  and the statistical properties of the fluctuating motion. Liverpool & Edwards [55] considered the evolution of a meandering river, set   and #  , and identified the right-hand side of (2.8.18) with the rate of displacement normal to the centerline. A similar choice was made more recently by Stelitano and Rothman [92] in their numerical study of membrane fluctuations in an ambient viscous fluid.

2.8.3 Stationary equilibrium shapes Consider the equilibrium shape of a two-dimensional capsule resembling a cylindrical tube in hydrostatics, in the absence of significant gravitational forces. Working as in Section 2.6.4, we find that the equilibrium equations (2.8.4) and (2.8.5) combined with the constitutive equation (2.8.8) yield



 

4

 

 



#





 



(2.8.19)

where 4 is the transmural pressure. Solving the first equation in (2.8.19) for , substituting the result into the second equation, and then using the third equation to eliminate  in favor of the curvature, we find


  
     



4



#


     

(2.8.20)

which is consistent with the asymptotic form (2.6.37) for axisymmetric shapes. Next, we recasting (2.8.20) into the form


  
   4
   
#


 







 



(2.8.21)

and integrate once with respect to  to derive the integro-differential equation


 


  
0 4
 
  # 








 ¼

¼ 

 ¼  ¼

(2.8.22)

where 0 is an integration constant with dimensions of inverse squared length, and  is an arbitrary arc length. Equation (2.8.22) describes the equilibrium shape of

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Capsules and Cells

a cylindrical capsule in terms of the curvature; the unstressed shape corresponds to   and 4 . Deformed inflated or buckled shapes arise, respectively, for positive or negative values of the transmural pressure [73]. Using the first and third equations in (2.8.19) in conjunction with (2.8.22), we find that the in-plane tension is given by

  
0   #


 ¼



 ¼   ¼

(2.8.23)

It is important to emphasize that, unlike (2.8.8), expression (2.8.23) is not a constitutive equation relating the in-plane tension to deformation, but rather expresses an equilibrium condition. An additional constitutive equation may be imposed, and its role will be to determine the total length of the membrane and the relative distribution of point particles along the deformed shape with respect to the resting configuration. When the unstressed membrane has a flat or circular shape,  is constant and + and any value of 4 , where + is the constant (2.8.21) is satisfied with  curvature of the rolled up or resting shape. In this case, equation (2.8.22) admits an obvious solution corresponding to uniform curvature  + and 0 
+ 4 # + . Substituting these expressions in the right-hand side of (2.8.23), we find that the uniform in-plane tension is given by 4 + expressing the Young–Laplace law. To describe small deformations from the circular cross-section, we perturb the +
- ¼ , where - is a dimensionless number uniform curvature by setting  whose magnitude is much less than unity. Substituting this expression in (2.8.22), and linearizing with respect to - while holding 0 constant, we find


¼ 
where we have defined




+


¼

4

# +

(2.8.24)



(2.8.25)

Without loss of generality, we may take the solution of (2.8.24) to be a cosine wave ; , and require the periodicity condition of arbitrary amplitude ;, set ¼ to find  6 + , where 6 is an integer representing the circumferential mode. Rearranging, we find that small deformations are possible only when the reduced pressure difference takes the values

4
4

# +

*
6




(2.8.26)

in agreement with classical results on the buckling of cylindrical shells (e.g., [36], p. 177; [97]) These critical values mark the location of bifurcation points in the solution space [73]. Figure 2.8.2 shows the shape of a buckled cylindrical membrane with circular resting shape in the 6  mode, for 4
4 # +  = 3.1, 3.2, 3.3, 3.4, 3.5,

Membrane theory for capsules and cells

85

3.5

3

2.5

y

2

1.5

1

0.5

0

−0.5

−2

−1.5

−1

−0.5

0 x

0.5

1

1.5

2

Figure 2.8.2 Buckled shape of a cylindrical membrane with circular resting shape in the 6  mode for dimensionless transmural pressure 4 = 3.1, 3.2, 3.3, 3.4, 3.5, 4.0, 4.5, 5.0, and 5.247 where touching occurs.

4.0, 4.5, 5.0, and 5.247 where touching occurs, computed by solving the boundaryvalue problem described by (2.8.22) using the shooting method [73]. The biconcave shape is reminiscent of the resting shape of red blood cells. , computed Figure 2.8.3 shows buckled shapes with three-fold symmetry, 6 using the numerical method developed by Blyth & Pozrikidis [12]. As the transmural pressure is lowered to negative values, point contact occurs at a critical threshold. When the curvature at the point of contact vanishes, segment contact is observed over a length that must be computed as part of the solution. The preceding formulation may be extended to describe the shape of cells resting on a horizontal or inclined support, buckling under the combined influence of a negative transmural pressure and their own weight [12]. Figure 2.8.4 shows the shape of a membrane with a circular unstressed shape resting on a horizontal support, for a range of increasing membrane material densities. As the membrane becomes heavier, point contact occurs at critical conditions. When the curvature at the contact point vanishes, the shell starts spreading over the support and contact over a segment occurs over a length that must be computed as part of the solution.

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Capsules and Cells

Figure 2.8.3 Buckled shapes of a cylindrical membrane with circular resting shape in the 6  mode [12].

2.9 Incompressible interfaces Biological membranes consisting of lipid bilayers have a large modulus of dilatation, that is, they behave like two-dimensional nearly-incompressible media. To account for the membrane incompressibility, we add a position-dependent isotropic tension  playing the role of surface pressure to the in-plane elastic stress resultants. The introduction of a scalar surface function provides us with a degree of freedom that allows the satisfaction of the incompressibility constraint at every point over the membrane. The new contribution is expressed by the isotropic tension tensor

 , where   is the tangential projection operator. In global Cartesian coordinates, the incompressibility constraint for an evolving membrane is expressed by the equation







<

<

     

(2.9.1)

where  is the rate of surface dilatation, << is the material derivative,  is the surface metric associated with the convected surface curvilinear coordinates  , and  is the membrane point particle or fluid velocity [85]. More explicitly,







<

<

     


      

(2.9.2)



      
   

= 
= 
   



Membrane theory for capsules and cells

87

Figure 2.8.4 Shapes of a heavy elastic shell with a circular resting shape sitting on a horizontal surface for a range of increasing shell densities [12].

where  is the mean curvature ([69], p. 21; [96], eq. 3.2b; [101]), and a vertical bar denotes the covariant derivative with respect to the subscripted variable defined in terms of the Christoffel symbols (e.g., [3]). The last expression in (2.9.2) also follows from the convection–diffusion equation for a uniformly distributed insoluble surfactant, by requiring that the surfactant concentration at the position of interfacial point particles moving with the fluid velocity remains constant in time. The numerical implementation of expressions (2.9.2) for a capsule deforming under the influence of a simple shear flow is discussed by Zhou & Pozrikidis [101].

2.9.1 Axisymmetric membranes In the case of an axisymmetric membrane, it is convenient to express the azimuthal and meridional tensions in terms of the isotropic and deviatoric components. The former is computed by requiring the incompressibility constraint (2.9.2), while the latter derives from a surface strain energy function. Pozrikidis [66] used this formulation to simulate the transient deformation of an axisymmetric capsule enclosed by

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Capsules and Cells

an incompressible membrane under the influence of an elongational flow. Requiring

<   <



(2.9.3)

where << is the material derivative, and   are the principal extension ratios, we obtain a scalar constraint on the distribution of the membrane velocity ,

 




 

%



(2.9.4)

where  is the unit vector tangential to the membrane in a meridional plane, and % is the unit vector normal to the  axis. It can be shown by straightforward rearrangement that condition (2.9.4) is consistent with the more general expressions (2.9.1) and (2.9.2).

2.9.2 Two-dimensional membranes In the case of a two-dimensional (cylindrical) membrane, the incompressibility constraint simplifies to



 

   
 



(2.9.5)

where  is the unit vector tangential to the membrane,  is the arc length measured in the direction of , and  is the curvature of the membrane in the  plane. The numerical implementation of (2.9.5) is discussed by Zhou & Pozrikidis [101].

2.10 Membrane viscoelasticity Impurities, surfactants, adsorbed macromolecules, and molecular layers generated by chemical reactions are responsible for interfaces that resemble two-dimensional Newtonian or more general non-Newtonian viscoelastic fluids embedded in threedimensional space [9], [28], [60], [81], [88], [96]. The concept of an interface whose rheological properties are distinct from those of the bulk fluids was first submitted by Boussinesq (e.g., [28]). Scriven [81] proposed a constitutive equation for the in-plane Newtonian interfacial tension tensor written in surface curvilinear coordinates. Secomb & Skalak [85] (see also [9, 67]) noted that the coupling between the interfacial dynamics and the hydrodynamics on either side of an interface is facilitated by working in global Cartesian coordinates, and expressed the Newtonian surface tension tensor in the form

 
7¼

7


7 



(2.10.1)

Membrane theory for capsules and cells

89

where  is the isotropic tension,  is the rate of surface dilatation given in equations (2.9.1) and (2.9.2),   is the tangential projection operator, 7 and 7¼ are two physical constants expressing the interface shear and dilatational viscosity, and  is the Cartesian surface rate-of-deformation tensor with components

 

 4 4  =,
=   ,   ,

(2.10.2)

where  is the membrane or fluid velocity. The two projections on the right-hand side of (2.10.2) remove derivatives of the velocity normal to the interface, as well as derivatives of the normal component of the velocity in directions that are tangential to the interface. Thus, the right-hand side of (2.10.2) may be computed from knowledge of the velocity distribution over the interface. The rate of surface dilation  is equal to the trace of  . In the case of an axisymmetric membrane, the principal viscous tensions corresponding to the last term on the right-hand side of (2.10.1) are given by

"

"

 7 6       7 Æ

<Æ  <

 7 



  7 Æ <Æ < 



 7     

<  <

  7  < < 

(2.10.3)

where << is the material derivative,  and  are the principal stretches, and the rest of the variables are defined in Figure 2.6.1 (see also Section 1.3.4). A similar expression can be written for the viscous tension developing along a two-dimensional (cylindrical) membrane. If the membrane is viscoelastic, an elastic tension tensor involving in-plane and transverse shear tensions is added to the right-hand sides of (2.10.1), (2.10.2), and (2.10.3). The combined effect of viscoelasticity on capsule deformation is discussed by Barth`es-Biesel [4, 5] (see also Chapter 1).

2.11 Discrete models and variational formulations Previously in this chapter, we have discussed the mathematical modeling of interfacial tensions and bending moments in the context of continuum mechanics, working under the auspices of the theory of thin shells and interfacial rheology. The molecular nature of biological interfaces suggests that an alternative formulation that regards a membrane as a network of generally viscoelastic molecular links defined by computational nodes may also be appropriate. An example of this approach can be found in the numerical studies of Hansen et al. [40, 41, 42], who developed a network model based on random Delaunay triangulation representing the red blood cell cytoskeleton. Their simulations provided estimates for the macroscopic elastic shear modulus and modulus of areal expansion.

90

Capsules and Cells

2.11.1 Variational formulation An alternative method of computing the membrane load hinges on the concept of configurational energy playing the role of an effective Hamiltonian (e.g., [2, 19, 63]). Canham [20] and Helfrich [44] proposed that the energy of a membrane consisting of a lipid bilayer can be expressed in the form of a functional defined with respect to the principal, mean, and Gaussian curvatures, as discussed in Section 2.5.2. At thermal equilibrium, and in the absence of flow-induced deformation, a vesicle enclosed by a membrane assumes a shape with minimum bending energy, subject to constraints on the membrane surface area and capsule volume. The basic formulation can be extended to account for chemical asymmetries in the bilayer and for changes in the molecular surface density due to bending [83]. The variational formulation has the advantage of being able to predict the static stability of computed equilibria by considering the second variation in energy around the equilibrium shapes. However, it is unable to describe the stress resultants and bending moments developing over the deformed membrane, and implies a unknown set of constitutive equations pertinent to the properties of the membrane. More important, the variational formulation relies on a somewhat arbitrary and seemingly unphysical choice for the bending energy functional. Indeed, a noticeable weakness of the Helfrich form (2.5.12) is apparent at a saddle point where the mean curvature vanishes. If the spontaneous curvature is zero, the integrand of the first term in (2.5.12) predicts zero contribution to the bending energy, which seems physically implausible. Notwithstanding this limitation, numerous papers have adopted (2.5.12) as a point of departure in the study of equilibrium shapes (e.g., [83]). To establish the shape equation governing axisymmetric equilibrium shapes, as illustrated in Figure 2.2.4, previous authors added to the right-hand side of (2.5.12) appropriate Lagrange multipliers expressing constraints on the surface area and internal volume, and invoked the Euler-Lagrange equations of variational calculus to derive a first-order system of ordinary differential equations with appropriate boundary conditions [82]. This approach was criticized by subsequent authors who argued that the earlier work had not correctly allowed for variations in the meridional arc length [45]. In terms of the angle subtended between the tangent to the cell contour in a meridional plane and the axis of symmetry, > , the equilibrium shapes are described by the corrected equation

 " >  > dd > dd>
  > dd >

  " >  > d> 
 >   > dd>  d

0  0 " > " >   >
  > d>   # d 
  " > 0 " >
" >
" >  >  # #  





>

d > d 


















































(2.11.1)






which is identical to equation (7) presented in [45] if only the apparently misprinted term "
>? is replaced by the corrected term "  >? . The constant 

Membrane theory for capsules and cells

91

acts as a Lagrange multiplier and has no apparent physical interpretation except in the case of a spherical shell where it represents half the uniform in-plane tension. To permit comparison with an alternative formulation based on shell theory, we recast (2.11.1) in the equivalent form

   
 
    

 0
  #  #  ¼



¼¼









 0
 0   
  #       


¼











¼








(2.11.2)

where  > ¼ is the meridional curvature,  is the azimuthal curvature, and a prime denotes a derivative with respect to meridional arc length . The governing equation derived by Seifert et al. [82] is different from that shown in (2.11.1). Naito & Okuda [62] verified that the two equations are distinct by showing that an exact solution of (2.11.1) does not fit the earlier formulation. In later work, Zheng & Liu [99] noted that equation (2.11.1) can be written as

 d   >  d

(2.11.3)

where






" > d> 
 > d> " >  d d  
 " > " > 0   " >  #  >   > #  >

>

d
> d
















(2.11.4)

Integrating once, we obtain

 >



(2.11.5)

where  is a constant of integration. However, this argument is incomplete – because the left-hand side of (2.11.3) is allowed to become infinite at , it can be set equal to a delta function Æ  , in which case the function within the square brackets in (2.11.3) does not assume the simple form suggested by (2.11.5), but reduces instead to a generalized Green’s function. Consequently, for cells that are topologically equivalent to the sphere, Zheng & Liu’s argument that  may be set to zero in (2.11.5) leaving  as the appropriate equation for such shapes is inaccurate. J¨ulicher & Seifert [46] attempted to clarify the confusion over the correct shape equation and refute the criticism. Working with the usual bending energy functional (2.5.12), they now derived the same equation as (2.11.1), and indicated that in the  in special case of cells with spherical topology, this equation reduces to agreement with Zheng & Liu’s proposal. Unfortunately, for the same reasons given above, this argument is also erroneous, and the assertion that the earlier work is correct is itself unfounded. It follows that the corresponding results reviewed in [83] are also born from a misconceived set of equations.

92

Capsules and Cells

Additional evidence as to why the whole or part of the earlier reasoning [46, 82, 99], is fallacious emerges by allowing to become large. This limit permits comparison with well-known equilibrium equations for a two-dimensional shell. In this eventuality, we see that equations (3.5a) through (3.5d) of [82], as well as the equivalent equation , all yield 
  at leading order, which is nonsensical. On the other hand, it may be confirmed that, as  , (2.11.1) reduces to



¼¼
 

0





(2.11.6)


where 0 0
 and

 # , which is precisely equation (2.8.20)
 describing the deformation of a two-dimensional shell with a circular undeformed shape. The fact that  does not produce the same relation in the limit of large is the result of the misguided step (2.11.3). In light of this asymptotic check, it seems in all likelihood that the correct shape equation derived from the variational approach based on the Helfrich functional is equation (2.11.1). Now, a sphere of radius + and curvature + + is always a solution of the correct equation (2.11.1) provided that

+ 0
  +

 

+ 0







#

#

(2.11.7)

By establishing the vanishing points of the second variation of the bending energy functional (2.5.12), Zhong-Can & Helfrich [100] showed that this solution loses stability to nonspherical shapes at the discrete points




 66


+ # 

0 +

(2.11.8)

for any integer 6. Thus, as the pressure outside a spherical capsule is increased, the membrane buckles at a sequence of critical points corresponding to increasingly high-order axisymmetric modes. A variety of shapes have been computed based on variational formulation using asymptotic and numerical methods. Unfortunately, the majority of these studies are based on erroneous derivations, as discussed previously in this section. Solution branches based on the correct equations were recently computed recently by Blyth & Pozrikidis [13].

2.11.2 Virtual displacements A generalization of the Helfrich energy functional shown in (2.5.12) allows us to express the instantaneous configurational energy of a membrane consisting of a symmetric bilayer in terms of two surface integrals in the form



-




 #

-


 

(2.11.9)

Membrane theory for capsules and cells

93

where  is a position-dependent in-plane tension developing to ensure membrane incompressibility, # is the modulus of bending, and  is the membrane mean curvature [83]. Let the instantaneous shape of the membrane be described by an equation of the form @
 , where @ is a suitable function, and express the membrane energy in  @
 evaluated at @
 , where   is a nonlinear integrothe form  differential functional defined over all possible membrane configurations. The membrane load  may be found in terms of   @
 using the principle of virtual displacements, as follows (e.g., [63]). Consider an infinitesimal incremental deformation of the membrane from the current configuration # , inducing the infinitesimal energy variation Æ . The principle of virtual displacements provides us with an integral equation of the first kind for the membrane load or hydrodynamic traction discontinuity,

Æ



  Æ




(2.11.10)

Assume, for illustration, that the membrane energy is proportional to the instantaneous surface area multiplied by a constant and uniform surface tension  ,







 

(2.11.11)

Using elementary differential geometry, we find

Æ

Æ














  Æ




(2.11.12)

Comparing the right-hand sides of (2.11.10) and (2.11.12), we derive the well-known relationship for the jump in traction across an interface with constant surface tension,    , expressing Laplace’s law. Kraus et al. [48] and Kern & Fourcade [47] discretized the membrane of a vesicle into flat triangles defined by computational nodes, and represented the flow due to the membrane deformation by a superposition of elementary flows induced by point forces located at the vertices. Applying the principle of virtual displacements to the discrete model, they computed the strength of the point force located at the A th node,
 , as

$  

   

(2.11.13)

which is the differential statement of (2.11.10). Although computationally convenient, discrete models are sensitive to the particulars of surface discretization – flat versus curved triangulation. In a related effort, Boey et al. [15, 27] (see also Reference [84]) developed a coarse-grained molecular model that permits the direct coupling of classical hydrodynamics and the dynamics of the molecular layers and networks comprising the membrane, in a manner that circumvents the explicit use of a macroscopic constitutive equation.

94

Capsules and Cells

2.12 Numerical simulations of flow-induced deformation A number of authors have studied the flow-induced deformation and rheological properties of suspensions of capsules enclosed by elastic membranes, using analytical and numerical methods for homogeneous and wall-bounded flow. The pioneering theoretical investigation of Barth`es-Biesel [4] and Barth`es-Biesel & Rallison [7, 8] first illustrated the effect of interfacial elasticity on the capsule deformation and its significance on the rheology of dilute suspensions in linear shear flow, for small deformations from the spherical unstressed shape. Subsequently, Barth`es-Biesel & Sgaier [9] investigated the effect of interfacial viscosity. Numerical studies of moderate and large capsule deformations were conducted by Pozrikidis and coworkers [67, 68, 75], Eggleton & Popel [29], Barth`es-Biesel and coworkers [6, 25, 26, 51, 52], and Navot [63]. The effect of bending moments was included in the simulations of Pozrikidis [49, 71], Kraus et al. [48], and Kern & Fourcade [47]. Parallel laboratory studies were conducted by Chang & Olbricht [21, 22], and more recent observations were reported by Walter et al. [95]. Breyiannis & Pozrikidis [16] performed numerical simulations of the flow of nondilute suspensions of two-dimensional capsules, and found that the rheological properties are intermediate of those of suspensions or rigid particles and deformable liquid drops. The numerical studies reviewed in the previous paragraph were conducted using the boundary-integral method for Stokes flow (e.g., [72]) or Peskin’s immersed interface method [80] (see Section 3.3). In both cases, the membrane is discretized into a network of surface elements defined by a collection of surface nodes. As an example, the illustrations on the cover of this book depict the instantaneous shape of a deforming capsule with biconcave resting shape, evolving under the action of a simple shear flow directed from left to right [71]. The mathematical formulation incorporates in-plane elastic tensions and bending moments. This work was supported by a grant provided by the National Science Foundation.

Membrane theory for capsules and cells

95

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[15] B OEY, S. K., B EAL , D. H., & D ISCHER , D. E., 1998, Simulations of the erythrocyte cytoskeleton at large deformation; I microscopic models, Biophys. J., 75, 1573-1583. [16] B REYIANNIS , G. & P OZRIKIDIS , C., 2000, Simple shear flow of suspensions of elastic capsules, Theor. Comp. Fluid Dyn., 13, 327-347. [17] B UDIANSKY, B., 1968, Notes on nonlinear shell theory, J. Appl. Mech., 35, 393-401. [18] B UDIANSKY, B. & S ANDERS , J R ., J. L., 1963, On the “best” first-order linear shell theory, Progress in Applied Mechanics, Prager Anniversary Volume, pp. 129-140, McMillan, New York. [19] C AI , W. & L UBENSKY, T. C., 1995, Hydrodynamics and dynamic fluctuations of fluid membranes, Phys. Rev. E, 52, 4252-4266. [20] C ANHAM , P. B., 1970, The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell, J. Theor. Biology, 26, 61-81. [21] C HANG , K. S. & O LBRICHT, W. L., 1993, Experimental studies of the deformation of a synthetic capsule in extensional flow, J. Fluid Mech., 250, 587-608. [22] C HANG , K. S. & O LBRICHT, W. L., 1993, Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow, J. Fluid Mech., 250, 609-633. [23] C ORNELIUSSEN , A. H. & S HIELD , R. T., 1961, Finite deformation of elastic membranes with application to the stability of an inflated and extended tube, Arch. Rational Mech. Anal., 7, 273-304. [24] D EFAY, R. & P RIGOGINE , I., 1966, Surface Tension and Adsorption, Wiley, New York. [25] D IAZ , A., BARTH E` S -B IESEL , D., & P ELEKASIS , N., 2001, Effect of membrane viscosity on the dynamic response of an axisymmetric capsule, Phys. Fluids, 13, 3835-3838. [26] D IAZ , A., P ELEKASIS N., & BARTH E` S -B IESEL , D., 2000, Transient response of a capsule subjected to varying flow conditions: Effect of internal fluid viscosity and membrane elasticity, Phys. Fluids, 12, 948-957. [27] D ISCHER , D. E., B OAL , D. H., & B OEY, S. K., 1998, Simulations of the erythrocyte cytoskeleton at large deformation; II micropipette aspiration, Biophys. J., 75, 1584-1597. [28] E DWARDS , D. A., B RENNER , H., & WASAN , D. T., 1991, Interfacial Transport Processes and Rheology, Butterworth Heinemann, Boston. [29] E GGLETON , C. D. & P OPEL , A. S., 1998, Large deformation of red blood cell ghosts in simple shear flow, Phys. Fluids, 10, 1834-1845.

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[30] E VANS , E. A. & F UNG , Y. C., 1972, Improved measurements of the erythrocyte geometry, Macrivasc. Res., 4, 335-347. [31] E VANS , E. A. & S KALAK , R., 1980, Mechanics and Thermodynamics of Biomembranes, CRC Press, Boca Raton. [32] E VANS , E.A. & Y EUNG , A., 1994, Hidden dynamics in rapid changes of bilayer shape, Chem. Phys. Lipids, 73, 39-56. [33] F LAHERTY, J. E, K ELLER , J. B., & RUBINOW, S. I., 1972, Post-buckling behavior of elastic tubes and rings with opposite sides in contact, SIAM J. Appl. Math., 23, 446-455. ¨ , W., 1973, Stresses in Shells, Springer-Verlag, Berlin. [34] F L UGGE [35] F UNG , Y. C., 1965, Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs. [36] F UNG , Y. C., 1984, Biodynamics: Circulation, Springer-Verlag, New York. [37] G REEN , A. E. & A DKINS , J. E., 1970, Large Elastic Deformations, Second Edition, Clarendon Press, Oxford. [38] G REEN , A. E. & Z ERNA , W., 1968, Theoretical Elasticity, Dover, New York. [39] G URTIN , M. E. & M URDOCH , A. I., 1975, A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 291-323. [40] H ANSEN , J. C., S KALAK , R., C HIEN , S., & H OGER , A., 1996, An elastic network model based on the structure of the red blood cell membrane skeleton, Biophys. J., 70, 146-166. [41] H ANSEN , J. C., S KALAK , R., C HIEN , S., & H OGER , A., 1997, Influence of network topology on the elasticity of the red blood cell membrane skeleton, Biophys. J. 72, 2369-2381. [42] H ANSEN , J. C., S KALAK , R., C HIEN , S., & H OGER , A., 1997, Spectrin properties and the elasticity of the red blood cell membrane skeleton, Biorheology, 34, 327-348. [43] H ARRIS , R. A. & H EARST, J. E., 1966, On polymer dynamics, J. Chem. Phys., 44, 2595-2601. [44] H ELFRICH , W., 1973, Elastic properties of lipid bilayers: Theory and possible experiments, Naturforsch, 28, 693-703. [45] J IAN -G UO , H. & Z HONG -C AN , O-Y., 1993, Shape equations of the axisymmetric vesicles, Physical Review E, 47, 461-467. ¨ , F. & S EIFERT, U., 1994, Shape equations for axisymmetric vesi[46] J ULICHER cles: A clarification, Phys. Rev. E, 49, 4728-4731. [47] K ERN , N. & F OURCADE , B., 1999, Vesicles in linearly forced motion, Europhys. Lett., 46(2), 262-267.

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[48] K RAUS , M., W INTZ , W., S EIFERT, U., & L IPOWSKY, R., 1996, Fluid vesicles in shear flow, Phys. Rev. Lett., 77, 3685-3688. [49] K WAK , S. & P OZRIKIDIS , C., 2001, Effect of bending stiffness on the deformation of liquid capsules in uniaxial extensional flow, Phys. Fluids, 13(5), 1234-1242. [50] L E D RET, H. & R AOULT, A., 1995, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl., 74, 549-578. [51] L EYRAT-M AURIN , A. & BARTH E´ S -B IESEL , D., 1994, Motion of a deformable capsule through a hyperbolic constriction, J. Fluid Mech., 279, 135163. [52] L I , X. Z., BARTH E´ S -B IESEL , D., & H ELMY, A., 1988, Large deformations and burst of a capsule freely suspended in elongational flow, J. Fluid Mech. 187, 179-196. [53] L IBAI , A. & S IMMONDS , J. D., 1998, The Nonlinear Theory of Elastic Shells, Cambridge University Press, Cambridge. [54] L IPOWSKY, R., 1991, The conformation of membranes, Nature, 349, 475-481. [55] L IVERPOOL , T. B. & E DWARDS , S. F., 1995, Dynamics of a meandering river, Phys. Rev. Lett., 75, 3016-3019. [56] M C D ONALD , P., 1996, Continuum Mechanics, PWS, Boston. [57] M ILLMAN R. S. & PARKER , G. D., 1977, Elements of Differential Geometry, Prentice-Hall, New Jersey. [58] M OHANDAS , N. & E VANS , E., 1994, Mechanical properties of the red cell membrane in relation to molecular structure and genetic defect, Annu. Rev. Biophys. Biomol. Struct., 23, 787-818. [59] M ØLLMANN , H., 1981, Introduction to the Theory of Thin Shells, John Wiley & Sons, New York. [60] NADIM , A., 1996, A concise introduction to surface rheology with applications to dilute emulsions of viscous drops, Chem. Eng. Comm., 148-150, 391407. [61] NAGHDI , P. M., 1972, Theory of shells and plates, In: Handbuch der Physik Vol. Vla/2, C. Truesdell (Edt.), pp. 435-640, Springer, Berlin. [62] NAITO , H. & O KUDA , M., 1993, Counter-example to some shape equations for axisymmetric vesicles, Physical Review E, 48, 2304-2307. [63] NAVOT, Y., 1998, Elastic membranes in viscous shear flow, Phys. Fluids, 10, 1819-1833. [64] O GDEN , R. W., 1984, Non-linear elastic deformations, Dover, New York.

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[65] P EARSON , J. R. A., 1981, Wider horizons for fluid mechanics, Ann. Rev. Fluid Mech., 106, 229-244. [66] P OZRIKIDIS , C., 1990, The axisymmetric deformation of a red blood cell in uniaxial straining flow, J. Fluid Mech., 216, 231-254. [67] P OZRIKIDIS , C., 1994, Effects of surface viscosity on the deformation of liquid drops and the rheology of dilute emulsions in simple shearing flow, J. NonNewt. Fluid Mech., 51, 161-178. [68] P OZRIKIDIS , C., 1995, Finite deformation of liquid capsules enclosed by elastic membranes in simple shear flow, J. Fluid Mech., 297, 123-152. [69] P OZRIKIDIS , C., 1997, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, New York. [70] P OZRIKIDIS , C., 1998, Numerical Computation in Science and Engineering, Oxford University Press, New York. [71] P OZRIKIDIS , C., 2001, Effect of bending stiffness on the deformation of liquid capsules in simple shear flow, J. Fluid Mech., 440, 269-291. [72] P OZRIKIDIS , C., 2001, Interfacial dynamics for Stokes flow, J. Comp. Phys., 169, 250-301. [73] P OZRIKIDIS , C., 2002, Buckling and collapse of open and closed cylindrical shells, J. Eng. Math., 42, 157-180. [74] P OZRIKIDIS , C., 2003, Deformed shapes of axisymmetric capsules enclosed by elastic membranes, J. Eng. Math., 45, 169-182. [75] R AMANUJAN , S. & P OZRIKIDIS , C., 1998, Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: Large deformations and the effect of fluid viscosities, J. Fluid Mech., 361, 117-143. [76] R EISSNER , E., 1949, On the theory of thin elastic shells, In: Contributions to Applied Mechanics, H. Reissner Anniversary volume, pp. 231-247, J.W. Edwards, Ann Arbor. [77] R EISSNER , E., 1950, On axisymmetrical deformations of thin shells of revolution, proceedings, In: Third Symposium in Applied Mathematics, pp.27-52, McGraw Hill, New York. [78] R EISSNER , E., 1963, On the equations for finite symmetrical deflections of thin shells of revolution, In: Progress in Applied Mechanics, Prager Anniversary Volume, pp. 171-178, McMillan, New York. [79] R EISSNER , E., 1969, On finite symmetrical deflections of thin shells of revolution, J. Appl. Mech., 36, Trans. ASME 91, Series E, 267-270. [80] ROMA , A. M., P ESKIN , C. S., & B ERGER . M. J., 1999, An adaptive version of the immersed boundary method, J. Comp. Phys., 153, 509-534.

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[81] S CRIVEN , L. E., 1960, Dynamics of a fluid interface, Chem. Eng. Sci., 12, 803-814. [82] S EIFERT, U., B ERNDL , K., & L IPOWSKY, R., 1991, Shape transformations of vesicles: Phase diagrams for spontaneous-curvature and bilayer-coupling models, Physical Review A, 44, 1182-1202. [83] S EIFERT, U., 1997, Configurations of fluid membranes and vesicles, Adv. Phys., 46, 13-137. [84] S EIFERT, U., 1998, Modelling nonlinear red cell elasticity, Biophys. J., 75, 1141-1142. [85] S ECOMB , T. W. & S KALAK , R., 1982, Surface flow of viscoelastic membranes in viscous fluids, Q. J. Mech. Appl. Math., 35, 233-247. ¨ ZKAYA , N., & S KALAK , T. C., 1989, Biofluid mechanics, [86] S KALAK , R., O Ann. Rev. Fluid Mech., 21, 167-204. ¨ [87] S KALAK , R., T OZEREN , A., Z ARDA , P. R., & C HIEN , S., 1973, Strain energy function of red blood cell membranes, Biophys. J., 13, 245-264 [88] S LATTERY, J. C., 1990, Interfacial Transport Phenomena, Springer-Verlag, Berlin. [89] S TEIGMANN , D. J., 1999, Fluid films with curvature elasticity, Arch. Rat. Mech., 150, 127-152. [90] S TEIGMANN , D. J. & O GDEN , R. W., 1997, Plane deformations of elastic solids with intrinsic boundary elasticity, Proc. R. Soc. London A, 453, 853877. [91] S TEIGMANN , D. J. & O GDEN , R. W., 1999, Elastic surface substrate interactions, Proc. R. Soc. London A, 455, 437-474. [92] S TELITANO , D. & ROTHMAN , D., 2000, Fluctuations of elastic interfaces in fluids: Theory, lattice-Boltzmann model, and simulation, Phys. Rev. E, 62, 6667-6680. [93] U GURAL , A. C. & F ENSTER , S. K., 1975, Advanced Strength and Applied Elasticity, Elsevier, New York. [94] VALID , R., 1995, The Nonlinear Theory of Shells through Variational Principles: From Elementary Algebra to Differential Geometry, Wiley, New York. [95] WALTER , A., R EHAGE , H., & L EONHARD , H., 2000, Shear-induced deformations of polyamide microcapsules, Coll. Pol. Sci., 278, 169-175. [96] WAXMAN , A. M., 1984, Dynamics of a couple-stress fluid membrane, Stud. Appl. Math., 70, 63-86. [97] YAMAKI , N., 1984, Elastic Stability of Circular Cylindrical Shells, NorthHolland, New York.

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[98] Z ARDA , P. R., C HIEN , S., & S KALAK , S., 1977, Elastic deformations of red blood cells, J. Biomech., 10, 211-221. [99] Z HENG , W.-M. & L IU , J., 1993, Helfrich shape equation for axisymmetric vessels as a first integral, Phys. Rev. E, 48, 2856-2860. [100] Z HONG -C AN , O.-Y. & H ELFRICH , W., 1989, Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders, Phys. Rev. A, 39, 5280-5288. [101] Z HOU , H. & P OZRIKIDIS , C., 1995, Deformation of liquid capsules with incompressible interfaces in simple shear flow, J. Fluid Mech., 283, 175-200.

Chapter 3 Multi-scale modeling spanning from cell surface receptors to blood flow in arteries

N. N’Dri, W. Shyy, H. Liu, and R. Tran-Son-Tay Cell biomechanics and blood flow encompass elementary processes involving a broad range of length scales, as illustrated in Figure 3.1. Applications range from situations involving molecular-scale cell surface receptors, to blood flow through large vessels and in the microcirculation. For example, the adhesion of a leukocyte to the endothelium wall involves the deformation of a cell whose size is on the order of micrometers. In contrast, adhesion is mediated by bonds whose length is on the order of the nanometers. Blood flows from large arteries whose diameter is on the order of centimeters to capillaries whose diameter can be as small as a few micrometers. The importance of coupling the different length scales is underscored by observing that plaque developed in the carotid can have a significant effect on the flow in the whole circulation. When plaque develops, the heart is required to pump a higher volume of blood to supply the amount necessary for the normal function of the brain. In the particular situation of leukocyte adhesion to a substrate, we encounter dimensions ranging from micrometers ( m) associated with the size of the cell to nanometers (nm) associated with the size of receptors. An analysis of the adhesive behavior of the cells is desirable not only for describing microcirculatory flow dynamics, but also for understanding the cell function and behavior in health and disease. A successful model of cell adhesion must incorporate molecular and cellular information, and a successful model of large-scale blood flow must take into account the small vessels and the presence of cells and receptors in the capillaries. To simulate cell adhesion and blood flow in the arteries, a multi-scale modeling approach is required, as illustrated in Figure 3.2. In this chapter, a detailed account of a multi-scale technique that is capable of addressing the entire spectrum of haemodynamics in a computationally feasible framework is provided with particular reference to the adhesion kinetics of bonds and to cell motion, deformation, and recovery following deformation. The general approach divides the computational work into elementary individual but related tasks. In addressing the various modeling issues, an extended framework spanning from cell surface receptors to blood flow in arteries is developed. At the cellular level, a continuum approach is employed based on the field equations of momentum and mass transfer. Computational fluid dynamics (CFD) offers

103

104

Capsules and Cells

Figure 3.1 Cell biomechanics and blood flow encompass processes that involve a wide range of disparate length scales.

an arsenal of powerful tools for solving problems involving fluid flow, possibly combined with mass and energy transport. In recent years, these methods have been generalized and made capable of solving a variety of problems in haemodynamics including cardiovascular and capillary fluid flow, cellular physics, and adhesion dynamics. The application of CFD methods to the problem of cell adhesion and deformation will be reviewed in this chapter. At the receptor-ligand level, molecular bonds

Multi-scale modeling

105

Figure 3.2 Illustration of the multi-scale nature of haemodynamics modeling; model of cell adhesion (left), and three-dimensional and lumped flow model of the circulation (right).

are represented by springs whose association and dissociation is described by a reversible two-body kinetics model. Upward scaling allows us to incorporate the effect of the individual cells by means of statistical averaging performed on the macroscopic transport equations. Moving further up, a lumped model is employed to derive boundary conditions pertinent to the interface between different flow regimes. The lumped model enables us to study the significance of the resistance of small vessels on the flow of blood through the aortic arch. Changes in resistance are caused either by disease or by the presence of adhering leukocytes. Multi-scale modeling of blood flow through arteries provides us with an effective means of understanding the various physiological functions of the circulatory system and the patho-physiology of disease, including atherosclerosis, hypertension, and diabetes. For example, in large arteries with diameters ranging from 3 cm to 1 mm, a change in the dicrotic wave due to reflection indicates changes in the vessel wall and peripheral elasticity, while the absence of the dicrotic wave indicates that a person may suffer from diabetes or hypertension. In the second example, the multi-scale technique allows us to assess the effect of a reflected wave on the flow pattern through the aorta. Thus, severe artery stenoses (constrictions) can be diagnosed by measuring the pulse wave propagation and reflection in terms of blood pressure or heart sounds. As another example, we mention that the location of artery disorders (atherogenesis) has been shown to correlate with the spatial or temporal distribution of the shear stress exerted by the macroscopic blood flow on the artery vessel wall [20, 22, 29, 35], as well as with the microscopic mass transport occurring across the arterial endothelium and inside the vessel wall. A direct computation incorporating all relevant scales from large vessels to receptor-ligand interaction is beyond the reach of current and foreseeable computational resources. As an alternative, an effective overall model can be devised globally using the lumped one-dimensional

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approach to compute the pressure and flow rate at a given location, and locally using the three-dimensional field equations to analyze the detailed fluid pattern in a given artery.

3.1 Cell adhesion Studying the effect of leukocyte adhesion to the walls of microvessels is imperative for understanding the significance of the hydrodynamic capillary resistance in the human circulation [48, 61, 70]. For example, adhesion of leukocytes to a blood vessel wall is recognized to be an important aspect of inflammation [34, 39], and has been shown to play an important role in the function of the immune system [43]. Cell adhesion involves receptors, cells, and vessels with respective dimensions on the order of nm, m, and cm. Knowledge of the effect of the adhesion parameters, cell viscosity, effective surface tension, and cell diameter is necessary for the optimal design of the particle size and receptor density distribution in drug manufacturing. The objective function is the residence time required for the drug to be effective. Rolling of leukocytes under physiological flow is the first step in the migration of cells toward an infection site. Adhesion of the cells to the substrate slows down but does not necessarily prevent the cell motion [2, 1]. Adhesion is mediated by a complex biochemical process involving receptor-ligand bond formation and dissociation. In particular, the initial adhesion is mediated by the P- and E-selectin found on the endothelium surface, and by the L-selectin found at the tip of leukocyte microvilli [25, 63]. Blood flow exerts a pulling hydrodynamic force on the adhesion bonds, which can shorten the adhesion bond lifetime or even extract receptor molecules from the cell surface. Early models of cell adhesion did not take into account the physical properties of the leukocytes [3, 13]. In recent years, these properties have been recognized to have a significant effect on the overall adhesion process [32]. Mathematical models of cell adhesion can be broadly classified into two categories according to whether equilibrium [3, 16, 17] or kinetics [3, 13, 26, 38] considerations are employed. The kinetics approach is better capable of handling the mechanics of cell adhesion and rolling. In this approach, formation and dissociation of bonds occur according to reverse and forward rate constants. For example, Hammer & Lauffenburger [26] used a kinetics model to study the effect of an external flow on cell adhesion. The cell was modeled as a rigid sphere, and the receptors on the surface of the sphere were assumed to diffuse and convect into the contact area, as reviewed in Reference [59]. The results showed that adhesion parameters including the reverse and forward reaction rates and the receptor number have a strong influence on the peeling of the cell from the substrate. Dembo et al. [13] developed an improved adhesion model based on earlier ideas put forward by Evans [17, 18] and Bell [3]. In this model, a pulling force is exerted on one end of a membrane attached to a wall, while the other end is held fixed.

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107

Subsequently, Cozens-Roberts et al. [12] incorporated a probabilistic formulation for the formation of bonds. Other authors used the probabilistic approach [27] as well as Monte Carlo simulations to study the adhesion process, as reviewed in Reference [11]. The basic model has been extended to account for the distribution of microvilli on the cell surface and to simulate the rolling and adhesion of a cell on a substrate under the action of a shear flow [25]. In this model, the cell is described as a hard sphere that is covered by adhesive springs representing microvilli. The binding and breakup of bonds and the distribution of the receptors at the tips of the microvilli are computed using a probabilistic approach. The model was used to investigate the significance of the number of receptors, density of ligands, rates of reaction between receptor and ligand, and stiffness of the receptor-ligand spring on the adhesion of the cell and on the process of peeling. However, in the past, the formulation did not take into consideration the deformability of the cells. This key issue will be addressed later in this chapter based on a recent computational investigation [46]. In particular, cell deformability will be shown to have an important effect on the magnitude of the adhesion forces. Shao & Hochmuth [55] used a micropipette suction technique to measure the adhesion bond force and study the formation of tethers from neutrophil membranes. A graph of the applied force against the tether velocity
reveals that a minimum force of 45 picoNewton (pN) is required for tether formation. The adhesion force increases linearly with respect to the rate of change of the tether length, defined as the tether velocity, as shown in Figure 3.1.1. Subsequently, Shao et al. [57] measured the static and dynamic length of the neutrophil microvilli. After adhering to the bead, the cell moves freely backward by a certain distance called the rebound length. Experiments showed that the rebound length is insensitive to the applied suction pressure, as shown in Figure 3.1.2. An explanation for this behavior is that the rebound length is, in fact, the natural or static tether length. After a microvillus has reached its natural length, it either extends under the influence of a small pulling force, or forms a tether under the influence of a high pulling force. Microvilli extension and tether formation were found to play an important role in the rolling of the cell on the endothelium by significantly affecting the magnitude of the bond force, as shown in Figure 3.1.2. The magnitude of the spring constant for microvillus extension was estimated by Shao et al. [57] under the assumption that a threshold force does not appear to be 43 pN/ m. At low Reynolds numbers, typical of blood flow through capillaries, inertial effects play a minor role in the motion of the fluid, and the solution for creeping flow past a sphere near a plane wall [23] can be used to estimate the force of a single bond, as depicted in Figure 3.1.3. Using elementary trigonometry, we find that the angle  determining the direction of the bond force is given by

  (3.1.1)      where  is the total length of the microvillus,  is the moment arm length, and  is 



  


the cell radius, as shown in Figure 3.1.3.

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Capsules and Cells 140 120 100

f (pN)

RBC Neutrophil NGC Anti-CD62L Anti-CD45 Anti-CD18

80 60 40 20 0 0

2

6

4

8

10

U t (mm/s) Figure 3.1.1 Dependence of the tether force on the tether velocity of a neutrophil membrane. (From Shao, J.-Y. & Hochmuth, R. M., 1996, Biophys. J., 71, 2892-2901. With permission from the Biophysical Society.)

Balancing now the forces exerted on the cell in the direction of the obtain




 



  

 axis, we (3.1.2)

where 
is the bond force defined as the force experienced by an adhesive bond,  is the hydrodynamic force, and is the imposed shear stress. A torque balance with respect to the point A shown in Figure 3.1.3 gives




  

   

    

(3.1.3)

where  is the torque imposed by the shear flow on the cell. Next, we assume that the bond force 
due to the extension of a microvillus is given by






 

(3.1.4)

where
is the bond spring constant, and  is the initial length of the microvillus; in the case of a neutrophil,
= 43 pN/ m [55]. In the process of tether formation, 
satisfies the balance equation




 

 



(3.1.5)

Multi-scale modeling

(a)

1.5

(b)

109

3 2.5

1

2 DD (mm) 1.5

0.5

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0.5 0

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2

2.5

3

0

0

0.5

1

1.5

2

2.5

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Time (s)

Figure 3.1.2 Motion of a neutrophil before and after a bond has been formed [57].  is the displacement of the neutrophil relative to the adhesion point before adhesion; after adhesion,  is the change of the microvillus length. The figure on the left shows the microvillus extension as a function of time; the slope of the dotted line is the velocity of the cell when it moves freely inside the micropipette under suction pressure of 0.5 pN/ m . The graph on the right illustrates the tether formation as a function of time; the slope of the dotted line is the velocity of the cell when it moves freely inside the micropipette under suction pressure of 1.0 pN/ m . (From Shao, J.-Y., Ping Ting-Beall, H., & Hochmuth, R. M., 1998, Proc. Natl. Acad. Sci., 95, 6797-6802. With permission from the National Academy of Sciences.)

Figure 3.1.3 Force balance on an adhered neutrophil with a single attachment;  is the total length of the microvillus,  is the moment arm length,  is the cell radius, 
is the bond force,  the hydrodynamic force, and  the shear stress exerted on the cell by the shear flow.

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Capsules and Cells 150

(a)

(b)

7 I L

6 5

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4 l&L (µm) 3 2 1

Fb (pN) 100 50

0

0

0 0

0.2

0.4

0.6

0.8

1

0.2

0.4

0.5

0.6

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Time (s)

Time (s)

Figure 3.1.4 Evolution of the bond force 
(left), and moment arm  and tether length  (right). (From Shao, J.-Y., Ping Ting-Beall, H., & Hochmuth, R. M., 1998, Proc. Natl. Acad. Sci., 95, 6797-6802. With permission from the National Academy of Sciences.)

where  is the initial bond force,   is the pulling velocity of the tether off the surface, and the coefficient is associated with changes of the spring constant in time. In the case of a neutrophil, we use the estimates  = 45 pN and = 11 pN s/ m [55]. The evolution of the bond force, 
, can be computed by solving the coupled system of governing equations (3.1.4) and (3.1.5). The predicted tether length  and moment arm  for shear stress of 0.08 pN/ m are plotted against time in Figure 3.1.4. The results show that the bond force decreases in time, and is reduced by 50% in 0.2 s before reaching a plateau. The moment arm and tether length increase rapidly over the first 0.2 s, and than at a slower rate. Shao & Hochmuth [56] studied the strength of anchorage of the trans-membrane receptors to the cytoskeleton; the latter are believed to be important in cell adhesion and migration. In their experiments, the micropipette suction method was used to apply a force to a human neutrophil adhering to a latex bead coated that is with CD626L (L-selectin), CD18 ( integrins), or CD45 antibodies, as shown in Figure 3.1.5. In particular, a human neutrophil was placed in a micropipette whose diameter is nearly equal to that of the cell, a latex bead 10  in diameter coated with antibodies was placed in a second micropipette, and suction pressure was applied to the neutrophil micropipette. The adhesion bond lifetime is defined as the time elapsed between the instant where the cell is pulled and the instant where the cell resumes its free motion. Figure 3.1.6 displays a graph of the pulling force  computed based on models developed in Reference [57], plotted against the bond lifetime . The data are well described by the exponential form



in the units shown in the figure.

 

 



(3.1.6)

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111

Figure 3.1.5 Microscopic view of the adhesion assay; the human neutrophil and bead position is shown in the top figure, and a schematic linkage of neutrophil adhering to an antibody-coated bead is shown at the bottom. (From Shao, J.-Y. & Hochmuth, R. M., 1999, Biophys. J., 77, 587-596. With permission from the Biophysical Society.)

Two mechanisms have been proposed for describing the rolling of cells over the endothelium. The first mechanism is based on intrinsic kinetics of bond dissociation in the absence of an external force, and the second mechanism is based on the concept of reactive compliance that takes into consideration the susceptibility of the dissociation reaction to an applied force. In the laboratory, leukocytes are observed to roll faster on L- than on E- or Pselectin. To determine which one of the two aforementioned mechanisms better describes this difference in behavior, Alon et al. [2] studied the kinetics of tethers and the mechanics of selectin-mediated rolling. Figure 3.1.7 illustrates the effect of shear

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Capsules and Cells

Figure 3.1.6 Dependence of the bond lifetime, , on the pulling force,  , for antiCD62L-coated beads (From Shao, J.-Y. & Hochmuth, R. M., 1999, Biophys. J., 77, 587-596. With permission from the Biophysical Society.)

Bell equation Bell equation  (s
)  (A ) L-selectin P-selectin E-selectin

                           

Spring model Spring model  (s
)  (N/m)

    – –

     – –

Table 3.1.1 Model parameters obtained using Bell’s equation, the spring model, and a linear model to fit the experimental data;  is the unstressed dissociation rate constant,  the reactive compliance,  the spring constant, and     , where   is the transition spring constant.




 in the figure), showing that the stress on the reverse reaction rate  (denoted as   value of  for the L-selectin is greater than that for the E- and P-selectins. Figure 3.1.7 also displays the best fit of the data to the spring model [13], the Bell equation model [3], and the linear model [2]. The predicted values of the reactive compliance,  , for the L-, P-, and E-selectin are found to be comparable, whereas those of the equilibrium reaction rate,  , for the L-selectin and the other two selectins show significant variations, as illustrated in Table 3.1.1.

Multi-scale modeling

113

Figure 3.1.7 Effect of shear stress and bond force (
) on the reverse reaction rate  , denoted in the text as . The solid and dashed lines represent the best    fit to existing models, as displayed in the legend. The squares correspond to the L-selectin, the circles correspond to the E-selectin, and the diamonds correspond to the P-selectin. (From Alon, R., Chen, S., Puri, K. D., Finger, E. B., & Springer, T. A., 1997, J. Cell Biol., 138, 1169-1180. With permission from Rockefeller University Press.)

