Modeling and Simulation of Capsules and Biological Cells

MODELING AND SIMULATION OF CAPSULES AND BIOLOGICAL CELLS CHAPMAN & HALL/CRC Mathematical Biology and Medicine Series ...

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MODELING AND SIMULATION OF CAPSULES AND BIOLOGICAL CELLS

CHAPMAN & HALL/CRC Mathematical Biology and Medicine Series Aims and scope: This series aims to capture new developments and summarize what is known over the whole spectrum of mathematical and computational biology and medicine. It seeks to encourage the integration of mathematical, statistical and computational methods into biology by publishing a broad range of textbooks, reference works and handbooks. The titles included in the series are meant to appeal to students, researchers and professionals in the mathematical, statistical and computational sciences, fundamental biology and bioengineering, as well as interdisciplinary researchers involved in the field. The inclusion of concrete examples and applications, and programming techniques and examples, is highly encouraged.

Series Editors Alison M. Etheridge Department of Statistics University of Oxford Louis J. Gross Department of Ecology and Evolutionary Biology University of Tennessee Suzanne Lenhart Department of Mathematics University of Tennessee Philip K. Maini Mathematical Institute University of Oxford Hershel M. Safer Informatics Department Zetiq Technologies, Ltd. Eberhard O. Voit Department of Biometry and Epidemiology Medical University of South Carolina

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Chapman & Hall/CRC Mathematical Biology and Medicine Series

MODELING AND SIMULATION OF CAPSULES AND BIOLOGICAL CELLS EDITED BY

C. POZRIKIDIS

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

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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

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-3596/03/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

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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. Barthès-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

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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

. . . . . . . . . . . . .

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. Barthès-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 Barthès-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.


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


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.


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


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., Barthès-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)


15

Figure 1.4.4 Theoretical predictions of the squeezing force for different membrane constitutive laws. (From Carin, M., Barthès-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., & Barthès-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


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., & Barthès-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-


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.


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)


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,


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, , .


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.


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.

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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, Birkhâuser, 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.


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.

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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)


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 Lamé 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)


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)


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)


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)


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


55

elastic tensions. Consequently, the membrane is nearly incompressible and the developing tensions are nearly isotropic. Barthès-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)

+ +


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)


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.


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’Hôspital 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 % % .


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


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


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


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


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


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.


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


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


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


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 : .


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)


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

84

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,


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.

86

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

> 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


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. Jülicher & 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.

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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)


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.

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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 Barthès-Biesel [4] and Barthès-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, Barthès-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], Barthès-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.


95

References [1] A DAMSON , A. W., 1990, Physical Chemistry of Surfaces, Fifth Edition, John Wiley & Sons, New York. [2] A NDERSON , D. M., M C FADDEN , G. B., & W HEELER , A. A., 1998, Diffuse-interface methods in fluid mechanics, Anni. Rev. Fluid Mech., 30, 139-165. [3] A RIS , R., 1962, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs. [4] 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. [5] 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. [6] 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. [7] BARTH E´ S -B IESEL , D. & R ALLISON , J. M., 1980, The time-dependent deformation of a capsule freely suspended in a linear shear flow, J. Fluid Mech., 113, 251-267. [8] 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. [9] 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. [10] B EATTY, M. F., 1987, Topics of finite elasticity: Hyperelasticity of rubber, elastomers, and biological tissues – with examples, Appl. Mech. Rev., 40, 1699-1734. [11] B EYER , W. H., 1984, Standard Mathematical Tables, CRC Press, Boca Raton. [12] B LYTH , M. G. & P OZRIKIDIS , C., 2002, Buckling and collapse of heavy tubes resting on a horizontal or inclined plane, Europ. J. Mech. A/Solids, 21, 831-843. [13] B LYTH , M. G. & P OZRIKIDIS , C., 2003, On the variational formulation for axisymmetric capsules enclosed by biological membranes. Submitted for publication. [14] B LYTH , M. G. & P OZRIKIDIS , C., 2003, Solution space of axisymmetric capsules enclosed by elastic membranes, Submitted for publication.

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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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Capsules and Cells

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.


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.

108

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)


(a)

1.5

(b)

109

3 2.5

1

2 DD (mm) 1.5

0.5

1

DD (mm)

0.5 0

0

0.5

1

1.5

Time (s)

2

2.5

3

0

0

0.5

1

1.5

2

2.5

3

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.

110


(a)

(b)

7 I L

6 5

200

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

1

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)


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

112

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.


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.


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.


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

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