Alon et al. [2] found that Bell’s equation and the spring model fit the data better than the linear relationship, and this suggests that  depends exponentially on 
. However, because the error incurred in the calculation of the force based on the assumption that the leukocyte behaves as a rigid body is on the order of 20%, the computed model constants must be regarded only as approximations. Alon et al. [1] also studied the kinetics of transient and rolling interactions of leukocytes with L-selectin immobilized on a substrate, and measured the rolling velocity for different values of ligand density consisting of L- and P-selectin. Selected results are presented in Figure 3.1.8. In the case of the L-selectin, increasing the ligand density reduces the rolling velocity. The rolling velocity over the P-selectin is smaller than that over the L-selectin, and there is a threshold in the shear stress, approximately equal to 0.4 dyn/cm , be-

114

Capsules and Cells 120 110

L-selectin 0.3 µg/ml

100

Rolling velocity (µm/s)

90

L-selectin 2 µg/ml

80

P-selectin 0.5 µg/ml

70 60 50 40 30 20 10 0

0

5 10 15 20 25 30 35 Wall shear stress (dyn/cm 2 )

40

Figure 3.1.8 Dependence of the rolling velocity of a leukocyte on the shear stress and ligand density. (From Alon, R., Chen, S., Fuhlbrigge, R., Puri K. D., & Springer, T. A., 1998, Proc. Natl. Acad. Sci., 95, 11631-11636. With permission from the National Academy of Sciences.)

low which rolling does not take place . Frame-by-frame observation shows that, in fact, leukocytes do not roll smoothly on the selectin substrate, but exhibit instead a jerky motion, as shown in Figure 3.1.9 [1]. Greenberg et al. [24] studied the interaction of microspheres coated with sialyl Lewisx and the rolling over E- and P-selectin substrates, and investigated the effect of ligand density on the rolling velocity. It was found that the rolling velocity increases as the applied shear stress is raised, and decreases as the ligand distribution becomes more dense. In the experiments, the mean rolling velocity was observed to lie between 25 and 225 m/sec. Smith et al. [62] found that the rolling velocity of a neutrophil over an immobilized L-selectin ranges from 50 to 125 m/sec, while the rolling velocity over the P- or E-selectin is 8 m/sec and 6 m/sec, respectively. Smith et al. [62] determined the effect of an external force on the duration of L-, P-, and E-selectin bonds using temporal resolution video microscopy in a parallel flow chamber. The goal was to measure and compare the dissociation rate constants,  , of bonds formed by the L-, P-, and E- selectin. In particular, the dissociation rate constants were determined from the distribution of pause times during leukocyte adhesion. Previously, Kaplanski et al. [34] had analyzed the pause times to quantify the effect of force on the bond lifetime. By comparing the pause times of neutrophils tethering on P-, E-, and L-selectin for estimated bond force ranging from 37 to 250 pN, Smith et al. [62] found that the pause times for neutrophils interacting with E- or P-selectin are significantly longer than those for the L-selectin. Figure 3.1.10 presents graphs of the pause times plotted against the shear stress. An increase in the shear stress reduces the pause time for all three selectins.

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115

Figure 3.1.9 Jerky motion of a leukocyte on a selectin substrate. (From Alon, R., Chen, S., Fuhlbrigge, R., Puri K. D., & Springer, T. A., 1998, Proc. Natl. Acad. Sci., 95, 11631-11636. With permission from the National Academy of Sciences.)

Figure 3.1.10 Effect of wall shear stress on the pause times of neutrophil tethers [62]: (A) P-selectin at 9 sites/ m , (B) E-selectin at 12 sites/ m , (C) Lselectin at 50 sites/ m . (From Smith, M. J., Berg, E. L., & Lawrence, M. B., 1999, Biophys. J., 77, 3371-3383. With permission from the Biophysical Society.)

Examining the dependence of the selectin dissociation rate constant,  , on the bond force, 
, allows us to compute the reaction compliance  and equilibrium reaction rate,  , using the Bell equation, as well as the parameters  and  associated with the spring model. Table 3.1.2 compares the results of Smith et al.

116

L-selectin [62] L-selectin [1] P-selectin [62] P-selectin [1] E-selectin [62] E-selectin [1]

Capsules and Cells Bell equation Bell equation  (s )  (A )

    7 - 9.7     0.93     0.5-0.7

                        

    

Spring model Spring model  s
)  (N/m)

        

    –

   –

             –      –

Table 3.1.2 Adhesion parameters computed using the Bell and Hookean spring models.

[62] with those Alon et al. [2, 1]. The first authors found that the models fit the experimental data for bond force smaller than 125 pN; for higher magnitudes, the models become inadequate. The differences observed in Table 3.1.2 can be attributed to the particulars of the experimental instrumentation. In the study of Smith et al. [62], high temporal and spatial resolution microscopy was used to capture previously inaccessible features [1, 2]. The dissociation constants for neutrophil tethering events at 250 pN/bond were found to be lower than those predicted by the Bell and Hookean spring models. The plateau observed in the graph of the shear stress versus the reaction rate  suggests a threshold above which the models become inadequate. Since leukocyte models used in various studies regard the cell as a rigid body, whether or not the plateau is due to molecular, mechanical, or cell deformation effects is not entirely clear. Jerky motion of leukocytes rolling over a selectin substrate was observed in several experiments (e.g., [9, 10]). Chen & Springer [10] analyzed the factors governing the formation of bonds between a cell moving freely over a substrate in shear flow, as well as the factors governing bond dissociation due to forces of hydrodynamic origin. It was found that bond formation is primarily determined by the shear rate, whereas bond breakage is primarily determined by the shear stress. The experimental data are well described by the Bell equation. Schmidtke & Diamond [54] studied the interaction of neutrophils with platelets and P-selectin in physiological flow using high-speed and high-resolution videomicroscopy. For wall shear rates ranging from 50 to 300 
, they observed an elongated tether being pulled out from the neutrophil causing the development of a tear drop shape, as depicted in Figure 3.1.11. The average length of the tether developing from a neutrophil interacting with a spread platelet was estimated to be    m, while the average tether lifetime was found to lie between 630 and 133 ms for shear rates ranging from 100 
for 
 pN, to 250 
for 
 pN, consistent with results of previous authors [3, 2, 59]. Figure 3.1.12 illustrates the dependence of the tether lifetime on the shear rate.



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117

Tether duration (sec)

Figure 3.1.11 Membrane tether and tear-drop shape deformation. (From Schmidtke, D. W. & Diamond, S. L., 2000, J. Cell Biol., 149, 719-729. With permission from Rockefeller University Press.)

1.2 1.0 0.8 0.6 0.4 (P<0.001)

0.2 0.0 0

50

100

150

200

250

300

Shear rate (s -1 ) Figure 3.1.12 Effect of wall shear rate on the bond lifetime;     marks significant changes in tether duration. (From Schmidtke, D. W. & Diamond, S. L., 2000, J. Cell Biol., 149, 719-729. With permission from Rockefeller University Press.)

Schmidtke & Diamond [54] also studied the rolling of neutrophils over P-selectin coated surfaces, and observed that, when a neutrophil rolls over the P-selectin, a tether develops as in the case of spread platelets, as depicted in Figure 3.1.13. The measured tether length and lifetime are     m and  s, respectively. Chang et al. [8] carried out numerical simulations of the initial tethering and rolling process based on Bell’s model, and constructed a state diagram for cell adhesion in viscous flow for an imposed shear rate of 100 
. This shear rate corre-





118

Capsules and Cells

Figure 3.1.13 Membrane tether developing on P-selectin-coated surfaces. (From Schmidtke, D. W. & Diamond, S. L., 2000, J. Cell Biol., 149, 719-729. With permission from Rockefeller University Press.)

sponds to the experimental value where rolling is observed to occur. To generate the state diagram, the ratio of the rolling velocity,  , to the hydrodynamic velocity of non-adherent cells translating near a plane wall,  , is computed as a function of the reverse reaction rate,  , for a given value of  , using the Bell equation










  


(3.1.7)

where  is the unstressed dissociation rate constant,
is Boltzmann’s constant,  is the absolute temperature,
 is the thermal energy,  is the reactive compliance, and  is the bond force. Other parameters of the simulations are listed in Table 3.1.3. Having prepared a graph of the reduced velocity  against the reaction rate  , we may identify conditions for  to have a certain value for a given constant  (denoted by  in Figure 3.1.14), and thereby infer the value of  , (denoted by  in Figure 3.1.14). The symbols in this figure correspond to experimental data obtained for different receptor-ligand pairs using the Bell equation, as shown in Table 3.1.4. The estimated values can be used to prepare a graph of  against  for a given ratio of  , and thereby identify four dynamic states of cell adhesion. In the no-adhesion regime, the cells move with a velocity that is greater than 95% the hydrodynamic velocity  . Rolling occurs when      , with the cells exhibiting fast or transient adhesion. Firm adhesion occurs when the rolling velocity is zero for a specified period of time.

Multi-scale modeling

Variable



        

119

Value Source

Cell radius 5.0 m Receptor radius 1.0 nm Number of receptors 25 000 Ligand density 3 600 cm Equilibrium bond length 20 nm Spring constant 100 dyne/cm Viscosity 0.01 g /(cm s) Shear rate 100 
Cut-off length for bond formation 40 nm Absolute temperature 310 Association rate 84 s


[6] [3] [7] [7] [3] [8]

Table 3.1.3 Parameter values used in the numerical simulations of cell rolling [8].

Receptor – Ligand E-selectin – neutrophil E-selectin –neutrophil P-selectin – neutrophil P-selectin – PSGL1 P-selectin – mutant PSGL-1 P-selectin – mutant PSGL-1 P-selectin – mutant PSGL-1 L-selectin – neutrophil L-selectin – neutrophil L-selectin – PSGL-1 L-selectin – mutant PSGL-1 L-selectin – mutant PSGL-1 L-selectin – mutant PSGL-1 PNAd – neutrophil PNAd – neutrophil

 (A )  
) Source 0.31 0.18 0.39 0.29 0.24 0.33 0.42 0.24 1.11 0.16 0.15 0.12 0.11 0.20 0.59

0.7 2.6 2.4 1.1 1.8 1.7 1.6 7.0 2.8 8.6 12.7 17.3 18.3 6.8 3.8

[2] [62] [62] [52] [52] [52] [52] [1] [62] [52] [52] [52] [52] [2] [62]

Table 3.1.4 Bell model parameters obtained by different experimental methods for different receptor-ligand pairs.

Adhesion is observed at high values of  and low values of  , as indicated by the wide margin between the no- and firm-adhesion zones in Figure 3.1.14. As  increases,  must decrease for adhesion to take place. In the simulations, the association rate  and wall shear rate are kept constant. Varying  does not alter

120

Capsules and Cells

Figure 3.1.14 State diagram for cell adhesion showing four possible states. The dotted curve corresponds to rolling velocity of   and the dashed curve corresponds to rolling velocity of   . The dots correspond to the data listed in Table 3.1.2 for E-selectin (squares, [2, 62]), P-selectin (triangles, [52, 62]), and L-selectin (diamonds, [1, 8, 52]. (From Chang, K. C., Tees, D. F. J., & Hammer, D. A., 2000, Proc. Natl. Acad. Sci., 97, 11262-11267. With permission from the National Academy of Sciences.)

the topology of the phase diagram, and only shifts the location of the rolling envelope in the   plane. The shear rate chosen by Chang et al. [8] lies in the range of the physiological flow for post-capillary venules, which is between 30 and 400 s
. As the shear rate increases, a sudden change occurs from firm to no adhesion without rolling motion. Figure 3.1.15 shows results of a typical simulation, and Figure 3.1.16 illustrates the effect of bond elasticity. For values of  less than 0.1 and high reverse rate constants  , the rolling velocity is insensitive to the spring constant. Tees et al. [66] adapted the approach of Chang et al. [8] to study the effect of cell size. Simulations were performed for three different spherical cells of radius 5.0, 3.75, and 2.5 m, and shear rate G = 100 s
; The spring constant and association rate constant were kept at fixed values, while either the number of receptors over the cells was held constant so that the receptor density varies with particle size, or the receptor density was held constant so that the number of receptors changes with particle size. The results were used to generate a state diagram similar to that presented in the earlier study [8], for  = 0.95, 0.5, 0.3, 0.1, 0.05, and 0.0. For high values of  , as  increases up to 0.1 AÆ , the graph of  against  remains essentially horizontal, and then plunges toward the  axis. The effect of the equilibrium reverse




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121

Figure 3.1.15 State diagram of cell adhesion for shear rate ranging in the range 30 to 400 s
. The dotted lines represent the boundaries of the rolling state for shear rate G = 100 s
. The rolling adhesion area is the regime where rolling occurs for shear rates in the range 30 to 400 s
. As the applied shear rate is raised from 100 to 400 s
in the bimodal adhesion regime, the cells display firm adhesion or no-adhesion without rolling motion. (From Chang, K. C., Tees, D. F. J., & Hammer, D. A., 2000, Proc. Natl. Acad. Sci., 97, 11262-11267. With permission from the National Academy of Sciences.)

reaction rate constant,  , on the rolling velocity is illustrated on the left diagram of Figure 3.1.17, corresponding to  = 1 AÆ . At small values of  , the cells adhere firmly without rolling. For  in the range 10 - 100 s
, the cells roll with the hydrodynamic velocity  regardless of the cell size. However, as  is reduced, the threshold where the cells start rolling occurs at higher values of  , as shown on the right diagram of Figure 3.1.17. Tees et al. [66] predicted that, as the cells become larger the rolling velocity becomes faster, in agreement with experimental observations [58]. In these studies, the Bell model was used to generate the state diagram. However, similar results are obtained by using the spring model [13]. In other studies [14, 15], the cell was modeled as a liquid drop encapsulated by an elastic ring, and the effect of the deformability and adhesion parameters on the interaction between the leukocyte and endothelium in shear flow was investigated. In the theoretical model, only a small portion of the adhesion length is allowed to peel off from the vessel wall, which does not seem physically sound. A more sophisticated model is required to provide a realistic description of the rolling process.

122

Capsules and Cells 10 4

s = 200 dyne cm-1

10 4

k ro (s -1 )

10 0 10 -2 10 -4 10 -6 10 -8 0.01

s = 200 dyne cm-1

0.1

1

g (Å)

Figure 3.1.16 Boundaries of the rolling states for two different spring constants,   and 200 dyn /cm. (From Chang, K. C., Tees, D. F. J., & Hammer, D. A., 2000, Proc. Natl. Acad. Sci., 97, 11262-11267. With permission from the National Academy of Sciences.)

3.2 Arterial blood flow The cardiovascular or circulatory system is composed of the pulmonary circuit and the systemic circuit. The former is responsible for circulating blood between the heart and the lungs, while the latter is responsible for supplying blood to the part of the body not serviced by the pulmonary circuit. Blood transports oxygen and nutrients to the body tissues, carries carbon dioxide from tissues to the lung, and removes waste produced by tissue cells to the kidneys for excretion. Other services provided by blood flow include mediating the body immune response, controlling fluid losses at injury sites, and regulating the body temperature. When the left ventricle of the heart starts pumping blood at the beginning of the cardiac ejection phase (systole), the pressure rises near the entrance of the aorta, as indicated by the curve labeled “0 cm” in Figure 3.2.1, and a certain volume of blood is ejected. Because of the blood vessel elasticity, the rise in pressure locally distends the vessel. As the distended segment starts contracting, blood accelerates a short distance downstream, thereby generating a propagating pulse wave. Further downstream, the motion is determined by the elasticity of the vessel wall and the inertia of blood flow. Certain salient features of the pressure wave and associated flow rate can be identified by interrogating the pressure measurements:

Multi-scale modeling

123

Figure 3.1.17 Effect of  on the reduced rolling velocity  for  = 1 AÆ (left) and 0.3 AÆ (right), cell radius 5 m (squares), 3.75 m (circles), and 2.5 m (triangles) and fixed number of receptors,  . (From Tees, D. F. J., Chang, K. C., Rodgers, S. D., & Hammer, D. A., 2002, Ind. Eng. Chem. Res., 41, 486-493. With permission from the American Chemical Society.)

1. The pressure wave propagates at the approximate speed of 5 m/s in the larger arteries. The wave amplitude increases away from the heart toward the periphery due to the development of a reflected wave. 2. The steepness of the pressure profile increases from the heart to the periphery due to changes in the wave-propagation velocity. The higher the wave pressure, the higher the wave speed. 3. The waveform of the flow velocity is different from that of the pressure. The amplitude of the wave velocity decreases from the heart to the periphery because of viscous resistance in the arteries. Theoretical and computational models can be developed to explain the propagation of the pressure and velocity waves. Blood flow in the large arteries has been shown to exhibit a rich variety of unsteady vortical flows, as illustrated by in vivo visualization of normal flow patterns in the thoracic aorta using four-dimensional magnetic resonance imaging (MRI), displayed in Figure 3.2.2 [5]. Such intricate flow patterns are commonly in the arterial system where the Reynolds number ranges from 10 to 10 . Dynamic blood flow pulsation, geometrical complexity of the vessel architecture, variations in the vessel cross-sectional area, vessel curvature and spatial conformation, are all significant in determining the dynamics of blood flow.

124

Capsules and Cells

Figure 3.2.1 Instantaneous blood pressure record at different sites in the arteries of a dog [44]. (From McDonald, D. A., 1974, Blood Flow in Arteries. With permission from Edward Arnold, London.)

Figure 3.2.2 Sagittal oblique four-dimensional reconstruction of the thoracic aorta blood flow displayed in two dimensions with velocities (left) and time of one heartbeat (right) of a 30-year-old normal male. (From Bogren, H. G. & Buonocore, M. H., 1999, J. Magn. Reson. Imag., 10, 861-869. With permission from Academic Press.)

Blood flow characteristics provide us with valuable diagnostics for detecting cardiovascular disease. For example, the wall shear stress has been found to play an important role in the onset and development of arteriosclerosis [20, 22, 36, 42, 45]. Systematic studies of the influence of the vessel geometry, waveform of the pulsating

Multi-scale modeling

125

Figure 3.2.3 Isovelocity surfaces and velocity vector fields in an aortic arch model. (From Liu, H., 2000, Fluids Engineering Division, FED, 253, 497-502. With permission from ASME.)

flow, swirling inlet boundary condition, and wall shear stress are of great importance in understanding and treating cardiovascular disease. The effect of geometry of blood vessels on blood flow has received considerable attention. Studies have shown that unsteady vortical flows are notably sensitive to the vessel curvature [36], nonplanarity expressed by the torsion [45], and variations in the vessel geometry such as those occurring in stenoses [42]. For example, unsteady vortex flow can develop in an asymmetric channel with a relatively simple geometry, causing significant fluctuations in the wall shear stress [42]. It was mentioned earlier that a main objective in the study of blood flow through arteries is to describe the temporal and spatial variation of the wall shear stress. Thanks to recent developments in computer-aided reconstruction techniques for faithfully generating the geometry of blood vessels from medical-imaging sources of computer tomography (CT) and magnetic resonance imaging (MRI), realistic geometric blood vessel models can today be utilized in theoretical and laboratory studies [21]. Most studies in vivo show the development of helical flow in the aorta, even in the ascending aorta. Liu [41] studied blood flow in the distorted aorta arch by imposing a swirling flow as an inlet boundary condition, and concluded that the observed helical flow in the arch is due to the swirling motion rather than to the aorta curvature and geometrical torsion. Figure 3.2.3 displays computed velocity vector fields of the secondary flow developing in the aortic arch at three time instants during a pulsating cycle, for a Reynolds number of 2000 and a Womersley number of 20. During early systole when the flow accelerates, the highest axial velocity occurs along the inner short path, while the influence of the swirling inflow on the overall flow pattern appears to be negligible. Approaching the mid-systole, the flow rapidly strengthens, and the fluid tends to migrate outward under the action of a developing secondary helical flow. This helical flow persists after the systole phase. The results of the simulation reveal features of

126

Capsules and Cells

Figure 3.2.4 The waveform of the physiological volumetric flow rate corresponding to fixed systolic and diastolic forms varies with systole-to-diastole time ratio,    . The illustration shows four waveforms for   (A), 1.0 (B), 0.33 (D), 0.2 (F), and 0.1 (G) [42]. (From Liu, H. & Yamaguchi, T., 2001, J. Biomech. Eng., 123, 88-96. With permission from the ASME.)

the helical flow in space and time that are similar to those observed in the laboratory using MRI techniques [35]. The effect of aorta dynamics on the vortical flow has been studied by a combination computational fluid dynamics (CFD) and MRI methods [40]. A helical flow was found to develop in the ascending aorta during late systole and most of the diastole. This feature has been further confirmed by the CFD study of unsteady flow in a human aortic arch with bifurcations [29]. Zhao et al. [73] compared pulsating flow in a realistic compliant human carotid bifurcation with corresponding flow through a rigid tube, and noted that quantitative differences exist in the wall shear stress distribution. Specifically, the wall shear stress in a compliant vessel is significantly reduced, and the region of reversed flow is larger and occurs over a longer period of time with respect to those in a rigid vessel. Studies of blood flow in large arteries have addressed the geometrical effects of curvature, non-planarity, and bifurcation, as well as inflow effects including waveform and three-dimensional velocity profiles. Because of difficulties in obtaining and interpreting non-invasive blood flow measurements, the influence of the threedimensional in-flow spatial structure has received only scant attention.

Multi-scale modeling

127

With regard to the important effect of pulsation on the structure of the vortical flow, a number of issues remain unclear. Even one-dimensional pulsating blood flow can show significant differences among different parts of arterial system and at different instants. Moreover, the dynamics is affected by a person’s physiological or pathological condition and state of activity such as still condition or exercise. Liu et al. [42] studied the effect of pulsating flow on the vortical fluid dynamics in a two-dimensional stenosed channel, and found that the inflow waveforms could significantly affect the vortical flow downstream; results are shown in Figures 3.2.4 and 3.2.5. The generation and development of vortex waves may be linked to the onset of an adverse pressure gradient along the systolic waveform.

Figure 3.2.5 Isovelocity contours for Reynolds number  = 750 and Strouhal number ! =0.024 at point " of the waveforms with  =2.0 (A), 1.0 (B), 0.33 (D), 0.2 (F), and 0.1 (G); the magnitude of the velocity increases from dark to gray. (From Liu, H. & Yamaguchi, T., 2001, J. Biomech. Eng., 123, 88-96. With permission from the ASME.)

3.3 Immersed boundary method To numerically simulate the deformation of a leukocyte, we may regard it as a liquid capsule suspended in an ambient viscous fluid. The motion of the fluid inside

128

Capsules and Cells

and outside the capsule is governed by the continuity equation expressing mass con, and the Navier-Stokes equation applicable to incompressible servation Newtonian fluids,



#

$ $




 


%   


(3.3.1)

where is the velocity, # is the density, is the viscosity, % is the pressure, and
is a distributed body force (e.g., [50]). At the interface, we require continuity of velocity 




 



(3.3.2)

and the dynamic condition



    

 & 

(3.3.3)

where the superscripts 1 and 2 denote, respectively, the ambient and capsule fluid;  is the Newtonian stress tensor,  is the unit vector normal to the interface pointing outward, and & is the curvature of the interface in the ' plane. The surface tension  is assumed to be uniform over the interface. The system of governing equations is solved on a fixed Cartesian grid, with the interface moving through the stationary mesh. The advantage of this mixed EulerianLagrangian approach is that the mesh topology remains simple, while large interfacial distortion is allowed to take place. Two methods are available for tracking the position of the interface: the immersed boundary method (IBM) developed by Peskin [49] to simulate blood flow in the heart and subsequently extended by others to various problems of two-phase flow [19, 30, 31, 67, 69], and the interface cut-cell method [60, 68, 71, 72]. The immersed boundary method incorporates the interfacial conditions (3.3.2) and (3.3.3) into the field equation, and thereby circumvents the need for explicitly tracking the interface. To demonstrate the implementation of the method, we consider a two-dimensional (cylindrical) interface denoted by ( , and represent the interface with a set of marker points   )   * , where  is the arc length, as shown in Figure 3.3.1. In the numerical implementation, the markers points are separated

  + and + is the grid size, and the interby the distance , where  + face is parameterized in terms of the arc length, , by fitting quadratic polynomials through three consecutive marker points. Once the position of the interface has been described, the components of the unit normal vector are calculated as



,




 '  ' 





,



 

  '



(3.3.4)

where the subscript  indicates a derivative with the respect to arc length, and the curvature is computed as

& where





is the surface divergence.

   

(3.3.5)

Multi-scale modeling

129

Figure 3.3.1 Illustration of a two-dimensional interface showing the distribution of marker points.

The discontinuous viscosity field is represented by the approximate form






 

  
  

(3.3.6)

where  is the interfacial node closest to the field point ; subject to this convention, the right-hand side of (3.3.6) is a discrete approximation to the two-dimensional Heaviside step function, as illustrated in Figure 3.3.2. The influence function     is defined as




 
 




   
  -  ' 
  -' when      when  .  





(3.3.7)







where      , ' ' '  ,    ' , +, and + is the uniform grid size. A similar approximation is used for approximating the density field. In the discrete representation, the source term on the right-hand side of (3.3.1) is given by









&  
   

(3.3.8)



where  is the unit vector normal to the interface at the )th node,  is the assigned interface arc length, and


 




¾



   -   -' when     when   

(3.3.9)

130

Capsules and Cells

Figure 3.3.2 The viscosity and density field are described using a discrete approximation of the two-dimensional Heaviside step function.







  ' , +, and + is the uniform where     , ' ' '  ,   grid size. According to (3.3.8), to compute the distributed force at the grid point  , we draw a circle of radius centered at the evaluation point  , as shown in Figure 3.3.3(a), and sum over all nodes that lie inside the circle. To compute the velocity at the interfacial nodes from grid values, we use the analogous formula 

+



 


 

(3.3.10)

where the sum is now computed over the grid points. Once the node velocity is available, the node position is advanced in time using the explicit Euler method (e.g., [51]).

3.4 Leukocyte deformation and recovery The numerical method described in Section 3.3 was applied by Kan et al. [31] to study the deformation of a compound liquid drop subjected to a uniaxial extensional flow, and the subsequent recovery of the drop when the imposed flow is stopped, as illustrated in Figure 3.4.1. The core fluid is labeled 3, the shell fluid is labeled 2, the inner interface is labeled 3, and the outer interface is labeled 2. The drop deformation is determined by the capillary number (/
  , where
is the suspending fluid viscosity,  is the rate of extension of the elongational flow,  is the undeformed spherical shell radius, and  is the shell interfacial tension.

Multi-scale modeling

131

(a)

(b)

Figure 3.3.3 (a) Computation of the distributed hydrodynamic force at the grid point  by summing contributions from the interfacial nodes. (b) The interfacial node velocity is computed by summing contributions from grid values.

The solid circles in Figure 3.4.2(a) illustrate the dependence of the deformation  0    0 on the capillary number at steady state for parameter  equal fluid viscosities,
, cor , vanishing Reynolds number, and  responding to a homogeneous drop with an inactive core-shell interface; the lengths  and 0 are defined in Figure 3.4.1. The critical capillary number beyond which the drop fails to reach a steady state yielding continuous elongation is found to be approximately 0.12, in agreement with earlier studies conducted by boundary-integral methods for Stokes flow [64]. Also shown in Figure 3.4.2(a) are results of numerical simulations for a compound  , and spherical undeformed core radius equal to drop with
,  half the drop radius. The upper solid curve corresponds to the whole drop, and the






132

Capsules and Cells

Coarse grid

W2 Fine grid Interface 3

W3 x Fluid 3

L3

j

L2 Fluid 2 Interface 2

Figure 3.4.1 Elongation of an axisymmetric compound drop consisting of a viscous core and a concentric shell.

lower solid curve corresponds to the core. Although the compound drop deforms markedly and exhibits a critical capillary number (/    that is lower than that of the homogeneous drop, the deformation of the core is substantially reduced. Moreover, the core deforms into an oblate spheroid while the shell deforms into a prolate spheroid, in agreement with the results of earlier studies [65]. The steadystate velocity vector field around the compound drop is shown in Figure 3.4.2(b) for a capillary number that is nearly equal to the critical value, (/ = 0.1. The recirculating flow inside the compound drop causes the oblate and prolate deformation of the inner and outer interface, respectively, at steady-state. Kan et al. [33] investigated the effect of core size, position, and physical properties of the cytoplasm and nucleus on the cell recovery dynamics. Figure 3.4.3 compares laboratory observations on a lymphocyte whose nucleus occupies 44.4% the total cell volume, with results of a numerical simulation. In the experiment, a lymphocyte with an undeformed diameter 8 m is held inside a micropipette for about 15 seconds before it is expelled. In the simulation, the cell is assigned the initial shape shown at the top illustration on the right column of Figure 3.4.3. The graphs in Figure 3.4.4 show that the cell and nucleus length predicted by the numerical simulations is in reasonable agreement with the experimental data up to the late stages of relaxation, whereupon it appears that capillary forces are stronger in the model than those developing in reality. A possible explanation is that, in the experiments, the cell is expelled at a nonzero velocity. The predictions of the simulations discussed in the section reconcile results published in the literature using different approaches, and this indicates that the compound drop model is an acceptable tentative model of leukocytes. A summary of laboratory conditions and further results obtained using the model is given in Table

Multi-scale modeling (a)

133

0.40 0.35

IV

0.30 III

0.25 III D 0.20

II 0.15 II 0.10 0.05 0.00 0.00

(b)

I

I

0.05

0.10

0.15

0.20 Ca

0.25

0.30

0.35

0.40

1.2 1.0 0.8

y 0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

X

Figure 3.4.2 Deformation of simple and compound drops in uniaxial extensional flow at steady state. (a) Deformation parameter  plotted against the capillary number (/ for a homogeneous drop (filled circles [64] and dashed line) and a compound drop (hollow circles [65] and solid line [31]); the shapes corresponding to the capillary numbers I to IV are shown on the left. (b) Velocity vector field around the compound drop at steady state for
,   , and (/ = 0.1; the undeformed core radius is half the whole drop radius.

3.4.1. The laboratory data suggest that, when the deformation ratio exceeds the approximate value 1.4, the cell no longer recovers like a Newtonian fluid. Accounting for the elastic properties of the nucleus is expected to further reconcile theoretical predictions and experimental observations.

134

Capsules and Cells

Figure 3.4.3 Comparison between laboratory observation of lymphocyte recovery visualized with fluorescent-illumination (left), and corresponding numerical simulation (right).

3.5 Rolling of adhering cells N’Dri et al. [47] developed a multi-scale approach that couples the macroscopic (continuum) and microscopic (ligand-receptor) scales, as illustrated on the left of Figure 3.2 in this chapter’s Introduction. The methodology allows the investigation of the effect of the physical properties of the cell and surrounding fluid on the cell deformation and adhesion. The coupling between the disparate length scales associated with different physical mechanisms is critical for a comprehensive description based on first principles that bypasses phenomenological modeling. Information between the macroscopic and microscopic scales is communicated as follows: first, an initial membrane shape is supplied as input to the microscale model; at the origin of time, the bond force is arbitrarily set to zero and the macroscopic model equations are solved to obtain the pressure and velocity fields inside and outside the cell; this information is communicated to the microscopic model along with the shape of the cell to calculate the bond density and adhesion bond force; subsequently, the macroscopic model uses this data to determine the next cell shape and position, and the computation is repeated. The algorithm is illustrated in Figure 3.5.1 in the form of a flow chart.

Multi-scale modeling

135

2.5 Solid line: cell length (simulation results) Dash line: nucleus length (simulation results) : lynphocyte length (Vigneron, 1998) : nucleus length (Vigneron, 1998)

2 Lo 1.5 Ln

1

0.5

0

0

10

20

40

30

50

60

70

t (s)

Figure 3.4.4 Recovery curves for cell and nucleus plotted against recovery time for the lymphocyte simulations. The solid and dashed lines represent computed recovery lengths for the cell and nucleus, respectively. The hollow circles and asterisks correspond to the experimental data for the cell and nucleus.

The macroscopic model considers fluid flow over a homogeneous or compound drop attached to a flat surface. The distributed body force is given by the following counterpart of (3.3.9),







 &     
    








(3.5.1)





where 
is the bond stress evaluated at the )th node. Adopting a model proposed by Dembo et al. [13], we set 


, where 
  , 
is the bond density,  is the spring constant, and  and  are the current and equilibrium bond lengths. To compute the bond density, 
, we use a balance equation for the formation and dissociation of bonds based on a simple kinetics relationship,









 


 
 
 









(3.5.2)




where  is the initial ligand density over the surface, and  is the initial density of receptors over the cell membrane. The reverse and forward reaction rates are given by



 

 
 
 








 


 
 






(3.5.3)

where the subscript 0 refers to the initial equilibrium state,   is the transition spring constant,
is Boltzmann’s constant, and  is the absolute temperature. The initial

136

Capsules and Cells

Experiment

Observation

Model

High deformation rate (high aspiration pressure)

Low apparent viscosity (shear thinning)

Cytoplasmic fluid responds immediately; nucleus deforms gradually

Small deformation (large pipette )

Low apparent viscosity (shear thinning)

Nucleus deforms only slightly due to resistance from cytoplasm

Aspiration experiment:

Large deformation (small pipette )

Nucleus deforms substantially

Recovery experiment: Short holding time

Fast recoil (non-Newtonian)

Nucleus does not have time to deform

Long holding time

Slow recovery (Newtonian)

Nucleus has time to deform

Small deformation

Low apparent viscosity

Nucleus is only slightly deformed

Large deformation

High apparent viscosity

Nucleus is highly deformed

Table 3.4.1 Predictions of the compound drop model for leukocyte behavior observed in aspiration and recovery experiments.

bond density, 
 , is found by solving the equilibrium equation

  


 
 
  












(3.5.4)

In the numerical simulations, equation (3.5.2) is integrated in time using the fourthorder Runge-Kutta method. The microscopic model has been used to analyze the behavior of a membrane pulled away from a surface at a constant velocity, and thereby assess the effect of the reaction rates, ligand density, and bond stiffness [47, 59]. N’Dri et al. [47] solved the macroscopic and microscopic model equations separately and investigated the

Multi-scale modeling

137

Receive parameters from the macro−model

Compute the bond density

Compute the bond force

Transfer the bond force to the macro−model

Apply interfacial conditions

Solve the continuum equations

Advect the interface

Figure 3.5.1 Algorithm illustrating the coupling of the micro-model (receptor scale) and the macro-model (cellular scale).

effect of the Reynolds number and capillary number. In the simulations, the initial membrane shape given by the macroscopic model is used as input to the microscopic model. Specifically, a pressure difference across the membrane is applied based on the macroscopic solution, and the new position of the membrane is calculated. In the first step of the computation, the bond model is not activated. In the second and subsequent steps, the bond density and force are computed based on the receptorligand model. Using this information, the new location of the interface is found at subsequent time instants. In the majority of the simulations, the following parameter   
 cm ,  = 5.0 dyn/cm,   = 4.5 dyn/cm, values are used: 
   dyne cm. Instantaneous membrane shapes subtended between the points A and B depicted on the left of Figure 3.2 after several steps are shown in Figure 3.5.2. The initial profile represents the original membrane segment prior to initiating the flow.





The results show that the membrane is initially pulled away from the substrate under the action of the hydrodynamic forces. Membrane movement activates receptorligand bonds, which limits the membrane deformation as a stronger force is required to deform the bonds. The jump between the initial and first time step, evident in Figure 3.5.2, occurs because the macroscopic and microscopic models are not started simultaneously. To explain the behavior described in Figure 3.5.2, the evolution of

138

Capsules and Cells

Figure 3.5.2 Illustration of instantaneous membrane profiles developing due to an applied pressure field, computed based on the macroscopic model with the bond force computed based on the receptor-ligand model. The vertical axis is normalized by the bond length, and the horizontal axis is normalized by the cell radius.

the reverse and forward reaction rates is presented in Figure 3.5.3. The two curves correspond to the quarter- and half-point of the membrane segment. Cursory inspection of equations (3.5.3) shows that the distance between the membrane and the wall has a strong influence on the rate constants. Accordingly, the initial upward movement of the membrane during the first time step, shown in Figure 3.5.2, causes an initial decrease in the forward rate constant,  , as shown in Figure 3.5.3(a). Subsequent downward movement of the membrane reverses this trend. This interpretation is applicable for both the forward and reverse reaction rates, except that the reverse coefficient,  , exhibits the converse behavior, as shown in Figure 3.5.3(b). The main findings of N’Dri et al. [47] regarding the kinetics model can be summarized as follows:

 The maximum bond length decreases as the reverse reaction rate is raised, and increases as the forward reaction rate is raised.

 As the cell velocity becomes higher during debonding process, the maximum bond length increases while the total peeling time is reduced.

 The rate of debonding decreases as the number of ligands increases.  The bonds strength can affect the force balance and, consequently, the local shape of the cell membrane.

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139

(a)

(b)

Figure 3.5.3 History of the (a) forward, and (b) reverse reaction rate at two different locations; 1 stands for the abscissa of the bond.

 The peeling time decreases with increasing the spring or slippage constant.  The peeling time is maximum when the number of ligands is equal to the number of its receptor counterparts.

3.5.1 Liquid drop model The peeling of a cell modeled as a liquid droplet attached to a vessel wall was investigated using the immersed boundary method. The parameters of the simulation are: tube diameter, 30 m; cell diameter, 6-8 m; tube length, 120 m; velocity at tube inlet, 50-3200 m/sec; plasma viscosity, 1.0 dyn sec/cm ; cell density, 1.0 g/cm ; plasma density, 1.0 g/cm ; interfacial tension, 10  -8 dyn/cm. Additional parameters are: cell viscosity, 10-1,000 dyn sec/cm [28];  = 2.0 - 5.0 10
 cm [4];   = 10
 cm /sec [26];  = 10

- 10 cm /sec [3];  = 0.5 - 10 dyn/cm [13, 15];   = 0.48 - 9.5 dyn/cm [13, 15];
 = 3.8 10 dyn cm [14];  = 5.0 10 cm [4];  = 4.0 10  cm [53];  = 0.04 - 0.1;  = 2.0-5.0 10
 cm [39].

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Capsules and Cells

 Figure 3.5.4 Instantaneous position of an interfacial marker point for  = 1.0,
= 0.1,  = 100, and  = 1.0;  is the tube radius. In all computations discussed in this section, unless specified otherwise, the values of  ,  , and   are set to 0.1 s
, 5 dyne/cm, and 4.5 dyne/cm. Variables and parameters with a caret are reduced by the following reference values: length, 30 m; velocity, 600 m/sec; spring constant, 5.0 dyn/cm; reverse reaction rate, 0.1 sec
; viscosity, 1 dyn sec/cm ; density, 1 g/cm ; bond density, 5 10
 cm ; tension, 5 dyn/cm. The schematic at the top of Figure 3.5.4 depicts the instantaneous position of a material point on the cell surface, illustrating that the cell translates along the wall while undergoing rotation. The combination of these two elementary motions amounts to a composite rolling motion. Sliding occurs when the bonds exhibit a lower resistance, as shown by the plateau of the graph at the bottom of Figure 3.5.4. This trend can be compared with that observed in Figures 3.1.2 and on the right of Figure 3.1.4, showing a rapid increase of the bond length followed by a slow increase and then a plateau [57]. The effect of the wall can be studied by deactivating the micro model, that is, by setting the bond force equal to zero. The significance of bond molecules is illustrated in Figure 3.5.5(a). The bond force is found to be strong enough to delay cell rolling and peeling off the vessel wall. Figure 3.5.5(b) illustrates the effect of the reverse reaction rate. Increasing  reduces the resistance due to the bond molecules. On the other hand, decreasing  allows the cell to roll for a longer distance and tends to increase the bond length of the adhesion molecule before breakage, as demonstrated by the upper asymptote.

Multi-scale modeling

141

The effect of the spring constant is depicted in Figure 3.5.5(c). Changing the spring constant by a factor of two has little effect on the rolling motion. Chang et al. [8] also found a weak dependence of the spring constant on the adhesion process for a certain range of adhesion parameters (see Figure 3.1.16). Dong et al. [14, 15] modeled the cell as a liquid drop enclosed by an elastic shell, and allowed only a small portion of the adhesion contact to peel away from the wall over an unspecified peeling zone. In the simulations presented in this section, the adhesion parameters and cell viscosity are the same as those used by Dong et al., but the bonds are assumed to break when the cell travels a distance equal to twice the cell radius. This assumption is based on the observation that a realistic cell shape is observed only after the cell has traveled by that distance. Kuo et al. [37] made a similar assumption in a study where the leukocytes are modeled as rigid bodies, reckoning that bonds are broken when the cell travels a distance equal to 10 cell radii. The only unknown parameters in Dong’s study are the cell surface tension, the contact length, and the number of bonds. In Figure 3.5.6, the results of Dong et al.  = 1.0, [15] are compared with the present results for  = 4.0, 5.0, and 10, and for
    = 0.4,  = 1.06,  = 0.1, and  = 0.04. The top curve, corresponding to the higher surface tension, is in good agreement with those of the earlier study. In Figure 3.5.7, the numerical results on the bond life time are compared with the experimental observations of Schmidtke & Diamond for a single bond [54]. Notwithstanding the lack of information on the reaction rates and spring constant, N’Dri et al. [46] showed that, in the range of adhesion parameters considered, these values do not significantly affect the rolling and displacement of the cell along the vessel wall. The key factors are the cell rheological properties, receptor density, and ligand density. In the work of N’Dri et al. [46], the number of ligands is assumed to be in excess of the number of receptors, the reduced cell surface tension is taken to be 1.2, and the reduced viscosity ranges from 50 to 300. The top curve in Figure 3.5.7, corresponding to the higher viscosity, is in good agreement with the laboratory data. An increase in the cell surface tension for a fixed value of cell viscosity shifts the simulation results upward. A reduced surface tension of 1.2 and a reduced viscosity of 200 provide the best fit.

3.5.2 Compound drop model Figure 3.5.8 depicts the compound drop configuration consisting of a nucleated cell that is attached to a vessel wall and subjected to a streaming flow at the inlet of the vessel tube. In this model, the nucleus is assumed to occupy 44% of the cell volume. In the simulations, the viscosity and surface tension of the nucleus are taken to be 10 times those of the cytoplasm and cellular membrane. The bond kinetics are the same as those discussed earlier for the homogeneous drop model. The graphs in Figure 3.5.9(a) show that the presence of the nucleus tends to delay cell rolling but does not affect the bond length before peeling. The graphs in Figure 3.5.9(b) show that the lower the cytoplasmic viscosity, the higher the cell deformation and the closer the cell remains to the surface, as illustrated in Figure 3.5.10. As the cell viscosity is reduced, adhesion bonds exhibit a higher resistance, and this

142

Capsules and Cells (a)

(b)

(c)

Figure 3.5.5 Effect of (a) bond molecules, (b) reverse reaction rate, and (c) spring  = 0.1,  = 100, and constant, on the rolling of the cell along the wall for
    ; in (c),    and  = 1.0.

Multi-scale modeling

143

Figure 3.5.6 Comparison of the rolling velocity obtained by different models. The symbols correspond to the results of Dong et al. [15], and the curves present results for  = 4.0, 5.0, and 10 from bottom to top, respectively. (From Dong, C. & Lei, X. X., 2000, J. Biomech., 33, 35-43. With permission from the ASME.)

Figure 3.5.7 Comparison of the bond lifetimes obtained by different models. The symbols corresponds to the experimental results of Schmidtke & Diamond  = 0.02,  = 1.00,  = 0.1,  [54], and the curves present results for  = 0.04, and  = 1.2. The solid, dashed, dotted, and dot-dashed lines correspond, respectively, to  = 300, 200, 100, and 50. (From Schmidtke, D. W. & Diamond, S. L., 2000, J. Cell Biol., 149, 719-729. With permission from Rockefeller University Press.)

144

Capsules and Cells

Figure 3.5.8 Schematic illustration of the compound drop model.

delays the separation of the cell from the wall. N’Dri et al. [46] found that a reverse reaction constant in the range   
has no significant effect on the position and displacement of the cell or nucleus. Graphs of the reduced lateral position ' against time for the compound drop are nearly identical to those for the homogeneous drop shown in Figure 3.3.11(b). The results shown in Figure 3.5.9(c) demonstrate that surface tension has a major effect on the mechanics of cell rolling. Lower surface tension allows the cell to undergo higher deformation, and the cell remains attached to the wall over a longer length, as illustrated in Figure 3.5.9(b). Consequently, lower surface tension delays the lifting of the cell from the wall. The graphs in Figure 3.5.11(a) demonstrate that, as the inlet velocity is increased, the motion of the cell along the vessel becomes faster, and the bonds break a shorter distance away from the wall. The reason is that, as the inlet velocity is increased, the cell becomes increasingly more deformed and stays closer to the wall, as depicted in Figure 3.5.11(b). Nondimensionalizing the governing equations shows that the behavior of the cells depends on the capillary number, (/
 , among other dimensionless numbers.  , viscosity , and Figure 3.5.12 illustrates the effects of the reduced inlet velocity
surface tension  on the rolling motion for a given capillary number. As expected, combinations of parameters that correspond to the same capillary number yield identical behavior. N’Dri et al. [46] computed the bond life time, cell rolling velocity, and force of a bond molecule for different cell properties corresponding to different viscosity ratios 2

  and    , where  is the plasma viscosity, is the cytoplasm viscosity, and  is the nucleus viscosity. Figure 3.5.13(a) illustrates the effect of the reduced inlet velocity  on the reduced cell peeling time  . Increasing the inlet velocity clearly reduces the peeling time. The data are well fitted to the curve drawn in the figure, described by the inverse function







 






(3.5.5)

Figure 3.5.13(b) illustrates the effect of the reduced inlet velocity on the reduced cell rolling velocity  , defined as the ratio of the distance traveled by the cell to

Multi-scale modeling

145

(a)

(b)

(c)

Figure 3.5.9 Effect of (a) nucleus, and (b) interior cell viscosity on cell rolling for  = 1.0,
 = 0.1, and   ; in (a)  = 100, and in (b)  = 10 or 100. (c)  = 0.1, and  = 100. Effect of the interfacial tension for  = 1.0,


146

Capsules and Cells (a)

(b)

Figure 3.5.10 Effect of (a) cytoplasmic viscosity, and (b) interfacial tension on the instantaneous cell shape. The horizontal lines mark the location of the substrate.

the peeling time. Here only the displacement of the cell along the  axis is shown. Increasing the inlet velocity raises the rolling velocity. The data are well fitted to the curve drawn in the figure, described by the function



 
 
 
   
   

(3.5.6)

Multi-scale modeling

147

(a)

(b)

 = 0.1, Figure 3.5.11 (a) Effect of the inlet velocity on cell rolling for  = 1.0,
  , and (b) instantaneous cell shapes for different values of inlet velocities. Receptor and ligand densities have been shown by several authors to affect the cell rolling velocity and peeling from the substrate [2, 1, 14]. Figure 3.5.14 shows that increasing the receptor density raises the peeling time and decreases the rolling velocity. Dong et al. [14] also found an increase of the peeling time with receptor density. The dependence of the peeling time and rolling velocity on the receptor density is well described by the functions



      

(3.5.7)

and



   



represented by the solid lines in Figure 3.5.14.





(3.5.8)

148

Capsules and Cells

 , viscosity , and surface tension Figure 3.5.12 Effect of the reduced inlet velocity
  on the rolling motion for (/ = 10 and  = 1.0. Figure 3.5.15 illustrates the effect of ligand density on the peeling time and cell   , raising the ligand density increases the peeling time rolling motion. When   .  , the peeling time and and decreases the rolling velocity, whereas when  rolling velocity remain nearly unchanged. A plateau is observed when receptors are not available for binding at higher ligand densities. Similar behavior was observed in the simulations of Dong et al. [14] and in the experiments of Alon et al. [1]. The results of the present simulations are well described by the functions



     





(3.5.9)

and



   







(3.5.10)

represented by the solid lines in Figure 3.5.15. Figure 3.5.16 illustrates the effect of surface tension on the peeling time and rolling velocity. The former is well described by the fitted function



  





(3.5.11)

and the latter is well described by the function

 for  for 

  , and



.  .


  
     

(3.5.12)

  
        

(3.5.13)

Multi-scale modeling

149

(a)

(b)

Figure 3.5.13 Effect of the inlet velocity on (a) the peeling time, and (b) the rolling  = 1.0,  = 1.0,  = 1.0,  = velocity of a leukocyte, for 2 = 100,  = 10,  0.2, and  = 1.0. Figure 3.5.17 illustrates the effect of the viscosity ratio  on the peeling time and rolling velocity. Raising  increases the peeling time and decreases the rolling velocity. The data are well described by the third-degree polynomials



  
          

(3.5.14)


      
      

(3.5.15)

and



Figure 3.5.18 illustrates the effect of the viscosity ratio 2 on the peeling time and rolling velocity. Raising 2 slows down the breakage of the forming bonds and

150

Capsules and Cells (a)

(b)

Figure 3.5.14 Effect of the receptor density on (a) the peeling time, and (b) the  = 0.1,  = 1.0,  = 1.0,  = 0.2, and rolling velocity for 2 = 100,  = 10,
 = 1.0.

decreases the cell rolling velocity. The data are well described by the third-degree polynomial




  2
  2    2   

(3.5.16)

and the power-law relation



   2





(3.5.17)

N’Dri et al. [46] found that increasing the contact length raises the initial bond number according to the approximate relation

,
   

(3.5.18)

Multi-scale modeling

151

(a)

(b)

Figure 3.5.15 Effect of the ligand density on (a) the peeling time, and (b) the rolling  = 0.1,  = 1.0,  = 1.0,  = 0.2, and  = 1.0. velocity for 2 = 100,  = 10,
where ,
is the number of bonds, and the angle  has been defined in Figure 3.5.19(a). An increase of the contact length results in a higher peeling time and a lower rolling velocity, as shown in Figures 3.5.19(b) and 3.5.19(c), respectively. The data are well described by the third-degree polynomials



         

(3.5.19)

and




     

(3.5.20)

where the angle  is measured in degrees. Figure 3.5.20 compares the present results based on the compound drop model with the experimental data of Schmidtke & Diamond [54] on the bond life times for several combinations of the viscosity ratios. In all cases, the reduced interfacial tension is   . The bond life time decreases as the applied shear rate is raised, as

152

Capsules and Cells (a)

(b)

Figure 3.5.16 Effect of surface tension on (a) the peeling time, and (b) the rolling  = 0.1,  = 1.0,  = 1.0,  = 1.0, and velocity for 2 = 100,  = 10,
  .

the cell cytoplasm becomes less viscous, and as the nucleus becomes more viscous. Results of the compound drop model for reduced cytoplasmic viscosity 50 provides a good fit to the experimental data. Results for reduced cytoplasmic viscosity of 100 provides a better fit for shear rates 
  up to 250 s
. The force of rupture of a single bond, 
 , has also been determined. The critical bond force computed using the analysis of Goldman et al. [23] and Dong et al. [14] is reported to range from 100 to 400 pN, and from 100 to 200 pN, respectively. Experimental measurements yield forces in the inclusive range of 37 to 250 pN [55, 62]. In the present analysis, the molecular bond force varies in the range 250 to 400 pN for shear stress varying from 2 to 20 dyne/cm , and for reduced spring constant   , as shown in Figure 3.5.21(b). The results are well described by the linear relation



  
 

in the units indicated in the figure.




(3.5.21)

Multi-scale modeling

153

(a)

(b)

Figure 3.5.17 Effect of the viscosity ratio  on (a) the peeling time, and (b) the  = 0.1,  = 1.0,  = 1.0,  = 1.0, and cell rolling velocity, for 2 = 100,
  . The value   = 0.1 corresponding to the dimensional value  = 0.5 dyne/cm was used in the computations of Dong & Lei [15]. Lower values for 
 can be achieved by decreasing the value of the spring constant. Lower rupture force means a shorter bond life time. N’Dri et al. [46] found that bond stiffness in the range     , corresponding to    dyn/cm. produces the range of bond forces reported in the literature. On the other hand, Shao & Hochmuth [55] reported a spring constant value of 43 pN/ m or 0.043 dyne/cm, which is much smaller than the values cited above. Figure 3.5.21(a) illustrates the dependence of the bond life time on the hydrodynamic force exerted on the cell, corresponding to the solid line in Figure 3.5.20. An increase of the pulling force results in a decrease in the bond life time, in agreement with the results of previous authors [62]. The exponential relation proposed by Shao

154

Capsules and Cells (a)

(b)

Figure 3.5.18 Effect of the viscosity ratio 2 on (a) the peeling time, and (b) the  = 0.1,  = 1.0,  = 1.0,  = 1.0, and  = 0.2. rolling velocity for  = 10,


& Hochmuth [56] can be used to describe the results shown on the left of Figure 3.5.21(a), and the best fit is found to be



  
   

(3.5.22)

in the units indicated in the figure. The inverse algebraic correlation



  



(3.5.23)

represented by the solid line on the right of Figure 3.5.21(a), provides us a nearly perfect fit in the units indicated in the figure. In summary, the multi-scale approach outlined in this section after N’Dri et al. [46] suggests that cell deformability significantly affects the rolling velocity along

Multi-scale modeling

155

(a)

(b)

(c)

Figure 3.5.19 (a) Definition of contact length expressed by the angle , and effect of the contact length on (b) the peeling time, and (c) the cell rolling velocity  = 0.1,  = 1.0,  = 1.0,  = 1.0,  = 0.2, and  = for 2 = 100,  = 10,
1.0. In (b) and (c), the angle  in the abscissa is expressed in degrees.

a vessel wall. Specifically, the higher the surface tension, the faster the rolling velocity. These results are in agreement with the findings of previous authors based on models where only a small section of the membrane was allowed to deform [15]. Moreover, the theoretical studies reveal that, as the shear rate is raised, the bond lifetime decreases, in agreement with laboratory observations [54].

156

Capsules and Cells

Figure 3.5.20 Comparison of the bond lifetimes with the experimental data of Schmidtke and Diamond [54] indicated by the symbols. In the simulations,  = 1.0,  = 1.0,  = 1.0,  = 0.2, and  = 1.2; solid line: 2 = 200,  = 10, dashed line: 2 = 200,  = 5, dotted line: 2 = 100,  = 10, dot-dashed line: 2 = 50,  = 10. (a)

(b)

Figure 3.5.21 Dependence of the bond lifetime on (a) the applied hydrodynamic force, and (b) bond force. In (a), the lines represent exponential (left) and inverse power fit (right).

Multi-scale modeling

157

References [1] A LON , R., C HEN , S., F UHLBRIGGE , R., P URI K. D., & S PRINGER , T. A., 1998, The kinetics and shear threshold of transient and rolling interactions of L-selectin with its ligand on leukocytes, Proc. Natl. Acad. Sci., 95, 1163111636. [2] A LON , R., C HEN , S., P URI , K. D., F INGER , E. B., & S PRINGER , T. A., 1997, The kinetics of L-selectin tethers and the mechanics of selectin-mediated rolling, J. Cell Biol., 138, 1169-1180. [3] B ELL , G. I., 1978, Models for the specific adhesion of cells to cells, Science, 200, 618-627. [4] B ELL , G., D EMBO , M. & B ONGRAND , P., 1984, Cell adhesion: Competition between nonspecific repulsion and specific bonding., Biophys. J., 45, 10511064. [5] B OGREN , H. G. & B UONOCORE , M. H., 1999, 4D magnetic resonance velocity mapping of blood flow patterns in the aorta in young vs elderly normal subjects, J. Magn. Reson. Imag., 10, 861-869. [6] B RUNK , D. K., G OETZ , D. J., & H AMMER D. A., 1996, Sialyl Lewis(x)/Eselectin-mediated mediated rolling in a cell-free system, Biophys. J., 71, 29022907. [7] B RUNK , D. K. & H AMMER , D. A., 1997, Quantifying rolling adhesion with a cell-free assay: E-selectin and its carbohydrate ligands, Biophys. J., 72, 28202833. [8] C HANG , K. C., T EES , D. F. J., & H AMMER , D. A., 2000, The state diagram for cell adhesion under flow: Leukocyte rolling and firm adhesion, Proc. Natl. Acad. Sci., 97, 11262-11267. [9] C HEN , S. & S PRINGER , T. A., 1999, An automatic breaking system that stabilizes leukocyte rolling by an increase in selectin bond number with shear, J. Cell Biol., 144, 185-200. [10] C HEN , S. & S PRINGER , T. A., 2001, Selectin receptor-ligand bonds: Formation limited by shear rate and dissociation governed by Bell model, Proc. Natl. Acad. Sci., 98, 950-955. [11] C HENG , Z., 2000, Kinetics and mechanics of cell adhesion, J. Biomechanics, 33, 23-33. [12] C OZENS -ROBERTS , C., L AUFFENBERG , D. A., & Q UINN , J. A., 1990, Receptors-mediated cell attachment and detachment kinetics, I. Probabilistic model and analysis. Biophys. J., 58, 841-856. [13] D EMBO , M., T ORNEY, D. C., S AXAMAN , K., & H AMMER , D., 1988, The reaction-limited kinetics of membrane-to-surface adhesion and detachment, Proc. Roy. Soc. Lond. B, 234, 55-83.

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[14] D ONG , C., C AO , J., S TRUBLE , E. J., & L IPOWSKY, H. H., 1999, Mechanics of leukocyte deformation and adhesion to endothelium in shear flow, Ann. Biomed. Eng., 27, 322-331. [15] D ONG , C. & L EI , X. X., 2000, Biomechanics of cell rolling: Shear flow, cell-surface adhesion, and cell deformability, J. Biomechanics, 33, 35-43. [16] E VANS , E. A., 1983, Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipette aspiration tests, Biophys. J., 43, 27-30. [17] E VANS , E. A., 1985, Detailed mechanics of membrane-membrane adhesion and separation. I. Continuum of molecular cross-bridges, Biophys. J., 48, 175-183. [18] E VANS , E. A., 1985, Detailed mechanics of membrane-membrane adhesion and separation. II. Discrete kinetically trapped molecular bridges, Biophys. J., 48, 185-192. [19] FAUCI , L. J. & P ESKIN , C. S., 1988, A computational model of aquatic animal locomotion, J. Comput. Phys., 77, 85-108. [20] F RY, D. L., 1987, Mass transport, atherogenesis, and risk, Arteriosclerosis, 7, 88-100. [21] F UJIOKA , H. & TANISHITA , K., 2000, Computational fluid mechanics of the blood flow in an aortic vessel with realistic geometry, In Yamaguchi, T., (Ed.), Clinical Application of Computational Mechanics to the Cardiovascular System, Springer-Verlag, Tokyo, pp. 99-117. [22] G IDDENS , D. P., Z ARINS , C. K., & G LAGOV, S., 1993, The role of fluid mechanics in the localization and detection of atherosclerosis., J. Biomech. Eng., 115, 588-594. [23] G OLDMAN , A. J., C OX , R. G. & B RENNER , H., 1967, Slow motion of a sphere parallel to a plane wall; II Couette flow, Chem. Eng. Sci., 22, 653-660. [24] G REENBERG , A. W., B RUNK , D. K., & H AMMER , D. A., 2000, Cell-Free rolling mediated by L-selectin and sialyl Lewisx reveals the shear threshold effect, Biophys. J., 79, 2391-2402. [25] H AMMER , D. A. & A PTE , S. M., 1992, Simulation of cell rolling and adhesion on surfaces in shear flow: General results and analysis of selectinmediated neutrophil adhesion, Biophys. J., 63, 35-57. [26] H AMMER , D. A. & L AUFFENBURGER , D. A., 1987, A dynamical model for receptor-mediated cell adhesion to surfaces, Biophys. J., 52, 475-487. [27] H AMMER , D. A. & T IRRELL , M., 1996, Biological adhesion at interfaces, Annu. Rev. Mater. Sci., 26, 651-691.

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[28] H OCHMUTH , R. M., T ING -B EALL , H. P., B EAT B. B., N EEDHAM , D., & T RAN -S ON -TAY, R., 1993, The viscosity of passive human neutrophils undergoing small deformations, Biophys. J., 64, 1596-1601. [29] J IN , S., O SHINSKI , J. & G IDDENS , D., 2001, Effects of inflow conditions and wall motion on flow in the ascending aorta, Proc. Bioeng. Conf. ASME, BED, vol. 50. [30] J URIC , D. & T RYGGVASON , G., 1996, A front tracking method for dendritic solidification, J. Comp Phys., 123, 127-148. [31] K AN , H.-C., U DAYKUMAR , H. S., S HYY, W., & T RAN -S ON -TAY, R., 1998, Hydrodynamics of a compound drop with application to leukocyte modeling, Phys. Fluids, 10, 760-774. [32] K AN , H.-C., U DAYKUMAR , H.S., S HYY, W. & T RAN -S ON -TAY R., 1999, Numerical analysis of the deformation of an adherent drop under shear flow, J. Biomech. Eng., 121, 160-169. [33] K AN , H.-C., U DAYKUMAR , H. S., S HYY, W., V IGNERON , P., & T RAN S ON -TAY, R., 1999, Effects of nucleus on leukocyte recovery, Annals Biomed. Eng., 27, 648-655. [34] K APLANSKI , G., FARNARIER , C., T ISSOT, O., P IERRES , A., B ENO LIEL , A.-M., A LESSI , M.C., K APLANSKI , S., & B ONGRAND , P., 1993, Granulocyte-endothelium initial adhesion. Analysis of transient binding events mediated by E-selectin in a laminar shear flow, Biophys. J., 64, 1922-1933. [35] K ILNER , P. J., YANG , G. Z., M OHIADDIN , R. H., F IRMIN , D. N., & L ONG MORE D. B., 1993, Helical and retrograde secondary flow patterns in the aortic arch studied by three-directional magnetic resonance velocity mapping, Circulation, 88, 2235-2247. [36] KOMAI , Y. & TANISHITA , K., 1997, Fully developed intermittent flow in a curved tube, J. Fluid Mech., 347, 263-287. [37] K UO , S. C., H AMMER , D. A., & L AUFFENBURGER , D. A., 1997, Simulation of detachment of specifically bound particles from surface by shear flow, Biophys. J., 73, 517-531. [38] L AUFFENBURGER , D. A. & L INDERMAN , J. J., 1993, Receptors: Models for Binding, Trafficking, and Signaling, Oxford University Press, New York. [39] L AWRENCE , M. B. & S PRINGER , T. A., 1991, Leukocytes roll on a selectin at physiologic flow rates: distinction from and prerequisite for adhesion through integrins, Cell, 65, 859-873. [40] L EUPRECHT, A., P ERKTOLD , K., KOZERKE , S. & B OESIGER , P., 2002, Combined CFD and MRI study of blood flow in a human ascending aorta model, Biorheology, 39, 425-429.

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[41] L IU , H., 2000, Computation of the vortical flow in human aorta, Proceedings of the ASME, Fluids Engineering Division, FED, 253, 497-502. [42] L IU , H. & YAMAGUCHI , T., 2001, Waveform dependence of pulsatile flow in a stenosed channel, J. Biomech. Eng., 123, 88-96. [43] L ONG , M. W., 1995, Tissue microenvironments, J. D. Bronzino, J. D. (Ed.), The Biomedical Engineering Handbook, CRC Press, Boca Raton, Chap. 114, pp. 1692-1709. [44] M C D ONALD , D. A., 1974, Blood Flow in Arteries, Second Edition, Edward Arnold, London. [45] M ORI , D., L IU , H., & YAMAGUCHI , T., 2000, Flow simulation of the aortic arch-effect of the 3D distortion on flows in the ordinary helix circular tube, in Yamaguchi, T. (Ed.), Clinical Application of Computational Mechanics to the Cardiovascular System, Springer-Verlag, Tokyo, 2000, pp. 132-135. [46] N’D RI , N., S HYY, W., & T RAN -S ON -TAY, R., 2003, Computational modeling of cell adhesion and movement using a continuum-kinetics approach, Submitted for publication. [47] N’D RI , N., S HYY, W., T RAN -S ON -TAY, R., & U DAYKUMAR , H. S., 2000, A multi-scale model for cell adhesion and deformation, ASME IMECE Conference, FED, 253, 205-213. [48] P EDLEY, T., 1995, Blood flow in arteries and veins, In Batchelor, G. K., Moffatt, H. K., & Worster, M. G., Perspectives in Fluid Dynamics; A collective Introduction to Current Research, Cambridge University Press, Cambridge. [49] P ESKIN , C. S., 1977, Numerical analysis of blood flow in the heart, J. Comput. Phys., 25, 220-252. [50] P OZRIKIDIS , C., 1997, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, New York. [51] P OZRIKIDIS , C., 1998, Numerical Computation in Science and Engineering, Oxford University Press, New York. [52] R AMACHANDRAN , V., N OLLERT, M. U., Q IU , H., L IU , W. J., C UMMINGS , R. D., Z HU , C., & M C E VER , R. P., 1999, Tyrosine replacement in P-selectin glycoprotein ligand-1 affects distinct kinetic and mechanical properties of bonds with P- and L-selectin, Proc. Natl. Acad. Sci., 96, 13771-13776. [53] S CHMID -S CHNBEIN , G. W., F UNG , Y., & Z WEIFACH , B. W., 1975, Vascular endothelium-leukocyte interactions: Sticking shear force in venules, Circ. Res., 36, 173-184. [54] S CHMIDTKE , D. W. & D IAMOND , S. L., 2000, Direct observation of membrane tethers formed during neutrophil attachment to platelets or P-selectin under physiological flow, J. Cell Biol., 149, 719-729.

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[55] S HAO , J.-Y. & H OCHMUTH , R. M., 1996, Micropipette suction for measuring piconewton forces of adhesion and tether formation from neutrophil membranes, Biophys. J., 71, 2892-2901. [56] S HAO , J.-Y. & H OCHMUTH , R. M., 1999, Mechanical anchoring strength of L-selectin,  integrins, and CD45 to neutrophil cytoskeleton and membrane, Biophys. J., 77, 587-596. [57] S HAO , J.-Y., P ING T ING -B EALL , H., & H OCHMUTH , R. M., 1998, Static and dynamic lengths of neutrophil microvilli, Proc. Natl. Acad. Sci., 95, 67976802. [58] S HINDE PATIL , V. R., C AMPBELL , C. J., Y UN , Y. H., S LACK , S. M., & G OETZ D. J., 2001, Particle diameter influences adhesion under flow, Biophys. J., 80, 1733-1743. [59] S HYY, W., F RANCOIS , M., U DAYKUMAR , H.S., N’D RI , N., & T RAN -S ON TAY R., 2001, Moving boundaries in micro-scale biofluid dynamics, Appl. Mech. Rev., 5, 405-452. [60] S HYY, W., U DAYKUMAR , H. S., R AO , M. M., & S MITH , R. W., 1996, Computational Fluid Dynamics with Moving Boundaries, Taylor & Francis, New York. [61] S KALAK , R., 1972, Mechanics of the microcirculation, In Fung, Y. C., Perrone, N., & Anliker, M., (Eds), Biomechanics: its Foundation and Objectives, Prentice-Hall, Englewood Cliffs. [62] S MITH , M. J., B ERG , E. L., & L AWRENCE , M. B., 1999, A direct comparison of selectin-mediated transient, adhesive events using high temporal resolution, Biophys. J., 77, 3371-3383. [63] S PRINGER , T. A., 1990, Adhesion receptors of the immune system, Nature, 346, 425-434. [64] S TONE , H. A. & L EAL , L. G., 1989, Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid, J. Fluid Mech., 198, 399-427. [65] S TONE , H. A. & L EAL , L. G., 1990, Breakup of concentric double emulsion droplets in linear flows, J. Fluid Mech., 211, 123-156. [66] T EES , D. F. J., C HANG , K. C., RODGERS , S. D., & H AMMER , D. A., 2002, Simulation of cell adhesion to bioreactive surfaces in shear: The effect of cell size, Ind. Eng. Chem. Res., 41, 486-493. [67] U DAYKUMAR , H. S., K AN , H.-C., S HYY, W., & T RAN -S ON -TAY, R., 1997, Multiphase dynamics in arbitrary geometries on fixed Cartesian grids, J. Comput. Phys., 137, 366-405. [68] U DAYKUMAR , H. S., M ITTAL , R. & S HYY, W., 1999, Computation of solidliquid phase fronts in the sharp interface limit on fixed grids, J. Comput. Phys., 153, 535-574.

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[69] U NVERDI , S. O. & T RYGGVASON , G., 1992, A front tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., 100, 25-37. [70] W EISS , L., 1990, Metastatic inefficiency, Adv. Can. Res., 54, 159-211. [71] Y E , T., M ITTAL , R., U DAYKUMAR , H. S., & S HYY, W., 1999, A Cartesian grid method for simulation of viscous incompressible flow with complex immersed boundaries, J. Comput. Phys., 156, 209-240. [72] Y E , T., S HYY, W. & C HUNG , J. N., 2001, A fixed-grid, sharp-interface method for bubble dynamics and phase change, J. Comput. Phys., 174, 781815. [73] Z HAO , S. Z., X U , X. Y., H UGHES , A. D., T HOM , S. A., S TANTON , A. V., A RIFF , B., & L ONG , Q., 2000, Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation, J. Biomechanics, 33, 975-984

Chapter 4 Mechanics of red blood cells and blood flow in narrow tubes

T.W. Secomb A mammalian red blood cell may be regarded as a capsule enclosed by a flexible membrane and containing an incompressible viscous fluid. The membrane, consisting of a lipid bilayer and a protein skeleton, strongly resists area changes and exhibits viscoelastic response to in-plane shear deformation. Knowledge of the mechanical properties of red blood cells has provided us with a basis for analyzing the cell motion and deformation in flow. Studies have focused mainly on two areas: single-file cell motion in capillary-sized tubes, and motion of dilutely suspended cells in shear flows. Although considerable progress has been made in both areas, the development of realistic models for multiple-file red blood cell motion in narrow tubes with diameters less than 1 mm, including the effects of cell-to-cell interactions, has remained rather elusive. In this chapter, a discussion of the mechanical properties of red blood cells is presented along with a review of theoretical methods for analyzing red blood motion and deformation in narrow tubes.

4.1 Introduction The primary function of the circulatory system is to transport materials throughout the body. This is achieved by the flow of blood through an extensive network of tubes with diameters ranging from 1 cm down to a few m, driven by the pumping action of the heart. Transport of oxygen from the lungs to all other parts of the body is a particularly crucial task performed by the circulatory system. Because the solubility of oxygen in water is relatively low, water would not be an effective medium for convective transport of oxygen. The presence of a high volume fraction (40-45%) of red blood cells (erythrocytes) in blood greatly increases the oxygen carrying capacity. Hemoglobin molecules inside red blood cells take up oxygen readily in the lungs and release it as needed in the body. A further consequence of the low aqueous solubility of oxygen is that the distance that oxygen can diffuse into an oxygen-consuming tissue is relatively short, typically of order 20-100 m. The circulatory system must

163

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Capsules and Cells

Figure 4.1.1 Human red blood cells flowing in glass tubes of approximate diameters 4.5 m (top), 7 m (middle), and 15 m (bottom). The flow direction is from left to right. (Reproduced by permission of Dr. Axel R. Pries, Free University of Berlin, Germany.)

therefore deliver blood within a short distance of every point in the tissue, and all oxygen-consuming tissues must be supplied with a dense network of very narrow blood vessels. These vessels, ranging in diameter from a few hundred m down to about 4 m, are known as the microcirculation. Freely suspended human red blood cells take the form of biconcave disks, about 8 m in diameter and 2 m in thickness. The cells are flexible enough to undergo large deformation as they pass through tubes with diameters typical of the microcirculation, as shown in Figure 4.1.1 (see also Reference [29]). The analysis of blood flow in microvessels presents intricate problems combining fluid and solid mechanics. In Section 4.2, the mechanical properties of red blood cells will be discussed, and a basis for analyzing the motion and deformation of red blood cells in flow through narrow tubes in subsequent sections of this chapter will be established.

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4.2 Mechanical properties of red blood cells Blood is a suspension. The suspending medium, called the plasma, is a solution of proteins, electrolytes, and other substances. From a macroscopic viewpoint, the plasma can be regarded as an incompressible Newtonian fluid whose viscosity is comparable to that of water, about 1 cP. The suspended elements include red blood cells (erythrocytes), white blood cells (leukocytes) of several different types, and platelets. Normal human blood has a hematocrit, defined as the volume fraction of red blood cells, of 40 to 45%. The mechanical properties of human red blood cells have been studied extensively by a variety of methods [35, 70]. The cell interior is a concentrated solution of the oxygen-binding protein hemoglobin that can be modeled as an incompressible fluid whose viscosity is elevated with respect to that of the plasma. Mature mammalian red blood cells do not contain a nucleus. The red cells are enclosed by a thin membrane consisting of a lipid bilayer and a cytoskeleton, which is a network of proteins. The cytoskeleton lies immediately inside the lipid bilayer, and its components project into the bilayer connecting the two structures together. One important property of the membrane is that it strongly resists changes in area. Considered as a thin shell, the membrane has an elastic modulus of isotropic dilation of about 500 dyn/cm, whereas the modulus of shear deformation is about 0.006 dyn/cm. The lipid molecules comprising the lipid bilayer can slide past each other relatively easily, but strongly resist being pulled apart. In this sense, the membrane behaves like a two-dimensional incompressible fluid. The cell membrane exhibits a relatively small flexural stiffness, i.e., resistance to bending [18] (see also Chapter 2). Bending moments become important only in regions where the membrane curvature is large. In addition to flexural stiffness, the membrane also exhibits viscous resistance to transient in-plane shear deformations. The combined viscoelastic response of the membrane in shear deformation can be represented by the Kelvin solid model. According to this model, the total shear stress is comprised of a viscous and an elastic component [19]. Physically, the viscous component arises from the fluid-like behavior of the lipid bilayer, whereas the elastic component arises from the stretching of the cytoskeleton. An implicit assumption of this model is that the bilayer and cytoskeleton undergo approximately the same shear deformation. Experimental evidence [15] suggests that relative motion may occur when part of the membrane is stretched into a narrow tongue. The effects of such behavior are not included in the models discussed below. Because of the fluidity of the cell interior and the low resistance of the membrane to shear and bending deformation, red blood cells can easily deform and squeeze through capillaries with diameters less than 8 m. However, the deformation is limited by the incompressibility of the interior fluid and the strong resistance of the membrane to area changes. These restrictions impose a critical minimum tube diameter for a red blood cell to pass through intact. For a typical human red blood cell, this diameter is about 2.8 m [6, 34].

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4.2.1 Calculation of membrane stresses in axisymmetric deformations A number of models of red blood cell motion in capillaries have been developed for axisymmetric cell shapes, that is, shapes with rotational symmetry about the tube axis (see Section 4.3). Although actual observed shapes of red blood cells in narrow tubes are rarely axisymmetric, as shown in Figure 4.1.1, models involving rotationally symmetric shapes provide us with useful insights into the mechanics of the three-dimensional motion. In the axisymmetric state, membrane stresses may be described explicitly in terms of membrane shape and deformation through simple relations expressing the mechanical properties previously described. To describe the shape of an axisymmetric red blood cell, it is convenient to identify the position of point particles along the membrane by the arc length
measured from the nose of the cell. Variables describing the cell shape are the distance from the axis, 

, and the angle between the normal to the membrane and the axis, 

, where   
comprise cylindrical polar coordinates, as shown in Figure 4.2.1 [68]. Dependence on time  can also be included to allow for transient axisymmetric deformations. When the initial shape of the membrane and the membrane load are also axisymmetric, the principal axes of the membrane stress and strain coincide with the coordinate directions
and . The membrane strain is characterized by the principal stretches or extension ratios   d
d
and 
  , defined with respect to the unstressed shape. The subscript
denotes the components in a meridional plane containing the axis of symmetry, the subscript  denotes the azimuthal components, and the subscript 0 denotes corresponding values of a material element in the unstressed configuration. The components of the in-plane membrane curvature are denoted by

and 

, the bending moments are denoted by

and 

, and the components of membrane tension are denoted by 

and 


. The shear force (transverse shear tension) per unit length is denoted by 

; consistent with the assumed symmetry, the  component vanishes. The bending and shear stresses in the membrane can be evaluated using constitutive relations proposed by Evans & Skalak [20] (pp. 77 and 109). The bending moments are assumed to be isotropic and proportional to the change in the total curvature, which is equal to twice the mean curvature [20],






 











(4.2.1)

where =  

 dyncm is the bending modulus. Viscous resistance to bending is assumed to be negligible. Because the principal axes of membrane stress coincide with the coordinate directions, the off-diagonal components of the two-dimensional tensor representing the in-plane stress in the membrane are zero. The principal tensions may be resolved into an isotropic (mean) and a deviatoric component,



    


 



(4.2.2)

Because of the high modulus of dilation, the membrane can be treated as an incompressible medium in a two-dimensional sense, so that the deformation is locally area

Red blood cell mechanics

ts

167

tf qs mf

ms ts

tf

s r z

q f

Figure 4.2.1 Variables describing geometry and stress resultants in an axisymmetric shell.

preserving, i.e.,  
 . A consequence of this assumption is that the isotropic component of the tension, 

, cannot be calculated explicitly in terms of membrane deformation, but should be determined instead by the forces acting on the membrane as is the hydrostatic pressure in incompressible fluid flow. The deviatoric component, 

, representing the shear stress in the membrane, is given by [60]

 





d d





















(4.2.3)

The first term on the right-hand side of (4.2.3) represents the viscous contribution to the membrane shear stress and depends on the rate of stretching of the membrane; the membrane shear viscosity is   0.001 dyns/cm [35]. The second term on the right-hand side of (4.2.3) represents the elastic component of membrane shear stress [20], where  = 0.006 dyn/cm is the membrane shear modulus. These two terms together describe the membrane viscoelastic behavior in transient shear deformation as a Kelvin solid [19]. The analysis of Secomb [60] showed that bending a membrane consisting of two coupled leaflets with a large resistance to area change induces anisotropic tension. This effect is represented by the third term on the right-hand side of (4.2.3), which is not included in the equations proposed by Evans & Skalak [20], but is implicit in the assumptions of their model. However, unless the radius of curvature is much less than 1 m, the effect of this term is small. The application of these equations to the simulation of red blood cell motion in capillaries will be discussed in Section 4.3.

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4.2.2 Mechanics of tank-treading membrane motion The flexibility of red blood cell membrane and the fluidity of the cell interior permits a type of continuous deformation known as “tank-treading.” In this mode, the shape and orientation of the cell may be constant in time, but the membrane undergoes a continuous “tank-treading” motion inducing rotational flow in the interior. Tank-treading motion has been observed for red blood cells suspended in shear flow [25], as well as in flow through narrow capillaries [30]. To analyze the tank-treading motion, Fischer et al. [25] developed an experimental system consisting of a fluid-filled, counter-rotating, cone-and-plate apparatus. The device permits the observation of nearly stationary red blood cells suspended in a linear shear flow subject to a prescribed shear rate. In the experiments, the viscosity of the suspending medium ranged from 11 to 59 cP, which is substantially higher than that of the plasma. Red blood cells were marked either with small latex spheres of diameter 0.8 m adhering to the cell membrane, or with chemically induced membrane-bound inclusions known as Heinz bodies. As the shear rate is increased, the shape of the red cells was observed to change from a biconcave disk, to an ellipsoid, and then to an elongated spindle-like shape with the long axis approximately aligned with the flow, as shown in Figure 4.2.2(a). Although these shapes appear stationary, steady tank-treading motion of the membrane around the interior is evident from the motion of the marker particles, as shown in Figure 4.2.2(b). The interfacial markers move along paths that are approximately parallel to the direction of the flow, from one end of the cell to the other and then back. This observation is a clear manifestation of continuous cyclic motion. The frequency of the tank-treading motion increases in proportion with the imposed shear rate and is insensitive to the suspending medium viscosity in the range considered. Gaehtgens & Schmid-Sch¨onbein [30] used a traveling capillary system to continuously observe red blood cells moving along narrow glass tubes with diameters ranging from 4 to 12 m. The cell membrane was labeled with small latex spheres or Heinz bodies. Periodic cyclic motion of the membrane was observed similar to that described in the previous paragraph, and was attributed to the asymmetry of fluid forces acting on each cell as a result of the eccentric position of the cell with respect to the tube axis, as seen in Figure 4.1.1. Because the tank-treading motion is fully three-dimensional, a theoretical analysis encounters considerable complications. As a step toward developing a quantitative description, Secomb & Skalak [66] considered the kinematics and dynamics of viscoelastic membranes executing surface flows. Because of the large modulus of dilatation of the membrane, the tank-treading motion must conserve membrane area locally at each point. It was shown that any steady area-conserving membrane motion can be represented in terms of a stream function  defined on the cell surface, u  n


(4.2.4)

where u is the membrane velocity, n is the unit vector normal to the surface, and the coordinate frame moves with the cell. In this way, the membrane motion is fully specified by a single scalar function  . As in the case of two-dimensional

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169

(a)

(b)

Figure 4.2.2 Untreated human red blood cells suspended in a high viscosity medium, subjected to a linear shear flow set up in a counter-rotating cone-andplate chamber. The flow direction is from top to bottom, the velocity gradient is perpendicular to the page, and the suspending medium viscosity is 13 cP. (a) Images obtained at several different shear rates (starting with upper left image: 0, 8.7, 18, 35, 70, 140, 270, 530, 1100, and 2200 s
). (b) Images of a single cell at time intervals of 20 ms, showing the motion of a latex sphere adhering spontaneously to the membrane and serving as a marker for the membrane motion, for shear rate 270 s
. (Reproduced by permission of Dr. Thomas M. Fischer, RWTH-Aachen, Germany.)

incompressible fluid flow, the lines of constant  are the streamlines of the surface motion. A further constraint on the kinematics is that the membrane motion must be synchronous, that is, all points in the membrane must execute cyclic motion with a common frequency  , as observed in the experiments [25]. Nonsynchronous motion

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Capsules and Cells

would result in an unbounded increase of membrane shear strain, which is not possible because of the nonzero elastic shear modulus of the membrane. Stream functions  corresponding to synchronous motions may be obtained as follows. Suppose that is any scalar function defined on the surface that does not necessarily yield a synchronous motion when substituted for  in equation (4.2.4). A synchronous tank-treading motion with frequency  and with the same streamlines can be obtained from by setting   
where, for each value of , 

is the period of the motion generated by the velocity n
 around the corresponding streamline. The function is chosen to give an approximately correct streamline pattern on the cell surface, and is conveniently represented as a polynomial in Cartesian coordinates. The application of this technique for analyzing the motion of nonsymmetric red blood cells moving through capillaries will be discussed in Section 4.3.2.

4.3 Single-file motion of red blood cells in capillaries The capillaries are the smallest blood vessels in the circulatory system, with diameters typically ranging from 4 to 10 m. These are the terminal branches of the arterial and venous vascular trees, and the primary sites of oxygen exchange with tissues. Red blood cells whose unstressed diameters are about 8 m must undergo large deformation in order to enter the smallest capillaries. In fact, the cells almost entirely fill a capillary and typically move in a single file, as illustrated in the upper two images of Figure 4.1.1. The effect of the cells on blood flow resistance through the capillaries has been considered in numerous investigations. The flow resistance of a single vessel or network of vessels is defined as the ratio of the driving pressure difference  to the induced volumetric flow rate . In the case of steady laminar flow of a Newtonian fluid in a cylindrical tube,  is related to  by Poiseuille’s law,





  

 





(4.3.1)

where  is the tube length,  is the tube diameter, and is the fluid viscosity. This relation may be used to estimate the flow resistance in blood vessels with diameter higher than 500 m, using values for obtained from bulk measurements. However, in microvessels, blood does not behave as a continuum, and the viscosity is an unknown constant. Accordingly, equation (4.3.1) merely provides us with the definition of the apparent viscosity,  

  

  



(4.3.2)

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171

The relative apparent viscosity is defined as 

 





(4.3.3)

where  is the viscosity of the plasma or other suspending medium. Based on laboratory observations such as those shown in Figure 4.1.1, it might be expected that the resistance to blood flow in capillaries would be very high as a result of the narrow spacing between red blood cells and the tube wall. However, experiments using glass tubes showed a marked decrease in the apparent viscosity as the tube diameter is reduced [22]. This phenomenon, known as the F˚ahraeusLindqvist effect, has been shown also to arise in tubes whose diameters lie in the capillary range [54]. Insights into the F˚ahraeus-Lindqvist effect can be obtained using the “axial-train” model developed by Whitmore [77], depicted in Figure 4.3.1, also referred to as the “stacked-coins” model. In this model, the red blood cells are confined in a cylindrical core of diameter  that is concentric with the tube and moves as a rigid body in a plug-flow mode, where  is the tube diameter. Such motion would occur if diskshaped red blood cells were stacked together to form a continuous column in the configuration of stacked coins. The spacing between the core containing the cells and the tube wall is assumed to be filled with plasma of viscosity  . Elementary analysis shows that the apparent viscosity of the suspension is given by  







(4.3.4)

According to this prediction, the presence of even a relatively thin layer of plasma in a capillary is sufficient to keep the apparent viscosity at a reasonably low level. For instance, a plasma layer 0.4 m wide in a capillary of diameter 8 m gives  = 0.9 and  = 2.9  , which is lower than the bulk viscosity of blood.

4.3.1 Axisymmetric models for red blood cell motion in capillaries In reality, the width of the plasma layer varies in space and time in a manner that is determined by the mechanical interaction of deformable red blood cells and the surrounding fluid flow. A number of theoretical models have been developed to describe this process, with the first studies appearing in the late 1960s [1, 43]. Most of these models rely on two key simplifying assumptions. Firstly, based on the observation that red blood cells have approximately axisymmetric shapes in narrow capillaries, as shown in Figure 4.1.1, the assumption of axisymmetric motion is invoked. Secondly, lubrication theory is used to describe the flow of plasma between the cell surface and the tube wall. The theory relies on the assumption that the spacing between the cell and the vessel wall is small relative to the physical dimension of the flow. The latter assumption allows for significant simplification of the equations governing fluid flow. The main deficiency of early models [1, 43] was the simplified description of the mechanical properties of the individual red blood cells. Lighthill [43] assumed that

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Capsules and Cells

r

z

Figure 4.3.1 Schematic illustration of the axial-train model of capillary blood flow. Shading indicates the core region containing red blood cells, moving as a rigid body in a plug-flow mode. The velocity profile is indicated by the arrows.

the undeformed cell shape is parabolic near wall, and the deformation is proportional to the local pressure. This approximation may be appropriate for an elastic solid material, but is not well suited to describing the mechanics of fluid-filled membranes. Barnard et al. [1] modeled the cell as a flexible circular sheet that deforms into a hollow “thimble” shape when subjected to flow. In a subsequent study, the cell was assigned a solid, bullet-like shape, and an isotropic tension was assumed to develop over the cell membrane [44]. This model provides us with a good approximation at relatively high cell velocities, where the external shear forces due to the flow are much larger than the tensions developing due to the membrane elastic resistance to bending and shear deformation [64]. A first attempt to incorporate the realistic mechanical properties of the red blood cell membrane in a realistic way was made by Zarda et al. [78]. Their formulation employed a residual minimization approach associated with a finite-element method to describe both the membrane deformation and the flow of the surrounding fluid, thereby relaxing the approximations of lubrication flow. The membrane mechanics was described using the equations of shell theory for axisymmetric shapes. The results illustrated the flow-induced changes in the red blood cell shapes and in the pressure drop across the cell surface. Secomb et al. [68] presented predictions of blood flow resistance in capillaries for a range of tube diameters and flow velocities, and compared the theoretical results with laboratory observations. As in the earlier study of Zarda et al. [78], the membrane was treated as an axisymmetric shell, and the effects of elastic resistance to membrane shear and bending were included using appropriate constitutive equations. Lubrication theory was used so that the governing equations could be formulated as a set of nonlinear ordinary differential equations. The main elements of this approach are next briefly outlined. The equations of membrane equilibrium for normal stress, tangential stress, and bending moments require, respectively,

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173


    

  d
 d 
   
   d
  d 
cos 
   d
 d

(4.3.5) (4.3.6) (4.3.7)

where  is the hydrostatic pressure difference across the membrane (external minus internal), and  is the fluid shear stress acting on the membrane produced by the motion of the external fluid. The membrane shear and bending stresses are computed from equations (4.2.1) through (4.2.3), where   
  . The key assumption of the lubrication theory is that the typical width of the gap between the cell and the vessel wall is small compared to the typical length of the gap. However, the gap is not necessary small relative to the vessel radius. Standard arguments can be made to show that, to leading order, the radial fluid velocity is small compared to axial velocity, and the pressure in the gap is independent of radial position (e.g., [5]). In cylindrical polar coordinates   
moving with the cell, the flow is steady and the axial velocity   
of the fluid in the gap satisfies the equation

      






d  d

(4.3.8)

where  is the plasma viscosity. The no-slip boundary condition requires that    on the cell surface,    
, and    at the wall,   , where  is the cell velocity, and  is the vessel radius. The “leakback,” defined as the volume flow rate of fluid relative to the cell per unit circumference, is given by











   
d 

(4.3.9)

Since the suspending fluid is incompressible,  is independent of  . Equations (4.3.7) and (4.3.8) may be solved [74] for the pressure gradient in terms of  
and  to give d d



 


  













   

 









    

  

 








(4.3.10) The shear stress acting on the cell surface is given by

 


 


      
 






   


(4.3.11)

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Capsules and Cells

The equations governing the elastic deformation and mechanical equilibrium of the membrane, (4.2.1) through (4.2.3) and (4.3.5) through (4.3.7), and the equations of lubrication theory, (4.3.10) through (4.3.11), may be combined to give a system of ordinary differential equations where the arc length
is the independent variable and , , ,  , , and  are the dependent variables. The internal pressure of the cell is unknown, and the solution is subject to the constraints that the cell volume and surface area are fixed, typically at 90 m and 135 m for human cells. Computing the membrane stresses on the basis of equations (4.2.1) through (4.2.3) requires specification of the unstressed shape of the membrane. This is taken to be a sphere with the same surface area as the red blood cell, giving a radius of 3.2776 m. The plasma viscosity is assigned the value  = 1 cP. The resulting system of six ordinary differential equations has been solved numerically using a multiple-shooting method [68]. Predicted red blood cell shapes are presented in Figure 4.3.2. The computed shapes are convex at the front and concave at the rear, in agreement with the experimental observations shown in Figure 4.1.1. An exception occurs when the tube diameter approaches the critical minimum diameter of about 2.8 m, as discussed in Section 4.2, whereupon the rear of the cell becomes convex, as shown in Figure 4.3.2 for tube diameter 3 m. Note that, near the trailing edge, the membrane bulges outward at the point of minimum gap. For a given tube diameter, the predicted cell shape depends on the flow velocity, as shown in Figure 4.3.2(b). As the velocity is raised, the cell contour becomes more streamlined. At high velocities, the full system of equations becomes increasingly difficult to solve by numerical methods. The results for velocity 0.125 cm/s and for the high-velocity limit were obtained using a version of the model in which membrane bending resistance is neglected. Consequently, a sharp cusp develops at the trailing edge of the membrane. Variations in the cell shape with velocity cause changes in flow resistance, defined as the driving pressure difference divided by flow rate. As the velocity decreases, the gap width in the lubrication region also decreases and the flow resistance increases. Flow resistance may be expressed in terms of apparent viscosity  defined in equation (4.3.2). In the theoretical model, because the interaction between neighboring cells is neglected,  depends linearly on the local tube hematocrit  , defined as volume fraction of red blood cells in the tube,  

 

 


(4.3.12)

where the apparent intrinsic viscosity,  , depends on the cell shape. To compute  , estimates of the pressure drop along the tube are required. The pressure drop  across each cell is obtained from the numerical computations. In the regions between cells, the pressure drop is assumed to be given by Poiseuille’s law, yielding







where  is the capillary radius, in the deformed state.

 



 











(4.3.13)

is the cell volume, and  is the length of the cell

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175

(a)

8 7 6 5 4 3

8

7

6

5

4

3

(b)

0.001

0.02

0.05

0.125

High velocity limit:

6 µm

Figure 4.3.2 Computed shapes of axisymmetric red blood cells in uniform tubes, modified after [59]. The flow direction is from right to left. (a) Cell velocity = 0.01 cm/s; the displayed numerical values denote tube diameters in m. (b) Vessel diameter = 6 m; the displayed numerical values denote velocities in cm/s.

Figure 4.3.3 displays theoretical and experimental results for  in a range of capillary diameters and cell velocities. Several trends are evident in the numerical results represented by the solid symbols. First, flow resistance is elevated at very low cell velocities; because of the reduced fluid forces acting on the cell, the membrane tends to bulge outward filling the tube, as shown in Figure 4.3.2. The effect is less pronounced for tube diameters of 7 or 8 m because the cells nearly fit inside the tube in the undeformed state. At very small tube diameters approaching the critical minimum diameter for the passage of intact cells, the flow resistance increases markedly. The results displayed in Figure 4.3.3 for diameters less than 4 m correspond to the high velocity limit where shear and bending forces in the cell membrane are neglected. However, similar results are obtained when these forces are included in the calculation [34].

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Capsules and Cells

Figure 4.3.3 Variation of  with tube diameter and red blood cell flow velocity. The solid circles represent numerical results with cell velocities indicated on the left in cm/s. The solid squares represent numerical results in the highvelocity limit [36, 59, 68]. The open symbols represent the experimental results of Lee & Fung [41], with scaled velocities indicated on the right in cm/s. The solid curve represents and empirical fit to experimental in vitro data [54].

The open symbols in Figure 4.3.3 represent results obtained by Lee & Fung [41] using macroscopic rubber models of red blood cells. The tube diameters used in their studies have been rescaled here according to cell dimensions. Moreover, the indicated red blood cell velocities have been rescaled so that the dimensionless parameter   ! matches that in the model experiments, where  is the cell velocity, and ! is the elastic modulus of the membrane in uniaxial in-plane stress [68]. Good overall agreement between these results and the theoretical predictions is observed. Pries et al. [54] compiled data on the apparent viscosity of suspensions of red blood cells flowing in narrow glass tubes from a variety of studies, and developed an empirical curve to describe the variation of viscosity with tube diameter. Values of  predicted by their empirical fit for diameters up to 8 m are shown with the solid curve in Figure 4.3.3. Red blood cell velocities included in this compilation are generally in the higher range, around 0.1 cm/s. The theoretical results shown here for the high-velocity limit generally agree well with the empirical result. For a tube diameter of about 4 m, the experimental data suggest higher than predicted values of  . It should be noted that the theoretical results assume a homogeneous population of red blood cells. In reality, because cell size varies, a few larger cells can

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177

contribute significantly to flow resistance. For a comparisons of model predictions with further experiments using red blood cell suspensions, see References [59, 68].

4.3.2 Nonaxisymmetric models for red blood cell motion in capillaries The actual shapes of red blood cells in single-file capillary flow are not axisymmetric, as illustrated in Figure 4.1.1. The question then arises as to whether asymmetry of shape leads to appreciably different predictions for the apparent viscosity. In particular, when the shape is not axisymmetric, continuous tank-treading motion of the membrane can occur relative to the cell [30]. Secomb & Skalak [67] developed a two-dimensional model in which tank-treading is described by the motion of the membrane around the cell perimeter at a constant rate. The results revealed the including the tank-treading motion significant decreases the pressure drop required to drive the flow. A limitation of the twodimensional analysis is the inability to account for the continuous membrane deformation exhibited in the physical three-dimensional space. To investigate this issue further, Hsu & Secomb [36] developed a three-dimensional model of nonaxisymmetric cells moving through cylindrical capillaries. The main goal of the analysis has been to compute the overall cell motion and estimate the membrane tank-treading frequency from hydrodynamics considerations. To simplify the analysis, the cell shapes and membrane motion associated with tank-treading are prescribed based on observation [30], as shown in the middle frame of Figure 4.1.1. In the mathematical description, the rear part of the axisymmetric cells shown in Figure 4.3.2(b), corresponding to the high-velocity limit, is truncated at an oblique angle and then deformed to a concave shape by fitting a part of an ellipsoid [36]. The modified shape is further smoothed to remove the cusp at the trailing edge. The membrane motion is specified using the approach described in Section 4.2.2, with being a suitably chosen polynomial function of Cartesian coordinates fixed on the cell, while the tank-treading frequency  is an unknown parameter to be determined as part of the solution. On the assumption that the gap width " is small compared to the radius  of the capillary, lubrication theory is applied to describe the motion of plasma in the narrow space between the red blood cell and the vessel wall. Lubrication analysis shows that the pressure field in the gap satisfies the Reynolds equation in three dimensions (i.e., [5], p. 61),

  "    







  "  
"        "
  "
     

 

(4.3.14)

where "   
is the gap width,    
is gap pressure,  is the velocity of the capillary relative to the stationary cell in the chosen frame of reference, and    
,    
are the components of the membrane tank-treading velocity

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Capsules and Cells

given in equation (4.2.4). The boundary conditions require that    at the front of the cell and    at the rear of the cell, where  is the driving pressure. The Reynolds equation can be solved by finite difference methods [36], and the results can be used to express the net force and torque exerted on the cell in terms of the driving pressure drop and the linear and angular velocities of the cell for any given cell position and orientation. The condition of zero net force and torque on the cell is then applied to obtain cell trajectories. The tank-treading frequency  is obtained by requiring that the net rate of working of external forces exerted on the cell is equal to the rate of energy dissipation due to the membrane. Using methods described in Reference [66], it can be shown that the internal fluid dissipation is smaller than the dissipation in the membrane [26] and internal fluid dissipation is therefore neglected. For capillary diameters ranging from 5 to 7 m, predicted tank-treading frequencies lie in the range 0.076 s
to 0.2 s
for cell velocity of 1 mm/s. The membrane velocity is thus much smaller than the cell velocity, and the effect of tank-treading on flow resistance is correspondingly small. Tank-treading reduces the flow resistance at most by a few percent with respect to that of a rigid cell with the same shape. This contrasts with the results of the two-dimensional model [67] where a relatively rapid tank-treading motion was found to occur, causing a significant decrease in flow resistance. In the three-dimensional model, viscous dissipation associated with membrane motion tends to inhibit tank-treading. Comparisons were also made with results of corresponding calculations for axisymmetric cell shapes. It was found that cell asymmetry and tank-treading do not have a significant effect on the resistance for single-file flow in capillaries ranging from 5 to 7 m in diameter. This finding is in line with the fact that axisymmetric models yield predictions for the apparent viscosity of blood flow in capillaries that agree well with experimental observations [68]. Overall, the results presented in Sections 4.3.1 and 4.3.2 indicate that theoretical models based on knowledge of the mechanical properties of individual red blood cells provide us with a good quantitative description of the mechanics of single-file red blood cell motion in capillary-sized glass tubes. However, subsequent work has shown that the rheological properties of blood in living capillaries differ substantially from those of blood in glass tubes with corresponding diameters. The major reason for this difference is the presence of a relatively thick layer of macromolecules adhering to the inner surface of living capillaries, up to about 1 m). which are responsible for a large increase in capillary flow resistance in vivo [55, 56]. In recent years, several theoretical analyses have been conducted to describe the motion of red blood cells in narrow tubes including the presence of a fluid-permeable layer lining the wall [14, 23, 65].

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179

4.4 Multi-file motion of red blood cells in microvessels Describing the micro-mechanics of blood flow in microvessels with intermediate diameters ranging from 8 to 300 m is particularly challenging. In single-file flow, the shape and motion of the red blood cells for a given driving pressure drop are largely determined by hydrodynamic interactions with the vessel walls, as already described. However, in vessels whose diameter is larger than 8 m, any cross-section through the vessel typically intersects multiple cells, as shown at the bottom frame of Figure 4.1.1, and the simplified analysis based on the motion of a single red blood cell is no longer appropriate. On the other hand, the red blood cells are not sufficiently small relative to the vessel diameter for a continuum representation of blood flow to be valid. Some of the factors influencing the motion of red blood cells under these conditions are shown schematically in Figure 4.4.1. In a cylindrical vessel, the average shear stress increases linearly with distance from the center-line to the vessel wall. Each red blood cell is therefore subjected to a level of shear stress that depends on the radial position. In addition, the cells experience forces originating from hydrodynamic interactions with neighboring cells as well as from interactions with vessel walls. Insights into the complex flow system can be obtained by examining the motion of red blood cells under idealized conditions, e.g., in a dilute suspension, far from the vessel walls, and for a linear velocity field with uniform shear stress. Relevant studies will be discussed in Section 4.5. In the remainder of this section, results of experimental studies of blood flow in narrow tubes will be described. Combining these with relatively simple theoretical arguments will reveal that the presence of a cell-free or cell-depleted layer near the vessel wall has a major effect on the global flow properties of blood in narrow tubes.

4.4.1 The F˚ahraeus-Lindqvist effect From a physiological standpoint, the most significant feature of blood flow mechanics in the range of diameters presently considered is the F˚ahraeus-Lindqvist effect [22]. Experiments using glass tubes show that the apparent viscosity of blood decreases when the tube diameter is reduced approximately below 1 mm. Pries et al. [54] compiled data from a number of sources and fitted a single empirical relationship giving the variation of relative apparent viscosity with the tube diameter for a fixed discharge hematocrit of 45%. Figure 4.4.2 reveals that the decrease in apparent viscosity with reduction of tube diameter extends to diameters as low as 8 m. Upon further decrease, the apparent viscosity increases, as discussed in Section 4.3.1. It should be emphasized that these results apply to blood flow velocities that lie in the high end of the physiological range. At low flow velocities, two factors contribute to raise the apparent viscosity of blood in bulk shear flow [9]: reduced deformation of red blood cells with decreasing levels of stress, and tendency of the cells to aggre-

180

Capsules and Cells

Figure 4.4.1 Schematic illustration of blood flow in a micro-vessel. The velocity profile indicates typical variation in the flow velocity. Each red blood cell is subjected to shear stresses that depend on the distance from the center-line, and to forces resulting from hydrodynamic interactions with other neighboring cells and vessel wall (short arrows.)

gate. In narrow tubes, cell deformability leads to elevated flow resistance at low flow velocities, as shown in Figure 4.3.1. However, red blood cell aggregation generally has the opposite effect on flow resistance in narrow tubes. Aggregated red blood cells are drawn toward the tube centerline, causing an increase in the width of the cell-depleted layer and a concomitant decrease in the flow resistance [11, 57]. Although several factors contribute to the F˚ahraeus-Lindqvist effect, the main physical reason for the reduction in apparent viscosity is the tendency of the cells to migrate away from the vessel wall, thereby causing the formation of a low hematocrit region near the wall [33]. A relatively thin layer of reduced hematocrit can have a substantial effect on flow resistance. To see this, consider a modified version of the axial-train model in which a core with a uniform viscosity  is surrounded by a cell-free annular wall layer with viscosity  , as depicted in Figure 4.4.3 [61, 76]. The fluid velocity is given by



 


 

 

¾ ¾



   
  ¾ ¾


where  is the tube radius and





    for    for

 d

  d

(4.4.15)

(4.4.16)

The apparent viscosity arises by integrating the velocity profile to compute the flow rate, and then applying equation (4.3.2) to obtain  
















(4.4.17)

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181

Figure 4.4.2 Variation of the relative apparent viscosity with tube diameter for discharge hematocrit of 45%. The solid curve represents an empirical fit to experimental in vitro data [54]. The solid circles represent numerical results in the high-velocity limit [36, 59, 68]. The dashed curve corresponds to the modified axial-train model. Results for diameters of 8 m or less correspond to those shown in Figure 4.3.3. The dependence of the apparent viscosity on hematocrit is nonlinear for larger diameters and cannot be expressed in terms of a single parameter  as in Figure 4.3.3.

Results of this model computed under the assumption that the effective width of the cell-free layer, Æ , is independent of the vessel diameter (   Æ), are shown in Figure 4.4.2 for Æ = 1.8 m and  /  = 3.3. These values were chosen empirically to provide a good fit to the experimental curve for diameters above 30 m where the assumption of a constant wall layer thickness is reasonable [61]. To this end, it should be noted that the assumption that the core viscosity is independent of the tube diameter disregards changes in the core hematocrit with diameter. More detailed calculations show that the expected change in hematocrit over the range of diameters considered is small and has little effect on the model predictions. Several important observations should be made regarding the model predictions and comparison with experimental data. Firstly, a thin cell-free layer only 1.8 m wide has a significant effect on the apparent viscosity even for diameters in the range 100 to 300 m. Secondly, the assumed width of the layer is comparable to the size of the red blood cells and lies within the range of experimental observations [4, 12]. Thirdly, the model provides a good fit to the measured apparent viscosity for diameters roughly above 30 m but underestimates the data for diameters below 30 m. This suggests that the effective width of the layer decreases with decreasing diameter below 30 m. Finally, it should be emphasized that the width Æ of the cell-free layer is an empirical parameter.

182

Capsules and Cells

r

z

Figure 4.4.3 Schematic illustration of the modified axial-train model for blood flow in a micro-vessel. Shading indicates core region containing red blood cells, with viscosity  . The velocity profile is indicated by row of arrows (cf. Figure 4.3.1.)

4.4.2 The F˚ahraeus effect The F˚ahraeus effect [21] is another important physiological consequence of the formation of a low-hematocrit region near the tube wall. Because red blood cells tend to accumulate at higher-velocity region of the flow field, their transit time through a tube is shorter on average than that of the plasma. Consequently, the hematocrit within the tube, termed the tube hematocrit and denoted by  , is reduced with respect to the hematocrit of the fluid exiting the tube, termed the discharge hematocrit and denoted by  . The two hematocrits are related by











(4.4.18)

where  is the mean velocity of the red blood cells, and is the overall mean flow velocity [73]. According to the modified axial-train model, the value of this ratio is given by









 



 







(4.4.19)

In Figure 4.4.4, this prediction is compared with an empirical relationship developed based on in-vitro measurements of the F˚ahraeus effect [55]. The modified axial-train model with a fixed plasma layer width of 1.8 m predicts a similar trend with tube diameter, although the reduction in hematocrit is smaller than observed experimentally. For diameters in the capillary range, the results of the single-file model agree well with the experimental data.

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183

Figure 4.4.4 Dependence of the F˚ahraeus effect expressed by the hematocrit ratio   on the tube diameter. The solid curve represents the empirical fit to in vitro experimental data [55]. The solid circles represent numerical results for cell velocity 0.1 cm/s [59, 68]. The dashed curve represents the modified axial-train model.

4.4.3 Hematocrit partition in bifurcations The presence of a cell-depleted wall layer is responsible for a third significant physiological phenomenon, which is the large degree of heterogeneity of the hematocrit seen in the microcirculation [53]. When blood flows through a diverging microvascular bifurcation, a disproportionate fraction of the red blood cells is generally observed to flow into the branch that receives a higher total flow rate, leading to a higher hematocrit in that branch than in the other [58]. In particular, if the flow into the low-flow branch is sufficiently small, red blood cells do not enter that branch. In effect, the low-flow branch “skims” the plasma from the peripheral layer of the flow. Experimental observations of the skimming phenomenon in vitro have been combined with a theoretical model which assumes that the red blood cells follow the streamlines of the underlying flow to provide estimates for the width of the plasma layer. For hematocrit 40%, this has been found to lie in the range 1.5 to 1.9 m [24]. Similar studies in vivo yield a width of 1.6 m [52]. These estimates are close to the value of 1.8 m obtained by comparing experimentally determined values of the apparent viscosity with the predictions of the modified axial-train model. The assumption that particles follow fluid streamlines in bifurcations has been critically examined for particles with the shape of spherical caps [17]. It was found that these particles may significantly deviate from the streamlines depending on the particle initial orientation. However, randomly oriented particles were found to follow on the average the streamlines of the unperturbed flow. This observation provides

184

Capsules and Cells

support for the estimate of the plasma layer width deduced by the models described in the previous paragraph.

4.5 Motion of red blood cells in shear flow Earlier in this chapter, we pointed out the complexity of blood flow in narrow tubes at normal hematocrit levels. A number of experimental and theoretical studies have been conducted under simplified conditions addressing dilute suspensions to bypass cell-cell interactions, simple shear flow to avoid the effects of nonuniform shear fields, and effectively infinite domains to eliminate the effect of walls. Several of these studies are reviewed in this section with emphasis on their implication on the formation of a cell-depleted layer near walls.

4.5.1 Uniform shear flow of dilute suspensions Unidirectional flow with a linear velocity profile and uniform shear stress, also called simple shear flow or viscometric flow, can be established in a cone-and-plane or concentric-cylinder Couette flow device. Fischer et al. [27, 28] used the counterrotating cone-and-plate system described in Section 4.2.2 to study the flow of a suspension of red blood cells diluted with plasma down to a volumetric cell concentration of 5%. Observations revealed that the cells assume irregular shapes and rotate continuously in the flow. In further experiments, the viscosity of the suspending medium was raised to much higher levels on the order of 11 cP, or even higher, by adding Dextran [28]. In this case, stable red blood cell orientations and tank-treading were observed, as shown in Figure 4.2.2. Two main modes of red blood cell motion are therefore possible in a dilute suspension: unsteady rotating or “flipping” motion for cells suspended in plasma, and “tank-treading” motion with steady orientation for cells suspended in high-viscosity media. Conditions for the onset of these two modes of motion were investigated by Keller & Skalak [38] using a theoretical model. A single ellipsoidal cell whose area and volume approximate those of human red blood cells was considered in infinite shear flow. The tank-treading motion of the membrane was prescribed in such a way that the membrane velocity is a linear function of spatial position. Although such an assumed membrane motion is not area-conserving, it has the advantage of allowing the resulting velocity field to be expressed analytically using a generalization of the analytical solution obtained by Jeffery for a rigid ellipsoid in a shear flow [37]. The interior fluid was assumed to be a viscous liquid with arbitrary viscosity. The effect of the membrane viscosity was not incorporated, though it can be included by introducing an effective internal viscosity that accounts for membrane dissipation.

Red blood cell mechanics

185

Theoretical analysis predicted and confirmed the two modes of red blood cell motion previously described. In the case of unsteady flipping motion, rotation of the cell about an axis perpendicular to the flow direction occurs along with a tank-treading motion whose magnitude and direction vary according to the cell orientation. In steady tank-treading motion, the long axis of the cell was found to be oriented at a small angle with respect to the flow direction, typically 5 to 20Æ . For any given cell shape, the prevailing mode of motion – steady tank-treading or flipping – is determined by the ratio of the effective internal viscosity to the suspending medium viscosity. Specifically, higher levels of external medium viscosity tend to lead to steady tank-treading. For red blood cells suspended in plasma, flipping motion was predicted as seen experimentally [27]. A limitation of this approach is that the cell shape is prescribed a priori instead of arising by considering the cell membrane mechanics [38]. Several subsequent studies examined the motion of flexible capsules with properties representative of those of red blood cells, as discussed in Chapters 1 and 2. For example, Barth`esBiesel & Sgaier [2] considered the small deformation of an initially spherical capsule and showed that both steady and flipping motions are possible depending on the membrane properties. Zhou & Pozrikidis [80] analyzed the motion of particles with area-conserving membranes and demonstrated the dependence of the tank-treading frequency and other observable properties on the undeformed shape. Eggleton & Popel [16] used the immersed-boundary method to examine the large deformation of red blood cell ghosts with equal internal and external viscosities using a realistic strain-energy function for the in-plane membrane elastic response. These studies contributed substantial advances toward the realistic simulation of red blood cell motion in a linear shear flow.

4.5.2 Uniform shear flow of non-dilute suspensions Using the counter-rotating cone-and-plate system described in Section 4.2.2, Fischer et al. [27, 28] observed the behavior of whole human blood at normal hematocrit (45%) in shear flow. At shear rates of 500 s
or higher, the red blood cells were seen to assume approximately ellipsoidal shapes with the long axes positioned approximately parallel to the flow. Tank-treading of the membrane around the cell was revealed by the motion of membrane-bound Heinz bodies [25]. Their observations suggest that tank-treading and alignment of elongated cells with the flow should be expected to occur in blood flow through narrow tubes at normal hematocrit. An interesting consequence of the deformability of red blood cells is that suspensions can be prepared at concentrations of 95% or even higher, and such dense suspensions may readily undergo shear flow. Secomb et al. [62, 63] developed a theoretical model of the flow under these conditions. Experimental observations show that the cells are packed closely together, with the cell membrane taking polyhedral shapes whose sides are separated by thin lubricating layers of plasma. Tank-treading membrane motion is accounted for using the techniques described in Section 4.2.2. The relative apparent viscosity of the suspension was predicted based on a lubrication analysis of the fluid motion inside the gaps between the cells, taking into

186

Capsules and Cells

account that layers of cells slide past adjacent layers. The theoretical results are in good agreement with experimental observations [10, 62].

4.5.3 Boundary effects in dilute suspensions Lateral migration of red blood cells toward the tube centerline causes the formation of a wall layer containing few or no red blood cells. Symmetry arguments can be made to show that no such migration is expected on average in a uniform unbounded shear flow for an infinitely dilute suspension of particles or a suspension with uniform concentration. In these cases, the flow field is invariant under a 180Æ rotation about an axis parallel to the vorticity vector of the flow in the particle frame of reference. Particle migration across streamlines would violate this symmetry. However, in a tube or channel flow, two additional factors can break the symmetry of the flow and allow for the possibility of net lateral migration. Firstly, the presence of walls or boundaries can influence the trajectories of the particles near the walls. Secondly, the velocity profile is nonlinear in the presence of a pressure gradient driving the flow. Further symmetry arguments can be made to define the conditions for lateral particle migration in tube and channel flow in the absence of appreciable inertial effects. The assumption of zero Reynolds number flow is appropriate for blood flow in microvessels. If the particle and the boundaries are rigid, then the governing equations are linear and invariant under a reversal of the velocity. According to this argument, a spherical particle suspended in a uniform tube or channel cannot migrate laterally: a reversal of the flow direction would imply opposite migration in an effectively identical flow system. The same argument applies to a nonspherical rigid particle undergoing continuous rotation (flipping), as long as the particle shape is symmetric with respect to a plane transverse the flow direction at some point during its rotation. For example, an rigid ellipsoidal particle satisfies this condition. In this case, the velocity reversal argument shows that no net lateral migration can occur during a complete cycle of particle rotation. It should be noted that this reasoning does not rule out the possibility of lateral migration in the case of rigid particles that assume a stable orientation [71] or rigid particles in nondilute suspensions [51]. The symmetry arguments indicate that particle deformability may play a crucial role in determining particle migration across streamlines in a zero Reynolds number flow, by allowing lateral drifting to occur in situations where rigid particles would not migrate. Goldsmith [32] examined the motion of rigid spheres, rigid disks, and deformable drops with diameters ranging from 0.2 to 1 cm in tube flow. A particle Reynolds number, #$ , was defined based on the particle size and the estimated deviation of the particle velocity from the local mean flow velocity. For #$ %   , rigid particles exhibited no net migration, but deformable drops migrated toward the tube axis. At higher values of #$ , above   , deformable particles migrated toward the axis whereas rigid particles migrated toward an equilibrium position located somewhere between the axis and the wall. The physical origin of migration to an equilibrium position, described as the “tubular pinch effect” is attributed to inertial forces [69].

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187

Goldsmith [32] also examined the motion of a dilute suspension of normal red blood cells for flow through glass tubes with diameters 60 to 200 m, and compared the motion with that of chemically rigidified cells. When suspended in low-viscosity media, cells of both types were observed to rotate continuously as they traveled along the tube at off-center positions. However, flexible cells spent more time oriented with the flow direction than rigid particles. At particle Reynolds numbers of about
  , normal red blood cells showed appreciable migration toward the axis at a distance of 1 cm downstream from the entry into an 83- m tube, whereas hardened red blood cells showed a fairly uniform distribution over the tube cross-section. When the cells were suspended in media with high viscosity, in the range 15 to 50 cP, rotation was no longer observed and the cells assumed stable orientations with the major axes oriented at 5 to 25Æ with respect to the flow direction. The similarity with the observations described in Section 4.4.1 for uniform shear flow appears to suggest that the cell membrane undergoes tank-treading membrane, although this was not directly observed in the experiments. Cell migration toward to tube axis was faster in the high- than low-viscosity suspending media. Overall, the experiments demonstrated that the deformability of red blood cells is an essential factor in determining migration toward the centerline of a micro-vessel in a dilute suspension. Theoretical studies of the lateral migration of deformable particles in dilute suspensions have considered simplified models involving liquid drops with isotropic or variable surface tension [13, 39, 40, 75]. Numerical and analytical studies have been successful in predicting drop migration drops toward the channel or tube centerline. In a recent series of papers, Olla [47, 48, 49, 50] examined the lateral migration of ellipsoidal particles in wall-bounded shear flow. The theory of Keller & Skalak [38] was used as a starting point, including a prescribed tank-treading membrane motion. The perturbation resulting from imposing the condition of no-slip on the plane boundary was calculated, and the transverse force exerted on the particle was estimated. For both nearly spherical particles [47] and particles with finite ellipticity [48], it was shown that stably oriented tank-treading particles exhibit a lateral drift away from the wall. This wall effect decays as the inverse square of the particle distance from the wall. In two further papers, the effect of a quadratic velocity profile on particle migration was considered. Olla [49] proposed that asymmetric deformation of a particle resulting from the quadratic component of the velocity profile is responsible for migration toward the channel centerline. More recently, a detailed analysis of the mechanics was introduced for nearly spherical vesicles enclosed by inextensible elastic membranes [50]. Drift of tank-treading particles with steady orientations toward the centerline was confirmed, and conditions under which a particle in the flipping-motion regime can show lateral migration were established. In summary, experimental and theoretical analyses have shown that deformable particles in a dilute suspension tend to migrate toward the centerline in tube and channel flow. This phenomenon seems to be more pronounced when the particle is tank-treading with a stable orientation, though it also occurs for particles exhibiting flipping motion. Such a migration tendency might reasonably be expected to occur also in blood flow through microvessels at normal hematocrit levels where the red

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blood cells are subject to strong inter-particle interactions, as will be discussed in the next section.

4.5.4 Factors determining the width of the cell-depleted layer It is evident from the discussion in Section 4.4 that the effective width of the cell-free or cell-depleted wall layer is a crucial determinant of blood flow in the microcirculation. As indicated previously, red blood cells in a dilute suspension undergoing tube flow tend to migrate toward the centerline as a result of the hydrodynamic interactions with the walls and because of the parabolicity of the velocity profile associated with an axial pressure gradient. In nondilute suspensions, this tendency is opposed by hydrodynamic interactions with other cells which become more pronounced at increased cell concentrations. It is reasonable then to propose that the width of the cell-free layer is determined by the balance between the various forces acting on cells at the outer edge of the red cell column, as shown in Figure 4.4.1. The effect of cell-to-cell interactions is expected to become more important with increasing hematocrit, resulting in a decrease of the width of the cell-free layer, as observed in the laboratory [4]. The hematocrit level also affects the type of motion that the cells undergo when suspended in plasma or another low-viscosity medium. At very low hematocrit, continuous rotation or flipping motion is observed [27, 32], whereas in the physiological hematocrit range the cells assume nearly steady orientations and exhibit continuous tank-treading motion [25]. A transition between these types of behavior presumably occurs at an intermediate hematocrit. The steady orientation seen at physiological hematocrit levels indicates that intercell hydrodynamic interactions stabilize red blood cell orientations in shear flow. This stabilization is not exclusively the result of the increased overall viscosity of the suspension at normal hematocrit. Indeed, the viscosity of whole blood is about 3 cP at high shear rates, which is much less than the minimum viscosity needed to stabilize red blood cells orientations in dilute suspensions, about 9 cP [3]. This effect of hematocrit on red blood cell motion may significantly affect the width of the cell-depleted layer and other flow characteristics. In principle, it should be possible to predict the profile of the hematocrit in a microvessel under given conditions based on a knowledge of the mechanical properties of individual red blood cells, by analyzing the motion and deformation of the cells. Such an analysis would, for example, permit the prediction of the effect of the tube diameter, hematocrit, flow velocity, and red blood cell properties on flow resistance in vessel segments and on hematocrit distribution in diverging bifurcations. Unfortunately, progress on these topics has so far been limited. Two broad strategies can be envisaged. One strategy is to simulate explicitly the motion of multiple individual cells in a vessel. This is likely to be more feasible for relatively small vessel diameters, about 8 to 30 m. An alternative strategy is based on a quasi-continuum approximation where the core region is treated as a continuum and a detailed analysis of cell dynamics is conducted near the walls. This strategy is appropriate for capillary diameters in the range above about 30 m.

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An example of the former approach can be found in the analysis of the two-file “zipper” flow [72]. This type of flow is seen in tubes with diameters of about 8 m, where two files of cells are arranged in an alternating, interdigitating pattern [29]. The model [72] uses a two-dimensional representation of the flow pattern where each cell has a prescribed triangular shape with rounded corners. Tank-treading is implemented by the continuous motion of the flexible membrane around the perimeter. A finite-element method was used to analyze the motion of the fluid around and within the cells, and thereby predict their trajectories, under the assumption that the cells remain in the periodic array. If the ratio of the internal viscosity to the external viscosity is below a critical value, a stable equilibrium zipper-type arrangement is observed, with the spacing between the cells and the walls being larger than the distance between adjacent cells. Increasing the viscosity ratio above this critical level results in a cyclic oscillation of the individual cells. These results indicate that tank-treading tends to stabilize the two-file flow of cells in ordered configurations. Extension of this approach involving the simulation of multiple discrete particles to account for realistic multi-file configurations is a major computational challenge. In recent years, numerical techniques have been developed to analyze the motion of multiple particles in shear and channel flow [45, 46, 51, 79] (see also Chapters 1 – 3). Rigid particles or droplets with isotropic surface tension have been mostly considered by previous investigators. As yet, application of these techniques to study the three-dimensional motion of random arrays of particles whose mechanical properties faithfully approximate those of red blood cells has not been reported. With increasing computing power, such computations may soon become feasible, although accounting for the exact cell mechanical properties is expected to contribute substantially to the computational complexity. The second strategy mentioned previously involves a quasi-continuum approach where the forces driving a cell toward the tube center-line are predicted by methods similar to those discussed in Section 4.5.3. A continuum description could be used to estimate the average force on a cell resulting from interactions with other cells. Such estimation could rely on the concept of shear-induced diffusion in a suspension, where particle-particle interactions resulting from the imposed shear flow lead to an overall net lateral particle migration in the direction of decreasing concentration [32, 42]. Carr et al. [7, 8] introduced an effective particle diffusion coefficient to predict the development of the hematocrit profile with distance along a micro-vessel. However, the width of the cell-free wall layer was a priori assumed in the transport equations. The quasi-continuum approach appears to be a promising direction for future development.

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4.6 Conclusions Experimental and theoretical efforts to understand the mechanics of blood flow in narrow tubes commenced in earnest in the 1960s [1, 12, 31, 43]. Many of the key phenomena involved were recognized within a period of only a few years. Since that time, accumulated knowledge of the mechanical properties of red blood cells and increasingly powerful computational techniques have allowed progress in developing quantitative models for blood flow in microvessels taking into account the mechanics of fluid flow and red blood cell deformation. Such models have been helpful for understanding, in some detail, the behavior of individual red blood cells flowing in single file in capillaries, and in shear flow when in dilute suspension in larger flow domains. However, detailed understanding of the mechanics of blood flow in microvessels larger than capillaries has presented formidable challenges, largely because of the inherent difficulties encountered in modeling the motion of multiple, interacting, highly deformable particles. Apart from its intrinsic interest, further work in this important field will have a direct bearing on understanding and ensuring the proper physiological functioning of the cardiovascular system. This work was supported by National Institutes of Health Grant HL34555. I thank Dr. A.W. El-Kareh for helpful comments on this chapter, and Drs. A.R. Pries and T.M. Fischer for providing previously unpublished photographs shown in Figures 4.1.1 and 4.2.2.

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References [1] BARNARD , A. C. L., L OPEZ , L., & H ELLUMS , J. D., 1968, Basic theory of blood flow in capillaries, Microvasc. Res., 1, 23-34. [2] BARTH E` S -B IESEL , D. & S GAIER , H., 1985, Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow, J. Fluid Mech., 160, 119-135. [3] B ESSIS , M. & M OHANDAS , N., 1975, Deformability of normal, shapealtered, and pathological red cells, Blood Cells, 1, 315-321. [4] B UGLIARELLO , G. & S EVILLA , J., 1970, Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes, Biorheology, 7, 85-107. [5] C AMERON , A., 1966, The Principles of Lubrication, Wiley, New York. [6] C ANHAM , P. B. & B URTON , A. C., 1968, Distribution of size and shape in populations of normal human red cells, Circ. Res., 22, 405-422. [7] C ARR , R. T., 1989, Estimation of hematocrit profile symmetry recovery length downstream from a bifurcation, Biorheology, 26, 907-920. [8] C ARR , R. T. & X IAO , J., 1995, Plasma skimming in vascular trees: numerical estimates of symmetry recovery lengths, Microcirculation, 2, 345-353. [9] C HIEN , S., 1975, Biophysical behaviour of red cells in suspensions, In: Surgenor, D. M. N. (ed.), The Red Blood Cell, Vol. II, Academic Press, San Diego, pp. 1031-1133. [10] C HIEN , S., U SAMI , S., TAYLOR , H. M., L UNDBERG , J. L., & G REGERSEN , M. I., 1966, Effects of hematocrit and plasma proteins on human blood rheology at low shear rates, J. Appl. Physiol, 21, 81-87. [11] C OKELET, G. R. & G OLDSMITH , H. L., 1991, Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates, Circ. Res., 68, 1-17. [12] C OPLEY, A. L. & S TAPLE , P. H., 1962, Haemorheological studies on the plasmatic zone in the microcirculation of the cheek pouch of Chinese and Syrian hamsters, Biorheology, 1, 3-14. [13] C OULLIETTE , C. & P OZRIKIDIS , C., 1998, Motion of an array of drops through a cylindrical tube, J. Fluid Mech., 358, 1-28. [14] DAMIANO , E. R., 1998, The effect of the endothelial-cell glycocalyx on the motion of red blood cells through capillaries, Microvasc. Res., 55, 77-91. [15] D ISCHER , D. E., M OHANDAS , N., & E VANS , E. A., 1994, Molecular maps of red cell deformation: hidden elasticity and in situ connectivity, Science, 266, 1032-1035.

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[16] E GGLETON , C. D. & P OPEL , A. S., 1998, Large deformation of red blood cell ghosts in a simple shear flow, Phys. Fluids, 10, 1834-1845. [17] E L -K AREH , A. W. & S ECOMB , T. W., 2000, A model for red blood cell motion in bifurcating microvessels, International Journal of Multiphase Flow, 26, 1545-1564. [18] E VANS , E. A., 1983, Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipette aspiration tests, Biophys. J., 43, 27-30. [19] E VANS , E. A. & H OCHMUTH , R. M., 1976, Membrane viscoelasticity, Biophys. J., 16, 1-11. [20] E VANS , E. A. & S KALAK , R., 1980, Mechanics and Thermodynamics of Biomembranes,. CRC Press, Boca Raton. ˚ [21] F AHRAEUS , R., 1928, Die Str¨omungsverh¨altnisse und die Verteilung der Blutzellen im Gef¨aßsystem. Zur Frage der Bedeutung der intravascul¨aren Erythrocytenaggregation, Klin. Wochenschr., 7, 100-106. ˚ [22] F AHRAEUS , R. & L INDQVIST, T., 1931, The viscosity of the blood in narrow capillary tubes, Am. J. Physiol., 96, 562-568. [23] F ENG , J. & W EINBAUM , S., 2000, Lubrication theory in highly compressible porous media: the mechanics of skiing, from red cells to humans, J. Fluid Mech., 422, 281-317. [24] F ENTON , B. M., C ARR , R. T., & C OKELET, G. R., 1985, Nonuniform red cell distribution in 20 to100 um bifurcations, Microvasc. Res., 29, 103-126. [25] F ISCHER , T. M., 1978, A comparison of the flow behavior of disc shaped versus elliptic red blood cells (RBC), Blood Cells, 4, 453-461. [26] F ISCHER , T. M., 1980, On the energy dissipation in a tank-treading human red blood cell, Biophys. J., 32, 863-868. ¨ , H., 1977, Tank tread motion of red [27] F ISCHER , T. & S CHMID -S CH ONBEIN cell membranes in viscometric flow: Behavior of intracellular and extracellular markers (with film), Blood Cells, 3, 351-365. ¨ , H., 1978, [28] F ISCHER , T. M., S TOHR -L IESEN , M. & S CHMID -S CH ONBEIN The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow, Science, 202, 894-896. ¨ [29] G AEHTGENS , P., D UHRSSEN , C., & A LBRECHT, K. H., 1980, Motion, deformation, and interaction of blood cells and plasma during flow through narrow capillary tubes, Blood Cells, 6, 799-812. ¨ [30] G AEHTGENS , P. & S CHMID -S CH ONBEIN , H., 1982, Mechanisms of dynamic flow adaptation of mammalian erythrocytes, Naturwissenschaften, 69, 294296.

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[31] G OLDSMITH , H. L., 1967, Microscopic flow properties of red cells, Fed. Proc., 26, 1813-1820. [32] G OLDSMITH , H. L., 1971, Red cell motions and wall interactions in tube flow, Fed. Proc., 30, 1578-1590. [33] G OLDSMITH , H. L., C OKELET, G. R., & G AEHTGENS , P., 1989, Robin Fahraeus: evolution of his concepts in cardiovascular physiology, Am. J. Physiol., 257, H1005. [34] H ALPERN , D. & S ECOMB , T. W., 1989, The squeezing of red blood-cells through capillaries with near-minimal diameters, J. Fluid Mech., 203, 381400. [35] H OCHMUTH , R. M. & WAUGH , R. E., 1987, Erythrocyte membrane elasticity and viscosity, Annu. Rev. Physiol., 49, 209-219. [36] H SU , R. & S ECOMB , T. W., 1989, Motion of nonaxisymmetric red blood cells in cylindrical capillaries, J. Biomech. Eng., 111, 147-151. [37] J EFFERY, G. B., 1922, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. Roy. Soc. Lond. A, 102, 161-179. [38] K ELLER , S. R. & S KALAK , R., 1982, Motion of a tank-treading ellipsoidal particle in a shear flow, J. Fluid Mech., 120, 27-47. [39] K ENNEDY, M., P OZRIKIDIS , C., & S KALAK , R., 1994, Motion and deformation of liquid drops, and the rheology of dilute emulsions in shear flow, Computers & Fluids, 23, 251-278. [40] L EAL , L. G., 1980, Particle motions in a viscous fluid, Ann. Rev. Fluid Mech., 12, 435-476. [41] L EE , J. S. & F UNG , Y. C., 1969, Modeling experiments of a single red blood cell moving in a capillary blood vessel, Microvasc. Res., 1, 221-243. [42] L EIGHTON , D. & ACRIVOS , A., 1987, The shear-induced migration of particles in concentrated suspensions, J. Fluid Mech., 181, 415-439. [43] L IGHTHILL , M. J., 1968, Pressure-forcing of tightly fitting pellets along fluidfilled elastic tubes, J. Fluid Mech., 34, 113-143. [44] L IN , K. L., L OPEZ , L., & H ELLUMS , J. D., 1973, Blood flow in capillaries, Microvasc. Res., 5, 7-19. [45] L OEWENBERG , M. & H INCH , E. J., 1996, Numerical simulation of a concentrated emulsion in shear flow, J. Fluid Mech., 321, 395-419. [46] N OTT, P. R. & B RADY, J. F., 1994, Pressure-driven flow of suspensions: simulation and theory, J. Fluid Mech., 275, 157-199. [47] O LLA , P., 1997, The role of tank-treading motions in the transverse migration of a spheroidal vesicle in a shear flow, Journal of Physics A - Mathematical and General, 30, 317-329.

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[48] O LLA , P., 1997, The lift on a tank-treading ellipsoidal cell in a shear flow, Journal de Physique II, 7, 1533-1540. [49] O LLA , P., 1999, Simplified model for red cell dynamics in small blood vessels, Physical Review Letters, 82, 453-456. [50] O LLA , P., 2000, The behavior of closed inextensible membranes in linear and quadratic shear flows, Physica A, 278, 87-106. [51] P OZRIKIDIS , C., 2002, Dynamical simulation of the flow of suspensions: Wall-bounded and pressure-driven channel flow, Ind. Eng. Chem. Res., 41, 6312-6322. [52] P RIES , A. R., L EY, K., C LAASSEN , M., & G AEHTGENS , P., 1989, Red cell distribution at microvascular bifurcations, Microvasc. Res., 38, 81-101. [53] P RIES , A. R., L EY, K., & G AEHTGENS , P., 1986, Generalization of the Fahraeus principle for microvessel networks, Am. J. Physiol., 251, H1324H1332. [54] P RIES , A. R., N EUHAUS , D., & G AEHTGENS , P., 1992, Blood viscosity in tube flow: dependence on diameter and hematocrit, Am. J. Physiol., 263, H1770-H1778. [55] P RIES , A. R., S ECOMB , T. W., G AEHTGENS , P., & G ROSS , J. F., 1990, Blood flow in microvascular networks. Experiments and simulation, Circ. Res., 67, 826-834. [56] P RIES , A. R., S ECOMB , T. W., G ESSNER , T., S PERANDIO , M. B., G ROSS , J. F., & G AEHTGENS , P., 1994, Resistance to blood flow in microvessels in vivo, Circ. Res., 75, 904-915. [57] R EINKE , W., G AEHTGENS , P., & J OHNSON , P. C., 1987, Blood viscosity in small tubes: effect of shear rate, aggregation, and sedimentation, Am. J. Physiol., 253, H540-H547. ¨ , G. W., S KALAK , R., U SAMI , S., & C HIEN , S., 1980, [58] S CHMID -S CH ONBEIN Cell distribution in capillary networks, Microvasc. Res., 19, 18-44. [59] S ECOMB , T. W., 1987, Flow-dependent rheological properties of blood in capillaries, Microvasc. Res., 34, 46-58. [60] S ECOMB , T. W., 1988, Interaction between bending and tension forces in bilayer membranes, Biophys. J., 54, 743-746. [61] S ECOMB , T. W., 1995, Mechanics of blood flow in the microcirculation, In: Ellington, C. P. and Pedley, T. J. (eds.), Biological Fluid Mechanics, Cambridge University Press, Cambridge, 305-321. [62] S ECOMB , T. W., C HIEN , S., JAN , K. M., & S KALAK , R., 1983, The bulk rheology of close-packed red blood cells in shear flow, Biorheology, 20, 295-309.

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[63] S ECOMB , T. W., F ISCHER , T. M., & S KALAK , R., 1983, The motion of close-packed red blood cells in shear flow, Biorheology, 20, 283-294. [64] S ECOMB , T. W. & G ROSS , J. F., 1983, Flow of red blood cells in narrow capillaries: role of membrane tension, Int. J. Microcirc. Clin. Exp., 2, 229-240. [65] S ECOMB , T. W., H SU , R., & P RIES , A. R., 1998, A model for red blood cell motion in glycocalyx-lined capillaries, Am. J. Physiol., 274, H1016-H1022. [66] S ECOMB , T. W. & S KALAK , R., 1982, Surface flow of viscoelastic membranes in viscous fluids, Q. J. Mech. Appl. Math., 35, 233-247. [67] S ECOMB , T. W. and S KALAK , R., 1982, A two-dimensional model for capillary flow of an asymmetric cell, Microvasc. Res., 24, 194-203. ¨ ZKAYA , N., & G ROSS , J. F., 1986, Flow [68] S ECOMB , T. W., S KALAK , R., O of axisymmetric red blood cells in narrow capillaries, J. Fluid Mech., 163, 405-423. [69] S EGR E´ , G. & S ILBERBERG , A., 1962, The behaviour of macroscopic rigid spheres in Poiseuille flow. II. Experimental results and interpretation, J. Fluid Mech., 14, 136-157. [70] S KALAK , R., 1976, Rheology of red blood cell membrane, In: Grayson, J. and Zingg, W. (eds.), Microcirculation, Vol. I, pp. 53-70. [71] S UGIHARA -S EKI , M., 1996, The motion of an ellipsoid in tube flow at low Reynolds numbers, J. Fluid Mech., 324, 287-308. [72] S UGIHARA -S EKI , M., S ECOMB , T. W., & S KALAK , R., 1990, Twodimensional analysis of two-file flow of red cells along capillaries, Microvasc. Res., 40, 379-393. [73] S UTERA , S. P., S ESHADRI , V., C ROCE , P. A., & H OCHMUTH , R. M., 1970, Capillary blood flow. II. Deformable model cells in tube flow, Microvasc. Res., 2, 420-433. ¨ , H. & S KALAK , R., 1978, The steady flow of closely fitting incom[74] T OZEREN pressible elastic spheres in a tube, J. Fluid Mech., 87, 1-16. [75] U IJTTEWAAL , W. S. J. & N IJHOF, E. J., 1995, The motion of a droplet subjected to linear shear flow including the presence of a plane wall, J. Fluid Mech., 302, 45-63. [76] VAND , V., 1948, Viscosity of solutions and suspensions. I. Theory, J. Phys. Colloid Chem., 52, 277-299. [77] W HITMORE , R. L., 1968, Rheology of the Circulation, Pergamon. [78] Z ARDA , P. R., C HIEN , S., & S KALAK , R., 1977, Interaction of viscous incompressible fluid with an elastic body, In: Belytschko, T. and Geers, T. L. (eds.), Computational Methods for Fluid-Solid Interaction Problems, American Society of Mechanical Engineers, New York, 65-82.

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[79] Z HOU , H. & P OZRIKIDIS , C., 1993, The flow of ordered and random suspensions of two-dimensional drops in a channel, J. Fluid Mech., 255, 103-127. [80] Z HOU , H. & P OZRIKIDIS , C., 1995, Deformation of liquid capsules with incompressible interfaces in simple shear flow, J. Fluid Mech., 283, 175-200.

Chapter 5 Capsule dynamics and interfacial transport

A. Nir and O. M. Lavrenteva Cells and capsules are fluid-like bodies enclosed by interfaces that are able to support tangential and normal force distributions, and thereby participate in active and passive motion. Such particles actively respond to an external stimulus associated with a temperature field or with the concentration gradient of a dissolved molecular species, and spontaneously interact when internal nonequilibrium thermodynamic conditions induce transport across the interfaces. In this chapter, theoretical models describing the response of solitary and interacting capsules and small vesicles with position dependent and time varying interfacial tension are reviewed. Single- and multi-body systems are discussed in the context of forced and spontaneous dynamics, and recent theoretical developments on the role of dual transport mechanisms, such as combined diffusion and convection, are considered with emphasis on the effect of interfacial deformation.

5.1 Introduction A capsule is a fluid or solid particle enclosed by a thin surface layer that separates it from the environment. The interfacial layer not only provides mechanical separation, but also serves as a barrier to heat transfer and mass transport of a dissolved chemical species between the encapsulated compartment and the ambiance. The role and significance of the interface are determined by the interface mechanical and rheological properties. Capsules can be broadly classified as natural or manufactured. Natural capsules abound in a variety of contexts. For example, in living tissues, we encounter cells and intra-cellular organelles such as vesicles spanning a wide range of sizes, from microns to nanometers. Encapsulated natural cells are mainly intended for immobilization purposes in bioprocesses [30]. Synthetic capsules are produced by the process of encapsulation or micro-encapsulation achieved by various techniques, including coacervation, condensation, polymerization, phase separation, and mechanical processing such as electrostatic and centrifugal encapsulation [32]. The product is used

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in a variety of industrial, agricultural, food-related, diagnostic, pharmaceutical, and health related applications. Following the advancement of techniques for preparing micro-emulsions of liposomes and niosomes [39], micro-capsules have gained increasing popularity as colloidal drug media in recent years. The structure of these fabricated capsules mimics that of natural capsules used in therapeutic applications. Examples include the deposition of drugs and diagnostic agents in the respiratory system and pulmonary tree [59], agents facilitating drug delivery to specific targets or sites through the cardiovascular system, and capsules serving as transdermal drag carriers [40]. The interfaces of natural capsules typically consist of a membrane that is composed of a phospholipid bilayer, and may also host other species such as proteins. Such membranes flow readily under the action of an external flow and exhibit elastic resistance to stretching and bending deformation. Artificial capsules are enclosed by a variety of coating materials with various physical and mechanical properties depending on the intended application. In some cases, the interfacial layer is a solidlike elastic membrane generated by surface polymerization with cross-linking constituents (e.g., [32]). In other cases, capsules are coated with a soft layer or gel that exhibits fluid-like, viscous or visco-elastic properties. In the preparation of microemulsions, liposomes, and niosomes, added surfactants are responsible for a variety of surface structures ranging from single unilamellar vesicles, to oligolamellar vesicles, to multivesicular bodies where vesicles are enclosed by others in a nested configuration [39]. Understanding the mechanical properties and transport behavior of capsules is of special interest for the optimal design and efficient use of particulate systems. Suspensions of soft-coated capsules are often modeled as emulsions of droplets whose interfaces exhibit isotropic interfacial tension [33]. More complex systems such as those associated with dividing cells are emulated by introducing interfacial layers with variable and anisotropic surface tension [86] and various degrees of viscoelastic behavior [87]. Liquid-drop-like models of capsules with complex interfacial rheological properties are reasonable in the initial stages of preparation of micro-emulsions prior to layer polymerization, and also during the final stages of disintegration of the coating barrier after the content of the micro-capsule has been released and massive transport across the interface has taken place. In these instances, the surface composition and properties are not uniform, and the interfacial tension varies with position over the interface. In addition, the tension varies with temperature and depends on the concentration of dissolved species surrounding and transported across the interface. In the work reviewed in this chapter, suspensions of micro-capsules are modeled as micro-emulsions of fluid particles coated with a thin layer of infinitesimal thickness exhibiting variable surface tension. In particular, the tension varies due to changes in the surface properties induced by an externally imposed field associated with transport of heat or a dissolved molecular species. In addition, the dynamics is influenced by spontaneous interfacial transport from capsules in close proximity. To isolate the effect of variability of the surface tension and the significance of inter-particle interaction, the elastic properties of the interfacial layer is not taken into account.

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The results of theoretical and numerical studies will show that capsules and drops responding to a stimulus in the presence of other particles exhibit a rich dynamics whose satisfactory understanding requires further investigation.

5.2 Model of fluid-particle motion with interfacial mass transport An interface between two immiscible fluids is a transition zone whose thickness spans a few molecular dimensions. Across the interfacial layer, the physical and thermodynamic properties of the medium, such as density, internal energy, and entropy, change rapidly but in a continuous fashion. Certain molecules residing in the interface are composed of segments that favor one of the fluids, while other segments favor the other. Such molecules tend to be adsorbed onto the interface making the composition of the interfacial layer significantly different from that in the bulk fluids. For example, alcohol and organic fatty acid molecules consisting of hydrophobic hydrocarbon chains and hydrophilic polar ends preferentially adsorb on a water-air or water-oil interface. From the macroscopic point of view of continuum mechanics, the effective thickness of the interfacial layer can be assumed to be infinitesimal, and the interface can be regarded as a surface of discontinuity of the physical and thermodynamic properties. Throughout this chapter, the simplest and the most popular mechanical model introduced by Gibbs [28] will be employed. In this model, the interface is regarded as a two-dimensional medium possessing its own free energy, internal energy, entropy, and surface concentration of chemical species. Thus, the interface is assumed to exhibit isotropic surface tension that depends on the temperature and local concentration of a surface-active species, and is the counterpart of the pressure developing in the bulk of a fluid. In further discussion, capsules whose internal viscosity is higher or comparable to that of the ambient fluid will be called drops, whereas capsules whose internal viscosity is much lower than that of the ambient fluid will be called bubbles.

5.2.1 Interface balances Consider mass transfer of a substance that is soluble in both phases and is adsorbed at the interface, under isothermal conditions. The overall mass transfer between the surface and the bulk phases is governed by the balance of the flux due to sorption kinetics, and the diffusive mass flux from the th bulk phase to the interface,






















(5.2.1)

where the index labels the two fluids, is the unit vector normal to the is the diffusivity in the th bulk phase, interface pointing into the fluid labeled 1, and is the species concentration in the th bulk phase. 















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In practice, the flux can be assumed to be proportional to the difference between the actual value of the surface concentration and the value at thermodynamical equilibrium,














(5.2.2)














The coefficient is inversely proportional to the characteristic adsorption/desorption relaxation time. Heretoforth, we shall adopt a simple linear law for the dependence of the equilibrium value of the surface concentration on the bulk concentration (adsorption isotherm), by setting 















(5.2.3)






The constants and are referred to as the adsorption and desorption coefficients, respectively. More involved nonlinear isotherms based on detailed microscopic analysis of adsorption-desorption are also available (e.g., [16, 25], see also Chapter 6). An adsorbed substance is redistributed over the interface by surface diffusion and convection. The surface concentration of the substance is governed by the conservation law [25]



















































 













(5.2.4)



where is the surface gradient, is the interface mean curvature, is the local liquid velocity, is the surface velocity, and is the surface diffusivity. The time derivative on the left-hand side of (5.2.4) expresses the rate of change of a quantity following interfacial marker points that move with the normal component of the fluid velocity. In the case of an insoluble surfactant, surfactant transport does not occur between the phases, and the surface . concentration is governed by equation (5.2.4) with Another important special case arises when a substance is not adsorbed onto the interface. Under these circumstances, the concentration in the bulk phases are in local thermodynamic equilibrium, and the diffusive flux satisfies the continuity conditions 































































 

$









"





(5.2.5)































In (5.2.5), is the ratio of the equilibrium concentrations in the two phases, termed the phase distribution coefficient. When the momentum of the interfacial layer is negligible, an interfacial force balance provides us with the dynamic boundary condition $







(

*

,





(

,

*







1










where tension. (





1



(5.2.6)



,

is the Cauchy stress tensor in the th bulk phase, and 

*



1





is the surface

Capsules and interfacial transport

201

The dependence of the surface tension on the surface concentration can be established experimentally or deduced theoretically from thermodynamic considerations (e.g., [16, 25]). In practice, this dependence is often linearized to yield 



 




(5.2.7) 

  








 










is a reference value. Normally, the interfacial tension decreases as the where in (5.2.7) termed the surface concentration is raised. Thus, the derivative interfacial elasticity is negative over a broad range of surface concentrations. When the surface concentration is in local thermodynamic equilibrium with the concentration in the bulk of the fluids, and when a linear adsorption isotherm applies, the surface tension depends linearly on the bulk concentration at the interface, 




















 






(5.2.8)



  













Surfactants adsorbed on an interface at high concentrations affect not only the surface tension, but also the interfacial mobility and other macroscopic mechanical properties of the interface. Advanced interfacial models designed to describe these dependencies are discussed in Chapters 1 and 2.

5.2.2 Bulk phase transport and Marangoni convection When the total mass of the solute substance is small compared to the mass of the solvent, the bulk phases are nearly incompressible, and the velocity field satisfies the continuity equation 











(5.2.9) 





, where is the velocity in the th phase (e.g., [61]). The kinematic for condition requires that the tangential and normal components of the velocity be continuous across the interface,





















(5.2.10)







Now, Cauchy’s equation of motion in the bulk of the fluids provides us with the evolution equation 















































(5.2.11) 



 

 



for , where is the density, and is the acceleration of gravity (e.g., [61]). If the fluids is Newtonian, the stress tensor is given by


































u

(5.2.12) 







where is the pressure, is the unit matrix, is the fluid viscosity, gradient tensor, and the superscript denotes the matrix transpose.









is the velocity

202

Capsules and Cells

The concentration field in the two fluids satisfies the convection–diffusion equation 































(5.2.13)






for , subject to appropriate boundary conditions. The temperature field in the two fluids is governed by a heat transport equation that is similar to (5.2.13). At the interface, the temperature is required to be continuous, while the combined heat flux from the bulk of the fluids is balanced by the heat produced by adsorption–desorption reaction and by changes in the internal energy of the interface (e.g., [25]). In general, the transport coefficients depend on the local temperature. In the special case of constant surface tension and in the absence of a body force, the governing equations are satisfied for stationary fluids and for interfacial shapes with constant mean curvature. The solution describes quiescent spherical drops and bubbles suspended in a still ambient fluid. The uniform distribution of concentration in each phase and over the interface satisfies 





















(5.2.14)





where , and is the phase distribution coefficient. If the interface is nonisothermal or populated by a surfactant, the surface tension varies with position. Tangential gradients of the surface tension (5.2.6) induce a type of fluid motion known as Marangoni convection. In the case of a suspended drop or bubble, Marangoni convection may have a considerable effect on the self-induced particle motion, causing particle migration even in the absence of a body force or an external flow. When interfacial mass transfer takes place through a collection of particles, concentration nonuniformities may be induced spontaneously as a result of a geometrical anisotropy. For example, consider the transport of a surfactant between two adjacent drops into the continuous phase. Because the concentration in the gap between the drops is higher than elsewhere around the drops, the local interfacial tension is lower than in the other regions over the interface. The developing surface tension gradients induce a flow in the surrounding fluid directed outward from the gap, thereby causing the drops to approach. 







































5.2.3 Marangoni migration of drops and bubbles Consider a swarm of fluid particles with radii suspended in a virtually unbounded ambient viscous fluid, . Let and denote, respectively, the domain occupied by the ambient fluid and the th particle. At the outset, we introduce dimensionless variables defined using as characteristic length the radius of the particle designated as the first particle, , as characteristic velocity the quantity 

































 

 

 



 



(5.2.15)

Capsules and interfacial transport

203 



, and as characteristic pressure and stress . In as characteristic time (5.2.15), is a designated difference of the surfactant concentration whose definition depends on the particular application. In dimensionless variables, the Navier-Stokes equation and continuity equation in the th fluid read






































 






 


 











 

 










 















,















(5.2.17) 

is the ratio of the viscosities of the th and







(5.2.16)





where the point lies in the ambient phase, and














(5.2.18)










is the Reynolds number of the flow in the th phase. The dimensionless concentration field satisfies the linear convection-diffusion equation 

















 



















(5.2.19) 




 



where 



 





(5.2.20) 




is the P`eclet number expressing the importance of convection relative to diffusion. is small, the left-hand side of (5.2.19) can be neglected, and the concentraWhen tion field may be assumed to be a harmonic function to leading-order approximation. In dimensionless variables, the kinematic boundary condition at the interface is expressed by (5.2.10), and the dynamic boundary condition takes the form 

















 























 



 







(5.2.21)




where 







 






(5.2.22) 

 






 









is the capillary number expressing the interface deformability. The problem formulation is completed by specifying the initial concentration and velocity fields, and the initial shape and relative position of the interfaces. In the majority of applications involving capsules and cells, the Reynolds number is much smaller than unity, and the creeping flow approximation can be invoked

204

Capsules and Cells

to replace the nonlinear unsteady Navier-Stokes equation with the linear and quasisteady Stokes equation
























(5.2.23) 

  



where the point lies in . In the limit of vanishing and , corresponding to the “quasi-steady approximation,” the concentration and velocity fields at any instant are computed by solving a linear stationary problem distinguished by the absence of time derivatives in the governing equations and boundary conditions. The physical reason underlying this assumption is that the velocity and concentration fields are established on a time scale that is much shorter than the time scale of geometrical change of the boundaries. The simplified description significantly facilitates the solution by permitting the use of powerful methods for solving linear differential equations, such as series expansions and representations in terms of surface potentials. , surface tension dominates, and the capsules remain spherIn the limit ical at all times. In this case, only the tangential component of the interfacial force balance is retained as a boundary condition, 























 





 



 









(5.2.24) 







where is the tangential projection operator. The complementary normal component is abandoned in favor of the condition of sphericity. The motion of the th particle is described by the evolution of the particle center of mass, , which is governed by Newton’s second law of motion, 









































 





(5.2.25) 













is the particle mass reduced by , the Reynolds number is defined where in (5.2.18), and represents the net force exerted on the particle. Geometrical parameters of the problem include the ratio of the individual particle radii to the , for , and the initial position of radius of the first particle, the particle centers, . When the motion of the th spherical capsule is analyzed in the context of Stokes , (5.2.25) takes the form flow, 
































































































(5.2.26)



with the understanding that the force is a linear function of all particle velocities. The condition of zero net force on the particle (5.2.26) is used to determine the instantaneous particle translational velocity.



Capsules and interfacial transport

205

5.3 Boundary-integral formulation Following Rallison & Acrivos [63], we find that a velocity field that satisfies the equations of Stokes flow and tends to a specified velocity field far from a collection of capsules is given by the boundary-integral representation (e.g., [51, 62])



















 







































 

 

























 





























(5.3.1) 



 

 













"











where the point

lies in the th domain

, and




























































(5.3.2) 





 

 



is the traction discontinuity across the th interface. The kernels of the single- and double-layer potential of Stokes flow are given by



















% 



(5.3.3)



 

  





 &









'











Physically, these tensors represent the velocity and stress fields due to a point force. When the point lies on the surface of the th capsule, , the velocity satisfies the integral equation 

















































 



 

 

)

 





















































 

1







  











"









.





(5.3.4)

0



3

where denotes the principal value of the double-layer potential (e.g., [60]). If the surfactant concentration is available, the interfacial traction jump can be determined from equation (5.2.6), and the result can be substituted into the single-layer potential of (5.3.4) to produce a Fredholm integral equation of the second kind for the interfacial velocity, known as the boundary-integral equation. When the viscosities of all particles are equal to the viscosity of the ambient liquid, the coefficients of the individual double-layer integrals vanish, and (5.3.4) provides us with an explicit representation for the interfacial velocity. Boundary-integral equations for motion in the presence of solid boundaries are identical to (5.3.4), provided that and are replaced by appropriate Green functions that vanish over the boundaries (e.g., [60]). To solve the integral equations, we require an accurate method for computing the improper single-layer potential and the principal value of the double-layer potential. 

206

Capsules and Cells

Note that, although the single-layer integral is bounded, the integrand tends to infin, and this degrades the ity as the integration point tends to the evaluation point, accuracy the computation. The issue of accuracy also arises in calculating a “nearly singular” integral arising a point that is close to, but not precisely on, an interface. Nearly singular integrals are encountered in many-particle interactions and in the case of highly deformable drops. The singularity or near-singularity of the double-layer potential can be removed using the integral identity 

 



























 





















(5.3.5) 



































where , when the point is located, respectively, inside, on, or outside . The integrand of the single-layer potential is proportional to the jump in traction across the interface, which can be resolved into a normal and a tangential component. Singularity and near-singularity subtraction for the part of the integral containing the normal component can be achieved using the identity 


























 


































































(5.3.6) For the part of the integral containing the tangential component of the traction jump, the method of singularity and near-singularity subtraction developed by Rother et al. [65] can be employed. In this approach, the contribution to the integral from a small region around the singular and near-singular points are subtracted out, and the integration of the regularized kernel is performed analytically. In contrast to the conventional singularity subtraction method where singular and near-singular integrands are treated in the same way, different methods are applied when the evaluation and integration points lie on the same or connected parts of the interface. , the cumbersome integration When the viscosities of all fluids are equal, technique may be avoided altogether if only the normal component of the interfacial velocity is required, 































































 































(5.3.7)



Note that the integrand of the second term inside the square brackets containing the tangential component remains bounded as the integration point approaches the evaluation point, [12]. Another practical difficulty concerns the computation of the mean curvature of a highly deformable interfaces. To overcome the pronounced numerical sensitivity due 



Capsules and interfacial transport

207

to the discretization error, algorithms based on a “curvatureless” formulation of the inhomogeneous term in the boundary-integral equation (5.3.4) have been developed by Zinchenko et al. [85] for interfaces with uniform surface tension. Pozrikidis [62] extended this approach to the case of position-dependent surface tension by showing that  





















































 



























 



 


























  

 












 







 



























 

 












 

 









  





























 

















(5.3.8) Each of the integrals in the last line of (5.3.8) is discontinuous at the interface, but the two discontinuities cancel one another and the sum remains continuous throughout vanishes the entire domain of the flow. Specifically, because the projection quadratically with distance, the kernel of the second term in (5.3.8) is bounded as , and the limiting value depends on the orientation of ( ). On the other hand, the kernel of the first integral diverges quadratically with respect to . Consequently, if the point is located at the interface, , this integral is regarded as a principal value. The reduction of the singular integral to a proper integral for constant and variable surface tension is discussed, respectively, by Zinchenko et al. [85] and Pozrikidis [62]. Desingularization is based on the identity 










































 




















 



 







 










































 









 













 



 















 

















 





 





 





















 




















































 






















(5.3.9)



In the limit of vanishing P`eclet number, the distribution of the surfactant concentration in the bulk of the fluids satisfies Laplace’s equation and may thus be represented in terms of boundary integrals involving the surfactant concentration and its normal derivative. Substituting the integral representation into the interfacial balance equations (5.2.1), reduces (5.2.4) to a system of integro-differential equations for the interfacial concentration of the surfactant. Because this equation contains the interfacial velocity, interfacial transport is strongly coupled with hydrodynamics. In certain simplified or asymptotic models of interfacial mass transfer, the interfacial concentration is found independent of the velocity field and then substituted into the right-hand side of (5.3.4) to produce the interfacial jump in traction.

208

Capsules and Cells

Figure 5.4.1 Schematic illustration of interfacial transport in a suspension of droplets.

5.4 Particle motion induced by interfacial mass transport Spontaneous Marangoni interaction of fluid particles suspended in a liquid with uniform concentration of a surfactant, denoted by , occurs when the continuous and dispersed phases are not in thermodynamic equilibrium. Specifically, if the initial concentration inside a particle is different than the value prevailing at equilibrium with the outer concentration,

 







"











#





(5.4.1)



mass transfer occurs between the two phases, as illustrated in the schematic of Figure 5.4.1. In the presence of other particles, mass transport induces a spatially inhomogeneous concentration field that causes surface tension gradients. In turn, these gradients induce a Marangoni flow in the vicinity of the interfaces, and consequently cause the migration of drops and bubbles relative to the initial position, as illustrated in Figure 5.4.2. In this section, several types of interaction between small particles will be discussed in the limit of small or large P`eclet number where, respectively, diffusion dominates or plays a secondary role.

Capsules and interfacial transport

209

a1 a2 V2

V1

W1

z

W2

W0 Figure 5.4.2 Spontaneous interaction of two drops induced by interfacial surfactant transport.

5.4.1 Quasi-steady interfacial transport Assume first that the adsorption/desorption mass flux at the interfaces is balanced by fluxes from the two phases, while diffusion and convection over the interfaces is negligible to leading-order approximation. Equations (5.2.3) and (5.2.4) show that the interface surfactant concentration is given by











(5.4.2)





  



For simplicity, we shall assume that all the particles consist of the same material and the properties of the interfaces regarding kinetics are identical. Extending the analysis to more general conditions requires only straightforward modifications. Assume further that most of the resistance to mass transfer occurs in the dispersed phase near the interface, . Physically, the characteristic diffusion time is much smaller than the adsorption/desorption relaxation time, and mass transfer between the interface and the dispersed phase is adsorption controlled. Under these circumstances, to leading-order approximation, the surfactant concentration . inside each drop retains its initial value, It is convenient to eliminate and from the governing equations by introducing the dimensionless scaled ambient concentration field 


































 











(5.4.3)



The denominator on the right-hand side of (5.4.3) is the difference between (a) the concentration in the continuous liquid that would be in equilibrium with the given initial concentration of the dispersed phase (see Section 5.2.2), and (b) the concentration far from the particles. When the characteristic diffusion time and adsorption/desorption relaxation time for mass transfer between the suspending fluid and the interface are of the same order of magnitude, we may substitute (5.4.2) into the mass balance (5.2.1) to derive

210

Capsules and Cells

a Robin boundary condition for the concentration in the ambient fluid [29], 



 





(5.4.4) 








where the point 



lies in 

, and  

 

(5.4.5)




is the Sherwood number. The minus sign on the right-hand side of (5.4.4) applies when the concentration inside the drop exceeds the equilibrium value, in which case mass transfer occurs from the drop to the continuous fluid; the plus sign applies otherwise. The scaled concentration field, , satisfies Laplace’s equation subject to the interfacial condition (5.4.4), and decays to zero at infinity. Since in the present model the interfacial concentration is a linear function of the ambient concentration at the interface, , the interfacial tension also depends linearly on the ambient concentration, and the interfacial stress balance reads 





















 

























(5.4.6)




The model of spontaneous thermo-capillary interaction of drops discussed in this section was initially developed by Golovin et al. [29] for studying the axisymmetric interaction of a pair of nondeformable droplets at moderate separations. The case of small separations was subsequently investigated by Berejnov et al. [13]. A summary of the analysis and results will be given in the remainder of this section. Spherical particles When the capillary number is sufficiently small, the drops retain the spherical , arise by first computing the ambient shape. The velocities of the drop centers, concentration field, and then solving the hydrodynamics problem for each instan, evolve according to the kinematic taneous configuration. The drop centers, condition 







































(5.4.7)

Consider the interaction of two unequal neutrally buoyant drops in the absence of an external flow, as illustrated in Figure 5.4.2, and select the radius of the larger drop as the characteristic length. Because the geometrical arrangement possesses axial symmetry, axisymmetric solutions are anticipated. In analyzing problems involving two spherical particles, it is convenient to introduce the axisymmetric Stokes stream function and work in orthogonal bi-spherical coordinates. The general solution for the Stokes stream function was given by Stimson & Jeffrey [69] as a Fourier series. The force exerted on each particle may be expressed in terms of the Fourier coefficients [69], which are evaluated by substituting the series representation into the boundary conditions [29].

Capsules and interfacial transport (a)

211 (b)

Figure 5.4.3 Streamline patterns of spontaneous Marangoni flow inside and around two spherical droplets. In (a), the drops are separated by a distance equal to half the drop radius; in (b), the interfaces are in contact.

Because of the linearity of the equations of Stokes flow, the stream function and force acting on the drops can be expressed as the sum of two terms that are proportional to the drop velocities and , and a third term that depends on the tangential concentration gradient [29]. The force along the axis connecting the centers of the two drops is given by 2

2









2


2 

(5.4.8)












, is the hydrodynamic resistance representing the force exerted on where , the th particle due to the th particle moving with unit velocity, and is the force exerted on the th particle by the Marangoni flow around two stationary particles. Setting the net force exerted on each particle to zero, we obtain a linear system of equations for the drop center velocities. Golovin et al. [29] investigated the dependence of the drop velocities on the Sherwood number, viscosity ratio, radii ratio, and interface separation. In the case of equal-sized drops and identical surfactant concentrations, the results showed that the drops approach one other when mass transfer occurs from the dispersed phase to the interfaces, and separate otherwise. The drop velocity reaches a maximum at a certain separation distance on the order of the drop radius. A typical streamline pattern for the case of approaching equal-sized drops separated by a distance that is equal to half the drop radius is shown in Figure 5.4.3(a). An unusual interaction pattern occurs when either the drop radii are unequal or the surface concentration fields are not identical. If two drops with different radii are initially well separated, and if mass transfer occurs toward the continuous phase, the drops begin moving into opposite directions toward one other, the larger drop moving at a slower rate. As the drops approach one another, the direction of motion 



















212

Capsules and Cells x 10

−3

5

V1 , V2

0 −5

−10 −15 −20 −4 10

10

−2

10

0

h

Figure 5.4.4 Dimensionless velocities of two unequal spherical drops with radii ratio equal to 2 during spontaneous interaction. The upper curve corresponds to the larger drop, and the lower curve corresponds to the smaller drop.

of the larger drop is reversed, and the drops start moving in the same direction. The dimensionless velocities of the individual drops are plotted in Figure 5.4.4 against the drop separation. Another type of nonsymmetric interaction occurs when the surfactant concentration inside one of the drops is lower than the equilibrium value, while the surfactant concentration inside the other drop is higher than the equilibrium value [29]. In this case, drops of equal size approach one another at any separation. The migration velocity increases monotonically as the drop separation decreases, and a maximum occurs when the drops are in contact. The description in bi-spherical coordinates becomes impractical as the separation distance tends to zero due to the slow convergence of the Fourier series. An appropriate method of analysis in this limit relies on the lubrication approximation for the fluid motion inside the narrow gap between the drops, coupled with an analysis of the thermo-capillary flow established inside and around the contacting pair of drops [10, 23, 50]. The results show that, in this limit, the dimensionless hydrodynamic resistance to the relative motion, , is given by  



 




 



 

$ 

 

(5.4.9) 













where is the dimensionless drop separation, and is the distance between the centers of the two drops. The thermo-capillary part of the force and the hydrodynamic resistance of two drops moving with equal velocities depend continuously on the separation and 











Capsules and interfacial transport

213

. In the case of touching drops, , may be approximated with the value at the force exerted on each one of the drops is balanced by the “contact force” acting on each drop, given by 
























(5.4.10)



 






The velocities of the individual drops are given by the approximate expressions


2

 

2



 








(5.4.11) 

2




2

 








where
























2

 

 

 



(5.4.12) 











 

are, respectively, the relative velocity of the drops and the migration velocity of a doublet of touching drops [23, 46]. The hydrodynamic resistance to translation of a drop in contact with another drop, , the thermo-capillary part of the force exerted on each drop, , and the interfacial concentration distribution can be determined working in orthogonal tangentspheres coordinates. The solution was computed numerically by Berejnov et al. [13] using a technique previously developed by Leshansky et al. [46]. Specifically, the force acting on each particle was calculated as an integral in tangent-sphere coordinates [50]. Because at small separations, , the lubrication resistance is of order while the rest of the terms are nonsingular, it follows from (5.4.12) that the evolution of the interface separation is governed, to leading order, by the equation 





























 












(5.4.13) 





where






&  
























  

 






 






(5.4.14)

 $



This expression shows that nondeformable drops touch at a finite time. After collision has taken place, the pair exhibits steady thermo-capillary migration with the velocity defined in the second of equations (5.4.12). In the case of equal-sized drops, symmetry requires . However, even though the doublet is stationary, the fluid inside and outside the drops undergoes a Marangoni flow, as illustrated in Figure 5.4.3(b). The depicted streamline pattern is similar to that of two approaching drops illustrated in Figure 5.4.3(a). However, since the drop interfaces are also 







214

Capsules and Cells U

0

Ucm

- 0.002

Ca=0

- 0.004

- 0.006

Ca=0.67

- 0.008 0

0.2

0.4

0.6

0.8

1

R

Figure 5.4.5 Migration velocity of a drop doublet in close proximity plotted against the drop radii ratio. The solid curve corresponds to touching spherical drops, . and the dashed curve corresponds to deformable drops at






"






stream surfaces, the stagnation rings are contained inside each drop, and the external streamlines close at infinity. Figure 5.4.5 illustrates the dependence of the migration velocity of the drop pair for the case of equal viscosities and on the drop radius ratio . The graph exhibits a maximum at , and vanishes in the limit , corresponding to vanishing radius of one drop, as well as in the limit , corresponding to equal-sized drops. 

























&






"








"





5.4.2 Convective transport effects at small P`eclet numbers When the P`eclet number defined in (5.2.20) is nonzero, the convective term in equation (5.2.13) couples hydrodynamics and species transport. For small , it is reasonable to anticipate that the solution will be close to the quasi-steady solution discussed in Section 5.4.1, and search for a small correction. The leading-order was computed by Lavrenteva et al. [43] for spherical drops in the correction in limit of Stokes flow, as summarized in this section. At zero , the concentration field, , is a harmonic function exhibiting the asymptotic behavior 























 

 



(5.4.15)








 













Expanding the concentration field in a regular perturbation series of the form 



 



 

 











 





(5.4.16)

Capsules and interfacial transport

215






we find that







and the correction






satisfies the Poisson equation 











 

(5.4.17)



 

 

 

For large 







, the right-hand side of (5.4.17) is given by 

    

















 



(5.4.18)

 



  

 









 


















 



Unfortunately, equation (5.4.17) does not admit a decaying solution. Worse, unless is constant, the equation does not admit a bounded solution. The the function failure of the regular perturbation expansion underscores the need for singular perturbation. Following the well-established procedure of matched asymptotic expansions (e.g., [74]), inner and outer expansions of the concentration field are introduced and then matched to satisfy appropriate conditions. The inner expansion satisfies the required boundary conditions at the interfaces, and is given by 












 















(5.4.19)





 






















The outer expansion decays at infinity, and is given by 










 














(5.4.20) 



















where . The small parameter requiring the matching condition [74], 






and the coefficients


















are determined by 

(5.4.21)





















At low Reynolds numbers, the velocity and associated pressure field satisfy the equations of Stokes flow, which do not contain a small parameter. The solution is on the order of the interfacial concentration and may be expanded in a regular perturbation series whose coefficients depend on the small parameter in a manner that is similar to that displayed in the inner expansion for the concentration. Similar expansions are anticipated for the drop velocities, . Substituting expressions (5.4.18) and (5.4.20) into the governing equations, boundand [43]. The ary, and matching conditions, we find relative approach velocity of two drops is given by


2













































2

2

(5.4.22)

2







 






where

2



is the velocity obtained by setting

 

, and





 2





2 








 

 

 

(5.4.23)

216

Capsules and Cells

Thus, the leading-order correction to the quasi-steady solution appears in the form . This term describes the inherent unsteadiness of a Basset history term of of the process due to temporal changes in the mutual position of the drops, and is important at moderate separations . As the separation increases, the coefficient decays, and when , the leading order correction term is downgraded to .











































5.4.3 Slightly deformable drops at small capillary numbers In this section, we proceed to investigate the significance of small interface deformability occurring when . To isolate this effect, we neglect the convective mass transport by setting . A correction to the solution for a spherical drop, , can be constructed using the methods of regular perturbation expansions. It is convenient to describe the interfaces in individual spherical polar with origin at the respective centers of the undeformed drops. coordinates If the arrangement is axisymmetric, the surface of each drop is described as












































 





(5.4.24) 





















, and is defined in (5.2.22). Other variables are assigned where analogous representations. To leading order, the normal stress balance (5.4.6) takes the form 






























 















(5.4.25) 







 




















where 





























(5.4.26) 












and is the jump of the normal traction across the interface evaluated at the location of the unperturbed interface, . In equation (5.4.25), is the rise in pressure inside the drop computed by requiring the condition of incompressibility. Once the solution for the spherical shape is available, the perturbation can be found independently in terms of the perturbations of other variables. Homogeneous solutions of (5.4.25) are given by 
































































(5.4.27) 




















where is the hypergeometric function [31]. Note that the first solution exhibits a logarithmic singularity at . The nonsingular general solution of (5.4.25) can be expressed in the form













 



 
























 










 








 













 



(5.4.28)

Capsules and interfacial transport

217


If the drops are well separated,  &









, the functions


 &




are approximated by






 




 













(5.4.29) 

 







 









and , which shows that the interfaces assume the shape of prolate for spheroids whose aspect ratio increases monotonically as the viscosity ratio is raised. The spheroidal shape is anticipated if one imagines that the deformation of a drop to the thermo-capillary flow is induced by a point source situated at the center of the second drop. When the distance between the drops is small, the deformation can be calculated using formula (5.4.28), where the zeroth-order approximation to the interface stress jump is computed working in bi-spherical coordinates. Figure 5.4.6 shows a graph for equal-sized drops and various interface separations and visof the function for unequal drops. cosity ratios. Figure 5.4.7 shows graphs of the functions Well separated drops are near prolate spheroids whose elongation in the gap region is slightly larger than that on the other side. At dimensionless separations on the order , the interfaces start flattening inside the gap. The maximum deviation from the spherical shape occurs in the region opposite to the gap. At separation distances of , the distortion from the spherical shape in the gap becomes negative, and a dimple develops at the drop axis. As the viscosity ratio is raised, the deformation becomes less pronounced, and the aforementioned changes near the gap occur at smaller separations. Modifications are required when the drop separation is comparable to the interface deformation and the drops nearly touch. The solution can be found using the method , following the analysis of Yiantsios of matched asymptotic expansions for small & Davis [80] for gravity-driven motion. The outer solution for two spherical droplets moving in apparent contact provides us with the value of the contact force, . The inner solution describing the flow and interface deformation in the gap is constructed using the lubrication approximation. The suitably scaled gap thickness , radial velocity, , and pressure, , in the gap satisfy a system of integrodifferential equations coupling the flow in the gap to the flow inside the drops, 



































































































 















 

   

 









 













(5.4.30) 



 

 

 &



 

 




















 











  



 





 


























 

The parameter is proportional to the contact force . Numerical solutions reported by Yiantsios & Davis [80] reveal that, initially, the separation behaves as it does in the case of undeformable drops. As time progresses, the interfaces flatten and a dimple forms. At long times, the dimple radius approaches a constant value while the gap 

218

Capsules and Cells (a)

−3

6

x 10

10

4

f(q)

2

f(q)

(b)

−3

x 10

5

0

0

−2

−5

0

p/3

2 p/3

q

p

0

p/3

q

(c) 0

0

−0.02

f(q)

f(q)

p

(d)

0.01

−0.01

−0.04

−0.02 −0.03

2 p/3

−0.06

0

p/3

q

2 p/3

−0.08

p

0

p/3

q

2 p/3

p

Figure 5.4.6 Deformation pattern of equal drops at small capillary numbers for var(dotted-dashed line); ious separations and viscosity ratios: (solid); (dashed line); (a) , (b) , (c) , and (d) . 
















&




























thickness at the axis and minimum gap thickness, occurring off the axis, behave like 









 











 














(5.4.31)

To apply this analysis to the present problem, we simply replace the gravity-induced contact force with the thermo-capillary-induced force. Perturbations of the spherical shape in the outer region are given by (5.4.28). Since the force exerted on each drop by the zeroth-order flow is balanced by contact force discussed in the previous section, the leading-order correction to the spherical shape (5.4.28) computed by this force exhibits a logarithmic singularity at the near-contact region. This singularity is relieved by matching the outer with an appropriate inner solution at the gap where deformation is important.

Capsules and interfacial transport (a)

−3

x 10

219 (b)

−3

x 10 10

5

5

f(q)

f(q)

10

0

0

−5

−5

0

p/3

2 p/3

q

p

0

p/3

2 p/3

q

p

Figure 5.4.7 Deformation pattern of unequal drops at small capillary numbers for , , and various separations: (dotted-dashed line); (solid line); (dashed line), (dotted line). (a) drop 1; (b) drop 2.




&

























&























5.4.4 Deformable drops at moderate capillary numbers To describe the motion at moderate capillary numbers where substantial interfacial deformation takes place, we use the boundary-integral method. Following standard potential-field theory (e.g., [34]), we express the concentration field as the sum of single- and a double-layer potential, 



















& 



























(5.4.32)





  









































for a point

in 

, and 

 





















 









(5.4.33)





  















 























for a point that lies at the interface . Substituting this representation into (5.4.4), we obtain the boundary-integral equation 





















 



 









 







































 







 



 





















 

(5.4.34) 



 

 
















 

 




























. where the point lies at the interface The integral equation (5.4.34) can be solved by the method of successive sub, stitutions. Because the kernel of the Laplace potential diverges weakly as 







220

Capsules and Cells 1 0.5

a

0.2

b

h

c

0.05

d

0.02 0.01

e 10

20

t

100

200

Figure 5.4.8 Evolution of the interface separation between two equal-sized drops. The curves labeled a, b, c, d, and e correspond, respectively, to =1, 0.8, 0.6, 0.4, and 0.2.




the corresponding integral may be computed with adequate accuracy using one of the methods described by Pozrikidis [62]. Once the interfacial concentration is available, the interfacial velocity can be determined by solving the boundary integral equations (5.3.4), as discussed in Section 5.3. Numerical simulations were performed by Berejnov et al. [13] for the axisymmet. The computations ric motion of a pair of drops with fluids of equal viscosity, show that two equal-sized drops approach one another with a velocity that depends strongly on the drop separation and capillary number. The evolution of the interface , is illustrated in Figure 5.4.8 for several separation along the axis of symmetry, capillary numbers. The drops are seen to approach one another during the initial period of evolution for all capillary numbers. However, as time progresses, the evolution for small capillary numbers differs from that at large capillary numbers where , the interface separation continues to delarge deformations take place. For small crease monotonically in time, whereas for larger , the interface separation reaches a minimum and then it increases slightly toward an apparent steady value. At the time of reversal, the velocity of the drop center of mass is still substantial, as shown in Figure 5.4.9(a). The graphs in this figure reveal that the approach velocity initially grows toward a maximum value, and then it starts declining. The rate of decline decreases as the capillary number is raised. Thus, while the main bodies of the drops move toward one another, points located on the axis of symmetry move farther apart. This behavior reflects the development of a dimpled interfacial shape near the axis of symmetry, as will be further described. A similar pattern was observed by Manga & Stone [51] in the case of buoyancy-driven motion. Figure 5.3.9(c) illustrates the evolution of the drop center-of-mass approach velocity with respect to the interface separation for various capillary numbers. The reversal of the sign of the rate of change of the interface separation at a nonzero approach velocities for large enough is apparent. Included in this figure are asymptotic and 




















Capsules and interfacial transport

221

. The numerical results were obtained by working in numerical results for bispherical coordinates, as discussed in Section 5.4.2, and by carrying out boundary integral calculations for spherical drops positioned at various separation distances (dashed curve). The asymptotic results are described by the equation [29]



"




  

2

(5.4.35)



































The evolution of the interfaces of equal-sized drops at is illustrated in Figure 5.4.10(a). When the drops are well separated, the interface deformation is similar to that observed in the case of slightly deformable drops at small capillary numbers, as discussed in Section 5.4.4, and the drops obtain slightly prolate shapes. As the drops approach one another, the interfaces begin to flatten at the gap, and a dimple develops at the axis. At the final stage of interaction, the rate of approach and interface deformation drastically slow down, and an apparent steady state is established. The qualifier apparent is used intentionally to convey the impression that the drops do not actually arrive at a steady state, but continue to approach one another with diminishing velocity. A typical velocity vector field developing at long times is shown in Figure 5.3.11(a) , corresponding to moderate interfacial deformation. In this illustrafor tion, the length of the arrows is proportional to the magnitude of the velocity. Inspection of the flow pattern inside the gap indicates that the drops keep approaching one another with a small velocity, as shown in Figure 5.4.9(a). The stagnation ring developing inside each drop is similar to that observed for contacting non-deformable drop, as shown in Figure 5.4.3. When the interfaces are deformable, an external stagnation ring also develops because of the counterflow of the fluid drawn from infinity due to the Marangoni flow and the fluid ejected from the gap due to the continuing . The streamapproach of the drops. An external ring does not appear when line pattern shown in Figure 5.4.3 suggests that, in this case, the external ring has collapsed to the origin. In summary, the numerical calculations on the interaction of equal-size drops show two different kinds of behavior arising below and above a critical capillary number. For small capillary numbers, the interfaces are concave, the drops approach one another monotonically, and the long-time behavior is adequately described by the asymptotic theory discussed in Section 5.3.4. The theory predicts the continuous thinning of the thin film developing in the gap between the interfaces and the eventual rupture due to instability or London-van der Waals intermolecular forces [45]. For moderate capillary numbers, the deformation of the interfaces in the gap is more pronounced. At the final stage of the motion, drop migration drastically slows down yielding a quasi-steady behavior. The shape and mutual position of the drops at the final stage of the motion are illustrated in Figure 5.4.12 for a broad range of capillary numbers, from nondeforming drops, , to the highest degree of deformability considered, . The illustrations show that, at small , the drops come to near contact while exhibiting small deformation. At larger , a liquid film whose radius and width increase with the capillary number develops between the interfaces. At , dimpled shapes develop. even larger values of










"
















"




















"

222

Capsules and Cells

(a)

(b)

0.006

.002

0.005

V0

0.004

- .002

c

V

a

0.001

- .01

100

c

- .008

0

50

b

- .006

b

0

a

- .004

0.002

200

150

0

50

100

150

200

250

t

t

(c)

(d) 0.01

0.01 0.005

V

0.005

e a

0

0.001 0.0005

V - 0.005

d

- 0.01

0.0001 0.00005

b

d1

a1

b1

c1 d2

b2 a2

c2

- 0.015

c

- 0.02

0.001

0.01

0.1

1

0.001

10

0.01

0.1

1

h

h

10

Figure 5.4.9 (a) Evolution of the velocity of the center of mass of two equal-sized drops for initial interface separation equal to the drop radius, . The curves labeled a, b, and c correspond, respectively, to 1, 0.6, and 0.2. (b) Evolution of the velocity of the center of mass of two unequal drops with , for . The curves labeled a, b, and c correspond, radii ratio respectively, to 0.3, 0.5, and 0.7. (c) Phase plane of the drop centerof-mass velocity plotted against the interface separation for equal-size drops. ; curves b The curve labeled a corresponds to non-deformable drops, and c correspond to = 0.2 and 1; the dashed curve labeled d corresponds to a computation where the interfaces are assumed to be spherical at every instant; the curve labeled e represents the asymptotic results of Golovin et al. described by equation (5.4.35). (d) Phase plane of the drop center of mass velocity plotted against the interfacial separation for unequal drops with radius . The subscripts 1 and 2 denote the large and small drop. The ratio ; curves b and curve labeled a corresponds to a non-deformable drop, c correspond to = 0.3 and 0.6; the dashed curve labeled d corresponds to a computation where the interfaces are assumed to be spherical at every instant. 

"

















"



"
















"












"





"







When the drops have different sizes, the interaction pattern is more complex and the individual drops exhibit distinct behavior. At large separations, the drops approach one another as described earlier for equal-sized drops, with the small drop moving faster than the large drop. The magnitude of the individual drop velocity

Capsules and interfacial transport (a)

223

(b)

t=0

t=50

t=100

t=150

t=200

Figure 5.4.10 Deformation and migration of two equal-sized drops for (a) and (b) , .






,










"











"






(a)

(b)

Figure 5.4.11 Velocity vector field around two interacting drops at close proximity , and (b) , . for (a)
















"






"








"






first grows and then decreases due to viscous retardation. At a certain separation distance, the direction of motion of the large drop is reversed, and the drops migrate in the same direction with the small drop pushing the large drop. Because the magnitude of the velocity of the trailing small drop exceeds that of the leading large drop, the drops continue to approach one another and the gap continues to shrink. At close proximity, the relative velocity becomes small, the doublet drifts in the direction of the large drop with an apparent constant velocity, and the small drop appears

224

Capsules and Cells Ca=0

Ca=0.2

Ca=0.4

Ca=0.6

Ca=0.8

Ca=1

Figure 5.4.12 Deformation of two equal-sized drops at close proximity at dimen, illustrating the effect of the capillary number. sionless time 

&



"

"

to slightly penetrate the larger drop. Moreover, because of the higher developing internal normal stresses, the deformation of the small drop is less pronounced than that of the large drop. This is in contrast to the buoyancy-driven deformation of unequal drops where the deformation of the small drop is more pronounced [51]. Typical evolutions of the velocity of the drop center of mass are displayed in Figure 5.4.9(b) for several capillary numbers. The dependence of the velocities on the interface separation, , is illustrated in Figure 5.4.9(d). As in the case of equal drops, the evolution corresponding to was calculated working in bi-spherical coordinates [29]. The results show that the migration velocity of the doublet at long times increases as the capillary number is raised. The dependence of the pair migration velocity on the drop radius ratio, , is illustrated in Figure 5.4.5. The solid , and the dashed curve correcurve corresponds to non-deformable drops, sponds to the intermediate case . In Figure 5.4.10(b) and 5.4.11(b), the dynamics of the drop interaction, deformation pattern, and structure of the velocity at , including the field are illustrated for drops with radii ratio initial approach, deformation at close proximity, and simultaneous migration of the doublet with the smaller drop pushing the larger drop.



"









"









"











"








"






5.4.5 Migration due to unsteady mass transfer of a non-adsorbing species Consider two adjacent drops suspended in an immiscible fluid with an initially uniform concentration of a weak soluble surfactant. The initial concentration inside the drops is also uniform but different than that at thermodynamic equilibrium with the solvent. The concentration and velocity fields can be calculated by solving the boundary-value problem outlined in Section 5.2. If the concentration inside the drop

Capsules and interfacial transport

225

exceeds the equilibrium value, mass transfer occurs from the dispersed phase to the continuous phase, the gap becomes rich in surfactant, and each drop is attracted to the other. On the other hand, if the initial concentration inside the drops is lower than the equilibrium value, the drops are repelled. Because the initial concentration in the two phases is not in thermodynamic equilibrium, “shock” boundary layers develop near the interfaces over an initial period of time, while the initial values of the concentration are preserved away from the interfaces. In what follows, the time period over which the boundary layer of the concentration field prevails inside the drops will be considered. Moreover, because we are interested mostly in changes in the relative drop position, the lifetime of the internal boundary layer will be assumed to exceed the time required for Marangoni forces to propel the drop a distance comparable to the drop radius. Since the appropriate time scale is the ratio of the length scale determined by the radius of one of the drops and the Marangoni velocity generated by the initial temperature difference, the characteristic P`eclet number in the dispersed phase is high. When the P`eclet numbers in the continuous and dispersed phase are comparable , concentration boundary layers develop along the in magnitude, inner and outer side. On the other hand, when resistance to mass transfer is mostly due to the dispersed phase, that is, the P`eclet number of the continuous phase is small, boundary layers develop inside the drops during an initial period of time, while the concentration field outside the drops is fully established. The analysis in this section follows the work of Lavrenteva & Nir [44] who studied these two asymptotic limits ) of spherical drops, . The for the thermo-capillary-induced motion ( results reveal qualitatively different patterns of mass transfer and drop dynamics. If the P`eclet number of the ambient medium is negligible, the concentration in the continuous phase is quasi-steady to leading-order approximation, and resistance and to mass transfer mostly occurs in the dispersed phase. At , mass transfer inside the drops is governed to leading order by equation (5.2.13) with , while the concentration remains at the initial value, . The plus and minus sign correspond to the cases of attraction and repulsion, respectively. In the continuous phase, the leading-order approximation of the concentration is a harmonic function that vanishes at infinity and satisfies the boundary conditions 













#







"










































"



















(5.4.36) 



 

where the point lies at the interface, . Note that depends on time parametrically through the interface geometry. Because the zeroth order solution does not satisfy the mass flux condition 





 

 

 











 





(5.4.37)

an internal boundary layer is established along each interface. In (5.4.37), the point lies in , and is the ratio of the diffusivities of the dispersed and continuous phase. 









226

Capsules and Cells

The solution of the boundary-layer problem derived by Lavrenteva & Nir [44] for the axisymmetric interaction of two drops reveals that the concentration field over the interfaces can be expanded as 

  

 








 

 

 





(5.4.38) 





















  


 

where 








 

 



(5.4.39) 



   



















. Note that the boundary layer arises as a result of a discontinuity in and the normal derivatives of the zeroth order approximation rather than a discontinuity in the functions themselves, and it is therefore a weak boundary layer. The resulting , which is still higher normal component of the concentration gradient is of tangential component, confirming the applicability of the boundary than the layer approximation. Once the concentration distribution along the interfaces is available, the thermocapillary flow and drift velocities of the drops can be evaluated. Since the concentra, the drop tion difference across the boundary layer and along the interfaces is of velocities are of the same order of magnitude and depend strongly on time. It was and , shown in [44] that time and interface separation scale as and the total approach time is . The motion of drops in close proximity is studied by combining the description in tangent sphere coordinates with the lubrication analysis in the gap region. Since, to leading order, for small separation the lubrication resistance diverges as whereas the driving thermo-capillary force remains finite, the evolution of the separation is governed by the equation 
















































































































 











(5.4.40) 
















where is a constant coefficient evaluated in [44]. Integration shows that the drops collide at a finite time. is sufficiently small, the In the case of attraction, when the initial separation evolution of the interface separation is found by solving the asymptotic equation , obtaining (5.4.40) subject to the initial condition 



























(5.4.41)












It is shown in [44] that a dimensionless time where the drops approach and collide is given by . The mean relative velocity is defined as 






























 





2

 



 








$

(5.4.42)

Capsules and interfacial transport

227

(a)

(b)

0.1

0.07 0.06

0.08

h =.01 0

e −2/3 V

e −2/3 V

0

0.06

h0=.1

0.05

h =1

0.04

h =2 0

0.04

0.03

h0=1

0.02 0.02

0.01

h0=.5 0

0

10

20

e

2/3

30

0 −1 10

40

0

1

10

e2/3 t

t

2

10

10

Figure 5.4.13 Evolution of the relative velocity of two equal drops for and for various initial separations; repulsion, and attraction. 















"












and the instantaneous relative velocity is given by 









 







 2

1 2









 







$ (







which vanishes at and at the moment of collision, reaches a maximum of 

(5.4.43)

"

. The magnitude of 

2



2 2 








& &

 

2 

 

(5.4.44) 

 


















, whereupon . It follows then that, at close at proximity, the relative approach velocities can be normalized to collapse to a single graph. For larger initial separations, the drop velocities are computed working in bispher, and under the auspices of the lubrication approxiical coordinates when mation for smaller separations. The threshold is evaluated from the condition that the velocity computed using the results displayed in Figure 5.4.13 represents the evolution of the relative velocity for several initial interface separations . The dashed curves represent asymptotic predictions for drops in near proximity. Although the self-induced capillary motion is weak, it causes nevertheless drop attraction and col, which is much shorter than lision after an evolution time on the order of the time period where the boundary layer approximation applies. If the P`eclet numbers in the two phases are large and comparable in magnitude, boundary layers develop simultaneously inside and outside. Substantial capillary motion occurs only after concentration disturbances produced by one drop reach the surface of the other drop. For a moderate separation distance, this time is long, on the order of . For small separations, , the interaction of the boundary layers induces substantial temperature gradients on the interface inside the that leads drop collision. The gap, thereby causing a thermo-capillary force of 







"



















$

























































228

Capsules and Cells

evolution of the separation distance is given by [44]










(5.4.45) 




 













is the time required for the drops to approach one anwhere other and collide after the initiation of the thermal boundary layers interaction in the gap. The proportionality coefficient is discussed in Reference [44]. In this case, the mean relative velocity is given by . The instantaneous relative velocity, decays linearly in time and vanishes . at the time of collision, 



























2








2




2































5.5 Locomotion induced by the internal secretion of a surfactant Living cells contain a variety of internal structures known as organelles. Some organelles, including nucleus, mitochondria, lysosomes, and golgi, produce surface active substances that alter the properties of the interfaces. If the source is not located precisely at the cell center, the concentration of the surfactant is higher in regions of the cell boundary close to the inclusion, and the induced nonuniformity in surface concentration is responsible for cell locomotion. We shall consider an idealized model of locomotion in which the cell is a viscous drop enclosing another nonconcentric smaller drop in the absence of intra-cell structures, as depicted in Figure 5.5.1. We shall further assume that the ambient fluid far from the drop is free of surfactants, and the concentration of the surfactant is uniform over the inner interface. Under the auspices of the nonadsorbing interfacial model , the mass flux and concentration are continuous across the interface. with At the outset, we introduce dimensionless variables defined with respect to the size of the small droplet and properties of the large drop, as discussed in Section 5.2. For simplicity, we consider conditions under which the quasi-steady approximation applies and the capillary numbers associated with both interfaces are negligibly small so that the drops maintain the spherical shape. For any given configuration, the concentration field is determined by solving Laplace’s equation
















(5.5.1) 



 

, where , and the domains for to the following conditions:














are defined in Figure 5.5.1, subject



when












. 







when of the drop and the ambient fluid.



































as 





.





, where 

is the ratio of conductivities

Capsules and interfacial transport

229

W1 W0

R

W2 c= 1

1

R2 h

Figure 5.5.1 Schematic illustration of a drop with an eccentrically located inclusion.

The velocity and pressure fields in each phase satisfy the equations of Stokes flow subject to the boundary conditions (5.2.10) and (5.2.24). Axial symmetry allows us to express the velocity field in terms of the Stokes stream function. It is convenient to work in bispherical coordinates and express the general solution of the Laplace and Stokes equations in the form of Fourier series (e.g., [72]). Substituting the series presentation for the concentration fields in and into the boundary conditions, we obtain an infinite system of linear equations for the Fourier coefficients, which we then solve for a specified level truncation error [29]. Once as the concentration field is available, the stream function can be determined using the algorithm discussed previously in Section 5.4 for computing the interaction of two drops. Isoconcentration contours are shown in Figure 5.5.2 for several diffusivity ratios, and streamline patterns are shown in Figure 5.5.3, after Tsemakh et al. [73]. The Marangoni flow induced around the surface of the large drop causes the drop center to drift in the direction of the enclosed droplet. Note that, although the tangential stresses are continuous at the surface of the internal drop, the flow generated by the interface of the larger drop causes the migration of the internal droplet in the same direction, as illustrated in Figure 5.5.3(a). An interesting feature of the motion is the onset of a region of reverse flow, similar to that observed by Morton et al. [57] for a compound drop subjected to an externally imposed temperature gradient, as will be discussed in more detail in Section 5.6. The streamline pattern of the induced flow, shown in Figure 5.5.3(b) in a frame of reference attached to the large drop, reveals that the migration velocity of the inner drop is higher than that of the outer drop. Consequently, the inner droplet approaches the outer interface. As the distance between the inner and outer drop centers increases, the relative drop velocity becomes higher, reaches a maximum, and declines due to strong hydrodynamic interactions. In the limit where the inner drop 





230

Capsules and Cells

(b)

(a)




Figure 5.5.2 Isoconcentration contours for , . The interfaces are drawn with dashed lines. 



and







$





"









, 





"



(a)

(b)




Figure 5.5.3 Streamline pattern for , , and , in the laboratory reference frame, and in a frame of reference attached to the larger drop. 















$







touches the surface of the outer large drop, the pair moves with a constant velocity. A graph illustrating the dependence of the velocity on the interface distance is shown in Figure 5.5.4.

5.6 Drop migration in an ambient concentration gradient Living cells are able to sense the direction of an external chemical source and respond by migrating toward chemo-attractants or away from chemo-repellents (e.g., [49]). This phenomenon, known as chemotaxis, is crucial for the function of single-

Capsules and interfacial transport

231

0 −0.02 −0.04

a

b

V −0.06 −0.08 −0.1 −4 10

−2

0

10

2

10

10

h


Figure 5.5.4 Dependence of the migration velocity on the interface distance for , , and . Upper curve - large drop; lower curve - inner drop. The dark dot shows the velocity of the touching drops. 













"









cell organisms as well as for the function of the nervous and immune system of complex multi-cellular organisms. A common feature of most chemotactic signaling systems is their ability to adapt to different levels of external stimuli, so that the gradient of concentration of the signaling molecule rather than the average signal value determines the response [49]. Similar behavior is observed in the case of Marangoni migration of drops in an external concentration gradient, which may then serve as a simple model of chemotaxis. The mathematical modeling of chemo-capillary induced drop migration in an external concentration gradient of a weak surfactant differs from that of spontaneous motion described in Section 5.4 only in the far-field condition for the concentration. Specifically, instead of approaching a given constant value at infinity, the surfactant concentration field tends asymptotically to a specified linear function, +



)











(5.6.1)







, where is a constant vector determining the far-field gradient. The as interface concentration is assumed to be at local equilibrium, , and the mass fluxes are required to be continuous across the interface. The characteristic variation of concentration over a length that is comparable to the particle size is defined in terms of the applied external gradient, 











#





 
















(5.6.2)

Dimensionless variables are defined as in Section 5.2. , the models of chemo-capillary and thermo-capillary migration are For identical. The latter has been studied extensively in recent years, in the context of #







232

Capsules and Cells

space applications involving micro-gravity technology. Results for undeformable spherical drops and bubbles are reviewed by Subramanian & Balasubramaniam [72], and will be briefly summarized in the remainder of this section along with recent results on the motion of deformable drops.

5.6.1 Single drop Consider an isolated drop suspended in a continuous phase of infinite extent, and assume that a constant concentration gradient is applied at infinity. The nonuniform concentration over the interface causes variations in surface tension, and these induce a Marangoni flow inside and outside the drop and lead to drop migration. The drop migration velocity, , is parallel to the direction of the applied concentration gradient. Recall that, in the case of spontaneous motion of a drop, migration can only be induced by the presence of neighboring or interior particles. It is convenient to work in a frame of reference attached to the drop, and introduce a Cartesian coordinate system with the axis directed along the applied constant gradient, which is defined by the vector . In this frame of reference, the dimensionless version of condition (5.6.1) takes the form 





 









 2







(5.6.3)

 



 









Note that, in the chosen frame of reference, the far-field concentration is timedependent. As the drop moves in the direction of the concentration gradient, the surrounding fluid becomes increasingly rich in surfactant. In the special case of a nondeformable drop, , and when all the physical properties except for the interfacial tension are independent of concentration, the gradient of the concentration field reaches a steady state whereupon the drop migrates at a constant velocity. The magnitude of the steady-state velocity was calculated by Young et al. [81] using the quasi-steady approximation, which is applicable in the limit of vanishing Reynolds and P`eclet number,



"






2 

(5.6.4) 

 













$





where and are, respectively, the ratios of the surfactant diffusivities and viscosities of the drop and ambient fluid. It follows from (5.6.4) that the physical (dimensional) migration velocity depends linearly on the concentration gradient and on the , and the perturdrop radius. The velocity field far from the drop decays like , decays like . Note that the bation in the surfactant concentration, velocity field decays much faster than in the case of gravity-induced motion where . the decay rate is It was shown in References [20, 7] that the velocity field constructed in Reference [81] accompanied by properly defined pressure field satisfies the full Navier-Stokes equation in both fluids. Consequently, the migration velocity given in (5.6.4) is valid for arbitrary values of the Reynolds number, provided that the P`eclet and capillary 











































Capsules and interfacial transport

233

numbers are infinitesimal. In the special case of a neutrally buoyant drop, the solution also satisfies the boundary condition for the normal stress jump on the spherical interface, and is thus valid for arbitrary values of Reynolds and capillary numbers if the P`eclet number is negligibly small. When the drop is heavy or light and the inertial correction to the pressure field is included, the drop shape is no longer spherical. It was shown by Balasubramaniam & Chai [7] that when the P`eclet number is zero and the capillary number is small, the drop takes the form of an oblate or prolate spheroid when the drop density is, respectively, lower or higher than that of the ambient fluid. The migration velocity can be expanded as 2

2








2









(5.6.5)











where 





2


$ 





 



(5.6.6) 





   





$

and is the ratio of the densities of the drop and ambient fluid. Bratukhin [14] studied the migration of a deformable viscous drop at small Reynolds , when the Prandlt number and P`eclet numbers, related by is on the order of unity. In spherical polar coordinates centered at the undeformed drop, the shape of the interface is described by the equation , where is the meridional angle, 






















































 
























(5.6.7) 









 











2 &

 

















 



2 





$ 






(5.6.8) 



&

 &

 

 











$ $





is the ratio of the diffusivities in the internal and surrounding fluid, is the dimensionless surface tension corresponding to the concentration at the drop center, and is the second-order Legendre polynomial. The first term on the right-hand side of (5.6.8) is associated with the inertial term in the Navier-Stokes equation, and the second term is associated with the convective term in the energy transport equation. It is interesting to note that the expansion , which vanishes for for the droplet deformation begins with a term of order , or in the absence of convective mass transport, . It equal densities, follows from (5.6.7) that the deformation is small for large values of the reference surface tension corresponding to small capillary numbers. The effect of weak convective transport occurring at small P`eclet number on the motion of an undeformable spherical drop at zero Reynolds number was studied by Subramanian [70, 71] using the method of matched asymptotic expansions for . The analysis shows that the steady migration velocity is given by 















































































2

2 





2




(5.6.9)

234

Capsules and Cells 2

2

is given in (5.6.4), and is calculated by Subramanian [71]. Note that where the correction is of , which is in contrast to the case spontaneous interaction where the leading order correction is of . A different version of the method of singular asymptotics was applied by Balasubramaniam & Subramanian [8, 9] to study the limiting case of predominant convective and . Their study reveals a complicated structure transport occurring at large for the concentration field, distinguished by the presence of interfacial boundary layers and internal and external diffusion wakes. As the drop viscosity is reduced, the scaled migration velocity approaches the limiting value 










































2  





"









"



(5.6.10) 

 $



$








$







where . More generally, the steady state migration velocity is a nonlinear function of the drop radius, applied gradient, and physical properties of the two media, 





&



 





2 

(5.6.11) 





 













 $





where . An explicit expression for the function can be found in Balasubramaniam & Subramanian [9, 72]. Although the migration velocity is constant in time for all cases considered, the concentration field is unsteady and so is the mean value of the interface tension and effective capillary number. Moreover, when the surface tension is a linear function of the surfactant concentration, the surface tension becomes negative after a certain evolution time. Under these conditions, the pseudo-steady solutions make physical sense only during a limited period of time that is much larger than the time it takes for a drop to reach a steady migration velocity. If a drop begins moving from the state of rest, the pseudo-steady state is attained after a certain period of transient motion, as discussed by Dill & Balasubramaniam [24] and by Antanovskii & Kopbosynov [4, 5, 6]. The latter authors also included the effect of a time-dependent mass force by working with the Laplace transform of the linearized governing equations. In most cases, the drop is predicted to reach a steady velocity after it has moved by a distance on the order of the drop radius. 



















5.6.2 Quasi-steady interaction of spherical drops in the presence of other particles and external boundaries In practice, the motion of suspended drops is influenced by the presence of other neighboring drops and flow boundaries. Most theoretical studies of bubble and drop interaction in thermo-capillary migration have been carried out under the quasisteady approximation, which is valid in the limit of vanishing Reynolds and P`eclet number, and under the assumption of nondeformable interfaces, which is valid in the limit of vanishing capillary number. A comprehensive discussion of the validity of these approximations can be found in Reference [72].

Capsules and interfacial transport

235

The analysis of axisymmetric pairwise drop interaction and of the interaction between a drop and a flat surface is expedited by introducing the Stokes stream function and working in bispherical coordinates, as discussed in Section 5.4. Alternative approaches, such as the method of reflections and multipole expansions, are useful in cases of multi-particle interaction. The multi-drop problem was initially studied in the limit of zero viscosity ratio and vanishing thermal conductivity ratio. Meyyappan et al. [56] and Feuillbois [27] investigated the Marangoni induced axisymmetric migration of two bubbles. Subsequently, Satrape [66] used the method of twin multi-pole expansions to extend the analysis to nonaxisymmetric configurations. Ensembles of more than two bubbles have been investigated by several authors (e.g., [1, 66, 76, 77]. A remarkable finding of these studies is that a bubble migrating in the presence of other bubbles of the same size translates as though it evolved in isolation. In the case of two unequal bubbles, the influence of the large bubble on the motion of a smaller bubbles is more pronounced than vice versa. Expansions of the migration velocity in power series was constructed by of the inverse of the distance between the bubble centers, Wang et al. [76] up to the twelfth power. Anderson [3] and Keh & Chen [38, 35, 36] extended the analysis to liquid drops, , Their results can be expressed in terms and constructed expansions up to of the migration velocity of each drop in isolation as in (5.6.4) as 















 









 



 

 
























(5.6.12)



 

 

 



 

The mobility tensor takes the form , where is the unit vector directed form the center of the first drop to the center of the second drop. The matrix coefficients and are given by





































































 



 











 


















 











 







(5.6.13)





 



 

 

 




  







 









 

 









  

 



where . The second and fourth expressions apply for . The influence of one drop on the migration velocity of the other drop is manifested at . Consideration of the thermo-capillary migration of an isolated drop reveals that the far from the drops. Knowledge of the rate of disturbance velocity decays as decay is important for studying the effect of weak convective transport, discussed in the next section. In contrast to a bubble, the migration velocity of a drop in the presence of another drop differs from its velocity in isolation even when the drop sizes are equal. When the arrangement is axisymmetric, two equal-sized drops migrate with equal velocities. When the drops have different sizes, the large drop exerts a stronger influence on the small drop than vice versa. In the case of equal-sized drops, a pseudo-steady state is achieved at long times, whereupon the concentration gradient and migration







































236

Capsules and Cells

velocities are constant but the solute concentration continues to change in time. More generally, although the motion is unsteady due to the changing drop separation, the quasi-stationary approximation discussed in Section 5.2 can be invoked to describe the dynamics. The thermo-capillary motion of two nonconducting drops in close proximity was studied by Loewenberg & Davis [50] working in tangent-sphere coordinates. The analysis yields expressions for the pair migration velocity and contact force between the drops. Knowledge of this contact force and of the lubrication resistance to the relative motion permits the computation of the relative drop velocity [23] (see also Section 5.4.3 and Reference [50] for details). The concentration field far from a pair of drops is similar to that prevailing far from a single drop. In contrast, the Stokes stream function decays at a lower rate, 










 











 




















(5.6.14) 






















 

 
















A special kind of interaction occurs when a small drop is enclosed by another larger drop. This compound configuration is encountered in three-phase emulsions, and serves as a theoretical model of eukaryotic cell configuration. If the capillary numbers pertinent to the two interfaces are small, the drops retain their spherical shape, and the analysis can be conveniently conducted working in spherical or bispherical coordinates. An approximate formula for the migration velocity in the limit where the inner drop is much smaller than the outer drop was derived by Morton et al. [57] using the method of reflections. Figure 5.6.1(a) illustrates a typical streamline pattern in the laboratory frame of reference, revealing the onset of a region of reverse flow region (see also References [57, 72]). The streamline pattern is qualitatively similar to that observed in the case of self-induced locomotion discussed in Section 5.5 (see Figure 5.5.3). The corresponding streamline pattern in a frame of reference attached to the drop, illustrated in Figure 5.6.1(b), reveals that the large drop migrates slower than the small drop. The thermo-capillary migration of a gas bubble normal to a plane surface was described by Meyyappan et al. [55] and Sadhal [71] working in bispherical coordinates. The analysis was subsequently extended by Barton & Subramanian [11] to the more general case of viscous drops. Chen & Keh [19, 17] accounted for the presence of rigid and free surfaces. When the drop to boundary separation exceeds three drop radii, the bounding surface has a negligible influence on the migration velocity. Meyyappan & Subramanian [54] studied the motion of a bubble in a temperature gradient oriented in an arbitrary direction with respect to the surface, while Chen [18] provided an approximate solution based on the method of reflections for the more general case of drops. The results showed that a neighboring rigid surface reduces the migration velocity, while a free surface increases the migration velocity. The motion of a droplet inside another stationary viscous drop was considered by Shankar et al. [67] and Shankar & Subramanian [68], respectively, for concentric and eccentric configurations. The thermo-capillary motion of a drop along the centerline

Capsules and interfacial transport

237 (b)

(a)




Figure 5.6.1 Streamline pattern around a compound drop for , , and , in (a) the laboratory frame of reference, and (b) in a frame of reference attached to the large drop. "




"











of a cylindrical tube was studied by Chen et al. [15], and the motion parallel to two plane walls was studied by Keh et al. [37].

5.6.3 Effect of convective transport on bubble interaction 



Consider the motion of two bubbles in the parametric region . It is convenient to work in spherical polar coordinates with origin at the center of one chosen bubble, and the axis directed along the applied concentration gradient. A correction to the quasi-steady solution can be constructed using the method of matched asymptotics expansions. Specifically, inner and outer expansions for the disturbance of the concentration field, , are introduced such that the inner field , satisfies the required boundary conditions on the bubble surfaces, and the outer field , decays at infinity, where , and is a small dimensionless parameter. The two expansions are required to match asymptotically. Leshansky et al. [47] showed that an appropriate choice for the small parameter is . In the outer region where , conduction is balanced by convection. The quasi-steady solution described in the previous section can be used as the zeroth-order expansion term only if both bubbles lie in the inner region, that is, . If this is not true, the if the reduced interface separation satisfies interaction of each bubble with the external thermo-capillary flow dominates over correction predicted by Subramanian [70] is the mutual interaction, and an expected to arise to leading order. The inner limit of the outer expansion has been shown to be "
















"
























































































































 



 









 

 














 












&


where







and

 

 






(5.6.15)

 















are functions of time [47]. The functional form

238

Capsules and Cells

displayed in (5.6.15) indicates that the second term of the inner expansion is of order , (5.6.16) 





















 

In the case of bubbles of equal size, the coefficients and independent, and the expansion of the outer solution simplifies to 














 



are time-

(5.6.17)














As the result of convective transport in the vicinity of the bubbles, the inner expansion still has the form displayed in (5.6.16) with a nonzero . The first term of the inner expansion is given by the quasi-steady concentration disturbance. The second term, , can be found by solving the Poisson equation 









 








(5.6.18) 









for

, where 



















(5.6.19) 










 

















subject to the boundary condition 











(5.6.20) 







and the far-field condition    







 

  







 






&









 





(5.6.21)

 

and in (5.6.21) express, respectively, the influThe terms proportional to ence of the temporal change in the mutual positions of the bubbles. and the effect of advection from the outer region. The right-hand side of (5.6.18) expresses the effect of convective transport in the vicinity of the bubbles. Unlike in the case of spontaneous motion, the problem for is quasi-stationary, and the solution depends on time parametrically through the right-hand side of (5.6.18), the condition at infinity expressed by (5.6.21), and the changing geometry of the solution domain. The problem expressed by (5.6.18) through (5.6.21) was solved numerically by Leshansky et al. [47]. Once the temperature distribution on the interface is available, the solution of the hydrodynamics problem can be obtained in a straightforward fashion. A detailed calculation shows that the motion of the leading bubble always becomes faster, whereas the motion of the trailing bubble is retarded. The relative velocity of equal-sized bubbles is plotted in Figure 5.6.2 against the interface separation . The relative velocity tends to zero at large separations as well as at close proximity, and reaches a maximum at . At large separations, the relative velocity of equal-sized bubbles decays as . Figure 5.6.3 shows the correction to the individual velocities of unequal bubbles, , plotted against the interface separation, for radii ratio = 1.2 and 0.6. Note that 


















2

2

2















2






Capsules and interfacial transport

239

0.02

Pe−1 V

0.015

0.01

0.005

0 −1 10

0

10

h

10

1

Figure 5.6.2 Graph of the relative velocity of two equal-sized bubbles plotted against the reduced interface separation. The dashed curve represents the asymptotic solution for large reduced separations . 














−3

x 10

5

0.02 0.015

0 1

V1 , V2

0.005

−5

1

V11 , V 12

0.01

0

−10 −0.005 −0.01 −1 10

0

h

−15

1

10

0

10

10

h

1

10

Figure 5.6.3 Correction to the individual velocity of unequal bubbles for radii ratio (a) and (b) =0.6. The upper and lower curves correspond, respectively, to the leading and trailing bubble. The dashed curves represent asymptotic predictions for large .












the larger bubble exerts a stronger influence on the motion of the smaller bubble than decays like . vice versa. The magnitude of the correction , can be expanded as The relative bubble velocity, 





2



2

2

2











2

2 












2






















(5.6.22)

240

Capsules and Cells 0.04 0.03

Attraction

0.02

Q 0.01

Repulsion

0 −0.01 −1 10

10

0

10

1

h Figure 5.6.4 Domains of attraction and repulsion of two nearly equal bubbles in the . parameter plane 






2

is the quasi-steady relative velocity. If the leading bubble is chosen as where the reference particle, it follows from (5.6.12) and (5.6.13) that is positive when , that is, when the small bubble is trailing, and negative when and . Expression (5.6.22) shows that, if the bubbles are significantly different in size, the first term on the right-hand side dominates and the second term provides only a small correction to the quasi-stationary velocity. In contrast, if the bubbles sizes are equal, the first term vanishes and the second term provides us is always positive, in this case with the leading-order correction. Because and the drop separation increases with time. In the case of nearly equal bubbles, the two leading terms may have comparable magnitudes. Expansion (5.6.22) can be recast into the form 2



















2





















2

2












2



 





 

 

2






 











 




















which shows that if 

2


 












(5.6.23) 








the relative velocity is positive and the bubbles are repelled. Note, that this condition is satisfied if . In the opposite case, the relative velocity is negative, and the bubbles are attracted. The domains of attraction and repulsion in the parameter are shown in Figure 5.6.4. The function has a maximum plane at . For large , decays like [47].











































&

$









$












Capsules and interfacial transport

241

When the bubbles are widely separated, the problem involves two small parameand . An asymptotic expansion in these two parameters was performed ters, by Leshansky & Nir [48] who found the leading-order corrections to the individual migration velocities to be 





'
















 



  2


 


































  
&


























(5.6.24) '











 

 

  

2

























 









 

 &


























for and , where is the angle subtended between the applied temperature gradient and the line connecting the bubble centers. At large separations, the bubble velocity given in (5.6.12) and (5.6.13) is close to the bubble velocity in isolation, which is proportional to the bubble radius. Thus, to , the relative velocity of a bubble pair is proportional to . leading order in When the bubble radius ratio is far from unity, the convective term provides only a small correction to . On the other hand, equal-sized bubbles do not exert a mutual influence at and remain stationary. Thus, convective transport may determine the interaction pattern when the radii ratio is close to unity. The effect can also be seen by examining the radial and tangential components of the relative velocity field around equal-sized bubbles. Expression (5.6.24) shows that, in spherical polar coordinates with the origin centered at the reference bubble labeled 1 so that is parallel to , the radial and tangential components are given by 

































































2 









 



 






(5.6.25) 







2





 












 &






Cursory inspection of (5.6.25) reveals that the axisymmetric alignment ( ) is unstable with respect to convection. Thus, a small perturbation in the bubble position will destroy this configuration. On the other hand, the parallel configuration ) is stable. The pattern of relative motion of two equal-sized bubbles, ( illustrated in Figure 5.6.5, reveals that the bubbles exhibit a tendency to align in the plane that is perpendicular to the applied temperature gradient. At long times, any two bubbles attract one another in the course of this alignment. In the case of two nearly equal bubbles, the relative velocity of the smaller bubble , labeled 2 with respect to the larger bubble labeled 1 contains a contribution from and another contribution from the correction term due to the thermal convection,





















2










 

 



2 









 



 




 





 









(5.6.26)

242

Capsules and Cells

Figure 5.6.5 Relative motion of two equal-sized bubbles driven by convective transport in an upward temperature gradient. The arrows indicate the velocity of a bubble relative to the reference bubble drawn with the dashed line, in a frame of reference where the reference bubble is stationary. The solid lines represent the trajectories of the relative motion.



 





2  



 



















 





&








where . In the vicinity of the reference bubble, the second term on the right-hand sides of (5.6.26) becomes comparable to the leading-order term . When the bubble separation is sufficiently large, the corresponding to terms proportional to in (5.6.26) make the dominant contribution. The trajectories described by (5.6.25), illustrated in Figure 5.6.5, are not perturbed substantially in the far region. However, the attraction regime that occupies the whole trajectory space in the case of equal-size bubbles shrinks to a finite size in the case of unequal bubbles. The equations governing the evolution of the mutual position of two unequal bubbles may be expressed in terms of stretched dimensionless time and separation, and , defined by the equations 























 









(5.6.27) 









where

 



 














, as 

  

















 






2







 









 








 










Capsules and interfacial transport

243

a d

e b c

Figure 5.6.6 Motion of a small bubble relative to a large bubble in a frame of reference moving with the large bubble, subject to an upward temperature gradient. The dotted circle is described by the equation . The arrows indicate the velocity of the small bubble, and the solid lines depict the bubble trajectories. Bold lines denote the limiting trajectories. Areas marked a, d, and e correspond to attraction, whereas areas marked b and c correspond to repulsion. The dark dots indicate saddle-point equilibrium positions. 



















(5.6.28) 

   








 

 

 





2 



 















 

















The interaction pattern of two nearly equal bubbles is illustrated in Figure 5.6.6. can be calculated for a Two distinct equilibrium nodes corresponding to given pair of values of and . The first node corresponds to the axisymmetric alignment, and , as previously described. The second node rad and , occurs for corresponding to asymmetric alignment relative to the external gradient around a ring of equilibrium positions in physical space. The volume enclosed by (a) the limiting trajectory originating from infinity in the upper half-plane and ending at the second equilibrium position, separating regions a and b in Figure 5.6.6, and (b) the limiting trajectory emanating from the trailing pole of the reference bubble and ending at the same equilibrium point, separating regions 














&












































&






























$

$




$












244

Capsules and Cells

e and c in Figure 5.6.6, defines the domain of bubble attraction. Trajectory calculations are important for determining the collision efficiency, , defined as the cross-sectional area enclosed by the surface of revolution formed by the limiting trajectory that escapes to infinity in the upper half-plane, normalized by . Bubbles that are initially outside the region of attraction move past one has the value of unity, and in another at long times. In the absence of interaction, diverges to infinity. In the case of two nearly the case of two equal-sized bubbles, equal bubbles, far from the reference bubble in the direction of upward infinity, the surface of revolution formed by the limiting trajectory is a cylinder with radius , and the collision efficiency is given by . In the case of nearly equal and well separated bubbles ( , ), the relative motion is determined by heat conduction, and the relative velocity is dominated by the leading terms in equation (5.6.26), being proportional to . In this case, the larger bubble overtakes the smaller one at long times. At closer separa, the interaction due to the convection becomes significant and tions, the relative motion is controlled by the combined action of conduction and convection. Far from the reference bubble, all trajectories diverge to infinity, whereas in the vicinity of the reference bubble, some trajectories form closed lines. , The rescaled velocity of the relative motion of two nearly equal bubbles, , where depends on two parameters: the stretched separation distance , and the angle subtended between and the direction of the temperature gradient, . Here, plays a role of a characteristic length scale of the interaction distance where convection effects balance the conduction driven mo. tion. The evolution of the bubble relative position occurs on a time scale Rescaling reveals a similarity in the relative motion. For any given value of the ratio , the trajectories described in terms of may be generated simply by an appropriate coordinate destretching of the given configuration. The pattern may also be at every produced by superimposing a constant velocity vector field equal to point in the trajectory space shown in Figure 5.6.5. Cursory inspection reveals that the pattern shown in Figure 5.6.5 is topologically unstable to slight disturbances in around . The union of regions a, d, and e in Figure 5.6.6 defines the domain of bubble attraction, while the union of regions b and c defines domain of bubble repulsion. Bubbles falling into region a will be attracted for any relative initial orientation, while bubbles falling in regions d and e may be initially repelled before being attracted at long times. The equilibrium state lying on the axis of symmetry is a saddle point, which is stable to axisymmetric perturbations of the bubble position (the bubble is attracted to this point along the axis) and unstable for any other orbital disturbance. The other equilibrium node lying at the intersection of two limiting trajectories that escape to infinity, as shown in Figure 5.6.6, is also an unstable saddle point. Since the , it is evident that, if , collision efficiency is given by then convection-driven interaction of slightly unequal bubbles is small, and the larger bubble overtakes the smaller bubble for almost every initial orientation. Thus, bubbles with uniform size are expected to align horizontally, and bubble with different sizes are expected to separate. 
























&
































$


























































































































































































Capsules and interfacial transport

245

5.6.4 Interaction of deformable drops Most studies on the Marangoni migration of interacting deformable drops consider the motion under a temperature gradient rather than a solute concentration gradient. Motivation has been provided by applications in materials processing technology in a micro-gravity environment. A brief overview will be given in this section with emphasis on recent boundary-integral simulations. The analysis and results are also applicable to the motion induced by the presence of weak surfactants subject to straightforward changes in notation. Manga & Stone [51] discussed the deformation of well separated drops undergo, ing thermo-capillary migration, and estimated the deformation rate to be where is the separation of the drop centers. Loewenberg & Davis [50] noted that the earlier lubrication analysis of Yiantsios & Davis [80] for buoyancy-driven motion of two drops in close proximity is directly applicable to the case of thermo-capillaryinduced motion of nonconducting drops. A discussion of the lubrication approach and results on the evolution of the film developing between the drops was given in Section 5.4 in the context of spontaneous interaction. Rother & Davis [64] extended the analysis to drops with arbitrary thermal conductivity ratio. In particular, it was observed that a dimple develops near the axis of symmetry, and the film thickness while the minimum separation of the interfaces decays decreases in time as as . When a drop migrates under the action of a temperature gradient, the temperature in the vicinity of the interfaces continues to rise, while the surface tension decreases and the effective capillary number is raised. If the surface tension is assumed to depend linearly on the temperature, the surface tension becomes zero at the front of the leading drop after a certain evolution time, and the analysis ceases to be physically relevant. On the other hand, when the surface tension depends on concentration of a solute, zero surface tension never arises. The boundary-integral approach allows us to study the thermo-capillary interaction of deformable viscous drops making no a priori assumptions regarding the magnitude of the interfacial deformation. As in the case of spontaneous interaction, the interfacial velocities satisfy boundary-integral equation (5.3.4), while the interfacial concentration is described by (e.g., Rother et al. [65]) 

















































 

 

















 



















































 






(5.6.29) where is the ratio of the diffusivities or thermal conductivities of the drop and the ambient fluid. Zhou & Davis [83] applied the boundary-integral method to study the axisymmetric thermo-capillary motion of two unequal drops, where the leading drop is smaller than the trailing drop. The viscosity of both drops was assumed to be equal to that of the ambient fluid. As mentioned in Section 5.3, if the interfacial temperature distribution is available, the boundary-integral approach provides us with an explicit formula for the interfacial velocities in terms of a single-layer Stokes potential. The numerical results show that drop deformability becomes significant only 

246

Capsules and Cells 6 5 4

z

3 2 1 0 

Figure 5.6.7 Deformation of equal-sized drops for 25 (from left to right).

, at times =0, 5, 19, and

when the drops are close together, and confirm the validity of the asymptotic models based on the lubrication approximation at small separations. To avoid negative surface tension, the temperature at the leading point of the smaller drop was assumed to remain constant in time. Unfortunately, because the external temperature field would be unsteady even in the absence of the drops, this assumption undermines the physical relevance of the simulations. Berejnov et al. [12] extended the analysis of Zhou & Davis [83] to the case of equal-sized drops, and considered cases where the leading drop is larger than the trailing drop. Moreover, their boundary conditions were stated with respect to a reference point that is fixed in space instead of moving with one of the drops. Numerical simulations for fluids of equal viscosity and thermal conductivity at moderate capillary numbers revealed a variety of patterns of interaction. Equal-sized drops were found to always approach one another, although the minimum interface separation first grows and then diminishes in time. If the leading drop is larger than the trailing drop, the interfaces initially approach and then separate as the drops move away. Berejnov et al. [12] discussed a case where the surface tension is a nonlinear function of the solute concentration. For some fluid-solute pairs, the interaction can be divided into two regimes. In the first regime, the linear model discussed previously in this section is used to describe the strong dependence of the surface tension on the solute concentration. In the second regime, the surface tension is approximated with a constant function [2]. This dual model can be implemented by allowing the surface tension to depend linearly on the concentration until the critical value,  0



0



"



"



  






(5.6.30)

and holding it constant thereafter. In (5.6.30), is the concentration gradient and is the radius of the undeformed spherical drops. Results of a numerical simulation based on this model are presented in Figure 5.6.7. As the evolution enters the low surface tension regime, the deformation pattern changes suddenly, and a pronounced 




Capsules and interfacial transport D

A= T

(b) (d)

2a

247 (e)

(a)

(c)

V b

L

Figure 5.6.8 Schematic illustration of a bubble moving through a capillary tube that is filled with a viscous fluid showing: the constant thickness film region (a), the capillary static regions (b, c), and the transition regions (d, e).

distortion of the spherical shapes accompanied by a reduction in the drop velocity is observed, halting the entire migration. Rother et al., [65] used the boundary-integral method to study three-dimensional, thermo-capillary, pairwise drop interactions for different drop size ratios, drop-tomedium thermal conductivity ratios, and viscosity ratios (see Section 5.3). Computations were performed for both a fixed temperature reference point and a reference point moving with a drop, and a nonlinear dependence of the surface tension on the temperature was employed. The simulations showed that interface deformation raises the minimum interface separation, inhibits coalescence, and prevents capture and breakup observed in buoyancy-driven interaction [22].

5.6.5 Thermo-capillary migration of long drops in capillary tubes In real life, capsules and cells move in confined spaces rather than in unbounded media. Drops and capsules in tube flow have been studied extensively as models of red blood cell motion in the systemic or pulmonary micro-circulation, as discussed in Chapters 1-4. In this section, we consider bubble motion driven by an axial temperature gradient, following Mazouchi & Homsy [52]. Consider a long bubble moving steadily in an otherwise quiescent fluid along the axis of an open circular cylindrical tube, as illustrated in Figure 5.6.8. A constant temperature gradient, , is imposed along the tube axis. At the outset, we introduce dimensionless variables defined using as characteristic length scale the tube radius , and as characteristic velocity scale the unknown constant velocity of the bubble, . When the capillary number, , is small, five distinct physical flow regions can be identified, as shown in Figure 5.6.8; in the definition of the capillary number, is a reference surface tension. Region a consists of the film developing between the wall and the bubble surface, having a constant a priori unknown thickness . If the length of the bubble is much , the temperature gradient is nearly conhigher than the film thickness, stant along the free surface. A simple shear flow is then established with the no-slip










0








0







248

Capsules and Cells

boundary condition satisfied at the wall, and a maximum thermo-capillary velocity, , occurring at the interface; the subscript denotes the derivative with respect to the temperature. In regions b and c, surface tension dominates viscous and inertial forces, and the two ends of the bubble are nearly hemispherical cups whose radius is be determined as part of the solution. In the transition regions d and e, the , and the axial length scale is . radial length scale is of order Because the film thickness varies only slowly with axial position, standard lubrication theory can be applied to derive the modified Landau-Levich equation for the , scaled film thickness 

0























































 













(5.6.31) 

















where and is the unknown ratio of the maximum thermocapillary velocity and the bubble velocity, to be determined by matching the solution of (5.6.31) with the solutions in regions a, b, and c, and by enforcing a global mass balance [52]. The latter requires that the volumetric flux of the fluid pumped by the Marangoni stresses be balanced by the volume of fluid swept by the bubble. To , leading order in














$






























(5.6.32)















Numerical solutions of the problem just formulated were presented by Mazouchi & Homsy [52]. These authors investigated the structure of the flow as a function of the modified capillary number defined in terms of the physical properties of the fluids and the tube radius, [52]. The computations show that




















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(5.6.33)




Wilson [79] performed a similar analysis of the thermo-capillary motion of a drop in a closed tube, and derived a relationship between the drop velocity and radius. More recently, Mazouchi & Homsy [53] extended the earlier analysis by considering the thermo-capillary migration of bubbles through tubes with a polygonal cross-section.

5.7 Combined effect of gravity and thermo-capillarity In all the cases discussed thus far, the flow is driven solely by Marangoni traction. In most applications, gravitational effects also play an important role. The relative importance of Marangoni traction and gravity is expressed by the Marangoni number 




 

 



  



















(5.7.1)

Capsules and interfacial transport

249

is the radius of a reference drop. The where is the acceleration of gravity, and deformability of the drop interface is determined by the Bond number, 





















(5.7.2) 






0

where is a reference surface tension. In defining these dimensionless numbers, the velocity of a drop in buoyancy-driven motion is used as a characteristic velocity scale. 0

5.7.1 Single spherical drop Consider first the motion of a single spherical drop in an unbounded ambient fluid subject to a linear distribution of a weak surfactant, and assume that the reference drop is heavier than the continuous phase, in which case gravity induces downward motion. In the creeping flow approximation, , and in the limit of conductive , the flow field generated by the steady migration of a single drop transport, under the combined action of gravity and thermo-capillarity can be superimposed. In the axisymmetric configuration where the undisturbed concentration gradient is parallel to the axis, the downward migration terminal velocity is given by








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(5.7.3)




 









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(e.g., [72]). If the concentration increases downward, , the Marangoni number is negative, and the buoyancy-driven migration accelerates the buoyancy-driven motion. On the other hand, if the concentration increases upward, the Marangoni number is positive, and equation (5.7.3) shows that the drop moves downward if

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(5.7.4)



it remains stationary if , and moves upward otherwise. Typical streamline patterns of the upward and downward motion are illustrated in in Figure 5.7.1. Circular dividing streamlines are observed inside the drop in the case of the downward motion, and in the outer fluid in the case of upward motion. The effect of weak convective transport was investigated by Zhang et al. [82] using the method of matched asymptotic expansions. It was shown that, in the Stokes flow approximation, the correction to the concentration grows logarithmically far from the drop, and is thus ill-behaved. The ill-posedness was traced to the interaction Stokes flow with the uniform concentration gradient far of the slowly decaying from the drop. The singular behavior can be removed by correcting the Stokes flow solution to account for inertial effects, resulting in a faster decay of the far velocity. Nonsingular outer expansions for the concentration field may then be constructed, and the correction to the migration velocity can be determined. 















250

Capsules and Cells

(a)

(b) 1 −0.03 2 −0.015 3 −0.005 4 0 5 0.01 6 0.02 7 0.03 8 0.04

1 2 3 4

1 −0.004 2 −0.0015 3 0 4 0.01 5 0.025 6 0.05 7 0.1 8 0.15

1 2 3

5 6 7 8

4 5 6

7

8

Figure 5.7.1 Streamlines in the laboratory reference frame for a liquid drop moving (a) upward and (b) downward . &





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5.7.2 Pairwise interaction of spherical drops As in the case of a single drop or bubble, the quasi-steady interaction of two spherical drops or bubbles under the combined action of gravity and Marangoni motion can be computed by superposing the individual solutions. The interaction pattern for drops is similar to that for bubbles. A detailed investigation of bubble interaction by Wei & Subramanian [78] reveals new features. Drops and bubbles of equal size migrate with the same velocity. For a given separation, there is a criti, where the drops remain stationary. The cal value of the Marangoni number, curve separating the domains of upward and downward motion, generated using data provided by Wacholder & Weihs [75] and Keh & Chen [38], is displaying in Figure 5.7.2. In the regime above the curve, heavy drops move upward; in the regime below the curve, heavy drops move downward.


















Theoretical analysis of the motion of unequal spherical drops predicts a similar critical separation where the drops move with equal velocities [78]. If the small drop lies above the large drop, the equilibrium configuration is unstable. If the initial separation exceeds the critical value, the drops keep moving farther apart; in the opposite case, the drops are attracted and eventually collide. On the other hand, if the larger drop lies above the small drop, the separation tends to the critical value, and the configuration is stable. Golovin et al. [29] studied the spontaneous motion of a pair of drops induced by the interfacial mass transfer in the gravitational field. Since in the course of spontaneous interaction the velocities of the drops point in opposite directions, gravity influences in different ways the motion of the two drops. If the drops are heavier than the ambient fluid, the lower drop slows down while the upper drop is sped up. When the capillary interaction is sufficiently strong, the lower drop may move against its own weight. Drops of equal size moving with equal velocities in the absence of Marangoni motion are attracted and eventually collide due to the capillary motion. When the lower drop is larger than the upper drop, the interaction may lead to a stationary configuration where the two drops move with equal velocities.

Capsules and interfacial transport

251

3

Ma 2.8

2.6

2.4

0

2

4

6

8

h Figure 5.7.2 Critical value of the Marangoni number plotted against the scaled sepand . aration of a pair of identical drops with 











5.7.3 Interaction of deformable drops Boundary-integral simulations of the interaction of two deformable drops under the combined action of gravity and thermo-capillarity were reported by Lavrenteva et al. [41]. The calculations showed that deformability promotes the relative motion of equal-sized drops. The drops approach one another for small initial separations , and move farther apart for large initial separations. If the pair lies close described in Figure 5.7.2, the buoyancy force is nearly to the curve compensated by thermo-capillarity, and this may lead to a change in the direction of the motion in the course of the interaction. For example, a pair of heavy drops may , while the drop separation increases initially migrate upward in the direction of because of the deformation. When the separation has become sufficiently large, the drops will begin to sink. Conversely, the drops may initially move in the direction of gravity while the drop separation decreases due to the deformation. When the separation has become sufficiently small, the drops begin to rise. When the Bond number is small, the motion is qualitatively similar to that of undeformable bubbles described in previous sections. Interaction patterns of equalsized drops are illustrated in Figure 5.7.3 for several Bond numbers. Frames (a, b) describe the evolution of the position of the center of mass of the upper drop, frames (c, d) describe the evolution of the position of the center of mass of the lower drop, and frames (e, f) describe the evolution of the interface separation along the vertical axis of symmetry. When initial separation is sufficiently small, =0.2, the drops initially move upward, as shown in frames (a, c, e). However, since the interface separation decreases monotonically in time, the direction of motion is reversed to =0.5, the downward after a certain evolution time. On the other hand, when direction of descending drops reverses while the interface separation keeps widening, as shown in frames (b, d, f). 




























252

Capsules and Cells (a)

(b) 4

3.6

3.8

3.5

Z

3.4 1

Z

3.3

3.6 1

3.4

3.2

3.2

3.1

3

3 100

200

300

0

400

100

200

300

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t

(c)

(d)

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400

500

400

500

400

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2

1.2

0.4

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0.2

100

200

300

0

400

100

200

300

t

t

(e)

(f)

0.2 0.8

0.15 0.7

0.1

h

h 0.6

0.05

0 0

0.5

200

100

300

t

400

500

0

100

200

300

t

Figure 5.7.3 Evolution of the positions of two equal drops. On the left column, and ; the dashed, solid, and dotted lines correspond to =0.043, 0.05, and 0.057. On the right column, and ; the = 0.066, 0.074, and 0.081. dashed, solid, and dotted lines correspond to 

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When the drops have different sizes and the radii ratio is sufficiently small, the motion is qualitatively similar to that of nondeformable spherical drops. When the radius ratio is close to unity, the interplay of the slight disparity in size and interfacial deformability described in Figure 5.7.2 results in a smaller value of the critical

Capsules and interfacial transport

253

0.08

0.075

h 0.07

0.065

0

100

200

300

400

500

t Figure 5.7.4 Evolution of the distance between two drops near the critical initial , , and . The dashed, solid, separation for and dotted lines correspond to =0.0716, 0.0714, and 0.0712.




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drop separation. If the initial separation exceeds this value, the drops separate and demonstrate a behavior that is qualitatively similar to that of undeformable spherical drops after a certain transient period. On the other hand, if the drops are initially sufficiently close, the approach takes place during a long time and may culminate in coalescence in cases where the theory of nondeformable drops predicts separation. These two types of interaction are illustrated in Figure 5.7.4 for several initial separations near the critical threshold. To demonstrate the influence of weak thermo-capillary motion on the behavior of unequal highly deformable drops, Lavrenteva et al. [41] considered the motion for values of the Bond number, radii ratio, and initial separation that are comparable to those used previously by Davis [22] to describe large deformation and subsequent breakup of the trailing drop in the absence of a Marangoni motion. Figure 5.7.5 il, lustrates the dynamics subject to an upward temperature gradient for (a) , and , and (b) , , at various values of . The sequences on the left correspond to the pure buoyancy-driven motion, , studied by Davis [22]. The results show that, for , although Marangoni effects have almost no influence on the net velocity of the drops, they drain the first matically affect the deformation of the interface. When case, or in the second case, the trailing drop relaxes after initial elongation and does not breakup. Surface flow driven by thermo-capillary stresses and drop migration toward a region of low surfactant concentration contribute a strong stabilizing influence that prevents drop breakup. The combined effect of gravity and




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Capsules and Cells (a)

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-20 -20

-30 -30 -40 0

10

20

30

40

5

0

15

10

20

25

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Figure 5.7.5 Evolution of two unequal drops for various Bond and Marangoni , , , and numbers: (a) (from left to right); (b) , , , and (from left to right).




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spontaneous thermo-capillarity of deformable drops is an important topic for further investigation.

5.8 Concluding remarks In this chapter, capsules and cells have been modeled as viscous drops with position dependent and time varying surface tension determined by a temperature field or by the concentration of a dissolved molecular species. Concentration nonuniformities may either be imposed externally or appear spontaneously as the result of internal secretion. Our focus has been on pairwise interactions, with particular attention on selected limits that put to the spotlight effect of the various physical factors such as convective mass and heat transport and interface deformability. A wide range of interaction patterns has been observed depending on the type of mass transfer and on the values of the dimensionless parameters, demonstrating that mass transfer may have a significant effect on the particle motion. For example, interfacial transport may enhance the collision rate of drops in the absence of external forcing, prevent drop collision, and retard coalescence. The model of a viscous drop with variable surface tension moving in the otherwise quiescent fluid considered in this chapter is highly idealized, in that it does not take into consideration several important factors including interaction with an external

Capsules and interfacial transport

255

flow, interfacial rheological properties, and the involved nature of the surfactant dynamics. Interfacial models with more advanced mechanical properties are discussed in Chapters 1 and 2 of this book. The studies reviewed in this chapter may be extended to include more complex interfacial dynamics involving insoluble, slightly soluble, or soluble surfactants whose concentration is affected by simultaneous diffusion, surface convection, and adsorption/desorption kinetics. Moreover, consideration must be given to situations where surfactants are swept by surface convection to form stagnant-caps that immobilize parts of the interface. Relevant studies for single drops employing a nonlinear dependence of the surface tension on the surfactant concentration are reported in References [16, 26] and reviewed in Reference [72]. Studies of the effect of a slightly soluble surfactant on the motion of spherical bubbles have confirmed the formation of a stagnant cup [21]. When a drop migrates in an applied temperature gradient in a liquid that is contaminated with a soluble surfactant, the accumulation of the surface-active substance at the rear causes substantial retardation of the thermo-capillary drift velocity. Several authors have considered the influence of surfactants on the propagation of long drops and bubbles in capillary tubes by techniques that are similar to those described in Section 5.6.5, as reviewed in Reference [58]. The idealized model of cell and capsule interaction discussed in this chapter takes only partially into account the important effect of surface tension variations and its dependence on solute concentration. The model allows us to study the dynamics of attraction, repulsion, aggregation, and structure formation, as well as chemotactic locomotion. Future extensions of the analysis in the aforementioned directions will surely reveal a richer dynamics and further modes of interaction.

256

Capsules and Cells

References [1] ACRIVOS , A., J EFFREY, D. J., & S AVILLE ., D. A., 1990, Particle migration in suspensions by thermocapillary or electrophoretic motion, J. Fluid Mech., 212, 95-110. [2] A DAMSON , A. W., 1990, Physical Chemistry of Surfaces, Fifth Edition, Wiley, New York. [3] A NDERSON , J. L., 1985, Droplet interaction in thermocapillary motion, Int. J. Multiphase Flow, 11, 813-824. [4] A NTANOVSKII , L., K., 1991, Influence of capillary forces on the transient fall of a drop in an unbounded liquid with surfactants, Zh. Prikl. Mech. Tekh. Fiz., 6, 60-65. [5] A NTANOVSKII , L., K., 1991, Symmetrization of interface dynamics equations, Eur. J. Appl. Math., 3, 60-65. [6] A NTANOVSKII , L., K. & KOPBOSYNOV, B. K., 1986, Non-stationary thermocapillary drift of a drop of viscous fluid, Zh. Prikl. Mech. Tekh. Fiz., 2, 59-64. [7] BALASUBRAMANIAM , R. & C HAI , A., 1987, Thermocapillary migration of droplets : An exact solution for small Marangoni numbers, J. Colloid Interface Sci., 119, 531-538. [8] BALASUBRAMANIAM , R., & S UBRAMANIAN , R. S., 1996, Thermocapillary bubble migration-thermal boundary layers for large Marangoni numbers, Int. J. Multiphase Flow, 22, 593-612. [9] BALASUBRAMANIAM , R. & S UBRAMANIAN , R. S., 2000, The migration of a drop in a uniform temperature gradient at large Marangoni numbers, Phys. Fluids, 12, 733-743. [10] BARNOCKY, G. & DAVIS , R. D., 1989, The lubrication force between spherical drops, bubbles and rigid particles in a viscous fluid, Int. J. Multiphase Flow, 15, 627-638. [11] BARTON , K. D. & S UBRAMANIAN , R. S., 1990, Thermocapillary migration of a liquid drop normal to a plane surface, J. Colloid Interface Sci., 137, 170-182. [12] B EREJNOV, V., L AVRENTEVA , O., M., & N IR , A., 2001, Interaction of two deformable viscous drops under external temperature gradient, J. Colloid Interf. Sci., 242, 202-213. [13] B EREJNOV, V., L ESHANSKY, A. M., L AVRENTEVA , O. M., & N IR , A., 2002, Spontaneous thermocapillary interaction of drops: effect of surface deformation at nonzero capillary number, Phys. Fluids, 14, 1326-1339.

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[14] B RATUKHIN , Y. K., 1973, Termokapillyarnyy dreyf kapel’ki vyazkoy zhidkosti, Izv. Akad. Nauk SSSR, Mekh., Zhidh. Gaza, 5, 156-161; Original in Russian, NASA Technical Translation 17093, June 1976. [15] C HEN , J., DAGAN , Z., & M ALDARELLI , C., 1991, The axisymmetric thermocapillary motion of a fluid particle in a tube, J. Fluid Mech., 233, 405-437. [16] C HEN , J. & S TEBE ., K. J., 1997, Surfactant-induced retardation of the thermocapillary migration of a droplet, J. Fluid Mech., 340, 35-59. [17] C HEN , S. H., 1999, Thermocapillary deposition of a fluid droplet normal to a planar surface, Langmuir, 15, 2674-2683. [18] C HEN , S. H., 1999, Thermocapillary motion of a fluid sphere parallel to an insulated plane, Langmuir, 15, 8618-8682. [19] C HEN , S. H. & K EH ., H. J., 1990, Thermocapillary motion of a fluid droplet normal to a plane surface, J. Colloid Interf. Sci., 137, 550-562. [20] C RESPO , A. & M ANUEL , F., 1983, Bubble motion under reduced gravity, Proceedings of the European Symposium on Material Science under Microgravity, (pp. 45-49), European Space Agency SP-333. 12-16 April 1983, Madrid, Spain. &



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[61] P OZRIKIDIS , C., 1997, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, New York. [62] P OZRIKIDIS , C., 2001, Interfacial dynamics for Stokes flow, J. Comput. Phys., 169, 250-301. [63] R ALLISON , J. M. & ACRIVOS , A., 1978, A numerical study of the deformation and burst of a drop in an extensional flow, J. Fluid Mech., 89, 191-200. [64] ROTHER M. A. & DAVIS , R. H., 1999, The effect of slight deformations on thermocapillary-driven droplet coalescence and growth, J. Colloid Interf. Sci., 214, 297-318. [65] ROTHER , M. A., Z INCHENKO , A. Z., & DAVIS , R. H., 2002, A threedimensional boundary-integral algorithm for thermocapillary motion of deformable drops, J. Colloid Interf. Sci., 245, 356-364. [66] S ATRAPE , J. V., 1992, Interaction and collision of bubbles in thermocapillary motion, Phys. Fluids A, 4, 1883-1900. [67] S HANKAR , N., C OLE , R. D., & S UBRAMANIAN , R. S., 1981, Thermocapillary migration of a fluid droplet inside a drop in a space laboratory, Int. J. Multiphase Flow, 7, 581-594. [68] S HANKAR , N. & S UBRAMANIAN , R. S., 1983,, The slow axisymmetric thermocapillary migration of an eccentrically placed bubble inside a drop at zero gravity, J. Colloid Interf. Sci., 94, 258-275. [69] S TIMSON , M. & J EFFREY, G. B., 1926, The motion of two spheres in a viscous fluid, Proc. Roy. Soc. Lond. A, 111, 110-116. [70] S UBRAMANIAN , R. S., 1981, Slow migration of a gas bubble in a thermal gradient, AIChE J., 27, 646-654. [71] S UBRAMANIAN , R. S., 1983, Thermocapillary migration of bubbles and droplets, in Advances in Space Research, ed. Y. Malmejac (vol 3, no 5, p. 145-153), Pergamon Press. [72] S UBRAMANIAN , R. S. & BALASUBRAMANIAM , R., 2001, The Motion of Bubbles and Drops in Reduced Gravity in Transport Processes in Bubbles, Drops, and Particles, Cambridge University Press, New York. [73] T SEMAKH , D., L AVRENTEVA , O. M., & N IR , A., 2003, On the locomotion of a drop, induced by the internal secretion of surfactant, Submitted for publication. [74] VAN DYKE , M., 1975, Perturbation methods in Fluid Mechanics, Parabolic Press, Palo Alto. [75] WACHOLDER , E. & W EIHS , D., 1972, Slow motion of a fluid sphere in the vicinity of another sphere or a plane boundary, Chem. Eng. Sci., 27, 1817-1828.

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[76] WANG , Y., M AURI , R., & ACRIVOS , A., 1994, Thermocapillary migration of a bidisprse suspension of bubbles, J. Fluid Mech. 261, 47-64. [77] W EI , H. & S UBRAMANIAN , R. S., 1993, Thermocapillary migration of a small chain of bubbles, Phys. Fluids A, 5, 1583-1595. [78] W EI , H. & S UBRAMANIAN , R. S., 1995, Migration of a pair of bubbles under the combined action of gravity and thermocapillarity, J. Colloid Interf. Sci., 172, 395-406. [79] W ILSON , S. K., 1993, The steady thermocapillary-driven motion of a large droplet in a closed tube, Phys. Fluids A, 5, 2064-2066. [80] Y IANTSIOS , S. G. & DAVIS , R. H., 1991, Close approach and deformation of two viscous drop due to gravity and van der Waals forces, J. Colloid Interf. Sci., 144, 412-433. [81] YOUNG , N. O., G OLDSTEIN , J. S., & B LOCK , M. J., 1959, The motion of bubbles in a vertical temperature gradient, J. Fluid Mech., 6, 350-356. [82] Z HANG , L., S UBRAMANIAN , R. S., & BALASUBRAMANIAM , R., 2001, Motion of a drop in a vertical temperature gradient at small Marangoni number – the critical role of inertia, J. Fluid Mech., 448, 197-211. [83] Z HOU , H. & DAVIS , R. H., 1996, Axisymmetric thermocapillary migration of two deformable viscous drops, J. Colloid Interf. Sci., 181, 60-72. [84] Z INCHENKO , A. Z., ROTHER , M. A., & DAVIS , R. H., 1997, A novel boundary-integral algorithm for viscous interaction of deformable drops, Phys. Fluids, 9, 1493-1511. [85] Z INCHENKO , A. Z., ROTHER , M. A., & DAVIS , R. H., 1999, Cusping, capture and breakup of interacting drops by a curvatureless boundary integral algorithm. J. Fluid Mech., 391, 249-293. [86] Z INEMANAS , D. & N IR , A., 1988, On the viscous deformation of biological cells under anisotropic surface tension, J. Fluid Mech., 193, 217-241. [87] Z INEMANAS , D. & N IR , A., 1990, Surface visco-elasticity effects in cell cleavage, J. Biomech., 23, 417-424.

Chapter 6 Capsule motion and deformation in tube and channel flow

A. Borhan and N. R. Gupta The motion of flexible particles in a confined domain, such as a fluid-filled channel or tube, is of interest to a variety of natural and biological processes and industrial applications. Much of early interest in this subject has been motivated by its relevance to the motion of blood cells through the capillaries of the human circulation. However, similar flow problems also arise in industrial applications involving the production of cosmetics and various pharmaceutical, agricultural, and food products. Aside from its significance in these practical applications, the motion of deformable particles through tubes with constant and variable cross-section has received special attention as a pore-scale model in the study of two-phase flow through porous materials. Surface elasticity resulting from gradients of the interfacial tension between two fluids can induce bulk fluid motion, referred to as a Marangoni flow. Variations in the interfacial tension are induced by a nonuniform distribution of temperature, surfactant concentration, or electric charge. Marangoni flow becomes significant when buoyancy effects are negligible, as in the case of small-scale systems or under microgravity conditions. The importance of this flow in systems with macroscopic dimension encountered in the space environment has been a main reason for a substantial growth in research activity in the last two decades. The dynamics of confined liquid capsules with nonuniform interfacial tension and capsules with more involved interfacial behavior has received particular attention. A fundamental goal of theoretical studies has been to describe the balance between the surface elasticity associated with an elastic membrane or by a nonuniform surface tension distribution, and viscous forces, for various flow configurations and under various flow conditions. The presence of confining solid boundaries in the vicinity of the interface of a deformable particle makes the confined flow configurations particularly interesting. It has been found that interfacial properties can have a striking influence and alter the dynamics by rendering surface and viscous effects comparable in magnitude. In this chapter, an overview of the theoretical advances concerning capsule motion and deformation in tube flow will be given, including results for deformable particles ranging from liquid drops with nonuniformities in interfacial temperature or surfactant distribution to liquid capsules enclosed by elastic membranes. Emphasis will be

263

264

Capsules and Cells

Sd

R

n

r

So

Si z 

St

Figure 6.1.1 Schematic illustration of a capsule suspended in a fluid-filled tube.

given in the effect of surfactant and temperature variation. A more extensive description of the hydrodynamics of capsules enclosed by elastic membranes can be found in Chapters 1, 2, and 4.

6.1 Capsule motion in tube flow Consider the axisymmetric motion of a liquid particle through a tube of radius R that is filled with another immiscible fluid, as shown in Figure 6.1.1. The exterior and interior phases, denoted by and , consist of two incompressible Newtonian fluids with densities and , and corresponding viscosities and , where is the density ratio, and is the viscosity ratio. If the capsule is a conventional drop or bubble, the interface exhibits isotropic interfacial tension, . The dynamics of drops and bubbles has been studied extensively by theoretical and laboratory methods (e.g., [14, 49, 72]). On the other hand, if the interfacial tension is nonuniform or the interior and exterior phases are separated by a thin membrane that exhibits elastic or more general properties, as discussed in Chapters 1 and 2, the fluid particle is a capsule. 























6.1.1 Governing equations In the following analysis, lengths will be made dimensionless by the tube radius , velocities will be made dimensionless by a characteristic velocity to be defined on a case-by-case basis, stresses will be made dimensionless by , and time will be reduced by . Throughout this chapter, unprimed and primed variables refer to the exterior and interior phase, respectively. Because the particle shape is a priori , will be used unknown, the radius of a spherical particle with the same volume, as the characteristic particle size, where is a dimensionless coefficient.

















Capsules in tube and channel flow

265

Most studies of capsule hydrodynamics in tubes have addressed situations where , is small so that inertial effects can be neglected the Reynolds number, and the motion of the fluid is either steady or quasi-steady. In the limit of vanishing Reynolds number, the flow in the two phases is described by the continuity equation and the Stokes equation whose dimensionless forms are






















































for 









(6.1.1)



for 







(6.1.2)











where and are the velocity and modified pressure incorporating pressure variations due to gravity. Consistent with our restriction of axisymmetric motion, we shall assume that the tube is vertical and the acceleration of gravity vector points along the tube axis, , where is the unit vector in the axial direction. In the polar cylindrical coordinates defined in Figure 6.1.1, the position vector is given by 

















#

$

!







%

'

#





(6.1.3)

!

and the unit vector normal to the capsule surface directed into the exterior phase is given by 

,







'

,

%



%

(6.1.4)

+





%

where is the unit vector in the radial direction. It is convenient to describe the motion in a frame of reference that is fixed at the center of mass of the particle and moves with the particle velocity, . Relative to this moving frame of reference, the far-field and boundary conditions are 3







7





as 

3











$





at 

3



$







 $

=



D

D



$



J

for



H H



A

(6.1.7)

B





A

B

(6.1.8)

E

+



(6.1.6)

for 

+

C



<







:

for





+

(6.1.5)

9

!





#

9





A

B

(6.1.9)

stands for the interface. The far-field where is the Newtonian stress tensor, and condition (6.1.5) states that, far upstream and downstream, the velocity field reduces . In the case of pressure-driven flow through a to that of the undisturbed flow, straight tube, the undisturbed velocity describes Poiseuille flow with average . Equation (6.1.6) imposes the no-slip boundary velocity , and condition at the tube surface. H

A



7





7



P

R

=









!

T



7

B

266

Capsules and Cells

Boundary conditions at the interface include continuity of velocity expressed by (6.1.7), the kinematic condition expressed by (6.1.8), and a dynamic condition involving the jump in interfacial traction due to the interfacial mechanics, expressed by (6.1.9). These conditions do not over-specify the problem, since the capsule shape is unknown a priori and must be determined as part of the solution. The form of the stress jump depends on the nature of the interface separating the two phases. For liquid capsules exhibiting nonuniform isotropic interfacial tension, 













































for











(6.1.10)



is the surface gradient. In (6.1.10), the interfacial tension, where , has been made dimensionless by a reference value to be defined. The capillary number, 


















(6.1.11)







expresses the ratio between the deforming viscous forces to interfacial tension forces resisting deformation. The Bond number, 


















 

(6.1.12)



expresses the relative importance of gravitational and interfacial forces. The first term on the right-hand side of (6.1.10) accounts for surface elasticity due to variations in interfacial tension, while the second term represents the jump in the normal stress across a curved interface caused by surface tension. In this chapter, we shall consider circumstances where nonuniformities in interfacial tension are induced by variations in either surfactant concentration distribution or temperature. For liquid capsules enclosed by elastic membranes, the traction jump is proportional to the membrane load, , 














 







 















(6.1.13)

where 




 





(6.1.14)

is the elasticity capillary number defined with respect to an appropriate surface mod. Physically, provides us with a measure of the relative ulus of elasticity magnitude of deforming viscous forces compared to the restoring elastic forces. As decreases, the membrane becomes stiffer and undergoes smaller deformations. The membrane load is determined by the membrane tensions and the bending moments, as will be described in Section 6.1.4 (see also Chapters 1 and 2). 













Capsules in tube and channel flow

267

6.1.2 Surfactant-induced elasticity Surfactants are present in most multiphase fluid systems either in the form of additives or in the form of contaminants and impurities that are extremely difficult to remove. As the concentration of a surfactant is raised, the interfacial tension is generally lowered. Since the surface concentration depends on the surface velocity field, the flow problem is strongly coupled with the mass transport problem determining the surface and bulk surfactant distribution. Accordingly, surfactants can have a profound effect on the dynamics of an interfacial flow. For example, surfactants have been shown to suppress convective instabilities [4] and to stabilize viscous film flows under certain conditions [76]. Surfactant molecules are amphiphiles consisting of a hydrophilic headgroup and a hydrophobic tail, which is typically a hydrocarbon chain. Because of the finite size of the headgroups and the excluded volume effect of the hydrophobic components, surfactants cannot accumulate at an interface beyond a certain limit corresponding to maximum surface packing. If the local surface concentration attempts to exceed the critical value due, for example, to convection, surface monolayers may saturate. This leads to the precipitation of a new surface phase, to the solubilization of the surfactant in the bulk liquid phase, or both. These transitions considerably complicate both the physical interpretation and the mathematical description of interface mass transport. Surfactant molecules adsorbed at an interface can interact either cohesively or repulsively. The adsorption/desorption kinetics, the partitioning of the surfactant between the various phases, and the dependence of the interfacial tension on the surface concentration are all strongly affected by these molecular interactions. Consequently, the physical chemistry of the surfactant significantly influences the dynamics of the induced flow. Surface transport In the absence of bulk fluid motion, surfactant molecules adsorb and diffuse along , which is in equilibrium with the interface to establish a uniform concentration, . The interfacial flow induced by the bulk fluid motion the bulk concentration, disturbs the equilibrium, and the new balance between the convective flux and the competing fluxes due to diffusion and adsorption/desorption leads to surfactant redistribution. , is govThe nonequilibrium surface concentration, , nondimensionalized by erned by the unsteady convection–diffusion equation [1, 47, 71, 79] 






























 

























 















(6.1.15)



, where is the rate of change following interfacial marker points for moving normal to the interface [79], and is the tangential or surface velocity. The first three terms on the left-hand side of (6.1.15) represent accumulation, surface convection, and diffusion, while the fourth term accounts for variations in surfactant concentration resulting from local interface stretching and 

































268

Capsules and Cells

deformation. The surface P`eclet number,







 

(6.1.16)



defined with respect to the surfactant surface diffusivity , expresses the relative importance of convective to surface diffusive transport. The sorption flux j on the right-hand side of (6.1.15), expressing the rate at which surfactant is supplied from the bulk of the fluids to the interface, has been nondimensionalized by . If the surfactant is insoluble in both bulk phases, bulk transport can be neglected. Accordingly, (6.1.15) is simplified by setting . Assuming that the adsorption rate kinetics is first order in the bulk surfactant concentration and in the available interfacial sites, corresponding to second-order overall adsorption kinetics, and that the desorption rate is first order in the surface concentration, we express the net adsorptive flux at the interface in the dimensionless form 





































(6.1.17) 









where is an adsorption coefficient – not necessarily a constant – expressing the ratio between the characteristic adsorption and desorption rates. The Biot number, 









 

(6.1.18)



expresses the ratio between the characteristic sorption rate to the rate of interfacial convection, where is the desorption rate constant. In (6.1.17), the sublayer concentration is in fact the surfactant concentration in the fluid immediately adjacent to the interface, nondimensionalized by the equilibrium bulk concentration . The quantity represents the maximum surface concentration for monolayer adsorption nondimensionalized by . This maximum value corresponds to a theoretical limit that normally will not be reached due to bulk constraints, such as the critical micelle concentration or the surfactant solubility limit. 


































Bulk transport If the surfactant is soluble in the suspending fluid alone, and if the bulk convective transport is negligible, surfactant transport occurs by a combination of adsorption/desorption kinetics and molecular diffusion. The bulk surfactant concentration is governed by the steady diffusion equation, that is Laplace’s equation,





 





for 



(6.1.19)



subject to: (a) the far-field condition






as 







(6.1.20)



(b) the no-flux condition on the tube wall, and (c) the interface flux condition 






 






































(6.1.21)

Capsules in tube and channel flow where 



269

. The Damkohler number, 

 






 

(6.1.22)









expresses the ratio between the characteristic adsorptive and diffusive flux. Condition (6.1.21) arises by requiring that the diffusive flux from the bulk to the sublayer be equal to the net adsorptive flux at the interface. , several mass transfer regimes can be identified Depending on the value of , the adsorpfor nonzero values of the Biot number . For small values of tion/desorption process between the sublayer and the interface is much slower than bulk-phase diffusion, and surfactant mass transfer is sorption-controlled. In this prevails everywhere in the suspendregime, a uniform bulk concentration , surfactant diffusion from the ing fluid. On the other hand, for large values of bulk to the sublayer is much slower than the adsorption/desorption process, and surfactant mass transfer is diffusion-controlled. In this regime, equilibrium between the sublayer and the interface is established instantaneously, and the surfactant is partitioned according to an equilibrium adsorption isotherm, as will be described later in , the flux of surfactant from the bulk to the sublayer this section. In the limit tends to zero and the mass transfer problem describes an insoluble surfactant. Since the interfacial tension gradient on the right-hand side of (6.1.10) is due to a nonuniform surfactant distribution, boundary condition (6.1.9) can be rewritten as


































 





















 

















(6.1.23)

 

for . In this equation, the interfacial tension of a monolayer of uniform concentration, , established in the absence of flow, has been used as the reference value, . To evaluate the term on the right-hand side of (6.1.23), we require a surface equation of state of the form , corresponding to an appropriate adsorption framework. The Frumkin adsorption framework accounts for the finite size of surface molecules as well as for nonideal behavior of the surface layer. The corresponding adsorption isotherm, commonly used to describe the equilibrium behavior of nonionic surfactants, can be written as [11] 











































 

 

(6.1.24)



 



 



 









where is the surface coverage, is the Frumkin equilibrium adsorption constant, and is a dimensionless interaction parameter determining the extent and nature of nonideal behavior. The parameters and and the equilibrium surface coverage, , are related by




























































 










(6.1.25)

270

Capsules and Cells 

, the Frumkin isotherm reduces to the Langmuir isotherm and the adWhen sorption coefficient in (6.1.17) and (6.1.21) becomes a constant. The interaction parameter is negative in the case of cohesive interactions among surface molecules, and positive in the case of repulsive interactions. Nonideal (repulsive or cohesive) interactions among adsorbed molecules can either raise or lower the activation energy of adsorption, and thereby render the adsorption coefficient in (6.1.17) a function of surface coverage. For example, in the case of long-chain saturated surfactants such as -alcohols, cohesive van der Waals interactions among saturated chains can raise the desorption activation energy relative to that for adsorption. The opposite effect is expected for surfactants with bulky side branches which tend to exhibit repulsive interactions [40, 41]. The effect of nonideal molecular interactions on the net adsorptive flux can be included by introducing a linear dependence of the adsorption/desorption activation energy on surface coverage (for in (6.1.17) and (6.1.21) nonzero ), and by replacing the adsorption coefficient to obtain an interface flux condition that is consistent with the with Frumkin adsorption framework [54]. 

























Surface equation of state The appropriate selection of a surface equation of state corresponding to an adsorption framework is dictated by interfacial thermodynamics. The dimensionless Gibbs adsorption equation states that quasi-static (equilibrium) changes under constant temperature are related by 











(6.1.26)





The surface elasticity, , is a measure of the sensitivity of the interfacial tension to variations in the surfactant monolayer concentration. A surface equation of state arises by integrating (6.1.26) in conjunction with a chosen adsorption isotherm. In turn, the interfacial tension is determined by assuming that the surface equation of state applies locally for nonuniform surfactant distributions. In the case of the Frumkin isotherm, we obtain the relation 


















 





 











 

 








(6.1.27)















, (6.1.27) provides us with a linear law. Because of its inIn the dilute limit ability to capture surface saturation effects, the linear equation of state over-predicts the reduction in interfacial tension at high surface coverage. Nonlinear effects can have a significant impact on the interfacial tension. To demonstrate this feature, in Figure 6.1.2 we show a graph of the dimensionless surface pressure 




































(6.1.28)

representing the work per unit area required to compress the monolayer in order to add more surfactant, for different types of surfactants. The surface pressure is nearly

Capsules in tube and channel flow

271

3

2.5

surface pressure

[=+4 2

[=0 1.5

[=-4

1

0.5

0 0

0.2

0.4

0.6

0.8

1

surface coverage 



Figure 6.1.2 Adsorption isotherms for surfactants with no interaction ( ), repulsive interaction ( ), and cohesive interaction ( ). (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.) 














zero for low surface coverage corresponding to the dilute limit, and diverges as the surface coverage approaches monolayer saturation. For any surface coverage, it is more difficult to add surfactant in the presence of repulsive molecular interactions than in the ideal case of the Langmuir framework which assumes no molecular interactions. This explains why the surface pressure for at any value of . On the other hand, positive values of is larger than that for surfactants with cohesive interactions corresponding to negative values of require less energy, that is, smaller surface pressure with respect to those of the Langmuir framework. According to (6.1.27), in the case of strongly cohesive surfactants with , the surface pressure has a critical point at 50% surface coverage such that any stronger cohesion would result in a surface phase change [18]. The surface pressure isotherm for such a surfactant exhibits a plateau that can be thought of as a coexistence region between a surface-expanded state at low surface saturation and a surface-condensed or aggregated state at high surface saturation. The plateau extends over the range , wherein the interfacial tension remains essentially independent of surface coverage. Thus, in the case of strongly-cohesive surfactants, Marangoni stresses are suppressed in regions of the interface where the surface coverage falls within the coexistence region. 





































272

Capsules and Cells

6.1.3 Temperature-induced surface elasticity When the interfacial tension gradient on the right-hand side of (6.1.10) is due to a nonuniform temperature distribution, the boundary condition (6.1.9) takes the form 
















 
















(6.1.29)


















where , and is the temperature nondimensionalized with respect to a reference value. An example of a reference temperature is the undisturbed temperature at the center of mass of the capsule in a translating frame of reference. In (6.1.29), the interfacial tension at the reference temperature is used as the reference value . To compute the term , we require a surface equation of state of the form . In the case of a liquid drop moving in an otherwise quiescent fluid under the action , the characteristic velocity is given of a constant axial temperature gradient by , and the dimensionless parameter 










































































(6.1.30) 










provides us a measure of the sensitivity of the interfacial tension to the temperature; the temperature in (6.1.29) has been nondimensionalized by . In general, the characteristic velocity scale is time dependent due to the variations in the reference temperature defined in a frame of reference fixed at the capsule. However, the quasisteady approximation can be invoked when the time scale of the capsule deformation is short compared to that of the capsule motion. Interfacial tension is usually a weakly decreasing function of temperature. Using a linear equation of state of the form 























(6.1.31)
















where the subscript ‘o’ signifies evaluation at the reference temperature, (6.1.29) can be written in the more specific form 






 






















(6.1.32)























where . The linearization implicit in the Gibbs equation of state (6.1.31) automatically restricts the range of physically relevant capillary numbers to small values, . Stated differently, a linear dependence of the interfacial tension on temperature is generally not realizable over an extended range of interfacial tensions, and can only be justified when the variation of interfacial tension across the capsule surface is small relative to the mean or reference value of the interfacial tension. Notwithstanding this physical limitation, computations have been performed with capillary 
















Capsules in tube and channel flow 

273




in order to mathematically explore the effect of this paramnumbers as high as eter on the capsule deformation [6]. In general, more realistic equations of state are needed to obtain physically relevant predictions of the capsule shape and migration velocity at large capillary numbers. Before we can solve equation (6.1.32), we must have available the temperature distribution over the interface. When thermal convection is negligible in the bulk of the fluids, the temperature distribution in the two phases satisfies Laplace’s equation,




in 








in 











(6.1.33)

The accompanying boundary conditions require no-flux condition at the insulated tube wall, continuity of temperature and heat flux at the interface, and specification of the undisturbed linear temperature distribution far from the capsule,





for















at











(6.1.35)

 



as



for 



(6.1.34)































(6.1.36) (6.1.37)



where is the ratio of the thermal conductivities of the interior and exterior phase. Equations (6.1.1) through (6.1.8) and (6.1.32) through (6.1.37) must be solved simultaneously to determine the quasi-steady migration velocity and capsule shape. 

6.1.4 Capsules enclosed by elastic membranes A Newtonian liquid drop enclosed by a thin deformable membrane is a reasonable model of biological cells, such as erythrocytes, and artificial cells encountered in industrial applications (see also Chapters 1 through 4). Studies of the motion and deformation of biological and artificial cells in pores provide us with valuable insight into the hydrodynamics of the microcirculation, and serve as a basis for the quantitative analysis of laboratory data on cell filtration. The important distinguishing feature of a large class of biological cells, as compared to artificial cells, is that the membrane is invariably composed of a lipid bilayer that strongly resists changes in surface area. For example, the modulus of shear elasticity of a human red blood cell is on the order of dyn/cm, which is much dyn/cm [62]. Thus, an isotropic tension smaller than the modulus of dilatation of that tends to increase the area of the membrane encounters a stiff elastic response, while the membrane can be easily sheared. The membrane bending modulus is on dyn cm, which is also very small. Consequently, resisthe order of tance to bending is significant only in regions of high curvature (see also Chapter 2). Overall, the surface-area preserving property of biological cells imposes limits on the possible modes of deformation. 


































274

Capsules and Cells

q m

ms



r







s



r=1 (rd,zd) rd

s

z

Figure 6.1.3 Geometry of an axisymmetric capsule membrane shape.

In the undeformed axisymmetric configuration, the contour of the membrane in a meridional plane is described by the position of material points with coordinates and arc length , with corresponding to the front point where . After deformation, the material points are displaced at the position corresponding to the deformed arc length . The principal extension ratios and in the azimuthal and meridional directions are defined as 







































































 





 





(6.1.38)







  

 

The membrane tensions in the azimuthal and meridional directions are denoted by and , the shear tension acting normal to the membrane cross-section is denoted by , and the azimuthal and the meridional bending moments are denoted by and , as shown in Figure 6.1.3. Equilibrium of normal stresses, tangential stresses, and surface torques requires 



































 

 



(6.1.39)



 













 

















 



 











 

  

 







 



 

(6.1.40)



 





 







 







(6.1.41)











 











 

 





, where is the unit tangent vector in the direction of for increasing arc length, . All tensions have been rendered dimensionless with an appropriate surface modulus of elasticity . The principal curvatures of in the 

























Capsules in tube and channel flow

275

meridional and azimuthal planes are given by 







































(6.1.42)



 







 



To complete the statement of the problem, we introduce constitutive laws for the ), and bending moments ( ). Several available membrane tensions, ( choices are next summarized (see also Chapters 1 and 2): 



















1. Linear elasticity: For small deformations, the membrane can be assumed to obey Hooke’s law for a two-dimensional continuum. The principal tensions are given by 

















 








































(6.1.43) 



































 
















 






where



is the surface Poisson ratio.

2. Neo-Hookean elasticity: The membrane is modeled as an infinitely thin threedimensional incompressible material that is allowed to undergo local changes in surface area [20]. The principal tensions are given by 












 

 




 



 

 



 















 

 





 







(6.1.44)





3. Red-blood-cell elasticity: This constitutive law applies to lipid bilayer membranes which are known to strongly resist local changes in the surface area [69]. The principal tensions are given by 















 











































(6.1.45) 





 







 



 





 











 













 






where is the ratio of the dilatational to the Young modulus of elasticity. Consistent with the requirement of near incompressibility, the value of is high. 














4. Evans-Skalak elasticity: This model is used to describe area incompressible [17]. The principal tensions are membranes obeying the restriction decomposed into a mean and a deviatoric component, 














































(6.1.46)

276

Capsules and Cells The deviatoric component is given by 




 






 

(6.1.47)



























is the ratio of the surface shear modulus, , to the where Young modulus, . The isotropic component, , is determined by requiring the condition of incompressibility locally on the membrane. 

















5. Viscoelasticity: Viscous behavior can be incorporated by adding to any of the preceding elastic constitutive laws viscous tensions with principal components  





 















(6.1.48)



























where






 









, and





is the surface viscosity. 



Evans & Skalak [17] assumed that the bending moments are isotropic and proportional to the increase in the total membrane curvature above that of resting state, 




































































(6.1.49)

where , is the reduced bending modulus, and the superscript ‘ ’ indicates the unstressed (reference) shape. More accurate and realistic constitutive equations have been developed in recent years under the framework of shell theory, as discussed in Chapter 2. The complete system of equations describing the motion of a capsule consists of the equations of motion (6.1.1) through (6.1.8) and (6.1.13), equations (6.1.39) through (6.1.41) describing the elastic response and mechanical equilibrium of the membrane, and the membrane constitutive equations. If the typical thickness of the gap between the capsule and the vessel wall is small compared to the length scale of the capsule, the lubrication approximation and the method of matched asymptotic expansions can be used to obtain solutions. More generally, the governing equations must be solved by numerical methods. Most theoretical studies of the hydrodynamics of biological and artificial cells have considered low-Reynolds-number flow, and presented solutions using the boundary-integral method described in Section 6.2. 



6.2 Numerical methods When inertial and convective effects in the hydrodynamics and surfactant transport equations are insignificant, a numerical solution of the governing equations may be computed efficiently using the boundary-integral method. This method is particularly well-suited for moving/deformable boundary problems, as it only requires

Capsules in tube and channel flow

277

the discretization of the boundaries of the computational domain. Thus, computing solutions to an axisymmetric problem requires solving one-dimensional integral equations defined over the contour of the boundaries in a meridional plane. A brief account of the solution procedure will be given in this section. Following the standard boundary-integral formulation for Stokes flow, (e.g,. [37, 55]), we derive a representation for the velocity field in terms of fundamental solutions, 















 















 




 





































 









































































 

 

(6.2.1)



































 































 

















where is the evaluation point, and is the integration point. The kernels and are the velocity and pressure fundamental solutions or Green’s functions of Stokes flow, given by 















 















 











 



(6.2.2)



















and . The domain of integration in (6.2.1) includes where and , the the inflow and outflow planes in the capsule-fixed reference frame, intervening section of the tube wall, , and the surface of the capsule , as shown in Figure 6.1.1. The unit normal vector points into the capsule exterior . The no-slip condition is imposed on the tube wall, while the disturbance velocity due to the capsule is required to vanish at the inflow and outflow. Given an instantaneous capsule shape, a standard boundary-element collocation method can be used to solve the integral equation for ), subject to a specified distribution of the load over the interface . The unknowns are the interfacial velocity on , and the traction distribution on . The computation of over for liquid capsules with surfactant- or temperature-induced elasticity and for liquid capsules enclosed by elastic membranes will be described in the following sections. 

















































































6.2.1 Surfactant-induced elasticity 

To compute the jump in stress , we require the instantaneous surfactant distribution on the capsule surface , which must be obtained by solving the surface convective-diffusion equation (6.1.15). In the case of an axisymmetric interface, (6.1.15) can be rewritten in terms of the distance from the axis and arc length along





278

Capsules and Cells

a meridional plane , as 









 




 















  







 



 










 











(6.2.3)







 



 

















 



 



where

and are the interfacial velocities tangent and normal to the interface, is the total curvature, equal to twice the mean curvature, and is the net sorptive flux of the surfactant at the interface. In the case of an insoluble monolayer, the interface flux is zero, and the solution of (6.2.3) can be marched in time using the unconditionally stable Crank-Nicolson method combined with a second-order central difference approximation for the spatial derivatives. The discretization is implemented on the interfacial grid used for solving the integral equations [34]. Specifically, given the surfactant distribution at , a tridiagonal system of linear algebraic time t and the capsule shape at time equations is solved to obtain the surfactant distribution at time . In the case of bulk-soluble surfactants, the interface flux on the right-hand side of (6.2.3) is given by (6.1.17). To compute the solution, we require the bulk concentration distribution at the edge of the sublayer, and this arises by solving equations (6.1.19) through (6.1.21) using numerical methods. Following the standard boundary-integral formulation for Laplace’s equation, we derive an integral representation for the bulk concentration of the surfactant, 






























































































































































 

























 

 











(6.2.4)


































where is the disturbance in the bulk concentration, and and fundamental solutions of Laplace’s equation in three dimensions, given by 













are















 



 









(6.2.5)



Specifically, is the free-space Green’s function of Laplace’s equation. Substituting the boundary conditions (6.1.20) and (6.1.21) into (6.2.4), we obtain an integral equation of the second kind for the unknown sublayer concentration distribution. Given an instantaneous capsule shape, this integral equation can be solved using a standard boundary-element collocation method.

Capsules in tube and channel flow

279

6.2.2 Temperature-induced elasticity In this case, to evaluate the interfacial traction jump , we require the instantaneous temperature distribution over the interface, which must be obtained by solving the thermal energy transport equations (6.1.33) through (6.1.37) using numerical methods. Applying the boundary-integral formulation for Laplace equation, we derive integral representations for the temperature field, 












































 



(6.2.6)








  













 











































 





















































(6.2.7)



 





















 



 

 

 



and are the fundamental solutions of Laplace’s equation where the kernels defined in (6.2.5). An integral identity requires 































(6.2.8)






 









where or . Applying the boundary conditions (6.1.34) through (6.1.37), we obtain an integral equation for the unknown surface temperature distributions,




































 


























 









 











 































 







(6.2.9)































 



A boundary-element collocation method can be used to solve these integral equations for the unknown interfacial temperature distributions.

6.2.3 Capsules enclosed by elastic membranes In the case of liquid capsules enclosed by elastic membranes, the traction jump is given by , where is the membrane load determined by the equilibrium equations (6.1.39) through (6.1.41) (see also Chapter 2). In the absence of bending moments, we find 













 



  





 






















 

 

 



 





















(6.2.10)

280

Capsules and Cells

. Given constitutive equations for the membrane material, the calculation for of the membrane tensions requires information on the unstressed and instantaneous membrane shape. The instantaneous position of material points on the membrane derives from a kinematic condition stating that membrane point particles move with the fluid velocity,














(6.2.11)





where


. 





6.2.4 Numerical algorithm When the capsule shape is axisymmetric, the azimuthal integrations in (6.2.1), (6.2.4), and (6.2.9) can be performed analytically working in cylindrical polar coordinates, thereby reducing the surface integrals to line integrals along the interface contour in the meridional plane. The integration contours are discretized into boundary elements, and the line integrals are evaluated by numerical methods. Although , the singularities are the kernels (6.2.2) and (6.2.5) become singular as integrable, and the integrals can be evaluated with adequate accuracy. Given the position of the capsule contour at any time , the local unit normal and tangent vectors as well as the surface curvature can be calculated using standard formulas of differential geometry. The solution involves the following steps: 





1. Specify the initial capsule shape, the capsule velocity , and the pressure drop between and . In the case of surfactant- or temperature-induced elasticity, an initial guess for the surface distribution of the surfactant concentration or temperature is also required. 













2. Determine the traction jump, , as described in sections 6.2.1 through 6.2.3. 3. Solve the integral equation (6.2.1) for the interface velocity and wall traction distributions. 4. Correct the value of





using a force balance on . 

5. Integrate the kinematic condition (6.1.8) or (6.2.11) to obtain the capsule shape at . 







6. Adjust the capsule velocity,

, as 





 



















 







where





is the axial displacement of the capsule center of mass.

(6.2.12)

Capsules in tube and channel flow

281

7. In the case of surfactant- or temperature-driven elasticity, determine the surover the face distributions of the surfactant or temperature at time updated location of the interface. This step requires the numerical solution of (6.2.3) and (6.2.4) in the case of surfactant-induced elasticity, or (6.2.9) in the case of temperature-induced elasticity.




8. Check one or both of the termination criteria: (a) the normal component of ); (b) the velocity at each node is less than a specified tolerance ( relative change in the normal velocity between successive steps is less than a specified tolerance. 











After each time step, the collocation points are redistributed to ensure interfacial elements of uniform size with respect to arc length, and the (in principle conserved) capsule volume is calculated to estimate the truncation error. The simulation is carried out until either a steady state is achieved, or the capsule appears to be on the verge of breakup. The step size, , and the number of elements necessary to obtain sufficiently accurate results depend on the values of the dimensionless parameters, particularly the dimensionless capsule size and capillary number. Typically, elements are sufficient to discretize the contour of a capsule with dimensionless size , although a larger number is required for highly deformed capsules. For a and , a typical dimensionless time step of capsule size with is sufficient to obtain accurate results. Larger time steps are used as the capsule size or the capillary number increase and the capsule becomes more deformable, though in these cases more time steps will be needed to reach the steady state. 














































6.3 Capsules with surfactant-induced elasticity in tube flow A number of theoretical studies on the motion of drops and bubbles through cylindrical tubes in the presence of surfactants have been conducted using asymptotic and numerical methods. Most of the earlier studies used asymptotic expansions to describe the motion of long bubbles through capillaries [3, 19, 31, 51, 58, 70]. A main goal has been to determine the influence of surfactants on the thickness of the liquid film separating the bubble from the tube wall. Ginley & Radke [19] found that, when surfactant transport is controlled by sorption kinetics and the surfactant concentration is uniform in the bulk phase, the film thickness is smaller than that in the case of surfactant-free flow. On the other hand, Ratulowski & Chang [58] showed that, if surfactant transport in the thin film is controlled by bulk diffusion from the liquid ahead of the bubble in the limit of asymptotically small bulk surfactant concentration, the film thickness can actually increase by compared to that of the “clean” system. Stebe & Barth`es-Biesel up to a factor of [70] discovered a similar film-thickening effect in the case of sorption-controlled mass transfer at elevated bulk surfactant concentrations. 








282

Capsules and Cells

The effect of the trailing end of a large bubble of finite length was taken into account by Park [51] who showed that, in the diffusion-controlled regime, the film thickening effect arises only when the bubble length is larger than a certain critical value, which is in line with the experimental observations of Schwartz et al. [61]. Park’s analysis was also able to explain the experimental results of Barth`es-Biesel et al. [3] and Marchessault & Mason [42], by showing that the total pressure drop required to drive a finite bubble through the tube increases with the length of the bubble, so long as the bubble length is smaller than the critical value. Once the critical bubble length has been exceeded, the pressure drop becomes constant. Few studies have accounted for the effect of surfactants on the motion of capsules whose size is comparable to the tube diameter [8, 29, 34, 75]. He et al. [29] studied the influence of surfactant adsorption on the pressure-driven motion of a neutrallybuoyant capsule in low-Reynolds-number tube flow, and verified the retarding effect of the surfactants on the capsule motion. In this study, a fixed spherical capsule shape was assumed, and the interfacial deformation was not considered. The increase in interfacial area due to deformation causes local dilution of the monolayer concentration which, in turn, affects the surface convection by modifying the Marangoni traction. Interface deformation is an important aspect of the motion that must be taken into account in order to determine the correct distribution of surfactant along the surface of the capsule, particularly when the capsule size is comparable to the tube diameter. Borhan & Mao [8] included the effect of interfacial deformation in boundaryintegral simulations of low-Reynolds-number tube flow in the presence of a bulkinsoluble monolayers. Surfactant accumulation at the rear end of the particle was shown to facilitate the formation of a re-entrant cavity. The penetration length of the cavity into the capsule increases as surface convection becomes more important, that is, as the surface P`eclet number is raised. The effect of an insoluble monolayer on the formation of a region of negative curvature at the rear end was further confirmed by Tsai & Miksis [75]. Borhan & Mao [8] and Tsai & Miksis [75] confined their attention to the special case of bulk-insoluble monolayers. In both cases, the surfactant monolayer was assumed to obey a linear surface equation of state applicable for dilute concentrations. When the monolayer concentration becomes appreciable, the linear model becomes inadequate due to monolayer saturation and nonideal – cohesive or repulsive – interactions among the surface molecules. Johnson & Borhan extended the earlier results by examining the effect of bulk-insoluble monolayers at high surface coverage using a surface equation of state based on the Frumkin adsorption framework [34], and by considering the effect of the bulk solubility of surfactants in the limit where bulk diffusion dominates convection [35]. In the remainder of this section, we describe the results of numerical simulations of capsule motion through a tube, concentrating on the effect of the relevant dimensionless parameters on the capsule shape, capsule velocity, and extra pressure loss due to the presence of the particle. In the case of pressure-driven motion of a capsule through a cylindrical tube, the surface flow pattern includes two stagnation rings and two stagnation points at the leading and trailing end of the capsule. The stagnation

Capsules in tube and channel flow

283

1

3 3

15 17

17

1

5 5

7

7

19 15

17

15

Level

C

3

19

1.0003 1.0002 1.0000 0.9999 0.9998 0.9997 0.9996 0.9994 0.9993 0.9992

17 15 13 11 9 7 5 3 1

1

19

5

11

13

C

1

15

17

11

13

19 17 15 13 11 9 7 5 3 1

11

13

19

Level

9 11 13

9

3

15

11

7.6458 6.7622 5.8785 4.9948 4.1112 3.2275 2.3439 1.4602 0.5766

1

1

15

17

17

15

(a)

(b)


Figure 6.3.1 Bulk concentration distribution for , , , , and . (b)
















































,












; (a)



















, and














,
















point at the trailing end and the stagnation ring near the leading end are both regions of diverging surface flow depleted of surfactant. On the other hand, the stagnation point at the leading end and the stagnation ring near its trailing end are both regions of converging surface flow where surfactants tend to accumulate. For finite values of , the surfactant concentration field mirrors the surface concentration distribution, which strongly depends on the surface flow pattern. Typical isocontours of the bulk concentration field are shown in Figure 6.3.1 at low and and [35]. The isoconcentraelevated surface saturations, for tion contours in this figure provide a qualitative description of mass transport in the suspending fluid. The surfactant distribution is much broader at higher surface saturations, with the bulk of the surfactant accumulating near the stagnation ring at the rear. The mobility of a capsule covered with an insoluble monolayer is illustrated in Figure 6.3.2 for several values of the surface P`eclet number. In general, increasing the surface P`eclet number reduces the migration velocity and raises the pressure . Because of the strong surface flow loss due to the presence of the capsule, in the convection-dominated regime corresponding to high , the interface supports large surfactant concentration and interfacial tension gradients, and the resulting Marangoni tractions are responsible for a significant retardation of the capsule , motion. In the case of a surfactant exhibiting strong cohesive interactions, the mobility of the capsule is insensitive to the surface P`eclet number up to a thresh; at higher values, the mobility monotonically decreases old value, 


























































284

Capsules and Cells 3.5

1.60

migration velocity, U

extra pressure loss, ∆P+

3.0 1.55

= -4

1.50

=0

= +4 1.45

= +4 2.5 =0 2.0 = -4 1.5

1.40

1.0 0.1

1

10

100

0.1

1

10

100

Peclet number, Pes

Peclet number, Pes

(a)

(b)

Figure 6.3.2 (a) Capsule velocity, and (b) extra pressure loss for insoluble mono), strong cohesive ( ), and no ( ) layers with repulsive ( molecular interactions, for , , , , and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.) 

&




















=









=












































2.0

0.8

1.1 Pes = 10

0.6

Pes = 10

1.0 Pes = 1.0 Pes = 1.0

0.8

0.9

Marangoni stress,

1.4

interfacial tension,

s

surfactant concentration,

Pes = 10

0.4

0.2

0.0

0.2 0.0

0.2

0.4

0.6

0.8

0.8 1.0

Pes = 1.0

-0.2 0.0

0.2

0.4

0.6

0.8

normalized arclength, s

normalized arclength, s

(a)

(b)

1.0

Figure 6.3.3 Distribution of (a) the monolayer concentration and interfacial tension, , , , , , and (b) Marangoni stress, for and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.) 

























































"

with ; whereas at lower values, surface concentration gradients are small and the surfactant distribution is contained entirely within the coexistence region evident in Figure 6.1.2, as shown by the surface distribution of the monolayer concentration for in Figure 6.3.3 [34]. #

#

%

'



%



'

Capsules in tube and channel flow

285

The normalized arc length, , in Figure 6.3.3 is measured along the top half of the capsule profile, with corresponding to the rear of the capsule and corresponding to the leading end. For , the interfacial tension is nearly uniform and the induced Marangoni stresses are negligible. In the surface-convectiondominated regime, the nonuniformity in the surfactant distribution is large enough for the coexistence region shown in Figure 6.1.2 to be fully spanned. Monolayer concentrations on both sides of the plateau are present on the interface, as is evident . In a small in the interfacial tension distribution shown in Figure 6.3.3 for , the interfacial tension is constant and equal to its equilibrium region near value at . The Marangoni stress is nearly zero in the small coexistence regime, but rises sharply in the transition zone around , particularly toward the rear end where the surfactant is in the ’surface-condensed’ state, as shown in Figure 6.3.3(b). The effect of surface P`eclet number on the surface velocity profile is illustrated in Figure 6.3.4. For monolayers with strong cohesive interactions, the velocity profiles are essentially identical to that in the absence of below the threshold value of , the magnitude of the surface velocity surfactants. Above the threshold value of is diminished, indicating a reduction in the capsule mobility. This reduction is consistent with the stagnant cap behavior postulated by Davis & Acrivos [16], and later investigated by Sadhal & Johnson [59], and He et al. [30]. These authors showed that, for negligible surface diffusion, surfactant mass balance at the interface requires the tangential surface velocity to vanish in a region of nearly uniform concentration at the rear. The stagnant region formed at the trailing end of a capsule translating with a terminal velocity acts like a no-slip surface, increasing the drag on the cap,a sule and reducing the mobility. For convection-dominated systems, region of near-zero surface velocity develops in the vicinity of the trailing end, as shown in Figure 6.3.4. For , the stagnant cap covers 30% of the capsule , leading to a 7% decrease in the capsule surface, roughly corresponding to velocity, and a 150% increase in the extra pressure loss compared to those of the surfactant-free case. In contrast to strongly cohesive monolayers, repulsive and Langmuir monolayers do not exhibit a threshold value for because of the absence of a coexistence plateau in their surface pressure isotherms. In the surface-convection dominated limit, capsules with either one of these monolayers approach a limit of decreased mobility corresponding to a fully-immobilized interface. However, they do not develop a stagnant cap observed for strongly cohesive monolayers where only a portion of the interface becomes completely immobilized. Instead, as the surface P`eclet number increases, the magnitude of the surface velocity decreases uniformly over the entire interface until it reaches nearly zero values. The uniform retardation of the surface flow for monolayers with strong repulsive interactions is evident in Figure , the entire interface essentially acts like a no-slip surface, 6.3.4. For maximizing the capsule drag and minimizing the migration velocity. The steady migration velocity of the capsule and extra pressure loss are plotted in . To allow for Figure 6.3.5 against the interaction parameter for three values of a quantitative assessment of the influence of molecular interactions on the surfactant 













=







=









=









=































=















=





















=



Capsules and Cells 0.2

0.2

0.0

0.0

surface velocity, us

surface velocity, us

286

Pes = 100

-0.2 10

Pes = 100

10 -0.2 1.0

1.0 -0.4

-0.4

0.1

0.1 clean

clean

-0.6 0.0

0.2

0.4

0.6

0.8

-0.6 0.0

1.0

0.2

0.4

0.6

0.8

normalized arclength, s

normalized arclength, s

(a)

(b)


1.0



Figure 6.3.4 Surface velocity profiles for , , , , and ;( ) and ( ) [- - -, surfactant-free interface]. (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.) 






















































transport and redistribution, the initial equilibrium interfacial tensions were chosen to be identical. For comparison, results for a surfactant-free case are also shown with the dotted line, corresponding to surface tension that is equal to that of the initial equilibrium value for the surfactant-laden interface. In all cases, the extra pressure loss is higher, and the migration velocity is smaller than those for a clean interface. The reduction in the capsule mobility is more pronounced for larger values of . As the value of is decreased at fixed surface P`eclet number, stronger cohesive interactions between surfactant molecules cause the surfactant to accumulate near the rear stagnation ring. The larger concentration variations between the front and rear stagnation rings result in higher Marangoni tractions and cause a reduction in the capsule mobility. Difference in the qualitative behavior of strongly cohesive monolayers and Langmuir or repulsive monolayers is evident in Figure 6.3.6. In the case of repulsive and Langmuir-type monolayers, the capsule velocity monotonically decreases, and the extra pressure loss monotonically increases with increasing equilibrium surface coverage. The retarding effect of increasing equilibrium surface coverage can be better understood from a closer examination of the surface distributions of the surfactant concentration, interfacial tension, and Marangoni stress shown in Figure 6.3.7 for , saturation plays an a Langmuir monolayer. At high surface coverage, important role in the development of the monolayer concentration profile, resulting . in a more uniform distribution of surfactant compared to the dilute case, However, the highly saturated interface develops larger interfacial tension gradients than the dilute interface. For , the Marangoni stress is given by . is raised, a smaller surface concentration gradient is Hence, as the value of needed to generate the same Marangoni traction.
































































Capsules in tube and channel flow 1.60

287

3.5 clean

extra pressure loss, ∆P+

migration velocity, U

3.0 Pes = 0.1

1.55

1 1.50

10

1.45

Pes = 10 2.5

2.0 1 1.5 0.1 1.0

1.40

clean

0.5 -3

-2

-1

0

1

2

3

-3

-2

-1

interaction parameter,

0

1

2

3

interaction parameter,

(a)

(b)

Figure 6.3.5 (a) Steady capsule speed, and (b) extra pressure loss for fixed equilibrium interfacial tension for , , and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.)




=













=













The aforementioned effect is clearly illustrated in Figure 6.3.7 where the maxioccurring at is seen to be much mum Marangoni stress for , even though the concentration gradient is larger in larger than that for the more dilute interface. A major consequence of the larger Marangoni stress for is a reduction in the magnitude of the interfacial velocity shown in Figure 6.3.8, causing the highly saturated interface to become more rigid compared to an interface with . Figure 6.3.8 also reveals a shift in the location of the stagnation rings on the surface of the capsule away from the ends, occurring as surface convection becomes weaker with increasing surface coverage. In the case of a strongly cohesive monolayer, , the capsule becomes more mobile with increasing the surface coverage, and exhibits maximum mobilinitial saturation. Beyond saturation, the capsule mobility ity at roughly decreases with increasing surface coverage, as it does in the case of repulsive and Langmuir monolayers. This unique behavior of strongly cohesive monolayers can be attributed to the existence of a phase transition-like region in the surface pressurface sure isotherm, identified earlier in Figure 6.1.2 by the inflection point at coverage. Since the surface pressure and hence the interfacial tension are essentially , independent of surface coverage in the coexistence region centered around the Marangoni stress is effectively reduced to zero for concentration variations inside the plateau, as shown in Figure 6.3.9. Although monolayer concentration variations for are larger than those for the repulsive monolayer, the interfacial tension is nearly uniform and virtually no . The resulting enhancement in interface Marangoni stresses are induced for mobility is confirmed by the surface velocity distributions shown in Figure 6.3.10. 






























=































=














































288

Capsules and Cells 2.5

1.60

extra pressure loss, ∆P+

migration speed, U

= -4 1.55

=0

1.50

= +4

2.0

= +4

1.5

=0

= -4 1.45

1.0 0.01

0.1

1

0.01

0.1

equilibrium surface coverage, 1/eq∞

1

equilibrium surface coverage, 1/eq∞

(a)

(b)

Figure 6.3.6 (a) Steady capsule speed, and (b) extra pressure loss for insoluble ), repulsive interacmonolayers with strong cohesive interactions ( tions ( ), and no interactions ( ), for , , , , and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.)































































1.2

1.05

0.3

=0.1 = 10 eq ∞

= 1.3 eq ∞ =0.77

s

surfactant concentration,




1.00

1.0 0.95

Marangoni stress,

= 10

1.1


0.4

interfacial tension,

∞ =0.1 eq


1.10 = 1.3 =0.77

∞ eq


0.9

0.1


0.90

0.8 0.0

0.2

0.4

0.6

0.8

= 1.3

∞ =0.77 eq


0.2

= 10 ∞ =0.1 eq

0.0

0.85 1.0

-0.1 0.0

0.2

0.4

0.6

0.8

normalized arclength, s

normalized arclength, s

(a)

(b)

1.0

Figure 6.3.7 Distribution of (a) the monolayer concentration and interfacial tension, , , , , and (b) the Marangoni stress, for , and in the dilute ( ) and concentrated ( ) limit. (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.)














































































Capsules in tube and channel flow

289

surface velocity, us

0.25

0.00

=0.77 eq ∞ = 1.3 -0.25

∞ =0.4 eq

= 2.5

=0.1 eq ∞ = 10 -0.50 0.0

0.2

0.4

0.6

0.8

1.0

normalized arclength, s








Figure 6.3.8 Surface velocity profiles for dilute ( ), moderately concentrated ( ), and concentrated ( ) monolayers, for , , , , , and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.)





























































































The dashed line represents the surface velocity profile for a clean drop with constant interfacial tension equal to the equilibrium value in the presence of a uniform monolayer with . Decreasing the value of , while keeping the equilibrium surface coverage of an insoluble monolayer fixed, eventually restores the free motion of the interface. When the strongly cohesive monolayer is soluble in the bulk phase, the effect of increasing equilibrium surface coverage on capsule mobility is similar to that shown in Figure 6.3.6 for the insoluble monolayer, with the maximum capsule mobility occurring at higher surface coverage, as shown in Figure 6.3.11. In the case of bulk-soluble surfactants, the primary dimensionless parameters governing the interfacial adsorption/desorption and bulk transport of surfactant are the , and the Damkohler number, . The effects of these two paBiot number, rameters on the capsule mobility are demonstrated in Figures 6.3.12 and 6.3.13 for low and moderate equilibrium surface saturations, respectively. In general, as the Damkohler number is increased, the sorptive flux of surfactant to or from the interface is enhanced relative to the bulk diffusive flux, and mass transfer becomes increasingly diffusion-controlled. Consequently, the concentration distributions be, as shown in Figure 6.3.14. For come more nonuniform with increasing , the surface region between and is nearly devoid of surfactant, while a large concentration gradient and hence a high Marangoni traction develops . at 


























































290

Capsules and Cells

1.2

0.4

1.05

0.3

= +4

s

= +4 1.00

1.0 0.95

Marangoni stress,

= -4

interfacial tension,

surfactant concentration,

1.1

1.10 = +4

= -4

0.9

0.2 =0 0.1

0.90

0.0 = -4

0.8 0.0

0.2

0.4

0.6

0.85 1.0

0.8

-0.1 0.0

0.2

0.4

normalized arclength, s

0.6

0.8

1.0

normalized arclength, s

(a)

(b)

Figure 6.3.9 Distribution of (a) the monolayer concentration and interfacial tension, ) and and (b) the Marangoni stress for monolayers with repulsive ( strong cohesive ( ) interactions, , , , , , and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.) 





































































0.2

surface velocity, us

0.0

-0.2

= +4

=0 -0.4 = -4

-0.6 0.0

0.2

0.4

0.6

0.8

1.0

normalized arclength, s





Figure 6.3.10 Surface velocity profiles for monolayers with repulsive ( ), ), and no ( ) molecular interactions, for , strong cohesive ( , , , , and . The dashed line corresponds to a surfactant-free interface with equilibrium interfacial tension . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid equal to that for Interf. Sci., 218, 184-200. With permission from Academic Press.) 











































































Capsules in tube and channel flow 1.57

291

2.4

2.2

migration velocity, U

P

+

1.55 

extra pressure loss



1.53

1.51

1.49

2.0

1.8

1.6

1.4

1.47 0.0

0.2

0.4

0.6

0.8

equilibrium surface eq coverage,

1.2 0.0

1.0

0.2

0.4

0.6

0.8

equilibrium surface eq coverage,

eq


(a)

1.0

eq


(b)

Figure 6.3.11 Dependence of (a) the steady capsule velocity, and (b) the extra pres, , , sure loss on the equilibrium surface coverage for , , , and . (From Johnson, R. A. & Borhan, A., 1999, J. Colloid Interf. Sci., 218, 184-200. With permission from Academic Press.)


































































 

, bulk-phase diffusion is so small that the surfactant lost from the For interface at the rear end accumulates within the adjacent sublayer. On the other hand, , the surfactant surface distribution is nearly uniform. The sensitivity for of the capsule mobility to and is weaker for the larger surface coverage. For low equilibrium surface coverage, capsule mobility approaches the bulk-insoluble and at large Damkohler numbers. monolayer limit with Because of the significant partitioning of the surfactant between the sublayer and the interface, capsules with the elevated equilibrium surface coverage achieve a higher mobility than that corresponding to the insoluble monolayer limit where and . The significant loss of surfactant from the interface to the sublayer is manifested in Figure 6.3.14 by the fact that the average surface saturation is much less than the equilibrium surface coverage of . As surfactants preferentially desorb from the interface at elevated surface coverage, the Marangoni stresses are reduced, and the capsule mobility is raised. is increased, the rate of sorption of the surfactant increases relative to the As rate of interfacial convection, and this results in a more uniform interfacial distribu, the rate of surfactant sorption to and from the tion of the surfactant. For interface is sufficiently high to cause a strong coupling between the interfacial and , the sorption rate diminishes and bulk surfactant distributions. In the limit the insoluble monolayer limit is recovered for low surface coverage. For capsules with high equilibrium surface coverage, the surface saturation profile becomes qualitatively similar to that for the insoluble monolayer represented by the dashed line
































































































292

Capsules and Cells 3.5

1.58

Bi = 0.1

+

3.0

1.50

extra pressure loss, P

migration velocity, U

1.54

Bi = 10

Bi = 0.1

1.46

2.5

Bi = 10 2.0

1.5

1.42 0.1

1

10

100

1000

1.0 0.1

10000

1

10

100

1000

DamkohlerDa number, Da

DamkohlerDa number, Da

(a)

(b)

10000

Figure 6.3.12 (a) Steady capsule speed, and (b) extra pressure loss in the low surface coverage limit ( ) for , , , , , and . (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.)














































































1.35

1.57

Bi = 0.001 1.30

0.1

+

extra pressure loss, P

migration velocity, U

Bi = 10




1.56

0.1 0.001 1.55

1.25

10

1.20

1.15

1.54 0.1

1

10

100

1000

1.10 0.1

10000

1

10

100

1000

DamkohlerDa number, Da

DamkohlerDa number, Da

(a)

(b)

10000

Figure 6.3.13 (a) Steady capsule speed, and (b) extra pressure loss in the moderate surface coverage limit ( ) for , , , , , and . (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.)














































































in Figure 6.3.15(a), but with a smaller average surface saturation due to the partitioning of the surfactant between the interface and the sublayer. For , the surfactant transport is sorption-controlled and the sublayer concentration is main. In this case, decreasing the Biot number leads to only minor tained at 




















Capsules in tube and channel flow 14

0.8

12

sublayer concentration, C

Da = 10000 0.6

surface saturation,

293

0.4

100 0.2

Da = 10000 10 8 6 4

0.0 0.0

100

2

0.1 0.2

0.4

0.6

0.8

0.1

0 0.0

1.0

0.2

0.4

normalized arclength, s

0.6

0.8

1.0

normalized arclength, s

(a)

(b)

Figure 6.3.14 Distribution of (a) the surface saturation, and (b) sublayer concentration, for , , , , , , , and . (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.)















































































, reflecting variations in the variations in the sublayer concentration around surface concentration, as shown in Figure 6.3.15(b). The dependence of the migration velocity, , and extra pressure loss, , on the capsule size is illustrated in Figure 6.3.16. In general, increases and decreases with increasing capsule size. These trends are qualitatively similar to those observed for clean liquid drops. An insoluble monolayer with strong cohesive interactions has a weaker retarding effect on capsule mobility than a repulsive or Langmuir monolayer. The migration velocity approaches a plateau as the capsule size in. The limiting value of creases, and eventually tends to a limiting value for the migration velocity for large capsules depends on the value of , with the largest value corresponding to strongly cohesive monolayers for which the extra pressure loss also appears to approach a plateau in the same manner as for a clean drop. For strongly cohesive monolayers, concentration variations on the capsule surface are restricted entirely within the coexistence region in the surface pressure isotherm where the interfacial tension is nearly independent of surface coverage. Thus, significant variations in surfactant concentration lead to virtually no Marangoni stresses on the capsule surface, as in the case of a surfactant-free interface. Surfactant transport to and from the bulk phase affects the capsule mobility in much the same manner as the presence of strongly cohesive interactions. Specifically, bulk solubility causes large capsules to behave more like clean drops, the physical reason being that surfactant transport to and from the interface acts to reduce the surface concentration gradients and hence the Marangoni stresses at the interface. A comparison of the capsule mobilities for bulk-soluble and insoluble Langmuir mono



























Capsules and Cells

1.0

1.04

0.8

1.03

0.6

sublayer concentration, C

surface saturation,

294

Bi = 0.1

0.4

0.2

Bi = 0.1 1.02

1.01

0.0 0.0

Bi = 10

1.00

Bi = 10

0.2

0.4

0.6

0.8

0.99 0.0

1.0

0.2

0.4

0.6

0.8

normalized arclength, s

normalized arclength, s

(a)

(b)

1.0

Figure 6.3.15 Distribution of (a) the surface saturation, and (b) the sublayer con, , , , , , centration, for , and . The dashed line corresponds to a bulk-insoluble monolayer. (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.)













































































2.2

10

= +4 8

extra pressure loss, ∆P+

migration velocity, U

2.0

1.8

= -4

1.6

=0 6

= -4 4

1.4

2

0

=0 1.2 0.0

0.2

0.4

= +4

0.6

0.8

1.0

-2 0.0

1.2

0.2

drop size,

0.4

0.6

0.8

1.0

1.2

drop size,

(a)

(b)






Figure 6.3.16 (a) Steady capsule speed, and (b) extra pressure loss for , , , , and . The dotted lines correspond to a surfactant-free interface with equilibrium interfacial tension equal to that for . (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.)



















































Capsules in tube and channel flow 2.0

295

10

1.9

extra pressure loss, P

migration velocity, U

+

8

1.8




1.7 1.6 1.5

6

4

2

0

1.4 1.3 0.0

0.2

0.4

0.6

drop size

0.8

1.0

1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

drop size, 

(a)

(b)

Figure 6.3.17 Dependence of (a) the steady capsule speed, and (b) the extra pressure loss on capsule size for , , , , , , and . The solid, dotted, and long-dashed lines correspond to a bulk-soluble surfactant, an insoluble monolayer, and a surfactantfree interface. (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.)































































layers is presented in Figure 6.3.17. The size dependence of the mobility of a clean is shown by the dashed line. The insoluble drop with interfacial tension equal to monolayer causes the capsule to behave like a slug, and this leads to an unbounded increase in the extra pressure loss with increasing capsule size. This is in contrast to the behavior of clean drops where the extra pressure loss reaches a plateau at large values of the size parameter . Even in the absence of cohesive interactions, the plateau in the extra pressure loss for large capsules is recovered when bulk-soluble surfactants are considered. We conclude this section by discussing the evolution of the capsule shape and the critical conditions for the occurrence of capsule breakup. Surface convection causes the rear of the capsule to flatten, and the front of the capsule to elongate relative to that of a clean drop. The extent of deformation increases as the surface P`eclet number becomes higher [8, 34]. The effect of surface convection on interface deformation is more pronounced in the uniform retardation regime developed by repulsive monolayers than in the stagnant cap regime corresponding to strong cohesive interactions. Transient shapes of a surfactant-laden capsule and of the corresponding cleaninterface drop are shown in Figure 6.3.18. In both cases, the value of the capillary , number based on the interfacial tension of a surfactant-free drop is held fixed at reported by Olbricht & Kung [50] which is well below the critical value for breakup of a clean drop with similar parameter values. Note that, because the



























296

Capsules and Cells

t=0

5

10

15

(a)

(b) Figure 6.3.18 Transient shapes of (a) a drop with a surfactant-free interface, and (b) , , , , , a capsule with , , and . (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.) 



























































































center of mass of each transient shape has been shifted by a uniform axial distance, the displayed position does not reflect any differences in migration velocity. Both simulations shown in Figure 6.3.18 are initiated with a spherical shape. Breakup occurs through the re-entrant cavity mechanism reported by Olbricht & Kung [50] for clean drops. The apparently faster rate of growth of the penetrating jet in Figure 6.3.18(b) suggests that the presence of surfactants has a destabilizing influence. The computation of critical conditions for the onset of capsule breakup requires careful consideration of the numerical protocol followed in the simulations. The experimental protocol followed by Olbricht & Kung [50] prescribes that the drop be injected into the capillary tube at low capillary numbers, that is, under low flow-rate conditions, and the capillary number be incrementally increased by raising the flow rate while allowing the drop to reach an equilibrium shape at each incremental value . The critical capillary number for drop breakup is taken to be the value of of beyond which a steady drop shape cannot be found. The experimental values of reported by Olbricht & Kung were underpredicted by the computational results of Tsai & Miksis [75] who performed several computations by incrementally increasing values of capillary number, each time using initially-spherical drop shapes. This numerical protocol is significantly different from the experimental protocol since it corresponds to situations where initially spherical drops are injected into increasingly stronger tube flow. Considering the nonlinear nature of the shape evolution problem, the disagreement between the nu


















Capsules in tube and channel flow

-2.0

-1.5

-1.0

-0.5

0.0

0.5

297

1.0

1.5

2.0

z 












 







Figure 6.3.19 Steady capsule shapes for , , and . The dotted contour corresponds to a drop with a surfactant-free interface, and the solid contour corresponds to an equivalent capsule with , , , , and . (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.) 






























































merically predicted and the experimentally observed values of for clean drops is thus not surprising. Recent computations [35] suggest that, not only the predicted value of , but also the breakup mechanism itself depends on the numerical protocol followed for an initially spherical drop in the simulations. For example, the value of with the clean interface shown in Figure 6.3.18 is much smaller than that reported by Olbricht & Kung for a similar system [50]. However, if the numerical protocol is altered to more closely resemble the experimental procedure, steady drop shapes can be found at much higher values of the capillary number. The steady capsule shapes shown in Figure 6.3.19 are found by performing a set of simulations where the capillary number is increased in an incremental fashion. Each simulation is initiated with the converged solution for the steady capsule shape obtained from the previous . The capillary number associated with simulation at a slightly smaller value of the steady capsule shapes shown in Figure 6.3.19 is the same as that for the unstable capsule shown in Figure 6.3.18. The same numerical protocol can be used to obtain steady capsule shapes at higher values of the capillary number. The resulting steady shapes for large capillary numbers are highly elongated, with a region of negative curvature at the rear stagnation point. The presence of surfactants leads to a more elongated steady shape compared to that of a clean drop, without significantly affecting the region of negative curvature at the trailing end, as can be seen by the comparison of the steady shapes shown in Figure 6.3.18. The increasingly elongated steady capsule shapes found at larger capillary numbers suggest a transition in the mechanism of capsule breakup from an “impinging jet mode” to “transient elongation.” The latter mode has been observed experimentally by Olbricht & Kung [50], and Borhan & Pallinti [9] for liquid drops 


























298

Capsules and Cells

with O(1) viscosity ratios. Consistent with the results of the numerical simulations, the critical capillary number for the occurrence of the transient elongation mode was found to be much larger than that for the impinging jet mode.

6.4 Capsules with temperature-induced elasticity in tube flow The thermo-capillary motion of a drop was first analyzed by Young et al. [80], who derived the following expression for the dimensionless migration velocity of placed in a constant temperature gradient, , in an a spherical drop of radius unbounded fluid,










(6.4.1)








 





















Equation (6.4.1) is valid so long as the drop is sufficiently small that convective transport of heat and momentum are negligible compared to molecular transport, , and Marangoni that is, for small values of the Reynolds number, , where is the thermal diffusivity of the surrounding fluid. number,

































In the vicinity of a solid boundary, the migration velocity of a drop is affected not only by the hydrodynamic resistance to forward motion due to presence of the wall, but also by the thermal interactions between the particle and the boundary. The latter determines the surface temperature distribution driving the particle motion. This realization has motivated a series of theoretical analyses that successfully predicted the thermo-capillary migration velocity of drops in the vicinity of other particles or walls. A summary of these analyses can be found in the review article by Subramanian [73] (see also Chapter 5). Hasan & Balasubramanian [28], investigated the thermo-capillary migration of a long capsule in a closed tube (see also References [77, 78]). Noting that large capsules can be significantly deformed and the motion of the meniscus results in an entrained film of suspending fluid near the wall, they focused their attention on the thin film region away from the ends, and derived a relationship between the migration velocity and the film thickness. However, the solution in the film region was not matched to that in the meniscus and cap regions to determine the migration velocity and film thickness. More recently, Mazouchi & Homsy [45, 46] used lubrication theory and matched asymptotic expansions to describe thermo-capillary motion of long capsules in cylindrical and polygonal tubes. In the case of motion through a cylindrical tube, the quasi-steady capsule shape and velocity were determined, and a nonlinear relation between the migration velocity and the capillary number was derived. The competition between thermal and hydrodynamic interactions was investigated by Chen et al. [13] in their study of the thermo-capillary migration of a small fluid particle in a tube subject to an imposed axial temperature gradient on the insulated

Capsules in tube and channel flow

299 (b)

Ca = 0.1 Ca = 0.5

0.20

Initial Shape Final Shape

deformation parameter,

0.15

r

0.10

0.05

0.00

-0.05

-0.10 -1.5

-1.0

-0.5

0.0

0.5

polar angle,

1.0

1.5






Figure 6.4.1 Definition of the the deformation parameter (left), and deformation of capsules with size for (right). (From Johnson, R. A & Borhan, A., 2003, J. Colloid Interf. Sci., In press. With permission from Academic Press.) 

=













=









wall, in the limit . It was found that the tube wall generally retards the motion of the particle, in some cases by as much as one order of magnitude. In this study, a fixed spherical shape for the fluid particle was assumed. This simplification is appropriate for high and nearly-uniform interfacial tension, that is, at vanishingly small capillary numbers; unfortunately, it is not certain that these conditions prevail during in capsule motion in the vicinity of solid boundaries. Consequently, the results of Chen et al. are expected to be applicable only for capsule sizes that are small compared to the tube diameter. The original analysis was extended by Borhan & Haj-Hariri [6], and Chen et al. [12], who used the boundary-integral method to describe large capsule deformations. In the case of thermo-capillary motion of a capsule in an unbounded domain of flow, the interface stress distribution is uniform and thus consistent with the spherical shape. The presence of a tube wall induces a nonuniform stress distribution and causes deformation. The deviation of the capsule shape from sphericity can be quantified by the deformation parameter, , defined as the distance of the interface from the capsule center, minus the radius of the undeformed spherical capsule, as shown on the left of Figure 6.4.1. Typical quasi-steady capsule deformations are shown on the right of Figure 6.4.1 , the capsule flattens and in Figure 6.4.2 for various capsule sizes [7]. When at the rear, while the front end elongates in the direction of the applied temperature gradient, thereby destroying the fore-and-aft symmetry. The deformation is qualitatively similar to that observed in the case of pressure-driven motion of drops in a tube, though the magnitude of the deformation is now much smaller. Capsule deformation becomes more pronounced as the capsule size increases due to the increase





















300

Capsules and Cells = 0.3 = 0.5 = 0.7 = 0.9

-3

-3

60x10

5

0

-5

-1.5

-1.0

-0.5

0.0

polar angle,

0.5

1.0

(b)

= 0.3 = 0.5 = 0.7 = 0.9

(a)

Deformation parameter,

Deformation parameter,

10x10

40

20

0

-20

1.5

-1.5

-1.0

-0.5

0.0

(a)

1.0

1.5




(b)


Figure 6.4.2 Deformation of capsules with [7].


0.5

polar angle,












for (a)

 





and (b)







in the viscous force exerted on the capsule by the tube wall relative to the restoring interfacial force. For capsules whose size is comparable to the tube radius, such as that shown on the right of Figure 6.4.1 for , both the rear and front elongate. As the capillary number increases, the elongation of the front in the direction of the applied temperature gradient exceeds that of the rear, and the capsule deformation , the trailing end of the capsule becomes becomes more pronounced. For more flattened as the capillary number increases. Thermo-capillary migration of a capsule is induced by nonuniformity in the interfacial tension associated with a hot leading end and a cold trailing end. According to (6.4.1), the migration velocity of a capsule translating in an unbounded medium is proportional to the capsule size. The dependence of the scaled migration velocity ( ) on the capsule size for tube flow is described in Figure 6.4.3 [7]. For , a substantial reduction in the relative capsule mobility is evident. Two competing forces determine the migration velocity: a thermo-capillary driving force propelling the capsule in the direction of the temperature gradient, and a hydrodynamic drag force retarding the motion. Both of these forces grow with increasing capsule size. However, the increase in the drag force arising from hydrodynamic interactions with the tube wall is substantially higher than the enhancement in the thermo-capillary driving force, and the capsule moves slower than in an unbounded fluid medium. Increasing the capillary number leads to a slight increase in the mo. Physically, larger capsule deformation tends to bility of large capsules with reduce the hydrodynamic interaction with the tube wall by increasing the thickness of the liquid film between the capsule and the wall. The dependence of the scaled migration velocity on the thermal conductivity ratio is illustrated in Figure 6.4.4 [7]. As the thermal conductivity of the interior phase 















































Capsules in tube and channel flow

301

scaled migration velocity, U/UYGB

1.0 Ca = 0.1 Ca = 0.5 0.8

0.6

0.4

0.2

0.0

0.2

0.4

0.6

0.8

1.0

drop size,

Figure 6.4.3 Dependence of the relative capsule mobility on the capsule size for equal viscosities and conductivities, =1 and [7]. 



=



increases from zero to an O(1) value, a significant reduction in the capsule mobility is observed. The reduction in relative mobility is more pronounced for larger capsules whose motion is substantially inhibited by the hydrodynamic interactions with the tube wall. The retardation of the migration velocity of large capsules with increasing the conductivity ratio is consistent with the smaller deformations experienced by these capsules as their thermal conductivity is increased, as shown in Figure 6.4.5. Decreasing the thermal conductivity ratio has a qualitatively similar effect on the capsule shape as that of increasing the capillary number. In both cases, flattening of the trailing end of the capsule accompanied by elongation of the front end are observed. Since the migration velocity of the capsule in Figure 6.4.4 is scaled by , the retarding effect of increasing thermal conductivity ratio evident in this figure is beyond that predicted by (6.4.1) for the motion of capsules in an unbounded domain, and is entirely due to the presence of the tube wall and its effect on the temperature distribution. When the thermal conductivities of the interior and exterior phases are equal, , the temperature fields inside and outside the capsule are unified, and a linear temperature field prevails for any capsule size. For , the thermal resistance is smaller in the exterior phase, and a greater fraction of thermal energy is conducted through the thin fluid layer between the capsule and the tube wall. Consequently, the isotherms are bent toward the interface, thereby increasing the effective temperature . Since the driving force of the thermo-capillary difference across the capsule, motion is proportional to , capsule mobility is increased as is reduced. The opposite trend is observed for , that is, a greater portion of thermal energy is transported through the capsule causing the isotherms to bend away from the interface and reducing the driving force of the thermo-capillary motion. 







3



=



=











=

302

Capsules and Cells 1.0

scaled migration velocity, U/UYBG

(b)

0.8

= 0.1 = 0.3 = 0.5 = 0.7 = 0.9

0.6

0.4

0.2

1

2

3

4

5

thermal conductivity ratio,

Figure 6.4.4 Effect of thermal conductivity ratio on the relative mobility of capsules with and [7].




=















= 0.1 = 0.5 = 1.0 = 5.0

0.10 0.08

deformation parameter,




(b)

0.06 0.04 0.02 0.00 -0.02 -0.04

-1.5

-1.0

-0.5

0.0

polar angle,

0.5

1.0

1.5



Figure 6.4.5 Effect of thermal conductivity ratio on the deformation of a capsule of , for and [7]. size














=

















The effect of on the effective temperature difference across the capsule is more pronounced for larger capsules, as described in Figure 6.4.6. As the thermal conductivity of a large capsule increases, the capsule exhibits smaller deformation, as shown in Figure 6.4.5, and is separated from the tube wall by a thinner liquid film. The smaller axial elongation causes even smaller effective temperature differences

Capsules in tube and channel flow

303 = 0.3 = 0.5 = 0.7 = 0.9 = 1.0

temperature difference across capsule,




6

5

4

3

2

1

1

2

3

4

thermal conductivity ratio,

5



Figure 6.4.6 Effect of the thermal conductivity ratio on the effective temperature difference across the capsule for and [7].




=













=



across the capsule. In addition to lowering the thermo-capillary driving force, smaller capsule deformation leads to larger drag by means of increased hydrodynamic interaction with the tube wall.

6.5 Capsules enclosed by elastic membranes in tube flow The axisymmetric motion of a capsule enclosed by an elastic membrane in a confined domain at low Reynolds numbers has been studied on several occasions using asymptotic methods based on lubrication theory [26, 27, 33, 63, 64, 65, 66, 67] and numerical methods based on the boundary-integral formulation [2, 38, 39, 56, 57]. Secomb and coworkers [26, 33, 63, 64] used the lubrication approximation to describe the axisymmetric motion of spherical capsules and red blood cells through narrow capillaries (see also Chapter 4). The Evans-Skalak law along with a viscous constitutive equation was adopted to describe the membrane tensions [63]. For high flow velocities, the external viscous stresses are balanced by the isotropic membrane tension, while the shear and bending resistance are negligible. In this limit, referred to as the ‘’high velocity limit,” the red blood cells obtain a bullet shape with a hemispherical nose and a uniformly small gap along the entire length. At lower flow velocities, shear tensions become important. Typical cell shapes for different vessel diameters and cell velocities are shown in Figure 6.5.1 [67]. Both shear and bending elasticity effects have been included in this computation.

304

Capsules and Cells

Figure 6.5.1 Cell shapes calculated using the lubrication approximation for (a) fixed cell volume and velocity and varying vessel diameters, and (b) fixed cell volume and vessel diameter and varying cell velocity. (From Secomb, T. W., ¨ Skalak, R., Ozkaya, N., & Gross, J. F., 1986, J. Fluid Mech., 163, 405–423. With permission from Cambridge University Press.)

A minimum tube diameter exists below which cells enclosed by incompressible membranes cannot pass through the tube. Halpern & Secomb [26] used the EvansSkalak elasticity model to analyze the motion of red cells in capillaries with nearminimal diameter. As in the high velocity limit, shear elastic tensions are found to be negligible. In this limit, the near-critical shapes are cylindrical with spherical end caps, and the shape in the transition region is determined in a manner that is similar to that developed by Bretherton for the motion of long bubbles through capillary tubes [10]. Shape variations at the front end of the cell is found to be monotonic, whereas the rear end exhibits a bulge. The results of this study have shown that nearcritical shapes occur over a small range of tube radii. As the radius of the capillary is increased, the rear end of the cell flattens and then obtains a concave shape. In vivo measurements of the apparent viscosities in microvessels are substantially higher than those in glass tubes, particularly for the narrowest microvessels (see also Chapter 3). Although the actual reason for this discrepancy is not known with certainty, several possible mechanisms have been proposed, including irregularity in the cross-section of the microvessel and the presence of the macromolecular glycocalyx layer in the interior of the capillaries. Secomb & Hsu [64] studied the motion of red cells through capillaries with tapered shapes, shapes with a neck geometry, and corrugated surfaces. The results of the simulations showed that the irregularity in the vascular geometry can lead to a much higher pressure drop compared to that realized in a tube with uniform cross-section and the same mean diameter. Damiano [15] investigated the motion of red blood cells through glycocalyx-lined capillaries by modeling the endothelial layer as a deformable porous surface and assuming that the cell membrane obeys the Evans-Skalak constitutive equation in the high velocity limit. The results showed that the cells are more elongated and

Capsules in tube and channel flow

305

develop a larger gap clearance compared to the case of motion through smoothwalled tubes. Elevated membrane tension and pressure occur over the entire length of the cell. If the glycocalyx is modeled as a porous matrix with a radially varying hydraulic resistivity [65], the thickness and hydraulic resistivity of the layer largely determines its effect on flow resistance and hematocrit. The effect of a permeable endothelial surface layer on the motion of biconcave disk-shaped cells was discussed by Secomb et al. [66]. In this model, the endothelial layer was assumed to be compressible and permeable to water. As the red blood cell flows through the capillaries, the endothelial layer is compressed and reacts by exerting a normal force on the cell membrane. Although the force applied by the endothelial layer on the cell membrane is expected to increase as the endothelial layer becomes more compressed, Secomb et al. [66] assume that the force is independent of the level of compression. The presence of the endothelial layer is found to substantially increase the flow resistance at low velocities, resulting in more slender cell shapes. Experiments have shown that the motion of red blood cells in narrow capillaries is in fact non-axisymmetric (see also Chapter 3). Hsu & Secomb [33] used the lubrication theory to investigate the effect of a prescribed asymmetry of the cell shape on the cell steady orientation and membrane tank-tread frequency. Their computations showed that the driving pressure required for sustaining a specified cell velocity is insensitive to the non-axisymmetry, and this provides partial justification for the use of axisymmetric models to predict the pressure drop across deformed cells moving through capillaries. Working on the numerical side, Barth`es-Biesel and coworkers [2, 57, 56] used the boundary-integral method to simulate the entrance of a cell enclosed by a neoHookean membrane through an axisymmetric pore. These studies are limited to the case where the viscosities of the internal and external liquid are equal. Since the internal phase is stationary at steady state, the assumption of unit viscosity ratio only affects transient shapes and inconsequential to the asymptotic shape established at steady state. Figure 6.5.2 illustrates the unsteady motion of an initially spherical cell entering a cylindrical pore under constant flow rate conditions. The disturbance flow due to the the cell decays to an insignificant level a distance of about one tube radius on either side of the cell. As the cell approaches a steady shape, the rear end becomes concave and assumes a “parachute” shape. The entrance length required for the cell to reach a steady shape depends on the cell size and elasticity capillary number. Increasing the capillary number causes a more pronounced deformation at steady state. However, there exists a critical capillary number beyond which a steady shape cannot be found. Since the surface area of a neo-Hookean membrane is unconstrained, increasing the capillary number beyond the critical value causes the cell to experience continuous transient deformation, eventually leading to breakup. The complementary process of a cell exiting a cylindrical pore was considered by Queguiner & Barth`es-Biesel [57]. In these simulations, the initial cell shape was the steady shape established in the pore entrance problem at steady state. Exiting cells were found to obtain a more pronounced parachute shape as they are “blown out” of

306

Capsules and Cells 2

(a)

1 0 -4

-3

-2

-1

0

1

2

3

4

5

6

7

-4

-3

-2

-1

0

1

2

3

4

5

6

7

2

(b)

1 0

Figure 6.5.2 Deformation of (a) a small, and (b) a large cell entering a cylindrical pore. (From Queguiner, C. & Barth`es-Biesel, D., 1995, Boundary Elem. XVII, 661–668, Brebbia C. A., Kim, S., Oswald, T. A., & Power, H. (Eds.) With permission from Computational Mechanics Publications.)

the pore, before finally relaxing to the unstressed spherical shape. It should be noted that, because the elasticity of the membrane allows energy to be stored, the entrance and exit problem are not identical even in the context of creeping flow. In related numerical studies, Barth`es-Biesel and coworkers [38, 39] investigated the deformation of a spherical cell passing through an axisymmetric hyperbolic pore, and found that the cells do not reach steady shapes. However, the assumption of unit viscosity ratio limits the range of shape deformations that can be realized. Two flow situations can arise depending on whether the flow rate or pressure drop is held constant. The evolution of capsule shapes for initially spherical capsules with and are shown in Figure 6.5.3 for constant pressure drop and constant flow rate conditions. The lubrication film separating the cell from the wall is thicker in the former case, resulting in a smaller apparent hydraulic resistance. Consequently, constant-pressure-drop systems better discriminate between normal and stiffened capsule membranes. The difference disappears as the capillary number is raised. The effect of the membrane constitutive equation has been considered by Queguiner & Barth`es-Biesel [57] who compare the behavior of spherical cells enclosed by neoHookean membranes to that of discoidal cells whose membranes are described by the red-cell-membrane elasticity model (see also Chapter 1). The constraint on constant surface area of the red-cell-membrane effectively lowers the cell deformability and prevents the development of concave shapes at the rear end. More important, a critical capillary number for the onset of continuous deformation and cell breakup does not arise. However, such cells may experience breakup when the tension level reaches a certain critical threshold. In practice, a certain amount of area extensibility must be allowed in the constitutive model to ensure numerical convergence. This numerical limitation renders the computations of fully area-incompressible membranes difficult, because the area dilatation modulus cannot be set to arbitrarily large values. In the case of the red






=















=

Capsules in tube and channel flow

307

z

z

(a)

(b)

Figure 6.5.3 Axisymmetric passage of capsules through constrictions with and under (a) constant pressure-drop, and (b) constant flow rate conditions. (From Leyratmaurin, A. & Barth`es-Biesel, D., 1994, J. Fluid Mech., 279, 135–163. With permission from Cambridge University Press.) 

=




















=

cell-membrane model, the results of Queguiner and Barth`es-Biesel [57] have shown that even though the maximum area dilatation is reduced with increasing values of , the extra pressure loss remains essentially unthe area dilatation modulus, changed. Consequently, whereas accurate predictions of the deformations of areato be as large as numerically incompressible membranes require the value of possible, the extra pressure loss due to the presence of a fully area-incompressible cell can be well estimated from computations where the membrane has a finite areal dilatation.

















6.6 Capsule motion in channels Capsule motion through a channel with parallel-sided walls has received relatively little attention compared to the akin motion through a cylindrical tube. Theoretical analyses have focused mostly on the most amenable case of flow through a channel with very small gap width constituting the Hele-Shaw cell. Motivation for these studies has been provided by fascinating observations of viscous fingering [32, 60] and bubble motion [36, 44, 53, 74]. The analysis is typically based on the local application of Darcy’s law relating the mean or depth-averaged velocity to the local two-dimensional pressure gradient parallel to the plates.

308

Capsules and Cells

b

b
1 a

W U s g 2a

y

x

Figure 6.6.1 Schematic of the model problem for a drop rising in a Hele-Shaw cell. (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2002, J. Colloid Interf. Sci., 252, 236–248. With permission from Academic Press.)

6.6.1 Governing equations 

Consider the motion of a capsule with equivalent radius in a Hele-Shaw cell with gap width that is much smaller than , as illustrated in Figure 6.6.1. The capsule translates with velocity , where and are the unit vectors in the and directions, and . The position of the interface is described by . The Hele-Shaw cell is assumed to extend to infinity in the the equation and directions, and to be held vertical so that the acceleration of gravity is given . by At low Reynolds numbers, the dimensionless equations and boundary conditions governing the fluid motion inside and outside the capsule are 




















































































for



























(6.6.1) 











for 



(6.6.2)















$









'








as





 







, 



,



$

&





(6.6.4)





for







(6.6.3)

%



for

'





%









(6.6.5)









/





 


















for













(6.6.6)

Capsules in tube and channel flow

309

where and are the dimensionless depth-averaged velocities in the interior ( ) and exterior ( ) phases, and are the corresponding dimensionless modified pressures, and is the interface velocity, with and denoting the unit normal and tangent vectors at the interface, respectively. All variables have been nondimensionalized using as characteristic length the equivalent capsule radius, , as characteristic velocity a velocity to be specified, as characteristic time , , where is the permeability of the and as characteristic pressure Hele-Shaw cell. denotes Equation (6.6.6) expresses the normal stress balance at the interface; the curvature of the capsule profile in the plane, and denotes the interfacial tension scaled with respect to a reference value . The capillary and Bond numbers and . Equation (6.6.6) is valid are defined as so long as remains small [52]. Because the Hele-Shaw equations resemble the potential flow equations, the computation of the tangential velocity is not required as part of the solution. However, to account for interfacial tension gradients, the tangential stress condition must be incorporated into the depth-averaged equations describing the Hele-Shaw flow. Nadim et al. [48] derived a suitable depth-averaged tangential stress condition by solving the complete microscale problem near a flat interface, and then depth-averaging the and exerted on resulting expressions. The dimensionless tangential stresses the interface, respectively, on the interior and exterior side, are given by



































































































































































(6.6.7) 

 



















where is the “slip layer” thickness in each of the two phases, and is a known constant [48]. The tangential velocities of the interior fluid, the exterior , , and , must be computed as part of fluid, and the interface, denoted by the solution. Using (6.6.7), we state the tangential stress balance in the form

















































 

(6.6.8)











 










for











, where is the surface gradient operator, and . The tangential velocities of the bulk phases evaluated at the interface can be determined by taking the dot product between the tangent unit vector at the interface, , and equations (6.6.1) and (6.6.2). Recognizing that , we write 




























































 













































 



 



 

(6.6.9)



for , where is the dimensionless arc length along the capsule contour. Substituting (6.6.9) into (6.6.8), we obtain the following depth-averaged tangential stress condition at the interface















 



















 

  
















(6.6.10)

310

Capsules and Cells

This equation can be used to determine the unknown tangential velocity at the inter, which is necessary for the solving the interfacial transport equation. face, 





6.6.2 Surfactant-induced elasticity When the interface is covered with an insoluble monolayer, the distribution of the , is governed by nonequilibrium surface concentration , nondimensionalized by the convection–diffusion equation






 






 





 








 



 





(6.6.11)





 













 













, where is the surface P´eclet number. Equations (6.6.1) for through (6.6.6), (6.6.10), (6.6.11), and (6.1.27) provide us with a complete set of equations describing the motion of surfactant-bearing capsules in a Hele-Shaw cell. Setting the divergence of the right-hand sides of (6.6.1) and (6.6.2) to zero to ensure are harmonic mass conservation, we find that the pressure distributions and functions satisfying the two-dimensional Laplace equation 









































(6.6.12)



The boundary-integral method may then be used to account for the strong coupling between the fluid flow and the surfactant transport equations on the deformable interface. It is convenient to decompose the interfacial pressures into a mean and a deviatoric component given by 


































(6.6.13)





























Following standard procedure, we derive the integral equations 






 







 




























(6.6.14)





  $



























# 






















 







 


















 














  



 







 









# 

$ 





(6.6.15)



where the subscript signifies a fixed point on the interface, length along the capsule profile, and

is the total arc


















 






 

 








 





 









(6.6.16)



 

 &





&















 

is the Green’s function of Laplace’s equation in two dimensions. Equations (6.6.14) and (6.6.15) are discretized in the spirit of the boundary-element method, and the resulting algebraic equations are combined to yield the following linear system for the unknown normal velocities 































 

 



 



 











(6.6.17)

Capsules in tube and channel flow

311





is a column vector containing the normal velocwhere is the identity matrix, ities at the collocation nodes, and the matrices and are the discretized representations of the integral operators with kernels and . The at the collocacolumn vector contains the values of the deviatoric pressure tion points, given by 































































































(6.6.18)

















where varies along the arc length , and the subscript denotes differentiation with respect to the arc length .







Starting with an initial capsule shape and a uniform distribution of surfactant in the monolayer, the normal velocities at the collocation points can be calculated from at the collocation points are then evaluated (6.6.17) and (6.6.18). The values of from (6.6.14), and the tangential stress balance (6.6.10) is used to determine the tangential velocity of the interface. Finally, the capsule shape is updated by using the interfacial kinematic condition, and the new distribution of surfactant along the interface is calculated by integrating equation (6.6.11). The procedure is repeated with the updated capsule shape and surfactant distribution until either a steady capsule shape is established, or capsule breakup becomes imminent. 





6.6.3 Temperature-induced elasticity In the absence of inertial effects, the migration velocity of the capsule arises by requiring the net force on the capsule to be zero. The drag force on the translating capsule can be written in dimensionless form as 







 


















(6.6.19)




 



 





The two terms inside the integral represent contributions from the exterior pressure and from the exterior tangential stress. The last term on the right-hand side expresses the force exerted on the capsule by the channel walls. Once the pressure distribution has been determined by solving the Hele-Shaw equations, the depth-averaged tangential stress balance (6.6.10) can be used to obon the tain the tangential velocity at the interface. The Marangoni stress right-hand side of (6.6.10) must be determined from a surface equation of state in conjunction with the solution of the temperature distribution, as described in Section 6.1.3 for capsule motion through a tube. Once the tangential interfacial velocity has can be calculated been determined, the exterior tangential stress on the capsule from the second equations in (6.6.7) and (6.6.9). The force-free condition may then be used to determine the capsule migration velocity. 













312

Capsules and Cells

6.7 Capsules with surfactant-induced elasticity in channel flow The influence of surfactants on the steady motion of a bubble in a Hele-Shaw cell was studied by Maruvada & Park [43]. Assuming that the ambient fluid wets the walls and the bubble surface is immobilized due to the surfactants, estimates for the translational velocity of elliptic bubbles were obtained. It was found that the increased drag due to the influence of the surfactant in the thin film region can reduce the bubble velocity by one order of magnitude below the Taylor & Saffman prediction [74]. The calculated bubble velocities were found to be in reasonable agreement with the experimental observations of Kopf-Sill & Homsy [36] and Park et al. [53]. Gupta et al. [24] investigated the stability of capsules whose surface is covered with an insoluble Langmuir monolayer, translating in a Hele-Shaw cell under the influence of buoyancy. In this case, the characteristic velocity is given by . In the analysis of these authors, the evolution of a perturbed shape was computed by numerical methods, and critical conditions for breakup were established. Similar computations were performed for the buoyancy-driven motion of clean liquid drops [23]. The flow induced by a circular drop moving due to buoyancy with velocity is consistent with the equations of steady flow, that is, it provides us with an exact solution. For nonzero interfacial tension corresponding to nonzero values of the Bond number, the circular shape is found to be stable to infinitesimal perturbations [22]. Prolate drops evolve to become circular for all Bond numbers, whereas oblate drops behave in a way that is a complex function of the initial eccentricity and Bond number. The initial shape perturbation can be described by the deforma, where and are the width and height of tion parameter the elliptical interface along the and axes. Prolate capsule shapes correspond to positive values of , and oblate capsule shapes correspond to negative values of . The results on the critical conditions for instability are summarized in the phase diagram shown in Figure 6.7.1. Each point in this figure describes the outcome of a numerical simulation, with solid circles representing the eventual formation of a steady shape, and symbols representing unstable drops that eventually break up. The solid lines are approximate boundaries of marginal stability, separating stable from unstable regions. Two stable branches of oblate drops can be identified in Figure 6.7.1. Along the first branch, corresponding to the marginal stability curve labeled M and hereafter referred to as the main branch, drops with fixed initial deformation become unstable as the Bond number exceeds a critical value. For sufficiently small Bond numbers, drops that are initially perturbed into an elliptical shape revert to a circular shape. At Bond numbers beyond the critical value, oblate drops continuously deform until drop breakup occurs through the formation and growth of a region of negative curvature at the leading end of the drop. A typical evolution of the shape of such drops is shown in Figure 6.7.2. The indentation developing at the front 











































































Capsules in tube and channel flow

313

0.1

Breakup

Steady

0.05

Stable

0

M

Initial Deformation,

A -0.05

S

B

-0.1

D Stable

-0.15

Unstable

-0.2

C

-0.25 -0.3 40

50

60

70

80

90

100

110

Bond number, Bo






Figure 6.7.1 Drop breakup diagram for a clean drop with . (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2000, J. Colloid Interf. Sci., 222, 107–116. With permission from Academic Press.)

stagnation point penetrates completely the drop, splitting it into two smaller drops. This behavior is qualitatively similar to the tip-splitting instability observed in the experiments of Maxworthy [44] on the buoyancy-driven motion of a train of bubbles in the Hele-Shaw cell. Along the main branch, the critical initial deformation for instability decreases as the Bond number is raised. This is not surprising, since a larger Bond number implies a lower resistance to interface deformation due to weaker interfacial tension forces, and a diminished capability to recover from shape perturbations. The Bond number dependence of the critical value of represented by the marginal stability curve M is consistent with the predictions of linear stability analysis for a planar interface oriented perpendicular to the gravitational field in a Hele-Shaw cell [23]. The second stable branch, bounded by the marginal stability curve labeled S, hereafter referred to as the secondary branch, is unusual in that it represents stabilization of moderately oblate drops over a range of relatively large Bond numbers. For a fixed Bond number that is larger than a limiting value of about 65, oblate drops are stabilized provided that the magnitude of the initial shape perturbation lies within a small range bounded by the marginal stability curve S. The window of stability in disappears altogether as the Bond number is reduced below the limiting value. Corresponding results on the stability of surfactant-bearing capsules are shown in Figure 6.7.3. As in Figure 6.7.1, the curves mark the approximate boundaries of marginal stability corresponding to the critical conditions for breakup. These 



314

Capsules and Cells t = 35.5

t = 18

t=0

t=0 ∆ = -0.02

∆ = -0.11

Figure 6.7.2 Evolution of the shape of rising drops for






and












.

results were obtained by performing a large number of numerical simulations in the parameter space, similar to those shown in Figure 6.7.1 for clean drops. For clarity of presentation, the actual data points describing the outcome of the numerical simulations are not shown in Figure 6.7.3. The stability diagram for capsules is similar to that for clean drops, in that both branches (M and S) of marginal stability are present. In fact, the marginal stability boundaries for clean drops are virtually indistinguishable from those for capsules . Surface convection has a stabilizing effect on the main branch in with the sense that, for a fixed value of , the critical Bond number for breakup is larger for than that for . On the other hand, as the P`eclet number increases, the secondary branch is shifted toward larger Bond numbers, causing the destabilization of the upper section. The effect of surface convection on the marginal stability boundaries for capsules is qualitatively similar to that of the viscosity ratio for a clean drop [23], as shown in Figure 6.7.4. Thus, the dominant influence of surfactants on the stability properties of translating capsules is mediated through the development of Marangoni stresses on the surface of the capsule, which act to reduce interface mobility and retard the overall motion. Increasing the surface P`eclet number leads to higher Marangoni stresses on the surfactant-laden interface and reduces the capsule mobility; the same effect is observed when increasing the viscosity ratio of clean drops. 








































Capsules in tube and channel flow

315

0.1 Pes==0.1 0.1 Pe Pes==10 10 Pe

0.05

Stable

Initial Deformation,

0

M

-0.05

S -0.1

Stable

-0.15

Unstable

-0.2 -0.25 -0.3 40

50

60

70

80

90

100

110

Bond Number, Bo

Figure 6.7.3 Effect of surfactant surface convection on capsule breakup for , , and . (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2002, J. Colloid Interf. Sci., 252, 236–248. With permission from Academic Press.) 

=











=


















0.1 = 0.5 = 1.0 = 2.0

0.05

Initial Deformation,

M

Stable

0 -0.05

S -0.1

Unstable Stable

-0.15 -0.2 -0.25 -0.3 40

50

60

70

80

90

100

110

Bond Number, Bo

Figure 6.7.4 Effect of viscosity ratio on the conditions for breakup of clean drops. (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2000, J. Colloid Interf. Sci., 222, 107–116. With permission from Academic Press.)

316

Capsules and Cells t = 43.5

t = 31

t = 15

t=0

t=0 Clean Clean

t=0 Pe ==11 Pe s

Pe = 10 Pe s = 10

Figure 6.7.5 Effect of surface convection on the shape evolution of capsules with , , , , and . (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2002, J. Colloid Interf. Sci., 252, 236–248. With permission from Academic Press.) 

















































The effect of surface convection on the evolution of capsules and equivalent clean drops represented by points A, B, and C in Figure 6.7.1 is shown in Figures 6.7.5 through 6.7.7. The clean drop corresponding to point B just below the upper section of the secondary branch is stable and eventually recovers the circular shape following the formation of a protrusion at its leading end, as shown in Figure 6.7.6. As the P`eclet number increases, the development of the protrusion at the front of the , the capsules is less pronounced. In the case of weak surface convection, capsule obtains a slightly prolate steady shape with a steady migration velocity that is slightly higher than that for a circular drop. As convective transport of the surfactant becomes more dominant, the retarding effect of Marangoni stresses on interface mobility eventually suppresses the protrusion and causes the flattening of the front end of the drop. Capsule breakup occurs through the development of a tip-splitting instability. The clean drop represented by point C in Figure 6.7.1, lying immediately below the lower section of the secondary branch, also develops a protrusion at the front, and then obtains a prolate shape as the protrusion grows, as shown in Figure 6.7.7. However, in this case, the drop further evolves into an oblate shape with a flattened front which eventually breaks up by the process of tip-splitting. For small P`eclet numbers, the shape evolution of the corresponding capsule is qualitatively similar to









Capsules in tube and channel flow

317

t = 36.5

t = 29.4

t = 19.5

t=0

t=0 Clean Clean

t=0

Pes == 0.1 Pe 0.1

Pes == 1.0 Pe 1.0

Figure 6.7.6 Effect of surface convection on the shape evolution of capsules with , , , , and . (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2002, J. Colloid Interf. Sci., 252, 236–248. With permission from Academic Press.) 

















































that of the clean drop. The presence of the insoluble monolayer merely causes an increase in the time required for breakup. As the P`eclet number is raised, the tipsplitting instability disappears altogether and the capsule obtains a slightly prolate shape at steady state. Thus, in this case, convective transport of the surfactant has a stabilizing influence on the capsule shape. The stabilizing effect of the Marangoni stresses on the tip-splitting instability is also observed for capsules described by point A in Figure 6.7.1, lying immediately below the main branch, as illustrated in Figure 6.7.5. A comparison between these transient shapes in this figure reveals that the tip-splitting instability is suppressed as the P`eclet number is raised. Surfactant transport not only affects in a quantitative sense the critical conditions for the onset of capsule breakup, but also alters qualitatively the capsule dynamics by determining the mode of breakup of unstable shapes. This is demonstrated by the transient shapes of capsules and their equivalent clean drop corresponding to point D in Figure 6.7.1, as shown in Figure 6.7.8. In the absence of an insoluble monolayer, the drop is unstable and breaks up into one large and two small daughter drops. The breakup process involves the formation and growth of a protrusion at the front stagnation point, followed by pinch-off at the wings. In the presence of weak surface convection, the capsule exhibits the mode of , the monolayer acts to retard breakup observed for the clean drop. For 









318

Capsules and Cells t = 31 t = 26.5 t = 19.5

t=0

t=0 Clean Clean

t=0

Pes==0.1 0.1

Pes = 1.0 Pe

Figure 6.7.7 Effect of surface convection on the shape evolution of capsules with , , , , and . (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2002, J. Colloid Interf. Sci., 252, 236–248. With permission from Academic Press.) 

















































the growth of the protrusion at the front end, causing the formation of larger wings; upon pinch-off, three capsules of nearly equal size arise. As the intensity of surface , the protrusion is completely suppressed and the convection increases, capsule breaks up through a tip-splitting process. Although the capsule is unstable indeed in this case, the mode of breakup is qualitatively different from that exhibited and by the equivalent drop in the absence of surby the same capsule at factants. These comparisons clearly demonstrate that the presence of the insoluble monolayer can affect the stability of capsules translating in a Hele-Shaw cell in a complex manner.



















6.8 Capsules with temperature-induced elasticity in channel flow The thermo-capillary migration of a circular capsule in a Hele-Shaw cell was investigated by Siekmann [68] and Nadim et al. [48] (see also Reference [5]). In the case of thermo-capillary motion, the characteristic velocity used to nondimensional. In a plane polar coordinate system , ize the flow variables is with being the radial distance from the capsule center and being the angle measured relative to the direction of the imposed temperature gradient, equations (6.6.1) 





















Capsules in tube and channel flow

t = 18

319

t = 18 t = 15.5

t=0

t=0

t=0

Clean Clean

Pes==0.1 Pe 0.1

Pes ==1.0 Pe 1.0

Figure 6.7.8 Effect of surface convection on the shape evolution of capsules with , , , , and . (From Gupta, N. R., Nadim, A., Haj-Hariri, H., & Borhan, A., 2002, J. Colloid Interf. Sci., 252, 236–248. With permission from Academic Press.) 














































and (6.6.2) lead to the following depth-averaged boundary-value problem for the flow field, 



for 





 



for 













(6.8.1)

subject to the kinematic condition   









 







at









(6.8.2)









is required to decay far from the capsule, and is required to be nonsingular as . Because the shape of the capsule has already been specified as circular, the normal stress balance is not imposed. Solving these equations for the pressure distribution in the two phases by analytical methods, we find [48] 
















(6.8.3) 















The tangential velocity at the interface can be determined from the depth-averaged tangential stress balance (6.6.10), 










 







 







 







(6.8.4)






 



where is the dimensionless ”slip-layer” thickness in the presence of Marangoni traction [48]. Substituting this result into the depth-averaged tangential stress balance, we determine the tangential stress exerted by the external fluid on the interface

320

Capsules and Cells

to be 



 










(6.8.5)










  












 



Substituting further this expression into (6.6.19), we find that the total drag force on the capsule is given by  





 




































 























(6.8.6) 

where is a unit vector in the direction of . It should be emphasized that (6.8.6) is an exact solution and does not require to be sufficiently small. If the interfacial tension variation along the interface is known, it therefore is a simple matter to account for the effect of Marangoni forces on the migration velocity. In the absence of external forces due to buoyancy or an external pressure gradient, the force-free condition requires the following expression for the capsule velocity in terms of the interfacial tension distribution,

























 











(6.8.7)














 











Nadim et al. [48] used this constraint to obtain an expression for the thermo-capillary migration velocity of a capsule whose motion is driven by a constant temperature gradient, , imposed along the entire length of the Hele-Shaw cell. Using a linear equation of state for the dependence of interfacial tension on temperature, they found 







 









(6.8.8)























In the case of a gas bubble, , this expression takes the particularly simple form ; in physical dimensional variables the bubble migration velocity is . Thus, the bubble migrates in the direction of increasing temperature at a speed that is proportional to the Hele-Shaw cell permeability and to the temperature gradient, and inversely proportional to the bubble radius and viscosity of the surrounding fluid. The dependence on capsule size is radically different from that of a three-dimensional spherical capsule translating in an unbounded fluid. Expression (6.4.1) shows that the thermo-capillary migration velocity of the latter is proportional to the capsule radius rather than the inverse. This difference can be explained by noting that the hydrodynamic drag exerted on a capsule in the Hele-Shaw cell is proportional to , as compared to for a spherical capsule suspended in an infinite fluid, whereas the driving Marangoni force is proportional to , as compared to for the spherical capsule. Based on this scaling argument, the presence of a thin wetting film of suspending fluid between the capsule and the walls in the Hele-Shaw cell is expected to lead to a migration velocity that is insensitive to capsule size. An implicit assumption is that the capsule radius must be large compared to the gap width of the Hele-Shaw cell in order for the analysis of Nadim et al. to be valid.










































Capsules in tube and channel flow

321

6.9 Capsules enclosed by elastic membranes in channel flow Halpern & Secomb [27] presented the only available theoretical analysis of creeping flow of red blood cells through a channel with parallel-sided walls. The mathematical formulation relies on the lubrication approximation to describe the fluid flow, and on the Evans-Skalak law to describe the mechanics of the membrane. A critical minimum channel gap width was found under which the cells are not able to pass intact through the channel. When the gap width is slightly larger than the critical value, the overall shape of the cell is a disk with an almost flat surface parallel to the walls (region I) and a semicircular rim (region II). As the cell contour transitions from region I to region II, the pressure distribution and membrane curvature undergo rapid changes. The separation between the cell and the channel wall increases monotonically in the transition region at the front end, and exhibits oscillations at the rear end of the cell. This behavior is qualitatively similar to that of long bubbles translating through a tube. The critical gap in channel flow is analogous to the critical capillary size in axisymmetric tube flow. However, the motion of cells in a channel is distinctly different from that in the axisymmetric geometry. In the case of tube flow, the cells translate with a velocity that is higher than the mean velocity of the external fluid. In contrast, the velocity of cells in channel flow can be substantially lower than the mean fluid velocity prevailing in the cell’s absence. An explanation is that in the axisymmetric configuration the cell almost completely fills the entire tube cross-section, whereas in channel flow the external fluid can flow around the cell. As a consequence, in the case of channel flow, the volume fraction of cells (tube hematocrit), can be higher than the flow fraction of cells (discharge hematocrit). The opposite is true in the case of tube flow (see Chapter 4).

6.10 Summary and outlook Confining surfaces and boundaries such as tube and channel walls can dramatically affect the capsule motion and deformation, particularly when the capsule size is comparable to the size of the confinement. In this chapter, we have discussed the motion of liquid capsules exhibiting surface elastic effects due to (a) variations in isotropic interfacial tension induced by a surfactant or a temperature field, or (b) the presence of an elastic membrane. An overview of computational studies on the motion of liquid capsules through tubes and channels was provided, with focus on the better understood case of axisymmetric flow. Capsule motion through channels with rectangular or other non-circular crosssection is largely unexplored. A few available computational studies are limited to the special case of two-dimensional capsules or filaments, and channels with small

322

Capsules and Cells

aspect ratios in the configuration of the Hele-Shaw cell for which depth-averaged equations can be applied. The analysis of capsule motion in a channel whose gap width is comparable to the capsule size remains to be addressed in future studies. These investigations will require three-dimensional solutions of the governing equations using appropriate numerical techniques such as the finite-element or the immersed-boundary method (e.g.,[21, 25] ). An additional limitation of previous studies, motivated by linearization, is the restriction to the creeping flow regime and to conditions of negligible mass and heat or larger values of the Reynolds and P`eclet convection. Capsule dynamics at number is important in applications related to food and polymer processing, and relevant to the motion of cells through large blood vessels. Extending the state of the art to account for inertial and convective effects will be a significant contribution. Finally, the interior of liquid capsules has been assumed to be an incompressible Newtonian fluid. This approximation is not generally accurate for biological cells whose intracellular matter, extracellular substance, or both, may exhibit viscoelastic behavior. Computational modeling of the confined dynamics of capsules when one or both phases are non-Newtonian is an important topic for further research. 

=



$

Capsules in tube and channel flow

323

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Index

adsorption, 200, 268 arteriosclerosis, 124 aspiration, 11, 78 atherogenesis, 105 axial-train model, 171 axisymmetric capsule, 264 extension, 65, 166 membrane, 274 stretch, 166 stretch ratio, 65

boundary-element method, 27, 94, 277– 280 boundary-integral method for Laplace’s equation, 219, 278, 279 for Stokes flow, 27, 94, 205, 277 Boussinesq interface, 88 bubble, 199 in a tube, 247 interaction, 234 migration of, 202

Basset history, 216 Bell equation, 112, 118 bending measure of strain, 65 modulus, 59 in two dimensions, 81 moments, 39, 60 axisymmetric, 166 of a membrane, 24, 37, 38, 41, 43, 45, 58, 60 bifurcation, 183 Biot number, 268 blood cell red, 37, 54, 67, 75, 89, 163 white (see leukocyte), 165 blood flow arterial, 122 through capillaries, 170 Boltzmann’s constant, 118 bond density, 135 force, 108 kinetics, 111 spring constant, 108 Bond number, 249, 266

capillary number, 130, 144, 203, 266 elasticity, 21, 266 capsule axisymmetric, 4, 42, 62, 264 cylindrical, 39 in tube flow, 281 liquid, 127 shapes in hydrostatics, 66 two-dimensional, 39 with isotropic tension, 67 cardiac systole, 122 Cauchy-Green deformation tensor, 3, 52 cell adhering to a surface, 106 adhesion, 106 states, 118 compression, 11 flipping, 184 peeling, 139 red, 273 red blood, 275 rolling, 113, 134, 140 CFD, 104 chemotaxis, 230 Codazzi’s equation, 62

329

330

Capsules and Cells

collision efficiency, 243 compound drop, 130, 141 computer tomography, 125 continuity equation, 128, 203, 265 convection–diffusion equation, 267 corrotational derivative, 22 creeping flow, 203 CT, 125 curvature mean, 47, 200, 278 normal, 49 of an axisymmetric surface, 62 principal, 46, 50 spontaneous, 61 tensor Cartesian, 46 surface, 49 total, 166, 278 cytoskeleton, 165 Damkohler number, 269 Darcy law, 308 Deborah number, 22 deformation gradient relative, 51 surface, 3, 52 of a capsule, 1 rate of a surface, 89 derivative corrotational, 22 covariant, 49 Jaumann, 22 desorption, 200, 268 deviatoric tension, 57, 166 diffusivity, 199, 298 surface, 200, 268 dilatation, 53 modulus, 10, 165 surface, 3, 53 drop compound, 130, 141 in a tube, 247 in extensional flow, 130 interaction, 234

migration of, 202 elastic modulus shear, 5, 11, 57, 77 elasticity interfacial, 201 equation of state surface, 269, 270, 272 erythrocyte (see red blood cell), 163 extension isotropic, 8 ratio axisymmetric, 65, 166, 274 in two dimensions, 81 principal, 3, 52 uniaxial, 7 flexural stiffness, 24 flipping motion, 184 flow resistance, 170 Frumkin adsorption isotherm, 269 fundamental form first, 48 second, 49 F˚ahraeus effect, 182 F˚ahraeus-Lindqvist effect, 171, 179 Gauss-Bonnet theorem, 61 Gibbs adsorption equation, 270 medium, 199 globulin, 2 glycocalyx, 304 gradient surface, 266 Green elastic medium, 53 Green’s function of Laplace’s equation, 278 of Stokes flow, 277 Green-Lagrange strain tensor 3D, 53 surface, 3, 53 Heinz body, 168

Index Hele-Shaw cell, 308 permeability, 309 Helmholtz free energy, 53 hematocrit, 165, 174, 305 discharge, 182, 321 partition, 183 tube, 182, 321 hemoglobin, 165 Hooke’s law, 27, 40, 275 hydrostatics, 66 hyperbolic flow, 19 hyperelastic material, 4, 56 hypergeometric function, 216 immersed boundary method, 127 incompressible membrane, 5, 65, 86, 275, 276, 304 insoluble surfactant, 200, 268 interface cut-cell method, 128 elasticity, 201 Newtonian, 88 interfacial transport quasi-steady, 209 Jaumann derivative, 22 Kelvin solid, 165 Landau-Levich equation, 248 Langmuir isotherm, 270 Laplace equation, 268, 273, 310 law, 42 lateral migration, 186 leakback, 173 leukocyte, 165 adhesion, 103, 106 deformation, 130 recovery, 130 rolling, 106 lipid bilayer, 30, 165 Love’s first approximation, 59, 66 lubrication theory, 173 lymphocyte, 132

331 magnetic resonance imaging, 123 Marangoni convection, 201 flow, 263 migration, 202 number, 249 membrane approximation, 39 aspiration, 11, 78 axisymmetric, 4, 42, 62, 166, 171 planar, 77 compression, 11 cylindrical, 39 elastic modulus, 167 incompressible, 5, 65, 86, 275, 276, 304 inflated, 12 load, 4 Newtonian, 88 properties experimental determination, 11 tank-treading, 22, 168 tension isotropic, 67 tether formation, 107 two-dimensional, 39, 79 viscoelastic, 7, 88 viscosity, 7, 22, 23, 89, 167 wrinkling, 24 microvessel, 164 modulus of bending, 59, 166, 273 of dilatation, 165 of elasticity, 40, 56 in two dimensions, 81 of a red blood cell, 79, 273 shear, 77, 165 surface, 5, 7, 23, 57, 266 Mooney–Rivlin law, 5 strain-energy function, 55, 56 MRI, 123 multi-scale modeling, 103 Navier-Stokes equation, 128, 203

332

Capsules and Cells past cells, 184 simple, 19, 23, 184 stress, 105 on blood vessel wall, 124 tension, 39 transverse, 39, 166

neo-Hookean material, 5, 56, 65 oxygen transport, 163 P`eclet number, 203 for an interface, 268 phase distribution coefficient, 200 plasma, 165 skimming, 183 Poiseuille flow, 265 law, 170 Poisson ratio, 59 surface, 5, 7, 10, 22, 24, 275 Prandtl number, 233 pressure arterial, 122 modified, 265 surface, 270 projection operator tangential, 4, 45 rate-of-deformation surface, 89 reactive compliance, 111 rebound length, 107 receptor-ligand bond, 106, 137 red blood cell, 37, 54, 67, 75, 89, 163, 273, 275 aggregation, 180 axisymmetric, 166, 171 membrane, 6 motion through capillaries, 170, 304 tank-treading, 168 Reynolds equation, 177 number, 20, 203, 265 rheometer, 16 rubber elasticity, 5 sea-urchin eggs, 12 selectins, 106 shear flow

shell axisymmetric, 4, 42, 62 planar, 77 cylindrical, 39 two-dimensional, 39 Sherwood number, 210 sorption, 268 spinning rheometer, 16 stacked-coins model, 171 Stokes equation, 265 equations, 20 flow, 20, 204, 265 strain -energy function, 38, 53, 56, 58 bending, 65 energy function surface, 4 hardening, 6 measure of bending, 65 straining flow axisymmetric, 20 two-dimensional, 19 stream function, 170 stress resultants, 3, 39 tensor Cauchy, 52 Lagrange, 52 Piola-Kirchhoff, 52, 53 stretch, 40 axisymmetric, 65, 166 in two dimensions, 81 principal, 3, 52 tensor, 52 surface elasticity, 263 thermal, 272 equation of state, 269, 270

Index fundamental form, 48, 49 gradient, 266 Poisson ratio, 275 pressure, 270 rate of deformation, 89 strain invariants, 3, 54 tension, 35, 128 viscosity, 7, 22, 89 surfactant, 35, 199, 267 insoluble, 200, 268 tank-treading, 168 tension axisymmetric, 166 deviatoric, 57, 166 membrane, 3, 39 tether formation, 107 thermo-capillary motion, 245, 298 transmural pressure, 42, 66, 83 tubular pinch effect, 186 velocity gradient, 201 virtual displacements, 82, 93 viscoelastic membrane, 7, 88 viscometric flow, 184 viscosity apparent, 171 intrinsic, 174 membrane, 23, 167 surface, 7, 22, 89, 276 wall cell-free layer, 179 shear stress, 124 wrinkling of a membrane, 24 Young -Laplace law, 84 modulus, 7, 23 zipper flow, 189

333